Charge-spring model for predicting the piezoelectric response of dielectric materials: Considering tetragonality extends validity to ferroelectric crystals

In the 2014 Whitehead Memorial Lecture, a straight-forward charge-spring model was introduced and shown to yield estimates for the piezoelectric thickness coefficients of ferroelectric polymers such as polyvinylidene fluoride (PVDF) and of polymer ferroelectrets such as cellular polypropylene or tubular-channel fluoroethylene-propylene copolymer systems. It was, however, observed that the predictions for ferroelectric perovskites such as lead zirconate titanate (PZT) led to a discrepancy of up to two orders of magnitude. Here, we try to resolve the mystery. In ferroelectric polymers, the relevant springs represent the soft amorphous phase between the crystalline dipole polarization. Consequently, the (secondary) piezoelectricity is mainly a dipole-density effect predicted from the rigid dipoles of the crystalline phase and the soft springs (elasticity) of the amorphous phase. In polymer ferroelectrets, the internally charged cavities are macroscopic dipoles and soft springs at the same time, and the (primary) piezoelectricity can be directly predicted from the charge densities and the elasticity in the cavities. In ferroelectric perovskites, the spontaneous polarization disappears when the tetragonal axis is compressed back into the cubic axis. Consequently, the tetragonality needs to be used for correcting the effective length of the relevant spring. With the corrected spring (or elasticity) constants, we find reasonable agreement between the predicted and the experimental values for the piezoelectric thickness coefficients also in perovskites.