High-Kappa Limits of the Time-Dependent Ginzburg–Landau Model

Abstract

The time-dependent Ginzburg-Landau model of superconductivity is examined in the high-x, high magnetic field setting. This work generalizes the previous result for the steady-state model with a constant applied magnetic field. The significance of this generalization lies in the ability to incorporate the effects of both the applied magnetic field and applied current or voltage. Thus, it is possible to use the simplified setting obtained in this paper to study the motion and "pinning" of vortices in the presence of an applied current and a variable applied field. The results within are obtained via a formal asymptotic expansion of the Ginzburg-Landau equations in terms of , which yields a simplified system of leading-order equations. The formal asymptotic expansion is then justified by showing the solution to the full time-dependent Ginzburg-Landau equations converges to the solution of the leading-order equations as c. Computational results are also given that show the simplified leading-order model is indeed an accurate approximation to the solution of the full system of equations, even for moderate values of .

The frozen-field approximation and the Ginzburg-Landau equations of superconductivity

“…The analysis given here differs at several points from the analysis of Ref. [10]. First, our scaling is slightly different and, we believe, more in tune with the physics; second, our regularity assumptions on the applied field are weaker; third, our proofs are more direct; and fourth, our results hold in a stronger topology.…”

“…This choice is realized by identifying the gauge χ with a solution of the linear parabolic equation 10) subject to the condition n · ∇χ = −n · A on the boundary. In [9], it was shown that the TDGL equations, subject to the constraint (2.9), define a dynamical system under suitable regularity conditions on H. (In the more general case, where H varies not only in space but also in time, the TDGL equations define a dynamical process.)…”

“…The dynamics of the GL equations have been studied by several authors; see [7,8,9] and the references cited therein. The present investigation is closely related to the work of Du and Gray [10]. Section 2 introduces the Ginzburg-Landau equations, Section 3 contains the numerical results and Section 4 the analysis.…”

Simplified models of superconducting-normal-superconducting junctions and their numerical approximations

“…In this regime, much of the discussion we have already examined can be further simplified, similar to the work of Chapman et al [9] and Du & Gray [11].…”

“…To give a simple illustration, let us follow the discussion of Du & Gray [11] and choose the length scale to be ξ, the coherence length, and assume that the penetration depth λ (thus κ) is large with respect to the size of the superconducting sample. Let H ext = κH 0 , where H 0 is independent of κ, and define…”