Edge States Of Topological Insulator Research Articles

Phase transitions, conductance fluctuations and distributions in disordered topological insulator stanene

It is essential to understand to what extent the protected edge states of topological insulators (TIs) can survive against the degradation of the ubiquitous disorders in realistic devices. From a different perspective, disorders can also help to enrich the applications by modulation of the phases in TIs. In this work, the phases and phase transitions in stanene, a two-dimensional TI, have been investigated via the statistical approach based on the random matrix theory. Using a tight binding model with Aderson disorder term and the Landauer–Büttiker formalism, we calculated the conductance of realistic stanene ribbons of tens of nanometers long with random disorders. The calculated phase diagram presents TI in the gap, metal in high energy and ordinary insulator in large disorder region. Increasing the width of the ribbon can significantly enhance the robustness of TI phase against disorders. Due to different underlying symmetries, the metallic phase can be further categorized into unitary and orthogonal classes according to the calculated universal conductance fluctuations. The local density of states is calculated, showing characteristic patterns, which can facilitate the experimental identification of the phases. It is found that different phases have distinguishing statistical distribution of conductance. Whereas at the phase boundary the distribution exhibits intermediate features to show where the phase transition occurs. To reveal the phase evolution process, we further studied the effects of the disorders on respective transmission channels. It is found that when phase transition takes place, the major transmission channels of the old phase are fading and the new channels of the new phase are emerging.

Chern Numbers Associated with the Periodic Toda Lattice

The periodic Toda lattice is solved by exploiting the spectral properties of the Lax operator, in which the boundary states play an important role. We show that these boundary states have a topological origin similar to that of the edge states in topological insulators, and consequently, that the bulk wave functions of the Lax operator yield nontrivial Chern numbers. This implies that the periodic Toda lattice belongs to the same topological class as the Thouless pump. We demonstrate that the cnoidal wave of the Toda lattice exhibits a Chern number of $-1$ per period.

Nonlinear optical absorption and photocurrents in topological insulators

Theory of optical absorption of linearly and circularly polarized light in surface and edge states in topological insulators is developed for a nonlinear in light intensity regime. The absorbance for surface states and the absorption width for the edge states both decease as $1/\sqrt{I}$ at high light intensity $I$. Elastic scattering of the photoexcited electrons and holes is taken into account. The absorption bleaching at high intensity is shown to be strongly suppressed by the elastic scattering. The linear-circular dichroism of one- and two-photon absorption in surface states, emerging in the nonlinear in intensity regime, decreases due to the elastic scattering of photocarriers. Absorption in the edge states of two-dimensional topological insulators also depends on the elastic scattering rate being suppressed when it is stronger that the energy relaxation rate. The linear dichroism - a dependence of the absorption length on the polarization plane orientation - is studied at arbitrary light intensities. We show that a degree of the linear dichroism is governed by the ratio of the elastic and inelastic relaxation times. The photocurrents generated in the edge states by both linearly and circularly polarized light change their intensity dependencies from the linear one at low $I$ to the $\propto \sqrt{I}$ at high intensities. The variation of the photocurrents with both polarization and intensity are strongly dependent on the elastic scattering rate. The considered effects can be observed in THz frequency range at laser intensities used in modern experiments.

Coherent spin transport through helical edge states of topological insulator

We study coherent spin transport through helical edge states of topological insulator tunnel-coupled to metallic leads. We demonstrate that unpolarized incoming electron beam acquires finite polarization after transmission through such a setup provided that edges contain at least one magnetic impurity. The finite polarization appears even in the fully classical regime and is therefore robust to dephasing. There is also a quantum magnetic field-tunable contribution to the polarization, which shows sharp identical Aharonov-Bohm resonances as a function of magnetic flux—with the period hc/2e—and survives at relatively high temperature. We demonstrate that this tunneling interferometer can be described in terms of ensemble of flux-tunable qubits giving equal contributions to conductance and spin polarization. The number of active qubits participating in the charge and spin transport is given by the ratio of the temperature and the level spacing. The interferometer can effectively operate at high temperature and can be used for quantum calculations. In particular, the ensemble of qubits can be described by a single Hadamard operator. The obtained results open wide avenue for applications in the area of quantum computing.

