Study of the enhanced electronic and thermoelectric (TE) properties of Zr_xHf_1−x−yTa_yNiSn: a first principles study

Abstract

A density functional theory (DFT) approach employing generalized gradient approximation (GGA) and the modified Becke Johnson (TB-mBJ) potential has been used to study the electronic and thermoelectric (TE) properties of Zr_xHf_1−x−yTa_yNiSn. The presence of an indirect band gap at E_F in the parent compound predicts this material to be a small band gap insulator. The substitution of Ta atoms at the Hf site increases the density of states (DOS) at E_F which facilitates charge carrier mobility. The influence of Ta content increases the Seebeck coefficient and electrical conductivity, and suppresses the thermal conductivity; as a result the figure of merit ZT is enhanced. We report an increment in ZT value of 36% over the undoped system. The theoretical data were compared with the experimental results.

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1 Introduction

The Heusler compounds have been at the centre of scientific research since the time NiMnSb was first predicted to be a half metallic ferromagnet.¹ Due to their half metallic properties the use of Heusler compounds is challenging in the field of spintronics.^2–4 Other than the half metallicity of Heusler compounds, they also have many interesting qualities such as high Curie temperatures (730 K for NiMnSb and 985 K for Co₂MnSi), the most stable zinc-blende structure,⁵ thermoelectric properties,⁶ etc. Among all other Heusler compounds, MNiSn (M = Ti, Hf, Zr) a type of half Heusler (HH) compound is of particular interest because of its narrow semi-conducting band gap.⁷ Unlike other transition metal based HH compounds, MNiSn (M = Hf, Zr) has no significant signature of high spin polarization (spin transport) at E_F. The chemical formula of HH alloys is XYZ, where X, Y, and Z can be selected from many different elemental groups (for example X = Ti, Zr, Hf, V, Mn, Nb; Y = Fe, Co, Ni, Pt; Z = Sn, Sb). The crystal structures of the ternary inter-metallic compounds formed are usually of the cubic MgAgAs type (space group F43̄m).⁸ Some of these HH compounds with a valence electron count of 18 per formula unit have a narrow semiconducting band gap while metallic and sometimes magnetically ordered systems are found at both higher and lower electron counts.⁹ This type of material is of great interest because of the large mobility of charge carriers as compared to other semi-conductors. This unique characteristic leads to a high Seebeck coefficient and moderate electric resistivity.^10–12 The higher value of the Seebeck coefficient and low resistivity are evidence for these compounds being promising thermoelectric materials (TE-materials).^13–20,36 In recent years, TE-materials have attracted much attention because they possess all the qualities to be an alternative source of energy, as they play an important role in energy conversion between heat and electricity. So far, several materials have been investigated to achieve the desired level of thermoelectric properties such as Heusler compounds, derivatives of HH compounds, skutterudites, zintl compounds, Ca₃Co₄O₉ and BiCuSeO.^21–28 However, the energy conversion efficiencies of TE-materials are limited for commercial purposes. Highly efficient TE-materials are therefore in urgent demand. Thermoelectric efficiency is measured by a dimensionless figure of merit (ZT) which is connected to the Seebeck coefficient (S), the electrical conductivity (σ), the thermal conductivity (κ) and the absolute temperature T. A ZT value around unity or more is considered to be good for TE-materials.²⁹ Typical TE-materials based on tellurium, lead, antimony and selenium with ZT values 0.85–1.20, are considered to be more efficient, however they are not safe to handle due to their toxicity.^30–32 Therefore, searching for new low cost and environmentally friendly materials with high values of ZT poses a great challenge. Mostly, studies of the thermoelectric properties of HH compounds were focused on the bulk materials.^33–35 An unannealed ZrNiSn shows a ZT value around 0.64 at 800 K for an undoped system.¹⁶ The development of nanoparticles and thin films is advantageous for enhancing thermoelectric performance, but they are very expensive computationally and experimentally. Thermoelectric efficiency was successfully enhanced in a layered structure BiCuOCh (Ch = S, Se and Te), a nano-structure, a doped structure etc.^36–38 The room temperature power factors for nanosized TiNiSn³⁹ and thin film HfNiSn⁴⁰ are 2.5 mW mK⁻² and 1.3 µW K⁻² cm respectively. Nowadays, band structure engineering is considered to be an effective method to improve the ZT of bulk TE-materials. Thermoelectric properties of bulk semiconductors are in close relation with the electronic structure, which is sensitive to the stoichiometric composition. Yang et al. in their study have shown the relationship between the electronic structure and thermoelectric properties.⁴¹ Similarly, the theoretical study conducted by Ye and his group using first principles calculations has also come across a similar kind of relationship.⁴² The doping of heavy elements in HH alloys plays a key role in reducing the thermal conductivity (κ) and band gap (E_g), giving dense energy bands near E_F, facilitating electron transport and even inducing half-metallicity in some cases.^43–47 The advancement of doping a thallium impurity in PbTe resulted in an increased ZT value above 1.5 at 773 K.²⁹ Other than the conventional TE-materials a high value of ZT, 1.5 at 700 K, has also been obtained in Ti doped HH Ti_x(Zr_0.5Hf_0.5)_1−xNiSn.³³ The Bi-doped superlattice of ZrNiSn/ZrNi₂Sn shows an enhanced power factor of 3.3 mW mK⁻² as compared to 1.6 mW mK⁻² of Bi-doped bulk ZrNiSn.⁴⁸ This paper attempts to study the preparation of the TE-material Zr_xHf_1−x−yTa_yNiSn with an enhanced electrical conductivity and Seebeck coefficient by modifying the band energies near the Fermi level with suitable doping. The optimized figure of merit, ZT, of Zr_xHf_1−x−yTa_yNiSn is enhanced with Ta doping.

