Excellent Thermoelectric Performance in KBaTh (Th = Sb, Bi) Based Half-Heusler Compounds and Design of Actuator for Efficient and Sustainable Energy Harvesting Applications

Abstract

To examine the structural, optoelectronic, thermodynamic, and thermoelectric properties of KBaTh (Th = Sb, Bi) half-Heuslers, we used the full potential, linearized augmented plane wave (FP_LAPW) approach as in the Wien2K simulator. Generalized gradient approximation (GGA), technique, was used for the structural optimization. Mechanical stability and ductility were inherent characteristics of the studied KBaTh (Th = Sb, Bi). Having band gaps of 1.31 eV and 1.20 eV for the KBaTh (Th = Sb, Bi) compounds, they have a semiconducting character. The KBaTh (Th = Sb, Bi) compounds are suggested for use in optoelectronic devices based on studies of their optical characteristics. Thermoelectric properties were investigated using the Boltzmann transport provided by the BoltzTraP software. Since the acquired figures of merit (ZT) values for the KBaTh (Th = Sb, Bi) compounds are all almost equal to one at room temperature, this demonstrates that these substances can be used in thermoelectric devices. Additionally, we used the Slack method to determine the lattice thermal conductivity of KBaTh (Th = Sb, Bi). Our research shows that the half-Heusler compounds under investigation increase actuator response time and hence can be considered as good materials for actuators.

1. Introduction

In recent years, various techniques for converting energy have been the focus of extensive research due to the increasing need for sustainable energy sources. Thermoelectric materials have drawn increasing interest for their potential use in energy harvesting since they can directly transform waste heat into usable electricity [1,2,3,4,5]. The figure of merit (ZT) of the material indicates its viability for usage in thermoelectric devices. ZT = S²σT/κ, where S, σ, T, and κ stand for Seebeck coefficient, electrical conductivity, temperature, and thermal conductivity, respectively [6,7]. It requires a significant power factor and a low thermal conductivity to operate at peak efficiency. Therefore, the emphasis of several theoretical and experimental investigations was the examination of these material transport features. First-principle computation has been considered by researchers to describe the optical as well as the thermoelectric features of non-magnetic HHs [8,9]. A semiconducting nature is predicted for those with an 8-valence electron count. Based on the atoms that make up its structure, HHs (those with eight valence electrons) may have a broad bandgap. The optoelectronics features of the 20 Hhs were studied compressively by T. Gruhn [10]. HHs of I-III-IV and I-II-V were explored by P.S. Kacimi et al., who demonstrated their structural, optoelectronic, and transport properties [11]. Y. Cherchab et al. investigated the stability of the structure of the KLaX (X = C, Si, and Sn) HHs with their physical characteristics [12]. Spin-orbit interaction has a greater impact on the valence band of KLaX than on the conduction band. Due to their high merit factors and thermopower values, these alloys are considered suitable for thermoelectric usage. Kamlesh et al. investigated the optical as well as the transport features of YScP (Y = Li, Na, P = Ge, Si) HHs [13]. They concluded that these can be utilized for solar technologies and are good candidates for thermoelectric usage because of their direct bandgap, excellent optical activity in the visible region, and high figure of merit. The LiMgN, NaMgN, and KMgN HHs have ZT values nearly equal to unity, as reported by Ahmed et al. suggesting that these are useful for thermoelectric applications [14]. The thermoelectric properties of KScX (X = C, Ge) HHs have recently been reported [15]. Compared to Bi₂Te₃, this response was 1.5 times higher. KYX (X = Si, Ge) possesses a higher power factor revealing optimum thermoelectric properties [16].

Additionally, the HHs are reported to be a promising candidate for the actuators used in technological applications of microelectromechanical actuators (MEMS). The amazing technology known as “Micromachines”—microelectromechanical systems, or MEMS, and nanoelectromechanical systems, or NEMS—offers special advantages like the expanding miniaturization of the electronics industry, which is required for the creation of high-precision and high-performance devices. Since 1987, MEMS have played a significant role in nanoscale devices such as switches, optical attenuators, pumps, valves, etc. Microelectromechanical actuators (MEMS) are the subject of current scientific research. MEMS are widely sought-after instruments because of their operational theory, which depends on the application of high-frequency electrostatic forces and can give actuation modes such as electrostatic, magnetostatic, piezoelectric, and thermal expansion [17,18]. Recently, there has been a lot of interest in polysilicon electrothermal microactuators, whose operation is based on directed thermal expansion brought on by the Joule action. Unfortunately, this chemical is in high demand due to its exclusivity and expanding use, which puts it at risk of exhausting resources and increasing price. According to statistics, the globe utilized 2.034 million m² of silicon wafers in 2020, an increase of 8% from the 1.884 million m² consumed in the first quarter of the year. It is challenging for researchers to find other chemicals with almost equal physical properties to polysilicon since it was 6% more than that of the second quarter of 2019 [19]. As a result, the goal of this article is to choose a new compound that is comparable to polysilicon and to develop a numerical simulation that will enable the performance of an actuator made of the new material to be anticipated before it is manufactured.

