Introduction

At least as much as the energy we used on earth is lost in the form of waste heat1. Thermoelectric (TE) power generators that enable the direct conversion from heat to electricity have been studied for a long time, much earlier than the claimed energy crisis2,3,4,5,6,7,8,9,10,11,12. A good TE material should have high figure of merit ZT = (S2σ/κtot)T, where S, σ, κtot (=κl + κe), T, represent the Seebeck coefficient, electrical conductivity, total thermal conductivity and absolute temperature and κl and κe are the lattice and electronic components, respectively. The thermally-driven electrical performance of a TE material is measured by the power factor (PF = S2σ), while a high heat-to-electricity conversion efficiency is scaled by ZT. Conceptually, in order to possess a large ZT, the PF must be large and the total thermal conductivity κtot should be minimized. Good electrical conduction usually corresponds to high thermal conductivity and a counteracted relationship between the S and σ is often observed, resulting in the complex relationships between these physical parameters (S, T, σ, κl and κe). Given this dilemma, an optimization of all these properties so that the largest PF and ZT can be obtained simultaneously is far beyond fast-track experimental explorations. By the way, technically, a reliable measurement of the κtot and evaluation of its two components (κl, κe) seem to be tricky and thus the reported data are sometimes authors-dependent. These issues are thus appealing materials computation and property design as a pre-requisite for exploring TE materials for favorable applications. As a result, a theoretical prediction from first-principles calculations and other methods has been of interest for a long time13,14,15,16. For example, Yan et al.15 developed a scheme to calculate the carrier mobility, effective mass and lattice thermal conductivity related to TE performances and Sparks et al.16 proposed an approach of data mining to search for novel TE materials, both of which are of significance. Nevertheless, developing a full-scale computation scheme for the TE properties of a material to guide the experimental search is still appealed.

However, such a scheme remains yet to be found, in particular for carrier-doped TE materials. There exist two major challenges for an accurate calculation of these TE properties. First of all, given reliable knowledge on electronic structure as produced by the ab-initio calculations and/or experimental probing using techniques like angle-resolved photoelectron spectroscopy for a TE compound, indeed the semi-classic Boltzmann transport theory can be employed to predict the σ, S and κe. Even though, earlier calculations on these parameters (σ, S, κe) utilizing this semi-classic theory relies on a constant relaxation time (τ) or non-constant relaxation time obtained from the experimental data12,17,18 which are certainly questionable for many cases. This drawback damages the reliability of the predicted properties. In our study, we use the deformation potential (DP) theory combined with the effective mass approximation to calculate the relaxation time (τ) which is no longer treated as a constant. In this method, the effective mass is very important to the calculation of relaxation time (τ). The effective carrier masses are often calculated by a fitting of energy bands along the high symmetry lines, which is not accurate. In our work, we calculate the effective carrier masses at all k points in the first-Brillouin zone and then obtain the average of the effective masses at all specific energies. For the p-type and n-type semiconductors, we respectively take the average effective masses at the top of the valence band and the bottom of the conduction band. If the k-point-mesh is dense enough, the effective carrier mass calculation using our method is accurate.

Second, it is known that the lattice thermal conductivity κl is mainly determined by the three-phonon processes (corresponding to the intrinsic lattice thermal conductivity denoted by κl-in), impurity scattering and boundary scattering. The latter two processes make the calculation particularly difficult at the current stage. On one hand, a prediction of ZT is a formidable task because it requires accurate carrier relaxation time for evaluating σ and κe. On the other hand, a first-principles computation of the κl, especially for doped TE materials would demand computational resource that is too big to access in the material design routine. Along this line, an alternative computation scheme partially free of this difficulty would be favorable. In fact, Bardeen and Shockley proposed in 1950s the deformation potential (DP) theory to interpret the carrier transport in semiconductors where the carriers are mainly scattered by acoustic phonons. Consequently, the density functional perturbation theory (DFPT) combined with the quasi-harmonic approximation (QHA) can be used to calculate the κl of a stoichiometric compound. Within the framework of this modified density functional theory, the κl of a doped compound can be evaluated by the Klemens’ equation. To this end, a full-scale computation of the whole set of TE parameters for doped TE compounds is thus possible.

