Editing Shot noise (section)

===Electronic devices===
Shot noise in electronic circuits consists of random fluctuations of [[direct current|DC current]], which is due to [[electric current]] being the flow of discrete charges ([[electron]]s). Because the electron has such a tiny charge, however, shot noise is of relative insignificance in many (but not all) cases of electrical conduction. For instance 1 [[ampere]] of current consists of about {{val|6.24|e=18}} electrons per second; even though this number will randomly vary by several billion in any given second, such a fluctuation is minuscule compared to the current itself. In addition, shot noise is often less significant as compared with two other noise sources in electronic circuits, [[flicker noise]] and [[Johnson–Nyquist noise]]. However, shot noise is temperature and frequency independent, in contrast to Johnson–Nyquist noise, which is proportional to temperature, and flicker noise, with the spectral density decreasing with increasing frequency. Therefore, at high frequencies and low temperatures shot noise may become the dominant source of noise.

With very small currents and considering shorter time scales (thus wider bandwidths) shot noise can be significant. For instance, a microwave circuit operates on time scales of less than a [[nanosecond]] and if we were to have a current of 16 [[nanoampere]]s that would amount to only 100 electrons passing every nanosecond. According to [[Poisson statistics]] the ''actual'' number of electrons in any nanosecond would vary by 10 electrons [[root mean square|rms]], so that one sixth of the time fewer than 90 electrons would pass a point and one sixth of the time more than 110 electrons would be counted in a nanosecond. Now with this small current viewed on this time scale, the shot noise amounts to 1/10 of the DC current itself.

The result by Schottky, based on the assumption that the statistics of electrons passage is Poissonian, reads<ref name="buttiker">{{cite journal|last=Blanter|first=Ya. M.|author2=Büttiker, M.|year=2000|title=Shot noise in mesoscopic conductors|journal=[[Physics Reports]]|publisher=[[Elsevier]]|location=Dordrecht|volume=336|issue=1–2|pages=1–166|doi=10.1016/S0370-1573(99)00123-4|arxiv = cond-mat/9910158 |bibcode = 2000PhR...336....1B |s2cid=119432033}}</ref> for the spectral noise density at the frequency <math>f</math>,
:<math>
S (f) = 2e\vert I \vert \ ,
</math>
where <math>e</math> is the electron charge, and <math>I</math> is the average current of the electron stream. The noise spectral power is frequency independent, which means the noise is [[White noise|white]]. This can be combined with the [[Landauer formula]], which relates the average current with the [[transmission eigenvalue]]s <math>T_n</math> of the contact through which the current is measured (<math>n</math> labels [[transport channel]]s). In the simplest case, these transmission eigenvalues can be taken to be energy independent and so the Landauer formula is 
:<math>
I = \frac{e^2}{\pi\hbar} V \sum_n T_n \ ,
</math>
where <math>V</math> is the applied voltage. This provides for 
:<math>
S = \frac{2e^3}{\pi\hbar} \vert V \vert \sum_n T_n \ ,
</math>
commonly referred to as the Poisson value of shot noise, <math>S_P</math>. This is a [[Classical physics|classical]] result in the sense that it does not take into account that electrons obey [[Fermi–Dirac statistics]].  The correct result takes into account the quantum statistics of electrons and reads (at zero temperature)
:<math>
S = \frac{2e^3}{\pi\hbar} \vert V \vert \sum_n T_n (1 - T_n)\ .
</math>
It was obtained in the 1990s by [[Viktor Khlus (physicist)|Viktor Khlus]], [[Gordey Lesovik]] (independently the single-channel case), and [[Markus Büttiker]] (multi-channel case).<ref name="buttiker"/> This noise is white and is always suppressed with respect to the Poisson value. The degree of suppression, <math>F = S/S_P</math>, is known as the [[Fano factor]]. Noises produced by different transport channels are independent. Fully open (<math>T_n=1</math>) and fully closed (<math>T_n=0</math>) channels produce no noise, since there are no irregularities in the electron stream.

At finite temperature, a closed expression for noise can be written as well.<ref name="buttiker"/> It interpolates between shot noise (zero temperature) and Nyquist-Johnson noise (high temperature).

====Examples====
* [[Tunnel junction]] is characterized by low transmission in all transport channels, therefore the electron flow is Poissonian, and the Fano factor equals one.
* [[Quantum point contact]] is characterized by an ideal transmission in all open channels, therefore it does not produce any noise, and the Fano factor equals zero. The exception is the step between plateaus, when one of the channels is partially open and produces noise.
* A metallic diffusive wire has a Fano factor of 1/3 regardless of the geometry and the details of the material.<ref>{{cite journal|last=Beenakker|first=C.W.J.|author2=Büttiker, M.|year=1992|title=Suppression of shot noise in metallic diffusive conductors|journal=[[Physical Review B]]|volume=46|issue=3|pages=1889–1892|bibcode = 1992PhRvB..46.1889B |doi = 10.1103/PhysRevB.46.1889 |pmid=10003850|hdl=1887/1116|url=https://openaccess.leidenuniv.nl/bitstream/handle/1887/1116/172_088.pdf?sequence=1|hdl-access=free}}</ref>
* In [[2DEG]] exhibiting [[fractional quantum Hall effect]] electric current is carried by [[quasiparticles]] moving at the sample edge whose charge is a rational fraction of the [[electron charge]]. The first direct measurement of their charge was through the shot noise in the current.<ref>
{{cite journal 
 |author=V.J. Goldman, B. Su
 |year=1995
 |title=Resonant Tunneling in the Quantum Hall Regime: Measurement of Fractional Charge
 |journal=[[Science (journal)|Science]]
 |volume=267 |pages=1010–1012
 |doi=10.1126/science.267.5200.1010
|bibcode = 1995Sci...267.1010G
 |issue=5200
 |pmid=17811442|s2cid=45371551
 }} See also [http://quantum.physics.sunysb.edu/index.html Description on the researcher's website] {{webarchive|url=https://web.archive.org/web/20080828090752/http://quantum.physics.sunysb.edu/index.html |date=2008-08-28 }}.</ref>

