# Possible Three-Dimensional Topological Insulator in Pyrochlore Oxides

^{*}

## Abstract

**:**

_{2}B

_{2}O

_{7}(A = Sn, Pb, Tl; B = Nb, Ta), which are properly described by this model, and found that heavily hole-doped Sn

_{2}Nb

_{2}O

_{7}is a good candidate. Surprisingly, an effective spin–orbit coupling constant λ changes its sign depending on the composition of the material. Furthermore, we calculated the band structure of three virtual pyrochlore oxides, namely In

_{2}Nb

_{2}O

_{7}, In

_{2}Ta

_{2}O

_{7}and Sn

_{2}Zr

_{2}O

_{7}. We found that Sn

_{2}Zr

_{2}O

_{7}has a band gap at the

**k**= 0 (Γ) point, similar to Sn

_{2}Nb

_{2}O

_{7}, though the band structure of Sn

_{2}Zr

_{2}O

_{7}itself differs from the ideal nearest-neighbor tight-binding model. We propose that the co-doped system (In,Sn)

_{2}(Nb,Zr)

_{2}O

_{7}may become a candidate of the three-dimensional strong topological insulator.

## 1. Introduction

**σ**

_{αβ}denotes the spin αβ component of the Pauli matrices. In the pyrochlore lattice there are four sites in the primitive cell, and we obtain four bands. Here,

**d**

_{ij}

^{1,2}denote the nearest-neighbor vectors traversed between the second neighbor sites i and j. The summation <<ij>> is taken for these second neighbor sites. In case of λ = 0, this Hamiltonian is rigorously diagonalized and the eigenvalues are

^{3}and E

^{4}do not depend on the wave vector

**k**, so they are called flat bands (FBs). These energy eigenvalues are shown in Figure 1b. Note that these FBs touch the dispersive band ${E}_{k}^{2}$ at

**k**= 0. In other words, the state at

**k**= 0 is three-fold (six-fold if we consider spin) degenerated. Therefore, in case of half-filling, i.e., two of the four bands are filled, the system is a zero-gap semiconductor. FB itself induces many attractive physical properties [7,8,9,10], but in this paper we focus on the topological aspect of this model.

_{7}) and four-fold (Γ

_{8}) degenerated states. Which of these states have higher energy depends on the sign of λ. A schematic picture is shown in Figure 1c,d. When λ > 0, the energy of Γ

_{7}is higher than that of Γ

_{8}, i.e., E(Γ

_{7}) > E(Γ

_{8}). In this case, four-fold degenerated FBs split into Γ

_{7}and Γ

_{8}at the Γ point, but are still degenerated at the zone boundary points X and L. Therefore, the system remains gapless when λ > 0. On the contrary when λ < 0, the energy of Γ

_{7}is lower than that of Γ

_{8}, i.e., E(Γ

_{7}) < E(Γ

_{8}). In case of half-filling and λ < 0, the system has a band gap Δ = 24 |λ|. Guo and Franz [6] have shown that this is a 3D strong TI having topological indices (1;000), where (ν

_{0};ν

_{1}ν

_{2}ν

_{3}) denote the four-component Z

_{2}topological indices [11,12]. Each ν

_{i}has a binary value 0 or 1, and if all ν

_{i}are zero, i.e., (0;000), the system is an ordinary band insulator; if ν

_{0}= 1 it is a strong TI, which means topologically protected surface states exist on all surfaces and they are robust with regard to non-magnetic disorder.

_{2}B

_{2}O

_{7}include two independent pyrochlore sublattices [13]. However, even though there are hundreds of pyrochlore oxides [13], there are very few compounds whose electronic states near the Fermi level (E

_{F}) can be approximated by the GF model. The reason for this is as follows: In the GF model, the transfer integral t is isotropic, while in most of the pyrochlore oxides the relevant orbital is an anisotropic d- or f- orbital.

