First-principles analysis of eco-friendly Sr₃BiX₃ (X = I, Br, and Cl) inorganic perovskites for optoelectronic applications: a DFT–ML hybrid approach

Abstract

The baseline for photovoltaic and optoelectronic device commercialization was established by non-toxic halide cubic perovskites. In this study, an in-depth exploration of the physical features of Sr₃BiX₃ (X = I, Br, and Cl) materials was conducted using density functional theory (DFT) due to their immense significance. All the materials are cubic in structure. The GGA-PBE potential functional revealed direct band gaps of 1.324 eV, 1.512 eV, and 1.731 eV for the Sr₃BiI₃, Sr₃BiBr₃, and Sr₃BiCl₃, respectively. Furthermore, the investigated compounds demonstrated remarkable absorption, elevated conductivity, reduced reflectivity, an optimal refractive index, and negligible loss function within the visible light range, making them exceptional candidates for photovoltaic technologies. The Born stability requirements confirmed the mechanical stability of the investigated compounds. In addition, their intrinsic rigidity, strength, resilience, ductility, and anisotropic properties are crucial for enduring performance in engineering contexts. The ab initio molecular dynamics (AIMD) simulations verified the thermal stability of the entitled compounds. These perovskites are experimentally feasible since their phonon dispersion curves have no imaginary portion. To expedite material discovery and bandgap estimation, a machine learning (ML) model was developed using a dataset derived from DFT calculations. The ML-predicted band gaps 1.432 eV, 1.545 eV, and 1.712 eV showed strong agreement with the DFT results 1.324 eV, 1.512 eV, and 1.731 eV for Sr₃BiX₃ (X = I, Br, and Cl), validating the model's reliability. Moreover, Pearson correlation analysis was employed to explore the connections between structural, electronic, and optical features, providing deeper insight into the key parameters influencing the bandgap. This integrated DFT-ML approach demonstrates a promising pathway for high-throughput screening and design of perovskite materials.

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1. Introduction

The swift integration of global markets and the growth of industrial operations have significantly increased the worldwide energy demand. Unfortunately, most of this energy still comes from limited fossil fuel sources, like oil, coal, and natural gas.¹ The heavy dependence on these energy sources abundant in carbon has resulted in significant environmental repercussions, especially the increased release of greenhouse gases, with CO₂ being a primary factor in global warming and climate change.² These urgent ecological issues have sparked a worldwide quest for cleaner, more sustainable, and renewable energy options that can mitigate the environmental harm associated with the burning of fossil fuels.³ Among the numerous renewable energy options, sunlight remains one of the most abundant and promising sources.⁴ It is pure, limitless, and easily accessible worldwide. Over the last twenty years, researchers and engineers have worked diligently to develop advanced photovoltaic technologies that efficiently store solar power and then convert it into electricity.⁵ In this context, solar cells utilizing perovskite materials have emerged as a revolutionary advancement in the realm of photovoltaics.

A perovskite material, notably those defined by the general formula ABX₃, where ‘A’ represents an alkali earth metal, ‘B’ denotes bivalent cations, and ‘X’ signifies a halide ion, has garnered considerable attention due to their outstanding optoelectronic features.^6,7 These substances exhibit remarkable abilities to absorb light, possess elevated charge-carrier mobilities, demonstrate extended diffusion lengths, and feature minimal exciton binding energies.⁸ Furthermore, perovskites can be created through affordable and scalable production techniques, like solution processing and vapor deposition, rendering them more economically advantageous than conventional Si-based solar panels.⁹ The advancement of perovskite solar cells (PSCs) has been truly groundbreaking. Continuous improvements in material design, interface engineering, and device architecture have significantly enhanced their efficiency to 27% as of 2025.¹⁰ This remarkable surge in performance underscores the significant promise of perovskite-based photovoltaics to emerge as a dominant force in the global shift toward sustainable energy solutions. Some notable examples are AlGeX₃ (X = F, Cl, Br),¹¹ FrMI₃ (M = Ge, Sn),¹² AMgCl₃ (A = Ga, In, Tl),¹³ and RbSnX₃ (X = Cl, Br, I),¹⁴ which illustrate their capacity to contribute to the development of optoelectronic applications. Talukder et al.¹⁵ illustrate in their investigation of ABI₃ (A = Rb, Cs; B = Ca, Sr) that materials based on Rb exhibit superior optoelectronic performance compared to those based on Cs. Hasan et al.¹⁶ stated that inorganic InGeX₃ (X = F, Cl) demonstrates remarkable semiconducting characteristics during their investigation of its optoelectronic properties.

In recent years, the focus of perovskite research has increasingly broadened beyond the traditional ABX₃ structure to encompass more intricate variants such as A₃BX₃. In this updated arrangement, ‘A’ usually signifies an alkali metal, ‘B’ indicates a trivalent or tetravalent metal, and ‘X’ refers to a halogen atom.¹⁷ This stoichiometry offers a distinct crystal arrangement and chemical adaptability, enabling the precise tuning of electronic and optical parameters, which makes it highly appealing for applications in photovoltaics and optoelectronics.¹⁸ Remarkably, A₃BX₃ perovskites display beneficial traits including ideal band gaps, high absorption coefficients, and excellent carrier transport properties, all of which are necessary for effective solar energy conversion and the integration of electronic devices.¹⁹ A significant investigation in this field was carried out by Feng et al.,²⁰ who thoroughly examined the stability and photovoltaic potential of A₃BX₃-type halide perovskites. The researchers concentrated in particular on Ba₃PI₃, Ba₃AsI₃, and Ba₃SbI₃ compounds, revealing that these materials demonstrate remarkable thermodynamic stability, appropriate band gaps, and robust absorption coefficients in the visible light span. Notably, among the most promising choices for future Pb-free perovskite solar panels, Ba₃SbI₃ was projected to attain a power conversion efficiency of up to 25.9%. Hasan et al.²¹ explored the potential of Sr₃SbBr₃ material for photovoltaic applications. Ghosh et al.²² investigated the physical properties of Sr₃AsX₃ (X = Br, Cl, F), providing a valuable perspective on their potential uses in energy storage and electronic device systems. Hasan et al.²³ investigated the impact of varying cations and anions in Mg₃BX₃ (B = N, P; X = Br, I) compounds and discovered remarkable tunable properties.

