The piezoelectric effect, which consists in the dielectric polarization of non-centrosymmetric crystals under a mechanical stress, was discovered by the Curie brothers in 18801. The following year, from thermodynamic considerations, G. Lippmann predicted the converse effect, i.e. that a piezoelectric material would be mechanically strained by an applied electric field2 and the Curies readily measured it3. These findings spawned more research which eventually led to the discovery of ferroelectricity in polar piezoelectrics4. Since those early discoveries, the unique ability of piezoelectrics and ferroelectrics for interconverting mechanical and electrostatic energies5 has endlessly inspired technological developments and these materials, which represent nowadays a billion euro industry, are found in many everyday applications6–12: ultrasound generators for echography scanners, shock detectors within airbags, accelerometers, diesel injection valves, tire pressure sensors, vibration dampers, oscillators, improved capacitors, or new dynamic access random memories, to just cite a few. Moreover, the prospects for future applications in new markets are bright, including energy harvesting, CMOS replacement switches, or photovoltaics and photocatalysis13–16. Yet, in spite of such industrial relevance and the amount of past and present research, the basic understanding of piezoelectricity and ferroelectricity is challenged and reshaped by findings that come along with new developments in the characterization of materials. This is well illustrated by the advances in atomic force microscopy, which brought a new perspective of ferroelectric domain walls17– 20. The development of new modes and an improved spatial resolution have revealed the domain wall complexity and its intrinsic properties 21–23 and have also opened the door to get more insight in longdate issues such as the extrinsic contributions to dielectric permittivity and piezoelectricity due to domain wall pinning at dislocations and grain24. In this direction, Piezoresponse Force Microscopy (PFM) is the most widely used technique for the nanoscale and mesoscale characterization of ferroelectric and piezoelectric materials25–28. PFM method is based on the converse piezoelectric effect and consists in measuring the material deformation under an AC electric field applied through the contacting AFM tip. In this technique the sample vibration is determined by an optical beam deflection system, which is an indirect measurement29, making the accurate determination of the piezoelectric coefficient challenging. Moreover, the quantitative piezoelectric measurements by PFM30, are further complicated by the difficulty of disentangling, from the electromechanical response, the contributions of the piezoelectric response and other physical phenomena such as, ionic motion and charging, electrostatic or thermal effects18,31–33. Indeed, the increasing awareness about these issues among the scientists of the field34 prompts the need for new developments in scanning probe microscopies, which remain a unique tool for the characterization of piezoelectric and ferroelectric materials at the nanoscale.
