Journal of Physics D: Applied Physics Related content PAPER Topological phase transition in layered XIn2P2 (X = Ca, Sr) - Prediction of topological insulators in supercubane-like materials based on firstprinciples calculations Guo-Xiang Wang, Shuai Dong and JingMin Hou To cite this article: Zhenwei Wang et al 2017 J. Phys. D: Appl. Phys. 50 465304 - Topological semimetals predicted from first-principles calculations Hongming Weng, Xi Dai and Zhong Fang - Topological nodal line semimetals* Chen Fang, Hongming Weng, Xi Dai et al. View the article online for updates and enhancements. This content was downloaded from IP address 129.59.95.115 on 29/10/2017 at 02:26 Journal of Physics D: Applied Physics J. Phys. D: Appl. Phys. 50 (2017) 465304 (5pp) https://doi.org/10.1088/1361-6463/aa8eaf Topological phase transition in layered XIn2P2 (X = Ca, Sr) Zhenwei Wang, Guangtao Wang , Xianbiao Shi, Dongyang Wang and Xin Tian College of Physics and Electronic Engineering, Henan Normal University, Xinxiang, Henan 453007, People’s Republic of China E-mail: wangtao@htu.cn Received 18 July 2017, revised 8 September 2017 Accepted for publication 25 September 2017 Published 26 October 2017 Abstract Based on fully relativistic first-principles calculations, we studied the topological properties of layered XIn2P2 (X = Ca, Sr). Band inversion can be induced by strain without SOC, forming one nodal ring in the kz = 0 plane, which is protected by the coexistence of time reversal and glide mirror symmetries. Including SOC, a substantial band gap is opened along the nodal line and the line-node semimetal would evolve into a topological insulator. These results reveal a category of materials showing quantum phase transition from trivial semiconductors and topologically nontrivial insulators by tuneable elastic strain engineering. Our investigations provide a new perspective about the formation of topological line-node semimetal under stain. Keywords: first-principles calculations, topological insulator, line-node semimetal (Some figures may appear in colour only in the online journal) 1. Introduction band structure has been measured by the angle-resolved photoelectron spectroscopy (ARPES) in PbTaSe2 [24] and ZrSiS [25]. Interestingly, in addition to these materials, researchers also found node line structures in the photonics crystals [26] and spin liquids [27]. Their intriguing properties characterizing topological nodal line semimetals include drumhead like nearly flat surface states [23], unique Landau energy levels [22], long-range Coulomb interactions [28], which open an important route to achieve high-temperature superconductivity [29–31]. The topological line-node semimetal could be driven into 3D TI or Dirac semimetal by the SOC and strain. For example, CaTe [32] is a TSM, possessing the nodal rings without SOC, exhibiting Dirac semimetal behavior when SOC is included. When it was applied by appropriate strain, it becomes a strong topological insulator [32, 33]. However, the CaAgX (X = P, As) can be driven into a Tl phase from line-node semimetal by taking into account of SOC [34]. Bulk CaIn2P2 and SrIn2P2 are layered semiconductors with indirect-gap and direct-gap respectively. They have been investigated with their electronic and optical properties [35]. Up to now, their topological properties have not been studied. In this study, we find that the indirect-gap semiconductor can be changed into ‘direct’-gap Topological insulator (TI) [1, 2] is a new kind of material possessing gapped bulk states and exotic metallic surface states. The robust surface state in a 3D system, sheltered against backscattering from nonmagnetic impurities as long as the bulk gap remains open and the time-reversal symmetry is preserved [2–8]. The unique feature of the surface state has attracted enormous attention in theoretical calculations and experimental observations [3, 8], not only because of their theoretical importance but also because of their great potential applications [9–11]. Recently, the topological properties have been extended into a variety of three dimensional (3D) topological semimetal (TSM) systems [12–16]. In most Weyl/Dirac semimetals, conduction bands overlap with valence bands at certain momentum points. For instance, the topological properties of Na3Bi [17, 18] and Cd2As3 [19, 20] semimetals have been experimentally confirmed. However, in line-node semimetals, the band crossing points around the Fermi level (EF) form a closed loop. Many systems have been