Structural Analysis, Phase Stability, Electronic Band Structures, and Electric Transport Types of (Bi₂)_m(Bi₂Te₃)_n by Density Functional Theory Calculations

Abstract

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 (Bi₂)_m(Bi₂Te₃)_n homologous series can aid the development of efficient thermoelectric materials. However, to the best of our knowledge, (Bi₂)_m(Bi₂Te₃)_n has been studied through experimental synthesis and measurements only. Therefore, we performed density functional theory calculations of nine members of (Bi₂)_m(Bi₂Te₃)_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 (Bi₂)_m(Bi₂Te₃)_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 (Bi₈Te₉, 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 Bi₂Te₃, Bi₈Te₉, and BiTe are candidates for thermoelectric materials for automobile energy harvesting technologies because they are semiconducting.

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 = α²σT/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].

Meanwhile, (Bi₂)_m(Bi₂Te₃)_n homologous series have been investigated as layer-stacked structures consisting of Bi bilayers and Bi₂Te₃ 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 (Bi₂)_m(Bi₂Te₃)_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 (Bi₂)_m(Bi₂Te₃)_n.

First-principles density functional theory calculations of the (Bi₂)_m(Bi₂Te₃)_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 (Bi₂)_m(Bi₂Te₃)_n. Further, it was revealed that the (Bi₂)_m(Bi₂Te₃)_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 (Bi₂)_m(Bi₂Te₃)_n were determined from electronic band structure calculations. Consequently, electric transport types were obtained, and the (Bi₂)_m(Bi₂Te₃)_n considered here were semimetallic or semiconducting. In particular, Bi₂Te₃, Bi₈Te₉, and BiTe among the (Bi₂)_m(Bi₂Te₃)_n homologous series were semiconducting. Therefore, Bi₂Te₃, Bi₈Te₉, and BiTe are expected to be feasible thermoelectric materials for applications in automobile energy harvesting.

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_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 $12\times 12\times k_3$ $\mathsf{\Gamma }$-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 (Bi₂)_m(Bi₂Te₃)_n homologous series considered here were obtained from Bos’ research [14]. Their end members were semiconducting Bi₂Te₃ and semimetallic Bi₂; nine members of the series were studied: Bi₂Te₃, Bi₄Te₅, Bi₆Te₇, Bi₈Te₉, BiTe, Bi₄Te₃, Bi₂Te, Bi₇Te₃, and Bi₂. Here, the ratio of Bi₂ to Bi₂Te₃ increased in the following order from Bi₂Te₃ to Bi₂ 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 Bi₂Te₃ 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 (Bi₂)_m(Bi₂Te₃)_n materials [14,15,16,17] (see

Table 1. Phase properties, structural parameters, intralayer thickness, and interlayer distance of (Bi₂)_m(Bi₂Te₃)_n homologous series depending on the E_xc used for our calculations.

