Functionalized Thallium Antimony Films as Excellent Candidates for Large-Gap Quantum Spin Hall Insulator

Group III-V films are of great importance for their potential application in spintronics and quantum computing. Search for two-dimensional III-V films with a nontrivial large-gap are quite crucial for the realization of dissipationless transport edge channels using quantum spin Hall (QSH) effects. Here we use first-principles calculations to predict a class of large-gap QSH insulators in functionalized TlSb monolayers (TlSbX2; (X = H, F, Cl, Br, I)), with sizable bulk gaps as large as 0.22 ~ 0.40 eV. The QSH state is identified by Z2 topological invariant together with helical edge states induced by spin-orbit coupling (SOC). Noticeably, the inverted band gap in the nontrivial states can be effectively tuned by the electric field and strain. Additionally, these films on BN substrate also maintain a nontrivial QSH state, which harbors a Dirac cone lying within the band gap. These findings may shed new light in future design and fabrication of QSH insulators based on two-dimensional honeycomb lattices in spintronics.

characteristic of III-V films is that its unsaturated p z orbital is chemically active, due to the weak π -π interaction as caused by the bond length between III-V atoms (~3 Å). This feature, together with the out-of-plane orientation of p z orbital, facilitates strong orbital interaction with external environments, and thus its electronic properties are easily affected by adsorbates and substrates, unfavorable for practical applications in spintroncis.
As a representative, here we provide a systematical study on structural and topological properties of 2D TlSb monolayers functionalized with hydrogen and halogens, namely TlSbX 2 (X = H, F, Cl, Br, I). We find that the surface functionalization on TlSb, i.e., saturating the p z orbital composed of TISb with hydrogen or halogens, can stabilize the 2D TlSb, according to the calculated phonon spectrum of TlSbX 2 films. All the systems are found to be QSH insulators, with the bulk gap in the range of 0.22 ~ 0.40 eV, tunable by external strain and electric field. A single pair of topologically protected helical edge states is established for these systems with the Dirac point locating in the bulk gap, and the odd numbers of crossings between edge states and Fermi level prove the nontrivial nature of these TlSbX 2 films. These findings may provide a new platform to design large-gap QSH insulator based on group III-V films, which is important for device application in spintronics.

Results and Discussion
The geometric structure of TlSbX 2 (X = H, F, Cl, Br, I) are displayed in Fig. 1(a), in which the Tl or Sb atoms are saturated with X atoms on both sides of the plane in an alternating manner along the hexagonal axis, and thus breaks IS of TlSbX 2 . Table 1 lists the calculated equilibrium lattice constants, buckling heights, as well as Tl-Sb, Tl-X, and Sb-X bond lengths after structural optimization. In comparison to pristine TlSb, the TI-Sb bonds in TlSbX 2 slightly expand, while the buckling changes differently due to the weakly hybridization between π and σ orbitals, stabilizing these structures. The stability of functionalized TlSbX 2 is studied through the formation energy defined as where E TlSbX2 , E TISb and E X are the total energy of double-side functionalized TlSbX 2 , pristine TlSb, and molecule X 2 , respectively. N X is the number of X atoms. The calculated formation energies for TlSbH 2 , TlSbF 2 , TlSbCl 2 , TlSbBr 2 , and TlSbI 2 , are − 1.862, − 2.997, − 1.613, − 1.567, and − 1.420 eV, respectively, suggesting that hydrogen  or halogens are chemically bonded to TlSb, indicating a higher thermodynamic stability relative to their elemental reservoirs. The dynamic stability of TlSbH 2 , as an example, is further confirmed by the phonon dispersion curves in Fig. 1(b), in which all branches have positive frequencies and no imaginary phonon modes, confirming the stability of TlSbH 2 . Figure 2(a,d) display the calculated band structure for TlSbH 2 and TlSbF 2 as representative examples, in which the red and blue lines correspond to band structures without and with SOC. In the absence of SOC, they are both gapless semimetal with the valence band maximum (VBM) and conduction band minimum (CBM) degenerate at the Fermi level. When takes SOC into account, the band structures of TlSbH 2 and TlSbF 2 produce a semimetal-to-semiconductor transition, with sizeable bulk-gaps of 0.22 eV and 0.40 eV, respectively. As observed in previously reported 2D TIs like ZeTe 5 30 , HfTe 5 31 , and GaSe 32 , graphene-like materials [33][34][35] , the SOC-induced band-gap opening at the Fermi level is a strong indication of the existence of topologically nontrivial phases.
