Supported data and manuscript "Unveiling the structural, optical coating and thermoelectric characteristics of kesterite-quaternary chalcogenides Ag2InGaX4 (X = S, Se, Te) via DFT study"

Unveiling the structural, optical coating and thermoelectric characteristics of kesterite-quaternary chalcogenides Ag 2 InGaX 4 (X =S, Se, Te) via DFT study C. Bourahla a , F. Chiker a , H. Khachai a , R. Khenata b,c,* , A. Bouhemadou d , Devraj Singh e , S. Bin-Omran f , R.D. Eithiraj g,** , Hamad R. Jappor h , Saleem A. Khan i a Laboratory for the Study of Materials and Optical Instrumentations-Faculty of Exact Sciences, Djillali Liab` es University, 22000, Sidi Bel Abb` es, Algeria b Laboratoire de Physique Quantique de la Mati` ere et de Mod´ elisation Math´ ematique, Universit´ e de Mascara, 29000, Mascara, Algeria c Algerian Academy of Sciences and Technologies, Rais Hamidou VillaEl Madania, Algiers, 16000, Algeria d Laboratory for Developing New Materials and Their Characterizations, Department of Physics, Faculty of Science, Ferhat Abbas University - Setif 1, 19000, Setif, Algeria e Department of Physics, Prof. Rajendra Singh (Rajju Bhaiya) Institute of Physical Sciences for Study and Research, Veer Bahadur Singh Purvanchal University, Jaunpur, 222003, India f Department of Physics and Astronomy, College of Science, King Saud University, P.O. Box 2455, Riyadh, 11451, Saudi Arabia g Department of Physics, School of Advanced Sciences, Vellore Institute of Technology (VIT), Chennai, Tamil Nadu, 600127, India h Department of Physics, College of Education for Pure Sciences, University of Babylon, Hilla, Iraq i New Technologies Research Centre, University of West Bohemia, CZ-301 00, Pilsen, Czech Republic ARTICLE INFO Keywords: Ag 2 InGaX 4 (X=S Se and Te) FP-LAPW method Kesterite structure Optical thin-films Thermoelectric properties ABSTRACT This study explores the distinctive features of novel kesterite-type chalcogenide semiconductor materials through a new scheme designated as I 2 -III-III-VI 4 , focusing on Ag 2 InGaX 4 (X =S, Se, Te). The investigation employs density functional theory (DFT) using the advanced all-electron full potential linear augmented plane wave(FPLAPW) method. The exchange-correlation potential is assessed through the Perdew–Burke–Ernzerhof (PBE) parameterization, complemented by the Tran–Blaha modified Becke–Johnson (TB-mBJ) exchange potential estimation.Furthermore, thermodynamic parameters are analyzed in relation to temperature and pressure for the selected materials, utilizing the quasi-harmonic model. The electronic structure analysis reveals that Ag 2 InGaX 4 (X =S, Se, Te) materials display semiconducting behavior, with direct band gaps measured at 1.9 eV, 1.1 eV, and 0.86 eV, respectively.Moreover, the predicted refractive index, absorption coefficient, dielectric function, absorbance, transmittance and reflectance revealed that Ag 2 InGaX 4 (X =S, Se, Te) are promising materials for photovoltaic and optoelectronic devices. Furthermore, the analysis of thermoelectric properties considering the Seebeck coefficient, thermal conductivity, electronic conductivity, and highly valued figures of merit showed that the studied kesterite-type compounds have strong potential for applications in the fields of thermoelectric power energy.Finally, all these results are considered favorable and appropriate as per the characteristics mentioned earlier, and their potential advantages and applications in advanced hybrid photovoltaic and thermoelectric systems have been highlighted. It has been declared that this study’s attained results were considered a prediction in this kesterite family. 