Huge power factor in p-type half-Heusler alloys NbFeSb and TaFeSb

NbFeSb is a promising thermoelectric material which according to experimental and theoretical studies exhibits a high power factor of up to 10 mW m−1 K−2 at room temperature and ZT of 1 at 1000 K. In all previous theoretical studies, κlatt is calculated using simplified models, which ignore structural defects. In this work, we calculate κlatt by solving the Boltzmann transport equation and subsequently including the contributions of grain boundaries, point defects and electron–phonon interaction. The results for κlatt and ZT are in excellent agreement with experimental measurements. In addition, we investigate theoretically the thermoelectric properties of TaFeSb. The material has recently been synthesised experimentally, thus confirming the theoretical hypothesis for its stability. This encourages a full-scale computation of its thermoelectric performance. Our results show that TaFeSb is indeed an excellent thermoelectric material which has a very high power factor of 16 mW m−1 K−2 at room temperature and ZT of 1.5 at 1000 K.


Introduction
NbFeSb is a half-Heusler intermetallic compound which has recently attracted a lot of attention as a potential thermoelectric material due to its ecologically friendly properties and the relatively high earth abundance of Nb and Fe. NbFeSb alloys are reported to have a large power factor of up to 10 mW m −1 K −2 [1], beating some of the best thermoelectrics, e.g. Bi 2 Te 3 . However, their thermal conductivity is also a lot higher than Bi 2 Te 3 [1][2][3]. The high thermal conductivity of NbFeSb is phonon dominated and this provides much room for improvement of the current thermoelectric figure of merit maximum of ZT=1 at 1000 K.
The thermoelectric figure of merit is given by the equation ZT=S 2 σT/κ and several theoretical and experimental studies which aim to optimise the thermal conductivity (κ) as well as the Seebeck coefficient (S) and electrical conductivity (σ) have been conducted in the past couple of years [1,2,[4][5][6][7][8][9][10]. This optimisation is done by p-type doping with Ti, Hf and Zr for Nb or Sn for Sb. Such an approach maximises the power factor by fine tuning of the doping levels and decreases the lattice thermal conductivity by enhancing the phonon scattering due to the mass difference between the dopant and host atoms. To date, the best NbFeSb results are obtained by Ti doping [1] due to the large mass difference between Ti and Nb. The mass difference can be further enhanced if Nb is substituted with a heavier but chemically similar element like Ta, which is something that has not yet been thoroughly investigated.
The first aim of this study is to compute the lattice thermal conductivity (κ latt ) of NbFeSb using the semiclassical Boltzmann transport equation (BTE) and compare the obtained theoretical thermoelectric (TE) results to experimental measurements. The second aim is to use the same approach and calculate the TE properties of a compound very similar to NbFeSb, namely TaFeSb. A theoretical study by Bhattacharya and Madsen [9] reports that TaFeSb is a stable compound which can also be doped with Ti in a similar way to NbFeSb. A very recent experimental study by Zhu et al [11] investigates extensively the phase stability of the compound and provides an XRD pattern after the successful experimental synthesis of TaFeSb. The main interest in TaFeSb comes from the fact that it has the same number of valence electrons as NbFeSb, while Ta has almost twice the mass of Nb. This suggests that TaFeSb should have the same good electronic TE properties as NbFeSb. In addition, the heavier Ta should also lead to an increase in the scattering strength in doped TaFeSb due to point defects (PD) and thus decrease κ latt . As a result, TaFeSb may be expected to have a significantly higher ZT than NbFeSb but until now there have been no full-scale theoretical studies on the pure TaFeSb compound to confirm this hypothesis. Zeeshan et al [12] investigates the thermoelectric properties of TaFeSb but without computing the electron relaxation time or including the additional phonon scattering mechanisms. Another recent study conducted by Yu et al [8] investigates the effect of Ta but in NbFeSb systems. Hence, this is clearly a very hot topic and there is a strong need for a full study of the thermoelectric properties of TaFeSb.

Methodology and theory
We split our calculations into two stages. We solve the electron BTE in the first stage and the phonon BTE in the second one. The energy distribution of the charge carriers and phonons is computed from first-principles.

