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Superconductivity

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In my notes from University the reason a material can exhibit superconducting properties is at really low temperatures electrons form Cooper pairs of which the electrons of the Cooper pairs have lower energy than if they were in the conduction band so which means that they dont collisions with the cations of the lattice which creates resistance.

However im not very convinced since in p type doping ,the electrons are moving inside the valence band and there is a band gap between valence band and conduction band so by the definition these materials must also be superconducting but this is clearly not the case.

What is wrong/incomplete from the notes of the class?

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