Strain induced band inversion and topological phase transition in methyl-decorated stanene film

The researches for new quantum spin Hall (QSH) insulators with large bulk energy gap are of much significance for their practical applications at room temperature in electronic devices with low-energy consumption. By means of first-principles calculations, we proposed that methyl-decorated stanene (SnCH3) film can be tuned into QSH insulator under critical tensile strain of 6%. The nonzero topological invariant and helical edge states further confirm the nontrivial nature in stretched SnCH3 film. The topological phase transition originates from the s-pxy type band inversion at the Γ point with the strain increased. The spin-orbital coupling (SOC) induces a large band gap of ~0.24 eV, indicating that SnCH3 film under strain is a quite promising material to achieve QSH effect. The proper substrate, h-BN, finally is presented to support the SnCH3 film with nontrivial topology preserved.


Results and Discussion
(a) presents the geometric structure of SnCH 3 film from top and side views. The CH 3 groups are adsorbed on Sn atoms on both sides of stanene in an alternating manner. There are ten atoms per unit cell, including two Sn atoms, two C atoms and six H atoms. Via geometric relaxation, the lattice parameters can be obtained. The lattice constant (a) and Sn-Sn bond length (d) are 4.74 Å and 2.87 Å, respectively, which are slightly expanded than the case of pure stanene. The buckling height (h), Sn-C bond and C-H bond length are 0.86 Å, 2.20 Å and 1.10 Å, respectively. The hexagonal lattice and structural inversion symmetry are still maintained.
The kinetic stability of SnCH 3 film is confirmed by calculating the phonon spectrum. There is only minor imaginary frequency appearing near the Γ point as observed in Fig. 1(b). In fact, such small imaginary frequency is a common issue in first-principles calculation for 2D materials, such as germanene, stanene 8,27,28 , monolayer of group-V binary compounds 29 , fluorinated monolayer As and AsSb 30 , where the ZA branch (out of plane acoustical modes) becomes soft and get imaginary frequencies near the Γ point. It is believed such instability can be removed by defects, such as ripples, or finite size sheets. On the other hand, the ZA imaginary frequencies around Γ point also depends on the mesh size used in the calculations. It may be an artifact of the mesh size since the interatomic forces related with ZA modes decay rapidly. One way to get rid of them is to use a finer mesh grid in the DFT calculation. Therefore, SnCH 3 film is also dynamically stable. Generally, the buckled configuration can endure a larger mechanical distortion than planar one. Hence the modulation of the structure and electronic properties can be realized by the strategy of external in-plane strain. For instance, a reasonable strain can induce topological phase transition in functionalized germanene 21,31 , group-IV and V monolayers [32][33][34] . To well understand the structural variation of SnCH 3 film under a large strain without dissociation, the evolutions of the buckling height (h), Sn-Sn bond length (d) and buckling angle (θ) as a function of tensile strain are analyzed as plotted in Fig. 1(c). Under the tensile strain, the Sn-Sn bond length is changed slightly with respect to the variations of the buckling height and angle. For instance, under tensile strain of 6% the Sn-Sn bond length is only stretched by 4.5%, whereas the buckling height and angle are reduced by 11.7% and 15.9%, respectively. Consequently, the total energy is slightly increased as shown in Fig. 1(d). Within this tensile strain range the Sn-Sn bonds of SnCH 3 film are covalently preserved. Figure 2(a) displays the band structure of SnCH 3 film at the equilibrium state without considering SOC. Compared with the case of pure stanene where two energy bands cross linearly at the K point 12 , a large band gap is substantially opened up at the K point by CH 3 group decoration due to the saturation of the π orbital. Consequently, both the conduction band minimum (CBM) and valence band maximum (VBM) are shifted to the Γ point. Such similar feature is also observed for the halogenation of stanene. Significant differences between the two systems exist at the Γ point when excluding SOC. In SnCH 3 film, a direct energy gap of about 0.356 eV appears with the valence and conduction bands separated at the Γ point. By projecting the bands onto different atomic orbitals, one can clearly see that the CBM is mainly occupied by s orbitals, whereas the twofold degenerate VBM is mostly dominated by p xy orbitals. As we know, the SOC plays a key role in achieving QSH insulators. When the SOC is included, the band structure is calculated as shown in Fig. 2(b). We can find that the degenerate VBM is lifted out and split into two single states with the second valence band moved down. Moreover, the conduction bands are shifted downward close to Fermi level, leading to the band gap decreasing to 0.153 eV. From the partial orbital projection, the s-p xy -p xy orbital order from top to bottom near the Fermi level at the Γ point is not varied by the SOC. This feature can also be found in hydrogenated stanene and GeCH 3 films 35 , but different from halogenated stanene with p xy -p xy -s orbital order. It is quite clear that SnCH 3 film belongs to trivial phase in topology, which is in good agreement with previous work 36 .
