Abstract
The discovery of new topological electronic materials brings a chance to uncover new physics. Up to now, many materials have been theoretically proposed and experimentally proved to host different kinds of topological states. Unfortunately, there is little convincing experimental evidence for the existence of topological oxides. The reason is that oxidation of oxygen leads to ionic crystal in general and makes band inversion unlikely. In addition, the realization of different topological states in a single material is quite difficult, but strongly needed for exploring topological phase transitions. In this work, using firstprinciples calculations and symmetry analysis, we propose that the experimentally tunable continuous solid solution of oxygen in pyrochlore Tl_{2}Nb_{2}O_{6+x} (0 ≤ x ≤ 1.0) leads to various topological states. Topological insulator, Dirac semimetal, and triply degenerate nodal point semimetal can be realized in it via changing the oxygen content and/or tuning the crystalline symmetries. When x = 1, it is a semimetal with quadratic band touching point at Fermi level. It transits into a Dirac semimetal or a topological insulator depending on the inplane strain. When x = 0.5, the inversion symmetry is spontaneously broken in Tl_{2}Nb_{2}O_{6.5}, leading to triply degenerate nodal points. When x = 0, Tl_{2}Nb_{2}O_{6} becomes a trivial insulator with a narrow band gap. These topological phase transitions driven by solid solution of oxygen are unique and physically plausible due to the variation of valence state of Tl^{+} and Tl^{3+}. This topological oxide will be promising for studying correlation induced topological states and potential applications.
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Introduction
The remarkable discoveries of various quasiparticles in solids with or without the counterpart in highenergy physics have inspired intensive studies on topological electronic materials (TEMs).^{1,2,3,4,5} They are promising for future applications, owing to lowdissipation transport property and intrinsic insensitivity to environment perturbations. TEMs are characterized as having electronic structures with nontrivial 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 doubledegenerate 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 eightfold 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 doubledegenerate 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 now^{12,30,31,32,33,34} and their properties are to be extensively explored once they are available experimentally.
Tl_{2}Nb_{2}O_{6+x} is in pyrochlore structure,^{35} which has been known since 1960s. The ideal pyrochlore Tl_{2}Nb_{2}O_{6}O′_{x}_{=}_{1} was first discovered, and then Fourquet et al. demonstrated that there exist continuous solid solutions Tl_{2}Nb_{2}O_{6}O′_{x} (0 ≤ x ≤ 1.0) via thermogravimetric analysis (TGA), chemical analysis, and Xray thermodiffractometry.^{36} Interestingly, with the removal of (1x) O′ out of the Tl_{2}Nb_{2}O_{7}, 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 Tl_{2}Nb_{2}O_{6+x} can have several attractive topological features as x changes. DSM, TI, and TDNP semimetal states all can be realized in Tl_{2}Nb_{2}O_{6+x} series via tuning the crystalline symmetry or oxidation level. When x = 1, Tl_{2}Nb_{2}O_{7} is cubic and is a zerogap semimetal similar to HgTe with quadratic contact point (QCP) at Γ.^{37} Inplane compressive strain can drive it into DSM, while inplane tensile strain can drive it into TI. When x = 0.5, Tl_{2}Nb_{2}O_{6.5} has no inversion symmetry and is a TDNP semimetal. When x = 0, Tl_{2}Nb_{2}O_{6} is a trivial insulator with narrow band gap. Because strain engineering greatly contributes to exploring physics^{37,38,39,40} and quite a small strain is introduced here, it is feasible for experimental observation of the topological states in Tl_{2}Nb_{2}O_{6+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, multiphase control, and potential applications.
Results and discussion
Crystal structure
The ternary oxide Tl_{2}Nb_{2}O_{6+x} belongs to the pyrochlore structure. The ideal structure of x = 1 is in space group Fd3m (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 NbO_{6} octahedra. The experimental lattice constant is a_{0} = 10.622 Å and is used for the calculations in the present paper.^{36}
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 Tl_{2}Nb_{2}O_{7}. 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 Tl_{2}Nb_{2}O_{7}
The 5d orbitals of Tl^{3+} atom split into e_{g} and t_{2g} orbitals due to the crystal field formed by oxygen hexagonal bipyramid. Without considering the SOC, Tl_{2}Nb_{2}O_{7} 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 t_{2g} 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 t_{2g} orbitals.^{44} Therefore, the final effective SOC of the Γ point is determined by the competition between Tl t_{2g} and O p spinorbit splitting.^{44} With SOC, the QCP at Γ (Fig. 1d) splits into a double degenerate Γ_{7}^{+} band and a fourfold degenerate Γ_{8}^{+} states (Figs 1e and 2c). Γ_{7}^{+} is higher than Γ_{8}^{+}, which indicates that the effective SOC in these bands is negative due to the dp hybridization as discussed in TlN.^{44} The fourfold degenerate Γ_{8}^{+} is half occupied and becomes another QCP similar to HgTe.^{37}
Band structure and topological property of strained Tl_{2}Nb_{2}O_{7}
The QCP at Γ is protected by O_{h} point group. Breaking the O_{h}, 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 Tl_{2}Nb_{2}O_{7} system with strain (positive strain refers to expansion, while negative strain refers to compression). The related space group is changed from Fd3m (No. 227) to I4_{1}/amd (No. 141). A top view of the structure without strain is shown in Fig. 2a, while its nonSOC 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 Γ_{8}^{+}, which is also similar to the case of Cu_{2}Se.^{45} In Tl_{2}Nb_{2}O_{7}, Γ_{7}^{+} states are higher than Γ_{8}^{+} states, while in Cu_{2}Se, Γ_{8}^{+} states are higher than Γ_{7}^{+} states.
