ECE594I Notes set 8: Diffusion Noise - University of California

ECE594I notes, M. Rodwell, copyrighted

ECE594I Notes set 8:Diffusion 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


Diffusion Noise: After van der Ziel

apply.not does analysis noise diodeSchottky preceding thehence emission, nicby thermionot and diffusion,carrier minority by governed is conduction diodes, PN In

noise. diffusioncreate diffusion withassociated nsfluctuatio theand diffusion, from arise currents Average

fewamakewillweinstead;rigorouslylimitationthisaddressto)backgroundtheI(nor time thehavet don' We driven. thermallyis diffusion yet process, malinfinitesi thein

assumed is noiseShot .satisfyingfully not but short is Ziel,der vanfrom taken,derivation The

s.adjustmentfew a makewillweinstead, ;rigorouslylimitation thisaddress to)background theI(nor



blli tdi id dti dflC id ΔΔΔ

)(ilh)(d)(bdjh

).,,( counts ),,(by Index them. boxessmallintodividedtor semiconduc of lumpa Consider ΔΔΔ

lklklk

mlkzyxzyx

ons.distributi velocity thermalhave carriers These ).,,1( &),,( ionsconcentratelectronhave),,1(and),,( boxesadjacent twoThe

++

mlknmlknmlkmlk

.determined be oconstant t some is where , ),,1( :1 to)1(box from electrons ofFlux

and ,),,( :)1( tobox from electrons ofFlux

1

1

ΔΔΔ⋅+⋅=++ΔΔΔ⋅⋅=+

→+

+→

kk

kk

azyxmlknawkk

zyxmlknawkk

surebeingNot.2~thisFromt.independenllystatisticaarecrossingsboundary

electron that thei.e. noise,shot of statistics thehas and that asserts Zielder van

11

11

=→++→

kkkk

kkkk

wS

ww

be.might later whatdetermine can weif see and ,2~ writeand betsour hedge instead uslet this,of

surebeingNot .2 thisFromt.independenlly statistica are crossingsboundary

11

11

γγ⋅= +→+→

+→+→

kkkk

kkkk

wS

wS


Diffusion Noise: After van der Ziel:fluxNet

[ ]

),,1(),,(

2

11

zyxxnazyxx

xna

zyxmlknmlknawww kkkk

ΔΔ⋅Δ⋅∂∂⋅−=ΔΔΔ⋅⎥⎦

⎤⎢⎣⎡ Δ⋅∂∂

⋅−=

ΔΔΔ⋅+−⋅=−= →++→

( )

