60
Chapter
4:
Models and Theories
A number of related formulas for the dependence of Tc on 2 and #* have
appeared in the literature (e.g., McMillan, 1968).
The BCS theory predicts that at T~ there is a jump in the electronic specific
heat from its normal state value Ce = 7T to its superconducting state value C s
given by
Cs -TT~
= 1.43. (31)
In the free electron approximation the electronic specific heat coefficient 7
depends on the Fermi temperature T F and the gas constant R through the
expression
-- ~z2R/2TF .
(32)
Below To, the BCS theory predicts that the specific heat Cs(T ) depends
exponentially on the inverse temperature,
Cs(T )
-- a exp[-A/kBT ],
(33)
where A - 1.76 k B T c, and a is a constant.
Mechanisms for Cooper Pairing
There are three levels of explanation of the nature of superconductivity that are
commonly called BCS. One is the general formulation that does not specify
particular interactions. The second is the phonon-mediated version of the theory,
in which phonons play the role of bringing about the coupling together of two
electrons to form Cooper pairs. The third level, which was described in the
original formulation of the theory and provided the results summarized in the
previous section, further assumes the simplification of an isotropic, homogeneous
material with a square-well electron-electron interaction potential involving a
phonon coupling mechanism and s-wave singlet-state pairing. The superconduct-
ing elements, which are almost all Type I, as well as many classical Type II
superconductors, are looked upon as phonon-mediated s-state types. Pairing
mechanisms involving the exchange of particles other than phonons, such as
excitons or antiferromagnetic spin fluctuations, have been proposed for non-
classical superconductors. In particular, no consensus exists yet about the pairing
mechanism of the cuprates, although d-wave pairing seems to be favored (Annett
et al., 1996). A handbook is probably not the appropriate forum for elaborating
on these matters.