New three-dimensional flat band candidate materials Pb₂As₂O₇ and Pb₂Sn₂O₇

Abstract

Energy dispersion of electrons is the most fundamental property of the solid state physics. In models of electrons on a lattice with strong geometric frustration, the band dispersion of electrons can disappear due to the quantum destructive interference of the wavefunction. This is called a flat band, and it is known to be the stage for the emergence of various fascinating physical properties. It is a challenging task to realize this flat band in a real material. In this study, we performed first-principles calculations on two compounds, Pb$_2$As$_2$O$_7$ and Pb$_2$Sn$_2$O$_7$, which are candidates to have flat bands near the Fermi level. Both compounds have electronic states close to flat bands, but the band width is significantly larger than that of Pb$_2$Sb$_2$O$_7$ shown in previous research. Nevertheless, the density of states at the Fermi level of Pb$_2$As$_2$O$_7$ is large enough to cause the system to undergo a ferromagnetic transition. In the case of Pb$_2$Sn$_2$O$_7$, pseudo-gap behavior near the Fermi level was observed. These findings underscore the importance of investigating the influence of flat bands on electronic energy dispersion, providing a crucial step toward understanding the emergence and characteristics of flat bands in novel materials.

Introduction

Flat band (FB) refers to a situation where the energy band has no dispersion, and is an important concept in the study of strongly correlated electron systems, inducing fascinating properties like perfect ferromagnetism, superconductivity, and various topological effects^{1,2,3,4,5,6,7,8,9,10,11,12,13,14}. Therefore, there have been many attempts to realize flat bands in real materials. Most of them are 2D materials, such as twisted bilayer graphene¹⁵, herbertsmithite and other compounds having a kagome lattice^{16,17,18,19,20}, and metal-organic framework^21,22. On the other hand, there are not many studies on 3D materials. There are theoretical calculations on WN$_2$²³, calculations on several pyrochlore oxides^24,25,26, and the Laves phase CaNi$_2$²⁷.

The flat band is most simply described by the following tight-binding model:

$$\begin{aligned} H = t\sum _{\langle i,j \rangle }(c^{\dagger }_{i}c_{j}+h.c.). \end{aligned}$$

(1)

Here $c^{\dagger }_{i}$ and $c_{j}$ are the creation operator at the i site and the annihilation operator at the j site. And here $\langle i,j \rangle$ denotes the nearest-neighbor sites of the lattice, and this lattice belongs to a special class so-called line-graph lattice. For example, the Kagome lattice is the line graph of the honeycomb lattice, and the pyrochlore lattice is that of the diamond lattice.

In case of pyrochlore lattice, there are four sites in the fcc primitive unit cell, $\textbf{r}_1 = (1/4,0,1/4)a, \textbf{r}_2 = (0,1/4,1/4)a, \textbf{r}_3 = (0,0,0), \textbf{r}_4 = (1/4,1/4,0)a$, where a is the lattice parameter. In this case, Bloch Hamiltonian reads²⁸

$$\begin{aligned} H_\textbf{k} = 2t\begin{pmatrix} 0 & \cos {(k_x-k_y)} & \cos {(k_x+k_z)} & \cos {(k_y-k_z)} \\ & 0 & \cos {(k_y+k_z)} & \cos {(k_x-k_z)}\\ & & 0 & \cos {(k_x+k_y)} \\ & & & 0 \end{pmatrix} ,\end{aligned}$$

(2)

where the wave vector $\textbf{k}$ is scaled by 4/a. The eigenvalues of this Bloch Hamiltonian can be analytically calculated as follows:

$$\begin{aligned} E_\textbf{k}^{(3,4)} = -2t, \end{aligned}$$

(3)

and

$$\begin{aligned} E_\textbf{k}^{(1,2)} = 2t[1\pm \sqrt{1+A_\textbf{k}}], \end{aligned}$$

(4)

with $A_\textbf{k}=\cos {(2k_x)}\cos {(2k_y)}+\cos {(2k_y)}\cos {(2k_z)}+\cos {(2k_z)}\cos {(2k_x)}$. The former two bands $E_\textbf{k}^{(3,4)}$ do not have any dependence on $\textbf{k}$, so they called as FBs.

