Michele Ruggeri

Multiscale modeling in micromagnetics: Existence of solutions and numerical integration

Various applications ranging from spintronic devices, giant magnetoresistance sensors, and magnetic storage devices, include magnetic parts on very different length scales. Since the consideration of the Landau–Lifshitz–Gilbert equation (LLG) constrains the maximum element size to the exchange length within the media, it is numerically not attractive to simulate macroscopic parts with this approach. On the other hand, the magnetostatic Maxwell equations do not constrain the element size, but cannot describe the short-range exchange interaction accurately. A combination of both methods allows one to describe magnetic domains within the micromagnetic regime by use of LLG and also considers the macroscopic parts by a nonlinear material law using the Maxwell equations. In our work, we prove that under certain assumptions on the nonlinear material law, this multiscale version of LLG admits weak solutions. Our proof is constructive in the sense that we provide a linear-implicit numerical integrator for the multiscale model such that the numerically computable finite element solutions admit weak H1-convergence (at least for a subsequence) towards a weak solution.

Spin-polarized transport in ferromagnetic multilayers: An unconditionally convergent FEM integrator

We propose and analyze a decoupled time-marching scheme for the coupling of the Landau–Lifshitz–Gilbert equation with a quasilinear diffusion equation for the spin accumulation. This model describes the interplay of magnetization and electron spin accumulation in magnetic and nonmagnetic multilayer structures. Despite the strong nonlinearity of the overall PDE system, the proposed integrator requires only the solution of two linear systems per time-step. Unconditional convergence of the integrator towards weak solutions is proved.

A three-dimensional spin-diffusion model for micromagnetics.

We solve a time-dependent three-dimensional spin-diffusion model coupled to the Landau-Lifshitz-Gilbert equation numerically. The presented model is validated by comparison to two established spin-torque models: The model of Slonzewski that describes spin-torque in multi-layer structures in the presence of a fixed layer and the model of Zhang and Li that describes current driven domain-wall motion. It is shown that both models are incorporated by the spin-diffusion description, i.e., the nonlocal effects of the Slonzewski model are captured as well as the spin-accumulation due to magnetization gradients as described by the model of Zhang and Li. Moreover, the presented method is able to resolve the time dependency of the spin-accumulation.

A self-consistent spin-diffusion model for micromagnetics.

We propose a three-dimensional micromagnetic model that dynamically solves the Landau-Lifshitz-Gilbert equation coupled to the full spin-diffusion equation. In contrast to previous methods, we solve for the magnetization dynamics and the electric potential in a self-consistent fashion. This treatment allows for an accurate description of magnetization dependent resistance changes. Moreover, the presented algorithm describes both spin accumulation due to smooth magnetization transitions and due to material interfaces as in multilayer structures. The model and its finite-element implementation are validated by current driven motion of a magnetic vortex structure. In a second experiment, the resistivity of a magnetic multilayer structure in dependence of the tilting angle of the magnetization in the different layers is investigated. Both examples show good agreement with reference simulations and experiments respectively.

Mixed formulation for interface problems with distributed Lagrange multiplier

We study a mixed formulation for elliptic interface problems which has been recently introduced when dealing with a test problem arising from fluid-structure interaction applications. The formulation, which involves a Lagrange multiplier defined in the solid domain, can be approximated by standard finite elements on meshes which do not need to fit with the interface. In this paper we discuss a modification of the original formulation involving a different approach for the analysis and the numerical implementation of the Lagrange multiplier. New two-dimensional numerical results confirm the good performances of the proposed schemes.

Convergence of an implicit–explicit midpoint scheme for computational micromagnetics

Abstract Based on lowest-order finite elements in space, we consider the numerical integration of the Landau–Lifschitz–Gilbert equation (LLG). The dynamics of LLG is driven by the so-called effective field which usually consists of the exchange field, the external field, and lower-order contributions such as the stray field. The latter requires the solution of an additional partial differential equation in full space. Following Bartels and Prohl (2006), we employ the implicit midpoint rule to treat the exchange field. However, in order to treat the lower-order terms effectively, we combine the midpoint rule with an explicit Adams–Bashforth scheme. The resulting integrator is formally of second-order in time, and we prove unconditional convergence towards a weak solution of LLG. Numerical experiments underpin the theoretical findings.