Abrikosov’s Flux-Line Lattice

  • Takafumi Kita
Part of the Graduate Texts in Physics book series (GTP)


Superconductors can be classified into two types according to their response to applied magnetic fields. Whereas type-I superconductors exclude the magnetic field completely from the bulk due to the Meissner effect, type-II superconductors can retain quantized magnetic fluxes in the bulk over a certain range of magnetic field. In 1957, Abrikosov solved the Ginzburg–Landau equations analytically for a couple of limiting cases to predict that type-II superconductors can form a lattice of quantized flux lines between lower critical field Hc1 and upper Hc2, which was later confirmed by experiments. In this chapter, we elaborate on this flux-line lattice.


Flux Density Landau Level Critical Field Landau Gauge London Penetration Depth 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    M. Abramowitz, I.A. Stegun (eds.), Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables (Dover, New York, 1965)Google Scholar
  2. 2.
    A.A. Abrikosov, J. Exp. Theor. Phys. 32, 1442 (1957). (Sov. Phys. JETP 5, 1174 (1957)) Corrections are summarized as follows. First, the square lattice given by Eqs. (20)–(22) is metastable, and the equilibrium is given by the hexagonal lattice with β = 1. 16 [12]. See Eq. (15.59) in the text. Second, numerical coefficients of Eqs. (36) and (37) should be corrected as 0. 08 → 0. 497 and 0. 18 → 0. 282 [9].Google Scholar
  3. 3.
    H.M. Adachi, M. Ishikawa, T. Hirano, M. Ichioka, K. Machida, J. Phys. Soc. Jpn. 80, 113702 (2011)CrossRefADSGoogle Scholar
  4. 4.
    G.B. Arfken, H.J. Weber, Mathematical Methods for Physicists (Academic, New York, 2012)Google Scholar
  5. 5.
    E. Brown, Phys. Rev. 133, A1038 (1964)CrossRefADSGoogle Scholar
  6. 6.
    I.M. Gelfand, S.V. Fomin, Calculus of Variations (Prentice-Hall, Englewood Cliffs, 1963)Google Scholar
  7. 7.
    V.L. Ginzburg, L.D. Landau, J. Exp. Theor. Phys. 20, 1064 (1950)Google Scholar
  8. 8.
    J.L. Harden, V. Arp, Cryogenics 3, 105 (1963)CrossRefADSGoogle Scholar
  9. 9.
    C.-R. Hu, Phys. Rev. B 6, 1756 (1972)CrossRefADSGoogle Scholar
  10. 10.
    T. Kita, J. Phys. Soc. Jpn. 67, 2067 (1998)CrossRefADSGoogle Scholar
  11. 11.
    T. Kita, M. Arai, Phys. Rev. B 70, 224522 (2004)CrossRefADSGoogle Scholar
  12. 12.
    W.H. Kleiner, L.M. Roth, S.H. Autler, Phys. Rev. 133, A1226 (1964)CrossRefADSGoogle Scholar
  13. 13.
    L.D. Landau, E.M. Lifshitz, Quantum Mechanics: Non-relativistic Theory, 3rd edn. (Butterworth-Heinemann, Oxford, 1991)Google Scholar
  14. 14.
    A.B. Pippard, Proc. R. Soc. Lond. A 216, 547 (1953)CrossRefADSGoogle Scholar
  15. 15.
    J.J. Sakurai, Modern Quantum Mechanics, rev. edn. (Addison-Wesley, Reading, 1994)Google Scholar

Copyright information

© Springer Japan 2015

Authors and Affiliations

  • Takafumi Kita
    • 1
  1. 1.Department of PhysicsHokkaido UniversitySapporoJapan

Personalised recommendations