Anharmonicity-induced multiphonon processes in high-temperature superconductors

Electron–two-phonon interactions are considered on the basis of an extended BCS-type Hamiltonian. An equation for the superconducting transition temperature ${\mathit{T}}_{\mathit{c}}$ and the energy gap 2${\mathrm{\Delta }}_0$ is derived in the weak-coupling limit. Instead of a Debye frequency a reduced optic-acoustic mode frequency, together with a coupled effective electron-phonon interaction, determines the magnitude of ${\mathit{T}}_{\mathit{c}}$. This leads to a distinctly different isotope effect from BCS, which is compared with data on ${\mathrm{La}}_{2\mathrm{-}\mathit{x}}$ ${\mathrm{Sr}}_{\mathit{x}}$ ${\mathrm{CuO}}_4$. The energy gap is predicted to be anisotropic and 2${\mathrm{\Delta }}_0$/${\mathit{kT}}_{\mathit{c}}$ to be compound dependent.