## 3 Results and Discussions
Compared to the convex hull in LDA + SOI, E\({}_{\rm Mixing}\) differs by 0.0019 eV for Bi\({}_{4}\)Te\({}_{5}\), 0.0021 eV for Bi\({}_{6}\)Te\({}_{7}\), 0.0027 eV for Bi\({}_{8}\)Te\({}_{9}\), 0.0006 eV for BiTe, and 0.0067 eV for Bi\({}_{2}\)Te. In LDA, the difference between the convex hull and E\({}_{\rm Mixing}\) was, on average, about two times larger than that in LDA + SOI except for Bi\({}_{8}\)Te\({}_{9}\) and Bi\({}_{4}\)Te\({}_{3}\); in particular, that of BiTe was \(\sim\)3.8 times larger. The differences were less than 0.0035 eV, except for those of Bi\({}_{2}\)Te in LDA (0.0098 eV) and LDA + SOI (0.0067 eV). In addition, considering the SOI, overall E\({}_{\rm Mixing}\), on average, decreased by 0.0091 eV; the maximum decrease was 0.0135 eV for Bi\({}_{4}\)Te\({}_{3}\), and the minimum was 0.0046 eV for Bi\({}_{4}\)Te\({}_{5}\). Thus, this demonstrates that LDA overestimates the phase stabilities of (Bi\({}_{2}\))\({}_{\rm m}\)(Bi\({}_{2}\)Te\({}_{3}\))\({}_{\rm n}\) compared to PBE. Figures 3-6 show the electronic band structures with their DOS for (Bi\({}_{2}\))\({}_{\rm m}\)(Bi\({}_{2}\)Te\({}_{3}\))\({}_{\rm n}\) obtained using PBE or LDA calculations, without and with SOI, respectively. All the DOS comprised small components of s- and d-orbital as well as that of large p-orbital in the whole energy range of electrons. It is clear that the orbitals of Bi 6p and Te 5p mainly contribute to the dominant p-orbital component. Note that the shapes of the electronic band structures are more complicated than the N\({}_{\rm Atom}\) increases due to supercell calculations. Meanwhile, for the convenience of band structure analysis, the VBM and CBM of each phase were used and the band topology near the band edge (VBM, CBM) was investigated. The E\({}_{\rm gap}^{\rm d}\) and E\({}_{\rm gap}^{\rm i}\) of all the phases and their reciprocal positions were obtained. From this, the electrical transport types (semiconductor, semimetal, or metal) of all the phases were investigated. The electric transport types with the band structures using PBE and LDA are summarised in Tables 2 and 3, respectively. Figure 2: Layer mixing energy (E\({}_{\rm Mixing}\)) of (Bi\({}_{2}\))\({}_{\rm m}\)(Bi\({}_{2}\)Te\({}_{3}\))\({}_{\rm n}\) for **(a)** PBE and **(b)** LDA as a function of N\({}_{\rm BL}\)/(N\({}_{\rm BL}\) + N\({}_{\rm QL}\)). The black dashed line shows E\({}_{\rm Mixing}\) for PBE, and the black line represents the E\({}_{\rm Mixing}\) for PBE with SOI (PBE + SOI), where the red solid and dashed lines indicate a convex hull of E\({}_{\rm Mixing}\) for PBE and PBE + SOI, respectively. The green dashed line displays E\({}_{\rm Mixing}\) for LDA, and the green dashed line represents E\({}_{\rm Mixing}\) for LDA with SOI (LDA + SOI), where the red solid and dashed lines indicate the convex hull of E\({}_{\rm Mixing}\) for LDA and LDA + SOI. Figure 3: Electronic band structures with density of states along the k-point paths 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 using DFT calculations with PBE. Figure 4: Electronic band structures with the density of states along the k-point paths 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\({}_{6}\)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 PBE including SOI.