First-Principles Calculation of the Phonon and Thermoelectric Properties of NiZnSn Half Heusler Alloy

Umukoro JO, Omagbemi OG and Osafile OE

Published on: 2022-06-28

Abstract

The search for viable thermoelectric materials is on the front burner for researchers in material science due to the world energy crisis. Several materials have proved to be promising at both high and low-temperature regimes; however, to correctly predict the thermoelectric properties of a material, the relationship among the parameters of the dimensionless figure of merit needs to be understood and correctly analysed. In this work, we have adopted a first-principles method to investigate the structural, electronic, phonon, and transport properties of NiZrSn based on the density functional theory using the generalized gradient approximation implemented in the quantum espresso suite. The band gap, lattice constant and other structural and electronic property results obtained in this work compare well with experimental and theoretical results from previous work. Furthermore, we studied the phonon and transport properties of the alloy. The phonon dispersion and phonon density of states prove that NiZrSn is stable. The LO-TO splitting in the acoustic and optical phonon branches supports covalent and ionic bonding in the alloy. The thermoelectric properties with hole and electron doping and carrier concentrations suggest that NiZrSn will be a better thermoelectric material as an n-type semiconductor.

Keywords

Density functional theory Density functional perturbation theory Half-Heusler alloys Phonon properties Thermoelectric properties

Introduction

The accurate prediction of the properties of materials paves the way for the possible experimental simulation of such materials. The multilateral needs for materials in a technology-driven world are nearly insatiable, and the rate at which novel viable materials are discovered seems to be lagging behind the demand. The need for smarter materials in electronic devices, agriculture, medicine, the environment, and energy is growing astronomically. Among these needs, energy is like a lifeline without which the other needs might not be effectively met. This background justifies the need for continued research for affordable, available, environmentally friendly, and easily fabricated materials to supply the growing energy demand. Many materials have competed in the energy supply chain [1-3]; however, most of these materials present flaws that leaves a lot to be desired. These flaws are especially true in the face of the threat to life arising from climate change. The impact of global warming from pollution has necessitated a global need for an energy shift from the conventional carbon-emitting sources.

One such energy source that has proved to be sustainable is solar energy. One of the materials of interest in solar energy generation is the semiconductor materials that facilitate electron transport in solar systems [4,5]. A lot of semiconductor materials have been reviewed in the literature, among which are silicon and its alloys [6], germanium and its alloys [7,8], chalcogenides [9], pnictides [10], carbon nanotube and its variants [11], MXenes [12,13], and Heusler alloys [14-16].

Heusler alloy was discovered in 1903 [17] and has since presented remarkable properties that have found economic applications on a commercial scale in many devices such as energy harvesters [18] and magnetic read access memories (MRAM) [19]. Other areas of application of Heusler alloys include sensors, spintronics [20,21], actuators [21], thermoelectric materials [22-25], solar cells [26,27], and as semiconductors materials [28]. The Heusler family of alloys is attractive because a simple valence count can give valuable details of the possible property of such alloys. In addition, the alloy can be easily tuned to achieve the desired property. Heusler alloys comprise two transition metals and the main group element. There are three fundamental variants of Heusler alloys: full Heusler alloys, Half Heusler alloys, and quaternary Heusler alloys. The half Heusler alloys are ternary alloy that leaves one interpenetrating layer vacant, and they combine the zinc-blende and the rocksalt structure. There are reports in the literature on the uses of half Heusler alloys ranging from magnetic to ferromagnetic and non-magnetic. The 18-valence electrons are predicted to be semiconductors. These semiconductors find applications in electronic devices such as piezoelectric materials, spintronics, thermoelectric materials and generators [29-31], etc.

Miyazaki et al. [32] conducted an experimental investigation on the crystal structure around atomic defects of NiZrSn by comparing observed and theoretical X-ray absorption fine structure (XAFS) spectra of the crystal structure. The results of both Zr and Ni K-edge XAFS spectra verified the existence of atomic defects at the vacant sites distorting the C1b-type crystal structure. Miyazaki et al. concluded that the distortion of the atoms around the interstitial Ni disorder could be the probable reason for the observed lower thermal conductivity values than that predicted theoretically in half-Heusler alloys. Fiedler and Kratzer, in 2016 [33], carried out the first-principles calculation using the Purdue-Burke-Ernzerhof (PBE) and HSE06 exchange functional on ternary NiZrSn and CoZrBi semiconductors with C1b crystal structure. They calculated the basic structural, electronic, and phonon properties using density functional theory. They reported NiZrSn as a small indirect band-gap semiconductor (Eg=0.46 eV in PBE and 0.60 eV in HSE06) and exhibited its maximum thermopower in the n-type regime.

