What happens to the resistance of a superconducter when its temperature drops below the critical temperature?

Answer 1

It becomes exactly zero, so long as certain conditions are met. Namely no strong external magnetic fields.

Some superconducting materials exhibit a transition region in between super- and normal-conducting modes, although resistance here is still very close to 0; crossing the critical temperature is still accompanied by a discontinuous drop in resistance.

Answer 2

The zero resistance of a superconductor is a quantum effect.

Traditional superconductivity was explained in 1956 by Bardeen, Cooper Schrieffer and is known as the BCS theory. (The more recent form of superconductivity found since 1986 is called high temperature superconductivity and the theory for that is still undergoing development and so can not be discussed here.)

BCS theory begins with the normal band theory of a metal where electrons have wave functions that extend throughout the conductor. In BCS theory, at very low temperatures, there is an electronic state that can be formed between two electrons of opposite spin and the name given to such a pair of electrons is a Cooper pair. That Cooper pair then decreases the energy of all the other conduction electrons in the metal. That lower energy creates a new electronic state and there is an energy gap between that state and the other possible (unoccupied) electronic states in the metal.

This new electronic state can have a finite momentum, i.e. velocity, and so these electrons can move. Because of the energy gap, they can not lose energy through scattering which is the usual cause of resistance. Thus, once started, a current in a superconducting loop can exist forever.

Caveat: The originators of this theory received a Nobel prize, so the reader can not expect that all the details are properly included in these few paragraphs.