Comparative studies for determining the optical band-gap energy of the Novel polycrystalline thin ZnGa 2 S 4 films sprayed at different film thicknesses

This article has dedicated to studying some structural features and optical characteristics of the Novel polycrystalline ZnGa 2 S 4 (ZGS) thin films utilizing spray pyrolysis process at different thicknesses (293, 375, 452, and 517nm). The microstructural properties and crystal defects of these films have been studied in previous work. While in this work, the crystallinity degree and crystalline volume fraction have been studied using X-ray diffractograms. The stoichiometry of these ZnGa2S4 films has been checked using the energy dispersive x-ray analysis. The field-emission-scanning-electron microscope has been utilized to investigate the morphology of ZGS films' surfaces. Optical properties have been studied via transmittance and reflectance spectra in the range 300nm - 2500nm. Some important optical parameters such as absorption coefficient, skin depth, Urbach's energy, steepness parameters, and electron-phonon interactions have been extensively studied. The direct and indirect gap energy were determined by different four models and compared with Tauc’s model. Optical data analysis revealed that; all studied properties are strongly dependent on the film thickness. The optical band gap values were slightly decreased from 3.7eV to 4.1eV with increment of the film thickness owing to improving the crystallization process. These obtained results confirm that these films are wide band gap semiconductors, which makes them recommended for use in many solar cell applications as a window layer.


Introduction
In the last few decades, scientists, researchers and engineers have focused their efforts on studying manufactured novel materials that are used to convert photovoltaic energy into reliable electrical energy.
So, there is an increased recall to use abundant, inexpensive, and environmentally friendly materials in thin-film solar cells. CuInGaSe2 (CIGS) has been used as an absorbent in thin-film solar cells since it achieved the highest efficiency, which may reach about 20 % [1][2][3]. The search for toxic elements, scarcity, and the need to reduce the cost of mass production are driving chalcogenide-based solar cells as one of the next generations of thin-film solar cells [3]. Recently, the chalcogenides (ChG) thin films have received much attention in the past years owing to their diverse and numerous applications in electronic devices and memory switching, as well as in the nonlinear optical devices [4][5][6][7][8]. These ChG materials are characterized by their low phonon energy and their transparency in visible and infrared regions [9][10][11][12]. Also be doped by some rare earth elements, like Pr, Er, Nd, ... etc. Moreover, these ChGglasses are highly optical nonlinear materials and hence can be used for the optical-switching devices, OS. These glasses are also sensitive to the absorption of electromagnetic radiation and show a variety of photoinduced effects because of illumination.
The ChG materials are varied and found in abundance in the earth and have distinct and unique characteristics such as good optical and electrical properties, as well as their cheap prices, ease of production, and chemical stability. Hence, and owing to these wonderful properties, they are very suitable for producing window layers and economical absorbers for the solar cells [13]. It is worth noting that, the fabrication of the window layers require a semiconducting material that has a wide band gap ( 3.5 eV -3.8 eV) and of higher transmittance [14]. These unique conditions have been exhibited in the metal oxides, such as ZnO, SnO2, and In2O3, which have good efficiency to make as window layers of solar cells [15]. Moreover, the production of the absorbing layer of electromagnetic waves requires a semiconducting material that has a narrower energy gap (1.0 eV-1.5 eV), and a distinguished absorption of light of a wavelength arranged between 350 nm and 1000 nm, i.e., more than 10 4 cm -1 [16].
Thin Cu-ChG films like either Cu-Zn-Sn-S, or Cu-Zn-Sn-Se, CZTS(Se) exhibited better efficiency in solar cells. The scientists and researchers deposited thin CZTS films utilizing the sputtering technique and then they have measured their p-type conductivity. The electronic transition was found to be direct band gap about 1.45 eV, and their absorption coefficient was more than 10 4 cm -1 in the visible region [17]. The CZTS compositions and their films can be crystallized as a bland-zinc crystalline structure similar to other semiconducting materials such as Cu-In-Ga-Se, CIGSe, and Si. Generally, CZTS compositions can have two other forms; the first one is the stannite crystal structure, which has the same tetragonal coordination, but with unlike symmetry owing to the alternate situation of cations in the main crystal lattice. While the second is the wurtzite-derived crystal structure, which is a hexagonal close-packed array [18]. The three crystal structures have a formal Cu2ZnSnS4 stoichiometric, although the kesterite structure is the most often used in solar cells. The experimental procedures of manufacturing CZTS(Se) solar absorber layers could be classified into two different categories, they are: The vacuumbased techniques include flash evaporation, thermal evaporation, sputtering, and atomic layer deposition techniques. While the non-vacuum-based techniques (or the solution-based) processes include all chemical deposition processes, such as spray pyrolysis, sol-gel, electro-deposition, molecular precursors, and nano-crystal techniques [17][18][19][20][21].
The II-III2-VI4 compounds like ZnIn2S4, CdIn2S4, ZnGa2S4, and CdAl2S4 are also considered as significant ChG materials as they showed the n-type conductivity, higher transparency, and a wide gap ranged between 3.4 eV and 3.8 eV [19]. The importance of these ChG materials encourages the researchers to look for synthesizing of new compositions from this group, for the possibility of their using in solar cells as new optical windows [20]. The present work has focused on the novel thin ZnGa2S4 films owing to their non-toxic, low economical price, earth-abundant, and stability, along with their wide band gap and higher thermal stability.
The aim of the present work is to continue the previous study that presented a thorough study of the synthesis of the Novel nanocrystalline thin ZnGa2S4 films of various thicknesses by an inexpensive spray pyrolysis technique. In addition, the authors have studied the effect of film thickness on the surface morphology, crystal structure, crystal defects, and microstructure properties of these novel films [21]. The X-ray diffractions, XRD, Field emission-scanning electron microscope, FESEM, and the energy dispersive X-ray spectroscopy, EDX techniques have been employed to inspect the structural characteristics of the ZnGa2S4 thin films. Consequently, the authors of the current article will focus on studying the optical properties of these film samples. The optical characteristics of the films have been studied via studying the transmission, reflection, and absorption spectra and determining the parameters that describe the electronic transitions, such as: absorption coefficient, absorbance or optical density, the skin depth, band gap energy, Urbach energy, and valence band tails. Along with some important optical parameters of the ZnGa2S4 thin films that related to the principal optical transitions in the UV, Vis, and IR regions have been investigated and discussed. Moreover, the optical band-gap energy values will be determined by various methods, such as Tauc's plots, the absorption-spectra fitting curves, ASF, Cody representations, and Davis-Mott model.

