Temperature fluctuations in freely suspended tin films at the superconducting transition

Abstract

We have used a superconducting-quantum-interference-device voltmeter to measure the spectral density $S_v\mathrm{}\left(f\right)\mathrm{}$ of the voltage noise across current-biased tin films at the superconducting transition. Each film was freely suspended between two thermal clamps a distance $L$ apart in a vacuum can. The transverse dimensions of the films were small compared with ${\left(\frac{D}{f}\right)}^{\frac{1}{2}}$ at the frequencies of interest ($D$ is the thermal diffusivity), so that the heat flow was one dimensional. A thin layer of lead was evaporated on some of the tin films to leave an uncoated middle region of length $l$. $S_v\mathrm{}\left(f\right)\mathrm{}$ was proportional to the square of the bias current and to the square of the temperature coefficient of resistance, indicating that the noise was generated by temperature fluctuations. $S_v\mathrm{}\left(f\right)\mathrm{}$ was flat at frequencies below $f_L\approx \frac{D}{L^2}$. For samples without the lead overlay, $S_v\mathrm{}\left(f\right)\mathrm{}$ typically varied as $f^{-1.3\mathrm{}}$ at frequencies above $f_L$. For samples with the lead overlay, a second knee was usually observed at $f_l\approx {\left(\frac{L}{l}\right)}^2{f}_L$. At frequencies between $f_L$ and $f_l$ the slope was typically -0.8, while at frequencies above $f_l$ the slope was somewhat less steep than -1.5. This behavior is in reasonable agreement with the predictions of an equilibrium temperature-fluctuation model in which the equal-time temperature fluctuations are spatially uncorrelated. The magnitude of $S_v\mathrm{}\left(f\right)\mathrm{}$ is within a factor of 2 or 3 of the model predictions. The autocorrelation function of the voltage noise was obtained from the time derivative of the response of the film to a step function in power. The cosine transform of the autocorrelation function was in excellent agreement with the measured spectral density. These results are in marked contrast with those obtained for normal and superconducting films supported by substrates, for which a model is required with spatially correlated fluctuations. We conclude that the $\frac{1}{f}$ noise for films on substrates is mediated by an interaction between substrate and film.