We also note that the transfer integral t is isotropic, i.e., independent of the direction of the i–j bond. If the orbital on this lattice is s-orbital, this condition is naturally satisfied. Usually, anisotropic orbitals do not generate FBs because they result in anisotropic $t_{ij}$, but in some special cases, FBs may be generated by orbitals other than s-orbitals. For example, $p_x$ and $p_y$ orbitals on the honeycomb lattice form a FB⁶. However, in this paper we only consider the simplest case: the relevant orbitals are the s-orbitals on the pyrochlore lattice.

In order to realize the fascinating physical properties described above, not only the existence of flat bands but also their positions are important. Since these properties have relatively low energy scale, the Fermi level ($=E_\textrm{F}$) must be directly above or near the flat band.

For example, Sn$_2$Nb$_2$O$_7$ is a wide-gap insulator and the top of the valence band becomes a FB consisting mainly of Sn-s orbitals. Therefore, external hole doping is required to settle $E_\textrm{F}$ on FB. However, this external hole doping is very difficult experimentally because the oxygen deficiency occurs inevitably and the system tends to become n-type²⁹.

Therefore, we have recently proposed Pb$_2$Sb$_2$O$_7$ as another candidate compound in which a flat band appears. In addition to the pyrochlore-lattice band (= PB1) mainly composed of Pb-s and O$^{\prime }$-p orbitals, this compound also has another pyrochlore-lattice band (= PB2) mainly composed of Sb-s orbitals, and holes are doped into PB1 by the overlap of the two (self-doping). In this case, ferromagnetism is expected to appear without external doping, and this compound is very interesting in that it does not contain any magnetic elements. However, the weberite structure has been found to be energetically more stable than the pyrochlore structure³⁰. For this reason, it is very important to further explore flat band materials.

In this study, we performed first-principles calculations on two candidate compounds, Pb$_2$As$_2$O$_7$ and Pb$_2$Sn$_2$O$_7$ for realizing flat bands near $E_\textrm{F}$.

We found that both Pb$_2$As$_2$O$_7$ and Pb$_2$Sn$_2$O$_7$, a flat band appears at the top of the valence band, as previously reported Pb$_2$Sb$_2$O$_7$³⁰. In both of these compounds, we found the flat band is maintained over most of the $\textbf{k}$-space, but is significantly modulated near the $\Gamma$ point. In Pb$_2$As$_2$O$_7$, similar to Pb$_2$Sb$_2$O$_7$, the density of states at $E_\textrm{F}$ becomes very large, stabilizing the ferromagnetic state. Additionally, Pb$_2$Sn$_2$O$_7$ exhibits a pseudogap state. These findings highlight the importance of the constituent elements in flat band materials and are an important step toward understanding the emergence and properties of flat bands in new materials.

The remainder of this paper is organized as follows. The calculation methods are described in Section II. Results and discussions are shown in Section III. Conclusions are given in Section IV.

Methods

We computed the electronic structure of Pb$_2$As$_2$O$_7$ and Pb$_2$Sn$_2$O$_7$ from first-principles calculations based on the density functional theory^31,32. We used a full-potential augmented plane-wave scheme, and the exchange-correlation potential was constructed within the general gradient approximation³¹.

Experimental results show that Pb$_2$As$_2$O$_7$ has a complex K$_2$Cr$_2$O$_7$ type triclinic structure³³. The space group is P$\bar{1}$ (No. 2). We used the experimental values for the lattice parameters (a, b, c, $\alpha$, $\beta$, $\gamma$)³³. Since this structure contains 44 atoms (twice as many as the cubic pyrochlore structure), we used $\sim 500$ k-points in the first Brillouin zone with nearly equal interval in the self-consistent calculation.

In order to investigate the stability of this structure, we calculated both of this triclinic structure and cubic pyrochlore structure. As for the cubic pyrochlore structure, we optimized both lattice parameter (a) and internal atomic parameter (u) of the 48f oxygen site (u, 1/8, 1/8) with space group Fd$\bar{3}$m (No. 227). The $\textbf{k}$-point mesh was selected so that the total number of mesh points in the first Brillouin zone was approximately 1000 in the self-consistent calculation. The convergence of the atomic positions was assessed to be less than 1.0 mRy/a.u. based on the forces acting on each atom.

