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Electronic structures and thermoelectric properties of polytype phases of bismuth
Journal of Physics and Chemistry of Solids 134 (2019) 52–57
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Electronic structures and thermoelectric properties of polytype phases of bismuth
T
C.Y. Wua,b, L. Suna, C.P. Lianga, H.R. Gonga,∗, M.L. Changc, D.C. Chenc,∗∗ a
State Key Laboratory of Powder Metallurgy, Central South University, Changsha, Hunan, 410083, China Department of Educational Science, Hunan First Normal University, Changsha, Hunan, 410205, China c School of Materials Science and Energy Engineering, Foshan University, Foshan, 528000, Guangdong, China b
A R T I C LE I N FO
A B S T R A C T
Keywords: Bismuth Electronic structure First principles calculation Seebeck coefficient Thermoelectric property
In this study, first principles calculations demonstrated that the cohesive energies, atomic volumes, or bulk moduli of the rhombohedral (A7), monoclinic (M), and triclinic (T1 and T2) structures of bismuth are very close to each other, thereby suggesting that these four structures of bismuth will probably coexist with each other, which agrees well with previously reported experimental and theoretical observations. The band structures showed that the A7, M, and T1 structures are semimetals with a small overlap, whereas T2 is a direct semiconductor with a narrow band gap. We also found that the T2 structure has much higher Seebeck coefficients than the A7, M, and T1 structures, mainly because of its larger effective mass and lower carrier density. In addition, the figure of merit was determined as highest for T2 among the four structures, and thus the T2 structure of bismuth should be preferred in actual situations in terms of the thermoelectric properties.
1. Introduction During the past decades, the element bismuth (Bi) and its alloys have attracted much interest as a prototype semimetal, mainly because of peculiar features such as its low carrier density, small effective mass, weak overlap, and band structure close to the Fermi level [1–5]. Moreover, pure bismuth has been used extensively in thermoelectric devices because of its excellent thermoelectric properties since the wellknown Seebeck effect was first determined for bismuth in 1826 [6,7]. In addition, the monolayers of Bi and its alloys are two-dimensional materials that exhibit better thermoelectric properties than their corresponding bulks, and they may have potential applications in the new field of thermoelectric devices [8–11]. It is well known that the ground state structure of Bi is a rhombohedral A7 structure (space group No. 166, R3¯m) with trigonal symmetry [12]. This structure can be derived from the simple cubic structure through two separate distortions, i.e., trigonal shear and relative displacement along the trigonal axis in a Peierls distortion of the simple cubic structure [13–15]. Interestingly, the mosaic structure of Bi is considered to exist where it differs slightly from the A7 structure and it may possess unusual physical properties [16–18]. Recently, the mosaic structures of Bi were shown to comprise one monoclinic (M) and two triclinic structures (T1 and T2) [19]. These
∗
three polytype phases coexist with the A7 structure and they may be formed via minute lattice distortions of the A7 structure [19,20]. The crystal structure, lattice constants, and atomic positions of these polytype phases have been identified by high-resolution transmission electron microscopy, X-ray diffraction, and using the CALYPSO software [19,21]. However, to the best of our knowledge, the detailed electronic structures and thermoelectric properties of the polytype phases of Bi have not been reported previously. In the present study, based on highly accurate first principles calculations and Boltzmann transport theory [22,23], we systematically investigated the lattice structure, electronic structure, and thermoelectric properties of the polytype phases of Bi. In particular, were derived the band structures, Seebeck coefficients, electronic conductivity, electric thermal conductivity, and figure of merit for the A7, M, T1, and T2 structures of bismuth, and compared them with each other. The calculated results are discussed in terms of the effective mass and carrier density, which agree well with previously reported experimental observations and theoretical results, and they also provide a deeper understanding of the intrinsic relationships between the band structure and the thermoelectric properties of bismuth.
Corresponding author. Corresponding author. E-mail addresses: [email protected] (H.R. Gong), [email protected] (D.C. Chen).
∗∗
https://doi.org/10.1016/j.jpcs.2019.05.042 Received 9 January 2019; Received in revised form 23 May 2019; Accepted 24 May 2019 Available online 25 May 2019 0022-3697/ © 2019 Elsevier Ltd. All rights reserved.
Journal of Physics and Chemistry of Solids 134 (2019) 52–57
C.Y. Wu, et al.
electric conductivity, and thermal conductivity tensors) were obtained with the following formulae [32]:
να (i, k ) =
1 ∂εi, k , ℏ ∂ka
(1)
σαβ (i, k ) = e 2τi2, k να (i, k ) νβ (i, k ),
σαβ (ε ) =
1 N
∑i,k σαβ (i, k )
(2)
δ (ε − εi, k ) , dε
∂fμ (T ; ε ) ⎤ ⎥ dε , ∂ε ⎣ ⎦
σαβ (T , μ) =
1 Ω
0 (T , μ) = καβ
1 e 2TΩ
ναβ (T , μ) =
1 eTΦ
∫ σαβ (ε ) ⎡⎢−
∂fμ (T ; ε ) ⎤ ⎥ dε , ∂ε ⎣ ⎦
∫ σαβ (ε )(ε − μ)2 ⎡⎢−
∂fμ (T ; ε ) ⎤ ⎥ dε , ∂ε ⎣ ⎦
∫ σαβ (ε − μ) ⎡⎢−
Sij = Ei (∇j T )−1 = (σ −1)αi ναi,
(3)
(4)
(5)
(6) (7)
where να (i, k ) is the αth component of the group velocity of the carriers with the wave vector of k , σαβ (i, k ) are the conductivity tensors, and τ is the relaxation time of the carrier. The transport properties were determined with respect to the temperature and chemical potential using the so-called constant relaxation time approximation (CRTA) [33,34]. It should be noted that the transport properties of many thermoelectric materials, such as degenerately doped semiconductors, Zintl-type phases, oxides, and tellurium, have all been obtained successfully with the CRTA [35,36]. The Seebeck coefficient can be evaluated directly from the first principles band structures using the CRTA with equation (7). The effects of the temperature and carrier density (n) were simulated using the rigid band approximation [37], which assumes that the effects do not change the shape of the band structure, but instead they only shift the Fermi energy.
