Abstract
In this article, we predict the emergence of nontrivial band topology in the family of XX′Bi compounds having \(P\overline{6}2m\) (# 189) space group. Using first principles calculations within hybrid functional framework, we demonstrate that NaSrBi and NaCaBi are strong topological insulator under controlled band engineering. Here, we propose three different ways to engineer the band topology to get a nontrivial order: (i) hydrostatic pressure, (ii) biaxial strain (due to epitaxial mismatch), and (iii) doping. Nontriviality is confirmed by investigating bulk band inversion, topological Z_{2} invariant, surface dispersion and spin texture. Interestingly, some of these compounds also show a three dimensional topological nodal line semimetal (NLS) state in the absence of spin orbit coupling (SOC). In these NLS phases, the closed loop of band degeneracy in the Brillouin zone lie close to the Fermi level. Moreover, a drumhead like flat surface state is observed on projecting the bulk state on the [001] surface. The inclusion of SOC opens up a small band gap making them behave like a topological insulator.
Introduction
Symmetry protected nontrivial band topology has become an area of paramount research interest for unravelling novel dimensions in condensed matter physics^{1,2}. The time reversal invariant topological insulator (TI) has stimulated intense interests due to their intriguing properties, such as gapless boundary states, unconventional spin texture and so on^{3,4,5}. The recent years have witnessed a series of theoretical developments which have enabled us to classify the Z_{2} type nonmagnetic band insulators. For example, the Z_{2} even (ordinary) and Z_{2} dd (topological) states are separated by a topological phase transition, where the bulk gap diminishes during the adiabatic deformation between these two states^{6}. In twodimensional (2D) systems, Z_{2} odd class can be distinguished by the odd number of Kramer’s pairs of counter propagating helical edge states, whereas in threedimensional (3D) systems, it can be characterized by the odd number of Fermi loops of the surface band that encloses certain high symmetry points in the Brillouin zones (BZ)^{3}. Soon after the experimental realization of quantum spin hall effect in 2D HgTe quantum wall^{4}, a number of 2D and 3D TI systems have been theoretically predicted and experimentally verified^{7,8,9,10}. In fact, the search for new TI has been extended to zintl compounds^{11,12}, antiperovskites^{13}, and heavy fermion felectron Kondo type of systems^{14}.
With the conceptual development in the topological field, research on topological material has been extended from insulators to semimetals and metals^{15,16,17}. In topological semimetals, symmetry protected band crossing or accidental band touching leads to a nontrivial band topology in 3D momentum space. The topological properties of such semimetals mainly depend on the degeneracy of the bands at the crossing/touching point. A zero dimensional band crossing with two and four fold band degeneracy defines the Dirac^{15} and Weyl semimetal^{16}, respectively, which are quasiparticle counterparts of Dirac and Weyl fermions in high energy physics. Low energy Dirac fermions in condensed matter are essentially protected by time reversal symmetry (TRS), inversion symmetry (IS) and certain crystal symmetry. Quasiparticle Weyl fermion state can be realized by breaking either space inversion or time reversal of crystal^{16}. On the contrary, in quantum field theory, Dirac and Weyl fermions are strictly restricted by Lorentz invariance. However, in case of nodal line semimetal (NLS), the conduction and valance band touches along a line to form a one dimensional close loop^{17}. The characteristic feature of Dirac semimetal (DS) is a point like Fermi surface (FS) at the crossing point^{15}, whereas it is 1D circle for NLS^{17}. But for Weyl semimetal (WS), FS forms an arc like surface, instead of closed^{16}. Due to the nontrivial FSs, all the topological semimetals show some exotic phenomena, such as quantum magnetoresistance^{18}, chiral anomaly^{19} etc.
NLS are the precursor states for other topological phases. In general, spinful nodal lines are not robust in the presence of a mass term in Hamiltonian^{20}, which can be explained by simple codimensional analysis. Thus, inclusion of spin orbit coupling (SOC) can convert the NLS state to DS, WS or TI by opening up a gap around the nodal loop. However, in the presence of an extra crystalline symmetry, nodal line can be robust^{20}. Owing to the unique properties such as torusshaped Fermi surface, relatively higher density of states and interaction induced instability of the FS, NLS can provide a unique playground for the quasiparticle correlations and unusual transport studies^{17}.
