Introduction

Topologically protected electronic states hold great promise for the development of next-generation electronic and spintronic devices1,2,3. For topological insulators (TI), such as the well-known case of Bi2Se34, the carriers associated with the topologically-protected surface states have the unique properties of spin-momentum locking and suppression of backscattering, making the surfaces of such TIs highly attractive platform for spin-transport applications. In the case of topological semimetals, topologically protected crossings occur in the bulk band structure, leading to the formation of bulk Dirac points. In the vicinity of these points, the electronic bands cross linearly, opening up the potential for carriers with ultrahigh-mobility, such as are found in Cd3As25.

The class of transition metal dichalcogenides (TMDs), already well-known for hosting an array of correlated electronic phenomena, has recently been identified as a materials system which also hosts topologically-protected states6,7,8,9,10,11. Density functional theory (DFT) calculations for a wide range of compounds predict a set of bulk Dirac points and band inversions (associated with topological surface states, TSS), as the chalcogen orbital symmetries combined with a natural hierarchy of energy scales ensure that these states exist rather generically. However, the energies at which they occur is not guaranteed by any physical constraint, and many are predicted to exist in the unoccupied states, where they are hard to verify experimentally. On the other hand, topological states below the Fermi level (EF) can be readily verified and scrutinised by angular resolved photoemission spectroscopy (ARPES) and spin-ARPES. When these states lie at high binding energies however, they do not affect the transport properties, limiting any potential applications.

Given that TMDs are being increasingly incorporated into device structures12,13,14,15,16,17, it is highly desirable to identify a candidate material whose transport properties might be predominantly derived from topologically protected states. An intriguing candidate in this regard is NiTe2. Recent transport measurements have confirmed a substantial magnetoresistance effect, the response becoming large and linear in magnetic field, a characteristic feature of topological semimetals18. The Berry phase of the observed quantum oscillations is close to π, often interpreted as a signature of the contribution of a topologically non-trivial state. Moreover, DFT calculations have suggested that one of the bulk Dirac points of NiTe2 may indeed lie much closer to the Fermi level than in other members of the family where the equivalent states are much further away and thus less relevant to transport (or superconductivity)6. These calculations strongly motivate an experimental survey of the electronic structure of NiTe2 using ARPES.

In this paper, we demonstrate with a combination of ARPES and comprehensive DFT calculations that the band structure of NiTe2 shows the formation of type-II Dirac fermions near the Fermi level and a set of spin polarised topological surface states. A Fermi surface consisting of electron pockets associated with type-II Dirac fermions and topological surface states is predicted. The band crossing leading to type-II Dirac fermions is dominated by Te p-orbitals, consistent with the mechanism of band inversion from the chalcogen p-orbital manifold shown in other TMDs6,7. Our ARPES study shows the formation of electron pockets hosting Dirac carriers at the Fermi level. Topological surface states observed through ARPES agree well with our DFT calculations. One of the surface states is found to intersect the Fermi level, forming electron pockets. The measured Fermi surface map matches well with our calculations and the observation of electron pockets implies finite contribution of Dirac and surface carriers to the transport properties.

Results and discussion

In what follows, we adopt the tight-binding-based analysis introduced in Refs6,7 for understanding the band structure of NiTe2, which is distinct from previously known cases of isovalent and isostructural transition-metal dichalcogenides TMX2 (TM = Pd, Pt; X = Te, Se). Our tight-binding model incorporates two chalcogen sites (Fig. 1a), and captures the manner in which the chalcogenide p orbital manifold develops into dispersive bands which generically possess topological characteristics: bulk Dirac points, inverted band gaps (IBGs), and topologically protected surface states (TSS).

Figure 1
figure 1

(a) Crystal structure of 1T-NiTe2 (space group: P-3m1), composed of hexagonal basal planes (ab-planes) of Ni-atom (red sphere) coordinated to the Ni-atom at the centre and triangular layers with inequivalent Te atomic site (blue sphere) above (Te-1) and below (Te-2) the basal plane along the (001)-direction18. The hopping parameters between 2-site Te p-orbitals are categorised as intra-layer hopping (t1 = t2), interlayer hopping (t3) within the unit cell and between two unit cells (t4). (b) Brillouin zone. (c) Hierarchy of p-orbital derived energy levels at Γ and A-point showing the inverted band gaps (IBG), Dirac points (BDP: circled) and topological states. The symmetry of the states are labelled with IREPs (Γi, Aj, Δk) and parity (+/−). Inspired from Ref.7,10. Figure made using Adobe Illustrator 30.