As for Pb$_2$Sn$_2$O$_7$, previous experimental results have shown that it has a cubic pyrochlore structure³⁴. Therefore, we performed calculations only for the pyrochlore structure. In this calculation, atomic positions were optimized using the same procedure as for Pb$_2$As$_2$O$_7$.

To determine where the FB(s) exist, we performed an analysis using the maximally localized Wannier functions (MLWFs)³⁵ for Pb$_2$As$_2$O$_7$ and Pb$_2$Sn$_2$O$_7$ with pyrochlore structure. For the MLWFs, we used localized orbitals centered at the A-site and B-site with s-symmetry. These MLWFs were constructed from the ab initio eigenvalues and eigenvectors on $8\times 8\times 8$ unshifted $\textbf{k}$-mesh points.

Results and discussions

Tight-binding model

Before discussing the calculation results, we will briefly introduce the pyrochlore crystal structure. Pyrochlore oxide has the formula A$_2$B$_2$O$_7$, but since oxygen has two types of positions, it can also be written as A$_2$B$_2$O$_6$O$^{\prime }$.

A$_2$B$_2$O$_6$O$^{\prime }$ can be decomposed into a corner-shared A$_4$O$^{\prime }$ tetrahedral network (A$_2$O$^{\prime }$ unit) and corner-shared BO$_6$ octahedral network (B$_2$O$_6$ unit)²⁴. We can construct the group of bands PB1 in the following way. Let us consider only the A-s orbitals, and introduce a transfer integral (t in Eqn.1) between the nearest-neighbor A–A sites. This A–A transfer integral actually occurs along the path A–O$^{\prime }$–A, but the degree of freedom of the O$^{\prime }$ site can be integrated out using a procedure similar to that used to construct an effective model of only the Cu site in a copper oxide with a CuO$_2$ plane³⁶.

We should also note that there is not only the nearest-neighbor A–A transfer integral, but also second-nearest and further transfer integrals in real material. These terms can bend the FBs, but the bandwidth of the “FBs” in PB1 is not so large in Pb$_2$X$_2$O$_7$ (X = As, Sb, Sn) as shown in the following results.

Similarly, for the B site, we can integrate out the O site from the BO$_6$ network to obtain the group of bands PB2. However, while the O$^{\prime }$ site is at the center of the A$_4$ tetrahedron, the O site is not at the center of the B$_4$ tetrahedron. Therefore, the geometrical frustration is incomplete in PB2, and the top of PB2 has a large dispersion. However, in the compounds discussed in this paper, $E_\textrm{F}$ is near the top of PB1 and is far from the top of PB2. Therefore, the large dispersion of the top of PB2 does not affect the low energy physical properties.

The Hamiltonian which describes both PB1 and PB2 can be written as follows:

$$\begin{aligned} \begin{aligned} H = \sum _{i, \alpha }\epsilon ^{\alpha }c^{\dagger }_{i\alpha }c_{i\alpha }&+\sum _{\langle i,j \rangle , \alpha }t_{1}^{\alpha \text {-}\alpha }(c^{\dagger }_{i\alpha }c_{j\alpha }+h.c.) \\&+\sum _{\langle i,j \rangle _\textrm{AB}}t_{1}^{\mathrm {A\text {-}B}}(c^{\dagger }_{i\textrm{A}}c_{j\textrm{B}}+h.c.) \\&+\sum _{\langle \langle i,j \rangle \rangle , \alpha }t_{2}^{\alpha \text {-}\alpha }(c^{\dagger }_{i\alpha }c_{j\alpha }+h.c.) \\&+\sum _{\langle \langle i,j \rangle \rangle _\textrm{AB}}t_{2}^{\mathrm {A\text {-}B}}(c^{\dagger }_{i\textrm{A}}c_{j\textrm{B}}+h.c.) + ... \end{aligned} \end{aligned}$$

(5)

This is a natural extension of Eq. (1). $\epsilon ^\alpha$ in the first term denotes the on-site energy with sublattice $\alpha$ = A and B. The second term is the same with Eq. (1), including the nearest-neighbor transfer integral in the $\alpha$ sublattice. These two terms form two band groups with FB at the top. The third to fifth terms represent the nearest-neighbor transfer integral across sublattices $t_1^{\mathrm {A\text {-}B}}$, next nearest-neighbor transfer integral within the same sublattice $t_2^{\mathrm {A\text {-}A}}$, $t_2^{\mathrm {B\text {-}B}}$ and the next nearest-neighbor transfer integral across sublattices $t_2^{\mathrm {A\text {-}B}}$. These terms deform the PB1 and PB2 bands formed by the previous two terms.

