First-Principles Calculations of the Structural, Mechanical, Optical, and Electronic Properties of X₂Bi₄Ti₅O₁₈ (X = Pb, Ba, Ca, and Sr) Bismuth-Layered Materials for Photovoltaic Applications

Abstract

For the first time, density functional theory (DFT) calculations have been employed for the measurement of the structural, mechanical, optical, and electrical properties of a bismuth-layered structure ferroelectrics (BLSFs) family member possessing an orthorhombic structure with Cmc2₁ space group. Based on the exchange–correlation approximation, our calculations show that Pb₂Bi₄Ti₅O₁₈ possesses an indirect band gap, while the materials X₂Bi₄Ti₅O₁₈ (X = Ba, Ca, and Sr) demonstrate direct band gap, where the estimated density functional fundamental band gap values lie between 1.84 to 2.33 eV, which are ideal for photovoltaic applications. The optical performance of these materials has been investigated by tuning the band gaps. The materials demonstrated outstanding optical characteristics, such as high absorption coefficients and low reflection. They exhibited impressive absorption coefficient (α = 10⁵ cm⁻¹) throughout a broad energy range, especially in the visible spectrum (10⁵ cm⁻¹ region). The findings show that the compounds demonstrate lower reflectivity in the visible and UV regions, making them suitable for single-junction photovoltaic cells and optoelectronic applications. The Voigt–Reuss–Hill averaging technique has been employed to derive elastic parameters like bulk modulus (B), Young’s modulus, shear modulus (G), the Pugh ratio (B/G) and the Frantesvich ratio (G/B) at 0.1 GPa. The mechanical stability of the compounds was analyzed using the Born stability criteria. Pugh’s ratio and Frantesvich’s ratio show that all the compounds are ductile, making them ideal for flexible optical applications.

1. Introduction

The continuously growing demands of energy storage applications in society, along with the impacts of global warming and the depletion of petroleum reserves, have compelled scientists to explore alternative fuel sources capable of performing efficiently under fatigue conditions. The most viable options to overcome such issues are the development of renewable energy sources, which include biomass, solar, wind, and other sources [1,2,3]. The content of solar energy that reaches the Earth’s surface is far more than the amount of energy used worldwide today. Solar energy is regarded as one of the finest energy technology solutions owing to its cheap cost, ease in accessibility, renewable nature, and lack of an environmental damage effect. One significant strategy for providing energy to the world is to develop solar photovoltaic (PV) and photocatalytic (PC) technologies, which can convert sunlight into chemical and electrical energy [4,5,6,7]. To accomplish this, semiconductor materials with the appropriate band gaps must be engineered to absorb light and create free electrons. It is the desire of the time to miniaturize the size of devices to near-atomically thin sizes while maintaining the high semiconductor characteristics required for today’s PV and PC applications. In semiconductor-based PV cells, photon absorption in the donor material generates excitation (a pair of electrons and holes), which may be separated at a D–A (donor–acceptor) interface. The created hole goes through the material and is collected at the anode. When an electron is separated, it may be transferred to an acceptor substance and transported to the cathode [8,9,10,11,12].

The scientific community has paid limited attention to bismuth-layered structure ferroelectrics (BLSFs) for optoelectric and photovoltaic devices because of their other multiple potential applications, which include photo–ferroelectric, ferroelectric–mechanical, and ferroelectric–elastic applications. For many decades, BLSFs have been significantly utilized as a host material for their ferroelectric and up-conversion luminescence (UCL) capabilities, allowing the integration of optoelectronic devices. These BLSFs are perfect as UC host materials because, unlike glasses and halides, they have stable mechanical, thermal, and chemical characteristics as well as a high Curie temperature (T_C). BLSFs with the general formula (Bi₂O₂)²⁻ (A_m−1B_mO_3m+1)²⁺ belong to the aurivillius family [12,13,14,15,16]. The (Bi₂O₂)²⁻ layer is positioned between the perovskite blocks (A_m−1B_mO_3m+1)²⁺. The A site might be monovalent, divalent, trivalent, or a combination of these, where tetravalent, pentavalent, and hexavalent cations often occupy the B site, and m represents the octahedral layer, which influences the dielectric and ferroelectric properties of BLSFs.

