ECE594I notes, M. Rodwell, copyrighted
ECE594I Notes set 7:Shot Noise
Mark RodwellUniversity of California, Santa Barbara y
[email protected] 805-893-3244, 805-893-3262 fax
ECE594I notes, M. Rodwell, copyrighted
References and Citations:
DevicesState-Solid in Noise :Zielder Van PhysicsThermal : Kroemerand Kittel
:Citations / Sources
ng.EngineeriionsCommunicatofsPrinciple:Jacobs&Wozencraftory)(introduct s PrincipleSignal RandomVariables, Randomty, Probabili: PeeblesZ. Peyton
ive)comprehens(hard, Variables Randomandity Probabil:Papoulis
1982circaStanfordHellmanMartin:noteslectureyProbabilit1982circa Stanford, Cover, Thomas :notes lecture theory nInformatio
Designc ElectroniNoise Low :erMotchenbakng.EngineeriionsCommunicatofs Principle:Jacobs & Wozencraft
t dffS t d
circuits. in Noise :Notes nsApplicatio tor LinearSemiconduc National1982circa Stanford,Hellman, Martin:notes lecturey Probabilit
design)receiver (optical Personik & Smith noise), (device by Fukui Papers Kroemer and Kittel Peebles,Jacobs, & Wozencraft Ziel,der Van
study.for references Suggested
Theory nInformatio of Elements:Williams andCover )(!Notes App. Semi. National
ECE594I notes, M. Rodwell, copyrighted
Shot Noise ( idealized)
processidealizedonfirstFocusevents. emissiont independen with
processidealized onfirst Focus
ionsapproximatsomewiththisrelatewillWetors.semiconduc inansport carrier tr to
ions,approximatsome with this,relate willWe
2.1 region from emission of y probabilita is there,period timesmallsomeinobserved that,Suppose
1
1
→pτ
/flux electron average 11pr == τ
charge electron writehere clarity we notationalfor /current DCaverage 11
qpqI
=⋅== τ
).1(for shorthanda as usenot and pq −
ECE594I notes, M. Rodwell, copyrighted
Shot Noise (idealized)
)1( period theis period Time.incrementstimesmallconsider Now
tittitt
i Δ⋅+
ECE594I notes, M. Rodwell, copyrighted
Shot Noise (idealized)
)1(1)/(herecasesonlconsidereIf t Δ
)/(varianceand)/(meanwithGaussian)( : workgsimplifyin Gaussian,a toconverge willondistributi the
)1(,1)/(wherecasesonly consider weIf
ττ
τ
tptpkp
pptp
Δ⋅Δ⋅=
−Δ⋅
)/( varianceand)/(mean withGaussian)( ττ tptpkpit
Δ⋅Δ⋅=Δ
:nsfluctuatioandvaluesmeantheseparatelyConsider
tkkEkEkk
Δ==
=−=Δ
. influx electron mean][ where flux, in nfluctuatio][
kk k =Δ2 variance withGaussian is σ
ECE594I notes, M. Rodwell, copyrighted
Shot Noise
[ ][ ]
jiji
kkE
jikkEtt2:periodtimesamefor theBut
for 0 : and steps timegdiffereninFor
σ=ΔΔ
≠=ΔΔΔΔ
[ ]
[ ]
kii
trkkE
kkE
2 )(generalinSo
:period timesame for theBut
δδσ
σ
⋅Δ⋅=⋅=ΔΔ
=ΔΔ
[ ]jiji
jijikji
jitrkkE
,,
,,
otherwise. 0 ,for 1 here w )( general in So
δδ
δδσ
===
⋅Δ⋅=⋅=ΔΔ
jijikk trttR )()(function ationautocorrel time-discrete withsequencea is This
δ⋅Δ⋅=ΔΔ jijikk trttR ,)()( δΔΔΔ
).0(limit time-continuousthetowardsmust work We →Δt )(
ECE594I notes, M. Rodwell, copyrighted
Shot Noise: Computing Power Spectrum
:follows asView
ttttf
tr
Δ
=
0db tb dil t#)( times.randomat impulses of seriesa ...clearly
second)electrons/(#boundary acrossflux electron)(
ththtrtf
ttttf
=
=Δ−==
:responseimpulseanis)(where)(*)()(Then
.0and betweenbondary crossing electrons #)(
thttthththtrtf
=
ECE594I notes, M. Rodwell, copyrighted
Shot Noise: Computing Power Spectrum
trff
ΔΔΔ
andisofvariancethe(1) because shown as bemust of function ationautocorrel The
ttrtftf
trf
Δ+ΔΔ
ΔΔ
thanlessisifonlyoverlapwindows-timeaveraging whose)( of averages-timerepresent )( and )( (2)
andis of variance the(1)
ττ
tΔthanlessis ifonly overlapwindows timeaveraging τ
ECE594I notes, M. Rodwell, copyrighted
Shot Noise: Computing Power Spectrum
( )ttrjSf ⎥⎤
⎢⎡ Δ
Δ=Δ2
2 2/sin)()(:ofdensityspectralPower ωω ( )( )( ) ttjH
ttrjSf ff
⎥⎤
⎢⎡
Δ⋅Δ
=
⎥⎦
⎢⎣ Δ⋅Δ⋅=Δ ΔΔ
