# Phase-Pure Epitaxial b-Axis-Oriented Bronze TiO\({}_{2}\) Films
## 3 Results and Discussion
### Optical Properties  
Variable-angle spectroscopic ellipsometry was used to demonstrate the different optical properties of the TiO\({}_{2}\) films. The energy-dependent ellipsometric angles \(\Psi\) and \(\Delta\) were recorded for all three single-layer samples. The data were then fit to a simple two-layer model consisting of the substrate and film in order to determine the optical properties of  
Figure 4: Calculated total energies per formula unit of TiO\({}_{2}\) as a function of the in-plane lattice parameter \(a_{\text{sub}}\). The epitaxial relationships considered are biaxial strains within in the anatase (001) plane, bronze (010) plane, rutile (001) plane and rutile (110) plane. Pseudocubic lattice parameters of standard single-crystal substrates are marked for reference. The total energies are given with respect to the ground state energy of the anatase bulk structure. the film. The substrate and the TiO\({}_{2}\) layer are approximated by a B-spline with 38 data points over the full energy range. Here, the optical properties of the substrates are first determined independently by fitting the VASE data of a bare substrate. Next, the recorded film-substrate VASE data were simulated by fitting the TiO\({}_{2}\) optical properties and leaving the substrate properties fixed. Figure 5 shows a Tauc plot of the absorption coefficients \(\alpha\) of the three single-layer films. Here, the square of \(\alpha\) multiplied by the photon energy is drawn as a function of the photon energy. This Tauc-plot representation is typically used to determine direct optical band gaps by extrapolating the linear part to zero. For indirect band-gap materials, the square root instead of the square of the absorption coefficient would be used. It should be mentioned that our DFT calculations (not shown) as well as previous work indicate that bronze TiO\({}_{2}\) has a direct band gap. The same is true for rutile. Anatase is considered to be an indirect band-gap material. This fact indicates that the Tauc plot in Figure 5 is technically not suited to determine the exact band gap of the anatase film. However, we decided to present all the data in one single Tauc plot for comparability. It should be stressed that our main goal is not to determine exact band-gap values, but to highlight the strong differences in the optical properties between the thin film samples. Our experimental data show that, for the bronze film, the band gap (-4.3 eV) is at notably larger energies than that for anatase (3.7 eV) and rutile (3.8 eV). Little is known about the optical band gap of bronze TiO\({}_{2}\). First-principle calculations have placed the electronic band gap of the bronze phase at about 4.1 eV, and more than 0.5 eV above that of anatase and rutile [24]. Our data are therefore in good agreement with the theory. In the literature, the experimentally determined optical band gaps of bulk TiO\({}_{2}\) polymorphs vary quite a bit, with the values for anatase and rutile typically ranging between 3.1-3.4 eV and 3.0-3.3 eV, respectively [25]. Our experimental values on thin films are about 0.5 eV higher, mainly due to the method that was used to determine the band gaps [26]. Additionally, band gaps of thin films determined using ellipsometry have consistently been shown to be 0.1-0.2 eV higher than those determined using transmission spectroscopy [27], the standard technique used for bulk materials. Our data are, however, consistent with the work of [PERSON] et al., the only experimental study known to us that reports on the optical properties of bronze-type epitaxial films [14]. In this report the optical band gap of bronze-type TiO\({}_{2}\) was also found to be about 0.5 eV higher than that of anatase. Figure 5 clearly demonstrates that the film on MAO has vastly different optical properties than the other two films and  
Figure 5: Tauc plot of the absorption coefficient \(\alpha\) as function of the photon energy h\(\
u\) for the single-layer TiO\({}_{2}\) films. The band gaps were determined by extrapolating the linear parts of the curves to zero.