computational materials discovery Acta Crystallographica Section C
Structural Chemistry ISSN 2053-2296
Manipulating topological phase transition by strain Junwei Liu,a Yong Xu,a Jian Wu,a Bing-Lin Gu,a S. B. Zhangb and Wenhui Duana* a
Department of Physics and State Key Laboratory of Low-Dimensional Quantum Physics, Tsinghua University, Beijing 100084, People’s Republic of China, and b Department of Physics, Applied Physics, and Astronomy, Rensselaer Polytechnic Institute, Troy, New York 12180, USA Correspondence e-mail: [email protected] Received 19 September 2013 Accepted 21 October 2013
First-principles calculations show that strain-induced topological phase transition is a universal phenomenon in those narrow-gap semiconductors for which the valence band maximum (VBM) and conduction band minimum (CBM) have different parities. The transition originates from the opposite responses of the VBM and CBM, whose magnitudes depend critically on the direction of the applied strain. Our work suggests that strain can play a unique role in tuning the electronic properties of topological insulators for device applications, as well as in the achievement of new topological insulators.
as graphene (Pereira & Castro Neto, 2009), carbon nanotubes (Lu et al., 2003; Minot et al., 2003), GaN (Kisielowski et al., 1996), GaAs (Christensen, 1984) and HgTe (Bru¨ne et al., 2011). The strain can be hydrostatic, uniaxial or biaxial. The nonhomogeneous strain in epitaxial films could be the result of a mismatch with the substrate or merely the result of applying external pressure. The strain anisotropy offers an additional degree of freedom to tune the electronic structure of TIs. Strain may impact the electronic properties of TIs or even induce topological phase transition (Young et al., 2011; Liu et al., 2011; Zhao et al., 2012). The underlying mechanism (especially the interplay between spin–orbit coupling and strain) is, however, still not clear. In this work, we developed a generic road map for the effects of uniaxial strain on the topological and electronic properties of TIs by performing systematic first-principles calculations. It was found that a uniaxial strain can critically impact the topological state to make the transition from a TI to a non-TI or, alternatively, from a TI to a metal, and vice versa. More importantly, such a transition is universal for narrow-gap semiconductors with a valence band maximum (VBM) and conduction band minimum (CBM) of different parities. Furthermore, the strain-induced effect is at a maximum for strain applied along the bonding direction (i.e. the maximum overlap direction of atomic orbitals), while the effect is at a minimum for strain perpendicular to the bonding direction. To demonstrate this, we show that Bi2Se3, Bi2Te3
Keywords: computational materials discovery; strain-induced topological phase transition; valence band maximum; conduction band minimum.
The prediction and discovery of topological insulators (TIs) prepared the way for the study of new fundamental physics in condensed matters, in areas such as topological magnetoelectric effects, axions, Majorana fermions and topological quantum computing (Hasan & Kane, 2010; Qi & Zhang, 2011). The control of TIs, and in particular the creation and annihilation of TI states, offers a rare opportunity both for investigating the critical phenomena associated with the onset of TI transition and for giving a detailed account of their interplay with exciting but little-explored new physical phenomena. In this regard, Bi2Se3, Bi2Te3, Sb2Te3 and Sb2Se3 are of particular interest as they are at the border between topological insulators and normal insulators, viz. the first three have topological surface states, but Sb2Se3 does not (Zhang et al., 2009). There are different approaches for altering the electronic states of TIs, thereby creating or annihilating the TI states (Xu et al., 2011). Among these approaches, applying a strain is, perhaps, the most practical. As a matter of fact, a mechanical strain could significantly alter the electronic properties (e.g. open up a band gap) of semiconductors and semi-metals, such
118
# 2014 International Union of Crystallography
Figure 1 (a) Minimum gap and -point gap versus strain for Bi2Se3. The diagram is divided into three regions (metal, topological insulator and normal insulator) by two critical points where the minimum gap is zero. (b) Band structures of Bi2Se3 under different strains. Negative and positive values represent compressive and tensile strains, respectively. The Fermi level is set to zero.
doi:10.1107/S2053229613032336
Acta Cryst. (2014). C70, 118–122
computational materials discovery
Figure 2 (a) Distances between adjacent atomic layers versus strain for Bi2Se3. Here, d1 and d2 are the distances between layers within a single QL and d5 is the distance between two adjacent QLs, as shown in the inset. (b) The wavefunctions under zero strain for VBM and CBM, respectively, in a (100)-plane view. The VBM is mainly composed of the Se 4p antibonding state with odd parity ( Se), whereas the CBM is mainly composed of the Bi 6p bonding state with even parity ( +Bi ). (c) Se relative to +Bi as a function of strain. (d) The corresponding wavefunctions under 9.2% strain. To show the effect of strain, SOC is not included here.
