Cracks and critical current

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Abstract
In superconductors, cracks (of width w ≳ ξs) are effective in enhancing local supercurrent, and we calculate the size of these supercurrent hot spots as a function of the crack length a and the London penetration depth λ using the two-dimensional (2D) London theory. In the λ→∞ limit and constant injected current density, we show that this 2D solution is also exact for the current flow in a thin film containing a through crack. We argue that large local supercurrents near a surface crack nucleate vortex creation. If flux pinning is weak enough these vortices flow under the influence of the large local Lorentz force. The dissipation so produced can lead to a reduction in observed critical current. If flux pinning is moderate, the first additional vortices nucleated near the crack tip are pinned, in a region we label ρ, the flux pinning zone. In the case of a through crack in a thin film, we then argue that jc(a) reduces as jc(a)/jc(0) ∼ (ρ/a)x (for a/L≪1), where L is the film width and x =1/2 for the simplest London theory. We compare this theory with the Bean (critical state) model, which predicts that the critical current is (approximately) determined by the cross section available for supercurrent, so that in a film containing a crack, jc(a)/jc(0) ∼ 1 − a/L (ignoring self-field effects). We argue that superconductors with sufficiently weak pinning should obey the hot-spot theory, while sufficiently hard superconductors should obey the critical-state model, and suggest experiments that should illustrate these two limiting cases.