Néel and Cross-Tie Wall Energies for Planar Micromagnetic Configurations

François Alouges; Tristan Rivière; Sylvia Serfaty

ESAIM: Control, Optimisation and Calculus of Variations (2010)

  • Volume: 8, page 31-68
  • ISSN: 1292-8119

Abstract

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We study a two-dimensional model for micromagnetics, which consists in an energy functional over S2-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.

How to cite

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Alouges, François, Rivière, Tristan, and Serfaty, Sylvia. "Néel and Cross-Tie Wall Energies for Planar Micromagnetic Configurations." ESAIM: Control, Optimisation and Calculus of Variations 8 (2010): 31-68. <http://eudml.org/doc/90651>.

@article{Alouges2010,
abstract = { 
We study a two-dimensional model for micromagnetics, which consists in an energy functional over S2-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. },
author = {Alouges, François, Rivière, Tristan, Serfaty, Sylvia},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Micromagnetics; thin films; cross-tie walls; Gamma-convergence.; micromagnetics; Gamma-convergence},
language = {eng},
month = {3},
pages = {31-68},
publisher = {EDP Sciences},
title = {Néel and Cross-Tie Wall Energies for Planar Micromagnetic Configurations},
url = {http://eudml.org/doc/90651},
volume = {8},
year = {2010},
}

TY - JOUR
AU - Alouges, François
AU - Rivière, Tristan
AU - Serfaty, Sylvia
TI - Néel and Cross-Tie Wall Energies for Planar Micromagnetic Configurations
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2010/3//
PB - EDP Sciences
VL - 8
SP - 31
EP - 68
AB - 
We study a two-dimensional model for micromagnetics, which consists in an energy functional over S2-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.
LA - eng
KW - Micromagnetics; thin films; cross-tie walls; Gamma-convergence.; micromagnetics; Gamma-convergence
UR - http://eudml.org/doc/90651
ER -

References

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