Introduction

Alkalines transition metal oxide has specific physical properties, particularly in the transport and magnetic fields. Alkaline earth iron oxide (Ba, Ca)FeO3 perovskites are interesting materials because theircontrollable ferromagnetic domains. These materials exhibit multiple oxidation states, likely to induce mixed electronic and ionic conductivity. The chemical composition, the nature of cations, as well as the preparation methods influence their physical properties such as electrical conductivity, crystallographic structure, inducing significant modifications of their electrochemical behavior. Calcium-iron oxide is promising for energy storage in so called rechargeable oxide batteries. It possesses favorable properties like good storage capacity for oxygen ions and slow microstructural degradation during repeated reduction and oxidation. Barium iron oxide is highly insoluble, thermally stable, and suitable for glass, optical, and ceramic applications. Inorganic calcium ferrite (CaFeO3) oxide perovskite solar cells material used as hole transport layer provides good stability and high-power conversion efficiency1. Thesuper-exchange interactionFe+ 4–O2–Fe+ 4is responsible for ferromagnetism in CaFeO3 is due to the Fe+ 4–O2–Fe+ 4, and its compression to a transition lattice constant causes the loss of its entire magnetic moment and become metallic2. The paramagnetic Curie temperature of CaFeO3 is evaluated to be78°K3. The bulk modulus of a material can be expressed in terms of the product of ionic charge of their constituents with its lattice parameter4. First-principles calculations of the cubic perovskite oxide BaFeO3, performed using density functional theory (DFT) combined with the Hubbard correction (DFT + U) within the generalized gradient approximation, reveal that the material exhibits a half-metallic and ferromagnetic state5. The physical properties of both BaFeO3 and CaFeO3 are influenced by their oxygen content, with charge disproportionation occurring through the splitting of Fe4+ ions into Fe5+ and Fe3+ oxidation states6,7. A metastable cubic perovskite structure BaFeO3 is considered as a network of [FeO6] octahedra, with linear Fe–O–Fe arrangements8. The structure of BaFeO3 determined using the Jahn-Teller effect illustrates a transition from cubic to tetragonal symmetry9,10. The cubic and hexagonal phases of BaFeO3 studied by means of first-principle methods show ferromagnetic order11. Inorganic CaFeO3 is used as a hole transport layer, offers superior stability and theoretical analysis validates the experimental results with a power conversion efficiency of ∼16%1. Recent advancements have focused on enhancing the electrochemical and structural performance of BaFeO₃-based materials for energy applications. For instance, Co-doped Ba₀.₉₅La₀.₀₅FeO₃−δ (BLF) has recently been investigated as a promising cathode for protonic ceramic fuel cells (PCFCs) using a combined machine learning, Bayesian optimization, experimental, and DFT-based framework12. The study revealed that controlled doping can tune the oxygen vacancy formation energy and improve the material’s electrochemical behavior and structural stability highlighting the multifunctional potential of BaFeO₃ derivatives. Similarly, CaFeO₃-based materials have shown excellent performance in energy storage applications. A recent study demonstrated that Mo-doped CaFeO₃ synthesized via a hydrothermal route exhibited a specific capacitance of 1722.5 F/g and strong cyclic retention over 5000 cycles when used in supercapacitor electrodes13. These findings emphasize the versatility of CaFeO₃ and motivate further theoretical investigation into its intrinsic electronic, magnetic, and structural characteristics an effort undertaken in the present study using first-principles methods. The selection of BaFeO₃ and CaFeO₃ arises from their rich physical phenomena involving Fe⁴⁺/Fe³⁺ charge ordering, magnetic transitions, and perovskite distortions. These materials are known for their spin polarization, optical activity, and mechanical tunability. The motivation behind revisiting BaFeO₃ and CaFeO₃ lies in the lack of a comprehensive and comparative study employing multiple exchange-correlation functionals (GGA, mBJ-GGA, and YS-PBE0) to simultaneously evaluate structural, mechanical, magnetic, optical, and electronic properties. While these materials have been studied individually in the literature, there is a scarcity of integrated analysis correlating their half-metallicity, optical response, and elastic behavior, especially under hybrid functionals like YS-PBE0. Furthermore, this study explores their potential in spintronic and optoelectronic applications by examining their behavior up to high photon energies, which has not been previously reported in detail. This approach provides a broader understanding of their suitability for multifunctional device applications. We report in this investigation the ground state structural parameters, electronic nature, magnetic phase, optical and thermoelectric characteristics of BaFeO3 and CaFeO3 for advanced technological applications.