1. Introduction In recent years, quaternary chalcogenides of the I 2 -II-IV-VI 4 family have been extensively studied, especially in the optoelectronic field, due to their peculiarity [1–4]. In this paper, the new chalcogenide semiconductor compounds have been explored with a new scheme I 2 -III-III-VI 4 [I: Ag, III: In, Ga and VI: S, Se, andTe], which represents a deviation from the traditional framework I 2 -II-IV-VI 4 compounds. Mostquaternary chalcogenide compounds have most of the earth-abundant components and these materials have precisely tunable * Corresponding author. Laboratoire de Physique Quantique de la Mati` ere et de Mod´ elisation Math´ ematique, Universit´ e de Mascara, 29000, Mascara, Algeria. ** Corresponding author. E-mail addresses: [email protected] (R. Khenata), [email protected] (R.D. Eithiraj). Contents lists available at ScienceDirect Journal of Physics and Chemistry of Solids journal homepage: www.elsevier.com/locate/jpcs https://doi.org/10.1016/j.jpcs.2025.112970 Received 16 March 2025; Received in revised form 14 May 2025; Accepted 20 June 2025 Journal of Physics and Chemistry of Solids 207 (2025) 112970 Available online 21 June 2025 0022-3697/© 2025 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY license ( http://creativecommons.org/licenses/by/4.0/ ). physical properties for which they have been coveted in various fields and have been proven to be good candidates in both scientific [5–11] and industrial applications [12–27]. Hence, numerous quaternary chalcogenide compounds that belong to the I 2 -II-IV-VI 4 family have been investigated, allowing the formation of thousands of quaternary chalcogenide compounds [1]. Among them, specifically,copper-zinc-tin selenide Cu 2 ZnSnSe 4 (CZTSe) and copper-zinc-tin-sulfide Cu 2 ZnSnS 4 (CZTS), which were fascinated by tremendous observation in solar cells implementation due to their suitable optical features such as a desirable direct band gap of around 1.5 eV and tunability of the semiconducting properties by alloying various elements [28–31].Most efforts have been made to design the optimized device structure that can accommodate current CZTS absorbers and help the development of sustainable photovoltaic technology [32,33]. Despite their impressive efficiencies, CZTS compounds suffer from stability issues that require further research. A series of other quaternary chalcogenide compounds with the formula I 2 -III-III-VI 4 have been suggested numerically by MengweiGao et al. [34], and they showed elemental properties extracted from Wolfram programming language, as well as 2180 quaternary semiconductor compounds with a band gap energy array from 1.6 eV to 3.2 eV. In addition, they calculated the electronic features, including the density of states and band structure for the Ag 2 InGaS 4 compound, using the HSE06-DFT method,and found bandgap energy equal to 1.72eV. This family of compounds is close to CZTS. On the other hand, these materials have been found in several polymorph crystal structures such as stannite (I4‾2 m), kesterite (I4‾), sphalerite (zinc blende, Fm) and PMCA (primitive mixed CuAu-like, P4‾2 m) at high temperatures. According to research related to the structural stability of CZTS, kesteritehas been found the most stable phase [1,36–41]. This motivates us to investigate the different physical properties of three quaternary chalcogenides Ag 2 InGaX 4 (X =S, Se,Te) in the kesterite structure using the abinitio FP-LAPW method [42,43] implemented in the Wien2k code [44]. For the exchange-correlation potential, the Perdew–Burcke–Ernzerhof(PBE) generalized gradient approximation (GGA) [45] and the Trans Blaha modified Becke-Johnson (TB-mBJ) approach [46] has been chosen for more electronic eigenvalues accuracy [47,48]. Nevertheless,due to a lack of studies and research on these types of compounds I 2 -III-III-VI 4 in kesterite structure, a comparative study together has been performed with their homologous succession of kesterite light absorber materials CZTX (X =S, Se, andTe) [36–41] to better understand their properties. The obtained results of the optical thin film characteristics and thermoelectric parameters description have been considered valuable additions and are of considerable significance for the area of hybrid photovoltaic thermal systems, photovoltaic systems, and optoelectronics devices’ applications. 