Shot noise

[Christian Rausch]

Hello everybody,

Q1.
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..."

Does anybody know what the exact criterion for this 'independence of the
charge carriers' is?
Are there situations with 'partial' shot noise?

Q2.
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?
Anybody who can give me a reference on how to calculate the output noise
current of a transistor current source?

Thanks for any advice!

Christian

 

[Winfield Hill]

Christian Rausch wrote...

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? Anybody who
can give me a reference on how to calculate the output noise current of a
transistor current source?

Christian, after reading the AoE material, you're supposed to be able to
evaluate and calculate it yourself! For a current source consider the
voltage across the resistor in determining the current, and then the sum
of the transistor's e_n and the resistor's Johnson noise, divided by the
total resistance, i.e. the transistor's r_e plus the emitter resistor,
to get the current noise. If the voltage across the resistor is high
enough you can create a very quiet current indeed, far below shot noise.

 

[Mike]

Hello everybody,

Q1.
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..."

Does anybody know what the exact criterion for this 'independence of the
charge carriers' is?
Are there situations with 'partial' shot noise?

This page describes noise in metal resistors of various lengths:

http://www.ilorentz.org/beenakkr/mesoscopics/topics/noise/noise.html

From the paper:
------------
Shot noise results from the fact that the current is not a continuous flow
but the sum of discrete pulses in time, each corresponding to the transfer
of an electron through the conductor. Its spectral density is proportional
to the average current, I, and is characterized by a white noise spectrum
up to a certain cut-off frequency, which is related to the time taken for
an electron to travel through the conductor. In contrast to thermal noise,
shot noise cannot be eliminated by lowering the temperature.

In devices such as tunnel junctions the electrons are transmitted randomly
and independently of each other. Thus the transfer of electrons can be
described by Poisson statistics, which are used to analyse events that are
uncorrelated in time. For these devices the shot noise has its maximum
value at 2eI, where e is the electronic charge.

However, shot noise is absent in a macroscopic, metallic resistor because
the ubiquitous inelastic electron-phonon scattering smoothes out current
fluctuations that result from the discreteness of the electrons, leaving
only thermal noise. But recent progress in nanofabrication technology has
revived the interest in shot noise, particularly since nanostructures and
"mesoscopic" resistors allow measurements to be made on length scales that
were previously inaccessible experimentally.

 

[Phil Hobbs]

This page describes noise in metal resistors of various lengths:

http://www.ilorentz.org/beenakkr/mesoscopics/topics/noise/noise.html

From the paper:
------------
Shot noise results from the fact that the current is not a continuous flow
but the sum of discrete pulses in time, each corresponding to the transfer
of an electron through the conductor. Its spectral density is proportional
to the average current, I, and is characterized by a white noise spectrum
up to a certain cut-off frequency, which is related to the time taken for
an electron to travel through the conductor. In contrast to thermal noise,
shot noise cannot be eliminated by lowering the temperature.

In devices such as tunnel junctions the electrons are transmitted randomly
and independently of each other. Thus the transfer of electrons can be
described by Poisson statistics, which are used to analyse events that are
uncorrelated in time. For these devices the shot noise has its maximum
value at 2eI, where e is the electronic charge.

However, shot noise is absent in a macroscopic, metallic resistor because
the ubiquitous inelastic electron-phonon scattering smoothes out current
fluctuations that result from the discreteness of the electrons, leaving
only thermal noise. But recent progress in nanofabrication technology has
revived the interest in shot noise, particularly since nanostructures and
"mesoscopic" resistors allow measurements to be made on length scales that
were previously inaccessible experimentally.

Shot noise in metallic resistors is suppressed by a factor of the mean free
path of the electrons in the metal divided by the physical length of the
resistor. Most of the time this is a very small number.


And c'mon, Win, even you guys got shot noise wrong in the first edition. ;-)

Cheers,

Phil Hobbs

 

[Winfield Hill]

Phil Hobbs wrote...

And c'mon, Win, even you guys got shot noise wrong in the
first edition. ;-)

Really? That was 25 years ago, I've forgotten, wha'd we say?

 

[Joerg]

Hi Winfield,

Christian, after reading the AoE material, you're supposed to be able to
evaluate and calculate it yourself!

I always thought you also had to own a rifle to understand shot noise....

