A DFT Study on the Structural, Electronic, Optical, and Elastic Properties of BLSFs XTi₄Bi₄O₁₅ (X = Sr, Ba, Be, Mg) for Solar Energy Applications

Abstract

For the first time, a theoretical investigation has been conducted into the structural, electrical, elastic, and optical properties of innovative bismuth-layered structure ferroelectric (BLSF) materials XTi₄Bi₄O₁₅ (where X = Sr, Ba, Be, and Mg). For all of the calculations, PBE-GGA and the ultra-soft pseudopotential plane wave techniques have been implemented with the DFT-based CASTEP simulation tool. Based on the exchange correlation approximation, the calculations reveal that XTi₄Bi₄O₁₅ (X = Sr, Ba, Be, and Mg) materials demonstrate direct band-gap semiconductor behavior with an estimated density functional fundamental gap in the range from 1.966 eV to 2.532 eV. The optical properties of these materials exhibit strong absorption and low reflection in the visible range. Moreover, the estimations of the elastic properties of the materials have shown mechanical stability and ductile behavior (due to B/G > 1.75), where G and B denote the shear modulus and the bulk modulus. Based on the above-mentioned highlights, it can be confidently stated that these materials are promising potential candidates for photovoltaic applications and solar cells due to their suitable direct band gap and high absorption coefficient.

1. Introduction

Presently, with the urgent growing demands of green energy production, the trend of investigating potential materials to be utilized in solar cells and photoelectric applications is rising [1,2,3]. The concept of developing smart cities with pollution-free environments with the help of renewable energy devices has gained specific importance. Hence, there comes the urgent need to establish a strong linkage between the chemical compositions and stable semiconductor structures of the materials possessing inherently photophysical properties [3,4,5]. Semiconductors serve as charge carrier transporters/photon absorbers to produce clean energy. This is important for a semiconductor’s capacity to produce energy efficiently for solar cell applications, which depends on the various parameters operating simultaneously [5,6,7,8,9]. Like dye-sensitized solar cells and photoelectrochemical water splitting processes, TiO₂ is one of the semiconductors that is most frequently utilized for this purpose. To improve the performance of TiO₂-based devices, it is necessary to focus on improving exceptional photophysical properties such as carrier diffusion, light absorption, and charge carrier extraction.

Due to their high Curie temperatures and exceptional resistance against thermal depoling, BLSFs have gained attention as prospective functional materials for high-temperature multifunctional applications. With the general formula of BLSFs (Bi₂O₂)²⁺(A_m−1B_mO_3m+1)²⁻, they possess an orthorhombic structure, with the space group Cmc2_1, where A and B stand for two different types of cations that occupy the centers of the dodecahedron and octahedron, respectively, and m is the number of the stacked oxygen octahedron along the direction perpendicular to the layers. BLSFs, like Bi₂Ti₄O₁₁, Bi₈TiO₁₄, Bi₂Ti₂O₇, Bi₁₂TiO₂₀, Bi₄Ti₃O₁₂, etc., are rarely reported materials for solar energy applications. They have attracted a lot of attention due to their attractive band-gap range which can be tuned by substituting various suitable elements at their lattice sites [10,11,12,13,14].

By replacing the A/B sites depending on the suitable ionic radii and valance in the XTi₄Bi₄O₁₅, it is possible to control the energy gap, enhance the optical performances, extend the lifetime of photogenerated electrons, and lower the rate of electron–hole recombination. Xue et al. employed the density function theory and calculated the Schottky (hole) barriers for abrupt interfaces, which are 1.58 eV (2.50 eV) and 2.06 eV (1.41 eV) for SrBi₂Ta₂O₉ and Bi₄Ti₃O₁₂, respectively [15]. Noureldine et al. used both theoretical and experimental approaches to investigate the optoelectronic characteristics of the nonstoichiometric Bi_2xTi₂O₇ structure (x = 0.25). The authors reported a computed band gap of 3.3 eV, and high values of up to 80.1 were observed in the static dielectric constants for Bi_1.75Ti₂O_6.6, which may be used in a range of optoelectronic applications [16]. Lardhi et al. performed a computational study on sillenite Bi₁₂TiO₂₀ and bismuth-layered Bi₄Ti₃O₁₂ materials and measured quite large optical band gaps of 3.1 eV for sillenite and 3.6 eV for bismuth-layered structures [17]. Among various BLSF family members, XTi₄Bi₄O₁₅ possesses the significant importance of having m = 4, an orthorhombic structure, and the A₂₁am space group. Hussain and his co-workers have performed comprehensive experimental research work on the BLSF family, which has compelled us to analyze the theoretical perspectives of these materials as well [18,19,20]. Recently, the research group led by Jabeen has performed theoretical investigations on BLSF materials, which have provided comprehensive explanations of hidden merits of the materials [21,22].