2. Computational methods The first-principles evaluations were attempted to thoroughly examine the geometry optimization, structural, thermodynamic, optoelectronic and thermoelectric characteristics of the kesterite structure for the selected chalcogenide semiconductor compounds with the new scheme I 2 -III-III-VI 4 [I:Ag, III:In, Ga and VI: S, Se, andTe]. All the calculations have been performed with the help of the full-potential linearaugmented-plane-wave (FP-LAPW) plus local-orbital technique inside the density functional theory framework [42,43], prosecuted in the WIEN2K code [44]. The exchange-correlation issues were considered with the help ofthe GGA, assumingPBE parameterization [45]. In addition, we include the Tran–Blaha modified Becke–Johnson (TB-mBJ) exchange potential approximation [46] to get more precise electronic wave functions [47,48]. Moreover, to attain a reasonable convergence, the keystone set for self-consistent field evaluations is within 10 −5 Ry of the entire energy of the system, and a plane wave cut-off of kinetic energy R mt ×K max =7.0 has been nominated for Ag 2 InGaX 4 (X =S, Se, Te). The muffin tin radii for Ag, In, and Ga have been disposed to 2.0, 2.2, and 2.0 Bohr, respectively, whereas for S, Se, and Te, the muffin tin radii are taken to 1.6, 2.0, and 2.2 Bohr, respectively. For the K-blending over the Brillouin zone [49],a mesh of 6 ×6 ×6 has been selected in the Brillouin zone, which corresponds to 35 k-points for structural and electronic calculations, while a mesh of 2562 k-points (27 ×27 ×27) in the complicated illustrations have been utilized for the calculation of the optical and thermoelectric features. Finally, the valence electrons contain the 4d and 5s orbitals for Ag and In and the open-shell s and p orbitals for S, Se and Te. 3. Results and discussion 3.1. Structural features It is known that the Ag 2 InGaX 4 (X: S, Se, Te) compounds belong to the CZTS family of compounds. The first principles demonstrated that these compounds are the most stable in kesterite structure, which has the lowest binding energy [1,28–31]. Therefore, we focus on the physical features of the suggested materials Ag 2 InGaS 4 , Ag 2 InGaSe 4 , and Ag 2 InGaTe 4 in kesterite structure that possesses tetragonal unit cell with space group I-4 (space group:0.82) (displayed in Fig. 1),where the Wyckoff locations in the conventional unit cell have been optimized as presented in Table 1. In this research, the calculations performed by the PBE-GGA approximation [45] as implemented within Wien2k [44].Our results of the structural properties have been provided in the non-magnetic state for three chosen quaternary chalcogenide compounds Ag 2 InGaX 4 (X: S, Se, Te).The values of the c/a ratio have been found for selected materials Ag 2 InGaX 4 ; after that,the volume optimization of the kesterite unit cell has been investigated for Ag 2 InGaX 4 . The fluctuation of the entire energy-dependent volume is carried out and the curve is fitted to the equation of state (EoS) of Birch-Murnaghan [50–52] for finding the different equilibrium parameters of the unit cell like: lattice parameters a and c, equilibrium bulk modulus B 0 and pressure derivative of bulk modulus B ′ , equilibrium volume V 0 and the equilibrium total energy (E 0 ). All optimized lattice parameters have been