Chemical Formula				Bi2Te3	Bi4Te5	Bi6Te7	Bi8Te9	BiTe	Bi4Te3	Bi2Te	Bi7Te3	Bi2
Phase properties		Hexagonal compositions		Bi6Te9	Bi12Te15	Bi18Te21	Bi24Te27	Bi6Te6	Bi12Te9	Bi6Te3	Bi42Te18	Bi6
		NBL: NQL		0:3	1:5	2:7	3:9	1:2	3:3	2:1	15:6	3:0
		NBL/(NBL + NQL)		0	0.167	0.222	0.250	0.333	0.500	0.667	0.714	1
Structural parameters		Volume/NAtom (Å3)		33.892	33.974	33.986	34.015	33.887	34.224	35.093	34.352	35.383
		aHex (Å)		4.388	4.415	4.424	4.410	4.423	4.451	4.490	4.472	4.546
		cHex (Å)		30.488	54.330	78.200	103.000	24.002	41.890	18.090	119.000	11.862
		cHex/NAtom (Å)		2.033	2.012	2.005	2.020	2.000	1.995	2.010	1.983	1.977
PBE	w/o SOI	Intralayer thickness (Å)	BL	-	1.697	1.693	1.701	1.692	1.681	1.671	1.674	1.977
			QL	7.529	7.472	7.453	7.492	7.449	7.415	7.390	7.358	-
		Interlayer distance (Å)	BL-BL	-	-	-	-	-	-	2.464	2.375	1.977
			BL-QL	-	2.436	2.428	2.453	2.421	2.434	2.447	2.363	-
			QL-QL	2.634	2.600	2.586	2.626	2.570	-	-	-	-
	w/ SOI	Intralayer thickness (Å)	BL	-	1.758	1.753	1.762	1.748	1.729	1.704	1.702	1.977
			QL	7.623	7.560	7.538	7.579	7.525	7.480	7.454	7.409	-
		Interlayer distance (Å)	BL-BL	-	-	-	-	-	-	2.468	2.375	1.977
			BL-QL	-	2.412	2.400	2.423	2.386	2.377	2.380	2.303	-
			QL-QL	2.539	2.488	2.465	2.494	2.433	-	-	-	-
LDA	w/o SOI	Intralayer thickness (Å)	BL	-	1.661	1.657	1.66	1.657	1.645	1.632	1.638	1.977
			QL	7.454	7.403	7.384	7.419	7.383	7.346	7.310	7.291	-
		Interlayer distance (Å)	BL-BL	-	-	-	-	-	-	2.525	2.426	1.977
			BL-QL	-	2.474	2.469	2.492	2.467	2.486	2.495	2.404	-
			QL-QL	2.709	2.677	2.663	2.714	2.645	-	-	-	-
	w/ SOI	Intralayer thickness (Å)	BL	-	1.718	1.714	1.722	1.710	1.693	1.661	1.661	1.977
			QL	7.565	7.509	7.489	7.526	7.476	7.421	7.381	7.345	-
		Interlayer distance (Å)	BL-BL	-	-	-	-	-	-	2.549	2.439	1.977
			BL-QL	-	2.466	2.450	2.476	2.434	2.425	2.419	2.338	-
			QL-QL	2.598	2.534	2.510	2.540	2.471	-	-	-	-

). 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 (Bi₂)_m(Bi₂Te₃)_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 (Bi₂)_m(Bi₂Te₃)_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 (Bi₂)_m(Bi₂Te₃)_n when mixing BL and QL, the layer mixing energy of each phase per atom was defined as

$${\mathrm{E}}_{\mathrm{Mixing}}=\frac{{\mathrm{E}}_{\mathrm{Total}}^{{({\mathrm{Bi}}_2)}_{\mathrm{m}}{({\mathrm{Bi}}_2{\mathrm{Te}}_3)}_{\mathrm{n}}}-{ \mathrm{m} \mathrm{E}}_{\mathrm{Total}}^{{\mathrm{Bi}}_2}-{ \mathrm{n} \mathrm{E}}_{\mathrm{Total}}^{{\mathrm{Bi}}_2{\mathrm{Te}}_3}}{{\mathrm{N}}_{\mathrm{atom}}}$$

(1)

