Large-band-gap non-Dirac quantum spin Hall states and strong Rashba effect in functionalized thallene films

The quantum spin Hall state materials have recently attracted much attention owing to their potential applications in the design of spintronic devices. Based on density functional theory calculations and crystal field theory, we study electronic structures and topological properties of functionalized thallene films. Two different hydrogenation styles (Tl2H and Tl2H2) are considered, which can drastically vary the electronic and topological behaviors of the thallene. Due to the C3v symmetry of the two systems, the px and py orbitals at the Γ point have the non-Dirac band degeneracy. With spin–orbit coupling (SOC), topological nontrivial band gaps can be generated, giving rise to non-Dirac quantum spin Hall states in the two thallium hydride films. The nontrivial band gap for the monolayer Tl2H is very large (855 meV) due to the large on-site SOC of Tl px and py orbitals. The band gap in Tl2H2 is, however, small due to the band inversion between the Tl px/y and pz orbitals. It is worth noting that both the Tl2H and Tl2H2 monolayers exhibit strong Rashba spin splitting effects, especially for the monolayer Tl2H2 (αR = 2.52 eVÅ), rationalized well by the breaking of the structural inversion symmetry. The Rashba effect can be tuned sensitively by applying biaxial strain and external electric fields. Our findings provide an ideal platform for fabricating room-temperature spintronic and topological electronic devices.

Since Kane and Mele first proposed the quantum spin Hall (QSH) effect in graphene 1,2 , QSH insulators, also known as one type of two-dimensional (2D) topological insulators (TIs), have attracted great attention in condensed matter physics and material science due to their wide potential applications in spintronics and topological quantum computation [3][4][5] .QSH insulators are characterized by an insulating band gap in bulk and non-dissipative, fully spin-polarized gapless helical edge states at the sample boundary, which are protected by time inversion symmetry.As the first predicted 2D TI, graphene 1 opens a topologically nontrivial band gap at the Dirac point, with the consideration of the spin-orbit coupling (SOC) effect.However, besides the factor of the light carbon element, the horizontal mirror symmetry in the graphene structure inhibits the first-order SOC effect between the nearest neighbor carbon atoms.Therefore, the weak second-order SOC makes the QSH effect only appear at an unrealistically low temperature 6 .Till now, the experimental observations of quantized Hall conductance through the QSH effect are only reported in few systems including Bismuthene 7 , WTe 2 8 , and HgTe/CdTe 9,10 and InAs/GaSb 11,12 quantum-wells with an ultralow temperature (< 10 K), which limits their tempting applications in room temperature spintronic devices.
To achieve QSH effects at room temperature, 2D materials with strong SOC interactions and large topologically nontrivial band gaps are very desired.Some graphene-like 2D layered materials have been proposed to own QSH effects with relatively large band gaps, including group-IVA monolayers for the low-pucker silicene 13,14 , germanene 13,15 , stanene 16 , and group-VA bismuthene 7 .Based on the existing 2D materials, chemical functionalization has been found being a valid tactic to design the QSH effects with pretty large band gaps 14,16,17 .These researches indicate that two types of schemes may be employed to produce the QSH states with large band gaps.One is designing the materials containing heavy elements which can bring strong SOC effects.Another approach is through chemical functionalization.Their combination should be a more effective tactic.
Besides the groups IVA Pb and VA Bi, group IIIA Tl can also be regarded as a heavy element, whose graphenelike monolayer structure is called thallene 18,19 .The pristine 2D honeycomb-like thallene is a topological trivial semiconductor.The phase transition from a semiconductor to a QSH insulator can be achieved under the application of large biaxial strain and its nontrivial topology comes from a band inversion between p x/y and p z orbitals induced by SOC 18 .The thallene monolayer has been successfully prepared in experiments by cooling the 2/3 monolayer of mobile Tl atoms on a single-layer NiSi 2 atop a Si(111) substrate below ~ 150 K 19 .Compared with the theoretically assumed free-standing thallene, the thallene structure frozen on the NiSi 2 /Si(111) substrate in the experiments undergoes strong tensile strain (~ 27%) and then enters the topological phase 19 .Very recently, high-quality large-scale thallene monolayers with exotic electron bands demonstrating colossal spin-polarization have been fabricated through the decoration of thallene/NiSi 2 interface by Sn interlayers 20 .Thus, Tl monolayers actually become a new heavy-element material platform, which may be designed to explore the interesting 2D topological electronic states etc.
In this work, based on first-principles calculations, two different configurations of hydrogenated thallene Tl 2 H and Tl 2 H 2 are built.The electronic structures and topological properties for the two monolayers are systematically studied.Due to the C 3v symmetry of the structures, the Tl p x and p y orbitals of the Tl 2 H and Tl 2 H 2 monolayers form quadratic non-Dirac bands.A topologically nontrivial band gap (up to 855 meV) can be generated by the SOC interaction, giving rise to a non-Dirac quantum spin Hall state in the materials.The variation between the electronic states of the Tl 2 H and Tl 2 H 2 monolayers can be comprehended through the crystal field splitting and band inversion.Particularly, due to the absence of the structural inversion symmetry, the Tl 2 H and Tl 2 H 2 monolayers both exhibit marked Rashba spin splitting effect, especially in the monolayer Tl 2 H 2 .The Rashba splitting can be tuned sensitively by applying biaxial strain and external electric fields, beneficial to spintronic applications.

