1 Introduction

Topological quantum materials with novel surface states protected by different symmetries have attracted enormous interests in the last decade. Topological insulators (TIs) [1–6] were first realized, prompting the emergence of multiple types of topological semimetal phases, such as Dirac semimetal [7, 8], Weyl semimetal [9–12], nodal and nodal-line semimetal [13–16] and so on [17–20]. In real materials, non-trivial topology is mainly induced by the band inversion [21–23]. For example, Bi2Se3 or Bi2Te3, the prototypical TI, is topologically non-trivial because \(p_{z}\) orbitals of Bi and Se (Te) have opposite parity at the Γ point [24]. In Na3Bi, the first confirmed Dirac semimetal, Na 3s and Bi \(6p_{x,y}\) bands invert [25]. The band inversion can occur not only between two different atomic manifolds, but also between the same orbital manifold in binary compounds. For example, in PdTe2, R4 and R4′ bands from Te atoms invert, which drives PdTe2 into a topological semi-metallic phase [26]. Besides the binary compounds, some ternary compounds were also confirmed as topological materials, such as GeBi2Te4 [27], SnBi2Te4 [28], TaNiTe5 [29] and so on. The band inversion in Ge(Sn)Bi2Te4 is similar to Bi2Te3. In TaNiTe5, it is more complicated because all bands strongly mixed. In our previous report, we observed Dirac-like surface state (SS) that is immune to the surface relaxation of atoms in TaNiTe5 and proposed a topological origin due to the band inversion between three bands [29]. TaPdTe5 has the similar crystal structure to TaNiTe5. The de Haas-van Alphen oscillations in magnetoresistance measurements showed a possible non-trivial Berry phase in TaPdTe5 [30, 31]. It is interesting to check our previous proposal by looking for the possible SS in TaPdTe5 and its relation to the band inversion. In this work, by combining ARPES and first principles calculations, we studied the low energy electronic structure near the Fermi level (\(E_{F}\)) in TaPdTe5. Two inverted band gaps were determined following the parity evolution of the bulk bands by artificially varying spin-orbital coupling (SOC) strength. Although the inverted bulk bands in TaPdTe5 are different from that in TaNiTe5, two sets of surface bands were observed within the gaps and nicely agree with the calculations. Our findings unambiguously reveal the topological origin of the SS. In addition, we also observed strong surface resonant states that hybridize with the topologically protected SS in TaPdTe5.

2 Materials and methods

2.1 Experiments

TaPdTe5 single crystals were synthesized by the self-flux method [31, 32]. High-purity Ta pieces, Pd shots, and Te ingots were mixed in a molar ratio of 1:1:10 and sealed in an evacuated quartz tube. The quartz tube was loaded into a furnace and heated to 1000°C. Samples were kept at 1000°C for 24 hours and then slowly cooled to 500°C at a rate of 5°C/h. After being centrifuged at 500°C and cooled to room temperature, needle-shaped TaPdTe5 single crystals were obtained. ARPES measurements were performed at beamlines BL03U and BL09U in Shanghai Synchrotron Radiation Facility (SSRF). Samples were cleaved in situ at 20 K to obtain the fresh surface. All ARPES spectra were collected at 20 K in the ultrahigh vacuum chamber with base pressure better than 5 × 10−11 Torr. Scienta analyzer (DA30-L) was used in the experiments. Angular resolution is ∼0.05° and energy resolution is ∼ 10 meV.

2.2 Calculations

First-principles calculations based on density-functional theory (DFT) were performed using the Vienna ab initio simulation package (VASP) [33, 34]. Plane waves basis with an energy cutoff of 500 eV were used to expand the electronic states. The exchange-correlation functional was given by the generalized gradient approximation parameterized by Perdew, Burker and Ernzerhof (GGA-PBE) [35]. We first relaxed the bulk TaPdTe5 to optimal geometry for both atomic positions and lattice constants until the residual force on each atom is smaller than 0.001 eV/Å. Structural optimization and electronic properties calculation were performed by using a 9 × 3 × 3 k-grid. To investigate the surface states, we employed the Wannier90 code [36] to construct the tight-binding Hamiltonian from the VASP self-consistent calculations. The tight-binding Hamiltonian was used to obtain the surface electronic states based on a Green function scheme [37], as implemented in the Wannier tools code [38].

