Spin Chern numbers and time-reversal-symmetry-broken quantum spin Hall effect

The quantum spin Hall （QSH） effect is considered to be unstable to perturbations violating the time-reversal （TR） symmetry. We review some recent developments in the search of the QSH effect in the absence of the TR symmetry. The possibility to realize a robust QSH effect by artificial removal of the TR symmetry of the edge states is explored. As a useful tool to characterize topological phases without the TR symmetry, the spin-Chern number theory is introduced.

Quantum spin Hall and quantum valley Hall effects in trilayer graphene and their topological structures

The present study pertains to the trilayer graphene in the presence of spin orbit coupling to probe the quantum spin/valley Hall effect. The spin Chern-number Cs for energy-bands of trilayer graphene having the essence of intrinsic spin-orbit coupling is analytically calculated. We find that for each valley and spin, Cs is three times larger in trilayer graphene as compared to single layer graphene. Since the spin Chern-number corresponds to the number of edge states, consequently the trilayer graphene has edge states, three times more in comparison to single layer graphene. We also study the trilayer graphene in the presence of both electric-field and intrinsic spin-orbit coupling and investigate that the trilayer graphene goes through a phase transition from a quantum spin Hall state to a quantum valley Hall state when the strength of the electric field exceeds the intrinsic spin coupling strength. The robustness of the associated topological bulk-state of the trilayer graphene is evaluated by adding various perturbations such as Rashba spin-orbit （RSO） interaction αR, and exchange-magnetization M. In addition, we consider a theoretical model, where only one of the outer layers in trilayer graphene has the essence of intrinsic spin-orbit coupling, while the other two layers have zero intrinsic spin-orbit coupling. Although the first Chern number is non-zero for individual valleys of trilayer graphene in this model, however, we find that the system cannot be regarded as a topological insulator because the system as a whole is not gaped.

Quantum spin Hall insulators in chemically functionalized As(110)and Sb(110)films

We propose a new type of quantum spin Hall （QSH） insulator in chemically functionalized As （110） and Sb （110） film. According to first-principles calculations, we find that metallic As （110） and Sb （110） films become QSH insulators after being chemically functionalized by hydrogen （H） or halogen （C1 and Br） atoms. The energy gaps of the functionalized films range from 0.121 eV to 0.304 eV, which are sufficiently large for practical applications at room temperature. The energy gaps originate from the spin-orbit coupling （SOC）. The energy gap increases linearly with the increase of the SOC strength λ/λ0. The Z2 invariant and the penetration depth of the edge states are also calculated and studied for the functionalized films.

First-Principles Calculation on Geometric,Electronic and Optical Properties of Fully Fluorinated Stanene:a Large-Gap Quantum Spin Hall Insulator

The searches for large-gap quantum spin Hall insulators are important for both practical and fundamental inter- ests. In this work, we present a theoretical observation of the two-dimensional fully fluorinated stanene （SnF） by means of density functional theory. Remarkably, a significant spin-orbit coupling is observed for the SnF monolayer in the valence band at the F point, with a considerable indirect band gap of 278 meV. The direct gap of the SnF monolayer is at the F point, which is slightly larger by as much as 381 meV. In addition, the elastic modulus of the SnF monolayer is about 20J/m^2, which is comparable with the in-plane stiffness of black phos- phorus monolayer along the x-direction （~28.94 J/m^2）. Finally, the optical properties of stanene, SnF monolayer and stanene/SnF bilayer are calculated, in which the stanene/SnF bilayer is supposed to be an attractive sunlight absorber.

Finite size effects on the quantum spin Hall state in HgTe quantum wells under two different types of boundary conditions

The finite size effect in a two-dimensional topological insulator can induce an energy gap Eg in the spectrum of helical edge states for a strip of finite width. In a recent work, it has been found that when the spin--orbit coupling due to bulk-inversion asymmetry is taken into account, the energy gap Eg of the edge states features an oscillating exponential decay as a function of the strip width of the inverted HgTe quantum well. In this paper, we investigate the effects of the interface between a topological insulator and a normal insulator on the finite size effect in the HgTe quantum well by means of the numerical diagonalization method. Two different types of boundary conditions, i.e., the symmetric and asymmetric geometries, are considered. It is found that due to the existence of the interface between topological insulator and normal insulator this oscillatory pattern on the exponential decay induced by bulk-inversion asymmetry is modulated by the width of normal insulator regions. With the variation of the width of normal insulator regions, the shift of the Dirac point of the edge states in the spectrum and the energy gap Eg closing point in the oscillatory pattern can occur. Additionally, the effect of the spin-orbit coupling due to structure-inversion asymmetry on the finite size effects is also investigated.