In this work, we will demonstrate this scheme by applying it to doped half-Heusler alloys which have received much attention for their good TE performance at the intermediate temperature, less than 900 K. These alloys offer high PF and high thermal stability. For instance, the XNiSn (X = Zr, Hf) compounds possess large Seebeck effect (S ~ 100–500 μV/K at T ~ 300 K) and moderately low electrical resistivity. An optimization by microstructure engineering and doping allows a reproducible ZT ~ 1.0 at T ~ 900 K19,20,21,22,23,24,25,26,27,28,29. For MgAgSb-based half-Heusler alloys, a proper optimization allows the ZT value as large as ~1.0 at room temperature and ~1.4 at T ~ 475 K30. In particular, the p-type NbFeSb-based half-Heusler alloys have their ZTs up to ~1.1 at T ~ 1100 K31. It is shown that these materials are always doped and experiments revealed the substantial change of thermal conductivity upon varying carrier density (n). Therefore, the calculated κl from the measured total κtotusing the Wiedemann-Franz relation is much less reliable, which on the contrary enhances the significance of the present work.

Our calculations start from the half-Heusler NbFeSb compound and consider the Ti substitution for Nb as dopant. Since this compound has the face-centered cubic structure with only three atoms in the primitive cell, a calculation of phonon dispersion behavior becomes possible. A description of the scheme and procedure of the full-scale computations is presented in the Methods section below.

Results

Electronic structures

The crystal structure of NbFeSb belongs to the #216 space group31, as illustrated in Fig. S1 of the supplementary materials. The calculated electronic dispersion relations along the high symmetry lines and thus evaluated density of states (DOS) are shown in Fig. 1(a,b). It is seen that the NbFeSb lattice has the indirect band gap of ~0.55 eV and band degeneracy at the valence band maximum (VBM). The major DOS contributions to the VBM come from the Fe atoms, while almost identical contributions of the Fe and Nb atoms to the DOS near the conduction band minimum (CBM) are identified. For both cases, the contributions from the Sb atom around the VBM and CBM are quite weak. These imply that the Seebeck effect is mainly induced by the Fe and Nb atoms rather than the Sb atoms.

Figure 1
figure 1

The calculated band structure (a) and DOS spectra for Fe atoms, Nb atoms and Sb atoms, as well as the total DOS (b) for NbFeSb compound. The projected DOS for Nb atoms (c), Fe atoms (d) and Sb atoms (e) in NbFeSb compound are plotted too.

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For further illustrating the bonding characteristics, one looks at the bonding peaks between Fe and Nb atoms. The projected DOS profiles for the s-, p-, d-orbitals of these atoms in the energy interval between −5.0 eV and 5.0 eV are presented in Fig. 1(c,d) and the electronic density distribution on the (011) plane is shown in Fig. S2 of the supplementary materials. The d-orbitals from the Nb and Fe atoms contribute mostly to the total DOS, while the DOS contribution from the Sb atoms mainly comes from the p-orbitals, which shows that there is the p-d hybridization between the Sb-Fe pairs and Sb-Nb pairs.

Electrical transport

For stoichiometric NbFeSb compound and its Ti-doped counterparts Nb1−xTixFeSb, the available physical parameters are shown in Table S1 of the supplementary materials. The calculated elastic constants cij, bulk modulus B and shear modulus GH are listed in Table S2 of the supplementary materials. Given these constants and the obtained electronic band structure, the electrical transport behaviors are characterized by parameters (carrier effective mass m, carrier mobility μ, carrier relaxation time τ) for electrons and holes. Figure 2(a) presents the band edge energy (Eedge) values at the CBM and VBM for electrons and holes as a function of the uniaxial strain δβ assigned along the a-axis, respectively, exhibiting good linear dependence. Here, the average electrostatic potential32 is set as a reference to obtain the absolute band edge shifts. The DP constant λβ values, as listed in Table 1, are similar for electrons (λβ = −15.94 eV) and holes (λβ = −14.51 eV). The calculated m at zero temperature and mobility μ and relaxation timeτ at T = 300 K for electrons and holes, are shown in Table 1. It is seen that τ = 245.04 fs for electrons is almost 12 times longer than that for the holes, while the mobility for electrons is 1018.46 cm2 V−1 s−1, ~66 times larger than that for the holes. We also investigated the energy (ε) - dependent effective carrier mass mε in the valence band and conduction band, as shown in Fig. 2(b) where me is the electron mass. In the valence band, the effective mass is positive, indicating the hole-like behavior of the energy band due to the concave curvatures, while the carriers in the conduction band show electron-like behaviors. The effective mass near the VBM is ~1.87 me, much heavier than ~0.35 me near the CBM. This difference results in the huge discrepancies of the carrier relaxation time and mobility for holes and electrons.