These Bi-based ternary metal oxides are direct bandgap semiconductors with a very small energy gap (2.4–2.8 eV); hence, they can absorb light and hold significant potential for solar energy applications. They are also utilized as light-carrier masses, which should aid in the efficient extraction and separation of photo-excited charge carriers. They are also utilized as photocatalytic materials under visible light irradiation to split water into O₂ and H₂, as well as to degrade organic pollutants [17,18,19,20,21,22,23].

Among the family of BLSFs, SrBi₂B₂O₉ with B = Nb and Ta have been explored in theoretical studies like Shu et al., where they performed the first principal study and measured the band gaps for SrBi₂Ta₂O₉ (2.44 eV) and SrBi₂Nb₂O₉ (2.35 eV) [24]. Li et al. performed the theoretical calculations for the measurements of the band gap for another BLSF family member Na_0.5Bi_2.5Nb_2−xW_xO_9+δ and found that it lies in the range of 1.61–1.99 eV [25]. Recently, Fatima et al. performed the theoretical DFT calculations for XNb₂Bi₂O₉ with X = Ca, Ba, Be, Mg, and Sr. The materials have presented bandgaps in the range of 1.8 to 2.6 eV for all compounds [26]. Li et al. calculated the electronic energy band structure of M-doped BaTiO₃ (M = Al, Fe, V, Nb) systems and studied the effect of different metal atoms (M) on the electronic structure of BaTiO₃ [27]. Wu et al. calculated the electronic energy band structure of (1 − x)BaTiO₃₋xBiScO₃ with the first-principles method and discovered factors related to the orbital hybridization and electrical properties [28]. Stambouli et al. performed a first-principles study on the SrBi₂B₂O₇ crystal for ultraviolet optoelectronic applications and measured a band gap range of 2.812–3.94 eV by different types of calculation approaches [29]. Cai et al. performed a first-principles study calculation within DFT on Bi₄Ti₃O₁₂ and measured an indirect band gap of 1.87 eV [30] experimentally. Han, J. Y. et al. were able to enhance the optical band gap from 2.69 eV to 2.75 eV for iron-doped lanthanum-modified Bi₄Ti₃O₁₂ thin films fabricated on a SrTiO₃ substrate using RF sputtering [31]. Zhang et al. synthesized Sr₂Bi₂Nb₂O₉-xEr multifunctional ceramics, which exhibited strong luminescence and improved temperature stability, with band structure and density of states analysis indicating effective photon energy absorption [32]. The literature survey compelled us to explore the hidden merits of BLSF materials for optoelectronic applications.

In this work, we investigated the effects of substituting X with Pb, Ba, Ca, and Sr on the structural, electrical, optical, and mechanical attributes of X₂Bi₄Ti₅O₁₈. A local density approximation (LDA) using the Ceperleye–Alder and Perdewe–Zunger (CA-PZ) functional technique built within the CASTEP algorithm and density functional theory (DFT) calculations were performed to explore the comprehensive energy calculations. This research also studied the band structure, partial density of states (PDOS), and total density of states (TDOS) of the compounds. Reflectivity, absorption, refractive index, extinction coefficient, dielectric function, optical conductivity, and loss function were also investigated to optimize the optical properties of the compounds. Furthermore, using the Forcite module, mechanical properties were explored, including elastic constants, young modulus, shear modulus, bulk modulus, B/G, and G/B ratio.

2. Methodology

Cambridge Serial Total Energy Package (CASTEP) computer code was utilized at 400 eV cut-off plane-wave energy for the first-principles measurements (based on the plane-wave pseudopotential approach of DFT); measurements were performed by employing LDA and CA-PZ functions [33]. The atomic positions and the lattice parameters of all the X₂Bi₄Ti₅O₁₈ unit cells were optimized with the CASTEP code. The geometrical optimization and structural stability of the compounds were calculated by the Broyden–Fletcher–Goldfarb–Shanno (BFGS) algorithm technique, keeping the total energy fixed to 2.0 × 10⁻⁵ eV/atom, self-consistent field (SCF) at 1.0 × 10⁻⁶ Å, maximum force of 0.05 eV/Å, maximum stress of 0.1 GPa, and maximum displacement at 0.002 Å. The structural, electronic, and optical attributes of X₂Bi₄Ti₅O₁₈ were calculated using exchange–correlation by LDA with CA-PZ functions [34,35] to provide a better outlook for a system with a larger number of atoms.

