Hysteresis in layered spring magnets

This article addresses a problem of micromagnetics: the reversal of magneticmoments in layered spring magnets. A one-dimensional model is used of a filmconsisting of several atomic layers of a soft material on top of several atomiclayers of a hard material. Each atomic layer is taken to be uniformlymagnetized, and spatial inhomogeneities within an atomic layer are neglected.The state of such a system is described by a chain of magnetic spin vectors.Each spin vector behaves like a spinning top driven locally by the effectivemagnetic field and subject to damping (Landau-Lifshitz-Gilbert equation). Anumerical integration scheme for the LLG equation is presented that isunconditionally stable and preserves the magnitude of the magnetization vectorat all times. The results of numerical investigations for a bilayer in arotating in-plane magnetic field show hysteresis with a basic period of $2\pi$at moderate fields and hysteresis with a basic period of $\pi$ at strongfields.