In this article, we explore the possibility of controlling topological order in a series of ternary compounds XX′Bi (X = Na, K, Rb, Cs and X′ = Ca, Sr). The XX′Bi compounds have a noncentrosymmetric hexagonal structure with \(P\overline{6}2m\) (# 189) space group as shown in Fig. 1(a). The theoretically relaxed lattice parameters and formation energy of these systems are provided in supplementary material (SM)^{21}. These compounds show interesting topological properties (including NLS state) which can be tuned under various external factors. Recently, NaBaBi has been theoretically studied and predicted to be a topological insulator under hydrostatic pressure^{22}.
Computational Details
All the calculations were carried out using projector augmented wave^{23,24} formalism based on Density Functional Theory (DFT) as implemented in the Vienna Ab Initio Simulation Package (VASP)^{25}. The PerdewBurkeErnzerhof (PBE)^{26} type functional with generalizedgradient approximation (GGA)^{27} was employed to describe the exchange and correlation effects. All the structures are fully relaxed until the Hellmann  Feynman forces on each atom are less than 0.01 eV/Å and the total energy converge up to 10^{−6} eV. An energy cutoff of 500 eV is used to truncate the planewave basis sets for the representation of Kohn  Sham wave functions. The BZ is integrated over 7 × 7 × 11 gamma centered kmesh in all the electronic calculations. Hybrid functional (HSE06)^{28,29} level of calculations is further carried out to verify the accuracy of PBEresults for electronic structure calculations. Tightbinding (TB) Hamiltonians are constructed using wannier90 package^{30} based on the maximally localized Wannier functions^{31} (MLWFs). The topological properties including surface spectrum and Fermi surface were analyzed based on the iterative Green’s function method^{32}. The formations energy and phonon dispersion calculations show that the studied systems are chemically and dynamically stable (Table SI and Figs S1–S3)^{21}.
Topological Insulator
Topological insulating state can be predicted in a material if it shows band inversion driven by spinorbit coupling^{5}. Since the strength of spinorbit coupling increases with heavy elements, we have systematically studied XX′ elements from group IAIIA of the periodic table. The electronic structures of all these compounds have been performed using the GGA level of theory. Here, we have mainly discussed NaSrBi and NaCaBi compounds. Our findings related to all other compounds are given in SM^{21}. Figure 1(c–f) presents the electronic structures of NaSrBi and NaCaBi compounds. In the absence of SOC, the conduction band minima (CBM) and valence band maxima (VBM) at Γ point are dominated by Sr/Ca s and Bi p_{z} orbital as shown in Fig. 1(c,e). However, inclusion of SOC results in an inverted band order between Sr/Ca s and Bi p_{z} orbitals at Γ point with a direct band gap of ~80(100) meV at Γ point for NaSrBi(NaCaBi) as shown in Fig. 1(d,f) which clearly indicate the nontrivial band topology in these systems.
The nontrivial band topology suggests an interesting surface state^{2}. Henceforth, we have studied surface electronic structure of NaSrBi and NaCaBi compounds. Initially, we constructed the slab Hamiltonian from maximally localized wannier functions (MLWF) for Na s, Ca/Sr s and Bi p orbitals. Then we have projected the band structure onto the (001) surface by using the iterative Green’s function method as implemented in Wanniertool^{32}. The surface spectra of the slab with a thickness of 200 unit cells are shown in SM (Fig. S4)^{21}.
Since the GGA method underestimates the band gap and overestimates the band inversion, we have used the hybrid functional HSE06^{28,29} to confirm the predicted nontrivial topology. It turns out that the band inversion between Sr/Ca s orbital and Bi p_{z} orbital disappears at HSE06 level and both of these materials show trivial band order as shown in Fig. 2(a,d). In order to check the evolution of nontrivial band order, we have applied external effects such as pressure, strain and doping. We find that NaSrBi (NaCaBi) system shows a topological insulating behaviour under strain (both hydrostatic as well as biaxial strain induced by epitaxial mismatch). Topological nontrivial properties also emerge in these materials if we partially/fully substitute Na by K, Rb, and Cs in these compounds.
Hydrostatic Pressure
We have performed electronic structure calculations on NaSrBi and NaCaBi systems under hydrostatic expansion. A trivial to nontrivial phase transition occurs at ~−2 GPa (~1% expansion in lattice parameter) and both the materials sustain nontrivial band order at higher expansion, as shown in Fig. 3. Since the calculated bulk modulus for NaSrBi and NaCaBi are 21 and 22 GPa respectively, it ensures that such nontrivial band ordering could be realized under low strain. Interestingly, hydrostatic compression also gives nontrivial band ordering (band inversion between Bip and Ca/Sr d bands) in these systems. Our calculations show that pd band inversion can be realized under a large hydrostatic compression (around ~20 GPa). The detailed informations of compressive strain and the associated bands are given in SM^{21} (Fig. S7). These pressure, however, are quite large and may not be easy to realize. Hence we consider only the hydrostatic expansion and investigated the nontrivial properties of both materials under 3% lattice expansion. Detailed bulk band structure and surface dispersions for both the systems with GGA and HSE06 level are shown in Sec. IV of SM^{21}.
