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Topological Insulator in Two-Dimensional SiGe Induced by Biaxial Tensile Strain Tamiru Teshome and Ayan Datta* Department of Spectroscopy, Indian Association for the Cultivation of Science, 2A and 2B Raja S. C. Mullick Road, Jadavpur, Kolkata 700032, West Bengal, India S Supporting Information *

ABSTRACT: Strain-engineered two-dimensional (2D) SiGe is predicted to be a topological insulator (TI) based on first-principle calculations. The dynamical and thermal stabilities were ascertained through phonon spectra and ab initio molecular dynamics simulations. 2D SiGe remains dynamically stable under tensile strains of 4 and 6%. A band inversion was observed at the Γ-point with a band gap of 25 meV for 6% strain due to spin−orbit coupling interactions. Nontrivial of the TI phase was determined by its topological invariant (υ = 1). For SiGe nanoribbon with edge states, the valence band and conduction band cross at the Γ-point to create a topologically protected Dirac cone inside the bulk gap. We found that hexagonal boron nitride (h-BN) with high dielectric constant and band gap can be a very stable support to experimentally fabricate 2D SiGe as the h-BN layer does not alter its nontrivial topological character. Unlike other heavymetal-based 2D systems, because SiGe has a sufficiently large gap, it can be utilized for spintronics and quantum spin Hall-based applications under ambient condition.

1. INTRODUCTION Topological insulators (TIs) are novel quantum spin Hall (QSH) insulators that are characterized by boundary states within a bulk insulating gap and maintain time-reversal symmetry due to spin−orbit coupling (SOC).1−4 Thus, the low-energy scattering of the edge states is robust by timereversal symmetry, which leads to dissipationless transport edge channels. TIs have many potential applications, such as in spintronics and quantum devices.5 TI phase in graphene has been predicted due to very weak spin−orbital coupling (SOC) that opens a gap at the Dirac point.6 In the last decade, several new two-dimensional (2D) TIs have been theoretically predicted and also experimentally confirmed mostly in elements with strong spin−orbit coupling.7−11 However, these materials have very small bulk band gaps and hence experimental observation of a QSH insulator has been rather difficult other than at ultralow temperature. Consequently, there is a strong interest to search and design suitable 2D TIs at the normal temperature with large bulk gaps. Very recently, theoretical calculations have predicted 2D TIs phase.12−32 Unfortunately, most of these theoretically predicted structures would be extremely difficult for translating into devices due to the presence of the heavy atoms, which are known to be toxic. Among these systems, Bi4Br4 and ZrTe5/HfTe5 appear to be highly promising because they exist as layered crystals and have significant band gaps. Also, IV−VI 2D materials seem attractive as calculations indicate TI in their rocksalt structures.33−36 To design a large-band-gap two-dimensional topological nontrivial phase, different techniques have been utilized to © 2018 American Chemical Society

achieve QSH effect. One widely used approach is applying a perpendicular external electric field, that is, applying an electric field to multilayers of black phosphorus to convert it from a normal insulator (NI) to TI. The nontrivial phase originates from the external electric field, whereas SOC opens the band gap at the Dirac cone to realize TI phase.31 Strain modification is another approach for switching a NI in TI.37,38 Furthermore, low-buckled silicene and germanene also show QSH effect and band inversion due to the effect of spin−orbit coupling.39 Herein, using density functional theory, we predicted that 2D SiGe undergoes phase transition from NI to TI under a strain of 6%. The dynamical stability of SiGe was proven by phonon dispersion calculations. The band structures and the parity of occupied bands under strain confirmed trivial and nontrivial topological phases before and after the band inversion, respectively. The gapless bands of the topological edge states are achieved within a zigzag nanoribbon (with L = 8.837 nm) of SiGe. Most interestingly, hexagonal boron nitride (h-BN) is shown as an excellent substrate for supporting 2D SiGe and yet preserving its nontrivial topological phase. Hence, SiGe/h-BN is predicted as a room-temperature TI for QSH phase applications. Received: December 8, 2017 Accepted: December 19, 2017 Published: January 2, 2018 1

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Figure 1. Top and side views of (a) 2D SiGe monolayer. (b) Total energy as a function of different stains with and without SOC. Phonon spectra under (c) 4% and (d) 6% tensile strain for 2D SiGe.

