Session B29: Quantum Spin Hall Effect

Quantum spin Hall effect in two-dimensional transition-metal dichalcogenide Haeckelites

The Quantum Spin Hall (QSH) effect, discovered nearly ten years ago, is such a promising option, because it can be viewed as the time-reversal-invariant version of the QH effect, which does NOT need any external magnetic field and can be in principle realized at room temperature. So far, QSH effect has only been observed in HgTe/CdTe and InAs/GaSb quantum wells. Both of them require precisely controlled MBE growth and ultralow temperature. The study of 2D TI has been seriously hampered due to lack of proper materials with large band gap, stable structure, and easy fabrication. In this report, I will introduce a family of the single layer 2D transition metal dichalcogenide (TMD) Haeckelites MX2 (M=W or Mo, X=S, Se or Te), which can host QSH effect. The phonon spectra indicate that these Haeckelites are dynamically stable. Further, a simple tight-binding model based on square-like lattice has been established to uncover the underlying mechanism. This will extend further studies from graphene-based hexagonal lattice to square-like lattice and broad the range for searching topological materials largely.   

Quantum Spin Hall Effect in thin films of topological crystalline insulators.

The quantum spin Hall effect (QSHE) is predicted to exist in topological crystalline insulator materials [1]. Using a tight-binding approach we demonstrate that in (111)-oriented thin films of SnSe and SnTe the energy gaps depend in an oscillatory fashion on the layer thickness. The calculated topological invariant indexes and edge state spin polarizations show that in the negative energy gaps regions (\textasciitilde 20--40 monolayers) a 2D topological insulator phase appears. In this range of thicknesses in both SnSe and SnTe, edge states are obtained with Dirac cones having opposite spin polarization in their two branches. While in SnTe layers a single Dirac cone appears at the projection of $\Gamma $ point of the 2D Brillouin zone, in SnSe layers three Dirac cones at $M $points projections are obtained. Unfortunately, in SnSe films an overlapping of bands at $\Gamma $ and $M$ diminishes the final band gap in the vicinity of all $M$ points and the edge states appear either against the background of the bands or within a very small energy gap. We show that this problem can be removed by applying to the layers a biaxial strain [2]. This should enable observation of the QSHE also in SnSe layers. 1. S. Safaei, et al., New J. Phys. 17, 063041 (2015). 2. S. Safaei et al, arXiv: 1508.01364 [cond-mat.mtrl-sci].   

Transport in quantum spin Hall systems in parallel magnetic fields

Edge states in quantum spin Hall (QSH) systems are protected by time-reversal symmetry, resulting in a qunatized conductance. A magnetic field breaks that protection, and should lead to a deviation from perfect quantization. We will discuss generic features of semiconductor-based QSH systems (such as HgTe/CdTe and InAs/GaSb) that affect the magnetic field dependence of edge state conductance, focusing on the effect of an in-plane field.   

Giant Rashba spin splitting with unconventional spin texture in a quantum spin Hall insulator

We propose a non-centrosymmetric honeycomb-lattice quantum spin Hall effect family formed by atoms of the groups IV, V and VII of the periodic table. We make a structural analysis, a $Z_2$ characterization. According to our ab-initio phonon calculations, the system formed by Bi, Pb and I atoms is only mechanically stable system. This material presents a Rashba-type spin-splitting and a hexagonal warping effect, which lead to an unusual spin texture. Due to this spin texture, the backscattering is forbidden for both edge conductivity channels and bulk conductivity channels. This suggests that, contrary to what happens in most systems with nontrivial topological phases, the bulk states would not pose a problem for spintronic devices. The value of the spin-splitting due to the Rashba effect is about 60 meV, which is huge compared with the values found in 2D systems and surprisingly is on the order of the highest found in 3D systems.   

Large band gap quantum spin hall insulators of fluorinated Pb-X (X= C, Si, Ge, Sn)

The Quantum Spin Hall Insulating (QSHI) phase was first observed in the HgTe/CdTe quantum well structure. However, the observed band gap of 5 meV is too small for practical applications. Other materials have also been proposed for the observation of the QSHI phase, such as silicene, germanene, stanene, and its halogenated phases. The spin-orbit interaction is a key feature in topological insulators, raising the interest in heavy elements, such as Bismuth. In fact, Bi is responsible for the high spin-orbit coupling that drives the band inversion in Bi$_{2}Se$_{3}$ and Bi$_{2}Te$_{3}$. Another element that also has a large spin-orbit interaction is Lead (Pb). Here we present a set of 2D QSH insulators with a very large band gap based on fluorinated Pb-X (X= C, Si, Ge, Sn). First-principles phonon dispersion calculations indicate that these systems are structurally and mechanically stable. By performing DFT-based electronic structure calculations we show that 2D Pb-X functionalized with fluorine are topological insulators with very large band gaps (over 0.7 eV). Addtional calculations, for nanoribbons structures, show the presence of a Dirac cone at the center of the Brillouin zone. These results can establish a new route to the observation of QSHI phase at room temperature.   

