×

Proof of the De Gennes formula for the superheating field in the weak \(\kappa\) limit. (English) Zbl 0889.34010

Summary: In continuation of our preceding paper [Rev. Math. Phys. 8, No. 1, 43-83 (1996; Zbl 0864.35097)] concerning superconducting films, we present in this article new estimates for the superheating field in the weak \(\kappa\) limit. The principal result is the proof of the existence of a finite superheating field \(h^{sh,+}(\kappa)\) (obtained by restricting the usual definition of the superheating field to solutions of the Ginzburg-Landau system \((f, A)\) with \(f\) positive) in the case of a semi-infinite interval. The bound is optimal in the limit \(\kappa\to 0\) and permits to prove (combining with our previous results) the De Gennes formula \[ 2^{-{3\over 4}}= \lim_{\kappa\to 0} \kappa^{{1\over 2}} h^{sh,+}(\kappa). \] The proof is obtained by improving slightly the estimates given in the paper of the authors, where an upper bound was found but under the additional condition that the function \(f\) was bounded from below by some fixed constant \(\rho>0\).

MSC:

34A34 Nonlinear ordinary differential equations and systems
82D55 Statistical mechanics of superconductors

Citations:

Zbl 0864.35097
PDFBibTeX XMLCite
Full Text: DOI Numdam EuDML

References:

[1] Bolley, C.; Helffer, B., Rigorous results on the Ginzburg-Landau models in a film submitted to an exterior parallel magnetic field, Nonlinear Studies, Vol. \(3, n^o 2, 1-32 (1996)\), Part II · Zbl 0857.34006
[2] Bolley, C.; Helffer, B., Sur les asymptotiques des champs critiques pour l’équation de Ginzburg-Landau, (Séminaire Equations aux dérivées partielles de l’Ecole Polytechnique (November 1993)) · Zbl 0877.35120
[3] Bolley, C.; Helffer, B., Rigorous results for the Ginzburg-Landau equations associated to a superconducting film in the weak κ-limit, Reviews in Math. Physics, Vol. \(8, n^o 1, 43-83 (1996)\) · Zbl 0864.35097
[4] Bolley, C.; Helffer, B., Superheating in a film in the weak κ limit: numerical results and approximate models (Dec. 1994), (Part I to appear in \(M^2 AN )\)
[5] C. BOLLEY and B. HELFFER, In preparation.; C. BOLLEY and B. HELFFER, In preparation.
[6] Chapman, S. J., Asymptotic analysis of the Ginzburg-Landau model of superconductivity: reduction to a free boundary model (1992), Preprint · Zbl 0848.35130
[7] Galaiko, V. P., Superheating critical field for superconductors of the first kind, Soviet Physics JETP, Vol. \(27, n^o 1\) (July 1968)
[8] de Gennes, P. G., Superconductivity, selected topics in solid state physics and theoretical Physics, (Proc. of 8th Latin american school of physics. Proc. of 8th Latin american school of physics, Caracas (1966)) · Zbl 0138.22801
[9] Ginzburg, V. L., On the theory of superconductivity, Nuovo Cimento, Vol. 2, 1234 (1955) · Zbl 0067.23504
[10] Ginzburg, V. L., On the destruction and the onset of superconductivity in a magnetic field, Soviet Physics JETP, Vol. 7, 78 (1958) · Zbl 0099.44703
[11] Landau, L. D., (TerHaar, D., Men of Physics (1965), Pergamon: Pergamon oxford), 138-167, English translation
[12] Hastings, S. P.; Kwong, M. K.; Troy, W. C., The existence of multiple solutions for a Ginzburg-Landau type model of superconductivity (May 1995), Preprint
[13] Saint James, D.; de Gennes, P. G., Onset of superconductivity in decreasing fields, Phys. Lett., Vol. 7, 306 (1963)
[14] Saint James, D.; Sarma, G.; Thomas, E. J., Type II Superconductivity (1969), Pergamon Press
[15] Parr, H., Superconductive superheating field for finite κ, Z. Physik, Vol. B25, 359-361 (1976)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.