ECE594I notes, M. Rodwell, copyrighted
ECE594I Notes set 7:
Shot Noise
Mark Rodwell
University
y of California, Santa Barbara
rodwell@ece.ucsb.edu 805-893-3244, 805-893-3262 fax
References and Citations:
ECE594I notes, M. Rodwell, copyrighted
Sources / Citations :
Kittel and Kroemer : Thermal Physics
Van der Ziel : Noise in Solid - State Devices
Papoulis : Probabil ity and Random Variables (hard, comprehens ive)
Peyton Z. Peebles : Probabili ty, Random Variables, Random Signal Principle s (introduct ory)
Wozencraft & Jacobs : Principle s of Communicat ions Engineeri ng.
Motchenbak er : Low Noise Electroni c Design
Informatio n theory lecture notes : Thomas Cover, Stanford, circa 1982
Probabilit y lecture notes : Martin Hellman,
Hellman Stanford,
Stanford circa 1982
National Semiconduc tor Linear Applicatio ns Notes : Noise in circuits.
Suggested
S
t d references
f
f study.
for
t d
Van der Ziel, Wozencraft & Jacobs, Peebles, Kittel and Kroemer
Papers by Fukui (device noise), Smith & Personik (optical receiver design)
National Semi. App. Notes (! )
Cover and Williams : Elements of Informatio n Theory
Shot Noise ( idealized)
Focus first on idealized process
with independen t emission events.
We will relate this,
this with some approximat ions,
ions
to carrier tr ansport in semiconduc tors.
Suppose that, observed in some small time period τ 1 ,
there is a probabilit y p1 of emission from region 1 → 2.
average electron flux = r = p1 / τ 1
average DC current = I = q ⋅ p1 / τ 1
for notational clarity we here write q = electron charge
and not use q as a shorthand for (1 − p ).
ECE594I notes, M. Rodwell, copyrighted
Shot Noise (idealized)
Now consider small time increments Δt.
Time period Δti is the period i ⋅ Δt < t < (i + 1) ⋅ Δt
Then
1) The
Th # off electrons
l
emitted
i d in
i eachh Δt is
i independen
i d
d t
of the # of electrons emitted in any other period Δt .
2) pΔti ( k electrons) = pΔti ( k ) = p ( k )
⎛ Δt / τ ⎞ k
= ⎜⎜
⎟⎟ ⋅ p1 (1 − p1 ) ( Δt / τ −1)
⎝ k ⎠
because there are ( Δt / τ ) trials each of probabilit y p1.
ECE594I notes, M. Rodwell, copyrighted
Shot Noise (idealized)
ECE594I notes, M. Rodwell, copyrighted
If wee consider only
onl cases where
here p ⋅ ( Δt / τ ) >> 1, p << (1 − p )
the distributi on will converge to a Gaussian, simplifyin g work :
pΔti ( k ) = Gaussian with mean p ⋅ ( Δt / τ ) and variance p ⋅ ( Δt / τ )
Consider separately the mean values and fluctuatio ns :
Δk = k − E [k ] = fluctuatio n in flux,
where E [k ] = k = mean electron flux in Δt.
Δk is Gaussian with variance σ k2 = k
Shot Noise
ECE594I notes, M. Rodwell, copyrighted
For differenin g time steps Δti and Δt j : E [Δki Δk j ] = 0 for i ≠ j
But for the same time period : E [Δki Δki ] = σ k2
So in general E [Δki Δk j ] = σ k2 ⋅ δ i , j = ( r ⋅ Δt ) ⋅ δ i , j
w here δ i , j = 1 for i = j , δ i , j = 0 otherwise.
This is a sequence with discrete - time autocorrel ation function
Rkk ( Δti Δt j ) = ( r ⋅ Δt ) ⋅ δ i , j
We must work towards the continuous - time limit ( Δt → 0)).
Shot Noise: Computing Power Spectrum
ECE594I notes, M. Rodwell, copyrighted
View as follows :
r ( t ) = electron flux across boundary (# electrons/ second)
...clearly a series of impulses at random times.
f ( t ) = # electrons
l t
crossing
i bondary
b d
b t
between
t = − Δt and
d t = 0.
Then f (t ) = r ( t ) * h ( t ),
) where h ( t ) is an impulse response :
h (t ) = 1 for − Δt < t < 0 , h (t ) = 0 otherwise.
