High-Throughput Exploration of Half-Heusler Phases for Thermoelectric Applications

Abstract

As a result of the high-throughput ab initiocalculations, the set of 34 stable and novel half-Heusler phases was revealed. The electronic structure and the elastic, transport, and thermoelectric properties of these systems were carefully investigated, providing some promising candidates for thermoelectric materials. The complementary nature of the research is enhanced by the deformation potential theory applied for the relaxation time of carriers (for power factor, PF) and the Slack formula for the lattice thermal conductivity (for figure of merit, ZT). Moreover, two exchange-correlation parametrizations were used (GGA and MBJGGA), and a complete investigation was provided for both p- and n-type carriers. The distribution of the maximum PF and ZT for optimal doping at 300 K in all systems was disclosed. Some chemical trends in electronic and transport properties were discussed. The results suggest TaFeAs, TaFeSb, VFeAs, and TiRuAs as potentially valuable thermoelectric materials. TaFeAs revealed the highest values of both PF and ZT at 300 K (PF${}_p$ = 1.67 mW/K${}^2$m, ZT${}_p$ = 0.024, PF${}_n$ = 2.01 mW/K${}^2$m, and ZT${}_p$ = 0.025). The findings presented in this work encourage further studies on the novel phases, TaFeAs in particular.

1. Introduction

The wide application of thermoelectric (TE) materials is desirable due to the accumulating heat losses of various origins (e.g., energy and automotive industries). Despite numerous comprehensive studies [1,2,3,4,5,6,7], the subject of TE performance among half-Heusler (hH) phases is still an interesting area of research. Many alloys and whole families of compounds (e.g., tellurides) were not investigated up to now [8,9,10,11]. The experimental reports for known systems are generally focused on one regime of carriers [12,13,14,15,16,17,18,19,20,21,22,23,24]. In many theoretical studies, the predictions of thermoelectric properies are based on the standard exchange-correlation functional (XCF) [25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53], which may be insufficient for an accurate description of electronic structures of semiconducting materials. Recent studies showed new directions for a search of novel hH-based systems with complex crystal structures [54,55].

In this study, the continuation of previous investigations of electronic structures in over 150 hH phases with eighteen valence electrons [11] is presented. The set of 34 novel and stable (according to the Born’s criteria and thermodynamics) potentially valuable TE materials, previously predicted, is a subject of careful examination of TE properties by ab initio calculations. The results were obtained with two XCF parametrizations. Electron- and hole-like carriers are considered. The relaxation time of the carriers and lattice thermal conductivity are approximated in physical units, which provides a limited number of reasonable candidate systems for further efforts in the experimental synthesis of novel hH phases.

2. Computational Details

Electronic structure calculations with spin-orbit coupling included, based on the density functional theory (DFT), were performed with the use of VASP [56,57,58,59]. The generalized gradient approximation (GGA [60]) and modified Becke–Johnson GGA (MBJGGA [61]) approaches were selected for XCF. The cut-off energy for a plane-wave basis was set to 500 eV. The meshes of k-points for electronic structure and transport calculations were set to ${20}^3$ and ${50}^3$ points in the Brillouin zone, respectively. The equilibrium lattice parameters and elastic constants were only obtained with the GGA calculations. The relaxation time of carriers was approximated with the deformation potential theory [62]. The lattice thermal conductivity calculations were performed following the Slack formula [63,64,65]. Transport analysis were carried out with the Boltzmann transport equation, employing the BoltzTraP2 package [66]. The diagram of the complete calculation scheme is shown in Figure 1. The open quantum materials database was employed for studies of thermodynamic stability [67].

