On some recent results concerning the superconducting transition temperature

Abstract

The Eliashberg gap equations relate the transition temperature T c of an isotropic superconductor to its electron-phonon spectral function α 2 F( ω) and Coulomb pseudopotential parameter μ ∗ . Recently the Eliashberg theory has been used to derive some supposedly rigorous results bearing on the problem of attaining higher superconducting transition temperatures: Bergmann and Rainer derived an expression for the functional derivative δT c δα 2F(ω) ; Allen and Dynes showed that in the asymptotic limit of very large λ(λ≲10)k BT c=f(μ ∗)(λ〈ω 2〉) 1 2 and Leavens proved that for any isotropic superconductor k B T c ⩽0.2309 A, where A is the area under its electron-phonon spectral function. In this letter we show that the result of Allen and Dynes is not compatible with the other results and is, in fact, incorrect.