*f*or "flicker" noise, and it's something of a special case. It's intrinsic in the sense that it originates with the material whose conductance or resistance is being measured, but it's usually treated as extrinsic, in the sense that its physical mechanism is not what's of interest and in the limit of an "ideal" sample it probably wouldn't be present. Consider a resistance measurement (that is, flowing current through some sample and looking at the resulting voltage drop). As the name implies, the power spectral density of voltage fluctuations,

*S*

_{V}, has a component that varies approximately inversely with the frequency. That is, the voltage fluctuates as a function of time, and the slow fluctuations have larger amplitudes than the fast fluctuations. Unlike shot noise, which results from the discrete nature of charge, 1/

*f*noise exists because the actual resistance of the sample itself is varying as a function of time. That is, some fluctuation d

*V*(

*t*) comes from

*I*d

*R*(

*t*), where

*I*is the average DC current. On the bright side, that means there is an obvious test of whether the noise you're seeing is of this type: real 1/

*f*noise power scales like the square of the current (in contrast to shot noise, which is linear in

*I*, and Johnson-Nyquist noise, which is independent of

*I*).

The particular 1/

*f*form is generally thought to result from there being many "fluctuators" with a broad distribution of time scales. A "fluctuator" is some microscopic degree of freedom, usually considered to have two possible states, such that the electrical resistance is different in each state. The ubiquitous two-level systems that I've mentioned before can be fluctuators. Other candidates include localized defect states ("traps") that can either be empty or occupied by an electron. These latter are particularly important in semiconductor devices like transistors. In the limit of a single fluctuator, the resistance toggles back and forth stochastically between two states in what is often called "telegraph noise".

A thorough bibliography of 1/

*f*noise is posted here by a thoughtful person.

I can't leave this subject without talking about one specific instance of 1/

*f*noise that I think is very neat physics. In mesoscopic conductors, where electronic conduction is effectively a quantum interference experiment, changing the disorder seen by the electrons can lead to fluctuations in the conductance (within a quantum coherent volume) by an amount ~

*e*

^{2}/

*h*. In this case, the resulting 1/

*f*noise observed in such a conductor actually

*grows*with

*decreasing*temperature, which is the opposite of, e.g., Johnson-Nyquist noise. The reason is the following. In macroscopic conductors, ensemble averaging of the fluctuations over all the different conducting regions of a sample suppresses the noise; as

*T*decreases, though, the typical quantum coherence length grows, and this kind of ensemble averaging is reduced, since the sample contains fewer coherent regions. My group has done some work on this in the past.

## 3 comments:

Nice series, Doug.

You didn't mention the usual argument for the exponent (1) in 1/f, though: that if the fluctuators have a uniform distribution of some parameter that enters exponentially into the frequency at which they fluctuate, they will have a uniform distribution in the log of frequency, which means their density in frequency space will go as 1/f.

In the case of MOSFETs, the parameter that enters exponentially into the frequency is the tunneling distance to a trap. In other cases one imagines a distribution of activation energies.

At one point it seemed plausible that 1/f noise might be directly related to two-level-systems phenomena like linear specific heat. How did that work out?

Thanks, Don. I meant to make this one a little longer, but pre-holiday-travel kind of ate my time.

Regarding your last point, yes, the general assumption (unproven as far as I know) is that 1/f noise in, e.g., polycrystalline metals is caused by exactly the infamous two-level systems. At least, the numbers (how many TLS do you need for the noise, and how many for the TLS-like contributions to, e.g., the sound speed) are consistent.

The other thing I wanted to mention is that these fluctuators and associated 1/f noise are a big concern in the superconducting qubit community as sources of decoherence.

My feeling is that 1/f noise is more about a mathematical interpretation of resistance "drifting". I remember vaguely that if a linear increase term or a log term is added to the resistance then after FFT will get 1/f. The exponent might can also be changed.

I am not totally convinced by the interpretation that enlists a broad band distribution of Lorentzian spectra of TLS. It is just difficult to image there is high amplitude at extremely low frequency. I remember somewhere people discussed that the power is also not conserved.

The TLS explanation for linear specific heat Don mentioned, may be related to the Cornell group work on glass at low temperature.

For Doug's group's work of TDUCF. I think it is very solid and systematic, although not very fashionable. I thought it is difficult to get money for this kind of research.

There is one technical issue I am very interested to know: Is there any difference between using Si/SiO2 substrate and GaAs substrate as in these papers? My experience is that Ag, Au wires deposited on Si/SiO2 substrate will recrystalize, which leads to a slight decrease of resistance after two weeks and then increase afterwards, and broken after a longer time. It seems to be surface energy issue but I don't know whether GaAs is better or not.

One interesting thing is that the fluctuators the Q computing people worry about seem to be originated in the substrate (Martini's group), not on the surface.

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