.),,1( and ),,(

so , so ,iprelationsh diffusion thehave also But we

11

2

xzymlknDw

xzymlknDw

xDazyxnDwxx

nkknkk

nn

ΔΔΔ

⋅+⋅=ΔΔΔ

⋅⋅=

Δ=ΔΔ⋅∂∂⋅−=

∂⎦⎣∂

→++→

:densities spectral noiseflux calculatenow can we,2~ From 11 wS

xx

kkkk

ΔΔΔΔ⋅=

ΔΔ

+→+→ γ

, ),,1(2~ , ),,(2~11 x

zymlknDSx

zymlknDS nkknkk

ΔΔ

⋅ΔΔΔ

⋅+⋅=⋅ΔΔΔ

⋅⋅= →++→ γγ

henceiscurrentdiffusionnettheFinally

)(4~ thereforeisflux net theofdensity spectal noise The

qwI

xzyxnDS nww

=

⋅ΔΔΔ

⋅⋅= γ

hence,iscurrent diffusionnet theFinally, qwI =

γμγ ⋅ΔΔΔ

⋅⋅=⋅ΔΔΔ

⋅⋅=x

zyxnqkTx

zyxnDqS nnII )(4)(4~ 2



carriersminorityormajoritywithworkingareewhether wstatednothaveWe

zyx

xnqR

ΔΔΔ

⋅=Δ ,)(

1 thenis boxesadjacent between resistance The carriers.majority now Consider

carriers.minority or majority withworkingaree whether wstatednot have We

μ

RkTzyxnqkTS

zyxnq

nII

n

Δ=⋅ΔΔΔ

⋅⋅=

ΔΔ

4)(4~ thereforeisdensity spectal noise theand

)(

γγμ

μ

xn

⎤⎡

Δ

that derivation noise lour therma fromknow But we

hfhf

RkT

kThfhfhf

RS

kThf

kThfII nn

⎯⎯ →⎯⎥⎤

⎢⎡

+⋅=

⎯⎯ →⎯⎥⎦

⎤⎢⎣

⎡−

+=

<<

<<

11factorcorrectionourfoundhavewehence

,41)/exp(2

*4~

γkThfkT

→⎥⎦

⎢⎣ −

+=

:found thusis termnoise diffusion The

.11)/exp(2

factor,correctionour found have wehence γ

xzyxnqkT

kThfhfhf

xzyxnqS n

kThfnII Δ

ΔΔ⋅⋅⎯⎯ →⎯⎥

⎦

⎤⎢⎣

⎡−

+⋅ΔΔΔ

⋅= << )(41)/exp(2

)(4~ μμ


Noise of a PN junction

junction.a PNinnoise)(diffusion"noiseshot"computeNow

regionsneutralquasitheinlevelFermiquasitheindropsproducesthusvoltageappliedTheregion. depletion theacross mequilibriu-near in carriers withon,approxmatiShockley theRecall

junction.a PN innoise)(diffusion noiseshot computeNow

idd l tit thit tii itthdib dthfDi tl

layer. depletion theacross drops negligible withregionsneutral-quasitheinlevel Fermi-quasitheindropsproduces thus voltageapplied The

( )m.equilibriu in ionconcentratcarrier minority theis where

,/exp)0(isedgedepletionat theionconcentratcarrier minority thediagram, band thefromDirectly

po

fpop

nkTqVnxn ==



diff il tit tth tdi d/ PNC id

region,- Pneutral-quasi theIn

diffusion.electrononionconcentratcanthat wesodiode,- / PN annow Consider +

1' '

'

n

nn

iA

nnxnADqi

∂∂⋅−−=

∂∂

∂∂⋅⋅⋅=

. lifetime ionrecombinat the andarea junction theis R

R

AxqAt

ττ ∂∂

equationsdiffusionthesetotermsnoise addnow must We


Generation / Recombination Noise

./current current generation a thermal also is There./current ionrecombinat electron an is there, a volume In

vnqIvnqIzyxv

RpoG

RR

Δ⋅⋅=Δ⋅⋅=ΔΔΔ=Δ

ττ

( ) /'/ thusiscurrent ionrecombinat-generationnet The

g

vnqvnnqIII

q

RRpoRGGR

RpoG

Δ⋅⋅−=Δ⋅−⋅−=−= ττ

( ) ( )~ then, ion)approximat noise(shot

eventst independenlly statistica of composed are processes 2 that theassume weIf

22 ( ) ( )

area, junction theis that Noting

./2'2/222 22

zyA

vnnqvnnqqIqIS RpoRpoRGII GRGR

ΔΔ=

Δ⋅+⋅=Δ⋅+⋅=+= ττ

. 2'

2~ 2 xAnn

qSR

poRII GRGR

Δ⋅⋅⎥⎦

⎤⎢⎣

⎡ +=

τ


Diffusion Noise in a PN junction

)(4)(4)(4~ is noise diffusion that thefoundearlier had We

22 ADzyDzykTS ΔΔΔΔ

current diffusion in nsfluctuatio model could we, that Given

)(4)(4)(4

diffusion

22

IxnAqDI

xxnDq

xyxnDq

xyxnqkTS

diffn

nnnII

ΔΔ⋅=

Δ⋅=

Δ⋅=

Δ⋅⋅=

δ

μ

( )

betheno ldnoisediff sionfromnsfl ct atiodensitelectrontheseofdensitspectralThe

/density electron in nsfluctuatioby driven being as 1

diffusion

xAqDIn ndiff

diffn

Δ⋅= −δδ

4~bethenwouldnoisediffusionfrom nsfluctuatiodensity electron theseofdensity spectral The

xAD

nSn

nn Δ⋅=

ion.recombinat and diffusion coupledrepresent circuit to equivalent an drawn have weonce clarified be willonsubstituti hoc-adsomewhat This