listed in Table 2. The fluctuation of the entire energy as a function of the volume of the selected kestrites is visualized in Fig. 2. Because there are no studies on the growth mechanism available for the selected kesterite Ag 2 InGaX 4 (X: S, Se, Te) in the literature, our results on these materials are a fulfilled prediction. However, this does not prevent us from comparing results between the compounds. As Fig. 1. Crystal structure of the kesterite Cu 2 ZnSnS 4 (CZTS). C. Bourahla et al. Journal of Physics and Chemistry of Solids 207 (2025) 112970 2 commonly seen in chalchogenide type compounds, like CZTX (with X = S, Se and Te) the popular used materials in thin film solar cell energy, it is noteworthy that the rise in the total number of electrons, as well as the difference in the atomic radii of S, Se, and Te atoms, contributed to the increase in the volume of the crystal unit cell (see Table 2). As a result, due to the decrease in bulk modulus (B 0 ), it can be said that the results of the present investigation are in the right direction because it is familiar that the volume (V) is inversely proportionate to the bulk modulus (B 0 ). 3.2. Thermodynamic features To design materials with enhanced thermodynamic features it is significant to check the temperature and pressure-dependent thermal parameters, like bulk modulus B 0 , heat capacity (C V ) and thermal expansion coefficient ( α ).In this section, the Debye’s quasi-harmonic model has been applied in the Gibbs code [52–54] to enumerate the thermodynamic features of the Ag 2 InGaX 4 (X: S, Se, Te) materials over a temperature array of 0–550 K and in the pressure array of 0–12 GPa as given in Figs. 3–6.The calculated results of volume-dependent temperature have been shown in Fig. 3 in the pressure range 0–12 GPa. One can see that the volume of the elementary cell is almost constant below 50 K, after this temperature, it increases monotonously for all studied compounds Ag 2 InGaS 4 , Ag 2 InGaSe 4 and Ag 2 InGaTe 4 . This indicates that at high temperatures, anharmonic effects play a prominent role, whereas the volume of the elementary cell decreases with an increase in pressure, as shown in Fig. 3So, we can say that these are the three selected kestrite compounds Ag 2 InGaX 4 , which expand easily on heating.On the other hand, the results of temperature and pressure dependence of the bulk modulus B for all three selected compounds have been plotted in Fig. 4. The examination of this curve shows that the bulk modulus B remains constant below the temperature of 40 K. But above this temperature, the bulk modulus reduces with an increase in temperature at a specified pressure, while B 0 rises with the increase in pressure P at a specified temperature. However, in the case of these selected materials, the influence of pressure on B 0 is much more significant than that of Table 1 Wyckoff positions of Ag 2 InGaX 4 (X: S, Se and Te). Atom Wyckoff positions x y z In 2b0 0 0.5 Ga 2c0 0.5 0.25 Ag1 4a000 Ag2 4f0 0.5 0.75 S8e0.2224 0.2646 0.3688 Se 8e0.2352 0.2637 0.3691 Te 8e0.2456 0.2621 0.3698 Table 2 The calculated lattice constants a and c, volume V 0 , bulk modulus B 0 , pressure derivative of bulk modulus B ′ , and the total energy E 0 for Ag 2 InGaX 4 compared to CZTX type compounds (with X: S, Se and Te). Lattice parameters Ag 2 InGaS 4 Cu 2 ZnSnS 4 Ag 2 InGaSe 4 Cu 2 ZnSnSe 4 Ag 2 InGaTe 4 Cu 2 ZnSnTe 4 a (Å) 5.8551 5.427 [39] 6.0903 5.606 [39] 6.5190 6.1834 [40] c (Å) 11.1235 10.854 [39] 11.6824 11.212 [39] 12.4273 12.363 [40] B 0 (GPa) 58.214 72.802 [40] 50.6864 66.757 [40] 39.9977 41.552[40] B’ (GPa) 4.8502 –5.6403 –5.0126 – V 0 (Å 3 ) 95.3783 –108.3320 –132.0352 – E (Ryd) −40117.925 –−56365.985 –−91299.242 – Fig. 2. The variation of total energy with volume of kesterites (a) Ag 2 InGaS 4 , (b) Ag 2 InGaSe 4 and (c) Ag 2 InGaTe 4 using GGA approximation. C. Bourahla et al. Journal of Physics and Chemistry of Solids 207 (2025) 112970 3 temperature. The heat capacity of a substance represents a strong point for understanding the thermal reactivity and intermolecular vibrations of substances, as well as numerous uses. The variations of the heat capacities Cv with the temperature at 0–12 GPa pressures have been presented in Fig. 5 for all quaternary chalcogenide compounds Ag 2 InGaX 4 (X: S, Se, Te). is understandable from Fig. 5 that temperature and pressure have reverse effects on heat capacity. The influence of Fig. 3. The cell volume change with temperature at 0–12 GPa pressures for Ag 2 InGaX 4 (X =S, Se and Te). Fig. 4. The variation of the bulk modulus B 0 with temperature at different pressures for Ag 2 InGaX 4 (X =S, Se and Te). C. Bourahla et al. Journal of Physics and Chemistry of Solids 207 (2025) 112970 4 Fig. 5. The changes of constant volume heat capacity (Cv) with temperature at 0–12 GPa pressures for Ag 2 InGaX 4 (X =S, Se and Te). Fig. 6. The variation of Debye temperature θ D with temperature at different pressures for Ag 2 InGaX 4 (X: S, Se and Te). C. Bourahla et al. Journal of Physics and Chemistry of Solids 207 (2025) 112970 5 temperature on heat capacity is more important than pressure’s. Further, the constant volume heat capacities Cv are proportional to T 3 , which follows the Debye model. Note that at T =0 K, the constant volume heat capacity (Cv) is zero for these three kesterite compounds and the constant volume heat capacity (Cv) rises rapidly with the increase of the temperature until about T =500 K, in which the curves approach and tend towards the Dulong–Petit limit [55], The value of Cv is 197 J/molK for Ag 2 InGaS 4 . In contrast,Cv is 198 J/molK for other two kesterites Ag 2 InGaSe 4 and AgInGaTe 4 . This indicates that the thermal energy stimulates all phonon modes at high temperatures, which is mutual to entire solids at high temperatures. The Debye characteristic temperature θ D is considered a thermal parameter characterized by the behavior of the thermal capacity of the studied solids. We can say that Debye temperature is defined as the maximum temperature that causes the normal vibration of the lattice atoms. The Debye characteristic temperature D changes with the temperature and pressure have been visualized in Fig. 6 for Ag 2 InGaX 4 (X: S, Se and Te) compounds. The heat capacity and Debye characteristic temperature of the selected Ag 2 InGaX 4 (X: S, Se and Te) are tabulated in Table 3 at room temperature (T =300 K) and zero (0) pressure. The Debye characteristic temperature decreases as the temperature rises, but it rises with the increase of pressure. Additionally, the results of Debye temperature remain constant at low temperatures (T <50 K) for the selected kestrites. Ultimately, these results indicate that the Debye quasi-harmonic model confirms efficient calculation of the contribution of thermal vibrations to the thermodynamic function of a crystal. In addition, the behavior of the Debye characteristic temperature curves is similar to that of the compressibility modulus for their fluctuations with temperature and pressure. 