ZnGa2S4 film formation
In this research work, an inexpensive spray pyrolysis method has employed to obtain the ZnGa2S4 thin films. At first, we make substrate cleaning process to get the ZnGa2S4 thin films of better quality.
The used substrates are ordinary micro slides of soda lime glass of dimension 26×76×2 mm 3 which have good optical characterizations. The cleaning process was carried out as follows; first, the glass sheets have been dipped in freshly equipped chromic acid, which has been heated up to 60°C. The glass sheets have been removed from the acid and washed with a distilled water. Subsequently they were placed in the solution of an alkaline soap for at least ten minutes. Thereafter, they have been washing with running water, where the process of washing was repeated. Next, the substrates have been washed using doubledistilled water then using the ultrasonic waves. Finally, the glass sheets have been dried in a furnace of hot air. This removes any dust particles that may be sticking to a surface, ensuring maximum hygiene and making the film stickier to the substrate. Chemicals in the powder form have been used to synthesize the ZnGa2S4 solution. This solution was used to precipitate thin films using the spray pyrolysis process. These chemicals are (GaCl3), (Zn(CH3COO)2·2H2O) and (Na2S2O3.5H2O) of molecular weights 281.26, 219.51 and 248.18, respectively. These chemical powders have high purity degrees (99.99%) and were purchased from the company Sigma-Aldrich. These chemicals were utilized without any further purification.
The precursor solutions for fabricating the novel ZnGa2S4 films were synthesized using: (i) 0.1 M from Zn (CH3COO)2·2H2O, the zinc acetate dihydrate, (ii) 0.2 M from GaCl3, gallium trichloride, which is used as a gallium source and (iii) 0.4 M from Na2S2O3, sodium thiosulfate, as a sulfur source.
The ZnGa2S4 solution was stirred well for one hour to produce a yellow solution. These molarity values were chosen to ensure that thin film having the stoichiometric ratio 1: 2: 4 were obtained. The ZnGa2S4 solution was sprayed from a nozzle on the pre-cleaned glass substrates via the spray pyrolysis technique, where the substrate temperature was fixed at 300 ⁰ C, the rate of flow was adjusted at 10 ml/min, and the airflow pressure from the compressor was fixed invariant at 3 bar. While the distance between the spray nozzle and the substrate was fixed at 30 cm, for all films. Furthermore, the deposition time during the spray process was 5, 10, 15, and 20 min to get thin films having these thicknesses: 293 nm, 375 nm, 452 nm, 517 nm, respectively, as measured by the alpha step D-500 stylus profilometer. It is worthy to mention that the thickness of each film was measured multiple times, and mean values were taken into account. For more specifics on the used spray pyrolysis method and the experimental procedures that have been exercised during the preparation of film samples, refer to the authors' previous work [22][23][24][25].

