Electronic circuit reliability modeling

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1 Þ where#failuresand#testedarethenumberofactualfail- uresthatoccurredasafractionofthetotalnumberofunits subjectedtoanacceleratedtest.Theaccelerationfactor, A F ,mustbesuppliedbythemanufacturerssinceonlythey knowthefailuremechanismsthatarebeingacceleratedin thehightemperatureoperatinglife(HTOL)anditisgener- allybasedonacompanyproprietaryvariantoftheMIL- HDBK-217approachforacceleratedlifetesting.Thetrue taskofreliabilitymodeling,therefore,istochoosean appropriatevaluefor A F basedonthephysicsofthedom- inantdevicefailuremechanismsthatwouldoccurinthe Xeld. TheHTOLqualiXcationtestisusuallyperformedasthe XnalqualiXcationstepofasemiconductormanufacturing process.Thetestconsistsofstressingsomenumberof parts,usuallyabout100,foranextendedtime,usually 1000h,atanacceleratedvoltageandtemperature.Two featuressheddoubtontheaccuracyofthisprocedure. 2 J.B.Bernsteinetal./MicroelectronicsReliabilityxxx(2006)xxx 3xxx ARTICLEINPRESS Onefeatureisthelackofsu cientstatisticaldataandthe secondisthatcompaniesgenerallypresentzerofailures resultsoftheirqualiXcationtestsandhencestresstheir partsunderrelativelylowstresslevelstoguaranteezero failuresduringqualiXcationtesting. Unfortunately,withzerofailures,nostatisticaldataare acquired.Anotherfeatureistheircalculationoftheaccel- erationfactor A F .IfthequaliXcationtestresultsinzero failures,whichallowstheassumption(withonly60%con- Xdence!)thatnomorethan1/2afailureoccurredduring theacceleratedtest.Thiswouldresult,basedontheexam- pleparameters,inareportedFIT=5000/ AF ,whichcanbe almostanyvaluefromlessthan1FITtomorethan500 FIT,dependingontheconditionsandmodelusedforthe voltageandtemperatureacceleration. TheacceptedapproachformeasuringFITwould,in theory,bereasonablycorrectiftherewasonlyasingle dominantfailuremechanismthatwasexcitedequallyby eithervoltageortemperature.Forexample,electromigra- tionisknowntofollowBlack 9sequation(describedlater) andisacceleratedbyincreasedcurrentstressinaconduc- tororbyincreasingthedevicetemperature.If,however, multiplefailuremechanismsareresponsiblefordevicefail- ures,eachfailuremechanismshouldbemodeledasanindi- vidual 8 8element 9 9inthesystemandthecomponentsurvival ismodeledasthesurvivalprobabilityofallthe 8 8elements 9 9 asafunctionoftime. Ifmultiplefailuremechanisms,insteadofasinglemech- anism,areassumedtobetime-independentandindepen- dentofeachother,FIT(constantfailurerate approximation)shouldbeareasonableapproximation forrealisticXeldfailures.Undertheassumptionofmultiple failuremechanisms,eachwillbeaccelerateddi erently dependingonthephysicsthatisresponsibleforeachmech- anism.If,however,anHTOLtestisperformedatanarbi- traryvoltageandtemperatureforaccelerationbasedonly onasinglefailuremechanism,thenonlythatmechanism willbeaccelerated.Inthatinstance,whichisgenerallytrue formostdevices,thereportedFIT(especiallyonebasedon zerofailures)willbemeaninglesswithrespecttootherfail- uremechanisms. 1.3.Competingmechanismtheory 1.3.1.Multiplefailuremechanismmodel WhereasthefailureratequaliXcationhasnotimproved overtheyears,thesemiconductorindustryunderstanding ofreliabilityphysicsofsemiconductordeviceshas advancedenormously.Everyknownfailuremechanismis sowellunderstoodandtheprocessesaresotightlycon- trolledthatelectroniccomponentsaredesignedtoperform withreasonablelifeandwith nosingledominantfailure mechanism .StandardHTOLtestsgenerallyrevealmultiple failuremechanismsduringtesting,whichsuggestsalsothat nosinglefailuremechanismdominatestheFITrateinthe Xeld.Therefore,inordertomakeamoreaccuratemodel forFIT,apreferableapproximationisthatallfailures are equallylikely andtheresultingoverallfailuredistribu- tionresembles constantfailurerateprocess thatisconsis- tentwiththemil-handbook,FITrateapproach. Theaccelerationofasinglefailuremechanismisa highlynon-linearfunctionoftemperatureand/orvoltage. Thetemperatureaccelerationfactor( AF T )andvoltage accelerationfactor( AF V )canbecalculatedseparately andarethesubjectofmoststudiesofreliabilityphysics. Thetotalaccelerationfactorofthedi erentstresscombi- nationsaretheproductoftheaccelerationfactorsoftem- peratureandvoltage: AF ¼ k ð T 2 ; V 2 Þ k ð T 1 ; V 1 Þ ¼ AF T Á AF V ¼ exp E a k 1 T 1 À 1 T 2 exp ð c 1 ð V 2 À V 1 ÞÞ . ð 2 Þ Thisaccelerationfactormodeliswidelyusedastheindus- trystandardfordevicequaliXcation.However,itonly approximatesasingledielectricbreakdowntypeoffailure mechanismanddoesnotcorrectlypredicttheacceleration ofothermechanisms. Tobeevenapproximatelyaccurate,electronicdevices shouldbeconsideredtohaveseveralfailuremodesdegrad- ingsimultaneously.Eachmechanism 8competes 9withthe otherstocauseaneventualfailure.Whenmorethanone mechanismexistsinasystem,thentherelativeacceleration ofeachonemustbedeXnedandaveragedunderthe appliedcondition.Everypotentialfailuremechanism shouldbeidentiXedanditsuniqueAFshouldthenbecal- culatedforeachmechanismatgiventemperatureandvolt- agesotheFITratecanbeapproximatedforeach mechanismseparately.Then,theXnalFITisthesumof thefailureratespermechanism,asdescribedby FIT total ¼ FIT 1 þ FIT 2 þÁÁÁþ FIT i ; ð 3 Þ whereeachmechanismleadstoanexpectedfailureunitper mechanism,FIT i .Unfortunately,individualfailuremecha- nismsarenotuniformlyacceleratedbyastandardHTOL test,andthemanufacturerisforcedtomodelasingleaccel- erationfactorthatcannotbecombinedwiththeknown physicsoffailuremodels. 1.3.2.Accelerationfactor ThequaliXcationofdevicereliability,asreportedbya FITrate,mustbebasedonanaccelerationfactor,which representsthefailuremodelforthetesteddevice.Ifwe assumethatthereisnofailureanalysis(FA)ofthedevices aftertheHTOLtest,orthatthemanufacturerdoesnot reportFAresultstothecustomer,thenamodelshould bemadefortheaccelerationfactor, AF ,basedonacombi- nationofcompetingmechanisms.Thiswillbeexplainedby wayofexample.SupposetherearetwoidentiXable,con- stantratecompetingfailuremodes(assumeanexponential distribution).Onefailuremodeisacceleratedonlybytem- perature.Wedenoteitsfailurerateas k 1 ( T ).Theotherfail- uremodeisonlyacceleratedbyvoltage,andthe correspondingfailurerateisdenotedas k 2 ( V ).Byperform- J.B.Bernsteinetal./MicroelectronicsReliabilityxxx(2006)xxx 3xxx 3 ARTICLEINPRESS ingtheaccelerationtestsfortemperatureandvoltagesep- arately,wecangetthefailureratesofbothfailuremodes attheircorrespondingstressconditions.Thenwecancal- culatetheaccelerationfactorofthemechanisms.Iffor theXrstfailuremodewehave k 1 ( T 1 ), k 1 ( T 2 ),andforthe secondfailuremode,wehave k 2 ( V 1 ), k 2 ( V 2 ),thenthetem- peratureaccelerationfactoris AF T ¼ k 1 ð T 2 Þ k 1 ð T 1 Þ ; T 1 < T 2 ð 4 Þ andthevoltageaccelerationfactoris AF V ¼ k 2 ð V 2 Þ k 2 ð V 1 Þ ; V 1 < V 2 . ð 5 Þ Thesystemaccelerationfactorbetweenthestresscondi- tionsof( T 1 , V 1 )and( T 2 , V 2 )is AF ¼ k 1 ð T 2 ; V 2 Þþ k 2 ð T 2 ; V 2 Þ k 1 ð T 1 ; V 1 Þþ k 2 ð T 1 ; V 1 Þ ¼ k 1 ð T 2 Þþ k 2 ð V 2 Þ k 1 ð T 1 Þþ k 2 ð V 1 Þ . ð 6 Þ Theaboveequationcanbetransformedtothefollowing twoexpressions: AF ¼ k 1 ð T 2 Þþ k 2 ð V 2 Þ k 1 ð T 2 Þ AF T þ k 2 ð V 2 Þ AF V ð 7 Þ or AF ¼ k 1 ð T 1 Þ AF T þ k 2 ð V 1 Þ AF V k 1 ð T 1 Þþ k 2 ð V 1 Þ . ð 8 Þ ThesetwoequationscanbesimpliXedbasedondi