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Antiadiabatic Phonons and Superconductivity in Eliashberg-McMillan Theory

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 Eliashberg - McMillan approach in 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 $\tildeλ$, 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 $Ω_0\gg E_F$, the new small parameter of perturbation theory is $λ\frac{E_F}{Ω_0}\simλ\frac{D}{Ω_0}\ll 1$ ($D$ 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 $μ^{\star}$ and show, that antiadiabatic phonons do not contribute to Tolmachev's logarithm and its value is determined by partial contributions from adiabatic phonons only.