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.

From magnetically doped topological insulator to the quantum anomalous Hall effect

Quantum Hall effect （QHE）, as a class of quantum phenomena that occur in macroscopic scale, is one of the most important topics in condensed matter physics. It has long been expected that QHE may occur without Landau levels so that neither external magnetic field nor high sample mobility is required for its study and application, Such a QHE free of Landau levels, can appear in topological insulators （TIs） with ferromagnetism as the quantized version of the anomalous Hall effect, i.e., quantum anomalous Hall （QAH） effect. Here we review our recent work on experimental realization of the QAH effect in magnetically doped TIs. With molecular beam epitaxy, we prepare thin films of Cr-doped （Bi,Sb）2Te3 TIs with well- controlled chemical potential and long-range ferromagnetic order that can survive the insulating phase. In such thin films, we eventually observed the quantization of the Hall resistance at h/e2 at zero field, accompanied by a considerable drop in the longitudinal resistance. Under a strong magnetic field, the longitudinal resistance vanishes, whereas the Hall resistance remains at the quantized value. The realization of the QAH effect provides a foundation for many other novel quantum phenomena predicted in TIs, and opens a route to practical applications of quantum Hall physics in low-power-consumption electronics.

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.

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.

Spin direction dependent quantum anomalous Hall effect in two-dimensional ferromagnetic materials

We propose a scheme for realizing the spin direction-dependent quantum anomalous Hall effect(QAHE)driven by spin-orbit couplings(SOC)in two-dimensional(2D)materials.Based on the sp^(3)tight-binding(TB)model,we find that these systems can exhibit a QAHE with out-of-plane and in-plane magnetization for the weak and strong SOC,respectively,in which the mechanism of quantum transition is mainly driven by the band inversion of p_(x,y)/p_(z)orbitals.As a concrete example,based on first-principles calculations,we realize a real material of monolayer 1T-SnN_(2)/PbN_(2)exhibiting the QAHE with in-plane/out-of-plane magnetization characterized by the nonzero Chern number C and topological edge states.These findings provide useful guidance for the pursuit of a spin direction-dependent QAHE and hence stimulate immediate experimental interest.

Progress on 2D topological insulators and potential applications in electronic devices

Two-dimensional topological insulators(2DTIs)have attracted increasing attention during the past few years.New 2DTIs with increasing larger spin-orbit coupling(SOC)gaps have been predicted by theoretical calculations and some of them have been synthesized experimentally.In this review,the 2DTIs,ranging from single element graphene-like materials to bi-elemental transition metal chalcogenides(TMDs)and to multi-elemental materials,with different thicknesses,structures,and phases,have been summarized and discussed.The topological properties(especially the quantum spin Hall effect and Dirac fermion feature)and potential applications have been summarized.This review also points out the challenge and opportunities for future 2DTI study,especially on the device applications based on the topological properties.

Classification of Topological Insulators with Time-Reversal and Inversion Symmetry

In this paper, we find that topological insulators with time-reversal symmetry and inversion symmetry featuring two-dimensional quantum spin Hall （QSH） state can be divided into 16 classes, which are characterized by four Z2 topological variables ζk =0, 1 at four points with high symmetry in the Brillouin zone. We obtain the corresponding edge states for each one of these sixteen classes of QSHs. In addition, it is predicted that massless fermionic excitations appear at the quantum phase transition between different QSH states. In the end, we also briefly discuss the threedimensional case.

Mn-doped topological insulators: a review

Topological insulators (TIs) host robust edge or surface states protected by time-reversal symmetry (TRS), which makes them prime candidates for applications in spintronic devices. A promising avenue of research for the development of functional TI devices has involved doping of three-dimensional (3D) TI thin film and bulk materials with magnetic elements. This approach aims to break the TRS and open a surface band gap near the Dirac point. Utilizing this gapped surface state allows for a wide range of novel physical effects to be observed, paving a way for applications in spintronics and quantum computation. This review focuses on the research of 3D TIs doped with manganese (Mn). We summarize major progress in the study of Mn doped chalcogenide TIs, including Bi2Se3, Bi2Te3, and Bi2(Te,Se)3. The transport properties, in particular the anomalous Hall effect, of the Mn-doped Bi2Se3 are discussed in detail. Finally, we conclude with future prospects and challenges in further studies of Mn doped TIs.