and Sb2Te3 become non-TIs by applying a tensile strain, whereas Sb2Se3 becomes a TI by applying a compressive strain. The calculations have been performed in the framework of density functional theory (DFT) using the Perdew–Burke– Ernzerhof (PBE) generalized gradient approximation (Perdew et al., 1996) and the projector-augmented wave potential (Blo¨chl, 1994), as implemented in the Vienna ab initio simulation package (Kresse & Furthmu¨ller, 1996). The energy cutoff of the plane-wave basis was 340 eV and the 9 9 9 and 9 9 1 Monkhorst–Pack k points were used for bulk and surface calculations, respectively. Structural relaxations were performed with forces converged to less than ˚ 1. The spin–orbit coupling (SOC) has been 0.01 eV A included (Hobbs et al., 2000). The lattice constants for Bi2Se3, Bi2Te3 and Sb2Te3 were taken from experiments (Wyckoff, 1964), while for Sb2Se3 they were obtained by structural optimization (Zhang et al., 2009), due to the lack of experimental data. The approach adopted here has been widely used in theoretical studies of these materials (Zhang et al., 2009; Yazyev et al., 2010; Jin et al., 2011; Wang et al., 2011) to yield reasonable band gaps, effective masses (Larson et al., 2000) and Seebeck coefficients (Mishra et al., 1997). Acta Cryst. (2014). C70, 118–122
To demonstrate the strain-induced effect on TIs, we first studied bulk Bi2Se3, a model TI system, under different uniaxial strains. Herein, the uniaxial strain " is applied in the [111] direction by changing the lattice parameter from the equilibrium lattice constant c0 to c. The corresponding strain is defined by " = (c c0)/c0. As shown in the phase diagram (Fig. 1a), both the minimum gap (i.e. the energy gap) and the -point gap do not change monotonously with the strain, and there are two critical points (i.e. " = 12.7 and 4.7%), where the minimum band gap is zero. When applying a compressive strain (" < 0), bands broaden and band dispersion changes significantly, accompanying a decreasing minimum band gap (Fig. 1b). Meanwhile, the VBM at ( VBM) and the CBM at ( CBM) move away from each other and the band gap at the point therefore increases. By continuously increasing the compressive strain to a critical value (" = 12.7%), the minimum band gap gradually closes due to overlap of the VBM and CBM, indicating that Bi2Se3 undergoes a TI-tometal transition (Wilson-type transition). The topology of the band structure should be unchanged, since there is no direct gap closure and band inversion. In contrast, applying a tensile strain (" > 0) has a quite different effect on the system. The band gap first decreases to Liu et al.
Manipulating topological phase transition by strain
119
computational materials discovery
Figure 3 -point gap with SOC ( SO) and without SOC ( NSO) as a function of strain for (a) Bi2Se3, (b) Sb2Te3 and (c) Bi2Te3. In contrast to Fig. 1, here we show the -point gap of the topological phase as being negative to facilitate an easy comparison between ‘with SOC’ and ‘without SOC’.
zero and then increases (Fig. 1a). For small tensile strains, the VBM and CBM shift toward each other, which changes the band gap from indirect to direct. Interestingly, when the tensile strain reaches a critical value of " = 4.7%, the energy gap shrinks to zero (see Fig. 1b) and an anisotropic threedimensional Dirac cone appears. Further increasing the tensile strains shifts the VBM and CBM away from each other and enlarges the band gap. More importantly, the ‘band inversion’ at the point disappears, indicating that a TI-tonormal insulator transition occurs. This is further confirmed by calculating the Z2 topological invariant (Fu & Kane, 2007). Our calculations show that the value of Z2 jumps from 1 to 0 at the critical strain of 4.7% because the parity of the VBM states changes from 1 to +1. We also observe that the metallic Dirac fermion surface states vanish when the strain is larger than the critical strain. All these results unambiguously confirm that Bi2Se3 undergoes a strain-induced topological phase transition. In the TI region, the system has a maximum energy gap of 0.33 eV under a strain of 4.0%. Near the critical strain of 4.7%, the band gap at the point exhibits a linear dependence on strain with a slope of 60 meV per 1% strain. It is known that the small gap of TIs hinders the use of the TIs magic surface states. Our results suggest that applying strain is an alternative way of tuning and increasing the band gap of TIs.
It is important to uncover the underlying mechanism responsible for the creation and annihilation of TI states by strain. To focus on the effect of strain, here we omit SOC first. Fig. 2(a) shows that the strain effect is highly anisotropic, with the most significant effect on the inter-QL spacing (i.e. d5), but a very small effect on the intra-QL spacing (i.e. d1 and d2), since the QLs bind to each other via a weak van der Waals (vdW) interaction. Fig. 2(b) shows that the Se 4p antibonding state with odd parity ( Se ), corresponding to the VBM state at zero strain, is mainly located between adjacent QLs, + ) is whereas the Bi 6p bonding state with even parity ( Bi mainly located within a single QL. Therefore, strain has a + significant effect on Se but a much smaller effect on Bi . More importantly, the opposite parity or bonding character + (i.e. bonding and antibonding) of the Bi and Se states make their energies move in opposite directions in response to the applied strain. In particular, a compressive strain will decrease the intralayer distances and hence increase the overlap between the wavefunctions. In a tight binding picture, the antibonding character of the Se state will cause it to move up, + state will cause it to while the bonding character of the Bi move down in energy. Fig. 2(c) shows that this relative movement leads to a band inversion at " 5%. This inversion can also be verified by the change in the wavefunctions of the VBM and CBM states. Figs. 2(b) and 2(d)
Figure 4 Schematic diagrams showing the effect of strain on the topological phase transition. (a) A normal insulator with different parities for the valence band (VB) and conduction band (CB); VB is the antibonding state whereas CB is the bonding state of p orbitals, as indicated by the schematic charge contours. (b) Upon applying a compressive uniaxial strain (" < 0) along the maximum overlap direction of the wavefunctions (i.e. horizontal direction here), VB moves up while CB moves down relative to each other, resulting in closure and inversion of the gap. (c) SOC opens up the gap at the crossing points, making the material a topological insulator. Conversely, a tensile strain (" > 0) will result in a transition from TI to normal insulator.
120
Liu et al.
