# What are Cooper pairs

1. Jul 24, 2014

### Greg Bernhardt

Definition/Summary

A bound state of two electrons having equal and opposite momentum and spin. These states can form in a normal metal due to: 1) an attractive phonon-mediated interaction, and 2) the presence of the Fermi surface. The formation of Cooper pairs in a normal metal will occur if the temperature is low enough and will cause the metal to become a superconductor.

Equations

$$\psi({\bf r_1},\sigma_1;{\bf r_2},\sigma_2)= \left(\alpha(\sigma_1)\beta(\sigma_2)-\beta(\sigma_1)\alpha(\sigma_2)\right)\sum_{{\bf k},k>k_F}\frac{\cos({\bf k}\cdot({\bf r_1}-{\bf r_2}))}{2\epsilon_k-E}\;,$$
where the first factor is the spin-singlet part of the wave function, and the next factor is the symmetric real-space part of the wave function.

In the above equation $\epsilon_k=k^2/2m$ and $E$ is the energy of the Cooper pair, given in the weak coupling approximation by $E=2E_F-2\hbar\omega_c e^{-2/(\nu(0)V)}$, where $E_F$ is the Fermi energy, $\hbar$ is the reduced Planck constant, $\omega_c$ is an effective cutoff energy specifying over what interval the effective attraction is non-zero and equal to $-V$, and $\nu(0)$ is the density of states at the Fermi energy.

Extended explanation

One of the obvious problems with the idea of a bound state of two electrons is that the electrons have the same charge. Thus, in addition to an attractive phonon-mediated force there is a repulsive Coulomb force. The way around this problem is the effect of screening. In a metal there are many mobile electrons in addition to the two electrons making up the Cooper pair. It is possible that these "other" electrons screen the Coulomb repulsion enough that the phonon-mediated attraction is dominant.

But, even if the net force is attractive between electrons, this does not necessitate the existence of a bound state. Only in two or fewer dimensions will an arbitrarily weak attraction lead to a bound state. The way out of this dilemma is the existence of a Fermi surface. The fact that the two electrons making up a Cooper pair are above the filled "Fermi Sea" of the normal metal make the density of states appear two dimensional and allows for the existence of a bound state.

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