Tuesday, December 15, 2009

Noise III

While Johnson-Nyquist noise is an equilibrium phenomenon, shot noise is a nonequilibrium effect, only present when there is a net current being driven through a system.  Shot noise is a consequence of the fact that charge comes in discrete chunks.  Remember, current noise is the mean-square fluctuations about the average current.  If charge was a continuous quantity, then there wouldn't be any fluctuations - the average flow rate would completely describe the situation.  However, since charge is quantized, a complete description of charge flow would instead be an itemized list of the arrival times of each electron.  With such a list, a theorist could calculate not just the average current, but the fluctuations, and all of the higher statistical moments.  This is called "full counting statistics", and is actually achievable under certain very special circumstances.

Schottky, about 90 years ago, worked out the expected current noise power spectral density, SI, for the case of independent electrons traversing a single region with no scattering (as in a vacuum tube diode, for example).  If the electrons are truly independent (this electron doesn't know when the last electron came through, or when the next one is going through), and there is just some arrival rate for them, then the electron arrivals are described by Poisson statistics.  In this case, Schottky showed that SI = <(I - < I >)2> = 2 e < I > Amps2/Hz.  That is, the current noise is proportional to the average current, with a proportionality constant that is twice the electronic charge.

In the general case, when electrons are not necessarily independent of each other, it is more common to write the zero temperature shot noise as SI = F 2 e < I >, where F is called the Fano factor.  One can think if F as a correction factor, but under sometimes it's better to think of F as describing the effective charge of the charge carriers.  For example, suppose current was carried by pairs of electrons, but the pair arrivals are Poisson distributed.  This situation can come up in some experiments involving superconductors.  In that case, one would find that F = 2, or you can think of the effective charge carriers being the pairs, which have charge 2e.  These deviations away from the classical Schottky result are where all the fun and interesting physics lives.  For example, shot noise measurements have been used to show that the effective charge of the quasiparticles in the fractional quantum Hall regime is fractional.  Shot noise can also be dramatically modified in highly quantum coherent systems.  See here for a great review of all of this, and here for a more technical one.

Nanostructures are particularly relevant for shot noise measurements.  It turns out that shot noise is generally suppressed (F approaches zero) in macroscopic conductors.  (It's not easy to see this based on what I've said so far.  Here's a handwave:  the serious derivation of shot noise follows an electron at a particular energy and looks to see whether it's transmitted or reflected from some scattering region.  If the electron is instead inelastically scattered with some probability into some other energy state, that's a bit like making the electrons continuous.)  To see shot noise clearly, you either need a system where conduction is completely controlled by a single scattering-free region (e.g., a vacuum tube; a thin depletion region in a semiconductor structure; a tunnel barrier), or you need a system small enough and cold enough that inelastic scattering is rare.

The bottom line:  shot noise is a result of current flow and the discrete nature of charge, and deviations from the classical Schottky result tell you about correlations between electrons and the quantum transmission properties of your system.  Up next:  1/f noise.   


Don Monroe said...

When I was trying to measure shot noise in grain-boundary Josephson junctions in high-Tc films 20 years ago, I was surprised how few physicists knew that shot noise isn't present in ordinary resistors. The theoretical argument seemed so fundamental, many people simply assumed it had to be there.

I heard two other hand-waving explanations for this absence:

(1) Correlations: The derivation depends on the charge quanta arriving independently. If instead each one has to wait for the one before to get out of the way, the fluctuations will be reduced.

In the case of a normal resistor, one imagines a large number of electrons filling the resitor material to capacity, creating a space charge that admits new electrons only as fast as others leave.

(2) Low-pass filtering: How long does it take a single electron to show up as a pulse of current (with net area one electron charge) in an external circuit? In a tunnel junction, it's very fast. In a resistor, one imagines that the electron enters and diffuses and drifts for a long time before leaving. It induces a displacement current as long as its moving between the electrodes.

The shot noise formula applies only for frequencies that are low compared to the inverse of this time. Practically speaking, this is such a low frequency that it's irrelevant, and low-pass filtered shot noise is gone at ordinary frequencies.

I think these two explanations are actually the same in some cases, but I don't remember seeing them developed formally. However, in both cases it makes sense that nano-scale devices, where transit times are fast and there are only a few carriers, would make it easier to see.

Douglas Natelson said...

Hi Don - I was also surprised by this. I even had a conversation with a theorist who has published on the low-T shot noise in nanostructures, and this person didn't seem to get this point. FWIW, in the Blanter/Buttiker review article that I linked, the discussion of this takes place on pp. 37-39. I don't see how their explanation jibes with your low pass filtering picture; they do mention the overall charge neutrality issue, though. Still, it seems like in their derivation inelastic scattering has the critical role.

Dale Ritter said...

