3.1. Structural, Electronic, and Magnetic Properties

The perovskite structure can manifest in various crystallographic arrangements [2]. In this investigation, we focus on the three space groups, cubic (221-Pm-3m), tetragonal (127-P4/mbm), and orthorhombic (62-Pnma), as illustrated in Figure 1. For the cubic phase, we performed volume optimization, while for the tetragonal phase, optimization was conducted for both volume and c/a ratio, interchanging parameters to achieve the most stable configuration. Similarly, in the case of the orthorhombic phase, we optimized volume, c/a ratio, and b/a ratio by sequentially fixing two parameters in each trial and determining the third, as outlined in Table 1. In Figure 2, we present the calculated energy plotted against the volume per formula unit, fitted to the Murnaghan equation of state. The energetically favorable phases, which exhibit the lowest energy, are the orthorhombic phase for GeScS₃ and GeZnS₃, and the tetragonal phase for GeFeS₃.

Due to the absence of available experimental data, we assume an equal likelihood for each phase to occur naturally, pending experimental evidence to validate our predictive findings.

Our analysis of the studied compounds considers each structural space group in three magnetic phases: ferromagnetic (FM), antiferromagnetic (AFM), and nonmagnetic (NM) phases. Table 2 illustrates the E_For values for each structure. It is evident that the most stable structure (highlighted in yellow) and the corresponding (nonmagnetic) 62-Pnma for both GeScS₃ and GeZnS₃ and the magnetic phase occur at (FM) 127-P4/mbm for GeFeS₃.

We have reevaluated our GeFeS₃ calculations by consistently applying Hubbard’s correction through the mBJ + U approach across all space groups for the most favorable magnetic phases. The effective potential, sourced from literature and set at 2.2 eV [28], has been implemented at the iron (Fe) atom site. Despite the use of mBJ + U, the overall conclusion regarding this compound remains unchanged. The marginal variations observed in the total energy and formation energy of the compound are minimal, essentially rounding to zero. Additionally, all other deductions related to the band structure and density of states remain consistent (refer to Tables 3 and 4).

In circumstances of exceedingly low temperature (absolute zero) and fixed N, as is the case here, dG/dP = V. From Figures 3(a) and 3(c), it becomes evident that the orthorhombic phase is the most stable for GeScS₃ (GeZnS₃). Conversely, for GeFeS₃, the tetragonal phase exhibits the highest stability (Figure 3(b)). This assertion is derived from the fact that these phases correspond to the shallowest lines. Notably, the calculations of Gibbs free energy align seamlessly with the Murnaghan equation of state, in regard to the stability of the structure.

The spin magnetic moments have been documented in Table 4. The magnetic behavior of the material is predominantly governed by the presence of the 3d-TM atom in site B, and this behavior is intricately tied to factors such as the unoccupied 3d orbitals, the local environment of neighboring atoms, and bond lengths [18]. The examined compounds undergo analysis in three magnetic phases for each space group: ferromagnetic (FM), antiferromagnetic (AFM), and nonmagnetic (NM) as outlined in Tables 2 and 3. As previously mentioned, the most reliably predicted stable phases manifest as ferromagnetic (FM) in 127-P4/mbm for GeFeS₃ and nonmagnetic (NM) in 62-Pnma for both GeScS₃ and GeZnS₃. Notably, the antiferromagnetic (AFM) calculations reveal a complete disappearance of magnetic moment in all compounds. The unfavorable nature of the AFM phase is evident through the formation energy calculations.

In GeScS₃, the magnetic properties are absent for both the cubic (221-Pm-3m) and orthorhombic (62-Pnma) phases. However, in the tetragonal phase (127-P4/mbm), magnetism is observable. Note that the magnetism at the interstitial region is higher than that at the atomic sites; this can be attributed to the hybridization of orbitals, especially involving transition metal elements like Sc, which can contribute to magnetic moments. The interaction of the atomic orbitals in the interstitial region may lead to localized magnetic effects.

Conversely, in the GeFeS₃ compound, all phases exhibit fairly comparable ferromagnetic behavior, as anticipated, and this behavior is predominantly attributed to iron atom sites. The contributions from Ge and S atoms are minimal and bear negligible values. The contribution of interstitial magnetic moments varies mainly due to fluctuations in the local environment of nearest neighbors. The calculation of the total magnetic moment using mBJ + U at the Fe atom site shows an insignificant variation, as mentioned earlier. It is worth noting that the mapping of magnetic moments can be complex and is often linked to the intricate interactions that transition metal ions exhibit within specific coordination environments involving chalcogenide atoms, such as sulfur in this case [30]. For the GeZnS₃ compound, magnetism is absent across all phases and scenarios.

The energy bandgap (Eg) and electronic density of states (DOS) have been thoroughly examined for all structures across various phases. GeScS₃ displays metallic behavior in all phases. However, an interesting conduction band intersection with E_F is observed in the spin-up channel of the tetragonal phase, categorizing the compound as a semimetal (Figure 4).

For GeFeS₃, the energy band structure unveils a direct (Γ − Γ) bandgap of 0.235 eV solely in the orthorhombic phase within the spin-up channel (Figure 5(c)); meanwhile, this compound shows metallic behavior in the other space groups like cubic and tetragonal (Figures 5(a) and 5(b)). Conversely, a semiconducting characteristic is demonstrated by an indirect (X − Z) bandgap of 0.827 eV within the tetragonal phase of GeZnS₃ (Figure 6(b)) and metallic behavior in the other phases (Figures 6(a) and 6(c)). In order to ensure the accuracy of band structure calculations, specific paths were adopted, such as (Γ-X-M-Γ-Z-R-A-Z-Γ) and (Γ-X-S-Γ-Y-S) for the tetragonal and orthorhombic phases, respectively, as depicted in Figure 7.

