Strain Effects in Low-Dimensional Silicon MOS and AlGaN/GaN HEMT Devices

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Title:
Strain Effects in Low-Dimensional Silicon MOS and AlGaN/GaN HEMT Devices
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1 online resource (192 p.)
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english
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Baykan, Mehmet Onur
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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:
Nishida, Toshikazu
Committee Co-Chair:
Thompson, Scott
Committee Members:
Guo, Jing
Ural, Ant
Gila, Brent P

Subjects

Subjects / Keywords:
100 -- 110 -- algan -- ballistic -- binding -- body -- confinement -- consistent -- coulomb -- electron -- finfet -- gan -- hole -- interface -- mobility -- mosfet -- nanowire -- phonon -- poisson -- quantum -- quasi -- roughness -- scattering -- schrodinger -- self -- silicon -- strain -- stress -- surface -- thin -- tight -- transport -- ultra
Electrical and Computer Engineering -- Dissertations, Academic -- UF
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Electrical and Computer Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

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Abstract:
Strained silicon technology is a well established method to enhance sub-100nm MOSFET performance. With the scalability of process-induced strain, strained silicon channels have been used in every advanced CMOS technology since the 90nm node. At the 22nm node, due to the detrimental short channel effects, non-planar silicon CMOS has emerged as a viable solution to sustain transistor scaling without compromising the device performance. Therefore, it is necessary to conduct a physics based investigation of the effects of mechanical strain in silicon MOS device performance enhancement, as the transverse and longitudinal device dimensions scale down for future technology nodes. While silicon is widely used as the material basis for logic transistors, AlGaN/GaN HEMTs promise a superior device platform over silicon based power MOSFETs for high-frequency and high-power applications. In contrast to the mature Si crystal growth technology, the abundance of defects in the GaN material system creates obstacles for the realization of a reliable AlGaN/GaN HEMT device technology. Due to the high levels of internal mechanical strain present in AlGaN/GaN HEMTs, it is of utmost importance to understand the impact of mechanical stress on AlGaN/GaN trap generation. First, we have investigated the underlying physics of the comparable electron mobility observed in (100) and (110) sidewall silicon double-gate FinFETs, which is different from the observed planar (100) and (110) electron mobility. By conducting a systematic experimental study, it is shown that the undoped body, metal gate induced stress, and volume-inversion effects do not explain the comparable electron mobility. Using a self-consistent double-gate FinFET simulator, we have showed that for (110) FinFETs, an increased population of electrons is obtained for the Delta_2 valley due to the heavy nonparabolic confinement mass, leading to a comparable average electron transport effective mass for both orientations. The width dependent strain response of tri-gate p-type FinFETs are experimentally extracted using a 4-point bending jig. It is found that the low-field piezoresistance coefficient of p-type FinFETs can be modeled by using a weighted conductance average of the top and sidewall bulk piezoresistance coefficients. Next, the strain enhancement of p-type ballistic silicon nanowire MOSFETs is studied using sp3d5s* basis nearest-neighbor tight-binding simulations coupled with a semiclassical top-of-the-barrier transport model. Size and orientation dependent strain enhancement of ballistic hole transport is explained by the strain-induced modification of the 1D nanowire valence band density-of-states. Further insights are provided for future p-type high-performance silicon nanowire logic devices. A physics based investigation is conducted to understand the strain effects on surface roughness limited electron mobility in silicon inversion layers. Based on the evidence from electrical and material characterization, a strain-induced surface morphology change is hypothesized. To model the observed electrical characteristics, we have employed a self-consistent MOSFET mobility simulator coupled with an ad hoc strain-induced roughness modification. The strain induced surface morphology change is found to be consistent among electrical and materials characterization, as well as transport simulations. In order to bridge the gap between the drift-diffusion based models for long-channel devices and the quasi-ballistic models for nanoscale channels, a unified carrier transport model is developed using an updated one-flux theory. Including the high-field and carrier confinement effects, a surface-potential based analytical transmission expression is obtained for the entire MOSFET operation range. With the new channel transmission equation and average carrier drift velocity, a new expression for channel ballisticity is defined. Impact of mechanical strain on carrier transport for both nMOSFETs and pMOSFETs in both linear and saturation regimes is explained using the new channel transmission definitions. To understand the impact of mechanical strain on AlGaN/GaN HEMT trap generation, we have devised an experimental method to obtain the photon flux-normalized relative areal trap density distribution using photoionization spectroscopy technique. The details of the trap extraction method and the experimental setup are given. Using this setup, the trap characteristics are extracted for both ungated transmission line module (TLM) and gated HEMT devices from both Si and SiC substrates.  The changes in the device trap characteristics are emphasized before and after electrical stressing. It is found through the step-voltage stressing of the AlGaN/GaN HEMT gate stack that the device degradation is due to the near bandgap trap generation, which are shown to be related to the structural defects in GaN.
General Note:
In the series University of Florida Digital Collections.
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Includes vita.
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Includes bibliographical references.
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Description based on online resource; title from PDF title page.
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This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility:
by Mehmet Onur Baykan.
Thesis:
Thesis (Ph.D.)--University of Florida, 2012.
Local:
Adviser: Nishida, Toshikazu.
Local:
Co-adviser: Thompson, Scott.
Electronic Access:
RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2013-08-31

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lcc - LD1780 2012
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STRAINEFFECTSINLOW-DIMENSIONALSILICONMOSANDALGAN/GANHEMTDEVICESByMEHMETONURBAYKANADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2012

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c2012MehmetOnurBaykan 2

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Tomymother,myfather,mybrotherandmybabynephew,Efe 3

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ACKNOWLEDGMENTS Firstofall,Iwouldliketothankmyadvisor,Dr.ToshikazuNishida,towhomIamindebtedforhisinvaluableguidanceandsupportthroughoutmyentiregraduateresearch.Also,Iwouldliketothankmyco-advisor,Dr.ScottThompson,forhisinsightfulcontributionsandinspirationalperspectiveonresearchproblems.IalsowouldliketothankmyPh.D.committeemembers,Dr.JingGuo,Dr.AntUralandDr.BrentGilafortheirsuggestionsandinterestsinmywork.Inaddition,IwouldliketothankAirForceOfceofScienticResearchfortheirsupportofmygraduateresearchthroughMURIprojectfunding.IalsowouldliketothankmycolleaguesatSematech,Dr.ChrisHobbs,Dr.KeremAkarvardar,Dr.ChadwinYoung,andDr.PrashantMajhifortheopportunitiesduringmyinternshipin2010-2011,whichgavemeaprofessionalperspectiveofthestateoftheartsemiconductorindustry.Also,Iwouldliketothankallmylabmates:Amit,Andy,Ashish,David,Hyunwoo,Jingjing,Min,Nicole,Sagar,Sri,Tony,Ukjin,Uma,Xiaodong,YongkeandYounsung,fortheirhelp,supportandfriendshipwhichmakeseverydayatworkajoyfulandaproductiveone.IwouldliketogivespecialthankstoTheScienticandTechnologicalResearchCouncilofTurkey(TUBITAK)fortheirgeneroussupportofmydoctoralstudies.Finally,Iwouldliketothankmymother,myfather,mybrotherandmydearHelenfortheireverydaysupportandencouragement. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 8 LISTOFFIGURES ..................................... 9 ABSTRACT ......................................... 13 CHAPTER 1INTRODUCTION ................................... 16 1.1Motivation .................................... 16 1.2ObjectiveandOrganization .......................... 19 2BACKGROUNDONCLASSIFICATIONOFDEVICESINTERMSOFCARRIERDEGREEOFFREEDOM .............................. 21 2.1Motivation .................................... 21 2.2ClassicationofDevicesinTermsofCarrierDegreeofFreedom ..... 22 2.2.1Unconned/3DOFBulkDevices .................... 22 2.2.2DeviceswithPredominantElectrostaticConnement(EC) ..... 25 2.2.2.11DEC/2DOFsurface-inversionMOSFET ......... 25 2.2.2.21Dpolarization-inducedEC/2DOF2DEGAlGaN/GaNHEMTdevices ........................ 27 2.2.2.31DEC/2DOFvolume-inversionMuGFETdevices ..... 28 2.2.2.42DEC/quasi-1DOFvolume-inversiondevices ....... 29 2.2.3DeviceswithPredominantSpatialConnement ........... 32 2.2.3.11DSC/2DOFultra-thin-body(UTB)devices ........ 32 2.2.3.22DSC/1DOFnanowires(NW) ............... 34 2.3Summary .................................... 38 3BACKGROUNDONTRAPCHARACTERIZATIONMETHODS ......... 39 3.1Motivation .................................... 39 3.2OverviewofTrapCharacterizationMethods ................. 40 3.2.1PulsedI-V ................................ 40 3.2.2ConductanceDispersion ........................ 41 3.2.3SubthresholdSwing .......................... 43 3.2.4CurrentTransients ........................... 43 3.2.5DeepLevelTransientSpectroscopy(DLTS) ............. 44 3.2.6PhotoionizationSpectroscopyandLuminescence .......... 45 3.3Summary .................................... 46 5

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4INSIGHTSONTHE(100)AND(110)DOUBLE-GATEFINFETELECTRONTRANSPORTANDTHESTRAINRESPONSEOFP-TYPETRI-GATEFINFETS 48 4.1PhysicalInsightsonComparableElectronTransportin(100)and(110)Double-gateFinField-effectTransistors ................... 48 4.1.1Motivation ................................ 48 4.1.2Experimental .............................. 49 4.1.3ResultsandDiscussion ........................ 51 4.1.4Conclusion ............................... 54 4.2StrainResponseofp-typeTri-GateFinFETs ................. 54 4.3Summary .................................... 58 5SIZEANDORIENTATIONDEPENDENTSTRAINEFFECTSONBALLISTICSIP-TYPENANOWIREFIELDEFFECTTRANSISTORS ............ 60 5.1Motivation .................................... 60 5.2Method ..................................... 61 5.3ResultsandDiscussion ............................ 66 5.3.1UnstrainedDevices ........................... 68 5.3.2StrainedDevices ............................ 71 5.4Summary .................................... 75 6PHYSICALINSIGHTSONSTRAINENHANCEDSURFACEROUGHNESSLIMITEDELECTRONMOBILITYINSILICONINVERSIONLAYERS ...... 77 6.1Motivation .................................... 77 6.2Experimental .................................. 79 6.3Simulation .................................... 81 6.3.1TemperatureDependence ....................... 81 6.3.2StrainModication ........................... 82 6.3.3MobilityModel .............................. 83 6.3.3.1Intravalleyphononscattering ................ 84 6.3.3.2Intervalleyphononscattering ................ 85 6.3.3.3Surfaceroughnessscattering ................ 86 6.4ResultsandDiscussion ............................ 88 6.5Summary .................................... 97 7UNIFIEDQUASI-BALLISTICCOMPACTMODELFORSTRAINEDNANOSCALEMOSFETSUSINGANUPDATEDONE-FLUXTHEORY ............. 99 7.1Motivation .................................... 99 7.2Fundamentals ................................. 101 7.2.1Natori'sOriginalBallisticTransportModel .............. 101 7.2.2CurrentControlinNatori'sModel ................... 103 7.2.3One-uxTheory:Basics ........................ 104 7.2.4RevisitingtheLundstromModel .................... 108 7.2.5CurrentControlinLundstromModel ................. 112 6

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7.2.6IssuesintheLundstromModel .................... 114 7.2.7RevisitingtheGildenblatModel .................... 117 7.2.8RevisitingtheOriginalShockleyWork ................ 120 7.3UFCompactModel:Theory .......................... 121 7.3.1IncludingHighFieldEffects ...................... 122 7.3.2IncludingConnementEffects ..................... 125 7.3.3RevisitingtheNatori'sHighFieldTransportModel .......... 126 7.3.4UnderstandingFeedbackinNatori'sHighFieldModel ....... 130 7.3.5PhysicalSignicanceofEffectiveVelocityTerm ........... 136 7.3.6ExpandingNatori'sHighFieldModelforMOSFETs ......... 139 7.4QualitativeUnderstandingofStrainEffects ................. 140 7.5Summary .................................... 142 8OPTICALCHARACTERIZATIONOFALGAN/GANHEMTTRAPGENERATION 144 8.1Motivation .................................... 144 8.2OverviewofHEMTDevice ........................... 146 8.3Method ..................................... 148 8.4ExperimentalSetup .............................. 150 8.5ResultsandDiscussion ............................ 157 8.6Summary .................................... 165 9SUMMARYANDRECOMMENDATIONFORFUTUREWORK ......... 166 9.1ResearchSummary .............................. 166 9.2RecommendationforFutureWork ...................... 168 REFERENCES ....................................... 170 BIOGRAPHICALSKETCH ................................ 192 7

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LISTOFTABLES Table page 3-1Summaryofdifferenttrapcharacterizationmethodsintermsofcapabilitiesofdensity,timeconstantandactivationenergyextraction,andchallenges. .... 47 5-1Theactualatomisticheightandwidthofthenanowiresusedinthisstudy. ... 64 6-1Thevaluesofphonondeformationpotentials,phononenergies,andsurfaceroughnessparametersusedformobilitycalculationsareshown. ........ 90 8

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LISTOFFIGURES Figure page 1-1Theevolutionofthecostoflithographytoolswithtime. .............. 17 1-2TheillustrationofthespontaneousandpiezoelectricpolarizationinAlGaN/GaNheterostructureisshown. .............................. 19 2-1Classicationofsemiconductordevicesbasedonthetypeanddimensionalityofcarrierconnement. ................................ 22 2-2Various2DOFsilicondevicesareshown. ..................... 23 2-3Electricallyconnedquasi-1DOFvolume-inversionandspatiallyconned1DOFnanowiresiliconMOSFETchannelsareshown. .................. 24 2-4Conductionbandminimafornon-spatiallyconnedunstrainedsilicondevicesareshown. ...................................... 25 2-5Spatiallyconned2DOFUTBsiliconand1DOFsiliconnanowiredevicesareshown. ........................................ 31 2-6Thedirectbandgaptransitionfrom3DOFbulksilicontoto(001)2DOFUTBSi(left)andh100ichannel1DOFSiNW(right)areillustrated. .......... 33 2-7Thevalencebandsforunstrained2.1nmX2.1nmsiliconnanowiresareshown. 35 3-1ThepulsedI-Vmethodforhigh-siliconMOSFETisshown. .......... 41 3-2Equivalentcircuitmodelsareshownfortheconductancedispersionmethod. 42 3-3Photoemissionisshownforpre-stressandpost-stressconditionsunderforwardandreversebias. ................................... 46 4-1Effectiveelectronmobilityisextractedfrom(100)and(110)sidewallFinFETsusingsplit-CVmethod,andshownasafunctionofNinv. ............. 50 4-2FlowchartofmodiedSchrodinger-Poissonself-consistentsimulationproceduretotakenon-parabolicconnementmassintoaccount. .............. 51 4-3Energy-wavevectordispersionrelationofthetransverseelectronmassandtheextractedconnementmass/quantizationenergypairsareshown. ..... 52 4-4Evolutionof(110)CBvalleydegeneraciesisshownwithandwithoutnon-parabolicconnementmassapproach. ............................ 52 4-5Averageelectronconductivityeffectivemassvs.Ninvisshownforboth(100)and(110),usingbothparabolicandnon-parabolicconnementmass. ..... 53 9

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4-6Tri-gatep-typeFinFETsamplecleavedfroma300mmwaferisunder1GPauniaxialtension. ................................... 55 4-74-pointuniaxialwaferbendingisillustrated. .................... 56 4-8TheexternalparasiticresistanceRSDextractedfromn(blue)andp-type(red)tri-gateFinFETsusingshiftandratiomethod. ................... 56 4-9Experimentallyextractedpiezoresistancecoefcientsfromp-typetri-gateFinFETsareshown. ...................................... 57 5-1Straighth110iand45orotatedh100inanowiredevicesareshownonconventional(001)silicon-on-insulatorwafer. ........................... 62 5-2ForaconstantnumberofmobilechargesinaunitvolumeowingalongL,theequivalent3D,2Dand1Dchargedensitiesareillustrated. .......... 67 5-3Thevalencebandstructureisshownforunstrainednarrowest(3nmx3nm)andwidest(7nmx7nm)nanowiresinbothh110iandh100idirections. ..... 68 5-4Thesizeandorientationdependentunstrainedballisticaverageinjectionvelocityandholedraincurrentareshown. ......................... 70 5-5Thebandstructureholedistributionisshownfor3nmX3nmh100inanowireunder0.8%tension,unstrainedand0.8%compressionatjVovj=0.6V. .... 72 5-6Thebandstructurecurrentdistributionisshownfor3nmX3nmh100inanowireunder0.8%tension,unstrainedand0.8%compressionatjVovj=0.6V. .... 73 5-7Thebandstructureholedistributionisshownfor3nmX3nmh110inanowireunder0.8%tension,unstrainedand0.8%compressionatjVovj=0.6V. .... 74 5-8Thebandstructurecurrentdistributionisshownfor3nmX3nmh110inanowireunder0.8%tension,unstrainedand0.8%compressionatjVovj=0.6V. .... 74 5-9Sizedependentgaugefactorisshownforbothstraightandrotatedp-typenanowiresatanoverdrivevoltageof0.6V. ..................... 76 6-1Thecryogenictemperaturewaferbendingsetupandbendingapparatusareshown. ........................................ 79 6-2ThestrainenhancementofnMOSFETlineardraincurrentisshownasafunctionofsampletemperaturefortwodifferentstressconditions. ............ 80 6-3AFMsystemcoupledtomechanicalbendingjigandthemeasuredroughnessamplitudeareshown. ................................ 80 6-4Double-gateandsingle-gateoperationoftheFinFETsimulatorisdemonstrated. 81 10

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6-5Theeffectivemobilityanditsconstituentsforasingle-gatebulkMOSFETareplottedvs.effectiveelectriceldat300Kandnostress. ............. 89 6-6Electronmobilityvs.inversionelectrondensityisshownforabulksingle-gate(100)MOSFETanda20nmthickdouble-gate(100)MOSFETat300K. .... 91 6-7Electronmobilityvs.effectiveelectriceldisshownforabulksingle-gate(100)MOSFETanda20nmthickdouble-gate(100)MOSFETat300K. ........ 92 6-8Simulatedsingle-gateMOSFETlow-eldelectronmobilityvs.gatevoltageisshownforatemperaturerangeof400Kto100K. ................. 93 6-9ForthesinglegateMOSFET,simulatedinversioncarrierdensityvs.gatevoltageisshownfortemperaturesrangingfrom400Kto100Kwithdecrementsof50K. 94 6-10Simulatedsingle-gateMOSFETlineardraincurrentvs.gatevoltageisshownforatemperaturerangeof400Kto100K. ..................... 94 6-11Simulatedsingle-gatenMOSFETnormalizedchannelresistancechangevs.uniaxialstressandtheexpectedchangefromthe-coefcientareshown. ... 95 6-12Theadhocsimulationresultsareshownforthe75MPacondition. ....... 96 6-13Normalizedadhocttingerrorisshownfor25MPa,50MPaand75MPaconditionsforagivengatevoltage. ............................... 97 7-1SchematicofconductionorrstsubbandprolealongthelengthofaMOSFETchannelfordevicebiasedinsaturationconditions. ................ 102 7-2Incomingandoutgoingcarrieruxesareshownforaslabofthicknessdx. ... 106 7-3DenitionoftheKT-layerintheLundstromModel,transmittedandbackscatteredcarriersnearthecriticalregionfortransportareshown. ............. 109 7-4Natori'shigheldtransportmodelshowingtheconceptualInitialElasticZoneandOpticalPhononEmissionZoneregionsinadevicecarryingcurrent. ... 128 7-5Ourderivationoftheoveralltransmissioncoefcientusinganinniteseriessummationapproach. ................................ 132 7-6EquivalentnegativefeedbacksystemrepresentationsoftheNatoriModelareshown. ........................................ 132 7-7EvolutionofMOSFETconductionbandandchanneltransmissioncharacteristicsisshownfordifferenttransistoroperationconditions. ............... 136 8-1ThesampleinformationisprovidedfortheAlGaN/GaNHEMTstructuresusedinthisstudy. ...................................... 146 11

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8-2Experimentallymeasuredcapacitance-voltageandextracted2DEGdensityareshownforCV-dotsonbothSiandSiCsubstrates. .............. 148 8-3Thecarrierdynamicsofthephotoionizationspectroscopymethodareillustrated. 149 8-4Theexperimentalsetupusedinthisstudyisshown.BothelectricalandmechanicaldegradationexperimentsareintegratedintothePSmeasurements. ...... 151 8-5Thelineardraincurrentresponsefroma30mlongTLMdeviceisshownforbothwithandwithoutfocusinglenses(i.e.highandlowintensity)650nmillumination. 152 8-6ThetransmissioncharacteristicsoftheUV-rejectionhigherorderlterisshown. 153 8-7CrosscalibrationcurvesbetweenSpectra-650colorimeterandOriel70124pyroelectricdetector. ................................. 155 8-8Thenormalizedphotonuxwithrespectto360nmilluminationvs.incidentphotonwavelengthisshown. ............................ 156 8-9Thedensitynormalizationcoefcientisshownfordifferentphotonwavelengthsandenergies. ..................................... 157 8-10Thecurrent-voltagecurvesareshownfor30mTLMstructuresonSiandSiCsubstrates. .................................... 158 8-11Atypicalphotoionizationspectroscopydatafroma5mTLMsiteonSisubstrateisshown. ....................................... 159 8-12Extractedphotonux-normalizedrelativearealtrapdensityisshownfora30mlongTLMonSiCanda5mlongTLMonSisubstrate. .............. 160 8-13TherepeatabilityofthePSexperimentisdemonstratedonavirgin100nmoff-centergateHEMTonSiCsubstrate. ...................... 161 8-14DatacollectedfromtwoconsecutiverunsofOFF-stateelectricalgatestressexperimentsisshownfora100nmoff-centergateHEMTonSiCsubstrate. .. 162 8-15Gatecurrentdensityisshownforpre-stressandpost-OFF-stateelectricalstressconditions. ................................... 163 8-16Widthnormalizeddraincurrentisshownforpre-stressandpost-OFF-stateelectricalstressconditions. ............................. 164 8-17Extractedphotonux-normalizedrelativeelectronandholearealtrapdensitiesareshownforbeforeandaftertheOFF-statestress. ............... 164 12

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophySTRAINEFFECTSINLOW-DIMENSIONALSILICONMOSANDALGAN/GANHEMTDEVICESByMehmetOnurBaykanAugust2012Chair:ToshikazuNishidaCochair:ScottE.ThompsonMajor:ElectricalandComputerEngineering Strainedsilicontechnologyisawellestablishedmethodtoenhancesub-100nmMOSFETperformance.Withthescalabilityofprocess-inducedstrain,strainedsiliconchannelshavebeenusedineveryadvancedCMOStechnologysincethe90nmnode.Atthe22nmnode,duetothedetrimentalshortchanneleffects,non-planarsiliconCMOShasemergedasaviablesolutiontosustaintransistorscalingwithoutcompromisingthedeviceperformance.Therefore,itisnecessarytoconductaphysicsbasedinvestigationoftheeffectsofmechanicalstraininsiliconMOSdeviceperformanceenhancement,asthetransverseandlongitudinaldevicedimensionsscaledownforfuturetechnologynodes. Whilesiliconiswidelyusedasthematerialbasisforlogictransistors,AlGaN/GaNHEMTspromiseasuperiordeviceplatformoversiliconbasedpowerMOSFETsforhigh-frequencyandhigh-powerapplications.IncontrasttothematureSicrystalgrowthtechnology,theabundanceofdefectsintheGaNmaterialsystemcreatesobstaclesfortherealizationofareliableAlGaN/GaNHEMTdevicetechnology.DuetothehighlevelsofinternalmechanicalstrainpresentinAlGaN/GaNHEMTs,itisofutmostimportancetounderstandtheimpactofmechanicalstressonAlGaN/GaNtrapgeneration. First,wehaveinvestigatedtheunderlyingphysicsofthecomparableelectronmobilityobservedin(100)and(110)sidewallsilicondouble-gateFinFETs,whichis 13

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differentfromtheobservedplanar(100)and(110)electronmobility.Byconductingasystematicexperimentalstudy,itisshownthattheundopedbody,metalgateinducedstress,andvolume-inversioneffectsdonotexplainthecomparableelectronmobility.Usingaself-consistentdouble-gateFinFETsimulator,wehaveshowedthatfor(110)FinFETs,anincreasedpopulationofelectronsisobtainedforthe2valleyduetotheheavynonparabolicconnementmass,leadingtoacomparableaverageelectrontransporteffectivemassforbothorientations. Thewidthdependentstrainresponseoftri-gatep-typeFinFETsareexperimentallyextractedusinga4-pointbendingjig.Itisfoundthatthelow-eldpiezoresistancecoefcientofp-typeFinFETscanbemodeledbyusingaweightedconductanceaverageofthetopandsidewallbulkpiezoresistancecoefcients. Next,thestrainenhancementofp-typeballisticsiliconnanowireMOSFETsisstudiedusingsp3d5sbasisnearest-neighbortight-bindingsimulationscoupledwithasemiclassicaltop-of-the-barriertransportmodel.Sizeandorientationdependentstrainenhancementofballisticholetransportisexplainedbythestrain-inducedmodicationofthe1Dnanowirevalencebanddensity-of-states.Furtherinsightsareprovidedforfuturep-typehigh-performancesiliconnanowirelogicdevices. Aphysicsbasedinvestigationisconductedtounderstandthestraineffectsonsurfaceroughnesslimitedelectronmobilityinsiliconinversionlayers.Basedontheevidencefromelectricalandmaterialcharacterization,astrain-inducedsurfacemorphologychangeishypothesized.Tomodeltheobservedelectricalcharacteristics,wehaveemployedaself-consistentMOSFETmobilitysimulatorcoupledwithanadhocstrain-inducedroughnessmodication.Thestraininducedsurfacemorphologychangeisfoundtobeconsistentamongelectricalandmaterialscharacterization,aswellastransportsimulations. Inordertobridgethegapbetweenthedrift-diffusionbasedmodelsforlong-channeldevicesandthequasi-ballisticmodelsfornanoscalechannels,auniedcarriertransport 14

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modelisdevelopedusinganupdatedone-uxtheory.Includingthehigh-eldandcarrierconnementeffects,asurface-potentialbasedanalyticaltransmissionexpressionisobtainedfortheentireMOSFEToperationrange.Withthenewchanneltransmissionequationandaveragecarrierdriftvelocity,anewexpressionforchannelballisticityisdened.ImpactofmechanicalstrainoncarriertransportforbothnMOSFETsandpMOSFETsinbothlinearandsaturationregimesisexplainedusingthenewchanneltransmissiondenitions. TounderstandtheimpactofmechanicalstrainonAlGaN/GaNHEMTtrapgeneration,wehavedevisedanexperimentalmethodtoobtainthephotonux-normalizedrelativearealtrapdensitydistributionusingphotoionizationspectroscopytechnique.Thedetailsofthetrapextractionmethodandtheexperimentalsetuparegiven.Usingthissetup,thetrapcharacteristicsareextractedforbothungatedtransmissionlinemodule(TLM)andgatedHEMTdevicesfrombothSiandSiCsubstrates.Thechangesinthedevicetrapcharacteristicsareemphasizedbeforeandafterelectricalstressing.Itisfoundthroughthestep-voltagestressingoftheAlGaN/GaNHEMTgatestackthatthedevicedegradationisduetothenearbandgaptrapgeneration,whichareshowntoberelatedtothestructuraldefectsinGaN. 15

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CHAPTER1INTRODUCTION 1.1Motivation Siliconbasedmetal-oxide-semiconductoreld-effecttransistors(MOSFET)havebeentheprimarychoiceofmainstreamintegratedcircuit(IC)technologyforover40years.ThecontinuousgeometricscalingofplanarsiliconMOSFETsenabledICswithsuperiorperformance,density,speed,cost,functionalityandpowergurescomparedtotheirpredecessors.However,asthechannellengthoftheplanarMOSFETsshrinktodeepsubmicronregime,thelimitationsduetotheincreasedlithographycostshavesloweddownthegeometricscaling(Figure 1-1 ).Inthemeantime,strainengineeringhasemergedasacheapersolutiontosiliconMOSFETenhancementcomparedtotheexpensivegeometricscalingatthe90nmtechnologynode[ 1 ].Process-inducedstrainhasincreasedMOSFETcarriermobilitythroughthereductionofconductivityeffectivemassandscatteringrates[ 2 ].Duetothescalabilityoftheprocess-inducedstrain,strainedsiliconchannelshavebeenusedinalltechnologynodessinceitsdemonstrationin2002.WiththefurtherscalingofthestrainedplanarMOSFETs,deleteriousshort-channeleffectshasledtoincreasedoff-stateleakageandpowerdissipationduetothedecreasedgatecontrolonthechannelresistivity.Tomitigatetheseshortchanneleffects,higherdopingdensitiesareusedtoincreasethegatecontrolbyhigherverticalelectriceldinthechannel,whilesacricingthecarriermobility.Onesolutionhasemergedtotheshort-channeleffectsandloweredcarriermobilityinhighlydopedchannelswiththeintroductionofthemulti-gatenon-planarsiliconCMOSdesignwiththinundopedchannels[ 3 6 ].Infact,Intelhasrecentlyannouncedtheirtri-gatenon-planarstrainedsiliconCMOSbaselinefortheupcoming22nmnode[ 7 ]. Withthecontinuouschannellengthscalingforincreasedtransistordensityandperformance,thedominantmechanismforcarriertransporthasevolvedfromdrift-diffusionforlongchanneldevicestoquasi-ballisticforultrashortchanneldevices 16

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Figure1-1. Theevolutionofthecostoflithographytoolswithtime.Adaptedfrom[ 8 ]. [ 9 10 ],forwhichstrainisexpectedtocontinuebeingaviableperformancebooster[ 11 ].Forthedevicesapproachingthescalinglimitbeyondthe22nmnode,carriersareexpectedtotraversethetransistorchannelatnearballistictransportconditions[ 12 ].Withtheintroductionofnon-planarmulti-gatesiliconMOSFETs,atransversedimensionscalingisalsorequiredtocontroltheshortchanneleffectswhiledecreasingthechannellengthatsmallernodes.Hence,ultranarrowstrainedgate-all-aroundsiliconnanowireMOSFETsareexpectedtoemergetoovercomethestrongershort-channeleffectsinfurtherscaleddevicesbeyondthe22nmtechnologynode.Consequently,thereisaneedforunderstandingthephysicsofstrainenhancementoflow-dimensionalsiliconMOSdevicesasthetransverseandlongitudinaldimensionsscaledownforfuturelogicdevices. AlthoughthesiliconMOSFETtechnologyisthemainstreamforthestate-of-the-artlow-powerandhigh-performancelogicdevices,siliconbasedpowerMOSFETsareoutperformedbyAlGaN/GaNhighelectronmobilitytransistors(HEMTs),especiallyforhigh-powerandhigh-frequencyapplications[ 13 ].TherstAlGaN/GaNHEMTwasdemonstratedbyKhanetal.in1994[ 14 ].AlGaN/GaNHEMTsbenetfrommaterial 17

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advantagesoversiliconpowerMOSFETssuchashighbreakdownvoltage,duetoits3.4eVwidebandgap,andlowercapacitiveloadingathighfrequenciesduetosmallerdielectricconstantwithacomparableheatdissipation[ 15 16 ].Withthesematerialadvantages,theAlGaN/GaNHEMTsareshowntobecapableofimpressivehighfrequencyandhighpowercharacteristics[ 17 ]. WhilemechanicalstressisincorporatedinsiliconMOSdevicefabricationtoimproveperformancewithrespecttoitspredecessors,alargebuilt-inepitaxialstressintheAlGaNlayerisrequiredtorealizethewidebandgapGaNbasedHEMTs[ 18 ].ByepitaxiallygrowingathinAlGaNlayeronafullyrelaxedGaNsubstrate,thein-planebiaxialtension(3GPa)intheAlGaNbarriercreatesaxedpiezoelectricinducedpolarizationchargeaddedtothedifferencebetweenAlGaNandGaNspontaneouspolarization(Figure 1-2 ).Thexedpositivepolarizationinducedchargeenablestheaccumulationofatwo-dimensionalelectrongas(2DEG)viathedonorstatesontheAlGaNlayer[ 19 ].Inadditiontothebuilt-inepitaxialstress,thestressgeneratedbytheinversepiezoelectriceffectintheAlGaNlayerathighgateelectriceldsisshowntobeontheorderof500MPaunderhighpoweroperatingconditions[ 20 ].Clearly,mechanicalstrainisanessentialcomponentoftheAlGaN/GaNHEMToperation. Despitetheimpressivehighfrequencyandhighpowerperformance,AlGaN/GaNHEMTsarefarfromreachingtheirfullpotentialduetodetrimentalreliabilityissues[ 21 ].PointandstructuraldefectsinthebulkGaNregion,intheAlGaNbarrier,attheAlGaN/GaNinterface,attheNi/AlGaNgatemetalinterface,andattheungatedregionoftheAlGaNsurfacecreatetrappingcenters[ 21 23 ].Thesetrapsleadtodegradationinthedeviceperformancethroughvariousmechanismssuchascurrentcollapse,drainlagandgatelag[ 21 25 ].Moreover,thestressduetotheinversepiezoelectriceffectishypothesizedtoleadtostructuraldefectsintheAlGaNbarrier[ 20 26 ].AllofthesedefectsresultsinachallengetoachievetherealizationofreliableAlGaN/GaNHEMTsforlong-termapplications.Sincemechanicalstressisinherentlyincorporated 18

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Figure1-2. TheillustrationofthespontaneousandpiezoelectricpolarizationinAlGaN/GaNheterostructureisshown.Thenetxedpositivecharge()ontheAlGaNsideoftheinterfacecreatesatwo-dimensionalelectrongasontheGaNsideoftheboundary. inAlGaN/GaNHEMToperation,itisrequiredtounderstandtheimpactofmechanicalstrainonAlGaN/GaNHEMTdegradationandreliability. 1.2ObjectiveandOrganization TheaimofthisworkistounderstandthestraineffectsontheperformanceofthecontemporaryandfuturelowdimensionalsiliconMOSdevices,aswellastheimpactofthemechanicalstrainonAlGaN/GaNHEMTdevicereliability.Toachievethisgoal,asystematicapproachofphysicsbasedmodels,appropriatesimulationmethodsandwaferbendingexperimentswereconducted.ThecontributionsofthisworkincludepredictingthestrainenhancementoffuturestrainedsiliconMOSFETsthroughanimprovedunderstandingofcontemporarydevicesandidentifyingtheeffectsofmechanicalstrainonAlGaN/GaNHEMTdevicetrapgeneration,degradationandreliability.TheseresultsmaybeusedtoguidepathsforchoosingdevicedesignsforfutureSilogictechnologynodesandimprovedAlGaN/GaNHEMTdesignsforachievingreliablelong-termoperation. 19

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Inthenexttwochapters( 2 and 3 ),wewillrstdiscusstheclassicationofthesiliconMOSdevicesintermsofcarrierdegreeoffreedom.Thedevicegeometrydependenteffectsofelectrostaticandspatialconnementonthesiliconbandstructureareprovided.Next,anoverviewofthewidelyusedtrapcharacterizationmethodswillbepresentedfromtheperspectiveoftheirapplicationtoAlGaN/GaNdevices,whileemphasizingthemeritofeachtechnique. Chapter 4 presentsthephysicalinsightsontheexperimentallyobservedcomparableelectronmobilityin(100)and(110)sidewallsilicondouble-gateFinFETs.Supportedwithasystematicalsetofexperiments,apossiblephysicalmechanismisdiscussedastheresultofamodiedself-consistentSchrodinger-Poissondouble-gateFinFETsimulation.Then,thewidthdependenceofthestrainresponseofp-typetri-gateFinFETsisinvestigatedwith4-pointbendingexperimentsandweightedaverageofbulkpiezoresistivecoeffcients. InChapter 5 ,thestrainenhancementofballisticsiliconnanowireholetransportwillbepresented.Themodicationofthestrain-inducedballisticcurrentimprovementasafunctionofnanowirechanneldirectionandsizeisstudiedusingansp3d5sbasisnearest-neighbortight-bindingsimulations. PhysicalinsightsonthestrainenhancementofsurfaceroughnesslimitedmobilityinsiliconinversionlayersarepresentedinChapter 6 .Thefundamentalphysicsareinvestigatedusinganadhocsimulationmethodsupportedbyacombinationofelectricalandmaterialscharacterization. Chapter 7 demonstratesthedevelopmentandthedetailsofauniedquasi-ballisticcompactmodelforstrainednanoscaleMOSFETsusingone-uxtheory. TheopticalcharacterizationofstraineffectsonAlGaN/GaNHEMTtrapgenerationwillbepresentedinChapter 8 .Detailsonthesamples,experimentalsetup,experimentsandresultswillbediscussed. Chapter 9 providestheoverallsummaryandfuturework. 20

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CHAPTER2BACKGROUNDONCLASSIFICATIONOFDEVICESINTERMSOFCARRIERDEGREEOFFREEDOM 2.1Motivation Classicationofdevicesas3DOF,2DOF,and1DOFfocusesontheeffectsofthepredominanttypeoftransversequantumconnement(unconned,electrostaticorspatialconnement)anddevicegeometry(planarornon-planar)onthedegreesoffreedomofthecarriermomentum.Inadditiontothetransversedimensionsofthedevices(connement),effectsofthelongitudinaldimensionofthedevice(long/shortchannel)willbeconsideredfromtheperspectiveofthedominantcarriertransportcharacteristics.TheclassicationofsemiconductordevicesbasedoncarrierconnementisshowninFigure 2-1 .First,devicesaredividedintotwohigh-levelgroups,unconnedandconneddevices,basedonthedegreesoffreedomofthecarriermomentum.Unconneddevicesreferto3DOFbulkdevicessuchaspiezoresistorswhileconneddevicesincludebothdeviceswith1-dimensionalconnement/2DOFcarriermomentumand2-dimensionalconnement/1DOFcarriermomentum.Next,conned2DOF(Figure 2-2 )and1DOF(Figure 2-3 )devicesaredividedintoelectricallyconned(EC)andspatiallyconned(SC)devices.Theelectricallyconneddevicesmaybefurthersubdividedintermsofsemi-innitesubstrate2DOFplanarMOSFET,2DOFAlGaN/GaNHEMT,undoped2DOFvolume-inversiondouble-gateFET,andquasi-1DOFvolume-inversiongate-all-aroundFET.Likewise,thespatiallyconneddevicesinclude2DOFultra-thin-body(UTB)siliconand1DOFnanowiresilicon(SiNW). PartsarereprintedwithpermissionfromMehmetO.Baykan,ScottE.Thompson,andToshikazuNishida,Straineffectsonthree-dimensional,two-dimensional,andone-dimensionalsiliconlogicdevices:Predictingthefutureofstrainedsilicon,J.Appl.Phys.108,093716(2010),DOI:10.1063/1.3488635.Copyright2010,AmericanInstituteofPhysics. 21