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

(1)

$${\epsilon }_2\left(\omega \right)={\displaystyle \frac{4{\pi }^2{\mathrm{e}}^2}{m^2{\omega }^{2}V}}{\displaystyle \sum }{{\displaystyle \int }}_{Bz}{|\langle {\psi }_k^v\left|\overrightarrow{p_i}\right|{\psi }_k^C\rangle |}^2\delta \left(E_{{\psi }_k^c}-E_{{\psi }_k^v}-\mathrm{ħ}\omega \right)$$

(2)

The real part ${\epsilon }_1\omega$ in Equation (1) is defined as the electronic contribution to the dielectric constant. The Kramers–Kronig dispersion allows the real part ${\epsilon }_1\omega$ of the frequency-dependent dielectric function to be deduced from the imaginary part ${\epsilon }_2\omega$, where P denotes the principal value of the integral and the imaginary part ${\epsilon }_2\omega$ of the dielectric function is expressed as the momentum matrix elements between the occupied and unoccupied electronic states. Here, ${\epsilon }_2\omega$ can be calculated directly using Equation (2), where e is the electronic charge, ${\psi }_k^C \mathrm{and} {\psi }_k^v$ are the conduction and valence band wave functions at k, respectively. V is the unit cell volume, and ω is the frequency of the incident photon [36]. All the properties were calculated using the OTFG ultrasoft pseudopotentials plane-wave method of USP. The Monkhorst pack grid was used for the k-integration, with a 4 × 4 × 4 k-point mesh throughout the Brillouin zone. The values of elastic constants were calculated using the Forcite module in the software using a universal force field, and charges as assigned at fine quality. In the case of the electrostatic summation method, “Ewald” was used. Van der Waals’ “atom-based” summation was chosen during the calculation of the mechanical properties of all compounds.

3. Results and Discussion

3.1. Structural Properties

X₂Bi₄Ti₅O₁₈ (X = Pb, Ba, Ca, Sr) possesses the orthorhombic structure with Cmc2₁ space group symmetry, having a total of 116 atoms, in which X (X = Pb, Ba, Ca, Sr) contains 08, Bi has 16, Ti has 20 atoms, and O has a total of 72 atoms as shown in Figure 1. Five inequivalent Ti⁴⁺ sites are present in the structure, so five corner-sharing TiO₆ octahedrals are formed when five Ti⁴⁺ sites are linked to six O²⁻ ions in a 6-coordinate geometry. Two X²⁺ locations are not fixed, as both X²⁺ locations are associated with 12 O²⁻ ions in a 12-coordinate geometry. There exist 4 lattice sites for Bi³⁺ that are inequivalent. The first Bi³⁺ site is bound to seven O²⁻ ions in a 7-coordinate geometry. The second Bi³⁺ site is bound to seven O²⁻ ions in a seven-coordinate configuration. Six O²⁻ ions and Bi³⁺ are bound in a 6-coordinate geometry at the third and fourth Bi³⁺ sites.

