Journal of Nonlinear Science

, Volume 21, Issue 6, pp 921–962 | Cite as

Domain Structure of Bulk Ferromagnetic Crystals in Applied Fields Near Saturation



We investigate the ground state of a uniaxial ferromagnetic plate with perpendicular easy axis and subject to an applied magnetic field normal to the plate. Our interest is in the asymptotic behavior of the energy in macroscopically large samples near the saturation field. We establish the scaling of the critical value of the applied field strength below saturation at which the ground state changes from the uniform to a multidomain magnetization pattern and the leading order scaling behavior of the minimal energy. Furthermore, we derive a reduced sharp interface energy, giving the precise asymptotic behavior of the minimal energy in macroscopically large plates under a physically reasonable assumption of small deviations of the magnetization from the easy axis away from domain walls. On the basis of the reduced energy and by a formal asymptotic analysis near the transition, we derive the precise asymptotic values of the critical field strength at which non-trivial minimizers (either local or global) emerge. The non-trivial minimal energy scaling is achieved by magnetization patterns consisting of long slender needle-like domains of magnetization opposing the applied field.


Magnetic domains Self-similarity Ansatz-free analysis Variational bounds 

Mathematics Subject Classification (2000)

35B36 49S05 35B40 49J40 


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  1. Aftalion, A., Serfaty, S.: Lowest Landau level approach in superconductivity for the Abrikosov lattice close to \(H_{c_{2}}\). Sel. Math. 13, 183–202 (2007) MathSciNetMATHCrossRefGoogle Scholar
  2. Brown, W.F.: Thermal fluctuations of a single-domain particle. Phys. Rev. 130, 1677–1686 (1963) CrossRefGoogle Scholar
  3. Cape, J.A., Lehman, G.W.: Domain nucleation and boundary effects in thin uniaxial plates. J. Appl. Phys. 42, 5732–5756 (1971) CrossRefGoogle Scholar
  4. Choksi, R.: Scaling laws in microphase separation of diblock copolymers. J. Nonlinear Sci. 11, 223–236 (2001) MathSciNetMATHCrossRefGoogle Scholar
  5. Choksi, R., Kohn, R.V.: Bounds on the micromagnetic energy of a uniaxial ferromagnet. Commun. Pure Appl. Math. 51, 259–289 (1998) MathSciNetMATHCrossRefGoogle Scholar
  6. Choksi, R., Kohn, R.V., Otto, F.: Domain branching in uniaxial ferromagnets: a scaling law for the minimum energy. Commun. Math. Phys. 201, 61–79 (1999) MathSciNetMATHCrossRefGoogle Scholar
  7. Choksi, R., Kohn, R.V., Otto, F.: Energy minimization and flux domain structure in the intermediate state of a Type-I superconductor. J. Nonlinear Sci. 14, 119–171 (2004) MathSciNetMATHCrossRefGoogle Scholar
  8. Choksi, R., Conti, S., Kohn, R.V., Otto, F.: Ground state energy scaling laws during the onset and destruction of the intermediate state in a Type-I superconductor. Commun. Pure Appl. Math. 61, 595–626 (2008a) MathSciNetMATHCrossRefGoogle Scholar
  9. Choksi, R., Peletier, M.A., Williams, J.F.: On the phase diagram for microphase separation of diblock copolymers: an approach via a nonlocal Cahn-Hilliard functional. SIAM J. Appl. Math. 69, 1712–1738 (2008b) MathSciNetCrossRefGoogle Scholar
  10. Dacorogna, B.: Direct Methods in the Calculus of Variations, 2nd edn. Applied Mathematical Sciences, vol. 78. Springer, New York (2008) MATHGoogle Scholar
  11. De Giorgi, E.: Nuovi teoremi relativi alle misure (r−1)-dimensionali in uno spazio ad r dimensioni. Ric. Mat. 4, 95–113 (1955) MATHGoogle Scholar
  12. DeSimone, A., Kohn, R.V., Müller, S., Otto, F.: Magnetic microstructures—a paradigm of multiscale problems. In: ICIAM 99 (Edinburgh), pp. 175–190. Oxford University Press, London (2000) Google Scholar
