## 2 Thin Film Deposition and Characterization by X-ray Diffraction
### Characterization of Films by Optical Methods and Atomic Force Microscopy  
In the range where absorption in the film can be neglected, typically above 700 nm for BFO [41; 42; 43; 44], the modulation of reflectance is accurately defined. On each film deposited on Si, a fairly clear interference modulation is observed in spectral reflectance measurements, and the periodicity allows a reliable estimation of the optical thickness \(\left(nH\right)_{\text{opt}}\) (refractive index, \(n\), times the geometrical thickness, \(H\)) to be obtained. However, very poor agreements are obtained between the experimental reflectance ratios (\(\approx\) intensities \(\approx\)) and the fitted ones in the framework of theoretical models. Additionally, the reflectograms of the BPFMO samples collected at different places of the sample were very different. Since the refractive index is related to the periodicity, there are no chances to measure them reliably. Only the optical thickness that depends on the pseudo-periodicity of the diagrams can be deduced. This situation could be attributed to the fact that a part of the light is scattered instead of being specularly reflected. Such a scattering is presumably due to the inhomogeneities of the film surfaces observed under optical microscope and bulk inhomogeneities revealed by the TDBS technique (Figure 3). The birefringent character of the polycrystalline films could be an additional reason for strong diffusive scattering. This prevented separate measurements of the refractive indices and the geometrical thicknesses of the films, requiring different approaches for the measurements of the refractive index. At the same time, the measurements of the optical thickness at 830 nm optical wavelength were highly precise and reliable (see the fifth column in Table 2). In order to obtain the required values of the refractive indices from these measurements, the thickness of the films should be measured independently. We scratched the BPFMO films with an AFM tip down to the Si substrate by applying a controlled force [45]. Since it had previously been checked that the applied force was not sufficient to scratch the harder silicon substrate, the measurement of the resulting step height gives access to the BPFMO film thickness. Unfortunately, the film with 5% Pr substitution was too hard for scratching with this method. The results for the measured thicknesses \(H_{\text{AFM}}\) are presented in the fourth column in Table 2. Combining this data with those presented in the fifth column in the same Table provides refractive indices presented in the sixth column in Table 2. Finally, using the obtained refractive indices and BFs meas \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Pr Substitution**} & \multicolumn{2}{c}{**Average BFs for Two Angles of**} & \multirow{2}{*}{\(n_{\text{TDBS}}=\frac{f_{\text{s}}\sin\phi}{\sqrt{6}-f_{\phi}^{*}}\)} & \multirow{2}{*}{\(v_{\text{TDBS}}=\frac{\lambda_{0}\sqrt{f_{\phi}-f_{\phi}^{*}}}{2\text{sin}\phi}\)} \\  & \(0^{\circ}\) (\(\boldsymbol{\
ot{V}}_{0}\)) & \(52^{\circ}\) (\(\boldsymbol{\
ot{V}}_{2}\)) & & **[m/s]** \\ \hline
5\% & 24.9 \(\pm\) 1.6 & 24.8 \(\pm\) 0.3 & 8.00 \(\pm\) 84.0 & \(1143\)\(\pm\) 10980 \\
10\% & 17.0 \(\pm\) 0.7 & 16.9 \(\pm\) 0.9 & 7.27 \(\pm\) 58.6 & \(944\)\(\pm\) 7639 \\
15\% & 17.7 \(\pm\) 0.7 & 16.7 \(\pm\) 0.4 & 2.38 \(\pm\) 1.45 & 3007 \(\pm\) 1954 \\
20\% & 25.1 \(\pm\) 0.3 & 24.1 \(\pm\) 2.7 & 2.82 \(\pm\) 4.40 & 3596 \(\pm\) 5650 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Average Brillouin frequency (BF) values measured for two angles of incidence, \(\psi\) = 0\({}^{\circ}\) and \(\psi\) = 52\({}^{\circ}\) for the different concentration of Pr substitution in the (Bi\({}_{1-x}\)Pr\({}_{x}\))(Fe\({}_{0.95}\)Mn\({}_{0.05}\))O\({}_{3}\) thin films deposited on Si. The calculated optical indices (\(n_{\text{TDBS}}\)) and acoustic velocity (\(\
u_{\text{TDBS}}\)) using the measured BFs are also reported.