Quantum Spin Hall State Research Articles

Circular photogalvanic effect from third-order nonlinear effect in 1T’-MoTe2

The two-dimensional layered material MoTe2 has aroused extensive research interests in its rich optoelectronic properties in various phases. One property of particular interest is the circular photogalvanic effect (CPGE): a conventional second order nonlinear optical effect that is related to the chirality of materials. It has been demonstrated in Td-MoTe2, a type-II topological Weyl semimetal candidate, while it has been unclear so far whether it exists in the semimetallic 1T’ phase, another interesting phase that hosts a quantum spin hall state. In this article, we report a clear experimental observation of in-plane CPGE in 1T’-MoTe2. The observation is confirmed under various experimental designs with excitation by normally incident mid-infrared laser, and we find it to be related to an in-plane internal DC electric field. We attribute the circular photogalvanic response to a third-order nonlinear optical effect involving this DC electric field, which is consistent with the crystal symmetry of the lattices and present in both the 1T’ and Td phases of the material.

Computational characterization of thermal and mechanical properties of single and bilayer germanene nanoribbon

Two-dimensional germanene has provided a cornucopia of new functionalities in the field of nanotechnology owing to its remarkable electronic and thermoelectric attributes. The robust spin–orbit coupling and high carrier mobility give rise to many salient features including non-trivial topological properties, quantum spin-Hall state near room temperature, and topological superconductivity, rendering it an excellent contender for valleytronics, spintronics, and quantum computation. As such, an in-depth characterization of thermal and mechanical properties of germanene is crucial for its practical implication and efficient operation, which remains elusive. Here, we employed equilibrium molecular dynamics simulations utilizing Stillinger Weber potential to reveal the mechanical strength, melting temperature, and phonon thermal conductivity (PTC) of single-layer germanene nanoribbon (SLGeNR) and bilayer germanene nanoribbon (BiLGeNR). Effects of temperature, biaxial tensile and compressive strain, monovacancy defects, length and width of the nanoribbon on the PTC have been rigorously investigated. It has been found that PTC of SLGeNR could be substantially reduced by BiLGeNR. Our simulation results suggest that PTC of SLGeNR demonstrates an inverse relation with temperature, biaxial compressive strain, and monovacancy defects while biaxial tensile strain, length and width of the nanoribbon increases the PTC of SLGeNR significantly. To understand the PTC more profoundly, phonon density of state (PDOS) profiles have been studied. The BiLGeNR demonstrates more tensile strength as well as melting temperature compared to SLGeNR. This study offers a comprehensive guideline for engineering the TC as well as discloses important mechanical and melting characteristics of the SLGeNR and BiLGeNR for a wide range of applications in flexible nano-electronics and thermoelectric nanodevices.

Exotic Topological Bands and Quantum States in Metal-Organic and Covalent-Organic Frameworks.

ConspectusMetal-organic and covalent-organic frameworks (MOFs/COFs) have been extensively studied for fundamental interests and their promising applications, taking advantage of their unique structural properties, i.e., high porosity and large surface-to-volume ratio. However, their electronic and magnetic properties have been somewhat overlooked because of their relatively poor performance as conductive and/or magnetic materials. Recent experimental breakthroughs in synthesizing two-dimensional (2D) π-conjugated MOFs/COFs with high conductivity and robust magnetism through doping have generated renewed and increasing interest in their electronic properties. Meanwhile, comprehensive theoretical studies of the underlying physical principles have led to discovery of many exotic quantum states, such as topological insulating states, which were only observed in inorganic systems. Especially, the diversity and high tunability of MOFs/COFs have provided a playground to explore novel quantum physics and quantum chemistry as well as promising applications.The band theory has empowered us to understand the most fundamental electronic properties of inorganic crystalline materials, which can also be used to better understand MOFs/COFs. The first obvious difference between the two is that instead of atomic orbitals residing at lattice sites of inorganic crystals, molecular orbitals of organic ligands are predominant in MOFs/COFs. The second key difference is that usually all atomic orbitals in an inorganic crystal are subject to one common group of lattice symmetry, while atomic orbitals of metal ion and molecular orbitals of different organic ligands in MOFs/COFs belong to different subgroups of lattice symmetries. Both these differences will impact the band structure of MOFs/COFs, in particular making it more complex. Consequently, which subset of bands are of most importance depends strongly on the location of Fermi level, i.e., electron counting and charge doping. Furthermore, there are usually two types of characteristic electrons coupled in MOFs, i.e., strongly correlated localized d and f electrons and diffusive s and p electrons, which interplay with lattice, orbital, and spin degrees of freedom, leading to more exotic topological and magnetic band structures.In this Account, we present an up-to-date review of recent theoretical developments to better understand the exotic band structures of MOFs/COFs. Starting from three fundamental 2D lattice models, i.e., honeycomb, Kagome, and Lieb lattices, exotic Dirac and flat bands as well as the intriguing topological quantum states they host, e.g., quantum spin Hall and quantum anomalous Hall states, are outlined. In addition to the single-lattice models, we further elaborate on combined lattice model Hamiltonians, which give rise to overlapping bands hosting novel quantum states, such as nodal-line Dirac semimetal and unconventional superconducting states. Also, first-principles predictions of candidate MOFs/COFs that host these exotic bands and hence quantum phases are reviewed, which greatly extends the pool of materials beyond inorganic crystals for hosting exotic band structures.