Computational details

Calculations have been performed using the density functional theory as implemented in the WIEN2K code14 were explored for obtaining all results. The electronic exchange-correlation energy in the optimization of structural, electronic, optical, thermoelectric and vibrational properties of (Ba, Ca)FeO3 perovskites is treated using the generalized gradient approximation of Perdew, Burke, and Ernzerhof (PBE-GGA)15 and the modified Becke-Johnson exchange potential (mBJ-GGA)16. The computational parameters reported in Table 1 include the plane wave cutoff parameter in the interstitial region (RMT×Kmax), the muffin-tin radius (RMT), the maximum plane wave vector (Kmax), and the k-point mesh used for integrations within the irreducible Brillouin zone. Self-consistency is achieved with an energy convergence criterion of 10− 5 Ry. The use of multiple exchange-correlation functionals was essential to ensure accurate and reliable predictions. GGA was employed as a baseline functional for structural and mechanical properties. The mBJ-GGA functional is known for improving the estimation of electronic band gaps in semiconductors, while the YS-PBE0 (a hybrid functional similar to HSE06) incorporates non-local exchange, offering more accurate magnetic and electronic structure predictions. This comparative approach allows us to assess the sensitivity of results to the functional and strengthens the reliability of the conclusions. the charge convergence was set to 10⁻⁴ e. The muffin-tin radii (RMT) were carefully selected to minimize sphere overlap and were as follows: Ba = 2.50 Å, Ca = 2.30 Å, Fe = 2.00 Å for BaFeO₃ and 1.82 Å for CaFeO₃, and O = 1.72 Å and 1.57 Å respectively. The plane wave cutoff parameter RMT×Kmax was chosen as 8.5 for BaFeO₃ and 9.0 for CaFeO₃. Wavefunctions inside the muffin-tin spheres were expanded in spherical harmonics up to lmax = 10. Valence electrons were treated semi-relativistically, and core electrons were treated fully relativistically. All calculations were carried out until total energy differences were less than 10⁻⁵ Ry between successive iterations. Inside the muffin-tin spheres, wave functions, electron charge densities, and potentials are expanded in spherical harmonics up to angular momentum lmax = 10, whereas plane waves are used in the interstitial regions.

Table 1 RMTxKmax, RMT for each component, and k-point using GGA for BaFeO3 and CaFeO3.
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Results and discussions

Structural analysis

(Ca, Ba)FeO3 perovskites crystallize in the cubic \(Pm\;\overline{3} m\) space group. (Ca²⁺, Ba²⁺) are bonded to twelve equivalent O²⁻ atoms to form (Ca, Ba)O₁₂ cuboctahedra that share corners with twelve equivalent (Ca, Ba)O₁₂ cuboctahedra, faces with six equivalent (Ca, Ba)O₁₂ cuboctahedra, and faces with eight equivalent FeO₆ octahedra. (Ca, Ba) atoms occupy 1b (1/2, 1/2, 1/2), Fe atom are 1a (0, 0, 0) and O are located at 3d (1/2, 0, 0). The schematic crystal structure of (Ba, Ca)FeO3 is visualized in Fig. 1.

Fig. 1
figure 1

Schematic crystal structure of BaFeO3 perovskite.

