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We know that like charges repel each other. But my professor claimed that two electrons can attract each other as well. What he said was that due to screening an electron traveling at some speed wont repel another electron but will in some case they attract each other due to weak phonon exchange. What does that mean? What is phonon exchange? Do two electrons really attract each other?

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From the Wiki article Cooper pair:

In condensed matter physics, a Cooper pair or BCS pair is two electrons (or other fermions) that are bound together at low temperatures in a certain manner first described in 1956 by American physicist Leon Cooper. Cooper showed that an arbitrarily small attraction between electrons in a metal can cause a paired state of electrons to have a lower energy than the Fermi energy, which implies that the pair is bound. In conventional superconductors, this attraction is due to the electron–phonon interaction. The Cooper pair state is responsible for superconductivity, as described in the BCS theory developed by John Bardeen, Leon Cooper, and John Schrieffer for which they shared the 1972 Nobel Prize.

Although Cooper pairing is a quantum effect, the reason for the pairing can be seen from a simplified classical explanation. An electron in a metal normally behaves as a free particle. The electron is repelled from other electrons due to their negative charge, but it also attracts the positive ions that make up the rigid lattice of the metal. This attraction distorts the ion lattice, moving the ions slightly toward the electron, increasing the positive charge density of the lattice in the vicinity. This positive charge can attract other electrons. At long distances this attraction between electrons due to the displaced ions can overcome the electrons' repulsion due to their negative charge, and cause them to pair up. The rigorous quantum mechanical explanation shows that the effect is due to electron–phonon interactions.

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I have been reading this article since few hours and something is making me confused. I dont know how can an electron distort the lattice site even though the positive ion has much greater charge than a single electron? –  AaKASH Jun 12 '13 at 1:48
@A4KASH, I think it may be the case that you're failing to distinguish between the case of "not distorting the lattice at all" and the case of "minutely distorting the lattice". The next sentence after the quote of the article I provide is: The energy of the pairing interaction is quite weak, of the order of 10−3eV, and thermal energy can easily break the pairs. So only at low temperatures are a significant number of the electrons in a metal in Cooper pairs. –  Alfred Centauri Jun 12 '13 at 1:53
@AlfredCentauri The most important point of the Alfred Centauri's answer is obviously that the second paragraph of the Wikipedia description is a simplified classical explanation of the Cooper pairing. The Cooper pairing is better explained (to my mind) as an instability of the Fermi sea caused by electron-phonon coupling, as the first paragraph of his quotation tells you. You need a quantum mechanical treatment to understand the Cooper pairing. By the way, neither a phonon nor an electron is a classical object :-) –  FraSchelle Jun 12 '13 at 18:01
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