ISSN:
1432-0673
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
,
Physics
Notes:
Abstract. We consider the Ginzburg‐Landau system with an applied magnetic field and analyze the behavior of solutions when the domain is a cylinder (of radius $\overline{r}$ ) and the applied field is parallel to the axis. It is shown that there is an upper critical value $\overline{h}$ such that if the modulus of the applied field is greater than $\overline{h}$ , the normal (nonsuperconducting) state (in which the order parameter is identically zero) is stable and if the modulus of the applied field is slightly below $\oh$ , the normal state is unstable. In addition, it is shown that there is a positive lower critical value $\underline{h}\leq\overline{h}$ such that the normal state is unstable if the modulus of the applied field is less than $\underline{h}$ and stable if the modulus is slightly above $\underline{h}$ . In the case of type‐II materials for whic h the Ginzburg‐Landau constant κ is large, it is shown that there is a discrete set of radii ℬ(κ) such that if $\overline{r}\notin\cal{B}(\kappa)$ and $\kappa\overline{r}$ is sufficiently large, then for each applied field of modulus slightly less than $\overline{h}$ (or slightly more than $\underline{h}$ ) there is precisely one small superconducting solution (up to a gauge transformation) which is stable. Moreover for this solution, the complex‐valued order parameter ψ is zero only on the axis of the cylinder, and its winding number is proportional to the product of κ2 and the cross‐sectional area of the cylinder. In addition, the solution exhibits “surface superconductivity” as predicted by the physicists de Gennes and St. James.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/s002050050082
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