Spectroscopic Evidence on Realization of a Genuine Topological Nodal Line Semimetal in LaSbTe
Yang Wang, Yuting Qian, Meng Yang, Hongxiang Chen, Cong Li, Zhiyun Tan, Yongqing Cai, Wenjuan Zhao, Shunye Gao, Ya Feng, Shiv Kumar, Eike F. Schwier, Lin Zhao, Hongming Weng, Youguo Shi, Gang Wang, Youting Song, Yaobo Huang, Kenya Shimada, Zuyan Xu, X. J. Zhou, Guodong Liu
SSpectroscopic Evidence on Realization of a Genuine TopologicalNodal Line Semimetal in LaSbTe
Yang Wang , (cid:93) , Yuting Qian , (cid:93) , Meng Yang , . (cid:93) , Hongxiang Chen , (cid:93) ,Cong Li , (cid:93) , Zhiyun Tan , Yongqing Cai , , Wenjuan Zhao , , ShunyeGao , , Ya Feng , Shiv Kumar , Eike F. Schwier , , Lin Zhao , HongmingWeng , , , , Youguo Shi , , , Gang Wang , , , Youting Song , Yaobo Huang ,Kenya Shimada , Zuyan Xu , X. J. Zhou , , , ∗ and Guodong Liu , ∗ Beijing National Laboratory for Condensed Matter Physics,Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China University of Chinese Academy of Sciences, Beijing 100049, China Center of Materials Science and Optoelectronics Engineering,University of Chinese Academy of Sciences, Beijing 100049, China School of Materials Science and Engineering,Fujian University of Technology, Fuzhou 350118, China School of Physics and Electronic Science,Zunyi Normal College, Zunyi 563006, China Beijing Academy of Quantum Information Sciences, Beijing 100193, China Hiroshima Synchrotron Radiation Center, Hiroshima University,Higashi-Hiroshima, Hiroshima 739-0046, Japan Experimentelle Physik VII, Universit ¨ a t W ¨ u rzburg,Am Hubland, D-97074 W ¨ u rzburg, Germany, EU Songshan Lake Materials Laboratory, Dongguan 523808, China CAS Center for Excellence in Topological Quantum Computation,University of Chinese Academy of Science, Beijing 100190, China Shanghai Synchrotron Radiation Facility,Shanghai Advanced Research Institute,Chinese Academy of Sciences, Shanghai 201204, China Technical Institute of Physics and Chemistry,Chinese Academy of Sciences, Beijing 100190, China (cid:93)
These people contributed equally to the present work. a r X i v : . [ c ond - m a t . m t r l - s c i ] F e b Corresponding authors: [email protected] and gdliu [email protected] (Dated: February. 22, 2021)
Abstract
The nodal line semimetals have attracted much attention due to their unique topological elec-tronic structure and exotic physical properties. A genuine nodal line semimetal is qualified by thepresence of Dirac nodes along a line in the momentum space that are protected against the spin-orbit coupling. In addition, it requires that the Dirac points lie close to the Fermi level allowingto dictate the macroscopic physical properties. Although the material realization of nodal linesemimetals have been theoretically predicted in numerous compounds, only a few of them havebeen experimentally verified and the realization of a genuine nodal line semimetal is particularlyrare. Here we report the realization of a genuine nodal line semimetal in LaSbTe. We investigatedthe electronic structure of LaSbTe by band structure calculations and angle-resolved photoemission(ARPES) measurements. Taking spin-orbit coupling into account, our band structure calculationspredict that a nodal line is formed in the boundary surface of the Brillouin zone which is robust andlies close to the Fermi level. The Dirac nodes along the X-R line in momentum space are directlyobserved in our ARPES measurements and the energies of these Dirac nodes are all close to theFermi level. These results constitute clear evidence that LaSbTe is a genuine nodal line semimetal,providing a new platform to explore for novel phenomena and possible applications associated withthe nodal line semimetals. [46], CaAgX (X=P, As) [47–49], Ta SiTe [50], monolayer Cu Si [51], XB (X=Zr,Ti, Al, Mg) [52–56], RAs (R=Ca,Sr) [57, 58], Pt HgSe [59], Co MnGa [60], ZrSiS family[61–73], single-layer GdAg [74], InBi [75], PbTaSe [76], IrO [77] and RuO [78].Strictly speaking, two requirements are necessary for realizing a genuine nodal linesemimetal. One