where ${\mathrm{E}}_{\mathrm{Total}}^{{({\mathrm{Bi}}_2)}_{\mathrm{m}}{({\mathrm{Bi}}_2{\mathrm{Te}}_3)}_{\mathrm{n}}}$, ${\mathrm{E}}_{\mathrm{Total}}^{{\mathrm{Bi}}_2}$, and ${\mathrm{E}}_{\mathrm{Total}}^{{\mathrm{Bi}}_2{\mathrm{Te}}_3}$ are the total energies of ${({\mathrm{Bi}}_2)}_{\mathrm{m}}{({\mathrm{Bi}}_2{\mathrm{Te}}_3)}_{\mathrm{n}}$, ${\mathrm{Bi}}_2$, and ${\mathrm{Bi}}_2{\mathrm{Te}}_3$, 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 (Bi₂)_m(Bi₂Te₃)_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] based on our DFT calculations. In our band calculations, the electronic band structures of the full k-point path (Γ−M−K−Γ−A−L−H−A, L−M, K−H) were obtained with 51 mesh grids in the first Brillouin zone of hexagonal lattices. From this, it was found that all the valence band maximum (VBM) and conduction band minimum (CBM) of the (Bi₂)_m(Bi₂Te₃)_n considered in this study were located in the k-point path of K−Γ−M or L−A−H for our calculation setting: K (0.333, 0.333, 0), Γ (0, 0, 0), M (0.5, 0, 0), L (0.5, 0, 0.5), A (0, 0, 0.5), H (0.333, 0.333, 0.5). Consequently, only electronic band structures in K−Γ−M and L−A−H are presented for the E_xc used in our calculations. In addition, our electronic band structures were projected onto atomic elements (Bi: blue, Te: red), except for Bi₂, and our DOS is projected onto electronic orbitals (s: blue, p: red, d: green, f: violet). For the electronic band topology of (Bi₂)_m(Bi₂Te₃)_n phases, the direct bandgap was defined as ${\mathrm{E}}_{\mathrm{gap}}^{\mathrm{d}}$ (the minimum of the difference between CBM and VBM at the same k-point) and the indirect bandgap as ${\mathrm{E}}_{\mathrm{gap}}^{\mathrm{i}}$ (the difference between CBM and VBM). Electric types were defined to determine the electric transport characteristics of (Bi₂)_m(Bi₂Te₃)_n: semiconductors if both ${\mathrm{E}}_{\mathrm{gap}}^{\mathrm{d}}$ and ${\mathrm{E}}_{\mathrm{gap}}^{\mathrm{i}}$ were positive, semimetals if ${\mathrm{E}}_{\mathrm{gap}}^{\mathrm{d}}$ was positive and ${\mathrm{E}}_{\mathrm{gap}}^{\mathrm{i}}$ negative, and metals if both ${\mathrm{E}}_{\mathrm{gap}}^{\mathrm{d}}$ and ${\mathrm{E}}_{\mathrm{gap}}^{\mathrm{i}}$ were negative.

3. Results and Discussions

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 Bi₂Te₃ to Bi₂. In addition, for each (Bi₂)_m(Bi₂Te₃)_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 ų. The average lattice parameters along the c-axis with respect to the number of layers were 1.977–2.033 Å.