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The formation energy as a function of volume in ferromagnetic state for (Ca, Ba)FeO3 (Fig. 2), present in the negative formation energy indicates that these perovskites are energetically stable and could be synthesized experimentally under ambient conditions. The lattice constant, bulk modulus and its pressure derivative is estimated from the optimum volume of the unit cell as shown in Table 2. The pressure derivative of the bulk modulus (B′) provides insight into how the material’s resistance to compression evolves under external pressure. A higher value of B′ indicates that the material becomes significantly stiffer when compressed, reflecting a strong interatomic interaction response to pressure. In contrast, a lower B′ implies a more gradual change in stiffness with pressure. This parameter is essential in understanding the mechanical resilience of perovskite structures under high-pressure conditions. The optimized lattice constants of the cubic unit cell CaFeO3 and BaFeO3 are in good agreement with the experimental values of 3.77 Å and 3.971 Å3,8,17,18. Our lattice constants of CaFeO3 and BaFeO3 agree well with theoretical values reported in the literature2,19. Bond lengths in molecular crystals depend primarily on the nature of the elements involved and the number of bonding electron pairs minus the number of antibonding electron pairs. It should be noted that the present study assumes ideal stoichiometric structures for BaFeO₃ and CaFeO₃. The potential influence of oxygen vacancies and Fe⁴⁺/Fe³⁺ charge disproportionation, which are known to affect structural and electronic behavior in perovskites, was not considered in the current calculations. The calculated bond lengths between different atoms for CaFeO3 and BaFeO3 are listed in Table 2. The Goldschmidt tolerance factor t was calculated using the conventional expression20:

$$t = \frac{{r_{A} + r_{o} }}{{(r_{B} + r_{o} ) \times \sqrt 2 }}$$

where rA_​, rB​, and rOare the ionic radii of the A-site cation (Ba²⁺ or Ca²⁺), B-site cation (Fe⁴⁺), and the oxygen anion (O²⁻), respectively. This factor provides a qualitative assessment of the geometric stability of the perovskite structure. The calculated tolerance and octahedral factors for CaFeO3 and BaFeO3 are (1.0002 and 0.349) and (1.0004 and 0.416). The tolerance and octahedral factors are lyingin the range (1.0004–1.04) and 0.47, which confirm their structural stability21,22. The chemical reaction of formation of (Ba, Ca)FeO3 from their elementary components can be defined as:

$$(Ba,Ca)_{{(s)}} + Fe_{{(s)}} + \frac{3}{2}O_{{2_{{(g)}} }} \to (Ba,Ca)FeO_{{3_{{(s)}} }}$$
(1)
Fig. 2
figure 2

Calculated energy vs. volume curves for optimizing BaFeO3 and CaFeO3 using GGA.

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Table 2 Lattice constant, bulk modulus and its pressure derivative, the minimum energy E0 and bond lengths for BaFeO3 and CaFeO3.
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The Gibbs free energy of reaction of equation:

$$\Delta G_{f}^{0} = G_{{(BaFeO_{3} )}}^{0} - G_{{(Ba)}}^{0} - G_{{Fe}}^{0} - \frac{3}{2}G_{{O_{2} }}^{0}$$
(2)
$$\Delta G_{f}^{0} = G_{{(CaFeO_{3} )}}^{0} - G_{{(Ca)}}^{0} - G_{{Fe}}^{0} - \frac{3}{2}G_{{O_{2} }}^{0}$$
(3)

where G0 is the standard Gibbs free energy of the substances involved in the reaction and is expressed as:

$$G^{0} = E^{{DFT}} + ZPE - TS^{0}$$
(4)

where EDFT is the total energy obtained from the DFT calculations, ZPE is the zero-point energy estimated from vibration analyses, and TS0 is the product of temperature and entropy at T = 298.15 K. The formation energy of BaFeO3 and CaFeO3 per atom is -5.56 eV and − 5.71 eV respectively. Although phonon dispersion calculations were not performed in this study, the significantly negative formation energies obtained for BaFeO₃ (− 5.56 eV/atom) and CaFeO₃ (− 5.71 eV/atom) provide strong evidence of their thermodynamic stability under equilibrium conditions. These values suggest that both materials are energetically favorable and could be synthesized experimentally. Similar computational approaches involving formation energy and phonon dispersion to assess the structural and dynamic stability in oxide compounds have been successfully demonstrated in recent studies, such as that by Amira Nour Asfora et al.23. The relevance of phonon dispersion and cohesive energy analysis in determining structural stability has also been highlighted in similar oxide systems such as ZrSi₃O₈ and Zr₃SiO₈, as reported by M. Ould Moussa et al.24. In the context of rare-earth and transition-metal perovskites, first-principles studies like that of Slimane Haid et al.25 have shown how combining formation energy with GGA + U corrections can reveal both structural and magnetic stability. Furthermore, the integration of phonon analysis, tolerance factor, and structural parameters to assess the thermodynamic stability of double perovskites has also been effectively demonstrated in the work on Ba₂DyTaO₆ by Slimane Haid et al.26.

