A First-Principles Investigation on the Structural, Optoelectronic, and Thermoelectric Properties of Pyrochlore Oxides (La2Tm2O7 (Tm = Hf, Zr)) for Energy Applications

A first-principles calculation based on DFT investigations on the structural, optoelectronic, and thermoelectric characteristics of the newly designed pyrochlore oxides La2Tm2O7 (Tm = Hf, Zr) is presented in this study. The main quest of the researchers working in the field of renewable energy is to manufacture suitable materials for commercial applications such as thermoelectric and optoelectronic devices. From the calculated structural properties, it is evident that La2Hf2O7 is more stable compared to La2Zr2O7. La2Hf2O7 and La2Zr2O7 are direct bandgap materials having energy bandgaps of 4.45 and 4.40 eV, respectively. No evidence regarding magnetic moment is obtained from the spectra of TDOS, as a similar overall profile for both spin channels can be noted. In the spectra of ε2(ω), it is evident that these materials absorb maximum photons in the UV region and are potential candidates for photovoltaic device applications. La2Tm2O7 (Tm = Hf, Zr) are also promising candidates for thermoelectric device applications, as these p-type materials possess ZT values of approximately 1, which is the primary criterion for efficient thermoelectric materials.


Introduction
Pyrochlore oxides with the formula A 2 B 2 O 7 (A 2 B 2 O 6 O') have emerged as potential multiferroic materials due to the tunability of their structures and local displacement of their B-site atoms inside the original structure of these materials [1]. The stability of the pyrochlore oxides depends on the size ratio of the two cations occupying the A and B sites [2]. The ideal pyrochlore structures have eight molecules per unit cell and belong to the Fd-3m space group. In the pyrochlore structures, pentavalent/tetravalent cations can occupy the B-site; divalent/trivalent cations can occupy the A-site [3]. Every atom in a crystalline structure occupies a distinct position known as the Wyckoff position [4]; therefore, A-site, B-site, O'-and O-atoms are located at the 16c, 16d, 48f and 8a sites, respectively [3]. However, the 8b-site remains vacant. Due to their remarkable ferroelectric properties, many pyrochlore materials, including Nd 2 Zr 2 O 7 , Bi 2 T i2 O 7 , Cd 2 Re 2 O 7 , and La 2 Zr 2 O 7 , have been reported in the literature [5,6]. Ekaterina A. et al. [7] presented a series of newly synthesized pyrochlore oxides La 2-x Sr x Zr 2 O 7-δ (x = 0, 0.05, 0.1, 0.15 and 0.5) by the citrate-nitrate procedure. Amalesh K. et al. [8] reported the effect of variable doping concentrations of Pr 3+ ions on the photoluminescence properties of La 2 Zr 2 O 7 :Pr 3+ phosphor materials. Pyrochlore oxides have been employed for various technological applications such as antierosion of Ag [9], magnetic devices [10], photoluminescence devices [11], thermal barrier coatings of diesel engines [12], gas sensors [13], high-temperature catalysts [14], enhanced

Results and Discussion
The calculated results for the optoelectronic, thermoelectric and structural properties of La 2 Tm 2 O 7 (Tm = Hf, Zr) are presented and discussed in this manuscript section. We explored the capability of these pyrochlore oxides for optoelectronic and thermoelectric device applications to produce clean energy. The structural stability of La 2 Tm 2 O 7 (Tm = Hf, Zr) is determined using the calculated structural properties of these compounds. The potential of the pyrochlore oxides for optoelectronic applications such as solar cells can be investigated using calculated optoelectronic properties. Furthermore, the usability of these materials in thermoelectric devices such as thermal generators can be determined using their thermoelectric properties. Based on the presented results, these pyrochlore oxides are promising materials for thermoelectric and optoelectronic device applications.

Structural Properties
In this manuscript section, we discuss essential parameters to check the structural stability of La 2 Tm 2 O 7 (Tm = Hf, Zr) using the calculated structural properties. The structural optimization parameters for La 2 Tm 2 O 7 (Tm = Hf, Zr) are calculated using the Birch-Murnaghan equation of state (EOS) presented in Equation (1) [22]. The optimized volume versus energy curves for the unit cells of pyrochlore oxides is presented in Figure 1. The ground state energy of the material corresponds to the lowest point in the volume-energy spectra, and the volume at that point is optimized. The lattice parameters corresponding to the equilibrium volume are known as optimized lattice parameters. Tables 1 and 2 present geometric and optimized parameters calculated for pyrochlore oxides La 2 Tm 2 O 7 (Tm = Hf, Zr).

Structural Properties
In this manuscript section, we discuss essential parameters to check the structural stability of La2Tm2O7 (Tm = Hf, Zr) using the calculated structural properties. The structural optimization parameters for La2Tm2O7 (Tm = Hf, Zr) are calculated using the Birch-Murnaghan equation of state (EOS) presented in Equation (1) [22]. The optimized volume versus energy curves for the unit cells of pyrochlore oxides is presented in Figure 1. The ground state energy of the material corresponds to the lowest point in the volume-energy spectra, and the volume at that point is optimized. The lattice parameters corresponding to the equilibrium volume are known as optimized lattice parameters. Tables 1 and 2 present geometric and optimized parameters calculated for pyrochlore oxides La2Tm2O7 (Tm = Hf, Zr).

