- Email: [email protected]
Thermoelectric properties of monolayer α-Te: Low lattice thermal conductivity and extremely high dimensionless figure of merit
JID:PLA AID:126222 /SCO Doctopic: Condensed matter
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Thermoelectric properties of monolayer α -Te: Low lattice thermal conductivity and extremely high dimensionless figure of merit
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School of Physics and Wuhan National High Magnetic Field Center, Huazhong University of Science and Technology, Wuhan 430074, China
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Xia Jiang, Lin Zhu ∗ , Bowen Li, Kailun Yao
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Article history: Received 21 October 2019 Received in revised form 13 December 2019 Accepted 20 December 2019 Available online xxxx Communicated by R. Wu Keywords: Thermoelectric properties Monolayer α -Te Semiconductor Electrical conductivity Lattice thermal conductivity Seebeck coefficient Dimensionless figure of merit
We have investigated the thermoelectric properties of monolayer α -Te by the first-principles and the Boltzmann transport theory. The results show that monolayer α -Te is an indirect bandgap semiconductor with a moderate bandgap of 0.63 eV. The value of the Seebeck coefficient for monolayer α -Te is larger than 200 μVK−1 , the electrical conductivity has the magnitude of 1 × 106 . The lattice thermal conductivity is smaller than 4 Wm−1 K−1 , which is extremely low. Instructively, the dimensionless figure of merit for monolayer α -Te is more than 4, which is markedly high. The above phenomena imply that monolayer α -Te can be served as a considerable thermoelectric material with excellent thermoelectric conversion efficiency. © 2020 Elsevier B.V. All rights reserved.
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1. Introduction
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The world energy demand shows an increasing trend year by year, and 90% of human total energy consumption comes from fossil fuels, which are non-renewable energy sources and the usage rate is very low. Almost 70% of fossil fuels are released into the atmosphere in the form of waste heat and exhaust gas, which causes serious air pollution [1]. Therefore, it is imperative to explore new energy sources to improve energy efficiency. Thermoelectric materials can realize the conversion of heat and electric energy, and have the advantages of small size, light weight, noise-free, pollution-free and long service life, etc. [2]. Then thermoelectric materials is very promising energy materials, and the thermoelectric properties have been extensively studied in recent years [3–6]. Yet to date, thermoelectric materials have been well used in waste heat recovery, aerospace, microelectronics, biomedicine and other fields. Unfortunately, the conversion efficiency of traditional thermoelectric materials is very low, they can’t be used commercially on a large scale. Based on the above discussion, it is essential to find new materials with higher thermoelectric conversion efficiency, improving the performance of traditional thermoelectric materials. Generally, the dimensionless figure of merit Z T = S 2 σ T /k is considered to be used to measure the thermoelec-
60 61 62 63 64 65 66
*
Corresponding author. E-mail address: [email protected] (L. Zhu).
https://doi.org/10.1016/j.physleta.2019.126222 0375-9601/© 2020 Elsevier B.V. All rights reserved.
tric conversion performance of materials, the larger the Z T , the better the thermoelectric conversion performance [7,8]. Obviously, large thermoelectric conversion efficiency implies large S 2 σ and small k. Among them, S 2 σ is usually considered as power factor (PF), S represents the thermoelectric power or Seebeck coefficient, σ stands for electrical conductivity. And k is the total thermal conductivity of the material, which comes from the contribution of electronic thermal conductivity (kele ) and lattice thermal conductivity (klatt ). The electrical conductivity (σ ) and electrical thermal conductivity (kele ) are associated by the Wiedemann–Franz law: kele = L σ T , where L is the Lorentz number, its most typical value is 2.45 × 10−8 WK2 for metals and degenerate semiconductors [9,10]. In fact, L varies with material and temperature [11], but we often overlook its changes. Recently, Heremans et al. found that the tellurium (Te) thin film not only showed high flexibility but also had a large power factor of 321 μWm−1 K−2 at room temperature [12]. Previous research pointed out that the thermoelectric properties of materials could be improved by reducing the dimensionality of the materials [1, 13–18]. Based on the above research background, we predict that the thermoelectric properties of Te thin film can also be enhanced by making it into monolayer Te. But up to present, there have not reports on thermoelectric properties of monolayer Te. A recent theoretical work proposed three types of structure for two-dimensional monolayer Te, that is tellurene, which are stable 1T-MoS2 -like (α -Te), metastable tetragonal (β -Te) and 2H-MoS2 -like (γ -Te) structures, among which monolayer α -Te and
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Table 1 The elastic modulus C 2D , effective mass m∗ , deformation potential E ι , the product of relaxation time and temperature τ (s) T of the electron and hole, which are obtained from the SCF-SPB model.
