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Electronic Band Structures of the Possible Topological Insulator Pb_{2}BiBrO_{6} and Pb_{2}SeTeO_{6} Double Perovskite: An Ab Initio Study

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## Abstract

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## Featured Application

**This work offers a detailed description of the topological insulator behavior and the crystal structure of the Pb2AA’O6 double perovskites.**

## Abstract

_{2}BiBrO

_{6}and Pb

_{2}SeTeO

_{6}double perovskites. Our ab initio theoretical calculations show that Pb

_{2}BiBrO

_{6}and Pb

_{2}SeTeO

_{6}are Z

_{2}nontrivial, and their bandgaps are 0.390 eV and 0.181 eV, respectively. The topology comes from two mechanisms. Firstly, the band inversion occurs at $\mathsf{\Gamma}$ point in the absence of spin-orbit interactions and secondly, the bandgap is induced by the SOC. This results in a larger bandgap for this new class of topological insulators than conventional TI. In Pb

_{2}BiBrO

_{6}double perovskites, our slab calculations confirm that the topology-protected surface metallic bands come from the BiBrO

_{4}surface which means that one can build a transport device using Pb

_{2}BiBrO

_{6}double perovskites with a PbO layer as an outmost protection layer. The mechanical stabilities such as bulk, shear, Young’s moduli, Poisson’s and Pugh’s ratio, longitudinal, transverse, and average sound velocity, together with Debye temperature are also studied. Our results show that these Pb

_{2}AA’O

_{6}(A = Sb and Bi; A’ = Br and I) and Pb

_{2}SeTeO

_{6}are mechanically stable.

## 1. Introduction

_{2}BB’O

_{6}made of two single perovskites ABO

_{3}and AB’O

_{3}are potential materials for many applications and attract both experimental synthesis and theoretical predictions in the past few decades. For example, Sr

_{2}FeMoO

_{6}[20] and Sr

_{2}FeOsO

_{6}[21] double perovskites are candidates for spintronic devices for their half metallicity at room temperature. Moreover, Sr

_{2}CrOsO

_{6}[22] is a ferrimagnetic insulator with a very high Curie temperature of 720 K. The ideal structure of the double perovskites is cubic, however, they might undergo a temperature-induced structural phase transition. For example, Pb

_{2}CoTeO

_{6}[23,24] double perovskites are synthesized with monoclinic, rhombohedral, and cubic structures in different temperature ranges. Beyond the experimental preparations, a lot of theoretical predictions for the emerging double perovskites are made. For example, halfmetallic, topological and multiferroic double perovskites are often mentioned in theoretical predictions. We, therefore, believe that there is possible topological insulator in double perovskites.

_{2}Se

_{3}[13,25], Bi

_{1−x}Sb

_{x}[26] are shown to be topological insulators. Furthermore, it is theoretically predicted that CsPbI

_{3}[27,28] and A

_{2}BiXO

_{6}(A = Ca, Sr, Ba; X = I, Br) [29] are also possibly TIs. Typically, these materials are ordinary metals or insulators, but become insulators by including spin-orbit coupling (SOC) effect in the calculations. In other words, the topological behavior cannot be discovered without including SOC in theoretical calculations. The CsPbI

_{3}perovskite is an insulator, it may become a topological insulator by applying an external axial strain. Large bandgap topological insulators are particularly of interest because they have a strong ability to overcome thermal fluctuation. Unlike 2D TIs, where the bandgap in these materials can be tuned in several ways, the bandgap of 3D TIs is hard to increase. Several ways to tune the bandgap of 3D TIs. The first approach is to find a material composed of heavy elements, such as Bi or Te, to provide a stronger SOC [30]. The second approach is to find specific crystal structures, such as perovskite and double perovskite. The last way is to apply external fields. More recently, Pi et al. proposes a new class of large bandgap of 0.55 eV topological insulators in the A

_{2}BiXO

_{6}(A = Ca, Sr, Ba; X = I, Br) [30] double perovskites structures. Most recently, Lee et al. also predicts A