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In NiTe2, the triply degenerate energy levels of the Te px,y,z-orbitals split into pz (upper-state) and px,y (lower-state) manifolds due to crystal-field splitting (CFS) (Fig. 1c). Additionally, spin–orbit coupling (SOC) causes the px,y orbitals to split into singlets (R4′ and R5,6 levels) and modifies the energetic separation between pz and px,y. The pz orbital transform according to the irreducible representation (IREP) R47. The two Te sites in the unit cell imply that the p-orbitals are split into bonding (B) and antibonding (AB) levels at the Γ (k = (0, 0, 0)) and the A (k = (0, 0, π/c))-points of the Brillouin zone7,10. Symmetry wise, the p-bands associated with B and AB levels at Γ and A points transform as the IREPs of the double space groups \(\overline{\Gamma }_{8}\) (\({\overline{\text{A}}}_{8}\)), \(\overline{\Gamma }_{4} \overline{\Gamma }_{5} \left( {{\overline{\text{A}}}_{4} {\overline{\text{A}}}_{5} } \right) \;{\text{and}}\;\overline{\Gamma }_{9} \left( {{\overline{\text{A}}}_{9} } \right)\)19,20. Because of the phase introduced, the B-AB splitting of the pz-states decreases from the Γ to the A point13. This induces strong kz dispersion and a large bandwidth for pz-derived bands as compared to planar px,y orbitals. For TMDs, the approximation of pz-derived bands having larger bandwidth is valid considering the expected large inter-layer-hopping (t3, t4 in Fig. 1a) along the c-axis for pz states as compared to the px,y states.

When the bandwidth of pz-derived states become greater than the combination of CFS and SOC, a crossing between the pz and px/y states occurs as a function of kz, resulting in band inversion. The states formed due to the B-AB splitting of p-orbitals pick up even (+) or odd (−) parity since the Te-atom sites at the Γ and A points are located across the crystal inversion point. Therefore, kz-dispersion of a single orbital (p) manifold leads to band parity inversion. Figure 1c, shows the symmetry correspondence between IREPs of the double group at the Γ and A point, and along the Δ symmetry line for all the k = (0, 0, kz) points with 0 < kz < π/c. A crossing between \(R_{{4^{\prime } }} \)(\(\pm\)) and \(R_{{4}} \) (\(\mp\)) causes hybridisation as seen from the identical IREP, \({\overline{\Delta }}_{6}\) , for these two levels along the Δ point. This leads to a parity inverted band gap (IBG) (See Fig. 1c). In addition, the opposite parity of \(R_{{4^{\prime } }} \) ( ±) and \(R_{{4}} \) (\(\mp\)) suggests that the IBG will have a ℤ2 topological order and are capable of hosting topological surface states7. On the other hand, \(R_{{4}} \) (\(\pm\)) and \(R_{{5, 6}} \)(\(\mp\)) have different symmetry properties along the Δ line, as they transform differently under threefold rotation. This allows band crossing between \(R_{{4}} \) and \(R_{{5, 6}}\) leads to the formation of the bulk Dirac points (BDPs) along the kz direction (valid for 0 < kz < π/c). The location of the BDP in k-space is determined by the band width of \(R_{{4}} \) (\(\pm\))-derived bands and the strength of CFS.

The discussion above is based on symmetry, yielding a generic and phenomenological viewpoint. To obtain a more quantitative prediction of the location of the expected bulk Dirac points and IBGs, we turn to ab-initio DFT calculations. Our calculated band structure showing band dispersion and the orbital character of bands is shown in Fig. 2a. The density of states (DoS) confirms that the bands close to the Fermi level, EF, are dominated by Te p-orbitals. Ni d-orbital derived bands appear away from the Fermi level at E–EF ~ − 2 eV (Fig. 2b). The px/y bands show significantly stronger dispersion along the in-plane high symmetry directions Γ(A)-M (L) and Γ(A)-M (K), compared to the out-of-plane Γ-A direction. However, consistent with the tight-binding analysis, the pz bands have a large dispersion along Γ-A with wide band width and cross the px/y bands as a function of kz.

