Exploring Layered Ruddlesden-Popper Structures for High-Performance Energy Devices

Abstract

This study presents comprehensive DFT calculations to determine the structural, electronic, mechanical, and optical properties of the Ruddlesden–Popper Phase family member, La₂XO₄, which has an orthorhombic crystal structure with a Cmce space group. Ultrasoft pseudopotential plane wave and PBE-GGA approaches have been implemented using the CASTEP tool. The exchange–correlation approximation calculations show that the La₂XO₄ (where X = Ni, Fe, Ba, and Pb) compounds possess no band gap. The results indicate that the compounds are metallic, which are ideal for supercapacitor (SC) applications. The compound’s optical conductivity, dielectric function, extinction coefficients, absorption refractive index, loss function, and reflectivity are also analyzed for SC applications. UV spectra of the compounds observed high absorption coefficient (10⁵ cm⁻¹), dielectric function (9–10), optical conductivity (7 fs⁻¹), and refractive index (4) values. Furthermore, as B/G > 1.75, the mechanical (elastic) properties have shown ductile behavior and mechanical stability. Using the Born stability criteria, the mechanical stability of the compounds is examined. All of the compounds are ductile, according to Pugh’s and Frantesvich ratios. Finally, time-simulations-dependent temperature stability plots for the compounds are computed by employing dynamical stability with norm-conserved pseudopotential, which confirm their potential for SC applications.

1. Introduction

The scientific community is discovering and investigating a wide range of combinations and alternative energy resources that can improve the effectiveness of energy storage materials [1,2,3]. Supercapacitors are the most interesting electrical energy storage devices because of their significant advantages, which include high power density, long life cycle, and excellent stability [4]. For supercapacitor (SC) research, to examine the properties of the entire system, it appears that multiple DFT versions should be combined. To estimate the capacitance of the electrical double layer (EDL), classical density functional theory (cDFT) offers insights into the interface between the electrode and the electrolyte. Meanwhile, electronic density functional theory (eDFT) is useful for determining the quantum capacitance and conductivity of the electrode materials [5,6]. eDFT serves as an effective method for predicting electrical properties, including quantum capacitance and band gap energy, by analyzing the density of states (DOS) of the materials [7]. The structural and electronic properties of these systems can be studied with the help of DFT. Well-established codes have made it possible to simulate electronic structures using large memories and fast supercomputers [5]. Theoretical understanding and experimental measurements can be combined to create effective SCs with increased energy and power densities. In addition to providing insights into the mechanism through orbital interactions and electronic properties, theoretical simulations can corroborate experimental data. Numerous studies that used extensive DFT simulations to support experimental results have been published [8,9,10]. The increasing need for energy storage applications have forced researchers to look at alternate fuel sources that may function well under fatigue conditions [11]. The most practical solution to deal with these problems are the establishment of renewable energy sources, such as wind, solar, biomass, etc. The amount of solar energy reaching the Earth’s surface is significantly greater than what is now used globally. On the other hand, SCs are the devices which work on the redox reaction, and they are based on an outstanding energy storage [12,13].

Ruddlesden-Popper phases (RPPs) are one of the most significant types of layered perovskite structures and are also made up of alternating 2D perovskite layers with cations as spacers. Their usual chemical formula is A_n+1B_nX_3n+1 with n = 1, 2, … [14,15]. Superconductivity, magnetoresistance, orbital ordering transition, catalytic activity, and ferroelectricity are just a few of the intriguing characteristics that the RPP structure has shown in the extensively studied oxide perovskites [16]. The RPP can shift between hyperstoichiometric and hypostoichiometric states due to variations in oxygen stoichiometry, leading to the formation of oxygen defects. Consequently, oxygen diffusion within the RPPs may be influenced by mechanisms involving oxygen interstitials, oxygen vacancies, or a combination of both [17]. RPP-based materials possess the exceptional capacity to store a sizable amount of interstitial oxygen in their structure resulting the exceptional oxide ion conductivity, making them mixed ionic–electronic conductors (MIEC) at intermediate temperatures [18,19,20] and making them suitable candidates for SCs, which is the main motivation behind conducting this research. Many theoretical (DFT-based) studies have been performed to analyze the properties and estimation of the SCs calculations. Using DFT, the electronic structure, charge storage mechanisms, and defect energetics of transition metal oxides, such as MnO₂ and NiCo₂O₄, are explored. The analysis reveals how oxygen vacancies and surface terminations influence the capacitance performance, offering strategies for material optimization [21,22]. Using DFT, the interplay between electronic conductivity and ion adsorption is quantified. The study emphasizes the tunability of MOFs through ligand design and explores the charge transfer between conductive polymers and adsorbed ions during operation, aiding in the design of next-generation SCs [23,24]. Recent theoretical investigations have demonstrated that substituting A- or B-site cations in RP structures can lead to improved stability and enhanced conductivity, which are essential for practical applications. Furthermore, the layered nature of these materials facilitates the accommodation of strain and lattice distortions, potentially broadening the operational voltage range and improving cyclability [25,26]. Berri et al. have performed predictions on half-metallic ferromagnetism in double perovskite [27,28,29]. Wei et al. have studied the Ruddlesden–Popper-type La₂NiO_4+δ oxide coated by Ag nanoparticles as an outstanding anion intercalation cathode for hybrid SCs [30]. Sang et al. have been investigated that La₂NiO_4+δ oxides work as a pseudocapacitor [31]. The ability to predict and tailor these features using DFT calculations makes RP phases a promising platform for advancing SC technology.