erent assumptions. When k 1 ( T 1 )= k 2 ( V 1 )(i.e.equalprobabilityundernor- maloperatingconditions): AF ¼ AF T þ AF V 2 . ð 9 Þ Therefore,unlessthetemperatureandvoltageiscare- fullychosensothat AF T and AF V areveryclose,withina factorofabout2,thenoneaccelerationfactorwillover- whelmthefailuresattheacceleratedconditions.Similarly, when k 1 ( T 2 )= k 2 ( V 2 )(i.e.equalprobabilityduringacceler- atedtestcondition)thenAFwilltakethisform: AF ¼ 2 1 AF T þ 1 AF V ð 10 Þ andtheaccelerationfactorappliedtonormaloperating conditionswillbedominatedbytheindividualfactorwith thegreatestacceleration.Ineithersituation,theaccelerated testdoesnotaccuratelyreYectthecorrectproportionof accelerationfactorsbasedontheunderstoodphysicsof failuremechanisms. Thisdiscussioncanbegeneralizedtoincorporatesitua- tionswithmorethantwofailuremodes.Supposeadevice has n independentfailuremechanisms,and k LT FM i repre- sentsthe i thfailuremodeatacceleratedcondition, k use FM i representsthe i thfailuremodeatnormalcondition,then A F canbeexpressed.Ifthedeviceisdesigned,suchthat thefailuremodeshaveequalfrequencyofoccurrencedur- ingnormaloperating conditions : AF ¼ k use FM1 Á AF 1 þ k use FM2 Á AF 2 þÁÁÁþ k use FM n Á AF n k use FM1 þ k use FM2 þÁÁÁþ k use FM n ¼ P n i ¼ 1 AF i n . ð 11 Þ Ifthedeviceisdesigned,suchthatthefailuremodes haveequalfrequencyofoccurrenceduringthe test conditions : AF ¼ k LT FM1 þ k LT FM2 þÁÁÁþ k LT FM n k LT FM1 Á AF À 1 1 þ k LT FM2 Á AF À 1 2 þÁÁÁþ k LT FM n Á AF À 1 n ¼ n P n i ¼ 1 1 AF i . ð 12 Þ Fromtheserelations,itisclearthatonlyiftheacceleration factorsforeachmodearealmostequal,i.e., AF 1 % AF 2 ,the totalaccelerationfactorwillbe AF = AF 1 = AF 2 ,andcer- tainlynottheproductofthetwo(asiscurrentlythemodel usedbyindustry).If,however,theaccelerationofonefail- uremodeismuchgreaterthanthesecond,thestandard FITcalculation(Eq. (2) )couldbeincorrectbymanyorders ofmagnitude. Duetotheexponentialnatureoftheaccelerationfactor asafunctionof V or T ,ifonlyasingleparameterischan- ged,thenitisnotlikelyformorethanonemechanismtobe acceleratedsigniXcantlycomparedtotheothersforany given V and T .Inthenextsection,atleastfourmecha- nismsshouldbeconsidered.Also,thevariousvoltage andtemperaturedependenciesmustbeconsideredinorder tomakeareasonablereliabilitymodelforelectrondevices. Theassumptionofequalfailureprobabilityundernormal operatingconditionsisthemostconservativeandprobably themostaccurate.Infact,theexactproportionswillnot altertheresultsigniXcantlysincetheproportionalfactor isonlylinearlyrelatedtotheXnalaccelerationfactor,as comparedtotheexponentialandpower-lawfactorsof therelatedphysicsmodels. 2.MOSfailuremechanismsmodels Themajorwearoutmechanismsofsemiconductor-based micro-electronicdevicesareelectromigration(EM),gate oxidebreakdownalsoknownastimedependentdielectric breakdown(TDDB),hotcarrierinjection(HCI)andnega- tivebiastemperatureinstability(NBTI).Thesemecha- nismsarebrieYyreviewedbelow. 2.1.Electromigration Electronspassingthroughaconductortransfersomeof theirmomentumtoitsatoms.Atsu cientlyhighelectron currentdensities(greaterthan10 5 A/cm 2 [6] ),atomsmay shifttowardstheanodeside.Thematerialdepletionat thecathodesidecausescircuitdamageduetodecreased 4 J.B.Bernsteinetal./MicroelectronicsReliabilityxxx(2006)xxx 3xxx ARTICLEINPRESS electricalconductanceandeventualformationofopencir- cuitconditions.Thisiscausedbyvoidsandmicro-cracks, whichmayincreasetheconductorresistanceasthecross- sectionalareaisreduced.Increasedresistancealonemay resultindevicefailure,yet,theresultingincreaseinlocal currentdensityandtemperaturemayleadtothermalrun- awayandcatastrophicfailure [7] ,suchasanopencircuit failure.Alternatively,shortcircuitconditionsmaydevelop duetoexcessmaterialbuildupattheanode.Hillocksform wherethereisexcessmaterial,breakingtheoxidelayer, allowingtheconductortocomeincontactwithother devicefeatures.Othertypesofdamageincludewhiskers, thinning,localizedheating,andcrackingofthepassivation andinter-leveldielectrics [8] . Thisdi usiveprocess,knownaselectromigration,isstill amajorreliabilityconcerndespitevastscientiXcresearchas wellaselectricalandmaterialsengineeringe orts.Electro- migrationcanoccurinanymetalwhenhighcurrentdensi- tiesarepresent.Inparticular,theareasofgreatestconcern arethethin-Xlmmetallicinterconnectsbetweendevicefea- tures,contactsandvias [8] . 2.1.1.EMphysics Athighcurrentdensities,theforceexertedbyelectrons scatteringo thepositivelychargedmetalionsbecomes strongerthantheelectrostaticpullforcetowardthecath- ode.Thus,thedi usionoftheionsisbiasedinthedirection oftheelectronYow,leadingtoelectromigration.Itse ects areexpectedtobecharacteristicofthematerial,suchthat theactivationenergyforelectromigrationisdependenton thematerialtype,thesizeandorientationofthegrains, stress,temperatureandeventhelengthoftheconductor. Evenlowconcentrationdopingmayhavegreatimpact ontheEMfeatures.Asanexample,theEMactivation energyofbulkAlis1.4eV,whileaddingsmallamounts (0.3 35%)ofCureducesthisactivationenergybyabout 0.5 30.8eV [8] . Grainsizeandpatternalsohavesubstantialimpacton thee ectiveEMactivationenergyofthemetal.For instance,theactivationenergyrangesbetween1and2eV forthinXlmswithlargegrainsizes.ForveryXnegrained samples,theactivationenergymaybeaslowas0.4 3 0.6eV.Thus,masstransport-induceddamageismore severeatgrainboundariesandisgreatestwherethreeor moregrainsmeet.Whensmalldimensionconductorsare used,columnargrowthofthemetallowersthegrain boundarydensityandincreasestheelectromigration lifetime. Stressgradientsalsoa ectelectromigrationsincethey caninduceatomicmotionwithinthemetal.Atomsmigrate fromregionsofcompressivestresstoregionsoftensile stress.Whenaconductorisshorterthanacriticallength, L c ,knownasthe 8BlechLength 9,thestress-inducedYow ofatomscounterstheEMdrivingforceandEMiselimi- nated [9] . Temperaturegradients,causedbyhighcurrentJoule heatingalsoa ectelectromigration.Whilethesegradients mayonlyspanatemperaturechangeofsometensof degrees,thetemperaturechangeoverafewmicronsresults inlargegradients [9] .SinceEMisathermallyactivated process,thetemperaturegradientsproduceYuxdiver- gencessuchasthosefoundatcontactsorotherdevice features. Increasingly,lowresistivityCuinterconnectshavebeen madeuseinICssinceCuhasaloweratomicdi usivity thanAl.However,thesurfaceself-di usionincopper appearstobefasterthangrain-boundaryself-di usion. Thus,Cudoesnotprovidethedesiredsolutionandthereli- abilityofCuinterconnectsmaybeimprovedbysuppress- ingtheinterfaceandsurfacedi usion [10] . 2.1.2.Lifetimeprediction Modelingelectromigrationmediantimetofailure (MTTF)fromtheXrstprinciplesofthefailuremechanism isdi cult.Whiletherearemanycompetingmodels attemptingtopredicttime-to-failurefromXrstprinciples, thereisnouniversallyacceptedmodel. Currently,thefavoredmethodtopredicttimetofailure isanapproximatestatisticalonegivenbythe Black 9s equa- tion,whichdescribestheMTTFby MTTF ¼ A ð j e Þ À n exp ð E a = kT Þ ; ð 13 Þ where j e isthecurrentdensityand E a istheEMactivation energy.Failuretimesaredescribedbythelognormaldistri- bution [11] .Thesymbol A isaconstant,whichdependson anumberoffactors,includinggrainsize,linestructureand geometry,testconditions,currentdensity,thermalhistory, etc.Blackdeterminedthevalueof n toequal2.However, n ishighlydependentonresidualstressandcurrentdensity [8] anditsvalueishighlycontroversial. ArangeofvaluesfortheEMactivationenergy, E a ,of aluminum(Al)andaluminumalloysisalsoreported.The typicalvalueis E a =0.6±0.1eV.Theactivationenergy canvaryduetomechanicalstressescausedbythermal expansion.Introductionof0.5%CuinAlinterconnects mayresultin n =2.63andanactivationenergyof E a =0.95eV.Formulti-levelDamasceneCuinterconnects, anactivationenergyof E a =0.94±0.11eVata95% conXdenceinterval(CI)andavalueofthecurrent densityexponentof n =2.03±0.21(95%CI)werefound [12] . 