Fractal Dimensions in Switching Kinetics of Ferroelectrics

Abstract

Early work by the author with Prof. Ishibashi [Scott et al., J. Appl. Phys. 64 , 787 (1988); 61 , 5467 (1987)] showed that switching kinetics in ferroelectrics, particularly lead zirconate titanate (PZT), 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 β characterizing the order parameter P(T) can be written as a continuous function of dimension d as β(d) = [ν (d)/2] [d+η(d)−2], which is exact within hyperscaling; here ν and η 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 β is approximately 1/4.

Keywords:

• Fractals
• critical exponents
• phase transitions

Acknowledgment

I thank P. Zubko for helpful discussions.