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Rui Yu, Hongming Weng, Zhong Fang, Xi Dai, [Xiao Hu](https://orcid.org/0000-0001-6880-402X)

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[Topological Node-Line Semimetal and Dirac Semimetal State in Antiperovskite  Cu 3 PdN](https://mdr.nims.go.jp/datasets/57529c06-e037-4d99-854e-bc92f8ac4c29)

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untitledTopological Node-Line Semimetal and Dirac Semimetal State in Antiperovskite Cu3PdNRui Yu,1 Hongming Weng,2,3,* Zhong Fang,2,3 Xi Dai,2,3 and Xiao Hu1,†1International Center for Materials Nanoarchitectonics (WPI-MANA),National Institute for Materials Science, Tsukuba 305-0044, Japan2Beijing National Laboratory for Condensed Matter Physicsand Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China3Collaborative Innovation Center of Quantum Matter, Beijing 100190, China(Received 22 April 2015; published 17 July 2015)Based on first-principles calculation and effective model analysis, we propose that the cubicantiperovskite material Cu3PdN can host a three-dimensional (3D) topological node-line semimetal statewhen spin-orbit coupling (SOC) is ignored, which is protected by the coexistence of time-reversal andinversion symmetry. There are three node-line circles in total due to the cubic symmetry. Drumheadlikesurface flat bands are also derived. When SOC is included, each node line evolves into a pair of stable 3DDirac points as protected by C4 crystal symmetry. This is remarkably distinguished from the Diracsemimetals known so far, such as Na3Bi and Cd3As2, both having only one pair of Dirac points. Once C4symmetry is broken, the Dirac points are gapped and the system becomes a strong topological insulatorwith (1;111) Z2 indices.DOI: 10.1103/PhysRevLett.115.036807 PACS numbers: 73.20.At, 71.55.Ak, 73.43.−fIntroduction.—Band topology in condensed-mattermaterial has attracted broad interest in recent years. Ithas benefitted from the discovery of two-dimensional (2D)and three-dimensional (3D) topological insulators (TIs)[1,2]. These materials exhibit a bulk energy gap betweenthe valence and conduction bands, similar to normalinsulators, but with gapless boundary states that areprotected by the band topology of bulk states.Topological classification has also been proposed for 3Dmetals [3]. The topological invariant, so called Fermi Chernnumber, can be defined on a closed 2D manifold, such asthe Fermi surface, in 3D momentum space [3–5]. This isessentially the same as the Chern number defined in thewhole 2D Brillouin zone for insulators. Up to now, threetypes of nontrivial topological metals have been proposed.They are the Weyl semimetal [6–9], Dirac semimetal[4,10], and node-line semimetal (NLS) [11,12]. All ofthem have band crossing points due to band inversion [13].For Weyl and Dirac semimetals, the band crossing points,which compose the Fermi surface, are located at differentpositions. For the NLS, the crossing points around theFermi level form a closed loop. The breakthrough intopological semimetal research happened in the materialrealization of the Dirac semimetal state in Na3Bi andCd3As2, which were first predicted theoretically [4,10]and then confirmed by several experiments [14–17].Starting from the Dirac semimetal, one can obtain theWeyl semimetal by breaking either time-reversal [7,8,18] orinversion symmetry [19–22]. Among them, the predictionof the Weyl semimetal state in the nonmagnetic andnoncentrosymmetric TaAs family [23,24] has been verifiedby experiments [25–30].The existence of the NLS has been proposed in Bernalgraphite [31,32] and other all-carbon allotropes, includinghyperhoneycomb lattices [33] and the Mackay-Terronescrystal (MTC) [12]. In Ref. [12], it was discussed that whenspin-orbit coupling (SOC) is neglected and band inversionhappens, the coexistence of time-reversal and inversionsymmetry protects the NLS in 3D momentum space. It isheuristic that compounds with a light element can alsoexhibit nontrivial topology, which is totally different fromthe common wisdom that strong SOC in heavy-elementcompounds is crucial for topological quantum states. TheNLS state was also proposed in photonic systems with time-reversal and inversion symmetry [34]. It was also shown thatmirror symmetry, instead of inversion symmetry, togetherwith time-reversal symmetry protects the NLS when SOC isneglected [35,36]. TaAs [23], Ca3P2 [37], and LaX (X ¼ N,P, As, Sb, Bi) [38] belong to this type.In the present work, based on first-principles calculationsand effective model analysis, we