There exist 18 O²⁻ sites, which are inequivalent. O²⁻ is twistedly bent with two Ti⁴⁺, one X²⁺, and one Bi³⁺ ions at the initial O²⁻ site. O²⁻ is bound in a 4-coordinate geometry to two Ti⁴⁺, one X²⁺, and one Bi³⁺ ions at the second O²⁻ site. O²⁻ is bound in a 2-coordinate geometry to two equivalent Ti⁴⁺, two equivalent X²⁺, and one equivalent Bi³⁺ ions at the third O²⁻ site. O²⁻ is bound in a 2-coordinate geometry to two equivalent Ti⁴⁺, two equivalent X²⁺, and one equivalent Bi³⁺ ions at the fourth O²⁻ site. O²⁻ is bound to four Bi³⁺ ions in the fifth O²⁻ site to produce a combination of edge-sharing and corner-sharing O-Bi⁴⁺ tetrahedral. O²⁻ is bound to four Bi³⁺ ions at the sixth O²⁻ site to produce a mixture of edge-sharing and corner-sharing O-Bi⁴⁻ tetrahedral. O²⁻ is bound in a 4-coordinate geometry to two Ti⁴⁺, two X²⁺, and one Bi³⁺ ions at the seventh O²⁻ site. O²⁻ is bound to two Ti⁴⁺, two X²⁺, and one Bi³⁺ ions in a 2-coordinate geometry for the eighth O²⁻ site. O²⁻ is twistedly bent to two Ti⁴⁺, two X²⁺, and one Bi³⁺ ions at the ninth O²⁻ site. O²⁻ is bound to two Ti⁴⁺, two X²⁺, and one Bi³⁺ ions in a 4-coordinate geometry at the tenth O₂ site. For the eleventh O²⁻ site, O²⁻ is coupled to two Ti⁴⁺, two equivalent X²⁺, and one Bi³⁺ ions in a 4-coordinate geometry. O²⁻ is linked in a 3-coordinate geometry to two Ti⁴⁺, two equivalent X²⁺, and one Bi³⁺ ions at the twelfth O²⁻ site. One Ti⁴⁺ and two Bi³⁺ ions are bound to O²⁻ in a deformed single-bond geometry at the thirteenth O²⁻ site. One Ti⁴⁺ and two Bi³⁺ ions are bound to O²⁻ in a deformed single-bond geometry at the fourteenth O²⁻ site. O²⁻ is bound in a 3-coordinate geometry to two Ti⁴⁺, one X²⁺, and one Bi³⁺ ions at the fifteenth O²⁻ site. O²⁻ is bound in a 2-coordinate geometry to two Ti⁴⁺, one X²⁺, and one Bi³⁺ ions at the sixteenth O²⁻ site. At the seventeenth O²⁻ site, O²⁻ is bound to two Ti⁴⁺, two equivalent X²⁺, and one Bi³⁺ ions in a 1-coordinate geometry. At the eighteenth O²⁻ site, O²⁻ is bound to two Ti⁴⁺, two equivalent X²⁺, and one Bi³⁺ ions in a 1-coordinate geometry.

3.2. Band Structure

A significant source of information regarding the possible energy states that electrons may occupy (i.e., energy bands) is the electronic band structures of all the compounds, which are presented in Figure 2a–d. Results are computed using DFT with LDA calculations. The monoelectronic Khon-Sham equations can be used to generate these energies. The properties of all under observation compounds, such as whether they are insulators, conductors, or semiconductors, can be determined by their energy band structures. The valence band (VB) maximum and the conduction band (CB) minimum are subtracted to obtain the insulator and semiconductor natures of the compounds. Our calculations indicate that, for X₂Bi₄Ti₅O₁₈ (where X = Ba, Ca, and Sr), the VB maxima and CB minima are located at the same k-point, resulting in direct band gaps of 2.319 eV, 2.30 eV, and 2.311 eV, respectively, confirming that these compounds are direct band gap semiconductors. In contrast, Pb₂Bi₄Ti₅O₁₈ exhibits VB maxima and CB minima at different points in the Brillouin zone, which results in an indirect band gap of 1.848 eV, making it an indirect band gap semiconductor. An indirect band gap indicates a weak spin-orbit coupling effect near the Z symmetry points [37].

3.3. Density of States

We have calculated the TDOS and PDOS of all the compounds to better understand their electronic band structures. The vertical dashed line in Figure 3 represents the Fermi level (EF) at zero eV. In Figure 3 TDOS data of all the compounds are displaying their electron distribution’s energy spectra. The energy range, from −6 eV to 6 eV, is selected to analyze the total and partial density of states. A maximum of 3.2 states/eV in the CB and −1.1 states/eV in the VB at Fermi level EF display the total density of the states.