Functionalization induced quantum spin Hall to quantum anomalous Hall phase transition in monolayer jacutingaite.

As novel states of quantum matter, quantum spin Hall (QSH) and quantum anomalous Hall (QAH) states have attracted considerable interest in condensed matter and materials science communities. Recently, a monolayer of the naturally occurring mineral jacutingaite (Pt2HgSe3), was theoretically proposed to be a large-gap QSH insulator and experimentally confirmed. Here, based on first-principles calculations and tight-binding modeling, we demonstrate QSH to QAH phase transition in jacutingaite by chemical functionalization with chalogen. We show that two-dimensional (2D) chalogenated jacutingaite, Pt2HgSe3-X (X = S, Se, Te), is ferromagnetic with Curie temperature up to 316 K, and it exhibits QAH effect with chiral edge states inside a sizeable topological gap. The physical mechanism lies in the adsorption induced transformation of the original Kane-Mele model into an effective four-band model involving (px, py) orbitals on a hexagonal lattice, so that the topological gap size can be controlled by spin-orbit coupling strength of the chalogen (0.28 eV for Pt2HgSe3-Te). These results not only show the promise of functionalization in orbital-engineering of 2D functional structures, but also provide an ideal and practical platform for achieving exotic topological phases for dissipationless transport and quantum computing.

Atoms to topological electronic materials: a bedtime story for beginners

In this review, We discussed the theoretical foundation and experimental discovery of different topological electronic states of material in condensed matter. At first, we briefly reviewed the conventional electronic states, which have been realized in band theory of solid. Next, the simplest non-trivial insulating phase (Integer Quantum Hall State) and the concept of topological order in condensed matter electronic system have been introduced. In the following sections, we discussed Quantum Spin Hall (QSH) State in two dimensions (2D), and reviewed the theoretical and experimental developments from 2D QSH state to 3D topological insulators (TI). Subsequently, we gave a brief overview on theoretical and experimental understanding on recently discovered topological Dirac semimetals, Weyl semimetals, three-, six- , and eight-fold degenerate semimetals, and Nodal line semimetals. Then, topological crystalline insulator, which can not be considered as a descendent of Quantum Spin Hall or Integer Quantum Hall insulator, has been introduced. Finally, we discussed the presence of magnetism in some topological materials and its consequence on electronic band structure.

Nonvolatile tuning of the spin–orbit coupling in graphene by a ferroelectric dipole

Spin–orbit coupling (SOC) offers an alternative technique for generating pure spin currents in non-magnetic materials and controlling spin precessions for spin-field effect transistors. In addition, introducing SOC into graphene causes pristine graphene to evolve into a new condensed matter phase, such as the topological insulator state (quantum spin Hall state). Thus, the control of SOC in graphene is essential for its functional spin-orbitronic applications. Here, we report the nonvolatile tuning of SOC in graphene through the proximity effect from a ferroelectric substrate, Pb(Zr52,Ti48)O3 (PZT). Ferroelectric poling by applying a gate voltage induces a change in the SOC strength in addition to shifting the charge neutral point in graphene. The variations in SOC were extracted from weak localization within the quantum interference theory of graphene. Our analyses show that the dipole moments from the PZT polarization significantly enhance the z → −z asymmetric and symmetric SOCs of graphene. Unlike the impurity doping and/or gating, our methodology leads to the nonvolatile electrical control of SOC, thereby paving the way for versatile spin-orbitronic applications of graphene.

