Self-consistent domain theory in soft ferromagnetic media. I. Solenoidal distributions in elliptical thin-film elements

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Abstract
The basic concepts of a self-consistent domain theory for the class of ideal soft ferromagnetic media is unfolded. The treatment is based on micromagnetic equilibrium and stability equations which are simplified by removing the contributions due to the intrinsic anisotropy and the spatial variation term in the exchange energy density. Only the solenoidal magnetization distributions in thin film objects are investigated. This limitation makes confinement to two-dimensional dipole distributions possible. Cauchy’s method of characteristics is employed to derive dipole distributions that satisfy the relation ∇⋅M=0, where M is the magnetization, and the constitutive equation. It is proved that the characteristic base curves are straight lines perpendicular to the magnetization vector. In an ellipse, four different characteristic base curves that do not coincide intersect at one single point in certain regions, thus generating ambiguities in the magnetization direction. It is demonstrated that these ambiguities originate in the incompatibility of the magnetization distribution that is imposed by the various segments of the edge of the ellipse. Based on differential geometrical principles, a partitioning of the edge into segments is introduced, and along with it two adjoining regions in which the magnetization direction is uniquely determined are defined. It is proved that a domain wall is required in the cross section of both regions to accomplish a dipole configuration in stable equilibrium. Experimental confirmation is given by means of the ferrofluid pattern of a 3500-Å-thick Permalloy layer with elliptical geometry.