Ba₃PX₃ are metal halide compositions that crystallize in the cubic Pm-3m space group. Each formula unit contains three Ba atoms, one P atom, and three halide atoms X (F, Cl, Br, or I). Ba²⁺ cations occupy the 3c Wyckoff sites (0.5, 0, 0) near the cube corners, P³⁻ anions sit at the 1a Wyckoff site (0, 0, 0) at the center of the cube, and halide X⁻ anions reside at the 3d Wyckoff sites (0, 0.5, 0.5) on the faces, forming corner-sharing PBa₆ and PX₆ octahedra. To ensure the lowest ground energy and highest structural stability, the equilibrium lattice constant was optimized to yield the most stable structure (Fig. 1). The specific values of the optimized structural properties are shown in Table 1, such as the unit cell volume (in ų), equilibrium lattice constant a₀ (in Å), and ground state energy (in eV), all obtained from the PBE functional. The data from this table can be used to see that as the size of the halide anion grows (F, Cl, Br, or I), the lattice constants also increase. This trend is in good agreement with the fact that I has a larger ionic radius than F. In addition, the ground-state energy of Ba₃PX₃ (X = F, Cl, Br, I) decreases with increasing anion size, indicating that Ba₃PI₃ has the most stable configuration among all the compositions investigated.

The Bulk Modulus (B₀) quantifies resistance to hydrostatic compression and decreases from 31.985 GPa (Ba₃PF₃) to 23.166 GPa (Ba₃PI₃). These lower B₀ values illustrate the reverse trend of lattice expansion and indicate that as the atomic size of the halogen increases, the corresponding compounds become softer and more compressible. The observed decrease of B₀ points to a decrease in the average strength of inter-atomic bonding because the Ba–X and P–X bond lengths become longer in heavier halide systems. The positive First Pressure Derivative of Bulk Modulus (B₀^′) values illustrate that with the application of pressure, the materials under consideration will become less compressible, reflecting an increase in resistance to compression. This behaviour is consistent with the overall mechanical stability of the compounds. The Ground State Energy (E₀) values become more negative from −50,120.673 eV (Ba₃PF₃) to −92,235.682 eV (Ba₃PI₃), showing that there is a greater number of electrons and a greater total amount of binding energy for the heavier halide compound systems. The consistently negative values for the E₀ confirm the presence of thermodynamic stability for all of the investigated phase equilibrium points. The computed lattice parameters and their trend are in good agreement with the previously reported studies [22,33,34], which confirms the reliability of the adopted computational methodology.

To realize the electronic behaviour of the compositions considered and their uses in different areas, we calculated their electronic band structures using PBE functional along the high-symmetry directions R-Γ-X-M-Γ, with the Fermi level (E_F) set to 0 eV in the Brillouin zone. The calculated band-gap values obtained using the PBE functional, as shown in Table 2, are in good agreement with previously reported theoretical results [22,33,35], with minor variations arise from differences in computational parameters, structural optimization conditions, and methodological choices across different studies. To address the problem of the consistent underestimation of the bandgap by the traditional PBE functional [26] and achieve more accurate electronic bandgap values, the modified Becke-Johnson (mBJ) potential was used. The modified badstructures have been plotted and presented in Fig. 2. These bandstructures clearly demonstrate that the VBM and CBM of all the compounds are at the Γ-point, indicating a direct relationship between the structure and bandgap. The calculated PBE+mBJ bandgaps (Table 2) show a systematic decrease from 2.373 eV (Ba₃PF₃) to 1.480 eV (Ba₃PI₃). As the halide changes from F to I, a reduction in the bandgap occurs because the covalent nature of the Ba-X and P-X bonds is more pronounced. It decreases the energy difference between VBM & CBM.

The systematic decrease in the band gap from Ba₃PF₃ to Ba₃PI₃ can be explained by halide p-orbital energetics, electronegativity, and orbital overlap. The energy of the halide p orbitals decreased with a variation in the halogen from F to I because of the reduction in effective nuclear charge and principal quantum number. Simultaneously, the electronegativity of the halogen decreases, limiting the ionic nature of the P-X and Ba-X bonds and enhancing the covalent interaction. This decrease in electronegativity enhances the spatial overlap between the halide p and phosphorus p states, resulting in an increase in the maxima of the valence band. Consequently, the energy gap between the valence and conduction bands narrows as the series progresses. Additional evidence of the enlarged orbital overlap is that the bandwidth and dispersion are higher in the valence bands of Br- and I-based compounds, which confirms that halide substitution is a dominant factor in the electronic structure tuning of Ba₃PX₃ materials.