Crystal structures and stability
Different from other heavy element monolayers, the thallene monolayer is a fully flat 2D honeycomb lattice without buckling 18,19 .In other words, the Tl atoms in thallene are aligned in the same plane, with two Tl atoms in each unit cell.The structure has a space group symmetry of P6/mmm (No. 191) with a space inversion symmetry.On this basis, we construct two types of functionalized thallene structures to explore the unique electronic structures: unilateral semi-saturation and unilateral full saturation with hydrogen atoms, which were named as Tl 2 H and Tl 2 H 2 , respectively, as shown in Fig. 1a,b.To simulate the experimental configuration of thallene grown on a substrate 19 , merely the unilateral saturation is considered.Due to the unilateral hydrogenation, structural inversion symmetry of the two materials is broken and the space groups of Tl 2 H and Tl 2 H 2 are P3m1 (No.156) and P6mm (No.183), respectively.After hydrogenation, the Tl atoms still own a honeycomb lattice (see the top views of Fig. 1a,b) for Tl 2 H and Tl 2 H 2 , and the Tl atomic layer of monolayer Tl 2 H is not a completely planar structure, but a low-buckled structure with a vertical height h between the two Tl atomic layers (see the side view of Fig. 1a).In Tl 2 H 2 , the Tl atomic layer still maintains a planar structure (see the side view of Fig. 1b).The buckled structural characteristic in Tl 2 H is attributed to the different chemical environments around the www.nature.com/scientificreports/two types of Tl atoms located in different positions of the honeycomb lattice.In the unit cell of Tl 2 H (Fig. 1a), the left Tl atom is not only bonded to three in-plane neighbor Tl atoms, but also saturated by one H atom.The interaction between Tl and H leads to the Tl atom (bonded to the H atom) approaching the H atom. Thus, a low-buckling structure appears for the Tl atoms in Tl 2 H (Fig. 1a).Dissimilarly, in Tl 2 H 2 , both of the Tl atoms in the unit cell bond with three neighbor Tl atoms and one H atom, resulting in a planar structure of the Tl atoms in the monolayer (Fig. 1b).
The optimized lattice constants of the Tl 2 H and Tl 2 H 2 are a = 5.24 Å and 5.28 Å (Fig. 1c,d), respectively, which together with other structural parameters, the binding energies, and the formation energies, are listed in Table 1.The positions of the atoms in the unit cells for the two monolayers are given in the Supplementary Material as Table SI.Compared to the Tl-Tl bond length ( d Tl−Tl ) in pristine thallene (3.01 Å 18 and 3.03 Å 19 ), the Tl-Tl bond lengths in Tl 2 H (3.04 Å) and Tl 2 H 2 (3.05 Å) are increased slightly by the hydrogenation.The trend is reasonable because the repulsive force between the H-H and Tl-H ions appears in the monolayer with H adsorbed.And it becomes larger with the increase of the H concentration.The binding energies and formation energies of the two monolayers are also calculated to investigate their structural stability.The binding energies are calculated by , where E Tl 2 H /E Tl 2 H 2 is the total energy per unit cell of the Tl 2 H /Tl 2 H 2 monolayer and E Tl /E H is the total energy of one Tl/H atom.The positive and relatively large values (> 4.5 eV) of the obtained E b for Tl 2 H and Tl 2 H 2 indicate very strong bonding between the atoms.The formation energies for the two monolayers are calculated with , where E Tl ′ is the total energy per unit cell of the pristine thallene monolayer and E H 2 is the total energy of one H 2 molecule.The obtained negative formation energies for the monolayers (Table 1) show exothermic reactions from the thallene monolayer and H 2 molecules, guaranteeing the feasibility of the experimental synthesis for these functionalized materials.
To explore the dynamical stability of the two monolayers, we calculate the phonon spectra for them.The results are displayed in Fig. S1(a) and (b).There are some negative frequencies in phonon spectra for the both Tl 2 H and Tl 2 H 2 monolayers.The results are not surprising because the phonon spectrum of the pristine thallene has also negative frequencies, as reported in Ref. 18.To eliminate the negative frequencies, the two monolayers are deposited to SiC substrates.The geometries of the Tl 2 H 2 /SiC heterostructure are displayed in Fig. S2 (a) and  (b).The substrates are helpful to stabilize the dynamical stability for the two materials.There are, however, still some negative frequencies for the Tl 2 H/SiC heterostructure, with the lowest value of − 0.7 THz (larger than that of the material without the substrate, − 2.0 THz).All the soft modes in the Tl 2 H 2 monolayer are removed by the SiC substrate, as indicated in Fig. S2(c).Thus, to experimentally observe the interesting electronic states and topological behaviors in the materials, the two monolayers should be placed on the SiC substrates.There are rich examples that the materials are not completely dynamically stable (with some soft vibration modes), which could be, however, fabricated successfully in experiments, for such as stanene 21,22 and plumbene 23,24 .

