Pinning effects and their breakdown for a Ginzburg–Landau model with normal inclusions

We study a Ginzburg–Landau model for an inhomogeneous superconductor in the singular limit as the Ginzburg–Landau parameter κ=1∕ϵ→∞. The inhomogeneity is represented by a potential term V(ψ)=14(a(x)−∣ψ∣2)2, with a given smooth function a(x) which is assumed to become negative in finitely many smooth subdomains, the “normally included” regions. For bounded applied fields (independent of the Ginzburg–Landau parameter κ=1∕ϵ→∞) we show that the normal regions act as “giant vortices,” acquiring large vorticity for large (fixed) applied field hex. For hex=O(∣lnϵ∣) we show that this pinning effect eventually breaks down, and free vortices begin to appear in the superconducting region where a(x)>0, at a point set which is determined by solving an elliptic boundary-value problem. The associated operators are strictly but not uniformly elliptic, leading to some regularity questions to be resolved near the boundaries of the normal regions.

Pinning effects and their breakdown for a Ginzburg–Landau model with normal inclusions Journal Articles