Flicker ($\frac{1}{f}$) noise: Equilibrium temperature and resistance fluctuations

We have measured the $\frac{1}{f}$ voltage noise in continuous metal films. At room temperature, samples of pure metals and bismuth (with a carrier density smaller by ${10}^5$) of similar volume had comparable noise. The power spectrum $S_V\mathrm{}\left(f\right)\mathrm{}$ was proportional to $\frac{{\overline{V}}^2}{\Omega f^{\gamma }}$, where $\overline{V}$ is the mean voltage across the sample, $\Omega$ is the sample volume, and $1.0\lesssim \gamma \lesssim 1.4$. $\frac{S_V\mathrm{}\left(f\right)\mathrm{}}{{\overline{V}}^2}$ was reduced as the temperature was lowered. Manganin, with a temperature coefficient of resistance ($\beta$) close to zero, had no discernible noise. These results suggest that the noise arises from equilibrium temperature fluctuations modulating the resistance to give $S_V\mathrm{}\left(f\right)\mathrm{}\propto \frac{{\overline{V}}^2{\beta }^2{k}_B{T}^2}{C_V}$, where $C_V$ is the total heat capacity of the sample. The noise was spatially correlated over a length $\lambda \mathrm{}\left(f\right)\mathrm{}\approx {\left(\frac{D}{f}\right)}^{\frac{1}{2}}$, where $D$ is the thermal diffusivity, implying that the fluctuations obey a diffusion equation. The usual theoretical treatment of spatially uncorrelated temperature fluctuations gives a spectrum that flattens at low frequencies in contradiction to the observed spectrum. However, the empirical inclusion of an explicit $\frac{1}{f}$ region and appropriate normalization lead to $\frac{S_V\mathrm{}\left(f\right)\mathrm{}}{{\overline{V}}^2}\propto \frac{{\beta }^2{k}_B{T}^2}{C_V}[3+2\mathrm{}\ln (\frac{l}{w}\left)\right]f$, where $l$ is the length and $w$ is the width of the film, in excellent agreement with the measured noise. If the fluctuations are assumed to be spatially correlated, the diffusion equation can yield an extended $\frac{1}{f}$ region in the power spectrum. We show that the temperature response of a sample to $\delta$- and step-function power inputs has the same shape as the autocorrelation function for uncorrelated and correlated temperature fluctuations, respectively. The spectrum obtained from the cosine transform of the measured step-function response is in excellent agreement with the measured $\frac{1}{f}$ voltage noise spectrum. Spatially correlated equilibrium temperature fluctuations are not the dominant source of $\frac{1}{f}$ noise in semiconductors and discontinuous metal films. However, the agreement between the low-frequency spectrum of fluctuations in the mean-square Johnson-noise voltage and the resistance fluctuation spectrum measured in the presence of a current demonstrates that in these systems the $\frac{1}{f}$ noise is also due to equilibrium resistance fluctuations.