Fig. 1. Four polytype phases (A7, M, T1, and T2) of Bi and their corresponding first Brillouin zones.
2. Ii. Theoretical methods 3. Results and discussion First principles calculations were conducted using the well-established Vienna ab initio simulation package with the density functional theory (DFT) and projector-augmented wave (PAW) method [24–26]. After several test calculations, the local density approximation (LDA) combined with the spin-orbit coupling (SOC) was selected as the exchange-correlation functional for Bi with several structures in the present study [27,28]. The A7, M, T1, and T2 structures of Bi are shown in Fig. 1, where the unit cells of A7, M, T1, and T2 were set to two, four, two, and eight atoms, respectively. The space groups of A7, M, T1, and T2 structures were determined as No. 166, No. 12, No. 2, and No. 2 (R3¯m, C2/m, P1¯, and P1), respectively [19,29], and the Wyckoff atom positions for A7, M, T1, and T2 as 2c, 4i, 2i, and 8a, respectively. After the test calculations, k-meshes of 15 × 15 × 15, 9 × 15 × 11, 18 × 18 × 16, and 9 × 9 × 6 were employed for the relaxation calculations of the A7, M, T1, and T2 structures, and 30 × 30 × 30, 17 × 27 × 21, 27 × 27 × 23, and 21 × 21 × 13 for the static and electronic structure calculations. The energy criteria were selected as 0.001 and 0.01 meV for the electronic and ionic relaxations, and the unit cell was allowed to fully relax [30,31]. The Boltzmann transport theory and rigid band approach were employed for calculating the transport properties in the BoltzTraP program [32]. The energy eigenvalues of the A7, M, T1, and T2 structures of Bi obtained from the self-consistent converged electronic structure calculations were employed on very dense nonshifted k-point meshes of 405000, 385560, 503010, and 171990 in the full Brillouin zone, respectively. The thermoelectric properties (Seebeck coefficient,
3.1. Band structures of polytype phases of Bi In order to study the band structures of the A7, M, T1, and T2 structures of Bi, the structural parameters (a, b, c, α, β, and γ) were first calculated for the M, T1, and T2 structures of Bi using the PAW-LDASOC method, because our recent study demonstrated that this method is more accurate for describing the A7 structure of Bi than the generalized gradient approximation of Perdew et al. and Perdew-Burke-Ernzerhof for the exchange-correlation functions with the PAW method [27,38,39]. The crystal structures of the four polytype phases of Bi were fully optimized to obtain the lattice parameters and unit volume of Bi [40,41]. In addition, the cohesive energies of the four polytype structures of Bi were calculated in order to compare their structural stability [42–44]. Based on various calculations, Table 1 summarizes the derived structural parameters, cohesive energy, and unit volume for the A7, M, T1, and T2 structures. Table 1 shows that the optimized lattice parameters (a, b, c, α, β, and γ) for the A7, M, T1, and T2 phases of Bi obtained with the PAW + LDA + SOC method are in good agreement with the previously reported calculated results [19,45]. In addition, Table 1 shows that the cohesive energies (E) are almost the same for the four phases, thereby suggesting that the four structures of Bi probably coexist with each other in terms of their thermodynamic stability, which agrees well with previous experimental and theoretical observations [19]. Interestingly, the atomic volumes or bulk moduli (B) of A7, M, T1, and T2 phases are also very close to each other. In addition, the coordinates of the high symmetry k-paths of M, T1, 53
Journal of Physics and Chemistry of Solids 134 (2019) 52–57
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Table 1 Optimized lattice constant, unit volume (V), bulk modulus (B), and cohesive energy (E) values for the A7, M, T1, and T2 structures of bismuth. The theoretical values from previous studies are also listed for comparison. Structure
A7
a(Å) b(Å) c(Å) α β γ E(eV/atom) V(Å3/atom) B (GPa) Method Ref.
4.540 4.540 11.577 90.00° 90.00° 120.00° −2.845 34.44 45.25 LDA + SOC This work
M 4.544 4.544 11.588 90.00° 90.00° 120.00°
LDA + SOC [45]
7.849 4.532 6.505 90.00° 90.00° 143.50° −2.844 34.42 45.12 LDA + SOC This work
T1 7.884 4.557 6.584 90.00° 90.00° 143.00°
GGA [19]
Fig. 2. Band structure of the M phase of Bi (corresponding to symmetry group C2/m) close to the Fermi level (Ef). εh and εe are the energy differences in the valence band top and conduction band bottom at points Y and E as a reference for Ef, respectively.