2.1.3.Lifetimedistributionmodel Traditionally,theEMlifetimehasbeenmodeledbythe lognormaldistribution.MosttestdataappeartoXtthelog- normaldistribution,butthesedataaretypicallyforthe failuretimeofasingleconductor [13] .Throughthetesting ofover75,000Al(Cu)connectors,Galletal. [13] showed thattheelectromigrationfailuremechanismdidfollow thelognormaldistribution.ThisisvalidfortheTTFof theXrstlinkwiththeassumptionthattheXrstlinkfailure willresultindevicefailure.Thelimitationisthatalognor- maldistributionisnotscalable.Adevicewithdi erent numbersoflinksfailswithadi erentlognormaldistribu- J.B.Bernsteinetal./MicroelectronicsReliabilityxxx(2006)xxx 3xxx 5 ARTICLEINPRESS tion.Thus,ameasuredfailuredistributionisvalidonlyfor thedeviceonwhichitismeasured.Galletal.alsoshowed thattheWeibull(andthustheexponential)distributionis notavalidmodelforelectromigration. EventhoughthelognormaldistributionisthebestXtfor predictingthefailureofanindividualdeviceduetoEM, theexponentialmodelisstillapplicableformodelingEM failureinasystemofmanydeviceswherethereliabilityis determinedbytheXrstfailureofthesystem. 2.1.4.Lifetimesensitivity Thesensitivityoftheelectromigrationlifetimecanbe observedbyplottingthelifetimeasafunctionoftheinput parameters.ForEM,themostsigniXcantinputparameters correspondingtolifetimearethetemperature( T )andcur- rentdensity( j e ).Thelifetimemaybenormalizedusingan accelerationfactor. Substituting Black 9s equationandassuminganexponen- tialfailuredistributioninto A f ¼ k rated = k ð 14 Þ providestheaccelerationfactorforEM A f ; EM ¼ð j e = j e ; rated Þ À n exp ½ð E a ; EM = kT Þð 1 = T À 1 = T rated Þ? . ð 15 Þ Obviously, T hasamuchgreaterimpacton A f than j e . Asdevicefeaturescontinuetoshrinkandinterconnect currentdensitiesgrow,EMwillremainaconcern.New technologiesmayreducetheEMimpactofincreasingden- sitiesbutnewperformancerequirementsemergethat requireincreasedinterconnectreliabilityunderconditions ofdecreasedmetallizationinherentreliability [9] .Thus, EMwillremainadesignandwearoutissueinfuturesemi- conductordesigns. 2.2.Timedependentdielectricbreakdown Timedependentdielectricbreakdown(TDDB),also knownasoxidebreakdown,isasourceofsigniXcantreli- abilityconcern.Whenasu cientlyhighelectricXeldis appliedacrossthedielectricgateofatransistor,continued degradationofthematerialresultsintheformationofcon- ductivepaths,whichmayshorttheanodeandcathode [14] . Thisprocesswillbeacceleratedasthethicknessofthegate oxidedecreaseswithcontinueddevicedown-scaling. TheTDDBprocesstakesplaceintwostages [15] .Inthe Xrststage,theoxideisdamagedbythelocalizedholeand bulkelectrontrappingwithinitandatitsinterfaces.The secondstageisreachedwhentheincreasingdensityoftraps withintheoxideformapercolation(conduction)path throughtheoxide.Thisshortcircuitbetweenthesubstrate andgateelectroderesultsinoxidefailure.Thisprocesshas beensuccessfullymodeledusingMonteCarlosimulations. Theformationofapercolationpathmayresultinoneof twotypesoffailure.Onceaconductionpathforms,current Yowsthroughthepathcausingasuddenenergyburst, whichmaycauserunawaythermalheating.Theresult maybea softbreakdown ifthedevicecontinuestofunction. Localmeltingoftheoxidewilldestroythegateandisthus denotedas hardbreakdown .Ithasbeenspeculatedthatsoft breakdowndoesnotevensigniXcantlya ecttransistor operation,althoughitmaystillleadtothefailureofshort channeldevices.Whilethechangeinboththresholdvolt- ageandleakagefromsoftbreakdownissmallandinitially doesnota ectdeviceoperation,thee ectsarecumulative. Multiplesoftbreakdownsmayresultinanincreaseinleak- agecurrenttounacceptablelevels [16] . 2.2.1.TDDBphysics Trapgenerationisthekeyfactordeterminingoxidedeg- radationandbreakdown.Threegeneralmodelsaredis- cussedintheliteraturefortrapgeneration.Thesemodels arethe 8 8anodeholeinjection 9 9(AHI)model,the 8 8thermo-chemical 9 9model,andthe 8 8anodehydrogen release 9 9(AHR)model. TheAHImodel(1/ E model)wasproposedbySchuegraf andHu [17] andstudiedbymanyresearchers.In thismodel,electronsinjectedfromthegatemetalcathode intotheoxideundergoimpactionizationevents,which generateholesintheprocess.Someoftheseholestunnel backintothecathodeandcreateelectrontrapsinthe oxide.Thephysicsofthetrapcreationprocessisstillspec- ulative. Thethermo-chemicalmodel( E model)isanotherwidely citeddielectricbreakdownmodel.McPhersonandMogul [18] reviewedthedevelopmentofthismodelandproposed aphysicalexplanation.Thismodelproposesthatdefect generationisaXeld-drivenprocessandthecurrentYowing throughtheoxideplaysasecondaryroleatmost.The interactionoftheappliedelectricXeldwiththedipole momentsassociatedwithoxygenvacanciesleadstoacon- ductionsub-bandformationandtosevereJouleheatingat thestageofoxidebreakdown. IntheAHRmodel,theenergyreleaseoftheincoming electronsattheanodemayactivatehydrogenreleaseat theanode,besidescreatingholes.Thereleasedhydrogen di usesthroughtheoxideandcangenerateelectrontraps. TherehavebeencontradictingopinionsontheexactXeld accelerationlawoftime-to-breakdown 3 t BD .According totheAHImodel(1/ E model)theXelddependenceof the t BD takestheform: t BD ð t Þ¼ s 0 exp G E OX ; ð 16 Þ where E OX istheelectricXeldacrossthedielectricand s 0 and G areconstants. Accordingtothethermo-chemicalmodel( E model)the Xelddependenceofthe t BD isoftheform: t BD ð t Þ¼ t 0 exp ðÀ c E OX Þ ; ð 17 Þ where t 0 and c areconstants. Thereisstillnoconsensusonthecorrectacceleration lawandthediscussionofthe E and1/ E modelscontinues. 6 J.B.Bernsteinetal./MicroelectronicsReliabilityxxx(2006)xxx 3xxx ARTICLEINPRESS Thedebateabout E vs.1/ E modelsismostapplicablefor thickoxides.Forultra-thinoxidesevidenceshowsthatgate voltageistheprimarydriverofthebreakdownprocess [19] . Additionally,thereisevidencethatthetemperature dependenceofultra-thinoxidesisnon-Arrhenius,but ratherthetemperatureaccelerationfactorislargerat highertemperatures.Toaccountfortheseobservations, Wuetal. [19] haveproposedarelationshipintheform of MTTF ¼ T BD0 ð V Þ exp ð a ð V Þ = T þ b ð V Þ = T 2 Þ ; ð 18 Þ where T BD0 ð V Þ , a and b arevoltagedependentfactors.The secondorderterm, b / T 2 ,isincludedinordertoaccountfor anynon-Arrheniustemperaturee ects. 2.2.2.Lifetimedistributionmodel Thetime-to-breakdown( t BD )isastatisticallydistrib- utedparameter.AthighXelds,averywidedistributionof breakdowntimesisfound.Itiscommonlyassumedtobe distributedaccordingtotheWeibullstatistics,whichistyp- icalfor 8weakestlinkprocesses 9.Thecumulativedistribu- tionfunctioncanthenbedescribedas F ð t Þ¼ 1 À exp À t g b "# ; ð 19 Þ where b istheshapefactorofthedistribution,oftencalled theWeibullslope,and g isascalefactor.Eq. (19) ,canbe rearrangedsuchthat ln ½À ln ð 1 À F ð t ÞÞ?