Computational micromagnetics with Commics

We present our open-source Python module Commics for the study of the magnetization dynamics in ferromagnetic materials via micromagnetic simulations. It implements state-of-the-art unconditionally convergent finite element methods for the numerical integration of the Landau–Lifshitz–Gilbert equation. The implementation is based on the multiphysics finite element software Netgen/NGSolve. The simulation scripts are written in Python, which leads to very readable code and direct access to extensive post-processing. Together with documentation and example scripts, the code is freely available on GitLab. Program summary: Program title: Commics Program Files doi: http://dx.doi.org/10.17632/29wv9h78h7.1 Licensing provisions: GPLv3 Programming language: Python3 Nature of problem: Numerical integration of the Landau–Lifshitz–Gilbert equation in three space dimensions Solution method: Tangent plane scheme [1]: original first-order version, projection-free version, second-order version, efficient second-order IMEX version; Midpoint scheme [2]: original version, IMEX version; Magnetostatic Maxwell equations are treated by the hybrid FEM–BEM method [3] Additional comments including restrictions and unusual features: An installation of the finite element software Netgen/NGSolve and an installation of the boundary element library BEM++ are required. References [1] F. Alouges. A new finite element scheme for Landau–Lifchitz equations. Discrete Contin. Dyn. Syst. Ser. S, 1(2):187–196, 2008. [2] S. Bartels and A. Prohl. Convergence of an implicit finite element method for the Landau–Lifshitz–Gilbert equation. SIAM J. Numer. Anal., 44(4):1405–1419, 2006. [3] D. R. Fredkin and T. R. Koehler. Hybrid method for computing demagnetization fields. IEEE Trans. Magn., 26(2):415–417, 1990

Theory and Simulation of the Microwave Response of Concentric Ferromagnetic Shells

Structured ferromagnetic metal-based metamaterials comprised of spherical particles exhibit properties that are attractive for microwave applications, such as a broad frequency bandwidth and higher working frequencies when compared with bulk ferrimagnetic oxides. In this thesis, the dynamical properties of ferromagnetic spherical shells are studied using a combination of analytical and numerical methods, to further understanding and enhance the permeability of these materials towards higher frequencies. Using linearised micromagnetic equations, saturated spherical shells are investigated in the exchange-dominated regime when assuming that surface anisotropy is present at both the inner and outer boundaries. This configuration is amenable to exact solutions for the resonance eigenvalues and to investigate the size/thickness dependence of the resonance frequencies. It is found that the mode frequency can increase with decreasing shell thickness or is driven rapidly towards the ferromagnetic resonance frequency depending on the choice of the surface anisotropy constant at each boundary. Moreover, surface anisotropy introduces a dependence of the zeroth mode on shell thickness, removing the degeneracy with the ferromagnetic resonance and leading to a pronounced size dependence of this mode for thin shells. A generalised resonance theory is further outlined for a multilayered spherical nanoparticle comprised of exchange-coupled concentric layers. It can be used to compute the resonance spectra of core-shell nanoparticles, as in the case of a solid spherical ferromagnetic core surrounded by an outer oxide shell. Detailed micromagnetic modelling of two- and three-dimensional ferromagnetic particles was carried out to study the role of long-range magnetostatic interactions between concentric rings and the influence of realistic domain structures on the dynamic susceptibility. Micromagnetic modelling of such structures demonstrates that a family of higher-order flexural modes is present for spherical shells relaxed into the vortex state, which can reach high-frequencies 20-25 GHz under weak-field excitations. These simulations provide an alternative and more plausible interpretation of observed high-frequency resonance modes in measured permeability spectra of spherical shell particle composites, and aid in the design of high-frequency, light-weight composite materials. The dynamical properties of three-dimensional permalloy elements supporting vortex domain structures were also investigated with micromagnetic simulations and compared with experiment. This is to study the influence of nonuniform field gradients and three-dimensional static magnetisation configurations on 1 the dynamical behaviour. It is found that the permalloy elements support domain walls with perpendicular out-of-plane components which can be switched dynamically in response to specific magnetic pulse parameters. This project further aimed to incorporate the fundamental nonlinear micromagnetic and electromagnetic details, including exchange and magnetocrystalline anisotropy, within the finite-difference time-domain (FDTD) method. This is to study the interaction between magnetic materials and electromagnetic waves in the presence of current and magnetic sources at microwave frequencies. Results are presented for conducting semi-infinite permalloy pillars in the micrometer and sub-micrometer size range. It is found that microwave absorption results primarily from edge modes localised at the boundaries of the pillar in accordance with the skin depth, which appear at a lower frequency than the ferromagnetic resonanceEngineering and Physical Sciences Research Council (EPSRC