Our desire to learn more about HH compounds was sparked by the results described above. Here, we explore the physical characteristics of KBaTh (Th = Sb, Bi) HHs employing first-principle simulations with an attempt to recommend and defend the usage of half-Heusler KBaTh (Th = Sb, Bi) in place of polysilicon in the manufacture of MEMS. Ansys software will then employ the physical properties of this unique compound in a simulation model to predict and monitor the condition of an actuator. The paper introduction is covered in the first section, while the computational method and variables are covered in the second section. In the third section, the findings are presented along with an analysis and justification. The final portion includes a summary of the major achievements as well as a conclusion.

2. Computational Technique

The FP-LAPW approach [20] as armed by the WIEN2k [21,22] was used for the prediction of physical features of KBaTh (Th = Sb, Bi) HHs. The structural and elastic characteristics of the KBaTh (Th = Sb, Bi) system were ascertained using the generalized gradient potential (GGA) [23]. Modified Becke–Johnson potential (mBJ), an approach to the issues with electronic properties, was adopted in the current study [24]. The valence wave function expanded within the sphere to a maximum value of l_max = 10. R_MT × K_max is set at 7 as converge criterion. The k mesh about 16 × 16 × 16 was taken into consideration. The BoltzTrap function [25] implemented in the Wien2K code was used to determine the transport qualities of KBaTh (Th = Sb, Bi) HH. An extremely dense grid (50 × 50 × 50 k-points) was used to calculate the transport parameters. Using the Vienna ab initio simulation package (VASP), we were able to determine the dynamical stability of the KBaTh (Th = Sb, Bi) [26].

3. Findings and Analysis

3.1. Structural Properties

Analysis of the structural characteristics of KBaTh (Th = Sb, Bi) is the main goal of this section. Analysis of structural features is particularly fascinating because it offers the chance to learn more about the microscopic structure of materials and has a big impact on the predictions of other properties. According to Figure 1, the investigated KBaTh (Th = Sb, Bi) HHs possess a non-centrosymmetric cubic crystal structure with space group 216, F-43m, structure. The K atom is placed at (0.25, 0.25, 0.25), the Ba atom at (0.5, 0.5, 0.5), and the Sb/Bi atom at (0, 0, 0), respectively. The obtained lattice parameters are 7.91 Å and 8.02 Å for KBaTh (Th = Sb, Bi) HHs respectively. The calculated lattice parameters agree with previous theoretical and experimental findings [15]. The Birch–Murnaghan equation of state [27] was used to determine the structural features of the KBaTh (Th = Sb, Bi) HHs at equilibrium. The energy and volume optimization graph are represented in Figure 2a,b. In addition to lattice parameters,

Table 1. Calculated lattice parameter a (Å), volume (V), bulk modulus (B), its derivative B_P, minimum energy (E_tot), enthalpy of formation E_f and bond length (Å) for KBaTh (Th = Sb, Bi).

HH	a (Å)	V (a.u.)3	B	Bp	Formation Energy (Ef) eV/atom	Bond Length (Å)
KBaSb	7.91	894.25	17.19	3.69	−0.506	K-Ba = 3.28 Ba-Sb = 3.63
KBaBi	8.02	945.06	16.18	3.75	−0.440	K-Ba = 3.21 Ba-Bi = 3.74
Ref [16]	7.10			4.57

also provides other structural information. The formation energy of the examined KBaTh (Th = Sb, Bi) HHs was calculated as follows to determine their thermodynamic stability.

$${{\mathrm{E}}_{\mathrm{f}}=\mathrm{E}}_{\mathrm{T}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}^{\mathrm{K}\mathrm{B}\mathrm{a}\mathrm{T}\mathrm{h}}{-\mathrm{E}}_K^{\mathrm{b}\mathrm{u}\mathrm{l}\mathrm{k}}{-\mathrm{E}}_{\mathrm{B}\mathrm{a}}^{\mathrm{b}\mathrm{u}\mathrm{l}\mathrm{k}}{-\mathrm{E}}_{\mathrm{T}\mathrm{h}}^{\mathrm{b}\mathrm{u}\mathrm{l}\mathrm{k}}$$

(1)

It can be identified that KBaTh (Th = Sb, Bi) has estimated formation energies of −0.506 eV/atom, and −0.440 eV/atom, respectively. The E_f being negative indicates that the examined HHs are stable and can be prepared experimentally [28].

For KBaTh (Th = Sb, Bi) to be dynamically stable, we approximated the phonon spectrum using VASP. Although the plot’s negative frequency guarantees dynamical instability, the material under study is stable, as shown by the positive phonon spectra [29,30]. The phonon dispersion along M-K-Γ-A-L in the first Brillouin zone was estimated (Figure 3). The investigated KBaTh (Th = Sb, Bi) HH dynamical stability was demonstrated by the absence of an imaginary frequency at the Γ-point. The results attained were found to be consistent with past findings [12].