Band structures and strain tuning
Figure 2 shows band structures of the hydrogenated Tl 2 H and Tl 2 H 2 monolayers without the consideration of SOC.For convenient comparison, the bands of the pristine 2D thallene are also displayed and discussed first.Similar to the case of the low-buckled plumbene 17 , a linear Dirac cone composed of Tl p z orbitals is found around the K point (at about 1.4 eV in Fig. 2a,d).Due to the C 3v symmetry of the honeycomb structure, another set of twofold degenerate bands appear at the Γ point around 1.0 eV in the planar thallene, composed of Tl p x and p y orbitals (Fig. 2a,d).The dispersion of these p x and p y orbitals belongs to quadratic non-Dirac bands since the Γ point is a time-reversal invariant point and the linear k term in the energy eigenvalue at such point is forbidden 25 .This band feature with two twofold degenerate points respectively at Γ and K also presents in Pb monolayers 17 , which happen at the Fermi level (E F ).They are, however, located above the E F in the Tl monolayer due to the less valence electron number of a Tl atom, compared to a Pb atom.In Tl 2 H, since partial Tl p z orbitals are saturated by hydrogen atoms, the linear Dirac degenerate bands around the K point are opened and become two relatively flat bands (the blue curves in Fig. 2e).The tendency is the same as those in half-hydrogenated Bi honeycomb monolayers in nonmagnetic states 26 .The flat bands of the Tl p z orbitals in Tl 2 H are, however, not located around the E F .Thus, spontaneous spin polarization does not happen in Tl 2 H, different from many other half-hydrogenated Ge 27 , Sn 27 , and Bi 26 monolayers etc.The quadratic non-Dirac bands (composed of Tl p x and p y orbitals) at the Γ point still exist because of the C 3v symmetry unbroken by the hydrogenation.For the electronic structure of Tl 2 H 2 (Fig. 2c,f), it can be intuitively understood that the dangling bonds of Tl p z are all saturated by H atoms now, which makes the upper flat p z orbitals move down in energy.
The orbital-resolved band structures with SOC interactions for the Tl 2 H and Tl 2 H 2 monolayers are shown in Fig. 3.As displayed in Fig. 3a,b, the bands around the E F in Tl 2 H are primarily made up of the Tl p orbitals (and also some s orbitals below the E F ).When the SOC interaction is taken into account, the twofold degeneracy at the Γ point is lifted and a direct band gap ΔE d = 1.19 eV is opened at the Γ point (Fig. 3(b)).This large band gap can be ascribed to the strong on-site SOC interaction contributed by Tl p x and p y orbitals 28 .Due to the band dispersion, a relatively small global band gap ΔE g = 854 meV is obtained for Tl 2 H, which is pretty large and close to the record value of QSH effects reported in the F-decorated Bi monolayer 29 .For Tl 2 H 2 , the quadratic non-Dirac degenerate point (red and green colors) is located at 1.3 eV at the Γ point (Fig. 3c), below which there is an energy band primarily composed of the Tl p z orbitals (blue color).Under the influence of SOC, not only the quadratic non-Dirac point is broken with a band gap opened, but also the band inversion between Tl p x/y and Tl p z orbitals is induced in Tl 2 H 2 .Due to the large on-site SOC, the SOC-induced direct band gap of the Tl p x/y is still very large (ΔE d = 1.32 eV, as marked in Fig. 3d), within which there is the band of the Tl p z orbitals hybridizing with Tl p x/y orbitals.Thus, the band inversion between Tl p x/y and Tl p z orbitals causes the global band gap (334 meV) for the monolayer Tl 2 H 2 much less than that (854 meV) of Tl 2 H.These SOC-induced global band gaps in the two monolayers are expected to be topologically nontrivial 30 , to be discussed below.Compared to the pristine thallene with a topologically trivial band gap (98 meV) 18 , the hydrogenation causes the thallene becoming a metal.Especially, for the full passivated thallene, more bands cross the E F and the Tl 2 H 2 monolayer turns to be a typical metal.This trend is opposite to the full-passivation effect in group IVA monolayer.For example, the full-hydrogenation makes graphene change from a gapless semiconductor to a semiconductor with a large band gap of 3.49 eV 31 due to the Dirac cone composed of C p z orbitals located right at E F in the pristine graphene.
