A.J.vanGenderen,N.P.vanderMeijsandT.Smedes*
DelftUniversityofTechnology,Mekelweg4,2628CDDelft,TheNetherlands*nowwith:PhilipsSemiconductors,Gerstweg2,6534AENijmegen,TheNetherlandse-mail:arjan@cas.et.tudelft.nl,nick@cas.et.tudelft.nl,smedes@nlnmg04.seminy.philips.nlAbstract
Inthispaper,wedescribeamethodtoquicklyandac-curatelyestimatesubstratecouplingeffectsinanalogandmixeddigital/analogintegratedcircuits.Unlikenumericalmethods,thatcanbeusedforcircuitscontainingonlyafewhundredsofsubstrateterminals,thenewmethodcanquicklyextractcircuitscontainingmanythousandsofsubstrateter-minals.Examplesaregiventhatshowthatthemethodissufficientlyaccurateforpracticalcircuitverification.Themethodhasbeenimplementedinthelayout-to-circuitex-tractorSpace.
1Introduction
Inmodernanalogcircuitsandmixeddigital/analogcir-cuits,couplingeffectsviathesubstratecanbeanimportantcauseofmalfunctioningofthecircuit.Thisproblembe-comesmoreprominentas(1)thereisatrendtointegratemoreandmoredifferentcomponentsonachip,(2)thede-creaseofwirewidthandincreaseofwirelengthcausestheinterconnectparasiticsandhencethelevelofnoiseonthechiptoincrease,and(3)theuseoflowersupplyvoltagesmakesthecircuitsmoresensitivetointernalpotentialvaria-tions.
Thesubstratecouplingeffectsinintegratedcircuitscanbeverifiedbycomputingthesubstrateresistancesbetweenallcircuitspartsthatinjectnoiseintothesubstrateand/orthataresensitivetoit.Thenoiseinjectorsaremainlythecon-tactsthatconnectthesubstrateandthewellstothesupplyvoltages.Thecurrentvariationsinthesupplylinescausefluctuatingpotentialsovertheirresistancesandinductances,thatareinjectedintothesubstrateviathesubstratecontactsandthewellcontacts.Thenoisereceiversareoftenthebulkconnectionsofthetransistors.Otherpartsthatmaygener-atenoiseand/orthataresensitivetoitare(1)drain/sourceareasoftransistors,(2)on-chipresistorsandcapacitors,and(3)interconnectwiresthatarecoupledtothesubstrateviaa(large)substratecapacitance.Inthefollowing,wewillcallthepartsofthecircuitthatgeneratenoiseand/orthataresen-sitivetoit,thesubstrateterminalsofthecircuit.
Severalpublicationsalreadydescribehowsubstratecou-
plingeffectscanbeverifiedpriortothefabricationofthecircuit.In[1]and[2],a2Ddevicesimulatorisusedtosim-ulatethesubstratecouplingeffects.Thismethodallowstoinvestigatethegeneraleffectsofguard-rings,substratecon-tacts,etc.,butitisnotwellsuitedforpracticalcircuitde-sign.Moregeneralmethods,thatgeneratea3Dmeshforthesubstratetodeterminethecouplingeffects,aredescribedine.g.[3]and[4].Methodsthatusea3Dboundary-elementmethodandthatgeneratemuchfewerelementsarefoundin[5,6]and[7].However,becauseofhighmemoryusageandlongcomputationtimes,thesenumericalmethodsdonothandlecircuitscontainingmorethanafewhundredsofsub-strateterminals.
Althoughthenumericalmethodsthatarementionedabovecanadvantageouslybeusedtoverifysmallcircuitsorlocaleffectsinlargecircuits,inpractice,substratecouplingeffectsoftenoccurforrelativelylargecircuits.Hencethereisaneedformethodsthatcanquicklyestimatesubstratere-sistancesforlargecircuits.Attemptstospeed-upthecompu-tationofsubstrateresistancesarefoundin[8]and[9].In[8],parameterizedlumpedmodelsaregivenforseveraldiffer-entisolationschemesusingguard-rings.In[9],thespeed-upisobtainedbyprecomputingpoint-to-pointimpedances,whicharethenusedtofindtheadmittancematrixfortheac-tualterminalconfiguration.Alsohierarchyanddelimitationareusedin[9]toreducecomputationcomplexity.However,thelattermethodstillrequiresmatrixinversion.
Inthispaper,wedescribeanewmethodforsubstratere-sistancecomputationthatissimple,fastandgeneral,andthathasmoreoverbeenimplementedinalayout-to-circuitextractortoextractthesubstrateresistancesincombinationwiththerestofthecircuit,includinginterconnectparasitics.Theoutputoftheextractorcandirectlybeverified,e.g.us-ingacircuitsimulator.