Table 1 The calculated DP constant λβ, carrier effective mass m at zero temperature and carrier mobility μ and relaxation time τ at T = 300 K for electrons and holes.
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Figure 2
figure 2

The edge energy shifts (Eedge) for the conduction band (CBM) and valence band (VBM) with respect to the lattice dilation along the a-axis (a) and the energy ε-dependent effective masses for holes (b) and electrons (c). The red lines are the fitting curves.

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We present extensive calculations on these TE properties (σ, S, κe), given either the p-type doping or the n-type doping with a broad range of carrier density n. The results are summarized in Fig. 3 where the parameters in the (T, n) plane are plotted on the right column for hole-doping and on the left column for the electron-doping. In the overall sense, these parameters all show non-trivial variations. First, the counter-acting relationship between S and σ (or κe) can be established for both doping cases. The large |S| appears in the region of low n and intermediate T, but both σ and κe are low in this region. Second, in the low n region, one observes only insignificant variation of σ over the T-range covered here, but κe exhibits remarkable increasing with increasing T. This remarkable increasing is much more than the linear T-dependence as predicted by the Wiedemann-Franz law κe = LσT where L = (1.49~2.45) × 10−8 W K−2 is the Lorenz constant. Third, these non-trivial dependences of (σ, S, κe) on (T, n) suggest that an optimized design in terms of the TE performance by carrier doping strategy is necessary.

Figure 3
figure 3

The calculated S (top row), logσ (middle row) and logκe (bottom row) on the (T, n) plane for carrier-doped NbFeSb.

Left panel: n-type and right panel: p-type.

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Lattice thermal transport

The more critical issue is to evaluate accurately parameter κl-in (intrinsic thermal conductivity) for NbFeSb compound and κl for the doped Nb1−xTixFeSb. First, the phonon band structures are shown in Fig. 4(a,c), respectively, for FeNbSb lattice without and with non-analytical term correction along the representative symmetry lines within the first Brillouin zone of the primitive cell with three atoms. The three atoms give rise to nine phonon branches, i.e., one longitudinal acoustic (LA) mode, two transverse acoustic (TA) modes, two longitudinal optical (LO) modes and four transverse optical (TO) modes. The two TA modes along the Γ-L and Γ-X directions are two-fold degenerate. The acoustic and optical branches overlap near the L point.

Figure 4
figure 4

The calculated phonon spectra (a) and phonon DOS (b) without the LO-TO splitting for NbFeSb compound. The corresponding spectra and DOS with the LO-TO splitting are plotted in (c,d) respectively.

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It is known that the electric dipoles caused by displacement of charges of long-wavelength LO modes can lead to internal electric field. The phonon frequencies for the LO modes at the Γ point will be up-shifted by this induced electric field. Thus, the LO-TO splitting is an important parameter to evaluate the strength of ionicity. A comparison of Fig. 4(a,c) shows the clear non-vanishing LO-TO splitting at the zone-center of the Brillouin zone, which implies the existence of the ionic bonding. It is seen from Fig. 4(a) that the frequencies of the two triply degenerate optical phonons at the Γ point are 6.07 THz and 7.80 THz, respectively. Surely, one of the two triply degenerate optical phonons is split into two-fold degenerate and single-fold degenerate optical phonons when the LO-TO splitting effect is taken into account. Subsequently, we can obtain the phonon density of states by taking the LO–TO splitting into account, as shown in Fig. 4(d) evaluated from Fig. 4(c). It is seen that the low-frequency branches up to 5.6 THz are mainly from the Sb atomic vibrations, while the frequency branches between 5.6 THz and 7.1 THz are mainly from the Nb atomic vibrations. The Fe atomic vibrations contribute to the high-frequency branches above 7.1 THz.

