G.W. Roberts

Geometric integration of the Gilbert equation

Geometric integration refers to numerical methods which aim to preserve the qualitative and geometric features of a differential equation after discretization. In micromagnetics, the magnetization vector M represents a statistical average of magnetic moments, the magnitude of which should be conserved in time. In general, the midpoint scheme preserves the modulus of solutions on the sphere due to intrinsic quadratic invariance. This method has previously been implemented in various forms to solve the Landau–Lifshitz and Landau–Lifshitz–Gilbert equations within finite-difference formulations. In this article, it is shown that an explicit Euler method will overestimate |M| while an implicit Euler method will make an underestimate, whereas the midpoint method conserves |M| up to round-off error. The midpoint scheme is then utilized within a variational finite-element formulation of the Gilbert equation. Numerical stability and error control are discussed using the reversal of a cobalt nanoparticle as an exam...

Computational and experimental micromagnetics of arrays of 2-D platelets

2-D regular nanoelements are of interest as micromagnetic model systems and in a number of sensor applications. In this paper we concentrate on a recent development in the form of experimental structures of arrays of small nanoelements which are 300 nm long and between 50-80 nm wide in small arrays which are amenable to computational studies. A direct comparison of theoretical and experimental hysteresis loops gives good quantitative agreement and suggests that both interactions and variations in intrinsic properties contribute significantly to the width of the loops. The experimental samples were produced by electron beam lithography and consisted of either a 6/spl times/3 array or a 6 element row. The intra-row spacing was 50 nm or 80 nm and the inter-row spacing was 100 nm. Magnetic images were obtained by Lorentz microscopy, from which the magnetization curves were determined. Computational studies were carried out using a finite element method with magnetostatic field calculations via the maximization of the scalar potential. The technique is computationally efficient and allows the calculation of the properties of interacting elements.

Finite element modelling of nanoelements

Abstract The numerical simulation of the magnetisation structure of two-dimensional permalloy nanoelements is described. The nanoelements have a uniform magnetisation structure and consist of bars with flat and/or pointed ends. The Landau–Lifschitz–Gilbert equation of motion is used to described the time evolution of the system and the numerical simulation consists of a finite element spatial discretisation coupled with a time-stepping scheme. The Poisson equation for the demagnetising field scalar potential φ is solved by a standard finite element variational or stationary functional formulation. The jump condition in ∇ φ at a nanoelement boundary is integrated into the finite element formulation so that φ is obtained globally, both inside and outside the nanoelement, thus allowing interactions between two or more nanoelements to be modelled easily. The work will be compared to previously published experimental and numerical results for single nanoelements.

Modelling the evolution of domains in nanoelements using finite elements

Investigation into the theoretical magnetic behaviour of permalloy is described. The material is discretised into a structure of nanoelements, so that we may employ micromagnetic simulations in order to investigate the material behaviour. In this paper we describe the formation and structure of domains in an array of interacting nanoelements with varying space between them. The simulations begin at saturation and end when the nanoelements are in a zero field equilibrium state. The time evolution of the system is described by the Landau-Lifschitz equation of motion and the field calculations are carried out by the use of a finite element spatial discretisation scheme.

Simulation of the micromagnetic behavior of arrays of interacting nanoelements

In this work we investigate the behavior of small arrays of interacting permalloy particles at the submicron level. Each individual particle is termed a nanoelement and is rectangular in form with varying elongation. The interest in such structures of magnetic material is increasing, due mainly to the possible potential use in future high-density magnetic storage media applications. To carry out our investigations we have developed a dynamical micromagnetic model based on the use of the finite element method. For our results we investigate the effects of misaligned and aligned anisotropy distributions on arrays with varying size and space of nanoelement. We observe that the reversal mechanism of the arrays is very sensitive to the disorder of the intrinsic material properties. In the case of aligned uniaxial anisotropy a highly symmetric cooperative switching mechanism is observed. The larger anisotropy has the effect of stabilizing states during the reversal process, leading to distinctive switching alon...