_{2}TePoO

_{6}(A = Ca, Sr, Ba) [31] double perovskites as TIs which also have high bandgap as high as 0.4 eV. Typically, it can be achieved to theoretically predict a topological insulator by observing the band inversion with and without SOC in the first Brillouin zone in the theoretical calculations. In other words, the SOC introduces band inversion and bandgap in the conventional TIs. Pi et al. [30] reported a tight-binding model in the rock-salt structure, showing that if certain s- and p-orbital orderings are satisfied in the absence of SOC, the material can be a topological insulator. In addition, a useful and practical topological number Z

_{2}was introduced by Fu and Kane [32], which is calculated by analyzing the bulk band structure.

_{2}AA’O

_{6}(A = Sb and Bi; A’ = Br and I) and Pb

_{2}SeTeO

_{6}under density functional theory. The crystal structures and computational methods are described in Section 2. In Section 3.1, we present the band structures of the double perovskites with and without SOC. We show that in the Pb

_{2}BiBrO

_{6}and Pb

_{2}SeTeO

_{6}double perovskites, there is a band contact at $\mathsf{\Gamma}$ point along with s- and p- band inversion. Mechanical properties and stability are also discussed in Section 3.2. In Section 3.3, we present the metallic surface band of the Pb

_{2}BiBrO

_{6}double perovskites as an example. Finally, we draw conclusions in Section 4.

## 2. Structures and Computational Methods

_{2}AA’O

_{6}is a body-centered tetragonal with a space group of I4/mmm. Plotted in Figure 1 is the primitive unit cell of the Pb

_{2}AA’O

_{6}double perovskite. Listed in Table 1 are the lattice constants a, b, c, unit cell volume, atomic coordination O

_{1z}, O

_{2x}, and O

_{2y}. In the Pb

_{2}AA’O

_{6}double perovskite, the two Pb atoms are located in (0.75, 0.25, 0.5) and (0.25, 0.75, 0.5), A in (0.0, 0.0, 0.0) and B in (0.5, 0.5, 0.0). The six oxygen atoms can be divided into two groups. The first group consists of two oxygen whose positions are (O

_{2y}, O

_{2y}, O

_{1z}), ($-$O

_{2y}, $-$O

_{2y}, $-$O

_{1z}) where O

_{2y}and O

_{1z}are shown in Table 1. The second group had four oxygen located at (O

_{2x}, O

_{2x}, 0), ($-$O

_{2x}, $-$O

_{2x}, 0), ($-$O

_{2y}, O

_{2y}, 0), and (O

_{2y}, $-$O

_{2y}, 0), see also Table 1.

^{−6}eV. To calculate the surface band structure, we prepared a large supercell containing 12 layers of Pb

_{2}AA’O

_{6}double perovskites, i.e., 6 formula units, with a vacuum of 20$\u212b$ at least. In this supercell, one PbO and one AA’O

_{4}surface are in contact with the vacuum. We also prepare another two supercells that contain the same surface, PbO or AA’O

_{4}surface, on both sides by cutting the specific layers to understand the contribution of the surface band structures.

## 3. Results and Discussion

#### 3.1. Structural, and Electronic Band Structure

_{0}, atomic coordination O

_{1z}, O

_{2x}, O

_{2y,}and bandgap are listed in Table 1. It is obvious that the c/a ratio for the Pb

_{2}AsIO

_{6}, Pb

_{2}SbIO

_{6}, Pb

_{2}BiBrO

_{6}, Pb

_{2}BiIO

_{6}, and Pb

_{2}SeTeO

_{6}is 1.413 which is very close to the ideal value of $\sqrt{2}$. The c/a ratio of Pb

_{2}AsBrO

_{6}and Pb

_{2}SbBrO

_{6}were 1.282 and 1.286, respectively. We also found that Pb

_{2}AsIO

_{6}has the smallest equilibrium volume whilst Pb

_{2}BiIO

_{6}has the largest equilibrium volume. It is related to the difference in wigner-seitz radius of the As and Bi atom. The calculated bandgap is all zero without SOC, while it becomes nonzero in Pb

_{2}BiBrO

_{6}and Pb

_{2}SeTeO

_{6}double perovskites with SOC. The bandgap for the Pb

_{2}BiBrO

_{6}and Pb

_{2}SeTeO

_{6}double perovskites is indirect and the magnitude is 0.390 eV and 0.181 eV.

