The secrets of shot noise

[Christian Rausch]

Help!

Shot noise, as stated in Horowitz&Hill, ch7.11, p.432, shows a noise current
of

Inoise(rms) = sqrt(2*q*Idc*B)
with q:electron charge, B:bandwidth and Idc:the DC current,

but this formula "assumes that the charge carriers making up the current act
independently. This is indeed the case for charges crossing a barrier, as
for example the current in a junction diode...but is not true for the
important case of metallic conductors, where there are long-range
correlations between charge carriers..."

My questions are the following, and I hope that there is somebody out there
who can answer them:

1.
Assume we have a battery with a resistor connected between the poles. Is
there a shot noise current flowing in the resistor? Is the battery a
potential barrier like the one mentioned by H&H?

2.
What about a (charged) capacitor discharging into a resistor? Shot noise
current flowing or not? The dielectric between the plates seems to be quite
a huge barrier, but there are no charge carriers crossing it; only a
displacement current is flowing?

3.
Assume a battery which is shorted for AC via a large capacitor, that means
that any shot noise current that the battery may produce runs through the
capacitor and not elsewhere. Across it is a resistor. Does it really depend
on the composition of this resistor (metallic conductor, i.e. metal film vs.
metal oxide, thick film etc.) if there is a shot noise current flowing or
not?

4.
At the bottom of the first column of p.432, H&H mention that the standard
transistor current source runs quieter than shot-noise-limited. Anybody out
there who knows a little more (literature, math) about this?

Thanks for any advice!


cnhcr

 

[Jonathan Kirwan]

On Mon, 30 Aug 2004 16:27:03 +0200, "Christian Rausch"
<Christian.Rausch@toptica.com> wrote:

1.
Assume we have a battery with a resistor connected between the poles. Is
there a shot noise current flowing in the resistor?

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.

Is the battery a
potential barrier like the one mentioned by H&H?

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.

2.
What about a (charged) capacitor discharging into a resistor? Shot noise
current flowing or not? The dielectric between the plates seems to be quite
a huge barrier, but there are no charge carriers crossing it; only a
displacement current is flowing?

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 <V^2> 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 <A> 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!

3.
Assume a battery which is shorted for AC via a large capacitor, that means
that any shot noise current that the battery

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

may produce runs through the
capacitor and not elsewhere. Across it is a resistor. Does it really depend
on the composition of this resistor (metallic conductor, i.e. metal film vs.
metal oxide, thick film etc.) if there is a shot noise current flowing or
not?

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.

4.
At the bottom of the first column of p.432, H&H mention that the standard
transistor current source runs quieter than shot-noise-limited. Anybody out
there who knows a little more (literature, math) about this?

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.

Thanks for any advice!

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

 

[Christian Rausch]

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

 

[Roy McCammon]

Fred Chen wrote:

"Christian Rausch" <Christian.Rausch@toptica.com> wrote in message news:<2pj5jjFl0fa1U1@uni-berlin.de>...

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

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