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Figure2-1. Classicationofsemiconductordevicesbasedonthetypeanddimensionalityofcarrierconnement.CarriersinthesemiconductorlatticeobeytheUniversalRuleofDimensionalityforwhichn+m=3,wherenisthedegreeoffreedom(DOF)ofthecarriermomentumandmisthedimensionalityofconnement.Inthissense,3DOFcarriershavenoconnement(0C),2DOFcarriershave1Dconnement(1C)and1DOFcarriershave2Dconnement(2C).c2010,AmericanInstituteofPhysics. Inthefollowingsub-sections,theelectricalcharacteristicsofunstrained3DOFto1DOFdevicesarereviewedintheframeworkoftheclassicationscheme.Wewilldiscusstheeffectsofcarrierconnementontheunconned3DOFbulksemiconductorbandstructureforconneddevicesbymeansoftheeffectivemassofcarriers,degeneracyofbands,andenergysplittingbetweenthelowestconductionandvalencebands.Thedevicesaregroupedas:(A)Unconned3DOFBulkDevices,(B)ElectricallyConned2DOF/quasi-1DOFFETsand(C)SpatiallyConned2DOFUTBand1DOFNanowiresilicondevices. 2.2ClassicationofDevicesinTermsofCarrierDegreeofFreedom 2.2.1Unconned/3DOFBulkDevices Anexampleofapredominantlyunconnedbulkdeviceisaresistorwherethemajoritycarrierspossessthreedegreesoffreedomandarethusunconnedexceptattheboundaries.Hence,mostofthemobilecarriersdonotinteractwithasurface,andcarriertransportcanbeassumedtobefreeofanysurfaceeffects.Anotable 22

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Figure2-2. Various2DOFsilicondevicesareshown.Thecarrierdistributioninsidethedevicechannelsisshownalongwiththeconductionbandedgeproleforgate-oxide/silicon-channelheterostructure(black)andthelowestenergysubband(red).(a)Electricallyconned2DOFplanarMOSFEThassurface-inversioncarriersconnedintheelectricalwellduetogatebiasinducedbandbending.(b)Forelectricallyconned2DOFdouble-gatevolume-inversionFET,carrierspopulatethebulkpartsofthesilicon,asthewavefunctionsfromthefrontandbackgatesinteract.Thelowestenergysubbandstillliesintheelectricalwelldenedbythegatebiasinducedbandbending.(c)Spatiallyconned2DOFUTBdouble-gateFEThasthelowestenergysubbandconnedinthesiliconthicknessdenedspatialquantumwell,andthepeakofspatialcarrierdistributionoccursinthemiddleofthesiliconslab.c2010,AmericanInstituteofPhysics. strain-sensitive3DOFdeviceisapiezoresistor,astress-dependentresistorwidelyusedasastraintransducerinMEMS(Micro-Electro-MechanicalSystems).Thesedevicestransducestraininthemechanicaldomainbyexhibitingastrain-dependentresistancewhichisconvertedintoavoltagesignalbybiasingthepiezoresistorinabridgecircuit.Piezoresistancecoefcientsin3DOFbulksiliconandgermaniumwererstreportedbySmithin1954[ 27 ]. Asabaselineforcomparisonwithconneddevices,thebandstructureofunconned3DOFbulksiliconisrstreviewed.Inanunconnedbulksemiconductor,thecarriermomentumiscontinuousinallthreeorthogonaldirectionsink-space.Neglecting 23

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Figure2-3. (ColorOnline)Electricallyconnedquasi-1DOFvolume-inversionandspatiallyconned1DOFnanowiresiliconMOSFETchannelsareshown.Similartotheir2DOFcounterparts,strongervolume-inversionisseeninthesedevices,asthepredominanttypeofcarrierconnementchangesfromelectricaltospatial.c2010,AmericanInstituteofPhysics. theconductionbandedgeenergyandk-valueattheminima,thekineticenergyofanelectronundertheparabolicbandapproximationisgivenby, E=~2k2x 2mx+~2k2y 2my+~2k2z 2mz,(2) wheremx,my,andmzaretheeffectivemassesinthethreedirections.Theconductionband(CB)ofunconned3DOFbulksiliconconsistsofsix-folddegenerate(6)prolatespheroidalequienergysurfacesalongsixequivalent)]TJ /F1 11.955 Tf 6.78 0 Td[(-Xdirections(Figure 2-4 ),withminimaat0.83(2=a).Foreach6energy-wavevector(E-k)valley,themassalongthemajoraxisisml=0.916mo(longitudinalmass)andthemassesalongtheminoraxesaremt=0.191mo(transversemass),wheremoistheelectronrestmass.Six-folddegeneracyoftheCBedgeimpliesequalelectronoccupancyofthe6valleys. Thevalencebandcanbeapproximatedbythehighlyanisotropicheavy-hole(HH)band,therelativelyisotropiclight-hole(LH)band,andthesplit-off(SO)band,allhavingmaximumatthe)]TJ /F1 11.955 Tf 10.1 0 Td[(point.TheHHbandexhibitstheheaviestmassalongtheh110idirectionwithmHH,h110i=0.61mo,whiletheLHbandisfairlyisotropicwithmLH=0.15mo 24

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Figure2-4. Conductionband(CB)minimafornon-spatiallyconnedunstrainedsilicondevicesareshown.Eachisoenergyellipsoidhaslongitudinalmassml=0.92m0andtransversemassmt=0.19m0.(a,d)Unconned3DOFbulksiliconCBhas6-folddegenerateellipsoidvalleysinthe3DBrillouinZone.(b)1Dconned(1C)bandstructurehasquantizedkz(nkz0)andthecorresponding2DisoenergycontoursdenotestheCBedgesforelectricallyconned2DOFplanarMOSFET(e)and2DOFvolume-inversionFET(f).(c)2Dconned(2C)CBhasbothkyandkzwavevectorsquantized.Theisoenergyellipsoidsarereducedtoisoenergydotsduetothequantizationinbothyandzdirections,anddenotetheCBedgesforanelectricallyconnedquasi-1DOFvolume-inversiondevice(g).(d-g)TheenergylevelsandshiftsinCBedgesduetoelectricalconnementisalsoshownfordifferentdeviceswiththeircontributorvalleys,valleyconnementmassesandlevelsofdegeneracy.ArrowsexplaintheunderlyingphysicsforthechangesinCBvalleysplittingduringtransitionfromonedevicetotheother.c2010,AmericanInstituteofPhysics. [ 2 ].Inunstrained3DOFbulksilicon,theHHandLHbandsaredegenerateatthe)]TJ /F1 11.955 Tf -428.92 -23.91 Td[(point,whiletheSObandmaximumisapproximately44meVlowerthantheHHandLHbands. 2.2.2DeviceswithPredominantElectrostaticConnement(EC) 2.2.2.11DEC/2DOFsurface-inversionMOSFET AsshowninFigure 2-2 -a,thesurfacepotentialinducedtransversebandbendingoftheconductionbandedge(valencebandedge)inn-type(p-type)inversionchannelmetal-oxide-semiconductoreld-effecttransistors(MOSFETs)fabricatedonbulksiliconsubstrateconneselectrons(holes)toaverynarrowsurfacelayerquantumwell, 25

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betweentheoxidepotentialbarrierandthesiliconbandedge.ProvidedthatthewidthofthisquantumwelliscomparabletothedeBrogliewavelengthoftheelectron,theelectronwavevectornormaltotheSi=SiO2interfaceisnolongercontinuous,buthasdiscretevalues.ThisquantizationinthewavevectorleadstoformationofisoenergybandcontoursasshowninFigure 2-4 -b.Althoughthemobilecarriersarestillallowedtofreelymovewithintheplaneunderthegate,motionofthesecarriersisrestrictedinthedirectionofconnement.Inthiscase,theenergyofachargecarrierisafunctionofthetwocontinuouswavevectorsandthediscreteenergyleveldenedbythequantizedwavevectoralongtheconnementdirection.Assumingthatthegatebias-inducedsurfaceelectriceldconnementoccursalongthez-axis,theE-kdispersionrelationcanbedenedby, En=Ezn+~2k2x 2mx+~2k2y 2my,(2) wheretheenergyofthenthinversionlayersubband,Ezn,underthedepletionassumptionisapproximatedby[ 28 ], Ezn=[3hq"s 4p 2mz(n+3 4)]2 3.(2) Here"s,mzandEznarethesurfaceelectriceld,connementmass,anddiscreteenergylevelsduetothequantizedwavevectorinthezdirectionrespectively[ 28 ].Asseenin( 2 ),thediscreteenergylevelsarefunctionsoftheeffectivemassofthevalleyalongtheconnementdirectionandthesurfaceelectriceld.ThisimpliesdifferentsubbandenergylevelsforvalleysofdifferentconnementmassasshowninFigure 2-4 -c. Comparedtounconned3DOFbulksilicon,inversioncarriersinelectricallyconned2DOFsurface-inversionMOSFETsexperienceamodiedbandstructureduetothegate-inducedsurfaceconnementelectriceld("s),whichshiftstheenergylevelsofthevalleysdifferentlydependingontheirconnementeffectivemass.This 26

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liftsthedegeneracyandchangestheoccupationoftherespectivevalleys.On(001)surfacewafers,unlikethelargedifferencebetweenmtandmloftheconductionband,theout-of-planemassesoftheHHandLHvalencebandsareveryclosetoeachother,0.29moand0.20morespectively[ 2 ].Hence,attheinversionchargedensitiesofcontemporary2DOFplanarCMOSdevices(1013cm)]TJ /F9 7.97 Tf 6.58 0 Td[(3),theenergysplittingbetweenthelowestsubbandsofthetwoground-mostvalleyson(001)substrateare95meVfornMOS[ 29 ]butonly20meVforpMOS[ 2 ].Asaresult,inunstrainedplanar2DOFpMOSFETdevices,boththeHHband(top-most)andtheLHband(second)contributetocarriertransport,unlikethemostlyisolated2valleyinn-typedevices.Incontrast,for(110)substrate,theconnementmassvaluesfortheHHandLHbandsare0.58moand0.15morespectively[ 30 ],whereas,theconnementmassfor2ismtandfor4ismh110i=0.315mo[ 31 ],possiblyleadingtoaslightlygreaterbandsplittingofVBthanCB. 2.2.2.21Dpolarization-inducedEC/2DOF2DEGAlGaN/GaNHEMTdevices Similarto1DEC/2DOFsurface-inversionsiliconMOSFETs,the1DpolarizationinducedEC/2DOFAlGaN/GaNHEMTdevicesexhibitsurfaceinversionlayercharacteristics.Asbrieydiscussedin 1.1 ,thetotalinterfacexedchargeduetothespontaneousandpiezoelectricpolarizationoftheAlGaN/GaNheterostructureresultsinaquantumwellcreatedbythelargeconductionbandbendingattheAlGaN/GaNinterface.Asaresult,ahighdensityoftwo-dimensionalelectrongasaccumulatesinthehighelectronmobilityGaNlayer(Figure 1-2 ).Incontrasttotheenhancementmodeoperationofthe1DEC/2DOFsiliconMOSFETs,theAlGaN/GaNHEMTisadepletionmodedevicewithlowresistivity(highinversiondensity)stateat0Vgatebias.The2DEGisdepletedbyapplyinganegativevoltageonthegatebelowthethresholdvalue. Unlikethesix-folddegenerateconductionbandofsilicon,theGaNconductionbandiscomposedofasingleisotropicminimumatthe)]TJ /F1 11.955 Tf 10.1 0 Td[(point,withelectronmassof0.20m0[ 32 ].Therefore,theelectrostaticconnementeffectofvalleydegeneracyliftingisnotseenforGaNbaseddevices. 27

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2.2.2.31DEC/2DOFvolume-inversionMuGFETdevices Thefundamentaldifferencebetweenconventionalplanar2DOFsurface-inversionMOSFETdevicesandvolume-inversion2DOFdevicesliesinthethicknessofthesiliconlayer.Volume-inversiondeviceshaveaverythinsiliconlayerbetweentwoequipotentialgateswithathickness(20nm)comparabletothecumulativebulkdepletionwidthfromeachgatesurface.Whenthesiliconthicknessisinthisrange,theelectronwavefunctionsconnedbythetwogatesoverlapspatially.Asthemagnitudesquaredofthewavefunctiongivesthecarrierprobabilitydistributioninthedevicecrosssection,thecarriersstarttopopulatenotonlythenearSi=SiO2interfacialregion,butalsothesiliconregionawayfromthegate.AsshowninFigure 2-2 -b,whentheoverlapofthewavefunctionsissignicant,thecentervolumeofthethinsiliconlayerbecomesinverted,resultinginasignicantamountofinversionchargeawayfromthegatesurfaces.AnincreasednumberofinversioncarriersinthecenterofthesiliconslabreducesthebandbendingrelativetotheSi=SiO2interface,andhencereducestheoverallconningeffectiveelectriceld.Therst2DOFvolume-inversiondevicewasdemonstratedbyBalestraetal.[ 33 ].Generally,thedouble-gateMOSFETs(DGFET)fallunderthiscategory.Anotherdifferencebetweentheelectricallyconned2DOFplanarMOSFETandthe2DOFvolume-inversionDGFETisinthedopingofthesiliconsubstrate.IncontrasttotheplanarMOSFET,thechannelsofvolume-inversiondevicesaremostlyundopedandhavelesssurfaceelectriceldthan2DOFplanarMOSFETs.As( 2 )suggests,areducedsurfaceelectriceldin2DOFvolume-inversiondevicesleadstoasmallersubbandsplittingcomparedtoconventional2DOFplanarbulksubstratedevices,asillustratedinFigure 2-4 -f.Additionally,thewavefunctionsthatstemfromtheequipotentialgatescreateoverlappingenergybands.AlthoughtheseoverlappingbandssplitduetoamechanismsimilartothePauli'sExclusionPrinciple,theenergydifferenceisgenerallysmallerthankBTandtreatedasequallyoccupied[ 30 ].Underthiscondition,alargerdensity-of-statesforthegroundsubbandthatcan 28

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beoccupiedbyahigherdensityofmobileinversionchargesisobtainedin2DOFvolume-inversionMuGFETscomparedto2DOFplanarbulksubstrateMOSFETs. Asthesiliconlayerthicknessapproachesthepredominantlyspatialconnementlimitof4nm,furtherchangesareobservedinthebandstructureof2DOFvolume-inversiondevicescomparedtoconventional2DOFplanarsurface-inversiondevices.Whenthethicknessofthesiliconlayerbecomescomparabletothewidthofthewider,secondlowestenergyvalley/band;theenergyofthesecondlowestenergybandincreasesduetospatialconnement,resultinginanincreaseinthesplittingbetweenthetwoground-mostvalleys/bands.Fortheconductionband,thepartialspatialconnementincreasestheenergyleveloftheprimedladders(nomenclatureisafterSternetal.[ 34 ]).Sincetheunprimedvalleys/subbandshavesmallerwidth,electricalconnementremainsasthemainsourceofcarrierconnementontheunprimedvalleys,thusnoenergyincreaseisobservedfortheunprimedsubbandsunderpartialspatialconnement[ 35 ].Theresultisincreasedvalleysplittingwhilestillhavinganincreaseddensity-of-statesinthegroundsubbandduetowavefunctionoverlap.Underthiscondition,thegroundsubband,alreadyoccupiedatahighlevel,increasesitsoccupationfurtherduetotheonsetofspatialconnementoftheprimedvalleys/subbands.Likewiseforthevalencebandofpartiallyspatiallyconnedsilicon,holesmostlyresideinthetop-most(HH)bandclosetothe)]TJ /F1 11.955 Tf 10.1 0 Td[(pointduetoincreasedbandsplittingandincreasedDOSofthegroundsubband. 2.2.2.42DEC/quasi-1DOFvolume-inversiondevices Thicknanowire(widthsbetween4)]TJ /F3 11.955 Tf 12.13 0 Td[(20nm),Tri-GateFinFETandgate-all-aroundFinFET(GAA-FET)devicesareconsideredasquasi-1DOFvolume-inversiondevices.Thethicknessrangeis4)]TJ /F3 11.955 Tf 12.07 0 Td[(20nmsincebelowapproximately4nm,spatialconnementbecomesdominantwhileaboveapproximately20nm,wavefunctionoverlapisnegligible.Extendingthediscussionon2DOFvolume-inversiondevicestotheirquasi-1DOFcounterparts,weobservethatgate-all-arounddevicesdemonstratesimilarcharacteristics 29

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ofvolume-inversion,suchasdecreasedbandsplittingandincreasedDOSofthegroundvalley/subband.However,thedifferenceindeviceelectrostaticsissignicantin2DOFandquasi-1DOFvolume-inversiondevices.Ina1DEC/2DOF-surfacedevicewithaplanargate,thepotentialvariationoccursprimarilyalongoneaxis,resultingin1Delectrostatics.Thediscreteenergylevelsapproximatedby( 2 )areonlydependentonthe1Delectriceldandmassalongtheelectrostaticconnementdirection.Whileplanar2DOFdeviceshavegate(s)onlyinoneplaneanda1Delectriceld,gate-all-around1DOF-volumedeviceshavesurroundinggatesanda2Delectriceld.AsseeninFigure 2-4 -c,withthequantizationinbothkyandkzwavevectors,isoenergybandsurfacesof3DOFbulksiliconreducetoisoenergybanddots.TheinteractionoftheYandZconductionbandellipsoidsoftheh100ichannelquasi-1DOFvolume-inversiondevicewiththe2Dgateelectriceldinvolvesboththelongitudinalandtransversesiliconeffectivemass.Undertheassumptionthatthesurroundinggatesareequipotentialandcreatesymmetric2DelectriceldcontoursasillustratedinFigure 2-3 -a,theenergyshiftoftheYandZCBvalleys(EYnandEZn)canbeapproximatedbyusing( 2 ).Undertheisotropicconnementmassapproximation,theeffectivemassforvalleyalongtheyandzdirectionsisgivenby[ 36 ] 1 m,conf=Z20d 2(cos2 m,y+sin2 m,z),(2) wherem,yandm,zareeffectivemassvaluesforvalleyalongyandzdirectionsandisthepositiveanglefromky-axistokz-axis[ 36 ]. Now,thefundamentaldifferencebetween2DOFandquasi-1DOFvolume-inversiondevicesbecomesapparent.Whiletheelectriceldinducedvalleysplittingina(001)2DOFdeviceisimpliedbyconnementmassesofmt=0.191mo(forX,Yvalleys)andml=0.916mo(forZvalley),thesemassvalueschangetomt=0.191mo(forXvalley)and0.315mo(forY,Zvalleys)forah100ichannelquasi-1DOFvolume-inversiondevice.Similarlyforh110ichannelquasi-1DOFvolume-inversiondevices,wehaveaverage 30

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Figure2-5. Spatiallyconned2DOFUTBsiliconand1DOFsiliconnanowiredevicesareshowninthegure.Siliconatomsindarkcolorandinframesareconstructedunitcellsforsp3d5stight-bindingmethod.Inaccordancewiththereallatticeperiodicityofunitcellsfor2DOFUTBand1DOFSiNW,reallatticevectorsazinUTBanday,azinSiNWarenotdened.Theresultisundenedkzfor2DOFUTBandky,kzfor1DOFSiNWreciprocallattice.Bulk-SivalleysalongtheundenedreciprocalvectorsareprojectedtooriginofBrillouinZone()]TJ /F1 11.955 Tf 10.1 0 Td[(point),convertingmaterialfromindirecttodirectbandgapcharacteristics.Insetshowssimulatedbandstructureofdevicesandcorrespondingenergysplittinglevelsanddirectbandgapenergies[ 38 ].Highvalley/bandsplittinginthesedevicesensuresthatcarriersresideinthegroundmostsubband,andcarrieropticalphononscatteringissuppressed.c2010,AmericanInstituteofPhysics. connementmassvaluesof0.237mo(forX,Yvalleys)and0.315mo(forZvalley).Accordinglywhenweconsider( 2 ),asmallerdifferenceinconnementeffectivemassesofCBvalleysinaquasi-1DOFvolume-inversiondeviceisexpectedtoresultinasmallerconnementinducedCBsplittingthan2DOFsurfaceandvolume-inversiondevices,asillustratedinFigure 2-4 -g. Itshouldbepointedoutthatmulti-gatedeviceshavingthicknessesgreaterthanthecumulativedepletionwidth(nospatialwavefunction/inversion-layeroverlap),areessentiallycomprisedofmultipleindependentchannels[ 37 ].Hence,thesedevicesmaybeconsideredasmultiple2DOFplanarsurface-inversionMOSFETsoperatinginparallelwitheachotheranddonotexhibitthevolume-inversioncharacteristicsdescribedabove. 31

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2.2.3DeviceswithPredominantSpatialConnement WhenoneormoredimensionsofasemiconductorslabarescaleddowntobecomparabletotheBohrexcitonradiusofthematerial,spatialconnementofthecarriersreducesthedegreesoffreedomofthecarriermomentumandhencethereciprocalspace,leadingtoquantizationofthecarriermomentumandsplittingofthecarrierenergy.IllustratedinFigure 2-2 -candFigure 2-3 -b,thevolume-inversioneffectisstronglypresentinthesedevicesduetotheverystronginteractionofthemultiplegates.Forthesespatiallyconnedsilicondevices,exceptionallydifferentelectricalandopticalpropertiescomparedtonon-spatiallyconnedsiliconarepredictedandobserved[ 39 44 ].Forspatiallyconned2DOFultra-thin-body(UTB)and1DOFnanowire(NW)devices,thenitesiliconcrystalthatconstitutesthetransistorchannelisformedbyacountablenumberofatomiclayers,asshowninFigure 2-5 .Theatomicnatureofthesedevicesimpliesthatrstprinciplescalculationsarerequiredtocapturetheuniqueelectricalcharacteristicsofthesenoveldevices.Thesp3d5stight-bindingmethod(TB)anddensityfunctionaltheory(DFT)arethetwomostcommonmethodsemployedtocalculatetheatomicbandstructure.Thesemethodscapturethebandstructureeffectsofthesenoveldevicesmoreaccuratelythantheconventionaleffectivemassapproximation(EMA)[ 45 ].TheE-kdispersionrelationsfor2DOFUTBand1DOFSiNWdevicesarecalculatedbyTBmethod[ 38 ]andshowninFigure 2-5 .Qualitatively,thecrucialbandstructurepropertiesthatdenetheelectricalandopticalcharacteristics,suchastheeffectivemass,valley/bandenergysplitting,andlocationofbandextremashowremarkabledifferencescomparedtonon-spatiallyconnedsilicon. 2.2.3.11DSC/2DOFultra-thin-body(UTB)devices ShowninFigure 2-5 ,2DOFUTBSidevicesareextremelyconnedinonedirectionandareassumedtopossessanunperturbedlatticeperiodicityalongthein-planedirections.TheUTBsiliconlayercanbeconstructedfromasingleunitcell,extendingfromthebottomgatetothetopgateandbyrepeatingtheunitcellalongtheunitreal 32

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Figure2-6. Thedirectbandgaptransitionfrom3DOFbulksilicontoto(001)2DOFUTBSi(left)andh100ichannel1DOFSiNW(right)areillustrated.TheZ-valleysforUTBandtheY,Z-valleysforSiNWareshownastheyareprojectedontotheoriginofrstBrillouinZonebyredarrows.CBedgesatthe)]TJ /F1 11.955 Tf 10.1 0 Td[(pointarelessupshiftedbyconnementeffectsthanothervalleysduetotheirhigherconnementeffectivemass.Theseresultsofspatialconnementandconnementdirectionsleadtodirectbandgapcharacteristics.c2010,AmericanInstituteofPhysics. latticevectors^axand^ay.Thisapproachinherentlyleadstoa2Dreciprocalspacewithwavevectorsonlyalongkxandkydirections.Thewavevector,kz,isundenedanditsE-kdispersionisprojectedontothekx-kyplanewithkz=0,i.e.E(k)UTBE(kx,ky,kz)jkz=O,withinthe1stBrillouinZone(BZ)[ 43 ].Thisimpliesaspecialcasefor(001)surfacedeviceswherethebulkZ-valleyat[0,0,0.83](2=a0)ofthe3DBZappearsatthe)]TJ /F1 11.955 Tf -439.52 -23.91 Td[(pointofthe2DBZ(Figure 2-6 )asa)]TJ /F1 11.955 Tf 10.1 0 Td[(valley.Forothersurfaceorientations,noneoftheconductionbandedgescollapseontothe)]TJ /F1 11.955 Tf 10.1 0 Td[(pointofthe2DBZduetothepresenceofnon-zerocomponentsofthein-planewavevectorinthecoordinatesofthebulkCBminima. AsillustratedinFigure 2-6 ,spatialconnementof(001)surface2DOFUTBsilicondeviceresultsinadirectbandgapbandstructure,unliketheindirectbandgapofnon-spatiallyconnedsilicon.For(001)surfaceUTBSi,theoff-)]TJ /F1 11.955 Tf 10.1 0 Td[(valleysoriginatefromtheXandYbulk-valleys.Theoff-)]TJ /F1 11.955 Tf 10.1 0 Td[(valleyshavesmallerconnementmassand 33

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largertransportmass.Ontheotherhand,theaforementioned)]TJ /F1 11.955 Tf 10.1 0 Td[(valleyconverselyhasalargerconnementmassandasmallertransportmass.Consequently,higherlevelsofquantumconnement(achievedbyeitherdecreasingthesiliconthicknessorincreasingthegatevoltage)furthersplitthefour-foldoff-)]TJ /F1 11.955 Tf 10.1 0 Td[(valleysandtwo-fold)]TJ /F1 11.955 Tf 10.1 0 Td[(valleys.Theresultisasmallerupshiftofthe)]TJ /F1 11.955 Tf 10.1 0 Td[(valleycomparedtotheoff-)]TJ /F1 11.955 Tf 10.09 0 Td[(valley.Whenthe)]TJ /F1 11.955 Tf 10.1 0 Td[(valleybecomesthelowestedgeofthe(001)2DOFUTBconductionband,anindirecttodirectbandgaptransitionoccurs[ 43 ].Similarly,thevalencebandsaredownshiftedinenergywithincreasingconnementinaccordancewiththeirconnementmasses.Thus,anincreaseinforbiddenenergybandgapisobservedwithincreasedspatialconnement.Thebandsplittingenergiesanddirectbandgapfora2.1nmthick(001)UTBSidevicecalculatedusingtight-bindingmethodareshowninFigure 2-5 [ 38 ]. Finally,highlevelsofspatialconnementseenin2DOFUTBdevicesresultindeviationfromthewidelyacceptedmlandmtmassvaluesoftheCBelectrons,aswellastheVBholeeffectivemasses.Theunderlyingphysicsforthisobservation(changeincurvatureofbandedges)willbediscussedalongwithasimilardiscussionon1DOFSiNWswithinthenextsubsection. 2.2.3.22DSC/1DOFnanowires(NW) While2DOFUTBdevicesarespatiallyconnedinonedimension,carriersinsiliconnanowire(SiNW)devicespossessasingleDOFifthecarriersarespatiallyconnedintwodimensions.AsshowninFigure 2-5 ,theunitcellfora1DOFSiNWextendsonlyalongthe^axdirection.Duetothespatialconnementinthetransversedirections,forah100ichannelSiNW,onlyonewavevectorinthekxdirectionisdened.Theh100iSiNWunitcellhastwicethesizeofabulk-Siunitcellalongthedirectionofperiodicity(x-axis)[ 46 ].WhilethebulkSiBZboundariesoccurat2=a0,wherea0isthebulk-Silatticeconstant,theh100iSiNW1stBZoccursat=a0[ 46 ]. Similartothe2D(001)UTB,theYandZbulk-valleysareprojectedtothe)]TJ /F1 11.955 Tf 10.09 0 Td[(point,resultinginadirectbandgapsemiconductorforspatiallyconnedh100iSiNW(Figure 34

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Figure2-7. Thevalencebandsforunstrained2.1nmX2.1nmthick(a)h100i,(b)h110iand(c)h111isquaresiliconnanowiresaresimulatedthroughonlineresources[ 38 ].Atroomtemperature,smallernumberofdifferencevalencebandsareexpectedtobeoccupiedforh100i,h110iandh111irespectively,thusresultintheleastinterbandphononscatteringrateforh111i.Thecamelbackband(red)isclosesttotopofvalencebandforh100i,whichmayhaveeffectsonphotoniccharacteristicsofstrainedh100inanowires.c2010,AmericanInstituteofPhysics. 2-6 ).Inaddition,theXbulk-valleyat[0.83,0,0](2=a0)ofthe3DBZfoldsbackintothehalfsize1DBZ,appearingat0.34=a0ofthe1DSiNWBZ[ 47 ].Theresultingfourfold)]TJ /F1 11.955 Tf 10.09 0 Td[(valleyhaslargerconnementmassandsmallertransportmassalongthechannel.Fortheoff-)]TJ /F1 11.955 Tf 10.09 0 Td[(valleys,asmallerconnementmassandalargertransportmassareobserved.Asinthe2DOFUTBcase,increasedquantumconnementbythegatevoltage-inducedband-bendingorreducedsiliconthicknessshiftsthebandenergiesintheconductionandvalencebands.Consequently,higherconnementresultsinincreasedbandgapandbandsplitting. Likewise,spatiallyconnedh110ichannelsiliconnanowiresalsohaveaCBminimumatthe)]TJ /F1 11.955 Tf 10.09 0 Td[(pointoftheBZ.Incontrasttotheh100ichanneldevice,onlytwobulkCBellipsoidsalongtheconnedh001iandh00-1idirections,projectontotheoriginofthe1Dreciprocalspace,whilefourbulkCBellipsoidsappearatanon-zerokvalue.h110iSiNWsarereportedtohavesmallertransportmassatthe)]TJ /F1 11.955 Tf 10.09 0 Td[(andoff-)]TJ /F1 11.955 Tf 10.1 0 Td[(valleys 35

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comparedtothoseofh100iSiNWsofsamethickness[ 41 48 ].Inaddition,alargervalleysplittingfortheh110iNWtriggersrepopulationofcarrierstothegroundvalleysandresultsinasmalleraverageelectronconductivityeffectivemasscomparedtotheh100iNW[ 41 49 50 ].Therefore,ahigheron-currentisexpectedforah110ichannelSiNWcomparedtoah100ichannelSiNW[ 51 ]. Forhigherindexsiliconnanowires(h111i,h112i,etc.channel),adirectbandgapisunlikelytobeobserved[ 48 ].Inthesedevices,noneofthebulkoriginatedconductionbandellipsoidsresideinadirectionnormaltooneoftheconnementplanesofthedevice,hence,projectionofbulkvalley(s)ontotheoriginof1DBZisnotexpected.Instead,theseellipsoidsappearatnon-zerovaluesofthe1Dwavevector[ 48 ].Nonetheless,adirectbandgapcharacteristichasbeensimulatedinbandcalculationsofh111inanowiresthinnerthan2nm[ 41 52 53 ].Thedirectbandgappredictedinultra-thinh111isiliconnanowiresispostulatedtobetheresultofinteractionbetweentwo-setsofthree-folddegenerateCBvalleyssituatedatkCBmin,aboutthe)]TJ /F1 11.955 Tf 10.09 0 Td[(point.Asthethicknessofthenanowiredecreases,aatterdispersionrelationisseenonbothsidesofthe)]TJ /F1 11.955 Tf 10.1 0 Td[(point,correspondingtotheincreaseintheeffectivemassneartheCBedge,whicheventuallyresultsinamixingthetwosetsofthree-foldCBvalleystoformaminimaatthemiddle,whichisthe)]TJ /F1 11.955 Tf 10.1 0 Td[(pointoftheBZ[ 53 ].Possibleevidenceofthedirectbandgappropertyofsiliconnanowiresincludesameasuredincreaseinphotoluminescence(PL)intensityandefciencyandablueshiftinphotonwavelength(modeledbyanincreaseinsilicondirectbandgap)withdecreasingnanowirediameter[ 39 40 54 56 ]. Theeffectofspatialconnementonthevalencebandsmaybequalitativelyinvestigatedbyemployingasemi-analyticalconstructionfromthe3DOFbulkE-kdispersionrelation[ 57 ].Inthismethod,holesinspatiallyconnednanowiresaretreatedasparticlesinaboxintheconnementdirections.Inordertoapproximatethe1DOFnanowirevalencebandstructure,spatialconnementeffectsareincorporatedintothe 36

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3DOFbulksiliconbandstructurebymodelingthetwoconnementdirectionsasaninnite2Dpotentialwell.Thisresultsinthequantizationofthetwowavevectorsalongtheconnementplane,withquantizationintervalsofk1,2=n1,2=L1,2,wherenisaninteger,andListhethicknessofthenanowirealongthedirectionofthequantizedwavevector.Then,thequantizedvaluesofthetwoconnementwavevectorsareusedtosampleenergyvaluescorrespondingtothecontinuouswavevectorinthe3DOFbulkvalencebanddispersion.Hence,the3Dbulkvalencebandrelation,E(kx,ky,kz),becomesE(n1=L1,n2=L2,kz)whichresultsinindividualEn1,n2(kz)subbandsforagiven1DOFnanowirecrosssectionofL1byL2.Althoughnonlineareffectsofsubbandmixingandwarpingcannotbecapturedwiththismethod,knowledgeofthelinearcombinationsofn1,n2subbandscanprovideimportantinsightsonthevalencebandofsiliconnanowires,withoutthecomputationalcostsofTBorDFTmethods.Neophytouetal.alsoshowedthattheconductionbandof1DOFsiliconnanowirescanberelatedtothebulksilicondispersionrelationsthroughthesamesemi-analyticalapproach[ 51 ]andsuggestedsimilarapplicationstoestimationofextremelyconned2DOFUTBsiliconbandstructure[ 51 57 ]. Thedispersionrelationsforh100i,h110iandh111isiliconnanowirevalencebandsdifferfromeachotherintermsoftheenergyseparationbetweenthetopvalencebandsandtheircurvatures(Figure 2-7 ).Forh100iSiNWs,spatialconnementinducedsplittingofthetoptwovalencebandsisnotaspronouncedasintheh110iandh111ichanneldevices.Unliketheh100iSiNWVB,thelighthole(LH)-liketopvalencebandoftheh110iSiNWresidesatsignicantlylowerholeenergythanthesecondvalenceband.Inaddition,theh111isiliconnanowirevalencetopLH-likeVBisalmostfullyisolatedfromtheheavyhole(HH)-likebandbyseveralkBT.Moreover,thetopVBeffectivemassismuchlighterforh111iandh110ithanh100iSiNW.Therefore,alargerholecurrentisexpectedforh111iandh110icomparedtotheh100iSiNW[ 57 58 ].Furthermore,thesecondvalencebandoftheh100isiliconnanowirehasashapeofacamelbackwith 37

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anegativeeffectivemass(oppositebandcurvaturethanconventionalpositivemassdispersion)atthe)]TJ /F1 11.955 Tf 10.1 0 Td[(pointandtwopeaksatnon-zerokvalues. Forgatedsiliconnanowires,theconduction/valencebandsandelectron/holemobilitiesareexpectedtobemodiedthroughthecombinedspatialandelectrostaticconnementeffectsonthecarriereffectivemassandscatteringrates.Dependingonthethicknessofthenanowire(strengthofthespatialconnement),thegatebiasmaystillbeeffectiveinfurthersplittingofthebands/subbands,resultinginforexamplegreaterisolationofthelightermasstopband[ 53 ].Moreover,theappliedelectriceldatdifferentorientationsurfacegatesandtheanisotropicnatureoftheVBleadtopreferredlocalizationofcarrierswithinthecrosssectionofthedevice[ 53 ].Alongtheconnementdirectionsoflargermass,mobilecarrierstendtoresideclosertothesurface.Thisissimilartothedependenceofthecarriercentroidontheout-of-planeorconnementeffectivemassesin2DOFMOSFETs[ 2 ].Inthe1DOFnanowirecase,multiplegateorientationsintroducedifferentscatteringratesgovernedbytheroughnessoftheparticularsurfaceorientationandtheproximityofthecarrierstothegate. 2.3Summary Basedonthecarrierconnementtype(electrostaticandspatial)andthedegreesoffreedomofthemobilecarriers(3DOF,2DOF,and1DOF),semiconductordevicesareclassiedintermsoftheirbandstructuredependentelectricalproperties.Fromtheelectrostaticsperspective,asthedevicesevolvefrombulktospatiallyconnedstructures,thesurface-inversioncarriersspreadinsidethebodyofthecrystal,leadingtovolume-inversion.Forthequantumeffects,itisshownthatcarrierdegeneracy,electronandholeeffectivemasses,conductionandvalencebanddensity-of-states,scatteringratesandopticalpropertiesaremodiedwiththetypeandthedimensionalityoftheconnement. 38

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CHAPTER3BACKGROUNDONTRAPCHARACTERIZATIONMETHODS 3.1Motivation InordertounderstandtheoriginsoftheAlGaN/GaNdegradationmechanisms,thenatureoftheelectricallyactivetrapsandtrapgenerationmustbeknown.DuetothematerialchallengesinGaNbaseddevices,asignicantnumberofpointandstructuraldefectsexistinadditiontointentional/unintentionalimpurities[ 59 60 ].ThepointdefectsaredenedasavacancyofaGa(VGa)orN(VN)atominthelatticeorasubstitutionalatomattheselocations,aswellasaninterstitialatombetweenaGaandaNintheunitcell.Inadditiontothesefundamentalpointdefects,theircombinationsmayexistinasingleGaNunitcell.Forexample,VGaONisacomplexpointdefectforwhichtheGaatomismissingintheunitcellandanOatomhasreplacedaneighboringNsite.VGaONcomplexcanhavedifferentionchargesandisacandidatefortheyellowandgreenbandluminescencefromGaNcrystals[ 59 61 ].Inadditiontothepointdefects,structuraldefectssuchasplanestackingfaults,edgedislocations,threadingdislocations,etc.mayalsoadverselyaffectthedeviceperformanceandlongtermreliabilityofAlGaN/GaNHEMTs.FurtherdefectsthatcontributetodevicedegradationarethesurfacestatesontheAlGaNbarrier,anddefectsintheAlGaNandattheinterfaceofthegatemetalandthebarrier.Consequently,thecharacterizationofelectricallyactivetrapsandunderstandingthefundamentalsoftrapgenerationiscrucialfortherealizationofreliableGaNbasedHEMTdevices. Inthisregard,itisnecessarytostudytheadvantagesanddisadvantagesofthecommonlyacceptedtrapcharacterizationmethodsintermsoftheirmeasurementcapabilitiesandexperimentalchallenges. 39