Trochoidal motion and pair generation in skyrmion and antiskyrmion dynamics under spin-orbit torques

Skyrmions and antiskyrmions in magnetic ultrathin films are characterised by a topological charge describing how the spins wind around their core. This topology governs their response to forces in the rigid core limit. However, when internal core excitations are relevant, the dynamics become far richer. We show that current-induced spin-orbit torques can lead to phenomena such as trochoidal motion and skyrmion-antiskyrmion pair generation that only occurs for either the skyrmion or antiskyrmion, depending on the symmetry of the underlying Dzyaloshinskii-Moriya interaction. Such dynamics are induced by core deformations, leading to a time-dependent helicity that governs the motion of the skyrmion and antiskyrmion core. We compute the dynamical phase diagram through a combination of atomistic spin simulations, reduced-variable modelling, and machine learning algorithms. It predicts how spin-orbit torques can control the type of motion and the possibility to generate skyrmion lattices by antiskyrmion seeding

Study of the applied current asymmetry of the spin-torque interaction

This thesis describes experimental work that is designed to address unresolved issues in domain-wall dynamics that have emerged from recent scientific work on magnetic field and electric-current driven domain-wall dynamics in permalloy microstructures. A principle issue involves nonlinear effects in the electric-current driven domain-wall velocity under reversal of electric current. Existing theoretical models of the spin-torque effect (responsible for electric-current driven domain-wall motion) do not allow nonlinear current response. Other issues are related to magnetic-field and electric-current driven Barkhausen jumps in magnetic microstructure. Sample preparation and characterization studies reported provide the technical basis for improved experiments that should resolve the stated unsettled scientific issues.Physic

Multiscale and multimodel simulation of Bloch point dynamics

We present simulation results on the structure and dynamics of micromagnetic point singularities with atomistic resolution. This is achieved by embedding an atomistic computational region into a standard micromagnetic algorithm. Several length scales are bridged by means of an adaptive mesh refinement and a seamless coupling between the continuum theory and a Heisenberg formulation for the atomistic region. The code operates on graphical processing units and is able to detect and track the position of strongly inhomogeneous magnetic regions. This enables us to reliably simulate the dynamics of Bloch points, which means that a fundamental class of micromagnetic switching processes can be analyzed with unprecedented accuracy. We test the code by comparing it with established results and present its functionality with the example of a simulated field-driven Bloch point motion in a soft-magnetic cylinder

Numerical micromagnetism of strong inhomogeneities

The size of micromagnetic structures, such as domain walls or vortices, is comparable to the exchange length of the ferromagnet. Both, the exchange length of the stray field $l_s$ and the magnetocrystalline exchange length $l_k$ are material-dependent quantities that usually lie in the nanometer range. This emphasizes the theoretical challenges associated with the mesoscopic nature of micromagnetism: the magnetic structures are much larger than the atomic lattice constant, but at the same time much smaller than the sample size. In computer simulations, the smallest exchange length serves as an estimate for the largest cell size admissible to prevent appreciable discretization errors. This general rule is not valid in special situations where the magnetization becomes particularly inhomogeneous. When such strongly inhomogeneous structures develop, micromagnetic simulations inevitably contain systematic and numerical errors. It is suggested to combine micromagnetic theory with a Heisenberg model to resolve such problems. We analyze cases where strongly inhomogeneous structures pose limits to standard micromagnetic simulations, arising from fundamental aspects as well as from numerical drawbacks