Based on the above consideration, one can calculate a set of parameters for evaluating the thermal conductivity, including the Grüneisen parameter γ, isometric heat capacity CV, Debye temperature ΘD and intrinsic lattice thermal conductivity κl-in as a function of T for NbFbSb lattice. The results are summarized in Fig. S3 of the supplementary materials. The values of γ, CV and ΘD are 1.69, 69.03 J K−1 mol−1 and 384.90 K at T = 300 K. For T > 300 K, the calculated κl-in(T) curve is plotted in Fig. S3(d), which decreases monotonously with increasing T, from ~22.0 W K−1 m−1 at T = 300 K to ~6.0 W K−1 m−1 at T = 1000 K. Given the κl-in(T) data, the κl for Nb1−xTixFeSb with different doping level x is calculated by Eq. (20) in the Methods section. For this calculation, relationship xper = x/3 = nV/3ΔZ should be satisfied, where V is the volume of unit cell and ΔZ is the valence difference between the master atom (Nb) and substituting atom (Ti). The values of these parameters are given in Table S1 of the supplementary materials.

To this stage, with κtot = κl + κe, PF = σ·S2 and ZT = σ·S2·T/κtot, one reaches a full-scale computational scheme for the whole set of TE parameters for a TE compound or doped TE alloy. In the following section, we apply this scheme to the doped compounds Nb1−xTixFeSb with different x for TE performance optimization.

Optimization of TE performance in Nb1−xTixFeSb

We have performed extensive calculations on the whole set of TE parameters for a series of p-type Ti doped Nb1−xTixFeSb compounds, considering the nominal substitution of Nb5 +  $ by Ti4 + . Several representative sets of data on these parameters as a function of n at T = 600, 800 and 1000 K are summarized in Fig. 5(a–f). It is noted here that the calculated results represent a spatial averaging over the three major axes [100], [010] and [001]. Given the cubic lattice structure, these parameters along the three major axes are nearly identical with one and another. It is seen that a linear dependence of logσ on n is identified in the intermediate and high n ranges. A linear dependence of logσ on n is identified in the intermediate and high n ranges. The σ(n) is sensitive to T but show weak dependence on n in the low n ranges. The S(n) first increases and then decreases with increasing n, exhibiting a single-peaked pattern. With increasing T, the peak height of S(n) decreases, nevertheless, the peak location increases. As expected, the S(n) and σ(n) exhibit the opposite dependences and both are insensitive to T in the high n range. The PF(n) first increases and then decreases with increasing n, similar to the S(n) curves. The peak height becomes low and the peak location shifts to the high n range with increasing T.

Figure 5
figure 5

The calculated electrical conductivity σ(n) (a), Seebeck coefficient S(n) (b), power factor PF(n) (c), electronic thermal conductivity κe(n) (d), lattice thermal conductivity κl(n) (e) and total thermal conductivity κtot(n) (f) at three temperatures T = 600 K, 800 K and 1000 K, respectively, for the p-type Nb1−xTixFeSb alloys. The evaluated figure of merit ZT on the (T, n) plane is plotted in (g).

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More interested are the calculated κe(n), κl(n) and κtot(n). The κe(n) remains low in the low-n range (<1021 cm−3), beyond which a rapid increase of κe(n) is identified. The κl(n), instead, shows the saturated plateaus in the low-n range and then falls rapidly in the high-n range due to electron-phonon scattering. Due to the different n-dependences and similar magnitudes of κe(n) and κl(n), the κtot(n) exhibits strong n- and T-dependences: less sensitive to n in the low-n range but highly sensitive to n in the high-n range. Conclusively, we summarize the ZT(n, T) contour in Fig. 5(g). While the ZT is quite low over most of the region unfortunately, a large ZT ~ 0.86 is obtained at the optimal condition (n ~ 1.45 × 1021 cm−3 and T ~ 1000 K). The ZT values are ~0.39 and ~0.62 at T = 600 K and 800 K, given n ~ 1.45 × 1021 cm−3.