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Carriers
C 2D (J m−2 )
m∗ (me )
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τ (s) T
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hole electron
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β -Te are semiconductor with the band gap of 0.75 and 1.47 eV,
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respectively [19,20]. Previous studies have shown that monolayer α -Te has high mobility [20]. Considering the stability of monolayer α -Te structure, we speculate that it has a good thermoelectric performance. In this paper, we systematically study the structure and thermoelectric properties of monolayer α -Te, finding ultralow thermal conductivity, extremely high dimensionless figure of merit. Moreover, the p-type doping has better thermoelectric conversion performance.
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2. Computation details
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The structural optimization is calculated by first-principles density functional theory (DFT) [21,22] which is implemented in Vienna ab initio simulation package (VASP) [23–26]. The projectoraugmented-wave (PAW) [27] pseudopotentials and the Perdew– Burke–Ernzerhof (PBE) [28,29] exchange correlation functional are used. We set up an 11 × 11 × 1 Monkhorst-pack k-mesh. The kinetic-energy cutoff for the plane wave is chosen as 300 eV. The lattice parameters and atomic positions are fully optimized until the maximum Hellmann-Feynman forces exerted on each atom are less than 1E-6 eV/Å. The electronic band structure is calculated based on the optimized structure, and the k grid is set to 25 × 25 × 1. Taking into account the large atomic mass of Te, the spin-orbit coupling (SOC) has been considered [20,30,31]. We also consider the effect of hybrid functional (HSE) [20,32–34] on the band gap calculation. That is to say, we have calculated the energy bands with SOC and HSE06. After the structural optimization and band calculation, we perform electronic transport properties calculation by using BoltzTraP2 code [35]. As a result of electrical conductivity and the electron thermal conductivity in the output of BoltzTraP2 contains relaxation time, we utilize a self-consistent single parabolic band model (SCF-SPB) to calculate the electron relaxation time (τ ),
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τ=
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μ=
μm∗ e eh¯ 3 C 2D
k B T m∗md∗ E ι2
where C 2D (C 2D = 1/ S 0 ∂
2
ι 2 ∗ E /∂(
ι ) ), m 0
(m∗
=
2 h¯ / ∂∂ KE2 ), md∗
2
=
(m∗x m∗y )1/2 and E ι (E ι = E /( ι/ι0 )) are elastic constant, effective mass, average effective mass and deformation potential, respectively [36–40]. The calculated results are list in Table 1. We calculate the phonon spectrum of monolayer α -Te with harmonic approximation by Phonopy code [41]. And the lattice thermal conductivity is calculated by solving the phonon Boltzmann transport equation, which is carried out by ShengBTE code [42]. We adopt the finite displacement approach to calculate the second-order harmonic interatomic force constants (IFCs) and third-order anharmonic interatomic force constants (IFCs), and a 5 × 5 × 1 super cell containing 75 atoms based on the optimized structure is used. The convergence testing for k-mesh is carried out from 13 × 13 × 1 to 53 × 53 × 1 in the ShengBTE code.
Fig. 1. The optimal structure of monolayer α -Te, (a) top view, (b) side view. (c) The calculated band structure and density of states (DOS) of monolayer α -Te.
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Fig. 2. The variation of chemical potential with the change of carrier concentration under different temperature. (For interpretation of the colors in the figure(s), the reader is referred to the web version of this article.)
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3. Results and discussion
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From Figs. 1 (a) and (b), we notice that monolayer α -Te has a structure similar to 1T-MoS2 , there are three Te atoms per unit cell, and the structural parameters of the optimized structure are a = b = 4.23 Å, layer thickness is 3.623 Å. Looking closely at Fig. 1 (c), we observe that monolayer α -Te is an indirect bandgap semiconductor with a moderate bandgap of 0.63 eV. The highest point of the valence band is between G-M, and the lowest point of the conduction band is located at G point. This moderate band gap gives us such a signal that the thermoelectric performance can be optimized within a reasonable doping level. What’s more, we find that the valence bands exhibit much higher density of states than the conduction bands, which implies that p-type doping thermoelectric performance is superior to that of n-type doping. Generally, doping will lead the shift in the Fermi level, the appropriate doping concentration need the chemical potential be in the band gap between the conduction and the valence band. The relationship between carrier concentration and chemical potential under different temperatures is displayed in Fig. 2, which manifests that for n-type doing, the doping concentration should be in the range of 0 to 1019 cm−3 , and for p-type doing, the best doping concentration is in the range of 0 to 1021 cm−3 . So we have
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Fig. 3. The electrical conductivity (a), electronic thermal conductivity (b), Seebeck coefficient (c) and power factor (d) for both p-type and n-type monolayer of carrier concentration at temperatures of 300, 400, 500 and 600 K.
