JAMP  Vol.4 No.4 , April 2016
The Magnetic Field Distribution of Type II Superconductors Based on the Modified GL Equations

The standard Ginzburg-Landau (GL) equations are only valid in the vicinity of the critical temperature. Based on the Eilenberger equations for a single band and s-wave superconductor, we derive a modified version of the standard GL equations to improve the applicability of the standard formalism at temperature away from the critical temperature. It is shown that in comparison with previous studies, our method is more convenient to calculate and our modified equations are also compatible with a dirty superconductor. To illustrate the usefulness of our formalism, we solve the modified equations numerically and give the magnetic field distribution in the mixed state at any temperature. The results show that the vortex lattice could be still observed even away from the critical temperature (e.g., T/Tc = 0.3).

Cite this paper: Huang, R. and She, W. (2016) The Magnetic Field Distribution of Type II Superconductors Based on the Modified GL Equations. Journal of Applied Mathematics and Physics, 4, 669-676. doi: 10.4236/jamp.2016.44077.

[1]   Ginzburg, V.L. and Landau, L.D. (1950) On the Theory of Superconductivity. Sov. Phys. JETP, 20, 1064.

[2]   Abrikosov, A.A. (1957) Magnetic Properties of Superconductors of the Secondgroup. Sov. Phys. JETP, 5, 2274.

[3]   Gorkov, L.P. (1959) Microscopic Derivation of the Ginzburg-Landau Equations in the Theory of Super-conductivity. Sov. Phys. JETP, 36, 1364.

[4]   Abrikosov, A.A., Gorkov, L.P. and Dzyaloshinski, I.E. (1975) Methods of Quantum Field Theory in Statistical Physics. Dover Publications.

[5]   Eilenberger, G. (1968) Transformation of Gorkov’s Equation for Type II Superconductors into Transport-Like Equations. Z. Phisik, 214, 195.

[6]   Klein, U. (1987) Microscopic Calculations on the Vortex State of Type II Superconductors. J. Low Temp. Phys, 69, 1.

[7]   Ichioka, M., Hayashi, N. and Machida, K. (1997) Local Density of States in the Vortex Lattice in a Type-II Superconductor. Phys. Rev. B, 55, 6565.

[8]   Shanenko, A.A., Milosevic, M.V., Peeters, F.M. and Vagov, A.V. (2011) Extended Ginzburg-Landau Formalism for Two-Band Superconductors. Phys. Rev. Lett, 106, 047005.

[9]   Vagov, A.V., Shanenko, A.A., Milosevic, M.V., Axt, V.M. and Peeters, F.M. (2012) Extended Ginzburg-Landau Formalism: Systematic Expansion in Small Deviation from the Critical Temperature. Phys. Rev. B, 85, 014502.

[10]   Orlova, N.V., Shanenko, A.A., Milosevic, M.V., Peeters, F.M., Vagov, A.V. and Axt, V.M. (2013) Ginzburg-Landau Theory for Multiband Superconductors: Microscopic Derivation. Phys. Rev. B, 87, 134510.

[11]   Kogan, V.G. (1985) Eilenberger Equations for Moderately Dirty Superconductors. Phys. Rev. B, 31, 1318.

[12]   Hao, Z., Clem, J.R., Elfresh, M.W.M., Civale, L., Malozemoff, A.P. and Holtzberg, F. (1991) Model for the Reversible Magnetization of High-Type-II Superconductors: Application to High-t c Superconductors. Phys. Rev. B, 43, 2844.

[13]   Brandt, E.H. (1997) Precision Ginzburg-Landau Solution of Ideal Vortex Lattices for Any Induction and Symmetry. Phys. Rev. Lett, 78, 2208.

[14]   Brandt, E.H. (1995) The Flux-Line Lattice in Superconductors. Rep. Prog. Phys, 58, 1465.

[15]   Kleiner, W.M., Roth, L.M. and Autler, S.H. (1964) Bulk Solution of Ginzburg-Landau Equations for Type II Superconductors: Upper Critical Field Region. Phys. Rev., 133, A1226.

[16]   Doria, M.M., Gubernatis, J.E. and Rainer, D. (1989) Virial Theo-rem for Ginzburg-Landau Theories with Potential Applications to Numerical Studies of Type-II Superconductors. Phys. Rev. B, 39, 9573.

[17]   Helf, K. and Werthamer, N.R. (1966) Temperature and Purity Dependence of the Superconducting Critical Field, hc2.iii. Electron Spin and Spin-Orbit Effects. Phys. Rev, 147, 288.