As the interval between s and p xy energy levels at the Γ point sensitively depends on the Sn-Sn bonding strength, the band inversion and the associated QSH states can be effectively tuned by applying external strain. Here, we impose biaxial tensile strain on the SnCH 3 film by turning the in-plane lattice parameter. The magnitude of strain is described by ε = (aa 0 )/a 0 , where a 0 and a denote the lattice constants of the unstrained and strained systems, respectively. With the increase of the tensile strain, the s-type CBM shifts down in energy and the band gap decreases. When the strain is about 6% as shown in Fig. 2(c), the SnCH 3 film becomes gapless when excluding SOC. More interestingly, the so-called s-p-type band inversion takes place at the Γ point. The two p xy bands touch each other at the Fermi level, whereas the s band moves down to the valence band region, resulting in p xyp xys orbital order. Such band inversion is also driven in many other QSH insulators 12,21,31,37-39 . When we turn on the SOC, the band structure of the stretched SnCH 3 film is plotted in Fig. 2(d). Unsurprisingly, the SOC opens a band gap in the gapless film. Furthermore, the VBM is transformed from the "Λ-shape" to "M-shape". The direct band gap and indirect band gap are 0.359 eV and 0.240 eV, respectively. The above results show that the tensile strain drives the trivial SnCH 3 film to a nontrivial QSH insulator. Such a large SOC gap is quite promising for achieving QSH states at room temperature. When the tensile strain is continually increased beyond 6%, the indirect band gap will be gradually enlarged. For instance, at the value of 12%, the energy gap reaches up to 0.262 eV. Within such strain range, the nontrivial topology feature is not damaged.
There are two strategies that have been widely used to confirm the topological nontrivial insulators. One is to calculate the topological invariant proposed by Fu and Kane 40 . For the 2D TI phase, the topological invariant is calculated from the parities of the Bloch wave functions for occupied bands at time-reversal invariant momenta (TRIM) points, one Γ and three M points, as where δ is the product of parity eigenvalues at the TRIM points, ξ = ±1 denotes parity eigenvalues and N is the number of the occupied bands. According to the Z 2 classification, v = 0 characterizes a trivial band topology, while v = 1 characterizes nontrivial phase. We calculate the parity eigenvalues of the Bloch wave function for the 11 occupied spin-degenerate bands at all TRIM points in stretched SnCH 3 film, due to the preserve of structural inversion symmetry. As expected, in the equilibrium state without strain, the products of the parity eigenvalues at symmetry points: Γ and M are both +1, yielding a trivial topological invariant v = 0. With the strain increasing larger than 6.0%, the s-p band order exchanges at the Γ point. The parity eigenvalue of the VBM changes sign from +to -, while that at the M point remain +. Thus, the products of the parity eigenvalues at these two points are now distinct and the system becomes QSH insulator with v = 1.
