J. Nig. Soc. Phys. Sci. 3 (2021) 121–130 Journal of the Nigerian Society of Physical Sciences Lattice Dynamics and thermodynamic Responses of XNbSn Half-Heusler Semiconductors: A First-Principles Approach O. E. Osafile∗, O. N. Nenuwe Department of Physics, Federal University of Petroleum Resources, PMB 1221, Effurun, Nigeria Abstract The need to predict the properties of new half Heusler alloys for possible technological applications from first principles cannot be overem- phasised. This need is driven by the huge industrial demand for desirable properties under the influence of temperature. In this work, XNbSn (X = Co, Ir, Rh) half-Heusler (hH) alloys were investigated for the structural, electronic, elastic, and thermodynamic properties using PBE- GGA exchange-correlation functional as implemented in the quantum espresso computational suite. XNbSn alloys crystallise in the face centred cubic (FCC) structure with space group F-43m and number 216. The lattice structures were optimised, and the lattice constants of CoNbSn alloy compares well with reports in literature, while IrNbSn and RhNbSn are reported on for the first time. XNbSn hH alloys are non-magnetic semiconductors with bandgaps 1.03, 0.72 and 0.78 for CoNbSn, IrNbSn and RhNbSn alloys, respectively. The splitting of the bandgap at the Fermi level is dominated by the d-d hybridisation between the d orbitals of Nb and Co/Ir/Rh. The negative formation energies obtained from computing the formation enthalpy support the structural stability and possible experimental simulation of the alloys. There is a clear bandgap across the Brillouin zones at a frequency of 150 cm−1; RhNbSn exhibits the largest bandgap while the bandgap is almost non-existent in CoNbSn. Non-negative frequencies in the phonon dispersion predict the thermodynamic stability of the alloys. The alloys Dulong-Petit law is obeyed at a temperature of about 600 K and heat capacity of 74 J mol −1K −1. The phonon dispersion and density of states show that the d orbitals of Co and Nb are the dominant contributors to the dispersions at both the acoustic and optical modes of the alloys. IrNbSn alloy has the least Debye temperature (low thermal conductivity), positioning it as the most promising material for possible thermoelectric applications. DOI:10.46481/jnsps.2021.174 Keywords: Density functional theory, Density functional perturbation theory, Half-Heusler compounds, phonon dispersions, Thermodynamic properties. Article History : Received: 15 March 2021 Received in revised form: 04 May 2021 Accepted for publication: 08 May 2021 Published: 29 May 2021 ©2021 Journal of the Nigerian Society of Physical Sciences. All rights reserved. Communicated by: A. H. Labulo 1. Introduction Heusler alloys are a large family of alloys with properties that can be modified to suit technological applications in sev- eral areas of scientific and engineering research. There are ∗Corresponding author tel. no: +2347033249124 Email address: osafile.omosede@fupre.edu.ng (O. E. Osafile ) two main classes of Heusler alloys; these are the full and half Heusler alloys. Albeit, there are several variants arising from the two main classes [1-5]. Heusler first announced the full Heusler alloys in 1903 [6,7]; it has the X2YZ (2:1:1) stoichiometry and generally crystallises in the FCC phase and L21 structure. The half Heusler alloy, on the other hand, was discovered in 1983 by De Groot et al. [8]. 121 Gemanam et al. / J. Nig. Soc. Phys. Sci. 3 (2021) 121–130 122 It has the XYZ (1:1:1) stoichiometry and crystallises mostly in the FCC phase. The X and Y elements are transition or rare earth metals, while the Z element is a main group (or a P- block) element. The properties of the Heusler alloy can be tuned via dop- ing, strain & stress application, vacancies, impurities, disloca- tions, and heat treatments to achieve the desired property for technological applications [9-12]. In recent times, the ther- moelectric properties of materials became desirable research focus primarily due to their applications in power generation and refrigeration from supposed waste materials. A material is deemed useful for thermoelectric application based on the efficiency, and the efficiency is parametrised by the Seebeck coefficient, electrical conductivity, and thermal conductivity. The efficiency is measured relative to the temperature change using the three parameters mentioned above. Fermi energy plays a prominent role in the Seebeck coefficient and electri- cal conductivity