Reported DFT-based studies on layered and double perovskite materials, for investigation of their electronic, magnetic, and thermoelectric properties highlighting their potential for various technological applications. One of the materials, RBaMn₂O_6−δ (R = Nd, Pr, La; δ = 0.1), was investigated using an FP-LAPW method with PBE-GGA, showing half-metallic ferromagnetism optimized with lattice parameters, which is consistent with experiments, and half-metallic behavior with magnetic moments of 10, 9, and 7 μB/fu. These findings suggest their suitability for spintronic and magnetoelectronic applications [27]. Similarly, another material, Sr₂GdReO₆, was also studied using spin density functional theory, which shows its semi-metallic behavior with a magnetic moment of 9 μB/fu and a half-metallic flip gap of 1.82 eV. The ferromagnetic phase was found to be more stable as compared to the paramagnetic phase, making it a promising material for spintronic applications [28]. Additionally, investigations into X₂MnUO₆ (X = Sr, Ba) revealed semi-metallic ferromagnetism with an integer magnetic moment of 5.00 μB, credited to the double-exchange interaction via Mn 3d–O 2p–U 5f hybridization. This highlights the potential of these materials for spintronic applications [29]. Furthermore, the structural, electronic, and thermoelectric properties of Sr₂EuReO₆ were examined using the GGA + U approach. The material demonstrated semi-metallic behavior with a magnetic moment of 8.00 μB, resulting from Re(5d)–O(2p)–Eu(4f) hybridization. Boltzmann transport calculations further indicated promising thermoelectric performance, with a Seabeck coefficient of 297.97 μV/K and a figure of merit (ZT) of 1 at 300 K, making Sr₂EuReO₆ suitable for both spintronic and thermoelectric applications [32].

Building on previous theoretical investigations of half-metallic and spintronic materials, this study focuses on these La₂XO₄ (X = Ni, Fe, Ba, and Pb) compounds for investigating their mechanical, optical, electronic, and structural properties. Time-simulation-dependent temperature stability plots for the compounds are also studied. To investigate the comprehensive energy estimations, DFT computations were performed by applying the generalized gradient approximation (GGA) and using the Perdewe–Zunger (CA-PZ) functional approach integrated into the CASTEP algorithm. Band structure, partial density of state (PDOS), and total density of state (TDOS) were also examined in this research. It has been found that La₂XO₄ has a metallic material; it has no band gap. So, this material has shown great potential to be utilized in SC applications. Mechanical properties, such as elastic constants, young modulus, shear modulus, bulk modulus, Pugh’s ratio (B/G), and Frantesvich ratio (G/B), were also investigated by using a Forcite module tool.

2. Methodology

Theoretical calculations (plane-wave pseudopotential approach of DFT), by using the Cambridge Serial Total Energy Package (CASTEP) code, have been performed at a cut-off plane-wave energy of 500 eV [33]. The CASTEP code was used to optimize the atomic positions and lattice parameters of each La₂XO₄ (X = Ni, Fe, Ba, and Pb) unit cell. Numerous physical properties were studied by using the most widely used first principles method, such as structural, electronic (band structure, density of states (DOS), and partial density of states (PDOS)), mechanical, and optical (extinction coefficient, refractive index, reflectivity, absorption, real and imaginary conductivity, loss function, real and imaginary dielectric function) properties [34,35]. The generalized gradient approximation (GGA), the Perdew–Burke–Ernzerhof (PBE) exchange–correlation functional, and the periodic ambient condition were applied alongside using the Scalar–relativistic OTFG ultrasoft pseudopotentials of DFT by using the CASTEP code [36]. In simplified form, the local density approximation (LDA) effects’ precision is considered reasonable. However, it becomes inaccurate when there are abrupt variations in the density [37]. In the presence of density gradients, the GGA approach has a higher accuracy than the LDA approach, as seen by Equation (1):

$${\mathrm{E}}_{\mathrm{X}\mathrm{C}}={\mathrm{E}}_{\mathrm{X}}^{\mathrm{G}\mathrm{G}\mathrm{A}}+{\mathrm{E}}_{\mathrm{C}}^{\mathrm{L}\mathrm{D}\mathrm{A}}+{\mathrm{E}}_{\mathrm{C}}^{\mathrm{n}\mathrm{I}}$$

(1)

The OTFG ultrasoft pseudopotentials plane-wave method was employed to calculate all of the measurements [38]. The k-integration was performed by using a Monkhorst pack grid, which used a 2 × 4 × 4 k-point mesh across the Brillouin zone (BZ) [39]. We calculated elastic constants by employing the software’s Forcite module, which charges at a custom quality using a universal force field [40]. Density mixing was used as an electronic minimizer to conclude the electronic and optical characteristics of the specified structures, and 2.0 × 10⁻⁶ was the SCF tolerance [41].

