The secrets of shot noise

I have been studying shot noise characteristics in transistors myself. Also came across the cited passage above.

It depends on the resistance, but a metallic wire resistor falls in line with the example of Horowitz and Hill. The way to think of it here is the electric field inside the metal wire is not large compared to the interactions between the electrons. These interactions therefore correlate the electrons fairly significantly, so there is negligible shot noise.

If the wire were a semi-insulator, the correlations would not be so strong compared to the electric field responsible for drift current.

Here is a reference to study further:

It's the same as above. We're considering the shot noise of the current in the current-flowing parts.

So your current consists of electrons going thru the resistor. As in #1, if the resistance is high enough, the field inside will make the correlations between the electrons relatively negligible. This would allow a shot noise description.

According to the passage, the feedback in the current source circuit helps to quiet the shot noise. The base junction is usually so thin it is effectively ignored, because the base current is negligibly small, even though a shot description is appropriate there. The collector-emitter current path is considered almost like a single wire (current~constant), so probably in that sense, the shot noise is small.

No, and I believe that AofE (H&H) states this somewhere. Resistors do have several other sources of noise, one of which operates without regard to the fabrication itself and is tied directly to the resistance itself, Johnson noise. AofE talks about this, too.

I can't say there is no noise process in battery chemistry. But I don't believe that any noise process in battery chemistry is similar to that across a PN junction. If there is, I don't think it wouldn't be described by this same equation. In the case of the PN junction, it's a statistical process where electrons move from valence into the conduction band (and back) and there is a small 'step' to it, in the neighborhood of 1+ eV, or so. Perhaps someone can be more 'sure' about this or correct me.

My 'understanding' of shot noise is that it is always associated with cases where discrete quanta of charge (e) or quanta of energy (hv) are emitted randomly in time by some source. The 'independence' part that AofE talks about here is that each of these individual, elemental 'emitters' do so (roughly) independently of each other. If there's a large number of these emitters and there is a very small probability of emission then the overall emission rate is (or, at least, theoretically should be) Poisson distributed. It's this that is talked about when saying 'shot noise,' I think.

The charge on the capacitor has an energy distribution that has to obey Boltzmann statistics for the probability density.

There is some noise associated with capacitance, kT/C, based on Boltzman statistics. In fact, from this you can derive the Johnson noise equation for resistance. In Boltzmann's equipartition theorem, the mean internal energy associated with each degree of freedom in a system is (1/2)kT (Joules.) In the case of electrostatic energy in a capacitor, this becomes:

(1/2) C V^2 = (1/2) k T

or, the geometric mean of the V^2's is:

V^2 = kT/C

Or, another way... The energy distribution [E=(1/2)*C*V^2] must obey Boltmann statistics for the probability density, such that:

p(E) = (1/(k*T))*e^(-E/(k*T))

Then, the probability density of V^2 is:

p(V^2) = p(E) * dE/d(V^2)

Since dE/d(V^2) of [E=(1/2)*C*V^2] is just (1/2)C, we can substitute for E and for dE/d(V^2) to get:

p(V^2) = (C/(2kT))*e^(-CV^2/(2kT))

By this, you can arrive at the geometrical mean of must be 0 or kT/C.

Or it can be taken from the point of view of a statistical thermodynamics theorem from Einstein, which is that "every state function A that can be expressed in terms of the system entropy S has a mean value that is obtained by finding the maximum of S, solving for dS/dA=0 (well, often the partials instead of the derivatives.) By solving this, we also find that the variance is again, kT/C.

In the case of an RC discharge, the noise is just Johnson and not shot. In fact, it's the kT/C realization that generates the formula for Johnson noise, as any stray capacitance C that is inevitably present across an R, gives a cutoff frequency that is B=1/(2*PI*RC) and the voltage's spectral density must be:

Integral(Sv(f) df) = kT/C

substituting Sv(f) with Sv0/(1+(f/B)^2), where Sv0 is the DC value of spectral density, we can then combine to yield:

kT/C = 2*Integ(Sv0/[1+(f/B)^2] df) = 2*Sv0*|atn(f/B)| at 0,B or, kT/C = 2*(Sv0/(2*PI*RC))*(PI/4)

Solving for Sv0, we find that Sv0 = 4kTR!

assumes a fact not in evidence -- is there shot noise in batteries?

The composition of resistors does have an impact on 1/f noise (which is sometimes called 'shot noise' in SPICE simulators, I gather), but the shot noise I've been talking about is white, not 1/f. White shot noise doesn't depend on the composition. But I'm not sure of your question.