_{2}B

_{2}O

_{7}(A = Sn, Pb, Tl; B = Nb, Ta) have band structures that are well described in the GF model. In those papers, we mainly focused on the possible ferromagnetism induced by the quasi-FB. We have already shown [14] that Sn

_{2}Nb

_{2}O

_{7}has E(Γ

_{7}) < E(Γ

_{8}), i.e., λ < 0. On the other hand, Sn

_{2}Ta

_{2}O

_{7}has λ > 0. It means that Sn

_{2}Nb

_{2}O

_{7}is a good candidate for a strong TI when the holes are sufficiently doped (Sn

_{2}Nb

_{2}O

_{7}itself is away from half-filling), but Sn

_{2}Ta

_{2}O

_{7}is not. After this work, Zhou et al. proposed a novel topological state for ferromagnetic Sn

_{2}Nb

_{2}O

_{7}induced by quasi-FB [17], and Zhang et al. proposed that Tl

_{2}Nb

_{2}O

_{6+x}(0 ≤ x ≤ 1) leads to various topological phases accompanied with lattice distortion [18].

_{F}is approximated by the GF model. We calculated the band structure of six pyrochlore oxides A

_{2}B

_{2}O

_{7}(A = Sn, Pb, Tl; B = Nb, Ta) from first principles. Additionally, we calculated the band structure of three hypothetical compounds, In

_{2}Nb

_{2}O

_{7}, In

_{2}Ta

_{2}O

_{7}and Sn

_{2}Zr

_{2}O

_{7}, which may be potential candidates for a TI. We found that only Sn

_{2}Nb

_{2}O

_{7}and Sn

_{2}Zr

_{2}O

_{7}gives λ < 0, though the band structure of Sn

_{2}Zr

_{2}O

_{7}cannot fit very well with the GF model.

## 2. Materials and Computational Methods

_{2}B

_{2}O

_{7}(A = Sn, Pb, Tl; B = Nb, Ta) from first principles. We used the density-functional theory (DFT) and a linearized augmented plane wave with the addition of a local orbital (LAPW+lo) scheme (WIEN2k code [19]). The exchange-correlation potential was constructed within the general gradient approximation [20]. The k-point mesh was set so that the total number of the mesh was about 1000 in the first Brillouin zone. The parameter RK

_{max}, which is a product of the maximum radius of the muffin-tin spheres R and the cut-off wave number of the plane-wave basis K

_{max}, was set to be 7.0. For simplicity, we assumed that the calculated compounds all have an ideal A

_{2}B

_{2}O

_{6}O’ pyrochlore structure with the space group Fd-3m (#227). Since oxygen atoms occupy two crystallographic sites, we named these sites as O and O’ to distinguish between them. The atomic positions are A(0, 0, 0), B(1/2, 1/2, 1/2), O(u, 1/8, 1/8) and O’(1/8, 1/8, 1/8). This pyrochlore structure contains 88 atoms in the conventional cubic unit cell. In this work we focus on the A atoms, which form the pyrochlore lattice shown in Figure 1a. O’ atoms are at the center of the A

_{4}tetrahedron shown by a circle in Figure 1a. We optimized this parameter u by minimizing the Hellmann–Feynman force $\mathit{F}=-\u27e8{\psi}_{\mathit{R}}\left|\frac{\partial H}{\partial \mathit{R}}\right|{\psi}_{\mathit{R}}\u27e9$, where H denotes the one-electron Hamiltonian for the atomic coordinates

**R**, and ${\psi}_{\mathit{R}}$ is its eigenfunction. The convergence of the atomic positions was judged by this Hellmann–Feynman force working on each atom, which was to be less than 1.0 mRy/a.u. We also optimized the lattice constant a for Tl

_{2}Ta

_{2}O

_{7}. The optimized value a = 10.716 Å well agreed with an experimental value a = 10.56 Å [21] or 10.651 Å [22]. For the other five compounds, we used the experimental lattice constant.

_{SO}= E(Γ

_{7}) – E(Γ

_{8}) and use the relation Δ

_{SO}= 24 λ [6].