Since the distributions of the featured bands for the Tl 2 H and Tl 2 H 2 monolayers are different, the mechanisms of the band evolution for the two materials are now analyzed.For Tl 2 H and Tl 2 H 2 monolayers, the evolutions of the atomic orbitals around the Γ point are both mainly triggered by two aspects: one is chemical bonding and the other is SOC.The difference is that the two Tl p z bands in the monolayer Tl 2 H are far away from the concerned energy degenerate point at Γ (Fig. 2e).Since we mainly focus on the bands near the two-fold degenerate point, only the Tl p x/y orbitals are discussed in Tl 2 H.As shown in Fig. 4a, in the absence of SOC, the chemical bonding between Tl-Tl atoms forms bonding and anti-bonding states for the p x/y orbitals.These states are labeled as |p ± x/y � , where the superscript (+ / − ) denotes the bonding/anti-bonding states and the concerned degenerate energy level www.nature.com/scientificreports/ is mainly contributed by the p + x/y orbitals.When SOC is turned on, |p + x/y � splits into two energy levels.Analogous to the cases of stanene 15 and germanene films 16 , the two SOC-induced splitting energy levels can be expressed as p + x+iy,↑ , p + x−iy,↓ � and p + x−iy,↑ , p + x+iy,↓ � (Fig. 4a), with p + x+iy,↑ , p + x−iy,↓ � moving up and the other set of states moving down.Thus, a band gap is opened at the Γ point.Note that the role of SOC here is only to lift the degeneracy or open a band gap.It does not induce band inversion, as displayed in Fig. 4b.The status in Tl 2 H 2 is, however, dissimilar to that of Tl 2 H.The effect of the Tl p z orbital cannot be ignored in Tl 2 H 2 due to some of its locations close to the two-fold degenerate point at the Г point (Fig. 2f).As illustrated in Fig. 4c, the chemical bonding between Tl-Tl atoms makes all atomic orbitals split into bonding and anti-bonding states, labeled as |p ± x/y � and |p ± z � , respectively.Owing to the increase of the H concentration in Tl 2 H 2 , more |p − z � states are passivated.Hence, the |p − z � state moves down in energy and is lower than the two-fold degenerate |p + x/y � states (Fig. 4c).When the SOC is taken into account, |p + x/y � splitting is similar to that of Tl 2 H, where p + x+iy,↑ , p + x−iy,↓ � moves down and the anti-bonding state of the p z orbitals (labeled as p − z,↑ , p − z,↓ � ) moves up.The large SOC splitting of the |p + x/y � states leads to the band inversion between p x/y and p z states (Fig. 4d), resulting in a decrease of the band gap in Tl 2 H 2 (Fig. 4d), compared to that of Tl 2 H.To confirm the results, the electronic structures of the two monolayers are also calculated with the hybrid density functional (HSE06).Similar band structures with the indirect band gaps of 924 and 233 meV are obtained for Tl 2 H and Tl 2 H 2 , respectively (Fig. S3).
Applying strain is generally an effective strategy for making controls to the structures and electronic states of the monolayer materials.Here, we apply biaxial strain in the range of − 5 to 5% to the two monolayers.To check the structural stability of the monolayers under strain, the binding energies of the two structures under strain are calculated.As illustrated in Fig. S4, the strain does not decrease much the binding energies, indicating the structural stability of the monolayers under strain.The trend is consistent with the results reported in Ref. 32 that thallene can undergo very large tensile strain (~ 27%) in experiments.The band gaps opened at the twofold degenerate point of the non-Dirac bands as a function of the biaxial strain for the monolayers are shown in Fig. 5a.The strain is defined as (a′-a)× 100% /a, where a′ stands for the lattice constant with strain applied and a stands for the equilibrium lattice constant.Due to the different orbital components near the concerned band gaps for the Tl 2 H and Tl 2 H 2 , the responses to the external tuning are various.
The global band gaps for Tl 2 H exhibit roughly linear and saturated trends with respect to the compressive and tensile strain, respectively (Fig. 5a), comprehended as following.As indicated in Fig. S5, with the strain varying from -3 ~ 3%, the p x/y states of Tl 2 H move down in energy while the p z states (at about 1.6 eV) move up.And a band inversion occurs with p x/y and p z orbitals around the Г point at -1% strain (Fig. S5).During this process, the buckled height (h) decreases (Fig. 5b), which can be straightforward ascribed to the increase of the in-plane lattice constants.The h decrease makes the p z orbitals of the Tl atoms without H atoms more isolated.The interaction between this Tl atom and its three neighbor Tl atoms more tends to form planar sp 2 hybridization instead of three dimensional sp 3 hybridization, giving rise to the Tl p z orbitals moving up in energy during the process (especially around Γ point).This tendency causes the band inversion between the p x/y and p z orbitals at − 1% strain, as illustrated in the inset of Fig. 5b.Since the SOC split of p x/y orbitals does not change much with the strain (see the black arrows in Fig. S5), the band gap hardly changes with the increase of the tensile strain (Fig. 5a).
Different from the case of Tl 2 H, the band gap of Tl 2 H 2 increases linearly with the biaxial strain varying from − 5 to 5% (Fig. 5a).The overall tendency is, however, relatively weak.The increase of the band gap for Tl 2 H 2 can be primarily attributed to two factors.One is that the band dispersion tends to weaken with the increase of the atomic distance.The other is that the on-site SOC of p x/y orbitals is enhanced slightly by the gradual decrease of the p z state involved in the concerned bands with the strain varying from − 5 to 5% (Fig. S6).

