Stanane on InSe and GaTe

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Quasi-Free-Standing Features of Stanene/Stanane on InSe and GaTe Nanosheets: A Computational Study Yi Ding, and Yanli Wang J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.5b08946 • Publication Date (Web): 16 Nov 2015 Downloaded from http://pubs.acs.org on November 18, 2015

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The Journal of Physical Chemistry

Quasi-free-standing Features of Stanene/Stanane on InSe and GaTe Nanosheets: A Computational Study Yi Ding∗,† and Yanli Wang∗,‡ †Department of Physics, Hangzhou Normal University, Hangzhou, Zhejiang 310036, People’s Republic of China ‡Department of Physics, Center for Optoelectronics Materials and Devices, Zhejiang Sci-Tech University, Xiasha College Park, Hangzhou, Zhejiang 310018, People’s Republic of China E-mail: [email protected](Y.Ding); [email protected](Y.Wang) Phone: +086057128865297 (Y. Ding); +086057186843226 (Y. Wang)

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Abstract Two-dimensional tin and its iodized derivative, called stanene and stanane, are intriguing nanomaterials as quantum spin Hall (QSH) insulators. A recent experiment on stanene has found that the strong interaction from substrates will disturb the Dirac cones and cause a metallicity problem into stanene [F. Zhu et al. Nat. Mater. 14, 1020 (2015)]. Based on van der Waals density functional calculations, we find that stanene and stanane can form commensurate van der Waals heterosheets with InSe and GaTe layers, which exhibit quasi-free-standing features with comparable spinorbit coupling gaps to isolate values. Particularly, the supported stanane is still a large-gap QSH insulator, for which the substrates induce a tunable Rashba splitting that can be modulated by changing the interlayer distance of heterosheets. Our study demonstrates that layered III-VI chalcogenides are promising substrates for stanene and stanane, on which their intrinsics features can be well preserved for potential applications in topological materials and quantum manipulated devices.

Introduction The experimental discovery of silicene and germanene enriches the family of carbon group nanomaterials, 1–4 which inspires researchers to explore more elemental nanosheets composed of heavy group-IV atoms. Two-dimensional Sn sheet has a buckled honeycomb lattice akin to silicene and germanene, which also exhibits a Dirac-like electronic property owing to the half-filled pz orbitals. 5–7 In analogy to graphene, silicene and germanene, this Sn nanosheet has been called stanene from the Latin word ”stannum” of tin. 8 The fully iodized stanene (SnI) sheet, i.e. stanane, has been found to possess a large spin-orbit coupling (SOC) gap up to 0.3 eV, which is a prominent host to realize the quantum spin Hall effect at room temperature. 8–11 Very recently, the stanene nanosheet has been successfully synthesized on the Bi2 Te3 substrate by molecular beam epitaxy. 12 The scanning tunneling microscopy and 2 ACS Paragon Plus Environment

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angle-resolved photoemission spectroscopy measurements show that the deposited Sn atoms form a buckled hexagonal monolayer as stanene, but its Dirac cones are disturbed by the strong interaction with substrates. Besides that, the interlayer charge transfer shifts the Fermi level out of the SOC gap of stanene, making it metallic. 12 Similar metallicity phenomena also exist in stanene on the CdTe, InSb, Si and Au surfaces, 13,14 and a recent theoretical work has proposed that the Fermi level can be restored into the SOC gap by half-iodization when stanene is chemically adsorbed on appropriate substrates. 15 For practical applications in modern semiconductor technology and industry, it is required to isolate and transfer the Sn and SnI nanosheets onto various substrates. An appropriate nonpoplar substrate, which has a normal van der Waals interaction with stanene and stanane, will not only avoid the metallic trouble in supported systems but also provide a convenient way to manipulate their intrinsic properties. In this work, we pay main attention to the possible integrations of Sn/SnI and III-VI chalcogenide layers. These gallium and indium chalcogenide nanosheets are semiconductors, 16–18 which are ideal substrates to retain the quasi-free-standing features of graphene, 19 silicene, 20,21 and germanene sheets. 22 Experimentally, GaTe and InSe nanosheets have already been synthesized, 23,24 which can form van der Waals heterostructures with graphene sheets. 25 So far, the exploration about the stanene/stanane on InSe and GaTe layers is still lacking. Some essential questions arise for these hybrid systems: Can they form stable van der Waals heterosheets? Will these chalcogenide substrates overcome the metallicity problem? What are the peculiar electronic properties of them?

