Topological crystalline insulator state with type-II Dirac fermions in transition metal dipnictides
Baokai Wang, Bahadur Singh, Barun Ghosh, Wei-Chi Chiu, M. Mofazzel Hosen, Qitao Zhang, Li Ying, Madhab Neupane, Amit Agarwal, Hsin Lin, Arun Bansil
TTopological crystalline insulator state with type-II Dirac fermions in transition metaldipnictides
Baokai Wang, Bahadur Singh ∗ ,
1, 2
Barun Ghosh, Wei-Chi Chiu, M. Mofazzel Hosen, Qitao Zhang, Li Ying, Madhab Neupane, Amit Agarwal, Hsin Lin, and Arun Bansil ∗ Department of Physics, Northeastern University, Boston, Massachusetts 02115, USA SZU-NUS Collaborative Center and International CollaborativeLaboratory of 2D Materials for Optoelectronic Science & Technology,Engineering Technology Research Center for 2D MaterialsInformation Functional Devices and Systems of Guangdong Province,Institute of Microscale Optoelectronics, Shenzhen University, Shenzhen, 518060, China Department of Physics, Indian Institute of Technology Kanpur, Kanpur 208016, India Department of Physics, University of Central Florida, Orlando, Florida 32816, USA Institute of Physics, Academia Sinica, Taipei 11529, Taiwan
The interplay between topology and crystalline symmetries in materials can lead to a variety oftopological crystalline insulator (TCI) states. Despite significant effort towards their experimentalrealization, so far only Pb − x Sn x Te has been confirmed as a mirror-symmetry protected TCI. Here,based on first-principles calculations combined with a symmetry analysis, we identify a rotational-symmetry protected TCI state in the transition-metal dipnictide RX family, where R = Ta orNb and X = P, As, or Sb. Taking TaAs as an exemplar system, we show that its low-energyband structure consists of two types of bulk nodal lines in the absence of spin-orbit coupling (SOC)effects. Turning on the SOC opens a continuous bandgap in the energy spectrum and drives thesystem into a C T -symmetry-protected TCI state. On the (010) surface, we show the presence ofrotational-symmetry-protected nontrivial Dirac cone states within a local bulk energy gap of ∼ materials familyprovides an ideal setting for exploring the unique physics associated with type-II Dirac fermions inrotational-symmetry-protected TCIs. I. INTRODUCTION
Topological insulators (TIs) represent a new state ofquantum matter which is described by the topology ofthe bulk bands instead of a local order parameter withinthe Landau paradigm . These materials support anodd number of metallic surface states with linear energydispersion while remaining insulating in the bulk. Thetopological surface states are protected by time-reversalsymmetry and are immune to nonmagnetic impurities.Soon after the realization of TIs, the topological classi-fication of insulating electronic structures was extendedbeyond the time-reversal symmetry protected states toencompass crystalline symmetries. The topological na-ture of topological crystalline insulators (TCIs) arisesfrom the bulk crystal symmetries . Owing to the richnessof crystal symmetries many possible TCI states are possi-ble. The first TCI state was predicted theoretically in theSnTe materials class , which was subsequently verified inexperiments by observing surface Dirac-cone states .The nontrivial topology in SnTe appears due to the pres-ence of mirror-symmetry and it is manifested by the ex-istence of an even number of Dirac-cone states over thesurface. ∗ Corresponding authors’ emails: [email protected],[email protected]