α -Te as a function
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Fig. 4. (a) The phonon dispersion relations along high symmetry direction, (b) the temperature dependence of phonon thermal conductivity along x and y axis.
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studied the thermoelectric transport properties of monolayer α -Te under light doping below. Figs. 3 (a) to (d) present the variation of thermoelectric coefficient with doping concentration and temperature. The good thermoelectric materials should have electrical conductivity with the magnitude of 105 −1 m−1 [11]. The electrical conductivity calculated by us is always in such a favorable range. Comparing the electrical conductivity and electronic thermal conductivity, we note that their trend is similar, which confirms that the electrical conductivity and electrical thermal conductivity are associated with the Wiedemann–Franz law: kele = L σ T , and the difference is because L varies with temperature and material, which has been mentioned in many literatures [10,11]. Besides, the Seebeck coefficient of many thermoelectric materials with excellent thermoelectric conversion performances is around 230 μV/K, which has been confirmed by experiments and theoretical studies [43,44]. In our research, the Seebeck coefficients in most part of the doping concentration region are greater than 200 μV/K. Therefore, we deduce that monolayer α -Te should have good thermoelectric conversion performance. By analyzing the power factor, we observe that the maximum power factor exceeds 1.1 × 10−3 Wm−1 K−1 , and that n-type doping is better than p-type doping when the sample con-
cm−3 , whereas p-type doping is
centration is below 1 × 10 better than n-type doping, as is depicted in Fig. 3 (d). Since the thermoelectric conversion performance is ultimately measured by Z T , in order to analyze the thermoelectric conversion performance of monolayer α -Te more accurately, it is necessary to calculate the lattice thermal conductivity of monolayer α -Te. The phonon spectrum is described in Fig. 4 (a), where the phonon frequency ranges from 0 to 5 THZ. There is a slight coupling between the acoustic and optical phonon modes at the high symmetry point M-L-H. Therefore, we predict that the thermal conductivity of the monolayer α -Te should be relatively small. Fig. 4 depicts the change of lattice thermal conductivity with temperature, which shows that the lattice thermal conductivity is always less than 3 W/K in the temperature range of 150 to 1500 K, and decreases with the increase of temperature. Based on above calculations, we analyze the variation of ZT with doping type and doping concentration at temperatures: 300 K, 400 K, 500 K and 600 K, as presented in Fig. 5. The maximum value of Z T is more than 4, which is very large. And the variation of ZT with carrier concentration and carrier species is similar to that of power factor. We predict that monolayer α -Te 19
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[3] Y.S. Liu, X.Y. Shao, T. Shao, J.Y. Zhang, Y.W. Kuang, D.B. Zhang, Z.G. Shao, H.L. Yu, X.K. Hong, J.F. Feng, X.F. Yang, X.S. Chen, X.F. Wang, Carbon 109 (2016) 411–417. [4] X.F. Yang, Y.W. Kuang, Y.S. Liu, D.B. Zhang, Z.G. Shao, H.L. Yu, X.K. Hong, J.F. Feng, X.S. Chen, X.F. Wang, Nanoscale 8 (2016) 15712–15719. [5] X.F. Yang, F.X. Tan, Y.J. Dong, H.L. Yu, Y.S. Liu, Phys. Chem. Chem. Phys. 21 (2019) 5243–5252. [6] Y.S. Liu, X. Zhang, J.F. Feng, X.F. Wang, Appl. Phys. Lett. 104 (2014) 242412. [7] X. Jiang, L. Zhu, K.L. Yao, J. Alloys Compd. 764 (2018) 505–511. [8] L. Zhu, X. Jiang, G.Y. Gao, H.H. Fu, K.L. Yao, ACS Omega 3 (2018) 13630–13635. [9] M. Jonson, G.D. Mahan, Phys. Rev. B 21 (1980) 4223.