Elastic constants and mechanical characteristics

The three elastic constants C11, C12 and C44and bulk modulus for BaFeO3 and CaFeO3 with cubic symmetryare calculated using the nonlinear Charpin method27. We note that all elastic moduli verify the stability criteria, hence their mechanical stability20.

$$0 \prec C_{{11}} ,0 \prec C_{{44}} ,0 \prec C_{{11}} - C_{{12}} ,0 \prec C_{{11}} + 2C_{{12}} ,C_{{12}} \prec B \prec C_{{11}}$$

The mechanical properties of BaFeO3 and CaFeO3, including shear modulus, bulk modulus, anisotropy factor, Young’s modulus, Poisson’s ratio, and the BH/GH ratio calculated using the Voigt, Reuss, and Hill approximations, are presented in Table 3. The shear and bulk moduli of BaFeO3 are higher than those of CaFeO3, indicating that BaFeO3 is both harder and stiffer. Both perovskites exhibit significant anisotropy. Additionally, Young’s modulus confirms that BaFeO3 has a greater resistance to uniaxial deformation compared to CaFeO3. Poisson’s ratio, which measures the ratio of transverse to longitudinal strain, has values greater than 0.25 for both materials, suggesting that the dominant interatomic forces are of a central nature. The B/G ratio serves as an indicator of ductility, where values above 1.75 correspond to ductile behavior and values below indicate brittleness. Based on this criterion, BaFeO3 is ductile, while CaFeO3 is brittle. It is also noteworthy that the bulk modulus lies between the shear modulus and Young’s modulus for both compounds, implying that these materials accommodate uniaxial deformation more readily than volumetric compression. Figures 3 and 4 display the 3D and 2D distributions of Young’s modulus (E) for BaFeO3 and CaFeO3. In both materials, Young’s modulus reaches its maximum along the z-axis (Emax = 125 GPa for BaFeO3 and 100 GPa for CaFeO3) and its minimum along the x-axis (Emin = 80 GPafor BaFeO3 and 50 GPa for CaFeO3). This results in elastic anisotropy factors (AE = Emax/Emin) of 1.56 and 2.

Table 3 Elastic constants, shear moduli, bulk modulus, anisotropy factor, young’s moduli, poisson’s ratios and BH/GH ratio for BaFeO3 and CaFeO3.
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Fig. 3
figure 3

3D surface of Young’s modulus for BaFeO3 and CaFeO3.

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Fig. 4
figure 4

2D surface of Young’s modulus for BaFeO3 and CaFeO3.