3.3. Electronic features Investigation of the crystal and electronic structures of Ag 2 InGaX 4 is needed to understand the origin of the physicochemical features and investigate the efficiency of their optoelectronic properties for use in solar cells and thermal power applications. Therefore, in this work, the electronic structure evaluations along with band structure and density of states for all the selected materials were implemented with the help of the generalized gradient approximation (GGA) within the modified Becke-Johnson (mBJ) potential (mBJ-GGA) [45,46]. It later proved to be a promising tool for mapping vacancies of electronic structures. These gaps are close to the experimental gaps, at low cost [56], for a large family of semiconductors and insulators [57]. The band path for specific symmetry points in k-space has been extracted from the suggested Brillouin zone of the kesterite structure as interpreted in Fig. 7 [58]. The calculated electronic band structures are illustrated in Fig. 8 [(a), (b) &(c)]for all selected compounds Ag 2 InGaS 4 and Ag 2 InGaSe 4 and Ag 2 InGaTe 4 . However, these materials present semiconductor behavior with a direct bandgap at Γ−Γ, with 1.9 eV, 1.1 eV, and 0.86 eV values for Ag 2 InGaS 4 , Ag 2 InGaSe 4 , and Ag 2 InGaTe 4 , respectively. It is well known that Te and Se atoms are more negative than S atoms, which is why the band value of Ag 2 InGaS 4 is higher than that of the other two compounds, Ag 2 InGaSe 4 and Ag 2 InGaTe 4 . As clearly stated, there is no data available to compare with the current studied compounds as these are new studies, only for Ag 2 InGaS 4 our calculated bandgap energies (1.9 eV) match strongly with the theoretical predictions (1.72 eV) given recently by Gao et al. [34]. Nevertheless, it is reliable to choose the homologous array of kesterite light absorber compounds Cu 2 ZnSnX 4 (CZTX; X =S, Se,andTe) as an example for comparison. Because of their similar crystalline structure, these compounds represent the. Majority of candidates used to fabricate multi-junction solar cells. Table 4 shows the presently calculated results at different k-points of the valence band maxima (VBM) and conduction band minima (CBM) values, as well as the band gap energies of the Ag 2 InGaS 4 , Ag 2 InGaSe 4 and Ag 2 InGaTe 4 compounds compared to available experimental and theoretical data of (CZTX; X =S, Se,andTe) semiconductors. The band gap energies obtained using the GGA-mBJ method closely align with previous experimental predictions and theoretical findings regarding CZTX compounds. Notably, in the case of the studied selenium kesterite phase Ag 2 InGaSe 4 , the band gap has been found around 1.1 eV paired with a significant absorption coefficient of approximately 10,000 cm −1 . These results indicate that Ag 2 InGaSe 4 holds considerable potential for use as the absorber layer in kesterite solar cells, whereas for sulfide kesterite phase Ag 2 InGaS 4 the energy gap is equal to 1.9 eV which is a more promising candidate than the first one to be used in the top cell of a multiple junctions thin film solar cell in which the ideal band gap for the top cell is situated between 1.6 eV up 2 eV. However, it is well known that the calculated band gaps from the TB-mBJ functional are more accurate than the GGA-PBE functional and agree with experiments with a typical discrepancy of less than 10 % for certain semiconductors and insulators, but some limitations can also seen on simple applications of the TB-mBJ method to other materials that still encounter significant difficulties because of the insufficient treatment of the localized d electrons [36,40]. To describe the chemical bonding behavior and to elucidate the participating orbitals of compounds, it is crucial to discuss the electronic density of states (total density TDOS and partial PDOS) as a representative case given in Fig. 9(a) and (b) and (c), for kestrite-type Ag 2 InGaS 4 , Ag 2 InGaSe 4 & Ag 2 InGaTe 4 semiconductors, respectively. It was found from the electronic band structure that all chalcogenides quaternary compounds Ag 2 InGaX 4 (X =S, Se, Te) have a semiconductor nature, with the band gaps energies of 0.9 eV, 1.1 eV, and 1.9 eV for Ag 2 InGaS 4 , Ag 2 InGaSe 4 & Ag 2 InGaTe 4 , respectively. At low-level energies of the DOS, the band