<%BANNER%>

Simulation of Noise Mechanisms in Scaled Bulk and Partially Depleted Silicon-on-Insulator Field-Effect Transistors


PAGE 1

SIMULATIONOFNOISEMECHANISMSINSCALEDBULKANDPARTIALLY DEPLETEDSILICON-ON-INSULATORFIELD-EFFECTTRANSISTORS By DEREKO.MARTIN ADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOL OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT OFTHEREQUIREMENTSFORTHEDEGREEOF DOCTOROFPHILOSOPHY UNIVERSITYOFFLORIDA 2007 1

PAGE 2

c r 2007DerekO.Martin 2

PAGE 3

ToKris,Will,Kate,andEmily 3

PAGE 4

ACKNOWLEDGMENTS Iwouldliketothankmysupervisorycommitteechair,Profes sorGijsBosman,for hisguidanceandsupport.Ialsowouldliketoexpressmygrat itudetoProfessorsMark Law,JerryFossum,andLiShenforservingonmysupervisoryc ommitteeandfortheir interest.IthankQiXiangandRainerThomaforservingashos tsandmentorsduring myinternships.Iwouldalsoliketothankmyformerprofesso rs,mentorsandteachersin researchlabsandinclassrooms,incollegeandinsecondary school.Thisworkwouldnot havebeenpossiblewithoutasolidcollectiveofknowledgea ndexperience. IwouldalsoliketothanktheSemiconductorResearchCompan y,theNationalScience Foundation,AdvancedMicroDevices,Motorola,Agilent,and theUniversityofFlorida fortheirnancialsupport.IamgratefultoConcettaRiccob eneofAMDforproviding thebulkandSOItransistorsmeasuredandsimulatedinthisw ork,aswellastheprocess descriptionsrequiredforprocesssimulation. MysincerestthanksgotoProfessorLaw,ProfessorBosman,J uanSanchez,andFrank HoufordevelopingtheplatformforsimulatingnoiseinFLOO DS.Withoutit,thiswork wouldnothavebeenpossible.IalsothankfellowstudentsLi saKoreandJong-HwanLee fortheircollaborationandtechnicaldiscussions. Finally,Iamdeeplygratefultomywife,Kris,andmychildre nfortheirsupport, patience,andperseveranceasIhavepursuedthiswork. 4

PAGE 5

TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................4 TABLE ...........................................7 LISTOFFIGURES ....................................8 ABSTRACT ........................................11 CHAPTER 1INTRODUCTION ..................................13 1.1FluctuationPhenomena ............................14 1.2NumericalDeviceSimulation ..........................16 1.3TransportModels ................................16 1.4NoiseSimulation ................................19 1.5SOIDevicesandNoise .............................19 2TRANSPORTMODELSANDPHYSICALPROPERTIES ............22 2.1AdvancedTransportModelsforPDE-basedDeviceSimulatio n .......22 2.1.1Poisson'sEquation ...........................23 2.1.2MomentsoftheBoltzmannTransportEquation ...........23 2.1.2.1Zeroethordermoment:carriercontinuityequation s ....25 2.1.2.2Firstordermoment:momentumbalanceequations ....25 2.1.2.3Secondordermoment:energybalanceequations .....26 2.1.2.4Thirdordermoment:heatrowbalanceequations .....27 2.1.3ClosureandReductionRelations ....................27 2.1.3.1Momentumrelatedreductionrelations ...........28 2.1.3.2Energyrelatedreductionrelations ..............28 2.1.3.3Heatrowrelatedreductionandclosurerelations ......28 2.1.3.4PEandrstfourBTEmoments,closedandsummarized .29 2.1.4HydrodynamicModel ..........................29 2.1.5ReducedHydrodynamic,or\EnergyBalance"Model ........30 2.1.6Drift-DiusionModel ..........................32 2.2ImplementationofSurfaceMobilityModelsinFLOODS ..........32 2.2.1LowKineticEnergyMobilityFormulation ..............34 2.2.2HotCarrierMobility ..........................36 2.2.3Implementation-SpecicDetails ....................37 2.3ImplementationoftheEnergyBalanceTransportEquatio nsinFLOODS .38 2.4ImpactIonization ................................40 2.5ImpedanceFieldSimulationofVelocityFluctuationNoi se .........42 2.5.1VelocityFluctuationNoiseSimulation .................42 2.5.1.1Hydrodynamicmodelwithvelocityructuationnoise ...46 5

PAGE 6

2.5.1.2Energybalancemodelwithvelocityructuationnois e ...48 2.5.1.3Drift-diusionmodelwithvelocityructuationnois e ....51 2.5.2VelocityFluctuationsinEBandDDModels .............52 2.5.2.1OneDimensional n + =n=n + ResistorSimulations ......52 2.5.2.2nMOSFETSimulations ...................53 2.6ImpedanceFieldSimulationofNumberFluctuationNoise .........54 2.6.1NumberFluctuationNoiseSimulation .................54 2.6.1.1Bulkorsurfacetrapcaptureandemission .........57 2.6.1.2DirectCarriertunneling ...................58 2.6.1.3ImpactIonization .......................61 2.6.2NumberFluctuationsinEBandDDModelsfor n + =n=n + Resistors 62 2.7QuantizationEectsandNoise ........................63 3NOISEMEASUREMENTS .............................78 3.1MeasurementSetup ...............................78 3.1.1ResistorNetworkDesign ........................78 3.1.2PracticalConsiderations ........................79 3.2MeasuredDevices ................................80 4SIMULATIONSOFMEASUREDDEVICES ...................101 4.1The90nmBulkMOSFET ...........................102 4.2The90nmSOIMOSFET ............................107 5CONCLUSION ....................................138 5.1Summary ....................................138 5.2RecommendationsforFutureWork ......................138 AAPPENDIX:SIMULATIONFILES .........................140 BIOGRAPHICALSKETCH ................................171 6

PAGE 7

TABLE Table page 4-1Finalimpactionizationratesandcarrierlifetimes .................123 7

PAGE 8

LISTOFFIGURES Figure page 1-1Componentsofasignalincludingdeterministicandstoc hasticparts .......21 2-1Electrontemperatureina0 : 1 mn + =n=n + resistor ................66 2-2Lowbiasvelocityructuationsina0 : 1 mn + =n=n + resistor ............67 2-3Highbiasvelocityructuationsina0 : 1 mn + =n=n + resistor ...........68 2-4Velocityructuationnoisefor n + =n=n + resistorsofvaryinglengths. .......69 2-5EBandDDmodelsandvelocityructuationsinresistor ..............70 2-60 : 25 m nMOSFETdopingconcentraton ......................71 2-7VelocityructuationsinDDmodelfor0 : 25 m nMOSFET .............72 2-8VelocityructuationsinEBmodelfor0 : 25 m nMOSFET .............73 2-9Lowbiasnumberructuationsina0 : 1 mn + =n=n + resistor ............74 2-10Numberructuationsnoisefor n + =n=n + resistorsofvaryinglengths .......75 2-11EBandDDmodelsandnumberructuationsinresistor ..............76 2-12Probabilitydistributionsandenergiesinaninnitew ell ..............77 3-1CircuitusedforbulkandSOIdevicenoisemeasurements. ............82 3-2Physicallayoutofcircuitshowingthecriticalpathofe lectricalconnections. ...83 3-3Connectionschemetoensuretheavoidanceofdielectric breakdown. .......84 3-4Linearscale I D vs. V GS for90 nm bulknMOSFET .................85 3-5Logarithmicscale I D vs. V GS for90 nm bulknMOSFET .............86 3-6 I D vs. V DS for90 nm bulknMOSFET ........................87 3-7Draincurrentnoiseofthe3 : 2 m x90 nm n-channelbulkMOSFET .......88 3-8Linearscale I D vs. V GS for90 nm SOInMOSFET .................89 3-9Logarithmicscale I D vs. V GS for90 nm SOInMOSFET ..............90 3-10 I D vs. V DS for90 nm SOInMOSFET ........................91 3-11Draincurrentnoiseofthe4 : 8 m x90 nm n-channelSOIMOSFET .......92 3-12Linearscale I D vs. V GS for2 : 33 m bulknMOSFET ................93 8

PAGE 9

3-13Logarithmicscale I D vs. V GS for2 : 33 m bulknMOSFET ............94 3-14 I D vs. V DS for2 : 33 m bulknMOSFET .......................95 3-15Draincurrentnoiseofthe3 : 2 m x2 : 33 m n-channelbulkMOSFET ......96 3-16Linearscale I D vs. V GS for2 : 33 m SOInMOSFET ................97 3-17Logarithmicscale I D vs. V GS for2 : 33 m SOInMOSFET .............98 3-18 I D vs. V DS for2 : 33 m SOInMOSFET .......................99 3-19Draincurrentnoiseofthe0 : 6 m x2 : 33 m n-channelSOIMOSFET ......100 4-1AbsolutevalueofbulknMOSFETdoping .....................113 4-2Linearscalematchof I D vs. V GS in90 nm bulkdevice. ..............114 4-3Logarithmicscalematchfor I D vs. V GS ......................115 4-4Matchofsimulateddraincurrentforbulkdevice ..................116 4-5Matchofnoisefor90 nm bulknMOSFETwithcontanttrapdensity. .......117 4-6Draincurrentnoisecontributionalongchannel/oxidesu rface ...........118 4-7Simulatedoxidetraplocations ............................119 4-8Finaltofmeasuredandsimulatednoiseforthe90nmBulk nMOSFET. ....120 4-9Quantizedn(x)undertrap1 .............................121 4-10Quantizedn(x)undertraps2and3 .........................122 4-11AbsolutevalueofSOInMOSFETdoping ......................124 4-12Logarithmicscalematchof I D vs. V GS .......................125 4-13MatchofsimulateddraincurrentforSOIdevice ..................126 4-14MathematictofLorentziaforplotting .......................127 4-15Oxidenoisewithoutexcessnoise ..........................128 4-16Quantizedn(x)undertrap1 .............................129 4-17Quantizedn(x)undertraps2and3 .........................130 4-18Velocityructuationnoisesimulationsfor V GS =1 : 25 V ..............131 4-19Excessnoiseat100 Hz ................................132 4-20Linearscalematchof I D vs. V DS withadjustedimpactionizationrate ......133 9

PAGE 10

4-21Zero-frequencyvalueofLorentzia ..........................134 4-22Cut-ofrequencyofLorentzia ............................135 4-23Fitofsimulatednoisewithcorrectedinterfacecharge ...............136 4-24Finaltofsimulatednoise,withmodiedbacksidebody doping .........137 10

PAGE 11

AbstractofDissertationPresentedtotheGraduateSchool oftheUniversityofFloridainPartialFulllmentofthe RequirementsfortheDegreeofDoctorofPhilosophy SIMULATIONOFNOISEMECHANISMSINSCALEDBULKANDPARTIALLY DEPLETEDSILICON-ON-INSULATORFIELD-EFFECTTRANSISTORS By DerekO.Martin May2007 Chair:GijsBosmanMajor:ElectricalandComputerEngineering Thefocusofthisdissertationisonthesimulationandmeasu rementoflowfrequency noiseinhighlyscaledbulkandSilicon-On-Insulator(SOI) metal-oxide-semiconductor eld-eectdevices(MOSFETs).Tofurtherthecapabilityof simulatingnoiseinsuch devices,theFloridaObject-OrientedDeviceSimulator(FLO ODS)hasbeenextendedto beabletousetheEnergyBalancesystemofequations.Anadva ncedsurfacemobility modelisadded,andthenoisecapabilityofthisnumericalde vicesimulatorismodied toincludethemechanismofdirectbandtooxidetraptunneli ngforthesimulationof degeneratedevices. ThederivationoftheEnergyBalancemodelandotherhigh-or dermodelsispresented, andthecorrectinsertionofnoisesourcesintothesemodels isdescribed.Simpletestcases areusedtodemonstratethesimulationofnoiseusingthemod elimplementationandthe eectsofthismodelascomparedtotheDrift-Diusionmodel. NoisemeasurementsaretakenforbulkandSOIMOSFETs,andsi mulationsare performedtomatchtheconditionsofthemeasurements.Thes hapeofthelowfrequency noiseisusedtoproletheactivetrapsinthegateoxides.Wh ilehigherordermoments thanDrift-DiusiongaveanomalousDCsimulationresultsfort hedevicesmeasured, Drift-DiusionresultsarecomparedtothemeasuredDCandnois edatawithgood agreement,anderrorcorrectionforquantizationeectsis computed. 11

PAGE 12

ThoughtheoriginalintendedusageoftheEnergyBalancemod elwasnotperformed, theDriftDiusionsimulationsgavegoodagreementtothemeas ureddata,andthiswork representsthemostcompletePDE-basednumericalsimulatio nofnoiseinsuchdevicesto date.Theanalysisprovidesreverse-engineeringofthenum berandthelocationoftraps intheoxideofthedevicesmeasured,whichwasaprimarygoal forthiswork.Theerror correctionforquantizationeectsvalidatesthatquasi-c lassicalnumericalsimulationis applicabletohighlyscaleddevices.Thisworkalsoprovide sencouragementthatfuture workcanextendtheuseofthismethodologytoemergingdevic etechnologies. 12

PAGE 13

CHAPTER1 INTRODUCTION Silicon-On-Insulator(SOI)deviceshavegainedacceptanc easawaytofabricate semiconductorintegratedcircuitsthatarefasterthanthe irbulksiliconcounterparts, andthatallowhigherdensityduetotheirDClatchupimmunity .ThishigherICdensity attributeisbestutilized,however,whenthebodyoftheSOI deviceisnotcontacted,and lefttoroat.Specicnoisecharacteristicsuniquetosuchp artiallydepleteddeviceswere recentlystudied[ 1 ],suchasexcessLorentzian-shapedcomponentspresentath ighdrain bias.Ourstudyfurtherinvestigatedtheseandothernoisec haracteristicsexhibitedby modernscaledpartiallydepletedandbulkMOSFETs. Chapter2givesaderivationofmomentsoftheBoltzmannTran sportEquation, whichyieldtheenergybalanceanddriftdiusionmodels.Te rmswhichcauseructuations andnoiseinsemiconductordevicesaretrackedthroughthed erivationtoshowthe correctplacementofLangevinnoisesourcesinthetranspor tmodels.Thedetailsof theimplementationofthesetransportmodelsinFLOODSarede scribed,including enhancementstoallowaccuratesimulationofscaleddevice s.Amoreadvancedmobility modelisimplementedfortheaccuratesimulationofoxidetr apping-associatednoise. Directband-to-traptunneling,necessaryforthedegenerat echannelregionofthe scaledMOSFETs,isadded.Examplesofnoisesimulationsins impledevicesusing high-ordertransportmodelsforbothvelocityructuationn oiseandnumberructuation noisearedemonstrated.Inchapter3,themeasurementsofth eDCcharacteristics andnoiseofbulkandSOIdevicesarepresented.Asthedevice swerenotprotected fromESDandarediculttohandle,aconnectionschemeforth eexecutionofnoise measurementsminimizinghightransientvoltagesisdiscus sed.Chapter4describesthe reverseengineeringandsimulationofthedevicesmeasured ,includingallassumptions madeandparametersused.Goodagreementbetweensimulatio nandmeasurementis acheivedforthenoiseduetodirectoxidetunnelinginbothd evices.Theexcesslow frequencynoiseexhibitedbytheSOIdevicemeasurementsgi vesgoodagreementtoboth 13

PAGE 14

themagnitudeandcutofrequenciesofthesimulatedLorent zia,oncethexedinterface chargeisconsideredinitspropercontextforquantization eects.Numericalsimulation withthesemi-classicaldrive-diusionmodelisshowntobe usefulfortheprolingof trapsinthegateoxideofthesedevices,thesimulationofth ediusionnoiseinSOI devicesisshowntogivegoodagreementwithmeasurednoisec haracteristics,andinsight intotherelationshipsbetweenvarioussimulationparamet ersandtheresultingnoiseis acheived.Chapter5summarizeswhathasbeenaccomplishedi nthiswork,andgives recommendationsforfuturework. Thefollowingsectionsofthischaptergivebackgroundinfo rmationonthenatureof ructuationsinsemiconductors,numericalsimulationofde vices,simulationofnoise,and semiconductordevicesingeneral.Thisisintendedtoprovi deabasisfordiscussionof implementationchoicesforsimulatingnoiseinSOIdevices 1.1FluctuationPhenomena Itishelpfultorstdiscussthenatureofnoise.Voltagesan dcurrents(andnumber ofcarriers,carriervelocities,evenpositionofindividu alatoms)insemiconductorsare timedependentsignalsinanon-linear,causalsystem.Thes ystemconsistsofthematerial propertiesofthesemiconductordevice(itscrystallinema keup,dopingconcentrations, andgeometricaldescription)andthephysicallawsthatgov ernthebehaviorofallparts ofthesystem.Knowledgeofthenon-linearsystemitselfand itsstimuli(inputstothe system,consistingofappliedvoltagesandcurrents,heat, light,sound,mechanicalforces, electricandmagneticelds,gravity,radiation,etc.)all owspredictionofmanycomponents ofthesignalsinthesystem.TheDC,ormean,componentofasig nalisequaltoits timeaverage, R T s ( t ) dt=T ,where t istime, s ( t )isthesignal,and T isthelengthofthe observation.Anotherpossiblecomponentisthatwhichisze ro-meanandperiodic,usually duetoperiodicstimuli.Finally,theremayalsobeothercom ponentswhicharenot periodic,butcanbedeterminedfromchangestootherpartso fthesystemorinputs,such ascapacitiveormagneticcouplingtoothervoltagesandcur rents.Thesecomponents 14

PAGE 15

(whethertheyaredesiredornot)aredeterministic,andcan beentirelypredicted fromcompleteknowledgeofthesystemanditsstimuli.Howev er,evenwhenallthese componentsareaddedtogether,theydonotcompletelydescr ibeasignal(Figure 1-1 ). Someinteractionsinthehighlycomplexnon-linearsystema restochastic(randomand unpredictable,evenwithcompletesystemandstimulusknow ledge).Thestochastic componentsarethesubjectofthisstudy,andwhiletheirtim e-dependentvaluecannot bepredicted,someoftheirphenomenologicalaspects(e.g. powerspectraldensityandits biasdependencies)canbepredicted(givencompleteknowle dgeofthesystemandstimuli). Inaddition,itisthepremiseofthisworkthatknowledgeoft hesystem(device)thatis limitedtoknowledgeofitsfabricationmethods,currently understoodphysicsandDCand ACcharacteristicsisincomplete,andthemeasurementands tudyofthenoiseinadevice canfurtherknowledgeofthesystem. Commonlyobservedtypesofnoiseincludewhitenoise(forwh ichthepowerspectral densityisindependentoffrequency,atleastforobservabl efrequencies),1 =f or1 =f -like noise(inwhichthepowerspectraldensityisapproximately inverselyproportionalto frequency),andLorentzianshapednoise(forwhichthepowe rspectraldensityiswhite uptoacutofrequency,abovewhichitdecreasesinverselyp roportionaltofrequency squared).Whitenoiseinsemiconductordevicesisgenerall ydue(directlyorindirectly) toructuationintheaveragevelocityofchargecarriers.Lo rentzianshapednoisehas beenattributedtoructuationsinnumbersoftrappedcarrie rs,andtowhitenoisethatis lteredbythenon-linearsystem.1 =f noiseissomewhatmorecontroversial,andhasbeen attributedtotraps,aswellasructuationinotherquantiti es. Astheknowledgeofrealsemiconductordevicesisincomplet e(theexactgeometries andmaterialpropertiesofadevicecanonlybeestimatedfro mknowledgeofits fabrication,andthecompletephysicsisapproximatedbyou rlimitedunderstanding), apseudo-realisticmodelofanydeviceconsideredinthiswo rkanditsphysicsissetup fornumericalsimulation,andthephysicsofructuationsin thepseudo-realisticsimulated 15

PAGE 16

devicearethenusedtopredictthemeasurableattributesof thenoiseoftherealdevice. Thisrequiresanadequatelyrealisticmodelofthedevicean ditsphysicsforsimulation, accuratemeasurementoftheDCbehaviorofthedevicetosuppl ementtheunderstanding ofthedeviceitself,andnally,rigorousmeasurementofth enoiseofthedevice. 1.2NumericalDeviceSimulation Partialdierentialequations'(PDEs)numericalsolutiona ndsubsequentsemiconductor devicesimulationhavebeenusedintheanalysisanddesigno fsemiconductordevices fornearlyhalfacentury[ 2 ].Whilephysics-basedcompactmodels(closed-formor near-closed-formsolutionsbasedonsimplifyingapproxim ationssuchasquasineutrality anddepletion)havebeenshown[ 3 4 ]tobecapableofaccuratelysimulatingawide varietyofphysicaleects(someofwhichareverydicultto modelmicroscopicallyfor PDE-basednumericalsimulation),andwhilerecentPDE-based modelshavebeenshown tobediculttodiscretizeforstablenumericalsolution[ 5 ],numericaldevicesimulation remainsaveryhelpfultoolfortheevaluationofphysicale ectsinsemiconductordevices. Thisisparticularlytrueincaseswherethephysicsofdevic esdoesnotlenditselfeasilyto simplifyingassumptionsorincaseswhereobtainingthedet ailedgeometricaldependence ofcertaineectsisdesired. 1.3TransportModels Thederivationofmodelsforchargetransportinsemiconduc tordevicestypically beginswithPoisson'sEquationandtheBoltzmanTransportE quation,andthen usesthesetoderivethePDEswhichwillbesolvedinthesemico nductor.Onesuch derivationisverynicelygivenbyLundstrom[ 6 ].Thisderivationisfollowedtoobtain thehydrodynamicmodelimplementedintheFloridaObject-O rientedDeviceSimulator (FLOODS)asdiscussedinChapter 2 ,thoughthenotationusedinthisworkmoreclosely followsthatusedbyTang[ 7 ]. Thedrift-diusionsystemofequationsincludesPoisson's Equationandthe zeroeth-ordermomentsoftheBoltzmannTransportEquation forelectronsandholes. 16

PAGE 17

Therst-ordermomentsareusedtogiverelationsforthepar ticleruxes.Themobilityhas beenmodeledempirically,basedonandcalibratedtomacros copicaveragescomputedfrom measurementdata.Klaassengaveamodelwhichisusedforthe computationofdoping[ 8 ] andtemperature[ 9 ]dependencesofcarriermobilitiesinbulksilicon.Darwish etal.[ 10 ] gaveamodelwhichisusedtofurthercomputethedegradedmob ilityinsurfaceinversion layersinsiliconMOSFETs. Thehydrodynamicandenergybalancesystemsofequationsfo rtransportin semiconductordevicesarederivedfromconsiderationofhi gher-ordermomentsof theBoltzmannTransportEquation.Manydierentmodelshav ebeenderived(i.e., Stratton[ 11 ]andBltkejr[ 12 ]).Thomaetal.derivedamorephysicallycomplexsetof equationswhichdonotincludeanyconstant-eective-mass approximationsandaremore consistentwiththeresultsofMonteCarlosimulations[ 13 ].Tang[ 7 ]derivedageneralized modelwhichcanbemade,throughjudiciouschoiceofclosure relations,tomatchseveral othermodelsasdemonstratedbyIeongandTang[ 14 ]. QuantizationoftheinversionlayerinMOSFETshastheeect sofreducingthe overallinversionlayercharge(comparedtowhataclassica lmodelwouldpredict),and changingtheshapeofthedistributionofcarriersthrought hechannel,suchthatthe peakofthedistributionisdisplacedfromthesurface.Inth edevicesconsideredforthis work,thecarriersintheinversionlayerofthedeviceareco nnedinthevertical(gateto substrate)direction,andassucharesubjecttoquantizati oneects. Thesimulationsinthisworkemployaroughsemi-classicala pproach,assuming thatthereductionintheinversionlayerchargeisapproxim atedbyxedchargeat theinterface.Thisapproachisnotdissimilartothatemplo yedbytherecentmodel presentedbyLiandYu[ 15 ],whichcorrectsclassicalmodelsbyintroducingaxedcha rge distributedthroughthechannel.Theremainingerrorliesi ntheshapeoftheinversion layerdistributionandthepositionfromwhichcarrierscan tunnelintotheoxidetogive risetolowfrequencynoise. 17

PAGE 18

Theseeectsare,however,shownnottobedirectlyimportan ttothetunneling rates(andthereforetherandomcharginganddischargingra tes)associatedwithan oxidetrap.Sincemodifyingthephysicsinthenumericaldev icesimulationtoinclude quantizationeectsdoesnotgreatlyaectthesimulatedno iseassociatedwiththeoxide traps,quantizationeectsarenotdirectlyincludedinthe simulation,thoughanerror correctionforthecomputedtrappositionsandenergiesisc omputedandgiven. Whilethestrengthsofhigherordertransportmodelsinclud emoreaccurateprediction ofimpact-ionizationrates[ 16 ]andsimulationofsomesmalldeviceeectssuchasvelocity overshoot,thesemodelshavealsobeenshowntocauseundesi rableresults.Egleyet al.[ 17 ]demonstratedanoverestimationofhotchannelcarrierdi usionintoroating bodies,leadingtoincorrectpredictionofnegativediere ntialdrainconductance(or inversekinkeect)insimulationsofpartiallydepletedSO IMOSFETs.Thisresulthas alsobeenreportedbyothers.ToquotePolskyetal.[ 18 ,page504],\hydrodynamic transportproducesnegativedierentialresistance(NDR). ..nootherexperimental publicationinthisareahasconrmedtheobservationofNDR. ..Therefore,NDRisnow consideredbymanytobeanartifactofthehydrodynamictran sportmodel."Toquote MunteanuandLeCarval[ 19 ,pageG574],\Weclearlyshowthatanomalousbehaviors existinallmajorcommercialsimulationcodesandconcernb othpartiallydepleted silicon-on-insulatordevices(inversekinkeect)andbul ktransistors(positivesubstrate currenteect)."PolskynotesthattheNDReectisduebothto overestimationof inversioncarrierdiusionintotheroatingbodyandtofail uresinthehydrodynamic model'shandlingofShockleyReedHallrecombination.Seve ralmethodshavebeen employedtotrytoreduceoreliminatetheNDReect,includin gmodifyingthediusivity modeling[ 19 { 21 ],modifyingthemodelingoftheenergyrelaxationtimes[ 19 ],andby usingtrap-assisted-tunnelinggeneration-recombinatio ntoosetthenegativecharging[ 19 ]. However,nodenitivetheoryhasbeenputforwardthatsatis fyinglyresolvestheproblem inawaythatgivesgreatcondenceintheaccuracyofthemode ling,orwouldworkfor 18

PAGE 19

awiderangeofstructuresandcarrierrecombinationlifeti mes.Further,workbyBanoo andLundstrom[ 22 ]hassuggestedthatthehydrodynamicmodelpredictsoutput currents whicharehigherthanderivedballisticlimits.Therefore, thesemodelsshouldbeused primarilywhentheaccuratecomputationofimpact-ionizat ionisrequiredorwhenspecic qualitativeresults{ratherthanstrictlyquantitativere sults{aresought. 1.4NoiseSimulation Theimpedanceeldmethodforcomputingnoiseinsemiconduc tordeviceswas rstproposedbyShockleyetal.[ 23 ].Thiswaslongusedinthecontextofclosed-form analysis,butwasalsointroducedasamethodologyusefulin theforumofnumerical devicesimulationbyBonanietal.[ 24 ].ThishasalsobeenimplementedinFLOODSfor diusionnoisebySanchezandHou[ 25 ],andforgeneration-recombinationnoisebyHou etal.[ 26 ].SimulationoftheFouriercoecientsofthesolutionvari ablesbythemethodof harmonicbalanceandsubsequentsimulationoftheinter-fr equencymixingofnoisehave beenimplementedbySanchez[ 27 ].RecentworkbyHou[ 28 29 ]hasaddedthecapability tosimulate1 =f -likenoisebyconsideringinterfacetrappingandtunnelin gofcarriersto andfromoxidetraps. Analternatemethodofnoisesimulationbasedonthehydrody namicmodelwas demonstratedbyGooetal.[ 30 ],usingthemodelingofnoisesourcesassmallcurrent sourcesinalossytransmissionline.Thesimulationofnois einhydrodynamicmodelsusing theimpedanceeldmethodwasconsideredbyBonanietal.[ 31 ];however,detailswere notgivenregardinghoworwhethertheructuationsofphysic alquantitieswerecoupled intotheenergybalanceequations.Morerecentwork[ 32 ]hasdetailedthecouplingofnoise ructuationsintohigher-ordertransportmodels,andthisw orkfurthersthisgoal. 1.5SOIDevicesandNoise TheUFSOIcompactmodels[ 3 ]havebeendevelopedbyprofessorFossumand hisresearchgroup.Thecharge-basedcouplingofthefronta ndbacksurfaceswas characterizedbySuhandFossum[ 4 ]andLimandFossum[ 33 ].Non-localimpact 19

PAGE 20

ionizationwasmodeledbyKrishnan[ 16 ].Noisemodelingwasaddedtothemodelby Workman[ 1 ]andwasshowntoaccuratelycomputetheexcesslowfrequenc yLorentzian shapeobservedinpartiallydepleteddevices. ThisexcessnoiseinpartiallydepletedSOItransistorshas beenmeasuredand publishedbySimoenetal.[ 34 ].Separatemeasurementdatacanbefoundin[ 1 ].Low frequencynoisemeasurementshavealsobeenmadeonburiedchannelSOIp-MOSFETs byLukyanchikova[ 35 ].Thisworkcontributesnewnoisemeasurementsinsimilarl y processedlongandshortchannelbulkandSOItransistorspr ovidedbyaSemiconductor ResearchCorporationmembercompany. Afteradiscussionofthenumericalsimulationmethodsused ,theanalysisofthisdata usingthesimulationswillbegiven. 20

PAGE 21

0 5e-061e-051.5e-052e-05 t (seconds) 0 1 2 3s(t) (arbitrary units) a b c d e Figure1-1.Componentsofasignal(a)inanonlinearsystem, includingtheDC component(b),ACcomponent(c),anaperiodicdeterministi ccomponent (d),andastochasticcomponent(e). 21

PAGE 22

CHAPTER2 TRANSPORTMODELSANDPHYSICALPROPERTIES Thenumericalsimulationofscaledmoderndevicescanyield questionableresults, iftheuserdoesnotadequatelyunderstandthephysicalmode ls(includingboundary conditions)andassumptionsmadeforthesimulationsbeing performed.Furthermore,the modelingofcertainpropertiesasappropriateforthenewer andmoreadvancedtransport modelsisnotfullymature,andthedicultyofnumericalsim ulationmakesnecessary assumptionsforthesemodelswhicharequestionable.Never theless,highlyscaleddevices exhibiteectsthatcanonlybeadequatelycapturedbytrans portmodelsthatpossess sucientcomplexitytocaptureobservedbehavior.Itisdes irabletobalancetheneedfor increasingcomplexitywithskepticismwithregardtocerta insimulationeects.What followsisadetaileddiscussionoftheoriginofeachofthet ransportmodelsandthe correspondingusageofimpedance-eldnoisesimulation,a swellasthemodelingofthe physicalpropertiesofthematerialsusedforthesimulatio n. 2.1AdvancedTransportModelsforPDE-basedDeviceSimulat ion TheSchrodingerwaveequation(SWE),Poisson'sequation( PE),andtheBoltzmann transportequation(BTE)togetherdescribethetransporto fchargethroughany semiconductordevice(theSWEandPEtogetherdenetheallo wedstatesinthesystem andtheBTEandPEtogetherdescribehowthecarriersoccupyt hosestates,reactto electriceldsandpotential,andtransportchargethrough thedevice).Transportmodels aregenerallysimplicationsoftheseequationswhichcanb esolvednumericallyorin closedform. Generally,insemiconductorssuchasSilicon,thecarriers arenearlyfree,andthe onlywaytheSWEandPEaecttransportisinthedenitionofp arameterssuchas mobilityandintrinsiccarrierdensities,sinceelectrons intheconductionbandandholes inthevalencebandarenearlyfreecarriers.However,intig htlyconnedareassuchasthe higheldregionunderthegateoxideinahighlyscaledandst ronglyinvertedMOSFET, 22

PAGE 23

thesolutionsoftheSWEandPEgivequantizedstatesorsub-b andsthatalterthese properties. Thefollowingderivationsofvarioustransportmodelsuset heBTEandPEtodene themotionofsemi-classicalcarriersmovinginasemicondu ctordevice.TheSWEdoesnot enterthederivation,savethroughmodelingofband-relate dparameters. Theimplicationsofneglectingthequantizationeectinth edevicesmeasuredinthis workwillbediscussedafterpresentingthetransportmodel sandthecomputationofnoise associatedwitheachone.2.1.1Poisson'sEquation Poisson'sEquationrelateschargedensitytotheelectric eldandpotentialinthe semiconductor,andisgivenasfollows: r 2 = q p n + N + D N A t (2{1) where istheelectrostaticpotential, q isthefundamentalelectroncharge, isthe permittivityofthesemiconductor,and N + D and N A aretheionizeddonorandacceptor concentrations.Theholeandelectronconcentrationsare p and n .Thevolumecharge densityduetochargedelectronandholetrapsis t .Thesemaybeneutraldefectstates withtrappedcarriersordonororacceptorlikestateswhich arechargedandempty. 2.1.2MomentsoftheBoltzmannTransportEquation TheBoltzmannTransportEquationkeepstrackofalltransit ionsbetweenallowed carrierstatesinasemiconductor,conservingbothenergya ndmomentum.Theoccupancy function f ( ~r;~p;t )givesthefractionofallowedstateswhichareoccupied.In generalit isafunctionofposition,momentum,andtimebutforelectro nsinasemiconductorin electricandthermalequilibriumitistheFermi-Diracdistr ibution.Innon-equilibriumthe BoltzmannTransportEquationdenes f asfollows: df dt + ~v r r f ~ F r p f = s ( ~r;~p;t )+ @f @t Coll (2{2) 23

PAGE 24

where ~v isthevelocityofthecarriersoccupyingstatesattheconsi deredmomentum, positionandtime.Theforceimpingingonsuchcarriersis ~ F .Theterm s ( ~r;~p;t )accounts forthechangein f duetointerbandcarriertransitionstoandfromtheconside red positionandmomentum,and @f @t j Coll accountsforchangesin f duetointraband transitionsfromonemomentumtoanother.TheBTEisarrange dhereinthiswayto highlightthefactthattermsonthelefthandsidearedeterm inisticquantitiesgoverned bytheelectrongasandelectrostatics.Thetermsontherigh thandsideinvolvecollisions withphonons,photonsandothernon-carrierparticlesanda rethereforestochastic.These right-hand-sidetermsprovidetheructuatingquantitiesw hichcauseintrinsicdevicenoise. BymultiplyingtheBTEbysome n -orderedfunctionsofmomentum ( ~p )and summingoverallallowedmomenta,balanceequationsarede nedwhichnearlyaccount forallofthechargetransporteects(butalwayssomenon-p hysicalassumptionmust bemadetoclosethesystemofequationssuchthattheycanbes olved).Considering higher-ordermomentsoftheBTE,moreoftheeectsofthephy sicscanbecaptured, butthephysicalmodelingofsometermsbecomesmoreuncerta inbecausetheseterms aremorediculttovisualize.Asaresult,itisdesirable(f orclarityofresultsaswellas forcomputationaleciency)tokeepthenumberofmomentsco nsideredtotheminimum neededtoclosethesystemofequationsandtodescribethepe rtinentphysics. Generally,amomentoftheBTEyieldsabalanceequationofth efollowingform. d dt 1 n X p f # + r r 1 n X p ~vf # ~ F 1 n X p f r p # = 1 n X p s + 1 n X p @f @t Coll (2{3) Thevalidityofmovingthe d dt and r r operationsoutsidethersttwotermsandofthe chainrulingofgradientstotransformthethirdtermisdisc ussedinLundstrom[ 6 ].The lasttermisgenerallyexpressedbyarelaxationtimeapprox imationasfollows: 1 n X p @f @t Coll = hh 1 ii 1 n X p ( f f 0 ) (2{4) 24

PAGE 25

wheretherelaxationtimeisdenedbytheensembleaverage hh 1 = ii ofthefollowing function. 1 ( ~p ) = X p 0 1 ( ~p 0 ) ( ~p ) S ( ~p;~p 0 )(2{5) Thebalanceequationsassociatedwiththesemomentshavebe enrigorouslyderived previously[ 6 7 ],butthisbearsrepeatingasthephysicaloriginsofeachte rminthese equationsshouldbeunderstoodinordertoproperlyconside rtherandomructuationof physicalquantitiesinsuchsystems(resultingfromthesto chastictermsontherighthand sideofEq.( 2{3 )). 2.1.2.1Zeroethordermoment:carriercontinuityequation s Thecarriercontinuityequationsresultfromconsideringt hezeroeth-ordermoment ( ( ~p )=1)oftheBTEforelectronsandholes.Theelectroncontinu ityequationresults fromthismoment(asinEq.( 2{3 ))asfollows: d dt h n i + r r h n ~ V n i 0=[G n R n ]+0(2{6) where ~ V n istheaverageelectronvelocity h ~v i ,andG n andR n aretheelectrongeneration andrecombinationrates,respectively.Thethirdtermisze robecausethegradient r p = r p 1iszero.Thesecondtermontherighthandsideiszerobecaus e ( ~p 0 )= ( ~p )=1 andtherefore1 ( ~p 0 ) ( ~p ) =0forall( ~p;~p 0 ).Thisequation,condensed,andthecorresponding equationforholesaregivenasfollows: dn dt = r ( n ~ V n )+G n R n (2{7) dp dt = r ( p ~ V p )+G p R p (2{8) 2.1.2.2Firstordermoment:momentumbalanceequations Themomentumbalanceequationsresultfromconsideringthe rst-ordermoment ( ( ~p )= ~p = m ~v )oftheBTEforelectronsandholes.Theyarevector-valued. The 25

PAGE 26

momentumbalanceequationforelectronsresultsfromthism omentasfollows: d dt h n ~ P n i + r r h n ^ U n i ( q ~ En )=0+ n ~ C P n (2{9) where ~ P n istheaverageelectronmomentum h ~ ~ k i ^ U n isthekineticenergytensor h ~v ~ ~ k i ~ E istheelectriceld,and ~ C P n isthechangeduetocollisionsinmomentumperelectron. Thersttermontherighthandsideiszerobecausetherandom velocityofgenerated orrecombinedcarriersisdistributedwithsphericalsymme trysuchthatthestatistical averageof ~vs ( ~p )iszero.Thisequation,condensed,andthecorrespondinge quationfor holesaregivenasfollows: d ( n ~ P n ) dt = r ( n ^ U n ) nq ~ E + n ~ C P n (2{10) d ( p ~ P p ) dt = r ( p ^ U p )+ pq ~ E + p ~ C P p (2{11) 2.1.2.3Secondordermoment:energybalanceequations Theenergybalanceequationsresultfromconsideringthese cond-ordermoment ( ( ~p )=E( ~p )= j ~p j 2 2 m )oftheBTEsforelectronsandholes.Theenergybalanceequa tionfor electronsresultsfromthismomentasfollows: d dt h nW n i + r r h n ~ S n i ( q ~ E ) ( n ~ V n )=[G W n R W n ]+ nC W n (2{12) where W n istheaverageelectronkineticenergy h E i ~ S n istheaverageenergyrux percarrier h E ~v i ,andG W n andR W n arethechangesinkineticenergyduetoparticle generationandrecombinationevents,respectively.Thech angeduetocollisionsinkinetic energyperelectronis C W n .Thisequation,condensed,andthecorrespondingequation for holesaregivenasfollows: d ( nW n ) dt = r ( n ~ S n ) q ~ E ( n ~ V n )+G W n R W n + nC W n (2{13) d ( pW p ) dt = r ( p ~ S p )+ q ~ E ( p ~ V p )+G W p R W p + pC W p (2{14) 26

PAGE 27

2.1.2.4Thirdordermoment:heatrowbalanceequations Theheatrowbalanceequationsresultfromconsideringthet hird-ordermoment ( ( ~p )=E( ~p ) ~p )oftheBTEsforelectronsandholes.Theheatrowbalanceequ ationfor electronsresultsfromthismomentasfollows. d dt h n ~ Q n i + r r h n ^ R n i ( q ~ E ) h nW n ^ I + n ^ U n i =0+ n ~ C Q n (2{15) where ~ Q n istheaverageheatrowperelectron h E ~ ~ k i ^ R n isthefourth-ordermoment tensor h ~v E ~ ~ k i ^ I istheunittensor,and ~ C Q n isthechangeduetocollisionsinheat rowperelectron.Thersttermontherighthandside,simila rtotheoneinthe rst-ordermoment,iszeroonstatisticalaverageforcarri ersgeneratedorrecombining withsphericallysymmetricdistributedmomenta.Thisequa tion,condensed,andthe correspondingequationforholesaregivenasfollows: d ( n ~ Q n ) dt = r ( n ^ R n ) q ~ E ( nW n ^ I + n ^ U n )+ n ~ C Q n (2{16) d ( p ~ Q p ) dt = r ( p ^ R p )+ q ~ E ( pW p ^ I + p ^ U p )+ p ~ C Q p (2{17) 2.1.3ClosureandReductionRelations Thesystemofequationsderivedthusfarisnotsolvable,ast herigorouscomputation of ^ R n wouldrequireafth-ordermoment,whichwouldinvolveterm srequiringevenhigher ordermoments,andsoon.Also,itisdesirabletoreducethen umberofsolutionvariables totheminimumrequiredtosolvethesystem,inordertoreduc ethecostofsolvingthe system.Therefore,closurerelationsareusedtoclosethes ystem,andreductionrelations areusedtoeliminatevariables.Theclosureandreductionr elationsusedinthisworkare givenasfollows: 27

