Optical and fractal properties of sputter deposited TiO2 films

In this study, TiO2 films were deposited on the glass substrates at different pressures using a DC magnetron sputtering system. The surface topography, fractality, particle size, transparency spectra, and roughness of the deposited films were analyzed through atomic force microscopy in non-contact mode and spectrophotometer (UV–visible) analysis. The Gwyddion open source software was used to analyze the AFM results. The surface morphology, absorption spectra, transparency, and energy band gap of the films were studied through analytical and mathematical relations, including fractal and multifractal dimensions of the films. The pressure influenced the particle size leading to the changes in the surface roughness and fractality of the films. When the working pressure increased above 8 × 10−3 Torr, the nonuniformity in the distribution of surface properties and multifractality increased. Additionally, using optical data and the Tauc’s method, their energy gap was calculated. It was revealed that the TiO2 coatings have a relatively large energy gap of ~ 4 eV, probably due to their small grain size.


Introduction
Thin films have unique properties that are significantly different from their corresponding properties in the bulk state. This difference is often due to their physical dimensions (surface to volume ratio), geometric shape, and microstructure. However, a thin film with leveraged characteristics can be synthesized by modifying its properties to a great extent, which is the basis for developing thin film applications in various industries (Mirzaei et al. 2020a). Submicron thin films are widely used in Micro-Electro-Mechanical Systems (MEMS), microelectronic integrated circuits, and semiconductors, as well as optical, magnetic, and quantum interference devices. On the other hand, thin films with a large surfaceto-volume ratio are used in solar cells for photovoltaic conversion, shielding layers, inactivation layers, etc. Therefore, it is expected that the application of thin films will extend even further by developing many products in the form of the thin film in the near future.
Different methods exist for the mass and easy production of thin films and nanostructures. The differences between these methods are mainly in the mechanism of material transfer from source to the substrate, transfer medium, working pressure and deposition sensitivity, and simplicity of work (Das et al. 2021;Astinchap et al. 2020;Sadeghi et al. 2020;Ghobadi et al. 2019;Salehi et al. 2018;Shakoury et al. 2020;Yin et al. 2020). In recent years there has been a surge of interest in nanostructured films of metals and their oxides due to their unique mechanical, chemical, optical, and structural properties (Mirzaei et al. 2020b;Astinchap and Laelabadi 2019;Shakoury et al. 2021;Astinchap 2019;Ahmadpourian et al. 2014;Deymehkar et al. 2018;Liu et al. 2019;Fathinezhad et al. 2020). Meanwhile, titanium and its associated oxides have been reported to be environmentally friendly and cost-effective materials that exhibit good chemical and optical stability, high corrosion resistance in 3.5% NaCl solution, and insolubility in the water (Zare et al. 2019;Rezaee et al. 2020;Achour et al. 2017;Altaf et al. 2021). Titanium dioxide (TiO 2 ), or titania, shows n-type semiconductor properties at room temperature because of the formation of oxygen voids or titanium atoms within the lattice; each of those defects causes the formation of donor-type semiconductors (Blumenthal et al. 1967;Nowotny et al. 2007;Tan et al. 2021).
The TiO 2 exists in nature with three different crystal structures; anatase, rutile, and brookite, where anatase and rutile have a tetragonal structure while brookite has an orthorhombic structure (Korpi et al. 2017;Achour et al. 2018;Asadzadeh et al. 2021;Shan et al. 2020). Among the three natural mineral phases of titanium dioxide, the rutile is the most stable and dense phase, while the anatase phase is the least dense. The rutile structure is found naturally in volcanic rocks or rocks exposed to high temperatures and pressures. Rutile is more likely to be found in such environmental conditions due to its low molecular volume than the other two structures. Natural rutile contains about 10% iron, a large amount of niobium, and tantalum. In fact, anatase and brookite are polymorphs of rutile formed by the cooling of igneous rocks (Tao et al. 2021).
