Pressure enhanced thermoelectric properties in Mg₂Sn

Abstract

The pressure dependence of the electronic structure and thermoelectric properties of Mg₂Sn are investigated by using a modified Becke and Johnson exchange potential, including spin–orbit coupling. The corresponding value of spin–orbit splitting at the Γ point is 0.47 eV, which is in good agreement with the experimental value of 0.48 eV. With increasing pressure, the energy band gap first increases, and then decreases. In certain doping range, the power factor for n-type has the same trend with energy band gap, when the pressure increases. Calculated results show that the pressure can lead to a significantly enhanced power factor in n-type doping at the critical pressure, which can be understood by the pressure inducing accidental degeneracy of the conduction band minimum (CBM) at the critical pressure. It is also found that the corresponding lattice thermal conductivity near the critical pressure shows a relatively small value. These results make us believe that thermoelectric properties of Mg₂Sn can be improved in n-type doping by pressure.

---

Introduction

Thermoelectric materials can use the Seebeck effect to convert waste heat directly to electricity to solve energy problems. Conversely, they also can be applied in the cooling field using the Peltier effect. The performance of thermoelectric materials can be characterized by the dimensionless figure of merit,^1,2 ZT = S²σT/(κ_e + κ_L), where S, σ, T, κ_e and κ_L are the Seebeck coefficient, electrical conductivity, absolute temperature, and the electronic and lattice thermal conductivities, respectively. Bismuth–tellurium systems,^3,4 silicon–germanium alloys,^5,6 lead chalcogenides^7,8 and skutterudites^9,10 have been identified as excellent thermoelectric materials for thermoelectric devices. For thermoelectric research, the main objective is to search for high ZT materials, which has proven to be interesting and challenging. To enhance ZT, it is possible to improve the power factor without increasing the thermal conductivity or to reduce the lattice thermal conductivity without affecting the electrical conductivity.¹¹

The thermoelectric material Mg₂X (X = Si, Ge, Sn) composed of abundant, low-cost elements and their alloys have attracted much recent attention,^12–14 and various doping strategies have been adopted to attain high ZT.^15–17 Pressure by tuning the electronic structures of materials can accomplish many interesting phenomena like recent pressure-induced high-T_c superconductivity in (H₂S)₂H₂.^18,19 The thermoelectric power factor S²σ can be improved dramatically under compression^20,21 in some thermoelectric materials. Here, we use first-principle calculations and Boltzmann transport theory to address the pressure dependence of thermoelectric properties in the Mg₂Sn. Calculated results show that the pressure dependence of energy band gap with a modified Becke and Johnson (mJB) exchange potential with spin–orbit coupling (SOC) is consistent with one with mBJ,²² and first increases, and then decreases. Pressure can significantly improve the power factor in n-type doping below the critical pressure. It is found that pressure can reduce the lattice thermal conductivity in a certain pressure range. These lead to enhanced ZT, and make Mg₂Sn more efficient for thermoelectric applications in n-type doping by pressure. So, pressure tuning offers a very effective method to search for materials with enhanced thermoelectric properties.

The rest of the paper is organized as follows. In the next section, we shall give our computational details. In the third section, we shall present our main calculated results and analysis. Finally, we shall give our conclusion in the fourth section.

Computational details

We use a full-potential linearized augmented-plane-waves method within the density functional theory (DFT),²³ as implemented in the package WIEN2k.²⁴ We use Tran and Blaha’s mBJ exchange potential plus local-density approximation (LDA) correlation potential for the exchange–correlation potential²⁵ to do our main DFT calculations. The full relativistic effects are calculated with the Dirac equations for core states, and the scalar relativistic approximation is used for valence states.^26–28 The SOC was included self-consistently by solving the radial Dirac equation for the core electrons and evaluated by the second-variation method.²⁹ We use 6000 k-points in the first Brillouin zone for the self-consistent calculation. We make harmonic expansion up to 1_max = 10 in each of the atomic spheres, and set R_mt × k_max = 8. The self-consistent calculations are considered to be converged when the integration of the absolute charge-density difference between the input and output electron density is less than 0.0001|e| per formula unit, where e is the electron charge. Transport calculations are performed through solving Boltzmann transport equations within the constant scattering time approximation as implemented in BoltzTrap,³⁰ which has been applied successfully to several materials.^31–33 To obtain accurate transport coefficients, we use 200 000 k-points in the first Brillouin zone for the energy band calculation. The lattice thermal conductivities are calculated by using Phono3py + VASP codes.^34–37 For the third-order force constants, 2 × 2 × 2 supercells are built, and reciprocal spaces of the supercells are sampled by 2 × 2 × 2 meshes. To compute lattice thermal conductivities, the reciprocal spaces of the primitive cells are sampled using the 13 × 13 × 13 meshes.

