Abstract
The coexistence of nontrivial topology and giant Rashba splitting, however, has rare been observed in twodimensional (2D) films, limiting severely its potential applications at room temperature. Here, we through firstprinciples calculations to propose a series of inversionasymmetric groupIV films, ABZ_{2} (A ≠ B = Si, Ge, Sn, Pb; Z = F, Cl, Br), whose stability are confirmed by phonon spectrum calculations. The analyses of electronic structures reveal that they are intrinsic 2D TIs with a bulk gap as large as 0.74 eV, except for GeSiF_{2}, SnSiCl_{2}, GeSiCl_{2} and GeSiBr_{2} monolayers which can transform from normal to topological phases under appropriate tensile strain of 4, 4, 5, and 4%, respectively. The nontrivial topology is identified by Z_{2} topological invariant together with helical edge states, as well as the berry curvature of these systems. Another prominent intriguing feature is the giant Rashba spin splitting with a magnitude reaching 0.15 eV, the largest value reported in 2D films so far. The tunability of Rashba SOC and band topology can be realized through achievable compressive/tensile strains (−4 ~ 6%). Also, the BaTe semiconductor is an ideal substrate for growing ABZ_{2} films without destroying their nontrivial topology.
Introduction
Topological insulators (TIs) are new states of quantum matter interesting for fundamental condensed matter physics and important for practical applications^{1,2,3}. In particular, for twodimensional (2D) TIs, all the scatterings of conducting edge states are completely forbidden, while the surface states of threedimensional (3D) TIs are not protected against scattering at any angles other than 180°. The helical edge states of nanoribbons made from 2D TIs provide massless relativistic carriers with intrinsic spinmomentum lock and negligible backscattering, ideal for the realization of conducting channels without dissipation in electronic devices^{4,5}. However, quantum spin Hall (QSH) effect has only been experimentally observed in HgTe/CdTe^{6,7} and InAs/GaSb^{8,9} quantum wells at very controlled and low temperature. In spite of this successful progress, the experimental realizations of these topological quantum states in 2D films are still challengeable.
Graphene^{1,2}, the first 2D TI model, supports an unobservable small bulk gap due to its extremely weak spinorbit coupling (SOC). Afterwards, several candidate materials are proposed to overcome the issue in graphene for the observation of QSH effect. These samples include silicene^{10}, germanene^{11}, stanene^{12}, plumbene^{13}, bismuth^{14}, and arsenene^{15}, in which they exhibit the topology properties alike to grapheme, but their nontrivial bulk gaps are still small. Motivated by the experimental synthesis of hydrogenated germanene (GeH)^{16}, the chemical functionalization on honeycomb structure, e.g., Bi/Sb/As^{17,18,19,20} and Ge/Sn/Pb films^{21,22,23,24,25,26,27}, has been proposed to host QSH effect with enhanced band gaps. For example, Xu et^{12} reported that the nontrivial band gap of pristine stanene is merely 0.1 eV, while it is enlarged to 0.3 eV under fully halogenation^{12,23,24,25,26}. More interestingly, the bulk gaps of 2D plumbene films with chemical decoration, PbX (X = H, F, Cl, Br and I) monolayers, have been elevated to a remarkable 1.34 eV^{13} the largest gap of all the reported 2D TIs as we know. All these results provide a great potential for the realization of QSH effect at room temperature in device paradigms for quantum information processing.
Rashba SOC, on the other hand, generally leads to spinpolarized band dispersion curves with inplane opposite helical spin texture^{28}. This feature originates from structural inversionsymmetry breaking at the surface/interface system, allowing the control of spin direction through an electric field^{29}. As compared with inversionsymmetric TIs, these systems are called 2D inversionasymmetric TIs (IASTIs), which are more promising due to their perfect performance in realizing new topological phenomena, such as crystallinesurface dependent topological electronic states^{30}, pyroelectricity^{31}, natural topological pn junctions^{32} and so on. In conventional semiconductor materials^{33,34,35,36}, Rashba SOC can only be tuned in a relatively limited range under electric field since it is generally rather weak, so it would be one of the important reasons to interpret the great attention the DattaDas spin FET^{37}, which still has not been realized in experiments. Recently, Arguilla et al.^{38} synthesize 2D honeycomb Ge_{1−x}Sn_{x}H_{1−x} (OH)_{x} film from the topochemical deintercalation of CaGe_{2–2x}Sn_{2x}. In comparison with inversionsymmetric systems^{15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31}, the introduction of tin breaks the inversionsymmetry of 2D germanene, which may induce a significant large Rashba SOC. Motived by above suppose, we attempt to extend and advance the groupIV 2D films to realize giant Rashba SOC accompanied with nontrivial topology, which may our proposal achievable experimentally accessible.
