Temperature of superconducting transition for very strong coupling in antiadiabatic limit of Eliashberg equations

TL;DR

This paper revises the known limit for the superconducting transition temperature in the strong coupling, antiadiabatic regime of Eliashberg theory, providing a new upper bound and asymptotic behavior.

Contribution

It introduces a modified asymptotic limit for $T_c$ in the antiadiabatic strong coupling regime, extending the understanding beyond the Allen-Dynes limit.

Findings

The traditional Allen-Dynes limit is replaced by a new asymptotic expression.

The upper limit for $T_c$ in this regime is derived as proportional to $rac{2}{ extpi^2} imes ext{(coupling and bandwidth terms)}$.

The results clarify the behavior of $T_c$ in the antiadiabatic, strong coupling limit of Eliashberg equations.

Abstract

It is shown that the famous Allen -- Dynes asymtotic limit for superconducting transition temperature in very strong coupling region $T_{c}>\frac{1}{2\pi}\sqrt{\lambda}\Omega_0$ (where $\lambda\gg 1$ - is Eliashberg - McMillan electron - phonon coupling constant and $\Omega_0$ - the characteristic frequency of phonons) in antiadiabatic limit of Eliashberg equations $\Omega_0/D\gg 1$ ($D\sim E_F$ is conduction band half-width and $E_F$ is Fermi energy) is replaced by $T_c>(2\pi^4)^{-1/3}(\lambda D\Omega_0^2)^{1/3}$, with the upper limit for $T_c$ given by $T_c<\frac{2}{\pi^2}\lambda D$.