## 2 Materials and Methods  
All the chemicals used in this work were of analytical grade and used as purchased from Sigma-Aldrich. Poly (vinylidene fluoride) with a molecular weight of 180,000 phosphonic acid molecules was used for the grafting, and a N,N-dimethyl formamide (DMF) solution, acetone, isopropanol, tetrahydroforurane (THF) solution was thus used for the fabrication of the polymer-based films. The silicon substrates were pretreated before the film deposition in order to remove all contaminants using a standard clean room procedure. All details for the thin films' elaboration can be found in our previous work [42] and are summarized briefly in Figure 1.  
All the films presented in this work were deposited by a standard spin coating technique on conductive substrates in order to properly perform the local piezoelectric characterization. In the case of neat PVDF film, a specific number of PVDF pellets (MPVDF = 180,000), Mw PVDF = 180,000 g\(\cdot\)mol\({}^{-1}\), were completely dissolved in DMF for 3 h at 70 \({}^{\circ}\)C before adding acetone. This PVDF solution was then drop casted onto the pretreated substrate and put in rotation at various speeds to allow for the spreading of the solution [42]. Very flat neat PVDF films with a mean roughness between 3 and 14 nm and homogeneous microstructures were obtained after the optimization of the spin coating process and parameters (see Supplementary Materials Figures S1 and S2, Table S1). In the case of the hybrid films, we decided to use a low concentration of pellets (2% weight of PVDF) in order to elaborate very thin films with a similar approach used for the neat PVDF ones. To control the dispersion of the nano-inclusions inside the PVDF matrix as well as the connectivity between the two phases, we used 2 categories of nanoparticles: functionalized and non-functionalized one [42]. The Ni\({}_{0.5}\)Zn\({}_{0.5}\)Fe\({}_{2}\)O\({}_{4}\) nanoparticles (NZFO-NPs) used to elaborate our hybrid thin films were synthesized using a soft chemistry polyol route in 1,2-propandiol as reported elsewhere [43; 44; 45]. In order to avoid their agglomeration as much as possible and to promote their dispersion in the hydrophobic matrix, we functionalized them with specific molecules. The procedure for the functionalization of the nanoparticles was conducted on the basis of a previously reported process [46; 47]. Again, very flat films were obtained with a mean roughness of 5 nm and homogeneous microstructures (see Supplementary Materials Figures S1 and S2, Table S1). All the films studied in this work presented a mean thickness between 75 and 100 nm, and they were not poled during the elaboration process. In the case of the hybrid films, the nanoparticles content was kept at 0.5% wt for both the functionalized and non-functionalized cases (see Supplementary Materials Table S1). More details on the optimization of the films can be found in [42]. The X-ray diffraction patterns acquired from the four samples studied are reported in Figure S3 in the Supplementary Materials. The main Bragg peaks of the PVDF crystalline phases were evidenced and clearly showed the presence of several phases (\(\alpha\), \(\beta\), and \(\gamma\)). The diffraction lines associated with the nanoparticles' crystalline peaks are not reported here, as they have already been analyzed in Reference [42] for both F-PVDF and NF-PVDF thin films. The spherical morphology, mean size, and magnetization behavior of the nanoparticles are reported in Figure S4a-c of the Supplementary Materials. Infrared (IR) spectroscopy (Thermo Nicolet, AVATAR 370 FTIR) was carried out over a range of 700-1500 cm\({}^{-1}\) and used to quantify the electroactive phase content of PVDF. The surface structures and film roughness were analyzed by statistical studies of high-resolution AFM images (AFM, Brucker D3100).