PAGE 28

2.1.3.1Momentumbalanceequationandrelatedreductionre lations ~ P n = m n ~ V n ; ~ C P n = q ~ V n n + P n r ^ U n ; ^ U n = U n ^ I = kT n ^ I ~ P p = m p ~ V p ; ~ C P p = q ~ V p p + P p r ^ U p ; ^ U p = U p ^ I = kT p ^ I (2{18) where m n and m p aretheelectronandholeconductivityeectivemasses.The electron andholemobilitiesare n = q h P n i m n and p = q h P p i m p ,where h P n i and h P p i aretheelectron andholemomentumrelaxationtimes.Thevalues P n and P p arefactorlessempirical parametersthatmodelthebehaviorofthesystemininhomoge neouseldconditions, consistentwithMonteCarlocomputations[ 7 ].Therelationsinvolving ^ U n and ^ U p rerect theequipartitionenergyapproximationanddenethecarri ertemperatures. 2.1.3.2Energybalanceequationandrelatedreductionrela tions W n = 3 2 kT n ;C W n = ( W n W 0 n ) h W n i ;W 0 n = 3 2 kT L W p = 3 2 kT p ;C W p = ( W p W 0 p ) h W p i ;W 0 p = 3 2 kT L (2{19) Therelationsinvolving W n and W p rerecttheapproximationthatthecarrierkinetic energiesareentirelyduetorandomthermalmotionratherth anorganizeddriftingmotion; i.e., ~v = h ~v i + ~v and W n = h 1 2 m j ~v j 2 ih 1 2 m j ~v j 2 i = 3 2 kT n W n and W p aretheelectron andholeenergyrelaxationtimes.2.1.3.3Heatrowbalanceequationandrelatedreductionand closurerelations ~ Q n = m n ~ S n ; ~ C Q n = q ~ S n S n + Q n r ^ R n ; ^ R n = R n ^ I = 10 9 W 2 n ^ I ~ Q p = m p ~ S p ; ~ C Q p = q ~ S p S p + Q p r ^ R p ; ^ R p = R p ^ I = 10 9 W 2 p ^ I (2{20) Theelectronandholeheatrowmobilitiesare S n = q h Q n i m n and S p = q h Q p i m p ,where h Q n i and h Q p i aretheelectronandholeheatrowrelaxationtimes.Thevalu es Q n and Q p are factorlessempiricalparametersthatmodelthebehaviorof thesystemininhomogeneous 28

PAGE 29

eldconditions,consistentwithMonteCarlocomputations [ 7 ].Therelationsinvolving ^ R n and ^ R p rerecttheequipartitionenergyapproximationandcloseth esystemofequations. TheirvalidityhasalsobeenveriedbyMonteCarlocomputat ions[ 7 ]. 2.1.3.4PEandrstfourBTEmoments,closedandsummarized Thesystemofequationsthusderivedandclosedcanbesummar izedasfollows: r 2 = q ( p n + N + D N A ) t (2{21) dn dt = r ( n ~ V n )+G n R n (2{22) dp dt = r ( p ~ V p )+G p R p (2{23) m n d ( n ~ V n ) dt = r ( nkT n ) qn ~ E n q ~ V n n P n r ( kT n ) (2{24) m p d ( p ~ V p ) dt = r ( pkT p )+ qp ~ E p q ~ V p p P p r ( kT p ) (2{25) 3 2 k d ( nT n ) dt = r ( n ~ S n ) q ~ E ( n ~ V n )+G W n R W n 3 2 nk T n T L h W n i (2{26) 3 2 k d ( pT p ) dt = r ( p ~ S p )+ q ~ E ( p ~ V p )+G W p R W p 3 2 pk T p T L h W p i (2{27) m n d ( n ~ S n ) dt = r 5 2 nk 2 T 2 n q ~ E 5 2 nkT n n q ~ S n S n Q n r 5 2 k 2 T 2 n (2{28) m p d ( p ~ S p ) dt = r 5 2 pk 2 T 2 p + q ~ E 5 2 pkT p p q ~ S p S p Q p r 5 2 k 2 T 2 p (2{29) Thisfour-momentsystemisnotcommonlysolved,asitisvery complex,andthenumber ofvariablestobesolvedisstillquitehigh(9variablesfor 1-Dproblems,13for2-D problemsand17for3-Dproblems).Furtherapproximationsa rethereforemadetoget tothehydrodynamicsystemsthatarecommonlysolved.Howev er,thissystemwillbe usefulfordeterminingwhereructuationscoupleintothesy stemandhownoisesimulations shouldbeextendedtoincludehigher-ordermoments.2.1.4HydrodynamicModel Therstreductiontotheequationsderivedthusfarinvolve sthequasi-stationary approximationwithregardtotheheatrowbalanceequations ,namelythatthetime 29

PAGE 30

derivativetermsonthelefthandsidearenegligiblecompar edwithanyofthetermson theridehandside.Theseequationsarethenreducedfrompar tialdierentialequations tovectorexpressionsfortheheatrow.Thiseliminates2to6 variables(dependingon thedimensionalityofthesolutionspace)fromthesolution set.Theremainingsystem constitutesthehydrodynamicmodelandconsistsoftherst sevenpreviousEqs.( 2{21 { 2{27 )withtheremainingtwoauxiliaryrelationsforheatrow,as follows: r 2 = q ( p n + N + D N A ) t dn dt = r ( n ~ V n )+G n R n dp dt = r ( p ~ V p )+G p R p m n d ( n ~ V n ) dt = r ( nkT n ) qn ~ E n q ~ V n n P n r ( kT n ) m p d ( p ~ V p ) dt = r ( pkT p )+ qp ~ E p q ~ V p p P p r ( kT p ) 3 2 k d ( nT n ) dt = r ( n ~ S n ) q ~ E ( n ~ V n )+G W n R W n 3 2 nk T n T L h W n i 3 2 k d ( pT p ) dt = r ( p ~ S p )+ q ~ E ( p ~ V p )+G W p R W p 3 2 pk T p T L h W p i ~ S n = S n q 5 2 kT n q ~ E + 5 2 k 2 T 2 n r n n +(1 Q n ) r 5 2 k 2 T 2 n (2{30) ~ S p = S p q 5 2 kT p q ~ E + 5 2 k 2 T 2 p r p p +(1 Q p ) r 5 2 k 2 T 2 p (2{31) Thusthesystemtobesolvedisreducedtohaveasolutionseto fupto11variables: n p V nx V ny V nz V px V py V pz T n ,and T p 2.1.5ReducedHydrodynamic,or\EnergyBalance"Model Thenextanalogousreductiontotheequationsderivedthusf arinvolvesthe quasi-stationaryapproximationwithregardtothemomentu mbalanceequations,namely thatthetimederivativetermsonthelefthandsidearenegli giblecomparedwithanyof thetermsontheridehandside.Theseequationsarealsoredu cedfrompartialdierential equationstovectorexpressionsforthecarriervelocities .Thiseliminatesanadditional2to 30

PAGE 31

6variables(dependingonthedimensionality)fromthesolu tionset.Theremainingsystem constitutesthe\energybalance"modelandconsistsofPois son'sequation( 2{21 ),the electronandholecontinuityequations( 2{22 and 2{23 )andtheelectronandholeenergy balanceequations( 2{26 and 2{27 ),withtheauxiliaryrelationsforheatrow(Eqs.( 2{30 and 2{31 ))andtheadditionaltwoauxiliaryrelationsforcarrierve locities. r 2 = q ( p n + N + D N A ) t dn dt = r ( n ~ V n )+G n R n dp dt = r ( p ~ V p )+G p R p ~ V n = n q q ~ E + kT n r n n +(1 P n ) r ( kT n ) (2{32) ~ V p = p q q ~ E + kT p r p p +(1 P p ) r ( kT p ) (2{33) 3 2 k d ( nT n ) dt = r ( n ~ S n ) q ~ E ( n ~ V n )+G W n R W n 3 2 nk T n T L h W n i 3 2 k d ( pT p ) dt = r ( p ~ S p )+ q ~ E ( p ~ V p )+G W p R W p 3 2 pk T p T L h W p i ~ S n = S n q 5 2 kT n q ~ E + 5 2 k 2 T 2 n r n n +(1 Q n ) r 5 2 k 2 T 2 n ~ S p = S p q 5 2 kT p q ~ E + 5 2 k 2 T 2 p r p p +(1 Q p ) r 5 2 k 2 T 2 p Thusthesystemtobesolvedisreducedtohaveasolutionseto f5variables: n p T n and T p Toavoidthedirectcomputationofthevectorparticleveloc ity,theJouleheating termsoftheenergybalanceequations(suchas q ~ E ( n ~ V n ))aretypicallyrearrangedas follows[ 2 ]. q ~ E ( n ~ V n )= q r ( n ~ V n ) q r ( n ~ V n )= q r ( n ~ V n )+ q dn dt G n +R n (2{34) q ~ E ( p ~ V p )= q r ( p ~ V p )+ q r ( p ~ V p )= q r ( p ~ V p ) q dp dt G p +R p (2{35) The r ( n ~ V n )termscanbediscretizedusingSharfetter-Gummeltypemet hods[ 2 ]. 31

PAGE 32

2.1.6Drift-DiusionModel Furtherreductionofthesystemcanbemadebyconsideringth eelectronsandholes tobenearthermalequilibrium,suchthat T n T p T L .Forthiscase,theenergy balanceequationscanbeeliminatedandsometermscanbedro ppedfromtheexpressions forcarriervelocities(Eqs.( 2{32 )and( 2{33 )).Theremainingsystemconstitutesthe drift-diusionmodel,andconsistsoftherstthreeequati onsofthepreviouslyderived systems,andthesimpliedauxiliaryrelationsforthecarr iervelocities,asfollows. r 2 = q ( p n + N + D N A ) t dn dt = r ( n ~ V n )+G n R n dp dt = r ( p ~ V p )+G p R p ~ V n = n q q ~ E + kT L r n n +(1 P n ) r ( kT L ) (2{36) ~ V p = p q q ~ E + kT L r p p +(1 P p ) r ( kT L ) (2{37) Thusinthemostsimplecaseofbipolartransportthesystemt obesolvedisreducedtoa solutionsetof3variables: n and p 2.2ImplementationofSurfaceMobilityModelsinFLOODS InordertosimulatethenoiseinSOIdevices,itisrstneces sarytobeableto simulatetheDCandACcharacteristicsofthedevicewithreas onableaccuracy.This requirestheuseofamobilitymodelthattakesintoaccountt hedegradationofcarrier mobilityinsemiconductordevices(suchasSOIMOSFETs)inw hichthecarriersare tightlyboundnearaninsulatorsurface. Inaddition(andmoreimportantly),theimpedanceeldnois esimulationavailable inFLOODShasrecentlybeenextended[ 28 ]toincludetrappingatsilicon/silicon dioxideinterfacesandtunnelingbetweentheseinterfacet rapsandtrapslocatedinthe oxide.Thisnoisyprocessprovidesslowenoughtransitions toproducelowfrequency Lorentzian-shapednoisecomponents,andthedistribution ofoxidetrapsinpositioncanbe 32

PAGE 33

produceanexponentialdistributionoftimeconstantssuit ablefor1 =f or1 =f -likenoise. While1 =f -likenoiseisunderstoodtobecausedbythefundamentalnoi semechanism ofnumberructuation,itistruethatcarrierstrappedinthe oxidecauseructuationsin theelectriceldwhichbindsthecarriersinthechanneltot hesurface.Therefore,the eectivemobilityofthecarriersinthechannelmayructuat easasecondaryeect.While thisisnotexpectedtodominatethenoisespectrum,itmayha veaneectontheGreen's functionsproducedbythesystemofequationsinthatitgive sanextracouplingpath betweenthetrappedcarriercontinuityequationintheoxid enodeandtheexternaldevice contact(thechargingofthetrapnowcouplestoPoisson'sEq uationand,inaddition,the carriercontinuityequationattheinterfacenode). Itisimportanttonotethateventhoughthesurfacemobility modelsthathavebeen publishedarebasedonsomeexpectationoftheirphysicalfo rm,theyareempirically shapedtotmacroscopic\eective"mobilitieswhichareob tainedfromcompact model-typeexpressionsmatchedtomeasureddata.Therefor e,thecouplingfromthe trapsintheoxidetothemobilitymodeledinthechannelshou ldbeviewedasatoolto givepossiblephysicalinsightandqualitativeunderstand ing,ratherthanarigorously accuratequantitativeresult.Thesurfacemobilitymodelp arameterspublishedmayvary wildlyfromthoserequiredtomatchthedependenceofcarrie rmobilitiesonelectriceldin dierentprocesseswithdieringoxidationrecipesorimpu ritiespresentneartheinterface. Also,themodelisempiricallydeterminedforsingle-gateb ulkSiliconMOSFETsand whileitisexpectedtoapplytothepartiallydepletedSOIde vicesinthisstudy,results maydeviateifthemodelisappliedtohighlyscaledfullydep letedSOIMOSFETsor double-gateMOSFETswherethecarriersareexpectedtointe ractwithfrontandback interfacesorwheretheelectricelddistributionisveryd ierentintheinversionlayer(as insymmetricdouble-gatestructures).Thesemodelsalsowo uldlikelygiveerrantresultsin asystemsuchasthedensitygradientsystem[ 5 ]foraccountingforquantumconnement eectswherethecarriersaredisplacedadistancefromthei nterface. 33

PAGE 34

Earlysurfacemobilitymodelsusedthedistancefromasemic onductorinterfaceto modelthedecreaseinmobility[ 36 ].Thisisnon-physical,asitisexpectedthattighter electrostaticbindingtoasurfaceshouldfurtherincrease thescatteringofcarriersby theroughnessofthesurfaceandbysurfaceacousticphonons .Modernsurfacemobility models([ 37 ],[ 10 ])empiricallymodelthesurfacemobilitybydegradingthem obilityas afunctionofelectriceldperpendiculartotheinterface. Someimplementationshave usedanon-locallycomputed\eective"surfaceeldcomput edthroughouttheinversion layer.Thiswouldbeexpectedtogivethebestttothemacros copic\eective"mobility data,butcausessomedicultyinimplementation,asityiel dsnonzeroderivativeterms whicharefarothediagonaloftheJacobianmatrix,increas ingthecostofsolvingthe numericalsystemandrequiringdicultimplementationfor ageneralizedscript-based simulatorsuchasFLOODSforproblemsinwhichsmall-signalA Csolutionsarerequired. Someimplementationshaveusedthelocaleldperpendicula rtothenearestinterface (the\vertical"eldissometimesused,limitingtheapplic abilityoftheimplementation toplanarMOSFETs).Somehaveusedtheelectriceldperpend iculartotheedgesinthe mesh(theassumptionbeingthatifthereismuchcurrentrowi ngthroughtheedgethenit mustbealignedcloselywiththedirectionofthecurrentrow ,andiftheperpendiculareld ishighforthiscase,thentheedgemustbelocatedneartoasu rface-bindingeld).While thismaynotbeasaccurateasusingtheverticaleld,itdoes allowthemobilitymodelto beusedinawidervarietyofgeometries.Theexactimplement ationdoesnotseemtobe thatimportant[ 38 ]asinthelimitsofdensemeshesandhighsurfaceelds(whic harethe casesthatareofmostconcern)theseassumptionsbecomemor evalid. 2.2.1LowKineticEnergyMobilityFormulation ThesurfacemobilitymodelimplementedinFLOODSforthiswor kisthatpresented byDarwish,et.al[ 10 ].ThismodelwasformulatedasanupdatetotheLombardi model[ 37 ]whichissimilarinformbutunder-predictsthedegradatio nofthemobility atveryhighelectriceldsinscaledprocesseswhichareexp ectedtohavesimilarsurface 34

PAGE 35

mobilitymodelparameterstothoseoftheirlesshighlyscal edpredecessors,duetosimilar oxidationprocessing.TheDarwishmodeldegradesthebulkmo bility( b )byusing Mathiessen'sRuletoreciprocallycombineitwithtwoother mobilityterms,asfollows. 1 0 ( ~r ) = 1 b ( ~r ) + 1 ac ( ~r ) + 1 sr ( ~r ) (2{38) Theresult 0 isthelowkineticenergymobilitywhichwillinturnbedegra dedforhot carriereects,aswillbediscussedlater.Thevalues ac and sr aremobilityterms whichaccountforsurfaceacousticphononscatteringandsu rfaceroughnessscattering, respectively.Thesurfaceacousticphononscatteringterm iscomputedasfollows. ac ( ~r )= B E ? ( ~r ) T L 300 + CN I ( ~r ) E 1 = 3 ? ( ~r ) 300 T L (2{39) where B and C arebasedonphysicallyderivedquantities,butareexpecte dtovary withxedinterfacechargeandarethereforetreatedastpa rameters.Thevalue isan empiricaltparameter,and isthetemperaturedependenceoftheprobabilityofsurface phononscattering(1.7forelectronsand0.9forholes).The value N I = N + D + N A isthe totaldensityofionizedimpurities.Thesurfaceroughness scatteringterm(dominantfor highperpendicularelds)iscomputedasfollows. sr ( ~r )= E r ? ( ~r ) (2{40) where isatparameterwhichdependsontheoxidationprocess,and isexpectedto bevarywildlyfromprocesstoprocess.Theparameter r isgivenaweakdependenceon inversioncharge(itisheldconstantintheLombardiformul ation)asfollows. r = A + n ( ~r ) N I ( ~r ) (2{41) where A ,and arettingparameters. 35

PAGE 36

2.2.2HotCarrierMobility ElectronsandholesinSiliconarecharacterizedbyvelocit ythatsaturatesunderthe conditionofhighhomogeneouseld,orhighaveragekinetic energy.Indrift-diusion simulations,theacceleratingelectriceld(paralleltot hedirectionofcurrentrow)isused tomodelthevelocitysaturationeect,thoughinhighlysca leddevicestheeldisfarfrom homogeneousandthekineticenergyoftheelectrondistribu tionisamoreappropriate andphysicalmeasureformodelingthiseect.TheDarwishmob ilitymodelwasoriginally developedforuseindrift-diusionsimulations,andtheve locitysaturationexpressionused wasthatofHansch,asfollows. ( ~r )= 2 0 ( ~r ) 1+ 1+4 0 ( ~r ) E k ( ~r ) v sat 2 1 2 (2{42) Inthecontextofahydrodynamicsimulation,thevelocitysa turationeectshould ratherbemodeledasafunctionofaveragekineticenergyore lectrontemperature.A commonapproachusedforthisistoconsidertheelectronorh oleenergybalanceequation undertheconditionofalong,homogeneouseld.Thisyields anexpressionforcarrier temperatureasafunctionofsuchalongstationaryhomogene ouseld E s .Theinverseof thisfunctionisexpressedasfollows(forelectrons). E s = s 3 2 k ( T n T L ) q n E n (2{43) Sincethisexpressionisafunctionoftheoverallmobility, thevelocitysaturation expressionmustberearrangedalgebraically.Thevelocity saturationexpressionformulated byCanaliismoreeasilyrearranged,andisexpressedfordri ft-diusionandforhydrodynamic simulationasfollows. ( ~r )= 0 ( ~r ) 1+ 0 ( ~r ) E k ( ~r ) v sat 1 = (2{44) 36

PAGE 37

( ~r )= 0 ( ~r ) r 1+ 1 4 3 2 0 k ( T n T L ) q E n v 2 sat + r 1 4 3 2 0 k ( T n T L ) q E n v 2 sat 1 = (2{45) where istypicallychosenas2forelectronsand1forholes. 2.2.3Implementation-SpecicDetails Intheinterestofusingthismobilitymodelforsmall-signa lACorimpedanceeld noisesimulation,itisimportantonceagaintonotethat exactderivativesofthepartial dierentialequationswithrespecttothesolutionvariabl esarerequired .Theutilityof theAlagatorscriptingformatforFLOODSisthattheequation parserknowshowto takeanalyticalderivativesofsimpleexpressions.Theref ore,theonlyprimaryconcernis thatofcomputingthetransverseelectriceld,whichisnot aruxtermbutdependson thepotentialatmorethantwonodes.Thisproblemhasbeenad dressedbymodifying FLOODStoallowtheassemblyoftheJacobianmatrixbytriangu larfacesor3-Delements ratherthanbynodesandedgesonly. Anoperator\ trans(a) "hasbeenwrittenwhichcomputesthemagnitudeofthe electriceldperpendiculartomeshedgesinanelement,asa functionofthepotentialat theothernodesintheelement.Avectoreldisrstcomputed fortheelementusingthe methodofleast-squares,asfollows. 266666664 P Ni =1 x 2i P x i y i P x i z i P x i P x i y i P y 2 i P y i z i P y i P x i z i P y i z i P z 2 i P z i P x i P y i P z i N 377777775 266666664 E x E y E z j ~r =0 377777775 = 266666664 x 1 x 2 :::x N y 1 y 2 :::y N z 1 z 2 :::z N 11 ::: 1 377777775 266664 1 ... N 377775 (2{46) where N isthenumberofnodesintheelement,and x i y i z i ,and i arethe x y ,and z coordinatesandpotentialatnode i ,respectively.The4x4matrixonthelefthandsideis invertedandmultipliedbythe4x N matrixontherighthandside,andtherst3rowsare storedasa3x N matrixwhichdependsonlyonthegeometryoftheelementandt herefore onlyneedstobecomputedonceperelement.Thismatrix,mult ipliedbythe N x1vector 37

PAGE 38

ofanysolutionvariable'svaluesatthe N nodestoyieldthe x y ,and z componentsofthe vectorgradientfortheelement( E x E y ,and E z ,for psi astheargument).Thecomponent perpendiculartoanedgeisthencomputedasfollows. E ? ij = q (1 x ij =l ij ) E 2 x +(1 y ij =l ij ) E 2 y +(1 z ij =l ij ) E 2 z (2{47) Ifitisdesiredtocompute E ? astheeldperpendiculartothenearestinterfacerather thananedge,thesamevectoreldfortheelementcanbeused, andthecomponentis computedinthedirectionofthenearestinterfacenode(int hiscasetheresultwouldbe thesameforeachedgeintheelement). 2.3ImplementationoftheEnergyBalanceTransportEquatio nsinFLOODS FLOODSusesthestandardgeneralizedboxdiscretizationtec hniquetosolvethe PDEswhicharespeciedinstringspassedtotheAlagatorpars erinTclscripts.The Sharfetter-Gummeldiscretizationmethod[ 39 ]providesanumericallystableexpression foranyedge-evaluatedruxoftheform ~ f =( x r y c r x )where c isconstantalongthe edge, y varieslinearlyalongtheedge,and x variesnon-linearly.Theelectronrowina drift-diusionsystem n ~ V n inEquation 2{36 tsthisform(andisindeedtheexpression thiswasoriginallyderivedfor).Thediscretizedruxalong anedgebetweennodes i and j isthenexpressedasfollows. f ij = c l ij x i B ( y i y j ) c x j B ( y i y j ) c (2{48) B ( z )= z= ( exp ( z ) 1)istheBernoullifunction.Thisexpressionhasbeenusedf ordrift diusiontransport,scriptedas\ c*sgrad(x,y/c) "butintheenergybalancetransport modeltheelectronrow n ~ V n inEquation 2{32 includestheelectrontemperature T n ,which varieslinearlyacrossanedge.Fortunately,aSharfetterGummel-likediscretizationhas beenderived[ 40 ]whichprovidesanumericallystableexpressionforthoser uxeswherethe 38

PAGE 39

c termvariesalongtheedge,expressedasfollows. f ij = c if l ij x i B ( y i y j ) c ij x j B ( y i y j ) c ij (2{49) c ij = c i c j ln( c i =c j ) (2{50) Anoperatornamed\ hdsgrad(x,y,c) "hasbeenimplementedwhichevaluatesthis discretizedexpressionandprovidesitsfullandcompleted erivativeswithrespect toanyarbitraryvariable,provided(byFLOODS)thederivati vesofthearguments withrespecttothesamearbitraryvariable.Thefunction f ij canthenbescriptedas \ hdsgrad(x,y,c) ,"andtheelectronrowintheenergybalanceformulationcan be scriptedas\ ($mun/$q)*hdsgrad(Elec,$q*DevPsi-(1-$lamp)*($Un),$U n) ."\ $mun "is theTclvariablerepresentingtheelectronmobiliy,\ $q "isthefundamentalcharge,\ $Un istheelectronthermalenergy kT n ,\ Elec "and\ DevPsi "are n and ,respectively,and \ $lamp "isfromthevalue P n fromtheenergybalanceclosurerelations. Theelectronheatrux n ~ S n isrearrangedtofollowthisSharfetter-Gummellikeformas followsandsimilarlyscripted. n ~ S n = S n q 5 2 kT n n r q +(2 Q n 1) 5 2 kT n 5 2 kT n r 5 2 kT n n (2{51) TheJouleheatingterm q r ( n ~ V n )isrecastas r ( q n ~ V n ) q ( dn dt G n + R n ).The rsttermrequiresthevector q n ~ V n tobecomputedalongedges,andcanbediscretized bymultiplyingtheparticlerowexpressionbytheaveragepo tentialalongtheedge.Thisis scriptedas\ Devpsi*$mun*hdsgrad(Elec,$q*DevPsi-(1-$lamp)*$Un,$U n) ." Finally,theenergyrelaxationtermsareobservedtobeprob lematic,particularlyin depletionregionswherethecarrierdensitygetsverysmall .Toavoidtheseconvergence problems,asmalldensity(ontheorderof10 3 cm 3 )isaddedtothecarrierdensityinthe energyrelaxationterms.Thisgreatlyimprovestheconverg encebyensuringaverysmall butniteenergyrelaxationindepletionregions. 39

PAGE 40

2.4ImpactIonization InordertoaccuratelycapturetheroatingbodyeectinSOId eviceswhichgives risetotheexcessnoise[ 1 ],itisnecessarytomodeltheelectronimpactionizationwh ich generatesholestoelevatethebiasoftheroatingbody. Themostaccuratemethodsofcomputingimpactionizationra tesinvolvedirect solutionoftheBoltzmannTransportEquation,orindirects olutionviaShockley's LuckyElectronmodelinordertoaccuratelycomputethenumb erofcarriersthat havesucientenergyandmomentumtoexciteanelectronfrom thelocaldistribution ofvalencebandelectronstotheconductionband.However,w ithinthescopeofthis work(numericalsimulationviaBTEmoments)someassumptio nmustbemadeto modeltheimpactionizationrate.Commonassumptionsareth atthelocalelectron distributionisfunctionallysimilartotheequilibriumdi stribution,displacedinenergy, andthatthedisplacementofthisfunctioninenergyisrelat edonlytothelocalelectric eld.Therstassumptionisallthatisrequiredforsimulat ionwithinthehydrodynamic orenergy-balancemodels,astheenergydisplacementisdir ectlyrelatedtotheelectron temperature.Fordrift-diusionsimulationitisnecessar ytodoextranon-localcomputation suchasshootingmethodsorlucky-electronpost-processin gtoascertainthekineticenergy ofthecarriersifthesecondassumptionisnotmade.Theseme thodsarenotnecessarily appropriatefornoisesimulation,however,asthederivati vesofthesetermsarenot availableforinclusionintheACJacobianmatrix.Thisleav esuswiththetraditional assumptionthattheimpactionizationratescanbemodelled asafunctionoflocalelectric eld(paralleltocurrentrow). Sinceimpactionizationiscausedbyafractionofthecarrie rsmovingthrougha localityinthesemiconductor,it'srateisusuallymodeled asfollows. G II = n ~ J n + p ~ J p (2{52) 40

PAGE 41

where n and p aretheelectronandholeimpactionizationcoecients,res pectively. Inn-channelFETs,electronimpactionizationdominates,a ndholeimpactionizationis usuallyatmostasecond-ordereect;thesecondtermcangen erallybeignored(orthe rstterm,inp-channelFETs).Forlonghomogeneoussemicon ductorsinwhichthecarrier distributionisallowedtoreachaquasi-equilibriuminaco nstantelectriceld[ 2 ],the coecientsarefoundtofollowtheChynowethrelation[ 41 ],asusedbyvanOverstraeten [ 42 ]. ( E k )= Ae ( B=E k ) (2{53) where A and B aretparametersthathavebeenpreviouslycharacterizedf orthe homogeneouseldcase. Theextensionofthisimpactionizationmodeltoenergy-bal anceorhydrodynamic simulationinvolvestheassumptionthatinascaleddevicet herelationshipofimpact ionizationratetoaveragecarrierkineticenergyisthesam easthatinalongsemiconductor withhomogeneouseld(consistentwiththepreviousassump tions).Theequationrelating ahomogeneouselectriceldtocarriertemperature(Equati on 2{43 )isusedtoderivethe followingrelationship. n ( T n )= Ae B r 2 q n E n 3 k ( T n T L ) (2{54) Sinceinhighlyscaleddevicesthecarrierdistributiondoe snotreachanequilibrium andinlightoftheassumptionsmadeinderivingtheseexpres sionsaswellasthe dependenceoncarrierenergyondefect-relatedscattering rates,itisnotexpectedthat theparameters A and B shouldagreebetweenaccuratelycalibrateddrift-diusio nand hydrodynamicmodels,norshouldtheynecessarilyagreewit hpreviouslydetermined empiricalvalues.Therefore,forthepurposesofmatchingm easurementstosimulationsin thisworktheyaretreatedastparameters. 41

PAGE 42

2.5ImpedanceFieldSimulationofVelocityFluctuationNoi se Theimpedanceeldmethodhasbeenusedpreviously[ 26 27 ]tosimulatenoisein semiconductordevicesusingFLOODS[ 43 44 ].Localnoiseructuationsaremodeledusing rst-principlephysicsandtheirmappingtotheexternalci rcuitarecomputedusingthe Green'sfunctions,whichdescribetheeectofasmallsigna lAC-ructuationinoneofthe system'sPDEsataninternalnodeonthepotentialorcurrenta tacontact. Tosimulatephysicalructuationmechanismsusingadvanced transportmodelssuch asthoseinsection 2.1 ,itisnecessarytotracetheructuationtermsthroughtheir origins intheBTE(Eq.( 2{2 ))toeachcontributiontothesystemofequations.Thisisdo nefor velocityructuationnoiseinthefollowingsection.2.5.1VelocityFluctuationNoiseSimulation Velocityructuationsarecausedbyintrabandtransitionso fparticlesfromone momentumstatetoanotherduetocollisionswithphonons,ra ndomlylocatedionized impurities,andotherparticles.Electronsscatter,gaini ngorlosingmomentum.Microscopically, thecollisiontermoftheBTEcanbedescribedbyastatistica lmeancomponentanda stochasticcomponent,whichructuatesrandomlyandwithaz erostatisticalmean. @f @t Coll ( ~p )= X p 0 ( ~p 0 ) ( ~p ) 1 S ( ~p;~p 0 ) f ( ~p )+ r f = f ( ~p ) ( ~p ) + r f (2{55) where r f isaLangevinnoisesourcewhichrepresentsaructuationin @f @t Coll .These ructuationscanalsobecharacterizedbyaructuationovera shorttime intheaverage electronandholevelocitiesaroundastatisticalmeanvalu e. ~ V n = 1 n X p ~vf = ~ V n + 1 n X p ~vr f = ~ V n + ~r v n (2{56) Sincetheseructuationsarecausedbycollisionswithphono nsandotherparticles, theirassociatedLangevinnoisesourcesareinsertedintot hesystem'spartialdierential equationsthroughthosetermsderivedfromthecollisionte rmoftheBTE(Eq.( 2{2 )). Thisshouldbeconsideredforeachmomentindividually.For thezeroeth-ordermoment, 42

PAGE 43

thecollisiontermoftheBTEiszero,sinceforeverycollisi onapositive @f=@t atone momentum ~p isosetbyanequalnegative @f=@t atadierentmomentum ~p 0 .Forthe rst-ordermoment,thecollisiontermischaracterizedasf ollows. 1 n X p ~p @f @t coll = n ~ C P n + 1 n X p ~pr f = n ~ P n h P n i P n r ( kT n ) + m n n~r v n (2{57) Forthesecond-ordermoment,thistermischaracterizedasf ollows. 1 n X p j ~p j 2 2 m n @f @t coll = nC E n + 1 n X p j ~p j 2 2 m n r f = 3 2 nk T n T L h W n i + nr w n (2{58) Ifthecollisionsareconsideredtobenearlyelasticsuchth at j ~p jj ~p 0 j r w n canbeassumed tobezero.Forthethird-ordermoment,thistermischaracte rizedasfollows. 1 n X p j ~p j 2 2 m n ~p @f @t coll = n ~ C Q n + 1 n X p j ~p j 2 2 m n ~pr f = n ~ Q n h Q n i Q n r ( 5 2 k 2 T 2 n ) + m n n~r s n (2{59) Byexaminationofthederivationofthethird-ordermoment, comparedtothederivationof therst-ordermoment,itbecomesobviousthat ~r s n canberelatedto ~r v n asfollows. ~r s n = 5 2 kT n ~r v n (2{60) Therefore,thevelocityructuationLangevinsources ~r v n and ~r v p enterthesystem ofequationsderivedfromtherstfourmomentsoftheBTEasf ollows.Notethatthe asteriskstodenoteeectivemassesandmobilitiesareomit tedsotheywillnotbeconfused 43

PAGE 44

withcomplexconjugatesinthelaterphasorexpressions. r 2 = q ( p n + N + D N A ) t dn dt = r ( n ~ V n )+G n R n dp dt = r ( p ~ V p )+G p R p m n d ( n ~ V n ) dt = r ( nkT n ) qn ~ E n q ~ V n n P n r ( kT n ) + m n n~r v n (2{61) m p d ( p ~ V p ) dt = r ( pkT p )+ qp ~ E p q ~ V p p P p r ( kT p ) + m p p~r v p (2{62) 3 2 k d ( nT n ) dt = r ( n ~ S n ) q ~ E ( n ~ V n )+G W n R W n 3 2 nk T n T L h W n i 3 2 k d ( pT p ) dt = r ( p ~ S p )+ q ~ E ( p ~ V p )+G W p R W p 3 2 pk T p T L h W p i m n d ( n ~ S n ) dt = r 5 2 nk 2 T 2 n q ~ E 5 2 nkT n n q ~ S n S n Q n r 5 2 k 2 T 2 n + 5 2 kT n m n n~r v n (2{63) m p d ( p ~ S p ) dt = r 5 2 pk 2 T 2 p + q ~ E 5 2 pkT p p q ~ S p S p Q p r 5 2 k 2 T 2 p + 5 2 kT p m p p~r v p (2{64) Theructuationofvoltage ~ V c attheconsideredexternalcontactisthengivenasfollows. ~ V c = ZZZ r 0B@ ~ G P n ( m n n~r v n )+ ~ G P p ( m p p~r v p ) + ~ G Q n 5 2 kT n m n n~r v n + ~ G Q p 5 2 kT p m p p~r v p 1CA d 3 r (2{65) where ~ G P n ~ G P p ~ G Q n and ~ G Q p aretheGreen'sfunctionsassociatedwiththevector-value d electronandholemomentumbalanceequationsandelectrona ndholeheatrowbalance equations,respectively.Thepowerspectraldensity S V =2 T ~ V c ~ V c iscomputedasfollows. S V =2 T ZZZ r i 0B@ ~ G P n ( m n n~r v n )+ ::: ::: ~ G Q p 5 2 kT p m p p~r v p 1CA d 3 r i ZZZ r j 0B@ ~ G P n ( m n n~r v n )+ ::: ::: ~ G Q p 5 2 kT p m p p~r v p 1CA d 3 r j (2{66) 44

PAGE 45

where T isthelengthofthetimewindowofthemeasurement.Iftheimp licitapproximation ismadethatthenoisesources ~r v n and ~r v p arestatisticallyindependentinposition,the followingapproximationcanbemade. 2 T ~r v n ( ~r i ) ~r v n ( ~r j )= ^ K v n v n ( ~r i ~r j )(2{67) where ^ K v n v n isthelocalnoisestrengthtensorofelectronvelocityruct uations,and ( ~r )is theDiracdeltafunction.Thelocalnoisestrengthscanbeeva luatedusingMonteCarlo computationstodirectlycomputethevariouslocalnoisest rengthtensorsasFourier transformsofthevelocity-velocity,velocity-energyand energy-energycorrelationfunctions, ashasbeenshownpreviously[ 32 45 46 ].Theadvantageofthisisthat r w n and r w p can becomputedforinelasticscatteringmechanismswithoutan yfurtherphysicalmodeling. ThedownsideisthattheMonte-Carlocomputationscanonlyb efeasiblyperformedatRF frequenciesandhigher.Analternativemethodtoevaluatet heseexpressionsisusingthe localnoisestrengthspectraltensorsthatfollow,consist entwithMilatz'theorem. ^ K v n v n = 4 D n n h P n i 2 ^ I (2{68) ^ K v p v p = 4 D p p h P p i 2 ^ I (2{69) ~r v n ( ~r i ) ~r v p ( ~r j )= ~r v p ( ~r i ) ~r v n ( ~r j )=0(2{70) where D n and D p aretheelectronandholediusivities.Thesediusivities aretypically modeledusingEinstein'srelation D = kT q witheitherthecarriertemperaturesor thelatticetemperature.Recentwork[ 45 ]hasshownthatneithertemperaturegives strictlyaccuratelocalnoisestrengths,butthatthecorre ctlycomputedvalues,usingthe Fouriertransformofthevelocityautocorrelationfunctio ncomputedfromMonteCarlo simulations,issomewhereinbetweenthelatticeandcarrie rtemperatures.Thepublished guresofthatworksuggestthatusinganaverageofthetwote mperatureswillbecloser thanusingeitherone,andthisissatisfyingsincethescatt eringprocessinvolvescollisions 45

PAGE 46

betweencarriers(characterizedbythecarriertemperatur e)andphonons(characterizedby thelatticetemperature). Withthe ~r ~r termsmodeledasabove,thepowerspectraldensityexpressi oncanthen besimpliedasfollows. S V = ZZZ r 0B@ m 2n n 2 ~ G P n ^ K v n v n ~ G P n + 5 2 kT n m 2n n 2 ~ G P n ^ K v n v n ~ G Q n + ::: ::: + 5 2 kT p m 2p p 2 ~ G Q p ^ K v p v p ~ G P p + 25 4 k 2 T 2 p m 2p p 2 ~ G Q p ^ K v p v p ~ G Q p 1CA d 3 r (2{71) Asmentionedpreviously,thiscompletefour-momentBTE-ba sedsystemofequations isnottypicallysolvedduetoitscomplexity,soforthesimp lertransportmodelsthe Langevinnoisesourcetermsmustbeappropriatelyfoldedin totheremainingPDEs. 2.5.1.1Hydrodynamicmodelwithvelocityructuationnoise TocorrectlyplacetheLangevinnoisetermsinthehydrodyna micmodel,theLangevin termsarefoldedintotheremainingPDEsfromtheheatrowbala nceequations.Firstthe quasi-stationaryexpressionsforthecarrierheatrowsare rearrangedasfollows. ~ S n = S n q 5 2 kT n q ~ E + 5 2 k 2 T 2 n r n n +(1 Q n ) r 5 2 k 2 T 2 n + 5 2 kT n h Q n i ~r v n = ~ S n ~ s n n (2{72) ~ S p = S p q 5 2 kT p q ~ E + 5 2 k 2 T 2 p r p p +(1 Q p ) r 5 2 k 2 T 2 p + 5 2 kT p h Q p i ~r v p = ~ S p + ~ s p p (2{73) where ~ s n = 5 2 kT n S n n h P n i n~r v n and ~ s p = 5 2 kT p S p p h P p i p~r v p aretheLangevintermsrecast asructuatorsforthetotalcarrierheatrow n ~ S n or p ~ S p 46

PAGE 47

Thehydrodynamicsystemofequationswithvelocityructuat ionLangevinnoise sourcescanthenbeexpressedasfollows. r 2 = q ( p n + N + D N A ) t dn dt = r ( n ~ V n )+G n R n dp dt = r ( p ~ V p )+G p R p m n d ( n ~ V n ) dt = r ( nkT n ) qn ~ E n q ~ V n n P n r ( kT n ) + m n n~r v n m p d ( p ~ V p ) dt = r ( pkT p )+ qp ~ E p q ~ V p p P p r ( kT p ) + m p p~r v p 3 2 k d ( nT n ) dt = r ( n ~ S n ) q ~ E ( n ~ V n )+G W n R W n 3 2 nk T n T L h W n i + r ( ~ s n )(2{74) 3 2 k d ( pT p ) dt = r ( p ~ S p )+ q ~ E ( p ~ V p )+G W p R W p 3 2 pk T p T L h W p i r ( ~ s p )(2{75) ~ S n = S n q 5 2 kT n q ~ E + 5 2 k 2 T 2 n r n n +(1 Q n ) r 5 2 k 2 T 2 n ~ S p = S p q 5 2 kT p q ~ E + 5 2 k 2 T 2 p r p p +(1 Q p ) r 5 2 k 2 T 2 p Theructuationofvoltageattheconsideredexternalcontac tisthengivenasfollows. ~ V c = ZZZ r ~ G P n ( m n n~r v n )+ ~ G P p ( m p p~r v p )+( r G E n ) ( ~ s n ) ( r G E p ) ( ~ s p ) d 3 r (2{76) where G E n and G E p arethescalar-valuedGreen'sfunctionsassociatedwithth eelectron andholeenergybalanceequations,respectively.Thepower spectraldensityofthenoiseat thecontactisgivenasfollows. S V =2 T ZZZ r i 0B@ ~ G P n ( m n n~r v n )+ ::: ::: ( r G E p ) ( ~ s p ) 1CA d 3 r i ZZZ r j 0B@ ~ G P n ( m n n~r v n )+ ::: ::: ( r G E p ) ( ~ s p ) 1CA d 3 r j (2{77) 47

PAGE 48

Theadditionallocalnoisespectraltensorscanbeexpresse dasfollows,inaccordancewith Milatz'theorem,asintheprevioussection. ~r v n ( ~r i ) ~ S n ( ~r j )= ~ S n ( ~r i ) ~r v n ( ~r j )= 10 kT n S n n D n h P n i ^ I ( ~r i ~r j ) 2 T = ^ K v n S n ( ~r i ~r j ) 2 T (2{78) ~r v p ( ~r i ) ~ S p ( ~r j )= ~ S p ( ~r i ) ~r v p ( ~r j )=10 kT p S p p D p h P p i ^ I ( ~r i ~r j ) 2 T = ^ K v p S p ( ~r i ~r j ) 2 T (2{79) ~r v ( ~r i ) ~ S ( ~r j )= ~ S ( ~r i ) ~r v ( ~r j )=0(2{80) ~ S n ( ~r i ) ~ S n ( ~r j )=25 k 2 T 2 n 2S n 2n D n n ^ I ( ~r i ~r j ) 2 T = ^ K S n S n ( ~r i ~r j ) 2 T (2{81) ~ S p ( ~r i ) ~ S p ( ~r j )=25 k 2 T 2 p 2S p 2p D p p ^ I ( ~r i ~r j ) 2 T = ^ K S p S p ( ~r i ~r j ) 2 T (2{82) ~ S n ( ~r i ) ~ S p ( ~r j )= ~ S p ( ~r i ) ~ S n ( ~r j )=0(2{83) where representseither n or p ,and representstheoppositechoice.Thisresultsinthe followingsimpliedexpressionfor S V S V = ZZZ r 0B@ m 2n n 2 ~ G P n ^ K v n v n ~ G P n + m n n ~ G P n ^ K v n S n ( r G E n )+ ::: ::: m p p ( r G E p ) ^ K v p S p ~ G P p +( r G E p ) ^ K S p S p ( r G E p ) 1CA d 3 r (2{84) Thistypeofimpedanceeldnoisesimulationhasbeenperfor medpreviouslyasin[ 32 46 ] andhasbeenreferredtoas\accelerationructuation"noise sincethenoisesourcesappear oppositethe d ( n ~ V n ) dt terms. 2.5.1.2Energybalancemodelwithvelocityructuationnois e TocorrectlyplacetheLangevinnoisetermsintheenergybal ancemodel,the LangevintermsarefoldedintotheremainingPDEsfromthemom entumbalance equations.Firstthestationaryexpressionsforthecarrie rvelocitiesarerearrangedas follows. ~ V n = n q q ~ E + kT n r n n +(1 P n ) r ( kT n ) + h P n i ~r v n = ~ V n ~ n qn (2{85) ~ V p = p q q ~ E + kT p r p p +(1 P p ) r ( kT p ) + h P p i ~r v p = ~ V p + ~ p qp (2{86) 48

