Electronic structure and thermoelectric properties of half-Heusler alloys NiTZ

In the past few decades, researchers have been focused on the investigation of multi-functional materials, which can be used in various applications such as in spintronics, optoelectronics, thermoelectrics, and so on. With the surge in demand for green energy sources, TE materials are extensively taken into considerations for their ability to convert relatively small and waste heat into useful energy at the time of energy consumption. A wide range of materials have been explored for potential half-metals and thermoelectric (TE) devices such as organic thermoelectric materials,¹ chalcogenides,^2,3 skutterudites,^4–6 oxides,^7–12 hybrid perovskites,^13–15 triple-point metals,¹⁶ ternary compounds,¹⁷ and half-Heusler (hH) alloys.^18–29 Among them, Heusler compounds have gained much more attention since their discovery in 1903 due to their simple crystalline structure with fascinating properties such as magnetism, half-metallicity, superconductivity, optoelectronics, piezoelectric semiconductors, thermoelectricity, topological insulators, and semimetals.^30–38 

Thermoelectric materials are found to be applicable in day-to-day lives to fulfill the increasing demand of energy of the globalized society. The highly efficient TE devices (cooler, power generator, temperature sensors, and so on) can utilize a large amount of wasted thermal energy to generate electricity and vice versa.^39,40 For this, the device needs a larger figure of merit (ZT), which depends on the transport properties^41,42 defined by

where α (V K⁻¹) is the Seebeck coefficient, σ (S m⁻¹) is the electrical conductivity, κ = κ_e + κ_l (W m⁻¹ K⁻¹) is the thermal conductivity, and T(K) is the absolute temperature. α²σ is defined as the power factor (PF). The symbols κ_e and κ_l represent the electronic and lattice thermal conductivity, respectively. The materials having a high value of PF along with the low value of κ are suitable for the efficient TE devices.⁴³ 

Among others, most of the cubic hH alloys with an 18 valence electron count (VEC) exhibits high Seebeck coefficients and are reported as promising materials for TE applications due to high electrical conductivity and narrow bandgap semiconductors with novel electrical and mechanical properties even at high temperatures.^19–22,44 In addition to it, hH alloys contain non-toxic and readily available elements, making them environmental-friendly and more cost effective.

Recent experimental and theoretical investigations on hH alloys are mainly focused on improving their thermoelectric efficiency ZT by tuning the power factor and thermal conductivity. Bandgap engineering and fluctuation of carrier concentration around the Fermi level (E_F) in the Z position are the widely used methods to enhance the power factor, whereas the thermal conductivity can be decreased by alloying or by doping on the X or Y site to fluctuate the mass of the carriers introducing impurities and nanostructuring.^45–48 

From the literature, we noticed that Ni-based hH alloys with 18 VEC are less investigated. Following Slater–Pauling’s rule, the total magnetic moment for these types of hH alloys should be zero. Thus, the zero moment on Ti or Sc at the Y site and P or As or Sb at the Z site will give rise to zero moment for the Ni atom at the X site resulting in a non-magnetic system.^49,50 This motivates us to explore the electronic, TE, and other related properties to confirm if these groups of materials could be suitable for TE devices.

The cubic hH alloys NiTZ (T = Sc and Ti; Z = P, As, Sn, and Sb) belong to the Cl_b structure with space group $F4̄3m$⁠. It contains three in-equivalent atoms forming inter-penetrating fcc sublattices with the Wyckoff positions Ni (1/4, 1/4, 1/4), T (1/2, 1/2, 1/2), and Z (0, 0, 0), respectively, as shown in Fig. 1. The iso-structural NiTiSn is used here to validate our calculations based on the earlier reported results (both theoretical and experimental).

The density functional (DF) calculations have been performed using the full-potential linear augmented plane wave (FP-LAPW) method as implemented in the WIEN2k code.⁵¹ We double checked some parts of our calculations using the plane-wave based pseudopotential Quantum Espresso (QE) package.⁵² The standard generalized-gradient approximation (GGA) in the parameterization of Perdew, Burke, and Ernzerhof (PBE)⁵³ was used in the scalar-relativistic mode. The modified Becke–Johnson (mBJ) potential⁵⁴ was further included to check the accuracy of the bandgaps. The self-consistency convergence criteria for charge and energy were set to 10⁻⁴e and 10⁻⁵ Ry, respectively.

