Optical Properties and Analysis of OJL Model’s Electronic inter-band Transition Parameters of TiO2 Films

Titanium dioxide is a wide band gap semiconductor responsible for the bright white appearance in most substances. This material has many unique properties due to its extra-ordinary chemical stability. TiO2 has a conduction band that closely matches the excited energy level of organic dyes hence it is used in fabrication of photo-anode electrode of dye sensitized solar cell. However, the optical properties and the density of states of TiO2 thin films determine the performance of dye sensitized solar cell fabricated from TiO2 photo-anode electrode. For this reason, the purpose of this study was to investigate the optical properties and the OJL electronic inter-band transition analysis of TiO2 nanoparticle thin films. Under the OJL model, the expressions of density of states were specified for the optical transition from the valence band to the conduction band. The TiO2 nanoparticles were prepared using sol-gel and hydrothermal methods and deposited on a conductive glass substrate by screen printing and spray pyrolysis techniques. SEM analysis revealed that TiO2 nanoparticles were spongy and had unevenly sphere-shaped profile while TiO2 nanotubes had a skein-like morphology with abundant number of nanotubes intertwined together. This study showed that TiO2 thin films have both direct and indirect band-gaps. The OJL Gap energy (E0) values were observed to be between 30273.2356 and 31072.0000 wavenumbers which translated to band-gap energies between 3.744 and 3.843 eV. From these findings showed that TiO2 films prepared could be used in the fabrication of high performing dye-sensitized solar cell.


INTRODUCTION
TiO2 is a semiconductor material that is cheap and easily available [1]. TiO2 is widely used in pigments and paints. Since it is non-toxic, it is used in food production industries [2], cosmetics and hygienic products [3]. TiO2 occurs in three main crystalline forms namely anatase, rutile and brookite [4]. For DSSCs fabrication, the anatase crystalline structure is the most suitable [5]. The usual TiO2 nanoparticle dimension to be used as DSSC's photo-anode electrode film is 10-30 nm [6].
Several deposition techniques have been used to prepare TiO2 films which include screen printing [7], sol-gel [8], sputtering [9], chemical vapor deposition [10] and electrophoretic [11]. Sol-gel technique is the utmost popular since it's easy in the preparation of the nanoxide films. In addition, TiO2 powders prepared though sol-gel have high catalytic activity owing to their large surface area, fine structure and high porosity [12].
The optical properties and the density of states of TiO2 thin films, among other factors, determine the performance of dye sensitized solar cell fabricated from TiO2 photo-anode electrode. On this basis, the main aim of this study was to report on the optical properties, OJL model parameters and SEM analysis of TiO2 thin films synthesized using sol-gel and hydrothermal methods. The fabricated thin films were optically characterized from which the OJL parameters were determined from transmittance data obtained. SEM analysis was done so as to confirm their morphology.

Optical transmission and reflectance
Photons of selected wavelengths and beam intensity Io (photons/cm 2 ) are directed at the film.
Those with energies greater than band gap (Eg) are absorbed while those with energies less than Eg are transmitted. This is because photons with energy less than band gap do not excite electrons in the valence band to higher states, hence transmitted. The comparison between intensities of incident (Io) and transmitted (It) photons is given by transmittance. Transmittance is given by Equation 1: When light is passed across a material, it's expected that it does not get scattered. However, internal reflections of the light take place on the medium surface. For this reason, sampling of gases, liquids and solids is performed by transmission spectroscopy where the ratio of the transmitted light against the incident light is considered. This ratio is dependent on several factors like light path-length (thickness of the sample), the sample's reflectivity, the sample's absorption coefficient, the orientation of the particles in the sample, sizes of the particles in the sample and the light's incidence angle.
Reflectance is the percentage measure of ratio of intensity of incident light to that of reflected light. In the spectrophotometer, incident light of known wavelength and intensity is shone onto the surface of thin film and the intensity of reflected light measured. The two intensities are compared by reference called reflectance (R) as given in Equation 2: (2) where IR and Io are intensities of the reflected and incident beams respectively.

