# Band Structures and Transport Properties of High-Performance Half-Heusler Thermoelectric Materials by First Principles

^{*}

## Abstract

**:**

## 1. Introduction

^{2}σT/(κ

_{e}+ κ

_{L}), where α, σ, T, κ

_{e}, and κ

_{L}are the Seebeck coefficient, electrical conductivity, absolute temperature, and electronic and lattice contributions to the total thermal conductivity κ, respectively [1].

_{e}are related to the electronic structure of the material. The three parameters are intercorrelated and cannot be optimized independently. For instance, increasing α is usually accompanied by a decreasing σ; an increase in σ concomitantly leads to an increase in κ

_{e}via the Wiedemann-Franz law κ

_{e}= LσT (L is the Lorenz number). One possible way to optimize zT is to maximize the power factor (α

^{2}σ), which can be achieved by the band engineering [2]. The band structure is one of the basic characteristics of materials, as well as the vital tool in understanding, optimizing, and even designing novel functional materials [3]. Once the electronic structure calculation is done, the electrical transport properties can be effectively tuned according to the band structure–related parameters. Additionally, new TE materials with high power factors can also be screened using the band structures combined with Boltzmann transport theory.

^{*}[5]. According to the formula m

^{*}= N

_{v}

^{2/3}m

_{b}

^{*}(N

_{v}is the band degeneracy and m

_{b}

^{*}is the band effective mass), increasing both N

_{v}and m

_{b}

^{*}contributes to a enhanced m

^{*}and, consequently, α [6]. However, a high m

_{b}

^{*}always leads to a low carrier mobility μ due to μ $\propto $ 1/m

_{b}

^{*}. It has been proved that increasing N

_{v}is beneficial to large m

^{*}without deterioration of μ, and is an efficient strategy to improve TE performance for many materials [7,8,9,10].

_{L}, related to its phonon vibration, is more or less independent of the electronic transport properties. However, defects that reduce κ

_{L}, such as forming solid solutions or making composite structures, are likely to reduce μ [4]. Nevertheless, the differences between the mean free paths of phonons and electrons open the window for nanostructuring technology at a length scale that scatters phonons, but not electrons [11]. Alternatively, seeking promising TE materials with low thermal conductivity attracts much attention [12,13,14]. Only phonon-phonon Umklapp scattering is considered, κ

_{L}$\propto $ MV

^{1/3}θ

_{D}

^{3}/γ

^{2}(M is the average mass per atom, V is the average atomic volume, θ

_{D}is the Debye temperature and γ is the Grüneisen parameter) [15]. Accordingly, low M, V, θ

_{D}, and high γ contribute a low κ

_{L}. Starting with this viewpoint, many good TE materials with intrinsically low κ

_{L}have been reported [16,17,18].

_{2}X is obtained by filling the vacant sublattice with B atoms, as shown in Figure 1b. The most electronegative element X and the most electropositive element A usually form the NaCl sublattice with octahedral coordination, leaving the all-etrahedral site to the intermediate electronegative element B [21]. The properties of HH compounds depend strongly on the valence electron count (VEC) of the constituent elements. HH compounds, with VEC = 8 or 18, are usually semiconductors with excellent potential as TE materials.

## 2. Manipulating the Band Structures of HH Compounds

#### 2.1. Band Structures and Atomic Disorders in N-Type MNiSn

_{1+x}Sn is reduced from 0.45 to 0.12 eV due to the Ni interstitial, which is consistent with the experimental gap of 0.12 eV for TiNiSn [24]. Recently, Zeier et al. ascribed the in-gap band to that Ni vacancies or excess Ni should be largely electron neutral (effectively Ni

^{0}by assigning bonding (NiSn)

^{4−}orbitals to Sn as Sn

^{4−}). In this case, non-stoichiometry in the form of ZrNi

_{1+x}Sn may be expected to not move the Fermi level outside the band gap [46].

_{0}method are 0.45, 0.62, and 0.75 eV, respectively [47]. The differences are mainly due to that the observed relative positions of the d levels in the transition metal atoms vary among the different methods. However, considering that calculations using the GW

_{0}method are very computationally expensive for defect calculations due to the large supercells, more precise and computationally-tractable methods should be applied to the MNi

_{1+x}Sn systems.

#### 2.2. Performance Optimization of P-Type Heavy-Band HH Using Band Engineering

^{−1}at 300 K) and a large power factor (4.5 × 10

^{−3}W m

^{−1}K

^{−2}at 300 K) [48]. However, due to a relatively high lattice thermal conductivity (10 W m

^{−1}K

^{−1}at 300 K), earlier studies have been focused on improving the TE performance of n-type FeRSb by alloying [49,50] or nano-structuring [51], but only a marginal improvement in zT (≈0.33 at 650 K) was obtained.

_{v}= 3. In comparison, the valence band maximum of FeRSb lies in point L with a higher band degeneracy of N

_{v}= 8 [52,53], which is beneficial for TE performance as a large DOS effective mass m

^{*}is desired for good TE materials [2,7]. According to the formula m

^{*}= N

_{v}

^{2/3}m

_{b}

^{*}and µ = 1/m

_{b}

^{*5/2}, large N

_{v}is beneficial for large m

^{*}without deterioration of µ. Therefore, increasing N

_{v}is an effective way to improve TE performance of a material without deteriorated side effects. TE properties of p-type Ti-doped FeV

_{0.6}Nb

_{0.4}Sb solid solutions were first investigated [28]. Combined with the high N

_{v}of 8 and heavy m

_{b}

^{*}of 2.5 m

_{e}, a high m

^{*}of 10 m

_{e}was obtained in the p–type Fe(V

_{0.6}Nb

_{0.4})

_{1−x}Ti

_{x}Sb compounds, which resulted in a high Seebeck coefficient. Although the heavy m

_{b}

^{*}led to a low µ, the low deformation potential and alloy scattering potential were both beneficial for a reasonably high mobility in this system. Therefore, a high power factor of about 3 × 10

^{−3}W m

^{−1}K

^{−2}was available at 900 K for Fe(V

_{0.6}Nb

_{0.4})

_{0.8}Ti

_{0.2}Sb. Mainly due to the high power factor, in addition to the relatively low lattice thermal conductivity among the FeRSb system, a high zT of ≈0.8 at 900 K was achieved.