In present work, we report on the electronic and topological properties of a series of 2D IASTIs in groupIV binary films, which are called as ABZ_{2} monolayers, where A and B belong to groupIV elements (Si, Ge, Sn, Pb), as well as Z is the decorated halogen atoms (F, Cl, and Br), as illustrated in Fig. 1 (a). We find that most of these groupIV binary films are demonstrated as 2D IASTIs possess sizeable band gaps, reaching a maximum value of 0.74 eV for PbSnCl_{2}, which exceed the thermal energy at room temperature. Though GeSiF_{2}, SnSiCl_{2}, GeSiCl_{2} and GeSiBr_{2} monolayers are normal insulators (NI), they can transform 2D TI phases under tensile strain of 4, 4, 5, 4%, respectively. The nontrivial quantum states are identified by Z_{2} topological invariant together with helical edge states induced by SOC, as well as berry curvature of these systems. The most prominent feature in these systems is the giant Rashba splitting energy, for example, a value of 0.15 eV for PbSiCl_{2}, which is the largest value reported in 2D films so far. Furthermore, a favorable tunability of Rashba SOC and nontrivial topology can be obtained through achievable compressive/tensile strains (−4 ~ 6%). We also find that the Te (111)terminated surface in semiconductor BaTe is an ideal substrate for experimental realization of these monolayers, without destroying their nontrivial topology. The coexistence of giant Rashba SOC and band topology in these films may bring rich spinrelated phenomena in 2D IASTIs, which offers a special contribution to nanoelectronics and spintronics.
Calculated methods and details
All calculations are carried out using the plane wave basis Vienna ab initio simulation pack (VASP) code^{39,40} implementing density functional theory (DFT). The projectoraugmented wave (PAW) method^{41} is used to describe electronion potential. The exchangecorrelation potential is approximated by generalized gradient approximation (GGA) in the PerdewBurkeErnzerhof (PBE)^{42} form. Considering the possible underestimation of GGA method, the HeydScuseria Ernzerhof (HSE06) hybrid functional is employed to check the band topology^{43}. We use an energy cutoff of 450 eV and maximum residual force less than 0.001 eV/Å. Periodic boundary conditions are employed to simulate these 2D systems, and the Brillouin zone is sampled by using a 17 × 17 × 1 Gammacentered MonkhorstPack grid. Moreover, SOC is taken into account in terms of the secondvariational procedure. The phonon dispersion calculations are carried out by employing the PHONOPY code^{44} through the densityfunctional perturbation theory (DFPT) approach.
Results and Discussion
Figure 1(a) displays the honeycomb lattice of ABCl_{2} (A ≠ B = Si, Ge, Sn, Pb) monolayers, accompanied with its Brillouin zone in Fig. 1(b). All these configurations are isostructural to inversionsymmetric Ge/Sn/Pb monolayers with C_{3v} symmetry^{19,20,21,22,23,24,25}, but the A and B sublattices are occupied with different groupIV atoms, resulting in inversionsymmetry breaking. To verify the geometric stability of these structures, the total energies ABCl_{2} monolayers with respect to lattice constant a are illustrated in Fig. 1(c), where a is varied to identify structural ground state. Interestingly, the doublewell energy variation curves occur in all cases, namely, the highbuckled (HB) and lowbuckled (LB) states, respectively. In HB structure, the vertical layer distance lies in the range of 3.58 − 5.15 Å, while the distance in LB state is smaller within 1.42−2.51 Å, as listed in Table 1. The LB structure here, however, is more stable by at least 1.04 eV per unit cell than HB structure, different form the cases of IIIBi films, where the HB structure is the ground state. Thus, we only focus on stable LB chlorinated monolayers in the following. Furthermore, the energetically structural stability is evaluated by the formation energy expressed as
Where E(ABCl_{2}) and E(AB) are the total energies of chlorinated and pristine AB monolayers, respectively, while μ(Cl) is the chemical potential of chlorination atoms. As listed in Table 1, the calculated formation energies are in the range of −2.284 ~ −3.146 eV per atom, indicating no phase separation emerges in these systems. For practical application as a nanodevice, the 2D films should be chemically inert and environmentally friendly. Thus, the phonon spectrum dispersions are calculated and plotted in Fig. 2, which demonstrates that all ABCl_{2} monolayers are dynamically stable since all the vibrational models are real in the whole Brillouin zone.