of the material. The Fermi level is defined by the electron and hole concentrations of the alloy, effec- tive mass, and temperature. Suggestions in literature indicate that decoupling the Seebeck coefficient and electrical con- ductivity without compromising efficiency can be achieved by employing the transport functions of the alloys [13]. The most efficient materials are expected to exhibit high electri- cal conductivity, high Seebeck coefficient and a relatively low thermal conductivity to minimise thermal losses [14-16]. Despite the gains from other materials as thermoelectric materials, their setback positions the half Heusler alloys as promising materials in thermoelectric applications. In 2016 Huang et al. reported a ZT of ≈ 1 in NbCoSn, NbFeSn and VFeSb in the cubic MgAgAs FCC phase [17]. Liu et al. recorded a ZT of ≈ 1 and 0.7 for p- and n-type materials at a tempera- ture of 978 K respectively by combining 17-electron TiFeSb and 19-electron TiNiSb HH alloys, based on the same parent TiFe0.5Ni0.5Sb alloy [18]. An experimental report by Xia et al. showed that ZT = 0.9 was obtained for Nb0.83CoSb at 1123 K compared to the pristine n-type NbCoSb and ZT peak of ap- proximately 0.4 at 973 K [19, 20]. Also, the p-type FeNbSb doped by Hf exhibited a ZT 0f 1.5 at a temperature of 1200 K putting its performance ahead of other materials at such high temperature [21]. The beauty of the half Heusler alloy is that elements can be interchanged, replaced, and doped to achieve properties desired for specific applications. Many n-type and p-type half Heusler semiconductor ma- terials have been predicted from first-principles calculations and experiments for application as thermoelectric materials. These half Heusler alloys have been reported to exhibit sub- stantial TE properties comparable to conventional Bi2Te3 and PbTe based materials [22-27]. Poon et al. reported that the n-type Hf0.6 Zr0.4 NiSn0.995 Sb0.005 hH alloy and p-type Hf0.3 Zr0.7 CoSn0.3 Sb0.7 /nano-ZrO2 composites achieved ZT = 1.05 and 0.8 near 900 – 1000 K, respectively [28] . In another work, Joshi et al. reported a high ZT value of ∼1 at 700 ◦C for a nanostructured p-type Nb0.6Ti0.4FeSb0.95Sn0.05 composition [25]. Another attraction of the half-Heusler alloys is the possi- bility of tuning the bandgap. The bandgap of a thermoelec- tric material plays a vital role in defining the scope of ap- plication in a thermoelectric device. Recall that the dimen- sionless figure of merit ZT is given as Z T = σS 2T /κ where σ is the electrical conductivity, S is the Seebeck coefficient, T is temperature, and κ is the thermal conductivity. The ther- mal conductivity is given by the sum of contributions from the electronic carriers (κe l ) and the lattice (κl ) carriers. ZT is enhanced when σ and S increase as κ decreases. However, the σ, S & κ parameters are interdependent, such that it be- comes a challenge to increase ZT. If the thermal conductiv- ity must decrease, the lattice carrier κl is expected to domi- nate. One way to increase the electrical conductivity σ is to increase the carrier concentration. Unfortunately, increasing carrier concentration compromises the Seebeck coefficient and increases the electronic carrier so that the thermal con- ductivity increases. The Half Heusler semiconductors have come in handy in solving this problem because the tuning of the bandgap (that is, increasing or decreasing the size of the bandgap) can help separate the n-type and p-type carri- ers as the concentration increases, hence, having a single car- rier type without compromising charge mobility. Tuning the bandgap also provides the opportunity of reducing the ther- mopower as the temperature increases. Thermopower reduc- tion happens when the bandgap is narrow enough; hence, at an excited state, n-type carriers can cross to the conduction band and p-type carriers to the valence band, resulting in a cancelling effect that creates a balance [29-32]. The present work is focused on investigating the lattice and thermodynamic responses of XNbSn (X = Co, Rh, Ir) Half- Heusler Semiconductors considering the Lattice dynamics and Phonon Dependent Parameters in the cubic (F-43m) space groups via first-principles calculations. This work is moti- vated by suggestions that these 18-valence half Heusler semi- conductors are promising materials for thermoelectric appli- cations [13]; these alloys are known to have closed-shells non- magnetic and semiconducting [33]. There are no substantive claims