Kawaharada et al. [34] conducted experimental and theoretical calculations of antimony (Sb) doped and pristine NiZrSn alloys in another work. The pristine NiZrSn showed decreased electrical resistivity as the temperature increased. They recommended that the thermoelectric properties be enhanced for NiZrSn to be effective as a thermoelectric material. However, their results from doping with Sb did not improve the thermoelectric properties. Andrea et al. [35] calculated the phonon properties of NiZrSn using the finite size displacements method and many-body perturbation theory. Andrea et al. reported the thermal conductivity of NiZrSn to be 13.3 W m−1 K−1 at 300 K. Alsobhi [36] investigated the structural, electronic, thermal, and elastic properties of the half-Heusler alloy, NiZrSn using full-potential local orbital minimum-basis (FPLO) code and the WIEN2k software package. They reported the bulk modulus and first pressure derivative to be 141.014 GPa and 4.2, respectively, using FLPO and 118 GPa and 5.1, respectively, using the WIEN2K package. They reported NiZrSn to be brittle under ambient conditions and pressure.

Musari et al. [37] predicted the NiZrSn compound to be stable using the Density-functional perturbation theory. They studied thermodynamic properties between 0 K and 1000 K and reported that the internal energies and entropies increase while the vibrational energies decrease with increasing temperature. Yang et al. [38] said on thermoelectric-related electrical transport properties of 36 half-Heusler (HH) compounds, including ZrNiSn, they report the alloy to be an n-type semiconductor

The reviews indicate that report from first principles on the critical properties of NiZrSn is scanty in literature. Furthermore, the disorder issue at the Ni site is also raised in the experimental report but not substantiated. Hence, in this work, we employ first-principles calculation to compute the structural, electronic, thermoelectric and phonon properties of the NiZrSn compound using the quantum espresso package.

Computational Method

Using the quantum espresso code (QE) [39], we studied the structural, electronic, thermoelectric, and phonon properties of NiZrSn half Heusler alloy from the first principles. The potentials for calculations were constructed using the projector augmented wave (PAW) [40, 41] method, and the generalized gradient approximation (GGA) was employed to treat the Perdew-Burke-Erzhenhof exchange and correlation between electrons [42]. We used the Monkhorst-Pack scheme [43] to construct an 8×8×8 grid for the electronic structure calculation. A converged value of 70 eV was used as the kinetic energy cutoff of the plane-wave basis function. We relaxed each atom in the unit-cell using the Broyden–Fletcher–Goldfarb–Shanno (BFGS) to obtain the equilibrium configuration for the atomic positions. A denser k-point mesh of (20×20×20) grid with a tetrahedra occupation was used to calculate states' electronic density. We also sampled the alloys' behaviour at equilibrium temperature as functions of the volume and pressure by fitting the result obtained from the total energy calculation to the Birch-Murnaghan equation of state [44-46]. As reported in the literature, we treated the compound as a ferromagnetic system in the fcc crystallized phase. We selected dense high symmetry k-points for the FCC structure using (X-window) Crystalline Structures and Densities (XCrySDen) package [47] for the band structure calculations. We carried out the phonon calculations at the harmonic (volume) and quasiharmonic (volume change) levels using the density functional perturbation theory (DFPT) as described by Togo and Tanaka [48]. A plane-wave energy cutoff of 300 eV and an energy convergence criterion of 108 eV was used. A  k-point sampling mesh was used for the unit cell, and the equivalent density mesh was used for the supercells together with a 0.2 eV smearing width of the Methfessel–Paxton scheme [49]. Supercell and finite displacement approaches were used for the phonon calculations with  supercells of the primitive unit cell without atomic displacements. We used the Fermi-Dirac distribution to obtain the Fermi level from the electronic density of states distribution and computed the transport coefficients according to the Boltzmann equation. The relaxation time approximation is used, which assumes that the relaxation time is constant regardless of energy. 

