Cylinder stress in nanostructures: effect on domains in nanowires, nanotubes, and nano-disks

Since the work of Landau-Lifshitz in 1935, Kittel in 1946 and by Roytburd and Arlt more recently, we have understood that the width w of magnetic or ferroelectric or elastic domains and twins is proportional to the square root of the characteristic length d, which is thickness in a thin film or diameter in a small grain. This square root relationship is derived by balancing stress: larger-area domains have larger stress, which can be minimized by having adjacent domains of reversed orientation, but at the cost of wall energy. Three-dimensional objects undergo three kinds of stress: axial, radial, and azimuthal ('hoop stress'), the last of which has previously been ignored. Unlike axial stress, it is proportional to d, not d(2), and we show that it leads to w linear in d.