Recently a new TCI state protected by N -fold ro-tational symmetries was proposed in time-reversal-invariant systems with spin-orbit coupling (SOC) .Such rotational-symmetry-protected TCIs are distinctfrom mirror-symmetry protected TCIs and support N Dirac cones on the surface normal to the rotationalaxis. Their Dirac cone states are not restricted to high-symmetry points and can appear at generic k points inthe Brillouin zone (BZ). Also, the rotational-symmetry-protected TCIs evade the fermion multiplication theoremto drive a rotational anomaly, and harbor helical edgestates on the hinges of their surfaces parallel to the ro-tational axis. This class of TCIs can support anomaloustransport properties and it could provide a basis for re-alizing Majorana zero modes through proximity inducedsuperconductivity .Monoclinic lattices with C h point-group symmetryare ideally suited in connection with materials discov-ery of rotational-symmetry protected TCIs . C h symmetry includes a mirror plane M [010] , a two-fold ro-tational axis C , and the space inversion symmetry I . Although this symmetry group can support both amirror-symmetry protected TCI (Fig. 1(a)) as well asa rotational-symmetry protected TCI (Fig. 1(b)), theassociated gapless surface states are located on differentsurfaces. In particular, the (010) surface preserves thetwo-fold rotational symmetry C , and therefore thissurface can support rotational-symmetry protected sur-face states. Rotational-symmetry-protected TCIs have a r X i v : . [ c ond - m a t . m e s - h a ll ] N ov been predicted recently in α − Bi Br , Ca As , andBi .In this paper, we discuss the existence of a rotational-symmetry-protected TCI phase in the transition-metaldipnictides RX (R = Ta or Nb and X = P, As, or Sb)materials class with C h lattice symmetries. RX materi-als in which many intriguing properties are observed havebeen realized in experiments. For example, NbAs showsa large, non-saturating transverse magnetoresistance anda negative longitudinal magnetoresistance, which may bereflective of its non-trivial bulk band topology . Otherexperiments show that at low temperatures, thermal con-ductivity of TaAs scales with temperature as T , whilethe resistivity is independent of T , indicating breakdownof the Weidemann-Franz law and possible presence of anon-Fermi liquid state . Notably, the RX materialshave been predicted as nearly electron-hole compensatedsemimetals in which a continuous SOC driven bandgapbetween the valence and conduction bands leads to weaktopological invariants ( ν ; ν ν ν ) = (0; 111) . Re-gardless, to the best of our knowledge, a rotational-symmetry-protected TCI phase has not been discussedpreviously in the literature in the RX materials family.Our analysis reveals that the RX materials realize the C rotational-symmetry-protected TCI state. Tak-ing TaAs as an example, we show in-depth that it sup-ports two types of nodal lines in the absence of theSOC effects. Inclusion of the SOC gaps out the nodallines and drives the system into a topological state withweak topological invariants (0;111) and symmetry in-dicators ( Z Z Z ; Z ) = (111; 2). A careful inspec-tion of the topological state shows that TaAs harborsa C T symmetry-protected TCI state. To highlightthe associated nontrivial band topology, we present the(010)-surface electronic spectrum and show the existenceof rotational-symmetry-protected nontrivial Dirac conestates at generic k points within a local bulk energygap of ∼
300 meV. The Dirac cones are found to ex-hibit a unique type-II energy dispersion. In this way,our study demonstrates that transition metal dipnictidesRX could provide an experimentally viable platformfor exploring rotational-symmetry-protected TCIs withtype-II Dirac cones.The organization of the remainder of the paper is asfollows. In Sec. II, we provide the methodology andstructural details of the RX compounds. The bulk topo-logical electronic structure is explored in Sec. III. In Sec.IV, we discuss surface electronic structure and C T sym-metry protected Dirac cone states in TaAs . Finally, wesummarize conclusions of our study in Sec. V. II. COMPUTATIONAL DETAILS