The band structures of ABCl_{2} monolayers are shown in Fig. 3, with the orbitalprojected components displayed in Fig. S1 in the Supplementary Information. In the absence of SOC, the PbSnCl_{2}, PbGeCl_{2} PbSiCl_{2} and SnGeCl_{2} monolayers show gapless semiconductor feature with valence band maximum (VBM) and conduction band minimum (CBM) degenerated at the Γ point. The Fermi level locates exactly at the degenerated point with a zero density of states. Notably, the slopes of the valence band and conduction band in such a touching point show the parabolic dispersion, instead of a linear one, as illustrated by a 3D band structure for PbSnCl_{2} monolayer (Fig. 4 (a)). This scenario for the electronic properties is quite similar to Ge/Sn/Pb monolayers^{21,22,23,24,25,26,27}. Upon including SOC, the band structures produce a transition from semimetal to semiconductor, as the degenerated states at the touching point separate from each other, see Figs 3(a–d) and 4(b). As observed in previously reported Bi/Sb/As^{17,18,19,20}, the SOCinduced bandgap opening at the Fermi level is a strong indication of the existence of topologically nontrivial phases. Furthermore, from Si to Pb atom, the band gap opening gradually increases (Table 1), since the SOC effect gets stronger due to increasing atomic numbers of heavier atoms. Here we label this SOCinduced band gap at the Γ point as E_{Γsoc}, to be distinguished from whole gap (E_{gsoc}), as listed in Table 1. To eliminate the possible underestimation of the band gap, we employ additional HSE06^{37} to confirm the existence of band inversion for all these systems. As expected, the values of E_{Γ} are enhanced to 0.74, 0.49, 0.41 and 0.23 eV, respectively, for PbSnCl_{2}, PbGeCl_{2}, PbSiCl_{2} and SnGeCl_{2} monolayers.
To confirm the nontrivial topological nature, we introduce the evolution of Wannier Center of Charges (WCCs)^{45} to calculate Z_{2} topological invariant due to inversionsymmetry breaking. The Wannier functions (WFs) related with lattice vector R can be written as:
Here, a WCCs can be defined as the mean value of , where the is the position operator and is the state corresponding to a WF in the cell with R = 0. Then we can obtained
Assuming that with S = I or II, where summation in α represents the occupied states and A is the Berry connection. So we have the format of Z_{2} invariant:
The Z_{2} invariant can be obtained by counting the even or odd number of crossings of any arbitrary horizontal reference line. Taking PbSnCl_{2} monolayer as an example, we present the evolution lines of WCCs in Fig. 5(a). One can see that the WCCs evolution curves cross any arbitrary reference lines odd times, yielding Z_{2} = 1.
The most important character of 2D TIs is the appearance of helical edge states with spinpolarization protected by TRS. Here we compute the edge Green’s functions using the ab initio TB model. The ab initio TB model is constructed by downfolding the bulk bands obtained by DFT calculations, using the recursive method in terms of maximally localized Wannier functions (MLWFs)^{46}. Figure 4(c) and (d) show the orbitaldecomposed band structures, in which the red and blue circles represent groupIV s and p_{x,y} states, respectively, and the symbol size is proportional to the orbital weight. Obviously, the bulk energy bands near the Fermi level are predominantly from p_{x,y} orbitals, thus the MLWFs are derived from atomic plike orbitals. In this case, the TB bulk bands perfectly reproduce the DFT bands around the Fermi level up to ± 1.0 eV, as illustrated in Fig. 5(b). Figure 5(c) displays the local density of states (LDOS) of the calculated edge states. Indeed, all the edge bands for PbSnCl_{2} monolayer connect completely the conduction and valence bands and span 2D bulk gap, yielding a 1D gapless edge states, in good agreement with the Z_{2} invariant. By identifying spinup and spindown contributions in edge spectral function, the counterpropagating edge states exhibit opposite spinpolarization (Fig. 5(d)), which benefit from their robustness against nonmagnetic scattering.