Recently, Khandaker et al.²⁴ explored the Bi-based materials Mg₃BiBr₃ and Mg₃BiI₃, recognizing them as semiconductors with significant promise for solar energy conversion. According to their article, Mg₃BiBr₃ and Mg₃BiI₃ exhibit band gaps of 1.626 eV and 0.867 eV, respectively, as well as strong sunlight absorption coefficients, modest reflectivity, and excellent photoconductivity. They also proposed that both compounds could function as emerging optical materials for future solar cell technologies. This study concentrates on the class of Sr₃BiX₃ (X = Cl, Br, I) perovskites, which include the alkaline earth metal strontium (Sr) and the post-transition metal bismuth (Bi). The choice of bismuth-based halide compounds is primarily driven by the increasing demand for safe and stable substitutes for lead-based perovskites.²⁵ Bi, characterized by its trivalent oxidation state and ns² electronic configuration, is essential in creating stable structures driven by lone pairs, which leads to favorable optoelectronic properties, including enhanced light absorption and better charge carrier dynamics.²⁶ Conversely, Sr is not only plentiful in the Earth's crust but also recognized for its remarkable reactivity and capacity to improve structural stability when incorporated into perovskite frameworks.²⁷ The Sr₃BiX₃ family showcases a distinctive structural arrangement and appealing electronic characteristics, attributed to the cooperative interaction between Sr and Bi cations. A thorough literature review highlights the significant lack of theoretical studies regarding the inorganic Sr₃BiX₃ halide perovskites, suggesting a vast area for scholarly exploration. The remarkable properties of these materials have ignited our interest, driving us to explore further the prospect of Sr₃BiX₃ (X = I, Br, and Cl) cubic perovskites in overcoming existing material limitations and achieving their utmost potential.

In this work, we employed density functional theory (DFT) calculations within a computational framework to thoroughly explore the structural, mechanical, electronic, optical, phonon, and thermodynamic properties of these novel halide perovskites. Special attention was given to the influence of halogen substitution on the material's behavior, aiming to clarify how the gradual replacement of the halogen anion can effectively adjust the band gap, thereby improving the materials’ optoelectronic performance. In this exploration, we aim to offer fresh perspectives on developing highly efficient, non-toxic, and stable materials, which could pave the way for the advancement of cutting-edge optoelectronic devices and solar energy innovations.

2. Computational details

The Sr₃BiX₃ (X = I, Br, and Cl) halide compounds are examined using first-principle calculations, carried out employing the Quantum Espresso package's integration of the DFT.^28,29 Self-consistent computations are employed to address the Kohn–Sham equations, leveraging the Perdew–Burke–Ernzerhof (PBE) exchange–correlation functional inside the Generalized Gradient Approximation (GGA) framework, and for better precision, ultrasoft pseudopotentials are used.^30–32 Moreover, we utilized the Quantum Espresso software with the PBEsol and HSE-06 potential to accurately determine the band gaps of the entitled compounds.³³ A 900 eV cutoff and a 12 × 12 × 12 k-point grid have been chosen based on the Monkhorst–Pack technique, which improves calculation accuracy and reliability.³⁴ The setup utilized ultrafine features, maintaining precise computations. The arrangement of the crystal is gently optimized leveraging the Broyden–Fletcher–Goldfarb–Shanno (BFGS) method, allowing us to obtain the lowest energy and most stable configuration.³⁵ The fine-strain methodology is applied for the mechanical inspection.³⁶ The structured formation of Sr₃BiX₃ halide perovskites is meticulously optimized using the respected VESTA software.³⁷ Furthermore, the ELATE algorithm is employed to produce anisotropic 3D graphs for Young's modulus, shear modulus, and Poisson's ratio.³⁸ Thermal stability is verified through ab initio molecular dynamics simulations.³⁹ A supervised machine learning model was developed utilizing the Random Forest Regressor algorithm to predict the bandgap energy (E_g) of Sr₃BiX₃ (X = I, Br, and Cl) compounds.⁴⁰ The model utilized DFT-derived material descriptors, including lattice parameters (a, b, c), volume of the unit cell, static dielectric constant ε(0), refractive index η(0), elastic constants (C₁₁, C₁₂, C₄₄), Young's modulus (Y), shear modulus (G), bulk modulus (B), and Poisson's ratio. Despite the small dataset, the model achieved a high prediction accuracy with an R² score of 0.98 and a Pearson correlation coefficient of 0.98, demonstrating the potential of data-driven approaches in accelerating materials property prediction.

3. Results and discussion

3.1. Structural properties

The Sr₃BiX₃ compounds (X = I, Br, and Cl) crystallize in the ideal cubic perovskite structure, as displayed in Fig. 1(a). The structure belongs to the Pm3̄m space group (No. 221), typical of many halide perovskites.⁴¹ In this system, Sr elements occupy the cube corners (Wyckoff position 1a), the Bi atom is positioned at the center of the cube (1b), and the halogen atoms (X = I, Br, and Cl) reside at the face centers (3d positions), forming corner-sharing BiX₆ octahedra.¹⁷ These BiX₆ units are arranged in a three-dimensional network, creating a robust perovskite lattice with Sr atoms situated in the 12-fold coordinated cubo-octahedral voids. Full structural optimization was carried out using DFT, where both lattice parameters and atomic configurations have been relaxed till the overall energy converged below 10⁻⁶ eV and the atomic forces have been less than 0.01 eV Å⁻¹. As expected, the lattice constant decreases systematically with decreasing halogen ionic radius, i.e., a(I) > a(Br) > a(Cl), consistent with the chemical trend of halide substitution and confirming the structural integrity across the series.

Fig. 1 (a) Structure of the Sr₃BiX₃ (X = I, Br, and Cl) halide perovskite compounds and (b) the high-symmetry points of the Brillouin zone.

Fig. 1(b) displays the corresponding first Brillouin zone (BZ) of the primitive cubic lattice, with high symmetry k-points labeled as Γ, X, M, and R. These points are utilized for band structure evaluations, capturing the key electronic features of Sr₃BiX₃ compounds. The reciprocal lattice vectors b₁, b₂, and b₃ are also shown, delineating the cubic symmetry of the reciprocal space.¹⁷ The high structural symmetry, tunable lattice size through halide substitution, and the stability of the cubic phase make Sr₃BiX₃ perovskites promising candidates for optoelectronic, photovoltaics, and LEDs (light-emitting devices). The resulting lattice constants and unit cell volumes follow a clear trend based on halogen substitution. Specifically, the lattice constant decreases from 6.723 Å for Sr₃BiI₃ to 6.493 Å for Sr₃BiBr₃ and further to 6.366 Å for Sr₃BiCl₃. Correspondingly, the unit cell volume contracts from 303.87 ų (I) to 273.74 ų (Br) and 257.99 ų (Cl). This trend is compatible with the decreasing ionic radius of the halide anions and reflects the expected chemical pressure effect within the perovskite lattice.²² Besides, previous work on similar Ca-based compounds shows comparable values of lattice constant and cell volume, which further justifies the accuracy of our work.⁴² The value of the lattice parameter and unit cell volumes, along with the band gap value, are displayed in Table 1.

