Strain Effects in Long to Short Channel MOSFETs

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Title:
Strain Effects in Long to Short Channel MOSFETs From Drift-Diffusion to Quasi-Ballistic Transport
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1 online resource (185 p.)
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english
Creator:
Parthasarathy, Srivatsan
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University of Florida
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Gainesville, Fla.
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Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Electrical and Computer Engineering
Committee Chair:
Thompson, Scott
Committee Co-Chair:
Nishida, Toshikazu
Committee Members:
Guo, Jing
Hershfield, Selman P
Majhi, Prashant

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Subjects / Keywords:
finfet -- nanoscale -- one-flux -- parasitic -- quasi-ballistic -- rsd -- strain -- temperature -- transmission -- transport -- uniaxial
Electrical and Computer Engineering -- Dissertations, Academic -- UF
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Electrical and Computer Engineering thesis, Ph.D.
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government publication (state, provincial, terriorial, dependent)   ( marcgt )
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Abstract:
In the last decade, process induced uniaxial strain has emerged as a proven technique for improving CMOS performance and is projected to be useful for future scaling.  Transistor performance has consistently improved with significant innovations in novel materials and device design. However, the scaling benefits are slowing down as the industry faces significant technological challenges in terms of fundamental limitations and process integration.  In this dissertation, a comprehensive experimental and theoretical study is presented towards understanding the impact of uniaxial strain in emerging nanoscale devices where the carrier transport becomes quasi ballistic. To verify the fundamental mechanisms of strain-induced mobility enhancement in long channel MOSFETs, drive current enhancement in n-type and p-type planar MOSFETs is characterized at temperatures from 300K to 80K.  For p-type MOSFETs, it is found that for low stress levels, the conductivity mass reduction with compressive strain is the major contributor to the observed mobility enhancement at all temperatures.  For n-type MOSFETs, our experimental results suggest that the surface roughness reduction with uniaxial tensile strain is a possible mechanism for the observed electron mobility enhancements at high transverse fields.   The benefits of process induced stress in FinFETs are understood through detailed electrical characterization of devices having stress introduced by a contact etch stop layer.  The impact of underlapped architecture on the parasitic source/drain resistance is explained with a simple model.  The physics of stress transfer in FinFETs is qualitatively studied with  a thorough review of existing literature along with experiments and simulation of mechanical wafer bending.  To clarify the physics of quasi-ballistic carrier transport and the impact of strain, an updated one-flux theory based transport model is developed.  It is verified that the high field optical phonon scattering plays an important role in determining the overall saturation current due to the inherent feedback between device electrostatics and carrier transport.  A surface-potential based analytical formula for the nanoscale transmission coefficient including high field and quantum confinement effects is developed.  The differences between the strain-induced linear and saturation current enhancements for electrons and holes is qualitatively explained.
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In the series University of Florida Digital Collections.
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Includes vita.
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Statement of Responsibility:
by Srivatsan Parthasarathy.
Thesis:
Thesis (Ph.D.)--University of Florida, 2012.
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Adviser: Thompson, Scott.
Local:
Co-adviser: Nishida, Toshikazu.
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RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2013-08-31

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STRAINEFFECTSINLONGTOSHORTCHANNELMOSFETS:FROMDRIFT-DIFFUSIONTOQUASI-BALLISTICTRANSPORTBySRIVATSANPARTHASARATHYADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2012 1

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c2012SrivatsanParthasarathy 2

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Tomybelovedfatherandmother 3

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ACKNOWLEDGMENTS Iamdeeplyindebtedtomyadvisor,Prof.ScottThompson,forgivingmeagreatopportunitytoworkintheareaofdevicephysicsandsupervisingmyresearchwork.IreallyappreciatethefreedomandtimehehasgivenmeinthecourseofmyPhDandhisvaluableadviceonapproachingproblemsfromapracticalandlogicalviewpoint.Iamalsoobligedtomyco-advisorProf.ToshikazuNishidaforhisverypatientadviceandguidancewithbothexperimentalandtheoreticalworkthroughtheyears.Hisadvicehashelpedmebecomeconsistentandmethodicalindoingresearchwork.BothmyadvisorsarewonderfulexamplesofdedicationandhardworkthatIhopetoemulateinmylifeandcareer.IthankProf.JingGuoandProf.SelmanHersheldforservingonmycommitteeandfortheirvaluableclassesthatIhavehadtheopportunitytotake.IthankProf.JerryFossumforhisclassesonsemiconductordevicetheoryandFinFETsthatgreatlyhelpedmyfundamentalunderstandingofthesubject.IamverygratefultomycommitteememberDr.PrashantMajhi(Intel)fortheinternshipopportunityatSEMATECH(Austin)during2008-2009.Ireallyappreciatethepersonalinteresthetookinmyresearch.IamalsogreatlyindebtedtoDr.RustyHarrisfromSEMATECH(currentlyatTAMU)forhissupportandtoDr.CaseySmithfromSEMATECH(currentlyatKAUST)forprovidingmewithsamplesandguidingmethroughexperiments.IwouldliketothankallofmyformerandcurrentcolleaguesatUF{Dr.ToshiNumata,Dr.YongkeSun,Dr.Ji-songLim,Dr.GuangyuSun,Dr.SagarSuthram,Dr.YounsungChoi,AndrewKoehler,Dr.DanielCummings,Dr.HyunwooPark,Dr.UmaAghoram,Dr.MinChu,Dr.SaurabhMorarka,TonyAcosta,OnurBaykan,AshishKumarandAmitGuptaforallthehelpandsupportthroughtheyears.I'dalsoliketothankmyfriendsSudarshanBahukudumbi,KaushikSridharandRamachandranNarayanaswamyforalltheirsupportandforstandingbymepatientlyinthepastfewyears.Idedicatethisdissertationtomyparents.Withouttheirlove,blessingsandnumeroussacricesformyeducation,Icouldnothavecomethisfar. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS ................................. 4 LISTOFTABLES ..................................... 7 LISTOFFIGURES .................................... 8 ABSTRACT ........................................ 11 CHAPTER 1INTRODUCTION .................................. 13 1.1Background ................................... 13 1.2Drift-DiusiontoBallisticTransport ..................... 14 1.3OverviewoftheThesis ............................. 15 2STRAINEFFECTSINLONGCHANNELPLANARMOSFETS ........ 17 2.1StrainEngineering:OriginandState-of-the-Art ............... 18 2.2StrainCharacterization ............................. 22 2.3StrainEectsonCarrierMobility ....................... 25 2.3.1N-typeBulkSiandN-MOSFET .................... 27 2.3.2P-typeBulkSiandP-MOSFET .................... 29 2.4PMOSLowTemperatureStudy ........................ 31 2.5NMOSLowTemperatureStudy ........................ 39 2.6Summary .................................... 42 3STRAINEFFECTSINSOIFINFETS ....................... 43 3.1EvolutionofSOIDevices ............................ 44 3.2FinFETTechnology ............................... 46 3.3StudyonStrainEectsinFinFETs ...................... 49 3.3.1DeviceDetails .............................. 50 3.3.2ExperimentalResultsforSiFinFETs ................. 53 3.3.2.1UnderstandingFinFETRSD ................. 59 3.3.2.24-pointwaferbending .................... 65 3.3.3PhysicsofStressTransfer ........................ 66 3.3.4ExperimentalResultsforSi/SiGeFinFETs .............. 73 3.4Summary .................................... 80 4HIGHFIELDTRANSPORTINSILICONMOSFETS .............. 81 4.1Semi-ClassicalTransportModels ....................... 82 4.1.1Drift-DiusionApproach ........................ 85 4.1.2VelocitySaturationandVelocityOvershoot .............. 87 4.1.3HydrodynamicModel .......................... 87 5

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4.2MOSFETCurrent-VoltageexpressionsinDDandHDmodels ....... 90 4.3NanoscaleTransportModels .......................... 97 4.3.1Natori'sBallisticMOSFETModel ................... 97 4.3.1.1Physicsofquantumballistictransport ........... 98 4.3.1.2CurrentcontrolinNatori'smodel .............. 99 4.3.2Lundstrom'sQuasi-BallisticKBT-layerModel ............ 102 4.3.2.1PhysicsofscatteringinKBT-layermodel .......... 106 4.3.2.2IssuesintheKBT-layermodel ............... 107 4.4UnderstandingStrainEectsonQuasi-BallisticTransport ......... 117 4.4.1OriginofSaturationVelocity ...................... 120 4.4.2StudyingtheImpactofStrain ..................... 124 4.5Summary .................................... 128 5UFCOMPACTMODELFORTRANSMISSIONCOEFFICIENT ........ 129 5.1One-uxTheory:Fundamentals ........................ 129 5.2Gildenblat'sTransmissionModel ....................... 134 5.3RevisitingtheOriginalShockleyWork .................... 138 5.4UFCompactModel:Theory .......................... 139 5.4.1IncludingHighFieldEects ...................... 140 5.4.2IncludingConnementEects ..................... 143 5.4.3RevisitingNatori'sHighFieldTransportModel ........... 145 5.4.4UnderstandingFeedbackinNatori'sHighFieldModel ........ 148 5.4.5PhysicalSignicanceofEectiveVelocityTerm ........... 154 5.4.6ExpandingNatori'sHighFieldModelforMOSFETs ......... 157 5.5QualitativeUnderstandingofStrainEects .................. 159 5.6Summary .................................... 161 6CONCLUSIONSANDFUTUREWORK ...................... 162 6.1Summary .................................... 162 6.2RecommendationsforFutureWork ...................... 163 REFERENCES ....................................... 165 BIOGRAPHICALSKETCH ................................ 185 6

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LISTOFTABLES Table page 2-1-coecientsforsiliconn-MOSFETs:2Dvs. bulk ................. 23 2-2-coecientsforsiliconp-MOSFETs:2Dvs. bulk ................. 23 7

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LISTOFFIGURES Figure page 1-1Strainengineeringinhighvolumeproduction ................... 13 1-2Modesofcarriertransport .............................. 14 2-1Moore'slawbasedscaling .............................. 17 2-2Classicationofmethodstoimprovecarriermobility ............... 19 2-3State-of-the-artdeviceusingseveralstresstechniques ............... 20 2-4Gatelastprocessow ................................ 21 2-54-pointwaferbending ................................ 24 2-6Uniaxialtensilestressappliedtoasiliconsample ................. 25 2-7Lowtemperaturecryogenicprobestation ...................... 26 2-8Lowtemperature4-pointbendingapparatus .................... 26 2-9Strainapparatusinsidethecryogenicprobestation ................. 27 2-10Unstrainedandstrainedn-typebulkSiliconbandstructure ............ 28 2-11Unstrainedandstrainedp-typebulkSiliconbandstructure ............ 30 2-12Subbandsplittingdiagramsforholes ........................ 31 2-13Experimentalresultsoneectofuniaxialstressonelectronandholemobility .. 32 2-14Experimentalprocedureforlowtemperaturestressmeasurements ........ 33 2-15Experimentalresultsforp-channelMOSFETs ................... 34 2-16Contributionofscatteringchangeinoverallmobilityenhancement ........ 35 2-17Contributionofmasschangeinoverallmobilityenhancement .......... 35 2-18Warpingofthevalencebandwithuniaxialcompressivestrain .......... 36 2-19Holerepopulationinunstrainedcase ........................ 37 2-20Holerepopulationinthepresenceofstrain ..................... 37 2-21Simulatedconductivitymasschangeforstraininducedmasschangeasafunctionoftemperature .................................... 38 2-22Experimentalresultsforn-channelMOSFETs ................... 39 2-23Amplitudechangesofthemicro-scaleroughnessofp-Siunderuniaxialtension 41 8

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3-1TimelineofSOItechnologies ............................. 45 3-2ScalingrulesforbulkMOSFETalternatives .................... 47 3-3KeystepsinsiliconFinFETfabricationprocess .................. 51 3-4SiFinFETProcessFlow ............................... 51 3-5DetailsoffabricatedsiliconFinFET ......................... 52 3-6TEMcrosssections .................................. 52 3-7Fabricated-gatep-channelSi/SiGeFinFETs ................... 53 3-8MeasuredID-VGcharacteristicsof30nmgatelengthFinFETs .......... 54 3-9MeasuredVTandDIBLvariationasfunctionofchannellength .......... 54 3-10ComparisonofVTandDIBLbetweenunstrainedandCESLstrainedFinFETs 55 3-11MeasuredION-IOFFcharacteristicsshowingexcellento-statebehavior ..... 55 3-12Measuredlong-channelelectronandholemobility ................. 56 3-13MeasuredgateleakagecharacteristicsofsiliconFinFETdevices ......... 56 3-14n-channelION-IOFFplotshowingimprovedshortchannelcurrent[1]c2009IEEE. ......................................... 57 3-15ION-IOFFplotshowingsubstantiallyimprovedshortchannelcurrentp-channeldeviceswithcompressiveCESL[1]c2009IEEE. ................. 57 3-16EvolutionofID-VGcharacteristicsforincreasingnwidths ............ 58 3-17ObservedRSDreductionforn-channelFinFETswithCESLstress ........ 62 3-18ObservedRSDreductionforp-channelFinFETswithCESLstress ........ 63 3-19ComparisonofnormalizedRSDreductionbetweenn-channelandp-channelFinFETswithCESLsstress .................................. 64 3-20Measuredlinearcurrentenhancementwith4-pointwaferbending ........ 66 3-21StresstransfermechanisminaplanarbulkMOSFET ............... 67 3-22MobilityenhancementasafunctionofchannellengthforSiFinFETs ...... 69 3-23OnepossiblemechanismofCESLstresstransferinFinFETs. .......... 69 3-24MetalgateinducedstressinFinFETs ........................ 71 3-25SimulationstructureusedforABAQUSsimulations ................ 71 9

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3-26ResultsfromABAQUSsimulations ......................... 72 3-27ComparisonbetweenCESLstressandS/DstressinSOIFinFETs ........ 73 3-28MeasuredID{VGcharacteristicsof(110)SiGeFinFETs .............. 74 3-29ThresholdvoltageshiftduetobiaxialstraininSiGelayer ............. 75 3-30MeasuredION-IOFFcharacteristicsof(110)SiGeFinFETs ............ 75 3-31Mobilitycomparisonbetween(110)SiGeandSiFinFETs ............. 76 3-32Mobilitycomparisonbetween(100)SiGeandSiFinFETs ............. 77 3-33SiGemobilityat300K ................................ 78 3-34SiGemobilityat87K ................................. 78 3-35EvolutionofSiGeFinFETmobilityasafunctionoftemperature ......... 79 4-1ConductionbandprolealongthelengthofaMOSFETchannel. ........ 99 4-2QuasiballistictransportinananometricMOSFET ................ 104 4-3Carrierbackscatteringinquasi-ballistictransport ................. 108 4-4ElectrondistributionintherstBrillouinzone ................... 118 4-5Variationofelectrondriftvelocityandelectronenergy .............. 119 4-6Ryder'svelocitysaturationdatafrom1953 ..................... 123 4-7Khakiroozmodelforquantifyingstraineects .................. 126 4-8Extractedballisticeciencyforn-channelMOSFETsforchannellengthsintherangeof130nmto45nm,atVG)]TJ /F3 11.955 Tf 11.95 0 Td[(VT(sat)=1VandVD=1:2V. ......... 127 5-1Incomingandoutgoingcarrieruxesinaslabofthicknessdx .......... 130 5-2Includingconnementeectsfortransmissionmodel ............... 143 5-3Eectsofpositiondependenceonoverallaveragedriftvelocity .......... 144 5-4Natori'shigheldtransportmodelshowingtheconceptual\InitialElasticZone"and\OpticalPhononEmissionZone"regionsinadevicecarryingcurrent. ... 147 5-5Newderivationofoveralltransmissioncoecient ................. 149 5-6TwoequivalentnegativefeedbacksystemrepresentationoftheNatorimodel .. 153 5-7Evolutionofconductionbandproleandcorrespondingtransmissioncharacteristicsofthechannel ..................................... 158 10

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophySTRAINEFFECTSINLONGTOSHORTCHANNELMOSFETS:FROMDRIFT-DIFFUSIONTOQUASI-BALLISTICTRANSPORTBySrivatsanParthasarathyAugust2012Chair:ScottE.ThompsonCochair:ToshikazuNishidaMajor:ElectricalandComputerEngineeringInthelastdecade,processinduceduniaxialstrainhasemergedasaproventechniqueforimprovingCMOSperformanceandisprojectedtobeusefulforfuturescaling.Transistorperformancehasconsistentlyimprovedwithsignicantinnovationsinnovelmaterialsanddevicedesign.However,thescalingbenetsareslowingdownastheindustryfacessignicanttechnologicalchallengesintermsoffundamentallimitationsandprocessintegration.Inthisdissertation,acomprehensiveexperimentalandtheoreticalstudyispresentedtowardsunderstandingtheimpactofuniaxialstraininemergingnanoscaledeviceswherethecarriertransportbecomesquasiballistic.Toverifythefundamentalmechanismsofstrain-inducedmobilityenhancementinlongchannelMOSFETs,drivecurrentenhancementinn-typeandp-typeplanarMOSFETsischaracterizedattemperaturesfrom300Kto80K.Forp-typeMOSFETs,itisfoundthatforlowstresslevels,theconductivitymassreductionwithcompressivestrainisthemajorcontributortotheobservedmobilityenhancementatalltemperatures.Forn-typeMOSFETs,ourexperimentalresultssuggestthatthesurfaceroughnessreductionwithuniaxialtensilestrainisapossiblemechanismfortheobservedelectronmobilityenhancementsathightransverseelds.ThebenetsofprocessinducedstressinFinFETsareunderstoodthroughdetailedelectricalcharacterizationofdeviceshavingstressintroducedbyacontactetchstop 11

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layer.Theimpactofunderlappedarchitectureontheparasiticsource/drainresistanceisexplainedwithasimplemodel.ThephysicsofstresstransferinFinFETsisqualitativelystudiedwithathoroughreviewofexistingliteraturealongwithexperimentsandsimulationofmechanicalwaferbending.Toclarifythephysicsofquasi-ballisticcarriertransportandtheimpactofstrain,anupdatedone-uxtheorybasedtransportmodelisdeveloped.Itisveriedthatthehigheldopticalphononscatteringplaysanimportantroleindeterminingtheoverallsaturationcurrentduetotheinherentfeedbackbetweendeviceelectrostaticsandcarriertransport.Asurface-potentialbasedanalyticalformulaforthenanoscaletransmissioncoecientincludinghigheldandquantumconnementeectsisdeveloped.Thedierencesbetweenthestrain-inducedlinearandsaturationcurrentenhancementsforelectronsandholesisqualitativelyexplained. 12

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CHAPTER1INTRODUCTION 1.1BackgroundVeryLargeScaleIntegration(VLSI)inmicroelectronicswouldbenon-existentifnotfortheaggressivescalingofmetal-oxide-semiconductoreld-eecttransistors(MOSFETs)thatstartedinthe1960s[ 2 ].Today,weareinthesub-50nmregimeforchannellengthsandsiliconstillremainsthebestoptionfortheindustry,thoughthesearchforasuitablereplacementisunderway.Whilefuturescalingposesinterestingandchallengingquestionssuchasusingnovelchannelmaterials(III-Vsemiconductors,graphene,nanowires)andnoveldevicearchitectures(3DFinFETs,tunnelFETsandFeFETs),strainalteredmaterialsoermanybenecialtransportpropertieswhichrequireinvestigationatshorterchannellengths. Figure1-1.Strainengineeringinhighvolumeproduction.(Photosfrom:[ 3 ][ 4 ][ 5 ][ 6 ][ 7 ]cIEEE2004-2012) OneofthebiggestbreakthroughsinrecenthistoryofMOSFETshasbeentheintroductionofstraintoenhancetransistorperformance.Strain-enhancedtransistorsarecurrentlyinthe5thgenerationofproductionnowwiththerecentannouncementofhighvolume22nmnon-planartransistors(Fig. 1-1 ).Thecarriertransportmechanismintheseshortchannellengthdeviceshasbecomeincreasinglyquasi-ballistic.Towardsextendingthesiliconroadmap,the2011ITRS[ 8 ]liststwoimportantchallengesforemergingnanoscaledevices(1)inducingadequatestressinthetransistorchanneland(2)understandingstrainenhancedquasiballistictransport.Inthisdissertation,weinvestigate 13

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therelevanceandimportanceofstrainengineeringthroughexperiments,simulationandmodeling. 1.2Drift-DiusiontoBallisticTransportCarriermotioninsemiconductorscanbeoftwotypes:deterministicandrandom.Deterministicmotionisonethatiscausedbysomeforce(e.g.,electriceld)therebyfollowingNewton'slawsandisshownasarrowsinFig. 1-2 .Randommotionresultsduetoscattering,shownastheabruptchangesbetweenarrowsinFig. 1-2 .Ifthelengththroughwhichthecarrierpropagatesismuchgreaterthanthemeanfreepathbetweencollisions,transportissaidtobedrift-diusiondominated.Ifthetransportlengthismuchsmallerthanthemeanfreepath,wehaveballistictransport(noscattering).Ifthetransportlengthisthesameorderofmagnitudeasthemeanfreepath,thetransportregimeisquasi-ballistic.Currentdaydevicesarethoughttobeinthisregime.Verygoodtransportmodelsexistforthecompletelydrift-diusiondominatedtransportandforcompletelyballistictransport.However,modelingtheintermediatequasi-ballistictransportregimeiscomplicatedbecauseofthehighlyo-equilibriumnatureofthetransportprocess. Figure1-2.Modesofcarriertransport Devicedimensionscontinuetoshrinkwithcontinuedscaling.Assuch,operationneartheballisticlimitisbecomingabetterandbetterpossibilitywitheverynode.Inthesecircumstances,thequestionofwhethercommonlyusedmacroscopicmodels,especiallyfordescribingtransportinhigheldconditions,losevaliditybecomesimportant.Althoughitseemsclearthatmacroscopictransportequationsoftheconventionalkindwillfailinthe 14

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ballisticlimit,theysomehowcontinuetobeusefulinnanoscaledevices.Notonlydoesthiscomplicateunderstandingoftheeectoffactorssuchasstrain,italsobringsintoquestionourunderstandingofthefundamentalphysicsoftransportitself.StrainhasproventobeextremelybenecialinimprovingMOSFETperformance,especiallyforholes.Thereisexcellentliteraturepublished,forexample,[ 9 ][ 10 ]onthissubjectbuttherestillexistsonetopicthatislessexplored:straineectsinhigheldconditionswhichistheactualoperatingmodeofaMOSFETindigitalcircuits.ForlongchannelMOSFETs,themechanismforcurrentsaturationiswellunderstood.WhiletherehasbeenplentyofspeculativepapershintingatwhythecurrentsaturatesinashortchannelMOSFET,itisonlyveryrecentlythatamorefundamentalunderstanding[ 11 ][ 12 ]hasbeenproposed.Evenso,thereisstillamuchdebateabouttheactualmodeoftransport(Drift-Diusion/Quasi-Ballistic)thatexistsinastate-of-the-artlogicMOSFET. 1.3OverviewoftheThesisAsillustratedintheprevioussection,theeectofstrainonthesaturationcurrentofultra-scalednanoscaletransistorsisnotsoclear.Thedicultyarisesbecauseofthequasi-ballisticnatureofcarriertransportinstate-of-the-artdevices,whichisnolongernegligible.Inthepresenceofstrain,thisproblembecomesevenmorecomplicated.Hence,thepurposeofthisdissertationistobringclarityinunderstandingstraineectsinsaturationinmoredetail,specicallyfortheseshortdevices.Theinsightwegainfromthisworkwillnotonlyberelevanttocurrentdaytransistorsbutalsotofuturedevicesthatareevenmoreclosertotheultimateperformancelimits.Thisstudyiscomposedofthreeparts:(1)understandingthestraineectsinlongchannelMOSFETs(bothplanarandnon-planar)inmoredetail(2)identicationofthelimitationsofcommonlyusedtransportmodelsinassessingstraineectsonnanoscaledevices,and(3)explorationofstraineectsonultra-scalednanoscaledevicesattheballisticlimit.Priortounderstandingstresseectsinscaleddevices,aclearunderstandingofstraineectsinlongchanneldevicesisneeded.Areviewofthestresseectsonsilicon 15

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bandstructureanduniaxialstressmeasurementsperformedonlongchannelplanarbulktransistorsatvarioustemperaturesissummarizedinChapter2.Theinsightgainedfromtheseexperimentsmotivatesthestudyonshortchanneldevicesinlaterchapters.InChapter3,ashortreviewofstate-of-the-artFinFETsisprovidedwithexperimentaldataonbothSi(bothn-FinFETandp-FinFET)andSiGe(p-FinFETonly)non-planardevices.TheimportanceofunderstandingtheparasiticRSDintheseunderlappedFinFETsandthephysicsofstresstransferisdiscussedindetail.FEMsimulationsonungatednsprovideinsightintothedierencesbetweenprocessinducedandmechanical4-pointwafer-bendinginducedstressinFinFETs.InChapter4,abriefsummaryoftheshortcomingsofthetraditionaldrift-diusionmodelfornanoscaledevicemodelingispresented.Thisisfollowedbyathoroughreviewofstate-of-the-arttransportmodelsformodelingsaturationcurrentinshortchanneldeviceshavingquasi-ballistictransport.Understandingthevariousassumptionsmadeineachofthesemodelsiscrucialforevaluatingtheapplicabilityofeachmodelforunderstandingstresseectsonshortchannelsaturationcurrent.InChapter5,wepresentthedevelopmentofanew,unied,surfacepotentialbasedmodelforevaluatingthetransmissioncoecientinaMOSFETchannel.Thismodelisdevelopedbycarefullyaddinghigheldeectsandconnementeectstothebasicone-uxtheory.Wealsoshowthatthismodelisapplicableforallregionsofdeviceoperationunlikeexistingmodels.Natori'shigheldtransportmodelisreinterpreted,expandedandlinkedtoourwork.Anewdenitionofballisticeciencyisdeveloped,whichexplicitlyincludestheimpactofchannelscattering.Usingourtransmissionmodel,aqualitativeexplanationofstraineectsonsaturationcurrentforbothtypeofcarriersispresented.Thisstrainmodelhighlightsthestronginterplaybetweencarriermass/scatteringanddeviceelectrostaticsinthepresenceofstrain.Finally,Chapter6providesasummaryofthedissertationanddiscussesfuturework. 16

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CHAPTER2STRAINEFFECTSINLONGCHANNELPLANARMOSFETSMoore'slawistheempiricalobservationthatcomponentdensityandperformanceofintegratedcircuitsdoubleseveryyear,whichwasthenrevisedtodoublingeverytwoyears[ 13 ][ 14 ].GuidedbythescalingrulessetbyDennard[ 15 ]in1974,smartoptimization,timelyintroductionofnewprocessingtechniques,devicestructures,andmaterials,Moore'slawhascontinuedunabatedformorethan40years(Fig 2-1 ).Thepredicted\endofscaling"hasbeenpushedtothefutureineachgeneration,thankstodesignandmaterialinnovationsthathavebeendevelopedwheneverneeded.Drivenbytremendousadvancesinlithography,the22nmlogictechnologynodefeaturing3Dtransistorswith5thgenerationofprocessstrainiscurrentlyinhighvolumeproduction. Figure2-1.Moore'slawbasedscaling[ 16 ] Dennardscaling[ 15 ]wasthede-factoscalingrulefortransistorstilljustafewyearsago,whereoxidethickness,channellengthandwidthwerescaledbyaconstantfactor(1=k).Inthisscalingmethodology,thepowerdensityremainsconstantwhiledelayimprovementhappensby1=k.The130nmnodewasthelastgenerationwithconstanteldscaling,beyondwhichitwasnotpossibletodelivertherequiredperformancewith 17

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sizescalingonly.Feature-enhancedCMOSscalingstartedwiththeintroductionofprocess-inducedstraininplanarSiMOSFETtransistorsin2003[ 3 ]atthe90nmnode.Insubsequentgenerationsadditionalenhancerswereneeded,suchasstrainandhigh-K-metalgateinthe45nmand32nmnodes[ 5 ][ 6 ]tocontinuescaling.Someoftheimportantquestionsthatarebeingdiscussedtodayforfuturetransistorsare: Whatarchitecture?(planarornon-planar) Whatmaterial?(SiorIII-Vorother) Willstrainstillberelevant?TheimpactofstrainonprolongingtheCMOSscalingroadmaphasbeensignicant.Futuretransistorarchitecturesolutions(planarormulti-gateorotherhybridorientationschemes)willprobablycontinuetousestrainforimprovingperformance.Areviewofthehistoryofstraintechnologyprovidesinsightonhowithasbeenusefulandhowitcanbeexploitedforfuturegenerationsofdevices.Inthischapter,wereviewthestraineectsinlongchannelMOSFETs.Abriefreviewofpastworkispresented,bothexperimentalandtheoretical.Wethenshowresultsfromthelowtemperaturestrainmeasurementsperformedinthisworkanddiscusshowthesenewndingsfurtherenhanceourunderstandingofstraineects. 2.1StrainEngineering:OriginandState-of-the-ArtFig. 2-2 showsthreetechniquesthatarecurrentlyinusetodayforimprovingthecarriermobility.HybridOrientationTechnology(HOT)involvesusingn-typeandp-typetransistorsindierentchannel/surfaceorientationsmanufacturedonthesamewafertoexploitthecarriermobilitydierences.ChannelengineeringinvolvestheincorporationofchannelmaterialswithcarriermobilityhigherthanSi,forexample,InGaAsforelectrons,inthechannelregionofthetransistor.Strainengineeringinvolvesinducingtheproperstraininthetransistorchannelregionwhichisknowntoimproveelectronmobility(tensilestrain)andholemobility(compressivestrain).Ofthethreemethodsshown, 18

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strainengineeringhasbeenchosenbytheindustryduetoitscosteectivenessandeaseofintegration.Fig. 2-2 alsoprovidesahighlevelclassicationofthevariousstrainengineeringtechniques.Localstraintechniquesareusedtoinduceuniaxialstraininthechannelandacombinationofdierentmethodsareusedtointroducethenecessaryamountsofstraininstate-of-the-arttransistorstoday,asdepictedinFig. 2-3 .Newstrainmethods[ 17 ][ 18 ]arebeingdevelopedandintroducedeveryyearforuseinemergingdevices.Webrieydiscussthehistoryofstraintechnologyinthissection. Figure2-2.Classicationofmethodstoimprovecarriermobility TheinitialworkinstrainedSiCMOStechnologywasinspiredbytheresultsinIII-Vsemiconductors.StrainintheMOSFETchannelwasintroducedduetothelatticemismatchproducedbygrowingSionSiGe.Welserin1992[ 20 ]experimentallydemonstratedthestrainenhancementinstrainedSionrelaxedSiGe.J.Hoyt[ 21 ]explainedtheeectofverticaleectiveeldandchanneldopingontheenhancement 19

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Figure2-3.State-of-the-artdeviceusingseveralstresstechniques,fromChapter2in[ 19 ],c2011Springer-Verlag/Wien(withkindpermissionfromSpringerScience+BusinessMediaB.V) seenwithstrain.Inthesameyear,Rim[ 22 ]publishedhisworkexplainingthethresholdvoltage(VT)reductionthatwasexperimentallyseeninshortchannelNMOSdevices.Itodevelopedanewway[ 23 ]toincorporatestrainintothechannel.HeusedaSiNcontactetchstoplayer(CESL)toinducehighlevelsofchannelstressinn-typedevices,whichwaslatershowntobecompatibleforhighvolumemanufacturing.Thisworkwasimportantbecauseitshowedthatstraincanbeintroducedwithouthavingtoentirelyre-engineerthechannel.Pidin[ 24 ],extendedIto'sworkin2004forbothn-typeandp-typedevicesonthesamewaferwhereeachtypeofdevicehadadierentlystrainedCESL.Mayuzumi[ 25 ]reportedthatitispossibletointegratedualstresslinersintoahigh-kmetalgateprocessin2007.In2002,Thompson[ 26 ]presentedadierentsolutionforp-typedevices.ByembeddingSiGeinthesource/drainregions,highlevelsofcompressivestrainwasmadepossibleinthedevicechannel.Thompson'smethodwasimportantbecauseheshowedthatembeddedSiGestressmethodiscompatiblewithCESLstressorforp-typedevices.GhaniandChidambaramextendedthisworkinlateryears[ 3 ].TheapplicabilityofembeddedSiGe(e-SiGe)techniqueforregularSOIbyLee[ 27 ]andthinbodySOIby 20

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Zhang[ 28 ]happenedin2005.Theimportanceofe-SiGeproleengineeringwasresearchedbyOhta[ 29 ].In2007-2008,Wang[ 30 ]andAuth[ 31 ]demonstratedyetanothercompatiblemethodtoinduceadditonalstressinthechannel.Byfollowingagate-lastprocess(Fig. 2-4 )whereintherealmetalgatereplacedadummygateattheendoftheprocessow,itwasshownthatmorestrainonp-typedevicesispossible. Figure2-4.Gatelastprocessowtoincorporatehigherstressinchannel,fromChapter2in[ 19 ],c2011Springer-Verlag/Wien(withkindpermissionfromSpringerScience+BusinessMediaB.V) Stressmemorizationisanothertechniquethathasrecentlybeenestablishedasasuccessfulidea.Inthismethod,afterimplantingandcappingthegate,anadditionalannealingstepisaddedtointroducechannelstress.Thiswasrstdemonstratedforn-typedevicesbyOta[ 32 ]in2002.Chen[ 33 ]reportedlargeenhancementswiththistechniquein2004.In2007,Wei[ 34 ]showedthatwhenthestressmemorizationprocessisrepeated,additionalmobilityenhancementisobserved.SMTwasdemonstratedonagatestackwithhigh-k/metalgatebyKubicek[ 35 ]in2008.In2004,Ang[ 36 ]reportedthatusingSiCinsource/drainenhanceselectronmobility,similartowhatembeddedSiGedoesforholemobility.Liu[ 37 ]showedSiCenhancementwithimplantandsolidphaseepitaxy.Metalgateinducedstressisalsobeinginvestigated 21