Herein, the pseudopotential plane wave approach has been employed to conduct a theoretically systematic examination of the influence of suitable cations (X = Sr, Ba, Be and Mg) in XT_i4Bi₄O₁₅ using first-principles calculations. The substitution of cations has influenced the band gap, electronic structure, and optical properties of XTi₄Bi₄O₁₅ (X = Sr, Ba, Be, Mg) in order to forecast the properties of these materials in the field of solar cell and photovoltaic applications.

2. Methodology

A theoretical study was performed by using the first principle of density functional theory to investigate the optical, electronic, elastic, and phonon dispersion properties of XTi₄Bi₄O₁₅ (X = Sr, Ba, Be, Mg) [23,24]. The electronic and optical properties of the reported BLSF materials were analyzed by using the CASTEP (Cambridge Sequential Total Energy Package) Module within Materials Studio Software 2020 [25]. One of the well-developed exchange correlation functions for more accurate approximation employed was the GGA (Generalized Gradient Approximation) of the PBE (Perdew Burke Ernzerhof) function, which is used in these calculations [26,27,28]. The precision of the local density approximation (LDA) effects is considered reasonable in the simplified version of the information. However, it fails to be accurate when the density undergoes sudden changes. When density gradations are taken into consideration, the Generalized Gradient Approximation (GGA) exceeds the LDA in accuracy, as shown in Equation (1).

$$E_{xc}^{GGA}\left(\rho \uparrow \left(r\right), \rho \downarrow \left(r\right)\right)=\int f_{xc}^{GGA}\rho \left(r\right)\mathrm{}\mathsf{\Delta }\rho \left(r\right)\mathrm{d}{r}^3$$

(1)

To gain insight into materials regarding their electronic structure, the Koelling–Harmon relativistic treatment was implemented. Pseudopotential parameters were set in reciprocal space by keeping spin–orbit coupling OFF [29]. During calculations of electron spectroscopy, a general parameter of output verbosity was set as normal. The cut-off energy value used in the calculations was set to 1200 eV; also, smearing was set to Gaussian in this work. In the context of density mixing parameter Pulay (named after John R. Pulay, a pioneer in quantum chemistry), we used the density -mixing scheme. The Pulay density mixing scheme is a procedure for figuring out the best mixing parameter for each iteration. The new density’s contribution to the total density in each iteration is modified by the mixing parameter. The K-points utilized for SCF (self-consisted field) calculations was 4 × 4 × 4, proposed by Monkhorst-Pack k-Point sampling [30]. The mechanical properties of the materials were investigated using the Forcite tool. In the calculations, a universal forcefield was used by setting the summation method as “atomic based” for electrostatic and Van der Waals forces (also known as Ewald). Calculations were performed under the constant strain, the algorithm used was “smart”, and the external pressure was set to 0 GPa. In order to completely realize the intricate interactions of phonon dispersion, dynamic features, thermodynamics, and vibrational properties in crystalline materials were measured by using the Wien2K framework with the support of DFT.

3. Results and Discussions

In this work, the optical and electronic properties of XTi₄Bi₄O₁₅ (X = Sr, Ba, Be, Mg) are calculated. Electronic properties including the materials’ band structure, total density of state (TDOS), and partial density of state (PDOS) have been measured. Moreover, the optical properties such as absorption, the optical conductivity loss function, the dielectric function, and reflectivity are analyzed for each composition. Additionally, this article has reported the elastic properties including Poisson’s ratio, Cauchy pressure, and Pugh’s ratio.