Electronic Properties
The calculated electronic properties, i.e., energy band structures and the density of states (DOS) for La 2 Tm 2 O 7 (Tm = Hf, Zr), are presented and discussed in this manuscript. The energy band structures of the compounds mentioned above are plotted along the high symmetric axis of the IBZ (irreducible Brillouin zone) on a continuous energy range of −3 to 5 eV. DOS spectra for both compounds are presented and discussed on a continuous energy range of −6 to 6 eV to obtain insight into the most feasible electronic transitions between CB and VB.

Energy Band Structures
The valuable information regarding ground state properties of pyrochlore oxides La 2 Tm 2 O 7 (Tm = Hf, Zr), such as optical and charge transport properties, can be obtained from the calculated energy band structures. A continuous energy range of −3 to 5 eV is used to plot the calculated energy band structures by taking the Fermi level (E F ) at 0 eV (presented in Figure 2). High symmetry points in IBZ are used to calculate energy band structures using GGA + U approximation in the DFT. In Figure 2a,b, the calculated energy band structures using GGA + U approximation for La 2 Hf 2 O 7 and La 2 Zr 2 O 7 , respectively, are presented. This study shows that La 2 Tm 2 O 7 (Tm = Hf, Zr) are wide-bandgap semiconductors. The calculated values of energy band gaps for La 2 Hf 2 O 7 and La 2 Zr 2 O 7 are approximately 4.45 and 4.40 eV, respectively. Both pyrochlore oxides La 2 Tm 2 O 7 (Tm = Hf, Zr) are direct band semiconductors, as both VBM (valence band maxima) and CBM (conduction band minima) occur at the same symmetric point Γ. The potential of the materials for technical applications such as thermoelectric, magneto-electronic, and optoelectronic devices can be analyzed using in-depth information regarding the energy band structures and gaps.

Density of States
Deep knowledge regarding the nature of energy states and the various atomic orbitals that are contributing to these states can be obtained from the calculated density of states (DOS) for that compound. Detailed analyses of TDOS (total density of states) and

Density of States
Deep knowledge regarding the nature of energy states and the various atomic orbitals that are contributing to these states can be obtained from the calculated density of states (DOS) for that compound. Detailed analyses of TDOS (total density of states) and PDOS (partial density of states) are required to understand the composition of VB and CB. The spectra of TDOS for La 2 Tm 2 O 7 (Tm = Hf, Zr) are presented in Figure 3. The energy range of −6.0 to 6.0 eV is used to plot the investigated results of TDOS and PDOS. From Figure 3, we can state that there is no noticeable magnetic moment in La 2 Tm 2 O 7 (Tm = Hf, Zr), as the spectra of TDOS are identical in both spin channels over the entire energy range. Hence, only results for the spin-up channel are presented in this study. The Fermi level in all electronic results is set at 0 eV. Deep knowledge regarding the nature of energy states and the variou als that are contributing to these states can be obtained from the calcula states (DOS) for that compound. Detailed analyses of TDOS (total density PDOS (partial density of states) are required to understand the composition The spectra of TDOS for La2Tm2O7 (Tm = Hf, Zr) are presented in Figure  range of −6.0 to 6.0 eV is used to plot the investigated results of TDOS an Figure 3, we can state that there is no noticeable magnetic moment in La2Tm Zr), as the spectra of TDOS are identical in both spin channels over the entir Hence, only results for the spin-up channel are presented in this study. The all electronic results is set at 0 eV.  The spectra of PDOS for La 2 Hf 2 O 7 are presented in Figure 4. The PDOS spectra can validate the results obtained from the energy band structure evaluation. In the valence band of La 2    The spectra of PDOS for La 2 Zr 2 O 7 are presented in Figure 5  The spectra of PDOS for La2Zr2O7 are presented in Figure 5. In the valence band of La2Zr2O7, significant contributions from O-atoms are evident from the spectra of PDOS

Optical Properties
The reaction of the crystalline materials while interacting with electromagnetic (EM waves at different energies ( = ℎ ) can be explained using optical properties. Deep knowledge of the optical characteristics, such as the dispersion and absorption of the ma terial, is necessary while designing photovoltaic devices such as solar cells, LEDs, optica fibers, etc. [22]. Dielectric function ( ) presented in Equation (2) is a vital parameter to model the impact of EM radiation on a crystalline material [23]. ( ) is the fundamenta input parameter for computing the remaining optical characteristics. The dielectric func tion ( ) consists of two parts: 2 ( ) (imaginary/absorptive) and 1 ( ) (real/disper sive) part.