Numerous thermodynamic characteristics at certain temperatures and pressures must be researched to develop KBaTh (Th = Sb, Bi) HH-dependent devices and other industrial applications [31]. The thermodynamic properties of the KBaTh (Th = Sb, Bi) HH under standard pressure and temperatures ranging from 0 to 1000 K were determined using a quasi-harmonic model. The C_v discusses the phase transition, displays molecular motion, and describes the lattice vibration shown in Figure 4. It has been found that specific heat rises with temperature while pressure remains constant. Compounds in the cubic phase experience a considerable volumetric change below 400 K, which causes the specific heat to increase quickly. The specific heat starts to get close to the Dulong–Petit limit at 875 K [32,33]. Entropy S is once more depicted in Figure 4 as it varies with pressure and temperature. It has been found that entropy grows uniformly with temperature. Entropy achieves its maximum value at standard pressure and temperature, which is 251 J/molK. It can be shown that, in accordance with earlier results, the free energy falls as temperature rises. Figure 5 and Figure 6 illustrate 3D representations of the crystal direction dependence of Young’s modulus, shear modulus G, and Poisson ratio for KBaSb and KBaBi, respectively.

3.2. Elastic Property

The elastic properties of KBaTh (Th = Sb, Bi) were determined using the IRelast software [34], as outlined in the Wien2K code. Its elastic characteristics can be described by the various constants C₁₁, C₁₂, and C₄₄. These various elastic coefficients can be used to determine stability. Where C₁₁ denotes opposition to strains, C₁₂ denotes shear stress; also C₄₄ provides details on the resistance provided for shear deformation [35]. The calculated elastic constants for HH KBaTh (Th = Sb, Bi) are shown in

Table 2. Values of elastic constants (C_ij), bulk modulus (B), shear modulus (G), Young’s modulus (E), Poisson’s ratio (σ), Pugh ratio (B/G), shear anisotropy factor (A) Cauchy pressure C^P, sound velocities (m/s), Debye Temperature Θ_D (K) for KBaTh (Th = Sb, Bi).

Material Property	KBaSb	KBaBi	Ref [38]
C11 (GPa)	35.23	40.07	54.41
C12 (GPa)	13.87	18.54	16.98
C44 (GPa)	5.00	5.48	11.66
S11 (GPa−1)	0.03650451
S12 (GPa−1)	0.01031196
S44 (GPa−1)	0.2
Bulk modulus, B (GPa)	20.99	25.72	29.46
Shear modulus, G (GPa)	7.27	7.59
Young modulus, E (GPa)	19.56	20.74
Poisson ratio, σ (GPa)	0.34	0.36	0.29
Pugh ratio, B/G (GPa)	2.88	3.38
Cauchy pressure CP (GPa)	8.87	13.06
Anisotropy factor (A)	0.46	0.51	0.62
Transverse sound velocity (m/s)	1254	1398
Longitudinal sound velocity(m/s)	2634	3095
Average sound velocity (m/s)	1410	1576
Temperature ΘD (K)	142.4	150.8
Melting temperature ${\mathrm{T}}_{\mathrm{m}}$ (K)	471.6	588.9

. Furthermore, the Born–Huang stability condition [36,37] was employed to assess the mechanical stability of the examined HHs:

(C₁₁ − C₁₂)/2 > 0, (C₁₁ + 2C₁₂) > 0, C₄₄ > 0

(2)

According to our calculations, KBaTh (Th = Sb, Bi) satisfies the Born–Huang stability criterion, which means that they are stable. Other elastic constants including the bulk modulus (B), shear modulus (G), Young modulus (Y), Poisson’s ratio (v), and anisotropy (A), among others, were calculated using the formulas below:

$$\mathrm{B}=\frac{C_{11}+2C_{12}}{3}$$

(3)

$$\mathrm{G}=\frac{C_{11}-C_{12}}{2}$$

(4)

$$\mathrm{Y}=\frac{9GB}{3B+G}$$

(5)

$$\mathsf{\nu }=\frac{3\mathrm{B}-2\mathrm{G}}{2\left(3\mathrm{B}+\mathrm{G}\right)}$$

(6)

$$\mathrm{A}=\frac{2C_{44}}{C_{11}-C_{12}}$$

(7)

The bulk modulus (B) reflects how robust it is to volumetric alteration when compressed high B values suggest strong crystals, while high shear modulus (G) values imply crystals that are resistant to plastic deformation [39]. The computed B for KBaTh (Th = Sb, Bi) is 18.1 GPa and 18.07 GPa, respectively. Because KBaSb has a greater B value than KBaBi, it is more resistant to volume fluctuation. Additionally, the fact that KBaSb has a larger G value than KBaBi (12.17) shows that the former exhibits greater transverse resistance. Pugh’s ratio (B/G) [40] can be used to measure ductility or brittleness. The substance under investigation is naturally ductile if it is more than 1.75 and otherwise brittle. The predicted B/G for KBaTh (Th = Sb, Bi) in Table 2 illustrates the brittleness of the investigated HHs.