Furthermore, molecular dynamics (MD) simulations were also performed to check the dynamical stability and temperature dependent behavior of these compounds under conditions. These conditions include an NVT ensemble (constant number of particles, volume, and temperature), which was performed by using a Nose–Hoover thermostat in order to maintain temperature stability. Also, the initial temperature was adjusted at 323 K, with a time step of 1 femtosecond, adjusted via the DMol3 module of Material studio by using a norm-conserved pseudopotential, which is the default in this code.

3. Results and Discussions

3.1. Structural Properties

La₂XO₄ possesses an orthorhombic crystal structure with a total number of 28 atoms, out of which 8 are La, 4 are X (X = Ni, Fe, Ba, and Pb), and 16 are O atoms. It is a part of the space group symmetry of Cmce. Here, the pre-calculated optimal structure (mp-20143) of the material (La₂NiO₄) has been used for this study. With α = β = γ = 90°, the lattice constants for La₂NiO₄ orthorhombic structures are a = 12.44 Å, b = 5.51 Å, and c = 5.68 Å. All other materials parameters, which were obtained after optimization, are provided in

Table 1. Comparison of obtained optimized lattice parameters of La₂XO₄ compounds, where X = Ni, Fe, Ba, and Pb.

Compound	a (Å)	b (Å)	c (Å)	α (°)	β (°)	γ (°)	Volume (Å3)	Space Group
La2NiO4	5.5117	12.4381	5.6765	90	90	90	389.15	Cmce (No. 64)
La2FeO4	5.4821	12.5427	5.6081	90	90	90	385.61	Cmce (No. 64)
La2BaO4	6.9688	13.8669	6.5513	90	90	90	633.09	Cmce (No. 64)
La2PbO4	6.6865	13.4659	6.3709	90	90	90	573.64	Cmce (No. 64)

. The lattice parameters of La₂BaO₄, which is presented in the table, are much wider, as it has the highest unit cell volume among all four compounds. This is due the fact that the ionic radium of Ba is much greater than others, which made this structure expand. The structure’s side view shows the layer of X atoms, which is separated by the atoms from the A and B sites. The unit cells are shown in Figure 1.

The calculated tolerance factors for La₂XO₄ compounds are 0.9365 for La₂NiO₄, 0.8979 for La₂FeO₄, 0.7118 for La₂BaO₄, and 0.756 for La₂PbO₄, highlighting the structural strength. For all the compounds, the values fall within the typical stability range for layered perovskite-related structures (0.7 to 1.00), supporting the observed orthorhombic (Cmce) crystal symmetry. The relatively small ionic radii of Ni²⁺ and Fe²⁺ help maintain a more stable and compact octahedral framework. In contrast, the lower tolerance factors seen in La₂BaO₄ and La₂PbO₄ point to a minute deviation from ideal layered perovskite geometry. The larger sizes of Ba²⁺ and Pb²⁺ create a mismatch with the surrounding oxygen atoms, which can introduce structural strain, promote tilting of the octahedra, and contribute to layered arrangements within the lattice.

3.2. Electronic Properties

3.2.1. Band Structure

The band structure for La₂XO₄ (X = Ni, Fe, Ba, and Pb) compounds is computed with GGA potentials, as shown in Figure 2. In the energy band configuration separation, the dashed line represents the Fermi level or zero energy. By marking the Fermi level on the band structure, you can directly observe if the conduction band intersects with the Fermi level (a characteristic of metals). If the band structure shows a band crossing or touching the Fermi level, it further confirms metallic behavior.

For computed results, La₂XO₄ indicates that the bands are above the Fermi level since they are metallic in nature, and there is no band gap. Different electronic transitions between the conduction band (CB) and valance band (VB) of the electronic band gap structure define the electronic properties of the materials. Each compound’s valence band maxima (VBM) and conduction band minima (CBM) are located along high-symmetry positions Y and G, and VBM and CBM are overlapping with each other. In the absorption spectra, various peaks are produced due to the occurrence of electronic transitions between VB and CB sub-energy levels [42,43]. According to the nature of these materials, these materials can prove to be applicable for SC applications.