DFT calculations and electronic TE properties
The first-principles calculations were performed with the CASTEP [13] code and the generalised gradient approximation Perdew-Burke-Ernzerhof (GGA-PBE) exchange-correlation functional [14]. On-the-fly ultrasoft pseudopotentials (C9 set) [15] were used with a plane-wave cut-off energy of 700 eV with a grid scale of size 2.0. A cubic unit cell, corresponding to four elementary rhombohedral cells, was used for all simulations. The Brillouin zone was sampled using a Monkhorst-Pack grid [16] with an 8×8×8 k  -points mesh (equivalent to k  -points spacing of 0.021 2πÅ −1 ). The structure was fully optimised until pressure and energy were converged to 0.1 GPa and 0.02 meV/atom, respectively. Density of states (DOS) and partial density of states were analysed using the OptaDOS code [17].
Electronic transport properties were calculated using the semi-classical Boltzmann transport formalism as implemented in the BoltzTraP code [18]. The electronic eigenenergies required for the transport properties were calculated with CASTEP on a 48×48×48 k  -points mesh, which was later interpolated on a 5 times denser mesh in BoltzTraP. The simulated half-Heusler alloys are isotropic and the Seebeck coefficient S, electrical conductivity σ and electron thermal conductivity κ el can be evaluated as the average of the trace of the respective tensors. The final results are obtained as a function of the temperature (T) for 37 fixed doping levels from n h =10 18 cm −3 to n h =10 22 cm −3 . BoltzTraP calculates both electrical and electron thermal conductivity as s t and κ el /τ where τ is the relaxation time. We use the deformation potential (DP) theory to compute τ [19]. A more detailed explanation of the steps needed for calculating τ is provided in the supplementary materials.

Lattice thermal conductivity modelling 2.2.1. ShengBTE and thirdorder programs
The lattice thermal conductivity was calculated by solving the phonon BTE in ShengBTE, which as inputs requires the second order force constants (usually just called the 'force constants') and the anharmonicity (third order force constants) of the system. The second order force constants were obtained with CASTEP using density-functional perturbation theory for the phonons [20]. The calculations used the GGA-PBE exchangecorrelation functional [14], on-the-fly norm-conserving pseudopotentials (NCP17 set) and a plane-wave cut-off energy of 2000 eV with a grid scale of size 2.0. The Brillouin zone was sampled using a Monkhorst-Pack [16] grid with an 5×5×5 k  -points mesh (equivalent to k  -points spacing of 0.034 2πÅ −1 ). A q  -point grid of the same size and spacing was used for calculating the second order force constants. The third order force constants were calculated using the finite-displacement supercell approach. The set of supercells and the reconstruction of the force constants was performed by the thirdorder.py script that is provided as part of the ShengBTE package. The ab initio calculations were done using CASTEP. The settings for these runs included: a 2×2 × 2 cubic supercell, on-the-fly ultrasoft pseudopotentials (C9 set), a plane-wave cut-off energy of 600 eV with a grid scale of size 2.0 and a very fine energy per atom convergence tolerance of 2×10 −10 eV.
ShengBTE computes the intrinsic lattice thermal conductivity κ int due to 3P (three-phonon) processes. We have also included the effect of grain boundaries (GB), PD and electron-phonon (EP) interaction to the lattice thermal conductivity. More details on how this is done are given in the supplementary materials.

Results
The results are split into two subsections. The first one presents the calculations on the TE properties of NbFeSb. We start by following the well-established procedure of using BoltzTraP [18] to obtain the electronic properties of the material and then solve the phonon BTE using ShengBTE [21]. Furthermore, we build upon the method proposed by Hong et al [5] for the inclusion of PD and introduce the contributions of GB and EP interaction to the lattice thermal conductivity of NbFeSb. To the best of our knowledge, this is the first instance when the lattice thermal conductivity of NbFeSb is calculated by solving the BTE and including all these additional contributions. For this reason, the results are thoroughly compared to the available experimental data. The second section follows a similar layout but is focused on TaFeSb and the observed improvements in TE properties with respect to NbFeSb.
It is worth pointing out that BolzTraP calculates the TE properties at different doping levels by changing the chemical potential implicitly and hence the dopant atoms are not explicitly included. For this reason, the p-type compounds in electronic properties section are referred simply as NbFeSb and TaFeSb. However, the computation of the change in the lattice thermal conductivity due to PD requires knowledge of the atomic mass of the dopant atoms. In this case, the structures are referred as Nb 1−x Ti x FeSb and Ta 1−x Ti x FeSb, with Ti being used for the p-type doping.

