BIBECHANA Vol. 21, No. 1, April 2024, 12–22 ISSN 2091-0762 (Print), 2382-5340 (Online) Journal homepage: http://nepjol.info/index.php/BIBECHANA Publisher: Dept. of Phys., Mahendra Morang A. M. Campus (Tribhuvan University)Biratnagar Lattice thermal conductivity in half-Heusler Model compounds XNiSn (X=Ti, Zr, Hf) using Slack’s model Prakash Khatri1,2, N. P. Adhikari1,∗ 1Central Department of Physics, Institute of Science and Technology, Tribhuvan University, Kirtipur, Kathmandu, Nepal 2Department of Physics, Siddhanath Science Campus, Mahendranagar, Tribhuvan University, Nepal ∗Corresponding author. Email: narayan.adhikari@cdp.tu.edu.np Abstract Reducing lattice thermal conductivity (κl) that reflects the material’s heat-carrying capacity through lattice phonon vibrations is crucial for optimizing the figure of merit (zT). Using density functional theory (DFT) and density functional perturbation theory (DFPT), present work considers the structural, electronic, magnetic, and phonon properties of the XNiSn (X=Ti, Zr, Hf) half Heusler (hH) compounds. TiNiSn, ZrNiSn, and HfNiSn are identified as semiconductors with indirect band gaps of 0.43 eV, 0.47 eV and 0.39 eV, respectively, ex- hibiting dynamical stability. The temperature- dependent κl of hH XNiSn are compared using Slack’s model with two approaches for analyzing phonon velocity: elastic constants from quasi- harmonic approximation (QHA) as implemented in the Thermo_pw code and slope of phonon bands based on DFPT. At room temperature, TiNiSn, ZrNiSn and HfNiSn have κl values of 28.61 Wm−1K−1, 34.61 Wm−1K−1 and 29.85 Wm−1K−1 respectively using phonon velocity from slopes of phonon bands based on DFPT. These values show deviation of 1.48%, 6.29%, and 5.82% to those κl values 29.01 Wm−1K−1’, 36.79 Wm−1K−1 and 31.59 Wm−1K−1 for TiNiSn, ZrNiSn and HfNiSn respectively using QHA. Keywords Density functional theory, Lattice thermal conductivity, Slack’s model, Phonon bands. Article information Manuscript received: September 16, 2023; Revised: December 5, 2023; Accepted: December 16, 2023 DOI https://doi.org/10.3126/bibechana.v21i1.58405 This work is licensed under the Creative Commons CC BY-NC License. https://creativecommons. org/licenses/by-nc/4.0/ 12 http://nepjol.info/index.php/BIBECHANA narayan.adhikari@cdp.tu.edu.np https://doi.org/10.3126/bibechana.v21i1.58405 https://creativecommons.org/licenses/by-nc/4.0/ https://creativecommons.org/licenses/by-nc/4.0/ Prakash Khatri and N. P. Adhikari/ BIBECHANA 21 (2024) 12-22 13 1 Introduction The rising energy demands but limited fossil fu- els have led to a worldwide energy crisis. Waste heat resulting from industrial processes, residen- tial heating and vehicle exhaust affects the environ- ment. The potential of thermoelectric (TE) mate- rials to utilize such waste heat energy can play a vital role on the reduction of fossil fuel usage and global greenhouse emissions [1–3]. The thermoelec- tric figure of merit (zT) is employed as a means to quantify the conversion efficiency of thermoelectric materials. It is expressed as [4], zT = s2σT κ (1) It depends on the material’s transport coefficients, including the Seebeck coefficient (S), electrical conductivity(σ), absolute temperature (T) and to- tal thermal conductivity κ = κele +κl, which is the sum of electronic component (κele) and lattice com- ponent of total thermal conductivity (κl). Thermal conductivity (κ) is crucial for various technologi- cal advancements, including thermoelectricity and heat management applications [5,6]. The transport coefficients S, σ and κele are interdependent due to the influence of carrier concentration and effec- tive mass. The interdependence between the Kele and σ is established by the Wiedemann-Franz law (κele = σLT). A zT in TE materials corresponds to a high power factor (PF = S2σ) and a low κl. Strong electron-phonon coupling in bulk materials like half-Heusler (hH) alloys imposes restrictions on simultaneous optimization of PF and reduction of K [7,8]. Nowadays the primary focus is on reducing κl since efforts to manipulate κele are ineffective due to its direct correlation with σ. It is worth noting that the reduction of κl has shown limited impact on the PF. As a result, reducing κl independently of the parameters S, σ , and κele becomes crucial in optimizing zT value [9]. The lattice thermal con- ductivity of solid [10], κl = 1/3Cvvsl, is directly influenced by the phonon velocity (vs) and mean free path (MFP) (l). By increasing the scattering of phonons, the heat flux can be reduced hindering MFP and resulting in lower κl [11]. Multiple meth- ods like grain boundary scatterings, point