Why the cooper pair do not obey the exclusion principle?

Question

In superconductivity, which occurs in certain materials at very low temperature, electrons are linked together in cooper pair. And why the cooper pair do not?

Thank you in advance.

Answer 1

In superconducting materials two electrons are bound together such that they have up and down spin and make usually a spin 0 (sometimes spin 1 as in Helium-3 super fluid) particle (a boson), this bonded pair of electrons is known as cooper pair. This pairing is due to electron-phonon interaction in which at low temperature the positive lattice of ions is actually cancelling the repulsion force between two electrons.

Since cooper pair is a boson hence it follows Bose-Einstein statistics, which greatly reduces the scattering of electrons and this is the reason behind superconductivity.

Answer 2

The standard approach is that there is bosonization. Suppose that we have the fields $\psi_1(x)$ and $\psi_2(x')$ for two fermions, or usually electrons. Let us consider the product of states $\bar\psi_1\psi_2$ and $\bar\psi_2\psi_1$ $=~(\bar\psi_1\psi_2)^\dagger$ with $\psi_i~=~\psi_i^\dagger\gamma^0$, and the positions implicit. The anticommutator of products $\{\bar\psi_1\psi_2,~\bar\psi_2\psi_1\}$ can be computed by shifting the position of the fields with a minus sign so that $$ \{\bar\psi_1\psi_2,~\bar\psi_2\psi_1\}~=~\bar\psi_1\psi_2\bar\psi_2\psi_1~+~\bar\psi_2\psi_1\bar\psi_1\psi_2 $$ $$ =~2\bar\psi_1\psi_2\bar\psi_2\psi_1, $$ where several commutation operations were performed to get to the last step. The commutator is $$ [\bar\psi_1\psi_2,~\bar\psi_2\psi_1]~=~0. $$ This suggests this product of fields has bosonic quantum commutator properties

We then write the products as $$ \bar\psi_1\psi_2~=~:e^{i\phi}:,~\bar\psi_2\psi_1~=~:e^{-i\phi^\dagger}:, $$ where the colons mean normal ordering. Now evaluate the commutator $$ [\bar\psi_1\psi_2,~\bar\psi_2\psi_1]~=~:[e^{i\phi},~e^{-i\phi^\dagger}]:~=~:[\phi,~\phi^\dagger]: $$ where the normal ordering means this commutator is zero. This means the fermions have been paired into bosons. This is the basic mechanism behind superconductivity.