Molecular beam epitaxy growth and strain-induced bandgap of monolayer 1T′-WTe2 on SrTiO3(001)

A monolayer 1T′-WTe2 film is grown on SrTiO3(001) with in-plane tensile strain. A height of ∼0.7 nm, obvious charge transfer, and incommensurate charge fluctuations in 1T′-WTe2 suggest strong coupling to the STO substrate. Scanning tunneling spectroscopy on the surface reveals that a large energy gap opens at the Fermi level with nearly zero conductance. The opened energy gap decreases with the increase in the WTe2 island size. The lack of the metallic edge state on monolayer 1T′-WTe2/SrTiO3(001) indicates the absence of the quantum spin Hall (QSH) state. Our study here demonstrates that the energy gap of monolayer 1T′-WTe2 can be tuned by lattice strain and illustrates the importance of interface coupling to realize the metallic edge state and QSH in monolayer 1T′-WTe2.

Electronic Band Topology of Monoclinic MoS2 Monolayer: Study Based on Elementary Band Representations for Layer Groups

A monoclinic monolayer is demonstrated to be a topological semimetal (spinless), or zero‐gap semiconductor (spinful, without spin–orbit coupling), or topologically trivial insulator (finite spin–orbit coupling). The latter contradicts a previous prediction that exhibits quantum‐spin Hall (QSH) effect, and the reported value of ‐invariant, calculated from inversion‐parity values at four Brillouin‐zone high‐symmetry points. Namely, as the electronic states of two of these points are double degenerate and transform according to irreducible representations carrying no parity, the band topology of is investigated using elementary band representations (EBRs) for layer groups (LGs). Novel subroutines, which output EBRs and Wilson loop operators for low‐dimensional systems, are incorporated into POLSym code. Based on the calculated band structure decomposition onto EBRs of the relevant symmetry group and Wilson loop eigen‐spectra, it is revealed that valence‐band Wannier functions do not break the symmetry, and that Wannier centers are localized within a unit cell. Moreover, the QSH state is proved to be not realizable within ‐phase group‐VI transition‐metal dichalcogenides, because EBRs for the relevant symmetry groups are topologically trivial. However, it is predicted that among compounds that form other monoclinic structure, with symmorphic LG symmetry, there may exist topologically nontrivial phases, including the QSH state.

Crossover between weak antilocalization and weak localization in few-layer WTe2 : Role of electron-electron interactions

We report electron transport studies in an encapsulated few-layer WTe$_2$ at low temperatures and high magnetic fields. The magnetoconductance reveals a temperature-induced crossover between weak antilocalization (WAL) and weak localization (WL) in quantum diffusive regime. We show that the crossover clearly manifests coexistence and competition among several characteristic lengths, including the dephasing length, the spin-flip length, and the mean free path. In addition, low temperature conductance increases logarithmically with the increase of temperature indicating an interplay of electron-electron interaction (EEI) and spin-orbit coupling (SOC). We demonstrate the existences and quantify the strengths of EEI and SOC which are considered to be responsible for gap opening in the quantum spin hall state in WTe2 at the monolayer limit.

Band bending and zero-conductance resonances controlled by edge electric fields in zigzag silicene nanoribbons

We study the band structure and transport property of a zigzag silicene nanoribbon when electric fields are applied to the edges. It is found that band bending could be induced and controlled by the antisymmetric edge fields, which can be understood based on the wave functions of the edge states. The highest valence band and the lowest conduction band coexist in the band-bending region. With the narrowing of edge potentials, the bending increases gradually. When the edge fields become symmetric, an asymmetric band gap at the Dirac points can be obtained due to the intrinsic spin-orbit interaction, suggesting a valley-polarized quantum spin Hall state. The gap could reach a maximum value rapidly and then decrease slowly as the electric fields increase. Due to the combining effect of the band bending, band-selective rule, and resonant states, many zero-conductance resonances and resonance peaks appear in different regions, which could be described by the Fano resonance effect. Furthermore, the band bending and zero-conductance resonances are robust against the Hubbard interaction. The Hubbard interaction could work as a spin-dependent edge field, together with the edge electric fields, leading to a spin-dependent band gap and various quantum phases such as metal and half-metal.