PAGE 49

where ~ n = qn h P n i ~r v n and ~ p = qp h P p i ~r v p aretheLangevintermsrecastasructuators forthetotalcarriercurrentdensity ~ J n = qn ~ V n or ~ J p = qp ~ V p .Thetotalcarrier heatrowructuatorscanbeexpressedrelativetothecurrent densityructuatorsas ~ S n = 5 2 kT n q S n n ~ n and ~ S p = 5 2 kT p q S p p ~ p .Theenergybalancemodel'ssystemof equationswithvelocityructuationLangevinnoisesources canthenbeexpressedas follows. r 2 = q ( p n + N + D N A ) t dn dt = r ( n ~ V n )+G n R n + 1 q r ( ~ n )(2{87) dp dt = r ( p ~ V p )+G p R p 1 q r ( ~ p )(2{88) ~ V n = n q q ~ E + kT n r n n +(1 P n ) r ( kT n ) ~ V p = p q q ~ E + kT p r p p +(1 P p ) r ( kT p ) 3 2 k d ( nT n ) dt = r ( n ~ S n )+ q r ( n ~ V n )+ q dn dt G n +R n + G W n R W n 3 2 nk T n T L h W n i + r ( ~ s n ) r ( ~ n )(2{89) 3 2 k d ( pT p ) dt = r ( p ~ S p ) q r ( p ~ V p ) q dp dt G p +R p + G W p R W p 3 2 pk T p T L h W p i r ( ~ s p )+ r ( ~ p )(2{90) ~ S n = S n q 5 2 kT n q ~ E + 5 2 k 2 T 2 n r n n +(1 Q n ) r 5 2 k 2 T 2 n ~ S p = S p q 5 2 kT p q ~ E + 5 2 k 2 T 2 p r p p +(1 Q p ) r 5 2 k 2 T 2 p Theructuationofvoltageattheconsideredexternalcontac tisthengivenasfollows. ~ V c = ZZZ r 1 q r G n ~ n 1 q r G p ~ p + r G E n ( ~ s n ~ n ) r G E p ( ~ s p ~ p ) d 3 r (2{91) where G n and G p arethescalar-valuedGreen'sfunctionsassociatedwithth eelectronand holecontinuityequations,respectively.Thepowerspectr aldensityofthenoiseatthe 49

PAGE 50

contactisgivenasfollows. S V =2 T ZZZ r i 0B@ 1 q r G n ~ n ::: ::: r G E p ( ~ s p p ) 1CA d 3 r i ZZZ r j 0B@ 1 q r G n ~ n ::: ::: ( r G E p ) ( ~ s p p ) 1CA d 3 r j (2{92) Theadditionallocalnoisestrengthspectraltensorscanbe expressedasfollows,alsoin accordancewithMilatz'theoremasintheprevioussections ~ n ( ~r i ) ~ n ( ~r j )=4 q 2 nD n ^ I ( ~r i ~r j ) 2 T = ^ K j n j n ( ~r i ~r j ) 2 T (2{93) ~ p ( ~r i ) ~ p ( ~r j )=4 q 2 pD p ^ I ( ~r i ~r j ) 2 T = ^ K j p j p ( ~r i ~r j ) 2 T (2{94) ~ n ( ~r i ) ~ p ( ~r j )= ~ p ( ~r i ) ~ n ( ~r j )=0(2{95) ~ n ( ~r i ) ~ S n ( ~r j )=10 kT n q S n n q 2 nD n ^ I ( ~r i ~r j ) 2 T = ^ K j n S n ( ~r i ~r j ) 2 T (2{96) ~ p ( ~r i ) ~ S p ( ~r j )=10 kT p q S p p q 2 pD p ^ I ( ~r i ~r j ) 2 T = ^ K j p S p ( ~r i ~r j ) 2 T (2{97) ~ ( ~r i ) ~ S ( ~r j )= ~ S ( ~r i ) ~ ( ~r j )=0(2{98) ~ S n ( ~r i ) ~ S n ( ~r j )=25 k 2 T 2 n 2S n 2n D n n ^ I ( ~r i ~r j ) 2 T = ^ K S n S n ( ~r i ~r j ) 2 T ~ S p ( ~r i ) ~ S p ( ~r j )=25 k 2 T 2 p 2S p 2p D p p ^ I ( ~r i ~r j ) 2 T = ^ K S p S p ( ~r i ~r j ) 2 T ~ S n ( ~r i ) ~ S p ( ~r j )= ~ S p ( ~r i ) ~ S n ( ~r j )=0 Itisimportanttonotethatsince ~ n ~ p ~ S n ,and ~ S p originateinthesamecollisionterm oftheBTE(Eq.( 2{2 ))as ~r v n intheaccelerationructuationscheme,theapproximation thatthesesourcesarestatisticallyindependentofpositi onhasnotlostanyvalidity, andthismodelisasaccuraterelativetotheaccelerationsc hemeasthequasi-stationary approximationfor ~ V n and ~ V p areappropriate.Whiletheructuationinvelocityofa singlecarrierrowingthroughasemiconductor(asinMonteC arlocomputation)maybe correlatedovershortdistances,theructuationsinaverag ecarrierdistributionvelocity ~ V n arenot. 50

PAGE 51

Theaboverelationsgivethefollowingsimpliedexpressio nforthepowerspectral density. S V = ZZZ r 0B@ 1 q 2 r G n ^ K j n j n r G n + 1 q r G n ( ^ K j n S n ^ K j n j n ) r G E n + ::: ::: + r G E p ( ^ K S p S p 2 ^ K j p S p + 2 ^ K j p j p ) r G E p 1CA d 3 r (2{99) 2.5.1.3Drift-diusionmodelwithvelocityructuationnoi se TheplacementoftheLangevinnoisetermsinthedrift-dius ionmodeliswellknown, andiseasilyobtainedfromthepreviousformulationsimply byignoringtheenergybalance andheatrowbalanceequationsandtaking T n and T p as T L .Thedrift-diusionmodel's systemofequationswithvelocityructuationLangevinnois esourcescanthenbeexpressed asfollows. r 2 = q ( p n + N + D N A ) t dn dt = r ( n ~ V n )+G n R n + 1 q r ( ~ n ) dp dt = r ( p ~ V p )+G p R p 1 q r ( ~ p ) ~ V n = n q q ~ E + kT n r n n +(1 P n ) r ( kT L ) ~ V p = p q q ~ E + kT p r p p +(1 P p ) r ( kT L ) Theructuationofvoltageattheconsideredexternalcontac tisthengivenasfollows. ~ V c = ZZZ r 1 q r G n ~ n 1 q r G p ~ p d 3 r (2{100) Thepowerspectraldensityofthenoiseatthecontactisgive nasfollows. S V =2 T ZZZ r i 1 q r G n ~ n 1 q r G p ~ p d 3 r i ZZZ r j 1 q r G n ~ n 1 q r G p ~ p d 3 r j (2{101) 51

PAGE 52

ThisexpressionisevaluatedusingMilatz'theoremwhichgi vesthefollowinglocalnoise strengthspectraltensors,asshownpreviously. ~ n ( ~r i ) ~ n ( ~r j )=(4 q 2 D n n ) ^ I ( ~r i ~r j ) 2 T = ^ K j n j n ( ~r i ~r j ) 2 T ~ p ( ~r i ) ~ p ( ~r j )=(4 q 2 D p p ) ^ I ( ~r i ~r j ) 2 T = ^ K j p j p ( ~r i ~r j ) 2 T ~ n ( ~r i ) ~ p ( ~r j )= ~ p ~ n =0 Theseexpressionsinturngivethegreatlysimpliedexpres sionforpowerspectraldensity, asfollows. S V = ZZZ r 1 q 2 r G n ^ K j n j n r G n + 1 q 2 r G p ^ K j p j p r G p d 3 r (2{102) 2.5.2ComparisonofVelocityFluctuationNoiseSimulation withEnergy BalanceandwithDrift-DiusionModels 2.5.2.1OneDimensional n + =n=n + ResistorSimulations Velocityructuationnoisesimulationswereperformedon n + =n=n + resistors,using theenergybalanceanddrift-diusionmodelssimulatedbyF LOODS.Thethicknessofthe n + regionsconsideredwere0 : 5 m .Thedopingdensitywas10 18 cm 3 inthe n + regions and10 17 cm 3 inthelowerdopedregions.Thelow-eldcarriermobilities usedweregiven bytheKlaassenModel[ 8 9 ],andthevelocitysaturationrelationsusedfollowtheCan ali modelforDrift-DiusionanditsEnergyBalanceequivalent[ 38 ],withelectronandhole saturationvelocitiesof1 : 07 10 7 cm = sand8 : 37 10 6 cm = s,respectively.Theratioofheat rowmobilitytocarriermobilitywaschosentobe0.8andthei nhomogeneityparameters P n and Q n werechosentobezero.Thediusivityusedtocomputelocaln oisestrength fortheenergybalancemodelusesEinstein'srelationwitht helowkineticenergymobility andtheaverageofthecarrierandlatticetemperaturesasdi scussedpreviously. Fig. 2-1 showsthedopingconcentrationandtheelectrontemperatur ecomputed usingtheenergybalancemodelfora0 : 1 m resistorwithabiasof25 mV .Theeectsof 52

PAGE 53

PeltiercoolingandJouleheatingastheelectronsmovefrom lefttorightcanclearlybe seen.Fig. 2-2 showsthecontributionstovelocityructuationnoise(theq uantityinside theintegralofEqs.( 2{101 )and( 2{92 )).Thisresultisinqualitativeagreementwiththe resultspublishedforthefullhydrodynamicanddrift-diu sionmodelsbyJungemann, et.al.[ 46 ].Theeectoftheenergybalancemodelonthelocalnoisecon tributionisto enhanceitwherethetemperatureincreasesinthedirection ofaveragecarriervelocity (theforceduetothetemperaturegradientassiststherowof carriers)andtodiminish thenoisecontributionwhenthetemperaturedecreasesinth edirectionoftheaverage carriervelocity(theforceduetothetemperaturegradient opposestherowofcarriers). Fig. 2-4 showsthetotalvelocityructuationnoiseforseverallengt hsofthelowlydoped region,forsimulationsusingboththeenergybalanceanddr ift-diusionmodelsandattwo biaspoints.Thepointsforthe0 : 1 m resistorat25 mV or0 : 5 V biasaretheintegralof thecurvesinFigs. 2-2 and 2-3 .Thisgureshowsthattheenergybalancemodelpredicts loweroverallvelocityructuationnoisethandrift-diusi on,andthiseectisenhanced athigherbias.ThisisfurtherillustratedbyFig. 2-5 ,whichshowstheratioofthenoise predictedbytheenergybalancemodeltothatpredictedbyth edrift-diusionmodelfor thesameresistorsandatthesamebiaspoints.2.5.2.2nMOSFETSimulations Velocityructuationnoisesimulationswerealsoperformed ona0 : 25 m n-channel MOSFET,usingtheenergybalanceanddrift-diusionmodels simulatedbyFLOODS. Theconsidereddevicehasa30 A gateoxide,andistypicalofrecenttransistordesigns, with100 nm shallowsourceanddrainextensionstoreduce2Dshortchann eleects.The transportparametersusedinthedrift-diusionandenergy balancemodelswerethesame asthoseusedintheresistorsimulations.TheDarwishmobili tymodelwasincludedto degradethemobilityinthechannelduetosurfacescatterin g. Fig. 2-6 depictsthedopingproleofthedevice,onalogscale.Thisp rovidesa pictureofthegeometryofthedevicewithshallow n + source/drainextensions.Fig. 2-7 53

PAGE 54

showsthecontributionofthelocalvelocityructuationsto thedraincurrentnoiseforthe drift-diusionmodelinthelinearregionofoperation.Fig 2-8 showsthecontributionof thelocalvelocityructuationstothedraincurrentnoisefo rtheenergybalancemodel, forthesamebias.Bothcasesclearlyshowthatvelocityruct uationsonlycoupleout fromthechannel.Also,thetotalsimulatednoiseinbothcas esisingoodagreement withthetheoreticalresultof4 kTg d 0 where g d 0 wascomputedataverylowdrainbias bythesimulator.Theseresultswerealsoinqualitativeagr eementwiththoseexhibited intheresistorresults,inthattheenergybalancemodelpre dictsslightlylowervelocity ructuationnoise. 2.6ImpedanceFieldSimulationofNumberFluctuationNoise Inthissectiontheructuationtermsfornumberructuations aretracedthrough theiroriginsintheBTE( 2{2 )toeachcontributiontothesystemofequationsforeach transportmodel.2.6.1NumberFluctuationNoiseSimulation Numberructuationsarecausedbyeitherinterbandtransiti onsofparticlesor tunnelingfromonelocationtoanother.Thesetransitions( exceptforthecaseofcarrier tunneling)involvecollisionswithphonons,photons,orot hercarriers.Aswithanyother transition,bothenergyandmomentummustbeconserved.Mic roscopically,theinterband carriertransitiontermoftheBTEcanbedescribedbyastati sticalmeancomponentand astochasticcomponent,whichructuatesrandomlyandwitha zerostatisticalmean. s ( ~r;~p;t )= s ( ~r;~p;t )+ r f (2{103) where r f isaLangevinnoisesourcewhichrepresentsaructuationin s ( ~r;~p;t ).Langevin noisesourcesmustbeinsertedintothePDE'softhetransport modelthroughthoseterms derivedfromtheinterbandtransitiontermoftheBTE.Thiss houldbeconsideredforeach momentindividually. 54

PAGE 55

Forthezeroeth-ordermoment,theinterbandtransitionter mischaracterizedas follows. 1 n X p s ( ~r;~p;t )=[G n R n ]+ 1 n X p r f =G n R n + r n (2{104) Inthecaseofbulkorsurfacetraps,therewillalsobearstordermomentandcorresponding termfortrappedelectrons.Fortherst-ordermoment,this termischaracterizedas follows. 1 n X p ~ps ( ~r;~p;t )=0+ 1 n X p ~pr f = ~r v n (2{105) Sincethedirectionofthemomentumofgeneratedorrecombin ingcarriersisrandom (resultinginthezerostatisticalmeancomponent),itfoll owsthattheeectofnumber ructuationsontheaveragemomentumofthecarrierdistribu tionissmall,and ~r v n is usuallyassumedtobenegligible.Thisassumptionisvalid, sinceintrabandtransitions haveamuchgreatereectonthemomentumdistribution.Howe ver,thisisanassumption anditmaynotbeappropriateinsomesituationswherethecar rierconcentrationisvery low(suchthatvelocityructuationsdonotdominatethister m).Forthesecond-order moment,thistermischaracterizedasfollows. 1 n X p j ~p j 2 2 m n s ( ~r;~p;t )=[G W n R W n ]+ 1 n X p j ~p j 2 2 m n r f =G W n R W n + r w n (2{106) Forthecaseoftransitionsthatcanbeconsideredtoonlyinv olvecarrierswithlowkinetic energy(i.e.tunnelingorband-traptransitions)thisterm canbeassumedtobezero. However,forthecaseofimpactionizationorAugerrecombin ationthistermshouldbe considered,andcanonlybeignorediftheeectsofthisterm arestudiedandshown tobenegligible.Forthethird-ordermoment,theLangevint erm ~r s n isrelatedto ~r v n (Equation 2{60 )andisthereforeassumedtobezero. Therefore,thenumberructuationLangevinsources r n r p r n T (trappedcarriers,if necessary), r w n ,and r w p enterthecompletesystemofequationsderivedfromtherst four momentsoftheBTEasfollows.Aspreviously,theasteriskst odenoteeectivemassesand 55

PAGE 56

mobilitiesareomittedforclarity. r 2 = q ( p n + N + D N A ) t dn dt = r ( n ~ V n )+G n R n + r n (2{107) dp dt = r ( p ~ V p )+G p R p + r p (2{108) dn T dt =G p +R n G n R p + r n T (2{109) m n d ( n ~ V n ) dt = r ( nkT n ) qn ~ E n q ~ V n n P n r ( kT n ) m p d ( p ~ V p ) dt = r ( pkT p )+ qp ~ E p q ~ V p p P p r ( kT p ) 3 2 k d ( nT n ) dt = r ( n ~ S n ) q ~ E ( n ~ V n )+G W n R W n 3 2 nk T n T L h W n i + r w n (2{110) 3 2 k d ( pT p ) dt = r ( p ~ S p )+ q ~ E ( p ~ V p )+G W p R W p 3 2 pk T p T L h W p i + r w p (2{111) m n d ( n ~ S n ) dt = r 5 2 nk 2 T 2 n q ~ E 5 2 nkT n n q ~ S n S n Q n r 5 2 k 2 T 2 n m p d ( p ~ S p ) dt = r 5 2 pk 2 T 2 p + q ~ E 5 2 pkT p p q ~ S p S p Q p r 5 2 k 2 T 2 p Theructuationofvoltage ~ V c attheconsideredexternalcontactisthengivenasfollows. ~ V c = ZZZ r G n r n + G p r p + G n T r n T + G E n r w n + G E p r w p d 3 r (2{112) where G n G p G n T G E n ,and G E p aretheGreen'sfunctionsassociatedwiththe scalar-valuedelectron,hole,andtrappedelectroncarrie rcontinuityequationsandelectron andholeenergybalanceequations,respectively.Thepower spectraldensity S V =2 T ~ V c ~ V c iscomputedasfollows. S V =2 T ZZZ r i 0B@ G n r n + G p r p + G n T r n T + G E n r w n + G E p r w p 1CA d 3 r i ZZZ r j 0B@ G n r n + G p r p + G n T r n T + G E n r w n + G E p r w p 1CA d 3 r j (2{113) 56

PAGE 57

Asinthesectiononvelocityructuations,thereare,depend ingonthenumberructuation mechanismconsidered,approximationsthatthenoisesourc esarestatisticallyindependent inposition,asfollows. r ( ~r i ) r ( ~r j )= K ( ~r i ~r j ) 2 T (2{114) Thisisvalidsincerandomructuationsingenerationratesa renotexpectedtobestrongly correlatedthroughposition,exceptforthecaseofcarrier tunneling.Thenoisestrengths canbeevaluatedusingMonteCarlocomputations,butthisis notfeasibleforlow frequencycomputations.Thealternativeistoevaluatethe localnoisestrengthsusing theclosedformexpressionssuchasthosederivedbyvanVlie t[ 47 ].Termsinvolving r w n mustbeevaluateddierentlyforeachtypeofmechanism. Noneofthenumberructuationtermsareinvolvedinback-sub stitutedexpressions forheatroworparticlecurrent,asinthecaseofvelocityru ctuations,soextendingthis tothehydrodynamic,energybalance,anddrift-diusionca sesistrivial.Theeectsof independentmechanismsareaddedusingthesuperpositionp rincipal,asfollows. S V j total = S V j trapping + S V j tunneling + S V j impact (2{115) 2.6.1.1Bulkorsurfacetrapcaptureandemission Forthecaseofatrapspecieslocatedataparticularenergyl evel E T relativetothe intrinsicenergylevel E I ,thestatisticalmeancaptureandemissionratesofelectro nsand holesaregovernedbyShockley-Read-Hallrecombinationan dgenerationratesR nT ,R pT G nT ,andG pT ,asfollows. R nT = c n n ( N T n T )(2{116) G nT = e n n T = c n n 1 n T (2{117) R pT = c p pn T (2{118) G pT = e p ( N T n T )= c p p 1 ( N T n T )(2{119) 57

PAGE 58

where c n e n c p ,and e p aretheelectronandholecaptureandemissioncoecients,a nd n 1 and p 1 aretheequivalencecarrierconcentrations,i.e.theelect ronandholeconcentrations forthecasewhentheFermilevelisthesameasthetraplevel. Theyarecomputedusing theMaxwell-Boltzmannapproximationasfollows. n 1 = n i exp E T E I kT L (2{120) p 1 = n i exp E I E T kT L (2{121) where E T isthetrapenergylevel, E I istheintrinsicenergylevel,and n i istheintrinsic electronconcentration.Thecorrespondinglocalnoisestr engthsarecomputedasfollows [ 29 47 ]. r n ( ~r i ) r n ( ~r j )= r n ( ~r i ) r t ( ~r j )=2(G nT +R nT ) ( ~r i ~r j )) 2 T = K nn ( ~r i ~r j )) 2 T (2{122) r p ( ~r i ) r p ( ~r j )= r p ( ~r i ) r t ( ~r j )=2(G pT +R pT ) ( ~r i ~r j )) 2 T = K pp ( ~r i ~r j )) 2 T (2{123) r t ( ~r i ) r t ( ~r j )=2(G nT +R nT +G pT +R pT ) ( ~r i ~r j )) 2 T = K tt ( ~r i ~r j )) 2 T (2{124) Notethatthegenerationandrecombinationratesusedinthe seexpressionsareonlythe componentsduetothistrapspeciesanditsassociatedinter actions.Thetermsinvolving r w n and r w p areassumedtobezerosincetheprobabilityofcaptureoremi ssionishighest fromlowkineticenergystates.Thesimpliedpowerspectra ldensityisthengivenas follows. S V = ZZZ r 0BBBBB@ G n K nn G n G n K nn G n T + G p K pp G p G p K pp G n T + G n T K tt G n T G n T K nn G n G n T K pp G p 1CCCCCA d 3 r (2{125) 2.6.1.2DirectCarriertunneling Previously[ 29 ],noiseduetocarriertunnelingimplementedinFLOODSwasmo delled ascaptureatfastsurfacestatescoupledwithtrap-to-trap tunnelingfrominterface trapstotheoxidetraps.Thiswasappropriateforthedevice andbiasconditionsin 58

PAGE 59

previousworks,sincetheFermilevelatthesurfacewasinth eenergygap.However, DCsimulationsofthedevicesmeasuredforthiswork,andfora nysuchhighlyscaled MOSFETs,showthattheelectrondistributionisdegenerate atthesurface.Therefore, directtunnelingbetweenthestatesintheconductionbanda tthesilicon/oxidesurfaceand trapsintheoxidehasbeenimplementedinFLOODSasapartofth iswork. Directband-to-trapcarriertunnelingisuniqueamongtheme chanismsconsidered inthiswork,inthatitisanonlocalmechanisminvolvingruc tuationsintwolocations. Thecaseofinterestforthedevicesinvolvedinthisworkist hatoftunnelingfromthe conductionbandatthesiliconsurfaceandatrapintheoxide .AsintheworkofHou[ 29 ], trappedcarrierstunnelprimarilytothenearestsiliconno de,andtransitionsbetweenthe gateandoxidetrapsareneglected.Whilethelatterassumpt ionisdiculttojustify,it greatlysimpliestheimplementationandisthereforeused .Theinteractionsarebetween thetrappedcarriercontinuityequation dn iT dt =R ijnt G ijnt + r i n T = r i n T atsomeoxidenodeat position r i andtheelectroncontinuityequationatsomeinterfacesili connodeatposition r j .ThetunnelingcaptureandemissionratesR ijnt andG ijnt aregivenasfollows. R ijnt = c ijnt n j ( N i T n iT )(2{126) G ijnt = e ijnt n iT (2{127) where c ijnt and e ijnt arethecaptureandemissioncoecients.Sincethereisnost eady-state changeintheconcentrationoftrappedcarriers,ifthereis notunnelingtothegatethen theseratesareequal.Thecaptureandemissioncoecientsa regivenasfollows. e ijnt = c ijnt n j ( N i T n iT ) n iT (2{128) c ijnt = T 0 exp ( d )(2{129) where T 0 isthetunnelingcoecient, istheattenuationconstantoftheelectronwave functions,and d isthedistancebetweennode i andnode j .Theattenuationconstant,for 59

PAGE 60

atriangularbarrier,isgivenasfollows[ 29 ]. = 4 3 p 2 m ox ~ 1 : 5 itf 1 : 5 ox itf ox (2{130) where m ox istheeectivemassofthetunnelingelectronsintheoxide, and itf isthe barrierheightattheinterface,equaltothedierenceinth eelectronanitiesofthebulk siliconandthegateoxide,4 : 05 eV .Thebarrierheightatthetrapis ox =4 : 05 eV + q ( j i ).Inthelimitthattheeldintheoxideiszero, is1 : 3 10 8 cm 1.The tunnelingcoecientforsuchband-to-traptransitions T 0 isnotknown.Thisvalueis arbitraryhowever,asthesimulationshaveshownthattheox idenoiseisnotsensitiveto thisparameter(thereiseectivelyafunctionaldependenc ein G nt G nt of1 =T 0 ,suchthat thetotalnoisepowerspectraldensityisinsensitiveto T 0 ). Thecontributiontoructuationinacontactvoltageduetosu chtransitionsisgivenas follows. ~ V ij c = G n T ( ~r i ) r n T ( ~r i ) V i + G n ( ~r j ) r n ( ~r j ) V j (2{131) where V i and V j arethenodalvolumesattheoxideandinterfacenodes,respe ctively.The powerspectraldensityisthencomputedasfollows. S ij V =2 T 0B@ G n T ( ~r i ) r n T ( ~r i ) r n T ( ~r i ) G n T ( ~r i ) V 2 i + G n T ( ~r i ) r n T ( ~r i ) r n ( ~r j ) G n ( ~r j ) V i V j + G n ( ~r j ) r n ( ~r j ) r n T ( ~r i ) G n T ( ~r i ) V j V i + G n ( ~r j ) r n ( ~r j ) r n ( ~r j ) G n ( ~r j ) V 2 j 1CA (2{132) Theructuationsattheinterfaceandatthetraparealsorela ted,asdictatedby conservationofparticles. r n V j = r n T V i (2{133) 60

PAGE 61

Thisresultsinthefollowingrelations. r n ( ~r j ) r n ( ~r j )=2(G nt +R nt ) 1 2 TV j = K tt 1 2 TV j (2{134) r n T ( ~r i ) r n T ( ~r i )= V 2 j V 2 i r n ( ~r j ) r n ( ~r j )= K tt V j 2 TV 2 i (2{135) r n ( ~r j ) r n T ( ~r i )= V j V i r n ( ~r j ) r n ( ~r j )= K tt 1 2 TV i (2{136) where r n representsructuationsinelectronconcentrationatthesu rface,and r n T representsructuationsintheconcentrationoftrappedcar riersatalocationinthe oxide.Thetermsinvolving r w n areassumedtobezerosincethecarrierdistributionis closetotheconductionbandedgeunlessthesurfaceisstron glydegenerate,inwhichcase acarrierinvolvedinthetransitionisoneofmany.Thus,the averagekineticenergyofa degeneratecarrierdistributiondoesnotructuatemuchdue tothismechanism. Finally,thesimpliedpowerspectraldensitythenreduces tothefollowing. S ij V = V j 0B@ G n T ( ~r i ) K tt G n T ( ~r i ) G n T ( ~r i ) K tt G n ( ~r j ) G n ( ~r j ) K tt G n T ( ~r i )+ G n ( ~r j ) K tt G n ( ~r j ) 1CA (2{137) 2.6.1.3ImpactIonization Animpactionizationeventresultsinthegenerationofanel ectronandahole, accompaniedbyalossinkineticenergyofasinglecarrierth atissucienttocausethe interbandtransition.Thecarrierdensityandkineticener gyrateructuationsassociated withelectronionizationeventsarethenrelated,asfollow s. r p = r n (2{138) r w n = E r n (2{139) where E representstheaverageenergylostbyelectronsinimpactio nizationevents. Thisisexpectedtobenearlyequaltotheminimumkineticene rgyrequiredtoexcite anelectronfromthevalencebandtotheconductionbandwhil econservingenergyand 61

PAGE 62

momentum,whichfortheminimumUmklappprocessisslightly higherthan E G .The approximation E = E G isassumed.Thelocalnoisestrengthsarethencomputedas follows. r n ( ~r i ) r n ( ~r j )= r p ( ~r i ) r p ( ~r j )= r n ( ~r i ) r p ( ~r j )=2G ii ( ~r i ~r j ) 2 T = K nn ( ~r i ~r j ) 2 T (2{140) r w n ( ~r i ) r w n ( ~r j )=2G ii E 2 ( ~r i ~r j ) 2 T = E 2 K nn ( ~r i ~r j ) 2 T (2{141) r w n ( ~r i ) r n ( ~r j )= r w n ( ~r i ) r p ( ~r j )= 2G ii E ( ~r i ~r j ) 2 T = E K nn ( ~r i ~r j ) 2 T (2{142) whereG ii istheimpactionizationrate,computedasinSection 2.4 Thepowerspectraldensityisthengivenasfollows. S V = ZZZ r 0BBBBB@ G n K nn G n G n K nn G p E G n K nn G E n G p K nn G p G p K nn G n E G p K nn G E n E 2 G E n K nn G E n E G E n K nn G n E G E n K nn G p 1CCCCCA d 3 r (2{143) 2.6.2ComparisonofNumberFluctuationNoiseSimulationwi thEnergy BalanceandDrift-DiusionModelsin n + =n=n + Resistors Numberructuationnoisesimulationswereperformedonthes ameresistorstructures thatwerepreviouslyusedinsection 2.5.2.1 ,withthesamechoicesfortransport parameters.Anelectrontrap( c n =10 9 cm 3 s 1 >c p =10 11 cm 3 s 1 )wasusedto simulatebulkgeneration-recombinationnoisewithaconst anttrapdensityof10 15 cm 3 andatrapenergylevelneartotheequilibriumFermilevelin thelowlydopedregion. Figure 2-9 showsthecontributionstonumberructuationnoise(thequa ntityinside theintegralofEq.( 2{113 ).Clearly,theeectofthehydrodynamicmodelonnumber ructuationnoiseisnotasstrongasthatinthecaseofveloci tyructuations.Figure 2-10 showsthetotalnoiseintegratedacrossthelengthofthedev ices.Forlongresistors, thebiasdependenceoftheNoiseis I 2 ,consistentwiththatoftrappinginquasineutral regions.Figure 2-11 showstheratioofthenoisepredictedbytheEnergyBalancem odel tothatpredictedbytheDrift-Diusionmodel. 62

PAGE 63

2.7QuantizationEectsandNoise TheSchrodingerwaveequationallowstheevaluationofall owedstatesforelectrons, giventhepotentialenergythatconnestheelectronsinast ructure(forexample,inthe attractiveelectriceldthatsurroundspositivelycharge datomicnucleii).TheSWEis givenasfollows. ~ 2 2 m r 2 + V ( ~r;t ) ( ~r;t )= E ( ~r;t )(2{144) V ( ~r;t )isthepotentialenergyseenbyanelectron,asafunctionof positionandtime. ( ~r;t )and E arethewavefunctionandassociatedenergyofanyallowedel ectronstate thatisasolutionofthisEigenproblem.ThesolutionoftheS WEinaninnitewelliswell known,asillustratedinFigure 2-12 .Notethattheplottedfunctionsarethewaveforms timestheircomplexconjugate,givingtheshapeoftheproba bilityfunctionsforelectrons intheseoddandevenstates.Theoddsolutionshaveanoddnum berofnodes,andthe evensolutionshaveanevennumber. InsemiconductorssuchasSilicon,theperiodicityofthepo sitionsofatomicnucleii givesrisetoaperiodicpotentialenergy,yieldingmanyall owedstatesthatareclusteredin energywithinbands.Thecoreelectrons(inatomsthathavet hem)aretightlyboundin energybandsthathaveelectronsineachandeveryallowedst ate.Thevalenceelectrons arelesstightlyboundinthevalenceband,withafewenerget icelectronsthatareexcited fromthevalencebandorfromboundstatessuchasdonorimpur itiesorelectrontrap defectstotheconductionbandortoboundstatesatacceptor impuritiesandholetrap defects.Aslongasthecrystalisperiodicforlongdistance s,andtheelectricelds associatedwithappliedbiasornonhomogeneityaresignic antlylowerthanthebinding forcesintheperiodicstructureofthecrystallattice,the SWEsolutioncanbeassumedto givestatesconsistentwiththenearlyfreeelectronandhol emodelspreviouslydescribed. However,inhighlyscaledandstronglyinvertedMOSFETssuc hasthedevices consideredinthisstudy,theelectriceldunderthegateox ideconnescarrierstightly totheSilicon/Oxidesurface.Thisconnementisinonlyone dimension(incidenttothe 63

PAGE 64

surface),sotheallowedstatesintheSiliconnearthesurfa ceformquantizedsubbandsfor whichcarriersconductcurrentintheunconneddirections (alongthesurface). Thisquantizationeectgivesrisetodisplacementofthein versioncarriersfromthe surfaceaswellasalocalincreaseintheeectiveenergygap ofthesemiconductor.These eectsaremanifestedintheMOSFETasanincreaseintheeec tivethresholdvoltageand adecreaseinthenumberofcarriersinthechannelcomparedt owhatarepredictedby classicalmodels. Severalmethodshavebeenemployedtoapproximatetheeect sofquantizationin simulationframeworksoriginallydevelopedforsemi-clas sicaltransport.Someinvolve thecomputationalofadeformationpotentialtoreduceandd isplaceinversionlayers[ 48 ] .Thisrequiresadjustmentofthemobilitymodel[ 49 ].However,thegoalofthisstudy istoaccuratelysimulatethenoiseintheseMOSFETs.Thevel ocityructuationnoise forcurrentrowtowardthedrainisnotexpectedtodependgre atlyontheshapeofthe distributionofconductivecarriersindirectionsperpind iculartothiscurrentrow,andis notobservedwithinthemeasurementtimewindow.Theonlyve locityructuationnoise observedwithinthemeasurementtimewindowinthisworkisi ntheSOIdevices,where itiscausedbymajoritycarriersintheroatingbody.Thenum berructuationnoise,in themeasurementtimewindow,isprimarilymanifestedbytun nelingofinversionlayer carriersintoandoutoftrapsintheoxide.Thisnumberructu ationnoiseisprimarily aectedbythenumberofcarriersintheinversionlayer,and thetunnelingcaptureand emissionratesoftrappedcarriers,andisagainnotgreatly aectedbytheshapeofthe inversionlayercarrierdistribution.Itisimportanttono tethatthetunnelingprobabilites ofcarriersintheinteractingtrapsandinversionlayerare afunctionofthetunneling distance.Theyarenotasstronglydependentonpositionasa nexponentialfunctionofthe tunnelingdistance(suchasthosecomputedformodelswhere thecarriersareassumedto tunnelfromtheoxideedgeandforwhichthebarrierheightis integratedthroughoutthe tunnelingdistance),becausetheerrordisplacementintun nelingpositionisintheregion 64

PAGE 65

wheretheelectronenergyisgreaterthanthepotentialener gy.Lateranalysisshowsthis eectindetail,butfornowthequantizationeectsarelump edintotheeectivetunneling distanceandbarrierheight,similartothewaythatquantiz ation-aectedscattering islumpedintothemobilitymodels.Therefore,quantizatio neectsinhighlyscaled MOSFETswhosebiasislimitedtostronginversioncanbeappr oximated,forthepurpose ofnoisesimulation,byalargexedsurfacecharge.Thisrai sesthethresholdvoltagein stronginversionandreducethemobileinversioncharge,wi thoutadverselyaectingthe simulationofnoiseinthedevice.Thispointisillustrated forthesimulateddevicesandis presentedinChapter 4 65

PAGE 66

290 295 300 305 310 -0.2 -0.1 0 0.1 10 16 10 17 10 18 10 19 Electron Temperature (K) Donor Concentration (cm -3 )x ( m m) T n 0.025V bias N D Figure2-1.Electrontemperatureasafunctionofpositionf ortheenergybalancemodelin a0 : 1 m n + =n=n + resistorwithabiasof25 mV 66

PAGE 67

-0.5 -0.4-0.3-0.2-0.100.10.20.30.4 0.50.6 x ( m m) 0 5e-27 1e-26 1.5e-26 2e-26Noise Contribution (A 2 Hz -1 cm -3 ) Figure2-2.Velocityructuationnoisecontributionasafun ctionofpositionina0 : 1 m n + =n=n + resistorforabiasof25 mV 67

PAGE 68

-0.5 -0.4-0.3-0.2-0.100.10.20.30.4 0.50.6 x ( m m) 0 1e-26 2e-26 3e-26 4e-26 5e-26 6e-26Noise Contribution (A 2 Hz -1 cm -3 ) Figure2-3.Velocityructuationnoisecontributionasafun ctionofpositionina0 : 1 m n + =n=n + resistorforabiasof0 : 5 V 68

PAGE 69

10 -32 10 -31 10 -30 0.01 0.1 1 10 Total Noise (A 2 Hz -1 cm -2 )L R ( m m) DD, 0.025V bias EB, 0.025V bias DD, 0.5V bias EB, 0.5V bias Figure2-4.Velocityructuationnoisefor n + =n=n + resistorsofvaryinglengths. 69

PAGE 70

0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 0.01 0.1 1 10 Ratio (#)L R ( m m) EB/DD, 0.025V bias EB/DD, 0.5V bias Figure2-5.Ratioofvelocityructuationnoisepredictedby EnergyBalancemodeltothat predictedbyDrift-Diusionfor n + =n=n + resistorsofvaryinglengths. 70

PAGE 71

-0.2 0 0.2 0.4 -0.4 -0.2 0 0.2 0.4 0.6 10 16 10 18 10 20 y ( m m) x ( m m) Net Majority Doping Concentration (cm -3 ) Figure2-6.Thenetmajority-typedopingconcentrationplo ttedfora0 : 25 mnMOSFET. Thegateislocatedtotherear,withthesourceontheleftand thedrainon theright. 71

PAGE 72

-0.2 0 0.2 0.4 -0.4 -0.2 0 0.2 0.4 0.6 0 0.5 1 1.5 2 x 10 -15 y ( m m) x ( m m) Contribution of Local Noise (A 2 Hz -1 cm -3 ) Figure2-7.Contributionoflocalvelocityructuationdrai ncurrentnoiseasafunction ofpositioninthe0 : 25 mnMOSFETassimulatedusingthedrift-diusion model,with V GS =1 : 5Vand V DS =10mV.Thetotalsimulated noisewas1 : 89109 10 23 A 2 Hz 1 cm 1 ,comparedtothetheoreticalvalue 4 kTg d 0 =1 : 7675 10 23 A 2 Hz 1 cm 1 72

PAGE 73

-0.2 0 0.2 0.4 -0.4 -0.2 0 0.2 0.4 0.6 0 0.5 1 1.5 2 x 10 -15 y ( m m) x ( m m) Contribution of Local Noise (A 2 Hz -1 cm -3 ) Figure2-8.Contributionoflocalvelocityructuationdrai ncurrentnoiseasafunctionof positioninthe0 : 25 mnMOSFETassimulatedusingtheenergybalance model,with V GS =1 : 5Vand V DS =10mV.Thetotalsimulated noisewas1 : 75531 10 23 A 2 Hz 1 cm 1 ,comparedtothetheoreticalvalue 4 kTg d 0 =1 : 7805 10 23 A 2 Hz 1 cm 1 73

PAGE 74

-1e-05 0 1e-052e-05 x( m m) 0 1e-27 2e-27 3e-27 4e-27Noise Contribution (A 2 Hz -1 cm -3 ) DD EB Figure2-9.Numberructuationnoisecontributionasafunct ionofpositionina0 : 1 m n + =n=n + resistorwithabiasof25 mV 74

PAGE 75

0.010.1110 L R ( m m) 1e-37 1e-36 1e-35 1e-34 1e-33 1e-32 1e-31Total Noise (A 2 Hz -1 cm -2 ) DD, 0.025V bias EB, 0.025V bias DD, 0.5V bias EB, 0.5V bias Figure2-10.Numberructuationnoisefor n + =n=n + resistorsofvaryinglengths. 75

PAGE 76

0.010.1110 L R ( m m) 1 1.02 1.04 1.06 1.08 1.1Ratio (#) EB/DD, 0.025V bias EB/DD, 0.5V bias Figure2-11.Ratioofnumberructuationnoisepredictedbyt heEnergyBalancemodelto thatpredictedbyDrift-Diusionfor n + =n=n + resistorsofvaryinglengths. 76

PAGE 77

-4e-09-2e-0902e-094e-09 position (m) 0 0.1 0.2 0.3 0.4 0.5 0.6eigenvalue(eV) + Y Y (0.1eV representation) Figure2-12.SolutionofSchrodinger'sEquation(probabi litydistributionsandassociated energies)forallowedstatesinaninnitepotentialwell. 77

PAGE 78

CHAPTER3 NOISEMEASUREMENTS Themeasurementofnoiseinasemiconductordevicemustbeco nductedwithcare, becausealldevicesinthemeasurementsystemgeneratenois ewhichmustbeseparated outfromormadenegligiblecomparedtothenoisegeneratedw ithinthedeviceunder test.Asaresult,noisemeasurementsetupsgenerallyfocus onlow-noisecomponents, cleanDCpowersupplies,andqualityelectricalcontacts.Tr aditionallyusedmethodsfor protectingadevicefromhigh-voltagetransientssuchasth euseofreedrelayswitchesand programmablevoltagerampsfromswitchingpowersuppliesc annotbeusedduetothe noisysignalstherequiredcomponentsgenerate. 3.1MeasurementSetup Thenoisemeasurementsinthisstudyweretakenusingthecir cuitdepictedin Figure 3-1 .ThemetallmresistorsweresolderedinplaceandBNCcable swereused toconnectthecircuittotheamplierandthepowersupply(w hichconsistsoftwoDC batteries,one12.5Vandone2V).TheamplierusedistheBro okdeal5004,thespectrum analyzerisaHewlett-Packard3561AandtwoAgilent34401Ad igitalmultimeterswere usedtomeasuregateanddrainDCvoltages.3.1.1ResistorNetworkDesign Thegatebiasisappliedusingasimpleresistordividercirc uit.Theappliedgate voltageisthengivenbythefollowingequation. V G = R 1 R 1 + R 2 V DD (3{1) Theresistorvalues( R 1 and R 2 )shouldbechosentobeassmallaspossibletolimit theircontributionstothethermalnoiseatthedrain,ampli edfromthegate,buttaking intoconsiderationtheirpowerdissipationlimitsandthec urrentsupplyingabilityofthe batteries.Theresistorvaluesof1 : 5 k nand16 k nwerechosenfor R 1 and R 2 ,respectively. Thedrainbiasisappliedusingthethree-resistornetworkc onsistingofresistors R 3 R 4 ,and R 5 .Thedrainbiasisdeterminedbysolvingthefollowingequat ion,usingthe 78

