As touched upon earlier, conventional BCS superconductors
are characterized by a standard *s**-wave, spin-singlet* pairing state with
*S*=0, *L*=0 Cooper pairs. The two electrons of a pair have
equal and opposite momenta and ,
so that the centre-of-mass momentum of a Cooper pair is zero.
For an attractive electron-electron interaction, the bound state is
symmetric upon exchange of electron positions, so it must be an antisymmetric
singlet upon exchange of electron spins to satisfy the Pauli exclusion
principle. Thus, at any instant of time,
one can think of the electrons in a Cooper pair as being
in a state
,
and the
wavefunction describing the pair consists of all states ``*i*'' occupied by
the pair during
its lifetime. An attractive interaction between two electrons
results in a potential energy contribution which is negative, and thus
lowers the total energy of the electron system.
The negative potential energy associated
with a Cooper pair is the *binding energy* of that pair.

In normal metallic superconductors, the attractive interaction between electrons originates from the electron-phonon interaction, which is short range and retarded in space-time [43]. The lattice deformation resulting from the first electron takes a finite time to relax, and thus a second electron can be influenced by the lattice deformation at a later time. The attraction arising from the electron-phonon interaction overcomes the screened Coulomb repulsion between electrons due to the presence of other electrons and ions in the solid [26]. This screening aids in reducing the natural Coulomb repulsion between two electrons, leading to an effective interaction which is relatively short range compared with the unscreened Coulomb potential [38].

The
net effect of the attractive interaction on all the other electrons in the
material renders the normal Fermi liquid state unstable.
Consider the normal ground state of a metal in the absence of an attractive
electron-electron interaction at *T*=0K.
The kinetic energy (and hence the total energy) of the system is minimized
by requiring that the momenta of the plane wave states of the electrons
fill up a sphere of radius
(*i.e.*
a Fermi sphere) in three-dimensional
momentum space.
Fig. 2.6(a)
illustrates a Fermi sphere of radius *k*_{F} in three-dimensional *k*-space
for a free electron gas
at *T*=0K.
The corresponding normal density of states *N*(*E*), where *N*(*E*)*dE* is defined
as the number of electron states with energy between *E* and *E*+*dE*, is
shown in Fig. 2.6(b). *N*(0) denotes the density of states
at the Fermi surface at absolute zero.

Original BCS theory
described superconductivity in metals, so that the Fermi surface was
assumed spherical in the normal state.
In the BCS ground state (*i.e.* the state of the superconductor at *T*=0K),
some of the electron states just outside the normal
Fermi surface are occupied, and some just inside the Fermi surface are
unoccupied. Certainly such an arrangement has a higher kinetic energy
than does the normal state of the metal at *T*=0K. However, the BCS
ground state
is in part comprised of Cooper pairs, the formation of which lowers
the potential energy of the system. In the BCS ground state the arrangement
of Cooper pairs is such that lowering of the potential energy outweighs
the increase in kinetic energy, so that the BCS ground state has
a lower total energy than that of the normal ground state [44].

As mentioned, the decrease in potential energy is due to the phonon-induced, electron-electron attractive interaction. The phonon-induced attractive force scatters Cooper pairs from one state to another state . An electron occupying state near the Fermi surface vibrates the lattice resulting in the emission of a phonon of wave vector . The electron is thereby scattered to a state , where . A second electron occupying state , absorbs the phonon thus scattering to a state , where . The total centre-of-mass momentum in the final state is thus , which is unaltered from before the scattering process. The two electrons forming a Cooper pair are continually scattered between states with equal and opposite momentum. Since only the total centre-of-mass momentum is conserved in such scattering processes, the momenta of the individual electrons is continually changing, so that one cannot explicitly assign a single momentum to each electron in the pair.

Each scattering process reduces the potential energy of the electron
system further.
The negative potential energy contribution is greatest for scattering
between states of equal and opposite momentum [39].
The range of momenta available to the scattered electrons
in the ground state is dictated
by the energy of the phonon and the Pauli exclusion principle.
The phonon-induced
attractive interaction can only affect those electrons in the vicinity
of the Fermi surface. Electrons further inside the Fermi sphere cannot
scatter to other levels because of the Pauli exclusion principle. To
scatter electrons well below the Fermi level into unoccupied states would
require phonon frequencies much larger than that generated from the
electron interaction with the lattice. Electrons closer to the Fermi
surface may form Cooper pairs; however, as they do so
the number of states available to the scattering electrons decreases.
As the number of scattering events decreases, so does the maximum amount
by which the potential energy of the system can be lowered. The formation
of Cooper pairs must then cease at the moment the increase in kinetic
energy due to moving electrons above the Fermi level exceeds the amount
by which the potential energy is lowered. This limit then specifies
the arrangement of the BCS ground state.
One of the most distinctive consequences of BCS theory is that
an energy gap opens
up between the ground state and the lowest excited state.
As shown in Fig. 2.7(a), the energy
gap
has the same symmetry as the Fermi surface of
the normal state.
The important feature in *s*-wave pairing
is that the wave vector dependence of the energy gap
is finite everywhere on the Fermi surface at
temperatures below *T*_{c}.

As the temperature is increased above *T*=0K, thermally excited phonons
become available to
scatter the electron pairs. Because of the energy gap in the excitation
spectrum, excitations cannot occur with an arbitrarily small amount of
energy as in the case of a normal metal.
Phonons with energy comparable to twice the
energy gap ()
will scatter electrons in a Cooper pair to states above the gap.
The electrons from the pair will no longer have equal and opposite
momentum, so that their interaction potential becomes negligible and
the Cooper pair is destroyed.
Of course at low temperatures, the density
of phonons with this much energy is small. Near *T*_{c}on the other hand, phonons with energy on the order of the energy gap
are plentiful, and pair breaking is greatly enhanced.
In addition, the gap itself
is temperature dependent and can be well approximated by [15]:

(30) |

except near

Fig. 2.7(b) and Fig. 2.8 show the associated
density of quasiparticle states *N*(*E*)at *T*=0K and *T* slightly above absolute zero, respectively.
The states which are no longer occupied
between
and
*E*_{F}+
pile up on
the edges of the gap region. Above the energy gap,
the electrons are often referred to as *normal electrons*
or *quasiparticles*. That is, if an electron occupies a state
then the state
need not be occupied. The
corresponding density of quasiparticles
(*i.e.* the *normal fluid density*) is usually denoted
*n*_{n}. Below the energy gap the electrons
are paired. These electrons are referred to as *superconducting electrons*,
and their associated density
(*i.e.* the *superfluid density*) is
denoted *n*_{s}. At *T*=0K all the electrons are
superconducting, while for
they are all normal.
For temperatures in between, the system is a mixture of superconducting
and normal electrons.

2001-09-28