Journal of Magnetism and Magnetic Materials 468 (2018) 279–286 Contents lists available at ScienceDirect Journal of Magnetism and Magnetic Materials journal homepage: www.elsevier.com/locate/jmmm Research articles Hydrostatic strain-induced topological phase of KNa2Sb ⁎ T Shahram Yalameha, Zahra Nourbakhsh , Aminollah Vaez Department of Physics, Faculty of Sciences, University of Isfahan, Isfahan, Iran A R T I C LE I N FO A B S T R A C T Keywords: Topological insulators Band inversion Density functional theory Elastic properties Electronic properties Topological insulators are novel state of quantum matter that have a bulk band gap like an ordinary insulator, but have protected conducting states by time reversal symmetry on their edge or surface. The spin-orbit coupling can play an important role in these materials, resulting in a band inversion at time reversal invariant momenta (TRIM) points. The topological phase and the eﬀect of the hydrostatic pressure on the electronic structure and topological phase of the KNa2Sb compound are investigated by using both ﬁrst-principles calculations and abinitio based tight-binding computations. Under hydrostatic lattice strain until 5.6%, the KNa2Sb compound is semimetal with zero energy band gap and has an inverted band order. In this pressure, the Z2 invariants of this compound are calculated using the parity analysis at TRIM points and evolution of wannier charge centers at the ¯ line at six TRIM plane. The calculated surface states at (0 0 1) surface show a single Dirac cone exists on the X¯ Γ̄W the surface Brillouin zone. To investigate the stability of KNa2Sb compound the phonon dispersions and elastic tensors of this compound in the cubic structure are calculated. 1. Introduction In 2007, it was theoretically realized that the Z2 topological invariant in a Quantum spin Hall insulator can be generalized to threedimensions, thus realizing the ﬁrst three dimensional topological phase of matter [1–3]. Although discovered only in the recent few years, topological insulators (TIs) have attracted intensive interest among the community of condensed matter physics and material science. TIs are insulating in the bulk but have conductive gapless edge or surface states on the boundaries, which have their origin in the nontrivial bulk band topology that is induced by the strong spin-orbital interactions in the materials. Time-reversal symmetry is required to specify the class of topological insulators. It can be visualized in the surface state point of view, if this symmetry is broken, the surface state degeneracy at the Kramers point can be lifted. In three dimensions, there are eight Kramers points, which leads to four independent Z2 topological invariants (ν0; ν 1,ν2, ν3), where ν0 is the strong invariant and (ν1, ν2, ν3) are the weak invariants. Liang Fu and C.L. Kane [4] showed that the presence of inversion symmetry simpliﬁes the calculation of topological invariants, which can be determined from the knowledge of the parity of the occupied electronic states at the time reversal invariant momenta (TRIM) points in the Brillouin zone (BZ). A band inversion is a key ingredient of a topologically nontrivial material and it is needed to change the topological band order [4]. The band inversion is usually result from the band splitting at the high symmetry points in BZ caused ⁎ by SOC. Zhiyong Zhu et al., determined that scalar relativistic, nonrelativistic and fully-relativistic (FR) eﬀects are contributing to the band inversion [5]. Although the topological band inversion is usually due to SOC, it has been speciﬁed that the band inversion of HgTe does not depend on the SOC strength [5,6]. In recent years, to control topological band order, the electromagnetic ﬁelds [7,8], composition [9,10], pressure [11–14], strain [15–18] and temperature eﬀects [19] are used. In this paper, we study the topological band order of KNa2Sb under hydrostatic lattice strain. The alkali antimonides compounds, such as X3Sb (X = Na, K, Cs, Li), KNa2Sb, RbNa2Sb, CsNa2Sb, RbK2Sb, CsK2Sb and CsRb2Sb, are widely used as photocathode materials [20,21]. Their characteristics such as photon absorption and work function make these compounds suitable candidates for electron emission devices applications [22]. In the alkali antimonide compounds there isn't any comprehensive study to examine the topological band order characteristics, regardless of their