Quantum Spin Hall Regime Research Articles

Quantitative theory of backscattering in topological HgTe and (Hg,Mn)Te quantum wells: Acceptor states, Kondo effect, precessional dephasing, and bound magnetic polaron

We present the theory and numerical evaluations of the backscattering rate determined by acceptor holes or Mn spins in HgTe and (Hg,Mn)Te quantum wells in the quantum spin Hall regime. The role of anisotropic s-p and sp-d exchange interactions, Kondo coupling, Luttinger liquid effects, precessional dephasing, and bound magnetic polarons is quantified. The determined magnitude and temperature dependence of conductance are in accord with experimental results for HgTe and (Hg,Mn)Te quantum wells.

Local equilibrium charge and spin currents in two-dimensional topological systems

We study the equilibrium and nonequilibrium electronic transport properties of multiprobe topological systems using a combination of the Landauer-B\"uttiker approach and nonequilibrium Green's functions techniques. We obtain general expressions for both nonequilibrium and equilibrium local electronic currents that, by suitable projections, allow one to compute charge, spin, valley, and orbital currents. We show that external magnetic fields give rise to equilibrium charge currents in mesoscopic system and study the latter in the quantum Hall regime. Likewise, a spin-orbit interaction leads to local equilibrium spin currents, that we analyze in the quantum spin Hall regime. We show that an accurate theoretical assessment of the equilibrium currents is quite challenging and propose a two-measurement protocol that facilitates a comparison between experiment and theory.

Topological viewpoint of two-dimensional group III–V and IV–IV compounds in the presence of electric field and spin–orbit coupling by density functional theory and tight-binding model

Using the tight-binding (TB) model and density functional theory, the topological invariant of the two-dimensional (2D) group III–V and IV–IV compounds are studied in the absence and the presence of an external perpendicular electric field and spin–orbit coupling. It will be recognized that a critical value of these parameters changes the topological invariant of 2D graphene-like compounds. The significant effects of an external electric field and spin–orbit coupling are considered to the two-center overlap integrals of the Slater–Koster model involved in band structures, changing band-gap, and tuning the topological phase transition between ordinary and quantum spin Hall regime. These declare the good consistency between two theories: TB and density functional. So, this study reveals topological phase transition in these materials. Our finding paves a way to extend an effective Hamiltonian, and may instantly clear some computation aspects of the study in the field of spintronic based on the first-principles methods.

Topological phase transitions induced by disorder in magnetically doped (Bi,Sb)2Te3 thin films

We study disorder induced topological phase transitions in magnetically doped (Bi, Sb)$_2$Te$_3$ thin films, by using large scale transport simulations of the conductance through a disordered region coupled to reservoirs in the quantum spin Hall regime. Besides the disorder strength, the rich phase diagram also strongly depends on the magnetic exchange field, the Fermi level, and the initial topological state in the undoped and clean limit of the films. In an initially trivial system at non-zero exchange field, varying the disorder strength can induce a sequence of transitions from a normal insulating, to a quantum anomalous Hall, then a spin-Chern insulating, and finally an Anderson insulating state. While for a system with topology initially, a similar sequence, but only starting from the quantum anomalous Hall state, can be induced. Varying the Fermi level we find a similarly rich phase diagram, including transitions from the quantum anomalous Hall to the spin-Chern insulating state via a state that behaves as a mixture of a quantum anomalous Hall and a metallic state, akin to recent experimental reports.

Local Chern marker of smoothly confined Hofstadter fermions

The engineering of topological non-trivial states of matter, using cold atoms, has made great progress in the last decade. Driven by experimental successes, it has become of major interest in the cold atom community. In this work we investigate the time-reversal invariant Hofstadter model with an additional confining potential. By calculating a local spin Chern marker we find that topologically non-trivial phases can be observed in all considered trap geometries. This holds also for spin-orbit coupled fermions, where the model exhibits a quantum spin Hall regime at half filling. Using dynamical mean-field theory, we find that interactions compete against the confining potential and induce a topological phase transition depending on the filling of the system. Strong interactions furthermore yield a magnetic edge, which is localized through the interplay of the density distribution and the underlying topological band structure.

Bias-voltage-induced topological phase transition in finite size quantum spin Hall systems in the presence of a transverse electric field

Using the tight-binding BHZ model and Landauer-B\"uttiker formalism, the topological invariant of the finite width of ribbons of HgTe/CdTe quantum well is studied in the absence and presence of an external transverse electric field. It will be recognized that a critical current changes topological invariant of ribbons of quantum well. This topological phase transition, which occurred by adjustment of the bias voltage, depends on the width of the sample and the gate voltage. The profound effects of an external transverse electric field are considered to the separation of spin-up and spin-down band structures, decreasing band gap and tuning the topological phase transition between ordinary and quantum spin Hall regime. These declares the transverse electric field amplifies the quantum spin Hall regime and causes inducing the topological phase transition in ribbons of quantum well. Our finding may instantly clear some practical aspects of the study in the field of spintronic for employment in spin-based devices.