Methods To address these issues, we perform a comprehensive first-principles investigation by the VASP code, 26 which adopts the plane-wave basis sets with a cut-off energy of 500 eV and Perdew-Burke-Ernzerhof (PBE) projector augmented wave pseudopotentials. The non-local



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van der Waals (vdW) density functional has been used to account for the interlayer dispersion interaction, which adopts the optB86b functional form in the calculations. 27 A vacuum layer more than 15 ˚ A is used to simulate the isolate bilayer structures and a 9 × 9 × 1 k-mesh is utilized to sample the Brillouin zone. The atomic structures are fully optimized until the maximum residual force is less than 0.02 eV/˚ A. Following the previous work, 11,14,28,29 we have performed the convergence tests on the used cut-off energy and k-mesh in the calculations. As shown in Figs. S1 and S2 of Supporting Information, the convergence tests have verified that the adopted parameters in our calculation are of sufficient precision for the investigated systems.

Results and discussion Based on optB86b-vdW calculations, the lattice constants of Sn and SnI sheets are obtained to 4.63 and 4.79 ˚ A, and the corresponding values for InSe and GaTe monolayers are 4.04 and 4.10 ˚ A, respectively. These results agree well with the previous studies, 7–9,16–18 and indicate √ √ that a 3 × 3 supercell of Sn/SnI sheet will match the 2 × 2 units of InSe/GaTe layer. The corresponding lattice mismatches are -0.7% and -2.2% for Sn−InSe and Sn−GaTe systems, and they become 2.6% and 1.2% in SnI−InSe and SnI−GaTe cases. All of them are smaller than the value of 6.3% between the Sn sheet and used Bi2 Te3 substrate in experiments, 12 which suggests the feasibility of commensurate Sn/SnI−InSe/GaTe supercells. In the calculations, considering the flexibility of grown nanosheets, the used lattice constants of heterostructures are adopted to those in InSe and GaTe layers. This is consistent with the previous studies, 20–22 which show the buckled group-IV sheets can be easily altered to suit the substrates. In order to obtain the most stable stacking between Sn/SnI and InSe/GaTe layers, we firstly scan a rough potential energy surface from a 6 × 6 mesh of fixed configurations with a vertical interlayer distance of 3 ˚ A. Then, we sort out the plausible minimum energy configurations and fully re-optimize them to get the lowest energy stacking structures.


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Since the atomic structures of Sn/SnI sheets on the GaTe substrate are similar to those on the InSe one, Fig. 1 just depicts the Sn-InSe and SnI-GaTe heterosheets as representatives. In the primitive cell of heterosheets, the three bottom Sn atoms (Sn-I pairs) of Sn (SnI) sheet are located near the top sites of one In atom, one Se atom, and the hollow site of an InSe hexagon. As shown in Figs. 1(a) and (b), the Sn atoms (Sn-I pairs) experience a small sliding from the high symmetric points, which lowers the total energies of Sn−InSe and SnI−InSe heterosheets by 23 and 9 meV, respectively. The binding energies Eb , which are calculated as the energy differences between the heterosheets and isolated layers, are -0.167 and -0.162 eV per Sn atom for the Sn−InSe and Sn−GaTe heterosheets, respectively. For the SnI−InSe and SnI−GaTe ones, the binding strengths are a little weaker with the Eb of -0.134 and -0.136 eV per Sn atom, respectively. These binding energies are in the range of physical adsorption and comparable to the value of silicene on gallium chalcogenides (about -0.126 eV per Si atom 20 ). This suggests that layered III-VI chalcogenides will be general vdW-type substrates for group-IV nanosheets. The phonon dispersions of isolated stanene and stanane sheets are displayed in Fig. S6 of Supporting Information, which demonstrate their dynamical stabilities with no soft modes. Since InSe and GaTe layers are vdW-type substrates for Sn and SnI sheets, it can be expected that the dynamical stabilities of supported systems are well preserved in the heterosheets. However, due to the low symmetry and large supercell of heterosheets, the direct phonon calculations on them are beyond the affordable computational cost. Thus, to verify the stability of supported stanene sheet, ab initio molecular dynamics (AIMD) simulations are carried out on a 2×2 supercell of Sn-InSe/GaTe heterosheets. During the AIMD simulations, a Nosé thermostat of 300 K and a simulated step of 1 fs are used. As shown in Fig. S7 of Supporting Information, after 1500 steps, the whole structure keeps the integrity despite of some distortions in the Sn layer. Stanene sheet is still well located on the InSe and GaTe surfaces after AIMD simulations, which approves the stabilities of heterosheets. Figures 2(a) and (b) depict the variations of Eb versus the interlayer distance h in these



