The surface features strongly influence the optical transmission and absorption processes and explain their behavior in the wide interval of wavelengths. On the other hand, the TiO2 energy gap value substantially affects the optical processes for higher energies of photons (for wavelengths below 200 nm). The combination of multifractal analysis, as well as optical transmission and absorption processes, provides valuable information to explain the importance of optical processes in the growth of TiO2 layers.
Figure 2 shows the AFM images of titanium dioxide films recorded at a 5×5 µm2 scale. The AFM images show that the TiO2 nanoparticles were formed, and the growth of the layers followed the island growth model (Volmer–Weber). It can be seen that by increasing the input flux, the particle size distribution was changed, leading to changes in the surface roughness. Figure 3 shows the particle size distribution diagram. We know that as the working pressure increases, the density of the plasma on the substrate surface increases, which in turn increases the rate of nucleation and the rate of growth of the nuclei. At the same deposition time, increasing the nucleation rate resulted in an increase in the thickness of the layers, which was confirmed by the thickness measurement results reported by the quartz crystal. When the working pressure was 6×10− 3 Torr, the average particle size was measured to be 52 nm while with increasing the working pressure to 8×10− 3, and then 1.0×10− 2, and finally 1.2×10− 2, the particle size was increased to 58, 60, and 62 nm, respectively. This indicates that the increase in the growth rate of the nuclei leads to a slight increase in the particle size formed in the plasma and accumulated on the surface, which was confirmed by the particle size distribution diagrams obtained from the images.
The roughness values were varied between 1.59 nm and 1.73 nm; as for the lowest pressure, the average roughness was 1.7 nm, while it decreased to 1.65 and then to 1.58 nm with increasing the working pressure. On the other hand, further increasing the working pressure to 1.2×10− 2 led to an increase in the roughness to 1.73 nm. Therefore, no clear relation between the roughness and working pressure was found. However, surface parameters such as maximum height and the total height of roughness profile as well as particle distribution of surface particles might lead to such a result .
The results of fractal analysis of the TiO2 films are presented in Fig. 4. The multifractal singularity spectra f(α) (Fig. 4(a)) and generalized fractal dimension Dq (Fig. 4(b)) indicate well developed multifractal properties of the h(x,y) function determined from the AFM images for all TiO2 films. The profile of the f(α) function indicates a decreasing trend in the surface fractality (over) with increasing the working pressure from 6×10− 3 Torr to 8×10− 3 Torr (film #1 and #2). On the other hand, the surface fractality was increased dramatically by further increasing the working pressure from 8×10− 3 to 1.2×10− 2 Torr (films #2 - #4). The width of the f(α) function corresponds to a broader range of surface features (see. Figure 4(a)). Therefore, comparing the multifractal spectra of deposited TiO2 films reveals a broadening of f(α) with the working pressure above 8×10− 3 Torr, which can be attributed to the increasing non-uniformity in the distribution of surface features at higher pressures.
As shown in Fig. 4(a), for the working pressure of 6.0×10− 3 Torr, the left arm of the curve is longer, which indicates that larger nanoparticles were identified on the surface of this layer. However, as the working pressure increased, the curves changed, and the right arm of the curves became taller, indicating that smaller nanoparticles outperformed larger nanoparticles in determining the surface behavior of the layers. As shown in Fig. 4(b), an identical conclusion can be deduced from the trend of generalized fractal dimension Dq. Here, the slope of the Dq function is associated with the fractality of the surface. Increasing the working pressure from 6×10− 3 Torr to 8×10− 3 Torr (films #1 and #2) led to a slight decreasing fractality, while a further increase in the pressure resulted in a significant increase in multifractality (films #2 - #4). These results are in good agreement with the values of (αmax-αmin) and (Dqmax - Dqmin) in which the minimum was assigned to film #2 (Fig. 4(c)), whereas, for the higher working pressures above 8×10− 3 Torr, an increase in the multifractality was observed. The bigger the value of (αmax- αmin), the higher the h(x,y) fluctuations are expected. The h(x,y) fluctuations detected by the multifractal singularity spectra were also the smallest for film #2.