An important character of the QSH insulator is helical edge states which is key to spintronic applications due to the ability to conduct dissipationless currents. Thus, we calculate the topological edge states by the Wannier90 package 36 . We construct the maximally localized Wannier functions (MLWFs) and fit a tight-binding Hamiltonian with these functions. The calculated edge Green's function 37 of semi-infinite TlSbX 2 (X = H, F) is shown in Fig. 3(a,d). One can see that all the edge bands connect completely the conduction and valence bands and span 2D bulk band gap, yielding a 1D gapless edge states. Besides, the counter-propagating edge states exhibit opposite spin-polarization, in accordance with the spin-momentum locking of 1D helical electrons. In addition, the Dirac point located at the band gap are calculated to have a high velocity of ~2.0 × 10 5 m/s, comparable to that of 5.5 × 10 5 m/s in HgTe/CdTe quantum well 7,8 . All these consistently indicate that TlSbX 2 (X = H, F) are ideal 2D TIs. The topological states can be further confirmed by calculating topological invariant Z 2 after the band inversion. Due to IS breaking in TlSbX 2 , the method proposed by Fu and Kane 38 cannot be used to calculate the Z 2 invariant. Thus, a method independent of the presence of IS is needed. As reported by Yu et al. 39 , we employ a recently proposed equivalent method for the Z 2 topological invariant based on the U(2N) non-Abelian Berry connection. This approach allows the identification of the topological nature of a general band insulator without any of the gauge-fixing problems that plague the concrete, previous implementation of invariants. Here, we introduce the evolution of Wannier Center of Charges (WCCs) 39 to calculate the Z 2 invariant, which can be expressed as: which indicates the change of time-reversal polarization ( θ P ) between the 0 and T/2. The evolution of the WCC along ky corresponds to the phase factor, θ, of the eigenvalues of the position operator,X, projected into the occupied subspace. Then the WFs related with lattice vector R can be written as: Here, a WCC x n can be defined as the mean value of n X n 0 0 , where theXis the position operator and n 0 is the state corresponding to a WF in the cell with R = 0. Then we can obtain 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. In Fig. 3(c,f), we display the evolution lines of Wannier function centers (WFC) for TlSbH 2 and TlSbF 2 , respectively. It can be seen that the WFC evolution curves cross any arbitrary reference lines odd times, thus yielding Z 2 = 1. Now, we turn to the physics of QSH effect in TlSbX 2 . Since the decorated atoms hybridizes strongly with the dangling bonds of p z orbital in TISb, it effectively removes the p z bands away from the Fermi level, leaving only the s and p x,y orbitals, as displayed in Fig. 2(b,c,e,f). However, through projecting the bands onto different atomic orbitals, we find that there are two scenarios for the effect of SOC on the bands around the Fermi level, in which the s and p x,y band inversion are different from each other. For TlSbH 2 , at the Γ point, the two p x,y orbitals from TI and Sb atoms are energy degenerate, while the bands away from the Γ point are well separated due to orbital splitting. The Fermi level is located between one s and two p x,y orbitals, rendering the s above p x,y orbitals in energy, thus forming a normal band order, similar to the cases in conventional III-V semiconductors. While for TlSbF 2 , the band structures are changed drastically, as shown in Fig. 2(d-f). In sharp contrast to TlSbH 2 , the band order at the Γ point is inverted, i.e., the s is shifted below two p x,y orbitals. These two different band orders may be attributed to the chemical bonding and orbital splitting between TI and Sb atoms. To further understand the physics of band inversion, we display in Fig. 4 the schematic of orbital inversion at the Γ point around the Fermi level in TlSbH 2 and TlSbF 2 films. On can see that, the chemical bonding and crystal field splitting between TI and Sb atoms make the s and p x,y orbital split into the bonding and anti-bonding states, i.e., s ± and p ± x,y , which the superscripts + and − represent the parities of corresponding states, respectively. As displayed in Fig. 4(a), the s + orbital for hydrogenated one is significantly higher somewhere above p − x,y orbital of Tl and Sb atoms under the effect of crystal field. The inclusion of SOC makes the degeneracy of p − x,y orbital split into p − x + iy,↑ & p − x−iy,↓ and p − x−iy,↑ & p − x + iy,↓ , leading s + locate in between them. On the other hand, for TlSbF 2 in Fig. 4(b), the larger lattice constant results in a weaker s-p hybridization, and accordingly a smaller energy separation between the bonding and anti-bonding states. Thus, the s − orbital is downshifted while the p + x,y is upshifted, i.e., the s − will be occupied, while the degenerate P + x,y is half occupied, resulting in semi-metallic character (Fig. 2(d)).Though the inclusion of SOC make also the degeneracy of p + x,y orbital split into p + x + iy,↑ & p + x−iy,↓ and p + x−iy,↑ & p + x + iy,↓ , but its s-p band order are not changed. As a result, the mechanism of QSH effect can be roughly classified into two categories: i.e., type-I: SOC-induced p-s-p TI (TlSbH 2 ), and type-II: Chemical bonding induced p-p-s TI (TlSbF 2 ). Obviously, it is the s orbital insertion into p + x + iy,↑ & p + x−iy,↓ and p + x−iy,↑ & p + x + iy,↓ that the topological bulk-gap (0.22 eV) of TlSbH 2 is smaller twice than that (0.40 eV) of TlSbF 2 film.
