## 2 Methods and Calculations Details  
First-principles density functional theory (DFT) [19; 20] calculations were performed. The DFT calculations were implemented using the Vienna Ab initio simulation package [21; 22; 23]. A plane-wave basis set with an energy cut-off of 300 eV was used with the Perdew-Burke-Ernzerhof (PBE) exchange correlational functional (E\({}_{\text{xc}}\)) [24], and local density approximation (LDA) [23] and projector augmented wave (PAW) pseudopotentials [25; 26] were used: PAW_PBE Bi_d_GW 14 Apr2014, PAW_PBE Te_GW 22 Mar2012 for PBE and PAW Bi_d_GW 14 Apr2014, PAW Te_GW 22 Mar2012 for LDA. A \(12\times 12\times k_{3}\)\(\Gamma\)-centered k-point mesh grid was used to sample the k-point in the Brillouin zone, where \(k_{3}\) is a subdivision of the third direction along the third reciprocal lattice vector. To consider the relativistic correction of heavy elements such as Bi and Te, the spin-orbit interaction (SOI) was employed [27; 28; 29]. The structures of the \((\text{Bi}_{2})_{\text{m}}(\text{Bi}_{2}\text{Te}_{3})_{\text{n}}\) homologous series considered here were obtained from [PERSON] research [14]. Their end members were semiconducting \(\text{Bi}_{2}\text{Te}_{3}\) and semimetallic Bi\({}_{2}\); nine members of the series were studied: \(\text{Bi}_{2}\text{Te}_{3}\), \(\text{Bi}_{4}\text{Te}_{5}\), \(\text{Bi}_{6}\text{Te}_{7}\), \(\text{Bi}_{8}\text{Te}_{9}\), BiTe, \(\text{Bi}_{4}\text{Te}_{3}\), \(\text{Bi}_{2}\text{Te}\), \(\text{Bi}_{7}\text{Te}_{3}\), and \(\text{Bi}_{2}\). Here, the ratio of Bi\({}_{2}\) to Bi\({}_{2}\text{Te}_{3}\) increased in the following order from \(\text{Bi}_{2}\text{Te}_{3}\) to Bi\({}_{2}\) as the ratio of m to n: 0.3, 1.5, 2.7, 3.9, 1.2, 3.3, 2.1; 15.6, and 3.0 (see Figure 1). In the figure, the side view shows the atomic structure viewed from the \(a\)-axis direction, and the top view shows the atomic structure viewed from the \(c\)-axis direction. The purple spheres represent Bi atoms and the gold spheres are Te atoms. The red blocks are the Bi bilayers (BLs) and the blue blocks the \(\text{Bi}_{2}\text{Te}_{3}\) quintuple layers (QLs). The lattice parameters of the \(a\)-axis and the \(c\)-axis in the hexagonal supercells were obtained from previous experimental studies of \((\text{Bi}_{2})_{\text{m}}(\text{Bi}_{2}\text{Te}_{3})_{\text{n}}\) materials [14; 15; 16; 17] (see Table 1). In particular, we referred to the layer-stacking method of BL and QL in [PERSON] study [14]; for convenient calculations, the different stacking methods of BL and QL were not considered. Furthermore, the optimised atomic structure of each material of \((\text{Bi}_{2})_{\text{m}}(\text{Bi}_{2}\text{Te}_{3})_{\text{n}}\) was obtained while lowering the total force acting on each atom below \(10^{-3}\) eV/A, where the initial atomic structure had fixed lattice parameters with all the atoms arrayed equidistantly in the direction of the \(c\)-axis. Figure 1: Atomic structures of hexagonal (Bi\({}_{2}\))\({}_{m}\)(Bi\({}_{2}\)Te\({}_{3}\))\({}_{n}\): (**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\({}_{2}\). The violet and gold balls indicate Bi and Te, respectively. The red and blue blocks display the Bi\({}_{2}\) bilayer (BL) and Bi\({}_{2}\)Te\({}_{3}\) quintuple layer (QL), respectively. The side and top view indicate the atomic structure seen from the vector direction of \(\overrightarrow{a}\) and \(\overrightarrow{c}\) in each respective (Bi\({}_{2}\))\({}_{m}\)(Bi\({}_{2}\)Te\({}_{3}\))\({}_{n}\) lattice. In the atomic structure of the (Bi\({}_{2}\))\({}_{\text{m}}\)(Bi\({}_{2}\)Te\({}_{3}\))\({}_{\text{n}}\) homologous series, the number of BLs or QLs was defined as N\({}_{\text{BL}}\) or N\({}_{\text{QL}}\), respectively. The total number of atoms in each phase was defined as N\({}_{\text{Atom}}\).