Observation of nodal line in non-symmorphic topological semimetal InBi

TNS is a newly discovered gapless-type quantum material with numerous intriguing physical properties [1, 2]. The first kind of TNS is where the conduction and the valence band touch in zero-dimension forming either quadruply degenerate points (topological Dirac semimetal) [3–6] or pairs of doubly degenerate points with opposite chirality (topological Weyl semimetal) [7, 8]. Apart from being protected due to the topological nature, these points have bands disperse through them linearly in three-dimensional (3D) momentum space, which brings interesting phenomena such as unusually high carrier mobility [9, 10], large intrinsic spin Hall effect [11], giant diamagnetism [12], and chiral anomaly in magnetoresistance [13, 14]. Recent experimental works have successfully demonstrated the existence of 3D Dirac point in materials e.g. Na₃Bi [5, 15, 16], Cd₃As₂ [4, 17, 18], ZrTe₅ [19–21], etc, while the Weyl semimetal phase is also confirmed in materials such as TaAs [6, 13, 22–26], NbAs [25–27], NbP [10, 24–26], TaP [24, 26, 28], MoTe₂ [29–32], TaIrTe₄ [33–35], etc.

If the linear touching of the bands extend from (Dirac or Weyl) points to one-dimension (1D) lines, the TNLS emerges [1]. Comparing to discrete degenerate nodes, 1D nodal line (NL) structure forms a more significant feature in the electronic band structure (figure 1(a)) and can have stronger contribution to the physically observable properties of these materials. To realize the NL phase, there are several paths including the band inversion of opposite parity bands that occurs at the high symmetry point without spin–orbit coupling (SOC) [2], inversion symmetry and TR consequence without SOC [36], mirror symmetry and TR consequence without SOC [37, 38], and mirror symmetry and TR with SOC [39]. However, as there is no guarantee that the NL will lie non-dispersively at equal energy, let alone at fermi level even if those discrete symmetries are imposed [1], observing the NL is a challenging task for experiment.

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While several compounds have been predicted to be TNLS (e.g. Cu₃PdN [36], Ca₃P₂ [40], 3D graphene network [41], and hyperhoneycomb lattice [42]), only few observations have been attempted, such as in PbTaSe₂, PtSn₄, and ZrSiS. Among them, PbTaSe₂ is inversion asymmetric and hosts nodal loops protected by reflection symmetry under SOC [39]; PtSn₄ has broken nodal-arcs but its topological character is still under investigation [43]. Furthermore, it is not easy to identify the nodal-lines in PbTaSe₂ and PtSn₄ convincingly due to the complicated Fermi surface (FS) of the two compounds. More recently, non-symmorphic compounds have been predicted to host nodal-line structure which are robust against the SOC due to the protection of the non-symmorphic symmetry as reported on the non-symmorphic ZrSiS family compounds where the NLs along two paths along the Brillouin zone (BZ) boundary were observed [44].

In this work, we explored the electronic structure of another non-symmorphic compound InBi and demonstrated it as a new member compound of TNLS. By using angle resolved photoemission spectroscopy (ARPES) we demonstrate that InBi possesses quadruply degenerate bands along the high symmetry lines of ${k}_{z}$ (X–R–X) directions (as shown in red line figure 1(b)). The band structure and the nodal-line position show good consistency with the ab initio calculation and proved InBi as a TNLS. As the spin–orbit coupling strength is large due to the presence of heavy Bi element, our observation of the TNLS phase in this compound further proves the protection of the non-symmorphic symmetry.

Our ab initio calculation as shown in figure 1(b) demonstrates how each band evolves with contribution from Bi ${p}_{z}$ (blue color) to Bi ${p}_{x,y}$ (green color) emphasized. A closer investigation reveals that the bands involved in the crossing along the ${\rm{\Gamma }}{\rm{Z}}$ direction exchanges orbital (Bi ${p}_{z}$ to Bi ${p}_{x,y}$ and In ${p}_{x,y},$ and vice versa) and is not gapped when SOC is included, establishing InBi to possess a type-II Dirac point (see appendix A), similar to that in AMgBi (A = K, Rb, Cs) compounds [45]. Its non trivial topology was also confirmed by the evolution of the Wannier charge centers, which is using non-Abelian Berry connection [46, 47]. In addition to that, there are also quadruply degenerates NLs along XRX and AMA direction which are formed due to the glide symmetry and the screw axis symmetry intrinsic to the crystal structure of InBi (see appendix B). Since the signature Dirac semimetal has been attempted in several studies mentioned before, in this work we then focus on these topologically trivial degenerate lines along XRX (check the BZ in figure 1(b)) to demonstrate the NL and its dispersion in energy.

