The electromechanics of piezoresponse force microscopy for a transversely isotropic piezoelectric medium

Piezoresponse force microscopy (PFM) has emerged as one of the most powerful tools for characterizing and manipulating electromechanical responses of piezoelectric and ferroelectric materials at the nanoscale, yet the interpretation and quantitative analysis of PFM data remains difficult and is not well established. In this paper, we develop a rigorous analysis of PFM using the Hankel integral transform and the effective point charge model, which accounts for the electromechanical coupling of the transversely isotropic piezoelectric medium, the concentrated electric field induced by scanning probe microscopy tip, and the coupling of the electromechanical field at the interface of air and piezoelectric half-space. Using this method, the contact mode is first considered in comparison with the decoupled method, followed by a detailed analysis of the non-contact mode considering the effects of experimental conditions. We note that the decoupled method is only valid for materials with weak piezoelectricity, and the effective piezoelectric coefficient derived from PFM shows nonlinear dependence on the complete set of specimen electromechanical moduli, as well as on the experimental conditions. The analysis thus sheds considerable insight into PFM and could enable determination of intrinsic piezoelectric coefficients through PFM measurement using inverse calculations.