proposed as line-node semimetals including Mackay– Terrones crystal (MTC) [21], Bernal stacked graphene bilayer [22], antiperovskite Cu3PdN [23] and so on. Such line-node 1361-6463/17/465304+5$33.00 1 © 2017 IOP Publishing Ltd Printed in the UK Z Wang et al J. Phys. D: Appl. Phys. 50 (2017) 465304 Figure 1. (a) The crystal structure of XIn2P2 with P63/mmc symmetry. (b) Brillouin zone of bulk and the projected surface Brillouin zones of (0 0 1) plane, as well as high-symmetry points. There is a line-node ring that lies in the kz = 0 plane. Figure 2. Electronic band structures of unstrained CaIn2P2 with SOC. (b) Electronic band structures of band-inverted CaIn2P2 (a/a0 = 1.05 ) with SOC. The details of band-inverted CaIn2P2 around EF are shown in the inset. Firstly, we perform density functional calculations by using the WIEN2K package [37], with the modified Becke–Johnson exchange potential (mBJ) to get the accurate band gap [38]. The valence configurations of the Ca, Sr, P and In atoms are as Ca 3s2 3p6 4s2, Sr 4s2 4p6 5s2, P 3s2 3p3 and In 4d10 5s2 5p1, respectively. The plane-wave cutoff parameter RMT kmax is set to 7 and a 12 × 12 × 3 K-mesh is used for the BZ integral. The SOC interaction is included by using the second-order variational procedure. The tight binding model based on maximally localized Wannier functions (MLWF) method [39, 40] has been constructed in order to investigate the surface states. The surface Green’s function of the semi-infinite system, whose imaginary part is the local density of states to obtain the dispersion of the surface states, can be calculated through an iterative method [41–43]. To confirm the compounds are dynamicly stable, the phonon spectrum was calculated by using the PHONON code [44]. by applying uniaxial strain slightly. Further increasing strain, the compound undergoes a transition from trivial insulator to a line-node semimetal with a nodal ring in the kz = 0 plane. When SOC is taken into account, the compounds become a nontrivial topological insulator with Z2 index (1000). Our results provide a new perspective to understand the effect of strain and SOC on the formation of topological line-node semimetal and insulator. 2. Crystal structure and method The ternary compound CaIn2P2 is isostructural with SrIn2P2 and crystallizes in a hexagonal structure with the space group P63/mmc [36]. The alkaline earth cations X (Ca or Sr) are located at a site with 3 m symmetry; In and P are located at sites with 3 m symmetry. The lattice constants of CaIn2P2 and SrIn2P2 are a = b = 4.022 Å, c = 17.408 Å and a = b = 4.094 Å, c = 17.812 Å, respectively [24]. The corre sponding unit cell of the XIn2P2 (X = Ca, Sr) compounds is depicted in figure 1(a). It contains two chemical formula units, 2− the layers of X2+ cations are separated by [In2P2] layers. The atoms positioned at the following Wyckoff positions: X (Ca, Sr): 2a (0, 0, 1/2), In: 4f (2/3, 1/3, 0.329) and P: 4f (1/3, 2/3, 0.396). The bulk Brillouin zone (BZ) and projected surface Brillouin zones of (0 0 1) plane are illustrated in figure 1(b). 3. Results and discussion The electronic band structures of the unstrained and 5% strained CaIn2P2 (a/a0 = 1.05 ) with SOC are depicted in figures 2(a) and (b). The unstrained CaIn2P2 has an indirect band gap about 0.5 eV, which agrees with previous studies [35]. In figure 2(a), the Γ8+ band was pushed down, while the Γ8− was pushed up, when we induced tensile stress in the a-b 2 Z Wang et al J. Phys. D: Appl. Phys. 50 (2017) 465304 Table 1. The parities of time-reversal invariant k points with SOC and their corresponding topological invariant. SOC, strain = 0% SOC, strain = 5% Γ M A L Z2 + − + + + + + + 0 1 characteristic band structure (around the Γ and near Fermi energy) of SrIn2P2 under the different strains. To quantify the effects of strain and SOC on the band inversion in SrIn2P2, we temporally switch off the SOC in figures 4(a) and (b). For the unstrained compound (figure 4(a)), there is a direct band gap about 0.3 eV, with the In-s state above the P-p state, indicating a trivial topological insulator. When we induce the strain with a = 5%a0 and without SOC, the In-s state was pushed down the P-p state, resulting a band inversion. The band inversion happens along the M–Γ–K direction, resulting in the nodal ring from the crossing between the valence band and the conduction band in the kz = 0 plane. The glide symmetry Gz = Mz | (0,0,1/2) is the combination of mirror reflection symmetry Mz: (x, y, z) −→ (x, y, −z) and a translation by a half lattice vector τz = (0,0,1/2). Thus, the eigenvalue of Gz is the product of the eigenvalue of Mz and the phase factor induced by τz . The irreducible representations of the little group of k points in the kx-ky plane are determined by the eigenvalue of Mz . As the Hamiltonian without SOC is spinrotation invariant, i.e. Mz2 = 1, the eigenvalue of Mz is +1 or −1. According to this argument, the two bands with different representations will not induce a band gap when they cross each other. Therefore, the appearance of the nodal ring in the absence of SOC is protected by the glide mirror symmetry. Therefore, the strained SrIn2P2 formed a line-node band structure (figure 4(b)), which indicates that the band inversion is caused by the strain rather than the SOC. For the 5% strained compound, when we include SOC, there is a band gap about 35 meV exactly at the Fermi energy. Comparing the three figures in figure 4, we can draw the conclusion that lattice strain induces the band inversion while SOC opens the band gap. To gain further insight into the mechanism of the strain induced band inversion, we study the electronic structure of unstrained SrIn2P2 around EF in more detail. Without SOC, the projected band structure of SrIn2P2 has been shown in figure 5(a). It shows that the valence bands Γ6 and Γ4 are dominated by the P-pz states, while the conduction bands Γ2 and Γ3 are mainly composed of In-s state. Both the valence bands and conduct bands are doubly degenerated at the A-point, and they split up into Γ− /Γ+ and Γ− /Γ− bands, respectively. As a result, the Γ+ and Γ− bands give rise to the conduction band minimum and valence band maximum, respectively. The relative position of the Γ+ and Γ− bands decides whether a band inversion occurs or not. We have determined the four param eters ∆E(A), ∆E (Γ4,6), ∆E (Γ2,3), and ∆Eg as functions of the uniaxial strain (a − a0 )/a0 (volume is fixed). In the unit cell, there are two In-P-In-P slabs in XIn2P2. The ‘inter-slab interactions’ come from the van der Waals forces between the two slabs. The ‘intra-slab interactions’ refers to the covalent bonds between atoms in one slab. The energy difference ∆E(A) is determined by the intra-slab covalent s-pz hybridization, while Figure 3. Phonon dispersion of 5% strained CaIn2P2. plane and compressive stress along c-axis, i.e. increasing the a-axis and decreasing the c-axis with fixed cell volume. When the a 1%a0, the compound becomes a direct band gap insulator. At the critical point a = 2 %a0, the Γ8+ and Γ8− touch each other. When we further increase the a-axis to a = 5%a0 the Γ8+ dropped below Γ8− band in figure 2(b). Comparing figures 2(a) and (b), we find that it is the strain rather than the SOC that induces the ‘band-invertion’ [45]. From the inset in figure 2(b), we can see a band gap about 35 meV. It is well known that the inversion of bands with opposite parity is a strong indication of the formation of topologically nontrivial phases. This suggests that the strain induces a topological phase transition in CaIn2P2. Additionally, the stability of unstrained and 5% strained compound CaIn2P2 has been checked by computing the phonon dispersion, as illustrated in figure 3. Here, we did not find imaginary frequency neither in the unstrained nor in the 5% strained CaIn2P2. To confirm the topological nature of CaIn2P2, all four Z2 topological invariants (v0; v1 v2 v3) [46] were calculated before (zero strained) and after (5% strained) the band inversion structures. Following the method developed by Fu and Kane [46], we calculate directly the Z2 topological invariants from knowledge about the parity of each pair of Kramer’s degenerate occupied energy bands at the eight time-reversal momenta (1Γ, 3M, 1A, and 3L), because of the existence of inversion symmetry in our compounds. Before the band inversion, the products of parities δi are identical, yielding a value of zero for all four Z2 topological invariants, i.e. a topologically trivial state (0;000). When the bands with Γ8+ and Γ8− symmetries switch around EF, see figures 2(a) and (b), δΓ changes its sign while all other seven δi remain unchanged. The parities of time-reversal momenta are listed in table 1. It gives rise to a topological state with Z2 class (1;000) and confirms that the strain induces a topological phase transition. Generally, the band inversion is generated by SOC, formatting the topologically nontrivial state, similar to the proto typical 3D topological insulator