2 Computational details

The electronic structures are calculated by adopting the full potential linearized augmented plane wave (FPLAPW) method for KS-DFT, as implemented in the WIEN2K package.^49,52 Two methods, GGA⁵⁰ and mBJ^51,52 are used to describe the electron exchange and correlation. Nonspherical contributions to the charge density and potential within the muffin tin (MT) spheres are considered up to l_max = 10 (the highest value of angular momentum functions). The cut-off parameter is R_MT × K_max = 7 where K_max is the maximum value of the reciprocal lattice vector in the plane wave expansion and R_MT is the smallest atomic sphere radius of all atomic spheres. In the interstitial region the charge density and potential are expanded as a Fourier series with wave vectors up to G_max = 12 a.u.⁻¹. 286 special k points in the irreducible Brillouin zone are used for the selfconsistent DFT calculation. The convergence criteria for the selfconsistency are set to be 0.0001 Ry in the total energy. However the semicore states are treated semi-relativistically, i.e. by ignoring the spin–orbit (SO) coupling. The experimental lattice constants, 6.113 Å for ZrNiSn and 6.083 Å for HfNISn⁸ were used for the ground-state structure optimization. The crystal structure of the HH alloy MNiSn having space group F43̄m and atomic positions M (1/4, 1/4, 1/4), Ni (1/2, 1/2, 1/2) and Sn (0, 0, 0) were considered. The calculated lattice constants are 6.45 Å for HfNiSn and 6.24 Å for ZrNiSn, larger than the available experimental data usual for GGA. The crystal structures of Zr_xHf_1−x−yTa_yNiSn were constructed by a supercell method with a 2 × 2 × 2 fcc cell along the [1, 1, 1] direction and eight Ni/Hf atoms were generated. The composition x = 0.375 was generated by replacing three of the eight Hf atoms with three Zr atoms, as shown in Fig. 1. The crystal structures for the other compositions were constructed by replacing the remaining Hf atoms with Ta atoms as y = 0.125 (1/8), 0.250 (2/8), 0.375 (3/8), 0.500 (4/8) and 0.625 (5/8). The relaxed structure for each of the compositions was obtained from the total energy plot as a function of unit cell volume (volume optimization).⁵⁶ The crystal structures of each composition of Zr_xHf_1−x−yTa_yNiSn along with the volume optimization are presented in Fig. 1. Theoretically determined lattice constants for each of these Zr_xHf_1−x−yTa_yNiSn compounds are used for the calculation of electronic properties. For investigating the thermoelectric transport properties, we make use of the BoltzTraP⁵⁷ based on Boltzmann semiclassical theory with a 24 × 24 × 24 k-mesh inside the Brillouin zone. The Fermi energy at zero temperature (T = 0 K) is taken as the chemical potential in the transport calculation.

