Abstract
Topological nodalline semimetals are characterized by linecontact bulk band crossings and topological surface states. Breaking certain protecting symmetry turns this system into a Dirac semimetal or Weyl semimetal that hosts zerodimensional isolated nodal points. Recent advances in band theory predicted a topological nodalline semimetal state possessing a new type of nodal line in AlB_{2}type diborides. Here, we report an experimental realization of nodalline fermions and associated surface states near the Fermi energy in ZrB_{2} by angleresolved photoemission spectroscopy combined with firstprinciples calculations. The Dirac nodal lines in ZrB_{2} wind into two groups of nodal rings, which are linked together along the ΓK direction. We further observe a distinct surface state connecting to each nodal line, indicative of the nontrivial topological nature of the bulk nodal lines. Therefore, our results provide convincing experimental evidence of nodalline semimetal states in ZrB_{2} both in the bulk and on the surface, suggesting ZrB_{2} as a remarkable platform for discovering unique phenomena induced by nodalline fermions.
Introduction
The realization of novel quantum states of matter with nontrivial topology beyond topological insulators has become a significant objective in current condensedmatter physics research.^{1,2,3} Very recently, the discovery of topological semimetals has achieved this goal, which ignites extensive work focusing on the exotic topological properties and their underlying connection with the electronic structure.^{4} The topological semimetal states host nontrivial bulk bandcrossing points in crystal momentum space.^{5,6} Characterized by the degeneracy, distribution of the bandcrossing points in the Brillouin zone (BZ), and the associated topological boundary states, the topological semimetals can be classified into three categories: Dirac, Weyl, and nodalline semimetals. In Dirac and Weyl semimetals, the bulk nodes are discrete in the BZ and their surface projections are connected by surface Fermi arcs.^{4} While in nodalline semimetals, the bulk nodes extend along onedimensional curves and the corresponding surface states are flat in dispersion according to the previous nodalline modelings,^{7} where the band crossings of a nodal line should occur at zero energy with a constraint chiral symmetry. Hence the flat surface bands are dubbed the drumhead states. However, the chiral symmetry is not exact in a real crystal, resulting in the nodal line does not generally occur at a constant energy, thus the associated topological surface states are not flat either.^{8,9}
The Dirac semimetal and Weyl semimetal states have been theoretically predicted and experimentally verified in various families of compounds.^{10,11,12,13,14,15,16,17,18,19,20,21} Although there have been several theoretical proposals for the material realization of topological nodalline semimetal states,^{4,8,22,23,24} the conclusive experimental proof remains absent until recent angleresolved photoemission spectroscopy (ARPES) measurements on TiB_{2},^{25} showing a tangible realization of bulk nodalline fermions. Since the surface states associated with the nodal lines are not observed in TiB_{2},^{25} the confirmation of the topological nodalline semimetal state by the coexistence of bulk evidence and surface signature is still desired. In this work, we investigate the electronic structure of ZrB_{2}, which is predicted to host similar nodalline configurations and surface states to that of TiB_{2}.^{26,27} By using ARPES and firstprinciples calculations, we clearly observe two groups of nodal rings embedded in different mirror planes. These rings are further found to be linked with each other along the ΓK direction. More importantly, we identify distinct surface states emanating from the bulk nodal lines, the surface signatures demonstrate the nontrivial topology of the nodalline semimetal states in ZrB_{2}.
Compared with the previous nodalline candidates CaAgAs, PbTaSe_{2}, and the ZrSiS family,^{28,29,30,31,32,33} ZrB_{2} has the following advantages: (1) the whole bandcrossing features forming the nodal lines are clearly resolved below the Fermi energy (E_{F}) in ZrB_{2}, while the nodes of CaAgAs,^{28} PbTaSe_{2},^{29} and the ZrSiS family^{30,31,32,33} cannot be observed by ARPES with the crossings located above E_{F}; (2) the nodalline fermions and their connections with the surface states are observed without any interference from other bands in ZrB_{2}, while in PbTaSe_{2}^{29} and the ZrSiS family,^{30,31,32,33} the observation of bulk nodal lines is seriously obstructed by surface states. Thus, our experimental discovery in ZrB_{2} establishes a unique system that has the conclusive evidence of topological nodalline semimetal states both in the bulk and on the surface.
