Theoretical exploration of inherent electronic, structural, mechanical, thermoelectric, and thermophysical response of KRu4Z12 (Z = As12, Sb12) filled skutterudite materials

Using the density functional theory methodology, we have thoroughly examined KRu4As12 and KRu4Sb12 skutterudites, including their structural, electronic, mechanical, transport, and thermodynamic properties. First and foremost, using the Birch–Murnaghan equation of state, the structural stability has been calculated in terms of their total ground state and cohesive energies. With the use of the approximation approaches GGA and GGA + mBJ, the electrical structure and density of the states reveal their metallic nature. This demonstration predicts the dominant ferromagnetic spin configuration of materials by considering their electronic behavior and magnetic interactions. The ductile behavior of these alloys is also addressed by their mechanical qualities, which indicate how they might be used in engineering and industrial settings. Moreover, the semi-classical Boltzmann transport theory has been employed to examine the Seebeck coefficient as well as the electric and thermal conductivities. The general tendency of these compounds demonstrates their various potential uses as electrode materials. The quasi-harmonic Debye approximation is a method used to analyze the stability of a system under high pressures and accounts for the temperature dependency of thermodynamics. It combines the quasi-harmonic approximation, which considers the anharmonicity of vibrations, with the Debye model, which describes the vibrational modes of a solid. This approach allows for a more accurate representation of the system's behavior at different temperatures and pressures. By implementing this approximation, researchers can gain insights into the stability and thermodynamic properties of materials under extreme conditions.


Introduction
CoSb 3 , with its thermal stability and high Seebeck coefficient at room temperature, emerges as an attractive candidate for midtemperature thermoelectric (TE) applications.CoSb 3 is a compound composed of cobalt (Co) and antimony (Sb) elements, considered less toxic compared to other compounds and is relatively abundant in nature. 1 Unlled and doped thermoelectric (TE) skutterudites have garnered considerable attention, especially in the last decade, offering advantages like low cost due to the absence of rare earths, ease of processing, and the ability to be synthesized as both n-and p-type TE materials. 2In recent decades, there has been renewed interest in developing novel enhanced thermoelectric substances for cooling and power generation applications, driven by consumption habits and growing energy demand.Reports indicate that nearly 66% of primary energy is wasted as heat, with only 33% being used for actual work. 3Thermoelectric materials provide a solution by directly converting waste heat into electricity through the Seebeck effect.Solid-state thermoelectric devices, acting as thermoelectric generators, can generate a voltage potential by applying a temperature difference, or as refrigerators, functioning as micro-coolers.
Thermoelectric generators (TEGs) can convert waste heat generated by various sources, such as solar irradiation, heat generated in car exhaust, or industrial processes, into useable electricity.Moreover, these thermoelectric materials retain mechanical stability, high reliability, long-term equipment, and low environmental impact, offering practical advantages without the need for moving parts. 4,5Similarly, skutterudite compounds represent a promising class of materials with excellent thermoelectric properties at high temperatures.In thermoelectric materials, the gure of merit (ZT) is a key parameter that determines performance.It is calculated using the equation (ZT = S 2 sT/k), where S represents the Seebeck coefficient, s is the electrical conductivity, k denotes the thermal conductivity, and T is the absolute temperature.These properties are interconnected and play a crucial role in determining the efficiency of thermoelectric materials.Researchers aim to enhance the performance of thermoelectric materials for various applications such as waste heat recovery and solid-state cooling by optimizing these factors.For a material to exhibit good thermoelectric performance, it must satisfy several requirements.Not only should its Seebeck coefficient and electrical conductivity be high, but it should also possess a lower thermal conductivity (k) value.A material's thermal conductivity (k) is determined by the sum of electronic (k e ) and lattice (k l ) components.Low k and high S are required to achieve high ZT.Despite this, the band structure and scattering mechanisms are tightly coupled and mutually constrain these parameters, making it challenging to separate the electrical and thermal performance parameters.According to the Weidmann-Franz law, k e = LsT, where k e is closely related to s, while k l is unrelated to the electrical properties.][8][9][10][11][12][13] The efficiency of a thermoelectric (TE) device hinge signicantly on the choice of p-and n-type TE materials.Recent research has explored various TE materials, investigating their electrical and thermal properties. 14The article titled "vibrational and structural properties of the RFe 4 Sb 12 (R = Na, K, Ca, Sr, Ba) lled skutterudites" discusses the vibrational and structural characteristics of lled skutterudites with different R elements (Na, K, Ca, Sr, Ba) through experimental and theoretical analysis. 15High spin polarization in the ferromagnetic lled skutterudites KFe 4 Sb 12 and NaFe 4 Sb 12 " explores the spin polarization properties of the ferromagnetic lled skutterudites KFe 4 Sb 12 and NaFe 4 Sb 12 , investigating their electronic structure and magnetic properties using experimental techniques and theoretical calculations.Both compounds exhibit high spin polarization, making them promising candidates for spintronic applications.16 "Insightful analysis of magneto-electronic, mechanical, and thermophysical properties of novel lled skutterudites LiFe 4 X 12 (X = As, Sb) through ab initio calculations" explores the properties of lled skutterudites LiFe 4 X 12 (X = As, Sb) using ab initio calculations, focusing on magnetoelectronic, mechanical, and thermophysical properties.17 "High-temperature electrical and thermal transport properties of fully lled skutterudites RFe 4 Sb 12 (R = Ca, Sr, Ba, La, Ce, Pr, Nd, Eu, and Yb)" by Qiu et al. 18 studied the high-temperature electrical and thermal transport properties of fully lled skutterudites RFe 4 Sb 12 (R = Ca, Sr, Ba, La, Ce, Pr, Nd, Eu, and Yb).The lled skutterudite materials NaFe 4 Sb 12 and KFe 4 Sb 12 exhibit itinerant electron ferromagnetism characterized by high spin polarization.19 Conversely, the alkaline-earth-lled skutterudites, specically AFe 4 Sb 12 (A = Ca, Sr, Ba), manifest properties of nearly ferromagnetic systems.[20][21][22][23]