Results

Structural Properties

NiZrSn crystallizes in the face-centred cubic structure in the space group  and space number 216 as a non-magnetic semiconductor. The electronic configuration of the elements investigated is 3d84s2 and 4d25s2 for nickel and zirconium, respectively, while Sn is 5s²5p². The alloys crystallize in the MgAgAs structure with X, Y, Z atoms occupying the atomic positions (0, 0, 0), (0.25, 0.25, 0.25) and (0.75, 0.75, 0.75) respectively. The equilibrium lattice parameter in Angstrom is 6.15. As expected, NiZrSn obeys the Slater-Pauling rule for hH alloys, and it crystallizes as a non-magnetic semiconductor. We confirmed this by using the rule for 18 valence electrons Mt = Zt – 18 as proposed by Slater & Pauling, where Zt is the alloy's valence number, and Mt is the magnetic moment per unit formula of the alloy. In our case, Zt is 18.

The lattice parameter computed for NiZrSn agrees with the value reported by Musari et al. [37] and other researchers. However, the experimental report for the lattice constant, as noted by Ö?üt and Rabe [50], has a discrepancy of about 1%; the difference can be attributed to the DFT-GGA, which is known to overestimate lattice parameters in solids [51].

The results obtained during lattice optimization for energy and volume are fitted to the Murnaghan equation of state using:

Where E0 and V0 equilibrium values of energy and volume, respectively, without pressure, while B and B denote the bulk modulus and its derivative. The relationship between the energy and volume is shown in Fig. 1. The bulk modulus value, pressure derivative of bulk modulus, volume, ground state energy, energy bandgap, and lattice constant using PBE-GGA are presented in Table 1. The compound's magnetic moments are zero in all three interacting elements confirming its non-magnetic nature.

One test for ascertaining an alloy's structural stability and establishing the possibility of experimental simulation of the alloy is deduced using the compound's formation energy. Negative formation energy supports simulation, while positive formation energy prohibits experimental simulation. We calculated the formation energy by subtracting the energy of each element from the energy of the bulk compound using the equation; The formation energy obtained for NiZrSn is -0.33 eV showing that the alloy is structurally stable while establishing the possibility of experimental simulation.

Figure 1: Lattice optimization for the minimum energy versus volume as fitted to the Murnaghan equation of state.

Table 1:   Lattice constant (a0), ground state energy E, bulk modulus , pressure derivative of the bulk modulus , volume V0, formation energy   and energy bandgap (Eg) of NiZrSn compound in the most stable state using PBE-GGA.

compound

a0

 (Å)

E (Ry)

 

(GPA)

  

V0 (a.u.)3

 

(eV)

Eg (eV)

NiZrSn

6.15

6.05 32

-523.99

120.8

4.48

392.18

-0.33

0.489

Electronic Properties

We present results for the electronic structure of the NiZrSn compound, and the results include the density of states (DOS) and electronic band structure calculations using PBE-GGA. The FCC structure's band structure calculation is along the high symmetry path K→Γ→X→W→K→Γ→L with a dense k-point. From the band structure and DOS in Figure 2, we observe steep slopes at the Γ point, the centre of the crystal momentum space or k = 0 space and X (the edge π/L of the first Brillouin zone in the 001 direction) symmetric point. The steep dispersion indicates a stronger orbital interaction arising from the d-orbitals of Ni and Zr, resulting in a possible increase in the mobility of charges. The steeper slope also suggests a covalent bonding between the d8 orbital of Ni and the d3 orbital of Zr.

Furthermore, Γ→X. the bandgap shows that Ni's d states are delocalized in the region of the Fermi energy. On the other hand, we observe shallow or flat bands between W→K, indicating ionicity and low group velocity resulting from the s and p orbitals' localization. The flat bands in the conduction band occur around 1 eV and 2 eV, while the flat bands appear in the valence band around -1 eV to -3 eV. Figure 2 shows the electronic band structure and electronic density of states for NiZrSn alloy in the non-magnetic (paramagnetic) state. NiZrSn exhibits an indirect bandgap of approximately 0.498 eV between  and X high symmetry points. The band gap overestimates the experimental report of 0.18 eV [52]. Although the overestimation of the experimental bandgap is contrary to the prediction that PBE-GGA underestimates the bandgap, the results suggest a structural defect around Ni, as suggested in the experimental work of Miyazaki et al. [32]. The conduction band minimum CBM is 13.9265 eV, while the valence band maximum is 13.4374 eV. The narrowness of the bandgap supports charge mobility.