for the mechanical stability of XNbSn alloys; hence, their viability as thermoelectric materials are unsubstantiated. Though, Mubarak et al. reported on the thermal, electro - magnetic and thermoelectric properties of CoNb1− x Tix Sn (x= 0, 0.75, 0.5, 1) hH alloy using the DFT Full-Potential Linearized Augmented Plane Wave (FP-LAPW ) method as implemented in the WIEN2k code [34], there are no experimental reports to compare with. There are no reports in the literature for lat- tice dynamics and thermodynamic properties of IrNbSn and RhNbSn compounds to the best of our knowledge. The lattice dynamical parameters and phonon properties currently ab- sent in literature play a vital role in understanding the phys- ical properties; hence in this work, we investigate the struc- tural, electronic properties, formation energies, mechanical stability, and lattice dynamical properties of the alloys in de- tail. The rest of this article is organised as follows, and section 2 reports the computational method while results are pre- sented and discussed in section 3, followed by the conclusion. 122 Gemanam et al. / J. Nig. Soc. Phys. Sci. 3 (2021) 121–130 123 2. Computational Method 2.1. Structural and Electronic Properties We have used the FCC primitive unit cell for structural and electronic property calculations with three atoms (X, Nb, Sn) in this investigation, where X is Co, Rh, or Ir. The first principles investigations are based on the Quantum Espresso (QE) computational suite [35, 36]. The structural and elec- tronic property calculations are computed within the frame- work of the density functional theory (DFT) implemented in QE. The projector augmented wave (PAW ) functional and the generalised gradient approximation (GGA) with the Perdew- Burke-Ernzerhof (PBE) exchange and correlation between elec- trons [37-39] was adopted for all calculations. An 8×8×8 grid in the Monkhorst-Pack scheme [40] was constructed for the electronic structure calculation and convergence of 70 eV ki- netic energy cut-off of the plane-wave as the basis function. To ascertain the convergence of the results, we set a 10−12 Ry criterion for convergence of the total energy, and a Marzari- Vanderbilt [41] smearing of 0.02 thickness was applied to the alloys. Before the computation of the structural and electronic properties, the alloys’ unit-cells were fully relaxed to obtain the equilibrium configuration for the atomic positions. The converged values for the smearing, lattice constant, k-point grid, and atomic positions were inputted for the electronic structure calculation. A denser k-point mesh of (14×14×14) grid was used to calculate states’ electronic density (DOS) cal- culation with a tetrahedra occupation. At equilibrium tem- perature, the relevant information about the structural be- haviour of the alloys was extracted by fitting the result ob- tained from the total energy calculation to the Birch-Murnaghan Equations of state [42-44]. For the band structure calcula- tion, dense high symmetry k-points were selected along W → L →G → X→W in units of 2p/a. Using Crystalline Struc- tures and Densities (XCrySDen) package [45] in the FCC crys- tallised phase for visualisation. 2.2. Mechanical and Phonon Properties After computing the structural and electronic properties of the systems, the elastic, mechanical, thermodynamic, and phonon dispersion properties were determined using the den- sity functional perturbation theory (DFPT). The linear response method in the quasi-harmonic approximation as implemented in the thermo-pw package [46,47] was explored as a post-proc in QE. For convergence, a 4×4×4 uniform supercell was used with symmetry implemented. Thermodynamic quantities im- mediately obtained from the approximation includes the vi- brational energy Ei = ∑ q ,ν ( nq ,ν + 12 ) }ωq ,ν of the phonon modes where nq ,ν is the number of phonons and ωq ,ν is the phonon frequency. Others include the isochoric heat capacities, elas- tic constant and compliances, bulk modulus, and thermal ex- pansion using Voigt, Reuss, and Hills [48-50] relations. Table 1: Equilibrium lattice constant a0 (Å) and Energy (Ry) of the three pos- sible configurations of CoNbSn, RhNbSn and IrNbSn half Heusler alloys us- ing PBE-GGA approximation. Compound Eg (Ry) a0 (Å) XYZ (0.0, 0.5, 0.25) CoNbSn -938.227 6.188 RhNbSn -1246.416 6.310 IrNbSn -1497.761 6.299 YZX (0.25, 0.0, 0.5) CoNbSn -938.260 6.225 RhNbSn -1246.457 6.340 IrNbSn -1497.839 6.322 ZXY (0.5, 0.25, 0.0) CoNbSn -938.451 5.981 RhNbSn -1246.597 6.214 IrNbSn -1497.984 6.246 3. Results and Discussion 3.1. Structural and Electronic Properties The electronic configuration for Co, Rh and Ir transition metals are (3d7 4s2), (4d8 5s1), and (5d7 6s2) with atomic mass units of 58.3195u, 102.9055u, and 192.217u, respectively. In comparison, Nb and Sn have electronic configurations of (4d4 5s1) and (5s2 5p2), respectively. The three alloys are 18 va- lence electron hH non-magnetic semiconductors. Fundamen- tally, there are three possible configurations for hH alloys. We investigated the three possible configurations of the alloys to establish the equilibrium lattice constant and the most stable structural state of the alloys. In table 1, Results for the atomic positions, energy and lattice constants for the three config- urations of the alloys are presented, and the results for the variation of energy with the lattice constant are represented in Figure I. Figure I shows that the most stable state of the three alloys is the ZXY configuration. The hH alloy crystallises in the C1b structure belonging to the F 43m group with space group number 216. The result shows that IrNbSn has the low- est ground state energy of 1497.984 Ry. It also exhibits the largest volume while CoNbSn has the highest ground state energy and the smallest volume; this is expected considering the atomic mass of Ir compared to Co and Rh. The relevant information about the structural behaviour of the alloys was extracted by fitting the result obtained from the energy-volume optimisation calculation to the Birch-Murnaghan Equation of state at 0 K and 0 GPA as given in Eqn. (1). E (V ) = E0 + 9V0B 16   [( V0 V ) 2 3 − 1 ]3 B ′ + [( V0 V ) 2 3 − 1 ]2 [ 6 − 4 ( V0 V ) 2 3 ]  (1) The equilibrium lattice constant, total energy, bulk modulus, pressure, and pressure derivatives are documented in Table 2. Results for the lattice constants are in good agreement with 123 Gemanam et al. / J. Nig. Soc. Phys. Sci. 3 (2021) 121–130 124 Table 2: Equilibrium lattice constant a0 (Å), bulk modulus B0 (G P a), pressure derivative of the bulk modulus B ′ 0 , formation energy E f (Ry), and bandgap Eg (eV) of CoNbSn, RhNbSn and IrNbSn half Heusler alloys using PBE-GGA approximation compared with existing results where available. Compound a0 B B’ E f E g CoNbSn 5.98 5.9783 4 5.9475 4 5.905 5 152.7 158.193 4 170.555 5 4.34 -3.05 -3.083 4 1.03 1.025 5 0.983 4 IrNbSn 6.21 153.1 4.37 -2.02 0.72 RhNbSn 6.24 168.6 4.53 -3.189 0.78 Figure 1: Energy-lattice parameter relationship of (a) CoNbSn alloy (b) RhNbSn alloy (c) IrNbSn alloy with XYZ (0.0, 0.5, 0.25), YZX (0.25, 0.00, 0.5), and ZXY (0.5, 0.25, 0.0) atomic arrangements. results reported in literature. In addition, we observe that the lattice parameter increases from Co 3d to Rh 4d and Ir 5d electronic configurations, showing that the lattice param- eter increases with volume. Table 2 also shows results for the formation energy (E f ) of each alloy. The formation energy predicts the possibility of synthesis; robust negative forma- tion energy supports synthesis while positive formation en- ergy does not. The formation energy (Ef ) is computed by sub- tracting the energy (EX, ENb, ESn) of the constituent elements in the unit cell of the alloy from the total energy (EX N bSn ) of the alloy using Eqn. (2). E f = E X N bSn − E X − E N b − ESn , (2) X is either Co, Rh, or Ir. The negative formation energy ob- tained for the three compounds shows they can be synthe- sised experimentally. The band structure and the partial density of states (Pdos) from the electronic structure calculations are presented in Figs 2 (a-f ). The bandgaps are 1.02, 0.72 and 0.78 eV for CoNbSn, IrNbSn and RhNbSn alloys, respectively. The band structure shows a denser particle concentration in the valence band in comparison with the conduction band. The compounds dis- play indirect bandgaps between X and W with the conduction band minimum (CBM) at X and the valence band maximum (VBM) at W. In IrNbSn, however, the VBM has electron con- centration at both L and W. From Fig. 2b, the Co, Ir, Rh (d) orbital, Nb (d) orbital and the Sn (p) orbital contributes most to the bandgap around the Fermi level in the valence band. However, in the conduction band, the d orbitals of Co and Nb dominates, the d-d hybridi- sation between the orbitals accounts for the splitting that re- sults in the bandgap. High electron concentration is observed in CoNbSn alloy at 1.9 eV, and the electron concentration can be adduced to the hybridisation of the d orbitals of Co and Nb below the Fermi level (valence band), the concentration at 1.0 eV results from the hybridisation of the d orbitals of Co and Nb and the p orbitals of Nb and Sn. The d orbitals of Nb and Ir/Rh contributes strongly to the bandgap in the valence band of IrNbSn and RhNbSn, while the p orbitals of Ir/Rh and Sn contributes weakly; the dominant contributor in the con- duction band is the d orbital of Nb for both alloys. The alloys are all p-type semiconductors. 