prospective technological applications [23,24]. Consequently, the focus on topological phase of these materials can lead to various emergent photonics applications due to creation of surface states [25]. Also, the bi-alkali antimonides compounds are highly quantum eﬃcient semiconductors [26]. Unlike the alkali bismuthides compounds, A3Bi (A = Na, K, Rb), where the valence and conduction bands touch at discrete points at the Fermi level [27], the alkali and bi-alkali antimonides compounds usually have non-zero gaps. In the alkali bismuthides, KNa2Bi, theoretically demonstrated that this compound is a Corresponding author. E-mail address: z.nourbakhsh@sci.ui.ac.ir (Z. Nourbakhsh). https://doi.org/10.1016/j.jmmm.2018.07.086 Received 4 May 2018; Received in revised form 15 June 2018; Accepted 30 July 2018 Available online 01 August 2018 0304-8853/ © 2018 Elsevier B.V. All rights reserved. Journal of Magnetism and Magnetic Materials 468 (2018) 279–286 S. Yalameha et al. Table 1 Positions of atoms, experimental and calculated lattice parameter and elastic constants of the cubic KNa2Sb compound. Position Lattice constant (Å) Elastic constants (GPa) This work Other Na1 Na2 K Sb a0 aexpt C11 C12 C44 C11 C12 C44 (1/4, 1/4, 1/4) (3/2, 3/3, 3/2) (1/2, 1/2, 1/2) (0, 0, 0) 7.79 7.74a 30.18 12.82 20.27 22.28b 19.3b 17.59b a b Ref. [47] Experimental. Ref. [26] Ab initio. electron charge density and the maximum angular momentum are optimized as Gmax = 12 Ry1/2 and lmax = 10 Ry1/2, respectively. In addition, 9 × 9 × 9 k-point meshes are employed in the reciprocal space. To identify the Z2 topological invariant, the Wannier center evolutions is calculated in the whole BZ for six TRIM planes (k1 = 0.0, k1 = 0.5; k2 = 0.0, k2 = 0.5; k3 = 0.0, k3 = 0.5) [33]. The Bloch spectral functions (BSFs) deﬁned by the imaginary part of Green’s function, is diagonal in the momentum space. They have been carried out using the fully-relativistic Korringa-Kohn-Rostoker (KKR) Green’s function method [34]. The topological surface states were calculated using ab initio based tight-binding formalism [35,36] after obtaining the tightbinding Hamiltonian from the MLWF [37] as implemented in the Wien2k package. The phonon spectrum is calculated using the all-electron FHI-aims code [38] with the phonopy code [39] within the GGA as parametrized by Perdew-Burke-Ernzerhof (PBE). The reciprocal space integration was carry out over an 8 × 8 × 8 Monkhorst-Pack grid. In additions, the FHI-aims “tight” default option has been used, which speciﬁes a preconstructed high-accuracy all-electron basis set of numerical atomic orbitals. The convergence criteria of total atomic forces for the supercells, that made by the phonopy code, was less than 10−5 eV/Å. zero-gap semimetal with an inverted band structure at the BZ center. It can be driven into nontrivial topological phases by applying hydrostatic pressure and uniaxial strain [13]. In this paper, we show that the ternary alkali antimonides KNa2Sb compound under hydrostatic strain until 5.6%, a zero-gap semimetal with an inverted band structure at the BZ center, because the inverted Γ8 and Γ6 bands in near the Γ point. In addition, after applying pressure, the parity eigenvalues in the Γ point are changed with and without considering spin-orbit coupling (SOC). Spin-orbit coupling and relativistic eﬀects are enhanced in materials with heavy elements [5,6]. Nevertheless, the Bloch spectral functions (BSFs) of KNa2Sb were obtained from FR calculations show that, both FR eﬀects and SOC are play a central role in band inversion. Therefore, to identify the topological properties, we have to calculate the Z2 topological invariants. The Z2 topological invariants for 3D bulk system can be obtained from the calculation of the wannier charge center (WCC) for the six time-reversal invariant momentum plane; we have compared the topological invariants with and without SOC. Also, with respect to two important symmetries, time reversal symmetry (TRS) and inversion symmetry (IS), we calculated the topological invariants (ν0;ν1,ν2,ν3) with determined from the knowledge of the parity of the occupied band eigenstates at the eight TRIM points in the BZ. The results of both methods show that the most important invariant is ν0; ν0 = 1 representing a strong topological insulator (STI), while weak invariants are ν1 = ν2 = ν3 = 0. Thus, an STI display topological surface states (TSS) with