The Distribution of Superconductivity Near a Magnetic Barrier

  • Wafaa Assaad
  • Ayman Kachmar
  • Mikael Persson-SundqvistEmail author
Open Access


We consider the Ginzburg–Landau functional, defined on a two-dimensional simply connected domain with smooth boundary, in the situation when the applied magnetic field is piecewise constant with a jump discontinuity along a smooth curve. In the regime of large Ginzburg–Landau parameter and strong magnetic field, we study the concentration of the minimizing configurations along this discontinuity by computing the energy of the minimizers and their weak limit in the sense of distributions.



We would like to thank Jacob Christiansen for his insightful comments on the manuscript and Virginie Bonnaillie-Noël for the numerical computations and Fig. 5. We would also like to acknowledge the constructive reviews by the anonymous referees, which led to substantial improvements of this manuscript. The research of the second author was partially supported by a grant from the Lebanese University.


  1. AH07.
    Almog, Y., Helffer, B.: The distribution of surface superconductivity along the boundary: On a conjecture of X.B. Pan. SIAM J. Math. Anal. 38(6), 1715–1732 (2007)Google Scholar
  2. AK16.
    Assaad W., Kachmar A.: The influence of magnetic steps on bulk superconductivity. Discrete Contin. Dyn. Syst. Ser. A 36, 6623–6643 (2016)MathSciNetzbMATHGoogle Scholar
  3. Ass.
    Assaad W.: The breakdown of superconductivity in the presence of magnetic steps. In progressGoogle Scholar
  4. Att15a.
    Attar K.: Energy and vorticity of the Ginzburg–Landau model with variable magnetic field. Asymptot. Anal. 93(1–2), 75–114 (2015)MathSciNetzbMATHGoogle Scholar
  5. Att15b.
    Attar K.: The ground state energy of the two dimensional Ginzburg–Landau functional with variable magnetic field. Ann. Henri Poincaré 32(2), 325–345 (2015)ADSMathSciNetzbMATHGoogle Scholar
  6. BNF07.
    Bonnaillie-Noël V., Fournais S.: Superconductivity in domains with corners. Rev. Math. Phys. 19(06), 607–637 (2007)MathSciNetzbMATHGoogle Scholar
  7. CDR17.
    Correggi M., Devanarayanan B., Rougerie N.: Universal and shape dependent features of surface superconductivity. Eur. Phys. J. B 90(11), 231 (2017)ADSMathSciNetGoogle Scholar
  8. CFKS09.
    Cycon H.L., Froese R.G., Kirsch W., Simon B.: Schrödinger Operators: With Application to Quantum Mechanics and Global Geometry. Springer, Berlin (2009)zbMATHGoogle Scholar
  9. CG17.
    Correggi M., Giacomelli E.L.: Surface superconductivity in presence of corners. Rev. Math. Phys. 29(02), 1750005 (2017)MathSciNetzbMATHGoogle Scholar
  10. CR14.
    Correggi M., Rougerie N.: On the Ginzburg–Landau functional in the surface superconductivity regime. Commun. Math. Phys. 332(3), 1297–1343 (2014)ADSMathSciNetzbMATHGoogle Scholar
  11. CR16a.
    Correggi M., Rougerie N.: Boundary behavior of the Ginzburg–Landau order parameter in the surface superconductivity regime. Arch. Ration. Mech. Anal. 219(1), 553–606 (2016)MathSciNetzbMATHGoogle Scholar
  12. CR16b.
    Correggi M., Rougerie N.: Effects of boundary curvature on surface superconductivity. Lett. Math. Phys. 106(4), 445–467 (2016)ADSMathSciNetzbMATHGoogle Scholar
  13. DH93.
    Dauge M., Helffer B.: Eigenvalues variation. I.: Neumann problem for Sturm–Liouville operators. J. Differ. Equ. 104(2), 243–262 (1993)ADSMathSciNetzbMATHGoogle Scholar
  14. DHS14.
    Dombrowski N., Hislop P.D., Soccorsi E.: Edge currents and eigenvalue estimates for magnetic barrier Schrödinger operators. Asymptot. Anal. 89(3-4), 331–363 (2014)MathSciNetzbMATHGoogle Scholar