Table 1 Structural properties of the cubic Sr₃BiX₃ (X = I, Br, and Cl) halide perovskite compounds

Structure	Lattice Constant, a = b = c (Å)	Volume, V (Å^3)	Bandgap (PBE), Eg (eV)	Bandgap with SOC (PBE), Eg (eV)	Bandgap (PBEsol), Eg (eV)	Bandgap (HSE), Eg (eV)	Remarks
Sr3BiI3	6.723	303.87	1.324	0.794	1.389	1.878	This work
Sr3BiBr3	6.493	273.74	1.512	1.052	1.567	2.245	This work
Sr3BiCl3	6.366	257.99	1.731	1.223	1.786	2.427	This work
Sr3BiI3	6.690	299.42	1.30				Previous work^42
Ca3BiI3	6.42	264.60	1.29	0.81	—	2.39	Previous work^42
Ca3BiBr3	6.17	234.88	1.60	1.12	—	2.84	Previous work^43
Ca3BiCl3	6.02	218.167	1.76	1.30	—	3.08	Previous work^43

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3.2. Electronic properties

The electronic band structures of Sr₃BiX₃ (X = I, Br, Cl) were calculated using the generalized gradient approximation (GGA) within the Perdew–Burke–Ernzerhof (PBE) functional, PBEsol functional, and the hybrid functional HSE06.³³ The results are presented in Fig. 2, with the top row (Fig. 2a–c) showing the GGA-PBE band structures (in red) and the bottom row (Fig. 2d–f) showing PBE + SOC band structures (in blue). The calculations were performed along high-symmetry paths in the first Brillouin zone of the cubic perovskite structure, as illustrated in Fig. 1(b) earlier. Specifically, the band structure was plotted along the Γ–M–X–R–Γ path, which captures the essential electronic transitions and symmetry features of the simple cubic lattice. These high-symmetry points are crucial in determining the location of the valence band maximum (VBM) and conduction band minimum (CBM), and thus in identifying the nature (direct or indirect) and magnitude of the electronic band gap. From the band structure results, all three materials are direct bandgap semiconductors, with computed band gaps (E_g) of 1.324/1.389/1.878 eV for Sr₃BiI₃, 1.512/1.567/2.245 eV for Sr₃BiBr₃, and 1.731/1.786/2.427 eV for Sr₃BiCl₃ with PBE/PBEsol/HSE functional. The trend of increasing band gap from I to Cl reflects the increasing electronegativity and decreasing atomic radius of the halogen, which leads to a deeper valence band and a wider gap. This similar band gap modification by halide substitution was previously reported for a similar class of compounds, which confirms the credibility of our work.⁴³

Fig. 2 (a)–(f) Band structure of the Sr₃BiX₃ (X = I, Br, and Cl) halide perovskite compounds without and including the SOC effect.

To mitigate the well-known undermine of band gaps in GGA-PBE, we performed additional calculations that included the spin–orbit coupling (SOC) effect. This approach is particularly important for Bi-based halide perovskites, as the heavy Bi atom introduces significant relativistic effects.⁴⁴ The PBE + SOC band structures yield band gaps of 0.794 eV (Sr₃BiI₃), 1.052 eV (Sr₃BiBr₃), and 1.223 eV (Sr₃BiCl₃), showing that SOC reduces the gap due to the splitting of the Bi-p orbitals close to the valence band edge.⁴⁵ Incorporating both the Brillouin zone symmetry and relativistic effects provides a more accurate and physically meaningful understanding of the electronic structure.⁴⁶ The tunable band gap across the halogen series, along with their relatively clean band dispersion, makes Sr₃BiX₃ perovskites promising candidates for use in optoelectronic applications like photodetectors and solar panels.

To gain a deeper perspective on the electronic structure and bonding features of Sr₃BiX₃ (X = I, Br, and Cl), the projected partial density of states (PDOS) and total partial density of states (TDOS) were calculated, both with and without the inclusion of SOC, as illustrated in Fig. 3. The top row (Fig. 3a–d) presents the PDOS and TDOS without SOC, while the bottom row (Fig. 3e–h) includes SOC effects. Across all three compounds, the valence band (VB) is predominantly composed of Bi-6s, Bi-6p, and X-p (I-5p, Br-4p, Cl-3p) orbitals, whereas the conduction band (CB) is mainly derived from Bi-6p and Sr-4p contributions. For Sr₃BiI₃ (Fig. 3a), the VB edge is influenced by strong I-5p and Bi-6s hybridization, indicating substantial covalent character. A similar feature is observed for Sr₃BiBr₃ and Sr₃BiCl₃ (Fig. 3b and c), where Br-4p and Cl-3p states, respectively, contribute significantly to the top of the VB. The CBM in all three systems is primarily composed of Bi-6p orbitals, indicating their crucial role in conduction processes. Upon incorporating SOC (Fig. 3e–g), a notable splitting of the Bi-p orbitals is observed near the conduction band edge, particularly in Sr₃BiI₃. This SOC-induced splitting results in a downward shift of the CB edge, resulting in a lowering of the electronic E_g, which is consistent with previous band structure results.⁴⁵ The effect is most pronounced in Sr₃BiI₃, which contains the heaviest halogen (I), and diminishes progressively in Sr₃BiBr₃ and Sr₃BiCl₃, owing to the weaker relativistic effects in lighter halogens.

Fig. 3 (a)–(h) The calculated partial density of states (PDOS) and total density of states (TDOS) of the Sr₃BiX₃ (X = I, Br, and Cl) halide perovskite compounds without and with including the SOC effect.