Additionally, metastable anatase and brookite phases can be formed at temperatures above 600 °C. The brookite phase is thermodynamically unstable, and it cannot be found independently, though it exists alongside other titanium dioxide phases. The two most commonly used titanium dioxide crystal structures are the rutile and anatase structures. In the rutile structure, the recombination rate of the charges created by light activity is much higher than that of anatase. This causes the low charge transfer of the rutile structure to the reactants and low efficiency, which is the reason for using this material in color compounds. In contrast, the anatase structure with low recombination, high charge transfer, and consequently, high efficiency can be used as a material with suitable optical activity (CarpO and Reller 2004). Therefore, the application and efficiency of TiO 2 depends on its crystal structure, as well as its particle size and shape Tao et al. 2021). Qiu et al. (2008) reported the enhanced photocatalytic efficiency and optical absorption of TiO 2 films through doping with ZrFe 2 O 4 by the sol-gel method. When studying the thermal stability of e-beam deposited TiO 2 coatings, it has been shown that the optical loss of coatings synthesized by e-beam deposition decreased with annealing temperature (Yao et al. 2009). Accordingly, great efforts are made to synthesize TiO 2 nanoparticles with controlled size, shape, and porosity to be used in thin films, ceramics, composites, and catalysts Gao et al. 2004;Kim et al. 2003;Zare et al. 2018;Zhang et al. 2021). Other applications of TiO 2 include a wide energy gap in all its crystalline forms, a good position of the conduction and valence bands, and chemical stability, which is widely used in photocatalytic and water decomposition applications. Since it has hydrophilic properties, this material can also be used to prepare self-cleaning surfaces (Das et al. 2021;Korpi et al. 2017;Zare et al. 2018;Hoseinzadeh et al. 2018).
Moreover, the ability of TiO 2 to absorb ultraviolet light has led to its application in cosmetics, especially sunscreens, to prevent skin damage. Owing to its unique optical and electrical properties, TiO 2 is used in the manufacturing of solar cells, chemical sensors, and optical coatings. It has also been used to extend medical applications such as artificial heart valves and dental implants (Hoseinzadeh et al. 2018;Astinchap et al. 2017). This substance causes the uniformity of bone cells between medical implants and bone. TiO 2 in solution or suspension can also break down proteins containing the amino acid proline in places where proline is present (Jones et al. 2007).
However, no studies have been performed on the effect of working pressure on the surface morphology of thin films by the multifractal method and its relationship with the optical energy gap. Therefore, the present study is primarily based on atomic force microscopy (AFM) measurements of magnetron sputtered TiO 2 films. We utilized AFM and spectrophotometer measurements to study the effect of inward gas flux changes on the structural, morphological, and optical properties of the TiO 2 films. Furthermore, optical data and the Tauc's method were used to calculate the energy gap of deposited films. Moreover, the effect of working pressure on the optical constants and the thickness of the TiO 2 films has been investigated.

Materials and methods
A DC magnetron sputtering system with a cylindrical glass chamber was used to deposit TiO 2 films. The chamber consisted of two electrodes facing each other, one at the bottom with a larger radius (location under the films) and the other at the top with a radius of 9 cm connected to the DC source. The distance from the target to the substrate was 8 cm. For deposition of TiO 2 films, the glass (BK7) substrates were used in this work. A schematic of the deposition chamber is shown in Fig. 1. Glass substrates were first cleaned in soap solution and then in an ultrasonic bath by immersing in acetone for 5 min. First, the chamber pressure was reduced to less than 10 −5 Torr by a rotary and turbo molecular pump. Then oxygen and argon gases were introduced into the chamber. A mass flow controller (MFC) was used to adjust the working pressure of the chamber. The ratio of oxygen to gas argon flow was kept constant during deposition (O 2 /Ar:1/4), but the flow of each gas changed so that the pressure was kept constant throughout the deposition time. The deposition time of all samples was set to 60 min. The film thickness was measured by monitoring a quartz crystal gauge, installed inside the chamber. A pure titanium target was sputtered by applying a constant power of 400 W in the medium of argon and oxygen to reactively deposit titanium oxide thin films. Detailed information on deposition parameters is listed in Table 1.
