## 3 Results and Discussion
### Substrate Stage Rotation Rate Effects
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Samples/Parameters** & **S\({}_{\text{S1}}\)** & **S\({}_{\text{S2}}\)** & **S\({}_{\text{S3}}\)** \\ \hline Substrate stage rotation rate (rpm) & 7.5–8 & 15.5–16 & 25.5–26 \\ Substrate temperature & RT & RT & RT \\ Process gas (Ar) pressure & 2 mTorr & 2 mTorr & 2 mTorr \\ RF power densities at target & 3.8 W/cm\({}^{2}\) & 3.8 W/cm\({}^{2}\) & 3.8 W/cm\({}^{2}\) \\ Deposition run time & 62 min & 62 min & 62 min \\ Deposited layer thickness & 364 nm & 376 nm & 352 nm \\ Annealing process parameters & 6 h at 580 \({}^{\circ}\)C or & 6 h at 580 \({}^{\circ}\)C or & 6 h at 580 \({}^{\circ}\)C or \\
1 h at 670 \({}^{\circ}\)C & 1 h at 670 \({}^{\circ}\)C & 1 h at 670 \({}^{\circ}\)C & 1 h at 670 \({}^{\circ}\)C \\ \hline \multirow{3}{*}{
\begin{tabular}{c} MO parameters measured at \\ 532 nm \\ \end{tabular} } & Specific Faraday rotation (deg/\(\mu\)m) & 6.03 & 6.36 & 6.27 \\  & Figure of merit (degrees) & 11.9 & 15.1 & 11.2 \\ \cline{1-1}  & Coercivity (H\({}_{\text{c}}\)) (Oe) & \textasci{}130 & \textasci{}130 & \textasci{}160 \\ \cline{1-1}  & Saturation magnetization (H\({}_{\text{sat}}\)) (Oe) & \textasci{}230 & \textasci{}250 & \textasci{}300 \\ \hline \multicolumn{3}{c}{Average crystallite size (nm)} & 36.6 nm & 36.8 nm & 31.3 nm \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Summary of the process parameters and the corresponding materials’ properties of Bi\({}_{2.1}\)Dy\({}_{0.9}\)Fe\({}_{3.9}\)Ga\({}_{1.1}\)O\({}_{12}\) thin films fabricated using different substrate stage rotation rates. be related to the variations in the state of mechanical stress in films subjected to different thermal processing regimes, which could affect the lattice parameters. The intensity of diffraction peaks from the samples prepared using the 15.5-16 rpm stage rotation rate (S\({}_{\rm S2}\)) was higher than that seen in samples S\({}_{\rm S1}\) and S\({}_{\rm S3}\), which implies that the volumetric contents of garnet phase within the films' volume differed. The measured average crystallite size (derived using Scherrer's formula) was in the range of 31-37 nm. The larger (average) crystallite size was observed (\(\sim\)36.8 nm) also in films prepared using the 15.5-16.0 rpm substrate stage rotation rate, compared to the films grown at either lower or higher substrate stage rotation rates. The values of average crystallite size are also listed in Table 2.  
Using slower rotation rate helps to spontaneously grow porous films on the substrates and also increases the surface roughness, which affects the entropy of the garnet films and the MO performance [22], while the higher substrate stage rotation rates also contribute to the nucleation of pores in garnet layers. Increasing the surface roughness of the film reduces the garnet film quality in terms of the optical and MO properties. The EDS elemental compositional analysis also showed that the samples sputtered using the rotation rate of around 15.5-16.0 rpm had a slightly higher Bi content compared with other samples (Table 2). Figure 3d presents the measured elemental concentrations (in atomic %) obtained from the as-deposited films prepared using different substrate stage rotation rates. The substrate stage rotation rate is one of the physical parameters affecting the initial stages of the thin film growth process, most likely by influencing the uniformity of adatom growth and nucleation across the substrate surface. This is because the time scale (of the order of 1 s) related to the formation of ultrathin (adatom-scale) initial film layers is, for the usual sputtering deposition rates of a few nanometers per minute, very similar to the periodicity of substrate stage rotation used in most deposition experiments. More studies are necessary to uncover the physical phenomena and processes linking the substrate stage rotation rate to the properties of ultrathin transitional layers forming near the film-substrate interface, including other process details, such as Thornton zone-based growth-model-related parameters [32; 33].