Impurity-Controlled Crystal Growth in Low-Dimensional Bismuth Telluride

Topological insulators, such as layered Bi2Te3, exhibit extraordinary properties, manifesting profoundly only at nanoscale thicknesses. However, it has been challenging to synthesize these structures with controlled thicknesses. Here, control over the thickness of solvothermally grown Bi2Te3 nanosheets is demonstrated by manipulating the crystal growth through select and controlled impurity atom addition. By a comprehensive analysis of the growth mechanism and intentional addition of Fe impurity, we demonstrate that the nucleation and growth of few-layer nanosheets of Bi2Te3 can be stabilized in solution. Via optimization of the Fe concentration, nanosheets thinner than 6 nm, and as thin as 2 nm, can be synthesized. Such thicknesses are smaller than the anticipated critical thickness for the transition of topological insulators to the quantum spin Hall regime.

Gapless Andreev bound states in the quantum spin Hall insulator HgTe.

In recent years, Majorana physics has attracted considerable attention because of exotic new phenomena and its prospects for fault-tolerant topological quantum computation. To this end, one needs to engineer the interplay between superconductivity and electronic properties in a topological insulator, but experimental work remains scarce and ambiguous. Here, we report experimental evidence for topological superconductivity induced in a HgTe quantum well, a 2D topological insulator that exhibits the quantum spin Hall (QSH) effect. The a.c. Josephson effect demonstrates that the supercurrent has a 4π periodicity in the superconducting phase difference, as indicated by a doubling of the voltage step for multiple Shapiro steps. In addition, this response like that of a superconducting quantum interference device to a perpendicular magnetic field shows that the 4π-periodic supercurrent originates from states located on the edges of the junction. Both features appear strongest towards the QSH regime, and thus provide evidence for induced topological superconductivity in the QSH edge states.

Probing topological transitions in HgTe/CdTe quantum wells by magneto-optical measurements

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 for possible signatures of the transition between the QSH and QH regimes. In the absorption spectrum, several peaks arise due to nonequidistant Landau levels in both regimes. However, in the QSH regime, we find an additional absorption peak at low energies in the finite-strip geometry. This peak arises due to the presence of edge states in this geometry and persists for any Fermi level in the QSH regime, while in the QH regime the peak vanishes if the Fermi level is situated in the bulk gap. Thus, by sweeping the gate voltage, it is possible to experimentally distinguish between the QSH and QH regimes due to this signature. Moreover, we investigate the effect of spin-orbit coupling and finite temperature on this measurement scheme.

Multiple Quantum Phases in Graphene with Enhanced Spin-Orbit Coupling: From the Quantum Spin Hall Regime to the Spin Hall Effect and a Robust Metallic State

We report an intriguing transition from the quantum spin Hall phase to the spin Hall effect upon segregation of thallium adatoms adsorbed onto a graphene surface. Landauer-Büttiker and Kubo-Greenwood simulations are used to access both edge and bulk transport physics in disordered thallium-functionalized graphene systems of realistic sizes. Our findings not only quantify the detrimental effects of adatom clustering in the formation of the topological state, but also provide evidence for the emergence of spin accumulation at opposite sample edges driven by spin-dependent scattering induced by thallium islands, which eventually results in a minimum bulk conductivity ∼4e²/h, insensitive to localization effects.

Nonlocal transport in a hybrid two-dimensional topological insulator

We study nonlocal resistance in an H-shaped two-dimensional HgTe/CdTe quantum well consisting of an injector and a detector, both of which can be tuned in the quantum spin Hall or metallic spin Hall regime. Because of strong spin-orbit interaction, there always exists the spin Hall effect and nonlocal resistance in the HgTe/CdTe quantum well. We find that when both the detector and the injector are in the quantum spin Hall regime, the nonlocal resistance is quantized at $0.25\frac{h}{{e}^{2}}$, which is robust against weak disorder scattering and small magnetic field. When the detector or injector is beyond this regime, the nonlocal resistance decreases rapidly and will be strongly suppressed by disorder and magnetic field. In the presence of a strong magnetic field, the quantum spin Hall regime will be switched into the quantum Hall regime, and the nonlocal resistance will disappear. The nonlocal signal and its various manifestations in different hybrid regimes originate from the special band structure of the HgTe/CdTe quantum well, and they can be considered as the fingerprint of the helical quantum spin Hall edge states in a two-dimensional topological insulator.