Regards, Joerg

 

[Phil Hobbs]

Phil Hobbs wrote...



Really? That was 25 years ago, I've forgotten, wha'd we say?

I'll have to go check, but I think you said that all currents had full shot
noise. If not, I will abase myself suitably. (No wise suggestions from the
peanut gallery, please.)

Cheers,

Phil Hobbs

 

[Mike Engelhardt]

Phil,

I'll have to go check, but I think you said that all
currents had full shot noise.

I think you're right, Phil, on page 289 of the 1st
edition AoE does make it sound(incorrectly) like shot
noise is associated with any old current. It reads,

"An electric current is the flow of discrete
electronic charges, not a smooth fluidlike flow.
The finiteness of the charge quantum results in
statistical fluctuations of the current..."

This is true for electrons flowing, e.g., in a vacuum
and the application of this to all currents was a
common error in some physics crowds. At least I had
one thesis advisor that professed that. But the truth is
that the electrons' wavefunctions overlap in a conductors,
so in fact current flow is more like a fluid-like flow
than discrete electrons.

--Mike

 

[John Larkin]

Shot noise in metallic resistors is suppressed by a factor of the mean free
path of the electrons in the metal divided by the physical length of the
resistor. Most of the time this is a very small number.

OK, humor me here: hang a metal-film resistor across a power supply,
and we get a current with low shot noise. What happens if we put two
equal-value resistors in series across the supply, one metal-film and
one something crummy, carbon film or something? Will this just act
like a voltage divider between a noisy resistor and a quiet one,
giving half the shot noise current as an all-carbon circuit?

John

 

[Fred Chen]

Christian Rausch wrote...

Christian, after reading the AoE material, you're supposed to be able to
evaluate and calculate it yourself! For a current source consider the
voltage across the resistor in determining the current, and then the sum
of the transistor's e_n and the resistor's Johnson noise, divided by the
total resistance, i.e. the transistor's r_e plus the emitter resistor,
to get the current noise. If the voltage across the resistor is high
enough you can create a very quiet current indeed, far below shot noise.

I took a look at the current-source passage referred by Christian, and
thought that the correlations among the electrons could be suppressed
due to the lack of any significant barrier (base is thin) as long as
resistance is also reasonably low and electron density is high. Is
there anything wrong with this intuition?

 

[Bill Sloman]

Phil,


I think you're right, Phil, on page 289 of the 1st
edition AoE does make it sound(incorrectly) like shot
noise is associated with any old current. It reads,

"An electric current is the flow of discrete
electronic charges, not a smooth fluidlike flow.
The finiteness of the charge quantum results in
statistical fluctuations of the current..."

This is true for electrons flowing, e.g., in a vacuum
and the application of this to all currents was a
common error in some physics crowds. At least I had
one thesis advisor that professed that. But the truth is
that the electrons' wavefunctions overlap in a conductors,
so in fact current flow is more like a fluid-like flow
than discrete electrons.

Actually even electrons flowing in a vacuum can interact - though when
it happens people talk about "space charge". This creates several
forms of detectable non-linearity in photomultipliers - the one that
impressed me was due to the "space charge" from a single electron in
the space between the photo-cathode and the first dynode. Since the
space is about a cm long and the electron is travelling at about 10%
of the speed of light at roughly 3cm/nsec you need of the order of a
nanoamp of photo-current to see it (which is quite a lot).

The electron sources (guns) in electron microscopes produce a slightly
less noisy beam current than you''d expect from straight shot noise
statistics, which s presumably due to electron-electron interaction as
well.

 

[Phil Hobbs]

OK, humor me here: hang a metal-film resistor across a power supply,
and we get a current with low shot noise. What happens if we put two
equal-value resistors in series across the supply, one metal-film and
one something crummy, carbon film or something? Will this just act
like a voltage divider between a noisy resistor and a quiet one,
giving half the shot noise current as an all-carbon circuit?

You can find this by invoking linearity. The power supply plus quiet
resistor makes a Thevenin source. If the noisy resistor R_noisy draws a current

i~ = I_dc + i_noise,
then the quiet resistor's voltage drop will change in such a way as to oppose
the noise. Since the Thevenin resistance of the combination is R_quiet ||
R_noisy, the voltage change won't be as large as i_noise*R_noisy, so that the
noise current won't be completely suppressed. This is exactly the way that
an emitter resistor will make a BJT's collector current quieter than full
shot noise. A sufficiently large, quiet resistor will dominate the noisy
one, because R_quiet || R_noisy ~ R_noisy, making the noise suppression
almost perfect.