Here, we wish to point out that fluorination in TlSb is not the only way to achieve large-gap QSH state, the same results can be obtained by decorating the surface with otherwise halogens, such as Cl, Br, and I. We thus performed calculations for TlSbX 2 (X = Cl, Br, I) films to check their topological properties, as illustrated in Fig. S1. Table 1 summarizes their lattice constants, Sb-X and Tl-X bond lengths, and nontrivial QSH bulk-gaps at their equilibrium states. The results demonstrate that the electronic structures of all these TlSbX 2 films are similar to TlSbF 2 , and exhibit nontrivial topological states (Fig. S2). Interestingly, as can be seen in Fig. 4(c) and Fig. S1, the global gaps in QSH state are obtained to be 0.34, 0.32, and 0.29 eV for TlSbCl 2 , TlSbBr 2 and TlSbI 2 , respectively, which are sufficient for practical applications at room temperature. However, when comparing the band gaps with each other, we can find some fascinating phenomena that the global band gaps of these systems monotonically decrease in the contrary order of TlSbF 2 > TlSbCl 2 > TlSbBr 2 > TlSbI 2 . It is known that, from F to I, the SOC becomes stronger in the order of F < Cl < Br < I, thus the SOC-induced bulk-gap should be increased correspondingly. This interesting contradiction can be attributed to the variation of band components of Tl and Sb atoms near the Fermi level, as the band splitting driven by SOC can directly determine QSH gap. As shown in Fig. 4(c), the ratio (R 1 ) from the Sb-p x,y to X-p x,y orbitals at Γ point can be established by R 1 = Sb-p x,y /X-p x,y , which decreases in the order of TlSbF 2 > TlSbCl 2 > TlSbBr 2 > TlSbI 2 . Similar results are obtained for the ratio R 2 = TI-p x,y /X-p x,y in Fig. 4(d). Considering that the Tl and Sb atoms exhibits stronger SOC strength than halogens, it is expected that the larger the ratio is, the larger the contribution to the states near the Fermi level, and thus the larger the SOC strength.
Strain engineering is a powerful approach to modulate electronic properties and topological natures in 2D materials, and thus it is interesting to study these effects in TlSbX 2 films. We employ an external strain on these monolayers maintaining the crystal symmetry by changing its lattices as ε = (a-a 0 )/a 0 , where a (a 0 ) is the strained (equilibrium) lattice constants. As shown in Fig. 5(a,b), the magnitude of nontrivial bulk-gaps of TlSbH 2 and TlSbF 2 can be modified significantly by strain, implying the interatomic coupling can modulate the topological natures of these systems. For TlSbH 2 , with increasing the strain, the CBM is continuously to shift downward to the Fermi level, while the VBM increases reversibly, leading the band gap to decrease significantly ( Fig. 5(a)). When the critical value reaches up to − 3.8%, a semi-metallic state with zero density of states at the Fermi level occurs. If increases the strain beyond − 3.8%, a trivial topological phase appears. While for TlSbX 2 (see also Fig. 5(b) and Fig. S3), both the direct and indirect band gaps decreases steadily with respect to tensile strain. Especially, these QSH states are robust with the strain in the range of − 8 ~ 16%. Such robust topology against lattice deformation makes TlSbX 2 easier for experimental realization and characterization on different substrate.