3.1. Structural Properties

The orthorhombic crystal structure of XTi₄Bi₄O₁₅ belongs to the space group Cmc2_1, consisting of a layered structure of O and Bi atoms; herein, we have used the pre-calculated optimized structure of the material (mp-1217847). The lattice constants for these orthorhombic structured BLSF materials are a = 5.50 Å, b = 41.21 Å, and c = 5.56 Å, with α = β = γ = 90°. The unit cell is displayed in Figure 1. X is bonded in a 10-coordinate geometry to 10−O atoms. There are two asymmetric Ti sites; Ti is bonded with 6−O atoms to form a corner-sharing TiO⁸⁺ octahedral. In the second Ti sites, Ti is bonded in a 6-coordinate geometry to 6−O atoms. There are also two inequivalent Bi sites. The first site of Bi is bonded to 6−O atoms, and similar to this, there are eight inequivalent O sites. There exists a total of 96 atoms, with 4 atoms of X²⁺, 16 atoms of Ti⁴⁺, 16 atoms of Bi³⁺, and 60 atoms of O²⁻ in the crystal structure.

3.2. Electronic Properties

Figure 2 shows the band structures with the energy range from −2.0 to 4.0 eV that are calculated using the GGA for analyzing the electronic characteristics along the high-symmetry directions of the irreducible Brillion zone (IBZ). The dashed line represents the energy band configurations, where zero energy is also known as the Fermi level. For all of these materials, the conduction band minima (CBM) and valance band maxima (VBM) are found at high-symmetry points Y and G. Electronic transitions between sub-energy levels of the valence band (VB) and the conduction band (CB) produce different peaks in the absorption spectra. When an electron shifts from the VBM to CBM at the same high-symmetry point, it is referred to as a direct electronic transition. In addition, in the situation where the high-symmetry point is different, it is called an indirect electronic transition. From Figure 2, Y and G represent the VBM and CBM, which confirms the direct energy band gaps of 2.532, 2.551, 1.966, and 2.480 eV for SrTi₄Bi₄O₁₅, BaTi₄Bi₄O₁₅, BeTi₄Bi₄O₁₅, and MgTi₄Bi₄O₁₅, respectively. The energy band gaps are tuned as a result of more electrons in the elements providing more electronic states (in both valence and conduction bands). The band gaps of XTi₄Bi₄O₁₅ compounds (X = Sr, Ba, Be, and Mg) are in the visible spectrum, suggesting that these compounds can be good choices for optoelectronic device manufacturing.

The electronic properties of materials are dependent on different electronic transitions between the CB and VB. Energy in the sub-bands in the CB and VB can be studied using spectra of the total density of states (TDOS) and partial density of states (PDOS). Figure 3 displays the calculated TDOS spectra using the GGA. The computed spectra of the TDOS for every molecule provide information about the number of energy levels at a given energy. Plots of the TDOS for XTi₄Bi₄O₁₅ (X = Sr, Ba, Be, and Mg) in the energy range from −6 to 6 eV confirm no existence of states at the Fermi level (0 eV) for XTi₄Bi₄O₁₅ (X = Sr, Ba, Be, and Mg) compounds.

At the VB’s edge, close to the Fermi level, sharp peaks are observed. The CB states are exhibiting greater energies. The greatest VB energies and lowest CB energies exhibit a noticeable band gap. A similar kind of behavior is observed in all of the compositions under study; the estimated band gaps displayed in Figure 3 are consistently accurate with those in Figure 2. The density of states D(E) represents the accessible electronic states at each energy level. It is determined using the eigenvalues derived from the Kohn–Sham equation shown in Equation (2):

$$D\left(E\right)={\sum }_i \delta \left(E-{\epsilon }_i\right)$$

(2)

where δ is the Dirac delta function E and ${\epsilon }_i$ is the eigenvalue.