Work with fluctuatio ns Δr = r − r , Δf = f − f
Δf is Gaussian with variance σ 2f = rΔt
Shot Noise: Computing Power Spectrum
ECE594I notes, M. Rodwell, copyrighted
The autocorrel ation function of Δf must be as shown because
(1) the variance of Δf is rΔt and
(2) Δf ( t ) and Δf ( t + τ ) represent time - averages of r (t ) whose
averaging time - windows overlap only if τ is less than Δt
Shot Noise: Computing Power Spectrum
ECE594I notes, M. Rodwell, copyrighted
⎡ sin (ωΔt / 2 )⎤
Power spectral density of Δf : S ΔfΔf ( jω ) = r ⋅ ( Δt ) 2 ⋅ ⎢
⎥
(
)
Δ
ω
t
/
2
⎣
⎦
⎡ sin (ωΔt / 2 )
⎤
Filter Transfer function : H ( jω ) = ⎢
⋅ ( Δt ) ⎥
⎣ (ωΔt / 2 )
⎦
But S ΔfΔf ( jω ) = H ( jω ) ⋅ S ΔrΔr ( jω )
2
Power spectral density of Δr :
S Δ rΔ r ( j ω ) = S Δ fΔ f ( j ω ) H ( j ω )
−2
=r
2
Shot Noise: Computing Power Spectrum
ECE594I notes, M. Rodwell, copyrighted
This simple calculatio n has shown 2 things :
1) The shot noise process has a power spectrum,
constant over all frequencie s, equal to the flux. The
spectrum * does not * decrease at frequencie s higher
than the inverse of the average emission time.
2) If we filter shot noise with some impulse response Δt ,
such
h that
h r Δt >> 1 , then
h the
h filtered
fil d process is
i Gaussian.
G
i
If we filter instead with r Δt
Δt not >> 1,
1 the process is no longer
Gaussian, but it remains constant - power with frequency (white).
DC current and Noise Current
To convert from electron fluxes to currents :
I = q⋅r
→ I = I DC = qr
→ σ I2 = q 2σ r2
S I ( j ω ) = q 2 S r ( jω ) = q 2 r = q ( q r ) = q I
Using instead single - sided Hz - based spectra :
~
S I ( jf ) = 2 qI
ECE594I notes, M. Rodwell, copyrighted
Resistors don't have shot noise
ECE594I notes, M. Rodwell, copyrighted
Note we have assumed independen t emissions
As electrons move in resistors, electric fields are produced.
These fields produce forces on other nearby electrons, thereby
modifying their motions.
Electron passages through resistors during DC current flow
is therefore not a sequence of independen t emission events.
Resistor shot noise ? Reductio ad absurdum
ECE594I notes, M. Rodwell, copyrighted
~
Assume : resistors have shot noise : S I sn I sn = 2 q (1 mA )
Following the illustrate d manipulati ons,
~
~
I SN ,tot = ( I sn1 + I sn 2 ) / 2 where S I sn 1I sn 1 = S I sn 2 I sn 2 = 2 q (1 mA )
~
~
~
So S I sn ,tot I sn ,tot = S I sn 1I sn 1 + S I sn 2 I sn 2 4 = q (1 mA )
(
)
~
~
S I sn I sn ≠ S I sn ,tot I sn ,tot
Breaking a resistor into a vast # of small resistors, a similar argument
shows * how * shot noise is suppressed in resistors.
Shot Noise Example: Optical Receiver
Photocurre nt is amplified and filtered over a time Δt = Tbit .
Assume a receiver limited by detector leakage shot noise;
not a typical situation.
I leak = 100 nA,, 40 Gb/s data rate.
~
Input referred noise S I leak = 2 q( 100 nA) = 3.2(10 -16 ) A 2 /Hz.
Δt ⋅ r = (100 nA) / q ⋅ (1 / 40 Gb/s) = 16 electrons
We expect 16 ± 4 (mean
(
± one std.
d deviation)
d i i ) leakage
l k
electrons
l
per
bit period, but the distributi on will be Poisson, not Gaussian.
ECE594I notes, M. Rodwell, copyrighted
Shot noise in a thin Schottky diode
ECE594I notes, M. Rodwell, copyrighted
Current in thermioni c limit, approximat e analysis.
I ms = I 0
I sm = I 0 ⋅ exp( qV f / kT )
We postulate that thes e are independen t currents, each of which arises from electron being
thermally excited above the barrier energy wit h probabilit y p ( Ei ) ∝ exp( − Ei / kT )
Total current in thermioni c limit, approximat e analysis.
