Maximum T_C for the phonon mediated mechanism, with forward electron-phonon scattering

We estimate the maximum superconducting transition temperature (T_C)_max that can be expected from the BCS phonon-mediated interaction, in high-T_C cuprates. The large polarizability of the oxygen ions plays a critical role, by replacing the conduction electrons as the main screening medium. As a result the static electronic Thomas-Fermi screening parameter κ_TF is very small. This causes the electron-ion potential V(q, ω) to be very large at small values of q(q ≈ κ_TF) and very small values of ω(ω < ξ₁, where ξ₁ characterizes the dispersion of the ionic dielectric constant). This, in turn, causes the electron-phonon matrix element I(q, ω) to be very strong for these (q, ω) values, and as a result T_C is very high, particularly in a thin 2-D electron gas. We estimate (T_C)_max ≈ 200 K for parameters characteristic of the cuprates.