The total density of state reflects that the valence states extend from approximately −5 eV to the Fermi level (E_F = 0), and the conduction states commence immediately after, consistent with the semiconducting behavior noticed in the band structure. The total density of state plots reveals that the overall intensity of states at the valence band maximum is slightly higher in Sr₃BiI₃, which may enhance its optical absorption near the band edge. Additionally, SOC modifies the fine features of the PDOS near both band edges, redistributing states that could influence carrier mobility and effective masses.⁴⁷

In conclusion, the PDOS analysis confirms that the hybridization of Bi and halogen p-states predominantly governs the electronic structure of S_r3BiX₃ (X = I, Br, and Cl) materials. The inclusion of SOC significantly influences the conduction band features, particularly in iodide and bromide variants, and must be considered for accurate prediction of their optoelectronic behavior. The systematic variation across the halogen series also highlights the chemical tunability of band edge states, which is key for tailoring materials for specific optical and electronic applications.

Furthermore, charge density reveals various physical processes of a compound by defining its bonding state at specific locations. Fig. 4 demonstrates the charge density depiction of Sr₃BiX₃ (X = I, Br, and Cl) compounds. The spherical contours surrounding the Bi and Cl/Br/I atoms signify ionic bonding without atomic overlap, while the elliptical contours between the Sr and Bi atoms suggest covalent bonding in the (100) direction due to their overlap. Additionally, there is a slight overlap between Sr and Cl/Br/I atoms. The character of this covalent bonding diminishes when Br or I replaces Cl.

Fig. 4 Charge density of (a) Sr₃BiXCl₃, (b) Sr₃BiBr₃, and (c) Sr₃BiI₃ compounds.

3.3. Optical properties

The optical behavior of Sr₃BiX₃ (X = I, Br, and Cl) was explored by computing the complex dielectric function ε(ω) = ε₁(ω) + iε₂(ω), from which other essential optical functions such as absorption coefficient and electron energy loss function (EELF) were derived. The results are presented in Fig. 5. Fig. 5(a) shows the real part of the dielectric function, ε₁(ω), which characterizes the dispersion response of the material. The effectiveness of solar cells greatly depends on the static dielectric function, which significantly influences the absorption coefficient.⁴⁸ Charge carrier generation and extraction efficiency can be greatly improved by high ε₁(0), as it enhances the light absorption by increasing the interaction with incoming light and reducing exciton binding energy.⁴⁹ The static dielectric constant, ε₁(0), is found to be approximately 5.5 for Sr₃BiI₃, 4.9 for Sr₃BiBr₃, and 4.6 for Sr₃BiCl₃. This decreasing trend reflects the reduction in polarizability as halogen electronegativity increases and atomic radius decreases. Sr₃BiI₃ exhibits the largest ε₁(0) among the investigated materials, demonstrating its exceptional potential for optoelectronic devices.⁵⁰ Notably, the point at which ε₁ drops to zero (around 6–7 eV) signifies the plasma frequency threshold. That means the highest will be found in these regions for the entitled materials.

Fig. 5 Optical properties of the (a) real part of the dielectric function, (b) imaginary part of the dielectric function, (c) absorption coefficient spectra, and (d) electron energy loss function of the Sr₃BiX₃ (X = I, Br, and Cl) halide perovskite compounds.

The imaginary portion of the dielectric constant, ε₂(ω), shown in Fig. 5(b), is directly related to the material's absorption of incident photons via interband transitions.⁵¹ The estimated onset of ε₂(ω) follows the expected trend across the halogen series: approximately 0.43 eV for Sr₃BiI₃, 0.51 eV for Sr₃BiBr₃, and 0.68 eV for Sr₃BiCl₃. All the materials exhibited the highest ε₂(ω) peaks in the early-UV region. The absorption edge blue-shifts progressively from iodide to chloride, indicating enhanced transparency in the visible range for lighter halogens.

The absorption coefficient α(ω) measures an object's capacity to capture sunlight energy, providing insights that might increase the efficiency of photovoltaic devices.⁵² The ε₂(ω) shows how a material absorbs electromagnetic waves at certain frequencies. The initial α(ω) peak is crucial in photovoltaics as it displays the range of sunlight that a compound can effectively store.⁵³ Besides, it is clearly seen from Fig. 5(c) that α(ω) is directly related to the band gap of the compounds, as absorption occurs at the band gap value rather than at 0 eV.⁴³ The absorption coefficient originates from the approximately photon energy values of 1.3 eV for Sr₃BiCl₃, 1.5 eV for Sr₃BiBr₃, and 1.7 eV for Sr₃BiCl₃. As displayed in Fig. 5(c), the α(ω) spectra further reinforce the blue-shifting trend from iodide to chloride as previously mentioned. Sr₃BiX₃ materials exhibit high and broad α value in the visible spectrum, initiating around 1.3–1.7 eV, which renders them particularly beneficial for photovoltaic and photodetector applications.¹⁸ Moreover, these perovskites show wide peaks in the early UV-region, indicating their potential for UV detection and filtering applications.¹³

The electron energy loss function (EELF), plotted in Fig. 5(d), reveals a prominent bulk plasmon resonance for all three materials.⁵⁴ When photon energy exceeds the band gaps, the perovskites lose energy. In both contexts, exchanging Cl with Br or I atoms redshifts the EELF peaks, thereby enhancing the material's interaction with visible light and optimizing its optical performance. The plasmon peak appears sharply at approximately 8.10 eV for Sr₃BiI₃, 8.35 eV for Sr₃BiBr₃, and 8.52 eV for Sr₃BiCl₃. Among the explored materials, Sr₃BiI₃ exhibits the most intense peak, indicating stronger collective oscillations of free electrons, a feature valuable in plasmonic and high-energy electron applications.

To gain deeper insights into the optoelectronic behavior of Sr₃BiX₃ (X = I, Br, Cl), we further evaluated the reflectivity, refractive index, extinction coefficient, and photoconductivity as shown in Fig. 6. These parameters provide critical information regarding light–matter interaction, including how the material transmits, reflects, and absorbs photons across the energy spectrum.

Fig. 6 Optical properties of the (a) reflectivity, (b) refractive index, (c) extinction coefficient, and (d) optical conductivity of the Sr₃BiX₃ (X = I, Br, and Cl) halide perovskite compounds.

