## 4 Recoveries and Robustness of Local Piezoelectric Properties  
Comparison between the functionalized and non-functionalized hybrid films allowed us to study the effect of the dispersion of the nanoparticles inside the matrix, not only over the optimization of the interface between the two phases but also over the local piezoelectric properties of the polymer. As we reported before, to efficiently stress the magnetostrictive phase of the NPs, thus controlling their moment orientation by applying an electric field on the piezoelectric polymer, is a major key parameter for using these hybrid thin films in new magnetic sensors. In this frame, the study of the local piezoelectric response of the polymer by piezoelectric force microscopy is of primary importance. In Figure 5, we show the piezoelectric response, which was measured by applying an alternative electric bias field through the use of a nanometric-probing tip. The latter is in contact with the surface of the polymer and applies the field through the thickness of the film (i.e., the conductive substrate was maintained at 0 V). This electric bias resulted in sample deformation due to the converse piezoelectric effect. Piezoelectric surface displacement was detected by the reflection of a laser beam on the photodiode of the microscope that records the cantilever vibration/position. The out-of-plane piezoelectric response was thus measured when an AC voltage was applied through the tip with an amplitude vibration in the range of 0-10 V. The voltage was increased by a step of 2 V as can be seen in Figure 5a. Figure 4: SEM images of (**a**) a neat PVDF film; (**b**) non-functionalized NPs embedded in the PVDF film (NF-PVDF); (**c**) functionalized NPs embedded in the PVDF film (F-PVDF); (**d**) cross-sectional image of the NF-PVDF sample. The scale bar in (**a–c**) is \(1~{}\upmu\)m.  
plane piezoelectric response for the whole set of samples. These results confirm a linear piezoelectric effect in the neat PVDF films as well as in the hybrid ones. This is in agreement with the presence of a polar phase in all our samples. However, in the case of hybrid-PVDF films (i.e., NF-PVDF and F-PVDF), the vibration amplitude was very small, showing a large loss of the piezoelectric response compared to the neat PVDF films. Figure 5b shows the measured displacement vs. the applied AC voltage of the analyzed samples. As can be seen, we obtained good linear behavior that is in agreement with the converse piezoelectric effect theory, according to which the displacement of the material is proportional to the applied AC voltage. The d\({}_{33}\) value was estimated using a periodic poled lithium niobate (PPLN) test sample. The obtained d\({}_{33}\) values are reported in the inset of Figure 5b. First, we can observe that the slight difference between the linear piezoelectric responses of the two neat PVDF films did not present a higher slope in the case of the PVDF-20 which had higher \(\beta\)-phase content (see Figure 3). This can be attributed to its local roughness of one order of magnitude higher than the one for PVDF-80 and which is well known to affect the PFM response (see Supplementary Materials Table S1). Surprisingly, NP inclusions strongly affected the local piezoelectric response of the hybrid films in contrast with the constant \(\beta\)-phase content measured. Locally, in the case of the NF-PVDF sample, the agglomerate nano-fillers destroyed the charge polarization of the PVDF and prevented us from measuring the piezoelectric coefficient. Thanks to the dispersion of the functionalized NPs in the F-PVDF film, we could recover 10% of the piezoelectric response observed in the neat PVDF films. In Figure 5a, the tip vibration presented a better-defined resonance in the case of the F-VDF film compared to the NF-PVDF (see the zoomed in areas in 5a).