For GeSiCl_{2} and SnSiCl_{2} monolayers, on the other hand, they exhibit semiconductor feature with direct band gap of 0.285 and 0.265 eV in the absence of SOC, as shown in Fig. 3(e) and (f). As we know that external strain is an effective avenue to tune the band gap, this may introduce a band inversion by SOC. To simulate strain effects on the electronic structures, we employ the external strain to these systems which is defined as ε = (aa_{0})/a_{0}, where a and a_{0} are lattice constants with and without strains, respectively. In Fig. 6 we present the band structures with respect to external strains of GeSiCl_{2} as an example, where the red and blue colors represent s and p_{x,y} states, respectively. One can see that the band structure of GeSiCl_{2} is sensitive to the lattice expansion. Without SOC, the CBM (s) and VBM (p_{x,y}) move toward each other, leading to a reduction of band gap. The band gap closes at the Γ point when ε = 5%, as illustrated in Fig. 6(c), suggesting semimetallic character at the Fermi level. When the tensile strains are beyond 5%, the p_{x,y} orbitals shifted above s orbital, see Fig. 6(d). As a consequence, the opposite spatial shift of VBM and CBM decreases the band gap, which is essential to topological phase transitions. With increasing biaxial strain, the energy difference (E_{sp}) between the s and p_{x,y} orbitals at the Γ point near the Fermi level increase monotonically, but the system remains the semimetallic feature. When SOC is included, a sizable band gap opens, see Fig. 6(e–h). Similar results are also found in SnSiCl_{2} monolayer in Fig. 7, but its critical point of elastic strain is 4%, smaller than GeSiCl_{2} monolayer. Such an SOCinduced band gap reopening usually corresponds to a NI  TI transition, as illustrated in Fig. 8.
To gain a physical understanding of QSH effect, we analyze the orbital contribution around the Fermi level for GeSiCl_{2} monolayer. Figure 9 presents the band evolution at the Γ point, in which the energy levels near the Fermi level are mainly composed of s and p_{x,y} orbitals from groupIV elements. The chemical bonding between SiGe atoms makes it split into the bonding and antibonding states, i.e., s^{±} and p_{x,y}^{±}, respectively. At ground state in Fig. 9(a), the bands near the Fermi level are contributed by the p_{x,y}^{+} and s^{−}, with the s^{−} being above p_{x,y}^{+}, indicating a normal band order. When considering the external tensile strains, as illustrated in Fig. 9(b), the enlarged lattice constant weakens the interactions between A and B sublattices, decreasing the splitting between the bonding and antibonding states, which lowers s^{−} and raises p_{x,y}^{+} level accordingly. Thus, depending on the strength of interatomic coupling, the band gap of GeSiCl_{2} can be continuously reduced, with the band order being reversed at critical point (ε = 5%), making it a 2D TI with Z_{2} = 1. In another words, the level crossing leads to a parity exchange between occupied and unoccupied bands, inducing a NITI phase transition. While for PbSnCl_{2}, PbGeCl_{2} PbSiCl_{2} and SnGeCl_{2} monolayers, the lattice constant (See Fig. 1(c) and Table 1) is enlarged significantly at equilibrium state, equal to straininduced GeSiCl_{2}, thus they are native 2D TIs without any strain. Herein, we must point out that, although the sp_{x,y} band inversion is caused mainly by the strength of interatomic coupling, the SOC is still indispensable because it makes the p_{x,y}^{+} orbital split into p^{+}_{x+iy,↑}& p^{+}_{xiy,↓} and p^{+}_{xiy,↑}& p^{+}_{x+iy,↓}, opening a larger band gap. In previous works, the topological feature mainly originates from p_{z} states such as graphene and silicene^{1,2,3}, but here the band topology is attributed to the contribution of p_{x,y} orbitals on groupIV elements, thus their band gap is enhanced significantly.
Furthermore, to clearly show the QSH effect in these systems and confirm its nontrivial topology, we also calculate the intrinsic spin Hall conductivity of PbSnCl_{2} monolayer. Using the Kubo formula^{47,48}, the spin Hall conductivity tensor is expressed as,
Here the spin Berry curvature is given by
where n is the band index, ψ_{n,k} is the eigenstate, v_{y} is the velocity operator, and j_{x} is the spin current operator defined as (s_{z}v_{x} + v_{x}s_{z})/2. Also, a factor of 2e/ћ in σ^{SH} is multiplied to convert its unit into the conventional electron conductivity e^{2}/h. Figure 10(a) displays the kresolved spin Berry curvature at the Fermi level. One can clearly see the positive peak of Ω^{SH}(k) around the Γ point, which is consistent with its band topology. The spin Hall conductance via its chemical potential shows a nearly quantized value 2e^{2}/h of conductance within the energy window of the SOC gap ranging from about − 0.2 eV to the Fermi energy, demonstrating that the QSH effect could be easily observed experimentally at room temperature.
