## 3 Results and Discussion
### Mechanical Properties  
Listed in Table 2 is the calculated elastic constants and their related properties calculated from these elastic constants. In the tetragonal structure, there are six nonzero elastic constants \(C_{11}\), \(C_{12}\), \(C_{13}\), \(C_{33}\), \(C_{44}\), and \(C_{66}\). The elastic constant matrix can be written as:  
\[ \begin{bmatrix}C_{11}&C_{12}&C_{13}&0&0&0\\ C_{12}&C_{11}&C_{13}&0&0&0\\ C_{13}&C_{13}&C_{33}&0&0&0\\ 0&0&0&C_{44}&0&0\\ 0&0&0&0&C_{44}&0\\ 0&0&0&0&C_{66}\end{bmatrix}  \tag{1}\]  
Our calculated non-zero elastic constants \(C_{ij}\) are all positive. Therefore, the mechanical stabilities [41] criteria of the Pb\({}_{2}\)AA\({}^{\prime}\)O\({}_{6}\) double perovskites can be simplified as \(C_{11}>C_{12}\), \((C_{11}+C_{33}-2C_{13})>0\), and \(C_{33}(C_{11}+C_{12})>2C_{13}^{2}\). The Pb\({}_{2}\)AA\({}^{\prime}\)O\({}_{6}\) 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]:  
\[M=\frac{(M_{R}+M_{V})}{2},\ M=B,\ G \tag{2}\]  
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:  
\[B_{V}=\frac{2(C_{11}+C_{12})+C_{33}+4C_{13}}{9} \tag{3}\] \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & \multicolumn{8}{c}{**Pb\({}_{2}\)AA\({}^{\prime}\)O\({}_{6}\)**} \\ \hline  & **AA\({}^{\prime}\) = AsBr** & **AsI** & **SbBr** & **SbI** & **BiBr** & **BiI** & **SeTe** \\ \hline C\({}_{11}\) (GPa) & 185.28 & 188.62 & 176.62 & 160.63 & 180.21 & 166.60 & 199.12 \\ \hline C\({}_{12}\) (GPa) & 90.70 & 106.72 & 101.72 & 105.93 & 103.57 & 112.36 & 109.61 \\ \hline C\({}_{13}\) (GPa) & 71.75 & 97.33 & 62.39 & 85.14 & 73.44 & 65.18 & 96.05 \\ \hline C\({}_{33}\) (GPa) & 236.41 & 199.08 & 254.26 & 186.08 & 211.17 & 214.27 & 214.78 \\ \hline C\({}_{44}\) (GPa) & 45.56 & 48.74 & 44.39 & 47.43 & 67.98 & 74.31 & 56.89 \\ \hline C\({}_{66}\) (GPa) & 47.11 & 41.39 & 36.53 & 27.50 & 38.17 & 26.98 & 44.81 \\ \hline B\({}_{\rm R}\) (GPa) & 118.99 & 131.01 & 117.23 & 117.74 & 119.16 & 114.77 & 135.15 \\ \hline B\({}_{\rm V}\) (GPa) & 119.48 & 131.01 & 117.83 & 117.75 & 119.16 & 114.77 & 135.16 \\ \hline B (GPa) & 119.24 & 131.01 & 117.53 & 117.74 & 119.16 & 114.77 & 135.16 \\ \hline G\({}_{\rm R}\) (GPa) & 50.71 & 44.26 & 43.40 & 32.22 & 46.40 & 36.26 & 49.26 \\ \hline G\({}_{\rm V}\) (GPa) & 52.81 & 44.63 & 48.89 & 35.89 & 50.27 & 45.97 & 50.06 \\ \hline G (GPa) & 51.76 & 44.45 & 46.15 & 34.56 & 48.34 & 41.12 & 49.66 \\ \hline E (GPa) & 135.65 & 119.79 & 122.42 & 94.43 & 127.73 & 110.19 & 132.72 \\ \hline \(\
u\) & 0.31 & 0.35 & 0.33 & 0.37 & 0.32 & 0.34 & 0.34 \\ \hline \(A^{U}\) & 0.21 & 0.04 & 0.64 & 0.40 & 0.42 & 1.34 & 0.08 \\ \hline \(B/G\) & 2.30 & 2.95 & 2.54 & 3.41 & 2.47 & 2.79 & 2.72 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Calculated single crystal elastic modulus \(C_{11}\), \(C_{12}\), \(C_{13}\), \(C_{44}\), \(C_{66}\), bulk modulus \(B\), shear modulus \(G\), [PERSON]’s modulus \(E\), [PERSON]’s ratio \(\
u\), universal anisotropy index \(A^{U}\) and Pugh’s \(B/G\) ratio of the Pb2 AA’O6 double perovskites. \[B_{R}=\frac{(C_{11}+C_{12})C_{33}-2C_{13}^{2}}{C_{11}+C_{12}+2C_{33}-4C_{13}} \tag{4}\]  
\[G_{V}=\frac{2C_{11}-C_{12}+C_{33}-2C_{13}+6C_{44}+3C_{66}}{15} \tag{5}\]  
\[\frac{15}{G_{R}}=\frac{18B_{V}}{(C_{11}+C_{12})C_{33}-C_{13}^{2}}+\frac{6}{C_{ 11}-C_{12}}+\frac{6}{C_{44}}+\frac{3}{C_{66}} \tag{6}\]  
In terms of the Voigt-Reuss-Hill approximations, the [PERSON]'s modulus \(E\) and [PERSON]'s ratio \(\
u\) can be calculated from the formulas:  
\[E=\frac{9 BG}{3B+G} \tag{7}\]  
\[\
u=\frac{3B-2G}{6B+2G} \tag{8}\]  
It is well known that [PERSON] introduces an anisotropy factor of a crystal as \(A=2C_{44}/(C_{11}-C_{12})\).