Manipulating topological phase transition by strain
Acta Cryst. (2014). C70, 118–122
computational materials discovery
Figure 5 (a)/(b) Band structures of Sb2Se3 at different strains. Panel (b), with " = 0.9%, is the critical strain for topological phase transition. (c) The band structure of a six-QL Sb2Se3 slab under a compressive strain of 6.7%, showing a metallic Dirac cone crossing bulk band gap. The shaded areas are the projected bulk band structure. (d) Parities of the wavefunctions at the point for the nth band with " = 0 and 6.7%. The products of the parities are given in red on the right hand side for each row.
show that the VBM at " = 9.2% corresponds to the CBM at " = 0, and vice versa. Conversely, a tensile strain will reduce the overlap between wavefunctions. As a result, the VBM with antibonding character will move down, whereas the CBM with bonding character will move up in energy, causing an increase in the band gap. On the other hand, the strength of the band inversion caused by the SOC is mainly determined by the mass of the atoms, which depends little on the distance between atomic layers. Indeed, Fig. 3 shows that the relative change of the gap (with and without SOC) with respect to the strain is nearly a constant: 0.4, 0.5 and 0.9 eV for Sb2Te3, Bi2Se3 and Bi2Te3, respectively. Therefore, under a tensile strain, when the gap is too large to be inverted by the SOC, the bands restore their non-SOC order. In other words, a phase transition from TI to normal insulator takes place. Evidently, topological phase transition depends critically on the direction of the applied uniaxial strain. The optimal direction should be the one along the bonding direction (i.e. the direction with maximum atomic wavefunction overlap). For Bi2Se3, this should be the [111] direction (as we have used in our calculation). As a comparison, we also applied a uniaxial strain perpendicular to the [111] direction. Here, despite the large variation of the strain from 15 to +15%, we did not observe any sign of topological phase transition. Instead, some other states, which are much more sensitive to strain than the VBM and CBM states, appear in the gap. Consequently, the TI is driven to a metallic phase before the bands restore the non-SOC order. Our extensive calculations establish the basic conditions for a material to exhibit strain-induced transition: (i) the VBM and CBM must have different parities, and (ii) the material must have sufficiently large SOC. Fig. 4 shows schematically the physical picture. Note that for the materials studied here, VBM and CBM of different parities typically correspond to Acta Cryst. (2014). C70, 118–122
antibonding and bonding states. With a compressive strain (" < 0), the energy of the bonding state (i.e. the CBM) decreases while the energy of the antibonding state (i.e. the VBM) increases due to the different signs of the overlap integrals (see Figs. 4a and 4b). The role of the SOC is to cause a strong enough coupling and splitting between the original bonding and antibonding states, resulting in the inversion of bands with different parities (see Fig. 4c). On the other hand, applying a tensile strain (" > 0) will result in a transition from TI to normal insulator state, as shown by our first-principles calculations on Bi2Se3. Moreover, the transition depends mainly on the different parities of the VBM and CBM, but not on the details of the atomic orbitals or bond types (for example, - or -type). Different parities here imply different signs of overlap integrals of the atomic orbitals, resulting in opposite movements of the VBM and CBM under strain. This suggests that straininduced topological phase transition is universal for narrowgap semiconductors with VBM and CBM of different parities. Bi2Te3 and Sb2Te3 are just two examples reflecting the generality of the mechanism. For Bi2Te3, the two critical strains are found to be 7 and 9% for metal–insulator and TI–normal insulator transitions, respectively. For Sb2Te3, the values are 6 and 6%, respectively. The universal strain mechanism provides a guide for tuning the properties of TI states. It also offers a means of searching for new TIs with superior properties. As an example, we calculated the electronic properties of rhombohedral Sb2Se3 under uniaxial strain along the [111] direction (i.e. the bonding direction of VBM and CBM), as shown in Fig. 5. It can be seen that unstrained Sb2Se3 is a normal insulator with normal band order (Fig. 5a), as confirmed by the calculated parity (Fig. 5d). When a compressive strain is applied, the energy gap first reduces to zero at " = 0.9% (Fig. 5b) and then increases [see the shadow regions in Fig. 5(c) for the projected bulk Liu et al.
Manipulating topological phase transition by strain
121
computational materials discovery band structure]. A band inversion at the point appears accompanied by a parity change. The Dirac–fermion metallic surface states embedded in the bulk band gap (Fig. 5c) also indicate that Sb2Se3 has transformed from a normal insulator to a TI. The small critical strain (" = 0.9%) suggests that the TI transition in Sb2Se3 may be easily achieved experimentally, which makes it an ideal system for the study of the novel three-dimensional Dirac cone and related critical phenomena. By applying uniaxial strain along the proper direction, TIs may also form in narrow-gap semiconductors. This would include PbX (X = S, Se, Te), XTe (X = Sn, Ge), transition metal oxides like Na2IrO3 or ternary Heusler compounds, for which the VBM and CBM have different parities and the SOC is sufficiently large. Here, strain should play two roles: (i) inducing band inversion, and (ii) breaking the symmetry of the originally equivalent TRIM (time-reversal invariant momenta) points. A recent study observed a strain-induced topological phase transition in a ternary half-Heusler compound, LaPtBi (Zhang et al., 2011). So far, we have only focused on those systems with space inversion symmetry. In fact, even for those systems without space inversion symmetry, our conclusion still holds. Due to the absence of space inversion symmetry, we cannot connect the bonding types to the parity, while under strain the responses of states of different bonding types are still different. That is to say, strain can still induce band inversion between bonding and antibonding states (VBM and CBM) at TRIM. As is well known, if the gap can close and reopen by the tuning of a single parameter, there will be a Z2 topological transition (Murakami, 2007; Murakami et al., 2007). Therefore, strain can also induce topological phase transition for those systems without space inversion symmetry. In addition, our conclusion can also be applied to the newly discovered threeand two-dimensional topological crystalline insulators (Fu, 2011; Hsieh et al., 2012; Liu et al., 2014b), where the topological protected boundary states are very different from TIs, protected by mirror symmetry but not time-reversal symmetry (Liu et al., 2014a). In conclusion, systematic first-principles calculations reveal that TI states can be controlled by applying a uniaxial strain. The calculated strain phase diagram and wavefunction analysis for Bi2Se3, Bi2Te3, Sb2Te3 and Sb2Se3 show that strain can induce reversible metal-to-TI and TI-to-normal insulator transitions for existing TIs and narrow-gap semiconductors with VBM and CBM of different parities and having large enough SOC. The topological phase transition is unambiguously confirmed by examining the changes in the band structure (such as band inversion and gap closing), by the application of the parity criterion and by the observation of the creation of surface states. It is found that, due to the different parities of the VBM and CBM, the two states move in opposite directions in energy under uniaxial strain. This, combined with SOC, results in phase transitions. The direction
122
Liu et al.