Comparison with experiments

Finally, we compare our calculated data quantitatively with measured data on Nb1−xTixFeSb31, noting that no data for x = 0.0 itself are available. The measured S, σ and κtot data for polycrystalline samples at x = 0.04, 0.06 0.08, are taken from ref. 31 with the n of ~6 × 1020 cm−3, ~9 × 1020 cm−3 and ~12 × 1020 cm−3, respectively, giving the ratio x/n of ~0.67 × 10−22 cm3. By inputting these parameters we obtain σ(T), S(T), κl(T), κtot(T) and ZT(T) data as plotted in Fig. 6(a–l), where the dots are the measured data and the solid lines are from calculations. Experimentally, the κl(T) data are extracted from the κtot(T) data by excluding the κe(T) estimated by the Wiedemann-Franz relation. Even so, the calculated κl(T) is still consistent well with the extracted ones.

Figure 6
figure 6

The evaluated electrical conductivity σ(T), Seebeck coefficient S(T), otal thermal conductivity κtot(T) and ZT(T) for Nb1−xTixFeSb alloys at x = 0.04 (a–d), x = 0.06 (e–h), x = 0.08 (i–l), respectively. The solid lines are the calculated results and the dots are measured data extracted from ref. 31.

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For all the three doped cases, the calculated S(T) and κtot(T) match with measured data nicely over the whole T-range. It indicates that our computational scheme works well for predicting the S and κl and suggests that the imposed approximations with this scheme don’t induce remarkable uncertainties to the two parameters at least. However, the measured σ is lower than the calculated values in the low T-range. This discrepancy can be partially ascribed to the microstructural defects like grain boundaries, impurities and other defects etc, scattering the carriers and decreasing the electrical conductivity particularly in the low T-range where the carrier scattering by these defects and impurities is important. In the high T-range, the carrier scattering from optical phonons may be neglected and this may partially explain the slightly higher calculated σ than the measured one. The measured thermal conductivity is bigger than the calculated in the high T-range which is due to that the contribution from optical phonon to thermal conductivity is neglected in the calculation. Consequently, the difference between the calculated κ(T) and measured ones makes the calculated ZT values larger than measured ones, as shown in Fig. 6(d,h,l). It is noted that recent studies15,33 incorporated a minimum optical contribution (a constant) to the lattice thermal conductivity at high T and obtained good agreement with measured lattice thermal conductivity for materials with big number of atoms in the unit cell. In this study, the NbFeSb compound has only three atoms in the unit cell, a minimum optical contribution (a constant) to the lattice thermal conductivity is negligible. In addition, the NbFeSb compound is believed to be polar to some extent. Although the carrier relaxation time τ is less affected by the polarization field, the induced contribution may be one of the origins for the difference in σ(T).

Discussion

The present computational scheme has been demonstrated for the whole set of TE properties on half-Heusler NbFeSb compound. This scheme combines efficiently the multi-scale computation techniques, but the imposed several approximations may introduce substantial discrepencies in some cases such as synthesis-dependent polycrystalline samples. They are deserved for additional discussions.

First, the TE parameters depend substantially on the microstructures and associated defects. The carrier scattering from electron-phonon interaction, polarization electric field, impurities, defects and grain boundaries, etc., may be important. However, only the electron-phonon interaction is considered in the present scheme. Substantial uncertainties may arise and in particular the electrical conductivity does show deviations from measured results. Based on the constant relaxation time approximation, it is believed that the Seebeck effect is independent of carrier relaxation τ and determined by the Fermi level Ef (or n) and electronic structure, allowing much better consistency of the calculated values with measured ones.

Second, the calculated parameters are obtained by spatially averaging the data on single crystal along the three main axes. Both the lattice symmetry and anisotropy make this averaging inaccurate. Fortunately, the NbFeSb compound has cubic structure, reasonably insuring the applicability of the present computational scheme. Surely, this scheme can be easily extended to those systems of high lattice symmetry and low polar nature and the extension is straightforward, noting that the Debye approximation which ignores the optical phonons should be modified for calculating the anisotropic κl.

Third and surprisingly, it is interesting to note that the calculated κtot data are also in good agreement with measured ones for the polycrystalline samples synthesized by ball-milling method offering fine grain size. These fine-structured features, not considered in the present scheme, may add additional scattering processes to the phonon transport. The good agreement seems to suggest the dominant role of the long-wavelength phonons in the lattice thermal conductivity and therefore these multi-scale microstructural features may work for further reducing the lattice thermal conductivity.