Tuning topological states in a Ni-hexaaminobenzene framework by NO2 adsorption

Exploring a quantum topological phase, such as the quantum spin Hall (QSH) effect and quantum anomalous Hall (QAH) effect in two-dimensional metal-organic frameworks (MOFs), has been attracting significant attention. In this study, we propose to realize a topological phase transition via NO2 adsorption in a recently synthesized two-dimensional Ni-hexaaminobenzene [Ni3(HAB)2]. First-principles calculations show a QSH state of the pristine Ni3(HAB)2 nanosheet. However, when NO2 molecules are adsorbed on the Ni3(HAB)2 surface, the system changes to a ferromagnetic state and the time-reversal symmetry is broken. We find that the NO2-adsorbed Ni3(HAB)2 exhibits an intrinsic QAH state with the Fermi level exactly located within a topologically nontrivial bandgap. The calculated gapless edge states and quantized Hall conductance further confirm the QSH-to-QAH phase transition. Our finding offers an alternative approach of tuning the topological phase transition in a real MOF material.

Persistent currents and spin torque caused by percolated quantum spin Hall state

Motivated by recent experiments, we investigate the quantum spin Hall state in 2D topological insulator/ferromagnetic metal planar junctions by means of a tight-binding model and linear response theory. We demonstrate that whether the edge state Dirac cone is submerged into the ferromagnetic subbands and the direction of the magnetization dramatically affect (i) how the edge state percolates into the ferromagnet, and (ii) the spin-momentum locking of the edge state. Laminar flows of room temperature persistent charge and spin currents near the interface are uncovered. In addition, the current-induced spin polarization at the edge of the 2D topological insulator is found to be dramatically enhanced near the impurities. The current-induced spin polarization in the ferromagnet is mainly polarized in the out-of-plane direction ${\hat{\bf z}}$, rendering a current-induced spin torque that is predominantly field-like $\propto {\bf S}\times{\hat{\bf z}}$.

Emergent quantum Hall effects below 50 mT in a two-dimensional topological insulator.

The realization of the quantum spin Hall effect in HgTe quantum wells has led to the development of topological materials, which, in combination with magnetism and superconductivity, are predicted to host chiral Majorana fermions. However, the large magnetization in conventional quantum anomalous Hall systems makes it challenging to induce superconductivity. Here, we report two different emergent quantum Hall effects in (Hg,Mn)Te quantum wells. First, a previously unidentified quantum Hall state emerges from the quantum spin Hall state at an exceptionally low magnetic field of ~50 mT. Second, tuning toward the bulk p-regime, we resolve quantum Hall plateaus at fields as low as 20 to 30 mT, where transport is dominated by a van Hove singularity in the valence band. These emergent quantum Hall phenomena rely critically on the topological band structure of HgTe, and their occurrence at very low fields makes them an ideal candidate for realizing chiral Majorana fermions.

Signature of Large-Gap Quantum Spin Hall State in the Layered Mineral Jacutingaite.

Quantum spin Hall (QSH) insulators host edge states, where the helical locking of spin and momentum suppresses backscattering of charge carriers, promising applications from low-power electronics to quantum computing. A major challenge for applications is the identification of large gap QSH materials, which would enable room temperature dissipationless transport in their edge states. Here we show that the layered mineral jacutingaite (Pt2HgSe3) is a candidate QSH material, realizing the long sought-after Kane–Mele insulator. Using scanning tunneling microscopy, we measure a band gap in excess of 100 meV and identify the hallmark edge states. By calculating the invariant, we confirm the topological nature of the gap. Jacutingaite is stable in air, and we demonstrate exfoliation down to at least two layers and show that it can be integrated into heterostructures with other two-dimensional materials. This adds a topological insulator to the 2D quantum material library.

Prediction of tetragonal monolayer CuN with a quantum spin Hall state

Abstract Two-dimensional (2D) quantum spin Hall (QSH) insulator are promising candidates for miniaturized electronic and dissipationless transport devices. The bottleneck preventing applications from QSH phase, however, is lack of feasible 2D materials in experiments so far. By using first-principles calculations, here we propose a realistic 2D QSH phase in tetragonal CuN, which is dynamically and mechanically stable. Band structure analysis reveals that it is a nontrivial QSH phase robust against external strains. The origin of QSH effect can be identified by a px-pz band inversion, topological invariant Z2, and helical edge states within the bulk gap. Also, a heterostructure composed of CuN deposited on semiconducting AlAs substrate remains topologically nontrivial with a sizable gap. These findings provide a promising platform to design new QSH insulators in tetragonal lattices, which show potential applications in multifunctional spintronics devices.