The TDOS results therefore validate the accuracy of the prediction concerning the semiconductor nature of XTi₄Bi₄O₁₅ (X = Sr, Ba, Be, and Mg). The sub-TDOS can be observed in Figure 3, with the green, blue, red, and pink lines showing the behavior of the X-cations which are Sr, Ba, Be, and Mg, respectively. In Figure 4, the peaks that appeared in the VB for all materials under study are at the Fermi level, which is due to the substituted cation atoms. The O, Sr, and Ba atoms are responsible for the high peaks seen in the energy range from −1 to −2 eV of the VB for SrTi₄Bi₄O₁₅ and BaTi₄Bi₄O₁₅, whereas Mg and Be have very little influence on the formation of the VB for MgTi₄Bi₄O₁₅ and BeTi₄Bi₄O₁₅. Ti, Bi, Be, and Mg atoms exhibit prominent peaks in the CB developments, but Sr does not.

➢

SrTi₄Bi₄O₁₅

Figure 4a shows the PDOS graph for SrTi₄Bi₄O₁₅. In the VB, Sr and O atoms are dominant, with a minor contribution of Bi and Ti. O-2p⁴ and Sr-3d¹⁰ have a significant effect on the material’s VB. The VB and CB have a sufficient gap to anticipate the material’s semiconductor properties. Ti and Bi contribute more than the rest of the elements to the CB. Sr and O atoms are of little significance in the CB.

➢

BaTi₄Bi₄O₁₅

In Figure 4b, Ba and O atoms are showing strong dominance in the VB. The O-2p⁴, Ba-4d¹⁰, and Ba-5p⁶ have a significant influence, while the presence of Bi-3p⁶ is small. The presence of forbidden energy gaps indicates the material’s capacity to be operated as a semiconductor. The energy band gap of BaTi₄Bi₄O₁₅ is 2.551 eV. Ba and Bi atoms also have a relatively small effect on the CB compared to Ti.

➢

BeTi₄Bi₄O₁₅

BeTi₄Bi₄O₁₅ displays an extremely narrow energy band gap, indicating its significant potential for use in semiconductor devices. In the VB, Ti-3d² and O-2p⁴ contribute most, with little contribution from Bi-3p⁶ (Figure 4c). Be and Ti atoms contribute significantly to the CB, while the influence of the other elements is insignificant.

➢

MgTi₄Bi₄O₁₅

Figure 4d displays a band gap for MgTi₄Bi₄O₁₅ that is almost identical to that of the SrTi₄Bi₄O₁₅ and BaTi₄Bi₄O₁₅ materials. O-2p⁴ and Bi-3P⁶ make up the majority of the contribution in its VB. However, in its CB, Ti has a significant influence, whereas Mg and Bi atoms have minor contributions.

3.3. Optical Properties

For the utilization of BLSFs in solar energy applications, it is important to measure their electronic properties and optical characteristics to understand the potential optical response of the substituted cations in XTi₄Bi₄O₁₅. When photons with a sufficient frequency $(E=h\upsilon )$ operate with the material, the material experiences an electronic transition from the occupied states in the VB to the unoccupied states in the CB. These kinds of transitions give meaningful information about the optical properties of materials. The optical dispersion data are displayed in Figure 5 and Figure 6, showing how the interband/intraband transitions affect the dielectric function.

To understand the optical responses and transitions, the following parameters are often investigated: refractive index $n\left(\omega \right)$, reflectivity $R\left(\omega \right)$, dielectric constant $\epsilon \left(\omega \right)$, optical loss $L\left(\omega \right)$, and absorption coefficient $I\left(\omega \right)$. The photon–electron interaction and the system’s response to electromagnetic radiation are explained by the dielectric function $\epsilon \left(\omega \right).$ The sum of the real ${\epsilon }_1\left(\omega \right)$ and imaginary ${\epsilon }_2\left(\omega \right)$ components yields the dielectric function $\epsilon \left(\omega \right)$.

One should be well aware of the transitions between occupied and unoccupied states, which can occur when a photon interacts with the electrons in a semiconductor device. The combined densities of states between the VB and CB can be used to characterize the spectra that arise from these excitations. The imaginary part of the dielectric function ${\epsilon }_2\left(\omega \right)$ is derived from the appropriate momentum matrix elements between the occupied and unoccupied wave functions within the selection rules over the Brillouin zone.