A compressive strain of −1% in xyplane is applied (Fig. 2d) and the lattice constants become a = b = 0.99 a_{0}, and c = 1.02 a_{0}. 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 O_{h} to D_{4h} at Γ point. This leads to orbitals like d_{xz} and d_{yz} extending in z direction and having different onsite energy from that of inplane orbital like d_{xy}. The valenceband maximum and conductionband minimum are degenerate at Γ point, and the third band crosses them along ΓZ direction. Wavefunction analysis explains that the triply degenerate point is protected by the C_{4v} symmetry since these three bands are described by different irreducible representations (IRs) of the C_{4v} point group. The IR of the band shown in black is B_{2}, while that of the doubledegenerate bands shown in red and blue is E.^{46} Thus, in the nonSOC case without the spin degree of freedom, two TDNPs related with inversion or timereversal 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 wavefunction analysis shows the bands forming Dirac cone in black/red and blue/green belong to different IRs of C_{4v} point group: E_{1/2} and E_{3/2.}^{46} In other words, the Dirac point is protected by the C_{4v} symmetry. To further prove the topological properties of this phase, Wilson loop method^{47} is leveraged to trace the evolution of the Wannier charge centers in g_{3} = 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 g_{3} = 0 plane, but not for g_{3} = π plane, confirming that Tl_{2}Nb_{2}O_{7} with strain of −1% is a topological Dirac semimetal with Z_{2} = 1. Moreover, surface states on (010) surface are calculated based on the tightbinding Hamiltonian constructed with Wannier functions using Green’sfunction method. Both bulk 3D Dirac point and gapless nontrivial surface states are clearly present, which makes the Tl_{2}Nb_{2}O_{7} 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.
An expansion strain of 1% in the xyplane is applied to the system (Fig. 2g), and the related lattice constants are changed to a = b = 1.01 a_{0}, and c = 0.98 a_{0}. The nonSOC band structure is shown in Fig. 2h. There exists one band intersection along the XΓ, which is formed by bands in red and blue. The two bands belong to IRs of the C_{2v} point group: A_{2} and B_{2}, respectively.^{46} To be specific, the band intersection is protected by the C_{2v} symmetry. In fact, this nodal point is on a nodal line in k_{x}–k_{y} plane, which is protected by the coexistence of inversion and timereversal 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 firstprinciples calculations.
TDNPs in noncentrosymmetric Tl_{2}Nb_{2}O_{6.5}
Compared with the case of x = 0, the extra O′ (in the network of NbO_{6}) 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 ΓL ([111] axis) host massless fermions, which appear in a pair due to timereversal symmetry. To shed light on the forming mechanism of TDNPs, wavefunction analysis is performed. These bands belong to IRs of the C_{3v} point group: the IRs of the black/green, blue, and red bands are E_{1/2}, 2E_{3/2}, and 1E_{3/2}, respectively.^{46} Thus, band crossings can form two TDNPs near Γ point, protected by C_{3v} and timereversal symmetries. HSE06 calculation without SOC also confirms the existence of TDNP along ΓL 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}
To understand the mechanism forming TDNPs in Tl_{2}Nb_{2}O_{6.5}, 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 k_{a} and k_{c} axis, respectively. The little group at Γ point is C_{3v}, which includes two generators: a C_{3} rotation about k_{c} axis and a mirror symmetry M_{100} with the normal of the mirror plane in k_{a} direction. The Hamiltonian of k·p model keeping invariant under C_{3} and M_{100} symmetries is obtained as
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_{±} = k_{a} ± ik_{b}. The coefficients e_{s} and h_{s} are parameters that can be obtained by fitting the firstprinciples results, listed as Supplementary Table 1. The bands comparison between the k·p model and the firstprinciples calculations is shown in Supplementary Fig. 7. Now it is easy to check that, along the k_{c} axis, the  ± 1/2 〉 band is twofold 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 Tl_{2}Nb_{2}O_{6+x} (0 ≤ x ≤ 1.0) under oxidation, the TDNPs exist at a highsymmetry line and can move along the line via tuning the oxidation level. The space group of Tl_{2}Nb_{2}O_{6+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 highsymmetry point, which are protected by nonsymmorphic symmetries.^{24}
Finally, we study the SOC band structure and evolution of the Wannier charge centers of Tl_{2}Nb_{2}O_{6} (R3m, No. 160) to make a comparison with those of Tl_{2}Nb_{2}O_{6.5} (see Supplementary Figs 8 and 9). Owing to the lacking of extra O′, all Tl in Tl_{2}Nb_{2}O_{6} are in +1 valence state. The Tl_{2}Nb_{2}O_{6} system shows trivial insulating state with an indirect narrow band gap.