_{2}B

_{2}O

_{7}system, we also performed band calculations on the virtual compounds In

_{2}Nb

_{2}O

_{7}, In

_{2}Ta

_{2}O

_{7}and Sn

_{2}Zr

_{2}O

_{7}. For these compounds, we relaxed both of the lattice constant and the atomic positions. The obtained lattice constant is a = 10.611, 10.631 and 10.618 Å for In

_{2}Nb

_{2}O

_{7}, In

_{2}Ta

_{2}O

_{7}and Sn

_{2}Zr

_{2}O

_{7}, respectively. They are all within the normal range of the lattice constants of pyrochlore oxides [13].

## 3. Results and Discussions

_{2}Ta

_{2}O

_{7}in Figure 2a. Similar to other quasi-FB oxides [14,15,16,17,18], this compound also shows a characteristic quasi-FB just below E

_{F}. The origin of this quasi-FB is explained as follows. The formal chemical valences of Pb

_{2}Ta

_{2}O

_{7}are denoted as Pb

^{2+}

_{2}Ta

^{5+}

_{2}O

^{2−}

_{7}. Since Ta

^{5+}and O

^{2−}form closed shells, we see that only Pb

^{2+}is chemically active in the first approximation. The electron configuration of Pb

^{2+}is (6s)

^{2}, so we can expect that the band gap is opened between the Ta5d band and Pb6s band. This is verified by density-of-states (DOS) analysis shown in Figure 2b. We see that the sharp peak coming from quasi-FB mainly consists of the Pb-s and O’-p components.

_{2}Ta

_{2}O

_{7}and the simple GF model is quite good, considering that the GF model includes only one parameter, t (in Figure 2c we set λ = 0). Since the s-orbital is isotropic, the hopping integral t in Equation (1) is also isotropic. Note that in most of the pyrochlore oxides the relevant (frontier) orbitals are d- or f-orbitals, which are anisotropic. We can also show that the more precise 10-orbital model, including four Pb-s and 6 O’-p orbitals, also have a FB [14]. However, we deal with the minimal 4-orbital model because we can utilize the results for the GF model.

_{2}Ta

_{2}O

_{7}is not a complete FB but a quasi-FB. This small dispersion (about 0.2 eV) is attributed to the hopping terms other than the nearest neighbor one, such as next-nearest-neighbor hopping. However, this small dispersion does not change the topology of the band structure and we can use the result for the GF model.

_{2}Ta

_{2}O

_{7}is slightly split due to the SOI. Other bands, e.g., the bands at energy ~1.5 eV, split more than the quasi-FB at the valence band. This is because the conduction band mostly consist of Ta5d orbitals in which the SOI is large. If the valence band purely consists of Pb6s orbitals, the SOI should be zero. Small splitting of the quasi-FB in Pb

_{2}Ta

_{2}O

_{7}is due to the hybridization of the other orbitals. In Figure 2d we show a blow-up of the band structure of the Pb

_{2}Ta

_{2}O

_{7}near E

_{F}(shown by a circle in Figure 2a).

_{2}indices (1;000). On the other hand, it does not have a band gap when λ > 0. We see that E(Γ

_{7}) is larger than E(Γ

_{8}), i.e., λ > 0 in Pb

_{2}Ta

_{2}O

_{7}. This situation corresponds to the case in Figure 1c. Note that these Z

_{2}indices are purely topological ones and robust for the small perturbations. Therefore, we can expect that these indices are unchanged in the real compounds we calculated, unless there are extra band crossings. Above these reasons, we can investigate the topological properties of the A

_{2}B

_{2}O

_{7}compounds by calculating the energy difference Δ

_{SO}= E(Γ

_{7}) − E(Γ

_{8}) at the Γ point (

**k**= 0).

_{SO}= E(Γ

_{7}) − E(Γ

_{8}) in eight pyrochlore oxides A

_{2}B

_{2}O

_{7}(A = In, Sn, Tl, Pb; B = Nb, Ta), including two virtual oxides, In

_{2}Nb

_{2}O

_{7}and In

_{2}Ta

_{2}O

_{7}. Apparently, only Sn

_{2}Nb

_{2}O

_{7}has a band gap, and the other seven oxides do not have a band gap. This result shows that Sn

_{2}Nb

_{2}O

_{7}may become a strong TI when the chemical potential is put in this band gap by heavily doping the holes. At present, hole doping in Sn

_{2}Nb

_{2}O

_{7}has been successful [25], but such a heavy doping is difficult. As for Sn

_{2}Nb

_{2}O

_{7}, combining the topological band structure and the possible ferromagnetism due to quasi-FB, the existence of a novel three-dimensional Weyl point is suggested [17].

