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Communications in Mathematical Physics

, Volume 367, Issue 1, pp 317–349 | Cite as

On the First Critical Field in the Three Dimensional Ginzburg–Landau Model of Superconductivity

  • Carlos RománEmail author
Article
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Abstract

The Ginzburg–Landau model is a phenomenological description of superconductivity. A crucial feature of type-II superconductors is the occurrence of vortices, which appear above a certain value of the strength of the applied magnetic field called the first critical field. In this paper we estimate this value, when the Ginzburg–Landau parameter is large, and we characterize the behavior of the Meissner solution, the unique vortexless configuration that globally minimizes the Ginzburg–Landau energy below the first critical field. In addition, we show that beyond this value, for a certain range of the strength of the applied field, there exists a unique Meissner-type solution that locally minimizes the energy.

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Notes

Acknowledgements

I am very grateful to my former Ph.D. advisors Etienne Sandier and Sylvia Serfaty for suggesting the problem and for useful comments. I also thank the anonymous referees for helpful comments. Most of this work was done while I was a Ph.D. student at the Jacques-Louis Lions Laboratory of the Pierre and Marie Curie University, supported by a public grant overseen by the French National Research Agency (ANR) as part of the “Investissements d’Avenir” program (reference: ANR-10-LABX-0098, LabEx SMP). Part of this work was supported by the German Science Foundation DFG in the context of the Emmy Noether junior research group BE 5922/1-1.

Conflict of interest

The author declares that they have no conflict of interest.