In Figure 5a, the calculated values of ${\epsilon }_1\left(\omega \right)$ are shown as a function of energy (0–20 eV). The estimated ${\epsilon }_1\left(\omega \right)$ provides information about the dispersion of incoming photons. It is shown that for XTi₄Bi₄O₁₅ (X = Sr, Ba, Be, and Mg), significant dispersion of incoming photons takes place between 2.0 and 4.0 eV. For XTi₄Bi₄O₁₅ (X = Sr, Ba, Be, Mg), the static values of ${\epsilon }_1\left(0\right)$ can be found at zero frequency or a constant electric field. For XTi₄Bi₄O₁₅ (X = Sr, Ba, Be, and Mg), the static values of ${\epsilon }_1\left(0\right)$ are 6.88, 6.98, 7.25, and 6.93, respectively, which correspond to the plasmon frequency.

In Figure 5b, the computed values of ${\epsilon }_2\left(\omega \right)$ are shown as a function of energy (0–20 eV). Electronic transitions between the VB (occupied states) and CB (unoccupied states) have produced a variety of peaks in ${\epsilon }_1\left(\omega \right)$. After a strong rise at lower energy, the peaks for XTi₄Bi₄O₁₅ (X = Sr, Ba, Be, and Mg) approach their maximum positions. These compounds have clearly absorbed a significant amount of incoming photons in the lower UV and IR regions. The maximum absorption power is found in XTi₄Bi₄O₁₅ (X = Sr, Ba, Be, and Mg), with values of 5.46, 5.23, 5.48, and 4.59 eV, respectively. Their broad range of absorption makes them ideal candidates for tunable devices.

The acceptability of a material for optical devices is assessed by its optical characteristics. Like with ${\epsilon }_1\left(\omega \right)$, the spectra of $n\left(\omega \right)$ can provide extensive information about the dispersion of incident photons. Figure 5c shows the estimated spectra of $n\left(\omega \right)$ for XTi₄Bi₄O₁₅ (X = Sr, Ba, Be, and Mg). The static values of $n\left(0\right)$ for XTi₄Bi₄O₁₅ (X = Sr, Ba, Be, and Mg) are 2.62, 2.64, 2.63, and 2.69, respectively. When a material’s refractive index falls between 2.0 and 3.0, it is regarded as an active optical material with reflectivity and the ability to bend light. Peaks then climb to their maximum levels, which are around 4.0 eV. For XTi₄Bi₄O₁₅ (X = Sr, Ba, Be, and Mg), the $n\left(\omega \right)$ spectra suddenly decline and become less than unity at 8.74, 8.76, 8.38, and 8.56 eV, respectively. Metals behave like this in the area where $n\left(\omega \right)$ is less than unity. These materials show the phenomenon of superluminality, which may be seen both theoretically and empirically when ($n<1$). This unusual phenomenon indicates that light in a vacuum travels slower than light in a medium, which is not feasible. Photovoltaic device applications can make use of materials with high optical conductivity values and refractive indexes and low emissivity absorption coefficients. The extinction coefficient $k\left(\omega \right)$ is an optical characteristic that indicates how effectively a material can absorb incident photons or radiations at a certain frequency. Like ${\epsilon }_2\left(\omega \right)$, the spectra of $k\left(\omega \right)$ can provide profound information about the incoming photon absorption. Figure 5d displays the estimated $k\left(\omega \right)$ spectra for XTi₄Bi₄O₁₅ (X = Sr, Ba, Be, and Mg). It is clear from the spectra of $k\left(\omega \right)$ that the values are initially zero until the peaks for XTi₄Bi₄O₁₅ (X = Sr, Ba, Be, and Mg) begin to appear at 1.9, 1.98, 1.36, and 1.82 eV, respectively. These recognized threshold energies of $k\left(\omega \right)$ are the values at which the peaks appear. Figure 5d shows that the $k\left(\omega \right)$ spectra suddenly increase after the threshold energies and approach their maximum values in the UV region.

