Electronic structure and thermoelectric properties of p-type half-Heusler compound NbFeSb: a first-principles study

Abstract

The electronic structure and thermoelectric (TE) properties of p-type NbFeSb are studied by first-principles calculations and the Boltzmann transport equation under the constant relaxation time approximation. The carrier concentration dependences of TE properties of p-type NbFeSb are calculated to be well agreement with the experimental data. When the minimum lattice thermal conductivity is obtained, the maximum ZT of 1.4 is achievable at 800 K, ∼40% higher than the value of 1 for the best p-type Nb_1−xHf_xFeSb compounds. To further evaluate the optimal electrical transport properties of p-type NbFeSb at higher temperatures, the maximum power factors and corresponding optimal carrier concentrations are calculated. The temperature dependence of Seebeck coefficient and power factor are also studied based on the estimated optimal doping levels, which indicates that further composition optimization can't improve the power factors when the carrier concentration reaches ∼2.6 × 10²¹ cm⁻³.

---

1. Introduction

Thermoelectric (TE) materials are of current interest for direct thermal-to-electric energy conversion with reliable, silent, vibration-free operation and no moving parts.¹ The main hurdle for the widespread application of TE materials is their low efficiency compared to conventional technologies.² The efficiency of TE materials is gauged by the dimensionless figure of merit, ZT = S²σT/κ, where S, σ, T, and κ are respectively Seebeck coefficient, electrical conductivity, absolute temperature, and thermal conductivity. The thermal conductivity (κ = κ_e + κ_L) consists of those from electrons (κ_e) and lattice parts (κ_L).³ In order to obtain a higher ZT value, one needs to maximize the power factor S²σ, while keeping the thermal conductivity κ suppressed.

Half-Heusler (HH) compounds, with a valence electron count (VEC) of 18, have been extensively studied as promising TE materials for high temperature power generation due to their good electrical properties and high temperature stability.^4–8 Up to now, the state-of-the-art HH compounds are MNiSn (M = Ti, Zr, Hf)^9–11 for n-type and MCoSb (M = Ti, Zr, Hf)^12–14 for p-type. The maximum ZT values for both n- and p-type are close to unity. At the same time, one prominent problem of these HH compounds for large scale application is the cost if a lot of Hf is needed.¹⁵

Recently, NbFeSb-based HH compounds, with abundantly available constituent elements, have attracted great attentions due to the excellent thermoelectric performances at high temperatures.^16–18 Fu et al.¹⁶ reported that the maximum ZT of 0.8 is achieved for p-type Ti doped FeV_0.6Nb_0.4Sb at 900 K due to a high band degeneracy of 8. It was found experimentally¹⁷ that increasing Nb content in the FeV_1−yNb_ySb solid solutions can reduce the optimal carrier concentration, which is favorable for p-type Fe(V_1−yNb_y)_1−xTi_xSb due to the limited solubility of Ti, leading to a higher ZT value of 1.1 at 1100 K for FeNb_0.8Ti_0.2Sb. Then Fu et al.⁷ reported a record-high ZT of ∼1.5 at 1200 K for the p-type FeNbSb heavy-band half-Heusler alloys by changing the dopant from Ti to Hf. Joshi et al.¹⁸ also achieved a high ZT of ∼1 at 973 K in a nanostructured p-type Nb_0.6Ti_0.4FeSb_0.95Sn_0.05 composition.

On the theoretical side, Yang et al.¹⁹ calculated the electronic transport properties of 36 semiconducting HH compounds including NbFeSb, and estimated their optimal doping concentrations. However, most of the theoretical researches focused only on the electronic band structures,^16–18,20 only limited research has been conducted on the thermoelectric performance of NbFeSb compounds. In this paper, based on the band structure calculations, the semi-classic Boltzmann transport theory is used to investigate the thermoelectric performance of p-type NbFeSb. The results are compared with the available experimental data. The electrical transport properties of p-type NbFeSb at high temperatures are also calculated based on the optimal doping concentrations corresponding to the maximum power factors. Our results can offer useful guidance on the experimental investigations.