¼ b ln ð t ÞÀ b ln ð g Þ ; ð 20 Þ whichimpliesthataplotofln[ À ln(1 À F )]asafunctionof thelogarithmof t yieldsastraightlinewithaslopeof b . Lognormaldistributionhasalsobeenusedtoanalyze acceleratedtestdataofdielectricbreakdown.Althoughit mayXtfailuredataoveralimitedsampleset,ithasbeen demonstratedthattheWeibulldistributionmoreaccu- ratelyXtslargenumbersofTDDBfailures [20] .Animpor- tantdisadvantageoflognormaldistributionisthatitdoes notpredicttheobservedareadependenceof t BD forultra- thingateoxides. 2.2.3.Thebreakdownevent Thebreakdowneventitselfisusuallydescribedusinga 8 8weakestlink 9 9model.Gateoxidefailureisaweakestlink typeofproblembecausethewholechipfailsifanyone devicefails,andadevicefailsifanysmallportionofthe gateareaofthedevicebreaksdown. TheXrst 8weakestlink 9modelwasformulatedbySune etal. [21] intheearly1990sanddescribedoxidebreakdown anddefectgenerationviaaPoissonprocess.Inthismodel, acapacitorisdividedintoalargenumberofsmallcells.It isassumedthatduringoxidestressing,neutralelectron trapsaregeneratedatrandompositionsonthecapacitor area.Thenumberoftrapsineachcelliscounted.Once thenumberoftrapsinacellreachesacriticalvalue,break- downwilloccur. ThedisadvantageofSune 9smodelisitstwo-dimensional nature.Anewthree-dimensionalmodel,basedontheper- colationconcepthasbeensuggestedinRef. [22] andhas beenthoroughlyelaboratedinRefs. [23,24] .Themodel assumesthatelectrontrapsaregeneratedinsidetheoxide atrandompositionsinspace.Aroundthesetraps,asphere isdeXnedwithaXxedradius r ,whichistheonlyparameter ofthemodel.Ifthespheresoftwoneighboringtrapsover- lap,conductionbetweenthesetrapsbecomespossibleby deXnition. Thismechanismoftrapgenerationcontinuesuntilacon- ductingpathiscreatedfromoneinterfacetotheother, whichdeXnesthebreakdowncondition.Thepercolation modelforoxidebreakdownisabletoquantitativelyexplain twoimportantexperimentalobservations:(i)astheoxide thicknessdecreases,thedensityofoxidetrapsneededto triggerbreakdowndecreases [23 325] ,and(ii)astheoxide thicknessdecreases,theWeibullslopeofthebreakdowndis- tributiondecreasesandapproachesunity,i.e.alarger spreadofthe t BD -valuesisobserved [23,24,26,27] . Oxidethicknesswillcontinuetobescaledinfuture devicesbecauseoftheneedtoimproveandoptimizecircuit performance.Thee ectofTDDBinthecaseofultrathin oxidesisstillarguable.Thecontrastingviewpointsindicate theneedforbetterunderstandingofTDDBasdevicefea- turesshrink. 2.3.Hotcarrierinjection Hotcarriersinthesemiconductordevicearethecauseof adistinctwearoutmechanism,thehotcarrierinjection (HCI).Hotcarriersareproducedwhenthesource 3drain currentYowingthroughthechannelattainshighenergy beyondthelatticetemperature.Someofthesehotcarriers gainsu cientenergytobeinjectedintothegateoxide, resultinginchargetrapandinterfacestategeneration.The lattermayleadtoshiftsintheperformancecharacteristics ofthedevice,e.g.,thethresholdvoltage,transconductance, orsaturationcurrent,andeventuallytoitsdegradation. Therateofhotcarrierinjectionisdirectlyrelatedtothe channellength,oxidethicknessandtheoperatingvoltage ofthedevice.Sincethelatterareminimizedforoptimal performance,thescalinghasnotkeptpacewiththereduc- tioninchannellength.Currentdensitieshavebeen increasedwithacorrespondingincreaseindevicesuscepti- bilitytohotcarriere ects. 2.3.1.HCIphysics Hotcarriersaregeneratedduringtheoperationofsemi- conductordevicesasitswitchstates.Ascarrierstravel throughthechannelfromsourcetodrain,thelateralelec- tricXeldnearthedrainjunctioncausescarrierstobecome hot [28] .Asmallpartofthesehotcarriersgainsu cient energy 4higherthantheSi 3SiO 2 energybarrierofabout 3.7eV 4tobeinjectedintothegateoxide.InnMOS(neg- ative-channelmetal-oxidesemiconductor)devices,hotelec- tronsaregeneratedwhilehotholesareproducedinpMOS J.B.Bernsteinetal./MicroelectronicsReliabilityxxx(2006)xxx 3xxx 7 ARTICLEINPRESS (positive-channelmetal-oxidesemiconductor)devices. Injectionofeithercarrierresultsinthreeprimarytypesof damage:trappingofelectronsorholesinpre-existingtraps, generationofnewtraps,andthegenerationofinterface traps [29] .ThesetrapsmaybeclassiXedbylocation [30] whiletheire ectsvary. InterfacetrapsarelocatedatorneartheSi 3SiO 2 inter- faceanddirectlya ecttransconductance,leakagecurrent andnoiselevel.Oxidetrapsarelocatedfurtherawayfrom theinterfaceanda ectthelongtermMOSFETstability, speciXcallythethresholdvoltage.E ectsofdefectgenera- tionincludethresholdvoltageshifts,transconductancedeg- radationanddraincurrentreduction [28] .NBTIseemsto havesimilardegradationpatterns,exceptforpMOS,so bothwillbetreatedsimilarlyhere. Huetal. [31] proposedthe 8lucky 9electronmodelforhot carriere ects.Thisisaprobabilisticmodelproposingthat acarriermustXrstgainenoughkineticenergytobecome 8hot 9,andthenthecarriermomentummustbecomeredi- rectedperpendicularlysothecarriercanentertheoxide. Thecurrentacrossthegateisdenotedby i gate andduring normaloperationitsvalueisnegligible.Degradationdue tohotcarriersisproportionalto i gate ,makingthelattera goodmonitoroftheformer.Iftherateofchangeofthe HCI-inducesdamage,designatedby D ,isproportionalto i gate ,then d D = d t $ i gate ¼ A ð D Þ = W Á i drain Áð i sub = i drain Þ m ; ð 21 Þ where W isthewidthoftheMOSFET.Byletting B = A ( D )/ W ,andknowingthatMTTFdependsonthere- ciprocalofd D /d t ,thefailurerateisfoundfrom k ¼ B Á i drain Áð i sub = i drain Þ m . ð 22 Þ Thisequationassumesstatic(dc)voltagesandcurrents.To accountfordynamicdegradation k hastobeintegrated overafullcycletime. Temperatureplaysaninteresting,thoughsmallrolein hotcarrierinjection,sincetheactivationenergyisnegative, implyingthatHCIdiminisheswithincreasingtemperature. Atlowtemperatures,thesubstratecurrentincreases becausethedraincurrentincreases.AccordingtoAcovic etal. [32] ,thee ectsofoxidedegradationarestrongerat lowtemperaturesbecausetheelectrons,havinglowerther- malenergy,aremoreconXnedwithinthenegatively chargeddegradedzone.Anotherpossibilityisthatfreeze- outofimpuritiesinthedrainatlowtemperaturesmakes n-MOSFETsmoresensitivetoelectronstrappedinthe drainregion,increasingdegradation.Degradation decreasesathightemperaturesbecausethedraincurrent andthemeanfreepathdecrease. 2.3.2.Lifetimeprediction Theluckyelectronmodeldoesnotfullypredicthotcar- rierinjectionlifetime.Sincethereisnodirectmethodof measuringdevicelifetime,theArrheniusrelationship remainsafavoredlifetimepredictiontool.Thefollowing modelsarefromJEP-122A [14] .Itcontainstwomodels. The N-Channel modelisfornMOSdevices.Inthese devicesthesubstratecurrentisanindicatorofhotcarriers. TheMTTFequationis MTTF ¼ B ð i sub ÞÀ N exp ð E a = kT Þ ; ð 23 Þ where B isascalefactor,whichisafunctionofdopingpro- Xles,sidewallspacing,dimensions,etc., i sub isthesubstrate current, N rangesfrom2to4,and E a istheactivationen- ergyintherangeof À 0.1eVto À 0.2eV. InpMOSdevices,hotholesdonotshowupassubstrate current.However,thegatecurrentcanserveasanindica- torofhotcarriers.Thusthe P - Channel modelis MTTF ¼ B ð i gate Þ À M exp ð E a = kT Þ ; ð 24 Þ where B and E a arethesameasbeforewhile i gate isthepeak gatecurrentduringstressingand M rangesfrom2to4. However,theArrheniustermisnotnecessarilyappropriate forthesemechanisms. 