Charge Transport Study of in-Situ Cesium Doped Monolayer Graphene

Modifying graphene with surface adsorbates is a widely studied way of modifying graphene’s electronic properties. Different graphene/adatom systems have shown that it is possible to open a band gap in graphene1, engineer topological insulators2 and create magnetic moments3. The atomic thickness of graphene maximizes the effect of adsorbates, enabling chemical sensors that achieve single molecule sensitivity4. Alkali metal adatoms are a particularly efficient dopant for graphene due to their high electronegativity. Lithium/graphene systems have been studied for in the context of low-dimensional superconductivity5 and achieving ultrafast diffusion of lithium6 for battery applications.However, doping graphene with alkali metal adatoms bears challenges. Experiments with alkali metals must be conducted in an ultra-high vacuum or inert gas environment due to the high reactivity of the alkali. Alkali metal adsorbed graphene is not air stable, which imposes limits on the available characterization methods. An impurity free graphene surface must be achieved prior to adsorption to achieve efficient adsorption and avoid unwanted parasitic reactions. This problem is particularly acute for graphene samples prepared by the use of polymer handles. Alkali metal atoms can also intercalate under the graphene layer depending on substrate and sample preparation conditions. Moreover, it is possible for adatoms to form clusters on graphene surface7which negatively impacts both doping uniformity and efficiency.In this work, we report a new method of alkali doping of graphene to reach ultra-high doping (~1014cm-2) for charge transport studies beyond the limit of ionic liquid gating. We work with chemical vapor deposition grown graphene transferred onto quartz substrates. Quartz is the substrate of choice due to its chemical inertness and optical transparency, with the latter enabling non-invasive Raman spectroscopy through the substrate. Micron scale graphene devices were prepared using lithography-based methods and the samples were thermally annealed in a nitrogen glove box environment to desorb water prior to alkali doping. The graphene was subsequently exposed to cesium vapor at different temperatures using a flip-chip method that allows hermetic sealing of the air sensitive samples in an inert gas environment with a liquid cesium source of cesium vapour. We measured the in-situ variation of graphene resistivity as cesium atoms are adsorbed onto the graphene surface, and resistivity is modulated by charge transfer doping. Evidence of ultra-high doping is shown via Raman spectroscopy performed through the quartz substrate window, with G-peak shifts from 1589 cm-1 up to 1608 cm-1. Hall measurements in-situ further confirm the strong doping. We measured the electronic transport properties of cesium doped graphene at temperatures as low as 1.2 K and under magnetic fields up to 7 T. Weak-localization, magnetoresistance and Hall resistance are measured and analyzed. 1. Elias, D. C.; Nair, R. R.; Mohiuddin, T. M.; Morozov, S. V.; Blake, P.; Halsall, M. P.; Ferrari, A. C.; Boukhvalov, D. W.; Katsnelson, M. I.; Geim, A. K.; Novoselov, K. S., Control of graphene's properties by reversible hydrogenation: evidence for graphane. Science 2009, 323 (5914), 610-3.2. Weeks, C.; Hu, J.; Alicea, J.; Franz, M.; Wu, R., Engineering a Robust Quantum Spin Hall State in Graphene via Adatom Deposition. Physical Review X 2011, 1 (2).3. Hong, X.; Zou, K.; Wang, B.; Cheng, S. H.; Zhu, J., Evidence for spin-flip scattering and local moments in dilute fluorinated graphene. Phys Rev Lett 2012, 108 (22), 226602.4. Schedin, F.; Geim, A. K.; Morozov, S. V.; Hill, E. W.; Blake, P.; Katsnelson, M. I.; Novoselov, K. S., Detection of individual gas molecules adsorbed on graphene. Nat Mater 2007, 6 (9), 652-5.5. Profeta, G.; Calandra, M.; Mauri, F., Phonon-mediated superconductivity in graphene by lithium deposition. Nat Phys 2012, 8 (2), 131-134.6. Kühne, M.; Paolucci, F.; Popovic, J.; Ostrovsky, P. M.; Maier, J.; Smet, J. H., Ultrafast lithium diffusion in bilayer graphene. Nat Nanotechnol 2017, 12, 895.7. Fan, X.; Zheng, W. T.; Kuo, J. L.; Singh, D. J., Adsorption of single Li and the formation of small Li clusters on graphene for the anode of lithium-ion batteries. ACS Appl Mater Interfaces 2013, 5 (16), 7793-7.