Figure 4a shows that Pb₂Bi₄Ti₅O₁₈ has a narrow energy band gap. In the VB, Ti-3d², Bi-5p⁶, and O-2p⁴ display most of the contribution, while Pb-6p² is contributing the minute portion. Forbidden energy gaps are a sign of a material’s semiconductor ability. The CB is mostly influenced by the atoms Bi-6s² and Ti-3d², with very little effect from other elements. Figure 4b demonstrates the high dominance of the Ba²⁺ and O²⁻ ions in the VB area of Ba₂Bi₄Ti₅O₁₈. O-2p⁴, Ba-4d¹⁰, and Ba-5p⁶ demonstrate a strong impact, while Bi-3p⁶ is not very prominent. The Ba₂Bi₄Ti₅O₁₈ compound has an energy band gap of 2.319 eV, consistent with the electronic band structure value. The impact of Ba²⁺ and Bi³⁺ ions on the CB is comparatively lower than that of Ti⁴⁺ ion, which has a significant influence. The VB of Ca₂Bi₄Ti₅O₁₈ in Figure 4c is mostly contributed by O-2p⁴ and Bi-3P⁶ states, whereas the CB is primarily influenced by Ti⁴⁺ ion, with small contributions from Bi³⁺ and Ca²⁺ ions as well. The partial density of states for Sr₂Bi₄Ti₅O₁₈ is displayed in Figure 4d, where Sr²⁺ and O²⁻ ions are dominant in the VB, with Bi³⁺ making a negligible contribution. The VB of the compound is significantly influenced by O-2p⁴ and Sr-3d¹⁰ states. In the CB, Ti⁴⁺ and Bi³⁺ ions make stronger contributions than the other elements. The O²⁻ and Sr²⁺ ions in the CB are not contributing prominently.

3.4. Dielectric and Optical Properties

Figure 5a,b shows the imaginary and real parts of the dielectric functions ${\epsilon }_1\left(\omega \right)$ and ${\epsilon }_2\left(\omega \right)$ of X₂Bi₄Ti₅O₁₈ (where X = Pb, Ba, Ca, and Sr). For X₂Bi₄Ti₅O₁₈ (X = Pb, Ba, Ca, and Sr), the highest values of ${\epsilon }_1\left(\omega \right)$ are 11.81, 10.99, 10.97, and 10.92, respectively. As seen in Figure 5a, these values are found in the electromagnetic spectrum’s visible range. Based on the evaluation of ${\epsilon }_1\left(\omega \right)$, Pb₂Bi₄Ti₅O₁₈ (7.45) has a higher static dielectric constant than Ba₂Bi₄Ti₅O₁₈ (6.84), Sr₂Bi₄Ti₅O₁₈ (6.68), and Ca₂Bi₄Ti₅O₁₈ (6.67). This static dielectric function data reveals that the analyzed compounds would be ideal candidates for solar energy applications. In the ultraviolet region of the spectrum, the distribution of ${\epsilon }_1\left(\omega \right)$ clinches negative values at a certain energy range.

The dielectric and metallic natures at the specified energies are shown by the positive and negative values of ${\epsilon }_1\left(\omega \right)$ which can be calculated by Equation (1). Metallic nature has been observed for the compounds at different energies, like Pb₂Bi₄Ti₅O₁₈ exhibits between 6.56 and 10.86 eV, Ba₂Bi₄Ti₅O₁₈ between 6.56 and 11.04 eV, Sr₂Bi₄Ti₅O₁₈ between 6.71 and 10.80 eV, and Ca₂Bi₄Ti₅O₁₈ between 6.61 and 10.91 eV. The energies at which these compounds change from metallic to dielectric and where ${\epsilon }_1\left(\omega \right)$ approaches to zero, the phenomenon is known as plasmon excitations. A semiconductor’s optical properties are highly controlled by plasmon excitations. Plasmon energies for Pb₂Bi₄Ti₅O₁₈, Ba₂Bi₄Ti₅O₁₈, Sr₂Bi₄Ti₅O₁₈, and Ca₂Bi₄Ti₅O₁₈ are 4.09, 4.26, 4.18, and 4.09 eV, respectively.

Figure 5b displays the imaginary part ${\epsilon }_2\omega$ of the dielectric function, which is drawn against the photon energy (Equation (2)). Dielectric function distribution spectra are used to illustrate the main optical transitions between VBs and CBs. It is evident in Figure 5b that the transitions of X₂Bi₄Ti₅O₁₈ (X = Ba, Pb, Ca, and Sr) occur in the UV region. For X₂Bi₄Ti₅O₁₈ (X = Ba, Pb, Ca, and Sr), the optical transitions between the VB maxima and CB minima at the mid-UV range occur at 4.74, 4.62, 4.64, and 4.66 eV, respectively.