Electronic structure calculations were performedwithin the framework of the density functional the-ory (DFT) using the projector augmented wave (PAW)method as implemented in the VASP suite of codes . (a) (b)(c) (d) a b c As Ta M y C [001] M y [100] [010] C [100] [001][010] FIG. 1: Schematic illustration of the Dirac-cone surface statesof (a) a mirror-symmetry protected topological crystalline in-sulator (TCI) and (b) a C rotational-symmetry protectedTCI with C h point-group symmetry. The (010) mirror plane( M y , shaded gray plane) and high-symmetry crystal axes areshown. The Dirac cones in a mirror-symmetry protectedTCI are pinned to the mirror-invariant line over the surfacewhereas they lie at generic k points on the surface normal tothe rotational axis in a rotational-symmetry-protected TCI.(c) The conventional unit cell of RX compounds. The M y (010) mirror-invariant plane and C y ( C ) rotational axisare shown. (d) The primitive Brillouin zone and its projectionon the (010) surface. The generalized-gradient-approximation (GGA) wasused to incorporate exchange-correlation effects . Anenergy cutoff of 350 eV was used for the plane-wave basisset and a Γ-centered 12 × × k -mesh was used for BZintegrations. We started with experimental lattice pa-rameters and relaxed atomic positions until the residualforces on each atom were less than 0.001 eV/˚A. We con-structed a tight-binding model with atom-centered Wan-nier functions using the VASP2WANNIER90 interface .The surface energy spectrum was obtained within the it-erative Green’s function method using the Wanniertoolspackage .Transition metal dipnictides RX crystallize in a mon-oclinic Bravais lattice with space-group C /m (No.12) . The crystal structure is shown in Fig. 1(c).The primitive unit consists of two transition-metal (R)atoms and four pnictogen (X) atoms. This crystal struc-ture supports a two-fold rotational axis C , mirror-plane symmetry M [010] , and inversion symmetry I . TheRX materials are nonmagnetic and respect time-reversalsymmetry T . The bulk BZ and the associated (010) sur-face BZ are shown in Fig. 1(d) where the high-symmetrypoints are indicated. A Y M A L V M-1-0.500.51 E - E F ( e V ) A Y M A L V M-1-0.500.51 E - E F ( e V ) As: p x As: p y Ta: d xy Ta: d x2-y2 Ta: d xz Ta: d yz Ta: d Γ VLA YM + ++ + - -- ℎ " 𝑒 $ - (a) (b)(c) (d) A M E – E F ( e V ) E – E F ( e V ) FIG. 2: Bulk band structure of TaAs (a) without and (b)with spin-orbit coupling (SOC). The orbital compositions ofbands are shown using various colors. The band crossings in(a) are seen resolved along the Γ − Y and M − A directions;these are gapped in the presence of the SOC. (c) Fermi surfaceof TaAs with electron (cyan) and hole (purple) pockets. (d)Parity eigenvalues of the valence bands at eight time-reversal-invariant momentum points in the BZ. III. BULK ELECTRONIC STRUCTURE ANDTOPOLOGICAL INVARIANTS
The bulk band structure of TaAs without and with in-cluding SOC is presented in Figs. 2(a) and 2(b), respec-tively. The orbital character of the bands (color coded)shows that bands near the Fermi level mainly arise fromthe Ta d and As p states. Without the SOC, the valenceand conduction bands are seen to cross along the Γ − Y and M − A high-symmetry directions. On the inclusionof SOC, a bandgap opens up at the band-crossing points,separating the valence and conduction bands locally ateach k point. This separation leads to well-defined bandmanifolds and facilitates the calculation of topologicalinvariants as in the insulators. Since the TaAs crystalrespects inversion symmetry, it is possible to