InBi is known to have a simple tetragonal structure [48]. As shown in figure 1(c), the crystal has a layered structure in general with cleaving plane along the [001] direction due to its relatively weak bonding between the neighboring sublayer of Bismuth [49]. Our 3D XRD results (figure 1(d)) confirm the crystal structure (P4/nmm space group) with high quality (lattice constant a = 5.014 ± 0.002 Å and c = 4.784 ± 0.003 Å). After cleaving, flat and shiny (001) surface is exposed, making it suitable for ARPES measurement (figure 1(d)). The core level photoemission spectra in figure 1(e) reveals characteristic 4d_5/2 and 4d_3/2 peaks for In and 5d_5/2 and 5d_3/2 peaks for Bi, further confirming the elemental composition of the sample. The broad FS map obtained by ARPES measurements in figure 1(f) shows the square BZ with four-fold symmetry, proving the tetragonal crystal structure and (001) cleavage plane.

We then systematically studied the 3D band structure of InBi as illustrated in figure 2(a) where both the FS and the band dispersions along high symmetry directions are shown. The overall electronic structures appear more complex than the ab initio calculation of the bulk bands in figure 1(b), indicating that many bands have surface origin. We thus performed slab calculations for both bulk and surface electronic states (see appendix C.2 for detailed description) and plot them along the side of each corresponding measurement (e.g. $\bar{{\rm{X}}}\bar{{\rm{\Gamma }}}\bar{{\rm{X}}},\bar{{\rm{M}}}\bar{{\rm{\Gamma }}}\bar{{\rm{M}}},{\rm{and}}\,\bar{{\rm{M}}}\bar{{\rm{X}}}\bar{{\rm{M}}}$ in figures 2(b)–(d), respectively). The comparisons between experiment and calculations now show nice agreement, confirming the important contribution of surface states in the photoemission intensity of InBi. In figure 2(e), the equal energy contours of the electronic structures (from the 3D ARPES data in figure 2(a)) and their evolution also shows the significance of the surface states, which again in excellent consistency with our ab initio calculation in figure 2(f).

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The existence of the surface states may prevent us from observing the crossing of the bulk bands (and thus the line-node) clearly. For example, a closer investigation on the calculated surface state at $\bar{{\rm{X}}}$ on higher binding energy reveals several surface states band crossings sitting within the energy range (Eb = 0.71–1.26 eV), making it more difficult to resolve the bulk NLs along XRX as predicted in figure 1(b).

To distinguish the contribution of the bulk bands from the complex surface states, we performed photon energy dependent measurement (30–240 eV) on the electronic structure of InBi. From the FS measured at different photon energies (figure 3(a)), we could always identify a 'flower' shape feature near the $\bar{{\rm{\Gamma }}}$ point (labeled with SS), indicating it is originated from the surface states which is further proved by our ab initio calculations. In addition, some other features on the FS, including the circular pocket around $\bar{{\rm{\Gamma }}}$ (labeled with BS1) and the 'bridge' shape at FS along $\bar{{\rm{\Gamma }}}-\bar{{\rm{X}}}$ (labeled with BS2) change in the shape and intensity at different photon energies, suggesting their bulk nature. We further prove the surface and bulk nature of different bands by showing the constant energy contours at FS and $E-{E}_{F}=-0.75\,{\rm{eV}}$ on the ${k}_{x}-{k}_{z}$ plane $({\rm{\Gamma }}\mathrm{XRZ})$ in figures 3(b) and (c). This was done by taking a photon energy dependence data for each photon energy probes different ${k}_{z}$ position. From the constant energy maps, we could identify two straight lines in the middle of ${\rm{\Gamma }}$ and ${\rm{X}}.$ These two lines come from the 'flower' shape in figure 3(a) and the lack of k_z variation proves their surface nature. In contrast, the feature around ${\rm{X}}$ (corresponding to the 'bridge' shape in figure 3(a)) shows strong variation with different k_z value and agree nicely with the calculation result of the bulk bands (see the schematic on top of figure 3(b)), proving its bulk nature. The band structure along the XRX line also show a visible periodicity following the calculated result in figure 3(d).

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In figure 4(a), we show detailed 3D band structures at 100 eV photon energy, where the surface states are greatly suppressed and better cross section was observed for the bulk band [50]. As shown by the ab initio calculations, a (2D) Dirac point formed by the bulk band is seen (figures 4(a) and (b)), and the calculated band structures can be nicely overlapped on top of the ARPES data.

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We then study the evolution of the Dirac point in energy along the NL (along the XRX direction). Figure 4(d) shows cuts at different ${k}_{z}$ positions (from ${\rm{\Gamma }}{\rm{X}}{\rm{\Gamma }}$ to ${\rm{ZRZ}}$) while the results from calculation (${\rm{\Gamma }}{\rm{X}}{\rm{\Gamma }}$ to ${\rm{ZRZ}}$) are overlapped at figure 4(c). From each cut in figure 4(d), we observe band crossing with fair agreement with the bulk calculation indicating the existence of Dirac point for each cut, hence NL for XRX. However, the ${k}_{z}$ broadening and the increasing scattering rate at high binding energy make it harder to pin the Dirac point for each cut [51]. Thus, from the cut at ${\rm{\Gamma }}{\rm{X}}{\rm{\Gamma }},$ the momentum distribution curve (MDC) and energy distribution curve (EDC) are plot at figure 4(e) to demonstrate that the crossing is indeed gapless (within our resolution). Afterwards, figure 4(f) demonstrates the EDC X through R from the cuts given in figure 4(d) which shows how the extracted peak with fitting error (indicating the Dirac point) moves creating the NL. It can also be seen that the ${k}_{z}$ broadening effect severely broaden the EDC e.g. the second EDC from the left shows a relatively broad peak due to the contribution from the neighboring ${k}_{z}$ positions. However, there is a fair agreement in the shape of the NL between the overlapped calculated result (figure 4(g)) with peaks movement of the measured ARPES cut which broadly demonstrates how the NL dispersing.

