Fluctuation Effects in the ac Conductivity of Thin Superconducting Lead Films at Microwave Frequencies

The microwave transmission and reflection coefficients have been measured for thin ($T_c \left[\right(T_c-T\left)\le 0.2 \mathrm{to} 0.3 \mathrm{K}\right]$, the temperature and frequency dependence of these ratios are in good agreement with theoretical values calculated using the complex conductivities derived by Mattis and Bardeen. Best fits with the experimental results were obtained using complex conductivities corresponding to a superconducting energy gap of $(4.5\mathrm{}\pm 0.2)\times k_B{T}_c$. Above $T_c$ both the microwave and dc data implied an excess conductivity for each film studied. This excess conductivity has been attributed to thermodynamic fluctuations in the Ginzburg-Landau order parameter for temperatures near the normal-to-superconducting phase transition. In agreement with the predictions of Aslamazov and Larkin for the two-dimensional limit, the dc measurements indicate a value of 0.152 × ${10}^{-4\mathrm{}}$ ${\mathrm{\Omega }}^{-1\mathrm{}}$ for $\frac{{\tau }_0}{R_n}$. Both the temperature and frequency dependence of the implied excess ac conductivity above $T_c$ are accounted for by fluctuation-effect calculations of Schmidt. In the temperature region just below $T_c \left[\right(T_c-T\left)\sim 0.1 \mathrm{K}\right]$, the experimental results indicate an excess ac conductivity compared to the predictions of the Mattis and Bardeen (MB) theory. Recently, Schmidt has extended his theory of fluctuation effects in the ac conductivity to temperatures below $T_c$; combining these fluctuation-induced conductivities with those of MB (due to quasiparticles) permits the calculation of transmission coefficients in good agreement with experiment. We believe this to be direct evidence for fluctuation effects in a two-dimensional superconductor in the presence of quasi-long-range order. The deposition of a manganese overlay of approximately 2 Å upon a 150-Å lead film resulted in a 46% reduction in the microwave-measured energy gap. Essentially no charge was observed in the dc transition temperature.