Néel and Cross-Tie wall energies for planar micromagnetic configurations

Alouges, François; Rivière, Tristan; Serfaty, Sylvia

ESAIM: Control, Optimisation and Calculus of Variations, Tome 8 (2002), pp. 31-68.

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 $\pi /2$ 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.

[1] G. Anzelotti, S. Baldo and A. Visintin, Asymptotic behavior of the Landau-Lifschitz model of ferromagnetism. Appl. Math. Optim. 23 (1991) 171-193. | MR | Zbl

[2] L. Ambrosio, C. De Lellis and C. Mantegazza, Line energies for gradient vector fields in the plane. Calc. Var. Partial Differential Equation 9 (1999) 327-355. | MR | Zbl

[3] P. Aviles and Y. Giga, A mathematical problem related to the physical theory of liquid crystals configurations. Proc. Centre Math. Anal. Austral. Nat. Univ. 12 (1987) 1-16. | MR

[4] P. Aviles and Y. Giga, 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. | MR | Zbl

[5] L. Ambrosio, M. Lecumberry and T. Rivière, A Viscosity Property of Minimizing Micromagnetic Configurations. Preprint. | MR | Zbl

[6] N. André and I. Shafrir, On nematics stabilized by a large external field. Rev. Math. Phys. 11 (1999) 653-710. | MR | Zbl

[7] F. Bethuel, H. Brezis and F. Hélein, Ginzburg-Landau vortices. Birkhauser (1994). | MR | Zbl

[8] F. Bethuel and X. Zheng, Density of smooth functions between two manifolds in Sobolev spaces. J. Func. Anal. 80 (1988) 60-75. | MR | Zbl

[9] W. Brown, Micromagnetics. Wiley, New York (1963).

[10] G. Carbou, Regularity for critical points of a non local energy. Calc. Var. Partial Differential Equation 5 (1997) 409-433. | MR | Zbl

[11] S. Conti, I. Fonseca and G. Leoni, A $\Gamma$-convergence result for the two-gradient theory of phase transitions. Preprint. | MR | Zbl

[12] A. Desimone, Energy minimizers for large ferromagnetic bodies. Arch. Rational Mech. Anal. 125 (1993) 99-143. | MR | Zbl

[13] B. Dacorogna and I. Fonseca, Minima absolus pour des énergies ferromagnétiques. C. R. Acad. Sci. Paris 331 (2000) 497-500. | MR | Zbl

[14] A. Desimone, R.V. Kohn, S. Müller and F. Otto, A compactness result in the gradient theory of phase transitions. Proc. Roy. Soc. Edinburgh 131 (2001) 833-844. | MR | Zbl

[15] A. Desimone, R.V. Kohn, S. Müller and F. Otto, Magnetic microstructures, a paradigm of multiscale problems. Proceedings of ICIAM. | Zbl

[16] A. Desimone, R.V. Kohn, S. Müller and F. Otto, A reduced theory for thin-film micromagnetics. Preprint (2001). | MR | Zbl

[17] A. Desimone, R.V. Kohn, S. Müller and F. Otto (in preparation).

[18] C. Evans and R. Gariepy, Measure theory and fine properties of functions. CRC Press, Boca Raton, FL, Stud. Adv. Math. (1992). | MR | Zbl

[19] R. Hardt and D. Kinderlehrer, Some regularity results in ferromagnetism. Comm. Partial Differential Equation 25 (2000) 1235-1258. | MR | Zbl

[20] F. Hang, Ph.D. Thesis. Courant Institute (2001).

[21] A. Hubert and R. Schäfer, Magnetic Domains. Springer (1998).

[22] W. Jin and R. Kohn, Singular Perturbation and the Energy of Folds. J. Nonlinear Sci. 10 (2000) 355-390. | MR | Zbl

[23] R.D. James and D. Kinderlehrer, Frustration in ferromagnetic materials, Continuum Mech. Thermodynamics 2 (1990) 215-239. | MR

[24] P.E. Jabin, F. Otto, and B. Perthame, Ginzburg-Landau line energies: The zero-energy case (to appear).

[25] P.E. Jabin and B. Perthame, Compactness in Ginzburg-Landau energy by kinetic averaging. Comm. Pure Appl. Math. 54 (2001) 1096-1109. | Zbl

[26] M. Lecumberry and T. Rivière, Regularity for micromagnetic configurations having zero jump energy. Calc. Var. Partial Differential Equations (to appear). | MR | Zbl

[27] L. Modica and Mortola, Il limite nella $\Gamma$-convergenza di una famiglia di funzionali ellittici. Boll. Un. Mat. Ital. A (5) 14 (1977) 526-529. | MR | Zbl

[28] Y. Nakatani, Y. Uesaka and N. Hayashi, Direct solution of the Landau-Lifshitz-Gilbert equation for micromagnetics. Japanese J. Appl. Phys. 28 (1989) 2485-2507.

[29] W. Rave and A. Hubert, The Magnetic Ground State of a Thin-Film Element. IEEE Trans. Mag. 36 (2000) 3886-3899.

[30] T. Rivière and S. Serfaty, Limiting Domain Wall Energy for a Problem Related to Micromagnetics. Comm. Pure Appl. Math. 54 (2001) 294-338. | MR | Zbl

[31] T. Rivière and S. Serfaty, Compactness, kinetic formulation and entropies for a problem related to micromagnetics. Comm. in Partial Differential Equations (to appear). | MR | Zbl

[32] A. Visintin, On Landau-Lifschitz equations for ferromagnetism, Japanese J. Appl. Math. 2 (1985) 69-84. | Zbl

[33] E. Sandier, preprint (1999) and habilitation thesis. University of Tours (2000).

[34] P. Sternberg, The effect of a singular perturbation on nonconvex variational problems. Arch. Rational Mech. Anal. 101 (1988) 209-260. | MR | Zbl

[35] H.A.M. Van Den Berg, Self-consistent domain theory in soft micromagnetic media, II, Basic domain structures in thin film objects. J. Appl. Phys. 60 (1986) 1104-1113.