The transversal propagation of the edge states in a two-dimensional quantum spin Hall system are classified by decay characteristic quantity $λ$. Two different modes of the helical edge states exhibit distinct behaviors. The penetration depth is momentum dependent in normal edge states. The finite size gap decays monotonously with sample width, leading to the normal size effect. In contrast, the penetration depth maintains a uniform minimal value in the special edge states with much shorter length. The finite size gap decays non-monotonously with width, leading to the anomalous finite size effect. Real materials are compared in the phase diagram, which explicitly demonstrates their differences. We also propose an intuitive way to search the special edge states in two-dimensional quantum spin Hall system.
The thermoelectric properties of a three terminal quantum spin Hall (QSH) sample are examined. Inherent helicity of the QSH sample helps to generate a large charge power efficiently. Along with charge the system can be designed to work as a highly efficient spin heat engine too. The advantage of a helical over a chiral sample is that, while a multiterminal quantum Hall sample can only work as a quantum heat engine due to broken time reversal(TR) symmetry, a multiterminal QSH system can work effectively both as a charge/spin heat engine as well as a charge/spin refrigerator as TR symmetry is pr…
In this work, we study the disorder effect on topological metals that support a pair of helical edge modes deeply embedded inside the gapless bulk states. Strikingly, we predict that a quantum spin Hall (QSH) phase can be obtained from such topological metals without opening a global band gap. To be specific, disorder can lead to a pair of robust helical edge states which is protected by an emergent $Z_2$ topological invariant, giving rise to a quantized conductance plateau in transport measurements. These results are instructive for solving puzzles in {various transport experiments on QSH mat…
We consider the proximity effect between a singlet s-wave superconductor and the edge of a Quantum Spin Hall (QSH) topological insulator. We establish that Andreev reflection at a QSH edge state/superconductor interface is perfect while nonlocal Andreev processes through the superconductor are totally suppressed. We compute the corresponding conductance and noise.
Although indications are that a single chiral quantum anomalous Hall(QAH) edge mode might have been experimentally detected. There have been very many recent experiments which conjecture that a single chiral QAH edge mode always materializes along with a pair of quasi-helical quantum spin Hall (QSH) edge modes. The reason for this seems to lie in the origin of QAH edge modes. These evolve from QSH edge modes via suppression of one of the spin edge modes by application of a ferromagnet or magnetic impurity. In this work we deal with a substantial 'What If ?' question- in case the QSH edge modes…
Monolayer WTe2 is predicted to be a quantum spin Hall insulator (QSHI) and electron transport along its edges has been experimentally observed. However, the 'smoking gun' of QSHI, spin momentum locking of the edge electrons, has not been experimentally demonstrated. We propose a model to establish the relationship between the anisotropic magnetoresistance (AMR) and spin orientation of the helical electrons in WTe2. Based on the predictions of the model, angular dependent magnetoresistance measurements were carried out. The experimental results fully supported the model and the spin orientation…
The edge states of a two-dimensional quantum spin Hall (QSH) insulator form a one-dimensional helical metal which is responsible for the transport property of the QSH insulator. Conceptually, such a one-dimensional helical metal can be attached to any scattering region as the usual metallic leads. We study the analytical property of the scattering matrix for such a conceptual multiterminal scattering problem in the presence of time reversal invariance. As a result, several theorems on the connectivity property of helical edge states in two-dimensional QSH systems as well as surface states of t…
We measure microwave photocurrent in devices made from InAs/GaInSb bilayers where both insulating bulk state and conducting edge state were observed in the inverted-band regime, consistent with the theoretical prediction for a quantum spin Hall (QSH) insulator. It has been theoretically proposed that microwave photocurrent could be a unique probe in studying the properties of QSH edge states. To distinguish possible photoresponse between bulk state and helical edge state, we prepare Hall bar and Corbino disk from the same wafer. Results show that the Corbino disk samples have a negligible phot…