Strong Rashba effect
Considerable Rashba SOC interactions 32 exist in the monolayer Tl 2 H and Tl 2 H 2 due to the breaking of vertical reflection symmetry.The strength of the Rashba SOC generally can be manipulated markedly by an external electric field.And the Rashba SOC effect has been employed to fabricate all-electric spintronic devices, such as spin field effect transistors and spin valves (without magnetic fields needed) [33][34][35] , promoting the spintronic applications.Very large Rashba spin splitting has been reported for the deep-energy π states (~ − 8 eV below E F ) of graphene deposited on metallic substrates 36 .Here, we merely discuss the Rashba effect of Tl 2 H 2 since it is stronger than that of Tl 2 H (Fig. 3b,d).The orange bands of Tl 2 H 2 in Fig. 6 are focused on.The magnified bands around the Г point (Fig. 6b) exhibit an obvious Rashba splitting phenomenon similar to that of semiconductor quantum wells and heavy metal surfaces [37][38][39][40][41] .To examine the Rashba effect, the spin textures in the k x -k y plane for Tl 2 H 2 are calculated.The spin-projected constant-energy contour plots for the spin textures calculated in the k z = 0 plane are shown in Fig. 7a.The flower-like spin textures appear in Tl 2 H 2 .For both S x and S y spin components, the pair of spin-splitting bands have the same spin orientation.However, for pure 2D Rashba spin splitting, the pair of spin-splitting bands for the both S x and S y spin components generally have opposite spin orientations 42 .For Tl 2 H 2 , the SOC effect not only opens the nontrivial band gap, but also induces the band inversion near E F (Fig. 4c,d), resulting in an anomalous spin texture.To verify this point, the p z state is moved down in energy to the position far from the concerned energy point by applying 20% biaxial tensile strain.In this case, the SOC is not sufficient to reverse the bands and the corresponding spin structures of the Tl 2 H 2 monolayer are displayed in Fig. 7b.Clearly, the pair of spin-splitting bands for both S x and S y spin components have opposite spin orientations.Due to the disappearance of the S z component, the spin moments of the two rings shown in Fig. 7b have opposite chirality.The large ring is anticlockwise, while the small ring is clockwise.This large Rashba effect has recently been observed in experiments in the thallene 20 with a different forming mechanism.
The Rashba coefficient α R is also calculated to describe the strength of the Rashba SOC effect.The Rashba coefficient can be obtained from the formula α R = 2E R /K R 41 , where E R and K R are defined in Fig. 6b.From Fig. 6b, the E R , K R , and α R are estimated to be 68 meV, 0.054 Å −1 , and 2.52 eVÅ, respectively.The obtained α R  is significantly higher than those of many heterostructures or surface states of 2D TIs, such as InGaAs/InAlAs (0.07 eVÅ) 34 , Au/W(110) (0.16 eVÅ) 35 , and CSb 3 (0.83 eVÅ) 43 .In Fig. 8, we show the Rashba energy E R , the momentum offset K R , and Rashba parameter α R under different biaxial strain (− 5 to 5%) and external electric fields (− 0.5 to 0.5 V/Å).With the increase of the biaxial strain and external electric field, the values of E R and K R increase almost linearly and are very sensitive to the biaxial strain.On the contrary, the values of α R shows a decreasing trend with the increase of the biaxial strain and external electric field, indicating that the application of compressive strain and electric fields along −z axis are more conducive to the Rashba effect of Tl 2 H 2 .Under compressive strain, the Tl p z component of the concerned bands becomes more (Fig. S6), which together with the slight increase of Tl-H bond strength (the bond length becomes short) and the Tl p x/y asymmetric distribution about the Tl-Tl plane results in the enhancement of the Rashba effect with the compressive strain in Tl 2 H 2 .As shown in Fig. 1b, the hydrogen atoms are located above the Tl atoms.When a negative electric field (along −z axis) is applied, the electrons of thallene tend to move toward the H atoms.And the charge densities between the Tl-H atoms become more, leading to the increase of the Rashba effect.