Figure 2
figure 2

(a) Electronic DFT bulk band structures with orbital character of bands. Type-II Dirac fermion formed by crossing of band 1 and 2 near Fermi level. (b) Density of states (DoS) showing dominant contribution of Te p-bands at the Fermi Level compared to Ni d-bands. Above Fermi level, the DoS spectra is scaled up by factor of 2. (c) Band dispersion along the Γ-A direction with inverted band gaps (IBG) and bulk Dirac points (BDP) marked, (d) and (e) Zoomed-in dispersion and measured photon-energy dispersion with linear horizontal polarization along the Γ-A direction. A non-dispersive feature is marked with an arrow, which does not match with bulk DFT calculation. Figures made using Igor29 and Adobe Illustrator30.

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Our DFT calculations indicate several important differences between NiTe2 and the other TMX2 compounds. Weakly dispersive bands of predominantly d-character are found at E–EF ~ − 2 eV (Fig. 2a, c). The location of these bands is almost 1.5 eV closer to the Fermi level compared to PdTe2, a significant difference on replacing Pd with Ni. In addition, at K close to the Fermi level (E–EF ~ 0 to − 250 meV), the electron pockets are predominantly formed by bands derived from d-orbitals.

The large shift of TM-derived bands towards the Fermi level in NiTe2 increases the hybridisation between Ni-d and the Te-p bands. For example, the crossing of Te pz and px,y bands occurs near the A point around E–EF ~ − 1 eV, but the pz bands in this energy range show a significant hybridisation with Ni d-orbitals, unlike in other TMX2 compounds. However, this hybridisation does not disrupt the universal mechanism of formation of the inverted band gaps, Dirac fermions and topological surface states in TMX2. A type-I bulk Dirac point (BDP-I) appears at E–EF ~ 1.5 eV, which is followed by a type-II protected bulk Dirac point (BDP-II) at E–EF ~ 76 meV and kz = \(\pm\) 0.36c* (c* = 2π/c). The bands forming the BDP-II are labelled as 1 and 2 (Fig. 2d). These bands cross EF and form electron pockets. This type-II Dirac fermion has been found in other TMX2; however, the proximity of BDP-II to the Fermi level is unique to NiTe26,7,8,9,10,18. In NiTe2, the location of BDP-II in momentum space is slightly closer to A-point than PtTe2 (kz = \(\pm\) 0.346c*) but further away from A than in PdTe2 (kz = \(\pm\) 0.40c*)7,9,10,21. This implies that the strength of CFS and bandwidth of \(R_{{4}} \)(\(\pm\)) derived bands is intermediate between PtTe2 and PdTe2. From the calculated band structure we found that close to BDP-II, the pz derived bands form inverted band gap IBG-I at E–EF ~ − 0.65 eV (smaller gap ~ 200 meV), which is accompanied by IBG-II at E–EF ~ − 1.5 eV (larger gap ~ 1 eV). We want to point out that the inverted band gaps (IBGs) at the A-point have inverted parity, implying the existence of topological states.

With these predictions in hand, we now turn to the experimental measurements of the bulk and surface electronic structure of NiTe2. First, we focus on tracking the BDP-II and surface states along Γ-A using photon-energy-dependent ARPES and measure the occupied states below the Fermi level (Fig. 2e). The band features are broadened due to the finite kz resolution of photoemission but the states are in good agreement with calculated bulk bands. Both IBGs, near Fermi level (E–EF ~ − 0.65 eV) and away from it (E–EF ~ − 1.5 eV) are found. The BDP-II lying above the Fermi level is not accessible by ARPES, but the bands 1 and 2 forming the BDP-II are observed at the A-point (kz = π/c). Interestingly, a non-dispersive two-dimensional band is found at E–EF ~ − 1.35 eV close to A-point which does not match any calculated bulk band (marked in Fig. 2e).