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3.2OverviewofTrapCharacterizationMethods 3.2.1PulsedI-V ThepulsedI-Vmethodisbasedonmeasuringthepulseddraincurrentwhilesimultaneouslychangingthegatevoltagefromthedepletionconditiontoinversionconditionandbacktodepletionfasterthanthetrappingtimeofthemobilechargecarriersinthedielectriclayer[ 62 ].Incontrasttopulsedbiasconditions,staticDCdraincurrentmeasurementssufferfromanincreaseinthethresholdvoltageduetothetrappedinversionlayerelectronsinthedielectriclayer.Thisinreturnleadstoadecreaseinthedraincurrent.PulsedI-Vmeasurementsonahigh-siliconMOSFETisshowninFigure 3-1 .Byadjustingthepulsewidthlongerthanthetrappingtime,adecreaseinthedraincurrentisobservedbecauseofthefasttransientchargingeffects(right).Duetothedecreaseinthedraincurrentduringthelongpulse,ahysteresisvoltagedifferencebetweentheriseandfallportionsofthegatepulseisobserved(left).TheincreaseinthethresholdvoltagebetweentheriseandfallID-VGisusedtoextracttheinterfaceanddielectrictrapdensitiesinsiliconMOSFETs. ForAlGaN/GaNHEMTs,pulsedgatevoltagehasbeenusedtoanalyzethetrapsintheNi/AlGaNinterface,AlGaNbarrier,andthesurfacestatesattheungatedregionbetweenthegateandthedrain[ 63 64 ].UnlikeSiMOSFETs,alowerdraincurrentthanthestaticDCmeasurementsisobservedforpulsedgateAlGaN/GaNHEMTs.ItisshownthatthedispersionbetweenthestaticDCandpulsedI-Vdraincurrentdecreaseswithincreasingpulsewidth.Thisisexplainedbythetime-dependentcontributionofemittedelectronsfromtheNi/AlGaNandAlGaNbarriertrapstothe2DEG,aswellasbyovercomingthetime-dependentgatelagcausedbychargedsurfacestates.Bygraduallyincreasingthepulsewidth,trapswithemissiontimeshorterthanthepulsewidtharereleased,increasingthedraincurrent.Theratioofthetrapdensityforthecorrespondingemissiontimetothe2DEGisfoundbythechangesinthedraincurrent. 40

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Figure3-1. ThepulsedI-Vmethodforhigh-siliconMOSFETisshownfrom[ 62 ].Thegatevoltage(left)andtime(right)dependentdraincurrentisplotted. 3.2.2ConductanceDispersion ThefrequencyandbiasdependentconductancedispersionisawellestablishedinterfacetrapcharacterizationmethodforbothsiliconMOSFETandAlGaN/GaNHEMTdevices.OriginallydevelopedforsiliconMOSCAPs[ 65 ],theeffectsofinterfacetrapsaremodeledasanadditionalcapacitance(Cit)duetothepresenceoftrappedchargesandasaresistance(Rit)fortheenergylossduringthecapture-emissionprocesses.Theequivalentcircuitmodeloftheoxidecapacitance(Cox),semiconductorcapacitance(CS),interfacetrapcapacitance,andcapture/emissionlossisshowninFigure 3-2 a.ForacontinuumoflevelsoftrapstatesofdensityDit,theelementsoftheequivalentcircuitaregivenby[ 66 ], CP=Cs+Cit !ittan(!it) (3) GP !=qDit 2!itln[1+(!it)2], (3) whereCit=q2Dit,!istheangularfrequencyofcapacitancemeasurement,andit=RitCitistheinterfacetraptimeconstant.Bymeasuringcapacitanceatdifferentfrequencies,thecapacitanceandconductancedispersioncurvesareobtained.Fromthese,theinterfacetrapdensityisextractedfromthemaximumoftheGP=!andthe 41

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Figure3-2. (a)SiliconMOSCAPcapacitanceismodeledwiththeoxidecapacitance,semiconductorcapacitance,interfacetrapcapacitanceandcapture/emissionloss.(b)Equivalentcircuitwithinterfacetrapconductance.(c)Circuitofthemeasuredcapacitanceandparallelconductance. interfacetraptimeconstantisfoundfromtheangularfrequencyforwhichthepeakconductanceismeasured. Theconductancedispersionmethodhasbeenusedtoextractthetrapdensity,timeconstant,andactivationenergyforGaNbasedHEMTdevices[ 67 70 ].ThedifferencesbetweensiliconMOSCAPsandGaNbasedHEMTdevicesarerepresentedasadditionalelementstotheequivalentcircuitsfortheAlGaNsurfacetrapstates,seriesresistanceandmultipleinterfaceandbulkGaNtraps.Bymodifyingthebiasconditions,depletionorenhancementofthe2DEGisachievedforextractionoftrapsfromdifferentspatiallocationsoftheHEMTstructure(i.einterfaceorbulk).Activationenergiesandcapturecrosssectionsofthemodeledtrapsareobtainedbyvaryingthetemperatureduringthedispersionexperiments.Howeverwithanincreasingnumberofresistiveandcapacitiveelementsinthemodelcircuits,theaccuracydecreasesduetothefactthatcapacitancemeasurementscanonlyprovidethetotaleffectivecapacitanceandconductance.Thisleadstoanequationsetwithtoomanyunknownsandtwomeasureddata.Theindividualtrapparametersareapproximatedbymathematicalttingmodelswhichmaybeeithertoocomplicatedorinaccurate. 42

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3.2.3SubthresholdSwing ThesubthresholddraincurrentmeasurementfortrapcharacterizationprovidesanaverageinterfacetrapdensityforMOSFETs[ 66 ].Thesubthresholdswingismodiedbytheamountofchargetrappedattheinterfacestatesandisgivenby, S=KT qln(10)Cb+Cit Cox,(3) whereCbisthespace-chargeregioncapacitance.Inordertoextracttheinterfacetrapcapacitance(i.edensity),theexactvaluesoftheoxideanddepletioncapacitanceisrequired.Hence,thesubthresholdswingmethodisgenerallyemployedasasimpledifferentialtrapcharacterizationmethodfordevicesbeforeandafterdegradation.Howeverbyvaryingthetemperature,Chungetal.hasdemonstratedthatthesubthresholdswingmethodcanprovideaquickestimateoftheaverageinterfacetrapdensityforAlGaN/GaNHEMTs[ 71 ]. 3.2.4CurrentTransients Investigationoftrappinganddetrappingbehaviorsthroughcurrenttransientsisoneoftheearliestmethodsoftrapparameterextractionfromsemiconductordevices[ 72 73 ].AlthoughthemethodwasdevelopedbyC.T.Sahalmost40yearsago,itisstillcurrentlyusedasafasttrapcharacterizationtechniqueforAlGaN/GaNHEMTdevices[ 25 ].BulkGaNtraps,AlGaNbarriertraps,andAlGaNsurfacestatesmaytrapenergeticchannelelectronswhichthenleadtopartialdepletionofthechannel,andhencetoacurrentcollapse[ 22 23 ].JohanddelAlamoexaminedthetrappinganddetrappingbehaviorofthecurrentcollapseanditsrecoveryusingcurrenttransients[ 25 ].Bothmeasuredtransientswerettedtoalinearcombinationofexponentialswithdifferenttimeconstants.Byexploringthetemperaturedependentchangesinthettedtimeconstants,trapactivationenergieswereestimatedusingArrheniusplots.Additionally,bychangingthedeviceoperationcondition,informationonthespatiallocationofthetrapisestimated. 43

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Currenttransientsisasimpleandfasttrapcharacterizationtechnique,whichisusefulforsemi-quantitativemeasurements.Thedisadvantagesofthismethodaretherequirementofacryogenictemperaturerangeandtheinabilitytoextracttrapdensity. 3.2.5DeepLevelTransientSpectroscopy(DLTS) ThefundamentalsforthedeepleveltransientspectroscopymethodweredevelopedbyC.T.Sahandhisstudentsintheearly70's[ 72 74 75 ].Later,Langhasimplementedthemethodwithadoubleboxcarintegrator[ 76 ].InDLTS,thetransientcapacitance,charge,orcurrentfromareversebiaspulsedpnjunctionoraSchottkydiodeismonitoredatadesiredpairofsamplingtimes(t1andt2)underaslowlyvaryingtemperature.Whenplottedwithrespecttothemeasurementtemperature(T),theresultingdifferentialcapacitance(orcharge,current)ofthetransientatt1andt2formsalocalmaximumforeachtrap.Theemissiontime(e)isxedforeachchosenpairoft1andt2,anddeterminesthetemperatureforwhichthepeakisobserved.Ateachlocalmaxima,thedensityofthetrapisobtainedwiththehelpofthevalueofthetransientpeak.Bychangingt1andt2,andperforminganewtemperaturesweepwhilemeasuringthetransients,anewdifferentialpeakisobservedatadifferenttemperature.Hence,anewsetofeandTisfoundforeachtrap.Consequently,theln(eT2)vs1/T(Arrhenius)plotsareobtainedforindividualtraps.UsingtheArrheniusplots,thetrapactivationenergyandcapturecrosssectionsarefound. TheDLTSbasedapproacheshavebeenusedfortrapcharacterizationinGaNbasedmaterialsystems[ 77 81 ].Okinoetal.haveinvestigatedthetrapcharacteristicsofametal/SiN/AlGaN/GaNHEMTthroughcurrent-DLTSmeasurementswhilesweepingthetemperature[ 81 ].Faqiretal.havealsoemployedthetemperaturedependentcurrent-DLTSmethodfortheanalysisoftraprelatedeffectsinAlGaN/GaNHEMTs[ 80 ].Itwasalsoshownthatacombinationofphotonenergy(deeplevelopticalspectroscopy-DLOS)andtemperaturesweepcanbeusedinacomplementarymannertoscandifferentpartsoftheGaNenergygapduringcapacitanceorcurrentmeasurements 44

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[ 77 79 ].In[ 79 ],thegateanddrainvoltagesweredynamicallycontrolledtoachieveaconstantdraincurrentatdifferentoperationregimesduringDLOSandDLTSexperiments.Bymonitoringthechangesinthefeedbackresponseofthegateanddrainvoltages,thedistributionoftrapsunderthegateandnearthedrainaccessregionwasestimated[ 79 ]. DLTSmeasurementsarepowerfultoolstoinvestigatetrapsinAlGaN/GaNdeviceshowever,therequirementofacryogenictoelevatedtemperaturerangewithopticalilluminationwhereneededresultsinaverychallengingexperimentalsetupwhencombinedwithmechanicalwaferbendingexperiments. 3.2.6PhotoionizationSpectroscopyandLuminescence Thedensity,energy,andtrappingtimeofsubbandgaptrapsinAlGaN/GaNHEMTdevicescanbedeterminedbyphotoionizingthetrappedelectronswithavaryingopticalexcitationwavelengthandmeasuringthechangesinthetimedependentdraincurrent.PhotoionizationspectroscopyhasbeenemployedfortheinvestigationoftrapsviathereversalofdraincurrentcollapseinGaNdevicesduetothecapturedhotelectronsinslowbulkGaNtrapsandimpuritysites[ 82 86 ].ForatrapstatewithanactivationenergyofEA,opticalilluminationfromanincidentphotonwithanenergyequaltoorgreaterthanEAresultsintheemissionofthetrappedelectrontotheGaNconductionband.WiththeassumptionthatthetimerequiredtotraversetheHEMTchannelisshorterthanthetrapcapturetime,thetrapstateatEAbelowtheconductionbandiscontinuouslyemptiedtoanon-equilibriumsteady-state,underasufcientlyhighdensityofincidentphotonux.Hence,theelectronsresidingatmultipletrapswithactivationenergylowerthantheincidentphotonenergycontributetothe2DEGandincreasethedraincurrentduringtheopticalillumination.Assumingthatthecarriermobilityremainsunchanged,theratioofthetotalexcitedtrapdensitytothe2DEGisobtainedbycomparingtheincreaseddraincurrentunderilluminationtothedarkcurrent.Usingarangeofdifferentsubbandgaplightwavelengths,theenergydistributionofthearealtrap 45

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Figure3-3. Thephotoemissionisshownforpre-stressandpost-stressconditionsunderforwardandreversebias[ 93 ]. densitymaybeextracted.Afterthelightsourceisswitchedoff,theemittedelectronsarecapturedbytheemptiedtrapstates,leadingtoadecayingtransientinthedraincurrent.Byttingexponentialfunctionstothedecaytransient,trapcapturetimescanbeextracted. TheluminescencepropertiesoftheGaNpointdefects,structuraldefects,andimpuritieshavebeenstudiedthoroughlyintheliterature[ 59 61 87 93 ].Thepreviousinvestigationsofthedefectsresponsibleforred,yellow,green,blueandnearbandgapopticalemissionbandsprovidesinvaluableinformationonthenatureofthetrapsobtainedfromopticalexcitationexperiments.Moreover,sincethetrapsareextractedbyopticalexcitation,thespatiallocationofthetrapscanbedeterminedthroughthephotoluminescence,cathodoluminescenceandelectroluminescenceresponseofthedeviceforthegivenradiativetrapenergy(Figure 3-3 ). 3.3Summary Basedonthemeasurementcapabilitiesandexperimentalchallengesoftrapcharacterizationmethods,Table 3-1 summarizesacomparisonofdifferenttechniques.Thetrapactivationenergy(EA),correspondingenergydistributionofarealtrapdensity(Dit),andtrappingtimeconstant()arerequiredtoquantifythetrappingbehaviorinAlGaN/GaNHEMTs.Inthisregard,conductancedispersion,DLTSandphotoionizationspectroscopycanprovidemostoftherequiredparameters.Howeveraswehave 46

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Table3-1. Summaryofdifferenttrapcharacterizationmethodsintermsofcapabilitiesofdensity(Dit),timeconstant(),andactivationenergy(EA)extraction,andexperimentalchallenges. MethodCanMeasureCannotMeasureExperimentalChallenges PulsedI-VDit,EAnanosecondRFprobingConductanceDit,,EAN/AcryogenictoelevatedTrangeDispersionSubthresholdDavgitEA,elevatedTrangeSlopeCurrent,EADitcryogenictoelevatedTrangeTransientsDLTSDit,,EAN/AcryogenictoelevatedTrange+opticalilluminationPhotoionizationDit,,EAN/AopticalilluminationSpectroscopy discussed,theconductancedispersionmethodsuffersfromthelimitednumberofmeasuredquantities(totalconductanceandcapacitance)comparedtothenumberofcapacitiveandresistivetrapelementstobemodeled.Also,therequirementofcryogenictemperaturerangewhileconductingexternalwaferbendingincreasesthecomplexityoftheexperimentalsetupsignicantly.Inaddition,theDLTStechniquerequiresbothcryogenictemperaturerangeandopticalilluminationduringwaferbending,whichfurtherincreasesthecomplexityoftheexperimentalsetup.Ontheotherhand,photoionizationspectroscopycanbecarriedoutbyilluminatingthesampleinthewaferbendingjig,withouttheneedforacryogenicchamber.Thus,photoionizationspectroscopyisanexperimentallyfeasiblemethodforthecharacterizationofstraineffectsonAlGaN/GaNHEMTtrapgeneration. 47

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CHAPTER4INSIGHTSONTHE(100)AND(110)DOUBLE-GATEFINFETELECTRONTRANSPORTANDTHESTRAINRESPONSEOFP-TYPETRI-GATEFINFETS 4.1PhysicalInsightsonComparableElectronTransportin(100)and(110)Double-gateFinField-effectTransistors 4.1.1Motivation Asthechannellengthofcomplementarymetal-oxide-semiconductor(CMOS)transistorsisscaleddowntosub-100nmlengthrange,detrimentalshortchanneleffects(SCE)havemitigateddeviceperformance.Toovercometheseeffects,non-planarsiliconchannelswithmultiplegatesareintroducedtorestorethegatecontrolonshort-channelon/offresistance.Infact,Intelhasrecentlyannouncedtheir22nmbaselinecomprisedofTri-GateFinFETs[ 7 ]. InadditiontobetterSCEcontrol,thenon-planartransistorapproachprovidesboth(100)and(110)sidewalltransportonthesameconventional(100)siliconwafer,avoidingmoreexpensivesolutionssuchasHybridOrientationTechnology(HOT)[ 94 ].Double-gateFinFETs(DG-FinFETs)fabricatedon(100)wafershave(110)-surface/h110i-channelor(100)-surface/h100i-channelcongurations.Forconventionalplanarmetal-oxide-semiconductoreldeffecttransistors(MOSFETs)fabricatedonseparatewafers,(100)ismorefavorableforelectronconduction,whereas(110)ispreferableforholetransport. Contrarily,wehaveexperimentallyobservedcomparableelectronmobilityin(110)and(100)sidewallDG-FinFETs[ 95 ],while(110)isstillfavorableforholetransportover(100).Therefore,aphysicsbasedinvestigationisverytimelyforselectionof Section 4.1 isreprintedwithpermissionfromMehmetO.Baykan,ChadwinD.Young,KeremAkarvardar,PrashantMajhi,ChrisHobbs,PaulKirsch,RajJammy,ScottE.Thompson,andToshikazuNishida,Physicalinsightsoncomparableelectrontransportin(100)and(110)double-gateneld-effecttransistorsAppl.Phys.Lett.100,123502(2012),DOI:10.1063/1.3696038.Copyright2012,AmericanInstituteofPhysics. 48

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thenextgenerationCMOSintegrationschemeon(110)DG-FinFETplatform.Inthissection,weinvestigateoneofthepossibleunderlyingphysicalreasonsforcomparableelectronmobilityseenin(100)/h100iand(110)/h110iFinFETs,fromaperspectiveofnon-parabolicdispersionrelationoftransverseelectronmass(mt)intheh110idirection. 4.1.2Experimental Silicon-on-insulator(SOI)DG-FinFETswerefabricatedwithchannelsformedon(100)and(110)planes.Neutralstress(100)intrinsic(ND2x1015cm)]TJ /F9 7.97 Tf 6.58 0 Td[(3)SOIsubstrateswerepatternedusing193nmlithographytoproducearraysof200nswith10mchannellength.Hafniumbasedhigh-k(2nm)andmid-gapmetal(10nm)gatestackwasformed.Theresultingnshavecrosssectiondimensionsof40nmheightand20nmwidth.Splitcapacitance-voltage(C-V)measurementswereconductedtoextracteffectiveelectronmobility(n,e). Figure 4-1 showstheextractedn,evs.inversionelectrondensity(Ninv)fromatotalof25sitesontwodifferent200mmand300mmwafersmanufacturedatdifferentfacilities.Unliketheconventionalplanarn-typeMOSFETsfabricatedonseparate(100)and(110)wafers,nodegradationinmobilityisobservedasthesidewallischangedfrom(100)to(110).Previously,itwassuggestedthathigh-/metal-gateinducedstresscouldbethereasonfortheobservedanomaly[ 96 ].However,wehaverecentlyshownthatmobilityextractedfromSiO2/poly-SigateFinFETsalsodemonstratecomparableelectronmobilityfor(100)and(110)sidewalls[ 95 97 ].Similarly,wehaveobservedthesamephenomenonfromnswithvaryingthicknessfrom20nmto45nm[ 95 ],aswellashighdosePorBdoping[ 97 ].Hence,thereasonforthecomparableelectronmobilityin(100)and(110)sidewallnscouldnotbeexplainedbyanycombinationofthemetal-gateinducedstress,volume-inversion,andundopedbodyeffects. Tounderstandtheunderlyingphysicalmechanisms,wehaverevisitedtheconductionbandof(100)and(110)FinFETs.TheinsetofFigure 4-1 showstheconductionband(CB)ellipsoidsandtheirconductivityeffectivemass(mcond)for 49

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Figure4-1. Effectiveelectronmobilityisextractedfrom(100)and(110)sidewallFinFETsusingsplit-CVmethod,andshownasafunctionofNinv.Insetshowstheconductionband(CB)ellipsoidsandtheirconductivityeffectivemassesfor(100)and(110)inversionlayer.Inaddition,thedensity-of-states(mD)andconnementmass(mZ)valuesarealsotabulated.c2012,AmericanInstituteofPhysics. both(100)and(110)inversionlayers.The2valleymcondis0.19m0forbothsidewallorientations.Ontheotherhand,4valleysof(100)havetwodifferenttransportmasseswithequaloccupancy,resultinginaveragemcond=0.315m0.Similarly,themcondof4valleysin(110)isfoundtobeequalto0.55m0.ThetableshownintheFigure 4-1 insetliststhedensity-of-states(DoS)andconnementeffectivemassvalues(mDandmzrespectively).Althoughgenerallytakenasparabolic,withmt=0.19m0,onlythemzof(110)2isnon-parabolicandhassignicanteffectsonelectrontransportin(110).Theeffectsofnon-parabolicconnementmass(mz,NP)for(110)havebeenconsideredbyUchida[ 98 ],Silvestri[ 99 ]andShimizu[ 100 ].Yet,noneoftheseeffortscouldexplainourexperimentalresultsshowninFig. 4-1 Inordertoobtaintheaverageconductivityeffectivemass(mcond,avg)forboth(100)and(110)FinFETs,aself-consistentsingleelectroneffectivemassSchrodinger-PoissonDG-FinFETsimulatorisused.Additionally,toinvestigatetheeffectsofmz,NP,wehaveemployedanearest-neighborsp3d5stight-binding(TB)formalism[ 101 102 ]toobtaintheE-krelationof20nmthick(110)and(100)FinFETsupercells.Itisveriedthatonly 50

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Figure4-2. StandardSchrodinger-Poissonself-consistentsimulationprocedureismodiedtotakenon-parabolicconnementmass(mz,NP)of2valleyon(110)intoaccount.Theinnerloopensuresthatthecorrectmz,NP/quantizationenergypairconverges.c2012,AmericanInstituteofPhysics. 2of(110)FinFEThasnon-parabolicmz.UsingtheTBresults,weobtainedsubbandquantizationenergydependentmz,NP.Figure 4-2 showstheprocedurefollowedduringtheself-consistentSchrodinger-Poissonsimulation.Forthe(110)FinFET,wehavemodiedtheprocedurewithaninnercontrollooptoselectthecorrect(mz,NP/E0,2)pairobtainedfromtheTBcalculations. 4.1.3ResultsandDiscussion Conventionally,aparabolictisusedtoapproximate(110)2connementmassasmt=0.19m0.Sincethevalleysubbandenergiesareinverselyrelatedtomz,andtheelectronspreferablyoccupylowerenergystates,theparabolicconnementmassapproach(PCM)assumesthatthemajorityofelectronsoccupy2for(100)and4for(110).Thisapproachinherentlyleadstoheaviermcond,avgfor(110),andhence,muchlowern,ecomparedto(100)sidewallDG-FinFETs. However,theE-kdispersionofmtintheh110idirectionisstronglynon-parabolic,whichresultsinanon-parabolicconnementmassforthe(110)2valley.AsshowninFigure 4-3 -a,theE-kcurvedeviatesfromtheparabolicbandastheenergyincreases. 51

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Figure4-3. (a)Energy-kdispersionrelationofmtinh110idirectionisshownforparabolicandnon-parabolicbandmodels.(b)Thesolidlinerepresentsthemz,NPof(110)2isobtainedfromtheinversesecondderivativeofthetight-bindingE-krelation.Circlesaretheconverged(mz,NP/E0,2)pairsintheinnerloopofFig. 4-2 .c2012,AmericanInstituteofPhysics. Figure4-4. (110)CBvalleyenergyanddegeneraciesareshown.UnderPCM,connementsplitsvalleysand4becomesgroundvalleyduetotheheaviermz.Whennon-parabolicityisincluded,themzof2becomesmuchheavierandswitchesthevalleysover.c2012,AmericanInstituteofPhysics. Infact,themz,NPquicklybecomesheavierthanmtwithincreasingquantizationenergy(Figure 4-3 -b).SeenasthecirclesonFigure 4-3 -baretheconverged(mz,NP/E0,2)pairsintheinnercontrolloopofthesimulator.Consequently,averygoodagreementbetweenTBandSchrodinger-Poissondomainshasbeenachieved. Heaviernon-parabolicconnementmassresultsinlower(110)2energyforagivenNinv,thusincreasingitsoccupancy.Withtheincreasedoccupancyofthe2valley,thepercentageofelectronswiththelightermtconductivityeffectivemassincreases. 52

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Figure4-5. Inversionelectrondensitydependentmcond,avgisshownfor(100)and(110)forbothparabolicandnon-parabolicconnementmassmethods.Undernon-parabolicmodel,themcond,avgforboth(100)and(110)aresimilar,whichsuggeststhecomparableelectronmobilityshowninFig. 4-1 .c2012,AmericanInstituteofPhysics. Therefore,the(110)averagemconddecreaseswithstrongernon-paraboliceffectsontheconnementmass.Figure 4-4 demonstratestheeffectofnon-parabolicconnementmassfor(110)CB.Previousreportsonthemz,NPfor(110)haveindeedshowedhigher2occupancy[ 98 100 ].However,theseprioreffortsresultedinhigher2valleyenergythan4,indicatingmt
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Infact,athighNinv,the(110)massislowerthan(100)duetolargerDoSat2valleyof(110)comparedto(100).Yet,higherDoSof(110)wouldincreasethescatteringrateswithrespectto(100),counteractingthelightermass.Consequently,(100)and(110)DG-FinFETelectronmobilityareexpectedtobecomparable,qualitativelyexplainingtheexperimentaldatashowninFigure 4-1 4.1.4Conclusion Byincludingthenon-paraboliceffectsontheconnementmassof2valleyof(110)conductionband,wehaveexplainedapossiblephysicalmechanismgoverningthecomparableelectronmobilityin(100)and(110)DG-FinFETs.AlthoughtheshortcomingsinexplainingplanarMOSFETbehaviorareyettobeinvestigated,itisshownthatmcond,avgfor(100)/h100iand(110)/h110iFinFETsbecomessimilarwiththeinclusionofnon-parabolicityofconnementmass.Consequently,withoutdegradationinelectrontransport,the(110)sidewallFinFETisacheaperalternativetoHOTforhighperformancedeviceintegrationatandbeyondthe22nmnode. 4.2StrainResponseofp-typeTri-GateFinFETs Followingtheeffortsofunderstandingtheelectrontransportindouble-gateFinFETsintheprevioussection,wecontinuewiththeinvestigationofthestrainenhancementofp-typeFinFETs,particularlytoachievebalancedpull-upandpull-downelementsintheCMOSimplementationofthisnon-planartechnology.Duetothenon-planarstructureofthetri-gatedevice,carrierstransportthroughdifferentsurfaceorientationssuchas(110)forthesidewalland(100)forthetopsurface.Inthisregard,wehaveconductedanexperimentalstudyof4-pointexternalmechanicalbendingofp-typetri-gateFinFETswithaxedheight(25nm)andavaryingwidthfrom25nmto500nm(Figure 4-6 ).Figure 4-7 showsthefundamentalsofthe4-pointuniaxialwaferbendingapproach.Itisshownthattheportionofthesamplebetweentheinnerrodsisunderaconstantuniaxialstress 54

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Figure4-6. Tri-gatep-typeFinFETsamplecleavedfroma300mmwaferisunder1GPauniaxialtension. givenby[ 103 ], =Eyt 2a(L 2)]TJ /F3 11.955 Tf 13.15 8.09 Td[(2a 3),(4) whereistheuniaxialstress,EistheYoung'sModulusofthesubstrate,tisthewaferthickness,yistheverticaldisplacement,aistheinnerrodspacingandListheouterrodspacing.InadditiontothecalculatedvaluefromEq. 4 ,theamountofappliedstrainisalsomeasuredbyastraingageattachedtothewafersampleduringtheapplicationofexternalmechanicalbending. Thestrainresponseofthetri-gateFinFETsischaracterizedbythepiezoresistance()coefcient.The-coefcientisdenedasthenormalizedchangeintheresistanceperstress,i.e. R R1 =,(4) whereistheappliedmechanicalstress.Tomeasurethepiezoresistancecoefcients,lowelddraincurrentismeasuredfrom11deviceswithvaryingwidthsfrom25nmto500nm,underVDS=0.05VandVGS=)]TJ /F3 11.955 Tf 9.3 0 Td[(0.5Vto1.5V,upto250MPalongitudinal 55

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Figure4-7. 4-pointuniaxialwaferbendingisillustrated. Figure4-8. TheexternalparasiticresistanceRSDextractedfromn(blue)andp-type(red)tri-gateFinFETsusingshiftandratiomethod. uniaxialcompression.ThelineardevicecurrentiscorrectedusingtheextractedRSDcontributionshowninFigure 4-8 .Thepiezoresistancecoefcientsareextractedatthestronginversionlimit(VGS)]TJ /F4 11.955 Tf 11.95 0 Td[(VTH=0.5V)andareshownassymbolsinFigure 4-9 Inordertomodeltheexperimentallyobservedwidthdependenceofthe-coefcientofthetri-gateFinFETs,wehaveconsideredthebulkMOSFET-coefcientsofthetopandsidewallsurfaces[ 104 ].AswillbediscussedinChapter6,volume-inversionrelated 56

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Figure4-9. Experimentallyextractedpiezoresistancecoefcientsfromp-typetri-gateFinFETsareshown(symbols).Solidlinedenotesthecalculated-coefcientusingEq. 4 .Closeagreementbetweenexperimentalandcalculatedvaluesisobtainedfor(100)=6510)]TJ /F9 7.97 Tf 6.59 0 Td[(11Pa)]TJ /F9 7.97 Tf 6.59 0 Td[(1,(110)=3010)]TJ /F9 7.97 Tf 6.59 0 Td[(11Pa)]TJ /F9 7.97 Tf 6.59 0 Td[(1and(110)=(100)=2.InsetshowsthecrosssectionTEMofa30nmwidetri-gatedevice. effectsareminimalforrelativelythickFinFETsof20nmorwider,whichjustiestheuseofbulksurfaceinversionMOSFET-coefcients.Hence,thetri-gateFinFETisassumedtoconsistof2parallelchannels,namelythe(100)topsurfaceandthe(110)sidewallsurface.FortheFinFETwithheightH,widthWandgatelengthL,thewidthofthe(100)and(110)channelsareequaltoWand2Hrespectively.Thentheconductancesofthetopandsidewallchannelsaregivenas, G(100)=W(100)Cox,100(VG)]TJ /F4 11.955 Tf 11.96 0 Td[(VT,100) L (4) G(110)=2H(110)Cox,110(VG)]TJ /F4 11.955 Tf 11.95 0 Td[(VT,110) L (4) GTG=G(100)+G(110), (4) 57

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whereisthesurfacemobility,Cox(VG)]TJ /F4 11.955 Tf 12.91 0 Td[(VT)=qNinvdenotestheinversionarealcarrierdensity,andGTGthetotalFinFETconductance.Intermsofconductance,the-coefcientcanbedenedas, G0)]TJ /F4 11.955 Tf 11.95 0 Td[(G G=,(4) whereG0andGaretheunstrainedandstrainedconductancesrespectively.Thenthetri-gate-coefcient,TGcanbewrittenas, TG=1 W(100)qN(100)+2H(110)qN(110) L)]TJ /F16 11.955 Tf 11.95 16.86 Td[(W(100)qN(100) L(1+(100))+2H(110)qN(110) L(1+(110)) W(100)qN(100) L(1+(100))+2H(110)qN(110) L(1+(110))(4) whereN(100)andN(110)aretheinversioncarrierdensitiesfortopandsidewallsurfaces,whichareassumedtobesimilar.Forlowstressconditionsupto200MPa,both(110)and(100)termsaremuchsmallerthan1,andthenthetri-gate-coefcientisfoundby, TG=W(100)+2HK(110) W+2HK,(4) whereK=(110)=(100).UsingEq. 4 ,wehaveplottedthecalculatedtri-gate-coefcientasafunctionofnwidthusing(100)=6510)]TJ /F9 7.97 Tf 6.59 0 Td[(11Pa)]TJ /F9 7.97 Tf 6.59 0 Td[(1,(110)=3010)]TJ /F9 7.97 Tf 6.59 0 Td[(11Pa)]TJ /F9 7.97 Tf 6.59 0 Td[(1and(110)=(100)=2.The(100)and(110)valuesthatprovidethebestttoexperimentaldataarewellwithinthepublishedexperimentalvalues[ 104 ].Consequently,fornon-planarstructureswiderthan20nm,thepiezoresistancecoefcientscanbeobtainedbyusingaweightedconductanceaverageofthebulkMOSFET-coefcientsofthecontributingtransportsurfaces. 4.3Summary Wehaveconductedacoupledexperimentalandmodelingstudytoinvestigatetheunderlyingphysicsofcomparableelectrontransportin(100)and(110)double-gateFinFETsandthewidthdependenceofthestrainresponseofp-typetri-gateFinFETs.Onepossiblereasonforthecomparableelectrontransportisshowntobethenonparabolicconnementeffectivemassof2CBvalleyin(110)FinFETs,resultinginasimilar 58

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averageelectronconductivityeffectivemasswithrespectto(100)FinFETs.Ontheotherhand,thestrainresponseofp-typetri-gateFinFETsisshowntobedependentonnwidth,andcanbeaccuratelymodeledbyusingtheconductivityweightedaverageofthetopandsidewallsurfacebulkMOSFETpiezoresistancecoefcients. 59

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CHAPTER5SIZEANDORIENTATIONDEPENDENTSTRAINEFFECTSONBALLISTICSIP-TYPENANOWIREFIELDEFFECTTRANSISTORS 5.1Motivation Thecontinuousscalingdownofplanarsilicontransistordimensionsforoverfourdecadeshasintroducedanunavoidablebarrierofdetrimentalshort-channeleffects.Asthedevicesapproachthequasi-ballistictransportlimit,gatecontrolofthechannelresistancehasdegradednotably.Onesolutiontoovercomethisproblemisthenon-planarmultiple-gatetransistordesign,whichhelpstorestorethegatecontroloverthechannelresistance[ 105 ].Duetotheirexcellentshort-channelcontrolcapability,siliconbasedFinFETandnanowire(NW)deviceshavebeenconsideredasthenextpossiblecandidatesforthenearfuturetechnologynodes.Infact,Intelhasrecentlyannouncedtheir22nmbaselinecomprisedofTri-GateFinFETs[ 7 ].Fordevicesatthescalinglimitbeyondthe22nmnode,carriersareexpectedtotraversethetransistorchannelatnearballistictransportconditions[ 12 ].Hence,thenanowirebaseddevicedesignmayovercomethestrongershort-channeleffectsinfurtherscaleddevicesbeyondthe22nmtechnologynode. Inrecentyears,processinducedexternalmechanicalstrainhasbecomethemostestablisheddeviceperformanceboostersinceIntel's90nmtechnology[ 1 ].StrainedadvancedCMOSdeviceshaveenjoyedsuperiorperformancecomparedtounstrainedonesthroughstrain-inducedreductionofthecarriereffectivemassandscatteringrates[ 2 106 ].Notsurprisingly,strainisalsoexpectedtobeemployedasaperformanceboosterinthefutureballisticdevices[ 11 107 ].Theincorporationofmechanicalstrain ReprintedwithpermissionfromMehmetO.Baykan,ScottE.Thompson,andToshikazuNishida,SizeandOrientationDependentStrainEffectsonBallisticSip-typeNanowireFieldEffectTransistorsIEEETransactionsonNanotechnology,2012.Copyright2012,IEEE. 60

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intotheatomisticnanowiredevicesimulationsisofutmostimportanceforperformanceevaluationofnearfuturedevicedesigns. InordertomodelandpredictfuturestrainedsiliconCMOSplatforms,thereisaneedforadetailedphysicsbasedinvestigationofthestrainenhancedballisticp-typenanowiretransport.Priorworkshowedthroughextensiveself-consistentatomicsimulationsthattheh110iorientationprovidesthehighestcurrentforbothpandn-typeunstrainedballisticnanowiretransistors[ 51 57 108 ].Toincreasecomputationalefciency,compactelectrostaticmodelswerecoupledtoatomicbandstructuresimulationstoinvestigateballisticnanowireelectroncurrent[ 109 110 ].Recently,astudyofstraineffectsonn-typeballisticnanowiretransportwaspublishedbyZhangetal.[ 111 ].Therefore,mergingatomisticbandstructurecalculationswithrecentdevelopmentsincompactballistictransportmodels[ 112 113 ]isverytimelyforunderstandingtheballisticholecurrentenhancementinstrainedp-typenanowires.Fromamanufacturingpointofview,straincouldalsomitigatetheeffectsofprocess-inducedthicknessvariationforp-typeballisticnanowires[ 114 ].Duetothedependenceofspatialconnementeffectsonthedevicesizeandorientation,physicsofthestrainenhancementofballisticp-typenanowiretransistorperformanceneedstobeevaluatedandcomparedasafunctionofsizeanddeviceorientationforfuturehigh-performanceCMOSintegrationplatforms. Inthischapter,weexplainthephysicalmechanismswhichgovernthelongitudinaluniaxialtensileandcompressivestrainresponsesofh110iandh100iSiballisticnanowirep-typeeld-effecttransistorsofthreedifferentdimensions:3nm3nm,5nm5nmand7nm7nm. 5.2Method Here,weconsiderstraightnanowiredevicesextendingintheh110idirectionwith(001)and(110)gatesurfaces,whiletherotateddevicechannelisintheh100idirectionwithequivalent(100)gates(Figure 5-1 ).Conventional(100)wafersareassumedforboth 61

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Figure5-1. Straighth110iand45orotatedh100idevicesareshownonconventional(001)silicon-on-insulatorwafer.Theunitsupercellisrepresentedusingballandstickmodelofcovalentbonds.c2012,IEEE. straighttransistorsintheh110idirectionalongthewafercutand45orotatedtransistorsintheh100idirection.Duetothequantizedspacingbetweensiliconatomsindifferentcrystalplanes,actualdevicedimensionsareslightlydifferentthantheconsideredvalues.TheactualatomisticdevicedimensionscanbeseeninTable 5-1 .Finally,alltypesofdevicesconsideredinthisworkhavesurroundinggateswithaneffectiveoxidethicknessof1.1nm. Duetotheatomisticnatureofnanowiredimensions,nearest-neighborsp3d5stight-binding(TB)formalismisemployedforstrainedandunstrainedbandstructurecalculations[ 115 ]andtheupdatedSitight-bindingparametersfromBoykin'sworkareused[ 116 117 ].Thenanowireunitsupercelliscreatedforgivenmaterial,orientationandcross-sectionaldimensions.Theresultingsupercellincludesalltheatomsinthe2-dimensionalcrosssectionofthenanowireandallowsustoassume1-dimensionalplane-wavepropagationinthechannellengthdirection.Hence,thecalculatedbandstructureleadstoanenergyvs.1Dwavevectordispersionrelationinthetransportdirection.Thechoiceofsp3d5sbasissetwithspinorbitcouplingimpliesa20N20N 62