Electronic structure
NbFeSb is a half-Heusler compound, which has a composition of XYZ, where X and Y are transition metals and Z is a main group element. The crystal structure is face-centred cubic, having space group F m 43 (216). The lattice constant is calculated to be 5.96 Å, which agrees well with the experimental value of 5.95 Å [4]. The band structure and DOS are presented in figure 1. The figure shows that the conduction band minimum is at the Γpoint, whereas the valence band maximum is positioned at the L-point. The magnitude of the formed indirect band gap (ò g ) is 0.53 eV, which is in an excellent agreement with other theoretical [1, 5, 6] (ò g =0.52 and 0.53 eV) and experimental [1] (ò g =0.51 eV) studies. The partial DOS show that Fe and Nb are the main contributors to states around the Fermi level. This means that the power factor is mainly affected by Fe and Nb rather than Sb.

Electronic TE properties
The parameters needed to calculate the electron relaxation time for bulk NbFeSb are given in table 1. These include the DP (V DP ), effective mass of the charge carriers (m * ), carrier mobility (μ) and relaxation time (τ). The elastic constants are given in table S1 in the supplementary materials available online at stacks.iop.org/ JPMATER/2/035002/mmedia. The values of the parameters obtained for holes are slightly higher, but within the margin of error, than the ones obtained experimentally by He et al [1] and Fu et al [4]. The experimental measurements have been performed on doped systems which exhibit structural defects. Therefore, a slight overestimate is to be expected when the results are compared to the modelled perfect bulk system. To the best of our knowledge there are no experimental results on the electron parameters. However, the electron values presented in table 1 agree extremely well with the theoretical prediction of Hong et al [5]. The magnitude of the DP constant for holes (V 13.98 eV DP = -) is lower than the one for electrons (V DP = −14.53 eV). This can be =´-) the Seebeck coefficient becomes 129 μV K −1 and 266 μV K −1 at 300 K and 1000 K, respectively. These values are slightly lower than the experimental results obtained by Fu et al [4] (S=150 μV K −1 and 285 μV K −1 at 300 K and 1000 K, respectively) and He et al [1] (S=175 and 300 μV K −1 at 300 and 1000 K, respectively). It is worth mentioning, however, that a lower doping level in the theoretical model of n 6 10 cm The electrical conductivity σ and electronic thermal conductivity κ el also agree very well with other theoretical studies [5,6], but are slightly larger than found in experiment [1,4]. There are a few reasons for this discrepancy. As already mentioned, the carrier mobility of the perfect crystal is expected to be higher than μ of the doped compounds, hence τ and σ are also larger. Second, the temperature dependence of τ is no longer proportional to T −3/2 at temperature <450 K [1,4]. Finally, the constant relaxation time approximation lacks dependence on the chemical potential, which means that additional scattering events are not captured when the doping levels are increased. Thus, σ and κ el tend to be overestimated at high doping levels. Nevertheless, the current model for τ is a computationally inexpensive approach that allows us to calculate values for σ and κ el , which agree relatively well with both theoretical and experimental studies.
The highest value of the power factor PF=S 2 σ is obtained at n 7 10 cm =´-(x=0.1). Beyond that value, the theoretical prediction starts to overestimate the experimental results by values up to Table 1. Parameters needed for electron and hole τ calculations of NbFeSb. These include the deformation potential (V DP ), effective mass of charge carriers (m * ), carrier mobility (μ) and relaxation time (τ) at 300 K for electrons and holes.  ≈2 mW m −1 K −2 when one reaches x=0.3. Such behaviour is also noticed by the other theoretical studies mentioned before. The reason for this could be either the constant relaxation time approximation, or the fact that the heavy doping significantly changes the electronic structure of the system. However, as shown experimentally, NbFeSb exhibits its best thermoelectric performance at around x=0.05, and this region is accurately modelled by the current theoretical approach.