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Fig. 5. The variation of Z T with doping concentration for both p-type and n-type monolayer α -Te at temperatures: 300 K, 400 K, 500 K and 600 K.
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is a very promising candidate material for thermoelectric applications.
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4. Conclusions
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To summary, our research reveals that monolayer α -Te has a structure similar to 1T-MoS2 , containing three Te atoms per unit cell, and the structural parameters of our optimized structure are a = b = 4.23 Å, layer thickness d z = 3.623 Å. The monolayer α -Te is an indirect bandgap semiconductor with a moderate bandgap of 0.63. Further analysis of the results shows that the electrical conductivity value has 105 orders of magnitude, the Seebeck coefficients in a large part of doping concentration region are greater than 200 μV/K. The maximum power factor exceeds 1.1 × 10−3 Wm−1 k−1 , and n-type doping is better than p-type doping when the sample concentration is below 1 × 1019 cm−3 , whereas p-type doping is better than n-type doping. What’s more, we observe that the lattice thermal conductivity is always less than 3 W/K in the temperature range of 150 to 1500 K, and decreases with the increase of temperature. The most exciting result is that the maximum value of ZT is more than 4, which is colossal for the thermoelectric material. The above results show that monolayer α -Te is a very promising candidate material for thermoelectric applications.
42 43
Declaration of competing interest
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The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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Acknowledgements
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This work was supported by the National Natural Science Foundation of China under the Grant No. 11374111 and 11874159.
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References
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[10] N. Stojanovic, D.H.S. Maithripala, J.M. Berg, M. Holtz, Phys. Rev. B 82 (2010) 075418. [11] R.S. Joseph, Y.C. Duck, G.K. Mercouri, Angew. Chem., Int. Ed. 48 (2009) 8616–8639. [12] J.Y. Zhou, H. Wang, D. He, Y. Zhou, W. Peng, F. Fan, H.H. Huang, Appl. Phys. Lett. 112 (2018) 243904. [13] L.D. Zhao, H.J. Wu, S.Q. Hao, C.I. Wu, X.Y. Zhou, K. Biswas, J.Q. He, T.P. Hogan Wolverton, V.P. Dravid, M.G. Kanatzidis, Energy Environ. Sci. 6 (2013) 3346–3355. [14] L. Zhu, F. Zou, G.Y. Gao, K.L. Yao, Sci. Rep. 7 (2017) 497. [15] G. Chen, M.S. Dresselhaus, G. Dresselhaus, J.P. Fleurial, T. Caillat, Int. Mater. Rev. 48 (2003) 45–66. [16] G.Q. Ding, G.Y. Gao, Z.S. Huang, W.X. Zhang, K.L. Yao, Nanotechnology 27 (2016) 375703. [17] K.H. Park, P.N. Martin, U. Ravaioli, Nanotechnology 27 (2016) 03540. [18] L. Zhu, R.M. Li, K.L. Yao, Phys. Chem. Chem. Phys. 19 (2017) 4085–4092. [19] X.C. Huang, J.Q. Guan, Z.J. Lin, B. Liu, S.Y. Xing, W.H. Wang, J.D. Guo, Nano Lett. 17 (2017) 4619–4623. [20] Z.L. Zhu, X.L. Cai, S. Yi, J.L. Chen, Y.W. Dai, C.Y. Niu, Z.X. Guo, M.H. Xie, F. Liu, J.H. Cho, Y. Jia, Z.Y. Zhang, Phys. Rev. Lett. 119 (2017) 106101. [21] X. Gonze, B. Amadon, P.M. Anglade, J.M. Beuken, F. Bottin, P. Boulanger, F. Bruneval, D. Caliste, R. Caracas, M. Côté, T. Deutsch, L. Genovese, Ph. Ghosez, M. Giantomassi, S. Goedecker, D.R. Hamann, P. Hermet, F. Jollet, G. Jomard, S. Leroux, M. Mancini, S. Mazevet, M.J.T. Oliveira, G. Onida, Y. Pouillon, T. Rangel, G.M. Rignanese, D. Sangalli, R. Shaltaf, M. Torrent, M.J. Verstraete, G. Zerah, J.W. Zwanziger, Comput. Phys. Commun. 