Biaxial Strain
Next, we have investigated the electronic properties of these materials under biaxial strain (BAS). Experimentally, biaxial strain can be realized by substrate induced lattice mismatch. Accordingly, we have applied biaxial strain along [110] direction to observe the band evolution around the Fermi level. Figure 3 shows the change in the band gap and trivial to nontrivial transform under biaxial strain for NaSrBi and NaCaBi. The trivial and nontrivial regions are mentioned in the plots using arrowheads. Above 1.6% (1.4%) biaxial strain, band inversion occurs in NaSrBi (NaCaBi), which sustains its nontrivial band ordering even at higher strain. Furthermore, we have simulated the bulk band structure for NaSrBi and NaCaBi at +3% biaxial strain as shown in Fig. 2. Figure 2(b,e) clearly shows band inversion between Sr/Ca s and Bi p_{z} orbitals at Γ point. To further confirm the topological nontrivialness, we have calculated the topological Z_{2} invariant. Owing to the inversion asymmetry in the crystal structure, the parity is not a good quantum number of the Bloch eigenstates. Therefore, parity counting method proposed by Fu and Kane is not applicable here^{33}. As such we have adopted the method of Wannier charge center (WCC) evolution in half BZ to calculate the Z_{2} invariant along the k_{2} direction, as shown in Fig. 2(c,f). It is clear from the figure that the WCC evolution lines cut the reference line one (odd) and zero (even) times in the k_{3} = 0 and π planes respectively, for both the systems. Thus the pressureinduced band inversion exhibits a topological phase transition from a trivial insulator to TI.
To see the topological features, we have calculated surface spectra for NaSrBi and NaCaBi at +3% BAS along [110]. The calculated bulk electronic structures using GGA and HSE06 show similar band ordering for both the systems (see Fig. S8 of SM)^{21}. Hence it is reasonable to expect similar surface dispersion at GGA and HSE06 level of calculations. Therefore, we took the GGA functional to construct the MLWFs and then simulated the surface dispersions for TI phases of two compounds at +3% BAS along [110]. The surface dispersion is shown in Fig. 4. Since the slab calculation involves two surfaces, the corresponding surface bands and spectral intensity maps for both surface (top and bottom) are given.
In the slab model, the top surface is terminated by a XBi layer, while the bottom surface is truncated at X′Bi layer. The asymmetric surface truncation leads to different surface potentials which in turn results into two nonidentical Dirac cones lying at different energy as shown in Fig. 4. Another characteristic feature of topological surface state is the helical spin texture. To address this, we have projected the spin directions on the FS of the slab, which is located just above the DP and we find a spin momentum locking feature as shown in Fig. 4. This again confirms the topological nontrivial behavior in both the systems. Similar to most other TI materials, the surface Dirac Cone of both the systems exhibits lefthanded spin texture for top surface states (TSS). The bottom surface states, however, exhibit righthanded spin texture for the Dirac Cone in both the materials.
Doping
Doping or alloying is a promising strategy for hydrostatic expansion/compression of lattice parameters. Therefore, we have doped K, Rb, and Cs at the Na site. Doping with bigger atoms leads to an expansion of lattice parameters, which in turn naturally causes a band inversion instead of a physical hydrostatic expansion imposed on the material. A detailed analysis of such findings, by doping K, Rb or Cs at Na sites in both NaSrBi and NaCaBi are discussed in SM^{21} (see Fig. S9).
Nodal Line Semimetal
In a topological nodal line semimetal, the bands cross each other due to band inversion and they form a closed loop instead of discrete points around the Fermi level. In contrast to WSs, which have an open arc like FS^{16}, NLSs are characterized by the 1D closed ring (a line shape FS) and 2D topological drumhead surface state^{17}. The distinguishing characteristic of these drumhead surface state is that they are nearly dispersionless and therefore, have a large density of states near E_{F}. Such flat bands and large density of states could provide a potential play ground for the high temperature superconductivity, magnetism, and other related phenomenons.