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currently.Othernewtechniqueslikeintentionaldefectengineeringinthesource/drainregiontointroduceadditionalstressinthechannel[ 18 ]arealsobeinginvestigated. 2.2StrainCharacterizationPiezoresistancemeasurementshasbeenshowntobeaveryeectivetechniquetoexperimentallypredictthestraininducedperformanceenhancement.Thepiezoresistancecoecient(-coecient)isdenedasthenormalizedchangeinresistivitywithappliedstress.Whenastressisappliedonasamplewithresistivity,weseeachangeinresistivity.Perdenition,thepeizoresistancecoecientis: = ()(2{1)Theresistivityofthesamplecanbesimplyestimatedfrom=1=(qnn+pp),wheren(p)denotetheelectron(hole)mobilityandn(p)denotetheelectron(hole)concentration.UndersteadystatecurrentconditionsinaMOSFET,theelectron/holedensitiesarenearlyconstant;therefore,the-coecientcanbedirectlyrelatedtothechangeincarriermobilitywithstress.Thelineardraincurrentisrelatedtotheloweldcarriermobilityas: IDLin=W L:Qinv:VDS(2{2)whereQinv'Cox:(VGS)]TJ /F1 11.955 Tf 11.95 0 Td[(VT)(thetermsCox;VGS;VDShavetheirusualmeanings).Sincethemobilityisdirectlyproportionaltothelineardraincurrent,the-coecientgivesastraightforwardideaabouthowmuchdrivecurrentenhancementcanbeachievedunderaparticularstress.Thisisthereasonwhy-coecientsarewidelyusedintheindustrytocharacterizestraineects.In1954,Smithreported[ 38 ]therstexperimentaldataon-coecientsforn-andp-typebulkSiandGe.Thesebulk-coecientscannotbedirectlyusedforMOSFETs 22

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Table2-1.-coecientsforsiliconn-MOSFETs:2Dvs. bulk ,from[ 39 ]c2008JAP Type ltB (001)Surface <100>channel 47(7.7)22(4)50(2.3) -102 53 -49 <110>channel 32(7.4)15(6.4)47(3.2) -31 -18 -49 (110)Surface <100>channel 24(1)25(1)10(2.4) -102 53 -49 <110>channel 37(1.8)11(2.4)-7(3.9) -31 53 -49 Table2-2.-coecientsforsiliconp-MOSFETs:2Dvs. bulk ,from[ 39 ]c2008JAP Type ltB (001)Surface <100>channel 15(6.4)9(4.3)11(3.7) 6.6 -1.1 -5.5 <110>channel 71(15.6)32(7.7)16(5.8) 71.8 -66.3 5.5 (110)Surface <100>channel 31.3(1)-9.5(1)25.4(2.4) 6.6 -1.1 5.5 <110>channel 27(8.8)-5(3)25.8(2.2) 71.8 -1.1 5.5 duetothegateeldwhichconnestheelectronsnearthesurface.In2008,Min[ 39 ]reportedacomprehensivesetoftwo-dimensional(2D)inversionlayer-coecientson(001)and(110)surface,<110>and<100>channel,n-andp-typeSiMOSFETs.ThevaluesarereproducedinTable 2-1 (n-MOSFET)andTable 2-2 (p-MOSFET).Unitsarein10)]TJ /F5 7.97 Tf 6.58 0 Td[(11Pa)]TJ /F5 7.97 Tf 6.59 0 Td[(1.Inbothtablesabove,Smith'sbulk-coecientsarealsonoted(bluetext).Itcanbeclearlyseenthattheexperimental2Dinversionlayer-coecientsdonotmatchwiththebulkvaluesasmentionedabove.l;tandBrefertouniaxiallongitudinal(i.e.alongtransportdirection),uniaxialtransverse(i.e.perpendiculartotransportdirection)andbiaxial-coecientsfortheappropriatebenecialstressineachcase(i.etensileforn-MOSFETandcompressiveforp-MOSFET).Thevalueintheparenthesisrefertotheestimatedexperimentaluncertainty. 23

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Typically,the-coecientsaremeasuredbyapplyingexternalmechanicalstressonsamplesandmeasuringthechangeinresistanceofthedeviceundertest.Therearemanymethodsavailableforapplyingstress(directlyhangingweightsthatSmithfollowed,3-pointbending,cantileverbending)but4-pointwaferbendinghasbeenproventobeoneofthemoreaccuratemethods.Wu[ 40 ]usedabendingapparatustoapplyupto500MPa(whichwasthehighestatthattime).Subsequently,Sagar[ 41 ]developedaexure-based4-pointbendingsetupwhichcanbeusedtoapplyupto1GPaofuniaxialstress.Theschematicfora4-pointwaferbendingsetupisshowninFig 2-5 (sampleundertension).Thestressonthetopsurfaceofthewaferisestimatedbyusing =Eyt 2aL 2)]TJ /F1 11.955 Tf 13.15 8.08 Td[(2a 3(2{3)whereEistheYoungsmodulus,tisthethickness,yistheverticaldisplacementatthepointofcontactoftheinnerrods,andaandLareasshownFig. 2-6 showsanactualsamplebeingbentundertheappliedtensilestress. Figure2-5.4-pointwaferbending Inadditiontoa4-pointbendingsetup,acustomlowtemperaturesetuphasalsobeenusedformanyofthemeasurementsinthisdissertation.Thissetuphasthe 24

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Figure2-6.Uniaxialtensilestressappliedtoasiliconsample.Photocourtesy:Suthram[ 42 ] capabilityofbeingabletosimultaneouslyvarybothstressandtemperatureforelectricalcharacterizationofsamples.ThesetupconsistsofaLakeshorecryogenicprobestationmodiedtohousea4-pointwafersetupthatiscapableofapplyinguniaxialstress(bothtensileandcompressive)ontypical600-700msiliconwafers.Byusingliquidnitrogeninconjunctionwithtwodedicatedtemperaturecontrollers(thatsenseandcontrolthetemperatureofvariousstagesintheprobestation)andpoweredexternalheaters,weareabletocontinuouslyvarythetemperaturefrom300Kto77Kinthissetup.ADT-670diodesensorisdirectlyattachedtosamplestoaccuratelymonitorthesampletemperature.Figs. 2-7 to 2-9 showdetailsofthelowtemperaturesetup. 2.3StrainEectsonCarrierMobilityAsimplepictureforunderstandingstraineectsonbothelectronandholemobilityinSiispresentedhereasanintroductionwithoutgettingintothecomplicatedphysics.WerstdiscussbulkSiandthentheMOSFETcase,explainingwhyitisdierentfrombulk.Wediscussthetechnologicallyrelevant(001)/<110>caseonly. 25

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Figure2-7.Lowtemperaturecryogenicprobestation.Photocourtesy:UFStrainGroup Figure2-8.Lowtemperature4-pointbendingapparatus.Photocourtesy:UFStrainGroup 26

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Figure2-9.Strainapparatusinsidethecryogenicprobestation.Photocourtesy:UFStrainGroup 2.3.1N-typeBulkSiandN-MOSFETIntheconductionbandofbulkSi,therearesixdegeneratevalleyswiththeminimumenergylocatedneartheXpoint(Fig. 2-10 ).Whenanexternalstressisapplied,itcausesshiftingandsplittingofthesesubbands(2and4).On(001)typeSi,longitudinaltensilestressappliedin<110>directioncausestheenergyofthe2subbandtoshiftdownandtheenergyofthe4toshiftup.Thisresultsinelectronsrepopulatingfromthe4valleytothe2valley,asshowninFig. 2-10 .Theconductivityeectivemassinthe2valley(0.19m0)issmallerthanthatofthe4valley(0.315m0).Thustherepopulationfrom4to2causestheaverageconductivitymassin<110>directiontodecrease,therebyincreasingthecarriermobility.Inadditiontocausingmasschange,the2-4splittingalsocauseschangesinscatteringrates.TherearetwodominantscatteringmechanismsinstrainedSi,namelysurfaceroughnessscatteringandintervalleyphononscattering[ 43 ][ 44 ].Whenthe 27

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conductionbandsplits,theintervalleyscatteringratebecomeslower[ 45 ]owingtothesmallerdensity-of-states[ 46 ].Thisalsohelpstoincreasethemobility. Figure2-10.Unstrainedandstrainedn-typebulkSiliconbandstructure A2Dinversionlayerinastrainedn-MOSFEThasaverydierentresponsewithstraincomparedton-typebulkSi.Duethetheappliedgateeld,the2and4valleysarealreadysplit(i.e.non-degenerate)evenbeforeanystrainisapplied.Thedierenceinenergiesbetweenthevalleyswilldependon(a)thetransverseelectriceldand(b)thedierenceintheirout-of-planeeectivemass.For(100)surfacen-MOSFETswithoutstrain,electronsmostlyoccupythe2valley.Thus,withstress,thechangeineectivemasschangeismuchsmallerthanthatforbulkSi.Inaddition,thecarriertransportisalsoverydierentina2Dinversionlayer.Inhightransverseeldconditions(typicalfor 28

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MOSFEToperation),surfaceroughnessscatteringisthedominantscatteringmechanism.However,westillnoticeelectronmobilityenhancementintheseconditions.Ithasbeenpostulated[ 47 ]thatstrainactuallyhelpsreducetheamountofsurfaceroughnessscatteringbychangingtheinterfaceroughnessproperties.Temperaturebasedexperimentsdoneinthisdissertationprovidesclearevidenceforthisphenomenon.Thiswillbediscussedindetailinsection 2.5 2.3.2P-typeBulkSiandP-MOSFETThevalencebandminimuminbulkSiislocatedatthe)-326(point,wheretheheavy-holeandlight-holebandsaredoublydegenerate.Thespin-orbitsplit-obandislocated44meVbelowthesetwobandsanddoesnotcontributetoholetransport.InunstrainedSi,80%oftheholesoccupytheheavy-hole(HH)band,whichhasaneectivemassof0:59m0alongthe<110>direction.Thelight-hole(LH)bandhasamassof0:15m0.Underauniaxialcompressivestrainalong<110>direction,thedegeneracyoftheLH/HHbandsislifted.Thebandsbecomeheavilywarped(Fig. 2-11 )whichcasestwothingstohappen: Masschange(holerepopulation) Scatteringrate(bandsplitting)Under1GPaofuniaxialcompressivestress,thesplittingbetweenthetopandthebottombandsissmallerthantheopticalphononenergyinSi(whichis61.3meV).Therefore,theconductivityeectivemasschangeinthe<110>directionduetheappliedstressismainlyresponsiblefortheholemobilityenhancement.Similartothen-MOSFET,evenwithnostrain,thedegeneracyoftheheavyholeandthelightholebandsisliftedbythegateelectriceld,asshowninFig. 2-11 .ThesplittingbetweentheHHandtheLHbandsincreaseswithincreasinggateeld.Strainalsoactstoliftthisdegeneracy.However,dependingonthetypeofstrainapplied,theeectcanbeadditiveorsubtractive.For(001)surface,<110>channelp-MOSFETs,theeectofuniaxialcompressivestressisadditivetotheeectofthegateeld. 29

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Figure2-11.Unstrainedandstrainedp-typebulkSiliconbandstructure Forusualdeviceoperationconditions,thesumofconnement-inducedbandsplittingandstrain-inducedbandsplittingcanbebiggerthantheopticalphononenergy.Thismeansthatwecannotignorethecontributionofscatteringratechangetothe2Dcarriermoblity(likewedidforbulkp-typeSi).Simulationsindicatethatmasschangeisdominantonlyforstresseslessthan500MPa(comparedto1GPainthebulkcase).Temperaturebasedexperimentsdoneinthisdissertationverifythesendingsandisdiscussedindetailinsection 2.4 .Fig. 2-12 schematicallyshowsthestraineectsonbulkand2Dn/p-typetogetherforquickreference.Theeectsofuniaxialstresshasbeenexperimentallystudiedbynumerousauthors.Electonmobilityenhancementinn-MOSFETsuptonearly50%for1.5GPaofuniaxialtensilestressandholemobilityenhancementinp-MOSFETsuptonearly200%havebeenexperimentallyreportedwithmechanicalwaferbendingasshowninFig. 2-13 from[ 42 ].Whilebiaxialtensilestresshasbeenexperimentallyandtheoreticallyshowntobemorebenecialforenhancingelectronmobility,biaxialcompressivestressisnotasbenecialforholemobilityinthe(100)=<110>orientation.Inaddition,uniaxialtensilestressonn-MOSFETsanduniaxialcompressivestressonp-MOSFETscanbesimultaneously 30

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(a)Conductionbandunderuniaxialtension (b)Valencebandunderuniaxialcompression Figure2-12.Subbandsplittingdiagramsforholes(from[ 48 ]c2006IEEE)for1GPaappliedstressand1MV/cmelectriceldfor(001)/<110>Si.Bothstrainandquantumconnementsplitthebands. introducedondevicesinthesamewaferwiththeadditionofonlyafewprocesssteps.Thisfact,combinedwiththehigherenhancementforholemobilityhasmadeuniaxialstressthemethodofchoiceintheindustryforimprovingtransistorperformancesincethe90nmnode. 2.4PMOSLowTemperatureStudyExperimentalandtheoreticalstudiesofthestraineectsonmobilityatroomtemperaturehaveprovidedalotofusefulinsightsintothestrainenhancementofcarriermobility.Todate,temperatureeectonstrainenhancedelectronmobilityhavebeen 31

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Figure2-13.Experimentalresultsoneectofuniaxialstressonelectronandholemobility,from[ 42 ] widelyreportedwithdiscrepanciesstillexistingbetweenexperimentalandtheoreticalresults.Fortemperaturedependentholemobility,somedatahavebeenreportedforbiaxiallystresseddevices.Thereisverylittletemperaturedependentexperimentaldataforp-channelMOSFETsavailableintheliteratureforuniaxialstress.Inthissection,weexplainexperimentalresultsfromcontrolledmechanicalwaferbendingexperimentswhereweapplieduniaxialstresstop-channelMOSFETs.Usingasimpleconceptualmodel,weexplaintheobservedenhancementsoflineardraincurrentasafunctionoftemperature.Withtheconceptualmodelandsupporting6bandkpsimulations,wemodelthestrainenhancementofcarriermasstogiveinsightintothephysicalmechanismsresponsibleformobilityenhancement.Experimentsareperformedonindustrial(100)/<110>orientedSip-channelMOSFETsthatwere10mwideand10mlong.Thetransistorgatestackconsistedofaborondopedp+poly-Sigateelectrodewitha1.2nmthicknitridesSiO2gateoxide.Thewelldopingwas51017cm3n-typeinthedevice.Mechanicallongitudinalcompressivestressalongthe<110>directionisintroducedintotheMOSFETusingafour-point 32

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bendingsetupshowninFig. 2-9 .ElectricalcharacterizationwasdoneusingaKeithley4200semiconductorparameteranalyzer.Thetemperaturerangeusedinthestudywasfrom300Kto80K.TheexperimentalprocedurewefollowedisoutlinedinFig. 2-14 .ID)]TJ /F3 11.955 Tf 12.12 0 Td[(VGandC)]TJ /F3 11.955 Tf 12.12 0 Td[(Vmeasurementsweretakenatroomtemperaturewithnoappliedstressforuseasbaselinedata.Next,uniaxialstresswasappliedtothesampleatroomtemperatureandthelineardraincurrent(IDlin)wasmonitored.Atthesamestresslevel,temperaturewasloweredfrom300Kto80Kandatregularintervals,theIDlinandC)]TJ /F3 11.955 Tf 11.61 0 Td[(Vweremonitored.Oncethetemperaturerunwascomplete,thestresslevelwasincreasedandthewholemeasurementprocesswasrepeated. Figure2-14.Experimentalprocedureforlowtemperaturestressmeasurements .Thelinearcurrentenhancementwasextractedasfunctionoftemperatureforthreedierentstresslevels(20,45and60MPa),andisshowninFig. 2-15 .Atroomtemperature,theenhancementseenwiththeappliedstresswasconsistentwiththewell-established)]TJ /F1 11.955 Tf 9.3 0 Td[(coecientnumbersthatwasdiscussedinearliersections.However,asseeninthegure,themeasuredenhancementwithstressishigheratlowertemperaturescomparedtoroomtemperature.Ourexperimentalresultcontrastswithbiaxiallystresseddevicesinwhichmobilityenhancementdecreasesorremainsconstantatlowertemperatures[ 49 ][ 50 ].Tounderstandtheexperimentalresults,werstprovideasimpleconceptualmodelusingFigs. 2-16 to 2-20 ,followedbysupportingkpsimulationsthatvalidateourconceptualmodel. 33

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Figure2-15.Experimentalresultsforp-channelMOSFETsfromlowtemperaturestresexperiments Itwasshownin[ 51 ]withdetailedbandstructuresimulationsthatforuniaxialstressbelow500MPa,thestrain-enhancedmobilityresultsmainlyfrombandwarpingatalltemperatures.Figs. 2-16 and 2-16 reproducedfrom[ 51 ]showtheindividualcontributionsofmasschangeandscatteringchangetothemobilityenhancement.Itisclearlyseenthatforsmallstresslevels,mostofthemobilityenhancementisduetothemasschangeresultingfromstrain-inducedbandwarping.Thebandwarpingcreatesstrongin-planeenergyanisotropyinthetopbandwithalighthole-likemassoccurringalongthe<110>channeldirection,especiallynearthebandedge.Connedtwodimensionalenergycontoursfrom[ 19 ]areshownininFig. 2-18 forunstressed(leftmost)tohighlystressed(rightmost)cases.Theenergycontouriscomposedoffourwingsofwhichtwolongitudinalwingscontributelargerconductivityeectivemassandtwotransversewingscontributesmallermass.Thetotalconductivityeectivemassistheaveragecontributionbythefourwings.Asseeninthegure,withtheapplicationof 34

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Figure2-16.kpsimulationresultsshowingcontributionofscatteringchangeinoverallmobilityenhancementfrom[ 52 ].Forlowstresslevels,thecontributionfromscatteringchangeisminimal. Figure2-17.kpsimulationresultsshowingcontributionofmasschangeinoverallmobilityenhancementfrom[ 52 ].Forstresslevelsuptoa500MPa,mostofthestraininducedmobilityenhancmentcomesmasschange. 35

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Figure2-18.Warpingofthevalencebandwithuniaxialcompressivestrainfrom[ 19 ]from[ 19 ],c2011Springer-Verlag/Wien(withkindpermissionfromSpringerScience+BusinessMediaB.V).Withincreasingcompressivestrain,thecurvatureinthe<110>directionincreases,reducingthetransportmassinthisdirection. compressivestress,thecurvatureofthebandinthe<110>directionisgreatlyaected.Thestatesnearthetopbandedgehavealowerconductivitymassduetothechangeincurvature.Intheabsenceofstress,lowertemperaturedecreasestheFermienergylevelandsharpenstheFermidistributionstep,resultinginmoreholesconcentratedatstatesnearthetopbandedgeforaspecicelectricaleldasshownconceptuallyinFig. 2-19 .Inthepresenceofstrain,thebandstructurenearthe-pointiswarpedatbothroomandlowtemperatures.Atlowertemperatures,duetotheadditionalrepopulationofholestothetopbandinadditiontotheeectofstrain,agreaterpercentageofholesisnowaectedbythestraininducedbandwarping.ThisistheunderlyingcausefortheincreasedmobilityenhancementseeninourexperimentalworkinFig. 2-15 ,notingthattheappliedstressrangeislessthan100MPa.Tovalidateourconceptualmodel,strain-inducedchangesintheSisubbandenergylevel,valencebandwarping,andrepopulationbetweenthetopthreesubbandsaremodeledusingthekpmethod.Usingtheobtainedsubbandstructure,the2-Ddensityofstatesofeachsubbandisevaluatednumerically.Theholemobilityis 36

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determinedaftercalculatingtheholeconductivityeectivemassamdtheholescatteringrateincludingphononscatteringandsurfaceroughnessscatteringmechanisms.Chargedandneutralimpurityscatteringisneglectedsinceonlyhighelectriceld(0.8MV/cm)isconsidered.Moredetailsofthesimulationworkcanbefoundin[ 53 ]. Figure2-19.Conceptualgureshowingholerepopulationwithoutstrain.Asthetemperatureisreduced,holesrepopulatetoheavyholeband. Figure2-20.Conceptualgureshowingholerepopulationinthepresenceofstrain.Thebandsarealreadysplitandwarpedduetostrain.Atlowtemperature,agreaterpercentageoftheholesareaectedbybandwarping. 37

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ThesimulatedchangeintheconductivitymasschangewithstressatvarioustemperaturesisshowninFig. 2-21 .Theeectivemassisseentobedecreasingatlowertemperaturesforagivenstress.ThemassreductionisveryclosetotheexperimentallyobservedchangeinholemobilityinFig. 2-15 .Thesesimulationresultsareincompletecontrasttothelittlegainobservedinholemobilityforbiaxiallystresseddevicesin[ 49 ]and[ 50 ]. Figure2-21.SimulatedconductivitymasschangeforstraininducedmasschangeatroomtemperatureconrmingtheexperimentalresultsinFig.( 2-15 ). Insummary,withappliedcompressiveuniaxialstress,lowertemperaturescausemoreholestopopulatestatesnearthebandedge.Agreaterpercentageoftheholesareaectedbythestraininducedbandwarping,whichresultsinalighterholeconductivitymassinthechanneldirection.Theexperimentaldataandmodelingsuggestthatsmalllevelsofuniaxialstressdoesnotsignicantlyaltertheholescatteringrates. 38

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2.5NMOSLowTemperatureStudyTheexperimentalprocedureoutlinedinFig. 2-14 wasusedtoperformasimilartemperaturestudyonuniaxiallystressedn-channelplanarMOSFETs.Thedevicesusedwerefromthesamesampleasintheprevioussection.Thelinearcurrentenhancementextractedasfunctionoftemperaturefortwodierentstresslevels(45and80MPa)isshowninFig. 2-15 .Atroomtemperature,theenhancementseenwiththeappliedstresswasconsistentwiththewell-established)]TJ /F1 11.955 Tf 9.3 0 Td[(coecientnumbersthatwasdiscussedinearliersections.Similartotheresultsforp-typedevices,themeasuredenhancementwithstressishigheratlowertemperaturescomparedtoroomtemperature.Theseexperimentalresultcontrastswithpreviouslyreportedexperimentaldatafromprocess-stressedn-typeMOSFETs. Figure2-22.Experimentalresultsforn-channelMOSFETsfromlowtemperaturestressexperiments Thecompletephysicalmechanismofthemobilityenhancement,especiallyunderhightransverseeldconditionsfortensilestrainedn-MOSFETisstillnotclear.In2002,basedonverydetailedsimulationsFischetti[ 54 ]theorizedthatthehighlevelsof 39

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electronmobilityenhancementobservedwithtensilestresscouldonlybeexplainedbyanad-hocassumptionthatstrainmodulatestheroughnessoftheSi-SiO2interface.In2008,usinghighresolutionatomicforcemicroscopymeasurementsBonnoreported[ 55 ]thatthesurfacemorphologyparametersnamely,RMSheight()andthecorrelationlength(),arebothreducedwhenabiaxialtensilestrain(0.8%)isapplied.In2009,Zhao[ 56 ]reportedsimilarresultsusingtransmissionelectronmicroscopymeasurementsonstrainedSilayers.Theappliedrangeofsrainwasbetween0.4%and1.6%.Thereisverylimitedworkavailableinthepublishedliteratureonactualtransistorsinvestigatingthissurfaceroughnessreductionlevelwithapplieduniaxialstress.Veryrecently,Cousinreported[ 57 ]thereductionoftheroughnessparametersandwithanexperimentalstudyonuniaxiallyloadedstrainedSibeam.AMEMSbasedsystemwasusedtouniaxiallyloadtheSibeam.TheauthorsusedaveryhighresolutionAFMmeasurementsetuptotoprobethesurfaceroughnessofthetheSibeambothwithandwithoutstress.Areasof750nm2and95nm2(fromwithinthebigger750nm2)wereprobedunderstrainlevelsrangingbetween0:2%and2:8%(correspondingtonearly340MPaand2.5GPaofuniaxialtensilestress).Theauthorsreportareductionfrom0.29nm(unstrained)to0.07nm(2.8%strain)andareductionfrom5.3nm(unstrained)to4.3nm(2.8%strain).AFMmeasurementsperformedbyourresearchgroupconrmtheresultspublishedin[ 57 ].Roh[ 58 ]monitoredsurfacemorphologychangesof(100)Siundercontrolledbendingmeasurementsona4-pointbendingsetupcombinedwithAFMcharacterization.Thebenetofthisstudy,likeCousin's,isthatroughnessparametersareextractedfromthesamesamplewiththeonlyvariablebeingtheexternalstressapplied.InRoh'swork,ap-typeSi(100)sampleswascleanedandalayerof15nmthickSiO2isgrownusingwetoxidation.TheSiO2layerisremovedusingawetetchandAFM/strainmeasurementswereperformeduptostresslevelsof200MPa.Byanalyzingtherecordedpowerspectraldensityoftheroughnessvariationalongthe<110>directiononthesample,theRMS 40

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heightparameterofthesurfaceroughnesswasextractedwithandwithoutstress.Fig. 2-23 showstheresultsfromthisanalysis. Figure2-23.Amplitudechangesofthemicro-scaleroughnessofp-SiunderuniaxialtensionfromAFM/strainmeasurementsfrom[ 58 ]. Thenormalizedchangeinroughnessamplitudefromourgroup'sworkis1.7%for100MPa,anditmatchesreasonablywellwithCousin'srecentworkin[ 57 ].Theresultsofthestressexperimentsatlowtemperatureseemtosupportthistheory.Atlowertemperatures,thecarriersintheinversionlayerareconnedclosertotheinterface.Ifuniaxialstressreducesthesurfaceroughnessscatteringratebymakingtheinterfacesmoother,itcouldexplaintheincreasedenhancementsseeninFig. 2-22 ,especiallyconsideringthehighoverdriveconditionsunderwhichthecurrentenhancementwascalculated.Thelowtemperatureexperimentalresults,inconjunctionwithresultsofAFMmeasurementsfromourgroupandtherecentreportsonreducedsurfaceroughnessparametersreportedbyCousinin[ 57 ]seemtovalidateFischetti'sad-hocassumptionforsurfaceroughnessscatteringreductionwithtensilestrain. 41

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2.6SummaryInthischapter,wepresentedabriefreviewoftheeectsofstrainonthesiliconbandstructure.LowtemperaturestressexperimentswereperformedonlongchannelplanarbulkMOSFETs(bothn-typeandp-type)togiveinsightintothefundamentalsofuniaxialstrain.Forbothn-typeandp-typeMOSFETs,weobservedthat,foragivenstresslevel,themobilityenhancementwithstressincreasesatlowertemperaturecomparedtoroomtemperature.Fromtheexperimentalresultsforp-typeMOSFETs,weveriedthatthecompressivestrain-inducedreductionofconductivityeectivemassofholesintheinversionlayeristhemajorcontributortothemobilityenhancementatlowstresslevels.Forn-typeMOSFETs,ourexperimentalresults,inconjunctionwithotherrecentexperimentalworksonAFMbasedsurfaceroughnessmeasurements,supportFischetti'smodelofad-hocsurfaceroughnessreductionwithtensilestrain. 42

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CHAPTER3STRAINEFFECTSINSOIFINFETSComplementaryMOSdevicesfabricatedonbulksiliconsubstrateshavebeenthesemiconductorindustrysworkhorsetechnologyfordecades.Bulkdevicesaremadeonwafersthatareapproximately600-800mthick,butonlythetop1or2micronsareusedforactivedevices.TokeepupwiththepaceofMoore'slawscaling,thegatelengthofthetransistorsscaleapproximatelyeverytwoyears.However,asthedevicesgetsmaller,theparasiticsbecomemoreandmoredetrimentalforperformance.Specically,thecapacitancebetweenthesource/drainandthesubstrateincreaseswithsubstratedoping.Sourceanddraincapacitancehastwocomponents,namely,thecapacitanceassociatedwiththeassociatedp-njunctions,andthecapacitancebetweenthejunctionsandtheheavilydopedchannel-stopregionthatisformedbelowtheeldoxideisolatingadjacentdevices.LatchupisalsoanotherissuethatarisesduetounwantedtriggeringofthePNPNstructureinherenttoCMOSdevices.Severalfabricationtechniquesareutilizedtoreducetheseparasiticsinmoderntechnologies.Forexample,latchupisreducedbyusingdeeptrenchisolationorbyusingepitaxialsubstratesandsource/drainjunctionareacanbeminimizedbuyusinglocalinterconnects.Suchmethodsnecessitateadditionalprocessingstepsthatimpactsnotonlycost,butalsoyieldofthedevices.Tosolvesomeoftheseproblems,theideaofmakingtransistorsinathinsiliconlmwhichisformedonainsulatingsubstratewasformulateddecadesago.Suchtransistorswouldbethreeterminaldevicesonly,withoutasubstratecontact.Thecompletedielectricisolationofadjacentdevicesissupposedtodrasticallyreducetheeectsduetounwantedparasiticcapacitances.Silicon-on-Insulator(SOI)CMOStechnologyisalsobenecialforthereasonthatrealizingshallowjunctionsiscomparativelyeasiercomparedtobulktechnology.SomeoftheotherbenetsofSOItechnologyareimprovedradiationhardnessandimprovedSCEs(sharpersubthresholdslopeandimprovedtransconductance). 43

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SOItechnologyhasfoundgoodcommercialsuccesstoday.IBMandAMDmakemicroprocessorswithsub-100nmtransistorsusingadvancedSOItechnologies. 3.1EvolutionofSOIDevicesScalingofbulktransistorbeyond20nmnodeisprojectedtobeabigchallengeduetodicultiesincontrollingshortchanneleects(SCEs)thatconsiderablydegradedeviceperformance.SCEsarisewhengatecontrolofthechannelregionisaectedbyelectriceldlinesemanatingfromsource/drainregionspropagatingthroughtheassociateddepletionregionsoneachside.Onewaytocontroltheinuenceoftheeldlinesonthechannelregionostoincreasethedopinglevelinthechannelregion.However,thismethodisnotafeasiblesolutionforshortchanneldevicessincemobilityisadverselyaected.InafullydepletedSOI(FDSOI)device,mostoftheeldlinespropagatetroughtheburiedoxide(BOX)beforereachingthechannelregionandthustheirinuenceondeviceoperationisminimized.Severaltechniqueshavebeenproposedtofurtherreducetheencroachingelectriceldfromsource/drainregions.Useofagroundplanebeneathathinnerburiedoxidelayerwasproposed[ 59 ]withtheideaofterminatingtheE-eldlinesonthegroundplaneinsteadofthechannel.However,thismethodhadsomeissues,mostnotablyanincreasedbodyeect.AverycomprehensivereviewofSCEcontrolandperformancebenettrade-osinFD-SOIdeviceswasreportedbyFossumandcoworkers[ 60 ][ 61 ][ 62 ][ 63 ][ 64 ][ 65 ][ 66 ].Theideaofusinganadditional,2ndgatetoreducetheSCEshasbeenprevalentasearlyas1984[ 67 ].Withthisdoublegatedarchitecture,thethresholdvoltageroll-owasshowntobereducedinshortchanneldevices.Typically,inadouble-gatedevice,bothgatesaretiedtogether,andtheelectriceldlinesfromsource/drainunderneaththedeviceterminateonthebottomgateelectrodeanddonotreachthechannelregion.WiththeuseofadoublegateandathinSilm,theSCEsareconsiderablyreducedcomparedtoabulkdevice.Fig. 3-1 from[ 68 ]showstheevolutionofSOItechnologies,frompartiallydepleted(thicklm)andfullydepleted(thinlm)singlegateSOIdevicestofullydepleted 44

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Figure3-1.TimelineofSOItechnologies,from[ 68 ] 45

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double/triplegatestructuresthatarebecomingcommerciallyavailabletoday.PartiallydepletedSOIdevices(PDSOI)wereinitiallyusedinnicheapplicationslikeradiation-hardeneddevicesforouterspaceuse.Inthe2000s,PDSOIwereusedincommerciallyavailablehighperformancemicroprocessors.FDSOItechnologyiscurrentlyusedinanumberofsectorsincludinglow-voltage,low-powerapplicationstoRFintegratedhighperformanceapplications.Therstfabricateddouble-gateSOIMOSFETwasthe\fullyDEpletedLean-channelTrAnsistortheDELTAtransistorin1990[ 69 ].TwoimportantfeaturesoftheDELTAMOSFETmadeitasignicantstructureworthconsideration:(1)Self-alignedfrontandbackgateand(2)channelsformedonthesidewallofthesiliconbody/n.Doublegate(FinFET,MIGFET),Triplegate(TrigateFET,=FET)andSurroundinggate(Cylindrical,Gate-all-around,Multi-bridged/stackeddevices)thatareinvestigatedtodayallevolvedfromtheDELTAtransistorstructure.AswithplanarMOSFETs,SOIdevicesalsohavesomescalingrulesrelatedtogoodSCEcontrol.OneofthekeyscalingrulerelatestotheSi-lmthickness/gatelengthratio[ 70 ].AsshowninFig. 3-2 ,thisratiodependsonthedevicearchitecturechosen.Inthisstudy,wefocusonadoublegateFinFETstructuresinceitisonetheverycompetitivealternativesavailabletodayforextendingthescalingroadmap. 3.2FinFETTechnologyTherstfabricatedn-channelFinFETswithathinnatopaBOXlayerwerereportedin1998fromHisamoto'sgroup[ 71 ]andtherstn-channelFinFETswasreportedbyDr.ChenmingHu'sgroupfromUCBerkeley[ 72 ]in1999.Deviceswithchannellengthsaslowas17nmand18nmwithaWfintoHfinaspectratiosof2=5inthesepapers.Inearly2000s,therewasvestedinterestfromtheindustryinthesedevicesforpotentialplanarMOSFETreplacementforadvancednodes.OneofthemainchallengesthattheFinFETtechnologyfacedwasrelatedtolithography,tohavenswithanaspectratioof<2=3toadequatelysuppressDIBL.Sub-lithographicnpatterningtechniqueswas 46

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Figure3-2.ScalingrulesforbulkMOSFETalternatives developedtomeetthischallenge.Twospacerlithographytechniques,namelySidewallImageTransfer(SIT)[ 73 ]andSelf-AlignedDoublePatterning(SADP)[ 74 ]wereamongstthekeytechniquesthatmadehighaspectrationspossibleforhighvolumeproduction.Oneofmainbenetsofspacerlithographyisthatitprovidesabettercontrolofthencriticaldimensionandauniformnwidth[ 75 ].SpacerdenedFinFETswerereportedbyChenmingHu'sgroup[ 76 ]thathadaWfin=Hfinratioof2/3withexcellento-statecontrolandanNMOSdrivecurrentof650A=matagatelengthof60nm.AMDreported10nmgatelengthFinFETsin2002[ 77 ].FurtherrenementsinFinFETprocessingrelatedtoimprovingcarriermobilitiesandthresholdvoltageadjustmentwasreportedin2002[ 78 ].Inthispaper,Hydrogenannealingofthenswasshowntoimprovethesmoothnessofthensidewall,and 47