Lattice thermal conductivity
The phonon DOS are presented in figure 3. The data is in a good agreement with the results obtained by Hong et al [5] and Zeeshan et al [12] and as there are no imaginary frequencies the structure is mechanically stable. The phonon DOS can be split into three regions. The first one is at low frequency, ω<170 cm −1 where the lattice vibrations are primarily due to Sb atoms. The dominant contributor to the phonon DOS for 170<ω<230 cm −1 is Nb, whereas for ω>230 cm −1 lattice vibrations are predominantly due to Fe with a small contribution from Nb. The Nb atomic vibrations have the biggest frequency spread among the constituents of the material. In addition, the mass difference between Nb and the dopant atoms (here assumed to be Ti) leads to an increase in the scattering strength. Thus, the lattice thermal conductivity κ latt of NbFeSb can be reduced significantly by doping. Our results show the clear presence of a phonon gap at ω≈275 cm −1 , something which is not observed either by Hong et al or Zeeshan et al [12]. The reason for this discrepancy comes from the choice of the q  -point grid for the phonon calculations. The phonon DOS converges slowly and the gap only becomes apparent when the q  -point mesh is at least 3×3×3 or equivalently a spacing of 0.056 2πÅ −1 .
Next we focus on the estimated value for the lattice thermal conductivity and how different contributions affect it. The intrinsic value of κ latt obtained from ShengBTE is 21.82 and 6.49 W m −1 K −1 at 300 and 1000 K, respectively. This agrees very well with the theoretical result obtained by Hong et al [5] but is a bit higher than the experimental measurements [1,4]. The main reasons for this discrepancy is the fact that there are no defects such as GB, PD or dopant atoms in the modelled structure. To correct this, we include the effect of all mentioned impurities by using Klemens' model [22] and calculating the impact on the intrinsic value obtained from ShengBTE.
The study conducted by He et al reports that the size of the GB in NbFeSb varies between 0.3 and 4.5 μm, depending on the hot pressing temperature. Figure 4 shows how the lattice thermal conductivity of phonons with a given mean free path changes at room temperature when GB are included in the theoretical model. The graph illustrates the effect of GB (L GB ) by considering two different average sizes of L 4.5 GB = and 0.5 μm. Blue circles represent the intrinsic values of κ latt and it can be seen that L 4.5 GB = μm, illustrated with black and white squares, have an almost negligible impact on κ latt . However, there is a noticeable change in κ latt when the size of the GB is reduced to 0.5 μm (orange triangles), and the accumulated value of κ latt becomes 18.84 W m −1 K −1 . For completeness, L GB =0.3 μm was also tested and yielded a result of κ latt =17.59 W m −1 K −1 at 300 K. Both results for L GB =0.3 and 0.5 μm are within the margin of error of the experimental value of κ latt ≈17 W m −1 K −1 (undoped NbFeSb, 12% relative error).
To complete the calculation, we include the effect of PD and EP interaction to κ latt . The computation of the EP interaction requires knowledge of the electron τ. The lack of doping level dependence in the constant relaxation time approximation makes it unsuitable for calculating the EP contribution. The experimental data from the He et al study, including the temperature and doping dependencies, was used in accordance to the theoretical model and is discussed in more details in the supplementary materials. Figure 5(a) shows how the lattice thermal conductivity of Nb 1−x Ti x FeSb is reduced when all contributions are included. The results are presented for doping x=0.05 and the best match to the experimental data is obtained with L GB =0.5 μm.  values for the lattice thermal conductivity agree very well with the experimental study, particularly with the He et al study at temperatures of up to 700 K. There is a slight underestimate of the theoretical value of κ latt at higher temperature for x=0.04 and x=0.05. This can be explained with the lack of a bipolar thermal conductivity (κ bip ) term in the calculations. In order to compute that, one needs to calculate a value for the electron relaxation time which depends on the doping level. Therefore, using the constant relaxation time approximation to compute κ bip would yield inaccurate results. However, as it can be seen in figure 5(b), the contribution of κ bip is sufficiently small that the computed values are still in a good agreement with the experimental measurements.

Figure of merit
The final results on the thermoelectric figure of merit ZT for the p-type Nb 1−x Ti x FeSb are presented in figure 6. A comparison between ZT values obtained in this study and the experimental data is shown in figure 6(a). There is a good agreement up to T=650 K between our results and the measurements conducted by He et al. The overestimate of ZT above this temperature for x=0.04 and 0.05 can be explained by the missing κ bip term in the lattice thermal conductivity. This has already been discussed in the previous section and explains why the agreement between the experimental and theoretical results at high temperatures improves with the increase of the doping concentration. Additionally, the limitations of the constant relaxation time approximation, e.g. no dependence on the chemical potential and no inclusion of the extrinsic scattering mechanisms, can easily add up and lead to the observed discrepancies at lower temperatures. The results in this study slightly overestimate ZT when compared to Fu et al [4]. However, as with the lattice thermal conductivity results, there is a mismatch between the experimental results presented by He et al and Fu et al. The latter uses a much lower annealing  temperature, and so the density of the GB in the sample is expected to be higher. This further confirms that the constant relaxation time approximation could play a major role along with the bipolar term in the discrepancy between the theoretical and experimental results. The sample preparation in the Fu et al study influences both the electrical and thermal conductivity, and as a consequence, the measured ZT values are expected to be a bit lower than the ones obtained in our calculations.
The colour map in figure 6(b) shows that NbFeSb remains most efficient at high temperature, despite the big power factor of PF=9.3 mW m −1 K −2 at 300 K. The p-doped NbFesb displays its best figure of merit (ZT≈1.0) at T=1000 K and high doping levels between x=0.05 and 0.10, corresponding to n h =1×10 21 and 2×10 21 cm −3 . This result is typical for half-Heusler alloys [23] and illustrates that a reduction of κ latt can significantly enhance the thermoelectric performance of similar half-Heusler alloys.