Fig. 1

Energy band dispersion of (a) Pb$_2$As$_2$O$_7$, (b) Pb$_2$Sb$_2$O$_7$ and (c) Pb$_2$Sn$_2$O$_7$ from first-principles calculations. The purple crosses are the energy bands obtained by the first-principles calculations, and the green solid line are the bands constructed from the Wannier functions. The orange hatched area contains PB1, and the blue hatched area contains PB2. In the panel (c), a quadratic band touching occurs at $E_\textrm{F}$, shown by a red circle. $E_\textrm{F}$ is set to 0 eV.

Table 1 Tight-binding parameters for pyrochlore Pb$_2$X$_2$O$_7$ (X = As, Sb, Sn).

Pb$_2$As$_2$O$_7$ and Pb$_2$Sb$_2$O$_7$

Now we show our calculated results. Figure 1 shows the band dispersion in the pyrochlore structure of Pb$_2$As$_2$O$_7$, Pb$_2$Sb$_2$O$_7$, and Pb$_2$Sn$_2$O$_7$.

We also show the tight-binding parameters using the MLWFs in Table 1. In Table 1, the parameter with the largest absolute value is the nearest-neighbor $t_1^{\mathrm {X\text {-}X}}$, followed by the nearest-neighbor $t_1^{\mathrm {Pb\text {-}Pb}}$. These determine the bandwidth of PB2 and PB1, respectively (if only the nearest-neighbor $t_1$ exists in the model, the bandwidth would be $8|t_1|$). The nearest-neighbor $t_1^{\mathrm {Pb\text {-}X}}$ is much smaller than those, indicating that the coupling between PB1 and PB2 is small as a first approximation. The next nearest-neighbor transfer integral $t_2^{\mathrm {Pb\text {-}Pb}}$ in Pb$_2$Sb$_2$O$_7$ is much larger than that in Pb$_2$As$_2$O$_7$ and Pb$_2$Sn$_2$O$_7$, which indicates that the band width of the top of PB1 in Pb$_2$Sb$_2$O$_7$ is much smaller than that in Pb$_2$As$_2$O$_7$ and Pb$_2$Sn$_2$O$_7$.

Let us focus on Pb$_2$Sb$_2$O$_7$ as a reference compound, shown in Fig. 1b. The details of the calculation results for this compound have been shown in Ref.³⁰. The band near $E_\textrm{F}$ of Pb$_2$Sb$_2$O$_7$ consists of two groups of bands, PB1 and PB2. PB1 is mainly composed of Pb-s and O$^{\prime }$-p orbitals with energy spread from -2.6 eV to 0 eV (= $E_\textrm{F}$), and PB2 is mainly composed of Sb-s and O-p orbitals with energy spread from -0.8 eV to 3.0 eV. PB1 and PB2 overlap each other, and holes are introduced into PB1 by some electrons flowing in from PB2 (self-doping). The top of PB1 is almost flat and has a very large density of states (DOS), so $E_\textrm{F}$ is pinned to the top of PB1.

The electronic state of Pb$_2$As$_2$O$_7$ is basically similar to that of Pb$_2$Sb$_2$O$_7$ as shown in Fig. 1a. The bands near $E_\textrm{F}$ consist of two flat bands, PB1 and PB2. PB1 is mainly composed of Pb-s and O$^{\prime }$-p orbitals with energy spread from -2.1 eV to 0 eV (= $E_\textrm{F}$), and PB2 is mainly composed of As-s and O-p orbitals with energy spread from -1.4 eV to 2.4 eV. The difference is that the overlap between PB1 and PB2 is much larger than that of Pb$_2$Sb$_2$O$_7$. This is a consequence of the fact that the energy of the As-s orbital is much lower than that of the Sb-s orbital.

In addition, the flatness of the top band of PB1 is reduced compared to Pb$_2$Sb$_2$O$_7$ as a hump is observed near the $\Gamma$ point. As a result, the DOS at $E_\textrm{F}$ is reduced, but it is still large enough to induce magnetic transition.