Manipulating topological phase transition by strain
of the applied strain is also crucial for driving the topological phase transitions. Evidently, the optimal direction of the applied strain should be along the bonding direction. This work was supported by the Ministry of Science and Technology of China (grant Nos. 2011CB921901 and 2011CB606405) and the National Natural Science Foundation of China. SBZ was supporded by the US Defense Advanced Research Project Agency (DARPA) (award No. N66001-12-14034).
References Blo¨chl, P. E. (1994). Phys. Rev. B, 50, 17953–17979. Bru¨ne, C., Liu, C. X., Novik, E. G., Hankiewicz, E. M., Buhmann, H., Chen, Y. L., Qi, X. L., Shen, Z. X., Zhang, S. C. & Molenkamp, L. W. (2011). Phys. Rev. Lett. 106, 126803. Christensen, N. E. (1984). Phys. Rev. B, 30, 5753–5765. Fu, L. (2011). Phys. Rev. Lett. 106, 106802. Fu, L. & Kane, C. L. (2007). Phys. Rev. B, 76, 045302. Hasan, M. & Kane, C. (2010). Rev. Mod. Phys. 82, 3045–3067. Hobbs, D., Kresse, G. & Hafner, J. (2000). Phys. Rev. B, 62, 11556–11570 Hsieh, T. H., Lin, H., Liu, J., Duan, W., Bansil, A. & Fu, L. (2012). Nature Commun. 3, 982. Jin, H., Song, J.-H., Freeman, A. J. & Kanatzidis, M. G. (2011). Phys. Rev. B, 83, 041202. Kisielowski, C., Kru¨ger, J., Ruvimov, S., Suski, T., Ager, J. W., Jones, E., Liliental-Weber, Z., Rubin, M., Weber, E. R., Bremser, M. D. & Davis, R. F. (1996). Phys. Rev. B, 54, 17745–17753. Kresse, G. & Furthmu¨ller, J. (1996). Phys. Rev. B, 54, 11169–11186. Larson, P., Mahanti, S. D. & Kanatzidis, M. G. (2000). Phys. Rev. B, 61, 8162– 8171. Liu, J., Duan, W. & Fu, L. (2014a). Phy. Rev. B, 88, 241303(R). Liu, J., Hsieh, T. H., Wei, P., Duan, W., Moodera, J. & Fu, L. (2014b). Nature Mat. In the press. doi:10.1038/nmat3828. Liu, W., Peng, X., Tang, C., Sun, L., Zhang, K. & Zhong, J. (2011). Phys. Rev. B, 84, 245105. Lu, J.-Q., Wu, J., Duan, W., Liu, F., Zhu, B.-F. & Gu, B.-L. (2003). Phys. Rev. Lett. 90, 156601. Minot, E. D., Yaish, Y., Sazonova, V., Park, J.-Y., Brink, M. & McEuen, P. L. (2003). Phys. Rev. Lett. 90, 156401. Mishra, S. K., Satpathy, S. & Jepsen, O. (1997). J. Phys. Condens. Matter, 9, 461–470. Murakami, S. (2007). New J. Phys. 9, 356. Murakami, S., Iso, S., Avishai, Y., Onoda, M. & Nagaosa, N. (2007). Phys. Rev. B, 76, 205304. Perdew, J. P., Burke, K. & Ernzerhof, M. (1996). Phys. Rev. Lett. 77, 3865– 3868. Pereira, V. M. & Castro Neto, A. H. (2009). Phys. Rev. Lett. 103, 046801. Qi, X.-L. & Zhang, S.-C. (2011). Rev. Mod. Phys. 83, 1057–1110. Wang, Y.-L., Xu, Y., Jiang, Y.-P., Liu, J.-W., Chang, C.-Z., Chen, M., Li, Z., Song, C.-L., Wang, L.-L., He, K., Chen, X., Duan, W.-H., Xue, Q.-K. & Ma, X.-C. (2011). Phys. Rev. B, 84, 075335. Wyckoff, R. W. G. (1964). In Crystal Structures, Vol. 2. New York: John Wiley and Sons. Xu, S.-Y., Xia, Y., Wray, L. A., Jia, S., Meier, F., Dil, J. H., Osterwalder, J., Slomski, B., Bansil, A., Lin, H., Cava, R. J. & Hasan, M. Z. (2011). Science, 332, 560–564. Yazyev, O. V., Moore, J. E. & Louie, S. G. (2010). Phys. Rev. Lett. 105, 266806. Young, S. M., Chowdhury, S., Walter, E. J., Mele, E. J., Kane, C. L. & Rappe, A. M. (2011). Phys. Rev. B, 84, 085106. Zhang, H., Liu, C.-X., Qi, X.-L., Dai, X., Fang, Z. & Zhang, S.-C. (2009). Nature Phys. 5, 438–442. Zhang, X. M., Wang, W. H., Liu, E. K., Liu, G. D., Liu, Z. Y. & Wu, G. H. (2011). Appl. Phys. Lett. 99, 071901. Zhao, L., Liu, J., Tang, P. & Duan, W. (2012). Appl. Phys. Lett. 100, 131602.