For the practical applications of these compounds in optoelectronic devices, the refractive index $n\left(\omega \right)$ information plays a crucial role (Equation (3)), which indicates the data pertaining to material transparency; as $n\left(\omega \right) \ge 1$ designates an optical substance of transparent nature. Figure 5c illustrates the $n\left(\omega \right)$ analysis for X₂Bi₄Ti₅O₁₈ (X = Pb, Ba, Ca, and Sr), where $n\left(\omega \right)$ values reach the maximum limit at the end of the visible range and at the start to decrease in the UV region. Pb₂Bi₄Ti₅O₁₈ (3.51) has the highest $n\left(\omega \right)$ value than Ba₂Bi₄Ti₅O₁₈ (3.38), Ca₂Bi₄Ti₅O₁₈ (3.37), and Sr₂Bi₄Ti₅O₁₈ (3.32). Due to the relatively narrower band gap of Pb₂Bi₄Ti₅O₁₈ (n = 2.73), it demonstrates the highest $n\left(\omega \right)$ value than Ba₂Bi₄Ti₅O₁₈, Ca₂Bi₄Ti₅O₁₈, and Sr₂Bi₄Ti₅O₁₈ (n = 2.61, 2.57, and 2.58). Our results demonstrate that these compounds are suitable choices for solar cell and photovoltaic applications. These compounds reach the highest values in the visible range of the light spectrum for their refraction spectra. As photon energy increases in the ultraviolet (UV) region, $n\left(\omega \right)$ begins to decrease and reaches a value of less than 1 between 8 and 9 eV. According to these values, the compounds would be transparent in the visible range and UV region, but if the photon energy ranges above 1, the transparency would disappear.

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

(3)

Similar profiles may be observed in the imaginary part ${\epsilon }_2\left(\omega \right)$ and extinction coefficient $k\left(\omega \right)$ of the dielectric constant, as shown in Figure 5b and Figure 5d, respectively. Similar to the imaginary fraction ${\epsilon }_2\left(\omega \right)$ of the dielectric loss, the extinction coefficient $k\left(\omega \right)$ also originates at a certain threshold energies. For X₂Bi₄Ti₅O₁₈ (X = Pb, Ba, Ca, and Sr), the threshold energy values for extinction coefficient $k\left(\omega \right)$ are around 2.0, 2.30, 2.33, and 2.31 eV, respectively. Extinction coefficient $k_{max}\left(\omega \right)$ are 2.086 at 6.89 eV, 1.81 at 6.31 eV, 1.87 at 6.94 eV, and 1.85 at 6.85 for Pb₂Bi₄Ti₅O₁₈, Ba₂Bi₄Ti₅O₁₈, Ca₂Bi₄Ti₅O₁₈, and Sr₂Bi₄Ti₅O₁₈ compounds, respectively. The extinction coefficient’s $k\left(\omega \right)$ spectrum then falls towards unity after reaching its maximum values.

The reflectivity $R\left(\omega \right)$ of these compounds can be determined using Equation (4) by utilizing the calculated values of the extinction coefficient $k\left(\omega \right)$ and refractive index $n\left(\omega \right)$ [38].

$$R\left(\omega \right)={\displaystyle \frac{{\left(1-n\right)}^2+k^2}{{\left(1+n\right)}^2+k^2}}$$

(4)

Frequency-dependent reflectivity $R\left(\omega \right)$ for X₂Bi₄Ti₅O₁₈ (where X = Pb, Ba, Ca, and Sr), the calculated spectra are displayed in Figure 6a. The values of static reflectivity (zero frequency limit) for X₂Bi₄Ti₅O₁₈ (X = Pb, Ba, Ca, and Sr) are 0.21, 0.199, 0.195, and 0.194, respectively. According to the spectrum, Pb₂Bi₄Ti₅O₁₈ has the highest reflectivity compared to X₂Bi₄Ti₅O₁₈ (where X = Ba, Ca, and Sr). The frequency-dependent reflectivity $R\left(\omega \right)$ spectra of X₂Bi₄Ti₅O₁₈ (where X = Pb, Ba, Ca, and Sr) display their highest peaks in the energy range of 4 to 10 eV. For X₂Bi₄Ti₅O₁₈ (X = Pb, Ba, Ca, Sr), the maximum reflectivity $R\left(\omega \right)$ values are 0.477, 0.349, 0.404, and 0.356, respectively.

