Flicker noise

Flicker noise, also called 1/f noise, is noise whose power spectral density varies as 1/f^α with α close to 1. It is found in vacuum tubes, resistors, transistors and the ionic current through a nanopore, and it dominates every one of them at low enough frequency. In a nanopore recording it sets the floor from the lowest frequencies up to a few hundred hertz or a few kilohertz, depending on the pore, where the white thermal and shot terms take over (see Nanopore noise).^[1]

J. B. Johnson reported it in 1925, as an excess of noise at low frequencies in vacuum tubes that the theory of the shot effect did not predict.^[2] Walter Schottky, who had explained shot noise in 1918, discussed it in 1926 under the name that has stuck, the flicker effect.^[3] In 1975 Voss and Clarke reported that the loudness and pitch of music and speech fluctuate with spectra of the same form.^[4]

No single mechanism explains all of this, and the standard account is a family of them. Suppose many independent processes, such as charges hopping on and off traps, each switch with a time constant τ, and each by itself gives a spectrum that is flat below 1/(2πτ) and falls as 1/f² above it. If the time constants are spread evenly in their logarithm, the sum of all these spectra is 1/f over the range that the spread covers. Dutta and Horn reviewed the evidence for such a distribution in 1981.^[5]

A spectral density S(f) = K/f puts the same power into every factor of two, and into every decade, of frequency. The mean-square noise between f₁ and f₂ is K ln(f₂/f₁). Extending a recording to lower frequencies, which means making it longer, adds noise, and averaging does not remove it. For an ideal 1/f spectrum from 1 Hz up to 10 kHz, a recording a day long, which reaches down to 12 µHz, has 1.5 times the rms noise of a recording a second long.^[a] White noise behaves otherwise: its standard error falls as the square root of the averaging time.

In 1969 F. N. Hooge proposed that the noise of a homogeneous conductor carrying a current I has the spectral density

where N_c is the number of charge carriers and α_H a dimensionless constant, which he put at about 2 × 10⁻³ for many metals and semiconductors.^[6] The constant did not prove constant. Later measurements, summarized in a review of 1981, gave values that differ by orders of magnitude between materials and devices.^[7] The relation still has two uses. It says that a smaller sample is noisier, because it has fewer carriers, and that the noise power is proportional to I².

The second point has a consequence for the pore. The signal of a molecule is a fraction of the current, and so is the flicker noise, so raising the voltage makes both larger together and leaves the signal-to-noise ratio set by flicker where it was. Only a quieter pore improves it.

Smeets and colleagues found that Hooge's relation fits silicon nitride pores, with α_H = 1.1 × 10⁻⁴ for pores that behaved and noise power inversely proportional to the number of ions in the pore.^[8] Take a pore 10 nm wide through a membrane 20 nm thick in 1 M potassium chloride. It holds about 1,900 ions and passes 5.9 nA at 200 mV. The relation gives a noise of 1.4 pA/√Hz at 1 Hz, against 22 fA/√Hz of thermal noise from the pore's conductance, and the two are equal at about 4 kHz. Between 1 Hz and 10 kHz the flicker term would contribute 4.3 pA rms and the thermal term 2.2 pA. These are calculated figures that take the constant for a well-behaved pore. In the same study two pores about 21 nm wide differed in 1/f power by nearly three orders of magnitude.

What moves the noise in a real pore is its surface. In 2009 a Harvard group found an excess noise in solid-state pores that exceeded all other sources between 0.1 and 10 kHz. Its dependence on pH and salt concentration pointed to fluctuations in the charge of the pore wall, from the protonation and deprotonation of surface groups, and the noise could be minimized by operating at a chosen pH.^[9] A study of pores 7 to 200 nm wide found that both the bulk current and the surface current contribute, and that the silver chloride electrodes can dominate the noise when the pore is large or the salt concentration high.^[10]

The origin is not settled. Measurements across two-dimensional channels in 2023 found spectra that scale as 1/f^1+a with a between 0 and 0.5 in pristine channels, and a theory in which the 1/f fluctuations arise in the reservoirs on either side of the channel and the channel multiplies them by a function of its own ion dynamics. The authors called Hooge's law "long-standing yet poorly understood".^[11]

Flicker noise appears once in the open design of the MR1 Nanopore Reader, in the docstring of a mock serial port in the host software, which lists among the things the class does: "Emulates realistic baseline ionic pore current (~1.2 nA) with 1/f noise." The code that generates the samples adds to each one the term math.sin(self._frame_index * 0.1 + i * 0.05) * 120, a sine wave with a peak of 120 counts. On the scale the code uses for the baseline, one count is 125 fA, so the peak is 15 pA.^[b][12] A sine wave concentrates its power at one frequency, where 1/f noise spreads it equally over every decade.

• Johnson-Nyquist noise
• Dielectric noise
• Signal-to-noise ratio
• Whisper in a thunderstorm problem