Frantsevich et al. [41] employed Poisson’s ratio (v) to distinguish between brittle and ductile materials. A material is ductile if the v value is more than 0.26; otherwise, it is brittle. The computed values for v for the KBaTh (Th = Sb, Bi) are, respectively, 0.20 and 0.22, which indicate their brittle nature. The aforementioned results are supported by the Cauchy pressure (Cp = C₁₂ − C₄₄) [42]. Ionic (covalent) bonding is present when the Cauchy pressure has a positive (negative) magnitude. The negative Cauchy pressure was used to create the covalent bonding. Debye temperature (θ_D) [43] and melting point (T_m) were two more thermodynamic variables that were investigated and expressed as follows:

$${\theta }_D={\frac{h}{K_B}\left[\frac{3n}{4\pi }(\frac{N_{A\rho }}{M})\right]}^{-1/3}{\nu }_m$$

(8)

where N_A is Avogadro’s number, k is Boltzmann’s constant, h is Plank’s constant, n is the number of atoms in a single cell, and M is its molecular weight. The calculated Debye temperature is listed in Table 2. Utilizing below the investigated material mean sound speeds

$${\nu }_{\mathrm{m}}={\left[\frac{1}{3}(\frac{2}{{\mathsf{\theta }}_{\mathrm{t}}^3}+\frac{1}{{\mathsf{\theta }}_{\mathrm{t}}^3})\right]}^{-(1/3)}$$

(9)

where the shear modulus and bulk modulus, respectively, are used to determine the longitudinal and transverse components of sound velocity gives

$${\mathsf{\theta }}_{\mathrm{t}}=\sqrt{\frac{\mathrm{G}}{\mathsf{\rho }}}\mathrm{and}\mathsf{\theta }=\sqrt{\frac{3\mathrm{B}+4\mathrm{G}}{3\mathsf{\rho }}}$$

(10)

Table 2 shows the computed values of υ_t, υ_l, V_m, θ_D, and T_m.

The anisotropic behavior of the mechanical modulus is visualized and fully explained by the study of the elastic characteristics of three-dimensional (3D) surfaces [38]. The following formulas are used to compute the bulk modulus B, Young’s modulus E, and shear modulus G crystallographic direction dependence:

$$\frac{1}{B}=\left(S_{11}+2S_{12}\right)\left(l_1^2+l_2^2+l_3^2\right)\frac{1}{E}=S_{11}-2\left(S_{11}-S_{12}-\frac{1}{2}{S}_{44}\right)\left(l_1^2{l}_2^2+l_2^2{l}_3^2+l_3^2{l}_1^2\right)$$

$$\frac{1}{\mathrm{G}}=S_{44}-4\left(S_{11}-S_{12}-\frac{1}{2}{S}_{44}\right)\left({sin}^2\theta .{cos}^2\theta +0.125{sin}^4\right)\left(1-cos4\phi \right)$$

(11)

$S_{ij}$ are the contents of the elastic compliance constants matrix, which are obtained from the inverse of the elastic constant matrix $S_{ij}=C_{ij}^{-1}$, and their values are represented in matrix form below. l₁, l₂, and l₃ are the x, y, and z-axis director cosines, respectively.

$$S_{ij}=\left(\begin{array}{cccccc}{S}_{11}& S_{12}& S_{12}& 0& 0& 0\\ S_{12}& S_{11}& S_{12}& 0& 0& 0\\ S_{12}& S_{12}& S_{11}& 0& 0& 0\\ 0& 0& 0& S_{44}& 0& 0\\ 0& 0& 0& 0& S_{44}& 0\\ 0& 0& 0& 0& 0& S_{44}\end{array}\right)$$

(12)

The linear compressibility obtained 3D directional dependency and cross-section in several reticular planes; the degree of divergence from a spherical shape typically reflects the degree of anisotropy. The spherical shape of the linear compressibility is confirmed by the fact that Y and G deviate significantly from the sphere, confirming the considerable anisotropy of KBaTh (Th = Sb, Bi). Furthermore, it can be seen from Figure 7 that the stress along the [100], [010], and [001] axes produce the maximum Young’s modulus value for both compounds under consideration. Conversely, the external stress applied along the [111] direction results in the maximum shear modulus value, indicating both strong stiffness and weak shear deformation resistance along the [100], [010], and [001] directions.

3.3. Electronic Characteristics

To investigate and illustrate the use of our compounds for electronic devices, we approximated the band structures of KBaTh (Th = Sb, Bi). Figure 8 displays the band structure speculated via the PBE-GGA along with the mBJ method. Using the (GGA-PBE) method, the band gaps for KBaTh (Th = Sb, Bi) are 0.79 eV and 0.63 eV, respectively. Considering mBJ the band gap values for KBaTh (Th = Sb, Bi) are 1.31 eV and 1.20 eV, respectively. By dividing the valence and conduction states by the Fermi level, the band structure can be represented for the investigated KBaTh (Th = Sb, Bi). At the top of the curve are the conduction states, while near the bottom are the valence levels. Between the two locations, there is an energy bandgap through which the Fermi level passes. The studied compounds exhibit semiconductor behavior with an indirect bandgap, as observed in Figure 8a–d. The electronic bandgap computed from mBJ is in good agreement with earlier reports as shown in

Table 3. Computed bandgap using differently an exchange-correlation functional, effective mass of the electron (me*) and holes (mh*), Bader charges (using PBE-GGA approach) for KBaTh (Th = Sb, Bi).

