A least upper bound on the superconducting transition temperature

Abstract

It is rigorously shown that the superconducting transition temperature of any material for which the Eliashberg theory is valid must satisfy kBTc ⩽ 0.2309 A, where A is the area under its electron-phonon spectral function α2F(ω). This relation is a least upper bound, not just an upper bound, in the sense that there is an optimal situation in which the equality holds. This occurs when the Coulomb pseudopotential parameter μ∗ is zero and the spectral function is the Einstein spectrum Aδ(ω − 1.750 A). These results are generalized in an approximate, but sufficiently accurate, way to the case μ∗ ≠ 0 to obtain the more useful least upper bound kBTc ⩽ c(μ∗) A and the corresponding optimal spectrum Aδ[ω − d(μ∗)A]. Numerical results for the functions c(μ∗) and d(μ∗) are presented for 0 ⩽ μ∗ ⩽ 0.20. It is shown that the Tc's of many materials (including Nb3Sn), for which experimental values of A and μ∗ are available, do not lie very far below the upper bound.