Fractal Dimensions in Switching Kinetics of Ferroelectrics

TL;DR

This paper investigates the fractal nature of switching kinetics in ferroelectrics, providing evidence that the domain structures exhibit a true fractal dimension of approximately 2.5, which influences the critical behavior of the order parameter.

Contribution

The study offers new insights into the fractal dimensionality of ferroelectric switching domains and relates this to the critical exponent eta, establishing a theoretical framework within hyperscaling.

Findings

Switching kinetics in ferroelectrics exhibit a true fractal dimension of about 2.5.

The critical exponent eta is approximately 1/4 for d=2.5.

Theoretical relations connect the fractal dimension to the order parameter's critical behavior.

Abstract

Early work by the author with Prof. Ishibashi [Scott et al., J. Appl. Phys. 64, 787 (1988)] showed that switching kinetics in ferroelectrics satisfy a constraint on current transients compatible with d = 2.5 dimensionality. At that time with no direct observations of the domains, it was not possible to conclude whether this was a true Hausdorff dimension or a numerical artefact caused by an approximation in the theory (which ignored the dependence of domain wall velocity upon domain diameter). Recent data suggest that the switching dimensionality is truly fractal with d = 2.5. The critical exponent \beta characterizing the order parameter P(T) can be written as a continuous function of dimension d as \beta(d)= [\nu(d)/2] [d+\eta(d)-2], which is exact within hyperscaling; here \nu and \eta are the exponents characterizing the pair correlation function G(r,T) and the structure factor S(q,T). For d=2.5 the estimate is that \beta is approximately 1/4.