2. Computational method

All initio calculations including band structure and DOS are performed by the projector augmented wave (PAW) method^21,22 as implemented in the Vienna ab-initio simulation package (VASP).²³ The exchange–correlation energy is in the form of Perdew–Burke–Ernzerh of with generalized gradient approximations (GGA).²⁴ A plane-wave cutoff energy of 400 eV and an energy convergence criterion of 10⁻⁶ eV for self-consistency are adopted throughout our calculations. Spin–orbit coupling is not considered. A 15 × 15 × 15 Monkhorst–Pack uniform k-point sampling is used for the self-consistency calculations.

We extended our band structure studies to calculate the transport properties, which are calculated employing Boltzmann theory with constant scattering time approximation (CSTA) as implemented in the BoltzTraP code.²⁵ In order to get reasonable transport properties, a 31 × 31 × 31 Monkhorst–Pack k-point sampling is used. In the CSTA, an energy-independent constant τ is adopted in all our calculations and Seebeck coefficient S can be directly calculated with no adjustable parameters. Thus the temperature and carrier concentration dependence of S discussed in this article should be reasonably realistic. In addition to the CSTA, we also treat doping within the rigid band approximation (RBA).²⁶ According to the RBA, doping a system does not change its band structure but only the chemical potential. This approximation is widely used in theoretical calculations of transport properties of many thermoelectric materials, including HH compounds.^19,27–30

3. Results and discussion

3.1 Crystal structure and stability

NbFeSb HH compound crystallizes in the cubic MgAgAs-type structure (space group F4̄3m), which is shown in Fig. 1(a). Elements Nb and Sb form a rock salt structure and Fe is located at one of the two body diagonal positions (1/4, 1/4, 1/4) in the cell, leaving the other one (3/4, 3/4, 3/4) unoccupied. The lattice constant is calculated to be 5.968 Å, comparable to the experimental value of 5.949 Å.¹⁷ For potential TE applications at high temperatures, it is worth discussing the stability of NbFeSb-based HH compound. As known, thermal stability at high temperatures is one of the biggest advantages for HH compounds,^5,7,8 and no imaginary phonon frequency appears in the phonon dispersion curves of NbFeSb compound, indicating that it is dynamically stable.^18,31 Accordingly, we only focus on the stability of doped systems here. Previous experimental work^7,17,18 showed that Ti, Zr and Hf doping at Nb site can simultaneously optimize the electrical power factor and suppresses thermal conductivity, and Hf dopant is more efficient in supplying carriers than Zr and Ti. To explain the doping efficacy, we calculated the formation energies (E_f) for NbFeSb using a R (R = Ti, Zr or Hf) atom to replace Nb in the conventional cell. The formation energy was calculated using the following formula:

E_f = E_doped − E_bulk − E_Nb + E_R (1)

where, E_doped and E_bulk are the total energies for the conventional cell containing the dopant R and the same bulk NbFeSb conventional cell, respectively. E_Nb and E_R are the host and doping atom energies in the bulk phase. At the doping content of 25%, the formation energies of Hf, Zr and Ti-doping NbFeSb are 0.209, 2.953 and 4.345 eV, respectively. Therefore, Hf-doping NbFeSb is the most stable among the considered doping systems, which is an important reason making Hf more efficient to supply carriers than Zr or Ti.

Fig. 1 (a) The conventional cell of NbFeSb and (b) the band structure of NbFeSb.

3.2 Electronic structures

The band structure of NbFeSb in Fig. 1(b) shows an indirect band gap of 0.51 eV from L → X, which is well agreement with the previous calculated results.^16–19 Unfortunately, as far as we know, there is no available experimental band gap to validate the calculated value. The bands near the valence-band maximum (VBM) and conduction-band minimum (CBM) are both approximately parabolic. But the characteristic of valence band shows great disparity from the conduction band. Because of the combination of heavy and light bands, band degeneracy N_v for VBM is 8,¹⁶ while N_v is only 3 for CBM with one band. According to the formula m* = N_v^2/3m^*_b and μ ∝ 1/m^*_b^5/2 (where m* is the density-of-states (DOS) effective mass, m^*_b is band effective mass, and μ is the mobility of carriers),³¹ large N_v is beneficial to large m* without deterioration of μ. As known, the optimal ZT depends primarily on the TE quality factor B ∝ μm*^3/2/κ_L.^32,33 Therefore, the high band degeneracy N_v for VBM implies that NbFeSb is one of the good p-type TE materials.