calculatethe Z invariants ( ν ; ν ν ν ) from the parity eigenvaluesof the valence bands at the time-reversal-invariant mo-mentum points . In Fig. 2, we present these results andfind Z invariants as (0; 111). These agree well with theearlier studies and indicate that TaAs is a weak TI .We emphasize that despite the opening of a localbandgap between the valence and conduction states,TaAs preserves its semimetal character with the pres-ence of electron and hole pockets. This is seen clearlyin the Fermi surface plot of Fig. 2(c). We find one holeand four electron pockets in the BZ. The volume of theelectron and hole pockets is roughly the same, indicat-ing that TaAs is nearly an electron-hole compensatedsemimetal. This feature of the electronic spectrum coulddrive the large, nonsaturating transverse magnetoresis-tance observed experimentally in TaAs . -101 E - E F ( e V ) -101 E - E F ( e V ) -101 E - E F ( e V ) -101 E - E F ( e V ) -101 E - E F ( e V ) -101 E - E F ( e V ) Y AAY MM
SOC
ΓΓΓΓ MM LL AA (a) (b) (c)(d) (e) (f)(g) (h) E – E F ( e V ) E – E F ( e V ) E – E F ( e V ) E – E F ( e V ) E – E F ( e V ) E – E F ( e V ) FIG. 3: (a)-(f) Bulk band structure of TaAs around theselected time-reversal-invariant momentum points Y , A , and M where band inversion takes place. The top row shows bandstructure without SOC whereas the middle row shows bandstructure including SOC. (g) Two different views of the nodal-line structures in the BZ. The top figure highlights nodal linesextending across the BZs and the bottom figure highlightsnodal rings formed near the M point. (h) Schematic illus-tration of the formation of nontrivial insulating states withSOC. In order to characterize the nodal-line and TCI statesof TaAs , we examine the band-crossings in Fig. 3. In theabsence of SOC, there are band crossings along the Γ − Y and M − A symmetry lines, see Figs. 3(a)-(c). A carefulinspection shows that these band-crossings form Diracnodal lines in the BZ, see Fig. 3(g). There are two typesof nodal lines. The first type includes two non-closed spi-ral nodal-lines extending across the BZ through point A whereas the second type includes two nodal-loops nearthe M point. When SOC is included, these nodal linesare gapped (see Figs. 3(d)-(f)), and lead to a band inver-sion at the Y and A points. The band inversion at thesepoints primarily involves Ta d yz and d z orbitals. Sincethe SOC separates valence and conduction states by a lo-cal bandgap, the symmetry indicators for identifying spe-cific topological states for gapped systems now becomewell defined . In particular, the symmetry indica-tors are obtained from the full set of eigenvalues of thespace-group symmetry operators of the occupied bandsat high-symmetry points. Following Ref. , the topologi-cal phase in space group No. Z Z Z ; Z ). The computed symmetry in-dicators and topological invariants for TaAs are listed inTable 1. The results of Table 1 show that the topologicalstate of TaAs is described by weak topological invari-ants (111), non-zero rotational invariant n = 1, glideinvariant n g = 1 and inversion invariant n i = 1 . TABLE I: Calculated symmetry indicators and topological in-variants for TaAs . ( ν ; ν , ν , ν ) are the Z invariants for athree-dimensional TI, n m is the mirror-chern number for(010) plane, and n is an invariant for the two-fold rota-tional symmetry C . n g , n i and n are topologicalinvariants associated with glide symmetry, inversion symme-try and screw symmetry, respectively.( Z , Z , Z , Z ) ( ν ; ν , ν , ν ) n m n n g n i n (1,1,1,2) (0;111) 0 1 1 1 0 IV. SURFACE ELECTRONIC STRUCTURE
𝐸 = 𝐸