Fig. 1 Energy versus volume and crystal structure of Zr_xHf_1−x−yTa_yNiSn: (a) x = 0.00, y = 0.00, (b) x = 0.375, y = 0.125, (c) x = 0.375, y = 0.250, (d) x = 0.375, y = 375, (e) x = 0.375, y = 0.500 and (f) x = 0.375, y = 0.625 (Hf-blue, Ni-red, Sn-green, Zr-black and Ta-yellow).

3 Results and discussion

In order to examine the alloying stability of Zr_xHf_1−x−yTa_yNiSn structures (at 0 K), we have calculated the energy of formation (ΔE_f). The formation energy gives an idea about the existence of a stable crystal. Furthermore, negative values of ΔE_f indicate stronger bonding between the atoms and more alloying stability of the crystal.⁵³ The energy of formation (ΔE_f) of a compound Hf_xNi_ySn_z (say) is calculated by subtracting the sum of the energies (xE_Hf + yE_Ni + zE_Sn) of pure constituent elements in their stable crystal structures from the total energy (E_f) of the compound. Therefore, the ΔE_f of the compound Hf_xNi_ySn_z is calculated using the following expression:^54,55

Here, E_f is the total energy of the compound, E_Hf, E_Ni and E_Sn denote the total energy per atom of the pure elements Hf, Ni and Sn where x, y, z are the numbers of Hf, Ni and Sn atoms in the primitive cell, respectively. The calculated total energy, formation energy and the individual energies of the constituent atoms are presented in Table 1.

Table 1 The total number of atoms (n) in a compound, total energy (E_f in Ry) of the compound, total energies of the individual atoms E_M (M = Hf, Ni, Sn, Zr, Ta) in Ry, formation energy (ΔE_f in Ry) and formation energy per unit cell (ΔH_f in Ry/a.u.³)

Compound	n	Ef (Ry)	EHf (Ry)	ENi (Ry)	ESn (Ry)	EZr (Ry)	ETa (Ry)	ΔEf (Ry)	ΔHf (Ry/a.u.^3)
Hf1Ni1Sn1	3	−45 595.516	−7949.873	−801.581	−3281.838	0.000	0.000	−11 187.408	−24.534
Zr3Hf4Ta1Ni8Sn8	24	−296 830.385	−7949.873	−801.581	−3281.838	−1910.277	−8228.033	−9100.195	−2.903
Zr3Hf3Ta2Ni8Sn8	24	−297 885.921	−7949.873	−801.581	−3281.838	−1910.277	−8228.033	−9132.585	−2.947
Zr3Hf2Ta3Ni8Sn8	24	−298 945.624	−7949.873	−801.581	−3281.838	−1910.277	−8228.033	−9165.149	−2.748
Zr3Hf1Ta4Ni8Sn8	24	−299 998.754	−7949.873	−801.581	−3281.838	−1910.277	−8228.033	−9197.441	−3.044
Zr3Hf0Ta5Ni8Sn8	24	−301 388.321	0.000	−801.581	−3281.838	−1910.277	−8228.033	−9243.749	−2.940
Zr1Ni1Sn1	3	−22 598.345	0.000	−801.581	−3281.838	0.000	0.000	−5534.883	−12.30

