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The article on Wikipedia on BCS theory mentions

In conventional superconductors, an attraction is generally attributed to an electron-lattice interaction. The BCS theory, however, requires only that the potential be attractive, regardless of its origin.

But an attractive potential between two electrons could also be the gravity between electrons, correct? So could it be possible that Cooper-Pairs of electrons form purely by their gravitational interaction?

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    $\begingroup$ No, Cooper pair formation is due to an effective potential and symmetry breaking. The force of gravity between two electrons is negligible for Cooper pair formation. $\endgroup$
    – user106422
    Commented Oct 27, 2016 at 20:22
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    $\begingroup$ The electric force between 2 electrons is always $4.1 × 10^{42}$ times as great as the gravitational force between them, at any artibrary distance. $\endgroup$
    – user108787
    Commented Oct 27, 2016 at 22:27

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The positive potential requirement mentioned above refers to the 'net' potential after inclusion of the repulsive potential term due to electric forces. Sure gravitational force offers an attractive potential but it is not enough to overcome the great repulsive potential offered by the electric forces. It is however not possible to form Cooper-Pairs due to gravitational effects.

Also, I should add to this: the binding of two electrons to form a Cooper-Pair takes place in k-space which is mediated by phonons. This binding is NOT in real space! One, therefore, should not think of this phenomenon as 'attraction' even though it näively seems so from the intuitive (but wrong!) picture of lattice polarisation.

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