_{2}AA’O

_{6}double perovskites without, upper panel, and with, bottom panel, spin-orbit coupling. Let us take Pb

_{2}BiBrO

_{6}, see Figure 2e, as an example. Without SOC, the band structures of Pb

_{2}BiBrO

_{6}at $\mathsf{\Gamma}$ point is contributed by the Bi-p and Br-p orbitals. The Bi-p and Br-p orbitals are separated into p

_{z}and p

_{y}(p

_{x}) groups. About 1eV at $\mathsf{\Gamma}$ point, the bands are mainly contributed from the Bi-s and Br-s orbital. It satisfies the first condition given by Pi et al. [30] that there is a s- and p-band inversion that occurs at the $\mathsf{\Gamma}$ point without SOC. When the SOC is taken into account, see Figure 2l, the p-orbital of the Bi and Br atom split to −0.6 eV and 0.55 eV with respect to the Fermi level. The orbital contribution at $\mathsf{\Gamma}$ point is still contributed from the p-orbital of the Bi and Br atom. Away from $\mathsf{\Gamma}$ along $\mathsf{\Gamma}$-X direction, the band’s contribution transferred from p- to s-orbital At X point, the bands become mostly from the s-orbital of the Bi and Br atom. To make it clearer, we demonstrate the Bi (Se) and Br (Te), s- and p-orbital projected band structures in Figure 3. Therefore, we conclude that Pb

_{2}BiBrO

_{6}and Pb

_{2}SeTeO

_{6}are possible topological insulators. The SOC-induced bandgap for the Pb

_{2}BiBrO

_{6}is 0.390 eV, being higher than most 3D TIs. We also found that the Z

_{2}index [32] (${\nu}_{0}$; ${\nu}_{1}$ ${\nu}_{2}$ ${\nu}_{3}$) of Pb

_{2}BiBrO

_{6}and Pb

_{2}SeTeO

_{6}are (1; 0 0 0) showing that they are strong topological insulators.

_{x}-, p

_{y}- and p

_{z}-orbitals. In Pb

_{2}BiBrO

_{6}double perovskite, the density of states of Bi’s s-orbital is larger than Br’s s-orbital. In contrast, the density of states for Br’s p orbitals is larger than Bi’s p orbital. The peak for both Bi’s and Br’s s-orbital lies between −1 eV and −2 eV. It is quite different compared with the density of states of Se’s and Te’s s-orbital. The Te’s s-orbital is closer to the Fermi level than Se’s s-orbital. We also note that Br and Te’s p

_{x}- and p

_{y}-orbital completely degenerate between −6 eV to 2 eV.

_{2}BiBrO

_{6}and Pb

_{2}SeTeO

_{6}are possible topological insulators, we present in detail our calculations for Z

_{2}invariants. We first calculate the bulk band structure using the maximally localized Wannier functions as implemented in the WANNIER90 package [37]. The obtained bulk band structures are compared with our DFT band results are shown in Figure 5a which are in excellent agreement. The wannier orbitals for Pb, Bi, Br, and O atoms are s, p

_{x}, p

_{y}and p

_{z}. With the wannier orbitals, a tight-binding model [38] is used for obtaining the Z

_{2}invariants. The results are shown in Figure 5b–g. Figure 5b–g are wannier charge center as a function of momentum k for the k

_{1}= 0.0, k

_{1}= 0.5, k

_{2}= 0.0, k

_{2}= 0.5, k

_{3}= 0.0, and k

_{3}= 0.5 momentum plane, respectively. In practice, the Z

_{2}number can be identified by counting the number of times for an arbitary reference lines parallel to k that cross the evolution lines. [39] The results are 1, 0, 1, 0, 1, and 0 for Figure 5b,c,d,e,f,g, respectively. Another way to determine the Z

_{2}trivial or nontrival can be done by counting the number of crossings between the largest gap and the wannier charge center. [40] Although one can use the obtained wannier functions combined with the tight-binding hamiltonian to explore the surface band structures. We calculate the surface band structures using a large supercell that contains 12 layers and the results are shown in the Section 3.3.