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3.1 Electronic properties

For calculating the electronic structure, the optimal lattice constants were used. The previous reports revealed that Hf/ZrNiSn are semiconductors with an indirect band gap of ∼0.50 eV.^6,7,13,58 Our results are in good agreement with the indirect band gap between the Γ–X points in the Brillouin zone as shown in Fig. 3(a and b). The band gap is mainly formed by the hybridization of M (d-e_g, d-t_2g) and Sn-p orbitals with some Ni (d-t_2g) orbitals as shown in Fig. 2(a and b). As shown in Fig. 3(a and b) the top of the valence band (M-d-t_2g) degenerates into three sub-bands along the Γ point; the two with heavier masses are at a lower energy (below −2.0 eV). Meanwhile the lowest conduction band is formed by Ni (d-e_g) and is non-degenerated. In most cases LDA/GGA band gaps are underestimated.^48,59 Whereas in the case of HH MNiSn the theoretical band gaps are an overestimate of the experimental values obtained from resistivity measurements (E_g = 0.18 eV for Zr and 0.18 eV for Hf).^6,7,13,58 Our calculation of electronic structure with the modified Becke Johnson (mBJ) potential^51,52 is almost ineffective as there is no improvement in the band gap [see Fig. 3(a and b)]. Do et al.⁴⁸ have reported that even an implementation of a highly sophisticated hybrid functional proposed by Heyd–Scuseria–Ernzerhof (HSE06)⁶⁰ shows a negligible effect on its band gap. Thus we employed a computationally cheap exchange correlation (GGA) to treat these kinds of system. On the other hand the doping of Ta atoms has shifted the band gap towards a lower energy and as a result the Ni-d states disperse around the E_F and metallic character predominates (Fig. 2(c)), similar behaviour was reported in (Zr, Hf)Ni_1+xSn with increasing Ni doping.²⁰ With Ta doping the systems are metallic and hence non-local effects are not considered. Hence, all the systems with increasing doping concentrations are treated within GGA.

Fig. 2 Partial DOS of (a) HfNiSn/ZrNiSn (black line represent HfNiSn and red line ZrNiSn) and (b) total DOS of Zr_xHf_1−x−yTa_yNiSn.

Fig. 3 Band structures of (a) HfNiSn, (b) ZrNiSn and (c) Zr_xHf_1−x−yTa_yNiSn (x = 0.375, y = 0.625).

3.2 Thermoelectric properties

In this section the temperature dependent Seebeck coefficient S, thermal conductivity κ and electrical conductivity divided by the scattering time σ/τ are calculated from the semiclassical transport equation as implemented in a computational code called BoltzTrap.⁵⁷ Eqn (2) interprets electrical conductivity tensors :⁶¹where α and β are the tensor indices, v_α and v_β are the group velocities, e is the electron charge and τ_k is the relaxation time. The electron contribution remains near the chemical potential (µ) in a narrow range of µ − k_BT < ε < µ + k_BT, where k_B is the Boltzmann constant.⁶² The transport distribution is written as⁶³which is the kernel of all transport coefficients. From the rigid band approach, the electrical conductivity, thermal conductivity and Seebeck coefficient can be written as a function of temperature (T) and chemical potential (µ) by integrating the transport distribution.⁶⁴

Here F₀ is a Fermi-Dirac distribution function. The thermoelectric efficiencies of TE-materials are in close relation with their electronic band structure and thermal conductivity. The code BoltzTraP includes only the electronic thermal conductivity (κ) whereas the phonon contribution is neglected. The Seebeck coefficient is a sensitive test of the electronic structure at the vicinity of the E_F. Thus it can be increased by increasing the DOS (in relation with effective mass) near the edge of E_F by doping with heavy elements (see Fig. 3(c)). The charge transport is due to the two narrow bands along the Γ symmetry near E_F between M-d_t2g and Ni-d_eg. The lower bands at −1.3 eV (M-d_t2g) are more flat as compared to the upper bands at −0.6 eV (Ni-d_eg), see Fig. 3(c) (inset). The dense bands near E_F facilitated the transport of charge carriers.⁶¹ The chemical potential is equivalent to the Fermi energy at T = 0 K.

The thermoelectric efficiency denoted as ZT (figure of merit) has been calculated from eqn (7).