On the other hand, from the perspective of potential device applications, the ability to control topological electronic properties via the vertical electric field (E-field) is highly desirable. Thus, we study the change of band gaps of TlSbH 2 and TlSbF 2 under different vertical E-field, as shown in Fig. 5(c,d). One can see that the band gaps increase monotonically with increasing E-field strength for both cases. For TlSbF 2 (Fig. 5(d)), under − 0.8 V Å −1 ≤ E-field ≤ 0.8 V Å −1 , the trend of band gaps increase monotonically from 0.34 eV to 0.41 eV, with E Γ larger than E g significantly. While for TlSbH 2 , when E-field ≤ − 5.5 V Å −1 , it is a normal metal. But for large E-field (> − 0.4 V Å −1 ), it becomes a QSH insulator, along with E Γ being almost equal to E g . Noticeably, if E-field is in the range of ± 8%, the nontrivial bulk-gaps of other TlSbX 2 (X = Cl, Br, I) are still very large (~0.2− 0.5 eV) (Fig. S3), allowing for viable applications at room temperature. The predicted QSH insulators tuned by vertical E-field may provide a platform for realizing topological field-effect transistor (TFET).
The substrate materials are inevitable in device application, thus a free-standing film must eventually be deposited or grown on a substrate. As a 2D large-gap insulator with a high dielectric constant, the BN sheet has been successfully used as the substrate to grow graphene or assemble 2D stacked nanodevices 40,41 . Thus, we use it as a substrate to support TlSbX 2 films. Figure 6(a,b) show the geometrical structures of TlSbH 2 and TlSbF 2 on (2 × 2) BN sheet, where the lattice mismatch is only about 1.68% and 2.80%, respectively. After full relaxation with van der Waals (vdW) forces 42 , they almost retain the original structure with a distance between the adjacent layers of 3.35 Å. The calculated binding energy is about − 69, − 87, − 98, − 108, and − 114 meV for TlSbH 2 , TlSbF 2 , TlSbCl 2 , TlSbBr 2 , and TlSbI 2 per unit cell, respectively, showing that they are typical van der Waals heterostructures. The calculated band structure with SOC is shown in Fig. 6(c,d). In these weakly coupled systems, TlSbH 2 on the BN sheet remains semiconducting, there is essentially no charge transfer between the adjacent layers, and the states around the Fermi level are dominantly contributed by TlSbH 2 . If we compare the bands of TlSbH 2 with and without BN substrate, little difference is observed. Similar results are also found for all halogenated TlSbX 2 films on BN substrate (see also Fig. S4), except that TlSbF 2 on the BN sheet exhibits a metallic state. Evidently, all TlSbX 2 films on BN substrate are robust QSH insulators.

Conclusions
To conclude, on the basis of first-principles calculations, we predict a class of new QSH insulator of TlSbX 2 (X = H, F, Cl, Br, I) films, with a sizable bulk gap (0.22 ~ 0.40 eV), allowing for viable applications in spintronic devices. Two mechanisms, type-I: SOC-induced p-s-p type TI (TlSbH 2 ), and type-II: the chemical bonding induced p-p-s type TI (Halogenated ones) are obtained, significantly different from one in TISb monolayer. The topological characteristic of TlSbX 2 films are confirmed by the Z 2 topological order and topologically protected edge states. Furthermore, the band gap and topological phase transition could be tuned by the external strain and vertical E-field. When TlSbX 2 deposited on BN substrate, both the band gaps and low-energy electronic structures are only slightly affected by the interlayer coupling from the substrate. These predicted QSH insulators and their vdW heterostructures may provide a platform for realizing low-dissipation quantum electronics and spintronics devices.
Computational method and details. To study the structural and electronic properties of TlSbX 2 (X = H, F, Cl, Br, I) films, our calculations were performed using the plane-wave basis Vienna ab initio simulation package known as VASP code 43,44 . We used the generalized gradient approximation (GGA) for the exchange and correlation potential, as proposed by Perdew-Burk-Ernzerhof (PBE) 45 , the projector augmented wave potential (PAW) 46 to treat the ion-electron interactions. The energy cutoff of the plane waves was set to 500 eV with the energy precision of 10 −6 eV. The Brillouin zone was sampled by using a 21 × 21 × 1 Gamma-centered Monkhorst-Pack grid. The vacuum space was set to 20 Å to minimize artificial interactions between neighboring slabs. SOC was included by a second vibrational procedure on a fully self-consistent basis. The phonon spectra were calculated using a supercell approach within the PHONON code 47 .