3.2. Elastic and Mechanical Properties The elastic constants, Cij, are essential parameters in de- scribing the behaviour of alloys under applied stress, and it 124 Gemanam et al. / J. Nig. Soc. Phys. Sci. 3 (2021) 121–130 125 (a) (b) (c) provides information on the elastic stability and rigidity of the alloys. In this section, we study the mechanical stability of the compounds. To do this, we calculated the three independent elastic constants using the thermo-pw package in interface with Quantum Espresso by the quasi-harmonic approxima- tion. The values of the elastic constants C11, C12, and C44 ob- tained from the coefficient of the linear term are presented in Table 3. The elastic stability of a cubic structure is determined (d) (e) (f ) Figure 2: (a) Band structure of CoNbSn (b) Partial density of states of CoNbSn (c) Band structure of IrNbSn (d) Partial density of states of IrNbSn (e) Band structure of RhNbSn (f ) Partial density of states of RhNbSn alloys using the computed equilibrium lattice constants. by the following rules as proposed by Born and Huang [51]. C11 > 0; C44 > 0 (C11 −C12) > 0 (C11 + 2C12) > 0   (3) These equations are often referred to as the Born stability cri- 125 Gemanam et al. / J. Nig. Soc. Phys. Sci. 3 (2021) 121–130 126 teria. However, these criteria do not entirely constitute a nec- essary and sufficient condition for stability in cubic crystalline compounds; therefore, Mouhat and Coudert stated in their article that a crystalline structure could also be classified as stable if the elastic energy is a positive definite(E > 0,∀ε,0) in the absence of external load in the harmonic approxima- tion [52]; this is denoted mathematically as E = E0 + 1 2 V0 6∑ i , j =1 Ci , j εi εj + 0 ( ε3 ) (4) This condition is termed the elastic stability criterion. An al- loy must satisfy the necessary and sufficient stability condi- tions that the matrix C and the eigenvalues of C are favourable [52]. Against this background, we present in Table 3 the re- sults of the three independent elastic constants of the alloys; the results show that the alloys are stable. Other properties calculated from the elastic constants are Young’s modulus (Y ), the Poisson ratio (v), bulk modulus (B) and shear modulus (G). They were evaluated using the Voigt- Reuss-Hill (VRH) average approximation. The result of the VRH and relevant derivatives are tabulated in Table 4. The derivatives include the Pugh’s ratio, anisotropy, Vicker’s hard- ness, and Cauchy’s pressure. For crystals, the bulk modulus B defines the ability of the material to resist fracture, while the shear modulus measures resistance to plastic deformation. B and G can be deducted from VRH approximations usingB = 12 (BV + BR )G = 12 (GV +GR ); B is the bulk modulus, BV and BR are the Voigt and Reuss bound of the bulk modulus. G is the shear modulus, while GV and GR are the Voigt and the Reuss bound of the shear mod- ulus. The Debye temperatureΘD is crucial to predicting the thermal properties; it separates the high-temperature region of a crystal alloy from the low-temperature region. By default, a higher ΘD defines a material to have a higher thermal con- ductivity [53]. The Debye temperature can be obtained us- ing the average sound velocity and density from the relation h KB [ 3n 4p N Aρ M ]1/3 νm where h is Planck’s constant, KB is Boltz- mann’s constant, n is the number of atoms per unit cell, NA is Avogadro’s number, ρ its density, and M is the molecular weight. νm is the sound velocity at zero pressure. The result ΘD is recorded in Table 4, with CoNbSn having the highest De- bye temperature. TheΘD report by Ref. [34] is larger than our report. Ref. [34] calculated the three independent (C11, C12 and C44) elastic constants via the stress tensor matrix, under small strains δ using the ElaStic code by Charpin integrated in the WIEN2k code. The methodology and convergence of the parameters used by the authors probably account for the dis- crepancy between the results presented. We also