Domain Wall Switching: Optimizing the Energy Landscape

Abstract

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 <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">b</sub> /mu <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> m <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">s</sub> H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">sw</sub> , 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 <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">z</sub> , in which both the switching field [the maximum slope of E(m <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">z</sub> )] and the stability (determined by the energy barrier DeltaE) 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 also show that a quadratic anistropy suggested by Suess comes very close to this limit.