The most prominent feature in inversionasymmetric systems is the existence of giant Rashba SOC in valence band edge rather than Dresselhaus splitting^{49}. These can be confirmed by the spin texture of Rashba spin splitting in Fig. 10(c), in which the spins of the electronic states are oppositely aligned within the k_{x} and k_{y} plane, and are normal to the wave vector k. Here, we magnify the Rashba splitting energy E_{R} and the momentum offset k_{R} in Fig. 10(b), which corresponds to the splitting of the VBM along momentum direction at the Γ point, indicated by the rectangular box in Fig. 3(c). According to the linear Rashba model of the 2D systems, the dispersion of Rashba splitting can be expressed as with k = (k_{x}, k_{y}), where the m^{*} is effective mass and is the Rashba constant. Thus we can obtain a giant Rashba parameter eV/Å than the wellknown oxide interface LaAlO_{3}/SrTiO_{3} (0.01–0.05 eV/Å)^{50,51,52}, quantum well InGaAs/InAlAs (0.07 eV/Å)^{53} and Au (111) surface^{54}. These great Rashba splitting energy E_{R} and momentum offset k_{R} are desired for stabilizing spin and achieving a significant phase offset for different spin channels.
It is noteworthy that strain engineering can not only tune the topological phase, but also change the Rashba spin splitting significantly in these monolayers. In Fig. 11, we present three parameters (E_{R}, k_{R}, α_{R}) under different biaxial strains by solid triangles for PbSiCl_{2} as an example. These three parameters decrease monotonically with the increasing lattice constants, demonstrating that a compressive/tensile strain can enhance/decrease the Rashba SOC strength. For example, the Rashba splitting can reach up to 0.182 eV under −4% strain. Thus, by biaxial strain engineering, we can enhance or reduce the SOC strength according to the experimental demands. The Rashba spin splitting of energy bands obtained here provides a chance for spintronic device applications without magnetic field, for instance, spin fieldeffect transistor (FET).
Finally, we must point out that that the chlorination in groupIV monolayers is not the only way to achieve QSH effect, the same results can be obtained by decorating the surface with other halogens, such as F and Br atoms. As evidenced in Fig. S2 in the supplementary information, the doublewell energy curve still exists, with the LB structure being a ground state. The corresponding band structures of ABF_{2} and ABBr_{2} monolayers with and without SOC are further plotted in Fig. 12. Most of halogenated ABX_{2} monolayers are similar to chlorinated ones with a topological invariant Z_{2} = 1, except for GeSiF_{2} and GeSiBr_{2} with Z_{2} = 0. However, they can transform into QSH insulators under external strain of 4%, as illustrated in Fig. 8(a) and (c). In addition, Fig. 8 presents the calculated band gaps at equilibrium states, accompanied with the lattice parameters listed in Tables SI and II. Interestingly, the nontrivial band gaps can reach as large as 0.11–0.63 eV (Fig. 10(d)), which is comparable or larger than stanene (0.1 eV)^{12} or bismuth (0.2 eV)^{14}. Such large nontrivial band gaps are capable of stabilizing the boundary current against the influence of thermally activated bulk carriers, and thus are beneficial for hightemperature applications.
Experimentally, choosing a suitable substrate is a key factor in device application, since a freestanding film must be eventually deposited or grown on a substrate. Considering the small lattice constant mismatch between PbSnCl_{2} and semiconductor BaTe^{55}, we select Teterminated (111) surface of semiconductor BaTe to support PbSnCl_{2} monolayer, as illustrated in Fig. 13. In the equilibrium state, the bottom (top) Sn atoms in PbSnCl_{2} are located at top (hcp) sites of the substrate, while the top Pb atoms are located at hcp sites, see Fig. 13(a) and (b). On this Teterminated surface, Sn atoms bind preferably on top of Te atoms by forming chemical bonds, so the p_{z} orbital of groupIV atoms and dangling bond of the substrate are both fully passivated. Figure 13(c–f) present the band structures with and without SOC, respectively. One can see that a few bands appear within the bulk gap of substrate around the Fermi level, which is mostly contributed by groupIV atoms according to wavefunction analysis. By projecting Bloch wave functions onto atomic orbitals of PnSnCl_{2}, Bloch states contributed by the s orbital of groupIV atoms are visualize by red dots. These states shift to valence bands around the Γ point, suggesting a clear sp band order. This SOCinduced band gap is topologically nontrivial as explained by the band inversion, thus it is a robust 2D TI on Teterminated substrate of BaTe semiconductor.