Next we show the results for ferromagnetic states. In the complete flat band model, in the half-filled case, it is strictly shown that the ground state is a ferromagnetic state when on-site Hubbard U is added^1,2. Though there is no exact solution for the quasi-flat band case, there are some numerical results that the ground state becomes ferromagnetic when U is larger than the width of the quasi-flat band³⁷. Our calculations do not fully include many-body effects, but they still provide an overview of the electronic state. Many-body calculation for realistic three-dimensional material is a future task.

When calculations are performed taking into account spin polarization, the ferromagnetic state becomes stable and a magnetic moment $M=0.446 \mu _\textrm{B}$ (per Pb atom) is obtained in Pb$_2$As$_2$O$_7$. This value is larger than that of Pb$_2$Sb$_2$O$_7$, $M=0.307 \mu _\textrm{B}$. This is because of the large overlap between PB1 and PB2 in Pb$_2$As$_2$O$_7$. More holes are self-doped into PB1, and induces large magnetic moment in Pb$_2$As$_2$O$_7$.

We show the energy dispersion of this ferromagnetic state in Fig. 2. Comparing the up-spin band and the down-spin band, it can be seen that PB2 is hardly split, whereas PB1 is split by about 0.4 eV. This behavior is consistent with that the obtained magnetic moment is mostly consist of Pb-s and O$^{\prime }$-p orbitals. As in ordinary magnetic metals, this exchange splitting of the valence band ($\sim$ 0.4 eV) is almost independent of the wavenumber. The quasi-flat band feature is almost retained in this ferromagnetic phase. This quasi-flat band feature is also seen as a sharp peak near $E_\textrm{F}$ in the DOS curve shown in Fig. 3.

Fig. 2

Energy band dispersion of Pb$_2$As$_2$O$_7$ in the ferromagnetic state. The orange hatched area contains PB1, and the blue hatched area contains PB2. $E_\textrm{F}$ is set to 0 eV.

Fig. 3

Density of States (DOS) of Pb$_2$As$_2$O$_7$ in the ferromagnetic state. The orange hatched area is PB1, and the blue hatched area is PB2. $E_\textrm{F}$ is set to 0 eV.

Since Pb$_2$As$_2$O$_7$ does not contain any so-called magnetic elements, it is very interesting that a ferromagnetic state appears in the calculations likewise Pb$_2$Sb$_2$O$_7$³⁰.

However, this is the calculation result assuming a pyrochlore structure. Experimentally, it is known that Pb$_2$As$_2$O$_7$ has a triclinic K$_2$Cr$_2$O$_7$-type structure. We performed first-principles calculations of Pb$_2$As$_2$O$_7$ using this crystal structure and found that the K$_2$Cr$_2$O$_7$-type structure is stable by approximately 0.294 eV per atom. This value is sufficiently larger than the energy difference of 0.055 eV/atom between the weberite and pyrochlore phases in Pb$_2$Sb$_2$O$_7$. Therefore, it is considered very difficult to obtain pyrochlore phase Pb$_2$As$_2$O$_7$ under normal synthesis conditions.

Nevertheless, if pyrochlore phase Pb$_2$Sb$_2$O$_7$ is obtained, it may be possible to dissolve a small amount of As in it. In this case, it is considered that the overlap between PB1 and PB2 can be gradually increased as shown in Fig. 1. In other words, the amount of carriers due to self-doping can be controlled by the solid solution of As. This type of carrier control is considered to be a promising method of carrier control in flat band systems because it does not disturb the Pb$_4$O$^{\prime }$ network, which is the conductive path.

Pb$_2$Sn$_2$O$_7$

Next, we focus on Pb$_2$Sn$_2$O$_7$ shown in Fig.1c. Because the energy of the Sn-s orbital is very high, only PB1 is visible at this energy scale. PB1 is mainly composed of Pb-s and O$^{\prime }$-p orbitals, with energy spread from -1.6 eV to 0.4 eV, though the lower part of this band is overlapped with the band mainly composed of O-p orbitals. In the case of Pb$_2$Sn$_2$O$_7$, carriers are not introduced into PB1 because there is no overlap with PB2, but since there is originally one less valence electron per Sn (four less valence electrons per the primitive unit cell) than in Pb$_2$Sb$_2$O$_7$ and Pb$_2$As$_2$O$_7$, PB1 is exactly half-filled. This is similar to Tl$_2$Nb$_2$O$_7$ reported previously^25,38. The shape of PB1 is significantly distorted compared to Pb$_2$Sb$_2$O$_7$, especially in the vicinity of the $\Gamma$ point, but the band topology is unchanged.

