Abstract
The Ginzburg-Landau (GL) equations of superconductivity provide a computational model for the study of magnetic flux vortices in type-II superconductors. In this article it is shown through numerical examples and rigorous mathematical analysis that the GL model reduces to the frozen-field model when the charge of the Cooper pairs (the superconducting charge carriers) goes to zero while the applied field stays near the upper critical field.
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D.W. Braun, G.W. Crabtree, H.G. Kaper, A.E. Koshelev, G.K. Leaf, D.M. Levine and V.M. Vinokur, Structure of a moving vortex lattice. Phys. Rev. Lett. 76 (1996) 831–834.
G.W. Crabtree, D.O. Gunter, H.G. Kaper, A.E. Koshelev, G.K. Leaf and V.M. Vinokur, Numerical solution of driven vortex systems. Phys. Rev. B 61 (2000) 1446–1455.
I. Aranson and V. Vinokur, Surface instabilities and plastic deformation of vortex lattices. Phys. Rev. Lett. 77 (1996) 3208–3211.
M. Tinkham, Introduction to Superconductivity (2nd ed.). New York: McGraw-Hill (1996) xxiC454 pp.
V.L. Ginzburg and L.D. Landau, On the theory of superconductivity. Zh. Eksp. Teor. Fiz. (USSR) 20 (1950) 1064–1082; Engl. transl. in: D. ter Haar, L.D. Landau; Men of Physics (Vol.1). Oxford: Pergamon Press (1965) pp. 138-167.
Q. Du, M.D. Gunzburger and J.S. Peterson, Analysis and approximation of the Ginzburg-Landau model of superconductivity. SIAM Rev. 34 (1992) 54–81.
Q. Du, Global existence and uniqueness of solutions of the time-dependent Ginzburg-Landau model for superconductivity. Appl. Anal. 53 (1994) 1–18.
Q. Tang and S. Wang, Time-dependent Ginzburg-Landau equations of superconductivity. Physica D 88 (1995) 139–166.
J. Fleckinger-Pellé, H.G. Kaper and P. Takáč, Dynamics of the Ginzburg-Landau equations of superconductivity. Nonlin. Anal.: Theory, Methods & Applic. 32 (1998) 647–665.
Q. Du and P. Gray, High-kappa limits of the time-dependent Ginzburg-Landau model. SIAM J. Appl. Math. 56 (1996) 1060–1093.
A. Schmid, A time dependent Ginzburg-Landau equation and its application to a problem of resistivity in the mixed state. Phys. Kondens. Materie 5 (1966) 302–317.
L.P. Gor'kov and G.M. Eliashberg, Generalizations of the Ginzburg-Landau equations for non-stationary problems in the case of alloys with paramagnetic impurities. Zh. Eksp. Teor. Fiz. 54 612426 (1968); Soviet Phys.-JETP 27 (1968) 328–334.
L.P. Gor'kov and N. Kopnin, Vortex motion and resistivity of type-II superconductors in a magnetic field. Soviet Phys. - Usp. 18 (1976) 496–516.
V.L. Ginzburg, On the destruction and the onset of superconductivity in a magnetic field. Soviet Phys. - JETP 34(7) (1958) 78–87.
S.J. Chapman, Superheating field of type-II superconductors. SIAM J. Appl. Math. 55 (1995) 1233–1258.
V. Georgescu, Some boundary value problems for differential forms on compact Riemannian manifolds. Ann. Mat. Pura Appl. 122(4) (1979) 159–198.
D. Henry, Geometric Theory of Semilinear Parabolic Equations. Lecture Notes in Mathematics, Vol. 840. New York: Springer-Verlag (1981) 348 pp.
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Kaper, H.G., Nordborg, H. The frozen-field approximation and the Ginzburg-Landau equations of superconductivity. Journal of Engineering Mathematics 39, 221–240 (2001). https://doi.org/10.1023/A:1004854616387
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DOI: https://doi.org/10.1023/A:1004854616387