Observation of Dirac-like energy band and ring-torus Fermi surface associated with the nodal line in topological insulator CaAgAs

One of key challenges in current material research is to search for new topological materials with inverted bulk-band structure. In topological insulators, the band inversion caused by strong spin-orbit coupling leads to opening of a band gap in the entire Brillouin zone, whereas an additional crystal symmetry such as point-group and nonsymmorphic symmetries sometimes prohibits the gap opening at/on specific points or line in momentum space, giving rise to topological semimetals. Despite many theoretical predictions of topological insulators/semimetals associated with such crystal symmetries, the experimental realization is still relatively scarce. Here, using angle-resolved photoemission spectroscopy with bulk-sensitive soft x-ray photons, we experimentally demonstrate that hexagonal pnictide CaAgAs belongs to a new family of topological insulators characterized by the inverted band structure and the mirror reflection symmetry of crystal. We have established the bulk valence-band structure in three-dimensional Brillouin zone, and observed the Dirac-like energy band and ring-torus Fermi surface associated with the line node, where bulk valence and conducting bands cross on a line in the momentum space under negligible spin-orbit coupling. Intriguingly, we found that no other bands cross the Fermi level and therefore the low-energy excitations are solely characterized by the Dirac-like band. CaAgAs provides an excellent platform to study the interplay among low-energy electron dynamics, crystal symmetry, and exotic topological properties.


Introduction
Topological insulators (TIs) exhibit a novel quantum state with metallic edge or surface state (SS) within the bulk band gap generated by the strong spin-orbit coupling (SOC). The topological SS in three-dimensional (3D) TIs is characterized by a linearly dispersing Dirac-cone energy band, 1-3 which hosts massless Dirac fermions protected by the time-reveal symmetry (TRS). The discovery of TIs triggered the search for new types of topological materials containing surface or bulk Dirac-cone bands protected by crystal symmetries, as represented by topological crystalline insulators (TCIs) with the Dirac-cone SSs protected by mirror symmetry, [4][5][6] as well as 3D Dirac semimetals (DSMs) with bulk Dirac-cone bands protected by rotational symmetry (such as Cd3As2 and Na3Bi). [7][8][9][10][11] While the Dirac cone in DSMs is spin degenerate, breaking the TRS or space-inversion symmetry (SIS) leads to the Weyl-semimetal (WSM) phase with pairs of spin-split Dirac (Weyl) cones, as recently verified in transition-metal monopnictides. [12][13][14] Dirac-cone states are known to provide a platform to realize outstanding physical properties such as extremely high mobility, gigantic linear magnetoresistance, and chiral anomaly. [15][16][17][18][19][20][21][22] While the DSMs and WSMs are characterized by the crossing of bulk bands at the discrete points in k space (point nodes), there exists another type of topological semimetal characterized by the band crossing along a one-dimensional curve in k space (line node), called line-node semimetal (LNSM). The LNSMs are expected to show unique physical properties different from the DSMs and WSMs, such as a flat Landau level, the Kondo effect, long-range Coulomb interaction, and peculiar charge polarization and orbital magnetism. [23][24][25][26] Despite many theoretical predictions of LNSMs in various material platforms,  experimental studies on the LNSMs are relatively scarce. [55][56][57][58][59][60][61][62] Recently, it was theoretically proposed by Yamakage et al. that noncentrosymmetric ternary pnictides CaAgX (X = P, As) are the candidate of LNSM and TI. 41 These materials crystalize in the ZrNiAl-type structure with space group P 6 ___ 2m (No. 189) 63 (for crystal structure, see Fig. 1a). First-principles band-structure calculations have shown that, under negligible spin-orbit coupling (SOC), CaAgX displays a fairly simple band structure near the Fermi level (EF) with a ring-like line node (nodal ring) surrounding the G point of bulk hexagonal Brillouin zone (BZ) (bulk BZ is shown in Fig. 1b). The line node is associated with the crossing of bulk conduction band (CB) and valence band (VB) with Ag s and P/As p character, respectively, and is protected by the mirror reflection symmetry of crystal. When the SOC is included in the calculation, CaAgP still keeps the line node due to the very small spin-orbit gap (~ 1 meV) while a relatively large spin-orbit gap (~ 75 meV) opens along the line node in CaAgAs to make this material a narrow-gap TI. 41 The SOC thus plays a crucial role in switching the LNSM and TI phases in CaAgX. Transport measurements on the CaAgP and In this work, we report the ARPES study of CaAgAs. By utilizing bulk-sensitive soft-x-ray photons from synchrotron radiation, we established the bulk VB structure in the 3D bulk BZ. We suggest that CaAgAs is a narrow-gap TI with an ideal band structure suitable to study the low-energy excitations linked to the bulk Dirac-like band arising from the line node. This is demonstrated by observing the bulk Fermi surface which is solely derived from the VB and CB associated with the single line node, consistent with our first-principles band structure calculations. We discuss the consequence of our observation in relation to the exotic physical properties.