4.537 4.541 4.659 90.31° 118.20° 60.05° −2.844 34.49 45.23 LDA + SOC This work
T2 4.574 4.579 4.809 90.31° 118.25° 60.04°
GGA [19]
4.537 4.667 13.300 87.07° 90.02° 90.02° −2.845 34.47 44.98 LDA + SOC This work
4.577 4.779 13.030 86.07° 90.02° 90.02°
GGA [19]
Fig. 4. Band structure of the T2 phase of Bi (corresponding to symmetry group P1) close to the Fermi level (Ef). εh and εe are the energy differences in the valence band top and conduction band bottom at point Y as a reference for Ef, respectively.
and the location of the free electron pocket is at point E (0.5, 0.5, 0.5) along the C-E-M1 high symmetry direction in the Brillouin zone. The valence band maximum (VBM) is 0.07679 eV relative to the Fermi level (Ef) near the high symmetric point of Y and the conduction-band minimum (CBM) is 0.00459 eV above Ef at the high symmetric point of E, which indicates that the M phase of Bi has a small number of free holes and no free electrons. Furthermore, the band gap (Eg) and the band overlap (Et) of the M phase are −0.0722 eV and 0.0722 eV, respectively, thereby denoting that the M phase of Bi is a semimetal in a similar manner to the A7 phase of Bi [2,3,47,48]. Second, for the T1 phase, Fig. 3 shows that the position of the free hole pocket is at point Z (0.0, 0.0, 0.5) along the Γ-Z high symmetry direction and the location of the free electron pocket is at point N (0.5, 0.0, 0.5) along the Γ-N high symmetry direction. The VBM (εh) value is 0.08176 eV relative to Ef near the high symmetric point of Z and the CBM (εe) value is −0.01384 eV below Ef at the high symmetric point of N, thereby indicating that the number of free holes for the T1 phase is slightly more than the number of free electrons. In addition, the band gap (Eg) and band overlap (Et) of the T1 phase are −0.0956 eV and 0.0956 eV, respectively, which suggests that the T1 phase is also a semimetal. Third, for the T2 phase, Fig. 4 shows that the positions of the free hole and electron pocket are at point Y (0.0, 0.5, 0.0) along the Γ-Y high symmetry direction in the Brillouin zone. The VBM and CBM are 0.09679 eV and 0.01738 eV relative to Ef near the high symmetric point of Y, respectively. Apparently, the T2 phase of Bi is a direct semiconductor with a narrow band gap of 0.07941 eV. Thus, the A7, M, and
Fig. 3. Band structure of the T1 phase of Bi (corresponding to symmetry group P1¯) close to the Fermi level (Ef). εh and εe are the energy differences in the valence band top and conduction band bottom at points Z and N as a reference for Ef, respectively.
and T2 structures of Bi were determined based on the developed framework [46], and the derived band structures are shown in Figs. 2–4, respectively. It should be noted that the band structure of the A7 structure of Bi was reported in our recent study [27]. First, Fig. 2 shows that for the M phase, the position of the free hole pocket is in the vicinity of point Y (0.0, 0.0, 0.5) along the Γ-Y-H high symmetry direction 54
Journal of Physics and Chemistry of Solids 134 (2019) 52–57
C.Y. Wu, et al.
T1 phases of Bi are semimetals, whereas T2 is a semiconductor. In the following, we show that the T2 phase possesses much better thermoelectric properties. 3.2. Thermoelectric properties of polytype phases of Bi It is generally considered that differences in the crystal structure and energy bands play important roles in determining the thermoelectric properties of various thermoelectric materials. Thus, according to Boltzmann transport theory and the rigid band approach included in the BoltzTraP program [32], we systematically compared the thermoelectric properties of the four polytype structures of Bi. First, we calculated the Seebeck coefficients (S) for the holes or electrons in the A7, M, T1, and T2 phases of Bi as functions of the chemical potentials (μ) according to equation (7). The corresponding effective mass (m∗) and carrier density (n) were then derived in order to obtained a deeper understanding of the Seebeck coefficient. Next, the electric conductivity (σ) and thermal conductivity of the carriers (κ) were obtained using equations (4) and (5) as a function of the carrier density. Finally, the maximum thermoelectric figure of merit (ZhT) was derived for each of the four polytype structures and these values were then compared with each other [49]. After several calculations, the temperature-dependent Seebeck coefficients (S) of the electrons (Se) and holes (Sh) were calculated as functions of the chemical potential (μ) for the four polytype structures of Bi. As a typical example, Fig. 5 shows the values of Se and Sh obtained for the A7, M, T1, and T2 phases of Bi as functions of the chemical potential at a temperature of 300 K. Fig. 5 shows that the differences in the values of Se for the four structures of Bi are small and that the Sh value for each phase is much larger than the corresponding Se value. In addition, further calculations indicated that the electric conductivity (σ) and thermal conductivity of the carriers (κ) of electrons for the A7, M, T1, and T2 phases of Bi are also much smaller than the corresponding values for the holes (figures not shown). Therefore, the holes play a dominant role in determining the thermoelectric properties of each phase, and thus only the properties of the holes for each phase are presented in the following text and figures. Fig. 6 shows the Seebeck coefficients (Sh) of the holes (P type) in bismuth with the A7, M, T1, and T2 structures as a function of the carrier density (n) at 300 K. It should be noted that the carrier density (n) was calculated based on the chemical potential (μ) using the following formula:
n=
∫ g (ε ) e(ε−μ)/k1 T + 1 dε. B
Fig. 6. Seebeck coefficients (Sh) for the holes (P type) in bismuth with the A7, M, T1, and T2 structures as a function of the carrier density (n) at 300 K.
