## 1 Introduction  
Gasoline and hybrid electric vehicles cause waste heat energy to be released into the environment because of inefficient conversion of approximately 25% from chemical energy into mechanical energy through exhaust gases and coolants [1; 2; 3]. Automobile energy harvesting technology has attracted significant attention from the automobile industry to reduce the fuel consumption of vehicles [3; 4]. In addition, automobile waste heat recovery has been intensively investigated to alleviate global warming and improve the efficiency of vehicles [1; 2; 4; 5; 6; 7; 8; 9]. Meanwhile, thermoelectric generators are eco-friendly and durable owing to direct power conversion into electricity from automobile waste heat [1; 2; 4; 5; 6; 8; 9]. As a result, energy harvesting using thermoelectric generators in automobiles has been considered as an option to recover automobile waste heat.  
The Seebeck effect and conversion efficiency are important in thermoelectric energy harvesting technology. The thermoelectric dimensionless figure of merit (ZT) is defined as ZT \(\rm{\alpha^{2}\sigma T/\kappa}\), where \(\rm{\alpha}\) is the Seebeck coefficient, \(\rm{\sigma}\) is the electrical conductivity, \(\rm{\kappa}\) is the thermal conductivity, and \(\rm{T}\) is the absolute temperature [9; 10]. The thermoelectric conversion efficiency (\(\rm{\eta}\)) is defined as \(\rm{\eta}=P_{out}/Q_{h}\), where \(\rm{P}_{out}\) is the electric output power and \(\rm{Q}_{h}\) is the heat input [9; 10]. Many researchers have investigated thermoelectric materials with high ZT and \(\rm{\eta}\) for applications in thermoelectric energy harvesting [6; 11; 12; 13].  
Meanwhile, \((\text{Bi}_{2})_{\text{m}}(\text{Bi}_{2}\text{Te}_{3})_{\text{n}}\) homologous series have been investigated as layer-stacked structures consisting of Bi bilayers and \(\text{Bi}_{2}\text{Te}_{3}\) quintuple layers. In addition, they have been synthesised experimentally and their structural (lattice parameters, volume, etc.) and thermoelectric properties (resistivity, Seebeck coefficient, etc.) have been measured [14; 15; 16; 17; 18]. In particular, it has been demonstrated that \((\text{Bi}_{2})_{\text{m}}(\text{Bi}_{2}\text{Te}_{3})_{\text{n}}\) exhibits intrinsically low lattice thermal conductivity due to chemical bond softening and lattice anharmonicity [18]. However, to the best of our knowledge, no research has been conducted on the structural analysis and electronic band structures of \((\text{Bi}_{2})_{\text{m}}(\text{Bi}_{2}\text{Te}_{3})_{\text{n}}\).  
First-principles density functional theory calculations of the \((\text{Bi}_{2})_{\text{m}}(\text{Bi}_{2}\text{Te}_{3})_{\text{n}}\) homologous series were performed to investigate their internal structures, phase stability from mixing energies, and electronic band structures. From the structural analysis, it was found that exchange-correlation functionals affect both the intralayer thickness and interlayer distances of \((\text{Bi}_{2})_{\text{m}}(\text{Bi}_{2}\text{Te}_{3})_{\text{n}}\). Further, it was revealed that the \((\text{Bi}_{2})_{\text{m}}(\text{Bi}_{2}\text{Te}_{3})_{\text{n}}\) homologous series can be metastable from a comparison between their layer mixing energies and the convex hull. In addition, the energy gaps and band edges of \((\text{Bi}_{2})_{\text{m}}(\text{Bi}_{2}\text{Te}_{3})_{\text{n}}\) were determined from electronic band structure calculations. Consequently, electric transport types were obtained, and the \((\text{Bi}_{2})_{\text{m}}(\text{Bi}_{2}\text{Te}_{3})_{\text{n}}\) considered here were semimetallic or semiconducting. In particular, \(\text{Bi}_{2}\text{Te}_{3}\), \(\text{Bi}_{8}\text{Te}_{9}\), and BiTe among the \((\text{Bi}_{2})_{\text{m}}(\text{Bi}_{2}\text{Te}_{3})_{\text{n}}\) homologous series were semiconducting. Therefore, \(\text{Bi}_{2}\text{Te}_{3}\), \(\text{Bi}_{8}\text{Te}_{9}\), and BiTe are expected to be feasible thermoelectric materials for applications in automobile energy harvesting.