The intralayer thicknesses of the atomic structure of (Bi₂)_m(Bi₂Te₃)_n obtained through the DFT calculations were investigated, except for Bi₂ and Bi₂Te₃. 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 (Bi₂)_m(Bi₂Te₃)_n were 1.697 Å and 7.472 Å for Bi₄Te₅, 1.693 Å and 7.453 Å for Bi₆Te₇, 1.701 Å and 7.492 Å for Bi₈Te₉, 1.692 Å and 7.449 Å for BiTe, 1.681 Å and 7.415 Å for Bi₄Te₃, 1.671 Å and 7.390 Å for Bi₂Te, and 1.674 Å and 7.358 Å, Bi₇Te₃, 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 (Bi₂)_m(Bi₂Te₃)_n were 1.661 Å and 7.403 Å for Bi₄Te₃, 1.657 Å and 7.384 Å for Bi₆Te₇, 1.664 Å and 7.419 Å for Bi₈Te₉, 1.657 Å and 7.383 Å for BiTe, 1.645 Å and 7.346 Å for Bi₄Te₃, 1.632 Å and 7.310 Å for Bi₂Te, and 1.638 Å and 7.291 Å for Bi₇Te₃, 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 (Bi₂)_m(Bi₂Te₃)_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 (Bi₂)_m(Bi₂Te₃)_n were also analysed. Depending on the layer stacking in (Bi₂)_m(Bi₂Te₃)_n, there were BL–BL, BL–QL, and QL–QL interlayer distances; BL–BL interfaces were only observed in Bi₂Te and Bi₇Te₃; BL–QL interfaces were present in the remaining phases except for Bi₂Te₃ and Bi₂; and QL–QL interfaces were observed in Bi₂Te₃, Bi₄Te₃, Bi₆Te₇, Bi₈Te₉, 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 Bi₂Te₃. 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 (Bi₂)_m(Bi₂Te₃)_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 Bi₂Te₃ and Bi₂, 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 Bi₂Te₃–Bi₂Te–Bi₂. Compared to the convex hull, the E_Mixing differs by 0.0017 eV/atom for Bi₄Te₅, 0.0022 eV/atom for Bi₆Te₇, 0.0025 eV/atom for Bi₈Te₉, 0.0045 eV/atom for BiTe, 0.0003 eV/atom for Bi₄Te₃, and 0.0027 eV/atom for Bi₇Te₃, 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 Bi₄Te₃ and Bi₇Te₃, whereas in PBE, the differences of Bi₄Te₃ and Bi₇Te₃ 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 (Bi₂)_m(Bi₂Te₃)_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 Bi₄Te₃, and the minimum was 0.0048 eV for Bi₄Te₅. 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 Bi₂Te₃–Bi₇Te₃–Bi₂ and that of LDA + SOI through Bi₂Te₃–Bi₄Te₃–Bi₇Te₃–Bi₂. Compared to the convex hull in LDA + SOI, E_Mixing differs by 0.0019 eV for Bi₄Te₅, 0.0021 eV for Bi₆Te₇, 0.0027 eV for Bi₈Te₉, 0.0006 eV for BiTe, and 0.0067 eV for Bi₂Te. 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 Bi₈Te₉ and Bi₄Te₃; in particular, that of BiTe was ~3.8 times larger. The differences were less than 0.0035 eV, except for those of Bi₂Te 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 Bi₄Te₃, and the minimum was 0.0046 eV for Bi₄Te₅. Thus, this demonstrates that LDA overestimates the phase stabilities of (Bi₂)_m(Bi₂Te₃)_n compared to PBE.

Figure 3, Figure 4, Figure 5 and Figure 6 show the electronic band structures with their DOS for (Bi₂)_m(Bi₂Te₃)_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 ${\mathrm{E}}_{\mathrm{gap}}^{\mathrm{d}}$ and ${\mathrm{E}}_{\mathrm{gap}}^{\mathrm{i}}$ 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

Table 2. Determination of electric transport types (Type) of (Bi₂)_m(Bi₂Te₃)_n from the energy gap in electronic band structure depending on the exclusion (X) and inclusion (O) of SOI using the E_xc of PBE. E_VBM and E_CBM are the energy of VBM and CBM relative to the Fermi level of each (Bi₂)_m(Bi₂Te₃)_n, respectively. ${\mathrm{E}}_{\mathrm{gap}}^{\mathrm{i}}$ and ${\mathrm{E}}_{\mathrm{gap}}^{\mathrm{d}}$ are the indirect and direct energy gaps, respectively. k_VBM, k_CBM, and ${\mathrm{k}}_{\mathrm{gap}}^{\mathrm{d}}$ are reciprocal positions of VBM, CBM, and ${\mathrm{E}}_{\mathrm{gap}}^{\mathrm{d}}$ in the first Brillouin zone, respectively. SC and SM represent semiconductors and semimetals, respectively.

