Hostname: page-component-586b7cd67f-tf8b9 Total loading time: 0 Render date: 2024-11-28T14:50:34.292Z Has data issue: false hasContentIssue false
  • English
  • Français

Néel and Cross-Tie Wall Energiesfor Planar Micromagnetic Configurations

Published online by Cambridge University Press:  15 August 2002

François Alouges ,
Tristan Rivière  and
Sylvia Serfaty
François Alouges
Affiliation:
Département de Mathématiques, Université de Paris-Sud, bâtiment 425, 91405 Orsay Cedex, France; [email protected].
Tristan Rivière
Affiliation:
Department of Mathematics, ETH Zürich, 8092 Zürich, Switzerland; [email protected].
Sylvia Serfaty
Affiliation:
Courant Institute, 251 Mercer St., New York, NY 10012, USA, and supported by CNRS; [email protected].
Get access

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 “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.

Keywords

Micromagneticsthin filmscross-tie wallsGamma-convergence.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 8: A tribute to JL Lions , 2002 , pp. 31 - 68
Copyright
© EDP Sciences, SMAI, 2002

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Anzelotti, G., Baldo, S. and Visintin, A., Asymptotic behavior of the Landau-Lifschitz model of ferromagnetism. Appl. Math. Optim. 23 (1991) 171-193. CrossRef
Ambrosio, L., De Lellis, C. and Mantegazza, C., Line energies for gradient vector fields in the plane. Calc. Var. Partial Differential Equation 9 (1999) 327-355. CrossRef
Aviles, P. and Giga, Y., A mathematical problem related to the physical theory of liquid crystals configurations. Proc. Centre Math. Anal. Austral. Nat. Univ. 12 (1987) 1-16.
Aviles, P. and Giga, Y., On lower semicontinuity of a defect obtained by a singular limit of the Ginzburg-Landau type energy for gradient fields. Proc. Royal Soc. Edinburgh Sect. A 129 (1999) 1-17. CrossRef
L. Ambrosio, M. Lecumberry and T. Rivière, A Viscosity Property of Minimizing Micromagnetic Configurations. Preprint.
André, N. and Shafrir, I., On nematics stabilized by a large external field. Rev. Math. Phys. 11 (1999) 653-710. CrossRef
F. Bethuel, H. Brezis and F. Hélein, Ginzburg-Landau vortices. Birkhauser (1994).
Bethuel, F. and Zheng, X., Density of smooth functions between two manifolds in Sobolev spaces. J. Func. Anal. 80 (1988) 60-75. CrossRef
W. Brown, Micromagnetics. Wiley, New York (1963).
Carbou, G., Regularity for critical points of a non local energy. Calc. Var. Partial Differential Equation 5 (1997) 409-433. CrossRef
S. Conti, I. Fonseca and G. Leoni, A Γ-convergence result for the two-gradient theory of phase transitions. Preprint.
DeSimone, A., Energy minimizers for large ferromagnetic bodies. Arch. Rational Mech. Anal. 125 (1993) 99-143. CrossRef
Dacorogna, B. and Fonseca, I., Minima absolus pour des énergies ferromagnétiques. C. R. Acad. Sci. Paris 331 (2000) 497-500. CrossRef
DeSimone, A., Kohn, R.V., Müller, S. and Otto, F., A compactness result in the gradient theory of phase transitions. Proc. Roy. Soc. Edinburgh 131 (2001) 833-844. CrossRef
A. DeSimone, R.V. Kohn, S. Müller and F. Otto, Magnetic microstructures, a paradigm of multiscale problems. Proceedings of ICIAM.
A. DeSimone, R.V. Kohn, S. Müller and F. Otto, A reduced theory for thin-film micromagnetics. Preprint (2001).
A. DeSimone, R.V. Kohn, S. Müller and F. Otto (in preparation).
C. Evans and R. Gariepy, Measure theory and fine properties of functions. CRC Press, Boca Raton, FL, Stud. Adv. Math. (1992).
Hardt, R. and Kinderlehrer, D., Some regularity results in ferromagnetism. Comm. Partial Differential Equation 25 (2000) 1235-1258. CrossRef
F. Hang, Ph.D. Thesis. Courant Institute (2001).
A. Hubert and R. Schäfer, Magnetic Domains. Springer (1998).
Jin, W. and Kohn, R., Singular Perturbation and the Energy of Folds. J. Nonlinear Sci. 10 (2000) 355-390. CrossRef
James, R.D. and Kinderlehrer, D., Frustration in ferromagnetic materials, Continuum Mech. Thermodynamics 2 (1990) 215-239. CrossRef
P.E. Jabin, F. Otto, and B. Perthame, Ginzburg-Landau line energies: The zero-energy case (to appear).
Jabin, P.E. and Perthame, B., Compactness in Ginzburg-Landau energy by kinetic averaging. Comm. Pure Appl. Math. 54 (2001) 1096-1109. CrossRef
M. Lecumberry and T. Rivière, Regularity for micromagnetic configurations having zero jump energy. Calc. Var. Partial Differential Equations (to appear).
Modica, L. and Mortola, Il limite nella Γ-convergenza di una famiglia di funzionali ellittici. Boll. Un. Mat. Ital. A (5) 14 (1977) 526-529.
Nakatani, Y., Uesaka, Y. and Hayashi, N., Direct solution of the Landau-Lifshitz-Gilbert equation for micromagnetics. Japanese J. Appl. Phys. 28 (1989) 2485-2507. CrossRef
Rave, W. and Hubert, A., The Magnetic Ground State of a Thin-Film Element. IEEE Trans. Mag. 36 (2000) 3886-3899. CrossRef
Rivière, T. and Serfaty, S., Limiting Domain Wall Energy for a Problem Related to Micromagnetics. Comm. Pure Appl. Math. 54 (2001) 294-338. 3.0.CO;2-S>CrossRef
T. Rivière and S. Serfaty, Compactness, kinetic formulation and entropies for a problem related to micromagnetics. Comm. in Partial Differential Equations (to appear).
Visintin, A., Landau-Lifschitz, On equations for ferromagnetism, Japanese J. Appl. Math. 2 (1985) 69-84. CrossRef
E. Sandier, preprint (1999) and habilitation thesis. University of Tours (2000).
Sternberg, P., The effect of a singular perturbation on nonconvex variational problems. Arch. Rational Mech. Anal. 101 (1988) 209-260. CrossRef
Van den Berg, H.A.M., Self-consistent domain theory in soft micromagnetic media, II, Basic domain structures in thin film objects. J. Appl. Phys. 60 (1986) 1104-1113. CrossRef