compared the Debye temperature from this work with results obtained by Carrette at al. [54] using machine learning. The anisotropic factor (A), which as opposed to isotropy, describes the properties of the alloys in different directions and is evaluated using the relation 2C44/(C11 −C12). Follow- ing the rule that any alloy whose value deviates from one (1) is anisotropic, the alloys are all anisotropic, having values of 0.69, 1.18, and 1.09 for CoNbSn, IrNbSn, and RhNbSn alloys, respectively. CoNbSn deviates most from isotropy. The ratio of stress to strain as given by the young’s modu- lus (E) measures the material’s stiffness. A stiff material has a high young’s modulus meanwhile a ductile material has a low young’s modulus. E is derived using 9B G3B+G , where the parame- ters retain their usual meaning. The Poisson ratiov measures the absolute value of the ratio of transverse contraction strain to the longitudinal extension strain and is given as 3B−2G2(3B+G ) . Since no force or constraint was applied to the crystal sur- faces, the Poisson ratios predict the stability of the alloys. The Poisson’s ratio can also be used as an indicator in bond sort- ing. For a typical covalently bonded compound, the value of the Poisson ratio should be lower than 0.25, while that of a typical ionic compound is nearly 0.25 or more [55]. The val- ues of the ratio show that ionic bonding is preferred and that the materials are ductile. Pugh rule (B /G ) [56] defines the brittleness or ductility of a material. Pugh proposes that the critical value between brittleness and ductility is 1.75; below this value, the material is brittle, and above it, the material is ductile; results in Table 4 predicts the alloys to be ductile. The Cauchy’s pressure (Cp ) (C12 −C44) predicts that materi- als with negative values are brittle, while positive values are ductile. A material favouring covalent bonding has a negative Cauchy pressure while it is positive for the compounds with dominant ionic bonds [57]. The results in Table 4 shows that ionic bonding is favoured in the crystals. Considering hardness as a measure of the resistance that a lattice offers to the motion of dislocations, it becomes nec- essary to predict the hardness of these materials to enable correct recommendations for technological application. Some properties that predict the level of hardness of a crystal com- pound includes the microhardness parameter (H ), Kleinman’s parameter (ς), and the Lame’s coefficients (λ& µ). The parametersλ and µalso known as the Lame constants predict the hardness of a compound. λ, which is the first Lame constant, describes the possibility of compressibility in compounds, and the sec- ond Lame constant µdescribes the shear stiffness of compounds, these constants are obtained using λ = Eν(1+ν)(1−2ν) ,µ = E2(1+ν) as proposed by Matori et al. [58]. The result obtained for the Lame constants is presented in Table 5, and further predicts the alloys to be anisotropic since for isotropic systemsλ and µare expected to be equivalent to C12 and C44. Also, µit is expected to be equal to the shear modulus B; for all com- pounds, µit is in agreement with the shear modulus as ex- pected. Kleinman parameter ςshows the bending and stretch- ing capability of a compound; it is computed using C11+8C127C11+2C12 . The operational principle of the Kleinman capacity is that at a value approaching 1 (0), minimum bond stretching (bend- ing) is attained. The values reported in Table 5 shows that the alloys maintain a balance between bond stretching and bending. For a layered crystal, compounds are not expected to be isotropic; therefore, the anisotropy elasticity was calculated. We used the universal anisotropy index AU proposed by Ran- ganathan and Ostoja-Starzewski [59], which accounts for both compression and shear contributions 5 GVGR + BV BR − 6, GV , BV , 126 Gemanam et al. / J. Nig. Soc. Phys. Sci. 3 (2021) 121–130 127 GR , and BR are the bulk modulus and shear modulus of Voigt & Reuss approximations, respectively. If Au is not zero, the compound is not isotropic, and the extent of deviation from zero is a measure of the elastic anisotropy of the compound. The universal anisotropy indices are 0.16, 0.03, and 0.009 for CoNbSn, IrNbSn, and RhNbSn half Heusler alloys. In this case, CoNbSn alloy exhibits more anisotropic behaviour, while the others are almost negligible. Some authors have proposed re- lations (Vickers hardness and elastic moduli) for computing the microhardness (Hv ), also known as Vicker’s hardness of a cubic crystal solid; they can be predicted using the following relations: Hv = 0.1769G − 2.899 [60], Hv = 0.1475G [61], and Hv = 0.0607E [61]. 