odd number of Dirac cones on any surface, while a weak topological insulator (WTI) determines TSS with even number of Dirac cones only at speciﬁc surface planes [28]. These results and the TSS calculated using the maximally localized wannier functions (MLWFs) on (0 0 1) surface, clearly conﬁrms the Bulk-Boundary Correspondence (BBC) [29]. The ﬁrst good of this paper is to investigate the stability of KNa2Sb compound using the calculated elastic constants (which show mechanical stability criteria) and phonon spectrum (which show the dynamical stability) and to study the electronic band structure and energy band gap of this compound. The second good of this paper is to study the topological phase of KNa2Sb compound using the calculated band inversion strength between s and p orbitals, Z2 topological invariants and topological surface states. The third good of this paper is to investigate the eﬀect of pressure on the topological phase and band order of this compound. 3. Results and discussion 3.1. Structural and electronic properties The atomic positions and the experimental and calculated optimal lattice parameter (a0) of the KNa2Sb cubic structure with Fm3̄m space group are given in Table 1. The calculated lattice parameter is in acceptable agreement with the experimental value [39]. Fig. 1 (a) displays the band structures of KNa2Sb compound using GGA in the presence of SOC; the calculated direct energy band gap (0.51 eV) is smaller than the experimental value (1.0 eV) [40]. The DFT with GGA approach usually underestimate the energy band gap. But, the EV-GGA approach with optimize exchange potential improves some electronic properties, such as energy band gap. Thus, to improve the calculated energy band gap, the EV-GGA approach is used. The calculated energy band gap of this compound, within EV-GGA (0.9 eV) is in good agreement with experimental value. To investigate the dynamical stability of KNa2Sb in the cubic structure, the phonon dispersion curve at the equilibrium volume (V0) is calculated (Fig. 1(b)). Overall, the acoustic phonon frequencies of this compound over the whole BZ are positive that showing the dynamical and the internal stability of the atomic positions. There is no imaginary frequency at hydrostatic lattice strain, which reveals the stability of this compound. The mechanically stability can be described by the elastic constants (Cij). The KNa2Sb compound with cubic structure has three independent elastic constant components (C11, C12 and C44). The calculated elastic constant of this compound are given in Table 1. The mechanical stability criteria for cubic phase are given by Eq. (1) and according to the calculated results of Table 1, this stability is well represented. 2. Method and computational details The structural and electronic properties and the elastic constants calculations were performed using full potential linearized augmented plane-wave (FP-LAPW) method [30] within the generalized gradient approximation (GGA) and Englo–Vosko GGA (EV-GGA) [31], as implemented in the Wien2k package [32]. In the calculations, an energy threshold of -7 Ry is used to separate the valence and the core states. The wave functions in the interstitial region were expanded in the terms of plane waves with a k vector cut-oﬀ Kmax = 9/RMT (a.u.)−1, where RMT denotes the smallest muﬃn-tin sphere radius and Kmax is the largest k vector in the plane wave expansion. The Fourier expansion of the C11 > 0, C11 > |C12|, (C11 + 2C12 ) > 0, C44 > 0 . 280 (1) Journal of Magnetism and Magnetic Materials 468 (2018) 279–286 S. Yalameha et al. Fig. 1. (a) The dispersion of the energy bands and (b) the phonon band structure of KNa2Sb using GGA. Orientation dependence of Poisson’s ratio in (c) (1 0 0) and (d) (0 1 1) planes. Blue color is maximum, green color is minimum positive value of Poisson’s ratio and red color is minimum when it is negative. (For interpretation of the references to color in this ﬁgure legend, the reader is referred to the web version of this article.) their interior but can support the ﬂow of electrons on their surface, while conventional insulators do not have this feature. Band order is the most important condition for identifying topological phase of materials, which is the most practically useful guideline for the initial screening of possible topological phase transitions of materials. The band inversion is a phenomenon in the electronic