Antiadiabatic Phonons and Superconductivity in Eliashberg-McMillan Theory

Abstract

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 ${\Omega }_0$ 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 $\lambda$, 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 $\tilde{\lambda }$, 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 ${\Omega }_0\gg E_F$, the new small parameter of perturbation theory is $\lambda \frac{E_F}{{\Omega }_0}\sim \lambda \frac{D}{{\Omega }_0}\ll 1$ ($D$ 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 $\lambda$. 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 ${\mu }^{\star }$ and show, that antiadiabatic phonons do not contribute to Tolmachev's logarithm and its value is determined by partial contributions from adiabatic phonons only.

8 pages, 2 figures, to be published in a special issue of Journal of Superconductivity and Novel Magnetism in honor of Ted Geballe on the occasion of his 100th birthday, reviews the results of arXiv:1809.02531, arXiv:1811.10184

References in corpus (3)

• High Pressure and Road to Room Temperature Superconductivity
• Breakdown of Migdal-Eliashberg theory; a determinant quantum Monte Carlo study
• The electron-phonon interaction with forward scattering peak is a relevant approach to high Tc superconductivity in FeSe films on SrTiO3 and TiO2

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