Main calculated results and analysis

The electronic structures, optical properties and thermoelectric properties of Mg₂Sn at hydrostatic pressure have been investigated by the mBJ exchange-potential.²² However, SOC is very important for power factor calculations.³⁸ Here, we investigate the electronic structures and thermoelectric properties by using mBJ + SOC. First, the energy band structures of Mg₂Sn (the experimental crystal structure, with a corresponding pressure of 0.9 GPa) with mBJ and mBJ + SOC are shown in Fig. 1. It is found that the SOC has little effect on the conduction bands, and has an obvious influence on valence bands. The SOC splits the valence band at the Γ point, and the corresponding value of spin–orbit splitting, 0.47 eV, is in good agreement with the experimental value 0.48 eV.³⁹ As expected, the SOC reduces the energy band gap due to the CBM moving toward lower energy. The energy band structures of Mg₂Sn for the considered pressure are also shown in Fig. 2, and the energy band gap and value of spin–orbit splitting at the Γ point as a function of pressure by using mBJ + SOC are present in Fig. 1. The trend of energy band gap with mBJ + SOC is consistent with that with mBJ,²² and first increases, and then decreases with increasing pressure. The explanation of this trend of the energy band gap is that the Mg-s character near the high symmetry X point transforms from the first conduction band to the second one (the detailed discussions can be found in ref. 22). In this transformation process, the first two conduction bands produce accidental degeneracy, which has an important influence on the power factor of Mg₂Sn. The spin–orbit splitting monotonically increases with increasing pressure, but has little change about 0.05 eV with pressure varying from 0 GPa to 17.2 GPa.

Fig. 1 Left: The energy band structures by using mBJ (red lines) and mBJ + SOC (black lines). Right: The energy band gap (Gap) and the value of spin–orbit splitting at the Γ point (Δ_SO) as a function of pressure by using mBJ + SOC.

Fig. 2 The energy band structures of Mg₂Sn with pressure being 0.9, 3.0, 5.6, 8.6, 12.5 and 17.2 (unit: GPa) calculated by using mBJ + SOC.

Mg₂Sn based thermoelectric materials are considered as potential candidates for efficient thermoelectricity. The pressure dependence of the semi-classical transport coefficients as a function of doping level is investigated within constant scattering time approximation Boltzmann theory. We firstly consider the SOC effects on the Seebeck coefficient S; electrical conductivity with respect to scattering time σ/τ and power factor with respect to scattering time S²σ/τ as a function of doping levels at the temperature of 300 K by using mBJ and mBJ + SOC are presented in Fig. 3. It is clearly seen that the negative doping levels (n-type doping) show the negative Seebeck coefficient, and the positive doping levels (p-type doping) imply the positive Seebeck coefficient. Calculated results show that SOC has a detrimental influence on S, σ/τ and S²σ/τ in p-type doping, but has a negligible effect in n-type doping. These can be explained by the larger influence of SOC on the valence bands than on the conduction bands. Similar SOC-induced detrimental effects on the power factor can be found in half-Heusler ANiB (A = Ti, Hf, Sc, Y; B = Sn, Sb, Bi).⁴⁰ When SOC is absent, the best p-type power factor is larger than the best n-type one. However, including SOC, the power factor in n-type doping is larger than that in p type doping in the considered doping range, which agrees with the experimental results reporting high ZT values for n-type than for p-type.¹⁵ Therefore, it is crucial to include SOC effects for related theoretical analysis and prediction of the thermoelectric properties of Mg₂Sn.

Fig. 3 At temperature of 300 K, transport coefficients as a function of doping levels (electrons [minus value] or holes [positive value] per unit cell): Seebeck coefficient S (left), electrical conductivity with respect to scattering time σ/τ (middle) and power factor with respect to scattering time S²σ/τ (right) calculated with mBJ (black solid line) and mBJ + SOC (red dotted line).