2.3.3.Lifetimedistributionmodel Thereislittlediscussioninliteratureaboutapropersta- tisticallifetimedistributionmodelforhotcarrierinjection. Alogicalhypothesisforthelifetimedistributionwouldbe theexponentialone.Thisisagoodassumptionbecauseas adevicebecomesmorecomplex,withmillionsofgates,it maybeconsideredasasystem. Thefailureprobabilityofeachindividualgateisnot mostlikelyanexponentialdistribution.However,the cumulativee ectofearlyfailuresandprocessvariability, ensuringeachgatehasadi erentfailurerate,widensthe spreadofthedevicefailures.Theendresultisthatintrinsic hotcarrierinjectionbecomesstatisticallymorerandomas thefailuresoccurataconstantrate. 2.3.4.Lifetimesensitivity AsforEM,hotcarrierinjectionlifetimeissensitiveto changesintheinputparameters.Theaccelerationfactor forhotcarrierinjectionis A F ; HCD ¼ exp ð B ð 1 = V dd À 1 = V dd ; max ÞÞ . ð 25 Þ HCIcontinuestobeareliabilityconcernasdevicefeature sizesshrink.HCIisafunctionofinternalelectricXeldsin thedeviceandassuchisa ectedbychannellength,oxide thicknessanddeviceoperatingvoltage.Shorterchannel lengthsdecreasereliabilitybuttheoxidethicknessand thevoltagemayalsobereducedtohelpalleviatethereduc- tioninreliability.Anotherwayofimprovinghotcarrier reliabilitymaybebyshiftingthepositionofthemaximum drainsoitisdeeperinthechannel [32] .Thiswouldresultin hotcarriersbeinggeneratedfurtherawayfromthegate andSi 3SiO 2 interface,reducingthelikelihoodofinjection intothegate.Anothermethodistoreducethesubstrate currentbyusingalightlydopeddrain(LDD)wherepart ofthevoltagedropisacrossalightlydopeddrainextension notcoveredbythegate.AnnealingtheoxidesinNH 3 ,N 2 O orNOorgrowingthemdirectlyinN 2 OorNOimproves 8 J.B.Bernsteinetal./MicroelectronicsReliabilityxxx(2006)xxx 3xxx ARTICLEINPRESS theirresistancetointerfacestategenerationbythehot carriers. 2.4.Negativebiastemperatureinstability NBTIdi ersfromhotcarrierinjectioninthatNBTI causesashiftinthedevicethresholdvoltage.Themecha- nismforNBTIdamageareholestrappedwithintheinter- facebetweentheSiO 2 gateinsulatorandtheSisubstrate. NBTIdamageismostprevalentinp-MOSFETdevices whereholesarethermallyactivatedandgainsu cient energytodisassociatetheinterface/oxidedefectsnearthe lightlydopeddrain(LDD)regions.Thishappensatthe LDDregionsbecauseofthehigherholeconcentrations nearthegateedge. NBTImainlyoccursinp-channelMOSdevicesstressed withnegativegatevoltagesatelevatedtemperatures.It appearstobenegligibleforpositivegatevoltageandfor eitherpositiveornegativegatevoltagesinn-MOSFETs. NBTImanifestsitselfasdecreaseinabsolutedraincurrent I Dsat andtransconductance g m whiletheabsolute 8 8o 9 9cur- rent I o andthresholdvoltage V th increase. ThetypicalstressconditionsofNBTIaretemperatures of100 3250 ° CandoxideelectricXeldsbelow6MV/cm. Thesestressconditionsaretypicalduringburn-in.The thresholdvoltage V th andYat-bandvoltage V FB ofaMOS- FETaregivenby V th ¼ V FB À 2 / F À j Q B j C ox ; ð 26 Þ V FB ¼ / MS À Q f C ox À Q it ð / s Þ C ox . ð 27 Þ TheXxedoxidecharge Q f andtheinterface-trappedcharge density Q it arethetwofactorsdeterminingthethreshold voltageshift.Positiveincreasesofthesetwoparameters leadtoanegativethresholdvoltageshift: D V th ¼À D Q it ð / s Þ C ox À D Q f C ox . ð 28 Þ DuringtheNBTIdegradation,thethresholdvoltage shiftstomorenegativedirection,a ectingeithertheinter- facetrapsortheXxedoxidecharges. Thesimplestformoftheon-statedrivingcurrent I Dsat andtransconductance g m ofaMOSFETisgivenby I Dsat ¼ W 2 L l eff C ox ð V gs À V th Þ 2 ; ð 29 Þ g m ¼ W L l eff C ox ð V gs À V th Þ . ð 30 Þ Theseequationsshowthattheparametersleadingto I Dsat and g m degradationarethethresholdvoltageandthe mobility l e .Themobilitydegradationismostlyinduced byinterfacetrapgeneration,leadingtoadditionalsur- face-relatedscattering. 2.4.1.NBTIphysics SinceXrstobservedbyDealetal. [33] in1967,NBTIhas beenintensivelyinvestigatedandmanymodelsforitsphys- icalmechanismshavebeenproposed.Thethreemost prominentmodelsintheliteraturefeatureholesinjection intotheoxide,electrontunnelingandelectrochemical reactions. Thehole-trappingmodelisbasedonavalanchehole injectionmeasurementsofunstressedMOScapacitors andNBTItests [34 336] .Thismodelproposesthattheneg- ativemidgapvoltageshift(whichisbelievedtobeamon- itorofthepositiveoxidechargewithnocontribution frominterfacestates),isduetopopulationofintrinsichole traps.Allthepositivechargegeneratedbyprecedingnega- tivebiasstressescanberemovedbythepositivebiasstress. However,theexactmechanismforholeinjectionintothe oxideisstillunknown. Thethermallyassistedelectrontunnelingmodelwas establishedbyBreed [37,38] .Accordingtothismodel,the neutralorpositivecenters,whicharethecharge traps,arelocatedneartheoxideinterface.Undernegative biasstress,thecentersareexcited.Theelectronstunnel fromtheexcitedstatesintoemptystatesoftheSiconduc- tionband.Thisisathermallyassistedtunnelingprocess. Severalauthorsproposedtheelectrochemicalreaction modelorthereaction-di usionmodel,whichhasrecently beenacceptedbymanyresearchers [39 342] .Thismodel explainstheNBTIe ectintermsofelectrochemical reactions. NBTIbecameevidentwiththeadventof0.13 l m processesasdevicesrequiredmuchthinnergateoxides andintroducednitridesintheSiO 2 topreventboron penetrationintothegate.Anothersourceofconcernis plasma-induceddamageduringinterconnectdeposition resultingindrivinghydrogenatomsintotheSi 3SiO 2 interface. 2.4.2.Lifetimeprediction Generally,athresholdvoltageshift D V th causedby NBTIcanbeexpressedas D V th ¼ Af 1 ð t Þ f 2 ð V g Þ exp À E a kT . ð 31 Þ Here f 1 and f 2 arefunctionsaccountforthetimedepen- denceandgatevoltagedependence.Basedonthephysical mechanismsandexperimentaldata,severalmodelsforthe timedependencehavebeensuggested. 2.4.2.1.Logarithmictimedependence D V th ¼ A log ð t Þ . ð 32 Þ Thismodelwasestablishedontheidealofchargetrapping, whereincarrierstunnelintoexistingtraps [33] .According tothismodel,theNBTIisXeldaccelerated,andthereislit- tleornotemperatureactivation.Thesaturationbehavioris duetotheXnitetrapdensity.ThereissigniXcantdeviation atlongtimewhenusingthismodel.However,itisfre- quentlyobservedinrecenthigh- k gatedielectricsexperi- ments. J.B.Bernsteinetal./MicroelectronicsReliabilityxxx(2006)xxx 3xxx 9 ARTICLEINPRESS 2.4.2.2.Exponentialtimedependence D V th ¼ A exp t s \x2\x3 ð 33 Þ or D V th ¼ A exp t s 1 þ B exp t s 2 . ð 34 Þ Asingleexponentialtimedependencemodelwasestab- lishedatXrstbasedontheXrstorderreaction,whichwas limitedbytheholeconcentration [39] .Di erentfromthe logarithmictimedependence,thismodelsuggestedthetem- peratureactivationfromthereactions.Later,atwo-expo- nentialmodelwasfurthersuggested [43,44] . 2.4.2.3.Powerlawtimedependence. D V th ¼ At n . ð 35 Þ Thepowerlawmodelwasbasedonthereactiondi usion mechanism.Accordingtothismodel,thehydrogenproXle determinesthetimedependency [40,41] .Thetemperature dependencearisesfromthereactionanddi usionprocesses andthesaturationbehaviorcomesfromthedi usionbar- rierortheavailableSi 3Hbonds.Comparedwiththeother twomodels,thismodelhasthemostobservedfeaturesand iswidelyaccepted. 