Electronic Band Structures of the Possible Topological Insulator Pb₂BiBrO₆ and Pb₂SeTeO₆ Double Perovskite: An Ab Initio Study

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

Abstract

Using the frameworks of density functional theory, we found a new class of three-dimensional (3D) topological insulators (TIs) in Pb₂BiBrO₆ and Pb₂SeTeO₆ double perovskites. Our ab initio theoretical calculations show that Pb₂BiBrO₆ and Pb₂SeTeO₆ are Z₂ 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 Pb₂BiBrO₆ double perovskites, our slab calculations confirm that the topology-protected surface metallic bands come from the BiBrO₄ surface which means that one can build a transport device using Pb₂BiBrO₆ 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 Pb₂AA’O₆ (A = Sb and Bi; A’ = Br and I) and Pb₂SeTeO₆ are mechanically stable.

1. Introduction

The discovery of topological insulators [1,2] stems from the studies on the quantum Hall effect [3,4] and the quantum spin Hall effects [5,6,7,8,9] over the past few decades, thus becoming a successful area of research in condensed matter physics. Topological insulators are insulators that have a specific bandgap in bulk form. Nonetheless, unlike ordinary insulators, TIs have metallic Dirac states on the surface [10,11,12,13,14,15]. The coexistence of the bulk insulating and surface conducting states leads to TIs with unusual electronic characteristics. For example, TIs are promising materials for spin-dependent applications [2]. So far, several topological materials, such as topological crystalline insulators [16], topological insulators, Dirac [17], and Weyl [18,19] semimetals have received particular attention and thus have attracted intensive research [1]. It is found that the metallic surface state protected by the time-reversal invariants can always be found in three-dimensional TIs. Furthermore, the metallic surface states of TIs allow only for a specific spin-momentum state, i.e., spin-momentum-locking transport [2]. Therefore, compared with the electron transport of ordinary metals, TIs have great potential for spintronic applications.

Double perovskites A₂BB’O₆ made of two single perovskites ABO₃ and AB’O₃ are potential materials for many applications and attract both experimental synthesis and theoretical predictions in the past few decades. For example, Sr₂FeMoO₆ [20] and Sr₂FeOsO₆ [21] double perovskites are candidates for spintronic devices for their half metallicity at room temperature. Moreover, Sr₂CrOsO₆ [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, Pb₂CoTeO₆ [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.

Experimentally, Bi₂Se₃ [13,25], Bi_1−xSb_x [26] are shown to be topological insulators. Furthermore, it is theoretically predicted that CsPbI₃ [27,28] and A₂BiXO₆ (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 CsPbI₃ 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 A₂BiXO₆ (A = Ca, Sr, Ba; X = I, Br) [30] double perovskites structures. Most recently, Lee et al. also predicts A₂TePoO₆ (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 Z₂ was introduced by Fu and Kane [32], which is calculated by analyzing the bulk band structure.

In this study, we investigated the electronic band structure of Pb₂AA’O₆ (A = Sb and Bi; A’ = Br and I) and Pb₂SeTeO₆ 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 Pb₂BiBrO₆ and Pb₂SeTeO₆ 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 Pb₂BiBrO₆ double perovskites as an example. Finally, we draw conclusions in Section 4.

2. Structures and Computational Methods

The double perovskite Pb₂AA’O₆ is a body-centered tetragonal with a space group of I4/mmm. Plotted in Figure 1 is the primitive unit cell of the Pb₂AA’O₆ double perovskite. Listed in

Table 1. Structural parameters of the fully optimized Pb₂AA’O₆ 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 $Å$), c/a ratio, the volume of the primitive cell V₀ (${Å}^3/\mathrm{f}.\mathrm{u}.$ ) and bandgap (in eV). The bandgap in the brackets is the SOC results.

