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.

Abstract. We develop an approach to treat magnetic energy of a ferromagnet for arbitrary curved wires and shells on the assumption that the anisotropy contribution much exceeds the dipolar and other weak interactions. We show that the curvature induces two effective magnetic interactions: effective magnetic anisotropy and effective Dzyaloshinskii-like interaction. We derive an equation of magnetisation dynamics and propose a general static solution for the limit case of strong anisotropy. To illustrate our approach we consider the magnetisation structure in a ring wire and a cone surface: ground states in both systems essentially depend on the curvature excluding strictly tangential solutions even in the case of strong anisotropy. We derive also the spectrum of spin waves in such systems.

Topologically stable structures include vortices in a wide variety of matter, such as skyrmions in ferro-and antiferromagnets, and hedgehog point defects in liquid crystals and ferromagnets. These are characterized by integer-valued topological quantum numbers. In this context, closed surfaces are a prominent subject of study as they form a link between fundamental mathematical theorems and real physical systems. Here we perform an analysis on the topology and stability of equilibrium magnetization states for a thin spherical shell with easy-axis anisotropy in normal directions. Skyrmion solutions are found for a range of parameters. These magnetic skyrmions on a spherical shell have two distinct differences compared to their planar counterpart: (i) they are topologically trivial, and (ii) can be stabilized by curvature effects, even when Dzyaloshinskii-Moriya interactions are absent. Due to its specific topological nature a skyrmion on a spherical shell can be simply induced by a uniform external magnetic field.

Magnetic helix wire is one of the most simple magnetic systems which manifest properties of both curvature and torsion. There exist two equilibrium states in the helix wire with easy-tangential anisotropy: a quasi-tangential magnetization distribution in case of relatively small curvatures and torsions, and an onion state in opposite case. In the last case the magnetization is close to tangential one, deviations are caused by the torsion and curvature. Possible equilibrium magnetization states in the helix magnet with different anisotropy directions are studied theoretically. The torsion also essentially influences the spin-wave dynamics, acting as an effective magnetic field. Originated from the curvature induced effective Dzyaloshinskii interaction, this magnetic field leads to the coupling between the helix chirality and the magnetochirality, it breaks mirror symmetry in spinwave spectrum. All analytical predictions on magnetization statics an dynamics are well confirmed by the direct spin lattice simulations.

Properties of magnon modes localized on a ferromagnetic skyrmion are studied. Three types of possible asymptotic behavior of the modes eigenfrequencies are found for the case of large skyrmion radius Rs, namely ω0 ∼ R −2 s for the breathing mode, ω −|µ| ∼ R −1 s and ω |µ| ∼ R −3 s for modes with negative and positive azimuthal quantum numbers, respectively. A number of properties of the magnon eigenfunctions are determined. This enables us to demonstrate that the skyrmion dynamics based on the traveling wave model is described by the massless Thiele equation.

The interplay of topological defects with curvature is studied for
out-of-surface magnetic vortices in thin spherical nanoshells. In the case of
easy-surface Heisenberg magnet it is shown that the curvature of the underlying
surface leads to a coupling between the localized out-of-surface component of
the vortex with its delocalized in-surface structure, i.e. polarity-chirality
coupling.Comment: 6 pages, 4 figure

The control of the vortex state magnetic nanoparticle by ultrafast magnetic fields is studied theoretically. Using the micromagnetic simulations for the Permalloy nanodisk we demonstrate that the vortex core magnetization can be irreversible switched by the alternating field, rotating in the disk plane, with the frequency about 10 GHz and intensity about 20 mT. We propose an analytical picture of such phenomena involving the creation and annihilation of vortex-antivortex pairs and calculate the phase diagram of the fields parameters leading to the switching.

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