Energetic Analysis of Magnetic Transitions in Ultra-small Nanoscopic   Magnetic Rings
  In this article, we report on experimental and theoretical investigations of magnetic transitions in cobalt rings of size (diameter, width and thickness) comparable to the exchange length of cobalt. Magnetization measurements were performed for two sets of magnetic ring arrays: ultra-small magnetic rings (outer diameter 13 nm, inner diameter 5nm and thickness 5 nm) and small thin-walled magnetic rings (outer diameter 150 nm, width 5 nm and thickness 5 nm). This is the first report on the fabrication and magnetic properties of such small rings. Our calculations suggest that if the magnetic ring's sizes are comparable to, or smaller than, the exchange length of the magnetic material, then only two magnetic states are important - the pure single domain state and the flux closure vortex state. The onion-shape magnetic state does not arise. Theoretical calculations are based on an energetic analysis of pure and slightly distorted single domain and flux closure vortex magnetic states. Based on the analytical calculations, a phase diagram is also derived for ultra-small ring structures exhibiting the region for vortex magnetic state formations as a function of material parameter. 
Deepak K. Singh, Robert V. Krotkov and Mark T. Tuominen