Structural Analysis, Phase Stability, Electronic Band Structures, and Electric Transport Types of (Bi 2 ) m (Bi 2 Te 3 ) n by Density Functional Theory Calculations

: Thermoelectric power generation is a promising candidate for automobile energy harvesting technologies because it is eco-friendly and durable owing to direct power conversion from automobile waste heat. Because Bi − Te systems are well-known thermoelectric materials, research on (Bi 2 ) m (Bi 2 Te 3 ) n homologous series can aid the development of efﬁcient thermoelectric materials. However, to the best of our knowledge, (Bi 2 ) m (Bi 2 Te 3 ) n has been studied through experimental synthesis and measurements only. Therefore, we performed density functional theory calculations of nine members of (Bi 2 ) m (Bi 2 Te 3 ) n to investigate their structure, phase stability, and electronic band structures. From our calculations, although the total energies of all nine phases are slightly higher than their convex hulls, they can be metastable owing to their very small energy differences. The electric transport types of (Bi 2 ) m (Bi 2 Te 3 ) n do not change regardless of the exchange–correlation functionals, which cause tiny changes in the atomic structures, phase stabilities, and band structures. Additionally, only two phases (Bi 8 Te 9 , BiTe) became semimetallic or semiconducting depending on whether spin–orbit interactions were included in our calculations, and the electric transport types of the other phases were unchanged. As a result, it is expected that Bi 2 Te 3 , Bi 8 Te 9 , and BiTe are candidates for thermoelectric materials for automobile energy harvesting technologies because they are semiconducting.


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 = α 2 σT/κ, where α is the Seebeck coefficient, σ is the electrical conductivity, κ is the thermal conductivity, and T is the absolute temperature [9,10]. The thermoelectric conversion efficiency (η) is defined as η = P out /Q h , where P out is the electric output power and Q h is the heat input [9,10]. Many researchers have investigated thermoelectric materials with high ZT and η for applications in thermoelectric energy harvesting [6,[11][12][13].

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 xc ) [24], and local density approximation (LDA) [23] and projector augmented wave (PAW) pseudopotentials [25,26] were used: PAW_PBE Bi_d_GW 14Apr2014, PAW_PBE Te_GW 22Mar2012 for PBE and PAW Bi_d_GW 14Apr2014, PAW Te_GW 22Mar2012 for LDA. A 12 × 12 × k 3 Γ-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 (Bi 2 ) m (Bi 2 Te 3 ) n homologous series considered here were obtained from Bos' research [14]. Their end members were semiconducting Bi 2 Te 3 and semimetallic Bi 2 ; nine members of the series were studied: Bi 2 Te 3 , Bi 4 Te 5 , Bi 6 Te 7 , Bi 8 Te 9 , BiTe, Bi 4 Te 3 , Bi 2 Te, Bi 7 Te 3 , and Bi 2 . Here, the ratio of Bi 2 to Bi 2 Te 3 increased in the following order from Bi 2 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 Bi 2 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 (Bi 2 ) m (Bi 2 Te 3 ) n materials [14][15][16][17] (see Table 1). In particular, we referred to the layer-stacking method of BL and QL in Bos' 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 (Bi 2 ) m (Bi 2 Te 3 ) n was obtained while lowering the total force acting on each atom below 10 −3 eV/Å, where the initial atomic structure had fixed lattice parameters with all the atoms arrayed equidistantly in the direction of the c-axis.  In the atomic structure of the (Bi 2 ) m (Bi 2 Te 3 ) n homologous series, the number of BLs or QLs was defined as N BL or N QL , respectively. The total number of atoms in each phase was defined as N Atom . Furthermore, the average thickness of BL or QL was defined as the intralayer thickness of BL or QL. Additionally, the average distance between BL and BL, BL and QL, or QL and QL was defined as the interlayer distances of BL−BL, BL−QL, or QL−QL, respectively (see Table 1 and Figure 1).
To investigate the relative stability of (Bi 2 ) m (Bi 2 Te 3 ) n when mixing BL and QL, the layer mixing energy of each phase per atom was defined as Total are the total energies of (Bi 2 ) m (Bi 2 Te 3 ) n , Bi 2 , and Bi 2 Te 3 , respectively. Note that E Mixing is normalised by the total atomic number in the supercell (N atom ) to reflect only the structural differences between the (Bi 2 ) m (Bi 2 Te 3 ) n phases. In addition, note that only zero-temperature E Mixing is displayed here because the temperature dependence of E Mixing is not of interest.
Our electronic band structures with density of states (DOS) were generated using pymatgen [30]  our electronic band structures were projected onto atomic elements (Bi: blue, Te: red), except for Bi 2 , and our DOS is projected onto electronic orbitals (s: blue, p: red, d: green, f: violet). For the electronic band topology of (Bi 2 ) m (Bi 2 Te 3 ) n phases, the direct bandgap was defined as E d gap (the minimum of the difference between CBM and VBM at the same k-point) and the indirect bandgap as E i gap (the difference between CBM and VBM). Electric types were defined to determine the electric transport characteristics of (Bi 2 ) m (Bi 2 Te 3 ) n : semiconductors if both E d gap and E i gap were positive, semimetals if E d gap was positive and E i gap negative, and metals if both E d gap and E i gap were negative. Table 1 shows the chemical formula, hexagonal composition, ratio between the number of BLs and QL (N BL :N QL ), ratio between N BL and the total number of layers (N BL /(N BL + N QL )), lattice volume per atom (Volume/N Atom ), lattice constants in a hexagonal supercell (a Hex , c Hex ), c Hex per atom (c Hex /N Atom ), average intralayer thickness, and average interlayer distance depending on E xc with (w/) or without (w/o) SOI used in our calculations. Note that N 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 N BL /N 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 ) m (Bi 2 Te 3 ) n , the lattice parameters in the a vector direction were 4.388-4.546 Å, the lattice parameters in the c vector direction were 11.862-119 Å, and the volume per atom (Volume/N Atom ) was 33.887-35.383 Å 3 . The average lattice parameters along the c-axis with respect to the number of layers were 1.977-2.033 Å.

