Magnetic Normal Mode Calculations in Big Systems: A Highly Scalable Dynamical Matrix Approach Applied to a Fibonacci-Distorted Artificial Spin Ice

Loris Giovannini,Lance E De Long,Justin S Woods,Ali Frotanpour,Federico Montoncello,Barry W Farmer

doi:10.3390/magnetochemistry7030034

Loris Giovannini, Lance E De Long + Show 4 more

Open Access

https://doi.org/10.3390/magnetochemistry7030034

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Abstract

We present a new formulation of the dynamical matrix method for computing the magnetic normal modes of a large system, resulting in a highly scalable approach. The motion equation, which takes into account external field, dipolar and ferromagnetic exchange interactions, is rewritten in the form of a generalized eigenvalue problem without any additional approximation. For its numerical implementation several solvers have been explored, along with preconditioning methods. This reformulation was conceived to extend the study of magnetization dynamics to a broader class of finer-mesh systems, such as three-dimensional, irregular or defective structures, which in recent times raised the interest among researchers. To test its effectiveness, we applied the method to investigate the magnetization dynamics of a hexagonal artificial spin-ice as a function of a geometric distortion parameter following the Fibonacci sequence. We found several important features characterizing the low frequency spin modes as the geometric distortion is gradually increased.

Highlights

A good preconditioning matrix is found for this specific problem, which improves the computational efficiency by improving the convergence, and opens the way to the use of an additional important class of eigensystem solvers, as will be discussed below. In this way we show that the aforementioned symmetry properties can be transposed to the dynamical matrix method, obtaining an approach that combines the ability to deal with big systems, as in standard micromagnetic packages, with that of naturally obtaining the eigenmodes of the structure, degenerate or not, and without the need of any external excitation
In order to get a deeper understanding of the behavior of spin modes in the hexagonal structure, we have calculated the spin wave dispersion of a single, infinite permalloy strip 40 nm wide, with the same thickess of the Kagome artificial spin ice (ASI) (15 nm) and infinite length using the standard dynamical matrix method with periodic boundary conditions [39], taking into account dipolar, ferromagnetic exchange and Zeeman contributions
We illustrated a new method for studying the dynamics of large single magnetic particles and arrays, overcoming size limitations related to the computation time and the memory space needed for such systems by direct solvers

Summary

Introduction

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. A good preconditioning matrix is found for this specific problem, which improves the computational efficiency by improving the convergence, and opens the way to the use of an additional important class of eigensystem solvers, as will be discussed below In this way we show that the aforementioned symmetry properties can be transposed to the dynamical matrix method, obtaining an approach that combines the ability to deal with big systems, as in standard micromagnetic packages, with that of naturally obtaining the eigenmodes of the structure, degenerate or not, and without the need of any (specially shaped) external excitation. A system of this type has been recently studied with a different method [29]

Theoretical Approach

System Structure

Calculation Methods

Unperturbed System

Perturbed System

Conclusions