3.3. Thermodynamic Stability To correctly recommend materials for an experimental pro- cedure, it is essential to ascertain the thermodynamic stabil- ity. Hence, in this section, we present results and analyse the behaviour of the alloys at low and high temperatures. The cal- culations were done using the quasi-harmonic approxima- tion (QHA) rather than the harmonic approximation. The QHA is preferred because the harmonic approximation does not account for thermal expansion and thermal transport as tem- perature increases; it also does not take phonon interactions into account; hence the phonon lifetime is interpreted as in- finite. Effective implementation of the QHA accommodates the possible change in lattice parameter/volume with tem- perature. This addition is termed the X-factor in the vibra- tional Helmholtz energy, where X represents the lattice pa- rameter or the volume. The relation for the QHA from the vibrational Helmholtz energy (F v i b ) is given in Eqn. (5) as: F v i b (X , T ) = 1 2 ∑ −→ q ,υ }ω (−→ q ,υ, X ) +kB T ∑ −→ q ,υ I n [ 1 − e x p ( −}ω (−→ q ,υ, X ) kB T )] (5) Where q is the wave-vector, the vibrational frequency is ω(q), kB is the Boltzmann constant, T is temperature, } is the re- duced Planck’s constant, and υ is the frequency. We, however, do not consider the effect of pressure on the crystal. The pa- rameters analysed and reported include the Debye tempera- ture ΘD , specific heat capacity at constant volume Cυ ( its re- lation to the Dulong-Petit law), and the average mean veloc- ity. The Debye temperature provides information on the ther- mal properties of the material at ambient and high tempera- tures. Table 6 presents the longitudinal and transverse sound velocities and average sound velocity. The average sound ve- locity (υm ) can be calculated using the compressional /lon- gitudinal (Vl ) and shear/transverse (Vt ) sound velocities ob- tained from the modulus known as the Navier’s equation [62] as υm = [ 1 3 ( 2 V 3 l + 1 V 3t )]−1/3 (6) Figure 3: Heat capacity Cv of CoNbSn, IrNbSn, and RhNbSn hH alloy The longitudinal Vl and transverse Vt velocities can be ob- tained using Equations (7) and (8) Vl = [( B + 4 3 G ) 1 ρ ]1/2 (7) Vt = √ G ρ (8) The parameters in Eqn. (7) retain their usual meanings, ρ is the density of the alloy. The average sound velocity (Debye temperatures) of the alloys are 3.354 km/s (382.44 K), 2.757 km/s (300.93 K), and 2.792 km/s (306.47 K) for CoNbSn, IrNbSn, and RhNbSn alloy, respectively. At 0 GPa and constant vol- ume, the compound does not obey the Dulong Petit law, as shown in the heat capacity in Fig. 3. The heat capacity can be defined by the relation Cυ = ∂U∂T |V . The Dulong-Petit limit is achieved at a temperature of about 600 K and Cυ of 74 J mol −1K −1 in agreement with Einstein’s contribution to the heat theory at high temperature. The Debye temperature and heat ca- pacity behaviour with the Dulong-Petit law is consistent with other half Heusler compounds with similar atomic mass units. 3.4. Dynamical Stability and Phonon Dispersions The Lattice vibrations and behaviour of the phonon dis- persions contribute to establishing phase stability in crystalline solids. We have calculated the phonon dispersion and the vibration of the lattices with temperature increase using the density functional perturbation theory (DFPT) and QHA im- plemented in Quantum ESPRESSO. The calculations were per- formed on a 2×2×2 mesh in the first Brillouin zone in the re- ciprocal lattice space, and force constants in real space de- rived from the computations are used to interpolate between the q-points and to obtain the continuous branches of the phonon band structure [33]. We present the phonon disper- sion curves along Γ→X→K→Γ→L→W→X symmetry directions based on results from a 2 × 2 × 2 cubic supercell. The results compare well with Ref. [33] for the phonon dispersions of 127 Gemanam et al. / J. Nig. Soc. Phys. Sci. 3 (2021) 121–130 128 Table 3: Elastic constants for the Born elastic stability criteria of CoNbSn, IrNbSn, and RhNbSn half Heusler compound using the quasi-harmonic approxima- tion. Compound C11 C12 C44 (C11−C12) (C11+2C12) CoNbSn 273.50 301.663 4 82.58 81.173 4 66.27 110.073 4 190.92 438.66 IrNbSn 256.72 136.12 71.18 120.60 529.96 RhNbSn 191.54 88.25 56.37 103.29 368.04 Table 4: Computed values of