In the plane-wave pseudopotential approach, we used the norm-conserving pseudopotentials with plane wave cut-off energy for wave function set to 90 Ry. The full Brillouin Zone (BZ) was sampled with an optimized 10 × 10 × 10 mesh of Monkhorst–Pack k− points. To check the dynamical stability, phonon spectrum calculations have been performed with a 4 × 4 × 4 q− mesh in phonon BZ, which is based on the DF perturbation theory (DFPT) implemented in the QE package.⁵² 

The TE properties were calculated using the Boltzmann semi-classical transport equation and constant relaxation time approximation based on a smoothed Fourier interpolation of the bands implemented in the BoltzTraP code.⁵⁵ The full BZ was sampled with a 50 × 50 × 50 k− mesh for the calculation of the transport properties. The electrical conductivity and PF were calculated under constant relaxation time approximation (τ) using the BoltzTraP code based on Boltzmann theory. τ is approximated by fitting the experimental data from the work of Kim et al.¹⁸ The lattice thermal conductivity was obtained by solving the linearized Boltzmann transport equation (BTE) within the single-mode relaxation time approximation (SMA) using the thermal2 code implemented in the QE package.⁵² 

We started our calculations by optimizing the cubic hH alloys with $F4̄3m$ symmetry. Our calculated values of lattice parameters and the bandgap within GGA and GGA + mBJ are listed in Table I. These values are found to be in fair agreement with the earlier reports of Ma et al.⁴⁹ for the GGA case.

The calculated phonon dispersion curves along the high-symmetry points shown in Fig. 2 depict that the proposed hH alloys are thermally stable. This is evidenced by the absence of imaginary phonon frequencies throughout the whole BZ, as expected for dynamic stability.⁵⁶ We observed three acoustic (low-frequency region) and six optical phonon (high-frequency region) branches due to three atoms per unit cell. The majority of the lattice contribution to the thermal conductivity arises from the acoustic part as it has high group velocity compared to the optical part. We found that the acoustical phonon branches of NiScP and NiScAs extend nearly to 200 cm⁻¹, while NiScSb and NiTiSn lie within 150 cm⁻¹ in frequency. The observation of dynamical stability and preferable energy gap in our proposed hH alloys motivates us to explore the electronic and transport properties for their potential application as TE materials.

To understand the ground state electronic properties of the material, the total and partial density of states (DOS) are shown in Fig. 3. The proposed systems are found to be semiconducting with an energy gap lying within ∼0.32 eV to 0.62 eV, in fair agreement with the earlier report.⁴⁹ As seen in the PDOS, the main contribution to the total DOS at and around E_F is from the 3d-orbitals of Ni and Sc atoms, while the contributions from the atom in the Z site are negligible (see Fig. 3). This is an indication that doping onto the Z site may improve the carrier concentration.

It is interesting to note that with an increase in the atomic radius of atoms at the Z site, say, from P to Sb, the bandgap reduces gradually, which further leads to the decrease in the hybridization of Ni-3d and Sc-3d states. An indirect bandgap is observed in the band structures for hH alloys (see Fig. 4) with their valence band maximum (VBM) lying at Γ and conduction band minimum (CBM) at X in the BZ. The VBM for the hH alloys is threefold degenerate comprising heavy and light bands. From the observed band structure in Fig. 4, the scenario of heavy bands can enhance the Seebeck coefficient, whereas the light band can facilitate the mobility of charge carriers.^57–59 Thus, the combination of heavy and light bands is preferable for increasing the TE performance. The band structure shown in Figs. 4(a)–4(c) dictates the effective mass to be more at X −Γ in CBM than that of VBM at Γ (i.e., the effective mass of electron at CBM is greater than that of the hole at VBM), which plays a significant role in TE properties. As seen in NiTiSn [Fig. 4(d)], the VBM (at Γ) is flatter than the CBM (at X) indicating that the effective mass of holes at VBM is more than that of electrons on CBM.