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selectiveimplantationofMolybdenumwasusedtoengineerthegateworkfunctionforthresholdvoltagecontrol.ReliabilitystudiesonCMOSFinFETs[ 79 ]showedthatanarrowernimprovedhotcarrierimmunityandthatthehotcarrierlifetimewasimprovedwithsmootherns.Tri-gateFinFETswithanadditionalchannelatthetopsurfaceofthenswitha1:1aspectratioat60nmchannellengthswerereportedbyIntelin2003[ 80 ].However,Fossum[ 81 ]showedthat1:1aspectratioFinFETswillnothavetherequiredSCEcontrol.TrivediandFossumalsohighlighted[ 82 ]theimportanceofanunderlappedarchitectureshowingitsimportanceincontrollingSCEs.ThedoublegateFinFEThasahardmaskonthetopofthenanddoesnotrequireaselectiveetchforthegate,whichreducesprocesscomplexity.However,fringinggatecapacitanceislowerinatri-gatestructuresincethereisacurrentcarryingchannelatthetopnsurface.Freescalereported[ 83 ]thefeasibilityofaback-gatedFET,whereonegateisusedforswitchingthetransistoronando,andtheothergatewasusedforcontrollingthethresholdvoltage.In2004,Samsungreported[ 84 ]lowcostFinFETsmadeonbulksiliconwafersat90nmchannellength.ThesedeviceshadalowdefectdensitywithouttheoatingbodyeectassociatedwithSOIdevices.Inaddition,theheattransferratewasmuchhighercomparedtoSOIFinFETsimprovingthermalperformanceandsinceitwasmadeonbulkwafers,itwascompatiblewithstandardCMOSprocess.ThensinSamsung'sworkhadatotalheightof100nm.ArecentpaperfromIBM[ 85 ]providesanin-depthreviewofthetrade-osbetweenbulkvsSOIFinFET.SinceFinFETsaretypicallymadewithanunderlap,SiGeembeddedinthesource/drainregionswasinvestigatedtowardsreducingthecontactresistanceforinp-channelFinFETs.SiGehasalowerSchottkybarrierheightforholesthanSi.Nearlt25%improvementinsaturationcurrentwasshown[ 86 ]withembeddedSiGe,partlyduetoparasiticresistancereductionandpartlyduetotheaddedcompressivestraininthedevicechannel.Anothertechniquetoreducethecontactresistancethatisapplicableforbothtypeofcarriersintheuseofraisedsource/drainregions.Thisisdonebywidening 48

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thenoutsidethegateareabyselectiveepitaxialgrowth.Theadvantageoftheraisedsourceanddrainstructureistwofold;itreducestheparasiticsource/drainresistancewhilesimultaneouslyincreasingthecontactareaformetalcontacts.Theraisedsource/drainstructurehowevercomeswithsomeprocesscomplicationsrelatedtoremovalofthesidewallspacerspriortoepitaxialgrowthofsource/drainregions.Kedzierski[ 87 ]discussedonepossiblemethodtoovercomethisprobleminvolvingadryetchprocessforspacerremoval.Kaneko[ 88 ]proposedasidewalltransferprocesstechniquewhereintheformationofparasiticnsidewallspaceriscompletelyavoided.Anotherissuerelatedtotheselectiveepitaxialgrowthistheformationoffacetsonthedierentsurfacesofthen.Depedingonwhetherthenhasa(110)or(100)sidewall,theshapeofthefacetswillbedierent.For(100)sidewallsurface,selectiveepitaxialgrowthresultsinarectangularshapedn,butfor(110)sidewallns,thefacetwillbecomediamondshapedduetoadditionalgrowthin45planes.Bridgingofgatemetalandthesefacetsshouldbeavoidedforimprovingdeviceyield.StrainisanintegralcomponentofCMOStechnologytoday,andeectsofprocessinducedstrainonFinFETperformancewithdierentstressors(strainedSiNlayers,strainedSOIsubstrates,embdeedSiGesource/drainsandstrainfrommetalgate)havebeenreportedbyseveralauthors[ 89 ][ 86 ][ 90 ][ 91 ][ 92 ][ 93 ][ 94 ][ 95 ][ 96 ][ 97 ][ 98 ][ 99 ].Inourwork,weinvestigatetheeectsofprocessinducedSOIFinFETsusingtwomethods(1)uniaxialstrainintroducedintothechannelbyCESLlayersinSin-channelandp-channelFinFETsand(2)biaxialstrainedSi/SiGe-gatep-channelFinFETs. 3.3StudyonStrainEectsinFinFETsInorderformulti-gateMOSFETtechnologytobesuccessful,itmustbecompatiblewithprovenperformanceboosterslikeuniaxialstrain.UnderstandingstraineectsinFinFETsischallengingduetotworeasons,bothrelatedtothe3Dnatureofthedevice.Forthetechnologicallyrelevant(100)Si,thesurfaceorientationandchannelowdirectionsofthefabricatedFinFETcanbeeither(110)/<110>or(100)/<100>.Other 49

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surface/channelcombinationsarepossibleifthestartingsurfaceorientationisdierent.InSi,eachsurfaceorientationandchanneldirectionhasadierentresponsetoagivenstress.ForaproperunderstandingoftheimpactofstressinaFinFET,afaircomparisonwithanidenticalsurface/channelorientationinaplanardeviceisnecessary.Also,dependingonthenatureofappliedstress(processinducedvsmechanical),thestresstransfermechanismcouldbeverydierentandneedstobeproperlyunderstood. 3.3.1DeviceDetailsNeutralstress(100)intrinsicSOIsubstrateswerepatternedusing193nmlithographytoproducearraysofnswithanaspectratioof2:1.Hafniumbasedhigh-k(2nm)andmidgapmetalgate(10nm)withpolycapformedthegatestack.Spacerdepositionimmediatelyfollowedgatepatterningwithoutextensionimplants.nMOSandpMOSdeviceswereformedbyseparateimplantandannealprocesseswithNibasedsalicidecompletingthefront-endprocessing.Si3N4lmsservedascontactetchstoplayers.Twotypesoflmswereused,withtensileandcompressivelmstressesof1.1GPaand-1.4GParespectivelyasdeterminedbywaferbowmeasurement.StandardBEOLusingTEOSILD,Tungstencontactplugs,andAlCumetalizationcompletedthefabricationsequence(Fig. 3-3 andFig. 3-4 ).Uniaxialstresswasappliedusingaexurebased4-ptbendingapparatusdescribedinearlierchapters.Fig. 3-5 andFig. 3-6 areTEMimagesofthefabricateddevicesshownwidht/nheight/gatelengthdetails.ThehardmaskonthetopofthenispartiallyliftedasseeninFig. 3-6 .Wespeculatethedevicecurrentissomewhathigherthanastrictdoublegatedeviceduetoconductionfromthepartofthetopsurfaceofthen,butsincetheaspectratiois2:1,theincreaseisnotmuch.TheSiGe/Si-gatetransistorswereformedbystandardgate-rstCMOSow.Thin(15nm)epitaxialSiGe0:3isselectivelygrownbychemicalvapordepositiononSinsthatwereetchedonSOIsubstrates.Uniformepitaxyon(110)and(100)orientednsisobservedasshowninFig. 3-7 .HfSiOxdielectricandTiNmetalgatewasdepositedusingatomiclayerdeposition.Both(110)and(100)orientednsshowgreaterthan90%gate 50

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Figure3-3.KeystepsinsiliconFinFETfabricationprocess[ 1 ]c2009IEEE Figure3-4.SiFinFETProcessFlow[ 1 ]c2009IEEE 51

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coverage.Theoverallheightofthenswas95nm,withtheSicoreofthenbeing80nmtall.Finwidthwas30nmfortheSicoreand60nmoverall.ThechannelofthedevicewasintheSiGeoutershellonthreesidesofthen. Figure3-5.DetailsoffabricatedsiliconFinFET[ 1 ]c2009IEEE Figure3-6.FinFETTEMcrosssectionsshowing(a)gatecrosssectionthroughnand(b)Fincrosssectionthroughgate[ 1 ]c2009IEEE 52

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Figure3-7.Fabricated-gatep-channelSi/SiGeFinFETs.Thedierentfacetsareclearlyseenforthe(110)and(100)sidewalls[ 100 ]c2009IEEE. 3.3.2ExperimentalResultsforSiFinFETsMeasurementresultsfromelectricalcharacterizationofthefabricatedSiFinFETsareshowninFig. 3-8 throughFig. 3-15 .SiFinFETsusedinthisstudyhaveminimumgatelengthof32nmandnwidthof18nm.Drivecurrentsof725A=mand660A=mwereachieved(Fig. 3-8 )fornMOSandpMOSdevicesrespectivelyatIOFFlevelsoflessthan1nA/m,showingexcellentLowOperatingPower(LOP)characteristics(Fig. 3-11 ).SymmetricVTH,lowDIBL,andnearidealsub-thresholdslopeobservedareattributedtotheunder-lap(UL)devicestructureforbothunstrainedandCESLstraineddevicesasseeninFig. 3-10 .Longchannelmobilitieswasextractedusinglongchannelcurrentandcapacitancemeasurementsshowexpectedresultsforthe(110)sidewallsurfaceasseeninFig. 3-12 .Additionally,thereisnoindicationofSCEorgateleakage(JG)degradationasafunctionoftheCESLstresspolarityasseeninFig. 3-13 .Ion-Ioffcomparisonofn-channelandp-channeldevicesbetweenwaferswithtensileandcompressiveCESLshowsabout17%enhancementinsaturationcurrentatshortchannellengths(Figs. 3-14 and 3-15 ).Fig. 3-16 showstheimpactofthenthickness/gatelengthscalingrulesthatwasexplainedin[ 81 ]andshowninFig. 3-2 .ThisgurepresentstherawID)]TJ /F3 11.955 Tf 12.64 0 Td[(VGdatafor 53

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Figure3-8.MeasuredID-VGcharacteristicsof30nmgatelengthn-andp-channelFinFETs[ 1 ]c2009IEEE Figure3-9.MeasuredVTandDIBLvariationasfunctionofchannellengthshowinglowVTroll-oandwellcontrolledDIBLresponseforunstraineddevices[ 1 ]c2009IEEE. 54

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Figure3-10.ComparisonofVTandDIBLbetweenunstrainedandCESLstrainedFinFETs[ 1 ]c2009IEEE. Figure3-11.MeasuredION-IOFFcharacteristicsshowingexcellento-statebehavior[ 1 ]c2009IEEE 55

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Figure3-12.Measuredlong-channelelectronandholemobilitycomparedtouniversal(100)mobilitycurves[ 1 ]c2009IEEE. Figure3-13.MeasuredgateleakagecharacteristicsofSiFinFETdeviceswithandwithoutstress[ 1 ]c2009IEEE. 56

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Figure3-14.n-channelION-IOFFplotshowingimprovedshortchannelcurrent[ 1 ]c2009IEEE. Figure3-15.ION-IOFFplotshowingsubstantiallyimprovedshortchannelcurrentp-channeldeviceswithcompressiveCESL[ 1 ]c2009IEEE. 57

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32nmchannellengthp-typeFinFETsforfourdierentnwidths.Asthenwidthisincreasedfrom20nmto100nm,theprogressivelossofgatecontrolwithincreaseinthetSi=LGratiofrom2/3tonearly3isclearlyseen.Thesubsthresholdslopedegradesandthesource-to-drainleakagecurrentincreasesquickly,evenwithunderlappedarchitecture. (a) (b) (c) (d) Figure3-16.EvolutionofID-VGcharacteristicsforincreasingnwidthsshowinglossofgatecontrol. 58

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3.3.2.1UnderstandingFinFETRSDExtractingtheparasiticRSDinFinFETsiscomplicatedduetothepresenceofunderlapinthesedevices.Theunderlapregionsgetfullyinvertedonlyunderhighoverdriveconditions,andthereforecontributetotheparasiticresistanceofthedevice.Sincethisresistanceisinherentlylinkedtothechannelresistance,anewmethodologywasdevelopedtoextracttheRSDinthesedevices.WestartwiththestandardMOSFETintrinsiclinearcurrentequation IDlin=W LCox(VG)]TJ /F3 11.955 Tf 11.95 0 Td[(VTH)VD(3{1)Intheaboveequation,mobilityisafunctionoftheoverdrivevoltage.Thetotaldeviceresistanceconsistsoftwoparts:thechannelresistanceandtheparasiticsource/drainresistance. RTOT=RCH+RSD(3{2)ThetotalresistancecanbesimplyobtainedfromEq. 3{1 as RTOT=VD IDlin=L W1 1 VG)]TJ /F3 11.955 Tf 11.95 0 Td[(VTH(3{3)Ifagoodrangeofchannellengthsareavailable,oneofthesimplestmethodstoextracttheparasiticRSDistosimplyplotthetotalresistance(inlinearregime)asafunctionofchannellengthforafamilyofoverdrivevoltages.Ifthemobilitydependenceontheoverdrivevoltagecanbeneglected,thentheseplotswillbelinearwillintersectsomewhereinthe1stquadrant.They-valueattheintersectionpointgivestheRSDandthex-valueattheinterceptgivesthedierencebetweenthemasklengthandtheactualchannellength.WhilethismethodgivesaroughideaofRSDforplanardevices,usingthismethodforextractingtheparasiticresistancefromshortchanneldevicesiswroughtwithproblems.Notonlydoesthemobilitystronglydependonoverdrivevoltage,italsodegradeswithchannellength.ThesetwoeectsmaketheRTOT-Lplotsnon-linear 59

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atshortchannellengthsandmakesitimpossibletondasingleintersectionpointfordierentoverdriveconditionsinevenforplanardevices.ForFinFETs,theproblemiscompoundedduetotheunderlapeect.Whilethespacerwidthisknownfromprocessdata,theactualunderlaplengthafterdeviceformationisnotaccuratelyknown.Inaddition,theunderlapregionsbecomesprogressivelymoreinvertedasthegatevoltageonaFinFETisincreased,whichmakesmodelingtheunderlapresistanceaverycomplextask.Whilethereareseveralalternativemethodstoextracttheparasiticresistancemoreaccuratelyinplanardevicesliketheshift-and-ratiomethod[ 101 ],Y-functionmethod[ 102 ]andparasiticjunctioncurrentmeasurementmethod[ 103 ]tonameafew,forFinFETsthereareonlyfewmethodsavailableinliterature,forexample[ 104 ][ 105 ][ 106 ].SomeofthesetrytoextendtheY-functionmethodforFinFETsbringinginempiricalparametersandsomeofthemignoretheeectofunderlap.Thereforetounderstandourexperimentalresults,wehaveformulatedanewtechniquetoinvestigatewhetherthestrainedCESLhadanyimpactonthedeviceRSD.LetV0DandV0GdenotethevoltagesattheterminalsoftheintrinsicMOSFET.ThesevoltagesarerelatedtotheexternalappliedvoltagesVDandVGas V0D=VD)]TJ /F3 11.955 Tf 11.95 0 Td[(IDRSD (3{4a) V0G=VG)]TJ /F1 11.955 Tf 13.15 8.09 Td[(1 2IDRSD (3{4b) ThelinearcurrentintheintrinsicMOSFETis IDlin=W LCox(V0G)]TJ /F3 11.955 Tf 11.95 0 Td[(VTH)V0D(3{5)Thechannelresistanceis RCH=L W1 1 V0G)]TJ /F3 11.955 Tf 11.96 0 Td[(VTH(3{6) 60

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WeconsiderthatmobilityisafunctionoftheoverdrivevoltrageV0G)]TJ /F3 11.955 Tf 12.09 0 Td[(VTH.Thetotalresistanceis RTOT=L W1 1 V0G)]TJ /F3 11.955 Tf 11.96 0 Td[(VTH+RSD(3{7)Nowconsiderthederivativeoftotalresistancew.r.ttothetermVG)]TJ /F3 11.955 Tf 12.66 0 Td[(VTH.Usingchainrule,wecanshowthat @RTOT @(VG)]TJ /F3 11.955 Tf 11.96 0 Td[(VTH)=@RCH @(V0G)]TJ /F3 11.955 Tf 11.96 0 Td[(VTH)@(V0G)]TJ /F3 11.955 Tf 11.95 0 Td[(VTH) @(VG)]TJ /F3 11.955 Tf 11.95 0 Td[(VTH)+@RSD @(VG)]TJ /F3 11.955 Tf 11.96 0 Td[(VTH)(3{8)Wewillrstevaluatetheterm@(V0G)]TJ /F3 11.955 Tf 11.96 0 Td[(VTH) @(VG)]TJ /F3 11.955 Tf 11.96 0 Td[(VTH).FromEq. 3{4 VG)]TJ /F3 11.955 Tf 11.95 0 Td[(VTH=V0G)]TJ /F3 11.955 Tf 11.96 0 Td[(VTH+1 2IDRSD (3{9) @(VG)]TJ /F3 11.955 Tf 11.96 0 Td[(VTH) @(V0G)]TJ /F3 11.955 Tf 11.96 0 Td[(VTH)=1+1 2 ID@RSD @(V0G)]TJ /F3 11.955 Tf 11.95 0 Td[(VTH)+RSD@ID @(V0G)]TJ /F3 11.955 Tf 11.95 0 Td[(VTH)! (3{10) =1+1 2 ID@RSD @(V0G)]TJ /F3 11.955 Tf 11.95 0 Td[(VTH)+RSDW LCOXV0D(V0G)]TJ /F3 11.955 Tf 11.95 0 Td[(VTH)+! (3{11) whererepresentsthederivativeofw.r.tV0G)]TJ /F3 11.955 Tf 11.95 0 Td[(VTH.FromtheRCHexpression,itiseasytoshowthat @RCH @(V0G)]TJ /F3 11.955 Tf 11.95 0 Td[(VTH)=L W1 COX )]TJ /F1 11.955 Tf 13.78 8.09 Td[( 21 V0G)]TJ /F3 11.955 Tf 11.95 0 Td[(VTH)]TJ /F1 11.955 Tf 13.74 8.09 Td[(1 !(3{12)CombiningEqs. 3{11 and 3{12 ,wecanshowthat @RTOT @(VG)]TJ /F3 11.955 Tf 11.96 0 Td[(VTH)=2666664L W1 COX )]TJ /F1 11.955 Tf 13.78 8.09 Td[( 21 V0G)]TJ /F3 11.955 Tf 11.96 0 Td[(VTH)]TJ /F1 11.955 Tf 13.75 8.09 Td[(1 1+1 2 ID@RSD @(V0G)]TJ /F3 11.955 Tf 11.96 0 Td[(VTH)+RSDW LCOXV0D(V0G)]TJ /F3 11.955 Tf 11.96 0 Td[(VTH)+!3777775+@RSD @(VG)]TJ /F3 11.955 Tf 11.96 0 Td[(VTH) (3{13) 61

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ThelargetermwithinthesquarebracketsinEq. 3{13 vanishesasL!0.Wecanusethisconcepttounderstandtheimpactofunderlapresistanceandit'smodulationwithstress.Werstplottedthetotalresistancevschannellengthforafewchannellengths(ignoringverylargeLGs)foranumberofoverdrivevoltagesasintheRTOT{Lmethod.They-interceptpointofthesecurvesrepresentsL!0,andthereforeisameasureofRSDatthatparticularoverdrivecondition.WethenplottheextractedRSDagainstoverdrivevoltageasshowninFig. 3-17 andFig. 3-18 .Thisgureshowsthedierencesbetweenn-channelandp-channeldeviceswithtensileandcompressiveCESLrespectivelyforthreedierentnwidths.Forbothn-andp-channelFinFETs,theplotsindicatethattheRSDvaluereducesforincreasingoverdrivevoltages,andtendstosaturate.Thisisexplainedbytheunderlapaction.Underlowappliedgatevoltage,theunderlapregionsarenotinvertedandhaveahighresistancethatisinserieswiththechannelresistance.Asthegatevoltageisincreased,theunderlapresistancedecreases,andeventuallybecomesnegligible. Figure3-17.ObservedRSDreductionforn-channelFinFETswithCESLstressfordierentnwidths[ 1 ]c2009IEEE. 62

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Figure3-18.ObservedRSDreductionforp-channelFinFETswithCESLstressfordierentnwidths[ 1 ]c2009IEEE. Thesaturatedvalueintheaboveplotscorrespondstothegate-voltageindependentpartoftheparasiticresistance.Veryinterestingly,weseethateventhesaturatedRSDvaluereduceswiththerightkindofstress(tensileforn-channelandcompressiveforp-channel)forbothn-andp-channeldevices.InFig. 3-19 ,weshowthereductionintheobservedgate-voltage-independentcomponentoftheparasiticresistanceagainstoverdrivevoltage.ItisseenthatthenormalizedRSDreductionobservedbetweenn-channeldeviceswithtensileCESLandcompressiveCESLisinvariantoftheoverdrivevoltage.Comparingthesamequantityforp-channeldeviceswithcompressiveCESLandtensileCESL,theRSDreductiondropswithoverdrivevoltage(notenegativey-axis).OnepossiblereasonfortheobservedRSDreductioncouldbeduetothestressmodulationofthesilicide-siliconcontactresistance.InFig. 3-19 ,wealsoseethatwidernsshowalargerreductionbenet,whichcouldbeduetotheincreasedcontactareabetweenthesilicideandaheavilydopedSiregion.Thisincreasedcontactcanoccurwith 63

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Figure3-19.ComparisonofnormalizedRSDreductionbetweenn-channelandp-channelFinFETswithCESLsstress[ 1 ]c2009IEEE. thepresenceofaresidualgatespaceralongthebaseofthenbeforethesalicidationstep.WhileSchottkybarrierheightmodicationwithbiaxialstresshasbeenreportedintheliterature[ 107 ],theyhavemostlybeenforthecaseofin-planestrainedSiwithlowdoping,whichisnotthecaseinourdevices.DopantsegregationintheareaofthenwhereCESLispresentcouldalsobeacontributingfactor.Furtherinvestigationisnecessarytocompletelyunderstandwhetherthecontactresistanceismodiedinthedevice.AnotherfactorcouldbecontributingtotheobservedRSDreduction.Themobilityofcarriersintheunderlapregionischangingwithstress,andwhilethecontributionofthisresistancetothetotalresistanceissmall,itispossiblethatitisnotanegligiblefraction.ThecorrelationbetweenthetrendsinFig. 3-19 asafunctionoftheoverdrivevoltageforn-typeandp-typeFinFETsandthemobilitydependenceofn-typeandp-typeFinFETsasafunctionofinversioncarrierdensity(whichdependsontheoverdrivevoltage)inFig. 3-12 givestrengthtotheaboveobservation. 64

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ThusweconcludethatpossiblereductioninRSDwithstressandcarriermobilityenhancementwithstressbothcontributetothe17%enhancementthatwasobservedforthedeviceswithstrainedCESLlmsasreportedinFigs. 3-14 andFig. 3-15 ). 3.3.2.24-pointwaferbendingTofurtherunderstandtheeectofuniaxialstraininFinFETs,4-pointwaferbendingmeasurementswereperformedonbothlongandshortchanneldevicesusingaexurebased4-pointwaferbendingsetup[ 41 ].SincemechanicalwaferbendingisnotthoughttoaectthedeviceRSD,thisexperimentcomplementsouranalysisintheprevioussection.Theobservedlinearcurrentenhancementforbothlongchannel(LG=2m)andshortchannel(LG=32nm)isshowninFig. 3-20 .Theenhancementwasmeasuredattheapplieddrainvoltageofj50jmVforbothchannellengthsatagateoverdrivevoltageofj0:5jV(thresholdvoltagewas0:4V)forupto300MPaofstress.Forplanardevices,ithasbeenreportedthat[ 41 ]theshortchannellinearcurrentenhancementissimilartothatofthelongchanneldevicesonceparasiticRSDisaccountedfor.WehavenotcorrectedforRSDinFig. 3-20 duetothegatebiasdependencereportedintheprevioussection.TheenhancementsreportedinFig. 3-20 arefor<110>channeldirectionFinFETsfromacontrolwaferwithnoCESLstress.Toruleoutmeasurementerror,multiplemeasurementsweretakenonthe3samplesfromthesamewaferontwodierentexuresetups(oneatSEMATECHandoneatUF).Theopenandtheclosedsymbolsshowtheaveragedresultsofmeasurementsperformedoneachsetup.Weseethatlongchanneln-typeFinFETsshownearly14%enhancementinIDlinat300MPaoftensilestress,whilethep-typeFinFETsshowabout10%enhancementforthesameamountofcompressivestress.Theextracted-coecientfromthismeasurementare)]TJ /F1 11.955 Tf 9.3 0 Td[(47:241:4forn-typeand33:451:21forp-typedevices(inunitsof1011Pa)]TJ /F5 7.97 Tf 6.58 0 Td[(1).Comparedtothe-coecientsfrom<110>=(110)orientedplanarMOSFETs()]TJ /F1 11.955 Tf 9.3 0 Td[(371:8forn-nypeand278:8forp-type,[ 39 ]fromourgroup),our-coecientsarehigherforbothtypesofdevices.-coecientssimilartoourresultswasreportedinciteSuthram2008and[ 108 ].The 65

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Figure3-20.Measuredlinearcurrentenhancementwith4-pointwaferbendingonlongandshortn-andp-channelFinFETs. formertheorizedthatstrongelectricalandphysicalconnementcouldbethecaseofthedierencesfromplanar<110>=(110)values.AlltheFinFETsusedthisstudyweremadeon20nmwidenswhichprecludesvolumeinversionofcarriersandtherelatedphysicalconnementeects,whichindicatesthattheremaybeotherpossiblemechanismsinaction.Togainamoredetailedunderstandingofbothprocessstressandmechanicalstress,wediscussthephysicsofstresstransferinFinFETsinthenextsection. 3.3.3PhysicsofStressTransferWithbothprocessinducedandexternalmechanicalstress,therearethreestresscomponents(1)Longitudinalcomponentxxisthestressinthedirectionparalleltochannelcurrentow(2)Transversecomponentyyisintheplaneofthechannelconductionperpendiculartothecurrentow(3)Outofplanecomponentzzisinthe 66

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directionverticaltotheplaneofthecurrentow.Understandingwhatcomponentsareinvolvedforagivenstressoriscrucialtowardsexplainingstressbenetsondeviceperformance.OneofthemostwidelyusedindustrytechniqueforintroducinguniaxialstraininaMOSFETchannelisbyusingaSiNContactEtchStopLayer(CESL)thatisengineeredwithabuilt-instress[ 23 ][ 24 ][ 109 ].IntheCMOSprocessow,thislayerisdepositedafterthesource/drainandgatesilicidationstep,servingasastoppinglayerforthecontactetchbetweenthemetal-1laterandthesource/drain/gateregions.Typically,aCESLisabout20nmthickandcanhave2-3GPaoftensileorcompressivestress,dependingonhowitisdeposited.TheintrinsicbiaxialstressfromtheCESLtranslatesintoauniaxialstressinsidethechannel.SincethedevicesusedinthisstudyusedCESLasthestressor,wewilldiscussthestresstransfermechanismwithCESLinthissection. Figure3-21.StresstransfermechanisminaplanarbulkMOSFETwithacontactetchstopliner,from[ 110 ]c2008IEEE. Thephysicsofstresstransferinwellunderstoodforplanardevicesfordierentstressors.Forinstance,inbulkPMOSdevices,aCESLlmwithbuilt-intensilestressisusedtointroducecompressivestressinthechannel.TheoppositeistrueforNMOS.A 67

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tensilelmstendstocompress(likeanexpandedspringlikestocompress),sowhenitisconformallydepositedontopofthegateandoverthesource/drainregions,theregionsimmediatelyincontactwiththelmalltendtocompress.Sincethelmisincontinuouscontactwiththegatesidewallregions,theregionunderneaththechannelexperiencesanettensilestressintheout-of-planedirectionasshowninFig. 3-21 .TheCESLontopofthegatecompressesthetoppartofthegateinthelongitudinaldirectionwhichtransferstothechannelregionunderneath.Electronmobilityin(100):<110>Sichannelsisenhancedbytensilestressinthexxdirectionandcompressivestressintheyydirection,thereforeanetgaininelectronmobilityisachievedwithatensileCESL(whichhasbuiltincompressivestress).Holemobilityisnearlyinsensitivetostressintheyydirection,buttheresponsetocompressivestressinthexxdirectionislargeandthereforeacompressiveCESL(withbuiltintensilestress)isveryusefulforimprovingperformance.Eneman[ 111 ]providesacomprehensivesimulationstudyonthescalabilityofCESLeectivenessinplanardevices.Ingeneral,ithasbeenfoundthatthenetstressinthechannelwithCESLdependson(indecreasingorderofsignicance) 1. Stresslevelbuiltintothelm 2. Widthofspacerysed 3. Totalheightofgate 4. Source/drainregionsize 5. ChannellengthNarrowspacersarebenecialbecausetheyallowtheCESLtobeplacednearertothegate.Atallgateisbenecialbecauseitallowsmorestressintheyydirection.CESLiseectiveforshorterdevicesbecausethestressproleinthechannelbecomesmoreuniform.Inlongerdevices,sincethelmislocatedfartherfromthecenterofthechannel,thestressproleisnotuniformreducingtheCESLeectiveness.WeobservedthisinourdevicesaswellasshowninFig. 3-22 68

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Figure3-22.Mobilityenhancementasafunctionofchannellength.IntrinsicmobilityasdeterminedbyslopeofID=p gmfromourstrainedFinFETsindicatesthatthestrainbenetswithCESLislostatlongerchannellengths[ 1 ]c2009IEEE. Figure3-23.OnepossiblemechanismofCESLstresstransferinFinFETsproposedin[ 93 ]c2009IEEE. 69

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InFinFETs,stresstransfermechanismwithCESLisnotcompletelyclear.WhilesomeauthorsbelievethatCESLintroducesthesametypeofstressproleinaFinFETcomparedtoplanar[ 112 ],othersarguethatthestressproleisdierent.Suthram[ 93 ]observedthatCESLinducedstressindoublegateFinFETsishigherthaninplanarMOSFETs(forthesameCESLstresslevel)andthatlongchannelFinFETsshowedlittletonoenhancementcomparedtoshorterchannellengths.Hetheorizedthatstresscouplingintothechanneloccursthroughthenextension(Fig. 3-23 ).andnotthroughthegateastypicallyhappensinplanardevices.Usinganovelholographicinterferometrymeasurementtechnique,Conzatti[ 99 ]showedthatthemetalgateusedinFinFETsinducestressinthesiliconn,evenwithoutintentionalstressfromtheCESL.Themetalgateinducedstresswaspresentnotonlyinlongitudinalandtransversedirections,butalsointhenwidthdirection.Alargetransversecompressivestresswasobservedinthetransistorn,whichwouldimproveelectronmobilitywithoutimpactingholemobility.Sincethemetalgatewrapsaroundthenonthreesides,therewasacompressivestressinthenwidthdirectionaswell.Fig. 3-24 reproducedfromConzatti'sworkshowthesetwostresscomponentsveryclearly.InthepresenceofbothCESLandmetalgate,theresultingstressprolecouldbeverycomplicatedandthenetstressinthelongitudinaldirectioncouldbehigherthanexpected.SuchamechanismcanpossiblyexplaintheresultsinourCESLstraineddevices.TounderstandthedierencesbetweenprocessstressandmechanicalstressinFinFETs,weperformed3DsimulationsofstressprolesinSiFinsusingtheniteelementmodelingtoolABAQUS.ThesimulationstructureshowninFig. 3-25 consistedofa\dogbone"structurewithtwolarge50m2padsand20nsconnectedinparallelbetweenthepads.Thens10mlong,20nmwide,40nmtallandwereungated.Thesimulationstructuresatatopofa2mthickSiO2layer,thatwasconstructedontopofa650mthickSibaselayersimilartoatypicalFinFET.A4-pointbendingmomentof290MPawasapplied,makingsurethattherewasenoughclearancebetweenthepoints 70

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Figure3-24.MetalgateinducedstressinFinFETs,measuredusinganovelholographicinterferometrytechnique[ 99 ]c20011IEEE.Largecompressivestressisseeninthetransversedirectioninthen. wheremomentwasapplied.Thegoalwastosimulateamomentthathappenswithourmechanicalwaferbendingsetup.The290MPastresslevelwaschosentomatchthestress(300MPa)appliedonrealdevicesusingtheexurebasedapparatusinourwaferbendingmeasurements. Figure3-25.SimulationstructureusedforABAQUSsimulations.Gatestackwasnotincluded. Fig. 3-26 showstheresultantstressproleinthelongitudinaldirectionofthenfromABAQUS.Thestressinthisdirectionismostlyuniaxial.Theaveragestresslevelsinthelongitudinaldirectionisnearly400MPa,whichishigherthantheapplied290MPabendingmoment.Thecenterandtheedgesofthenshowamuchhigherstresslevelofnearly600MPa.Therewasnegligiblestressinthetransverse(i.e.vertical)and 71

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Figure3-26.ResultsfromABAQUSsimulationsinthelongitudinaldirection.Therewasnegligiblestressintransverseandwidthdirectionsofthen. 72

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thewidthdirectionsofthen.ItisveryinterestingtonotethedierencesbetweenthestressprolesinFig. 3-26 comparedtostressproleswithaCESLlmfromsimulationspublishedveryrecentlybyNuin[ 113 ].BasedonSentaurussimulations,thedierencesbetweenstressinducedbyCESLandstressinducedbysource/drainstressorsinSOIFinFETsandbulkFinFETswasclearlyexplainedin[ 113 ].Itwasshown(Fig. 3-27 )thatatensileCESLstressoronaSOIFinFETintroducesacompressivestressinthetransverse(vertical)directionoftheninagreementwithConzatti's[ 99 ]holographicinterferometrymeasurements.Atensilesource/drainstressoralsointroducesatransversecompressivestress,butthestresslevelsbothinthelongitudinal/transversedirectionsaremuchhigherandmoreuniformcomparedtotheCESLstressorforthesamestresslevelsinthestressor(1GPa)andsamechannellength.XualsoconcludedthattheincreasedgapbetweentheCESLandthechannelduetothepresenceofelevatedsource/drainregionsreducestheeectivenessoftheCESL. Figure3-27.ComparisonbetweenCESLstressandS/DstressinSOIFinFETsfrom[ 113 ]c2012IEEE.Bothstressorhavethesamestressof1GPa(tensile). 3.3.4ExperimentalResultsforSi/SiGeFinFETsHolemobilityinSiGewithbiaxialcompressivestrainhasbeenknowntobehigherthanthatinSilicon[ 114 ].Severalreportsofp-channelMOSFETsincorporatingaSiGe 73