defects, dislocations, impurity scattering [12], nanostructur- ing [13], alloy scattering [14] leading anharmonicity reduce κl by scattering phonons. The dominant scattering mechanism in a material is temperature dependent. At low temperatures, grain boundary scattering are significant whereas near to Debye temperature, defects act as scattering centers and above room temperature phonon-phonon scatter- ing dominates [15, 16]. Phonon Boltzmann trans- port equation (PBTE) [17] involving large num- bers of linear equations are challenging to solve for κl due to high dimensionality and complex- ity of determining reliable II and III-order force constants based on atomic interactions [18]. It- erative solutions based on DFT calculations, such as ShengBTE software [19], temperature gradient molecular dynamics (MD) simulations based soft- ware tools like LAMMPS [20] require a high com- putational cost [20]. The Debye-Callaway model offers a method to estimate κl using Debye temper- ature (θD), phonon velocity (vs), and the Grüneisen parameter (Υ) [21]. Slack developed a model [22] (Equation 3) to express the κl in semiconductors, considering the significant role of acoustic modes in heat transfer. Half-Heusler compounds display a wide range of thermal conductivities [7]. The presence of symme- try in hHs contributes to a decreased computational cost in comparison to more intricate systems like layered compounds or those with distorted symme- tries [23]. The κl of the 18 valence electron system XNiSn (X=Ti, Zr, Hf) has been reported in many experimental and computational studies [8, 24–28]. Instead of computationally expensive models, the Slack model is more efficient and feasible to predict the lattice thermal conductivity (κl) and provid- ing insights into thermal transport. To our best knowledge, a comprehensive comparison of the κl values for these compounds using Slack’s model is still not available. Such a comparison could be ben- eficial in the field of high-throughput computations to seek for the materials with ultralow or ultrahigh κl efficiently. It could eliminate the need for exten- sive efforts in calculating the Debye temperature and Grüneisen parameter, as required by alterna- tive methods. In the present work, our primary focus is on calculating κl of hH XNiSn (X=Ti, Zr, Hf) using Slack’s equation. We obtained the re- quired parameters for the equation from two dif- ferent approaches: from elastic constants by QHA in thermo_pw code [29] implemented in Quantum ESPRESSO and the slopes of acoustical branches of phonon bands using DFPT [30]. The paper is structured as follows: second sec- tion discusses the computational details, the third section focuses on the main results and discussion and the last section presents the conclusions and concluding remarks of present work. 2 Computational Details We used DFT [31] in Quantum ESPRESSO (ver- sion 7.1) [32] with a plane wave basis set and gener- alized gradient approximation (GGA) method with Perdew, Burke and Ernzerhof (PBE) exchange- correlation functional [33]. The RRJK model [34] of ultrasoft pseudopotentials was used for calculations on structural optimization, electronic and magnetic properties. We incorporated a ki- Prakash Khatri and N. P. Adhikari/ BIBECHANA 21 (2024) 12-22 14 netic energy cutoff of 70 Ry and a charge den- sity cutoff of 700 Ry with Brillouin zone inte- gration utilized the Monkhorst–Pack scheme [35] with a 10 × 10 × 10 k-mesh in the first irre- ducible Brillouin zone. The "david" diagonaliza- tion method with a mixing factor of 0.3 is em- ployed in the self-consistent field (SCF) calcula- tions, with a convergence threshold of 10−6 Ry. For band structure calculations, k-points were chosen along Γ→X→W→K→Γ→L→U→W→L→K, high symmetry path (Figure 1(c)), in the first Brillouin zone of reciprocal lattice. The density of states and partial density of states were computed through non-self consistent field (nscf) calculations with a denser mesh of 20 × 20 × 20 k-points. Marzarri- Vanderbilt smearing [36] of broadening width of 0.005 Ry was employed in all calculations. The compounds’ dynamical stability was checked via phonon dispersion calculations using density func- tional perturbation theory (DFPT) with a 2 × 2 × 2 q-point mesh by computing second-order derivative of the total energy with respect to atomic displace- ments. The Slack’s equation was used for calcu- lating κl with velocity of phonon determined using two different techniques. The first method involved the utilization the elastic parameters obtained from QHA implemented in thermo_pw codes. On the other hand, the second approach employed the slopes of the longitudinal and transverse compo- nents of the acoustic branches in the phonon dis- persion curve. 