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Hamiltonianmatrix,whereNisthenumberofatomsinthe2Dnanowiresupercell.Thischoiceallowsustocapturethedetailedbandstructureofthesenanowiresforevaluationofthestraineffectsontheballisticholetransport.Duetotheveryhighsurfacetovolumeratiopresentinnanowirestructures,thedanglingsurfacebondsofthenitecrystalarepassivatedtoavoidsignicantamountofmidgapeigenvaluesinthebandstructure[ 118 ]. Themechanicalstrainisincorporatedviaelasticdeformationofthetwo-dimensionalnanowiresupercell.AlthoughLeuetal.showedthatthesiliconYoung'sModulusmaychangeasafunctionofthenanowiresizeandorientation[ 58 ],weassumebulkcompliancecoefcientsholdforournanowiredevices[ 119 ].Thepositionvectorofeachatomisrecalculatedundergivenastrainconditionvia, r=(I+")r0,(5) where"isthe3x3straintensor,randr0arethepositionvectorsoftheatominthestrainedandunstrainedcrystalrespectively.Foragivenscalaruniaxialstrainalongthenanowirechanneldirection,,theorientationspecic"canbeshownas[ 120 ], "100=0BBBB@1000S12 S11000S12 S111CCCCA, (5) "110=0BBBBBBBBBBBB@1 1+S44 2(S11+S12)S44 2(S11+S12) 1+S44 2(S11+S12)0S44 2(S11+S12) 1+S44 2(S11+S12)1 1+S44 2(S11+S12)0002S12 (S11+S12) 1+S44 2(S11+S12)1CCCCCCCCCCCCA 63

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Table5-1. Theactualatomisticheightandwidthofthenanowiresusedinthisstudy. Straighth110iRotatedh100iRoundedAtomisticAtomistic 3nmX3nm3.07nmX2.72nm2.72nmX2.72nm5nmX5nm4.99nmX4.75nm4.75nmX4.75nm7nmX7nm6.91nmX6.78nm6.78nmX6.78nm whereSijarethecompliancecoefcients.Oncetheatomicpositionsaresetinthestrainednanowiresupercell,thedevicetight-bindingHamiltonianissolvedforeigenvaluesandeigenvectors.Thestrainmodiestheoverlapanglesbetweenneighboringatomicorbitals,whicharecapturedbythedirectionalcosinesoftheSlaterandKostertwo-centerintegrals[ 121 ].Moreover,thestrainchangesinteratomicbondlengthswhichmodiesthetight-bindingenergyterms.Theeffectsofbondlengthchangeonsame-atomandtwo-centerenergytermsarecalculatedbyusingBoykin'smodicationofthegeneralizedHarrison'srule[ 122 123 ].Additionally,theeffectofnearestneighbordisplacementsduetoh110ishearstrainiscapturedbyincorporatingageneralizedLodwin'smethod[ 117 ].Finally,thestrainednanowirebandstructureisobtainedforthegivenstraincondition. Theelectrostaticsandself-consistencyofthegatebiasareintroducedthroughNatori'scompactnanowiremodel[ 112 ].ThismodelcapturestheeffectsofquantumcapacitanceandadjuststhenanowirebandedgeaccordingtothesourceFermilevel,withoutextensivecomputationalcostsoftheself-consistentsolutiontothe2DPoisson'srelationinthedevicecrosssectionandthetight-bindingbandstructure[ 109 110 ].However,thegainofcomputationalefciencyresultsinalossintheaccuracyintheself-consistentbandstructuresfornanowireswithlargerdimensions[ 51 57 ].Foreachstraincondition,thecrystalbandstructureiscalculatedforeachdimensionandorientation,andtheE-krelationisassumedtobeunchangedunderinducedoverdrivegatevoltage.UsingNatori'scompactmodel[ 112 ],theinversioncarrierdensityforagivengateoverdrivevoltageisfoundby, 64

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Q=CejVg)]TJ /F4 11.955 Tf 11.96 0 Td[(Vthj=CgjVg)]TJ /F4 11.955 Tf 11.95 0 Td[(Vthj)]TJ /F4 11.955 Tf 19.13 8.08 Td[(E0,max)]TJ /F5 11.955 Tf 11.95 0 Td[(s q(5) whereE0,maxisthebandedgeenergyoftheground-mostvalley,sisthesourceFermienergy,andCeisthetotaleffectivegatecapacitance.TheCeismodeledastheoxide(Cg)andquantumcapacitancesinseries,withouttheeffectsofinversionandparasiticcapacitances.Forsurroundingrectangulargates,theoxidecapacitanceisgivenby[ 124 ], Cg=25 40 ln1+5 4tox WNW+25 40 ln1+5 4tox HNW,(5) whereistherelativepermittivityofthegateinsulator,WNWisthewidth,andHNWistheheightofthesquarenanowire.Simultaneously,usingthesemiclassicaltop-of-the-barrierapproach[ 125 ],thecarrierdensityoftheithband(Qi)isfoundbyllingtheelectronicstatesobtainedfromtheTBbandstructureasafunctionofsthrough, Qi=q Z=ax01 1+exp[(s)]TJ /F4 11.955 Tf 11.95 0 Td[(Ei(k))=kBT]+1 1+exp[(d)]TJ /F4 11.955 Tf 11.96 0 Td[(Ei(k))=kBT]dk, (5) whered=s)]TJ /F4 11.955 Tf 13.16 0 Td[(qVdsisthedrainFermilevel.Finally,thetotalcarrierdensityiscalculatedbysummingthecontributionofallbands, Q=XiQi.(5) Theself-consistencybetweenthegateelectrostaticsandthebandstructuredensity-of-states(DoS)isachievedbyiterativelycomparingthetotalchargeobtainedfrom( 5 )and( 5 ). IncontrasttothebandedgelimitedLandauer'sapproachusedinNatori'scompactmodelforelectrontransport,weuseafull-bandapproachtocalculatetheholecurrent 65

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inthesemiclassicaltop-of-the-barrierballistictransport.Comparedtothenanowireconductionbands,thehighercomplexityofthevalencebandstructurewithmanylocalextremaisinconsistentwiththeassumptionsusedinthebandedgelimitedcurrentcalculation[ 112 ].ThepointchargeiscalculatedforeachpointintheE-krelationandthepointbandstructuregroupvelocityisobtainedvia, (E0,k0)=1 ~@E @kE0,k0.(5) Thenasshownin( 5 ),theballisticcurrentisfoundthroughthemultiplicationofchargeandvelocityateach(E,k)pointandintegratedforthecompleteEandkranges.Thisalsoensuresthatwedonotoverlooktheeffectsofthestrongnon-parabolicityseeninnanowirebandstructures.Additionally,byusingthefull-bandcurrentmethod,thepotentialerrorsinthebandedgelimitedLandauer'sapproachduetothecomplexityofthenanowirevalencebandsareavoidedintheballisticholecurrentcalculations. Ibal=q ~XiZ=ax0@E @k1 1+exp[(s)]TJ /F4 11.955 Tf 11.96 0 Td[(Ei(k))=kBT])]TJ /F3 11.955 Tf 76.75 8.09 Td[(1 1+exp[(d)]TJ /F4 11.955 Tf 11.96 0 Td[(Ei(k))=kBT]dk (5) Finally,theaverageinjectionvelocityisfoundby, hinji=Ibal=Q(5) 5.3ResultsandDiscussion Inthissection,wepresenttheresultsanddiscussionsofthestraineffectsontheballisticholecurrentofstraightandrotatedSinanowireeld-effecttransistors.TheballisticholecurrentiscalculatedatadrainbiasofjVdsj=j1Vjandanoverdrivevoltage 66

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Figure5-2. ForaconstantnumberofmobilechargesinaunitvolumeowingalongL,differentrepresentationsoftheequivalent3D,2Dand1Dchargedensitiesareillustrated.c2012,IEEE. (jVovj=jVgs-Vthj)rangeof-0.4Vto1V.Inordertomakeafaircomparisonbetweennanowiresofdifferentsizes,theoncurrentisnormalizedwiththeperimeter(i.ethegatewidthinplanardevices).Forbenchmarkpurposeswithconventional2Dplanardeviceswithsurfaceinversion,the1Dlinecarrierdensityofnanowiresisconvertedtoanequivalent2Dinversioncarrierdensitywhererequired,foraconstantamountofmobilecharge,Q,owingalongL,asshowninFigure 5-2 .Withoutchangingthetotalnumberofmobilecarriers,wecanconvertthe1Dlinecarrierdensityinto2Dsidewallsurfaceand3Dvolumechargedensitiesasshownbelow, Q=qN3DWHL=qN2D(2HL+2WL)=qN1DL,(5) andthus, N2D=N1D 2(H+W),(5) whereQisthetotalmobilecharge,qistheelementarycharge,whileN3D,N2DandN1Daretheequivalent3Dvolume,2Dsidewallsurface,and1Dlinecarrierdensities,respectively. Thesizeandorientationdependentcharacteristicsofunstrainednanowireswillbediscussedrst.Then,themodicationofthesecharacteristicsareevaluatedundertensileandcompressivelongitudinaluniaxialstrainconditions. 67

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Figure5-3. Thevalencebandstructureisshownforunstrainednarrowest(3nmx3nm)andwidest(7nmx7nm)nanowiresinbothh110iandh100idirections.Itisseenthattheholemassdecreasesaswemovefromh100idirectiontoh110i,aswellasfromaweakerstructuralconnementtostronger.c2012,IEEE. 5.3.1UnstrainedDevices Theunstrainedvalencebandstructureofwidestandnarrowestnanowiresaresimulatedforbothstraightandrotateddevices.AsshowninFigure 5-3 ,changesinsizeandorientationmodifythecurvatureanddensityofthebands.Independentofcrystalsize,straightnanowireshavelargerbandcurvaturecomparedtorotateddevices,indicatingalighterholeeffectivemass[ 57 ].Asthedevicesscaledown,thestructuralconnementincreases.Increasedquantumconnementwarpsandshiftsthebands.Asaresult,narrowerdeviceshavelighterholeeffectivemassduetosharperbands.Whilelightereffectivemassindicatesbetterholetransport,italsoleadstoasmaller1DDoSduetothesingularityoftheeffectivemass.Inotherwords,unlike2DMOSFETs,carriersintrue1Dcrystalshaveonlyoneeffectivemassalongtheonlycontinuouswavevector,andthus,boththetransportandDoSeffectivemassvaluesarethesame.Consequentlyfornarrowerdevices,asmaller1DDoSisobtainedduetoacombinationofmoresparsebandsandlightereffectivemass. 68

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Thelimited1DDoSofnanowirescoupledwiththin-oxidesurroundinggatesresultinaneffectivegatecapacitanceloss[ 109 ],similartotheDOSbottleneckshownbyFischettietal.[ 126 ].Sincetheeffectivegatecapacitanceismodeledasaseriesofoxideandquantumcapacitances,ananalyticalexpressionofthequantumcapacitance(Cq)canbeobtainedfromEq. 5 as, Cq=CgjVg)]TJ /F4 11.955 Tf 11.95 0 Td[(Vthj)]TJ /F4 11.955 Tf 19.13 8.08 Td[(E0,max)]TJ /F5 11.955 Tf 11.95 0 Td[(s q E0,max)]TJ /F5 11.955 Tf 11.96 0 Td[(s q.(5) Theterm(E0,max)]TJ /F5 11.955 Tf 12.15 0 Td[(s)=qrepresentshowdegeneratethebandstructurehastobecomeinordertosupporttheamountofchargerequiredbythegateelectrostatics.Inotherwords,theFermilevelhastobepushedmuchlowerthanthevalencebandedgeduetolimitedDoSinthenanowire.Hence,thesmallertheDoS,thelargerthe(E0,max)]TJ /F5 11.955 Tf 12.23 0 Td[(s)=qtermis,leadingtoasmallerCq.Consequentlyfornanowires,theCebecomessmallerthantheoxidecapacitanceduetolimitedDoS,leadingtoadecreaseintheinversioncarrierdensityatagivenoverdrivevoltage.ForexampleatVov=0.6V,theratioofquantumcapacitancetotheoxidecapacitanceforthenarrowesth110inanowireisaround5,whichleadstoanalmost20%decreaseintheCe.Thiseffectbecomeslesspronouncedforthewidesth110inanowireforareductionofapproximately7%.Ontheotherhand,theh100inanowireCq=Cgratioisrelativelyinsensitivetoscalingatavalueofaround13.5.Inthelimitingcase,thiseffectisnotseenforabulk2Dinversiondeviceforwhichthesinglegateoxidecapacitanceismuchlowerthantheverylargequantumcapacitanceduetoplentyofavailable2Ddensity-of-states. ThedecreaseinDoSleadstoadrasticenhancementinbandstructurecarriervelocity,incontrasttothereductionofinversioncarrierdensity.InordertocompensateforthelimitedDoS,carriersoccupystatesatmuchhigherenergiesinthebandstructure,wheretheFermivelocityishigh.Thisresultsinasignicantincreaseinaveragecarrierbandstructurevelocity.SeeninFigure 5-4 (a),theaverageholevelocityfor 69

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Figure5-4. Thesizeandorientationdependentunstrainedballisticaverageinjectionvelocity(a)andholedraincurrent(b)areshownatanequivalent2DcarrierdensityofN2D,eq1.21013cm)]TJ /F9 7.97 Tf 6.58 0 Td[(2andanoverdrivevoltageof0.6V,respectively.Forafaircomparison,thedraincurrentisnormalizedwithtransistorwidth(i.e.circumference).c2012,IEEE. h110inanowiresmorethandoublesasthesizechangesfromthewidest7nmdevice(0.89107cm=s)tothenarrowest3nmdevice(1.83107cm=s).SincethenegativeeffectofthelimitedDoSintotalcarrierdensityismostlybufferedwiththerelativelysmalloxidecapacitance,theincreaseinFermivelocitybecomesthedominantfactorindeterminingtheeffectofdecreasingDoSforholecurrent.Consequently,theballisticcurrent,calculatedastheproductofvelocityandcarrierdensity,increaseswithdecreasingDoS.TheballisticholecurrentiscalculatedatjVovj=0.6VandshowninFig 5-4 (b).InaccordancewiththemodicationoftheDoS,sizeandorientationhavedirectimpactonthecurrenttransportcharacteristicsofthenanowires.Whiletheh110iNWsshowdrasticincreaseinbothballisticcurrentandvelocityasthedevicebecomesnarrower,therotateddevicesremainrelativelyinsensitivetocrosssectionscaling.Thesetrendsareinaccordancewiththeself-consistentresultsof[ 108 ]andjustifythechoiceofthecompactelectrostaticsmodelmethodusedinthisworktoincreasethecomputationalefciency. 70

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5.3.2StrainedDevices Inthissubsection,theeffectsofstrainonholetransportinballisticnanowiredevicesareinvestigatedthroughthemodicationsonnanowirebandstructureandDoS.UsingtheintegrandtermsinEq. 5 and 5 ,wecanexaminethe1Dbandstructureholedistribution(P(E,k))andthe1Dbandstructureholecurrentdistribution(J(E,k))underbothstrainedandunstrainedconditions. Thestrain-inducedmodicationsofthebandstructureandDoSaresimulatedforthestrained3nmX3nmh100inanowiredevice.ThebandstructureandP(E,k)areshownfordifferentstrainconditionsinFigure 5-5 ,atanoverdrivevoltageof0.6V.OnlythepositivewavevectorportionofthecompletebandstructureisplottedsincetheE-kdispersionrelationisanevenfunctionofwavevectork.Asseenfromthegure,uniaxialcompressionincreasestheenergyseparationbetweenthetopandthesecondvalencebands,whilewarpingthetopbandintoasharperdispersionrelation.Ontheotherhand,tensionlowersthebandsplittingenergyandshapesthetopbandintoaatterdispersion.Consequently,the1DDoSaroundthevalencebandmaximumdecreaseswithcompressionandincreaseswithtension.Thisleadstoalowerinversioncarrierdensityforacompressivelystraineddevice,withholesmostlylocalizingnearthelightmassregionatthetopvalenceband.Contrarily,fortensilestresseddevices,theincreaseinDoSleadstohigherinversioncarrierdensitywhicharenowmostlyresidingataatterbandstructuredispersion.Hence,uniaxialstressleadstooppositesignmodicationsoftotalholecarrierdensityandbandstructurevelocityin1Dnanowires.ThiseffectisnotobservedinconventionalMOSFETs,forwhichthechangesininversioncarrierdensityduetotheuniaxialstrain-inducedmodicationsinDoSisshowntobenegligibleforbulk2Dinversiondevices[ 127 ]. Usingtheproductoftheholedistributionandthe(E,k)atagivenpointinthestrainedbandstructure,wecaninvestigatethestrainmodicationoftheholecurrent.Theeffectofstrainon3nmX3nmh100inanowireJ(E,k)canbeseeninFigure 5-6 71

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Figure5-5. (Coloronline)Thebandstructureholedistributionisshownfor3nmX3nmh100inanowireunder(a)0.8%tension,(b)unstrainedand(c)0.8%compressionconditionsatjVovj=0.6V.c2012,IEEE. Comparedtotheunstrainedcase,under0.8%compression,theincreaseintheportionoftotalcarriersresidingatthesteepsloperegionofthetopvalencebandresultsinmorethan8%increaseincurrent,despitethereductionofinversioncarrierdensitybyanamountof3%.Ontheotherhandfortensilestrain,anincreaseof3%inholedensityisnotenoughtocompensateforthelossingroupvelocityatthemuchattertopregionofthevalenceband,leadingtoan8%lossinballisticholecurrent.Thus,theeffectofstrain-inducedreductioninDoSforballisticinjectionvelocityimprovementintrue1DnanowiresisinaccordancewiththeexpectationsstatedinRef.[ 106 ]. Similartotheh100idevices,compressivestressisalsobenecialforholetransportinh110inanowires.Aswehavementionedbefore,evenintheunstrainedcase,theh110inanowirebandstructureismuchmoredesirableforholetransportthannanowirespatternedintheh100idirection.Additionally,h110inanowiresaremoresensitivetouniaxialstraincomparedtoh100idevices.Thisisexplainedbythegreaterdeformationofthecrystalsymmetrybytheshearcomponentoftheh110istraintensor,leadingtosignicantchangesinthebandstructure[ 2 ].Thestraineffectson3nmX3nmh110i 72

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Figure5-6. (Coloronline)Thebandstructurecurrentdistributionisshownfor3nmX3nmh100inanowireunder(a)0.8%tension,(b)unstrainedand(c)0.8%compressionconditionsatjVovj=0.6V.c2012,IEEE. nanowireP(E,k)andJ(E,k)areshowninFigure 5-7 andFigure 5-8 .Under0.8%uniaxialcompression,almostalloftheholesarerepopulatedtothetopmostvalencebandwithdecreasedeffectivemassduetobandwarping.Theresultisa14%increaseinballisticcurrent.Fordeviceswith0.8%tensilestrain,holesoccupytheheaviermassregionsofthebandstructure,leadingtoatotalcurrentdecreaseofalmost20%. InordertonormalizethedeviceperformanceenhancementforNWswithdifferentsizeandorientations,thegaugefactor(GF)iscalculatedforalldevicesusingEq. 5 .TheGF,normalizedchangeinresistanceperunitstrain,forasemiconductorisobtainedby, GF=Rs)]TJ /F4 11.955 Tf 11.95 0 Td[(R0 R01 =I0)]TJ /F4 11.955 Tf 11.95 0 Td[(Is Is1 ,(5) whereRsandR0arethestrainedandunstrainedchannelresistances,whileIsandI0arethestrainedandunstraineddraincurrentsrespectively.Thegaugefactorsfornanowiresarecalculatedat0.8%compressionandVov=0.6V,andshowninFigure 5-9 .Theh110iNWGFisfoundtobemuchlargercomparedtotheh100idevices,becausetheshearstraincomponentoftheh110iuniaxialstresseffectivelybreaksthe 73

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Figure5-7. (Coloronline)Thebandstructureholedistributionisshownfor3nmX3nmh110inanowireunder(a)0.8%tension,(b)unstrainedand(c)0.8%compressionconditionsatjVovj=0.6V.c2012,IEEE. Figure5-8. (Coloronline)Thebandstructurecurrentdistributionisshownfor3nmX3nmh110inanowireunder(a)0.8%tension,(b)unstrainedand(c)0.8%compressionconditionsatjVovj=0.6V.c2012,IEEE. crystalsymmetry.ItisalsoseenthattheGFforh110inanowiresdecreasesasthenanowirecrosssectiondimensionsshrink.Thisisexplainedbyamechanismsimilartothesmallerstrain-inducedenhancementof2DMOSFETelectronlow-eldmobilityathighinversioncarrierdensities.Bothincreasedspatialconnementanduniaxial 74

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compressionenhancestheh110iNWholetransportviaareductionintheDoS.Hence,thestrain-inducedcurrentimprovementforthenarrowest3nmh110inanowireislessthanwider5nmand7nmdevices.Ontheotherhand,thewidest7nmh100idevicehasacomparabletrade-offbetweenstrain-inducedinversioncarrierdensitylossandgroupvelocityincrease,andhencehasaverylowGF.Astheh100idevicesshrink,thevelocityenhancementincreaseswhilethecarrierdensitylossremainsalmostconstant,leadingtohigherGFfordownscaledh100inanowires. Inadditiontothedrivecurrentenhancementduetostraineffectsonthebandstructure,mechanicalstrainmayfurtherimprovetransistorperformanceifemployedduringnanowirefabricationtoreducesurfaceatomicdisorders.Itisobservedthroughmoleculardynamicsstudiesthatveryhighlevelsofradialcompressionismemorizedinthenanowirecore,asaresultofthethermaloxidationprocessestothinthenanowirechannel[ 128 ].Recently,itisshownthatthisradialcompressionleadstoalargescaleofspatialdisorderatthenearsurfaceatomiclocationsinnanowiressubjectedtothermaloxidation[ 129 ].Thisinreturnresultsinlocalizedstateswithinthebandsandincreasedcarriermass,especiallyfornarrownanowireswithhighsurfacetovolumeratio.Therefore,asignicantamountof(upto2times)reductioninballisticcurrentisexpectedinnanowiretransistorsduetoatomicdisorder[ 129 ].Inthiscase,awaferleveluniaxial4-pointbendingtechniquecanbeusedtoinduceuniaxialcompressionalongthenanowirechannelduringthethermaloxidationprocesstorelaxtheradialcompressionbuildup.Consequently,theatomicdisorderattheinterfaceandtherelatedcurrentdegradationmaybeminimized. 5.4Summary Theunderlyingphysicsofstraineffectsonballisticholetransportinnanowireeld-effecttransistorsareexplainedbythestrain-inducedmodicationofthedensity-of-the-statesforthersttime.Thestrainresponseofballisticp-typeNWsisinvestigatedasafunctionofchannelorientationandcrosssectiondimensions.Itisfoundthatthe 75

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Figure5-9. Sizedependentgaugefactorisshownforbothstraightandrotatedp-typenanowiresatanoverdrivevoltageof0.6V.c2012,IEEE. h110inanowiresrespondtouniaxialstrainmorethantheirh100icounterparts,astheshearstraincomponentinh110istressbreaksthecrystalsymmetryandleadstosignicantchangesinbandstructure.Alsoforh110inanowires,theeffectsofcompressivestrainandstructuralconnementareshowntobesimilar,whichresultsinadecreaseinthestrain-inducedenhancementofballisticholecurrentatsmallerdimensions.Itisalsoshownthatunlikethebulk2Dinversiondevices,thereductionofthenanowire1DDoSunderuniaxialcompressiondecreasesthetotalinversionholedensityandhasanegativeeffectonthetotalcurrentenhancement.Nevertheless,thehighunstrainedballisticdrivecurrentcapabilityandlargeimprovementwithuniaxialcompressionindicatethatnarrowh110inanowirep-typeeld-effecttransistorsarepotentialcandidatesforhigh-performancelogicdevices,whilewiderh110ideviceswithlargerGFareexpectedtobesuitableforsensorapplications.Finally,itissuggestedthatintroducingawaferleveluniaxialtensionduringthethermaloxidationprocessmayimprovethedrivecurrentofnanowiretransistorsbyreducingtheatomicdisordersattheinterface. 76

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CHAPTER6PHYSICALINSIGHTSONSTRAINENHANCEDSURFACEROUGHNESSLIMITEDELECTRONMOBILITYINSILICONINVERSIONLAYERS 6.1Motivation Thedeviationofthesilicon/oxideboundaryfromanidealinterfaceanditseffectsoncarriertransportinsiliconinversionlayershavebeenthoroughlyinvestigatedsincetheearly1980s'pioneeringworksbyGoodnicketal.[ 28 ]andAndoetal.[ 130 ].Thestatisticaldistributionofthesilicon/oxideinterfacepositionintroducesaperturbationintheself-consistentpotential,whichresultsinthescatteringofMOSFETinversioncarriersnearthesurface.Especiallyforcontemporaryplanardeviceswithveryhighlevelsofdoping(1017)]TJ /F3 11.955 Tf 12.51 0 Td[(1018cm)]TJ /F9 7.97 Tf 6.59 0 Td[(3)andoperatingathighinversioncarrierdensities(1013cm)]TJ /F9 7.97 Tf 6.59 0 Td[(2),theverticalelectriceldwhichconnesthecarriersinthequantumwellnearthesurfacegoesuptotheorderof1MV=cm.Attheseveryhigheldconditions,carriersresidesignicantlyclosetotheoxide,resultinginveryhighsurfaceroughnessscatteringrates.Infact,evenatroomtemperature,surfaceroughnessscatteringisthedominantscatteringmechanismthatlimitsmobilityatONstate[ 131 132 ].Therefore,understandingtheeffectsofmechanicalstressonthesurfaceroughnesslimitedcarriermobilityiscrucialtomodelandpredicttheperformanceofmoderndaystrainedlogicdevices. Onewaytoinvestigatethestraineffectsonsurfaceroughnesslimitedmobilityistomeasurethelow-eldeffectivemobilityatlowtemperaturesunderdifferentstrainconditions.Sinceatlowtemperaturesphononscatteringbecomeslesspronouncedandcarriersarefurtherconnedtothesilicon/oxideinterface,surfaceroughnessscatteringbecomesthedominantlimitingmechanismforlow-eldelectronmobility.Hence,theresultantstrain-inducedchangeinlineardraincurrentormobilitycanbemostlyattributedtothemodicationinsurfaceroughnesslimitedmobility.Theeffectsofstressonsurfaceroughnesslimitedmobilityhavebeensystematicallyinvestigatedforp-typesingle-gateMOSFETs[ 133 134 ].Itwasfoundthattheincreasedstrain-induced 77

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enhancementofholemobilityatlowertemperaturesisaresultofthegreaterimpactofbandwarpingonholeeffectivemassreduction.However,itisknownthatelectrostaticconnementinducedvalleysplittinginsiliconconductionbandleadstoalmostcompleterepopulationofelectronstolower2valleywithlightereffectivemass.Thusathighverticalelds,furtherstrain-inducedreductionofelectronmasscannotbethereasonofelectronmobilityenhancementalone. InordertounderstandthediscrepancybetweenthetheoreticalexpectationsandexperimentalresultsofsurfaceroughnessmobilityinstrainednMOSFETs,anadhocstrain-inducedmodicationofsurfacemorphologyhasbeenproposedbyFischettietal[ 29 ].Itisfoundunderbiaxialtensilestressthatbydecreasingthermsvalueoftheroughnessamplitudeandincreasingthecorrelationlengthoftheroughnessdistribution,themodelcanaccuratelypredicttheexperimentalresultsofenhancedelectronmobilityathighinversioncarrierdensity[ 29 ].Later,Bonnoetal.providedfurtherexperimentalsupporttotheadhocmodicationofthesurfaceroughnessviaAFMmeasurementsonepitaxially0.8%strainedSOIdevices[ 135 ].AdditionalexperimentalsupporttoFischetti'smodelhasbeenpublishedbyZhaoetal.throughthecharacterizationoftheeffectsofprocessinducedbiaxialstrainonsurfaceroughnessmobilityatcryogenictemperatures[ 136 137 ]. ToextendthesystematicalapproachemployedforpMOSFETs,wehavemeasuredthestrain-inducedlineardraincurrentenhancementasafunctionoftemperature,whileapplyingcontrolledlongitudinaluniaxialstressusinga4-pointbendingsetup.Also,tocharacterizethemorphologicalchangeswithstrain,asimilar4-pointbendingsetupisusedtocontrolthemagnitudeofappliedstressduringAFMmeasurements. Theunderlyingphysicsofincreaseduniaxialstrain-inducedenhancementoflineardraincurrentinnMOSFETsareinvestigatedbyaself-consistentMOSFETsimulatorandamobilitymodelusingKubo-Greenwoodformulawithvaryingsurfaceroughnessconditionsunderstress. 78

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Figure6-1. (a)Thecryogenictemperaturewaferbendingbendingsetupisshownontheleftand(b)thestressjigusedinthecryogenicstationisshownonright. 6.2Experimental Inordertoinvestigatethestresseffectsonsurfaceroughnesslimitedmobilityinn-typesiliconinversionlayers,thelineardraincurrentismeasuredunderuniaxialtensilestressconditionsatlowtemperatures,forwhichtheinuenceofphononscatteringisreduced.ThelowtemperaturestrainmeasurementswereconductedonacustombuiltLakeshoresystemscryogenicstationwithliquidnitrogen(77K)andheliumcapabilities(4K).ThecryogenicstationandthelowtemperaturestressjigareshowninFigure 6-1 Longchannel(Lgate=10m)nMOSFETlineardraincurrentismeasuredatVDS=50mVforatemperaturerangevaryingfrom293Kdownto80K.Atthesetemperatureconditions,uniaxialtensilestressisappliedandthestrain-inducedlineardraincurrentenhancementismeasuredatanoverdrivevoltageof0.5V.ThelineardraincurrentenhancementasafunctionoftemperatureisshowninFigure 6-2 .Atroomtemperature,thestrain-inducedlineardraincurrentenhancementismeasuredtobeequivalenttothepiezoresistancecoefcientof(100)/h110inMOSFETs[ 104 ].Asthetemperaturedecreasesdownclosetotheliquidnitrogenlevel,thestrainenhancementoflineardraincurrentmonotonicallyincreases. 79

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Figure6-2. ThestrainenhancementofnMOSFETlineardraincurrentisshownasafunctionofsampletemperaturefortwodifferentstressconditions. Figure6-3. (a)AFMsystemisshownwiththemechanicalbendingjig.(b)Themodicationoftheroughnessamplitudeisshownat200MPauniaxialtensilestress.[ 138 ] Atthesametime,atomicforcemicroscopy(AFM)scanofthebaresiliconsurfaceisdoneundermechanicalstressjustaftertheremovalofthenativeoxide[ 138 ].AsshowninFigure 6-3 ,itisfoundthatuniaxialtensilestressresultsinadecreaseintheroughnessamplitudeofthebaresiliconsurface. 80

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Figure6-4. (a)Double-gateoperationofundoped20nmthick(100)MOSFETconductionbandproleisshown.(b)Withhighdopingleveland100nmthicksilicon,eachgateisisolatedandoperatesasindividualsingle-gateMOSFETs. 6.3Simulation Inordertosimulatethestraineffectsonsurfaceroughnesslimitedmobilityinsiliconinversionlayers,weemployamodiedversionoftheself-consistentdoublegateFinFETsimulatordevelopedinChapter 4 .Bysettingtheseparationbetweentwogateslargerthanthecumulativedepletionwidthofbothgates,wecaneffectivelyisolatetheindividualinversionlayersundereachgateandobtainasinglegateoperationfortwoparalleldistinctMOSFETstructures.Theconductionbandprolesofthemetal/oxide/siliconheterostructureforsingle-gateMOSFETanddouble-gateFinFEToperationsareshowninFigure 6-4 .Forsingle-gateoperation,thepotentialdeepinsidethengoestozero,indicatingaclearseparationofeachinversionlayer.Ontheotherhandforthinnswithlowdopingdensity,theinversionlayersmergeatthecenterofthenandresultinanon-zeropotentialprolealongthenthickness. 6.3.1TemperatureDependence Thetemperatureeffectsarecapturedwithintheself-consistentSchrodinger-PoissonsimulatorthroughthevalueofthethermalvoltagekBT=qandthetemperature 81

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dependentsiliconbandgapenergy,bulkconduction,andvalencebandeffectivedensity-of-states.Thetemperaturedependentsiliconbandgapiscalculatedby[ 139 ], Eg(T)=1.17)]TJ /F3 11.955 Tf 11.96 0 Td[(4.7310)]TJ /F9 7.97 Tf 6.59 0 Td[(4T2 T+636.(6) Similarly,theconductionandvalencebandeffectivedensityofstatesasafunctionoflatticetemperaturearegivenas, Nc,e(T)=6.21015T3=2[cm)]TJ /F9 7.97 Tf 6.58 0 Td[(3] (6) Nv,e(T)=3.51015T3=2[cm)]TJ /F9 7.97 Tf 6.58 0 Td[(3]. UsingEq. 6 and 6 ,wecancalculatethetemperaturedependentintrinsiccarrierdensityinsilicon, ni(T)=p Nc,e(T)Nv,e(T)exp()]TJ /F4 11.955 Tf 9.3 0 Td[(Eg(T) 2kBT)[cm)]TJ /F9 7.97 Tf 6.59 0 Td[(3].(6) 6.3.2StrainModication Energybandsplittingandbandwarpingeffectsofmechanicalstressonthesiliconconductionbandiscalculatedusingadeformationpotentialbasedmodel[ 140 ].Inthismodel,aparabolicconductionbandvalleyE-kdispersionrelationisgivenby, E=~2 2(k2x mx+k2y my+k2z mz)+d(xx+yy+zz)+uii+mjkkjkk, (6) wheredenotesthedoublydegenerateX-Y-Zvalleys,sarethedeformationpotentials[ 141 ],isthemechanicalstrain,subscriptiisthedirectionalongthemajoraxisofthevalley,andnally,subscriptsj,kdenotetheout-of-planedirectionforthesamevalley[ 142 ].Usinga3x3straintensorasinEq. 5 ,wecancalculatethestrain-induced 82

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energyshiftsineachoftheX,YandZvalleysby, EX=d(XX+YY+ZZ)+uXX, (6) EY=d(XX+YY+ZZ)+uYY,EZ=d(XX+YY+ZZ)+uZZ. Duringeachpassoftheself-consistentloop,thestrain-inducedenergyshiftsofconductionbandvalleysaresuperposedonthesubbandenergiesobtainedfromtheSchrodinger'sequation.TheoccupancyofeachsubbandladderineachvalleyiscalculatedwiththesuperposedvalleyenergiesandtheresultingspatialcarrierdistributionisinputinthePoissonsolver.Withtheincorporationofthebandshifteffectsofmechanicalstrain,thestrainedeigenenergiesandeigenvectorsareself-consistentlycalculated. Similarly,theeffectofshearstraininducedbandwarpingisobtainedviathemjkkjkkterminEq. 6 .For(100)/h110iinversionlayers,thetensileshearstraindecreasesthetransportmassof2valley.Thebandwarpingdependenttransporteffectivemassof2isgivenby[ 143 ], m2,110=mt 1+XYmt,(6) whereisaconstantexperimentallyfoundtobe86.8=m0[ 144 ]. 6.3.3MobilityModel Oncetheself-consistentvalleyenergiesandwavefunctionsarefoundforthegivenbias,temperatureandstrainconditions,thelow-eldeffectiveelectronmobilityiscalculatedusingtheKubo-Greenwoodformula[ 145 146 ].Sinceweareinterestedinthestraineffectsathighinversioncarrierdensities,theCoulombscatteringpotentialofionizedimpuritiesandinterfacetrapsisassumedtobeeffectivelyscreenedbymobileelectrons.Hence,themechanismschosentocontributetoelectronscatteringareacousticphononscattering(ac),opticalphononscattering(op),andsurfaceroughness 83

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scattering(sr).Basedontheenergyandwavevectortransfercharacteristicsbetweenthescatteringpotentialandtheelectron,thetransitionofthecarriermayhappenwithinthesamevalley(intravalley)orbetweendifferentvalleys(intervalley).ThecumulativescatteringrateofdifferentmechanismsisfoundbytheMathiessen'sRule, 1 =1 1+1 2+1 3+...,(6) wheren'saremobilityvaluesresultingfromandlimitedbyindependentscatteringevents. 6.3.3.1Intravalleyphononscattering Electronstravelingthroughthesemiconductorchannelinteractwithquantizedlatticevibrations(phonons).Asaresultofthisinteraction,electronsmayscattertoadifferentstatewithinthesamevalleyviaanelasticintravalleyscatteringevent.Forcubicsemiconductors,elasticintravalleytransitionsviaopticalphononsareneglectedandonlyacousticphononsareconsidered[ 147 ].Theelasticnatureoftheacousticphononinteractionresultsinchangesinelectronwavevectorwithoutalteringthescatteredelectron'senergy.Duetotheshortwavevectorsoftheacousticbranchofthephonondispersion,acousticphononinteractionofelectronsonlyresultinscatteringwithinthesamevalley.However,scatteredelectronsmaytransferwithinthesubbandsoftheinitialvalley.Thus,acousticphononscatteringisanintravalley/intersubbandprocess.Theinuenceoftheacousticphononsonelectrontransportinsiliconinversionlayersismodeledusinganisotropicdeformationpotentialapproximationfortheelasticintravalleyscatteringprocess.Thephononnumberisfoundbytheequipartitionapproximation.Foranelectronattheithsubbandofvalley,theenergydependentacousticscatteringrateisthesummationofallpossiblescatteringeventstothejthsubbandofthesamevalley.Theintravalleyacousticphononscatteringrateisgivenby[ 148 ], 1 ,iac(E)=XjmdD2ackBT ~3s2lFi,jU(E)]TJ /F4 11.955 Tf 11.96 0 Td[(Ej),(6) 84