3. Results and Discussion

A primary set of potentially valuable hH alloys was created based on the valence electron count (VEC) of eighteen (due to the expected stability of such systems [68]). The elastic stability and thermodynamic stability of these phases were carefully examined in the previous study [11]. It is worth recalling that the criteria of mechanical stability (related to elastic constants, i.e., born conditions) are easy to meet in cubic hH materials. The thermodynamic stability of the phases considered here was evaluated based on analyses of formation energies (convex hull and hull distance, $\Delta E_{HD}$). The phases with zero hull distance, which means that their ground state crystal structures are expected to be cubic, were selected for further investigations of electronic structure. One may consider that other systems with formation energies close to the convex hull may also be stable or metastable in some nonequilibrium conditions. Only systems far from the most favorable composition should decompose into another phase or a mixture of various phases. Some additional hH compounds with very small hull distances (below 50 meV with respect to the lower crystal symmetry Pnma), e.g., TiNiPb, NbRuAs, and VRhGe, are not considered in this work. Although the results of ab initio calculations are usually reasonable, the synthesis of novel materials may be challenging even in the case of selected 34 phases. It is worth recalling that the thermal properties of similar materials may also be successfully studied with the use of molecular dynamics [69,70].

Only the semiconducting hH phases are selected here for further investigations. The band structures of the systems considered are depicted in Figure 2. One may find some general trends in electronic structure of hH materials, e.g., the band gaps ($E_g$) are narrow for relatively heavy elements (when comparing TiCoAs and TiCoBi, VCoGe and VIrGe, NbCoGe, and NbIrGe). Indirect band gaps are found in most systems (with the MBJGGA approach in particular). Direct $E_g^{\mathsf{\Gamma }-\mathsf{\Gamma }}$ is predicted with GGA for ScPdBi, ZrRuTe, HfPtGe, and TaRuAs, whereas the MBJGGA approach yields indirect $E_g^{\mathsf{\Gamma }-X}$ in these compounds. Only HfIrSb exhibits a direct $E_g$ within both XCFs. The use of more than one XCF and the effects of spin-orbit coupling on the electronic structures of hH systems were extensively discussed in previous reports (e.g., for YPdAs [11] and ScPtSb [8]). As presented in

Table 1. A comparison of band gaps E${}_{XCF}$ (eV), effective mass m${}_{XCF}^{p/n}$ (m${}_e$), and relaxation time values ${\tau }_{XCF}^{p/n}$ (fs) calculated for hH phases.