Results and Discussions
The intralayer thicknesses of the atomic structure of (Bi 2 ) m (Bi 2 Te 3 ) 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 ) m (Bi 2 Te 3 ) n were 1.697 Å and 7.472 Å for Bi 4  , 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 ) m (Bi 2 Te 3 ) n . In addition, considering the SOI, the intralayer thickness of BL for PBE increases more than that of BL for LDA, although that of QL for PBE increases less than that of QL for LDA. This indicates that the E 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 ) m (Bi 2 Te 3 ) n were also analysed. Depending on the layer stacking in (Bi 2 ) m (Bi 2 Te 3 ) n , there were BL-BL, BL-QL, and QL-QL interlayer distances; BL-BL interfaces were only observed in Bi 2 Te and Bi 7 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 BL-QL, when considering SOI, their interlayer distances decrease by more than 1.3% when N BL /N Layer ≥ 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 Mixing ) of (Bi 2 ) m (Bi 2 Te 3 ) n as a function of N BL /N Layer = N BL /(N BL + N QL ). Because there is no layer mixing between QL and BL in Bi 2 Te 3 and Bi 2 , their E Mixing is zero. Figure 2a shows the convex hull and E Mxing in PBE and PBE + SOI. In PBE + SOI, E Mxing exhibits a convex hull through Bi 2 Te 3 -Bi 2 Te-Bi 2 . Compared to the convex hull, the E 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 8 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 Mixing of PBE is similar to that of PBE + SOI with a few exceptions: in PBE, the difference between the convex hull and E Mxing was, on average,~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~13.6 and~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 ) m (Bi 2 Te 3 ) n phases can exist energetically because their E Mixing is very close to their convex hulls. Meanwhile, considering the SOI, overall E 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 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 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 . Compared to the convex hull in LDA + SOI, E 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 Mxing 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 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 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 ) m (Bi 2 Te 3 ) n compared to PBE. Figures 3-6 show the electronic band structures with their DOS for (Bi 2 ) m (Bi 2 Te 3 ) 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 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 d gap and E i gap 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. band structures are more complicated than the NAtom 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 gap d and E gap 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 3. The band structure of Bi 2 Te 3 is shown in Figure 3a. There is a VBM of −0.136 eV at Γ and a CBM of 0.136 at Γ, indicating that Bi 2 Te 3 is a semiconductor as previously reported [31][32][33]. Note that the experimental energy gap is about 0.13 eV. Figure 3d shows that Bi 8  Note that our result for Bi 2 is consistent with the experimental reports that bulk Bi is semimetallic [34] since the supercell of Bi 2 is to be bulk Bi. Additionally, the other phases (Bi 4 Te 5 , Bi 6 Te 7 , Bi 4 Te 3 , Bi 2 Te, Bi 7 Te 3 ) were semimetallic, as shown in Figure 3 and Table 2. Note that PBE showed that the electric transport types of (Bi 2 ) m (Bi 2 Te 3 ) n were semimetallic because of positive E d gap and negative E i gap except for Bi 2 Te 3 . In addition, the electric transport types of (Bi 2 ) m (Bi 2 Te 3 ) n from LDA are the same as those from PBE, although the detailed values (E VBM , k VBM , E CBM , k CBM , E i gap , k i gap ) of LDA were different from those of PBE (see Figures 3-5 and Tables 2 and 3).