Fig. 3a–d shows the total and partial density of states (TDOS and PDOS) for the lead-free all-inorganic compounds Ba₃PF₃, Ba₃PCl₃, Ba₃PBr₃, and Ba₃PI₃, calculated via first-principles DFT. The DOS plots span an energy range of −5 to +5 eV with respect to the Fermi level (EF = 0 eV), highlighting the valence band and the conduction band. Each of these compounds has a clear forbidden band around the Fermi energy level, confirming their semiconducting nature. The valence band maxima are dominated by P-p states, with smaller participation from Ba-p/d and X-p states. This hybridization evolves systematically across the series, reflecting changes in P–X bonding character and halide p-orbital contributions. The evolution of hybridization is reflected in the band structures by the flattening of the valence bands for heavier halides, particularly in Ba₃PI₃. As the halogen atom changes from F to I, the crystal field and bonding environment around the Ba-P framework are modified, leading to a gradual upward shift of the Ba-P-dominated valence band edge, which is responsible for the progressive reduction of the band gap across the series. The CBM primarily arises from Ba-d states, with minor contributions from P-p orbitals. It indicates ionic Ba-X interactions and covalent P-X bonding, with the conduction band edge showing relatively similar dispersion across all four compounds, indicating that the conduction band position is only weakly affected by halogen substitution. The dominance of Ba-d orbitals near the CBM also accounts for the comparatively strong dispersion of the conduction bands, suggesting relatively moderate electron effective masses.

As a measure of carrier transport, the effective masses of the electrons (m_e*) and holes (m_h*) were determined at the Γ-point and are provided in Table 2. For the studied materials, the electron effective mass lies in the range 0.561–0.686 m₀ and hole effective mass in the range 0.698–0.956 m₀. The non-monotonic variation is a result of the interplay between halide orbital and band hybridization. The effective mass of electrons is relatively low in the case of Ba₃PCl₃ and Ba₃PI₃, which implies that these elements have good electron transport, whereas Ba₃PBr₃ exhibits comparatively lower hole effective mass, suggesting improved hole mobility. Generally, m_h* is greater than m_e* of all compounds, and this means that they transport electrons more efficiently. These moderate values of effective mass justify the prospective application of Ba₃PX₃ compounds in optoelectronic and photovoltaic processes.

This section provides a thorough analysis of the various optical characteristics of Ba₃PX₃, such as dielectric function, absorption spectra, optical conductivity, energy-loss function, reflectivity & refractive index over an energy range of 0–13 eV. The aim is to understand how these compounds behave when exposed to solar or other high energy sources. Information contained in optical spectra can be utilized to extract information about the induced polarization capacity and internal structure of the materials. For example, their bond types and band structure, as well as empty states can all be determined from an analysis of the material’s optical spectra [36]. All optical properties were computed using the electronic structure obtained through the TB-mBJ potential to make them consistent with the better band-gap determination.

A material’s optical properties can be described using the complex dielectric function, which is defined as [37]:(1)ε(ω)=ε1(ω)+iε2(ω)where, ω is the angular frequency of the electromagnetic (EM) radiation incident on the material. The real part of the complex dielectric function, ε₁(ω), describes electronic polarization & anomalous dispersion, and the imaginary part, ε₂(ω), describes the optical absorption of the compounds. The imaginary part of the complex dielectric function, ε₂(ω), can be expressed in terms of ε₁(ω) [37]:(2)ε2(ω)=4e2π2ω2m2∑i,j∫|⟨i|M|j⟩|2fi(1−fi)Xδ(Ef−Ei−ћω)d3khere, the dipole matrix is represented by the symbol M, where i and j denote the initial state (valence band) and the final state (conduction band). The function f_i is the Fermi distribution function associated with the valence band. The expression δ(E_f − E_i − ћω) indicates the energy difference between the valence band and conduction band at a specific k point due to the absorption of a photon with an energy of ћω. The other variables are as follows: e = electron charge, ω = angular frequency of the photon, ћ = reduced Planck’s constant (where ћ = h/2π), and m = electron mass. To calculate ε₁(ω), the Kramers-Kronig transformation is used [37], as given below:(3)ε1(ω)=1+2πP∫0∞ω′ε2(ω′)ω′2−ω2dω′

The letter P denotes the integral’s principal value. To analyse how the compounds we are studying respond to incoming light, we have measured their dielectric function, denoted as ε(ω).