3 Results and Discussion 3.1 Structural Properties The XNiSn (X=Ti, Zr, Hf) half-Heusler com- pounds are extensively studied ternary inter- metallics,having an edge-centered cubic structure with a fundamental composition of 1:1:1. The structure has the specific symmetry of F 4̄ 3m with X element is the most electropositive. Inside the unit cell of these compounds, three atoms (X, Ni, and Sn) are situated at specific in-equivalent Wyck- off positions: X at 4a (0, 0, 0), Ni at 4b (0.5, 0.5, 0.5), and Sn at 4c (0.25, 0.25, 0.25) as shown in figure 1(a) and 1(b). We have calculated the optimized lattice param- eters of XNiSn (X=Ti, Zr, Hf) by minimizing to- tal energy of crystal with Murnaghan’s equation of state [37], E(V ) = E0 + BV B′ [ 1 B′ − 1 ( V0 V )B′ + 1 ] (2) In the equation B is bulk modulus, B’ is first pres- sure derivative of the bulk modulus that corre- sponds to pressure of P=0, E0 is the equilibrium energy and V0 is the unit-cell volume that corre- sponds to zero pressure. The optimized lattice constant represents the minimum ground state energy, signifying the crys- tal’s most stable configuration. Figure 2(a), 2(b) and 2(c) shows the total energy vs lattice param- eter for TiNiSn, ZrNiSn and HfNiSn respectively. The optimized lattice constant for TiNiSn, ZrNiSn and HfNiSn are 11.259 Bohr (5.958Å), 11.629 Bohr (6.154Å) and 11.552 Bohr (6.113Å) respectively. The HfNiSn compound exhibits smaller lattice pa- rameters compared to ZrNiSn and this difference can be attributed to the lanthanide contraction phe- nomenon. Table 1. presents the optimized lattice constants for XNiSn (X= Ti, Zr, and Hf) along with the previously reported experimental and computa- tional data. The deviation of our calculated lattice parameter values from that in experimental reports by Aliev et al. [38] are 0.63%, 0.71% and 0.76% for TiNiSn, ZrNiSn and HfNiSn respectively. The cal- culated lattice parameter values in the present work agree with the values in computational report by Öğüt, S., Rabe [39] within 1.36%-2.66%, J. Yang et al. [40] within 0.47%-0.66% and Muta et al. [26] within 0.47%-0.71%. All the variation in lattice pa- rameters are ≤ 2.66% and the variation could be attributed due to the choice of exchange-correlation functional utilized in the studies. Figure 1: (a) Conventional unit cell of XNiSn (X=Ti, Zr, Hf), (b) Primitive unit cell of XNiSn (X=Ti, Zr, Hf). The blue, red, and green colors in both figures represent X, Ni, and Sn, respectively visualized by XcrySDen. (c) Brillouin zone locating symmetric points. Prakash Khatri and N. P. Adhikari/ BIBECHANA 21 (2024) 12-22 15 Figure 2: Total energy (in Rydberg) VS lattice parameter (in Bohr) for (a) TiNiSn (b) ZrNiSn (c) HfNiSn. Table 1: Comparison of calculated lattice parameter (in Å) of XNiSn (X=Ti, Zr, Hf) with experimental and computational results in literature. Compound Our Work Aliev et al.[38] Öğüt, S., & Rabe[39] J. Yang et al.[40] Muta et al.[26] TiNiSn 5.958 5.920 5.814 5.921 5.930 ZrNiSn 6.154 6.110 6.070 6.113 6.110 HfNiSn 6.113 6.066 5.950 6.084 6.080 3.2 Electronic and Magnetic Properties We have studied the band structures of XNiSn (X=Ti, Zr, Hf) using GGA-PBE exchange- correlation functional as shown in figures 3(a), 3(b) and 3(c). From computed band structure of XNiSn compounds, all the compounds exhibit a semicon- ductor nature with an indirect band gap. The po- sition of the conduction band minimum (CBM) is at the X point of the Brillouin zone (BZ), while the valence band maximum (VBM) is at the Γ point of the BZ. The TiNiSn, ZrNiSn, and HfNiSn com- pounds exhibit band gaps (Eg) of 0.43 eV, 0.47 eV, and 0.39 eV, respectively. The calculated band gap values agree with the values in computational re- port by Öğüt, S., & Rabe [39] within 8.51%-23.07%, J. Yang et al. [40] within 0.00%-8.51% and Zou et al. [41] within 7.69%-10.64% as shown in Table 2. These deviations might be because of the different exchange correlation functional used in the calcula- tions in different studies. In general, DFT under- estimates the band gaps. However, in these com- pounds, the experimental band gap value reported in Aliev et al. [38] is smaller than the calculated value. This difference is because of the formation of an impurity band within the gap, which occurs due to the presence of Ni-interstitial atoms [42]. Density of states (DoS) and partial density of states (PDoS) plots revealing the distribution of electrons among atomic orbitals are crucial tools for understanding the electronic and magnetic proper- ties of materials. The