Introduction

The remarkable discoveries of various quasiparticles in solids with or without the counterpart in high-energy physics have inspired intensive studies on topological electronic materials (TEMs).1,2,3,4,5 They are promising for future applications, owing to low-dissipation transport property and intrinsic insensitivity to environment perturbations. TEMs are characterized as having electronic structures with non-trivial topology in momentum space. Typically, TEMs can be classified into topological insulator (TI),6,7,8 topological semimetal (TSM)9,10,11,12,13,14, and topological superconductor. The initial impetus originates from the TI, which exhibits linear dispersive surface/edge states and can make new quantum electronic devices compatible with current electronic technologies. Moreover, magnetically doped TIs are proved to hold quantum anomalous Hall effect.15,16 Recently, research focus of TEMs has shifted towards TSMs, which have exotic transport properties.17,18,19 TSMs are special metals with Fermi surfaces composed of and only of nodal points. They include four members, namely Weyl semimetal (WSM),20,21 Dirac semimetal (DSM),10,11 nodal line semimetal (NLSM),22,23 and triply- or multiply- degenerate nodal point (TDNP or MDNP) semimetal.24,25,26,27,28,29 These TSMs are distinguished from each other by the degeneracy of the nodal points and the topological protection mechanism. WSM has isolated double-degenerate nodal points at or close to the Fermi level and is topologically robust as long as the translation symmetry of lattice is preserved, while DSM has isolated fourfold degenerate nodal points and is protected by proper crystalline symmetries. NLSM contains continuous nodal points forming lines, while MDNPs host three-, six-, or eight-fold degenerate nodal points. Both of them need some proper crystalline symmetries, such as rotation, mirror, and/or nonsymmorphic translation. The TDNP in WC family is a crossing point formed by a nondegenerate band and a double-degenerate band.25,27 It is identified as an intermediate state between Weyl and Dirac TSM, bringing in new interesting physics. However, there have been quite few reports on TEMs discovered in oxide materials till now12,30,31,32,33,34 and their properties are to be extensively explored once they are available experimentally.

Tl2Nb2O6+x is in pyrochlore structure,35 which has been known since 1960s. The ideal pyrochlore Tl2Nb2O6O′x=1 was first discovered, and then Fourquet et al. demonstrated that there exist continuous solid solutions Tl2Nb2O6O′x (0 ≤ x ≤ 1.0) via thermogravimetric analysis (TGA), chemical analysis, and X-ray thermodiffractometry.36 Interestingly, with the removal of (1-x) O′ out of the Tl2Nb2O7, the Tl atoms could shift along [111] axis and be away from the central symmetric position, leading to spontaneous inversion symmetry breaking, which brings a very unique way to systematically tune the topological phases in it.

In this work, we propose that Tl2Nb2O6+x can have several attractive topological features as x changes. DSM, TI, and TDNP semimetal states all can be realized in Tl2Nb2O6+x series via tuning the crystalline symmetry or oxidation level. When x = 1, Tl2Nb2O7 is cubic and is a zero-gap semimetal similar to HgTe with quadratic contact point (QCP) at Γ.37 In-plane compressive strain can drive it into DSM, while in-plane tensile strain can drive it into TI. When x = 0.5, Tl2Nb2O6.5 has no inversion symmetry and is a TDNP semimetal. When x = 0, Tl2Nb2O6 is a trivial insulator with narrow band gap. Because strain engineering greatly contributes to exploring physics37,38,39,40 and quite a small strain is introduced here, it is feasible for experimental observation of the topological states in Tl2Nb2O6+x. Moreover, many intriguing phenomena and rich physics have been found in pyrochlore oxides, such as complex magnetic phases, superconducting, and multiferroics. Thus, our studies may provide a unique platform for investigating the strongly correlated topological phases, multi-phase control, and potential applications.