is the presence of Dirac nodes along a line and the nodes retain gap-less under the spin-orbit coupling (SOC). The other is that the Dirac points should lie closeto the Fermi level allowing to produce exotic physical phenomena. Many of the nodal linesemimetals proposed without considering SOC may not fall into the category because theDirac points become unstable and get gapped when SOC is considered, as seen in CaAgX(X=P, As) [47–49], Ta SiTe [50], monolayer Cu Si [51], XB (X=Zr, Ti, Al, Mg )[52–56],RAs (R=Ca, Sr) [57, 58], Pt HgSe [59] and Co MnGa [60]. The genuine topological nodalline semimetals that have been proposed and can satisfy the two criteria are rare [77, 78].The layered ternary WHM system (W represents transition metal Zr, Hf, or rare earth el-ement La, Ce, Gd; H represents group IV or group V element Si, Ge, Sn, or Sb, and Mrepresents group VI element O, S, Se, and Te) [79] has become an important candidatefamily of topological nodal line semimetals. Without SOC or when the SOC is negligible,this system can host nodal lines and nodal surface, which have been confirmed experimen-tally by ARPES measurements in HfSiS [61], ZrSiX (X=S, Se, Te) [62–66], ZrGeTe [67, 68]3nd GdSbTe [69]. When the SOC is taken into account, however, most of the nodal linesand nodal surface become gapped, leaving only two topological nodal lines in the bound-ary surface of Brillouin zone [71, 72]. Meanwhile, the robust nodal lines observed so far inthis system stay away from the Fermi level, contributing little to the macroscopic physicalproperties. Finding genuine topological nodal line semimetals is essential for exploring newphenomena and realizing exotic properties.In this paper, we report identification of LaSbTe as a genuine topological nodal linesemimetal. Compared to the ZrSiS system in which the robust Dirac nodal lines existat the Brillouin zone boundary but their energy position stays away from the Fermi level( ∼ α radiation ( λ =0.71073 ˚ A ), indicating that our LaSbTe samples crystallizein a nonsymmorphic space group P /nmm (no.129) with PbFCl-type tetragonal structure:a=b=4.399 ˚ A and c=9.566 ˚ A . The measured crystal parameters of LaSbTe are summa-rized in Table I-III in Supplementary Materials[80]. This structure is identical to the WHMfamily. The Sb atoms constitute a two-dimensional square net sandwiched in between twoLaTe layers (Fig. 1a). ARPES measurements were performed at Hiroshima SynchrotronRadiation (HiSOR) BL-1 Beamline [81, 82], the ‘Dreamline’ Beamline of the Shanghai Syn-chrotron Radiation Facility (SSRF) and the ARPES system in our lab [83, 84]. The energy4esolution is ∼
30 meV and the angular resolution is ∼ in situ and measured at different photon energiesin ultrahigh vacuum with a base pressure better than 5.0 × − mbar. The first-principlesband-structure calculations based on the density functional theory (DFT) have been doneusing the Vienna ab initio simulation package (VASP) [85] within the generalized gradientapproximation (GGA) of Perdew-Burke-Ernzerhof type [86]. We take 300 eV as the cut-offenergy for plane wave expansion. The 8 × × A , c=9.659˚ A are employed. The Wannier90 [87] and WannierTools [88] are used to calculate the Fermisurface and find nodes based on the maximally localized Wannier functions which consist ofp orbits of Sb and Te, and d orbits of La.We start our discussion with bulk electronic structure calculations of LaSbTe. Symmetriesare crucial to understand the electronic structure of materials as they protect the bandcrossings in the Brillouin zone in high-symmetry planes, lines or points[26, 29]. For thetetragonal LaSbTe crystallized in nonsymmorphic space group P /nmm , the glide mirror˜ M z = (cid:110) M z |
12 12 (cid:111) , the mirror ˜ M x = (cid:110) M x | (cid:111) , ˜ M y = (cid:110) M y | (cid:111) , ˜ M xy = (cid:110) M xy |