Tosimplifythecomputationofthesubstrateresistancesandtoreducethecomplexityoftheoutputcircuit,themethodusesthenotionofa“substratenode”towhichallsubstrateterminalsareconnectedviaaresistance.Directcouplingresistancesbetweensubstrateterminalsareonlycomputedbetweenterminalsthatare“neighbors”ofeach
d
A
B
Rab
Ra
Rbsubstrate node
Figure1:Substratemodelforaconfigurationwithtwosub-strateterminals.
other.Whetherornottwoterminalsareconsideredneigh-borsisdeterminedbyaDelaunaytriangulationoftheareabetweentheterminals.Thespeed-upisfurtherobtainedbyusinginterpolationtechniquesincombinationwithresultsforstandardterminalconfigurations.
Wecomparethenewmethodwithanumericalmethodandshowthatthemethodissufficientlyaccurateforprac-ticalcircuitverification.Wealsoshowthatthemethodcanquicklyextractcircuitscontainingmanythousandsofsub-strateterminals.
Thestructureofthispaperisasfollows.First,inSection2,wedescribethemodelthatisusedtocomputesubstrateresistances.Next,inSection3,wediscusstheselectionoftheterminalpairsforwhichdirectcouplingresistancesarecomputed.Then,inSection4,wedescribethecomputationofthevaluesofthesubstrateresistances.InSection5,wepresentresultsofthemethod.Finally,inSection6,wegiveadiscussion.
2TheSubstrateModel
ThesubstratemodelthatweusetocomputethesubstrateresistancesisillustratedinFigure1.Thefigureshowstworectangularsubstrateterminals,representingsubstratecon-tactsortransistorbulkconnections,etc.
Inthesubstratemodel,wedefineacommonsubstratenodetowhichallsubstrateterminalsaredirectlyconnectedviaaresistance.Forexample,inFigure1,terminalAiscon-nectedtothesubstratenodeviaresistanceRa,andterminalBisconnectedtothesubstratenodeviaresistanceRb.Thesubstratenodeisidenticaltothereferencenodeorgroundnodethatisfoundwiththeboundary-elementmethod[5–7].Usuallythesubstratenodecanbeassumedtobepresentatinfinity.However,forsubstratesthathaveawell-conductingbottomlayerorametalbackplane,thesubstratenodeaccu-ratelyrepresentsthispartofthecircuit[1].
Thevalueoftheresistancebetweenaterminalandthesubstratenodeisprimarilydeterminedbythepropertiesofthesubstrateandthegeometryoftheterminal.
Inthesubstratemodel,aresistanceisalsocomputedbe-tweenterminalsthatare“neighbors”ofeachother(seeSec-tion3forthedefinitionof“neighbor”terminals).InFig-
ure1,suchadirectcouplingresistancehasbeencomputedforterminalAandterminalBandiscalledRab.Thedirectcouplingresistancebetweentwoterminalscarriesthecur-rentbetweenthoseterminalsthatisnotflowingviathesub-stratenode.Itsvalueisdependentonthepropertiesofthesubstrate,onthegeometriesoftheterminalsandonthepo-sitionoftheterminalswithrespecttoeachother.Thevalueofthedirectcouplingresistanceislargeiftheterminalsarefarapartanditbecomessmallerwhenthedistancebetweentheterminalsbecomessmaller.
Todemonstratethevalidityoftheabovesubstratemodel,weconsidertheconfigurationthatisshowninFigure2.Itconsistsofaheavilydopedsubstrateof300µ(resistivity0.05Ω·cm)withalightlydopedepitaxiallayerof7µ(re-sistivity15Ω·cm)grownonit.Thedimensionsofthesub-strateandtheepi-layerinhorizontaldirectionsareconsid-eredinfinite.Ontopoftheepi-layertherearetwoterminalsofsizeW×Wthatareatadistanced.Substrateresistanceshavebeencomputedforthisconfigurationusingthe3Dsub-strateresistancecomputationprogramdescribedin[7].
d
WABW15 Ohm cm
7µ
0.05 Ohm cm
300µ
Figure2:Heavilydopedsubstratewithalightlydopedepi-layerandtwoterminals.
InFigure3,theresistancebetweenthetwoterminalsinFigure2isshownasafunctionoftheirdistance,fordiffer-entsizesoftheterminals.Fromtheresultswenotethattheresistancebetweentheterminalsapproachesanasymptoticvalue—dependingonthegeometryoftheterminals—asthedistancebetweentheterminalsisincreased.Thiscanbeexplainedbythefactthatwhenthedistancebetweentheter-minalsislargecomparedtothethicknessoftheepi-layer,almostthecompletecurrentbetweentheterminalswillflowviathewell-conductingbottomlayer(seealso[1]).Theto-talresistanceisthenprimarilydeterminedbytheresistancebetweenterminalAandthebottomlayer,andtheresistancebetweenterminalBandthebottomlayer,whichcorrespondtorespectivelyresistanceRaandresistanceRbinthemodelinFigure1.