Crystallographic structure of the TiO 2 films was assessed by X-ray diffraction (XRD) analysis utilizing an STOE STADI MP diffractometer. This diffractometer was equipped with a Cu-Kα radiation source operated at wavelength of 1.54056 Å. In order to statistically study the surface topography and roughness of the deposited films, an atomic force microscope (NT-MDT model BL022) in non-contact mode was used. The measurements were conducted on the 5 × 5 µm 2 area of the films. Using the spectrophotometer (Hitachi, model U-3501), the transparency spectra of the films were measured in the range of 300-900 nm. Subsequently, using this data, the absorption spectra of the films were drawn, then the energy gap was calculated using the Tauc's method (Tauc 1970) as follows; When a light beam enters a matter, the intensity of the output light is expressed by the Beer-Lambert law, as follows; Fig. 1 The sketch of the deposition chamber where I is the intensity of the transmitted light, I 0 is the intensity of the incident light, d is the thickness of the layer, and α is the linear absorption coefficient of the material. From the above equation, the absorption coefficient can be calculated, which is used to obtain the energy gap in the Tauc's method.
The T is the transmittance which is defined as the fraction of transmitted intensity to incident intensity. The relationship between the absorption coefficient and the photon energy is expressed as follows: where E g is the energy gap of matter, h is the photon energy, and A and n are constant coefficients. It is n = 2 for indirect transitions and n = 0.5 for direct circuits. By plotting the variation curve n ( h ) in terms of h and fitting the straight line in the linear range up to ( h ) = 0, the energy gap value can be estimated. A well-known approach to obtaining optical constants (refractive index n and extinction coefficient k) and film thickness is Swanepoel's method (Swanepoel 1983). This method is based on the maximum and minimum points of the sample transmittance spectrum. Essential Macleod software (https:// thinfi lmce nter. com) first obtains an initial estimate of the optical constants and film thickness with Swanepoel's method, then improves the accuracy of the optical constants and thickness by comparing the measured transmittance and retrieved transmittance and repeating the calculations. We have used this software to obtain optical constants and film thickness.

Results and discussion
Figure 2 depicts the XRD pattern of deposited TiO 2 at different pressures. The substantial peaks from anastase and rutile phases of TiO 2 was inserted from JCPDS Card 21-1272 and JCPDS Card 21-1276, respectively. As can be seen, TiO 2 films exhibited a quasi-amorphous structure wherein, the decreasing of working pressure from 6 × 10 −3 to 1.2 × 10 −2 Torr had insignificant effect on the crystallinity of the coatings. Such broad peaks also can be observed where there is a short range atomic ordering. Therefore, the broad peak, centered at 26° and 60° can be assigned to the mixture of anastase/rutile phases, however, the explicit identification of the incorporated phases is impossible due to broad diffraction patterns.
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 TiO 2 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 TiO 2 layers. Figure 3 shows the AFM images of titanium dioxide films recorded at a 5 × 5 µm 2 scale. The AFM images show that the TiO 2 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 4 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 Torr, 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 varied between 1.59 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 Torr 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 (Sridhar et al. 2017).
The results of fractal analysis of the TiO 2 films are presented in Fig. 5. The multifractal singularity spectra f(α) (Fig. 5a) and generalized fractal dimension Dq (Fig. 5b) indicate well-developed multifractal properties of the h(x,y) function determined from the AFM images for all TiO 2 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 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 Fig. 5a). Therefore, comparing the multifractal spectra of deposited TiO 2

Fig. 2 XRD patterns of TiO 2 coatings deposited at different pressures
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. 5a, 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. 5b, an identical conclusion can be deduced from the trend of generalized fractal dimension Dq. Here, the slope of the Dq functions is associated with the fractality of the surface. Increasing the working pressure from 6 × 10 −3 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 (α max − α min ) and (Dq max − Dq min ), in which the minimum was assigned to film #2 (Fig. 5c), whereas, for the higher working pressures above 8 × 10 −3 Torr, an increase in the multifractality was observed. The bigger the value of (α max − α min ), the higher the h(x,y) Fig. 3 The AFM images of TiO 2 films a #1, b #2, c #3 and d #4 recorded at 5 × 5 µm 2 surface area 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. 6. 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  . 6 The results of Minkowski Functionals analysis obtained from AFM images of TiO 2 films S1, S2, S3, and S4: a Minkowski boundary (S); b Minkowski connectivity (χ) the high connection occurs for larger nanoparticles at the surface of samples #1 and #2 compared to samples #3 and #4.