A Kramers pair of helical edge states in quantum spin Hall effect (QSHE) is robust against normal dephasing but not robust to spin dephasing. In our work, we provide an effective spin dephasing mechanism in the puddles of two-dimensional (2D) QSHE, which is simulated as quantum dots modeled by 2D massive Dirac Hamiltonian. We demonstrate that the spin dephasing effect can originate from the combination of the Rashba spin-orbit coupling and electron-phonon interaction, which gives rise to inelastic backscattering in edge states within the topological insulator quantum dots, although the time-re…
Following the recent observation of the quantum spin Hall (QSH) effect in HgTe quantum wells, an important issue is to understand the effect of impurities on transport in the QSH regime. Using linear response and renormalization group methods, we calculate the edge conductance of a QSH insulator as a function of temperature in the presence of a magnetic impurity. At high temperatures, Kondo and/or two-particle scattering give rise to a logarithmic temperature dependence. At low temperatures, for weak Coulomb interactions in the edge liquid the conductance is restored to unitarity with unusual …
It is well known that quantum spin Hall (QSH) edge modes being helical are immune to backscattering due to non-magnetic disorder within the sample. Thus, quantum spin Hall edge modes are non-localized and show a vanishing Hall resistance along with quantized 2-terminal, longitudinal and non-local resistances even in presence of sample disorder. However, this is not the case for contact disorder. This paper shows that when all contacts are disordered in an N-terminal quantum spin Hall sample, then transport via these helical QSH edge modes can have a significant localization correction. All the…
Motivated by a recent experiment (Sanchez-Yamagishi et.al, arXiv:1602.06815) reporting evidence of helical spin-polarized edge states in layer-biased twisted bilayer graphene under a magnetic flux, we study the possibility of stabilising a Quantum Spin Hall (QSH) phase in such a system, without Zeeman or spin-orbit couplings, and with a QSH gap induced instead by electronic interactions. We analyse how magnetic flux, electric field, interlayer rotation angle, and interactions (treated at a mean field level) combine to produce a pseudo-QSH with broken time-reversal symmetry, and spin-polarized …
We theoretically manifest that the edge of a quantum spin Hall insulator (QSHI), attached to an insulating ferromagnet (FM), can realize a highly efficient spin-to-charge conversion. Based on a one-dimensional QSHI-FM junction, the electron dynamics on the QSHI edge is analyzed, driven by a magnetization dynamics in the FM. Under a large gap opening on the edge from the magnetic exchange coupling, we find that the spin injection into the QSHI edge gets suppressed while the charge current driven on the edge gets maximized, demanded by the band topology of the one-dimensional helical edge states…
We investigate one-dimensional (1D) Majorana bound states (MBSs) realized in terms of the helical edge states of a 2D quantum spin-Hall insulator (QSHI) in a heterostructure with a superconducting substrate and two ferromagnetic insulators (FIs). By means of Bogoliubov-de Gennes approach we demonstrate that there is a helical spin texture in the MBS wave function with a pitch proportional to the Fermi momentum of the helical edge states of QSHI. Moreover, simultaneous detection on local density of states by scanning tunneling microscopy and spectroscopy at a position close to one FI edge and a…
The discovery of the Quantum Spin Hall state, and topological insulators in general, has sparked strong experimental efforts. Transport studies of the Quantum Spin Hall state confirmed the presence of edge states, showed ballistic edge transport in micron-sized samples and demonstrated the spin polarization of the helical edge states. While these experiments have confirmed the broad theoretical model, the properties of the QSH edge states have not yet been investigated on a local scale. Using Scanning Gate Microscopy to perturb the QSH edge states on a sub-micron scale, we identify well-locali…
Inverted-gap GaSb/InAs quantum wells have long been predicted to show quantum spin Hall insulator (QSHI) behavior. The experimental characterization of the QSHI phase in these systems has relied on the presence of quantized edge transport near charge neutrality. However, experimental data showing the presence of edge conductance in the \emph{trivial} regime suggest that additional experimental signatures are needed to characterize the QSHI phase. Here we show that electric field- induced gap oscillations can be used as an indicator of the presence of helical edge states in system. By studying …
