Relationship between direct and converse piezoelectric effect in a nanoscale electromechanical contact

Linear piezoelectric coupling between mechanical and electrical phenomena is extremely common in inorganic and biological materials and constitutes the basis for multiple applications. In the macroscopic case, the coupling coefficients between electric displacement and strain (direct piezoeffect) and stress and electric field (converse piezoeffect) are equal, the symmetry stemming from the existence of the corresponding thermodynamic potential. Hence, studies of electromechanical coupling provide information on strain-induced polarization change, and vice versa. This is not necessarily the case for the electromechanical coupling (or any other cross-coupled property) in the contact geometry of a scanning probe microscopy or nanoindentation experiment. In this local case, the hypothetical (unknown) thermodynamic potential $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{G}(P,\ensuremath{\psi},\dots{})$ depends not only on conventional variables (e.g., load $P$ and bias $\ensuremath{\psi}$), but also on additional free length parameters, e.g. radius of contact, $a$, or indentation depth, $h$. Here we derive the relationship between the direct and converse piezoelectric effects in the contact geometry. The implications of the established relationships for nanoscale electromechanical and piezoelectric measurements are analyzed.