_{b}

^{*}leads to a low µ in Fe(V

_{0.6}Nb

_{0.4})

_{1−x}Ti

_{x}Sb. Based on the band structures, the m

_{b}

^{*}of 0.16 m

_{e}for p-type FeNbSb is lower than that of 0.25 m

_{e}for p-type FeVSb, indicating that increasing Nb content may lead to a lower m

_{b}

^{*}and, hence, higher µ (Figure 4a). Moreover, the m

_{b}

^{*}decrease can lower optimal carrier concentration [54]. The solubility limit of Ti in Fe(V

_{0.6}Nb

_{0.4})Sb was about 20%. The optimized power factor may be realized within the solubility limit of Ti by decreasing the optimal carrier concentration (Figure 4b). The band gap of 0.54 eV for FeNbSb is also larger than that of 0.34 eV for FeVSb, meaning that higher Nb content in Fe(V

_{0.6}Nb

_{0.4})

_{1−x}Ti

_{x}Sb will broaden the band gap and consequently increases the temperature at which bipolar diffusion begins to diminish TE performance. The enhanced carrier mobility and reduced optimal carrier concentration result in the optimal power factor. The power factor of p-type FeNb

_{0.8}Ti

_{0.2}Sb was about 4.5× 10

^{−3}W m

^{−1}K

^{−2}at 1100 K, ~50% higher than the Fe(V

_{0.6}Nb

_{0.4})

_{1−x}Ti

_{x}Sb solid solutions. A higher zT value of 1.1 at 1100 K was achieved for FeNb

_{0.8}Ti

_{0.2}Sb due to the enhanced power factor [29].

^{*}of 6.9 m

_{e}for FeNb

_{1−x}Ti

_{x}Sb is also higher than that of 0.3 m

_{e}and 1.3 m

_{e}for conventional PbTe-based and Bi

_{2}Te

_{3}-based materials, respectively [54,55]. The large m

_{b}

^{*}in these Fe-containing p-type TE materials is due to the spatially localized nature of transition metal 3d orbitals [56]. It has been reported that p-type skutterudites containing 4d transition metal, such as Ru, possess low m

_{b}

^{*}at the valence band [57], which is beneficial for high carrier mobility. Therefore, Ru-based HH alloys may also be promising TE materials with low band effective mass. The calculated band structures showed that the valence bands of Ru-based HH alloys are lighter than that of Fe-based compounds (Figure 5a) [58]. The power factors of p-type RuNbSb and RuTaSb are about 100% higher than that of p-type FeNbSb due to the lower m

_{b}

^{*}and hence higher µ (Figure 5b). Moreover, the lattice thermal conductivities of RuNbSb and RuTaSb are also lower than FeNbSb, exhibiting high potential for high-temperature TE power generation.

^{*}of these heavy-band materials are in the range of 2 m

_{e}–10 m

_{e}[59] (Figure 6a). Accordingly, higher carrier concentrations, which demands for higher contents of dopants, are necessary to optimize the power factors. Even though these heavy-band TE materials have low µ, their optimal power factors are 2–3 times higher than that the state-of-the-art light-band PbTe (Figure 6b). Combined with the strong point-defect phonon scattering due to a high content of dopant, high power factors make these heavy-band TEs promising for power generation. Therefore, the tradeoff between the band effective mass and carrier mobility is crucial to the TE performance of these heavy-band semiconductors.

## 3. Electronic Transport Properties of HH Compounds

_{e}) are ultimately determined by the transport distribution function [61], Σ(E) = v

^{2}τg (where E is the energy, v, τ, and g are the group velocity, the relaxation time, and the density of states, respectively). Currently, the greatest challenge for computations is to capture the relaxation time τ, which is affected by many scattering mechanisms and difficult to calculate accurately [62]. The energy dependence of τ impedes the direct evaluation for large sets of TE materials. Thus, several approximations for τ have been proposed for quantifying the TE performance of materials.

#### 3.1. Constant Relaxation Time Approximation

^{21}cm

^{−3}. Based on the estimated n-type power factors, IIIB-(Ni, Pd) and some Co-containing HH compounds show reasonable n-type performance and their carrier concentrations fall in the range of 10

^{20}–10

^{21}cm

^{−3}(Figure 7b). Recently, Fu et al. have proved that p-type FeNb(V)Sb are promising HH-based TE materials, verifying the CRT approximation in the study of HH compounds.

#### 3.2. Constant Mean Free Path Approximation

_{h}

^{*}) is higher than that of electrons (m

_{e}

^{*}) according to their band structure calculations. Using the Spearman rank correlation coefficient Σ [67], power factor is found to be a better predictor of zT than lattice thermal conductivity. At both room and high temperatures, the power factor depends most markedly on the band effective mass and band gap. Carrete et al. also provided simple rules to determine if nanograined HH compounds are likely to be good TE materials through machine learning techniques. In this work, they only considered a few elements’ properties, such as atomic numbers and masses, positions in the periodic table, atomic radii, Pauling electronegativities [68], and Pettifor’s chemical scales [69]. Five promising HH compounds were ultimately identified for room temperature and high-temperature applications. BiBaK, AuAlHf, and CoBiZr are the best candidates at both temperatures. Unfortunately, the accuracy of this approach cannot be verified since the predicted candidates have not been investigated experimentally.

#### 3.3. Calculations of Relaxation Times

_{e}are both directly proportional to τ. An actual value of τ is still needed to calculate the power factor and zT. The simplified zT (zT

_{e}= α

^{2}σT/κ

_{e}) has been used to identify candidate TE materials [70]. This method does not require knowledge of the value of τ, but zT

_{e}is always greater than zT, since the lattice thermal conductivity κ

_{L}is treated as zero. Alternatively, an approximate value τ from the experimental electrical conductivity can also be used to compute zT, assuming that τ is direction independent and a constant at a certain specific temperature and carrier concentration [71].