Exc	Phase	SOI	EVBM	kVBM	ECBM	kCBM	${\mathrm{E}}_{\mathrm{g}\mathrm{a}\mathrm{p}}^{\mathrm{i}}$	${\mathrm{E}}_{\mathrm{g}\mathrm{a}\mathrm{p}}^{\mathrm{d}}$	${\mathrm{k}}_{\mathrm{g}\mathrm{a}\mathrm{p}}^{\mathrm{d}}$	Type
PBE	Bi2Te3	X	−0.136	0.000 0.000 0.000	0.136	0.000 0.000 0.000	0.272	0.272	0.000 0.000 0.000	SC
		O	−0.061	0.140 0.000 0.500	0.068	0.090 0.000 0.000	0.129	0.135	0.130 0.000 0.500	SC
	Bi4Te5	X	0.006	0.033 0.033 0.500	−0.059	0.000 0.000 0.500	−0.065	0.005	0.010 0.000 0.500	SM
		O	0.012	0.170 0.000 0.000	−0.035	0.090 0.000 0.000	−0.047	0.004	0.010 0.000 0.500	SM
	Bi6Te7	X	0.002	0.033 0.033 0.000	−0.043	0.007 0.007 0.000	−0.045	0.029	0.020 0.000 0.000	SM
		O	0.006	0.180 0.000 0.500	−0.018	0.000 0.000 0.500	−0.024	0.008	0.020 0.000 0.000	SM
	Bi8Te9	X	−0.004	0.033 0.033 0.500	−0.030	0.007 0.007 0.500	−0.026	0.027	0.020 0.000 0.500	SM
		O	0.001	0.170 0.000 0.500	−0.019	0.000 0.000 0.500	−0.020	0.001	0.000 0.000 0.500	SM
	BiTe	X	0.017	0.060 0.000 0.000	−0.023	0.013 0.013 0.500	−0.040	0.035	0.060 0.000 0.000	SM
		O	−0.064	0.180 0.000 0.500	0.053	0.027 0.027 0.500	0.117	0.152	0.040 0.000 0.500	SC
	Bi4Te3	X	0.030	0.110 0.000 0.000	−0.036	0.027 0.027 0.500	−0.066	0.021	0.027 0.027 0.500	SM
		O	0.108	0.000 0.000 0.000	−0.084	0.500 0.000 0.000	−0.192	0.001	0.020 0.000 0.000	SM
	Bi2Te	X	0.025	0.067 0.067 0.000	−0.051	0.033 0.033 0.000	−0.076	0.038	0.033 0.033 0.000	SM
		O	0.138	0.000 0.000 0.000	−0.041	0.500 0.000 0.500	−0.179	0.030	0.030 0.000 0.000	SM
	Bi7Te3	X	0.077	0.150 0.000 0.500	−0.068	0.027 0.027 0.500	−0.145	0.021	0.027 0.027 0.500	SM
		O	0.041	0.000 0.000 0.500	−0.093	0.500 0.000 0.500	−0.134	0.025	0.020 0.020 0.500	SM
	Bi2	X	0.066	0.047 0.047 0.500	−0.197	0.480 0.000 0.500	−0.263	0.020	0.047 0.047 0.500	SM
		O	0.113	0.000 0.000 0.500	−0.048	0.500 0.000 0.500	−0.161	0.079	0.500 0.000 0.500	SM

and

Table 3. Determination of electric transport type (Type) of (Bi₂)_m(Bi₂Te₃)_n from the energy gap in the electronic band structure depending on the exclusion (X) and inclusion (O) of SOI using the E_xc of LDA. E_VBM and E_CBM are the energy of VBM and CBM relative to the Fermi level of each (Bi₂)_m(Bi₂Te₃)_n, respectively. ${\mathrm{E}}_{\mathrm{gap}}^{\mathrm{i}}$ and ${\mathrm{E}}_{\mathrm{gap}}^{\mathrm{d}}$ are the indirect and direct energy gaps, respectively. k_VBM, k_CBM, and ${\mathrm{k}}_{\mathrm{gap}}^{\mathrm{d}}$ are reciprocal positions of VBM, CBM, and ${\mathrm{E}}_{\mathrm{gap}}^{\mathrm{d}}$ in the first Brillouin zone, respectively. SC and SM represent semiconductors and semimetals, respectively.