ranges from −13eV to −11.6eVand is dominated by the s states of the S and Se atoms for three compounds Ag 2 InGaS 4 , Ag 2 InGaSe 4 and Ag 2 InGaTe 4 . The band found in the interval [−6.50 eV; −5.60 eV] is strongly composed of the satomic states of the In and Ga atoms for the three studied quaternary compounds. Then, the valence band (VB) situated at the range [−5.5; E F ], contains mainly a hybridization of the Table 3 The heat capacity at constant volume Cv and the Debye characteristic temperature θ D at T =300 K and at zero (0) pressure for Ag 2 InGaX 4 (X: S, Se and Te). Material Cv θ D Ag 2 InGaS 4 189.83761 301.27 Ag 2 InGaSe 4 193.35407 239.05 Ag 2 InGaTe 4 195.25911 198.23 Fig. 7. Brillouin zone of the space group II-4 for the Cu 2 ZnSnS 4 -like kesterite unit cell with special high symmetry points highlighted in orange [58]. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.) C. Bourahla et al. Journal of Physics and Chemistry of Solids 207 (2025) 112970 6 d states of the Ag atom and p states of the X atoms (X =S, Se,andTe). Finally, at high-level energies, the conduction bands (CB) primarily contribute by hybridizingthes and p states of the In and Ga atoms. It is worth noting that most of the research work has been done on homologous quaternary chalcogenide materials of type I 2 -II-IV-VI 4 (CZTX (X =S, Se, andTe)). Consequently, the comparison to most research and work given on homologue quaternary chalcogenide compounds is crucial because these materialsare similar to selected compounds Ag 2 InGaX 4 (X =S, Se, and Te). This is confirmed by the results of our findings and as mentioned by others studies [34,36–41] that going from the tellurides to the selenides with larger lattice constants and higher p orbital energies, the band gap is generally much smaller than for the sulphides [see Table 4]. 3.4. Optical and optical thin film features Recently, numerous academics have examined and optimized the optical and electrical features of photovoltaic systems, where the best efficiencies have been achieved using good absorbent materials by improving the essential constraints including the band gap of semiconductors with good absorption coefficients, attractive attention to the solar spectrum and good stability in contrast to solar irradiation [12]. To achieve these results, the investigation of semiconductors occupying the kesterite type chalcogenides quaternary compounds Ag 2 InGaX 4 (X =S, Se, and Te) is of relevant concern for their utilizations in photovoltaic systems and optoelectronic devices. In particular, these materials displays timulating features such as calculated band gaps around 1.5 eV, a value that improved matches with the extreme solar spectrum Fig. 8. The band structures of (a) Ag 2 InGaS 4 , (b) Ag 2 InGaSe 4 and (c) Ag 2 InGaTe 4 using GGA +mBJ approximations. Table 4 The valence band maxima (VBM) as well as the conduction band minima (CBM) positions of Ag 2 InGaX 4 (X =S, Se, Te). k-points Ag 2 InGaS 4 Ag 2 InGaSe 4 Ag 2 InGaTe 4 VBM CBM VBM CBM VBM CBM Γ−0.0076 1.9090 0.0065 1.1180 −0.0002 0.8678 X−0.3014 2.9124 −0.3416 2.1122 −0.4479 1.3554 Z−0.6317 3.4322 −0.7352 2.6976 −0.9395 1.7062 N−0.1740 3.2262 −0.2860 2.6024 −0.4602 2.0359 P−0.3484 3.0204 −0.4583 2.3227 −0.6561 1.6044 E g (eV) Present work 1.9166 1.1115 0.868 DFT 1.72 [34]– – CZTX→Cu 2 ZnSnS 4 Cu 2 ZnSnSe 4 Cu 2 ZnSnTe 4 Expt 1.46, 1.50, 1.51 [36] 1.0, 1.40–1.65 [36] 0.84–0.88 [37] DFT 0.91 [36], 1.47 [38] 0.54 [36], 0.90 [38] 0.47 [36], 0.32 [40] 1.50 [39], 0.65 [40] 0.96 [39], 0.28 [40] C. Bourahla et al. Journal of Physics and Chemistry of Solids 207 (2025) 112970 7 conversion for photovoltaic equipment. To complement these findings, we have extensively tested, performed and discussed the optical properties as well as optical thin film