Samples and experimental
High-quality single crystals of CaAgAs were grown on the sintered pellets of CaAgAs (for details, see Method). A typical photograph of our single crystal is shown in Fig. 1c. ARPES measurements were performed with synchrotron light at BL2 in Photon Factory, KEK. Samples were cleaved in situ along the (11 2 ___ 0) crystal plane (a shiny mirror plane in Fig. 1c) as confirmed by the Laue x-ray diffraction measurement on the cleaved surface (typical Laue pattern is shown in Fig. 1d) and the photon-energy (hn) dependence of the band dispersion. This indicates that the cleaved plane is the ky -kz plane in the hexagonal BZ (Fig. 1b). Figure 1e displays the energy distribution curve (EDC) in the wide energy region measured at hn = 580 eV. One can recognize several core-level peaks originating from the Ca (3s, 3p), Ag (4s, 4p, 4d), and As (3s, 3p, 3d) orbitals. No other core-level peaks were found in this energy range, confirming the clean sample surface.

Valence-band structure
First, we present the overall VB structure of CaAgAs. We found that soft-x-ray photons are useful for revealing the bulk electronic states of CaAgAs as in the case of noncentrosymmetric Weyl semimetals such as TaAs, [12][13][14] although we need to sacrifice the energy/momentum resolution compared to the vacuum ultraviolet (VUV) photons. In fact, the obtained VUV data were found to suffer large broadening along wave vector perpendicular to the surface probably because of the final-state effect and rather rough nature of the cleaved surfaces, and therefore we concluded that the VUV photons are not best suited for resolving 3D electronic states of CaAgAs. Figure  this confirms that the cleaving plane is (112 ___ 0). The holelike dispersion approaching EF around the G point is well reproduced by the calculations, and therefore it is assigned as the topmost VB with the As 4p orbital character. Moreover, a good agreement of the band width between experiment and calculation signifies no apparent band-renormalization effect, suggesting the weak electron correlation. It is noted that we observe a single holelike band within 1.5 eV of EF in the experiment, while the calculation predicts two holelike bands. Such difference may be due to the finite k/energy-broadening effect as well as the matrix-element effect of photoelectron intensity which turned out to be rather strong in this material.
We comment here that our Hall conductivity measurement of CaAgAs single crystal suggests the existence of hole carriers with carrier concentration of ~ 1.6 ×  Fig. 2f (cut C in Fig. 2c) where the topmost VB always stays below EB ~ 1 eV without crossing EF.