#### 3.2. Mechanical Properties

_{11}, C

_{12}, C

_{13}, C

_{33}, C

_{44}, and C

_{66.}The elastic constant matrix can be written as:

_{2}AA’O

_{6}double perovskites can be simplified as ${C}_{11}>{C}_{12}$, $\left({C}_{11}+{C}_{33}-2{C}_{13}\right)>0$, and ${C}_{33}\left({C}_{11}+{C}_{12}\right)>2{C}_{13}^{2}$. The Pb

_{2}AA’O

_{6}double perovskites are mechanically stable if all criteria are met.

_{R}(B

_{V}) is the bulk modulus of the Reuss (Voigt) bound. Typically, the Reuss bound is obtained from a uniform stress assumption and is the lower limit of the actual modulus. In contrast, the Voigt is the upper bound on the actual modulus. In the tetragonal phase, the bulk modulus and shear modulus can be represented as:

_{2}BiBrO

_{6}double perovskite. The results show that Pb

_{2}BiBrO

_{6}double perovskite is unstable.

#### 3.3. Band Structure of the Supercell

_{2}BiBrO

_{6}and Pb

_{2}SeTeO

_{6}are topological insulators, Figure 6 shows the electronic band structures of the Pb

_{2}BiBrO

_{6}and Pb

_{2}SeTeO

_{6}supercell. We constructed a supercell containing 12 layers of Pb

_{2}BiBrO

_{6}or Pb

_{2}SeTeO

_{6}(6 formula units) with a vacuum of 20$\u212b$. The calculations were done without structure relaxation, including surface atomic reconstruction. There are two kinds of surface layers in our Pb

_{2}BiBrO

_{6}or Pb

_{2}SeTeO

_{6}supercell, one is the PbO layer and the other is the BiBrO

_{4}(SeTeO

_{4}) layer. We denote these two surface layers as top PbO and top BiBrO

_{4}(SeTeO

_{4}). We also denote the second nearest BiBrO

_{4}(SeTeO

_{4}) layer to the surface as top2 BiBrO

_{4}and top2 SeTeO

_{4}. The calculated electronic band structure of the Pb

_{2}BiBrO

_{6}and Pb

_{2}SeTeO

_{6}supercell are shown in Figure 6a,e. Clearly, we observe metallic bands between $\mathsf{\Gamma}$-X and M-$\mathsf{\Gamma}$. It theoretically confirms that Pb

_{2}BiBrO

_{6}and Pb

_{2}SeTeO

_{6}are topological insulators. It is also important to verify the origin of these metallic bands. To investigate, we illustrate the project band structures in Figure 5b–d,f–h. In Figure 6b,f, the red solid circles represent the energy band due to top PbO layer. Very interestingly, the top PbO layer gives no contributions to the metallic surface bands in Pb

_{2}BiBrO

_{6}. We found that the surface BiBrO

_{4}layer, Figure 6c, and nearest surface BiBrO4 layers, Figure 6d, give contributions to the metallic surface bands. Unlike Pb

_{2}BiBrO

_{6,}both PbO and SeTeO

_{4}surface give contribution to the metallic bands. The surface energy can be calculated from the equation

_{2}BiBrO

_{6}in the supercell, and ${E}_{bulk}$ is the bulk total energy. Our calculated surface energy for Pb

_{2}BiBrO

_{6}is 0.56 J/m

^{2}. In Pb

_{2}BiBrO

_{6}, although the PbO layer gives no contribution to the surface conduction bands, the surface PbO layer is still useful because the surface PbO layer can act as a protective layer when building the actual devices. It can prevent the topological protected surface metallic bands damaged from the experimental defects, such as gas and atom adsorption, surface disorders, and dangling bonds.