3.Reviewofexistingreliabilitysimulationtools ICreliabilitysimulationisnotanewconceptanda numberofreliabilitymodelsandsimulationmethodologies suchasBERT [45] andARET [46] havebeendeveloped duringthepastdecade.Moststate-of-the-artreliability simulationmethodstrytoemulatethedegradationprocess ofageddevicesinarepetitivescheme.Theyarebasedon thephysicalfailuremechanismsandcontainthemajor wearoutmodelsforEM,HCI,NBTIandTDDB.Aset ofparametersforeachofthesefailuremechanismsare identiXedandthealgorithmsofextractingtheseparameters foragiventechnologyaredevelopedbyacceleratedtests onteststructures.Acircuitsimulator,suchasSPICE,is employedtocalculatetheelectricalparametersoffresh anddegradeddevicestopredicttheirdegradationorfailure fromtheseparameters. Thisreliabilitysimulationmethodcanhelpdesigners understandhowthedevicesdegradeovertime,identify thereliabilitybottleneckswithinthecircuitsandmake designtradeo sbetweenperformanceandreliabilityin theproductdesignstage.Itcanalsohelpmanufacturers buildtheircircuitssuchthatnoknownwearoutmechanism willdominateoverthelifeofanoperatingdeviceand assureadequatereliabilityfortheproduct.Inwhatfollows, twocommercialstate-of-the-artreliabilitysimulation methodsarereviewed,thenasetoffailureequivalentcir- cuitsforthemostimportantintrinsicsiliconwearoutmech- anismsincludingHCI,TDDBandNBTIarereviewed. Finally,inthenextsection,anewfailurerate-basedSPICE reliabilitysimulationmethodologyisintroducedtoinvesti- gateproductreliabilityindi erentways. 3.1.Degradation-basedreliabilitysimulationtools Hotcarrierreliabilitysimulationmodelsandmethods havebeenimplementedandwidelyusedinthesemiconduc- torindustryformanyyears.Tosomeextent,theaccuracy ofhotcarrierreliabilitysimulationrepresentstherobust- nessande ciencyoftheentirereliabilitysimulator,there- fore,forthepurposeofsimplicity,HCIsimulationis employedasthevehicletodeliverthebasicconceptsand Yowsrealizedinsomecommercialdegradation-basedreli- abilitysimulationmethods. 3.1.1.HotcarrierreliabilitysimulationinVirtuosoUltraSim VirtuosoUltraSimistheCadenceFastSPICEcircuit simulatorcapableofpredictingandvalidatingtiming, powerandreliabilityofmixed-signal,complexdigitaland System-on-Chip(SoC)designsinadvancedtechnologyof 0.13 l mandbelow.Ithasasetofspecializedreliability models(AgeMos)forHCDandNBTIsimulation [47] .In thesimulation,anAgeparameteriscalculatedforeach nMOSdevicewiththefollowingformula: Age ð s Þ¼ Z t ¼ s t ¼ 0 I sub I ds m I ds WH d t ; ð 36 Þ where W referstothewidthofthetransistor; m and H are technologydependentparametersanddeterminedfrom experiments; I sub isthesubstratecurrent; I ds isthedrain current; s isthetimeforstress.ForpMOSdevices,thegate current I gate isusedinsteadof I sub todeterminetheAge parameter.ThedegreeofMOSdevicedegradationhas beenexperimentallyfoundtobeafunctionofthis Age parameterforwiderangesofchannellengthandstresscon- ditionsandtherelationshiphasaplausibletheoreticalbasis [48] . Thesimulationstartswithdeviceparameterextraction andmodeling.FromtheSPICEmodelparametersoffresh devices,someotherdeviceparametersareaddedtoaccu- ratelymodel I sub .Saturationcurrent I dsat ,thresholdvolt- age, V th ,orthemaximumtransconductance, g max ,canbe usedasadegradationmonitoringparameter. I dsat isagood degradationmonitorfordigitalcircuits,while V th issuit- ableforanalogapplications.Normally,thestresstime resultingin10%decreaseofoneofthesedegradationmon- itoringparametersisarbitrarilysetasthedevicelifetime. TheXnalstepis AgeMos extraction.Basedonthe Age parametercalculatedafterthefreshsimulation,the Age- Mos appliesthedegradationmodels,whichcanbeinput tomostSPICE-likesimulators,fortheagedcircuitsimula- tion.ReliabilitysimulationwithVirtuosoUltraSimisan iterativeprocess,inwhichseveraliterationsareoften neededinordertogetaccuratemodeling.Thesimulation cancalculateandoutputthedegradationresultstopredict thelifetimeofeachMOSinstance [49] .Theoverallsimula- tionYowisdepictedin Fig.1 . 10 J.B.Bernsteinetal./MicroelectronicsReliabilityxxx(2006)xxx 3xxx ARTICLEINPRESS Thefundamentalmodelsandmethodologyofreliability simulationwasXrstproposedandimplementedinBERT (BerkeleyReliabilityTools),introducedbyChenmingHu in1992 [48] .Thisapproachtoreliabilityequivalentcircuits wascommerciallyrealizedinVirtuosoUltraSimandsimi- larlyinELDO,thesimulatordevelopedbyMentorGraph- ics.ThemainadvantagesoftheBERTsimulation methodologiesareaccuracyandSPICEmodelingtechnol- ogycompatibility.However,itimposesaburdenon designerstocorrectlyextractthedevicefreshanddegraded parametersandleadstonon-physicaltrends,whichpre- ventsitspopularityinreliabilitydesignprocess.Thesetools areveryimportantfortheICdesigners,butonceacircuitis produced,noYexibilityremainstoalterthereliabilityif anewapplicationorsetofoperatingparametersis applied. 3.1.2.HotcarrierreliabilitysimulationinEldo EldoandUltraSimbothdeliverallthecapabilityand accuracyofSPICE-levelsimulationforcomplexanalogcir- cuitsandSoCdesigns.Howevertherearesomesubtledif- ferences.InEldo,thesubstratecurrent I sub isnotselected astheprimaryreliabilityparameterasinUltraSim.Ingen- eral,thedraincurrent I d ,thresholdvoltage V t ortranscon- ductance g m isoftenusedasadegradationmonitoring parameter,andthestresstimeresultingin10%decrease ofoneofthesemonitoringparametersisarbitrarilysetto thedevicelifetime.Hotcarrierreliabilitysimulationin Eldoadopts I d asthedegradationmonitoringparameter andcharacterizesitwithacompact D I d model,which directlymodelsthedi erenceofdraincurrentsbetween freshandageddevices. Thereexisttwocompetingmechanisms,whichleadto theobvioushotcarrierinduceddraincurrentvariations betweenfreshanddegradeddevices:thedeviationof I d fromitslineardependenceon V ds duetovelocitysatura- tione ectsandthedecreasingof D I d / I d duetothereduc- tionofchargedinterfacestates [51] .InEldo,the D I d is modeledwithEq. (41) to (40) ,whichunifythesubthresh- old,linearandsaturationregionswithasimplerelation forbothforwardandreverseoperationmodes [50] : D I d I d ¼ B 6 ð 1 À e À B 1 V gs Þþ B 2 1 þ B 5 ð V gs À B 3 V th Þ N it L it L eff Â 1 1 þ a ð V ds À V low Þþ b V ds ; ð 37 Þ V low ¼ A 3 V dsat ; ð 38 Þ a ¼ A 1 1 þ A 4 ð V gs À V th Þ A 2 ; ð 39 Þ b ¼ A 5 V gs þ A 6 ; ð 40 Þ where N it istheinterfacetrapdensity, L it istheextensionof thedamagewithinthechannel, L e isthee ectivechannel length, V gs isthegate-to-sourcevoltage, V t isthethreshold voltage, V ds isthedrain-to-sourcevoltage, V dsat isthedrain saturationvoltage, A 1 to A 6 and B 1 to B 6 aremodelXtting parameters. Thesame Age parameterdeXnedbyEq. (36) isincorpo- ratedtomodelthe 8 8age 9 9ofeachtransistor.TheHCIaging processissimulatedinaniterativewayasdepictedsche- maticallyin Fig.2 . Theperiod, T age ,atwhichthecircuitperformanceisto betestedisdividedintosmallertimeintervals, T 1 .The Age tableiscalculatedattheendofeachtimeintervalanda newsimulationwithEldoiscarriedforward.Thisprocess isrepeateduntil T age isreached.Thisiterativeschemecan accountforthegradualchangeofbiasconditionsasa resultofdevicewearout. The D I d modelingapproachprovidesthepossibilityto havearelativelysimplerparameterextractionprocess.It issuitabletomodelbi-directionalstressandasymmetrical draincurrentbehavior.However,becausethisapproach alsoadoptsboth Age parameterandsmall-stepiterative algorithminthedegradationsimulationprocess,itinherits