2. COMPUTATIONAL DETAILS All calculations were carried out using the Vienna ab initio simulation package (VASP).40 The exchange-correlation term was described within the generalized gradient approximation (GGA) parameterized by the Perdew−Burke−Ernzerhof (PBE) functional. Vacuum region was set as 20 Å along zdirection of the sheets to minimize the interaction between the monolayers using periodic boundary conditions. Kinetic energy cutoff for plane-wave expansion was 500 eV. Atoms in the unit cell are relaxed until the force per atom falls below 0.001 eV/Å. A 11 × 11 × 1 Monkhorst−Pack41 k-point mesh is used to sample the Brillouin zone. The modified tensile strain is described as ε = Δa/ao, where ao is the equilibrium lattice and Δa + ao is the strained-modified lattice. Spin−orbit coupling (SOC) effect was included consistently up to the second-order (LS coupling).42 The phonon spectra were implemented by the PHONOPY code43 using density functional perturbation theory method in VASP. Molecular dynamics (MD) simulations were carried out to determine the room-temperature stability of 2D SiGe. MD simulation was achieved for 10 ps with a time step of 1.0 fs at T = 300 and 500 K.

energies are compared without and with SOC at different strains, as shown in Figure 1b. The dynamical stability of 2D SiGe was confirmed by the absence of imaginary frequencies in their phonon dispersions. The computed phonon dispersions for tensile strains of 4 and 6% corresponding to the cases prior and subsequent to band inversion are shown in Figure 1c,d respectively. Absences of imaginary frequencies indicate that the SiGe monolayer at 4 and 6% strains was dynamically stable. The thermal stability of 2D SiGe was further investigated by performing ab initio molecular dynamics simulations. We utilized a sufficiently large 4 × 4 supercell for thermal equilibration. After heating at 300 and 500 K for 10 ps with a time step of 1 fs, we observed no structural reconstruction, indicating that 2D SiGe is thermally robust. The snapshots of atomic configurations of 2D SiGe structure for MD simulation with a time step of 1 fs at T = 300 and 500 K are reported in Figure S1, and the corresponding total energy, average bond length, and temperature fluctuation with time during the simulation are illustrated in Figure 2a−c. The average Si−Ge bond length is 2.41 Å and average dihedral angle ⟨ϕ⟩ is 42° (cf. ⟨ϕ⟩ ≈ 47.1° at 0 K for GGA−PBE calculations), which clearly indicate that the 2D SiGe remains thermally stable at room temperature (i.e., T = 300 K). 3.2. Band Structures of Strain-Modified SiGe. The electronic band structures of 2D SiGe can be tuned through an external tensile strain as plotted in Figure 3a−h without and with SOC. This is indeed gratifying considering that doping with Ge atoms does not alter the band gap in silicene.44 As shown in Figure 3a,e without and with SOC, respectively, the π and π* bands cross the Fermi level at the Γ-point. The Fermi level is dominated mainly by the pz atomic orbitals of Si and Ge atoms. Analogous Dirac cones at K point have been observed in pristine silicene, germanene, and graphene. The origin of the band gap in 2D SiGe was the effect of hybridization of the orbitals under strain with SOC. Because of strain, bands gaps of 47 and 89 meV are observed without SOC for ε = 2 and 4%, respectively, at Γ-point, as presented in Figure 3b,c. When SOC

3. RESULTS AND DISCUSSION 3.1. Dynamical Stability and Electronic Structures. The 2D SiGe monolayer has an out-of-plane puckered geometry within a hexagonal structure, in which the height of buckling, Si−Ge bond length, and bond angle (θ) are 0.58, 2.36 Å, and 113.9°, respectively, as shown in Figure 1a (see Table ST1). We have previously shown that doping of silicene with Ge atoms increases its buckling height to about 0.69 Å, and also the lattice constant increases with increasing doping concentrations in agreement with Vegard’s law.44 Interestingly, 2D SiGe has two polymorphic forms: (i) phase-separated lattice, in which the Si and Ge atoms are localized in opposite sides. This form shows quantum confinement and band offset for nanowires.45,46 (ii) Honeycomb lattices for which Si and Ge are alternatively bonded, as shown in Figure 1a. The total 2

