MOS Electrostatics

8/12/2019 MOS Electrostatics

1/8

MOSDeviceModeling Electrostatics Dr.RamaKomaragiri 1of8

MOSElectrostaticsinThermalEquilibrium:AcrosssectionofMOSCinthermalequilibriumisshowninfig.1.Thesubstrateisptype.Thegateis

dopedn+

type

with

1020

cm3.

Fig.1:CrosssectionofaMOSCinthermalequilibrium

0 0 0B a d GqN XQ Q (1.1)

Where

Na

is

substrate

doping,

tox

is

oxide

thickness

1

.

In

the

analysis,

we

fix

the

origin

of

ordinate

at

the

gateoxidesubstrateinterfacesothatthegateelectrodegateoxideinterfaceisattox.Theboundary

conditionontheelectricfieldattheoxidesubstrateinterfaceresultsin

0 0( ( )) 0ox sx oE E x (1.2)

ByapplyingKirchhoffsvoltagelawtotheMOScapacitorinthermalequilibrium(showninfig.1),

,0 0 0mn ox B pmV V (1.3)

mn+andpmaretheworkfunctionsofMScontactsonthegate.Assumingthatthemetalonthegate

electrodeand

semiconductor

substrate

is

same,

the

MS

contacts

potential

drops

have

no

significance.

Solvingfortheinternaldropfromn+polysilicongatetotheptypebulk,

,0 0ox B pm pmn nV V (1.4)

Inordertoquantifythepotentialvariation0(x)andthewidthofthedepletionregionXd0,westartwithGaussslawinintegralform,whichconnectstheelectricfieldjustinsidethesilicontothedepletion

chargeQB0(C/cm2)2.Thus

1Thesubscript0indicatesthermalequilibrium.


2/8


0 00 0 =B a ds s

XQE

qN

(1.5)

Sincethereisnochargelayerattheoxidesiliconinterface,theboundaryconditionthatconnectsthe

siliconfieldtotheoxidefieldisgiveby

12

00E E

(1.6)

Usingeqn.(1.5)andeqn.(1.6)weget

0 00 0 s a d a d oxxo ox s

E qN X qN X

E

(1.7)

Thepotentialintheoxidewhere 0oxt x isgivenby

0

ox

a dox ox ox oxn

x

o

oxt

qN Xx dx x tE E x t

(1.8)

Thepotentialattheoxide/siliconinterfacecanbeevaluatedbysubstitutingx=0ineqn. (1.8).Thispo

tentialisthevalueofsurfacepotentialinthermalequilibrium.Substitutingfortheoxidecapacitance

ox ox oxC t [F/cm2]

000 = a d

o s nox

qN Xx

C (1.9)

TheresultcanbeexpressedintermsofVox,0whichisthedropacrosstheoxide.

0 0,0 0 a d G

ox onox ox

qN QXV

C C (1.10)

Toevaluatedepletionregioninthesiliconsubstrate,fromtheintegralfromofGausslaw,theelectric

fieldEo(x)inthechargedregion

00

d

x X isgivenby

0

0

x

s o s o a aqN qE E Nx dx x

(1.11)

SolvingforEo(x)andsubstitutingforEo(0+),theelectricfieldattheinterface,fromeqn.(1.5),wefind

2Inthisanalysis,weconsiderchargeperunitarea.Togetcharge,wehavetomultiplywithgateareawhichisWL


3/8


0a d

o

s

qN XE

xx

(1.12)

Integratingtheelectricfieldfromeqn. (1.12)tofindthepotentialo(x)inthechargedregion

00 dx X

0

2

002

x

ao o o d

s

E dx qN x

x x X x

(1.13)

Substitutingforo(0)fromeqn.(1.9)thepotentialinthechargedregionisgivenby

2

0

0 2

a d a

o dnox s

qN X qN xx X x

C

(1.14)

Sincethepotentialatx=Xd0is 0Xo d p ,thethermalequilibriumbuiltinpotentialacrossthepolysi

liconoxidesiliconsandwichisgivenby

2

0 0a d a d pn

ox s

qN X qN X

C

(1.15)

SolvingforXd0resultsin

20

21 1

ox pnsd ox

ox s a

CX t

q N

(1.16)

Itcanbeseenbycomparingeqn.(1.15)witheqn.(1.10)andeqn.(1.13)thefirsttermisequilibriumpo

tentialdropacrosstheoxideVox,0andthesecondtermispotentialdropacrossthechargedregionVB0.