Computational methodology
The present calculations used the density functional theory calculation implemented in the WIEN2k code. 24The code is highly reliable and precise, thus providing better accuracy of results.In the present set of calculations, we have performed the exact computation.The calculation process begins with the simple generalized gradient approximation (GGA) 25 to verify the intimate electronic structure and the density of states of KRu 4 As 12 and KRu 4 Sb 12 .The main drawback of GGA is that it underestimates the electronic structure, especially in systems containing d/f electrons.Therefore, in response to this issue, we have adopted a modied Becke-Johnson (mBJ) 26 to handle the exchange-correlation function.The calculations have been further extended by adopting the R MT K MAX = 6.0, which controls the interstitial atomic size.The term "R MT " represents the smallest muffin-tin radius, and "K MAX " represents the maximum reciprocal lattice vector used in the plane wave expansion.The potential and charge density non-spherical contributions to muffin-tin (MT) spheres were expanded to l max = 10, and the convergence criteria for energy and charge were set to 10 −4 Ry and 10 −4 eV.The tetrahedral method and a k-mesh of 1000 points in the Brillion zones (BZ) have been adopted.We have calculated the elastic properties by cubic elastic code. 27In addition to this, we have calculated the thermodynamic properties based on a quasi-harmonic Debye model using Gibbs2 code. 28The calculation of transport properties, such as the Seebeck coefficient (S), electrical conductivity (s), thermal conductivity (k), and power factor (PF), has been carried out by using the BoltzTraP code 29,30 under constant relaxation time approximation.