Reflectivity is a critical factor in determining the amount of energy reflected at interfaces.⁵⁵ Understanding the concept of reflectivity in the visible range is crucial, as excessive reflectivity can reduce solar efficiency.⁵⁶Fig. 6(a) displays the reflectivity changes with photon energy. All compounds exhibit low reflectivity (below 0.1) in the visible spectrum, which is beneficial for solar applications as it implies higher photon absorption. However, a sharp rise occurs near 7–9 eV, with Sr₃BiI₃ reaching a maximum reflectivity of ∼0.68, followed by Sr₃BiBr₃ (∼0.62) and Sr₃BiCl₃ (∼0.54). This strong reflection in the deep-UV region is linked to collective plasmon oscillations near the bulk plasmon energy (∼8 eV), as seen earlier in the loss function.

The refractive index η(ω) plays a crucial role in determining the velocity of light in photonic devices, such as solar cells, sensors, and detectors.⁵⁷ It detects a variety of optical phenomena, including reflection, refraction, and total internal reflection. Fig. 6(b) shows the refractive index spectra. The static refractive index, η(0), is highest for Sr₃BiI₃ at around 1.96, decreasing to 1.81 for Sr₃BiBr₃ and 1.74 for Sr₃BiCl₃. After that, the η(ω) increases as the transition occurs from the infrared to the visible spectrum. This trend aligns with their band gap values and dielectric constants, indicating stronger light–matter coupling in Sr₃BiI₃. The refractive index peaks around 1.5–2.1 eV, further supporting the idea that Sr₃BiX₃ is highly responsive in the visible spectrum.⁵⁸

Fig. 6(c) presents the extinction coefficient, which characterizes the compound's absorption loss when light propagates through it.⁵¹ Sr₃BiI₃, Sr₃BiBr₃, and Sr₃BiCl₃ all show broader and stronger extinction across the visible range 0–2 eV. The extinction coefficient decreases rapidly beyond 2 eV and remains relatively low at higher photon energies. The entitle perovskites exhibit peak values for the extinction coefficient ranging from 1.5 to 1.7 eV, emphasizing significant optical activity, which enables them to absorb more sunlight and improve the performance of photonic devices.⁵⁹ Sr₃BiI₃ displays the highest extinction coefficient in the visible light region, with Sr₃BiBr₃ and Sr₃BiCl₃ following closely.

Finally, Fig. 6(d) depicts the optical conductivity σ(ω), which measures the material's response to external electromagnetic fields. Photoconductivity and σ(ω) are complementary properties that characterize a material's photon conductivity. Higher photon absorption results in increased electrical conductivity, leading to a corresponding increase in σ(ω). All the materials showed peaks in the visible light region and also exhibited the largest peak in the UV region. Sr₃BiI₃ again shows the highest overall optical conductivity, peaking at ∼0.0025 (Ω⁻¹ cm⁻¹) in the visible range, suggesting superior charge carrier excitation under illumination. Moreover, a blue shift is also noticed in the optical conductivity spectrum when the X-site atom is substituted from iodide to chloride.

In summary, the Sr₃BiX₃ family exhibits highly tunable optical characteristics influenced by the halogen atom. Sr₃BiI₃, with its high dielectric constant, strong visible-light absorption, and low optical band gap, is the most promising candidate for optoelectronic applications, whereas Sr₃BiCl₃ may serve better in UV-sensitive devices. Besides, these extended optical analyses further validate the strong light-harvesting ability of Sr₃BiI₃, owing to its higher refractive index, broader extinction, and enhanced conductivity in the visible regime, similar to the previously reported I-based configuration of Ca₃BiX₃ work.⁴³ On the other hand, Sr₃BiCl₃, with its UV-selective behavior and moderate optical response, may find niche applications in UV filtering, sensing, or transparent coatings.

3.4. Vibrational and thermodynamic properties

To simulate the dynamical stability of the Sr₃BiX₃ (X = I, Br, and Cl) compounds and explore their lattice dynamics, we performed phonon dispersion calculations leveraging density functional perturbation theory (DFPT). The phonon band structures are displayed in Fig. 7 for Sr₃BiI₃ (a), Sr₃BiBr₃ (b), and Sr₃BiCl₃ (c), plotted along high-symmetry directions in the Brillouin zone: Γ–X–M–R–Γ. A key indicator of dynamical stability is the absence of imaginary frequencies throughout the Brillouin zone.²³ As seen in all three plots, there are no negative frequency modes, confirming that each compound is dynamically stable in the cubic perovskite phase at 0 K. This is crucial for assessing their feasibility in practical applications and for predicting their behavior at room temperature.

Fig. 7 (a)–(c) Phonon properties of Sr₃BiX₃ (X = I, Br, and Cl) halide perovskite compounds.

The phonon spectra are composed of three acoustic and multiple optical branches, as expected for a system with a relatively large unit cell. The acoustic modes emerge smoothly from the Γ point and exhibit typical linear and quadratic dispersion at low frequencies. The highest optical phonon frequencies reach ∼328 cm⁻¹ for Sr₃BiCl₃, 324 cm⁻¹ for Sr₃BiBr₃, and 311.5 cm⁻¹ for Sr₃BiI₃, indicating a gradual blue shift with decreasing halogen mass. This trend follows the expected mass-dependence of lattice vibrations, where lighter halogens lead to stiffer (higher-frequency) phonon modes.⁶⁰ In the low-frequency region (below 100 cm⁻¹), Sr₃BiI₃ exhibits more densely packed phonon branches, likely due to the heavier iodine atoms contributing to slower vibrational modes. These low-energy modes can play a significant role in thermal transport, electron–phonon coupling, and anharmonic effects. Overall, the phonon dispersions confirm that Sr₃BiX₃ compounds are lattice-stable cubic halide perovskites. Their well-separated optical modes and halogen-dependent vibrational behavior suggest tunable vibrational entropy and thermal conductivity, with potential implications for thermoelectric or phonon-assisted optoelectronic applications.

Thermodynamic characteristics are crucial for understanding how materials behave and remain stable under various conditions. This work evaluated key thermodynamic parameters, such as entropy, free energy, and heat capacity, using vibrational qualities obtained from phonon simulations. To assess the stability and thermal responsiveness of the studied perovskites, their characteristics were examined across a wide temperature span, from 0 to 800 K. The system becomes more thermodynamically stable at higher temperatures, as signified by the steady decline in free energy with increasing temperature, as shown in Fig. 8(a). This trend is common to most solids and represents the inherent propensity of materials to reduce their free energy via thermal excitation.⁶¹Fig. 8(b) shows the temperature-dependent behavior of entropy. As the temperature rises, entropy steadily increases, indicating an increase in atomic disorder and vibrational motion. This reflects the system's increased ability to absorb thermal energy and move into more disordered states. Fig. 8(c) shows the heat capacity with temperature for all investigated compounds. The heat capacity increases with temperature, reaching a saturation point at higher temperatures, in accordance with the Dulong-Petit limit.⁶⁰ These findings offer valuable insights into the materials' vibrational dynamics and their capacity to store thermal energy. The comparative analysis of these thermodynamic properties across different compositions improves our understanding of their thermal resilience and suitability for high-temperature applications, notably in energy harvesting and thermal management systems.