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wheremdisthedensityofstateseffectivemassforthevalley,Dacistheacousticdeformationpotential,andandslarethemassdensityandlongitudinalspeedofsoundofthecrystal,respectively.ThetermFi,jdenotestheelectron-phononwavefunctionoverlaportheformfactorofthescatteringeventandisgivenby, Fi,j=Z 2i,(z) 2j,(z)dz,(6) where i,and j,aretheenvelopefunctionsoftheithandjthsubbandsalongtheconnementaxis,U(E)istheHeavisidestepfunction,andEjistheenergyofthesubband. 6.3.3.2Intervalleyphononscattering Inadditiontointravalleytransitionsbyacousticphonons,electronsmayalsoscattertodifferentvalleys.Theintervalleytransitionsareassistedbylargewavevectoracousticandopticalphononsatrelativelyhighenergies.Thisleadstoinelasticscatteringofelectrons.Duringaninelasticscatteringevent,electronsmayloseenergywhilethelatticeemitsaphonon,orelectronsmayabsorbenergyfromthesuppressionofaphononwithinthelattice.Dependingontheangleoftheelectronwavevectorchange,intervalleyscatteringeventscanbef-typeorg-typetransitions[ 147 149 ].Transverseoptical,transverseacousticandlongitudinalacousticphononsmayassistanf-typetransition.Similarly,forag-typescatteringevent,longitudinaloptical,transverseacousticandlongitudinalacousticphononsareinvolved.Foranelectronattheithsubbandofvalley,theenergydependentintervalleyphononscatteringrateisthesummationofallpossiblescatteringeventstothejthsubbandofthevalleywiththeassistanceofthephononm.Theintervalleyphononscatteringrateisgivenby[ 150 ], 1 ,iiv(E)=XXjXmn,m,jdD2iv,m 2~Eiv,mF,i,j(Niv+1 21 2)1)]TJ /F4 11.955 Tf 11.96 0 Td[(f(EEiv,m) 1)]TJ /F4 11.955 Tf 11.95 0 Td[(f(E)U(EEiv,m)]TJ /F4 11.955 Tf 11.95 0 Td[(Ej), (6) 85

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wheren,isthedegeneracyleveloftheprocess,m,jdisthedensityofstateseffectivemassforthenalvalley,Div,mistheintervalleydeformationpotentialforphononm,Eiv,mistheenergyoftheintervalleyphononm,Niv,mistheBose-EinsteinoccupationnumberoftheintervalleyphononmodeofenergyEiv,m,(Niv,m+1 21 2)isthemodicationforphononemissionandabsorption,andU(E)istheHeavisidestepfunctionforagivenenergy.Theselectionrulesforallowedintervalleytransitionscanbefoundin[ 149 ]. 6.3.3.3Surfaceroughnessscattering Astheelectronsinsiliconinversionlayersareconnedtoaverynarrowquantumwellatthesilicon/SiO2interface,theirregularitiesatthecrystal-oxideboundarycreatesaperturbationtotheself-consistentpotential,whichleadstosurfaceroughnessscattering(SRS).Thesurfaceroughnessscatteringisanelasticprocesswhichresultsonlyinthemodicationoftheelectronwavevector.Hence,surfaceroughnessscatteringisanintravalley/intersubbandprocess. Foradeviationfromtheidealinterfaceboundary((r))atthecoordinater,thepotentialneartheinterfaceisgivenby[ 151 152 ], V(z+(r))=V(z)+(r)@V(z) @z,(6) whereV(z)istheunperturbedpotentialandzistheaxisnormaltothegateplane.UsingEq. 6 ,theperturbationpotentialatlocation(z,r)canbeapproximatedby, HSR(z,r)=e(r)E(z),(6) whereE(z)isthetransverseelectriceld.Then,thescatteringmatrixelementfromthsubbandtothsubbandisgivenby, jMi,j(q)j=e2Z i(z)E(z) j(z)dz2j(q)j2,(6) whereq=k)]TJ /F11 11.955 Tf 12.45 0 Td[(k'isthedifferencebetweentheinitialandscatteredwavevectors,and(q)istheFouriertransformofthe(r),whichisdescribedbystatisticalformulae 86

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ratherthandeterministicmodels[ 28 130 ].AsseeninEq. 6 ,theFouriertransformoftheautocovariancefunctionofthedeviationfromtheidealinterfaceisemployedforcarriertransportanalysisinthewavevectorspace.Withintheliterature,itiscommonlyacceptedthatthedeviationfromtheidealinterfacecanbeapproximatedbyeitheraGaussianoranexponentialdistributionfunctionwhoseFouriertransformisgivenby, j(q)j2=2mL2m (1+q2L2m=2)1+n,(6) wheremisthermsamplitudeofthesurfaceroughnessandLmisthecorrelationlengthofthedistribution.Theparameterndeterminesthecharacteristicsofthedistributionandisgenerallytakenasn=2[ 153 ]orn=1=2[ 152 ],wherethelattervalueresultsinawellagreedexponentialdistributionsuggestedbyAndo,FowlerandStern[ 28 ].UsingtheperturbationHamiltonian(HSR)andthewellacceptedexponentialroughnessdistribution,thesurfaceroughnessscatteringrateforanelectronintheithsubbandtothejthsubbandisgivenby[ 151 ], 1 ,iSR(kk)=m,jde2E2e2mL2m 2~3Z20d (q)1+L2mq2 23=2,(6) where(q)isthestaticdielectricfunctionwithscreeningeffectsfrommobilecarriersincluded.Theeffectsofscreeningisconsideredonlyforintrasubbandtransitionsandthescreenedstaticdielectricfunctionisgivenby, (q)=1+e2 2Si01 qm,jd ~2F(q),(6) forwhichF(q)isthescreeningoverlapfactorandiscalculatedthrough, F(q)=XiZdzZdz0j i(z)j2j i(z0)j2e()]TJ /F6 7.97 Tf 6.59 0 Td[(qjz)]TJ /F6 7.97 Tf 6.59 0 Td[(z0j).(6) Whileforbulksingle-gateMOSFETsitisshowninEq. 6 thatthesurfaceroughnessperturbationpotentialcanbeestimatedbyextendingtheidealboundary 87

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electriceldalongthedisplacementoftheroughinterface,thisapproachleadstoaninaccuraterepresentationoftheHSRforultra-thin-bodysingle-gateMOSFETs[ 152 ]anddouble-gateFinFETs[ 154 ].Inordertocalculatethesurfaceroughnessperturbationpotentialaccuratelyinnon-bulkdevices,theapproximationinEq. 6 isreplacedbythegeneralizeddenitionofthepotentialperturbationgivenby[ 152 ], HSR(z,r)=)]TJ /F4 11.955 Tf 9.29 0 Td[(e(r)Vm(z) m,(6) whereVm(z)isthedifferencebetweentheself-consistentpotentialsobtainedfortheidealinterfaceconditionandfortheconditionofidealinterfacedisplacedbym.Consequently,thesurfaceroughnessscatteringratefornon-bulkplanardevicesisnallygivenby, 1 ,iSR(kk)=m,jde22mL2m 2~3R i(z)Vm(z) m j(z)dz2 (6) XR20d (q)1+L2mq2 23=2. 6.4ResultsandDiscussion Theroomtemperature(300K)low-eldelectronmobilityiscalculatedforanunstrained10mlongsingle-gate,10mwide(100)/h110inMOSFETdevicewithNA=21018cm)]TJ /F9 7.97 Tf 6.59 0 Td[(3dopingdensity.Duringthemobilitycalculation,theacousticphonondeformationpotentialistakenas12eV[ 155 ]andtheintervalleyphononenergies,deformationpotentials,andselectionrulesaretakenfrom[ 149 ].Thesurfaceroughnessparametersaretakenasm=0.4nmandLm=2.4nmfrom[ 133 ],whereitisshownthatthesevaluesaccuratelytthepMOSFETstakenfromthesameCMOSwaferlotwiththenMOSFETsusedinthisstudy.ThesummaryofthemobilityparameterscanbeseeninTable 6-1 .Thesimulatedlow-eldeffectivemobilityanditsconstituentsareshownwithrespecttoeffectiveelectriceld(Ee)inFigure 6-5 .Astheeldincreases, 88

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Figure6-5. Theeffectivemobilityanditsconstituentsareplottedvs.effectiveelectriceldat300Kandnostress.Almostcompletelog-logdependenceoftotalmobilitytoeldindicatesthesignicanceofroughnessscatteringevenatroomtemperature. theintervalleyphononlimitedmobilitydecreaseswiththeincreaseintheoccupancyofthelowestsubbandofthe4valley,whichhasahigherintervalleyscatteringratethanthegroundsubbandof2.However,afurtherincreaseintheeldleadstolargersplittingof04and12,andtheoverallintervalleyphononscatteringdecreases.Atthesametime,ahighereldresultsinfurtherconnementofcarrierstoanarrowerquantumwell.Thisresultsinanincreasedformfactorandtheintravalleyacousticphononlimitedmobilitymonotonicallydecreasesathigherelds.Similarlyathigherelectricelds,thestrongerconnementofelectronstowardstheinterfacedrasticallydecreasesthesurfaceroughnessscatteringlimitedmobility.Infact,thetotaleffectivelow-eldmobilitycalculatedforagatebiasrangeof0.3Vto1.3Vshowsalmostlinearlog-logdependencewiththeeld,whichindicatesthesignicanceofsurfaceroughnessscatteringevenatroomtemperature. Theroomtemperatureelectronmobilityofa20nmthickundoped(NA,DG=21015cm)]TJ /F9 7.97 Tf 6.59 0 Td[(3)double-gateMOSFETwithmidgap-metal(m=4.62eV)isalsosimulated 89

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Table6-1. Thevaluesofphonondeformationpotentials[ 149 155 ],phononenergies[ 149 ]andsurfaceroughnessparameters[ 133 ]usedformobilitycalculationsareshown. ParameterSymbolValueUnits AcousticphonondeformationpotentialDac12eVSiliconcrystaldensity2.329g=cm3Siliconsoundvelocitysl9.037104cm=sf-typeTAphonondeformationpotentialD(f,TA)iv0.3108eV=cmf-typeLAphonondeformationpotentialD(f,LA)iv2108eV=cmf-typeTOphonondeformationpotentialD(f,TO)iv2108eV=cmg-typeTAphonondeformationpotentialD(g,TA)iv0.5108eV=cmg-typeLAphonondeformationpotentialD(g,LA)iv0.8108eV=cmg-typeLOphonondeformationpotentialD(g,LO)iv11108eV=cmf-typeTAphononenergyE(f,TA)iv19meVf-typeLAphononenergyE(f,LA)iv47.4meVf-typeTOphononenergyE(f,TO)iv59meVg-typeTAphononenergyE(g,TA)iv12meVg-typeLAphononenergyE(g,LA)iv18.5meVg-typeLOphononenergyE(g,LO)iv61.2meVSurfaceroughnessamplitudem0.4nmSurfaceroughnesscorrelationlengthLm2.4nm forgatebiasconditionsofupto1.3V.Thecomparisonofelectronmobilitywithrespecttothesingle-gatebulkMOSFETisseeninFigure 6-6 and 6-7 .Forthesameinversionelectrondensity,theundopeddouble-gatemobilityistwicethemobilityforthedopedbulksingle-gatedevice(Figure 6-6 ).AsseenintheFigure 6-7 ,theeffectiveelectriceldismuchlowerfortheundopeddouble-gatedevice,whichgreatlyincreasesthesurfaceroughnesslimitedmobility.Infact,uptoVG=0.8V,themainlimitingmechanismforthedouble-gatetransistoristhephononscattering.However,athigheldconditions(closetothedopedsingle-gateoperatingconditions),surfacescatteringbecomesthelimitingfactorinthedouble-gatedeviceandthemobilitydecreaseslinearlywitheldinthelog-logscale.Still,theslopeofthelineardecreaseinthelog-logscaleisslightlylower 90

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Figure6-6. Electronmobilityvs.inversionelectrondensityisshownforabulksingle-gate(100)MOSFETwithdopingdensityofNA,SG=2x1018cm)]TJ /F9 7.97 Tf 6.59 0 Td[(2andfora20nmthickdouble-gate(100)MOSFETwithdopingdensityofNA,DG=2x1015cm)]TJ /F9 7.97 Tf 6.58 0 Td[(2at300K. forthedouble-gatedevicecomparedtothebulksingle-gatedevice,whichisexplainedbythenon-zerodensityofinversionelectronsdeepinsidethenawayfromtheroughsurface. Thesingle-gateMOSFETeffectivelow-eldmobilityissimulatedforatemperaturerangeof400Kto100Kwithdecrementsof50KandisshowninFigure 6-8 .Atelevatedtemperatures,independentoftheverticalelectriceld,theincreaseinphononscatteringdegradestheelectronmobilitysignicantly.Thetotalphononlimitedmobilityincreasesfrom500cm2=V.sectoaround4000cm2=V.secasthetemperaturedecreasesfrom400Kto100K.Hence,thesurfaceroughnessscatteringlimitedmobilitybecomesthedominantscatteringmechanismforlowtemperatures.The100Ksurfaceroughnesslimitedmobilityatlowgatebias(Ee=0.8MV=cm)isaround420cm2=V.secandsetsthelevelofthetotalmobilityatthistemperature.Sincethecarriersspreaddeepintothebulkathightemperatures,thesurfaceroughnesslimitedmobilityatlowgatebiasincreases60%to690cm2=V.secat400K.Howeverathigheffectiveverticalelds,the 91

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Figure6-7. Electronmobilityvs.effectiveelectriceldisshownforabulksingle-gate(100)MOSFETwithdopingdensityofNA,SG=2x1018cm)]TJ /F9 7.97 Tf 6.58 0 Td[(2andfora20nmthickdouble-gate(100)MOSFETwithdopingdensityofNA,DG=2x1015cm)]TJ /F9 7.97 Tf 6.59 0 Td[(2at300K. temperaturedependenceofsurfaceroughnessscatteringisnegatedsincethestrongelectriceldbringselectronsclosetotheinterfaceatalltemperatures.RoughlyaroundEe=1.4MV=cm,thesurfaceroughnesslimitedmobilitybecomesalmostthesameforalltemperatures.Consequentlyathighgatebias(higherelds),thesurfaceroughnesslimitedmobilitysetsthelimitforthetotalmobilityandthetemperaturedependenceofthephononlimitedmobilitydenesthedifferencebetweenthetotalmobilityatdifferenttemperatures. Thetemperaturedependenceoftheinversionelectrondensityandthenormalizedlineardraincurrentvs.gatevoltagearesimulatedandshowninFigure 6-9 andFigure 6-10 .Forthesamedopingdensityandgateworkfunction,thethresholdvoltageshiftduetotemperatureisfoundtobearound)]TJ /F3 11.955 Tf 9.3 0 Td[(0.6mV=K,whichisconsistentwiththeliterature[ 156 ].ThenormalizedlineardraincurrentatVDS=0.05Viscalculatedby, Ilin W=qNinvVDS L.(6) 92

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Figure6-8. Simulatedsingle-gateMOSFETlow-eldelectronmobilityvs.gatevoltageisshownforatemperaturerangeof400Kto100K. Asexpected,atlowtemperatures,thethresholdvoltageisshiftedtohighervalues,andthelineardraincurrentexhibitsasteeperincreasecomparedtothatatelevatedtemperatures. Thenormalizedstrain-inducedchangeinlineardraincurrentissimulatedat300Kforauniaxialstressrangeof75MPatensionto75MPacompression,withouttheinclusionofanadhocmodicationofthesurfaceroughnessparameters.Thehydrostaticanduniaxialdeformationpotentialsforstraineffectsonconductionbandstructurearetakenfromthepseudo-potentialcalculationsof[ 140 ].In[ 140 ],theshearstraindeformationpotentialisempiricallychosentomatchthenMOSFETpiezoresistancecoefcient,whichmayovershadowthepossibleeffectsofsurfaceroughnessmorphologymodicationandattributetheenhancedmobilitychangetoalargerbandwarpinginducedeffectivemassreduction.Thus,wehavemodeledthebandwarpinginducedmassreductionusingtheexperimentalresultsofHenseletal.[ 144 ],asshowninEq. 6 .ThesimulatedchangeinchannelresistanceatEe=1MV=cmisshowninFigure 6-11 ,alongwiththeexpectedchangecalculated 93

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Figure6-9. ForthesinglegateMOSFET,simulatedinversioncarrierdensityvs.gatevoltageisshownfortemperaturesrangingfrom400Kto100Kwithdecrementsof50K.Thethresholdvoltageshiftduetotemperatureisfoundtobearound-0.6mV/K. Figure6-10. Simulatedsingle-gateMOSFETlineardraincurrentvs.gatevoltageisshownforatemperaturerangeof400Kto100K. 94

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Figure6-11. Simulatedsingle-gatenMOSFETnormalizedchannelresistancechangevs.uniaxialstress(symbols)andtheexpectedchangefromnMOSFET-coefcient(line)areshownforEe=1MV=cm.Forthesimulation,adhocmodicationofsurfacemorphologyisnotincluded.Thesimulated-coefcient()]TJ /F3 11.955 Tf 9.3 0 Td[(2210)]TJ /F9 7.97 Tf 6.58 0 Td[(11Pa)]TJ /F9 7.97 Tf 6.58 0 Td[(1)issmallerthanexpected()]TJ /F3 11.955 Tf 9.3 0 Td[(3210)]TJ /F9 7.97 Tf 6.59 0 Td[(11Pa)]TJ /F9 7.97 Tf 6.59 0 Td[(1). bythenMOSFET-coefcient[ 104 ].Withoutanadhocmodicationofthesurfacemorphologywithstrain,thesimulated-coefcient()]TJ /F3 11.955 Tf 9.3 0 Td[(2210)]TJ /F9 7.97 Tf 6.58 0 Td[(11Pa)]TJ /F9 7.97 Tf 6.58 0 Td[(1)isfoundtobesmallerthantheexpectedvalueof)]TJ /F3 11.955 Tf 9.3 0 Td[(3210)]TJ /F9 7.97 Tf 6.59 0 Td[(11Pa)]TJ /F9 7.97 Tf 6.59 0 Td[(1.Consequently,amodicationofthesurfaceroughnessparametersundermechanicalstressisnecessarytounderstandthegoverningphysicsofstrain-inducedenhancementofelectronmobilityinsiliconinversionlayers. Tomatchtheexperimentallyextractedpiezoresistancecoefcient,wehavesimulatedalargematrixofandLpairsforthreestressconditions(25,50and75MPa).Onceacloseagreementisobtainedforagivenstresscondition,theadhocchangesinthesurfacemorphologyislinearlyinterpolatedtootherstressconditionsandthe-coefcientsarecalculatedundertheseconditions.Forsimulationsofallthreestressconditions,a1.7%decreaseinanda1.7%increaseinLper100MPauniaxialtensionresultedinaconstant-coefcientof)]TJ /F3 11.955 Tf 9.3 0 Td[(3210)]TJ /F9 7.97 Tf 6.58 0 Td[(11Pa)]TJ /F9 7.97 Tf 6.58 0 Td[(1.Figure 6-12 showsthedetails 95

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Figure6-12. Thesimulationresultsareshownforthe75MPacondition.Forthesimulatedunstrainedlineardraincurrentshownwithblueline,theexpecteddraincurrentat75MPausingtheexperimental-coefcientisshownwithsymbols.Thesimulatedcurrentat75MPawithandwithoutanadhocstrain-inducedsurfacemorphologyareshownwithblackandredlines. ofthesimulationresultsforthe75MPacondition.Theunstrainedlineardraincurrent(blueline)andtheexpected75MPadraincurrentusingtheexperimental-coefcient(symbols)areshown.Simulatedcurrent-voltagecurvesat75MPawithandwithoutanadhocsurfacemorphologychangeareshownwithblackandredcurves.Whiletheabsenceofthesurfacemodicationleadstoanunderestimateddraincurrentat75MPa,theagreementbetweentheexpectedandsimulatedcurrentisverycloseusingamodiedsetofandLparameters.ThenormalizederrorintheadhocttingisshowninFigure 6-13 forallthreestressconditions,usinga-1.7%/100MPachangeinand+1.7%/100MPachangeinL.Theerrorinallstressconditionsislessthan0.05%atthestronginversionoperation. 96

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Figure6-13. Normalizedadhocttingerrorisshownfor25MPa,50MPaand75MPaconditionsforagivengatevoltage. Thettedvalueofa1.7%magnitudechangeinbothandLper100MPauniaxialtensionisquitelarge.Infact,itisalmostequivalenttothesiliconYoung'smodulusloweredto6GPa.However,theAFMmeasurementscarriedoutinourgroupalsoindicateanequivalentreductionof3.4%for200MPauniaxialtension[ 138 ].Recently,asimilarAFMdataontheroughnessofsiliconsurfaceswithuniaxialtensionispublished,claiminga1.9%decreaseinper100MPaofuniaxialtension[ 157 ].Althoughtheseindependentresultsareconsistent,thesuggestedchangeinthelatticeforthegivenstresslevelsisenormous.However,ithasbeenshownthroughmoleculardynamicssimulationsthattheYoung'smodulusofsiliconnanowiresdecreasessignicantlyatverysmallthicknessesduetoreductionofcrystalrigidity[ 58 ].Therefore,asimilarlocalreductioninthesiliconYoung'smoduluswithinacoupleofmonolayersneartheoxideboundarycouldbethereasonfortheobservedstrain-inducedsurfacemorphologychange. 6.5Summary TheexperimentalevidencefromelectricalandmaterialscharacterizationsuggestsreducedinterfaceroughnessattheSi=SiO2boundary.Byusingaself-consistent 97

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Schrodinger-Poissonsimulatorandanadhocassumptionofstrain-inducedsurfacemorphologychange,wehaveshownthattheimprovedelectrontransportseenathighinversioncarrierdensitiesandtensilestressislikelytobearesultofasmootherSi=SiO2interface.Whiletheclaimedchangeinthesurfacemorphologyisquitelargeforthegivenamountofmechanicalstress,consistentresultsareobservedfrombothourgroupandotherresearchersintheliterature.ApossiblelowerYoung'sModulusatthesiliconcrystalsurfaceisexpectedtobethereasonfortheenhancedstrain-inducedmorphologychange,andthusthecarriertransportathighinversioncarrierdensities. 98

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CHAPTER7UNIFIEDQUASI-BALLISTICCOMPACTMODELFORSTRAINEDNANOSCALEMOSFETSUSINGANUPDATEDONE-FLUXTHEORY 7.1Motivation TheincessantpursuitofMoore'slaw[ 158 ][ 159 ]andtransistorscalingoverthepastfourdecadeshasledthesemiconductorindustrytowardshighperformancetransistorarchitecturesatsub-100nmchannellengthstoday[ 160 ].Asdevicedimensionsapproachtechnologicalandfundamentallimits[ 161 ],asolidunderstandingofthecarriertransportmechanismisofparamountimportance.Recently,therehasbeensignicantinterestinre-evaluatingandre-examiningthetraditionalcarriertransportmodelsinMOSFETstowardscapturingandexplainingoff-equilibriumtransportphenomenalikevelocityovershoot[ 162 ][ 163 ]observedexperimentally.Itisofinteresttonotethatthisefforthasbeenongoingatalllevelsofmodelingnumerical,empirical,compactandphysics-based.Theconictingrequirementsofaccuracy,generalityandcomputationalaccuracyonlymakesthistaskmoredifcultespeciallywithshrinkingchannellengths,wherethecarriertransportisthoughttobecomemoreandmoreballistic. Carriertransportinsemiconductordevicesisanalyzedusingawidespectrumofapproaches.TheserangefromthemacroscopicDrift-Diffusion(DD)[ 164 ]approach,tothesemi-classicalHydrodynamic(HD)andenergy-transportmodels[ 165 ]employinghighermomentsoftheBoltzmannTransportEquation(BTE),totheveryrigorousnumericalsolutionsoftheBTEusingMonteCarlotechniques[ 166 ][ 167 ].However,inthelastdecade,methodsbasedonMcKelvey/Shockley'salternativeone-uxapproach[ 168 ][ 169 ][ 170 ][ 171 ]haveemergedasasimpliedalternativeforunderstandingthephysicsoftransportinnanoscaledevices.Especially,theone-uxtheorybasedLundstrommodel(LM)[ 172 ]hasfoundalotoftractionowingtoitssimpleexpressionsgivingcalculatedcurrentsclosetothosepredictedbyMontecarlomethods. BoththeDDandone-uxmethodshavebeenusedwellbeyondtheirnominalrangeofapplicability,andthereasonforthissuccesswasinvestigatedbyLundstrom 99

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andco-workers[ 173 ].Theyreportedthattheone-uxtheoryisstillapplicableinanalyzingtransport,eventhoughtheassumptionofcarriersmovingatequilibriumthermalvelocityneednotnecessarilyhold.Theyattributedthistothepresenceofathecurrent-limitingbottleneckinatransistor(controlledbyagateorabase)andthatthisbottleneckregionispresentinalow-eldregionadequatelydescribedbytheDD/one-uxapproaches.TheLMadoptsatransmissionpictureofthetransistor.Inthispicture,carriersareinjectedfromathermalequilibriumsourcereservoir,acrossapotentialenergybarrier,intothechannelandnallyexitthroughthedrain.Theheightofthepotentialbarrierismodulatedbytheappliedvoltageongatethatiselectrostaticallycoupledtothechannel.Thoughcurrentowisdependentonthevoltageappliedonthedrain,thedensityofcarriersinthechannelaredeterminedbythedeviceelectrostaticsthatcomeintoplaytomaintainchargebalance. Usingtheone-uxtheoryapproach,theLMmakesitpossibletoquantifydeviceperformanceintermsoftwokeyparametersnamely,theballisticefciencyB(ameasureofhowthecurrentinagivendevicecomparestothatofanideal,scatteringfreeballistictransistorofthesamedimensions)andthecarrierinjectionvelocityvinj(theaveragevelocityofthecarriersatthetopofthesource-channelpotentialbarrierinthesamedevice).Severalnumerical[ 174 ][ 175 ][ 176 ][ 177 ][ 178 ]andexperimentalstudies[ 179 ][ 180 ][ 181 ][ 182 ][ 183 ][ 184 ][ 185 ][ 186 ][ 187 ][ 188 ][ 189 ]thatsupporttheoveralltrendsdiscussedintheLMhavebeenpublished.Someauthorshavealsoquestioned[ 190 ][ 191 ]thevalidityoftheLM,pointingoutafewunderlyingtheoreticalassumptions.Recently,Natoriproposedanewsolution[ 192 ][ 193 ]forunderstandingthehigh-eldtransportinbulksemiconductors,thatcouldbeapplicableforunderstandingquasi-ballistictransport. Sinceitsintroductionin2004[ 194 ],uniaxialstresshasbeenaprovenindustrytechniqueforenhancingtransistorperformance.Thebenetsofuniaxialstresscontinuestobeexploitedforcurrenttechnologies[ 195 ]andisprojectedtobean 100

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importantfactorevenatsub-10nmchannellengths[ 196 ].Whiletheeffectsofstressoncarriermobilityhasbeenwellstudied,understandingitsimpactonquasi-ballistictransportislistedasoneofthechallengestheITRS[ 197 ]foremergingdevices.Therehavealsobeenseveralexperimentalworkspublished,forexample[ 198 ][ 199 ][ 200 ][ 201 ],thatattempttoextendtheLMtowardsexplainingtheexperimentallyobservedresultsontheballisticefciencyofstraineddevices.MostoftheseworkssimplyextendtheLMempiricallytoexplaintheobservedeffects,anditisdifculttogetaclearunderstandingofthestraineffects.Thisisbecauseofthreereasons(1)fundamentalassumptionswithintheLMframeworkforhigheldtransport(2)inherenteffectofstrainoncarriermassandcarrierscattering(3)couplednatureofcarriertransportanddeviceelectrostatics. Thepurposeofthischapteristwofold:(1)AthoroughreviewandanalysisoftheunderlyingassumptionsandshortcomingsintheLMandLMbasedmethodsforevaluatingtheballisticefciencyofananoscaleMOSFET(2)Presentinganupdated,surface-potentialbased,uniedmodeloftransmissioncoefcientusinganenergyaveragedone-uxtheory.Thistransmissioncoefcientcouldtobeusedtobetterstudytheeffectofstrainincomparisontoexistingmethods. 7.2Fundamentals 7.2.1Natori'sOriginalBallisticTransportModel In1994,Natori[ 202 ]rstproposedandstudiedtransportinacompletelyballisticMOSFET.athinSilmconnectedtotworeservoirsofcarriers(sourceanddrain)oneitherside.ThegateelectrodeiselectricallyseparatedfromtheSilmbyadielectric.TheSilmisleftundopedtoreducedscatteringinthechannel.Inthetransportmodeldescribed,theMOSFETIVrelationisexpressedintermselementaryparameterswithoutanydependenceonthecarriermobilitybecauseoftheabsenceofscattering.Thecurrentisindependentofthechannellengthandisproportionaltothechannel 101

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Figure7-1. SchematicofconductionorrstsubbandprolealongthelengthofaMOSFETchannelfordevicebiasedinsaturationconditions. width.ThecurrentvaluesaturatesasthedrainvoltageisincreasedandthelinearandsaturationoperationsarespeciedasinaconventionalMOSFET. Forpurelyballistictransport,modelingthetransportinvolvesthestudyofcarriertransmissionoverthesource-to-drainpotentialbarrier(Fig.( 7-1 )).ElectronspossessingenergieshigherthanthebarrierheightEmaxaretransmittedfromsourcereservoirintothechannelbythermionicemission.Similarphenomenonhappensatthedrainreservoirhowever,onceabiasisappliedatthedrain,thebarrierheightforthedraininjectedcarriersbecomesreallyhighasshowninthegure,anddraininjectionissuppressed.ThestartingassumptionsmadebyNatoriare: 1. ChannelpotentialvariesgraduallyalongthechannellengthLfromahighvalueatthesourcetoalowvalueatthedrain.AmaximumvalueEmaxoccursatx=xmax. 2. ThecarriersareconnedalongthechannelwidthWbythesteepbarrierattheedges.Thepotentialprolealongthewidthisapproximatedbyasquarepotentialwellforchannelwidthslargerthantherangeofpotentialvariation. 3. Carrierconnement(y-direction)takesplaceduetoasharptriangularpotentialwellwiththeelectronicstatesexpressedbydiscreteenergylevels,similartoaregularMOSFET. 102

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NatorialsoderivedacompactequationfortheI)]TJ /F4 11.955 Tf 11.95 0 Td[(Vrelation: ID=Wp 2q(KBT)3=2Mvmt 2~2F1=2(u))-221(F1=2(u)]TJ /F4 11.955 Tf 11.96 0 Td[(vD)(7) wheremtisthetransverseeffectivemassoftheelectron,Mv=(lowestvalleydegeneracy)(totalpopulation/populationoflowestlevels).F1=2istheFermi-Diracintegraloforder1/2denedas:F1=2(x)=Z10p xdy 1+exp(y)]TJ /F4 11.955 Tf 11.96 0 Td[(x) (7) TheotherparametersinEq.( 7 )aredenedasbelow:u=ln[((1+exp(vD))2+4exp(vD)exp()]TJ /F3 11.955 Tf 11.95 0 Td[(1))0.5)]TJ /F3 11.955 Tf 11.96 0 Td[((1+exp(vD))])]TJ /F3 11.955 Tf 11.95 0 Td[(ln(2)vD=qVD KT=2~2 qKBTmtMv(CG(VG)]TJ /F4 11.955 Tf 11.96 0 Td[(VTH)) (7) Sincethismodeldescribespurelyballistictransportwithnoscatteringofanykindinsidethechannel,theresultingcurrentisthemaximumcurrentobtainableforaparticulargeometry.Itiswellknownthattransportinsub-50nmMOSFETsarequasi-ballisticandNatori'soriginalmodel,therefore,isnotdirectlyapplicablefordescribingtransportwhereonlyafractionofthecarriersreachthedrainwithoutundergoingscattering. 7.2.2CurrentControlinNatori'sModel InconventionalMOSFETtheory,thedraincurrentisgovernedbythecarriervelocityinthechannel.Thiscarriervelocityisexpressedasaproductofthelateralelectriceldandthecarriermobilitywhentheelectriceldisnotaslarge.Whentheelectriceldisincreased,however,thecarriervelocitysaturatestoavaluearound107cm/sduetoenergyrelaxationviaopticalphononscattering.InthefullyballisticMOSFETsinNatori'sdiscussion,carriersinthechannelarefreefromscatteringandallcarrierspropagating 103

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towardsthedrain,passingthroughthebottleneckaroundxmax,reachthedrainwithoutscatteringbacktothesource,ifthereectionatthedrainedgeisneglected.Thechannelcurrentisgovernedbythecurrentatthebottleneck.Incurrentsaturation,thenumberofcarriersinjectedfromthesourcetothebottleneckdominatesthetotalcurrent,andthecarriervelocityinthebackwardchannelhaslittletodowiththecurrentvalue.Fordegeneratecarriers,themeancarriervelocity(knownastheinjectionvelocity)atthebottleneckisexpressedas: vinj=8~p Q 3mtp qMv'8~p CG(VG)]TJ /F4 11.955 Tf 11.96 0 Td[(VTH) 3mtp qMv (7) Toquantifytheballistictransportindevices,NatorialsocameupwiththeBallisticEfciencyparameterBthatisdenedastheratiooftheexperimentalID(sat)tothetheoreticalballisticID(sat(bal)).Sincethecurrentdependsontheinjectionvelocity,wehaveB=vinj vinj(bal)=ID(sat) ID(sat(bal)) (7) NatoripredictedthatBwillincreaseasthecarrierscatteringisreducedandwilleventuallyreachunity,theoretically,whenballistictransportisrealized. 7.2.3One-uxTheory:Basics FollowingMcKelvey'sapproach[ 168 ][ 169 ][ 171 ],thebasicformalismoftheone-uxtheorycanbeunderstoodbyconsideringFigure 7-2 ,wheretheposition(andtime)dependentcarrieruxesareshownataslabofthicknessdxforapositivelychargedcarrier.Thechannelofatransistorcanbethoughtofacascadeofmanysuchslabs,withtheoutputofoneslabbecomingtheinputofthenextinacontinuousfashion.Forsimplicity,weignoreboththegeneration/recombinationprocessesinsidethedeviceandthetimedependenceofthevariousparameters.Theslabitselfcanbecharacterizedbyascatteringmatrix,whichlinkstheincomingandoutgoinguxesas 104

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264J+(x+dx)J)]TJ /F3 11.955 Tf 7.08 -4.34 Td[((x)375=264t+(x)r)]TJ /F3 11.955 Tf 7.09 -4.34 Td[((x)r+(x)t)]TJ /F3 11.955 Tf 7.08 -4.34 Td[((x)375264J+(x)J)]TJ /F3 11.955 Tf 7.09 -4.34 Td[((x+dx)375 Herer(x)andt(x)representtheratioofcarriersreectedandtransmittedinagivendirection.Iftheexpressionsforthesecoefcients(intermsofpositionandeld)andtheeldproleinchanneltheareknown,itispossibletocalculatethescatteringmatrixfortheentirechannel.Fromthatmatrix,theoveralltransmissioncoefcienttandreectioncoefcientsroftheentiredevicecanbecalculatedforafamilyofgate/drainvoltagesappliedtothedevice.Sinceweignoregenerationandrecombination,itfollowsthatt+r=1. Thismethodiscalledtheone-uxmethodbecauseonlyonecarrieruxisconsideredineachdirection.Inreality,sincecarriersenteringthechannelhaveadistributionofincomingenergies,eachenergylevelwillhaveitsownuxcomponentsandtherewouldbeintermixingoftheuxesbetweendifferentenergylevelsaftermomentumalteringscatteringevents.Theoretically,itispossibletodoaenergysummationoftheuxestoarriveatanoverallone-uxequation.AssadandLundstrom[ 173 ]formulatedaM-uxtheorythatreformulatedtheBTEasMcoupledcontinuityandDDequationsintheenergy-momentumspace.Aswouldbeexpected,thedetailsoftransportoverthebarrierusingthisapproachisentirelynumericalandwhileaccurate,itisalsoverycomplextoimplement.Itwasresolvedintheirworkthatthesimpleanalyticalexpressionsoftheone-uxtheory(discussedbelow)failinthehigheldconditions,andthattheyshouldbeusedforonlyrst-ordercalculationpurposes.However,owingtothesimplicityofapproach,thequasi-ballistictransportmodelformulatedlaterbyLundstromforunderstandingtheessentialphysicsofcarriertransport[ 203 ]utilizestheone-uxtheoryassumptionsandapproximations.Sinceourgoalistoimproveupontheone-uxtheorytowardsabetterunderstandingofstraineffectswithamoreaccurateanalytical 105

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Figure7-2. Incomingandoutgoingcarrieruxesinaslabofthicknessdx.r+(x)andr)]TJ /F3 11.955 Tf 7.09 -4.34 Td[((x)arethereecteduxesintheforward(paralleltoappliedeld)andreverse(antiparalleltoappliedeld)directionsrespectively.Carriergeneration/recombinationwithintheslabisignoredforsimplicity. expressionforthetransmissioncoefcient,wefocusontheone-uxtheoryapproachinthischapter. Thedifferentialequationsfortheuxmethod,ignoringrecombination/generation,areasbelow(detailedderivationcanbefoundin[ 168 ][ 170 ]). dJ+ dx=)]TJ /F4 11.955 Tf 9.3 0 Td[(r+J++r)]TJ /F4 11.955 Tf 7.09 -4.94 Td[(J)]TJ /F3 11.955 Tf 10.4 -4.94 Td[(=dJ)]TJ ET q .478 w 279.52 -373 m 299.49 -373 l S Q BT /F4 11.955 Tf 283.1 -384.19 Td[(dx (7) HereJ+andJ)]TJ /F1 11.955 Tf 10.41 -4.34 Td[(denotethepostion/elddependentpositive(left-directed)andnegative(right-directed)uxes.r+andr)]TJ /F1 11.955 Tf 10.41 -4.34 Td[(arethepostion/elddependentbackscatteringcoefcientsintheappropriatedirections. Thenetuxowingleftcanbewrittenas J=J+)]TJ /F4 11.955 Tf 11.95 0 Td[(J)]TJ /F1 11.955 Tf 180.75 -4.94 Td[((7) Wehavedroppedthesubscriptsforsimplicityi.e.J+(x,Ex)issimplywrittenasJ+.Exdenotestheelectriceldintheleftdirection. 106

