## 3 Results and Discussion  
The powder XRD patterns of the MAPbI3 3D perovskite, after selected high-energy ball-milling times (0-240 h), are shown in Figure 1a. As can be seen, even after 240 h of the external treatment, the sample maintains a good crystallinity, with only a slight peak broadening with respect to the starting pattern, probably originating from strain-induced defects due to the mechanical treatment. The XRD data of the as-prepared (0 h of ball-milling) MAPbI3 that overlapped with the peaks of the tetragonal crystal structure (vertical red bars) are reported in the bottom part of Figure 1b. As can be appreciated, all the expected peaks are present in the experimental pattern, confirming the typical tetragonal structure of the synthesized MAPbI3 perovskite [25]. On the other hand, already considering the pattern of the sample after 72 h of milling (top part of Figure 1b), a clear phase transition occurs. The matching of the obtained diffraction peaks with those of the cubic phase (blue vertical bar) confirms the stabilization of this symmetry in the treated MAPbI3 crystals. The clear sign of tetragonal symmetry [25,43], namely, the two peaks around 25\({}^{\circ}\), are not present in the milled sample, suggesting that the high-energy mechanical treatment induces the structural phase transition of the 3D perovskite [32]. The cubic symmetry is also found in all the samples by further increasing the milling time, as displayed in Figure 1a. In addition, as reported in Table 1, the treatment has an effect on the lattice parameters, leading to a slight cell volume expansion as a function of time. The Le Bail approach was applied to calculate these parameters with a zero-shift value of \(-\)0.0651 [44]. The XRD patterns, as a function of the ball-milling time (0-420 h) for the PEA\({}_{2}\)PbI\({}_{4}\) 2D perovskite, are displayed in Figure 2. The crystal structure of the as-prepared material is in agreement with the monoclinic space group, and the pattern is dominated by the 001 peaks, as is common in 2D metal halide perovskites [45,46]. A significant crystallinity reduction is found in the samples, already after 120 h, and it is progressively increased by raising the milling treatment time, as can be appreciated by the peak's broadening. In particular, after 420 h of ball-milling, most of the intensity is lost and the peaks are very broad. The FWHM moves from about 0.05\({}^{\circ}\) for the as-prepared sample, to about 0.49\({}^{\circ}\) after 420 h of milling time. At the same time, the first peak, corresponding to the 001 reflection, shifts to a lower angle by increasing the treatment, from 5.413\({}^{\circ}\) (0 h) to 5.238\({}^{\circ}\) (420 h), indicating an expansion of the c-axis. However, no sign of amorphization is found, suggesting the relevant robustness of the 2D sample structure. Furthermore, contrary to what was assessed for MAPbI\({}_{3}\), no evidences of phase transition are found in PEA\({}_{2}\)PbI\({}_{4}\) perovskite crystals as a consequence of the ball-milling treatment at various times. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Crystalline** & **Ball-Milling** & **A** & **C** & **Cell Volume (Å)** \\
**Phase** & **Time (h)** & **(Å)** & **(Å)** & **Cell Volume (Å)** \\ \hline Tetragonal & 0 & 8.868 \(\pm\) 0.001 & 12.628 \(\pm\) 0.001 & 248.300 \\ \hline \multirow{4}{*}{Cubic} & 72 & 6.291 \(\pm\) 0.001 & – & 248.977 \\  & 120 & 6.293 \(\pm\) 0.001 & – & 249.214 \\ \cline{1-1}  & 192 & 6.295 \(\pm\) 0.001 & – & 249.452 \\ \cline{1-1}  & 240 & 6.301 \(\pm\) 0.001 & – & 250.166 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Lattice parameters and cell volume for MAPbI\({}_{3}\) at different ball-milling times. Figure 1: (a) X-ray diffraction (XRD) patterns of MAPbI\({}_{3}\) (MA = methylammonium) as a function of ball-milling time; (b) bottom: XRD pattern of as-prepared MAPbI\({}_{3}\) overlapped with the expected pattern for the tetragonal unit cell; top: XRD pattern of 72 h-milled MAPbI\({}_{3}\) overlapped with the expected pattern for the cubic unit cell.