Materials characterization
The polycrystalline nature of ZnGa2S4 thin films was examined using a Philips-X'Pert X-ray diffractometer, with CuK radiation. The X-ray diffractograms were published elsewhere [21]. The crystallographic studies showed that the samples the tetragonal crystal nanostructure of the lattice constants equal a = 0.5272 nm and c = 1.0451 nm that consistent with the following XRD Cards: JCPDS 89-4207, 80-1707, and 40-1462. Moreover, the average crystallite size of films increased from 14 nm to 40 nm as the thickness of the film increased [21]. The compositional element percentages and the surface morphological features of the ZnGa2S4 thin films were characterized via using the Quanta-FeG-250 USA field-emission-scanning-electron microscope, FE-SEM. The energy-dispersive X-ray spectroscopy technique, EDAX was also employed to investigate the elemental compositions and the finding confirmed that all thin-films have a good stoichiometry, along with in a good match with the theoretical computations. For more details about these EDAX characteristics and FE-SEM micrographs, anyone can refer to the previous work [21]. Optical properties of the ZnGa2S4 films were carried out via measuring the reflectance, R and transmittance, T of films in the spectral range 400 nm -2500 nm. A double beam UV-Vis-NIR-Shimadzu spectrophotometer of the model UV-310-PC was used to record the R-and Tspectra of the film samples. All optical measurements are performed at room temperature.

FE-SEM and EDS investigations
Optical microscopes are effective tools used to specify the morphology, microscopic structural imperfections, and macro-surface defects that may manifest in the crystalline films, especially the optically transparent microscopes. Where the electron beam interacts with the atoms of the film sample, to produce various signals. These signals provide good information about the morphology, topography, and compositional elements of the surface. The electron beam is generally scanned using raster scanning and the location of the beam is combined with the signal to produce an image [21]. By scanning the sample and collecting the secondary emitted electrons with a special detector, an image showing the surface morphology is created.
The morphological features of ZnGa2S4 film surface have been scanned by the field-emissionscanning-electron microscope, FE-SEM of the type Quanta-FeG-250 USA. Figures (1-a) and (1-b) depict the obtained FE-SEM micrographs of nano-dimensions ZnGa2S4 films for the smallest thickness (293 nm) and largest thickness (517 nm) samples, respectively, as typical samples of ZnGa2S4 films. These FE-SEM micrographs show that the polycrystalline ZnGa2S4 thin films have a good surface shape and that the particle sizes are almost homogeneous and uniform. These FE-SEM-micrographs also illustrate that the particle size of the film sample is almost increasing as the film thickness increases. This affirms the improvement of the crystallization of film and crystallinity degree of ZnGa2S4 film samples. These results are also in good matching with those obtained from the study of the X-ray diffraction.
On the other hand, the apparatus of FE-SEM has been equipped with interface equipment for the energy-dispersive x-ray spectroscopy investigations, EDS to check and analyze the percentage of the compositional elements of the studied thin films. The EDS spectra have been shown in Fig. (1-c) and  Table 1. The analysis of the EDS data and their chart spectra of the polycrystalline ZnGa2S4 thin films confirmed that all synthesized films are consisting of Zn, Ga, and S elements, only, and no others detected. It is worthy to mention that the atomic fractional percentages of the three constituent elements (Zn, Ga, and S) are almost stoichiometry, where their percentages were close to the ratio (1: 2: 4) for all films. Furthermore, it is observed that the percentage of the S-element is slightly increasing with the increment of the film thickness. This increase is occurring on the account of the Sipercentage.
The thicker the film, the greater the distance traveled by the incident beam of electrons through the film compared to thinner films. Therefore, the higher the thickness of the film, the more accurate the percentages of the thin-film elements obtained for the film itself. At the same time as the thickness of the sample increases, it is found that the silicon percentage is decreasing. On the other hand, when excluding the percentage of Si originated from the glass substrate used, it turns out that the percentages of the three elements are almost invariant unaffected as the film thickness increases (maybe varied very slightly). This is because the solution used to prepare all samples has not changed, but what has changed is the deposition time that increases the film thickness.