Results and discussion

Crystal structure

The ternary oxide Tl2Nb2O6+x belongs to the pyrochlore structure. The ideal structure of x = 1 is in space group Fd-3m (No. 227) (Fig. 1a), whose first Brillouin zone (BZ) is shown in Fig. 1b. Tl and O′ atoms are located at 16d (1/2, 1/2, 1/2) and 8b (3/8, 3/8, 3/8) positions, respectively. Four Tl atoms will form a tetrahedra with an O′ atom at the center (Fig. 1c). Nb and O atoms are in 16c (0, 0, 0) and 48f (0.2925, 1/8, 1/8) positions, respectively, forming NbO6 octahedra. The experimental lattice constant is a0 = 10.622 Å and is used for the calculations in the present paper.36

Fig. 1
figure 1

Tl2Nb2O7 system. a Conventional unit cell constructed by a three-dimensional corner-sharing network of NbO6 octahedra (left panel). The related primitive unit cell (right panel). b Schematic of the bulk first Brillouin zone (BZ). c Tetrahedron constructed by four Tl atoms surrounding O′ atom. The tetrahedron corners represent position 16d, while asterisks represent position 32e. d Non-SOC band structures calculated within GGA (black solid lines) and hybrid functional HSE06 (red dotted lines). e SOC band structure calculated within GGA

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Compared with the case of x = 1, the missing of O′ makes the same number of Tl ions change from +3 to +1 and shift away from the centrosymmetric position (Fig. 1c). Though the distribution of O′ vacancy and Tl+ ions is somehow random in x = 0.5 case, we take away half of O′ atoms in the primitive unit cell (Z = 2), and Tl atoms are shifted away from 16d to the 32e (0.507, 0.507, 0.507) positions (see Supplementary Fig. 1). The lattice constant is taken as 10.6397 Å according to the experimental value in x = 0.490 case, which is the closest to 0.5.36 The crystal structure symmetry becomes R3m (No. 160) without inversion center, being different from that of Tl2Nb2O7. When x is reduced to 0, all the Tl atoms become +1 and stay on the noncentral position. The lattice constant is taken as 10.6829 Å, which is the experimental value when x = 0.070.36

Band structure of bulk Tl2Nb2O7

The 5d orbitals of Tl3+ atom split into eg and t2g orbitals due to the crystal field formed by oxygen hexagonal bipyramid. Without considering the SOC, Tl2Nb2O7 is a QCP semimetal with a triply degeneracy at Γ point (Fig. 1d), which is also verified by the hybrid functional HSE06 calculation (red color bands in Fig. 1d). This is the same as the results in Materiae, an online database of topological materials, and other similar databases.41,42,43 The states at Γ point mainly come from t2g orbitals composed by hybridization of Tl 5d and O 2p orbitals. When SOC is taken into consideration, SOC splitting among p orbitals is opposite to that among t2g orbitals.44 Therefore, the final effective SOC of the Γ point is determined by the competition between Tl t2g and O p spin-orbit splitting.44 With SOC, the QCP at Γ (Fig. 1d) splits into a double degenerate Γ7+ band and a fourfold degenerate Γ8+ states (Figs 1e and 2c). Γ7+ is higher than Γ8+, which indicates that the effective SOC in these bands is negative due to the d-p hybridization as discussed in TlN.44 The fourfold degenerate Γ8+ is half occupied and becomes another QCP similar to HgTe.37

Fig. 2
figure 2

Strained Tl2Nb2O7 system. a Top view of the crystal structure, b non-SOC and c SOC energy bands for the structure without strain. d Top view of the crystal structure, e non-SOC and f SOC energy bands for the structure with strain of −1% (inplane compression). g Top view of the crystal structure, h non-SOC and i SOC energy bands for the structure with strain of 1% (inplane expansion)

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Band structure and topological property of strained Tl2Nb2O7

The QCP at Γ is protected by Oh point group. Breaking the Oh, this fourfold degeneracy will be lifted, and thus topological insulating states or topological semimetal states are formed.37 In this section, we consider the topological phase transition in Tl2Nb2O7 system with strain (positive strain refers to expansion, while negative strain refers to compression). The related space group is changed from Fd-3m (No. 227) to I41/amd (No. 141). A top view of the structure without strain is shown in Fig. 2a, while its non-SOC and SOC bands are shown in Fig. 2b, c for comparison. From the pictures, we can see when SOC is included, the gapless semimetal is formed owing to the fourfold degeneracy of Γ8+, which is also similar to the case of Cu2Se.45 In Tl2Nb2O7, Γ7+ states are higher than Γ8+ states, while in Cu2Se, Γ8+ states are higher than Γ7+ states.