Optical Properties
The reaction of the crystalline materials while interacting with electromagnetic (EM) waves at different energies (E = hυ) can be explained using optical properties. Deep knowledge of the optical characteristics, such as the dispersion and absorption of the material, is necessary while designing photovoltaic devices such as solar cells, LEDs, optical fibers, etc. [22]. Dielectric function ε(ω) presented in Equation (2) is a vital parameter to model the impact of EM radiation on a crystalline material [23]. ε(ω) is the fundamental input parameter for computing the remaining optical characteristics. The dielectric function ε(ω) consists of two parts: ε 2 (ω) (imaginary/absorptive) and ε 1 (ω) (real/dispersive) part.
The material exhibits both interband and intraband electronic transitions. Interband transitions can be direct or indirect and typically occur in semiconductor materials, whereas intraband transitions are typically seen in metallic compounds. The phonon scattering effect is related to indirect transitions. The complex dielectric function ε(ω) can be determined using the calculated energy band structure for La 2 Tm 2 O 7 (Tm = Hf, Zr). The imaginary part ε 2 (ω) involves all probable band transitions (between populated valence states (VS) and vacant conduction states (CS)), and Equation (3) can be used to calculate ε 2 (ω) [24].
The calculated values of ε 2 (ω) can be used to calculate ε 1 (ω) using the Kramers-Kronig relation shown in Equation (4) [25]. The ε 1 (ω) and energy bandgap are in inverse relation with one another.
The spectra of ε 1 (ω) can be used to explain a significant case of the dispersion of incident photons. The calculated spectra of ε 1 (ω) for La 2 Tm 2 O 7 (Tm = Hf, Zr) are presented in Figure 6a. An energy range of 0 to 14 eV is used to plot all the studied optical parameters. At zero energy (ω = 0), we can observe definite values of ε 1 (ω), known as zero frequency limit or static values of ε 1 (0). Penn's model equation shown in Equation (5) can be used to establish a correlation between the energy bandgap and ε 1 (0).
The calculated values of 2 ( ) can be used to calculate 1 ( ) using the Kramers-Kronig relation shown in Equation (4) [25]. The 1 ( ) and energy bandgap are in inverse relation with one another.
The spectra of 1 ( ) can be used to explain a significant case of the dispersion of incident photons. The calculated spectra of 1 ( ) for La2Tm2O7 (Tm = Hf, Zr) are presented in Figure 6a. An energy range of 0 to 14 eV is used to plot all the studied optical parameters. At zero energy ( = 0), we can observe definite values of 1 ( ), known as zero frequency limit or static values of 1 (0). Penn's model equation shown in Equation (5) can be used to establish a correlation between the energy bandgap and 1 (0). From Figure 6a, we can note that the static values of 1 (0) are 2.24 and 2.31 for La2Hf2O7 and La2Zr2O7, respectively. After that, peaks rise and reach the maximum value of around 5.0 eV. The spectra of 1 ( ) then decrease sharply and enter the negative region at 11.82 and 11.5 eV for La2Hf2O7 and La2Zr2O7, respectively. The region where 1 ( ) is negative can be used to characterize screened plasma, and plasmon plasma is the From Figure 6a, we can note that the static values of ε 1 (0) are 2.24 and 2.31 for La 2 Hf 2 O 7 and La 2 Zr 2 O 7 , respectively. After that, peaks rise and reach the maximum value of around 5.0 eV. The spectra of ε 1 (ω) then decrease sharply and enter the negative region at 11.82 and 11.5 eV for La 2 Hf 2 O 7 and La 2 Zr 2 O 7 , respectively. The region where ε 1 (ω) is negative can be used to characterize screened plasma, and plasmon plasma is the frequency where ε 1 (ω) intersects the zero level (the dotted line). Materials show metallic and dielectric behavior below and above this dotted line [4].
The spectra of ε 2 (ω) can be used to explain a significant case of the absorption of incident photons. The calculated spectra of ε 2 (ω) for La 2 Tm 2 O 7 (Tm = Hf, Zr) are presented in Figure 6b. In the spectra of ε 2 (ω), it is evident that initially, values of ε 2 (ω) are zero, and then peaks start to emerge from 3.28 and 3.17 eV for La 2 Hf 2 O 7 and La 2 Zr 2 O 7 , respectively. The value from where the peak emerges is known as the optical bandgap of the material or threshold energy of ε 2 (ω). Both energy bandgaps and threshold energies show good agreement with each other. From Figure 6b, we can note a sudden increase in the spectra of ε 2 (ω) after threshold energies and move to the maximum values around 7.0 eV. Therefore, we can conclude that La 2 Tm 2 O 7 (Tm = Hf, Zr) are efficient photon absorbers in the near UV region.
The ratio of the speed of light in vacuum (c) to the speed of light in a medium (v) is known as the refractive index, n = c/v. This optical parameter is used to determine whether a material is acceptable for technological optical devices. Similar to ε 1 (ω), deep knowledge regarding the dispersion of incident photons can also be obtained from the spectra of n(ω). The calculated spectra of n(ω) for La 2 Tm 2 O 7 (Tm = Hf, Zr) are presented in Figure 6c. At zero energy (ω = 0), we can observe definite values of n(ω), known as zero frequency limit or static values of n(0). Penn's model equation shown in Equation (5) can be used to establish a correlation between the energy bandgap and ε 1 (0). Equation (8) can verify static values of n(0).
From Figure 6a, we can note that the static values of n(0) are 1.5 and 1.52 for La 2 Hf 2 O 7 and La 2 Zr 2 O 7 , respectively. The material is considered to be an active optical material when the value of its refractive index is between 1.0 and 2.0. After that, peaks rise and reach the maximum values of around 5.0 eV. The n(ω) spectra then decrease sharply and become less than unity at 10.19 and 11.1 eV for La 2 Hf 2 O 7 and La 2 Zr 2 O 7 , respectively. In the region where n(ω) is less than unity, materials show metallic behavior. When (n < 1), these materials exhibit the superluminal phenomenon that can be theoretically and experimentally observed. This unnatural phenomenon communicates that the speed of light in a vacuum is less than that of light in the medium, c < v g , which is impossible [27]. Material with high optical conductivity values, refractive index, and absorption coefficients with low emissivity can be utilized in photovoltaic device applications.