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Sn/SnI−InSe/GaTe heterosheets. Note that the Eb − h curves exhibit typical intermolecular interaction features, which can be well described by an empirical Buckingham (Exp-6) h i 6 h h0 6 30 exp α 1 − − . Here, the parameters and h0 potential as E(h) = 1−6/α α h0 h correspond to the depth and site of energy well, and α is a dimensionless quantity standing for the steepness of repulsive part. The fitted parameters of Buckingham potential are listed in Tab. 1. It shows the optimum interlayer distances h are as large as 3.07 ∼ 3.48 ˚ A in the Sn/SnI−InSe/GaTe heterosheets, which approximate to the equilibrium interlayer distance in graphite. 31 The interlayer elastic constants, calculated as C33 =

h ∂2E | 0 , 31 S ∂h2 h=h

are small in

these heterosheets. As shown in Tab. 1, the C33 of Sn/SnI−InSe/GaTe systems are only 12∼14 GPa, which are about 40% of the value in graphite (30∼40 GPa for different stackings 31 ). It indicates that the interlayer distance of these heterosheets can be easily enlarged through mechanical and chemical exfoliation, which facilitates the isolation and transfer of Sn/SnI nanosheets. Besides the optB86b-vdW functional, additional calculations with different functionals are also performed to verify the weak vdW interactions in Sn/SnI−InSe/GaTe heterosheets. As shown in Figs.

2(c) and (d), the PBE functional is poor to describe vdW interac-

tion, which obtains improper small binding strength for the heterosheets. The Grimme’s D2 (PBE+D2) method 32 obtains similar results to the optB86b ones, and the calculation by second version of vdW-DF (vdW-DF2) 33 gives slightly lower Eb values for the heretrosheets. Thus, despite of pure PBE case, all the calculations show the binding energies of Sn/SnI−InSe/GaTe heterosheets are in the magnitude of -0.10∼-0.175 eV per Sn, which correspond to 10∼19 meV/˚ A2 in these heterosheets. These values are close to the typical van der Waals bonding strength in layered compounds (∼20 meV/˚ A2 ), 34 indicating that the Sn/SnI−InSe/GaTe heterosheets are also representative vdW systems. It would be noted that due to the strong interaction with substrates, the grown silicene sheets on Ag(111) surface will possess different super-structures depending on the deposition parameters. 35,36 A √ √ 3 × 3 Si super-structure is found to keep the linear band dispersion of silicene, 37 while a


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3×3 one opens a band gap in the Si nanostructure. 38 When on the vdW-type GaS substrate, the silicene sheet prefers the simple 1 × 1 chair-like buckle akin to the free-standing case. 20 Here, on the InSe and GaTe layers, the stanene sheet is also weakly bound to the substrates via vdW interaction. Thus, the supported Sn layer would exhibit the quasi-free-standing feature and keep the chair-like buckle in the isolated case. When the substrate materials are varied, on which stanene is strongly bonded, different super-structures would appear as in the case of silicene sheets. 36 Due to the weak vdW interactions, the band structures of heterosheets can be regarded as a superpostion of bands from Sn/SnI and InSe/GaTe layers. Figure 3 depicts the band structures of Sn/SnI−InSe/GaTe heterosheets without the SOC. It would be noted that for the Sn sheet, its Dirac cones, which are constructed by out-of-plane pz orbitals, are folded to √ √ the Γ point of 3 × 3 supercell. In the Sn−InSe heterosheet, the Dirac cone touches the bottom of conduction band of InSe layer, and the zero-band-gap characteristic is retained for the Sn sheet. While in the Sn−GaTe heterostructure, the Dirac cone is located in the middle of band gap of GaTe layer, which opens a tiny gap of 0.075 eV due to the substrate effect. For the SnI sheet, different from the Sn case, its Dirac cone is composed of in-plane s and px,y oribitals, which is individually located at the Γ point regardless of the supercell size. On the InSe and GaTe substrates, the SnI layer exhibits a zero-band-gap behaviour akin to the free-standing case as shown in Figs. 3(b) and (d). In the heterosheets, the Dirac cone of SnI sheet stays in the middle of band gap of InSe layer, while it is close to the top valence bands of GaTe substrate. For all the Sn/SnI−InSe/GaTe systems, the Fermi level is just across the Dirac points of Sn/SnI layers. Hence, there is no metallicity existing in stanene and stanane on the InSe and GaTe substrates. Since in the vicinity of Fermi level the bands of stanene/stanane are significantly affected by the spin-orbit coupling, 8,9 we further calculate the band structures of Sn/SnI−InSe/GaTe heterosheets including the SOC as shown in Fig. 4. It is worth nothing that the SOC gap of stanene is foled to the Γ point, which is hidden by other bands. Considering that the