Here we demonstrate that the materials NaSrBi (NaCaBi) can be transformed into a nodal line semimetal by complete replacement of Na atom by Rb or Cs. Our detailed calculations predict that the class of systems XX′ Bi (X = Rb, Cs and X′ = Ca, Sr) are NLS and show drumheadlike surface flat band. Bulk band structures for all these systems are shown in SM^{21} (see Sec. VIII). Of these, we have chosen RbCaBi for detailed analysis here. Figure 5(a,c) shows the band structure of RbCaBi with an inverted band order and 1D torus like bulk Fermi surface (where conduction and valence band crosses each other along a line) respectively in the absence of SOC. At the Γ point, CBM and VBM have \({A^{\prime\prime} }_{2}\) and \({A^{\prime} }_{1}\) representation of D_{3h}. Along ΓM, it becomes \(A^{\prime\prime} \) and A′ representation of C_{s} where as it takes B_{2} and A_{1} representation of C_{2v} along ΓK, as indicated in Fig. 5(a). The D_{3h} little group at Γ point ensures the presence of σ_{h} mirror plane perpendicular to the C_{3} principle axis. For spinless case, if two bands belonging from two different irreducible representations (IRs) (here IRs are differed by the eigen value of σ_{h} symmetry) cross each other in the σ_{h} plane (σ_{h} plane contains M, Γ and K high symmetry points in the BZ), then the band hybridization will be prohibited due to the point group symmetry protection. All the crossing points on σ_{h} plane will have now band degeneracy along a one dimensional loop. Hence the two intercrossing bands form the nodal loop structure in the BZ. Therefore, in RbCaBi compound the lowest conduction band and highest valence band cross each other along the nodal line and protected from opening up a gap along the nodal loop. Other systems also show similar nature of band structure, confirming the NLS behaviour (see SM Fig. S10)^{21}.
From the perspective of bulk boundary correspondence, topologically nontrivial drumheadlike surface states are expected to appear either inside or outside the projected nodal loop on the surface of NLS RbCaBi. In order to calculate the surface states, we have constructed tight binding Hamiltonian using the method of MLWFs and the surface states are projected onto (001) surface using the iterative scheme of Green’s function technique. Interestingly, we found a nearly flat surface band which is nestled between two bulk Dirac cones on the (001) surface, as shown in Fig. 5(d,e).
Further, we take SOC effect into consideration and found that a little gap is opened along the nodal line in the bulk band structure (see Fig. 5(b)). Our first principle calculations show that two set of bands with same IRs Λ_{5}, which hybridize along ΓK and open up ~10 meV gap at nodal point. However, along ΓM direction the hybridized gap between the pair of IRs Δ_{4}, is almost zero. Similar gap opening along the nodal line is also observed in several other NLS systems e.g. Cu_{3} PdN(~60 meV)^{34,35}, ZrSiS(~20 meV)^{36,37}, TiB_{2} (~25 meV)^{38,39}, CaAgBi(~80 meV)^{40}, CaPd (~27 meV)^{41} and so on. We have also carried out the surface density of states calculation for our material RbCaBi under SOC effect. Indeed we have found drumhead like surface states in (001) surface as shown in Fig. 5(f,g). This, however, is a common feature in almost all the existing NLS compounds^{36,37,38,39,41,42}, whenever SOC effect is taken into account in any first principles calculation. Even the experimentally reported other NLS systems, such as TiB_{2}, ZrSiS, show somewhat concave surface bands as obtained in our case for RbCaBi [Fig. 5(f,g)]. As such, we believe that RbCaBi can be a promising candidate for NLS and worthy of careful experimental investigation.
Conclusion
In summary, using the first principles calculations, we have predicted topologically nontrivial phases including nodal line semimetal states in a series of compounds belonging to the class of XX′ Bi (X = Na, K, Rb, Cs; X′ = Ca, Sr). We closely engineer the topology of the bands by applying hydrostatic compression/expansion, biaxial strain and external doping, which in turn helps to achieve nontrivial band order. Nontriviality is further confirmed by investigating Wannier charge center, surface dispersion and spin texture. NaSrBi and NaCaBi are found to be strong TI under hydrostatic and biaxial strain. Doping or alloying is another efficient way to control the nontrivial order. Partial or complete replacement of Na by Rb, Cs or K in the compound NaX′ Bi (X′ = Sr, Ca) helps to intrigue the TI or the NLS phase. We have also studied 1D bulk Fermi surface and the topological flat surface band properties of systems showing NLS behavior. Possibility of experimental synthesis is confirmed by presenting the chemical stability of all the compounds. We endorse a much higher predictability power of the present report due to the use of HSE06 functionals compared to most of the similar previous reports based on GGA functional. Such accurate abinitio predictions serve as a guiding path for the discovery of new novel materials.