HHs	GGA Band Gap (eV)	mBJ Band Gap (eV)	Effective Mass	Bader Charge
KBaSb	0.79	1.31	me* = 0.28 mh* = 0.36	K = 0.72 Ba = 1.21 Sb = −1.78
KBaBi	0.63	1.20	me* = 0.24 mh* = 0.35	K = 0.62 Ba = 1.10 Bi = −1.69
Ref [13].	0.64	1.05

.

We calculated the associated density of states (DOS) for KBaTh (Th = Sb, Bi), HH displayed in Figure 9 to gain an understanding of the band structure. In contrast, the higher part of the valence band, which runs from around 2.55 eV to 0 eV, is mainly caused by hybridization between the Ba-p and Bi/Sb-p states. The orbitals associated with the K atom may be shown in the image to have a minimal effect on the energy range being studied. The orbitals associated with the K atom may be shown in the image to have a minimal effect on the energy range being studied. Bi-d, Sb-d, Ba-s, and K-s states are the contributions to the valance band. The bandgap is found to be in accordance with the computed density of states.

3.4. Electron Density

Along the (110) plane electron density (ED) for the studied KBaTh (Th = Sb, Bi) can be computed as shown in Figure 10 [44]. On the Sb atom’s left and right positions, a blue zone can be seen in the (110) plane. The (110) plane’s lowest ED, or minimum, is denoted by blue. Both compound blue zones on the (110) plane are surrounded by a thin yellow coronal and a green coronal. The (100) plane shows a thin green line connecting the K- and Ba atoms and a faint yellow line connecting the Sb/Bi atoms. It is conceivable to conclude that the K-K bonds connecting the two Sb/Bi atoms on the right side of the Ba atom are weak based on the color scheme and the atomic locations in the (100) plane in Figure 8. Sb-Sb/Bi-Bi, as well as Ba-Ba bonds, are ductile in the (100) plane [45]. As indicated in Table 3, the atomic charge transfer distributions were also calculated using the Bader charge analysis. According to statistics, Sb atoms gain electrons while K and Ba atoms lose them. The sharing of electrons between Sb/Bi and K atoms, respectively, establishes their covalent connections.

Finding the effective mass (electron m_e* and hole m_h*) is necessary to comprehend the photovoltaic properties, which are greatly impacted by resistivity [46], and can be computed as follows:

$$\frac{\mathbf{1}}{{\mathit{m}}^{\mathbf{*}}}=\frac{\mathbf{1}}{{\mathbf{ħ}}^{\mathbf{2}}}\frac{{\mathit{d}}^{\mathbf{2}}{\mathit{E}}_{\mathit{n}}\left(\mathit{k}\right)}{\mathit{d}{\mathit{k}}^{\mathbf{2}}}$$

(13)

The expected effective mass values for KBaTh (Th = Sb, Bi) HHs are shown in Table 3.

The results show that the HHs under investigation had a small effective carrier mass (electron and hole). Materials with low effective mass are regarded as being especially beneficial for solar cell applications because they facilitate carrier transmission. This suggests that the KBaTh (Th = Sb, Bi) may prove to be useful in photovoltaics.

3.5. Optical Characteristics

The outstanding electronic features of KBaTh (Th = Sb, Bi) led to the exploration of its optical and electronic transport characteristics. The energy-dependent optical features were established using the research on the dielectric functions. The dielectric function ε(ω), [47] is represented as follows:

ε = ε₁(ω) + iε₂(ω)

(14)

The momentum matrix P can be represented as the imaginary ε₂(ω) component of the inter-band and intra-band transition, which offers information about electron transitions over the Brillouin Zone (BZ) that are dependent on the density of states (DOS) and can be expressed as below:

$${\epsilon }_2\left(\omega \right)=\frac{e^2\hslash }{\pi m^2{\omega }^2}\sum \int {\left|M_{\nu ,c}\left(k\right)\right|}^2\delta [{\omega }_{\nu c}\left(k\right)-\omega ]d^{3}k$$

(15)

ε₁(ω), expressed as

$${\epsilon }_1(\omega )=1+\frac{2}{\pi }P{\int }_0^{\infty }\frac{{\omega }^{\prime }{\epsilon }_2(\omega \prime )d\omega \prime }{({{\omega }^{\prime }}^2-{\omega }^2)}$$

(16)

The ε₁(ω) and ε₂(ω) help in the computation of other parameters such as refractive index $n\left(\omega \right)$, extinction coefficient $k\left(\omega \right)$, reflectivity $R\left(\omega \right)$, optical absorption coefficient $\alpha \left(\omega \right),$ and optical conductivity $\sigma \left(\omega \right)$ Figure 11a depicts the variation of ε₁(ω) and ε₂(ω) with reference to incident energy (0 to 10 eV) for KBaTh (Th = Sb, Bi). At zero frequency the ε₁(ω) is termed a static dielectric constant ε₁(0). For KBaTh (Th = Sb, Bi), the magnitude of ε₁(0) is represented in

Table 4. Computed optical (at zero energy) and transport properties (at 300 K) for KBaTh (Th = Sb, Bi).