	a = b $(Å)$	c/a	V0 $({Å}^3/\mathbf{f}.\mathbf{u}.)$	O1z	O2x	O2y	Bandgap (eV)
Pb2AsBrO6	6.003	1.282	138.69	0.4371	0.2529	0.2185	0 (0)
Pb2AsIO6	5.905	1.413	145.50	0.4942	0.2473	0.2471	0 (0)
Pb2SbBrO6	6.151	1.286	149.61	0.4599	0.2586	0.2299	0 (0)
Pb2SbIO6	6.059	1.413	157.15	0.4981	0.2512	0.2510	0 (0)
Pb2BiBrO6	6.011	1.413	153.48	0.4715	0.2644	0.2642	0 (0.390)
Pb2BiIO6	6.165	1.413	165.63	0.4775	0.2612	0.2612	0 (0)
Pb2SeTeO6	5.894	1.413	144.70	0.4904	0.2451	0.2452	0 (0.181)

are the lattice constants a, b, c, unit cell volume, atomic coordination O_1z, O_2x, and O_2y. In the Pb₂AA’O₆ 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.

In order to find an equilibrium crystal structure, we performed a structural optimization using the conventional cell where all atoms were free to move until the force acting on each atom was smaller than 0.002 eV/$Å$. The structure optimization was done without considering SOC. After structural optimization, the space group of all considered double perovskites are still I4/mmm. All calculations use the plane-wave basis set as implemented in the Vienna ab-initio simulation package (VASP) [33,34]. The kinetic energy cutoff was chosen to be 520 eV and a $12\times 12\times 12$ Monkhorst-Pack k mesh sampling was used for the electronic density of state integration with the $\mathrm{\Gamma }$ point is included. To include the electronic exchange-correlation functions, a generalized gradient approximation (GGA) with the Perdew-Burke-Ernzerhof (PBE) correction was used [35,36]. The self-consistent procedures is complete when the total energy differences is less than 10⁻⁶ eV. To calculate the surface band structure, we prepared a large supercell containing 12 layers of Pb₂AA’O₆ double perovskites, i.e., 6 formula units, with a vacuum of 20$Å$ at least. In this supercell, one PbO and one AA’O₄ surface are in contact with the vacuum. We also prepare another two supercells that contain the same surface, PbO or AA’O₄ 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

The calculated equilibrium lattice constants, a, b, c/a ratio, unit cell volume V₀, 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 Pb₂AsIO₆, Pb₂SbIO₆, Pb₂BiBrO₆, Pb₂BiIO₆, and Pb₂SeTeO₆ is 1.413 which is very close to the ideal value of $\sqrt{2}$. The c/a ratio of Pb₂AsBrO₆ and Pb₂SbBrO₆ were 1.282 and 1.286, respectively. We also found that Pb₂AsIO₆ has the smallest equilibrium volume whilst Pb₂BiIO₆ 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 Pb₂BiBrO₆ and Pb₂SeTeO₆ double perovskites with SOC. The bandgap for the Pb₂BiBrO₆ and Pb₂SeTeO₆ double perovskites is indirect and the magnitude is 0.390 eV and 0.181 eV.

Displayed in Figure 2 is the electronic band structures of the Pb₂AA’O₆ double perovskites without, upper panel, and with, bottom panel, spin-orbit coupling. Let us take Pb₂BiBrO₆, see Figure 2e, as an example. Without SOC, the band structures of Pb₂BiBrO₆ 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 Pb₂BiBrO₆ and Pb₂SeTeO₆ are possible topological insulators. The SOC-induced bandgap for the Pb₂BiBrO₆ is 0.390 eV, being higher than most 3D TIs. We also found that the Z₂ index [32] (${\nu }_0$; ${\nu }_1$ ${\nu }_2$ ${\nu }_3$) of Pb₂BiBrO₆and Pb₂SeTeO₆ are (1; 0 0 0) showing that they are strong topological insulators.

We also demonstrate the atom and orbital decomposed density of states in Figure 4. We focus on the Bi, Br, Se, and Te atoms’ s-, p_x-, p_y- and p_z-orbitals. In Pb₂BiBrO₆ 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.

To further confirm that Pb₂BiBrO₆ and Pb₂SeTeO₆ are possible topological insulators, we present in detail our calculations for Z₂ 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 Z₂ invariants. The results are shown in Figure 5b–g. Figure 5b–g are wannier charge center as a function of momentum k for the k₁ = 0.0, k₁ = 0.5, k₂ = 0.0, k₂ = 0.5, k₃ = 0.0, and k₃ = 0.5 momentum plane, respectively. In practice, the Z₂ 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 Z₂ 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

Listed in

Table 2. Calculated single crystal elastic modulus C₁₁, C₁₂, C₁₃, C₄₄, C₆₆, 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.