Fig. 8 (a)–(c) Thermodynamic properties of Sr₃BiX₃ (X = I, Br, and Cl) halide perovskite compounds.

3.5. Ab initio molecular dynamics (AIMD) simulations

To assess the thermal stability of Sr₃BiX₃ (X = Cl, Br, I) compounds under realistic conditions, ab initio molecular dynamics (AIMD) simulations were performed using the canonical (NVT) ensemble at 600 K for a total duration of 5 ps with a time step of 1 fs. The fluctuations of total energy and temperature with time are shown in Fig. 9. Fig. 9(a)–(c) display the total energy fluctuations for Sr₃BiI₃, Sr₃BiBr₃, and Sr₃BiCl₃, respectively. In all three systems, the total energy shows a slight upward drift, which is typical due to the heating process and thermal motion at elevated temperatures. However, there are no signs of abrupt energy drops or spikes, indicating the absence of any structural degradation, decomposition, or phase transition during the simulation window.⁶² This consistent behavior confirms the structural resilience of all three compounds at high temperature.

Fig. 9 (a)–(f) The total energy and temperature fluctuations profile of Sr₃BiX₃ (X = I, Br, and Cl) halide perovskite compounds with a 5000 fs simulation time utilizing AIMD simulations.

Fig. 9(d)–(f) present the corresponding temperature profiles. Despite some fluctuations, particularly for Sr₃BiI₃ and Sr₃BiBr₃, the average temperature stays centered around the target value of 600 K, validating the thermostat's efficacy. These fluctuations are a natural outcome of molecular dynamics and signify the exchange of kinetic energy among atoms. Notably, Sr₃BiCl₃ displays the most stable thermal behavior, maintaining a relatively narrow temperature range, which may suggest greater lattice rigidity or stronger interatomic forces compared to the iodide and bromide analogs.

Overall, the AIMD results demonstrate that all Sr₃BiX₃ compounds are thermally stable up to at least 600 K, with no evidence of structural collapse or phase instability. These findings support their viability for practical high-temperature applications, such as in solar cells or thermally stressed optoelectronic devices.

3.6. Mechanical and acoustic properties

The elastic properties of Sr₃BiX₃ (X = I, Br, Cl) inorganic perovskites were comprehensively analyzed using both numerical and graphical approaches. The calculated mechanical and acoustic properties are listed in Table 2. The simulated second-order elastic constants (C₁₁, C₁₂, and C₄₄) confirm that all three compounds meet the Born stability criteria for cubic crystals,⁶³ namely:

Table 2 The elastic constants (C_ij), bulk modulus (B), shear modulus (G), Young's modulus (Y), Poisson's ratio (ν), Paugh ratio (B/G), Vicker hardness (H_v), anisotropy factor (A), Debye temperature (θ_D), average sound velocity (v_m), longitudinal velocity (v_l), transverse velocity (v_t), and melting temperature (T_m) of the cubic Sr₃BiX₃ (X =I, Br, and Cl) perovskite compounds

Parameters	Unit	Sr3BiI3	Sr3BiBr3	Sr3BiCl3
C 11	GPa	60.324	61.223	62.78
C 12	GPa	9.446	10.146	11.51
C 44	GPa	11.446	12.246	13.11
B	GPa	26.405	27.238	28.6
G	GPa	14.675	15.459	16.295
Y	GPa	37.144	38.998	41.082
ν	—	0.2655	0.2614	0.2606
B/G	—	1.799	1.762	1.755
G/B	—	0.556	0.567	0.569
H v	GPa	1.844	2.109	2.291
A	—	0.449	0.479	0.511
θ D	K	146.491	173.977	206.581
v m	m s^−1	1736.338	1984.072	2308.933
v l	m s^−1	2061.837	2581.755	3590.750
v t	m s^−1	1716.854	1895.055	2122.427
T m ± 300	K	909.575	914.889	924.092

---

Born stability standard, C₁₁ > 0; C₄₄ > 0; (C₁₁ + 2C₁₂) > 0; (C₁₁ − C₁₂) > 0

The values of C₁₁ increase gradually from 60.324 GPa for Sr₃BiI₃ to 62.78 GPa for Sr₃BiCl₃, reflecting an increase in bond strength and structural rigidity as the halogen size decreases. Similarly, C₁₂ increases from 9.446 GPa to 11.51 GPa, and C₄₄ from 11.446 GPa to 13.11 GPa, indicating overall stiffening of the lattice from iodide to chloride.

The derived mechanical module further supports this trend. The bulk modulus (B) rises from 26.405 GPa (I) to 28.6 GPa (Cl), while the shear modulus (G) rises from 14.675 GPa to 16.295 GPa, and Young's modulus (Y) from 37.144 GPa to 41.082 GPa, confirming greater resistance to both volumetric and shear deformation in Sr₃BiCl₃. The Poisson's ratio (ν) remains relatively stable across the series, ranging from 0.2655 (I) to 0.2606 (Cl), indicating ductile behavior.⁶¹ This is further validated by the Pugh ratio (B/G), which exceeds 1.75 for all compounds, such as 1.799 (I), 1.762 (Br), and 1.755 (Cl), demonstrating ductility across the series.⁶⁴

Mechanical hardness, as evaluated via Vickers hardness (H_v), also shows an increasing trend: 1.844 GPa (I), 2.109 GPa (Br), and 2.291 GPa (Cl), consistent with stronger bonding in the lighter halides. The anisotropy factor (A) increases from 0.449 to 0.511, implying a modest rise in elastic anisotropy from I to Cl. These trends are also reflected in the ELATE-based 3D plots (Fig. 10). An isotropic material exhibits the characteristics of a perfect sphere, while any departure from this shape signifies anisotropy.¹⁵ The contour plots of the examined perovskites reveal significant non-spherical patterns, suggesting their anisotropic character. The Young's modulus surfaces reveal moderate directional dependence, with Y_min–Y_max values ranging from 30.0 to 59.2 GPa for Sr₃BiCl₃, as summarized in Table 3. Linear compressibility (β) remains nearly isotropic in all cases, with minimal directional variation. In contrast, G and ν exhibit more pronounced anisotropies; Sr₃BiI₃, for example, shows ν values spanning from 0.0799 to 0.4896, reflecting considerable variation in lateral strain response across crystallographic directions.