characteristics on three above mentioned compounds Ag 2 InGaX 4 (X =S, Se, and Te) given in the following section. The frequency-dependent complex dielectric function ε ( ω ) = ε 1 ( ω ) + i ε 2 ( ω ) is a well-established model for describing the optical properties of a material. The imaginary portion ε 2 ( ω ) has been resolved by computing the electronic structure employing the joint density of states and the optical matrix elements [59–64]. The real part of the dielectric tensor ε 1 ( ω ) can calculted from ε 2 ( ω ) using the Kramers–Kr¨ onig relationship [65,66]. Then, the other optical properties can be borrowed from ε 1 ( ω ) and ε 2 ( ω ), such as the absorption coefficient ( ω ), refractive index n( ω ), and the reflectivity R( ω ) [59–64]. Due to the tetragonal structure of the kesterite structure (a =b∕=c), all-optical spectra have been enumerated at the equilibrium lattice constant using the GGA-mBJ method along both directions of light polarization (xx) and (zz) as presented in Figs. 10–15. Fig. 10 shows the calculated dispersive real dielectric function ε 1 for the energy range up to 16 eV for all Ag 2 InGaS 4, Ag 2 InGaSe 4 and Ag 2 InGaTe 4 compounds. From these curves, one can notice that they have almost the same contrast. The calculated static dielectric constant values are ε xx 1 (0) =5.193 and ε zz 1 =5.11 for Ag 2 InGaS 4 and ε xx 1 (0) = 6.736 and ε zz 1 =6.639 for Ag 2 InGaSe 4 and ε xx 1 (0) =9.084 and ε zz 1 = 8.826 for Ag 2 InGaTe 4 along both directions of light polarization (xx) and (zz) (Table 5). Since, going from Ag 2 InGaS 4 →Ag 2 InGaSe 4 → Ag 2 InGaTe 4 and regarding their calculated band gaps of 1.916eV, 1.111 eV, and 0.868 eV, respectively, we can confirm that the calculated static dielectric functions tend to increase with decreasing band gap. Further, we can see that below the frequency 6eV for Ag 2 InGaS 4 , 5eV for Ag 2 InGaSe 4 and 3.5 eV for Ag 2 InGaTe 4 , the compounds present a high permittivity, then above these frequencies, one can notice a rapid decrease in the result of the real portion of real dielectric constant which it exceeds zero starting from 8.2 eV for Ag 2 InGaS 4 , 7.7 eV for Ag 2 InGaSe 4 and 4.8 eV for Ag 2 InGaTe 4 . Fig. 9. Total and projected electron densities of states of quaternary chalcogenides type kesterite compounds (a) Ag 2 InGaS 4 , (b) Ag 2 InGaSe 4 and (c) Ag 2 InGaTe 4 using GGA approximation. C. Bourahla et al. Journal of Physics and Chemistry of Solids 207 (2025) 112970 8 The imaginary dielectric function enumerated with GGA-mBJ is shown in Fig. 11 for the selected materials. To explain the absorptions that have been depicted in these spectra, it is necessary to deliberate transitions from occupied to unoccupied bands in electronic energy band structure, expressly at high symmetry points in the Brillouin zone. Where the first one starts from the energy gap values called threshold energy E 0 , approximately at 1.97 eV, 1.12 eV, and 0.857eV for Ag 2 InGaS 4 , Ag 2 InGaSe 4 , and Ag 2 InGaTe 4 , respectively, which originates predominantly from the transitions of electrons in Γ −Γ directions. In the absorptive part, several stronger peaks appeared at 5.9eV and 7 eV Fig. 10. Real dielectric function along xx and zz directions for Ag 2 InGaX 4 (X = S, Se and Te). Fig. 11. The imaginary dielectric function along xx and zz directions for Ag 2 InGaX 4 (X =S, Se, Te). Fig. 12. The absorption coefficient α ( ω ) along xx and zz directions for Ag 2 InGaX 4 (X =S, Se, and Te). Fig. 13. The extinction coefficient k ( ω ) and refractive index n ( ω )using an effective polycrystalline dielectric constant approach for quaternary chalcogenide semiconductors Ag 2 InGaX 4 (X =S, Se and Te). C. Bourahla et al. Journal of Physics and Chemistry of Solids 207 (2025) 112970 9