Nanostructures with implications for shot noise have definitions, which atomic topological function modeling can specify. Recent advancements in quantum science have produced the picoyoctometric, 3D, interactive video atomic model imaging function, in terms of chronons and spacons for exact, quantized, relativistic animation. This format returns clear numerical data for a full spectrum of variables. The atom's RQT (relative quantum topological) data point imaging function is built by combination of the relativistic Einstein-Lorenz transform functions for time, mass, and energy with the workon quantized electromagnetic wave equations for frequency and wavelength.

The atom labeled psi (Z) pulsates at the frequency {Nhu=e/h} by cycles of {e=m(c^2)} transformation of nuclear surface mass to forcons with joule values, followed by nuclear force absorption. This radiation process is limited only by spacetime boundaries of {Gravity-Time}, where gravity is the force binding space to psi, forming the GT integral atomic wavefunction. The expression is defined as the series expansion differential of nuclear output rates with quantum symmetry numbers assigned along the progression to give topology to the solutions.

Next, the correlation function for the manifold of internal heat capacity energy particle 3D functions is extracted by rearranging the total internal momentum function to the photon gain rule and integrating it for GT limits. This produces a series of 26 topological waveparticle functions of the five classes; {+Positron, Workon, Thermon, -Electromagneton, Magnemedon}, each the 3D data image of a type of energy intermedon of the 5/2 kT J internal energy cloud, accounting for all of them.

Those 26 energy data values intersect the sizes of the fundamental physical constants: h, h-bar, delta, nuclear magneton, beta magneton, k (series). They quantize atomic dynamics by acting as fulcrum particles. The result is the exact picoyoctometric, 3D, interactive video atomic model data point imaging function, responsive to keyboard input of virtual photon gain events by relativistic, quantized shifts of electron, force, and energy field states and positions.

Images of the h-bar magnetic energy waveparticle of ~175 picoyoctometers are available online at http://www.symmecon.com with the complete RQT atomic modeling manual titled The Crystalon Door, copyright TXu1-266-788. TCD conforms to the unopposed motion of disclosure in U.S. District (NM) Court of 04/02/2001 titled The Solution to the Equation of Schrodinger.

Anonymous said...

Dale Ritter:

Jumping Jesus! What was THAT???

Don Monroe said...

Thanks for directing me to the proper point in the paper, Doug. Of course it's more than I can justify understanding in detail.

Nonetheless, I don't buy it. I realize it is ridiculously arrogant for a dabbler like me to challenge the giants, but I don't see why inelastic scattering changes anything about the quantized charge on electrons. The electrons aren't going away.

It's true that the external current reflects the combined contributions of many independent electrons. But the authors simply assert that at some scale(the inelastic scattering length) they can ignore the charge of the electrons and switch to a formalism that treats the charge as continuous. That may be a good approximation or not, but it's an assumption, not a proof.

For normal resistors, it may not matter much how you get to the answer that the shot noise is too small to measure. It will be interesting to see the ideas put to the test in mesoscopic (oops-I guess I'm supposed to say nanoscale) systems.

Unknown said...

I used to read van der Ziel's books for understanding the noises from a technical view point.
author = {Aldert van der Ziel},
title = {Noise in solid state devices and circuits},
publisher = {Wiley},
year = {1986},

Also there is a more theoretical book by Kogan, though later I think the math part of it is not as clear (or should say easy to understand) as some introductory probability book,
author = {Sh. Kogan},
title = {Electronic Noise and Fluctuations in Solids},
publisher = {Cambridge University Press},
year = {1996}

As far as I remember the key point is the ratio between the energy that gained by the electrons before getting scattered, and k_B T.
And shot noise is always related to I, so it is not an equilibrium noise.

As far as the shot noise measurement for a nanoscale/meso conductor, there are already too many. The one that is often cited is
author = "A. H. Steinbach and J. M. Martinis and M. Devoret",
title = "Observation of hot-electron shot noise in a metallic resisto",
journal = "Phys. Rev. Lett.",
volume = "76",
number = "20",
pages = "3806",
year = "1996",
detail = "Phys. Rev. Lett., vol.76, no.20, 13 May 1996, pp.3806-9. Publisher: APS,",
note = "",
abstract = "\\{\bf Abstract: }We have measured the current noise of silver thin-film resistors as a functionof current and temperature and for resistor lengths of 7000, 100, 30, and 1 mum. As the resistor becomes shorter than the electron-phonon interaction length,the current noise for large current increases from a nearly current independentvalue to the interacting hot-electron value ( square root (3)/4)2eI. However,further reduction in length below the electron-electron interaction lengthdecreases the noise to a value approaching the independent hot-electron value(1/3)2eI first predicted for mesoscopic resistors. (17 References",

Btw, according to the Basel group's work, (check Oberholzer thesis at http://www.nanoelectronics.ch/publications/theses.php) shot noise can be affected by partition and other things.