thesamelimitationsoftheBERT-liketoolsasdiscussed before. Fig.1.HotcarrierreliabilitysimulationYowchartinVirtuosoUltraSim [45] . Fresh Simulation ( n =0) Calculate Age-table Aged Simulation ( t = n · T 1 ) t = T age ? End n = n+ 1 N Yes Fig.2.HCIreliabilitysimulationinEldo [50] .AlargenumberofSPICE simulationiterationshavetobecarriedouttoobtainaccuracy. J.B.Bernsteinetal./MicroelectronicsReliabilityxxx(2006)xxx 3xxx 11 ARTICLEINPRESS 3.2.Failuremechanismequivalentcircuits Inordertoaccountforthee ectofthefailuremecha- nismsoncircuitfunctionalityandreliability,thedevice- levellifetimemodelshavetobeextendedtocircuit-level applications.Thebridgeconnectingthegapbetweendevice wearoutdegreeandcircuitperformancedriftisnodoubt thecircuitmodels.Theunderlyingconceptofthecircuit modelsismodelingdegradationofdeviceparameterswith someadditionallumpedcircuitelements(resistors,transis- torsordependentcurrentsources,etc.)tocapturethe behaviorofadamagedMOSFETincircuitoperationenvi- ronment.Thevaluesoftheseadditionallumpedelements aredeterminedbydevicewearoutparameters(suchas D N it whicharetimedependentandbydeviceterminalvoltage andcurrentwaveforms,therefore,atanytime t ,valuesof theselumpedelementscanbepredictedaccuratelyand theirmagnitudesreYectthedevicewearoutdegree.Thelar- gerthemagnitudeofthesevalues,themoresevereisthe damagetocircuitfunctionality.Asaresult,circuitdesign- erscanquicklyanalyzecircuitreliabilitybehavioratany giventimewiththesecircuitmodels. 3.2.1.HCI SeveralHCIcircuitmodelshavebeendevelopedinthe pastyearsandsomeofthemhavebeenbuiltintocommer- cialreliabilitysimulationtools.Inthissection,someof thesecircuitmodelsarebrieYyreviewed,followedbythe introductionoftheHCIcircuitmodelanditsimple- mentation. BERThasbeenthemostsuccessfulcircuitreliability simulationtool.BERTdirectlymodelsn-MOSFEThot carrierdamageindraincurrentdegradation.Thedrain currentdegradation, D I d ,resultsfromchannelmobility degradation,whichagainresultsfromHCI-inducedinter- facetraps D N it . D N it ismodeledintermsofthefamous Age parameterintroducedintheprevioussection.In BERT, D I d isimplementedasanasymmetricalvoltagecon- trolledcurrentsourceinparallelwiththeoriginaln-MOS- FET.Thep-MOSFETHCIe ectismodeledwiththe conceptofchannelshorteninganddrainresistanceincrease [45] .TheBERT D I d modelisshownin Fig.3 .Hereonecan seeasymmetryintheforwardandreverse I 3 V characteris- tics,allowingthesimulationofdevicesundergoingbi-direc- tionalstresses(suchasdevicesinatransmissiongate). Thedetailed D I d modelequationsandparametersare deXnedin [52] .ThemaincontributionofBERT D I d model istheabilitytocharacterizebi-directionalhotcarrierstress e ects,howeveritrequiresextractionofsixprocessparam- etersfromdevicetesting,whichisanon-trivialwork. ExperimentshaveproventhatHCI-inducedinterface trapsinn-MOSFETarelocalizedabovethechannelnear thedrainjunction.MorespeciXcally,theseinterfacetraps arelocalizedwithin100nmfromthedrain [53] .Basedon thisobservation,Leblebicietal.atUIUC [54,55] developed atwo-transistorHCIcircuitmodel,whichconsistedofan HCIdamagedparasitictransistorwithXxedchannellength L 2 ( L 2 % 0.1 l m)inseriesconnectionwiththeoriginal transistorwhosechannellengthwasshrunkto L À L 2 . Theprimaryassumptionforthismodelisthatallgenerated interfacetrapsareoccupiedwithelectrons,whichequalsto consideringonlynegativeXxedcharge.Themodelisillus- tratedin Fig.4 . From Fig.4 a,theinterfacetrappedcharge Q it dueto HCIcanbereadilyderivedas when(0 6 x < L 1 ): Q it ð x Þ¼ 0 ð 41 Þ orwhen( L 1 6 x < L ): Q it ð x Þ¼ Q M L 2 ð x À L 1 Þ ; ð 42 Þ where Q M denotesthelargestinterfacecharge, L 1 = L À L 2 ,and L 2 representsthelengthofthedamaged channelregion.Thistwo-transistormodelcharacterizesthe amountofhotcarrierdamagewithonlytwoparameters Q M and L 2 ,therefore,themodelparameterextraction workisgreatlyreduced.Thedrawbacksofthismodel are:thatthetriangularchargedensitydistributionisover simpliXed,andthatanaccurate Q M valueisdi cultto extrapolate. ThesimplestHCIcircuitmodelhasbeenthehotcarrier inducedseriesresistanceenhancementmodel(HISREM), alsonamed D R d model,whichisproposedbyHwang etal.atOregonStateUniversity [58] .Basedonthefactthat theincreaseofHCI-inducedseriesdrainresistanceisdueto theinjectionofhotcarriersclosetothedrainedge,aseries resistance D R d addedtothedrainofthen-MOSFETcan Fig.3.BERTn-MOSFETHCIcircuitmodel.(a)Bidirectionalinterface trapgenerationnearbothdrainandsource. L f and L r representforward andreversehotcarrierdamagedregions.(b)HCIdraincurrent D I d circuit model [51] . 12 J.B.Bernsteinetal./MicroelectronicsReliabilityxxx(2006)xxx 3xxx ARTICLEINPRESS reYecttheprocessofhotcarrierinducedinterfacetrapgen- erationandthereforeaccountsforthechannelmobility reductionandthresholdvoltagedrifts.HISREMconsists ofavoltagedependentdrainresistor D R d connectedinser- ieswiththeoriginaln-MOSFET. D R d isafunctionofthe appliedvoltagesandthehotcarrierinducedinterface trappedcharge D N it . Thebehaviorofthedamagedn-MOSFETisemulated bytheoriginalundamageddeviceoperatedwithareduced drain-to-sourcevoltage,whichiscontrolledbythisaddi- tionaldrainresistor D R d .Because D N it isatimedependent parameter, D R d modelisabletopredictdraincurrentdeg- radationatanygiventime.HISREMisalsocapableof modelingself-limitinge ectsofhotcarrierdamagebecause theincreaseinseriesdrainresistanceofann-MOSFET suppresseshotcarrierstress.Themostadvantageousfea- tureoftheHISREMmodelisthatonlyoneparameter, D N it ,needstobeextrapolatedfromdevicetestingwork. Consequently,HISREMcanbeeasilyusedbycircuit designerstoperformanexpeditiousreliabilityanalysis. TheHCIcircuitmodelisbasedontheabove D R d model withsomeimprovements.Themajorimprovementisthat D R d valueisconsideredtobedeterminedbybothinterface trappedcharge D N it andoxidetrappedcharge D N ox .The contributionof D N ox todevicewearoutisoftenneglected, butrecentexperimentalworkrecognizesthattheycan accountforsomeoftheobservedenhanceddegradation e ectsinn-MOSFETswhichcouldnotbeexplainedsolely by D N it generation. TheHCIcircuitmodelisillustratedin Fig.5 .Thederi- vationof D R d iscarriedoutassumingthat(1)allinterface trapsareacceptor-likeandoccupiedbyelectrons,and(2) thechannelmobilitydegradation, l ,iscausedbyboth D N it and D N ox .Theassumption(1)meansthenetchargein interfacetrapsisaXxednegativechargeforn-MOSFET instronginversionoperation.Assumption(2)leadsto: l ¼ l 0 ð 1 þ a Á D N Þ ; ð 43 Þ where D N = D N it + D N ox (inunitcm À 2 ), l 0 istheoriginal channelmobility, a isaprocessdependentconstantand a % 2.4 · 10 À 12 cm 2 [58] . ThedraincurrentdrainYowingthroughanundamaged nMOS(when D N =0at t =0)isdeXnedas I ds0 : I ds0 ¼ l 0 C ox W L V gs À V t À V ds 2 V ds . ð 44 Þ When D N issmall,therelationbetweenfreshanddegraded drain-to-sourcecurrentis I ds ¼ I ds0 1 þ a Á D N . ð 45 Þ Thisleadstoanexpressionof D R d whichisdeterminedby D N andtheterminalvoltagesandcurrents D R d ¼ V R d I R d ¼ 1 þ a Á D N I ds0 V R d ; ð 46 Þ where I ds0 isgivenbyEq. (43) and V R d iscalculatedusing [59] : Fig.5.HCIcircuitmodel.Inthemodel: V gd x = V gs À V t À V ds and V R d ¼ I ds D R d . V t isthresholdvoltageand I ds isthecurrentfromnode D to S . Fig.4.UIUCn-MOSFETHCItwo-transistorseriesmodel.