demonstrate that theantiperovskite Cu3PdN is a new candidate for realizingthe NLS and drumheadlike surface flat bands [12,39–43],which may open an important route to achieving high-temperature superconductivity [44–46]. Strong SOC willdrive the NLS in Cu3PdN into a Dirac semimetal state withthree pairs of Dirac points, leading to exotic surface Fermiarcs which can be observed on various surfaces of thismaterial. This is very unique since the Dirac semimetalsknown so far, Na3Bi and Cd3As2, have only one pair ofDirac points. When the C4 crystal symmetry in Cu3PdNis broken, the Dirac points are gapped and the systembecomes a strong TI with Z2 indices (1;111). It is wellknown that the cooperative interactions among lattice,charge, and spin degrees of freedom make antiperovskitesPRL 115, 036807 (2015) P HY S I CA L R EV I EW LE T T ER Sweek ending17 JULY 20150031-9007=15=115(3)=036807(5) 036807-1 © 2015 American Physical Societyhttp://dx.doi.org/10.1103/PhysRevLett.115.036807http://dx.doi.org/10.1103/PhysRevLett.115.036807http://dx.doi.org/10.1103/PhysRevLett.115.036807http://dx.doi.org/10.1103/PhysRevLett.115.036807exhibit a wide range of interesting physical properties, suchas superconductivity [47], giant magnetoresistance [48],negative thermal expansion [49], and magnetocaloric effect[50]. Our prediction of a 3D NLS and Dirac semimetalstate in antiperovskites provides a promising platform formanipulating these exotic properties in the presence ofnontrivial topology.Crystal structure and methodology.—Perovskite has aformula ABX3, where A and B are cations and X is ananion. The antiperovskite is similar to perovskite butswitching the position of anion and cation, namely, inABX3 X stands for an electropositive cation, while A for ananion as shown in Fig. 1(a) for Cu3PdN. Here we performdensity functional calculations by using the Vienna ab initiosimulation package [51] with generalized gradient approxi-mation [52] and the projector augmented-wave method[53]. The surface band structures are calculated in a tight-binding scheme based on the maximally localized Wannierfunctions (MLWFs) [54], which are projected from the bulkBloch wave functions.Electronic structures.—The band structure of Cu3PdN isshown in Fig. 2. The fat-band structure calculations suggestthat the valence and conduction bands are dominated byPd 4d (blue) and Pd 5p (red) states. They also indicate bandinversion at R point, where the Pd 5p is lower than Pd 4dby about 1.5 eV. To overcome possible overestimation ofband inversion [55], we employ hybrid density functionals[56] to confirm the existence of band inversion, while theenergy gap at R point is slightly reduced to 1.1 eV.Without SOC the occupied and unoccupied low energybands are triply degenerate at the R point (six-folddegenerate if spin is considered). These states belong tothe 3D irreducible representations Γ−4 and Γþ5 of the Ohgroup at the R point, respectively. We emphasize that,unlike the situation in a typical TI such as the Bi2Se3 familycompounds [57], the band inversion in Cu3PdN is not dueto SOC. To illustrate the band inversion process in Cu3PdNexplicitly, we calculated energy levels of Γþ5 and Γ−4 bandsat the R point for Cu3PdN under different hydrostaticstrains. As presented in Fig. 2(b), the band inversionhappens at a ¼ 1.11a0 and the inversion energy increaseswhen further compressing the lattice. The intriguing pointof the Cu3PdN band structure without SOC is that the bandcrossings due to the band inversion form a node-line circlebecause of the coexistence of time-reversal and inversionsymmetry as addressed in Ref. [12].The protection of a node line can be inferred from thetopological number of the form [39–43]γ ¼ICAðkÞ · dk; ð1Þwhere AðkÞ is the Berry connection of the occupied states,C is a closed loop in the momentum space. If C is piercedby the node line, one has γ ¼ π, otherwise γ ¼ 0. Below weprove the existence of the node line from the argument ofcodimension. A general 2 × 2 Hamiltonian is enough fordescribing the two crossing bands,HðkÞ ¼ gxðkÞσx þ gyðkÞσy þ gzðkÞσz; ð2Þwhere k ¼ ðkx; ky; kzÞ, gx;y;zðkÞ are real functions and σx;y;zare Pauli matrices characterizing the space of the twocrossing bands, which are mainly from pz and dxy orbitalsof Pd in Cu3PdN. The coexistence of time-reversal andinversion symmetry leads to gy ¼ 0, gx and gz being oddand even functions of k [12]. Up to the lowest order of k,gxðkÞ and gzðkÞ are given asFIG. 