An optical parameter that determines how effectively a material can absorb incident photons/radiations at a specific frequency is the extinction coefficient K(ω). Similar to ε 2 (ω), deep knowledge regarding the absorption of incident photons can also be obtained from the spectra of K(ω). The calculated spectra of K(ω) for La 2 Tm 2 O 7 (Tm = Hf, Zr) are presented in Figure 6d. In the K(ω) spectra, it is evident that initially, K(ω) values are zero, and then, peaks start to emerge from 2.33 and 3.43 eV for La 2 Hf 2 O 7 and La 2 Zr 2 O 7 , respectively. The value from where the peak emerges is the known threshold energy of K(ω). From Figure 6b, we can note a sudden increase in the K(ω) spectra after threshold energies and move to the maximum values in the UV region. Therefore, we can conclude that La 2 Tm 2 O 7 (Tm = Hf, Zr) are efficient photon absorbers in the near UV region.
Some electrons undergo inelastic scattering when a beam of electrons is incident on the material. The energy lost by these electrons in the material can be interpreted as energy loss function L(ω). It is evident from Equation (9) that there exists an inverse relation between ε 2 (ω) and L(ω). The calculated spectra of L(ω) for La 2 Tm 2 O 7 (Tm = Hf, Zr) are presented in Figure 7a. In the L(ω) spectra, it is evident that initially, values of L(ω) are zero, and then, peaks start to emerge from 3.92 and 4.11 eV for La 2 Hf 2 O 7 and La 2 Zr 2 O 7 , respectively. From Figure 7a, we can note a gradual increase in the L(ω) spectra after threshold energies and move to the maximum values around 13.11 and 13.28 eV for La 2 Hf 2 O 7 and La 2 Zr 2 O 7 , respectively. These energies are also known as points of plasmon resonance for the compounds mentioned above.
Some electrons undergo inelastic scattering when a beam of electrons is incident on the material. The energy lost by these electrons in the material can be interpreted as energy loss function ( ). It is evident from Equation (9) that there exists an inverse relation between 2 ( ) and ( ). The calculated spectra of ( ) for La2Tm2O7 (Tm = Hf, Zr) are presented in Figure 7a. In the L(ω) spectra, it is evident that initially, values of ( ) are zero, and then, peaks start to emerge from 3.92 and 4.11 eV for La2Hf2O7 and La2Zr2O7, respectively. From Figure 7a, we can note a gradual increase in the L(ω) spectra after threshold energies and move to the maximum values around 13.11 and 13.28 eV for La2Hf2O7 and La2Zr2O7, respectively. These energies are also known as points of plasmon resonance for the compounds mentioned above. The reflected-to-incident photons ratio is known as the reflectivity coefficient ( ). The calculated spectra of ( ) for La2Tm2O7 (Tm = Hf, Zr) are presented in Figure 7b. At zero energy ( = 0), we can observe definite values of ( ) known as zero frequency limit or static values of (0). From Figure 7b, we can note that the static values of (0) are 0.039 and 0.043 for La2Hf2O7 and La2Zr2O7, respectively. It is evident from the spectra of ( ) that these materials reflect a negligible number of incident photons (∼20%) up to 12.0 eV. La2Tm2O7 (Tm = Hf, Zr) reflects a maximum number of incident photons The reflected-to-incident photons ratio is known as the reflectivity coefficient R(ω). The calculated spectra of R(ω) for La 2 Tm 2 O 7 (Tm = Hf, Zr) are presented in Figure 7b. At zero energy (ω = 0), we can observe definite values of R(ω) known as zero frequency limit or static values of R(0). From Figure 7b, we can note that the static values of R(0) are 0.039 and 0.043 for La 2 Hf 2 O 7 and La 2 Zr 2 O 7 , respectively. It is evident from the spectra of R(ω) that these materials reflect a negligible number of incident photons (∼20%) up to 12.0 eV. La 2 Tm 2 O 7 (Tm = Hf, Zr) reflects a maximum number of incident photons (∼55%) above 13.0 eV. Based on R(ω) spectra, it can also be concluded that these materials are promising candidates for absorption-related applications.
The property of the material to conduct electricity when exposed to light is known as optical conductivity σ(ω). The calculated spectra of σ(ω) for La 2 Tm 2 O 7 (Tm = Hf, Zr) are presented in Figure 7c. In the spectra of σ(ω), it is evident that initially, values of σ(ω) are zero, and then, peaks start to emerge from 3.62 and 3.75 eV for La 2 Hf 2 O 7 and La 2 Zr 2 O 7 , respectively. From Figure 7c, we can note a sudden increase in the spectra of σ(ω) after threshold energies and move to the maximum values around 7.44 and 9.13 eV for La 2 Hf 2 O 7 and La 2 Zr 2 O 7 , respectively. It is evident from the spectra of σ(ω) that these materials are excellent conductors around the energies mentioned above.
Information regarding the penetration length of the incident photon in the material can be explained using absorption coefficient I(ω) when the energies of incident photons are greater than the energy band gap of the material. The calculated spectra of I(ω) for La 2 Tm 2 O 7 (Tm = Hf, Zr) are presented in Figure 7d. In the I(ω) spectra, it is evident that initially, values of I(ω) are zero, and then, peaks start to emerge from 4.18 and 4.25 eV for La 2 Hf 2 O 7 and La 2 Zr 2 O 7 , respectively. The value from where the peak emerges is known as the threshold energy of I(ω). From Figure 7d, we can note a sudden increase in the I(ω) spectra after threshold energies and moving to the maximum values in the UV region. Therefore, we can conclude that La 2 Tm 2 O 7 (Tm = Hf, Zr) are efficient photon absorbers in the near UV region and can be utilized in photovoltaic devices working in the UV region.