In Fig. 7a, b, the transmission and absorption spectra of TiO 2 films were plotted in the wavelength region of 300-900 nm. Figure 7a also shows the transmission spectrum of uncoated glass. 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, Tauc's equation was used to determine the magnitude of the bandgap in TiO 2 films. This is possible by analyzing the data in Fig. 8, 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 energy band gap; thus, the better the semiconductor or non-conductive properties of the material. The results of this analysis are given in Fig. 8. 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 TiO 2 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, Fig. 7 Transmittance (a) and absorbance (b) spectra of TiO 2 thin films deposited at different pressures fractality, particles shape, and particles quantum size), the thickness of layers, and chemical composition of the material. Here, we aimed 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 and the small nanoparticle size (see particle size distribution diagram in Fig. 4).
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. 7b for all structures, which was closely related to Eg = 4 eV determined by the Tauc's method for the TiO 2 layers. The development of surface features shows significant sensitivity of multifractal behavior on working pressure (see Fig. 5c, 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. 7a) and absorbance (Fig. 7b) 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.
As mentioned in the previous section, the Essential Macleod software obtains the refractive index n, extinction coefficient k, and film thickness d with good accuracy. As an example, in Fig. 9, the measured transmittance spectrum of the #1 sample and the retrieved transmittance spectrum is plotted. The retrieved curve is obtained by the calculated values of n, k, and d. It can be seen that the fit between the measured curve and the retrieved curve is acceptable. Table 2 shows the refractive index, extinction coefficient, and film thickness for all four samples. The highest thickness corresponds to a pressure of 8 × 10 −3 Torr, and the lowest thickness corresponds to the pressure of 1.2 × 10 −2 Torr. Although the deposition time is the same for all samples (one hour), with increasing pressure, the thickness initially increases and then decreases. This seems reasonable because the deposition rate is highly dependent on the pressure in the sputtering process. At high pressures, target bombardment is more intense, however, a sputtered Titanium has a lower chance of reaching the substrate. On the other hand, at low pressures, the bombardment of the target is done by ions with less intensity. Due to the low pressure, a sputtered titanium is more likely to reach the substrate and experience fewer collisions in the path of the target to the substrate. As a result, there must be an optimal pressure at which the deposition rate is maximum. Moreover, the changes in the refractive index of the samples with pressure changes are interesting. The second sample, which is the thickest, has the largest refractive index. Furthermore, the sample that has a higher refractive index has a larger extinction coefficient k, as shown in Table 2. More deposition rate on the surface of the substrate means a higher ratio of titanium to oxygen on the surface, that in turn leads to an increase in the extinction coefficient. Fig. 9 Transmittance spectrum of #1 sample (measured) and retrieved to calculate n, k, and d

Conclusion
In this research, titanium dioxide films were deposited at different working pressures utilizing a magnetron sputtering system. The surface morphology and optical properties of TiO 2 films were investigated. According to the measurement of the crystal thickness gauge installed on the device, it was shown that the thickness of the TiO 2 films varied with the pressure where the optimum pressure of 8 × 10 −3 Torr was found for film #2 with the maximum thickness. Also, multifractal properties synergistically interacted with the development of the thickness of the films during the deposition. Investigation of AFM results revealed that particle size was changed with the pressure leading to the changes in the surface roughness and fractality of the films. Additionally, the nonuniformity in the distribution of surface properties and multifractality increased by raising working pressures.
The changes in the thickness of the films were also evident in the optical spectra, where film #2 exhibited the lowest transmission while films #1 and #4 showed the highest transmission values. Also, it was found that by increasing the working pressure, absorption edges shift to higher wavelengths and possess high values for wavelengths greater than 350 nm. The energy gap and minimum and maximum transparency of the TiO 2 layers were less sensitive to changes in the working pressure. Using the Tauc's method, the energy gap of the films was calculated to be ~ 4 eV. Such a large energy gap can be attributed to the small grain size of TiO 2 coatings. Eventually, the obtained results from the multifractal analysis showed that at the working pressure of 8.0 × 10 −3 Torr, the layers have the lowest ∆α and, therefore, the least complexity, as well as the lowest energy band gap.