Structural stability
All the selected compounds of the family adopt the CoAs 3 type skutterudite structure, lled by alkaline earth metal atoms potassium (K), and based on one primitive cell per formula unit.In the early period, Oedal described the structure of CoAs 3 in 1928 (ref.31) as a cubic structure with 32 atoms corresponding to the Im 3 space group.However, the primitive unit cell contains seventeen atoms of three different types.The unit cell is composed of eight cubes of the transition metal (T = Ru) occupying the 8c sites (1/4, 1/4, 1/4), and 6 of these cubes are lled with square planar rectangles of the pnictogen (P n = As, Sb) occupying the 24g (0, y, z) sites.The generic chemical formula for the lled skutterudites is MT 4 X 12 .The "ller" atom M consists of Ca, Sr, Ba, Hf, La, Nd, and Sm.Additionally, the transition metals T can be Fe, Ru, Rh, Os, while the pnictogens consist of P, As, and Sb.The rare earth atoms are positioned at the origin with coordinates (0, 0, 0), the transition metal is positioned at (0.25, 0.25, 0.25), and the pnictogen is positioned at (0, 0.35, 0.16) with varying parameters depending on the chemical composition.In the unit cell, the remaining two voids (2a at (0, 0, 0) or 1/2, 1/2, 1/2) can be lled by atoms whose ionic radii are smaller than the cage.An alkaline atom resides in position (2a) (0, 0, 0), four T (Ru) atoms at position (8c) (1/4, 1/4, 1/4), and twelve (pnictogen) X atoms at position (24g) (0, y, z).In the [T 4 X 12 ] polyanion, y and z inner coordinates indicate the relative positions of X atoms.As a result, the structure of these alkaline-lled skutterudites is dened by one lattice parameter, a (nm), and two inner coordinates (y and z).It is commonly referred to as the half of the unit cell, T 4 Pn 12 , which has 88 valence electrons and is isoelectronic.The bond in the skutterudites is mainly covalent, as the distance between transition metal atoms is too great to form a bond.The interaction occurs between Pn-Pn (the pnictogen atom), forming Pn 4 rings, and T-Pn bonds (metallic atom utilizing the pnictogen).The valence electron arrangement of the pnictogen atoms is ns 2 np 3 , each with 5 electron bands. 32The calculated total energies are plotted in Fig. 2 as a function of the unit cell volume for KRu 4 Sb 12 and KRu 4 As 12 .We t the total energy versus unit cell volume curve to Murnaghan's equation of state (EOS) to optimize both the compounds with two methods, ferromagnetic (FM) and non-magnetic (NM) phases.In addition to this, specic ground state properties such as volume (nm 3 ), bulk modulus B 0 (in GPa), its pressure derivative B ′ , and the minimum energy at the equilibrium E 0 (in eV) parameters have been obtained and collected in Table 1.Fig. 1 shows the crystal structure of KRu 4 Sb 12 and KRu 4 As 12 .
The assessment of chemical stability and the potential experimental feasibility of the proposed KRu 4 Z 12 (Z = As, Sb) compound involves an evaluation of the formation energy per formula unit cell (E For ) and the cohesive energy (E Coh ) in the subsequent stage. 33The equations utilized for these computations are as follows: By employing the provided formula, the enthalpy of formation energy (denoted as DE) is utilized to evaluate the stability of the compounds. 34The calculation is represented as follows: here, E total signies the overall energy of KRu   the negative enthalpy of formation energy reinforces the stability of the compounds.

Electronic properties
The electronic properties of both KRu 4 As 12 and KRu 4 Sb 12 have been investigated by analyzing their band structures and density of states (DOS).However, a thorough discussion of the electronic properties of any material plays a signicant role in realizing its advantages in various research areas.In addition, the two-dimensional band structures and density of states (DOS) of the material provide essential information about the electrical properties of the material.In the present investigation, we have used GGA and GGA + mBJ approximation to predict the electronic properties.Fig. 3 and 4 show the spinpolarized band structures using the GGA and GGA + mBJ approximations.It can be observed that the energy bands in both the spin channels cross over the Fermi level (E F ), indicating the metallic character.Additionally, implementing the potential over GGA describes the comparative shi in energy states due to including potential over GGA + mBJ (Fig. 5).Further interpretation can be conducted by analyzing the density of states.The metallic character is also reected in the TDOS plots for both spin orientations of KRu 4 As 12 and KRu 4 Sb 12 .The geometric plotting in Fig. 6 and 7 illustrates that ruthenium (Ru) is more competent in describing electronic properties.The elements (arsenic, antimony) X-p states play a minor role in this conductor behavior but are primarily dominant in Ru-d orbitals.In the range of −5 to 5 eV, the Rud states and X-p states are dominant, and the Ru-d states strongly hybridize with the X-p states.Since potassium (K)-s states do not exist near the Fermi level, they provide minor information about the electronic structure.Below the Fermi level, the Ru-d states dominate, while above the Fermi level, the X-p states dominate. 46he charge density calculation is oen presented in a plane, where it tells us the direction of the charge transfer and nature of bonding in the material, namely, whether it is the ionic or covalent type of bond.The charged density plots are typically used to analyze electrons accumulating around atoms.It is challenging to analyze bonding if there is signicant charge accumulation between two atoms.Still, if the contour around each atom is not symmetric, there will be a complex type of interaction.Fig. 8    transition metals and Sb appears to be covalent.Therefore, the illustration of the electron charge density graph suggests that these skutterudite materials preserve both ionic and covalent bonds. 47

