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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


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


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 −


Shot Noise (idealized)

)1( period theis period Time.incrementstimesmallconsider Now

tittitt

i Δ⋅+<<Δ⋅ΔΔ

i d dihii dlf#Th)1Then

Δ. periodother any in emitted electrons of # theof

tindependen iseachinemittedelectrons of # The )1t

tΔ

Δ

/

)()(electrons) ( 2)

t

kpkpkp tt ii

τ ⎞⎛Δ

== ΔΔ

yprobabilitofeachtrials)/(aretherebecause

)1(/

)1/(11

pt

ppkt tk

τ

τ τ

Δ

−⋅⎟⎟⎠

⎞⎜⎜⎝

⎛Δ= −Δ

.y probabilitofeachtrials)/(are therebecause 1pt τΔ


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 =Δ 2 variance withGaussian is σ


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 )(


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

=<<Δ−==

otherwise. 0)( , 0for 1)( :responseimpulseanis)(where),(*)()( Then

fffrrr −=Δ−=Δ , nsfluctuatio Work with

trf f Δ=Δ 2 variance withGaussian is σ



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



( )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 )()()(



: 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


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)( =


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


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 IIII SS,,

~~ ≠

resistors. in suppressed is noiseshot *how* showsargumentsimilar a resistors, small of #a vast intoresistor a Breaking


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 ±±


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 qIqISnn

22~ +=


Shot noise in a thin Schottky diode

whileVolts, 0at // :bias zeroconsider First

00 IIIVkTqIdVdIg smmsf ====≡

4422~

:So

0 kTgqIqIqISnn 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 >


High-frequency diode shot noise : UV crisis again

kTkThf

hfhfdf

dP kThfavail ⎯⎯ →⎯⎥⎦

⎤⎢⎣

⎡−

+= <<

1)/exp(2

bemust biaszeroat power noise available The

kTdPavail f illt

derived have Yet we

kTdf

avail =

be shoulddensity spectralnoiseshot either the that suggests This

sfrequencie allat

kThfkThf

kThfqIjfSII ⎥

⎦

⎤⎢⎣

⎡−

+=1)/exp(

/2

2)(~

ypgg

0

kThf >for increases econductanc bias-zero diode eor that th...



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



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


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


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


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

noise notes set7.ppt - ece.ucsb.edu · peyton z. peebles : probability, random variables, random...

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