Finally, for a general sense, we have to remind that a capricious utilization of the present computation scheme to predict a potential TE material should be anyhow cautious. We list several of these considerations: (1) electronic and phononic structures are in fact dependent of carrier doping/element substitution; (2) mixing of covalent bonding and ionic bonding makes the relation between substitution level (x, xper) and carrier density (n) complicated; (3) the applicability of the Slack’s equation and the Klemen’s equation should be concerned. Indeed, an optimal prediction/design of a TE material is a collection of many physical parameters which are inter-related and a reliable design remains to be challenging.

Methods

Electrical transport

For the electronic structure, we employed the density functional theory (DFT) with full-potential linearized augmented plane-wave (LAPW) method implemented in WIEN2k code34 which can offer enough dense k-mesh to make transport coefficients converge. The exchange and correlation interactions are described using generalized gradient approximation (GGA) and Perdew-Burk-Ernzerhof (PBE) functional35. For practical computation, the muffin-tin radii can be set as 2.0 a.u. for all the atoms and the plane-wave cut-off is defined by RMT·Kn = 8.0, while an energy threshold of −8.0Ry is usually used in order to separate core and valence states. It is noted that we don’t include the spin-related contribution for transport properties of NbFeSb compound. It is usually assumed that the magnetic order has a slight impact on the electro-transport above 300 K.

Given the electronic structure, the electrical transport parameters at finite T are obtained by solving the Boltzmann transport equation36. In our calculation, the calculated transport coefficients are well converged using a shifted 43 × 43 × 43 k mesh. The original k-mesh is interpolated onto a mesh 5 times as dense. The electrical conductivity tensors σαβ and electronic thermal conductivity tensors at non-zero electric current are given36:

where subscripts α and β stand for the two axis directions in the momentum space (or corresponding real space), parameters e, Ω, ε and f represent the electron charge, unit cell volume, carrier energy and Fermi distribution function, respectively and Ef is the Fermi energy. The σαβ can also be expressed as a function of ε:

where τ, N and να (νβ) are the carrier relaxation time, number of sampled k-points and carrier group velocity along the α(β) direction. Subscripts i and k are the band index and the wave-vector. In the standard procedure, Seebeck coefficient S and electronic thermal conductivity tensors καβ at zero electric current can be given by36:

where subscripts (l, m, n) stand also for the axis directions in the momentum space, while the main axes (a, b, c) in real space are parallel to those axes in the momentum space. The coefficient tensor ηαβ at arbitrary two directions (α, β) are written as:

For an approximately isotropic system like NbFeSb compound, the σ, S and κe can be directly evaluated from σ = (σ11 + σ22 + σ33)/3, S = (S11 + S22 0 S33)/3 and κe = (κ11 + κ22 + κ33)/3, which all are the functions of T and Ef (or n).

Based on the constant relaxation time approximation, the Seebeck effect is determined mainly by the band structure and roughly irrelevant with relation time τ. Nevertheless, both the σ and κtot are strongly dependent on τ. An evaluation of τ alone from the ab-initio data is inaccurate since it depends on phonon scattering and polarization electric field if the lattice is polar and also on impurity and defects. For simplified consideration, the influences from impurity and defects can be safely neglected in high T range and we only need to consider the phonon scattering. The effective mass approximation based on the DP theory is used to evaluate the τ. Accordingly, the carrier relaxation time τ and mobility μ defined specifically at the conduction (valence) band edge for a three-dimensional lattice can be expressed as37:

where kB is the Boltzmann constant, cii is the lattice elastic constant (i = 1, 2, 3) and λβ is the DP constant defined as:

where δβ is the uniaxial strain along the β direction and this strain is defined by the variation of lattice constant a with respect to equilibrium lattice constant a0 along the β direction. In Eqs (7) and (8), m is the isotropic effective carrier mass at the conduction (valence) band edge, which can be calculated by:

where is the energy-dependent effective mass tensor and tensor mαβ is given by

with the Planck constant. It is noted that the effective carrier mass is obtained by averaging the mαβ over the whole momentum space.

Finally, the carrier density n is obtained via:

where n0 is the valence electron number and D(ε) is the total density of states (DOS) as a function of ε, evaluated from the electronic structure. There is the one-to-one correspondence between n and Ef at a given T.

