A survey of some new results in ferromagnetic thin films
Radu Ignat · French digital mathematics library (Numdam) · 2008
Ferromagnetic materials are widely used in nowadays as technological tools, especially for magnetic data storage. The modelling of very small ferromagnetic particules is based on the micromagnetic theory. The micromagnetic model states that ferromagnetic materials can be described by a 3−D vector-field distribution, called magnetization, where the stable configurations correspond to (local) minimizers of the micromagnetic energy. The associated variational problem is nonconvex and nonlocal. Moreover, it is a multi-scale system involving both intrinsic parameters (depending on the nature of the ferromagnetic material) and extrinsic parameters (coming from the geometry of the sample). According to the relative smallness of these parameters, different asymptotic regimes appear and lead to the formation of various magnetization patterns. The qualitative and quantitative analysis of the magnetization patterns is an extensively explored topic. Generically, a pattern (stable state) consists in large uniformly magnetized 3−D regions (magnetic domains) separated by narrow transition layers (magnetic walls) where the magnetization varies very rapidly. Depending on the scales of the system, the experiments predict different type of magnetic walls : 2−D wall defects (Neel walls, asymmetric Bloch wall), 1−D vortex-lines (Bloch lines) or a mixed type of vortex-wall defects (cross-tie walls). The main goal is to give a mathematical justification of the physical prediction on the formation and characterization of these defects. Classical methods of functional analysis are often insufficient to detect these phenomena of loss of regularity. New approches need to be developed in order to implement geometric measure theory contributing to the analysis of partial differential equations. In this survey, we focus on pattern formation in very thin films. The ferromagnetic samples are assumed to be cylinders with a very small thickness. In our regime, two types of magnetic walls are expected to be observed : Neel walls and Bloch lines. Moreover, there exists a physical prediction on global configurations of the magnetization : as stated by van den Berg [29], the observed magnetizations at the “mesoscopic” level are 2−D unit-length vector fields of distributionally vanishing divergence. A special configuration is the Landau state that corresponds to the viscosity solution of the eikonal equation. It fits perfectly to the magnetization pattern observed in experiments on rectangular thin films (see Hubert and Schafer [15]). Our aim is to present some results that rigourously prove the van den Berg conjecture at least in a special regime. For this purpose, we discuss the properties of Neel walls and Bloch lines. These defects give the leading order term of the energy of the Landau state. The main result shows compactness of configurations energetically close to the Landau state in the case where a Bloch line is energetically more expensive than a Neel wall. Consequently, their limiting pattern satisfies the van den Berg prediction. The paper is organized as follows : we start with a mathematical description of the 3−D micromagnetic model. Then we discuss a thin film regime where the 3−D model can be asymptotically ∗Laboratoire de Mathematiques, Universite Paris-Sud 11, Bât. 425, 91405 Orsay, France (e-mail: [email protected])