Compound	E${}_{\mathit{GGA}}$	E${}_{\mathit{MBJ}}$	m${}_{\mathit{GGA}}^{\mathit{p}}$	m${}_{\mathit{GGA}}^{\mathit{n}}$	m${}_{\mathit{MBJ}}^{\mathit{p}}$	m${}_{\mathit{MBJ}}^{\mathit{n}}$	${\mathit{\tau }}_{\mathit{GGA}}^{\mathit{p}}$	${\mathit{\tau }}_{\mathit{GGA}}^{\mathit{n}}$	${\mathit{\tau }}_{\mathit{MBJ}}^{\mathit{p}}$	${\mathit{\tau }}_{\mathit{MBJ}}^{\mathit{n}}$
ScPdBi	0.0705	0.1211	0.34	4.80	0.19	0.31	21.5	0.3	46.3	21.5
HfNiSn	0.3239	0.2705	0.65	0.75	0.59	0.78	18.5	13.1	18.5	10.8
HfPdSn	0.3791	0.3321	0.68	0.6	0.77	0.63	11.2	11.6	8.4	10.0
ZrPdSn	0.4373	0.4168	0.52	0.82	0.51	0.96	13.9	6.3	13.2	4.7
NbRuSb	0.3475	0.483	0.32	0.21	0.36	0.22	31.6	64.8	6.1	19.2
TiNiGe	0.6307	0.5812	1.37	1.20	1.23	1.05	5.6	5.9	5.9	6.5
VIrGe	0.2685	0.6402	0.61	0.54	0.46	0.38	14.0	18.1	20.6	32.6
VFeSb	0.3436	0.6569	0.65	0.24	0.49	0.41	14.2	61.5	2.8	2.5
NbFeAs	0.5731	0.6987	0.32	1.43	0.33	0.31	39.0	4.2	4.4	6.0
NbIrGe	0.6031	0.7214	0.82	0.38	0.82	0.36	8.6	29.3	8.0	30.3
ZrPtPb	0.6675	0.7294	0.48	0.19	0.47	0.37	14.8	53.4	14.0	18.0
NbIrSn	0.6251	0.7294	0.81	0.39	0.81	0.37	7.8	23.5	7.4	24.1
HfIrSb	0.6582	0.7538	0.92	0.59	0.24	1.54	9.7	6.1	64.1	1.3
TiCoBi	0.8786	0.779	0.54	0.56	0.54	1.13	14.2	11.8	13.3	3.8
TaRuAs	0.3657	0.7818	0.36	0.23	0.44	0.24	33.7	28.6	23.3	68.9
TiPtSn	0.665	0.8063	0.53	0.53	0.65	0.77	13.0	11.6	9.1	6.0
TiRhBi	0.6574	0.8138	0.39	0.58	0.55	1.18	18.5	9.1	10.3	2.8
VFeAs	0.3614	0.8625	0.39	0.33	0.39	0.36	30.6	39.6	27.1	35.6
TaFeSb	0.8135	0.8708	0.38	0.33	0.39	0.4	45.3	51.4	48.6	37.3
TiPtGe	0.7232	0.9088	0.44	0.42	0.48	0.47	19.6	19.0	16.0	15.2
HfCoBi	0.9763	0.9221	0.54	0.59	0.58	0.40	18.8	13.8	15.2	22.3
TiRhAs	0.7684	0.9571	0.32	0.41	0.32	0.42	31.6	20.2	29.8	19.4
TiIrSb	0.6813	0.9716	0.35	0.28	0.36	0.27	32.4	41.7	28.8	42.4
TaFeAs	0.879	0.9801	0.34	0.29	0.35	3.01	52.5	65.9	45.6	1.9
VCoGe	0.6832	0.9926	0.79	0.46	0.82	0.49	11.0	22.5	9.3	20.1
HfPtGe	0.9256	1.0236	0.56	0.44	0.25	1.36	17.6	8.4	53.0	3.4
TaCoSn	1.0083	1.0329	0.7	0.44	1.32	0.65	15.7	27.4	5.6	14.0
ZrPtGe	1.0119	1.0807	0.47	9.06	0.47	3.83	19.8	0.2	18.3	0.6
NbCoGe	1.0887	1.1272	0.58	0.34	0.60	0.35	17.2	33.6	14.0	30.1
TiFeTe	0.984	1.184	0.56	0.38	0.57	0.5	12.6	20.2	1.8	1.6
TaCoGe	1.1601	1.1881	0.69	0.45	0.78	0.46	20.0	32.2	14.2	26.7
TiCoAs	1.3001	1.241	0.70	1.70	0.83	1.57	14.0	2.9	10.3	3.0
ZrRuTe	0.932	1.247	0.51	0.34	0.24	0.28	11.4	9.8	37.6	29.1
HfCoAs	1.2861	1.3607	1.67	1.10	2.84	3.03	4.3	6.6	2.1	1.4

, the effect of MBJ potential on band gaps may be surprising, i.e., the direct-indirect transitions of $E_g$ result in narrower band gaps in some cases. Nevertheless, the general comparison between GGA- and MBJGGA-derived $E_g$, depicted in Figure 3a, clearly indicates wider band gaps yielded by this approach.

The thermoelectric performance of materials is characterized by two quantities: the power factor (PF) and the figure of merit (ZT), which are defined as follows:

$$\left\{\begin{array}{c}PF=S^2\sigma \left(\tau \right)\hfill \\ ZT=\frac{PF}{{\kappa }_e+{\kappa }_L}T.\hfill \end{array}\right.$$

where S is Seebeck coefficient, $\sigma$($\tau$) is electrical conductivity ($\tau$ is relaxation time of carriers), ${\kappa }_e$ is thermal conductivity of carriers, ${\kappa }_L$ is lattice thermal conductivity, and T is temperature. The values of the Seebeck coefficient, $\sigma$, and ${\kappa }_e$ are calculated from first principles with the Boltzmann transport equation. The theoretical estimations of $\tau$ and ${\kappa }_L$ require careful discussion. According to the deformation potential theory [62], the relaxation time of carries may be computed with the formula given as follows:

$${\tau }_{\beta }=\frac{2\sqrt{2\pi }{C}_{\beta }ħ}{3{\left(k_{B}T{m}^{\ast }\right)}^{3/2}{E}_{\beta }^2}$$

where $C_{\beta }$ is the elastic constant in the $\beta$ direction, $m^{\ast }$ is the effective mass, and $E_{\beta }$ is the deformation potential in the $\beta$ direction. The effective masses of electron- and hole-like carries, determined in the vicinity of the valence band maximum (VBM) and conduction band minimum (CBM), are gathered in Table 1. The relatively low values of $m^{\ast }$ are desirable for TE performance due to the long relaxation time of carriers. The strong variations between the p- and n-type $m^{\ast }$ occurred in some cases (ZrPtGe, TiCoAs), which promotes particular carrier regimes. The use of two XCF also may lead to similar discrepancies in $m^{\ast }$. The relaxation time of carriers is also strongly dominated by the deformation potential and elastic constant (not shown here), which generally results in good accordance between ${\tau }_{GGA}$ and ${\tau }_{MBJ}$ for a particular carrier type. Investigations of the deformation potential $E_{\beta }^{MBJ}$ for VBM and CBM revealed remarkably high values for some compounds, i.e, VFeSb: 101.00 and 121.66 eV; NbFeAs: 113.89 and 102.74 eV; and TiFeTe: 101.86 and 119.86 eV, respectively. These results are clearly larger than those for similar systems (e.g., HfCoAs: 31.11; and 36.20 eV, for VBM and CBM). One may consider some limitations of the MBJ functional in the case of non-equillibrium conditions and prefer the GGA deformation potentials for VFeSb (36.17 and 36.70 eV), NbFeAs (39.31 and 39.20 eV), and TiFeTe (39.29 and 41.40 eV, for VBM and CBM, respectively). This issue is crucial for the further discussion of TE performance, which could be strongly overestimated due to the extraordinarily long relaxation time of carriers.

The total thermal conductivity of semiconducting materials is dominated by the lattice contribution (${\kappa }_L$), which may be approximated with the Slack’s formula [63]. This approach also employs elastic constants of materials. The DFT-derived data are presented in

Table 2. Equilibrium lattice parameters (Å), elastic constants $C_{12}$ (GPa), unit cell volumes (Å${}^3$), and values of lattice thermal conductivity (W/mK).

Compound	a	${\mathit{C}}_{12}$	Volume	${\mathit{\kappa }}_{\mathit{L}}$
ScPdBi	6.525	65.93	69.45	72.20
HfNiSn	6.111	73.72	57.05	113.52
HfPdSn	6.360	93.09	64.31	99.54
ZrPdSn	6.396	86.55	65.41	91.21
NbRuSb	6.187	125.22	59.21	76.27
TiNiGe	5.668	95.40	45.52	61.53
VIrGe	5.818	169.88	49.23	38.92
VFeSb	5.788	98.83	48.48	45.43
NbFeAs	5.689	115.02	46.03	63.29
NbIrGe	6.010	156.94	54.27	62.45
ZrPtPb	6.508	90.27	68.91	95.69
NbIrSn	6.230	132.14	60.45	76.60
HfIrSb	6.333	112.19	63.50	112.06
TiCoBi	6.033	75.00	54.90	58.22
TaRuAs	5.972	158.49	53.25	52.81
TiPtSn	6.231	108.25	60.48	65.36
TiRhBi	6.280	93.95	61.92	50.51
VFeAs	5.496	126.62	41.50	44.79
TaFeSb	5.960	98.45	52.93	97.50
TiPtGe	5.991	126.35	53.76	51.50
HfCoBi	6.188	65.63	59.24	102.54
TiRhAs	5.889	128.96	51.06	43.12
TiIrSb	6.165	119.58	58.58	73.10
TaFeAs	5.692	128.43	46.10	69.27
VCoGe	5.512	121.07	41.87	47.04
HfPtGe	6.171	124.52	58.75	72.45
TaCoSn	5.962	96.36	52.98	88.66
ZrPtGe	6.200	119.11	59.58	66.56
NbCoGe	5.698	116.33	46.25	69.23
TiFeTe	5.864	67.43	50.41	60.65
TaCoGe	5.715	125.23	46.74	79.43
TiCoAs	5.605	104.92	44.02	57.09
ZrRuTe	6.298	82.02	62.45	86.20
HfCoAs	5.783	99.48	48.35	76.60