Growth and properties of HgTe quantum wells -A topic review

The properties of HgTe/Hg1−x Cdx Te superlattices (SLs) and quantum wells, (QWs), are of fundamental interest; SLs are potentially useful for infrared opto-electronic applications and QWs have aroused much interest due to recent evidence that they are topological insulators with dissipation less transport in edge states in the absence of a magnetic field. Depending on the HgTe thickness , the band structure has either the normal sequence of quantum well states or an inverted one when nm, in which the electron E1 subband is a valence subband and lies below the heavy hole H1 subband. This has been shown to lead to a very large Rashba spin–orbit (s–o) splitting of up to meV in HgTe/Hg1−x Cdx Te QWs with an inverted band structure. This as well as other transport results will be presented. These properties of the HgTe/Hg1−x Cdx Te QW, large s–o splitting and an inverted band structure, have recently been shown to result in the quantum spin Hall effect. In the quantum spin Hall effect, carriers with opposite spin move in opposite directions on a given edge in an insulator in the absence of a magnetic field. Convincing evidence for non-local edge channel transport in HgTe based QWs has been recently published. The results demonstrate that quantum transport through helical edge channels is dissipation less at zero magnetic field and agree quantitatively with the theory of the quantum spin Hall effect. Images of current in the edge channels by means of scanning gate microscopy have revealed well localized scattering sites which perturb the quantum spin Hall edge states on a sub-micrometer scale. In the micrometer sized regions between the scattering sites, current appeared to propagate unperturbed Recently, images of edge channels in HgTe QWs in the quantum spin Hall regime have been obtained by means of their magnetic fields with a scanning superconducting quantum interference device (SQUID). The edge channels were observed to dominate transport in devices with edges much longer than the scattering length, and to persist even in the presence of bulk conduction.

Revivals of Zitterbewegung in the Bernevig—Hughes—Zhang model in the quantum spin Hall regime with a laser pulse

We theoretically investigate the Zitterbewegung (ZB) behavior of electrons in the Bernevig—Hughes—Zhang model with a short laser pulse. To obtain a steady picture, we fix the electron on the Landau levels with a magnetic field. The ZB motion and the electromagnetic radiations in the quantum spin Hall regime are given. We find that over a shorter time, the electromagnetic radiations show a quasi-classical cyclotron oscillation, while over a longer time, they exhibit a clear revival picture. The resulting revival time and excited electric field are large enough to make experimental detection accessible.

Imaging currents in HgTe quantum wells in the quantum spin Hall regime

The quantum spin Hall (QSH) state is a state of matter characterized by a non-trivial topology of its band structure, and associated conducting edge channels. The QSH state was predicted and experimentally demonstrated to be realized in HgTe quantum wells. The existence of the edge channels has been inferred from local and non-local transport measurements in sufficiently small devices. Here we directly confirm the existence of the edge channels by imaging the magnetic fields produced by current flowing in large Hall bars made from HgTe quantum wells. These images distinguish between current that passes through each edge and the bulk. On tuning the bulk conductivity by gating or raising the temperature, we observe a regime in which the edge channels clearly coexist with the conducting bulk, providing input to the question of how ballistic transport may be limited in the edge channels. Our results represent a versatile method for characterization of new QSH materials systems.

Fabrication of samples for scanning probe experiments on quantum spin Hall effect in HgTe quantum wells

We present a fabrication process for devices on HgTe quantum wells through which the quantum spin Hall regime can be reached without the use of a top-gate electrode. We demonstrate that a nominally undoped HgTe quantum well can be tuned from p-type to n-type, crossing through the quantum spin Hall regime, using only a back-gate hundreds of microns away. Such structures will enable scanning probe investigations of the quantum spin Hall effect that would not be possible in the presence of a gate electrode on the surface of the wafer. All processes are kept below 80 °C to avoid degradation of the heat-sensitive HgTe quantum wells.

Spin polarization of the quantum spin Hall edge states

While the helical character of the edge channels responsible for charge transport in the quantum spin Hall regime of a two-dimensional topological insulator is by now well established, an experimental confirmation that the transport in the edge channels is spin-polarized is still outstanding. We report experiments on nanostructures fabricated from HgTe quantum wells with an inverted band structure, in which a split gate technique allows us to combine both quantum spin Hall and metallic spin Hall transport in a single device. In these devices, the quantum spin Hall effect can be used as a spin current injector and detector for the metallic spin Hall effect, and vice versa, allowing for an all-electrical detection of spin polarization.

Chiral heat transport in driven quantum Hall and quantum spin Hall edge states

We consider a model for an edge state of electronic systems in the quantum Hall regime with filling $\ensuremath{\nu}=1$ and in the quantum spin Hall regime. In both cases, the system is in contact with two reservoirs by tunneling at point contacts. Both systems are locally driven by applying an ac voltage in one of the contacts. By weakly coupling them to a third reservoir, the transport of the generated heat is studied in two different ways: (i) when the third reservoir acts as a thermometer, the local temperature is sensed and (ii) when the third reservoir acts as a voltage probe, the time-dependent local voltage is sensed. Our results indicate a chiral propagation of the heat along the edge in the quantum Hall and in the quantum spin Hall cases (if the injected electrons are spin polarized). We also show that a analogous picture is obtained if instead of heating by ac driving the system is put in contact to a stationary reservoir at a higher temperature. In both cases, the temperature profile shows that the electrons along the edge thermalize with the closest ``upstream'' reservoir.

Kondo Effect in the Helical Edge Liquid of the Quantum Spin Hall State

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 power laws characteristic of a "local helical liquid," while for strong interactions, transport proceeds by weak tunneling through the impurity where only half an electron charge is transferred in each tunneling event.