To see clearly the states near the Fermi level, the total density of states (DOS) of NbFeSb and the projected density of states (PDOS) of atoms are shown in Fig. 2(a). It can be seen that the total DOS near the VBM is higher than that near the CBM. The conduction band and valence band of NbFeSb are mostly composed of localized transition metal d states. The conduction band is dominated by Nb states and the valence band is derived primarily from Fe states. The total DOS of Nb_0.75R_0.25FeSb (R = Ti, Zr and Hf) are also shown to investigate the effect of doping on electronic structure (Fig. 2(b)). The shift of the Fermi level upon doping is significant, and at such a high doping content, the Fermi level enters the valence band. Despite the doping content of R is up to 25%, the unchanged DOS profiles near the Fermi level indicates that the contribution of the doping elements to the DOS is almost negligible (ESI Fig. 1†). This can be understood from the fact that the valence band structure is dominated by the electronic character of the Fe element, and is also consistent with the previous experimental results.¹⁷ Therefore, the RBA can be reasonably employed for investigating the p-type transport properties with heavily doping at Nb site in NbFeSb.

Fig. 2 (a) Total and projected DOS of atoms for NbFeSb and (b) total DOS of Nb_1−xR_xFeSb (R = Ti, Zr and Hf) compared with NbFeSb with conventional cell.

3.3 Thermoelectric properties

The calculated thermoelectric properties versus carrier concentration n for p-type NbFeSb are presented in Fig. 3, and compared with the attainable experimental data.⁷ The Seebeck coefficient S decreases with increasing carrier concentration (Fig. 3(a)). A good agreement between the calculated values and experimental data indicates that the valence band structure has weak dependence on doping, consistent with the discussion relevant to Fig. 2(b). Within the framework of the Boltzmann transport equation in CSTA, the electrical conductivity σ is expressed in the form of the ratio σ/τ (where τ is the electronic relaxation time). According to the standard electron–phonon dependence on temperature T and carrier concentration n for τ, namely, τ = CT⁻¹n^−1/3,³⁴ we determined τ = 3.12 × 10⁻¹²n^−1/3 and 2.82 × 10⁻¹²n^−1/3 at 300 and 800 K, respectively. σ increases linearly with the increasing carrier concentration and shows a metal-like behaviour (Fig. 3(b)). The carrier concentration dependence of power factor for NbFeSb are shown in Fig. 3(c). The maximum power factor and corresponding optimum carrier concentration increase with increasing temperature. It is noteworthy that the experimental maximum carrier concentrations of Ti and Hf doping samples at 800 K is almost optimal based on our calculations.

Fig. 3 Carrier concentration dependence of (a) Seebeck coefficient S, (b) electrical conductivity σ, (c) power factor and (d) ZT for p-type NbFeSb at 300 K, and 800 K.

The total thermal conductivity in a material includes both electronic thermal conductivity κ_e and lattice thermal conductivity κ_L. Generally, the maximum ZT value is achievable when the minimum lattice thermal conductivity is reached. The minimum lattice thermal conductivity κ_min can be approximately calculated using the Cahill's model,³⁵

where V is the average volume per atom, k_B is the Boltzmann constant, v_s and v_l are the shear and longitudinal velocities, respectively. The calculated κ_min of NbFeSb is 0.96 W m⁻¹ K⁻¹. This represents a 60% reduction compared to the value of 2.5 W m⁻¹ K⁻¹ for the best p-type NbFeSb-based compound (Nb_0.86Hf_0.14FeSb, with the highest ZT of ∼1.5 (ref. 7)). However, as a heavy-band material, NbFeSb system needs a high carrier concentration to optimize the power factor. Such a high carrier concentration will also deduce a high κ_e. Typically, when the optimal power factors are reached, the experimental κ_e of Nb_1−xR_xFeSb (R = Ti, Zr and Hf) are in the range of 2.7–4.2 and 1.9–2.5 W m⁻¹ K⁻¹ at 300 and 800 K, respectively. Here the minimum κ_e (2.7 W m⁻¹ K⁻¹ for 300 K and 1.9 W m⁻¹ K⁻¹ for 800 K) are considered to calculated the maximum ZT value, which is shown in Fig. 3(d). At 800 K, when the minimum κ_L is reached, the maximum ZT can be up to 1.4, ∼40% higher than the value of 1 for Nb_0.86Hf_0.14FeSb at the same temperature.