Conclusions
In summary, by means of firstprinciples calculations, we predict a series of inversionasymmetric groupIV monolayers, ABZ_{2} (A≠B = Si, Ge, Sn, Pb; Z = F, Cl, Br), allows for the simultaneous presence of topological order and Rashba SOC. The phonon spectrum calculations reveal that they are dynamically stable. The analyses of electronic structures reveal that most of these monolayers are native 2D TIs with a bulk gap as large as 0.74 eV, while GeSiF_{2}, SnSiCl_{2}, GeSiCl_{2} and GeSiBr_{2} can transform from normal to topological phases under tensile strain of 4, 4, 5 and 4%, respectively. The nontrivial topology is identified by Z_{2} topological invariant together with helical edge states, as well as the berry curvature of these systems. The most intriguing feature is their giant Rashba spin splitting with a magnitude reaching 0.15 eV for PbSiCl_{2}, the largest value reported in 2D films so far. A large tunability of Rashba SOC and band topology can be realized through achievable compressive/tensile strains (−4 ~ 6%). In addition, the Te(111)terminated surface in semiconductor BaTe is predicted to be an ideal substrate for experimental realization of these systems, without destroying their nontrivial topology. Considering their simple crystal structure as well as large nontrivial band gap and giant Rashba splitting energy in these monolayers, these findings present a platform to explore 2D inversionasymmetric TIs for roomtemperature device applications.
Additional Information
How to cite this article: Zhang, S. et al. TwoDimensional Large Gap Topological Insulators with Tunable Rashba SpinOrbit Coupling in GroupIV films. Sci. Rep. 7, 45923; doi: 10.1038/srep45923 (2017).
Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
References
 1.
Kane, C. L. & Mele, E. J. Quantum spin Hall effect in grapheme. Phys. Rev. Lett. 95, 226801 (2005).
 2.
Kane, C. L. & Mele, E. J. Z2 topological order and the quantum spin Hall effect. Phys. Rev. Lett. 95, 146802 (2005).
 3.
Moore, J. E. Majorana’s wires. Nature Nanotech 8, 194–198 (2013).
 4.
Hsieh, D. et al. A tunable topological insulator in the spin helical Dirac transport regime. Nature (London) 460, 1101 (2009).
 5.
Qi, X.L. & Zhang, S.C. Topological insulators and superconductors. Rev. Mod. Phys. 83, 1057 (2011).
 6.
Bernevig, B. A. et al. Quantum spin Hall effect and topological phase transition in HgTe quantum wells. Science 314, 1757 (2006).
 7.
König, M. et al. Quantum spin Hall insulator state in HgTe quantum wells, Science 318, 766 (2007).
 8.
Du, L. et al. Robust helical edge transport in gated InAs/GaSb bilayers. Phys. Rev. Lett. 114, 096802 (2015).
 9.
Liu, C. et al. Quantum spin Hall effect in inverted typeII semiconductors. Phys. Rev. Lett. 100, 236601 (2008).
 10.
Zhang, R. W. et al. Siliconbased Chalcogenide: Unexpected Quantum Spin Hall Insulator with Sizable Band Gap. Appl. Phys. Lett. 109, 182109 (2016).
 11.
Liu, C. C. et al. Quantum spin Hall effect in silicene and twodimensional germanium. Phys. Rev. Lett. 107, 076802 (2011).
 12.
Xu, Y. et al. Largegap quantum spin Hall insulators in tin films. Phys. Rev. Lett. 111, 136804 (2013).
 13.
Zhao, H. et al. Unexpected GiantGap Quantum Spin Hall Insulator in Chemically Decorated Plumbene Monolayer. Sci. Rep. 6, 20152 (2016).
 14.
Wada, M. et al. Localized edge states in twodimensional topological insulators: Ultrathin Bi films. Phys. Rev. B 83, 121310 (2011).
 15.
Wang, Y. P. et al. Tunable quantum spin Hall effect via strain in twodimensional arsenene monolayer. J. Phys. D: Appl. Phys. 49, 055305 (2016).
 16.
Bianco, E. et al. Stability and exfoliation of germanane: a germanium graphane analogue. ACS Nano 7, 4414 (2013).