Ring-torus Fermi surface
Having established the overall VB structure, a next important issue is the electronic structure in the vicinity of EF responsible for the physical properties. Figure 3a displays the ARPES intensity at EF as a function of kx and ky (the GKM plane). One immediately finds a bright intensity pattern surrounding the G point, in particular in 13th BZ, confirming the absence of additional Fermi surface away from the G point.
It is also obvious from Fig. 3b that no Fermi surface exists away from G in the ky-kz (GAKH) plane. As shown in Figs. 3a and 3b, when we overlaid the calculated Fermi surface (green curves) onto the ARPES intensity (note that we assumed the location of EF to be 0.05 eV below the VB top in the calculation to account for a small but finite hole-doping effect in experiment), the high-intensity region coincides with the k region where the calculated Fermi surface exists.
To gain further insight into the Fermi-surface topology, we show in the top panels of Figs. 3c and 3d the ARPES intensity near EF and the intensity obtained by taking second derivative of the EDCs, respectively, along the k cut nearly crossing the G point (cut A in Fig. 3a). One finds a linearly dispersive holelike band originating from the As 4p states which is better visualized in the second-derivative plot in Fig. 3d (by a linear extrapolation of the band dispersion around EF, we have estimated the Fermi velocity to be vF = 2.1 ± 0.1 eVÅ). This band is reproduced by our calculation as shown by red curves in Fig. 3c, and is responsible for the outer ring in Fig. 3a. As shown in cut A of Fig. 3c, there exists another electronlike band in the calculation which originates from the Ag 5s orbital; this band forms the inner ring in Fig. 3a. While the intensity of the electronlike band seems weak in the original intensity (Fig. 3c), the second-derivative image in Fig. 3d shows a finite spectral weight likely arising from the electronlike band.
As shown in cut A of Fig. 3c, the calculated electronlike band intersects the holelike band at ~ 0.1 eV above EF, and forms the nodes at ky ~ ± 0.15 Å -1 under negligible SOC, since cut A is on the (0001) mirror plane (the kx-ky plane) and the nodes are protected by mirror reflection symmetry. 41 This indicates that the electronic states within two opposite nodes across the G point have an inverted band character. Thus, our observation of electronlike feature can be regarded as a hallmark of the band inversion, which is a prerequisite for realizing LNSM or TI. It is remarked that with a finite SOC, an energy gap of 75 meV opens in the caluclation, as can be seen from a difference in the band dispersion with (solid curves) and without (dashed curves) SOC in Fig. 3c. 41 The opening of a spin-orbit gap at the node is also seen in some other LNSM candidates such as Cu3(Pd,Zn)N, 34,35 Ca3P2, 37,49 ZrSiS, 40 CaTe, 50 and fcc alkaline-earth metal 51 .
To clarify whether the node-like feature in CaAgAs is seen at a point or on a line in k space, it is necessary to measure the band dispersion along different k slices around the Fermi surface. For this sake, we show in the middle and bottom panels of Figs. 3c and 3d the intensity for cuts slightly away from the G point (cuts B and C in Fig. 3a) obtained with different hn's. One can recognize that overall band structure along cut B is similar to that along cut A regarding the EF crossing of holelike band and the presence of electronlike feature. This is reasonable since cut B is also on the mirror plane and still crosses the calculated nodal points. On the other hand, along cut C, the holelike band moves downward and shows no EF crossing. These behaviors are consistent with the presence of a ring-shaped nodal feature (nodal ring) on the mirror plane shown by a dashed curve in Fig. 3a.
It should be stressed again that there exists a spin-orbit gap along the nodal line in the calculation. Since the gap is almost isotropic (75±1 meV) along the nodal ring (not shown), the low-energy excitations in CaAgAs are characterized by the excitations across the band gap in the k region involving the entire line node.
Unfortunately, such a band gap (as well as the topological SSs) was not resolved in the ARPES experiment, likely due to the slightly hole-doped nature of crystal.
Considering the fact that (i) the calculated spin-orbit gap is not so small compared to other TIs and (ii) the ARPES-derived band dispersion shows a reasonable agreement with the calculation near EF, it would be more reasonable to regard CaAgAs as a narrow-gap TI, rather than a LNSM. It is also emphasized here that the TI nature of CaAgAs should be distinguished from that of prototypical TIs such as Bi2Se3 since the node never shows up even without SOC in Bi2Se3 unlike CaAgAs.

Sample preparation
High-quality single crystals of CaAgAs were synthesized by the following procedure. An equimolar mixture of calcium chips, silver powder, and arsenic chunks were put in an alumina crucible and sealed in an evacuated quartz tube. The tubes were kept at 773 K for 12 h and then at 1273 K for 12 h, followed by furnace cooling to room temperature. The obtained samples were pulverized, pressed into pellets, and sealed in quartz tubes. The pellets were sintered at 1173 K for 2 h and cooled to room temperature at a rate of 30 K h -1 , resulting in that shiny hexagonal-prismatic single crystals of CaAgAs were grown on the pellets. The quality of the crystal was checked by X-ray diffraction technique using a RIGAKU R-AXIS IP diffractometer.

Calculations
Electronic band-structure calculations were carried out by means of first-principles band structure calculations by using WIEN2k code 65 with the full-potential linearized augmented plane-wave method within the generalized gradient approximation. We used the experimental structural parameters for the calculations. 63 24 × 24 × 36 k-points sampling was used for the self-consistent calculations. 41

Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.