The OJL model
OJL inter band model has been used to describe inter band transition in crystalline materials. It provides the equations for the density of states for the optical transition from the valence band to conduction band. Parabolic bands are assumed with tail states exponentially decaying into the band gap. The original parameters of OJL density of states model are energy, Ev and Ec, and the 'damping constants' of the conduction and valence bands, γc and γv. In addition, the conduction band mass, mc and the valence band mass, mv are included (Equations 3, 4): Its represent the mobility edges of conduction and the valence bands.
The band gap, Eo, is consequently specified by [14]: The band gap energy (Eo) is calculated by scout programme which converts the wave numbers to energy (eV). The other OJL electronic inter-band transition parameters are the OJL decay, OJL Mass, the OJL Gamma valence band, and the OJL Ratio.

Experimental procedure
The TiO2 organic solution for TiO2 nanoparticle films was synthesized at room temperature using sol-gel method where 1 ml of 2-propanol (Aldrich anhydrous 99.9 %) was mixed with 3.7 ml titanium (IV) isopropoxide (Aldrich 99.9 %) in volume. 2-propanol solution was added dropwise for over 30 minutes into a mixture of 25 ml of water and 8 ml glacial acetic acid (Aldrich 99.9%) and stirred vigorously for 1 hour. The mixture was there-after heated to 80 °C, stirred magnetically for 8 hours and then allowed to cool. Commercial TiO2 nanotube powder was purchased from Sigma Aldrich and dissolved in ethanol to form a paste.
TiO2 thin films were coated on FTO glass by screen printing and spray pyrolysis methods and dried in an oven at 80 °C for 10 minutes. Some TiO2 nanotubes films were treated with titanium tetrachloride. The films were sintered in a furnace at 450 °C for 30 minutes. After annealing TiO2 films at 450 °C, they were mounted on a holder inserted in the scanning electron microscope (Carl Zeiss Sigma VP FE-SEM). The voltage was set at 6 kV and the working distance was set at 5.1 mm for the nanoparticle films and 10 kV and 3 mm for the nanotube film.
Shimadzu 3700 DUV UV-VIS-NIR Spectrophotometer was used to determine the optical characteristics of thin films. Transmittance and reflectance data at 200 nm to 1200 nm wavelength was collected. Drude model was used in optical constants analysis where the adjusted parameters included plasma frequency, damping constant and charge carrier density. For inter-band transitions, the OJL model as decay, gap energy and gamma were the varied parameters. Harmonic Oscillator was used to obtain the plasma frequency [18]. Transmittance measurements were fixed to the simulated spectra where film thickness, absorption coefficients and energy band gaps of thin films were obtained. The film thicknesses obtained from the scout software were compared to the thicknesses determined by the KLA-Tencor Alpha-Step IQ Surface Profilometer.

RESULTS AND DISCUSSION
SEM analysis. The micrograph of TiO2 compact layer is represented in Figure 1. It was noted that the film had a relatively uniform thickness owing to the deposition method used. The film was relatively smooth because of the slight movement of the glass substrate during the spray pyrolysis deposition process, which was caused by the pressure of the spray. It was also noted that the TiO2 compact layer coated fully the FTO glass which was desirable in the fabrication of high efficient dye-sensitized solar cell. Figure 2 shows the micrograph of TiO2 nanoparticle film. From this figure, the nanoparticles were spongy and had unevenly sphere-shaped profile, which is an important factor to high performing DSSC fabricated out of these films. Due to the screen printing method of film deposition, was little film uniformity which is evident due to dark and bright areas. Figure 3 shows that TiO2 nanotubes had a skein-like morphology with abundant number of nanotubes intertwined together such that their surface area is really large.

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Electronic copy available at: https://ssrn.com/abstract=3310510 These skein-like nanotubes were produced due to the heat treatment of the precursor solutions at 110 °C for 20 hours.
Transmission and reflectance characteristics of TiO 2 thin films. Figure 4 represents transmission characteristics of the prepared TiO2 films.