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tensile strains, as shown in Figure S4. Interestingly, both the maximum of valance band and the minimum of conduction band get localized and charge-transfer states are observed. Evidently, s-orbital crosses the Fermi level as the tensile strain increases and the pz orbital is lifted to the conduction band as a result of SOC. Further increase in strain to 6% makes the sorbital lower in energy than the pz orbital, as observed by projecting the density of states and bands onto different atomic orbitals. This explains the inversion of the s−p orbitals under the tensile strain due to the presence of SOC, as presented in Figure 5e−h. We now focus on optical spectra because the dielectric function depends on the electron band structure of the crystal. The origins of optical transitions are traced to core excitation from valance band to high-energy conduction band. The prominent optical transitions are obtained at 7.83, 8.02, 8.12, and 8.23 eV for ε = 0, 2, 4, and 6%, respectively, as shown in Figure 4 for the polarization of light perpendicular to the SiGe monolayer. Clearly, the absorption spectrum is blue-shifted with increased tensile strain vis-à-vis unstrained SiGe. 3.3. Nontrivial Topological Phase in Strain-Modified SiGe. To unambiguously determine the nontrivial topological features of 2D SiGe, we investigated the topological Z2 invariant (ν) during band inversion and prior to it. At a strain of 6%, we find the Z2 invariant ν = 0 (trivial phase) without SOC and ν = 1 (nontrivial phase) with SOC. We have used the parity criteria predicted by Fu and Kane;1−3 the presence of inversion symmetry helps calculate the Z2 parameter directly from the parities of eight occupied bands. For SiGe, there are four time-reversal invariant momenta (TRIM) points in the 2D Brillouin zone, one Γ-point and three M points (see Figure S4). The Z2 topological invariant, ν, is calculated as

Figure 2. Time evolution of (a) total energy, (b) average bond length of Si−Ge, and (c) temperature fluctuation in the molecular dynamics simulation at 300 K.

is considered, band gaps of 28 and 69 meV are observed at the Γ-point for ε = 2 and 4%, respectively, as shown in Figure 3f,g. Clearly, even a small tensile strain (ε = 2−4%) is sufficient to open the band gaps at Γ-point. As shown in Figure 3d, at ε = 6% without SOC, the valance band and conduction band (CB) come close to each other at Γ-point, thereby demonstrating a semimetal behavior. We then turn on the SOC, which lifts the degeneracy of orbitals, resulting in an indirect band gap of 25 meV. This causes the nature of the wave function of the valence band (VB) and CB to invert, as shown in Figure 3h. Therefore, the observation of “m” shape in the VB at the Γ-point for 6% strain arises entirely due to SOC. To further understand the electronic properties, the partial density of states of 2D SiGe is also computed under different

N

δ(K i) =

∏ ξ2in , n=1

4

( −1)ν =

∏ δ(K i) = δ(Γ) δ(M)3 i=1

where δ(Ki) is the product of parity eigenvalues at TRIM points, ξ = ±1 denotes parity eigenvalues, and N is number of the occupied bands. The parities of eight occupied bands at time-reversal momenta are shown in Figure 5a,b. When 6%

Figure 3. Band structures without and with SOC, respectively, at (a, e) 0% strain, (b, f) 2% strain, (c, g) 4% strain, and (d, h) 6% strain in 2D SiGe. K (1/3, 1/3, 0), Γ (0, 0, 0), M (1/2, 0, 0), and K (1/3, 1/3, 0) refer to special points in the first 2D Brillouin zone in reciprocal space. 3