Coexistence of giant Rashba spin splitting and quantum spin Hall effect in H–Pb–F

Rashba spin splitting(RSS)and quantum spin Hall effect(QSHE)have attracted enormous interest due to their great significance in the application of spintronics.In this work,we theoretically proposed a new two-dimensional(2D)material H–Pb–F with coexistence of giant RSS and quantum spin Hall effec by using the ab initio calculations.Our results show that H–Pb–F possesses giant RSS(1.21 eV·A)and the RSS can be tuned up to 4.16 e V·A by in-plane biaxial strain,which is a huge value among 2D materials.Furthermore,we also noticed that H–Pb–F is a 2D topological insulator(TI)duo to the strong spin–orbit coupling(SOC)interaction,and the large topological gap is up to 1.35 e V,which is large enough for for the observation of topological edge states at room temperature.The coexistence of giant RSS and quantum spin Hall effect greatly broadens the potential application of H–Pb–F in the field of spintronic devices.

The robustness of the quantum spin Hall effect to the thickness fluctuation in HgTe quantum wells

The quantum spin Hall effect （QSHE） was first realized in HgTe quantum wells （QWs）, which remain the only known two-dimensional topological insulator so far. In this paper, we have systematically studied the effect of the thickness fluctuation of HgTe QWs on the QSHE. We start with the case of constant mass with random distributions, and reveal that the disordered system can be well described by a virtual uniform QW with an effective mass when the number of components is small. When the number is infinite and corresponds to the real fluctuation, we find that the QSHE is not only robust, but also can be generated by relatively strong fluctuation. Our results imply that the thickness fluctuation does not cause backscattering, and the QSHE is robust to it.

Quantum spin Hall effect in a square-lattice model under a uniform magnetic field

We study a toy square-lattice model under a uniform magnetic field. Using the Landauer Biittiker fornmla, we calculate the transport properties of the system on a two-terminal, a four-terminal and a six-terminM device. W＇e find that the quantum spin Hall （QSH） effect appears ill energy ranges where the spin-up and spin-down subsystems have different filling factors. We also study the robustness of the resulting QSH effect and find that it is robust when the Fermi levels of both spin subsystems are far away from the energy plateaus but is fragile when the Fermi level of any spin subsystem is near the energy plateaus. These results provide an example of the QSH effect with a physical origin other than time-reversal （TR） preserving spin-orbit coupling （SOC）.

Discovery of Two-Dimensional Quantum Spin Hall Effect in Triangular Transition-Metal Carbides

Though the quantum spin Hall effect（QSHE） in two-dimensional（2 D） crystals has been widely explored, the experimental realization of quantum transport properties is only limited to HgTe/CdTe or InAs/GaSb quantum wells. Here we employ a tight-binding model on the basis of d（z^2）, d（xy）, and d（x^2-y^2） orbitals to propose QSHE in the triangular lattice, which are driven by a crossing of electronic bands at the Γ point. Remarkably, 2 D oxidized Mxenes W2 M2 C3 are ideal materials with nontrivial gap of 0.12 eV, facilitating room-temperature observations in experiments. We also find that the nontrivially topological properties of these materials are sensitive to the cooperative effect of the electron correlation and spin-orbit coupling. Due to the feasible exfoliation from its 3 D MAX phase, our work paves a new direction towards realizing QSHE with low dissipation.

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.

Hydrogenated antimonene as quantum spin Hall insulator:A first-principles study

Using first-principles calculations based on density functional theory(DFT), the structural and electronic properties of hydrogenated antimonene have been systematically investigated. Phonon dispersion and molecular dynamics(MD)simulation reveal that fully hydrogenated(FH) antimonene has high dynamic stability and could be synthesized. A newσ-type Dirac cone related to Sb-px,y orbitals is found in FH antimonene, which is robust to tensile strain. Noticeably, the spin orbital coupling(SOC) opens a quantum spin Hall(QSH) gap of 425 meV at the Dirac cone, sufficiently large for practical applications at room temperature. Semi-hydrogenated antimonene is a non-magnetic metal. Our results show that FH antimonene may have great potential applications in next generation high-performance devices.

