Top electrode size effects in the piezoresponse force microscopy of piezoelectric thin films attached to a rigid substrate

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In order to avoid the highly concentrated electric field induced beneath the sharp tip, the technique using a top coating electrode in the piezoresponse force microscopy (PFM) has been developed to detect the piezoelectric coefficients. Reliable theory should be erected to explain the broadly reported top electrode size effects and relate the responses with material constants. In this paper, the surface displacement, electric potential inside the film, electric charge and effective piezoelectric coefficient are expressed as a set of integral equations. Analytical solutions are obtained for two limiting cases, i.e., half space (HS) and infinitely thin film (IT). The effective piezoelectric coefficient of the HS case is proved to be the same as that from the PFM of a piezoelectric half plane without a top coating. For the IT case, it is identical to the well-known piezoelectric coefficient result of piezoelectric thin film clamped between flat rigid electrodes subject to homogeneous external electric field. For PZT4 thin layer, numerical results reveal that the surface displacement obviously decreases and the electric potential distributions inside the film become more and more homogeneous as the electrode radius to film thickness ratio (a/t) enlarges. The electric charge dramatically increases while the effective piezoelectric coefficient evidently decreases and they both transfer from the HS solutions to the IT results when a/t varies from 0.001 to 20. The transition occurs at about a/t = 1 in agreement with the experimental observations. A critical top electrode size, i.e., a/t > 10, is obtained and applicable to other piezoelectric materials. Under such circumstances, one can readily gain the piezoelectric coefficients e33, d33 and the dielectric coefficient if other mechanical coefficients and one piezoelectric constant are known a prior.