Figure 4 -Transmission characteristics of TiO2 films prepared
This figure shows that treating titanium dioxide nanotubes with TiCl4 slightly improves the transmission properties. This was attributed to the nanotube diameter since the treated nanotube has smaller diameter than that of untreated nanotube which allows more light to be transmitted in the treated nanotube than in the untreated nanotube. Nanotube diameter is determined by the water content in the TiO2 nanotubes and treating them with TiCl4 ensures that most of the water content is used up in the hydrolysis reaction that takes place. This minimizes the nanotube diameter consequently increasing the fraction of light transmitted [19].
In addition, introducing TiO2 compact layer to both screen printed TiO2 nanoparticle and TiO2 nanotube films (whether treated or untreated) was found to reduce the transmittance as shown in Figure 4. Compact layer increased the film thickness which resulted to a larger fraction of light not being transmitted. As film thickness increases, the transmittance decreases because there are excess atoms which are introduced in the structure of the film. Increase in the number of atoms causes darkening of the interior part of the film and therefore lesser portion of light photons can pass through the thick film as compared to the thin film [20]. Figure 5 shows the reflectivity characteristics of the prepared TiO2 thin films where it was observed that all the films have low reflectance (less than 15% in the visible region on average). Moreover, it was observed that there were several peaks in most of the curves which were attributed to the multiple reflection effect the three interfaces of the air/thin film/transparent substrate bi-layer [18]. Multiple reflection happens because the overall reflectance is described by taking into account the product of the reflection co-efficient multiplied by the complex conjugate of the reflection. Figure 5 also shows that at low wavelengths (not exceeding 250 nm), there was notably high reflectance than at all other wavelength marks which was attributed to high absorption at these wavelengths. This therefore meant that these thin films exhibited high absorption behavior which makes the material a good absorber material for solar cell applications.

Figure 5 -Spectral plot of reflectance characteristics of TiO 2 thin films
Film thickness determination and comparison. The experimental data was fitted in to the simulated data to obtain the film thickness. At the point when the simulated curves were in good agreement with the experimental curves, it was possible to obtain the film thickness directly from the fit parameters section in the Scout software. In this study, the experimental value of the film thickness was also compared with the film thickness value obtained from the simulation using Scout software models. Figure 6 shows the plots for the transmission characteristics and the simulated graphs against wavelength for all the titanium dioxide films fabricated.

Figure 6 -Combined fitting of simulated versus experimental transmittance data
Electronic copy available at: https://ssrn.com/abstract=3310510 There was very good agreement between the simulated and experimental graphs and hence film thickness values tabulated in Table 1 were got. Table 1 includes the film thicknesses ob-tained by using Scout software which have been compared (with their percentage error) from the film thickness obtained from the KLA-Tencor Alpha-Step IQ Surface Profilometer. Absorption coefficient. Figure 7 shows the plots of absorption coefficient against the incident radiation energy. At low energies (high wavelengths) TiO2 films had very low absorption coefficients and at high light energies, there was high absorption coefficients of these films.

Figure 7 -Plots for absorption coefficient against energy
This meant that the films were translucent to light at wavelengths not exceeding 370 nm because the energy possessed by the light had not exceeded the band gap energy of TiO2 and therefore electrons could not be excited from the valence band to the conduction band in the TiO2 band structure. Moreover, this physical phenomenon was attributed to the light absorption at the grain boundaries [21].
Optical band gap. TiO2 has been known to be a semiconductor with has got both indirect and direct band gaps [22]. The direct optical band gap for the TiO2 films fabricated was obtained by plotting (αhν) 2 against energy of the incident light and extrapolating (αhν) 2 to zero. In determination of direct band gap, direct transitions were assumed and therefore Equation (6) was used [22]. Figure 8 shows that TiO2 films had a direct band gap energy of 3.81 eV which agreed closely to literature [23]. However, the value reported for anatase TiO2 band gap energy is 3.39 eV [24] meaning that the band gap energy value obtained had a red-shift of 0.42 eV. This was attributed to the quantum-size (Q-size) effect on the TiO2. As the size of the TiO2 particles became small, the valence band was lowered and the conduction band was raised thereby widening the band gap. The TiO2 band gap blue-shift was in the acceptable range to the reported range of 0.1 to 0.6 eV [25]. Figure 9 represents a plot of (αhʋ) 1/2 against hʋ for the determination of the indirect band gap which was found to be 3.64 eV. This band gap energy value was very close to the reported value in literature [24].