. The elastic constants $C_{12}$ are inversely proportional to the unit cell volumes of particular compounds. As depicted in Figure 3b, the obtained values of $k_L$ exhibit a pronounced dependence on the cubic lattice parameter. Compounds with relatively small lattice parameters and high values of C${}_{12}$ are characterized by low lattice thermal conductivity. Furthermore, there is no clear interplay between $k_L$ and the mass of the particular constituent elements. For some systems, the big summary ionic mass of the phase leads to the high $k_L$ (e.g., HfCoAs and HfCoBi), and for others, the relation is the opposite (e.g., HfNiSn and HfPdSn). One may only consider some general trends, i.e., the presence of heavy X and Z elements leads to increased lattice thermal conductivity in $XYZ$ hH systems, whereas for the heavy Y ions an opposite relation is seen.

Figure 4 and Figure 5 depict distributions of the PF and ZT at 300 K. The values of the carrier concentrations considered are limited to the range from ≈10${}^{19}$ to ≈10${}^{21}$, which seems to be reasonable according to the experimental results for hH phases [71]. The optimal values of carrier concentrations, maximalizing PF and ZT, are presented in the bottom panels of Figure 4 and Figure 5, respectively. As seen if Figure 4, the low values of PF (below ≈0.5 mW/K${}^2$m) are found for most systems for both p- and n-type regimes. The phases with the highest PF are TaFeAs, TaFeSb, VFeAs, and TiRuAs. As discussed above, the MBJGGA results for other materials may be biased by the artificially long relaxation time of carries. Namely, the high values of PF for ZrRuTe may be questionable because the corresponding GGA results are significantly smaller. The careful analysis of MBJGGA and GGA band structures for this material, seen in Figure 2, suggests only minor change in $E_g$ and similar shapes of VBM and CBM. The relatively high values of the PF in compounds gathered in

Table 3. Compounds with the best predicted thermoelectric performance, based on PF and ZT at 300 K (see also Figure 4 and Figure 5). With indexes ${}^{p/n}$, types of carriers are denoted. XCF used for presented results were GGA and MBJGGA (MBJ).

Comp.	PF${}_{\mathit{GGA}}$	PF${}_{\mathit{MBJ}}$	ZT${}_{\mathit{GGA}}$	ZT${}_{\mathit{MBJ}}$
TaFeAs	1.67${}^p$, 2.01${}^n$	1.48${}^p$, 1.17${}^n$	0.024${}^p$, 0.025${}^n$	0.021${}^p$, 0.015${}^n$
TaFeSb	1.49${}^p$, 1.15${}^n$	1.63${}^p$, 1.06${}^n$	0.015${}^p$, 0.011${}^n$	0.017${}^p$, 0.010${}^n$
VFeAs	1.10${}^p$, 0.95${}^n$	0.98${}^p$, 1.25${}^n$	0.024${}^p$, 0.017${}^n$	0.021${}^p$, 0.025${}^n$
TiRhAs	0.95${}^p$, 0.69${}^n$	0.80${}^p$, 0.96${}^n$	0.018${}^p$, 0.014${}^n$	0.014${}^p$, 0.021${}^n$

are obtained with both XCF approaches, which may confirm an optimal electronic structure for TE performance in these phases.

