ESAIM: Control, Optimisation and Calculus of Variations

Research Article

Néel and Cross-Tie Wall Energies for Planar Micromagnetic Configurations

Alouges, Françoisa1, Rivière, Tristana2 and Serfaty, Sylviaa3

a1 Département de Mathématiques, Université de Paris-Sud, bâtiment 425, 91405 Orsay Cedex, France;

a2 Department of Mathematics, ETH Zürich, 8092 Zürich, Switzerland;

a3 Courant Institute, 251 Mercer St., New York, NY 10012, USA, and supported by CNRS;


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 “exchange-length” 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.

(Received January 18 2002)

(Online publication August 15 2002)

Key Words:

  • Micromagnetics;
  • thin films;
  • cross-tie walls;
  • Gamma-convergence.

Mathematics Subject Classification:

  • 35J20;
  • 35J60;
  • 35Q60;
  • 49S05;
  • 49K20