Results
Structure and transport properties of ZrB_{2}
As illustrated in Fig. 1a, ZrB_{2} crystallizes in a hexagonal lattice system with the space group P6/mmm (No. 191).^{34} The corresponding bulk and (001)projected surface BZs are shown in Fig. 1b, where the ΓKM and ΓAH planes are two mirror planes of D_{6h} group. The flat and shiny surface (inset of Fig. 1c) for our ARPES measurements is the (00l) plane, demonstrated by the single crystal xray diffraction (XRD) pattern in Fig. 1c. To further check the chemical composition of our samples, we display the corelevel photoemission spectrum in Fig. 1d, where the characteristic peaks of Zr4p, Zr5s, and B2p orbitals are clearly revealed.
The magnetic field dependences of the resistivity (ρ_{xx}) and Hall resistivity (ρ_{xy}) at T = 2.5 K are depicted in Fig. 1e and the inset of Fig. 1e, respectively. By fitting the experimental curves with the twocarrier model,^{35} the estimated densities of electrontype and holetype carriers are 1.156 (5) × 10^{21} and 1.153 (5) × 10^{21} cm^{−3}, respectively, comparable to that of TiB_{2} (n_{e}, n_{h} ~ 10^{21} cm^{−3})^{25} and the ZrSiS family (n_{e}, n_{h} ~ 10^{20} cm^{−3}).^{36,37} Figure 1f presents an overview of the calculated bulk band structure for ZrB_{2} without the spinorbit coupling (SOC) effect. Four Diraclike bandcrossing features, which are denoted as α, β, γ, and δ, respectively, give rise to two groups of nodal rings embedded in different mirror planes. When the SOC is included in the calculation, small energy gaps (~40 meV) open at these band crossings.^{38} We will proceed to a detailed discussion on the nodalline semimetal states by systematic electronic structure investigations in the following.
Fermi surface (FS) topologies in four highsymmetry planes
We present the measured and calculated FS topologies in Fig. 2. Figure 2d–f shows the FSs recorded with three different photon energies, hν = 132, 94, and 70 eV, which are close to the k_{z} ~ π (FS3), ~0 (FS1), and ~−π (FS2) planes, respectively. By continuously varying the photon energy, we are able to map out the FS in the ΓAH plane (FS4), as displayed in Fig. 2b. From the FS1 in the k_{z} ~ 0 plane (Fig. 2e), the r_{1} nodal rings formed by α and β are clearly observed surrounding K points. From the FS2 and FS3 in the k_{z} ~ ±π planes (Fig. 2d, f), triangular FSs are resolved at H points, which are attributed as the surface states and discussed later, but no sign of the r_{1} nodal ring is revealed. Another major contrast between FS1 and FS2/FS3 is the absence of intensity around Γ point for the former, and the presence of a circular FS centered at A point for the latter. This difference could be further illuminated by the FS4 in the ΓAH plane (Fig. 2b). The FS around A point, which agreeing well with the greencolored FS sheet in the calculation (Fig. 2c, g), arises from a holelike band presented in Fig. 4. Characterized by these features in experiment, the electronic structure exhibits a prominent threedimensional (3D) nature, as confirmed by the calculated 3D FSs (Fig. 2g) and their projections on the ΓAH plane (Fig. 2c). Since the r_{2} nodal ring in the ΓAH plane formed by γ and δ also pass through the Fermi wave vectors of β, thereby the β bandcrossing feature belongs to both the r_{1} and r_{2} nodal rings. Consequently, the 3D FS is a nodallink structure composed of these two nodal rings. Due to the k_{z} broadening effect, which is prominent in ARPES measurements with vacuum ultraviolet light,^{31,39,40} the ARPES spectra reflects the electronic states integrated over a certain k_{z} region of bulk BZ. Therefore, the (001)projected r_{2} nodal rings around Γ/A points are clearly resolved in FS1FS3 (Fig. 2d–f). Furthermore, the nodallink point (J) of the r_{1} and r_{2} nodal rings along the ΓK direction is unambiguously recognized in FS1 (Fig. 2e). All the above observations are consistent with the calculated bulk FSs shown in Fig. 2c, g, h.