Figure 2: Electronic band structure and density of states of NiZrSn using PBE-GGA.

We investigated the phonon properties and some thermodynamic properties of NiZrSn using the PHONOPY code. The PHONOPY code uses real space supercells to calculate the interatomic force constants. The interatomic force constants are calculated from the ab-initio results using the density functional perturbation theory and the linear response approach implemented in the quantum espresso simulation code. We have investigated the phonon dispersion and the phonon density of states, as shown in Figure 3, for the NiZrSn half-Heusler alloy using a 2x2x2 supercell. The phonon dispersion explains the relationship between the frequencies and the wave vector k. The results show three acoustic modes, six (6) optical modes, and no negative frequencies; hence, NiZrSn is a stable alloy in the hH FCC phase with infinitesimal distortions arising from the vibrations. The phonon dispersion spectra are shown along the Γ→X→W→K→Γ high symmetry Brillouin zone path for the alloy. The six (6) optical modes are in the frequency range of 4.5–7.0 THz. In addition, we observe a minor LO-TO splitting between Γ and X in the optical phonon branches. The split size is likely a product of the level of displacement between the NiSn and the Zr position. The splitting at Γ also suggests a covalent and an ionic bonding with significant contributions from the interactions between the nickel and zirconium 3d and 4d orbitals, respectively. Figure 4 shows results for the entropy, heat capacity and Gibbs vibrational free energy representing the thermo-dynamical properties of the alloy. We observed that the behaviour of the properties is consistent with reports in the literature and expected behaviour at increasing temperatures. The free energy decreases steadily as temperature increases to a minimum of about 140 kJ/mol at 1000 K while the entropy increases. On the other hand, the heat capacity attains a steady state at about 260 K; this compares well with Ozisik et al., who predicted the stability at 250 K [53].

Figure 4: Free energy, heat capacity (Cv), and entropy of NiZrSn alloy.

Thermoelectric Properties

Thermoelectric materials (TEM) are expected to convert heat to electricity, and the material's performance index (efficiency) is measured by the rate at which it carries out this conversion. The efficiency of a TEM material is computed using the dimensionless figure of merit described by the relation  S and T represent the Seebeck coefficient and absolute temperature, respectively, while σ and k are the electrical, thermal conductivity and lattice thermal conductivity, respectively. The mutual dependence of the parameters requires that the material have a high-power factor and a low thermal conductivity for a TEM to perform optimally. Results for the power factor per unit relaxation time, electrical, thermal conductivity per unit relation time, and Seebeck coefficients at a carrier concentration of   as functions of temperature are shown in figure 5 for both n-type and p-type phases. We observed that for both n and p-type, the Seebeck coefficient and power factor per unit relaxation time increase with temperature. Conversely, the electrical and thermal conductivity decreases with temperature increase. However, the n-type has a lower power factor and electrical thermal conductivity, positioning it as a better thermoelectric material. In figure 6, we observe a shift around the Fermi level for the n-type doping, likely resulting from the investigated low carrier concentration.

 We investigated the behaviour of the alloy for electron and hole doping for the alloy; from the result in Figure7, for electron doping at a temperature of 800 K and 0.00584/uc doping level, the Seebeck coefficient increases from 295  whereas there is a steady decrease for hole doping from 282 However, the power factor and thermal conductivity increase with hole doping and decrease with electron doping. The power factor and electrical thermal conductivity behaviour support n-type NiZrSn alloy as a better thermoelectric material than the p-type NiZrSn; this result is consistent with Zou et al. [54] for pristine NiZrSn and Hf doped NiZrSn alloys.

Figure 5: the power factor per unit relaxation time, electrical thermal conductivity per unit relation time, and Seebeck coefficients at a concentration of  as functions of temperature for p-type and n-type NiZrSn hH alloy.

