Screening effects in superconductors

The partition function of the Hubbard model with local attraction and long range Coulomb repulsion between electrons is written as a functional integral with an action $A$ involving a pairing field $\Delta$ and a local potential $V$. After integration over $V$ and over fluctuations in $|\Delta|^{2}$, the final form of $A$ involves a Josephson coupling between the local phases of $\Delta$ and a "kinetic energy" term, representing the screened Coulomb interaction between charge fluctuations. The competition between Josephson coupling and charging energy allows to understand the relation between $T_{C}$ and composition in high-$T_{C}$ materials, in particular superlattices, alloys and bulk systems of low doping.