## 4. Conclusions

_{2}BiBrO

_{6}and Pb

_{2}SeTeO

_{6}double perovskites by first-principles theoretical calculation. Our first-principles theoretical calculations show that Pb

_{2}BiBrO

_{6}and Pb

_{2}SeTeO

_{6}are Z

_{2}nontrivial and their bandgaps are high at 0.390 eV and 0.181 eV, respectively. The topology comes from the band contact and band inversion at $\mathsf{\Gamma}$ point in the absence of spin-orbit interactions. This results in a larger bandgap for this new class of topological insulators than conventional TI. We also use a large supercell to calculate the surface metallic bands and we confirm that the topology-protected surface metallic bands come from the BiBrO

_{4}or SeTeO

_{4}surface. It means that one can build a transport device using Pb

_{2}BiBrO

_{6}or Pb

_{2}SeTeO

_{6}double perovskites with a PbO layer as the outmost surface protection layer. The mechanical stabilities such as bulk, shear, Young’s moduli, Poisson’s and Pugh’s ratio, longitudinal, transverse, and average sound velocity, together with Debye and melting temperatures are also studied.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**(

**a**) Primitive and (

**b**) conventional crystal structure of the Pb

_{2}AA’O

_{6}(A = Sb and Bi; A’ = Br and I) and Pb

_{2}SeTeO

_{6}double perovskites.

**Figure 2.**Scalar relativistic, upper panel (

**a**–

**g**), and fully relativistic, bottom panel (

**h**–

**n**), electronic band structures of the Pb

_{2}AA’O

_{6}(A = Sb and Bi; A’ = Br and I) and Pb

_{2}SeTeO

_{6}double perovskites. The Fermi energy is set to be zero. The orange colors in (

**l**,

**n**) represents the bandgaps.

**Figure 3.**Scalar relativistic, (

**a**,

**b**), and fully relativistic, (

**c**,

**d**), electronic band structures of the Pb

_{2}BiBrO

_{6}and Pb

_{2}SeTeO

_{6}double perovskites projected onto the s-(red) and p-(blue) orbitals. Note that only BiBr (SeTe) s- and p-orbitals are projected. The Fermi energy is shifted to zero as a reference.

**Figure 4.**Scalar relativistic, (

**a**,

**b**), and fully relativistic, (

**c**,

**d**), atom and orbital decomposed densities of states of the Pb

_{2}BiBrO

_{6}and Pb

_{2}SeTeO

_{6}double perovskites. projected onto the s-(red) and p

_{x}-(blue), p

_{y}-(green) and p

_{z}-(orange) orbitals. Note that only BiBr(SeTe) s- and p-orbitals are projected. The Fermi energy is shifted to zero as a reference.

**Figure 5.**(

**a**) Band structures and (

**b**–

**g**) Wilson loop for the six time-reversal invariant momentum plane for the Pb

_{2}BiBrO

_{6}double perovskites.

**Figure 6.**Surface electronic band structure of the Pb

_{2}BiBrO

_{6}and Pb

_{2}SeTeO

_{6}supercell, (

**a**,

**e**) total band structures, (

**b**–

**d**) project band structures of the top PbO, BiBrO

_{4}and top2 BiBrO

_{4}layer, (

**f**–

**h**) project band structures of the top PbO, SeTeO

_{4}and top2 SeTeO

_{4}layer, respectively. Note that only the Pb, Bi/Se, and Br/Te atoms’ orbitals are projected. The Fermi energy is set to be zero as a reference.