Fully and partially iodinated germanane as a platform for the observation of the quantum spin Hall effect.

The Quantum Spin Hall Effect (QSHE) proposed in 2005 by Kane and Mele for graphene and by S.-C. Zhang et al. in 2006 for the HgTe/CdTe, became a very exciting area of condensed matter physics. Several materials have been proposed to overcome the issue of the small SOC band gap presented by the graphene and HgTe/CdTe structures, such the elemental materials germanene, stanene and others binary compounds with band gaps that goes to several meV to few eV. Motivated by the recent isolation of the trivial insulator germanane, a fully hydrogenated germanene, we show that a partially substitution of the hydrogen atoms in only one side of the material by iodine, creates a two dimensional topological insulator with a large band gap of 0.49 eV. This functionalization opens up new routes for the observation of the quantum spin Hall effect in a fully two-dimensional material. We also show that creating nanoroads or nanoribbons with the pattern functionalization of germanane by iodine in a ordered or disordered way, topologically protected interfaces states arises at the boundary of germanane/iodinated germanane.   

Spin Hall Conductivity and Spin Chern Number for Dirac Systems

A semiclassical differential form formalism of the spin Hall effect for Dirac systems is presented. In this formalism, space coordinates and momenta are usual dynamical variables, whereas spin is not a dynamical degree of freedom. Spin depicts itself in the matrix-valuedness of equations of motion. We demonstrate that the main contribution to the spin Hall conductivity is given by the spin Chern number whether the spin is conserved or not at the quantum level. We illustrated the formulation within the Kane-Mele model of graphene in the absence and in the presence of the Rashba spin-orbit coupling term. Kane-Mele Model of graphene, which incorporates intrinsic spin-orbit interaction, constitutes the first example of a two dimensional topological insulator. We established the anomalous Hall conductivity as well as the spin Hall conductivity from the term linear in the electric field and the Berry curvature in the the anamolous velocity term. In a basis where the component of spin under consideration is diagonal this term is diagonal. We argue that this semiclassical procedure of calculating the spin Hall conductivity can be generalized to any dimension.   

Room Temperature Quantum Spin Hall Insulators with a Buckled Square Lattice

Two-dimensional (2D) topological insulators (TIs), are excellent candidates for coherent spin transport related applications Currently, most known 2D TIs are based on a hexagonal lattice. Here, we propose that there exists the quantum spin Hall effect (QSHE) in a new tight-binding (TB) model for a two-orbital system with the buckled square lattices. We show that the band inversion is due to the hybridization between the$ p_{x\thinspace }$and$ p_{y} $orbitals, while the spin-orbit coupling (SOC) induced nearest-neighbor effective hopping is responsible for a band gap opening at the Dirac cone. Through performing global structure optimization, we predict a new three-layer quasi-2D (Q2D) structure which has the lowest energy among all structures with the thickness less than 6.0 Å for the BiF system. It is identified to be a Q2D TI with a large band gap (0.69 eV). The electronic states of the Q2D BiF system near the Fermi level are mainly contributed by the middle Bi square lattice, which are sandwiched by two inert BiF$_{\mathrm{2}}$ layers. This is beneficial since the interaction between a substrate and the Q2D material may not change the topological properties of the system, as we demonstrate in the case of the NaF substrate. Our analysis shows that the low-energy physics of the Q2D BiF system can be qualitatively described by our newly proposed two-orbital TB model. Our study not only predicts a Q2D QSH insulator for realistic room temperature (RT) applications, but also provides a new lattice system for engineering topological states such as quantum anomalous Hall effect.   

Superconducting quantum spin-Hall systems with giant orbital g-factors

Topological aspects of superconductivity in quantum spin-Hall systems (QSHSs) such as thin layers of three-dimensional topological insulators (3D Tis) or two-dimensional Tis are in the focus of current research. Here, we describe a novel superconducting quantum spin-Hall effect (quantum spin Hall system in the proximity to the s-wave superconductor and in the orbital in-plane magnetic field), which is protected against elastic backscattering by combined time-reversal and particle-hole symmetry [1]. This effect is characterized by spin-polarized edge states, which can be manipulated in weak magnetic fields due to a giant effective orbital g-factor, allowing the generation of spin currents. The phenomenon provides a novel solution to the outstanding challenge of detecting the spin-polarization of the edge states. Here we propose the detection of the edge polarization in the three-terminal junction using unusual transport properties of superconducting quantum Hall-effect: a non-monotonic excess current and a zero-bias conductance splitting. [1] R. W. Reinthaler, G. Tkachov, and E. M. Hankiewicz Phys. Rev. B 92, 161303(R) (2015)   

Effective spin dephasing mechanism in confined two-dimension 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-dimension 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-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/AlSb QSH systems.   

