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

The electronic structure of the high-${\mathit{T}}_{\mathit{c}}$ superconductors is that of a disordered metal with the Fermi energy (${\mathit{E}}_{\mathit{F}}$) relatively close to the mobility edge (${\mathit{E}}_{\mathit{c}}$); ${\mathit{E}}_{\mathit{F}}$\ensuremath{\sim}0.1--0.2 eV from ${\mathit{E}}_{\mathit{c}}$, lying in the extended-state region in the O 2p band. In such a metal, the polarizability of the localized states near ${\mathit{E}}_{\mathit{c}}$ can lead to an indirect attractive interaction between electrons (or holes) in extended states near ${\mathit{E}}_{\mathit{F}}$. 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 ${\mathit{E}}_{\mathit{F}}$ in analogy with the x-ray-absorption-edge problem. Thus, ${\mathit{V}}_{\mathrm{indirect}}$=-2${\mathit{f}}_{\mathit{L}}$(\ensuremath{\delta}/${\mathit{E}}_{\mathit{F}}$) (${\mathit{U}}_{\mathit{L}}$${)}^{2}$\ensuremath{\Vert}${\mathit{E}}_{\mathit{F}}$-${\mathit{E}}_{\mathit{c}}$${\mathrm{\ensuremath{\Vert}}}^{\mathrm{\ensuremath{-}}1}$, where ${\mathit{U}}_{\mathit{L}}$ is the screened Coulomb interaction between electrons in extended states near ${\mathit{E}}_{\mathit{F}}$ and electrons in localized states near ${\mathit{E}}_{\mathit{c}}$, ${\mathit{f}}_{\mathit{L}}$ is the fraction of the total number of states in the band which is localized, and \ensuremath{\delta} is an energy interval near ${\mathit{E}}_{\mathit{F}}$ (\ensuremath{\delta}${\mathit{E}}_{\mathit{F}}$) in which the exchange scattering is suppressed. Using ${\mathit{V}}_{\mathrm{indirect}}$ as the basis for pairing in BCS theory, one obtains 2${\mathrm{\ensuremath{\Delta}}}_{0}$=2\ensuremath{\Vert}${\mathit{E}}_{\mathit{F}}$-${\mathit{E}}_{\mathit{c}}$\ensuremath{\Vert}exp {-2/N(${\mathit{E}}_{\mathit{F}}$)${\mathit{V}}_{\mathrm{indirect}}$}, where N(${\mathit{E}}_{\mathit{F}}$) is the density of states at ${\mathit{E}}_{\mathit{F}}$. Since 2\ensuremath{\Vert}${\mathit{E}}_{\mathit{F}}$-${\mathit{E}}_{\mathit{c}}$\ensuremath{\Vert}\ensuremath{\sim}0.2--0.4 eV, the transition temperature can in principle be quite high.Moreover, as \ensuremath{\Vert}${\mathit{E}}_{\mathit{F}}$-${\mathit{E}}_{\mathit{c}}$\ensuremath{\Vert} is increased from zero by doping beyond the metal-insulator transition, the gap (and ${\mathit{T}}_{\mathit{c}}$) would initially increase, and then decrease when the exponential factor begins to dominate, in qualitative agreement with the phase diagram of the high-temperature superconducting cuprates and with the overdoping phenomenon. The interaction between electrons near ${\mathit{E}}_{\mathit{F}}$ will cause scattering and the dissipation of momentum, thereby making a contribution to the normal-state resistivity, which is linear in temperature, in agreement with experiment. We show that ${\mathit{T}}_{\mathit{c}}$ can be estimated directly from the magnitude of the linear term in the Drude scattering rate, with results that are also in agreement with experiment.