	Material Property	KBaSb	KBaBi	Ref [13]
Optical properties	ε1 (0)	8.19	9.51	13.62
	n (0)	2.86	3.08	3.69
	R (0)	0.23	0.26	0.32
Transport properties (300 K)	S (µV/K)	229	231	181
	σ/τ (Ωms)−1 (1019)	0.43	0.41	2.45
	k/τ (W/mKs) (1014)	0.9	1.7	2.0
	PF (1011) (W/K2ms)	2.71	2.62

. The effective transport of electrons from the valence to the conduction band caused the photon energy to show these multiple peaks. The imaginary part of the dielectric ε₂(ω) that quantifies the absorption process is shown in Figure 11b. The lack of absorption for KBaTh (Th = Sb, Bi) up to 1.13 and 1.04 eV suggests that the optical and electronic bandgaps are nearly identical. The inter-band transitions between the valance and conduction bands cause the absorption to rise after the threshold value, culminating at higher energies [48]. The refractive index n(ω) for the examined KBaTh (Th = Sb, Bi), as a function of photon energy is depicted in Figure 11c. Absolute refractive index, or n(0), is the refractive index at zero energy. Following n(0), the magnitude increases and reaches its maximum value for examined KBaTh (Th = Sb, Bi), at 2.1 ev and 1.92 eV, respectively. As demonstrated in Figure 11c, n (ω) begins to decline after reaching its greatest value. Figure 11d shows the reflectivity R(ω) of the KBaTh (Th = Sb, Bi) over the energy range from 0 eV to 10 eV. With reflectivity of less than 0.6, KBaTh (Th = Sb, Bi) demonstrates typical semiconducting characteristics. Absolute reflectivity R(0) is the measure of reflectance at zero energy. Table 4 represents the magnitude of R(0). The reflectivity rises after R(0) and falls when it reaches its maximum value.

In Figure 12a the variation of the extinction coefficient, k(ω) is represented. It was found that the pattern of k(ω) resembles that of ε₂(ω). According to the graph, oscillations begin as invisible phenomena but become evident as the photon energy rises. Another essential optical attribute to the optical properties is the absorption coefficient α(ω), which indicates the light absorption through the investigated substance. Figure 12b shows the absorption spectra for KBaTh (Th = Sb, Bi) HHs. As demonstrated, considerable absorption is observed in UV and visible light by the investigated KBaTh (Th = Sb, Bi) HHs. The fact that α(ω), has significant intensity and can be achieved at energies higher than 2.21 eV suggests that the chemicals under investigation can absorb both visible and UV light. As a result, for KBaTh (Th = Sb, Bi), the largest value of the α(ω) is 980 (cm)⁻¹, followed by 940 (cm)⁻¹, 920 (cm)⁻¹, and 895 (cm)⁻¹. The researched KBaTh (Th = Sb, Bi) has good absorption spectra that point to its use in optoelectronics. Figure 12c depicts the KBaTh (Th = Sb, Bi) HH optical conductivity σ(ω) variation. When an object is exposed to electromagnetic radiation, a covalent bond ruptures, causing an electric current to flow. As a result, a phenomenon called optical conductivity σ(ω) converts solar light into electricity. Figure 12d shows the relationship between optical conductivity and photon energy. The early peaks are located between 2.41 eV and 3.53 eV. Figure 12d shows the loss function L(ω) with photon energy. The largest loss is found in higher energies. Our results show the material adaptability to a range of optoelectronic applications [49]. The KBaTh (Th = Sb, Bi) HHs can be viewed as a potential choice for optoelectronic applications due to their significant optical absorption, low optical loss, and presence of a bandgap in the visible range.