Triplon band splitting and topologically protected edge states in the dimerized antiferromagnet

Search for topological materials has been actively promoted in the field of condensed matter physics for their potential application in energy-efficient information transmission and processing. Recent studies have revealed that topologically invariant states, such as edge states in topological insulators, can emerge not only in a fermionic electron system but also in a bosonic system, enabling nondissipative propagation of quasiparticles. Here we report the topologically nontrivial triplon bands measured by inelastic neutron scattering on the spin-1/2 two-dimensional dimerized antiferromagnet Ba2CuSi2O6Cl2. The excitation spectrum exhibits two triplon bands that are clearly separated by a band gap due to a small alternation in interdimer exchange interaction, consistent with a refined crystal structure. By analytically modeling the triplon dispersion, we show that Ba2CuSi2O6Cl2 is the first bosonic realization of the coupled Su-Schrieffer-Heeger model, where the presence of topologically protected edge states is prompted by a bipartite nature of the lattice.

Generalized nonequilibrium quantum transport of spin and pseudospins: Entanglements and topological phases

General nonequilibrium quantum transport equations are derived for a coupled system of charge carriers, Dirac spin, isospin (or valley spin), and pseudospin, such as either one of the band, layer, impurity, and boundary pseudospins. Limiting cases are obtained for one, two or three different kinds of spin occurring in a system. We show that a characteristic integer number Ns determines the formal form of spin quantum transport equations, irrespective of the type of spins or pseudospins, as well as the maximal entanglement entropy. The results may shed a new perspective on the mechanism leading to zero modes and chiral/helical edge states in topological insulators, integer quantum Hall effect topological insulator (QHE-TI), quantum spin Hall effect topological insulator (QSHE-TI) and Kondo topological insulator (Kondo-TI). It also shed new light in the observed competing weak localization and antilocalization in spin-dependent quantum transport measurements. In particular, a novel mechanism of localization and delocalization, as well as the new mechanism leading to the onset of superconductivity in bilayer systems seems to emerge naturally from torque entanglements in nonequilibrium quantum transport equations of spin and pseudospins. Moreover, the general results may serve as a foundation for engineering approximations of the quantum transport simulations of spintronic devices based on graphene and other 2-D materials such as the transition metal dichalcogenides (TMDs), silicene, as well as based on topological materials exhibiting quantum spin Hall effects. The extension of the formalism to spincaloritronics and pseudo-spincaloritronics is straightforward.

Heating up the quantum spin Hall effect

Topological Matter Taking practical advantage of the topologically protected conducting edge states of topological insulators (TIs) has proven difficult. Semiconductor systems that have been identified as two-dimensional TIs must be cooled down to near liquid helium temperatures to bring out their topological character. Wu et al. fabricated a heterostructure consisting of a monolayer of WTe2 placed between two layers of hexagonal boron nitride and found that its topological properties persisted up to a relatively high temperature of 100 K. Engineering this so-called quantum spin Hall effect in a van der Waals heterostructure makes it possible to apply many established experimental tools and functionalities. Science , this issue p. [76][1] [1]: /lookup/doi/10.1126/science.aan6003

Observation of the quantum spin Hall effect up to 100 kelvin in a monolayer crystal.

A variety of monolayer crystals have been proposed to be two-dimensional topological insulators exhibiting the quantum spin Hall effect (QSHE), possibly even at high temperatures. Here we report the observation of the QSHE in monolayer tungsten ditelluride (WTe2) at temperatures up to 100 kelvin. In the short-edge limit, the monolayer exhibits the hallmark transport conductance, ~e2/h per edge, where e is the electron charge and h is Planck's constant. Moreover, a magnetic field suppresses the conductance, and the observed Zeeman-type gap indicates the existence of a Kramers degenerate point and the importance of time-reversal symmetry for protection from elastic backscattering. Our results establish the QSHE at temperatures much higher than in semiconductor heterostructures and allow for exploring topological phases in atomically thin crystals.