For an efficient TE material, a high value of α and σ with a low κ is expected, as depicted in Eq. (1). The dimensionless figure of merit ZT can be optimized when these parameters are optimum. However, these parameters are inter-related with themselves. Thus, obtaining a high value of ZT is in-sufficient just by tuning one or two parameters. To get insight into the TE properties of hH alloys, we calculate the Seebeck coefficient α, electrical conductivity σ/τ, thermal conductivity (κ = κ_e + κ_l), power factor (PF), and ZT by using constant relaxation time approximation and rigid band approximation.

We first initiate our calculations for NiTiSn by validating the theoretical results, such as PF and thermal conductivity with the reported experimental measurements.¹⁸ From the comparison of the calculated and experimental electrical conductivity, we approximated the relaxation time τ = ∼2 × 10⁻¹⁵ s. In the whole process, we use the constant relaxation time, even though it depends on the doping level and temperature, obtained for NiTiSn to implement for all the iso-electronic systems.

The PF of NiTiSn was reported to be ∼16 μW cm⁻¹ K⁻² at 700 K, which upon electron doping (by 1% of the Sb atom to the Sn site) rises to ∼30 μW cm⁻¹ K⁻². When the temperature rises above 700 K, PF is found to decrease in both cases. Comparing these values, we estimate that PF may range between 10 μW cm⁻¹ K² and 15 μW cm⁻¹ K² at the 0.04–0.06 doping level of electron per unit cell in the same temperature range. In the case of hole doping, PF lies within 17 μW cm⁻¹ K⁻²–23 μW cm⁻¹ K⁻² at the same temperature range when the dopant is 0.1–0.2 hole per unit cell. This indicates that hole doping is more appropriate than the electrons for PF. The calculated total thermal conductivity 21 Wm⁻¹ K⁻¹–10 Wm⁻¹ K⁻¹ [see Fig. 5(b)] was slightly higher than the earlier report (i.e., 7 Wm⁻¹ K⁻¹–10 Wm⁻¹ K⁻¹), which is mainly due to the electronic contribution found prominent at higher temperatures. Our calculated results are comparable with the experimental measurements.¹⁸ 

The Seebeck coefficient [(a), (c), and (e)] and the PF [(b), (d), and (f)] for different levels of doping are shown in Fig. 6 for NiScP, NiScAs, and NiScSb, respectively. Around E_F (i.e., at μ = 0), the Seebeck coefficient is large (>±150µV/K), which on doping to either side falls off significantly. This is evident from its inverse relation with the carrier concentration.

The optimum values of the doping levels and the corresponding TE parameters for 1200 K are listed in Table II.

PF is another parameter to check the reliability of TE materials. As observed in Fig. 6, the PF value for the p or n− type is significant within the doping range of ±0.3. To be specific, at 1000 K, the calculated values are approximately 15 μW cm⁻¹ K⁻², 12 μW cm⁻¹ K⁻², and 13 μW cm⁻¹ K⁻² for NiScP, NiScAs, and NiScSb within 0.02–0.04 hole per unit cell reaching its maximum value at 1200 K. Similarly, for doping range 0.06–0.07 electron per unit cell, PF rises to ∼27 μW cm⁻¹ K⁻², 25 μW cm⁻¹ K⁻²,, and 20 μW cm⁻¹ K⁻² at 1000 K, respectively. The sizable value of PF within the doping range 0.07–0.08 electron per unit cell suggests that these materials could be good TE materials.

We further show the variation of PF with the chemical potential, μ, in Fig. 7. The peak values of PF noted in the chemical potential range between 0.4 eV and 0.7 eV for NiScP, NiScAs, and NiScSb. In contrast, the peak value of PF is around −0.2 eV for NiTiSn. From the above scenario, electron doping is found to be more suitable for NiScP, NiScAs, and NiScSb due to the larger effective mass of electrons to get a better TE performance. This indicates the presence of larger electron pockets resulting in the dense carriers, which are confined to the CBM along Γ − X [Figs. 4(a)–4(c)]. On the other hand, in NiTiSn, hole doping is much more favorable due to the higher effective mass of holes resulting from the nearly flatter band in the VBM and CBM along Γ − X [see the band structure in Fig. 4(d)].