Fig. 4a,b correspondingly demonstrates the real and imaginary parts of the ε(ω) as a function of the incoming EM energy. The static value of the ε₁(ω), i.e., ε₁(0), also called the dielectric constant for the given sample, occurs at the low-end of our measurements. From the measurements, it can be observed that the ε₁(0) was measured to be 2.75, 3.18, 3.57, and 4.35, respectively, for Ba₃PF₃, Ba₃PCl₃, Ba₃PBr₃, and Ba₃PI₃, respectively. Therefore, we have observed that as we increase the atomic size of the element in the X-position, the dielectric constant values also increase. This behaviour arises from an increase in polarizability, which increases the involvement of the lattice to the overall polarization. In particular, Ba₃PI₃ exhibits the highest ε₁(0), indicating strong polarizability and a high optical permittivity. The negative values of ε₁(ω) after 7.06 eV of the compounds in the ultraviolet (UV) region indicate a plasmonic or metallic-like response, where the induced polarization oscillates out of phase with the incident EM field. This behaviour suppresses light propagation and results in strong reflection, associated with plasma resonance and intense interband electronic transitions. Fig. 4b represents the computed threshold energy values of ε₂(ω) for Ba₃PX₃ (X = F, Cl, Br, and I). Ba₃PF₃ does not exhibit any sharp peak in ε₂(ω); its maximum value occurs at 5.26 eV, after which it decreases, followed by a secondary increase and subsequent decline after reaching another maximum at 9.08 eV. Whereas for Ba₃PCl₃, Ba₃PBr₃, and Ba₃PI₃, ε₂(ω) shows sharp peaks at 8.59, 7.74, and 6.73 eV, correspondingly. The main peak in ε₂(ω) occurs in 5.26–8.59 eV energy range, indicating strong interband optical transitions in this region. Among all compounds, Ba₃PI₃ exhibits the highest intensity peak in ε₂(ω), suggesting enhanced optical absorption in the near-UV region.

Fig. 5a presents curves for σ(ω), which represents the real part of optical conductivity for compositions studied. For all four compounds, σ(ω) has the highest values at energy of 9.08 eV for Ba₃PF₃, 8.59 eV for Ba₃PCl₃, 7.74 eV for Ba₃PBr₃, and 6.73 eV for Ba₃PI₃. As the anion changes from F to I, there is a general decrease in the corresponding energy value for peak values of σ(ω), indicating that the compounds have been further shifted towards lower photon energies. The pattern exhibited by σ(ω) is similar to that of the α(ω) (Fig. 5b) and ε₂(ω) (Fig. 4b), indicating that Ba₃PI₃ and Ba₃PBr₃ have higher optical conductivity overall than Ba₃PF₃ and Ba₃PCl₃. Fig. 5b presents the optical absorption profiles of Ba₃PX₃ compounds. The absorption coefficient (α(ω)) defines the efficiency of light absorption, is a key parameter in evaluating solar energy conversion efficiency [38–40]. The value of α(ω) was found to be the highest for Ba₃PBr₃ at 7.88 eV. After that, the peak value of α(ω) for Ba₃PCl₃ and Ba₃PI₃ was observed as 8.78 eV and 6.84 eV, respectively. α(ω) for Ba₃PF₃ was found to be the lowest compared to Ba₃PCl₃, Ba₃PBr₃, and Ba₃PI₃. The absorption peak of Ba₃PF₃ was found at 9.21 eV. The strong peaks in the ε₂(ω), σ(ω) and α(ω) spectra are mainly due to interband transitions of the form P-p/X-p → Ba-d. The fact that these peaks shift to lower energies, for variation from Ba₃PF₃ to Ba₃PI₃, is in line with the gradually narrowing band gap and the increased P-p hybridization. The absorption edge of each compound is observed to be very similar to the values of the calculated direct band gap, which confirms the nature of direct transition in these materials. The systematic red-shift of the absorption edge of Ba₃PF₃ to Ba₃PI₃ is correlated with the decrease in band gap, and the heavier halide compositions absorb the visible light better. Increased polarizability and orbital overlap contribute to the increased intensity of optical peaks in Ba₃PI₃. These observations are consistent with previously reported halide perovskite systems, where band gap reduction, enhanced P-p/P-d hybridization, and increased polarizability with heavier halides lead to a systematic red-shift and stronger optical response [22,35].