' :thuswritten-rebecanequations tranport free-noise The

AqDi

xn n=∂∂

''tnqAnqA

xi

AqDx

R

n

n

∂∂⋅−⋅−=

∂∂∂

τ

'. termsnoise theaddnow can we theseTo

n(x)AD

in n Δ+=∂∂

)('' xItnqAnqA

xi

( )AqDx

GRR

n

n

+∂∂⋅−⋅−=

∂∂∂

τ

2'2~and4~

densities spectral have terms2last These

2 xAnn

qSxnS po Δ⎥⎤

⎢⎡ +

=Δ= .2 and xAqSxAD

SR

pRII

nnn GRGR

Δ⋅⋅⎥⎦

⎢⎣

=Δ⋅=τ


PN junction: Equivalent Circuit of Diffusion & its Noise

''' nniin ∂∂∂ . )( xItnqAnqA

xin(x)

AqDi

xn

GRR

n

n

n +∂∂⋅−⋅−=

∂∂

Δ+=∂∂

τ

. 2'

2~ & 4~ 2 xAnn

qSxAD

nSR

poRIInn GRGR

Δ⋅⋅⎥⎦

⎤⎢⎣

⎡ +=Δ⋅=

τAD Rn ⎦⎣ τ

( ) /quantity physical

belowencecorrespondthe withcircuits,equivalentby drepresente be can equations These1−

⎬⎫

⎨⎧ qAqAqADIn nnn τ( )

( ) rightthewhile/onsubstitutithefromresultsdiagramleftThe

circuit equivalentq yp y

1−Δ

⎭⎬

⎩⎨

xAqDIn

CgRIVqqq

n

nnn

δδ ( )

ation. transformThevenin-Nortona wasnsustitutio thee,perspectivcircuit a Frommade. beennot had nsubsitutio that if resulted have woulddiagram

right thewhile,/onsubstitutithefrom results diagramleft The Δ⋅= xAqDIn ndiffδδ



and elements series ofnetwork ddistributea theory line-ion transmissFrom s xZ Δ

2/10 ,)/(// isimpedancesticcharacteri thewhere

,)( and )(

as wavespropagates elementsshunt

ps

xxxx

p

YZIVIVZeIeIxIeVeVxV

xYγγγγ

===

−=+=

Δ

−−++

+−−++−−+

2/1

0

analogydirectBy

.)( isconstant npropagatio theand

,)(p

ps

ps

YZγ =

2/1

)1(// where

)( and )( analogy,direct By

R

xxxx

jDACjGRInIn

eIeIxIenenxn

τ

γγγγ

=⎟⎟⎠

⎞⎜⎜⎝

⎛==

−=+=

−−++

+−−++−−+

2/1

2/12/12/1

)()1()( and

)1(

Rn

R

Rn

DjCjGR

jDqACjG

τωτωγ

ωτω

+=+=

+⎟⎠

⎜⎝ +



:esequivalencwithaboveasistionrepresentaintuitive/usefulmoreA

( )circuit equivalent

/'quantity physical

:esequivalenc withabove,as istionrepresentaintuitive / useful more A1

nnnpo

CgRIVxqAxqAqADxInnn

⎭⎬⎫

⎩⎨⎧ Δ⋅Δ⋅⋅Δ−= − τ

sideright on 2/ roughcurrent th ionrecombinat-generation of noiseshot sideleft on 2/ roughcurrent th ionrecombinat-generation of noiseshot

2

1

gr

gr

gIgI

~densityspectralwithnoisediffusionis

2)2'(

2~ and 2

)2'(2~ 21

2211 grgrgrgrn

poII

n

poII

g

SI

xAnnqqSxAnnq

qS Δ⋅+=

Δ⋅+=

ττ

s.review thi usLet

density spectral withnoise diffusion is diffdiff IIdiff SI



21 are fluxes diffusion the, and densities electron having 2 and 1 points BetweenqADqADqAD

nn

1221

211221212121

hencenoise,shothavingasmodelledareand,For

)(current totalwith and →→→→

<<

−⋅Δ

=−=⋅Δ

=⋅Δ

=

IIkThf

nnx

qADIIInx

qADInx

qADI nnn

2211

21

2

21

2

1221

1221

''beforeaswhere

),2''(2)(222~

hencenoise,shot havingasmodelledare and ,For

→→

→→

−=−=

++⋅Δ

=+⋅Δ

=+=

<<

nnnnnn

nnnxADqnn

xADqqIqIS

IIkThf

diffdiff ponn

II

2211

is 2 and 1 points between resistance that theNote

, before, as where,

Δ

nnnnnn popo

( )