Large band-gap quantum spin Hall state
The topological properties of the concerned band gaps for the monolayer Tl 2 H and Tl 2 H 2 are investigated.Figure 9a,b show the band structures of the monolayer Tl 2 H and Tl 2 H 2 with SOC.The acquired bands from the Wannier interpolation method are also displayed, which are in very good agreement with those obtained  from the DFT calculations.Figure 9c,d give the edge states of the semi-infinite Tl 2 H and Tl 2 H 2 samples by using the Green's function method and the semi-infinite MLWF Hamiltonian.Obviously, the topological protected gapless helical edge states connect the bands at the two sides of the band gaps, as expected.To further identify the topological properties of the gapped states of the monolayer Tl 2 H and Tl 2 H 2 , the topological invariant Z 2 is obtained by using the Wannier charge center (WCC) method 44 .The calculated Z 2 = 1 for the monolayer Tl 2 H and Tl 2 H 2 with the E F set to be located inside the band gaps indicates that the gaped states are QSH states with topological nontrivial band gaps induced by SOC (Fig. 9e-f).The E F movement can be achieved experimentally through such as carrier doping with the carrier concentration of 4.1 × 10 14 and 4.2 × 10 14 cm −2 for Tl 2 H and Tl 2 H 2 , respectively, which can be realized in experiments via current advanced gating technologies 45 .Our research results indicate that the QSH states with large band gaps can be achieved in the thallene film by functionalization.No strong tensile strain is needed to acquire the QSH state in Tl 2 H and Tl 2 H 2 , superior to the pristine thallene 18 .The obtained QSH states are actually very robust against the strain.As indicated in Fig. 5a, the global band gaps for Tl 2 H and Tl 2 H 2 are both not closed under the strain varying from − 5 to 5%.Thus, the QSH states are well maintained even under 5% tensile or compressive strain.
To check the accessibility of the topological states for Tl 2 H and Tl 2 H 2 in experiments, we deposit Tl 2 H and Tl 2 H 2 monolayers to a substrate material of SiC (0001).The built geometrical structures of the Tl 2 H/SiC and Tl 2 H 2 /SiC heterostructures are displayed in Fig. 10a,c.Note that the bottom and top surfaces of the SiC substrate are saturated with H atoms to remove the dangling bonds of the surface atoms for the SiC substrates.The lattice mismatch between the Tl 2 H /Tl 2 H 2 sample and the SiC (0001) substrate is very small (about 1.82%/0.96%).The optimized interface distance between the sample and the substrate is 2.32 Å/2.77Å, indicating the van der Waals (vdW) interactions in the interface.The achieved band structures are shown in Fig. 10b,d.The bands near the E F of Tl 2 H and Tl 2 H 2 are obviously not affected much by the SiC substrate, due to the weak vdW interactions from the substrates.Therefore, the QSH states displayed in Fig. 9a,b keep well in the samples when depositing on the SiC substrate.The acquired SiC substrate together with the large binding energies (> 4.5 eV) and the negative formation energies for Tl 2 H and Tl 2 H 2 indicates the experimental accessibility of the unique electronic and topological states in the two materials.