𝐸 = 𝐸 $ (c)(f) DP (a) (b)(d) (e) E - E F ( e V ) E - E F ( e V ) FIG. 4: (a) Surface band structure of TaAs along the high-symmetry lines in the (010) surface BZ. Constant energy con-tours at (b) E = E f and (c) E = E D , where E D = 110 meVdenotes the energy of the Dirac point (DP). Dirac points aremarked with white arrows. (d) Band structure along the k -path marked with a dashed white line in (c). (e) Constantenergy contours near the DP. The electron and hole pock-ets are noted in the plots. (f) Schematic illustration of therotational-symmetry protected Dirac cone states on the (010)surface of TaAs . The nontrivial value of n dictates that the (010)surface normal to the two-fold C rotational axis sup-ports two Dirac-cone state. In order to showcase thesestates, we present the (010) surface band structure inFig. 4, where a topological surface state centered atthe Γ point connecting the bulk valence and conductionbands can be seen clearly in Fig. 4(a), confirming thenontrivial topological nature of the material. The associ-ated Fermi surface contours are shown in Fig. 4(b). TheDirac-cone states associated with rotational-symmetry-protected TCIs generally lie at generic k points on therotationally invariant surface. Therefore, we scanned the entire (010) surface BZ and found these cones to lie at(0.1248, -0.0974) ˚ A − and (-0.1248, 0.0974) ˚ A − close tothe Γ − L direction as seen in the constant energy contoursof Fig. 4(c). Energy dispersion of these Dirac cone statesalong the path marked by the white dashed line in Fig.4(c) is shown in Fig. 4(d). The states lie within a localbandgap of ∼
300 meV and exhibit a tilted type-II energydispersion as seen from the constant-energy contours nearthe Dirac points in Fig. 4(e). The Dirac cones appear atthe touching point between the electron and hole pock-ets. At energies below the Dirac cone energy E D , thesize of the hole pockets increases while that of the elec-tron pockets shrinks. This behavior is reversed as we goto energies above E D . These results clearly indicate thatTaAs harbors a type-II rotational-symmetry-protectedDirac cone state as schematically shown in Fig. 4(f). V. SUMMARY AND CONCLUSIONS
The type-II Dirac cone state we have delineated inthis study will be interesting for exploring exotic prop-erties of TCIs. Notably, type-II Dirac cone states areprohibited in strong topological insulators with time-reversal symmetry . In contrast, they have been pre-dicted in the mirror-symmetry protected TCI family ofantipervoskites . Such type-II Dirac cones exhibit char-acteristic van Hove singularities in their density of surfacestates and have a Landau level spectrum that is distinctfrom type I Dirac states, leading to unique electronic andmagnetotransport properties . Since TaAs has beenrealized experimentally, our theoretically predicted type-II Dirac cone states in this material and the associatednontrivial properties would be amenable to experimentalverification.Our calculations on the entire family of transitionmetal dipnictides RX show that their band structureclosely resembles that of TaAs (see the Appendix A fordetails). This family of materials supports a nodal-linestructure without the SOC and transitions to a TCI statewith SOC. The symmetry indicators and topological in-variants reveal that these compounds are characterizedby weak topological invariants (111) and a nonzero rota-tional invariant n = 1. The RX family thus realizesthe rotational-symmetry-protected TCI state.In conclusion, we have identified the presence ofa rotational-symmetry-protected TCI state in thetransition-metal dipnictides materials family. TakingTaAs as an exemplar system, we show that it sup-ports a nodal-line semimetal state with two distinct nodallines without the SOC. Inclusion of the SOC gaps outthe nodal-lines and transitions the system into a topo-logical state with