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Electronic properties

Spin gapless semiconductors are a class of materials, where band structure presents no band gap for one spin channel while there is a finite band gap in another spin channel. Although the concept of spin gapless semiconductors (SGS) refers to materials that exhibit a true zero band gap in one spin channel and a finite gap in the other, the present results do not strictly fulfill this criterion. In our calculations using mBJ-GGA and YS-PBE0, BaFeO₃ and CaFeO₃ show a metallic character in the spin-up channel and a narrow indirect band gap (~ 1.0–1.7 eV) in the spin-down channel. Therefore, these compounds can be more appropriately described as exhibiting spin-polarized semiconducting behavior, rather than being classified as ideal SGS materials. Figures 5 and 6 show spin-up (left panel) and spin-down (right panel) band structure of BaFeO3and CaFeO3 using YS-PBE0 and mBJ-GGA approximations. For the spin-up and using YS-PBE0 and mBJ-GGA, both perovskites under study have a metallic character, while, they show an indirect M-Γ band gap for the spin-dn. The electronic and magnetic characteristics of a ferromagnetic material depend on the spin states, and the energy difference is adopted in the electronic transitions between orbits. The different band gaps of BaFeO3 and CaFeO3 perovskites are reported in Table 4. In the case of the spin-up state, the valence band crosses the Fermi level and enters into the conduction band, so BaFeO3and CaFeO3 are metallic. For the spin-down, a narrow indirect M-Γ band gap of 1.03 eV and 1.01 eV (1.74 eV and 1.55 eV) is found for mBJ-GGA (YS-PBE0(HSE06)). In the GGA case, both compounds are metallic in both spins. The electronic contribution in the band structure is explained by the total and partial spin-polarized densities of states (DOS and PDOS), as illustrated in Fig. 7. The electronic contribution in the valence band located between (-0.5 eV and − 2.5 eV) and (-4.5 eV and − 6 eV) is mainly due to Fe-3d and O-2p electrons. Hybridization of the O-2p and Fe-3d states is noted in the upper of the valence band in the spin-up states, as well as in the spin-down state, this makes BaFeO3 and CaFeO3 metallic. The conduction band is empty for the spin-up. In the case of spin-down and mBJ-GGA functional, polarization occurs at the Fermi level and Fe-3d jumps into conduction band, accordingly a narrow band gap of 1.03 eV and 1.01 eV exists between the valence and conduction band for BaFeO3and CaFeO3. For the spin-down channel of CaFeO₃, a narrow indirect band gap (~ 1.01–1.55 eV) appears between the M and Γ points under the mBJ-GGA and YS-PBE0 functionals, while the spin-up channel remains metallic. This indicates spin-polarized semiconducting behavior rather than a fully metallic state. Figures 8 and 9 show the electronic densities of states of Ba (s, p), Ca (p, d, d-eg, d-t2g), Fe (s, p, d, d-eg, d-t2g) and O (s, p) for BaFeO3and CaFeO3with both spin and using mBJ-GGA approach. Local octahedral coordination of Fe (3d) and Ca (3d) for both spin give two degenerate orbitalsd-eg and d-t2g. The tetrahedral and octahedral symmetries play a crucial role in stabilizing the ordering of the transition metal (Fe) in BaFeO3 and CaFeO3. The resulting crystal field causes the Fe 3d orbitals to split into two distinct sets of non-degenerate states, t2g and e.g., as depicted in Figs. 8 and 9. In this splitting, the e.g. doublet lies at a lower energy level than the t2g triplet. The six electrons are arranged such that the wave functions of the 3d e.g. doublet exhibit weak hybridization with the oxygen 2p orbitals. In contrast, the 3d t2g states strongly hybridize with the oxygen 2p orbitals, forming bonding and antibonding states. The Fe 3d states are occupied so that the t2g orbitals are filled for both spin-up and spin-down electrons, while the e.g. and t2g orbitals remain unoccupied for both spin channels. The bonding nature in both BaFeO₃ and CaFeO₃ exhibits a mixed ionic–covalent character. The Fe–O bonds show significant covalent behavior due to the overlap between Fe 3d and O 2p orbitals, as reflected in the electronic density of states (DOS) and partial charge density maps. On the other hand, Ba–O and Ca–O bonds possess more ionic features, attributed to the large electronegativity difference between Ba/Ca and O. This hybrid bonding nature contributes to the structural flexibility and multifunctional electronic behavior of these perovskite oxides. It is important to emphasize that under GGA functional, both spin channels demonstrate metallic behavior, which further supports the interpretation that these materials are not true spin gapless semiconductors. Instead, their spin-resolved electronic features suggest potential use in spintronic devices due to their asymmetrical spin band structures under advanced functionals. Recent studies have demonstrated that including spin–orbit coupling (SOC) and modified Becke–Johnson (TB-mBJ) functionals significantly alters the electronic structure and band gaps of perovskite materials. A comprehensive investigation on 40 novel AMX₃ compounds revealed how the combined influence of SOC and TB-mBJ can tailor the band structure, which is essential in tuning perovskite-based materials for electronic and optoelectronic applications. These insights support our use of hybrid and mBJ functionals in exploring the electronic behavior of (Ba, Ca)FeO₃28.

Fig. 5
figure 5

Band structure of BaFeO3 using spin-up and spin-dn using mBJ-GGA and YS-PBE0.

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Table 4 Band gap of BaFeO3 and CaFeO3 using GGA, Mbj-GGA and YS-PBE0(HSE06).
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Fig. 6
figure 6

Band structure of CaFeO3using spin-up and spin-dn using mBJ-GGAandYS-PBE0.

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Fig. 7
figure 7

The density of states for BaFeO3 and CaFeO3 using mBJ-GGA.

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Fig. 8
figure 8

Partial density of states (PDOS) for BaFeO3using mBJ-GGA.

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Fig. 9
figure 9

Partial density of states (PDOS) for CaFeO3 using mBJ-GGA.