Conclusions
In summary, we have built two hydrogenated thallene Tl 2 H and Tl 2 H 2 monolayers and explored systematically their electric structures and topological properties based on density functional and crystal field theories.Our results indicate that non-Dirac quantum spin Hall states with large band gaps can be achieved in thallene by unilateral hydrogenation.The topological nontrivial band gap can be up to 855 meV, much higher than that of most of the proposed quantum spin Hall insulator.The QSH states in both monolayers are very robust and well maintained even under 5% tensile or compressive strain.Due to the absence of structural inversion symmetry, the Tl 2 H 2 monolayer exhibits very strong Rashba spin splitting characteristics.The non-Dirac quantum spin Hall state and Rashba effect can be tuned efficiently by applying biaxial strain and external electric fields.Our results provide an excellent material platform for realizing room-temperature topological electronic devices.

Models and methods
The geometry optimization and electronic structure calculations of the functionalized thallene are performed based on density functional theory (DFT) with the projector augmented wave method as implemented in the Vienna ab initio simulation package (VASP) 46 .The Perdew-Burke-Ernzerhof generalized gradient approximation (GGA-PBE) is adopted for the exchange-correlation functional 47 .The cutoff energy is set as 550 eV for the plane-wave basis and the vacuum space along the c axis is set to about 15 Å to avoid the interactions between the two adjacent slabs.The first Brillouin zone (BZ) is sampled with k meshes of 15 × 15 × 1 by using Gamma- centered Monkhorst-Pack method 48 .The convergence threshold for the total energy is set to 1 × 10 −6 eV.All the atoms are allowed to relax until the force on each atom is less than 0.01 eV/Å, which is calculated according to the Hellmann-Feynman theorem.The structural optimization is performed without the symmetry constraint.The maximally localized Wannier functions (MLWFs) are constructed by employing the WANNIER90 code 49 , in which the states are calculated with an iterative Green function method 50,51 .In addition, the screened exchange hybrid density functional by Heyd-Scuseria-Ernzerhof (HSE06) 52 is also adopted for confirming the electronic structures.Table 1.The calculated equilibrium lattice constants (a), Tl − Tl bond lengths ( d Tl−Tl ), Tl − H bond lengths ( d Tl−H ), vertical distances between the two Tl atomic planes (h), the binding energies ( E b ), the formation energies ( E f ), and the global band gaps (ΔE g ) for the monolayer Tl 2 H and Tl 2 H 2 .