21

22

2121

),(24444

hence2/)( where,−

+⋅Δ

=⋅Δ

=⋅⋅⋅Δ

=⋅Δ

=

+=Δ

=

nnADqnDqAnkTqDqkTAnqkT

RkT

nnnAnq

xR

nnnn

n

μ

μ

21

2121

of noise thermal theis ~ that again see wehence

),(

−

− ΔΔΔΔ

RS

xxkTq

xq

xR

diffdiff II

nμ



nxqAnxqA )2/()2/(

are (currents) fluxes thecurrents,r -g For the

⎫⋅Δ⋅⋅Δ⋅

n

poG

nR

n

poG

nR

xxx

nxqAInxqAI

nxqAInxqAI

ττ

ττ )2/( widthof pieces equal 2 into separating are webecuase )2/(

)2/( and )2/(

)2/(and )2/(

22

2

11

1

⎩⎨⎧

ΔΔΔ

⎪⎪⎭

⎪⎪⎬

⎫

⋅Δ⋅=

⋅Δ⋅=

Δ=

Δ=

poRG

poRG

nn

nnxqAII

nnxqAII

ττ)()2/(

and )()2/(

thereforeiscurrent ionrecombinat-generationnet The

222

111

−⋅Δ⋅=−

−⋅Δ⋅=−

⎪⎭

( ) ( )popoII

nn

xAnnqq

xAnnqqS

kThf

grgr

ττ

and 2)2'(

22)(

2~so spectra, noiseshot having as processes se treat thecan we,for but

1111

Δ⋅⋅+⋅=

Δ⋅⋅+⋅=

<<

( ) ( )n

po

n

poII

nn

xAnnqq

xAnnqqS

grgr

grgr

ττ

ττ2)2'(

22)(

2~ 1111

11

Δ⋅⋅+⋅=

Δ⋅⋅+⋅=


Impedance and Noise Analysis of Short-base Diode

( )fpo kTqVnn −= 1)/exp()0(' :edge depletionAt

DCfdcff nnnVVV

Wbn

δδ +=+=

=

)0(')0(')0('and :analysis signal Small

.0)(' :edge metalAt

( ) ( ) ffdcfpoDC

DCfdcff

qn

VdVdnnkTqVnn δδ ⋅=−=

)0('

/)0(')0(' and 1)/exp()0(':Where)()()(yg

,

,

fVkT

qnn δδ ⋅=)0()0(' :functions lexponentia of sderivative Taking



element; finite singlea as region model small Assume WxpW bb =Δ−→

22

22

2221

2'

:diagram diffusion from Working bias. forward strong toanalysisRestrict qAWfj

WqADqAWfj

WqADqAWCfj

Rg

nI b

b

nb

b

n

R

b ⋅+≅

⋅++=

⋅++=

∂∂ ππ

τπ

:admittancediode thenow write can We

./2 if holds ionapproximat theWhere 2 DW Rnb << τ

)(

22)0(')0('

''

'qAWfj

WqAD

kTqn

kTqn

nI

Vn

nI

VIY b

b

n

ffdiode

⎤⎡

⎥⎦

⎤⎢⎣

⎡ ⋅+⋅=

∂∂

=∂∂

∂∂

=∂∂

≡π

.)(!)0(')0( :at DC But,kT

qIW

qADkT

qnfY DC

b

ndiode =⎥

⎦

⎤⎢⎣

⎡⋅=→



:ttensimply wri more be can admittance diode theSo,22 IWIWI ⎤⎡

e.capacitanc diffusion and resistance junction withmodel diodefamiliar theis This