_{ii}/(m

_{b}

^{*3/2}Ξ

^{2}) (where c

_{ii}and Ξ are elastic constant and DP constant, respectively), in which m

_{b}

^{*}is very important to calculate τ. In their work, the m

_{b}

^{*}at all k points in the first Brillouin zone has been calculated, and then the average of the effective masses at all specific energies can be obtained. The calculated m

_{b}

^{*}near the VBM is ~1.87 m

_{e}, a little higher than the experimental value of 1.6 m

_{e}[29]. The calculated values of σ are a little higher than the experimental values of FeNb

_{1−x}Ti

_{x}Sb (x = 0.04, 0.06 and 0.08) at low temperatures (Figure 8a). This discrepancy can be partially ascribed to other scattering processes (grain boundary scattering and so on), which cannot be ignored at low temperatures. A good agreement between the calculated and experimental σ at high temperatures indicates that the carrier scattering processes can be neglected. Due to the overestimation of σ, the calculated zT values are larger than the experimental ones at low temperatures (Figure 8b). However, the difference in zT between the calculation and experiment is even greater at high temperatures, which may be due to the underestimation of κ

_{L}. The calculated κ

_{L}is smaller than the measured one at high temperature, which results in the contribution from optical phonon to thermal conductivity being neglected in the calculation. As is known, the TE properties depend substantially on the microstructures and associated defects. There are many carrier and phonon scattering processes in p–type FeNbSb-based samples [73]. Only the electron-phonon interaction for the carrier and phonon-phonon Umklapp and point-defect scatter for the phonon were considered in their scheme. Substantial uncertainties may arise and, predictably, the electrical conductivity and thermal conductivity show deviations from the measured results. However, accurate descriptions for a variety of scattering processes is difficult. Considering the approximations used and the uncertainties in experimental data, the agreement between calculated and experimental results is reasonable and acceptable.

## 4. Lattice Thermal Conductivities of HH Compounds

_{L}is the Debye-Callaway model [74]. In this model, Grüneisen parameter (γ), which describes the strength of lattice anharmonicity, is set to be a constant. The predicted κ

_{L}using this model showed fairly good agreement with experimental values at room temperature [75].

_{L}can be calculated via solving the phonon Boltzmann transport equation. Using this fully ab initio approach, Andrea et al. reported the κ

_{L}of 15.4, 13.3, and 15.8 W m

^{−1}K

^{−1}at 300 K for TiNiSn, ZrNiSn and HfNiSn, respectively [76]. The calculated values of κ

_{L}were different from the experimental ones, which may be due to the different defects within the samples. Katre et al. revealed that Ni/vacancy antisites are the dominant defects affecting thermal transport in ZrNiSn [77]. The calculated temperature and concentration dependence of thermal conductivities were in quantitative agreement with the published experimental results.

_{L}of 75 thermodynamically-stable ordered HH alloys based on a combination of machine learning algorithms and automatic ab initio calculations [78]. Three approaches were used in the calculations of κ

_{L}. The first method is based on the empirical observation that the force constants show a high degree of transferability between compounds sharing a crystal structure [79]. They calculated approximated κ

_{transf}with anharmonic force constants from Mg

_{2}Si, since it shares the HH lattice with sites A and B occupied by Mg atoms. For cross-validation, the anharmonic force constants and κ

_{ω}of 32 HH systems were also computed using the fully ab initio approach. The second proposed approach is calculating κ

_{forest}via random-forest regression algorithm by leveraging the fully calculated κ

_{ω}of 32 HH compounds as a training set. The third method presents a new machine-learning descriptor of κ

_{ω}that integrates only the crucial pieces of the anharmonic properties of the solid. They calculated κ

_{anh}with four exact anharmonic force constants and a linear model for the rest. The lattice thermal conductivities calculated with different methods for TiNiSn, ZrNiSn and HfNiSn are shown in Table 1. It is clear that κ

_{forest}and κ

_{anh}quantitatively agree well with κ

_{ω}and κ

_{L}calculated using the fully ab initio approach, while κ

_{transf}shows an obvious discrepancy with κ

_{ω}and κ

_{L}.

_{L}

^{AGL}of 75 thermodynamically-stable ordered HH alloys using a quasiharmonic Debye model, which includes anharmonic contributions to a certain extent [80]. The difference between κ

_{L}

^{AGL}and κ

_{L}

^{anh}(κ

_{ω}in Table 1) of 32 HH compounds is acceptable except the values of FeNbP and NiPbTe (Figure 9a), and the corresponding Spearman correlation is 0.810. However, the values of κ

_{L}

^{AGL}do not agree with those of κ

_{L}

^{ML}(κ

_{anh}in Table 1), as shown in Figure 9b. The corresponding Spearman correlation is 0.706. Additionally, the predicted values of FeVSb and CoZrSb show deviations of almost an order of magnitude from the measured room-temperature values. Typically, the κ

_{L}

^{AGL}of TiNiSn, ZrNiSn, and HfNiSn are 10.7, 10.22, and 12.97 W m

^{−1}K

^{−1}, respectively. These values are much lower than κ

_{ω}and κ

_{L}in Table 1. Therefore, the accuracy of high-throughput calculations varies with the different models used. Although the fully ab initio approach to calculating lattice thermal conductivity is more accurate, the calculation of third-order force constants is computationally costly. Moreover, the accuracy of calculations using some simpler models is comparable to that of ab initio calculations. Therefore, high-throughput calculations can guide the search on new TE materials with low lattice thermal conductivity to some extent.

## 5. Conclusions and Outlook

_{0.84}CoSb using synchrotron X-ray diffraction and DFT calculations [85]. After that single-phased HH compounds Nb

_{0.8+δ}CoSb (0 ≤ δ < 0.05), with a remarkable enhancement on TE performance, have been successfully synthesized by levitation melting [86]. Recently, Anand et al. proposed a valence-balanced rule to understand the ground state stability of HH compounds. In other words, their ground state structures always have a common net valence of 0, regardless of stoichiometry and nominal electron count (8, 18, or 19) [87]. Using this rule, 16 nominal 19-electron HH compounds, which have not been reported previously, were predicted. The newly-predicted off-stoichiometric HH compound Ti

_{0.75+x}PtSb was successfully synthesized and confirmed using X-ray studies. The work on nominal 19-electron compounds opens a new avenue to search for potential HH-based TE materials theorically and experimentally. Also worth noting is that Tang et al. reported a narrow solubility range on the Ti-Ni-Sn phase diagram primarily in the range of TiNi

_{1+x}Sn (0 ≤ x ≤ 0.06) at 1223 K using phase boundary mapping, which explains the large discrepancy of the literature data on the thermoelectric properties of TiNiSn within a unified phase diagram framework [88]. This work also suggests a direction of research on HH thermoelectric materials by the interplay of theory and experiment.