Refractive index n(ω) and extinction coefficient k(ω) of Ba₃PX₃ compounds where X = F, Cl, Br and I are shown in Fig. 6a,b, respectively. The static value of n(ω) i.e., n(0), for Ba₃PX₃ compounds is 1.65, 1.78, 1.89 and 2.08, respectively, which increases with decreasing wavelength in the infrared and visible sections until it reaches a peak value of 2.09, 2.06, 2.15 and 2.47 at around 4.56, 2.08, 1.89 and 4.48 eV for Ba₃PF₃, Ba₃PCl₃, Ba₃PBr₃, and Ba₃PI₃, respectively. Following this point, the value of n(ω) continues to decrease; however, it will never be negative, which indicates that the compounds will still be transparent as the incident EM energy increases. Ba₃PI₃ had the highest value of n(0), which indicates that it will bend light more and allow for more potential applications with high refractive optical coatings. In Fig. 6b, the k(ω) value was found to be highest at 6.82 eV for Ba₃PI₃. The maximum k(ω) value was found to be at 7.85 eV for Ba₃PBr₃ and 8.69 eV for Ba₃PCl₃. The value of k(ω) for Ba₃PF₃ was found to be the lowest as compared to Ba₃PBr₃, Ba₃PCl₃, and Ba₃PI_3. The extinction coefficient peak of Ba₃PF₃ was found at 5.62 eV.

Reflectivity R(ω), which plays a crucial role in computing the reflected energy at interfaces, is presented in Fig. 7a. The zero-frequency reflectivity values are found to be 6.14% for Ba₃PF₃, 7.94% for Ba₃PCl₃, 9.48% for Ba₃PBr₃, and 12.40% for Ba₃PI₃. All compounds exhibit low reflectivity in the infrared and visible sections, demonstrating good transparency, while a noteworthy growth in reflectivity is found in the UV region, particularly for Ba₃PI₃, due to enhanced polarizability. In Fig. 7b, the peaks of the energy loss function E_loss(ω) for cubic Ba₃PX₃ structures are found in between 10–13 eV energy. These pronounced peaks correspond to plasma resonance arising from the collective oscillations of valence electrons, confirming the plasmonic behaviour of the materials in the UV region. E_loss(ω) defines the energy dissipated by fast electrons passing through the compound, rather than by direct interband transitions. The onset values of the dielectric constant, reflectivity, and refractive index for all the studied compounds have been mentioned in Table 3.

While the optical spectra of Ba₃PX₃ compounds are presented over a broad energy range extending into the UV region to capture all possible interband transitions, their photovoltaic relevance is primarily associated with the visible region of the spectrum (≈1.6–3.2 eV). In contrast, spectral features in the higher-energy UV region (>3.2 eV) arise from deeper interband transitions and are less relevant for solar energy harvesting. In the visible region, Ba₃PBr₃ and Ba₃PI₃ exhibit comparatively stronger absorption coefficients and higher dielectric response, indicating efficient photon harvesting within the solar spectral range. The relatively low reflectivity and moderate refractive indices further suggest favorable light coupling and reduced optical losses, which are advantageous for thin-film photovoltaic architectures where the absorber thickness is limited. Owing to their direct bandgap nature and strong visible-light absorption, efficient charge generation can be achieved at sub-micrometer thicknesses. Additionally, all optical properties reported here correspond to the isotropic response of the cubic crystal structure, as expected for the Pm-3m symmetry, and therefore do not exhibit directional anisotropy. This isotropic optical behavior is beneficial for device fabrication, as it ensures uniform light absorption irrespective of crystal orientation.

It should be noted that the current optical computations are based on the independent-particle DFT approach and do not include excitonic effects arising from electron–hole interactions. In practical substances, these effects may cause some small changes in the absorption onset and peak intensities, especially around the band edge. However, to perform comparative analysis across the Ba₃PX₃ series, the present method provides reliable trends in optical response. Similar approaches relating electronic structure to optical transitions have been reported in recent studies on semiconductor materials [41].