**Table 1.**Structural parameters of the fully optimized Pb

_{2}AA’O

_{6}structures where Pb (x, y, z) = (0.75, 0.25, 0.5), or (0.25, 0.75, 0.5), A (x, y, z) = (0, 0, 0), A’ (x, y, z) = (0.5, 0.5, 0), O1 (x, y, z) = (O

_{2y}, O

_{2y}, O

_{1z}), ($-$O

_{2y}, $-$O

_{2y}, $-$O

_{1z}) and O2 (x, y, z) = (O

_{2x}, O

_{2x}, 0), ($-$O

_{2x}, $-$O

_{2x}, 0), ($-$O

_{2y}, O

_{2y}, 0), and (O

_{2y}, $-$O

_{2y}, 0). The calculated lattice constant a (in $\u212b$), c/a ratio, the volume of the primitive cell V

_{0}(${\u212b}^{3}/\mathrm{f}.\mathrm{u}.$ ) and bandgap (in eV). The bandgap in the brackets is the SOC results.

a = b $(\u212b)$ | c/a | V_{0}$({\u212b}^{3}/\mathbf{f}.\mathbf{u}.)$ | O_{1z} | O_{2x} | O_{2y} | Bandgap (eV) | |
---|---|---|---|---|---|---|---|

Pb_{2}AsBrO_{6} | 6.003 | 1.282 | 138.69 | 0.4371 | 0.2529 | 0.2185 | 0 (0) |

Pb_{2}AsIO_{6} | 5.905 | 1.413 | 145.50 | 0.4942 | 0.2473 | 0.2471 | 0 (0) |

Pb_{2}SbBrO_{6} | 6.151 | 1.286 | 149.61 | 0.4599 | 0.2586 | 0.2299 | 0 (0) |

Pb_{2}SbIO_{6} | 6.059 | 1.413 | 157.15 | 0.4981 | 0.2512 | 0.2510 | 0 (0) |

Pb_{2}BiBrO_{6} | 6.011 | 1.413 | 153.48 | 0.4715 | 0.2644 | 0.2642 | 0 (0.390) |

Pb_{2}BiIO_{6} | 6.165 | 1.413 | 165.63 | 0.4775 | 0.2612 | 0.2612 | 0 (0) |

Pb_{2}SeTeO_{6} | 5.894 | 1.413 | 144.70 | 0.4904 | 0.2451 | 0.2452 | 0 (0.181) |

**Table 2.**Calculated single crystal elastic modulus C

_{11}, C

_{12}, C

_{13}, C

_{44}, C

_{66}, bulk modulus B, shear modulus G, Young’s modulus E, Poisson’s ratio $\mathsf{\nu}$, universal anisotropy index ${A}^{U}$ and Pugh’s B/G ratio of the Pb2AA’O6 double perovskites.