Single crystals of InBi were grown by a modified horizontal Bridgman technique. In and Bi were mixed at a molar ratio of 1:1. The mixture was sealed in an evacuated quartz tube, pre-reacted by heating until it completely melted. Using the obtained ingot, the crystal growth was carried out in a homebuilt horizontal moving tube furnace. The heating unit was kept at 200 °C, and moved in a lateral direction of the sealed quartz tube with a speed of 1 mm h⁻¹. The obtained single crystals had good cleavage along the ab-plane (001), producing flat surfaces of ∼1 × 1 cm². The in-plane resistivity, measured by the four-probe method, was very low (<500 nΩ cm) for T < 10 K and the residual resistivity ratio (2 K versus 300 K) was over 300, indicating a high quality of the crystals.

ARPES measurement were performed at i05 Diamond Light Source and at BL 5-4 Stanford Synchrotron Radiation Lightsource both equipped with Scienta R4000 analyzers. The sample temperature and the pressure during measurement were 10 K and lower than 1.5 × 10⁻¹⁰ Torr, respectively. The angle resolution was 0.2° and the overall energy resolutions were 15 meV. The samples were cleaved in situ along the (001) plane.

Tunable soft x-ray from synchrotron light source ranged from 30 to 240 eV is used to get the complete band structure image along the ${k}_{x},{k}_{y},$ and ${k}_{z}$ directions and also to distinguish the bulk and the surface band structure contribution [50]. The electron momentum parallel to the surface can be calculated from the relation ${k}_{\parallel }=\tfrac{\sqrt{2{m}_{e}{K}_{{\rm{o}}{\rm{u}}{\rm{t}}}}}{\hslash }\,\sin \,\theta ,$ ${k}_{x}=\tfrac{\sqrt{2{m}_{e}{K}_{{\rm{o}}{\rm{u}}{\rm{t}}}}}{\hslash }\,\sin \,\theta \,\cos \,\phi ,$ and ${k}_{y}=\tfrac{\sqrt{2{m}_{e}{K}_{{\rm{o}}{\rm{u}}{\rm{t}}}}}{\hslash }\,\sin \,\theta \,\sin \,\phi$ where ${m}_{e}$ is the mass of free electron, ${K}_{{\rm{o}}{\rm{u}}{\rm{t}}}$ is the kinetic energy of electron outside of the material, $\theta$ is the polar angle, $\phi$ is the azimuthal angle, ${k}_{\parallel }$ is the momentum parallel to the material with ${k}_{x}$ and ${k}_{y}$ are the components of it along the $x$ and $y$ directions. Even though the perpendicular momentum is not conserved, the perpendicular momentum inside the material can be predicted with the relation ${k}_{{z}_{{\rm{i}}{\rm{n}}}}=\sqrt{\tfrac{2{m}_{e}^{* }}{{\hslash }^{2}}({K}_{{\rm{o}}{\rm{u}}{\rm{t}}}+{V}_{{o}})-\tfrac{2{m}_{e}}{{\hslash }^{2}}{K}_{{\rm{o}}{\rm{u}}{\rm{t}}}{\sin }^{2}\phi }$ with ${V}_{o}$ as the inner potential for the material and ${m}_{e}^{* }$ as the effective mass of the electron [53]. The more complete explanation of ARPES technique can be found in other publications [54–57].

The bulk band calculation was performed by using density-functional theory (DFT) with the projected augmented wave potential as implemented in the Vienna ab initio simulation package [58, 59] with spin–orbit coupling calculation included, generalized gradient approximation [60] exchange-correlation functional employed. The maximally localized Wannier functions [61] constructed from DFT were used to calculate bulk 2D FSs with a dense k-point grid.

The surface state associated with (001) surface were calculated by first reconfirming that the cleaving plane terminates in between the Bismuth layer which showed formation energy approximately 1.3 eV per unit cell lower than other possible planes. Then, a slab model with a 36-atomic-layer structure was applied. The top and bottom surfaces were both terminated by Bi, and a vacuum layer of over 20 Å was added in z direction to avoid the coupling between top and bottom surfaces. The weighted wave function contributions from the outermost tri-layer (3-atomic-layer) are shown in color gradation (100% in red to 0% in white) in order to distinguish the surface states contribution from bulk states (of 36 atomic layers) contribution. The force convergence criterion of 0.01 eV Å⁻¹ was adopted for structural optimization.