SR2Pb and Bi2Te3 [6, 47]. However, in our study, the band inversion was induced by the strain only, without considering any SOC. Then, what role does the strain and SOC play in the topological phase transformation of XIn2P2? In figure 4, we take SrIn2P2 as the example to answer the question. Figure 4 shows the orbital 3 Z Wang et al J. Phys. D: Appl. Phys. 50 (2017) 465304 Figure 4. The orbital characteristic band structures (along the M–Γ–K direction, near Fermi energy) of SrIn2P2 under the different strains. (a) Zero strain without SOC, (b) 5% of uniaxial strain without SOC, (c) 5% of uniaxial strain with SOC. The weight of atomic orbital In(s) and P(p) is proportional to the radius of the green (blue) circle. Band inversion could be seen clearly around the Γ point. Figure 5. (a) The band structure of unstrained SrIn2P2 without SOC, weighted with the In-s and P-Pz characters. In-s and P-Pz states are distinguished by red and green colors. (b) ∆E(A), ∆E(Γ4,6 ), ∆E(Γ2,3 ) and ∆Eg as functions of the inter-slab tensile strain δ = (a − a0 )/a0 (volume is fixed), respectively. Figure 6. Surface density of states (SDOS) in SrIn2P2 (a) without and (b) with SOC. The yellow lines highlight the topologically protected metallic surface states for the (0 0 1) surface. ∆E (Γ4,6) and ∆E (Γ2,3) are related to inter-slab van der Waals pz–pz and s–s interactions. This observation is in line with findings for layered transition metal dichalcogenide semiconductors, such as MoS2 [48, 49]. The results are presented in figure 5(b). With increasing tensile strain in the ab-plane and compressive strain along the c-axis, the intra-slab interaction decreases, resulting in the increment of the energy spilt ∆E(Γ4,6 ). However, the interslab interaction increase with the c-axis decreasing, resulting in a strongly reduced ∆E(A), indicating strongly intra-slab covalent s-pz hybridization. The energy level EΓ4 (derived from P-pz orbit) increases with c-axis decreasing. So the band gap Eg decreases with the strain increment, and the band gap closes at the critical point δ = 0.02 . With further increasing the strain, the band inversion takes place and the compound becomes a topological nontrivial line-node semimetal. SOC opens the band gap along the nodal ring and drives the compound into topological nontrivial insulator. 4 Z Wang et al J. Phys. D: Appl. Phys. 50 (2017) 465304 Qi X-L, Li R, Zang J and Zhang S-C 2009 Science 323 1184 [11] Essin A M, Moore J E and Vanderbilt D 2009 Phys. Rev. Lett. 102 146805 [12] Wan X, Turner A M, Vishwanath A and Savrasov S Y 2011 Phys. Rev. B 83 205101 [13] Balents L 2011 Physics 4 36 [14] Ando Y 2013 J. Phys. Soc. 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So we calculate the (0 0 1) surface of SrIn2P2 by preforming the WANNIE R_TOOLS package [50] in a tight-binding (TB) scheme based on the maximally localized Wannier functions (MLWFs) [39]. The surface state of SR2In2P2 (0 0 1) surface without and with SOC have been shown in figures 6(a) and (b), respectively. When SOC is ignored, the system is a linenode semimetal, with the bulk bands touched along K̄ − Γ̄ , Γ̄ − M̄ lines and a surface state lining two touching points. This is similar to CaAgAs [34], protected by the coexistence of time-reversal and glide mirror symmetry. When SOC is included, a gap opened at the two touching points, and the compound became an topological nontrivial insulator, with two surfaces states connecting bulk valance and conduction bands (see figure 6(b)), coinciding with the previous calculated Z2 (table 1). 4. Conclusion In conclusion, by the first-principles calculations, we find the quantum phase transition in XIn2P2 from conventional semiconductor into the line node semimetal by strain engineering only without SOC. While SOC takes the role as opening band gap along the line node and gives rise to the nontrivial topological insulator state. Acknowledgments The authors acknowledge support from the NSF of China (No.11274095, No.10947001) and the Program for Science and Technology Innovation Talents in the Universities of Henan Province (No.2012HASTIT009, No.104200510014, and No.114100510021). This work is supported by The High Performance Computing Center (HPC) of Henan Normal University. ORCID iDs Guangtao Wang https://orcid.org/0000-0001-5331-3754 References [1] Moore J 2009 Nat. Phys. 5 378 [2] Hasan M Z and Kane C L 2010 Rev. Mod. Phys. 82 3045 [3] Qi X L and Zhang S C 2011 Rev. Mod. 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