A compressive strain of −1% in xy-plane is applied (Fig. 2d) and the lattice constants become a = b = 0.99 a0, and c = 1.02 a0. The band structures without and with SOC are calculated and compared in Fig. 2e, f. When SOC is neglected, the application of strain breaks the system symmetry and results in the point group changing from Oh to D4h at Γ point. This leads to orbitals like dxz and dyz extending in z direction and having different on-site energy from that of in-plane orbital like dxy. The valence-band maximum and conduction-band minimum are degenerate at Γ point, and the third band crosses them along Γ-Z direction. Wave-function analysis explains that the triply degenerate point is protected by the C4v symmetry since these three bands are described by different irreducible representations (IRs) of the C4v point group. The IR of the band shown in black is B2, while that of the double-degenerate bands shown in red and blue is E.46 Thus, in the non-SOC case without the spin degree of freedom, two TDNPs related with inversion or time-reversal symmetry can be formed in the −Z to Γ and Γ to Z directions, respectively. The energy band with SOC in strain of −1% case is also calculated (Fig. 2f). SOC drives a phase transition from the QCP semimetal to Dirac semimetal, where two fourfold degenerate Dirac points are on the path −Z to Γ and Γ to Z, respectively. Detailed wave-function analysis shows the bands forming Dirac cone in black/red and blue/green belong to different IRs of C4v point group: E1/2 and E3/2.46 In other words, the Dirac point is protected by the C4v symmetry. To further prove the topological properties of this phase, Wilson loop method47 is leveraged to trace the evolution of the Wannier charge centers in g3 = 0 and π planes. As shown in Supplementary Fig. 2, there exists one crossing of Wannier center (black lines) and the reference line (red line) in g3 = 0 plane, but not for g3 = π plane, confirming that Tl2Nb2O7 with strain of −1% is a topological Dirac semimetal with Z2 = 1. Moreover, surface states on (010) surface are calculated based on the tight-binding Hamiltonian constructed with Wannier functions using Green’s-function method. Both bulk 3D Dirac point and gapless non-trivial surface states are clearly present, which makes the Tl2Nb2O7 with −1% strain fantastic for exploring the coupling between Dirac point and topological insulator states (Fig. 3a). There are two branches of surface states emerging in the gap and touching at \({\bar{\mathrm X}}\) and \({\bar{\mathrm{Z}}}\) points due to Kramer’s degeneracy. One branch connects to the conduction bulk bands, while the other one links the valence bulk bands. Moreover, the Fermi surface at Dirac point is calculated (Fig. 3b). There exists a pair of surface Fermi arcs connecting two projected Dirac nodes.

Fig. 3
figure 3

Topological properties of strained Tl2Nb2O7 with SOC on (010) surface. Structure with inplane strain of −1% (compression): a 3D Dirac point and topological surface states both appear. Bulk states and surface states near Fermi level are enlarged and shown on the right panel. b Fermi surface of surface states with energy level at −0.7 meV where Dirac points locate. Two project Dirac nodes are represented as black dots. Structure with inplane strain of 1% (expansion): c Calculated topological surface states. Bulk states and surface states near Fermi level are enlarged and shown on the right panel. d Fermi surface of surface states with energy level at −6 meV