The proportion of refracted photons to incoming photons is known as the refraction coefficient, denoted by $R\left(\omega \right)$. In Figure 6a, the computed plots of $R\left(\omega \right)$ are shown as a function of energy (0–20 eV). For XTi₄Bi₄O₁₅ (X = Sr, Ba, Be, and Mg), the tremendous dispersion of incoming photons happens at 9.0 eV. Similarly, it is evident from the plot that static values of $R\left(0\right)$ at a constant electric field are 0.201, 0.203, 0.21, and 0.202, respectively, and the maximum reflectivity is found in the upper UV region (~9 eV). In the upper UV region, these substances can be used as reflective coatings.

Figure 6b shows the estimated absorption spectra for XTi₄Bi₄O₁₅ (X = Sr, Ba, Be, and Mg), plotted as a function of photon energy. One of the most important factors in determining a solar device’s competency is the absorption coefficient $\alpha \left(\omega \right)$. Since both spectra are directly reliant on the electronic band structures, the $\alpha \left(\omega \right)$ behaved consistently with the ${\epsilon }_2\left(\omega \right)$. For XTi₄Bi₄O₁₅ (X = Sr, Ba, Be, Mg), the greatest edges of absorption in the UV range are measured as follows: 2.9 × 10⁵ cm⁻¹ at 8.5 eV, 2.7 × 10⁴ cm⁻¹ at 8.27 eV, 2.4 × 10⁵ cm⁻¹ at 8.38 eV, and 2.35 × 10⁵ cm⁻¹ at 8.28 eV, respectively. The visible range is where the first portion of the absorption spectra originates. The electromagnetic spectrum’s ultraviolet region is where the absorption peaks are closest to reaching their maximum values.

Figure 6c displays the computed spectra of the real portion of optical conductivity $\sigma \left(\omega \right)$ for XTi₄Bi₄O₁₅ (X = Sr, Ba, Be, and Mg) in the energy range of 0–20 eV. Information about the conduction current of charge carriers coming from the breakage of atomic bonds under the impact of incident photon energy is obtained from the real part of the optical conductivity $\sigma \left(\omega \right)$. The $\sigma \left(\omega \right)$ spectra show that the highest peaks occur in the UV range (1.5–6.2 eV), with smaller peaks occurring at higher photon energies.

The energy loss function $L\left(\omega \right)$ may be used to understand the material’s loss of rapidly moving electrons. A graph of the computed values of $L\left(\omega \right)$ versus energy is displayed in Figure 6d. The peaks for XTi₄Bi₄O₁₅ (X = Sr, Ba, Be, and Mg) start appearing at 2.92, 2.68, 2.46, and 2.42 eV, respectively. The peaks for XTi₄Bi₄O₁₅ (X = Sr, Ba, Be, and Mg) exhibit a progressive rise until they reach their highest values at 11, 10.7, 11, and 10.9 eV, respectively. The point of plasmon resonance is when the peaks begin to decline from their highest positions.

3.4. Mechanical Properties

To calculate the bulk modulus $\left(B\right),$ shear modulus $\left(G\right)$, Young’s modulus $\left(Y\right),$ and Poisson’s ratio $\left(\nu \right)$ for each composition of XTi₄Bi₄O₁₅, the elastic stiffness constants $\left(C_{ij}\right)$ are determined (Figure 7). Unfortunately, experimental and theoretical elastic constants for the BLSFs are not known to the best of our knowledge. So, our calculations are the first to assess the elastic properties of this class of materials and cannot be compared with previous research. Using the Born elastic stability criterion, we have assessed the mechanical stability of our compounds using $C_{ij}$ [31,32,33]. Based on crystal symmetry, fewer elastic constants are needed. There are just nine independent elastic constants in the case of an orthorhombic system: $C_{11}, C_{22}, C_{33}, C_{44}, C_{55}, C_{66}, C_{12}, C_{13}, \mathrm{a}\mathrm{n}\mathrm{d} C_{23}$. $C_{ij}$ has to fulfill the well-known Born stability (Figure 7a) requirements for an orthorhombic structure [34]:

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

(3)

The Born stability criteria given by Eq 3 are satisfied by the orthorhombic XTi₄Bi₄O₁₅ (X = Sr, Ba, Be, and Mg) elastic constants. Thus, at atmospheric pressure, the orthorhombic XTi₄Bi₄O₁₅ (X = Sr, Ba, Be, and Mg) is mechanically stable. The obtained elastic constants $C_{ij}$ of orthorhombic XTi₄Bi₄O₁₅ (X = Sr, Ba, Be, and Mg) are listed and compared in

Table 1. Mechanical properties (bulk modulus, shear modulus, Young’s modulus, and Poisson’s ratio) through elastic constants c_ij.