)(2/sin)(:functionTransferFilter
2/)()(:ofdensity spectralPower
ωω
ωω
( )jSjHjS
tt
jH
rrff ⋅=
⎥⎦
⎢⎣
Δ⋅Δ
=
ΔΔΔΔ2 )()()( But
)(2/
)(:functionTransfer Filter
ωωω
ωω
rjHjSjS
r
ffrr
ff
==
Δ−
ΔΔΔΔ2)()()(
: ofdensity spectralPower
ωωω jjj ffrr ΔΔΔΔ )()()(
ECE594I notes, M. Rodwell, copyrighted
Shot Noise: Computing Power Spectrum
: things2 shown has ncalculatio simple This
The flux. the toequal s,frequencie allover constant spectrum,power a has process noiseshot The 1)
time.emission average theof inverse thethanhigher sfrequencieat decrease *not does* spectrum
G iifil dhh1hh, response impulse some withnoiseshot filter weIf 2)
ΔΔt
longernoisprocessthe1notwithinsteadfilterweIf
Gaussian.is processfilteredthe then, 1 that such
>>Δ
>>Δ
tr
tr
(white).frequency power with-constant remainsit but Gaussian,longernoisprocessthe1,not withinsteadfilter weIf >>Δtr
ECE594I notes, M. Rodwell, copyrighted
DC current and Noise Current
III:currents tofluxes electron fromconvert To
222
IqrqqrqjSqjS
qrqIIrqI
rI
rIDC
)()()(
22
222
====
=→==→⋅=
ωω
σσ
IqjfS 2)(~:spectra based- Hzsided-single instead Using
IqjfSI 2)( =
ECE594I notes, M. Rodwell, copyrighted
Resistors don't have shot noise
emissionst independen assumed have weNote
therebyelectrons,nearby other on forces produce fields Theseproduced. are fields electric resistors, in move electrons As
motions. their modifying
events. emissiont independen of sequencea not thereforeisflowcurrent DCduring resistors throughpassages Electron
ECE594I notes, M. Rodwell, copyrighted
Resistor shot noise ? Reductio ad absurdum
ons,manipulati dillustrate theFollowing
)mA 1(2~ :noiseshot have resistors :Assume qSsnsn II=
( ) )mA1(4~~~ So)mA 1(2~~ where2/)(
221121,
qSSS
qSSIIIsnsnsnsn
IIIIII
IIIIsnsntotSN
=+=
==+=
( ) )(So2211,,
qSSSsnsnsnsntotsntotsn IIIIII
totsntotsnsnsn IIIISS
,,
~~ ≠
resistors. in suppressed is noiseshot *how* showsargumentsimilar a resistors, small of #a vast intoresistor a Breaking
ECE594I notes, M. Rodwell, copyrighted
Shot Noise Example: Optical Receiver
noise;shot leakagedetector by limitedreceiver a Assume. a timeover filtered and amplified isnt Photocurre =Δ Tt bit
rate.data Gb/s 40 nA, 001
situation. a typicalnot
leak =I
/Hz.A)3.2(10nA) 100 (2~ noise referredInput
,
216-
leak
== qSleakI
ll kd i i )d(416
electrons 16Gb/s) 40/1(/nA) 100( =⋅=⋅Δ qrt
Gaussian.not Poisson,be willondistributi but the period,bit perelectronsleakagedeviation)std.one (mean 4 16expect We ±±
ECE594I notes, M. Rodwell, copyrighted
Shot noise in a thin Schottky diode
)/exp( analysis. eapproximat limit, c thermioniinCurrent
00 kTqVIIII fsmms ⋅==
)/exp()(y probabilit henergy witbarrier theabove excited thermally being electron from arises whichof each currents,t independen are e that thespostulate We
kTEEp ii −∝
[ ]fmssm kTqVIIII 1)/exp( analysis. eapproximat limit, c thermioniincurrent Total
0 −⋅=−=
mssm II thenemissions, electron
tindependen from driven processest independen as and both treat weIf
smmsII qIqIS nn 22~ +=
ECE594I notes, M. Rodwell, copyrighted
Shot noise in a thin Schottky diode
whileVolts, 0at // :bias zeroconsider First
00 IIIVkTqIdVdIg smmsf ====≡
4422~
:So
0 kTgqIqIqIS nn smmsII ==+=
~)4/1(/
:power available theHence
kTSgfPnnIIavail=⋅=∂∂
resistorawithparallelindiodeeconnect thtowereweifpowernoiseofflowneta have would weotherwise amics; thermodynfrom required is This
nn IIavail
?for happens what : wrongis something Yet,
resistor.a withparallel indiodeeconnect thtowere weifpower noise offlow
kThf >
ECE594I notes, M. Rodwell, copyrighted
High-frequency diode shot noise : UV crisis again
kTkThf
hfhfdf
dP kThfavail ⎯⎯ →⎯⎥⎦
⎤⎢⎣
⎡−
+= for increases econductanc bias-zero diode eor that th...