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the calculated Z2 topological invariant is a trivial phase (ν = 0), topologically speaking. Additionally, for 2D SiGe as topological insulator, a unique property is the presence of a single of Dirac-like edge state (one-dimensional gapless edge) within the bulk gap. For further understanding the nature of nontrivial TI at ε = 6%, we analyze the existence of edge states. A symmetric edge within a zigzag nanoribbon is constructed, and the dangling bonds at the edges were passivated by hydrogen atoms. The width of the SiGe nanoribbon is 8.837 nm, which is sufficiently large to avoid electron correlations between the two edges on either ends, as shown Figure 6a. We explicitly recognize that the Dirac-like edge states (green lines) cross at the Γ-point linearly within the bulk gap, thereby confirming unequivocally the existence of a topologically nontrivial phase at ε = 6% strain with SOC, as shown in Figure 6b. To have practical application as an electronic device, pristine forms of silicene and germanene are too unstable and airsensitive. Therefore, monoatomic 2D layers of Si and Ge are typically deposited on substrates. Recently, experimental syntheses of hydrogenated graphene,47 germanene,48−50 and silicene51−55 have been successfully undertaken on various substrates. Hence, it is imperative that the topological nontrivial phase has to be fabricated over substrates for their eventual experimental observation. h-BN monolayer is a large-band-gap insulator with Eg = 4.6 eV with a high dielectric constant and has been experimentally well characterized as a substrate.56−59 Therefore, we choose h-BN as a substrate support for 2D SiGe monolayer. The lattice mismatch between SiGe and h-BN is quite small (Δ = 0.04 Å), and the interlayer distance, dSiGe/h‑BN = 4.12 Å, results in a stable van der Waals heterostructure. The binding energy is 0.02 eV per unit cell, which again confirms the weak interaction between SiGe and h-BN monolayer, as presented in Figure 6c. Interestingly, the band structure for SiGe/h-BN as shown in Figure 6d, which clearly indicates that the “m”-shape of the valance band at the Γ-point for ε = 6% remains unperturbed. Therefore, there is no charge transfer between the h-BN and SiGe monolayer, which is also supported from the Bader charge analyses. Generally, most of the 2D TIs predicted to date are with heavy metals, which have strong SOC effects, including Pb, Hg, Sn, Sb, and Bi. Although, in some light elements, QSH effect has also been observed, the very small gaps induced by SOC are insignificant for thermally mobilizing the edge states. For example, the gaps are ∼10−3 meV for graphene,60,61 silicene (1.55−2.9 meV),39 germanene (Eg = 23.9−108 meV),29,32,39 arsenene (Eg = 696 meV),13 and few-layered phosphorene (Eg = 1.5−2.2 eV).31 In this context, we predicted the 2D SiGe band gap of 25 meV at ε = 6%. Interestingly, the gap for SiGe is also comparable to thermal energy at room temperature.

Figure 4. Imaginary component of the dielectric frequency of 2D SiGe for ε = 0% (black line), ε = 2% (red line), ε = 4% (blue line), and ε = 6% (green line). Note that the polarization of light is perpendicular to 2D SiGe monolayers.

Figure 5. Parity of occupied states: (a) trivial state without SOC, (b) nontrivial state for 6% strain with SOC. Band structure (c) without SOC and (d) at 6% strain with SOC. Partial charge density of (e) conduction band minimum (CBM) without SOC, (f) VBM without SOC, (g) CBM with SOC, and (h) VBM with SOC. The isosurface refers to isovalues of 2.279 × 10−3 electrons/bohr3.

strain was applied without SOC, SiGe behaves as a semimetal, as presented in Figure 5c. The s-orbital touches the Fermi level and the pz orbital dominates the valence band maximum (VBM), as shown in Figure 5e,f. In the presence of SOC, the band inversion observed, as presented in Figure 5d, switches normal insulator to topological insulator as a result of strain increasing the greater hybridization of atomic orbitals (s, px, py, pz), as shown in Figure 5g,h. Consequently, SiGe undergoes trivial to nontrivial phase transition as the strain is increased to 6% in presence of SOC. We have verified the parities of occupied bands of SiGe, which proves a nontrivial phase (ν = 1). The products of the parity eigenvalues at the Γ and M points are +1 and −1, respectively, for 6% strain. When strain is applied without SOC,

4. CONCLUSIONS In summary, we have predicted a new 2D TIs for SiGe through tensile strain engineering. The absence of imaginary frequency at 6% strain indicates dynamical stability. Thermal stability for the SiGe monolayer is also confirmed by ab initio MD at 300 K. A band inversion has been observed at the Γ-point at ε = 6% due to spin−orbit coupling interaction, which opens a significant band gap of 25 meV. The nontrivial phase was additionally verified by the topological Z2 invariant, υ = 1. The topologically protected edge states in SiGe nanoribbon cross at the Γ-point inside the bulk gap. We also predict that hexagonal 4

