Curvature Effects in Thin Magnetic Shells

A magnetic energy functional is derived for an arbitrary curved thin shell on the assumption that the magnetostatic effects can be reduced to an effective easy-surface anisotropy; it can be used for solving both static and dynamic problems. General static solutions are obtained in the limit of a strong anisotropy of both signs (easy-surface and easy-normal cases). It is shown that the effect of the curvature can be treated as the appearance of an effective magnetic field, which is aligned along the surface normal for the case of easy-surface anisotropy and is tangential to the surface for the case of easy-normal anisotropy. In general, the existence of such a field excludes the solutions that are strictly tangential or strictly normal to the surface. As an example, we consider static equilibrium solutions for a cone surface magnetization.

Figure 1

Onion state of a cone surface. a) Geometry and notations. Panels b) and c) show the onion solution (11) for $\psi =0$ and $\psi =\pi /4$, respectively, including the in-surface magnetization distribution (by streamlines) and the normal component $m_n$, normalized by $m_n^c={\sigma }^2/R^2$ (by color scheme). d) Variation of $m_n$ along the azimuthal direction ${\mathit{e}}_{\chi }$ for a cone with $\psi =\pi /4$. e) Schematic distributions of the magnetization $\mathit{m}$ and effective curvature field ${\mathbf{F}}^{\mathrm{t}}$ within the cut plane $z0y$.

Figure 2

a) Energies of the onion (thick line) and axial (thin line) solutions. The magnetization distributions of the axial-state cone with $\psi =\pi /3$ are shown precisely and schematically in panels b) and c), respectively. The other notations are the same as in Fig. 1.

Figure 3

Schematic representation of the magnetization $\mathit{m}$ and curvature-induced effective field ${\mathbf{F}}^{\mathrm{n}}$ distributions for the case of a strong easy-normal anisotropy, with a) inward and b) outward magnetization orientations.