# Magnetization in narrow ribbons: curvature effects

## TL;DR

This paper investigates how curvature and torsion influence magnetization states and spin wave excitations in narrow magnetic ribbons, revealing transitions in ground states and asymmetric dispersion laws due to geometry and chirality.

## Contribution

It derives a reduced one-dimensional micromagnetic model for narrow ribbons incorporating curvature and torsion effects, and analyzes specific geometries like helicoid and M"obius ribbons.

## Key findings

- Ground state magnetization transitions depend on tangential anisotropy.
- Helicoid ribbon exhibits a uniform perpendicular ground state below zero anisotropy.
- M"obius ribbon shows a nonuniform ground state at positive critical anisotropy.

## Abstract

A ribbon is a surface swept out by a line segment turning as it moves along a central curve. For narrow magnetic ribbons, for which the length of the line segment is much less than the length of the curve, the anisotropy induced by the magnetostatic interaction is biaxial, with hard axis normal to the ribbon and easy axis along the central curve. The micromagnetic energy of a narrow ribbon reduces to that of a one-dimensional ferromagnetic wire, but with curvature, torsion and local anisotropy modified by the rate of turning. These general results are applied to two examples, namely a helicoid ribbon, for which the central curve is a straight line, and a M\"obius ribbon, for which the central curve is a circle about which the line segment executes a $180^\circ$ twist. In both examples, for large positive tangential anisotropy, the ground state magnetization lies tangent to the central curve. As the tangential anisotropy is decreased, the ground state magnetization undergoes a transition, acquiring an in-surface component perpendicular to the central curve. For the helicoid ribbon, the transition occurs at vanishing anisotropy, below which the ground state is uniformly perpendicular to the central curve. The transition for the M\"obius ribbon is more subtle; it occurs at a positive critical value of the anisotropy, below which the ground state is nonuniform. For the helicoid ribbon, the dispersion law for spin wave excitations about the tangential state is found to exhibit an asymmetry determined by the geometric and magnetic chiralities.