Fig. 6 demonstrates that the highest Seebeck coefficient values for holes in the A7, M, T1, and T2 structures were determined as 119.4, 139.7, 126.8, and 213.2 μV K−1 at the corresponding optimized carrier densities of 8.67 × 1019, 3.32 × 1019, 7.37 × 1019, and 19 −3 1.65 × 10 cm , respectively. Thus, the T2 structure has the largest Sh value and the lowest carrier density among the four structures of Bi. It is important to understand the fundamental reasons why the Seebeck coefficient T2 is the highest among the four structures of Bi. Thus, the Seebeck coefficients (S) were estimated for the four polytype Bi using a relatively simple model of electron transport for metals or degenerate semiconductors (parabolic band) as follows [50,51]:
S=
8π 2κB2 π 2/3 m ∗ T⎛ ⎞ , 2 3eh ⎝ 3n ⎠
(9)
where m* is the effective mass of carriers at the high symmetry points of the band as mentioned above. Clearly, the Seebeck coefficient (S) for Bi in equation (9) is proportional to the effective mass (m*), but inversely proportional to the carrier density (n2/3). The effective mass (m*) was computed directly using BoltzTraP with the CRTA according to the following equation [52,53]:
m ∗ (T ; μ) = n (T ; μ) ×
(8)
e 2τ , σ (T ; μ)
(10)
where σ is the electric conductivity and τ is the user-specified constant relaxation time. As a typical example, Fig. 7 shows the effective mass (m*h) obtained for the holes (P type) in bismuth with the A7, M, T1, and T2 structures as a function of the carrier density (n) at 300 K. Fig. 7 clearly shows that the m*h values are significantly higher for the T2 structure than the other structures, and that the m*h values for T2, A7, M, and T1 corresponding to the optimized carrier density (1.65 × 1019, 8.67 × 1019, 3.32 × 1019, and 7.37 × 1019 cm−3) are 0.120m0, 0.0547m0, 0.0638m0, and 0.0588m0, respectively. Therefore, it follows that the highest effective mass and the lowest carrier density fundamentally lead to the highest Seebeck coefficients for the T2 structure. In addition, the temperature-dependent electric conductivity (σ) and thermal conductivity (κ) of Bi were derived as functions of the carrier density. As typical examples, Figs. 8 and 9 shows the values of σh/τ and κh/τ, respectively, obtained for Bi with the A7, M, T1, and T2 structures at a temperature of 300 K. It should be noted that σh and κh are the electric and thermal conductivities of the holes, respectively, and τ is the relaxation time of the carriers, which should be a constant at a certain temperature. Clearly, Figs. 8 and 9 show that both σh/τ and κh/τ increase for the four structures as the carrier density increases, and that the shapes of the σh/τ and κh/τ curves are very similar to each other, thereby agreeing well with the predictions obtained using the
Fig. 5. Seebeck coefficients for bismuth with the A7, M, T1, and T2 structures as a function of the chemical potential (μ) at 300 K. 55
Journal of Physics and Chemistry of Solids 134 (2019) 52–57
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Fig. 7. Effective mass (m*h) of the holes (P type) in bismuth with the A7, M, T1, and T2 structures as a function of the carrier density (n) at 300 K.
Fig. 10. Figure of merit (ZhT) values for the holes (P type) in bismuth with the A7, M, T1, and T2 structures as a function of the carrier density (n) at 300 K.
(σh/τ), and electric thermal conductivity (κh/τ) were used to calculate the well-known measure of the performance of the thermoelectric properties of Bi expressed by the figure of merit (ZhT) according to the following formula [49]:
Zh T = (S 2σ / τ )(κh/ τ )−1T .
(11)
Fig. 10 shows the figure of merit (ZhT) values calculated for the holes (P type) in bismuth with the A7, M, T1, and T2 structures as a function of the carrier density (n) at 300 K. The maximum figure of merit values for the A7, M, T1, and T2 structures of Bi are 0.326, 0.378, 0.36, and 0.58, respectively, at the corresponding carrier densities of 1.32 × 1020, 5.64 × 1019, 1.367 × 1020, and 3.027 × 1019 cm−3. Thus, the T2 structure of Bi has a much higher figure of merit than the A7, M, and T1 structures, and it would be preferred in actual situations. We hope that the theoretical predictions obtained in the present study will stimulate relevant experimental studies regarding the band structure and thermoelectric properties of the T2 structure of Bi in the near future.
Fig. 8. σh/τ values for the holes (P type) in bismuth with the A7, M, T1, and T2 structures as a function of the carrier density (n) at 300 K.
4. Conclusions In this study, highly accurate first principles calculation and Boltzmann transport theory were combined to investigate the crystal structure, band features, and thermoelectric properties of the A7, M, T1, and T2 structures of Bi. We found that the M, T1, and A7 structures are semimetals with an overlap, whereas T2 is a direct narrow band-gap semiconductor in terms of its band structure. The thermoelectric properties (S, σ/τ, κ/τ, and ZhT) of the holes in the T2, M, T1, and A7 structures were derived as functions of the temperature and carrier density, which showed that the highest effective mass and lowest carrier density fundamentally lead to the highest Seebeck coefficients and figure of merit for the T2 structure. The results obtained in the present study agree well with previously reported experimental observations. We hope that these findings will stimulate relevant experimental studies regarding the band structure and thermoelectric properties of the T2 structure of Bi in the near future.
Fig. 9. κh/τ values for the holes (P type) in bismuth with the A7, M, T1, and T2 structures as a function of the carrier density (n) at 300 K.
