  Detection of dynamic surface displacements associated with local changes in
material strain provides access to a number of phenomena and material
properties. Contact resonance-enhanced methods of Atomic Force Microscopy (AFM)
have been shown capable of detecting ~1-3 pm-level surface displacements, an
approach used in techniques such as piezoresponse force microscopy, atomic
force acoustic microscopy, and ultrasonic force microscopy. Here, based on an
analytical model of AFM cantilever vibrations, we demonstrate a guideline to
quantify surface displacements with a high accuracy by taking into account the
cantilever shape at the first resonant contact mode depending on the tip-sample
contact stiffness. The approach has been experimentally verified and further
developed for the piezoresponse force microscopy (PFM) using well-defined
ferroelectric materials. These results open up a way to accurate and precise
measurements of the surface displacement as well as piezoelectric constants at
the pm-scale with nanometer spatial resolution and will allow avoiding
erroneous data interpretations and measurement artefacts. This analysis is
directly applicable to all cantilever-resonance-based SPM techniques.

Materials Science