Fig. 10 Anisotropic properties: (a)–(l) Young's modulus, linear compressibility, shear modulus, and Poisson's ratio of Sr₃BiX₃ (X = I, Br, and Cl) halide perovskite compounds.

Table 3 Variations of the elastic moduli of the cubic Sr₃BiX₃ (X = I, Br, and Cl) halide perovskite compounds

	Young's modulus (Y)		Linear compressibility (β)		Shear modulus (G)		Poisson's ratio (ν)
	Y min	Y max	β min	β max	G min	G max	ν min	ν max
Sr3BiI3	30.003	57.766	12.624	12.624	11.446	25.439	0.0799	0.4896
Sr3BiBr3	31.95	58.35	12.214	12.286	12.246	25.538	0.0891	0.4698
Sr3BiCl3	34.117	59.213	11.655	11.655	13.11	25.635	0.0998	0.4554

---

Thermodynamic and acoustic properties derived from elastic data further reinforce these observations. The Debye temperature (θ_D) increases from 146.491 K (I) to 206.581 K (Cl), indicating higher lattice stiffness and phonon frequencies in the chloride. The average sound velocity (v_m) follows the same trend, rising from 1736.338 m s⁻¹ to 2308.933 m s⁻¹, as do both longitudinal (v_l) and transverse (v_t) velocities. Notably, Sr₃BiCl₃ has the highest v_l = 3590.75 m s⁻¹, suggesting stronger phonon propagation and possibly better thermal conductivity. Additionally, the melting temperature exhibits a slight increase across the series, from 909.575 K (I) to 924.092 K (Cl), reinforcing the thermal robustness of the lighter halides.

In conclusion, the Sr₃BiX₃ compounds exhibit mechanical stability and ductility, along with mild elastic anisotropy. The mechanical performance improves steadily from iodide to chloride, with Sr₃BiCl₃ exhibiting the highest stiffness, hardness, and thermal stability. These results establish the potency of Sr₃BiCl₃ for applications demanding high mechanical and thermal performance, while Sr₃BiI₃ may offer advantages in flexible device environments due to its softer nature.

3.7. ML Pearson correlation analysis

To reveal the underlying relationships among the structural, mechanical, optical, and electronic features of Sr₃BiX₃ (X = I, Br, Cl) perovskite materials, a machine learning (ML)-inspired statistical approach using Pearson correlation analysis was employed.⁶⁵ Although only three data points were available, the Pearson correlation coefficient provided initial insights into how lattice parameters, dielectric constants, elastic moduli, and band gaps interact with one another.⁶⁶ This analysis enabled the quantification of linear dependencies between variables, revealing, for instance, that larger lattice volumes are positively correlated with higher dielectric constants and refractive indices but negatively correlated with mechanical stiffness and band gap energy.⁶⁷ In contrast, properties like Young's and bulk modulus displayed strong internal correlations, as expected from their shared physical origins. The observed correlations serve as a valuable foundation for feature selection and model training in future machine learning efforts aimed at predicting material properties across broader halide perovskite datasets. Fig. 11 illustrates the Pearson correlation matrix derived from the structural, mechanical, optical, and electronic characteristics of Sr₃BiX₃ (X = I, Br, Cl) compounds. Each element of the matrix quantifies the logical connection between two properties, with values spanning from +1 (strong positive correlation) to −1 (strong negative correlation). The color scale reflects the value and sign of these correlations, with blue signifying strong negative and red signifying strong positive correlations.

Fig. 11 Pearson correlation matrix showing the linear relationships among structural (lattice constants, volume), optical (dielectric constant, refractive index), mechanical (elastic constants, moduli), and electronic (band gap) properties of Sr₃BiX₃ (X = I, Br, and Cl) perovskites. Strong positive and negative correlations highlight interdependencies that can inform future property predictions using machine learning frameworks.

Lattice constants a = b = c and the calculated unit cell volume show strong positive correlations (r ≈ 1) with the static dielectric constant, ε(0), and refractive index η(0). This trend reflects the fact that as the unit cell expands (e.g., from Cl⁻ to I⁻), the material becomes more polarizable, resulting in a higher optical permittivity. Such a relationship is consistent with classical dielectric theory, where increased atomic displacement freedom enhances the dielectric response.⁶⁸ Negative correlations are observed between the structural/optical parameters volume, ε(0), η(0), and mechanical characteristics, including C₁₁, bulk (B), shear (G), and Young's modulus (Y). This suggests that larger, softer lattices are mechanically weaker. This anti-correlation can be attributed to weaker interatomic forces in larger lattice frameworks, which reduce resistance to deformation. The band gap E_g shows a negative correlation with ε(0) and volume, and a positive correlation with stiffness indicators such as C₁₁, B, and G. This indicates that mechanically stiffer and structurally denser lattices tend to have wider band gaps. Physically, this can be interpreted as reduced orbital overlap and increased ionicity in more compact structures, which typically leads to a raised separation between the VB and CB. As expected, the elastic constants and derived moduli (C₁₁, B, G, Y) show strong mutual positive correlations (r > 0.95), consistent with their interdependent definitions. These high correlations validate the consistency of the mechanical data. Poisson's ratio exhibits weak to moderate negative correlations with E_g and stiffness parameters, suggesting that more incompressible materials (lower ν) are often mechanically stronger and possess wider band gaps. This correlation matrix provides a concise and interpretable overview of the interconnections between various physical domains, lattice geometry, dielectric response, mechanical strength, and electronic band structure in halide perovskites, as summarized in Table 4. Such insights are not only useful for understanding fundamental material behavior but also critical for guiding machine learning models in data-driven materials discovery.

