1. Cooper Pairs
In the 1950s it was becoming clear that the superelectrons were paired
ie there must be a positive interaction that holds a pair of electrons together despite Coulomb repulsion
The ground state of a free electron gas is represented by the complete filling of the one-electron energy levels of wave vector k and energy E=ħ2k2/2m EF E
N(E)
Cooper, in 1956, considered what would happen if two electrons were added to the Fermi sea at zero temperature, interacting with each other, but not with the sea except via the Pauli exclusion principle.
He found that the presence of an attractive potential between even just one pair of electrons makes this state unstable
Lecture 10

2. Two-particle wave function
Cooper began to look for a two-particle wave function to describe the pair of electrons added just above the Fermi surface.
By a relatively simple (Bloch) arguments it can be shown that the lowest energy state of a pair of electrons is when the pair has zero momentum.
ie when the two electrons have equal and opposite momenta.
This suggests
where gk is the probability amplitude of finding one electron with momentum ħk and the corresponding electron with momentum -ħk
Of course gk = 0 for |k| < kF where kF is the Fermi momentum, as all states below kF are already filled.
Because electrons are fermions the wavefunction must be antisymmetric with respect to the exchange of the two electrons
The two electron wavefunction can be made asymmetric by considering the spatial and spin components
Lecture 10

3. The antisymmetric wave function
To account for the antisymmetry can be converted to a sum of products of either:
with an antisymmetric spin singlet (12-12), ie 
(a) or
with a symmetric spin triplet (1 2, 12 + 12, 1 2 ), ie  
(b)
Anticipating an attractive interaction we expect the spin singlet to have the lower energy, (it obeys the Pauli exclusion principle) because the cosine dependence of the wavefunction on (r1-r2) gives a larger probabilty of electrons being near each other.
So we have
and this can be substituted into the two electron Schrodinger equation
Lecture 10

4. The two electron Schrödinger equation
Writing the spatial part of the wave function as
where r=r1-r2
We can write the Schrodinger equation as
where
12 and 22 operate on the coordinates of electron 1 and electron 2, giving 12  = -k2  and 22  = -k2 
V(r) is the attractive potential between the two electrons
EF is the Fermi energy and E is the change in energy due to the attractive force
as the free electron energy of one of the states
We can write 2EF+ E =E, the total energy of the electron pair and also
Lecture 10

5. The two electron Schrödinger equation
Substituting for o(r) the Schrodinger equation becomes
We can solve this equation for gk by first multiplying both sides by e-ik´.r
then integrating over the volume of the sample s
We know that the wave function is spatially symmetric (ie a cosine) so the integral on the left must be zero unless k=k´ (and the cosin=1), in which case it is simply the volume of the sample s
Strength of potential to scatter a pair of electrons from (k’,-k’) to (k,-k)
We therefore have
with
Lecture 10

6. The attractive potential
If a set of gk satisfying the equation
exists for EF < EF then a bound pair exists - the two electrons will have a total energy less than the energy of the free electrons
To solve this equation for any generalised Vkk’ is very difficult
Cooper simplified the problem by making the approximation that:
Vkk’ = -V for all k-states out to a cut off energy of ħc away from EF
Vkk’ = 0 for all k-states out beyond an energy of ħc away from EF so
Lecture 10

7. The attractive potential
Summing both sides of the equation
over all k-states we have
and
Replacing the summation by an integration, with N(EF) denoting the density of states at the Fermi energy for electrons with one of the two spin orientations so
Lecture 10

8. The Bound State Evaluating
in the weak coupling limit, where N(EF)V<<1 the energy E of the electron pair can be written as
Evaluating
So there is a bound state with negative energy with respect to the Fermi surface made up entirely of electrons with k>kF, ie with kinetic energies in excess of EF
The contribution of the attractive potential to the energy outweighs the kinetic energy of the electrons, leading to a binding irrespective of however small V might be
Note that this expression is not analytic at V=0, ie it cannot be expanded in powers of V, so it cannot be obtained by perturbation theory
Lecture 10

9. Cooper pairs - a summaryEF E
E<EF
In the weak coupling limit, where N(EF)V<<1 the energy E of the electron pair can be written as
and two electrons with equal and opposite momenta and opposite spin will form a pair with total energy less than the Fermi energy
Cooper pairs
So any attractive potential will cause electrons above the Fermi energy to condense into a more ordered (ie lower energy) state
Lecture 10