High-Tc superconductors: Disordered metals with pairing via polarizability from localized states near the mobility edge

The electronic structure of the high-Tc superconductors is that of a disordered metal with the Fermi energy (EF) relatively close to the mobility edge (Ec); EF∼0.1--0.2 eV from Ec, lying in the extended-state region in the O 2p band. In such a metal, the polarizability of the localized states near Ec can lead to an indirect attractive interaction between electrons (or holes) in extended states near EF. Although the direct and exchange terms cancel in second-order perturbation theory, we argue that because of the response of the many-electron wave function to the ''core hole'' in the intermediate state, exchange processes are turned off near threshold at EF in analogy with the x-ray-absorption-edge problem. Thus, Vindirect=-2fL(δ/EF) (UL)2|EF-Ec|-1, where UL is the screened Coulomb interaction between electrons in extended states near EF and electrons in localized states near Ec, fL is the fraction of the total number of states in the band which is localized, and δ is an energy interval near EF (δ<EF) in which the exchange scattering is suppressed. Using Vindirect as the basis for pairing in BCS theory, one obtains 2Δ0=2|EF-Ec|exp {-2/N(EF)Vindirect}, where N(EF) is the density of states at EF