Low-frequency noise in tin and lead films at the superconducting transition

We have measured the noise power spectra $S_V\mathrm{}\left(f\right)\mathrm{}$ of tin and lead films at the superconducting transition in the frequency range 0.1 Hz to 5 kHz. Two types of samples were made: Type $A$ were evaporated directly onto glass substrates, while type $B$ were evaporated onto glass and sapphire substrates with a 5-nm aluminum underlay. For type $A$ samples the spectra varied approximately as $f^{-1\mathrm{}}$ and were proportional to $\frac{{\overline{V}}^2{\beta }^2}{\Omega }$, where $\overline{V}$ was the mean voltage across the sample, $\beta$ was the temperature coefficient of resistance, and $\Omega$ was the sample volume. For type-$A$ tin samples, the noise in two regions a distance $d$ apart was correlated at frequencies $\lesssim \frac{D}{\pi d^2}$. These results are consistent with a thermal diffusion model of Clarke and Voss. The magnitude of the noise in type-$A$ tin samples was accurately predicted by the semiempirical formula $S_V\mathrm{}\left(f\right)=\frac{{\overline{V}}^2{\beta }^2{k}_B{T}^2}{C_V[3+2\mathrm{}\ln (\frac{l_1}{l_2}\left)\right]}$ where $l_1$ and $l_2$ are the length and width of the film; for lead samples, this formula overestimates the observed noise by a factor of 5. As the magnetic field perpendicular to the tin films was increased from zero to about 2 G, $\beta$ was reduced by about a factor of 5, and $S_V\mathrm{}\left(f\right)\mathrm{}$ was found to be proportional to ${\beta }^2$. The spectra of type-$B$ films were markedly different from those of type-$A$ films. In the case of the tin samples, the spectra became flat below about 30 Hz. The degree of spatial correlation of the noise was markedly reduced. In the case of the lead films, the spectra varied as about $f^{-1.1\mathrm{}}$ for type $A$ and as about $f^{-0.8\mathrm{}}$ for type $B$. These changes are ascribed to the enhancement of the thermal contact between the films and the substrate by the aluminum underlay. Separate experiments confirmed that the underlay decreased the thermal boundary resistance between the film and the substrate. Implications of this work for device applications are briefly discussed.