It is important to estimate the theoretical efficiency of a material prior to screening it for photovoltaic absorber applications. Absorber thickness, intrinsic defects, carrier recombination dynamics, temperature, and optical absorption characteristics strongly govern the efficiency of a photovoltaic device. A single-junction solar cell is reflected by the Spectroscopic Limited Maximum Efficiency (SLME) [30] as a complete and trustworthy indicator that assesses the upper-limit efficiency of both the absorber and the solar cell, including realistic absorption spectra and bandgap characteristics of the absorber. It follows that the values of SLME should be treated as theoretical screening limits rather than practically achievable device efficiencies, because they do not account for non-radiative recombination, defects, and interface losses. The SLME framework assumes radiative recombination as the dominant recombination mechanism, representing an upper theoretical limit for photovoltaic efficiency.

We have shown the SLME as a function of the absorber layer thickness of the studied Ba₃PX₃ (X = F, Cl, Br, I) compositions at 300 K in Fig. 8a. It is evident from the plot that the SLME rises rapidly as the film thickness increases within the sub-micrometer range, then reaches a saturation limit at thicker films. In particular, Ba₃PF₃ has the lowest efficiency of 2.43% at very thin layers, 17.73% at close to 1 μm, and then stabilizes. This thickness (~1 µm) is representative of typical thin-film photovoltaic absorbers and is therefore considered a practical reference for efficiency evaluation. Ba₃PCl₃ and Ba₃PBr₃, however, have a higher rate of 25.86% and 30.97%, respectively. The maximum performance is observed in Ba₃PI₃, where SLME rises sharply and stabilizes at 39.17%. This tendency shows that the replacement of the halides with heavier anions increases optical absorption and the theoretical efficiency that can be achieved. Unlike idealized models, the SLME formalism incorporates the calculated absorption coefficient, thereby accounting for realistic absorption losses. The variation in SLME across the Ba₃PX₃ series is therefore directly influenced by the computed absorption spectra, with higher absorption coefficients in Ba₃PBr₃ and Ba₃PI₃ contributing to their enhanced efficiency. The saturation behaviour at thickness above 1 μm indicates that an additional increase in thickness is not significantly helpful in increasing photon absorption, as the optical path length is already long.

Fig. 8b plots the temperature dependence of the SLME for the same compounds, with a 1 μm constant thickness of absorber. The decrease in the SLME with rising temperature from 300 to 900 K across all compositions is evident and shows that photovoltaic performance is highly sensitive to temperature. In the case of Ba₃PF₃, the efficiency decreases to 17.73% at 300 K and 8.95% at 900 K. Likewise, it decreases to 11.97% and 13.39% for Ba₃PCl₃ and Ba₃PBr₃, respectively. The strongest decrease is seen in the case of Ba₃PI₃, with the SLME at 300 K being 39.17%, followed by 14.57% at 900 K. This decrease may be explained by increased carrier recombination and by the thermally induced narrowing of the bandgap at high temperatures.

Ba₃PI₃ is the most efficient according to theory, though it is also relatively more sensitive to temperature changes than the other compounds being studied. It is worth noting that SOC, particularly relevant for Ba₃PI₃, may decrease the band gap and shift the onset of absorption, which could lead to a minor overestimation of the calculated values of the SLME. The total enhancement of SLME of F- to I-based compounds is indicative of the fact that the quality of absorber and optical response is enhanced with the heavier halide substitution. The materials are promising photovoltaic absorbers at the materials-screening level due to their relatively high values of SLME, especially in the case of Ba₃PBr₃ and Ba₃PI₃.

The systematic investigation of the elastic characteristics of Ba₃PX₃ (X = F, Cl, Br, I) compositions is done to realize their mechanical stability, bonding properties, anisotropy in elasticity, and ductile properties. The computed elastic parameters have been listed in Tables 4 and 5. Elastic constants in a single crystal and the mechanical parameters derived were obtained. These parameters define how the material responds to the external pressure and give an idea of the bonding nature of the material, directional anisotropic nature, and ductile nature. A cubic crystal is considered to be stable to elastic deformations when C₁₁ − C₁₂ > 0; C₄₄ > 0; C₁₁ + 2C₁₂ > 0 and C₁₁ > 0 [42]. Here, one defines Cauchy pressure as C^″ = C₁₂ − C₄₄. When the value is positive, this is a ductile behavior and when it is negative, this is a brittle behaviour [43]. Elastic constants, especially, C₁₁, C₁₂, and C₄₄ are important to explain how a crystal reacts to external forces. The elastic constants of the Ba₃PX₃ family show non-monotonic variation based on competing influences from lattice expansion, bond strengths and hybridization of orbitals. As the halides increase in ionic radius, interatomic interaction is weakened; however, the character of P-X bonds and their associated local structural environments cause a non-linear response of these bonds to elastic deformations. C₁₁ is usually the largest of the elastic constants, and is associated with longitudinal strain response in the primary crystallographic directions, whereas C₁₂ and C₄₄ are associated with compressive and shear effects, respectively. Even though C₁₁ reaches its maximum with Ba₃PF₃ and its minimum with Ba₃PI₃, its change across the Ba₃PX₃ series is not monotonic. C₁₁ and C₄₄ vary non-monotonically, whereas C₁₂ varies monotonically within the Ba₃PX₃ series; that is, the resistance increases monotonically with the increasing size of the halogen atom. The obtained values satisfy the Born stability criteria, indicating that the considered compositions are mechanically stable. The convergence of the elastic constants was validated by carefully controlling for computational parameters, while the optimized structures represent the stable energy minima; therefore, the trends are intrinsic and not a result of numerical artifacts.

