Asymptotic behavior of Eliashberg gap function in the complex plane and its implications for the Coulomb pseudopotential in superconductors

The energy gap function in the Eliashberg formalism draws a spiral in the complex plane. A simple, efficient numerical scheme is devised to extend that spiral towards the bottom of the Fermi sea for lead, mercury, tin, and a model superconductor. The spiral is confirmed to converge to a constant value with a finite negative real part. The asymptotic behavior is explored to reveal a V-shaped dependence of the Coulomb pseudopotential μ∗ on the electron cutoff frequency in Pb and Hg, with a strong growth tendency when the cutoff frequency is large. A reference formula due to Allen and Dynes correctly predicts the later growth trend of μ∗ at large cutoff frequencies but overestimates its rate. Previous values of μ∗ in the literature have to be viewed critically.