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Magnetic properties

Density functional theory (DFT) calculations were conducted using the GGA, mBJ-GGA, and YS-PBE0 approximations to investigate the magnetic properties of BaFeO3 and CaFeO3 perovskites. The magnetic moment originates from both the constituent atoms and the interstitial regions. Both BaFeO3 and CaFeO3 display ferromagnetic behavior, with the primary contribution to magnetization arising from the partially filled Fe 3d orbitals. Table 5 presents the interstitial, atomic, and total magnetic moments for these compounds calculated with the three functionals. Notably, the mBJ-GGA method yields higher magnetization values compared to YS-PBE0 and GGA. The total magnetic moments obtained with mBJ-GGA and YS-PBE0 are integer values (4 µB), consistent with the half-metallic nature of these materials. Half-metallic ferromagnetism and spin polarization effects, similar to those observed in Fe(Mn)-doped nitrides like CdN and ZnN, have also been reported using FP-LAPW methods29. These findings support our interpretation of strong p–d hybridization in (Ba, Ca)FeO₃ systems as a key factor driving their spin-polarized magnetic behavior. A similar trend was reported for PdO, where GGA predicted metallic behavior in both spin channels, but upon inclusion of a Hubbard U correction (GGA + U), half-metallic ferromagnetism emerged30. This highlights how strong electron correlation and exchange interactions, as modeled by GGA + U or hybrid functionals, can critically influence magnetic behavior and spin polarization consistent with our observations in (Ba, Ca)FeO₃. The O 2p bands at the Fermi level are partially unoccupied in the spin-up channel and fully unoccupied in the spin-down channel. It is observed that stronger hybridization between O 2p and Fe 3d orbitals tends to reduce the magnetic moment. Experimentally, the magnetic moment of Fe in CaFeO3 is reported as 3.5 µB at 15 K6. The direction of the interstitial magnetization aligns with that of Fe 3d and O 2p magnetizations for all functionals and both compounds, while for the mBJ-GGA functional, the magnetic moments of Ba and Ca are antiparallel to that of Fe. The calculated magnetic moments for Fe atoms (3.5–4.0 µB) are in good agreement with experimental reports, such as the 3.5 µB measured in CaFeO₃ at 15 K6. Additionally, the use of hybrid functionals improved the accuracy of band gap predictions, although direct experimental band gap values remain limited in the literature for a reliable one-to-one comparison. It is important to note that in half-metallic ferromagnets, the total magnetic moment per formula unit is expected to be an integer value, which is a signature of the complete spin polarization at the Fermi level. This behavior follows the Slater–Pauling rule, which relates the total magnetic moment (in µB) to the number of valence electrons (Zₜ) through the relation: Mₜ = Zₜ − 24, for transition-metal-based half-metallic systems. The integer nature of the magnetic moment arises because one spin channel (usually spin-down) has a semiconducting gap, while the other (spin-up) remains metallic, leading to complete spin asymmetry at the Fermi level. In our study, the calculated total magnetic moments of BaFeO₃ and CaFeO₃ (4.0 µB and 3.0 µB, respectively) conform to this expectation under the mBJ-GGA and YS-PBE0 functionals, reinforcing their half-metallic nature, For BaFeO₃, Zₜ is calculated as follows: Ba contributes 2 valence electrons (6s²), Fe contributes 8 (3d⁶ 4s²), and each oxygen atom contributes 6 electrons; hence, Zₜ = 2 (Ba) + 8 (Fe) + 3 × 6 (O) = 28. Applying the rule gives Mₜ = 28 − 24 = 4 µB, which is in excellent agreement with our calculated total magnetic moment. This confirms the half-metallic nature of BaFeO₃ and supports its potential use in spintronic applications. A similar analysis for CaFeO₃ yields Zₜ = 2 (Ca) + 8 (Fe) + 18 (O) = 28, and hence Mₜ = 4 µB as well, matching our computational results.