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An expansion strain of 1% in the xy-plane is applied to the system (Fig. 2g), and the related lattice constants are changed to a = b = 1.01 a0, and c = 0.98 a0. The non-SOC band structure is shown in Fig. 2h. There exists one band intersection along the X-Γ, which is formed by bands in red and blue. The two bands belong to IRs of the C2v point group: A2 and B2, respectively.46 To be specific, the band intersection is protected by the C2v symmetry. In fact, this nodal point is on a nodal line in kx–ky plane, which is protected by the coexistence of inversion and time-reversal symmetries (see Supplementary Fig. 3). When SOC is included, the gap is fully opened in the entire nodal line (Fig. 2i), generating a strong TI with global gap of ~13 meV. The same Wilson loop method is used here to identify the topological property of the structure with strain of 1% (see Supplementary Fig. 4). Topologically protected surface Dirac cone on (010) surface connecting the conduction and valence bands emerges inside the gap (Fig. 3c). These two branches of surface states also touch at \({\bar{\mathrm X}}\) and \({\bar{\mathrm Z}}\) points, similar to the case of −1% strain. Furthermore, isoenergy plot of surface states at energy of −6 meV in the gap is displayed (Fig. 3d).

To understand the phase transition mechanism under strain, an effective k·p model is constructed (see Supplementary Note 3). From Supplementary Fig. 5, we can see the band structures coincide with those from first-principles calculations.

TDNPs in noncentrosymmetric Tl2Nb2O6.5

Compared with the case of x = 0, the extra O′ (in the network of NbO6) oxidizes one of the nearest four Tl atoms to +3 and repels the other three monovalent Tl. Therefore, Tl atoms are away from the centrosymmetric positions. Such natural breaking of inversion symmetry provides a new material hosting intrinsic TDNPs, whose topological properties are studied by calculating the band structure with SOC (Fig. 4). We can see band crossings along Γ-L ([111] axis) host massless fermions, which appear in a pair due to time-reversal symmetry. To shed light on the forming mechanism of TDNPs, wave-function analysis is performed. These bands belong to IRs of the C3v point group: the IRs of the black/green, blue, and red bands are E1/2, 2E3/2, and 1E3/2, respectively.46 Thus, band crossings can form two TDNPs near Γ point, protected by C3v and time-reversal symmetries. HSE06 calculation without SOC also confirms the existence of TDNP along Γ-L direction (see Supplementary Fig. 6). These TDNPs are ~0.4 eV below Fermi level. They are higher than those in MoP,28 while lower than those in WC.29

Fig. 4
figure 4

Band structures of Tl2Nb2O6.5. a SOC band structure calculated within GGA. b Enlarged band structure along Γ-L around TDNP in a

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To understand the mechanism forming TDNPs in Tl2Nb2O6.5, we further introduce a k·p model around Γ point with SOC included. In order to make the discussions simple, we chose the [−110] and [111] direction (Fig. 1b) as ka and kc axis, respectively. The little group at Γ point is C3v, which includes two generators: a C3 rotation about kc axis and a mirror symmetry M100 with the normal of the mirror plane in ka direction. The Hamiltonian of k·p model keeping invariant under C3 and M100 symmetries is obtained as

$$H({\mathrm{k}}) = \left( {\begin{array}{*{20}{c}} {e_{10} + e_{11}k_ - k_ + + e_{12}k_c^2} & {h_{12}k_ - } & {{\mathop{\rm{i}}\nolimits} h_{13}k_ + } & {h_{14}k_c} \\ {} & {e_{20} + e_{21}k_ - k_ + + e_{22}k_c^2} & {h_{22}k_ - } & { - {\mathop{\rm{i}}\nolimits} h_{13}k_ + } \\ {} & {} & {e_{20} + e_{21}k_ - k_ + + e_{22}k_c^2} & {h_{12}k_ - } \\ \dagger & {} & {} & {e_{10} + e_{11}k_ - k_ + + e_{12}k_c^2} \end{array}} \right)$$

up to the first order of k with the basis set in order of | + 3/2 〉, | + 1/2 〉, |−1/2 〉, |−3/2 〉, which are the four eigenstates at Γ point near the Fermi level k± = ka ± ikb. The coefficients es and hs are parameters that can be obtained by fitting the first-principles results, listed as Supplementary Table 1. The bands comparison between the k·p model and the first-principles calculations is shown in Supplementary Fig. 7. Now it is easy to check that, along the kc axis, the | ± 1/2 〉 band is two-fold degenerate. Meanwhile, the | ± 3/2 〉 bands split into two bands which cross with | ± 1/2 〉 band, forming two TDNPs (Fig. 4b).