PAGE 79

draincurrentofthedeviceatthedesireddrainbias. V D = R 5 R 3 + R 5 V DD I D R 3 R 4 + R 4 R 5 + R 5 R 3 R 3 + R 5 (3{2) R 4 'sresistanceisdesiredtobeaslargeaspossibletomaximiz ethesignicanceofthe highfrequencyintrinsicdevicethermalnoisecomparedtot heresistorvoltagenoise(such that4 kT ( r d jj R L ) 4 kTr d ).However,exceptforverysmalldrainbiasesorveryquiet (noiseless)devices,thecornerfrequencywherelowfreque ncynoisebecomesinsignicant nexttohighfrequencythermalnoiseishigherthanthemaxim umfrequencyofthe spectrumanalyzer(100 kHz ).Inaddition,itisdesirabletoonlyvaryoneofthethree resistors( R 3 waschosenforthis)toapplydieringdrainbias.Themaximu mvoltage dropacross R 4 isthenlimitedandmustbesignicantlylessthan V DD V D toguarantee thatructuationswillnotbeclippedduetosupplycircuitli mitations.Sincethemaximum currentobservedinthemeasureddeviceswasapproximately 3 : 5 mA R 4 waschosento be2 k n,resultinginamaximumvoltagedropofaroundone-halfthe supplyvoltage.Itis desirabletochooseavalueof R 5 whichislessthan R 4 suchthattheeectofvarying R 3 willsetthevoltageoftheinternalnetworknode,independe ntoftheIVcharacteristicsof thedeviceundertest.Avalueof1 k nwaschosen. R 3 ischosenforeachdeviceandbias pointtomatchthedesireddrainvoltageandmeasuredIVchar acteristics. 3.1.2PracticalConsiderations ThephysicallocationofeachcircuitcomponentandtheBNCc ablesisdepicted inFigure 3-2 .Sinceswitchingpowersuppliesarenotused,theproblemis presentedof applyingthebiastothedeviceundertestwithoutapplyinga nyhigh-voltagetransients whichcausehighlyscaledandelectricallysensitiveMOSde vicestodegradeorsuergate dielectricbreakdown. ItwasfoundthattheparasiticcapacitanceoftheBNCcables wassignicantenough topresentenoughchargetobreakdownthegateoxideofthetr ansistorsmeasured. Therefore,itwasnecessarytouseapull-downresistortoli mitthechargingofthe 79

PAGE 80

parasiticcapacitanceswhilethegateanddrainbiaswereap plied.Theprocedureis depictedinFigure 3-3 .InitiallylocationsA,B,CandDarealldisconnected.TheB NC capacitancesarechargedto V DD ,andthegateanddrainvoltagesareresistivelytiedto ground.LocationAisconnectedrst,pullingthevoltageon thegate-sideBNCcable's parasiticcapacitancedownto 100n R 2 +100n V DD .LocationBisthenconnected,pullingthe voltagedownalittlefurther(atthispointthegatevoltage measuredwasabout84 mV ). LocationAisthendisconnected,andthegatevoltageincrea sestotheappliedbias ( 1 : 25 V ).Thesameprocedureisthenrepeatedonthedrainside,conn ectingrstD, thenC,andthendisconnectingD.Thegateanddrainvoltagesw eremoniteredatall timesduringtheconnectingprocedure.Breakdownofthegat eoxidecanbedetectedby anysignicantdropinthemeasuredgatevoltage.Afternois emeasurements,theorderof theprocedureisreversedtodisconnectthedevicefrombias Itwasalsoobservedthatdisconnectionofthebiasresistor networkonthedrain sidecancreateahighvoltagetransientonthegatesidewhic hissignicantenoughto breakdowntheoxide.Therefore,betweennoisemeasurement satdierentdrainbiasit isnecessarytoremovethebiasnetworkconnectionstobotht hedrainandgatebefore replacing R 3 3.2MeasuredDevices Fourdeviceswerecharacterizedinthiswork.Measuredwere long(2.33 mL G )and short(0.09 mL G )transistors,processedthesamewayinthesamefab,withbo thSOI startingsubstratematerialandbulksubstratematerial. Figures 3-4 and 3-5 showthegatecontrolledI-Vcharacteristicsoftheshortnchannel bulkdevice,asmeasuredusinganHP4145semiconductorpara meteranalyzer.Figure 3-6 showsthedraincontrolledI-Vcharacteristics.Figure 3-7 showsthedraincurrentnoise measuredatseveraldrainbiaspointswiththegatestrongly inverted. 80

PAGE 81

Figures 3-8 through 3-11 ,Figures 3-12 through 3-15 ,andFigures 3-16 through 3-19 showthesimilarmeasurementsfortheshortchannelSOIMOSF ET,andlongchannel bulkandSOIMOSFETs,respectively. 81

PAGE 82

R 2 R 1 R 3 R 5 B5004 DUT R 4 HP3561A Spectrum Analyzer V DD V DD -V DD Figure3-1.CircuitusedforbulkandSOIdevicenoisemeasur ements. 82

PAGE 83

B5004 V DD R 2 R 1 R 3 R 4 R 5 DUT HP3561A Spectrum Analyzer Figure3-2.Physicallayoutofcircuitshowingthecritical pathofelectricalconnections. 83

PAGE 84

B5004 DUT R 1 R 5 R 4 V DD R 3 R 2 A B CD C BNC C BNC 100 W 100 W Figure3-3.Connectionschemetoensuretheavoidanceofdie lectricbreakdown. 84

PAGE 85

-0.200.20.40.60.81 V GS (V) 0 0.0005 0.001 0.0015 0.002I DS (A) V DS = 1.2V V DS = 0.1V Figure3-4.GatecontrolledI-Vcharacteristicsofthe3 : 2 m x90 nm n-channelbulk MOSFET,inthelinearandsaturationconditions. 85

PAGE 86

-0.200.20.4 0.6 0.81 V GS (V) 1e-12 1e-11 1e-10 1e-09 1e-08 1e-07 1e-06 1e-05 0.0001 0.001I DS (A) V DS = 1.2V V DS = 0.1V Figure3-5.GatecontrolledI-Vcharacteristicsofthe3 : 2 m x90 nm n-channelbulk MOSFET,inthelinearandsaturationconditions,onalogari thmicscale. 86

PAGE 87

0 0.5 1 V DS (V) 0 0.0005 0.001 0.0015 0.002I DS (A) V GS = 1.254V V GS = 0.941V V GS = 0.628V Figure3-6.DraincontrolledI-Vcharacteristicsofthe3 : 2 m x90 nm n-channelbulk MOSFET,inthelinearandsaturationconditions. 87

PAGE 88

110100100010000 1e+05 frequency (Hz) 1e-21 1e-20 1e-19 1e-18 1e-17 1e-16 1e-15S I (A 2 /Hz) V DS = 0.88V V DS = 0.68V V DS = 0.54V V DS = 0.39V V DS = 0.29V V DS = 0.17V Figure3-7.Draincurrentnoiseofthe3 : 2 m x90 nm n-channelbulkMOSFET,with V GS 1 : 25V. 88

PAGE 89

0 0.5 1 V GS (V) 0 0.001 0.002 0.003 0.004I DS (A) V DS = 1.2V V DS = 0.1V Figure3-8.GatecontrolledI-Vcharacteristicsofthe4 : 8 m x90 nm n-channelSOI MOSFET,inthelinearandsaturationconditions. 89

PAGE 90

-0.200.20.40.60.81 V GS (V) 1e-13 1e-12 1e-11 1e-10 1e-09 1e-08 1e-07 1e-06 1e-05 0.0001 0.001I DS (A) V DS = 1.2V V DS = 0.1V Figure3-9.GatecontrolledI-Vcharacteristicsofthe4 : 8 m x90 nm n-channelSOI MOSFET,inthelinearandsaturationconditions,onalogari thmicscale. 90

PAGE 91

0 0.5 1 V DS (V) 0 0.001 0.002 0.003 0.004I DS (A) V GS = 1.25V V GS = 0.94V V GS = 0.63V Figure3-10.DraincontrolledI-Vcharacteristicsofthe4 : 8 m x90 nm n-channelSOI MOSFET,inthelinearandsaturationconditions. 91

PAGE 92

110100100010000 1e+05 frequency (Hz) 1e-21 1e-20 1e-19 1e-18 1e-17 1e-16 1e-15 1e-14S I (A 2 /Hz) V DS = 0.84V V DS = 0.69V V DS = 0.52V V DS = 0.38V V DS = 0.29V V DS = 0.17V V DS = 25mV Figure3-11.Draincurrentnoiseofthe4 : 8 m x90 nm n-channelSOIMOSFET,with V GS 1 : 25V. 92

PAGE 93

00.511.5 V GS (V) 0 5e-05 0.0001 0.00015 0.0002I DS (A) V DS = 1.2V V DS = 0.1V Figure3-12.GatecontrolledI-Vcharacteristicsofthe3 : 2 m x2 : 33 m n-channelbulk MOSFET,inthelinearandsaturationconditions. 93

PAGE 94

0 0.5 1 1.5 V GS (V) 1e-12 1e-11 1e-10 1e-09 1e-08 1e-07 1e-06 1e-05 0.0001I DS (A) V DS = 1.2V V DS = 0.1V Figure3-13.GatecontrolledI-Vcharacteristicsofthe3 : 2 m x2 : 33 m n-channelbulk MOSFET,inthelinearandsaturationconditions,onalogari thmicscale. 94

PAGE 95

0 0.5 1 1.5 V DS (V) 0 5e-05 0.0001 0.00015 0.0002I DS (A) V GS = 1.25V V GS = 0.94V V GS = 0.63V Figure3-14.DraincontrolledI-Vcharacteristicsofthe3 : 2 m x2 : 33 m n-channelbulk MOSFET,inthelinearandsaturationconditions. 95

PAGE 96

110100100010000 1e+05 frequency (Hz) 1e-23 1e-22 1e-21 1e-20 1e-19 1e-18 1e-17S I (A 2 /Hz) V DS = 1.70V V DS = 1.57V V DS = 1.43V V DS = 1.29V V DS = 0.90V V DS = 0.53V V DS = 0.28V Figure3-15.Draincurrentnoiseofthe3 : 2 m x2 : 33 m n-channelbulkMOSFET,with V GS 1 : 25V. 96

PAGE 97

00.511.5 V GS (V) 0 1e-05 2e-05 3e-05 4e-05I DS (A) V DS = 1.2V V DS = 0.1V Figure3-16.GatecontrolledI-Vcharacteristicsofthe0 : 6 m x2 : 33 m n-channelSOI MOSFET,inthelinearandsaturationconditions. 97

PAGE 98

0 0.5 1 1.5 V GS (V) 1e-13 1e-12 1e-11 1e-10 1e-09 1e-08 1e-07 1e-06 1e-05 0.0001I DS (A) V DS = 1.2V V DS = 0.1V Figure3-17.GatecontrolledI-Vcharacteristicsofthe0 : 6 m x2 : 33 m n-channelSOI MOSFET,inthelinearandsaturationconditions,onalogari thmicscale. 98

PAGE 99

0 0.5 1 1.5 V DS (V) 0 1e-05 2e-05 3e-05 4e-05 5e-05I DS (A) V GS = 1.25V V GS = 0.94V V GS = 0.63V V GS = 0.32V Figure3-18.DraincontrolledI-Vcharacteristicsofthe0 : 6 m x2 : 33 m n-channelSOI MOSFET,inthelinearandsaturationconditions. 99

PAGE 100

110100100010000 1e+05 frequency (Hz) 1e-24 1e-22 1e-20 1e-18S I (A 2 /Hz) V DS = 1.70V V DS = 1.57V V DS = 1.43V V DS = 1.29V V DS = 0.90V V DS = 0.53V V DS = 0.28V Figure3-19.Draincurrentnoiseofthe0 : 6 m x2 : 33 m n-channelSOIMOSFET,with V GS 1 : 25V. 100

PAGE 101

CHAPTER4 SIMULATIONSOFMEASUREDDEVICES Itisaprimarygoalofthisworktogaininsightintothephysi csofthenoiseofthe measureddevicesbysimulatingthenoiseusingFLOODSandtoc omparetothemeasured results.Inordertodothis,however,thestructureofthede vicemustbebroughtintothe simulationframeworkinsomemanner. ThedevicesusedforthesemeasurementswereprovidedbyaSe miconductorResearch Corporationmembercompany,andanexampleTSUPREMsimulat iondeckthatis meanttocreatethedopingprolesandoxidecontoursinthep rocessstructurewas givenalongwiththedevices.Thisinformationwasusedwith thecommercialDIOS simulator(FLOOPSwasnotusedduetoIndiumimplantsinther ecipe,andDIOSis capableofgeneratingmeshesthatcanbesimulatedbothwith thecommercialsimulator DESSISandwithFLOODS)togenerateadevicemeshthatisassume dtobeanaccurate representationofthedopingprolesandgeometricfeature softhedevicesmeasured. Theunknownsimulationparametersofthedevicesweretuned toachieveagreement betweensimulatedI-Vcharacteristicsandmeasuredresult s(subjecttothelimitations ofthephysicssimulated),andtogiveassurancethatthepro cess-simulateddevicemesh matchestherealdevices.Noisesimulationswerethenperfo rmedusingtheassumptions ofconstanttrapdensityintheoxide.Thetrapproleisthen adjustedbylookingatthe measurednoisedataandattemptingtoascertainthetrappro le,thenresimulatingto achievematchingresults.FortheSOItransistorstheveloc ityructuationnoisedominates overtheoxidetrappingnoiseovermuchofthemeasurementti mewindow,andtherefore thematchinoxidetrapsismadeoveranarrowerrangeoffrequ enciesandforfewerbias points. EnergybalancesimulationscouldnotbemadetomatchtheIVc haracteristicsofthe SOIdevices,duetotheextremenegativedrainresistanceof theresultingsimulations, asdiscussedintheChapter 1 .Asthereisnohopeofgettingnoisesimulationstocome 101

PAGE 102

closetomatchingthemeasurementsifDCsimulationscannotm atchtheDCcurrent, drift-diusionsimulationsareusedhereexclusively. 4.1The90nmBulkMOSFET Thenetdopingdensityofthedeviceresultingfromtheproce sssimulationsisplotted inFigure 4-1 .Thedrift-diusionsimulationsofthedeviceareplottedi nFigures 4-2 4-3 and 4-4 Thedevicestructurewasreverseengineeredandthesimulat ionsweretunedas follows,startingwiththeprocessrecipesuppliedbythepr ovidersofthedevices. First,thegateoxidethicknesswasmodiedsuchthatthesub thresholdslopeofthe I D V G characteristicsmatchesthatofthemeasurement.Thejusti cationforvariation fromtheprovidedrecipeforthisisthatthepreciseoxideth icknesscannotbenotknown, andthePhilipsuniedmobilitymodelwhichisusedforlow(a cceleration)eldmobility isusedinthesesimulationsandassumedtobeaccurate,espe ciallyinweakinversion (whereunknownfactorslikesurfaceroughnessscatteringd on'tmatter).Nextthepoly gatelengthisadjusteduntilthesimulatedI D V G characteristicsinlinearandsaturation regionsexhibitdrain-inducedbarrierlowering( V GS osetbetweensubthresholdlinearand saturationcurves)whichmatchesthatinthemeasuredresul ts.Itisappropriatetomodify thephysicalgatelength,sincetheprovideddescriptionof thedeviceswasthatthedrawn gatelengthwas0 : 13 m ,andwhiletheexpectedgatelengthafteragatetrimetchwas 70 nm to90 nm .Asquantizationeectsarenotincludedinthesenumerical simulations,it wasnecessarytondawaytosimulatethecurrentwiththecor rectamountofminority carriersintheinversionlayer.Thiswasdoneusingaxedch argeattheinterface,whichis adjustedtogivegoodagreementbetweensimulationandmeas urementatstronginversion andlowdrainbias,wherethequantizationeectisraisingt heeectivethresholdvoltage andreducingtheinversioncharge,butthelateraleldisno tsaturatingcarriervelocity. Thecoecientofthesurfaceroughnessscatteringterminth eDarwishmobilitymodel, whichisprocess-dependentandanunknownparameter,wasth enmodieduntiltheI-V 102

PAGE 103

characteristicsmatchbetweensimulationandmeasurement (bestobservedonalinear scale),fromtheonsetofstronginversiononward.Atthispo int,thephysicalstructure ofthedeviceanditsdoping,electriceldandoverallinver sioncharge,aswellasthe understoodmobilitymodel,matchtherealityofthestructu reandthephysicsinvolved withthenotableexceptionoftheshapeoftheinversionlaye r,duetotheincomplete modelingofthequantizaitoneects. Afterthedevicesimulations(processanddevice)weremodi edtomatchtheI-V characteristics,noisesimulationswereperformedforcom parisontothemeasuredresults. Acomparisonofthenoiseinthisdeviceversusthatsimulate dforuniformoxidetrap densityisgiveninFigure 4-5 .Forthissimulation,itisalsoclearfromtheplotsofnoise contributionduetotrapsalongtheoxideinFigure 4-6 thatwithincreasingdrainbias, thetrapsnearthesourceendofthechannelincreaseintheir eectonthenoise,where asthetrapstowardthedrainenddecreaseintheireects.To matchtheresultsofthe noisemeasurements,theslopeoftheoverall1 =f -likenoisewasmatchedbyadjustingthe parameter ofthefollowingoxidetrapequation[ 29 ]. N tox = N tox 0 exp ( x )+ N disc (4{1) where N tox 0 isthepeakvolumetricdensityoftrapsatthesurface, istheconstantof logarithmicproportionality, x isthedistancetotheinterface,and N disc representsthe eectivedensityofdiscretetrapsthatwereaddedtomatcht hebumpsinthemeasured characteristics,withtheirdistancefromtheinterfacese ttobringaboutamatchofthe cutofrequencyoftheresultingLorentzian,andthelatera lpositionsettomatchthe drainbiasdependenceoftheLorentzian.Thelocaleective trapdensityissettomatch thelowfrequencymagnitudeoftheLorentzian.Thereappear tobetrapsatthreegrid locationsnearthesourceendofthechannel.Finally,byobs ervingthedraindependenceof thepartsofthenoisemeasurementsnotdominatedbyanyobvi ousLorentzian,the N tox 0 parameterwasadjustedtogiveaGaussiandistributioninth elateraldirection,centered 103

PAGE 104

towardthedrainendwherethedrainbiasdependenceisdepre ssed.Thepeakvalueof N tox 0 usedis10 14 cm 3 .Thisbackgroundcomponentisinterpretedasafewtrapsint he drainregionforwhichthereisnotsucientinformationint hemeasuredcharacteristicsto discernanertrapplacement(themagnitudeofthenoiseint hesetrapsislowerthanthat ofthetrapsnearthesource,andthecutofrequencyoftheir associatedLorentziais notdiscernableinthemeasureddata).The\spreadout"Gaus siandistributioniny directionandexponentialincreaseinxdirectiontowardth egateapproximatestheeect ofdiscretetrapsnearthedrain.Thetrapdistributionused totthemeasuredresults isdepictedinFigure 4-7 .Trap1hasadensityof5 10 18 cm 3 ,correspondingtoatotal numberof4trapsinthe3 : 2 m widedevice.Traps2and3areinneighboringnodesand haveadensityof2 10 17 cm 3 ,correspondingtoatotalnumberof0 : 5trapseach.This isinterpretedasbeingonesingletrap,thoughthespreadof thebumpinthemeasured noisedataistoowidetobeasingleLorentzian.Thespreadis dueeithertothecarriers notalwaystunnelingdirectlyperpendiculartotheinterfa ce,ortocarrierstunnelingfrom somepositionintheinversionlayerotherthanrightatthes urface.Thistrapdistribution isunique,asthelateralpositionofthesetrapsdetermines thedrainbiasdependence ofthebump,andthepositionfromthechanneldeterminesthe cutofrequency.The positionofthesetrapsthereforeisasdeterminateasthemo delingofDrift-Diusionand processsimulationallows.Considerationofquantization eectsresultsintraplocations whicharealittleclosertothephysicalinterface,asissho wnlaterinthissection.The distributedtrapdistributionisagaussiandistributioni ntheydirection,withapeakand standarddeviation.Inthexdirectionitisexponential,as inEquation 4{1 .Ithasapeak densitynearthesurfaceof10 14 cm 3 andalateralspatialstandarddeviationof10 nm centeredalongthechannelbelowthecenteroftheovaldepic tedinthegure.Thefactor neededtomatchthegreaterthan1 =f dependenceinfrequencywas4 : 5 10 7 cm 1 .If theaveragetrapdensitythroughoutthecontributingregio nofthisoxideisassumedtobe approximately2 10 15 cm 3 throughoutthethicknessofthegateoxideover2 y oneach 104

PAGE 105

sideofthepeakvalue,thisgivesanequivalentofapproxima tely15additional,individually indistinguishabletraps.Theyaredescribedhereasbeingi ndividuallyindistinguishable becauseiftheyaredistributedinanyarrangementapproxim atingthisdistribution,they willprovidethebackground1 =f -likenoisenecessarytotthemeasurement,andthe individualLorentziafromeachtrapisnotsignicantenoug htobeindividuallyobservedin thenaltotalnoise. Thenaltofthesimulatedmeasurementstothemeasureddat aisshownin Figure 4-8 .Signicantdiusionnoisewasnotobservedwithinthemeas urementtime window. Itisofinteresttoconsiderwhattheerrorinthe3trapposit ionsandenergiesaredue toneglectingtheshapeofthecarrierdistributioninthein versionlayerassociatedwith thequantizationeect.Arst-order1-Dverticalsolution totheSWEwascomputedin theinversionlayerundereachofthesimulatedoxidetraps. Thissolutionwasobtainedby solvingthe1-DSWEusingtheshootingmethod.Inonedimensi ontheSWEequationis givenasfollows. 2 x 2 = 2 m ~ 2 ( E V ( ~r )) (4{2) Thisisintegratedtoget =x and asafunctionofposition,assumingsomevaluefor E .Theintegrationstartswithavalueofzerofor andasmallnite,arbitraryvaluefor =x atoneendofthesolution,where E C ishigherthan E and isexpectedtogoto zero.If E isanallowedEigenvalue, willgotozeroattheotherendaswell,andwill divergetoaverylargevalue(positiveornegative)if E istoohighortoolow. E isvaried andthesolutionistesteduntilavalidEigenvalueandEigen functionarefound.Unless E isveryclosetoavalidEigenvalue,divergenceofthesoluti onusuallycausesnumerical overrow,soanarbitraryprecisionsoftwarepackageforthe Perlprogramminglanguage wasusedtocomputethesolution,untilthesolutionwasclos eenoughtoswitchtofaster roatingpointxedprecisioncomputations.Ideally,theco mputedwavefunctionsand energyvalueswouldbeusedtoreshapetheinversionlayer,a ndthetransportmodelwould 105

PAGE 106

bere-solved,iteratinguntiltheSWEsolutionfortheinver sionlayerandthesimulated potentialwellandinverstioncarrierdistributionareinc ompleteagreement.However, asthexedinterfacechargehasadjustedtheelectriceldi ntheinversionlayertogive agreementfortheIVcurvesandtheoverallnumberofinversi oncarriersistherefore alreadyinagreement,theshapeofthepotentialwellshould notchangetoodramatically withiteration,andthissolutionshouldbecorrecttorsto rder,sucienttocomputean errorcorrectionforthesethreetraps. Thissolutionisrepresentedbythedistributionshapesdep ictedingures 4-9 and 4-10 .Theerrorinthetrapenergyisapproximatedbytheincrease intheminimum allowedelectronenergysincethecarriersaretunnelingfr omtherstsubbandrather thanfromtheclassicalconductionbandedge.Theerrorinth etrappositioncanbe approximatedbyassumingthatcarrierstunnelfromthemost likelyinversionelectron position,atthepeakoftheelectronwavefunction,rathert hantunnelingfromthesurface. Togetthesametunnelingprobabilities(andthesameLorent ziancutofrequency)the trapmustbeclosertothesurfacetogetthesameoveralldecr easeinwavefunctionvalue, asfollows. peak trap = classical surf classical trap (4{3) Thecomputederrorssuggestthetrappositionandenergyare 4 : 66 A closertothesurface and0 : 104 eV higherinenergyfortrap1,4 : 26 A closerand0 : 159 eV higherfortrap2and 4 : 29 A closerand0 : 159 eV higherfortrap3. Tosummarize,atrapdistributionhasbeenfoundforwhichth edrift-diusionmodel, augmentedwithdirectband-to-traptunnelingphysics,res ultsinnoisesimulationwhich matchesthemeasurednoisedatawithverygoodagreement.Sh ootingmethodsshow theerrorduetothesemi-classicalinversionlayershapean dclassicalenergydistribution, andcorrectiontermsarecomputedandwithingtherangeofex pectedvalues.From thiscomparisontheconclusioncanbedrawnthattheLorentz iaobservedinthenoise 106

PAGE 107

measurementsoriginatefromtrappingofcarriersatasmall numberoftrapsinthegate oxideinthevicinityofthesource/bodymetallurgicaljunc tion. 4.2The90nmSOIMOSFET Forthe90 nm SOIdevice,thenetdopingdensityisplottedinFigure 4-11 .The roatingbodyeectmakestheobservationofshortchannele ects,andthereforethe reverseengineeringofsuchadevice,moredicult.However ,sincethebulkandSOI deviceswerebothprocessedinthesamelotinthesamefabpro cess,andwithonly dierentstartingmaterial,itisassumedthatthepolygate lengthsandoxidethickness areidenticalbetweenthebulkandSOIdevices.Sincethesta rtingmaterialwasdierent, however,theremaybeslightdierencesinthequalityofthe oxideinterfaceunderthe gate,andtheresultingsurfacescatteringmobilitymodel, electriceldandthereforexed interfacechargetoapproximatequantizationeects.This appearstobethecase,asitwas necessarytomodifythesurfaceroughnessscatteringparam etersintheDarwishmodel andthexedinterfacechargetogetthe I D V G characteristicstomatchforthelinearcase aswellastheydidforthe90 nm bulktransistor.The I D V G characteristicsmatchthe measuredresultswellinstronginversion,asshowninFigur e 4-12 SincetheSOIdeviceishighlysensitivetothechargingofth eroatingbodybyimpact ionizationandthestandarddrift-diusionbasedVanOvers traetenimpactioinization model[ 42 ]isknowntooverestimatetheionizationrateinscaleddevi ces[ 16 ],itis necessarytomodifytheimpactionizationratecoecients, balancedwithShockleyReed Hallrecombination,inordertogettherightoverallamount ofroatingbodychargingasa functionofbiasandtogetthesimulated I D V D characteristicstomatchthemeasuredones withgoodagreement.Toobtainadecentmatchforthesehighl yscaleddevicesusingthis model,theBparameterwasincreasedfromtheclassical1 : 231 10 6 V=cm to3 : 8 10 6 V=cm andtheAparameterwasreducedfrom7 : 03 10 5 cm 1 to1 : 3 10 3 cm 1 .Thesizable reductionoftheAparameterisinparttothereductionofthe overallimpactionization rate,butlargelyalsoduetotheexponentialnatureofthede pendenceontheBparameter, 107

PAGE 108

whichwasincreasedtoraisetheonsetofimpactionizationt oahigherelectriceld(over shortdistances).The I D V D characteristicssimulatedusingdrift-diusionareshown in Figure 4-13 .Themodiedimpactionizationratesarealsoresponsiblef orthesaturation matchingofFigure 4-12 AfteragoodmatchwasobtainedfortheDCsimulations,noises imulationswerealso performed.Firstconsideredistheoxidenoise.Theoxideno iseinthemeasureddatais mostlyswampedoutbythediusionLorentzia,butsomeoxide noiseisapparentathigh drainbiasandlowfrequency(0 : 69 V and0 : 84 VV DS ,between1 Hz and100 600 Hz ),where theimpactionizationshiftstheexcessdiusionLorentzia tolowenoughzero-frequency magnitudethattheoxidenoisecanexceedit.Itisalsoappar entathighfrequency,for moderatedrainbias(0 : 27 V ,0 : 38 V ,and0 : 52 VV DS ,above10 kHz ),wheretheLorentzia rollsofastenough(approximately1 =f 2 )thatathighfrequenciesitisexceededby theoxidenoise.TheseLorentziaareatthesamecutofreque ncyastheLorentzia causedbythetrapsinthebulkdevice,sotheoxidenoiseisco mputedbyusingthetrap positionsusedforthebulkdevices(depictedinFigure 4-7 ),thentuningthetrapdensities andtheparameter tomatchtheincreasedoverallslopeoftheSOIdevice'soxid e noise,asplottedinFigure 4-15 .Trap1(asinFigure 4-7 )hasadensityof5 10 19 cm 3 correspondingtoatotalnumberof4trapsinthe4.8 m wideSOIdevice.Traps2and3 haveadensityof1 : 2 10 19 cm 3 ,correspondingtoatotalof1trap.Thepeakvalueof N tox 0 and usedtomatchthebackgroundnoiseasinthebulkdevicewere1 : 2 10 15 cm 3 and 2 : 6 10 7 cm 1 ,respectively.ThesimilaritiesbetweenthebulkandSOIde viceoxidenoise arestriking,andindicatethatthetrapsnearthesource/bo dyjunctionunderthegateare mostlikelyafunctionofthegateoxidationrecipe,and/ors ubsequentprocessingsteps. Thisdoesnotsayanythingaboutthetrapselsewhereintheox ide,however,asthisisthe regionofoxideinwhichthemeasuredoxidenoiseishighlyse nsitivetotrapswithinthe measurementtimewindow(1 Hz to100 kHz ). 108

PAGE 109

Asinthecaseofthebulkdevice'ssimulations,theshooting methodisusedto quantifytheerrorinthisresultduetotheshapeandenergyl eveloftheinversionlayer connedinthezdirection.Figures 4-16 and 4-17 depicttheshapeandenergylevelof therstsubbandintheinversionlayerunderthetraps.Thec arrierstunnelfromtherst subbandratherthanfromtheconductionbandedge,sotheses imulationssuggestthat trap1is134 meV higherinenergythantheclassicalmodelsuggests.Bymatch ingthe ratiosofthewavefunctionvalueatthepeakofthewavefunct ionandatthetrapposition tothatoftheclassicalmodel(asinEquation 4{3 ),thesesimulationsalsosuggestthat trap1is4 : 59 A closertothesurfacethansuggestedbytheclassicalmodel. Thesameerror correctionsarecomputedfortheothertrapsandsuggesttha ttraps2and3are196 meV higherinenergy(theyarelocatedatthesameyposition,soc arrierstunnelfromthesame rstsubband)and4 : 23 A and4 : 25 A closertothesurface,respectively. Forthevelocityructuationnoise,thesimulatedLorentzia areplottedinFigure 4-18 Thesourceofthisnoiseistheshotnoiseofthesource/bodyj unction,locatedprimarilyon thesourcesideofthemetallurgicaljunction,asdepictedi nFigure 4-19 .Hereitisuseful todiscusstherelationshipbetweenthebulkrecombination lifetimesofelectronsandholes andtheimpactionizationparameters,whichtogethersetth eIVcharacteristicsshownin Figure 4-13 .Iftherecombinationlifetimesofelectronsandholesareh alved,theparameter A ofEquation 2{53 canbedoubled,withoutaectingtheroatingbodypotential orthe kinkintheIVcurves.TheeectonthenoiseLorentziainstro ngsaturation(dominated bytheimpactionizationeect)however,istohalvetheirze ro-frequencyvalueanddouble theircutofrequency.Thisisinagreementwiththeclosedformequationsderivedby Workman[ 1 ]andcasthereasdraincurrentnoise,asfollow. S Lor I (0)= 4( mkTg db ) 2 q ( I GRtD + I Gi ) (4{4) f C = q ( I GRtD + I Gi ) 2 mkTC B (4{5) 109

PAGE 110

Theexpression S Lor I (0)isthezero-frequencyvalueoftheLorentzian,and f C isthe cutofrequency,setbytherstpoleintheimpedanceofthei nternalbodynode.The subthresholdslopefactorisgivenby m kT=q isthethermalvoltage,and g d b isthe draintobodyconductance.Theterms I GRtD and I Gi representthedrain/bodyjunction thermalgenerationandimpactionizationcurrents,respec tively. C B isthetotaleective capacitanceoftheinternalbodynode,includingstraycapa citancestogate,source,drain, andsubstratebeneaththeburiedoxide.Thiseect(scaling theimpactionizationrate inverselywiththerecombinationlifetimes)isheretakena dvantageoftoaectthenoise withoutaectingtheIVcurves,resultinginthetoftheLor entzias'zero-frequency valuesatlowdrainbias.However,thesimulatedvaluesathi gherdrainbiasdonott well.Therefore,theimpactionizationrateparameter B inEquation 2{53 isincreased slightlyto3 : 9 10 6 V=cm topushthiscurveouttotherightbyincreasingtheelectric eldrequiredtocauseagivenamountofimpactionization.W hilethisskewstheIV characteristicssomewhatasplottedinFigure 4-20 ,itseectonthesecurvesissignicantly outweighedbytheeectonthenoise.Anincreaseof40%issu cienttogreatlyincrease theagreementofthenoiseathigherdrainbias.Thistandth atoftheinitialcase areplottedinFigure 4-21 .Thecutofrequencyisstillextremelylow,asplottedin Figure 4-22 .Thoughthecutofrequencyqualitativelydecreasesandin creaseswithbias roughlyasthemeasureddatadoes,thevalueisalogarithmic averageof240toolow.Here considerationofthesimulatedproblemtodatemustbeconsi deredtoeectanimproved t.TheDESSIScommercialsimulatorusedinthisworkwaslimi tedtoconsiderasingle valueforthexedinterfacechargeusedatallSilicon/Oxid einterfaces,independentof position.Thexedinterfacechargeusedforthisworkisana rticiallyhighvalueused tocompensateforquantumconnement-relatedreductionin inversioncharge,andthisis onlyappropriateunderthethingateoxide,wherethehighbi ndingelectriceldreduces thenumberofinversioncarriers.Atthebackgate,thishigh xedchargehastheeectof accumulatingthep-typesurface,thinningthespacecharge regionsnearthesourceand 110

PAGE 111

drainjunctions,andincreasingthedrain/bodyandsource/ bodycapacitancesarticially. WhilethisbackgateaccumulationhaslittleeectontheDCIV characteristics,itis detrimentaltothetofthenoiseLorentzia,aspredictedby Equation 4{5 .Afterapplying aspatialdependencetothexedsurfacechargeinFLOODStodr ivethexedchargeto zeroatthebackgate,increasingtheimpactionizationrate byaconstantfactortoraise theimpactionizationtomaintainthetoftheIVcharacteri stics,andafterrebalancing theSRHrecombinationlifetimesandimpactionizationrate sasinTable 4-1 toobtaina goodtofthelow-frequencyLorentzia'smagnitudes,thelo wcutofrequencyhasbeen improvedtowithinanorderofmagnitude,asshowninFigure 4-23 Thereareunfortunatelynofurtherparametersthathavebee nconsideredarbitrary andwhichcanbetunedtoincreasethecutofrequencieswith outfurtheralteringthe tofthezero-frequencyvalue.Inordertomakethisparamet erhavebetteragreement, itisnecessarytodeviatefromthedopingprolegeneratedf romtheprovidedprocess description,orfromthephyicsofthetransportmodelprevi ouslypresented.Either isingeneralundesirableastheresultisthattheentiredev icestructurewouldthen beconsideredarbitrary.However,asmalldeviationisacce ptablebyaddingsome compensatingdonorconcentrationtotheregionatthebotto moftheactivedevicelm, betweenthemetallurgicalsourceanddrainp/njunctionsan dtheburiedoxide.Thisis acceptablebecauseitdoesnotaecttheDCIVcharacteristic softhedevice,whicharethe primaryconsiderationoftheprovidersofthedevice,whode signfordigitalapplications andwerenotoverlyconcernedwiththeaccuratemodelingoft hedopingprolesfornoise simulationpurposes.Reductioninthedopingatthebackoft helmcouldbecaused bytransientenhanceddiusionnearthebondedwaferoxide, andtheIVcharacteristics employedbyemployeesofthecompanythatdonatedthesedevi ceswouldbeinsensitive tosuchaneect.Thisreductionindopinghastheeectoffur therloweringthebulk capacitance,theonlytermintheclosedformequationforcu tofrequencynotpresent intheequationforzero-frequencyvalue.TheresultingLor entziaexhibitincreasedcuto 111

PAGE 112

frequencieswithoutalteredzero-frequencyvaluesandare plottedinFigure 4-24 .Thenal tisimprovedbothinlow-frequencyLorentziamagnitude,a ndincutofrequency. 112

PAGE 113

-4 -3 -2 -1 0 1 x 10 -5 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 x 10 -5 10 18 10 19 10 20 depth (cm) position (cm) Bulk doping, absolute value of net doping (cm ) -3 Figure4-1.AbsolutevalueofnetdopingdensityofthebulkM OSFETsimulatedinthis work.The"spikish"shapesbetweenthebodydopingandthest eeplyvarying dopinginthesource/drainregionsareduetothenetdopingz erocrossingnear themetallurgicaljunction,thesparsenessofthegrid,and theplottingonalog scale. 113

PAGE 114

00.20.40.60.81 V GS (V) 0 1e-05 2e-05I DS (A/ m m) Measurement: V DS = 1.2V FLOODS: V DS = 1.2V Measurement: V DS = 0.1V FLOODS: V DS = 0.1V Figure4-2.MatchofsimulatedgatecontrolIVcharacterist icsfor3 : 2 m x90 nm bulk device,for V DS =0 : 01 V and V DS =1 : 2 V 114

PAGE 115

-0.200.20.4 0.6 0.81 V GS (V) 1e-14 1e-13 1e-12 1e-11 1e-10 1e-09 1e-08 1e-07 1e-06 1e-05 0.0001 0.001I DS (A/ m m) Measurement: V DS = 1.2V FLOODS: V DS = 1.2V Measurement: V DS = 0.1V FLOODS: V DS = 0.1V Figure4-3.MatchofsimulatedgatecontrolIVcharacterist icsfor3 : 2 m x90 nm bulk device,for V DS =0 : 01 V and V DS =1 : 2 V ,onalogarithmicscale. 115

PAGE 116

00.20.40.60.81 V DS (V) 0 0.0001 0.0002 0.0003 0.0004 0.0005 0.0006 0.0007I DS (A/ m m) FLOODS: V GS = 1.25V Measurement: V GS = 1.25V Figure4-4.] MatchofsimulateddraincontrolIVcharacteristicsfor3 : 2 m x90 nm bulkdevice,for V GS =1 : 25 V 116

PAGE 117

110100100010000 1e+05 1e+06 frequency (Hz) 1e-21 1e-20 1e-19 1e-18 1e-17 1e-16 1e-15 1e-14S I (A 2 /Hz) V DS = 0.88V V DS = 0.68V V DS = 0.54V V DS = 0.39V V DS = 0.29V V DS = 0.17V lines: FLOODS, constant N tox Figure4-5.Matchofsimulatednoisecharacteristicsfor3 : 2 m x90 nm bulkdevicewith assumedconstanttrapdensity. 117

PAGE 118

-0.05 0 0.05 position along channel ( m m) 1e-39 1e-36 1e-33 1e-30 1e-27 1e-24 1e-21 1e-18 1e-15 1e-12 1e-09Noise Contribution ( (A 2 / Hz m m) cm -2 ) V DS = 0.17V V DS = 0.29V V DS = 0.39V V DS = 0.54V V DS = 0.68V V DS = 0.88V V DS Figure4-6.Contributiontodraincurrentnoiseofoxidetra ppingalongthechannelin 3 : 2 m x90 nm bulkdevice(sourceontheleftanddrainontheright)atthe depthofoxidecontributingmostat1kHz. 118

PAGE 119

Figure4-7.Graphicaldepictionoflocationsoftrapsonmes hforsimulatingthe90nm BulknMOSFET. 119

PAGE 120

110100100010000 1e+05 1e+06 frequency (Hz) 1e-22 1e-21 1e-20 1e-19 1e-18 1e-17 1e-16 1e-15 1e-14S I (A 2 /Hz) V DS = 0.88V V DS = 0.68V V DS = 0.54V V DS = 0.39V V DS = 0.29V V DS = 0.17V lines: FLOODS Figure4-8.Finaltofmeasuredandsimulatednoiseforthe9 0nmBulknMOSFET. 120

PAGE 121

-275 -270 -265-260-255-250-245 -240 -235 -230 x position (nm) 0 0.5 1 1.5 2 2.5 3Confinement Potential (eV) E C E C(surface) shape of n(x) 0.104eV first sub-band Figure4-9.Firstorderquantizedn(x)undertrap1,assumin gallcarriersareintherst subbandandusingthepotentialwellfromtheclassicalsimu lation. 121

PAGE 122

-270 -265-260-255-250-245 -240 -235 -230 x position (nm) 0 0.5 1 1.5 2 2.5 3Confinement Potential (eV) E C E C(surface) shape of n(x) 0.159eV first sub-band Figure4-10.Firstorderquantizedn(x)undertraps2and3,a ssumingallcarriersarein therstsubbandandusingthepotentialwellfromtheclassi calsimulation. 122

PAGE 123

Table4-1.Finalvaluesofimpactionizationratecoecient sandcarrierrecombination lifetimes parameter value A 1 : 3 10 3 cm 1 B 3 : 9 10 6 V=cm n 11 ns p 3 : 9 ns 123