References
 1.
Moore, J. E. The birth of topological insulators. Nature 464, 194–198 (2010).
 2.
Hasan, M. Z. & Kane, C. L. Colloquium: Topological insulators. Rev. Mod. Phys. 82, 3045 (2010).
 3.
Kane, C. L. & Mele, E. J. Z_{2} Topological Order and the Quantum Spin Hall Effect. Phys. Rev. Lett. 95, 146802 (2005).
 4.
König, M. et al. Quantum Spin Hall Insulator State in HgTe Quantum Wells. Science 318, 766–770 (2007).
 5.
Bernevig, B. A., Hughes, T. L. & Zhang, S.C. Quantum Spin Hall Effect and Topological Phase Transition in HgTe Quantum Wells. Science 314, 1757–1761 (2006).
 6.
Lin, H. et al. HalfHeusler ternary compounds as new multifunctional experimental platforms for topological quantum phenomena. Nature Materials 9, 546549 (2010).
 7.
Mondal, C., Kumar, S. & Pathak, B. Topologically protected hybrid states in graphene–stanene–graphene heterojunctions. J. Mater. Chem. C. 6, 1920–1925 (2018).
 8.
Xu, Y. et al. LargeGap Quantum Spin Hall Insulators in Tin Films. Phys. Rev. Lett. 111, 136804 (2013).
 9.
Hsieh, D. et al. Observation of TimeReversalProtected SingleDiracCone TopologicalInsulator States in Bi_{2}Te_{3} and Sb_{2}Te_{3}. Phys. Rev. Lett. 103, 146401 (2009).
 10.
Barman, C. K. & Alam, A. Topological phase transition in the ternary halfHeusler alloy ZrIrBi. Phys. Rev. B 97, 075302 (2018).
 11.
Sun, Y. et al. Straindriven onset of nontrivial topological insulating states in Zintl Sr_{2}X compounds (X=Pb, Sn). Phys. Rev. B 84, 165127 (2011).
 12.
Yan, B., Müchler, L., Qi, X.L., Zhang, S.C. & Felser, C. Topological insulators in filled skutterudites. Phys. Rev. B 85, 165125 (2012).
 13.
Sun, Y., Chen, X.Q., Yunoki, S., Li, D. & Li, Y. New Family of ThreeDimensional Topological Insulators with Antiperovskite Structure. Phys. Rev. Lett. 105, 216406 (2010).
 14.
Dzero, M., Sun, K., Galitski, V. & Coleman, P. Topological Kondo Insulators. Phys. Rev. Lett. 104, 106408 (2010).
 15.
Liu, Z. K. et al. Discovery of a ThreeDimensional Topological Dirac Semimetal, Na3Bi. Science 343, 864–867 (2014).
 16.
Xu et al. Discovery of a Weyl fermion semimetal and topological Fermi arcs. Science 349, 613–617 (2015).
 17.
Bian, G. et al. Topological nodalline fermions in spinorbit metal PbTaSe_{2}. Nat. Commun. https://doi.org/10.1038/ncomms10556 (2016).
 18.
Li, H. et al. Negative magnetoresistance in Dirac semimetal Cd3As2. Nat. Commun. https://doi.org/10.1038/ncomms10301 (2016).
 19.
Huang, X. et al. Observation of the ChiralAnomalyInduced Negative Magnetoresistance in 3D Weyl Semimetal TaAs. Phys. Rev. X 5, 031023 (2015).
 20.
Fang, C., Chen, Y., Kee, H.Y. & Fu, L. Topological nodal line semimetals with and without spinorbital coupling. Phys. Rev. B 92, 081201(R) (2015).
 21.
See Supplementary material for more details (2018).
 22.
Sun, Y. et al. Pressureinduced topological insulator in NaBaBi with righthanded surface spin texture. Phys. Rev. B 93, 205303 (2016).
 23.
Kresse, G. & Joubert, D. From ultrasoft pseudopotentials to the projector augmentedwave method. Phys. Rev. B 59, 1758 (1999).
 24.
Blöchl, P. E. Projector augmentedwave method. Phys. Rev. B 50, 17953 (1994).
 25.
Kresse, G. & Furthmüller, J. Efficient iterative schemes for ab initio totalenergy calculations using a planewave basis set. Phys. Rev. B 54, 11169–11186 (1996).
