Quantum Spin Hall Edge States Research Articles

Fraunhofer pattern in the presence of Majorana zero modes

We propose a new platform to detect signatures of the presence of Majorana bound states (MBSs) in the Fraunhofer pattern of Josephson junctions featuring quantum spin Hall edge states on the normal part and Majorana bound states at the NS interfaces. We use a tight-binding model to demonstrate a drastic change in the periodicity of the Fraunhofer pattern when comparing trivial and nontrivial regimes. We explain these results in terms of the presence of additional parallel-spin electron-hole reflections, which due to the spin-momentum locking, occur as cross Andreev reflections, accumulating a different magnetic flux and yielding a change in the Fraunhofer periodicity. We show that this detection scheme exhibits some advantages compared to previous ones as it is robust against disorder and finite temperature and works in equilibrium. Furthermore, we introduce a scattering model that captures the main results of the microscopic calculations with MBSs and extend our discussion to the main differences found using accidental zero-energy Andreev bound states. Published by the American Physical Society 2024

Transport properties of Hall-type quantum states in disordered bismuthene

Bismuthene, an inherently hexagonal structure characterized by a huge bulk gap, offers a versatile platform for investigating the electronic transport of various topological quantum states. Using nonequilibrium Green’s function method and Landauer–Büttiker formula, we thoroughly investigate the transport properties of various Hall-type quantum states, including quantum spin Hall (QSH) edge states, quantum valley Hall kink (QVHK) states, and quantum spin–valley Hall kink (QSVHK) states, in the presence of various disorders. Based on the exotic transport features, a spin–valley filter, capable of generating a highly spin- and valley-polarized current, is proposed. The valley index and the spin index of the filtered QSVHK state are determined by the staggered potential and the intrinsic spin–orbit coupling, respectively. The efficiency of the spin–valley filter is supported by the spacial current distribution, the valley-resolved conductance, and the spin-resolved conductance. Compared with a sandwich structure for QSVHK, our proposed spin–valley filter can work with a much smaller size and is more accessible in the experiment.

Broad and colossal edge supercurrent in Dirac semimetal Cd3As2 Josephson junctions

Edge supercurrent has attracted great interest recently due to its crucial role in achieving and manipulating topological superconducting states. Proximity-induced superconductivity has been realized in quantum Hall and quantum spin Hall edge states, as well as in higher-order topological hinge states. Non-Hermitian skin effect, the aggregation of non-Bloch eigenstates at open boundaries, promises an abnormal edge channel. Here we report the observation of broad edge supercurrent in Dirac semimetal Cd3As2-based Josephson junctions. The as-grown Cd3As2 nanoplates are electron-doped by intrinsic defects, which enhance the non-Hermitian perturbations. The superconducting quantum interference indicates edge supercurrent with a width of ~1.6 μm and a magnitude of ~1 μA at 10 mK. The wide and large edge supercurrent is inaccessible for a conventional edge system and suggests the presence of non-Hermitian skin effect. A supercurrent nonlocality is also observed. The interplay between band topology and non-Hermiticity is beneficial for exploiting exotic topological matter.

Thermal dissipation of the quantum spin Hall edge states in HgTe/CdTe quantum well

Quantum spin Hall effect is characterized by topologically protected helical edge states. Here we study the thermal dissipation of helical edge states by considering two types of dissipation sources. The results show that the helical edge states are dissipationless for normal dissipation sources with or without Rashba spin–orbit coupling in the system, but they are dissipative for spin dissipation sources. Further studies on the energy distribution show that electrons with spin-up and spin-down are both in their own equilibrium without dissipation sources. Spin dissipation sources can couple the two subsystems together to induce voltage drop and non-equilibrium distribution, leading to thermal dissipation, while normal dissipation sources cannot. With the increase of thermal dissipation, the subsystems of electrons with spin-up and spin-down evolve from non-equilibrium finally to mutual equilibrium. In addition, the effects of disorder on thermal dissipation are also discussed. Our work provides clues to reduce thermal dissipation in the quantum spin Hall systems.

Theory of glide symmetry protected helical edge states in a WTe2 monolayer

Helical edge states in quantum spin Hall (QSH) materials are central building blocks of topological matter design and engineering. Despite their principal topological protection against elastic backscattering, the level of operational stability depends on manifold parameters such as the band gap of the given semiconductor system in the 'inverted' regime, temperature, disorder, and crystal orientation. We theoretically investigate electronic and transport properties of QSH edge states in large gap 1-T' WTe$_{2}$ monolayers. We explore the impact of edge termination, disorder, temperature, and interactions on experimentally addressable edge state observables, such as local density of states and conductance. We show that conductance quantization can remain surprisingly robust even for heavily disordered samples because of an anomalously small edge state decay length and additional protection related to the large direct gap allowed by glide symmetry. From the simulation of temperature-dependent resistance, we find that moderate disorder enhances the stability of conductance by localizing bulk states. We evaluate the edge state velocity and Luttinger liquid parameter as functions of the chemical potential, finding prospects for physics beyond linear helical Luttinger liquids in samples with ultra-clean and well-defined edges.