In addition, we observe that some FS features in the k_{z} ~ ±π planes (FS2 and FS3) are absent in the calculated bulk FSs, i.e., the triangular FSs around H points, the FS sheets between two H points, and the hexagonal FS centered at A point (relatively weak in FS3 due to the matrix element effect and its presence can be proved in Fig. 2b). As illustrated in the hνk_{} [k_{} is oriented along the ΓK (AH) direction] ARPES intensity plot in Fig. 2b, the Fermi crossings of the hexagonal FS do not show noticeable photonenergy dependence over a wide range (65–140 eV), which is an indication of surface state. This result inspires us to carry out surface state spectrum calculations for a (001)oriented 20unitcellthick slab terminated by Zr layers. The calculated surface state FSs presented in Fig. 2i well reproduce these three experimental FS topologies, demonstrating their surface state origins.
Presence of bulk nodalline fermions below E _{F}
Based on the discussion above, the r_{1} nodal rings can only be observed in the k_{z} ~ 0 plane. This character establishes an excellent system to separately investigate the bulk nodal lines and electronic surface states. Now we record the nearE_{F} ARPES spectra along the ΓK and MK directions, which are indicated as cuts 1 and 2 in Fig. 2a, respectively, in the k_{z} ~ 0 plane (FS1) to prove the presence of bulk nodalline fermions in ZrB_{2}. The intensity plots and corresponding second derivative plots are shown in Fig. 3a–d. The electronic structure near E_{F} is only composed of the α and β bandcrossing features forming the r_{1} nodal rings, except a band dispersing from the crossing point of β along the ΓK direction, which is relatively weak when crossing E_{F} due to the matrix element effect and thus not resolved in the FS mapping data in Fig. 2e. This band is identified as the surface state and discussed later. We then plot the momentum distribution curves (MDCs) of β and α in Fig. 3e, f, respectively. The linear dispersions in a large energy range with the crossing points below E_{F} are unambiguously recognized. By comparing with the superimposed bulk band calculations in Fig. 3b, d one can obtain a high consistency between experiment and theory, which provides further evidence for the experimental realization of bulk nodalline fermions in ZrB_{2}. As for the holelike band ~2.5 eV below E_{F} at M point (Fig. 3c, d), due to the k_{z} broadening effect in ARPES, it derives from the projection of the band along the HLH direction with a similar feature in Fig. 4c, which is well reproduced by the calculation.
Electronic surface states associated with the bulk nodal lines
Next, from the perspective of surface signature, we demonstrate the nontrivial topology of the nodalline semimetal states in ZrB_{2} by investigating the surface states revealed in the k_{z} ~ −π plane (FS2). In Fig. 4a, c, e, we present the experimental band dispersions with overlapped bulk band calculations along the AH, LH, and AL directions, as indicated by cuts 3–5 in Fig. 2a, respectively. Besides the wellreproduced bulk bands, there are some extra bands (SS1–SS3) not existing in the bulk calculations. According to the experimental FSs measured in the k_{z} ~ ±π planes (FS2 and FS3), we can determine the origins of the crossingE_{F} ones among these bands, i.e., the SS1 corresponds to the triangular FS around H point, and the SS2 forms the hexagonal FS surrounding A point and the FS sheet between two H points, respectively. To further understand the three extra bands, we perform 20unitcellthick slab model calculations with Zrterminated layers. The calculated band dispersions along the \({\bar{\mathrm \Gamma }}\)\(\bar K\), \(\bar M\)\(\bar K\), and \({\bar{\mathrm \Gamma }}\)\(\bar M\) directions with spectral weight from the topmost unit cell are plotted in Fig. 4b, d, f, respectively, which can reproduce SS1–SS3 very well and confirm their surface state nature.
Discussion
In order to clarify the nontrivial topology of the bulk nodal lines realized in ZrB_{2}, which are protected by the mirrorreflection symmetries and the combination of spatialinversion symmetry (P) and timereversal symmetry (T), i.e., the P·T symmetry, here we illustrate the connection between the nodal lines and the surface states by combining both the bulk and surface observations. We plot the crossing points of β and α as the red and blue solid circles in Fig. 4a, c, whose positions are extracted from the MDCs in Fig. 3e, f, respectively. We can observe that the bulk nodes of β and α exactly locate on the loci of the surface states resolved in the k_{z} ~ −π plane. Along the AH direction (Fig. 4a), starting from the node of β, the SS2 disperses inwards with respect to A point. This behavior resembles that of the extra band observed along the ΓK direction in Fig. 3, showing its (the extra band around Γ point) surface state origin associated with the bulk nodal line. Along the LH direction (Fig. 4c), the SS1 passing through the node of α disperses outwards with respect to L point. The bulk nodes and the surface state spectra are obtained from two independent measurements carried out in different highsymmetry planes, therefore the discovery of their well match provides convincing evidence for the topological nature (the bulkboundary correspondence^{8,22,41}) of the nodalline semimetal states in ZrB_{2}.