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channelhavebeenpublishedintheliterature,forexample[ 115 ][ 116 ].SiGechannelp-channelMOSFETshavealsobeendemonstrated[ 117 ][ 118 ].ScalabilitystudiesofthebenetsofcompressivestrainedSiGechannelMOSFETshavebeenmadeevenupto20nmchannellengthplanardevices[ 119 ][ 120 ].Ontheotherhand,FinFETshavebeenshowntobeoneofthepotentialcandidatesforscalingtransistorstosub-20nmchannellengths.Inthissection,wediscusssomeexperimentalresultsonSiGe/SiFinFETs.Fabricationdetailscanbefoundinsection 3.3.1 andtheTEMsareseeninFig. 3-7 Figure3-28.MeasuredID{VGcharacteristicsof(110)orientedlongchannel(2m)SiGeFinFETs[ 100 ]c2009IEEE. AcomparisonbetweenthemeasuredID{VGofthe(110)orientedSiGeFinFETscomparedtoareferenceSiFinFETisshowninFig. 3-28 .ThestrainintheouterSiGelayeroftheFinFETsisbiaxialcompressionandcausesathresholdvoltageshift[ 121 ]asshowninFig. 3-29 .TheION{IOFFcharacteristicsinFig. 3-30 showanincreasedsaturationcurrentduetothebiaxialstraininducedholemobilityenhancements,butalsoshowanincreasedostateleakageattributedtotheunoptimizedprocessowused.Acomparisonofthe 74

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Figure3-29.ThresholdvoltageshiftduetobiaxialstraininSiGelayer[ 100 ]c2009IEEE. Figure3-30.MeasuredION-IOFFcharacteristicsof(110)orientedSiGeFinFETs.SiGedevicesshowincreasedo-stateleakage[ 100 ]c2009IEEE. 75

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measuredholemobilityfromlongchannelFinFETsinboth(110)and(100)sidewallorientationsispresentedinFigs. 3-31 and 3-32 ,showingimprovedholemobilityinbothcases. Figure3-31.Mobilitycomparisonbetween(110)SiGeandSiFinFETs.(100)universalSimobilityisshownforcomparison[ 100 ]c2009IEEE. Tofurtherunderstandtheimpactofthestress,weperformedlowtemperaturecurrentandcapacitanceonboth(110)SiGeFinFETsand(110)referenceSiFinFETsinthetemperaturerangeof300Kto80Kandextractedthemobilityforvarioustemperatures.Atroomtemperature,weobservedthat(110)SiGeFinFETshadahighermobilitythan(110)SiFinFETsatallinversioncarrierdensitiesasshowninFig. 3-33 .However,atlowtemperatures,themobilityof(110)Sideviceswashigherthanthe(110)SiGedevicesatallinversioncarrierdensities(Fig. 3-33 ),whichseemtoindicatethatthestrainbenetislostatlowertemperatures.Toinvestigatethisfurther,weplottedthevariationofmobilityasafunctionoftemperatureforbothlowandhighcarrierdensities.InFig. 3-35 weshowtheevolutionofloweld,higheldandpeakmobilityasafunctionoftemperature.We 76

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Figure3-32.Mobilitycomparisonbetween(100)SiGeandSiFinFETs.(100)universalSimobilityisshownforcomparison[ 100 ]c2009IEEE. noticeaveryinterestingtrendinthemobilityevolutionaswegofromloweldtohigheldconditions.TheenhancedholemobilityinSiGearisesduetothesamemechanismasinSilicon,namelychangeinconductivityeectivemassandthescatteringrateduetostrain-inducedbandwarpingandrepopulation.HoweverinSiGesystems,themobilityenhancementfromstrainiscounteractedbyanincreasedcoulombscatteringduetothepresenceofGe.Inaddition,aspointedoutbyFischetti[ 114 ]alloyscattering[ 122 ]candecreasetheholemobilityenhancementfromthebiaxialcompressivestrainforsystemswithlessthan40%Gecomposition.Fischettialsopointsoutthatat300K,bothphononandalloyscatteringhavethesamedependenceonthetransverseeldwhichmakesseparatingthesetwocomponentsdicult.However,atlowtemperatures,phononscatteringissuppressed,therebyreducingthebenetsofstraininducedsubbandsplittinginSiGe.Thismeansthemobilityenhancementatlowertemperatures,ifany,wouldarisesolelyfromconductivity 77

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Figure3-33.Mobilitycomparisonbetween(110)SiGeandSiFinFETsatroomtemperature.SiGeFinFetsshowhighermobilityatroomtemperature[ 100 ]c2009IEEE. Figure3-34.Mobilitycomparisonbetween(110)SiGeandSiFinFETsatlowtemperature(87K).SiFinFetsperformbetteratlowtemperature[ 100 ]c2009IEEE. 78

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(a)Loweldmobility (b)Peakmobility (c)High-eld Figure3-35.EvolutionofSiGeFinFETmobilitycharacteristicsasafunctionoftemperaturefordierentgateeldconditions. 79

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massreduction.Sincesurfaceroughnessscatteringscatteringisthedominantscatteringmechanismatlowtemperatureanditisnotastrongfunctionofstrainasforelectrons,wecanconcludefromthetrendsshowninFigs. 3-33 and 3-35 (higheld)thatalloyscatteringdoesplayadominantroleindeterminingtheoverallmobilityandispotentiallymodiedbythebiaxialstrain.Detailedsimulationsarenecessarytostudythisinmoredetail. 3.4SummaryInthischapter,wereviewedtheevolutionofFinFETtechnology.WealsostudiedtheimpactofprocessinducedstressonFinFETswithdetailedelectricalcharacterizationof(1)n-channelandp-channelSiFinFETswithandwithoutstrainedCESLlayersand(2)p-channelSiGe/SigateFinFETs.Wepresentedanewtechniquetostudytheextractandstudyimpactoftheparasiticsource/drainresistanceinFinFETsincludingtheeectofunderlapinprocessstrainedFinFETs.WithathoroughreviewofthephysicsofstresstransfermechanisminFinFETs,alongwithsimulationsperformedwithaFEMsimulationsoftware,wehighlightedthedierencesbetweenmechanicalstressandprocessinducedstressinFinFETs.ByanalyzingthemobilitycharacteristicsofSiGe/SiFinFETsatvarioustemperatures,wehighlightedthepotentialimpactofalloyscatteringbybiaxialcompressivestrain. 80

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CHAPTER4HIGHFIELDTRANSPORTINSILICONMOSFETSCarriertransportinsemiconductordevicesisanalyzedusingawidespectrumofapproaches.TheserangefromthemacroscopicDrift-Diusion(DD)[ 123 ]approachtothesemi-classicalHydrodynamic(HD)andenergy-transportmodelsemployinghighermomentsoftheBoltzmannTransportEquation(BTE)totheveryrigorousatomisticquantumtransportmodels1.ThesemethodsthemselvesarepartofaveryactiveareaofstudyknownasTechnologyComputerAidedDesign(TCAD).Inalltheapproachesformodelingcarriertransport,oneofthecommongoalsistoarriveatareasonablyaccurateexpressionforthecurrentthroughthedeviceasafunctionofappliedvoltage(s),beitabulkbarofsiliconoranintricatelydesignedMOSFET.Eachmethodarrivesatthisexpressiondierently.Irrespectiveofthecomplexityofcarriertransportthatactuallyhappensinthesemiconductor,twoequationsareanecessarypartofanymodel.TherstequationisthePoissonEquationwhichrelatesthedensityofchargedcarriersinthedevicetotheappliedelectrostaticpotential.TheotheristheContinuityEquation,whichrelatesthecurrentdensityandthechargedensity.Inordertoarriveatthenalexpressionforcurrent,thePoissonandContinuityequationshavetobesolvedsimultaneouslyandself-consistentlywiththeappropriateboundaryconditions.Forthistohappen,thissystemofequationswillalsoneedtoincludematerialparameters(suchasmobility,diusionconstantetc)thatrelateelectron/holeconcentrations,currentdensity,andtheappliedelectriceld.Anaccuratepictureoftherelevantphysicalmechanismsresponsibleformotionofchargedcarriedinsidethedeviceisrequiredforsuchanimplementation.Thedierencesinthevariousapproachesarisefromthedierentwaysinwhichthisphysicalpictureisdescribed. 1SeeRef.2in[ 124 ]foralistoffundamentalarticlesonthevariousquantumtransportmodels 81

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TCADisanenormouslylargeeldofresearchandcoveringitindetailisoutofthescopeofthisdissertation.However,toproperlyestimatetheimpactofstrainonemergingnanometricdevices,itisabsolutelyimportanttounderstandthediculties/challengesthatexistindescribingtransport(especiallyhigheldtransport).Tothisend,inthischapterwewilldiscussthesalientaspectsofthedierentapproachesthatpeopleusetodescribecarriertransport.Section 4.1 and 4.2 describesthesemi-classicalapproach,whichisjustiedwhenthecharacteristicdevicesizeismuchlargerthanthecorrespondingelectronwavelength.Sincetheunderlyingassumptionsofthisapproachbecomelessandlessvalidaswescaledowndevices,newertransportmodelsthatincludequantumeectsareneeded.Section 4.3 reviewsthesenewermodelsdescribingballisticandquasi-ballistictransport.InSection 4.4.2 wediscusstheapplicabilityofthesevariousmodelsspecicallyfromanexperimentalviewpointrelatedtostudyingthestraineects,followedbyasummaryinsection 4.5 4.1Semi-ClassicalTransportModelsInthesemi-classicalapproach,thecarriersarerepresentedasanensembleofclassicalparticles[ 125 ].2Thisensembleisdescribedbyaparticledistributionfunctionf(r;k;t)inthevectorspaceformedbyparticlepositionrandparticlemomentumkattimet.Integratingthedistributionfunctionoverspacegivesthetotaldensityoftheparticles.Chargecanneitherbecreatednordestroyed{whichmeansthatalongaparticletrajectory,thetimederivativeoff(r;k;t)iszero:df(r;k;t) dt=0 2Khannaprovidesanexcellentsummaryoftheevolutionofsemi-classicaltransportmodelsin[ 126 ].Someequationsinthissectionarereprintedfrom[ 126 ]withpermissionfromElsevierc2004. 82

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TheexpandedformoftheaboveequationistheBTE: df dt=@f @t+@f @k@k @t+@f @r@r @t(4{1)AnotherformofBTEis: @f @t+v@f @r+F ~@f @k=Zf(k0)P(k;k0))]TJ /F3 11.955 Tf 11.96 0 Td[(f(k)P(k;k0)dk0(4{2)wherevisthegroupvelocityoftheparticlesgivenbyv=dr dtandFisthetotalforceontheparticles,whichisthesuminternalforcesFIandtheexternalelectromagneticeldFE.P(k;k0)denotestheprobabilitythatacarrierwillbescatteredresultinginachangeofwavevectorktok0.Eq.( 4{2 )isnotasimpleequationtosolvenumericallyoranalyticallyintheformpresented.Itpresentsformidablemathematicalandcomputationalchallengesandthereforesimplifyingassumptionsarenecessaryeventogetanapproximatesolution.Ifwemakethefollowingassumptions,theBTEcanbesimpliedtoasolvableform: 1. Allscatteringmechanismsareelastic. 2. Externalforcesdonotaectthescatteringprobabilities. 3. Theintervalbetweencollisionsismuchlargerthanthetimedurationofthecollisions. 4. Constantexternalforcesoverlengthscalesofthecarrierwavepacketthatdescribesthecarriermotion. 5. Theelectriceldistheonlytheexternalforce. 6. Thefunctiondescribingthedistributionofparticlesissymmetricink-space.Undertheseassumptions,theBTEreducesto: @f @t+FE ~@f @k+v@f @r=@f @tcoll(4{3)whereFE=qEistheforcefromtotheappliedelectriceld.Theaboveequationisvalidforbothelectronsandholes.Thetermontherightrepresentsthecollisionoperator 83

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whichdescribesscatteringofparticlesduetovariousmechanisms.Itiscomprisedofthreeterms,namelyduetocarrierdensity,duetocarriervelocityandduetocarrierenergy. @f @tcoll=@n @tcoll+@v @tcoll+@ @tcoll(4{4)Thecollisiontermsforgeneration-recombinationandmomentum/energyrelaxationare dn dtcoll=)]TJ /F3 11.955 Tf 9.29 0 Td[(R (4{5a)dv dtcoll=)]TJ /F3 11.955 Tf 12.37 8.09 Td[(v p (4{5b)d dtcoll=)]TJ /F3 11.955 Tf 10.49 8.09 Td[(!)]TJ /F3 11.955 Tf 11.96 0 Td[(!0 p (4{5c)whereRisthegeneration-recombinationrate,pisthemomentumrelaxationtimeand!istheenergyrelaxationtime,!istheaverageelectronenergyand!0istheequilibriumelectronenergyforthetemperatureT0.TofurthersimplifytheBTE,wemaketwomoreassumptionsatthisstage,whichleadstotheRelaxationTimeApproximation(RTA): 1. Carriermotionisinequilibriumornearequilibriumconditions. 2. Thecarriervelocityisaninstantaneousfunctionofthelocalelectriceld.Usingtheseassumptions,thecollisiontermissimpliedto @f @tcoll=)]TJ /F3 11.955 Tf 10.5 8.09 Td[(f)]TJ /F3 11.955 Tf 11.96 0 Td[(f0 (4{6)wherewheref0denotesthespecial-casesolutionoftheBTEwhenthedistributionfunctionfissphericallysymmetricandissomeaveragerelaxationtime.TheoverallBTEsimpliesto: @f @t+qE ~@f @k+v@f @r=)]TJ /F3 11.955 Tf 10.5 8.09 Td[(f)]TJ /F3 11.955 Tf 11.96 0 Td[(f0 (4{7) 84

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ItisveryimportanttonotethatthisrelaxationtimeapproximationiswhatallowsustosolvetheBTEanalyticallyornumerically.WhenRTAbreaksdownsodothemodelsthatwederivefromtheBTEfordescribingtransport. 4.1.1Drift-DiusionApproachThemomentumandenergyconservationequationsforthecarrierensemblearederivedbyusingtherstthreemomentsofthesimpliedBTE(Eq.( 4{7 )).Typically,onlytheapproximatesolutionsofthersttwomomentsareusedindevicemodelsprevalenttoday.ThemeanoftheBTE,i.e.therstmomentgivesustheparticlecontinuityequation: @n @t=r(nv)=@n @tcoll(4{8)ThesecondmomentoftheBTEgivesthemomentumconservationequation.Assumingelectronsasthechargecarriers,themomentumconservationequationbecomes: @v @t=qE m+vrv+1 mnr(nkTn)=@v @tcoll(4{9)wheremistheelectroneectivemass,Edeontestheelectriceld,kisBoltzmannsconstantandmathrmTnistheelectrontemperature.Thetotalcurrentdensitycanbeobtainedfromaddingtheholeandelectroncurrentdensities:J=Jp+JnwhereJp=qnvpandJn=)]TJ /F3 11.955 Tf 9.3 0 Td[(qnvn(vpandvnarethehole/electrondriftvelocities).FromEqs.( 4{5 ),( 4{8 )and( 4{9 ),weget: rJn)]TJ /F3 11.955 Tf 11.95 0 Td[(qRdn dt=qR (4{10a)rJp+qRdp dt=qR (4{10b)AssumingthatthetemperatureTnissameasthetemperatureofthelatticeT,thetermvrvinEq.( 4{9 )becomesnegligiblysmall.Inaddition,thecollisiontermontherightof 85

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Eq.( 4{9 )isalsoassumedtobezero.ThususingEqs.( 4{5 )and( 4{9 )wearriveat: vn=)]TJ /F3 11.955 Tf 9.3 0 Td[(nE+Dn nrn(4{11)wheretheelectronmobilitynandthediusioncoecientDncanbedenedas: n=qp m (4{12a)Dn=kT qn (4{12b)Thecurrentdensityequationsbecome: Jn=qnnE+qDnrn (4{13a)Jp=qnnE)]TJ /F3 11.955 Tf 11.95 0 Td[(qDprp (4{13b)ThePoissonequationisgivenby: r2 = "0"Si(4{14)where andrepresenttheelectrostaticpotentialandthechargedensity(relatedtothedonor/acceptordopingdensitiesandelectron/holeconcentrations)."0isthepermittivityoffreespaceand"sisthesilicondielectricconstant.Thecontinuityequations(Eq.( 4{10 )),thecurrentdensityequations(Eq.( 4{13 ))andPoisson'sequation(Eq.( 4{14 )togethermakeupthecommonsemi-classicalcarriertransportequationsofcarriertransport.Thedrift-diusionequationsarecommonlyimplementedinTCADsimulationsoftwaresincethedierentialequationscanbeeasilyimplementedinanumericalsolverthatdividesthechannelintoadiscretizedmeshanditerativelysolvesfortheequationsateachpoint.Thevariousmaterialspecicpropertieslikemobilityanisotropycanbeimplementedinagenericmanner.2Dand3Ddevicesimulationsarealsopossibleusing 86

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suchanapproach.ThoughitincreasesthecomputationalcomplexityThesesimulations,accuracyisimprovedespeciallyconsideringadvanceddevicegeometries.However,sincetheDDmodelconsidersaconstantcarriertemperature,o-equilibriumeectsarenotaccountedforinthismodel.ThisisthereasonwhytheaccuracyoftheDDmodeldecreasesfortoday'sshortchanneldevices,becauseitdoesnotaccuratelycapturethenon-localnatureofthetransportinvolvingvelocitysaturationandovershoot. 4.1.2VelocitySaturationandVelocityOvershootForincreasingappliedelectriceldsinSi,thedriftvelocityofcarriersisinitiallyfoundtolinearlyincrease.Whentheelectriceldexceeds5103V/cm,thedriftvelocityisobservedtosaturate(i.e,becomesindependentofappliedeld).Whenthishappens,transportmodelsbasedonmobility-baseddescriptionsbegintobreakdown.Forlargeelds,thedriftvelocityvdapproachestherandomthermalvelocity.Thescatteringprocessesintensifyleadingtorapidcarrierenergyloss.Thehigh-eldlimiteddriftvelocityiscalledthesaturationvelocityvd(sat).Electronsinbulksiliconattainasaturationvelocityofabout1107cm/sat300Kwhentheelectriceldreachesthevalueof2104cm/s.Forholes,thevalueisalittlelower.Theholevelocitysaturatesat7106cm/sforeldvaluesof5104V/cmat300K.Whenthedistancetraveledbythecarriersismuchsmallerthantheaveragedistancebetweenscatteringevents,averagedriftvelocityandsaturationvelocityconceptslosemeaning.Whenalargefractionofthecarriersaremovefromthepointofinjectiontothepointofcollectionwithoutanycollisions,thetransportbecomesballistic. 4.1.3HydrodynamicModelAlotoftheassumptionsusedfortheDD-modelisnotapplicableforarealdevice.Forinstance,carriersdonotrespondinstantlytochangesinelectriceld.Mobilityanddiusioncoecientsaredependonparametersotherthantheappliedelectriceld.Inhighelds,asmentionedearlier,theRTAbreaksdown. 87

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AbettermodelcalledtheHydrodynamic(HD)modelofcarriertransporthasbeenproposedintheliterature.ThismodelisbasedonamorecompletesetoftransportequationsderivedfromBTEaccountingforenergyandmomentumrelaxation.TheHDmodelisstillamacroscopicapproximationtotheBTE,butitgoesbeyonddrift-diusionmodelandthereforeismoreaccurate.InsteadoftreatingthecarriersasaMaxwellianensemblewithaveragedproperties,intheHDmodel,carrierowinasemiconductoristreatedastheowofathermallyconductingandchargedgasmovingundertheinuenceofanelectriceld.Therelevantequationsforthismodelcanbeunderstoodbyremovingsomeofthelaterassumptionsintheprevioussection.SimilartotheDD-approach,theparticleconservationequationforelectronsstillremainssameasEq.( 4{8 ):Theparticlecontinuityequationis@n @t+r(nv)=)]TJ /F3 11.955 Tf 9.3 0 Td[(RRemovingtheassumptionthatelectronandlatticetemperatureshavetobethesameandnotneglectingthev:rvanddv dtterms,themomentumconservationequation(comparetoEq.( 4{9 ))becomes: qE m+vrv+1 mnr(nkTn)=)]TJ /F3 11.955 Tf 10.49 8.09 Td[(v(!) p=@v @t(4{15)where!isthesumofkineticandthermalenergycomponentsthatfactorintothecollisiontermontherightas:!=3 2KTn+1 2mv2Theenergyconservationequation(the3rdmomentofBTE)is: )]TJ /F3 11.955 Tf 13.15 8.08 Td[(!)]TJ /F3 11.955 Tf 11.96 0 Td[(!0 w=@! @t+vr!+qvE+1 nr(nvkTn)(4{16)ItispossibletoreducetheaboveequationsinaformsimilartotheDD-model,usingthestandardexpressionsforcurrentdensityandmobility. 88

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Theseare 1 qrJn=R (4{17a)Jn)]TJ /F3 11.955 Tf 13.15 8.09 Td[(pn qJnrJn n=qn(KTn qrn+nrKTn q)]TJ /F3 11.955 Tf 11.96 0 Td[(') (4{17b)rnTn+Jn q(!+KTn)=)]TJ /F3 11.955 Tf 9.3 0 Td[(R!+EJn)]TJ /F3 11.955 Tf 11.95 0 Td[(n!)]TJ /F3 11.955 Tf 11.95 0 Td[(!0 !n (4{17c)ItshouldbenotedthattheHDmodelalsousestheRTAformodelingtheeectsofcollisionsonenergy.momentumofthecarriers.Typically,themobilityismodeledasadecreasingfunctionofenergytakingintoaccountthescatteringratedependenceoncarrierenergy.SolvingtheHDequationsaspresentedaboveisstillverycomplicated.Toproperlyaccountforthetemporal/spatialvariationofcarrierenergyandtemperature,verysmalltimestepsareneededwhensolvingtheequationsnumericallytoensureconvergenceofthesolution.Acommontechniqueistousequasi-staticapproximationswheneverpossible.ThesearediscussedinSection 4.2 .WhyistheHDmodelbetterthanDD?TheHD-modelincludestheenergy-balanceequation(Eq.( 4{17c )).Thisisnothingbutthelawofconservationofenergyappliedtocarriertransport.TheLHSrepresentsvariationofenergyowwithrespecttopositioninsidethedevice.IntheRHSoftheequation, 1. Thersttermrepresentstheenergyabsorbedbythecarriersfromtheelectriceld 2. The2ndtermrepresentstheenergylostbythecarriersthroughcollisionswithopticalphononstothecrystallattice 3. Thethirdtermrepresentstheenergylossthroughrecombination.Thisequationisimportantbecauseinthehigheld(i.e\hot")regionswherethecarrierenergy(i.e.temperatureTn)islarge,alargerdiusionispredicted.Thisisdueto 89

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thenitevalueofthetimeforenergyrelaxation,whichisnotthecaseintheDD-model.Asaconsequence,!andTnbecomehigherinlargerelectriceldregionsascomparedtoequilibriumvalues.Inaddition,thetemperaturegradientdescribedrkTn qmakesthecarriersdiusefrom\hot"regionsto\cold"regions.VelocityovershootistreatedbetterintheHDmodel.Consideringa1Dcasewheretheelectriceldisincreasinginthedirectionoftransport,theenergybalanceequationimpliesthattheaverageenergy!islessthantheenergyvaluecorrespondingtothelocalelectriceldunderhomogeneousconditions.Sincemobilityisadecreasingfunctionof!andeld,thecarriervelocityintheHDmodelishigherthanthecarriervelocityobtainedfromtheDDmodel,wherethemobilityvalueisdependentonlythelocalelectriceld.Recently,a6-momenttransportmodelhasbeenproposed[ 127 ][ 128 ]formoreaccuracy.Thisisa2Dnon-parabolicmacroscopictransportmodelusingsixconsecutivemomentsoftheBTE,usinginterpolationofsomeoftheparametersusedfromstatisticalMonte-carlosimulations.Needlesstosay,thismodelissignicantlymorecomplex. 4.2MOSFETCurrent-VoltageexpressionsinDDandHDmodelsStartingwiththeDDequationsandusingtheGradualChannelApproximation(GCA),whereitisassumedthatthedepletionregionbetweensourceanddrainisaectedonlybythetransverseeldE(x)fromthegate(alongx-direction,perpendiculartothechannel)andnotbythelongitudinaleldE(y)(y-direction,fromsourcetodrain).i.e.,E(y)E(x),1Danalysiscanbeusedtodeterminedepletionwidth.The2DMOSFETstructurecanbedecomposedintotwo1DstructuresunderaboveassumptionsandwecanthenderivetheI)]TJ /F3 11.955 Tf 12.1 0 Td[(VequationsforaMOSFET.Itcanbeshownthat[ 2 ]thefundamentaldierentialequationthatrelatestheterminalvoltagesandthecurrentinaMOSFET(foran-typedevicebeyondthreshold,ignoringdiusion)is: )]TJ /F3 11.955 Tf 9.3 0 Td[(ID=Wn)]TJ /F1 11.955 Tf 10.49 8.08 Td[(dV(y) dyQch'Zxc0Wn)]TJ /F1 11.955 Tf 10.49 8.08 Td[(dV(y) dyqN(x)dx (4{18a)ID'WndV(y) dyCox(VG)]TJ /F3 11.955 Tf 11.95 0 Td[(VTH)]TJ /F3 11.955 Tf 11.95 0 Td[(V(y)) (4{18b) 90

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where xc isthethicknessoftheinversionchannel W isthegeometricwidthofthegate. ID isthecurrentowingintothedrainterminal. n istheaveragemobilityoftheelectronsinthen-typeinversionchannel.Thisisalsoknownastheconductivitymobility.Notethatnisassumedtobeaconstantindependentofbothpositionandelectriceld. Cox istheoxidecapacitanceperunitareagivenbyCox="=x0where"istheSidielectricconstantandx0isthethicknessofthegateoxide. VG istheDCvoltageappliedtothegatewithsubstrate/sourceasthecommonreference(i.e.VB=VS=0). VTH isthethresholdgatevoltagewhichistheminimumgatevoltagerequiredtoproduceasurfacechannelthatcausesacurrentowbetweensourceanddrain. V(y) isthepotentialattheoxidesemiconductorinterfaceinthechannelatadistanceyfromthesourceendofthechannel. dV(y) dy istheelectriceldalongthesurfacechanneloppositethetransportdirection.ThesolutiontothelongchannelMOSFETdierentialequation( 4{18 )canbesimplyobtainedfromintegratingfromsourcetothedrain(y=0toL)sincethevoltagesV(0)=0andV(L)=VDareknownattheboundaries. ZL0IDdy'ZVD0WnCox(VG)]TJ /F3 11.955 Tf 11.95 0 Td[(VTH)]TJ /F3 11.955 Tf 11.96 0 Td[(V(y))dV(y) (4{19a)IDL'WnCox(VG)]TJ /F3 11.955 Tf 11.95 0 Td[(VTH)VD)]TJ /F1 11.955 Tf 13.15 8.09 Td[(1 2V2D (4{19b)FromEq. 4{19b ,itiseasytoseethatIDbecomesconstantwhenwhenVD=VG)]TJ /F3 11.955 Tf 12.15 0 Td[(VTH.AthigherdrainvoltageVD>VG)]TJ /F3 11.955 Tf 12.14 0 Td[(VTH,thechannelcurrentisindependentofdrainvoltagei.eVDhasnocontroloverthecurrentsincetheelectrondensityatthedrainjunctionendofthechannelhasdroppedtozero.Thisiscalledthe`pinch-o'point.ItisveryimportanttonotethatinlongchannelMOSFETs,thesaturation(i.e.theconstancyandthusindependence)ofthedraincurrent(w.r.t.drainvoltage)isaresultofchannel 91

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pinch-oandnotvelocitysaturation.Thissaturateddraincurrentisgivenby:ID(SAT)=IDVD=VG)]TJ /F4 7.97 Tf 6.59 0 Td[(VTH=W 2LnCox(VG)]TJ /F3 11.955 Tf 11.95 0 Td[(VTH)2 (4{20)Thisisknownasthe`square-law'modelforsaturationcurrent.Itisapplicabletoonlylongchanneltransistorswherethevariousassumptions(e.g.GCA,almostconstantmobilityetc.)holdtrue.Whenwescaledownthetransistor,theelectriceldinsidethedeviceincreasesandthedecouplingofdrain/gatevoltagesusedintheGCAisnotpossibleanymore.Thethreeimportanthigheldeectsthathappeninadditionwhentheeldincreasesare: 1. Driftvelocitysaturation{Thishappensduetotheincreasingrateofelectronenergylossviaopticalphononscatteringinhighlongitudinalelds. 2. Carriermobilityreduction{Thishappensduetoscatteringbyinterfacedefectsandoxideionswhentheincreasinglyhightransverse(i.e.gate)eldpushestheinversionchargeclosertotheSi/SiO2interface. 3. Increasedgate(andsubstrate)currentduetoFowler-NordheimtunnelingthroughtheSi/SiO2barrieratincreasinggateeld.Itisimportanttodistinguishtheabovehigheldeectsfromhighvoltageeectsthathappenevenwhentheeldisnotsohigh.Themostimportantoftheseare: 1. Channellengthshorteningduetothickeningofthedrainjunctionspace-chargelayerwithincreasingdrainvoltages. 2. Channelwidthwideningduetofringinggateeldathighgatevoltages.Forthetimebeing,wewillignorechangeinVTHandeectofvarioustrapswhenthegateanddrainvoltagesincrease.Wewillalsonotdiscussthesub-thresholdbehaviorsincethefocusofthisdissertationistowardsunderstandingstraineectsintheactualoperatingconditionsofadigitaldevice,i.e.insaturationconditions.Inthefollowingsection,webrieyreviewhowtheDD/HDmodelsaccountforthehigheldandhighvoltageeects. 92

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CompactMobilityModelingTokeeptheformoftheMOSFETI-VsimilartoEq.( 4{19b ),evenforscaleddeviceswherehigheldandhighvoltageeectscannotbeignored,itiscommonpracticetoincludetheelddependenciesinthemobilityterm.Therefore,inbothDD/HDmodels,themobilityisactuallyaphenomenologicaltermwhichisgivenby: (Ey)=0 1+Ey ECL!1 (Longitudinaleld) (4{21a)(Ex)=0 1+Ex ECT!(Transverseeld) (4{21b)where0istheloweldmobilityandECL/ECTarethelongitudinal/transversecriticalelectricelds.andareempiricalparametersthatvarywithbothtemperatureandbothlongitudinal/transverseelds.Theseparameterscanbeobtainedonlybyttingtheorytoexperimentaldata.Theloweld(drift)mobility0itselfhasseveralcomponentsthataredeterminedbythescatteringprocessesinsidetheMOSFET.Inshort,therearetwofundamentaltypesofscatters{intrinsicandextrinsic.Theintrinsicscatterersaresimplythelatticeatomsthemselves,thatvibrateduetothermalenergytheyacquireatroomtemperature.Theextrinsicscatterersaretherandomlylocatedphysicaldefects,interfacialboundarylayerandtheintentionallyaddedchemicalimpurities(i.e.thedopants).Thescatteringfromintrinsicscatterersaremathematicallydescribedusingphonons.Phononsarethequantaofenergyfromquantizationofthelatticevibrationwaves.DuetothecomplexnatureoftheSibandstructure,manytypesofphononscatteringexist{acousticphononscattering,opticalphononscattering,polaropticalphononscatteringandintervalleyphononscattering.Theextrinsicscattererscanbeeitherneutralorionized,whichgivesrisetotwotypesofscattering. 93

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Eachofthescatteringmechanismsdescribedabovehavecomplexdependenceoneld(bothlongitudinalandtransverse),temperatureandcarrierenergy.Theloweldmobility0isacombinationofallthreescatteringmechanisms(acousticphonon,opticalphononandionizedimpurity).0isgivenby: 1 0=1 AP+1 OP+1 I(4{22)whereAP,OPandIaretheacousticphononscatteringlimited,opticalphononscatteringlimitedandimpurityscatteringlimitedmobilitiesrespectively.Eq.( 4{22 )isknownasMattheison'srule.ThelongitudinalcriticaleldECLisgivenbyvsat 0'107 0.Theempiricalparameter'2forelectronsand1forholes.ThetransversecriticaleldECTisapproximately100KV/cmandempiricalparametervariesbetween0.5to2.NotethatExinEq.( 4{21b )istheelectriceldattheSi/SiO2interface.InaMOSFET,themobilitydependsonbothlongitudinalandtransverseeldsi.e.n(x;y)=f(Ex;Ey).Tosimplifyanalysis,asimpleapproximationismade, n(x;y)=nfEx(x=0;y)g(4{23)Thatis,thetransverseelddependencecanbeanalyzedusingExdependentmultipliersintheloweldmobility0.ThisiswheretheDDandHDmodelsdierintheirdescription{theHDmodeltakingintoaccountthenon-localnatureoftransportasmentionedearlier.BothmodelsuseCGinsteadofthesimpleCoxtoaccountforthecapacitanceoftheinversionlayerthatisinserieswiththeoxidecapacitance.TheI)]TJ /F3 11.955 Tf 11.97 0 Td[(Vequationforan-MOSFET(abovethreshold)intheDDmodelissimplyanextensionofEq.( 4{19b )givenby:ID=2CGW Ln0 1+Ey ECL(VG)]TJ /F3 11.955 Tf 11.95 0 Td[(VTH)VD)]TJ /F1 11.955 Tf 13.16 8.08 Td[(1 2V2D=2CGW Ln0 1+nVDS(VG)]TJ /F3 11.955 Tf 11.95 0 Td[(VTH)VD)]TJ /F1 11.955 Tf 13.15 8.09 Td[(1 2V2D (4{24) 94

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wheren=n0 vsatL.IntheHDmodel,themobilityisexpressedintermsofthetemperatureproleofthecarriers[ 129 ]thatdependson(Tn)]TJ /F1 11.955 Tf 11.96 0 Td[(TL)whereTn=TLrepresentthecarrier/latticetemperaturerespectively.Theelectronmobilityisgivenby: n=n0 1+(Tn)]TJ /F1 11.955 Tf 11.96 0 Td[(TL)(4{25)Here,=2knn0 qwvsatwherewistheenergyrelaxationlengthandknistheelectronthermalconductivity.ItcanbeshownthattheoverallI)]TJ /F3 11.955 Tf 12.33 0 Td[(VequationisgiveninaformsimilartoEq.( 4{24 )as: ID=2CGW Ln0 1+nVDS(VG)]TJ /F3 11.955 Tf 11.96 0 Td[(VTH)VD)]TJ /F1 11.955 Tf 13.15 8.09 Td[(1 2V2D(4{26)wheren=n0 vsatL1+2w L)]TJ /F5 7.97 Tf 6.59 0 Td[(1.GeandFossum[ 130 ][ 129 ]extendedtheaboveequationstoincludeeectsofvelocityovershoot.Thisisdonebyintroducingatemperaturedependenttermintheexpressionfordriftvelocity:v=E=e(Ey)Ey(1+k qEydTn dy)=vD(1+k qEydTn dy) (4{27)wheree=e 1+eEy vsatandvD=e(Ey)Ey.(Note:eheredependsonlyonthetransverseeldEx.)Fossum's1Denergybalanceequationalongthechannellengthis: d dxTn(y))]TJ /F3 11.955 Tf 11.95 0 Td[(TL+Tn(y))]TJ /F3 11.955 Tf 11.96 0 Td[(TL 5v!=3=2qEy(y) 5k(4{28) 95