Quantum spin Hall effect in inverted InAs/GaSb quantum wells

We review the recent experimental progress towards observing quantum spin Hall effect in inverted InAs/GaSb quantum wells （QWs）. Low temperature transport measurements in the hybridization gap show bulk conductivity of a non-trivial origin, while the length and width dependence of con- ductance in this regime show strong evidence for the existence of helical edge modes proposed by Liu et al. [Phys. Rev. Lett., 2008, 100： 236601]. Surprisingly, edge modes persist in spite of compa- rable bulk conduction and show only weak dependence on magnetic field. We elucidate that seeming independence of edge on bulk transport comes due to the disparity in Fermi-wave vectors between the bulk and the edge, leading to a total internal reflection of the edge modes.

MoTe2 is a good match for Gel by preserving quantum spin Hall phase

Quantum spin Hall （QSH） insulator is a new class of materials that is quickly becoming mainstream in condensed-matter physics. The main obstacle for the development of QSH insulators is that their strong interactions with substrates make them difficult to study experimentally. In this study, using density functional theory, we discovered that MoTe2 is a good match for a GeI monolayer. The thermal stability of a van der Waals GeI/MoTe2 heterosheet was examined via molecular-dynamics simulations. Simulated scanning tunneling microscopy revealed that the GeI monolayer perfectly preserves the bulked honeycomb structure of MoTe2. The GeI on MoTe2 was confirmed to maintain its topological band structure with a sizable indirect bulk bandgap of 0.24 eV by directly calculating the spin Chern number to be -1. As expected, the electron mobility of the GeI is enhanced by MoTe2 substrate restriction. According to deformation- potential theory with the effective-mass approximation, the electron mobility of GeI/MoTe2 was estimated as 372.7 cm^2·s^-1·V^-1 at 300 K, which is 20 times higher than that of freestanding GeI. Our research shows that traditional substrates always destroy the topological states and hinder the electron transport in QSH insulators, and pave way for the further realization and utilization of QSH insulators at room temperature.

Gallium bismuth halide GaBi-X2 （X = I, Br, CI） monolayers with distorted hexagonal framework： Novel room- temperature quantum spin Hall insulators

Quantum spin Hall （QSH） insulators with a large topologically nontrivial bulk gap are crucial for future applications of the QSH effect. Among these, group III-V monolayers and their halides, which have a chair structure （regular hexagonal framework）, have been widely studied. Using first-principles calculations, we formulate a new structure model for the functionalized group III-V monolayers, which consist of rectangular GaBi-X2 （X = I, Br, C1） monolayers with a distorted hexagonal framework （DHF）. These structures have a far lower energy than the GaBi-X2 monolayers with a chair structure. Remarkably, the DHF GaBi-X2 monolayers are all QSH insulators, which exhibit sizeable nontrivial band gaps ranging from 0.17 to 0.39 eV. The band gaps can be widely tuned by applying different spin-orbit coupling strengths, resulting in a distorted Dirac cone.

Spin Polarization of Fractional Quantum Hall States with <i>ν</i><2

The spin polarization of a fractional quantum Hall state shows very interesting properties. The curve of polarization versus magnetic field has wide plateaus. The fractional quantum Hall effect is caused by the Coulomb interaction because the 2D electron system without the Coulomb interaction yields no energy gap at the fractional filling factor. Therefore, the wide plateau in the polarization curve is also caused by the Coulomb interaction. When the magnetic field is weak, some electrons have up-spins and the others down-spins. Therein the spin-exchange transition occurs between two electrons with up and down spins via the Coulomb interaction. Then the charge distribution before the transition is the same as one after the transition. So these two states have the same classical Coulomb energy. Accordingly, the partial Hamiltonian composed of the spin exchange interaction should be treated exactly. We have succeeded in diagonalizing the spin exchange interaction for the first and second nearest electron pairs. The theoretical results reproduce the wide plateaus very well. If the interval modulations between Landau orbitals are taken into the Hamiltonian, the total energy has the Peierls instability. We can diagonalize the Hamiltonian with the interval modulation. The results reproduce wide plateaus and small shoulders which are in good agreement with the experimental data.