The quantum spin Hall (QSH) state, observed in a zero magnetic field in HgTe quantum wells, respects the time-reversal symmetry and is distinct from quantum Hall (QH) states. We show that the QSH state persists in strong quantizing fields and is identified by counter-propagating (helical) edge channels with nonlinear dispersion inside the band gap. If the Fermi level is shifted into the Landau-quantized conduction or valence band, we find a transition between the QSH and QH regimes. Near the transition the longitudinal conductance of the helical channels is strongly suppressed due to the combi…
We study the electronic properties of the confined honeycomb lattice in the presence of the intrinsic spin-orbit (ISO) interaction and perpendicular magnetic field, and report on uncommon aspects of the quantum spin Hall conductance corroborated by peculiar properties of the edge states. The ISO interaction induces two specific gaps in the Hofstadter spectrum, namely the 'weak' topological gap defined by Beugeling et al [Phys. Rev. B 86, 075118 (2012)], and spin-imbalanced gaps in the relativistic range of the energy spectrum. We analyze the evolution of the helical states with the magnetic fi…
The xene family of topological insulators plays a key role in many proposals for topological electronic, spintronic, and valleytronic devices. These proposals rely on applying local perturbations, including electric fields and proximity magnetism, to induce topological phase transitions in xenes. However, these techniques lack control over the geometry of interfaces between topological regions, a critical aspect of engineering topological devices. We propose adatom decoration as a method for engineering atomically precise topological edge modes in xenes. Our first-principles calculations show …
The quantum spin Hall (QSH) effect is the property of a new state of matter which preserves time-reversal, has an energy gap in the bulk, but has topologically robust gapless states at the edge. Recently, it has been shown that HgTe quantum wells realize this novel effect. In this work, we start from realistic tight-binding models and demonstrate the existence of the helical edge states in HgTe quantum wells and calculate their physical properties. We also show that 3d HgTe is a topological insulator under uniaxial strain, and show that the surface states are described by single-component mass…
In two-dimensional topological insulators, such as inverted HgTe/CdTe quantum wells, helical quantum spin Hall (QSH) states persist even at finite magnetic fields below a critical magnetic field $B_\mathrm{c}$, above which only quantum Hall (QH) states can be found. Using linear-response theory, we theoretically investigate the magneto-optical properties of inverted HgTe/CdTe quantum wells, both for infinite two-dimensional and finite-strip geometries, and possible signatures of the transition between the QSH and QH regimes. In the absorption spectrum, several peaks arise due to non-equidistan…
We show that the Quantum Spin Hall Effect, a state of matter with topological properties distinct from conventional insulators, can be realized in HgTe/CdTe semiconductor quantum wells. By varying the thickness of the quantum well, the electronic state changes from a normal to an "inverted" type at a critical thickness $d_c$. We show that this transition is a topological quantum phase transition between a conventional insulating phase and a phase exhibiting the QSH effect with a single pair of helical edge states. We also discuss the methods for experimental detection of the QSH effect.
We propose to implement quantum computing based on electronic spin qubits by controlling the propagation of the electron wave packets through the helical edge states of quantum spin Hall systems (QSHs). Specfically, two non-commutative single-qubit gates, which rotate a qubit around z and y axes, can be realized by utilizing gate voltages either on a single QSH edge channel or on a quantum point contact structure. The more challenging two-qubit controlled phase gate can be implemented through the on-demand capacitive Coulomb interaction between two adjacent edge channels from two parallel QSHs…
Quantum spin Hall (QSH) insulators, a new class of quantum matters, can support topologically protected helical edge modes inside bulk insulating gap, which can lead to dissipationless transport. A major obstacle to reach wide application of QSH is the lack of suitable QSH compounds, which should be easily fabricated and has large size of bulk gap. Here we predict that single layer ZrTe$_5$ and HfTe$_5$ are the most promising candidates to reach the large gap QSH insulators with bulk direct (indirect) band gap as large as 0.4 eV (0.1 eV), and robust against external strains. The 3D crystals of…