main contribution to the DoS of XNiSn near the Fermi level arises from the 3d orbitals of Ni in the valence band (VB), while it comes from the 3d/4d/5d orbitals of Ti, Zr,and Hf, respectively in the conduction band (CB). The Sn- 5p orbitals play a minor role in both the VB and CB of the material. Symmetric and equal electron spin distribution in mentioned orbitals cancels magnetic moments, that explains the non-magnetic behavior in XNiSn compounds. The non-magnetic behav- ior in these 18 VEC hH is in agreement with the Slater-Pauling rule [43], which connects the mate- rial’s magnetic moment (M) to the number of va- lence electrons (Z) in the unit cell as M= (Z-18)µB . Prakash Khatri and N. P. Adhikari/ BIBECHANA 21 (2024) 12-22 16 Figure 3: The electronic band structure and PDoS plots of orbitals of constituent atoms (a) TiNiSn (b) ZrNiSn (c) HfNiSn with the Fermi level represented by a horizontal dotted line. Table 2: Comparison of band gap (Eg in eV) of XNiSn (X=Ti, Zr, Hf) with experimental and theoretical results reported in literatures. Compound Our Work Aliev et al.[38] Öğüt, S., & Rabe[39] J. Yang et al.[40] Zou et al.[41] TiNiSn 0.43 0.12 0.51 0.45 . . . ZrNiSn 0.47 0.18 0.51 0.51 0.52 HfNiSn 0.39 0.22 0.48 0.39 0.36 Figure 4: Phonon band diagram of (a) TiNiSn, (b) ZrNiSn and (c) HfNiSn compounds. Prakash Khatri and N. P. Adhikari/ BIBECHANA 21 (2024) 12-22 17 3.3 Phonon Spectrum and dynamical sta- bility This study presents the phonon dispersion of XNiSn (X=Ti, Zr, Hf) using DFPT as shown in figures 4(a), 4(b) and 4(c) respectively. The compounds were checked for dynamical stability using these curves. Stability is indicated by real frequencies, while instability is indicated by imaginary frequen- cies. Figure 4’s phonon dispersion curves confirms the dynamic stability of the compounds, as they lack imaginary phonon frequencies. With a primi- tive unit cell having 3 atoms, these compounds have three low-frequency acoustic modes and six high- frequency optical modes in their phonon dispersion curve. The equation vg = dω dk defines phonon group velocity, where optical dk branches exhibit a low group velocity due to their flat curvature, while acoustic branches show a high group velocity with a linear variation. Thus, the acoustic modes play a significant role in heat transport and κl, while the optical branch has a minor impact. There is a no- ticeable separation between the optical and acous- tic phonon modes, which decreases from TiNiSn to HfNiSn. The low κl might be achieved in the com- pounds having a small phonon gap or maximum overlapping because of enhanced phonon scattering [44]. The overlapping of phonon modes increases in the order of Ti < Zr < Hf, consequently strongly influencing the thermal transport of phonons in the same order. The maximum acoustic mode fre- quency for TiNiSn (156 cm−1), ZrNiSn (153 cm−1) and HfNiSn (140 cm−1). The low-energy acoustic phonons have limited energy transfer during heat transport, which ultimately leads to a low κl in these materials. Among these compounds, the low frequency for acoustic modes in the phonon is least for HfNiSn. 3.4 Estimation of lattice thermal conduc- tivity Slack’s equation [22] for calculating κl is, κl = V 1/3AMθ3D γ2Tn2/3 (3) The parameters in the equation are average atomic mass (M), volume per cell (V), number of atoms in the primitive unit cell (n), Grüneisen parameter (γ) and Debye temperature (θD). The Grüneisen parameter (γ) is a measure of the strength of lat- tice anharmonicity in a material that influences κl. It can be determined by considering the longitudi- nal (vl) and transverse (vt) velocities of the lattice vibrations. Our calculations are focused on deter- mining the γ through the analysis of elastic con- stants [45]. Grüneisen parameter (γ) is computed as, γ = 9− 12(vt/vl) 2 2 + 4(vt/vl)2 (4) The dimensionless constant A is computed as A = 2.43× 10−8 1− 0.514/γ + 0.228/γ2 (5) The vl and vt were calculated using two differ- ent techniques. The first method used the elas- tic parameters obtained from QHA implemented in thermo_pw codes. The thermo_pw algorithm calculates the various elastic constants and utilized them to derive various elastic properties, includ- ing bulk moduli, shear moduli and Young’s moduli, using the Voigt-Reuss-Hill approximation [46, 47]. These derived elastic properties were then employed to compute sound velocities. B = C11 + 2C12 3 (6) G = C11 − C12 + C13 5 + 5(C11 − C12) 3(C11 − C12 + 4C44) 2 (7) Y = 9BG 3B +G (8) The longitudinal and transverse velocity compo- nents of lattice vibration are, vt = √ G ρ (9) vl = √ 3B + 4G 3ρ (10) The average sound velocity, v = [ 