12 12 (cid:111) ,the spacial inversion {P| } symmetries and the two screw symmetries: ˜ C x = (cid:110) C x | (cid:111) and ˜ C y = (cid:110) C y | (cid:111) are especially important. Combining with time-reversal symmetry T ,these symmetries can lead to nonsymmorphically enforced degeneracy at the boundary ofBrillouin zone that give rise to multiple nodal lines. To clarify the topology of electronicstructure of LaSbTe protected by different symmetries, we calculated the Dirac nodal linesin three-dimensional (3D) Brillouin zone (Fig. 1c) and the band structures of bulk LaSbTewithout (Fig. 1d) and with SOC (Fig. 1e). Because the bands close to the Fermi level aremainly dominated by the α , β and γ in Fig. 1d and Fig. 1e, we will focus on the nodal linesformed by these three bands.Figure 1c shows the schematic of the calculated nodal line configurations formed by β - γ band (NL1-NL5) in the absence of SOC and by α - β band (NLE1 and NLE2) in the presenceof SOC in 3D Brillouin zone. They represent the locations of all the Dirac nodes in three-5imensional momentum space formed from α , β and γ bands although the energy of theDirac nodes can be different. The nodal lines formed by β - γ band can take various shapesunder different symmetry protections as shown by the light blue curves in Fig. 1c. Theyall lie inside the first Brillouin zone. We can find two diamond-shaped nodal lines: NL1 ink z =0 plane and NL2 in k z = π plane that are protected by the glide mirror symmetry ˜ M z . Inaddition, there are two vertical nodal lines, NL3 protected by ˜ M x and NL4 by ˜ M xy . Thesenodal lines form a 3D cage-like structure. They are 2-fold degenerate and not stable in thepresence of SOC which will open a gap at the Dirac point.The band crossings formed by α - β band form two nodal surfaces in the k x = π and k y = π planes due to the combined symmetry ˜ C y T as reported in WHM family [66, 89]. When SOCis taken into consideration, all band crossings are gapped except for the two lines along R-Xand A-M directions and some one-dimensional curves at generic k points in A-M-X-R-A planeowing to accidental degeneracy that are far away from Fermi level and will not be consideredfurther. The nodal lines along R-X and A-M high symmetry directions, as labeled by NLE1and NLE2 in Fig. 1c, host robust 4-fold degenerate massless Dirac nodal lines protectedby nonsymmorphic symmetry at the boundary of Brillouin zone [26, 64, 71, 90, 91] andcan be understood in the following way. Firstly, in the spinful case, it is well known thatthere exists Kramers’ pair due to ( PT ) =-1. Secondly, the two high symmetry lines R-X and A-M have higher symmetry than other points on the k y = π plane such that bandsalong these two lines are at least doubly degenerate ( | ψ (cid:105) , | ψ (cid:105) ) protected by ˜ M y . Thirdly,since the ˜ M y is a nonsymmorphic symmetry, anti-commutation relation {PT , ˜ M y } =0 issatisfied. Considering the above three conditions, there must exist a 4-fold degeneratestate ( | ψ (cid:105) , PT | ψ (cid:105) , | ψ (cid:105) , PT | ψ (cid:105) ) where each Kramers’ pair ( | ψ , (cid:105) , PT | ψ , (cid:105) ) belongs totwo orthogonal states with the same ˜ M y eigenvalues [26, 91]. Therefore, the NLE1 andNLE2 are symmetry enforced and appear along high symmetry lines at the boundary ofBrillouin zone.The formation of nodal lines in LaSbTe can be seen directly from the calculated bandstructures which show the overall α , β and γ bands, derived mainly from the Sb p x and p y orbits, in the entire 3D Brillouin zone. Without considering SOC, the location of the Diracpoints or nodal lines formed from α - β band (red arrows) and β - γ band (dashed black boxes)are marked in Fig. 1d. When SOC is taken into consideration as shown in Fig. 1e, all theDirac points formed by β - γ band are gapped. But some Dirac points formed by α - β band6emain intact. The SOC causes the band splitting along M-X (A-R) direction and turns theinitial Dirac line nodes (marked by black arrows in Fig. 1d) into topologically trivial 2-foldspin-degenerate bands. However, the two Dirac lines along the R-X and A-M lines surviveunder the protection of nonsymmorphic symmetry, as labeled by red arrows in Fig. 