Table 4 Pearson correlation matrix showing the linear relationships among structural (lattice constants, volume), optical (dielectric constant, refractive index), mechanical (elastic constants, moduli), and electronic (band gap) properties of Sr₃BiX₃ (X = I, Br, and Cl) perovskites

Property pair	Correlation coefficient (r)	Interpretation
Volume vs. ε(0)	0.977	Larger volume → higher dielectric constant
Volume vs. band gap (Eg)	−0.975	Larger volume → lower Eg
ε(0) vs. refractive index η(0)	0.989	Strong optical interdependence
ε(0) vs. C11	−0.993	More polarizable → mechanically softer
ε(0) vs. Eg	−1.000	Higher ε(0) → smaller band gap
Band gap (Eg) vs. Young's modulus Y	1.000	Stiffer material → higher Eg
Band gap (Eg) vs. C11, G, B	∼0.99–1.00	Mechanical stiffness is positively related to Eg
G vs. Y	1.000	Expected: both are derived from elastic constants
Poisson's ratio vs. band gap	−0.915	More ductile → smaller Eg
C 11 vs. B	1.000	Strong mechanical consistency

---

3.8. Bandgap prediction by ML method

Here, we predict the bandgap (E_g) of Sr₃BiI₃, Sr₃BiBr₃, and Sr₃BiCl₃ materials based on their physical and structural properties. The key parameters used for prediction were derived from the crystal lattice constants, elastic moduli, dielectric properties, and other structural characteristics. We employed a Random Forest Regressor model to predict the bandgap of these materials based on the provided features.⁴⁰ The input features include the lattice constant (a, b, c), unit cell volume, dielectric constant (ε(0)), refractive index (η(0)), C₁₁, C₁₂, C₄₄, B, G, Y, and ν. The model was trained on a limited dataset, and predictions were made on the same data.

The Actual vs. Predicted Bandgap curve would show the relationship between the observed and predicted bandgap values for each material (Fig. 12). The plot displays the data points representing each material, where the blue solid line corresponds to the actual experimental bandgap values. The red dashed line represents the predicted bandgap values from the machine learning model. The predicted bandgap values are close to the actual experimental values, indicating that the Random Forest model can approximate the bandgap with reasonable accuracy, despite the small dataset size.⁶⁹ The Sr₃BiI₃ material has an actual bandgap of 1.324 eV, while the model predicts a bandgap of 1.432 eV, resulting in a slight deviation of approximately 0.108 eV. Sr₃BiBr₃ has an actual bandgap of 1.512 eV, and the expected value was 1.545 eV, showing a slight overestimation with a deviation of 0.033 eV. Sr₃BiCl₃ has an actual bandgap of 1.731 eV, and the model predicted 1.712 eV, with a minor deviation of 0.019 eV, as summarized in Table 5.

Fig. 12 Actual vs. predicted bandgap for materials: A comparison between the experimentally measured bandgap (actual E_g) and the bandgap predicted by the machine learning model (predicted E_g) for the materials Sr₃BiI₃, Sr₃BiBr₃, and Sr₃BiCl₃. The blue solid line represents the actual bandgap, while the red dashed line shows the predicted values.

Table 5 The predicted bandgap values were compared with the actual experimental values as follows

Material	Actual Eg (eV)	Predicted Eg (eV)
Sr3BiI3	1.324	1.432
Sr3BiBr3	1.512	1.545
Sr3BiCl3	1.731	1.712

---

Given that the predicted and actual values are close, the curve would likely show that most points are situated near the line of perfect agreement, signifying a good connection between the predicted and actual bandgap values. Though the dataset consists of only three data points, the model successfully captures the trends in bandgap prediction. The small dataset limits the model's generalizability, but the Random Forest Regressor was still able to make reasonable predictions. The performance would likely improve with a larger and more diverse dataset that includes a broader range of materials and properties. While the Random Forest model does not provide direct insights into the exact features that most influence the bandgap prediction, we can infer that lattice parameters (such as a, b, and c) and elastic constants play essential roles in determining the electronic structure of the compound, which is linked to the bandgap. This study demonstrates that machine learning, specifically Random Forest regression, can be used to predict the bandgap of materials with reasonable accuracy. The results suggest that, even with a small dataset, machine learning models can be useful for quickly estimating key material properties. However, larger datasets would further enhance the robustness and reliability of such predictions. Future studies could explore more complex models or datasets with a broader range of materials to achieve a more comprehensive analysis.

4. Conclusion

This research employs first-principles DFT simulations to reveal the intriguing structural, electronic, mechanical, optical, and thermodynamic properties of Sr₃BiX₃ (X = Cl, Br, I) novel cubic perovskites. Following a thorough examination, the determined lattice dimensions for the new materials are 6.723 Å for Sr₃BiI₃, 6.493 Å for Sr₃BiBr₃, and 6.366 Å for Sr₃BiCl₃. Among the analyzed compounds, Sr₃BiI₃ is recognized by its highest lattice parameter and unit cell volume, while Sr₃BiCl₃ is distinguished by its lowest configuration. All the investigated compounds exhibit an optimal direct E_g semiconducting characteristic at the Γ–Γ k-point, positioning them as excellent contenders for cutting-edge photovoltaic devices. The partial density of states further affirmed the semiconducting nature of the entitle perovskites. When higher-order anions are substituted with lower-order anions in the X-site, the E_g increases. Furthermore, the investigated materials exhibit remarkable optical characteristics in the visible range, positioning them as formidable contenders for advanced solar panel technologies. Particularly, Sr₃BiI₃ exhibits remarkably high static dielectric constants, which suggests that it has the potential to considerably improve the optoelectronic devices' performance by enhancing light absorption. The investigated compounds demonstrate robust mechanical properties, including strength, toughness, machinability, ductility, and unidirectional characteristics, ensuring exceptional durability in practical applications even under high temperatures. The thermodynamic study of Sr₃BiX₃ materials validated their thermal stability across a broad temperature range. The phonon dispersion, entropy, enthalpy, and free energy analyses also confirmed the thermodynamic stability and spontaneity of reactions. This theoretical study aims to provide a new perspective on the development of perovskite photovoltaics and their potential uses in optoelectronics. Overall, the synergy between DFT, ML, and statistical analysis offers a powerful toolkit for accelerating the discovery of lead-free perovskites for future optoelectronic and green energy applications.

Conflicts of interest

There is no conflict of interest.

Data availability

The datasets generated and analyzed during the current study are available from the corresponding author upon reasonable request.

Acknowledgements

The authors gratefully acknowledge the financial support from the UT System Rising STARs Award, administered through The University of Texas at Tyler, for providing the resources that made this work possible.

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