OJL's model parameters.
The OJL Mass was taken as the square root of the third power of the valence and conduction band masses. This is a pa-rameter that acts as a scaling factor in the determination of the shape of the density of states and therefore small values of the OJL mass are the best. The smaller the OJL mass of electrons, the higher the electron mobility [26]. Table 2 shows that the higher the values of the OJL mass, the higher the values of OJL Gap energy (EO) which agreed well with literature [27]. The possible reason for this correlation was thought to be due to the decay function in the OJL model which minimized the spectral weight at higher energies in order to make the integral over the whole spectral range finite.
The difference between the energy at the conduction band and the energy at the valence band was taken to give the Gap energy (EO). The Gap energy was got when the damping constants of the conduction band (gC) and valence band (gV) are zero [16]. The OJL Gap energy (E0) values were obtained to be between 30273.2356 and 31072.0000 wavenumbers as seen in Table 2. These values translated to band-gap energies between 3.744 and 3.843 eV which were close to value reported in literature [27]. The Gamma valence (represented by gV) shows the extent to which damping at the valence band occurs when one assumes the parabolic bands for the crystalline material. In this case, the tail exponentially decays into the band gap [16]. From table 2, it is important to note that gV values' range is restricted since too small values would lead to difficulties in the dielectric function calculation.
The OJL Ratio is the ratio of gV to gC. This parameter is a constant as it shows the extent of damping for both the valence and conduction band exponential decays. As shown in Table 2, the Gamma valence band (gV) values showed that damping at the valence band took place at a rate between 490.1338 and 1587.8186. In this case, the TiO2 samples were taken to have parabolic bands in their structure. These values have a good range which is required in the dielectric function calculation.
The dielectric function usually has two parts; the imaginary and the real part. The density of states (DOS) of the CB and VB is proportional to the dielectric function's imaginary part. In cases where modification lacks, dielectric function's imaginary part increases to infinity as the energy increases. For this reason, the dielectric function's real part construction by the Fourier transforms becomes difficult and this demands a decay factor to be included to drag down the imaginary part to zero for high frequencies. The OJL Decay is a number that governs the way the imaginary part decays to zero for high frequencies [14].
From Table 2, the OJL decay number for the six TiO2 samples was found to be high enough (over 25000 1/cm) which indicated that the imaginary part of the dielectric function decayed to zero at high frequencies well. This is necessary for the construction of the real part of the dielectric function.
The ratios of the exponential decay of the valence band to conduction band are as shown under the OJL ratio column in Table 2. Ratios higher than 1 indicate that conduction band decay occurred at a faster rate than the valence band decay while ratios less than 1 show that the valence band decay was faster than the conduction band decay.

CONCLUSION
TiO2 thin films were successfully synthesized using sol-gel and hydrothermal methods. The SEM micrographs confirmed that spongy nanoparticles and skein-like nanotubes were formed, which is a good attribute of a film to be used in DSSC fabrication. The optical characterization done on the TiO2 thin films showed the effect of introducing compact layer and post-treatment of TiO2 nanotubes with TiCl4 to transmittance. In addition, this analysis showed that TiO2 thin films had both direct and indirect band-gaps. The OJL Gap energy (E0) values were obtained to be between 30273.2356 and 31072.0000 wavenumbers. Moreover, the Gamma valence band (gV) values showed that damping at the valence band took place at a rate between 490.1338 and 1587.8186.