Conductivity of Superconducting Films for Photon Energies between 0.3 and40kTc

Abstract

Far infrared and millimeter microwave transmission experiments through thin superconducting lead and tin films are reported. The frequency range covered corresponds to photon energies from 0.3 to $40k{T}_{c}$. The measurements include the previously unexplored frequency region in which a superconductor changes from an essentially lossless conductor to a normal one. By suitable analysis the effective complex conductivity of the films is obtained from the transmission data. For thin superconducting films it is shown that $\frac{[{\ensuremath{\sigma}}_{1}(\ensuremath{\omega})\ensuremath{-}i{\ensuremath{\sigma}}_{2}(\ensuremath{\omega})]}{{\ensuremath{\sigma}}_{N}}$ is, to a good approximation, a universal function of the reduced frequency ($\frac{\ensuremath{\hbar}\ensuremath{\omega}}{k{T}_{c}}$), being independent of film resistance, thickness, degree of anneal, and material (for the two metals tried). At $T=0$, ${\ensuremath{\sigma}}_{1}$ appears to be very small (or zero) for photon energies below roughly $3k{T}_{c}$. Starting at $3k{T}_{c}$, ${\ensuremath{\sigma}}_{1}$ rises rapidly and reaches its limiting value ${\ensuremath{\sigma}}_{N}$ at about $20k{T}_{c}$. This behavior suggests a gap of width $\ensuremath{\approx}3k{T}_{c}$ in the electronic excitation spectrum of the superconducting state. At $T=0$ and for photon energies considerably smaller than $3k{T}_{c}$, $\frac{{\ensuremath{\sigma}}_{2}}{{\ensuremath{\sigma}}_{N}}\ensuremath{\approx}\frac{\ensuremath{\alpha}k{T}_{c}}{\ensuremath{\hbar}\ensuremath{\omega}}$, with $\ensuremath{\alpha}=3.7\ifmmode\pm\else\textpm\fi{}0.7$ for both Sn and Pb. The frequency dependence is in agreement with the London theory. This theory, however, would not predict that the results for Pb and Sn should be the same and, moreover, it would give values of $\ensuremath{\alpha}\ensuremath{\sim}100$ times too large. The Pippard nonlocal theory predicts the type of universal dependence found, but with $\ensuremath{\alpha}=6.7$. For photon energies of the order of $3k{T}_{c}$, a polarizability term ${{\ensuremath{\sigma}}_{2}}^{\ensuremath{'}}(\ensuremath{\omega})$, required by the Kramers-Kronig relations because of the cutoff of ${\ensuremath{\sigma}}_{1}(\ensuremath{\omega})$ near $3k{T}_{c}$, becomes important and tends to cancel the $\frac{1}{\ensuremath{\omega}}$ term in ${\ensuremath{\sigma}}_{2}$. As a result, ${\ensuremath{\sigma}}_{2}$ is reduced to a small value for photon energies above $5k{T}_{c}$. Some information about the temperature dependence of the complex conductivity has also been obtained.