Thermoelectric Properties
In recent years, thermoelectric materials have attracted massive attention from scientists due to the global energy crisis. The phenomenon of directly converting temperature gradient into electrical energy is called the thermoelectric effect. The thermoelectric (TE) properties are also included in this study. The traditional Boltzmann kinetic transport theory and the rigid band approximation are employed to calculate thermoelectric properties of pyrochlore oxides La 2 Tm 2 O 7 (Tm = Hf, Zr) [28,29]. The TE behavior of pyrochlore oxides La 2 Tm 2 O 7 (Tm = Hf, Zr) can be characterized using basic TE parameters such as The ability of the material to conduct electricity can be explained using electrical conductivity . The calculated spectra of electrical conductivity for La2Tm2O7 (Tm = Hf, Zr) are presented in Figure 8a. A temperature range of 0 to 1000 K is used to plot calculated thermoelectric parameters. In the spectra of electrical conductivity , it is evident that initially, values of are zero, and peaks emerge from 650 and 600 K for La2Hf2O7 and La2Zr2O7, respectively. From Figure 8a, we can note an exponential increase in the spectra of with increasing temperature values. The semiconductor nature of the pyrochlore oxides La2Tm2O7 (Tm = Hf, Zr) is evident from the increasing trend in the spectra of . In the entire temperature range, it is evident from the spectra that values of for La2Hf2O7 are less than that of La2Zrf2O7. The maximum values of are 4.34 × 10 16 and 6.53 × 10 16 (Ω. . ) −1 for La2Hf2O7 and La2Zr2O7, respectively.
For semiconductors, thermal conductivity (κ) comes from electrons and lattice vibra- The ability of the material to conduct electricity can be explained using electrical conductivity σ. The calculated spectra of electrical conductivity σ for La 2 Tm 2 O 7 (Tm = Hf, Zr) are presented in Figure 8a. A temperature range of 0 to 1000 K is used to plot calculated thermoelectric parameters. In the spectra of electrical conductivity σ, it is evident that initially, values of σ are zero, and peaks emerge from 650 and 600 K for La 2 Hf 2 O 7 and La 2 Zr 2 O 7 , respectively. From Figure 8a, we can note an exponential increase in the spectra of σ with increasing temperature values. The semiconductor nature of the pyrochlore oxides La 2 Tm 2 O 7 (Tm = Hf, Zr) is evident from the increasing trend in the spectra of σ. In the entire temperature range, it is evident from the spectra that values of σ for La 2 Hf 2 O 7 are less than that of La 2 Zrf 2 O 7 . The maximum values of σ are 4.34 × 10 16 and 6.53 × 10 16 (Ω.m.s) −1 for La 2 Hf 2 O 7 and La 2 Zr 2 O 7 , respectively.
For semiconductors, thermal conductivity (κ) comes from electrons and lattice vibrations. The thermal conductivity (κ) can be calculated using Fourier law. The calculated spectra of electronic thermal conductivity κ e for La 2 Tm 2 O 7 (Tm = Hf, Zr) are presented in Figure 8b. In the spectra of thermal conductivity κ e , it is evident that initially, values of κ e are zero, and then, peaks start to emerge from 600 and 550 K for La 2 Hf 2 O 7 and La 2 Zr 2 O 7 , respectively. From Figure 8a, we can note an exponential increase in the spectra of κ e with increasing temperature values. At room temperature, the phonon wavelength is more significant than the crystal boundary, which results in an increasing trend in the spectra of κ e . The crystal boundary is greater than or equal to all contributions made by the phonons to κ e . Temperature and phonon wavelength are inversely proportional to each other. Furthermore, in the entire temperature range, it is evident from the spectra that values of κ e for La 2 Hf 2 O 7 are less than that of La 2 Zrf 2 O 7 . The maximum values of κ e are 1.94 × 10 13 and 2.56 × 10 13 W m.K.s for La 2 Hf 2 O 7 and La 2 Zr 2 O 7 , respectively. On joining two different materials, due to temperature gradient, electrons from the high-temperature region move toward the low-temperature region, establishing a potential difference (∆V) [4]. A ratio of ∆V to ∆T, known as the Seebeck coefficient S = ∆V ∆T can be used to determine the effectiveness of the thermocouple. The calculated spectra of the Seebeck coefficient (S) for La 2 Tm 2 O 7 (Tm = Hf, Zr) are presented in Figure 8c. The p-type nature of the studied pyrochlore oxides is evident from the positive values of the Seebeck coefficient (S). With the rising temperature, the spectra of S exhibit an exponential decline, a feature of semiconductor devices. The maximum values of S are 2.75 × 10 −3 (200 K) and 2.68 × 10 −3 (200 K) V K for La 2 Hf 2 O 7 and La 2 Zr 2 O 7 , respectively. Materials with S values greater than 200 µV K are considered suitable thermoelectric materials. The values of S for both pyrochlore oxides are close to the defined standard. Therefore, these materials can be considered potential thermoelectric materials for device applications. Thermoelectric parameters are also plotted against the chemical potential at 300 K. We can note peaks in both positive and negative regions of Figure 9a; therefore, these materials can be tuned as p-type or n-type depending on our requirement. On joining two different materials, due to temperature gradient, electrons from the high-temperature region move toward the low-temperature region, establishing a potential difference (∆ ) [4]. A ratio of ∆ to ∆ , known as the Seebeck coefficient ( = with S values greater than 200 are considered suitable thermoelectric materials. The values of S for both pyrochlore oxides are close to the defined standard. Therefore, these materials can be considered potential thermoelectric materials for device applications. Thermoelectric parameters are also plotted against the chemical potential at 300 K. We can note peaks in both positive and negative regions of Figure 9a; therefore, these materials can be tuned as p-type or n-type depending on our requirement. The calculated spectra of power factor for La2Tm2O7 (Tm = Hf, Zr) are presented in Figure 8d. In the spectra of power factor , it is evident that initially, values of are zero, and then, peaks start to emerge from 550 and 500 K for La2Hf2O7 and La2Zr2O7, respectively. From Figure 8d, we can note an exponential increase in the spectra of The calculated spectra of power factor PF for La 2 Tm 2 O 7 (Tm = Hf, Zr) are presented in Figure 8d. In the spectra of power factor PF, it is evident that initially, values of PF are zero, and then, peaks start to emerge from 550 and 500 K for La 2 Hf 2 O 7 and La 2 Zr 2 O 7 , respectively. From Figure 8d, we can note an exponential increase in the spectra of PF with increasing values of temperature. The maximum values of PF are 1.92 × 10 10 and 2.52 × 10 10 W m.K 2 .s for La 2 Hf 2 O 7 and La 2 Zr 2 O 7 , respectively. Finally, the calculated spectra of the figure of merit ZT for La 2 Tm 2 O 7 (Tm = Hf, Zr) are presented in Figure 8e The device's performance can be characterized by a thermoelectric quantity known as the figure of merit ZT. The ZT values start high and then gradually decrease as the temperature rises. The maximum value of ZT is 0.997 (200 K) for pyrochlore oxides La 2 Tm 2 O 7 (Tm = Hf, Zr). The materials employed in thermoelectric devices must have a ZT value of at least 1 or greater than 1. Pyrochlore oxides La 2 Tm 2 O 7 (Tm = Hf, Zr) have a ZT value of approximately 1. Hence, these compounds are potential candidates for thermoelectric devices. From the ZT spectra, La 2 Zrf 2 O 7 is the most promising candidate for thermoelectric device applications. We can predict that these materials are efficient thermoelectric materials by analyzing the figure of merit ZT spectra presented in Figure 9c.