Another method is to demonstrate the existence of helical gapless edge states in QSH insulators. In our work, we construct the zigzag-shaped-edges nanoribbon of SnCH 3 under strain of 6% as shown in Fig. 3(a). The width of the nanoribbon is selected to be large enough to avoid interactions between the edge states. The band structure is presented in Fig. 3(b). One can clearly see that the bands within bulk gap that connect the conduction and valence bands, cross linearly at the Γ point, forming helical edge states. Such edge states are protected from elastic backscattering by time-reversal symmetry, which are of significance to be used in electronics and spintronics. All the above results consistently indicate that stretched SnCH 3 film is an ideal candidate to realize QSH effect.
To physically understand the origin of the topological nature, here we start from atomic orbitals and consider the effect of chemical bonding and SOC on the energy levels at the Γ point for SnCH 3 film as presented schematically in Fig. 4(a and b). The energy levels around the Fermi level are mainly composed of Sn-5s and Sn-5p xy orbitals, where the p z orbitals are saturated by CH 3 group. According to the crystal field splitting theory in stage I, chemical bonding between Sn and Sn atoms makes the s and p xy orbitals split into the bonding and anti-bonding states, labeled as s ± and ± p xy , swhere the superscript (+,-) denotes the parities of corresponding Bloch states, respectively. In the equilibrium state, the bands close to the Fermi level are contributed by the sand + p xy , with sbeing above the + p xy in energy as shown in Fig. 4(a). Before turning on SOC, the + p xy orbitals are degenerate, and the system is a semiconductor. When the SOC is taken into account in stage II, the degeneracy of the + p xy levels is lifted, and the band gap is decreased. When considering the tensile strain as shown in Fig. 4(b), the interaction between the Sn atoms weakens because of the enlarged lattice constant, which makes the splitting between the bonding and anti-bonding states decreased, with slevel shifted down and + p xy shifted up. Therefore, the electronic structure can be continuously tuned with strain increased, and the order of sand + p xy is inverted at critical point of 6.0%. The slevel becomes fully occupied, whereas the quadruply degenerate + p xy is half occupied. Consequently the Fermi level stays at + p xy level, exhibiting a zero-gap semiconductor. As the SOC effect is turn on, the degeneracy of the + p xy orbital is lifted, opening a larger energy gap. To illustrate the band inversion process explicitly, we illustrate the s-p band inversion diagram in Fig. 4(c). With the tensile strain increased, the antibonding state sshifts downward with respect to the bonding state + p xy . A crossing between the sand + p xy level occurs at ~6%, which leads to a parity exchange between occupied and unoccupied bands. therefore, a topological phase transition from a trivial insulator to a nontrivial phase is induced. The mechanism is similar as for HgTe quantum well 4 , but different from the nontrivial topology in the previous works [31][32][33][34][35][36][37][38][39][40][41][42][43][44] , which originates from the massive Dirac cone, and there is no band inversion.
Since standard PBE calculation usually underestimates the band gap, we adopt the more reliable hybrid functional HSE06 (see Supplementary Figure S1). The results in Fig. S1(a) and (b) show that the band gap of SnCH 3 film without SOC is 0.837 eV with no strain, while the band gap with SOC decreases to 0.609 eV with the bands split. The band inversion associated QSH states occurs when the stretched strain is up to 9%, where the nontrivial indirect band gap is 0.314 eV as shown in Fig. S1(d), larger than that by PBE calculation. In addition, we find that as the stanene thickness increases (2-4 BLs), the topological phase transition from trivial to nontrivial appears compared to methyl-decorated 1 BL films (see Supplementary Figures S2 and S3). Figure S2 shows the atomic structure of methyl-decorated 4 BL stanene film seen from top and side. Equilibrium lattice constants for the methyl-decorated 2-4 BL stanene films are all 4.71 Å, quite close to the bulk value of 4.716 Å. The band structures in Fig. S3 show that 2 and 3 BL films are both insulator, while 4 BL film is semimetal with negative indirect energy gap. To examine their topology, the Z 2 invariants are calculated for 2-4 BL films. The results of parity analysis at the four time-reversal invariant symmetry points show that the Z 2 invariants in these films are all 1. Therefore, 2 and 3 BL films are nontrivial insulator, while 4 BL film is a nontrivial semimetal. In order to gain insight into the nature of band inversion in the films, we investigated orbital-projected band structures with SOC. Our analysis shows that the band inversion involved in the methyl-decorated 2-4 BL stanene films is of s-p type.