To better understand the band structure, we compared the experimental and calculated in-plane band dispersion along A–L and A–H in Fig. 3. The bulk bands show good agreement with bulk DFT calculations (Fig. 3a, b, d, e). Our DFT slab calculations (see “Methods” section for details) show a set of surface states which match well with the experimental data (Fig. 3c, f). The non-dispersive feature at E–EF ~ − 1.35 eV is reproduced as one of the surface states by the slab calculations. We assign the surface state as TSS2. At the A-point, TSS2 forms sharp and intense bands. This feature threads through the IBG-II and connects the time-reversal invariant momentum (TRIM) points (here A, L, H). TSS2 represents a topological non-trivial band but lies far away from the Fermi level.

Figure 3
figure 3

Spectral band dispersion along in-plane L′–A–L (a–c) and H′–A–H (d, e) direction, probed with photon energy hν = 99 eV (a) experimental ARPES data with linear horizontal polarization, supercell calculation integrated along kz with (b) bulk DFT and (c) slab surface. Arrows mark the position of the topological surface states (TSS). The overlaid dotted red lines in (a, d) represent the calculated bulk DFT bands. Figures made using Igor29 and Adobe Illustrator30.

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Slightly closer to the Fermi level, our DFT calculations show a topological surface state at E–EF ~ − 0.65 eV within the band gap IBG-I. This is named as TSS1 and lies within the manifold of bulk bands. TSS1 experiences strong interaction with the bulk states and is classified as a surface resonance state. In ARPES spectra, we observe weak spectral intensity of TSS1 at A, which matches well with the calculations (Fig. 3a, c). However, the mixing of TSS1 with bulk bands makes it difficult to resolve.

At the Fermi level, a surface state is visible in our DFT surface band structure along A–L (Fig. 3c). This is assigned as TSS0, which shows a large spin splitting of around 120 meV and lies between E–EF ~ 0 to − 200 meV. TSS0 is not present along A–H (Fig. 3f). Experimentally, TSS0 is found between A and L with in-plane momentum k||~ 0.5 (Å)−1. TSS0 is formed from two parabolic-like upper (labelled as ε) and lower (γ) branches which intersect EF and form electron pockets (see Fig. 4a).

Figure 4
figure 4

(a) In-plane dispersion of TSS0 forming electron pockets labelled as ε and γ and (b) the non-dispersive behaviour of TSS0 surface state close to the Fermi level in the photon-energy dependence. (c) Fermi surface measured with horizontal polarization of light and photon energy, hν = 23 eV, which corresponds to the A-plane. The region under the red dotted box is magnified by a factor of 5 to show the electron pockets α (Band 2), ε and γ (TSS0) and the bulk bands. (Inset: calculated bulk Fermi surface consisting of the circular α electron pocket and bands forming hexagonal contour, in good agreement with measured data). (d) and (e) Slab calculation with projected spin components of the topological surface states. TSS2 shows finite <Sx> and <Sy> components normal to H′–A–H and L′–A–L direction, respectively. TSS0 has finite spin component <Sy> normal to L′–A–L (note that TSS0 is hybridized with bulk in all other regions of the plots, so no further conclusions about surface state spin texture can be made). Figures made using Igor29 and Adobe Illustrator30.

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The kz—dependence of the electronic states spanning over a wide photon energy range (hν = 20–120 eV) shows that TSS0 is non-dispersive along Γ-A (Fig. 4b). This confirms that TSS0 is two-dimensional. The vicinity of TSS0 to EF in NiTe2 makes it unique compared to other surface states and implies finite contribution of topological surface carriers to the non-trivial transport properties. All topological surface states found previously in other TMX2 are also found in NiTe26,7,8,9,10, but notably TSS0 is absent in PtSe2.

To complete our studies, we map the Fermi surface of NiTe2 using photon energy tuned to the A-point (hν = 23 eV) (Fig. 4c). It is found that the Fermi surface is formed by TSS0 and the bulk bands crossing the Fermi level. The ε and γ bands of TSS0 are identified. These bands form electron pockets at the Fermi level. The pockets formed by ε are more circular than the pockets of the γ band. Identical to PdTe2, these bands form arc-like features and imply small projected bulk band gaps. At the A-point, band 2 forms an electron pocket (labelled as α) and the projected bulk bands form a hexagonal shape. We show that the calculated bulk Fermi surface agrees well with the experimental data (see Inset of Fig. 4c).

