Cooper Pairs

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
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.
where gk is the probability amplitude of finding
one electron with momentum hk and the
corresponding electron with momentum -hk
Of course gk 0 for k lt 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
or
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.
and this can be substituted into the two electron
Schrodinger equation
Lecture 10
4
The two electron Schrödinger equation
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
Lecture 10
5
The two electron Schrödinger equation
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 kk (and the cosin1),
in which case it is simply the volume of the
sample ?s
Lecture 10
6
The attractive potential
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 h?c away from EF
Vkk 0 for all k-states out beyond an energy of
h?c away from EF
Lecture 10
7
The attractive potential
Lecture 10
8
The Bound State
So there is a bound state with negative energy
with respect to the Fermi surface made up
entirely of electrons with kgtkF, 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 V0,
ie it cannot be expanded in powers of V, so it
cannot be obtained by perturbation theory
Lecture 10
9
Cooper pairs - a summary
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