1e. TheNLE2 nodal line lies about 1.6 eV above the Fermi level while the NLE1 nodal line lies rightat the Fermi level. These calculation results indicate that LaSbTe is a genuine nodal linesemimetal.Now we come to the electronic structure of LaSbTe from our ARPES measurements. Fig.2a-2c show the Fermi surface mapping and constant energy contours of LaSbTe measuredwith photon energies of 55 eV (Fig. 2a), 85 eV (Fig. 2b) and 95 eV (Fig. 2c). We canclearly see a large diamond-shaped Fermi surface centered around ¯Γ point. We also observestrong spectral weight at the ¯ X points. For different photon energies, the evolution ofconstant energy contours with binding energy is markedly different for the three differentphoton energies. Fig. 2d-2f show the calculated Fermi surface and constant energy contoursof LaSbTe at k z planes of 0.5 π/c (Fig. 2d), 0.75 π/c (Fig. 2e) and 0 π/c (Fig. 2f),corresponding to the photon energy of 55 eV, 85 eV and 95 eV, respectively. The diamond-shaped Fermi surface originates from β band and the feature around ¯ X point is mainly fromthe Dirac-like band formed from α or β bands, as shown in Fig. 1e. The measured resultsshow an overall agreements with the band structure calculations in terms of the observationof a large diamond-shaped Fermi surface and the feature at the ¯ X points.Figure 3 shows the band structure of LaSbTe measured along three high symmetry di-rections under different polarization geometries (more data measured at different photonenergies can be found in Supplementary Fig. S2 and Fig. S3[80]). We find that the observedband structures measured under different polarizations are quite different due to photoemis-sion matrix element effect. Therefore, our measurements under two distinct polarizationsare helpful to directly reveal the band structure depending on the orbital symmetry withrespect to the detection plane. To better identify the band structure, we also take secondderivative images (Fig. 3d-3f) of the original data (Fig. 3a-3c). For comparison, we putthe corresponding calculated bulk band in Fig. 3g-3i and carried out slab calculations (Fig.3j-3l) that can handle surface states.For the band structure along ¯Γ- ¯ M direction, a linearly dispersive band crosses the Fermilevel as shown in Fig. 3d marked by red and orange lines, which gives rise to the diamond-7haped Fermi surface. This band can extend to a high binding energy of ∼ ∼
12 eV · ˚ A − that is larger than those found in some typicalDirac materials like graphene (6.7 eV · ˚ A − ) [92] , SrMnBi (10.6 eV · ˚ A − ) [93] and ZrSiS (4.3eV · ˚ A − ) [64]. The measured band structures (Fig. 3d) areconsistent with the calculatedbands (Fig. 3g) in terms of both the number and the position of the bands. For the bandstructure along ¯Γ- ¯ X direction, a Dirac like band is observed crossing the Fermi level as shownin Fig. 3e marked by red and orange lines. The Dirac point lies exactly at ¯ X point andits energy position is rather close to the Fermi level. Again, the measured band structures(Fig. 3e) are consistent with the calculation (Fig. 3h) in terms of the number, shape andthe position of the bands. For the band structure along ¯ M - ¯ X direction, a tiny electron-likeband is observed that crosses the Fermi level at the ¯ X point as shown in Fig. 3f. In addition,two groups of bands are observed, labeled as SB and BB in Fig. 3f. Each group of bandis further composed of two bands. These results are well reproduced in the band structurecalculations (Fig. 3i and 3l), which indicate that the SB bands represent surface stateswhile the BB bands are bulk states. The SB band exhibits little change when measured atdifferent photon energies (Fig. S3 in Supplementary materials[80]), further confirming itssurface state nature.It is noted that we only observed single sheet of the diamond-shaped Fermi surface, evenat different binding energies (Fig. 2a-2c). According to the band structure calculations, twosheets of the Fermi surface are expected (Fig. 2d-2f). We also only observed a single bandover