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Shot noise

Discussion in 'Electronic Design' started by Christian Rausch, Aug 31, 2004.

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  1. 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
     
  2. 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.
     
  3. Mike

    Mike Guest

    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.
     
  4. Phil Hobbs

    Phil Hobbs Guest

    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
     
  5. Phil Hobbs wrote...
    Really? That was 25 years ago, I've forgotten, wha'd we say?
     
  6. Joerg

    Joerg Guest

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

    Regards, Joerg
     
  7. Phil Hobbs

    Phil Hobbs Guest

    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
     
  8. 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.

    --Mike
     
  9. John Larkin

    John Larkin Guest

    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
     
  10. Fred Chen

    Fred Chen Guest

    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?
     
  11. Bill Sloman

    Bill Sloman Guest

    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.
     
  12. Phil Hobbs

    Phil Hobbs Guest

    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
     
  13. Roy McCammon

    Roy McCammon Guest

    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".
     
  14. Bill,
    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
     
  15. I meant "...He went after..."

    --Mike
     
  16. Roy McCammon wrote...
    The phraseology was inspired by comments from another
    physics friend of ours, who should know...
     
  17. Fred Chen

    Fred Chen Guest

    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.
     
  18. Mike Engelhardt wrote...
    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 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.
     
  19. Win,
    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 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.
    You're welcome.

    Best Regards,

    --Mike
     
  20. John Larkin

    John Larkin Guest

    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
     
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