well-separated valence and conductionband manifolds. The symmetry indicators and topolog-ical invariants reveal that TaAs has symmetry indica-tor ( Z Z Z ; Z ) = (111; 2) and is a C rotational-symmetry-protected TCI. We further confirmed this TCIstate by calculating the (010)-surface band structure,which shows the existence of type-II Dirac-cone statesin a local bulk energy gap of 300 meV. Our study thusshows that the TaAs class of transition metal dipnictideswould provide an excellent platform for exploring TCIswith type-II Dirac fermions and the associated nontrivialproperties. ACKNOWLEDGEMENTS
The work at Northeastern University was supported bythe US Department of Energy (DOE), Office of Science,Basic Energy Sciences Grant No. DE-FG02-07ER46352,and benefited from Northeastern Universitys AdvancedScientific Computation Center and the National En- ergy Research Scientific Computing Center through DOEGrant No. DE-AC02-05CH11231. The work at Shen-zhen University was supported by the Shenzhen Pea-cock Plan (Grant No. KQTD2016053112042971) and theScience and Technology Planning Project of GuangdongProvince (Grant No. 2016B050501005). H.L. acknowl-edges Academia Sinica, Taiwan for support under In-novative Materials and Analysis Technology Exploration(AS-iMATE-107-11). BG acknowledges CSIR for the Se-nior Research Fellowship. The work at IIT Kanpur wasbenefited from the high-performance facilities of the com-puter center of IIT Kanpur. M.N. is supported by the AirForce Office of Scientific Research under award numberFA9550-17-1-0415 and the National Science Foundation(NSF) CAREER award DMR-1847962. A. Bansil, H. Lin, and T. Das, Rev. Mod. Phys. , 021004(2016). M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. , 3045(2010). X.-L. Qi and S.-C. Zhang, Rev. Mod. Phys. , 1057(2011). L. Fu, Phys. Rev. Lett. , 106802 (2011). T. H. Hsieh, H. Lin, J. Liu, W. Duan, A. Bansil, and L.Fu, Nat. Commun. , 982 (2012). P. Dziawa, B. Kowalski, K. Dybko, R. Buczko, A. Szczer-bakow, M. Szot, E. (cid:32)Lusakowska, T. Balasubramanian,B. M. Wojek, M. Berntsen, et al. , Nat. Mater. , 1023(2012). S.-Y. Xu, C. Liu, N. Alidoust, M. Neupane, D. Qian, I.Belopolski, J. Denlinger, Y. Wang, H. Lin, L. Wray, et al. ,Nat. Commun. , 1192 (2012). Y. Tanaka, Z. Ren, T. Sato, K. Nakayama, S. Souma, T.Takahashi, K. Segawa, and Y. Ando, Nat. Phys. , 800(2012). C. Fang and L. Fu, arXiv:1709.01929 (2017). Z. Song, Z. Fang, and C. Fang, Phys. Rev. Lett. ,246402 (2017). F. Schindler, A. M. Cook, M. G. Vergniory, Z. Wang,S. S. P. Parkin, B. A. Bernevig, and T. Neupert, Sci. Adv. , (2018). M. Cheng and C. Wang, arXiv:1810.12308 (2018). B. J¨ack, Y. Xie, J. Li, S. Jeon, B. A. Bernevig, and A.Yazdani, Science , 1255 (2019). Z. Song, T. Zhang, Z. Fang, and C. Fang, Nat. Commun. , 3530 (2018). C.-H. Hsu, X. Zhou, Q. Ma, N. Gedik, A. Bansil, V. M.Pereira, H. Lin, L. Fu, S.-Y. Xu, and T.-R. Chang, 2DMater. , 031004 (2019). X. Zhou, C.-H. Hsu, T.-R. Chang, H.-J. Tien, Q. Ma, P.Jarillo-Herrero, N. Gedik, A. Bansil, V. M. Pereira, S.-Y.Xu, H. Lin, and L. Fu, Phys. Rev. B , 241104 (2018). C.-H. Hsu, X. Zhou, T.-R. Chang, Q. Ma, N. Gedik, A.Bansil, S.-Y. Xu, H. Lin, and L. Fu, Proc. Natl. Acad. Sci. , 13255 (2019). B. Shen, X. Deng, G. Kotliar, and N. Ni, Phys. Rev. B ,195119 (2016). X. Rao, X. Zhao, X.-Y. Wang, H. Che, L. Chu, G. Hussain,T.-L. Xia, and X. Sun, arXiv:1906.03961 (2019). C. Xu, J. Chen, G.-X. Zhi, Y. Li, J. Dai, and C. Cao, Phys.Rev. B , 195106 (2016). Y. Luo, R. McDonald, P. Rosa, B. Scott, N. Wakeham,N. Ghimire, E. Bauer, J. Thompson, and F. Ronning, Sci.Rep. , 27294 (2016). D. Gresch, Q. Wu, G. W. Winkler, and A. A. Soluyanov,New J. Phys. , 035001 (2017). P. Hohenberg and W. Kohn, Phys. Rev. , B864 (1964). G. Kresse and J. Hafner, Phys. Rev. B , 558 (1993). G. Kresse and J. Furthmller, Comput. Mater. Sci. , 15(1996). G. Kresse and J. Furthm¨uller, Phys. Rev. B , 11169(1996). G. Kresse and D. Joubert, Phys. Rev. B , 1758 (1999). J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev.Lett. , 3865 (1996). A. A. Mostofi, J. R. Yates, Y.