1 (color online). (a) Crystal structure of antiperovskiteCu3PdN with Pm̄3m (No. 221) symmetry. A nitrogen atom is atthe center of the cube and is surrounded by octahedral Cu atoms.Pd is located at the corner of the cube. (b) Bulk and projected(001) surface Brillouin zone. The three node-line rings (orange)and three pairs of Dirac points (red), without and with SOCincluded, respectively, are schematically shown.FIG. 2 (color). (a) Electronic band structure without SOC,where the component of Pd 5p (4d) orbitals is proportional to thewidth of the red (blue) curves. Band inversion between p and dorbits happens at the R point. (b) Evolution of energy levelsof Γþ5 and Γ−4 at R under hydrostatic pressure. Band inversionhappens when a < 1.11a0. (c) Electronic band structure withSOC included. A small gap is opened in the R-X direction whilethe Dirac point in the R-M direction is stable and protected bycrystal symmetry C4 rotation.PRL 115, 036807 (2015) P HY S I CA L R EV I EW LE T T ER Sweek ending17 JULY 2015036807-2gxðkÞ ¼ γkxkykz; ð3ÞgzðkÞ ¼ M − Bðk2x þ k2y þ k2zÞ; ð4Þconsidering the crystal symmetry at the R point. Theeigenvalues of Eq. (2) are EðkÞ ¼ �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffig2xðkÞ þ g2zðkÞp.Node lines appear when gxðkÞ ¼ 0 and gzðkÞ ¼ 0. It iseasy to check that gzðkÞ ¼ 0 can be satisfied only in thecase of MB > 0, which is nothing but the band inversioncondition. When band inversion happens, there alwaysexists three closed node lines in momentum space as shownin Fig. 1(b) which are the solutions of gxðkÞ ¼ gzðkÞ ¼ 0.When SOC is considered, the six-fold degenerate statesat R are split into one fourfold and one twofold degeneratestate. As shown in Fig. 2(c), there are two fourfolddegenerate states close to the Fermi energy: the occupiedone with Γþ8 symmetry and the unoccupied one with Γ−8symmetry. Moving from R to X, the symmetry is lowered toC2v. The fourfold degenerate states at R are split intotwofold degenerate states along the R-X direction. Thefirst-principles calculation shows that two sets of bandswith the same Γ5 symmetry are close to the Fermi energyand a gap ∼0.062 eV is opened at the intersection as shownin the inset of Fig. 2(c). In order to reduce this SOCsplitting and thus achieving the NLS, one can replace Pdwith lighter elements, such as Ag and Ni. Along the R-Mdirection, the symmetry is characterized by the C4v doublegroup. As indicated in Fig. 2(c), the two sets of bands closeto the Fermi energy belong to the Γ7 and Γ6 representation,respectively. They are decoupled and the crossing point onthe R-M path is unaffected by SOC. They form a Diracnode near the Fermi energy as shown in Fig. 2(c), which isprotected by the crystal symmetry C4 rotation [4,10,58].Once the C4 rotational symmetry is broken, the Dirac nodewill be gapped, and the parities of the occupied states ateight time-reversal invariant momenta (TRIM) point areshown in Table. I. The Z2 indices are (1;111), indicating astrong TI. The band structure of Cu3PdN is different fromthat of antiperovskite Sr3PbO [59,60], where the bandinversion happens at Γ point and the involved bands belongto the same irreducible presentation, which leads to anti-crossing along the Γ-X direction.Effective Hamiltonian for the 3D Dirac fermions.—In order to better understand the band crossing and gapopening discussed above, we derive a low energy effectiveHamiltonian based on the theory of invariants in a similarway as that for the Bi2Se3 family [57].We first construct the model Hamiltonian along the R-Mdirection parallel to the kz axis. The symmetrical operationsin this direction contain the crystalline C4v symmetry andinversion with parity–time-reversal (PT) symmetry. Asdiscussed above, the wave functions of low-energy statesalong the R-M direction are Γ6 symmetry with angularmomentum jz ¼ �1=2 and Γ7 with jz ¼ �3=2. Therefore,the model Hamiltonian respecting both C4v and PTsymmetries can be written asHRM ¼26664M1 0 c2kþ c3k2− þ c4k2þM1 c4k2− þ c3k2þ −c2k−M2 0† M237775 ð5Þup to the second order of k in the basis offjjz¼ 12ip;jjz¼−12ip;jjz¼ 32id;jjz¼−32idg, where M1 ¼mp þm11kz þm12k2, M2 ¼ md þm21kz þm22k2, k2 ¼k2x þ k2y þ k2z and k� ¼ kx � iky. TheC4 rotation symmetryrequires the matrix element between jjz ¼ 12i and jjz ¼ − 32ito take the form of k2� in order to conserve the total angularmoment along the z direction. Because of PT symmetryand mirror symmetry, all parameters in Eq. (5) can bechosen as real. Their values can be derived by fitting thedispersions to those of first-principles calculations [61].On the kz axis, the effective Hamiltonian is diagonal andthe j�1=2i sets and j�3=2i sets are decoupled. Since theenergies of