## 3 Results and Discussions
Figure 5: Electronic band structures with density of states along the k-point path of M\(-\)\(-\)\(-\)K and L\(-\)A\(-\)H in hexagonal (Bi\({}_{2}\))\({}_{\text{m}}\)(Bi\({}_{2}\)Te\({}_{3}\))\({}_{\text{n}}\): (**a**) Bi\({}_{2}\)Te\({}_{3}\), (**b**) Bi\({}_{4}\)Te\({}_{5}\), (**c**) Bi\({}_{6}\)Te\({}_{7}\), (**d**) Bi\({}_{6}\)Te\({}_{9}\), (**e**) BiTe, (**f**) Bi\({}_{4}\)Te\({}_{3}\), (**g**) Bi\({}_{2}\)Te, (**h**) Bi\({}_{7}\)Te\({}_{3}\), and (**i**) Bi\({}_{2}\). All band structures were obtained by using DFT calculations with LDA. Figure 6: Electronic band structures with density of states along the k-point path of M\(-\Gamma-\)K and L\(-\)A\(-\)H in hexagonal (Bi\({}_{2}\))\({}_{\text{m}}\)(Bi\({}_{2}\)Te\({}_{3}\))\({}_{\text{n}}\): (**a**) Bi\({}_{2}\)Te\({}_{3}\), (**b**) Bi\({}_{4}\)Te\({}_{5}\), (**c**) Bi\({}_{6}\)Te\({}_{7}\), (**d**) Bi\({}_{8}\)Te\({}_{9}\), (**e**) BiTe, (**f**) Bi\({}_{4}\)Te\({}_{3}\), (**g**) Bi\({}_{2}\)Te, (**h**) Bi\({}_{7}\)Te\({}_{3}\), and (**i**) Bi\({}_{2}\). All band structures were obtained by using DFT calculations with LDA including SOI. \begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline \(\mathbf{E_{\mathrm{xc}}}\) & **Phase** & **SOI** & \(\mathbf{E_{\mathrm{VBM}}}\) & \(\mathbf{k_{\mathrm{VBM}}}\) & \(\mathbf{E_{\mathrm{CBM}}}\) & \(\mathbf{k_{\mathrm{CBM}}}\) & \(\mathbf{E_{\mathrm{gap}}^{\mathrm{I}}}\) & \(\mathbf{E_{\mathrm{gap}}^{\mathrm{d}}}\) & \(\mathbf{k_{\mathrm{gap}}^{\mathrm{d}}}\) & **Type** \\ \hline \multirow{4}{*}{B\({}_{\mathrm{2}}\)Te\({}_{3}\)} & X & \(-0.098\) & \(0.000\) & \(0.000\) & \(0.000\) & \(0.098\) & \(0.000\) & \(0.000\) & \(0.196\) & \(0.000\) & \(0.000\) & \(0.000\) & SC \\  & O & \(-0.055\) & \(0.140\) & \(0.000\) & \(0.500\) & \(0.056\) & \(0.130\) & \(0.000\) & \(0.500\) & \(0.111\) & \(0.115\) & \(0.130\) & \(0.000 PBE calculations give the electronic band structures shown in Figure 3. The band structure of Bi\({}_{2}\)Te\({}_{3}\) is shown in Figure 3a. There is a VBM of \(-\)0.136 eV at \(\Gamma\) and a CBM of 0.136 at \(\Gamma\), indicating that Bi\({}_{2}\)Te\({}_{3}\) is a semiconductor as previously reported [31; 32; 33]. Note that the experimental energy gap is about 0.13 eV. Figure 3d shows that Bi\({}_{2}\)Te\({}_{9}\) has a VBM of \(-\)0.004 eV on A\(-\)H (0.033, 0.033, 0.5), CBM of \(-\)0.03 eV on A\(-\)H (0.007, 0.007, 0.5), and F\({}_{\rm gap}^{\rm d}\) of 0.027 eV on A\(-\)L (0.02, 0, 0.5). The band structure of BiTe in Figure 3e shows that VBM is 0.017 eV on \(\Gamma\)\(-\)M (0.06, 0, 0), CBM is \(-\)0.023 eV on A\(-\)H (0.013, 0.013, 0.5), and F\({}_{\rm gap}^{\rm d}\) is 0.035 eV on \(\Gamma\)\(-\)M (0.06, 0, 0). The band structure of Bi\({}_{2}\) is shown in Figure 3i, where VBM is 0.066 eV on A\(-\)H (0.047, 0.047, 0.5), CBM is \(-\)0.197 eV on A\(-\)L (0.48, 0, 0.5), and F\({}_{\rm gap}^{\rm d}\) is 0.02 eV on A\(-\)H (0.047, 0.047, 0.5). Note that our result for Bi\({}_{2}\) is consistent with the experimental reports that bulk Bi is semimetallic [34] since the supercell of Bi\({}_{2}\) is to be bulk Bi. Additionally, the other phases (Bi\({}_{4}\)Te\({}_{5}\), Bi\({}_{4}\)Te\({}_{7}\), Bi\({}_{4}\)Te\({}_{3}\), Bi\({}_{2}\)Te\({}_{3}\)) were semimetallic, as shown in Figure 3 and Table 2. Note that PBE showed that the electric transport types of (Bi\({}_{2}\))\({}_{\rm m}\)(Bi\({}_{2}\)Te\({}_{3}\))\({}_{\rm n}\) were semimetallic because of positive F\({}_{\rm gap}^{\rm d}\) and negative F\({}_{\rm gap}^{\rm i}\) except for Bi\({}_{2}\)Te\({}_{3}\). In addition, the electric transport types of (Bi\({}_{2}\))\({}_{\rm m}\)(Bi\({}_{2}\)Te\({}_{3}\))\({}_{\rm n}\) from LDA are the same as those from PBE, although the detailed values (E\({}_{\rm VBM}\), kV\({}_{\rm VBM}\), E\({}_{\rm CBM}\), k\({}_{\rm CBM}\), E\({}_{\rm gap}^{\rm i}\), k\({}_{\rm gap}^{\rm i}\)) of LDA were different from those of PBE (see Figures 3-5 and Tables 2 and 3). Meanwhile, when considering the SOI in the PBE calculations, there are some changes in the band structures (see Figure 4 and Table 2).