Wiedemann–Franz law [53]. Interestingly, the values of σh/τ and κh/τ are lower for the T2 structure than the other three structures at the same temperature and carrier density, which implies that the effective mass of the T2 structure is the highest [53] and this is consistent with the results in Fig. 7 in terms of the electronic structure. Finally, the derived Seebeck coefficient (S), electric conductivity
Acknowledgments This study was supported by the State Key Laboratory of Powder Metallurgy, Central South University, Changsha, China, Key Project of Department of Education of Guangdong Province (2016GCZX008), and Project of Engineering Research Center of Foshan (20172010018). 56
Journal of Physics and Chemistry of Solids 134 (2019) 52–57
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57
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Journal of Physics and Chemistry of Solids journal homepage: www.elsevier.com/locate/jpcs
Electronic structures and thermoelectric properties of polytype phases of bismuth
T
C.Y. Wua,b, L. Suna, C.P. Lianga, H.R. Gonga,∗, M.L. Changc, D.C. Chenc,∗∗ a
State Key Laboratory of Powder Metallurgy, Central South University, Changsha, Hunan, 410083, China Department of Educational Science, Hunan First Normal University, Changsha, Hunan, 410205, China c School of Materials Science and Energy Engineering, Foshan University, Foshan, 528000, Guangdong, China b
A R T I C LE I N FO
A B S T R A C T
Keywords: Bismuth Electronic structure First principles calculation Seebeck coefficient Thermoelectric property
In this study, first principles calculations demonstrated that the cohesive energies, atomic volumes, or bulk moduli of the rhombohedral (A7), monoclinic (M), and triclinic (T1 and T2) structures of bismuth are very close to each other, thereby suggesting that these four structures of bismuth will probably coexist with each other, which agrees well with previously reported experimental and theoretical observations. The band structures showed that the A7, M, and T1 structures are semimetals with a small overlap, whereas T2 is a direct semiconductor with a narrow band gap. We also found that the T2 structure has much higher Seebeck coefficients than the A7, M, and T1 structures, mainly because of its larger effective mass and lower carrier density. In addition, the figure of merit was determined as highest for T2 among the four structures, and thus the T2 structure of bismuth should be preferred in actual situations in terms of the thermoelectric properties.
1. Introduction During the past decades, the element bismuth (Bi) and its alloys have attracted much interest as a prototype semimetal, mainly because of peculiar features such as its low carrier density, small effective mass, weak overlap, and band structure close to the Fermi level [1–5]. Moreover, pure bismuth has been used extensively in thermoelectric devices because of its excellent thermoelectric properties since the wellknown Seebeck effect was first determined for bismuth in 1826 [6,7]. In addition, the monolayers of Bi and its alloys are two-dimensional materials that exhibit better thermoelectric properties than their corresponding bulks, and they may have potential applications in the new field of thermoelectric devices [8–11]. It is well known that the ground state structure of Bi is a rhombohedral A7 structure (space group No. 166, R3¯m) with trigonal symmetry [12]. This structure can be derived from the simple cubic structure through two separate distortions, i.e., trigonal shear and relative displacement along the trigonal axis in a Peierls distortion of the simple cubic structure [13–15]. Interestingly, the mosaic structure of Bi is considered to exist where it differs slightly from the A7 structure and it may possess unusual physical properties [16–18]. Recently, the mosaic structures of Bi were shown to comprise one monoclinic (M) and two triclinic structures (T1 and T2) [19]. These
∗
three polytype phases coexist with the A7 structure and they may be formed via minute lattice distortions of the A7 structure [19,20]. The crystal structure, lattice constants, and atomic positions of these polytype phases have been identified by high-resolution transmission electron microscopy, X-ray diffraction, and using the CALYPSO software [19,21]. However, to the best of our knowledge, the detailed electronic structures and thermoelectric properties of the polytype phases of Bi have not been reported previously. In the present study, based on highly accurate first principles calculations and Boltzmann transport theory [22,23], we systematically investigated the lattice structure, electronic structure, and thermoelectric properties of the polytype phases of Bi. In particular, were derived the band structures, Seebeck coefficients, electronic conductivity, electric thermal conductivity, and figure of merit for the A7, M, T1, and T2 structures of bismuth, and compared them with each other. The calculated results are discussed in terms of the effective mass and carrier density, which agree well with previously reported experimental observations and theoretical results, and they also provide a deeper understanding of the intrinsic relationships between the band structure and the thermoelectric properties of bismuth.
Corresponding author. Corresponding author. E-mail addresses: [email protected] (H.R. Gong), [email protected] (D.C. Chen).
∗∗
https://doi.org/10.1016/j.jpcs.2019.05.042 Received 9 January 2019; Received in revised form 23 May 2019; Accepted 24 May 2019 Available online 25 May 2019 0022-3697/ © 2019 Elsevier Ltd. All rights reserved.
Journal of Physics and Chemistry of Solids 134 (2019) 52–57
C.Y. Wu, et al.
electric conductivity, and thermal conductivity tensors) were obtained with the following formulae [32]:
να (i, k ) =
1 ∂εi, k , ℏ ∂ka
(1)
σαβ (i, k ) = e 2τi2, k να (i, k ) νβ (i, k ),
σαβ (ε ) =
1 N
∑i,k σαβ (i, k )
(2)
δ (ε − εi, k ) , dε
∂fμ (T ; ε ) ⎤ ⎥ dε , ∂ε ⎣ ⎦
σαβ (T , μ) =
1 Ω
0 (T , μ) = καβ
1 e 2TΩ
ναβ (T , μ) =
1 eTΦ
∫ σαβ (ε ) ⎡⎢−
∂fμ (T ; ε ) ⎤ ⎥ dε , ∂ε ⎣ ⎦
∫ σαβ (ε )(ε − μ)2 ⎡⎢−
∂fμ (T ; ε ) ⎤ ⎥ dε , ∂ε ⎣ ⎦
∫ σαβ (ε − μ) ⎡⎢−
Sij = Ei (∇j T )−1 = (σ −1)αi ναi,
(3)
(4)
(5)
(6) (7)
where να (i, k ) is the αth component of the group velocity of the carriers with the wave vector of k , σαβ (i, k ) are the conductivity tensors, and τ is the relaxation time of the carrier. The transport properties were determined with respect to the temperature and chemical potential using the so-called constant relaxation time approximation (CRTA) [33,34]. It should be noted that the transport properties of many thermoelectric materials, such as degenerately doped semiconductors, Zintl-type phases, oxides, and tellurium, have all been obtained successfully with the CRTA [35,36]. The Seebeck coefficient can be evaluated directly from the first principles band structures using the CRTA with equation (7). The effects of the temperature and carrier density (n) were simulated using the rigid band approximation [37], which assumes that the effects do not change the shape of the band structure, but instead they only shift the Fermi energy.