p and d orbitals are inverted at the R point,namely, mp < md and m22 < 0 < m12, the j�1=2i andj�3=2i bands cross at the kcz ¼ f½ðm11 −m21Þ2 − 4ðm12 −m22Þðmp −mdÞ�1=2 − ðm11 −m21Þg=½2ðm12 −m22Þ� point.The model Hamiltonian along the R-X direction can bederived in the same way with C2v and PT symmetries.From the first-principles calculations, the states near theFermi energy are characterized by Γ5 with jz ¼ j�1=2i.The model Hamiltonian can be written asHRX ¼26664M1 0 M3 c1k−M1 c1kþ −M3M2 0† M237775 ð6Þup to the second order of k in the basis set offjjz ¼ 12ip; jjz ¼ − 12ip; jjz ¼ 12id; jjz ¼ − 12idg. Here, kz isset as along the R-X direction. The term M3 ¼m3 þm31kz þm32k2 conserves the total angular momen-tum along the z direction. It couples the going-down Γ5states and going-up Γ5 states and opens a gap at thecrossing point shown in Fig. 2(c).Surface states.—The band inversion and the 3D Diraccones in Cu3PdN suggest the presence of topologicallynontrivial surface states. They are calculated based onthe tight-binding Hamiltonian from the MLWF [5,54].The obtained band structures and surface density ofstates (DOS) on semi-infinite (001) surface are presentedin Fig. 3.TABLE I. Parity product of occupied states at the TRIM points.The Z2 indices are (1;111).TRIM points R Γ M ð×3Þ X ð×3ÞParity product þ − − −PRL 115, 036807 (2015) P HY S I CA L R EV I EW LE T T ER Sweek ending17 JULY 2015036807-3Without SOC, the bulk state is the same as the MTC [12]and there exists surface flat bands nestled inside theprojected node-line ring on the (001) surface, namely,the “drumhead” states as shown in Figure 3(a). Thepeaklike DOS from these nearly flat bands is also clearlyshown. The small dispersion of this drumhead state comesfrom the fact that the node-line ring is not necessarily onthe same energy level due to the particle-hole asymmetry[11,12,46]. This result is consistent with the correspon-dence between the Dirac line in bulk and the flat band at theboundary as established in Refs. [39–41,43]. Such 2D flatbands and nearly infinite DOS are proposed as a route toachieving high-temperature superconductivity [44–46].In the presence of SOC, the node-line ring will be gappedin general. However, there is an exception. For example, inthe TaAs family each ring evolves into three pairs of Weylnodes [23]. In Cu3PdN, each ring is driven into one pair ofDirac nodes. The (001) surface state band structure inFig. 3(b) clearly shows the gapped bulk state along theΓ̄-M̄ direction and the existence of a surface Dirac cone dueto topologically nontrivial Z2 indices as seen in Na3Bi [4]and Cd3As2 [10]. The bulk band structure along R-X andR-M overlap each other when projected onto (001) surfacealong the X̄-M̄ path. The bulk Dirac cones are hidden byother bulk bands. Therefore, it is difficult to identify thedetailed connection of Fermi arcs in the Fermi surface plotas shown in Fig. 4, though some eyebrowlike Fermi arcscan be clearly seen around these projected Dirac nodes.Conclusion.—In summary, we propose that 3D topo-logical node-line semimetal states can be obtained in anonmagnetic and centrosymmetric system Cu3PdN. Thedrumheadlike surface flat bands nestled in a projectednode-line ring have been obtained. Including SOC willdrive each node-line ring into one pair of Dirac points tohost a Dirac semimetal state. The surface Dirac cone andthe Fermi arcs around the projected Dirac cones areobserved. The existence of multiple pairs of 3D Diracpoints distinguishes this system from other Dirac semi-metals with only one pair. The cubic antiperovskitestructure of Cu3PdN makes it a good platform for manipu-lating ferromagnetism, ferroelectricity, and superconduc-tivity realized in a broad class of materials with perovskitestructure in the presence of nontrivial topology.This work was supported by the WPI Initiative onMaterialsNanoarchitectonics, andGrant-in-Aid forScientificResearch under the Innovative Area “Topological QuantumPhenomena” (No. 25103723), MEXTof Japan. H.M., Z. F.,and X. D. were supported by the National NaturalScience Foundation of China, the 973 Program of China(No. 2011CBA00108 and No. 2013CB921700), and the“Strategic Priority Research Program (B)” of the ChineseAcademy of Sciences (No. XDB07020100).Note added.—Recently, we became aware of a similar workby Kim et al. [62].*hmweng@iphy.ac.cn†Hu.Xiao@nims.go.jp[1] M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045(2010).[2] X. L. Qi and S.-C. Zhang, Rev. Mod. Phys. 83, 1057 (2011).[3] P. Hořava, Phys. Rev. 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