Skip to main content
SpringerLink
Log in

Asymmetric Domain Walls of Small Angle in Soft Ferromagnetic Films

  • Published:
Archive for Rational Mechanics and Analysis Aims and scope Submit manuscript

Abstract

We focus on a special type of domain wall appearing in the Landau–Lifshitz theory for soft ferromagnetic films. These domain walls are divergence-free \({\mathbb{S}^2}\)-valued transition layers that connect two directions \({m_\theta^\pm \in \mathbb{S}^2}\) (differing by an angle \({2\theta}\)) and minimize the Dirichlet energy. Our main result is the rigorous derivation of the asymptotic structure and energy of such “asymmetric” domain walls in the limit \({\theta \downarrow 0}\). As an application, we deduce that a supercritical bifurcation causes the transition from symmetric to asymmetric walls in the full micromagnetic model.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Anzellotti G., Baldo S.: Asymptotic development by \({\Gamma}\)-convergence. Appl. Math. Optim. 27(2), 105–123 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  2. Arnol’d, V.I.: Ordinary differential equations. Springer Textbook. Springer, Berlin, 1992. (Translated from the third Russian edition by Roger Cooke)

  3. Berkov D.V., Ramstöck K., Hubert A.: Solving micromagnetic problems. Towards an optimal numerical method. Phys. Status Solidi (a) 137(1), 207–225 (1993)

    Article  ADS  Google Scholar 

  4. Bethuel, F., Brezis, H., Hélein, F.: Ginzburg–Landau Vortices. Springer, Berlin, 1994

  5. Boutet de Monvel-Berthier, A., Georgescu, V., Purice, R.: A boundary value problem related to the Ginzburg–Landau model. Commun. Math. Phys. 142(1), 1–23 (1991)

  6. Braides A., Truskinovsky L.: Asymptotic expansions by \({\Gamma}\)-convergence. Contin. Mech. Thermodyn. 20(1), 21–62 (2008)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  7. Brezis, H., Nirenberg, L.: Degree theory and BMO. I. Compact manifolds without boundaries. Sel. Math. (N. S.) 1(2), 197–263 (1995)

  8. Dacorogna, B.: Direct methods in the calculus of variations. Applied Mathematical Sciences. Springer, Berlin, 2007

  9. DeSimone A., Knüpfer H., Otto F.: 2-D stability of the Néel wall. Calc. Var. Partial Differ. Equ. 27(2), 233–253 (2006)

    Article  MATH  Google Scholar 

  10. DeSimone, A., Kohn, R.V., Müller, S., Otto, F.: Repulsive interaction of Néel walls, and the internal length scale of the cross-tie wall. Multiscale Model. Simul. 1(1), 57–104 (2003)

  11. Döring L., Ignat R., Otto F.: A reduced model for domain walls in soft ferromagnetic films at the cross-over from symmetric to asymmetric wall types. J. Eur. Math. Soc. (JEMS) 16(7), 1377–1422 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  12. Evans, L.C.: Partial differential equations. Graduate Studies in Mathematics, Vol. 19, 2nd edn. American Mathematical Society, Providence, 2010

  13. Hardy G.: Note on a theorem of hilbert. Math. Z. 6(3), 314–317 (1920)

    Article  MathSciNet  MATH  Google Scholar 

  14. Hubert A.: Stray-field-free magnetization configurations. Phys. Status Solidi. B Basic Res. 32(2), 519–534 (1969)

    Article  ADS  Google Scholar 

  15. Hubert, A., Schäfer, R.: Magnetic Domains—The Analysis of Magnetic Microstructures, 1st edn. Springer, Berlin, 1998

  16. Ignat, R.: A \({\Gamma}\)-convergence result for Néel walls in micromagnetics. Calc. Var. Partial Differ. Equ. 36(2), 285–316 (2009)

  17. Ignat R., Otto F.: A compactness result in thin-film micromagnetics and the optimality of the Néel wall. J. Eur. Math. Soc. (JEMS) 10(4), 909–956 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  18. Kurzke, M., Melcher, C., Moser, R., Spirn, D.: Vortex dynamics in the presence of excess energy for the Landau–Lifshitz-Gilbert equation. Calc. Var. Partial Differ. Equ. 49(3-4), 1019–1043, (2014)

  19. LaBonte A.E.: Two-dimensional bloch-type domain walls in ferromagnetic films. J. Appl. Phys. 40(6), 2450–2458 (1969)

    Article  ADS  Google Scholar 

  20. Lions, P.-L.: Generalized solutions of Hamilton–Jacobi equations. Research Notes in Mathematics, Vol. 69. Pitman (Advanced Publishing Program), Boston, 1982

  21. Melcher C.: The logarithmic tail of Néel walls. Arch. Ration. Mech. Anal. 168(2), 83–113 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  22. Melcher C.: Logarithmic lower bounds for Néel walls. Calc. Var. Partial Differ. Equ. 21(2), 209–219 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  23. Nirenberg, L.: Topics in Nonlinear Functional Analysis, Vol. 6. American Mathematical Soc., Providence, 1974

  24. Otto, F.: Cross-over in scaling laws: a simple example from micromagnetics. Proceedings of the International Congress of Mathematicians (Beijing, 2002), Vol. III, pp. 829–838. Higher Ed. Press, Beijing, 2002

Download references

Author information

Authors and Affiliations

  1. Lehrstuhl I für Mathematik, RWTH Aachen, Pontdriesch 14-16, 52056, Aachen, Germany

    Lukas Döring

  2. Institut de Mathématiques de Toulouse, Université Paul Sabatier, 31062, Toulouse, France

    Radu Ignat

Authors
  1. Lukas Döring
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Radu Ignat
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Lukas Döring.

Additional information

Communicated by A. Braides

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Döring, L., Ignat, R. Asymmetric Domain Walls of Small Angle in Soft Ferromagnetic Films. Arch Rational Mech Anal 220, 889–936 (2016). https://doi.org/10.1007/s00205-015-0944-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00205-015-0944-0

Keywords

Search

Navigation