*3.2. Optical Measurements*

Spectrophotometry was employed to determine the optical constants and film porosity. The corresponding transmittances at 500 nm were measured to be 74% and 41%, respectively, for the as-deposited films, with the lower value for the films prepared at low *P*O2, which appear dark, almost black (Figure 1a, inset). After calcination the reflectance increased to about 70%–80% for all films, and they all became transparent with a slight visible tint due to light interference.

The optical constants and thicknesses of the two sets were determined using the envelope method suggested by Swanepoel [36]. The maxima and minima of the interference fringes at the transmittance spectrum were fitted with a set of spline functions, enveloping the spectrum, as depicted in Figure 4. The refractive index was then calculated using Equation (4).

$$m = \left[ N + \left( N^2 - s^2 \right)^{\bigvee\_2} \right]^{\bigvee\_2} \tag{4}$$

where *s* is the refractive index of the substrate and *N* is defined as

$$N = 2s\frac{T\_M - T\_m}{T\_M T\_m} + \frac{s^2 + 1}{2} \tag{5}$$

where *TM* and *Tm* are the maximum and minimum of the transmittance at a given wavelength. The refractive index of the substrate was calculated using

$$\mathbf{L} = \frac{1}{T} + \left(\frac{1}{T} - 1\right)^{\frac{1}{2}}\tag{6}$$

where *T* is the transmittance measured at a given wavelength. For our glass substrates *s* was determined to be *s* = 1.39. The refractive indices were determined from the averaged refractive index over all visible maxima and minima, expect the ones near the bandgap, where the transmittance starts to decrease, and the error becomes larger. The average refractive indices for films sputtered at *P*O2 = 0.65 and *P*O2 = 1.3 mTorr obtained in this manner were determined to be *n* = 2.23 and *n* = 2.04, respectively, and did not vary much over the wavelength region depicted in Figure 4. Hence, the reported values of *n* were determined as an average of the positions marked with a dashed line in Figure 4.

**Figure 4.** Transmittance spectra for a (**a**) non-oriented film, and (**b**) <001> oriented film. Positions of maxima and minima of *T* used in the analysis of the optical data, as well as average *T* over the visible wavelength region, are denoted by dashed lines.

The thickness of the samples was determined using Equation (7):

$$d = \frac{\lambda\_m \lambda\_{m+1}}{2\left(\lambda\_m \lambda\_{m+1} - \lambda\_{m+1} n\_m\right)}\tag{7}$$

where Ȝ*m*, Ȝ*m+*1 and *nm*, *nm+*1 are the wavelength and the corresponding refractive index calculated by means of Equation (4) for any consecutive pair of maxima or minima in the UV-Vis transmittance spectra. The thicknesses calculated by means of Equation (7) were calculated to be *d* = 591 and 739 nm, respectively, for *P*O2 = 0.65 and 1.3 mTorr, in good agreement with the results obtained from profilometry.

Changes in the refractive index are most likely to be associated with changes in sample porosity due different deposition conditions. The packing densities, ȡ, of the two samples were therefore estimated using the Pulker equation [37].

$$\rho = \frac{\rho\_f}{\rho\_b} = \frac{n\_f^2 - 1}{n\_f^2 + 2} \cdot \frac{n\_b^2 + 2}{n\_b^2 - 1} \tag{8}$$

where ȡ is the packing density of the sample; ȡ*f* and ȡ*b* the film and the bulk density of the material; and *nf* and *nb* the sample and bulk refractive index, respectively. The packing densities, corresponding to the measured refractive indices were determined to be 0.88 and 0.79, respectively, for the samples sputtered at *P*O2 = 0.65 and 1.3 mTorr, respectively, which corresponds to a difference in porosity of about 10%, *i*.*e*., a slightly increasing porosity with larger *P*O2, *i.e*., a similar trend as for the surface roughness.

The optical bandgap, *E*g, of the two films were calculated from the special absorption according to Hong *et al.* [38]

$$a = \frac{1}{d} \ln \left( \frac{1 - R}{T} \right) \tag{9}$$

where *d* is the thickness, *T* is the transmittance and, *R* the reflectance.

Since anatase TiO2 is an indirect bandgap semiconductor, a plot of ξȽܧ as a function of photon energy, *E = h*Ȟ, should yield a linear dependence assuming parabolic band dispersion. This is a good approximation close to *Eg* (*E* > *Eg*). A linear region is discerned for both films in the region ~3.4 to ~3.6 eV, and extrapolation yields *Eg* § 3.3 eV for both samples in fair agreement with tabulated data of bulk anatase TiO2 (*Eg* = 3.2 eV). Corresponding Tauc plots of ξȽܧ versus *E* are shown in Figure 5.
