## 3 Results And Discussion
### Surface Band Structures  
The surface conduction bands for the topological insulators are the most essential properties because these metallic surface states allow electrons to travel insensitive to scattering by impurities. In other words, these metallic surface states are useful for spintronic applications. To investigate the behavior of the surface conduction bands, we prepared a supercell containing 12 layers of APoO\({}_{3}\) perovskites (six formula units) with a vacuum of  
\begin{table}
\begin{tabular}{l c c c c c} \hline  & \(\rho\) (g/cm\({}^{3}\)) & \(v_{l}\) (m/s) & \(v_{l}\) (m/s) & \(v_{m}\) (m/s) & \(\Theta_{D}\) (K) \\ \hline BePoO\({}_{3}\) & 5.7545 & 5298.19 & 2586.58 & 2905.61 & 348 \\ MgPoO\({}_{3}\) & 5.7976 & 5485.07 & 2716.10 & 3048.67 & 359 \\ CaPoO\({}_{3}\) & 5.9446 & 5623.44 & 2827.93 & 3171.33 & 370 \\ SrPoO\({}_{3}\) & 6.7427 & 5319.33 & 2809.97 & 3141.26 & 364 \\ BaPoO\({}_{3}\) & 7.4724 & 5102.84 & 2804.69 & 3126.34 & 358 \\ RaPoO\({}_{3}\) & 9.0001 & 4677.82 & 2601.91 & 2897.61 & 330 \\ \hline \end{tabular}
\end{table}
Table 3: Calculated density \(\rho\) (g/cm\({}^{3}\)), the longitudinal, \(v_{l}\) (m/s), transverse, \(v_{t}\) (m/s), average sound velocity, \(v_{m}\) (m/s), and Debye temperature \(\Theta_{D}\) (K) from the elastic moduli.  
20 (A). Plotted in Figure 6 is the band structures of the SrPoO\({}_{3}\) slab. Figure 6a shows the supercell band structure calculations. In this calculation, the supercell contains six formula units of SrPoO\({}_{3}\) with SrO and PoO\({}_{2}\) surfaces on each side. There were clear conduction bands near \(M\) points, indicating that SrPoO\({}_{3}\) is a conductor. These conduction bands are topologically protected. It is interesting to see where these surface conduction bands come from. To verify, we prepared two kinds of supercells: one contains the SrO surface on both sides, denoted as SrO plane, and the other includes the PoO\({}_{2}\) surface on both sides, denoted as PoO\({}_{2}\) plane. The band structures are shown in Figure 6b. We found that both SrO and PoO\({}_{2}\) surfaces gave rise to the contributions of the conduction bands; the red-dashed bands come from the SrO surface while green solid bands come from the PoO\({}_{2}\) surface.  
We studied the electric polarization if the Po atom is moved away from the center in bulk form. However, it is interesting to see if the surface also being electrically polarized. In the surface calculations, the electric polarization has two kinds of freedom: one is the electric polarization produced by the displacement of the Po atom along the \(x\)-direction, in-plane, and the other is along the \(z\)-direction, out-of-plane. Plotted in Figure 6c is the calculated surface band structures with electric polarization that is in-plane. This result was obtained with the Po atom moving in the \(x\)-direction about 0.02 (A). The result was almost identical to the corresponding one in the normal supercell calculations. If the Po atom moves out-of-plane about 0.02 (A) which produces an electric polarization, see Figure 6d, the topologically protected surface conduction bands still exist.  
Figure 6: (**a**) Calculated fully relativistic band structure for the SrPoO\({}_{3}\) supercell. (**b**) Fully relativistic band structures with PoO\({}_{2}\) surfaces on both sides, green solid-lines, and SrO surfaces on both sides, red-dashed lines. (**c,d**) Fully relativistic band structures with Po atoms moving in-plane and out-of-plane, respectively.