structure where the relative energy positions between the conduction and valence bands around a certain TRIM point in the BZ are reversed. From this point of view, to investigate the topological phase of this compound, the band structure in the presence of SOC and description distribution of s and p electron at Γ symmetry point are shown in Fig. 2(a-d). To study strain eﬀects on the band structure and band order of this compound, the external hydrostatic strain which is defend as ε = (a − a0)/a0, (where a and a0 are cubic lattice constants with and without strain, respectively) is applied. The distribution of s and p electrons of Bi atoms on the band structure at Γ symmetry point around the Fermi energy is only focused in Fig. 2. The p-type bands of K and Na atoms have negligible distributions around the Fermi level. In the equilibrium state, Fig. 2(a), the s-type band Γ6 (red points) sits above the p-type Γ8 (green points) of valence band. Therefore, this compound has normal band order with 0.51 eV direct energy band gap and it is a trivial topological phase. As the hydrostatic strain increases, the band splitting between Γ8 as the valance band and Γ6 as the conduction band decreases (Fig. 2(b, c)). When V/V0 = 1.18, the Γ6 state shifts downward with respect to the Γ8 state, where the direct band energy gap disappear and a new band gap between Γ6 and Γ8 is created, as illustrated in Fig. 2 (d). Therefore, a topological phase transition from a trivial insulator (TI) to a non-trivial phase topological is induced. To conﬁrm this topological phase transition, we illustrate the parities of the Bloch states at Γ point before and after the topological phase transition pressure in the presence and absence of SOC in Unfortunately, no pervasive experimental and theoretical study has been devoted to examine the elastic properties, especially Poisson’s ratio, ν, of the KNa2Sb compound. Some textbooks have expressed that negative values in Poisson’s ratio are impassible, while Negative Poisson’s ratio (NPR) is theoretically possible [41]. In a solid, ν has deﬁned as the transverse contraction strain (εtransverse ) divided by the longitudinal extension strain (εlongitudinal ). Poisson’s ratio is important because it inﬂuenced on the strain concentration, wave speeds in a material and interatomic forces [42,43]. For metrical with cubic lattice structure the anisotropy factor, A, is deﬁned as: A= 2C44 (C11−C12 ) (2) For isotropic materials, A = 1. Using the calculated elastic constants, the anisotropy factor of KNa2Sb compound is calculated (2.33), which shows that this compound is anisotropic. Indeed, for isotropic solids the range of ν is from −1 to 0.5, while in anisotropic solids is not as restricted as it is in isotropic solids. The calculated ν of this compound in diﬀerent planes, (1 0 0) and (0 1 1) are shown in Fig. 1(c-d), that performed using the Elam code [41]. The minimum (green color) and maximum (blue color) positive values of the ν in the (1 0 0) and (0 1 1) planes are 0.31 and 0.47, respectively. It shows that when the stress is applied in the direction perpendicular to the (1 0 0) and (0 1 1) plane, the minimum strain is emerged in the [0 0 1] (or [0 1 0]) direction and the maximum strain in the [0 1 1] direction, while in this direction, the ν has appeared negative (−0.12) (red color). This feature is also exists in the (0 1 1) plane. 3.2. Topological phase of KNa2Sb 3.2.1. Strain-induced s-p band-inversion Topological insulators are materials that are normal insulator in 281 Journal of Magnetism and Magnetic Materials 468 (2018) 279–286 S. Yalameha et al. Fig. 2. (a-h) Energy band structures, (i) the distributions of s and p orbitals of KNa2Sb at Γ symmetry point around the Fermi energy as a function of pressure. The red points shows s orbit composition and the green points shows p orbit. The symbols “+ ”and “−” label the parities of the bands at the Γ point. (For interpretation of the references to color in this ﬁgure legend, the reader is referred to the web version of this article.) (s−) with diﬀerent parities. So, the system is an insulator and the band gap is Eg > 0.51 eV, as shown the pink area in stage (I). When the SOC is taken into account in stage (II) of pink area, the p− state splits into the |p , ± 1/2〉 with a total angular momentum of j = 3/2 and j = 1/2. Also, the energy band gap decreases, which is converted into a semiconductor system