The high value of ZT is related with a high value of S and a low value of κ. A ZT ∼ 1 is considered to be a benchmark value for the practical application of TE-materials.²⁹ Subsequently, the thermoelectric properties of the systems were investigated in the range 50–800 K. The calculated S, σ/τ, κ and ZT are presented in Fig. 4. In our case we have achieved a dense band near E_F as compared to the undoped system by doping Ta atoms as shown in Fig. 3(a–c). In Fig. 4(a), we see the increasing magnitudes of S with increasing doping concentrations at all temperatures, indicating greater carrier concentrations. Our plot of S shows similar behaviour to that of the experimental plot (see Fig. 4(a)).⁶⁵ The absolute value of S increases above 200 µV K⁻¹ (x = 0.375, y = 0.250, 0.375 and 0.50) at 400 K and remains almost constant up to 650 K (see Table 2). The S values for the sample with the composition Zr_0.25Hf_0.25Ti_0.5NiSn⁶⁶ agree well with these results. In Fig. 4(b), the electrical conductivity (σ/τ) is seen to increase over the whole temperature range, showing semiconducting like behaviour. The values are comparable to the resistivity curve as it is just the inverse of (σ/τ) which means the resistivity decreases with the increase in temperature as reported for Zr_0.30Hf_0.70Ta_0.05NiSn (see Fig. 3(a)).⁶⁵ In our calculation the σ/τ is highest for the undoped system (x = 0.0, y = 0.0); as the doping concentration increases, the value decreases. The evolution of the thermal conductivity κ is plotted in Fig. 4(c). It shows that κ is linearly dependent on temperature (T), which shows similar behaviour to that of the experimental plot (see Fig. 4(c)).⁶⁵ The calculated κ value is highest for the undoped system which agrees well with the experimental results.⁶⁵ The dimensionless figure of merit, ZT, is calculated from the measured physical properties by using eqn (7) and is presented in Fig. 4(d). ZT reaches maximum values between 0.70–0.75 in the temperature range from 300 K to 800 K which are in qualitative agreement with the experimental results.^19,45,65,67 At higher temperatures the value slightly decreases, due to a decrease in S and a sharp increase in κ. The calculated ZT values and Seebeck coefficient (S) at 750 K for all compositions together with the experimental results are shown in Table 2.

Fig. 4 (a) Seebeck coefficient S (V K⁻¹), (b) electrical conductivity σ/τ (Ω ms)⁻¹, (c) thermal conductivity k (W m⁻¹ K⁻¹ s⁻¹) and (d) ZT of Zr_xHf_1−x−yTa_yNiSn.

Table 2 Comparison of figure of merit ZT of Zr_xHf_1−x−yTa_yNiSn with the experimental data

Present calculation (750 K)				Previous experimental data
x	y	S (µV K^−1)	ZT	x	y	ZT	References
0.000	0.000	165	0.55	0.00	0.000	0.42 (800 K)	36
0.375	0.125	185	0.68	0.30	0.010	0.70 (870 K)	65
0.375	0.250	230	0.70	0.25	0.060	0.30 (790 K)	67
0.375	0.375	190	0.75	0.35	0.020	0.50 (875 K)	45
0.375	0.500	200	0.75	0.50	0.010	1.00 (873 K)	19
0.375	0.625	195	0.73
1.000	0.000	170	0.52	1.00	0.000	0.64 (800K)	16
				0.40	0.020	0.70 (880 K)	19
				0.30	0.050	0.85 (870 K)	65
				0.30	0.000	0.30 (870 K)	65

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4 Conclusion

In most of the studied cases, the half Heusler (HH) compounds with exactly 18 valence electrons do not show spin polarization at E_F and unlike other Heusler compounds, they are semiconductors/semi-metals. The compound MNiSn (M = Hf, Zr) under our investigation with an exact count of 18 valence electrons shows semiconducting behaviour with a small band gap of around ∼0.50 eV. The character of optimized thermoelectric properties is possible in Zr_xHf_1−x−yTa_yNiSn with enhanced efficiency. Unfortunately, our system could not achieve the benchmark value, i.e. 1, for its commercial application. The maximum calculated ZT is 0.75 at 750 K. Though we could not mimic the exact experimental composition, the results follow a similar trend and are in good agreement with the previous experimental result of 0.85 at 870 K. In our case the ZT value for the doped system is enhanced by 36% over the undoped sample (ZT = 0.50). However higher values of ZT are expected if the S and κ are improved. For future work, all those related parameters can be optimized to enhance the ZT above unity by means of suitable doping with heavy atoms and improving the density of states near E_F. Further, some other theoretical techniques (thin film or nano structure) may be adopted for the improvement of ZT within a suitable temperature range.

Acknowledgements

DPR acknowledges research fellowship from Beijing Computational Science Research Center (Beijing, China). AS and RKT a research grant from UGC (New Delhi, India). MPG acknowledges the partial support from NIMS, Japan. SD a grant from DST, New Delhi, India under Dy No. SERB/F/3586/2013-14 dated 6.09.2013.

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