Figure 6: The (a) power factor per unit relaxation time, (b) electrical thermal conductivity per unit relation time and (c) thermopower at temperatures of 300 K, 600 K, and 800 K versus chemical potential dependence.

Figure 7: Power factor per unit relaxation time, electrical thermal conductivity per unit relaxation time, and thermopower for the hole and electron doping at 800 K.

Conclusion

We have investigated the structural, electronic, phonon, and transport properties of NiZrSn half Heusler alloy. We investigated the structural and electronic properties using the generalized gradient approximation associated with the density functional theory. The lattice parameter and the results of other structural and electronic parameters compare well with previous results. The phonon and thermodynamic properties were computed using the constant time relaxation approach based on the density functional perturbation theory in PHONOPY. From the results obtained, there are no negative frequencies observed, and the Debye temperature of about 260 K compares well with previous work. The transport properties calculated using the Bolztrap code conform with expected behaviour at low and high temperatures, suggesting that the n-type NiZrSn is a better thermoelectric material than the p-type. This conclusion is based on the lower electrical thermal conductivity and power factor per relaxation time observed for electron-doped material than hole doping. We believe that this work will spur further experimental reports on NiZrSn alloy.

Funding

None

Conflict of Interest

Authors declare that they do not have any conflict of interest.

References

  1. Zhan L, Bo Y, Lin T, Fan Z. Development and outlook of advanced nuclear energy technology. Energy Strategy Reviews. 2021; 34: 100630.
  2. Berndes G, Hoogwijk M, Van den Broek R. The contribution of biomass in the future global energy supply: a review of 17 studies. Biomass and bioenergy. 2003; 25: 1-28.
  3. Clark RJ, Farid M. Experimental investigation into the performance of novel SrCl2-based composite material for thermochemical energy storage. Journal of Energy Storage. 2021; 36: 102390.
  4. Zhu L, Shang X, Lei K, Wu C, Zheng S, Chen C, et al. Doping in semiconductor oxides?based electron transport materials for perovskite solar cells application. Solar RRL. 2021; 5: 2000605.
  5. García de Arquer FP, Talapin DV, Klimov VI, Arakawa Y, Bayer M, Sargent EH. Semiconductor quantum dots: Technological progress and future challenges. Science. 2021; 373: eaaz8541.
  6. Nijs J. Advanced silicon & semiconducting silicon-alloy based materials & devices. CRC Press. 2021.
  7. Chiara R, Morana M, Malavasi L. Germanium?Based halide perovskites: materials, properties, and applications. ChemPlusChem. 2021; 86: 879-888.
  8. Terna AD, Elemike EE, Mbonu JI, Osafile OE, Ezeani RO. The future of semiconductors nanoparticles: Synthesis, properties, and applications. Materials Science and Eng. 2021; 272: 115363.
  9. de Vrijer T, Parasramka H, Roerink SJ, Smets AH. An expedient semi-empirical modelling approach for optimal bandgap profiling of stoichiometric absorbers: A case study of thin-film amorphous silicon germanium for use in multijunction photovoltaic devices. Solar Energy Materials and Solar Cells. 2021; 225: 111051.
  10. Mathur AS, Upadhyay S, Singh PP, Sharma B, Arora P, Rajput VK, et al. Role of defect density in absorber layer of ternary chalcogenide Cu2SnS3 solar cell. Optical Materials. 2021; 119: 111314.
  11. McKeown Wessler GC, Wang T, Sun JP, Liao Y, Fischer MC, Blum V, et al. Structural, Optical, and electronic properties of two quaternary chalcogenide semiconductors: Ag2SrSiS4 and Ag2SrGeS4. Inorganic Chem. 2021; 60: 12206-12217.
  12. Elemike EE, Osafile OE, Omugbe E. New perspectives 2Ds to 3Ds MXenes and graphene functionalized systems as high-performance energy storage materials. J Energy Storage. 2021; 42: 102993.
  13. Gogotsi Y, Anasori B. The rise of MXenes. ACS nano. 2019; 13: 8491-8494.
  14. Osafile OE, Nenuwe ON. Lattice dynamics and thermodynamic responses of XNbSn Half-Heusler semiconductors: A First-Principles Approach. Journal of the Nigerian Society of Physical Sciences. 2021; 3: 121-130.
  15. Adebambo PO, Osafile OE, Laoye JA, Idowu MA, Adebayo GA. Electronic fitness function, effective mass and thermoelectric properties of Rh-based (-ScTe;-TiSb;-VSn) alloys for thermoelectric generator applications. Computational Condensed Matter. 2021; 26: e00523.
  16. Zhu T, Fu C, Xie H, Liu Y, Zhao X. High-efficiency half?Heusler thermoelectric materials for energy harvesting. Advanced Energy Materials. 2015; 5: 1500588.