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Theeffectivecarrierconcentrationisgivenby n=J+ v++J)]TJ ET q .478 w 256.08 -44.83 m 270.02 -44.83 l S Q BT /F4 11.955 Tf 256.08 -56.02 Td[(v)]TJ /F1 11.955 Tf 177.17 4.75 Td[((7) wherev+andv)]TJ /F1 11.955 Tf 10.41 -4.34 Td[(arethevelocitiesofthetwouxes.Assumingthateachoftheseisequaltotheequilibrium1-Dcarriervelocityofnon-degeneratesemiconductorwitheffectivemassm,givenby c=r 2kBT m(7) itisveryeasytoshowthat 2J=cnJ(7) Usingaboveequations,wecanderiveanexpressionforthenetuxJas J=)]TJ /F4 11.955 Tf 10.49 8.09 Td[(r+)]TJ /F4 11.955 Tf 11.96 0 Td[(r)]TJ ET q .478 w 183.39 -272.95 m 223.19 -272.95 l S Q BT /F4 11.955 Tf 183.39 -284.14 Td[(r++r)]TJ /F4 11.955 Tf 8.28 4.75 Td[(cn)]TJ /F4 11.955 Tf 29.73 8.09 Td[(c r++r)]TJ /F4 11.955 Tf 9.47 12.84 Td[(dn dx(7) Forsmallelectricelds,itisassumedthatthe(r+)]TJ /F4 11.955 Tf 12.38 0 Td[(r)]TJ /F3 11.955 Tf 7.09 -4.34 Td[()shouldbeproportionaltotheelectriceld(nearequilibriumconditions),andthat(r+)]TJ /F4 11.955 Tf 12.2 0 Td[(r)]TJ /F3 11.955 Tf 7.09 -4.34 Td[()beindependentoftheeld, r+)]TJ /F4 11.955 Tf 11.95 0 Td[(r)]TJ /F3 11.955 Tf 10.41 -4.94 Td[(=Ex(7) andtherefore,wecanwriteEq. 7 as J=)]TJ /F4 11.955 Tf 23.97 8.08 Td[(c r++r)]TJ /F4 11.955 Tf 8.28 4.75 Td[(nEx)]TJ /F4 11.955 Tf 29.73 8.08 Td[(c r++r)]TJ /F4 11.955 Tf 9.48 12.83 Td[(dn dx(7) ThisequationcanbewritteninafamiliarformsimilartotheDrift-Diffusionequation,ifwedenemobility-likeanddiffusion-constant-liketermsandD (x,Ex)=c r++r)]TJ ET BT /F1 11.955 Tf 426.8 -569.32 Td[((7a) D(x,Ex)=c r++r)]TJ ET BT /F1 11.955 Tf 426.8 -596.21 Td[((7b) D =1 (7c) 107

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toarriveat J=)]TJ /F5 11.955 Tf 9.3 0 Td[(nEx)]TJ /F4 11.955 Tf 11.95 0 Td[(Ddn dx(7) TheaboveequivalencebetweentheDDtheoryandtheone-uxequationswasdiscoveredbyShockleyin1962[ 170 ],wherehealsoobservedthatthediffusionconstantDofthecarrierstraversingainnitesimallysmalleld-freeregion(onethatissmallerthantheaveragemeanfreepath)isthesameasthatofthebulksemiconductor,D0. Iftheoneuxequationshavetobecompatiblewithintheframeworkofgeneralsemiconductortransportequations,itisclearthat limEx!0=0=q kBT (7a) limEx!0=0=0c 2r0=q kBT1 D0 (7b) where0,k0,D0denoteequilibriumquantities. McKelveyshowed([ 168 ],AppendixB)that r0=limEx!0r=3 4s(7) Heresdenotesthescatteringmeanfreepath.NotethatShockleyargues([ 170 ],Appendix1)thatMcKelvey'sexpressionforr0needsasmallcorrection. 7.2.4RevisitingtheLundstromModel Lundstromusedtheone-uxtheorytoarriveatacompactexpressionforthedraincurrentinaMOSFET.Westartthissectionwithadiscussionontheconceptofballisticmobilitywhichexplainsthelinkbetweentheone-uxtheoryandLundstrom'smodel. ThephenomenologicaleffectivemobilitythatistypicallyusedintheMOSFETI)]TJ /F4 11.955 Tf 12.06 0 Td[(Vequationswasthoughttobeindependentofchannellength.However,experimentalmeasurementofthisefromshortchannelMOSFETsisseentobeconsiderablysmallerthanthelongchannelmobility0.Shur[ 204 ]rstlinkedthisdegradationtothenitecarrieraccelerationtimeinthechannel.Toaccountforthis,heintroduceda 108

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Figure7-3. DenitionoftheKT-layerintheLundstromModel,showingtransmittedandbackscatteredcarriersnearthecriticalregionfortransport. ballisticmobilityB,whichhedenedas: B=qL mcvT(7) whereListhechannellength,mcistheconductivityeffectivemassandvT=p 2kBT=mcisthethermionicemissionvelocity(sameasthe1-Dvelocitycdescribedinthelastsection).TheshortchannelmobilitywasthenformulatedinaMattheissen'sruleformas: 1 e=1 B+1 0(7) ShurnotedthattheniteaccelerationtimeL=vTinEq.( 7 )describesthetimerequiredforacarriertotraveloveralengthLwiththenitevelocityvTgivenatthebeginningofthechannel.Shur'snalI)]TJ /F4 11.955 Tf 12.63 0 Td[(VexpressionissimplyanextensionoftheclassicDD-basedMOSFETequation.ID'W LCG1 B+1 0(VG)]TJ /F4 11.955 Tf 11.96 0 Td[(VTH)VD (7) 109

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WhentheratioB=0issmall,i.eforlongchanneldeviceswherethetransportisnotballisticatall,theaboveequationreducestothetraditionalDDequation. Inthelandmarkpaper[ 205 ]writtenin1997,Lundstromgaveare-interpretationofShur'sEq.( 7 )foraquasi-ballisticMOSFEToperatinginsaturationconditions.FollowingMcKelvey'sFlux-basedapproach,theMOSFETistreatedasacascadeofthreeregions:source-channel-drain.Thesourceisareservoirofthermalcarrierswhichinjectsauxofcarriersintothechanneloverapotentialbarrier.Thegatemodulatestheheightofthispotentialbarrier.Nearthebarrier,afractionrofthesourceuxbackscattersfromthechannelandre-entersthesource.Theremainingfraction1)]TJ /F4 11.955 Tf 12.48 0 Td[(rtransmitsandeventuallyentersthedrain.ThisisconceptuallyshowninFig.( 7.2.4 ). Itwasshownin[ 205 ]thatID(sat)canbecompactlyexpressedas: ID(sat)=Cox(VG)]TJ /F4 11.955 Tf 11.96 0 Td[(VTH)1)]TJ /F4 11.955 Tf 11.96 0 Td[(rsat 1+rsatvT(7) wherersatistheratioofthebackscatteredcarriers(i.e'backscatteringratio')undersaturationconditions.vtisthevelocityoftheforwardgoingelectrons(assumingnMOSFET)atthethetopofthesource-channelpotentialbarrier(i.ethe'virtualsource').Thebackscatteredelectronsareassumedtohavethisvelocityaswell.Thisvelocityissimplythe1Dthermalvelocityinthetransportdirection,givenbyr 2KBT mc.Theratio1)]TJ /F4 11.955 Tf 11.95 0 Td[(rsat 1+rsatisreferredtoastheBallisticEfciency,B.TheproductofBandvTiscalledtheinjectionvelocity.NotethatthisdenitionisslightlydifferentfromNatori'sdenition. TheappealofLundstrom'stheorystemsfromthesimplicityofEq.( 7 )thatdescribesthesaturationcurrent.FromMcKelvey'sscatteringtheory[ 168 ][ 170 ],the ID(sat) W=Qi(0)1)]TJ /F4 11.955 Tf 11.95 0 Td[(r 1+rvTF1=2(F) F0(F)8>>><>>>:1)]TJ 13.15 8.51 Td[(F1=2(F)]TJ /F4 11.955 Tf 11.95 0 Td[(UD) F1=2(F) 1+1)]TJ /F4 11.955 Tf 11.96 0 Td[(r 1+rF0(F)]TJ /F4 11.955 Tf 11.95 0 Td[(UD) F0(F)9>>>=>>>;(7) 110

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backscatteringratioisactuallywelldenedforzero(orverysmall)electriceldsas: r0=L L+(7) whereListhechannellengthisthemean-freepath(ormoreappropriatelythevelocity/momentumrelaxationlength)oftheelectronsjustnearthevirtualsource.Inthenon-degeneratelimit,theparameterisactuallyrelatedtotheloweld,longchannelmobilityas: =2KBT q0 vT(7) BasedonanearlierobservationbyPrice[ 206 ]thatanelectronwhichhastraveledafewtimes10)]TJ /F9 7.97 Tf 6.58 0 Td[(6cminaeldof104V/cm,sohasdescendedfartherthanthishaslittlechanceofgettingbackallthewayandwillalmostcertainlyreachthe[drain],(i.e.electronsthathavetraveledpastacriticaldistancefromthevirtualsourcecannotreturnbacktothesource)Lundstromspeculatedthatthebackscatteringratioinhigheldconditionscanbedenedas:rE=l l+ (7) Herelissomecriticaldimensionwhichisassumedtobethewidthoftheso-calledKBT-layerin[ 205 ].Torstorder,lKBT=qVD=L.ThewidthoftheKBT-layer,asshowninFig.( 7.2.4 ),issimplythedistancefromthevirtualsourceoverwhichthepotentialdropsbyKBT.Thus, rE=1 1+ l=1 1+20 vTVD L(7) CombiningEqs.( 7 )and( 7 ),amoregeneralexpressionforthebackscatteringratioinsaturationcanbewrittenas rsat=r0 1+20 vTVD L(7) 111

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ToestablishaconnectionbetweenthismodelandconventionalDD-basedmodels,consideradevicewithchannellengthLlongenoughsothatr0'1(i.ersat'rE).Then,usingEqs.( 7 )and( 7 ),Lundstromcameupwiththefollowingexpressionforthequasi-ballisticlinearMOSFETI)]TJ /F4 11.955 Tf 11.96 0 Td[(Vequation: ID(sat)=WCG8><>:1 rsat)]TJ /F3 11.955 Tf 11.96 0 Td[(1 1 rsat+19>=>;vT(VG)]TJ /F4 11.955 Tf 11.96 0 Td[(VTH) (7a)=WCG8>><>>:1 1 vT+L 0VD9>>=>>;(VG)]TJ /F4 11.955 Tf 11.95 0 Td[(VTH) (7b) ComparethisequationwithShur'sEquation( 7 ),whichcanberewrittenas: ID'WCG1 L VD1 B+1 0(VG)]TJ /F4 11.955 Tf 11.95 0 Td[(VTH)(7) ThisshowsthelinkbetweentraditionalmodelsandLundstrom'sKBT-layermodel.In[ 172 ][ 207 ],LundstromandRenfurtherclariedtheirmodelexplainingthescatteringmechanismandcurrentcontrolmechanisminmoredetailwithdetailedsimulationsbasedontheNon-equilibriumGreen'sFunctionformalism.Inthiswork,carrierbackscatteringintheKT-layerwasquantiedandshowntobecrucialinestimatinghowcloseagivendevicewasclosetotheballisticlimit.Webrieydiscusssomeaspectsofthephysicsinvolvedinthenextsection. 7.2.5CurrentControlinLundstromModel Lundstrompointedoutin[ 207 ]thatthatthemaximumaveragecarriervelocityatthebeginningofthechannelwastheequilibrium,uni-directionalthermalvelocity(asrstpointedoutbyNatori).Fordegenerateconditions,thisvelocityis: vT(VD)=s 2KBT mcF1=2(F) F0(F)(7) 112

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whereF=EF)]TJ /F4 11.955 Tf 11.95 0 Td[(E1 KBTwithEFbeingtheFermienergyandE1beingtheenergylevelofthelowestsubband.Forsimplicity,itisassumedthatmostofthecarriersresideintherstsubband. ThecompleteequationforID(sat)forthefullrangeofbiasconditionsonthegateandthedrainterminalsofaMOSFETisshownatthebottomofthepage(Eq.( 7 )).Here, a. ThersttermQi(0)istheinversionchargeatthevirtualsource.ForawelldesignedMOSFET,Qi(0)'CG(VG)]TJ /F4 11.955 Tf 11.95 0 Td[(VTH)abovethreshold. b. ThesecondtermistheballisticefciencyB,whichdescribesthereductionofID(sat)duetobackscattering.TheparameterrisafunctionofbothVGandVD. c. ThethirdfactoristhedegeneratethermalvelocitywhichdependsonQi(0)throughF. d. ThelastfactoraccountsforthedependanceonthedrainbiasVD.ThisfactorisproportionaltoUD=qVD KBTforlowVDandapproachesunityforhighVDconditions,therebyreducingtotheformofEq.( 7 ).Alsonotethatwhenr=0,thisequationreducestoNatori'sballisticMOSFETequation. Thebackscatteringratior(andultimatelyID(sat))dependsonthemeanfreepath()inthecriticalregionandthewidth(l)ofthecriticalregionitself.,inturndependsontheloweld,longchannelmobility.WhileEq.( 7 )isapplicableinnon-degenerateconditions,thecompleteexpressionforis[ 207 ] =2KBT q0 vTF0(F) F)]TJ /F9 7.97 Tf 6.59 0 Td[(1(F)(7) ThewidthoftheKBT-layerwasmodeledas l=LKBT qVDl(7) whereandlaresomeempiricalconstants.Thegatevoltagedependenceofthelayercomesthroughthelparameter.TheoriginalLMassumedl0.7and>1. ThusitwasarguedthatthelongchannelmobilityinagiventechnologyisstillanimportantparameterevenfornanometricMOSFETsinthesametechnology,thatare 113

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presumedtobequasi-ballistic.LundstromalsoshowedthatthevelocitysaturationoccursinaballisticMOSFETbutagaisttheconventionalpictureofsaturationthroughhigheldopticalphononemissionitwasindicatedthatthevelocityatthevirtualsource(i.ethetopofthebarrier)saturatesatthethermallimit.ForunstrainedSi,thesetwovelocities,i.e.theeldlimitedsaturatedvelocity(vsat)andthemaximumuni-directionalthermalvelocity(vinj=BvT,withB=1)bothhappentobereallyclosetoeachother(107cm/s). LundstomalsoarguedthatthebackscatteringthathappensintheKBT-layeristhemostcriticalscatteringintheMOSFET.Accordingto[ 172 ],onlythisbackscatteringdeterminestheballisticefciencyandtheID(sat).Thereasoninggivenisquitesimple:evenifbackscatteringoccursbeyondthecriticaldistancel,thecarrierswillnothaveenoughlongitudinalenergytosurmountthebarrierandre-enterthesource.Morelikely,thecarrierswillundergoseveralscatteringevents(phonon/impurity),whichwillfurtherreducethelongitudinalenergy,andtheywilleventuallyreachthedrain. 7.2.6IssuesintheLundstromModel AdvancedMonteCarlosimulationperformedbyPalestri[ 175 ]in2005broughtoutsomeimportantissuesintheKBT-layertheory.Someoftheseissuesstemfromthefundamentalassumptiononthenatureofbackscatteringratio(esp.inhighVD)anditsdependenceontheloweld,longchannelmobility.Meanwhile,basedonatemperature-basedexperimentalmethodpioneeredbyChen[ 179 ]in2002,severalauthors[ 176 180 181 183 184 186 189 199 201 208 212 ]publishedhavewidelyvaryingnumbersandinterpretationsfortheballisticefciencyBinamodernMOSFET,bothwithandwithoutstrain. Tosimplifythediscussion,wefollowtheterminologyusedbyPalestri[ 175 ].Atthevirtualsource,thecurrentduetobackscatteredcarriersisdenotedbyI)]TJ /F6 7.97 Tf -1.59 -7.29 Td[(onandthecurrentduetoforwardmovingcarriersisdenotedbyI+on.ThebackscatteringratiorissimplyI)]TJ /F6 7.97 Tf -1.59 -7.29 Td[(on I+on.Ifwecallthetotalcurrentreachingthedraininthecaseofcompleteballistictransportas 114

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IbalandinthepresenceofscatteringasIon,itfollowsthat Ion=1)]TJ /F4 11.955 Tf 11.95 0 Td[(r 1+rIbal(7) Onewayofinterpretingthisequationisasfollows:ifwecancalculatethecompleteballisticcurrentusingsayadevicesimulator,thenforarealdevicewhichhasscattering,allweneedistodeterminertoestimatetheoncurrent.However,theextractionofprecisenumbersforriscloudedbyanumberoftheoreticalandexperimentaluncertainties. ThefollowingisalistofimplicitandexplicitassumptionsintheKBT-layertheory,includingtheapplicabilityandthevalidityoftheassumptions. 1. Atthetopofthebarrier(virtualsource),therearetwostreamsofcarriers(Fig.( 7.2.4 )),oneforwardmoving(relatedtoI+on,eventuallycontributingtothetotalIon)andonebackwardmoving(i.etheKBT-layerbackscatteredcarriers,relatedtoI)]TJ /F6 7.97 Tf -1.59 -7.29 Td[(on).ThesetwostreamsareassumedtohaveequilibriumMaxwelliandistributionsandhavetheaveragevelocitiesv+andv)]TJ /F1 11.955 Tf 7.08 -4.34 Td[(,bothequaltothenon-degenerate,unidirectionalthermalvelocityvT. Natorirstdiscussedthevalidityofthisin[ 213 ].Morerecentfullbandself-consistentMonte-Carlosimulations[ 214 ][ 188 ]showthatwhilev+wasapproximatelyvT,v)]TJ /F1 11.955 Tf -419.34 -18.78 Td[(wasonly'0.7vT.Ifthisisnotaccountedfor,rsatisoverestimatedby10%(andtheIonby7)]TJ /F3 11.955 Tf 8.79 0 Td[(8%).Inaddition,theassumptionthatthepositiveandnegativevelocitycarriershaveaMaxwelliandistributionhasbeenshowntobeinaccurate,esp.underhighVDconditionswherestrongnon-equilibriumtransporttakesplace. 2. CarriersinjectedintothechannelfromthesourceoccupystatesatthetopofthebarrieraccordingtotheFermilevelofthesource.TheKBT-layermodelassumesthatthechargeatthetopofthebarrierisCG(VG)]TJ /F4 11.955 Tf 11.96 0 Td[(VTH). 3. Underlowbiasconditions,risgivenbyrlin=L=(L+),whereisthemomentumrelaxationlengthrelatedtotheloweldlongchannelmobilityandListhechannellength. Theexpressionforrlinactuallyhasasoundfundamentalbasis[ 215 ]inmesoscopicphysics.Itwasalsoshownin[ 178 ]thatthisrelationcanbeactuallyderivedfromtheDrift-Diffusionformalism.TheauthorsspecifythattheassumptionsoftheDDapproacharevalidforlowVDconditionseveninananoscaledevice.Thereissomedebateaboutthishowever,butforthemostpart,assumingthattherelaxationtimeapproximation(RTA)isvalidisnotoffthemark.Thususageofofarelaxationlengthrelatedto0andvTshouldberighttotherstorder. 115

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4. Underhighdrainbiasconditions,risgivenbyrsat=l=(l+),wherelisthelengthofthecriticalregionwherethebackscatteringhappens.Itwasassumedin[ 172 ]thatthislengthwasthewidthoftheKBT-layerlKT. In1979,Priceobserved[ 206 ]fromsomeMonteCarlosimulationsthatcarrierssurpassingapotentialbarrierandtravellingdownthesamepotentialwerenotlikelytoreturntotheinjectionpointatthetopofthebarrier.PriortoPrice'swork,Bethe[ 216 ]showedthatanearballisticcurrentoccurinaforwardbiasedmetal-semiconductorjunction,iftheapotentialdropequaltotheKBT=qisseenbeforecarrierscanscatter.Thermionicemissionisapplicableinthiscase,duetothemuchshortercriticaldistancecomparedtothewidthofthebarrier. LundstromreasonedthatthecriticallayerforbackscatteringinaMOSFETisalsoroughlythedistanceoverwhichthechannelpotentialdropsbyKBT=q,whichisatinyfractionofthechannellength.Thusheextendedtheexpressionforbackscatteringinaeld-freeslab(r=L=(L+)toaMOSFET(underahighdrainbiasVD)asrsat=l=(l+)wherelisthewidthoftheKBTlayer.MonteCarlosimulationsin[ 175 ]showthatactualcriticallayerisnotquitethedistanceneededtodropKBT=qofpotential,butisslightlyhigher.Lundstromalsolaterconrmedthisresult[ 217 ],notingthatitisprobablynotpossibletoaccuratelyestimatehowmuchthewidthishigherwithoutadvancedsimulations. Inadditontotheabovepoints,itwasalsoarguedin[ 172 ]thatbecausethecriticalbackscatteringoccursinaregionwherethecarriershavegainedlittleenergyfromthechanneleld,itwasappropriatetousethesamemean-free-pathasusedinthelinearcaseabove.ThevalidityofthisassumptionhasbeenquestionedbyFischetti(seeappendixAandBin[ 191 ]).Inthesamework,FischettialsodiscussessourcestarvationandlongrangeCoulombeffectsandquestionsifitwillbeeverpossibletoattainballistictransportproperinaMOSFETevenifwescaledowntosub-10nm. 5. Scatteringeventsdeepinsidethechanneldoesnotaffectr,andthusIon. Itwasarguedthatasacarriermovesdeeperintothechannel,itsenergyincreaseswhichraisesitsprobabilityofscatteringbyphononemission.WhenphononemissionhappensbeyondtheKT-layer,thecarrierenergyisloweredandmakesitlesslikelytoreturntothesource.Eveninthecaseofelasticscatteringdeepinsidethechannel,acarriermightundergoadditionalscatteringeventsonitsjourneytothesource.Asufcientnumberoftheseeventswilleffectivelythermalizethecarrierenergyandmakeitimpossibletosurmountthepotentialbarrier.Inaddition,theinterfaceroughnessscatteringisstrongnearthebeginningofthechannelandreducesasthecarriermovesclosertothesourcewhichmeansthatthecarriersfarawayfromthesourcehavealesserchancetobeelasticallybackscatteredbythismechanism.Thusitwasconcludedthatchannelscatteringdoesnotchangerasdenedearlier,andtherebydoesnothaveanimpactonIon.However,inouropinion,thisconclusionisinaccurate. InelasticscatteringinsidethechannelactuallyaffectstheheightofthepotentialbarrierbetweenthesourceandchannelthroughthecouplingviaPoisson's 116

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equation.Thepotentialbarrieratthevirtualsourceisnot`frozen'atitsequilibrium,zeroVDvalueitisaffectedbyscatteringinthechannel[ 218 ].Thebalancebetweenforward-moving(v+)andbackward-goingcarriers(v)]TJ /F1 11.955 Tf 7.09 -4.34 Td[()islostonceacurrentIonstartsowinginthedevice.Thesurfacepotentialatthevirtualsourcewillattempttocontinuoslyreadjust,butitwillneverbeabletocometoitsequilibriumvalue.Instead,anewequilibriumwillbeattaineddependingonthegateanddrainvoltages.Theresultofthisnewequilibriumisanincreaseinthebarrierheight(see,forinstanceFig.(10)in[ 172 ]andFig.(9)in[ 175 ]),whichwillresultinadifferentvalueforrcomparedtothefrozeneldcase.ThisfeedbackbetweenbarrierheightandscatteringisnottakenintoaccountintheKBT-layertheory. Westronglystressatthispointthatthiseffectduetochannelscatteringcannotbeignoredtoday'sMOSFETs.Inthepresenceofstrain,whichaffectsmaterialpropertiesinafundamentalway,thisfeedbackmechanismneedscarefulconsideration,Natorihascomeupwithanewtheoryexplaininghigheldtransportinsemiconductors[ 192 ][ 193 ],whichdelvesintotheoriginofthisfeedback/couplingbetweeninelasticscatteringandbackscattering.Inlatersectionsofthischapter,weshowhowthepresenceofthisfeedbackiscrucialinunderstandingandquantifyingthenatureofquasi-ballistictransportinadevice. 7.2.7RevisitingtheGildenblatModel In2002,Gildenblatpresented[ 219 ]aslightlydifferentapproachtomodelingtheballisticefciency.Startingfromthefundamentaldentionofthescatteringmatrix(Section 7.2.3 ),Gildenblatdevelopedaclosedformanalyticalexpressionfortheoveralltransmissioncoefcientforadevicehavinganarbitrarypotentialprole.Inthesamepaper,healsoshowedtheLMmodelexpressionforrsatcanberecoveredfromhisexpressionveryeasily.WerefertothistransmissionmodelastheGildenblatModel(GM),uponwhichourmodelisbased.Forcompleteness,weoutlinethekeypointsinGildenblat'smodelinthissection. ThestartingpointforGMaretheequationspresentedinsection 7.2.3 .Toarriveatananalyticalexpressionforthetransmissioncoefcientt,Gildenblatassumesthattheratioisindependentofposition(whilebothandDarepositionandeld 117

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dependent).StartingfromEq. 7 andusing Ex=)]TJ /F4 11.955 Tf 10.49 8.09 Td[(d dx (7a) dn dx=dn dd dx (7b) Wehave J D=)]TJ /F16 11.955 Tf 11.3 16.85 Td[()]TJ /F5 11.955 Tf 9.29 0 Td[(n+dn dd dx(7) Nowconsiderthequantityne)]TJ /F15 7.97 Tf 6.58 0 Td[(,whereindependentof.Differentiatingthisw.r.t,wehave, d d(ne)]TJ /F15 7.97 Tf 6.58 0 Td[()=e)]TJ /F15 7.97 Tf 6.58 0 Td[(dn d)]TJ /F5 11.955 Tf 11.95 0 Td[(e)]TJ /F15 7.97 Tf 6.59 0 Td[(n(7) or ed d(ne)]TJ /F15 7.97 Tf 6.58 0 Td[()=dn d)]TJ /F5 11.955 Tf 11.95 0 Td[(n(7) UsingEq.( 7 )inEq.( 7 ), J D=)]TJ /F4 11.955 Tf 9.3 0 Td[(ed dx(ne)]TJ /F15 7.97 Tf 6.58 0 Td[()(7) Rearranging,weget d dx(ne)]TJ /F15 7.97 Tf 6.58 0 Td[()=)]TJ /F4 11.955 Tf 9.3 0 Td[(e)]TJ /F15 7.97 Tf 6.58 0 Td[(J D(7) NotingthatnetuxJisaconstant,andintegratingbothsidesafteraddingappropriatelimits, Zn(x)e)]TJ /F20 5.978 Tf 5.75 0 Td[((x)n(0)e)]TJ /F20 5.978 Tf 5.76 0 Td[((0)d(ne)]TJ /F15 7.97 Tf 6.59 0 Td[()=)]TJ /F4 11.955 Tf 9.3 0 Td[(JZx0e)]TJ /F15 7.97 Tf 6.58 0 Td[(1 Ddx(7) Theresultoftheintegrationis n(x)e)]TJ /F15 7.97 Tf 6.59 0 Td[((x))]TJ /F4 11.955 Tf 11.95 0 Td[(n(0)e)]TJ /F15 7.97 Tf 6.58 0 Td[((0)=)]TJ /F4 11.955 Tf 9.3 0 Td[(JZx0e)]TJ /F15 7.97 Tf 6.59 0 Td[(dx D(7) Here=(x)denoteselectrostaticpotential 118

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Dening (x)=(x))]TJ /F5 11.955 Tf 11.96 0 Td[((0),itcanbeshownthat n(x)=n(0)e (x))]TJ /F4 11.955 Tf 11.96 0 Td[(Je (x)Zx0e)]TJ /F15 7.97 Tf 6.59 0 Td[( (x)dx Ddx(7) Eq.( 7 )isaimportantresultof[ 219 ].Oncen(x)isknown,theforwardandthereverseuxescanbeeasilyestimatedfromEq.( 7 )as 2J+(x)=cn(0)e+J1)]TJ /F4 11.955 Tf 11.95 0 Td[(ce (x)Zx0e)]TJ /F15 7.97 Tf 6.59 0 Td[( (x)dx D2J)]TJ /F3 11.955 Tf 7.08 -4.94 Td[((x)=cn(0)e)]TJ /F4 11.955 Tf 11.96 0 Td[(J1+ce (x)Zx0e)]TJ /F15 7.97 Tf 6.59 0 Td[( (x)dx D Itiswellknownfromtheuxmethodthatthereectionandtransmissioncoefcientsofthesystemaregivenby r=J)]TJ /F3 11.955 Tf 7.09 -4.33 Td[((0) J+(0) (7a) t=1)]TJ /F4 11.955 Tf 11.95 0 Td[(r=J+(L) J+(0) (7b) Bydeninganeffectivevelocityegivenby 1 e=ZL0e)]TJ /F15 7.97 Tf 6.58 0 Td[( (x)dx D(7) andusingtheexplicitequationsforJ+(x)andJ)]TJ /F3 11.955 Tf 7.09 -4.34 Td[((x)shownabove,itiseasytoshowthat J)]TJ /F3 11.955 Tf 7.08 -4.94 Td[((0)=cn(0))]TJ /F4 11.955 Tf 11.96 0 Td[(J (7a) J+(0)=cn(0)+J (7b) J+(L)=cn(0)e (L)+J1)]TJ /F4 11.955 Tf 11.96 0 Td[(ce (L)ZL0e)]TJ /F15 7.97 Tf 6.59 0 Td[( (x)dx D=cn(0)e +J1)]TJ /F4 11.955 Tf 15.21 8.08 Td[(c ee (7c) 119

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( = L)]TJ /F5 11.955 Tf 12.05 0 Td[( 0= (L)i.e.thesurfacepotentialdifferencebetweenthedrainandsourceendsforaparticularvoltageonthegateanddrainterminals.) Usingtherelationr+t=1,wecanshowthat cn(0)=Je)]TJ /F15 7.97 Tf 6.59 0 Td[( +c e(7) Usingtheaboveexpression,tcanbeexpressedas t=2 1+c e+e)]TJ /F15 7.97 Tf 6.59 0 Td[( (7) ThusifeasdenedbyEq.( 7 )isknownforasystem,thetransmissioncoefcientcanbeestimatedusingEq.( 7 ). 7.2.8RevisitingtheOriginalShockleyWork Letusdiscussthephysicsbehindone-uxtheoryassumptioninEq.( 7 ).Inessence,thisisaloweld,near-equilibriumassumption,whichisbasedonasimplelineardependenceofthebackscatteringcoefcientsontheappliedelectriceld.McKelvey[ 171 ]assumedthebelowexpressionsforthecoefcients, r=r0(1) (7a) r++r)]TJ /F3 11.955 Tf 17.05 -4.93 Td[(=2r0 (7b) r+)]TJ /F4 11.955 Tf 11.95 0 Td[(r)]TJ /F3 11.955 Tf 17.05 -4.93 Td[(=(2r0) (7c) ItisclearlystatedintheinitialMcKelveypapers[ 169 ][ 171 ]thatthemethodisapplicableforsmallappliedelectriceldsonly.wasshowntobeaquantitylinearlydependentontheelectriceldas =0 cEx(7) (0istheloweldmobilitydenedas0=et=m) McKelveyquantiedsmallelectriceldsby1.Theoretically,thismeansthatthecarrierdistributionfunctionisnotfarawayfromequilibriumandtheaveragecarrier 120

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velocityissmallcomparedtoc(i.e,velocitysaturation,anhigheldeffect,hasnothappenedyet). Thus,theassumptionthatr+)]TJ /F4 11.955 Tf 12.16 0 Td[(r)]TJ /F3 11.955 Tf 10.91 -4.34 Td[(=Ex,withbeingaconstantindependentofpositionandindependentofpotentialGildenblat'snalexpressionforthetransmissioncoefcientisessentiallynodifferentthanwhatcouldbeobtainedfromanDrift-Diffusionperspective.WereiteratethattheGildenblatresultissimplyaLorentzsolutionoftheBoltzmannTransportEquation.Thatis,thesolutionobtainedbythismethodissimplytheequilibriumdistributionfunctionuponwhichadriftvelocity(arisingfromthepresenceofasmallelectriceld)hasbeensuperposed.Byitsnature,thissolutioncannotdescribequasi-ballistictransportwithanymoreaccuracythanatraditionalDDsolution.TheLundstromquasi-ballistictransportmodelalsosuffersfromasimilarproblemthatshowsupadifferentway,becauseofthesameunderlyingassumptionaboutthebackscatteringdifferencesintheforwardandreversedirections.OtherauthorshavealsodiscussedthevalidityofLundstrom'sassumptionsfromadifferentperspective-VaidyanathanandPulfreypresentangooddiscussionoftheissuesin[ 190 ]andFischetti[ 214 ]presentsverycomprehensiveMonteCarlosimulationresultsofananoscaletransistor,showingtheoff-equilibriumnatureofthecarrierdistributionfunctioneveninthelinearrange,forvoltagesassmallas0.1V.NotethattheM-uxtheorydevelopedbyLundstrom'sgroupdoesnothavetheseproblem;thetrade-offismodelcomplexity. 7.3UFCompactModel:Theory Toovercometheissuespresentedintheprevioussection,wehaveextendedtheGildenblatapproachfortransmissionbyincludinghigheldtransporteffectsandconnementeffects(aswouldoccurintheinversionlayerofaMOSFET).Oneofthegoalsofourworkwastoretainanalyticalsimplicity.Webelievethatincludingtheconnementeffectsintotheframeworkoftheone-uxtheoryisaindirectwayofincorporatingenergyresolveduxesforcarriertransport.Inthissection,werstexplain 121

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theadditionofhigheldeffects,andthenweshowtheadditionofconnementeffectstotheresultingmodel.WealsopresentanewinterpretationofNatori'shigheldtransportmodel[ 192 ][ 193 ],provingtheequivalenceofourapproachandNatori'sapproach.Withthehelpofthisequivalence,wethenexplainthephysicalsignicanceofthemodiedeffectivevelocitytermusedinourmodel(inGM,itwassimplyusedasamathematicalsimplication).Finallywetouchuponusingourmodelforaclearerunderstandstraineffectsonquasi-ballistictransportfromanewperspective. 7.3.1IncludingHighFieldEffects OurmethodforincludinghigheldeffectsstemsfromasuggestionputforwardbyPulverandMcKelveyin1966[ 220 ]intheirpaperdiscussingtheapplicationoftheuxmethodforsystemswithnonconstantelectricelds.Intheirwork,theyproposeanempiricalexpressionforthevariationofrwith(andtherebyEx)thatisapplicableevenathighelds.Whilenotrigourouslyjustiedinafundamentalway,theirexpressionreproducesthecorrectbehaviorofratalllimits.Theexpressionsuggestedis r=r0e(7) Underincreasingelectricelds,=0E=cxincreasesinvalue(andwilleventuallysaturatewhencarriervelocitysaturates)andeventuallythecondition1becomesinvalid.Inthislimit,PulverandMcKelveyarguethat,forathinslabofthicknessdx,r+dx!0andr)]TJ /F4 11.955 Tf 7.09 -4.34 Td[(dx!1.Theynotethatinextendingtheseresultstothelimitdx!0,andthus,r+!1andr)]TJ /F4 11.955 Tf 7.08 -4.34 Td[(dx!1.itisalsonotedthattheconditionthatr+(E)=)]TJ /F4 11.955 Tf 9.3 0 Td[(r)]TJ /F3 11.955 Tf 7.09 -4.34 Td[(()]TJ /F4 11.955 Tf 9.3 0 Td[(E)shouldapplyfromsymmetryconsiderations.Theexponentialrelationsuggestedaboveisthesimplestmathematicalexpressionthatsatisesboththeseconditionssimultaneously. 122

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Withthisformforthebackscatteringcoefcients,themobilityandthediffusionexpressionbecome (x,Ex)=r+)]TJ /F4 11.955 Tf 11.96 0 Td[(r)]TJ ET q .478 w 217.07 -68.74 m 256.87 -68.74 l S Q BT /F4 11.955 Tf 217.07 -79.93 Td[(r++r)]TJ /F16 11.955 Tf 8.28 21.6 Td[(c Ex (7) =c Exe)]TJ /F4 11.955 Tf 11.95 0 Td[(e)]TJ /F15 7.97 Tf 6.59 0 Td[( e+e)]TJ /F15 7.97 Tf 6.59 0 Td[(=c Extanh() (7) (7) D(x,Ex)=c r++r)]TJ ET BT /F1 11.955 Tf 433.45 -179.02 Td[((7) =c r01 e+e)]TJ /F15 7.97 Tf 6.59 0 Td[(=c 2r01 cosh() (7) Theexpressionforposition/elddependentEinsteinrelationis, D =Ex 2r01 sinh() (7) (7) Attheloweldlimit,wecanverifythattheaboveexpressiontakesitdefaultvalue limEx!0D =Ex 2r01 (7) =c 2r01 0=D0 0=kBT q (7) TheexpressionforD=shownabovedependsontheelectriceld,whichisthederivativeoftheelectrostaticpotential,butnotontheelectrostaticpotentialitself.ThismeansthatGildenblat'smathematicalapproachisstillapplicablespecically,Eq. 7 stillworks.Thismeansthatwecanderiveanexpressionforn(x)usingthesametechniquewasasoutlinedinthelastsectionaftera(careful)integrationinvolvingapositiondependent.However,theeffectivevelocityeneedstoberedenedtoaccountforthepositiondependentterm.Oncethisisdone,wecanarriveatanexpressionforthetransmissioncoefcientthatreducestotheGildenblatresultintheloweldlimit.Thisprocedureisoutlinedinthenextsection. 123

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WestartfromEq. 7 andusetheterm==Dinsteadofforclarity.Wereiteratethatthepositionandelddependenceofcannotbeignored. J D=)]TJ /F4 11.955 Tf 9.3 0 Td[(ed dx(ne)]TJ /F15 7.97 Tf 6.58 0 Td[()(7) Rearranging,weget d dx(ne)]TJ /F15 7.97 Tf 6.59 0 Td[()=)]TJ /F4 11.955 Tf 9.3 0 Td[(e)]TJ /F15 7.97 Tf 6.58 0 Td[(J D(7) or Zd(ne)]TJ /F15 7.97 Tf 6.59 0 Td[()=Z)]TJ /F4 11.955 Tf 9.3 0 Td[(e)]TJ /F15 7.97 Tf 6.58 0 Td[(J Ddx(7) Sincedependsonposition,weaccountforthisinthelimitsofintegrationontherighthandside(0andx,whicharevaluesofwhenx=0andx=x,respectively). NotingthatnetuxJisaconstantandaddinglimits, Zn(x)e)]TJ /F20 5.978 Tf 5.76 0 Td[(x(x)n(0)e)]TJ /F20 5.978 Tf 5.75 0 Td[(0(0)d(ne)]TJ /F15 7.97 Tf 6.58 0 Td[()=)]TJ /F4 11.955 Tf 9.3 0 Td[(JZx0e)]TJ /F15 7.97 Tf 6.58 0 Td[(1 Ddx(7) Afterintegration,wehave, n(x)=ex(x)n(0)e)]TJ /F15 7.97 Tf 6.58 0 Td[(0(0))]TJ /F4 11.955 Tf 11.95 0 Td[(JZx0e)]TJ /F15 7.97 Tf 6.58 0 Td[(dx Ddx(7) Eq.( 7 )istheanalogofGildenblat'sEq.( 7 ).WeproceedinthesamewayasbeforetoestimateJ=cn(x)J. 2J+(x)=cn(0)ex(x)e0(0)+J1)]TJ /F4 11.955 Tf 11.96 0 Td[(cex(x)Zx0e)]TJ /F15 7.97 Tf 6.59 0 Td[((x)dx D2J)]TJ /F3 11.955 Tf 7.08 -4.94 Td[((x)=cn(0)ex(x)e0(0))]TJ /F4 11.955 Tf 19.27 0 Td[(J1+cex(x)Zx0e)]TJ /F15 7.97 Tf 6.59 0 Td[((x)dx D Redeningtheeffectivevelocityeasbelow, 1 e=ZL0e)]TJ /F9 7.97 Tf 6.59 0 Td[((x(x))]TJ /F15 7.97 Tf 6.58 0 Td[(0(0))dx D(7) 124