Low-dimensional electronic systems have traditionally been obtained by electrostatically confining electrons, either in heterostructures or in intrinsically nanoscale materials such as single molecules, nanowires, and graphene. Recently, a new paradigm has emerged with the advent of symmetry-protected surface states on the boundary of topological insulators, enabling the creation of electronic systems with novel properties. For example, time reversal symmetry (TRS) endows the massless charge carriers on the surface of a three-dimensional topological insulator with helicity, locking the orienta…
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 isolating a pair of helical edge states within the band gap of TMDs, with monolaye…
Understanding possible mechanisms, which can lead to suppression of helical edge transport in Quantum Spin Hall (QSH) systems, attracted huge attention right after the first experiments revealing the fragility of the ballistic conductance. Despite the very intensive research and the abundance of theoretical models, the fully consistent explanation of the experimental results is still lacking. We systematize various theories of helical transport with the help of the spin conservation analysis which allows one to single out setups with the ballistic conductance being robustly protected regardles…
The quantum spin Hall (QSH) state was recently demonstrated in monolayers of the transition metal dichalcogenide 1T'-WTe$_2$ and is characterized by a band gap in the two-dimensional (2D) interior and helical one-dimensional (1D) edge states. Inducing superconductivity in the helical edge states would result in a 1D topological superconductor, a highly sought-after state of matter. In the present study, we use a novel dry-transfer flip technique to place atomically-thin layers of WTe$_2$ on a van der Waals superconductor, NbSe$_2$. Using scanning tunneling microscopy and spectroscopy (STM/STS)…
The search for topologically non-trivial states of matter has become an important goal for condensed matter physics. Recently, a new class of topological insulators has been proposed. These topological insulators have an insulating gap in the bulk, but have topologically protected edge states due to the time reversal symmetry. In two dimensions the helical edge states give rise to the quantum spin Hall (QSH) effect, in the absence of any external magnetic field. Here we review a recent theory which predicts that the QSH state can be realized in HgTe/CdTe semiconductor quantum wells. By varying…
A graphene nano-ribbon with armchair edges is known to have no edge state. However, if the nano-ribbon is in the quantum spin Hall (QSH) state, then there must be helical edge states. By folding a graphene ribbon to a ring and threading it by a magnetic flux, we study the persistent charge and spin currents in the tight-binding limit. It is found that, for a broad ribbon, the edge spin current approaches a finite value independent of the radius of the ring. For a narrow ribbon, inter-edge coupling between the edge states could open the Dirac gap and reduce the overall persistent currents. Furt…
A Quantum Spin-Hall Insulator (QSHI) is implemented into a simple mass-spring Kagome lattice. The transition from the trivial state to the topological one is described by an invariant Chern number function of a contrast parameter. The band diagram and helical edge states characteristic of QSHI are obtained by a combination of numerical and analytical methods. In particular, these states are shown to be Stoneley wave solutions to a set of asymptotic continuous motion equations. Last, scatterless propagation of polarized topological edge waves around sharp corners is demonstrated and robustness …
We investigate transport through a normal-superconductor (NS) junction made from a quantum spin Hall (QSH) system with helical edge states and a two-dimensional (2D) chiral topological superconductor (TSC) having a chiral Majorana edge mode. We employ a two-dimensional extended four-band model for HgTe-based quantum wells in a magnetic (Zeeman) field and subject to s-wave superconductivity. We show using the Bogoliubov-de Gennes scattering formalism that this structure provides a striking transport signal of a 2D TSC. As a function of the sample width (or Fermi energy) the conductance resonanc…
We study the conductance afforded by a normal-metal probe which is directly contacting the helical edge modes of a quantum spin Hall insulator (QSHI). We show a $2e^2/h$ conductance peak at zero temperature in QSHI-based superconductor--ferromagnet hybrids due to the formation of a single Majorana bound state (MBS). In a corresponding Josephson junction hosting a pair of MBSs, a $4e^2/h$ conductance peak is found at zero temperature. The conductance quantization is robust to changes of the relevant system parameters and, remarkably, remains unaltered with increasing the distance between probe …