Fig. 1. Four polytype phases (A7, M, T1, and T2) of Bi and their corresponding first Brillouin zones.
2. Ii. Theoretical methods 3. Results and discussion First principles calculations were conducted using the well-established Vienna ab initio simulation package with the density functional theory (DFT) and projector-augmented wave (PAW) method [24–26]. After several test calculations, the local density approximation (LDA) combined with the spin-orbit coupling (SOC) was selected as the exchange-correlation functional for Bi with several structures in the present study [27,28]. The A7, M, T1, and T2 structures of Bi are shown in Fig. 1, where the unit cells of A7, M, T1, and T2 were set to two, four, two, and eight atoms, respectively. The space groups of A7, M, T1, and T2 structures were determined as No. 166, No. 12, No. 2, and No. 2 (R3¯m, C2/m, P1¯, and P1), respectively [19,29], and the Wyckoff atom positions for A7, M, T1, and T2 as 2c, 4i, 2i, and 8a, respectively. After the test calculations, k-meshes of 15 × 15 × 15, 9 × 15 × 11, 18 × 18 × 16, and 9 × 9 × 6 were employed for the relaxation calculations of the A7, M, T1, and T2 structures, and 30 × 30 × 30, 17 × 27 × 21, 27 × 27 × 23, and 21 × 21 × 13 for the static and electronic structure calculations. The energy criteria were selected as 0.001 and 0.01 meV for the electronic and ionic relaxations, and the unit cell was allowed to fully relax [30,31]. The Boltzmann transport theory and rigid band approach were employed for calculating the transport properties in the BoltzTraP program [32]. The energy eigenvalues of the A7, M, T1, and T2 structures of Bi obtained from the self-consistent converged electronic structure calculations were employed on very dense nonshifted k-point meshes of 405000, 385560, 503010, and 171990 in the full Brillouin zone, respectively. The thermoelectric properties (Seebeck coefficient,
3.1. Band structures of polytype phases of Bi In order to study the band structures of the A7, M, T1, and T2 structures of Bi, the structural parameters (a, b, c, α, β, and γ) were first calculated for the M, T1, and T2 structures of Bi using the PAW-LDASOC method, because our recent study demonstrated that this method is more accurate for describing the A7 structure of Bi than the generalized gradient approximation of Perdew et al. and Perdew-Burke-Ernzerhof for the exchange-correlation functions with the PAW method [27,38,39]. The crystal structures of the four polytype phases of Bi were fully optimized to obtain the lattice parameters and unit volume of Bi [40,41]. In addition, the cohesive energies of the four polytype structures of Bi were calculated in order to compare their structural stability [42–44]. Based on various calculations, Table 1 summarizes the derived structural parameters, cohesive energy, and unit volume for the A7, M, T1, and T2 structures. Table 1 shows that the optimized lattice parameters (a, b, c, α, β, and γ) for the A7, M, T1, and T2 phases of Bi obtained with the PAW + LDA + SOC method are in good agreement with the previously reported calculated results [19,45]. In addition, Table 1 shows that the cohesive energies (E) are almost the same for the four phases, thereby suggesting that the four structures of Bi probably coexist with each other in terms of their thermodynamic stability, which agrees well with previous experimental and theoretical observations [19]. Interestingly, the atomic volumes or bulk moduli (B) of A7, M, T1, and T2 phases are also very close to each other. In addition, the coordinates of the high symmetry k-paths of M, T1, 53
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C.Y. Wu, et al.
Table 1 Optimized lattice constant, unit volume (V), bulk modulus (B), and cohesive energy (E) values for the A7, M, T1, and T2 structures of bismuth. The theoretical values from previous studies are also listed for comparison. Structure
A7
a(Å) b(Å) c(Å) α β γ E(eV/atom) V(Å3/atom) B (GPa) Method Ref.
4.540 4.540 11.577 90.00° 90.00° 120.00° −2.845 34.44 45.25 LDA + SOC This work
M 4.544 4.544 11.588 90.00° 90.00° 120.00°
LDA + SOC [45]
7.849 4.532 6.505 90.00° 90.00° 143.50° −2.844 34.42 45.12 LDA + SOC This work
T1 7.884 4.557 6.584 90.00° 90.00° 143.00°
GGA [19]
Fig. 2. Band structure of the M phase of Bi (corresponding to symmetry group C2/m) close to the Fermi level (Ef). εh and εe are the energy differences in the valence band top and conduction band bottom at points Y and E as a reference for Ef, respectively.
