Néel and Cross-Tie Wall Energies for Planar Micromagnetic Configurations

Abstract

Abstract.We study a two-dimensional model for micromagnetics, which consists in an energy functional over S 2 -valued vector fields. Bounded-energy configurations tend to be planar, except in small regions which can be described as vortices (Bloch lines in physics). As the characteristic "exchangelength" tends to 0, they converge to planar divergence-free unit norm vector fields which jump along line singularities. We derive lower bounds for the energy, which are explicit functions of the jumps of the limit. These lower bounds are proved to be optimal and are achieved by one-dimensional profiles, corresponding to Néel walls, if the jump is small enough (less than π/2 in angle), and by two-dimensional profiles, corresponding to cross-tie walls, if the jump is bigger. Thus, it provides an example of a vector-valued phase-transition type problem with an explicit non-one-dimensional energy-minimizing transition layer. We also establish other lower bounds and compactness properties on different quantities which provide a good notion of convergence and cost of vortices.Mathematics Subject Classification. 35J20, 35J60, 35Q60, 49S05, 49K20.

Convergence of a ferromagnetic film model

“…We also force the configurations to be vertically invariant. This permits us to do an analysis similar to those of [9,11] at first order and moreover to go one step further in the spirit of [2,19,20]. We also deeply think that this model could be of interest for the thick films physical situation, as suggested by the numerical computations given in [18].…”

“…Un second modèle, moins proche de la physique a ensuiteétéétudié par T. Rivière et S. Serfaty [19,20] puis F. Alouges, T. Rivière et S. Serfaty [2] ; ici, l'épaisseur η est infinie et ce modèle rend compte de phénomènes physiques importants tels que les structures en "cross-tie". Il permet de plus d'assurer une convergence forte des minimiseurs.…”

“…The derivation of reduced models in which the thickness η vanishes has been the aim of a few mathematical papers in the last decade [2,6,8,20]. These models are usually analyzed by means of Γ-convergence from the original problem.…”

“…The following order in the energy is given by the behavior of the magnetization inside the walls. This has been studied by T. Rivière and S. Serfaty [19,20] and F. Alouges, T. Rivière and S. Serfaty [2], on less physical models corresponding to η = ∞ and where the magnetization does not depend on the vertical coordinate. In [19,20] the magnetization is moreover constrainted to be in the horizontal plane, which avoids the vortices.…”

“…In [19,20] the magnetization is moreover constrainted to be in the horizontal plane, which avoids the vortices. Contrarily, in [2], the vertical component of the magnetization is only penalized and so, one can observe vortices (Bloch lines).…”