# Phase-Pure Epitaxial b-Axis-Oriented Bronze TiO\({}_{2}\) Films
## 3 Results and Discussion
### Stability Range from First-Principle Calculations  
MAO appears to be an ideal substrate choice for the stabilization of bronze TiO\({}_{2}\). Its lattice only leads to a small mismatch of \(-0.2\%\) along the a-axis and \(+1.6\%\) and \(-2.2\%\) along the parallel and perpendicular components of the c-axis, respectively. An only slightly smaller or larger lattice parameter would result in a larger mismatch in one of the two lattice directions and would increase stress substantially. In order to demonstrate that the crystal lattice of MAO favors the stabilization of the b-oriented bronze phase, we have performed first-principle density functional theory (DFT) calculations. All calculations have been conducted with the Quantum Espresso software package [22; 23] and are based on Perdew-Burke-Ernzerhof (PBE) exchange functionals and a 4 \(\times\) 4 \(\times\) 4 Monkhorst k-point grid. Projector-augmented wave pseudopotentials with an energy cut-off of 680 eV were employed.  
Figure 3: (**a**) \(\varphi\)-scan at 20 = 25.1\({}^{\circ}\) and \(\phi\) = 18\({}^{\circ}\). At these conjointer parameters, the 110 reflection of a b-axis-oriented bronze film and the 101 reflection of an a-axis-oriented anatase film would be expected. The two epitaxial configurations are illustrated in (**b**,**c**), respectively. In both cases, twinning would lead to a double peak structure with 4-fold symmetry as seen in (**a**). The stabilization of the bronze phase is confirmed via 20 scans around the 022 and 310 bronze reflections, as shown in (**d**). (**e**) Scans along the 103 and 204 anatase reflections show no film peaks and indicate that no anatase is present.  
In order to study the effect of epitaxy, the unit cells of the three observed TiO\({}_{2}\) polymorphs are fixed to a square lattice with varying lattice parameters, as they would be forced by epitaxially strained growth on cubic single-crystal substrates. Unit cell planes are constrained to match the substrates' square lattice, while the out-of-plane lattice parameters and all internal coordinates are free to relax. For all phases, the epitaxial relationships revealed through XRD were addressed. Additionally, for rutile we have considered the case of biaxial strain within the a-b plane. All other epitaxial relationships have been ruled out due to their large lattice mismatches.  
The result of the calculations is shown in Figure 4. The bronze phase is most stable in a narrow lattice parameter range around that of the MAO substrate. For smaller lattice parameters, as, for example, for LAO and SrTiO\({}_{3}\), the anatase phase is more stable. This is in agreement with our experimental observation. According to our calculations, a rutile phase with the (110) face epitaxially strained to the substrate is the most stable for larger lattice parameter a\({}_{\text{sub}}\) > 4.19 A. This fact matches with our experimental observation of (110)-oriented rutile films on MgO as well. It should be mentioned that this epitaxial configuration favors twinning, and the lattice mismatch of rutile to MgO is still substantial. Film growth with minor structural quality can be expected. In conclusion, our DFT calculations are in good agreement with our experiments and provide justification for why the stabilization of the bronze phase has not been observed as single layer on any other substrate yet.