_{SO}(or effective SOC coefficient λ = Δ

_{SO}/24) is material-dependent, even in its sign. We can see some trends in this figure:

- When A is the period-6 element of the periodic table, Δ
_{SO}is large, and when it is a period-5 element, Δ_{SO}is small. The difference of Δ_{SO}is about 40–50 meV. - When A is the group-13 element of the periodic table, Δ
_{SO}is large, and when it is a period-14 element, Δ_{SO}is small. The difference of Δ_{SO}is about 3–15 meV. - Δ
_{SO}does not so much depend on the B element. The difference of Δ_{SO}is about −0.3–10 meV.

_{SO}. In order to realize a TI in pyrochlore oxide, we should also tune the chemical potential. Half-filling of the 4-orbital GF model means that two of the four bands are occupied, i.e., there are four electrons in the primitive unit cell. Since there are four atomic sites, it turns so that the outermost electron configuration in the real compound is (5s)

^{1}or (6s)

^{1}, which means there is one electron in the outermost s-orbital. We call this type of compound the “s1 compound”. This includes Tl

_{2}Nb

_{2}O

_{7}and so on. A famous mother compound of the high-T

_{c}superconductor BaBiO

_{3}is also an s1 compound [26,27,28,29], but Tl

_{2}Nb

_{2}O

_{7}has a quasi-FB due to the strong geometric frustration of the pyrochlore lattice. This strong frustration may prevent forming a charge-density wave, which is seen in BaBiO

_{3}. On the contrary, a compound containing two electrons in the outermost s-orbital is called the “s2 compound”. As for realizing the TI, the s1 compound is desirable in order to realize the TI, though in general the s1 compound is not so stable.

_{2}Zr

_{2}O

_{7}may be a good candidate for a TI because it is formally described as Sn

^{3+}

_{2}Zr

^{4+}

_{2}O

^{2−}

_{7}, having an s1 ion Sn

^{3+}and the other ions form a closed shell. This compound may also have small λ since it contains an Sn atom on the A site. The calculated band structure of the Sn

_{2}Zr

_{2}O

_{7}is shown in Figure 4a,b. As is expected, we found Δ = 2.5 meV, which means there is a band gap at the Γ point. However, the valence band Sn

_{2}Zr

_{2}O

_{7}is apparently not described well by the GF model, as shown in Figure 4a. The reason why Sn

_{2}Zr

_{2}O

_{7}has such a band structure is not clear, but we note that the band structure of the other pyrochlore oxides with a formal ionic valence A

^{3+}

_{2}B

^{4+}

_{2}O

^{2−}

_{7}(e.g., Bi

_{2}Ti

_{2}O

_{7}) are not described well by the GF model (or non-interacting Mielke model) [15].

_{2}Nb

_{2}O

_{7}, has a band structure like the GF model (see Figure 4c), but λ > 0 (though not too large, see Figure 3). Sn

_{2}Zr

_{2}O

_{7}has λ < 0 but the band structure is not like the GF model. Therefore, we propose that a co-doped system, (In,Sn)

_{2}(Nb,Zr)

_{2}O

_{7}, may have both of the desired properties for a TI, namely, a GF-model-like band and λ < 0.

_{2}Nb

_{2}O

_{7}there is no experimental signature of lattice distortion from the cubic phase, Pb

_{2}Nb

_{2}O

_{7}suffers a lattice distortion in low temperature [30]. Lattice distortion can greatly affect the topological properties. Zhang et al. have shown that Tl

_{2}Nb

_{2}O

_{7}can transform from a semi-metal to a TI by in-plane strain [18]. Since the topological properties are not dependent on the detail of the band structure, it is highly possible that the TI is also induced by in-plane strain in Tl

_{2}Ta

_{2}O

_{7}, In

_{2}Nb

_{2}O

_{7}and In

_{2}Ta

_{2}O

_{7}, which have similar band structures.