The elastic parameters of a given material can be used to compute the bulk modulus (B) and shear modulus (G) of that material using the Voigt, Reuss, and Hill (VRH) approximation. In the case of a cubic structure, the bulk modulus computed by the Voigt approximation and the shear modulus computed by the Reuss approximation are the following [44,45]. (4)BV=BR=(C11+2C123) (5)GV=(C11−C12+3C445) (6)GR=5(C11−C12)C444C44+3(C11−C12)

The actual values of the moduli, according to Hill [46], are calculated by the arithmetical mean of two separate moduli computed using the Voigt and Reuss approximations:(7)B=(BV+BR2)

And (8)G=(GV+GR2)

The bulk modulus (B) indicates the resistance of a specimen to uniform compression, while the shear modulus (G) indicates its stiffness against shear deformation. In the Ba₃PX₃ series, the values of these two moduli decreased from Ba₃PF₃ (B = 31.89 GPa, G = 17.34 GPa) to Ba₃PI₃ (B = 23.35 GPa, G = 9.3 GPa). This decrease indicates that the presence of heavy halide ions decreases the internal bond strength of the compounds, making them relatively more flexible. Therefore, Ba₃PF₃ was found to be the most rigid, and Ba₃PI₃ was found to be the most flexible. The brittleness or ductility of a substance can be judged by Pugh’s ratio (B/G). If this ratio is greater than 1.75, the material is considered ductile, otherwise brittle. This ratio was found to be greater than 1.75 for Ba₃PF₃ and Ba₃PI₃, indicating their ductile nature. In contrast, Ba₃PCl3 and Ba₃PBr₃ had values below 1.75, making them moderately brittle, consistent with previous calculations [47].

Shear constant (C′) and Cauchy Pressure (C′′) can be calculated as follows [48]:(9)C′=(C11−C122) (10)C′′=C12−C44

Analysis of Cauchy pressure (C′′) also gives the same conclusion that while its positive value shows ductility, a negative value shows brittleness. Positive values were obtained for Ba₃PF₃ (0.51 GPa) and Ba₃PI₃ (1.86 GPa), which confirm their ductility. On the other hand, negative values for Ba₃PCl₃ and Ba₃PBr₃ were found to be −6.35 and −6.09 GPa, respectively, which makes them brittle in nature.

The Poisson’s ratio (η) also remained within a stable range for all compounds, indicating mechanical stability and a predominantly ionic bonding character. Young’s modulus (Y) is a fundamental indicator of a material’s stiffness, representing its response to longitudinal strain. A higher value of Y corresponds to greater stiffness of the material. It can be calculated using the relation:(11)Y=9BG3B+G

For the Ba₃PX₃ (X = F, Cl, Br, I) compositions, the Young’s modulus varies from 24.62 GPa for Ba₃PI₃ to 49.55 GPa for Ba₃PCl₃, which indicates that Ba₃PCl₃ possesses higher stiffness, while Ba₃PI₃ exhibits greater elasticity. Additionally, values such as the Zener ratio (A_z) and the Kleinman parameter (ζ) were used to understand the isotropy and bonding behaviour of the compounds. A substance is considered isotropic when the Zener ratio is close to 1. As shown in Table 5, all compounds exhibit elastic anisotropy (A_Z ≠ 1), with the degree of anisotropy varying non-monotonically across the halide series. With an anisotropy value nearest to unity (A_Z = 0.66), Ba₃PCl₃ has a relatively more isotropic mechanical behavior, which is advantageous to the homogeneous distribution of strain and minimized formation of defects in thin-film devices. Ba₃PI₃ (A_Z = 0.14) on the other hand exhibits the greatest level of anisotropy, which implies a stronger directional-dependent mechanical behavior that could affect the distribution of stress in the film growth. The rest of the compounds are moderately anisotropic indicating a balanced mechanical response that is appropriate to prepare a device.