3 Results and discussions

The crystal structure of TaPdTe5 is shown in Fig. 1(a). TaPdTe5 has a layered orthorhombic structure (space group Cmcm) [39, 40] with the lattice constants of \(a= 3.70\) Å, \(b= 13.23\) Å and \(c = 15.67\) Å. TaPdTe5 layers stack along the b-axis. Each layer contains two kinds of atom chains: bicapped trigonal prismatic Ta chains and octahedral Pd chains along the a-axis. Figure 1(b) shows the bulk Brillouin zone (BZ) and its surface projection to the (010) surface. The x/y/z direction in Fig. 1(b) corresponds to the a/b/c-axis in Fig. 1(a). The calculated bulk electronic bands near the \(E_{F}\) along some high symmetry directions are shown in Fig. 1(c). Several bands cross the \(E_{F}\), indicating TaPdTe5 is metallic.

Figure 1
figure 1

(a) Crystal structure of TaPdTe5. Dashed lines indicate the unit cell. (b) Bulk BZ and its surface projection to (010) plane. (c) Calculated bulk electronic bands. (d) Fermi surface mapping on the (010) surface using incident photons of 26 eV. (e) Calculated bulk FS in the ΓZMX plane. Green rectangles label the position where the experimental data deviate from the bulk calculations. Black arrows indicate the positions where two Fermi sheets hybridize. (f) and (g) Sketches of an electron Fermi pocket hybridizes with bulk-like states

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ARPES measurements were carried out on the cleaved (010) surface (ac-plane). Figure 1(d) shows the two-dimensional (2D) Fermi surface (FS) mapping (constant energy contour at the \(E_{F}\)) using incident photon of 26 eV. Figure 1(e) is the calculated FS in the ΓZMX plane (\(k_{y} = 2n\pi \)). Blue dashed lines mark the boundaries of the surface BZ. Although we don’t know the value of \(k_{y}\) in Fig. 1(d), we find that the measured FS seems to match most features in Fig. 1(d) except an “elliptical Fermi pocket”, marked by green rectangle. Such kind of Fermi pocket was also observed in TaNiTe5 [29], which was found to come from the SS. With the high resolution of the spectra, we found that this “elliptical Fermi pocket” actually breaks when it crosses the quasi-one-dimensional (Q1D) open Fermi sheets. Two breakpoints are indicated by black arrows in Fig. 1(d). The break of the “elliptical Fermi pocket” is caused by its hybridization with those Q1D states. Figure 1(f) and (g) sketch part of the FS without and with hybridization based on Fig. 1(d), respectively. The Q1D open Fermi sheets become closed 2D Fermi pockets. For instance, there become three pockets around Z̅ point instead of two pockets in bulk calculations. It is interesting why there is strong hybridization between the SS and the bulk-like state. Actually, we figure out that the detected bulk-like states are the surface resonant state (SRS) instead of the real bulk states. SRS resembles the bulk band dispersions at some \(k_{y}\) and have enhanced spectral weight on the surface, therefore it can strongly hybridize with the SS and change the FS topology. The nature of the SRS as well as SS are determined by photon-energy dependent measurements (\(k_{y}\) dependence) in Fig. 2.

Figure 2
figure 2

(a)–(h) ARPES spectra taken with various incident photon energies from 20 eV to 48 eV with an interval of 4 eV along \(\overline{Z} - \overline{\Gamma} - \overline{Z}\) direction. (i) Calculated spectral function of bulk bands projection along \(\overline{Z} - \overline{\Gamma} - \overline{Z}\) direction. (j) Calculated spectra function of the surface states and the bulk states projection along \(\overline{Z} - \overline{\Gamma} - \overline{Z}\) direction. (k) ARPES spectra taken with 26 eV incident photon energies along \(\overline{X} - \overline{\Gamma} - \overline{X}\) direction. (l) Calculated spectral function of bulk bands projection along \(\overline{X} - \overline{\Gamma} - \overline{X}\) direction. (m) Calculated spectra function of the surface states and the bulk states projection along \(\overline{X} - \overline{\Gamma} - \overline{X}\) direction. (n) Measured band dispersion of SS1 band along two high-symmetry directions. The shadow region represents the bulk bands projection