References

  1. ABM06.
    Alama S., Bronsard L., Montero J.A.: On the Ginzburg–Landau model of a super-conducting ball in a uniform field. Ann. Inst. H. Poincaré Anal. Non Linéaire 23(2), 237–267 (2006)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  2. ABO03.
    Alberti G., Baldo S., Orlandi G.: Functions with prescribed singularities. J. Eur. Math. Soc. (JEMS) 5(3), 275–311 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  3. ABO05.
    Alberti G., Baldo S., Orlandi G.: Variational convergence for functionals of Ginzburg–Landau type. Indiana Univ. Math. J. 54(5), 1411–1472 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  4. BBO01.
    Bethuel F., Brezis H., Orlandi G.: Asymptotics for the Ginzburg–Landau equation in arbitrary dimensions. J. Funct. Anal. 186(2), 432–520 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  5. BCS57.
    Bardeen J., Cooper L.N., Schrieffer J.R.: Theory of superconductivity. Phys. Rev. 108, 1175–1204 (1957)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. BJOS12.
    Baldo S., Jerrard R.L., Orlandi G., Soner H.M.: Convergence of Ginzburg–Landau functionals in three-dimensional superconductivity. Arch. Ration. Mech. Anal. 205(3), 699–752 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  7. BJOS13.
    Baldo S., Jerrard R.L., Orlandi G., Soner H.M.: Vortex density models for superconductivity and superfluidity. Comm. Math. Phys. 318(1), 131–171 (2013)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  8. BOS04.
    Bethuel F., Orlandi G., Smets D.: Vortex rings for the Gross-Pitaevskii equation. J. Eur. Math. Soc. (JEMS) 6(1), 17–94 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  9. Chi05.
    Chiron D.: Boundary problems for the Ginzburg–Landau equation. Commun. Contemp. Math. 7(5), 597–648 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  10. Con11.
    Contreras A.: On the first critical field in Ginzburg–Landau theory for thin shells and manifolds. Arch. Ration. Mech. Anal. 200(2), 563–611 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  11. DG99.
    De Gennes P.G.: Superconductivity of Metals and Alloys. Advanced book classics, Perseus, Cambridge, MA (1999)zbMATHGoogle Scholar
  12. FH10.
    Fournais S., Helffer B.: Spectral methods in surface superconductivity, Progress in Nonlinear Differential Equations and their Applications, vol. 77. Birkhäuser Boston, Inc., Boston, MA (2010)zbMATHGoogle Scholar
  13. FK13.
    Fournais S., Kachmar A.: The ground state energy of the three dimensional Ginzburg–Landau functional Part I: Bulk regime. Comm. Partial Differential Equations 38(2), 339–383 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  14. FKP13.
    Fournais S., Kachmar A., Persson M.: The ground state energy of the three dimensional Ginzburg–Landau functional. Part II: Surface regime. J. Math. Pures Appl. (9) 99(3), 343–374 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  15. GL50.
    Ginzburg, V.L., Landau, L.D.: On the theory of superconductivity, Zh. Eksp. Teor. Fiz. 20 (1950), 1064–1082. English translation in: Collected papers of L.D.Landau, Edited by D. Ter. Haar, Pergamon Press, Oxford 1965, pp. 546–568Google Scholar
  16. GP99.
    Giorgi T., Phillips D.: The breakdown of superconductivity due to strong fields for the Ginzburg–Landau model. SIAM J. Math. Anal. 30(2), 341–359 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  17. Jer99.
    Jerrard R.L.: Lower bounds for generalized Ginzburg–Landau functionals. SIAMJ.Math. Anal. 30(4), 721–746 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  18. JMS04.
    Jerrard R., Montero A., Sternberg P.: Local minimizers of the Ginzburg–Landau energy with magnetic field in three dimensions. Comm. Math. Phys. 249(3), 549–577 (2004)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  19. KS91.
    Kozono H., Sohr H.: New a priori estimates for the Stokes equations in exterior domains. Indiana Univ. Math. J. 40(1), 1–27 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  20. Lon50.
    London F.: Superfluids, Structure of matter series. Wiley, New York (1950)Google Scholar
  21. LR01.
    Lin F.-H., Rivière T.: A quantization property for static Ginzburg–Landau vortices. Comm. Pure Appl. Math. 54(2), 206–228 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  22. LR99.
    Lin F., Rivière T.: Complex Ginzburg–Landau equations in high dimensions and codimension two area minimizing currents. J. Eur. Math. Soc. (JEMS) 1(3), 237–311 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  23. Riv95.
    Rivière, T.: Line vortices in the U(1)-Higgs model, ESAIM Contrôle Optim. Calc. Var. 1 (1995/96), 77–167Google Scholar
  24. Rom19.
    Román, C.: Three dimensional vortex approximation construction and \({\varepsilon}\)-level estimates for the Ginzburg–Landau functional, Arch. Ration. Mech. Anal. 231(3), 1531–1614 (2019).  https://doi.org/10.1007/s00205-018-1304-7
  25. Ser99a.
    Serfaty S.: Local minimizers for the Ginzburg–Landau energy near critical magnetic field. I. Commun. Contemp. Math. 1(2), 213–254 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  26. Ser99b.
    Serfaty S.: Stable configurations in superconductivity: uniqueness, multiplicity, and vortex-nucleation. Arch. Ration. Mech. Anal. 149(4), 329–365 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  27. SS00.
    Sandier E., Serfaty S.: Global minimizers for the Ginzburg–Landau functional below the first critical magnetic field. Ann. Inst. H. Poincaré Anal. Non Linéaire 17(1), 119–145 (2000)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  28. SS03.
    Sandier E., Serfaty S.: Ginzburg–Landau minimizers near the first critical field have bounded vorticity. Calc. Var. Partial Differential Equations 17(1), 17–28 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  29. SS07.
    Sandier E., Serfaty S.: Vortices in the magnetic Ginzburg–Landau model, Progress in Nonlinear Differential Equations and their Applications, vol. 70. Birkhäuser Boston, Inc., Boston, MA (2007)zbMATHGoogle Scholar
  30. SS17.
    Sandier E., Shafrir I.: Small energy Ginzburg–Landau minimizers in \({{\mathbb{R}}^3}\). J. Funct. Anal. 272(9), 3946–3964 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  31. Tin96.
    Tinkham M.: Introduction to superconductivity, Second. McGraw-Hill, New York (1996)Google Scholar
  32. Xia16.
    Xiang X.: On the shape of Meissner solutions to a limiting form of Ginzburg–Landau systems. Arch. Ration. Mech. Anal. 222(3), 1601–1640 (2016)MathSciNetCrossRefzbMATHGoogle Scholar

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© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Mathematisches InstitutUniversität LeipzigLeipzigGermany

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