Mechanical properties
Mechanical properties are dened as a material's physical properties that are exhibited when forces are applied.These properties include the modulus of elasticity, tensile strength, hardness, and fatigue limit.These properties are important because they describe the material's compressibility, strength, ductility, brittleness, etc.In order to dene these properties, we evaluated the elastic constants of KRu 4 As 12 and KRu 4 Sb 12 using the cubic elastic code developed by Murtaza Jamal. 18The elastic parameters obtained from this evaluation are listed in Table 2. Due to the highly cubic symmetry of KRu 4 As 12 and KRu 4 Sb 12 , only three independent elastic constants are required, namely C 11 , C 12 , and C 44 .Using Born-Huang stability criteria condition: 48,49 C 11 -C 12 > 0; C 44 > 0; +2C 12 > 0 Furthermore, the mechanical properties of these skutterudites are affected by several additional parameters.Using the Voigt-Reuss-Hill scheme, one can determine the bulk modulus (B), shear modulus (G), and Young's modulus (Y) to predict the hardness, compressibility, and stiffness of a material.For the cubic system, the Voigt bounds for bulk modulus (B) and shear modulus (G) are calculated as: 50 However, Hill dened that any material's B and G values should be averaged by Voigt and Reuss limits. 51 Young's modulus (Y) determines a material's strength, a ratio of linear stress and strain.The Young modulus (Y) denes the material's stiffness, while B means resistance to volumetric deformation, and G means resistance to shape deformation.Table 2 presents the calculated values of B and G. 52 The Young's modulus and Poisson ratio can be determined through mathematical formulation: The calculated data in Table 2 show that Y > B > G indicates the stiffness of these skutterudite materials.We have calculated these mechanical parameters to dene the ductility and brittleness of these alloys.Parameters like Pugh's ratio (B/G), Poisson's ratios, and Cauchy pressure are used to quantify these properties. 53If the value of B/G is more signicant than 1.75, it   55 can be calculated from elastic constants using the formula illustrated: The calculated melting temperature of KRu 4 As 12 and KRu 4 Sb 12 are 1644.83K and 1592.62K, respectively.The Debye temperature is calculated using the average sound velocity (V m ).
here, h and k are Plank's and Boltzmann's constants, respectively, and are given as the average sound velocity.The presented V t and V l are transverse and longitudinal velocities calculated using Navier's equation. 52 s Both V t and V l present velocities differ along different planes, namely [100], [110], and [111].There is a maximum longitudinal velocity of 4232 m s −1 along the [100] direction and a maximum transverse velocity of 3445 m s −1 along the [110] plane. 56Table 3 presents the calculated values of V l , V m , V t , and Debye temperatures.