Figure 6b presents the calculated absorption spectra for X₂Bi₄Ti₅O₁₈ (X = Pb, Ba, Ca, and Sr), which are plotted as a function of photon energy. Absorption coefficient $\alpha \left(\omega \right)$ is considered one of the important factors in determining the potential of solar devices (Equation (5)) [39]. Given that the spectra of the compounds strongly rely on the electronic band structures, the behavior of the $\alpha \left(\omega \right)$ is also comparable to the ${\epsilon }_2\omega$. For X₂Bi₄Ti₅O₁₈ (X = Pb, Ba, Ca, and Sr), the absorption edges are situated at 1.84, 2.31, 2.3, and 2.31 eV, respectively. The first part of the absorption spectra begins in the visible region, while the absorption peaks reach their maximum values in the UV region. For X₂Bi₄Ti₅O₁₈ (X = Pb, Ba, Ca, and Sr), the maximum values of absorptions in the UV range are measured as 2.585 × 10⁵ cm⁻¹ at 8.1 eV, 2.051 × 10⁵ cm⁻¹ at 7.6 eV, –2.340 × 10⁵ cm⁻¹ at 8.11 eV, and 2.082 × 10⁵ cm⁻¹ at 7.90 eV, respectively. The values at which some compounds begin to change from dielectric to metallic are associated with the maximum absorption values. These materials’ remarkably high and acceptable absorption properties highlight their potential for solar cell and photovoltaic applications.

$$\alpha ={\displaystyle \frac{2k\omega }{c}}$$

(5)

The semiconductor’s optical conductivity is associated with the distinct space filled by electrons in the energy band orbits and free electrons. The holes and free electrons in the crystal molecules also contribute to the conductivity of the compounds. The optical conductivity varies for X₂Bi₄Ti₅O₁₈ (X = Pb, Ba, Ca, and Sr) with increasing photon energy, which is evident in Figure 6c. Similar to the absorption coefficient, optical conductivity begins with values that are roughly equal since they are directly proportional to the optoelectronic properties.

Figure 6d presents the energy-dependent loss function of the compounds, which is influenced by heating, scattering, and other processes. The observed optical loss in the visible region has increased up to 0.07, which is extremely small and negligible. Therefore, the visible light region’s greatest attenuation and least energy loss indicate the compounds’ potential for usage in clean energy applications. The optical loss function is calculated by the Equation (6) [38].

$$L\left(\omega \right)=-Im\left({\displaystyle \frac{1}{\epsilon }}\right)={\displaystyle \frac{{\epsilon }_2\left(\omega \right)}{{\epsilon }_1^2\left(\omega \right)+{\epsilon }_2^2\left(\omega \right)}}$$

(6)

3.5. Mechanical Properties

Figure 7a presents the elastic constants for X₂Bi₄Ti₅O₁₈ (X = Pb, Ba, Ca, and Sr) compounds, which have not been reported experimentally or theoretically before. Our calculations represent the first attempt to investigate their elastic characteristics. The stability of these compounds is affected by several variables that impact cell degradation, stress, and strain in solar cells. The mechanical characteristics of solar cells can be affected by their design, manufacturing, and industrial processing. Thus, using Voigt–Reuss–Hill approximations, the anisotropy dependence on the elastic constants (bulk modulus B, shear modulus G, and Young’s modulus) for X₂Bi₄Ti₅O₁₈ (X = Pb, Ba, Ca, and Sr) are presented in Figure 7b (Equations (7) and (8)) [40].

$$B={\displaystyle \frac{1}{3}}\left(C_{11}+2C_{12}\right)$$

(7)

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

(8)

It is analyzed that the shear modulus $G$ has lower values than the bulk modulus $B$, which suggests that the compounds under study are more resilient to volume compression than shear deformation. A compound’s Young’s modulus $E$ is a measure of its stiffness. The stiffer the compound, the higher the Young’s modulus [41].