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- He, J.; Tritt, T.M. Advances in thermoelectric materials research: Looking back and moving forward. Science
**2017**, 357, eaak9997. [Google Scholar] [CrossRef] [PubMed] - Pei, Y.; Shi, X.; LaLonde, A.; Wang, H.; Chen, L.; Snyder, G.J. Convergence of electronic bands for high performance bulk thermoelectrics. Nature
**2011**, 473, 66–69. [Google Scholar] [CrossRef] [PubMed] - Page, A.; Poudeu, P.F.P.; Uher, C. A first-principles approach to half-heusler thermoelectrics: Accelerated prediction and understanding of material properties. J. Materiomics
**2016**, 2, 104–113. [Google Scholar] [CrossRef] - Yang, J.; Xi, L.; Qiu, W.; Wu, L.; Shi, X.; Chen, L.; Yang, J.; Zhang, W.; Uher, C.; Singh, D.J. On the tuning of electrical and thermal transport in thermoelectrics: An integrated theory-experiment perspective. NPJ Comput. Mater.
**2016**, 2, 114–130. [Google Scholar] [CrossRef] - Snyder, G.J.; Toberer, E.S. Complex thermoelectric materials. Nat. Mater.
**2008**, 7, 105–114. [Google Scholar] [CrossRef] [PubMed] - Zhu, T.; Fu, C.; Xie, H.; Liu, Y.; Zhao, X. High efficiency half-heusler thermoelectric materials for energy harvesting. Adv. Energy Mater.
**2015**, 5, 1500588. [Google Scholar] [CrossRef] - Xin, J.; Tang, Y.; Liu, Y.; Zhao, X.; Pan, H.; Zhu, T. Valleytronics in thermoelectric materials. NPJ Quantum Mater.
**2018**, 3, 9. [Google Scholar] [CrossRef] - Zaitsev, V.K.; Fedorov, M.I.; Gurieva, E.A.; Eremin, I.S.; Konstantinov, P.P.; Samunin, A.Y.; Vedernikov, M.V. Highly effective Mg
_{2}Si_{1−x}Sn_{x}thermoelectrics. Phys. Rev. B**2006**, 74, 045207. [Google Scholar] [CrossRef] - Zhang, J.; Liu, R.; Cheng, N.; Zhang, Y.; Yang, J.; Uher, C.; Shi, X.; Chen, L.; Zhang, W. High-performance pseudocubic thermoelectric materials from non-cubic chalcopyrite compounds. Adv. Mater.
**2014**, 26, 3848–3853. [Google Scholar] [CrossRef] [PubMed] - Murphy-Armando, F.; Fahy, S. First-principles calculation of carrier-phonon scattering in n-type Si
_{1−x}Ge_{x}alloys. Phys. Rev. B**2008**, 78, 035202. [Google Scholar] [CrossRef] - Zhu, T.; Liu, Y.; Fu, C.; Heremans, J.P.; Snyder, J.G.; Zhao, X. Compromise and synergy in high-efficiency thermoelectric materials. Adv. Mater.
**2017**, 29, 1605884. [Google Scholar] [CrossRef] [PubMed] - Morelli, D.T.; Jovovic, V.; Heremans, J.P. Intrinsically minimal thermal conductivity in cubic I−V−VI
_{2}semiconductors. Phys. Rev. Lett.**2008**, 101, 035901. [Google Scholar] [CrossRef] [PubMed] - Jana, M.K.; Pal, K.; Waghmare, U.V.; Biswas, K. The origin of ultralow thermal conductivity in InTe: Lone-pair-induced anharmonic rattling. Angew. Chem. Int. Ed.
**2016**, 55, 7792–7796. [Google Scholar] [CrossRef] [PubMed] - Lin, S.; Li, W.; Li, S.; Zhang, X.; Chen, Z.; Xu, Y.; Chen, Y.; Pei, Y. High thermoelectric performance of Ag
_{9}GaSe_{6}enabled by low cutoff frequency of acoustic phonons. Joule**2017**, 1, 816–830. [Google Scholar] [CrossRef] - Shi, X.; Chen, L.; Uher, C. Recent advances in high-performance bulk thermoelectric materials. Int. Mater. Rev.
**2016**, 61, 379–415. [Google Scholar] [CrossRef] - Zhao, L.D.; Lo, S.H.; Zhang, Y.; Sun, H.; Tan, G.; Uher, C.; Wolverton, C.; Dravid, V.P.; Kanatzidis, M.G. Ultralow thermal conductivity and high thermoelectric figure of merit in SnSe crystals. Nature
**2014**, 508, 373–377. [Google Scholar] [CrossRef] [PubMed] - Nielsen, M.D.; Ozolins, V.; Heremans, J.P. Lone pair electrons minimize lattice thermal conductivity. Energy Environ. Sci.
**2013**, 6, 570–578. [Google Scholar] [CrossRef] - Kurosaki, K.; Kosuga, A.; Muta, H.; Uno, M.; Yamanaka, S. Ag
_{9}TlTe_{5}: A high-performance thermoelectric bulk material with extremely low thermal conductivity. Appl. Phys. Lett.**2005**, 87, 061919. [Google Scholar] [CrossRef] - Yu, J.; Xia, K.; Zhao, X.; Zhu, T. High performance p-type half-Heusler thermoelectric materials. J. Phys. D Appl. Phys.
**2018**, 51, 113001. [Google Scholar] [CrossRef] - Xie, W.; Weidenkaff, A.; Tang, X.; Zhang, Q.; Poon, J.; Tritt, T.M. Recent advances in nanostructured thermoelectric half-heusler compounds. Nanomaterials
**2012**, 2, 379–412. [Google Scholar] [CrossRef] [PubMed] - Graf, T.; Felser, C.; Parkin, S.S.P. Simple rules for the understanding of Heusler compounds. Prog. Solid State Chem.
**2011**, 39, 1–50. [Google Scholar] [CrossRef] - Kimura, Y.; Tanoguchi, T.; Kita, T. Vacancy site occupation by Co and Ir in half-Heusler ZrNiSn and conversion of the thermoelectric properties from n-type to p-type. Acta Mater.
**2010**, 58, 4354–4361. [Google Scholar] [CrossRef] - Aliev, F.G.; Brandt, N.B.; Kozyr’Kov, V.V.; Moshchalkov, V.V.; Skolozdra, R.V.; Stadnyk, Y.V. Metal-insulator transition of RNiSn (R = Zr, Hf, Ti) intermetallic vacancy systems. JETP Lett.