1 3 ( 1 v3l + 2 v3t )]−1/3 (11) Another significant factor in computing κl is the Debye temperature θD, which considers the aver- age sound velocities. The θD corresponds to the state where all phonon modes in a material are fully excited and it plays a crucial role in γ and hence κl [47]. θD determined form the average sound ve- locities of acoustic phonon modes [46], θD = h̄ kB ( 6π2n V )2 ν (12) The second method involves the calculation of the vl and vt by analyzing the slopes of longitudinal and transverse components of the acoustic branches re- spectively in the phonon dispersion curves depicted in Figure 4(a), 4(b) and 4(c). When computing κl, the linear region near the Γ point in the acoustic branch of the phonon dispersion curve is selected Prakash Khatri and N. P. Adhikari/ BIBECHANA 21 (2024) 12-22 18 because it impacts the thermal transport signifi- cantly. The region is selected with care to ensure linearity. Table 3. presents the vl , vt, v, θD, γ and κl (at 300K) of XNiSn compounds. The κl of determined using velocity values from phonon slopes by DFPT and QHA model in thermo_pw codes, are TiN- iSn (28.61 Wm−1K−1 and 29.00 Wm−1K−1), Zr- NiSn (34.61 Wm−1K−1 and 36.79 Wm−1K−1) and HfNiSn( 29.85 Wm−1K−1 and 31.59 Wm−1K−1) re- spectively. The results obtained from both meth- ods show an agreement of 1.48%, 6.29%, and 5.82% for TiNiSn, ZrNiSn, and HfNiSn, respectively. The variation of κl calculated using both approaches for all compounds are as shown in Figure 5. Both ap- proaches calculate κl from elastic parameters but differ in the way of estimation. These approaches have varying accuracy in velocity estimations. For phonon velocity from phonon band slopes, the acoustic branches near the point in a specific and localized region of the phonon dispersion curve are analyzed. The another approach estimates the phonon velocities by calculatingthe macroscopic elastic parameters from elastic constants without explicitly considering individual phonon vibrational modes. Figure 5. demonstrates the temperature- dependent nature of κl. Our data shows the trend κl (TiNiSn) < κl (HfNiSn) < κl (ZrNiSn) through- out the considered temperature range. As the tem- perature increases, the κl decreases and becomes less temperature dependent at higher temperatures. This is because of the increased lattice scattering due to increased anharmonicity. The greatest dif- ference between values of κl from both approaches at room temperatures is for TiNiSn and ZrNiSn. The deviation in values of κl between TiNiSn and ZrNiSn is by 20.97% and 26.81% from the phonon band slope of DFPT and QHA model respectively. On increasing temperature the deviation in the val- ues of κl decreases. Despite having high κl at room temperature, these compounds exhibit low κl at ele- vated temperatures, making them suitable for high- temperature TE applications and can have better zT values. Our calculated values by both approach are compared to those reported in Quinn et al.[3] (Ref. [3])in table 3. The calculated values of κl in present work by DFPT approach agree within (1.36%- 8.06%) and values by QHA model agree within (0.13%-1.65%) to those values reported in Ref. [3]. The κl values reported in Ref. [3] used Debye Callaway’s model considering Umklapp scattering only. However, if grain boundaries, point defects, and electron-phonon interactions were accounted, κl values would be much lower. PHONO3PY, solving Boltzmann transport equation using the relaxation-time approximation (RTA-BTE), re- ports κl values < 10 Wm−1K−1 for these com- pounds by considering anharmonicity resulting from phonon-phonon scattering, mass disorder and grain boundary effects [48, 49]. The values κl are 14.5, 16.7 and 15.3 Wm−1K−1 calculated by RTA- PBE approach [24], 17, 18, 19 Wm−1K−1 by DFT and machine learning [23] and 16.8, 17.5 and 19.5 Wm−1K−1 by machine learning algorithm [25] for TiNiSn, ZrNiSn and HfNiSn respectively are lesser κl than our calculated values. Although our data agrees the trend with these literature but are highly overestimated. Slack’s model shows a similar ten- dency to overestimate κl values in comparison to ShengBTE for NbFeSb and TaFeSb. The κl for NbFeSb and TaFeSb at room temperature, as esti- mated by ShengBTE, are approximately 22 and 21 Wm−1K−1, while those predicted by Slack’s model are higher at 24 and 26 Wm−1K−1, respectively [45]. Slack’s model predicts higher κl values (12.69, 18.26, and 20.83) Wm−1K−1 for MCoBi (M = Ti, Zr, and Hf) in Ref. [50] compared to the values (11.02, 11.14, and 10.43) Wm−1K−1 in Ref. [51]. Slack’s method offers the main advantage of being computationally intensive, but gives