Thermoelectric properties
Boltzmann's theory has been applied to study the thermoelectric properties of KRu 4 As 12 and KRu 4 Sb 12 materials.8][59][60][61][62] These properties include electrical conductivity (s/k), electrical and thermal conductivity (k e ), Seebeck coefficient (S), and power factor (PF) under constant relaxation time approximations.The maximum conversion efficiency of the TE device is expressible as; 14 In thermoelectric (TE) systems, T h and T c denote the temperatures of the hot and cold sides, respectively, while ZT represents the gure-of-merit of the TE material.The conversion efficiency of TE devices improves with increasing ZT, making the development of both p-type and n-type materials with high ZT essential for creating efficient TE devices.It's important to note that besides ZT, the efficiency of thermoelectric generators (TEGs) also relies on selecting an optimal contact material.
3.4.1 Seebeck coefficient (S).The thermoelectric phenomenon in materials can be quantied using various parameters, among them the Seebeck coefficient, which is a vital parameter due to its voltage sensitivity for a given temperature gradient.Generally, a high Seebeck coefficient leads to good thermoelectric properties of materials.The Seebeck coefficient provides a sensitive measure of the electronic structure near Fermi energy.If the Seebeck coefficient is negative, it indicates that electrons are the majority carriers.The temperature dependence of S for the temperature range of 50 K to 800 K can be illustrated in Fig. 9(a).At room temperature (300 K); S acquires a value of 9 mV K −1 for KRu 4 As 12 and 14 mV K −1 for KRu 4 Sb 12 .At 800 K the value of S is 16 mV K −1 for KRu 4 As 12 and 15 mV K −1 for KRu 4 Sb 12 .
3.4.2Electrical conductivity (s/s).Electrical conductivity is also an important aspect of dening the transport properties.It is a measure of how easily charge carriers pass through a material and can be denoted as (s/s).The Wiedemann-Franz law is used to determine the conductivity.As shown in Fig. 9(b), the electrical conductivities of lled skutterudites KRu 4 As 12 and KRu 4 Sb 12 are temperature-dependent as illustrated in Fig. 9(b).The electrical conductivities over relaxation time (s/s) of the present investigated compounds decreased with increasing temperature.This means that the ller atoms contribute to the reduction of electrical resistivities.The electrical conductivity of these compounds is of the order of 10 20 U −1 m −1 s −1 .The calculated value of electrical conductivity at 50 K for KRu 4 As 12 is 2.36 × 10 20 U −1 m −1 s −1 and then decreased linearly to 2.31 × 10 20 U −1 m −1 s −1 at room temperature.As a result, the electrical conductivity decreased gradually until it reached a peak of 2.10      the static approximation was performed.Aerward, we applied Murnaghan's EOS to determine the structural parameters at zero pressure and temperature.Subsequently, we computed the macroscopic properties based on standard thermodynamic relations.Temperatures ranging between 0 and 900 K are used to determine the thermal properties.This study examines pressure effects ranging from 0 to 25 GPa.Specic heat capacity (C V ) helps understand heat movement in crystals, lattice vibrations, or phase transfer processes.A plot of the heat capacity C V versus temperature at 0 to 25 GPa is shown in Fig. 10.
From the gure, it is observed that C V values increase rapidly at low temperatures, then increase gradually at high temperatures, and follow Dulong Petit limits, 58 which are common characteristics for all solids at high temperatures.The increase in C V with temperature is primarily attributed to the rise in atomic vibrations, as shown in Fig. 10.Furthermore, the C V follows the Debye T3 law at the steep low-temperature slope. 65At high temperatures, C V approaches 409.617 kJ mol −1 K −1 and 400 kJ mol −1 K −1 for KRu 4 As 12 and KRu 4 Sb 12 , respectively.Based on the calculations at zero pressure and 300 K, the calculated value of C V are 310.08 kJ mol −1 K −1 for KRu 4 As 12 and 734.58 kJ mol −1 K −1 for KRu 4 Sb 12 materials.The effect of pressure and temperature on the Grüneisen parameter (g) has been studied and is shown in Fig. 11.The Grüneisen parameter (g) is an important thermodynamic property widely used to predict thermoelastic behavior in solids.It delivers the concept of anharmonicity in solids, and various physical properties such as bulk modulus, specic heat, and frequency of lattice vibrations are directly related to it.When examining the temperature rise, a slow increase in g can be observed, suggesting the presence of anharmonic effects.In these materials, the variation in temperature is noticeable with the application of the pressure, g decreases signicantly, as seen in the graphical plot. 66Fig. 12 shows our examination of the effect of temperature and pressure on thermal expansion, further contributing to our understanding of the materials' behavior.We nd that the thermal expansion coefficient increases as temperature increases for both compounds.At a given temperature, the thermal expansion coefficient a decreases rapidly as pressure increases and becomes smaller at higher temperatures and pressures.In a given pressure range, a increases rapidly with increasing temperature up to 150 degrees Celsius above the reference temperature.Above this threshold, a approaches linear behavior with further increases in temperature.Considering a pressure of zero and a temperature of 300 K, the thermal expansion valve is 1.63 × 10 −5 K −1 for KRu 4 As 12 and 2.20 × 10 −5 K −1 for KRu 4 Sb 12 .The crystals behave classically above this temperature because thermal vibrations become more signicant than quantum effects.The variation of the Debye temperature as a function of pressure and temperature is shown in Fig. 13.It can be seen that q D remains almost constant from 0 to 150 K.However, above this temperature, q D exhibits a regular and linear decrease with increasing temperature.For a given temperature, the Debye temperature increases almost linearly.The calculated value of q D at zero pressure and zero temperature is 554.68K for KRu 4 As 12 and 427.72 K for KRu 4 Sb 12 compounds.Table 5: the calculated value of thermal parameters at 300 K for different pressure range (0-25).