Acta Cryst. (2014). C70, 118–122
supplementary materials
supplementary materials
sup-1
Structural Chemistry ISSN 2053-2296
Manipulating topological phase transition by strain Junwei Liu,a Yong Xu,a Jian Wu,a Bing-Lin Gu,a S. B. Zhangb and Wenhui Duana* a
Department of Physics and State Key Laboratory of Low-Dimensional Quantum Physics, Tsinghua University, Beijing 100084, People’s Republic of China, and b Department of Physics, Applied Physics, and Astronomy, Rensselaer Polytechnic Institute, Troy, New York 12180, USA Correspondence e-mail: [email protected] Received 19 September 2013 Accepted 21 October 2013
First-principles calculations show that strain-induced topological phase transition is a universal phenomenon in those narrow-gap semiconductors for which the valence band maximum (VBM) and conduction band minimum (CBM) have different parities. The transition originates from the opposite responses of the VBM and CBM, whose magnitudes depend critically on the direction of the applied strain. Our work suggests that strain can play a unique role in tuning the electronic properties of topological insulators for device applications, as well as in the achievement of new topological insulators.
as graphene (Pereira & Castro Neto, 2009), carbon nanotubes (Lu et al., 2003; Minot et al., 2003), GaN (Kisielowski et al., 1996), GaAs (Christensen, 1984) and HgTe (Bru¨ne et al., 2011). The strain can be hydrostatic, uniaxial or biaxial. The nonhomogeneous strain in epitaxial films could be the result of a mismatch with the substrate or merely the result of applying external pressure. The strain anisotropy offers an additional degree of freedom to tune the electronic structure of TIs. Strain may impact the electronic properties of TIs or even induce topological phase transition (Young et al., 2011; Liu et al., 2011; Zhao et al., 2012). The underlying mechanism (especially the interplay between spin–orbit coupling and strain) is, however, still not clear. In this work, we developed a generic road map for the effects of uniaxial strain on the topological and electronic properties of TIs by performing systematic first-principles calculations. It was found that a uniaxial strain can critically impact the topological state to make the transition from a TI to a non-TI or, alternatively, from a TI to a metal, and vice versa. More importantly, such a transition is universal for narrow-gap semiconductors with a valence band maximum (VBM) and conduction band minimum (CBM) of different parities. Furthermore, the strain-induced effect is at a maximum for strain applied along the bonding direction (i.e. the maximum overlap direction of atomic orbitals), while the effect is at a minimum for strain perpendicular to the bonding direction. To demonstrate this, we show that Bi2Se3, Bi2Te3
Keywords: computational materials discovery; strain-induced topological phase transition; valence band maximum; conduction band minimum.
The prediction and discovery of topological insulators (TIs) prepared the way for the study of new fundamental physics in condensed matters, in areas such as topological magnetoelectric effects, axions, Majorana fermions and topological quantum computing (Hasan & Kane, 2010; Qi & Zhang, 2011). The control of TIs, and in particular the creation and annihilation of TI states, offers a rare opportunity both for investigating the critical phenomena associated with the onset of TI transition and for giving a detailed account of their interplay with exciting but little-explored new physical phenomena. In this regard, Bi2Se3, Bi2Te3, Sb2Te3 and Sb2Se3 are of particular interest as they are at the border between topological insulators and normal insulators, viz. the first three have topological surface states, but Sb2Se3 does not (Zhang et al., 2009). There are different approaches for altering the electronic states of TIs, thereby creating or annihilating the TI states (Xu et al., 2011). Among these approaches, applying a strain is, perhaps, the most practical. As a matter of fact, a mechanical strain could significantly alter the electronic properties (e.g. open up a band gap) of semiconductors and semi-metals, such
118
# 2014 International Union of Crystallography
Figure 1 (a) Minimum gap and -point gap versus strain for Bi2Se3. The diagram is divided into three regions (metal, topological insulator and normal insulator) by two critical points where the minimum gap is zero. (b) Band structures of Bi2Se3 under different strains. Negative and positive values represent compressive and tensile strains, respectively. The Fermi level is set to zero.
doi:10.1107/S2053229613032336
Acta Cryst. (2014). C70, 118–122
computational materials discovery
Figure 2 (a) Distances between adjacent atomic layers versus strain for Bi2Se3. Here, d1 and d2 are the distances between layers within a single QL and d5 is the distance between two adjacent QLs, as shown in the inset. (b) The wavefunctions under zero strain for VBM and CBM, respectively, in a (100)-plane view. The VBM is mainly composed of the Se 4p antibonding state with odd parity ( Se), whereas the CBM is mainly composed of the Bi 6p bonding state with even parity ( +Bi ). (c) Se relative to +Bi as a function of strain. (d) The corresponding wavefunctions under 9.2% strain. To show the effect of strain, SOC is not included here.