The pressure dependence of S, σ/τ and S²σ/τ with pressure being 0.9, 3.0, 5.6, 8.6, 12.5 and 17.2 (unit: GPa) calculated by using mBJ + SOC at temperature of 300 K are shown in Fig. 4. It is interesting that S (absolute value), σ/τ and S²σ/τ have the same trend with energy band gap in a certain doping range for n-type with increasing pressure. When the pressure reaches the critical value of the energy band gap, the S, σ/τ and S²σ/τ attain the corresponding extremum. For p-type, the σ/τ has an obvious dependence on pressure, but the S has a very weak dependence on pressure, which leads to a weak pressure dependence for S²σ/τ. The strong pressure dependence for S²σ/τ in n-type doping shows that the Mg₂Sn under pressure may become a more efficient thermoelectric material. To see clearly the interesting pressure dependence in n-type doping, S²σ/τ as a function of temperature with a doping concentration of 1 × 10²⁰ cm⁻³ is displayed in Fig. 5. In the considered temperature range, the power factor always has the same trend with energy band gap.

Fig. 4 At a temperature of 300 K, transport coefficients as a function of doping levels (electrons [minus value] or holes [positive value] per unit cell): Seebeck coefficient S (left), electrical conductivity with respect to scattering time σ/τ (middle) and power factor with respect to scattering time S²σ/τ (right) with pressure being 0.9, 3.0, 5.6, 8.6, 12.5 and 17.2 (unit: GPa) calculated by using mBJ + SOC.

Fig. 5 Power factor with respect to scattering time S²σ/τ as a function of temperature for n-type with pressure being 0.9, 3.0, 5.6, 8.6, 12.5 and 17.2 (unit: GPa) calculated by using mBJ + SOC with the doping concentration of 1 × 10²⁰ cm⁻³.

To explain the interesting pressure dependence of the power factor in n-type doping, the total densities of state (DOS) at pressures of 0.9, 3.0, 5.6, 8.6, 12.5 and 17.2 (unit: GPa) calculated by using mBJ + SOC are displayed in Fig. 6. The calculated results imply that the slope of the densities of state of the conduction bands near the energy band gap first increase with increasing pressure, and then decrease. The critical pressure happens to be the critical one for the power factor. The large slope of densities of state near the energy gap may induce a large Seebeck coefficient in narrow-gap semiconductors,⁴¹ leading to large power factor, which gives rise to the corresponding pressure dependence of the power factor in n-type doping. The trend also can be understood by the change of CBM, and pressure can induce accidental degeneracies of CMB at the critical pressure, which increases the slope of DOS. The power factor first increases, and then decreases by tuning the degeneracies of CMB (see Fig. 2).

Fig. 6 The total densities of state with pressure being 0.9, 3.0, 5.6, 8.6, 12.5 and 17.2 (unit: GPa) calculated by using mBJ + SOC. The Fermi level is defined as the conduction band minimum (CBM).

Finally, the electronic thermal conductivities with respect to scattering time κ_e/τ with a doping concentration of 1 × 10²⁰ cm⁻³ for the n-type and the lattice thermal conductivities κ_L as a function of temperature with pressure being 0.9, 3.0, 5.6, 8.6, 12.5 and 17.2 are shown in Fig. 7. The electronic thermal conductivity has the same trend as the energy band gap and power factor with increasing pressure. The lattice thermal conductivity is generally considered to be independent of doping, and typically results as 1/T. At high T, it can reach the so-called minimum thermal conductivity. For typical thermoelectrical materials, the thermal conductivity is dominated by the κ_L. Calculated results show that the κ_L near the critical pressure of the energy band gap has a relatively small value, and the corresponding power factor in n-type doping has a relatively large one. These results imply that pressure can induce larger ZT by reducing lattice thermal conductivity and enhancing the power factor for the n-type.

Fig. 7 Top: The electronic thermal conductivities with respect to scattering time κ_e/τ with a doping concentration of 1 × 10²⁰ cm⁻³ for the n-type calculated by using mBJ + SOC. Bottom: The lattice thermal conductivities κ_L calculated by using GGA. The thermal conductivities as a function of temperature with pressure being 0.9, 3.0, 5.6, 8.6, 12.5 and 17.2 (unit: GPa).

