  We have analyzed the morphology of ferroelectric domains in very thin films
of multiferroic BiFeO3. Unlike the more common stripe domains observed in
thicker films BiFeO3 or in other ferroics, the domains tend not to be straight,
but irregular in shape, with significant domain wall roughening leading to a
fractal dimensionality. Also contrary to what is usually observed in other
ferroics, the domain size appears not to scale as the square root of the film
thickness. A model is proposed in which the observed domain size as a function
of film thickness can be directly linked to the fractal dimension of the
domains.