	Pb2AA’O6
	AA’ = AsBr	AsI	SbBr	SbI	BiBr	BiI	SeTe
C11 (GPa)	185.28	188.62	176.62	160.63	180.21	166.60	199.12
C12 (GPa)	90.70	106.72	101.72	105.93	103.57	112.36	109.61
C13 (GPa)	71.75	97.33	62.39	85.14	73.44	65.18	96.05
C33 (GPa)	236.41	199.08	254.26	186.08	211.17	214.27	214.78
C44 (GPa)	45.56	48.74	44.39	47.43	67.98	74.31	56.89
C66 (GPa)	47.11	41.39	36.53	27.50	38.17	26.98	44.81
BR (GPa)	118.99	131.01	117.23	117.74	119.16	114.77	135.15
BV (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
GR (GPa)	50.71	44.26	43.40	32.22	46.40	36.26	49.26
GV (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

is the calculated elastic constants and their related properties calculated from these elastic constants. In the tetragonal structure, there are six nonzero elastic constants C₁₁, C₁₂, C₁₃, C₃₃, C₄₄, and C_66. The elastic constant matrix can be written as:

$$\left[\begin{array}{cccccc}{C}_{11}& C_{12}& C_{13}& 0& 0& 0\\ C_{12}& C_{11}& C_{13}& 0& 0& 0\\ C_{13}& C_{13}& C_{33}& 0& 0& 0\\ 0& 0& 0& C_{44}& 0& 0\\ 0& 0& 0& 0& C_{44}& 0\\ 0& 0& 0& 0& 0& C_{66}\end{array}\right]$$

(1)

Our calculated non-zero elastic constants $C_{ij}$ are all positive. Therefore, the mechanical stabilities [41] criteria of the Pb₂AA’O₆ double perovskites can be simplified as $C_{11}>C_{12}$, $\left(C_{11}+C_{33}-2C_{13}\right)>0$, and $C_{33}\left(C_{11}+C_{12}\right)>2C_{13}^2$. The Pb₂AA’O₆ double perovskites are mechanically stable if all criteria are met.

We further calculate the bulk modulus B and shear modulus G from Voigt-Reuss-Hill approximation [42]:

$$M=\frac{\left(M_R+M_V\right)}{2}, M=B, G$$

(2)

where B_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:

$$B_V=\frac{2\left(C_{11}+C_{12}\right)+C_{33}+4C_{13}}{9}$$

(3)

$$B_R=\frac{\left(C_{11}+C_{12}\right)C_{33}-2C_{13}^2}{C_{11}+C_{12}+2C_{33}-4C_{13}}$$

(4)

$$G_V=\frac{2C_{11}-C_{12}+C_{33}-2C_{13}+6C_{44}+3C_{66}}{15}$$

(5)

$$\frac{15}{G_R}=\frac{18B_V}{\left(C_{11}+C_{12}\right)C_{33}-C_{13}^2}+\frac{6}{C_{11}-C_{12}}+\frac{6}{C_{44}}+\frac{3}{C_{66}}$$

(6)

In terms of the Voigt-Reuss-Hill approximations, the Young’s modulus E and Poisson’s ratio $\nu$ can be calculated from the formulas:

$$E=\frac{9BG}{3B+G}$$

(7)

$$\nu =\frac{3B-2G}{6B+2G}$$

(8)

It is well known that Zener introduces an anisotropy factor of a crystal as $A=2C_{44}/\left(C_{11}-C_{12}\right)$. In Zener’s definition, the anisotropy factor is determined from the $C_{44}$ and $\left(C_{11}-C_{12}\right)$. The elastic constant $C_{44}$ represents the deformation resistance of a crystal against the shear stress applied across the (100) plane in the [010] direction. Therefore, Zener’s definition is limited to cubic system. For a locally isotropy crystal, A = 1. As suggested by Ranganathan et al. [43], we calculate the universal anisotropy index $A^U$ by:

$$A^U=5\frac{G_V}{G_R}+\frac{B_V}{B_R}-6$$

(9)

Note that $A^U=0$ is for a locally isotropy single crystal.