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Understeadystatethisbecomes:d dxTn(y))]TJ /F3 11.955 Tf 11.96 0 Td[(TL=0Tn(y))]TJ /F3 11.955 Tf 11.96 0 Td[(TL 5v!=3=2qEy(y) 5kTn(y)=TL+v! 32qEy(y) k (4{29)SubstitutingEq.( 4{29 )inEq.( 4{27 )wegetaquasi-steady-stateapproximationforthenon-localvelocityovershooteect: v=vD8<:1+2 3 vsat! Ey!dEy dy9=;(4{30)Itcanbefurthershownthattheaboveequationcanbereducedto: v=vD(1+kTL qEyn0 vsatdEy dy)(4{31)Fromtheseexpressions,abias-onlydependent(i.enotdependentonxory)eectivesaturateddriftvelocityvsat(e)hasbeendenedtoreplacevsat.WhenVD!VD(sat)andvD!vsat,vsat(e)isthevelocityatthedrainendofthechannel.Ifovershootoccurs,thenvsat(e)>vsat.Thefollowingempiricalexpressionhasbeenderivedforvsat(e): vsat(e)=vsat(1+kTe qvsatlcsinh(Ld=lc) cosh(Ld=lc))(4{32)wherelcandLdareparametersdependentonoxide/depletioncapacitanceandVD(sat)respectively.TheoverallMOSFETequationintheisimprovedHDmodelisgivenby:ID/e 1+eEx=vsat(e)(Linearregion) (4{33)ID'WQNvsat(e)(Saturationregion) (4{34)(QNistheinversionchargedensityinthechannel.) 96

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4.3NanoscaleTransportModelsEventhoughtheHDmodeldescribedintheprevioussectionaccountsforbothvelocitysaturationandvelocityovershoot,itisstillnotsuitablefornanoscaleMOSFETs.Thisisbecauseitdependsontherelaxationtimeapproximation(Sec.( 4.1 ))atitscore.Whileitisreasonablyaccurate,peoplehavebeenreportingissueswithsuchmodelsforthepastfewyears.Moreandmoreempirical(andentirelynon-physical)approximationsneedtobemadefortheelddependenceofmobilityinTCADmodelstogetagoodtbetweenexperimentaldataandtheory.Whenwehavesomethinglikestrainthatchangesthefundamentalbandstructure(andtherebytheconductivitymass,scatteringratesetc),itbecomesverydiculttoanalyzetheeectinhigheldtransportconditionsusingsuchmodelsforsub-100nmMOSFETs.Therefore,afundamentalshiftinthewaywelookathigheldcarriertransportisneeded.Thissectiontalksaboutrecentdevelopmentsonthisfront.Whendevicechannellengthbecomecomparabletotheelectronmeanfreepath,quantumeectsshouldbetakenintoconsideration,alongwiththeusualscatteringwithphonons,impurities,andsurfaceroughness.Tostartwith,wewillrstdiscusstransportinacompletelyballisticMOSFET(Natori),followedbyadiscussiononaquasi-ballisticmodelthatseemstobemorerealisticfortoday's45and32nmMOSFETs. 4.3.1Natori'sBallisticMOSFETModelIn1994,Natori[ 131 ]rstproposedandstudiedtransportinacompletelyballisticMOSFET.ThetransistoriscomprisedofathinSilmconnectedtotworeservoirsofcarriers(sourceanddrain)oneitherside.ThegateelectrodeiselectricallyseparatedfromtheSilmbyadielectric.TheSilmisleftundopedtoreducedscatteringinthechannel.Inthetransportmodeldescribed,theMOSFETI)]TJ /F3 11.955 Tf 12.23 0 Td[(Vrelationisexpressedintermselementaryparameterswithoutanydependenceonthecarriermobilitybecauseoftheabsenceofscattering.Thecurrentisindependentofthechannellengthandisproportional 97

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tothechannelwidth.ThecurrentvaluesaturatesasthedrainvoltageisincreasedandthelinearandsaturationoperationsarespeciedasinaconventionalMOSFET. 4.3.1.1PhysicsofquantumballistictransportUnderballistictransportconditions,Natorimodeledthetransportbystudyingcarriertransmissionoverthesource-to-drainpotentialbarrier(Fig.( 4-1 )).ElectronswithenergieslargerthanthebarrierheightEmaxaretransmittedfromsourceintothechannelbythermionicemission.Similarphenomenonhappensatthedrainreservoir{however,onceabiasisappliedatthedrain,thebarrierheightforthedraininjectedcarriersbecomesreallyhighasshowninthegure,anddraininjectionissuppressed.ThestartingassumptionsmadebyNatoriare: 1. ChannelpotentialvariesgraduallyalongthechannellengthLwiththesourcepotentialbeinghigherthanthedrainpotential.ThemaximumvalueisEmaxoccursnearthesourceatpositionx=xmax. 2. ThecarriersareconnedalongthechannelwidthWbythesteepbarrierattheedges.Thepotentialprolealongthewidthisassumedtobeasquarepotentialwell. 3. Carrierconnement(y-direction)ismodeledwithatriangularpotentialwellwithdiscreteenergylevels,similartoaregularMOSFET.NatorialsoderivedacompactequationfortheI)]TJ /F3 11.955 Tf 11.96 0 Td[(Vrelation: ID=Wp 2q(KBT)3=2Mvmt 2~2F1=2(u))-222(F1=2(u)]TJ /F3 11.955 Tf 11.95 0 Td[(VD)(4{35)wheremtisthetransverseeectivemassoftheelectron,Mv=lowestvalleydegeneracy)(totalpopulation/populationoflowestlevels).F1=2istheFermi-Diracintegraloforder1/2denedas:F1=2(x)=Z10p xdy 1+exp(y)]TJ /F3 11.955 Tf 11.95 0 Td[(x) (4{36)TheotherparametersinEq.( 4{35 )aredenedasbelow:u=lnnp (1+exp(vD))2+4exp(vD)exp()]TJ /F1 11.955 Tf 11.96 0 Td[(1))]TJ /F1 11.955 Tf 11.95 0 Td[((1+exp(vD)))]TJ /F1 11.955 Tf 11.95 0 Td[(ln(2)o 98

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Figure4-1.ConductionbandprolealongthelengthofaMOSFETchannel. vD=qVD KT=2~2 qKBTmtMv)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(CG(VG)]TJ /F3 11.955 Tf 11.95 0 Td[(VTH) (4{37)Sincethismodeldescribespurelyballistictransport(ienoscatteringofanykindinsidethechannel),thisballisticcurrentisthemaximumcurrentobtainableforaparticulargeometry.Forthistohappen,thechannelmustbeshorterthanthemeanfreepath,whichisonlyafewnanometersinSi.Today'ssub-100nmMOSFETsarenothereyetandthismodel,thereforeisnotdirectlyapplicablefordescribingtransportwhere<100%ofthecarriersreachthedrainwithoutundergoingscattering. 4.3.1.2CurrentcontrolinNatori'smodelIntraditionaldevicetheory,thedraincurrentisgovernedbythevelocityofcarriersinthechannel.Forsmallelectricelds,thecarriervelocityisshownasaproductofthe 99

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lateralelectriceldandthecarriermobility.Whentheelectriceldisincreased,however,thecarriervelocitysaturatestoavaluearound107cm/sduetoenergyrelaxationviaopticalphononscattering.InthefullyballisticMOSFETsdiscussedhere,carriersinthechannelarefreefromscatteringandallcarrierspropagatingtowardsthedrain,passingthroughthebottleneckaroundxmax,reachthedrainwithoutscatteringbacktothesource,ifthereectionatthedrainedgeisneglected.Thechannelcurrentisgovernedbythecurrentatthebottleneck.Incurrentsaturation,forexample,thecurrentinjectedfromthesourcetothebottleneckdominatesthetotalcurrent,andthecarriervelocityinthebackwardchannelhaslittletodowiththecurrentvalue.Themeancarriervelocityatthebottleneckas: =ID(sat) WCG(VG)]TJ /F3 11.955 Tf 11.96 0 Td[(VTH)(4{38)givesthemeancarriervelocityinjectedfromthesourcetothechannelinballisticMOSFETs,anditisknownastheinjectionveIocity.ComparethisequationwithFossum'sequationforvsat(e)(Eq.( 4{34 )).Natorishowsthattheinjectionvelocitycanbetheoreticallyexpressedas:vinj=8~p Q 3mtp qMv'8~p CG(VG)]TJ /F3 11.955 Tf 11.95 0 Td[(VTH) 3mtp qMv (4{39)whenthecarrierstowardsthedrainaredegenerate.Notethatonceagain,thisisquitedierentfromtheHDmodeldenitionforvsat(e)whichdoesnotdependonthecarrierconcentration.Becauseofthisdependenceonp Q,ID(sat)isproportionalto(VG)]TJ /F3 11.955 Tf 11.95 0 Td[(VTH)3=2.Forahypothetical,verylongchannelballisticMOSFET,thepotential,thechargedensity,aswellasthecarriervelocityareallconstantalongthechannel.Allcarriersinjectedfromthesourceintothechannelpropagatealongthechannelwiththesamevelocitywithoutscatteringoracceleration.However,forarealisticMOSFET,duetothethreedimensionalnatureofPoisson'sequation,thechannelpotentialisactuallyinuencedbythepotentialofthesourceorthedrainelectrodes(andscatteringinthechannel). 100

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Betweenthesourcepotentiallevelandthedrainlevel(whichislowerthansourcepotentialbyqVD),thereisapotentialbarrierinthechannelcontrolledbythegateelectrode.Thepotentialproleisbell-shapedwithamaximuminthechannel(Fig. 4-1 ).Thispotentialmaximumresultsinabottleneckforthecarriertransportbetweenthetwoelectrodes.Eithersideofthemaximumisundertheinuenceofthesourceorthedrainelectrode.Asdiscussedinearliersections,whenthekineticenergyofacarrierissmall,thecarriervelocityisgovernedbyacousticphononscattering,impurityscattering,andsurfaceroughnessscattering.Whenthekineticenergyislarge,however,opticalphononscatteringbecomesdominant.InballisticMOSFETswithlongerchannels,thepotentialisconstantthroughoutthechannelandmostcarriersdonothaveenoughkineticenergy(orgainenoughenergytraversingthechannel)toemitopticalphonons.Inultra-shortchannelMOSFETswiththebell-shapedpotentialprole,opticalphononemissionoccursinsidethechannelsincetheeldislargeandthecarrierscangainenoughenergy.However,Natoripostulatedthatoncethephononemissionhashappened,thesescatteredcarrierscannotgooverthepotentialmaximumbacktothesourceelectrodebecausetheirenergyisalreadylowered.Scatteringeventsfarfromthesourcemakeitevenmoreharderforthesecarrierstolongitudinallytraversethechannelbackwardsandgointothesourceagain.ThisisoneofthecrucialpointsofthistheoryandLundstrom'sbackscatteringtheorythatisdiscussedinSection 4.3.2 .WewillelaborateonthisassumptionanditsvalidityingreaterdetailinSection 4.3.2.2 .Toquantifytheballistictransportindevices,NatorialsocamewiththeBallisticEciencyparameterBthatisdenedastheratiooftheexperimentalID(sat)tothetheoreticalballisticID(sat(bal)).Sincethecurrentdependsontheinjectionvelocity,weget:B=vinj vinj(bal)=ID(sat) ID(sat(bal)) (4{40) 101

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NatoripredictedthatBwillincreaseasthecarrierscatteringisreducedandwilleventuallyreachunity,theoretically,whenballistictransportisrealized. 4.3.2Lundstrom'sQuasi-BallisticKBT-layerModelToday'snanometricMOSFETsarenotfullyballistic.Assuch,Natori'smodelisnotdirectlyapplicabletomodeltransportinthesedevices.TheHDmodelswhichpredictedtransportreasonablywellforsub-1mdevicesstartstohaveproblemsforsub-100nmdevicesbecauseofthebreakdownofthefollowingimportantassumptionsthatarefundamentaltoit.Thereasonforbreakdownoftheseassumptionsisthehighelectriceldspresentduetoverysmalldimensions,whichcausesnon-uniformandnon-localtransport.Sincetransportisneithercompletelydrift-diusionbasednorcompletelyballistic,LundstromcameupwithaQuasi-BallisticTransport(QBT)modelwhichisanextensionofNatori'stheory.ThephenomenologicaleectivemobilitythatisusedintheMOSFETequationsisthoughttobeindependentofchannellength.However,experimentalmeasurementofthisefromshortchannelMOSFETsisseentobeconsiderablysmallerthanthelongchannelmobility0.Shur[ 132 ]rstlinkedthisdegradationtothenitecarrieraccelerationtimeinthechannel.Toaccountforthis,heintroduceda\ballistic"mobilityB,whichhedenedas: B=qL mcvT(4{41)whereListhechannellength,mcistheconductivityeectivemassandvT=p 2kBT=mcisthethermionicemissionvelocity.TheshortchannelmobilitywasthenformulatedinaMattheison'sruleformas: 1 e=1 B+1 0(4{42)ShurnotedthattheniteaccelerationtimeL=vTinEq.( 4{41 )describesthetimerequiredforacarriertotraveloveralengthLwiththenitevelocityvTgivenatthebeginningofthechannel.Shur'snalI)]TJ /F3 11.955 Tf 12.27 0 Td[(VexpressionissimplyanextensionoftheclassicDD-based 102

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MOSFETequation(Eq.( 4{19b )).ID'W LCG1 1 B+1 0(VG)]TJ /F3 11.955 Tf 11.95 0 Td[(VTH)VD (4{43)WhentheratioB=0issmall,i.eforlongchanneldeviceswherethetransportisnotballisticatall,theaboveequationreducestothetraditionalDDequation.Inthelandmarkpaper[ 133 ]writtenin1997,Lundstromgaveare-interpretationofShur'sEq.( 4{43 )foraquasi-ballisticMOSFEToperatinginsaturationconditions.FollowingMcKelvey'sFlux-basedapproach,theMOSFETistreatedasacascadeofthreeregions:source-channel-drain.Thesourceisareservoirofthermalcarrierswhichinjectsauxofcarriersintothechanneloverapotentialbarrier.Thegatemodulatestheheightofthispotentialbarrier.Nearthebarrier,afractionrofthesourceuxbackscattersfromthechannelandre-entersthesource.Theremainingfraction1)]TJ /F3 11.955 Tf 12.58 0 Td[(rtransmitsandeventuallyentersthedrain.ThisisconceptuallyshowninFig.( 4-2 ).Itwasshownin[ 133 ]thatID(sat)canbecompactlyexpressedas: ID(sat)=Cox(VG)]TJ /F3 11.955 Tf 11.96 0 Td[(VTH)1)]TJ /F3 11.955 Tf 11.96 0 Td[(rsat 1+rsatvT(4{44)wherersatistheratioofthebackscatteredcarriers(i.e'backscatteringratio')undersaturationconditions.vtisthevelocityoftheforwardgoingelectrons(assumingnMOSFET)atthethetopofthesource-channelpotentialbarrier(i.ethe'virtualsource').Thebackscatteredelectronsareassumedtohavethisvelocityaswell.Thisvelocityissimplythe1Dthermalvelocityinthetransportdirection,givenbyr 2KBT mc.Theratio1)]TJ /F3 11.955 Tf 11.95 0 Td[(rsat 1+rsatisreferredtoastheBallisticEciency,B.TheproductofBandvTiscalledtheinjectionvelocity.NotethatthisdenitionisslightlydierentfromNatori'sdenition.TheappealofLundstrom'stheorystemsfromthesimplicityofEq.( 4{44 )thatdescribesthesaturationcurrent.FromMcKelvey'sscatteringtheory[ 134 ][ 135 ],the 103

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Figure4-2.QuasiballistictransportinananometricMOSFET backscatteringratioisactuallywelldenedforzero(orverysmall)electriceldsas: r0=L L+(4{45)whereListhechannellengthandisthemean-freepath(ormoreappropriatelythevelocity/momentumrelaxationlength)oftheelectronsjustnearthevirtualsource.Inthenon-degeneratelimit,theparameterisactuallyrelatedtotheloweld,longchannelmobilityas: =2KBT q0 vT(4{46)Wewilldiscusstheimportofthisrelationinlatersections.BasedonanearlierobservationbyPrice[ 136 ]thatanelectronwhichhastraveled\afewtimes10)]TJ /F5 7.97 Tf 6.59 0 Td[(6cm"inaeldof104V/cm,sohas\descendedfartherthanthishas 104

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littlechanceofgettingbackalltheway"andwill\almostcertainlyreachthe[drain]",(i.e.electronsthathavetraveledpastacriticaldistancefromthevirtualsourcecannotreturnbacktothesource)Lundstromspeculatedthatthebackscatteringratioinhigheldconditionscanbedenedas:rE=l l+ (4{47)Herelissomecriticaldimensionwhichisassumedtobethewidthoftheso-calledKBT-layerin[ 133 ].Torstorder,l=KBT qE(0+),whereE(0+)denotestheelectriceldjustaheadofthevirtualsource(i.eE(0+)'VD=L).ThewidthoftheKBT-layer,asshowninFig.( 4-2 ),issimplythedistancefromthevirtualsourceoverwhichthepotentialdropsbyKBT.Thus, rE=1 1+ l=1 1+20E(0+) vT(4{48)CombiningEqs.( 4{47 )and( 4{46 ),amoregeneralexpressionforthebackscatteringratioinsaturationcanbewrittenas rsat=r0 1+20E(0+) vT(4{49)ToestablishaconnectionbetweenthismodelandconventionalDD-basedmodels,consideradevicewithchannellengthLlongenoughsothatr0'1(i.ersat'rE).Then,usingEqs.( 4{49 )and( 4{44 ),Lundstromcameupwiththefollowingexpressionforthequasi-ballisticlinearMOSFETI)]TJ /F3 11.955 Tf 11.96 0 Td[(Vequation: ID(sat)=WCG8>><>>:1 rsat)]TJ /F1 11.955 Tf 11.96 0 Td[(1 1 rsat+19>>=>>;vT(VG)]TJ /F3 11.955 Tf 11.95 0 Td[(VTH) (4{50a)=WCG8>><>>:1 1 vT+1 0E(0+)9>>=>>;(VG)]TJ /F3 11.955 Tf 11.96 0 Td[(VTH) (4{50b) 105

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ComparethisequationwithShur'sEquation( 4{43 ),whichcanberewrittenas:ID'W LCG1 1 B+1 0(VG)]TJ /F3 11.955 Tf 11.95 0 Td[(VTH)VD (4{51)'WCG1 L VD1 B+1 0(VG)]TJ /F3 11.955 Tf 11.95 0 Td[(VTH) (4{52)ThisshowsthelinkbetweentraditionalmodelsandLundstrom'sKBT-layermodel.In[ 137 ][ 138 ],LundstromandRenfurtherclariedthismodelexplainingthescatteringmechanismandcurrentcontrolmechanisminmoredetail.Webrieydiscusssomeaspectsofthephysicsinvolvedinthenextsection. 4.3.2.1PhysicsofscatteringinKBT-layermodelLundstrompointedoutin[ 138 ]thatthatthemaximumaveragecarriervelocityatthebeginningofthechannelwastheequilibrium,uni-directionalthermalvelocity(asrstpointedoutbyNatori).Fordegenerateconditions,thisvelocityis: vT(VD)=s 2KBT mc(F1=2(F) F0(F))(4{53)whereF=EF)]TJ /F3 11.955 Tf 11.96 0 Td[(E1 KBTwithEFbeingtheFermienergyandE1beingtheenergylevelofthelowestsubband.ThecompleteequationforID(sat)forallVG;VDconditionswasshowntobe: ID(sat) W=Qi(0)1)]TJ /F3 11.955 Tf 11.96 0 Td[(r 1+r(vTF1=2(F) F0(F))8>>><>>>:1)]TJ 13.15 8.51 Td[(F1=2(F)]TJ /F3 11.955 Tf 11.96 0 Td[(UD) F1=2(F) 1+1)]TJ /F3 11.955 Tf 11.96 0 Td[(r 1+rF0(F)]TJ /F3 11.955 Tf 11.95 0 Td[(UD) F0(F)9>>>=>>>;(4{54)where a. ThersttermQi(0)istheinversionchargeatthevirtualsource.ForawelldesignedMOSFET,Qi(0)'CG(VG)]TJ /F3 11.955 Tf 11.96 0 Td[(VTH)abovethreshold. b. ThesecondtermistheballisticeciencyB,whichdescribesthereductionofID(sat)duetobackscattering.TheparameterrisafunctionofbothVGandVD. 106

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c. ThethirdfactoristhedegeneratethermalvelocitywhichdependsonQi(0)throughF. d. ThelastfactoraccountsforthedependenceonthedrainbiasVD.ThisfactorisproportionaltoUD=qVD KBTforlowVDandapproachesunityforhighVDconditions,therebyreducingtotheformofEq.( 4{44 ).Alsonotethatwhenr=0,thisequationreducestoNatori'sballisticMOSFETequation.Thebackscatteringratior(andultimatelyID(sat))dependsonthemeanfreepath()inthecriticalregion.,inturndependsontheloweld,longchannelmobility.ThusitwasarguedthatmobilityisstillanimportantparameterevenfornanometricMOSFETshavingquasi-ballistictransport.LundstromalsoshowedthatthevelocitysaturationoccursinaballisticMOSFET{butagainsttheconventionalpictureofsaturationthroughhigheldopticalphononemission{itwasindicatedthatthevelocityatthevirtualsource(i.ethetopofthebarrier)saturatesatthethermallimit.ForunstrainedSi,thesetwovelocities,i.e.theeldlimitedsaturatedvelocity(vsat)andthemaximumuni-directionalthermalvelocity(vinj=BvT,withB=1)bothhappentobereallyclosetoeachother(107cm/s).LundstromalsoarguedthatthebackscatteringthathappensintheKBT-layeristhemostcriticalscatteringintheMOSFET.Accordingto[ 137 ],onlythisbackscatteringdeterminestheballisticeciencyandtheID(sat).Thereasoninggivenisquitesimple:evenifbackscatteringoccursbeyondthecriticaldistancel,thecarrierswillnothaveenoughlongitudinalenergytosurmountthebarrierandre-enterthesource(Fig.( 4-3 )).Morelikely,thecarrierswillundergoseveralscatteringevents(phonon/impurity),whichwillfurtherreducethelongitudinalenergy,andtheywilleventuallyreachthedrain.ThiswasthereasoningbehindLundstrom'sargumentthatscatteringeventsfarawayfromthesourcedonotaect.ID(sat). 4.3.2.2IssuesintheKBT-layermodelAdvancedMonteCarlosimulationperformedbyPalestri[ 139 ]in2005broughtoutsomeimportantissuesintheKBT-layertheory.Someoftheseissuesstemfrom 107

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Figure4-3.Carrierbackscatteringinquasi-ballistictransport.DoscatteringeventsfarawayfromthecriticalregionaectID(sat)? thefundamentalassumptiononthenatureofbackscatteringratio(esp.inhighVD)anditsdependenceontheloweld,longchannelmobility.Meanwhile,basedonatemperature-basedexperimentalmethodpioneeredbyChen[ 140 ]in2002,severalauthors[ 141 { 156 ]havepublishedwidelyvaryingnumbersandinterpretationsfortheballisticeciencyBinamodernMOSFET,bothwithandwithoutstrain.Werstdiscussthetheoreticalissuesandthensummarizetheexperimentalresultstilldateanddiscussshortcomings.TheoreticalIssuesTosimplifythediscussion,wefollowtheterminologyusedbyPalestri[ 139 ].Atthevirtualsource,thecurrentduetobackscatteredcarriersisdenotedbyI)]TJ /F4 7.97 Tf -.94 -7.29 Td[(onandthecurrentduetoforwardmovingcarriersisdenotedbyI+on.ThebackscatteringratiorissimplyI)]TJ /F4 7.97 Tf -.94 -7.29 Td[(on I+on.IfwecallthetotalcurrentreachingthedraininthecaseofcompleteballistictransportasIbalandinthepresenceof\scattering"asIon,itfollowsthat Ion=1)]TJ /F3 11.955 Tf 11.95 0 Td[(r 1+rIbal(4{55)Onewayofinterpretingthisequationisasfollows:ifwecancalculatethecompleteballisticcurrentusingsayadevicesimulator,thenforarealdevicewhichhasscattering, 108

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allweneedistodeterminertoestimatetheoncurrent.However,theextractionofprecisenumbersforriscloudedbyanumberoftheoreticalandexperimentaluncertainties.ThefollowingisalistofimplicitandexplicitassumptionsintheKBT-layertheory,includingtheapplicabilityandthevalidityoftheassumptions. 1. Atthetopofthebarrier(virtualsource),therearetwostreamsofcarriers(Fig.( 4-2 )),oneforwardmoving(relatedtoI+on,eventuallycontributingtothetotalIon)andonebackwardmoving(i.etheKBT-layerbackscatteredcarriers,relatedtoI)]TJ /F4 7.97 Tf -.93 -7.29 Td[(on).ThesetwostreamsareassumedtohaveequilibriumMaxwelliandistributionsandhavetheaveragevelocitiesv+andv)]TJ /F1 11.955 Tf 7.08 -4.33 Td[(,bothequaltothenondegenerate,unidirectionalthermalvelocityvT.Natorirstdiscussedthevalidityofthisin[ 157 ].Morerecentfullbandself-consistentMonte-Carlosimulations[ 139 ][ 151 ]showthatwhilev+wasapproximatelyvT,v)]TJ /F1 11.955 Tf -412.88 -18.78 Td[(wasonly'0:7vT.Ifthisisnotaccountedfor,rsatisoverestimatedby10%(andtheIonby7)]TJ /F1 11.955 Tf 8.72 0 Td[(8%).Inaddition,theassumptionthatthepositiveandnegativevelocitycarriershaveaMaxwelliandistributionhasbeenshowntobeinaccurate,esp.underhighVDconditionswherestrongnon-equilibriumtransporttakesplace. 2. CarriersinjectedintothechannelfromthesourceoccupystatesatthetopofthebarrieraccordingtotheFermilevelofthesource.TheKBT-layermodelassumesthatthechargeatthetopofthebarrierisCG(VG)]TJ /F3 11.955 Tf 12.85 0 Td[(VTH).ThisisareasonableassumptionforawelldesignedMOSFET,providedshortchanneleects(DIBLandS-Ddrainleakage)areundercontrol. 3. Underlowbiasconditions,risgivenbyrlin=L L+,whereisthemomentumrelaxationlengthrelatedtotheloweldlongchannelmobilityandListhechannellength.Theexpressionforrlinactuallyhasasoundfundamentalbasis(McKelvey'sFluxappraoch[ 134 ]).Itwasalsoshownin[ 158 ]thatthisrelationcanbeactuallyderivedfromtheDrift-Diusionformalism.NotethattheassumptionsoftheDDapproacharevalidforlowVDconditionseveninananoscaledevice.Thereissomedebateaboutthishowever,butforthemostpart,assumingthattheRTAisvalidisnotothemark.Thususageofofarelaxationlength=2KBT q0 vTrelatedtomobilitythusmakessense. 4. Underhighdrainbiasconditions,risgivenbyrsat=l l+,wherelisthelengthofthecriticalregionwherethebackscatteringhappens.Itwasassumedin[ 137 ]thatthislengthwasthewidthoftheKBT-layerlKT.stillremainsasdenedearlier. 109

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In1979,Priceperformed[ 136 ]someMonteCarlosimulationsoncarriertransport.Heshowedthatthatwhencarriersmovedownapotentialbarrier,notmanyofthemreturnedtothetopofthebarrier(injectionpoint)eveniftheyscatteredwhile/aftermovingdownaveryshortdistancedownthebarrier.Price'sresultsweresimulatetothoseofBethewhoshowedthatthermioniccurrentlimitationinforwardbiasedmetal-semiconductorjunctionisdeterminedbya\zone"ofwidthoverwhichthepotentialdropsbyavalueofnearlyKBT=qwhichwasconrmedin1985byBerz[ 159 ].LundstromreasonedthatthecriticallayerforbackscatteringinaMOSFETisalsoroughlythedistanceoverwhichthechannelpotentialdropsbyKBT=q,whichisatinyfractionofthechannellength.Thusheextendedtheexpressionforbackscatteringinaeld-freeslab(r=L=L+)toaMOSFET(underahighdrainbiasVD)asrsat=l=l+wherelisthewidthoftheKBTlayer.MonteCarlosimulationsin[ 139 ]showthatactualcriticallayerisnotquitethedistanceneededtodropKBT=qofpotential{itisslightlyhigher.Itisnotpossibletoestimateexactlyhowmuchhigherwithoutadvancedsimulations.Thiswasacknowledgedinarecentpaper[ 160 ]byKimandLundstrom.Inadditiontotheabovepoints,itwasalsoarguedin[ 137 ]thatbecausethecriticalbackscatteringoccursinaregionwherethecarriershavegainedlittleenergyfromthechanneleld,itwasappropriatetousethesamemean-free-pathasusedinthelinearcaseabove.ThevalidityofthisassumptionhasbeenquestionedbyFischetti(seeappendixAandBin[ 161 ]).WetouchuponthisassumptionbrieyinSection 4.4.2 wherewediscussstraineectsinmoredetail.Inthesamework[ 161 ],FischettialsodiscussessourcestarvationandlongrangecoulombeectsandquestionsifitwillbeeverpossibletoattainballistictransportproperinaMOSFETevenifwescaledowntosub-10nm. 5. Scatteringeventsdeepinsidethechanneldoesnotaectr,andthusIon.Asmentionedearlier,thereasoningforthisassumptionisthatevenifcarriersundergoscatteringinsidethechannel,theydonothaveenoughlongitudinalenergytosurmountthebarrierandre-enterthesource.Tounderstandwhy,itwasarguedthatasacarriermovesdeeperintothechannel,itsenergyincreaseswhichraisesitsprobabilityofscatteringbyphononemission,whichlowersitsenergyevenmoreandmakesitlesslikelytoreturntothesource.Evenifitdoesgetelasticallybackscattereddeepinsidethechannel,itmightundergoadditionalscatteringeventsonitsjourneytothesource.Asucientnumberoftheseeventswilleectivelythermalizethecarrierenergyandmakeitimpossibletosurmountthepotentialbarrier.Inaddition,theinterfaceroughnessscatteringisstrongnearthebeginningofthechannelandreducesasthecarriermovesclosertothesource{whichmeansthatthecarriersfarawayfromthesourcehavealesserchancetobeelasticallybackscatteredbythismechanism.Thusitwasconcludedthatchannelscatteringdoesnotchangerasdenedearlier,andtherebydoesnothaveanimpactonIon.However,thisconclusionisnotright. 110

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InelasticscatteringinsidethechannelactuallyaectstheheightofthepotentialbarrierbetweenthesourceandchannelthroughthecouplingviaPoisson'sequation.Thepotentialbarrieratthevirtualsourceisnot`frozen'atitsequilibrium,zeroVDvalue{itisaectedbyscatteringinthechannel[ 162 ].Thebalancebetweenforward-moving(v+)andbackward-goingcarriers(v)]TJ /F1 11.955 Tf 7.09 -4.34 Td[()islostonceacurrentIonstartsowinginthedevice.Thesurfacepotentialatthevirtualsourcewillattempttocontinuouslyreadjust,butitwillneverbeabletocometoitsequilibriumvalue.Instead,anewequilibriumwillbeattaineddependingonthegateanddrainvoltages.Theresultofthisnewequilibriumisanincreaseinthebarrierheight(see,forinstanceFig.(10)in[ 137 ]andFig.(9)in[ 139 ]),whichwillresultinadierentvalueforrcomparedtothefrozeneldcase.ThisfeedbackbetweenbarrierheightandscatteringisnottakenintoaccountintheKBT-layertheory.Westronglystressatthispointthatthiseectduetochannelscatteringcannotbeignoredtoday'sMOSFETs.Inthepresenceofstrain,whichaectsmaterialpropertiesinafundamentalway,thisfeedbackmechanismneedscarefulconsideration,Natorihascomeupwithanewtheoryexplaininghigheldtransportinsemiconductors(discussedbelowinSection( 5.4.3 )),whichdelvesintotheoriginofthisfeedback/couplingbetweeninelasticscatteringandbackscattering.WethinkthatLundstrom'sandNatori'smodelstakentogetherwillhelpshedlightonstraineectsinMOSFETsunderhighelds.ExperimentalIssuesInthelastsectionwediscussedsomeunderlingissuesintheKBTlayertheory.Inthissection,werstsummarizethemeasurementmethod(s)thatexistforexperimentallyextractingthebackscatteringratiorinaMOSFET.Thenwepresentadetailedanalysisofseveralassumptions(implicitandexplicit)oftheexperimentalmethodandhowtheseassumptionsdistortthemeasuredquantity(i.er)itself.Wewillrevisitsomeofthesebrieyinthenextchapterwherewediscussthemainproposalofourresearchstudy.TherstattemptatexperimentallymeasuringtheBallisticEciencyBofaMOSFETusingtemperaturebasedmeasurementswasproposedbyChenin2002[ 140 ][ 141 ].Lee[ 142 ]andLin[ 145 ][ 146 ]appliedthistechniquetostrainedMOSFETsandobservedthatstrain(uniaxialtensionforn-MOSFETsanduniaxialcompressionforp-MOSFETs)seemedtohaveabenecialeectonquasi-ballistictransport.Furtheradvancementsintheexperimentalmodelsused[ 147 { 149 ]extendedthetechnique,butalsoproducedmoredebateontheunderlyingphysicsitself. 111