In summary, our direct experimental observation by ARPES presents conclusive evidence of the bulk nodalline fermions in ZrB_{2} with negligible SOC effect. Under the protection of the mirrorreflection symmetries and the P·T symmetry, the electronic structure hosts two groups of Dirac nodal rings, which are linked together along the ΓK direction. Meanwhile, we clearly resolve electronic surface state emanating from each nodal line, proving the nontrivial topology of the bulk nodal lines. With the experimental realization of topological nodalline semimetal states in ZrB_{2} powerfully supported by both the bulk evidence and the surface signature, we establish an ideal material platform for studying the novel physics and exotic properties related to nodal lines.
Methods
Sample synthesis
Highquality single crystals of ZrB_{2} were grown via the Fe flux method.^{38} The starting elements of Zr (99.95%), B (99.99%), and Fe (99.98%) were put into an alumina crucible, with a molar ratio of Zr:B:Fe = 3:6:17. The mixture was heated up to 1873 K in a highpurity argon atmosphere and then slowly cooled down to 1623 K at a rate of 4 K/h. The ZrB_{2} single crystals were separated from the Fe flux using the hot hydrochloric acid solution.
ARPES measurements
ARPES measurements were performed at the Dreamline beamline of the Shanghai Synchrotron Radiation Facility using a Scienta D80 analyzer and at the beamline 13U of the National Synchrotron Radiation Laboratory equipped with a Scienta R4000 analyzer. The energy and angular resolutions were set to 25 meV and 0.2°, respectively. Fresh surfaces for ARPES measurements were obtained by cleaving the samples in situ along the (001) plane. All spectra presented in this work were recorded at T = 20 K in a working vacuum better than 5 × 10^{−11} Torr.
Band structure calculations
Firstprinciples calculations were carried out with the projector augmentedwave method^{42,43} as implemented in the Vienna ab initio Simulation Package.^{44} The generalized gradient approximation of PerdewBurkeErnzerhof formula^{45} was adopted for the exchangecorrelation functional. The kinetic energy cutoff of the planewave basis was set to be 420 eV. A 20 × 20 × 20 kpoint mesh was utilized for the BZ sampling. The bulk FSs were investigated by adopting the maximally localized Wannier function method.^{46} A slab with thickness of 20 unit cells along the [001] direction was used in our surface state calculations. The slabs were separated by a vacuum layer of 20 Å, which was sufficient to avoid the interactions between different slabs. SOC effect^{38} was not taken into account in all the above calculations.
Data availability
All relevant data are available from the corresponding authors upon request.
Change history
12 October 2018
The original version of this Article did not acknowledge Qi Wang as an equally contributing author. This has now been corrected in the HTML and PDF versions of the Article.
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Acknowledgements
The work was supported by the National Key R&D Program of China (Grants No. 2016YFA0300504 and No. 2017YFA0302903), the National Natural Science Foundation of China (Grants No. 11774421, No. 11574394, No. 11774423, No. 11774424, No. 11227902, and No. 11704394), and the Chinese Academy of Sciences (CAS) (Project No. XDB07000000). R.L., K.L., and H.L. were supported by the Fundamental Research Funds for the Central Universities, and the Research Funds of Renmin University of China (RUC) (Grants No. 17XNH055, No. 14XNLQ03, No. 15XNLF06, and No. 15XNLQ07). Y.H. was supported by the CAS Pioneer Hundred Talents Program. Z.L. acknowledges Shanghai Sailing Program (No. 17YF1422900).
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Contributions
R.L., H.C.L., Y.B.H., H.D., and S.C.W. conceived the experiments. R.L. performed ARPES measurements with the assistance of M.L., Z.H.L., X.C.W., Z.L.W., Z.S., and D.W.S. P.J.G., K.L., and Z.Y.L. performed firstprinciples calculations. Q.W., S.S.S., C.H.L., and H.C.L. synthesized the single crystals. R.L., H.C.L., and S.C.W. analysed the experimental data. R.L. plotted the figures. R.L. and S.C.W. wrote the manuscript.
Corresponding authors
Correspondence to Yaobo Huang or Hechang Lei or Shancai Wang.
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