**1987**, 45, 684–687. [Google Scholar] - Aliev, F.G.; Brandt, N.B.; Moshchalkov, V.V.; Kozyrkov, V.V.; Skolozdra, R.V.; Belogorokhov, A.I. Gap at the fermi level in the intermetallic vacancy system RniSn (R = Ti, Zr, Hf). Z. Phys. B
**1989**, 75, 167–171. [Google Scholar] [CrossRef] - Qiu, P.; Yang, J.; Huang, X.; Chen, X.; Chen, L. Effect of antisite defects on band structure and thermoelectric performance of ZrNiSn half-heusler alloys. Appl. Phys. Lett.
**2010**, 96, 152105. [Google Scholar] [CrossRef] - Xie, H.H.; Mi, J.L.; Hu, L.P.; Lock, N.; Chirstensen, M.; Fu, C.G.; Iversen, B.B.; Zhao, X.B.; Zhu, T.J. Interrelation between atomic switching disorder and thermoelectric properties of ZrNiSn half-Heusler compounds. CrystEngComm
**2012**, 14, 4467–4471. [Google Scholar] [CrossRef] - Do, D.T.; Mahanti, S.D.; Pulikkoti, J.J. Electronic structure of Zr-Ni-Sn systems: Role of clustering and nanostructures in half-Heusler and Heusler limits. J. Phys. Condens. Matter
**2014**, 26, 275501. [Google Scholar] [CrossRef] [PubMed] - Fu, C.; Zhu, T.; Pei, Y.; Xie, H.; Wang, H.; Snyder, G.J.; Liu, Y.; Liu, Y.; Zhao, X. High band degeneracy contributes to high thermoelectric performance in p-type half-Heusler compounds. Adv. Energy Mater.
**2014**, 4, 1400600. [Google Scholar] [CrossRef] - Fu, C.; Zhu, T.; Liu, Y.; Xie, H.; Zhao, X. Band engineering of high performance p-ype FeNbSb based half-Heusler thermoelectric materials for figure of merit zT > 1. Energy Environ. Sci.
**2015**, 8, 216–220. [Google Scholar] [CrossRef] - Liu, Y.; Xie, H.; Fu, C.; Snyder, G.J.; Zhao, X.; Zhu, T. Demonstration of a phonon-glass electron-crystal strategy in (Hf,Zr)NiSn half-Heusler thermoelectric materials by alloying. J. Mater. Chem A
**2015**, 3, 22716–22722. [Google Scholar] [CrossRef] - Schwall, M.; Balke, B. Phase separation as a key to a thermoelectric high efficiency. Phys. Chem. Chem. Phys.
**2013**, 15, 1868–1872. [Google Scholar] [CrossRef] [PubMed] - Schwall, M.; Balke, B. On the phase separation in n-type thermoelectric half-Heusler materials. Materials
**2018**, 11, 649. [Google Scholar] [CrossRef] [PubMed] - Ouardi, S.; Fecher, G.H.; Balke, B.; Schwall, M.; Kozina, X.; Stryganyuk, G.; Felser, C.; Ikenaga, E.; Yamashita, Y.; Ueda, S.; et al. Thermoelectric properties and electronic structure of substituted heusler compounds: NiTi
_{0.3−x}Sc_{x}Zr_{0.35}Hf_{0.35}Sn. Appl. Phys. Lett.**2010**, 97, 8616. [Google Scholar] [CrossRef] - Xie, H.; Wang, H.; Pei, Y.; Fu, C.; Liu, X.; Snyder, G.J.; Zhao, X.; Zhu, T. Beneficial contribution of alloy disorder to electron and phonon transport in half-heusler thermoelectric materials. Adv. Funct. Mater.
**2013**, 23, 5123–5130. [Google Scholar] [CrossRef] - Mao, J.; Zhou, J.; Zhu, H.; Liu, Z.; Zhang, H.; He, R.; Chen, G.; Ren, Z. Thermoelectric properties of n-type ZrNiPb-based half-Heuslers. Chem. Mater.
**2017**, 29, 867–872. [Google Scholar] [CrossRef] - Toboła, J.; Jodin, L.; Pecheur, P.; Scherrer, H.; Venturini, G.; Malaman, B.; Kaprzyk, S. Composition-induced metal-semiconductor-metal crossover in half-Heusler Fe
_{1−x}Ni_{x}TiSb. Phys. Rev. B**2001**, 64, 155103. [Google Scholar] [CrossRef] - Inorganic Crystal Structure Database (ICSD). Available online: http://icsd.ill.eu/icsd/ (accessed on 8 May 2018).
- Yang, J.; Li, H.; Wu, T.; Zhang, W.; Chen, L.; Yang, J. Evaluation of half-Heusler compounds as thermoelectric materials based on the calculated electrical transport properties. Adv. Funct. Mater.
**2008**, 18, 2880–2888. [Google Scholar] [CrossRef] - Hohenberg, P.; Kohn, W. Inhomogeneous electron gas. Phys. Rev.
**1964**, 136, B864–B871. [Google Scholar] [CrossRef] - Kohn, W.; Sham, L.J. Self-consistent equations including exchange and correlation effects. Phys. Rev.
**1965**, 140, A1133–A1138. [Google Scholar] [CrossRef] - Öğüt, S.; Rabe, K.M. Band gap and stability in the ternary intermetallic compounds NiSnM(M = Ti,Zr,Hf): A first-principles study. Phys. Rev. B
**1995**, 51, 10443–10453. [Google Scholar] [CrossRef] - Larson, P.; Mahanti, S.D.; Kanatzidis, M.G. Structural stability of Ni-containing half-Heusler compounds. Phys. Rev. B
**2000**, 62, 12754–12762. [Google Scholar] [CrossRef] - Miyamoto, K.; Kimura, A.; Sakamoto, K.; Ye, M.; Cui, Y.; Shimada, K.; Namatame, H.; Taniguchi, M.; Fujimori, S.-I.; Saitoh, Y.; et al. In-gap Electronic states responsible for the excellent thermoelectric properties of Ni-based half-Heusler alloys. Appl. Phys. Express
**2008**, 1, 081901. [Google Scholar] [CrossRef] - Hazama, H.; Matsubara, M.; Asahi, R.; Takeuchi, T. Improvement of thermoelectric properties for half-Heusler TiNiSn by interstitial Ni defects. J. Appl. Phys.
**2011**, 110, 063710. [Google Scholar] [CrossRef] - Douglas, J.E.; Chater, P.A.; Brown, C.M.; Pollock, T.M.; Seshadri, R. Nanoscale structural heterogeneity in Ni-rich half-Heusler TiNiSn. J. Appl. Phys.