only a rough estimate and lacks accuracy for κl. The main draw- back is that κl is indirectly derived from either elas- tic constants or acoustic phonons [45, 49]. Despite having limited heat transfer capabilities, modes re- lated to the optical branch can influence κl by in- teracting with acoustic vibrations. Slack’s method efficiently predicts both extremely low and high κl for compounds in high-throughput computations, particularly in the identification of potential ther- moelectric materials. The experimental κl values are significantly lower because even if the impurity concentrations are very low, the presence of small amounts of interstitial can cause significant disor- der, leading to a substantial reduction in κl [2, 26]. In order to achieve closer agreement between com- putational results and experimental values for κl, the consideration of phonon scattering is essential. Prakash Khatri and N. P. Adhikari/ BIBECHANA 21 (2024) 12-22 19 Table 3: Comparison of physical properties of XNiSn (X=Ti, Zr, Hf) computed using phonon band slopes in DFPT and QHA in thermo_pw codes, along with a comparison to Ref.[3] that reported κl values for defect-free compositions based on the Debye-Callaway model considering Umklapp scattering. Compound Approach vl (m/s) vt (m/s) v (m/s) θD (K) γ κl (Wm−1K−1) TiNiSn Slopes 6060.54 3358.89 3741.69 428.51 1.64 28.61 thermo_pw 5672.68 3125.60 3467.67 397.13 1.47 29.04 Ref.[3] 5426.00 3062.00 3406.00 392.00 1.58 29.00 ZrNiSn Slopes 6021.41 3380.13 3761.50 417.54 1.60 34.61 thermo_pw 5103.90 3069.61 3394.99 376.85 1.37 36.79 Ref.[3] 5498.00 3149.00 3498.00 390.00 1.53 37.40 HfNiSn Slopes 4900.15 2815.22 3126.77 348.65 1.51 29.85 thermo_pw 4674.76 2752.94 3049.43 339.99 1.43 31.59 Ref.[3] 4191.00 2503.00 2770.00 311.00 1.38 31.40 Figure 5: Variation of lattice thermal conductivity of XNiSn compounds with temperature using Slack’s model with velocity determined from phonon bands by DFPT and QHA model in thermo_pw suite. 4 Conclusions and Concluding Remarks The properties of XNiSn compounds, such as their structural, electronic, magnetic and phonon char- acteristics, are computed through DFT and DFPT. The calculated equilibrium lattice constants for TiNiSn, ZrNiSn and HfNiSn are 5.958Å, 6.154Å, 6.113Å respectively. Our study reveals that TiN- iSn, ZrNiSn and HfNiSn are non magnetic semi- conductors with indirect bandgap of 0.43 eV, 0.47 eV and 0.39 eV, respectively. The DoS near the Fermi level is mainly from 3d-Ni orbitals in VB and Ti, Zr, and Hf ’s 3d/4d/5d orbitals, respec- tively in CB with Sn-5p orbitals having minor role in both bands. The phonon dispersion confirms the dynamic stability of the systems. The overlapping optical and acoustical phonon modes increase from TiNiSn to HfNiSn. The κl variation with tempera- ture was assessed using Slack’s model, with phonon velocity analyzed with QHA model and the slope of phonon bands by DFPT. The κl has observed to decrease as the temperature increases because of phonon scattering. The room temperature κl val- ues of 28.61 Wm−1K−1, 34.61 Wm−1K−1 and 29.85 Wm−1K−1 for TiNiSn, ZrNiSn and HfNiSn respec- tively using DFPT follow same trend to those val- ues 29.01 Wm−1K−1, 36.79 Wm−1K−1 and 31.59 Wm−1K−1 respectively by QHA model. The find- ings obtained through both approaches exhibit an agreement of 1.48%, 6.29%, and 5.82% for TiN- iSn, ZrNiSn and HfNiSn, respectively. The values in present study are overestimated in comparison to prior methods. The scattering effects are to be included for matching the computational and ex- Prakash Khatri and N. P. Adhikari/ BIBECHANA 21 (2024) 12-22 20 perimental values for κl. Although the compounds have high κl at low temperatures, the decreasing values of them at elevated temperatures make them promising for high-temperature TE applications. Acknowledgements PK acknowledges the PhD grants award no. PhD-79/80-S&T-14 of UGC Nepal and NT-14 of ICTP/EAU for partial support. NPA acknowledges TU-NPAR-077/78 ERG14. References [1] G. J. Snyder and E. S. Toberer. Complex thermoelectric materials. Nature Materials, 7(2):105–114, 2008. [2] H. J. Goldsmid. Introduction to Thermoelec- tricity, volume 121. Springer, 2016. [3] R. J. Quinn and J. W. G. Bos. Advances in half-heusler alloys for thermoelectric power generation. Materials Advances, 2(19):6246– 6266, 2021. [4] G. D. Mahan and J. O. Sofo. The best thermo- electric. Proceedings of the National Academy of Sciences, 93(15):7436–7439, 1996. [5] X. Zhang and L. D. Zhao. Thermoelectric ma- terials: Energy conversion between heat and electricity. Journal of Materiomics, 1(2):92– 105, 2015. [6] Y. Pei, X. Shi, A. LaLonde, H. Wang, L. Chen, and