3.6. Thermoelectric Properties

Thermoelectric (TE) conversion of energy, where it incorporates waste heat recovery, is a popular way to produce electricity. By employing the BoltzTrap [25] and the constant relaxation time τ approximation (τ ≈ 10⁻¹⁴ s), as well as at constant carrier concentration (10¹⁹ cm⁻³), the transport behavior of cubic KBaTh (Th = Sb, Bi) HHs can be investigated. Band structure and effective mass both influence transport qualities. The band gap of the carriers (holes and electrons) involved in the transportation event is the major determining factor. It has been found that mBJ generates a wider band gap than GGA. Setting the Seebeck coefficient (S) requires carefully tuning the band gap. Metals exhibit the lowest Seebeck coefficient, whereas insulators exhibit the highest Seebeck coefficient. The Seebeck coefficient, which similarly relies on charge carriers, is used to compute the thermoelectric voltage brought about by the temperature differential between two edges of a material. The Seebeck coefficient must be significant for the thermoelectric effectiveness to be preferred. S is positive for p-type materials and negative for n-type materials [35,39]. The estimates of the S in Figure 13a demonstrate a p-type trend with a positive magnitude for the Seebeck coefficient. For the examined KBaTh (Th = Sb, Bi), the value of S at room temperature was computed to be 229 μV/K and 231 μV/K, respectively. Table 4 shows the computed value of the S with similar reports. Table 4 represents the computed S with other reports. The HHs under investigation appear to have an excellent thermoelectric response since they have Seebeck values that are higher than 200 μV/K, which is a property of a perfect TE material. As seen in Figure 13a, the Seebeck coefficient has a direct relationship with the band gap and an inverse relationship with electrical conductivity (σ/τ). As the temperature rises, it can be seen that σ/τ increases, showing typical semiconducting properties [50]. KBaSb has considerable electrical conductivity because of its enormous density of charge carriers, which raises the electrical conductivity. The lattice thermal conductivity decreases sharply with temperature due to the scattering of phonons. The lattice contribution to the total thermal conductivity for both compounds was computed using BoltzTraP code, and the lattice contribution computed using the Slack equation as [51,52]:

$${\kappa }_l=A.\frac{\overline{M}{\theta }_D^3\delta }{{\gamma }^{2}T{n}^{2/3}}$$

(17)

where $\overline{M}$ represent the average atomic mass of all constituent atoms, A is a constant calculated as $A=\frac{2.43.{10}^{-8}}{1-\frac{0.514}{\gamma }+\frac{0.228}{{\gamma }^2}}$, n is the number of atoms in the primitive cell, δ is the average atomic volume, T is the absolute temperature, γ is the Grüneisen parameter, and θ_D is the Debye temperature. The estimated κ_l values for KBaTh (Th = Sb, Bi) compounds are depicted in Figure 13c. From this figure, it is well seen that κ decreases with an increase in temperature The total thermal conductivity (κ_e/τ) is influenced by both the lattice and electronic thermal conductivities. It was observed that κ_e/τ significantly increases with temperature as deduced from Figure 13d. The variation of total thermal conductivity (κ/τ) with temperature is represented in Figure 13e. Electrical conductivity and thermal conductivity are related by the Widemann–Franz law (σ/κ) [53]. The σ/κ was found to be in the 10⁻⁵ range, which denotes superior electrical conductivity and lower heat conductivity. Figure 13d displays the computed PF for the studied KBaTh (Th = Sb, Bi). The obtained PF at 300 K for KBaTh (Th = Sb, Bi). was found to be 2.71 × 10¹¹ W/K²ms, and 2.62 × 10¹¹ W/K²ms, respectively. The compounds are useful in thermoelectric devices because they have a high-power factor at high temperatures. Finally, the figure of merit (ZT) for KBaTh (Th = Sb, Bi) was calculated and is displayed in Figure 14. The ZT values for KBaTh (Th = Sb, Bi) HHs were computed to be 0.740 and 0.737, respectively, at 300 K. The highest ZT values at 300 K show how KBaTh (Th = Sb, Bi) HHs could be used in thermoelectric devices.

3.7. Modeling and Analysis in ANSYS

Here, an attempt was made to advocate for and defend the substitution of half-Heusler KBaTh (Th = Sb, Bi) for polysilicon in the production of MEMS. Then, Ansys software uses the physical characteristics of this special compound in a simulation model to predict and maintain monitoring of an actuator’s status. The input to the model consisted of the temperature-dependent material properties of each component of the device. The current modeling illustrates how an electrothermal actuator behaves when our unique half-Heusler compound, KBaTh (Th = Sb, Bi), is used in place of the actuator’s original material, such as polysilicon. We provided all the computed properties necessary to complete our modeling of the compound KBaTh (Th = Sb, Bi) in

Table 5. Young’s modulus in GPa; $\nu$: Poisson’s ratio; $\alpha$: Thermal expansion coefficient in °K⁻¹; $\varpi$: Electrical resistivity in Ωm; $K_L$: Thermal conductivity in W.m⁻¹ °K⁻¹.

Material	$\mathit{E}$	$\mathit{\nu }$	$\mathit{\alpha }$	$\mathit{\varpi }$	${\mathit{K}}_{\mathit{L}}$
KBaSb	19.56	0.34	5.4 × 10−5	0.239	1.474
KBaBi	20.74	0.36	4 × 10−5	0.238	1.962

, along with the well-known properties of polysilicon [19]. The actuators’ components and measurements are depicted in the accompanying figure. (Model Geometry 3.4.1). The accompanying diagram shows the shape and dimensions of the electrothermal actuator. The designs are simulated using Ansys. The actuator operates under the influence of three related physical principles: thermal expansion-induced pressures, electric current conduction, and heat conduction. The actuator is affected electrothermally [54]. Temperature gradient arises as electrical current passes through the suspended beams’ electrical resistance. This idea of conserving energy is employed. The steady-state energy equation, assuming a resistive heating source, can be expressed as follows:

$$\Delta \left(∆T\right)+\frac{F^2}{\varpi }=0$$

(18)

where, F, ϖ, and ΔT are the electric field, electric resistivity, and temperature difference, respectively.