In this work, we predict a hinged quantum spin-Hall (HQSH) effect featured by a pair of helical hinge modes in antiferromagnetic (AFM) topological insulator (TI) multilayers. This pair of helical hinge modes are localized on the hinges of the top and bottom surfaces of the AFM TI multilayers. Unlike the conventional QSH effect, the HQSH effect survives the breaking of time-reversal symmetry (TRS) and thus represents a different kind of topological phenomenon. The helical hinge modes are sustainable to inelastic scattering and TRS-breaking disorder, which can be observed in macroscopic samples.…
In this work, we study the transport through a quantum point contact for bosonic helical liquid that exists at the edge of a bilayer graphene under a strong magnetic field. We identify "smoking gun" transport signatures to distinguish bosonic symmetry protected topological (BSPT) state from fermionic two-channel quantum spin Hall (QSH) state in this system. In particular, a novel charge insulator/spin conductor phase is found for BSPT state, while either charge insulator/spin insulator or charge conductor/spin conductor phase is expected for the two-channel QSH state. Consequently, a simple tr…
Quantum spin Hall (QSH) effect with great promise for the potential application in spintronics and quantum computing has attracted extensive research interest from both theoretical and experimental researchers. Here, we predict monolayer Ta$_2$Pd$_3$Te$_5$ can be a QSH insulator based on first-principles calculations. The interlayer binding energy in the layered van der Waals compound Ta$_2$Pd$_3$Te$_5$ is 19.6 meV/A$^2$; thus, its monolayer/thin-film structures could be readily obtained by exfoliation. The band inversion near the Fermi level ($E_F$) is an intrinsic characteristic, which happe…
Quantum spin Hall insulators (QSHIs), also known as two-dimensional topological insulators, have emerged as an unconventional class of quantum states with insulating bulk and conducting edges originating from nontrivial inverted band structures, and have been proposed as a platform for exploring spintronics applications and exotic quasiparticles related to the spin-helical edge modes. Despite theoretical proposals for various materials, however, experimental demonstrations of QSHIs have so far been limited to two systems--HgTe/CdTe and InAs/GaSb--both of which are lattice-matched semiconductor…
It is a conventional wisdom that the helical edge states of quantum spin Hall (QSH) insulator are particularly stable due to the topological protection of time-reversal symmetry. Here, we report the first experimental observation of an edge-dependent quantum (pseudo-)spin Hall effect by employing two Kekule electric circuits with molecule-zigzag and partially-bearded edges, where the chirality of the circulating current in the unit cell mimics the electron spin. We observe a helicity flipping of the topological in-gap modes emerging in opposite parameter regions for the two edge geometries. Ex…
Quantum spin Hall (QSH) insulators are materials that feature an insulating bulk and host edge states protected by time-reversal symmetry. The helical locking of spin and momentum in these states suppresses backscattering of charge carriers, promising applications from low-power electronics to quantum computing. A major challenge for applications is the identification of large gap QSH materials, which would enable room temperature dissipationless transport in their edge states. Here we show that the layered mineral jacutingaite (Pt$_2$HgSe$_3$) is a candidate QSH material, realizing the long s…
Topological insulators (TIs) are a novel class of materials with nontrivial surface or edge states. Time-reversal symmetry (TRS) protected TIs are characterized by the Z2 topological invariant and their helical property becomes lost in an applied magnetic field. Currently there exist extensive efforts searching for TIs that are protected by symmetries other than TRS. Here we show, a topological phase characterized by a spin Chern topological invariant is realized in an inverted electron-hole bilayer engineered from indium arsenide-gallium antimonide (InAs/GaSb) semiconductors which retains rob…
We proposed that the simultaneous presence of both Rashba and band inversion can lead to a Rashba-like spin-splitting formed by two bands with the same in-plane helical spin texture. Because of this unconventional spin texture, the backscattering is forbidden in edge and bulk conductivity channels. We propose a new non-centrosymmetric honeycomb-lattice quantum spin Hall (QSH) insulator family formed by the IV, V, and VII elements with this property. The system formed by Bi, Pb and I atoms is mechanically stable and has both a large Rashba spin-splitting of 60 meV and a large nontrivial band ga…
Buried Dirac points in quantum spin Hall insulators: Implications for Majorana Kramers pair-based quantum computing
The University of Texas at El Paso（埃尔帕索德克萨斯大学）；Sandia National Laboratories（桑迪亚国家实验室）；The University of Texas at Dallas（达拉斯德克萨斯大学）；William & Mary（威廉玛丽学院）
