Atomistic-continuum multiscale modelling of magnetisation dynamics at non-zero temperature

In this article, a few problems related to multiscale modelling of magnetic materials at finite temperatures and possible ways of solving these problems are discussed. The discussion is mainly centred around two established multiscale concepts: the partitioned domain and the upscaling-based methodologies. The major challenge for both multiscale methods is to capture the correct value of magnetisation length accurately, which is affected by a random temperature-dependent force. Moreover, general limitations of these multiscale techniques in application to spin systems are discussed.Comment: 30 page

Moving toward an atomistic reader model

With the move to recording densities up to and beyond 1 Tb/in/sup 2/, the size of read elements is continually reducing as a requirement of the scaling process. The expectation is for read elements containing magnetic films as thin as 1.5 nm, in which finite size effects, and factors such as interface mixing might be expected to become of increasing importance. Here, we review the limitations of the current (micromagnetic) approach to the theoretical modeling of thin films and develop an atomistic multiscale model capable of investigating the magnetic properties at the atomic level. Finite-size effects are found to be significant, suggesting the need for models beyond the micromagnetic approach to support the development of future read sensors

Magnetic free energy at elevated temperatures and hysteresis of magnetic particles

We derive a free energy for weakly anisotropic ferromagnets which is valid in the whole temperature range and interpolates between the micromagnetic energy at zero temperature and the Landau free energy near the Curie point T_c. This free energy takes into account the change of the magnetization length due to thermal effects, in particular, in the inhomogeneous states. As an illustration, we study the thermal effect on the Stoner-Wohlfarth curve and hysteresis loop of a ferromagnetic nanoparticle assuming that it is in a single-domain state. Within this model, the saddle point of the particle's free energy, as well as the metastability boundary, are due to the change in the magnetization length sufficiently close to T_c, as opposed to the usual homogeneous rotation process at lower temperatures.Comment: 16 pages, 4 figure

Homogenization in magnetic-shape-memory polymer composites

Magnetic-shape-memory materials (e.g. specific NiMnGa alloys) react with a large change of shape to the presence of an external magnetic field. As an alternative for the difficult to manifacture single crystal of these alloys we study composite materials in which small magnetic-shape-memory particles are embedded in a polymer matrix. The macroscopic properties of the composite depend strongly on the geometry of the microstructure and on the characteristics of the particles and the polymer. We present a variational model based on micromagnetism and elasticity, and derive via homogenization an effective macroscopic model under the assumption that the microstructure is periodic. We then study numerically the resulting cell problem, and discuss the effect of the microstructure on the macroscopic material behavior. Our results may be used to optimize the shape of the particles and the microstructure.Comment: 17 pages, 4 figure

Stabilization methods in relaxed micromagnetism

The magnetization of a ferromagnetic sample solves a non-convex variational problem, where its relaxation by convexifying the energy density resolves relevant macroscopic information. The numerical analysis of the relaxed model has to deal with a constrained convex but degenerated, nonlocal energy functional in mixed formulation for magnetic potential u and magnetization m. In [C. Carstensen and A. Prohl, Numer. Math. 90 (2001) 65–99], the conforming P1 - (P0)d-element in d=2,3 spatial dimensions is shown to lead to an ill-posed discrete problem in relaxed micromagnetism, and suboptimal convergence. This observation motivated a non-conforming finite element method which leads to a well-posed discrete problem, with solutions converging at optimal rate. In this work, we provide both an a priori and a posteriori error analysis for two stabilized conforming methods which account for inter-element jumps of the piecewise constant magnetization. Both methods converge at optimal rate; the new approach is applied to a macroscopic nonstationary ferromagnetic model [M. Kružík and A. Prohl, Adv. Math. Sci. Appl. 14 (2004) 665–681 – M. Kružík and T. Roubíček, Z. Angew. Math. Phys. 55 (2004) 159–182 ]