Since the working temperature of half-Heusler thermoelectric modules is as high as 1000 K, it is urgent to investigate the optimal carrier concentration and corresponding electrical transport properties of p-type NbFeSb at higher temperatures. The carrier concentration dependence of power factor over relaxation time S²σ/τ for p-type doping is calculated to obtain the optimal carrier concentration n_opt at 700 K, 900 K, and 1100 K (ESI Fig. 1†). The values of maximum S²σ/τ and corresponding optimal n_opt are listed in Table 1. As aforementioned, the maximum S²σ/τ and corresponding optimum carrier concentration increase with increasing temperature. To further evaluate the thermoelectric performance of p-type NbFeSb at the n_opt shown in Table 1, S and S²σ/τ as a function of temperature are calculated. Fig. 4(a) shows that S increases with increasing temperature, and decreases with increasing carrier concentration. It is noteworthy S²σ/τ is almost independent of carrier concentration at a given temperature (Fig. 4(b)). Specifically, the S²σ/τ–T curves are nearly overlapping at three different carrier concentrations (2.6 × 10²¹, 3.2 × 10²¹ and 3.9 × 10²¹ cm⁻³). Unchanged S²σ/τ with increasing carrier concentration indicates that composition optimization can't further improve the S²σ/τ when the carrier concentration reaches ∼2.6 × 10²¹ cm⁻³. In order to validate the above analysis, temperature dependence of S and S²σ/τ at lower (1.5 × 10²¹ and 2.0 × 10²¹ cm⁻³) and higher (5.0 × 10²¹ cm⁻³) carrier concentrations for p-type NbFeSb are also calculated. As shown in Fig. 4(a), Seebeck coefficient S still decreases with increasing carrier concentration n. Nevertheless, the S²σ/τ–T curve at lower carrier concentrations (1.5 × 10²¹ and 2.0 × 10²¹ cm⁻³) is different from that at higher carrier concentrations (2.6 × 10²¹, 3.2 × 10²¹, 3.9 × 10²¹ and 5.0 × 10²¹ cm⁻³), which is shown in Fig. 4(b). For example, at n = 2.0 × 10²¹ cm⁻³, S²σ/τ is almost equal to that at other carrier concentrations as T increases from 300 to 800 K, but it is lower than that at higher carrier concentrations above 800 K, as plotted in the inset of Fig. 4(b). The S²σ/τ–T curves are nearly unchanged when carrier concentration is higher than 2.6 × 10²¹ cm⁻³, even up to 5.0 × 10²¹ cm⁻³. All of the abovementioned results undoubtedly show that further increasing carrier concentration can't improve the S²σ/τ of NbFeSb greatly when carrier concentration reaches ∼2.6 × 10²¹ cm⁻³. The TE performance may be further improved by grain refinement^36,37 and incorporating nanoscale precipitates^38–40 to reduce the lattice thermal conductivity.

Table 1 The optimal power factor over relaxation time S²σ/τ and the corresponding optimal carrier concentration n_opt for p-type NbFeSb at 700 K, 900 K, and 1100 K

	700 K	900 K	1100 K
S^2σ/τ (W m^−1 K^−2 s^−1)	11.8 × 10^11	16.0 × 10^11	19.6 × 10^11
nopt (cm^−3)	2.6 × 10^21	3.2 × 10^21	3.9 × 10^21

---

Fig. 4 Temperature dependence of p-type (a) Seebeck coefficient S, and (b) power factors with respect to relaxation time S²σ/τ for NbFeSb at the different optimal doping carrier concentrations. Inset of (b) shows magnified view of (b) for a temperature range of 800–1100 K.