Another technique that was used further to describe the surface morphology of the deposited layers was Minkowski Functionals analysis. Minkowski boundary and Minkowski connectivity curves for all samples were calculated using the Gwyddion software. The curves obtained from the analysis of 4 samples are illustrated in Fig. 5. The curves show that the Minkowski boundary and Minkowski connectivity are functions of working pressure. It was observed that the maximum boundary was obtained for sample 2. These results indicate the nanoparticles were placed on the surface so that they have the most boundaries. It is known that the Minkowski connectivity describes the measure of the number of connections in the nanoparticles pattern by analyzing the relationship between connected and disconnected pixels in an image. Also, as can be seen, the minimum and maximum of the Minkowski connectivity curve for sample #2 are higher than others. Therefore, the connection between nanoparticles in sample #2 was dominant, and the high connection occurs for larger nanoparticles at the surface of samples #1 and #2 compared to samples #3 and #4.
Figure 6. depicts the transmission and absorption spectra of the TiO2 films in the visible region, derived from the dominant relationships between the optical spectra (transmission, absorption, and reflection). The films exhibited high absorption edges for wavelengths above 350 nm, which can be confirmed by a shift in their optical spectra (absorption edges) to higher wavelengths by increasing the working pressure. The minimum and maximum transparency in the films were almost constant, about 50% and 90%, respectively. It is notable that the optical behavior of #1 and #4 films, which were prepared at the lowest and highest pressures, respectively, were almost identical, while there was a slight phase shift in the wavelength of films #2 and #3. Therefore, it can be concluded that they exhibited similar transmittance and absorbance. This is in accordance with the changes in the surface roughness obtained from the AFM data.
In a semiconductor material, the dependence of the absorption of the UV-visible radiation can be determined by the electronic transitions between the valence band and the conduction band. Therefore, the optical properties of semiconductors can be modified by controlling the band gap energy. On the other hand, the band gap energy in thin films strongly depends on the surface roughness and correspondingly on the fractal features of the coating.
As discussed above, the Tauc ́s equation was used to determine the magnitude of the bandgap in TiO2 films. This is possible by analyzing the data in Fig. 6, n (ahʋ) in terms of hʋ, and extrapolating the values obtained in the high energy region with the horizontal axis. The closer the value of this parameter to zero, the higher the conductivity; thus, the better the semiconductor or non-conductive properties of the material. The results of this analysis are given in Fig. 7. The energy gap of layers was found to be 3.98, 3.95, 4.03, and 4.00 eV for samples #1, #2, #3, and #4, respectively. It is notable that the energy gap of all TiO2 films was in the vicinity of 4.00 eV, while the slight variation can be attributed to the changes in the working pressure. The results show that the lowest energy gap is related to the layer with the lowest ∆α, indicating that the surface complexity affected the energy gap.
As we know, several factors can influence the energy gap of semiconductors. For instance, the energy gap value strongly depends on the crystal structure (such as defects, charged impurities, disorder at the grain boundaries), morphology (such as roughness, fractality, particles shape, and particles quantum size), the thickness of layers, and chemical composition of the material. Here, our aim was to study the effect of the surface morphology of the films on their energy gap. The results showed that increasing the multifractality of the films increased their energy gap. The high energy gap of the deposited layers can be ascribed to the multifractal properties of their surface, the small nanoparticle size (see particle size distribution diagram in Fig. 3).
The energy gap determined by the Tauc ́s method is not very sensitive to changes in working pressure, as the Eg value is influenced mainly by material properties. The high absorption of wavelengths under 200 nm can be observed in Fig. 6(b) for all structures, which was closely related to Eg = 4 eV determined by the Tauc ́s method for the TiO2 layers. The development of surface features shows significant sensitivity of multifractal behavior on working pressure (see Fig. 4(c and d). The modification of surface features by applied forming steps (revealed by multifractal methods) directly influenced the spectral transmission and absorption processes. The shifts of spectral transmittance (Fig. 6(a)) and absorbance (Fig. 6(b)) curves directly correlated with the trends of multifractal parameters determined from the AFM h(x,y) functions. The multifractal analysis provides valuable information about the surface properties and can directly explain the spectral absorption and transmission processes.