Discussions and conclusion

It has been proved that the mBJ gap value agrees well with the experimental value 0.3 eV,²² but the mBJ + SOC gap value is less than the experimental one. However, mBJ + SOC is more satisfactory than the usual GGA or LDA + SOC in calculating electronic structure of Mg₂Sn. It is very important for power factor calculations to consider SOC for MgSn, especially for p-type doping. When SOC is included, the n-type doping has a more excellent power factor than p-type doping, which agrees that the best p-type material reported so far has a lower ZT than the best n-type. Pressure has an obvious effects on the conduction bands, and has little influence on the valence bands, which leads to remarkable effects on the power factor for n-type and small effects on p-type doping. Symmetry driven degeneracy, low-dimensional electronic structures and accidental degeneracies are common mechanisms for the high power factor.⁴² Symmetry driven degeneracy can increase the DOS, which can induce a large power factor.^43,44 Reducing the dimensionality of a material leads to increased DOS, which also can enhance the power factor, and has been realized in SrTiO₃ superlattices.⁴⁵ Low-dimensional like electronic structures also can be achieved in three-dimensional thermoelectric materials.⁴⁶ Here, the accidental degeneracies of the CMB of Mg₂Sn can be induced by pressure at the critical pressure, which produces a larger power factor in certain doping ranges.

Although the specific ZT cannot be attained due to calculating τ from the first principles being challenging, our predicted pressure dependence of thermoelectric properties is based mainly on the reliable first-principle calculations, which should be reasonable to a large extent. In summary, we investigate the pressure dependence of the thermoelectric properties of Mg₂Sn by using mBJ + SOC. It is found that pressure can realize an enhanced power factor below critical pressure and reduced lattice thermal conductivity near the critical pressure, which leads to an improved ZT for efficient thermoelectric application. By choosing an appropriate doping concentration, Mg₂Sn under pressure can provide great opportunities for efficient thermoelectricity.

Acknowledgements

This work is supported by the National Natural Science Foundation of China (Grant No. 11404391). We are grateful to the Advanced Analysis and Computation Center of CUMT for the award of CPU hours to accomplish this work.