## 4. Conclusions

_{2}B

_{2}O

_{7}(A = Sn, Pb, Tl; B = Nb, Ta). They have quasi-flat valence bands that are characteristic of the Guo–Franz flat-band model. Spin–orbit interaction partially lifts the degeneracy at the Γ point. This results in a strong topological insulator in the s1 compound if the split energy levels satisfy E(Γ

_{7}) < E(Γ

_{8}). We found only Sn

_{2}Nb

_{2}O

_{7}satisfies E(Γ

_{7}) < E(Γ

_{8}) in the above six compounds, though this is an s2 compound. We propose that the virtual compound (In,Sn)

_{2}(Nb,Zr)

_{2}O

_{7}may have all of the desired properties for a TI, i.e., a Guo–Franz-model-like band, an s1 compound and E(Γ

_{7}) < E(Γ

_{8}).

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**(

**a**) Pylochlore lattice in the face-centered cubic Bravais lattice. The large circle shows the 4 atomic sites in the primitive unit cell, forming a regular tetrahedron. These tetrahedra shear corners and form a three-dimensional network; (

**b**) energy dispersion of the Guo and Franz (GF) model with λ = 0 (Equation (2)). Numbers 1–4 denote the band index. Energy offset is added to simulate the band structure of Pb

_{2}Ta

_{2}O

_{7}. Note that the total bandwidth is large (= 8t), while the flat bands (FBs) (E

^{3}and E

^{4}) does not have any dispersion; (

**c**) a schematic picture of the energy dispersion of the GF model for λ > 0. At the Γ point the degeneracy is partly lifted, but at the X and L points the degeneracy is protected by symmetry, so the system remains gapless; (

**d**) a schematic picture of the energy dispersion of the GF model for λ < 0. In this case the system opens a gap, and this becomes a strong TI [6].

**Figure 2.**(

**a**) Band structure of Pb

_{2}Ta

_{2}O

_{7}. (

**b**) DOS curve of Pb

_{2}Ta

_{2}O

_{7}. (

**c**) Energy dispersion of the GF model with λ = 0. (

**d**) Blow-up figure of the valence band of Pb

_{2}Ta

_{2}O

_{7}near Γ point, shown in the circle on Figure 1a. The units of the vertical axes are eV for (

**a**), (

**c**) and (

**d**), and eV

^{−1}for (

**b**).

**Figure 3.**Effective spin–orbit coupling constant λ for eight A

_{2}B

_{2}O

_{7}pyrochlore oxides (units are eV). The left panels are for B = Nb, and the right panels are for B = Ta. A-site elements are arranged in the order of the periodic table. For example, the upper-left column in the left panel means In

_{2}Nb

_{2}O

_{7}.

**Figure 4.**(

**a**) Band structure of Sn

_{2}Zr

_{2}O

_{7}and (

**b**) its blow-up figure of the valence band near the Γ point. (

**c**) Band structure of In

_{2}Nb

_{2}O

_{7}and (

**d**) its blow-up figure of the valence band near the Γ point. The units of the vertical axes are eV.

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Hase, I.; Yanagisawa, T.
Possible Three-Dimensional Topological Insulator in Pyrochlore Oxides. *Symmetry* **2020**, *12*, 1076.
https://doi.org/10.3390/sym12071076

**AMA Style**

Hase I, Yanagisawa T.
Possible Three-Dimensional Topological Insulator in Pyrochlore Oxides. *Symmetry*. 2020; 12(7):1076.
https://doi.org/10.3390/sym12071076

**Chicago/Turabian Style**

Hase, Izumi, and Takashi Yanagisawa.
2020. "Possible Three-Dimensional Topological Insulator in Pyrochlore Oxides" *Symmetry* 12, no. 7: 1076.
https://doi.org/10.3390/sym12071076