Domain structure and ambiguities in the dipole direction

Abstract

A self-consistent mathematical formulation of the magnetization distribution in ideal soft-magnitude media, which is based on the constitutive equation previously derived from micromagnetic principles by the author, is presented. We confine ourselves to two-dimensional solenoidal distributions in thin-film objects. In particular, we focus on the magnetization distribution in elliptical objects, and Lagrange's method is used to solve the boundary-value problem. A function of the fourth degree in the magnetization direction results that gives rise to more than one real root for each location. We make a systematic inventory of the meaning of the various roots and construct magnetization distributions that satisfy the boundary conditions. We prove that the requirement of continuity of the magnetization distributions leads to the conclusion that the magnetization direction is multiple-valued in some regions and that the domain structure is required to prevent these ambiguities from occurring. We demonstrate both theoretically and experimentally that the simplest domain structure in an ellipse is formed by a simple, straight domain wall along the longitudinal symmetry axis of the ellipse.