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Figure 6. (a) Structure of zigzag SiGe nanoribbon (width = 8.837 nm) with ε = 6%. The edge is passivated by H atoms. (b) Electronic band structure of SiGe nanoribbon with 6% SOC strain (the gapless edge states (green lines) can be clearly seen within the bulk gap). (c) Top and side views of the fully grown 2D SiGe on h-BN substrate. (d) Band structure of SiGe deposited on h-BN substrate with 6% SOC strain. (3) Fu, L.; Kane, C. L. Superconducting Proximity Effect and Majorana Fermions at the Surface of a Topological Insulator. Phys. Rev. Lett. 2008, 100, 096407. (4) Li, L.; Zhang, X.; Chen, X.; Zhao, M. Giant Topological Nontrivial Band Gaps in Chloridized Gallium Bismuthide. Nano Lett. 2015, 15, 1296−1301. (5) Nagaosa, N. A new state of quantum matter. Science 2007, 318, 758−759. (6) Kane, C. L.; Mele, E. J. Quantum Spin Hall Effect in Graphene. Phys. Rev. Lett. 2005, 95, No. 226802. (7) König, M.; Wiedmann, S.; Brüne, C.; Roth, A.; Buhmann, H.; Molenkamp, L. W.; Qi, X. L.; Zhang, S. C. Quantum Spin Hall Insulator State in HgTe Quantum Wells. Science 2007, 318, 766−770. (8) Knez, I.; Du, R. R.; Sullivan, G. Evidence for Helical Edge Modes in Inverted InAs/GaSb Quantum Wells. Phys. Rev. Lett. 2011, 107, No. 136603. (9) Kim, S. H.; Jin, K.-Y.; Park, J.; Kim, J. S.; Jhi, S. H.; Kim, T. H.; Yeom, H. W. Edge and Interfacial States in a Two-Dimensional Topological Insulator: Bi(111) bilayer on Bi2Te2Se. Phys. Rev. B 2014, 89, No. 155436. (10) Kim, S. H.; Jin, K.-Y.; Park, J.; Kim, J. S.; Jhi, S. H.; Yeom, H. W. Topological Phase Transition and Quantum Spin Hall Edge States of Antimony Few Layers. Sci. Rep. 2016, 6, No. 33193. (11) Wu, R.; Ma, J.-Z.; Nie, S.-M.; Zhao, L.-X.; Huang, X.; Yin, J.-X.; Fu, B.-B.; Richard, P.; Chen, G.-F.; Fang, Z.; Dai, X.; Weng, H.-M.; Qian, T.; Ding, H.; Pan, S. H. Evidence for Topological Edge States in a Large Energy Gap near the Step Edges on the Surface of ZrTe5. Phys. Rev. X 2016, 6, No. 021017. (12) Weng, H. M.; Ranjbar, A.; Liang, Y. Y.; Song, Z. D.; Khazaei, M.; Yunoki, S. J.; Arai, M.; Kawazoe, Y.; Fang, Z.; Dai, X. Large gap TwoDimensional Topological Insulator in Oxygen Functionalized MXene. Phys. Rev. B 2015, 92, No. 075436. (13) Zhang, H.; Ma, Y.; Chen, Z. F. Quantum Spin Hall Insulators in Strain Modified Arsenene. Nanoscale 2015, 7, 19152−19159. (14) Wang, Z. F.; Su, N. H.; Liu, F. Prediction of a Two Dimensional Organic Topological Insulator. Nano Lett. 2013, 13, 2842−2845. (15) Zhou, L.; Kou, L. Z.; Sun, Y.; Felser, C.; Hu, F. M.; Shan, G. C.; Smith, S. C.; Yan, B. H.; Frauenheim, T. New Family of Quantum Spin Hall Insulators in Two-Dimensional Transition Metal halide with large Nontrivial Band Gaps. Nano Lett. 2015, 15, 7867−7872. (16) Ma, Y. D.; Kou, L. Z.; Li, X.; Dai, Y.; Smith, S. C.; Heine, T. Quantum Spin Hall Effect and Topological Phase Transition in Two-

boron nitride (h-BN) with high dielectric constant and large band gap acts as an ideal substrate to support SiGe for the experimental detection of its topological nontrivial phase. Also, the absence of heavy-metal atoms and significant topological gap makes 2D SiGe an excellent candidate for exploring its TI property.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acsomega.7b01957. Details of molecular dynamics simulations, partial charge densities, optical properties, and time-reversal invariant momenta (TRIM) points in the 2D Brillouin zone (PDF)

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Tel: +91-33-24734971. ORCID

Ayan Datta: 0000-0001-6723-087X Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors acknowledge The World Academy of Sciences (TWAS), Indian Association for the Cultivation of Science (IACS) (FR number: 3240280472), for financial support. A.D. thanks DST, INSA, and BRNS for partial funding.

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REFERENCES

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DOI: 10.1021/acsomega.7b01957 ACS Omega 2018, 3, 1−7

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DOI: 10.1021/acsomega.7b01957 ACS Omega 2018, 3, 1−7