Table 5 Interstitial, atomic and total magnetic moment forBaFeO3 and CaFeO3using GGA, mBJ-GGA and YS-PBE0 functionals.
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Optical characteristics

The optical study of BaFeO3 and CaFeO3 perovskites through their characteristics such as absorption, energy loss, real and imaginary dielectric constants, refractive index, extinction coefficient, real optical conductivity and optical reflectivity are visualized in Fig. 10. Figure 10 consists of eight subplots labeled (a) to (h), showing: (a) absorption coefficient, (b) energy loss function, (c) real part of dielectric function, (d) imaginary part, (e) refractive index, (f) extinction coefficient, (g) real optical conductivity, and (h) reflectivity. All plots have their axes clearly labeled, and the photon energy is shown on the x-axis in units of eV. Parallel polarized absorption peaks are observed in these perovskites. For photon energy between 15 eV and 30 eV, the absorption is more pronounced, this results in a decrease in light transmission. BaFeO3 and CaFeO3 exhibit large absorption coefficients 350 × 104 cm− 1 and 500 × 104 cm− 1 in the ultraviolet range attributed to the infrared phonon mode.

A higher absorption coefficient (α) implies that incident light is strongly attenuated within a short distance inside the material. This means that a significant portion of the incident optical power is absorbed per unit thickness, making materials with large α values particularly suitable for energy harvesting and optoelectronic devices. The absorption coefficient is directly related to the imaginary part of the dielectric function and governs how quickly the intensity of light decays as it propagates through the medium, according to Beer–Lambert’s law\(\:I\left(x\right)={I}_{0}.{e}^{-\propto\:.x}\)where I(x) is the intensity at depth x, and I0​ is the initial intensity. The loss energy is not important at visible and ultraviolet light for perovskites under study. The high absorption coefficient at ultraviolet region, low reflectivity near infrared region, the narrow band gap semiconducting nature for the spin-dn make these materials suitable for optoelectronic applications. The real part specifies the degree of polarization, it is more important in the visible region. The static dielectric constant traduces its ability to store electrical energy in low frequency. The static real dielectric constant for BaFeO3 and CaFeO3 is 9 and 7.7. The real part of the dielectric function becomes negative between 3 and 3.5 eV photon energy. This explains that the displacement vector and the electric field vector are opposite. The large static dielectric constant translates the high degree of polarization and points to the ferroelectric behavior of these materials. The imaginary part is associated with dielectric losses. The first peak of the imaginary dielectric function located in the energy about 2.5 eV corresponds to the drop of the real part of the dielectric function. The refractive index of a material determines the extent to which light bends as it passes through the material. The high static refractive index for BaFeO3 and CaFeO3 (3 and 2.75) limits the efficiency of solar cells. The observed refractive index values below unity at high photon energies are attributed to artifacts of DFT-based optical calculations and should not be interpreted as evidence of superluminal behavior. The extinction coefficient determines the degree to which light is absorbed or reflected at a particular wavelength. The maximum extinction coefficient for these materials is about 2. The real part of the optical conductivity contains the same information as the imaginary part of the dielectric function. The maximum real optical conductivity for BaFeO3 and CaFeO3 is 12,500 (Ω. cm)−1 and 17,500 (Ω. cm)−1 obtained at ultraviolet region.The real optical conductivity exhibits a prominent sharp peak at around 2.5 eV, attributed to the transition from O-2p orbitals from the valence band to the conduction band. Optical reflectivity refers to the phenomenon where light waves bounce back and forth as they pass from one material to another with different properties. The static optical reflectivity for BaFeO3 and CaFeO3 is around 0.25 and 0.22. The optical reflectivity of a solar device is a key parameter of its efficiency. The optical response of BaFeO₃ and CaFeO₃ is closely related to their electronic structure. In particular, the strong absorption in the UV region and the high values of the real and imaginary parts of the dielectric function originate from interband transitions. These transitions primarily occur between the O 2p states in the valence band and Fe 3d states in the conduction band. The onset of optical absorption corresponds to the energy difference between these states, which explains the position of absorption edges. Moreover, the sharp peaks in the optical conductivity and refractive index spectra can be attributed to direct electronic transitions near high-symmetry points in the Brillouin zone. Such transitions are influenced by the density of states and the spin-polarized band structures, which reflect the strong hybridization between O and Fe orbitals. This light–matter interaction is essential for understanding the potential use of these materials in UV and spin-dependent optoelectronic applications. While experimental optical spectra for BaFeO₃ and CaFeO₃ are scarce, the calculated static dielectric constants (~ 9 and ~ 7.7) and high absorption coefficients (~ 10⁴–10⁵ cm⁻¹) are comparable in magnitude to values reported for similar perovskite oxides31. Further experimental studies would help to validate and refine these predictions. Similar DFT-based optical investigations on related telluride systems have shown that spin–orbit coupling significantly influences optical transitions near the Fermi level, as reported for GeTe and SnTe32. This further emphasizes the sensitivity of optical properties to band structure modifications, particularly in narrow-gap or spin-polarized materials. Similar optical spectral profiles have been observed in related oxide systems such as BiGaO₃, where both mBJ-GGA and EV-GGA functionals yielded comparable absorption features and enhanced band gap predictions33. These results reinforce the reliability of using mBJ-GGA for studying the optical transitions and electronic excitations in perovskite-like oxides.