Electrically detected single-spin resonance with quantum spin Hall edge states

Detection is most often the main impediment to reduce the number of spins in paramagnetic resonance experiments. Here we propose a new route to carry out electrically-detected spin resonance of an individual spin, placed at the edge of a quantum spin Hall insulator (QSHI). The edges of a QSHI host a one dimensional electron gas with perfect spin-momentum locking. Therefore, the spin relaxation induced by emission of an electron-hole pair at the edge state of the QSHI can generate current. Here we demonstrate that driving the system with an $AC$ signal, a nonequilibrium occupation can be induced in the absence of applied bias voltage, resulting in a $DC$ measurable current. We compute the $DC$ current as a function of the Rabi frequency $\Omega$, the spin relaxation and decoherence times, $T_1$ and we discuss the feasibility of this experiment with state-of-the-art instrumentation.

Field-Effect Tunneling between Quantum Valley Hall Edge States and Topological Transistors Based on Bilayer Graphene

We study the tunneling effect between quantum valley Hall edge states under an applied electric field in bilayer graphene containing gaps. A topological transistor based on this tunneling effect is proposed. In the on state of the transistor, the quantum valley Hall edge state along one bilayer-graphene zigzag edge can tunnel to the other edge under an electric field. In the off state without the electric field, the edge state cannot tunnel to the other edge and should be reflected by the top armchair edge and retrace the route along the same zigzag edge. The performance of the transistor is shown to be robust against long-range impurities. Compared with the previous transistor based on quantum spin Hall edge states [Xu et al. Phys. Rev. Lett. 123, 206801 (2019)], this transistor based on quantum valley Hall edge states has the advantages that it does not need a ferromagnetic gate to bridge the gap in the edge states and the conductance is robust against both nonmagnetic and magnetic impurities.

The photogalvanic effect induced by quantum spin Hall edge states from first-principles calculations.

Based on non-equilibrium Green's function combined with density functional theory (NEGF-DFT), we theoretically investigate the spin-related photogalvanic effect (PGE) in topological insulators BiBr and SbBr nanoribbons from atomic first-principles calculations. It is demonstrated that the PGE generated photocurrents by quantum spin Hall edge states (QSHES) are in general pure spin currents due to the presence of time reversal and mirror symmetries, which is independent of the photon energies, polarization, and incident angles. Although the QSHES are topologically protected and robust against defects and impurities during their transport, the spin photocurrent generated by these edge states via the PGE is particularly sensitive to defects. By tuning the defect position of the nanoribbons, the magnitude of spin related photocurrent generated by the PGE can be significantly increased compared with that in pristine nanoribbons. Our work not only reveals the defect effect of PGE but also demonstrates the great potential of defect engineered topological insulator nanoribbons for novel application in two-dimensional opto-spintronic devices.

Quantum Spin Hall Edge States and Interlayer Coupling in Twisted Bilayer WTe2.

The quantum spin Hall (QSH) effect, characterized by topologically protected spin-polarized edge states, was recently demonstrated in monolayers of the transition metal dichalcogenide (TMD) WTe2. However, the robustness of this topological protection remains largely unexplored in van der Waals heterostructures containing one or more layers of a QSH insulator. In this work, we use scanning tunneling microscopy and spectroscopy (STM/STS) to explore the topological nature of twisted bilayer (tBL) WTe2. At the tBL edges, we observe the characteristic spectroscopic signatures of the QSH edge states. For small twist angles, a rectangular moiré pattern develops, which results in local modifications of the band structure. Using first-principles calculations, we quantify the interactions in tBL WTe2 and its topological edge states as a function of interlayer distance and conclude that it is possible to engineer the topology of WTe2 bilayers via the twist angle as well as interlayer interactions.

Effective lifting of the topological protection of quantum spin Hall edge states by edge coupling

The scientific interest in two-dimensional topological insulators (2D TIs) is currently shifting from a more fundamental perspective to the exploration and design of novel functionalities. Key concepts for the use of 2D TIs in spintronics are based on the topological protection and spin-momentum locking of their helical edge states. In this study we present experimental evidence that topological protection can be (partially) lifted by pairwise coupling of 2D TI edges in close proximity. Using direct wave function mapping via scanning tunneling microscopy/spectroscopy (STM/STS) we compare isolated and coupled topological edges in the 2D TI bismuthene. The latter situation is realized by natural lattice line defects and reveals distinct quasi-particle interference (QPI) patterns, identified as electronic Fabry-Pérot resonator modes. In contrast, free edges show no sign of any single-particle backscattering. These results pave the way for novel device concepts based on active control of topological protection through inter-edge hybridization for, e.g., electronic Fabry-Pérot interferometry.

