Dielectricity and hard phonons

The maximum value of the superconducting transition temperature T_c 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 T_c. 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 delocalized nature of the conduction electrons). Moreover, the ionic dielectric matrix is large only at very low frequencies ω (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 T_c 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-T_c cooperates.