MOSElectrostaticsunderAppliedBias:TheelectrostaticsofMOScapacitordifferalotdependingupontheappliedgatetobulkbiasVGB.

Flatband:AMOScapacitorunderflatbandisshowninfig.2.Agoodstartingpointistoapplyagatevoltagethatis

oppositetothebuiltininternalpotentialdropn+p,whichdefinestheflatbandvoltageVFB.

Accumulation:Whenthepotentialonn+polysilicongateispulledlessthanthatoftheptypebulk,leadstoanegative

chargeongateandpositivechargeattheoxidesiliconinterface.Theptypesubstratehasahighcon

centrationofmobileholesthataccumulateattheoxidesiliconinterfaceduetoattractionofpositive


4/8


chargesbynegativegatecharges.Anexcessofholesovertheacceptorconcentrationresultsinnetneg

ativechargeatthesiliconsurface. Thesurfacepotentialispulledlowerduetothesurfaceholeconcen

trationpsexceedingthebulkdopingconcentration.

ln lns asi i

p NkT kT

q n q n

(1.17)

However,thelogarithmicfunctionisweakanditisreasonableapproximationthats p

inaccumula

tion.

Fig.2:MOSCapacitorunderappliedflatbandvoltage

Depletion:ThethermalequilibriumcasewestudiedinthisexampleiswithVGB>VFB(VGB=0andVFB


5/8


TheThresholdVoltage:Astheappliedbiasincreases,thesurfacepotentialincreasesaccordingtoeqn. (1.18).Atsomepointthe

assumptionsunderlying

the

depletion

region

charge

distribution

become

invalid.

For

example,

ifwe

go

onincreasingVGB,depletionregionwidthandsurfacepotentialgoesonincreasingandthesurfacepo

tentialalsogoesonincreasing.However,thesurfacepotentialhasamaximumpotentialmeaningthat

thepotentialatthesiliconinterfaceshouldnotbemorethanthesurfacepotential.Theelectroncon

centrationincreasesaccordingto

s

skT

ien n

(1.20)

Theelectronconcentrationincreasestoapointwhereiteventuallybecomesdominantcomponentof

thenegativechargeinthesiliconsubstrate.Afterthesurfacebecomesstronglyntype,thechargeden

sityin

the

silicon

must

be

modified

to

include

the

electron

contribution

0x

a a ikT

dx q N q Nn x n e x X

(1.21)

Substitutingeqn.(1.21)intoPoissonsequationleadsintoanonlineardifferentialequationforthepo

tential(x).

Theexponentialincreaseintheelectronconcentrationwithincreasingsurfacepotentialisimportantin

makinganapproximationtothechargedensity.Wecanidentifyacriticalsurfacepotential(s)below

whichn(x)

can

be

neglected

and

the

MOS

capacitor

considered

depleted.

The

voltage

till

which

n(x)

can

beneglectediscalledonsetofinversion. Thusatonsetofinversion,thesurfacepotentialisequaltothe

potentialofthebulkptype.Thus

'

s p (1.22)

Thesurfaceelectronconcentrationat'

s p is

'

pskT kT

s i i a d e en n n N p x X

(1.23)

Inotherwords,eqn.(1.23)statesthatatthecriticalsurfacepotential,theelectronconcentrationis

equaltoacceptorconcentration,orthatthesurfaceisasmuchasntypeasthebulkisptype. Insolving

theelectrostaticsfors=p,wewillneglectns.