PAGE 124

-4 -3 -2 -1 0 1 x 10 -5 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 x 10 -5 10 16 10 17 10 18 10 19 10 20 depth (cm) position (cm) SOI device, absolute value of net doping (cm ) -3 Figure4-11.NetdopingdensityoftheSOIMOSFETsimulatedi nthiswork.Therelative smoothnessofthisgurenearthemetallurgicaljunctionco mparedtothat inFigure 4-1 isduetothenermeshinthisregion,whichwasnecessaryto obtainconvergenceintheSOIdeviceover V DS 124

PAGE 125

-0.5-0.25 0 0.250.50.75 1 1.25 V GS (V) 1e-13 1e-12 1e-11 1e-10 1e-09 1e-08 1e-07 1e-06 1e-05 0.0001I DS (A/ m m) Measurement: V DS = 1.2V Simulation: V DS = 1.2V Measurement: V DS = 0.1V Simulation: V DS = 0.1V Figure4-12.MatchofsimulatedgatecontrolIVcharacteris ticsfor4 : 8 mx 90 nm SOI device,for V DS =0 : 01 V and V DS =1 : 2 V ,onalogarithmicscale. 125

PAGE 126

0 0.250.50.75 1 1.25 V DS (V) 0 0.0001 0.0002 0.0003 0.0004 0.0005 0.0006 0.0007 0.0008I DS (A/ m m) FLOODS: V GS = 1.25V Measurement: V GS = 1.25V Figure4-13.MatchofsimulateddraincontrolIVcharacteri sticsfor4 : 8 mx 90 nm SOI device,for V GS =1 : 25 V 126

PAGE 127

110100100010000 1e+05 frequency (Hz) 1e-21 1e-20 1e-19 1e-18 1e-17 1e-16 1e-15 1e-14S I (A 2 /Hz) V DS = 0.84V V DS = 0.69V V DS = 0.52V V DS = 0.38V V DS = 0.27V V DS = 0.17V V DS = 25mV lines = fit Lorenzia Figure4-14.Lorentziamathematicallyttothemeasuredda tainordertoplottheoxide noisewithoutthediusionnoiseinthemeasurements. 127

PAGE 128

110100100010000100000 frequency (Hz) 1e-20 1e-19 1e-18 1e-17 1e-16 1e-15 1e-14 1e-13S I (A 2 /Hz) V DS = 0.84V V DS = 0.38V V DS = 0.84V: FLOODS V DS = 0.38V: FLOODS Figure4-15.FitofOxidenoisesimulationstofeaturesinth e0.84Vand0.38Vcases,with theexcessnoiseremoved. 128

PAGE 129

-275-250-225-200-175 x position (nm) -1 0 1 2 3 4Confinement Potential (eV) E 0 = 134 meV above E C(surf) Figure4-16.Firstorderquantizedn(x)undertrap1,assumi ngallcarriersareintherst subbandandusingthepotentialwellfromtheclassicalsimu lation. 129

PAGE 130

-275-250-225-200-175 x position (nm) -1 0 1 2 3 4Confinement Potenial (eV) E 0 = 96 meV above E C(surf) Figure4-17.Firstorderquantizedn(x)undertraps2and3,a ssumingallcarriersarein therstsubbandandusingthepotentialwellfromtheclassi calsimulation. 130

PAGE 131

0.0010.010.1110100100010000 frequency (Hz) 1e-22 1e-21 1e-20 1e-19 1e-18 1e-17 1e-16 1e-15 1e-14S I (A 2 /Hz) V DS = 0.84V V DS = 0.69V V DS = 0.52V V DS = 0.38V V DS = 0.27V V DS = 0.17V V DS = 25mV Figure4-18.Velocityructuationnoisesimulationsfor V GS =1 : 25 V 131

PAGE 132

-4 -3 -2 -1 0 1 x 10 -5 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 x 10 -5 1 2 3 4 5 6 7 8 9 x 10 -25 depth (cm) position (cm) SOI device, Velocity Fluctuation Noise Spectral Density at 100Hz and VDS=0.84V (V cm /Hz) 2 3 Figure4-19.Spatialdistributionofcontributionstothep owerspectraldensityat100 Hz oftheholediusionnoiseinthe90nmSOIdeviceat V DS =0 : 84 V and V GS =1 : 25 V 132

PAGE 133

0 0.250.50.75 1 1.25 V DS (V) 0 0.0001 0.0002 0.0003 0.0004 0.0005 0.0006 0.0007 0.0008I DS (A/ m m) Measurement: V GS = 1.25V FLOODS: V GS = 1.25V FLOODS: V GS = 1.25V, 1.4X I-I parameter B Figure4-20.MatchofsimulateddraincontrolIVcharacteri stics,for V GS =1 : 25 V includingcasewithimpactionization B parameteradjustedby+40%. 133

PAGE 134

00.20.4 0.6 0.8 V DS (V) 1e-18 1e-17 1e-16 1e-15 1e-14S I(diff) 0 (A 2 /Hz) measurement fit IV data sim B*1.1 B*1.2 B*1.4 Figure4-21.Zero-frequencyvalueofLorentziainmeasurem entandinsimulationswith V GS =1 : 25 V .Thetwosimulationcasesarewiththeimpact-ionization parametersthatgaveagoodtfortheIVandthosethatgaveag oodtto thenoisemagnitude. 134

PAGE 135

00.20.4 0.6 0.8 V DS (V) 1e-01 1e+00 1e+01 1e+02 1e+03 1e+04 1e+05 1e+06f C (Hz) measurement IV-fit I-I values 1.4x B parameter Figure4-22.Cut-ofrequencyofLorentziainmeasurementa ndinsimulationswith V GS =1 : 25 V .Thetwosimulationcasesarewiththeimpact-ionization parametersthatgaveagoodtfortheIVandthosethatgaveag oodtto thenoisemagnitude. 135

PAGE 136

110100100010000 1e+05 frequency (Hz) 1e-21 1e-20 1e-19 1e-18 1e-17 1e-16 1e-15 1e-14S I (A 2 /Hz) V DS = 0.84V V DS = 0.69V V DS = 0.52V V DS = 0.38V V DS = 0.27V V DS = 0.17V V DS = 25mV lines: simulations Figure4-23.Fitofsimulatedoxidenoiseandsimulatedvelo cityructuationnoisewith correctedxedinterfacecharge,tothemeasureddata. 136

PAGE 137

110100100010000 1e+05 frequency (Hz) 1e-21 1e-20 1e-19 1e-18 1e-17 1e-16 1e-15 1e-14S I (A 2 /Hz) V DS = 0.84V V DS = 0.69V V DS = 0.52V V DS = 0.38V V DS = 0.27V V DS = 0.17V V DS = 25mV lines: simulations Figure4-24.Finaltofsimulatedoxidenoiseandsimulated velocityructuationnoise, withmodiedbacksidelmdoping,tothemeasureddata. 137

PAGE 138

CHAPTER5 CONCLUSION 5.1Summary Theworksubmittedinthisdissertationmakesseveralimpor tantcontributionstothe eldofsimulationsandnoiseinhighlyscaledsilicondevic es.TheFLOODSnumerical devicesimulatorwasenhancedwithoperatorsfortheAlagat orlanguage,whichallow thesimulationsofimpactionization,surfacemobilitymod elsthatdependontransverse electriceld,andhigherordermomentsforhydrodynamicmo dels.Highlyscaledbulkand partiallydepletedSOItransistorsweremeasuredandameth odforreverseengineering themforuseinnumericalsimulationswasdescribed.Thepre viousoxidenoisesimulation workwasextendedfordirecttunnelingbetweeninterfaceca rriersandoxidetraps.The oxidetrapsin90nmbulkandSOItransistorswereproled,an dtheresultsindicated thatspecictrapspeciesandpositionwerecharacteristic ofthegateoxidationprocess. Theexcesslowfrequencynoiseduetoimpactionizationands ubsequentsource/body forwardbiaswassimulatedandcomparedwithgoodagreement tothemeasureddata. TheshootingmethodwasimplementedtosolveSchrodinger' sEquation,givingarst ordercorrectionforthepositionandenergylevelofthetra pssimulatedinthedevices considered. 5.2RecommendationsforFutureWork Forsuchhighlyscaledtransistorsasthoseusedinthiswork ,quantizationofcarriers inthechannelshouldbeincluded.Thoughcurrentapproache ssuchastheVanDort modelortheDensityGradientmethodscanbediculttosimula teandaddcomplexityto analreadydicultsimulationproblem,asnumericalsolver sandmethodsimprove,these shouldbecomepossibleforthesimulationofsuchdevices.T hiswillmakeitunnecessary tousethexedchargetoreducetheinversionchargeinstron ginversion,andextend thecapabilityofthisworktosimulatenoiseinbothsubthre sholdconditionsandstrong inversioninasinglesimulation.Caremustbetakentoensur ethatthesmall-signal simulationsareaccurate,however,tomakenoisesimulatio npossiblewithinsucha 138

PAGE 139

framework.Deviceself-heating(latticeheatrow)isanothe rimportantcomponentthat shouldbeadded,thoughittooaddscomplexitytothesystemo fPDEs.Thiswould extendthisworkfortheapplicationofmodernSiGeBJTs,and improveaccuracyforSOI transistorsaswell. Inaddition,asgateoxidethicknessescontinuetoscale,an dashigh-dielectricconstant insulatorsareadoptedforgateoxides,theimplementation ofthephysicsofgateoxide trappingmustberened.Tunnelingtothegateshouldbeincl uded,andrecentwork showsthatinelastictunnelingmechanismssuchasthoseexp loredinworkbyLee[ 50 ] shouldbeconsideredaswell. Asthemodelingofconductionandheatrowmobilitymodelsim proveforhigher-order momentsthandrift-diusion,thesimulationofpartiallyd epletedSOIdevicesusingthese transportmodelsshouldbecomemoreaccurate,reducingsuc heectsasthenegativedrain conductance.Atthispoint,impedanceeldsimulationsofs caledSOIdevicesusinghigher ordertransportmodelswillbebenecialtogivemoreaccura teimpactionizationrates, sothatthesimulationofhighlyscaleddevicessensitiveto impactionizationwillbemore predictive. Finally,oxidetrappingsimulationshouldbeappliedtodou ble-gateandgate-all-around devices,astheprolingoftrapsinthesegateoxidesshould providemuchinsightinto thesedevices'fabrication.Waystodoquantumsimulations includingnoiseshouldbe investigatedforuseindevicessuchascarbonnanotubesand quantumdots.Thesedevices arelikelytorequiretheuseof3Dsimulationstocapturethe physicsassociatedwithnoise, whichwillrequirefurtherrenementoftheoxidetrappings imulationassumptions. 139

PAGE 140

APPENDIXA APPENDIX:SIMULATIONFILES Inthischapter,examplesaregivenofscriptsforcontrolli ngFLOODSthatwereused inthesimulationofthenoiseinthedevicesinthisdisserta tion.Muchofthecontentof codehereisleveragedfromthatofJuanSanchez,FrankHou,a ndProfessorMarkLaw. Thecharacters\ // "indicatethatalinewasbrokenhereforformattingpurpose s. noise7.dd #Simulationof90nmSOIDevicewithVDS=0.84Vsourcedd.physourcesoi.cktinitsilvaco=noise7.silsourcesoi.cnt#Usepreconfiguredscriptsforprintingcircuitvaluessourcepscriptssourcedmswp.tclsetVd0.84setVs0.0setVg1.25setVx0.0pdbSetDoubleMathupdateLimit1.0pdbSetDoubleMathiterLimit50pdbSetDoubleMathnoimpLimit10pdbSetBooleanMathresidCheck0selname=IntChargez=2.2e12selname=iifactz=1.0devicedevicedevicedevicedevicesourcetraps.oxsetf[open"noise7.dat"w]for{setfreq1.0e-3}\\ {$freq<=1.0e5}\\{setfreq[expr$freq*1.75]}{ puts$freqdevicenoise=Cur_Vdsfreq=$freqoxidepacsetN[expr{abs([circuitvalueSdiffname=Cur_Vds])}]setN2[expr{abs([circuitvalueSoxname=Cur_Vds])}]puts$f"$freq$N$N2"flush$f 140

PAGE 141

} dd.phy #Drift-Diffusionsolutionname=Potentialnosolvesolutionaddname=DevPsisolvedampnegativesolutionaddname=Elecsolve!negativesolutionaddname=Holesolve!negativesolutionaddname=ntisolve!negativesetTemp300sourcePhyParamsetni"[ni_cal$Temp]"setVt[expr$k*$Temp/$q]termname=Taddeqn="$Temp"sourceKlaassenNAs.tclsourceDarwishN.tclsourceKlaassenPB.tclsetmun"($mun/\\(sqrt(1.0+($mun*diff(DevPsi)/1.3e7)^2)))"setmup"($mup/\\(sqrt(1.0+($mup*diff(DevPsi)/8.37e6)^2)))"settaun1.11108333333333e-08settaup3.88891666666667e-09setU"($taup*(Elec+$n1)+$taun*(Hole+$p1))"setU"(Elec*Hole-$ni*$ni)/$U"setGII"iifact*eImpact"setesi[expr11.7*8.85418e-14]seteps[expr$esi/$q]seteqnP"$eps*grad(DevPsi)+Doping-Elec+Hole"seteqnE"ddt(Elec)-($mun/$q)*\\hdsgrad(Elec,$q*DevPsi,$k*T)+$U-$GII"seteqnH"ddt(Hole)-($mup/$q)*\\hdsgrad(Hole,-$q*DevPsi,$k*T)+$U-$GII"#Silicon/PolypdbSetDoubleSiliconDevPsiDampValue0.005pdbSetDoubleSiliconDevPsiAbs.Error1.0e-9pdbSetStringSiliconDevPsiEquation$eqnPpdbSetStringSiliconElecEquation$eqnEpdbSetDoubleSiliconElecAbs.Error1.0e-5pdbSetStringSiliconHoleEquation$eqnH 141

PAGE 142

pdbSetDoubleSiliconHoleAbs.Error1.0e-5pdbSetDoublePolySiliconDevPsiDampValue0.005pdbSetDoublePolySiliconDevPsiAbs.Error1.0e-9pdbSetStringPolySiliconDevPsiEquation$eqnPpdbSetStringPolySiliconElecEquation$eqnEpdbSetDoublePolySiliconElecAbs.Error1.0e-5pdbSetStringPolySiliconHoleEquation$eqnHpdbSetDoublePolySiliconHoleAbs.Error1.0e-5#Oxideseteso[expr3.9*8.85418e-14]setepo[expr$eso/$q]seteqnO"$epo*grad(DevPsi)-nti"pdbSetDoubleOxideDevPsiDampValue0.025pdbSetDoubleOxideDevPsiAbs.Error1.0e-9pdbSetStringOxideDevPsiEquation$eqnOseteqnsurf"DevPsi_Oxide-DevPsi_Silicon"pdbSetBooleanOxide_SiliconDevPsiFixedOxide1pdbSetBooleanOxide_SiliconDevPsiFluxConstant1pdbSetStringOxide_SiliconDevPsi\\Equation_Oxide$eqnsurfseteqnsurf"DevPsi_Oxide-DevPsi_Polysilicon"pdbSetBooleanOxide_PolysiliconDevPsiFixedOxide1pdbSetBooleanOxide_PolysiliconDevPsiFluxConstant1pdbSetStringOxide_PolysiliconDevPsi\\Equation_Oxide$eqnsurf#Nitridesetesn[expr7.0*8.85418e-14]setepn[expr$esn/$q]seteqnN"$epn*grad(DevPsi)"pdbSetDoubleNitrideDevPsiDampValue0.025pdbSetDoubleNitrideDevPsiAbs.Error1.0e-9pdbSetStringNitrideDevPsiEquation$eqnNseteqnsurf"DevPsi_Nitride-DevPsi_Oxide"pdbSetBooleanOxide_NitrideDevPsiFixedOxide1pdbSetBooleanOxide_NitrideDevPsiFluxConstant1pdbSetStringOxide_NitrideDevPsiEquation_Oxide$eqnsu rf #DiffusionNoisepdbSetStringSiliconElec\\ 142

PAGE 143

Noise.V_Elec"4.0*$Vt*$mun*Elec"pdbSetStringSiliconHole\\Noise.V_Hole"4.0*$Vt*$mup*Hole"#Interface/oxidenoisestuffsetz3.1setcni1.0e-7setcpi1.0e-7setn1i"cons(Elec_Silicon)"setp1i"cons(Hole_Silicon)"setzkt[expr{$z*$Vt}]setfkt[expr{4.0*$Vt}]setNti01.0e11setNti"($fkt*$Nti0*\\((cons(Elec_Silicon)/$ni)^$zkt))"setRni"($cni*Elec_Silicon*($Nti-nti))"setGni"($cni*$n1i*nti)"setRpi"($cpi*Hole_Silicon*nti)"setGpi"($cpi*$p1i*($Nti-nti))"setUni"($Rni-$Gni)"setUpi"($Rpi-$Gpi)"pdbSetBooleanOxide_SiliconElecFixedOxide0pdbSetBooleanOxide_SiliconElecFluxConstant0pdbSetStringOxide_SiliconElecEquation_Silicon"+$Uni pdbSetBooleanOxide_SiliconHoleFixedOxide0pdbSetBooleanOxide_SiliconHoleFluxConstant0pdbSetStringOxide_SiliconHoleEquation_Silicon"+$Upi pdbSetBooleanOxide_SiliconntiFixedOxide0pdbSetBooleanOxide_SiliconntiFluxConstant0pdbSetStringOxide_SiliconntiEquation\\"ddt(nti)-$Uni+$Upi"setintcharge"IntCharge"pdbSetStringOxide_SiliconDevPsi\\Equation_Silicon$intchargepdbSetStringOxide_PolySiliconDevPsi\\Equation_PolySilicon$intchargepdbSetStringOxide_SiliconElecNoise.N_Elec_SS\\"2.0*($Gni+$Rni)"pdbSetStringOxide_SiliconHoleNoise.N_Hole_SS\\"2.0*($Gpi+$Rpi)"pdbSetStringOxide_SiliconntiNoise.N_nti_II\\"2.0*($Gni+$Rni+$Gpi+$Rpi)"pdbSetStringOxide_SiliconElecNoise.N_nti_SI\\ 143

PAGE 144

"-2.0*($Gni+$Rni)"pdbSetStringOxide_SiliconHoleNoise.N_nti_SI\\"-2.0*($Gpi+$Rpi)"pdbSetStringOxide_SiliconntiNoise.N_Elec_IS\\"-2.0*($Gni+$Rni)"pdbSetStringOxide_SiliconntiNoise.N_Hole_IS\\"-2.0*($Gpi+$Rpi)"sourcedev.ox#drainpdbSetBooleanDElecFixed1pdbSetBooleanDHoleFixed1pdbSetBooleanDDevPsiFixed1pdbSetStringDElecEquation"Doping-Elec+Hole"pdbSetStringDHoleEquation"DevPsi+$Vt*log(Hole/$ni)D" pdbSetStringDDevPsiEquation"DevPsi-$Vt*log(Elec/$ni )-D" pdbSetStringDEquation"$q*(Flux_Hole-Flux_Elec)"#sourcepdbSetBooleanSElecFixed1pdbSetBooleanSHoleFixed1pdbSetBooleanSDevPsiFixed1pdbSetStringSElecEquation"Doping-Elec+Hole"pdbSetStringSHoleEquation"DevPsi+$Vt*log(Hole/$ni)S" pdbSetStringSDevPsiEquation"DevPsi-$Vt*log(Elec/$ni )-S" pdbSetStringSEquation"$q*(Flux_Hole-Flux_Elec)"#gatepdbSetBooleanGElecFixed1pdbSetBooleanGHoleFixed1pdbSetBooleanGDevPsiFixed1pdbSetStringGElecEquation"Doping-Elec+Hole"pdbSetStringGHoleEquation"DevPsi+$Vt*log(Hole/$ni)G" pdbSetStringGDevPsiEquation"DevPsi-$Vt*log(Elec/$ni )-G" pdbSetStringGEquation"$q*(Flux_Hole-Flux_Elec)"#bodypdbSetBooleanXElecFixed1pdbSetBooleanXHoleFixed1pdbSetBooleanXDevPsiFixed1pdbSetStringXHoleEquation"Doping-Elec+Hole"pdbSetStringXElecEquation"DevPsi-$Vt*log(Elec/$ni)X" pdbSetStringXDevPsiEquation"DevPsi+$Vt*log(Hole/$ni )-X" pdbSetStringXEquation"$q*(Flux_Hole-Flux_Elec)" 144

PAGE 145

hd.phy mathdevicedim=2umfgmresrowscalesolutionname=Potentialnosolvesolutionaddname=DevPsisolvedampnegativesolutionaddname=Elecsolve!negativesolutionaddname=Holesolve!negativesolutionaddname=Tnsolvedamp!negativesetTemp300sourcePhyParamsetni"[ni_cal$Temp]"setVt[expr$k*$Temp/$q]termname=Taddeqn="$Temp"sourceKlaassenNAs.tclsourceDarwishN.tclsourceKlaassenPB.tclsetmup"($mup/(sqrt(1.0+($mup*diff(DevPsi)/8.37e7)^2 )))" #grsettn2.0e-11settp6.0e-12setn1$nisetp1$nisetU"(Elec*Hole-$ni*$ni)/($tn*(Hole+$p1)+$tp*(Elec+ $n1))" setGI"eImpact"#ClosureExpressionssetUn"($k*Tn)"setUp"($k*T)"setWn"(1.5*$k*Tn)"setWno"(1.5*$k*T)"settaueE0.4e-12setdsquig[expr{4/3}]setlsquig0.0setlamp0.0setlamep0.0setDsquig[expr{4/3*(1-$lamp-$lamep)}]setW_U1.5setBeta2.0setmun"($mun/((sqrt(1.0+0.25*(1.5*$mun*$k*(Tn-$Temp )\\ 145

PAGE 146

/$q/$taueE/1.0e14)^2.0)+0.5*(1.5*$mun*$k*(Tn-$Temp) /$q/\\ $taueE/1.0e14))))"setmun"((f)*(eMob)+(1-f)*($mup))"setmup"((f)*(hMob)+(1-f)*($mup))"#setmun"eMob"#setmup"hMob"setesi[expr11.7*8.85418e-14]seteps[expr$esi/$q]seteqnP"$eps*grad(DevPsi)+Doping-Elec+Hole"seteqnE"ddt(Elec)-($mun/$q)*\\hdsgrad(Elec,$q*DevPsi-(1-$lamp)*($Un),$Un)+$U-$GI"seteqnH"ddt(Hole)-($mup/$q)*\\hdsgrad(Hole,-$q*DevPsi,$k*T)+$U-$GI"setDnSn"(($W_U*$mun/$q)*\\hdsgrad(Elec*($Un),$dsquig*$q*DevPsi-\\($Dsquig-$dsquig*$lamp-$lsquig)*($Un),\\($dsquig+$lsquig)*($Un)))"setJHe"-(DevPsi*($mun)*\\hdsgrad(Elec,$q*DevPsi-(1-$lamp)*($Un),$Un))-\\DevPsi*(ddt(Elec)*$q)"setnCen"((Elec+1.0e10)*($Wn-$Wno)/($taueE))"seteqnTE"ddt(Elec*$Wn)+fact*($DnSn)+fact*($JHe)-$nC en" #seteqnTE"ddt(Elec*$Wn)-$nCen"#SiliconpdbSetDoubleSiliconDevPsiDampValue0.005pdbSetDoubleSiliconDevPsiAbs.Error1.0e-9pdbSetStringSiliconDevPsiEquation$eqnPpdbSetStringSiliconElecEquation$eqnEpdbSetDoubleSiliconElecAbs.Error1.0e-5pdbSetStringSiliconHoleEquation$eqnHpdbSetDoubleSiliconHoleAbs.Error1.0e-5pdbSetDoubleSiliconTnDampValue300.0pdbSetStringSiliconTnEquation$eqnTEpdbSetDoubleSiliconTnAbs.Error1.0e-8#PolySiliconpdbSetDoublePolySiliconDevPsiDampValue0.005pdbSetDoublePolySiliconDevPsiAbs.Error1.0e-9pdbSetStringPolySiliconDevPsiEquation$eqnPpdbSetStringPolySiliconElecEquation$eqnEpdbSetDoublePolySiliconElecAbs.Error1.0e-5pdbSetStringPolySiliconHoleEquation$eqnH 146

PAGE 147

pdbSetDoublePolySiliconHoleAbs.Error1.0e-5pdbSetDoublePolySiliconTnDampValue300.0pdbSetStringPolySiliconTnEquation$eqnTEpdbSetDoublePolySiliconTnAbs.Error1.0e-8#Oxideseteso[expr3.9*8.85418e-14]setepo[expr$eso/$q]seteqnO"$epo*grad(DevPsi)"pdbSetDoubleOxideDevPsiDampValue0.025pdbSetDoubleOxideDevPsiAbs.Error1.0e-9pdbSetStringOxideDevPsiEquation$eqnOseteqnsurf"DevPsi_Oxide-DevPsi_Silicon"pdbSetBooleanOxide_SiliconDevPsiFixedOxide1pdbSetBooleanOxide_SiliconDevPsiFluxConstant1pdbSetStringOxide_SiliconDevPsiEquation_Oxide$eqnsu rf setintcharge"IntCharge"pdbSetStringOxide_SiliconDevPsiEquation_Silicon$int charge seteqnsurf"DevPsi_Oxide-DevPsi_Polysilicon"pdbSetBooleanOxide_PolysiliconDevPsiFixedOxide1pdbSetBooleanOxide_PolysiliconDevPsiFluxConstant1pdbSetStringOxide_PolysiliconDevPsiEquation_Oxide$e qnsurf #NoisepdbSetStringSiliconElecNoise.V_Elec"4.0*$k*Tn*$mun* Elec/$q" pdbSetStringSiliconElecNoise.V_Tn\\"-4.0*$k*Tn*$mun*Elec*(2.0*$k*Tn/$q-DevPsi)"pdbSetStringSiliconTnNoise.V_Tn\\"4.0*$k*Tn*$mun*Elec*$q*((2.0*$k*Tn/$q-DevPsi)^2.0) pdbSetStringSiliconTnNoise.V_Elec\\"-4.0*$k*Tn*$mun*Elec*(2.0*$k*Tn/$q-DevPsi)"pdbSetStringSiliconHoleNoise.V_Hole\\"4.0*$Vt*$mup*Hole"#GpdbSetBooleanGElecFixed1pdbSetBooleanGHoleFixed1pdbSetBooleanGDevPsiFixed1pdbSetBooleanGTnFixed1pdbSetStringGElecEquation"Doping-Elec+Hole"pdbSetStringGHoleEquation"DevPsi+$Vt*log(Hole/$ni)G" pdbSetStringGDevPsiEquation"DevPsi-$Vt*log(Elec/$ni )-G" 147

PAGE 148

pdbSetStringGTnEquation"Tn-T"pdbSetStringGEquation"$q*(Flux_Hole-Flux_Elec)"#SpdbSetBooleanSElecFixed1pdbSetBooleanSHoleFixed1pdbSetBooleanSDevPsiFixed1pdbSetBooleanSTnFixed1pdbSetStringSElecEquation"Doping-Elec+Hole"pdbSetStringSHoleEquation"DevPsi+$Vt*log(Hole/$ni)S" pdbSetStringSDevPsiEquation"DevPsi-$Vt*log(Elec/$ni )-S" pdbSetStringSTnEquation"Tn-T"pdbSetStringSEquation"$q*(Flux_Hole-Flux_Elec)"#DpdbSetBooleanDElecFixed1pdbSetBooleanDHoleFixed1pdbSetBooleanDDevPsiFixed1pdbSetBooleanDTnFixed1pdbSetStringDElecEquation"Doping-Elec+Hole"pdbSetStringDHoleEquation"DevPsi+$Vt*log(Hole/$ni)D" pdbSetStringDDevPsiEquation"DevPsi-$Vt*log(Elec/$ni )-D" pdbSetStringDTnEquation"Tn-T"pdbSetStringDEquation"$q*(Flux_Hole-Flux_Elec)"#BpdbSetBooleanBElecFixed1pdbSetBooleanBHoleFixed1pdbSetBooleanBDevPsiFixed1pdbSetBooleanBTnFixed1pdbSetStringBHoleEquation"Doping-Elec+Hole"pdbSetStringBElecEquation"DevPsi-$Vt*log(Elec/$ni)B" pdbSetStringBDevPsiEquation"DevPsi+$Vt*log(Hole/$ni )-B" pdbSetStringBTnEquation"Tn-T"pdbSetStringBEquation"$q*(Flux_Hole-Flux_Elec)" dev.ox #OxideNoiseStuff.setNT1.2e15setNC2.98e19setyc0.01085setyc0.0setsy0.01pdbSetBooleanOxide_SiliconElecdegenerate1pdbSetStringOxide_SiliconNoiseEqn"Elec" 148

PAGE 149

pdbSetStringOxideNoiseEqn"nti"pdbSetStringOxidentitau0"2.4e-11"pdbSetStringOxidentialpha"-1.4e8"pdbSetStringOxide_SiliconntiNtitf"1.0"pdbSetStringOxidentiNtox0\\"$NT*(exp(-(y-$yc)*(y-$yc)/$sy/$sy))*xf"pdbSetStringOxidentieta"2.6e7"pdbSetStringOxidentilambda"1e-47"pdbSetStringOxidentiNt0"ntrap"pdbSetStringOxide_SiliconElecNc"$NC" traps.ox #Oxidetrapstuffsetwin0.00002setnt12e17setx1-0.269562sety1-0.0217setx11-0.26965selname=NT1z=$nt1*(x>($x1+$win))*(x<($x1-$win))*\\(y>($y1+$win))*(y<($y1-$win))selname=NT11z=$nt1*(x>($x11+$win))*(x<($x11-$win))* \\ (y>($y1+$win))*(y<($y1-$win))setnt25e18setx2-0.268991sety2-0.026875selname=NT2z=$nt2*(x>($x2+$win))*(x<($x2-$win))*\\(y>($y2+$win))*(y<($y2-$win))selname=ntrapz=NT1+NT11+NT2 soi.ckt #SOIdevicecircuitcircuitclearcircuitaddname=Vgsfrom=Gvoltageeq={G-$Vg}circuitaddname=Vdsfrom=Dvoltageeq={D-$Vd}\\ac.r={1.0}ac.i={0.0}circuitaddname=Vssfrom=Svoltageeq={S-$Vs}circuitaddname=Vxsfrom=Xvoltageeq={X-$Vx} soi.cnt ################################################### ############# #Definitionofcontactregionsfordevice#################################################### ############# ##Definegeometricallocationsforcontactboxes 149

PAGE 150

#contactname=Dsiliconylo=0.24yhi=0.35xlo=-0.268xhi=0.264 contactname=Ssiliconylo=-0.35yhi=-0.24xlo=-0.268xhi =-0.264 contactname=Gpolysiliconylo=-0.4yhi=0.4xlo=-0.42xhi =-0.417 contactname=Xpolysiliconxlo=0.099xhi=0.101 KlaassenNAs.tcl #KlaassenMobilityParametervaluesforArsenicsetKEmumax1417.0setKEmumin52.2setKENref19.68e16setKEalph10.68setKEcD0.21setKENrefD4.0e20setKEcA0.50setKENrefA7.2e20setKEtheta_e2.285setKEtheta_h2.247#Klaassen'sfitparameterssetKEme1.0setKEmh1.258setKEfcw2.459setKEfbh3.828#LatticescatteringsetKEmuL"($KEmumax*(300.0/T)^$KEtheta_e)"#ImpurityandHoleScattering.setKEmuiN[expr"($KEmumax*$KEmumax/($KEmumax-$KEmumi n))"] setKEmuiN"($KEmuiN*(T/300.0)^(3.0*$KEalph1-1.5))"setKEmuiC[expr"($KEmumax*$KEmumin/($KEmumax-$KEmumi n))"] setKEmuiC"($KEmuiC*(300.0/T)^0.5)"setKENsc"(Donors+Acceptors+Hole)"setKEPBH"(1.36e20/Elec*$KEme*(T/300.0)^2.0)"setKEfoo"(1.0+1.0/($KEcA+($KENrefA/$KENsc)^2.0))"termname=KEZaddeqn=$KEfoosetKEPCW"(3.97e13*((1.0/((KEZ)^3.0*$KENsc))*\\(T/300.0)^3.0)^(2.0/3.0))"setKEfoo"(1.0/($KEfcw/($KEPCW)+$KEfbh/($KEPBH)))"termname=KEPaddeqn=$KEfoo 150

PAGE 151

setKEs10.89233setKEs20.41372setKEs30.19778setKEs40.28227setKEs50.005978setKEs61.80618setKEs70.72169setKEfoo\\"(1.0-$KEs1/($KEs2+((T/300.0/$KEme)^$KEs4)*(KEP))^$ KEs3+\\ $KEs5/((($KEme*300.0/T)^$KEs7)*KEP)^$KEs6)"termname=KE_Gaddeqn=$KEfoosetKEr10.7643setKEr22.2999setKEr36.5502setKEr42.3670setKEr5-0.01552setKEr60.678setKEfoo\\"(($KEr1*KEP^$KEr6+$KEr2+$KEr3*$KEme/$KEmh)/\\(KEP^$KEr6+$KEr4+$KEr5*$KEme/$KEmh))"termname=KE_Faddeqn=$KEfoosetKENsceff"(Donors+KE_G*Acceptors+Hole/KE_F)"setKEfoo"($KEmuiN*$KENsc/$KENsceff*\\($KENref1/$KENsc)^$KEalph1+$KEmuiC*(Elec+Hole)/$KEN sceff)" termname=KEmuDAhaddeqn=$KEfoo#CombineusingMathiesson'srulesetKEfoo"($KEmuL*KEmuDAh/($KEmuL+KEmuDAh))"termname=KEmobaddeqn=$KEfoosetmun"KEmob" DarwishN.tcl #DarwishElectronModel.Assumesalowfieldmobilityhas#alreadybeendefinedand$munistheelectronmobility.#ModelparameterssetDEB3.61e7setDEC1.70e4setDEtau0.0233 151

PAGE 152

setDEdelta1.2e10setDEA1.08setDEalpha6.85e-21setDEeta0.0767setDEkappa1.7#surfaceacousticphononscatteringsetDETpri"((T/300.0)^$DEkappa)"setDEfoo"(($DEB*$DETpri)/(trans(DevPsi)+1.0e-10))"setDEfoo"($DEfoo+(($DEC*(Donors+Acceptors+Hole)^$DE tau)/\\ ((trans(DevPsi)+1.0e-10)^([expr1.0/3.0]))))"setDEmuac"($DEfoo*(1.0/$DETpri))"#surfaceroughnessscatteringsetDEgamma"(($DEA+(($DEalpha*(Elec+Hole))/\\((Donors+Acceptors+Hole)^($DEeta)+1.0e-45))))"setDEmusr"($DEdelta/(((trans(DevPsi))+1e-10)^($DEga mma)+\\ 1.0e-10))"#CombineusingMathiesson'srulesetDEmob"(($mun)*($DEmuac)*($DEmusr)/(($mun)*($DEmu ac)+\\ ($DEmuac)*($DEmusr)+($DEmusr)*($mun)))"setmun"$DEmob" PhyParam #PhysicalConstantsUsedprocbandgap{T}{ setA11.17;setB11.059e-5;setC1-6.05e-7;setA21.1785;setB2-9.025e-5;setC2-3.05e-7;setA31.206;setB3-2.73e-4;setC30.0;if{$T<168.0}{ return[expr{$A1+$B1*$T+$C1*$T*$T}] }elseif{$T<300.0}{ return[expr{$A2+$B2*$T+$C2*$T*$T}] }else{ return[expr{$A3+$B3*$T+$C3*$T*$T}] } }setpi3.14159265358979 152

PAGE 153

setk1.38066e-23setq1.60218e-19setmo0.91093837e-30seth2[exprpow(6.6260755e-34,2.0)]procmdc_cal{T}{ setEg0"[bandgap0.0]"setEg"[bandgap$T]"setm_t[expr(0.1905*$Eg0/$Eg)]setm_l0.9163 setfoo1[expr(pow(6.0,(2.0/3.0)))]setfoo2[expr(pow($m_t,2.0)*$m_l)]setfoo3[expr(1.0/3.0)] return[expr($foo1*pow($foo2,$foo3))] }procmdv_cal{T}{ seta0.4435870;setb0.3609528e-2;setc0.1173515e-3;setd0.1263218e-5;sete0.3025581e-8;setf0.4683382e-2;setg0.2286895e-3;seth0.7469271e-6;seti0.1727481e-8; setnum[expr($a+$b*$T+$c*$T*$T+\\$d*$T*$T*$T+$e*$T*$T*$T*$T)]setden[expr(1.0+$f*$T+$g*$T*$T+\\$h*$T*$T*$T+$i*$T*$T*$T*$T)]setfoo[expr(2.0/3.0)] return[expr(pow(($num/$den),$foo))] }procNc_cal{T}{ globalpikmoh2setm_dc"[mdc_cal$T]" setfoo\\[expr2.0*pow((2.0*$pi*$k*$mo*300.0/($h2*1.0e4)),1.5 )] return[expr($foo*pow(($m_dc*$T/300.0),1.5))] }procNv_cal{T}{ globalpikmoh2setm_dv"[mdv_cal$T]" setfoo\\[expr2.0*pow((2.0*$pi*$k*$mo*300.0/($h2*1.0e4)),1.5 )] return[expr($foo*pow(($m_dv*$T/300.0),1.5))] }procni_cal{T}{ 153

PAGE 154

globalkqsetEg"[bandgap$T]"setNc"[Nc_cal$T]"setNv"[Nv_cal$T]" setfoo[expr($Nc*$Nv*exp(-$Eg*$q/($k*$T)))] return[exprsqrt($foo)] } dmswp.tcl #tclproceduresfordoingsweeps#requirespscriptsprocRamp1{varnameinitvfinalvmaxstepbody}{ globalVgVdVbVsVxupvar$varnamevsetv$initvdeviceuplevel$bodywhile{[exprabs($initv)]<[exprabs($finalv)]}{ setinitv[expr$initv+$maxstep] if{[exprabs($initv)]>[exprabs($finalv)]}{ setinitv$finalv} setv$initv device uplevel$body} } proclinsweep{varnameinitvfinalvinitstep\\minstepmaxstepfilenametext1text2body}{#Declareglobalvariableshereforcontactvariables#(veryimportant,anddiffersforeachdevicetype!) globalVgVdVxVsupvar$varnamev #WriteMCAfilepreamble puts"writingto$filename"setofile[open$filenamew]setns""setmsg"%MC-ASCIItext{$text1}{$text2}\n%\n%\\record$nsVd$nsId$nsVg$nsIg$nsVs$nsIs$ns\\Vb$nsIb\n"setun"%V\\8m\\5A/\\8m\\5m"appendmsg"%None$un$un$un$un\n"setns"%" 154

PAGE 155

appendmsg"%1.0$ns1.0$ns1.0$ns1.0$ns1.0\\$ns1.0$ns1.0$ns1.0$ns1.0"puts$ofile$msgflush$ofilesetrcd100 setv[expr$initv]setret[catch{device}]if{$ret}{ puts"Initialconvergencefailure!!!"break; }setstep$initstepsetintlv[expr{$initv+$step}]setv[expr$intlv]if{$finalv>$initv}{ setsign1 }else{ setsign-1 } setncvg0 while{[expr$sign*$intlv]<[expr$sign*$finalv]}{ setret[catch{device}] if{$ret}{ setintlv[expr{$intlv-$step}] setncvg0setstep[expr$step/1.8] if{$sign*$step<$minstep}{ puts"Toosmallofastep!!!!!" break; } puts[format"Non-Convergence!!\\ReducingStepto%1.7e"$step] setintlv[expr{$intlv+$step}]setv[expr$intlv] }else{ uplevel$bodysetmsg"$rcd"setfoo[circuitname=Dvalue]appendmsg[format"%1.7e"$foo]setfoo[expr[circuitname=Cur_Vdsvalue]*1e6]appendmsg[format"%1.7e"$foo]setfoo[circuitname=Gvalue]appendmsg[format"%1.7e"$foo] 155

PAGE 156

setfoo[expr[circuitname=Cur_Vgsvalue]*1e6]appendmsg[format"%1.7e"$foo]setfoo[circuitname=Svalue]appendmsg[format"%1.7e"$foo]setfoo[expr[circuitname=Cur_Vssvalue]*1e6]appendmsg[format"%1.7e"$foo]setfoo[circuitname=Xvalue]appendmsg[format"%1.7e"$foo]setfoo[expr[circuitname=Cur_Vxsvalue]*1e6]appendmsg[format"%1.7e"$foo]puts$ofile$msgflush$ofile #Seeifwecanincreasethestepsize setncvg[expr$ncvg+1]if{$ncvg>=3}{ #3convergencesinarow.Increasestepsize! setstep[expr$step*1.9]if{$step>$maxstep}{ setstep[expr$maxstep] } } setintlv[expr$intlv+$step]setv[expr$intlv] }incrrcd } #whileisexited,andintlvisjustbeyond$finalv.#revertintlvtolastvalue,andthendowhateverit#takestodriveitto$finalvbeforeexiting. setintlv[expr$intlv-$step] setstep[expr$finalv-$intlv] while{[expr$sign*$intlv]<[expr$sign*$finalv]}{ setintlv[expr$finalv]setv[expr$intlv] setret[catch{device}] if{$ret}{ setintlv[expr$intlv-$step] setstep[expr$step/1.7] if{$step<$minstep}{ puts"Toosmallofastep!!!!!" break; } puts[format"Non-Convergence!!\\ReducingStepto%1.7e"$step] 156

PAGE 157

setintlv[expr$intlv+$step]setstep[expr$finalv-$intlv]setv[expr$intlv] } } uplevel$bodysetmsg"$rcd"setfoo[circuitname=Dvalue]appendmsg[format"%1.7e"$foo]setfoo[expr[circuitname=Cur_Vdsvalue]*1e6]appendmsg[format"%1.7e"$foo]setfoo[circuitname=Gvalue]appendmsg[format"%1.7e"$foo]setfoo[expr[circuitname=Cur_Vgsvalue]*1e6]appendmsg[format"%1.7e"$foo]setfoo[circuitname=Svalue]appendmsg[format"%1.7e"$foo]setfoo[expr[circuitname=Cur_Vssvalue]*1e6]appendmsg[format"%1.7e"$foo]setfoo[circuitname=Xvalue]appendmsg[format"%1.7e"$foo]setfoo[expr[circuitname=Cur_Vxsvalue]*1e6]appendmsg[format"%1.7e"$foo]puts$ofile$msgflush$ofile puts"Finishedwithsweep.Closing$filename."close$ofile}procdiffnoisesweep{varnameinitvfinalvinitstep\\minstepmaxstepfilehandletext1text2body}{ globalVgVdVbVsVxupvar$varnamevputs"writingtooutput"setv$initvwhile{[exprabs($initv)]<\\[exprabs($finalv)+1.0e-4]}{ setret[catch{device}] if{$ret}{ setinitv[expr$initv-$initstep]setv$initvsetinitstep[expr$initstep*3.0/4.0]if{$maxstep<$minstep}{ 157

