Research Page: Magnetization of Very Thin Ferromagnetic Films

Research Page

Magnetization of Very Thin Ferromagnetic Films

Ferromagnetic thin films are used in numerous applications. We can instance magnetic tapes with continuous, very thin film media (as opposed to the traditional particulate media) used in recent years for very high density audio and video recording. In a classical paper (Phil. Trans. R. Soc. London 240, 1948), Stoner and Wohlfarth developed a theory of uniformily magnetized ferromagnetic ellipsoids. By considering the limit of very thin, penny-shaped ellipsoids, Stoner and Wohlfarth's theory has been widely used to understand the magnetization of thin films. The conclusions drawn for penny-shaped ellipsoids are known to work well when applied to thin films of constant thickness and arbitrary shape in plan. However, a theory of magnetization of thin ferromagnetic films of constant thickness and arbitrary shape in plan remains unavailable. Our research is aimed at developing such a theory based on scaling arguments and weak convergence methods.

Theory

We consider the standard micromagnetic energy of a very thin ferromagnetic film of thickness h, E^h(m). E^h(m) is a functional of the magnetization function, m; the minimizer m^h of E^h (m) is the solution of the micromagnetic problem. To study the structure of the energy functional in the limit of very small film thickness, we have mapped an arbitrary film of thickness h to a film of the same shape and unit thickness, see the figure. By means of a scaling calculation and weak convergence methods, we have shown that the minimizer m^h converges in some sense to a function m° as h->0, where m° is the minimizer of a certain limiting energy functional E°. We have obtained an expression for the limiting energy functional, and have shown that in the limit h->0 the variational problem is completely local -that is to say, that the magnetostatic equation which constrains the magnetization m in the original problem disappears from the limiting one. The minimizer m° is independent of the position normal to the thin film; thus, the limiting problem is two dimensional. Furthermore, we have shown that the thinness of the film imports an artificial anisotropy which disfavors out-of-plane magnetization.

Predictions

The limiting energy functional, E°, is such that if the applied field H is constant, then the energy minimizer is a single domain, i.e., the magnetization is uniform. When the material is isotropic, the magnetization m is contained in both i) the plane of the film and ii) the plane defined by the applied field H and the normal to the film; furthermore, m is such that H·m>0. When H=0, m is contained in the plane of the film, but its orientation is undetermined. Therefore, an alternating H perpendicular to the film does not lead to magnetic hysteresis, whereas an alternating H in the plane of the film causes rectangular hysteresis loops. These predictions coincide with those of Stoner and Wohlfarth for penny-shaped ellipsoids. Indeed, we have shown that our calculation extends the applicability of Stoner and Wohlfarth's classical results to thin films of arbitrary shape in plan.

References

Gioia, G., and James, R.D.
Micromagnetics of Very Thin Films
Proc. Royal Soc. London A 453, 213-223, 1997.
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