Coexistence of Robust Edge States and Superconductivity in Few-Layer Stanene.

High-quality stanene films have been actively pursued for realizing not only quantum spin Hall edge states without backscattering, but also intrinsic superconductivity, two central ingredients that may further endow the systems to host topological superconductivity. Yet to date, convincing evidence of topological edge states in stanene remains to be seen, let alone the coexistence of these two ingredients, owing to the bottleneck of growing high-quality stanene films. Here we fabricate one- to five-layer stanene films on the Bi(111) substrate and observe the robust edge states using scanning tunneling microscopy/spectroscopy. We also measure distinct superconducting gaps on different-layered stanene films. Our first-principles calculations further show that hydrogen passivation plays a decisive role as a surfactant in improving the quality of the stanene films, while the Bi substrate endows the films with nontrivial topology. The coexistence of nontrivial topology and intrinsic superconductivity renders the system a promising candidate to become the simplest topological superconductor based on a single-element system.

Robust all-electrical topological valley filtering using monolayer 2D-Xenes

We propose a realizable device design for an all-electrical robust valley filter that utilizes spin protected topological interface states hosted on monolayer 2D-Xene materials with large intrinsic spin–orbit coupling. In contrast with conventional quantum spin-Hall edge states localized around the X-points, the interface states appearing at the domain wall between topologically distinct phases are either from the K or K^{prime} points, making them suitable prospects for serving as valley-polarized channels. We show that the presence of a large band-gap quantum spin-Hall effect enables the spatial separation of the spin–valley locked helical interface states with the valley states being protected by spin conservation, leading to robustness against short-range nonmagnetic disorder. By adopting the scattering matrix formalism on a suitably designed device structure, valley-resolved transport in the presence of nonmagnetic short-range disorder for different 2D-Xene materials is also analyzed in detail. Our numerical simulations confirm the role of spin–orbit coupling in achieving an improved valley filter performance with a perfect quantum of conductance attributed to the topologically protected interface states. Our analysis further elaborates clearly the right choice of material, device geometry and other factors that need to be considered while designing an optimized valleytronic filter device.

Momentum relaxation effects in 2D-Xene field effect device structures

We analyze the electric field driven topological field effect transition on 2D-xene materials with the addition of momentum relaxation effects, in order to account for dephasing processes. The topological field effect transition between the quantum spin Hall phase and the quantum valley Hall phase is analyzed in detail using the Keldysh non-equilibrium Green’s function technique with the inclusion of momentum and phase relaxation, within the self-consistent Born approximation. Details of the transition with applied electric field are elucidated for the ON–OFF characteristics with emphasis on the transport properties along with the tomography of the current carrying edge states. We note that for moderate momentum relaxation, the current carrying quantum spin Hall edge states are still pristine and show moderate decay with propagation. To facilitate our analysis, we introduce two metrics in our calculations, the coherent transmission and the effective transmission. In elucidating the physics clearly, we show that the effective transmission, which is derived rigorously from the quantum mechanical current operator is indeed the right quantity to analyze topological stability against dephasing. Exploring further, we show that the insulating quantum valley Hall phase, as a result of dephasing carries band-tails which potentially activates parasitic OFF currents, thereby degrading the ON–OFF ratios. Our analysis sets the stage for realistic modeling of topological field effect devices for various applications, with the inclusion of scattering effects and analyzing their role in the optimization of the device performance.

Characterization of helical Luttinger liquids in microwave stepped-impedance edge resonators

Coulomb interaction has important consequences on the physics of quantum spin Hall edge states, weakening the topological protection via two-particle scattering and renormalizing both the velocity and charge of collective plasmon modes compared to that of free electrons. Despite these effects, interactions remain difficult to quantify. We propose here simple and robust edge resonator geometries to characterize Coulomb interaction by means of high-frequency measurements. They rely on a transmission line approach, and take advantage of the impedance mismatch between the edge states and their microwave environment.

Band structure and end states in InAs/GaSb core-shell-shell nanowires

Quantum wells in InAs/GaSb heterostructures can be tuned to a topological regime associated with the quantum spin Hall effect, which arises due to an inverted band gap and hybridized electron and hole states. Here, we investigate electron-hole hybridization and the fate of the quantum spin Hall effect in a quasi one-dimensional geometry, realized in a core-shell-shell nanowire with an insulator core and InAs and GaSb shells. We calculate the band structure for an infinitely long nanowire using $\mathbf{k \cdot p}$ theory within the Kane model and the envelope function approximation, then map the result onto a BHZ model which is used to investigate finite-length wires. Clearly, quantum spin Hall edge states cannot appear in the core-shell-shell nanowires which lack one-dimensional edges, but in the inverted band-gap regime we find that the finite-length wires instead host localized states at the wire ends. These end states are not topologically protected, they are four-fold degenerate and split into two Kramers pairs in the presence of potential disorder along the axial direction. However, there is some remnant of the topological protection of the quantum spin Hall edge states in the sense that the end states are fully robust to (time-reversal preserving) angular disorder, as long as the bulk band gap is not closed.