3.2.2. Density of States

Each compound’s TDOS and PDOS are computed and shown in Figure 3 and Figure 4, respectively, to provide a more thorough understanding of the compounds’ electronic properties. Figure 3 shows that the Fermi level (EF) at 0 eV is denoted by the upright dashed line. The improved states close to the Electrochemical Frontier (EF) can significantly enhance the performance of a hybrid storage system, such as SCs and batteries. These states refer to the specific conditions or configurations of the electrodes, electrolyte, or the system as a whole, where the charge storage and transfer processes are optimized. These states can be contributed to enhance SC performance and charge storage by EDLC. The appearance of a finite DOS at the EF signifies that La₂XO₄ (X = Ni, Fe, Ba, and Pb) compounds are metallic, in agreement with the computational work [44]. In the energy range of −6 eV to 6 eV, the TDOS of every compound is measured. The TDOS data show the energy spectrum of the electronic distribution of each compound. Because of the flatlands in the band structure, the VBs in DOS at the Fermi level rise sharply [45]. The total DOS plot will show the number of available electronic states at each energy level. For metallic behavior, there should be states available at the Fermi level, indicating that the material can conduct electricity. According to the La₂XO₄ supercapacitance performance, it shows that it is suitable for SC applications. The PDOS can help to identify which atomic orbitals (e.g., s, p, d) contribute to the conduction states at the Fermi level. This is useful for understanding the material’s bonding and the origin of its metallicity. For example, a material with significant d-band contribution near the Fermi level might exhibit interesting magnetic or structural properties in addition to its metallic behavior. The PDOS of all compounds are calculated and presented in Figure 4.

La₂NiO₄

The PDOS of La₂NiO₄ is displayed in Figure 4a. In VB, O ions show the dominance over La and Ni ions. O-2p⁴ contributes more as compared to La-5d¹ and Ni-3d⁸. The material’s VB is significantly impacted by Ni-3d⁸ and O-2p⁴. There is no space between VB and CB to predict semiconductor characteristics. So, this is valid for SC-based energy devices. In CB, La ions contribute more, while Ni and O ions show a minor significance.

La₂FeO₄

Figure 4b shows the PDOS plot of La₂FeO₄ compound, where the VB is dominated by the O-2p⁴ state. Even though O-2p⁴ has a greater influence than La-5d¹ and Fe-3d⁶, its contributions are quite small. The occurrence of the forbidden energy gap illustrates the compound’s potential as an SC, including as a conductive material. La₂FeO₄ has no band energy gap. So, this is metallic in nature. Compared to La-5d¹, the effects of Fe-3d⁶ state on the CB is minor, but O-2p⁴ shows just a little contribution.

La₂BaO₄

The PDOS of La₂BaO₄ compound is displayed in Figure 4c. The band gap of La₂BaO₄ suggests that it has great potential to show the usage of SC energy devices. Ba-2p⁶ and O-2p⁴ states make up the majority contribution in VB, while La-5d¹ has a minor contribution. On the other hand, La-5d¹ shows significant influence in CB, while Ba-2p⁶ contributes minor, and O-2s²/O-2p⁴ display just a minute portion contribution in CB.

La₂PbO₄

Figure 4d shows the PDOS graph of the La₂PbO₄ compound_. In this plot, the identical results of La₂BaO₄ are observed. The O-2p⁴ state contributes dominantly, while Pb-6s², Pb-6p², and La-5d¹ contribute less in VB. In CB, La-5d¹, Pb-6s², and Pb-6p² contribute dominantly, while O-2s² and O-2p⁴ contribute very little.

3.3. Optical Properties

With regard to the potential for the usage of such compounds in solar cell applications, it is crucial to examine the optical characteristics of the orthorhombic La₂XO₄ crystal structure. It is necessary to increase the number of k-points in the irreducible BZ to precisely evaluate the optical properties. The dielectric function has also been computed, and the frequency-dependent optical properties of the La₂XO₄ compounds are also calculated. The two parts of the dielectric function ε(ω) in the form of real ε₁(ω) and imaginary ε₂(ω) dielectric functions can be used to characterize optical properties. A complex dielectric function expression is presented in Equation (2) [46]:

$$\mathsf{\epsilon }\left(\mathrm{\omega }\right)={\epsilon }_1\left(\mathrm{\omega }\right)+\mathrm{i}{\epsilon }_2\left(\mathrm{\omega }\right)$$

(2)

The momentum matrix P and DOS determine the potential evolutions from occupied to unoccupied states with fixed k-vectors throughout the BZ, which are represented by the imaginary component, ε₂(ω). The ε₂(ω) can be obtained by utilizing Equation (3) [47]:

$${\epsilon }_2\left(\mathrm{\omega }\right)=\left({\displaystyle \frac{h^2{e}^2}{{\pi w}^2{m}^2}}\right)\sum ij\mathrm{\int }{d}^3\mathrm{k}〈i_k\left|p^{\alpha }\right|j_k〉〈j_k\left|p^{\beta }\right|i_k〉\mathcal{x}\mathsf{\delta }\left({\epsilon }_{i_k}-{\epsilon }_{j_k}-\mathrm{\omega }\right)$$