and Sb2Te3 become non-TIs by applying a tensile strain, whereas Sb2Se3 becomes a TI by applying a compressive strain. The calculations have been performed in the framework of density functional theory (DFT) using the Perdew–Burke– Ernzerhof (PBE) generalized gradient approximation (Perdew et al., 1996) and the projector-augmented wave potential (Blo¨chl, 1994), as implemented in the Vienna ab initio simulation package (Kresse & Furthmu¨ller, 1996). The energy cutoff of the plane-wave basis was 340 eV and the 9 9 9 and 9 9 1 Monkhorst–Pack k points were used for bulk and surface calculations, respectively. Structural relaxations were performed with forces converged to less than ˚ 1. The spin–orbit coupling (SOC) has been 0.01 eV A included (Hobbs et al., 2000). The lattice constants for Bi2Se3, Bi2Te3 and Sb2Te3 were taken from experiments (Wyckoff, 1964), while for Sb2Se3 they were obtained by structural optimization (Zhang et al., 2009), due to the lack of experimental data. The approach adopted here has been widely used in theoretical studies of these materials (Zhang et al., 2009; Yazyev et al., 2010; Jin et al., 2011; Wang et al., 2011) to yield reasonable band gaps, effective masses (Larson et al., 2000) and Seebeck coefficients (Mishra et al., 1997). Acta Cryst. (2014). C70, 118–122
To demonstrate the strain-induced effect on TIs, we first studied bulk Bi2Se3, a model TI system, under different uniaxial strains. Herein, the uniaxial strain " is applied in the [111] direction by changing the lattice parameter from the equilibrium lattice constant c0 to c. The corresponding strain is defined by " = (c c0)/c0. As shown in the phase diagram (Fig. 1a), both the minimum gap (i.e. the energy gap) and the -point gap do not change monotonously with the strain, and there are two critical points (i.e. " = 12.7 and 4.7%), where the minimum band gap is zero. When applying a compressive strain (" < 0), bands broaden and band dispersion changes significantly, accompanying a decreasing minimum band gap (Fig. 1b). Meanwhile, the VBM at ( VBM) and the CBM at ( CBM) move away from each other and the band gap at the point therefore increases. By continuously increasing the compressive strain to a critical value (" = 12.7%), the minimum band gap gradually closes due to overlap of the VBM and CBM, indicating that Bi2Se3 undergoes a TI-tometal transition (Wilson-type transition). The topology of the band structure should be unchanged, since there is no direct gap closure and band inversion. In contrast, applying a tensile strain (" > 0) has a quite different effect on the system. The band gap first decreases to Liu et al.
Manipulating topological phase transition by strain
119
computational materials discovery
Figure 3 -point gap with SOC ( SO) and without SOC ( NSO) as a function of strain for (a) Bi2Se3, (b) Sb2Te3 and (c) Bi2Te3. In contrast to Fig. 1, here we show the -point gap of the topological phase as being negative to facilitate an easy comparison between ‘with SOC’ and ‘without SOC’.
zero and then increases (Fig. 1a). For small tensile strains, the VBM and CBM shift toward each other, which changes the band gap from indirect to direct. Interestingly, when the tensile strain reaches a critical value of " = 4.7%, the energy gap shrinks to zero (see Fig. 1b) and an anisotropic threedimensional Dirac cone appears. Further increasing the tensile strains shifts the VBM and CBM away from each other and enlarges the band gap. More importantly, the ‘band inversion’ at the point disappears, indicating that a TI-tonormal insulator transition occurs. This is further confirmed by calculating the Z2 topological invariant (Fu & Kane, 2007). Our calculations show that the value of Z2 jumps from 1 to 0 at the critical strain of 4.7% because the parity of the VBM states changes from 1 to +1. We also observe that the metallic Dirac fermion surface states vanish when the strain is larger than the critical strain. All these results unambiguously confirm that Bi2Se3 undergoes a strain-induced topological phase transition. In the TI region, the system has a maximum energy gap of 0.33 eV under a strain of 4.0%. Near the critical strain of 4.7%, the band gap at the point exhibits a linear dependence on strain with a slope of 60 meV per 1% strain. It is known that the small gap of TIs hinders the use of the TIs magic surface states. Our results suggest that applying strain is an alternative way of tuning and increasing the band gap of TIs.
It is important to uncover the underlying mechanism responsible for the creation and annihilation of TI states by strain. To focus on the effect of strain, here we omit SOC first. Fig. 2(a) shows that the strain effect is highly anisotropic, with the most significant effect on the inter-QL spacing (i.e. d5), but a very small effect on the intra-QL spacing (i.e. d1 and d2), since the QLs bind to each other via a weak van der Waals (vdW) interaction. Fig. 2(b) shows that the Se 4p antibonding state with odd parity ( Se ), corresponding to the VBM state at zero strain, is mainly located between adjacent QLs, + ) is whereas the Bi 6p bonding state with even parity ( Bi mainly located within a single QL. Therefore, strain has a + significant effect on Se but a much smaller effect on Bi . More importantly, the opposite parity or bonding character + (i.e. bonding and antibonding) of the Bi and Se states make their energies move in opposite directions in response to the applied strain. In particular, a compressive strain will decrease the intralayer distances and hence increase the overlap between the wavefunctions. In a tight binding picture, the antibonding character of the Se state will cause it to move up, + state will cause it to while the bonding character of the Bi move down in energy. Fig. 2(c) shows that this relative movement leads to a band inversion at " 5%. This inversion can also be verified by the change in the wavefunctions of the VBM and CBM states. Figs. 2(b) and 2(d)
Figure 4 Schematic diagrams showing the effect of strain on the topological phase transition. (a) A normal insulator with different parities for the valence band (VB) and conduction band (CB); VB is the antibonding state whereas CB is the bonding state of p orbitals, as indicated by the schematic charge contours. (b) Upon applying a compressive uniaxial strain (" < 0) along the maximum overlap direction of the wavefunctions (i.e. horizontal direction here), VB moves up while CB moves down relative to each other, resulting in closure and inversion of the gap. (c) SOC opens up the gap at the crossing points, making the material a topological insulator. Conversely, a tensile strain (" > 0) will result in a transition from TI to normal insulator.
120
Liu et al.