4.537 4.541 4.659 90.31° 118.20° 60.05° −2.844 34.49 45.23 LDA + SOC This work
T2 4.574 4.579 4.809 90.31° 118.25° 60.04°
GGA [19]
4.537 4.667 13.300 87.07° 90.02° 90.02° −2.845 34.47 44.98 LDA + SOC This work
4.577 4.779 13.030 86.07° 90.02° 90.02°
GGA [19]
Fig. 4. Band structure of the T2 phase of Bi (corresponding to symmetry group P1) close to the Fermi level (Ef). εh and εe are the energy differences in the valence band top and conduction band bottom at point Y as a reference for Ef, respectively.
and the location of the free electron pocket is at point E (0.5, 0.5, 0.5) along the C-E-M1 high symmetry direction in the Brillouin zone. The valence band maximum (VBM) is 0.07679 eV relative to the Fermi level (Ef) near the high symmetric point of Y and the conduction-band minimum (CBM) is 0.00459 eV above Ef at the high symmetric point of E, which indicates that the M phase of Bi has a small number of free holes and no free electrons. Furthermore, the band gap (Eg) and the band overlap (Et) of the M phase are −0.0722 eV and 0.0722 eV, respectively, thereby denoting that the M phase of Bi is a semimetal in a similar manner to the A7 phase of Bi [2,3,47,48]. Second, for the T1 phase, Fig. 3 shows that the position of the free hole pocket is at point Z (0.0, 0.0, 0.5) along the Γ-Z high symmetry direction and the location of the free electron pocket is at point N (0.5, 0.0, 0.5) along the Γ-N high symmetry direction. The VBM (εh) value is 0.08176 eV relative to Ef near the high symmetric point of Z and the CBM (εe) value is −0.01384 eV below Ef at the high symmetric point of N, thereby indicating that the number of free holes for the T1 phase is slightly more than the number of free electrons. In addition, the band gap (Eg) and band overlap (Et) of the T1 phase are −0.0956 eV and 0.0956 eV, respectively, which suggests that the T1 phase is also a semimetal. Third, for the T2 phase, Fig. 4 shows that the positions of the free hole and electron pocket are at point Y (0.0, 0.5, 0.0) along the Γ-Y high symmetry direction in the Brillouin zone. The VBM and CBM are 0.09679 eV and 0.01738 eV relative to Ef near the high symmetric point of Y, respectively. Apparently, the T2 phase of Bi is a direct semiconductor with a narrow band gap of 0.07941 eV. Thus, the A7, M, and
Fig. 3. Band structure of the T1 phase of Bi (corresponding to symmetry group P1¯) close to the Fermi level (Ef). εh and εe are the energy differences in the valence band top and conduction band bottom at points Z and N as a reference for Ef, respectively.
and T2 structures of Bi were determined based on the developed framework [46], and the derived band structures are shown in Figs. 2–4, respectively. It should be noted that the band structure of the A7 structure of Bi was reported in our recent study [27]. First, Fig. 2 shows that for the M phase, the position of the free hole pocket is in the vicinity of point Y (0.0, 0.0, 0.5) along the Γ-Y-H high symmetry direction 54
Journal of Physics and Chemistry of Solids 134 (2019) 52–57
C.Y. Wu, et al.
T1 phases of Bi are semimetals, whereas T2 is a semiconductor. In the following, we show that the T2 phase possesses much better thermoelectric properties. 3.2. Thermoelectric properties of polytype phases of Bi It is generally considered that differences in the crystal structure and energy bands play important roles in determining the thermoelectric properties of various thermoelectric materials. Thus, according to Boltzmann transport theory and the rigid band approach included in the BoltzTraP program [32], we systematically compared the thermoelectric properties of the four polytype structures of Bi. First, we calculated the Seebeck coefficients (S) for the holes or electrons in the A7, M, T1, and T2 phases of Bi as functions of the chemical potentials (μ) according to equation (7). The corresponding effective mass (m∗) and carrier density (n) were then derived in order to obtained a deeper understanding of the Seebeck coefficient. Next, the electric conductivity (σ) and thermal conductivity of the carriers (κ) were obtained using equations (4) and (5) as a function of the carrier density. Finally, the maximum thermoelectric figure of merit (ZhT) was derived for each of the four polytype structures and these values were then compared with each other [49]. After several calculations, the temperature-dependent Seebeck coefficients (S) of the electrons (Se) and holes (Sh) were calculated as functions of the chemical potential (μ) for the four polytype structures of Bi. As a typical example, Fig. 5 shows the values of Se and Sh obtained for the A7, M, T1, and T2 phases of Bi as functions of the chemical potential at a temperature of 300 K. Fig. 5 shows that the differences in the values of Se for the four structures of Bi are small and that the Sh value for each phase is much larger than the corresponding Se value. In addition, further calculations indicated that the electric conductivity (σ) and thermal conductivity of the carriers (κ) of electrons for the A7, M, T1, and T2 phases of Bi are also much smaller than the corresponding values for the holes (figures not shown). Therefore, the holes play a dominant role in determining the thermoelectric properties of each phase, and thus only the properties of the holes for each phase are presented in the following text and figures. Fig. 6 shows the Seebeck coefficients (Sh) of the holes (P type) in bismuth with the A7, M, T1, and T2 structures as a function of the carrier density (n) at 300 K. It should be noted that the carrier density (n) was calculated based on the chemical potential (μ) using the following formula:
n=
∫ g (ε ) e(ε−μ)/k1 T + 1 dε. B
Fig. 6. Seebeck coefficients (Sh) for the holes (P type) in bismuth with the A7, M, T1, and T2 structures as a function of the carrier density (n) at 300 K.