Douglas Natelson said...

Hi Don - My sense is that there are more detailed treatments that really do show how inelastic scattering (into the phonons) smears out the noise. I'll try to find a reference. I agree that the review article makes it sound more like an assertion than a result.

Weijian, thanks for the references. I was perhaps too brief in my discussion. You can think of two sources of noise, if you like. One is caused by, at fixed energy, partitioning between transmitted and reflected particles. The other is caused by partitioning electrons among the allowed energies (channels). The former is the source of the tau(1-tau) term in the Fano factor, where tau is the transmission probability of a particular channel. The latter is the source of the Johnson-Nyquist contribution.

Theorist said...

I just looked at the discussion and tried to understand what Don meant by absence of shot noise in "ordinary resistors", but I am not completely sure I got his question right.

Shot noise is a non-equilibrium phenomenon and its power is proportional to the applied bias, eV. For the shot noise to be measured one has to ensure that energy acquired by electrons in external field (bias) is much bigger than kT. Otherwise, the noise is thermal (Johnson-Nyquist). That's why inelastic scattering must be suppressed. If phonon scattering is frequent, electrons will simply thermalize to the lattice temperature T and thermal noise is going to be all that is seen.
If, on the other hand, the question is why shot noise is not measured trivially during any measurement by virtue of charge quantization, then you have to rethink more carefully how you actually measure current fluctuations. If you make measurement with a typical galvanometer-based ammeter (voltmeter) you are not detecting arrival of single electrons. You are not making your measurement on the time scale that is less than the time it takes an electron to cross galvanometer. If, however, you'd devised a measurement method where your electron detector would be operational at shorter time scales then yes, you'd be able in principle to detect shot noise even under conditions of thermal equilibrium. However, additional suppression from electron correlations would still be in effect: just think about how (non)random detection of current would be in an ideal Wigner crystal of electrons sliding frictionlessly across the detector.

Unknown said...

Theorist, what you said about eV < kT is just what I mentioned in the previous post. Note that the energy gained for an individual electron is not eV when there is "inelastic" scattering between the two electrodes.

I am an experimentalist, so excuse me I did not get your point that with fast voltage probes it is possible to measure shot noise for normal resistors. You may check the Steinbach paper in my previous post. What you get for large resistor will be just heating effect. Btw, as far as I understand shot noise measurements do not require fast electron counting. In fact, I think one can only measure shot noise at frequencies BELOW the frequency of the pulse and the frequency corresponding to the inverse of crossing time, since ultimately the Poisson distribution is a statistical quantity.

Doug, what amazes me is that it seems shot noise, or correlation measurements, can be used to infer decoherence, entanglement etc as shown by recent work by Neder et al
author = {{Neder}, I. and {Marquardt}, F. and {Heiblum}, M. and {Mahalu}, D. and
{Umansky}, V.},
title = "{Controlled dephasing of electrons by non-gaussian shot noise}",
journal = {Nature Physics},
opteprint = {arXiv:cond-mat/0610634},

author = {{Neder}, I. and {Ofek}, N. and {Chung}, Y. and {Heiblum}, M. and
{Mahalu}, D. and {Umansky}, V.},
title = "{Interference between two indistinguishable electrons from independent sources}",
journal = {Nature},
year = 2007,
month = jul,
volume = 448,
pages = {333-337},

author = {I. Neder and M. Heiblum and D. Mahalu and V. Umansky},
collaboration = {},
title = {Entanglement, Dephasing, and Phase Recovery via Cross-Correlation Measurements of Electrons},

I kind of understand that shot noise can be used to infer fractional charge, but this group's recent work is not as straightforward or maybe I should say more interesting.

Theorist said...

"I did not get your point that with fast voltage probes it is possible to measure shot noise for normal resistors"

OK, the point is this. Before expecting effects of quantization to be seen in any measurement of current, think about the process of measurement. With a galvanometer you are not measuring transmission of individual electrons, you are measuring fluctuating magnetic fields created by electric currents a long distance away (longer that spatial separation) from electrons. If you managed to create a detector that detects individual electrons (like a microscopic coil that is sensitive to changing magnetic field of individual electron spins) then you will be able to observe Poissonian statistics of arriving electrons even when kT>>eV (of course as long as correlations from Coulomb and Pauli correlations are neglected -- e.g. in non-degenerate (no Pauli correlations) but doped (screened Coulomb) semiconductors). But the question is why one can not see see shot noise in ordinary resistors with ordinary devices.

Douglas Natelson said...

Weijian, you have a point about the recent results from Israel. This paper in particular: http://arxiv.org/abs/0911.3023 makes me wonder if something considerably more subtle is going on in these measurements than simply modified Fano factors due to changing effective charges.