 17.
Ma, Y. D. et al. Robust twodimensional topological insulators in methylfunctionalized bismuth, antimony, and lead bilayer films. Nano Lett. 15, 1083 (2015).
 18.
Zhou, M. et al. Epitaxial growth of largegap quantum spin Hall insulator on semiconductor surface. PNAS 111, 14378 (2014).
 19.
Song, Z. G. et al. Quantum Spin Hall and Quantum Valley Hall Insulators of BiX/SbX (X = H, F, Cl, and Br) Monolayer with a Record Bulk Band Gap, NPG Asia Mater. 6, e147 (2014).
 20.
Wang, Y. P. et al. Controllable band structure and topological phase transition in twodimensional hydrogenated arsenene. Sci. Rep. 6, 20342 (2016).
 21.
Si, C. et al. Functionalized germanene as a prototype of largegap twodimensional topological insulators. Phys. Rev. B 89, 115429 (2014).
 22.
Xu, Y. et al. Largegap quantum spin Hall states in decorated stanene grown on a substrate. Phys. Rev. B 92(8), 081112 (2015).
 23.
Zhang, R. W. et al. Ethynylfunctionalized stanene film: a promising candidate as largegap quantum spin Hall insulator. New. J. Phys. 17, 083036 (2015).
 24.
Zhang, R. W. et al. Room Temperature Quantum Spin Hall Insulator in EthynylDerivative Functionalized Stanene Films. Sci. Rep. 6, 18879 (2016).
 25.
Wang, Y. P. et al. Largegap quantum spin Hall state in functionalized dumbbell stanene, Appl. Phys. Lett. 108, 073104 (2016).
 26.
Zhang, R. W. et al. New family of room temperature quantum spin Hall insulators in twodimensional germanene film. J. Mater. Chem. C 4, 2088 (2016).
 27.
Zhao, H. et al. Firstprinciples prediction of a giantgap quantum spin Hall insulator in Pb thin film, Phys. Chem. Chem. Phys. 18, 31862 (2016).
 28.
Bychkov, Y. A. & Rashba, E. I. Properties of a 2D electron gas with lifted spectral degeneracy. JETP Lett. 39, 78 (1984).
 29.
Jia, Y. Z. et al. FirstPrinciples Prediction of InversionAsymmetric Topological Insulator in Hexagonal BiPbH Monolayer. J. Mater. Chem. C 4, 2243 (2016).
 30.
Bahramy, M. S. et al. Emergence of noncentrosymmetric topological insulating phase in BiTeI under pressure. Nat. Commun. 3, 679 (2012).
 31.
Wan, X. et al. Topological semimetal and Fermiarc surface states in the electronic structure of pyrochlore iridates. Phys. Rev. B 83, 205101 (2011).
 32.
Wang, J. et al. Topological pn junction, Phys. Rev. B 85, 235131 (2012).
 33.
Zhong, Z. et al. Theory of spinorbit coupling at LaAlO 3/SrTiO 3 interfaces and SrTiO 3 surfaces. Phys. Rev. B 87, 161102 (2013).
 34.
Nitta, J. et al. Gate Control of SpinOrbit Interaction in an Inverted In 0.53 Ga 0.47 As/In 0.52 A l0.48 As Heterostructure. Phys. Rev. Lett. 78, 1335 (1997).
 35.
Lashell, S. et al. Spin splitting of an Au (111) surface state band observed with angle resolved photoelectron spectroscopy. Phys. Rev. Lett. 77, 3419 (1996).
 36.
Tesch, U. et al. Structural and electronic properties of graphene nanoflakes on Au (111) and Ag (111). Sci. Rep. 6, 23439 (2016).
 37.
Datta, S. & Das, B. Electronic analog of the electrooptic modulator. Appl. Phys. Lett. 56, 665 (1990).
 38.
Arguilla, M. Q. et al. Goldberger, Synthesis and stability of twodimensional Ge/Sn graphane alloys. Chem. Mater. 26, 6941–6946 (2014).
 39.
Kresse, G. & Furthmuller, J. Efficient iterative schemes for ab initio totalenergy calculations using a planewave basis set. Phys. Rev. B 54, 11169 (1996).
 40.
Kresse, G. & Joubert, D. From ultrasoft pseudopotentials to the projector augmentedwave method. Phys. Rev. B 59, 1758 (1999).
 41.
Fisher, A. J. et al. Projector augmentedwave method. Phys. Rev. B 50, 17953 (1994).
