Flicker (1/f) noise

Flicker noise, or 1/f noise, is a fluctuation whose power spectral density varies as $S(f) \propto 1/f^{\alpha}$ with $\alpha$ close to 1, typically 0.8–1.2. It appears in resistors carrying current, transistors, op amps, photodetectors, the intensity and frequency of lasers, and the gain of almost any physical system. At high frequencies it falls below the white floors of thermal noise and shot noise; at low frequencies it rises above them, and it is the main reason that a DC measurement of a weak signal is noisier than its white-noise budget predicts. The frequency at which the two contributions are equal is the 1/f corner: a few hertz to 1 kHz for op amps, and from hundreds of hertz to the megahertz range for some field-effect transistors.

Spectrum and corner frequency

A common model writes the total noise density as a white floor $S_w$ plus a flicker term that equals it at the corner frequency $f_c$:

$$S(f) = S_w \left(1 + \frac{f_c}{f}\right).$$

Integrated over a band from $f_1$ to $f_2$, the variance is

$$\sigma^2 = S_w \left[ (f_2 - f_1) + f_c \ln\frac{f_2}{f_1} \right].$$

The flicker part grows with the logarithm of the band ratio, so each decade of frequency contributes the same noise power, $f_c \ln 10 \approx 2.3\,f_c$ in units of $S_w$. This makes 1/f noise scale-free: lengthening a measurement from 10 s to 100 s adds one more decade at the low end, and with it as much flicker variance as the previous decade contained. Averaging longer therefore does not reduce it the way it reduces white noise, and an Allan deviation plot shows a flat flicker floor where white noise would keep falling.

The spectrum is measured by recording the signal with no input (or a constant input), computing its spectral density with an FFT analyzer or electrical spectrum analyzer, and plotting on log-log axes, where the 1/f region is a straight line of slope −1 in power (−1/2 in amplitude density) that meets the flat floor at $f_c$.

Worked example: DC measurement against modulation

Consider a preamplifier with a white voltage noise of 10 nV/√Hz and a corner at 1 kHz. A DC measurement with a 1 Hz low-pass filter, recorded for 100 s so that the band runs from 0.01 to 1 Hz, would see 9.9 nV rms with white noise alone. With the flicker term included,

$$\sigma^2 = S_w\,[\,0.99 + 1000 \ln 100\,]\ \text{Hz},$$

which gives 0.68 µV rms, about 70 times larger. If instead the signal is chopped at 10 kHz and recovered with a lock-in amplifier whose equivalent noise bandwidth is also 1 Hz, the noise density at 10 kHz is $10\sqrt{1.1}$ = 10.5 nV/√Hz, and the noise in the measurement is 10.5 nV rms, 65 times lower than in the DC case. Modulation moves the signal to a frequency above the corner, where only the white floor remains, and the lock-in's narrow band then rejects everything else. The same logic sets chopping frequencies for bolometers and thermal detectors, and the choice of a high modulation frequency in pump-probe experiments, where laser intensity noise is also largest at low frequencies.

Physical origins

No single mechanism accounts for all 1/f noise. In semiconductors a widely used picture is the superposition of many generation-recombination or trapping processes, each with a Lorentzian spectrum, whose time constants are spread uniformly on a logarithmic scale; their sum approximates 1/f over the range of time constants present. In MOSFETs, carrier trapping at the oxide interface dominates, which is why their corners are high. In resistors the fluctuation is in resistance, so it appears only when a current flows, and its relative size follows Hooge's empirical relation, $S_V/V^2 = \alpha_H/(N f)$, with $N$ the number of carriers; thick-film and carbon resistors have far more excess noise than metal-film or wirewound ones. Laser frequency noise has a 1/f region from current, temperature and intrinsic sources, the low-frequency part of the spectrum discussed under phase noise and frequency noise.

Pitfalls

Quoted noise densities measured at 1 kHz or 10 kHz say nothing about performance near DC; the corner frequency or a low-frequency specification is needed. A detector or amplifier with no bias current, such as an unbiased photodiode or a thermopile, has little 1/f noise, and biasing it to gain speed can introduce some.

Common questions

Why is it called flicker noise?

The name dates from J. B. Johnson's 1925 observation of slow fluctuations, which he called the flicker effect, in the emission current of vacuum tubes.

Does 1/f noise diverge at zero frequency?

The integral of $1/f$ diverges logarithmically at $f \to 0$, but any real measurement has a finite duration that cuts off the band. Measurements of 1/f spectra have extended to very low frequencies without finding a flattening in many systems, so the lower cutoff is usually set by the observation time.

How does a lock-in amplifier avoid 1/f noise?

It measures only at the modulation frequency, in a narrow band. If that frequency is above the 1/f corner of the detector and preamplifier, the measurement sees only white noise.

References: P. Horowitz and W. Hill, The Art of Electronics, 3rd ed. (Cambridge University Press, 2015); F. N. Hooge, "1/f noise," Physica B+C 83, 14 (1976); M. B. Weissman, "1/f noise and other slow, nonexponential kinetics in condensed matter," Reviews of Modern Physics 60, 537 (1988); J. B. Johnson, "The Schottky effect in low frequency circuits," Physical Review 26, 71 (1925).