References

1. Y. Pei, X. Shi, A. LaLonde, H. Wang, L. Chen and G. J. Snyder, Nature, 2011, 473, 66.
2. A. D. LaLonde, Y. Pei, H. Wang and G. J. Snyder, Mater. Today, 2011, 14, 526.
3. W. S. Liu, Q. Y. Zhang, Y. C. Lan, S. Chen, X. Yan, Q. Zhang, H. Wang, D. Z. Wang, G. Chen and Z. F. Ren, Adv. Energy Mater., 2011, 1, 577.
4. D. K. Ko, Y. J. Kang and C. B. Murray, Nano Lett., 2011, 11, 2841.
5. M. Zebarjadi, et al., Nano Lett., 2011, 11, 2225.
6. B. Yu, et al., Nano Lett., 2012, 12, 2077.
7. Y. Z. Pei, X. Y. Shi and A. Lalonde, et al., Nature, 2011, 473, 66.
8. J. Q. He, J. R. Sootsman and S. N. Girard, et al., J. Am. Chem. Soc., 2010, 132, 8669.
9. A. C. Sklad, M. W. Gaultois and A. P. Grosvenor, J. Alloys Compd., 2010, 505, L6.
10. X. Shi, J. Yang and J. R. Salvador, J. Am. Chem. Soc., 2011, 133, 7837.
11. P. Vaqueiro and A. V. Powell, J. Mater. Chem., 2010, 20, 9577.
12. W. Liu, X. Tan, K. Yin, H. Liu, X. Tang, J. Shi, Q. Zhang and C. Uher, Phys. Rev. Lett., 2012, 108, 166601.
13. W. J. Luo, M. J. Yang, Q. Shen, H. Y. Jiang and L. Zhang, Adv. Mater. Res., 2009, 66, 33.
14. M. Yang, W. Luo, Q. Shen, H. Jiang and L. Zhang, Adv. Mater. Res., 2009, 66, 17.
15. V. K. Zaitsev, M. I. Fedorov, E. A. Gurieva, I. S. Eremin, P. P. Konstantinov, A. Y. Samunin and M. V. Vedernikov, Phys. Rev. B: Condens. Matter Mater. Phys., 2006, 74, 045207.
16. Q. Zhang, J. He, T. J. Zhu, S. N. Zhang, X. B. Zhao and T. M. Tritt, Appl. Phys. Lett., 2008, 93, 102109.
17. W. Liu, X. Tang and J. Sharp, J. Phys. D: Appl. Phys., 2010, 43, 085406.
18. D. F. Duan, Y. X. Liu, F. B. Tian, D. Li, X. L. Huang, Z. L. Zhao, H. Y. Yu, B. B. Liu, W. J. Tian and T. Cui, Sci. Rep., 2014, 4, 6968.
19. A. P. Drozdov, M. I. Eremets, I. A. Troyan, V. Ksenofontov and S. I. Shylin, Nature, 2015, 525, 73.
20. S. V. Ovsyannikov and V. V. Shchennikov, Appl. Phys. Lett., 2007, 90, 122103.
21. S. V. Ovsyannikov, V. V. Shchennikov, G. V. Vorontsov, A. Y. Manakov, A. Y. Likhacheva and V. A. Kulbachinskii, J. Appl. Phys., 2008, 104, 053713.
22. S. D. Guo, Europhys. Lett., 2015, 109, 57002.
23. P. Hohenberg and W. Kohn, Phys. Rev., 1964, 136, B864; W. Kohn and L. J. Sham, Phys. Rev., 1965, 140, A1133.
24. P. Blaha, K. Schwarz, G. K. H. Madsen, D. Kvasnicka and J. Luitz, WIEN2k, an Augmented Plane Wave + Local Orbitals Program for Calculating Crystal Properties, Karlheinz Schwarz Technische Universität Wien, Austria, 2001.
25. F. Tran and P. Blaha, Phys. Rev. Lett., 2009, 102, 226401.
26. A. H. MacDonald, W. E. Pickett and D. D. Koelling, J. Phys. C: Solid State Phys., 1980, 13, 2675.
27. D. J. Singh and L. Nordstrom, Plane Waves, Pseudopotentials and the LAPW Method, Springer, New York, 2nd edn, 2006.
28. J. Kunes, P. Novak, R. Schmid, P. Blaha and K. Schwarz, Phys. Rev. B: Condens. Matter, 2001, 64, 153102.
29. D. D. Koelling, B. N. Harmon and J. Phys, J. Phys. C: Solid State Phys., 1977, 10, 3107.
30. G. K. H. Madsen and D. J. Singh, Comput. Phys. Commun., 2006, 175, 67.
31. B. L. Huang and M. Kaviany, Phys. Rev. B: Condens. Matter Mater. Phys., 2008, 77, 125209.
32. L. Q. Xu, Y. P. Zheng and J. C. Zheng, Phys. Rev. B: Condens. Matter Mater. Phys., 2010, 82, 195102.
33. J. J. Pulikkotil, D. J. Singh, S. Auluck, M. Saravanan, D. K. Misra, A. Dhar and R. C. Budhani, Phys. Rev. B: Condens. Matter Mater. Phys., 2012, 86, 155204.
34. G. Kresse, J. Non-Cryst. Solids, 1995, 192–193, 222.
35. G. Kresse and J. Furthmüller, Comput. Mater. Sci., 1996, 6, 15.
36. G. Kresse and D. Joubert, Phys. Rev. B: Condens. Matter, 1999, 59, 1758.
37. A. Togo, L. Chaput and I. Tanaka, Phys. Rev. B: Condens. Matter Mater. Phys., 2015, 91, 094306.
38. K. Kutorasinski, B. Wiendlocha, J. Tobola and S. Kaprzyk, Phys. Rev. B: Condens. Matter Mater. Phys., 2014, 89, 115205.
39. F. Vazquez, A. R. Forman and M. Cardonna, Phys. Rev., 1968, 176, 905.
40. S. D. Guo, J. Alloys Compd., 2016, 663, 128.
41. M. Onoue, F. Ishii and T. Oguchi, J. Phys. Soc. Jpn., 2008, 77, 054706.
42. K. F. Garrity, 2016, arXiv:1601.01622.
43. K. Shirai and K. Yamanaka, J. Appl. Phys., 2013, 113, 053705.
44. H. Usui, S. Shibata and K. Kuroki, Phys. Rev. B: Condens. Matter Mater. Phys., 2010, 81, 205121.
45. H. Ohta, S. Kim, Y. Mune, T. Mizoguchi, K. Nomura, S. Ohta, T. Nomura, Y. Nakanishi, Y. Ikuhara and M. Hirano, et al., Nat. Mater., 2007, 6, 129.
46. D. Parker, X. Chen and D. J. Singh, Phys. Rev. Lett., 2013, 110, 146601.

---