ac dynamics of ferroelectric domains from an investigation of the frequency dependence of hysteresis loops

We investigated the pinning dominated domain-wall dynamics under an ac field by studying the frequency $(f)$ dependence of hysteresis loops of a uniaxial ferroelectric (FE) system. We measured the fully saturated polarization-electric field $(P\text{\ensuremath{-}}E)$ hysteresis loops of high-quality epitaxial 100-nm-thick ${\text{PbZr}}_{0.2}{\text{Ti}}_{0.8}{\text{O}}_{3}$ capacitors at various $f$ (5--2000 Hz) and temperatures $T$ (10--300 K). We observed that the coercive field ${E}_{C}$ is proportional to ${f}^{\ensuremath{\beta}}$ with two scaling regions, which was also reported earlier in magnetic systems [T. A. Moore and J. A. C. Bland, J. Phys.: Condens. Matter 16, R1369 (2004), and references therein]. In addition, we observed that the two scaling regions of ${E}_{C}$ vs $f$ exist at all measured $T$. We found that the existence of the two scaling regions should come from a dynamic crossover between the creep and flow regimes of the FE domain-wall motions. By extending the theory of Nattermann et al., which was originally proposed for impure magnet systems [T. Nattermann, V. Pokrovsky, and V. M. Vinokur, Phys. Rev. Lett. 87, 197005 (2001)], to the disordered FE systems, we obtained analytical expressions for the dynamic crossovers between the relaxation and creep, and between the creep and flow regimes. By comparing with the experimental data from our fully saturated $P\text{\ensuremath{-}}E$ hysteresis loop measurements, we could construct a $T\text{\ensuremath{-}}E$ dynamic phase diagram with $f$ as a parameter for hysteretic FE domain dynamics in the presence of an ac field.