====Effects of interactions====
While this is the result when the electrons contributing to the current occur completely randomly, unaffected by each other, there are important cases in which these natural fluctuations are largely suppressed due to a charge build up. Take the previous example in which an average of 100 electrons go from point A to point B every nanosecond. During the first half of a nanosecond we would expect 50 electrons to arrive at point B on the average, but in a particular half nanosecond there might well be 60 electrons which arrive there. This will create a more negative electric charge at point B than average, and that extra charge will tend to ''repel'' the further flow of electrons from leaving point A during the remaining half nanosecond. Thus the net current integrated over a nanosecond will tend more to stay near its average value of 100 electrons rather than exhibiting the expected fluctuations (10 electrons rms) we calculated. This is the case in ordinary metallic wires and in metal film [[resistor]]s, where shot noise is almost completely cancelled due to this anti-correlation between the motion of individual electrons, acting on each other through the [[coulomb force]].

However this reduction in shot noise does not apply when the current results from random events at a potential barrier which all the electrons must overcome due to a random excitation, such as by thermal activation. This is the situation in [[p-n junction]]s, for instance.<ref>Horowitz, Paul and Winfield Hill, The Art of Electronics, 2nd edition. Cambridge (UK): Cambridge University Press, 1989, pp. 431–2.</ref><ref>{{Cite web |url=http://www.analog.com/library/analogDialogue/Anniversary/8.html |title=Bryant, James, Analog Dialog, issue 24-3 |access-date=2008-07-24 |archive-date=2016-09-29 |archive-url=https://web.archive.org/web/20160929001716/http://www.analog.com/library/analogDialogue/Anniversary/8.html |url-status=dead }}</ref> A semiconductor [[diode]] is thus commonly used as a noise source by passing a particular DC current through it.

In other situations interactions can lead to an enhancement of shot noise, which is the result of a super-poissonian statistics. For example, in a resonant tunneling diode the interplay of electrostatic interaction and of the density of states in the [[quantum well]] leads to a strong enhancement of shot noise when the device is biased in the negative differential resistance region of the current-voltage characteristics.<ref>{{Cite journal|last=Iannaccone|first=Giuseppe|date=1998|title=Enhanced Shot Noise in Resonant Tunneling: Theory and Experiment|journal=Physical Review Letters|volume=80|issue=5|pages=1054–1057|doi=10.1103/physrevlett.80.1054|arxiv=cond-mat/9709277|bibcode=1998PhRvL..80.1054I|s2cid=52992294}}</ref>

Shot noise is distinct from voltage and current fluctuations expected in thermal equilibrium; this occurs without any applied DC voltage or current flowing. These fluctuations are known as [[Johnson–Nyquist noise]] or thermal noise and increase in proportion to the [[Kelvin]] temperature of any resistive component. However both are instances of white noise and thus cannot be distinguished simply by observing them even though their origins are quite dissimilar.

Since shot noise is a [[Poisson process]] due to the finite charge of an electron, one can compute the [[root mean square]] current fluctuations as being of a magnitude<ref>Thermal and Shot Noise. Appendix C. Retrieved from class notes of Prof. Cristofolinini, University of Parma. Archived on Wayback Machine. [url=https://web.archive.org/web/20181024162550/http://www.fis.unipr.it/~gigi/dida/strumentazione/harvard_noise.pdf]</ref>

:<math>
\sigma_i=\sqrt{2qI\,\Delta f}
</math>

where ''q'' is the [[elementary charge]] of an electron,  Δ''f'' is the single-sided [[Bandwidth (signal processing)|bandwidth]] in [[hertz]] over which the noise is considered, and ''I'' is the DC current flowing.

For a current of 100 mA, measuring the current noise over a bandwidth of 1&nbsp;Hz, we obtain

:<math>
\sigma_i = 0.18\,\mathrm{nA}  \; .
</math>
<!-- Comment:  previous version had incorrect units for \sigma. Should be Amperes, not Amperes / sqrt(Hz). -->

If this noise current is fed through a resistor a noise voltage of
:<math>
\sigma_v = \sigma_i \, R
</math>
would be generated. Coupling this noise through a capacitor, one could supply a noise power of
:<math>
P = {\frac 1 2}qI\,\Delta f R.
</math>
to a matched load.