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Can anyone please clarify my confusion?

Why does a superconducting state conducts electricity even if it has an energy gap?

The difference between metals and insulators is that there is a gap at the Fermi energy so electrons cannot be excited (i.e. the band is full). In superconductor, we have fully occupied lower band and a gap $2*\Delta$ and yet it still conducts electricity (with zero resistance of course).

I guess that these two things are not quite related where one concerns the state of single electrons and the other concerns pairs of electrons.

Thank you in advance!

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The first thing you have to realize is that the superconducting state is the ground state of the system. In this scenario, the Cooper pairs of electrons are basic entities with some binding energy.

The $2\Delta$ gap you mention doesn't refer to an excitation across the band gap of the material, but to to breaking of such Cooper pairs. This breaking of pairs is an excitation in the ground state of the electrons, which in this case they are delocalized.

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    $\begingroup$ In addition to their strong delocalisation, you may also say that Cooper pairs are bosons (or at least boson-like), and as such they evade the band theory which is only valid for fermions. $\endgroup$ – FraSchelle Mar 20 '15 at 9:08

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