To be well applied in electronic devices, the proper substrate materials are indispensable to support the nontrivial topology. Previous works indicate that the topologically insulating properties of silicene and germanene are easily destroyed by the substrates [45][46][47] , due to the lattice mismatch and interaction with substrates. Because the band inversion takes place at the Γ point rather than K point, the nontrivial nature in SnCH 3 would be quite robust when they are on the substrate. Furthermore, the full saturation of p z orbitals of Sn atoms ensures a weak interaction between SnCH 3 with the substrate. Given these factors, we take 1 BL stretched SnCH 3 film with strain of 6% for example where the film are placed on top of a 2 × 2 hexagonal BN (h-BN) substrate forming SnCH 3 /h-BN heterostructure, as shown in Fig. 5(a). In this case, lattice mismatch between SnCH 3 (5.0244 Å for ε = 6%) and h-BN substrate (5.02 Å for 2 × 2 supercell) is extremely small. To correctly describe the van der Waals interaction, we use a DFT-D2 method of Grimme 48 , which has been demonstrated to reliably describe 2D heterostructures. The optimized interlayer spacing between adjacent layers is about 2.824 Å. The binding energy is obtained to be 0.204 eV per unit cell, indicating a weak interaction between SnCH 3 and BN sheet. The band structures without and with the SOC are presented in Fig. 5(b and c), respectively. One can see that when excluding the SOC, the p xy type bands mainly contributed by SnCH 3 film are degenerate to form Dirac point at the Fermi level at the Γ point. The SOC band gap opened at the Dirac point and the nontrivial bands are intact in comparison to that in Fig. 2(d). These results demonstrate that it is feasible to deposit SnCH 3 film on the h-BN substrate so as to attain QSH states.
Finally, we want to point out that the SnCH 3 film is more promising than GeCH 3 film to realize QSH effect at room temperature, since stretched SnCH 3 film possesses a larger nontrivial energy gap. Moreover, the critical strain applied for SnCH 3 film is much smaller than that for GeCH 3 to induce topological phase transition, which is more flexible to carry out in experiment. Such difference can be construed as that topological property is closely related to the Sn-Sn or Ge-Ge bond lengths. Although GeCH 3 and SnCH 3 film shares similar geometric structure, the Ge-Ge bond lengths are relatively less than those Sn-Sn bonds due to the strong bonding. The difference between s and p xy orbitals energy level thus is very large, resulting in a more sizable band gap when excluding the SOC. Therefore, much larger critical tensile strain is needed to drive band inversion in GeCH 3 film. In view of experimental realization, the similar strategy that biaxial strain larger than 10% has been achieved in graphene 49 , may be helpful for SnCH 3 film.
In summary, based on the first-principles calculations, we proposed that SnCH 3 film can be tuned to a quantum spin Hall insulator by imposing biaxial tensile strain larger than 6%. The topological phase transition from a trivial to nontrivial insulator can be ascribed to the strain-induced s-p xy type band inversion at the Γ point. The role of the SOC in stretched SnCH 3 is to open up a relatively large energy gap. The value of band gap is about 0.240 eV at critical strain of 6%, which increases with continual increase of tensile strain, while the band gap using more reliable hybrid functional HSE06 is increase to 0.314 eV. These interesting results make SnCH 3 film a promising candidate to achieve quantum spin Hall effect at room temperature, which is significant for future electronic devices with low-power consumption.