Exc	Phase	SOI	EVBM	kVBM	ECBM	kCBM	${\mathrm{E}}_{\mathrm{g}\mathrm{a}\mathrm{p}}^{\mathrm{i}}$	${\mathrm{E}}_{\mathrm{g}\mathrm{a}\mathrm{p}}^{\mathrm{d}}$	${\mathrm{k}}_{\mathrm{g}\mathrm{a}\mathrm{p}}^{\mathrm{d}}$	Type
LDA	Bi2Te3	X	−0.098	0.000 0.000 0.000	0.098	0.000 0.000 0.000	0.196	0.196	0.000 0.000 0.000	SC
		O	−0.055	0.140 0.000 0.500	0.056	0.130 0.000 0.500	0.111	0.115	0.130 0.000 0.500	SC
	Bi4Te5	X	0.027	0.040 0.040 0.500	−0.078	0.007 0.007 0.500	−0.105	0.017	0.007 0.007 0.500	SM
		O	−0.004	0.180 0.000 0.000	−0.028	0.090 0.000 0.000	−0.024	0.005	0.000 0.000 0.500	SM
	Bi6Te7	X	0.021	0.040 0.040 0.500	−0.057	0.020 0.000 0.000	−0.078	0.007	0.013 0.013 0.000	SM
		O	−0.009	0.000 0.000 0.000	−0.013	0.100 0.000 0.000	−0.004	0.006	0.010 0.000 0.000	SM
	Bi8Te9	X	−0.004	0.070 0.000 0.500	−0.019	0.013 0.013 0.500	−0.015	0.007	0.013 0.013 0.500	SM
		O	−0.005	0.000 0.000 0.000	−0.002	0.000 0.000 0.000	0.003	0.003	0.000 0.000 0.000	SC
	BiTe	X	0.024	0.060 0.000 0.000	−0.028	0.020 0.020 0.500	−0.052	0.013	0.060 0.000 0.000	SM
		O	−0.062	0.060 0.000 0.000	0.048	0.027 0.027 0.500	0.110	0.134	0.040 0.000 0.500	SC
	Bi4Te3	X	0.025	0.110 0.000 0.000	−0.039	0.027 0.027 0.500	−0.064	0.036	0.027 0.027 0.500	SM
		O	0.081	0.000 0.000 0.000	−0.077	0.500 0.000 0.000	−0.158	0.001	0.020 0.000 0.500	SM
	Bi2Te	X	0.044	0.067 0.067 0.000	−0.083	0.033 0.033 0.000	−0.127	0.005	0.033 0.033 0.000	SM
		O	0.120	0.000 0.000 0.000	−0.045	0.500 0.000 0.500	−0.165	0.021	0.020 0.000 0.000	SM
	Bi7Te3	X	0.085	0.170 0.000 0.000	−0.068	0.033 0.033 0.500	−0.153	0.010	0.033 0.033 0.500	SM
		O	0.041	0.000 0.000 0.000	−0.095	0.500 0.000 0.500	−0.136	0.009	0.000 0.000 0.000	SM
	Bi2	X	0.073	0.053 0.053 0.500	−0.207	0.500 0.000 0.500	−0.280	0.011	0.500 0.000 0.500	SM
		O	0.076	0.000 0.000 0.500	−0.038	0.500 0.000 0.500	−0.114	0.089	0.500 0.000 0.500	SM

, respectively.

