A common misconception about superconductivity is that the presence of an energy gap at the Fermi energy is fundamental to the phenomenon. This is not correct. The key property is long range phase coherence.
I give several counter-examples to the necessity of an energy gap.
If in BCS theory you take an s-wave superconductor and add magnetic impurities there is a critical range of impurity concentration for which there is no energy gap (i.e. the density of states is non-zero at the Fermi energy) but superconductivity (i.e. a non-zero Cooper pairing amplitude and superfluid stiffness exists).
Superfluid 3He (p-wave triplet pairing) and high-Tc cuprates (d-wave singlet pairing) have nodes in the energy gap on the Fermi surface.
A gap is also not sufficient. A charge density wave state can have a gap at Fermi energy but is not be a superconductor.
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