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OneoftheimportanttopicsthatisstillbeingactivelyresearchedtodayisthenatureofthecurrentlimitingmechanisminananometricMOSFET.WhichofvsatandvinjlimitsIoninaMOSFET,andhowstrainaectshigheldtransportisstillunderconsiderabledebate.Severalauthors(see,forinstance,[ 153 ])indicatetransportisnotballisticbutstilldominatedbydiusivemechanismsevendownto45nm,whileothersreport[ 163 ][ 164 ][ 165 ]thatdevicestodayoperateanywherebetween50{80%oftheirballisticlimit.Wesummarizethemethodrstandthendiscusswhythereissomuchdiscrepancyreportedinliterature,especiallyforStrainedMOSFETs.LowTemperatureMethodforDeterminingBackscatteringRatioThecompleteMOSFETI)]TJ /F3 11.955 Tf 11.68 0 Td[(VequationaccordingtotheKBT-layermodelisrepeatedbelowas ID(sat) W=Qi(0)1)]TJ /F3 11.955 Tf 11.96 0 Td[(r 1+r(vTF1=2(F) F0(F))8>>><>>>:1)]TJ 13.15 8.52 Td[(F1=2(F)]TJ /F3 11.955 Tf 11.96 0 Td[(UD) F1=2(F) 1+1)]TJ /F3 11.955 Tf 11.96 0 Td[(r 1+rF0(F)]TJ /F3 11.955 Tf 11.95 0 Td[(UD) F0(F)9>>>=>>>;(4{56)Underfullsaturationconditions,thisequationbecomes ID(sat) W=CG(VG)]TJ /F3 11.955 Tf 11.95 0 Td[(VTH)1)]TJ /F3 11.955 Tf 11.96 0 Td[(rsat 1+rsatvtherm(4{57)wherevtherm=vTF1=2(F) F0(F).DierentiatingEq.( 4{57 )w.r.ttemperature, 1 ID(sat)@ID(sat) @T=)]TJ /F1 11.955 Tf 9.3 0 Td[(1 VG)]TJ /F3 11.955 Tf 11.96 0 Td[(VTH@VTH @T)]TJ /F1 11.955 Tf 56.94 8.09 Td[(2 (1+rsat)(1)]TJ /F3 11.955 Tf 11.96 0 Td[(rsat)@rsat @T+1 vtherm@vtherm @T(4{58)Tobeabletoestimatersatfromtheaboveequation,weneedtwoquantities,@vtherm @Tand@rsat @T.Therstofthesetwotermsisfairlyeasy: vtherm=vTF1=2(F) F0(F)=s 2kBT mcF1=2(F) F0(F)=T (4{59a) 112

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@vtherm @T=T)]TJ /F5 7.97 Tf 6.58 0 Td[(1 (4{59b)1 vtherm@vtherm @T= T (4{59c)Here,thetemperaturesensitivityofthedegeneratedthermalvelocity,issomenumberthatisdierentthan0.5(accountingforthetemperaturedependenceoftheFermi-Diracterm).Theratio@rsat @Tisnoteasytodetermine.rsatdependsonbothonandl,whichthemselveshavecomplicatedrelationsonmanytemperaturedependentterms.Wehave, rsat=1 1+ l (4{60a)@rsat @T=)]TJ /F1 11.955 Tf 9.29 0 Td[(1 1+ l2@(=l) @T (4{60b)Weneedanexpressionfor=l.ThemodelforasdiscussedinSection( 4.3.2 )dependsinmobilityandthedegeneratedthermalvelocityas: =2KBT q0 vthermF0(F) F)]TJ /F5 7.97 Tf 6.59 0 Td[(1(F)(4{61)Thecriticalwidthl,simplybeingthewidthoftheKBT-layerinLundstrom'smodel,woulddependontheslopeoftheconductionbandprolejustnearthevirtualsource.ThisslopecanbeexpressedasafunctionofchannellengthLandthedrainbiasVDThus,Lundstromexpresseditas: l=LKBT qVDl(4{62)whereandlaresomeempiricalconstants.Lundstromspeculatedthatthecurvatureofthepotentialproleatthebeginningofthechanneldependsontheparameterl,whichiscontrolledbythetransportmodelandbyself-consistentelectrostatics.(i.eV(x)/(x)1=l). 113

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Itisshownin[ 138 ]that0:66l0:75andissomewhatgreaterthan1.Mostauthorsassume=1.ThuscombiningEq.( 4{61 )andEq.( 4{62 )asimpleexpressionfor=lcanbewrittedas l'2KBT qL0 vthermF0(F) F)]TJ /F5 7.97 Tf 6.59 0 Td[(1(F)qVD KBTl (4{63a)/T1++)]TJ /F4 7.97 Tf 6.58 0 Td[(l)]TJ /F4 7.97 Tf 6.59 0 Td[( (4{63b)where isthetemperaturesensitivitiesofthedegeneratefactorF0(F) F)]TJ /F5 7.97 Tf 6.59 0 Td[(1(F) isthetemperaturesensitivityoftheloweldmobility(simeq)]TJ /F1 11.955 Tf 11.95 0 Td[(1:5) listhetemperaturesensitivityoftheltermasexplainedabove isthetemperaturesensitivityofthevthermterminthedenominatorGeneralizingtheexpressioninEq.( 4{63 )andusing=1++()]TJ /F3 11.955 Tf 11.96 0 Td[(l),weget l/T)]TJ /F4 7.97 Tf 6.58 0 Td[( (4{64a)@(=l) @T/()]TJ /F3 11.955 Tf 11.96 0 Td[()T)]TJ /F4 7.97 Tf 6.58 0 Td[()]TJ /F5 7.97 Tf 6.58 0 Td[(1 (4{64b)Itiseasytoshowthat @(=l) @T=)]TJ /F3 11.955 Tf 11.96 0 Td[( TT)]TJ /F4 7.97 Tf 6.59 0 Td[( l (4{65a)SubstitutingthisinEq.( 4{60 ), @rsat @T=)]TJ /F1 11.955 Tf 9.3 0 Td[(1 1+ l2@(=l) @T (4{66a)=)]TJ /F1 11.955 Tf 9.3 0 Td[(1 1+ l2)]TJ /F3 11.955 Tf 11.96 0 Td[( TT)]TJ /F4 7.97 Tf 6.59 0 Td[( l (4{66b) 114

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=)]TJ /F1 11.955 Tf 9.3 0 Td[(1 1+ l=l 1+ l)]TJ /F3 11.955 Tf 11.96 0 Td[( TT)]TJ /F4 7.97 Tf 6.59 0 Td[( (4{66c)=)]TJ /F3 11.955 Tf 9.3 0 Td[(rsat(1)]TJ /F3 11.955 Tf 11.95 0 Td[(rsat))]TJ /F3 11.955 Tf 11.96 0 Td[( TT)]TJ /F4 7.97 Tf 6.58 0 Td[( (4{66d))]TJ /F1 11.955 Tf 9.3 0 Td[(2 (1+rsat)(1)]TJ /F3 11.955 Tf 11.96 0 Td[(rsat)@rsat @T=2rsat (1+rsat))]TJ /F3 11.955 Tf 11.95 0 Td[( TT)]TJ /F4 7.97 Tf 6.59 0 Td[( (4{66e)=2 (2+=l))]TJ /F3 11.955 Tf 11.96 0 Td[( TT)]TJ /F4 7.97 Tf 6.59 0 Td[( (4{66f)SubstitutingEq.( 4{59 )andEq.( 4{66f )intheoriginalEq.( 4{58 )forthetemperaturedependenceofID(sat),wehave1 ID(sat)@ID(sat) @T=)]TJ /F1 11.955 Tf 9.3 0 Td[(1 VG)]TJ /F3 11.955 Tf 11.96 0 Td[(VTH@VTH @T+2 (2+=l))]TJ /F3 11.955 Tf 11.96 0 Td[( TT)]TJ /F4 7.97 Tf 6.59 0 Td[(+ T (4{67)Solvingfor=l,wegetananalyticalexpressiondependentonthetemperaturedependencesofthesaturationcurrentandsaturationthresholdvoltage, l=)]TJ /F1 11.955 Tf 9.3 0 Td[(2()]TJ /F3 11.955 Tf 11.96 0 Td[() )]TJ /F8 11.955 Tf 11.96 20.45 Td[( 1 ID(sat)@ID(sat) @T+)]TJ /F1 11.955 Tf 9.3 0 Td[(1 VG)]TJ /F3 11.955 Tf 11.96 0 Td[(VTH@VTH @T!)]TJ /F1 11.955 Tf 11.95 0 Td[(2 (4{68)Once=lismeasured,wecanveryeasilyestimatersatandtherebytheBallisticEciencyB=1)]TJ /F3 11.955 Tf 11.95 0 Td[(rsat 1+rsat.Eq.( 4{68 )isthebasisfortheBestimatesthathasbeenwidelyreportedinliteratureforbothstrainedandunstraineddevices.PotentialPitfallsThelowtemperaturemethodjustdescribedisveryattractivebecauseallitneedsasinputishowfasttheID(sat)andVTH(sat)changewithtemperature,whichcanbeobtainedfromasimplemeasurement.However,themeasuredvaluefor=ldependsontheassumptionswemakeforthe()]TJ /F3 11.955 Tf 10.79 0 Td[()factor.Rememberthat()]TJ /F3 11.955 Tf 10.79 0 Td[()=1++()]TJ /F3 11.955 Tf 10.79 0 Td[(l))]TJ /F3 11.955 Tf 10.79 0 Td[(.Thediscrepancyinthenumberspublishedbyvariousauthorsarisefromthedierencesinthevaluesfortheseparameters.Thereisnoclearconsensusonthepropervaluesfortheseparametersinliterature.However,someguidelinesexist: 115

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1. FromTakagi'swellknownworkonuniversalmobility[ 166 ][ 167 ],0/T)]TJ /F5 7.97 Tf 6.58 0 Td[(3=2whichimplies')]TJ /F1 11.955 Tf 21.92 0 Td[(1:5. 2. Torstorder,asLundstromshowed[ 138 ]that0:66l0:75.Mostauthorsassumel'0:7.Some[ 155 ]assumeavalueof1forsimplicitywithoutanyjustication.However,recentlyamoreaccuratemodelforlobtainedbyusingaparabolicpotentialprolehasbeenproposed[ 149 ].Thismodelindicatesthatlmightactuallybecloserto1thanthoughtbefore(seediscussionbelow). 3. Inthenon-degeneratelimit,thetemperaturesensitivityofthedegeneracyfactorF0(F)=F)]TJ /F5 7.97 Tf 6.59 0 Td[(1(F),'0.However,thisvaluecannotbeusedinfullysaturatedconditionsbutsomeauthorscontinuetouseitforsimplicity,whichleadstoanover-estimationofB. 4. decreasesfrom0.5fornon-degenerateconditions(F!0)to0forfullydegenerateconditions(F!1).Theintermediatevaluesareverydiculttopredictwithoutcomplexself-consistentMonte-Carlosimulations.Someauthorsuseavalueof0.5forsimplicity.Lee[ 155 ]proposedanadvancedself-consistentprocedureforestimatingboth()]TJ /F3 11.955 Tf 12.17 0 Td[()and=lsimultaneously.Inthiswork,itwasshownthatassumingconstantvaluesforandlleadstonon-physicalvaluesfor=l.HowevertheauthorsassumedthatthemodelforKBT-layerwidthistherstordermodelproposedbyLundstromthatwediscussedearlier.Thismakesusquestionthenumbersthatwasshownin[ 155 ].Whiletheoreticallysound,forthismethodtobeaccurate,weneedtouseapropermodelforKBT-layerwidthl.Chen[ 149 ]proposeda(inouropinion)veryaccuratemodelforlthatwedescribebelow.Weaddthismodeltotheself-consistentprocedureproposedbyLeeandexplainitsuseinamoreadvancedexperimentdescribedinthenextchapter.Lundstrom'smodelforlisasimpleequation{l=LKBT qVDl.Theslopeofthepotentialbarrier(thatdeterminesl)dependsbothonVGandVD.However,Lundstrom'sequationdoesnothaveadirectdependenceonthegatevoltage.Thedependenceactuallycomesthroughtheparameterlforwhichvariousgroupshavereportedvaluesbetween0.5and1.Chenproposed[ 149 ]acleverwaytodealwithanunknownl.InsteadofhavingtheVGdependencethroughl,itwasmovedtotheLterminthefrontandhavea 116

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constantvalueof0.5forli.e. l(VD;VG)=~LKBT qVD0:5(4{69)where~L=L(VG).ThejusticationgivenwasthatthechannellengthismodulatedbythegatevoltageandifwebringintheVGdependencethisway,itaccountsforthechangeinKBT-layerwidthduetoVGautomatically.Chenfurthergaveanexpressionfor~(L)as ~L=L(VD)0:25 (VG)]TJ /F3 11.955 Tf 11.95 0 Td[(VTH)0:5KBT q0:5(4{70)wherewasaconstantthatwasshowntobe4.1(unitsofV0:25)andLwasthechannellengthignoringVGeect(i.ethemasklength).Thusthenalexpressionforlwasgivenas: l=L(VD)0:25 (VG)]TJ /F3 11.955 Tf 11.95 0 Td[(VTH)0:5KBT q0:5KBT qVD0:5(4{71)Eq.( 4{71 )explainswhyatemperaturesensitivityofl=1usedbysomeauthorsgaveveryreasonableresults.Thisisthemodelwewillbeusingforourexperimentbutwewilldetermineforoursamples.Atthispoint,itisagoodexercisetopauseandtakestockofwherewestand.Wehaveareasonablyclearunderstandingoftheissuesinthetheoryandpotentialpitfallsintheexperimentalmethod.However,whenwethinkofstrainanditsimpactontransportinnanoscaleMOSFETs,someverypertinentquestionsstillremainunanswered. 4.4UnderstandingStrainEectsonQuasi-BallisticTransportItisclearfromthemodelthatloweldmobility0playsanindirectroleindeterminingB.Sincetheeectofstrainonloweldmobility(albeitinlowVDconditions)isclearlyunderstood,itisnotastretchtoextrapolatethatknowledgetosaythatbenecialstrain(uniaxialtensionforn-MOSFETanduniaxialcompressionforp-MOSFET(100)surface)willalsohaveabenecialimpactonB.Theproblemliesinknowinghowmuch. 117

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ForananoscaleMOSFET,itisnotpossibletoextractmobilityandinjectionvelocityindependently.However,sincetheKBT-layermodelusesonlylongchannelmobility,itispotentiallypossibletoestimatetheeectofstrain.Sincersatdependson=land/0=vT,experimentallymeasuringBforastraineddevicegivesastraightforwardmethodtounderstandtheeectonquasi-ballistictransport.However,inadditiontochangingthescatteringrates,strainwillalsomodulatethefeedbackbetweeninelasticscatteringandbackscatteringasnotedinprevioussections.Evenwithadvancedmodels,thetemperaturebasedexperimentalmethodcannotaccountforthisatall,sinceitisabsentintheunderlyingtheoryitself.Weneedabetter,clearermethodforunderstandingstraineects.BeforeLundstrom'stheorycameabout,higheldtransportwasthoughttobefullycontrolledbyvelocitysaturation,whichwasunderstoodasabulkmaterialpropertythatisdependentonmaterialparameters(e.gdoping,temperature).Mostresearcherstendtoconvergeonthethemethatsaturationvelocitydoesnotsensitivelydependontheloweldmobility.However,asmentionedinearliersections,thereseemstobenoconsensusonthefactorslimitingthisvelocityandhencethecurrent. Figure4-4.ElectronDistributionintherstBrillouinzonereproducedfrom[ 124 ]fora250nmMOSFET.Fromlefttoright,theguresshowdistributionat(a)x=35nm(nearthesource)(b)(a)x=150nm(middleofchannel)(c)x=265nm(insidethedrain).Thedistributionwouldbesimilarforasub100nmdevicec1988APS. 118

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(a) (b) Figure4-5.Variationof(a)electrondriftvelocityand(b)electronenergyinthedirectionoftransportforthesame250nmdevice,from[ 124 ]c1988APS. Ithasbeenquestionedwhetherstrainaltersthemagnitudeofthesaturationvelocityitself.Thegeneralconsensussofarseemstobenegative[ 42 ]withthereasoningalongfollowinglines.Theenergyofthecarrierswhentheyattainsaturationvelocityisverylargeandhencetheyarenolongerlocatedatthebandminimum.Notingthatthecarrierswouldbewellspreadinthek-pointsinthereciprocalspace(seeFig.( 4-4 )),theeectofstressisthoughttobeminimalandmostoftheobservedchangeinID(sat)isthoughttobesomehowrelatedtostrain-alteredloweldmobility.Webelievethatthispictureisnotcompletelycorrectfortworeasons: 1. Afteraccountingfortheparasiticsource-drainresistanceRSD,westillseeappreciableID(lin)andimportantlyID(sat)enhancementevenforMOSFETswithgatelengthsaslowas90nm[ 41 ].Notethatthesedeviceshavesignicantnon-localtransportandvelocitysaturation,ifthelimitingfactorforID(sat),wouldhavehappenedatmuchlowervoltagescomparedtoa10mdevice. 2. State-of-the-artMOSFETsarenotcompletelyballisticyet,andstraincontinuestoimproveperformanceateverynode.Unlessthereisverysignicantvelocityovershoot(seeFig.( 4-5 )from[ 124 ]thatshowstypicalovershootina250nmdevice),explainingstrainenhancementisnotatalleasy.Inthenextfewpages,wediscusstheoriginofsaturationvelocity,itslimitingfactors,itsroleinMOSFETtransportandhowitisrelatedtotheinjectionvelocity. 119

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4.4.1OriginofSaturationVelocityIntheparabolicbandapproximation,theenergyspectrumofasemiconductorisgivenby: Ek=Ec3+~2 2m(k2x+k2z+k2z)(3Dbulk) (4{72a)Ek=Ec2+~2 2m(k2x+k2z)(2DMOSFET) (4{72b)Ek=Ec1+~2 2m(k2x)(1Dnanowire) (4{72c)Whenthereisnoeldapplied,thecarriermotioniscompletelyrandom,beit3D,2Dor1D.Theindividualcarriervelocitiesarerandomlyorientedindirection(i.evectorsumiszero)sothatthereiszerocurrent.Themagnitudeofthisintrinsicvelocityisgivenby=r 2Ek mwiththeappropriateEk.TheweightedaverageofbymultiplyingwiththeFermi-Diracdistributionandtheappropriatedensityofstatesgiveswhatisknownasthe\ultimate"orthemaximumvelocityvuacarriercanhaveforeachofthesecases.Arora[ 168 ]arguedcarrierstravelwiththisultimate(orballistic)velocityin-betweenscatteringeventsandthatinthenon-degenerateregime,thesevelocitesaregivenbythefollowingequation(wherei=3;2;1fordimensionality): viu=viTFi)]TJ /F5 7.97 Tf 6.58 0 Td[(1 2(iF) Fi)]TJ /F5 7.97 Tf 6.58 0 Td[(2 2(iF)(4{73)where { viT=)]TJ /F8 11.955 Tf 9.31 9.68 Td[()]TJ /F4 7.97 Tf 6.67 -4.98 Td[(i+1 2 )]TJ /F8 11.955 Tf 9.3 9.69 Td[()]TJ /F4 7.97 Tf 7.35 -4.98 Td[(i 2r 2KBT m { )-326(istheGamma-functiondenedas\(n+1)=n!with\(1=2)=p { Fi(i)=1 \(i+1)R10xi 1+exp(x)]TJ /F3 11.955 Tf 11.96 0 Td[(i)dxwherej=EF)]TJ /F3 11.955 Tf 11.96 0 Td[(Eic KBT 120

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FromEq.( 4{73 )itiseasytoshowthatthe\ultimate"velocitiesforthe3D/2D/1Dcasesaregivenby: v3Du=2 p r 2KBT mF1(F) F1=2(F) (4{74a)v2Du=p 2r 2KBT mF1=2(F) F0(F) (4{74b)v1Du=1 p r 2KBT mF0(F) F)]TJ /F5 7.97 Tf 6.59 0 Td[(1=2(F) (4{74c)NotethatLundstrom'sdentionofnon-degenerateballisticmobilityr 2KBT mF1=2(F) F0(F)astheisnotthesameasArora'sdenitions.Infact,itseemstobeacombinationofEq.( 4{74a )orEq.( 4{74b ).ThisdiscrepancyarisesfromthefactthatLundstromuseshalfof2 r 2KBT mastheinjectionvelocitywhichwasderivedbyaveragingoverone-halfoftheMaxwelliandistributionofelectronsinjectedintothe2Dchannelfromthe3Dsource.Aroraassertsin[ 168 ]thatLundstrom'sdenitioniserroneousandhealsopointsoutusingthisincorrectvaluedoesnotchangethesaturationvelocity(andhenceID(sat))sincethesaturationvelocityisdeterminedbyscatteringprocesses.Arorafurthershowedthatinthefullydegenerateregime(ieF!1),thisultimatevelocitybecomesafractionoftheFermivelocityandisgivenby: viu(degenerate)=2i i+1~ mp ")]TJ /F8 11.955 Tf 9.31 16.86 Td[(i+2 2nd 2#1=i(4{75)Again,itiseasytoshowthatforthe3D/2D/1Dcases,thisbecomes v3Du(degenerate)=3 4h m3n 81=3 (4{76a)v2Du(degenerate)=2 3h mn 41=2 (4{76b)v1Du(degenerate)=h mn 8 (4{76c) 121

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However,whenthereisaeldapplied,theconductionsandthevalencebandstilt,andthecarriershaveanetvelocityinthetransportdirection.Iftherewerenoscatteringwhatsoever,thencarrierswouldtravelwiththeappropriatenon-degenerateordegenerate\ultimate"velocityshownabove.Thisiswhatisknownasballistictransportandtheresultantvelocityistheballisticvelocity.Inreality,eveninabulksemiconductor,therearevariousintrinsicandextrinsicscatteringmechanismspresentdependingonthebandstructure,dopingetc.Theseaectcarriertransportsignicantly.Itisobservedthat,forlowelds,thecarrierdriftvelocityisproportionaltotheappliedelectriceld.Astheeldisincreased,initiallythedriftvelocityalsoincreaseslinearly.Beyondacriticalvalue,thevelocitydoesnotincreaseanyfurther.Thismaximumvalueofthedriftvelocityiscalledthesaturationvelocity.Itisseenthatthissaturationvelocityisquitelowerthanthe\ultimate"ormaximumcarriervelocitydiscussedabove.Ryderrstpublished[ 169 ]experimentaldataonvelocitysaturationinSiandGein1953(SidataisreproducedinFig.( 4-6 ).Eventhoughthisdataismorethanhalfacenturyold,itisstillveryrelevant.ItisseenthatelectronsinSiattainasaturationvelocityofabout1107cm/sat300Kwhentheelectriceldreachesthevalueof2104cm/s.Theholevelocitysaturatesat7106cm/sforaeldof5104V/cmatroomtemperature.NotethatRyder'sdataisforbulksemiconductors.ItiswellknownthatforSi,theelectron-phononscatteringdominatesallotherscatteringmechanismsinhighelds.Intheclassicalpicture,inhighelds,onceacarriergainsenoughenergyfromtheeld(63meV),itemitsanopticalphononandrelaxes.Thisprocesscontinuestillitreachesthedraincontact.Sincethemaximumcarrierenergyistruncatedat63meV,themaximumvelocitycarrierscanattainwasthoughttobeapproximatelyvd(sat)=r 2Eop m'1:43107cm/sforelectronsand2107cm/sforholes,whichisnottoofarfromRyder's'experimentalvalues. 122

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(a) (b) Figure4-6.Ryder'svelocitysaturationdatafor(a)n-typeand(b)p-typeSifrom1953[ 169 ]. 123

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Itisonlyveryrecently(2009)thatacomprehensivehigh-eldtheoryhasbeenproposed(Natori[ 11 ][ 12 ])thatexplainsthefundamentalphysicsbehindvelocitysaturationinexcellentdetail.WesummarizeNatori'stheoryinthenextchapter,notingthatthistheoryiscrucialtounderstandstraineects. 4.4.2StudyingtheImpactofStrainAsseeninChapter2,strainaltersthebandstructureproducingchangesineectivemassandscatteringrates.InaMOSFET,thecombinedeectofthesechangescanbeexperimentallyveriedbyobservingthechangeineectivemobilitywhichcanbeextracteddirectlyfromID(lin).However,aswehavediscussedinprevioussections,inhigheldconditions(highVGandVD)wherethedeviceoperatesinstronginversionandstrongsaturationconditions,itisnoteasytoquantifystraineectssincecarriertransportissignicantlyoequilibrium.Lundstrom'sKBT-layermodelprovidestheballisticeciencyparameterBwhichcanbeusedasaparametertoquantifystraineects.Lundstrompresentedthefollowingequationsontherelationshipbetweenmobilityanddraincurrent[ 170 ] ID(lin) ID(lin)=0 0(1)]TJ /F3 11.955 Tf 11.95 0 Td[(Blin)'0 0 (4{77a)ID(sat) ID(sat)=0 0(1)]TJ /F3 11.955 Tf 11.95 0 Td[(Bsat) (4{77b)where0referstotheloweld,longchannelmobility.Khakirooz[ 164 ]presentedacomprehensiveoverviewofthehistoricalperformancetrendsofSiMOSFETsandhaveemphasizedthestrongconnectionbetweenmobilityandinjectionvelocityextractedfrompublishedexperimentaldata(seeFigs.7{9in[ 164 ]),notingthatthevelocityatthevirtualsourcesaturatesandevendecreases3forgatelengthssmallerthanabout30-40 3Thiswasattributedtoeithertoadditionalimpurityscatteringinheavily-dopedchannelsorhalo,ortohighersurfaceroughnessscatteringatthehigherinterfacialelds,ortotheincreasingimportanceofparasiticeects. 124

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nm.Khakiroozprovidesthefollowingexpressionsfortherelationbetweenvx0(injectionvelocityatvirtualsource),andmobilityinvolvingtheballisticeciencyvx0 vx0=[k1+(1)]TJ /F3 11.955 Tf 11.95 0 Td[(k1+k2)(1)]TJ /F3 11.955 Tf 11.96 0 Td[(B)] (4{78)wherek1andk2areconstants.NotethatKhakiroozusesthemobilityintheactualdeviceintheaboveexpressionviatheusualexpressionID=W LQinvVDunlikeLundstrom(whousesloweldlongchannelmobility).Fromtheanalysisofhistoricaldata,itwasfurthershown(Fig.( 4-7 ))thattheratiobetweenrelativechangeinvelocitytorelativechangeinmobilitywasabout0.85,whichwashigherthanthethenbelievedvalueof0.5forBsat.WhileFig.( 4-7 )seemstoprovideastrongsupporttoLundstrom'smodel,thisrelationistrivialasFischettishowsin[ 161 ].Fischettistressesthatthatthemobilityextractedinashortchanneldeviceusingtheusualmethodisaphenomenological4mobilitywhichistotallyunrelatedtothelow-eldlongchannelmobility.Asisextractedfromashort-channeldevice,theexistenceofequilibriumconditionsevenatlowVDisnotguaranteedduetotherelativelylargeelds.ThiswasprovedbyusingadvancedMonte-Carlosimulations(seeFig.(17)in[ 161 ]).Multipleauthors(mostrecently[ 152 ][ 155 ])haveusedLundstrom'smodeltounderstandeectofuniaxialstraininshortchanneldevicesandhavereportedthatstrainseemstoimprovetheballisticeciencyBsatbymodulationof/=vthermandmodulationofl(KBT-layerwidth).Toourknowledge,noneoftheseworkstaketheeectofchannelscatteringintoconsideration.StrainaltersthescatteringmechanismsconsiderablyasdiscussedinChapter2(especiallyforholes)andwithouttakingthisintoaccount,theresultsaboutBsatmodulationbystrainwouldbeinaccurate.Also,aswementionedearlier,Lundstrom'smodelusesadierentvalueforthe2D-thermalvelocity 4Actually,evenLundstrom'sandShur'sdenitionsofa\ballisticmobility"ispurelyphenomenological. 125

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Figure4-7.Khakiroozmodelforquantifyingstraineects.From[ 164 ]c2008IEEE. aspointedoutbyArora[ 168 ].Combinedwiththeinaccuraciesinthetemperaturebasedexperimentalmethod,thisleadstoalotofpossibleerrorandprobablemisinterpretationoftheeectsofstrain.Forexample,Saitoh[ 154 ]evenproposestheinclusionofanadditional\hot-carrier"termintheLundstromFrameworktoexplainhisstrainresults.ToverifytheapplicabilityoftheLundstromModelforstuydingstraineects,weextractedtheballisticeciencyfromunstraineddevices.WeusedtheupdatedlowtemperaturemethodusingtheLundstromModel.ID-VDmeasurementswereperformedonplanarn-channelMOSFETsatvarioustemperatures(300Kto100Kin50Ksteps)withoutanyappliedstress.FourtotalDUTsweremeasured(containingseveralchannellengths)intwoseparateexperiments.AMATLABscriptwasdevelopedforndingthetemperaturedependentcoecientsself-consistentlyasexplainedin[ 155 ].TheparameterinEq. 4{71 wasusedattingparameter.TheballisticeciencywasextractedatseveralbiaspointsbeyondonsetofinversionforafewdrainvoltagesbeyondtheobservedVDsatforeachchannellength.Theextractedballisticeciencyonn-channeldevicesisshowninFig. 4-8 (eachcolorrepresentsdatafromagivenDUT). 126

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Figure4-8.Extractedballisticeciencyforn-channelMOSFETsforchannellengthsintherangeof130nmto45nm,atVG)]TJ /F3 11.955 Tf 11.95 0 Td[(VT(sat)=1VandVD=1:2V. Forthechannellengthsmeasured,itcanbenoticedthatthereseemstobeanlinearincreaseintheobservedballisticeciency(forsomeofthemeasurements)consistentwithwhathasbeenreportedintheliterature,forexamplein[ 171 ].However,thespreadintheextractedvaluesneedscarefulthought.TheoreticalissueswithintheframeworkoftheLundstromModel(explainedelsewhere)combinedwiththeempiricalnatureofthetemperaturedependenciesofthevariousexponentsusedintheextractionmethodgreatlycontributetotheobservedspread.Whilewecannotcompletelyavoidexperimentalerrorarisingfromtemperaturesensing,device-to-devicevariation,RSDextraction,useoflongchannelC)]TJ /F3 11.955 Tf 12.36 0 Td[(Vforloweldmobilityextraction,weemphasizethatgreatcarewastakentominimizepossiblesourcesoferror.Fromexperienceonlowtemperaturewafer-bendingmeasurementsperformedatstresslevelscapablewithoursetup,webelievethatitisnotpossibletoclearlyelucidatetheeectofuniaxialstressonballisticeciency.Thepossibleenhancementduetostresswouldbecompletelymaskedbythespreadpresentinthedata, 127

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evenifweignoretheadditionalerrorcausedbythenatureofthestressmeasurementitself.Withtheseobservations,wethereforeconcludethatanewmodelandanewexperimentalmethodologyisrequiredtounderstandtheeectofuniaxialstress.Themodelneedstoaccountforconnementandhigheldeectsinaclearmannerwithminimalempiricalparameters(ifany)andthemethodologyshouldbesimpleenoughtobeabletoextracttheballisticeciencywithouthavingtoperformlowtemperaturemeasurementsjusttoextracttheB.E.atroomtemperature. 4.5SummaryWepresentedanextensivesummaryoftheconventionalmodelsforcarriertransportinSiandidentiedthereasonsthatcausebreakdownofthesemodelsinhighelectricelds.Wealsopresentedasummaryofnanoscaletransportmodels,specicallyLundstrom'sKBT-layertheoryandidentiedthetheoreticalshortcomingsandfactorsleadingtoexperimentalinaccuracies.Inthenextchapter,wepresentthedevelopmentofanewcompactmodelforthetransmissioncoecientinaMOSFETchannel.Thismodelisthenusedtoinvestigateandqualitativelyexplaintheeectsofstraininnanoscaledevices. 128

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CHAPTER5UFCOMPACTMODELFORTRANSMISSIONCOEFFICIENTInthischapter,thephysicsofquasi-ballisticnanoscalecarriertransportisexaminedusinganupdatedone-uxtheory.Byaddinghigheldeectsandconnementeectstotheone-uxtheory,theassumptionsleadingtoanoversimplicationofquasi-ballistictransportareremoved.Asurface-potentialbasedanalyticalformulaisdevelopedforthetransmissioncoecientinaMOSFETchannel,anditisshownthatthisformulaisapplicableinallrangesofdeviceoperation.Natori'shigheldtransportmodelisreinterpreted,expanded,andlinkedtoourwork.Usingtheshownequivalence,anewdenitionofBallisticEciencyisdeveloped,whichexplicitlyincludestheimpactofchannelscattering.Withtheimprovedunderstandingoftransport,thestronginterplaybetweenstraineectsoncarriermass/scatteringanddeviceelectrostaticsishighlighted. 5.1One-uxTheory:FundamentalsFollowingMcKelvey'sapproach[ 134 ][ 172 ][ 173 ],thebasicformalismoftheone-uxtheorycanbeunderstoodbyconsideringFig. 5-1 ,wheretheposition(andtime)dependentcarrieruxesareshownataslabofthicknessdxforapositivelychargedcarrier.Thechannelofatransistorcanbethoughtofasacascadeofmanysuchslabs,withtheoutputofoneslabbecomingtheinputofthenextinacontinuousfashion.Forsimplicity,weignoreboththegeneration/recombinationprocessesinsidethedeviceandthetimedependenceofthevariousparameters.Theslabitselfcanbecharacterizedbyascatteringmatrix,whichlinkstheincomingandoutgoinguxesas 264J+(x+dx)J)]TJ /F1 11.955 Tf 7.09 -4.34 Td[((x)375=264t+(x)r+(x)r)]TJ /F1 11.955 Tf 7.09 -4.34 Td[((x)t)]TJ /F1 11.955 Tf 7.08 -4.34 Td[((x)375264J+(x)J)]TJ /F1 11.955 Tf 7.08 -4.34 Td[((x+dx)375(5{1)Herer(x)andt(x)representtheratioofcarriersreectedandtransmittedinagivendirection.Iftheexpressionsforthesecoecients(intermsofpositionandeld)andtheeldproleinthechannelareknown,itispossibletocalculatethescatteringmatrixfortheentirechannel.Fromthatmatrix,theoveralltransmissioncoecienttandreection 129