a large energy range ( ∼ M direction (Fig.3a and Fig. 3d) although two linear bands are expected in this energy range from the bandcalculations (Fig. 1d and Fig. 1e). Most of the ARPES measurements on WHM system haveobserved double sheets Fermi surface [61, 62, 65, 66, 68, 70, 71]. A single sheet Fermi surfacehas also been reported before in GdSbTe but the origin is unclear [69]. By carefully examingthe measured band structures in Fig. 3d, we believe this is caused by the photoemissionmatrix element effect. As seen in the right panel of Fig. 3d, in this particular polarizationgeometry, the two linear bands can actually be observed at high binding energy between1.0 ∼ ∼ M direction using different photon energies (Fig. S2 in Supplementarymaterials[80]). We observed sharp single band over an energy range of 0 ∼ M direction. This is probably because the gap openinginduced by SOC is small that exceeds our present detection limit.As predicted by band structure calculations, LaSbTe is a nodal line semimetal becauseof the presence of the nodal line along X-R direcion that is robust against the SOC and liesclose to the Fermi level. To examine on the band structure along k z direction, we carriedout measurements along ¯Γ- ¯ X direction with different photon energies between 26 eV and 95eV. Fig. 4a shows the Fermi surface mapping in Γ-X-R-Z-Γ plane of LaSbTe crossing the ¯ X point along ¯Γ- ¯ X direction. A one-dimensional linear Fermi surface is clearly observed alongthe X-R momentum line. Fig. 4b displays representative band structures measured usingdifferent photon energies between 32 eV and 50 eV, that corresponds to a full k z period.Dirac-like structure can be observed in all the measurements using different photon energiesas marked by the red arrows in Fig. 4b. The Dirac cones lie at the ¯ X point along the X-Rline in momentum space and their energies are all close to the Fermi level. These resultsstrongly indicate that there is a nodal line formed along the X-R direction.In summary, by carrying out ARPES measurements combined with band structurecalculations, we have systematically investigated the electronic structures of LaSbTe. Ourband structure calculations predict the formation of five nodal lines derived from the β - γ band including two diamond-like nodal lines, and two nodal surfaces (k x = π and k y = π plane) from the α - β band without considering SOC. Taking SOC into account, bandcalculations indicate that those five nodal lines from β - γ band become unstable and aregapped at the Dirac points while the two nodal lines from α - β band are robust. Amongthese two nodal lines, the one along M-A direction stays far away from Fermi level whilethe other one along X-R direction lies close to the Fermi level. Our ARPES results are in9ood agreements with the band structure calculations. We observed a diamond-like Fermisurface. 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This work is supported by the National Key Research and Development Program of China(Nos. 2016YFA0300600, 2018YFA0305602 and 2018YFE0202600), the National NaturalScience Foundation of China (Nos. 11974404, U2032204 and 51832010), the Strategic Pri-ority Research Program (B) of the Chinese Academy of Sciences (Nos. XDB33000000and QYZDB-SSW-SLH043), and the Youth Innovation Promotion Association of CAS(No. 2017013). ARPES measurements at HiSOR were performed under the Proposal Nos.18AU001 and 17BU025. We thank the N-BARD, Hiroshima University for supplying liquidHe. The theoretical calculations is supported by the National Natural Science Foundationof China (Grant Nos. 11674369, 11865019 and 11925408), the Beijing Natural Science Foun-dation (Grant No. Z180008), Beijing Municipal Science and Technology Commission (GrantNo. Z191100007219013), the National Key Research and Development Program of China(Grant Nos. 2016YFA0300600 and 2018YFA0305700), the K. C. Wong Education Founda-tion (Grant No. GJTD-2018-01) and the Strategic Priority Research Program of ChineseAcademy of Sciences (Grant No. XDB33000000).