  13. DeSimone, A., Kohn, R.V., Müller, S., Otto, F.: A reduced theory for thin-film micromagnetics. Commun. Pure Appl. Math. 55, 1408–1460 (2002) MATHCrossRefGoogle Scholar
  14. Druyvesteyn, W.F., Dorleijn, J.W.F.: Calculations of some periodic magnetic domain structures; consequences for bubble devices. Philips Res. Rep. 26, 11–28 (1971) Google Scholar
  15. Evans, L.C., Gariepy, R.L.: Measure Theory and Fine Properties of Functions. CRC Press, Boca Raton (1992) MATHGoogle Scholar
  16. Gabay, M., Garel, T.: Phase transitions and size effects in the Ising dipolar magnet. J. Phys. 46, 5–16 (1985) CrossRefGoogle Scholar
  17. Hubert, A., Schäfer, R.: Magnetic Domains. Springer, Berlin (1998) Google Scholar
  18. Kaczér, J.: On the domain structure of uniaxial ferromagnets. Sov. Phys. JETP 19, 1204–1208 (1964) Google Scholar
  19. Kittel, C.: Theory of the structure of ferromagnetic domains in films and small particles. Phys. Rev. 70, 965–971 (1946) CrossRefGoogle Scholar
  20. Kohn, R.V.: Energy-driven pattern formation. In: International Congress of Mathematicians, vol. I, pp. 359–383. Eur. Math. Soc., Zürich (2007) Google Scholar
  21. Kohn, R.V., Müller, S.: Surface energy and microstructure in coherent phase transitions. Commun. Pure Appl. Math. 47, 405–435 (1994) MATHCrossRefGoogle Scholar
  22. Kohn, R.V., Strang, G.: Optimal design and relaxation of variational problems. I. Commun. Pure Appl. Math. 39, 113–137 (1986) MathSciNetMATHCrossRefGoogle Scholar
  23. Kooy, C., Enz, U.: Experimental and theoretical study of the domain configuration in thin layers of BaFe12O19. Philips Res. Rep. 15, 7–29 (1960) Google Scholar
  24. Landau, L.D., Lifshitz, E.M.: On the theory of the dispersion of magnetic permeability in ferromagnetic bodies. Phys. Z. Sowjetunion 8, 153–169 (1935) MATHGoogle Scholar
  25. Landau, L.D., Lifshits, E.M.: Course of Theoretical Physics, vol. 8. Pergamon, London (1984) Google Scholar
  26. Lifshits, E.M., Pitaevskii, L.P.: Course of Theoretical Physics, vol. 8. Pergamon, London (1980) Google Scholar
  27. Muratov, C.B.: Droplet phases in non-local Ginzburg-Landau models with Coulomb repulsion in two dimensions. Commun. Math. Phys. 299, 45–87 (2010) MathSciNetMATHCrossRefGoogle Scholar
  28. Otto, F., Viehmann, T.: Domain branching in uniaxial ferromagnets—asymptotic behavior of the energy. Calc. Var. Partial Differ. Equ. 38, 135–181 (2010) MathSciNetMATHCrossRefGoogle Scholar
  29. Privorotskii, I.A.: Thermodynamic theory of domain structures. Rep. Prog. Phys. 35, 115–155 (1972) CrossRefGoogle Scholar
  30. Prozorov, R.: Equilibrium topology of the intermediate state in type-I superconductors of different shapes. Phys. Rev. Lett. 98, 257001 (2007) CrossRefGoogle Scholar
  31. Prozorov, R., Giannetta, R.W., Polyanskii, A.A., Perkins, G.K.: Topological hysteresis in the intermediate state of type-I superconductors. Phys. Rev. B 72, 212508 (2005) CrossRefGoogle Scholar
  32. Shur, V.Ya., Rumyantsev, E.L., Nikolaeva, E.V., Shishkin, E.I.: Formation and evolution of charged domain walls in congruent lithium niobate. Appl. Phys. Lett. 77, 3636–3638 (2000) CrossRefGoogle Scholar
  33. Strukov, B.A., Levanyuk, A.P.: Ferroelectric Phenomena in Crystals: Physical Foundations. Springer, New York (1998) MATHCrossRefGoogle Scholar
  34. Williams, H.J., Bozorth, R.M., Shockley, W.: Magnetic domain patterns on single crystals of silicon iron. Phys. Rev. 75, 155–178 (1949) CrossRefGoogle Scholar

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© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Hausdorff Center for MathematicsBonnGermany
  2. 2.Department of Mathematical SciencesNew Jersey Institute of TechnologyNewarkUSA

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