(3)

where j_k and i_k are the crystal wave functions that correspond to the VB and CB with crystal momentum k, and the momentum matrix element (p) lies between the band α and β states with vector k of a crystal wave. The dielectric function’s real portion, (ε₁(ω)), can be expressed via Equation (4) [48]:

$${\epsilon }_1\left(\mathrm{\omega }\right)=1+{\displaystyle \frac{2}{\pi }}p{\int }_0^{\mathrm{\infty }}{\displaystyle \frac{{\omega }^{\prime }{\epsilon }_2\left({\omega }^{\prime }\right)}{{{\omega }^{\prime }}^2-{\left(\mathrm{\omega }\right)}^2}}\mathrm{d}\mathrm{\omega }$$

(4)

where p is the integral’s principal.

The following formulas can be used to compute the optical conductivity (Equation (5)), refractive index (Equation (6)), extinction coefficient (Equation (7)), absorption coefficient (Equation (8)), and reflectance (Equation (9)), {σ(ω), n(ω), k(ω), α(ω), and R(ω), respectively}, which are all directly correlated with the dielectric function [49,50]:

$$\mathsf{\sigma }(\mathrm{\omega })=-{\displaystyle \frac{i\omega }{4\pi }}\mathsf{\epsilon }(\mathrm{\omega })$$

(5)

$$\mathrm{n}(\mathrm{\omega })={\displaystyle \frac{1}{\sqrt{2}}}(\sqrt{{\epsilon }_1^2\left(\omega \right)+{\epsilon }_2^2\left(\omega \right)}+{\epsilon }_1{(\omega ))}^{1/2}$$

(6)

$$\mathrm{k}(\mathrm{\omega })={\displaystyle \frac{1}{\sqrt{2}}}(\sqrt{{\epsilon }_1^2\left(\omega \right)+{\epsilon }_2^2\left(\omega \right)}-{\epsilon }_1{(\omega ))}^{1/2}$$

(7)

$$\mathsf{\alpha }(\mathrm{\omega })=\sqrt{2\omega }(\sqrt{{\epsilon }_1^2\left(\omega \right)+{\epsilon }_2^2(\omega )}-{\epsilon }_1{(\omega ))}^{1/2}$$

(8)

$$\mathrm{R}(\mathrm{\omega })={\left|{\displaystyle \frac{\sqrt{\epsilon \left(\omega \right)-1}}{\sqrt{\epsilon \left(\omega \right)+1}}}\right|}^2$$

(9)

The material’s capacity to polarize under an external electric field is linked with its dielectric function. This function is frequently computed by using the response function formalism, in which the electronic structure of the material and its capacity to accumulate charge in response to an electric field define the dielectric response. The dielectric function can be broken into its real part (permittivity) and imaginary part (losses due to absorption and resistance). The real part is related to the material’s ability to store energy, while the imaginary part relates to energy dissipation.

The predicted optical properties can provide deep insights into La₂XO₄’s electric properties. The material’s capacity to store energy is revealed by the real part of the dielectric function. As determined by DFT, a material with a high dielectric constant can accumulate charge more effectively, directly increasing the energy density of the SC. The electronic component of the static dielectric function, the zero-frequency limit, or ε₁(0), is the utmost important value in the ε₁(ω) spectrum, shown in Figure 5a. The computed values of ε₁(0) for La₂XO₄ (X = Ni, Fe, Ba, and Pb) compounds are 9.73, 9.95, −5.02, and 14.01, respectively. From ε₁(0), the ε₁(ω) values began to rise, reaching their highest position, and then fell towards negative values for specific energy ranges. The incoming photon beam in the La₂XO₄ crystal structure was attenuated in these energy values. This shows that Eg and ε₁(0) have an inverse relationship. For multiple types of materials, a similar trend has been observed. The materials have exhibited a metallic nature when ε₁(ω) becomes negative or near to zero.

Energy dissipation is described by the imaginary part of the dielectric function; for a material with low absorption losses (low imaginary part), less energy will be wasted, leading to higher efficiency during charge/discharge cycles. DFT assists in locating materials with low dielectric losses, which enhances the performance of the supercapacitor by guaranteeing that more energy is stored. Additionally, ε₂(ω) is closely connected with the electronic band optimization of the compounds under study and defines their absorptive performance. The ε₂(ω) values for La₂XO₄ compounds demonstrate the interband transitions and indicate the complete response of compounds to turbulences instigated by electromagnetic radiation. The values of optical crucial points are 0.5, 1.01, 120.01, and 49.01 for La₂NiO₄, La₂FeO₄, La₂BaO₄, and La₂PbO₄, respectively, as presented in Figure 5b. These compounds tend to absorb interacting photons efficiently in the visible and microwave regions, as demonstrated by their respective energy values of 0.7 eV, 0.6 eV, 0.4 eV, and 0.9 eV, respectively.