This is the origin of the noise suppression in metallic resistors too, I
think--the electrons thermalize over about one mean free path, leading their
conductance fluctuations to be independent, and hence the noise power drops
as the length of the resistor.

(I don't think carbons have full shot noise at high frequency, but they're
definitely noisier than metal, especially at low frequencies where their
conductivity fluctuations occur.)

The Pauli exclusion principle makes the electrons highly correlated in
metals, which violates the random-arrival assumption of the shot noise model.

Cheers,

Phil Hobbs

 

[Roy McCammon]

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..."

Does anybody know what the exact criterion for this 'independence of the
charge carriers' is?
Are there situations with 'partial' shot noise?

I've asked about those "long range correlations" before,
but Uncle Win remains uncharacteristically silent. I
can only conclude that this passage must be the words
of the other H.

So I am left with my own thoughts.

My first thought is that shot noise requires a
more or less irreversible barrier. When an
electron crosses a junction, an "event" has
occurred. The event cannot be undone (mostly).
An electron crossing an arbitrary plane in a
metallic conductor can move back and forth
across that plane and can sit on the plane
with any proportion on one side or the other.
Because of the reversibility, there is no "event".
And of course there are "leaky" junctions where
you would have a blend of noises.


My second thought is that "long range correlations"
are an effect and not a cause. Its about as profound
as saying "its not rough, because its smooth".

 

[Mike Engelhardt]

Bill,

Actually even electrons flowing in a vacuum can interact -
though when it happens people talk about "space charge".
This creates several forms of detectable non-linearity in
photomultipliers - the one that impressed me was due to the
"space charge" from a single electron in the space between
the photo-cathode and the first dynode. Since the space is
about a cm long and the electron is travelling at about 10%
of the speed of light at roughly 3cm/nsec you need of the
order of a nanoamp of photo-current to see it (which is
quite a lot).

The electron sources (guns) in electron microscopes produce
a slightly less noisy beam current than you''d expect from
straight shot noise statistics, which s presumably due to
electron-electron interaction as well.

Sure. I just use the vacuum example because that's the
context that people dwell on the Poisson(counting) statistics
that's the basis for shot noise. Basically, you need the
complete vacuum limit, devoid even of space charge, to
perfectly exhibit Poisson statistics with charged particles.
Shot noise was a big thing in low-current vacuum tube
electronics, though they more often saw the partition noise
variation. Similarity, it's not an exact statement that
electrons in a conductor are completely devoid of shot noise,
electron flow isn't perfectly fluid, but the granularity is
far below what would would get from a quantum charge and
Poisson statistics. Basically, the statement I quoted from
the 1st Ed. of AoE is the opposite of the truth for general
currents in conductors and I've seen Win have trouble with
this idea in another post regarding the analog nature of
the charge on the gate of a MOSFET. We went after the number
of electrons it takes to charge the gate, but that quantization
limit didn't apply to his example, again because electron
current flow really is more like continuous flow of fluid-
like current than a stream of discrete quantized charges.

Regards,

--Mike

 

[Mike Engelhardt]

...We went after the number of electrons it takes to charge
the gate, but that quantization limit didn't apply to his example,
again because...

I meant "...He went after..."

--Mike

 

[Winfield Hill]

Roy McCammon wrote...

I've asked about those "long range correlations" before,
but Uncle Win remains uncharacteristically silent. I
can only conclude that this passage must be the words
of the other H.

The phraseology was inspired by comments from another
physics friend of ours, who should know...

 

[Fred Chen]

This page describes noise in metal resistors of various lengths:

http://www.ilorentz.org/beenakkr/mesoscopics/topics/noise/noise.html

From the paper:
------------
Shot noise results from the fact that the current is not a continuous flow
but the sum of discrete pulses in time, each corresponding to the transfer
of an electron through the conductor. Its spectral density is proportional
to the average current, I, and is characterized by a white noise spectrum
up to a certain cut-off frequency, which is related to the time taken for
an electron to travel through the conductor. In contrast to thermal noise,
shot noise cannot be eliminated by lowering the temperature.

