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I am confused about the derivation of the critical current density of a thin superconductor film. The derivation is given in Tinkham book (2nd Ed. Page 124). The critical current is identified as the current where the order parameter drops to 2/3 of its equilibrium value.

This, however, implies that the order parameter is discontinuous since once we cross the critical current density it should drop to zero. Of course, my conclusion is wrong since superconductivity is a second-order phase transition and the order parameter should be continuous.

What do I miss? and how to understand the behavior of the order parameter around the critical current density.

enter image description here Maybe the confusion comes from the the fact that the current density is an interplay between the order parameter and the superfluid velocity?

Thanks

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The superconducting transition in terms of temperature is a second order phase transition. As a function of current it is, as you concluded, a first order transition.

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