## 3 Results And Discussion  
### Structural and Electronic Band Structures  
Calculated equilibrium lattice constant \(a\) (A), bandgap (eV), and cohesive energy (eV) for the APoO\({}_{3}\) perovskite are listed in Table 1. The cohesive energy is the energy differences between bulk phase and its isolated atoms, i.e., \(\sum E_{atom}-E_{tot}\). Calculated cohesive energies were 16.73, 18.76, 22.43, 22.96, 23.70, and 23.03 eV for BePoO\({}_{3}\), MgPoO\({}_{3}\), CaPoO\({}_{3}\), SrPoO\({}_{3}\), BaPoO\({}_{3}\), and RaPoO\({}_{3}\), respectively. The equilibrium lattice constants increased monotonically with the atomic number. We also demonstrate the electronic band structures without and with the spin-orbit coupling effect in Figure 2. Without the spin-orbit coupling effect, we found that all APoO\({}_{3}\) perovskites were ordinary metals. The topology can not be verified merely from the band structure with the SOC effect. It is worth to see how SOC affects the APoO\({}_{3}\) perovskite. We found that CaPoO\({}_{3}\), SrPoO\({}_{3}\), BaPoO\({}_{3}\), and RaPoO\({}_{3}\) were possible topological insulators. There were clear Dirac corn-like bands at \(R\) points for the CaPoO\({}_{3}\), SrPoO\({}_{3}\), BaPoO\({}_{3}\), and RaPoO\({}_{3}\) perovskites, see Figure 2 upper panel. When the SOC effect was taken into account, BePoO\({}_{3}\) and MgPoO\({}_{3}\) were still metals, see Figure 2g,h. However, CaPoO\({}_{3}\), SrPoO\({}_{3}\), BaPoO\({}_{3}\), and RaPoO\({}_{3}\) were insulators, and the band was inverted at \(R\) point, indicating these perovskites are possible topological insulators. In previous studies of the YBiO\({}_{3}\) perovskite [18; 19; 41], the topological phase appeared at equilibrium. Halide perovskites such as CsPbI\({}_{3}\) and CsPbBr\({}_{3}\) can transform into TI under very high pressure. To investigate how the bandgap varies with lattice constants, plotted in Figure 3 is the bandgap as a function of its lattice constants, a\({}_{eq}\). The calculated bandgap \begin{table}
\begin{tabular}{l c c c} \hline \hline  & **Lattice Constant \(a\)** & **Bandgap** & **Cohesive Energy** \\  & **(Å)** & **(eV)** & **(eV)** \\ \hline BePoO\({}_{3}\) & 4.254 & 0 & 16.73 \\ MgPoO\({}_{3}\) & 4.323 & 0 & 18.76 \\ CaPoO\({}_{3}\) & 4.366 & 0.861 & 22.43 \\ SrPoO\({}_{3}\) & 4.398 & 0.871 & 22.96 \\ BaPoO\({}_{3}\) & 4.446 & 0.820 & 23.70 \\ RaPoO\({}_{3}\) & 4.471 & 0.810 & 23.03 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Calculated equilibrium lattice constant \(a\), fully relativistic bandgap and cohesive energy for APoO\({}_{3}\) perovskite. Figure 1: **(a)** Crystal structure of APoO\({}_{3}\) (A = Be, Mg, Ca, Sr, Ba, and Ra). **(b)** Supercell structure contains six formula units with vacuum for the surface band calculation. In a normal supercell, there are PoO\({}_{2}\) and AO surfaces on opposite sides. To investigate which surfaces give rise to the contributions on conduction bands, we prepared two kinds of supercells. One has PoO\({}_{2}\) surface on both sides, i.e., bottom AO layer is removed, and the other has AO surfaces on both sides, i.e., remove the top PoO\({}_{2}\) layer. was the largest at the equilibrium lattice constant for CaPoO\({}_{3}\) and SrPoO\({}_{3}\), and the bandgap decreases away from the equilibrium lattice constant. For BaPoO\({}_{3}\) and RaPoO\({}_{3}\), the largest bandgap occurred at smaller lattice constants. We found that the TI phase existed in a wide range of lattice constants, see Figure 4.  
To understand how the bandgap varies with lattice constant, we demonstrated the electronic band structure of BaPoO\({}_{3}\) as a function of lattice constant with and without the SOC effect in Figure 4. In the absence of SOC effect, BaPoO\({}_{3}\) was metal until the lattice constante increased to 4.864 (A). When the SOC effect was taken into account, see Figure 4h, the indirect bandgap came from the energy differences between R and M points.