Thermodynamic Properties
The thermodynamic stability of the pyrochlore oxides La 2 Tm 2 O 7 (Tm = Hf, Zr) was determined by investigating specific heat (C V ), thermal expansion, and Gibbs free energy (G). The Gibbs2 code is used to determine thermodynamic parameters using quasi-harmonic Debye approximation [30]. Information regarding the required heat energy for a specific material to raise its temperature by 1 K by keeping volume constant is known as specific heat capacity at constant volume (C V ). The calculated spectra of C V as a function of pressure for La 2 Tm 2 O 7 (Tm = Hf, Zr) are presented in Figure 10 on a continuous temperature range. Initially, an exponential increase in the values of C V is evident from Figure 10. At higher temperatures, the values of C V become constant with increasing temperatures. The classical behavior of both compounds is evident from the C V : T 3 and Dulong-Petit law spectra at high and low temperatures, respectively [31].

Thermodynamic Properties
The thermodynamic stability of the pyrochlore oxides La2Tm2O7 (Tm = Hf, Zr) was determined by investigating specific heat ( ), thermal expansion, and Gibbs free energy (G). The Gibbs2 code is used to determine thermodynamic parameters using quasi-harmonic Debye approximation [30]. Information regarding the required heat energy for a specific material to raise its temperature by 1 K by keeping volume constant is known as specific heat capacity at constant volume ( ). The calculated spectra of as a function of pressure for La2Tm2O7 (Tm = Hf, Zr) are presented in Figure 10   The information regarding the maximum amount of work that a closed thermodynamical system may perform at constant pressure and temperature can be obtained from the thermodynamical quantity known as Gibbs free energy ( ). The calculated spectra of as a function of pressure for La2Tm2O7 (Tm = Hf, Zr) are presented in Figure 11 on a continuous temperature range. An indirect and direct relation of Gibbs free energy is evident from Figure 11 with temperature and applied temperature. At constant pressure and temperature, the value of reversible work that a system can perform can also be obtained from the spectra that a system can perform . The information regarding the maximum amount of work that a closed thermodynamical system may perform at constant pressure and temperature can be obtained from the thermodynamical quantity known as Gibbs free energy (G). The calculated spectra of G as a function of pressure for La 2 Tm 2 O 7 (Tm = Hf, Zr) are presented in Figure 11 on a continuous temperature range. An indirect and direct relation of Gibbs free energy is evi-dent from Figure 11 with temperature and applied temperature. At constant pressure and temperature, the value of reversible work that a system can perform can also be obtained from the spectra that a system can perform G.  Figure 11. Calculated spectra of Gibbs free energy ( ) for La2Tm2O7 (Tm = Hf, Zr).
The information regarding the variations in the volume and shape of the material at constant pressure with changing temperature can be obtained from the thermodynamical parameter known as thermal expansion coefficient ( ). The calculated spectra of as a function of pressure for La2Tm2O7 (Tm = Hf, Zr) are presented in Figure 12 on a continuous temperature range. An exponential increase in the values of is evident from Figure 12 and shows that La2Tm2O7 (Tm = Hf, Zr) absorbs maximum heat in this region. The values of and applied pressure are inversely proportional to each other. From Figure 12, we can note that these materials absorb constant heat at higher temperatures, as the curves are parallel to the temperature axis. We can conclude that La2Tm2O7 (Tm = Hf, Zr) are thermodynamically stable compounds based on these thermodynamical parameters.