Electronic structure
The crystal structure of TaFeSb is very similar to NbFeSb with the only difference being the atomic species on the X-site. The lattice constant is calculated to be 5.95 Å. The band structure and DOS are presented in figure 7. The band gap of TaFeSb is calculated to be 0.86 eV, close to the value ò g =0.93 eV computed by Bhattachrya and Madsen [9]. It can be seen that the valence bands and DOS near the Fermi level remain almost unchanged when compared to NbFeSb. This suggests that the p-type S, σ and κ el should exhibit the same behaviour as in NbFeSb, leaving the relaxation time as the determining factor for any change in the electronic TE properties.

Electronic TE properties
The relaxation time along with the parameters necessary for its calculation are shown in table 2. There is a noticeable reduction in the DP values for both holes V 11.06 eV DP = -(−13.98 eV for NbFeSb) and electrons V DP =−11.81 eV (−14.53 eV for NbFeSb). This means that stress has less effect on the electronic structure of TaFeSb. In addition, a slight reduction in the effective mass is also observed, with m m 1.57 h e * = ( ). As a result, the mobility of the holes and relaxation time are increased to μ h =53.11 cm 2 V −1 s −1 and τ h =47.32 fs.
Next we present the electronic TE properties of TaFeSb in the form of colour maps in figure 8. The colour maps investigate a very wide doping and temperature range and might not be intuitive for comparison purposes. For that reason, we also provide 2D plots in figure 9, which compare the electronic properties of TaFeSb and NbFeSb for the common doping levels of x=0.04, 0.05 and 0.10. The value of the p-type Seebeck coefficient for  Subfigure (b) is a colour map which shows the ZT of p-type NbFeSb with respect to the charge carrier concentration and temperature, with a maximum ZT of 1 at n h =2×10 21 cm −3 (x=0.1) and T=1000 K.
x=0.05 is calculated to be 113.81 and 247.5 μV K −1 at 300 and 1000 K, respectively. The change in S with respect to the NbFeSb results for the same doping concentration is less than 1%, which is expected due to the similarity in the valence bands of both materials. On the other hand, the bigger band gap in TaFeSb results in a bigger p-type S at a very low doping concentration and temperature around 600 K. This is visualised with an increase of the red area in figure 8(a) when compared to NbFeSb in figure 2(a). The results confirm that not only does TaFeSb exhibit a competitive Seebeck coefficient around the experimentally investigated doping levels, but also shows a significant improvement at very low n h and moderate T.
The results obtained from BoltzTraP for σ and el k predict a behaviour analogous to the changes observed for p-type S. Therefore, the increase of τ (holes), which is ≈80%, yields a significant improvement in σ, and an increase in κ el . The increase of σ leads to an astonishing power factor of PF≈16 mW m −1 K −2 at room temperature and x=0.03-0.05. For comparison, the power factor of NbFeSb is estimated to be  These include the deformation potential constant (V DP ), effective mass of charge carriers (m * ), carrier mobility (μ) and relaxation time (τ) at 300 K for electrons and holes.
Carrier type  9-10 mW m −1 K −2 , and the maximum value for Fe 2 VAl is measured to be 5.5 mW m −1 K −2 [24]. The compounds based on the already established TE material Bi 2 Te 3 have a power factor between 1.5 and 6 mW m −1 K −2 [3,25,26]. The improvement in PF of TaFeSb over NbFeSb is maintained over a wide range of doping levels from n h =10 20 cm −3 to n h =2×10 21 cm −3 and at higher temperatures (compare figures 8(d) and 2(d) and note the unchanged ranges). In summary, TaFeSb has a significantly better electronic TE performance than NbFeSb due to the increased band gap and higher mobility of the charge carriers.

