Flicker (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi>f</mml:mi></mml:mrow></mml:mfrac></mml:math>) 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 ${S}_{V}(f)$ was proportional to $\frac{{\overline{V}}^{2}}{\ensuremath{\Omega}{f}^{\ensuremath{\gamma}}}$, where $\overline{V}$ is the mean voltage across the sample, $\ensuremath{\Omega}$ is the sample volume, and $1.0\ensuremath{\lesssim}\ensuremath{\gamma}\ensuremath{\lesssim}1.4$. $\frac{{S}_{V}(f)}{{\overline{V}}^{2}}$ was reduced as the temperature was lowered. Manganin, with a temperature coefficient of resistance ($\ensuremath{\beta}$) close to zero, had no discernible noise. These results suggest that the noise arises from equilibrium temperature fluctuations modulating the resistance to give ${S}_{V}(f)\ensuremath{\propto}\frac{{\overline{V}}^{2}{\ensuremath{\beta}}^{2}{k}_{B}{T}^{2}}{{C}_{V}}$, where ${C}_{V}$ is the total heat capacity of the sample. The noise was spatially correlated over a length $\ensuremath{\lambda}(f)\ensuremath{\approx}{(\frac{D}{f})}^{\frac{1}{2}}$, 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 $\frac{{S}_{V}(f)}{{\overline{V}}^{2}}\ensuremath{\propto}\frac{{\ensuremath{\beta}}^{2}{k}_{B}{T}^{2}}{{C}_{V}}[3+2\mathrm{ln}(\frac{l}{w})]f$, 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 $\ensuremath{\delta}$- 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.

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