**2014**, 116, 163514. [Google Scholar] [CrossRef] - Zeier, W.G.; Schmitt, J.; Hautier, G.; Aydemir, U.; Gibbs, Z.M.; Felser, C.; Snyder, G.J. Engineering half-Heusler thermoelectric materials using Zintl chemistry. Nat. Rev. Mater.
**2016**, 1, 16032. [Google Scholar] [CrossRef] - Zahedifar, M.; Kratzer, P. Band structure and thermoelectric properties of half-Heusler semiconductors from many-body perturbation theory. Phys. Rev. B
**2018**, 97, 035204. [Google Scholar] [CrossRef] - Fu, C.; Xie, H.; Liu, Y.; Zhu, T.J.; Xie, J.; Zhao, X.B. Thermoelectric properties of FeVSb half-Heusler compounds by levitation melting and spark plasma sintering. Intermetallics
**2013**, 32, 39–43. [Google Scholar] [CrossRef] - Fu, C.; Xie, H.; Zhu, T.J.; Xie, J.; Zhao, X.B. Enhanced phonon scattering by mass and strain field fluctuations in Nb substituted FeVSb half-Heusler thermoelectric materials. J. Appl. Phys.
**2012**, 112, 124915. [Google Scholar] [CrossRef] - Fu, C.; Liu, Y.; Xie, H.; Liu, X.; Zhao, X.; Jeffrey Snyder, G.; Xie, J.; Zhu, T. Electron and phonon transport in Co-doped FeV
_{0.6}Nb_{0.4}Sb half-Heusler thermoelectric materials. J. Appl. Phys.**2013**, 114, 134905. [Google Scholar] [CrossRef] - Zou, M.; Li, J.F.; Guo, P.; Kita, T. Synthesis and thermoelectric properties of fine-grained FeVSb system half-Heusler compound polycrystals with high phase purity. J. Phys. D Appl. Phys.
**2010**, 43, 415403. [Google Scholar] [CrossRef] - Jodin, L.; Tobola, J.; Pecheur, P.; Scherrer, H.; Kaprzyk, S. Effect of substitutions and defects in half-Heusler FeVSb studied by electron transport measurements and KKR-CPA electronic structure calculations. Phys. Rev. B
**2004**, 70, 184207. [Google Scholar] [CrossRef] - Fang, T.; Zheng, S.; Chen, H.; Cheng, H.; Wang, L.; Zhang, P. Electronic structure and thermoelectric properties of p-type half-Heusler compound NbFeSb: A first-principles study. RSC Adv.
**2016**, 6, 10507–10512. [Google Scholar] [CrossRef] - Pei, Y.; LaLonde, A.D.; Wang, H.; Snyder, G.J. Low effective mass leading to high thermoelectric performance. Energy Environ. Sci.
**2012**, 5, 7963–7969. [Google Scholar] [CrossRef] - Hu, L.P.; Zhu, T.J.; Wang, Y.G.; Xie, H.H.; Xu, Z.J.; Zhao, X.B. Shifting up the optimum figure of merit of p-type bismuth telluride-based thermoelectric materials for power generation by suppressing intrinsic conduction. NPG Asia Mater.
**2014**, 6, e88. [Google Scholar] [CrossRef] - Yang, J.; Qiu, P.; Liu, R.; Xi, L.; Zheng, S.; Zhang, W.; Chen, L.; Singh, D.J.; Yang, J. Trends in electrical transport of p-type skutterudites RFe
_{4}Sb_{12}(R = Na, K, Ca, Sr, Ba, La, Ce, Pr, Yb) from first-principles calculations and Boltzmann transport theory. Phys. Rev. B**2011**, 84, 235205. [Google Scholar] [CrossRef] - Yang, J.; Liu, R.; Chen, Z.; Xi, L.; Yang, J.; Zhang, W.; Chen, L. Power factor enhancement in light valence band p-type skutterudites. Appl. Phys. Lett.
**2012**, 101, 022101. [Google Scholar] [CrossRef] - Fang, T.; Zheng, S.; Zhou, T.; Yan, L.; Zhang, P. Computational prediction of high thermoelectric performance in p-type half-Heusler compounds with low band effective mass. Phys. Chem. Chem. Phys.
**2017**, 19, 4411–4417. [Google Scholar] [CrossRef] [PubMed] - Fu, C.; Bai, S.; Liu, Y.; Tang, Y.; Chen, L.; Zhao, X.; Zhu, T. Realizing high figure of merit in heavy-band p-type half-Heusler thermoelectric materials. Nat. Commun.
**2015**, 6, 8144. [Google Scholar] [CrossRef] [PubMed] - Askerov, B.M. Electron Transport Phenomena in Semiconductors; World Scientific Press: Singapore, 1994. [Google Scholar]
- Mahan, G.D.; Sofo, J.O. The best thermoelectric. Proc. Natl. Acad. Sci. USA
**1996**, 93, 7436–7439. [Google Scholar] [CrossRef] [PubMed] - Gorai, P.; Stevanović, V.; Toberer, E.S. Computationally guided discovery of thermoelectric materials. Nat. Rev. Mater.
**2017**, 2, 201753. [Google Scholar] [CrossRef] - Scheidemantel, T.J.; Ambrosch-Draxl, C.; Thonhauser, T.; Badding, J.V.; Sofo, J.O. Transport coefficients from first-principles calculations. Phys. Rev. B
**2003**, 68, 125210. [Google Scholar] [CrossRef] - Madsen, G.K.H.; Singh, D.J. BoltzTraP. A code for calculating band-structure dependent quantities. Comput Phys. Commun.
**2006**, 175, 67–71. [Google Scholar] [CrossRef] - Wang, S.; Wang, Z.; Setyawan, W.; Mingo, N.; Curtarolo, S. Assessing the thermoelectric properties of sintered compounds via high-throughput ab-initio calculations. Phys. Rev. X
**2011**, 1, 021012. [Google Scholar] [CrossRef] - Carrete, J.; Mingo, N.; Wang, S.; Curtarolo, S. Nanograined half-Heusler semiconductors as advanced thermoelectrics: An ab initio high-throughput statistical study. Adv. Funct. Mater.
**2014**, 24, 7427–7432. [Google Scholar] [CrossRef] - Spearman, C. The Proof and measurement of association between two things. Int. J. Epidemiol.