Crystal structural studies
X-ray diffractograms, XRD of the current novel prepared thin ZnGa2S4 films with different thicknesses have depicted that all films have the same polycrystalline nature, as shown in Fig  Further, these diffraction lines are also found to have strong intensity, which confirm that the film samples are well crystallized. These detected diffraction peaks are in good consistency with the following

Evaluation of the degree of crystallinity
The physical, mechanical, and morphological properties of thin films depend on the degree of crystallinity and the direction of the preferred orientation growth during the preparation of materials and the deposition of film samples. Usually, these two factors have changed according to the used preparation route and the followed preparative parameters (such as the temperature of the substrate, the rate of the deposition, time of deposition, and the solution molarity). This is because of the growing crystallites of the films are exposed to thermal and kinetic energy during the deposition process. This increases the microscopic stress, lattice strains, and internal pressures that prevent the molecules from naturally arranging to form a certain crystal structure or to stacking in the form of crystals. This means that the degree of crystallinity of the thin films depends primarily on the preparation properties of those materials.
Therefore, thin films are usually a mixture of crystalline and amorphous phases.
The XRD-charts of ZnGa2S4 thin film shows that the crystallization process of films improves with increasing film thickness, where the diffraction Bragg's peaks become sharper and more intense with respect to the diffraction line of the crystalline phase of the largest thickness. The XRD-data can be employed to compute the crystallinity degree of the partially crystalline or polycrystalline films [26][27][28][29][30].
For the currently studied ZnGa2S4 thin films, this procedure succeeded in integrating all diffraction peaks corresponding to the observed lines: (112), (103), (202), (310) and (206), where the intensity of the Xray diffraction lines is as depicted in Fig. (2). Subsequently, by summing these intensities into one intensity that represents the integration of the crystallized part. For the subsequent analysis, we adopted a two-phase concept typically applied to thin films in which the amorphous contribution to the spectrum, which occurs as a broad diffraction band has been approximated by the background curve separating the amorphous part from the crystalline portion. Then the degree of crystallinity of each sample was obtained as a ratio between the area under the crystalline peaks and the total area under the diffraction curve.
In this work, a typical fitting procedure was used to separate the crystalline peaks from the amorphous halo of the ZnGa2S4 thin films of different thicknesses, as shown in Fig. (2). The crystal fraction was detached from the non-crystalline one using a computer program (EVA program) using the Hermans-Weidinger method [31]. Subsequently the degree of crystallinity, XCryst of each sample has been determined from the ratio between the area under the crystalline peaks, ACryst and the whole area under the XRD curve, (ACryst + AAmorph) according to a simple formula: It is clear from the tabulated results of Table 2 that the degree of crystallinity and the total area under the curves increase gradually with increasing the film thickness. This is evidence of an improvement in crystallinity of films and an increase in x-ray scattering. This in turn leads to a decrease in the amorphous ratio of the ZnGa2S4 thin films. The crystalline percentage, XCryst of thin ZnGa2S4 films increases with the increment of the thickness of films from 9.204 to 16.921. This means that an increment in crystallinity of ZnGa2S4 films has been observed for all samples of different thicknesses due to improvement in the crystallization. Consequently, it can be concluded that the film thickness increasing has a pronounced effect on the degree of crystallinity of ZnGa2S4 thin films [32].