The first Lamé constant (λ) is related to the degree of material compressibility, while the second Lamé constant (μ) is indicative of the degree of shear stiffness in a particular material. These coefficients can be defined by the following expressions using the Young’s modulus and the Poisson’s ratio of the considered compositions:(12)λ=Yη(1+η)(1−2η) (13)µ=Y2(1+η)

Table 5 reports the calculated values of λ and μ for each of the investigated compositions. Only positive values of λ and μ exist for all compounds, as presented in Table 5. The positive value of λ and μ for each composition indicates that they possess mechanical stability, while variations in μ reflect differences in shear resistance across the Ba₃PX₃ series.

The Debye temperature (θ_D) is a key thermal property that is correlated to many different physical properties of a material, including the melting point and specific heat of the materials. The θ_D can be expressed as [49]:(14)θD=hk[(3n4π)(NAρM)]1/3vmwhere, vm is the average sound velocity, h is Planck’s constant, k is Boltzmann’s constant, n is the number of atoms in unit cell of a compound, N_A is Avogadro’s number, ρ is compound density, and M is molecular weight of a compound. In general, the higher the value of vm or θ_D, the greater the strength of the bond. In polycrystalline materials, the average sound velocity, vm, is defined by [50]:(15)vm=[13(2vt3+1vl3)]−1/3where vt and vl are transverse and longitudinal sound velocities, which can be defined by means of elastic constants (B and G) using Navier’s equation, as below:(16)vt=Gρ (17)vl=3B+4G3ρ

The Debye temperature (θ_D), derived from average sound velocities, decreases systematically across the halide series_, from 253.06 K (Ba₃PF₃) to 155.26 K (Ba₃PI₃). Such a decrease is an indication of weaker interatomic forces and higher atomic masses in the iodide member, which results in lower-lattice vibration frequencies. Altogether, the mechanical study proves that the Ba₃PX₃ compounds are elastically stable, with tuneable stiffness and ductility based on the halide component. The non-linear change in elastic parameters is observed to emphasize how halide substitution alters bond lengths, elastic anisotropy, and lattice dynamics in a complex manner.

The computed elastic properties indicate that Ba₃PX₃ have mechanical properties that are compatible with thin-film photovoltaic fabrication. The medium bulk and shear moduli indicate that it is mechanically sound enough to resist thermal cycling and residual stress in the deposits related to sputtering or evaporation operations or solution-based film growth. Specifically, the comparatively reduced stiffness and increased ductility of Ba₃PI₃ suggest improved strain resistance, which is desirable for reducing crack formation and delamination in flexible or large area devices. On the other hand, the increased rigidity of Ba₃PF₃ and Ba₃PCl₃ can be advantageous when structural stability and integrity are required. The flexibility-rigidity trade-off created by substituting halides offers a convenient design parameter to customize Ba₃PX₃ absorbers for desired photovoltaic and optoelectronic device architectures.

The determined elastic constants and mechanical parameters are reasonably consistent with previously determined values [22,34,35]. Minor discrepancies come as a result of the difference in the computational methods such as exchange-correlation functionals, k-point density, and structural optimization procedures. Notably, the uniform patterns that were identified throughout the Ba₃PX₃ series confirms the strength and stability of the current set of computations.