 26.
Perdew, J. P. & Zunger, A. Selfinteraction correction to densityfunctional approximations for manyelectron systems. Phys. Rev. B 23, 5084–5079 (1981).
 27.
Perdew, J. P., Burke, K. & Ernzerhof, M. Generalized Gradient Approximation Made Simple. Phys. Rev. Lett. 77, 3865 (1997).
 28.
Heyd, J., Scuseria, G. E. & Ernzerhof, M. Hybrid functionals based on a screened Coulomb potential. J. Chem. Phys. 118, 8207 (2003).
 29.
Peralta, J. E., Heyd, J., Scuseria, G. E. & Martin, R. L. Spinorbit splittings and energy band gaps calculated with the HeydScuseriaErnzerhof screened hybrid functional. Phys. Rev. B 74, 073101 (2006).
 30.
Mostofi, A. A. et al. wannier90: A tool for obtaining maximallylocalised Wannier functions. Comput. Phys. Commun. 185, 2309 (2014).
 31.
Marzari, N., Mostofi, A. A., Yates, J. R., Souza, I. & Vanderbilt, D. Maximally localized Wannier functions: Theory and applications. Rev. Mod. Phys. 84, 1419 (2012).
 32.
Wu, Q., Zhang, S., Song, H.F., Troyer, M. & Soluyanov, A. A. WannierTools: An opensource software package for novel topological materials. Computer Physics Communications, https://doi.org/10.1016/j.cpc.2017.09.033 (2017).
 33.
Fu, L. & Kane, C. L. Topological insulators with inversion symmetry. Phys. Rev. B 76, 045302 (2007).
 34.
Kim, Y., Wieder, B. J., Kane, C. L. & Rappe, A. M. Dirac Line Nodes in InversionSymmetric Crystals. Phys. Rev. Lett. 115, 036806 (2015).
 35.
Yu, R. et al. Topological NodeLine Semimetal and Dirac Semimetal State in Antiperovskite Cu3PdN. Phys. Rev. Lett. 115, 036807 (2015).
 36.
Schoop, L. M. et al. Dirac cone protected by nonsymmorphic symmetry and threedimensional Dirac line node in ZrSiS. Nat. Commun. 7, 11696 (2016).
 37.
Neupane, M. et al. Observation of topological nodal fermion semimetal phase in ZrSiS. Phys. Rev. B 93, 201104 (2016).
 38.
Liu, Z. et al. Experimental Observation of Dirac Nodal Links in Centrosymmetric Semimetal TiB2. Phys. Rev. X 8, 031044 (2018).
 39.
Zhang, X., Yu, Z.M., Sheng, X.L., Yang, H. Y. & Yang, S. A. Coexistence of fourband nodal rings and triply degenerate nodal points in centrosymmetric metal diborides. Phys. Rev. B 95, 235116 (2017).
 40.
Yamakage, A., Yamakawa, Y., Tanaka, Y. & Okamoto, Y. LineNode Dirac Semimetal and Topological Insulating Phase in Noncentrosymmetric Pnictides CaAgX (X = P, As). J. Phys. Soc. Jpn. 85, 013708 (2016).
 41.
Liu, G., Jin, L., Dai, X., Chen, G. & Zhang, X. Topological phase with a criticaltype nodal line state in intermetallic CaPd. Phys. Rev. B 98, 075157 (2018).
 42.
Yongping, D. et al. CaTe: a new topological nodeline and Dirac semimetal. Npj Quant. Mater. 2, 3 (2017).
Acknowledgements
This work is financially supported by DST SERB (EMR/2015/002057), India. We thank IIT Indore for the lab and computing facilities. C.M., C.K.B. and S.K. thank MHRD for research fellowship. AA acknowledges National Center for Photovo ltaic Research and Education (NCPRE), IIT Bombay for possible funding to support this research.
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All authors potentially contributed to the intellectual contents of this work. C.M. has designed the initial stage of this project. C.M. and C.K.B. have performed all the calculations. S.K. contributed in the technical part of the first principle calculations. The theoretical analysis and discussions were done by C.M., C.K.B., B.P. and A.A. All authors contributed in writing the manuscript. B.P. was responsible for the overall research plan and integraton among the different research units.
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Mondal, C., Barman, C.K., Kumar, S. et al. Emergence of Topological insulator and Nodal line semimetal states in XX′Bi (X = Na, K, Rb, Cs; X′ = Ca, Sr). Sci Rep 9, 527 (2019). https://doi.org/10.1038/s41598018368690
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