PAGE 158

puts"Toosmallofastep!!!!!"break;} puts[format"Non-Convergence!!\\ReducingStepto%1.7e"$initstep]}else{uplevel$body device setfoo[circuitname=Dvalue]setmsg[format"%1.7e"$foo] devicenoise=Cur_Vdsfreq=100.0setfoo[expr\\{abs(1.0e4*[circuitvalueSdiffname=Cur_Vds])}]appendmsg[format"%1.7e"$foo]puts$filehandle$msgflush$filehandle } setinitstep[expr$initstep*6.0/5.0]if{$initstep>$maxstep}{ setinitstep[expr$maxstep] } setinitv[expr$initv+$initstep] setv$initv#if{[exprabs($initv)]>[exprabs($finalv)]}{#setinitv$finalv#}} close$ofilesetv$finalv } pscripts ####################procpac{}{ puts"\nACSolution\n\tReal\t\tImaginary" foreachi[circuitnodes]{ setx[circuitvaluerealname=$i]sety[circuitvalueimagname=$i]puts[format"%s\t%1.5e\t%1.5e"$i$x$y] } }procpac2{}{ puts"\nACSolution\n\tMagnitude\t\tPhase" foreachi[circuitnodes]{ 158

PAGE 159

setx[circuitvaluemagname=$i]sety[circuitvaluephasename=$i]puts[format"%s\t%1.5e\t%1.5e"$i$x$y] } }procpdc{}{ puts"DCSolution" foreachi[circuitnodes]{ setx[circuitvaluename=$i]puts[format"%s\t%1.5e"$i$x] } }procphb{}{ puts"HBSolution" setf[hbcircuitfreq] puts[format"freq\t%s"$f]foreachi[circuitnodes]{ setxr[hbcircuitvaluename=$ireal]setxi[hbcircuitvaluename=$iimag]puts[format"%s\t%s"$i$xr]puts[format"\t%s"$xi] } }procphb2{}{ puts"HBSolution" setf[hbcircuitfreq] puts[format"freq\t%s"$f]foreachi[circuitnodes]{ setxr[hbcircuitvaluename=$imag]setxi[hbcircuitvaluename=$ipha]puts[format"%s\t%s"$i$xr]puts[format"\t%s"$xi] } }procphbac{}{ puts"HBSolution" setf[hbcircuitssfreq] puts[format"freq\t%s"$f]foreachi[circuitnodes]{ setxr[hbcircuitvaluename=$irealss] 159

PAGE 160

setxi[hbcircuitvaluename=$iimagss]puts[format"%s\t%s"$i$xr]puts[format"\t%s"$xi] } }procphbac2{}{ puts"HBSolution" setf[hbcircuitssfreq] puts[format"freq\t%s"$f]foreachi[circuitnodes]{ setxr[hbcircuitvaluename=$imagss]setxi[hbcircuitvaluename=$iphass]puts[format"%s\t%s"$i$xr]puts[format"\t%s"$xi] } }procmydump{ddir}{ catch{execmkdir$ddir}cd$ddirsetf[select]foreachi$f{ if{![stringcompare$iEdgeCouple]}continue#puts$iselectz=$iprint.dataoutf=$i.dat }cd.. } shoot #!/usr/bin/envperluseMath::BigFloat;subfstr{ my$l=shift@_;my$D=shift@_;my$m=Math::BigFloat->new($l);$m->bround($D);my$mm=$m->mantissa();my$MM=$mm->bfloor();$MM="$MM"."\.0";my$me=$m->exponent(); 160

PAGE 161

while(abs($MM)>=10.0){ $MM=$MM/10.0;$me+=1; }my$str="$MM"."e"."$me";return$str; }$file=$ARGV[0];$E=Math::BigFloat->new($ARGV[1]);$m=(0.26*9.10938188e-31);#kg$hbar=(1/2/3.141592654)*6.626e-34;#Js$K=-1.0*($hbar/($m))*$hbar;#J^2s^2/kgorJm^2$K=Math::BigFloat->new($K/1.602e-19);#ConvertfromJt oeV #Note,firstandlastpointsy-valuesdon'tmatter,astheya reconsideredinf. open(IF,$file);$i=0;while(){ if(/(\S+)\s+(\S+)/){ $dat[0][$i]=$1;$dat[1][$i]=$2;$i++; } }closeIF;$N=$i;$psi[0]=Math::BigFloat->new(0);$psi[1]=Math::BigFloat->new(1e-100);$max=Math::BigFloat->new(1e-100);$sign=1;for($i=2;$i<$N;$i++){ $d1=($dat[0][$i-1]-$dat[0][$i-2]); #if($d1<1e-16){print"$dat[0][$i]$d1!!\n";} $d2=($dat[0][$i]-$dat[0][$i-1]); #if($d2<1e-16){print"$dat[0][$i]$d2!!\n";} $A=$K/($d1+$d2)/$d2;$C=$K/($d1+$d2)/$d1;$B=-$A-$C;$D=$psi[$i-1]*($dat[1][$i-1]-$E);#A*Psi(i)+B*Psi(i-1)+C*Psi(i-2)+D=0$psi[$i]=(-$B*$psi[$i-1]-$C*$psi[$i-2]-$D)/$A;$psi[$i]->bround(25); 161

PAGE 162

#$es=fstr($psi[$i]);#print"$i$dat[0][$i]$es\n";if($psi[$i]=~/inf/){ $psi[$N-1]=$psi[$i];last; }if(abs($psi[$i])>$max){$max=abs($psi[$i]);}if($psi[$i]>0){ if($psi[$i-1]<0){ $sign=$sign*-1; } }else{ if($psi[$i-1]>0){ $sign=$sign*-1; } } }$sE=fstr($E,25);$se=fstr($psi[$N-1],25);$sm=fstr($max,25);print"$sE$se$sm\n"; shoot.fast #!/usr/bin/envperl$file=$ARGV[0];$E=$ARGV[1];$m=(0.26*9.10938188e-31);#kg$hbar=(1/2/3.141592654)*6.626e-34;#Js$K=-1.0*($hbar/($m))*$hbar;#J^2s^2/kgorJm^2$K=$K/1.602e-19;#ConvertfromJtoeV#Note,firstandlastpointsy-valuesdon'tmatter,astheya reconsideredinf. open(IF,$file);$i=0;while(){ if(/(\S+)\s+(\S+)/){ $dat[0][$i]=$1;$dat[1][$i]=$2;$i++; } }closeIF; 162

PAGE 163

$N=$i;$psi[0]=0;$psi[1]=1e-100;$max=1e-100;$sign=1;for($i=2;$i<$N;$i++){ $d1=($dat[0][$i-1]-$dat[0][$i-2]);if($d1<1e-16){print"$dat[0][$i]$d1!!\n";}$d2=($dat[0][$i]-$dat[0][$i-1]);if($d2<1e-16){print"$dat[0][$i]$d2!!\n";}$A=$K/($d1+$d2)/$d2;$C=$K/($d1+$d2)/$d1;$B=-$A-$C;$D=$psi[$i-1]*($dat[1][$i-1]-$E);#A*Psi(i)+B*Psi(i-1)+C*Psi(i-2)+D=0$psi[$i]=(-$B*$psi[$i-1]-$C*$psi[$i-2]-$D)/$A; #$es=fstr($psi[$i]);#print"$i$dat[0][$i]$es\n"; if($psi[$i]=~/inf/){ $psi[$N-1]=$psi[$i];last; }if(abs($psi[$i])>$max){$max=abs($psi[$i]);}if($psi[$i]>0){ if($psi[$i-1]<0){ $sign=$sign*-1; } }else{ if($psi[$i-1]>0){ $sign=$sign*-1; } } }$y=$psi[$N-1];print"$E$y$max\n"; quantlin #!/usr/bin/perluseMath::BigFloat;subfstr{ my$l=shift@_;my$m=Math::BigFloat->new($l);$m->bround(15); 163

PAGE 164

my$mm=$m->mantissa();my$MM=$mm->bfloor();$MM="$MM"."\.0";my$me=$m->exponent();while(abs($MM)>=10.0){ $MM=$MM/10.0;$me+=1; }my$str="$MM"."e"."$me";return$str; }$n=1.0;$sta=Math::BigFloat->new($ARGV[0]);$sto=Math::BigFloat->new($ARGV[1]);$N=$ARGV[2];$step=($sto-$sta)/$N;for($E=$sta->copy();$E<$sto;$E+=$step){ $sE=fstr($E);$l=`./shootwell.dat$sE`;$l=~/\S+\s+(\S+)/;$e=$1;print"$sE$e\n"; } quantize #!/usr/bin/perluseMath::BigFloat;subfstr{ my$l=shift@_;my$D=shift@_;my$m=Math::BigFloat->new($l);$m->bround($D);my$mm=$m->mantissa();my$MM=$mm->bfloor();$MM="$MM"."\.0";my$me=$m->exponent();while(abs($MM)>=10.0){ $MM=$MM/10.0;$me+=1; }my$str="$MM"."e"."$me";return$str; 164

PAGE 165

}$n=1.0;$IE=Math::BigFloat->new($ARGV[0]);$step=Math::BigFloat->new($ARGV[1]);$E=$IE;$flag=1; $l=`./shootwell.dat$E`;$l=~/(\S+)\s+(\S+)/;$e0=Math::BigFloat->new($2);$E=$E+$step;$i=1; $flag2=1;while($flag){ if($flag2){ $l=`./shootwell.dat$E`; }else{ $l=`./shoot.fastwell.dat$E`; }$l=~/(\S+)\s+(\S+)\s(\S+)/;$ma=Math::BigFloat->new($3);$me=$ma->exponent();$e1=Math::BigFloat->new($2);$sE1=fstr($E,25);$se1=fstr($e1,25);print"Iter$i:$E$se1\n";$i++;$step=-1.0*$e1*$step/($e1-$e0);$E=$E+$step;$e0=$e1->copy();;$f=$e1->copy();$f->babs();if($sE1eq$sE0){last;}$sE0=$sE1;if($me<100){$flag2=0;}#Don'tneedBigFloatanymoreinsh oot. }$sE=fstr($E,30);$se1=fstr($e1,30);print"$sE$se1\n"; 165

PAGE 166

REFERENCES [1]G.O.Workman, PhysicalModelingandAnalysisofDeep-submicronSiliconOnInsulatorCMOSDevicesandCircuits ,Ph.d.dissertation,UniversityofFlorida, Gainesville,FL,1999. [2]M.R.Pinto, ComprehensiveSemiconductorDeviceSimulationforSilico nULSI Ph.d.dissertation,StanfordUniversity,PaloAlto,CA,19 90. [3] UFSOIMOSFETModels,version6.0 ,UniversityofFlorida,Gainesville,FL,1997, revised2000. [4]DongwookSuhandJerryG.Fossum,\Aphysicalcharge-base dmodelfornon-fully depletedsoimosfet'sandit'suseinassessingroating-bod yeectsinsoicmos circuits," IEEETransactionsonElectronDevices ,vol.42,no.4,pp.728{737,1995. [5]M.G.AnconaandB.A.Biegel,\Non-lineardiscretizatio nschemeforthe density-gradientequations,"in ProceedingsoftheInternationalConferenceon SimulationofSemiconductorProcessesandDevices2000 .pp.6{9,IEEE. [6]MarkLundstrom, FundamentalsofCarrierTransport ,Addison-WesleyPublishing Company,Reading,MA,1990. [7]T.Tang,\Animprovedhydrodynamictransportmodelfors ilicon," IEEETransactionsonElectronDevices ,vol.40,no.8,pp.1469{1477,1993. [8]D.B.M.Klaassen,\Aunifedmobilitymodelfordevicesimu lation{i.modelequations andconcentrationdependence," Solid-StateElectronics ,vol.35,no.7,pp.953{959, 1992. [9]D.B.M.Klaassen,\Aunifedmobilitymodelfordevicesimu lation{ii.temperature dependenceofcarriermobilityandlifetime," Solid-StateElectronics ,vol.35,no.7, pp.961{967,1992. [10]MohamedN.Darwish,JanetLLentz,MarkR.Pinto,P.M.Zei tzo,ThomasJ. Krutsick,andHongHaVuong,\Animprovedelectronandholem obilitymodelfor generalpurposedevicesimulation," IEEETransactionsonElectronDevices ,vol.44, no.9,pp.1529{1538,sep1997. [11]R.Stratton,\Diusionofhotandcoldelectronsinsemic onductorbarriers," Physical Review ,vol.126,no.6,pp.2002{2014,1962. [12]K.Bltekjr,\Transportequationsforelectronsintw o-valleysemiconductors," IEEE TransactionsonElectronDevices ,vol.ED-17,no.1,pp.38{47,1970. 166

PAGE 167

[13]R.Thoma,B.Meinerzhagin,H.Peifer,andW.H.Engl,\Ag eneralized hydrodynamicmodelcapableofincorporatingmontecarlore sults,"in ProceedingsoftheInternationalElectronDevicesMeeting .1989,pp.89{139{89{142,IEEE. [14]M.IeongandT.Tang,\Inruenceofhydrodynamicmodelso nthepredictionof submicrometerdevicecharacteristics," IEEETransactionsonElectronDevices ,vol. 44,no.12,pp.2242{2251,1997. [15]YimingLiandShao-MingYu,\Auniedquantumcorrectio nmodelfornanoscale single-anddouble-gatemosfetsunderinversioncondition s," Nanotechnology ,,no.8, pp.1009{1016,2004. [16]S.KrishnanandJ.G.Fossum,\Compactnon-localmodeli ngofimpactionizationin soimosfetsforoptimalcmosdevice/circuitdesign," Solid-StateElectronics ,vol.39, no.5,pp.661{668,1996. [17]J.L.(Skip)Egley,B.Polsky,B.Min,E.Lyumkis,O.Penz in,andM.Foisy,\Soi relatedsimulationchallengeswithmomentbasedbtesolver s,"in Proceedingsofthe InternationalConferenceonSimulationofSemiconductorP rocessesandDevices 2000 .2000,IEEE. [18]B.Polsky,O.Penzin,K.E.Sayed,A.Schenk,A.Wettstei n,andW.Fichtner,\On negativedierentialresistanceinhydrodynamicsimulati onofpartiallydepletedsoi transistors," IEEETransactionsonElectronDevices ,vol.52,no.4,pp.500{506, 2005. [19]D.MunteanuandG.LeCarval,\Assessmentofanomalousbe haviorofcmosbulkand partiallydepletedsoidevices," JournaloftheElectrochemicalSociety ,vol.149,no. 10,pp.G574{G580,2002. [20]I.Bork,et.al.,\Inruenceofheatruxontheaccuracyof hydrodynamicmodelsfor ultrashortsimosfets," NUPADTechnicalDigest ,pp.63{66,1994. [21]M.Gritsch,etal.,\Revisionofthestandardhydrodyna mictransportmodelforsoi simulation," IEEETransactionsonElectronDevices ,vol.49,no.10,pp.1814{1820, 2002. [22]K.BanooandM.Lundstrom,\Electrontransportinamode lsitransistor," SolidStateElectronics ,2000. [23]W.Shockley,J.A.Copeland,andR.P.James,\Theimpeda nceeldmethodof noisecalculationinactivesemiconductordevices,"in QuantumTheoryofAtoms, Molecules,andtheSolid-State ,P.-O.Lowdin,Ed.,NewYork,1966,pp.537{563, Academic. 167

PAGE 168

[24]F.Bonani,G.Ghione,S.Donati,L.Varani,andL.Reggian i,\Ageneralframework forthenoiseanalysisofsemiconductordevicesoperatingi nnonlinear(large-signal quasi-periodic)conditions,"in 14thInt.Conf.onNoiseinPhys.Syst.and1/fFluctuations ,C.ClaeysandE.Simoen,Eds.,pp.144{147.WorldScientic ,Singapore, 1997. [25]J.E.Sanchez,F.-C.Hou,G.Bosman,andM.E.Law,\Physi csbasednoise simulationimplementedinroods,"in TECHON98 .1998,SemiconductorResearch Corporation. [26]F.C.Hou,GijsBosman,andMarkE.Law,\Characterizati onof generation-recombinationnoiseusingaphysics-baseddev icenoisesimulator,"in 15thInternationalConferenceonNoiseinPhysicalSystems and 1 =f Fluctuations Aug.1999. [27]J.E.SanchezandG.Bosman,\Frequencyconversionofri ckernoiseinbipolar junctiontransistors,"in Bipolar/BiCMOSCircuitsandTechnologyMeeting ,pp. 176{179.IEEE,Piscataway,NJ,1998. [28]F.Hou,\Unpublishedwork,"2001.[29]F.C.Hou, LowFrequencyBulkandSurfaceGeneration-RecombinationN oise SimulationsofSemiconductorDevices ,Ph.d.dissertation,UniversityofFlorida, Gainesville,FL,2002. [30]Jung-SukGoo,Chang-HoonChoi,EijiMorifuji,HisayoM omoseSasaki,Zhiping Yu,HiroshiIwai,ThomasH.Lee,andRobertW.Dutton,\Rfnois esimulationfor submicronmosfet'sbasedonhydrodynamicmodel," IEEETransactionsonElectron Device ,vol.41,no.3,pp.330{339,1994. [31]F.Bonani,G.Ghione,M.R.Pinto,andR.K.Smith,\Anec ientapproachtonoise analysisthroughmultidimensionalphysics-basedmodels, IEEETransactionson ElectronDevices ,vol.45,no.1,pp.261{269,1998. [32]E.Starikov,P.Shiktorov,andV.Gruzinkskis,\Theacc elerationschemefornoise modelingofdeepsubmicrondevices,"in 16thInt.Conf.NoisePhys.Syst.1/fFluct. G.Bosman,Ed.,pp.663{668.WorldScientic,2001. [33]Hyung-KyuLimandJerryG.Fossum,\Thresholdvoltageo fthin-lm silicon-on-insulator(soi)mosfet's," IEEETransactionsonElectronDevices ,vol. ED-30,no.10,pp.1244{1251,1983. [34]E.Simoen,UlfMagnusson,AntonioL.P.Rotondaro,andC orClaeys,\The kink-relatedexcesslow-frequencynoiseinsilicon-on-in sulatormosts," IEEETransactionsonElectronDevices ,vol.41,no.3,pp.330{339,1994. 168

PAGE 169

[35]N.Lukyanchikova,et.al.,\Backandfrontinterfacere latedgeneration-recombination noiseinburied-channelSOIpMOSFET's," IEEETransactionsonElectronDevices vol.43,no.3,Mar.1996. [36]S.Selberherr,W.Hansch,M.Seavey,andJ.Slotboom,\ Theevolutionofthe minimosmobilitymodel," Solid-StateElectronics ,vol.33,no.11,pp.1425{1436, 1990. [37]ClaudioLombardi,StefanoManzini,AntonioSaporito, andMassimoVanzi,\A physicallybasedmobilitymodelfornumericalsimulationo fnonplanardevices," IEEE TransactionsonComputerAidedDesign ,vol.7,no.11,pp.1164{1171,1988. [38] Dessis-ISE,ISETCADRelease6.1 ,1995{2000. [39]D.L.ScharfetterandH.K.Gummel,\Large-signalanalys isofasiliconreaddiode oscillator," IEEETransactionsonElectronDevices ,vol.ED-16,pp.66{77,1969. [40]T.-W.Tang,\Extensionofthescharfetter-gummelalgo rithmtotheenergybalance equation," IEEETransactionsonElectronDevices ,vol.ED-31,no.12,pp.1912{1914, 1984. [41]A.G.Chynoweth,\Ionizationratesforelectrionsandh olesinsilicon," Physics Review ,vol.109,pp.1537{40,1958. [42]R.V.OverstraetenandH.D.Man,\Measurementoftheioni zationratesindiused siliconjunctions," Solid-StateElectronics ,vol.13,pp.583{608,1970. [43]M.LiangandM.E.Law,\Inruenceoflatticeself-heatin gandhot-carriertransport ondeviceperformance," IEEETrans.ElectronDevices ,vol.41,no.12,pp. 2391{2398,Dec.1994. [44]M.LiangandM.E.Law,\Anobjectorientedapproachtode vicesimulation," IEEE Trans.Computer-AidedDesign ,vol.13,no.10,pp.1235{1240,Oct.1994. [45]S.Decker,C.Jungemann,B.Neinhus,andB.Meinerzhage n,\Anaccurateand ecientmethodologyforrfnoisesimulationsofnm-scalemo sfetsbasedona langevin-typedrift-diusionmodel,"in 16thInternationalConferenceonNoise inPhysicalSystemsand1/fFluctuations ,G.Bosman,Ed.,pp.659{662.World Scientic,2001. [46]C.Jungemann,B.Neinhus,S.Decker,andB.Meinerzhage n,\Hierarchical2drf noisesimulationofsiandsigedevicesbylangevin-typedda ndhdmodelsbasedon mcgeneratednoiseparameters,"in IEEEIEDMTech.Dig. ,pp.481{484.IEEE, Piscataway,NJ,2001. [47]K.M.vanVliet,\Noiseandadmittanceofthegeneration -recombinationcurrent involvingsrhcentersinthespace-chargeregionofjunctio ndevices," IEEETrans. 169

PAGE 170

ElectronDevices ,vol.ED-23,pp.1236{1246,Nov.1976. [48]C.Jungemann,C.D.Nguyen,B.Neinhus,andB.Meinerzha ngen,\Improved modiedlocaldensityapproximationformodelingofsizequ antizationinnmosfets," MSM ,vol.HiltonHeadIsland,pp.458{461,2001. [49]B.Neinhus,C.D.Nguyen,C.Jungemann,andB.Meinerzha gen,\Acpuecient electronmobilitymodelformosfetsimulationwithquantum correctedcharge densities," ESSDERC ,vol.Cork,pp.332{335,2000. [50]Jong-HwanLee, ComprehensiveCompactNoiseModelingofNanoscaleCMOS Devices ,Ph.d.dissertation,UniversityofFlorida,Gainesville, FL,2003. 170

PAGE 171

BIOGRAPHICALSKETCH DerekO.MartinwasborninColoradoSprings,CO,in1973.Here ceivedthe B.S.E.E.degree(withhonors)andtheM.S.degreefromtheUn iversityofFlorida, Gainesville,in1996and1998,respectively.Heiscurrentl ypursuingaPh.D.degreeat thesameuniversity.Thisendeavorhasbeensupportednanc iallybyfellowshipsfromthe NationalScienceFoundationandtheSemiconductorResearc hCompany. Duringthefallof1999heworkedasaninternatAdvancedMicro Devicesin Sunnyvale,CA.Duringthesummerof2001heworkedasanintern atMotorolainTempe, AZ.HehasrecentlybeenworkingatAgilentTechnologiesinC oloradoSprings,CO,where hewillcontinuehiscareeruponcompletionofhisPh.D.Hisre searchinterestslieindevice physics,modelinganddesign. 171


Permanent Link: http://ufdc.ufl.edu/UFE0006140/00001

Material Information

Title: Simulation of Noise Mechanisms in Scaled Bulk and Partially Depleted Silicon-on-Insulator Field-Effect Transistors
Physical Description: Mixed Material
Copyright Date: 2008

Record Information

Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
System ID: UFE0006140:00001

Permanent Link: http://ufdc.ufl.edu/UFE0006140/00001

Material Information

Title: Simulation of Noise Mechanisms in Scaled Bulk and Partially Depleted Silicon-on-Insulator Field-Effect Transistors
Physical Description: Mixed Material
Copyright Date: 2008

Record Information

Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
System ID: UFE0006140:00001


This item has the following downloads:


Full Text





LLHN OF NCscAi i i) B iK SCA B KAND PA1 i i :
DEPLE7 D SILICON-ON-INSULATOR Fli; 0 -EFFECT TR.A- "'"TOIRS




















By

DEGREE 0. MARLIN


A Di:' : NATION PiR ; 'TED TO THE (.i IADUIATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FUL: :
OF THE BK"T-RETRFA/:: : FOR THE DEGREE OF
DO( : C:: P( iii OSOPil/

UN1VER : : OF FLORIDA

2007



































-7 Derck 0. Martin


































To Kris, Will, Kate, Katnd Emily









AC i uOW Li i x ,'

I would to thank -. y- *. V '* committee chair, Professor C IBosman. for

his guidance and support. I also would like to express my gratitude to VF* .** Mark

Law, 7 :: Fossum, and Li .: fo>r serving on :l. supervisory committee and for their

interest. I thank (' Xiang and i i ilnma serving as hosts and mentors during

my internships. I would also like to thank ::: former professors, mentors and teachers in

research labs and in classrooms, in college and in secondary school. Ti'. work would not

have been i. :.I without a solid collective of knowledge and perience.

I would also like to thank the Semiconductor ch Company, the National Science

Foundation, Advanced Micro Devices, Motorola, Agilent, and the Uni---i'I of Florida

for their financial [*-. :. I aim grateful to Concetta Riccobene of AMD for ing

the bulk and SO1 transistors measured and simulated in this work, as well as the process

i: ions re ::': 1 for :. ess simulation.

My sincerest thanks go to Professor Law, Professor Bosman, Juan Sanchez, and Frank

Hou for deveb :. ; the :-I *F. ::.. for simulating noise in FLOODS. Without it, this work

would not have been possible. I also thank -. -i-.w students Lisa Kore and Jong-lHwan Lee

for their collaboration and technical discussions.

Finally, I am deeply l -..v to x--- wife, Kris, and '. '- '*-en for their support,

patience, and perseverance as I have pursued this work.









TABLE OF CONTENTS


page


ACKNOWLEDGMENTS ........

TABLE. ..................


LIST OF FIGURES . . . . . . . . .


ABSTRACT ...............


CHAPTER


1 INTRODUCTION .........


Fluctuation Phenomena
Numerical Device Simulation.
Transport Models .......
Noise Simulation .......
SOI Devices and Noise .


2 TRANSPORT MODELS AND PHYSICAL PROPERTIES .......

2.1 Advanced Transport Models for PDE-based Device Simulation .
2.1.1 Poisson's Equation. ......................
2.1.2 Moments of the Boltzmann Transport Equation . .


2.1.2.1
2.1.2.2
2.1.2.3
2.1.2.4
2.1.3 Closure
2.1.3.1
2.1.3.2
2.1.3.3
2.1.3.4


Zeroeth order moment: carrier continuity equations .
First order moment: momentum balance equations .
Second order moment: energy balance equations
Third order moment: heat flow balance equations . .
and Reduction Relations . . . . .
Momentum related reduction relations . . .
Energy related reduction relations . . . .
Heat flow related reduction and closure relations . .
PE and first four BTE moments, closed and summarized .


2.1.4 Hydrodynamic Model . . . . . . .
2.1.5 Reduced Hydrodynamic, or "Energy Balance" Model . .
2.1.6 Drift-Diffusion M odel . . . . . . .
2.2 Implementation of Surface Mobility Models in FLOODS . . .
2.2.1 Low Kinetic Energy Mobility Formulation . . . .
2.2.2 Hot Carrier M obility . . . . . . .
2.2.3 Implementation-Specific Details . . . . .
2.3 Implementation of the Energy Balance Transport Equations in FLOODS
2.4 Im pact Ionization . . . . . . . .
2.5 Impedance Field Simulation of Velocity Fluctuation Noise . . .
2.5.1 Velocity Fluctuation Noise Simulation . . . . .
2.5.1.1 Hydrodynamic model with velocity fluctuation noise .










2.5.1.2 Energy balance model with velocity fluctuation noise 48
2.5.1.3 Drift-diffusion model with velocity fluctuation noise . 51
2.5.2 Velocity Fluctuations in EB and DD Models .. . . 52
2.5.2.1 One Dimensional n/n/n+ Resistor Simulations . 52
2.5.2.2 nMOSFET Simulations . . . ...... 53
2.6 Impedance Field Simulation of Number Fluctuation Noise . . 54
2.6.1 Number Fluctuation Noise Simulation . . . . 54
2.6.1.1 Bulk or surface trap capture and emission . . 57
2.6.1.2 Direct Carrier tunneling . . . ..... 58
2.6.1.3 Impact Ionization ........ . . .... 61
2.6.2 Number Fluctuations in EB and DD Models for n /n/n Resistors 62
2.7 Quantization Effects and Noise .. . . . . . 63

3 NOISE MEASUREMENTS . . . . .. ......... 78

3.1 Measurement Setup ... . . .. . . .... 78
3.1.1 Resistor Network Design . . . ... . 78
3.1.2 Practical Considerations . . . . ... . 79
3.2 Measured Devices . . . . . . . . 80

4 SIMULATIONS OF MEASURED DEVICES . . . ..... 101

4.1 The 90nm Bulk MOSFET . . . . . . 102
4.2 The 90nm SOI MOSFET . . . . . . . 07

5 CONCLUSION . . . . . . . . . 138

5.1 Summary . . . . . . . . . 138
5.2 Recommendations for Future Work . . . . ... .. 138

A APPENDIX: SIMULATION FILES ... . . ... .... 140

BIOGRAPHICAL SKETCH . . . . . . . . 171









TABLE
Table page

4-1 Final impact ionization rates and carrier lifetimes . . . ... 123









LIST OF FIGURES
Figure page

1-1 Components of a signal including deterministic and stochastic parts . 21

2-1 Electron temperature in a 0.lpm n /n/n resistor . . . 66

2-2 Low bias velocity fluctuations in a 0.lpm n /n/n resistor . ... 67

2-3 High bias velocity fluctuations in a 0.lpm n/n/n resistor . . 68

2-4 Velocity fluctuation noise for n /n/n+ resistors of varying lengths. . 69

2-5 EB and DD models and velocity fluctuations in resistor .. . . 70

2-6 0.25pm nMOSFET doping concentration . . . . ... .. 71

2-7 Velocity fluctuations in DD model for 0.25pm nMOSFET . . 72

2-8 Velocity fluctuations in EB model for 0.25pm nMOSFET . . 73

2-9 Low bias number fluctuations in a 0.lpm n /n/n+ resistor . ... 74

2-10 Number fluctuations noise for n /n/n+ resistors of varying lengths . 75

2-11 EB and DD models and number fluctuations in resistor . . 76

2-12 Probability distributions and energies in an infinite well . . 77

3-1 Circuit used for bulk and SOI device noise measurements. . . . 82

3-2 Physical layout of circuit showing the critical path of electrical connections. 83

3-3 Connection scheme to ensure the avoidance of dielectric breakdown. . 84

3-4 Linear scale ID vs. VGS for 90nm bulk nMOSFET . . . 85

3-5 Logarithmic scale ID vs. VGS for 90nm bulk nMOSFET .. . . 86

3-6 ID vs. VDS for 90nm bulk nMOSFET . . . . .. . 87

3-7 Drain current noise of the 3.2pm x 90nm n-channel bulk MOSFET . 88

3-8 Linear scale ID vs. VGS for 90nm SOI nMOSFET . . . 89

3-9 Logarithmic scale ID vs. VGS for 90nm SOI nMOSFET . . 90

3-10 ID vs. VDS for 90nm SOI nMOSFET . . . . . . 91

3-11 Drain current noise of the 4.8pm x 90nm n-channel SOI MOSFET . 92

3-12 Linear scale ID vs. VGS for 2.33pm bulk nMOSFET . . . 93










3-13

3-14

3-15

3-16

3-17

3-18

3-19

4-1

4-2

4-3

4-4

4-5

4-6


4-7 Simulated oxide trap locations


Final fit of measured and simulated noise f

Quantized n(x) under trap 1 . .

Quantized n(x) under traps 2 and 3 .

Absolute value of SOI nMOSFET doping

Logarithmic scale match of ID vs. VGS. .

Match of simulated drain current for SOI c

Mathematic fit of Lorentzia for plotting.

Oxide noise without excess noise . .

Quantized n(x) under trap 1 . .


4-17 Quantized n(x) under traps 2 and 3 .


Velocity fluctuation noise simulations for V

Excess noise at 100Hz. . . .


Logarithmic scale ID vs. VGS for 2.33/m b

ID vs. VDS for 2.33/m bulk nMOSFET .

Drain current noise of the 3.2pim x 2.33/pm

Linear scale ID vs. VGS for 2.33pm SOI nN

Logarithmic scale ID vs. VGS for 2.33/m S

ID vs. VDS for 2.33pm SOI nMOSFET .

Drain current noise of the 0.6pim x 2.33pm

Absolute value of bulk nMOSFET doping

Linear scale match of ID vs. VGS in 90nm

Logarithmic scale match for ID vs. VGS.

Match of simulated drain current for bulk

Match of noise for 90nm bulk nMOSFET

Drain current noise contribution along chain


4-20 Linear scale match of ID vs. VDS with adjusted impact ionization rate . .


ulk nMOSFET .. . . 94

. . . . . . 9 5

n-channel bulk MOSFET . 96

4OSFET . . . 97

01 nMOSFET . . 98

. . . . . . 9 9

n-channel SOI MOSFET . 100

. . . . . . 1 1 3

bulk device. . . . 114

. . . . . . 1 15

device . . . . 116

with constant trap density. . 117

nnel/oxide surface . . 118

. . . . . . 1 19

or the 90nm Bulk nMOSFET. . 120

. . . . . . 12 1

. . . . . . 12 2

. . . . . . 12 4

. . . . . . 12 5

device . . . ..... 126

. . . . . . 12 7

. . . . . . 12 8

. . . . . . 12 9

. . . . . . 13 0

Gs = 1.25V. . . . 131

. . . . . . 13 2


4-8

4-9

4-10

4-11

4-12

4-13

4-14

4-15

4-16


4-18

4-19










4-21 Zero-frequency value of Lorentzia .. . . . ... ...... 134

4-22 Cut-off frequency of Lorentzia . . . . . . . 35

4-23 Fit of simulated noise with corrected interface charge . . . 136

4-24 Final fit of simulated noise, with modified backside body doping . . 137









Abstract of Dissertation Presented to the Graduate School
of the Uni of. :.1 in Partial Ftulfillment of the
Requirements the Degree of Doctor (. F"': :V /

SIMULATION OF NOVT'' MEC :I A"!'" IN SCALED BULK AND PARTIALLY
L I I )L ) 'II ICON, i N LA OR ii A i S

By

Derek 0. Martin

May 2007

C.* G: CGH' Bosnian
M.'. : Electrical and C :a te r Engineering

focus of this dissertation is on the simulation and measurement of low

noise in highly scaled : and Silicon-On-Insulator (SOI) metal-oxide-semiconductor

field-effect devices (MO i Ts). To further the t .. i.. of simulating noise in such

devices, the Florida Object-Oriented Device Simunlator (FLOODS) has been extended to

be able to use the Energy Balance system of equations. An advanced surface mobility

model is added, and the noise capability < this numerical device simulator is modified

to include the inechanism of direct band to oxide i. tunneling for the simulation of

degenerate devices.

1 t derivation of the Energy Balance model and other high-order inodels is presented,

and the correct insertion of noise sources into these models is described. Simple test cases

are used to demonstrate the simulation < noise using the model ':.. :] ** : and the

*- of this model as compared to the Drift-L'i'- -' ... model.

Noise measutrerments are taken for bulk and SOI MOSFETs. and simulations are

performed to match the conditions of the measurements. T I shape of the low frequency

noise is used to pri :: the active traps in the gate oxides. While higher order moments

than Drift-Diffusion gave anomalous DC simulation results for the devices measured,

D:: :-Diffusion results are < ::: : ed to the measured DC and noise data wN ith good

agreement, and error correction for I : .!' .. : < !- i is .










TiV. ugh the original intended usage of the En-' Balance model was not performed,

the ::: : Diffusion simulations gave good agreement to the measured data, and this work

represents the most ( ..: -.:.1 PDE-based numerical simulation of noise in such devices to

date. i: analysis provides reverse-engineering of the number and the location of traps

in the oxide of the devices measured, which was a primary goal for this work. -i cr ror

correction :' quantization effects validates that : : -lassical numerical simulation is

.11cable to 1:i 1 scaled devices. This work also provides : :: -ement that future

work can extend the use of this :. i1 ..."1. 7 to emerging device tl -









CHAPTER 1
INTRODUCTION

Silicon-On-Insulator (SOI) devices have gained acceptance as a way to fabricate

semiconductor integrated circuits that are faster than their bulk silicon counterparts,

and that allow higher density due to their DC latchup immunity. This higher IC density

attribute is best utilized, however, when the body of the SOI device is not contacted, and

left to float. Specific noise characteristics unique to such partially depleted devices were

recently studied [1], such as excess Lorentzian-shaped components present at high drain

bias. Our study further investigated these and other noise characteristics exhibited by

modern scaled partially depleted and bulk MOSFETs.

C'! Ipter 2 gives a derivation of moments of the Boltzmann Transport Equation,

which yield the energy balance and drift diffusion models. Terms which cause fluctuations

and noise in semiconductor devices are tracked through the derivation to show the

correct placement of Langevin noise sources in the transport models. The details of

the implementation of these transport models in FLOODS are described, including

enhancements to allow accurate simulation of scaled devices. A more advanced mobility

model is implemented for the accurate simulation of oxide trapping-associated noise.

Direct band-to-trap tunneling, necessary for the degenerate channel region of the

scaled MOSFETs, is added. Examples of noise simulations in simple devices using

high-order transport models for both velocity fluctuation noise and number fluctuation

noise are demonstrated. In chapter 3, the measurements of the DC characteristics

and noise of bulk and SOI devices are presented. As the devices were not protected

from ESD and are difficult to handle, a connection scheme for the execution of noise

measurements minimizing high transient voltages is discussed. C! Ilpter 4 describes the

reverse engineering and simulation of the devices measured, including all assumptions

made and parameters used. Good agreement between simulation and measurement is

acheived for the noise due to direct oxide tunneling in both devices. The excess low

frequency noise exhibited by the SOI device measurements gives good agreement to both









the magnitude and c .:- .: frequencies of the simulated Lorentzia, once the fixed interface

charge is considered in its : : context for quantization effects. Nuimerical simulation

with the semi-classical drive-i :-:: : ..: model is shown to be useful for the profiling of

traps in the gate oxide of these devices, the simulation of the diffusion noise in SO1

devices is shown to give good agreement with measured noise characteristics, and insight

into the relation. *: between various simulation parameters and the resulting noise is

acheived. ( : ter 5 sumrnnarizes what has been acc I in this work, and gives

recommendations for future work.

UI: following sections of this chapter give background information on the nature of

tuations in semiconductors, numerical simulation of devices, simulation of noise, and

semiconductor devices in general. This is intended to provide a basis for discussion c

S.:. choices for n oise in SOI devices.

1.1 Fluctuation Phenomena

It is 1 *'' 1 to first discuss the nature of noise. Voltages and currents (and number

of carriers, carrier velocities, even position of individual atoms) in semiconductors are

time dependent signals in a non-linear, causal systern. Ti: system consists of the material

properties < the semiconductor device (its (: i !!:: makeup, doping concentrations,

and geometrical description) and the i 1. Ical laws that govern the behavior of all parts

of the systern. i : 1 1 of the non-linear ::: itself and its stimuli (:: : to the

of pp:. 1' voltages and currents, heat, light, sound, mechanical forces,

electric and magnetic fields, gravity, radiation, etc.) allows prediction (rn many ** )onents

of the signals in the system. i : DC, or mean, <. :..*inent of a signal is equal to its

time average, LJ s(t)dt/T, where t is tirne, fs(t) is the signal, and T is the length of the

observation. Another possible ( .::: onent is that which is zero-rnean and periodic, usually

due to perio( c stiruli. Finally, there also be other components which are not

perio- :c, but can be determined from changes to other parts of the system or inputs, such

as ( .. :ve or magnetic coupling to other voltages and currents. T.. components









(whether they are desired or not) are deterministic, and can be entirely predicted

from complete knowledge of the system and its stimuli. However, even when all these

components are added together, they do not completely describe a signal (Figure 1-1).

Some interactions in the highly complex non-linear system are stochastic (random and

unpredictable, even with complete system and stimulus knowledge). The stochastic

components are the subject of this study, and while their time-dependent value cannot

be predicted, some of their phenomenological aspects (e.g. power spectral density and its

bias dependencies) can be predicted (given complete knowledge of the system and stimuli).

In addition, it is the premise of this work that knowledge of the system (device) that is

limited to knowledge of its fabrication methods, currently understood physics and DC and

AC characteristics is incomplete, and the measurement and study of the noise in a device

can further knowledge of the system.

Commonly observed types of noise include white noise (for which the power spectral

density is independent of frequency, at least for observable frequencies), 1/f or 1/f-like

noise (in which the power spectral density is approximately inversely proportional to

frequency), and Lorentzian shaped noise (for which the power spectral density is white

up to a cutoff frequency, above which it decreases inversely proportional to frequency

squared). White noise in semiconductor devices is generally due (directly or indirectly)

to fluctuation in the average velocity of charge carriers. Lorentzian shaped noise has

been attributed to fluctuations in numbers of trapped carriers, and to white noise that is

filtered by the non-linear system. 1/f noise is somewhat more controversial, and has been

attributed to traps, as well as fluctuation in other quantities.

As the knowledge of real semiconductor devices is incomplete (the exact geometries

and material properties of a device can only be estimated from knowledge of its

fabrication, and the complete physics is approximated by our limited understanding),

a pseudo-realistic model of any device considered in this work and its physics is set up

for numerical simulation, and the physics of fluctuations in the pseudo-realistic simulated









device are then used to predict the measurable attributes of the noise of the real device.

This requires an adequately realistic model of the device and its physics for simulation,

accurate measurement of the DC behavior of the device to supplement the understanding

of the device itself, and finally, rigorous measurement of the noise of the device.

1.2 Numerical Device Simulation

Partial differential equations' (PDEs) numerical solution and subsequent semiconductor

device simulation have been used in the analysis and design of semiconductor devices

for nearly half a century [2]. While physics-based compact models (closed-form or

near-closed-form solutions based on simplifying approximations such as quasineutrality

and depletion) have been shown [3, 4] to be capable of accurately simulating a wide

variety of physical effects (some of which are very difficult to model microscopically for

PDE-based numerical simulation), and while recent PDE-based models have been shown

to be difficult to discretize for stable numerical solution [5], numerical device simulation

remains a very helpful tool for the evaluation of physical effects in semiconductor devices.

This is particularly true in cases where the physics of devices does not lend itself easily to

simplifying assumptions or in cases where obtaining the detailed geometrical dependence

of certain effects is desired.

