Electric field dependence of piezoelectric coefficient in ferroelectric thin films

A simple model is developed to explain the hysteretic electric field dependence of the piezoelectric coefficient in ferroelectric thin films. The nonlinear susceptibility and hysteretic electrical polarization behavior can explain the piezoelectric hysteresis characteristics. An empirical model introducing a weighting factor is utilized to represent the electric field dependence of the nonlinear susceptibility. Experimentally, the magnitude of the maximum of the piezoelectric coefficient measured in the backward direction of the piezoelectric hysteresis loop is usually larger than that of the maximum measured in the forward direction. The proposed model shows that the weighting factor in modeling the nonlinear susceptibility may account for this observed phenomenon. The nonlinear susceptibility may also explain the peaked shape observed in piezoelectric coeffieint-electric field hysteresis loops. The approach in this work does not require the "hard ferroelectric approximation" of the polarization characteristics, and provides a method of predicting the piezoresponse from measured capacitance and polarization properties of the ferroelectric thin film.