Fig. 10
figure 10

Absorption coefficient (a), energy loss (b), real (c) and imaginary (d) components of the dielectric function, refractive index (e), extinction coefficient (f), the real optical conductivity (g) and reflectivity (h) as functions of photon energy for BaFeO3 and CaFeO3perovskites using mBJ-GGA.

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Phonon properties

To further confirm the dynamical stability of BaFeO₃ and CaFeO₃, we have calculated the phonon dispersion relations along the high-symmetry directions of the Brillouin zone together with the phonon density of states (PDOS), as shown in Fig. 11. For both compounds, the absence of imaginary frequencies throughout the Brillouin zone clearly indicates their dynamical stability. In BaFeO₃, the phonon spectrum extends up to ~ 17 THz, with the low-frequency region dominated by vibrations of the heavier Ba atoms, while the intermediate range is mainly contributed by Fe atoms. The high-frequency optical modes are primarily associated with the O vibrations, as evidenced by the PDOS. Similarly, for CaFeO₃, the phonon frequencies reach ~ 18 THz, with Ca contributing mainly in the low-frequency acoustic modes, while Fe and O dominate the mid- and high-frequency regions, respectively.

The separation between acoustic and optical branches is more pronounced in CaFeO₃ due to the lighter mass of Ca compared to Ba. The strong hybridization between Fe and O modes in the optical region highlights the covalent nature of the Fe–O bonding, which plays a key role in the electronic and magnetic properties of these perovskites. Overall, the phonon results complement the negative formation energies reported earlier, confirming both the thermodynamic and dynamical stability of BaFeO₃ and CaFeO₃. These findings reinforce the reliability of our theoretical predictions regarding their electronic, magnetic, and optical characteristics, and further support their potential for spintronic and optoelectronic applications.

Fig. 11
figure 11

Phonon dispersion curves and partial phonon density of states (PDOS) for BaFeO₃ and CaFeO₃.

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Conclusion

Theoretical investigations of the cubic perovskites BaFeO3and CaFeO3were carried out using the full-potential linearized augmented plane-wave method implemented in the Wien2k code. Analysis of the tolerance and octahedral factors, formation energies, phonon dispersion and elastic moduli confirms the structural, dynamical, and mechanical stability of both compounds. BaFeO3 and CaFeO3 display ferromagnetic behavior, primarily driven by the unfilled Fe 3d orbitals. The Fe 3d t2g states strongly hybridize with O 2p states, resulting in bonding and antibonding interactions. The total magnetic moment for both materials, calculated using mBJ-GGA and YS-PBE0 functionals, is an integer value of 4 µB, reflecting their half-metallic character. The Fe 3d orbitals are occupied such that the t2g states are filled in both spin-up and spin-down channels, while the e.g. and t2g states remain unoccupied for both spins. Additionally, these materials exhibit a high absorption coefficient in the ultraviolet region, low reflectivity in the near-infrared range, and a narrow band gap semiconductor behavior in the spin-down channel, making them promising candidates for optoelectronic applications.Based on the calculated magnetic, optical, and electronic properties, both BaFeO₃ and CaFeO₃ show promising potential for use in spintronic and ultraviolet optoelectronic applications. Their strong spin polarization, high absorption coefficients in the UV region, and robust structural stability suggest applicability in devices such as spin-filters, photodetectors, and memory elements. Future studies could explore the effects of oxygen vacancies, mixed valence states, and spin–orbit coupling (SOC) on the physical behavior of these compounds. Experimental validation of the electronic and optical predictions is also recommended to assess their practical viability.