Elastic properties

An evaluation of the τ and μ needs the cij of a material and they are calculated using the strain method (energy approach) in the WIEN2k code. For this specific computation, the special points sampling integration over the Brillouin zone is realized using the Monkhorst-Pack method with 10000 special k-points meshes. For cubic lattice such as NbFeSb compound, there are three independent elastic constants, i.e., c11, c12 and c44, whose evaluation requires three independent homogeneous distortion modes. The first is the orthorhombic distortion mode satisfying the volume-conservation, the second is to change the lattice parameter in the a-axis, given the conserved cubic symmetry and the third is the monoclinic distortion mode with varying lattice parameter along the a-axis satisfying the volume-conservation rule too. According to the Voigt’s and Reuss’s approximations, the shear modulus can be expressed as38:

Hill et al. λ proposed the mean arithmetic value of GV and GR to reflect the real properties of a material:

The bulk modulus B is defined as:

Lattice thermal conductivity

Now one can calculate the κl. We first discuss the κl-in. The DFPT39 combined with the QHA is employed. The DFPT is a combination of the standard DFT with a linear electron density response, which is highly favorable for calculating the phonon frequencies over the Brillouin zone efficiently. In the specific operation, we specified the volume changes in 3%, 2%, 1%, 0%, −1%, −2%, −3% for the QHA. The Slack’s equation which applies at high T > ΘD yields40:

where A is a dimensionless collection of physical constants (A ~ 3.1 × 10−6), ΘD the Debye temperature, Vper the volume per atom, ntot the number of atoms in the primitive unit cell, the average mass of all the atoms in the crystal, γ is the Grüneisen parameter defined as41:

where i and q are the band index and the phonon wave vector, CV and CV(i, q) are the isometric heat capacity and mode heat capacity, respectively, γ(i, q) is the mode Grüneisen parameter42, V0 is the equilibrium volume, ω(i, q) is the phonon frequency of the i-th branch at wave vector q, D(q) is the dynamical matrix and e(i, q) is the eigenvector. The CV(i,q) and then CV are calculated from the phonon dispersions

and in general, the CV and γ are for all the phonon modes including the acoustic and optical modes.

On the other hand, the ΘD is evaluated by calculating the ΘD/T truncated expression of the isometric heat capacity CV43:

where symbol z stands for ћω/kBT.

To this stage, the lattice thermal conductivity κl for a carrier-doped TE compound can be obtained directly by the Klemens’ equation44,45,46:

where ωm and ω0 are the Debye frequency and cut-off frequency, respectively, M is the mass of the master atom to be substituted, ΔM is the difference in mass between the master atom (Nb here) and substituting atom (Ti here) and xper is the dopant number fraction per unit cell. The carrier density n can be written as n ~ 3xper ΔZ/V where V is the volume of unit cell and ΔZ is the difference in valence between the master atom and substituting atom. The negative and positive ΔZ values represent respectively the n-type and p-type carriers. For Nb1−xTixFeSb, one can have xper = x/3. It is noted that the ratio ω0/ωm can be negative upon a negative ΔM, but the κl remains positive.

To this end, the κl can be self-consistently calculated once the phonon spectrum is available. Specifically, the phonon spectrum calculation is performed using the VASP (Vienna ab initio simulation package) code47,48. A supercell of 2 × 2 × 2 primitive cell containing three atoms is considered, which consists of a total of 24 atoms. The first-principles calculations based on the DFPT are performed using the VASP code under the generalized gradient approximation (GGA) Perdew-Becke-Erzenhof (PBE) functional. A 6 × 6 × 6 mesh for the first Brillouin-zone sampling and 500 eV for cutoff of the plane-wave basis set are used. It is noted that the longitude optical (LO)-transverse optical (TO) splitting effect is taken into account in the phonon calculation. With the obtained phonon spectrum, the κl-in and κl are obtained accordingly using the Slack’s equation and the Klemens’ equation.

Additional Information

How to cite this article: Hong, A. J. et al. Full-scale computation for all the thermoelectric property parameters of half-Heusler compounds. Sci. Rep. 6, 22778; doi: 10.1038/srep22778 (2016).