Crystalline volume fraction
The determination of the crystalline volume fraction, Vcryst of a substance can simply be considered as a quantitative phase analysis of two-phase materials (crystalline and amorphous phases).
To work on such a hypothesis, a common XRD-based procedure as hypothesized by B.D. Cullity (1978) [33,34]. Thereby, to determine the volume fraction of the crystalline phase of partially crystalline materials from the structural measurements using XRD and by comparing the integrated intensities of the peaks derived from the amorphous and crystalline phases according to Huang's proposal [35]. In this case, the volume fraction of crystalline phase, VCryst can be estimated from the following Eq. [35]: Where ICryst and IAm are the integral intensities of the diffraction lines from the crystalline and amorphous phases, respectively. However, the parameter (α) is unknown, and it is called the Huang parameter.
Indeed, for Fe73.5Cu1Nb3Si13.5B alloy, α = 1.05 [36], and for Al88Ni4Sm8, α = 0.37 [37]. To determine the value of this α-parameter, a series of diffraction patterns were measured which could be different for different systems. When subtracting the background of the diffraction pattern, the diffractograms could be decomposed into two components owing to the amorphous and crystalline phases and subsequently the integrated reflection intensities from each phase could be estimated.   [38,39]. To find integrated intensities, the diffractions obtained in this way by Gaussians were approximated with the least squares fit. In this work the graphical relationship between Icryst and (ICryst+IAm) was plotted versus the crystalline phase (C1) as a linear relationship and the slope was calculated so the value of (α) was equal to 0.9043. It has been shown that the volumetric fractions of the crystal phase derived from different techniques can vary significantly [40].

Transmittance and reflectance spectra
The optical parameters of the examined thin ZnGa2S4 films have been determined from the transmission and reflection data. The spectral variation of the normal transmittance Texp(λ) and reflectance Rexp(λ) due to the effect of the used substrate is given as follows [42,43]:  It can also be observed that the transmittance and absorption spectra of all thin films can be divided into three main special regions: (1) the strong absorption region, which extends to wavelengths smaller than 750 nm, (2) the transparent region, which is after the wavelength of 1750 nm; (3) The absorption region between these two regions (750 nm -1750 nm). Also, the spectral distribution of both transmittance T(λ) is decreasing while that of the reflectance R(λ) is gradually increasing as increasing the film thickness. This is due to improving the film crystallinity and minimizing the crystal defects [23], which in turn leads to the increase in absorption of the ZnGa2S4 thin films of different thicknesses.
Moreover, the absorption edge remains unchanged with increasing film thickness while the summation of T and R is less than the unity after the absorption edge due to the scattering of light produced by the roughness of the surface of the studied films [43][44][45][46].

Absorption coefficients
The absorption coefficient of materials plays a substantial role in choosing a specified material

Skin effects and skin depth
The absorption of electromagnetic waves in thin semiconductor films relies on many parameters, the most important of which are (1)  . √ Where ρopt and σopt are the optical resistivity and conductivity, respectively of the thin ZnGa2S4 film, f is the frequency. While μr and μo are the relative permeability which is usually considered to be the unity and the absolute permeability constant, respectively (μr = 1 and μo = 4π×10 -7 H/m). Using the spectra of (α) the depth, δd can be computed for ZnGa2S4 thin films. Figure (6

-A) exhibits the variation of (δd)
versus the photon energy (hν) for the thin ZnGa2S4 films of different thicknesses.  Fig. (7-b) and subsequently recorded in Table 2. It can also observe here that the (σopt)-values decrease as the film thickness increases. This is owing to several reasons, like the activated thermal transfer of electrons from VB to CB in the semiconducting materials, and the charge carriers short-range hopping at the grain boundaries, as well as the localized hopping of the charge carriers within the grains, too [59]. Additionally, the strong coupling between an electron and a phonon is formed by the vibrations of the ions of the crystal lattice at a limited temperature known as a polaron [59,60].

Urbach energy
The absorption coefficients of the thin ZnGa2S4 films are shown as the exponential rise, called the tail of Urbach which exists down the excitonic peaks. This tail, which has an exponential nature, appears in polycrystalline, partially crystalline, and non-crystalline materials, because of the existence of these localized states, which extended in the forbidden gap between VB and CB [61][62][63].  Table 2. Apparently, the values of both (EU) and αo increase with the increment of the thickness of the films. This points out that a significant improvement in the crystallinity degree of the thin ZnGa2S4 films. The literature of similar works shows that the obtained results have the same trend [66][67][68]. Further, these results have already been affirmed from XRD findings.