The B/G ratio, also known as the Pugh’s ratio [44], describes the ductility or brittleness of the crystal. In the 1950s, Pugh analyzed his experimental data and found that a higher Pugh’s ratio indicated the ductility of the material. A cutoff value of 1.75 is generally recommended for the Pugh’s ratio.

The Debye temperature is the key quantity in the quasiharmonic Debye model. It leads to an estimate of many physical properties, such as specific heat, elastic constants, thermal conductivity, and melting temperature. It also sets the scale for electron-phonon interaction in superconductors. We calculate the Debye temperature using the formula

$${\Theta }_D=\frac{h}{k_B}\sqrt[3]{\frac{3n{N}_A\rho }{4\pi M}}{v}_m$$

(10)

where $\rho$ is the density, M the molecular mass, $N_A$ Avogardo constant, and $v_m$ the average sound velocity. The average sound velocity can be calculated from the Navier’s equation

$$v_l=\sqrt{\frac{3B+4G}{3\rho }}, v_t=\sqrt{\frac{G}{\rho }}$$

(11)

and

$$v_m=\frac{1}{\sqrt[3]{\frac{1}{3}\left(\frac{1}{v_l^3}+\frac{2}{v_t^3}\right)}}$$

(12)

We also calculate the melting temperature $T_m$, using an empirical expression $T_m\left(\mathrm{K}\right)=354+4.50(2C_{11}+C_{33})/3\pm 300 \mathrm{K}$. The calculated melting and Debye temperature, along with density and sound velocities $v_l$, $v_t$, and $v_m$ are listed in

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 Pb₂AA’O6 double perovskites.

	Pb2AA’O6
	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

. We also calculated the phonon dispersion of the Pb₂BiBrO₆ double perovskite. The results show that Pb₂BiBrO₆ double perovskite is unstable.

3.3. Band Structure of the Supercell

To further confirm that Pb₂BiBrO₆ and Pb₂SeTeO₆ are topological insulators, Figure 6 shows the electronic band structures of the Pb₂BiBrO₆ and Pb₂SeTeO₆ supercell. We constructed a supercell containing 12 layers of Pb₂BiBrO₆ or Pb₂SeTeO₆ (6 formula units) with a vacuum of 20$Å$. The calculations were done without structure relaxation, including surface atomic reconstruction. There are two kinds of surface layers in our Pb₂BiBrO₆ or Pb₂SeTeO₆ supercell, one is the PbO layer and the other is the BiBrO₄ (SeTeO₄) layer. We denote these two surface layers as top PbO and top BiBrO₄ (SeTeO₄). We also denote the second nearest BiBrO₄ (SeTeO₄) layer to the surface as top2 BiBrO₄ and top2 SeTeO₄. The calculated electronic band structure of the Pb₂BiBrO₆ and Pb₂SeTeO₆ supercell are shown in Figure 6a,e. Clearly, we observe metallic bands between $\mathsf{\Gamma }$-X and M-$\mathsf{\Gamma }$. It theoretically confirms that Pb₂BiBrO₆ and Pb₂SeTeO₆ 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 Pb₂BiBrO₆. We found that the surface BiBrO₄ layer, Figure 6c, and nearest surface BiBrO4 layers, Figure 6d, give contributions to the metallic surface bands. Unlike Pb₂BiBrO_6, both PbO and SeTeO₄ surface give contribution to the metallic bands. The surface energy can be calculated from the equation

$$\gamma =\frac{E_{slab}-N{E}_{bulk}}{2A}$$

(13)

where $E_{slab}$ is the total energy of the supercell structure, N is the number of bulk Pb₂BiBrO₆ in the supercell, and $E_{bulk}$ is the bulk total energy. Our calculated surface energy for Pb₂BiBrO₆ is 0.56 J/m². In Pb₂BiBrO₆, 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

We found a new class of three-dimensional (3D) topological insulators (TIs) in Pb₂BiBrO₆ and Pb₂SeTeO₆ double perovskites by first-principles theoretical calculation. Our first-principles theoretical calculations show that Pb₂BiBrO₆ and Pb₂SeTeO₆ are Z₂ 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 BiBrO₄ or SeTeO₄ surface. It means that one can build a transport device using Pb₂BiBrO₆ or Pb₂SeTeO₆ 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.