with the band gap of 0.51 eV. It is observed that the crossing between the s+ and p− level do not occurs at V/V0 = 1, due to the sizeable energy band gap. When considering the hydrostatic strain (V/V0 = 1.18) as can be seen in stage (I) of the orange area, the energy gap is decreased. Thus, in the presence of SOC, the s+ and p− 3/2 levels crossing in stage (II), leads to a parity exchange near the Fermi energy and band inversion. Furthermore, we have found that FR eﬀects can also induce the band inversion to the KNa2Sb. The band structure based on BSFs is presented in Fig. 3. As we already mentioned, as the pressure increases, the energy band gap reduces signiﬁcantly and the topological phase transition conditions is provided. Fig. 3(a) shows BSFs at the equilibrium state include the FR eﬀects. The red and blue color are corresponding to BSF values greater and less than 300 and 0.1 a.u., respectively. Thus, the FR eﬀects using the BSF calculations show that the most electron distributions around the Γ symmetry point that is related to the conduction bands (circle in Fig. 3(a)). Under hydrostatic pressure, the energy band splits due to the FR eﬀects and the most contribution of BSFs are related to the valence bands (circle in Fig. 3(b)), Fig. 2(e-h). When V/V0 = 1, in the absence of SOC, the parities of Bloch states of the conduction and valence bands at Γ are “+” and “−”, respectively. In the presence of SOC, the p− state splits into the |p , j = ± 3/2〉 (green bands) and |p , j = ± 1/2〉 (blue band) with odd parities, while the s+ state has an even parity (Fig. 2(e, f)). At phase transition pressure (V/V0 = 1.18) in the absence of SOC, only the energy band gap is reduced (Fig. 2(g)). Nevertheless, in the presence of SOC, KNa2Sb compound converts into a zero-gap semimetal, in which the degenerate and parities of conduction and valence bands at the Γ point are changed (Fig. 2(h)). As a result, the s-type state (Γ6) is lower in energy than the p-type state (Γ8); show the topologically non-trivial band order. The behavior is similar to some half-heuslers TIs [44] and KNa2Bi [13]. To explain the physical mechanism underlying the band inversion and the parity exchange of this compound, the schematically evolution of the energy levels of the atomic levels at Γ point under hydrostatic strain, in the presence and absence of SOC is shown in Fig. 2(i). Fig. 2(i), divided into two regions: the pink and orange areas that displayed before and after the hydrostatic strain including, respectively. According the normal band ﬂling order near the Fermi energy level, the valence band should be formed of p-Sb orbitals and the conduction band should be composed of s-type states, which hybridise and split into the bonding and antibonding states in the absence of SOC (because of formation of chemical bond in this compound), i. e., p+ (s+) and p− 282 Journal of Magnetism and Magnetic Materials 468 (2018) 279–286 S. Yalameha et al. Fig. 3. The Bloch spectral functions (BSF) of KNa2Sb in high symmetry directions ( X −Γ−L ); (a)/(b) In the absence / present of pressure. The red color corresponds to BSF values greater than 120 a.u.. (For interpretation of the references to color in this ﬁgure legend, the reader is referred to the web version of this article.) points, at the zero pressure (V/V0 = 1) and topological phase transitions pressure (V/V0 = 1.18). As shown in Table 2, at the zero pressure the PIs at Γ, X and L are −1, −1 and +1 TRIM points, respectively. In the case, the topological invariants are (0; 0, 0, 0), which corresponds to a conventional insulator. As expected, at topological phase transitions pressure of KNa2Sb (V/ V0 = 1.18), the parities of valence bands at the Γ symmetry point change, due to the s-p band inversion. Thus, the sign of PI at Γ point changes from −1 to +1. As a result, the topological invariant indexes of this compound are (1; 0, 0, 0), which shows the strong topological insulator. However, the parity invariants method is eﬃcient method for topological phase investigation, but for comprehensive cases where inversion symmetry is absent, the evaluation of the topological invariants is diﬃcult. The Wannier charge centers (or Wilson loop) method, proposed by Soluyanov and Vanderbilt is better than the parity invariants method. The Wilson loop method not require a gauge-ﬁxing condition so that greatly simplifying the calculation. In addition, it can be