1.3 Transport Models

The derivation of models for charge transport in semiconductor devices typically

begins with Poisson's Equation and the Boltzman Transport Equation, and then

uses these to derive the PDEs which will be solved in the semiconductor. One such

derivation is very nicely given by Lundstrom [6]. This derivation is followed to obtain

the hydrodynamic model implemented in the Florida Object-Oriented Device Simulator

(FLOODS) as discussed in Chi lpter 2, though the notation used in this work more closely

follows that used by Tang [7].

The drift-diffusion system of equations includes Poisson's Equation and the

zeroeth-order moments of the Boltzmann Transport Equation for electrons and holes.









The first-order moments are used to give relations for the particle fluxes. The mobility has

been modeled empirically, based on and calibrated to macroscopic averages computed from

measurement data. Klaassen gave a model which is used for the computation of doping [8]

and temperature [9] dependence of carrier mobilities in bulk silicon. Darwish et al. [10]

gave a model which is used to further compute the degraded mobility in surface inversion

L.i-c is in silicon MOSFETs.

The hydrodynamic and energy balance systems of equations for transport in

semiconductor devices are derived from consideration of higher-order moments of

the Boltzmann Transport Equation. 1T "'v different models have been derived (i.e.,

Stratton [11] and Bl, tkej;tnr [12]). Thoma et al. derived a more physically complex set of

equations which do not include any constant-effective-mass approximations and are more

consistent with the results of Monte Carlo simulations [13]. Tang [7] derived a generalized

model which can be made, through judicious choice of closure relations, to match several

other models as demonstrated by leong and Tang [14].

Quantization of the inversion 1viv-. in MOSFETs has the effects of reducing the

overall inversion 1i-cir charge (compared to what a classical model would predict), and

changing the shape of the distribution of carriers through the channel, such that the

peak of the distribution is displaced from the surface. In the devices considered for this

work, the carriers in the inversion 1i-<-r of the device are confined in the vertical (gate to

substrate) direction, and as such are subject to quantization effects.

The simulations in this work employ a rough semi-classical approach, assuming

that the reduction in the inversion 1 -i,-; charge is approximated by fixed charge at

the interface. This approach is not dissimilar to that emplol- .1 by the recent model

presented by Li and Yu [15], which corrects classical models by introducing a fixed charge

distributed through the channel. The remaining error lies in the shape of the inversion

1 .--r distribution and the position from which carriers can tunnel into the oxide to give

rise to low frequency noise.









These effects are, however, shown not to be directly important to the tunneling

rates (and therefore the random charging and discharging rates) associated with an

oxide trap. Since modifying the physics in the numerical device simulation to include

quantization effects does not greatly affect the simulated noise associated with the oxide

traps, quantization effects are not directly included in the simulation, though an error

correction for the computed trap positions and energies is computed and given.

While the strengths of higher order transport models include more accurate prediction

of impact-ionization rates [16] and simulation of some small device effects such as velocity

overshoot, these models have also been shown to cause undesirable results. Egley et

al. [17] demonstrated an overestimation of hot channel carrier diffusion into floating

bodies, leading to incorrect prediction of negative differential drain conductance (or

inverse kink effect) in simulations of partially depleted SOI MOSFETs. This result has

also been reported by others. To quote Polsky et al. [18, page 504], hydrodynamicc

transport produces negative differential resistance (NDR)... no other experimental

publication in this area has confirmed the observation of NDR... Therefore, NDR is now

considered by many to be an artifact of the hydrodynamic transport model." To quote

Munteanu and Le Carval [19, page G574], \\ I clearly show that anomalous behaviors

exist in all in jrii commercial simulation codes and concern both partially depleted

silicon-on-insulator devices (inverse kink effect) and bulk transistors (positive substrate

current effect)." Polsky notes that the NDR effect is due both to overestimation of

inversion carrier diffusion into the floating body and to failures in the hydrodynamic

model's handling of Shockley Reed Hall recombination. Several methods have been

emploi-, 1 to try to reduce or eliminate the NDR effect, including modifying the diffEi- .lii

modeling [19-21], modifying the modeling of the energy relaxation times [19], and by

using trap-assisted-tunneling generation-recombination to offset the negative charging [19].

However, no definitive theory has been put forward that satisfyingly resolves the problem

in a way that gives great confidence in the accuracy of the modeling, or would work for









a wide range of structures and carrier recombination lifetimes. Further, work by Banoo

and Lundstrom [22] has -.-.ii --I. I that the hydrodynamic model predicts output currents

which are higher than derived ballistic limits. Therefore, these models should be used

primarily when the accurate computation of impact-ionization is required or when specific

qualitative results rather than strictly quantitative results are sought.

1.4 Noise Simulation

The impedance field method for computing noise in semiconductor devices was

first proposed by Shockley et al. [23]. This was long used in the context of closed-form

analysis, but was also introduced as a methodology useful in the forum of numerical

device simulation by Bonani et al. [24]. This has also been implemented in FLOODS for

diffusion noise by Sanchez and Hou [25], and for generation-recombination noise by Hou

et al. [26]. Simulation of the Fourier coefficients of the solution variables by the method of

harmonic balance and subsequent simulation of the inter-frequency mixing of noise have

been implemented by Sanchez [27]. Recent work by Hou [28, 29] has added the capability

to simulate 1/f-like noise by considering interface trapping and tunneling of carriers to

and from oxide traps.

An alternate method of noise simulation based on the hydrodynamic model was

demonstrated by Goo et al. [30], using the modeling of noise sources as small current

sources in a lossy transmission line. The simulation of noise in hydrodynamic models using

the impedance field method was considered by Bonani et al. [31]; however, details were

not given regarding how or whether the fluctuations of physical quantities were coupled

into the energy balance equations. More recent work [32] has detailed the coupling of noise

fluctuations into higher-order transport models, and this work furthers this goal.

1.5 SOI Devices and Noise

The UFSOI compact models [3] have been developed by professor Fossum and

his research group. The charge-based coupling of the front and back surfaces was

characterized by Suh and Fossum [4] and Lim and Fossum [33]. Non-local impact









ionization was modeled by Krishnan [16]. Noise modeling was added to the model by

Workman [1] and was shown to accurately compute the excess low frequency Lorentzian

shape observed in partially depleted devices.

This excess noise in partially depleted SOI transistors has been measured and

published by Simoen et al. [34]. Separate measurement data can be found in [1]. Low

frequency noise measurements have also been made on buried-channel SOI p-MOSFETs

by Lukyanchikova [35]. This work contributes new noise measurements in similarly

processed long and short channel bulk and SOI transistors provided by a Semiconductor

Research Corporation member company.

After a discussion of the numerical simulation methods used, the analysis of this data

using the simulations will be given.


















2- a
2 b




-1-












t (seconds)



component (b), AC component (c), an aperiodic deterministic )onent
(d), and a stochastic component (c).
(ci). and a st (c~last iC ofnoel)fet ( ).


IIIIIIIIIIIIIIIIIII









('iAP' i iit2
TRANSPORT MODELS A-D PHYSICAL PROPE1Rii

i numerical simulation of scaled modern devices can yield questionable results,

if the user does not ...1, i: i.1y understand the I Ical models (including boundary

conditions) and :*" tions made for the simulations being P, n t *[. Furthermore, the

modeling of certain properties as appropriate for the newer and more advanced transport

models is not : ;y mature, and the <( .:' culty of numerical simulation makes necessary

assumptions for these models which are i: !: :: .1 Nevertheless, highly scaled devices

exhibit that can b(* 1 -be adequately captured by transport models that possess

:: :: : : complexity to capture observed behavior. It. is desirable to balance the need for

increasing complexity with skepticism with regard to certain simulation effects. What

follows is a detailed discussion of the origin of each (. the transport models and the

corresponding :: of ':. .. v.1 ..:e-field noise simulation, as well as the modeling of the

p1: : ties of the materials used for the simulation.

2.1 Advanced Transport Models for PDE-based Device Simulation

T Schro wave ( .:TE), Poisson's : :! :n (PE), and the Boltzmann

transport equation (BTE) together describe the t- --ort of charge through any

semiconductor device (the I''. i and FI together define the allowed states in the system

and the BTE and PE together describe how the carriers c ..: : those states, react to

electric fields and potential, and transport charge through the device). T': : .:' models

are generally o:.. f:( .: .:; of these equations which can be solved numerically or in

closed .

Generally, in semiconductors such as : ::on, the carriers are nearly : and the

S1 -;- the SWE and PE affect 1 -, .ort is in the definition of parameters such as

rnmo : : and intrinsic carrier densities, since electrons in the conduction band and holes

in the valence band are nearly free carriers. However, in tightly confined areas such as the

high field region under the gate oxide in a highly scaled and strongly inverted MO : ; : ,









the solutions of the E *'. i and PE give ( !-..i' 1d states or sub-bands that alter these

properties.

'1II following derivations of various transport models use the BTE and PE to define

the motion of seni-classical carriers moving in a semiconductor device. 'i : '2.' I does not

enter the derivation, save through mo (::. of band-related parameters.

Silmplications of neglecting the quantization effect in the devices measured in this

work :i be discussed after presenting the transport models and the ( ::: :: : of noise

associated with each one.

2.1.1 Poisson's Equation

Poisson's Equation relates charge density to the electric field and potential in the

semiconductor, and is given as follows:


V (p-n+ ) (2 1)


where zt is the electrostatic u 1 ntial, -q is the fundamental electron charge, c is the

permittivity (. the semiconductor, and N7 and NT are the ionized donor and acceptor

concentrations. T hole and electron concentrations are p and n. T volume charge

density due to charged electron and hole traps is pt. I7: 1: :-- be neutral ( i states

with \ carriers or donor or acceptor like states which are charged and empty.

2.1.2 i'i moments of the Boltzmann Transport Equation

ITF. Boltzrnann Transport T. .. .... keeps track of all transitions between :. wed

carrier states in a semiconductor, conserving both energy and momentum. T: oT :::>ancy

function f(r. 1 ) gives the fraction of allowed states which are occupied. In general it

is a ::: :: : of position, momentum, and time but for electrons in a semiconductor in

electric and thermal e.i :11'1.:':: it is the Fermi-Dirac i : .: :.:.. In non- ':':: the

Boltzrnann T: : Equation defines f as follows:

r -.f
I V, f F V, = s(r" ;, t) (2-2)
i0 Colil









where U is the velocity of the carriers occupying states at the considered momentum,

position and time. The force impinging on such carriers is F. The term s(r',j, t) accounts

for the change in f due to interband carrier transitions to and from the considered

position and momentum, and 1 Icoil accounts for changes in f due to intraband

transitions from one momentum to another. The BTE is arranged here in this way to

highlight the fact that terms on the left hand side are deterministic quantities governed

by the electron gas and electrostatics. The terms on the right hand side involve collisions

with phonons, photons and other non-carrier particles and are therefore stochastic. These

right-hand-side terms provide the fluctuating quantities which cause intrinsic device noise.

By multiplying the BTE by some n-ordered functions of momentum b(pq and

summing over all allowed moment, balance equations are defined which nearly account

for all of the charge transport effects (but ahv-wi some non-physical assumption must

be made to close the system of equations such that they can be solved). Considering

higher-order moments of the BTE, more of the effects of the physics can be captured,

but the physical modeling of some terms becomes more uncertain because these terms

are more difficult to visualize. As a result, it is desirable (for clarity of results as well as

for computational efficiency) to keep the number of moments considered to the minimum

needed to close the system of equations and to describe the pertinent physics.

Generally, a moment of the BTE yields a balance equation of the following form.


d + [ t fY- jf j f hZY- +h Cof (2-3)
P J L J p p

The validity of moving the 4 and V,- operations outside the first two terms and of the

chain ruling of gradients to transform the third term is discussed in Lundstrom [6]. The

last term is generally expressed by a relaxation time approximation as follows:

1 of 0 (1) t p (f fP) (2-4)
P p









where the relaxation time is defined by the ensemble average ((t1/'r)) of the following

function.

1 t = 1 S p f) (2-5)

The balance equations associated with these moments have been rigorously derived

previously [6, 7], but this bears repeating as the physical origins of each term in these

equations should be understood in order to properly consider the random fluctuation of

physical quantities in such systems (resulting from the stochastic terms on the right hand

side of Eq. (2-3)).

2.1.2.1 Zeroeth order moment: carrier continuity equations

The carrier continuity equations result from considering the zeroeth-order moment

(Q(pq = 1) of the BTE for electrons and holes. The electron continuity equation results
from this moment (as in Eq. (2-3)) as follows:


d + V. ni 0 [G, R] + 0 (2-6)

where is the average electron velocity (U), and G, and RP are the electron generation

and recombination rates, respectively. The third term is zero because the gradient Vp, =

V,1 is zero. The second term on the right hand side is zero because Q(j) = <(p) = 1

and therefore 1 0 for all ({,[). This equation, condensed, and the corresponding

equation for holes are given as follows:

dn

dp VG R(28)
dt= -V (p,) + G, R, (2-8)

2.1.2.2 First order moment: momentum balance equations

The momentum balance equations result from considering the first-order moment

((pO = f = m*%) of the BTE for electrons and holes. They are vector-valued. The









nmomnentumn balance equation for electrons results from this moment as follows:


SP,] + V, [nl] (-qi) = 0 + nC0p (2 9)
dt

where F-, is the : : electron rnomnenturn (K h) 1, is the kinetic energy tensor (rf hk),

E is th electric field, and Cp, is the change due to collisions in momentum per electron.

first term on the right hand side is zero because the random velocity of generated

or recombined carriers is distributed with spherical -nmmetry such that the statistical

average of es( .': is zero. Ti. equation, condensed, and the corresponding equation for

holes are given as follows:


V= -V (n7,,) -7 + nCp (2-10)

-V ( *) + : .' +pp (2 11)


2.1.2.3 Second order moment: energy balance equations

I'P. energy balance :J: ..; result from considering the second-order moment

(o(pp = E( = 2n-) of the F : for electrons and holes. i :" energy balance equation foir
electrons results .. this moment as follows:


+ [i + Vr. [,] (-qli) (on I) [Gn, R,11 + noCwv (2 12)

where I,, is the average electron kinetic v *** (E). Sq, is the average energy

per carrier ( .r), and Gw,, and 1.,, are the changes in kinetic energy due to particle

generation and recombination events, respectively. Ti: change due to collisions in kinetic

energy per electron is Cu,n This equation, condensed, and the corresp:: equation for

holes are given as follows:

L -V (nSo) qE ( V) + Gw, Rw + nCw,, (2 13)

S-V ( ) I qE (pV,) I Gn, R1, / (2-14)
dtC1









2.1.2.4 Third order moment: heat flow balance equations

I: heat flow balance e. : result from : 1 the third-order 0 moment

(( E( '. of the BTEs for electrons and holes. Ti. heat flow balance equation for

electrons results from this moment as f i >ws.


+ [ U] I V [LnR,i ( (-, K) n [nW, + nU, ( = 0 + C (2-15)


where Q,, is the average heat flow per electron (E hk). R, is the fourth-order moment

tensor (ui hk), 1 is the unit tensor, and (Cq is the change due to collisions in heat

S>w electron. T first term oil the :' m 1 hand side, similar to the one in the

first-order moment, is zero on statistical average for carriers generated or recombining

wi th 1. I. ."y symncmtric distributed moment. Ti : .: '. : condensed, and the

corresponding equation for holes are given as *:'.>ws:

d( -^ }
(I -V (n,) qE (nl + nU ) + nCQ,, (2 16)

d(. ) -( I )
-V (pRp) I qE (pIWI pUp) I pC- (2-17)


2.1.3 Closure and Reduction Relations

TI: L ; ::: of equations derived thus : : is not solvable, as the rigorous computation

of R,, would : 'e a fifth-order moment, which would involve terms requiring even higher

order mornents, and so on. Also, it is desirable to reduce the numbrner of solution variables

to the minimum required to solve the t :.. in order to reduce the cost of solving the

system. Therefore, closure relations are used to close the system, and reduction relations

are used to ::-- :. -: : variables. Ti .- closure and reduction relations used in this work are

given as follows:









2.1.3.1 Momentum balance equation and related reduction relations


Pn my / Cp= Q +ApflV.Uf, U ,= = kTI
(2-18)
Ip ,vp, Cp, = +p Up-, Up= U = kTI

where mu and m* are the electron and hole conductivity effective masses. The electron

and hole mobilities are p* = (T) and 4, = where (rp,) and ('rpp) are the electron

and hole momentum relaxation times. The values Ap. and Ap are factorless empirical

parameters that model the behavior of the system in inhomogeneous field conditions,

consistent with Monte Carlo computations [7]. The relations involving U, and Up reflect

the equipartition energy approximation and define the carrier temperatures.

2.1.3.2 Energy balance equation and related reduction relations


14,- 3 U ( W. W,0 ) 3
3 -(WP- W WN) 0 3
(219)
W = kpT, Cwp (, = kTL
2 ('w) 2
The relations involving W, and Wp reflect the approximation that the carrier kinetic

energies are entirely due to random thermal motion rather than organized drifting motion;

i.e., 5= (5) + 65 and W1, = ({-m*|v 2 2) (m*v = -T. 'Trw, and rw, are the electron

and hole energy relaxation times.

2.1.3.3 Heat flow balance equation and related reduction and closure rela-
tions


-qS 10
Qn,= mu, CQ +AQV +QV Rn, R RI W -W0I
PS. 9(2-20)
-qS 10
Qp = m*Sp, CQ + AQV Rp, Rp= RpI = --W 21
Ss9 p

The electron and hole heat flow mobilities are = and p'p = where (rQ.)

and (rQp) are the electron and hole heat flow relaxation times. The values AQ. and AQP are

factorless empirical parameters that model the behavior of the system in inhomogeneous









field conditions, consistent with Monte Carlo computations [7]. The relations involving R,

and Rp reflect the equipartition energy approximation and close the system of equations.

Their validity has also been verified by Monte Carlo computations [7].

2.1.3.4 PE and first four BTE moments, closed and summarized

The system of equations thus derived and closed can be summarized as follows:


dn
dt
dp

d(nV)
Tn dt

,d(p1/)
r dt

3 d(nTn)
2k `
2 dt
3,kd(pTp)
2 dt

, d(nS,)
dt

Sd(pSt
p dt


qPt
-(p-n+Nj- NA})P

-V (nV) + G, R,

-V (ipj) + Gp R,


-V(nkT,) qun


-V(pkTp) + qpE -


-V (nSn) qE *

-V (pS,) + qE + q


-Va7 k2T) q


-V pk2) + q
(2^


- n (K ApV(kT,))


p Ap V(kT,)



3 T T
(p V) + Gw Rw, -pk
2 TWj

SnkTn) (n AQ, k2T


SppkT AQPV(-k2)T2


(2-23)

(2-24)


(2-25)


(2-26)

(2-27)


(2-28)


(2-29)


This four-moment system is not commonly solved, as it is very complex, and the number

of variables to be solved is still quite high (9 variables for 1-D problems, 13 for 2-D

problems and 17 for 3-D problems). Further approximations are therefore made to get

to the hydrodynamic systems that are commonly solved. However, this system will be

useful for determining where fluctuations couple into the system and how noise simulations

should be extended to include higher-order moments.

2.1.4 Hydrodynamic Model

The first reduction to the equations derived thus far involves the quasi-stationary

approximation with regard to the heat flow balance equations, namely that the time









derivative terms on the left hand side are negligible compared with any of the terms on

the ride hand side. These equations are then reduced from partial differential equations

to vector expressions for the heat flow. This eliminates 2 to 6 variables (depending on

the dimensionality of the solution space) from the solution set. The remaining system

constitutes the hydrodynamic model and consists of the first seven previous Eqs. (2-21

2-27) with the remaining two auxiliary relations for heat flow, as follows:


i(q n+N^-N Pt

.V (nVn) + Gn R,

-V (p1Q) +G,- R


V(nkT,) qnE


V(pkTp) + qpE -


= -- (nS-) qE

= -V (pSp) + qE


n= L- 5kTnqE
q 2

Sp = kTql


n V

-P Ap V(kT,)

3 T n T
(nVn,) + Gw. Rw, nk -
2 (w)
3 T TL
(pV)+ GwP R pk T
2 (rw )

+ k2T2 AQ)V k2T2)

+2 p k +( Xq)V( 2TP


(2-30)

(2-31)


Thus the system to be solved is reduced to have a solution set of up to 11 variables: i, n,

P, Vnx, Vny, Vnz, Vp,, Vpy, V,z, T,, and T,.
2.1.5 Reduced Hydrodynamic, or "Energy Balance" Model

The next analogous reduction to the equations derived thus far involves the

quasi-stationary approximation with regard to the momentum balance equations, namely

that the time derivative terms on the left hand side are negligible compared with any of

the terms on the ride hand side. These equations are also reduced from partial differential

equations to vector expressions for the carrier velocities. This eliminates an additional 2 to


dn
dt
dp
dt
d(nVW)
dt

d(pK)
mP dt
3k d(nT,)
-k
2 dt
3 d(pT,)
2 dt


;









6 variables (depending on the dimensionality) from the solution set. The remaining system

constitutes the ir igy balance" model and consists of Poisson's equation (2-21), the

electron and hole continuity equations (2-22 and 2-23) and the electron and hole energy

balance equations (2-26 and 2-27), with the auxiliary relations for heat flow (Eqs. (2-30

and 2-31)) and the additional two auxiliary relations for carrier velocities.


V (n,) + G, R,

V (pVp) + Gp Rp

FqEi+ UT-, + (1- Ap)V(kT,)

-q+ kT, VP (1- A)V(kTp)]
q p
3
-V -(nS) -qE- (nV +Gw Rw,


S-V (pS4) + qE (p1) + Gw R


= kTqE + k2T2+ (-
Vn

q 2 2 n
= 5 5 k2T2 17
5p q- -k~pqE + 2k -P + (t


k T. TL
k I \


3 T TL
-pk
2 (Tw)



_AQ)V(|k2T)


Thus the system to be solved is reduced to have a solution set of 5 variables: b, n, p, T,,

and T,.

To avoid the direct computation of the vector particle velocity, the Joule heating

terms of the energy balance equations (such as -qE (nVa)) are typically rearranged as

follows [2].


-qE (nV,) q= V (nV,) qbV (nV,) = V (pnV) + q G + R,

dt
qE (pV,) -V ( pV,) + q V (pV,) S -qV (pVm) q y Gm + R[]

The V ( ipnn) terms can be discretized using Sharfetter-Gummel type methods [2].


(2-34)

(2-35)


77 =
dn
dt
dp
dt



VP


3 d(nT,)
-k
2 dt
3, k d(p)
2 d-k
2 dt


(2-32)

(2-33)









2.1.6 Drift-Diffusion Model

Further reduction of the system can be made by considering the electrons and holes

to be near thermal equilibrium, such that T, Tp TL. For this case, the energy

balance equations can be eliminated and some terms can be dropped from the expressions

for carrier velocities (Eqs. (2-32) and (2-33)). The remaining system constitutes the

drift-diffusion model, and consists of the first three equations of the previously derived

systems, and the simplified auxiliary relations for the carrier velocities, as follows.

(p n + NG NA)


dp
dt -V.(pV)+G,-R,

-a -qE+ kTLn + (1 Ap,)V(kTL) (236)




Thus in the most simple case of bipolar transport the system to be solved is reduced to a

solution set of 3 variables: i, n and p.

2.2 Implementation of Surface Mobility Models in FLOODS

In order to simulate the noise in SOI devices, it is first necessary to be able to

simulate the DC and AC characteristics of the device with reasonable accuracy. This

requires the use of a mobility model that takes into account the degradation of carrier

mobility in semiconductor devices (such as SOI MOSFETs) in which the carriers are

tightly bound near an insulator surface.

In addition (and more importantly), the impedance field noise simulation available

in FLOODS has recently been extended [28] to include trapping at silicon/silicon

dioxide interfaces and tunneling between these interface traps and traps located in the

oxide. This noisy process provides slow enough transitions to produce low frequency

Lorentzian-shaped noise components, and the distribution of oxide traps in position can be









produce an exponential distribution of time constants suitable for 1/ f or 1/f-like noise.

While 1/f-like noise is understood to be caused by the fundamental noise mechanism

of number fluctuation, it is true that carriers trapped in the oxide cause fluctuations in

the electric field which binds the carriers in the channel to the surface. Therefore, the

effective mobility of the carriers in the channel may fluctuate as a secondary effect. While

this is not expected to dominate the noise spectrum, it may have an effect on the Green's

functions produced by the system of equations in that it gives an extra coupling path

between the trapped carrier continuity equation in the oxide node and the external device

contact (the charging of the trap now couples to Poisson's Equation and, in addition, the

carrier continuity equation at the interface node).

It is important to note that even though the surface mobility models that have been

published are based on some expectation of their physical form, they are empirically

shaped to fit macroscopic effectivee mobilities which are obtained from compact

model-type expressions matched to measured data. Therefore, the coupling from the

traps in the oxide to the mobility modeled in the channel should be viewed as a tool to

give possible physical insight and qualitative understanding, rather than a rigorously

accurate quantitative result. The surface mobility model parameters published may vary

wildly from those required to match the dependence of carrier mobilities on electric field in

different processes with differing oxidation recipes or impurities present near the interface.

Also, the model is empirically determined for single-gate bulk Silicon MOSFETs and

while it is expected to apply to the partially depleted SOI devices in this study, results

may deviate if the model is applied to highly scaled fully depleted SOI MOSFETs or

double-gate MOSFETs where the carriers are expected to interact with front and back

interfaces or where the electric field distribution is very different in the inversion l--r (as

in symmetric double-gate structures). These models also would likely give errant results in

a system such as the density gradient system [5] for accounting for quantum confinement

effects where the carriers are displaced a distance from the interface.









Early surface mobility models used the distance from a semiconductor interface to

model the decrease in mobility [36]. This is non-physical, as it is expected that tighter

electrostatic binding to a surface should further increase the scattering of carriers by

the roughness of the surface and by surface acoustic phonons. Modern surface mobility

models ([37], [10]) empirically model the surface mobility by degrading the mobility as

a function of electric field perpendicular to the interface. Some implementations have

used a non-locally computed "effecti.-e surface field computed throughout the inversion

1liv-r. This would be expected to give the best fit to the macroscopic "effective mobility

data, but causes some difficulty in implementation, as it yields nonzero derivative terms

which are far off the diagonal of the Jacobian matrix, increasing the cost of solving the

numerical system and requiring difficult implementation for a generalized script-based

simulator such as FLOODS for problems in which small-signal AC solutions are required.

Some implementations have used the local field perpendicular to the nearest interface

(the 1. I ical" field is sometimes used, limiting the applicability of the implementation

to planar MOSFETs). Some have used the electric field perpendicular to the edges in the

mesh (the assumption being that if there is much current flowing through the edge then it

must be aligned closely with the direction of the current flow,and if the perpendicular field

is high for this case, then the edge must be located near to a surface-binding field). While

this may not be as accurate as using the vertical field, it does allow the mobility model to

be used in a wider variety of geometries. The exact implementation does not seem to be

that important [38] as in the limits of dense meshes and high surface fields (which are the

cases that are of most concern) these assumptions become more valid.

2.2.1 Low Kinetic Energy Mobility Formulation

The surface mobility model implemented in FLOODS for this work is that presented

by Darwish, et. al [10]. This model was formulated as an update to the Lombardi

model [37] which is similar in form but under-predicts the degradation of the mobility

at very high electric fields in scaled processes which are expected to have similar surface









mo '.: model parameters to those of their less highly scaled predecessors, due to similar

oxidation : essing. i :i Darwish model degrades the bulk mobility (pb) by using

Mathiessen's Rule to reciprocally combine it with two other mobility terms, as follows.

I (2-38)


result po is the low kinetic energy mobility which will in turn be degraded for hot

carrier 4i : as will be discussed later. T: values pac a(nd ps,, are mobility terms

which account for surface acoustic .1.... ... scattering and surface roti

Sively. T1 surface acoustic phonon scattering term is computed as follows.



!I Krio \~j((2 :)
( E T 1 TL (r

where 13 and C are based on physically derived i i. :, but are expected to

with fixed ::: : charge and are therefore treated as fit parameters. i ::: value 7 is an

'cal fit .: : .. ter, and H is the : -.: e dependence of the probe : of surface

phonon scattering (1.7 i:' electrons and 0.9 i:. holes). value N = -N11 NI is the

total < :. of ionized ': ...aritics. Ti. surface roughness scattering term (dominant for

high .: dicular fields) is computed as follows.


E ((2

where 6 is a fit parameter which depends on the oxidation process, and is expected to

be vary w'1.11. from process to process. Ti.. r.ameter 7 is given a weak dependence on

inversion charge (it is held constant in the Lombardi : : :i : : ) as : i >ws.


A + (2 41)
M (;


where A, a, and *r are fi :-. parameters









2.2.2 Hot Carrier Mobility

Electrons and holes in Silicon are characterized by velocity that saturates under the

condition of high homogeneous field, or high average kinetic energy. In drift-diffusion

simulations, the accl : I'::- electric field (parallel to the .:: i : of current flow) is used

to model the velocity saturation i :, though in highly scaled devices the field is far

homogeneous and the kinetic energy of the electron distribution is a more appropriate

and physical measure for mno ...; this effect. Ti.. Darwish mobility model Nwas originally

developed for use in 1 : : :: simulations, and the velocity saturation expression used

was that of Ilinsch, as follows.


p(,) = 2-o(7 (2-42)

\( ( va 2 )2)

In the context of a lhydrodynamnic simulation, the velocity saturation effect should

rather be modeled as a function of average kinetic energy or electron temperature. A

common i o ...... used for this is to consider the electron or hole en, balance equation

under the condition of a lo. homogeneous field. !' -I : dis an expression ** carrier

S:t ::ature as a ::: :: : ( such a long stationary homogeneous field E,. ]T: r inverse of

this function is expressed as follows (for electrons).


SL) (2 43)
2 ..: TE

Since this expression is a function of the overall mobility, the v... sat ration

expression must be rearranged algebraically. i ::: velocity saturation expression formulated

!. Canali is more :c rearranged, and is expressed for drift- : :: : : and for 1: droclynamic

simulation as follows.

7(-3 (2 44)
(l.(-L))










0(rV (2 413)



where /3 is typically chosen as 2 :- electrons and 1 -* holes.

2.2.3 Implementation-Specific Details

In the interest < using this mo : :i model for ::: :i signal AC or impedance field

noise simulation, it is important once again to note that exact derivatives r the partial

'.'ferential equations with respect to the solution variables are required. The utility of

the Alagator scripting format for FLOODS is that the equation ;I : er knows how to

take analytical derivatives of simple expressions. .i '. the only primary concern is

that <( <. :: >uting the transverse electric field, which is not a :: term but depends on

the potential at rnore than two nodes. This problem has been addressed by1 moti.

FLOODS to allow the assembly of the Jacobian matrix by triangular faces or 3-D elements

rather than by nodes and edges (*1 .

An operator : :: .(a)"' has been written which computes the magnitude of the

electric field -d. dicular to mesh edges in an element, as a function of the potential at

the other nodes in the element. A vector field is first < ::: uted : : the element using the

method of least-squares, as follows.


^i-1' xi *i E x X1 X2 ... XN ~

Zxh Z (2 46)



z 1 1 ... I

where N is the number of nodes in the clement, and xi, zi, and are the x, y, and z

coordinates and .tential at node i, respectively. Ti .. 4x4 Imatrix on the left hand side is

inverted and :::: .1 by the 4xN matrix on the right hand side, and the first 3 rows are

stored as a 'T matrix which depends only on the geonm i of the element and therefore

(:1 needs to be computed once 1 : element. Ter :: matrix, i: :1 i d the Nxl vector









of any solution variable's values at the N nodes to yield the x, y, and z components of the

vector gradient for the element (Ex, Ey, and E,, for psi as the argument). The component

perpendicular to an edge is then computed as follows.


Ei = (i Ax,/lj)E2 + (1 Ayj/lj)EY + (1 Az/lj)E2 (2-47)

If it is desired to compute El as the field perpendicular to the nearest interface rather

than an edge, the same vector field for the element can be used, and the component is

computed in the direction of the nearest interface node (in this case the result would be

the same for each edge in the element).

2.3 Implementation of the Energy Balance Transport Equations in FLOODS

FLOODS uses the standard generalized box discretization technique to solve the

PDEs which are specified in strings passed to the Alagator parser in Tel scripts. The

Sharfetter-Gummel discretization method [39] provides a numerically stable expression

for any edge-evaluated flux of the form f = (xVy cVx) where c is constant along the

edge, y varies linearly along the edge, and x varies non-linearly. The electron flow in a

drift-diffusion system nV, in Equation 2-36 fits this form (and is indeed the expression

this was originally derived for). The discretized flux along an edge between nodes i and j

is then expressed as follows.


fij xB ( y) xB ( )]j (2-48)
lij c c

B(z) = z/(exp(z) 1) is the Bernoulli function. This expression has been used for drift

diffusion transport, scripted as "c*sgrad(x,y/c)" but in the energy balance transport

model the electron flow nV, in Equation 2-32 includes the electron temperature T,, which

varies linearly across an edge. Fortunately, a Sharfetter-Gummel-like discretization has

been derived [40] which provides a numerically stable expression for those fluxes where the









c term varies along the edge, expressed as follows.
f ^ "r B (^ ---^} B f^ ---^V r- -



ci (2-50)
in( /cj)

An operator named -- :-ad(x,y, c)" has been i- 1. .1 which evaluates this

: : :d expression and provides its ::: : and complete derivatives with r : i !

to any arbitrary variable, provided (by FLOODS) the derivatives of the arguments

with respect to the same ar1'. : : variable. The function fJy can then be scripted as

i ad(x,y,c)," and the electron flow in the en, balance formulation can be

scripted as "($mun'" ) *hdsgrad(Elec, $q*DevPsi-(1-$lamp)*(T7: ), t7::)." ;:: ::: is

the Tel variable : :: the electron mobiliy, "$q" is the fundamental char ge,

is the electron thermal energy kT, and i are n and an ad /?,', r vely, and

-' ::, is from the value Apr from the energy balance closure relations.

ITF. electron heat flux nS, is rearranged to f i. w this ,' *tter-Gurmmel like form as

follows and similarly :.ted.

q T V + (2A2, -l)) 2'V T j (2-51)


Ti V Joule heating term i (n ,) is recast as V (qin) R).

first term requires the vector q wnVC, to be ::: 1 along edges, and can be z cized

1 p ing the :-ticle i >w expression by the average potential along the e. This is

scripted as "Devpsi*$mun*hdsgrad(Elec, $q*DevPsi-(1-t .))*$Un, $Un)."

Finally, the energy relaxation terms are observed to be problematic, :. :ticularly in

depletion regions where the carrier density gets very small. To avoid these convergence

problems, a small I1 : r (on the order of 10"cm"3) is added to the carrier density in the

energy relaxation terms. Ti. greatly improves the convergence by ensuring a very small

but finite energy relaxation in depletion regions.









2.4 Impact Ionization

In order to accurately capture the floating body effect in SOI devices which gives

rise to the excess noise [1], it is necessary to model the electron impact ionization which

generates holes to elevate the bias of the floating body.

The most accurate methods of computing impact ionization rates involve direct

solution of the Boltzmann Transport Equation, or indirect solution via Shockley's

Lucky Electron model in order to accurately compute the number of carriers that

have sufficient energy and momentum to excite an electron from the local distribution

of valence band electrons to the conduction band. However, within the scope of this

work (numerical simulation via BTE moments) some assumption must be made to

model the impact ionization rate. Common assumptions are that the local electron

distribution is functionally similar to the equilibrium distribution, displaced in energy,

and that the displacement of this function in energy is related only to the local electric

field. The first assumption is all that is required for simulation within the hydrodynamic

or energy-balance models, as the energy displacement is directly related to the electron

temperature. For drift-diffusion simulation it is necessary to do extra non-local computation

such as shooting methods or lucky-electron post-processing to ascertain the kinetic energy

of the carriers if the second assumption is not made. These methods are not necessarily

appropriate for noise simulation, however, as the derivatives of these terms are not

available for inclusion in the AC Jacobian matrix. This leaves us with the traditional

assumption that the impact ionization rates can be modelled as a function of local electric

field (parallel to current flow).

Since impact ionization is caused by a fraction of the carriers moving through a

locality in the semiconductor, it's rate is usually modeled as follows.


GII anJn + aJ, (2-52)









where an and a, are the electron and hole impact ionization coefficients, respectively.

In n-channel FETs, electron impact ionization dominates, and hole impact ionization is

usually at most a second-order effect; the second term can generally be ignored (or the

first term, in p-channel FETs). For long homogeneous semiconductors in which the carrier

distribution is allowed to reach a quasi-equilibrium in a constant electric field [2], the

coefficients are found to follow the C'li ''. !! relation [41], as used by van Overstraeten

[42].

a(E) = Ae(-B/E ) (2-53)

where A and B are fit parameters that have been previously characterized for the

homogeneous field case.

The extension of this impact ionization model to energy-balance or hydrodynamic

simulation involves the assumption that in a scaled device the relationship of impact

ionization rate to average carrier kinetic energy is the same as that in a long semiconductor

with homogeneous field (consistent with the previous assumptions). The equation relating

a homogeneous electric field to carrier temperature (Equation 2-43) is used to derive the

following relationship.

an(T,) = Ae ( 3(TL) (2-54)

Since in highly scaled devices the carrier distribution does not reach an equilibrium

and in light of the assumptions made in deriving these expressions as well as the

dependence on carrier energy on defect-related scattering rates, it is not expected that

the parameters A and B should agree between accurately calibrated drift-diffusion and

hydrodynamic models, nor should they necessarily agree with previously determined

empirical values. Therefore, for the purposes of matching measurements to simulations in

this work they are treated as fit parameters.









2.5 Impedance Field Simulation of Velocity Fluctuation Noise

The impedance field method has been used previously [26, 27] to simulate noise in

semiconductor devices using FLOODS [43, 44]. Local noise fluctuations are modeled using

first-principle physics and their mapping to the external circuit are computed using the

Green's functions, which describe the effect of a small signal AC-fluctuation in one of the

system's PDEs at an internal node on the potential or current at a contact.

To simulate physical fluctuation mechanisms using advanced transport models such

as those in section 2.1, it is necessary to trace the fluctuation terms through their origins

in the BTE (Eq. (2-2)) to each contribution to the system of equations. This is done for

velocity fluctuation noise in the following section.

2.5.1 Velocity Fluctuation Noise Simulation

Velocity fluctuations are caused by intraband transitions of particles from one

momentum state to another due to collisions with phonons, randomly located ionized

impurities, and other particles. Electrons scatter, gaining or losing momentum. Microscopically,

the collision term of the BTE can be described by a statistical mean component and a

stochastic component, which fluctuates randomly and with a zero statistical mean.


t ) )f ) (+7f (255)


where 7f is a Langevin noise source which represents a fluctuation in c .C These

fluctuations can also be characterized by a fluctuation over a short time r in the average

electron and hole velocities around a statistical mean value.

1 1
Vn = --tVZ+-f f= T Vn fr + 7i r (2-56)
p p

Since these fluctuations are caused by collisions with phonons and other particles,

their associated Langevin noise sources are inserted into the system's partial differential

equations through those terms derived from the collision term of the BTE (Eq. (2-2)).

This should be considered for each moment individually. For the zeroeth-order moment,










the collision term of the BTE is zero, since for every collision a positive 0 f/1t at one

mornentum fi is i by an e : :1 negative f/9t at a i<' : rnonent urn p. i ar the

first-order moment, the c( term is characterized as follows.


1 f (= 0 n1C i 1( m7/i = A V(Tu) i (2-57)
Q Q

For the second-order moment, this term is characterized as follows.

I f ( 1 f t 3 (I I)
'n 2E1 + k +k (2 58)
1 0t7 U 2 (Ti
p P

If the collisions are considered to be nearly elastic such that ': can be assumed

to be zero. I ar the third-order moment, this term is characterized as follows.


-coili I P L-' nT
p p
(2-59)

By examination of the derivation of the third-order moment, compared to the derivation of

the first-order moment, it becomes obvious that, ^ can be related to as follows.


s, -2 A (2 60)


Ti .. the velocity i .***tuation Langevin sources and enter the systern

of equations derived from the first four moments <- the BTT as follows. Note that the

asterisks to denote : .-. i ve masses and mobilities are omitted so "' x will not be confused









with ---- .!.



dn
dt


d(i)


d(plV)



di
d(pjT,)

dt


dt


d1t


conjugates in the later phasor c----essions.


q (
C


n N' -- NA)


V (ni ) i, Rn

V ( *J) CG Rp

V(nkT,) .. n -(n-- A1,V(kT,,) I


V ( '") iiE p I -

V( q i ,V) (nI ) G


IV(kT7)) I .. -

3 T T,

3 T 7T
- R k n
2 ( I)2

A S ( 2
--p 1) II


+5.-77


P I
\\/1,p


Ac V


2


T ::m-tuation of voltage I ait the considered external contact is then given as ) o i >ws.


-(kT


(2-65)


K


where CGF, G.p, (GQ, and Gq0, are the ( 5 -'s functions associated with the vector-valued

electron and hole momentum balance :; and electron and hole heat flow balance


equations, respecti'- Ti ** power spectral c(-:I--' S-


2TV1 *V is .. ciiuted as follows.


(* G p (n
Sv =2TJJJ
s {CGQ(


) GP ( f


(2-61)


(2-62)


(2 )


(2 64)


(1 .


j)


(2 66)


.P;


+ q -, : p "


*( ) +( r,
+ GQ. ^Tkm


; Q,


0"'


2( -
VZ 2 )









where T is the length of the time window of the measurement. If the implicit approximation

is made that the noise sources t. and .. are statistically independent in position, the

following approximation can be made.


2T (r)t. (,) K,,6(K ) (2-67)


where K,,,, is the local noise strength tensor of electron velocity fluctuations, and 6(r) is

the Dirac delta function. The local noise strengths can be evaluated using Monte Carlo

computations to directly compute the various local noise strength tensors as Fourier

transforms of the veloci i-- 1 -i,city, velocity-energy and energy-energy correlation functions,

as has been shown previously [32, 45, 46]. The advantage of this is that -r.. and 2... can

be computed for inelastic scattering mechanisms without any further physical modeling.

The downside is that the Monte-Carlo computations can only be feasibly performed at RF

frequencies and higher. An alternative method to evaluate these expressions is using the

local noise strength spectral tensors that follow, consistent with Milatz' theorem.