ECE594I notes, M. Rodwell, copyrighted
High-frequency diode shot noise : UV crisis again
cooling. withoperated devices THzfor important bemust yet ,literaturetheinnotednot widely isy discrepanc This
.resolutionour not is thisflux; mean theof inverse thenhigher thasfrequenciefor decreasenot does that shownearlier have We SII
?statethefill-retoit takedoeslonghowlost,isandbarrierover thepasses thenelectron thisIf .~ is thisof probabilty mequilibriu theenergy, Fermi theabove meV 200 statea occupies electron an if :Think
7.7−e?state thefillretoit takedoeslonghow lost, is andbarrier over the
ECE594I notes, M. Rodwell, copyrighted
High-frequency diode shot noise : UV crisis again
not have we Yet,emissions. electront independen assumedfurther have We occupancy. state ofy probabilit theobtain order to in amics thermodynmequilibriu assumed had We
density.spectral a white and eventst independen claimingfor basis no have wehence emptied, and
filled be wouldstates such at which rate theamics thermodynmequilibriu from calculated
.//sfrequenciefor failwouldderivationour t expect tha therefore wouldWe./~ delays at time drepopulate be states that ,~ from expect, wouldWe
hh
hh
kTEfEttE≈Δ>ΔΔΔΔ
noiseshotemission)(thermalofspectrumpowerthecalculatethisfromandoccupancy, state of variation timeof statistics thecalculate a way to besurely must There
qp f
myself.it derive how to uncertainpresently am and ,literature thein thisseennot have I
noise.shot emission)(thermalofspectrumpower thecalculate thisfrom and
ECE594I notes, M. Rodwell, copyrighted
Shot noise in a forward biased diode
)/exp( :bias forward strongconsider Now
0 kTqVII f≅→
// le whi
2222~ and
kTqIdVdIg
qIqIqIqISnn smsmmsII
=≡
=≅+=
//1 and 2~
withshown, as modelled is diode theSo,
kTqIrgqISnn jII
===
2/~)4/1(/
:power available theHence
kTSgfPnnIIavail=⋅=∂∂
.mperatureambient te halfat resistor a as modelled be can diode biased The
ECE594I notes, M. Rodwell, copyrighted
Forward biased diode as a low-noise resistor
econductancequalofresistorathannoiselessproducesdiodebiased-forwardA
circuitequivalent:imageFinalload. simple withphotodiode of case simplified :images 2Next
amplifier. optical dance transimpeof case real :images 2Left e.conductancequalofresistor a thannoiselessproduces diode biasedforward A
260A100mV/ 26 giving A100 that so k100
circuit equivalent :image Final
Ω===Ω= rIR jbiasbias μμ
260/2 A) 100(2~ )k100/(4~,,,,
Ω==Ω= kTqSkTSshotnshotnrnrn IIII
μ
. 260 in photodiode theloading insteadby produced that half is This
260/2~ :Total,,
Ω
Ω≅ kTStotntotn II
ECE594I notes, M. Rodwell, copyrighted
Shot noise in a reverse biased diode
:biasreversestrongconsiderNow
2222~ and
:biasreverse strongconsider Now
00
0
d
qIqIqIqIS
II
smmsII nn=≅+=
≅→
[ ]
thdllbbitU d
)1)/(exp(/ le whi 0 kTqVIdVddVdIg f −⋅=≡
. exceedsfar power noise available theandsmall,very becomes,bias reverse strongUnder
kTg
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