 42.
Perdew, J. P. et al. Generalized gradient approximation made simple. Phys. Rev. Lett. 77, 3865 (1996).
 43.
Heyd, J. & Scuseria, G. E. Ernzerhof M. Erratum: “Hybrid functionals based on a screened Coulomb potential”. J. Chem. Phys. 124, 219906 (2006).
 44.
Togo, A. et al. Firstprinciples calculations of the ferroelastic transition between rutiletype and CaCl_{2}type SiO_{2} at high pressures. Phys. Rev. B 78, 134106 (2008).
 45.
Yu, R., Qi, X. L., Bernevig, A., Fang, Z. & Dai, X. Phys. Rev. B 84, 075119 (2011).
 46.
Klimeš, J., Bowler, D. R. & Michaelides, A. Phys. Rev. B 83, 195131 (2011).
 47.
Yao, Y. G. & Fang, Z. Sign changes of intrinsic spin Hall effect in semiconductors and simple metals: firstprinciples calculations. Phys. Rev. Lett. 95, 156601 (2005).
 48.
Guo, G. Y. et al. Ab initio calculation of the intrinsic spin Hall effect in semiconductors. Phys. Rev. Lett. 94, 226601 (2005).
 49.
Rashba, I. E. Properties of semiconductors with an extremum loop. 1. Cyclotron and combinational resonance in a magnetic field perpendicular to the plane of the loop. Sov. Phys. Solid State 2, 1109 (1960).
 50.
Caviglia, A. D. et al. Tunable Rashba spinorbit interaction at oxide interfaces. Phys. Rev. Lett. 104, 126803 (2010).
 51.
Fête, A. et al. Rashba induced magnetoconductance oscillations in the LaAlO_{3}SrTiO_{3} heterostructure. Phys. Rev. B 86, 201105 (2012).
 52.
Zhong, Z. et al. Theory of spinorbit coupling at LaAlO 3/SrTiO 3 interfaces and SrTiO_{3} surfaces. Phys. Rev. B 87, 161102 (2013).
 53.
Nitta, J. et al. Gate Control of SpinOrbit Interaction in an Inverted In 0.53 Ga 0.47 As/In 0.52 A l0.48 As Heterostructure. Phys. Rev. Lett. 78, 1335 (1997).
 54.
Lashell, S. et al. Spin splitting of an Au (111) surface state band observed with angle resolved photoelectron spectroscopy. Phys. Rev. Lett. 77, 3419 (1996).
 55.
Miller, E. et al. The Preparation and Properties of Barium, Barium Telluride, and Barium Selenide. Trans. Metall. Soc. AIME 218, 978 (1990).
Acknowledgements
This work was supported by National Natural Science Foundation of China (Grant Nos. 11434006 and 11304121).
Author information
Affiliations
School of Physics and Technology, University of Jinan, Jinan, Shandong, 250022, People’s Republic of China
 Shoujuan Zhang
 , Weixiao Ji
 , Changwen Zhang
 , Ping Li
 & Peiji Wang
Authors
Search for Shoujuan Zhang in:
Search for Weixiao Ji in:
Search for Changwen Zhang in:
Search for Ping Li in:
Search for Peiji Wang in:
Contributions
S. Z and C. Z conceived the study and wrote the manuscript. W. J performed the firstprinciples calculations and calculated the phonon spectrum. P. L and P. W. prepared figures. All authors read and approved the final manuscript.
Competing interests
The authors declare no competing financial interests.
Corresponding author
Correspondence to Changwen Zhang.
Supplementary information
PDF files
Rights and permissions
This work is licensed under a Creative Commons Attribution 4.0 International License. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in the credit line; if the material is not included under the Creative Commons license, users will need to obtain permission from the license holder to reproduce the material. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/
About this article
Further reading

Edge modes and FabryPerot plasmonic resonances in anomalousHall thin films
Physical Review B (2019)

Theoretical study of the bandgap regulation of a twodimensional GeSn alloy under biaxial strain and uniaxial strain along the armchair direction
Physical Chemistry Chemical Physics (2018)

FerromagnetFree AllElectric Spin Hall Transistors
Nano Letters (2018)

Spin–orbit effects on the spin and pseudospin polarization in acdriven silicene
Journal of Physics: Condensed Matter (2018)

Probing the (110)Oriented plane of rutile ZnF2: A DFT investigation
Journal of Physics and Chemistry of Solids (2017)
Comments
By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.