4. Conclusions

In conclusion, the electronic structure of half-Heusler compound NbFeSb is studied by ab initio density functional methods, and the thermoelectric properties are also investigated by combining the Boltzmann transport theory with the electronic structures. The high band degeneracy N_v for valence band maximum (VBM) implies that NbFeSb is a potentially good p-type thermoelectric material. The carrier concentration dependences of thermoelectric properties for p-type NbFeSb are calculated, and in good agreement with the experimental data. At 800 K, the ZT value can be further improved by 40% when the minimum κ_L is reached. Based on the optimal carrier concentration from the maximum power factor, the temperature dependences of Seebeck coefficient and power factor are investigated to indicate that further increasing carrier concentration through composition optimization can't improve the power factors when the carrier concentration reaches ∼2.6 × 10²¹ cm⁻³.

Acknowledgements

This research was supported by the National Natural ScienceFoundation of China (No. 51171208 and No. 51271201).

References

1. L. E. Bell, Science, 2008, 321, 1457.
2. J. S. Zhang, X. Feng, Y. Xu, M. H. Guo, Z. C. Zhang, Y. B. Ou, Y. Feng, K. Li, H. J. Zhang, L. L. Wang, X. Chen, Z. X. Gan, S. C. Zhang, K. He, X. C. Ma, Q. K. Xue and Y. Y. Wang, Phys. Rev. B: Condens. Matter Mater. Phys., 2015, 91, 075431.
3. G. J. Snyder and E. S. Toberer, Nat. Mater., 2008, 7, 105.
4. C. Yu, T. J. Zhu, R. Z. Shi, Y. Zhang, X. B. Zhao and J. He, Acta Mater., 2009, 57, 2757.
5. W. Xie, A. Weidenkaff, X. Tang, Q. Zhang, J. Poon and T. M. Tritt, Nanomaterials, 2012, 2, 379.
6. H. H. Xie, H. Wang, Y. Z. Pei, C. G. Fu, X. H. Liu, G. J. Snyder, X. B. Zhao and T. J. Zhu, Adv. Funct. Mater., 2013, 23, 5123.
7. C. G. Fu, S. Q. Bai, Y. T. Liu, Y. S. Tang, L. D. Chen, X. B. Zhao and T. J. Zhu, Nat. Commun., 2015, 6, 8144,  DOI:10.1038/ncomms9144.
8. T. J. Zhu, C. G. Fu, H. H Xie, Y. T. Liu and X. B. Zhao, Adv. Energy Mater., 2015, 5, 1500588.
9. X. Yan, W. S. Liu, H. Wang, S. Chen, J. Shiomi, K. Esfarjani, H. Z. Wang, D. Z. Wang, G. Chen and Z. F. Ren, Energy Environ. Sci., 2012, 5, 7543.
10. S. Chen, K. C. Lukas, W. S. Liu, C. P. Opeil, G. Chen and Z. F. Ren, Adv. Energy Mater., 2013, 3, 1210.
11. Y. W. Chai, T. Oniki and Y. Kimura, Acta Mater., 2015, 85, 290–300.
12. X. Yan, W. S. Liu, S. Chen, H. Wang, Q. Zhang, G. Chen and Z. F. Ren, Adv. Energy Mater., 2013, 3, 1195.
13. E. Rausch, B. Balke, S. Ouardi and C. Felser, Phys. Chem. Chem. Phys., 2014, 16, 25258.
14. R. He, H. S. Kim, Y. C. Lan, D. Z. Wang, S. Chen and Z. F. Ren, RSC Adv., 2014, 4, 64711.
15. S. Chen and Z. F. Ren, Mater. Today, 2013, 16, 387–395.
