# Dresden 2006 – wissenschaftliches Programm

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# SYMM: Multi-ferroic materials

## SYMM 2: Multiferroic Materials - Experiment

### SYMM 2.2: Hauptvortrag

### Dienstag, 28. März 2006, 16:45–17:15, HSZ 04

**Some Observations about Static Scaling: Domain Widths and Circular and Toroidal Ordering in Ferroelectrics, Ferromagnets, and Magnetoelectrics** — •J. Scott — Cambridge University, Cambridge, UK

In 1946 Charles Kittel derived a formula for the width w of 180-degree domains, varying as the square root of the sample thickness d, and in the same early paper showed that nano-structures (nano-wires and nano-dots) would have circular ordering of spins. Both ideas were later independently rediscovered, w being proportional to the square root of d by Mitsui and Furuichi (Phys. Rev. 1953), and circular or toroidal ordering in ferroelectrics by Ginzburg in 1981-1984. Very recently the latter idea was put on an atomistic ab initio level by Naumov et al. (Nature 2004). In this talk I combine Kittel’s equation with a less well known one of Zhirnov (1959), which shows that the domain wall thicknesses D also varies as the square root of d. The resulting formula is (w-squared)/Dd = G, where G is a dimensionless constant = 32 in barium titanate and 1 in Rochelle Salt. This formula fits barium titanate over six decades of thickness d, from 1.5 nm to 1.5 mm (figure below). Where this formula breaks down depends upon boundary conditions and size, which connects it to circular/toroidal ordering. Moreover, as Ginzburg first showed, the symmetry requirements for toroidal ordering are closely related to those for magnetoelectricity (Schmid, 1994).