180 (2009) 2582–2615. [22] X. Gonze, J.M. Beuken, R. Caracas, F. Detraux, M. Fuchs, G.M. Rignanese, L. Sindic, M. Verstraete, G. Zerah, F. Jollet, M. Torrent, A. Roy, M. Mikami, Ph. Ghosez, J.Y. Raty, D.C. Allan, Comput. Mater. Sci. 25 (2002) 478–492. [23] G. Kresse, J. Hafner, Phys. Rev. B 47 (1993) 558–561. [24] G. Kresse, J. Hafne, Phys. Rev. B 49 (1994) 14251–14269. [25] G. Kresse, J. Furthmüller, Phys. Rev. B 54 (1996) 11169–11186. [26] G. Kresse, J. Furthmiiller, Comput. Mater. Sci. 6 (1996) 15–50. [27] G. Kresse, D. Joubert, Phys. Rev. B 59 (1999) 1758–1775. [28] M. Ernzerhof, G.E. Scuseria, J. Chem. Phys. 110 (1999) 5029–5036. [29] J.P. Perdew, K. Burke, M. Ernzerhof, Phys. Rev. Lett. 77 (1996) 3865–3868. [30] S.D. Guo, J.L. Wang, Semicond. Sci. Technol. 31 (2016) 095011. [31] S.D. Guo, Comput. Mater. Sci. 123 (2016) 8–13. [32] J. Heyd, G.E. Scuseria, Z. Ernzerhof, J. Chem. Phys. 118 (2003) 8207. [33] A.V. Krukau, O.A. Vydrov, A.F. Izmaylov, G.E. Scuseria, J. Chem. Phys. 125 (2006) 224106. [34] A. Togo, F. Oba, I. Tanaka, Phys. Rev. B 78 (2008) 134106. [35] K.H.M. Georg, C. Jesús, J.V. Matthieu, Comput. Phys. Commun. 231 (2018) 140–145. [36] Y.C. Ding, B. Xiao, Gang Tang, J.W. Hong, J. Phys. Chem. C 121 (2016) 225–236. [37] L.C. Zhang, G.Z. Qin, W.Z. Fang, H.J. Cui, Q.R. Zheng, Q.B. Yan, G. Su, Sci. Rep. 6 (2016) 19830. [38] J. Bardeen, W. Shockley, Phys. Rev. 80 (1950) 72–80. [39] R.Q. Guo, X.J. Wang, Y.D. Kuang, B.L. Huang, Phys. Rev. B 92 (2015) 115202. [40] L.J. Wu, Y.Q. Zhao, C.W. Chen, L.Z. Wang, B. Liu, M.Q. Cai, Chin. Phys. B 25 (2016) 107202. [41] A. Togo, I. Tanaka, Scr. Mater. 108 (2015) 1–5. [42] W. Li, J. Carrete, N.A. Katcho, N. Mingo, Comput. Phys. Commun. 185 (2014) 1747. [43] E.S. Toberer, G.J. Snyder, Nat. Mater. 7 (2008) 105–114. [44] G. Yumnam, T. Pandey, A.K. Singh, J. Chem. Phys. 143 (2015) 234704.
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Thermoelectric properties of monolayer α -Te: Low lattice thermal conductivity and extremely high dimensionless figure of merit
16 17
81
School of Physics and Wuhan National High Magnetic Field Center, Huazhong University of Science and Technology, Wuhan 430074, China
83
82 84 85
19
a r t i c l e
i n f o
a b s t r a c t
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Xia Jiang, Lin Zhu ∗ , Bowen Li, Kailun Yao
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Article history: Received 21 October 2019 Received in revised form 13 December 2019 Accepted 20 December 2019 Available online xxxx Communicated by R. Wu Keywords: Thermoelectric properties Monolayer α -Te Semiconductor Electrical conductivity Lattice thermal conductivity Seebeck coefficient Dimensionless figure of merit
We have investigated the thermoelectric properties of monolayer α -Te by the first-principles and the Boltzmann transport theory. The results show that monolayer α -Te is an indirect bandgap semiconductor with a moderate bandgap of 0.63 eV. The value of the Seebeck coefficient for monolayer α -Te is larger than 200 μVK−1 , the electrical conductivity has the magnitude of 1 × 106 . The lattice thermal conductivity is smaller than 4 Wm−1 K−1 , which is extremely low. Instructively, the dimensionless figure of merit for monolayer α -Te is more than 4, which is markedly high. The above phenomena imply that monolayer α -Te can be served as a considerable thermoelectric material with excellent thermoelectric conversion efficiency. © 2020 Elsevier B.V. All rights reserved.