When materials are exposed to light, their ability to conduct electricity is known as optical conductivity (σ(ω)). The optical conductivity can be calculated by examining the material’s response to an external electromagnetic field. It involves calculating the material’s electrons’ intraband and interband transitions when exposed to light or an alternating electric field. A material with high optical conductivity has good electron mobility, which is essential for fast charging and discharging cycles in SC. The observed σ(ω) spectra as a function of energy (0–14 eV) are presented in Figure 5c for La₂XO₄ (X = Ni, Fe, Ba, and Pb) compounds. The graph shows that the σ(ω) was originally zero for all of the compounds and then began to develop for La₂NiO₄, La₂FeO₄, La₂BaO₄, and La₂PbO₄ compounds at 0.2, 0.15, 0.35, and 0.52, respectively. The σ(ω) spectra of La₂NiO₄, La₂FeO₄, La₂BaO₄, and La₂PbO₄ compounds peaked at 8.214 eV, 8.23 eV, 8.01 eV, and 8.72 eV, respectively. Compounds with the highest conduction response have been found in the UV range.

Figure 5d shows the electron energy loss function (L(ω)), which measures the transmission energy loss in the under-study compounds and is another useful parameter for examining how materials behave when exposed to light. The L(ω) spectra of La₂XO₄ show the loss of energy under the influence of incident photon energy (above the material’s E_g). The L(ω) spectra show the highest values for La₂NiO₄, La₂FeO₄, La₂BaO₄, and La₂PbO₄ compounds at 12.01 eV, 11.99 eV, 11.85 eV, and 12.03 eV, respectively. Plasma resonance is typically referred to as the peaks in the L(ω) spectra, and plasma frequencies are the frequencies that correspond to these peaks.

The refractive index (n(ω)) is a vital parameter in the design of industrial optical devices, including photonic crystals, detectors, solar cells, wave guides, and LED lights. Furthermore, it is a decisive characteristic of semiconductors that relates to microscopic atomic interactions and determines the quantity of light refracted or bent by them. Figure 6a shows the n(ω) spectra for La₂XO₄ (X = Ni, Fe, Ba, and Pb) compounds. The static values of n(ω = 0) for La₂XO₄ (X = Ni, Fe, Ba, and Pb) are 4.93, 4.90, 13.52, and 14.01, respectively. Such materials are regarded as active optical materials because of their ability to bend light and reflect light. All of the compounds’ peaks then rise to their highest points, which are approximately 5 eV. For La₂XO₄ (X = Ni, Fe, Ba, and Pb) compounds, the n(ω) spectra drop sharply and fall below 1 at 5.31 eV, 5.30 eV, 5.33 eV, and 5.327 eV, respectively, confirming the metallic nature of the compounds and making them ideal for SC applications. Both theoretical and experimental observations of the superluminality phenomena are possible for these materials when n < 1.

Another important parameter for optical applications is the extinction coefficient k(ω), which shows how well these compounds can be absorbent of interacting photons at specific frequencies. Figure 6b displays the plot of k(ω) for all compounds, showing a pattern that is identical to ε₂(ω). The values of k(ω) peaks are 0.93, 0.75, 6.25, and 4.01 eV for La₂XO₄ (X = Ni, Fe, Ba, and Pb) compounds, respectively. Threshold energies are the values of k(ω) at which the peaks emerge. The spectra rapidly rise beyond the threshold energies, reaching their largest positions in the microwave and ultraviolet regions.

Energy (0–14 eV)-dependent absorption (α(ω)) spectra for La₂XO₄ (X = Ni, Fe, Ba, and Pb) are plotted in Figure 6c. Because the electronic band structure directly impacts the α(ω) spectra and is dependent on ε₂(ω) and ε₁(ω), the α(ω) is one of the most important features when evaluating the competency of an SC device. Additionally, it explains how deeply the interacting photons penetrate the medium. For La₂XO₄ (X = Ni, Fe, Ba, and Pb) compounds, the measurements reveal the highest margins of absorption lie in the UV range. The highest absorption co-efficient for La₂XO₄ (X = Ni, Fe, Ba, and Pb) compounds are 2.5 × 10⁵ cm⁻¹, 2.45 × 10⁵ cm⁻¹, 2.35 × 10⁵ cm⁻¹, and 2.63 × 10⁵ cm⁻¹, respectively, at 8.95 eV, 9.11 eV, 9.01 eV, and 9.18 eV, respectively. The spectra were initially zero. These compounds can be used in SCs because they are capable of absorbing interacting photons in the microwave and near-UV regions.

The reflected-to-entering-photons ratio is expressed by the reflection coefficient R(ω). Energy (0–14 eV)-dependent R(ω) plots of La₂XO₄ (X = Ni, Fe, Ba, and Pb) compounds are shown in Figure 6d. Massive interacting photon dispersion takes place at an energy range of 8–10 eV for all the compounds. Similarly, the plot clearly shows that the higher UV region (9.5 eV) has the highest reflectivity values of 0.35, 0.37, 0.81, and 0.71, respectively, with constant electric fields. They can serve as reflective coatings in the upper UV spectrum.