used to a system with or without IS. Fig. 4 shows the evolution lines of WCCs along ky of KNa2Sb compound at six TRIM planes at normal conditions (V/V0 = 1) and topological phase transitions pressure (V/V0 = 1.18). Z2 topological invariants can be obtained using the numbers of crossing between any arbitrary horizontal reference line (purple line) and evolution of the WCCs. As shown in Fig. 4(a) in the equilibrium state (V/V0 = 1), it can be seen that the WCC lines cross the reference line an even number of times for all six planes (kx = 0, ky = 0, kz = 0, kx = 0.5, ky = 0.5 and kz = 0.5). So, this compound is topological trivial and its topological invariants are (0; 0, 0, 0), which conﬁrms the previous results. Corresponding to above analysis at topological phase transitions pressure V/V0 = 1.18, the WCC lines cross the reference line an odd number of times for all three planes (kx = 0, ky = 0, kz = 0) containing (0, 0, 0) point and even number of times for all three planes (kx = 0.5, ky = 0.5, kz = 0.5) containing (0.5, 0.5, 0.5) point (Fig. 4(b)). Therefore, under hydrostatic strain the KNa2Sb compound is a nontrivial topological insulator with (1; 0, 0, 0). that show a band inversion. Therefore, FR eﬀects like the SOC have a central role on the band order of in the KNa2Sb compound. 3.2.2. Z2 topological invariants After the topological phase investigations of materials, the classiﬁcation of topological insulator must be determined. The classiﬁcation of topological insulator can be identiﬁed by its Z2 topological invariants that are determined by four indexes. It is note that time-reversal symmetry requires in this classiﬁcation. In this subsection, two methods have been widely used to investigate the topological invariants. One is to calculate the topological invariants proposed by Fu and Kane [4] and another method is demonstrated the topological invariants proposed of materials by Soluyanov and Vanderbilt [33,45]. In the ﬁrst method, with consider two TRS and crystal-inversion symmetry (IS), the topological invariants can be determined by the parity eigenvalues of topmost isolated valence bands at the eight TRIM points. The face centered cubic structure of KNa2Sb with the Fm3̄m group, has three nonequivalent TRIM points (Γi) in the momentum space. These points are one Γ, three X (3X) and four L (4L) which deﬁned by −Γi = Γi + G, where G is a bulk reciprocal lattice vector. Accordingly, the strong topological invariant, ν0, is determined by [4] 8 N δi = ∏ ξ2m (Γ), i m=1 (−1) ν0 = ∏ δi (3) i=1 where the ξ2m (Γ) i = ± 1 are the parity eigenvalues of the N occupied bands at three dimensions (3D) TRIM points Γi and δi are parity invariants (PIs) for each occupied bands for each TRIM points. The other three weak topological invariant, (ν1, ν2, ν3) are given by ∏ (−1) vl = δi = (n1 n2 n3) , l = 1, 2, 3 nl = 1 nj ≠ 0,1 (4) Table 2 shows the parity eigenvalues of the Bloch wave function of 4 occupied bands of KNa2Sb compound at three non-equivalent TRIM Table 2 Parities of 4 occupied bands and parity invariants at the TRIM points (δ(Γi)) of the KNa2Sb for V/V0 = 1, V/V0 = 1.18. Positive and negative signs explain even and odd parities, respectively. V/V0 = 1 3.2.3. Topological surface states Calculations. After the investigation of the band inversion and topological invariants of this compound using three diﬀerent methods, the calculation of bound states (surface or edge states) of KNa2Sb compound becomes necessary. The bound states calculations are useful for experiments, including scanning tunneling microscope and angle-resolved photoemission spectroscopy (ARPES) which it furnish the essential information for direct comparison. Before discussing the bound states calculations, the theoretical predictions of the TI surface state V/V0 = 1.18 Γi Parity of 4 occupied bands δ(Γi) Γi Parity of 4 occupied bands δ(Γi) G 3X 4L −−+− −−−+ ++++ −1 −1 +1 G 3X 4L +−+− −−+− ++−− +1 −1 +1 283 Journal of Magnetism and Magnetic Materials 468 (2018) 279–286 S. Yalameha et al. Fig. 4. Evolution of the wannier charge centers (WCCs) along ky at six TRIM palnes (a) at normal conditions (V/V0 = 1) and (b) at topological phase transitions pressure (V/V0 = 1.18). Each of the six planes passing through four TRIM points. The purple lines are arbitrary reference lines. (For interpretation of the references to color in this ﬁgure legend, the reader is referred to the web version of this article.) Fig. 5. (a, b) The projected surface density of states (PSDS) and (c, d) band dispersions using the ab initio TB model with 41 atomic layers of [0 0 1] surface in consideration of SOC. (a) and (c), when V/V0 = 1.81 and (c) and (d), when V/V0 = 1. (e) Brillouin Zone of bulk and the projected surface Brillouin Zones of (0 0 1) plane. π (Λ a) = (−1)nc δ (Γ) i δ (Γj) electronic structure proposed by Teo et al [46] are explained. Accordingly, the TSS is dependence to the values of the surface fermion parity π(Λa) at the TRIMs surface. π(Λa) can be determined from the PIs (δ (Γi)) as (5) where nc is the number of occupied bulk bands, which is 4. 