  15. FH05.
    Fournais S., Helffer B.: Energy asymptotics for type II superconductors. Calc. Var. Partial Differ. Equ. 24(3), 341–376 (2005)MathSciNetzbMATHGoogle Scholar
  16. FH10.
    Fournais S., Helffer B.: Spectral Methods in Surface Superconductivity, vol. 77. Springer, Berlin (2010)zbMATHGoogle Scholar
  17. FK11.
    Fournais S., Kachmar A.: Nucleation of bulk superconductivity close to critical magnetic field. Adv. Math. 226(2), 1213–1258 (2011)MathSciNetzbMATHGoogle Scholar
  18. FK13.
    Fournais S., Kachmar A.: The ground state energy of the three dimensional Ginzburg–Landau functional. Part I: bulk regime. Commun. Partial Differ. Equ. 38(2), 339–383 (2013)MathSciNetzbMATHGoogle Scholar
  19. 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. 99(3), 343–374 (2013)MathSciNetzbMATHGoogle Scholar
  20. FLBP94.
    Foden C.L., Leadbeater M.L., Burroughes J.H., Pepper M.: Quantum magnetic confinement in a curved two-dimensional electron gas. J. Phys. Condens. Matter 6(10), L127 (1994)ADSGoogle Scholar
  21. GDMH+08.
    Ghosh T.K., De Martino A., Häusler W., Dell’Anna L., Egger R.: Conductance quantization and snake states in graphene magnetic waveguides. Phys. Rev. B 77(8), 081404 (2008)ADSGoogle Scholar
  22. GGD+97.
    Geim, A.K., Grigorieva, I.V., Dubonos, S.V., Lok, J.G.S., Maan, J.C., Filippov, A.E., Peeters, F.M.: Nature (London) 390, 259 (1997)Google Scholar
  23. GP99.
    Giorgi T., Phillips D.: The breakdown of superconductivity due to strong fields for the Ginzburg–Landau model. SIAM J. Math. Anal. 30, 341–359 (1999)MathSciNetzbMATHGoogle Scholar
  24. HFPS11.
    Helffer B., Fournais S., Persson Sundqvist M.: Superconductivity between HC2 and HC3. J. Spectr. Theory 1(3), 273–298 (2011)MathSciNetzbMATHGoogle Scholar
  25. HK15.
    Helffer B., Kachmar A.: The Ginzburg–Landau functional with vanishing magnetic field. Arch. Ration. Mech. Anal. 218, 55 (2015)MathSciNetzbMATHGoogle Scholar
  26. HK17.
    Helffer B., Kachmar A.: Decay of superconductivity away from the magnetic zero set. Calc. Var. Partial Differ. Equ. 56(5), 130 (2017)MathSciNetzbMATHGoogle Scholar
  27. HK18.
    Helffer, B., Kachmar, A.: The density of superconductivity in domains with corners. Lett. Math. Phys. 1–19 (2018)Google Scholar
  28. HPRS16.
    Hislop P.D., Popoff N., Raymond N., Sundqvist M.: Band functions in the presence of magnetic steps. Math. Models Methods Appl. Sci. 26(1), 161–184 (2016)MathSciNetzbMATHGoogle Scholar
  29. HS08.
    Hislop P.D., Soccorsi E.: Edge currents for quantum Hall systems I: one-edge, unbounded geometries. Rev. Math. Phys. 20(01), 71–115 (2008)MathSciNetzbMATHGoogle Scholar
  30. HS15.
    Hislop P.D., Soccorsi E.: Edge states induced by Iwatsuka Hamiltonians with positive magnetic fields. J. Math. Anal. Appl. 422(1), 594–624 (2015)MathSciNetzbMATHGoogle Scholar
  31. Iwa85.
    Iwatsuka A.: Examples of absolutely continuous Schrödinger operators in magnetic fields. Publ. Res. Inst. Math. Sci. 21(2), 385–401 (1985)MathSciNetzbMATHGoogle Scholar
  32. JBY+97.
    Johnson M, Bennett B.R., Yang M.J., Miller M.M., Shanabrook B.V.: Hybrid Hall effect device. Appl. Phys. Lett. 71(7), 974–976 (1997)ADSGoogle Scholar
  33. Kac06.
    Kachmar A.: The ground state energy of a magnetic Schrödinger operator and the effect of the de Gennes parameter. J. Math. Phys. 47, 072106 (2006)ADSMathSciNetzbMATHGoogle Scholar
  34. Kac07.