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Dirac cones of stanene are mainly contributed by the out-of-plane pz orbitals of Sn atoms, 8 the projection of Sn pz orbitals is marked in the band structures of Figs. 4(a) and (c). It is evident that on the InSe substrate, the Sn sheet opens a SOC gap of 0.02 eV, which is much smaller than the free-standing value of 0.1 eV. While on the GaTe substrate, the SOC gap becomes 0.08 eV, still comparable to the free-standing value. For the SnI sheet, the s − p band inversion induces a large SOC gap of 0.3 eV at the free-standing state. 8 Such band inversion is well preserved in SnI on the InSe and GaTe substrates. As shown in Figs. 4(b) and (d), the projection of Sn s orbital at the Γ point mainly appears at the top valence band instead of the bottom conduction band. This suggests that the intrinsic quantum spin Hall insulating characteristic is retained in the supported SnI sheet. 15 Compared to the freestanding case, the SOC gap of SnI layer is still 0.3 eV on the InSe substrate, but decreases to 0.19 eV on the GaTe one. Thus, from the view of larger SOC gaps, the InSe substrate is a better choice for SnI sheet, and the GaTe substrate is more suitable for Sn sheet. The topological insulating behaviour can be characterized by a topological invariant of Z2 in the system. The Z2=1 means a non-trivial state, while Z2=0 corresponds to a trivial case. For the structures with inversion symmetry, the Z2 value can be directly obtained from the product of parity eigenvalues for all the occupied bands at the time-reversal-invariant momenta points, 39 which has been widely in the literatures on 2D topological insulators. 8,10,40 While for the systems without inversion symmetry, the Z2 analysis needs some complicated methods. 41–43 Following the wannier function approach proposed by Soluyanov and Vanderbilt, 43 we have done a Z2 analysis by tracking the evolution of wannier charge centers (WCCs). As shown in Fig. S3 of Supporting Information, utilizing the z2pack code 1 , the calculations can successfully obtain Z2=1 for the isolated Sn and SnI sheets. For the Sn-GaTe and Sn-InSe heterosheets, due to their narrow band gaps, the z2pack calculations encounter a convergence problem. Thus, only the results of SnI-GaTe and SnI-InSe systems are provided in Fig. S4 of Supporting Information. The z2pack calculations show that the 1

http://physics.rutgers.edu/z2pack and http://z2pack.ethz.ch


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Z2 invariant remains to be one in these systems, which indicates the robust topological insulating features for them. Thus, on these vdW-type GaTe/InSe substrates, the intrinsic topological properties of supported layers are well preserved. We have also performed the Z2 analysis on the SnI-GaTe and SnI-InSe heterosheets by the pure PBE calculations. The Z2 invariant is still one for both systems, indicating that the van der Waals correction does not affect the topological phase of supported systems. It should be noticed that in Figs. 4(c) and (d), the top valence bands of SnI layer exhibit a noticeable Rashba splitting, which normally stems from the inversion asymmetry induced by substrates. 15 In the heterosheets, there is pronounced charge redistribution in the interlayer interval. The deformation charge densities, defined as ∆ρ = ρhereto −ρSnI −ρsub , are depicted in Figs. 4(e) and (f). Here, ρhereto is the total charge density of heterosheet, ρSnI and ρsub are the charge densities of isolated SnI layer and InSe/GaTe substrate at the same position of heterosheet. It can be seen that electrons are mainly accumulated in the interval between two layers, while the electron deficient areas appear around the bottom I atoms as well as the upper Se/Te atoms as shown in Figs. 4(e) and (f). Besides that, one of every six Sn-I bonds, which is just on top of substrate Se/Te atoms, also has a remarkable charge redistribution phenomenon. The corresponding plane-integrated deformation charge densities ∆ρz are plotted in Fig.