NAME	SrTi4Bi4O15	BaTi4Bi4O15	BeTi4Bi4O15	MgTi4Bi4O15
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	286.77	281.39	252.64	262.15
G	16.45	32.01	24.09	24.02
E	48.43	92.54	70.04	69.92
B/G	17.42	8.78	10.48	10.91
G/B	0.05	0.11	0.09	0.09
ῡ	0.47	0.44	0.45	0.45

. It is possible to describe the ductility and brittleness behaviors of materials using the modulus equations given below:

Bulk modulus:

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

(4)

Shear modulus:

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

(5)

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

(6)

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

(7)

Young’s modulus:

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

(8)

Poisson’s ratio:

$$\overline{v}={\displaystyle \frac{3B-2G}{6B+2G}}$$

(9)

The brittle or ductile natures of the materials can be evaluated by a variety of characteristics, including ($B/G$) the ratio of ductility, Poisson’s ratio $\left(\nu \right)$, and Cauchy pressure $(C_{12}-C_{44})$ (Figure 7b). These materials are therefore naturally ductile. If the Frantesvich ratio $(G/B)$ is less than 0.571, the material is ductile; if not, it is brittle [35,36]. Table 1 shows that the bulk modulus $\left(B\right)$ and shear modulus $\left(G\right)$ for XTi₄Bi₄O₁₅ (X = Sr, Ba, Be, and Mg) are greater. Moreover, the Frantesvich ratio $(G/B)$ of all of the compositions is well below 0.571, which confirms their ductile nature (Figure 7c). The $B/G$ ratio is also referred to as Pough’s ratio, which can be examined using the computed values of $B$ and $G$; such values will define the brittle or ductile natures of the materials under study. If the ratio of $(B/G)$ is above 1.75, the material is ductile; otherwise, it is brittle. For XTi₄Bi₄O₁₅ (X = Sr, Ba, Be, and Mg), the values of $(B/G)$ are 17.42, 8.78, 10.48, and 10.91, respectively (Figure 7d). The shear modulus thus has a significant influence on the stability of these materials and has a great capability against deformation. The compounds under investigation appear to be more resistant to volume compression than shear deformation, as presented in Table 1, by the fact that the shear modulus $G$ is smaller than the bulk modulus B.

It is crucial to stress the importance of understanding phonon dispersion in order to completely realize the intricate interactions, dynamic features, thermodynamics, and vibrational properties in crystalline materials [37]. To address this issue in the Wien2K framework, we employ DFT, a well-used and efficient method, and use it to examine the dynamic stability of XTi₄Bi₄O₁₅ (X = Sr, Ba, Be, and Mg) primitive unit cells, as presented in Figure 8a–d. Due to their non-zero frequencies, the remaining branches are identified as optical phonons. Specifically, we identify one distinct longitudinal acoustic (LA) mode and two transverse acoustic modes in acoustics. A crystal made up of a unit cell with N atoms has 3N optical modes. The arrangement of constituent atoms results in various frequency branches which are categorized as acoustic when they converge at the Γ-point frequency bands in which they occur; we discover that the optical branches contain a spectrum of silent, infrared modes. Importantly, the inherent dynamical stability of the systems is highlighted by the absence of imaginary or negative frequencies in the resulting band dispersions [38].

Table 2. Comparison analysis of demonstrated properties of XTi₄Bi₄O₁₅ (X = Sr, Ba, Be, and Mg) with literature.