Well, perhaps Win can answer what he meant here (more likely if you ask in sci.electronics.design, I think.) But I'd imagine as a hobbyist-guess that the "independence" mentioned earlier isn't entirely true for this case -- some of the current is dependent in some way or the probability of emission is high, so the integral over all the behavior is no longer quite Poisson.

Wish I could have done better. But that's all I can do without putting effort in.

Might want to ask these questions in sci.electronics.design. I'm no expert, but there are some over there.

Jon

Thanks, Fred and Jon,

your answers were very enlightening. Jon, I will follow your suggestion and ask my current-source-question in sci.electronics.design.

Christian

Another shot noise came to mind in the last couple of days. This is associated with charging of capacitors. Specifically, this is Poisson noise from electrons randomly arriving at the capacitor as it charges up. This is different from the formal shot noise in the other discussion but has the same origin in the charge discreteness. It is essentially counting (sqrt(N)) noise.

It might seem that way, but there is no abrupt change from the wire to the capacitor plate. The influence of the next electron on the charges already on the capacitor and the influence of those charges on the next electron are felt all along the way. However, if the electrons are being shot through a nonconductor like air or vacuum, then you might well hear some shot noise.

--
local optimization seldom leads to global optimization

my e-mail address is:   AT mmm DOT com

I am just wondering aloud. My thinking was that the electrons will have interactions but they also scatter along the way to the capacitor. This is what makes the exact arrival times random. Even if the flux or field varies as a well-known function of time, that describes the average electron behavior rather than the individual motion. The well-known behavior j(t) or E(t) or V(t) is independent of the scattering events. Even with feedback (thru heating or electric field) to V or E or j, you have to consider that feedback a macroscopic, average phenomenon, rather than describing the actual counting of electrons passing by. Q(t), charge on capacitor, is also macroscopic and well behaved, e.g., in an RC circuit, but there is also a microscopic level. That is where I am pondering.

It's simple enough, but a rather powerful result nonetheless. Ohms law says that current is voltage divided by the resistance, so current-source noise density is given by ac voltage-noise density divided by the ac resistance, namely i_n = (e_n + sqrt(4kTR)) / (R + r_e). Manipulating equations on AoE page 436, we see that e_n = (4kT r_e/2)^1/2 (ignoring r_bb). So if the dc voltage across the emitter resistor, Ie*R, is greater than 50 to 100mV, so I*R >> kT/qI, then the current-source noise density is largely determined by the bias resistor's Johnson noise density, i_n = sgrt(4kT/R), and not the transistor's shot noise. This can be used to create a nearly-perfect quiet current source, using a moderate to high bias voltage (even 100 to 500V), regulated from a modestly-quiet voltage source.

--
 Thanks,
    - Win

 (email: use hill_at_rowland-dotties-org for now)

Their "arrival" times may be random, but their effects do not arrive all at once. The "arrival" may be discrete, but the effect is not discrete.

--
local optimization seldom leads to global optimization

my e-mail address is:   AT mmm DOT com

It can also been shown that current mirror accuracy dramatically improves as the voltage drop greatly exceeds kT/q (compensates offset mismatch between transistors).

...Jim Thompson

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|  James E.Thompson, P.E.                           |    mens     |
|  Analog Innovations, Inc.                         |     et      |
|  Analog/Mixed-Signal ASIC's and Discrete Systems  |    manus    |
|  Phoenix, Arizona            Voice:(480)460-2350  |             |
|  E-mail Address at Website     Fax:(480)460-2142  |  Brass Rat  |
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I love to cook with wine.      Sometimes I even put it in the food.

The shot noise gets suppressed by 6 dB when the emitter resistor drops

25 mV, so using a 100V supply ought to suppress it by 4000 times (72 dB). Base current shot noise will be a problem, though, so there isn't much point in making |Vbias| more than a few times beta*kT/e. Darlingtons and FETs don't help much with this.

Another point is that for situations like current mirrors or current dividers, where you want to accommodate a wide current range without adding full shot noise, you can use diode-connected transistors as emitter degeneration. The shot noise contributions add in RMS, whereas the transconductance goes down linearly, so N diode-connected transistors suppresses the added shot noise by 10*log(N) dB. I needed to do this once for a variation of the laser noise canceller that suppresses the photocurrent excess noise by an additional 2 dB over the regular noise canceller.

Cheers,

Phil Hobbs