Electrical current passes through the actuator from anchor to anchor. The actuator heats up and expands more because of the hot beam’s higher current density compared to the cold beam, which results in a lateral arcing motion towards the cold beam side, as seen in Figure 15. The formula can be used to determine the axial deflection ΔL brought on by thermal expansion for a one-dimensional structural element:

$$\frac{∆L}{L}=\alpha ∆T$$

(19)

where α is the coefficient of thermal expansion of the material, L is the original length of the element, and ΔT is the temperature difference.

Between the bases of the heated arms’ anchors, an electric potential is applied. Electrical insulation is included on the cold arm anchor and every other surface. Three anchors, three cells, and their bases all have fixed temperatures that match the substrate’s constant temperature. The heat flux coefficient represents one over the thermal resistance, and this can be applied as either thermal contact conditions or convective heat flux conditions [55]. We chose the employment of a heat flux condition in this model. By dividing the air’s thermal conductivity by the system’s distance from its surrounding surfaces, one can get the heat transfer coefficient. Different heat transfer coefficients are used in this exercise for the actuator’s top surface and other surfaces. The heat flux is given by (20):

$$Heatflux\perp =h(T-Tamb)$$

(20)

The three arms are mechanically attached to the base of the three anchors. The dimples can move freely in the plane of the substrate (the XY plane in the figure) but do not move in the direction perpendicular to the substrate (the Z direction), as represented in Figure 16, Figure 17 and Figure 18.

We can quickly achieve ideal meshing on all types of models with the intelligent meshing tool provided by Ansys 2020R1 software. The generation of high-quality meshes is ensured by intelligent and autonomous algorithms. Furthermore, adding adjustment commands is simple when required. Ansys multiphysics module is used to model the actuator and enables the investigation of displacements and temperature fields. The developed finite element model comprises 16,325 nodes, each with five degrees of freedom, 10,993 triangular elements, and 10,993 triangular elements [56]. The actuator’s surface temperature distribution is shown in Figure 17. Figure 19 uses a color and warp plot to depict the displacement field. The findings of the maximum displacement and maximum temperature of the two materials are compared in

Table 6. Table of comparison between the results of displacement and temperature of the two materials.

Materials	Displacement (m)	Temperature (K°)
KBaSb	$\mathrm{DMX}=0.186\times {10}^{-3}$	SMN = 298 SMX = 8281.56
KBaBi	$\mathrm{DMX}=0.107\times {10}^{-3}$	SMN = 298 SMX = 6321.04

The displacement of KBaTh (Th = Sb, Bi) was found to be seven times greater than that of polysilicon at the same conditions, as can be shown in Figure 19 and Figure 20. Due to the significant deformation, the temperature convergence in KBaTh (Th = Sb, Bi) was 18.1 °K greater than in polysilicon. Figure 19 and Figure 20 demonstrate that the KBaTh (Th = Sb, Bi) curve slope is greater than that of polysilicon, indicating a lower response time for the alloy. The design of an electrothermal actuator was modeled to compare the results of the actuator in polysilicon base material and the actuator, when these qualities were replaced with the predicted properties of the KBaTh (Th = Sb, Bi) compound [57]. According to the ANSYS data, the new KBaTh (Th = Sb, Bi) is more sensitive because its displacement under identical conditions was seven times greater than that of polysilicon. Given that the new material is more sensitive, this is a good sign because it suggests that our compound will work, even in situations where the voltage is only 150 V, to produce the same amount of displacement. The current intensity can be decreased by adding a resistor. Finally, this study sheds light on using the proposed compound in solar cells, one of the most significant renewable energy sources [58].

4. Conclusions

We investigated the physical features of KBaTh (Th = Sb, Bi) HHs using first-principle computing and Boltzmann transport theory. The computed lattice parameters are in good agreement with previous research. The KBaSb has a bandgap of 1.31 eV, whereas the KBaBi has a bandgap of 1.20 eV. The elastic characteristics reveal their mechanical stability as well as the ductile nature of the investigated compounds. We also computed the optical features of KBaTh (Th = Sb, Bi) in terms of transition and absorption as a function of photon energy (eV). The studied KBaTh (Th = Sb, Bi) HHs may be used in optoelectronic devices due to their optical properties. The calculated S and σ yield gave remarkable results that satisfy the criteria for strong thermoelectric performance with low thermal conductivity. At a wide range of temperatures, KBaTh (Th = Sb, Bi) displays a higher power factor and figure of merit. The predicted ZT at ambient temperature and optical absorption demonstrated that the examined KBaTh (Th = Sb, Bi) HHs are more suitable for optical as well as thermoelectric device applications. The results from ANSYS showed that the displacement of the new KBaTh (Th = Sb, Bi) material was seven times higher than that of the polysilicon under the same conditions, indicating that it is more sensitive. This is a good sign because it means that our compound will be able to function, even under conditions of very low voltage of 150 V, to give the same required displacement (the current intensity can be reduced by adding a resistor), knowing that the new material is more sensitive.