  17. Heusler F 1903 Verh. D. DPG 5 219
  18. Bulman G, Cook B. High-efficiency energy harvesting using TAGS-85/half-Heusler thermoelectric devices. In energy harvesting and storage: Materials, Devices, and Applications V. 2014; 9115: 17-24. SPIE.
  19. Erkisi A, Surucu G. The investigation of electronic, magnetic, mechanical, and lattice dynamical properties of PdMX (M= Cr, Fe and X= Si and Ge) ferromagnetic half-Heusler metallics: an ab initio study. Materials Research Express. 2017; 4: 066504.
  20. Hirohata A, Sukegawa H, Yanagihara H, Žuti? I, Seki T, Mizukami S, et al. Roadmap for emerging materials for spintronic device applications. IEEE Transactions on Magnetics. 2015; 51: 1-11.
  21. Wang C, Meyer J, Teichert N, Auge A, Rausch E, Balke B, Felser C. Heusler nanoparticles for spintronics and ferromagnetic shape memory alloys. Journal of Vacuum Science & Technology B. 2014; 32: 020802.
  22. Jolayemi OR, Adetunji BI, Osafile OE, Adebayo GA. Investigation of the thermoelectric properties of Lithium-Aluminium-Silicide (LiAlSi) compound from first-principles calculations. Computational Condensed Matter. 2021; 27: e00551.
  23. Adebambo PO, Osafile OE, Laoye JA, Idowu MA, Adebayo GA. Electronic fitness function, effective mass and thermoelectric properties of Rh-based (-ScTe;-TiSb;-VSn) alloys for thermoelectric generator applications. Computational Condensed Matter. 2021; 26: e00523.
  24. Agbaoye RO, Adebambo PO, Adetunji BI, Osafile O, Adebayo GA. Thermoelectric properties, optimal doping levels and high figure of merit in cobalt-based half/full Heusler alloys by first-principles calculations. Materials Science and Engineering: B. 2019; 248: 114409.
  25. Chauhan NS, Miyazaki Y. Contrasting role of bismuth doping on the thermoelectric performance of VFeSb half-Heusler. Journal of Alloys and Compounds. 2022; 908: 164623.
  26. Xie W, Weidenkaff A, Tang X, Zhang Q, Poon J, Tritt TM. Recent advances in nanostructured thermoelectric half-Heusler compounds. Nanomaterials. 2012; 2: 379-412.
  27. Graf T, Parkin SS, Felser C. Heusler compounds - A material class with exceptional properties. IEEE Transactions on Magnetics. 2010; 47: 367-373.
  28. Osafile OE, Umukoro JO. Stability of X-IV-IV half Heusler semiconductor alloys: a DFT study. Molecular Physics. 2021; 119: e1936249.
  29. Osafile OE, Azi JO. Structural, electronic, elastic and mechanical properties of novel ZrMnAs half Heusler alloy from first principles. Physica B: Condensed Matter. 2019; 571: 41-49.
  30. Iyorzor B, Babalola M. Half-metallic Characteristics of the Novel Half Heusler Alloys XCrSb (X= Ti, Zr, Hf). Journal of the Nigerian Society of Physical Sciences. 2022; 138-145.
  31. Nakatani TM, Hase N, Goripati HS, Takahashi YK, Furubayashi T, Hono K. Co-based Heusler alloys for CPP-GMR spin-valves with large magnetoresistive outputs. IEEE transactions on magnetics. 2012; 48: 1751-1757.
  32. Miyazaki H, Ozkendir OM, Gunaydin S, Watanabe K, Soda K, Nishino Y. Probing local distortion around structural defects in half-Heusler thermoelectric NiZrSn alloy. Scientific reports. 2020; 10: 1-8.
  33. Fiedler G, Kratzer P. Ternary semiconductors NiZrSn and CoZrBi with half-Heusler structure: A first-principles study. Physical Review B. 2016; 94: 075203.
  34. Kawaharada Y, Uneda H, Muta H, Kurosaki K, Yamanaka S. High-temperature thermoelectric properties of NiZrSn half-Heusler compounds. Journal of alloys and compounds. 2004; 364: 59-63.
  35. Andrea L, Hug G, Chaput L. Ab initio phonon properties of half-Heusler NiTiSn, NiZrSn and NiHfSn. Journal of Physics: Condensed Matter. 2015; 27: 425401.
  36. Alsobhi BO. A first-principles investigation of the structural, electronic, elastic and thermal properties of NiZrSn under pressure. J Nano Adv Mat. 2019; 7: 37-48.
  37. Musari AA, Adetunji BI, Adebambo PO, Adebayo GA. Lattice dynamics and thermodynamic investigation of MNiSn (M= Hf, Ti and Zr) Half-Heusler compounds: Density functional theory approach. Materials Today Communications. 2020; 22: 100671.
  38. Yang J, Li H, Wu T, Zhang W, Chen L, Yang J. Evaluation of half?Heusler compounds as thermoelectric materials based on the calculated electrical transport properties. Advanced Functional Materials. 2008; 18: 2880-2888.
  39. Giannozzi P, Baroni S, Bonini N, Calandra M, Car R, Cavazzoni C, et al. QUANTUM ESPRESSO: a modular and open-source software project for quantum simulations of materials. Journal of physics: Condensed matter. 2009; 21: 395502.