Finite size effects on helical edge states in HgTe quantum wells with the spin orbit coupling due to bulk- and structure-inversion asymmetries

There is a quantum spin Hall state in the inverted HgTe quantum well, characterized by the topologically protected gapless helical edge states lying within the bulk gap. It has been found that for a strip of finite width, the edge states on the two sides can couple together to produce a gap in the spectrum. The phenomenon is called the finite size effect in quantum spin Hall systems. In this paper, we investigate the effects of the spin-orbit coupling due to bulk- and structure-inversion asymmetries on the finite size effect in the HgTe quantum well by means of the numerical diagonalization method. When the bulk-inversion asymmetry is taken into account, it is shown that the energy gap Eg of the edge states due to the finite size effect features an oscillating exponential decay as a function of the strip width of the HgTe quantum well. The origin of this oscillatory pattern on the exponential decay is explained. Furthermore, if the bulk- and structure-inversion asymmetries are considered simultaneously, the structure-inversion asymmetry will induce a shift of the energy gap Eg closing point. Finally, based on the roles of the bulk- and structure-inversion asymmetries on the finite size effects, a way to realize the quantum spin Hall field effect transistor is proposed.

冷原子中的光致量子自旋Hall效应

讨论在冷原子体系中利用光致规范场模拟量子自旋霍尔效应.通过对6Li这样的多能级原子施加适当的激光脉冲,构造出具二重兼并的能量暗态(能量为零的本征态).类似于自旋二分量体系,可以分别定义2个暗态为自旋向上及自旋向下态.当原子在空间分布的激光场中运动时,其所感受到的有效自旋相关规范场将导致可观测的自旋霍尔电流.

On the mystery of the absence of a spin-orbit gap in scanning tunneling microscopy spectra of germanene

Germanene,the germanium analogue of graphene,shares many properties with its carbon counterpart.Both materials are two-dimensional materials that host Dirac fermions.There are,however,also a few important differences between these two materials:(1)graphene has a planar honeycomb lattice,whereas germanene’s honeycomb lattice is buckled and(2)the spin-orbit gap in germanene is predicted to be about three orders of magnitude larger than the spin-orbit gap in graphene(24 meV for germanene versus 20μeV for graphene).Surprisingly,scanning tunneling spectra recorded on germanene layers synthesized on different substrates do not show any sign of the presence of a spin-orbit gap.To date the exact origin of the absence of this spin-orbit gap in the scanning tunneling spectra of germanene has remained a mystery.In this work we show that the absence of the spin-orbit gap can be explained by germanene’s exceptionally low work function of only 3.8 eV.The difference in work function between germanene and the scanning tunneling microscopy tip(the work functions of most commonly used STM tips are in the range of 4.5 to 5.5 eV)gives rise to an electric field in the tunnel junction.This electric field results in a strong suppression of the size of the spin-orbit gap.

Two-dimensional transport and strong spin-orbit interaction in SrMnSb_2

We have carried out magneto-transport measurements for single crystal SrMnSb2. Clear Shubnikov-de Haas oscil- lations were resolved at relatively low magnetic field around 4 T, revealing a quasi-2D Fermi surface. We observed a development of quantized plateaus in Hall resistance （Rxy） at high pulsed fields up to 60 T. Due to the strong 2D confine- ment and layered properties of the samples, we interpreted the observation as bulk quantum Hall effect that is contributed by the parallel 2D conduction channels. Moreover, the spin degeneracy was lifted leading to Landau level splitting. The presence of anisotropic g factor and the formation of the oscillation beating pattern reveal a strong spin-orbit interaction in the SrMnSb2 system.

掺杂量子自旋霍尔态后所形成超导的平均场计算

给出了对易面t-λ模型详细平均场理论和计算的结果,特别是给出了平均场相图.研究表明,掺杂纯的量子自旋霍尔(QSH)态后可获得2种结论:1)相互作用强度较弱时,QSH态和超导态之间可发生一级相变;2)相互作用强度较强时,系统经过一个连续相变进入2相共存区,并发生QSH态消失的一级相变.计算结果表明,平均场理论虽然体现了对称破缺相的特征,但是其忽略了涨落,致使其得到QSH态与超导态共存的错误结论,从而使得平均场理论无法准确描述因拓扑激发所引起的相变.