Vortex Mechanics in Thin Ferromagnetic Nanodisks

TL;DR

This paper calculates the magnetostatic energy and gyrotropic frequency of magnetic vortices in elliptical nanodisks, extending understanding from circular to elliptical geometries and providing analytical expressions for various ellipticities.

Contribution

It introduces a conformal mapping approach to derive magnetization and energy expressions for elliptical nanodisks, advancing vortex mechanics modeling in noncircular geometries.

Findings

Energy depends quadratically on vortex displacement, reflecting lower symmetry.

Derived general expressions for energy and gyrotropic frequency for ellipticities 1.0 to 0.3.

Numerical integration confirms analytical predictions across ellipticities.

Abstract

The magnetostatic energy is calculated for a magnetic vortex in a noncircular elliptical nanodisk. It is well-known that the energy of a vortex in the circular disk is minimized though an ansatz that eliminates the magnetostatic charge at the disk edge. Beginning with this ansatz for the circular disk, a conformal mapping of a circle interior onto the interior of an ellipse results in the magnetization of the elliptical disk. This magnetization in the interior of an ellipse also has no magnetostatic charge at the disk edge also minimizing the magnetostatic energy. As expected the energy has a quadratic dependence on the displacement of the vortex core from the ellipse center, but reflecting the lower symmetry of the ellipse. Through numerical integration of the magnetostatic integral a general expression for the energy is obtained for ellipticity values from 1.0 to about 0.3. Finally a general expression for the gyrotropic frequency as described by the Thiele equation is obtained.