TheappliedgatebiasVGBatwhichtheonsetofinversionoccursisaveryimportantquantityandiscalled

asthresholdvoltageVTnforMOScapacitorsonptypesubstrates.AtonsetofinversionthepotentialdropVBacrossthedepletionregionisaknownquantity

' ' 2pB s p p pV (1.24)


6/8


Atonsetofinversion,thedepletionregionwidthhasincreasedtoitsmaximumvalueXd,max.Integration

oftheconstantchargeinthedepletionregionqNaleadsfrom0toXd,maxleadsto

2

,

' 22

ad max p

s

B

qNV X

(1.25)

SolvingforXd,max,

,2

2sd max pa

XqN

(1.26)

Thechargeinthedepletionregionattheonsetofinversionisproductofchargedensityanddepletion

width

, , 2= 2qB max a d max s a pQ qN X N (1.27)

UsingGausssintegraltofindtheelectricfieldinoxideleads

' ,' 2 2qB max ox oxox s a pox o

ox o

x

x

Q tV E

tt N

(1.28)

Sincetheinternalvoltagedropacrosstheoxideis

' 'B oxGB FB Tn FB

V VV V V V (1.29)

Or

' '

Tn F oB B xV V V V (1.30)

SubstitutingforVBandVoxfromeqn.(1.25)andeqn.(1.28)intoeqn.(1.30)yields

1

2 + 2q 2Tn FB p s a pox

V VC

N (1.31)

Theaboveexpressionissumofthreesimpleterms,thefirsttermisflatbandvoltage,thesecondtermis

dropacrossthedepletionregionandthefinaltermismagnitudeofdepletionchargeatinversiondi

videdbyoxidecapacitanceatonsetofinversion.

Inversion:Whentheappliedgatevoltageisincreasedbeyondthresholdvoltage,smallchangesinthesurfacepo

tentialleadtolargeincreasesinsurfaceelectronconcentration.Toincreasetheelectronconcentration

byanorder,thesurfacepotentialshouldbeincreasedby60mV.Thismeansthatthesurfacepotential


7/8


slowlyincreasesasVGBisraisedbuttheelectronconcentrationincreaseslinearlywithVGB.Tosimplify

theelectrostatics,wemakeanapproximationcalleddeltadepletionapproximation'

s p Tns GBforV V (1.32)

Thismeansthatthesurfacepotentialispinnedatpininversion.Theapproximationofsurfacepotentialpinnedmeanseqn.(1.20)cannotbeusedtofindtheelectronconcentration. Ourgoalistofindthe

electronchargeininversionlayerortheinversionlayerelectronchargeQN[C/cm2]asafunctionofap

pliedgatebias.WhenVGBissuchthattheMOScapacitorisininversion,electronsattheoxidesilicon

surfaceconstituteasheetchargeQN[C/cm2]thatisrepresentedbydeltafunctionattheinterface.The

depletionregionwidthisXd,maxafterinversionbecausethedropacrossremainsfixedat2p.ThepotentialdropacrosstheoxideincreasessinceVGB>VTnand

2GB FB ox pV V V (1.33)

UsingtheintegralformofGaussslawforconnectingVoxtothechargeinsiliconthatconsistsofelectron

chargeQNandthemaximumdepletioncharge'

,B maxBQ Q

,1

ox ox ox B max N

ox

V E QC

t Q (1.34)

Thereareatleasttworeasonsfordiscontinuitiesintheelectricfieldattheinterface.Thefirstone

comesfromthefactthatfielddropsbyafactor3(duetoratiosindielectricconstantratiosinsilicon

oxideto

silicon).

The

electrons

in

the

inversion

layer

from

the

charge

sheet

located

at

the

oxide

interface

donotcontributetotheelectricfield.Thusattheinterface,

,03

N B max ox oxox

s s

Q EE E

Q

(1.35)

And

,

0 0B max N

s s

Q QE E

(1.36)

Thewidthofinversionlayeris5nmsothedropinelectricfieldisnotthatabrupt.Substitutingeqn.

(1.34)intoeqn.(1.33)leads

,2 21

GB FB ox p B max N p

ox

V V V QC

Q (1.37)

Solvingtheeqn.(1.37)forinversionlayercharge,


8/8

Documents

MOS Electrostatics