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.

Realization of quantum anomalous hall effect at tens of kelvin by n-p codoping of topological insulators

With the support by the National Natural Science Foundation of China,the research teams led by Prof.Xu Xiaohong（许小红）at the School of Chemistry and Materials Science,Shanxi Normal University and Prof.Zhang Zhenyu at ICQD,University of Science and Technology of China used vanadium-iodine（Ⅴ-Ⅰ）codoped Sb2Te3 to realize high-temperature quantum anomalous Hall effect（QAHE）,which was

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.

Effective spin dephasing mechanism in confined two-dimensional topological insulators

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 Hamiltouian. 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-reversal symmetry is preserved throughout. Finally, we discuss the tunneling between extended helical edge states and local edge states in the QSH quantum dots, which leads to backscattering in the extended edge states. These results can explain the more robust edge transport in InAs/GaSb QSH systems.

Thermal Hall effect and the Wiedemann–Franz law in Chern insulator

Thermal Hall effect, where a transverse temperature difference is generated by implementing a longitudinal temperature gradient and an external magnetic field in the perpendicular direction to systems, is a useful tool to reveal transport properties of quantum materials. A systematic study of the thermal Hall effect in a Chern insulator is still lacking. Here,using the Landauer–Büttiker formula, we investigated the thermal Hall transport of the Harper–Hofstadter model with flux φ= 1/2 and its generalizations. We demonstrated that the Wiedemann–Franz law, which states that the thermal Hall conductivity is linearly proportional to the quantum Hall conductivity in the low temperature limit, is still valid in this Chern insulator, and that the thermal Hall conductivity can be used to characterize the topological properties of quantum materials.

Room temperature quantum anomalous Hall insulator in honeycomb lattice, RuCS_(3), with large magnetic anisotropy energy

The quantum anomalous Hall(QAH) effect has attracted enormous attention since it can induce topologically protected conducting edge states in an intrinsic insulating material. For practical quantum applications, the main obstacle is the non-existent room temperature QAH systems, especially with both large topological band gap and robust ferromagnetic order. Here, according to first-principles calculations, we predict the realization of the room temperature QAH effect in a two-dimensional(2D) honeycomb lattice, RuCS_(3) with a non-zero Chern number of C = 1. Especially, the nontrivial topology band gap reaches up to 336 me V for RuCS_(3). Moreover, we find that RuCS_(3) has a large magnetic anisotropy energy(2.065 me V) and high Curie temperature(696 K). We further find that the non-trivial topological properties are robust against the biaxial strain. The robust topological and magnetic properties make RuCS_(3) have great applications in room temperature spintronics and nanoelectronics.

First-principles prediction of quantum anomalous Hall effect in two-dimensional Co2Te lattice

The quantum anomalous Hall effect(QAHE) has special quantum properties that are ideal for possible future spintronic devices. However, the experimental realization is rather challenging due to its low Curie temperature and small non-trivial bandgap in two-dimensional(2D) materials. In this paper, we demonstrate through first-principles calculations that monolayer Co2Te material is a promising 2D candidate to realize QAHE in practice. Excitingly, through Monte Carlo simulations, it is found that the Curie temperature of single-layer Co2Te can reach 573 K. The band crossing at the Fermi level in monolayer Co2Te is opened when spin–orbit coupling is considered, which leads to QAHE with a sizable bandgap of Eg= 96 me V, characterized by the non-zero Chern number(C = 1) and a chiral edge state. Therefore, our findings not only enrich the study of quantum anomalous Hall effect, but also broaden the horizons of the spintronics and topological nanoelectronics applications.

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 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.

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.