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andfollowingthesameprocedureasintheprevioussection,wecanderiveanexpressionforthetransmissioncoefcienttas t=2 1+c e+e)]TJ /F9 7.97 Tf 6.58 0 Td[((L(L))]TJ /F15 7.97 Tf 6.58 0 Td[(0(0))(7) Ifisindependentofpostion/eld(i.e.aconstant),Gildenblat'sEq.( 7 )isrecovered. 7.3.2IncludingConnementEffects Unlikebulkdevices,MOSFETswith2Dand1Dinversionlayersoperateundertheeffectofgateeldinducedquantumconnement[ 221 ].Theconnementofcarriersleadtoreductionofcarrierdegreeoffreedom.Thisinreturnresultsinconnementinducedvalley/bandsplitting,leadingtorepopulationofcarriersbetweenthebands.Additionally,thesplittingofthebandsmodiesthejointdensityofstatesforscatteringmatrixelements[ 222 ].Forexample,forSielectroninversionlayers,thesplittingof2and4valleysleadtoreductioninf-typeopticalphononscatteringrates.Movingfroma3Dto2Dbandstructurereducestheeffectivedensitystateswhichalsocontributestoadditionalscatteringratereduction. Thegate-to-channelpotentialdifferencevariesalongthechannelbasedontheconductionbandprole.Thisleadstoapositiondependentlevelofquantumconnementforthecarriers.Atthesourceend,wherethegate-to-channelpotentialdifferenceishighest,theconnementinducedbandsplittingisatitslargestvalue.Hence,thecarrierrepopulationandtheresultantconductivitymassmodication(comparedtobulk)ismorepronouncedcomparedtodrainend.Forinstance,in(100)-orientedSi,theeffectiveconductivitymassforelectronsatthesourceendislowercomparedtothedrainend. Withapositiondependentconductivitymass,wecandeneaposition-dependentthermalvelocity, 125

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c(x)=s 2kBT m(x)(7) AddingthepositiondependenceofthermalvelocitytotUF,weget tUF=2 1+c(0) c(L)c(L) e+e)]TJ /F9 7.97 Tf 6.58 0 Td[((L(L))]TJ /F15 7.97 Tf 6.58 0 Td[(0(0))(7) wherec(0)andc(L)representtheaveragethermalvelocityatthesourceanddrainendofthechannel. Withtheproperinclusionofhigh-eldeffectsandconnementeffectsEq.( 7 )becomesapplicablefordescribingcarriertransportfortheentirerangeofdeviceoperation(lowandhighelds)andforallsurfacechannel/orientations. 7.3.3RevisitingtheNatori'sHighFieldTransportModel In2009,Natoripresentedanewapproach[ 192 ][ 193 ]towardsunderstandinghigheldtransportinbulksemiconductors.Inthisapproach,ananalyticalsolutionofapseudo-one-dimensionalBTEwithaconstantelectriceldispresented,aftertransformingitintoapairofcarrieruxequations.Neithertherelaxationtimeapproximationnortheperturbationexpansionisusedinsolvingtheequations.Explicitexpressionsforaveragecarriervelocityanddensityasfunctionsofeldandpositionarepresented. ThemainconceptsinNatori'stheoryareschematicallypresentedinFigure 7-4 .Natoriconsidersasimpleresistorinhispapers,withalinearpotentialproleinthedevicechannel.Thecarriersenteringthechannelfromthesourcehasadistributionofenergies,andanaveragevalueofKBT=qisassumed.Intheregionofthechanneluptodistancex0fromthesource,thecarriersdonothaveenoughenergytoemitanopticalphonon.Inotherwords,theonlyscatteringeventsinthisregionareelasticinnature(surfaceroughness,ionicimpurityandacousticphononscattering).ThisregionisthereforenamedastheInitialElasticZone(IEZ).IntraversingthelengthoftheIEZ, 126

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thecarriersgainenergyfromtheeldandhaveenoughenergyforopticalphonon(OP)emissionaftercrossingthiszone. Beyondthepointx=x0,OPemissionhappensinaveryshorttimeandcarriersareallowedtorelaxtothelowerenergylevel")]TJ /F5 11.955 Tf 12.04 0 Td[(".Thisenergylevel")]TJ /F5 11.955 Tf 12.05 0 Td[("isknownastheIstRelaxedLevel,andthecarriersintherstrelaxedlevelareonlyallowedtopopulatexx0.Thecarrierscontinuetogainenergyfromtheeldonceagain,tilltheyreachthepoint(x0+x1)insidethechannelwherethecarrierenergyisequaltotheOPenergy.Theregionx0x(x0+x1)isknownastheIstRelaxedElasticZone. ItiseasytoseethattheentirechannelcanberepresentedasacascadeoftheserelaxedelasticzoneswithOPemissionhappeningateachzoneboundary.Thetotaltransmissionthroughthedeviceisthereforeaproductofthetransmissioncoefcientsthrougheachelasticzone.T=T0T1T2TN (7) However,thereisonedifferencebetweenT0andtherestofthetransmissioncoefcientsT1TN.Beforex0,energyrelaxationcanhappenonlythroughelasticscatteringprocesses(acousticphononscatteringprocessisconsideredtobeelastic).Beyondthisthispoint,carrierenergyrelaxationhappensthroughbothelasticandinelasticscatteringmechanisms,andtransportphenomenabecomesconceptuallyidenticalinallthezonesbeyondIEZ.Thus,T1TNcanbecombinedintoonefactor,andasingleequivalentOpticalPhononEmissionZonecanbevisualized. Rightnearthezoneboundayx0,aftertheinelasticscatteringwithOP,thereisachanceforthecarrierstobeelasticallybackscatteredintotheIEZ.Theprobabilityforre-entryintotheIEZreducesasthecarriermovesawayfromx0eventhoughitcanundergoanelasticscatteringeventatanyfurthertime.ThetotalnumberofbackscatteredcarriersintotheIEZdependsontheamountofelasticbackscatteringandtheenergyrelaxationviaOPemissionbeyondxx0.Natoridescribesthisthrough 127

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Figure7-4. Natori'shigheldtransportmodelshowingtheconceptualInitialElasticZoneandOpticalPhononEmissionZoneregionsinadevicecarryingcurrent. aparameterwhichrepresentstheelastic-backscattering/energyrelaxationtrade-off.Thepresenceofthistradeoffisverysignicantinunderstandingquasi-ballistictransportbothwithandwithoutstraininvolved. WewouldliketopointoutthattheLMpictureisslightlydifferentinthisregard,asthebackscatteringparameterrinthatmodelrepresentsthefractionofcarriersgettingbackintothesource,whileNatori'sparameteraccountsforbackscatteringfromallofOPEZintotherstelasticzone.Thisisansubtledifferencethathasstrongimplicationsintheunderstandingofeithermodel. Foragivenenergyoftheincomingcarriers,theIEZwidthx0(andthereforetheOPEZwidthL)]TJ /F4 11.955 Tf 12.36 0 Td[(x0)willdependontheappliedvoltageonthedrain.Sincethecarrierswillhaveadistributionofenergiesatthesource,eachenergywillhaveit'sownzonedenitionsforagivendrainvoltageaswell.ItispossibletoconceptuallythinkofaenergyaveragedIEZandOPEZforagivenbiascondition.Higheldtransportinthe 128

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deviceisanalyzedusingareection-transmissionmethodwiththesezones,similartotheone-uxapproach. Usingveryrigorousanalysisandcalculations,Natorishowedthatthecurrent-eldandthevelocity-eldrelationshipsinthisdevicewerecloselyrelatedtotheuxtransmissionthoroughtheIEZ.IfTand(1)]TJ /F5 11.955 Tf 12.98 0 Td[()representthezonetransmissionsoftheIEZandOPEZrespectively,onewouldexpecttoseetheoveralltransmissionoftheentiredevicetobetheproductofindividualtransmissions,i.e,T(1)]TJ /F5 11.955 Tf 12.59 0 Td[()fromtheone-uxtheory.However,Natori'sanalysisshowsthattheoveralltransmissionisoftheformshowninEq.( 7 ). TN=T(1)]TJ /F5 11.955 Tf 11.96 0 Td[() 1)]TJ /F3 11.955 Tf 11.95 0 Td[((1)]TJ /F3 11.955 Tf 13.94 2.66 Td[(T)(7) ThiswasattributedtotheinherentfeedbackpresentbetweencarriertransportandelectrostaticsthroughthePoisson'sequationinthedevice.Notethatthetransmissionwillbeastrongfunctionoftheresultanteldinthechannel.Usingthistransmissionexpression,Natoriexplainscurrentsaturationthathappensundereldconditions.Traditionally,currentisthoughttobetheproductoftotalcarrierchargeandthecarriervelocity.Natoriintroducesthetransmissionintothedevicecurrentequationas I/qn0T(1)]TJ /F5 11.955 Tf 11.95 0 Td[() 1)]TJ /F3 11.955 Tf 11.96 0 Td[((1)]TJ /F3 11.955 Tf 13.95 2.65 Td[(T)(7) Usinghistransmissionmodel,NatoripointsoutthatcurrentthroughthedevicewillsaturatewhenthetransmissionTbecomesequaltounity,whichhappenswhenzonewidthx0!0athigheldconditions.Whenthishappens,thetotaltransmissionTNsaturatestothevalue(1)]TJ /F5 11.955 Tf 12.25 0 Td[().Thelevelatwhichcurrentsaturates,therefore,dependsonthetransmission(1)]TJ /F5 11.955 Tf 12.51 0 Td[()oftheOPEZ(whichistheentirelengthofthechannelasx0!0).Wewillexplainwhatvelocityistobeusedinthecurrentexpressioninlaterparagraphs. 129

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7.3.4UnderstandingFeedbackinNatori'sHighFieldModel WerstshowthatitispossibletoarriveatNatori'sexpressionfortotaltransmissioninaverysimpleandelegantmannerusingtheone-uxtheory,withtheadditionofcarrierexchangebetweenthezones.Forsimplicity,weusethenotationR=1)]TJ /F3 11.955 Tf 14.03 2.65 Td[(Tfromhereon.Figure 7-5 showsapictorialrepresentationofourapproach.Theredarrowsrepresentthereecteduxesandthegreenarrowsrepresentthetransmitteduxesfrom/toeitherzoneinthedevice. Consideranincidentuxf0enteringthechannel;apartofthisux(Tf0)istransmittedthroughtheIEZ,andpartofitisreectedbackintothesource,namelyRf0.OftheuxenteringtheOPEZ,theportion(1)]TJ /F5 11.955 Tf 12.52 0 Td[()Tf0istransmittedtothedrain,andthefractionTf0isreectedbackintotheIEZ.NotethatthesebackreectionsfromtheOPEZincludesthecontributionfromalltherelaxedzonesbeyondtheIEZ.ApartoftheuxreectedfromtheOPEZtotheIEZ(1)]TJ /F3 11.955 Tf 13.42 2.65 Td[(Rf0)transmitsbacktothesource,andapartofit(RTf0)isreectedbackintotheOPEZ.Byconsideringcarrierexchangethroughinnitereection/transmissionprocessthatcouldhappenbetweenthetwozones,wetakeintoaccounttheeffectofinelasticscatteringonthetotaltransmissioninafundamentalway. Toderivetheexpressionforthetotaltransmission,weneedthreeterms:(1)incidentuxJ+(0)(2)totaltransmitteduxJ+(L)(3)totalreecteduxJ)]TJ /F3 11.955 Tf 7.09 -4.34 Td[((0).TheuxtermscanbeobtainedbysummingthecomponentsshowninFigure 7-5 as, 130

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J+(0)=f0J)]TJ /F3 11.955 Tf 7.08 -4.93 Td[((0)=Rf0+(1)]TJ /F3 11.955 Tf 13.32 2.66 Td[(R)Tf0+(1)]TJ /F3 11.955 Tf 13.31 2.66 Td[(R)R2Tf0+(1)]TJ /F3 11.955 Tf 13.32 2.66 Td[(R)R23Tf0+(1)]TJ /F3 11.955 Tf 13.31 2.66 Td[(R)R34Tf0+...J+(L)=(1)]TJ /F5 11.955 Tf 11.95 0 Td[()Tf0+(1)]TJ /F5 11.955 Tf 11.95 0 Td[()RTf0+(1)]TJ /F5 11.955 Tf 11.96 0 Td[()R22Tf0+... (7) SumminguptheinniteseriesforJ+(0)andJ)]TJ /F3 11.955 Tf 7.09 -4.34 Td[((0),wehave J)]TJ /F3 11.955 Tf 7.09 -4.93 Td[((0)=Rf0+(1)]TJ /F3 11.955 Tf 13.31 2.65 Td[(R)Tf0n=1Xn=0Rnn=Rf0+(1)]TJ /F3 11.955 Tf 13.31 2.66 Td[(R)Tf01 (1)]TJ /F3 11.955 Tf 13.31 2.66 Td[(R) (7) J+(L)=(1)]TJ /F5 11.955 Tf 11.95 0 Td[()Tf0n=1Xn=0Rnn=(1)]TJ /F5 11.955 Tf 11.95 0 Td[()Tf01 (1)]TJ /F3 11.955 Tf 13.31 2.66 Td[(R) (7) andnally, TN=J+(L) J+(0)=T 1+ (1)]TJ /F5 11.955 Tf 11.96 0 Td[()T(7) whichisidenticaltoNatori'sexpression(Eq.( 7 )).Asimilarresultisalsoseeninmanyotherelds,forexampleinthetreatmentofopticaltransmissionthroughmultilayeredstructures[ 223 ],wheninterferenceeffectsareneglectedandallmultipleinternalreectionsaretakenintoaccount,similartoourcase. TheformoftheexpressionforthetotaltransmissioninEq.( 7 )issimilartothetransferfunctionofanegativefeedbacksystemcommonlyencounteredincontroltheory[ 224 ].Thisanalogyisnotcoincidence,butrather,aresultoftheeffectofthefeedback 131

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Figure7-5. Ourderivationoftheoveralltransmissioncoefcientusinganinniteseriessummationapproach. A B Figure7-6. TwoequivalentnegativefeedbacksystemrepresentationoftheNatoriModelwith(a)FEZtransmissioninthefeed-forwardpathand(b)OPEZtransmissioninthefeed-forwardpathshowinghowthefeedbackfromtheOPEZregulatesthetotalcurrentthroughthedevice. betweencarriertransportandthepotentialinsidethedevice.In[ 193 ],Natoriexplainshowthisfeedbackrelatestothecurrentcontrolmechanisminthedevice,andultimatelyisresponsibleforvelocityandcurrentsaturationinthedevice. Ingeneral,thetransmissioncoefcientsoftheIEZandOPEZareinverselyrelatedtotheirrespectivewidths.Astheappliedeldincreases,theIEZwidthx0!0and 132

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theOPEZwidth!L,therebymakingTand(1)]TJ /F5 11.955 Tf 12.51 0 Td[()proportionaltotheappliedeld.Theexactnatureoftheelddependencewillberelatedthepotentialproleinsidethedevice.Withthisunderstanding,itiseasytoseethattheI)]TJ /F4 11.955 Tf 12.95 0 Td[(Vcharacteristicswillbecloselyrelatedtothecarriertransmissionthroughthesezones.ForalinearpotentialprolediscussedbyNatori,whentheappliedeldistherangesuchthatthetransmissioncoefcientT1,Eq.( 7 )resultsinthefamiliarOhm'slaw(currentlinearlyproportionaltoappliedeld). WhentheappliedeldisincreasedsuchthatT!1,thecurrentandtheaver-agecarriervelocitybothsaturate.Natori'stheoryanticipatesthattheinstantaneousvelocityofthecarrierisnotconstantfromitstransitfromthesourcetodrain.Instead,itperiodicallyoscillatesineachofthezonesasthecarrierissuccessivelytransmittedtothe1st,2ndandhigherrelaxedlevels.However,themeancarriervelocityacrossallthezonesisuniformandremainsproportionaltotheelectriceldupto104V/cm,beyondwhichitsaturates.Thecarrierdensity,ontheotherhand,alsotendstohaveperiodicnatureacrossthezonesonceacurrentstartsowing.Tomaintaincurrentcontinuity,then(x)v(x)productshouldbeconstantatallpositionsinsidethedevice.Sincethepositionaveragedcarriervelocityisfoundtobeaconstant(likeinRyder'swork[ 225 ]),thecarrierconcentrationmultiplyingthisvelocityshouldbeaconstanttoo.Natorishowsthatthecarrierconcentrationtobeusedinthecurrentequationistheequilibriumcarrierdensityn0,whichissetbythefeedbackprocessfromOPEZ.The(negative)feedbackactstominimizethetotalelectrostaticenergyofthesystemviaPoisson'sequation. Tounderstandthefeedback,werstremindthattheOPEZisarepetitionofthesameunitstructureofwidthx1asshowninFigure 7-4 .Forthegiveneldcondition,theresultantcurrentissetintheIEZ,theuxdistributioninthe1standotherhigherrelaxedlevelsaredeterminedinaccordancewiththecarriervelocityintherespectivezonessoastomaintaincurrentcontinuity.However,sinceacurrentisowing,the 133

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carrierdistributionalongthezonesarenotbeconsistentwiththeoriginalconstantelddistribution. Thecarrierchargeexcess(decit)pushesup(pullsdown)thepotentialproleoftheoriginalconstant-eldcurve,whichmodiestheprole(i.ethebarrierheight)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,thenmeancarriervelocitythroughtheentiredevicesaturatesandcausesthecurrenttosaturate.SaturationwouldhappenirrespectiveoftheeldvalueifsomehowweweretomakeT!1happen.Inotherwords,transmissionsaturationthroughtheIEZcausescurrentsaturation(totheleveldeterminedbyproductofqn0,and(1)]TJ /F5 11.955 Tf 12.24 0 Td[()).ThisisafundamentaldifferencebetweenconventionalmodelsandNatori'snewmodel. TwoequivalentrepresentationsofanegativefeedbacksystemwithatransferfunctionasEq.( 7 )areshowninFigure 7-6 .Withoutfeedback,thesystemrepresentationwouldbeasimplecascadeofTand(1)]TJ /F5 11.955 Tf 12.52 0 Td[().Thestabilizingnegative 134

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feedbackfromtheOPEZactingthroughhelpstoregulatethetotalcurrentthedevice.WhileFigure 7-6A describestheactualphysicalpicture,modelFigure 7-6B helpsunderstandthesaturationoftotaltransmissionwhenthelengthoftherstelasticzonebecomessosmallthattheTbecomesunity. Ourmethodofusingtheone-uxtheoryincludinginnitecarrierexchangebetweentheIEZandOPEZ(Figure 7-5 andEq.( 7 ))toderivetheNatorimodeltransmissionequationsfacilitateabetterlinktotheLundstromorGildenblatcompactmodels.WhileNatoridiscussedabulktransportcaseinhispaper,ourderivationofEq.( 7 )isquitegenericandthereisnothingprecludingitsapplicationstostudyingquasi-ballistictransportinananoscaleMOSFET.Sinceourupdatedtransmissioncoefcientincludinghigh-eldandconnementeffects(Eq.( 7 ))isasurfacepotentialbasedequation,wecannowequateittotheNatoriequationEq.( 7 ),notingtheinherentlinkandfeedbackpresentbetweenthePossionequation,BTEandtheresultantsurfacepotential. UsingthefeedbackrepresentationoftheNatoriexpressionshowninFigure 7-6A ,wehave tUF=TN=T 1+ (1)]TJ /F5 11.955 Tf 11.95 0 Td[()T (7) 1)]TJ /F5 11.955 Tf 11.96 0 Td[(=1+1 tUF)]TJ /F3 11.955 Tf 15.14 8.09 Td[(1 T)]TJ /F9 7.97 Tf 6.59 0 Td[(1 (7) IftUFcanbecomputedforarealdevice,thenEq.( 7 )givesusahandletoestimateforhighbiasconditionswhenTsaturates.OnepossibleschemeforevaluatingtUFwouldbethetechniqueproposedbyGildenblat[ 226 ]involvingthePSPcompactmodel[ 227 ][ 228 ],combinedwithsimulationresultsforc(0)andc(L)fromaSchroedinger-Poissonsolution. 135

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A B Figure7-7. Evolutionof(a)conductionbandproleand(b)correspondingtransmissioncharacteristicsofthechannelarequalitativelyshownforfourpossibledeviceoperatingconditions:(1)lowVG,lowVD(2)lowVG,highVD(3)highVG,lowVDand(4)highVG,highVD 7.3.5PhysicalSignicanceofEffectiveVelocityTerm InGildenblat'soriginalwork[ 226 ],theeffectivevelocityterm(Eq.( 7 ))wassimplyamathematicalsimplicationusedtosimplifythetransmissionexpression.However,ourunderstandingof(1)thefeedbackmechanismand(2)theequivalencebetweentheNatoriexpressionandourupdatedexpressionfortransmissionprovidesanewphysicalunderstandingoftheeffectivevelocityterm. Sincetheredenedeffectivevelocityisadeniteintegraloverthechannellength,wecansplitintotwoparts,withthesplittingpointbeingtheFEZwidthx0(forthegivenbiasconditions)as, 136

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1 e=ZL0e)]TJ /F9 7.97 Tf 6.59 0 Td[((x(x))]TJ /F15 7.97 Tf 6.59 0 Td[(0(0))dx D=Zx00e)]TJ /F9 7.97 Tf 6.58 0 Td[((x(x))]TJ /F15 7.97 Tf 6.58 0 Td[(0(0))dx D+ZLx0e)]TJ /F9 7.97 Tf 6.59 0 Td[((x(x))]TJ /F15 7.97 Tf 6.59 0 Td[(0(0))dx D (7) or, 1 e=1 e1+1 e2 (7) Thetwoequivalenttransmissionexpressionswehaveare, TN=1 T+ 1)]TJ /F5 11.955 Tf 11.96 0 Td[()]TJ /F9 7.97 Tf 6.59 0 Td[(1 (7) tUF=1 2+c(0) 2c(L)c(L) e+K)]TJ /F9 7.97 Tf 6.59 0 Td[(1 (7) whereK=e)]TJ /F9 7.97 Tf 6.58 0 Td[((L(L))]TJ /F15 7.97 Tf 6.58 0 Td[(0(0)). EquatingEqs.( 7 )and( 7 ),weget, 1 T+ 1)]TJ /F5 11.955 Tf 11.96 0 Td[(=c(0) 2e+K 2c(0) c(L)+1 2(7) Expandingthevelocityintegralterm,wehave, 1 T+ 1)]TJ /F5 11.955 Tf 11.95 0 Td[(=c(0) 21 e1+1 e2+K 2c(0) c(L)+1 2(7) Whentheappliedgateanddrainvoltagesarehigh,theKtermbecomesmuchsmallerthanunityandcanbeneglected(SectionIVof[ 219 ]).Sincex0!0underthesevoltageconditions,T!1(i.e.,e1termvanishes),makingetobeentirelycomposedofthee2term,and.Thereforewehave, 137

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1)]TJ /F5 11.955 Tf 11.96 0 Td[(=c(0) 21 e)]TJ /F3 11.955 Tf 13.15 8.08 Td[(1 2(7) Solvingfor(highbias),weget, =1)]TJ /F5 11.955 Tf 19.12 8.08 Td[(e c(0) 1+e c(0)(7) LookingattheformoftheeterminEq.( 7 ),wecanseethatitrepresentsadiffusion-velocitylikequantity.Intheabsenceofadrainbias,edescribeshowefcientlythecarrierswoulddiffuseduetoagivenconcentrationgradientalongthechannel.Themaximumdiffusionvelocityisobtainedunderthecompleteabsenceofscatteringevents,andisequaltothethermalvelocityofthecarriersc(0).e,therefore,canatthemostbec(0).Whene=c(0)becomeslessthan1,itindicatesthepresenceofthescatteringeventsinthechannel.Underanapplieddrainbias,theeffectivenessofthesuperimposeddriftresponseofthecarriersdependonthediffusibilityofthechannel,i.e.,theeasewithwhichcarriertransporthappens.Thisisdescribedbythee=c(0)ratio.Withthisinterpretation,wecanreadilyseehowscatteringinuencestheoveralldevicecurrent.Theimpactofscatteringeventsonthecurrentisexplainedthroughtheparameter. Inaquasi-ballisticdevice,thesaturationcurrentisproportionalto(1)]TJ /F5 11.955 Tf 12.61 0 Td[().FromEq.( 7 ),whenthee=c(0),theelastic-backscattering/energyrelaxationtrade-offbetweenthezonesvanishmaking=0.Inotherwords,bothpartsofthechannelhaveunitytransmission,makingthetransportcompletelyballistic.Thus,wecannowunderstandthephysicalsignicanceoftheeffectivevelocityterm.Usingthisconcept,wepresentanewdenitionofballisticefciencyinthedeviceas, B=(1)]TJ /F5 11.955 Tf 11.96 0 Td[() vF(7) 138

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whererepresentsthepositionandenergyaverageddriftvelocity, =xR0ER0n(x,E)v(x,E)dEdx xR0ER0n(x,E)dEdx(7) andvFrepresentstheFermivelocityatthebeginningofthechannel.Whenbecomeszero,carriersatthetopofthebarrierenterthechannelwithFermivelocityvF.Whentherearenoscatteringeventsalongthechannel,=vF,makingB=1.Notethatcurrentisstilllimitedbynumberofavailablestatesatthebeginningofthechannel. 7.3.6ExpandingNatori'sHighFieldModelforMOSFETs WhileNatori'sideaofdividingthechannelintotwozoneswithdistincttransmissionpropertiesisquitegeneric,theapplicationofthismodelfor2Dor1Dinversionlayersneedstheinclusionoftheeffectsofgatetochannelpotential.Figure 7-7A showstheevolutionoftheconductionbandproleforvariouscombinationsofappliedgateanddrainbiases.Atthetopofthebarrier,theE-Kdiagramandoccupationofelectronsareillustrated.Carrierscanattainkineticenergyduetodrainbiaswhiledriftinginthechannel,aswellasfromthedegenerateoccupationoftheconductionbandduetohighgatebias.AccordinglytheseparationofthechannelintermsoftransmissioncharacteristicsareshownforallfourcasesinFigure 7-7B Case1:Heretheappliedgateanddrainvoltagesarelow(weakinversion,linearregime).Thepotentialproleinthechannelisalmostlinearandthecarrierstatisticseverywhereinthechannelarenon-degenerate.Hence,thekineticenergyofthecarriersremainsconstant(nearthethermalenergy)makingOPscatteringdominantonlyatthedrainentrance.Thetransmissionpictureisthereforesimple,withtheFEZexpandingtocovernearlytheentirechannel. Case2:Whenthedrainvoltageisincreasedkeepingthegatevoltagestilllow(weakinversion,saturationregime),thekineticenergyofthenon-degeneratecarriersintheinversionlayerincreasesalongthechannel.Thisinreturn,increasestheOP 139

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scatteringprobabilityinsidethechannel,makingtheOPEZcoveralargerportionofthechannel. Case3:Hereweconsideralargegatevoltageandsmalldrainvoltageappliedtothedevice(stronginversion,linearregime).Thedegeneratecarriersatthetopofthebarrierhavekineticenergyfromthegatepotential,andafractionofthesecarrierswillrelaxthroughOPscatteringclosetothebeginningofthechannel.Theprimedquantities(T0and1)]TJ /F5 11.955 Tf 12.08 0 Td[(0)describethezonetransmissionsrepresentingenergyrelaxationprocessofdegeneratecarriersonly.Whengatetochannelpotentialdifferencereducestothenon-degenerateconditions,thecarrierkineticenergyreducestonearthermalenergies,andcarriertransportfromthispointresemblescase1. Case4:Withincreaseofdrainbias(stronginversion,saturationregime)fromcase3,thereductionofthekineticenergyofthedegeneratecarriersiscompensatedbytheenergygainedfromhighdraineld.TheOPEZzonesincase3merge(sincetransmissionthroughthemiddleTzonesaturates)andthechannelintheseconditionsisalmostentirelycomprisedofoneOPEZ.Thisleadstoatotaltransmissionof(1)]TJ /F5 11.955 Tf 12.34 0 Td[()aspredictedbyNatori. 7.4QualitativeUnderstandingofStrainEffects Strainchangesbondanglesandlengths,whichmodiestheconductionandvalencebands.Thesemodicationsincludebandwarpingandbandsplitting,resultinginmodicationofconductivitymassandscatteringrates[ 229 ][ 230 ].Forelectroninversionlayers,theuniaxialstresseffects[ 231 ]comesfrom:(1)conductivitymassdecreasefromtheelectronsrepopulatedfrom4to2valleys(2)OPscatteringsuppressionduetothedensityofstatesdecreasefrom4to2(3)conductivitymassdecreaseduetothebandwarping.Forholeinversionlayers,thestresseffects[ 232 ][ 233 ]arisesfromsplittingandwarpingofheavy-holeandlight-holebandsthatresultsin(1)signicantlyreducedholeconductivitymassand(2)signicantlyreducedOPscattering. 140

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Therehavebeenseveralexperimentalworks[ 234 ][ 235 ]publishedthatshowthedifferencebetweenlinearcurrentandsaturationcurrentenhancementwithuniaxialstressunderstronginversionconditions.Ithasbeenobservedthatforholes,theID(sat)enhancementdropsbynearly50%comparedtoID(lin),andwhileID(sat)enhancementislowerforelectrons,thediscrepancyisnotasmuch. LookingatthetransportcharacteristicsshowninFigure 7-7B ,thedifferencebetweencarriertransmissionsunderlinearandsaturationconditionsareimmediatelyseen.Sincethecurrentisproportionaltotheproductofaveragedriftvelocityandtotaltransmission,thesimultaneousimpactofstrainoneachofthesetwoparametersforlinear/saturationcasesneedtoconsidered.Thevelocitytermisinverselyproportionaltothesquarerootoftheconductivitymass.Theincreaseinthevelocitytermstrictlydependsonthestraininducedreductionofeffectiveconductivitymass.Thestraindependenceofthetransmissiontermissomewhatmorecomplex,sinceitisdependentonbothbothscatteringandmass. Underfavorablestressconditions,inversionlayerelectronsexperiencereducedconductivitymasswithminimalchangesonscattering.Butforholes,thefavorablestrainresultsinsignicantreductionofbothcarriermassandscattering.Fromthestrainperspective,thebenetsofimprovedaveragecarrierdriftvelocityisfeltinbothlinearandsaturationregimes.ThedifferencesinID(lin)andID(sat)enhancementstherefore,mustcomefromthedifferencesinhowstrainaffectsthetotaltransmissionfortherespectiveregimes,forbothtypesofcarriers. Letusconsiderthelinearregimerst.Forahighgatebiasinthelinearregime(Case3inprevioussection),theinitialFEZregionbecomesreallysmall(i.eT0!1).Thechanneltheniscomprisedofthreeregions,aninitialOPEZcharacterizedby1)]TJ /F5 11.955 Tf 12.06 0 Td[(0,themiddleelasticregioncharacterizedbyTandthenalinitialOPEZcharacterizedby1)]TJ /F5 11.955 Tf 11.96 0 Td[(.Carrierinterchangeoccursatbothzoneboundaries. 141

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Forholes,strainsignicantlyreducestheOPscattering,whichincreasesandreducestotaltransmission.Sincethemassreductionfromstraininducedbandwarpingissignicant,theincreaseindriftvelocityismuchbiggerandcompensatesthestraininducedtransmissionloss.MostoftheenhancementinholeID(lin),therefore,comesfromtheaveragedriftvelocityincrease.Forelectrons,theeffectofstrainonscatteringissmall,andthecorrespondingtransmissionlossisalsosmall.Mostofthelinearcurrentenhancementforelectronscomesfromtheaveragedriftvelocityincreaseaswell.Theamountofvelocityincrease,however,issmallerforholescomparedtoelectrons. Inthesaturationregime,themiddleelasticzonetransmissionTsaturatesandtheOPEZzonesmergetogether.Thestrainimpactonvelocityisstillpresentforbothtypeofcarriers.OPscatteringhappensinalargerportionofthechannel.Forholes,thereductionintransmissionfromstrainismuchmorecomparedtolinearregime,resultinginareducedenhancementinID(sat).Forelectrons,thescatteringreductionissmall(comparedtolinearregime)andit'seffectoncurrentisnotfeltasmuch.ThustheelectronID(sat)alsoreduces,butnotasmuchcomparedtoholeID(sat). 7.5Summary Inthischapter,wereviewedthefundamentalsofseveralsignicantmodelsdescribingquasi-ballistictransport.Theone-uxtheorybasedLundstromModelandGildenblatModelwereexaminedindetail.Byaddinghigheldeffectsandconnementeffects,leadingtoanew,surfacepotentialbased,analyticalexpressionforthetransmissioncoefcientinaMOSFETchannelwasestablished.Thisexpressionisapplicableforallconditionsofdeviceoperationandforallsuface/channelorientations.ThephysicalsignicanceoftheeffectivevelocitywashighlightedanditwasshowntobeanimportantgureofmeritfordescribingthetransportinaMOSFETchannel.TheessentialphysicsofNatori'shigheldtransportmodelwasexplainedusingasimplemathematicaltreatmentandthemodelwasextendedtoincludetheeffectofgateconnement.Bylinkingthesetwoupdatedmodels,experimentallyreporteddifferences 142

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betweenstrainenhancedlinearandsaturationcurrentsinshortchanneldeviceswasqualitativelyexplained. 143

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CHAPTER8OPTICALCHARACTERIZATIONOFALGAN/GANHEMTTRAPGENERATION 8.1Motivation Afteritsrstdemonstrationin1994[ 14 ],AlGaN/GaNHEMTdevicesgainedawelldeservedpopularityforitshighfrequency,highpoweroperationwithhightemperaturestability,especiallyforhighpowermicrowaveapplicationssuchascommunicationsandradars.In1996,therstreportedmicrowavepoweroperationoftheAlGaN/GaNHEMTwas1.1W/mmpowerdensityat2GHzoperationfrequency[ 236 ].Thecontinuousinterestonhighfrequency,highpowertransistorsboostedresearcheffortswitha30W/mmHEMToperatingat8GHzdemonstratedin2004[ 17 ].WhiletheprogressontheAlGaN/GaNHEMTperformanceimprovementhasbeensustained,highpowerandhigheldoperationhaveexacerbatedreliabilityissuesofGaNbasedHEMTdeviceswhichhaslimitedtheachievementofthecompletepotentialofthesedevices[ 21 ]. AswehavediscussedinChapter 3 ,theabundanceofdefectsintheGaNmaterialsystemcancreatetrappingcentersinthebulkGaNregion,intheAlGaNbarrier,attheAlGaN/GaNinterface,attheNi/AlGaNgatemetalinterface,andattheungatedregionoftheAlGaNsurface[ 21 23 ].Inadditiontothenativedefectsformedduringcrystalgrowthanddevicefabrication,additionaldefectscanbegeneratedduringdeviceoperation.OneofthetheoriesfordegradationofAlGaN/GaNHEMTdevicesisthestructuraldefectformationintheAlGaNbarrierduetotheadditionoftheinversepiezoelectricstressathighgate-to-drainelectriceldstothelargebuilt-inepitaxialstress[ 20 26 ].ThestructuraldefectintheAlGaNlayercausedbytheincreasedstressisbelievedtocreatealeakagepathfromgatetochannel.AnothertheoryfortheincreasedgatecurrentisthemetaldiffusionintotheAlGaNbarrierathighgateelds,whichisalsoknownasgatesinking[ 21 ].ThereducedthicknessintheAlGaNbarrierwherethegatesinkscreatesaleakagepathbynarrowingtheAlGaNpotentialbarrier.Furtherreliabilityandstabilityissuesarecurrentcollapse,drainlag,andgatelag.Currentcollapseoccurs 144

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duetothetrappingofhotelectronsathighVDSconditionsintheslowdeepbulktraps,intheAlGaNbarrierorattheAlGaNsurface[ 21 25 ].Thesetrappedelectronschangetheelectrostaticsofthechannelandlocallydepletethe2DEG.Thisinturnleadstoatimedependentdecreaseinthedraincurrent.Oncethedraincurrentiscollapsed,thethermalemissionofelectronsfromthetrapsitesisrequiredforrecoveryofthecurrentcollapse.Hence,ifthedrainbiasispulsedfrom0Vtoahighvoltageandbackto0V,thedraincurrentlagsduetothetrappinganddetrappingprocessesrelatedtocurrentcollapse(drainlag).Similarly,theelectronstrappedontheAlGaNsurfacestatescreatesanegativevoltageandactsasavirtualgateonthedevice.WhenthegatevoltageispulsedfromdepletiontoON-condition,thedraincurrentlagsduetotheemissiontimeconstantofthelledsurfaceAlGaNstates,i.e.theopeningofthevirtualgatenearthedrainsideofthechannelwhichresultsingatelag. Clearly,built-intraps,structuraldefectformation,andtrapgenerationinAlGaN/GaNHEMTsarestillimportantissuesthatneedtobestudiedandsolved.Inadditiontothebuilt-inepitaxialstress(3GPafor28%Alconcentration),theinternalmechanicalstresscreatedduringdeviceoperationduetothethermalexpansioncoefcientmismatchandbytheinversepiezoelectriceffectintheAlGaNbarriermayaddtothebuilt-instressandgeneratestructuraldefects[ 20 26 ].ItisalsoknownthatlargelocalhydrostaticstraineldsareinducedinthelatticebypointdefectsinGaNbasedmaterials[ 237 240 ].Therefore,internalmechanicalstresscanalterthegenerationrateofpointdefects,suchasGaandNvacancies(VGaandVN),interstitialGaandNatomsinthelattice(GaiandNi),substitutionalOonNsite(ON),aswellasthediffusionofthegatemetalintotheAlGaNlayer.Hence,electricallyactivetrapcharacteristicsareexpectedtochangeoverlongperiodsofnominalhighpoweroperationduetoacombinationofelectricalandinternalmechanicalstressingatelevatedtemperature.TodecouplethecontributionofmechanicalstressinAlGaN/GaNHEMTdegradation,asystematicstudyofappliedexternalmechanicalstressduringdevicedegradation 145

