Dynamic hysteresis dispersion scaling of ferroelectric Nd-substituted Bi₄Ti₃O₁₂ thin films

The dynamics of ferroelectric hysteresis for Nd-substituted Bi ₄Ti₃O₁₂ thin films as a function of frequency and amplitude of time-varying external electric field is measured utilizing a Sawyer-Tower (ST) circuit. The frequency ranges from 1 to 10⁶ Hz and the amplitude range is 101-400 kV cm^-1. Given a fixed field amplitude E₀, the hysteresis area A first grows and then decays with increasing frequency f. We study the characteristics of the dynamic order parameter Q against field amplitude E₀. The frequency dependence of the hysteresis area over the low-frequency range can be expressed as A ∝ E₀^2/3 f^1/3. However, over the high-frequency range, the frequency dependence takes the form A ∝ E₀ ² f^-2/3. We demonstrate that the hysteresis dispersion approaches its scaling state when the field amplitude E₀ is higher than 170 kV cm^-1, and a unique effective characteristic time τ₁ exists for the domain reversal that is inversely proportional to the field amplitude.