16. C. G. Fu, T. J. Zhu, Y. Z. Pei, H. H. Xie, H. Wang, G. J. Snyder, Y. Liu, Y. T. Liu and X. B. Zhao, Adv. Energy Mater., 2014, 4, 1400600.
17. C. G. Fu, T. J. Zhu, Y. T. Liu, H. H. Xie and X. B. Zhao, Energy Environ. Sci., 2015, 8, 216–220.
18. G. Joshi, R. He, M. Engber, G. Samsonidze, T. Pantha, E. Dahal, K. Dahal, J. Yang, Y. C. Lan, B. Kozinsky and Z. F. Ren, Energy Environ. Sci., 2014, 7, 4070.
19. J. Yang, H. M. Li, T. Wu, W. Q. Zhang, L. D. Chen and J. H. Yang, Adv. Funct. Mater., 2008, 18, 2880–2888.
20. D. P. Young, P. Khalifah, R. J. Cava and A. P. Ramirez, J. Appl. Phys., 2000, 87, 317.
21. P. E. Blöchl, Phys. Rev. B: Condens. Matter Mater. Phys., 1994, 50, 17953.
22. G. Kresse and D. Joubert, Phys. Rev. B: Condens. Matter Mater. Phys., 1999, 59, 1758.
23. G. Kresse and J. Furthmuller, Phys. Rev. B: Condens. Matter Mater. Phys., 1996, 54, 11169.
24. J. P. Perdew, K. Burke and M. Ernzerhof, Phys. Rev. Lett., 1996, 77, 3865.
25. G. K. H. Madsen and D. J. Singh, Comput. Phys. Commun., 2006, 175, 67.
26. M. G. Holland, Phys. Rev., 1963, 132, 2461.
27. J. M. Yang, Y. L. Yan, Y. X. Wang and G. Yang, RSC Adv., 2014, 4, 28714.
28. A. H. Reshak, RSC Adv., 2015, 5, 47569.
29. L. L. Wang, L. Miao, Z. Y. Wang, W. Wei, R. Xiong, H. J. Liu, J. Shi and X. F. Tang, J. Appl. Phys., 2009, 105, 013709.
30. B. Z. Sun, Z. J. Ma, C. He and K. C. Wu, RSC Adv., 2015, 5, 56382.
31. C. Çoban, K. Çolakoğlu and Y. Ö. Çiftçi, Phys. Scr., 2015, 90, 095701.
32. Y. Z. Pei, X. Y. Shi, A. Lalonde, H. Wang, L. D. Chen and G. J. Snyder, Nature, 2011, 473, 66.
33. Y. Z. Pei, H. Wang and G. J. Snyder, Adv. Mater., 2012, 24, 6125.
34. J. M. Yang, Y. L. Yan, Y. X. Wang and G. Yang, RSC Adv., 2014, 4, 28714.
35. D. G. Cahill, S. K. Watson and R. O. Pohl, Phys. Rev. B: Condens. Matter Mater. Phys., 1992, 46, 6131.
36. G. Joshi, X. Yan, H. Wang, W. Liu, G. Chen and Z. F. Ren, Adv. Energy Mater., 2011, 1, 643.
37. X. Yan, G. Joshi, W. S. Liu, Y. C. Lan, H. Wang, S. Lee, J. W. Simonson, S. J. Poon, T. M. Tritt, G. Chen and Z. F. Ren, Nano Lett., 2011, 11, 556–560.
38. W. J. Xie, J. He, S. Zhu, X. L. Su, S. Y. Wang, T. Holgate, J. W. Graff, V. Ponnambalam, S. J. Poon, X. F. Tang, Q. J. Zhang and T. M. Tritt, Acta Mater., 2010, 58, 4705–4713.
39. J. P. A. Makongo, D. K. Misra, X. Y. Zhou, A. Pant, M. R. Shabetai, X. L. Su, C. Uher, K. L. Stokes and P. F. P. Poudeu, J. Am. Chem. Soc., 2011, 133, 18843–18852.
40. W. J. Xie, Y. G. Yan, S. Zhu, M. Zhou, S. Populoh, K. Gałązka, S. J. Poon, A. Weidenkaff, J. He, X. F. Tang and T. M. Tritt, Acta Mater., 2013, 61, 2087–2094.

---

Footnote

† Electronic supplementary information (ESI) available. See DOI: 10.1039/c5ra23091h

---