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Figure8-1. AlGaN/GaNHEMTonSiCwaferpieceisattachedtoacarbonsteelplatewithepoxy(topleft).Thesubreticlesareshownontheright.TheTLMmoduleislocatedatthebottomofeachsubreticle.ThecrosssectionalsketchoftheAlGaN/GaNheterostructure,2DEGandsource/draincontactsareshownonthebottomleft. andtrapcharacterizationisrequired.Consequently,itiscrucialtoincorporateexternalmechanicalstressingintoestablishedtrapcharacterizationmethodstoinvestigatetheeffectofmechanicalstressonAlGaN/GaNtrapgeneration.Toachievethisgoal,wedevelopedatrapcharacterizationmethodbasedonthephotoionizationspectroscopytechnique,integratedintocombinedelectricalandelectricalandmechanicalstressingexperimentsforcontrolledtrapgenerationandcharacterizationinonesingledevice. 8.2OverviewofHEMTDevice Inthisstudy,AlGaN/GaNtestwafersareusedtoinvestigatethemechanicalstresseffectsonAlGaN/GaNHEMTtrapgeneration.Thetestwaferlayoutconsistsoffoursubreticles,AtoD.EachsubreticleispopulatedwithoneFATFET,onecentergateshortchannelHEMT,oneoff-centergateshortchannelHEMT,one1mlongchannelHEMT,oneungatedHEMTsite,isolationstructures,vanderPauwteststructuresandtransferlengthmodule(TLM)structures.Amongthesubreticles,thelengthoftheshortchannelHEMTvariesas(A)100nm,(B)125nm,(C)140nmand(D)170nm. 146

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InordertoeffectivelyphotoionizethetrapsintheAlGaN/GaNlayerswithoutblockingtheincidentphotonsbythegatemetal,wechoosetheTLMstructuresandshortestchanneloff-centergateHEMTdevicestobeincludedinourstudy.TheTLMstructuresarelocatedatthebottomofeachsubreticleinthecurrentlayout(Figure 8-1 ).TheTLMstructuresarepatternedina3-by-2layoutwitharowseparationof50m.ThelengthoftheTLMstructuresvaryfrom5mto30mwithincrementsof5m.AllTLMdeviceshaveaxedwidthof75m.Fortheshort-channeloff-centergateHEMTs,thegate-to-drainspacingis3mandthegate-to-sourcespacingis1m.Sincetheportionofthechannelcoveredbythegatemetalisminimal(0.1m),thesedevicescanalsobeeffectivelyphotoionizedbytheincidentlight.TheepitaxialAlGaN/GaNheterostructureisgrownuniformlythroughoutthewholewafer.Hence,boththeTLMandHEMTsitesarecomposedofa15nmthickAl0.28Ga0.72Nbarrierwitha2.25mthickFe-dopedGaNbufferlayerontopofanAlNnucleationlayer.Thebottomsubstrateiseitherasemi-insulatingSiorSiCsubstrate.ThesourceanddrainTi/Al/Ni/Auohmiccontactsareformedaftera20secondsannealat850C0.AlldeviceshaveawidebandgapSiNpassivationlayerontop,whichistransparenttothewavelengthsconsideredinthisstudy. Thebuilt-in2DEGdensityfortheSiCsubstratesamplesismeasuredfromVanderPauwstructuresandfoundtobe11013cm)]TJ /F9 7.97 Tf 6.58 0 Td[(2.Additionally,wehavealsoconductedcapacitance-voltagemeasurementsonthe125mdiametercircularCV-dots(Figure 8-2 -a).Byintegratingthecapacitanceforabiasrangeof-6V(depletion)to0V,wecanobtainthedensityofthe2DEG(Figure 8-2 -b).ItisfoundthattheSiCsubstratedeviceshaveahigher2DEGforagivengatebiascomparedtoSisubstratedevicesduetoalowerthresholdvoltage.The2DEGdensityobtainedfromtheCVmeasurementsat0VfortheSiCdevicesissimilartothevalueextractedfromtheHallmeasurements.Thisindicatesaminimaleffectofmetalworkfunctiononthesurface 147

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A B Figure8-2. Capacitance-voltagemeasurementresults(a)andtheextracted2DEGdensity(b)areshownforAlGaN/GaNHEMTonSiCandSiCV-dots.Thethresholdvoltagedifferencebetweentwosamplescreateasignicant2DEGdensityatagivengatebias. potenialattheAlGaN/GaNinterface.Hence,the2DEGdensityofungateddevicescanbeapproximatedbythe2DEGobtainedfromCV-dotsat0Vgatebias. 8.3Method TheAlGaN/GaNHEMTtrapgenerationisinvestigatedbyacombinationofcontrolledtrapgenerationandphotoionizationspectroscopytrapcharacterizationexperiments.Thecontrolledtrapgeneration(degradation)inthedeviceisperformedbystressingthedevicewhilemonitoringsuddenincreasesinthegatecurrentanddegradationinthedraincurrent.Theenergydistributionofarealtrapdensityisobtainedbymeasuringthedraincurrenttransientsduetotheadditionalphotoionizedelectronemissionfromthetrapstatesatdifferentsubbandgapwavelengths(365nm).ByrestrictingthephotonenergiesbelowtheGaNenergybandgap(3.4eV),theopticalexcitationislimitedtothesubbandgaptrapstatesandnonewelectron-holepairsaregenerated. AswehavediscussedinSection 3.2.6 ,photoionizationspectroscopypreviouslyhasbeenemployedfortheinvestigationofslowbulkGaNtrapsandimpuritysitesvia 148

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Figure8-3. Thecarrierdynamicsofthephotoionizationspectroscopytechniqueareillustrated.(a)Deviceisunderdarkconditionswithtrappedelectrons.(b)Devicein(a)isilluminatedwithphotonsofenergygreaterthantheelectrontrap.(c)Deviceisunderdarkconditionswithneutralholetraps.(d)Devicein(c)isilluminatedwithphotonsofenergygreaterthantheneutralholetrap. thereversalofdraincurrentcollapseinGaNdevices[ 82 86 ].ThecarrierdynamicsofphotoionizationspectroscopyareshowninFigure 8-3 .ForadeviceunderdarkconditionswithoccupiedelectrontrapsofactivationenergyEA(Figure 8-3 -a),opticalilluminationfromanincidentphotonwithanenergyequaltoorgreaterthanEAresultsintheemissionofthetrappedelectrontotheGaNconductionband.ThephotoionizationofthetrappedelectronisshownasinFigure 8-3 -b.WiththeassumptionthatthetimerequiredtotraversetheHEMTchannelisshorterthanthetrapcapturetime,thetrapstateatEAbelowtheconductionbandiscontinuouslyemptiedtoanon-equilibriumsteady-state,proportionaltothedensityofincidentphotonux.Hence,theelectronsresidingatmultipletrapswithactivationenergylowerthantheincidentphotonenergycontributetothe2DEGandincreasethedraincurrentduringtheopticalillumination. 149

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Assumingthatthecarriermobilityremainsunchangedandlightintensityishighenoughtoachievecompletephotoionizationofalltraplocationsforthegivenphotonenergy,theratioofthetotaltrapdensitytothe2DEGisobtainedbycomparingtheincreaseddraincurrentunderilluminationtothedarkcurrent.Forasmallincrementalchangeintheincomingphotonwavelength,thearealtrapdensityforthegivenenergyisapproximatedby, Dt(hf) N2DEG=I(hf))]TJ /F4 11.955 Tf 11.95 0 Td[(I(hi) Idark1 hf)]TJ /F4 11.955 Tf 11.96 0 Td[(hi,(8) wherefandiarethenalandinitialphotonfrequencies,Idarkisthedarkcurrent,andI(hf)andI(hi)arethestabilizeddraincurrentunderilluminationwithnalandinitialphotonfrequencies. Whiletheincreaseinthedraincurrentisobservedduetothephotoionizedtrappedelectrons,neutralholetrapswithasmalleractivationenergythantheincomingphotonenergydecreasesthedraincurrent.ThisphenomenoniscalledopticalquenchingeffectandisobservedinGaNbaseddevices,especiallyunderredandyellowbandillumination[ 241 242 ].ThefundamentalsofopticalquenchingisshowninFigs. 8-3 cand 8-3 d.Whenadeviceisexposedtoopticalilluminationwithanincomingphotonenergygreaterthantheactivationenergyofaneutralholetrap,anelectroninthevalencebandisexcitedtotheneutralholetraplocation,shownasinFigure 8-3 -d.Then,theresultingholeinthevalencebandrecombineswithanelectronfromthe2DEGanddecreasesthecurrent(inFigure 8-3 -d).Therefore,notonlyelectrontrapsbutalsoholetrapscanbecharacterizedusingthephotoionizationspectroscopytechnique.Consequentlybyusingasetofdifferentsubbandgaplightwavelengths,theenergydistributionofthearealelectronandholetrapdensitiesmaybecharacterized. 8.4ExperimentalSetup ToinvestigatethetrapgenerationbehaviorinAlGaN/GaNheterostructures,theexperimentalsetupshowninFigure 8-4 isdesignedtoconducttrapgenerationandcharacterizationmeasurement,viaelectricalstressingandphotoionizationspectroscopy 150

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Figure8-4. Theexperimentalsetupusedinthisstudyisshown.ThebroadspectrumradiationfromthearclampismonochromatedbyusingacomputercontrolledOrielMS257spectrometer.Then,themonochromatedlightisroutedandfocusedonthedevice-under-test(DUT).TheDUTiselectricallyprobedbytheKeithley4200-SCSsemiconductorcharacterizationsystemforbothtrapcharacterizationandtrapgenerationexperiments. (PS)techniquesrespectively.Forthecharacterizationofthetraps,acontinuouswavelengthsweepmonochromatorisused.Thisensuresthatasufcientresolutionintrapenergydistributioncanbeachieved.Thecontinuouswavelengthsweepcapabilityisrealizedbyusinga300WarclampcoupledwithacomputercontrolledMS257OrielspectrometerlocatedattheUniversityofFloridaMicrofabritechfacility.Thebeamatthemonochromatorapertureiscollimatedandfocusedusingtwoopticalneutraldensitylenses,whileitisshonedirectlyontheGaNsampleinthe4-pointbendingapparatususingametalmirror.Thesizeofthebeamspotonthesampleisassmallasapproximately3mmby1mm,whichinturnincreasesthephotonuxdensityincidentontheDUT.TheeffectsoftheincreasedincidentphotonuxdensityisshowninFigure 8-5 .Withafocusedbeam,theincreaseduxofincidentphotonsreducesthetransienttimetoreachthenon-equilibriumsteady-state,whileincreasingthelevelofthesteady-statecurrentduetoalargerpercentageofthephotoionizedelectronsatthistrapenergy. 151

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Figure8-5. Thelineardraincurrentresponsefroma30mlongTLMdeviceisshownforbothwithandwithoutfocusinglenses(i.e.highandlowintensity)650nmillumination. Byavoidinganopticalberlightguide,opticalintensitylossisminimized.DuringPSexperiments,aKeithley4200systemisusedinthesamplingmodetomeasurethelineardraincurrentasafunctionoftime.Themeasurementspeed,ltering,anddelaysettingsareoptimizedtoachieveadesirabletradeoffbetweenthesignaltonoiseratioandthesamplingratetocapturefasttrapping-detrappingbehaviors.ThedrainvoltageissettoavaluewithinthelinearoperationregimeofthedeviceduringPSmeasurementstoensurethatthecurrentforcedthroughthedevicedoesnotcausefurtherdegradationoradditionaltrapgeneration. Duringthewavelengthsweep,aUV-VISbandgratingwith1200lines/mmandablazingwavelengthof350nmisusedtodecomposethewavelengthsoftheradiationfromthearclamp.Inordertoavoidadditionalelectron-holepairphotogeneration,subbandgapilluminationisperformed,settingtheshortestphotonwavelengthtothe 152

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Figure8-6. ThetransmissioncharacteristicsoftheUV-rejectionhigherorderlterisshown. 365-360nmrange.Todeterminethevalueofthelongestincidentphotonwavelength,theeffectsofhigher-orderstraylightistakenintoconsideration.Foragivenincidentangleofthearclampradiationontoagratingpresettoawavelength,thehigher-orderphotonswithwavelengthsof=nleakfromthegratingatareducedintensity.Forexample,agratingsetfor680nmredilluminationwouldalsopassUVcomponentsat340nm,170nm,etc.witheverdecreasingintensity.Ifthehigher-orderwavelengthsarenotblocked,thiswouldleadtomultiwavelengthilluminationinsteadofthedesiredsetwavelengthandresultinanerroneoustrapcharacterizationeffort,especiallyinthecaseofhigher-orderaboveGaNbandgapphotonenergies.Tominimizetheexperimentalerrorsduetohigher-orderwavelengthsandtoincreasethemeasurementaccuracy,wehaveplacedaUV-rejectionlongpasslterwithacut-onwavelengthat345nm.ThemeasuredtransmissioncharacteristicsoftheUV-rejectionlterisshowninFigure 8-6 .Bynotingthelongestlteredwavelength(340nm),wesetthemaximumofthePSwavelengthsweepto680nm.Thisenabledustoperformahigher-orderfreeilluminationfrom680nmto360nmwithoutanyattenuationcausedbytheUV-rejectionlter. 153

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InSection 8.3 ,wehavediscussedthephysicsofthephotoionizationspectroscopytechnique.ItwasemphasizedthatEq. 8 isvalidoncethenon-equilibriumsteady-statephotoexciteddraincurrentisreachedforagivenwavelength.ThecarrierdynamicsshowninFigure 8-3 aremodiedwiththeincidentphotoncountperunitareaatagiventime.Forexampleataconstanttemperature,adeviceinthedarkwithtrappedelectronsisatthermalequilibriumwithequilibriumcarrierconcentrationdeterminedbytrappingandemissionofelectronsbetweentheconductionbandandtheelectrontraplevel.Oncethedeviceisilluminated,theincomingphotonschangetheequilibriumconditiontoanon-equilibriumstatewithamorefavorableemissionratethantrapping.Next,theemittedelectronsincreasethe2DEGdensityandmodifytheequilibriumtoamorefavorabletrappingrate.Withtheinterplaybetweentheincreasingdetrappingratebytheincidentphotonsandtheinstantaneousincreaseinthetrappingrateduetohigher2DEGdensity,thedevicereachestoasteady-stateconditionunderthenon-equilibriumopticalilluminationcondition.Foranincompletephotoionizationcase,thepercentageofdetrappedelectronsatthenon-equilibriumsteady-stateisincidentphotonuxdependent.ThereforetointerpretthemeasuredPSdatacorrectly,therelativeincidentphotonuxforeachwavelengthisnecessary. TheopticalintensityspectrumismeasuredusingacombinationofOriel70124pyroelectricdetector,Spectra-650colorimeter,andanOriel77346photomultipliertube.Thereasontousemultipledetectorsisthatnoneofthethreecoversthewavelengthrangeofinterestcompletely.Thepyroelectricdetectorhasatresponsivityforwavelengthslongerthanthecut-offat580nm.Hence,thepyroelectricdetectorisusedtoverifythereadingfromtheSpectra-650colorimeterforwavelengthslongerthan580nm(Figure 8-7 ).WhiletheSpectra-650isveriedabove580nm,ithasacut-offlowerthan380nm.Finally,byusingtheOriel77346photomultipliertubeandthecolorimeterforthe400nmto360nmrange,wewereabletoobtaintheopticalintensityfortheentirerangeofinterest.Therelativeopticalintensityisobtainedthrough 154

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Figure8-7. CrosscalibrationcurvesbetweenSpectra-650colorimeterandOriel70124pyroelectricdetector. theratiooftheentireintensityspectrumtothenearbandgapilluminationintensityat360nm,sinceatthiswavelengthallthesubbandgaptrapsenergylevelsareexcited.Then,bynormalizingtherelativeopticalintensitywiththeincidentphotonenergy,wehaveobtainedtherelativephotonuxforeachwavelengthasshowninFigure 8-8 .Sincetheincidentphotonuxisdifferentfordifferentwavelengths,themeasuredpercentchangeinlineardraincurrentwouldbeintensitydependentforanincompletephotoionizationassumption.Thenon-zeroslopeofthephotoexcitedcurrentversustimeatthenon-equilibriumquasi-steady-stateofthehighintensityilluminationshowninFigure 8-5 suggeststhatoursetuppartiallyphotoionizesthetrappedelectrons.Toaccountforthewavelengthdependentphotonux,Eq.( 8 )wasmodiedtoobtainaphotonux-normalizedrelativearealtrapdensitydistributiongivenby, RDt(hf) N2DEG=I(hf))]TJ /F4 11.955 Tf 11.96 0 Td[(I(hi) Idark1 (hf)]TJ /F4 11.955 Tf 11.96 0 Td[(hi)Cf,(8) whereCfistherelativephotonuxcomputedfromtheratiooftheentireintensityspectrumtothenearbandgapilluminationintensityat360nmperphotonenergy.Since 155

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Figure8-8. Thenormalizedphotonuxwithrespectto360nmilluminationvs.incidentphotonwavelengthisshown. theexposuretimeisthesameforeachwavelength,normalizationbyphotonuxisthesameasnormalizationbyphotoncount.Theterm1=[(hf)]TJ /F4 11.955 Tf 12.96 0 Td[(hi)Cf],adensitynormalizationcoefcient,representsthenormalizedeffectivetrapdensityperenergyatenergyhfcorrespondingtothedraincurrentchangeperincrementinwavelength.Forawavelengthsweepfrom650nmto360nmwithdecrementsof5nm,thedensitynormalizationcoefcientisshownwithrespecttobothphotonwavelengthandenergyinFigure 8-9 .Hence,thepercentchangeinthelineardraincurrentobtainedfromthePSmeasurementsarecalibratedwiththedensitynormalizationcoefcienttonormalizethemeasuredarealtrapdensityateachenergy.Althoughincompletephotoionizationoftrapshindersthecapabilityofthismethodtomeasuretheenergydistributionoftheabsolutearealtrapdensity(Dt(h)),underthesameilluminationconditions,therelativechangesofthemeasurednormalizedtrapdensity(RDt(h))beforeandafterdevicedegradationexperimentscanbemonitoredreliably,providingkeyinsightsintothetypeoftrapgenerationresponsiblefordevicedegradation. 156

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A B Figure8-9. Thedensitynormalizationcoefcientisshownfordifferentphoton(a)wavelengthsand(b)energiesforawavelengthsweepof5nmsteps. WhiletheKeithley4200-SCSsystemisusedforcharacterization,thehighpowerstateelectricalstressingofthedeviceswereperformedwiththecustombuiltelectricalstressandcharacterizationsystemlocatedattheNanoscaleResearchFacilityattheUniversityofFlorida[ 86 ].Thishighpowersystemiscapableofapplying60Vtoallthreeterminalsofthedevicewhilepassingseveralhundredmilliamperesforadesiredperiodoftimeandstresscondition,suchasstepstress,constantstress,step-recoverystress,orpulse-recoverystress. 8.5ResultsandDiscussion Inordertoavoidself-heatingeffects,thephotoinduceddraincurrenttransientsaremeasuredatthelowVDSregion,whenthedeviceisinthelinearoperationregime.Anothermotivationforconductingthephotoionizationspectroscopyexperimentsatlowdrainbiasesistoavoidfurthertrapgenerationduringcharacterizationoftheexistingtraps.Figure 8-10 showsthecurrent-voltagedatameasuredfrom30mTLMstructuresfromSiandSiCsubstratesamples.Thecloseagreementbetweenthepre(black)andpost(red)photoionizationspectroscopyI-Vcurvesunderdarkconditionindicatesthatthetrapdistributionofthemeasureddevicesarenotalteredduringdevice 157

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Figure8-10. Thecurrent-voltagecurvesareshownfor30mTLMstructuresonSiandSiCsubstrates.Measureddarkcurrentbefore(black)andafter(red)trapcharacterizationexperimentsareshown.ThebluecurvesdenoteI-Vobtainedunder365nmilluminationafterPSmeasurements. characterization.ItisimmediatelyseenthattheAlGaN/GaNHEMTonSiCdeviceshavelargerdrivecurrentwhichisattributedtothethresholdvoltagebaseddifferenceinthe2DEGdensityasshowninFigure 8-2 -b.AmuchlowersensitivitytonearbandgapilluminationisobservedinSiCbaseddevicecomparedtoSidevices,whichisexpectedduetothelargertrapdensityoftheSidevices. Atypicalphotoionizationspectroscopydatameasuredfroma5mTLMsiteonSisubstrateisshowninFigure 8-11 .Thewavelengthissweptfrom650nmto360nmwithdecrementsof5nmandanexposuretimeof60secondsateachwavelength.Theshutteristurnedoffafter60secondsofbandgapilluminationat360nm.Afterthelightisturnedoff,thelineardraincurrentdecreasesimmediatelyduetotherecombinationofthesmallnumberofphotoexcitedexcesselectron-holepairswith360nmillumination[ 66 ].TheinitialsharpdropduetorecombinationisaveryfasttransientduetotheverylowintrinsiccarrierconcentrationoftheGaN(10)]TJ /F9 7.97 Tf 6.59 0 Td[(10cm)]TJ /F9 7.97 Tf 6.59 0 Td[(3)[ 32 ].Aftertheinitialsharpdrop,thelineardraincurrentdecaysexponentially. 158

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Figure8-11. Atypicalphotoionizationspectroscopydatafroma5mTLMsiteonSisubstrateisshown.Forawavelengthsweepof650nmto360nmwithdecrementsof5nmandexposuretimeof60seconds,thephotoexciteddraincurrentevolutionwithchangingwavelengthsovertimeispresented. Theevolutionofthenearsteady-statenon-equilibriumphotoexciteddraincurrentwithtimeisusedtoobtaintherelativechangesbetweenincrementsinthephotonenergy.Figure 8-12 showstheextractedRDt(h)forTLMstructuresonSiandSiCsubstrates,respectively.ThetotalnumberoftrapsinSiCsubstratedevicesisfoundtobesmallercomparedtotheSisubstratedevices.ThemaindifferencebetweenSiCandSisubstratedevicesisobservedatthenearbandgaptrapdensities.ThenearbandgaptrapdensityinSiCbaseddevicesislessthehalfofthatinSisubstrate.IthasbeenshownthatthenearbandgaptrapsareattributedtostructuraldefectsinGaN[ 59 92 ].SinceSiChasacloserlatticeconstantmatchwithGaNcomparedtoSi(111)substrate,alowerstructuraldefectdensityisexpected,reducingthephotosensitivityofthedraincurrenttonearbandgapillumination. 159

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Figure8-12. Extractedphotonux-normalizedrelativearealtrapdensityisshownfora30mlongTLMonSiCanda5mlongTLMonSisubstrate.Thedarkcondition2DEGdensitiesareapproximatedusingtheCVdatafromtheCV-dotdevicesoneachsubstratetypes. Aftercharacterizationofthephotonux-normalizedrelativearealtrapdensitydistributiononvirginTLMstructuresonSiandSiCsubstrates,wehavedemonstratedtherepeatabilityofthephotoionizationspectroscopymethodusingavirgin100nmoff-centergateHEMTonSiCsubstrate.Toachievethis,wehaveconductedtwoconsecutivePSexperimentsonthesamevirgindevice.BetweeneachPSrun,thedevicewaskeptindarkandtheterminalcurrentsweremonitoredtoensurethatthedevicereturnstopre-PScondition.DuringthePSmeasurements,thedrainvoltagewaskeptwithinthelinearregimeat0.2V.InordertoensurethatthePSsignalwasobtainedfromthegatedportionofthechannel,thegatebiaswasadjustedsuchthatthegatedportionofthechanneldominatesthetotalsource-to-drainresistance.UsingtheCVdata,thegatevoltageforthePSexperimentswaschosentobe-3.3V,forwhichthe2DEGdensityunderthegateis11010cm)]TJ /F9 7.97 Tf 6.58 0 Td[(2.Hence,the0.1m-longgatedportionofthechannelhas3ordersofmagnitudelowerelectrondensitycomparedtothe3.9m-longungatedpartofthechannel.Therefore,thechangesinthetotalchannelresistancearesolelyduetothephotoionizationofthetrappedelectronsunderthegate.Thedata 160

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A B Figure8-13. TherepeatabilityofthePSexperimentisdemonstratedonavirgin100nmoff-centergateHEMTonSiCsubstrate.(a)ThepercentchangeindraincurrentisshownfortwoconsecutivePSruns.(b)Theextractedphotonux-normalizedrelativearealtrapdensityisshownusingthedatashowninleft. obtainedfromrepeatedPSmeasurementsisshowninFigure 8-13 .Theerrorinthepercentchangeinphotoionizeddraincurrentforrepeatedmeasurementsisfoundtobeontheorderof0.5%.Hence,agoodagreementisobservedintheenergyrelativedistributionofextractedarealtrapdensityforconsecutivemeasurements. Next,theeffectsofOFF-stateelectricalgatestressingontheHEMTtrapcharacteristicsareinvestigatedusingphotoionizationspectroscopy.A100nmoff-centerHEMTonSiCdevicewascontrollablydegradedbyelectricalstepstressingofthegatefromVG=-5Vwithdecrementsof1Vforaperiodof70secondseach.DuringtheOFF-stategatestress,bothdrainandsourceterminalsweregrounded.ThedeviceissubjectedtoOFF-statestresstwicetoinvestigatebothinitialandfurtherdegradationeffects.Intherstrun,similartothemethodin[ 20 ],thegatecurrentdensityismonitoredwhilesteppingthereversegatebiasuntilasuddenjumpisobserved.Oncethedrasticsuddenincreaseingatecurrentisdetected,theexperimentisautomaticallystoppedbeforethedeviceisfurtherdegraded.Forthesecondrun,thedeviceisfurtherstresseduntilthe 161

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Figure8-14. DatacollectedfromtwoconsecutiverunsofOFF-stateelectricalgatestressexperimentsisshownfora100nmoff-centergateHEMTonSiCsubstrate.Duringthe1ststressexperiment,thegatecurrentdensityjumpisobservedatVcrit=-11V.Forthe2ndrunoftheOFF-statestressing,thegateleakageisfoundtobemuchnoisierandoneorderofmagnitudelarger. gatecurrentdensityhitsthelimitof800A/cm2.TheelectricalOFF-statestressingdataisshowninFigure 8-14 .ThecriticalvoltageforwhichthesuddenjumpinthegatecurrentwasobservedisfoundtobeVcrit=-11V,comparabletotheresultsshownin[ 243 ].Forthesecondstressexperiment,thegatecurrentisfoundtobeanorderofmagnitudelargerwithasignicantnoisecomponent. Thepre-stressandpost-stressgatecurrentdensityandwidthnormalizeddraincurrentofthedeviceareshowninFigure 8-15 andFigure 8-16 .Comparedtothevirgindevice,thegateleakagehasincreasedbyanorderofmagnitudeaftertherststressrun.Whilethechangesinthegatecurrentaresignicant,thedifferencebetweenthepre-stressandpost-stresslineardraincurrentsisnegligible.Ontheother 162

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Figure8-15. Gatecurrentdensityisshownforpre-stressandpost-OFF-stateelectricalstressconditions.Withsecondstressrun,thegateleakagehasincreasedmorethantwoordersofmagnitude. handthesecondgatestressingresultedinasignicantdegradationofthedevicecharacteristics.Thegateleakagehasincreasedbymorethan2ordersofmagnitude,becomingcomparabletothedraincurrent.Whilethe2DEGdensitywasfoundtobeunchangedforthegivendegradationcondition,adecreaseinthedraincurrentisobservedforagivengatebias,indicatingamobilitydegradation.Yet,atlargegatevoltagerange(above-2.5V),theparasiticRSDdominatesthechannelresistanceandtheeffectofreducedmobilitybecomesnegligible. ThetrapcharacteristicsoftheHEMTdeviceareexperimentallyextractedbothbeforeandafterOFF-statestress,usingthephotoionizationspectroscopymethodoutlinedpreviously.Thepre-stressandpost-stressphotonux-normalizedrelativearealtrapdensitiesareshowninFigure 8-17 .Comparedtothevirgindevice,nearbandgapelectrontrapdensitieshaveincreasedduringeachrunofelectricalOFF-statestressing.Theincreaseinthenearbandgaptrapsindicatesanincreasedstructuraldefectdensity 163

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Figure8-16. Widthnormalizeddraincurrentisshownforpre-stressandpost-OFF-stateelectricalstressconditions.Whiletherststressingdoesnotchangethedraincurrentmuch,furtherstresseddeviceexperiencesareductionin2DEGmobility. Figure8-17. Extractedphotonux-normalizedrelativeelectronandholearealtrapdensitiesareshownforbeforeandaftertheOFF-statestress.Asignicantincreaseinthestructuraldefectdensity(nearbandgaptraps)isobserved. [ 59 92 ].ForagatevoltagestressatandbeyondtheVcrit,thisresultisinaccordancewiththecrackformationhypothesisduetotheinversepiezoelectriceffect[ 20 ]. 164

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8.6Summary Usingthephotoionizationspectroscopytechnique,wehavedevisedanexperimentalbaselinetomeasureandcharacterizetrapsinAlGaN/GaNbaseddevices.Thedetailsofthemethodandtheexperimentalsetupareprovided.TheenergyrelativedistributionofthearealelectronandholetrapdensitiesfromTLMandHEMTdeviceswithbothSiandSiCsubstratesareextracted.ItisfoundthattheSiCbaseddeviceshavemuchlowertrapdensitiesaroundtheGaNbandgap,whichisattributedtothestructuraldefectsintheGaN.ForHEMTdevicessubjectedtoelectricalOFF-stategatestressing,itisfoundthattheelectrontrapsrelatedtothestructuralGaNdefectshasincreased.Thissupportsthepreviousndingsintheliteratureonthehypothesisofcrackformationatthedrainedgeofthegateduetoinversepiezoelectriceffect. 165

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CHAPTER9SUMMARYANDRECOMMENDATIONFORFUTUREWORK 9.1ResearchSummary ThemainfocusofthisresearchistounderstandtheeffectsofmechanicalstrainonsiliconMOSdeviceperformanceenhancementasthetransverseandlongitudinaldevicedimensionsscaledownandonthereliabilityofAlGaN/GaNHEMTdevices. Physicalinsightsontheexperimentallyobservedcomparableelectronmobilityin(100)and(110)sidewallundoped20nmthicksiliconFinFETswithhigh-/TiNgateareprovided.(100)and(110)sidewallsiliconFinFETswithboronandphosphorusdopedchannels,SiO2/poly-Sigatestacks,andthickernswerefabricatedtoexplorethepotentialunderlyingreasonssuchasundopedchannel,metal-gateinducedstress,andvolume-inversion.Comparableelectronmobilityon(100)and(110)FinFETswasobservedfromalloftheseteststructures.Usingamodiedself-consistentdouble-gateFinFETsimulator,itwasshownfor(110)FinFETthattheeffectoftheheaviernonparabolicconnementmassofthe2valleyleadstoamuchreducedvalleyenergy,making2thegroundvalley.Hence,inversionelectronsmostlyoccupythe2valleyforbothsurfaceorientationsandthus,acomparableaverageelectrontransporteffectivemassisobtained. Followingthediscussionontheelectrontransportindouble-gateFinFETs,thewidthdependenceofthestrainresponseofp-typetri-gateFinFETswasinvestigated.Usinga4-pointwaferbendingjig,low-eldpiezoresistancecoefcientsoftri-gateFinFETswithvaryingwidthsfrom500nmto20nmwereextractedathighoverdriveconditions.Itwasfoundthatforthenwidthrangeinvestigated,volume-inversioneffectswerenegligible.Theextracted-coefcientsoftri-gateFinFETswereaccuratelymodeledusingaweightedconductanceaverageofthetopandsidewallsurfacebulkpiezoresistivecoefcients. 166

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WithfurtherscalingofthesiliconFinFETsbothintransverseandlongitudinaldirections,gate-all-aroundnanowiretransistorsoperatingatnearballisticlimitwereconsidered.Thestrainenhancementofp-typeballisticsiliconnanowireMOSFETswasstudiedusingsp3d5sbasisnearest-neighbortight-bindingsimulations.Bycouplingthetight-bindingbandstructuretoacompactelectrostaticmodel,highcomputationalefciencywasachievedwhileprovidingsimilartrendstoself-consistenttight-bindingsimulations.Theimpactofnanowirechanneldirectionandcrosssectionsizeonthestrainenhancementofballisticholetransportwasexplainedbythestrain-inducedmodicationofthe1Dnanowirevalencebanddensity-of-states.Furtherinsightswereprovidedforfuturep-typehigh-performancesiliconnanowirelogicdevices. Inordertounderstandthecompleteschemeofstraineffectsinn-typesiliconMOSdevices,aphysicsbasedinvestigationofthestrainenhancementofsurfaceroughnesslimitedmobilitywasconducted.Experimentalevidenceofapossiblestrain-inducedmorphologychangewasobtainedthroughelectricalcharacterizationofstrainednMOSdevicesatlowtemperaturesandAFMmeasurementsofsiliconsamplesunderexternalmechanicalbending.Theadhocmodicationofsurfacemorphologywasvalidatedbytheagreementbetweenelectricalandmaterialscharacterizationandthetransportsimulations. Tolinkthestraineffectsonlow-eldlinearcurrenttothequasi-ballisticsaturationcurrent,auniedcarriertransportmodelapplicabletoallrangeofMOSFEToperationconditionswasdevelopedusinganupdatedone-uxtheory.Anewsurface-potentialbasedMOSFETchanneltransmissionexpressionwasdevelopedwiththeinclusionofhigh-eldandcarrierconnementeffects.Theballisticityofthetransistorwasredenedusingthetransmissionexpressionandtheaveragecarrierdriftvelocity.Thetransmissionathigh-eldandstronginversionconditionswasshowntobedependentonthediffusibilityofthechannel,whichlinksthedrift-diffusiondominantlow-eldtransporttohigh-eldquasi-ballisticsaturationcurrentinoneuniedmodel.Byusing 167

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ouruniedmodel,MOSFETchanneltransmissioncharacteristicsweredeterminedundergivengateanddrainbiases.Thisinturnwasusedtounderstandthestrainmodicationoftheaveragecarrierdriftvelocityandchanneltransmission,throughaninterplayofstrain-inducedmodicationofcarriereffectivemassandscatteringrates. TounderstandtheimpactofmechanicalstrainonAlGaN/GaNHEMTtrapgeneration,anexperimentalmethodtoobtainthephotonux-normalizedrelativearealtrapdensityusingphotoionizationspectroscopytechniquewasdeveloped.Usingthismethod,thetrapcharacteristicswereextractedforbothungatedtransmissionlinemodule(TLM)andgatedHEMTdevicesfrombothSiandSiCsubstrates.ItwasfoundforSiCsubstratedevicesthatthebuilt-instructuraldefectdensitywasmuchlowercomparedtoAlGaN/GaNHEMTdevicesonSisubstrateduetoacloserlatticeconstantmatchbetweenSiCandGaN.TheeffectofstepelectricalOFF-stategatestressingwasinvestigatedforHEMTonSiCdevices.ForrepeatedgatestressingexperimentsonthesameHEMTdevice,itwasshownthatthestructuraldefectdensityincreasesattheendofeachsuccessivedegradationexperiment.ThestructuraldefectformationwaslinkedtoanincreaseinthebiaxialtensionintheAlGaNbarrierthroughtheinversepiezoelectriceffectathighreversebiases,similartoobservationsfromotherresearchgroups. 9.2RecommendationforFutureWork Aswehavediscussed,strainedsiliconnanowireeld-effect-transistorsareexpectedtobeintroducedintohigh-volumemanufacturingatsub-10nmCMOStechnologynodes.Inordertofurtherunderstandtheadvantagesandissuesrelatedtothistechnology,asizeandorientationdependentexperimentalstudyofstraineffectsonelectronandholetransportinsiliconnanowiresisstronglyrecommended.Inadditiontothese,aniteelementmethod(FEM)basedinvestigationofanidealprocess-inducedstressorschemeforNWFETsisrequiredtoexploitthebenetsofmechanicalstrainatdeviceintegrationlevel.Thecumulativendingsoftheseeffortsmaytranslateintoamoresuccessful 168

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andcost-effectiveprocessintegrationtechnologydevelopmentofSiNWFETbasedhigh-volumemanufacturingICtechnology. InordertoimprovetheunderstandingoftherootcausesofAlGaN/GaNHEMTreliabilityissues,asimulationbasedmodelingoftheOFF-statestressgateleakageanddevicedegradationisrecommended.Toachievethis,measuredchangesintheenergydistributionofthearealtrapdensitiesduetothedevicedegradationexperimentscanbeincorporatedtofull-scaledevicesimulatorssuchasFloridaObjectOrientedReliabilitySimulator(FLOORS).Asaresult,furtherinsightsonadditionaldegradationphysicsmaybeobtainedbysimulatingdifferentdevicestructureandmaterialchoicesforAlGaN/GaNHEMTdevices.Basedonthesesimulations,devicesshowingimproveddegradationcharacteristicsmaybefabricatedforexperimentalverication.Consequently,theseverieddevicestructuresmaybeusedtoimprovethereliabilityofAlGaN/GaNHEMTtechnology. 169

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BIOGRAPHICALSKETCH MehmetOnurBaykanwasborninAdana,Turkey.HehasreceivedtheB.S.degreefromtheDepartmentofElectricalandElectronicsEngineeringatMiddleEastTechnicalUniversity,Ankara/Turkeyin2007,andM.SandPh.D.degreesfromtheDepartmentofElectricalandComputerEngineeringatUniversityofFloridain2009and2012respectively.HewasaninternDeviceEngineeratSEMATECHfromJune2010toMay2011.HeiscurrentlywithIntelPortlandTechnologyDevelopmentatHillsboro,OR,workingon22nm,14nmand10nmtechnologynodeintegration,developmentandyieldenhancement.Hisresearchinterestsincludecarriertransportandreliabilityinultra-scaledstrainedsiliconnanodevices,non-planarsiliconandcompoundsemiconductoreld-effect-transistors. 192