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Figure5-1.Incomingandoutgoingcarrieruxesinaslabofthicknessdx.r+(x)andr)]TJ /F1 11.955 Tf 7.08 -4.33 Td[((x)arethereecteduxmagnitudesintheforward(paralleltoappliedeld)andreverse(anti-paralleltoappliedeld)directionsrespectively.Carriergeneration/recombinationwithintheslabisignoredforsimplicity. coecientroftheentiredevicecanbecalculatedforafamilyofgate/drainvoltagesappliedtothedevice.Sinceweignoregenerationandrecombination,itfollowsthatt+r=1.Thedierentialequationsfortheuxmethod,ignoringrecombination/generation,areasbelow(detailedderivationcanbefoundin[ 134 ][ 135 ]). dJ+ dx=)]TJ /F3 11.955 Tf 9.3 0 Td[(r+J++r)]TJ /F3 11.955 Tf 7.08 -4.93 Td[(J)]TJ /F1 11.955 Tf 10.4 -4.93 Td[(=dJ)]TJ ET q .478 w 280.18 -539.73 m 300.93 -539.73 l S Q BT /F3 11.955 Tf 284.19 -550.92 Td[(dx (5{2) HereJ+andJ)]TJ /F1 11.955 Tf 10.99 -4.34 Td[(denotetheposition/elddependentpositive(left-directed)andnegative(right-directed)uxes.r+andr)]TJ /F1 11.955 Tf 10.99 -4.34 Td[(aretheposition/elddependentbackscatteringcoecientsintheappropriatedirections.Wehavedroppedthesubscriptsforsimplicity 130

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{i.e.J+(x;Ex)issimplywrittenasJ+.Exdenotestheelectriceldinthelefttorightdirection.Thenetuxowingrightcanbewrittenas J=J+)]TJ /F3 11.955 Tf 11.96 0 Td[(J)]TJ /F1 11.955 Tf 180.8 -4.93 Td[((5{3)Theeectivecarrierconcentrationisgivenby n=J+ v++J)]TJ ET q .478 w 256.14 -164.37 m 270.8 -164.37 l S Q BT /F3 11.955 Tf 256.88 -175.56 Td[(v)]TJ /F1 11.955 Tf 178.37 4.75 Td[((5{4)wherev+andv)]TJ /F1 11.955 Tf 10.99 -4.34 Td[(arethevelocitiesofthetwouxes.Assumingthateachoftheseisequaltotheequilibrium1-Dcarriervelocityofnon-degeneratesemiconductorwitheectivemassm,givenby c=r 2kBT m(5{5)itisveryeasytoshowthat 2J=cnJ(5{6)Usingaboveequations,wecanderiveanexpressionforthenetuxJas J=)]TJ /F3 11.955 Tf 10.5 8.08 Td[(r+)]TJ /F3 11.955 Tf 11.96 0 Td[(r)]TJ ET q .478 w 183.45 -392.5 m 223.43 -392.5 l S Q BT /F3 11.955 Tf 183.55 -403.68 Td[(r++r)]TJ /F3 11.955 Tf 8.38 4.75 Td[(cn)]TJ /F3 11.955 Tf 30.53 8.08 Td[(c r++r)]TJ /F3 11.955 Tf 9.47 12.83 Td[(dn dx(5{7)Forsmallelectricelds,itisassumedthatthe(r+)]TJ /F3 11.955 Tf 11.73 0 Td[(r)]TJ /F1 11.955 Tf 7.08 -4.34 Td[()shouldbeproportionaltotheelectriceld(nearequilibriumconditions),andthat(r++r)]TJ /F1 11.955 Tf 7.09 -4.34 Td[()beindependentoftheeld, r+)]TJ /F3 11.955 Tf 11.96 0 Td[(r)]TJ /F1 11.955 Tf 10.4 -4.94 Td[(=Ex(5{8)andtherefore,wecanwriteEq. 5{7 as J=)]TJ /F3 11.955 Tf 24.76 8.09 Td[(c r++r)]TJ /F3 11.955 Tf 8.28 4.74 Td[(nEx)]TJ /F3 11.955 Tf 30.53 8.09 Td[(c r++r)]TJ /F3 11.955 Tf 9.48 12.83 Td[(dn dx(5{9) 131

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ThisequationcanbewritteninafamiliarformsimilartotheDrift-Diusionequation,ifwedenemobility-likeanddiusion-constant-liketermsandD (x;Ex)=c r++r)]TJ ET BT /F1 11.955 Tf 429.63 -95.63 Td[((5{10a) D(x;Ex)=c r++r)]TJ ET BT /F1 11.955 Tf 428.98 -122.53 Td[((5{10b) D =1 (5{10c) toarriveat J=)]TJ /F3 11.955 Tf 9.3 0 Td[(nEx)]TJ /F3 11.955 Tf 11.95 0 Td[(Ddn dx(5{11)TheaboveequivalencebetweentheDDtheoryandtheone-uxequationswasdiscoveredbyShockleyin1962[ 135 ],wherehealsoobservedthatthediusionconstantDofthecarrierstraversinganinnitesimallysmalleld-freeregion(onethatissmallerthantheaveragemeanfreepath)isthesameasthatofthebulksemiconductor,D0.Iftheoneuxequationshavetobecompatiblewithintheframeworkofgeneralsemiconductortransportequations,itisclearthat limEx!0=0=q kBT (5{12a) limEx!0=0=0c 2r0=q kBT1 D0 (5{12b) where0,k0,D0denoteequilibriumquantities.McKelveyshowed([ 134 ],AppendixB)that r0=limEx!0r=3 4s(5{13)Heresdenotesthe\scatteringmeanfreepath".NotethatShockleyargues([ 135 ],Appendix1)thatMcKelvey'sexpressionforr0needsasmallcorrection.Thismethodiscalledtheone-uxmethodbecauseonlyonecarrieruxisconsideredineachdirection.Inreality,sincecarriersenteringthechannelhaveadistributionof 132

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incomingenergies,eachenergylevelwillhaveitsownuxcomponents,andtherewouldbeintermixingoftheuxesbetweendierentenergylevelsaftermomentumalteringscatteringevents.Theoretically,itispossibletodoacontinuousordiscreteenergysummationoftheuxestoarriveatanoverallone-uxequation.In1995-1998,Bringuierdeveloped[ 174 ][ 175 ][ 176 ]adierentapproachcalledtheFokker-Planckapproachtoextendtheone-uxtechnique.Astatisticalmechanicsdescriptionofcarriertransportunderhigh,non-homogeneouseldconditionswasformulatedforasemiconductorwitharbitrarybandstructureandelectron-phononinteraction.ItwasshownthattheresultingelectrondistributionfunctionwasoftheFokker-PlancktypeandthatthetransportstudiesusingthistechniquematchedMonte-Carlonumericalsimulationresultsextremelywell.Essentially,inBringuier'smethod,themomentumvariablesintheBTEaretranslatedintotheenergyspaceforsolvingtheBTE.FollowingBringuier'sideas,in2003McKinnondeveloped[ 177 ]acontinuousenergyversionoftheone-uxtechnique.Heshowedthattheone-uxequationscanberecoveredfromhisE-uxmethodwhenitisassumedthatthecarrierdistributionsfollowathermaldistribution.Thereareseveralbenetstosuchanapproach.Specically,sinceenergyisacontinuousvariable,treatingdierentmaterialswithdierentE-kdispersionswithdierentdensity-of-stateequationsbecomesstraightforward.Asaresult,understandingtheeectsofbandstructurerelatedmaterialpropertiessuchasmobilityanisotropyisintegratedintothemethodinherently.Inaddition,itisalsopossibletotakeintoaccounttheeectsofcarrierdegeneracy.Carrieruxconcentration/velocityandmobility/diusiontermsareallexpressedascontinuousfunctionsofenergyinthismethod,makingitpossibletocompareagainstMonte-Carlolikenumericalmethodstostudyballistictransport.Whilethismethodislikelyasuperiorapproachtounderstandstraineects,duetothenumericalnatureofthemethod,itnotpossibletoarriveatcompact/analyticalexpressionsfortransportrelatedquantities. 133

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In1998,AssadandLundstrom[ 178 ]formulatedaM-uxtheorythatreformulatedtheBTEasMcoupledcontinuityandDDequationsintheenergy-momentumspace.Aswouldbeexpected,thedetailsoftransportoverthebarrierusingthisapproachisalsoentirelynumericalandwhileaccurate,itisalsoverycomplextoimplement.Itwasresolvedintheirworkthatthesimpleanalyticalexpressionsofthesimpleone-uxtheory(discussedbelow)failinthehigheldconditions,andthattheyshouldbeusedforonlyrst-ordercalculationpurposes.ThemaindierencebetweenLundstrom'sM-uxmethodandMcKinnon'sE-uxmethodishowtheenergyvariableishandled;discreteintheformerandcontinuousinthelatter.Thequasi-ballistictransportmodelformulatedlaterbyLundstromforunderstandingtheessentialphysicsofcarriertransport[ 179 ]utilizestheone-uxtheoryassumptionsandapproximationsowingtothesimplicityofapproach.Ourgoalistoimproveupontheone-uxtheorytowardsaqualitativeunderstandingofstraineectswithamoreaccurateanalyticalexpressionforthetransmissioncoecient. 5.2Gildenblat'sTransmissionModelIn2002,Gildenblatpresented[ 180 ]aslightlydierentapproachtomodelingtheballisticeciency.Startingfromthefundamentaldenitionofthescatteringmatrix(Section 5{1 ),Gildenblatdevelopedaclosedformanalyticalexpressionfortheoveralltransmissioncoecientforadevicehavinganarbitrarypotentialprole.Inthesamepaper,healsoshowedtheLMmodelexpressionforrsatcanberecoveredfromhisexpressionveryeasily.WerefertothistransmissionmodelastheGildenblatModel(GM),uponwhichourmodelisbased.Forcompleteness,weoutlinethekeypointsinGildenblat'smodelinthissection.ThestartingpointforGMaretheequationspresentedinsection 5.1 .Toarriveatananalyticalexpressionforthetransmissioncoecientt,Gildenblatassumesthattheratioisindependentofposition(whilebothandDarepositionandelddependent). 134

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StartingfromEq. 5{11 andusing2 Ex=)]TJ /F3 11.955 Tf 10.5 8.09 Td[(d dx (5{14a) dn dx=dn dd dx (5{14b) Wehave J D=)]TJ /F8 11.955 Tf 11.29 16.85 Td[()]TJ /F3 11.955 Tf 9.3 0 Td[(n+dn dd dx(5{15)Nowconsiderthequantityne)]TJ /F4 7.97 Tf 6.59 0 Td[(,whereindependentof.Dierentiatingthisw.r.t,wehave, d d(ne)]TJ /F4 7.97 Tf 6.59 0 Td[()=e)]TJ /F4 7.97 Tf 6.58 0 Td[(dn d)]TJ /F3 11.955 Tf 11.96 0 Td[(e)]TJ /F4 7.97 Tf 6.58 0 Td[(n(5{16)or ed d(ne)]TJ /F4 7.97 Tf 6.59 0 Td[()=dn d)]TJ /F3 11.955 Tf 11.95 0 Td[(n(5{17)UsingEq.( 5{17 )inEq.( 5{15 ), J D=)]TJ /F3 11.955 Tf 9.29 0 Td[(ed dx(ne)]TJ /F4 7.97 Tf 6.59 0 Td[()(5{18)Rearranging,weget d dx(ne)]TJ /F4 7.97 Tf 6.59 0 Td[()=)]TJ /F3 11.955 Tf 9.29 0 Td[(e)]TJ /F4 7.97 Tf 6.59 0 Td[(J D(5{19)or Zd(ne)]TJ /F4 7.97 Tf 6.59 0 Td[()=Z)]TJ /F3 11.955 Tf 9.3 0 Td[(e)]TJ /F4 7.97 Tf 6.59 0 Td[(J Ddx(5{20)NotingthatnetuxJisaconstant,andintegratingbothsidesafteraddingappropriatelimits, Zn(x)e)]TJ /F12 5.978 Tf 5.75 0 Td[((x)n(0)e)]TJ /F12 5.978 Tf 5.76 0 Td[((0)d(ne)]TJ /F4 7.97 Tf 6.59 0 Td[()=)]TJ /F3 11.955 Tf 9.3 0 Td[(JZx0e)]TJ /F4 7.97 Tf 6.59 0 Td[(1 Ddx(5{21) 2Here=(x)denoteselectrostaticpotential 135

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Theresultoftheintegrationis n(x)e)]TJ /F4 7.97 Tf 6.58 0 Td[((x))]TJ /F3 11.955 Tf 11.95 0 Td[(n(0)e)]TJ /F4 7.97 Tf 6.58 0 Td[((0)=)]TJ /F3 11.955 Tf 9.3 0 Td[(JZx0e)]TJ /F4 7.97 Tf 6.59 0 Td[(dx D(5{22)Rearranging, n(x)e)]TJ /F4 7.97 Tf 6.58 0 Td[((x)=n(0)e)]TJ /F4 7.97 Tf 6.59 0 Td[((0)=)]TJ /F3 11.955 Tf 9.29 0 Td[(JZx0e)]TJ /F4 7.97 Tf 6.59 0 Td[(dx D(5{23) n(x)=e(x)n(0)e)]TJ /F4 7.97 Tf 6.58 0 Td[((0))]TJ /F3 11.955 Tf 11.95 0 Td[(JZx0e)]TJ /F4 7.97 Tf 6.59 0 Td[(dx Ddx(5{24)Dening (x)=(x))]TJ /F3 11.955 Tf 11.96 0 Td[((0),itcanbeshownthat n(x)=n(0)e (x))]TJ /F3 11.955 Tf 11.95 0 Td[(Je (x)Zx0e)]TJ /F4 7.97 Tf 6.58 0 Td[( (x)dx Ddx(5{25)Eq.( 5{25 )isaimportantresultof[ 180 ].Oncen(x)isknown,theforwardandthereverseuxescanbeeasilyestimatedfromEq.( 5{6 )as 2J+(x)=cn(0)e+J1)]TJ /F3 11.955 Tf 11.95 0 Td[(ce (x)Zx0e)]TJ /F4 7.97 Tf 6.59 0 Td[( (x)dx D2J)]TJ /F1 11.955 Tf 7.09 -4.94 Td[((x)=cn(0)e)]TJ /F3 11.955 Tf 11.95 0 Td[(J1+ce (x)Zx0e)]TJ /F4 7.97 Tf 6.59 0 Td[( (x)dx DItiswellknownfromtheuxmethodthatthereectionandtransmissioncoecientsofthesystemaregivenby r=J)]TJ /F1 11.955 Tf 7.09 -4.34 Td[((0) J+(0) (5{27a) t=1)]TJ /F3 11.955 Tf 11.95 0 Td[(r=J+(L) J+(0) (5{27b) Bydeningan\eectivevelocity"egivenby 1 e=ZL0e)]TJ /F4 7.97 Tf 6.58 0 Td[( (x)dx D(5{28) 136

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andusingtheexplicitequationsforJ+(x)andJ)]TJ /F1 11.955 Tf 7.08 -4.34 Td[((x)shownabove,itiseasytoshowthat J)]TJ /F1 11.955 Tf 7.08 -4.94 Td[((0)=cn(0))]TJ /F3 11.955 Tf 11.96 0 Td[(J (5{29a) J+(0)=cn(0)+J (5{29b) J+(L)=cn(0)e (L)+J 1)]TJ /F3 11.955 Tf 11.96 0 Td[(ce (L)ZL0e)]TJ /F4 7.97 Tf 6.59 0 Td[( (x)dx D!=cn(0)e +J1)]TJ /F3 11.955 Tf 15.74 8.09 Td[(c ee (5{29c) ( = L)]TJ /F3 11.955 Tf 12.21 0 Td[( 0= (L)i.e.thesurfacepotentialdierencebetweenthedrainandsourceendsforaparticularvoltageonthegateanddrainterminals.)UsingEq. 5{27 andEq. 5{29 r=J)]TJ /F1 11.955 Tf 7.08 -4.34 Td[((0) J+(0) (5{30a) =cn(0))]TJ /F3 11.955 Tf 11.95 0 Td[(J cn(0)+J (5{30b) and t=J+(L) J+(0) (5{31a) =cn(0)e +J1)]TJ /F4 7.97 Tf 15.41 4.71 Td[(c ee cn(0)+J (5{31b) Usingtherelationr+t=1,wecanshowthat r+t=cn(0))]TJ /F3 11.955 Tf 11.95 0 Td[(J+cn(0)e +J1)]TJ /F4 7.97 Tf 15.4 4.71 Td[(c ee cn(0)+J1=cn(0)(e +1))]TJ /F3 11.955 Tf 11.96 0 Td[(Jc ee cn(0)+J (5{32a) or, cn(0)=Je)]TJ /F4 7.97 Tf 6.59 0 Td[( +c e(5{33) 137

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Usingtheaboveexpression,tcanbeexpressedas t=cn(0)e +J1)]TJ /F4 7.97 Tf 15.41 4.71 Td[(c ee cn(0)+J (5{34a) =Je e)]TJ /F4 7.97 Tf 6.59 0 Td[( +c e+J1)]TJ /F4 7.97 Tf 15.41 4.71 Td[(c ee Je)]TJ /F4 7.97 Tf 6.59 0 Td[( +c e+J (5{34b) =2J Je)]TJ /F4 7.97 Tf 6.59 0 Td[( +c e+J (5{34c) (5{34d) or t=2 1+c e+e)]TJ /F4 7.97 Tf 6.58 0 Td[( (5{35)ThusifeasdenedbyEq.( 5{28 )isknownforasystem,thetransmissioncoecientcanbeestimatedusingEq.( 5{35 ). 5.3RevisitingtheOriginalShockleyWorkLetusdiscussthephysicsbehindtheone-uxtheoryassumptioninEq.( 5{8 ).Inessence,thisisaloweld,near-equilibriumassumption,whichisbasedonasimplelineardependenceofthebackscatteringcoecientsontheappliedelectriceld.McKelvey[ 173 ]assumedthebelowexpressionsforthecoecients, r=r0(1) (5{36a) r++r)]TJ /F1 11.955 Tf 17.05 -4.94 Td[(=2r0 (5{36b) r+)]TJ /F3 11.955 Tf 11.96 0 Td[(r)]TJ /F1 11.955 Tf 17.05 -4.93 Td[(=(2r0) (5{36c) ItisclearlystatedintheinitialMcKelveypapers[ 172 ][ 173 ]thatthemethodisapplicablefor\smallappliedelectricelds"only.wasshowntobeaquantitylinearlydependentontheelectriceldas =0 cEx(5{37)(0istheloweldmobilitydenedas0=qt=m) 138

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McKelveyquantied\smallelectricelds"by1.Theoretically,thismeansthatthecarrierdistributionfunctionisnotfarawayfromequilibrium,andtheaveragecarriervelocityissmallcomparedtoc(i.e,velocitysaturation,anhigheldeect,hasnothappenedyet).Thus,theassumptionthatr+)]TJ /F3 11.955 Tf 12.2 0 Td[(r)]TJ /F1 11.955 Tf 11.03 -4.34 Td[(=Ex,withbeingaconstant{independentofpositionandindependentofpotential{Gildenblat'snalexpressionforthetransmissioncoecientisessentiallynodierentthanwhatcouldbeobtainedfromanDrift-Diusionperspective.WereiteratethattheGildenblatresultissimplyaLorentzsolutionoftheBoltzmannTransportEquation.Thatis,thesolutionobtainedbythismethodissimplytheequilibriumdistributionfunctionuponwhichadriftvelocity(arisingfromthepresenceofasmallelectriceld)hasbeensuperposed.Byitsnature,thissolutioncannotdescribequasi-ballistictransportwithanymoreaccuracythanatraditionalDDsolution.TheLundstromquasi-ballistictransportmodelalsosuersfromasimilarproblemthatshowsupadierentway,becauseofthesameunderlyingassumptionaboutthebackscatteringdierencesintheforwardandreversedirections.OtherauthorshavealsodiscussedthevalidityofLundstrom'sassumptionsfromadierentperspective-VaidyanathanandPulfreypresentangooddiscussionoftheissuesin[ 181 ]andFischetti[ 182 ]presentsverycomprehensiveMonteCarlosimulationresultsofananoscaletransistor,showingtheo-equilibriumnatureofthecarrierdistributionfunctioneveninthelinearrange,forvoltagesassmallas0.1V.NotethattheM-uxtheorydevelopedbyLundstrom'sgroupdoesnothavetheseproblem;thetrade-oismodelcomplexity. 5.4UFCompactModel:TheoryToovercometheissuespresentedintheprevioussection,wehaveextendedtheGildenblatapproachfortransmissionbyincludinghigheldtransporteectsandconnementeects(aswouldoccurintheinversionlayerofaMOSFET).Oneofthegoalsofourworkwastoretainanalyticalsimplicity.Webelievethatincludingtheconnementeectsintotheframeworkoftheone-uxtheoryisanindirectwayofincorporatingenergy 139

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resolveduxesforcarriertransport.Inthissection,werstexplaintheadditionofhigheldeects,andthenweshowtheadditionofconnementeectstotheresultingmodel.WealsopresentanewinterpretationofNatori'shigheldtransportmodel[ 183 ][ 184 ],provingtheequivalenceofourapproachandNatori'sapproach.Withthehelpofthisequivalence,wethenexplainthephysicalsignicanceofthemodiedeectivevelocitytermusedinourmodel(inGM,itwassimplyusedasamathematicalsimplication).Finallywetouchuponusingourmodelforaclearerunderstandingstraineectsonquasi-ballistictransportfromanewperspective. 5.4.1IncludingHighFieldEectsOurmethodforincludinghigheldeectsstemsfromasuggestionputforwardbyPulverandMcKelveyin1966[ 185 ]intheirpaperdiscussingtheapplicationoftheuxmethodforsystemswithnon-constantelectricelds.Intheirwork,theyproposeanempiricalexpressionforthevariationofrwith(andtherebyEx)thatisapplicableevenathighelds.Whilenotrigorouslyjustiedinafundamentalway,theirexpressionreproducesthecorrectbehaviorofratalllimits.Theexpressionsuggestedis r=r0e(5{38)Underincreasingelectricelds,=0E=cxincreasesinvalue(andwilleventuallysaturatewhencarriervelocitysaturates)andeventuallythecondition1becomesinvalid.Inthislimit,PulverandMcKelveyarguethat,forathinslabofthicknessdx,r+dx!0andr)]TJ /F3 11.955 Tf 7.08 -4.34 Td[(dx!1.Theynotethatinextendingtheseresultstothelimitdx!0,andthus,r+!1andr)]TJ /F3 11.955 Tf 7.08 -4.34 Td[(dx!1.itisalsonotedthattheconditionthatr+(E)=)]TJ /F3 11.955 Tf 9.3 0 Td[(r)]TJ /F1 11.955 Tf 7.09 -4.33 Td[(()]TJ /F3 11.955 Tf 9.3 0 Td[(E)shouldapplyfromsymmetryconsiderations.Theexponentialrelationsuggestedaboveisthesimplestmathematicalexpressionthatsatisesboththeseconditionssimultaneously. 140

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Withthisformforthebackscatteringcoecients,themobilityandthediusionexpressionbecome (x;Ex)=r+)]TJ /F3 11.955 Tf 11.96 0 Td[(r)]TJ ET q .478 w 216.91 -68.74 m 256.89 -68.74 l S Q BT /F3 11.955 Tf 217.01 -79.93 Td[(r++r)]TJ /F8 11.955 Tf 8.38 21.6 Td[(c Ex (5{39) =c Exe)]TJ /F3 11.955 Tf 11.96 0 Td[(e)]TJ /F4 7.97 Tf 6.58 0 Td[( e+e)]TJ /F4 7.97 Tf 6.58 0 Td[(=c Extanh() (5{40) (5{41) D(x;Ex)=c r++r)]TJ ET BT /F1 11.955 Tf 435.48 -179.02 Td[((5{42) =c r01 e+e)]TJ /F4 7.97 Tf 6.59 0 Td[(=c 2r01 cosh() (5{43) Theexpressionfortheposition/elddependentEinsteinrelationis, D =Ex 2r01 sinh() (5{44) (5{45) Attheloweldlimit,wecanverifythattheaboveexpressiontakesitdefaultvalue{ limEx!0D =Ex 2r01 (5{46) =c 2r01 0=D0 0=kBT q (5{47) TheexpressionforD=shownabovedependsontheelectriceld,whichisthederivativeoftheelectrostaticpotential,butnotontheelectrostaticpotentialitself.ThismeansthatGildenblat'smathematicalapproachisstillapplicable{specically,Eq. 5{16 stillworks.Thismeansthatwecanderiveanexpressionforn(x)usingthesametechniqueasoutlinedinthelastsectionaftera(careful)integrationinvolvingapositiondependent.However,the\eectivevelocity"eneedstoberedenedtoaccountforthepositiondependentterm.Oncethisisdone,wecanarriveatanexpressionforthetransmissioncoecientthatreducestotheGildenblatresultintheloweldlimit.Thisprocedureisoutlinedinthenextsection. 141

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WestartfromEq. 5{18 andusetheterm==Dinsteadofforclarity.Wereiteratethatthepositionandelddependenceofcannotbeignored. J D=)]TJ /F3 11.955 Tf 9.3 0 Td[(ed dx(ne)]TJ /F4 7.97 Tf 6.59 0 Td[()(5{48)Rearranging,weget d dx(ne)]TJ /F4 7.97 Tf 6.58 0 Td[()=)]TJ /F3 11.955 Tf 9.29 0 Td[(e)]TJ /F4 7.97 Tf 6.59 0 Td[(J D(5{49)or Zd(ne)]TJ /F4 7.97 Tf 6.59 0 Td[()=Z)]TJ /F3 11.955 Tf 9.3 0 Td[(e)]TJ /F4 7.97 Tf 6.59 0 Td[(J Ddx(5{50)Sincedependsonposition,weaccontforthisinthelimitsofintegrationontherighthandside(0andx,whicharevaluesofwhenx=0andx=x,respectively).NotingthatnetuxJisaconstantandaddinglimits, Zn(x)e)]TJ /F12 5.978 Tf 5.76 0 Td[(x(x)n(0)e)]TJ /F12 5.978 Tf 5.75 0 Td[(0(0)d(ne)]TJ /F4 7.97 Tf 6.59 0 Td[()=)]TJ /F3 11.955 Tf 9.3 0 Td[(JZx0e)]TJ /F4 7.97 Tf 6.59 0 Td[(1 Ddx(5{51)Afterintegration,wehave, n(x)=ex(x)n(0)e)]TJ /F4 7.97 Tf 6.59 0 Td[(0(0))]TJ /F3 11.955 Tf 11.96 0 Td[(JZx0e)]TJ /F4 7.97 Tf 6.58 0 Td[(dx Ddx(5{52)Eq.( 5{52 )istheanalogofGildenblat'sEq.( 5{25 ).WeproceedinthesamewayasbeforetoestimateJ=cn(x)J. 2J+(x)=cn(0)ex(x)e0(0)+J1)]TJ /F3 11.955 Tf 11.96 0 Td[(cex(x)Zx0e)]TJ /F4 7.97 Tf 6.59 0 Td[((x)dx D2J)]TJ /F1 11.955 Tf 7.08 -4.94 Td[((x)=cn(0)ex(x)e0(0))]TJ /F3 11.955 Tf 19.26 0 Td[(J1+cex(x)Zx0e)]TJ /F4 7.97 Tf 6.59 0 Td[((x)dx DRedeningthe\eectivevelocity"easbelow, 1 e=ZL0e)]TJ /F5 7.97 Tf 6.59 0 Td[((x(x))]TJ /F4 7.97 Tf 6.59 0 Td[(0(0))dx D(5{54) 142

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Figure5-2.Includingconnementeectsfortransmissionmodel andfollowingthesameprocedureasintheprevioussection,wecanderiveanexpressionforthetransmissioncoecienttas t=2 1+c e+e)]TJ /F5 7.97 Tf 6.59 0 Td[((L(L))]TJ /F4 7.97 Tf 6.59 0 Td[(0(0))(5{55)Ifisindependentofposition/eld(i.e.aconstant),Gildenblat'sEq.( 5{35 )isrecovered. 5.4.2IncludingConnementEectsUnlikebulkdevices,MOSFETswith2Dand1Dinversionlayersoperateundertheeectofgateeldinducedquantumconnement[ 186 ].Theconnementofcarriersleadstoareductionofthecarrierdegreesoffreedom.Thisinreturnresultsinconnementinducedvalley/bandsplitting,leadingtorepopulationofcarriersbetweenthebands.Additionally,thesplittingofthebandsmodiesthejointdensityofstatesforscatteringmatrixelements[ 187 ].Forexample,forSielectroninversionlayers,thesplittingof2and4valleysleadstoareductioninthef-typeopticalphononscatteringrates.Movingfroma3Dto2Dbandstructurereducestheeectivedensitystateswhichalsocontributestoadditionalscatteringratereduction.Thegate-to-channelpotentialdierencevariesalongthechannelbasedontheconductionbandprole.Thisleadstoapositiondependentlevelofquantumconnement 143

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Figure5-3.Eectsofpositiondependenceonoverallaveragedriftvelocity forthecarriers.Atthesourceend,wherethegate-to-channelpotentialdierenceishighest,theconnementinducedbandsplittingisatitslargestvalue.Hence,thecarrierrepopulationandtheresultantconductivitymassmodication(comparedtobulk)ismorepronouncedcomparedtothedrainend.Forinstance,in(100)-orientedSi,theeectiveconductivitymassforelectronsatthesourceendislowercomparedtothedrainend.Withapositiondependentconductivitymass,wecandeneaposition-dependentthermalvelocity, c(x)=s 2kBT m(x)(5{56)AddingthepositiondependenceofthermalvelocitytotUF,weget tUF=2 1+c(0) c(L)c(L) e+e)]TJ /F5 7.97 Tf 6.59 0 Td[((L(L))]TJ /F4 7.97 Tf 6.58 0 Td[(0(0))(5{57) 144

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wherec(0)andc(L)representtheaveragethermalvelocityatthesourceanddrainendsofthechannel.Withtheproperinclusionofhigh-eldeectsandconnementeects,Eq.( 5{57 )becomesapplicablefordescribingcarriertransportfortheentirerangeofdeviceoperation(lowandhighelds)andforallsurfacechannel/orientations. 5.4.3RevisitingNatori'sHighFieldTransportModelIn2009,Natoripresentedanewapproach[ 183 ][ 184 ]towardsunderstandinghigheldtransportinbulksemiconductors.Inthisapproach,ananalyticalsolutionofapseudo-one-dimensionalBTEwithaconstantelectriceldispresented,aftertransformingitintoapairofcarrieruxequations.Neithertherelaxationtimeapproximationnortheperturbationexpansionisusedinsolvingtheequations.Explicitexpressionsforaveragecarriervelocityanddensityasfunctionsofeldandpositionarepresented.ThemainconceptsinNatori'stheoryareschematicallypresentedinFig. 5-4 .Natoriconsidersasimpleresistorinhispapers,withalinearpotentialproleinthedevice\channel".Thecarriersenteringthechannelfromthesourcehasadistributionofenergies,andanaveragevalueofKBT=qisassumed.Intheregionofthechanneluptothedistancex0fromthesource,thecarriersdonothaveenoughenergytoemitanopticalphonon.Inotherwords,theonlyscatteringeventsinthisregionareelasticinnature(surfaceroughness,ionicimpurity,andacousticphononscattering).ThisregionisthereforenamedastheInitialElasticZone(IEZ).IntraversingthelengthoftheIEZ,thecarriersgainenergyfromtheeldandhaveenoughenergyforopticalphonon(OP)emissionaftercrossingthiszone.Beyondthepointx=x0,OPemissionhappensinaveryshorttimeandcarriersareallowedtorelaxtothelowerenergylevel")]TJ /F3 11.955 Tf 12.23 0 Td[(".Thisenergylevel")]TJ /F3 11.955 Tf 12.23 0 Td[("isknownastheIstRelaxedLevel,andthecarriersintherstrelaxedlevelareonlyallowedtopopulatexx0.Thecarrierscontinuetogainenergyfromtheeldonceagain,tilltheyreachthe 145

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point(x0+x1)insidethechannelwherethecarrierenergyisequaltotheOPenergy.Theregionx0x(x0+x1)isknownastheIstRelaxedElasticZone.ItiseasytoseethattheentirechannelcanberepresentedasacascadeoftheserelaxedelasticzoneswithOPemissionhappeningateachzoneboundary.Thetotaltransmissionthroughthedeviceisthereforeaproductofthetransmissioncoecientsthrougheachelasticzone.T=T0T1T2TN (5{58)However,thereisonedierencebetweenT0andtherestofthetransmissioncoecientsT1TN.Beforex0,energyrelaxationcanhappenonlythroughelasticscatteringprocesses(acousticphononscatteringprocessisconsideredtobeelastic).Beyondthispoint,thecarrierenergyrelaxationhappensthroughbothelasticandinelasticscatteringmechanisms,andtransportphenomenabecomesconceptuallyidenticalinallthezonesbeyondIEZ.Thus,T1TNcanbecombinedintoonefactor,andasingleequivalentOpticalPhononEmissionZonecanbevisualized.Rightnearthezoneboundaryx0,aftertheinelasticscatteringwithOP,thereisachanceforthecarrierstobeelasticallybackscatteredintotheIEZ.Theprobabilityforre-entryintotheIEZreducesasthecarriermovesawayfromx0eventhoughitcanundergoanelasticscatteringeventatanyfurthertime.ThetotalnumberofbackscatteredcarriersintotheIEZdependsontheamountofelasticbackscatteringandtheenergyrelaxationviaOPemissionbeyondxx0.Natoridescribesthisthroughaparamterwhichrepresentstheelastic-backscattering/energyrelaxationtrade-o.Thepresenceofthistradeoisverysignicantinunderstandingquasi-ballistictransportbothwithandwithoutstraininvolved.WewouldliketopointoutthattheLMpictureisslightlydierentinthisregard,asthebackscatteringparameterrinthatmodelrepresentsthefractionofcarriersgettingbackintothesource,whileNatori'sparameteraccountsforbackscatteringfromallof 146

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Figure5-4.Natori'shigheldtransportmodelshowingtheconceptual\InitialElasticZone"and\OpticalPhononEmissionZone"regionsinadevicecarryingcurrent. OPEZintotherstelasticzone.Thisisansubtledierencethathasstrongimplicationsintheunderstandingofeithermodel.Foragivenenergyoftheincomingcarriers,theIEZwidthx0(andthereforetheOPEZwidthL)]TJ /F3 11.955 Tf 12.38 0 Td[(x0)willdependontheappliedvoltageonthedrain.Sincethecarrierswillhaveadistributionofenergiesatthesource,eachenergywillhaveitsownzonedenitionsforagivendrainvoltageaswell.ItispossibletoconceptuallythinkofanenergyaveragedIEZandOPEZforagivenbiascondition.Thehigheldtransportinthedeviceisanalyzedusingareection-transmissionmethodwiththesezones,similartotheone-uxapproach. 147

