## 3 Results and Discussions
Depending on the layer stacking in (Bi\({}_{2}\))\({}_{\rm m}\)(Bi\({}_{2}\)Te\({}_{3}\))\({}_{\rm n}\), there were BL-BL, BL-QL, and QL-QL interlayer distances; BL-BL interfaces were only observed in Bi\({}_{2}\)Te and Bi\({}_{2}\)Te\({}_{3}\); BL-QL interfaces were present in the remaining phases except for Bi\({}_{2}\)Te\({}_{3}\) and Bi\({}_{2}\); and QL-QL interfaces were observed in Bi\({}_{2}\)Te\({}_{3}\), Bi\({}_{4}\)Te\({}_{3}\), Bi\({}_{6}\)Te\({}_{7}\), Bi\({}_{8}\)Te\({}_{9}\), and BiTe. PBE (LDA) calculation gives the interlayer distances, as shown in Table 1. For BL-BL in PBE, the interlayer distances are almost unaffected by considering SOI owing to their increase of less than 1%. However, for BLQL, when considering SOI, their interlayer distances decrease by more than 1.3% when N\({}_{\rm BL}\)/N\({}_{\rm Layer}\geq\) 0.333, although they remain almost constant for the other phases owing to their changes of less than 1.3%. For QL-QL, owing to SOI, their interlayer distances were reduced by more than 3.5%, unlike BL-BL and BL-QL; in particular, LDA decreased the interlayer distances of QL-QL by more than 1% compared to PBE, except for Bi\({}_{2}\)Te\({}_{3}\). Thus, employing SOI in our calculations caused greater changes in the interlayer distances of QL-QL than those of the other interfaces. This revealed that PBE underestimated the interlayer distances compared to LDA: over 2% for BL-BL, over 1.5% for BL-QL, and over 1.5% for QL-QL. Figure 2 shows the layer mixing energies (E\({}_{\rm Mixing}\)) of (Bi\({}_{2}\))\({}_{\rm m}\)(Bi\({}_{2}\)Te\({}_{3}\))\({}_{\rm n}\) as a function of N\({}_{\rm BL}\)/N\({}_{\rm Layer}\) = N\({}_{\rm BL}\)/(N\({}_{\rm BL}\) + N\({}_{\rm QL}\)). Because there is no layer mixing between QL and BL in Bi\({}_{2}\)Te\({}_{3}\) and Bi\({}_{2}\), their E\({}_{\rm Mixing}\) is zero. Figure 2a shows the convex hull and E\({}_{\rm Mixing}\) in PBE and PBE + SOI. In PBE + SOI, E\({}_{\rm Mixing}\) exhibits a convex hull through Bi\({}_{2}\)Te\({}_{3}\)-Bi\({}_{2}\)Te-Bi\({}_{2}\). Compared to the convex hull, the E\({}_{\rm Mixing}\) differs by 0.0017 eV/atom for Bi\({}_{4}\)Te\({}_{5}\), 0.0022 eV/atom for Bi\({}_{6}\)Te\({}_{7}\), 0.0025 eV/atom for Bi\({}_{6}\)Te\({}_{9}\), 0.0045 eV/atom for BiTe, 0.0003 eV/atom for Bi\({}_{4}\)Te\({}_{3}\), and 0.0027 eV/atom for Bi\({}_{7}\)Te\({}_{3}\), respectively. The E\({}_{\rm Mixing}\) of PBE is similar to that of PBE + SOI with a few exceptions: in PBE, the difference between the convex hull and E\({}_{\rm Mixing}\) was, on average, \(\sim\)1.7 times larger than that in PBE + SOI, except for Bi\({}_{4}\)Te\({}_{3}\) and Bi\({}_{7}\)Te\({}_{3}\), whereas in PBE, the differences of Bi\({}_{4}\)Te\({}_{3}\) and Bi\({}_{7}\)Te\({}_{3}\) were \(\sim\)13.6 and \(\sim\)0.4 times larger than those in PBE + SOI, respectively. In fact, the differences are less than 0.0045 eV/atom, except for that of BiTe in PBE (0.0070 eV/atom). This indicates that the (Bi\({}_{2}\))\({}_{\rm m}\)(Bi\({}_{2}\)Te\({}_{3}\))\({}_{\rm n}\) phases can exist energetically because their E\({}_{\rm Mixing}\) is very close to their convex hulls. Meanwhile, considering the SOI, overall E\({}_{\rm Mixing}\), on average, decreased by 0.0095 eV; the maximum decrease was 0.014 eV for Bi\({}_{4}\)Te\({}_{3}\), and the minimum was 0.0048 eV for Bi\({}_{4}\)Te\({}_{5}\). This indicates that E\({}_{\rm Mixing}\) was overestimated by the PBE calculations with SOI than those without SOI. On the other hand, Figure 2b exhibits a convex hull and E\({}_{\rm Mixing}\) in LDA and LDA + SOI. Note that the convex hulls of LDA functionals (LDA and LDA + SOI) have different shapes from those of PBE functionals (PBE and PBE + SOI). In particular, the convex hull of LDA passes through Bi\({}_{2}\)Te\({}_{3}\)-Bi\({}_{7}\)Te\({}_{3}\)-Bi\({}_{2}\) and that of LDA + SOI through Bi\({}_{2}\)Te\({}_{3}\)-Bi\({}_{4}\)Te\({}_{3}\)-Bi\({}_{7}\)Te\({}_{3}\)-Bi\({}_{2}\).