Fig. 6 demonstrates that the highest Seebeck coefficient values for holes in the A7, M, T1, and T2 structures were determined as 119.4, 139.7, 126.8, and 213.2 μV K−1 at the corresponding optimized carrier densities of 8.67 × 1019, 3.32 × 1019, 7.37 × 1019, and 19 −3 1.65 × 10 cm , respectively. Thus, the T2 structure has the largest Sh value and the lowest carrier density among the four structures of Bi. It is important to understand the fundamental reasons why the Seebeck coefficient T2 is the highest among the four structures of Bi. Thus, the Seebeck coefficients (S) were estimated for the four polytype Bi using a relatively simple model of electron transport for metals or degenerate semiconductors (parabolic band) as follows [50,51]:
S=
8π 2κB2 π 2/3 m ∗ T⎛ ⎞ , 2 3eh ⎝ 3n ⎠
(9)
where m* is the effective mass of carriers at the high symmetry points of the band as mentioned above. Clearly, the Seebeck coefficient (S) for Bi in equation (9) is proportional to the effective mass (m*), but inversely proportional to the carrier density (n2/3). The effective mass (m*) was computed directly using BoltzTraP with the CRTA according to the following equation [52,53]:
m ∗ (T ; μ) = n (T ; μ) ×
(8)
e 2τ , σ (T ; μ)
(10)
where σ is the electric conductivity and τ is the user-specified constant relaxation time. As a typical example, Fig. 7 shows the effective mass (m*h) obtained for the holes (P type) in bismuth with the A7, M, T1, and T2 structures as a function of the carrier density (n) at 300 K. Fig. 7 clearly shows that the m*h values are significantly higher for the T2 structure than the other structures, and that the m*h values for T2, A7, M, and T1 corresponding to the optimized carrier density (1.65 × 1019, 8.67 × 1019, 3.32 × 1019, and 7.37 × 1019 cm−3) are 0.120m0, 0.0547m0, 0.0638m0, and 0.0588m0, respectively. Therefore, it follows that the highest effective mass and the lowest carrier density fundamentally lead to the highest Seebeck coefficients for the T2 structure. In addition, the temperature-dependent electric conductivity (σ) and thermal conductivity (κ) of Bi were derived as functions of the carrier density. As typical examples, Figs. 8 and 9 shows the values of σh/τ and κh/τ, respectively, obtained for Bi with the A7, M, T1, and T2 structures at a temperature of 300 K. It should be noted that σh and κh are the electric and thermal conductivities of the holes, respectively, and τ is the relaxation time of the carriers, which should be a constant at a certain temperature. Clearly, Figs. 8 and 9 show that both σh/τ and κh/τ increase for the four structures as the carrier density increases, and that the shapes of the σh/τ and κh/τ curves are very similar to each other, thereby agreeing well with the predictions obtained using the
Fig. 5. Seebeck coefficients for bismuth with the A7, M, T1, and T2 structures as a function of the chemical potential (μ) at 300 K. 55
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C.Y. Wu, et al.
Fig. 7. Effective mass (m*h) of the holes (P type) in bismuth with the A7, M, T1, and T2 structures as a function of the carrier density (n) at 300 K.
Fig. 10. Figure of merit (ZhT) values for the holes (P type) in bismuth with the A7, M, T1, and T2 structures as a function of the carrier density (n) at 300 K.
(σh/τ), and electric thermal conductivity (κh/τ) were used to calculate the well-known measure of the performance of the thermoelectric properties of Bi expressed by the figure of merit (ZhT) according to the following formula [49]:
Zh T = (S 2σ / τ )(κh/ τ )−1T .
(11)
Fig. 10 shows the figure of merit (ZhT) values calculated for the holes (P type) in bismuth with the A7, M, T1, and T2 structures as a function of the carrier density (n) at 300 K. The maximum figure of merit values for the A7, M, T1, and T2 structures of Bi are 0.326, 0.378, 0.36, and 0.58, respectively, at the corresponding carrier densities of 1.32 × 1020, 5.64 × 1019, 1.367 × 1020, and 3.027 × 1019 cm−3. Thus, the T2 structure of Bi has a much higher figure of merit than the A7, M, and T1 structures, and it would be preferred in actual situations. We hope that the theoretical predictions obtained in the present study will stimulate relevant experimental studies regarding the band structure and thermoelectric properties of the T2 structure of Bi in the near future.
Fig. 8. σh/τ values for the holes (P type) in bismuth with the A7, M, T1, and T2 structures as a function of the carrier density (n) at 300 K.
4. Conclusions In this study, highly accurate first principles calculation and Boltzmann transport theory were combined to investigate the crystal structure, band features, and thermoelectric properties of the A7, M, T1, and T2 structures of Bi. We found that the M, T1, and A7 structures are semimetals with an overlap, whereas T2 is a direct narrow band-gap semiconductor in terms of its band structure. The thermoelectric properties (S, σ/τ, κ/τ, and ZhT) of the holes in the T2, M, T1, and A7 structures were derived as functions of the temperature and carrier density, which showed that the highest effective mass and lowest carrier density fundamentally lead to the highest Seebeck coefficients and figure of merit for the T2 structure. The results obtained in the present study agree well with previously reported experimental observations. We hope that these findings will stimulate relevant experimental studies regarding the band structure and thermoelectric properties of the T2 structure of Bi in the near future.
Fig. 9. κh/τ values for the holes (P type) in bismuth with the A7, M, T1, and T2 structures as a function of the carrier density (n) at 300 K.
Wiedemann–Franz law [53]. Interestingly, the values of σh/τ and κh/τ are lower for the T2 structure than the other three structures at the same temperature and carrier density, which implies that the effective mass of the T2 structure is the highest [53] and this is consistent with the results in Fig. 7 in terms of the electronic structure. Finally, the derived Seebeck coefficient (S), electric conductivity
Acknowledgments This study was supported by the State Key Laboratory of Powder Metallurgy, Central South University, Changsha, China, Key Project of Department of Education of Guangdong Province (2016GCZX008), and Project of Engineering Research Center of Foshan (20172010018). 56
Journal of Physics and Chemistry of Solids 134 (2019) 52–57
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