In devices such as tunnel junctions the electrons are transmitted randomly
and independently of each other. Thus the transfer of electrons can be
described by Poisson statistics, which are used to analyse events that are
uncorrelated in time. For these devices the shot noise has its maximum
value at 2eI, where e is the electronic charge.

However, shot noise is absent in a macroscopic, metallic resistor because
the ubiquitous inelastic electron-phonon scattering smoothes out current
fluctuations that result from the discreteness of the electrons, leaving
only thermal noise. But recent progress in nanofabrication technology has
revived the interest in shot noise, particularly since nanostructures and
"mesoscopic" resistors allow measurements to be made on length scales that
were previously inaccessible experimentally.

This paper's explanation seemed counterintuitive to me, because it
seems that the scattering mechanisms are what cause the electrons to
become random in passing a given point. It seems that what helps
metals to suppress shot noise is the large electron density allowing
stronger Coulomb interactions between electrons. A large enough
conduction barrier or high-enough resistance should violate this
condition, allowing shot noise to be valid.

Here is a reference that has experimental demonstrations:
http://arxiv.org/PS_cache/cond-mat/pdf/0406/0406484.pdf

Shot noise is observed in a CdTe resistor.

 

[Winfield Hill]

Mike Engelhardt wrote...

... Basically, the statement I quoted from the 1st Ed. of
AoE is the opposite of the truth for general currents in
conductors ...

Well, if so, our mistake 25 years ago matched the mainstream
of most thinking on the subject, and at least we did break
ground in solidly fixing the error 9 years later. :>)

I've seen Win have trouble with this idea in another post
regarding the analog nature of the charge on the gate of a
MOSFET. He went after the number of electrons it takes to
charge the gate, but that quantization limit didn't apply
to his example, again because electron current flow really
is more like continuous flow of fluid-like current than a
stream of discrete quantized charges.

I vaguely recall the discussion, concerning some conductance
measurements I had made on a small MOSFET with a floating
gate? (I was attempting to determine its gate leakage.)
But I don't recall your arguing that one can't calculate the
number of charges associated with a small change in a FET's
gate voltage. How does that argument go again? Presumably
the FET gate is sitting on an insulator next to the channel.

 

[Mike Engelhardt]

Win,

Well, if so, our mistake 25 years ago matched the mainstream
of most thinking on the subject, and at least we did break
ground in solidly fixing the error 9 years later. :>)

Yes, what you made was a common error, but I don't think that
it was the mainstream accepted theory. Between your two
editions I was designing a preamp and that's when I realized
that Poisson statistics(shot noise) didn't apply to current
in a wire. When I looked into it at the time, I realized
Poisson statics require all electrons to interact independently,
as was well known. The idea that Poisson statistics applied
to current in wires was just a common mistake, not the accepted
theory at the time. I think it got popular with hack electronics
people because it did apply to current in tubes(values) at low
current. Though a common error, it was not the mainstream
accepted theory. Heck, when I became aware of this while
doing that preamp, I realized at that time I understood for
the first time some comments made to me about shoot noise
in the 70's, that basically said it didn't apply to wires
because the electrons act in one larger wavefunction.

I vaguely recall the discussion, concerning some conductance
measurements I had made on a small MOSFET with a floating
gate? (I was attempting to determine its gate leakage.)
But I don't recall your arguing that one can't calculate the
number of charges associated with a small change in a FET's
gate voltage. How does that argument go again? Presumably
the FET gate is sitting on an insulator next to the channel.

I thought that it was about being able to smoothly turn a
MOSFET on or off due to the discreteness of electron charge.
You went after the number of electrons it took to drive the
gate voltage anywhere. The gate wasn't floating, so I just
noticed that again you seemed to be applying charge quantization
in a situation that it didn't apply, since you can basically
charge a gate with any fraction of an electron you want. I
just noticed it because it matched the erroneous statement
in AoE. I don't know if you still have trouble with the
concept or not.

Thanks,

You're welcome.

Best Regards,

--Mike

 

[John Larkin]

I thought that it was about being able to smoothly turn a
MOSFET on or off due to the discreteness of electron charge.

There are tiny fets, "SETs", that can resolve gate charge to a small
fraction of e. I think it may be possible to do a neat demonstration
using an eprom... possible student project maybe.

John