Using the Gibbs2 program [29], the thermodynamic parameters of the investigated compositions have been calculated. The thermodynamic properties were calculated for each of these compounds from 0 to 900 K, and the pressure effect calculations were done for the range 0–10 GPa based on the Quasiharmonic model. This approach accounts for volume-dependent phonon contributions on thermodynamic behavior and is widely used for studying temperature- and pressure-dependent properties of crystalline materials. Therefore, the obtained results provide reasonable representations of the thermal response and stability trends of the Ba₃PX₃ compounds over a broad temperature range. At higher temperatures, particularly for softer compounds such as Ba₃PI₃, minor deviations may arise due to increased anharmonic phonon interactions; however, the overall trends remain well described within the quasi-harmonic framework. The thermodynamic behaviour of Ba₃PX₃ (X = F, Cl, Br, I) at varying temperatures and pressures is illustrated in Figs. 9 and 10, which include the bulk modulus (B), the acoustic Debye temperature Θ_D, the coefficient of thermal expansion (α), the entropy (S), the specific heat at constant pressure (C_p), and the specific heat at constant volume (C_v). The bulk modulus (Fig. 9a–d) shows a monotonic change downward with temperature of all the compounds, and this implies that the lattice is softening as a result of progressive anharmonic vibrations of the atoms. Conversely, the applied pressure has a substantial positive effect on B, which indicates greater resistance to volume compression due to shorter interatomic distances. It is evident that there is a halogen-dependent trend with the stiffest being Ba₃PF₃ and the softest being Ba₃PI₃, as the ionic radius increases and the strength of PX bonds between F and I weaken. The same applies to the Θ_D (Fig. 9e–h), which declines with temperature since the phonons become soft, and also rises with pressure since phonons become hard. The progressively reduced Θ_D values of heavier halogens are another indication that the predominant factors controlling lattice vibrations are atomic mass and bond strength. The thermal expansion coefficient (Fig. 9i–l) shows a rapid increase at low temperatures and a slow increase at high temperatures, which is a characteristic feature of anharmonic lattice dynamics. The pressure is inhibitory to α in any system, with lattice compression limiting the anharmonic atomic mobility. The larger alpha values of Ba₃PBr₃ and Ba₃PI₃ are explained by their lower bulk moduli and acoustic Debye temperatures, indicating greater lattice flexibility.

All compositions and pressures show a monotonically increasing entropy (Fig. 10a–d) as the temperature increases, which is evidence of the gradual filling up of phonon modes. Entropy decreases with pressure at a specified temperature; compression decreases the available vibrational phase space. The more massive systems have greater entropy, due to both lower characteristic phonon frequencies and a stronger vibrational disorder. Both C_v (Fig. 10e–h) and C_p (Fig. 10i–l) increase steeply at low temperatures and tend towards saturation at high temperatures, as observed in Debye theory. At higher temperatures, C_v approaches the Dulong-Petit limit, verifying the classical vibrational behaviour of the lattice. That minor surplus of C_p over C_v rises with temperature and falls with pressure, and, in harmony with thermodynamic relations between heat capacity and thermal expansion, is negative. In this way, the calculated thermodynamic parameters reveal a strong interdependence among mechanical stiffness, lattice vibrations, and thermal response in Ba₃PX₃ compounds. The lattice is stabilized by pressure, which improves stiffness, raises Θ_D, and inhibits thermal expansion and entropy. The substitution of a halogen by a suitable element is found to be a viable path to achieving thermodynamic softness and tuning vibrational behaviour. This tunability is highly valuable when thermal and mechanical stability are required across different operating conditions.

The trend of the Grüneisen parameter (γ) that is used to gauge anharmonicity and phonon-volume coupling is Ba₃PF₃ < Ba₃PI₃ < Ba₃PBr₃ < Ba₃PCl₃. This observation (non-mass-monotonic) suggests that the atomic mass of halides, bonding properties, and lattice rigidity are reflected in the values of γ. The relatively higher γ values of Br-based and Cl-based compounds indicate more phonon volume coupling due to intermediate electronegativity and more flexible P-X bonds prevailing over mass-driven phonon softening. In terms of devices, the moderate values of γ (1.49–2.17) suggest that there is enough anharmonicity to allow thermal expansion and strain throughout the manner of operation, and the lattice is stable. Such a balance helps maintain the thermal stability of the Ba₃PX₃ material in photovoltaic applications under operating conditions. At 300 K, the computed thermodynamic parameters are represented in Table 6. The moderate values of γ also indicate that anharmonic effects remain within a manageable range, supporting the validity of the quasi-harmonic approximation over the studied temperature range, although minor deviations may arise at higher temperatures, particularly for softer compounds.

The computed thermodynamic parameters also contain significant information about the functional behaviour of Ba₃PX₃ materials with respect to temperature. The reduction in B and θ_D with increasing temperature indicates lattice softening that can lead to bandgap renormalization and increased electron-phonon interactions. The increase in carrier recombination and thermally induced broadening of the bandgap is consistent with the observed reduction in SLME at elevated temperatures. An increase in entropy and heat capacity indicates an increase in phonon population; increased phonons can affect carrier scattering and optical absorption close to the band edge. Heavier halide materials were observed to have lower θ_D, which indicates that stronger vibrations within the lattice structure which can lead to greater temperature sensitivity in their electronic/optical responses.