Figure 1 .
Figure 1.(Color online) The geometry structures of the monolayer (a) Tl 2 H and (b) Tl 2 H 2 from the top and side views.The buckled height between the Tl atoms is h.The inset shows the first BZ with the high symmetry points.The green and white balls represent Tl and H atoms, respectively.In (a) and (b), the corresponding xyz axis are shown.(c, d) The total energies as a function of the lattice constants for the monolayer Tl 2 H and Tl 2 H 2 , respectively.

Figure 2 .
Figure 2. Band structures for (a) Tl, (b) Tl 2 H, and (c) Tl 2 H 2 monolayers without SOC.(d-f) are orbital projections corresponding to (a-c) band structures, respectively.The insets in (a-c) are the side views of the corresponding structures.

Figure 3 .
Figure 3. (a) and (b) give the orbital-projected band structures of the Tl 2 H without and with SOC, respectively.(c) and (d) give the orbital-projected band structures of the Tl 2 H 2 without and with SOC, respectively.The dot size is proportional to the contribution of the corresponding orbitals.In (b) and (d), ΔE d represents a direct band gap at the Γ point and ΔE g represents the global band gap.

Figure 4 .
Figure 4. (Color online) (a) and (c) correspond to the Tl-orbital evolution diagrams at Γ point near the E F in the monolayer Tl 2 H and Tl 2 H 2 , respectively.The signs of ' + ' and '−' indicate the bonding and anti-bonding states, respectively.The green dotted lines represent the location of the concerned energy degenerate points.(b) and (d) show the schematic band dispersion of the energy regions marked by the rectangles in (a) and (c), respectively.In (b) and (d), the blue curves represent the bonding states of the p x/y orbitals and the red curves represent the anti-bonding states of the p z orbital.

Figure 5 .
Figure 5. (Color online) (a) The global band gaps as a function of the strain for the monolayer Tl 2 H and Tl 2 H 2 .The red and green colors give the Tl 2 H and Tl 2 H 2 results, respectively.(b) The buckled heights (h) between Tl atoms in the monolayer Tl 2 H as a function of the strain.The inset shows the band evolution, in which the blue/ red colors express the p x/y / p z bands.

Figure 6 .
Figure 6.(a) Band structure of the Tl 2 H 2 monolayer with SOC.(b) The magnified view of the band structure (in the red rectangle in (a)) of the Tl 2 H 2 monolayer.The orange color indicates the concerned bands.

Figure 7 .
Figure 7. (Color online) Spin textures centered at the Г point calculated in the k z = 0 plane for the Tl 2 H 2 monolayer with (a) the pristine structure and (b) 20% tensile strain, respectively.The red and blue colors show spin-up and spin-down states, respectively.The energy in (a) is set at 1.1 eV.

Figure 8 .
Figure 8.The Rashba energy E R (red color) and the momentum offset K R (green color) under biaxial strain (a) and external electric fields (b) in the monolayer Tl 2 H 2 .The Rashba parameter α R under different biaxial strain (c) and external electric fields (d) is also shown.

Figure 9 .
Figure 9. Band structures for the (a) Tl 2 H and (b) Tl 2 H 2 monolayers with SOC by using Wannier interpolation (red dotted curves) calculations.The DFT results (blue solid curves) are also shown.(c, d) The calculated edge states for the Tl 2 H and Tl 2 H 2 monolayers.Evolutions of the Wannier function centers for Tl 2 H (e) and Tl 2 H 2 (f) along k y , yielding Z 2 = 1.

Figure 10 .
Figure 10.The geometry structures of the (a) Tl 2 H and (c) Tl 2 H 2 monolayers placed on a threefold symmetric SiC (0001) substrate from the top and side views, respectively.The substrate is modeled as a SiC three-layer, saturated with H atoms in the bottom and top surfaces.The unit cell of the system is in a commensurate ( √ 3 × √ 3 ) reconstruction of SiC (0001). (b) and (d) give the band structures of the Tl 2 H and Tl 2 H 2 monolayers on the SiC (0001) substrate without and with SOC, respectively.