Effective Hamiltonian for protected edge states in graphene

Edge states in topological insulators (TIs) disperse symmetrically about one of the time-reversal invariant momenta $\Lambda$ in the Brillouin zone (BZ) with protected degeneracies at $\Lambda$. Commonly TIs are distinguished from trivial insulators by the values of one or multiple topological invariants that require an analysis of the bulk band structure across the BZ. We propose an effective two-band Hamiltonian for the electronic states in graphene based on a Taylor expansion of the tight-binding Hamiltonian about the time-reversal invariant $M$ point at the edge of the BZ. This Hamiltonian provides a faithful description of the protected edge states for both zigzag and armchair ribbons though the concept of a BZ is not part of such an effective model. We show that the edge states are determined by a band inversion in both reciprocal and real space, which allows one to select $\Lambda$ for the edge states without affecting the bulk spectrum.

Tunneling into and between helical edge states: Fermionic approach

We study the four-terminal junction of spinless Luttinger liquid wires, which describes either a corner junction of two helical edge states of topological insulators or the tunneling from the spinful wire into the helical edge state. We use the fermionic representation and the scattering state formalism, in order to compute the renormalization group (RG) equations for the linear response conductances. We establish our approach by considering a junction between two possibly nonequivalent helical edge states and find an agreement with the earlier analysis of this situation. Tunneling from the tip of the spinful wire to the edge state is further analyzed which requires some modification of our formalism. In the latter case we demonstrate (i) the existence of both fixed lines and conventional fixed points of RG equations, and (ii) certain proportionality relations holding for conductances during renormalization. The scaling exponents and phase portraits are obtained in all cases.

Probing topological protection using a designer surface plasmon structure.

Topological photonic states, inspired by robust chiral edge states in topological insulators, have recently been demonstrated in a few photonic systems, including an array of coupled on-chip ring resonators at communication wavelengths. However, the intrinsic difference between electrons and photons determines that the ‘topological protection' in time-reversal-invariant photonic systems does not share the same robustness as its counterpart in electronic topological insulators. Here in a designer surface plasmon platform consisting of tunable metallic sub-wavelength structures, we construct photonic topological edge states and probe their robustness against a variety of defect classes, including some common time-reversal-invariant photonic defects that can break the topological protection, but do not exist in electronic topological insulators. This is also an experimental realization of anomalous Floquet topological edge states, whose topological phase cannot be predicted by the usual Chern number topological invariants.

Band structure of topological insulators from noise measurements in tunnel junctions

The unique properties of spin-polarized surface or edge states in topological insulators (TIs) make these quantum coherent systems interesting from the point of view of both fundamental physics and their implementation in low power spintronic devices. Here we present such a study in TIs, through tunneling and noise spectroscopy utilizing TI/Al2O3/Co tunnel junctions with bottom TI electrodes of either Bi2Te3 or Bi2Se3. We demonstrate that features related to the band structure of the TI materials show up in the tunneling conductance and even more clearly through low frequency noise measurements. The bias dependence of 1/f noise reveals peaks at specific energies corresponding to band structure features of the TI. TI tunnel junctions could thus simplify the study of the properties of such quantum coherent systems that can further lead to the manipulation of their spin-polarized properties for technological purposes.

Topological classification of dynamical phase transitions

We study the nonequilibrium time evolution of a variety of one-dimensional (1D) and two-dimensional (2D) systems (including SSH model, Kitaev-chain, Haldane model, $p+ip$ superconductor, etc.) following a sudden quench. We prove analytically that topology-changing quenches are always followed by nonanalytical temporal behavior of return rates (logarithm of the Loschmidt echo), referred to as dynamical phase transitions (DPTs) in the literature. Similarly to edge states in topological insulators, DPTs can be classified as being topologically protected or not. In 1D systems the number of topologically protected nonequilibrium time scales are determined by the difference between the initial and final winding numbers, while in 2D systems no such relation exists for the Chern numbers. The singularities of dynamical free energy in the 2D case are qualitatively different from those of the 1D case; the cusps appear only in the first time derivative.

Enhanced thermoelectric performance and anomalous seebeck effects in topological insulators.

Improving the thermoelectric figure of merit zT is one of the greatest challenges in material science. The recent discovery of topological insulators (TIs) offers new promise in this prospect. In this work, we demonstrate theoretically that zT is strongly size dependent in TIs, and the size parameter can be tuned to enhance zT to be significantly greater than 1. Furthermore, we show that the lifetime of the edge states in TIs is strongly energy dependent, leading to large and anomalous Seebeck effects with an opposite sign to the Hall effect. These striking properties make TIs a promising material for thermoelectric science and technology.

