I have been reading about superconductors and emerge me a inquietude about the explanation of existence of currents inside the superconductor while their magnetic field (inside too) is zero. Based in Ampère's law, the enclosed current by the superconductor must be zero (is it correct?). Some people tell me that the Ampère's law could not be used for superconductors and others the opposite. Then, can I explain the supercurrents in superconductors with Ampère's law? How can I do that? Is it correct use this law with these kind of materials?
The currents responsible for the Meissner effect (the cancellation of magnetic field inside the superconductor) are only surface currents, and the cancellation of magnetic fields doesn't happen on the surface, although it decays exponentially as you move deeper inside according to the equation for the London penetration depth. See for example this page which says:
This Meissner effect happens when electric current loops spontaneously appear on the surface of a material that becomes superconducting in the presence of a magnetic field. These currents create a magnetic field, similar to that of an electromagnet. These currents adjust to exactly compensate the magnetic field applied inside the superconductor. The total magnetic field hence becomes equal to zero in the sample volume, and the superconductor is protected, except on its surface where the currents develop. Thanks to these supercurrents, the superconductor expels the magnetic field. Since there is no electric resistance in a superconductor, these currents can exist forever without consuming any energy.
So, I think Ampere's law would imply that there can't be any non-negligible current enclosed by a loop that lies inside the superconductor well past the London penetration depth, but a loop that lies near the surface can have non-negligible current because it can also have a non-negligible magnetic field around it. This page specifically mentions using Ampere's law in the context of superconductors, so it can't generally be true that superconductors violate Ampere's law, though it's possible there could be specific conditions involving superconductors where it's not safe to use Maxwell's laws, perhaps someone else can comment on this.