Meanwhile, when considering the SOI in the PBE calculations, there are some changes in the band structures (see Figure 4 and Table 2). In our results, SOI flattens the electronic band of (Bi 2 ) m (Bi 2 Te 3 ) n in the vicinity of the band edges (VBM and CBM) because dE/dk decreases near these edges. We expect that this is related to the change in the electronic effective mass near the VBM and CBM, although it was not quantitatively analysed. For Bi 2 Te 3 , the SOI calculations accompanied an increase in the energy level of VBM (−0.061 eV) and a decrease in that of CBM (0.068 eV), resulting in a slight decrease in energy gaps (E i gap = 0.129 eV, E d gap = 0.135 eV). For Bi 2 , the SOI calculation increases both VBM (0.113 eV) and CBM (-0.048 eV), leading to a small increase in the energy gap values (E i gap = −0.161 eV, E d gap = 0.079 eV). In addition, for BiTe, the SOI calculations decrease VBM (−0.064 eV) and increase CBM (0.053 eV), eventually opening E i gap (0.117 eV), unlike closed gaps from calculations without SOI (−0.04 eV). Note that the electric transport type of BiTe changes to a semiconductor of calculations with SOI unlike semimetals of the calculations without SOI. For the other phases except for BiTe, with SOI, the electric transport types do not change, although E VBM , k VBM , E CBM , k CBM , E i gap , and k i gap change slightly compared to the case without SOI. In addition, note that with SOI, LDA and PBE give the same electric transport types except for Bi 8 Te 9 and BiTe unlike the results of the band structures from the calculations without SOI, although the detailed values (E VBM , k VBM , E CBM , k CBM , E i gap , and k i gap ) changed (see Figures 4-6 and Tables 2 and 3). In detail, LDA with SOI changes the electric transport types of Bi 8 Te 9 and BiTe from semimetals to semiconductors owing to the decrease in VBM (−0.005 eV for Bi 8 Te 9 , −0.062 eV for BiTe) and the increase in CBM (−0.002 eV for Bi 8 Te 9 , 0.048 eV for BiTe). Note that although it is not possible to explicitly state the semiconductor type, DFT calculations can determine possible electric transport types. Therefore, for the (Bi 2 ) m (Bi 2 Te 3 ) n homologous series, only three phases (Bi 2 Te 3 , Bi 8 Te 9 , BiTe) can be semiconductors.

Conclusions
First-principles DFT calculations of (Bi 2 ) m (Bi 2 Te 3 ) n were performed with fixed experimental lattice parameters to estimate their phase stabilities and to analyse the properties of the atomic structures and electronic band structures. For E xc , PBE and LDA were considered with or without SOI. From the structural analysis, PBE overestimated the distances between adjacent atoms along the c-axis in the intralayers of (Bi 2 ) m (Bi 2 Te 3 ) n compared to LDA, and LDA overestimated the interlayer distances in (Bi 2 ) m (Bi 2 Te 3 ) n compared to PBE. From the electronic band structures, the electric transport types of (Bi 2 ) m (Bi 2 Te 3 ) n remained as semimetals or semiconductors regardless of whether the E xc type or SOI was considered, except for Bi 8 Te 9 and BiTe. Our calculations revealed that Bi 8 Te 9 , BiTe, and Bi 2 Te 3 are expected to be semiconductors. Consequently, Bi 8 Te 9 and BiTe, including the well-known thermoelectric Bi 2 Te 3 , are expected to be potential thermoelectric materials for automobile energy harvesting technologies.