Lattice thermal conductivity
The phonon DOS of TaFeSb, presented in figure 10, show a close resemblance to the NbFeSb results. There are no imaginary frequencies and so this structure is also mechanically stable. The data is again in a very good agreement with the results obtained by Zeeshan et al [12]. The low frequency region is up to 150 cm −1 and is dominated by Sb. The intermediate region between 150 and 220 cm −1 is due to Ta, instead of Nb. The last region is dominated by Fe atomic vibrations and occupies the high frequencies up to 350 cm −1 . It is also noticeable that a gap is formed in between the regions dominated by Ta and Fe. Our calculations show that the intrinsic value of latt k is 20.57 and 5.75 W m −1 K −1 at 300 and 1000 K, respectively. This is slightly lower than the NbFeSb results and can be accounted for by the gap between Ta and Fe in figure 10.
The effect of GB on κ latt of TaFeSb at 300 K is shown in figure 11. GB of size 4.5 μm have an almost negligible effect on the lattice thermal conductivity. When their size is reduced to 0.5 μm κ latt is computed to be 17.63 W m −1 K −1 . Although a similar behaviour was noticed in NbFeSb, the presence of an additional gap in the phonon DOS of TaFeSb leads to a different phonon mean free path λ mfp distribution. A common dip in the phonon thermal conductivity is observed for both TaFeSb and NbFeSb between 0.3 and 0.4 μm. This can be explained by the common gap in the phonon DOS at ω≈275 cm −1 . However, whilst the Ta-Fe gap in TaFeSb leads to an extra dip at 0.08 μm, this has a small effect as phonons with λ mfp less than 0.3 μm contribute less to the total lattice thermal conductivity. Despite this difference, GB of the same size seem to reduce κ latt in both  TaFeSb and NbFeSb by a similar amount. This means that the change in the phonon mean free path distribution has an effect only on the intrinsic value of κ latt but little impact on the effect of GB.
Next we proceed by adding the contribution of the PD due to Ti doping. The major difference between TaFeSb and NbFeSb is in the atomic mass of the X element. The mass of Ta is 180.95 amu, whereas Nb is significantly lighter with a mass of 92.906 amu. One of the crucial parameters in the Klemens model [22] for the thermal conductivity of systems with PD is the mass difference between the dopant atom (Ti) and the atoms which are substituted (Ta or Nb): a larger mass difference results in a greater reduction in the lattice thermal conductivity. Therefore, the lattice thermal conductivity of Ta 1−x Ti x FeSb is expected to be affected significantly by the Ti dopants. Figure 12(a) illustrates this point by comparing the Ta 1−x Ti x FeSb and Nb 1−x Ti x FeSb results. It is indeed seen that the reduction in κ latt of the Ta-based compound due to PD (Ti doping) is much more significant. For Nb 1−x Ti x FeSb the lattice thermal conductivity is reduced by 23% and 9% at 300 K and 1000 K, respectively, when the PD are included. For Ta 1−x Ti x FeSb these numbers increase to 37% and 18% at 300 K and 1000 K, respectively.
The last contribution which needs to be added is the EP interaction. As already described, it is meaningless to use the constant relaxation time approximation to compute the EP interaction. For that reason, experimental data was used earlier to obtain a value for the NbFeSb compound. Unfortunately, there are no experimental measurements which can be used to extract a value for the EP contribution in TaFeSb. For practical purposes and because of the similarity in the electronic structure and phonon DOS between TaFeSb and NbFeSb, we will  use the EP contribution which was extracted for NbFeSb. In the worst case, such an approximation would lead to an overestimate of κ latt and an underestimate of the ZT of TaFeSb rather than the opposite. Figure 12(b) shows the lattice thermal conductivity of Ta 1−x Ti x FeSb and Nb 1−x Ti x FeSb at different doping levels with all contributions included. The trend shows that κ latt of the Ta-based compound is lower at all doping levels. At x=0.05, κ latt is lower by 21% (κ latt =8.99 W m −1 K −1 ) and 15% (κ latt =4.04 W m −1 K −1 ) at 300 K and 1000 K, respectively. At x=0.10, the reduction is 23% (κ latt =8.43 W m −1 K −1 ) and 18% (κ latt =3.20 W m −1 K −1 ) at 300 K and 1000 K, respectively. The improvement of 15%-23%, as already discussed, comes from the slightly lower intrinsic value of κ latt for TaFeSb and the bigger mass difference between Ta and Ti. There is also a noticeable similarity of the lattice thermal conductivity of Ta 1−x Ti x FeSb at x=0.05 and that of Nb 1−x Ti x FeSb at x=0. 10. This hints that TaFeSb might require less doping than NbFeSb to reach its maximum ZT value.

