In 1957 the underlying microscopic theory of superconductivity in
metals was unveiled by J. Bardeen, L.N. Cooper and J.R. Schrieffer
[24], in the now famous *BCS theory*.
In normal metals, the situation
is well described by *free electron theory*, where the electrons
behave as free particles and the metallic ions play a limited role
in conductivity. BCS theory outlines how in the presence of an
attractive interaction between electrons (*Cooper pairs*), the
normal state of an otherwise free electron gas becomes unstable to
the formation of a coherent many-body ground state.
The mechanism behind the weak attractive force binding
the Cooper pairs was actually first suggested by Herbert Frölich
[25]. He proposed that the same mechanism responsible for
much of the electrical resistivity in metals (*i.e.* the
interaction of conduction
electrons with lattice vibrations) leads to a state of superconductivity.
This hypothesis of an electron-phonon interaction was born
out of experiments which found that the critical temperature *T*_{c}varied with isotopic mass. In simple terms, an electron interacts
with the lattice by virtue of the Coulomb attraction it feels for
the metallic ions. The result is a deformation of the lattice
(*i.e.* a phonon). A second electron in the vicinity of the deformed
lattice correspondingly lowers its energy, resulting in an
electron-electron attraction via a phonon. Viewed in this context, the
superconducting order parameter
from GL-theory
can be interpreted as a one-particle wave function describing the
position of the center of mass of a Cooper pair [26].

Despite being an extremely weak attraction, bound
pairs form in part because
of the presence of a Fermi *sea* of additional electrons. As a
result of the Pauli exclusion principle, electrons that would prefer
to be in a state of lower kinetic energy cannot populate
these states because they are already occupied by other electrons.
Thus a Fermi sea is required to ensure the formation
of bound pairs of electrons; otherwise an isolated pair of electrons
would just repel one another as a result of the Coulomb force between
them. The Fermi sea itself is comprised of other distinct bound
pairs of electrons. It follows that each electron is a member of both
a Cooper pair and of the Fermi sea which is necessary for the formation
of all Cooper pairs. The force of attraction between the electrons
which comprise a Cooper pair has a range equivalent to the coherence length.
It should be noted that the separation between
electrons in a Cooper pair (and thus the correlation length ),
for a type-I superconductor, is
large enough that millions of other pairs have their centers of mass
positioned between them. It is then assumed that the occupancy of a bound
pair is instantaneous and uncorrelated with the occupancy of
other bound pairs at an instant in time [27].
Armed with knowledge of the fundamental particles responsible for
superconductivity (*i.e.* Cooper pairs), the substitutions *q*=2*e* and
immediately transform the
Ginzburg-Landau result for the penetration depth
[*i.e.* Eq. (2.6)]
into the result predicted by London theory [*i.e.*
Eq. (2.1)].

One of the most remarkable
features emerging from BCS theory, is the existence of an
*energy gap*
between the BCS ground state and the first excited
state. It is the minimum energy required to create a single-electron (hole)
excitation from the superconducting ground state.
Thus the binding energy of a Cooper pair is two times the energy gap
.
BCS theory estimates the zero-temperature energy gap
(0) as [26]:

and near the critical temperature

so that the energy gap approaches zero continuously as . Superconductors which obey Eq. (2.17) are considered to be weakly-coupled, in reference to the weak interaction energy between electrons in a Cooper pair. Furthermore, the wave functions corresponding to electron pairs (Cooper pairs), are spatially symmetric like an atomic

for small

If the value of the energy gap is not constant over the entire Fermi surface, then the minimum value of the gap determines the density of quasiparticle excitations at these low temperatures. Hence the topology of the energy gap is crucial in deciding the low-temperature behaviour of . If the sample under investigation is riddled with impurities, then there will exist a broad range of transition temperatures [30]. Equations (2.17) and (2.18) suggest that one should anticipate a corresponding distribution of gap energies in such materials.

Some systems (e.g. lead, mercury) produce experimental results which
deviate substantially from the BCS results [26]. These materials
are more appropriately described by strong-coupling theory where
the coupling ratio
is greater than the
BCS prediction of 1.76.
Under certain conditions superconductivity can occur without an energy gap
in some materials. Tunneling experiments on superconductors with
specific concentrations of paramagnetic impurities show this to be
possible [31]. Theories exist which explain such anomalies,
and the nature of the gap as we will soon see is a vital property
to be considered in any
theory describing superconductivity in the high-*T*_{c} compounds.

2001-09-28