Conclusion
The structural, mechanical, electronic, thermal, and thermoelectric properties of lled skutterudite materials, specically KRu 4 As 12 and KRu 4 Sb 12 , were thoroughly investigated using the GGA and GGA + mBJ potentials.The calculated structural properties, including lattice parameter, bulk modulus, and energy, are compared with available theoretical data.The evaluated elastic parameters reect that these alloys are mechanically stable.The hardness of these compounds can be dened by considering the dependence of pressure on the elastic constant.The calculated data also revealed that KRu 4 As 12 behaves as a brittle material, while KRu 4 Sb 12 acts as a ductile material.By using spin-polarized band structure and DOS plots, KRu 4 As 12 and KRu4Sb12 are found to exhibit metallic behavior.Furthermore, the calculated thermal and thermoelectric parameters indicate that these materials are thermodynamically stable and show potential for thermoelectric applications.This represents a signicant nding, as it is the rst-ever observation of such behavior in these compounds.Consequently, the calculated values provide a promising route for experimentalists to synthesize these materials successfully.
Table 5 The calculated value of thermal parameters at 300 K for different pressure range (0-25)
4 X 12 (X = As, Sb), where K, Ru, and X are represented by E A , E B , and E X respectively.The computed energy values for KRu 4 X 12 (X = As, Sb) are −5.03Ry, and −4.46 Ry respectively.These values indicate that

Fig. 3
Fig. 3 These are the band structures of KRu 4 As 12 and KRu 4 Sb 12 within the schemes of the GGA method.The arrows indicating spin channels, representing spin-up ([) and spin-down (Y).

Fig. 4
Fig. 4 These are the band structures of KRu 4 As 12 and KRu 4 Sb 12 within the schemes of GGA and GGA + mBJ methods.The arrows indicating spin channels representing, spin-up ([) and spin-down (Y).

Fig. 5
Fig.5The total density of states of KRu 4 As 12 and KRu 4 Sb 12 via GGA and GGA + mBJ.
presents the two-dimensional (2-D) electron charge distributions for KRu 4 As 12 and KRu 4 Sb 12 in the (111) and (001) planes.The colour differences around the atoms clearly show charge shearing in both plots.From the interpretation of charges, the description of electronic charge density provides a better understanding of chemical bonding.However, it considers non-bonding states and delivers overall charge density inside the material.Compared with d-group elements, the transition metal exhibits the maximum contribution on the two-electron density plots.As the density plots for transition metals show, transition metals have the most signicant contribution, whereas p-group elements have the least.Typically, both the alloys KRu 4 As 12 and KRu 4 Sb 12 exhibit ionic bonding characteristics, while the bonding between the