Thanks to the unique property of the continuous solid solution in pyrochlore Tl2Nb2O6+x (0 ≤ x ≤ 1.0) under oxidation, the TDNPs exist at a high-symmetry line and can move along the line via tuning the oxidation level. The space group of Tl2Nb2O6+x is symmorphic (0 < x ≤ 0.5) and the TDNPs here are protected by the rotation symmetry. It is different from the TDNP emerging at high-symmetry point, which are protected by nonsymmorphic symmetries.24

Finally, we study the SOC band structure and evolution of the Wannier charge centers of Tl2Nb2O6 (R3m, No. 160) to make a comparison with those of Tl2Nb2O6.5 (see Supplementary Figs 8 and 9). Owing to the lacking of extra O′, all Tl in Tl2Nb2O6 are in +1 valence state. The Tl2Nb2O6 system shows trivial insulating state with an indirect narrow band gap.

It is noted that Tl2Ta2O6+x35 has the same chemical and physical properties as Tl2Nb2O6+x, as well as the band topology. La2Hf2O7 is found to be a QCP semimetal in GGA calculation and it becomes a topological crystalline insulator in GGA + SOC calculation, which are the same as those in topological material database Materiae, and other similar databases.41,42,43 The effective SOC splitting in La2Hf2O7 is found to be opposite to that of Tl2Nb2O7, while the absence of valence variation in La ions makes the oxygen content hard to be tuned in La2Hf2O7.

In summary, we propose that a pyrochlore oxide Tl2Nb2O6+x with continuous oxidation level x can host various topological phases, which is realized by a change of valence state of Tl from +1 to +3 and the displacement of its atomic position. Tl2Nb2O7 with x = 1 is a semimetal with QCP due to cubic symmetry. When a small in-plane tensile strain is applied, a nodal line appears in the non-SOC case, which is protected by the inversion and time-reversal symmetries. When SOC is taken into account, Tl2Nb2O7 could harbor bulk Dirac points with gapless topological surface states under a small compressive in-plane strain, or become a TI under the expansion case. For Tl2Nb2O6.5 with x = 0.5 where inversion symmetry is absent, a couple of intrinsic triply degenerate nodal points exist and are protected by time-reversal and C3v symmetries. For Tl2Nb2O6 with x = 0, it is a narrow gap semiconductor with trivial topology. On the experimental aspect, the successful fabrication of crystals of Tl2Nb2O6+x series35,36 makes it feasible to observe these topological features as we proposed. Our work can widen the knowledge of TEMs in oxide materials. Furthermore, the realization of different topological states in one series can stimulate the study on the coupling among them, which may generate new physics or interesting transport properties, as well as interacting topological phases.

Methods

We have performed first-principles calculations within density functional theory (DFT), using the Vienna ab initio simulation package (VASP).48,49 Exchange-correlation potential is treated within the generalized gradient approximation (GGA) in the form of Perdew-Burke-Ernzerhof (PBE).50 Cut-off energy for plane wave expansion is 350 eV. Methfessel-Paxton method is used for semimetal systems, while Gaussian method is used for insulating systems. The width of the smearing is 0.01 eV. BZ is sampled with k-point meshes of 12 × 12 × 12 for self-consistent electronic structure calculations. The spin-orbit coupling is included self-consistently. The simulation of uniaxial strain along [001] is simulated by fixing the experimental volume with the ratio a/c tuned. This assumption might be rough but it is enough to demonstrate the main physics on the symmetry breaking, which can drive the topological phase transition. Here, a is lattice constant along x/y direction, while c is the lattice constant along z direction ([001] direction). The nonlocal Heyd-Scuseria-Ernzerhof (HSE06) hybrid functional calculation is carried out to remedy the possible underestimation of band gap and overestimation of band inversion.51,52 To calculate Z2 invariant, surface states and nodal line states of the system, the maximally localized Wannier functions (MLWF)53,54 are introduced into WannierTools55 and a tight-binding model was constructed.