Comparison between ZT of p-type TaFeSb and NbFeSb
Finally, we present the results on ZT of Ta 1−x Ti x FeSb and compare them to the Nb 1−x Ti x FeSb results. Figure 13 shows that the maximum thermoelectric figure of merit is obtained at T=1000 K, x=0.05 and is equal to ZT=1.53. For comparison the maximum ZT value for Nb 1−x Ti x FeSb is only 1.01, and at x=0.10. Figure 13(a) shows that Ta 1−x Ti x FeSb exhibits higher ZT across the entire temperature range and at all doping levels. The main difference to Nb 1−x Ti x FeSb is that there is a 50% increase in ZT and that the peak is achieved at x=0.05 rather than x=0.10, which is in agreement with the prediction made in the lattice thermal conductivity section.
The colour map in figure 13(b) reveals a broad area between 800 and 1000 K, and x=0.02 and x=0.15 in which the ZT of Ta 1−x Ti x FeSb is higher than 1.2. At moderate temperature (500-700 K) the ZT value drops to ≈1, which is still considered as an excellent TE result. Even at room temperature, the TE figure of merit (ZT=0.3) is almost two times bigger than that of NbFeSb (ZT=0.17). The wide range of conditions, which result in a good ZT value, suggests that p-type TaFeSb can indeed be used as a novel material for efficient thermoelectric devices.

Conclusions
We have conducted a thorough study of the thermoelectric properties of p-type NbFeSb and TaFeSb. In addition to solving the BTEs for electrons and phonon with ab initio inputs, several approximations were also included in Figure 13. Comparison between p-type TaFeSb and NbFeSb at x=0.04, 0.05 and 0.10 (a). Subfigure (b) is a colour map which shows the ZT of p-type TaFeSb with respect to the charge carrier concentration and temperature, with a maximum ZT of 1.53 at n h =1×10 21 cm −3 (x=0.05) and T=1000 K. the process. These are the constant relaxation time approximation with no dependence on the chemical potential due to doping, the choice of GB size and the inclusion of the EP interaction based on experimental data. This multi-step procedure needs to be executed with caution, and so at each step the results have been thoroughly compared to the available experimental measurements. We would like to point out that although the results in this study look promising and are consistent with the expectations, one should not use the presented theoretical framework lightly on fully unknown compounds. The key feature of this study was to preserve the chemical environment of NbFeSb and change it slightly to TaFeSb in a way that the empirical Klemens' equation is still applicable.
In summary, the NbFeSb results agree extremely well with multiple theoretical and experimental studies. The same procedure was then used to perform a full-scale computation on the TE properties of TaFeSb. The results have shown that both compounds exhibit high power factor at room temperature and have a good thermoelectric figure of merit at high temperatures. At 1000 K we find PF=9 mW m −1 K −2 and ZT=1 for NbFeSb and PF=16 mW m −1 K −2 and ZT=1.5 for TaFeSb. The higher atomic mass of Ta (compared to Nb) increases the scattering strength in Ti-doped TaFeSb, which reduces the lattice thermal conductivity of the compound. At the same time, p-type charge carries in TaFeSb exhibit higher mobility and relaxation time, which increases the power factor. The net result is a material with an amazing power factor of 16 mW m −1 K −2 and ZT value which is approximately 50% better than that of NbFeSb.
In conclusion, TaFeSb not only appears to be a better TE material than NbFeSb, but it also opens a new path of TE optimisation of materials based on the two alloys. In theory, an alloy based on Nb 1−x Ta x FeSb should exhibit good electrical properties due to the similarities in the electronic structure of NbFeSb and TaFeSb. At the same time, the mass difference between Nb and Ta should create additional scattering centres which would suppress the lattice thermal conductivity even before doping, and so the final doped compound should exhibit an even higher ZT value. This is further hinted by a very recent experimental study by Yu et al [8], which reports the successful synthesis of Nb 1−x Ta x FeSb alloys and a measured ZT of up to 1.6.