88 89 90 91 92 93 94 95 96 97 98 99
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1. Introduction
38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59
The world energy demand shows an increasing trend year by year, and 90% of human total energy consumption comes from fossil fuels, which are non-renewable energy sources and the usage rate is very low. Almost 70% of fossil fuels are released into the atmosphere in the form of waste heat and exhaust gas, which causes serious air pollution [1]. Therefore, it is imperative to explore new energy sources to improve energy efficiency. Thermoelectric materials can realize the conversion of heat and electric energy, and have the advantages of small size, light weight, noise-free, pollution-free and long service life, etc. [2]. Then thermoelectric materials is very promising energy materials, and the thermoelectric properties have been extensively studied in recent years [3–6]. Yet to date, thermoelectric materials have been well used in waste heat recovery, aerospace, microelectronics, biomedicine and other fields. Unfortunately, the conversion efficiency of traditional thermoelectric materials is very low, they can’t be used commercially on a large scale. Based on the above discussion, it is essential to find new materials with higher thermoelectric conversion efficiency, improving the performance of traditional thermoelectric materials. Generally, the dimensionless figure of merit Z T = S 2 σ T /k is considered to be used to measure the thermoelec-
60 61 62 63 64 65 66
*
Corresponding author. E-mail address: [email protected] (L. Zhu).
https://doi.org/10.1016/j.physleta.2019.126222 0375-9601/© 2020 Elsevier B.V. All rights reserved.
tric conversion performance of materials, the larger the Z T , the better the thermoelectric conversion performance [7,8]. Obviously, large thermoelectric conversion efficiency implies large S 2 σ and small k. Among them, S 2 σ is usually considered as power factor (PF), S represents the thermoelectric power or Seebeck coefficient, σ stands for electrical conductivity. And k is the total thermal conductivity of the material, which comes from the contribution of electronic thermal conductivity (kele ) and lattice thermal conductivity (klatt ). The electrical conductivity (σ ) and electrical thermal conductivity (kele ) are associated by the Wiedemann–Franz law: kele = L σ T , where L is the Lorentz number, its most typical value is 2.45 × 10−8 WK2 for metals and degenerate semiconductors [9,10]. In fact, L varies with material and temperature [11], but we often overlook its changes. Recently, Heremans et al. found that the tellurium (Te) thin film not only showed high flexibility but also had a large power factor of 321 μWm−1 K−2 at room temperature [12]. Previous research pointed out that the thermoelectric properties of materials could be improved by reducing the dimensionality of the materials [1, 13–18]. Based on the above research background, we predict that the thermoelectric properties of Te thin film can also be enhanced by making it into monolayer Te. But up to present, there have not reports on thermoelectric properties of monolayer Te. A recent theoretical work proposed three types of structure for two-dimensional monolayer Te, that is tellurene, which are stable 1T-MoS2 -like (α -Te), metastable tetragonal (β -Te) and 2H-MoS2 -like (γ -Te) structures, among which monolayer α -Te and
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Table 1 The elastic modulus C 2D , effective mass m∗ , deformation potential E ι , the product of relaxation time and temperature τ (s) T of the electron and hole, which are obtained from the SCF-SPB model.
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Carriers
C 2D (J m−2 )
m∗ (me )
E ι (eV)
τ (s) T
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hole electron
69.79 69.79
0.177 0.094
7.007 7.904
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β -Te are semiconductor with the band gap of 0.75 and 1.47 eV,
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respectively [19,20]. Previous studies have shown that monolayer α -Te has high mobility [20]. Considering the stability of monolayer α -Te structure, we speculate that it has a good thermoelectric performance. In this paper, we systematically study the structure and thermoelectric properties of monolayer α -Te, finding ultralow thermal conductivity, extremely high dimensionless figure of merit. Moreover, the p-type doping has better thermoelectric conversion performance.