When PB1 is exactly half-filled, a 3D topological insulator may be realized if spin-orbit coupling is taken into account²⁸. In the absence of spin-orbit coupling, so-called quadratic band-touching occurs at the $\Gamma$ point, shown by a red circle in Fig. 1c. This is inevitable due to the strong geometric frustration of this system³⁹. The irreducible representation of the $\Gamma$ point is $\Gamma _{25}$, which is six-fold degenerate when spin is included. When spin-orbit coupling is added to this, it splits into a doubly degenerate $\Gamma _{7}$ state and a four-fold degenerate $\Gamma _{8}$ state. If the $\Gamma _{8}$ state has a higher energy, a gap opens at this point, and a 3D topological insulator is realized. However, in our calculations, the $\Gamma _{7}$ state has a higher energy, and no gap opens. In fact, the $\Gamma _{7}$ state has a higher energy in many other pyrochlore oxides²⁶. In pyrochlore oxides having FB, another interesting electronic state has been predicted. Exchange splitting caused by above-discussed ferromagnetic transition can induce a pair of Weyl points, i.e., a Weyl semimetal state. Actually, it is pointed out that hole-doped Sn$_2$Nb$_2$O$_7$ is a promising candidate for a Weyl semimetal created by a 3D FB⁴⁰. This Weyl semimetal state can be induced even in Pb$_2$Sn$_2$O$_7$, where there is no gap in the paramagnetic state, if the system becomes ferromagnetic by some carrier doping⁴¹.

Bond-valence sum analysis

Next, we consider the effect of structural relaxation. In the pyrochlore structure, the only free atomic position parameter is oxygen (u, 1/8, 1/8) located at 48f. The structure was relaxed by minimizing the forces at the atomic positions, resulting in u(As) = 0.31562, u(Sb) = 0.31759, and u(Sn) = 0.33189. Clearly, the oxygen positions in Pb$_2$Sn$_2$O$_7$ are significantly different from the “ideal positions” (where the SnO$_6$ forms a regular octahedron), i.e. $u_\textrm{oct} = 5/16 = 0.3125$, whereas in Pb$_2$As$_2$O$_7$ and Pb$_2$Sb$_2$O$_7$ they are located at $u \sim u_\textrm{oct}$.

This situation can be understood by considering the bond-valence sum⁴². Table 2 shows the bond-valence sums and representative bond lengths for Pb$_2$As$_2$O$_7$, Pb$_2$Sb$_2$O$_7$, and Pb$_2$Sn$_2$O$_7$. For Pb$_2$Sn$_2$O$_7$, the value when $u=0.31759$ (the same value as Pb$_2$Sb$_2$O$_7$, denoted “before relaxation”) is also shown for reference.

First, let us compare Pb$_2$As$_2$O$_7$ and Pb$_2$Sb$_2$O$_7$. In Pb$_2$Sb$_2$O$_7$, Pb is close to +2 valence and Sb is close to +5 valence, which is close to the simple ionic picture Pb$_2^{+2}$Sb$_2^{+5}$O$_7^{-2}$. However, in Pb$_2$As$_2$O$_7$, the valence of Pb increases significantly and the valence of As decreases significantly. This is consistent with the large overlap between PB1 and PB2 in Fig. 1a, and the transfer of electrons from Pb to As. Since the oxygen position parameter u is almost the same in the two compounds, it can be said that the difference in the bond-valence sum is caused almost entirely by the difference in the constituent elements.

Next, we compare Pb$_2$Sb$_2$O$_7$ and Pb$_2$Sn$_2$O$_7$. First, in the case of Pb$_2$Sn$_2$O$_7$ before relaxation, the value of the bond-valence sum is almost equal to that of Pb$_2$Sb$_2$O$_7$. If this is expressed in an ionic picture, it becomes Pb$_2^{+2}$Sn$_2^{+5}$O$_7^{-2}$, but this is clearly irrational because Sn has only four valence electrons. This means that the structure before relaxation is not stable. After relaxation, the valence of Sn is greatly reduced, and it is closer to a stable structure by decreasing from +4 valence.