Manipulating topological phase transition by strain
Acta Cryst. (2014). C70, 118–122
computational materials discovery
Figure 5 (a)/(b) Band structures of Sb2Se3 at different strains. Panel (b), with " = 0.9%, is the critical strain for topological phase transition. (c) The band structure of a six-QL Sb2Se3 slab under a compressive strain of 6.7%, showing a metallic Dirac cone crossing bulk band gap. The shaded areas are the projected bulk band structure. (d) Parities of the wavefunctions at the point for the nth band with " = 0 and 6.7%. The products of the parities are given in red on the right hand side for each row.
show that the VBM at " = 9.2% corresponds to the CBM at " = 0, and vice versa. Conversely, a tensile strain will reduce the overlap between wavefunctions. As a result, the VBM with antibonding character will move down, whereas the CBM with bonding character will move up in energy, causing an increase in the band gap. On the other hand, the strength of the band inversion caused by the SOC is mainly determined by the mass of the atoms, which depends little on the distance between atomic layers. Indeed, Fig. 3 shows that the relative change of the gap (with and without SOC) with respect to the strain is nearly a constant: 0.4, 0.5 and 0.9 eV for Sb2Te3, Bi2Se3 and Bi2Te3, respectively. Therefore, under a tensile strain, when the gap is too large to be inverted by the SOC, the bands restore their non-SOC order. In other words, a phase transition from TI to normal insulator takes place. Evidently, topological phase transition depends critically on the direction of the applied uniaxial strain. The optimal direction should be the one along the bonding direction (i.e. the direction with maximum atomic wavefunction overlap). For Bi2Se3, this should be the [111] direction (as we have used in our calculation). As a comparison, we also applied a uniaxial strain perpendicular to the [111] direction. Here, despite the large variation of the strain from 15 to +15%, we did not observe any sign of topological phase transition. Instead, some other states, which are much more sensitive to strain than the VBM and CBM states, appear in the gap. Consequently, the TI is driven to a metallic phase before the bands restore the non-SOC order. Our extensive calculations establish the basic conditions for a material to exhibit strain-induced transition: (i) the VBM and CBM must have different parities, and (ii) the material must have sufficiently large SOC. Fig. 4 shows schematically the physical picture. Note that for the materials studied here, VBM and CBM of different parities typically correspond to Acta Cryst. (2014). C70, 118–122
antibonding and bonding states. With a compressive strain (" < 0), the energy of the bonding state (i.e. the CBM) decreases while the energy of the antibonding state (i.e. the VBM) increases due to the different signs of the overlap integrals (see Figs. 4a and 4b). The role of the SOC is to cause a strong enough coupling and splitting between the original bonding and antibonding states, resulting in the inversion of bands with different parities (see Fig. 4c). On the other hand, applying a tensile strain (" > 0) will result in a transition from TI to normal insulator state, as shown by our first-principles calculations on Bi2Se3. Moreover, the transition depends mainly on the different parities of the VBM and CBM, but not on the details of the atomic orbitals or bond types (for example, - or -type). Different parities here imply different signs of overlap integrals of the atomic orbitals, resulting in opposite movements of the VBM and CBM under strain. This suggests that straininduced topological phase transition is universal for narrowgap semiconductors with VBM and CBM of different parities. Bi2Te3 and Sb2Te3 are just two examples reflecting the generality of the mechanism. For Bi2Te3, the two critical strains are found to be 7 and 9% for metal–insulator and TI–normal insulator transitions, respectively. For Sb2Te3, the values are 6 and 6%, respectively. The universal strain mechanism provides a guide for tuning the properties of TI states. It also offers a means of searching for new TIs with superior properties. As an example, we calculated the electronic properties of rhombohedral Sb2Se3 under uniaxial strain along the [111] direction (i.e. the bonding direction of VBM and CBM), as shown in Fig. 5. It can be seen that unstrained Sb2Se3 is a normal insulator with normal band order (Fig. 5a), as confirmed by the calculated parity (Fig. 5d). When a compressive strain is applied, the energy gap first reduces to zero at " = 0.9% (Fig. 5b) and then increases [see the shadow regions in Fig. 5(c) for the projected bulk Liu et al.
Manipulating topological phase transition by strain
121
computational materials discovery band structure]. A band inversion at the point appears accompanied by a parity change. The Dirac–fermion metallic surface states embedded in the bulk band gap (Fig. 5c) also indicate that Sb2Se3 has transformed from a normal insulator to a TI. The small critical strain (" = 0.9%) suggests that the TI transition in Sb2Se3 may be easily achieved experimentally, which makes it an ideal system for the study of the novel three-dimensional Dirac cone and related critical phenomena. By applying uniaxial strain along the proper direction, TIs may also form in narrow-gap semiconductors. This would include PbX (X = S, Se, Te), XTe (X = Sn, Ge), transition metal oxides like Na2IrO3 or ternary Heusler compounds, for which the VBM and CBM have different parities and the SOC is sufficiently large. Here, strain should play two roles: (i) inducing band inversion, and (ii) breaking the symmetry of the originally equivalent TRIM (time-reversal invariant momenta) points. A recent study observed a strain-induced topological phase transition in a ternary half-Heusler compound, LaPtBi (Zhang et al., 2011). So far, we have only focused on those systems with space inversion symmetry. In fact, even for those systems without space inversion symmetry, our conclusion still holds. Due to the absence of space inversion symmetry, we cannot connect the bonding types to the parity, while under strain the responses of states of different bonding types are still different. That is to say, strain can still induce band inversion between bonding and antibonding states (VBM and CBM) at TRIM. As is well known, if the gap can close and reopen by the tuning of a single parameter, there will be a Z2 topological transition (Murakami, 2007; Murakami et al., 2007). Therefore, strain can also induce topological phase transition for those systems without space inversion symmetry. In addition, our conclusion can also be applied to the newly discovered threeand two-dimensional topological crystalline insulators (Fu, 2011; Hsieh et al., 2012; Liu et al., 2014b), where the topological protected boundary states are very different from TIs, protected by mirror symmetry but not time-reversal symmetry (Liu et al., 2014a). In conclusion, systematic first-principles calculations reveal that TI states can be controlled by applying a uniaxial strain. The calculated strain phase diagram and wavefunction analysis for Bi2Se3, Bi2Te3, Sb2Te3 and Sb2Se3 show that strain can induce reversible metal-to-TI and TI-to-normal insulator transitions for existing TIs and narrow-gap semiconductors with VBM and CBM of different parities and having large enough SOC. The topological phase transition is unambiguously confirmed by examining the changes in the band structure (such as band inversion and gap closing), by the application of the parity criterion and by the observation of the creation of surface states. It is found that, due to the different parities of the VBM and CBM, the two states move in opposite directions in energy under uniaxial strain. This, combined with SOC, results in phase transitions. The direction
122
Liu et al.