-S. Lee, I. Souza, D. Vander-bilt, and N. Marzari, Comput. Phys. Commun. , 685(2008). Q. Wu, S. Zhang, H.-F. Song, M. Troyer, and A. A.Soluyanov, Comput. Phys. Commun. , 405 (2018). K. Wang, D. Graf, L. Li, L. Wang, and C. Petrovic, Sci.Rep. , 7328 (2014). Y. Li, L. Li, J. Wang, T. Wang, X. Xu, C. Xi, C. Cao, andJ. Dai, Phys. Rev. B , 121115 (2016). Y.-Y. Wang, Q.-H. Yu, P.-J. Guo, K. Liu, and T.-L. Xia,Phys. Rev. B , 041103 (2016). D. Wu, J. Liao, W. Yi, X. Wang, P. Li, H. Weng, Y. Shi,Y. Li, J. Luo, X. Dai, and Z. Fang, Applied Physics Letters , 042105 (2016). S. Rundqvist, Nature , 847 (1966). L. Fu and C. L. Kane, Phys. Rev. B , 045302 (2007). F. Tang, H. C. Po, A. Vishwanath, and X. Wan, Nat. Phys. , 470 (2019). H. C. Po, A. Vishwanath, and H. Watanabe, Nat. Com-mun. , 50 (2017). T. Zhang, Y. Jiang, Z. Song, H. Huang, Y. He, Z. Fang,H. Weng, and C. Fang, Nature , 475 (2019). M. G. Vergniory, L. Elcoro, C. Felser, N. Regnault, B. A.Bernevig, and Z. Wang, Nature , 480 (2019). F. Tang, H. C. Po, A. Vishwanath, and X. Wan, Nature , 486 (2019). C.-K. Chiu, Y.-H. Chan, X. Li, Y. Nohara, and A. P.
Schnyder, Phys. Rev. B , 035151 (2017). B. Singh, X. Zhou, H. Lin, and A. Bansil, Phys. Rev. B , 075125 (2018). B. Ghosh, S. Mardanya, B. Singh, X. Zhou, B. Wang,T.-R. Chang, C. Su, H. Lin, A. Agarwal, and A. Bansil,arXiv:1905.12578 (2019). Note that we have only considered the rotational symmetryprotected topological states on the (010) surface of TaAs in Sec. IV. Exploration of other nonzero invariants will beinteresting. Appendix A: Band structure of transition metaldipnictide materials family
In Figs. 5 and 6, we present band structure of vari-ous members of the transition metal dipnictide RX fam-ily without and with spin-orbit coupling (SOC), respec-tively. We consider six members of this family, whichare listed in Table II along with the lattice parametersused in the calculations. All these materials exhibit bandcrossings along the Γ − Y and M − A directions, whichform nodal lines in the Brillouin zone (BZ) without theSOC similar to the case of TaAs . Inclusion of SOC,gaps out the nodal lines and separates the valence andconduction bands locally at each k -point, and yields the C -protected TCI states. The size of the invertedbandgap increases in the order P < As < Sb and Nb < Ta, which is consistent with the increasing order of theintrinsic SOC strength of the constituents atoms. Wefind that all materials have the same symmetry indica-tors with ( Z Z Z ; Z ) = (111; 2) and that the associatedtopological invariants are similar to those of TaAs . TABLE II: Experimental lattice constants a , b , c , and β andrelaxed-internal ionic positions of six members of the RX materials family .NbP NbAs NbSb TaP TaAs TaSb a (˚A) 8.872 9.354 10.233 8.861 9.329 10.233 b (˚A) 3.266 3.381 3.630 3.268 3.385 3.645 c (˚A) 7.510 7.795 8.328 7.488 7.753 8.292 β x R z R x X z X x X z X A YM A L V M-1-0.500.51 E - E F ( e V ) A YM A L V M-1-0.500.51 E - E F ( e V ) A YM A L V M-1-0.500.51 E - E F ( e V ) A YM A L V M-1-0.500.51 E - E F ( e V ) A YM A L V M-1-0.500.51 E - E F ( e V ) A YM A L V M-1-0.500.51 E - E F ( e V ) TaP TaAs TaSb NbP NbAs NbSb E – E F ( e V ) E – E F ( e V ) E – E F ( e V ) E – E F ( e V ) E – E F ( e V ) E – E F ( e V ) FIG. 5: Calculated bulk band structures of the transitionmetal dipnictide family of materials without the SOC. Bluetext on each figure identifies the material.