**2010**, 39, 1137–1150. [Google Scholar] [CrossRef] [PubMed] - Kandpal, H.C.; Felser, C.; Seshadri, R. Covalent bonding and the nature of band gaps in some half-eusler compounds. J. Phys. D Appl. Phys.
**2006**, 39, 776–785. [Google Scholar] [CrossRef] - Pettifor, D.G. A chemical scale for crystal-structure maps. Solid State Commun.
**1984**, 51, 31–34. [Google Scholar] [CrossRef] - Bhattacharya, S.; Madsen, G.K.H. High-throughput exploration of alloying as design strategy for thermoelectrics. Phys. Rev. B
**2015**, 92, 085205. [Google Scholar] [CrossRef] - Zou, D.F.; Xie, S.H.; Liu, Y.Y.; Lin, J.G.; Li, J.Y. Electronic structure and thermoelectric properties of half-Heusler Zr
_{0.5}Hf_{0.5}NiSn by first-principles calculations. J. Appl. Phys.**2013**, 113, 193705. [Google Scholar] [CrossRef] - Hong, A.J.; Li, L.; He, R.; Gong, J.J.; Yan, Z.B.; Wang, K.F.; Liu, J.M.; Ren, Z.F. Full-scale computation for all the thermoelectric property parameters of half-Heusler compounds. Sci. Rep.
**2016**, 6, 22778. [Google Scholar] [CrossRef] [PubMed] - Yu, J.; Fu, C.; Liu, Y.; Xia, K.; Aydemir, U.; Chasapis, T.C.; Snyder, G.J.; Zhao, X.; Zhu, T. Unique Role of Refractory Ta Alloying in Enhancing the Figure of Merit of NbFeSb Thermoelectric Materials. Adv. Energy Mater.
**2018**, 8, 1701313. [Google Scholar] [CrossRef] - Morelli, D.T.; Heremans, J.P.; Slack, G.A. Estimation of the isotope effect on the lattice thermal conductivity of group IV and group III-V semiconductors. Phys. Rev. B
**2002**, 66, 195304. [Google Scholar] [CrossRef] - Miller, S.A.; Gorai, P.; Ortiz, B.R.; Goyal, A.; Gao, D.; Barnett, S.A.; Mason, T.O.; Snyder, G.J.; Lv, Q.; Stevanović, V.; Toberer, E.S. Capturing anharmonicity in a lattice thermal conductivity model for high-throughput predictions. Chem. Mater.
**2017**, 29, 2494–2501. [Google Scholar] [CrossRef] - Andrea, L.; Hug, G.; Chaput, L. Ab initio phonon properties of half-Heusler NiTiSn, NiZrSn and NiHfSn. J. Phys. Condens. Mat.
**2015**, 27, 425401. [Google Scholar] [CrossRef] [PubMed] - Katre, A.; Carrete, J.; Mingo, N. Unraveling the dominant phonon scattering mechanism in the thermoelectric compound ZrNiSn. J. Mater. Chem. A
**2016**, 4, 15940–15944. [Google Scholar] [CrossRef] - Carrete, J.; Li, W.; Mingo, N.; Wang, S.; Curtarolo, S. Finding unprecedentedly low-thermal-conductivity half-Heusler semiconductors via high-throughput materials modeling. Phys. Rev. X
**2014**, 4, 011019. [Google Scholar] [CrossRef] - Giannozzi, P.; de Gironcoli, S.; Pavone, P.; Baroni, S. Ab initio calculation of phonon dispersions in semiconductors. Phys. Rev. B
**1991**, 43, 7231–7242. [Google Scholar] [CrossRef] - Toher, C.; Plata, J.J.; Levy, O.; de Jong, M.; Asta, M.; Nardelli, M.B.; Curtarolo, S. High-throughput computational screening of thermal conductivity, Debye temperature, and Grüneisen parameter using a quasiharmonic Debye model. Phys. Rev. B
**2014**, 90, 174107. [Google Scholar] [CrossRef] - Bhattacharya, S.; Madsen, G.K.H. A novel p-type half-Heusler from high-throughput transport and defect calculations. J. Mater. Chem C
**2016**, 4, 11261–11268. [Google Scholar] [CrossRef] - Yu, Y.G.; Zhang, X.; Zunger, A. Natural off-stoichiometry causes carrier doping in half-Heusler filled tetrahedral structures. Phys. Rev. B
**2017**, 95, 085201. [Google Scholar] [CrossRef] - Gautier, R.; Zhang, X.; Hu, L.; Yu, L.; Lin, Y.; Sunde, T.O.L.; Chon, D.; Poeppelmeier, K.R.; Zunger, A. Prediction and accelerated laboratory discovery of previously unknown 18-electron ABX compounds. Nat. Chem.
**2015**, 7, 308–316. [Google Scholar] [CrossRef] [PubMed] - Huang, L.; He, R.; Chen, S.; Zhang, H.; Dahal, K.; Zhou, H.; Wang, H.; Zhang, Q.; Ren, Z. A new n-type half-Heusler thermoelectric material NbCoSb. Mater. Res. Bull.
**2015**, 70, 773–778. [Google Scholar] [CrossRef] - Zeier, W.G.; Anand, S.; Huang, L.; He, R.; Zhang, H.; Ren, Z.; Wolverton, C.; Snyder, G.J. Using the 18-electron rule to understand the nominal 19-electron half-Heusler NbCoSb with Nb vacancies. Chem. Mater.
**2017**, 29, 1210–1217. [Google Scholar] [CrossRef] - Xia, K.; Liu, Y.; Anand, S.; Snyder, G.J.; Xin, J.; Yu, J.; Zhao, X.; Zhu, T. Enhanced thermoelectric performance in 18-electron Nb
_{0.8}CoSb half-Heusler compound with intrinsic Nb vacancies. Adv. Funct. Mater.**2018**, 28, 1705845. [Google Scholar] [CrossRef] - Anand, S.; Xia, K.; Hegde, V.I.; Aydemir, U.; Kocevski, V.; Zhu, T.; Wolvertona, C.; Snyder, G.J. A valence balanced rule for discovery of 18-electron half-Heuslers with defects. Energy Environ. Sci.
**2018**. [Google Scholar] [CrossRef] - Tang, Y.; Li, X.; Martin, L.H.J.; Cuervoreyes, E.; Ivas, T.; Leinenbach, C.; Anand, S.; Peters, M.; Snyder, G.J.; Battaglia, C. Impact of ni content on the thermoelectric properties of half-Heusler TiNiSn. Energy Environ. Sci.
**2018**, 11, 311–320. [Google Scholar] [CrossRef]