3.4. Mechanical Stability

Researchers have shown the keen interest for measuring the mechanical characteristics of materials as these characteristics demonstrate the material’s possibilities for industrial utilization. The mechanical properties also affect many other aspects of the materials, including phonon spectra, melting temperatures, equations of states, and interatomic potentials. Anisotropic bonding, the material’s structural stability, and the bonding characteristics between neighboring atomic planes are studied through evaluating the elastic constants.

For each composition of La₂XO₄ (X = Ni, Fe, Ba, and Pb), the elastic stiffness constants (C_ij) are measured and plotted in Figure 7a. The bulk modulus of elasticity (B), shear modulus of elasticity (G), Young’s modulus of elasticity (E), and Poisson’s ratio (v) are computed by using these stiffness constants. The elastic constants (C_ij) and the Born Elastic Stability Model are employed to assess the mechanical stability of the compounds. There exist nine distinct elastic constants for the orthorhombic symmetry, which are C₁₁, C₂₂, C₃₃, C₄₄, C₅₅, C₆₆, C₁₂, C₁₃, and C₂₃.

Table 2. Calculated elastic constants (C_ij) and other mechanical properties (Poisson’s ratio, Young’s modulus, Shear modulus, and Bulk modulus).

NAME	La2NiO4	La2FeO4	La2BaO4	La2PbO4
C11	59.76	60.44	114.42	242.34
C12	33.43	33.94	76.67	149.38
C22	44.54	43.83	148.67	230.89
C33	37.26	36.02	130.47	180.91
C44	8.23	8.29	45.46	39.51
C55	16.78	16.83	20.46	−1.83
C66	17.37	17.82	17.84	23.56
B	36.02	36.61	95.78	179.27
G	10.16	9.30	13.55	36.66
E	27.85	25.73	38.81	102.96
B/G	3.55	3.94	7.07	4.89
G/B	0.28	0.25	0.14	0.20
ν	0.37	0.38	0.43	0.40

lists the computed elastic constants for La₂XO₄ (X = Ni, Fe, Ba, and Pb) compounds. The well-known Born criteria (Equation (10)) must be met by the elastic constants to identify a stable orthorhombic system.

$$\begin{gathered} C_{ij}>0,\left(C_{11}+C_{33}-2C_{23}\right)>0,\left(C_{11}+C_{33}-2C_{13}\right)>0, \\ \left[C_{11}+C_{22}+C_{33}+2\left(C_{12}+C_{13}+C_{23}\right)\right]>0,and\left(C_{11}+C_{22}-2C_{12}\right)>0 \end{gathered}$$

(10)

Elastic constants for orthorhombic La₂XO₄ (X = Ni, Fe, Ba, and Pb) satisfy the Born stability conditions of Equation (10). A mechanically stable orthorhombic structure at the atmospheric pressure of the La₂XO₄ (X = Ni, Fe, Ba, and Pb) compounds is observed. Among all the compounds, La₂PbO₄ has shown the greatest elasticity values (179.27, 36.66, and 102.96 GPa for B, G, and E, respectively) in Figure 7b, whereas La₂BaO₄ has shown the lowest response. The derived elastic constants (C_ij) for orthorhombic La₂XO₄ (X = Ni, Fe, Ba, and Pb) are shown and compared in Table 2.

Furthermore, the following modulus equations can be used to determine the compounds’ brittleness or ductility [51,52]:

Bulk Modulus

$$\mathrm{B}={\displaystyle \frac{1}{3}}(C_{11}+C_{12})$$

(11)

Shear Modulus

$$G_H={\displaystyle \frac{1}{2}}(G_{\nu }+G_R)$$

(12)

$$G_{\nu }={\displaystyle \frac{1}{5}}\left(C_{11}-C_{12}+3C_{44}\right)$$

(13)

$$G_R={\displaystyle \frac{C_{44}(C_{11}-C_{12})}{{4C}_{44}+3(C_{11}-C_{12})}}$$

(14)

Young’s Modulus

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

(15)

Poisson’s Ratio

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

(16)

After gathering the pertinent data from the mechanical calculations using various equations, our goal is to discuss the brittle or ductile responses of the under-study compounds in detail. Using specific hypothesized relationships, the brittleness and ductility behavior of compounds can be described. One of the many criteria that are utilized to evaluate the ductile or brittle natures of compounds is the ratio of ductility (B/G), Cauchy pressure (Figure 7c). Moreover, Pugh’s ratio (B/G) is considered best to examine such nature of the materials, which should be approximately 1.75. As in the case of the under-study compounds, the B/G ratios lie below 1.75, which suggests that the compounds are ductile or fragile.

