Magnetostatic Modes in Samples With Inhomogeneous Internal Fields

Magnetostatic modes in ferromagnetic samples have been studied systematically since the 1950s. They are very well characterized and understood in samples with uniform internal magnetic fields. More recently, interest has shifted to the study of magnetization modes in ferromagnetic samples with inhomogeneous internal fields. This paper shows that under the magnetostatic approximation and for samples of arbitrary shape and/or arbitrary inhomogeneous internal magnetic fields, the modes can be classified as elliptic or hyperbolic, and their associated frequency spectrum and spatial range can be delimited. This results from the analysis of the character of the second-order partial differential equation satisfied by the magnetostatic potential: i.e., if it is hyperbolic or elliptic (parabolic is a limiting case). In elliptic regions, the magnetostatic modes have a smooth monotonic character (with generally decaying or “tunneling” behavior from the surfaces) and in hyperbolic regions an oscillatory wave-like character. A simple local criterion distinguishes hyperbolic from elliptic regions: the sign of a frequency-dependent susceptibility parameter. This study shows that one may control to some extent magnetostatic modes via external fields or geometry. For example, one may imagine propagation along interior regions avoiding surfaces, as is suggested in one case presented here.