4D,
K,, (2-68)

4D
K, I (2-69)


5(4)K V.(j) i*(r()K. (,) 0 (2-70)

where D, and Dp are the electron and hole diffusivities. These diffusivities are typically

modeled using Einstein's relation D = kJT with either the carrier temperatures or
Pi q
the lattice temperature. Recent work [45] has shown that neither temperature gives

strictly accurate local noise strengths, but that the correctly computed values, using the

Fourier transform of the velocity autocorrelation function computed from Monte Carlo

simulations, is somewhere in between the lattice and carrier temperatures. The published

figures of that work si---_- -1 that using an average of the two temperatures will be closer

than using either one, and this is satisfying since the scattering process involves collisions










between carriers (characterized by the carrier i.- lerature) and phonons (characterized 1.

the lattice temperature).

With the ;*u terms modeled as above, the power 1 density expression cian then

be ::: Ki:: as follows.


Svi j7// Q
.


T G
K I'n G P k Tp -Qr P 4

.+ IT,, .. l

,, ,, GQ, ...

S.d(3r


(2-71)

n mentioned prevy i., this complete four-mornent BTE-based .. of equations

is not typically solved due to its complexity, so for the simpler transport models the

Langevin noise source terms must be .. 1.:..:. : i. ly folded into the remaining PDEs.

2.5.1.1 Hydrodynamic model with velocity fluctuation noise

To correctly place the Langevin noise terms in the "-- drodynanmic model, the Langevin

terms are folded into the remaining PDEs :: : the heat flow balance equations. First the

( : .:... y ---.essions for the carrier heat flows are rearranged as follows.



q 2 1n 2 P
'-'n~ ~ ~ ~ ~ h n-1 Tr' i., r/ vI -re I { "fln\'ln ->d'*


(2-72)


p
(2-73)


where I

as f I : :


g 7 ; o + A, VvP 5
ip q 2 ( 1o V 2' +) + i


Ir I F11 /I \ r p -
S p and ci.,1 p k (I, : (r are the Langevin terms recast

for the total carrier heat : w -nS or pS,.








Ti.. 4--drodynanmic system of equations Nwith velocity fluctuation Langevin noise

sources can then be expressed as follows.



o- V (7 ~ (v 4 -
dt
+ N' Nt
S= -V (pV) p R,

mn = -V(nkTn) qnE n ApV(kT)

d (P)= -V(pkT,) E p p- (kT,) I.

3 d(nT,) 3 T( TL
k = -V (S) E (A) I G, B nk V ((-) (2-74)
2 dt 2 (dwl)
dp7p) 3 Tp TI,
d3 k V ( )) Iq E ( ) u w T- V (),,) (2-75)

[ I S, 5 L 5 A,,, 1( '7
S,,js k1,f + a + (I A V,,)
q 2 "n 2G
T 0 ) v T12V p
S s' T -. '

'.. i. ::i: ....: of voltage at the considered external contact is then given as follows.

Vc /f, (HI (in ) (m (m ) I (VCJ (,) (VGE, (,)) 'r

(2-76)
where GE,, and G-E are the scalar-Nvalued Green's functions associated with the electron
and hole energy balance e .: :': : 1 vely. i : power spectral density of the noise at
the contact is given as follows.


Sv = 2T jj ri Gr1 /1/ d r, (2-77)









The additional local noise spectral tensors can be expressed as follows, in accordance with

Milatz' theorem, as in the previous section.


(Ok liT,. Dn Vi r*)
(K)P ()n/n (Tpj 2T

ri) -10kT rip D(pr () I ) -2T



22T
k25k22 lS V _
?) s(rj) 25k2 2. Ts Ks
P 2 n 2T
12
(r) s, (rj) = 25k22Dl 2T


2T






2T
ys ------


where a represents either n or p, and 03 represents the opposite choice. This results in the

following simplified expression for Sv.


r


-n P. (VGEP)* K G K v ,S, P (+( GEV) + d r (2-84)
S... m"pp(VGE* -ksP Ga, + (VGE)*. K- p (kGsps) -


This type of impedance field noise simulation has been performed previously as in [32, 46]

and has been referred to as I. 1 .1ii .' fluctuation" noise since the noise sources appear

opposite the dj terms.

2.5.1.2 Energy balance model with velocity fluctuation noise

To correctly place the Langevin noise terms in the energy balance model, the

Langevin terms are folded into the remaining PDEs from the momentum balance

equations. First the stationary expressions for the carrier velocities are rearranged as

follows.


Vn = qE+ kT,- + (1 AP)V(kTn)] + (Tp,).

-, L= -qE+kTpVP +(1 Ap,)V(kT,) + (rp).


qn

Vq+ p
qp


(2-85)

(2-86)


(2-78)

(2-79)


(2-82)

(2-83)


Vi)S. Vj)

,*P (ri) Sp (ri)


S*P









where = p,) and p = qp(Trp)^ are the Langevin terns recast as fluctuators

for the total carrier current density ,/ ., V or Jp :. The total carrier

heat flow fluctuators can be < .1 relative to the current density fluctuators as

A i, I' ) and i I). ip: energy balance model's of
: 'ns with vci 1 fluctuation Langevin noise sources can then be( : as

follows.


V (- Nj -N-) f
dn 1
S-V (n,) G,, R,, ->V () (2-87)
dp 1


+ kT, l + (1 A),V(,)(k

V I + -T 7, p + (1 A p k,)V(kT)

3 (n T,,) do
-V (nS,) iV (oVtC ) If q( G,, ,)1



:3 t' -dp V 'l.) -(, Ipd Ri,
2 dt dt
33 T --T
;-V)- 7 P +. *) + Rip +

2 -
I (I ()V 2

5 2
S= V" + tV A-Q,,)V *,



7.. i -tuation of voltage at the considered external contact is then given as follows.




(2 91)

where G',, and G, are the scalar-valued Green's functions associated with the electron and

hole continuity equations, respect T1: power spectral density of the noise at the









contact is given as follows.


-VGI *-
q d3rj
*- (VGE). (^p p)(-
(2-92)


The additional local noise strength spectral tensors can be expressed as follows, also in

accordance with Milatz' theorem as in the previous sections.


Sn1


p


)(s()4= 10 2 S'nD I6 ~- K 6(L' I '2
V 2T 2T
2 0pD -I( ) 04K r- j)



)(s, (4) 10 2 2q p D,p l jp 22
a(4) ( ) w3(4) (rJ) 0

2 5 k P S 2P rl) I ( K
q p 2n 2T 2

'^Sp (rj)- 0 -u P' qp pi j sjpp
q pp p 2T 2




2S 6 )( r,

t*P
S~nVi p j) S* V) SnVj -


- i)
T



- )


T


(2-97)

(2-98)


It is important to note that since p, p, and sp originate in the same collision term

of the BTE (Eq. (2-2)) as K. in the acceleration fluctuation scheme, the approximation

that these sources are statistically independent of position has not lost any validity,

and this model is as accurate relative to the acceleration scheme as the quasi-stationary

approximation for V, and V, are appropriate. While the fluctuation in velocity of a

single carrier flowing through a semiconductor (as in Monte Carlo computation) may be

correlated over short distances, the fluctuations in average carrier distribution velocity V,

are not.


Sv = 2T


vii
(*( vi
Sn (6










Ti. above relations give the ) a>wing 41*- 'c. d expression for the power spectral

density.


V2 GV, T + V 4G (K*j,, l,, Kj,,j,) VG +
Sv = -q r (2-99)
S ..+ VG .(Ks b 2is, + ). VGI,,


2.5.1.3 Drift-diffusion model with velocity fluctuation noise

T. -1 -.. :. of the Langevin noise terms in the drift-i : model is well known,

and is easily obtained from the t I Is :) a on simply by ignoring the ener-cv balance

and heat flow balance equations and taking T, and T, as TL. T_ : ::: :on model's

of equations with velocity fiuctiutation Langevin noise sources c(an then be expressed

as follows.


V (p I a- )-


ci Vq
dIp 1
7- (p) +G -R (V )
(it q

-- ;/ -< V ?7
dt


pp Vp
Vp= k-T, I (1 Ap,)V(7kTL)
q p

I... ::.:: ....: of voltage at the considered external contact is then given as follows.


SVG, VGp, pr (2 100)


power density of the noise at the contact is given as follows.


Sv 27' V 7G* r) -,7G,, .1, d
1 1 1 1
(2 101)
(2-101)









This expression is evaluated using Milatz' theorem which gives the following local noise

strength spectral tensors, as shown previously.



(4() (42Dpp)IJ(i K i6UN)
2T 2T




These expressions in turn give the greatly simplified expression for power spectral density,

as follows.


Sv = VG* ( Kl *VG, 1 + VG 3 k I VG) d3r (2-102)



2.5.2 Comparison of Velocity Fluctuation Noise Simulation with Energy
Balance and with Drift-Diffusion Models

2.5.2.1 One Dimensional n+/n/n Resistor Simulations

Velocity fluctuation noise simulations were performed on n+/n/n+ resistors, using

the energy balance and drift-diffusion models simulated by FLOODS. The thickness of the

n+ regions considered were 0.5/m. The doping density was 1018 cm-3 in the n+ regions

and 1017 cm-3 in the lower doped regions. The low-field carrier mobilities used were given

by the Klaassen Model [8, 9], and the velocity saturation relations used follow the Canali

model for Drift-Diffusion and its Energy Balance equivalent [38], with electron and hole

saturation velocities of 1.07 107 cm/s and 8.37 106 cm/s, respectively. The ratio of heat

flow mobility to carrier mobility was chosen to be 0.8 and the inhomogeneity parameters

Apn and AQn were chosen to be zero. The diffEi i-vi i- used to compute local noise strength

for the energy balance model uses Einstein's relation with the low kinetic energy mobility

and the average of the carrier and lattice temperatures as discussed previously.

Fig. 2-1 shows the doping concentration and the electron temperature computed

using the energy balance model for a 0.lpm resistor with a bias of 25mV. The effects of









Peltier cooling and Joule heating as the electrons move from left to right can clearly be

seen. Fig. 2-2 shows the contributions to velocity fluctuation noise (the quantity inside

the integral of Eqs. (2-101) and (2-92)). This result is in qualitative agreement with the

results published for the full hydrodynamic and drift-diffusion models by Jungemann,

et. al. [46]. The effect of the energy balance model on the local noise contribution is to

enhance it where the temperature increases in the direction of average carrier velocity

(the force due to the temperature gradient assists the flow of carriers) and to diminish

the noise contribution when the temperature decreases in the direction of the average

carrier velocity (the force due to the temperature gradient opposes the flow of carriers).

Fig. 2-4 shows the total velocity fluctuation noise for several lengths of the lowly doped

region, for simulations using both the energy balance and drift-diffusion models and at two

bias points. The points for the 0.lpm resistor at 25mV or 0.5V bias are the integral of

the curves in Figs. 2-2 and 2-3. This figure shows that the energy balance model predicts

lower overall velocity fluctuation noise than drift-diffusion, and this effect is enhanced

at higher bias. This is further illustrated by Fig. 2-5, which shows the ratio of the noise

predicted by the energy balance model to that predicted by the drift-diffusion model for

the same resistors and at the same bias points.

2.5.2.2 nMOSFET Simulations

Velocity fluctuation noise simulations were also performed on a 0.25pm n-channel

MOSFET, using the energy balance and drift-diffusion models simulated by FLOODS.

The considered device has a 30A gate oxide, and is typical of recent transistor designs,

with 1OOnm shallow source and drain extensions to reduce 2D short channel effects. The

transport parameters used in the drift-diffusion and energy balance models were the same

as those used in the resistor simulations. The Darwish mobility model was included to

degrade the mobility in the channel due to surface scattering.

Fig. 2-6 depicts the doping profile of the device, on a log scale. This provides a

picture of the geometry of the device with shallow n+ source/drain extensions. Fig. 2-7









shows the contribution of the local velocity fluctuations to the drain current noise for the

drift-diffusion model in the linear region of operation. Fig. 2-8 shows the contribution of

the local velocity fluctuations to the drain current noise for the energy balance model,

for the same bias. Both cases clearly show that velocity fluctuations only couple out

from the channel. Also, the total simulated noise in both cases is in good agreement

with the theoretical result of 4kTgdo where gdo was computed at a very low drain bias

by the simulator. These results were also in qualitative agreement with those exhibited

in the resistor results, in that the energy balance model predicts slightly lower velocity

fluctuation noise.

2.6 Impedance Field Simulation of Number Fluctuation Noise

In this section the fluctuation terms for number fluctuations are traced through

their origins in the BTE (2-2) to each contribution to the system of equations for each

transport model.

2.6.1 Number Fluctuation Noise Simulation

Number fluctuations are caused by either interband transitions of particles or

tunneling from one location to another. These transitions (except for the case of carrier

tunneling) involve collisions with phonons, photons, or other carriers. As with any other

transition, both energy and momentum must be conserved. Microscopically, the interband

carrier transition term of the BTE can be described by a statistical mean component and

a stochastic component, which fluctuates randomly and with a zero statistical mean.


s(, f, t) = s( t) + 7f (2-103)

where 7f is a Langevin noise source which represents a fluctuation in s(r, t). Langevin

noise sources must be inserted into the PDE's of the transport model through those terms

derived from the interband transition term of the BTE. This should be considered for each

moment individually.









For the zeroeth-order moment, the interband transition term is characterized as

follows.

s (, g, t) [G, R.] + = -R + (2-104)
P P
In the case of bulk or surface traps, there will also be a first-order moment and corresponding

term for trapped electrons. For the first-order moment, this term is characterized as

follows.

5 fis g, t) 0 + P (2-105)
P P
Since the direction of the momentum of generated or recombining carriers is random

(resulting in the zero statistical mean component), it follows that the effect of number

fluctuations on the average momentum of the carrier distribution is small, and K. is

usually assumed to be negligible. This assumption is valid, since intraband transitions

have a much greater effect on the momentum distribution. However, this is an assumption

and it may not be appropriate in some situations where the carrier concentration is very

low (such that velocity fluctuations do not dominate this term). For the second-order

moment, this term is characterized as follows.


Ss(r, ;, t) [Gw. R ] + R f = Gw, Rw + .. (2-106)
P n P n

For the case of transitions that can be considered to only involve carriers with low kinetic

energy (i.e. tunneling or band-trap transitions) this term can be assumed to be zero.

However, for the case of impact ionization or Auger recombination this term should be

considered, and can only be ignored if the effects of this term are studied and shown

to be negligible. For the third-order moment, the Langevin term 7, is related to

(Equation 2-60) and is therefore assumed to be zero.

Therefore, the number fluctuation Langevin sources 7,, 7,, 7,, (trapped carriers, if

necessary), r.. and r.. enter the complete system of equations derived from the first four

moments of the BTE as follows. As previously, the asterisks to denote effective masses and









lmo : are omitted for clarity.


V2
do


di
dt.
It


q f+ A 1+ A Pt

-V (n,) G,,- B,,


P-V + I- I G( ) P +
Gp R a, Go lRp +*n


d ( n V1)
dt

d(pVI,)
di

3, d (nT,,)

3 d(pT_)



dt


dt

S::"tuation of


T 7
V(nkT ) P, n I





V (nS) qE (0ni) 1 G


Vi0 >) + (-( ) iv


- Ap,,V(7kT,,)


ApV(kTI))

i T T,
, R, -nk T"' ,
2 (ny, )
3 T T,
2 (npP)


-7 q -2 n kT, T'


V2( 'T) q p(kT,) p K AQV (
voltage at the ed ext l contact then gien a

voltage f/ at. the considered external contact is then given as : : >ws.


SGE,


I ) d3r


where G,,,, GC, C,,, .: and GE, are the Green's functions associated with the

scalar-valued electron, hole, and trapped electron carrier continuity equations and electron

and hole energy balance ( :: vely. T power i. : ." density Sv 2TV

is computed as follows.



Sv 21 G T ?I 13 /i G d 3 T
ri E, E* r16i ( 1
(2-113)


(2-107)

(2-108)

(2 109)


(2-110)

(2-111)


V fffG (Gr, nI G I n


(2-112)









As in the section on velocity fluctuations, there are, depending on the number fluctuation

mechanism considered, approximations that the noise sources are statistically independent

in position, as follows.

t(K)'(Q) =j Koo a (2-114)
2T

This is valid since random fluctuations in generation rates are not expected to be strongly

correlated through position, except for the case of carrier tunneling. The noise strengths

can be evaluated using Monte Carlo computations, but this is not feasible for low

frequency computations. The alternative is to evaluate the local noise strengths using

the closed form expressions such as those derived by van Vliet [47]. Terms involving r.

must be evaluated differently for each type of mechanism.

None of the number fluctuation terms are involved in back-substituted expressions

for heat flow or particle current, as in the case of velocity fluctuations, so extending this

to the hydrodynamic, energy balance, and drift-diffusion cases is trivial. The effects of

independent mechanisms are added using the superposition principal, as follows.


SV total = SV trapping + SV tunneling + SV |impact (2-115)

2.6.1.1 Bulk or surface trap capture and emission

For the case of a trap species located at a particular energy level ET relative to the

intrinsic energy level El, the statistical mean capture and emission rates of electrons and

holes are governed by Shockley-Read-Hall recombination and generation rates RrT, RrT,

GnT, and GpT, as follows.


R.T = Cn(NT nr ) (2-116)

GnT=' I' =CnncnT (2-117)

RpT = CppnT (2-118)

GpT = ep(NT nu) = Cppi(NT nT) (2-119)









where c,, en, cp, and e, are the electron and hole capture and emission coefficients, and ni

and pi are the equivalence carrier concentrations, i.e. the electron and hole concentrations

for the case when the Fermi level is the same as the trap level. They are computed using

the Maxwell-Boltzmann approximation as follows.


ni = niexp T EI (2-120)
( kTL

pi niexp I T- ET) (2-121)

where ET is the trap energy level, El is the intrinsic energy level, and ni is the intrinsic

electron concentration. The corresponding local noise strengths are computed as follows

[29, 47].


7 )7(T ) -7t()7t() 2(GnT + RnTz) 2T K 2T- (2-122)
2T '" 2T
i) 7p j) 7 -7)7(r) 7 2(Gj +2 K 2T (2-213)

7t*(()7tyt() = 2(GnT + R.T + GT + R 2T 2T

Note that the generation and recombination rates used in these expressions are only the

components due to this trap species and its associated interactions. The terms involving

and r.. are assumed to be zero since the probability of capture or emission is highest

from low kinetic energy states. The simplified power spectral density is then given as

follows.
G* K G, G* KI, G, +



Sv KetG, G*KaG + -r (2-125)
nT nT nT

2.6.1.2 Direct Carrier tunneling

Previously [29], noise due to carrier tunneling implemented in FLOODS was modelled

as capture at fast surface states coupled with trap-to-trap tunneling from interface

traps to the oxide traps. This was appropriate for the device and bias conditions in









previous works, since the Fermi level at the surface was in the energy gap. However,

DC simulations of the devices measured for this work, and for any such highly scaled

MOSFETs, show that the electron distribution is degenerate at the surface. Therefore,

direct tunneling between the states in the conduction band at the silicon/oxide surface and

traps in the oxide has been implemented in FLOODS as a part of this work.

Direct band-to-trap carrier tunneling is unique among the mechanisms considered

in this work, in that it is a nonlocal mechanism involving fluctuations in two locations.

The case of interest for the devices involved in this work is that of tunneling from the

conduction band at the silicon surface and a trap in the oxide. As in the work of Hou [29],

trapped carriers tunnel primarily to the nearest silicon node, and transitions between the

gate and oxide traps are neglected. While the latter assumption is difficult to justify, it

greatly simplifies the implementation and is therefore used. The interactions are between

the trapped carrier continuity equation = R GX + 7 = at some oxide node at

position ri and the electron continuity equation at some interface silicon node at position

rj. The tunneling capture and emission rates RX$ and G" are given as follows.


RX$t = C n"(N4 nf) (2-126)

G" = e"t (2-127)

where c
change in the concentration of trapped carriers, if there is no tunneling to the gate then

these rates are equal. The capture and emission coefficients are given as follows.


ent = c (2-128)

ct = Toexp(-ad) (2-129)


where To is the tunneling coefficient, a is the attenuation constant of the electron wave

functions, and d is the distance between node i and node j. The attenuation constant, for









a triangular barrier, is given as follows [29].


a 12--o (2-130)
3 h pox

where mox is the effective mass of the tunneling electrons in the oxide, and '*,, is the

barrier height at the interface, equal to the difference in the electron affinities of the bulk

silicon and the gate oxide, 4.05eV. The barrier height at the trap is (ox = 4.05eV +

q(Qh i). In the limit that the field in the oxide is zero, a is 1.3 l0cm-1. The
tunneling coefficient for such band-to-trap transitions To is not known. This value is

arbitrary however, as the simulations have shown that the oxide noise is not sensitive to

this parameter (there is effectively a functional dependence in G*tG,t of 1/To, such that

the total noise power spectral density is insensitive to To).

The contribution to fluctuation in a contact voltage due to such transitions is given as

follows.

V7" = GnT,,()nT,()V, + G,(y)7.(F,)Vj (2-131)

where Vi and Vj are the nodal volumes at the oxide and interface nodes, respectively. The

power spectral density is then computed as follows.


SY 2T2 + G(G(T)7 (()7(()Tf)G((if)l( Gf+


(2-132)

The fluctuations at the interface and at the trap are also related, as dictated by

conservation of particles.

7nV, = -7,TV (2-133)









This results in the following relations.


1 1
(?/ (rj) 2(Gat + Rt T 2 (2 134)

______V____ 12* (2
Vi' ) ( 2 i j) ()j 135)

1i 1
'* (1) (iM) ( .7 Vj) -( ) i (2 136)

where represents fluctuations in cl ctron concentration at the ; :: e I and

represents fluc tuations in the concentration of trap carriers at a location in the

oxide. I: terns invol ':: aire assured to be zero since the carrier (: a:: nationn is

close to the conduction band edge unless the surface is strongly i. e. in which case

a carrier involved in the transition is one of rnany. T1:: the average kinetic ene: of a

degenerate carrier distribution does not fluctuate much due to this nmechanisnm.

F:::: i the :: : p 1 power spectral density then reduces to the i i :wing.



S1 1 Vi I G (r) G () (r)KG,(r)-Y (2-137)



2.6.1.3 Impact Ionization

An *:::. 1 ionization event results in the generation of an electron and a hole,

accompanied by a loss in kinetic energy of a single carrier that is sufficient to cause the

inter band transition. T-: carrier density and kinetic energy rate fluctuations associated

with electron ionization events are then related, as follows.


(2 138)

-Es .(21 .:)


where E6L represents the average energy lost by electrons in impact ionization events.

This is i 1 '. to be nearly equal to the minimum kinctic : : / required to excite

an electron :...... the valence band to the conduction band while conserving -..7 and









momentum, which for the minimum Umklapp process is slightly higher than EG. The

approximation Es = EG is assumed. The local noise strengths are then computed as

follows.


7,i )7,(j) 7*)7, ()- 7*)7,(i ) 2GV -E K (2-140)
2T 2T

7 (r .. (K,) 2G ii E62 K nn (2-141)
2T 2T

()' E K,) (r ( 2G KE60 G ) 2-142)
2T 2T

where Gi is the impact ionization rate, computed as in Section 2.4.

The power spectral density is then given as follows.

G*K, n G*Knnnp Ei6G*KnGEn

s An e I n t Gp cK- G;-nn9G EjKnGE, d 3 (2- 143)




2.6.2 Comparison of Number Fluctuation Noise Simulation with Energy
Balance and Drift-Diffusion Models in nt /njna Resistors

Number fluctuation noise simulations were performed on the same resistor structures

that were previously used in section 2.5.2.1, with the same choices for transport

parameters. An electron trap (cn = 109cm3s-1 > cp = 1011cm3sl1) was used to

simulate bulk generation-recombination noise with a constant trap density of 1015cm-3

and a trap energy level near to the equilibrium Fermi level in the lowly doped region.

Figure 2-9 shows the contributions to number fluctuation noise (the quantity inside

the integral of Eq. (2-113). Clearly, the effect of the hydrodynamic model on number

fluctuation noise is not as strong as that in the case of velocity fluctuations. Figure 2-10

shows the total noise integrated across the length of the devices. For long resistors,

the bias dependence of the Noise is 12, consistent with that of trapping in quasineutral

regions. Figure 2-11 shows the ratio of the noise predicted by the Energy Balance model

to that predicted by the Drift-Diffusion model.









2.7 Quantization Effects and Noise

The Schrodinger wave equation allows the evaluation of allowed states for electrons,

given the potential energy that confines the electrons in a structure (for example, in the

attractive electric field that surrounds positively charged atomic nucleii). The SWE is

given as follows.

V2 + V( t)) (, t) = E (, t) (2-144)

V(F, t) is the potential energy seen by an electron, as a function of position and time.

(r, ,t) and E are the wavefunction and associated energy of any allowed electron state
that is a solution of this Eigenproblem. The solution of the SWE in an infinite well is well

known, as illustrated in Figure 2-12. Note that the plotted functions are the waveforms

times their complex conjugate, giving the shape of the probability functions for electrons

in these odd and even states. The odd solutions have an odd number of nodes, and the

even solutions have an even number.

In semiconductors such as Silicon, the periodicity of the positions of atomic nucleii

gives rise to a periodic potential energy, yielding many allowed states that are clustered in

energy within bands. The core electrons (in atoms that have them) are tightly bound in

energy bands that have electrons in each and every allowed state. The valence electrons

are less tightly bound in the valence band, with a few energetic electrons that are excited

from the valence band or from bound states such as donor impurities or electron trap

defects to the conduction band or to bound states at acceptor impurities and hole trap

defects. As long as the ( i I is periodic for long distances, and the electric fields

associated with applied bias or nonhomogeneity are significantly lower than the binding

forces in the periodic structure of the crystal lattice, the SWE solution can be assumed to

give states consistent with the nearly free electron and hole models previously described.

However, in highly scaled and strongly inverted MOSFETs such as the devices

considered in this study, the electric field under the gate oxide confines carriers tightly

to the Silicon/Oxide surface. This confinement is in only one dimension (incident to the









surface), so the allowed states in the Silicon near the surface form quantized subbands for

which carriers conduct current in the unconfined directions (along the surface).

This quantization effect gives rise to displacement of the inversion carriers from the

surface as well as a local increase in the effective energy gap of the semiconductor. These

effects are manifested in the MOSFET as an increase in the effective threshold voltage and

a decrease in the number of carriers in the channel compared to what are predicted by

classical models.

Several methods have been emploi-- d1 to approximate the effects of quantization in

simulation frameworks originally developed for semi-classical transport. Some involve

the computational of a deformation potential to reduce and displace inversion 1l.r-iS [48]

. This requires adjustment of the mobility model [49]. However, the goal of this study

is to accurately simulate the noise in these MOSFETs. The velocity fluctuation noise

for current flow toward the drain is not expected to depend greatly on the shape of the

distribution of conductive carriers in directions perpindicular to this current flow, and is

not observed within the measurement time window. The only velocity fluctuation noise

observed within the measurement time window in this work is in the SOI devices, where

it is caused by i iiPi ily carriers in the floating body. The number fluctuation noise, in

the measurement time window, is primarily manifested by tunneling of inversion 1 -
carriers into and out of traps in the oxide. This number fluctuation noise is primarily

affected by the number of carriers in the inversion 1i-
emission rates of trapped carriers, and is again not greatly affected by the shape of the

inversion 1 i*.r carrier distribution. It is important to note that the tunneling probabilites

of carriers in the interacting traps and inversion 1 -
distance. They are not as strongly dependent on position as an exponential function of the

tunneling distance (such as those computed for models where the carriers are assumed to

tunnel from the oxide edge and for which the barrier height is integrated throughout the

tunneling distance), because the error displacement in tunneling position is in the region









where the electron energy is greater than the potential energy. Later analysis shows this

effect in detail, but for now the quantization effects are lumped into the effective tunneling

distance and barrier height, similar to the way that quantization-affected scattering

is lumped into the mobility models. Therefore, quantization effects in highly scaled

MOSFETs whose bias is limited to strong inversion can be approximated, for the purpose

of noise simulation, by a large fixed surface charge. This raises the threshold voltage in

strong inversion and reduce the mobile inversion charge, without adversely affecting the

simulation of noise in the device. This point is illustrated for the simulated devices and is

presented in C'i lpter 4.










310 1019



~ co
305 E .... -
1018
S100


.300..............
E 300 ...........----.. .... \ :--|
C O
o 17 o
7101
o o
-U 295 -
w 0

Tn, 0.025V bias ------.....

290 ND 1016
-0.2 -0.1 0 0.1

x (im)


Figure 2-1. i itron temperature as a :: : : : of position for the energy balance model in
a 0.1 pinm n ln/nl resistor with a bias of { V.










2e-26


Q 1.5e-26 -









0
N
















0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6


Figure 2-2. Veil fluctuation noise contribution as a function of. tion in (a 0.1-n
Sresistor for a bias of .
2 le-26 -




0 I I \







-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6










I I I I I I I I I I


5e-26 h


4e-26 h


3e-26 h


2e-26 F-


IIJ,


-0.4 -0.3 -0.2 -0.1 0 0.1
x (Pm)


Figure 2-3. Velocity fluctuation noise contribution as a
n /n/n I resistor for a, bias of 0.5V.


0.2 0.3 0.4 0.5 0.6


:: : -:: of position in a 0.1 pm


le-26 -


6e-26









10-30


10-31


10-32 L
0.01


0.1 1
LR (Rm)


Figure 2-4. Velocity : *tuation noise for n /n/n resistors of Vc'ing lengths.












0.95 ,---
1 *-I-I---------' --------- ...

0.95 -

0.9 -

0.85 -

0.8 -
o0
u 0.75 -

0.7

0.65 -

0.6 \ ..-*'" EB/DD, 0.025V bias ---- ----
EB/DD, 0.5V bias ........ a ....
0.55.
0.01 0.1 1 10
LR (Rm)
Figure 2-5. Ratio of velocity fluctuation noise predicted by Energy Balance model to that
: .l.ted i. D: : :-Diffusion foIr nI /n/n I resistors of : lengths.












4e-27





' 3e-27





2e-27



u
0

Z le-27


x(Pm)


le-05


1 'ure 2-9. .: : H ::tuation noise contribution as a
Ur/tu/rc4 resistor with a bias of : V.


:: :tion of position in a 0.1 m


DD
-- EB











-


u -
-le-05


2e-05












le-31 -


le-32


le-33


le-34 -


le-35 -


le-36 DD, 0.025V bias
---- EB, 0.025V bias
---* DD, 0.5V bias
le-37 -- EB, 0.5V bias


0.01 0.1 1 10
LR (Pm)


Figure 2-10. "* .1.* fluctuation noise for n^ /n/inf resistors of lengths.













x--x EB/DD, 0.025V bias
---- EB/DD, 0.5V bias


















- - -_ --- -----..
--.* ------ r----*----..


LR (mn)


.01



Figure 2-11. Ratio of
that p:


number fluctuation noise predicted by the Energy Balance model to
v by C: i -Diffusion for ni /n/n, resistors of varying lengths.


1.08 -


1.06 H


1.04 -


1.02 -











I I I I I I I I I


0.6


I 0.5-


S0.4


S0.3


S0.2


e 0.1



-4e-09 -2e-09 0 2e-09 4e-09
position (m)

Figure 2-12. Solution of Schr6dinger's Equation (probability distributions and associated
energies) for allowed states in an infinite potential well.









CHAPTER 3
NOISE MEASUREMENTS

The measurement of noise in a semiconductor device must be conducted with care,

because all devices in the measurement system generate noise which must be separated

out from or made negligible compared to the noise generated within the device under

test. As a result, noise measurement setups generally focus on low-noise components,

clean DC power supplies, and quality electrical contacts. Traditionally used methods for

protecting a device from high-voltage transients such as the use of reed relay switches and

programmable voltage ramps from switching power supplies cannot be used due to the

noisy signals the required components generate.

3.1 Measurement Setup

The noise measurements in this study were taken using the circuit depicted in

Figure 3-1. The metal film resistors were soldered in place and BNC cables were used

to connect the circuit to the amplifier and the power supply (which consists of two DC

batteries, one 12.5V and one 2V). The amplifier used is the Brookdeal 5004, the spectrum

analyzer is a Hewlett-Packard 3561A and two Agilent 34401A digital multimeters were

used to measure gate and drain DC voltages.

3.1.1 Resistor Network Design

The gate bias is applied using a simple resistor divider circuit. The applied gate

voltage is then given by the following equation.


VG = R VD D(31)
Rli + R2

The resistor values (RI and R2) should be chosen to be as small as possible to limit

their contributions to the thermal noise at the drain, amplified from the gate, but taking

into consideration their power dissipation limits and the current supplying ability of the

batteries. The resistor values of 1.5kQ and 16kQ were chosen for R1 and R2, respectively.

The drain bias is applied using the three-resistor network consisting of resistors R3,

R4, and R5. The drain bias is determined by solving the following equation, using the









drain current of the device at the desired drain bias.

R R3R4 + R4R5 + R5R3 (32)
VD VDD ID (3-2)
R3 + R5 R3 + R5

R4's resistance is desired to be as large as possible to maximize the significance of the

high frequency intrinsic device thermal noise compared to the resistor voltage noise (such

that 4kT(rd \RL) M 4kTrd). However, except for very small drain biases or very quiet

(noiseless) devices, the corner frequency where low frequency noise becomes insignificant
next to high frequency thermal noise is higher than the maximum frequency of the

spectrum analyzer (100kHz). In addition, it is desirable to only vary one of the three

resistors (R3 was chosen for this) to apply differing drain bias. The maximum voltage

drop across R4 is then limited and must be significantly less than VDD VD to guarantee

that fluctuations will not be clipped due to supply circuit limitations. Since the maximum

current observed in the measured devices was approximately 3.5mA, R4 was chosen to

be 2kQ, resulting in a maximum voltage drop of around one-half the supply voltage. It is

desirable to choose a value of R5 which is less than R4 such that the effect of varying R3

will set the voltage of the internal network node, independent of the IV characteristics of

the device under test. A value of IkQ was chosen. R3 is chosen for each device and bias

point to match the desired drain voltage and measured IV characteristics.

3.1.2 Practical Considerations

The physical location of each circuit component and the BNC cables is depicted

in Figure 3-2. Since switching power supplies are not used, the problem is presented of

applying the bias to the device under test without applying any high-voltage transients

which cause highly scaled and electrically sensitive MOS devices to degrade or suffer gate

dielectric breakdown.

It was found that the parasitic capacitance of the BNC cables was significant enough

to present enough charge to break down the gate oxide of the transistors measured.

Therefore, it was necessary to use a pull-down resistor to limit the charging of the









parasitic capacitances while the gate and drain bias were applied. The procedure is

depicted in Figure 3-3. Initially locations A, B, C and D are all disconnected. The BNC

capacitances are charged to VDD, and the gate and drain voltages are resistively tied to

ground. Location A is connected first, pulling the voltage on the gate-side BNC cable's

parasitic capacitance down to R2iOOVDD. Location B is then connected, pulling the

voltage down a little further (at this point the gate voltage measured was about 84mV).

Location A is then disconnected, and the gate voltage increases to the applied bias

(t 1.25V). The same procedure is then repeated on the drain side, connecting first D,

then C, and then disconnecting D. The gate and drain voltages were monitored at all

times during the connecting procedure. Breakdown of the gate oxide can be detected by

any significant drop in the measured gate voltage. After noise measurements, the order of

the procedure is reversed to disconnect the device from bias.

It was also observed that disconnection of the bias resistor network on the drain

side can create a high voltage transient on the gate side which is significant enough to

break down the oxide. Therefore, between noise measurements at different drain bias it

is necessary to remove the bias network connections to both the drain and gate before

replacing R3.

3.2 Measured Devices

Four devices were characterized in this work. Measured were long (2.33PmLG) and

short (O.O09mLc) transistors, processed the same way in the same fab, with both SOI

starting substrate material and bulk substrate material.

Figures 3-4 and 3-5 show the gate controlled I-V characteristics of the short n-channel

bulk device, as measured using an HP4145 semiconductor parameter analyzer. Figure 3-6

shows the drain controlled I-V characteristics. Figure 3-7 shows the drain current noise

measured at several drain bias points with the gate strongly inverted.









Figures 3-8 through 3-11, Figures 3-12 through 3-15, and Figures 3-16 through 3-19

show the similar measurements for the short channel SOI MOSFET, and long channel

bulk and SOI MOSFETs, respectively.







VD D


VD D


HP3561A
Spectrum
Analyzer


DUT


-VDD


Figure 3-1. Circuit used for bulk and 31 device noise measurements.


R1









VDD

r\---r


HP3561A
Spectrum
Analyzer


Figure 3-2. Physical 1*--mout of circuit showing the critical .'!. of electrical connections.



































Figure 3-3. Connection scheme to ensure the avoidance of dielectric breakdown.










0.002


0.0015




0.001
SVDS 1.2V




0.0005 -





-0.2 0 0.2 0.4 0.6 0.8 1
VGs (V)

Figure 3-4. Gate controlled I-V characteristics of the 3.2ipm x 90nm n-channel bulk
MOSFET, in the linear and saturation conditions.











0.001


0.0001 -

le-05 -

le-06

le-07 -

le-08 VDS 1.2V
--- VDs 0.1V
le-09

le-10 -

le-11 -

le-12 -

-0.2 0 0.2 0.4 0.6 0.8 1
VGS (V)

Figure 3-5. Gate controlled I-V characteristics of the 3.2/m x 90nm n-channel bulk
MOSFET, in the linear and saturation conditions, on a logarithmic scale.










0.002


0.0015 I uS




0.001




0.0005 -





0 0.5 1
VDS (V

Figure 3-6. Drain controlled I-V characteristics of the 3.2pm x 90nm n-channel bulk
MOSFET, in the linear and saturation conditions.












le-15


le-16 -


le-17 -
V VDs= 0.88V
SVDS-= 0.68V
le-18 I Vs=- 0.54V

VDs = 0.39V
le-19 VDs -=0.29V
oVDS=- 0.17V

le-20 -


le-21 -

1 10 100 1000 10000 le+05
frequency (Hz)

Figure 3-7. Drain current noise of the 3.2/m x 90nm n-channel bulk MOSFET, with
VGS t 1.25V.
















0.003


0.002 -S 1.2
V V=1.2V
---- VD= 0.1V


0.001 -




0----- I --- ------- -
0 0.5 1
VGs (V)

Figure 3-8. Gate controlled I-V characteristics of the 4.8pm x 90nm n-channel SOI
MOSFET, in the linear and saturation conditions.












0.001


0.0001 -

le-05 -

le-06 -

le-07 -

le-08 V 1.2V

le-09 Ds

le-10 -

le-11 -

le-12 -

le-13
-0.2 0 0.2 0.4 0.6 0.8 1
VGS (V)

Figure 3-9. Gate controlled I-V characteristics of the 4.8pm x 90nm n-channel SOI
MOSFET, in the linear and saturation conditions, on a logarithmic scale.
















0.003


0.002




0.001





0 0.5 1
VDs (V)

Figure 3-10. Drain controlled I-V characteristics of the 4.8pm x 90nm n-channel SOI
MOSFET, in the linear and saturation conditions.












le-14


le-15


le-16


N le-17


v le-18 VDs= 0.84V
SVDS = 0.69V

le-19 VDs 0.52V

SVDs = 0.29V
le-20 0.17V
VDs= 0.17V
VDs: V 25mV
le-21
1 10 100 1000 10000 le+05
frequency (Hz)

Figure 3-11. Drain current noise of the 4.8pm x 90nm n-channel SOI MOSFET, with
VGS t 1.25V.











0.0002


0.00015





0.0001 1.2
V =1.2V
V= 0.1V



5e-05 -





0 1 4 -------- -i----T---- -- -i 1 1
0 0.5 1 1.5
VGs (V)

Figure 3-12. Gate controlled I-V characteristics of the 3.2/m x 2.33/m n-channel bulk
MOSFET, in the linear and saturation conditions.













0.0001


le-05


le-06


I I I I I I I I I


VGs (V)


Figure 3-13. Gate controlled I-V characteristics of the 3.2pim x 2.33/m n-channel bulk

MOSFET, in the linear and saturation conditions, on a logarithmic scale.


-----------------








S/ -Vs-1.2
S DS 0.1
-- s 0



--/

--/


le-07


le-08


le-09


le-10


le-11


le-12


IIIIIIIIIIIIIIIII


I


i











0.0002


0.00015 I __




0.0001




5e-05 -





0 0.5 1 1.5
VDs (V)

Figure 3-14. Drain controlled I-V characteristics of the 3.2pm x 2.33/pm n-channel bulk
MOSFET, in the linear and saturation conditions.











le-17



le-18



le-19



le-20
l0VDS= 1.70V

Vs = 1.57V
le-21 V DS -=1.43V
VD = 1.29V
V = 0.90V
le-22 -VDs 0.53V
VDS = 0.28V

le-23
1 10 100 1000 10000 le+05
frequency (Hz)

Figure 3-15. Drain current noise of the 3.2pm x 2.33/m n-channel bulk MOSFET, with
VGS t 1.25V.











4e-05


3e-05 -





2e-05 -1.2V
V= 1.2V
DS = 0.1V



le-05 -




0 ----------\-< ----------i---- z ---- ------- ---
0 0.5 1 1.5
VGS (V)

Figure 3-16. Gate controlled I-V characteristics of the 0.6pm x 2.33/m n-channel SOI
MOSFET, in the linear and saturation conditions.











0.0001


le-05


le-06

le-07


le-08

1 V 12V
le-09 DS
S DS 0.1
le-10


le-ll


le-12

le-13
0 0.5 1 1.5
VGs (V)

Figure 3-17. Gate controlled I-V characteristics of the 0.6pm x 2.33/m n-channel SOI
MOSFET, in the linear and saturation conditions, on a logarithmic scale.












5e-05


4e-05 0.63V

S-- GS= 0.32V


3e-05




2e-05




le-05 -


0------------------------


0 0.5 1 1.5
VDs (V)

Figure 3-18. Drain controlled I-V characteristics of the 0.6pm x 2.33pm n-channel SOI
MOSFET, in the linear and saturation conditions.











le-18


le-20




V --VDS 1.70V
c VD = 1.57V
le-22 AVDS 1.43V
Vs = 1.29V
9 VDS = 0.90V
VDgS =- 0.53V
X-- VDS 0.28V


le-24
1 10 100 1000 10000 le+05
frequency (Hz)

Figure 3-19. Drain current noise of the 0.6pm x 2.33/m n-channel SOI MOSFET, with
VGS t 1.25V.