Although the values of the ZT presented in Figure 5 are proportional to the corresponding PF, the distribution of the ZT is slightly different from that of the PF. Namely, a relatively low lattice thermal conductivity of TiRhAs leads to the ZT comparable to those of VFeAs and TaFeAs. A similar effect is also found in MBJGGA results for VIrGe. The TE properties of compounds with the highest values of the PF and ZT, gathered in Table 3, are lower than those of the best hH TE systems (e.g., NbFeSb [36]). It is worth noting that the lattice thermal conductivity of hH materials may be poorly modeled with Slack’s formula, whereas the values of the PF predicted here suggest very good TE performance of selected phases.

Regarding available experimental data, Downie et al. disclosed the n-type PF of ≈0.3 mW/mK${}^2$ in HfNiSn at 330 K with the possibility of further enhancement for various dopings [72]. The accordance between the experimental and n-type PF of 0.34 mW/mK${}^2$ calculated within the MBJGGA approach for HfNiSn is very good. Although experimental studies reported extremely high ZT at temperatures above 800 K for various HfNiSn materials [73], the room temperature ZT in this system is negligible. While a good accordance between the experimental results and our predictions was obtained for TaFeSb [74], the systems based on this compound are also considered as promising only at high T. In the case of TaCoSn [75], the low experimental ZT at 300 K in the n-type regime is also predicted in the present work. However, the p-type TE performance of TaCoSn seems to be an interesting area for further research. For some compounds studied here, the available experimental reports were focused on properties other than TE performance, e.g., the electrical transport in ScPdBi [76]. Syntheses of most novel materials predicted in this work were not reported up to now. The generally good accordance between the theoretical and available experimental data for hH phases indicates the DFT-based methods combined with the transport modeling and approximations for the relaxation time of carriers and lattice thermal conductivity as a reasonable tool for materials science. Furthermore, an enhancement of TE in particular phases can be achieved with experimental methods, which are not considered in theoretical calculations, e.g., the modifications of a crystal microstructure cause a strong decrease in lattice thermal conductivity and higher ZT, as reported for VFeSb [77]. It is worth recalling that NbFeSb, which has been extensively studied in recent years, exhibits the PF and ZT of ≈33 $\mathsf{\mu }$W/cm K${}^2$ and the ZT ≈ 0.4 at 300 K, respectively [16]. Furthermore, some of the properties of the two systems considered here were reported in previous theoretical studies, i.e., HfPdSn (half-Heusler compounds having full-Heusler counterparts regarded in machine learning models [21]) and ZrPdSn (electronic properties calculated with ab initio methods [78]).

Among the novel hH systems regarded in this work, TaFeAs is the most promising TE material. Other hH arsenides were recently predicted to show interesting properties [79,80,81,82]. Interestingly, the band gaps of materials selected here are not optimal according to the ‘10 k${}_B$T rule’ [83]. It is worth recalling that the band structures of promising TE materials among hH alloys, i.e., LaPtSb [84], exhibit characteristic shapes of VBM or CBM, which results in very low effective masses of carriers. The Fe-based arsenides exhibit such features in both p- and n-type regimes. The best n-type TE performance of TaFeAs may be explained to some extent by the significant flattening of conduction bands in the $\mathsf{\Gamma }-X-L$ direction without the flattening of CBM (high S and low $m^{\ast }$). Furthermore, the Fe-bearing phases are also expected to be promising TE materials in the p-type regime, which is related to the relatively high carrier relaxation time and low lattice thermal conductivity in such systems.

4. Conclusions

Based on the high-throughput ab initio calculations, including the two XCFs regarded, the deformation potential theory applied for carrier relaxation time, and Slack’s formula for lattice thermal conductivity, the potential TE performance of 34 novel hH semiconductors was carefuly examined. TaFeAs, TaFeSb, VFeAs, and TiRuAs attract particular attention due to their relatively high PF and ZT at 300 K. The desired set of electronic properties for TE materials is difficult to obtain in a single hH phase. Such a complex examination of multiple parameters for each potentially valuable TE material reveals some interesting trends for predicting novel systems. The hH arsenides are expected to be a promising family of novel hH phases for further experimental research.