Josephson effect in Al/Bi2Se3/Al coplanar hybrid devices

The edge states of Topological Insulators (TI) are protected against backscattering, thanks to the topological properties arising from their band structure. Coupling a TI to a superconductor (S) can induce unconventional effects, including the creation of Majorana bound states (MBS). The fabrication of coplanar hybrid devices is a fundamental step to pave the way to the understanding of proximity effects in topologically non-trivial systems, and to a large variety of experiments aimed at the possible detection of MBS. We discuss the feasibility and some relevant properties of Al–Bi2Se3–Al coplanar proximity devices. Special attention is devoted to the design of the junction, aimed at enhancing the coupling between the electrodes and the TI.

Appearance and disappearance of edge states in topological insulators

We investigate the relationship between spin Chern numbers and edge state properties in general situations, where the time-reversal symmetry may be broken. As an example, we consider a thin film of three-dimensional topological insulators sandwiched between two ferromagnetic insulators with an antiparallel magnetization configuration. A topological quantum spin Hall phase with quantized spin Chern numbers C ± = ±1, and a trivial insulator with C ± = 0 are found in different parameter regions. With tuning parameters, the quantum phase transition between the two phases can occur through closing of the spin spectrum gap rather than energy gap. It is further shown that for a junction between samples with different parameters, appearance of edge states at the interface is always related to the mismatch of spin Chern numbers, independent of symmetries.

Renormalization group approach for the scattering off a single Rashba impurity in a helical liquid

The occurrence of two-particle inelastic backscattering has been conjectured in helical edge states of topological insulators and is expected to alter transport. Here, by using a renormalization group approach, we provide a microscopic derivation of this process, in the presence of a time-reversal invariant Rashba impurity potential. We are able to prove that such an effect only occurs in the presence of electron-electron interactions. Furthermore, we find that the linear conductance as a function of temperature exhibits a crossover between two scaling behaviors, ${T}^{4K}$ for $K>1/2$ and ${T}^{8K\ensuremath{-}2}$ for $K<1/2$, with $K$ the Luttinger parameter.

Continuum theory of edge states of topological insulators: variational principle and boundary conditions

We develop a continuum theory to model low energy excitations of a generic four-band time reversal invariant electronic system with boundaries. We propose a variational energy functional for the wavefunctions which allows us to derive natural boundary conditions valid for such systems. Our formulation is particularly suited for developing a continuum theory of the protected edge/surface excitations of topological insulators both in two and three dimensions. By a detailed comparison of our analytical formulation with tight binding calculations of ribbons of topological insulators modelled by the Bernevig–Hughes–Zhang (BHZ) Hamiltonian, we show that the continuum theory with a natural boundary condition provides an appropriate description of the low energy physics.

Robustness of helical edge states in topological insulators

Topological insulators (TIs) are materials having an energy band gap in the bulk and conducting helical electronic states on the surface. The helical states are protected by time-reversal symmetry and thus are expected to be robust against static disorder scattering. In this work we report an atomistic first principles analysis of disorder scattering in two-probe transport junctions made of three-dimensional TI material Bi${}_{2}$Se${}_{3}$. The robustness of the device against disorder scattering is determined quantitatively. Examining many different scattering configurations, a general trend emerges on how strong is the perturbing potential and how it is spatially distributed so that it can derail the helical states on the Bi${}_{2}$Se${}_{3}$ surfaces.

Conductance fluctuations induced by bulk state in quasi-one-dimensional strips of topological insulator

We numerically calculate the conductance of a topological insulator confined as quasi-one-dimensional strips using a four-band Hamiltonian. The conductance is nearly unchanged in the presence of a short-range disorder when the Fermi level is located in the bulk band gap. Helical edge states of topological insulators are no longer protected against the disorder, and scattering takes place if the bulk state coexists. Both the magnitude of conductance fluctuations and the average conductance are found to vary nonmonotonically in the latter regime with the strength of disorder, where the fluctuation amplitude is reduced when the average conductance is around $2{e}^{2}/h$. The scattering of the topological states is hence evidenced to be nontrivially affected by the coupling with the bulk state.