Fig. 3: Electrical conductivities for the Eliashberg and BCS theories.
a The Eth (λ = 0.3) and BCS conductivity ratios versus t ≡ T/Tc in the dirty limit. The orange curve is for Eth with ν/ωE = 10−4. The blue curve is BCS theory, where we also use ν/ωE = 10−4, which implies \(\nu /{{{\Delta }}}_{0}^{{{{{{{{\rm{BCS}}}}}}}}}=(\nu /{\omega }_{E})({\omega }_{E}/{{{\Delta }}}_{0}^{{{{{{{{\rm{BCS}}}}}}}}})\approx 3.81\times 1{0}^{-3}\). Here we use Eq. (2.3) for the BCS gap edge \({{{\Delta }}}_{0}^{{{{{{{{\rm{BCS}}}}}}}}}\). In the case of the blue data points, for the gap we use the weak-coupling result given in Eq. (2.4). This plot shows that, using a BCS gap with a weak-coupling correction of \(1/\sqrt{e}\) gives an electrical conductivity that agrees with the small-λ Eth result. b The frequency-dependent conductivity in Eth (λ = 0.3) and BCS theory in the dirty limit. The solid curves are for Eth and the points are for BCS theory. Here we define a dimensionless frequency by \(\widetilde{\nu }=\nu /\left(2{{{\Delta }}}_{0}\right)\). Note that we use different horizontal axes for the Eth and BCS curves.