× 10 20 U
−1 m −1 s −1 at 800 K for KRu 4 As 12 .At 50 K, the electrical conductivity of KRu 4 Sb 12 is 4.90 × 10 20 U −1 m −1 s −1 and reached 4.78× 10 20 U −1 m −1 s −1 at 300 K.At 800 K; the electrical conductivity decreased continuously up to 4.75 ×10 20 U −1 m −1 s −1 .3.4.3Electronic thermal conductivity (k e ).To calculate thermal conductivity, we sum electronic and lattice parts by k = k e + k l .Where k e and k l represent the electronic and vibration components, respectively.Only the electronic contribution of thermal conductivity has been considered in this study since BoltzTraP can only calculate the electronic part k e of thermal conductivity.In Fig. 9(c), we present the electronic thermal conductivity of the KRu 4 As 12 and KRu 4 Sb 12 metallic skutterudites.From the graphical plot of thermal conductivity, k e increases linearly with temperature mainly due to the thermal energy of the electrons.3.4.4Power factor (S 2 s).Power factor is a measure the thermoelectric efficiency of a compound.It is calculated using the equation PF = S 2 s, where S is the Seebeck coefficient, and s represents the material's electrical conductivity.The calculated power factors for the KRu 4 As 12 and KRu 4 Sb 12 skutterudite compounds are shown in Fig. 9(d).These materials demonstrate their suitability for high-temperature applications as the power factor increases with rising temperature.At the maximum, the power factor reaches 608.43 mW m −1 K −2 for KRu 4 As 12 and KRu 4 Sb 12 146.16mW m −1 K −2 .Calculation of different transport parameters at 300 K for KRu 4 As 12 and KRu 4 Sb 12 as represented in Table4.

Fig. 9
Fig. 9 Temperature-dependent properties of KRu 4 As 12 and KRu 4 Sb 12 materials.(a) Seebeck coefficient, (b) electrical conductivity, (c) electronic thermal conductivity and (d) power factor are plotted as functions of temperature for both materials.

Fig. 12
Fig. 12 The variation of thermal expansion with temperature and pressure for (a) KRu 4 As 12 (b) KRu 4 Sb 12 materials.

Table 1
3stimated lattice constant (a in nm), volume (V in nm3) bulk modulus (B in GPa), its pressure derivative B ′ , minimum energy (E 0 in eV), and the cohesive energy (E coh in eV per atom) for KRu 4 Sb 12 and KRu 4 As 12 compounds

Table 2
50lculations of different elastic parameters C 11 , C 12 , C 44 in GPa, bulk modulus (B in GPa), shear modulus (G in GPa), Young modulus (Y in GPa), Pugh's ratio (B/G), Cauchy pressure (C p in GPa), anisotropy factor (A), Poisson's ratio (n), melting temperature (T m in K) at 0 GPa and 0 K for KRu 4 As 12 and KRu 4 Sb 12 compounds.indicatesthatthematerialisductileorotherwise brittle.The B/ G (ref.53) value of KRu 4 Sb 12 is 1.17, which is less than 1.75, meaning it is brittle.The B/G value of KRu 4 As 12 is 2.57, which means ductile in nature.If the C p value is positive, these materials are ductile; otherwise, they are brittle.54Anegativevalue of C p supports the brittle nature of KRu4As 12 and a positive value of C p refers to the ductile nature of KRu 4 Sb 12 .We can also evaluate the Poisson's ratio (n); if it is below 0.33, it is brittle; otherwise, it is ductile.50TheKRu 4 Sb 12 presents a value of 0.16, indicating its brittleness, and KRu 4 As 12 presents a value of 0.31, its brittle one.The material has an isotropic response whenever the anisotropic factor (A) equals unity.Otherwise, the material shows an anisotropic response.The degree of anisotropy increases as the deviation from unity increases.The calculated values of the anisotropic factor (A) for KRu 4 As 12 and KRu 4 Sb 12 are 0.91 and 1.20, indicating that the materials are anisotropic.The melting temperature (T m ) © 2023 The Author(s).Published by the Royal Society of Chemistry RSC Adv., 2023, 13, 27873-27886 | 27879 Paper RSC Advances

Table 3
Velocities of directional elastic waves and the Debye temperature of KRu 4 As 12 and KRu 4 Sb 12 skutterudite

Table 4
Calculation of different transport parameters at 300 K for KRu 4 As 12 and KRu 4 Sb 12 Sb 12 (ref.17