## 3 Results and Discussion
### Band Structures
Figure 3: CsPbI\({}_{3}\) band structures corresponding to (**a**) the equilibrium lattice constant, (**b**) the \(a/a_{\text{theo}}\) ratio of \(-3.5\%\) with spin–orbit coupling (SOC) and (**c**) the \(a/a_{\text{theo}}\) ratio of \(-8\%\) without SOC; (**d**) the equilibrium lattice constant, (**e**) the \(a/a_{\text{theo}}\) ratio of \(-3.5\%\) with SOC and (**f**) the \(a/a_{\text{theo}}\) ratio of \(-8\%\) without SOC, respectively. The dashed line is the Fermi level. The green line represents Pb\(p\)- and Is- orbitals. The red line depicts Pbs- and Ip- orbitals. A detailed comparison of the calculated band gaps of CsPbBrI\({}_{2}\) perovskites, which are virtually identical to doping iodine atoms in CsPbBr\({}_{3}\), is also warranted. The band structures of CsPbBrI\({}_{2}\) perovskites under different lattice constants with and without SOC are shown in Figure 4. The calculated band gaps of CsPbBrI\({}_{2}\) were also similar in behavior to those of CsPbI\({}_{3}\), which show that the calculated band gaps of CsPbBrI\({}_{2}\) are direct without SOC and decrease monotonically as external pressure increases, as illustrated in Figure 4a-c. Figure 4c clearly indicates that Pb\(p\)- and Brs-orbitals are degenerate or close together in energy (0.2 eV), whereas Pb\(p\)- and Is-orbitals are degenerate in energy, which was 0.6 eV above the Fermi level under the \(a/a_{\text{theo}}\) ratio of \(-8\%\) at the R point. When SOC was considered, the calculated band gap of CsPbBrI\({}_{2}\) at the equilibrium lattice constant was approximately one-third of the corresponding one without SOC, as shown in Figure 4a,d. Figure 4e shows the calculated band gap of CsPbBrI\({}_{2}\) approaching zero under the \(a/a_{\text{theo}}\) ratio of \(-3.8\%\). When SOC was considered, band reversal still occurred at the R point, as shown in Figure 4f. Notably, the energy of the highest occupied band at the M point became higher than the energy band at the R point at the \(a/a_{\text{theo}}\) ratio of \(-8\%\), showing that the transition may not be useful for optical applications. A useful and practical topological number of Z\({}_{2}\) in terms of the Green function and corresponding to the charge density derived from VASP and the Wannier90 package [37] was used to determine the characteristics of TIs. As shown in Figure 5, the Wannier  
Figure 4: CsPbBrI\({}_{2}\) band structures corresponding to (**a**) equilibrium lattice constant, as well as (**b**) the ratio \(a/a_{\text{theo}}\) of \(-3.8\%\) with and (**c**) \(-8\%\) without the spin–orbit coupling; (**d**) equilibrium lattice constant, (**e**) the ratio \(a/a_{\text{theo}}\) of \(-3.8\%\) with and (**f**) \(-8\%\) without the spin–orbit coupling. The dashed line is the Fermi level. functions of the CsPbI3 perovskite were applied in the range of \(-6\) to 3 eV, which closely matched the band structure of the CsPbI3 perovskite derived from VASP. Here, we followed the work by [PERSON] et al. [10] and calculated the \(Z_{2}\) invariants from six time-reversal invariant planes in the Brillouin zone [38], which have four independent \(Z_{2}\) integers, to obtain the four indices (\(\
u 0\); \(\
u 1\), \(\
u 2\), \(\
u 3\)). Note that \(\
u 0\) is called the strong insulator index and \(\
u 1\), \(\
u 2\), and \(\
u 3\) are called weak TI indices. In addition, \(Z_{2}=1\) and \(Z_{2}=0\) correspond to topological and trivial insulators, respectively. Our calculations indicated that all indices of CsPbBrxI3-x (x = 0, 1, 2, and 3) with hydrostatic strain were (1,0,0,0), demonstrating that CsPbBrxI3-x are strong TIs.