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Usingveryrigorousanalysisandcalculations,Natorishowedthatthecurrent-eldandthevelocity-eldrelationshipsinthisdevicewerecloselyrelatedtotheuxtransmissionthoroughtheIEZ.IfTand(1)]TJ /F3 11.955 Tf 12.94 0 Td[()representthezonetransmissionsoftheIEZandOPEZrespectively,onewouldexpecttoseetheoveralltransmissionoftheentiredevicetobetheproductofindividualtransmissions,i.e,T(1)]TJ /F3 11.955 Tf 12.08 0 Td[()fromtheone-uxtheory.However,Natori'sanalysisshowsthattheoveralltransmissionisoftheformshowninEq.( 5{59 ). TN=T(1)]TJ /F3 11.955 Tf 11.96 0 Td[() 1)]TJ /F1 11.955 Tf 11.96 0 Td[((1)]TJ /F1 11.955 Tf 14.25 3.02 Td[(T)(5{59)ThiswasattributedtotheinherentfeedbackpresentbetweencarriertransportandelectrostaticsthroughthePoisson'sequationinthedevice.Notethatthetransmissionwillbeastrongfunctionoftheresultanteldinthechannel.Usingthistransmissionexpression,Natoriexplainscurrentsaturationthathappensunderhigheldconditions.Traditionally,currentisthoughttobetheproductoftotalcarrierchargeandthecarriervelocity.Natoriintroducesthetransmissionintothedevicecurrentequationas I/qn0T(1)]TJ /F3 11.955 Tf 11.95 0 Td[() 1)]TJ /F1 11.955 Tf 11.96 0 Td[((1)]TJ /F1 11.955 Tf 14.25 3.02 Td[(T)(5{60)Usinghistransmissionmodel,NatoripointsoutthatthecurrentthroughthedevicewillsaturatewhenthetransmissionTbecomesequaltounity,whichhappenswhenthezonewidthx0!0athigheldconditions.Whenthishappens,thetotaltransmissionTNsaturatestothevalue(1)]TJ /F3 11.955 Tf 12.53 0 Td[().Thelevelatwhichcurrentsaturates,therefore,dependsonthetransmission(1)]TJ /F3 11.955 Tf 12.71 0 Td[()oftheOPEZ(whichistheentirelengthofthechannelasx0!0).Wewillexplainwhatvelocityistobeusedinthecurrentexpressioninlaterparagraphs. 5.4.4UnderstandingFeedbackinNatori'sHighFieldModelWerstshowthatitispossibletoarriveatNatori'sexpressionfortotaltransmissioninaverysimpleandelegantmannerusingtheone-uxtheory,withtheadditionofcarrierexchangebetweenthezones.Forsimplicity,weusethenotationR=1)]TJ /F1 11.955 Tf 14.47 3.02 Td[(Tfromhereon. 148

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Fig.( 5-5 )showsapictorialrepresentationofourapproach.Theredarrowsrepresentthereecteduxesandthegreenarrowsrepresentthetransmitteduxesfrom/toeitherzoneinthedevice.Consideranincidentuxf0enteringthechannel;apartofthisux(Tf0)istransmittedthroughtheIEZ,andpartofitisreectedbackintothesource,namelyRf0.OftheuxenteringtheOPEZ,theportion(1)]TJ /F3 11.955 Tf 12.34 0 Td[()Tf0istransmittedtothedrain,andthefractionTf0isreectedbackintotheIEZ.NotethatthesebackreectionsfromtheOPEZincludesthecontributionfromalltherelaxedzonesbeyondtheIEZ.ApartoftheuxreectedfromtheOPEZtotheIEZ(1)]TJ /F1 11.955 Tf 15.24 3.02 Td[(Rf0)transmitsbacktothesource,andapartofit(RTf0)isreectedbackintotheOPEZ.Byconsideringcarrierexchangethroughaninnitereection/transmissionprocessthatoccursbetweenthetwozones,wetakeintoaccounttheeectofinelasticscatteringonthetotaltransmissioninafundamentalway. Figure5-5.Newderivationoftheoveralltransmissioncoecientusinganinniteseriessummationapproach. 149

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Toderivetheexpressionforthetotaltransmission,weneedthreeterms:(1)incidentuxJ+(0),(2)totaltransmitteduxJ+(L)and(3)totalreecteduxJ)]TJ /F1 11.955 Tf 7.09 -4.34 Td[((0).TheuxtermscanbeobtainedbysummingthecomponentsshowninFig.( 5-5 )as, J+(0)=f0J)]TJ /F1 11.955 Tf 7.09 -4.94 Td[((0)=Rf0+(1)]TJ /F1 11.955 Tf 14.51 3.02 Td[(R)Tf0+(1)]TJ /F1 11.955 Tf 14.51 3.02 Td[(R)R2Tf0+(1)]TJ /F1 11.955 Tf 14.51 3.02 Td[(R)R23Tf0+(1)]TJ /F1 11.955 Tf 14.51 3.02 Td[(R)R34Tf0+:::J+(L)=(1)]TJ /F3 11.955 Tf 11.95 0 Td[()Tf0+(1)]TJ /F3 11.955 Tf 11.96 0 Td[()RTf0+(1)]TJ /F3 11.955 Tf 11.95 0 Td[()R22Tf0+::: (5{61) SumminguptheinniteseriesforJ+(0)andJ)]TJ /F1 11.955 Tf 7.08 -4.34 Td[((0),wehave J)]TJ /F1 11.955 Tf 7.08 -4.93 Td[((0)=Rf0+(1)]TJ /F1 11.955 Tf 14.51 3.02 Td[(R)Tf0n=1Xn=0Rnn=Rf0+(1)]TJ /F1 11.955 Tf 14.51 3.03 Td[(R)Tf01 (1)]TJ /F1 11.955 Tf 14.5 3.02 Td[(R) (5{62) J+(L)=(1)]TJ /F3 11.955 Tf 11.96 0 Td[()Tf0n=1Xn=0Rnn=(1)]TJ /F3 11.955 Tf 11.96 0 Td[()Tf01 (1)]TJ /F1 11.955 Tf 14.51 3.02 Td[(R) (5{63) andnally, TN=J+(L) J+(0)=T 1+ (1)]TJ /F3 11.955 Tf 11.95 0 Td[()T(5{64)whichisidenticaltoNatori'sexpression(Eq.( 5{59 )).Asimilarresultisalsoseeninmanyotherelds,forexampleinthetreatmentofopticaltransmissionthroughmulti-layeredstructures[ 188 ],wheninterferenceeectsareneglectedandallmultipleinternalreectionsaretakenintoaccount,similartoourcase.TheformoftheexpressionforthetotaltransmissioninEq.( 5{64 )issimilartothetransferfunctionofanegativefeedbacksystemcommonlyencounteredincontroltheory[ 189 ].Thisanalogyisnotacoincidence,butrather,aresultoftheeectofthe 150

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feedbackbetweencarriertransportandthepotentialinsidethedevice.In[ 184 ],Natoriexplainshowthisfeedbackrelatestothecurrentcontrolmechanisminthedevice,andultimatelyisresponsibleforvelocityandcurrentsaturationinthedevice.Ingeneral,thetransmissioncoecientsoftheIEZandOPEZareinverselyrelatedtotheirrespectivewidths.Astheappliedeldincreases,theIEZwidthx0!0andtheOPEZwidth!L,therebymakingTand(1)]TJ /F3 11.955 Tf 12.49 0 Td[()proportionaltotheappliedeld.Theexactnatureoftheelddependencewillberelatedtothepotentialproleinsidethedevice.Withthisunderstanding,itiseasytoseethattheI)]TJ /F3 11.955 Tf 13.04 0 Td[(Vcharacteristicswillbecloselyrelatedtothecarriertransmissionthroughthesezones.ForalinearpotentialprolediscussedbyNatori,whentheappliedeldisintherangesuchthatthetransmissioncoecientT1,Eq.( 5{60 )resultsinthefamiliarOhm'slaw(currentlinearlyproportionaltoappliedeld).WhentheappliedeldisincreasedsuchthatT!1,thecurrentandtheaveragecarriervelocitybothsaturate.Natori'stheoryanticipatesthattheinstantaneousvelocityofthecarrierisnotconstantduringitstransitfromthesourcetodrain.Instead,itperiodicallyoscillatesineachofthezonesasthecarrierissuccessivelytransmittedtothe1st,2ndandhigherrelaxedlevels.However,themeancarriervelocityacrossallthezonesisuniformandremainsproportionaltotheelectriceldupto104V/cm,beyondwhichitsaturates.Thecarrierdensity,ontheotherhand,alsotendstohaveaperiodicnatureacrossthezonesonceacurrentstartsowing.Tomaintaincurrentcontinuity,then(x)v(x)productshouldbeconstantatallpositionsinsidethedevice.Sincethepositionaveragedcarriervelocityisfoundtobeaconstant(likeinRyder'swork[ 169 ]),thecarrierconcentrationmultiplyingthisvelocityshouldbeaconstanttoo.Natorishowsthatthecarrierconcentrationtobeusedinthecurrentequationistheequilibriumcarrierdensityn0,whichissetbythefeedbackprocessfromOPEZ.The(negative)feedbackactstominimizethetotalelectrostaticenergyofthesystemviaPoisson'sequation. 151

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Tounderstandthefeedback,werstremindthattheOPEZisarepetitionofthesameunitstructureofwidthx1asshowninFig.( 5-4 ).Forthegiveneldcondition,theresultantcurrentissetintheIEZ,theuxdistributioninthe1standotherhigherrelaxedlevelsaredeterminedinaccordancewiththecarriervelocityintherespectivezonessoastomaintaincurrentcontinuity.However,sinceacurrentisowing,thecarrierdistributionalongthezonesisnotconsistentwiththeoriginalconstantelddistribution.Thecarrierchargeexcess(decit)pushesup(pullsdown)thepotentialproleoftheoriginalconstant-eldcurve,whichmodiestheprole(i.ethe\barrierheight")atthebeginningofthechannel.Thisresultsinadecrease(increase)ofthecarrierinjection,andthuscompensatesfortheexcess(decit)ofthecharge.ThepotentialproleintheIEZismodiedbythefeedbackfromtherestofthechannel.Thesteady-statepotentialproleisobtainedbytheself-consistentsolutionofthecoupledsystemoftheBTEandthePoissonequation.Thisself-consistencysupports(i.e.,triestobringthesystembackto)theconstanteldconditionforcedexternallybytheappliedbias.Thecarrierdistributioninthebulkactsinconjunctionwiththedopantchargedistribution(ifthedeviceisdoped)tominimizethetotalelectrostaticenergythroughthefeedbackmechanism.Thefeedbackresultsinsettingtheaveragecarrierdistribution=n0andthecurrentbecomesproportionaltoqn0TN.Theconventionaltheoryofvelocitysaturationisbasedonthebalance-of-energyequationandpredictsthatthecarriervelocitywillsaturateiftheopticalphononscatteringdominatestheenergyrelaxation,whichthencausesthecurrenttosaturate.Natori'smodelshowsthatthevelocityisproportionaltoEifthetransmissionissmall,eveniftheopticalphononemissionisdominant.IfT!1,thenthemeancarriervelocitythroughtheentiredevicesaturatesandcausesthecurrenttosaturate.SaturationwouldhappenirrespectiveoftheeldvalueifsomehowweweretomakeT!1happen.Inotherwords,transmissionsaturationthroughtheIEZcausescurrentsaturation(tothelevel 152

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determinedbytheproductofqn0,and(1)]TJ /F3 11.955 Tf 12.54 0 Td[()).ThisisafundamentaldierencebetweenconventionalmodelsandNatori'snewmodel. (a) (b) Figure5-6.TwoequivalentnegativefeedbacksystemrepresentationoftheNatorimodelwith(a)IEZtransmissioninthefeed-forwardpathand(b)OPEZtransmissioninthefeed-forwardpathshowinghowthefeedbackfromtheOPEZregulatesthetotalcurrentthroughthedevice. TwoequivalentrepresentationsofanegativefeedbacksystemwithatransferfunctionequivalenttotheterminEq.( 5{64 )areshowninFig.( 5-6 ).Withoutfeedback,thesystemrepresentationwouldbeasimplecascadeofTand(1)]TJ /F3 11.955 Tf 10.8 0 Td[().ThestabilizingnegativefeedbackfromtheOPEZactingthroughhelpstoregulatethetotalcurrentthroughthedevice.WhileFig.( 5-6a )describestheactualphysicalpicture,themodelFig.( 5-6b )helpsunderstandthesaturationphenomenonoftotaltransmissionwhenthelengthoftherstelasticzonebecomessosmallthattheTbecomesunity.Ourmethodofusingtheone-uxtheoryincludinginnitecarrierexchangebetweentheIEZandOPEZ(Fig.( 5-5 )andEq.( 5{64 ))toderivetheNatorimodeltransmissionequationsfacilitateabetterlinktotheLundstromorGildenblatcompactmodels.WhileNatoridiscussedabulktransportcaseinhispaper,ourderivationofEq.( 5{64 )isquitegenericandthereisnothingprecludingitsapplicationstostudyingquasi-ballistic 153

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transportinananoscaleMOSFET.Sinceourupdatedtransmissioncoecientincludinghigh-eldandconnementeects(Eq.( 5{57 ))isasurfacepotentialbasedequation,wecannowequateittotheNatoriequationEq.( 5{64 ),notingtheinherentlinkandfeedbackpresentbetweenthePoissonequation,BTE,andtheresultantsurfacepotential.UsingthefeedbackrepresentationoftheNatoriexpressionshowninFig.( 5-6a ),wehave tUF=TN=T 1+ (1)]TJ /F3 11.955 Tf 11.96 0 Td[()T (5{65) 1)]TJ /F3 11.955 Tf 11.96 0 Td[(=1+1 tUF)]TJ /F1 11.955 Tf 14.47 8.09 Td[(1 T)]TJ /F5 7.97 Tf 6.59 0 Td[(1 (5{66) IftUFcanbecomputedforarealdevice,thenEq.( 5{66 )givesusahandletoestimateforhighbiasconditionswhenTsaturates.OnepossibleschemeforevaluatingtUFwouldbethetechniqueproposedbyGildenblat[ 190 ]involvingthePSPcompactmodel[ 191 ][ 192 ],combinedwithsimulationresultsforc(0)andc(L)fromaSchroedinger-Poissonsolution. 5.4.5PhysicalSignicanceofEectiveVelocityTermInGildenblat'soriginalwork[ 190 ],theeectivevelocityterm(Eq.( 5{28 ))wassimplyamathematicalsimplicationusedtosimplifythetransmissionexpression.However,ourunderstandingof(1)thefeedbackmechanismand(2)theequivalencebetweentheNatoriexpressionandourupdatedexpressionfortransmissionprovidesanewphysicalunderstandingoftheeectivevelocityterm.Sincetheredenedeectivevelocityisadeniteintegraloverthechannellength,wecansplittheeectivevelocityintotwoparts,withthesplittingpointbeingtheIEZwidthx0(forthegivenbiasconditions)as, 154

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1 e=ZL0e)]TJ /F5 7.97 Tf 6.58 0 Td[((x(x))]TJ /F4 7.97 Tf 6.58 0 Td[(0(0))dx D=Zx00e)]TJ /F5 7.97 Tf 6.58 0 Td[((x(x))]TJ /F4 7.97 Tf 6.58 0 Td[(0(0))dx D+ZLx0e)]TJ /F5 7.97 Tf 6.59 0 Td[((x(x))]TJ /F4 7.97 Tf 6.59 0 Td[(0(0))dx D (5{67) or, 1 e=1 e1+1 e2 (5{68) Thetwoequivalenttransmissionexpressionswehaveare, TN=1 T+ 1)]TJ /F3 11.955 Tf 11.95 0 Td[()]TJ /F5 7.97 Tf 6.59 0 Td[(1 (5{69) tUF= 1 2+c(0) 2c(L)c(L) e+K!)]TJ /F5 7.97 Tf 6.58 0 Td[(1 (5{70) whereK=e)]TJ /F5 7.97 Tf 6.58 0 Td[((L(L))]TJ /F4 7.97 Tf 6.59 0 Td[(0(0)).EquatingEqs.( 5{69 )and( 5{70 ),weget, 1 T+ 1)]TJ /F3 11.955 Tf 11.96 0 Td[(=c(0) 2e+K 2c(0) c(L)+1 2(5{71)Expandingthevelocityintegralterm,wehave, 1 T+ 1)]TJ /F3 11.955 Tf 11.96 0 Td[(=c(0) 21 e1+1 e2+K 2c(0) c(L)+1 2(5{72)Whentheappliedgateanddrainvoltagesarehigh,theKtermbecomesmuchsmallerthanunityandcanbeneglected(SectionIVof[ 180 ]).Sincex0!0underthesevoltageconditions,T!1(i.e.,e1termvanishes),makingetobeentirelycomposedofthee2term,andthereforewehave, 155

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1)]TJ /F3 11.955 Tf 11.96 0 Td[(=c(0) 21 e)]TJ /F1 11.955 Tf 13.15 8.08 Td[(1 2(5{73)Solvingfor(highbias),weget, =1)]TJ /F3 11.955 Tf 18.04 8.09 Td[(e c(0) 1+e c(0)(5{74)LookingattheformoftheeterminEq.( 5{67 ),wecanseethatitrepresentsadiusion-velocitylikequantity.Intheabsenceofadrainbias,edescribeshowecientlythecarrierswoulddiuseduetoagivenconcentrationgradientalongthechannel.Themaximumdiusionvelocityisobtainedunderthecompleteabsenceofscatteringevents,andisequaltothethermalvelocityofthecarriersc(0).e,therefore,canatthemostbec(0).Whene=c(0)becomeslessthan1,itindicatesthepresenceofthescatteringeventsinthechannel.Underanapplieddrainbias,theeectivenessofthesuperimposeddriftresponseofthecarriersdependsonthediusibilityofthechannel,i.e.,theeasewithwhichcarriertransporthappens.Thisisdescribedbythee=c(0)ratio.Withthisinterpretation,wecanreadilyseehowscatteringinuencestheoveralldevicecurrent.Theimpactofscatteringeventsonthecurrentisexplainedthroughtheparameter.Inaquasi-ballisticdevice,thesaturationcurrentisproportionalto(1)]TJ /F3 11.955 Tf 12.81 0 Td[().ThelimitingcaseofballistictransportisseenfromEq.( 5{74 ),i.e.,whenthee!c(0),theelastic-backscattering/energyrelaxationtrade-obetweenthezonesvanishesmaking=0.Inotherwords,bothpartsofthechannelhaveunitytransmission,makingthetransportcompletelyballistic.Thus,wecannowunderstandthephysicalsignicanceoftheeectivevelocityterm.Usingthisconcept,wepresentanewdenitionofballisticeciencyinthedeviceas, B=(1)]TJ /F3 11.955 Tf 11.96 0 Td[() vF(5{75) 156

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whererepresentsthepositionandenergyaverageddriftvelocity, =xZ0EZ0n(x;E)v(x;E)dEdx xZ0EZ0n(x;E)dEdx(5{76)andvFrepresentstheFermivelocityatthebeginningofthechannel.Whenbecomeszero,carriersatthetopofthebarrierenterthechannelwithFermivelocityvF.Whentherearenoscatteringeventsalongthechannel,=vF,makingB=1.Notethatthecurrentisstilllimitedbythenumberofavailablestatesatthebeginningofthechannel. 5.4.6ExpandingNatori'sHighFieldModelforMOSFETsWhileNatori'sideaofdividingthechannelintotwozoneswithdistincttransmissionpropertiesisquitegeneric,theapplicationofthismodelfor2Dor1Dinversionlayersneedstheinclusionoftheeectsofthegatetochannelpotential.Fig.( 5-7a )showstheevolutionoftheconductionbandproleforvariouscombinationsofappliedgateanddrainbiases.Atthetopofthebarrier,theE-Kdiagramandoccupationofelectronsareillustrated.Carrierscanattainkineticenergyduetodrainbiaswhiledriftinginthechannel,aswellasfromthedegenerateoccupationoftheconductionbandduetohighgatebias.Accordingly,theseparationofthechannelintermsoftransmissioncharacteristicsareshownforallfourcasesinFig.( 5-7b ).Case1:Heretheappliedgateanddrainvoltagesarelow(weakinversion,linearregime).Thepotentialproleinthechannelisalmostlinearandthecarrierstatisticseverywhereinthechannelarenon-degenerate.Hence,thekineticenergyofthecarriersremainsconstant(nearthethermalenergy)makingOPscatteringdominantonlyatthedrainentrance.Thetransmissionpictureisthereforesimple,withtheIEZexpandingtocovernearlytheentirechannel.Case2:Whenthedrainvoltageisincreasedkeepingthegatevoltagestilllow(weakinversion,saturationregime),thekineticenergyofthenon-degeneratecarriersinthe 157

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(a) (b) Figure5-7.Evolutionof(a)conductionbandproleand(b)correspondingtransmissioncharacteristicsofthechannelarequalitativelyshownforfourpossibledeviceoperatingconditions:(1)lowVG,lowVD(2)highVG,lowVD(3)lowVG,highVDand(4)highVG,highVD inversionlayerincreasesalongthechannel.Thisinreturn,increasestheOPscatteringprobabilityinsidethechannel,makingtheOPEZcoveralargerportionofthechannel.Case3:Hereweconsideralargegatevoltageandsmalldrainvoltageappliedtothedevice(stronginversion,linearregime).Thedegeneratecarriersatthetopofthebarrierhavekineticenergyfromthegatepotential,andafractionofthesecarrierswillrelaxthroughOPscatteringclosetothebeginningofthechannel.Theprimedquantities(T0and1)]TJ /F3 11.955 Tf 12.64 0 Td[(0)describethezonetransmissionsrepresentingtheenergyrelaxationprocessofdegeneratecarriersonly.Whenthegatetochannelpotentialdierencereducestothenon-degenerateconditions,thecarrierkineticenergyreducestonearthermalenergies,andcarriertransportfromthispointresemblescase1.Case4:Withincreaseofdrainbias(stronginversion,saturationregime)fromcase3,thereductionofthekineticenergyofthedegeneratecarriersiscompensatedbytheenergygainedfromhighdraineld.TheOPEZzonesincase3merge(sincetransmissionthrough 158

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themiddleTzonesaturates)andthechannelintheseconditionsisalmostentirelycomprisedofoneOPEZ.Thisleadstoatotaltransmissionof(1)]TJ /F3 11.955 Tf 12.76 0 Td[()aspredictedbyNatori. 5.5QualitativeUnderstandingofStrainEectsStrainchangesbondanglesandlengths,whichmodiestheconductionandvalencebands.Thesemodicationsincludebandwarpingandbandsplitting,resultinginmodicationofconductivitymassandscatteringrates[ 193 ][ 194 ].Forelectroninversionlayers,theuniaxialstresseects[ 195 ]comesfrom:(1)conductivitymassdecreasefromtheelectronsrepopulatedfrom4to2valleys,(2)OPscatteringsuppressionduetothedensityofstatesdecreasefrom4to2and(3)conductivitymassdecreaseduetothebandwarping.Forholeinversionlayers,thestressefects[ 196 ][ 52 ]arisesfromsplittingandwarpingofheavy-holeandlight-holebandsthatresultsin(1)signicantlyreducedholeconductivitymassand(2)signicantlyreducedOPscattering.Therehavebeenseveralexperimentalworks[ 197 ][ 198 ]publishedthatshowthedierencebetweenlinearcurrentandsaturationcurrentenhancementwithuniaxialstressunderstronginversionconditions.Ithasbeenobservedthatforholes,theID(sat)enhancementdropsbynearly50%comparedtoID(lin),whiletheID(sat)enehacementislowerforelectrons,thediscrepancyisnotasmuchbetweenlinearandsaturation.LookingatthetransportcharacteristicsshowninFig.( 5-7b ),thedierencebetweencarriertransmissionsunderlinearandsaturationconditionsareimmediatelyseen.Sincethecurrentisproportionaltotheproductofaveragedriftvelocityandtotaltransmission,thesimultaneousimpactofstrainoneachofthesetwoparametersforthelinearandsaturationcasesneedtoconsidered.Thevelocitytermisinverselyproportionaltothesquarerootoftheconductivitymass.Theincreaseinthevelocitytermstrictlydependsonthestraininducedreductionoftheeectiveconductivitymass.Thestraindependenceofthetransmissiontermissomewhatmorecomplex,sinceitisdependentonbothscatteringandmass. 159

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Underfavorablestressconditions,inversionlayerelectronsexperiencereducedconductivitymasswithminimalchangesinthescatteringrate.Butforholes,thefavorablestrainresultsinsignicantreductionofbothcarriermassandscattering.Fromthestrainperspective,thebenetsofimprovedaveragecarrierdriftvelocityisfeltinboththelinearandsaturationregimes.ThedierencesintheID(lin)andID(sat)enhancementstherefore,mustcomefromthedierencesinhowstrainaectsthetotaltransmissionfortherespectiveregimes,forbothtypesofcarriers.Letusconsiderthelinearregimerst.Forahighgatebiasinthelinearregime(Case3inprevioussection),theinitialIEZregionbecomessmallsincethecarrierdegeneracyishigh.Pastthiszone,therestofthechanneliscomprisedofthreeregions,aninitialOPEZcharacterizedby1)]TJ /F3 11.955 Tf 12.34 0 Td[(0,themiddleelasticregioncharacterizedbyTandthenalinitialOPEZcharacterizedby1)]TJ /F3 11.955 Tf 11.95 0 Td[(.Carrierinterchangeoccursatbothzoneboundaries.Forholes,strainsignicantlyreducestheOPscattering,whichincreasesandreducestotaltransmission.Sincethemassreductionfromstraininducedbandwarpingissignicant,theincreaseindriftvelocityismuchlargerandcompensatesthestraininducedboundarytransmissionloss.MostoftheenhancementinholeID(lin),therefore,comesfromtheaveragedriftvelocityincrease.Forelectrons,theeectofstrainonscatteringreductionissmall,hencethecorrespondingreductionintransmissionisalsosmall.Mostofthelinearcurrentenhancementforelectronscomesfromtheaveragedriftvelocityincreaseaswell.Theamountofvelocityincrease,however,issmallerforholescomparedtoelectrons.Inthesaturationregime,themiddleelasticzonetransmissionTsaturatesandtheOPEZzonesmergetogether.Thestrainimpactonvelocityisstillpresentforbothtypeofcarriers.OPscatteringhappensinalargerportionofthechannel.Comparedtoelectrons,forholes,thereductionintransmissionfromstrainismuchmorecomparedtothelinearregime,resultinginareducedenhancementinID(sat).Forelectrons,thescattering 160

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reductionissmall(comparedtolinearregime)anditseectoncurrentisnotfeltasmuch.ThustheelectronID(sat)alsoreduces,butnotasmuchcomparedtoholeID(sat). 5.6SummaryInthischapter,wepresentedthedevelopmentofanewcompactmodelfornanoscaletransmission.ByaddinghigheldeectsandconnementeectstotheGildenblatmodel,anew,surfacepotentialbased,analyticalexpressionforthetransmissioncoecientinaMOSFETchannelwasestablished.Thisexpressionisapplicableforallconditionsofdeviceoperationandforallsurface/channelorientations.ThephysicalsignicanceoftheeectivevelocitywashighlightedanditwasshowntobeanimportantgureofmeritfordescribingthetransportinaMOSFETchannel.TheessentialphysicsofNatori'shigheldtransportmodelwasexplainedusingasimplemathematicaltreatmentandthemodelwasextendedtoincludetheeectofthegateconnement.Bylinkingthesetwoupdatedmodels,experimentallyreporteddierencesbetweenstrainenhancedlinearandsaturationcurrentsinshortchanneldeviceswasqualitativelyexplained. 161

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CHAPTER6CONCLUSIONSANDFUTUREWORK 6.1SummaryInthisdissertation,theeectofuniaxialstresson(a)longchanneln-typeandp-typeplanarMOSFETs,(b)n-typeandp-typeSiFinFETsandp-typeSiGe/SiFinFETs,and(c)carriertransportinquasi-ballisticnanoscaleMOSFETsarestudiedusingdetailedexperimentaldevicecharacterizationandmodeling.Themainaccomplishmentsofthisresearcharedescribedinthissection.InChapter2,weexperimentallyinvestigatedtheunderlyingcausesofmobilityenhancementinlongchanneln-typeandp-typeplanarMOSFETs.Controlledwaferbendingexperimentswereperformedattemperaturesfrom300Kto80Ktostudytheholeandelectronmobilityenhancement.Forbothelectronsandholes,themobilityenhancementwithfavorableuniaxialstrainincreasesatlowertemperaturecomparedtoroomtemperature.Theuniaxialcompressivestrain-inducedreductionofconductivityeectivemassofholesintheinversionlayeristhemajorcontributortothemobilityenhancementatlowstrainlevelsinp-typeMOSFETs.Forelectrons,theexperimentalresultsshowthatuniaxialtensilestrainpossiblyreducesthesurfaceroughnessattheSi/SiO2interface,leadingtoenhancedelectronmobilityathightransverseeldsasproposedbyFischetti.InChapter3,processstrainedp-typeandn-typeSiFinFETshavingstrainedcontactetchstoplayerswerethroughlycharacterized.Wepresentedanewtechniquetostudytheextractionandimpactoftheparasiticsource/drainresistanceinFinFETsincludingtheeectofunderlapinprocessstrainedFinFETs.Theexperimentalresultsshowedthatthemodicationofunderlapresistancebystrainisapotentialcontributingfactorintheexperimentallyobserveddrivecurrentenhancement.AthoroughreviewofthephysicsofstresstransfermechanismhighlightedthedierencesbetweenmechanicalstressandprocessinducedstressinFinFETs.Thestressproleproducedinthenfromwafer 162

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bendingisverydierentfromthatproducedwithaCESLbecauseofhowthestressistransferredtothen.InChapter4wepresentedathoroughreviewoftheexistingmodelsandexperimentalmethodsforevaluatingquasi-ballistictransportinnanoscaletransistors.Wealsoevaluatedtheuseoftheupdatedlowtemperaturemethodforextractingtheballisticeciencyinstraineddevicesandconcludedthatanewmodelwasnecessarytogainfundamentalunderstandingoftheimpactofstrainonquasi-ballistictransport.InChapter5,wepresentedanewcompactmodelfornanoscaletransmissionsuitableforunderstandingstraineects.Thismodelwasdevelopedbyaddinghigheldeectsandconnementeectstotheone-uxtheorybasedGildenblatmodel.AsurfacepotentialbasedanalyticalexpressionforthetransmissioncoecientinaMOSFETchannelwasestablished,andtheeectivevelocitytermwasshowntobeanimportantgureofmeritfordescribingcarriertransportwithaclearexpressionofitsphysicalsignicance.AsimplemathematicaltreatmentwasdevelopedtoexplainthefundamentalphysicsinNatori'shigheldtransportmodel.TheeectsofgateconnementwasaddedtoextendtheNatorimodel,andtheextendedmodelwaslinkedtoourstoprovideinsightintostraineectsoncarriertransport.Theeectofopticalphononscatteringonquasi-ballistictransportinMOSFETswasclearlyexplainedusinganupdatedtransmissionpictureofthedevicechannel,andusingthispicture,theeectsofuniaxialstrainwasqualitativeexplained.Itwasfoundthatthedierencesinstrain-inducedchangesinopticalphononscatteringrateforelectronsandholesininversionlayersexplainstheexperimentallyobserveddierencesbetweenlinearandsaturationdrivecurrentenhancementforbothtypesofcarriers. 6.2RecommendationsforFutureWorkGildenblatshowed[ 180 ]apossiblemethodforexperimentalinvestigationofhisnanoscaletransmissionmodel.ThistechniqueinvolvedobtainingthesurfacepotentialatthesourceanddrainendsofthedeviceusingthePSPcompactmodel[ 191 ].Since 163

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ourmodelalsoisasurfacepotentialbasedmodelbasedonGildenblat'smodel,itshouldtopossibletoextractthetransmissioncoecientusingasimilartechnique,alongwithsimulationresultsfromaSchroedinger-Poissonsolverfortheconnedbandstructurevelocitiesatthesourceanddrainends.Theverilog-AcodeforthePSPcompactmodelisavailablepublicly,anditpossibletointerfacetheverilogcodetoadevicesimulatorsuchasCadenceSpectre.Weveriedthatitispossibletomodifytheverilogcodetooutputthesurfacepotentialasoneoftheoperatingpointvariablesforadevice.InvestigatingstraineectswiththeUFtransportmodelispossiblewithsuchanimplementation,afterttingtherealdevicedatausingthePSPcompactmodelwithinSpectreisrecommendedforfuturework.ThiswouldhelpverifyourqualitativemodelforstraineectspresentedinChapter5.ExtendingtheuniversalUFnanoscaletransmissionmodelfor1Dconneddevicessuchasnanowiresisalsorecommendedforfuturework.Thereduceddegreeofcarrierfreedominnanowiresleadtosomeveryinterestingpossibilitiesindevicephysics[ 194 ],andinvestigatingthetransmissioncoecientinthesequasi-ballisticnanowireswillhelpgainfundamentalinsightintodevicescaling.Onthetheoreticalfront,incorporatingstrainintoBringuier'sFokker-Planckapproach[ 174 ][ 176 ][ 176 ]couldenableaveryaccuratefundamentalstudyofstraineectsonhigheldtransportinquasi-ballisticdevices.ComparingresultsfromsuchatheoreticalimplementationagainsttheexperimentaltechniqueoutlinedaboveusingtheUFnanoscaletransmissioncoecienttogethernotonlymayverifythequalitativeexplanationinChapter5,butalsomayprovideverygoodinsightintotheimpactandecacyofstraininemergingnanoscaledevices. 164

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BIOGRAPHICALSKETCH SrivatsanParthasarathywasborninMadras,Indiain1982.HeobtainedaB.E.degreeinelectricalengineeringin2003fromtheUniversityofMadras.HeworkedasaAssistantSystemsEngineeratTataConsultancyServices(Chennai)during2004-2005.HeobtainedtheM.S.degreeinelectricalandcomputerengineeringfromtheUniversityofFloridain2007.HeworkedasagraduateresearchinternintheFrontEndProcessinggroupatSEMATECH(Austin)during2008-2009,whereheworkedonelectricalchatecterizationofshortchannelFinFETs.HerecievedhisPh.D.degreeinelectricalandcomputerengineeringfromtheUniversityofFloridain2012.Hisresearchisfocusedoninvestigatingstraineectsonquasi-ballistictransportemergingnanoscaletransistors.HecurrentlyworksforthePortlandTechnolgyDevelopmentgroupatIntelCorporation(Hillsboro). 185