It is noted that Tl_{2}Ta_{2}O_{6+x}^{35} has the same chemical and physical properties as Tl_{2}Nb_{2}O_{6+x}, as well as the band topology. La_{2}Hf_{2}O_{7} 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 La_{2}Hf_{2}O_{7} is found to be opposite to that of Tl_{2}Nb_{2}O_{7}, while the absence of valence variation in La ions makes the oxygen content hard to be tuned in La_{2}Hf_{2}O_{7}.
In summary, we propose that a pyrochlore oxide Tl_{2}Nb_{2}O_{6+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. Tl_{2}Nb_{2}O_{7} with x = 1 is a semimetal with QCP due to cubic symmetry. When a small inplane tensile strain is applied, a nodal line appears in the nonSOC case, which is protected by the inversion and timereversal symmetries. When SOC is taken into account, Tl_{2}Nb_{2}O_{7} could harbor bulk Dirac points with gapless topological surface states under a small compressive inplane strain, or become a TI under the expansion case. For Tl_{2}Nb_{2}O_{6.5} with x = 0.5 where inversion symmetry is absent, a couple of intrinsic triply degenerate nodal points exist and are protected by timereversal and C_{3v} symmetries. For Tl_{2}Nb_{2}O_{6} with x = 0, it is a narrow gap semiconductor with trivial topology. On the experimental aspect, the successful fabrication of crystals of Tl_{2}Nb_{2}O_{6+x} series^{35,36} 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 firstprinciples calculations within density functional theory (DFT), using the Vienna ab initio simulation package (VASP).^{48,49} Exchangecorrelation potential is treated within the generalized gradient approximation (GGA) in the form of PerdewBurkeErnzerhof (PBE).^{50} Cutoff energy for plane wave expansion is 350 eV. MethfesselPaxton 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 kpoint meshes of 12 × 12 × 12 for selfconsistent electronic structure calculations. The spinorbit coupling is included selfconsistently. 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 HeydScuseriaErnzerhof (HSE06) hybrid functional calculation is carried out to remedy the possible underestimation of band gap and overestimation of band inversion.^{51,52} To calculate Z_{2} invariant, surface states and nodal line states of the system, the maximally localized Wannier functions (MLWF)^{53,54} are introduced into WannierTools^{55} and a tightbinding model was constructed.
Data availability
Any data used to generate the results in this study can be obtained from the corresponding authors upon reasonable request.
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Acknowledgements
This work was supported by the National Key Research and Development Program of China (Nos. 2017YFA0304700, 2017YFA0303402, 2016YFA0300600, and 2018YFA0305700), the National Natural Science Foundation of China (Nos. 11974076, 11674077, 11604273, 11704117, and 11674369), and Natural Science Foundation of Fujian Province of China (No. 2018J06001). C. F. and H. W. are also supported by the Science Challenge Project (No. TZ2016004), Beijing Natural Science Foundation (Z180008), and the K. C. Wong Education Foundation (Grant No. GJTD201801).
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Contributions
W.Z. carried out the firstprinciples calculations; K.L. and R.Y. constructed the k·p effective model. Z.C., Z.Z., and C.F. contributed to analyzing the data. W.Z., R.Y., and H.W. wrote the paper. H.W. supervised the project. All authors participate in research discussion.
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Zhang, W., Luo, K., Chen, Z. et al. Topological phases in pyrochlore thallium niobate Tl_{2}Nb_{2}O_{6+x}. npj Comput Mater 5, 105 (2019). https://doi.org/10.1038/s4152401902455
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DOI: https://doi.org/10.1038/s4152401902455
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