22

22

2122

diffusionjDC

n

bDC

n

bDCdiode Cfjr

kTqI

DWfj

kTqI

DWfj

kTqIY ⋅+=⋅+=⎥

⎦

⎤⎢⎣

⎡⋅+⋅= πππ

.derivation the togenerators noise theaddnow must We



( )circuitequivalent

/'quantity physical 1nnnpo

CgRIVxqAxqAqADxInnn

⎭⎬⎫

⎩⎨⎧ Δ⋅Δ⋅⋅Δ−= − τ

)2''(2~

2)2'(

2~ and 2

)2'(2~

circuitequivalent

21

2

212211

pon

II

n

poII

n

poII

nnnxADqS

xAnnqqSxAnnq

qS

CgRIV

diffdiff

grgrgrgr

++⋅Δ

=

Δ⋅+=

Δ⋅+=

⎭⎩

ττ

diagram. oncircuit -short0density carrier minority the2,point At 2 →=n

1.point at circuit -short AC an as drepresente is This.0)0(' hence ,0 Therefore . bias applied

fixeda withcurrent)(noisecurrent diodeinnsfluctuatio theanalyze Now we

, == nVV fDCf δδ



( )circuitequivalent

/'quantity physical 1nnnpo

CgRIVxqAxqAqADxInnn

⎭⎬⎫

⎩⎨⎧ Δ⋅Δ⋅⋅Δ−= − τ

)2''(2~

2)2'(

2~ and 2

)2'(2~

circuitequivalent

21

2

212211

pon

II

n

poII

n

poII

nnnxADqS

xAnnqqSxAnnq

qS

CgRIV

diffdiff

grgrgrgr

++⋅Δ

=

Δ⋅+=

Δ⋅+=

⎭⎩

ττ

)2''(22

)2'(2~~~

:current noisecircuit -Short

21

21

2

11nnn

WADqWAnnq

SSS ponbpo

IIIIII diffdiffgrgrnoisenoise++⋅+

⋅+=+=

τ

22

2~

:hence bias, forward strong in ' and 0' But,2

1

21

2

12

nWADqWAnqS

nnnW

nbII

po

bn

ii⋅+

⋅=

>>=τ

so , 2

simply iscurrent DCBut the

2

11

1

nW

qADWAqnI

W

b

nb

nDC

bnII noisenoise

⋅+⋅

=τ

τ

DCII IqSnoisenoise

⋅= 2~


Noise Model of Short-Base Diode

/:found have We

IkT

2~2/ where/

/2

DCII

nbdiffjdiffdiff

DCj

IqS

DWrC

qIkTr

noisenoise⋅=

==

=

ττ

biasforwardstrongofcaseforpeformedwasAnalysis(b). largefor derive to)! hard(but leducationa bemight It (a) b

DCII

W

qnoisenoise

pointmootaisItdiffusionermalcarrier thminorityfromandtion/recombinageneration both from arising noise theis that Note

bias.forwardstrongofcasefor peformed wasAnalysis (b)

nI

noise. or thermal noise,shot noise, diffusion thiscall whether topointmoot a isIt diffusion.ermalcarrier th-minority from and


Impedance and Noise Analysis of *Long*-base Diode

( )

)2''(2~

2)2'(

2~ and 2

)2'(2~

circuit equivalent/'quantity physical

21

2

21

1

2211

pon

II

n

poII

n

poII

nnnpo

nnnxADqS

xAnnqqSxAnnq

qS

CgRIVxqAxqAqADxInnn

diffdiff

grgrgrgr

++⋅Δ

=

Δ⋅+=

Δ⋅+=

⎭⎬⎫

⎩⎨⎧ Δ⋅Δ⋅⋅Δ−= −

ττ

τ

xΔ

equations. aldifferenti diffusion thesolvemust oneor elements,finite of #larger a useeither must one thenlonger, is region diffusion theIf

fjgfjjBfjgfjY frequency. with varies)2( where),2()2()2( :finds Helong. is analysis his solves; Zielder van

+= ππππ

DCDCdcII qIfjkTggfjgkTIIqSnoisenoise

2)2(4))2((4)2(2~0 −=−++= ππ

ECE594I Notes set 8: Diffusion Noise - University of California

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Transcript of ECE594I Notes set 8: Diffusion Noise - University of California