 

  1. Kresse G, Joubert D. From ultrasoft pseudopotentials to the projector augmented wave method. Physical Review B. 1999; 59: 1758.
  2. Perdew JP, Burke K, Ernzerhof M. Generalized gradient approximation made simple. Physical review letters. 1996; 77: 3865.
  3. Monkhorst HJ, Pack JD. Special points for Brillouin-zone integrations. Physical review B. 1976; 13: 5188.
  4. Marzari N, Vanderbilt D. Maximally localized generalized Wannier functions for composite energy bands. Physical review B. 1997; 56: 12847.
  5. Murnaghan FD. The compressibility of media under extreme pressures. Proceedings of the National Academy of Sciences. 1944; 30: 244-247.
  6. Birch F. Finite elastic strain of cubic crystals. Physical review. 1947; 71: 809.
  7. Murnaghan FD. Finite deformations of an elastic solid. American Journal of Mathematics. 1937; 59: 235-260.
  8. Kokalj A. XCrySDen - a new program for displaying crystalline structures and electron densities. Journal of Molecular Graphics and Modelling. 1999; 17: 176-179.
  9. Togo A, Tanaka I. First-principles phonon calculations in materials science. Scripta Materialia. 2015; 108: 1-5.
  10. Methfessel MPAT, Paxton AT. High-precision sampling for Brillouin-zone integration in metals. Physical Review B. 1989; 40: 3616.
  11. Ö?üt S, Rabe KM. Band gap and stability in the ternary intermetallic compounds NiSnM (M= Ti, Zr, Hf): A first-principles study. Physical Review B. 1995; 51: 10443.
  12. Khein A, Singh DJ, Umrigar CJ. All-electron study of gradient corrections to the local-density functional in metallic systems. Physical Review B. 1995; 51: 4105.
  13. Aliev FG, Brandt NB, Moshchalkov VV, Kozyrkov VV, Skolozdra RV, Belogorokhov AI. Gap at the Fermi level in the intermetallic vacancy system RBiSn (R= Ti, Zr, Hf). Zeitschrift für Physik B Condensed Matter. 1989; 75: 167-171.
  14. Ozisik HB, Ateser E, Ozisik H, Colakoglu K, Deligoz E. Ab-initio calculations on half-Heusler NiXSn (X= Zr, Hf) compounds: electronic and optical properties under pressure. Indian Journal of Physics. 2017; 9: 773-778.
  15. Zou DF, Xie SH, Liu YY, Lin JG, Li JY. Electronic structure and thermoelectric properties of half-Heusler Zr5Hf0.5NiSn by first-principles calculations. Journal of Applied Physics. 2013; 113: 193705.