Thin cylindrical magnetic nanodots revisited: variational formulation, accurate solution and phase diagram

TL;DR

This paper develops a rigorous variational approach to analyze magnetization states in thin cylindrical magnetic nanodots, providing accurate phase diagrams and critical parameters by considering exchange and demagnetization energies.

Contribution

It introduces a precise variational formulation combining Ritz's method and Fourier series, improving upon heuristic methods for modeling nanodot magnetization states.

Findings

Derived an explicit expression for demagnetization energy in single-domain states.

Produced an accurate phase diagram for vortex and single-domain states.

Validated phase diagram with existing 2D and 3D model data.

Abstract

We investigate the variational formulation and corresponding minimizing energies for the detection of energetically favorable magnetization states of thin cylindrical magnetic nanodots. Opposed to frequently used heuristic procedures found in the literature, we revisit the underlying governing equations and construct a rigorous variational approach that takes both exchange and demagnetization energy into account. Based on a combination of Ritz's method and a Fourier series expansion of the solution field, we are able to pinpoint the precision of solutions, which are given by vortex modes or single-domain states, down to an arbitrary degree of precision. Furthermore, our model allows to derive an expression for the demagnetization energy in closed form for the in-plane single-domain state, which we compare to results from the literature. A key outcome of the present investigation is an accurate phase diagram, which we obtain by comparing the vortex mode's energy minimizers with those of the single-domain states. This phase diagram is validated with data of two- and three-dimensional models from literature. By means of the phase diagram, we particularly find the critical radius at which the vortex mode becomes unfavorable with machine precision. All relevant data and codes related to the present contribution are available at http://dx.doi.org/10.18419/darus-3103.