4(g), which clearly show the asymmetry of charge redistribution in the

heterosheets. As a result, the inversion symmetry is broken in the support SnI layer, which causes a noticeable Rashba splitting in the band structures. Since the Rashba splitting is related to the interlayer coupling, the splitting magnitude would be modulated by changing the interlayer distance, which can be achieved by applying pressure perpendicular to the heterostructure. To this end, we calculate the SOC band structures of SnI−GaTe heterosheet with different interlayer distances as shown in Fig. 5. We have also performed calculations on SnI-GaTe with larger ∆h. It is found that when ∆h ≥ 2 ˚ A, the SOC gap of SnI layer would recover the free-standing value of 0.31 eV. It is evident that the band splitting is increased when the interlayer distance is reduced, and vice



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versa. There is a linear relationship between the band splitting ∆S and the interlayer strain ∆h in Fig. 5(c). For the SnI−GaTe, when ∆h varies from -0.5 to 0.5 ˚ A, the ∆S is tuned from 75 to 14 meV. Based on the fitted Buckingham expression, this ∆h range corresponds to a perpendicular pressure of [4.36, -0.87] GPa on the heterosheet, which is affordable in the experiments. Similarly, a linear modulation of ∆S versus ∆h is also present in the SnI−InSe heterosheet, which can vary from ∆S =15 meV at ∆h =-0.5 ˚ A to ∆S =3 meV at ∆h =0.5 ˚ A. Compared to the strain-free values, the relative variations of ∆S are about 190∼30% in the ∆h range of [-0.5, 0.5] ˚ A for both heterosheets. Such large variation range indicates the Rashba splitting can be well tuned in the supported stanane system, enabling it potential applications in topological superconductivity and spintronics. 15 Previous studies have shown that on the transition metal dichalcogenide substrates, the electronic structure of supported silicene sheet can be altered by the layer number of substrates. 21,44,45 Here, for the Sn/SnI-InSe/GaTe heterosheets, additional calculations are also performed on the bilayer (2ML) substrates. As shown in Figs. S9-10 of Supporting Information, Sn-2ML InSe, Sn-2ML GaTe, and SnI-2ML InSe systems have similar electronic properties to those on the monolayer substrates. While for the SnI-2ML GaTe heterosheet, a layer-dependent behaviour is observed, which originates from the change of top valence bands from the GaTe bilayer. Thus, from the view of robust electronic performance, the InSe substrates will be better than the GaTe ones for the stanene and stanane systems.

Conclusion In summary, we have investigated the commensurate van den Waals heterosheets composed of stanene/stanane and InSe/GaTe layers, which can overcome the metallicity problem for the supported systems. On the InSe/GaTe substrates, the intrinsic Dirac-like electronic structures are preserved in Sn and SnI layers, whose Dirac points are located in the band gaps of substrates. In presence of spin-orbit coupling, the SOC band gaps are still opened in the


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supported Sn/SnI sheets, especially the SnI one, which is still a large-gap quantum spin Hall insulator. Due to the substrate effect, the top valence bands of SnI layer exhibit a noticeable Rashba splitting, which can be effectively modulated by changing the interlayer distance of heterosheets. Our study demonstrates that stanene and stanane exhibit quasi-free-standing features on layered III-VI chalcogenides substrates, which have potential applications in topological materials and quantum manipulated devices.

Acknowledgement Authors acknowledge the supports from National Natural Science Foundation of China (11474081), Zhejiang Provincial Natural Science Foundation of China (LY15A040008). Parts of the calculations were performed in the Shanghai Supercomputer Center of China.

Supporting Information Available Additional figures: Fig. S1 the convergence tests of cut-off energy, Fig. S2 the convergence tests of k-mesh, Fig. S3 the evolution of wannier charge centers (WCCs) for isolated Sn and SnI sheets, Fig. S4 the evolution of WCCs for SnI-GaTe and SnI-InSe heterosheets, Fig. S5 the evolution of WCCs for SnI-GaTe and SnI-InSe heterosheets by the PBE calculation, Fig. S6 the phonon dispersions of stanene and stanane sheets, Fig. S7 the final structures of Sn-InSe and Sn-GaTe heterosheets from the ab initio molecular dynamics simulation, Fig. S8 the structures of Sn/SnI sheets on the bilayer InSe and GaTe substrates, Fig. S9 the band structures of Sn/SnI sheet on the bilayer InSe substrate, Fig. S10 the band structures of Sn/SnI sheets on the bilayer GaTe substrate, and Fig. S11 the SOC band structures of SnI-GaTe system at the ∆h = 2 ˚ A. This material is available free of charge via the Internet at http://pubs.acs.org/.



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Notes The authors declare no competing financial interest.

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Table 1: The fitted Buckingham parameters for the binding energies and interlayer elastic constants of heterosheets.

