  A nanometric needle sensor mounted in an Atomic Force Microscopy allows
systematic picometer-range distance measurements. This force sensing device is
used in Piezoresponse Force Microscopy (PFM) as a distance sensor, by employing
the cantilever spring constant as the conversion factor opening a pathway to
explore the piezoelectric effect at the nanoscale. The force-distance
equivalence is achieved if the force sensor does not disturb the system to
study, solely. In this manuscript we report a systematic study in which
different Lead Zirconate Titanate (PZT) materials, having different d33 values,
are measured following the standard theory available for PFM. Both in resonance
and out of resonance measurements demonstrate that PFM cannot be considered
quantitative. After performing the measurements, we propose a correction of the
standard theory employed in PFM by considering the force exerted by the
material as a variable. The g33 parameter, inherent to piezoelectricity,
governs the amount of force available from the system. A comparison of
piezoelectric stiffness for the case of a nanoscale site contact region,
similar to the one it is found while performing PFM, is provided. Such
stiffness is well below the cantilever stiffness, limiting and diminishing the
material movement, as the piezoelectric material does not have enough stroke to
induce the intended displacement. A correction factor, named Open
Piezopotential Gauge, accounts for these effects, which is used to correct the
measurements carried out in PZT materials towards a real quantitative PFM.