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

(9)

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

(10)

$$G_H={\displaystyle \frac{1}{2}}\left(G_v+G_R\right)$$

(11)

Following certain mathematical procedures, the stability requirements for the orthorhombic structure of X₂Bi₄Ti₅O₁₈ (X = Pb, Ba, Ca, and Sr) are well met to the defined standards [42]:

$$\begin{gathered} C11+C22 - 2C12>0, C11+C33 - 2C13>0, C22+C33 - 2C23>0, C11+C22+C33+2C12+2C13+ \\ 2C23>0, C11>0, C22>0, C33>0, C44>0, C55>0, C66>0 \end{gathered}$$

(12)

We now aim to discuss more about the brittle or ductile natures of these compounds after collecting the relevant information from the mechanical calculations (Equations (9)–(12)). It is possible to describe the ductility and brittleness behaviors of compounds using certain proposed relationships (Equation (9) to Equation (12)). We can determine if a certain compound is brittle or ductile based on a variety of characteristics, including the ratio of ductility, i.e., $\left(B/G\right)$ Cauchy pressure $\left(Cij\right)$ (Figure 7c). Pugh’s ratio ($B/G$ ratio) [43] is also a helpful ratio that has been proposed as a standard for differentiating between brittle and ductile materials. The critical value that differentiates between brittle or ductile nature for materials is around 1.75. A $B/G$ ratio less than 1.75 indicates that the material is fragile (ductile). The Pugh’s ratios of the compounds are investigated, which are greater than the crucial value, confirming the ductile character of these compounds. Furthermore, for all the compounds, Frantesvich ratios $\left(G/B\right)$ are significantly lower than 0.571, indicating their ductile character as well (Figure 7d). Detailed analysis of the mechanical properties of all the compounds is provided in

Table 1. Elastic constant C_ij, and calculated values of bulk modulus B, shear modulus G, Young’s modulus E, Pugh ratio (B/G), Frantesvich ratio (G/B) for the X₂Bi₄Ti₅O₁₈ (where X = Pb, Ba, Ca, and Sr).

NAME	Pb2Bi4Ti5O18	Ba2Bi4Ti5O18	Ca2Bi4Ti5O18	Sr2Bi4Ti5O18
C11	308.61	309.02	278.90	285.10
C12	275.85	267.58	239.51	250.68
C22	325.12	321.62	259.26	257.94
C33	294.90	320.07	265.49	272.27
C44	35.83	80.45	56.41	58.66
C55	97.43	81.74	76.31	81.05
C66	51.01	64.48	58.86	61.49
B	342.673	278.48	230.343	248.34
G	16.45	93.64	67.972	102.72
E	33.4	32.42	23.392	35.891
B/G	10.259	8.587	9.847	7.5
G/B	0.09	0.11	0.1	0.144

for comparison analysis. Such ductile nature of the compounds makes them the ideal candidates for flexible solar devices.

4. Conclusions

In the current work, the LDA approach with Ceperleye–Alder and Perdewe–Zunger (CA-PZ) functional methods is employed to calculate the structural, optical, electronic, and mechanical properties of X₂Bi₄Ti₅O₁₈ (X = Pb, Ba, Ca, and Sr). Based on the electronic band structure calculations, it is confirmed that X₂Bi₄Ti₅O₁₈ (X = Pb, Ba, Ca, and Sr) compounds are semiconductors with bandgap values of 1.848 eV, 2.319 eV, 2.30 eV, and 2.311 eV, respectively. These compounds are recognized as potential candidates for solar applications due to their high absorption in the visible and UV spectra. These compounds show remarkable reflection in the higher UV region, suggesting that they are suitable semiconductor materials for optical devices. The $B/G$ ratios of these compounds are above 1.75, and the $G/B$ ratios are less than 0.571, which shows that all compounds exhibit exceptionally ductile behavior. These compounds with such significant properties are ideal choices for flexible photovoltaic applications.