As described above, the bond-valence sum is a simple index that assumes ionic bonds, but at least in this system, it can be seen that it can consistently describe the change in the B-site cationic element and the accompanying structural change.

Finally, we discuss the changes in electronic states in Pb$_2$As$_2$O$_7$ caused by differences in crystal structure. Figure 4 presents the DOS curves of Pb$_2$As$_2$O$_7$ for (a) the pyrochlore structure and (b) the experimental (K$_2$Cr$_2$O$_7$-type) triclinic crystal structure. As for the pyrochlore structure, we observe a sharp DOS peak at $E_\textrm{F}$ corresponding with FB1 discussed above. We can also see that the states with energy from 0 eV (= $E_\textrm{F}$) to 2.0 eV mainly consist of As and O$^{\prime }$ orbitals.

Fig. 4

DOS of Pb$_2$As$_2$O$_7$ with (a) pyrochlore structure and (b) K$_2$Cr$_2$O$_7$-type triclinic structure. Note that in (b), there are 22 different crystallographic sites, so the total contributions of 4 Pb sites, 4 As sites, and 14 O sites are shown.

On the other hand, we found that Pb$_2$As$_2$O$_7$ with the K$_2$Cr$_2$O$_7$-type crystal structure becomes an insulator with a large band gap $\sim 3.2$ eV. In K$_2$Cr$_2$O$_7$-type structure, the As–O bond length is between 1.657 Å  and 1.778 Å, which is in good agreement with the sum of the ionic radii ($r(\textrm{O}^{2-})\sim 1.4$ Å, $r(\textrm{As}^{5+})=0.335$ Å) when O$^{2-}$ and As$^{5+}$ (four-coordinated) are assumed. From this, As is considered to be almost +5 valence and Pb is almost +2 valence, which is consistent with this system being an insulator. Incidentally, the As–O bond length in the pyrochlore structure is 2.001 Å, which is slightly longer than the sum of the ionic radii ($r(\textrm{O}^{2-})=1.38$ Å, $r(\textrm{As}^{5+})=0.46$ Å) when O$^{2-}$ and As$^{5+}$ (six-coordinated) are assumed. This does not contradict the partial occupation of the As 4s band in the pyrochlore structure.

As described above, comparing the total energies, it was found that the K$_2$Cr$_2$O$_7$-type structure has a lower total energy of 0.294 eV per atom than the pyrochlore structure. Therefore, it is considered very difficult for Pb$_2$As$_2$O$_7$ to have a pyrochlore structure. In the first place, compounds with AsO$_6$ octahedra are quite rare, and AsO$_4$ tetrahedral coordination is generally found. Chemically, this may be the reason why the pyrochlore structure is less stable than the K$_2$Cr$_2$O$_7$-type structure.

Table 2 Bond-valence sum for pyrochlore Pb$_2$As$_2$O$_7$, Pb$_2$Sb$_2$O$_7$ and Pb$_2$Sn$_2$O$_7$.

Conclusions

First-principles calculations were performed on pyrochlore oxides Pb$_2$As$_2$O$_7$ and Pb$_2$Sn$_2$O$_7$ as candidates for three-dimensional flat band compounds. For Pb$_2$As$_2$O$_7$, a flat band with self-doped holes was obtained, similar to the previously calculated Pb$_2$Sb$_2$O$_7$. As a result of calculations allowing spin polarization, a magnetic moment of 0.446 $\mu _\textrm{B}$ per Pb atom was obtained. Although the pyrochlore structure is not stable for Pb$_2$As$_2$O$_7$ itself, the solid solution Pb$_2$(Sb,As)$_2$O$_7$ will provide a stage where the flatness of the flat bands can be controlled continuously. On the other hand, for Pb$_2$Sn$_2$O$_7$, pyrochlore structure has been observed experimentally. For Pb$_2$Sn$_2$O$_7$, a quadratic touching electronic state was obtained at the $\Gamma$ point. Although the spin-orbit interaction does not open a gap at this point, a pair of Weyl points are expected to emerge when the ferromagnetic transition occurs. These results will provide important insights for a deeper understanding of flat band materials.