## 3 Results and Discussions  
Table 1 shows the chemical formula, hexagonal composition, ratio between the number of BLs and QL (NBL:NQL), ratio between NBL and the total number of layers (NBL/(NBL + NQL)), lattice volume per atom (Volume/N\({}_{\rm Atom}\)), lattice constants in a hexagonal supercell (aHex, cHex), cHex per atom (cHex/N\({}_{\rm Atom}\)), average intralayer thickness, and average interlayer distance depending on E\({}_{\rm xc}\) with (w/) or without (w/o) SOI used in our calculations. Note that N\({}_{\rm Atom}\) is the same as the total number of layers in each phase because our unit cells contain only a single atom on one layer. Additionally, note that NBL/N\({}_{\rm Atom}\) increases from 0 to 1 as the proportion of BL increases from Bi\({}_{2}\)Te\({}_{3}\) to Bi\({}_{2}\). In addition, for each (Bi\({}_{2}\))\({}_{\rm m}\)(Bi\({}_{2}\)Te\({}_{3}\))\({}_{\rm n}\), the lattice parameters in the a vector direction were 4.388-4.546 A, the lattice parameters in the c vector direction were 11.862-119 A, and the volume per atom (Volume/N\({}_{\rm Atom}\)) was 33.887-35.383 A\({}^{3}\). The average lattice parameters along the c-axis with respect to the number of layers were 1.977-2.033 A. The intralayer thicknesses of the atomic structure of (Bi\({}_{2}\))\({}_{\rm m}\)(Bi\({}_{2}\)Te\({}_{3}\))\({}_{\rm n}\) obtained through the DFT calculations were investigated, except for Bi\({}_{2}\) and Bi\({}_{2}\)Te\({}_{3}\). Based on the intralayer thickness without considering the SOI, the rate of increase of each intralayer thickness was calculated depending on whether or not SOI was considered. Regardless of the type of Exc used, the intralayer thickness with respect to SOI increased significantly compared to the thickness without considering SOI. In the case of PBE, the intralayer thicknesses of BL and QL for (Bi\({}_{2}\))\({}_{\rm m}\)(Bi\({}_{2}\)Te\({}_{3}\))\({}_{\rm n}\) were 1.697 A and 7.472 A for Bi\({}_{4}\)Te\({}_{5}\), 1.693 A and 7.453 A for Bi\({}_{6}\)Te\({}_{7}\), 1.701 A and 7.492 A for Bi\({}_{8}\)Te\({}_{9}\), 1.692 A and 7.449 A for BiTe, 1.681 A and 7.415 A for Bi\({}_{4}\)Te\({}_{3}\), 1.671 A and 7.390 A for Bi\({}_{2}\)Te, and 1.674 A and 7.358 A, Bi\({}_{7}\)Te\({}_{3}\), respectively. In PBE with SOI, compared to PBE, the thickness of BL increases by 1.64-3.63% and the thickness of QL also increases by 0.69-1.25% depending on their atomic structures. In LDA, the intralayer thicknesses of BL and QL for (Bi\({}_{2}\))\({}_{\rm m}\)(Bi\({}_{2}\)Te\({}_{3}\))\({}_{\rm n}\) were 1.661 A and 7.403 A for Bi\({}_{4}\)Te\({}_{5}\), 1.657 A and 7.384 A for Bi\({}_{6}\)Te\({}_{7}\), 1.664 A and 7.419 A for Bi\({}_{8}\)Te\({}_{9}\), 1.657 A and 7.383 A for BiTe, 1.645 A and 7.346 A for Bi\({}_{4}\)Te\({}_{3}\), 1.632 A and 7.310 A for Bi\({}_{2}\)Te, and 1.638 A and 7.291 A for Bi\({}_{7}\)Te\({}_{3}\), respectively. In LDA with SOI, compared to LDA, the thickness of BL increases by 1.40-3.51% and that of QL by 0.75-1.49%. Consequently, all increase rates of the intralayer thickness have positive values regardless of the intralayer type and the thickness of BL increases by more than 1.8 times of QL if SOI is considered, regardless of the phase of (Bi\({}_{2}\))\({}_{\rm m}\)(Bi\({}_{2}\)Te\({}_{3}\))\({}_{\rm n}\). In addition, considering the SOI, the intralayer thickness of BL for PBE increases more than that of QL for LDA, although that of QL for PBE increases less than that of QL for LDA. This indicates that the E\({}_{\rm xc}\) of PBE (PBE w/o or w/SOI) overestimates the intralayer thickness of BL by more than 2% compared to that of LDA (LDA w/o or w/SOI). Furthermore, this indicates that the intralayer thickness of QL is overestimated by less than 1.1% by PBE compared with LDA. Thus, this shows that LDA underestimates the distances between adjacent atoms along the c-axis in the intralayers compared with PBE. The interlayer distances of (Bi\({}_{2}\))\({}_{\rm m}\)(Bi\({}_{2}\)Te\({}_{3}\))\({}_{\rm n}\) were also analysed.