LAUSR

The standard Eliashberg–McMillan theory of superconductivity is essentially based on the adiabatic approximation. Here we present some simple estimates of electron–phonon interaction within the Eliashberg–McMillan approach in a non–adiabatic and even antiadiabatic situation, when characteristic phonon frequency Ω 0 becomes large enough, i.e., comparable or exceeding the Fermi energy E F . We discuss the general definition of Eliashberg–McMillan (pairing) electron–phonon coupling constant λ , taking into account the finite value of phonon frequencies. We show that the mass renormalization of electrons is in general determined by different coupling constant λ ~ $\tilde \lambda $ , which takes into account the finite width of conduction band, and describes the smooth transition from the adiabatic regime to the region of strong nonadiabaticity. In antiadiabatic limit, when Ω 0 ≫ E F , the new small parameter of perturbation theory is λ E F Ω 0 ∼ λ D Ω 0 ≪ 1 $\lambda \frac {E_{F}}{{\varOmega }_{0}}\sim \lambda \frac {D}{{\varOmega }_{0}}\ll 1$ ( D is conduction band half-width), and corrections to electronic spectrum (mass renormalization) become irrelevant. However, the temperature of superconducting transition T c in antiadiabatic limit is still determined by Eliashberg–McMillan coupling constant λ . We consider in detail the model with discrete set of (optical) phonon frequencies. A general expression for superconducting transition temperature T c is derived, which is valid in situation, when one (or several) of such phonons becomes antiadiabatic. We also analyze the contribution of such phonons into the Coulomb pseudopotential μ ⋆ and show that antiadiabatic phonons do not contribute to Tolmachev’s logarithm and its value is determined by partial contributions from adiabatic phonons only.