Asymptotic Effective Piezoelectric Coefficient Solution of Piezoresponse Force Microscopy for a Transversely Isotropic Piezoelectric Film

Based on the coupled theory, a simple explicit solution of piezoresponse force microscopy (PFM) in determining the effective piezoelectric coefficient for an ultra-thin transversely isotropic piezoelectric film bonded to a rigid conducting substrate is obtained, using the Taylor expansion and homogeneous assumption. And it is found to be exactly the same as the well-known result for the case of piezoelectric thin film clamped between flat rigid electrodes for homogeneous external electric field. The electric charge and the distance from the image charge model are also derived and the influences of the film thickness and substrate permittivity on the effective piezoelectric coefficient are then discussed. The obtained results can be used to quantitatively interpret the PFM signals and directly detect the piezoelectric constant through PFM for an ultra-thin film or supply important information for constructing a reliable formula to describe the thickness effect.