## 4 Recoveries and Robustness of Local Piezoelectric Properties
This promoted the local dispersion of the nanoparticles as a possible way to reduce the observed piezoelectric losses due to the presence of nano-inclusions and to re-enhance coupling performances for the next generation of magnetoelectric nanostructures. It is important to stress at this point that the piezoelectric coefficient values of approximately 10 pm/V measured in our un-poled neat PVDF films were in the low range of the piezoelectric coefficient values reported in the literature (between 13 and 28 pm/V) for poled PVDF and its co-polymer [19; 48; 50; 51; 52]. The mixing of the non-electroactive phase (\(\alpha\)) with the electroactive one (\(\beta\)) was considered at the origin of an intrinsic polymer chain disorder. In the case of poled films, the application of an electric field during the elaboration process, can help orient (polarized) polymer along the electric field lines and, thus, giving rise to the higher piezoelectric coefficient reported in the literature (between 13 and 28 pm/V) [19; 48; 50; 51; 52]. In our case, the un-poled elaboration procedure as well as the presence of sub-micrometer agglomerates of metallic nanoparticles may have increased the polymer disorder and facilitated leakage of the current effect at the same time. The latter decreased the measurable piezoelectric coefficient value in the hybrid thin film. This can be observed in Figure 5b in which the NF-PVDF film shows a non-detectable d\({}_{33}\). The partial recovery of this d\({}_{33}\) coefficient can be attributed to the effect of the dispersion of the nano-inclusions by following the easy scheme reported in [52]. The application of an applied electric field gives rise to the possible mechanism of coupling between electric properties of the PVDF and the magnetic one of the nanoparticles. The electric field poling in the z-direction induced strain in the PVDF matrix in the \(x\)- and \(y\)-directions and, consequently, the magnetic moments were locked in the \(x\)- and \(y\)-directions. When the hybrid films were electrically poled, the electric dipoles inside the ferroelectric layer could align in the applied field direction only if the nanoparticles were dispersed and locked. This is why, although the amount of electroactive phase was always approximately 50%, the polarization of the F-PVDF sample was successful. We could thus measure a local piezoelectric coefficient and show its partial recovery compared to the NF-PVDF sample. To investigate the robustness of the polarization switching, a DC bias was applied while scanning the surface in order to polarize several confined areas. First, a squared area of 10 \(\times\) 10 \(\upmu\)m was polarized in one direction by the application of a DC bias while scanning 256 lines at 1 line/s. Then a smaller squared area of 5 \(\times\) 5 \(\upmu\)m was poled in the opposite direction by the application of a DC bias of opposite polarity, and the entire region was imaged using an AC bias. The PFM amplitude and phase thus obtained are shown in Figure 5c,d. The amplitude (top) and phase (bottom) images of the PVDF-20 film are shown in Figure 5c, while the ones for the F-PVDF sample are illustrated in Figure 5d. In both cases, we can clearly see domains of opposite polarization giving rise to bright and dark areas well delimited by the square profile. The regions of uniform polarization were rough and irregular, and they corresponded well to the topographic structure shown by the colored squares (i.e., AFM topographic images) reported on the bottom right side of the PFM amplitude images. These irregular PFM images suggest that polarization switching occurred one grain at a time rather than on a multiple grain basis. The less uniform pattern in Figure 5d is indicative of the presence of NPs, which not only affected the topography of the film but also the local piezoelectric contrast. These images demonstrate the possibility of nanoscale polarization control in un-poled neat PVDF and F-PVDF hybrid very thin films.