Figure 8 shows the calculated thermal conductivity as a function of temperature for NiScP, NiScAs, NiScSb, and NiTiSn, respectively. The total thermal conductivity consists of two components, viz., electronic (κ_e) and lattice (κ_l) parts. At low temperature (say, 300 K), the lattice part was found to be dominant over the electronic part, and with the rise in temperature (say, up to ∼900 K, except NiScP), the lattice thermal conductivity and the overall conductivity decrease uniformly.

Note that with an increase in temperature starting from 300 K, the carrier concentration increases resulting in higher electrical conductivity and, hence, the overall thermal conductivity. Similar features were observed in the recent report of CoMnSb.⁶⁰ The calculated lattice conductivity is 10.6 Wm⁻¹ K⁻¹, 19 Wm⁻¹ K⁻¹, and 18.5 Wm⁻¹ K⁻¹ at 300 K, which reduces abruptly to 2.5 Wm⁻¹ K⁻¹, 4.7 Wm⁻¹ K⁻¹, and 4.5 Wm⁻¹ K⁻¹ at 1200 K for NiScP, NiScAs, and NiScSb, respectively.

The figure of merit ZT for hH alloys is as shown in Fig. 9. With a low value (say, 0.05) of ZT at 300 K, it is found to rise linearly with the increase in temperature. At 1200 K, the calculated values are 0.30, 0.45, 0.35, and 0.29, respectively, for NiTiSn, NiScP, NiScAs, and NiScSb alloys. On the other hand, the values of total thermal conductivity are found to be minimum at ∼700 K for NiScP and ∼900 K for the remaining systems, which start increasing afterward due to the dominance of electronic part. The variation of ZT with temperature shows linear nature. PF is dominant at a higher temperature range due to the increase in carrier concentration along with the electrical conductivity. The observed ZT is low mainly due to a higher value of κ. Even if our ZT values are lower than the commercialized TE materials such as Bi₂Te₃ and PbTe, they can be enhanced by means of doping to any of the three atomic sites. Though the ZT value is low (∼0.05) for the pristine systems compared to the widely used doped Bi-based alloys, say, Bi₂Se₃ (∼0.01 to 0.05),⁶¹ Bi₂Se₃ at Bi₂Te₃ (∼0.7),⁶² and Bi–Sb alloys (∼0.4),⁶³ it can be enhanced by electron doping.⁶⁴ 

As observed from the calculations above, the ZT value can increase when PF is enhanced while minimizing the thermal conductivity. The possible route to tune this from DF is by proper tuning of the bandgap with appropriate electron/hole doping as discussed.

On the basis of density functional calculations, we investigate the half-Heuslers NiTiSn, NiScP, NiScAs, and NiScSb. Electronic properties reveal that these materials are semiconductors with an indirect bandgap. The narrow bandgap marks them as suitable candidates for TE performance. The calculated power factor shows a large value in both the electron and hole doping cases. Electron doping is found to be more preferable than hole doping for NiScP, NiScAs, and NiScSb, while hole doping is preferable for NiTiSn. Based on the constant relaxation time approximation and rigid band approximation with sizable ZT, these compounds are predicted as possible TE materials.

M.P.G. acknowledges the Department of Science and Technology, India, for awarding the India Science and Research Fellowship (ISRF-2019) with Grant No. DO/CCSTDS/201/2019 and the Alexander von Humboldt Foundation, Germany, for the equipment subsidy grants. Part of this work was performed at IIT Roorkee, India, during the ISRF-2019 program, and part of it was performed with the computational resources provided by the Kathmandu University Supercomputer Center established with the equipment donated by CERN. M.P.G. thanks H. C. Kandpal for fruitful discussions suggesting the half-Heusler group to explore and also for all the logistic and technical supports at IIT Roorkee.

There is no conflict of interest.

The data that support the findings of this study are available with the corresponding author and can be obtained upon reasonable request.