Domain wall switching: Optimizing the energy landscape

It has recently been suggested that exchange spring media offer a way to increase media density without causing thermal instability (superparamagnetism), by using a hard and a soft layer coupled by exchange. Victora has suggested a figure of merit xi = 2E(b)/mu(o) m(s)H(sw), the ratio of the energy barrier to that of a Stoner-Wohlfarth system with the same switching field, which is 1 for a Stoner-Wohlfarth (coherently switching) particle and 2 for an optimal two-layer composite medium. A number of theoretical approaches have been used for this problem (e.g., various numbers of coupled Stoner-Wohlfarth layers and continuum micromagnetics). In this paper we show that many of these approaches can be regarded as special cases or approximations to a variational formulation of the problem, in which the energy is minimized for fixed magnetization. The results can be easily visualized in terms of a plot of the energy E as a function of magnetic moment m(z), in which both the switching field [the maximum slope of E(m(z))] and the stability (determined by the energy barrier Delta E) are geometrically visible. In this formulation we can prove a rigorous limit on the figure of merit xi, which can be no higher than 4. We allso show that a quadratic anistropy suggested by Suess et al. comes very close to this limit.