G. J. Snyder. Convergence of elec- tronic bands for high performance bulk ther- moelectrics. Nature, 473(7345):66–69, 2011. [7] Z. Feng, Y. Fu, Y. Zhang, and D. J. Singh. Characterization of rattling in relation to ther- mal conductivity: Ordered half-heusler semi- conductors. Physical Review B, 101(6):064301, 2020. [8] G. Ding, G. Y. Gao, and K. L. Yao. Ex- amining the thermal conductivity of the half- heusler alloy tinisn by first-principles calcula- tions. Journal of Physics D: Applied Physics, 48(23):235302, 2015. [9] Y. Zhang. First-principles debye–callaway ap- proach to lattice thermal conductivity. Journal of Materiomics, 2(3):237–247, 2016. [10] J. S. Dugdale and D. K. C. MacDonald. Lat- tice thermal conductivity. Physical Review, 98(6):1751, 1955. [11] H. Kim, G. Park, S. Park, and W. Kim. Strategies for manipulating phonon transport in solids. ACS nano, 15(2):2182–2196, 2021. [12] P. Bagheri, P. Reddy, J. H. Kim, R. Rounds, T. Sochacki, R. Kirste, and Z. Sitar. Impact of impurity-based phonon resonant scattering on thermal conductivity of single crystalline gan. Applied Physics Letters, 117(8), 2020. [13] W. Kim. Strategies for engineering phonon transport in thermoelectrics. Journal of Ma- terials Chemistry C, 3(40):10336–10348, 2015. [14] H. Xie, H. Wang, Y. Pei, C. Fu, X. Liu, G. J. Snyder, and T. Zhu. Beneficial contribution of alloy disorder to electron and phonon trans- port in half-heusler thermoelectric materials. Advanced Functional Materials, 23(41):5123– 5130, 2013. [15] S. H. Zaferani, R. Ghomashchi, and D. Vashaee. Strategies for engineering phonon transport in heusler thermoelectric compounds. Renewable and Sustainable Energy Reviews, 112:158–169, 2019. [16] R. He, T. Zhu, Y. Wang, U. Wolff, J. C. Jaud, A. Sotnikov, and G. Schierning. Unveiling the phonon scattering mechanisms in half-heusler thermoelectric compounds. Energy & Environ- mental Science, 13(12):5165–5176, 2020. [17] J. M. Ziman. Electrons and Phonons: The Theory of Transport Phenomena in Solids. Clarendon, Oxford, 1960. [18] J. J. Gutiérrez Moreno, J. Cao, M. Fronzi, and M. H. N. Assadi. A review of recent progress in thermoelectric materials through computa- tional methods. Materials for Renewable and Sustainable Energy, 9:1–22, 2020. [19] W. Li, J. Carrete, N. A. Katcho, and N. Mingo. Shengbte: A solver of the boltzmann transport equation for phonons. Computer Physics Com- munications, 185(6):1747–1758, 2014. [20] A. J. McGaughey and M. Kaviany. Phonon transport in molecular dynamics simulations: formulation and thermal conductivity predic- tion. Advances in heat transfer, 39:169–255, 2006. [21] A. Petersen, S. Bhattacharya, T. M. Tritt, and S. J. Poon. Critical analysis of lattice thermal conductivity of half-heusler alloys using vari- ations of callaway model. Journal of Applied Physics, 117(3), 2015. [22] D. T. Morelli and G. A. Slack. High lattice thermal conductivity solids. High thermal con- ductivity materials, pages 37–68, 2006. Prakash Khatri and N. P. Adhikari/ BIBECHANA 21 (2024) 12-22 21 [23] R. Tranås, O. M. Løvvik, O. Tomic, and K. Berland. Lattice thermal conductivity of half-heuslers with density functional theory and machine learning: Enhancing predictiv- ity by active sampling with principal compo- nent analysis. Computational Materials Sci- ence, 202:110938, 2022. [24] S. N. H. Eliassen. A first principles study of lattice thermal conductivity in xnisn (x= ti, zr, hf) half-heusler alloys for thermoelectric appli- cations, 2016. [25] J. Carrete, W. Li, N. Mingo, S. Wang, and S. Curtarolo. Finding unprecedentedly low- thermal-conductivity half-heusler semiconduc- tors via high-throughput materials modeling. Physical Review X, 4(1):011019, 2014. [26] H. Muta, T. Yamaguchi, K. Kurosaki, and S. Yamanaka. Thermoelectric properties of zr- nisn based half heusler compounds. In ICT 2005. 24th International Conference on Ther- moelectrics, pages 351–354, 2005. [27] Q. Shen, L. Zhang, L. Chen, T. Goto, and T. Hirai. Thermoelectric properties of zrnisn- based half-heusler compounds by solid state re- action method. Journal of materials science letters, 20(24):2197–2199, 2001. [28] Y. Kimura and Y. W. Chai. Ordered structures and thermoelectric properties of mnisn (m= ti, zr, hf)-based half-heusler compounds affected by close relationship with heusler compounds. Jom, 67:233–245, 2015. [29] P. O. Adebambo, B. I. Adetunji, O. T. Uto, S. Kenmoe, and G. A. Adebayo. First- principles calculations of the phonon, me- chanical and thermoelectric properties of half- heusler alloy virsi alloys. Crystals, 12(12):1838, 2022. [30] S. Baroni, S. De Gironcoli, A. Dal Corso, and