Logarithmic frequency dependence of the piezoelectric effect due to pinning of ferroelectric-ferroelastic domain walls

Pinning of the ferroelastic-ferroelectric domain walls on randomly distributed defects in ferroelectric ceramics leads to a field dependence of the piezoelectric coefficient that is analogous to the Rayleigh law for magnetic susceptibility. It is shown in this paper that the piezoelectric coefficient of a lead zirconate titanate ferroelastic-ferroelectric system depends linearly on the logarithm of the frequency of the field. Both the reversible and irreversible components of the piezoelectric coefficient are found to be frequency dependent. A similar type of frequency dependence due to domain-wall pinning has been predicted for the magnetic susceptibility in disordered ferromagnets and dilute antiferromagnets. The presented results offer experimental evidence that the theoretical approach developed for domain-wall pinning effects in magnetic materials is generally valid for pinning processes in all ferroic systems, ferromagnetics, ferroelectrics, and ferroelastics.