Controlling the Quantum Spin Hall Edge States in Two-Dimensional Transition Metal Dichalcogenides.

Two-dimensional transition metal dichalcogenides (TMDs) of Mo and W in their 1T' crystalline phase host the quantum spin Hall (QSH) insulator phase. We address the electronic properties of the QSH edge states by means of first-principles calculations performed on realistic models of edge terminations of different stoichiometries. The QSH edge states show a tendency to have complex band dispersions and coexist with topologically trivial edge states. We nevertheless identify two stable edge terminations that allow isolation of a pair of helical edge states within the band gap of TMDs, with monolayer 1T'-WSe2 being the most promising material. We also characterize the finite-size effects in the electronic structure of 1T'-WSe2 nanoribbons. Our results provide guidance to the experimental studies and possible practical applications of QSH edge states in monolayer 1T'-TMDs.

Conductance of quantum spin Hall edge states from first principles: The critical role of magnetic impurities and inter-edge scattering

The outstanding transport properties expected at the edge of two-dimensional time-reversal invariant topological insulators have proven to be challenging to realize experimentally, and have so far only been demonstrated in very short devices. In search for an explanation to this puzzling observation, we here report a full first-principles calculation of topologically protected transport at the edge of novel quantum spin Hall insulators - specifically, Bismuth and Antimony halides - based on the non-equilibrium Green's functions formalism. Our calculations unravel two different scattering mechanisms that may affect two-dimensional topological insulators, namely time-reversal symmetry breaking at vacancy defects and inter-edge scattering mediated by multiple co-operating impurities, possibly non-magnetic. We discuss their drastic consequences for typical non-local transport measurements as well as strategies to mitigate their negative impact. Finally, we provide an instructive comparison of the transport properties of topologically protected edge states to those of the trivial edge states in MoS$_2$ ribbons. Although we focus on a few specific cases (in terms of materials and defect types) our results should be representative for the general case and thus have significance beyond the systems studied here.

Optimal Thermoelectricity with Quantum Spin Hall Edge States.

We study the thermoelectric properties of a Kramers pair of helical edge states of the quantum spin Hall effect coupled to a nanomagnet with a component of the magnetization perpendicular to the direction of the spin-orbit interaction of the host. We show that the transmission function of this structure has the desired qualities for optimal thermoelectric performance in the quantum coherent regime. For a single magnetic domain, there is a power generation close to the optimal bound. In a configuration with two magnetic domains with different orientations, pronounced peaks in the transmission functions and resonances lead to a high figure of merit. We provide estimates for the fabrication of this device with HgTe quantum-well topological insulators.

Custodial glide symmetry of quantum spin Hall edge modes in monolayer WTe2

A monolayer of WTe$_2$ has been shown to display quantum spin Hall (QSH) edge modes persisting up to 100~K in transport experiments. Based on density-functional theory calculations and symmetry-based model building including the role of correlations and substrate support, we develop an effective electronic model for WTe$_2$ which fundamentally differs from other prototypical QSH settings: we find that the extraordinary robustness of quantum spin Hall edge modes in WTe$_2$ roots in a glide symmetry due to which the topological gap opens away from high-symmetry points in momentum space. While the indirect bulk gap is much smaller, the glide symmetry implies a large direct gap of up to 1~eV in the Brillouin zone region of the dispersing edge modes, and hence enables sharply boundary-localized QSH edge states depending on the specific boundary orientation.

Microwave Signature of Topological Andreev level Crossings in a Bismuth-based Josephson Junction.

Demonstrating the topological protection of Andreev states in Josephson junctions is an experimental challenge. In particular the telltale 4π periodicity expected for the current phase relation has remained elusive, because of fast parity breaking processes. It was predicted that low temperature ac susceptibility measurements could reveal the topological protection of quantum spin Hall edge states by probing their low energy Andreev spectrum at finite frequency. We have performed such a microwave probing of a phase-biased Josephson junction built around a bismuth nanowire, a predicted second order topological insulator, and which was previously shown to host one-dimensional ballistic edge states. We find absorption peaks at the Andreev level crossings, whose temperature and frequency dependencies point to protected topological crossings with an accuracy limited by the electronic temperature of our experiment.