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However, after the topological phase transition (V/V0 = 1.18) the surface fermion parity on the Γ̄ and X̄ TRIM surface isπ (Γ̄) = −1 andπ (X¯ ) = 1, which predicts the existence of an odd number of closed Fermi contours or Dirac cone around the surface Γ̄ TRIM point; this is symbolized by the blue circle around the Γ point as display in Fig. 5(e). The calculated projected surface density of states (PSDS) at [0 0 1] surface using the Green’s function is displayed in Fig. 5(a) and (b). From Fig. 5(b), it can be seen that bulk states not connected by surface states at Γ̄ point at zero pressure. In fact, surface states there is not appeared in this case. Also, the band dispersions of this surface using the ab initio TB model with 41 atomic layers conﬁrms this issue (Fig. 5(d)). To conﬁrm the pressure induced topological phase transition of KNa2Sb compound, the PSDS and band dispersions of this compound using the ab initio TB model with 41 atomic layers of [0 0 1] surface are calculated and shown in Fig. 5(c) and (d). From this ﬁgure, we can see that bulk states are connected by the TSSs at Γ̄ point in this surface, which lead to the appearance of the Dirac cone. This results are conﬁrmed the BBC, which relates the boundary topological invariants to the diﬀerence in the bulk topological invariants [29]. As expected, the theoretical predictions discussed before, clearly show these results. 4. Conclusion The calculated phonon dispersions, elastic constants and anisotropy factor show that KNa2Sb compound is mechanically and dynamically stable and is anisotropic. The hydrostatic strain and the relativistic effects such as SOC and FR eﬀects can lead to a topological phase transition. The FR eﬀects can be necessary for the band inversion mechanism rather than the SOC. The band inversion in the band structure, the evolution lines of the WCCs, Z2 topological invariants (ν0=1) and topological surface states conﬁrm the topological phase transition of KNa2Sb compound under hydrostatic pressure. The ﬁrst-principles calculations show that the KNa2Sb compound can be tuned to a 3D TI by imposing hydrostatic strain until 5.6%. The KNa2Sb is a strong topological insulator with (1; 0, 0, 0) Z2 topological class. Interestingly, KNa2Sb compound has normal band order using density functional theory in the absence of spin-orbit coupling (SOC), while in the presence of SOC this compound possesses an inverted band order between the s-type Γ6 and p-type Γ8 states. The calculated band structure of this compound using the Bloch spectral functions within the fully relativistic calculation without SOC (i.e. SOC is excluded) shows band inversion at Γ point. The topological surface states of this compound show a Dirac cone at Γ̄ symmetry point. In sum, photocathode properties, along with a striking feature of mechanical properties (i.e. negative-Poisson’s ratio or auxetic materials) and topological features under presser, make this compound suitable for practical applications. Acknowledgement We would like to thank Parviz Saeidi and Mohammad hossein Shahidi kaviyany. 285 Journal of Magnetism and Magnetic Materials 468 (2018) 279–286 S. Yalameha et al. Tran-Blaha modiﬁed Becke-Johnson density functional, Phys. Rev. B 82 (23) (2010) 235121. [45] A.A. Soluyanov, D. Vanderbilt, Computing topological invariants without inversion symmetry, Phys. Rev. B 83 (23) (2011) 235401. [46] J.C. Teo, L. Fu, C. Kane, Surface states and topological invariants in three-dimensional topological insulators: Application to Bi1− xSbx, Phys. Rev. B 78 (4) (2008) 045426. [47] J. Scheer, P. Zalm, Crystal structure of sodium-potassium antimonide (Na2KSb), Philips. 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