Author Contributions
Y.W., Y.T.Q., M.Y., H.X.C. and C.L. contributed equally to this work. G.D.L., X.J.Z.,Y.W. and C.L. proposed and designed the research. M.Y., H.X.C., Y.G.S. and G.W. con-tributed to LaSbTe crystal growth. Y.T.S. contributed to the single-crystal diffraction mea-surements and crystal structure analysis. Y.T.Q., Z.Y.T. and H.M.W. contributed to theband structure calculations and theoretical discussion. Y.W. carried out the ARPES experi-ment at SSRF and HiSOR with the assistance from S.Y.G. (SSRF), Y.B.H. (SSRF), W.J.Z.(HiSOR), Y.F. (HiSOR), S.K. (HiSOR), E.F.S. (HiSOR) and K.S. (HiSOR). W.Y. and C.L.carried out the experiment at home-built ARPES system in our Lab. C.L., Y.Q.C., Y.W.,G.D.L., L.Z., Z.Y.X. and X.J.Z. contributed to the development and maintenance of ourARPES system. Y.W., C.L., G.D.L. and X.J.Z. analyzed the data. Y.W., C.L., G.D.L. and19.J.Z. wrote the paper with Y.T.Q. and H.M.W. All authors participated in discussion andcomment on the paper. 20
IG. 1:
Band structure calculations and the nodal line configuration of LaSbTe. (a)Crystal structure of LaSbTe. The Sb atoms shown with blue balls form square nets, which aresandwiched between the two La-Te layers. The grey arrow indicates the cleavage plane between theLaTe layers. (b) Schematic of 3D Brillouin zone of LaSbTe. (c) Calculated nodal lines formed by β - γ band (NL1-NL5) and α - β band (NLE1 and NLE2) in 3D Brillouin zone. (d-e) Calculated bulkband structures without (d) and with SOC (e) along high symmetry directions by first-principlescalculations. We label the two conduction bands and one valence band close to the Fermi levelwith α (red), β (purple) and γ (blue) respectively. The Dirac points are marked by red arrows. IG. 2:
Fermi surface of LaSbTe. (a-c) The Fermi surface and constant energy contours ofLaSbTe measured with photon energy of 55 eV (a), 85 eV (b) and 95 eV (c), respectively, under LH polarization. (d-f) The DFT calculated Fermi surface and constant energy contours of LaSbTewith k z corresponding to 0.5 π /c (d), 0.75 π /c (e) and 0 π /c (f), respectively. IG. 3:
Measured band structures of LaSbTe and their comparison with band structurecalculations. (a-c) Band structures of LaSbTe measured along ¯Γ- ¯ M (a), ¯Γ- ¯ X (b) and ¯ X - ¯ M (c)directions, respectively, with a photon energy of 55 eV. The left and right panels in (a-c) aremeasured under LV and LH polarization geometries, respectively. (d-f) The second derivativeimages of the original data corresponding to (a-c). The red and orange lines mark the two branchesof the linearly dispersed Dirac bands. (g-i) The calculated bulk band structures of LaSbTe withSOC along ¯Γ- ¯ M (g), ¯Γ- ¯ X (h) and ¯ X - ¯ M (i) momentum directions for k z =0.5 π /c plane. (j-l) Thecalculated band structures of LaSbTe with SOC along ¯Γ- ¯ M (j), ¯Γ- ¯ X (k) and ¯ X - ¯ M (l) directions fora seven-unit-cell-thick slab. The intensity of color bar represents the proportion of surface statescontributed by the topmost unit cell of LaSbTe. Here, the BS, BB and the DP indicate the bulkband, surface band and the Dirac point, respectively. IG. 4:
Dirac nodal line along X-R direction. (a) Photon energy dependent ARPES-intensitymeasurement of LaSbTe crossing the zone center along ¯Γ- ¯ X direction. (b) Band structures mea-sured along ¯Γ- ¯ X direction at different photon energies. To highlight the bands, these images arefrom the second derivative of the original data. The Dirac points are marked by red arrows.direction at different photon energies. To highlight the bands, these images arefrom the second derivative of the original data. The Dirac points are marked by red arrows.