Sn−InSe SnI−InSe Sn−GaTe SnI−GaTe

(eV/Sn) h0 (˚ A) α C33 (GPa) 0.167 3.07 2.07 14.4 0.133 3.27 2.38 12.6 0.162 3.31 2.20 13.8 0.136 3.48 2.55 13.1


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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Sn Se In In Se

(a) Stanene-InSe I Sn I Te Ga Ga Te

(b) Stanane-GaTe Figure 1: The top and lateral view of (a) Sn−InSe and (b) SnI−GaTe heterosheets.


0.00

-0.05

-0.05

-0.10

Eb (eV/Sn)

0.00

Sn-InSe SnI-InSe fit

2

3

4

5

6

7

8

-0.10

Sn-GaTe SnI-GaTe fit

2

9

h(Ang)

(a)

4

5

6

7

8

9

h(Ang)

4.00 3.75

h(Ang)

-0.15

3

(b) optB86b PBE+D2 vdw-DF2 PBE

-0.20

-0.10 -0.05 0.00

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-0.15

-0.15

Eb(eV/Sn)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Eb (eV/Sn)


optB86b PBE+D2 vdw-DF2 PBE

3.50 3.25 3.00

Sn-InSe

SnI-InSe

2.75

Sn-GaTe SnI-GaTe

(d)

(c)

Sn-InSe

SnI-InSe

Sn-GaTe SnI-GaTe

Figure 2: The variations of binding energies Eb as a function of interlayer distances h for the (a) Sn/SnI-InSe and (b) Sn/SnI-GeTe heterosheets. [(c), (d)] The obtained binding energies Eb and interlayer distances h from different functionals.


2

2

1

1

E(eV)

InSe

0

Sn

-1

-2

InSe

0

SnI

-1

M

-2

M

K

G

M

M

K

G

(a)

(b)

2

2

1

1 GaTe GaTe

0

E(eV)

E(eV)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


E(eV)

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Sn

-1

-1

-2

SnI

0

M

K

G

-2

M

M

(c)

K

G

M

(d)

Figure 3: The projected band structures of (a) Sn-InSe, (b) SnI-InSe, (c) Sn-GaTe, and (d) SnI-GaTe heterosheets without the SOC. In all the band structures, the Fermi level is set to the 0 eV.


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0.4

0.4

0.2

0.2

0.2

0.2

0.0

Sn PZ

-0.2

Sn s

0.0

0.0

-0.2

-0.4

M

G

M

G

-0.4

-0.4

M

K

(b) SnI-InSe

G

K

M

0.04 0.02 0.00 -0.02 -0.04

Se

In

(f) SnI-GaTe

In

0.04 0.02 0.00 -0.02 -0.04

Te

Ga Ga

5

(g)

G

K

(d) SnI-GaTe

(c) Sn-GaTe

0

(e) SnI-InSe

Sn s

0.0

-0.2

Drz (e/Ang)

(a) Sn-InSe

Sn PZ

-0.2

-0.4

K

E(eV)

0.4

E(eV)

0.4

E(eV)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

E(eV)


Se

Te

I

Sn

Sn

I

I

Sn

Sn

I

10

15

20

z (Ang)

Figure 4: The SOC band structures around the Fermi level for (a) Sn-InSe, (b) SnI-InSe, (c) Sn-GaTe, and (d) SnI-GaTe heterosheets. The distributions of deformation charge densities in (e) SnI-InSe and (f) SnI-GaTe heterosheets and (g) the corresponding plane-integrated deformation charge densities.


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0.4

0.4

SnI-GaTe

0.2

SnI-GaTe

0.2 Dh=-0.5A

0.0

E(eV)

E(eV)

DS

-0.2

-0.4

-0.4

(a)

M

G

Dh=0.5A

0.0

-0.2

(b)

K

M

G

K

80 DS(%)

175 100

60

25

DS (meV)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


-0.5

40

0.5

SnI-GaTe SnI-InSe

20

0

(c)

0.0

-0.50

-0.25

0.00

0.25

0.50

Dh (Ang)

Figure 5: The SOC band structures of SnI−GaTe heterosheets with the interlayer distance change of (a) ∆h = −0.5 and (b) ∆h = 0.5 ˚ A. (c) The variations of band splitting versus the change of interlayer distance for the SnI−GaTe and SnI−InSe heterosheets. The inset shows the relative variations of band splitting compared to the strain-free values of these heterosheets.



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Graphical TOC Entry


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Stanane on InSe and GaTe

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