    Kachmar A.: On the perfect superconducting solution for a generalized Ginzburg–Landau equation. Asymptot. Anal. 54(3-4), 125–164 (2007)MathSciNetzbMATHGoogle Scholar
  35. Kat66.
    Kato T.: Perturbation Theory for Linear Operators. Springer, New York (1966)zbMATHGoogle Scholar
  36. KN16.
    Kachmar A., Nasrallah M.: The distribution of 3D superconductivity near the second critical field. Nonlinearity 29(9), 2856 (2016)ADSMathSciNetzbMATHGoogle Scholar
  37. LG50.
    Landau L.D., Ginzburg V.L.: On the theory of superconductivity. J. Exp. Theor. Phys. 20, 1064 (1950)Google Scholar
  38. LM99.
    Lassoued L., Mironescu P.: Ginzburg–Landau type energy with discontinuous constraint. J. Anal. Math. 77(1), 1–26 (1999)MathSciNetzbMATHGoogle Scholar
  39. LP99.
    Lu K., Pan X.B.: Estimates of the upper critical field for the Ginzburg–Landau equations of superconductivity. Phys. D 127(1), 73–104 (1999)MathSciNetzbMATHGoogle Scholar
  40. MJR97.
    Monzon F.G., Johnson M., Roukes M.L.: Strong Hall voltage modulation in hybrid ferromagnet/semiconductor microstructures. Appl. Phys. Lett. 71(21), 3087–3089 (1997)ADSGoogle Scholar
  41. NSG+09.
    Ning Y.X., Song C.L., Guan Z.L., Ma X.C., Chen X., Jia J.F., Xue Q.K.: Observation of surface superconductivity and direct vortex imaging of a Pb thin island with a scanning tunneling microscope. EPL 85(2), 27004–30000 (2009)ADSGoogle Scholar
  42. ORK+08.
    Oroszlany L., Rakyta P., Kormanyos A., Lambert C.J., Cserti J.: Theory of snake states in graphene. Phys. Rev. B 77(8), 081403 (2008)ADSGoogle Scholar
  43. Pan02.
    Pan X.B.: Surface superconductivity in applied magnetic fields above HC2. Commun. Math. Phys. 228(2), 327–370 (2002)ADSzbMATHGoogle Scholar
  44. PK02.
    Pan X.B., Kwek K.H.: Schrödinger operators with non-degenerately vanishing magnetic fields in bounded domains. Trans. Am. Math. Soc. 354(10), 4201–4227 (2002)zbMATHGoogle Scholar
  45. PM93.
    Peeters F.M., Matulis A.: Quantum structures created by nonhomogeneous magnetic fields. Phys. Rev. B 48(20), 15166 (1993)ADSGoogle Scholar
  46. RP98.
    Reijniers J., Peeters F.M.: Hybrid ferromagnetic/semiconductor Hall effect device. Appl. Phys. Lett. 73(3), 357–359 (1998)ADSGoogle Scholar
  47. RP00.
    Reijniers J., Peeters F.M.: Snake orbits and related magnetic edge states. J. Phys. Condens. Matter 12(47), 9771 (2000)ADSGoogle Scholar
  48. RS72.
    Reed M., Simon B.: Methods of Modern Mathematical Physics I: Functional Analysis. Academic Press, New York (1972)zbMATHGoogle Scholar
  49. SJG63.
    Saint James D., de Gennes P.G.: Onset of superconductivity in decreasing fields. Phys. Lett. 7(5), 306–308 (1963)ADSGoogle Scholar
  50. SJST69.
    Saint-James, D., Sarma, G., Thomas, E.J.: Type II superconductivity (1969)Google Scholar
  51. SS03.
    Sandier E., Serfaty S.: The decrease of bulk superconductivity close to the second critical field in the Ginzburg–Landau model. SIAM J. Math. Anal. 34(4), 939–956 (2003)MathSciNetzbMATHGoogle Scholar
  52. SS07.
    Sandier E., Serfaty S.: Vortices in the Magnetic Ginzburg–Landau Model. Progress in Nonlinear Partial Differential Equations and Their Applications. Birkhäuser, Boston (2007)zbMATHGoogle Scholar
  53. STH+94.
    Smith, A., Taboryski, R., Hansen, L.T., Sørensen, C.B., Hedegård, P., Lindelof, P.E.: Phys. Rev. B 50(19), 14726 (1994)Google Scholar

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Authors and Affiliations

  1. 1.Department of MathematicsLund UniversityLundSweden
  2. 2.Department of MathematicsLebanese UniversityNabatiehLebanon

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