Figure 7d shows the Poisson’s ratio (ν) of the compounds, which describe the relative transverse compression–longitudinal extension ratio, its value mostly lies between 0 and 0.5. Also, higher values of this ratio describe the better plasticity of the materials. For La₂XO₄ (X = Ni, Fe, Ba, and Pb) compounds, the evaluated ν values are provided in Table 2, which are 0.37, 0.38, 0.43, and 0.40, respectively. These values are well within the range of 0-0.5. The values of ν vary steadily, indicating a plasticity increase. As there are no existing elastic calculations for these materials, experimentally or theoretically, it is not possible to provide a better comparison of these values with the literature.

3.5. Time Simulation

In Figure 8a, the simulation time versus temperature plot for the La₂NiO₄ compound is presented; the plot demonstrates the DFT-MD simulation starting from 323K with a sharp decrease in temperature with the passage of time steps [53]. In Figure 8b, MD simulation time versus temperature plot for La₂FeO₄ shows the stable behavior a minute unnoticeable variation with simulation time. Interestingly, in Figure 8c, the simulation time-dependent temperature plot of La₂BaO₄ is presented, plot shows that temperature increase rapidly from 323 K to ~4000 K with every interval of simulation time. The plot measured for La₂PbO₄ also shows the temperature rapid variation from 323 K to ~900 K with varying the simulation time (Figure 8d). The outcomes demonstrate the permanency/stability of the compounds after operating in the mentioned simulation time, which clearly shows the strong stability of the La₂NiO₄ and La₂FeO₄ compounds, while other two compounds are also operating, but they become heated with the fatigue conditions, which is in concord with outcomes from the manuscript clearly presenting the potential of compounds for the renewable energy storage applications.

The metallic behavior predicted by DFT calculations, particularly in the Fe- and Ni-based La₂XO₄ compounds, may not reflect the true electronic nature of these materials. This is likely due to the limitations of standard DFT methods, which often struggle to accurately describe systems with strongly correlated electrons—especially those involving partially filled d-orbitals, like in Ni and Fe. In such cases, the absence of corrections for on-site Coulomb interactions can result in an artificial metallic character. In contrast, for compounds where X is Ba or Pb, the metallicity could be more intrinsic. This may arise from substantial orbital mixing, such as between La 5d, O 2p, and the d-states of the X-site atom, which can lead to broad electronic bands crossing the Fermi level. Similar metallic features have also been reported in other structurally related materials, as summarized in

Table 3. Comparison of theoretical studies on structural, electronic, and magnetic properties of various materials.

Materials	Functional	Key Findings	References
La2XO4 (X = Ni, Fe, Ba, Pb)	DFT (CASTEP, PBE-GGA)	Metallic nature, ideal for SC applications, high optical conductivity, mechanical stability confirmed by Born criteria	This study
RBaMn2O6 (R = Nd, Pr, La)	FP-LAPW, PBE-GGA	Half-metallic ferromagnets, high magnetic moments (7–10 μB), potential for spintronics	[27]
Sr2GdReO6	FP-LAPW, Spin-DFT	Half-metallic nature (HM), ferromagnetic stability, magnetic moment = 9 μB	[54]
X2MnUO6 (X = Sr, Ba)	DFT	Half-metallic ferromagnetism, Mn 3d–O 2p–U 5f hybridization, potential for spintronics	[55]
FeCrRuSi	DFT	Half-metallic material with a total magnetic moment of 2.0 μB	[56]
K2NaMI6 (M = Mn, Co, Ni)	GGA-PBE	Half-metallic ferromagnetism, magnetic moment = 4 μB, thermoelectric	[57]
Cs2NpBr6	GGA-PBE, GGA+U	Half-metallic ferromagnetic, magnetic moment is 3 μB via spin polarization	[58]

.

4. Conclusions

The electronic, structural, optical, and mechanical properties of RPP family member La₂XO₄ (X = Ni, Fe, Ba, and Pb) compounds is optimized by using the CASTEP code using GGA and PBE functions. The band structure results of La₂XO₄ compounds show that these compounds have no band gap, so their nature is metallic, which is ideal for SCs. The time simulation versus temperature stability plots of the compounds are calculated, which show temperature stability for the La₂NiO₄ and La₂FeO₄ compounds and variations in every interval of time for La₂BaO₄ and La₂PbO₄ compounds. Herein, the optical properties of these compounds, such as reflectivity, absorption, dielectric function (real and imaginary), conductivity, refractive index, extinction coefficient, and loss function, also suggest the potential for utilizing these compounds for renewable energy storage applications. The imaginary component of dielectric function ε₂(ω) shows some behaviors, such as that of the absorption co-efficient in the range of the UV region. The calculated Pugh’s ratio, Poisson’s ratio, and Cauchy’s pressure values show that all materials behave in an incredibly ductile manner. These calculations show that these compounds exhibit elastic stability and that B/G is greater than 1.75, which supports the compounds for flexible SCs electrodes.