Theory of Magnetization Ripple in Ferromagnetic Films

Abstract

In a polycrystalline ferromagnetic film, the magnetization direction is not uniform but exhibits small wave-like fluctuations known as magnetization ripple. In this paper, a general theory of ripple is developed, extending previous treatments by the present author, Hoffmann, and others. The theory is applicable to almost any magnetic film without gross domain structure, containing randomly oriented local anisotropies arising from inhomogeneities on any scale and of any physical origin. The magnetization direction is assumed to fluctuate in the plane of the film only. In addition to local anisotropy fields, the theory includes the effects of magnetostatic and exchange interactions, and a uniform field that consists of uniform uniaxial anisotropy and external fields. Nonlinear magnetostatic and uniform fields, which in previous treatments have been either neglected or treated as small perturbations, are taken fully into account through third-order torque terms. The ripple spectrum is derived, and from it a physical picture of the ripple is obtained in terms of a coupled region which (except for very large-scale inhomogeneities) is an elongated rhombus with long axis perpendicular to the mean magnetization. The spectrum contains a nonlinear effective field, which is evaluated in several limiting cases determined by the scale of inhomogeneity, the film thickness, an exchange length, and a magnetostatic length. The magnetization dispersion (mean ripple amplitude) is also obtained for these limiting cases, but no attempt is made to calculate other measurable magnetic-film properties. The results show: a weakening of first exchange and then magnetostatic forces as the scale of inhomogeneity increases, with the consequence that the dispersion shows an increasing dependence on the uniform field; a weakened dependence of the dispersion on both the magnitude of the local anisotropy and the uniform field for large-amplitude, nonlinear ripple; and a thickness-independent ripple in the thick-film limit, where the magnetostatic field is given by its bulk value.