P. Giannozzi. Phonons and related crys- tal properties from density-functional pertur- bation theory. Reviews of modern Physics, 73(2):515, 2001. [31] P. Hohenberg and W. Kohn. Inhomogeneous electron gas. Physical review, 136(3B):B864, 1964. [32] P. Giannozzi, S. Baroni, N. Bonini, M. Calan- dra, R. Car, C. Cavazzoni, and R. M. Wentz- covitch. Quantum espresso: a modular and open-source software project for quantum sim- ulations of materials. Journal of physics: Con- densed matter, 21(39):395502, 2009. [33] J. P. Perdew, K. Burke, and M. Ernzerhof. Generalized gradient approximation made sim- ple. Physical review letters, 77(18):3865, 1996. [34] S. Grimme. Accurate description of van der waals complexes by density functional the- ory including empirical corrections. Journal of computational chemistry, 25(12):1463–1473, 2004. [35] H. J. Monkhorst and J. D. Pack. Special points for brillouin-zone integrations. Physical review B, 13(12):5188, 1976. [36] N. Marzari, D. Vanderbilt, A. De Vita, and M. C. Payne. Thermal contraction and disor- dering of the al (110) surface. Physical Review Letters, 82(16):3296, 1999. [37] F. D. Murnaghan. The compressibility of me- dia under extreme pressures. Proceedings of the National Academy of Sciences, 30(9):244–247, 1944. [38] F. G. Aliev, N. B. Brandt, V. V. Moshchalkov, V. V. Kozyrkov, R. V. Skolozdra, and A. I. Belogorokhov. Gap at the fermi level in the intermetallic vacancy system rbisn (r= ti, zr, hf). Zeitschrift für Physik B Condensed Mat- ter, 75:167–171, 1989. [39] S. Öğüt and K. M. Rabe. Band gap and sta- bility in the ternary intermetallic compounds nisnm (m= ti, zr, hf): A first-principles study. Physical Review B, 51(16):10443, 1995. [40] J. Yang, H. Li, T. Wu, W. Zhang, L. Chen, and J. Yang. Evaluation of half-heusler compounds as thermoelectric materials based on the calcu- lated electrical transport properties. Advanced Functional Materials, 18(19):2880–2888, 2008. [41] D. F. Zou, S. H. Xie, Y. Y. Liu, J. G. Lin, and J. Y. Li. Electronic structure and thermoelec- tric properties of half-heusler zr0.5hf0.5nisn by first-principles calculations. Journal of Applied Physics, 113(19), 2013. [42] J. Zhang, X. Zhang, and Y. Wang. Hf/sb co-doping induced high thermoelectric perfor- mance of zrnisn: First-principles calculation. Scientific Reports, 7(1):14590, 2017. [43] W. Y. S. Lim, D. Zhang, S. S. F. Duran, X. Y. Tan, C. K. I. Tan, J. Xu, and A. Suwardi. A systematic approach for semiconductor half- heusler. Frontiers in Materials, 8:745698, 2021. [44] A. J. Hong, L. Li, R. He, J. J. Gong, Z. B. Yan, K. F. Wang, and Z. F. Ren. Full-scale compu- tation for all the thermoelectric property pa- rameters of half-heusler compounds. Scientific Reports, 6(1):22778, 2016. Prakash Khatri and N. P. Adhikari/ BIBECHANA 21 (2024) 12-22 22 [45] G. A. Naydenov, P. J. Hasnip, V. K. Lazarov, and M. I. J. Probert. Huge power factor in p-type half-heusler alloys nbfesb and tafesb. Journal of Physics: Materials, 2(3):035002, 2019. [46] N. Soga and O. L. Anderson. Simplified method for calculating elastic moduli of ce- ramic powder from compressibility and debye temperature data. Journal of the American Ceramic Society, 49(6):318–322, 1966. [47] C. Malica and A. Dal Corso. Quasi-harmonic temperature dependent elastic constants: ap- plications to silicon, aluminum, and sil- ver. Journal of Physics: Condensed Matter, 32(31):315902, 2020. [48] M. Schrade, K. Berland, S. N. Eliassen, M. N. Guzik, C. Echevarria-Bonet, M. H. Sørby, and T. G. Finstad. The role of grain boundary scat- tering in reducing the thermal conductivity of polycrystalline xnisn (x= hf, zr, ti) half-heusler alloys. Scientific Reports, 7(1):13760, 2017. [49] G. Qin, A. Huang, Y. Liu, H. Wang, Z. Qin, X. Jiang, and M. Hu. High-throughput compu- tational evaluation of lattice thermal conduc- tivity using an optimized slack model. Materi- als Advances, 3(17):6826–6830, 2022. [50] H. Ma, C. L. Yang, M. S. Wang, X. G. Ma, and Y. G. Yi. Effect of m elements (m= ti, zr, and hf) on thermoelectric performance of the half- heusler compounds mcobi. Journal of Physics D: Applied Physics, 52(25):255501, 2019. [51] C. Toher, J. J. Plata, O. Levy, M. De Jong, M. Asta, M. B. Nardelli, and S. Curtarolo. High-throughput computational screening of thermal conductivity, debye temperature, and grüneisen parameter using a quasiharmonic de- bye model. Physical Review B, 90(17):174107, 2014. Introduction Computational Details Results and Discussion Structural Properties Electronic and Magnetic Properties Phonon Spectrum and dynamical stability Estimation of lattice thermal conductivity Conclusions and Concluding Remarks