## 3 Results and Discussion
### Elastic Properties  
For a mechanically stable triangular structure, these elastic constants in Table 3 must satisfy the Born-Huang stability criteria given by [38]:  
\[C_{11}-|C_{12}|>0,(C_{11}+C_{12})C_{33}>2C_{13}^{2},(C_{11}-C_{12})C_{44}>2C_{ 14}^{2} \tag{8}\]  
By substituting the elastic constants in Table 3 into the above criteria conditions, it can be seen that the calculated results meet the stability conditions under the hydrostatic pressures of 0 GPa and \(-0.5\) GPa, which indicates that the crystal structure of CsGeI\({}_{3}\) is stable. The stability tolerance factor of ABX\({}_{3}\) can be obtained by Goldschmidt criterion [39]:  
\[T=\frac{R_{A}+R_{X}}{\sqrt{2}(R_{B}+R_{X})} \tag{9}\]  
For organic-inorganic ABX\({}_{3}\) halides, [PERSON] et al. found that the stability conditions should meet \(0.81<\) T \(<\) 1.11 [40]. According to the calculated results in Table 4, the crystal structure of CsGeI\({}_{3}\) is stable, which is consistent with the calculated results above. Figure 11 shows phonon spectra of CsGeI\({}_{3}\) under both pressure conditions. It can be seen from the figure that there is no virtual frequency, which further indicates that their structure is stable. \begin{table}
\begin{tabular}{c c c c c} \hline  & **Cs\({}^{+}\)** & **Ge\({}^{2+}\)** & **I\({}^{-}\)** & **T** \\ \hline R (nm) & 0.167 & 0.073 & 0.22 & 0.93 \\ \hline \end{tabular}
\end{table} 
Table 4: The ionic radium of CsGeI\({}_{3}\). Figure 11: Phonon spectrum of CsGeI\({}_{3}\) under the pressure of **(a)** 0 GPa and **(b)** \(-0.5\) GPa. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline  & **C\({}_{11}\)** & **C\({}_{12}\)** & **C\({}_{13}\)** & **C\({}_{14}\)** & **C\({}_{33}\)** & **C\({}_{44}\)** & **B** & **G** & **A** \\ \hline
0 GPa & 60.07 & 48.61 & 32.61 & 22.32 & 74.74 & 23.03 & 46.95 & 10.54 & 4.02 \\ \(-0.5\) GPa & 48.52 & 38.88 & 25.83 & 17.21 & 61.03 & 13.64 & 37.68 & 15.11 & 2.83 \\ \hline \end{tabular}
\end{table} 
Table 3: Calculated elastic constant, Bulk modulus (B), Shear modulus (G), and elastic anisotropy (A) of CsGeI\({}_{3}\) under the pressure of 0 GPa and \(-0.5\) GPa. The Bulk modulus is a measure to determine the strength or hardness of a crystal and it represents the brittleness resistance of a crystal, so the greater the Bulk modulus, the greater the strength of the crystal. As can be seen from Table 3, the Bulk modulus of perovskite CsGeI\({}_{3}\) is 46.95 GPa when there is no hydrostatic pressure, and 37.68 GPa when the hydrostatic pressure is \(-\)0.5 GPa. To understand the strength of crystals, this value can be compared with the Bulk modulus of glass (35-50 GPa) and diamond (443 GPa), indicating that the perovskite CsGeI\({}_{3}\) is a soft material under both pressure conditions. We found that the Bulk modulus decreased after applied the hydrostatic pressure of \(-\)0.5 GPa. This is due to the application of tension, resulting in a decrease of the density and hardness of the material. Shear modulus represents the ability of a material to resist shear strain. It can be seen from Table 3 that the shear modulus increases after the tension is applied, indicating that its transverse resistance to strain is enhanced. Based on the elasticity coefficient, [PERSON] proposed a standard to judge the ductility or brittleness of materials [41]: when B/G \(>\) 1.75, the material is ductile, otherwise it is brittle. It can be seen from Table 3 that the corresponding ratios of 0 GPa and \(-\)0.5 GPa are 4.45 and 2.49, both of which are greater than 1.75. As such, this compound has the properties of softness and ductility, and can be used for automobiles and curved surfaces of glasses. In order to judge material anisotropy, [PERSON] and [PERSON] introduced the criteria for judging material anisotropy [42]:  
\[A=\frac{2C_{44}}{C_{11}-C_{12}} \tag{10}\]  
\(A\) is the elastic anisotropy coefficient. If \(A\) = 1, the material is isotropic, otherwise the material is anisotropic.