Pb_{2}AA’O_{6} | |||||||
---|---|---|---|---|---|---|---|

AA’ = AsBr | AsI | SbBr | SbI | BiBr | BiI | SeTe | |

C_{11} (GPa) | 185.28 | 188.62 | 176.62 | 160.63 | 180.21 | 166.60 | 199.12 |

C_{12} (GPa) | 90.70 | 106.72 | 101.72 | 105.93 | 103.57 | 112.36 | 109.61 |

C_{13} (GPa) | 71.75 | 97.33 | 62.39 | 85.14 | 73.44 | 65.18 | 96.05 |

C_{33} (GPa) | 236.41 | 199.08 | 254.26 | 186.08 | 211.17 | 214.27 | 214.78 |

C_{44} (GPa) | 45.56 | 48.74 | 44.39 | 47.43 | 67.98 | 74.31 | 56.89 |

C_{66} (GPa) | 47.11 | 41.39 | 36.53 | 27.50 | 38.17 | 26.98 | 44.81 |

B_{R} (GPa) | 118.99 | 131.01 | 117.23 | 117.74 | 119.16 | 114.77 | 135.15 |

B_{V} (GPa) | 119.48 | 131.01 | 117.83 | 117.75 | 119.16 | 114.77 | 135.16 |

B (GPa) | 119.24 | 131.01 | 117.53 | 117.74 | 119.16 | 114.77 | 135.16 |

G_{R} (GPa) | 50.71 | 44.26 | 43.40 | 32.22 | 46.40 | 36.26 | 49.26 |

G_{V} (GPa) | 52.81 | 44.63 | 48.89 | 35.89 | 50.27 | 45.97 | 50.06 |

G (GPa) | 51.76 | 44.45 | 46.15 | 34.56 | 48.34 | 41.12 | 49.66 |

E (GPa) | 135.65 | 119.79 | 122.42 | 94.43 | 127.73 | 110.19 | 132.72 |

$\nu $ | 0.31 | 0.35 | 0.33 | 0.37 | 0.32 | 0.34 | 0.34 |

${A}^{U}$ | 0.21 | 0.04 | 0.64 | 0.40 | 0.42 | 1.34 | 0.08 |

B/G | 2.30 | 2.95 | 2.54 | 3.41 | 2.47 | 2.79 | 2.72 |

**Table 3.**Density $\rho \left(\mathrm{g}/{\mathrm{cm}}^{3}\right)$, longitudinal ${v}_{l}$, transverse ${v}_{t}$, average ${v}_{m}$, sound velocity (m/s), Debye temperature ${\Theta}_{D}\left(\mathrm{K}\right)$ and melting temperature ${T}_{m}\left(\mathrm{K}\right)$ of the Pb

_{2}AA’O6 double perovskites.

Pb_{2}AA’O_{6} | |||||||
---|---|---|---|---|---|---|---|

AA’ = AsBr | AsI | SbBr | SbI | BiBr | BiI | SeTe | |

$\rho \left(\mathrm{g}/\mathrm{c}{\mathrm{m}}^{3}\right)$ | 7.96 | 8.13 | 7.94 | 8.02 | 8.65 | 8.48 | 8.23 |

${v}_{l}\left(\mathrm{m}/\mathrm{s}\right)$ | 4861.57 | 4838.29 | 4759.84 | 4519.42 | 4607.82 | 4470.86 | 4947.25 |

${v}_{t}\left(\mathrm{m}/\mathrm{s}\right)$ | 2549.19 | 2338.39 | 2416.37 | 2075.70 | 2364.18 | 2201.39 | 2456.78 |

${v}_{m}\left(\mathrm{m}/\mathrm{s}\right)$ | 2851.17 | 2628.24 | 2708.24 | 2338.91 | 2647.99 | 2471.74 | 2757.15 |

${\Theta}_{D}\left(\mathrm{K}\right)$ | 353 | 321 | 327 | 278 | 317 | 289 | 337 |

${T}_{m}\left(\pm 300\mathrm{K}\right)$ | 1264 | 935 | 1265 | 1115 | 1211 | 1175 | 1273 |

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## Share and Cite

**MDPI and ACS Style**

Tung, J.-C.; Lee, C.-H.; Liu, P.-L.; Wang, Y.-K.
Electronic Band Structures of the Possible Topological Insulator Pb_{2}BiBrO_{6} and Pb_{2}SeTeO_{6} Double Perovskite: An Ab Initio Study. *Appl. Sci.* **2022**, *12*, 5913.
https://doi.org/10.3390/app12125913

**AMA Style**

Tung J-C, Lee C-H, Liu P-L, Wang Y-K.
Electronic Band Structures of the Possible Topological Insulator Pb_{2}BiBrO_{6} and Pb_{2}SeTeO_{6} Double Perovskite: An Ab Initio Study. *Applied Sciences*. 2022; 12(12):5913.
https://doi.org/10.3390/app12125913

**Chicago/Turabian Style**

Tung, Jen-Chuan, Chi-Hsuan Lee, Po-Liang Liu, and Yin-Kuo Wang.
2022. "Electronic Band Structures of the Possible Topological Insulator Pb_{2}BiBrO_{6} and Pb_{2}SeTeO_{6} Double Perovskite: An Ab Initio Study" *Applied Sciences* 12, no. 12: 5913.
https://doi.org/10.3390/app12125913