Millimeter Wave Absorption in Superconducting Aluminum. II. Calculation of the Skin Depth

Abstract

The skin depth in superconducting aluminum is calculated from the measured frequency dependence of the surface resistance through the Kronig-Kramers integral transforms. At absolute zero, it is found that the skin depth $\delta$ is independent of frequency at low frequencies but begins to increase at higher frequencies. The maximum rate of increase of $\delta$ occurs when the photon energy equals the gap energy, $h\nu =3.2k{T}_c={\mathcal{E}}_g$; at this point $\frac{\delta (h\nu ={\mathcal{E}}_g)}{\delta (h\nu =0)}\approx 1.12$. The maximum value of $\delta$ occurs at $h\nu \approx 4k{T}_c$. The superconducting penetration depth $\lambda$ [i.e., $\delta (h\nu =0)$] is found to vary approximately as $\lambda \mathrm{}\left(t\right)=\lambda \mathrm{}\left(0\right)\mathrm{}{(1-t^4)}^{-\frac{1}{2}}$, with $\lambda \mathrm{}\left(0\right)=5.15\mathrm{}\times {10}^{-6\mathrm{}}$ cm and $t\equiv \frac{T}{T_c}$. The effects of changes in the skin depth have been eliminated from the determination of the energy gap by calculation of the real part of the complex conductivity, ${\sigma }_r$. The energy gap values deduced from the behavior of ${\sigma }_r$ differ only slightly from the results obtained directly from the surface resistance measurements.