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ARPES spectra taken with various incident photon energies from 20 eV to 48 eV with an interval of 4 eV along \(\overline{Z} - \overline{\Gamma} - \overline{Z}\) direction crossing the BZ center are presented in Fig. 2, covering more than one bulk BZ along \(k_{y}\) direction. Except the variation of intensity, the detected band dispersions are very similar. Three bands (band-1, 2, 3 indicated by black arrows) crossing the \(E_{F}\), forming three Fermi pockets around Z̅ point, are clearly observed in Fig. 2(c). Since no \(k_{z}\) dispersion is detected, we conclude that band-1, 2 and 3 are all 2D states. Similar to TaNiTe5 [29], two band crossing points (“\(X_{1}\)” and “\(X_{2}\)”) in band-1 and 2 are observed, indicated by blue arrows in Fig. 2(d). To understand the origin of band-1, 2 and 3, we calculated the spectral functions of bulk bands projection along \(\overline{Z} - \overline{\Gamma} - \overline{Z}\), shown in Fig. 2(i). The calculated bulk bands along \(\overline{Z} - \overline{\Gamma} - \overline{Z}\) are simple near the \(E_{F}\). There are only two bands crossing the \(E_{F}\). The experimental observed band-1 and 3 match these two calculated bands nicely, however we did not detect any \(k_{y}\) dispersion of band-1 and 3 in Figs. 2(a)–(h). Therefore, we suggest that band-1 and 3 are actually SRS. Band-2 and band crossing points \(X_{1}\), \(X_{2}\) do not exist in Fig. 2(i). They are reproduced in calculations with the SS, shown in Fig. 2(j), which indicates that band-2 is a SS (labeled as “SS1”) locating in the bulk band gap. The bulk band gap at the \(X_{2}\) point is much smaller than that at the \(X_{1}\) point. Figure 2(k) shows the ARPES spectra along \(\overline{X} - \overline{\Gamma} - \overline{X}\) direction. Calculated spectral functions of bulk bands projection without and with SS are presented in Fig. 2(l) and (m), respectively. Along this direction, the bulk bands are complicated. Except no bands were detected near the X̅ points, experimental spectra again match the calculations pretty well. The SS1 band was also resolved in Fig. 2(k), indicated by dash lines around 0.2 eV below the \(E_{F}\). We summary the measured band dispersion of SS1 band along two high-symmetry directions in Fig. 2(n). SS1 band is very anisotropic.

Furthermore, we checked the topological nature of SS1 and SS2 bands through the parity analysis of the bulk bands near the band gaps [29], shown in Fig. 3. To trace the evolution of the bands, we did DFT calculations by artificially varying SOC strength (λ). Figure 3(a)–(f) show the band structures with different λ along Γ-Y direction. \(\lambda = 0\) means that SOC is turned off in calculations, \(\lambda = 0.2\) means 20% of SOC strength is included and \(\lambda = 1\) means SOC is fully included. The energy of SOC is calculated in DFT calculations. It’s correction to the different bands is from the order of tens of meV to hundreds of meV. In Fig. 3(a), without SOC, several bands cross each other below the \(E_{F}\). We label eight bands from “\(B_{1}\)” to “\(B_{8}\)”. The calculated parity at Γ and Y points of these bands are indicated in Fig. 3(a). “+” means positive (even) parity and “−” means negative (odd) parity. By including 20% SOC, energy gaps open at three crossing points, indicated by arrows in Fig. 3(b). We relabel two bands above and below the gap (indicated by green arrow) as band-I and II in Fig. 2(b). At this gap, band inversion happens. The parity of band-I is positive at Γ and negative at Y point, while the parity of band-II is negative at Γ and positive at Y point. The band inversion will cause the appearance of the SS in the gap [6, 26]. The position of this gap is consistent with the energy position of band SS1 in Fig. 2. On the other hand, there is no band inversion near other two band gaps (indicated by the blue arrows). Now the question is whether the observed SS2 relate to any band inversion or not. By increasing λ, we find that band \(B_{7}\) shifts up and becomes very close to band-II in Fig. 3(d). When λ becomes 0.8, band \(B_{7}\) and band-II cross and open a gap with parity inversion near Γ point, indicated by the green arrow in Fig. 3(e). We relabel band \(B_{7}\) as band-III after gap opening. The position of this gap also agrees with the energy position of band SS2. We summary the parity of band-I, II and III after the gap opening with full SOC (\(\lambda =1\)) in Fig. 3(f). If we only look at the band II’s parity at Γ and Y points, they are all positive. We might get wrong indication that no band inversion happens between band-I and band II. However, it is not the case. As we discussed in Fig. 3(e), such positive parity of band-II at Γ point comes from the band crossing between band-II and band-III very close to Γ point, which will not change the band inversion near the band gaps between band-I and band-II far away from Γ point. Therefore, we can conclude that there are two inverted band gaps in TaPaTe5, which caused topologically protected surface bands SS1 and SS2.

Figure 3
figure 3

Calculated band structure along Γ-Y direction with different SOC strength

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4 Conclusion

In summary, we found two topologically protected SSs within the inverted bulk band gaps in TaPdTe5 by combining of ARPES measurements and DFT calculations. In addition, SRS was also detected experimentally. One topological SS hybridizes with SRS and strongly modifies the Fermi surface topology at the surface region. Our findings provide comprehensive understanding on the interesting SSs in the topologically nontrivial material TaPdTe5. The hybridization between SS and SRS provides some possibility that SS may play important role in electric or spin transport properties when TaPdTe5 can be fabricated to ultrathin flakes.