I = I sm − I ms = I 0 ⋅ [exp( qV f / kT ) − 1]
If we treat both I sm and I ms as independen t processes driven from independen t
electron emissions, then
~
S I n I n = 2 qI ms + 2 qI sm
Shot noise in a thin Schottky diode
First consider zero bias :
g ≡ dI / dV = qI 0 / kT at V f = 0 Volts, while I ms = I sm = I 0
So :
~
S I n I n = 2 qI ms + 2 qI sm = 4 qI 0 = 4kTg
Hence the available power :
~
∂Pavail / ∂f = (1 / 4 g ) ⋅ S I n I n = kT
This is required from thermodyn amics; otherwise we would have a net
flow of noise power if we were to connect th e diode in parallel with a resistor.
resistor
Yet, something is wrong : what happens for hf > kT ?
ECE594I notes, M. Rodwell, copyrighted
ECE594I notes, M. Rodwell, copyrighted
High-frequency diode shot noise : UV crisis again
The available noise power at zero bias must be
⎤ hf << kT
dPavail ⎡ hf
hf
=⎢ +
⎥ ⎯⎯ ⎯→ kT
df
hf
kT
−
2
exp(
/
)
1
⎣
⎦
Yet we have derived
dPavail
= kT att all
ll frequencie
f
i s
df
This suggests
gg
that either the shot noise spectral
p
densityy should be
⎡ hf
⎤
hf / kT
~
+
S II ( jf ) = 2 qI 0 ⎢
⎥
⎣ 2kT exp( hf / kT ) − 1 ⎦
...or that th e diode zero - bias conductanc e increases for hf > kT
ECE594I notes, M. Rodwell, copyrighted
High-frequency diode shot noise : UV crisis again
This discrepanc y is not widely noted in the literature , yet must
be important for THz devices operated with cooling.
We have earlier shown that S II does not decrease for frequencie s
higher tha n the inverse of the mean flux; this is not our resolution .
Think : if an electron occupies a state 200 meV above the Fermi energy,
the equilibriu m probabilty of this is ~ e −7.7 . If this electron then passes
over the barrier and is lost, how long does it take to re - fill the state ?
ECE594I notes, M. Rodwell, copyrighted
High-frequency diode shot noise : UV crisis again
We had assumed equilibriu m thermodyn amics in order to obtain the probabilit y of state
occupancy. We have further assumed independen t electron emissions. Yet, we have not
calculated from equilibriu m thermodyn amics the rate at which such states would be filled
and emptied, hence we have no basis for claiming independen t events and a white spectral
density.
We would expect, from ΔEΔt ~ h, that states be repopulate d at time delays Δt ~ h / ΔE .
We would therefore expect
p tha t our derivation would fail for frequencie
q
s f > ΔE / h ≈ kT / h.
There must surely be a way to calculate the statistics of time variation of state occupancy,
and from this calculate the power spectrum of (thermal emission) shot noise.
noise
I have not seen this in the literature , and am presently uncertain how to derive it myself.
Shot noise in a forward biased diode
Now consider strong forward bias :
→ I ≅ I 0 exp( qV f / kT )
~
and S I n I n = 2 qI ms + 2 qI sm ≅ 2 qI sm = 2 qI
whi le g ≡ dI / dV = qI / kT
So, the diode is modelled as shown, with
~
S I n I n = 2 qI and g = 1 / rj = qI / kT
Hence the available power :
~
∂Pavail / ∂f = (1 / 4 g ) ⋅ S I n I n = kT / 2
The biased diode can be modelled as a resistor at half ambient te mperature .
ECE594I notes, M. Rodwell, copyrighted
Forward biased diode as a low-noise resistor
ECE594I notes, M. Rodwell, copyrighted
A forward - biased diode produces less noise than a resistor of equal conductanc e.
e
Left 2 images : real case of transimpe dance optical amplifier.
Next 2 images : simplified case of photodiode with simple load.
Final image : equivalent circuit
Rbias = 100 kΩ so that I bias = 100 μA giving rj = 26 mV/100 μA = 260 Ω
~
S I n ,r I n ,r = 4kT /(100 kΩ )
~
S I n , shot I n , shot = 2 q (100 μA) = 2kT / 260 Ω
~
Total : S I n ,tot I n ,tot ≅ 2kT / 260 Ω
This is half that produced by instead loading the photodiode in 260 Ω.
Shot noise in a reverse biased diode
Now consider strong reverse bias :
→ I ≅ I0
~
and S I n I n = 2 qI ms + 2 qI sm ≅ 2 qI 0 = 2 qI 0
whi le g ≡ dI / dV =
d
[I 0 ⋅ (exp( qV f / kT ) − 1)]
dV
Under
U
d strong
t
reverse bias
bi , g becomes
b
very small,
ll andd the
th
available noise power far exceeds kT .
ECE594I notes, M. Rodwell, copyrighted