Fig. (7): Plotting of ln (α) versus (hυ) of the polycrystalline ZnGa2S4 thin films. (A) shows the low and high absorption range (Urbach and Tauc regions, respectively) and (B) shows linear relationship.
According to Urbach assumption, there is another formula that correlates the two parameters (α) and (Eg) according to his suggestion [67][68][69]: Where σ is a new optical constant called the steepness parameter, β is another pre-exponential parameter, Eo is the energy of the electronic transitions, its value depends upon the electronic transition, where: For direct transitions: Eo = Eg; While for the indirect transitions: Eo = Eg ± Eph. Where the energy parameter, Eph represents the energy that bounds the phonon. Hence, Eph = 0 for the current study. Hence, if the direct transition case is considered, then Eo = Eg. Thus, it can substitute and reformulate Eq. (10) to get the following form: From Eqns. (9) and (12), it can conclude that: Therefore, the estimation of the sharpness or steepness of the absorption edge, which is called the regression parameter (σ), and is calculated as follows: Where KB is the Boltzmann's constant, and T is the room temperature (300 K), KB = 8.6173×10 -5 eV/K.
Hence, the steepness parameter, of the ternary compound ZnGa2S4 thin films has been computed for all films of varying thicknesses and recorded in Table 2. It is clear that, the steepness parameter decreases with the increment of the film thickness, which is acceptable results due to the increased value of the Urbach band-tail energy. On the other hand, the strength of the interaction between the electron and the phonon (Ee-ph) is linked with the parameters (σ) by this simple form [63,69]: Consequently, the value of the strength of interactions (Ee-ph) could be determined for the present films and subsequently recorded in Table 2

Optical band-gap energy
As a result of the extreme importance of the optical band gap of the semiconducting materials and the energy value of this gap, there were many attempts to study this optical band gap and deducing the amount of energy required for the electronic transition from the level of the valence band to that of the conduction band. Thus, several models are used to evaluate the energy of the band gap, such as the absorption spectra fitting (ASF) procedure, Tauc's plots, Cody representations, and Davis-Mott model.
These assumptions and models are based on the value of the absorption coefficient, α which appears near the edge of the band in semiconductors as an exponential function of the photon energy according to the following empirical relationship [47]: Where B is a parameter associated with the structural order of the semiconductors, is the band-gap energy of the studied material, hν is the energy of the incident electromagnetic waves and the exponent (y) is a parameter determines the nature of the electronic transitions. Since the value of (y) is what determines the type of transition; if it is allowed or forbidden; direct or indirect. If y is equal to 1/2 then the transition is allowed direct, but if it is equal to 2 then the transition is allowed indirect transition [47,57,69].
In this study, the authors will infer the optical bandgap energy values using different models, and subsequently compare these results to judge which models give the best and most accurate result.

Absorption spectrum fitting (ASF) procedure
The absorption spectrum fitting (ASF) procedure gives relatively good results, where this method depends also on Eq. (17). This equation can be rewritten as a function of the wavelength (c = υλ), as follows [71]: Where λgap, h, and c are the wavelength corresponding to the optical bandgap value, Planck's constant, and the velocity of light, respectively. In the case of the allowed direct transition (y = 1/2); so, Eq. (18) can be reformulated to become as follows: Subsequently, squaring this equation and let ( ( ) − ) = , thereby, Thus, by representing a graphical relationship between ( ) on the abscissa versus ( ) 2 on the ordinate as illustrated in Fig. (8-a), by the linear extrapolation at = 0, we can obtain the value of , which expresses the wavelength corresponding to the direct band-gap energy of thin ZnGa2S4 films of different thicknesses and thereafter recorded in Table 2. On the other hand, for the case of indirect allowed transition, let y = 2, thus Eq. (18) becomes as follows: By taking the square root of both sides and let √ / = , thus: Consequently, by graphical representation of the relationship between the value (1/λ) on the horizontal axis versus √ / on the vertical axis, as shown in Fig. (8-b). The linear extension of the straight line intersects the abscissa at a wavelength value that gives λgap of the indirect band-gap energy of the ZnGa2S4 thin films of different thicknesses and then the values have been recorded in Table 2.    (Table 2). with the increase in the film thickness of the ternary compound ZnGa2S4. While the direct and indirect gap energy values decrease gradually as the thickness increases, where the direct energy gap decreases from 4.106 eV to 3.754 eV and the indirect one decreases from 3.594 eV to 3.221 eV, as recorded in Table 3. Moreover, the dependence of the direct energy gap upon the film thickness (t) measured in (nm) by using this method of the film samples of the ternary composition ZnGa2S4 is illustrated in Fig. (12-a). The figure is linearly fitted to get the following experimental Eq.:

Tauc's Plots
This most popular model allows us to derive the band gap energy Eg as a function of the incident photon energy (E = hν) according to Eq. (17) [71]. The Tauc's optical band-gap associated with the thin films can be determined by extrapolating the linear trend observed in the spectral dependence of (αhν) 2 and (αhν) 1/2 over a finite range of photon energies (hν) [72].   Table 3. It is noticeable that there is a slight change in the value of the two gap energies with the increase in thickness of the films. Where the direct band gap energy decreases from 4.001 eV to become 3.902 eV, while decreases from 3.153 eV to 3.051 eV. This deceasing in the energy gab values is due to the improvement in crystallization with the greater the thickness of the film. Moreover, the direct band gap energies are represented against the film thickness of ZnGa2S4 composition, as depicted in Fig. (12-b), and the resultant was fitted linearly to get an empirical equation of a straight line that is given as: Fig. (9-b): The graphical relationship between (αhν) 1/2 and (hν) according to Tauc's method for the ternary ZnGa2S4 thin film with different thicknesses.

Cody representations
Cody assumed that the increasing value of Tauc's optical gap that is associated with the decrease in the thickness of films takes place owing to the curvature of the dependence of the spectra of the function (αhυ) ½ upon the energy of the incident photon, f (hν) [74]. According to the model presented by Cody [75], the optical gap energy of the polycrystalline ZnGa2S4 thin films of different thicknesses can be evaluated from the cut-cross of the extension of the linear part observed in the figure of representation of the spectra of (α/hν) 2 Table 3 illustrates the estimated optical gap values using the Cody's model. The obtained results from these graphs showed that the energy of the allowed direct band gap decreases from 4.003 eV to 3.854 eV, while for the indirect one, it decreases from 3.033 to become 2.901 eV. Moreover, the direct gap energy is represented against the film thickness in Fig. (12-c), which is linearly fitted to get this linear Eq. between Eg values and the thickness of films (t) in nm:

Eg (eV) = 4.221 -6.88×10 -4 t (nm).
Although slight differences in the band gap energy values can be noticed when making a comparison between the results obtained from applying Tauc 'plots and Cody's representations, it can neglect these differences if we take the measurement error range into account. Thus, it can conclude that these differences are not really important, so that use it. Generally, either model aims to compute the energy band gap for inorganic and organic semiconductors.

Davis-Mott model
In the amorphous materials the density of hypothetical states is an unexpected equivalent, although it may be valid for states beyond Ec and Ev according to Davis and Mott model [76]. The density of state distributions may contribute to explaining the existence of three types of possible optical transitions according to the Mott-CFO model [77], which may contribute to the inter-band absorption.
The matrix elements for these transitions are related to the spatial overlap between initial and final state wavefunctions. Davis and Mott showed that the matrix elements for transitions between extended states and those between weakly localized states. Accordingly, the absorption coefficient ( ) is defined by the following relationship [77,78]: Here ( films. The estimated values that deduced from these figures have been recorded in Table 3 for the direct and indirect gap energies.  absolutely. Therefore, it can be concluded that these four methods used in determining the energy gap of semiconductors are all considered accurate and reliable and the differences between them are practically acceptable.

Conclusions
Novel nanocrystalline ZnGa2S4 thin films were fabricated with good quality using an inexpensive spray pyrolysis technique at different film thickness. XRD analysis showed that all samples are singlephase and have the polycrystalline quadrangular crystal structure. The degree of crystallinity and the volume fraction have been studied via x-ray diffractograms and the findings were that these parameters directly dependent on the thickness of the films.
The energy-dispersive x-ray spectroscopy, EDAX is employed to investigate the elemental compositions of films and the result confirmed that all films have a good stoichiometry, along with in a good match with the theoretical computations. The field-emission-scanning electron microscope, FE-SEM was used to examine the morphology of the films' surfaces.
The optical transmittance and reflectance have been employed to investigate some important optical parameters such as absorption coefficient, Urbach energy, steepness parameters, and electronphonon interaction have been extensively studied and discussed.
The direct and indirect gap energy were also determined by different four models and compared with Tauc's model. The optical band gap values were slightly decreased if the film thickness increased, this is owing to the improvement of the crystallization process of films. The optical band gap energy of these films ranges from 4.106 eV to 3.754 eV. This means than that these film samples are wide band gap semiconductors, which makes them strongly recommended for use in many optical applications as in thin-film solar cells and as a window layer.

Funding
The authors declare that this work does not receive any funding or financial support from any institution, but that they are who have borne all the expenses