  The maximum value of the superconducting transition temperature Tc due to a
phonon-mediated interaction was estimated by Cohen and Anderson in 1972 from ab
initio considerations, and found to be about 10 K. McMillan's semi empirical
estimate from 1968 gives a value of about 40 K. We consider these estimates on
the basis of subsequent theoretical and experimental evidence, and pay
attention in particular to the inhomogeneity of the electron gas. This
inhomogeneity gives rise to local-field effects, which are mentioned by Cohen
and Anderson, however without an explicit estimate of their effect on Tc. They
claim that strong local-field effects cause a transition to covalent bonds,
which inhibits superconductivity. We consider here strong local-field effects
by making use of the inverse dielectric matrix (in reciprocal space), which we
review in some detail. We distinguish between the electronic and ionic
dielectric matrices, and show that the later can give rise to much stronger
local fields, because of the inherent local nature of the ions (in contrast
with the delocalised nature of the conduction electrons). Moreover, the ionic
dielectric matrix is large only at very low frequencies omega (below the
dispersion frequency of the dielectric constant), therefore the strong
local-field effects do not cause the instability leading to the formation of
covalent bonds. As a result, the maximum Tc can be much higher than the value
estimated by Cohen and Anderson, and by McMillan, and can reach about 200 K in
nearly-ferroelectric materials like the high-Tc cuprates.