Influences of granular microstructure on the reversal mechanism of Co nanoelements

A two-dimensional micromagnetic simulation is developed using a spatial finite element method, with the dynamic Gilbert equations discretized by a Galerkin method. An algorithm is presented for the generation of granular microstructures via the Voronoi tessellation and rectangular nanoelements of 200×40×20 nm3 are constructed. Hysteresis simulations are performed on isolated nanoelements in order to determine the extent to which experimentally observed variations in coercivity are attributable to the microstructural properties rather than magnetostatic interactions with neighboring members of an array. Dependence of coercivity on grain size is examined, as well as the role of grain structure in the reversal process. Mean grain size is shown to dictate the mode of reversal, whereas grain irregularity is observed to influence specific intermediate magnetization configurations and therefore specific coercivity values. Low correlation between grain irregularity and coercivity indicates that the magnetization ...

Investigation of magnetization behavior in nanoelements using the finite element method

A model of thin film permalloy using an efficient finite element variational approach to the magnetostatic field calculation is described. The material is discretized into a nanoelement structure at the micron level which enables us to investigate material properties due to patterning. Predicted domain structures agree well with experimental data. Interactions between elements are significant and it will be shown that the domain structure in the central element differs from that of its neighbors. As expected, the addition of pointed ends stabilizes the single-domain state. Elements with two pointed ends exhibit pseudo-single-domain behavior. We have studied the single-domain/pseudo-single-domain transition for permalloy platelets, results for noninteracting platelets are given as a function of the elongation. Interactions are shown to increase or decrease the critical size depending on the geometry.

Recent studies in thermal activation and spin waves

Abstract Some recent results in computational approaches to thermally activated fast reversal in magnetic recording media are reviewed. In particular, recent results reported in the simulation of pulsed-field-induced magnetisation reversal and thermal activation of spin waves are described. The short time scale breakdown of the Arrhenius–Neel law for a single moment is demonstrated and explained in terms of the dynamics of the precessional motion. The variation in response as a function of the damping parameter is found to be an important factor determining the remanent magnetisation for a given pulse width. The effects of interactions between moments are described, including the apparent increase in effective damping. It is shown that interactions between moments can be described in terms of thermally excited spin waves. The spectrum of relaxation times for systems consisting of coupled moments is explained in terms of the thermal excitation of spin waves.

Gilbert damping in polycrystalline thin films

Damped gyromagnetic precession equations are discussed and solutions of a single-spin system are illustrated. A brief description of our variational finite element model of magnetisation dynamics is then given. The reversal of an isolated Voronoi grain is investigated and its response to the applied magnetic field is shown to vary with the damping parameter in a manner that is somewhat different to the single-spin system, thus highlighting the need for sub-grain discretisation. Implicit periodic boundary conditions are enforced around an ensemble of Voronoi grains to simulate the effects of damping in a polycrystalline thin film. This reveals that damping influences not only the speed, but also the mode of magnetisation reversal. The minimum of magnetisation reversal time is shown to occur at the same value of the damping parameter in all three systems.

Modeling a perpendicular recording medium using a variational method

A micromagnetic model is used to investigate the dynamic behavior of a perpendicular recording medium which consists of a hard data layer and a soft underlayer. A variational finite element form of the Gilbert equation of motion is used to simulate the magnetization dynamics without thermal fluctuations. A computationally efficient variational approximation is used for the magnetization dynamics. The demagnetizing field calculation uses a hybrid finite element/boundary element calculation or a hybrid wavelet/boundary element method. The variational scheme is particularly suited to reducing numerical errors in areas of the material where magnetic inhomogeneities occur such as in the data layer during the nucleation of domains. Perpendicular recording media presents a viable alternative to the conventional longitudinal type in extending the superparamagnetic limit. Higher field strengths allowing higher densities have been shown to be sensitive to the characteristics of the soft underlayer. The dynamic effe...