Examination of the total and partial density of states (TDOS and PDOS) reveals that the PDOS concentrated around or near E_F is primarily influenced by the S-p orbital and the TM-d orbital (Figures 8, 9–10). Eventually, GeFeS₃ exhibits half-metallic behavior in the orthorhombic phase, displaying 100% spin polarization at E_F, a feature that renders this material suitable for utilization in the spintronic industry. Conversely, the tetragonal phase of GeZnS₃ displays semiconducting behavior.

3.2. Optical Properties

The intricate dielectric function, denoted as ε(ω) and composed of its two constituent parts ε₁(ω) and ε₂(ω), is meticulously explored for the studied phases employing the Kramer-Kronig relations [31, 32]. In this context, ε₁(ω) and ε₂(ω) signify the real and imaginary components of ε(ω), respectively. By incorporating the function ε(ω) into the Kramer-Kronig relations, it becomes possible to compute a range of spectra including the refractive index n(ω), reflectivity R(ω), absorption I(ω), and energy loss E_loss(ω), utilizing equations 49 to 54 as referenced in [33].

These spectra are acquired along both the xx and zz directions for cubic and tetragonal phases, which are notably identical within the cubic phase. However, for the orthorhombic phase, the analysis extends along the xx, yy, and zz directions, showcasing an anisotropic behavior.

The imaginary component of the dielectric function, ε₂(ω), is depicted in Figure 11. Evidently, all studied structures across their various phases may exhibit characteristics of a light harvester, as the spectra consistently maintain positive and nonzero values throughout the entire energy range under investigation [34]. As expected, the material’s behavior is isotropic in the cubic phase, in contrast to the other phases.

Prominent peaks, identifiable across all phases, are positioned below the 4 eV energy threshold, primarily arising from interband transitions involving the p-orbital of the Ge atom and the d-orbital of the TM (Sc, Fe, Zn) atom. Static ε₂(0) values are presented in Tables 5–7. The highest ε₂(0) values are observed in the semiconductor phase, which is orthorhombic for both GeScS₃ and GeFeS₃ and tetragonal for GeZnS₃. This suggests that these structural configurations could potentially serve as efficient light harvesters.

Conversely, the real component, ε₁(ω), is illustrated in Figure 12. It is evident that all compounds exhibit metallic behavior as the spectrum transitions into negative values. This phenomenon is observable for GeScS₃ across both cubic and tetragonal phases within the energy range of 4 to 8 eV. Similarly, the same metallic trend emerges for the orthorhombic phase beyond 6 eV. In the case of GeFeS₃ and GeZnS₃, the manifestation of metallicity is observed in energy intervals of 3-6 eV, 3-8 eV, and beyond 8 eV for cubic, tetragonal, and orthorhombic phases, respectively. The static constant values of ε₁(0) are outlined in Tables 5–7, reflecting distinct values across various directions for each phase. Obviously, the highest value is anticipated in the zz-directions of both orthorhombic GeScS₃ and GeFeS₃. In the case of cubic GeZnS₃, the largest ε₁(0) is found.

The interaction of light with matter is closely linked to the refractive spectrum, which reveals how much light is bent as it traverses a material. It is important to note that the static refractive indices vary across phases, as detailed in Table 5. Notably, the highest static refractive indices are predominantly associated with cubic phases.

Furthermore, prominent peaks are consistently identified in various directions for different phases, predominantly occurring below the 4 eV energy threshold. The refractive spectra reveal resonance in this region, resulting from intraband transitions. As energy levels rise, refractivity decreases, corresponding to the onset of negative values in ε₁. This marks the initiation of metallic behavior, as depicted in Figures 12 and 13.

In Figure 14, the static reflectivity R(0) is computed, revealing values below unity for all phases, as indicated in Table 6. Remarkably, these static values experience a rapid decline within the low energy range, with the lowest points situated in this region. This characteristic renders the material a promising absorbent within this energy range, making it potentially suitable for photovoltaic applications. A significant surge in reflectivity is observed in higher energy regions and beyond, aligning with the manifestation of metallic behavior. This correspondence between the rapid increase in reflectivity and the emergence of negative values in ε₁ reaffirms the presence of metallicity [35].

The absorption process unfolds when the energy of the bandgaps falls below that of the incident photon. As illustrated in Figure 15, the spectral components of the absorption coefficient I(ω) are showcased for each material. Characterized by three components (x, y, and z), the overall profiles of these spectra exhibit a consistent pattern across our various compounds. Upon closer inspection of the curve features, a noticeable trend emerges—the initial stages of absorption witness a surge in spectra energies aligned with the optical gap in the orthorhombic configurations for GeScS₃ and GeFeS₃, as well as the tetragonal structure for GeZnS₃. Interestingly, the absorption spectra extend into both the visible and ultraviolet spectra. This behavior suggests the potential suitability of these materials, positioning them as promising candidates for applications such as the window layer in solar cells and flat-panel displays.

The quantification of energy absorbed through the material is effectively represented by the E_loss function, as depicted in Figure 16. Conspicuously, the minimum E_loss values are consistently situated below 4 eV for all phases, aligning with the anticipated maxima of the refractive function n(ω). As energy levels surpass 4 eV, E_loss experiences an increment, consistent with the concurrent rise in refractivity within that energy range.