Materials and Methods
The full potential linearized augmented plane wave (FP-LAPW) method [32] is used within the framework of density functional theory (DFT) [33,34] to minimize the forces acting on the atoms of the crystal. Using the relaxed geometry, the ground state properties The information regarding the variations in the volume and shape of the material at constant pressure with changing temperature can be obtained from the thermodynamical parameter known as thermal expansion coefficient (α). The calculated spectra of α as a function of pressure for La 2 Tm 2 O 7 (Tm = Hf, Zr) are presented in Figure 12 on a continuous temperature range. An exponential increase in the values of α is evident from Figure 12 and shows that La 2 Tm 2 O 7 (Tm = Hf, Zr) absorbs maximum heat in this region. The values of α and applied pressure are inversely proportional to each other. From Figure 12, we can note that these materials absorb constant heat at higher temperatures, as the curves are parallel to the temperature axis. We can conclude that La 2 Tm 2 O 7 (Tm = Hf, Zr) are thermodynamically stable compounds based on these thermodynamical parameters.  Figure 11. Calculated spectra of Gibbs free energy ( ) for La2Tm2O7 (Tm = Hf, Zr).
The information regarding the variations in the volume and shape of the material at constant pressure with changing temperature can be obtained from the thermodynamical parameter known as thermal expansion coefficient ( ). The calculated spectra of as a function of pressure for La2Tm2O7 (Tm = Hf, Zr) are presented in Figure 12 on a continuous temperature range. An exponential increase in the values of is evident from Figure 12 and shows that La2Tm2O7 (Tm = Hf, Zr) absorbs maximum heat in this region. The values of and applied pressure are inversely proportional to each other. From Figure 12, we can note that these materials absorb constant heat at higher temperatures, as the curves are parallel to the temperature axis. We can conclude that La2Tm2O7 (Tm = Hf, Zr) are thermodynamically stable compounds based on these thermodynamical parameters.

Materials and Methods
The full potential linearized augmented plane wave (FP-LAPW) method [32] is used within the framework of density functional theory (DFT) [33,34] to minimize the forces acting on the atoms of the crystal. Using the relaxed geometry, the ground state properties