Poisson ratio (v), bulk modulus (B in GPa), shear modulus (G in GPa), Pugh’s ratio (B/G), anisotropy (A), Young’s modulus (E in GPa), Cauchy’s pressure (CP i n G P a), density (ρ in g/cm 3 ), and Debye temperature (θ in K) using GGA-PBE compared with results from literature where available. Compound ν B G B /G θ A E H C P ρ CoNbSn 0.28 146.22 76.72 1.905 1.403 4 382.44 502.53 4 34357 0.69 0.9983 4 195.91 11.22 16.31 8.41 IrNbSn 0.33 176.32 66.61 2.64 300.93 33457 1.18 177.47 7.56 64.94 11.01 RhNbSn 0.31 122.68 54.43 2.25 306.47 39657 1.09 142.25 6.88 31.88 8.72 Table 5: Lame’s coefficients (λ and µ) in GPa, dimensionless Kleinman’s pa- rameter (ς), and the microhardness parameter (H) in GPa obtained using the relations in Ref. [60,61] separated by semicolons of CoNbSn, IrNbSn, and RhNbSn half Heusler alloys Compound λ µ ς Hv CoNbSn 97.39 76.52 0.45 10.67; 11.32; 11.89 IrNbSn 129.51 66.71 0.65 8.88; 9.82; 10.77 RhNbSn 88.58 54.29 0.59 6.72; 8.02; 8.63 Table 6: Results for the Voigt-Reuss-Hill average longitudinal and transverse sound velocities (Vl and Vt ) in km/s and average Debye sound velocity υm in km/s, of CoNbSn and IrNbSn, and RhNbSn alloys using QHA compound Vl Vt υm CoNbSn 5.4355.9873 4 3.0203.6183 4 3.354 IrNbSn 4.906 2.459 2.757 RhNbSn 4.731 2.498 2.792 CoNbSn alloy. However, up to now, there have no experimen- tal/other theoretical works exploring the lattice dynamics of the compound under zero pressure to compare with results for IrNbSn and RhNbSn alloys obtained in this work. The alloys investigated has three atoms in the unit cell; hence, the phonon dispersion spectra presented in Fig. 4(a- c) displays six (6) optical modes and three (3) acoustic modes with non-negative frequencies. The highest frequency gap between the acoustic modes and the optical mode is seen in RhNbSn, while the gap is almost non-existent in CoNbSn. The absence of soft (negative) frequencies further ascertains that the compounds are dynamically stable. The optical modes have frequencies 150 cm−1 and 250 cm−1 for the alloys: this put the alloys in the far-infrared region at wavelengths of about 30303 nm. The strongest dispersions occur at Γ both in the acoustic and optical modes, giving rise to one longitudinal optical (LO) phonon and two transverse optical (TO), we also observe a phonon splitting at Γ, and L. LO-TO splitting re- moves the degeneracy between the LO and TO phonons at the Brillouin zone centre. The LO-TO splitting is an essential parameter that aids in evaluating the strength of ionic bond- ing in a material. In the acoustic mode, the dispersion at Γ produces one longitudinal acoustic (LA) phonon and two transverse acoustic (TA) phonons at X and L. The d orbital of Co and Nb contributes to the dispersions both in the acous- tic mode and optical modes. This behaviour suggests a strong bonding between the atoms and the absence of rattling. 4. Conclusion We have investigated the structural, electronic, mechani- cal, lattice dynamics, and thermodynamic properties of XNbSn (X = Co, Ir, Rh) half Heusler alloys using the GGA-PBE density functional theory for structural and electronic properties and the density functional perturbation theory (DFPT) for the phonon and thermodynamic properties. Results for the lattice con- stants and elastic properties for CoNbSn are in reasonable agreement with results in existing literature. We also com- puted the formation energy for the materials; the negative formation energies support possible experimental simulation. The alloys are non-magnetic semiconductors. The elastic pa- rameters and the lattice dynamics suggest that the alloys are structurally, mechanically, and thermodynamically stable, as there are no negative frequencies in the phonon dispersions. The materials are ductile and ionic bonding is most favoured. The alloys comply with the Dulong-Petit law at heat capac- ities of 74 J/Kmol, and temperature 600 K. The phonon fre- quencies (250 cm−1) translate to the wavelength of the ma- terials being in the far-infrared region with a wavelength of about 30303 nm. IrNbSn/CoNbSn has the lowest/highest ΘD 128 Gemanam et al. / J. Nig. Soc. Phys. Sci. 3 (2021) 121–130 129 Figure 4: (a) Phonon dispersion of CoNbSn (b) Phonon dispersion of IrNbSn (c) Phonon dispersion of RhNbSn alloys using the density functional perturbation theory. of 300.93 K/382.44 K at zero pressure. This positions IrNbSn as a more promising material for thermoelectric applications. 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