Whenthedistancedbecomessmaller,thetotalresistancebetweenterminalAandterminalBismoreandmoredeter-minedbytheresistanceofthatpartoftheepi-layerthatisbetweenterminalAandterminalB.Thisresistanceisrepre-sentedinFigure1bytheresistanceRab.
150
W=1µ
100RinkΩ
50
W=2µ
W=4µ
0
110dinµ
1001000Figure3:ResistancebetweenthetwoterminalsinFigure2asafunctionoftheirdistanced,fordifferentterminalsizes.
150
W=1µ
100RinkΩ
50
W=2µ
W=4µ
0
110dinµ
1001000Figure4:Resistancebetweentwoterminalsontopofa300µthicklightlydopedsubstrate(resistivity15Ω·cm)asafunc-tionoftheirdistanced,fordifferentterminalsizes.AlthoughthevalidityofthemodelinFigure1isintu-itivelyverifiedforsubstrateswithawell-conductingbottomlayerasshowninFigure2,itappearsthatthemodelcanalsobeusedforothertypesofsubstrates.ThisisillustratedinFigure4bycomputingtheresistancebetweentwoterminalsontopofasubstratethatissimilartothefirsttypebutthathasnoepi-layerandconsistsofonlya300µthicklightlydopedsubstrate(resistivity15Ω·cm).TheresultsshowasimilarbehaviorasinFigure3.TheresultsinFigure4areconfirmedbytheoreticalworkthatshowsthattheresistancebetweentwoterminalsontopofaconductinghalf-space,fordistancesmuchlargerthanthesizesofthecontacts,isinde-pendentofthedistancebetweenthecontacts(see[10]).
3NetworkReduction
WhenNisthenumberofsubstrateterminalsandwhenadirectcouplingresistanceiscomputedforeachpairofsub-strateterminals,thetotalnumberofdirectcouplingresis-tancesis1
Figure6:ExampleofaDelaunaytriangulation.Thetermi-nalsaredrawninsolidlines.AdirectcouplingresistanceiscomputedbetweentwoterminalsifthereisatleastonelineoftheDelaunaytriangulationthatdirectlyconnectsthem.IfweconsidertheDelaunaytriangulationasagraphinwhichtheterminalsarethenodesofthegraphandthereisanedgewheneverthereisatleastonelineconnectingtheter-minals,thenthenumberofresistancesintheoutputnetworkcanbeincreasedbycomputingadirectcouplingresistancebetweeneachpairofterminalsiftheterminalsareatadis-tance≤Linthecorrespondinggraph.
4ResistanceComputation
Thevaluesoftheresistancesinthesubstratemodelarecomputedviainterpolationbetweenknownresistanceval-uesforsomestandardconfigurations.Theresistancevaluesforthestandardconfigurationscanbeobtainedviameasure-mentonrealcircuitsor—aswedid—byusinganumericalmethod[7].
InFigure7.a,thevalueoftheconductancebetweenater-minalandthesubstratenodeisshownasfunctionoftheareaoftheterminalforahomogeneoussubstrate.InFigure7.b,thevalueofthesameconductanceisshownasafunctionoftheperimeteroftheterminal.Notethatinbothcasesthereis(approximately)alineardependencybetweentheconduc-tanceandtheareaortheperimeter.Therefore,fortheresis-tancebetweenaterminalandthesubstratenode,thefollow-inginterpolationformulaisused(seealso[1])
Rsub
1
k1
k2P
k3A
(1)
wherePistheperimeteroftheterminal,Aistheareaoftheterminal,andk1,k2andk3areempiricalfittingparametersthatareobtainedfromtheresistancevaluesofatleast3dif-ferentconfigurations.
Thedirectcouplingresistancebetweentwoterminalsasafunctionofthedistancebetweentheterminals,fordiffer-entterminalgeometries,isplottedinFigure8.Basedalsoonotherexperiments,wehavefoundthatareasonablevalueforthedirectcouplingresistancebetweentwoterminalsisob-tainedviatheinterpolationformula
Rdir
Kdp
Aa
W=1µ
10000
W=2µR
W=4µinkΩ
1000
100
125dinµ
102050Figure8:Directcouplingresistancebetweentwoterminalsasafunctionofthedistance.
3
3
4
5
24
1
12
(a)(b)
Figure9:(a)Terminalconfiguration1.(b)Terminalconfig-uration2.