\textbf{Effects of magnetic impurities on transport in 2D topological insulators}

Understanding the transport properties of topological insulators could bring such materials from fundamental research to potential applications. Here we report on the theoretical investigations of the effects of magnetic impurities on transport properties of model two-dimensional (2D) topological insulators (TIs). We utilize the tight-binding form of the Bernevig-Hughes-Zhang model and investigate the transport properties by employing the Landauer-B\"{u}ttiker formalism. We explore the current distribution in 2D TIs resulting from scattering by a magnetic impurity which breaks time-reversal symmetry. We find that a magnetic impurity could drive anti-resonant behavior of the conductance, as revealed from full backscattering of the electron current flowing at one of the edges of the TI. This phenomenon occurs due to spin-flip scattering when the Fermi energy matches the impurity state and the magnetic moment of the impurity is aligned along the TI edge. Additionally, we explore the effect of an external magnetic gate attached to the system and show that changing the magnetization orientation within the gate allows the control of conductance. This geometric setup could be realized experimentally providing the opportunity to tune transport properties of 2D TIs by a magnetic gate.   

One-dimensional edge states in Bi(111) bilayer grown on Sb$_{\mathrm{2}}$Te$_{\mathrm{3}}$

Well-ordered Bi bilayer islands with zigzag edges are grown epitaxially on Sb2Te3(111) film by molecular beam epitaxy. Scanning tunneling microscopy imaging shows that the Bi film assumes the lattice of the Sb2Te3, thus is coherently strained. Tunneling spectroscopy further reveals robust edge states, confirming it as a two-dimensional topological insulator. This is consistent with first-principles calculations that indicate the preservation of the topological nature of the Bi bilayer and edge states with only an energy shift even in the presence of strong interaction between Bi and Sb2Te3. These findings suggest that the interface between 2D and 3D TIs can be a promising platform to synthesize new topological matter.   

Controlling the Flow of Spin and Charge in Nanoscopic Topological Insulators

Rapid advances in quantum computation and spin electronics, heralded by the discovery of topological insulators, have been hampered by the inability to control the flow of spin and charge currents at the nanoscale. In this talk, I will demonstrate that such control can be established in nanoscopic two-dimensional topological insulators (TIs) by breaking their time reversal symmetry via magnetic defects. This allows for the emergence of two novel phenomena: the creation of nearly 100% spin-polarized charge currents, and the design of tunable spin diodes. I discuss two mechanisms by which spin-polarized currents can be produced, and show that by superposing them, spin-diodes which are tunable via gate and bias voltage, can be constructed. Finally, I show that a proof-of-concept for the proposed effects can be realized in meso- or macroscale hybrid structures in which TIs interface with both ferro- and antiferromagnets.   

Hunting down magnetic monopoles in 2D topological insulators?

Contrary to the existence of electric charge, magnetic monopole does not exist in nature. It is thus extraordinary to find that magnetic monopoles can be pictured conceptually in topological insulators. For 2D topological insulators, the topological invariant corresponds to the total flux of an effective magnetic field (the Berry curvature) over the reciprocal space. Upon wrapping the 2D reciprocal space into a compact manifold as a torus, the non-zero total flux can be considered to originate from magnetic monopoles with quantized charge. We will first illustrate the intrinsic difficulty via extending a 2D problem to a 3D reciprocal space, and then demonstrate that analytical continuation to the complex momentum space offers a natural solution in which 1) the magnetic monopoles emerge naturally in pairs each forming a string above and below the real axis possessing opposite charge, and 2) the total charge below the real axis gives exactly the topological invariant. In essence, the robustness of the topology is mapped to the robustness of the total charge in the lower complex plan, a mapping intriguing even mathematically. Finally, we will illustrate the evolution across the topological phase transition, providing a natural description of the metallic nature in the phase boundary, and offering a clear explanation why a change of global topology can be induced via a local change in reciprocal space.   

Circle of crossings and Berry curvatures in 2D topological insulators

HgTe forms a two-dimensional topological insulator when sandwiched between CdTe barriers for a HgTe layer wider than the critical thickness. We derive single-particle and two-particle interaction Hamiltonians describing physics of these compounds by using $\textbf{k}$ · $\textbf{p}$ theory and extended Kane model. We include contributions from upper conduction bands with orbital states of p-symmetry that bring about the terms describing lack of inversion symmetry in host semiconductors. A crucial ingredient is hetero-interface contribution to intrinsic spin-orbit interactions that drives significant anticrossing gaps in spectra at zero wavevector, but results in a circle of spectral crossings at finite wavevectors. Single-particle Hamiltonian and two-particle Hamiltonian contain important spin-dependent terms. The spin-dependent interaction couples orbital motion of one particle with evolution of spin of the other particle. Such particle-particle interactions do not conserve spin and lower the symmetry of exchange interactions, leading, e.g., to Dzyaloshinskii-Moriya exchange term. We study the effects of new interactions on Berry curvature and spin-Hall conductance.