Tuneable quantum spin Hall states in confined 1T' transition metal dichalcogenides

Investigation of quantum spin Hall states in 1T' phase of the monolayer transition metal dichalcogenides has recently attracted the attention for its potential in nanoelectronic applications. While most of the theoretical findings in this regard deal with infinitely periodic crystal structures, here we employ density functional theory calculations and k.p Hamiltonian based continuum model to unveil the bandgap opening in the edge-state spectrum of finite width molybdenum disulphide, molybdenum diselenide, tungsten disulphide and tungsten diselenide. We show that the application of a perpendicular electric field simultaneously modulates the band gaps of bulk and edge-states. We further observe that tungsten diselenide undergoes a semi-metallic intermediate state during the phase transition from topological to normal insulator. The tuneable edge conductance, as obtained from the Landauer-Büttiker formalism, exhibits a monotonous increasing trend with applied electric field for deca-nanometer molybdenum disulphide, whereas the trend is opposite for other cases.

Proximity-induced superconducting gap in the quantum spin Hall edge state of monolayer WTe2

The quantum spin Hall (QSH) state was recently demonstrated in monolayers of the transition metal dichalcogenide 1T'-WTe$_2$ and is characterized by a band gap in the two-dimensional (2D) interior and helical one-dimensional (1D) edge states. Inducing superconductivity in the helical edge states would result in a 1D topological superconductor, a highly sought-after state of matter. In the present study, we use a novel dry-transfer flip technique to place atomically-thin layers of WTe$_2$ on a van der Waals superconductor, NbSe$_2$. Using scanning tunneling microscopy and spectroscopy (STM/STS), we demonstrate atomically clean surfaces and interfaces and the presence of a proximity-induced superconducting gap in the WTe$_2$ for thicknesses from a monolayer up to 7 crystalline layers. At the edge of the WTe$_2$ monolayer, we show that the superconducting gap coexists with the characteristic spectroscopic signature of the QSH edge state. Taken together, these observations provide conclusive evidence for proximity-induced superconductivity in the QSH edge state in WTe$_2$, a crucial step towards realizing 1D topological superconductivity and Majorana bound states in this van der Waals material platform.

Two-dimensional graphene-like Xenes as potential topological materials

Since the discovery of quantum spin Hall (QSH) effect in graphene, searching for two-dimensional (2D) QSH materials with larger bulk gap has been an active field in the past decade. As cousins of graphene, the elemental graphene-like 2D materials (Xenes, X refers to group-IV, group-V, or group-VI elements) have been particularly interested in search of the QSH state as they share a similar honeycomb lattice with graphene. For those Xenes with heavier elements, theory predicted an extraordinarily large bulk gap suitable for room-temperature applications that have been experimentally verified in recent epitaxially grown bismuthene and stanene samples. We briefly review the progress in theoretical and experimental investigations of 2D Xenes as possible QSH topological materials.

Quantization of spin Hall conductivity in two-dimensional topological insulators versus symmetry and spin-orbit interaction

The third-rank tensor of the static spin Hall conductivity is investigated for two-dimensional (2D) topological insulators by electronic structure calculations. Its seeming quantization is numerically demonstrated for highly symmetric systems independent of the gap size. 2D crystals with hexagonal and square Bravais lattice show similar effects, while true rectangular translational symmetry yields conductivity values much below the quantum $e^2/h$. Field-induced lifting the inversion symmetry does not influence the quantum spin Hall state up to band inversion but the conductivity quantization. Weak symmetry-conserving biaxial but also uniaxial strain has a minor influence as long as inverted gaps dictate the topological character. The results are discussed in terms of the atomic geometry and the Rashba contribution to the spin-orbit interaction (SOI). Translational and point-group symmetry as well as SOI rule the deviation from the quantization of the spin Hall conductance.