**Figure 2.**In-gap state formation via d orbitals from interstitial Ni. (

**a**) Schematic diagram of the density of states in MNiSn showing the in-gap Ni states (purple) resulting from intrinsic defects. These extra states in the band gap lead to a smaller observed optical band gap. (

**b**) Calculated band structure showing the band within the energy gap near the conduction band edge. Reproduced with permission [46]. Copyright 2016, Nature Publishing Group.

**Figure 3.**Electronic band structure for (

**a**) FeVSb and (

**b**) FeNbSb. Reproduced with permission [6]. Copyright 2015, Wiley.

**Figure 4.**Hall carrier concentration dependence of (

**a**) carrier mobility and (

**b**) power factor for p-type Ti doped Fe(V

_{1−y}Nb

_{y})

_{1−x}Ti

_{x}Sb HH compounds. Reproduced with permission [29]. Copyright 2014, Royal Society of Chemistry.

**Figure 5.**The calculated carrier concentration dependence of (

**a**) Seebeck coefficient and (

**b**) power factor for RuMSb (M = V, Nb, Ta) samples at 300 K [58]. Reproduced by permission of the PCCP Owner Societies.

**Figure 6.**(

**a**) The optimal carrier concentration versus the density of state effective mass for TE materials. The solid line is a guide for eyes; (

**b**) carrier concentration dependence of the power factor for the typical light-band PbTe, and the heavy-band system: n-type ZrNiSn, n-type filled CoSb

_{3}and p-type FeNbSb near 800 K. Reproduced from [59].

**Figure 7.**Maximum power factors vs the corresponding carrier concentrations for (

**a**) p-type and (

**b**) n-type cases. Reproduced with permission [38]. Copyright 2008, Wiley.

**Figure 9.**Thermal conductivities of HH semiconductors at 300 K compared to (

**a**) full anharmonic phonon ab initio parametrization and (

**b**) machine learning algorithm predictions from [78]. The green line represents that the value of the x-axis is equal to that of the y-axis. Reproduced with permission [80]. Copyright 2014, American Physical Society.

**Table 1.**The lattice thermal conductivities calculated with different methods for TiNiSn, ZrNiSn, and HfNiSn. Unit: W m

^{−1}K

^{−1}. κ

_{ω}: lattice thermal conductivity from fully ab initio calculation; κ

_{transf}: approximated κ

_{ω}with anharmonic force constants from Mg

_{2}Si; κ

_{forest}: κ

_{ω}obtained random-forest regression; κ

_{anh}: κ

_{ω}obtained with four exact anharmonic force constants and a linear model for the rest; κ

_{L}: lattice thermal conductivity form fully ab initio calculation in [76].

Materials | κ_{ω} | κ_{transf} | κ_{forest} | κ_{anh} | κ_{L} |
---|---|---|---|---|---|

TiNiSn | 17.9 | 57.1 | 20.3 | 16.8 | 15.4 |

ZrNiSn | 19.6 | 73.3 | 20.7 | 17.5 | 13.3 |

HfNiSn | - | 75.4 | 22.1 | 19.5 | 15.8 |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Fang, T.; Zhao, X.; Zhu, T.
Band Structures and Transport Properties of High-Performance Half-Heusler Thermoelectric Materials by First Principles. *Materials* **2018**, *11*, 847.
https://doi.org/10.3390/ma11050847

**AMA Style**

Fang T, Zhao X, Zhu T.
Band Structures and Transport Properties of High-Performance Half-Heusler Thermoelectric Materials by First Principles. *Materials*. 2018; 11(5):847.
https://doi.org/10.3390/ma11050847

**Chicago/Turabian Style**

Fang, Teng, Xinbing Zhao, and Tiejun Zhu.
2018. "Band Structures and Transport Properties of High-Performance Half-Heusler Thermoelectric Materials by First Principles" *Materials* 11, no. 5: 847.
https://doi.org/10.3390/ma11050847