Manipulating topological phase transition by strain
of the applied strain is also crucial for driving the topological phase transitions. Evidently, the optimal direction of the applied strain should be along the bonding direction. This work was supported by the Ministry of Science and Technology of China (grant Nos. 2011CB921901 and 2011CB606405) and the National Natural Science Foundation of China. SBZ was supporded by the US Defense Advanced Research Project Agency (DARPA) (award No. N66001-12-14034).
References Blo¨chl, P. E. (1994). Phys. Rev. B, 50, 17953–17979. Bru¨ne, C., Liu, C. X., Novik, E. G., Hankiewicz, E. M., Buhmann, H., Chen, Y. L., Qi, X. L., Shen, Z. X., Zhang, S. C. & Molenkamp, L. W. (2011). Phys. Rev. Lett. 106, 126803. Christensen, N. E. (1984). Phys. Rev. B, 30, 5753–5765. Fu, L. (2011). Phys. Rev. Lett. 106, 106802. Fu, L. & Kane, C. L. (2007). Phys. Rev. B, 76, 045302. Hasan, M. & Kane, C. (2010). Rev. Mod. Phys. 82, 3045–3067. Hobbs, D., Kresse, G. & Hafner, J. (2000). Phys. Rev. B, 62, 11556–11570 Hsieh, T. H., Lin, H., Liu, J., Duan, W., Bansil, A. & Fu, L. (2012). Nature Commun. 3, 982. Jin, H., Song, J.-H., Freeman, A. J. & Kanatzidis, M. G. (2011). Phys. Rev. B, 83, 041202. Kisielowski, C., Kru¨ger, J., Ruvimov, S., Suski, T., Ager, J. W., Jones, E., Liliental-Weber, Z., Rubin, M., Weber, E. R., Bremser, M. D. & Davis, R. F. (1996). Phys. Rev. B, 54, 17745–17753. Kresse, G. & Furthmu¨ller, J. (1996). Phys. Rev. B, 54, 11169–11186. Larson, P., Mahanti, S. D. & Kanatzidis, M. G. (2000). Phys. Rev. B, 61, 8162– 8171. Liu, J., Duan, W. & Fu, L. (2014a). Phy. Rev. B, 88, 241303(R). Liu, J., Hsieh, T. H., Wei, P., Duan, W., Moodera, J. & Fu, L. (2014b). Nature Mat. In the press. doi:10.1038/nmat3828. Liu, W., Peng, X., Tang, C., Sun, L., Zhang, K. & Zhong, J. (2011). Phys. Rev. B, 84, 245105. Lu, J.-Q., Wu, J., Duan, W., Liu, F., Zhu, B.-F. & Gu, B.-L. (2003). Phys. Rev. Lett. 90, 156601. Minot, E. D., Yaish, Y., Sazonova, V., Park, J.-Y., Brink, M. & McEuen, P. L. (2003). Phys. Rev. Lett. 90, 156401. Mishra, S. K., Satpathy, S. & Jepsen, O. (1997). J. Phys. Condens. Matter, 9, 461–470. Murakami, S. (2007). New J. Phys. 9, 356. Murakami, S., Iso, S., Avishai, Y., Onoda, M. & Nagaosa, N. (2007). Phys. Rev. B, 76, 205304. Perdew, J. P., Burke, K. & Ernzerhof, M. (1996). Phys. Rev. Lett. 77, 3865– 3868. Pereira, V. M. & Castro Neto, A. H. (2009). Phys. Rev. Lett. 103, 046801. Qi, X.-L. & Zhang, S.-C. (2011). Rev. Mod. Phys. 83, 1057–1110. Wang, Y.-L., Xu, Y., Jiang, Y.-P., Liu, J.-W., Chang, C.-Z., Chen, M., Li, Z., Song, C.-L., Wang, L.-L., He, K., Chen, X., Duan, W.-H., Xue, Q.-K. & Ma, X.-C. (2011). Phys. Rev. B, 84, 075335. Wyckoff, R. W. G. (1964). In Crystal Structures, Vol. 2. New York: John Wiley and Sons. Xu, S.-Y., Xia, Y., Wray, L. A., Jia, S., Meier, F., Dil, J. H., Osterwalder, J., Slomski, B., Bansil, A., Lin, H., Cava, R. J. & Hasan, M. Z. (2011). Science, 332, 560–564. Yazyev, O. V., Moore, J. E. & Louie, S. G. (2010). Phys. Rev. Lett. 105, 266806. Young, S. M., Chowdhury, S., Walter, E. J., Mele, E. J., Kane, C. L. & Rappe, A. M. (2011). Phys. Rev. B, 84, 085106. Zhang, H., Liu, C.-X., Qi, X.-L., Dai, X., Fang, Z. & Zhang, S.-C. (2009). Nature Phys. 5, 438–442. Zhang, X. M., Wang, W. H., Liu, E. K., Liu, G. D., Liu, Z. Y. & Wu, G. H. (2011). Appl. Phys. Lett. 99, 071901. Zhao, L., Liu, J., Tang, P. & Duan, W. (2012). Appl. Phys. Lett. 100, 131602.
Acta Cryst. (2014). C70, 118–122
supplementary materials
supplementary materials
sup-1