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2. Computation details
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The structural optimization is calculated by first-principles density functional theory (DFT) [21,22] which is implemented in Vienna ab initio simulation package (VASP) [23–26]. The projectoraugmented-wave (PAW) [27] pseudopotentials and the Perdew– Burke–Ernzerhof (PBE) [28,29] exchange correlation functional are used. We set up an 11 × 11 × 1 Monkhorst-pack k-mesh. The kinetic-energy cutoff for the plane wave is chosen as 300 eV. The lattice parameters and atomic positions are fully optimized until the maximum Hellmann-Feynman forces exerted on each atom are less than 1E-6 eV/Å. The electronic band structure is calculated based on the optimized structure, and the k grid is set to 25 × 25 × 1. Taking into account the large atomic mass of Te, the spin-orbit coupling (SOC) has been considered [20,30,31]. We also consider the effect of hybrid functional (HSE) [20,32–34] on the band gap calculation. That is to say, we have calculated the energy bands with SOC and HSE06. After the structural optimization and band calculation, we perform electronic transport properties calculation by using BoltzTraP2 code [35]. As a result of electrical conductivity and the electron thermal conductivity in the output of BoltzTraP2 contains relaxation time, we utilize a self-consistent single parabolic band model (SCF-SPB) to calculate the electron relaxation time (τ ),
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τ=
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μ=
μm∗ e eh¯ 3 C 2D
k B T m∗md∗ E ι2
where C 2D (C 2D = 1/ S 0 ∂
2
ι 2 ∗ E /∂(
ι ) ), m 0
(m∗
=
2 h¯ / ∂∂ KE2 ), md∗
2
=
(m∗x m∗y )1/2 and E ι (E ι = E /( ι/ι0 )) are elastic constant, effective mass, average effective mass and deformation potential, respectively [36–40]. The calculated results are list in Table 1. We calculate the phonon spectrum of monolayer α -Te with harmonic approximation by Phonopy code [41]. And the lattice thermal conductivity is calculated by solving the phonon Boltzmann transport equation, which is carried out by ShengBTE code [42]. We adopt the finite displacement approach to calculate the second-order harmonic interatomic force constants (IFCs) and third-order anharmonic interatomic force constants (IFCs), and a 5 × 5 × 1 super cell containing 75 atoms based on the optimized structure is used. The convergence testing for k-mesh is carried out from 13 × 13 × 1 to 53 × 53 × 1 in the ShengBTE code.
Fig. 1. The optimal structure of monolayer α -Te, (a) top view, (b) side view. (c) The calculated band structure and density of states (DOS) of monolayer α -Te.
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Fig. 2. The variation of chemical potential with the change of carrier concentration under different temperature. (For interpretation of the colors in the figure(s), the reader is referred to the web version of this article.)
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3. Results and discussion
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From Figs. 1 (a) and (b), we notice that monolayer α -Te has a structure similar to 1T-MoS2 , there are three Te atoms per unit cell, and the structural parameters of the optimized structure are a = b = 4.23 Å, layer thickness is 3.623 Å. Looking closely at Fig. 1 (c), we observe that monolayer α -Te is an indirect bandgap semiconductor with a moderate bandgap of 0.63 eV. The highest point of the valence band is between G-M, and the lowest point of the conduction band is located at G point. This moderate band gap gives us such a signal that the thermoelectric performance can be optimized within a reasonable doping level. What’s more, we find that the valence bands exhibit much higher density of states than the conduction bands, which implies that p-type doping thermoelectric performance is superior to that of n-type doping. Generally, doping will lead the shift in the Fermi level, the appropriate doping concentration need the chemical potential be in the band gap between the conduction and the valence band. The relationship between carrier concentration and chemical potential under different temperatures is displayed in Fig. 2, which manifests that for n-type doing, the doping concentration should be in the range of 0 to 1019 cm−3 , and for p-type doing, the best doping concentration is in the range of 0 to 1021 cm−3 . So we have
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Fig. 3. The electrical conductivity (a), electronic thermal conductivity (b), Seebeck coefficient (c) and power factor (d) for both p-type and n-type monolayer of carrier concentration at temperatures of 300, 400, 500 and 600 K.
α -Te as a function
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Fig. 4. (a) The phonon dispersion relations along high symmetry direction, (b) the temperature dependence of phonon thermal conductivity along x and y axis.
