The evolution of crystallite size D, microstrain ε, and dislocation density δ in FeS₂ films as a function of deposition time are given in Fig. 2. The most intense and well-resolved peak in the diffraction patterns for each sample was used for crystallite size, microstrain, and dislocation density calculations. These parameters can be understood in terms of classical XRD line-broadening analysis (Scherrer formula) and defect-density scaling.

The crystallite size (D) is estimated by using the well-known Scherrer’s formula [18]. (1) D = k λ β cos θ where K is the Scherrer constant (denotes the shape of the particle and its value is most commonly taken as 0.9), λ is wave length of the X-ray beam used (1.54184 Å), β is the Full width at half maximum (FWHM) of the peak and θ is the Bragg angle.

According to the Fig. 2, it can be noticed that D rises monotonically from ≈ 8 nm at 15 min to ≈ 38 nm at 40 min. Longer ultrasonic-spray durations deposit more material and allow grain–boundary migration under 350°C heat, promoting coalescence of nanocrystallites into larger coherent domains. This trend is habitually observed in spray-pyrolysis [19] and other vapor–deposition techniques.

Despite the enlargement of the crystallite size with deposition time elevation, their values remain in the nano-crystalline order and these small sizes of the grains have many advantages such as: the increase of exchange surfaces and reactivity.

Using the grain size values, the strain ε is assessed using the following relation [18]:

From the other side, the lattice strain extracted via Eq. (2), drops from ~0.08 at 15 min to ~0.03 at 40 min which is in good correlation with crystallite size variation. Small crystallites accommodate a high density of lattice defects (dislocations, surface stress), yielding larger peak broadening from microstrain. As grains grow, many of these defects annihilate at grain boundaries or reorganize into lower-energy configurations, so the average internal strain falls [17,20].

The dislocation density δ_(dis) developed of FeS₂ thin films were evaluated by using the Eq. (3) [18].

The calculated δ_(dis) decreases from ≈3.0 × 10⁻² nm⁻² at 15 min down to ≈6.9 × 10⁻⁴ nm⁻² at 40 min. This inverse-square relation is a standard approximation linking coherent-domain size to average dislocation density in polycrystalline films. Physically, as grains enlarge, the number of dislocations per unit volume drops dramatically because many dislocations either exit the grain or form low-energy networks at boundaries [16].

The optical transmission T(λ) of iron disulfide (FeS₂) thin films deposited on glass substrates via ultrasonic spray pyrolysis at a substrate temperature of 350°C are shown in Fig. 3.

It is well known that, in general films’ transmittance increases with the reduction of film thickness [19], which is the case of transmittance variation in FeS₂ layers. From Fig. 3, it is notable that the transmittance starts high at a shorter deposition time than decreases gradually with deposition time elevation, reaching almost 5% for 40 min, with oscillatory fringes. These fringes arise from interference effects in thin films, indicating uniform thickness and good surface quality.

Moreover, the inverse relationship between the transmittance and time is due to thicker films from prolonged deposition, which scatter and absorb more light. Comparable trends are observed in spray-pyrolyzed FeS₂ films, where increased thickness (correlated with deposition parameters like time or temperature) reduces transmittance in the visible region [21].

The absorbance was calculated from the measured transmission using: (4) A = − log 10 T 100 and the absorption coefficient was estimated using the experimentally measured film thickness according to: (5) α = 2.303 A t where: T is the transmission and t is the film thickness.

Around 725 nm, the 40 min film exhibits very low transmission (~4.8%), corresponding to a high absorbance (A ≈ 1.32) and a large absorption coefficient of approximately 1.3 × 10⁵ cm⁻¹.

In contrast, the other samples show significantly higher transmission and much lower α values (~10³–10⁴ cm⁻¹). Consequently, the condition αt ≫ 1, which leads to a square-like suppression of transmission, is fulfilled only for specific samples and spectral regions. This explains why the pronounced transmission drop is observed only in certain films and highlights the combined role of absorption strength and film thickness in governing the optical response. The calculated absorption coefficient values at representative wavelengths are summarized in Table 1.

As the deposition time increases from 20 to 40 min, a clear enhancement of optical absorption is observed, particularly in the long-wavelength region (500–800 nm). Pyrite FeS₂ is known to exhibit very high absorption coefficients in the visible range, with reported values on the order of 10⁴–10⁵ cm⁻¹ [22]. Such high absorption implies that even relatively thin films can absorb a large fraction of incident visible light, while thicker films obtained at longer deposition times become effectively non-transmitting. This behavior explains why the 40 min film shows the highest absorption response, whereas the 20-min film exhibits the lowest. In addition, the increase in deposition time leads to thicker films, which further enhances absorption through an increased optical path length [23].

Using the optical data, the envelope method is employed in order to calculate FeS₂ thin films thickness (t) from two minima or maxima using the expression (6) [24]. (6) t = M λ 1 λ 2 2 λ 2 n 1 − λ 1 n 2 where M is the number of the oscillations between two extrema (M = 1 between the two consecutive minima or maxima), λ₁, λ₂, n₁ and n₂ are the corresponding wavelengths and indices of refraction, respectively.

The optical band gaps (Eg) of FeS₂ thin films has been determined on the basis of UV-VIS-NIR spectrophotometer by plotting (αhν)² as a function of hν, which are linear over a wide range of photon energies indicating the direct type of transitions.

The optical band gap is obtained from the plot of the following relation [9]: (αhν)² = A (hν − Eg)(7) where α is the absorption coefficient, A is a constant, Eg is the optical band gap, ν is the frequency of the incident photon and h is the Planck’s constant. The band gap energies of these films are determined from the intercept of the tangent to the plot with the abscissa axis as indicated in Fig. 4.

The obtained thickness and optical band gap values of FeS₂ films deposited at different durations are summarized in Table 2. These values are in agreement with the reported values in the literature [6].

Table 2 illustrates the dependence of film thickness (t) and optical band gap (Eg) on deposition time for FeS₂ thin films prepared by ultrasonic spray pyrolysis at 350°C on glass substrates. The film thickness increases with deposition time, starting at 0.23 μm for 20 min and rising to 1.53 μm for 40 min, this increase is approximately linear. This correlation between longer time deposition and film thickness elevation is a standard outcome in time-controlled deposition techniques, where thickness scales with exposure duration until saturation or peeling effects occur. In spray pyrolysis [16,19], longer deposition times allow for greater accumulation of atomized precursor material (Fe and S sources) on the heated substrate, leading to thicker films.

Besides, the band gap decreases monotonically with increasing deposition time, from 2.34 eV at 20 min to 1.60 eV at 40 min. The high optical gap (2.34 eV) at 20 min is mainly due to iron oxide (Fe₂O₃) and amorphous intermediate phases, as shown by XRD. These phases exhibit wider band gaps than pyrite FeS₂, directly linking the structural and optical observations, in agreement with reported Fe–S–O thin-film studies [24]. As the deposition time rises (thickness rises) the films become more continuous, better sulfurized and more dominated by the pyrite phase as it have been noticed from the DRX and optical results, these characteristics make the optical gap moves toward the intrinsic pyrite value (which is lower). Several spray-deposition studies [25,26] of Fe–S films report high apparent Eg (>1.5–2.0 eV) for short/poorly sulfided films and lower Eg for thicker, better-crystallized pyrite films.

These obtained results confirms that due to absorption coefficients exceeding 10⁴–10⁵ cm⁻¹, FeS₂ requires only tens to a few hundreds of nanometers to absorb most incident visible light. This point is emphasized to reinforce FeS₂’s suitability as an ultra-thin absorber layer for photovoltaic devices, consistent with established literature [27].