Materials and Methods
The full potential linearized augmented plane wave (FP-LAPW) method [32] is used within the framework of density functional theory (DFT) [33,34] to minimize the forces acting on the atoms of the crystal. Using the relaxed geometry, the ground state properties such as structural, optoelectronic, and thermoelectric use first-principles-based WIEN2K code [35]. To analyze the physical properties of La 2 Tm 2 O 7 (Tm = Hf, Zr), the exchange and correlation potential are calculated by employing Hubbard correction (U)-added generalized gradient approximation (GGA) [24]. Generally, band gap values are underestimated by simple GGA approximation. To overcome this issue, we added a U correction to correct the estimated values of energy bandgaps. While employing the FP-LAPW method, the electrons in the cluster are clustered as valance (electrons in the interstitial region) and core (electrons in muffin-tin spheres) electrons. The plane wave basis set shown in Equation (13) is used to expand the wave function in the interstitial region.
On the other hand, the product of spherical harmonics (Y lm ) and the radial solution for the Schrodinger wave equation (V lm ) shown in Equation (14) is used to define potential inside the MT spheres.
The value of R MT × K max = 7.0 is set as the cut-off criteria for the convergence of energy eigenvalues. K max and R MT represent the largest Fermi vector and smallest muffintin radii, respectively. A dense k-mesh of 500 k-points is used in these calculations. For self-consistent field calculations, the convergence criterion is less than 10 −3 Ry. The effective Hubbard potential is the difference between Coulomb interaction (U) and exchange potential (J), i.e., U e f f = U − J. In this study, the value of effective U is set as 7.0 eV by taking U = 7.0 eV and J = 0 eV. Equation (15) can obtain total energy while using GGA + U formulism. where The electrical conductivity, Seebeck coefficient, thermal conductivity, power factor, and figure of merit are generally used to evaluate the thermoelectric performance of crystalline materials. Botlztrap code is employed to calculate thermoelectric parameters by combining DFT with Boltzmann transport theory [28]. The crystalline unit cell structures of pyrochlore oxides La 2 Tm 2 O 7 (Tm = Hf, Zr) are presented in Figure 13. such as structural, optoelectronic, and thermoelectric use first-principles-based WIEN2K code [35]. To analyze the physical properties of La2Tm2O7 (Tm = Hf, Zr), the exchange and correlation potential are calculated by employing Hubbard correction (U)-added generalized gradient approximation (GGA) [24]. Generally, band gap values are underestimated by simple GGA approximation. To overcome this issue, we added a U correction to correct the estimated values of energy bandgaps. While employing the FP-LAPW method, the electrons in the cluster are clustered as valance (electrons in the interstitial region) and core (electrons in muffin-tin spheres) electrons. The plane wave basis set shown in Equation (13) is used to expand the wave function in the interstitial region.
On the other hand, the product of spherical harmonics ( ) and the radial solution for the Schrodinger wave equation ( ) shown in Equation (14) is used to define potential inside the MT spheres.
The value of × = 7.0 is set as the cut-off criteria for the convergence of energy eigenvalues. and represent the largest Fermi vector and smallest muffintin radii, respectively. A dense k-mesh of 500 k-points is used in these calculations. For self-consistent field calculations, the convergence criterion is less than 10 −3 Ry. The effective Hubbard potential is the difference between Coulomb interaction ( ) and exchange potential ( ), i.e., = − . In this study, the value of effective U is set as 7.0 eV by taking = 7.0 and = 0 . Equation (15) can obtain total energy while using GGA + U formulism.
The electrical conductivity, Seebeck coefficient, thermal conductivity, power factor, and figure of merit are generally used to evaluate the thermoelectric performance of crystalline materials. Botlztrap code is employed to calculate thermoelectric parameters by combining DFT with Boltzmann transport theory [28]. The crystalline unit cell structures of pyrochlore oxides La2Tm2O7 (Tm = Hf, Zr) are presented in Figure 13.

Conclusions
This manuscript investigates pyrochlore oxides' structural, optoelectronic, and thermoelectric properties using first-principles-based DFT calculations. This study reveals

Conclusions
This manuscript investigates pyrochlore oxides' structural, optoelectronic, and thermoelectric properties using first-principles-based DFT calculations. This study reveals that the compounds mentioned above are promising candidates for photovoltaic and thermoelectric device applications. Analyzing the investigated structural properties, we have concluded, based on ground state energy, that La 2 Hf 2 O 7 is more stable than La 2 Zr 2 O 7 . Energy band structures exhibit that La 2 Hf 2 O 7 and La 2 Zr 2 O 7 are direct bandgap materials with 4.45 and 4.40 eV, respectively. It was established by exploring TDOS spectra that these pyrochlore oxides are nonmagnetic compounds. Based on ε 2 (ω), we can report that these pyrochlore oxides can efficiently absorb photons in the UV energy range. These materials are ranked as active optical materials, as their refractive index values are between 1.0 and 2.0. The values of n(ω) are 1.5 and 1.52 for La 2 Hf 2 O 7 and La 2 Zr 2 O 7 , respectively. It is evident from the R(ω) spectra that these compounds are suitable for photovoltaic device applications, as their reflectivity value is negligible (∼20%) in the entire energy range. Thermoelectric properties of La 2 Tm 2 O 7 (Tm = Hf, Zr) are calculated and discussed using the Boltzmann transport theory employed in the Boltztrap code. The Seebeck coefficient (S) positive values confirm the p-type nature of these semiconductors. The S values are in the range of defined standards for both compounds, confirming their potential to be used in thermoelectric devices. The calculated values of ZT are also around one, which means their conversion efficiency is also good.