To identify the topological character of the surface states, the associated spin texture is calculated as shown in Fig. 4d, e. In our description, the x- and y-axis are along A–L and A–H, respectively. TSS2 has a chiral spin texture in the x–y plane with finite <Sx> and <Sy> components. Negligible <Sz> component is found (see Supplementary Information). The strong interaction with bulk bands for TSS1 limits the estimation of spin polarization, and we do not include polarization for TSS1. The electron pockets formed by TSS0 along L–A–L (indicated in Fig. 4c) are well separated from the bulk and hence it is possible to resolve spin texture here. We find that TSS0 has non-zero spin component solely along the y-direction <Sy>, which is normal to the A–L direction. The branches of TSS0, ε and γ, show opposite spin polarization. We want to point out that the spin-split branches of the surface state TSS0 close to the Fermi level are connected to the conduction band and the valence band separately, which highlights the topological non-trivial character22. For completeness, we include spin polarization for TSS0 along the entire L′–A–L and H′–A–H paths, but note that the only location where the surface character has not significantly hybridized with bulk is at the electron pockets, so calculated spin texture at all other locations does not accurately represent surface state spin texture of TSS0.

Conclusion

We have shown that NiTe2 exhibits topological surface states and bulk type-II Dirac points, which are derived solely from the Te 5p orbitals, consistent with the generic band inversion mechanism for transition metal dichalcogenides. However, the choice of Ni as the transition metal leads to 3d states significantly closer to the Fermi level compared with (Pd, Pt)(Se, Te)2, allowing additional d–p hybridisations which tune the bulk Dirac point very close to EF. The band dispersions away from this BDP-II, which form the alpha electron pocket, thus have a topologically non-trivial character, likely to explain the non-trivial Berry phase of a small electron-like pocket observed in transport studies18. We have also shown that NiTe2 harbours a unique topological surface state, TSS0, with one of the largest spin-splitting of up to 120 meV reported for any Fermi-crossing surface state of a transition metal dichalcogenide. For future studies, an exciting approach would be to exfoliate individual monolayers or grow very thin films of NiTe2, with a significantly increased contribution of these topological surface carriers to the transport properties. Thus both the surface and bulk electronic structures of this material are candidates for engineering novel spintronics devices underpinned by a robust and generic topological mechanism.

Note added

The present work was made public through posting as an e-print (arXiv:1912.08535). We note that a photoemission studies, which reports some similar results as part of our findings was carried out independently by another group and was published: see ref.23

Methods

ARPES measurements

High quality single crystals of NiTe2 were grown by chemical vapour transport as discussed elsewhere18. We used the high-resolution vacuum ultraviolet (VUV) ARPES branch of I05 beam line at Diamond Light Source, UK24. The samples were cleaved at 10 K, in ultra-high vacuum chamber of I05. Linear horizontal and vertical polarization of photon at energies between hν = 20 and 120 eV is used. The inner potential V0 = 16 eV of NiTe2 is determined from the kz dispersion from photon-energy-dependent ARPES studies using a free electron final state model. We want to point out that the final state induced matrix element effect can lead to suppression of the spectral intensity. This might give an impression of asymmetric spectra in the experimental data.

First principle calculations

Our calculations were performed within density functional theory (DFT) with room-temperature lattice parameters (details on lattice parameters is discussed elsewhere18. We employ the Vienna ab-initio simulation package (VASP)25 with generalized gradient approximation (GGA) using the Perdew-Burke-Ernzerhof (PBE) functional26 and projector augmented-wave (PAW) method27. We treat 3p, 3d, and 4s, and 5s and 5p electrons as valence for Ni and Te, respectively. We use an energy cut off of 550 eV for our plane-wave basis set, with a Γ-centred k-point mesh of 14 × 14 × 14 for the primitive unit cell. For surface state features, we perform full self-consistent DFT calculations using a Te-terminated supercell with a slab of ten primitive unit cells of NiTe2 stacked along the [001] direction, with 15 Å of vacuum. We conducted calculations with U-parameter related to Ni-d orbitals and found that the best agreement with experimental data occurs at U = 0. Here, we do not show DFT calculations using GGA + U ≠ 0, since they fail to improve the agreement with the experimental data.