Table1:Resistances(inkΩ)fortheterminalconfigurationinFigure9.aonaheavilydopedsubstratewithepi-layer(lay.=2)andonalightlydopedsubstrate(lay.=1).BEM=methodin[7],interp.=methodinthispaper.
2BEM
85.451612613653.6197
%diff.5.58.5-4.85.8-9.1-8.6interp.90.866112114350.7178
Table2:AsinTable1butnowfortheterminalconfigurationinFigure9.b
2BEM
263412191530418188531172480123
%diff.2.72.2-3.1-3.42.6-4.8-1.5-4.7-1.74.8interp.258396202547418182586167519126
VccRcRf110410RoOut6InT1T2T3R1260Rb160Re12GndFigure10:Schematic(a)andsimplifiedlayout(b)ofabipo-laramplifier.Greyareasindicatethepositionofatransistor(fromlefttorightT1,T2,T3aandT3b),blackareasindicatethepositionofasubstratecontact.
conductingbottomlayerthanforhomogeneoussubstrates.Thisisbecausewithawell-conductingbottomlayertheconductancetothesubstratenodeismoredominantcom-paredtothecouplingconductancesandhencetheerrorthatismadebyindependentlycomputingthecouplingconduc-tancesislessimportant.
AnotherexampleisshowninFigure10.Westudythehighfrequencybehaviorofabipolaramplifieronasubstrateconsistingofa14µ015Ω·cmtoplayeranda300µ4Ω·cmbottomlayer.ThecircuitinFigure10wasextractedwithoutsubstrateresistances,usingthesubstrateresistancecompu-tationmethodin[7]andusingthemethoddescribedinthispaper.Inallcases,theresultingcircuitwassimulatedus-ingSpice.ThesimulationresultsarepresentedinFigure11.Theyshowthatthesubstratecouplingeffectsthatareesti-matedusingthenewmethodarealmostidenticaltothere-sultsthatareobtainedusingthemethodin[7].
OnanHP9000/735computer,extractionoftheampli-fier,usingthemethodin[7],took3minutesand4seconds(248elementswereused).Extractiononthesamecomputer,usingthenewmethod,tooklessthan1second.Moreper-formancefiguresforthenewsubstrateresistanceextractionmethodarepresentedinTable3.
6Discussion
Inthispaper,wehavedescribedamethodtoquicklycom-puteaccuratesubstrateresistancesforlargecircuits.Prob-lemsthatarecausedbysubstratecouplingareusuallyglobalproblemsthatrequirethesimulationofthecompletecir-cuitinordertouncoverthem.Therefore,wehaveaimedat
1.5
....................................................mag.1
indB
0.5
nosub.res.BEM
.........interpolated
........................................Table3:Totalextractiontimes(onanHP9000/735)forcir-cuitshavingdifferentnumbersofsubstrateterminals.
(a)
...................pla
processormemory32814676360418135770576.427.7320.1
0.010.11frequencyinGHz
..............................................10......................................150phase100
indeg.
50
00.01(b)
...nosub.res.BEM
.........interpolated
...................0.11frequencyinGHz
10Figure11:Simulatedmagnitude(a)andphase(b)ofthetransferfunctionsoftheamplifiervs.frequency.
thedevelopmentofamethodthatcomputesinareasonableamountoftimeallrelevantsubstrateresistancesofacom-pletecircuit.
Toefficientlycomputethesubstrateresistances,themethodusesthenotionofa(virtual)substratenodetowhichallsubstrateterminalsareconnected.Itcomputesdirectcouplingresistancesonlybetweensubstrateterminalsthatareclosetoeachother.Notethatthismodelismoreorlesssimilartothemodelthatisusedtocomputecapacitancesus-inganarea/perimetermethod:Thesubstratenodeinthesub-strateresistancemodelisequivalenttothegroundnodeinthecapacitancemodeland,inanalogytothecouplingcapac-itancesbetweenwiresinthecapacitancemodel,directcou-plingresistancesbetweensubstrateterminalsareonlycom-putedbetweenneighborterminals.
Becauseofthespeedofextractionmethod,thecircuitsim-ulationthatisperformedafterwardswill,ingeneral,requiremuchmoretimethanthecomputationofthesubstratere-sistances.Hence,itbecomesmoreandmoreimportanttoinvestigateotherverificationtechniquesthatcanbeusedincombinationwiththemethodforfastsubstrateresistanceex-traction.
Acknowledgements
ThisresearchissponsoredinpartbytheDutchTechnol-ogyFoundation(STW)undergrantnr.DEL11.2450andgrantnr.DEL22.2810,andhasbeencarriedoutinthecon-textofDIMES,theDelftInstituteofMicroelectronicsandSubmicronTechnology.
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