Materials	Functional	Electronic Properties		Optical Property Peak Values			Mechanical Property Comparison			References
		Band Gap (eV)	Band-Gap Type	ε1(ω)	α(ω) cm−1	n(ω)	Ν	B/G	Nature
SrTi4Bi4O15	GGA-PBE	2.53	Direct	11 at 3.3 eV	2.29 × 105 at 8.5 eV	3.38 at 3.45 eV	0.47	17.42	Ductile	This Work
BaTi4Bi4O15	GGA-PBE	2.55	Direct	11 at 3.3 eV	2.27 × 105 at 8.27 eV	3.38 at 3.50 eV	0.44	8.78	Ductile	This Work
BeTi4Bi4O15	GGA-PBE	1.96	Direct	11.1 at 3.1 eV	2.45 × 105 at 8.38 eV	3.39 at 3.29 eV	0.45	10.48	Ductile	This Work
MgTi4Bi4O15	GGA-PBE	2.48	Direct	11.1 at 3.4 eV	2.35 × 105 at 8.26 eV	3.41 at 3.38 eV	0.45	10.91	Ductile	This Work
Pb2Bi4Ti5O18	LDA-CA-PZ	1.84	Indirect	10.9 at 4.18 eV	2.08 × 105 at 7.90 eV	3.58 at 3.7 eV	--	10.25	Ductile	[21]
SrNb2Bi2O9	GGA-PBE	2.49	Direct	10.3 at 3.38 eV	1.99 × 105 at 9 eV	3.29 at 3.54 eV	Research Gap			[22]
Cs2Au2F6	HSE06	2.41	Direct	2.52 at 2.01 eV	7.6 × 104 at 8.7 eV	1.68 at 2.04 eV	0.26	1.756	Critical value	[39]
Cs2Au2Cl6	HSE06	2.01	Direct	3.8 at 1.76 eV	9.5 × 104 at 9 eV	1.96 at 1.82 eV	0.33	2.514	Ductile	[39]
Cs2AgBiCl6	GGA-PBEsol	1.61	Indirect	5.4 at 2.6 eV	22.0 × 105 at 16 eV	2.5 at 2.6 eV	0.35	2.96	Ductile	[40]
Cs2AgBiBr6	GGA-PBEsol	1.11	Indirect	6.5 at 2.5 eV	18.7 × 105 at 17 eV	2.7 at 2.5 eV	0.34	2.88	Ductile	[40]
Cs2PbBr6	GGA-TB-mbj	2.36	Direct	5.3 at 4.2 eV	5.8 × 105 at 4.5 eV	2.4 at 4.3 eV	0.33	2.6	Ductile	[41]

presents the comparison analysis of the presented BLSF-based XTi₄Bi₄O₁₅ (X = Sr, Ba, Be, and Mg) materials with the other similar kinds of materials or with the materials which are capable of being utilized for optoelectronic and photovoltaic applications.

4. Conclusions

In conclusion, DFT calculations have been implemented to estimate the structural, elastic, electrical, dielectric, optical, and phonon dispersion properties of the BLSF-based XTi₄Bi₄O₁₅ materials (X = Sr, Ba, Be, and Mg) for their potential applications in optoelectronic devices. It was performed using the (FP-LAPW) approach, the GGA, and TB-mBJ functions. It is evident from the results that XTi₄Bi₄O₁₅ materials (X = Sr, Ba, Be, and Mg) are semiconductors with energy band gaps ranging from 1.96 to 2.53 eV. The optical characteristics, dielectric function, reflectivity, energy loss function, absorption coefficient, refractive index, and extinction coefficient are also reported to support the potential of these materials for solar energy applications. These materials absorb a significant amount of photons in the UV region, depicted by the imaginary component dielectric function ${\epsilon }_2\left(\omega \right)$. The obtained $n\left(\omega \right)$ values confirm the optical activity of these materials in the UV region as well, confirming their potential for optical devices. According to the calculated elastic and energetic characteristics, the materials are mechanically and energetically stable due to the orthorhombic symmetry of XTi₄Bi₄O₁₅ (X = Sr, Ba, Be, and Mg). The results of the computed Poisson’s ratio, Pugh’s ratio, and Cauchy’s pressure indicate that all of the materials under analysis demonstrate extremely ductile behavior. According to the calculations, $B/G$ exceeds 1.75, and the elastic constants suggest that these materials have elastic stability, so they can operate in flexible solar energy applications.