(a)Triangu- laroxidechargedistributionproXleusedinmodelderivation.(b)Cross- sectionalviewofn-MOSFETwithhotcarrierdamage, L 2 isthedamaged channelregion.(c)Two-transistorseriescircuitmodel.Theparasitic transistorhasdi erentchannelmobilityandthresholdvoltagewiththe channellength L 2 setto0.1 l m [54,56,57] . J.B.Bernsteinetal./MicroelectronicsReliabilityxxx(2006)xxx 3xxx 13 ARTICLEINPRESS V R d ¼À V gd x þ \x4\x4\x4\x4\x4\x4\x4\x4\x4\x4\x4\x4\x4\x4\x4\x4\x4\x4\x4\x4\x4\x4\x4\x4\x4\x4\x4\x4\x4\x4\x4\x4\x4\x4\x4\x4\x4\x4\x4\x4\x4\x4\x4\x4\x4\x4\x4\x4\x4\x4\x4\x4\x4\x4\x4\x4\x4\x4\x4\x4\x4\x4\x4\x4\x4\x4\x4\x4\x4\x4\x4\x4\x4\x4\x4 V 2 gd x þ 2 V ds D N a V gd x þ V ds 2 ÀÁ 1 þ a Á D N þ q C ox "# v u u t ; ð 47 Þ where V gd x = V gs À V t À V ds forthelinearregionand V gd x =0forthesaturationregion. Inquasi-staticoperation, D N isatime-dependent parameter,therefore, D R d isalsotime-dependent.Atany giventime t ,if D N isknown, D R d willbeuniquelydeter- mined.Themodelsfor D N and D N ox havebeenwelldocu- mentedinliterature [53,54] . D N it canbecalculatedusing D N it ¼ C 1 I ds W exp À U it ; e q k e E m t \x7! n ; ð 48 Þ where W isthechannelwidth, U it,e isthecriticalenergyfor electronstocreateaninterfacetrap, k e isthehot-electron meanfreepathand C 1 isaprocessconstant. Themodelsandmodelparametersfor D N ox aregivenin [55] (onpp.59 366).Forconvenience,theyarerecapitulated as D N ox ¼ N 1 ½ 1 À exp ðÀ r 1 I e i t Þ?À N 2 ½ 1 À exp ðÀ r 2 I e i t Þ? . ð 49 Þ AsetoftypicalmodelXttingparameters,forEq. (49) ,have beengivenin [55] (p.65).Theabovenew D R d modelinher- itsallthemeritsofHISREMmodelanditisphysically morecomprehensiveincharacterizinghotcarrierdamages. Thedrawbackofthisimproved D R d modelistheinclusion ofonemoreparameter D N ox ,whichcomplicatesparameter extractionwork. 3.2.2.NBTI SimulatingtheimpactofNBTIatthecircuitlevelusing SPICEisveryimportant [60] .MostoftheworkonNBTI SPICEsimulationisperformedsuchthatthedegradedcir- cuitbehaviorissimulatedusingthetransistorparameter V t ,whichisshiftedbyaXxedvalue [61] .Thiskindofsim- ulationmethodcannotphysicallyrelatecircuitperfor- mancedegradationtoNBTIwearoutunderdynamic operationconditionssinceitdoesnotincludetheNBTI stresstimeasaparameter.Themoste ectivewayto developsucharelationisbydevelopingaNBTIcircuit model.However,tothebestofourknowledge,nosuch electricalmodelexistsinliterature.Stretchedexponential timedependencedescribesthethresholdvoltagedegrada- tionas D V t ð t Þ¼ D V max 1 À exp À t s \x2\x3 b \x7! . ð 50 Þ Thus,anewNBTIcircuitmodelisproposed,whichisan electricalmodelrelatingthetimedependentNBTIphysical degradationparameter D V t tolumpedelectricalmodelele- ments,therebyenablinge ectiveandquickNBTIcircuit reliabilitysimulation. ThemostsevereNBTIe ectisp-MOSFETthreshold voltageincrease 4 D V t ,whichisequivalenttoadecrease inthep-MOSFETabsolutegate-to-sourcevoltage.This isequivalenttoaddingavoltagesourceatthegate.How- ever,wechosetosplitthep-MOSFETgateconnectionand addagateresistance R G betweentheoriginalgatebiasing point G andthep-MOSFETgateterminal G 0 .Thisallows inclusionofagateleakagecurrentYowingmechanism (voltagecontrolledcurrentsourcesbetweengateanddrain andbetweengateandsource),sothattheNBTImodelcan beincludedwiththeTDDBmodelwithoutaconYictdevel- opingbyhavingavoltagesourceandcurrentsourceatthe samenode.Infact,thereshouldbenodi erencebetween usingavoltagesourceoracurrentsourcetorepresent D V t ,wherebythegateleakagecurrentYowsthroughthe gateresistance R G andincreasesthep-MOSFETe ective gatevoltageatpoint G 0 .Thisgatecurrentwoulddepend onthegatevoltagebecauseof R .Thecurrentmaybesome- whatun-physicalsoasubstratebiassourcemaybesubsti- tutedinstead,butmanypossibilitieshaveyettobe exploredinthisarea. Sincethep-MOSFETsourceisheldXxedatitshighest potential,theinclusionof R G andgateleakagecurrent leadstoadecreaseofthep-MOSFETabsolutegate-to- sourcevoltage,therebyimitatingtheNBTIthresholdvolt- agedegradation.Basedonthisconcept,theNBTIcircuit modelisconstructedandshownin Fig.6 .Theadvantage ofthisconXgurationwillbecomeclearwhencomparedto themodelforTDDB,presentednext. Inthismodel, R G isavoltagedependentresistance becausegateleakagecurrentsarevoltagedependent. R G isalsoatimedependentresistancebecausevoltagedrop across R G atanyspeciXctime t isequaltothresholdvolt- ageshift D V t whichistimedependent.Accordingto [62] , thegateleakagecurrentduetooxidebreakdowncanbe Fig.6.NBTIcircuitmodel.NBTI-inducedp-MOSFETthresholdvoltage increaseismodeledasabsolutegate-to-sourcevoltagedecrease.Gate tunnelingcurrentYowingthroughthegateresistance R G leadstothe increaseofvoltageatpoint G 0 .Thiscorrespondstothedecreaseofp- MOSFETabsolutegate-to-sourcevoltageandthereforemimicsthe thresholdvoltagedegradatione ect.Gatetunnelingcurrentismodeled withtwovoltagecontrolledcurrentsourceswhichfollowtheformofa powerlawrelationas: I = KV p . 14 J.B.Bernsteinetal./MicroelectronicsReliabilityxxx(2006)xxx 3xxx ARTICLEINPRESS modeledasagate-to-di usionleakagecurrent,witha powerlawdependenceoftheformula I = KV p (where K and p areXttingparameters).Thesamepowerlawvoltage dependency,shownin Fig.6 ,isadoptedinmodelinggate leakagecurrents.Asaresult,forthegate-to-drainleakage current, I GD = K ( V GD ) p ,andforthegate-to-sourceleakage current, I GS = K ( V GS ) p .Thedefaultvalueof p issetto5, andthedefaultvalueof K is3 · 10 À 6 [62] . In Fig.6 ,thevoltagedropacross R G is V R G ð t Þ¼ V G 0 À V G ¼ D V G ð t Þ¼ð I GD þ I GS Þ R G . ð 51 Þ Thresholdvoltagedegradation D V t duetoNBTIisalready givenbyEq. (50) .Therefore,fromtherelation D V G ( t )= D V t ( t ),weobtainananalyticalsolutionfor R G : R G ¼ D V max KV p GD þ KV p GS 1 À e À t s ðÞ b \x7! . ð 52 Þ Thetypicalvaluesandextractionmethodsforthemodel parameters D V max , K , p , s and b havebeengivenanddis- cussedduringtheprocessofderivingEq. (52) . Oneofthemostimportantpointsshownin Fig.6 isthat thisnewmodelismuchbetterthanasimplemodelwhich onlyinsertsavoltagesourcebetween G and G 0 representing thresholdvoltageshiftinthatitinherentlyincorporates bothNBTIandpossibleoxidebreakdowne ects. FornMOSpositivebiastemperatureinstability(PBTI) circuitmodel,asimilarstructuretothatforpMOSNBTI shownin Fig.6 canbeconstructed,exceptthatallcurrent YowingdirectionsarereversedandthemodelXtting parametersofthethresholdvoltagemodel D V t (Eq. (50) ) aredeterminedfromnMOSPBTIstresstesting.Forthe twocurrentsources I GD and I GS innMOSPBTIcircuit model,abettergateleakagemodel,proposedbyLee etal. [63] isadopted: I GS ¼ 1 2 AL exp ð a V GS À b t À c ox Þð 53 Þ and I GD ¼ 1 2 AL exp ð a V GD À b t À c ox Þ ; ð 54 Þ where I GS and I GD arein l A, L ise ectivechannellength innanometer, t ox isoxidethicknessinnanometer, A =127.04, a =5.61, b =10.6and c =2.5.Thesetypical valuesforn-MOSFETswereobtainedbyXttingindustrial dataandfoundtobegoodfortechnologiesacrossmany generationsupto0.13 l m.Thesenewleakagemodelsare abletomaintaingoodstabilityinSPICEsimulation [63] . 3.2.3.TDDB Itisonerousworktodevelopane ectivecircuitmodel forgateoxidebreakdownbecausedevicepost-breakdown beha

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