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studied the thermoelectric transport properties of monolayer α -Te under light doping below. Figs. 3 (a) to (d) present the variation of thermoelectric coefficient with doping concentration and temperature. The good thermoelectric materials should have electrical conductivity with the magnitude of 105 −1 m−1 [11]. The electrical conductivity calculated by us is always in such a favorable range. Comparing the electrical conductivity and electronic thermal conductivity, we note that their trend is similar, which confirms that the electrical conductivity and electrical thermal conductivity are associated with the Wiedemann–Franz law: kele = L σ T , and the difference is because L varies with temperature and material, which has been mentioned in many literatures [10,11]. Besides, the Seebeck coefficient of many thermoelectric materials with excellent thermoelectric conversion performances is around 230 μV/K, which has been confirmed by experiments and theoretical studies [43,44]. In our research, the Seebeck coefficients in most part of the doping concentration region are greater than 200 μV/K. Therefore, we deduce that monolayer α -Te should have good thermoelectric conversion performance. By analyzing the power factor, we observe that the maximum power factor exceeds 1.1 × 10−3 Wm−1 K−1 , and that n-type doping is better than p-type doping when the sample con-
cm−3 , whereas p-type doping is
centration is below 1 × 10 better than n-type doping, as is depicted in Fig. 3 (d). Since the thermoelectric conversion performance is ultimately measured by Z T , in order to analyze the thermoelectric conversion performance of monolayer α -Te more accurately, it is necessary to calculate the lattice thermal conductivity of monolayer α -Te. The phonon spectrum is described in Fig. 4 (a), where the phonon frequency ranges from 0 to 5 THZ. There is a slight coupling between the acoustic and optical phonon modes at the high symmetry point M-L-H. Therefore, we predict that the thermal conductivity of the monolayer α -Te should be relatively small. Fig. 4 depicts the change of lattice thermal conductivity with temperature, which shows that the lattice thermal conductivity is always less than 3 W/K in the temperature range of 150 to 1500 K, and decreases with the increase of temperature. Based on above calculations, we analyze the variation of ZT with doping type and doping concentration at temperatures: 300 K, 400 K, 500 K and 600 K, as presented in Fig. 5. The maximum value of Z T is more than 4, which is very large. And the variation of ZT with carrier concentration and carrier species is similar to that of power factor. We predict that monolayer α -Te 19
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[3] Y.S. Liu, X.Y. Shao, T. Shao, J.Y. Zhang, Y.W. Kuang, D.B. Zhang, Z.G. Shao, H.L. Yu, X.K. Hong, J.F. Feng, X.F. Yang, X.S. Chen, X.F. Wang, Carbon 109 (2016) 411–417. [4] X.F. Yang, Y.W. Kuang, Y.S. Liu, D.B. Zhang, Z.G. Shao, H.L. Yu, X.K. Hong, J.F. Feng, X.S. Chen, X.F. Wang, Nanoscale 8 (2016) 15712–15719. [5] X.F. Yang, F.X. Tan, Y.J. Dong, H.L. Yu, Y.S. Liu, Phys. Chem. Chem. Phys. 21 (2019) 5243–5252. [6] Y.S. Liu, X. Zhang, J.F. Feng, X.F. Wang, Appl. Phys. Lett. 104 (2014) 242412. [7] X. Jiang, L. Zhu, K.L. Yao, J. Alloys Compd. 764 (2018) 505–511. [8] L. Zhu, X. Jiang, G.Y. Gao, H.H. Fu, K.L. Yao, ACS Omega 3 (2018) 13630–13635. [9] M. Jonson, G.D. Mahan, Phys. Rev. B 21 (1980) 4223.
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Fig. 5. The variation of Z T with doping concentration for both p-type and n-type monolayer α -Te at temperatures: 300 K, 400 K, 500 K and 600 K.
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is a very promising candidate material for thermoelectric applications.
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4. Conclusions
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To summary, our research reveals that monolayer α -Te has a structure similar to 1T-MoS2 , containing three Te atoms per unit cell, and the structural parameters of our optimized structure are a = b = 4.23 Å, layer thickness d z = 3.623 Å. The monolayer α -Te is an indirect bandgap semiconductor with a moderate bandgap of 0.63. Further analysis of the results shows that the electrical conductivity value has 105 orders of magnitude, the Seebeck coefficients in a large part of doping concentration region are greater than 200 μV/K. The maximum power factor exceeds 1.1 × 10−3 Wm−1 k−1 , and n-type doping is better than p-type doping when the sample concentration is below 1 × 1019 cm−3 , whereas p-type doping is better than n-type doping. What’s more, we observe that the lattice thermal conductivity is always less than 3 W/K in the temperature range of 150 to 1500 K, and decreases with the increase of temperature. The most exciting result is that the maximum value of ZT is more than 4, which is colossal for the thermoelectric material. The above results show that monolayer α -Te is a very promising candidate material for thermoelectric applications.
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Declaration of competing interest
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The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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Acknowledgements
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This work was supported by the National Natural Science Foundation of China under the Grant No. 11374111 and 11874159.
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