PBE calculations give the electronic band structures shown in Figure 3. The band structure of Bi₂Te₃ 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 Bi₂Te₃ is a semiconductor as previously reported [31,32,33]. Note that the experimental energy gap is about 0.13 eV. Figure 3d shows that Bi₈Te₉ has a VBM of −0.004 eV on A−H (0.033, 0.033, 0.5), CBM of −0.03 eV on A−H (0.007, 0.007, 0.5), and ${\mathrm{E}}_{\mathrm{gap}}^{\mathrm{d}}$ of 0.027 eV on A−L (0.02, 0, 0.5). The band structure of BiTe in Figure 3e shows that VBM is 0.017 eV on Γ−M (0.06, 0, 0), CBM is −0.023 eV on A−H (0.013, 0.013, 0.5), and ${\mathrm{E}}_{\mathrm{gap}}^{\mathrm{d}}$ is 0.035 eV on Γ−M (0.06, 0, 0). The band structure of Bi₂ is shown in Figure 3i, where VBM is 0.066 eV on A−H (0.047, 0.047, 0.5), CBM is −0.197 eV on A−L (0.48, 0, 0.5), and ${\mathrm{E}}_{\mathrm{gap}}^{\mathrm{d}}$ is 0.02 eV on A−H (0.047, 0.047, 0.5). Note that our result for Bi₂ is consistent with the experimental reports that bulk Bi is semimetallic [34] since the supercell of Bi₂ is to be bulk Bi. Additionally, the other phases (Bi₄Te₅, Bi₆Te₇, Bi₄Te₃, Bi₂Te, Bi₇Te₃) were semimetallic, as shown in Figure 3 and Table 2. Note that PBE showed that the electric transport types of (Bi₂)_m(Bi₂Te₃)_n were semimetallic because of positive ${\mathrm{E}}_{\mathrm{gap}}^{\mathrm{d}}$ and negative ${\mathrm{E}}_{\mathrm{gap}}^{\mathrm{i}}$ except for Bi₂Te₃. In addition, the electric transport types of (Bi₂)_m(Bi₂Te₃)_n from LDA are the same as those from PBE, although the detailed values (E_VBM, k_VBM, E_CBM, k_CBM, ${\mathrm{E}}_{\mathrm{gap}}^{\mathrm{i}}$, ${\mathrm{k}}_{\mathrm{gap}}^{\mathrm{i}}$) of LDA were different from those of PBE (see Figure 3, Figure 4 and Figure 5 and Table 2 and Table 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 (Bi₂)_m(Bi₂Te₃)_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 Bi₂Te₃, 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 (${\mathrm{E}}_{\mathrm{gap}}^{\mathrm{i}}$ = 0.129 eV, ${\mathrm{E}}_{\mathrm{gap}}^{\mathrm{d}}$ = 0.135 eV). For Bi₂, the SOI calculation increases both VBM (0.113 eV) and CBM (-0.048 eV), leading to a small increase in the energy gap values (${\mathrm{E}}_{\mathrm{gap}}^{\mathrm{i}}$ = −0.161 eV, ${\mathrm{E}}_{\mathrm{gap}}^{\mathrm{d}}$ = 0.079 eV). In addition, for BiTe, the SOI calculations decrease VBM (−0.064 eV) and increase CBM (0.053 eV), eventually opening ${\mathrm{E}}_{\mathrm{gap}}^{\mathrm{i}}$ (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, ${\mathrm{E}}_{\mathrm{gap}}^{\mathrm{i}}$, and ${\mathrm{k}}_{\mathrm{gap}}^{\mathrm{i}}$ 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 Bi₈Te₉ 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, ${\mathrm{E}}_{\mathrm{gap}}^{\mathrm{i}}$, and ${\mathrm{k}}_{\mathrm{gap}}^{\mathrm{i}}$) changed (see Figure 4, Figure 5 and Figure 6 and Table 2 and Table 3). In detail, LDA with SOI changes the electric transport types of Bi₈Te₉ and BiTe from semimetals to semiconductors owing to the decrease in VBM (−0.005 eV for Bi₈Te₉, −0.062 eV for BiTe) and the increase in CBM (−0.002 eV for Bi₈Te₉, 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 (Bi₂)_m(Bi₂Te₃)_n homologous series, only three phases (Bi₂Te₃, Bi₈Te₉, BiTe) can be semiconductors.

4. Conclusions

First-principles DFT calculations of (Bi₂)_m(Bi₂Te₃)_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 (Bi₂)_m(Bi₂Te₃)_n compared to LDA, and LDA overestimated the interlayer distances in (Bi₂)_m(Bi₂Te₃)_n compared to PBE. From the electronic band structures, the electric transport types of (Bi₂)_m(Bi₂Te₃)_n remained as semimetals or semiconductors regardless of whether the E_xc type or SOI was considered, except for Bi₈Te₉ and BiTe. Our calculations revealed that Bi₈Te₉, BiTe, and Bi₂Te₃ are expected to be semiconductors. Consequently, Bi₈Te₉ and BiTe, including the well-known thermoelectric Bi₂Te₃, are expected to be potential thermoelectric materials for automobile energy harvesting technologies.