<%BANNER%>

Carbon Nanotube and Graphene Device Modeling and Simulation

expEgqVD whereEgisthechangeofbandgapafterstrain,isaveragecurrenttransmissioncoecient,kBisBoltzmannconstant,Tistemperature,qiselectronchargemagnitude,hisPlanckconstant,andVDisapplieddrainvoltage.Forexample,thesimpleestimationindicatesthattheminimalcurrentdecreasesbyafactorof53after2%oftensilestrainisappliedtoa(16;0)CNT,andthenumericalsimulationindicatesthatthecurrentdecreasesbyafactorof75.Thesimpleestimationgivesavalueinreasonablygoodagreementwiththenumericalsimulation,whichindicatesthattheincreaseofthebandgapandthethermalbarrierheightinthebulkpartofthechannelaremostlyresponsible 29

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AsmallamountoftorsionalstrainalsosignicantlychangesImin.A5ooftorsionalstrainona(16;0)CNTchanneldecreasesIminbyafactorofabout12,andthesamestrainona(17;0)CNTchannelresultsinsmallerbandgapandtheincreasedIminbyafactorof15.SimpleestimationsusingEqn.( 2{8 )alsogivesagoodagreementofIminvariationwiththedetailednumericalsimulations. 2-6 showsIonvs.(a)uniaxialstrainand(b)torsionalstrain.Thesolidlinesarefor(16;0)CNTFETsandthedashedlinesarefor(17;0)CNTFETs.Inordertoperformafaircomparison,acommono-currentofIoff=107Aisspecied.Fromthen-typeconductionbranchofasimulatedID-VGcharacteristic,thegatevoltageatwhichthespeciedIoffisachievedisidentiedasVoff,andtheon-currentIonisfoundatthegatevoltageofVG=Von=Voff+VDD,whereVDDisthepowersupplyvoltage(AcompletecontrolofthegateworkfunctioncanshiftthegatevoltagerangefromVoffVGVoff+VDDto0VGVDD).AlargerbandgapresultsinalargerSchottkybarrierheight,poorertransmission,andsmallercurrentspectrum.Thecurrentspectrumisgivenby h(f1f2)Tr(E); wheref1andf2areFermifunctionforsourceanddrain,respectively,Tr(E)istransmissionatacertainenergylevel.Spindegeneracyandvalleydegeneracyareconsideredwithafactorof4.Figure 2-7B showscurrentspectrumvs.energyaton-state.Thesolidlineisforunstrainedandthedottedlineisfor2%strained(16;0)CNTFET.Theoscillationiscausedbythequantuminterferenceatthesource/draincontactandthecurrentspectrumof2%strainedCNTFETisloweratoverallenergyrangeduetoitslargerSchottkybarrier 30

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2-8 showsintrinsicdelayversusIon=Ioff,whichisgeneratedusingthecomparisonmethoddescribedindetailinRef.[ 39 ].Abriefsummaryisasfollows.Foraspeciedpowersupplyvoltage,theon-currentandtheo-currentareobtainedbyreadingthevaluesattheedgeofthegraywindowinFig. 2-3A fora(16;0)CNTFETwithuniaxialstrain.Wheno-stateisdenedattheleftedgeofthiswindowwithawidthofVDD,on-statecanbegivenattherightedgesinceVon=Voff+VDD.Theintrinsicdelayiscalculatedby=(QonQoff)/Ionandonepointisobtained.Byassumingthetotalcontrolofthresholdvoltage,thegraywindowcanbesweptalongtheVGaxisandvs.Ion=Ioffcurveisgenerated(Onlyn-typeconductionbranchisexaminedbecausethen-typeconductionandp-typeconductionbranchesaresymmetric).ByrepeatingthisoneachCNTFETwithdierentstrains,thecomparisonofintrinsicdelayvs.Ion=Ioffispossible.Thisapproachallowsonetofairlycomparetheintrinsicspeedofadeviceatthesameon-oratioandpowersupplyvoltage[ 40 ]. 31

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2-8 .ThisnonmonotonicbehaviorcanbeexplainedusingFig. 2-9 ,whichsketchestheconductionbandprolevs.channelpositionfor2%strained(17;0)CNTFETatahighgateoverdrive.Ahighon-oratioisreachedwhenthegraywindowinFig. 2-3A isplacedatalowgatevoltagerangeofthen-typeconductionbranch,andtheon-oratiodecreasesasthegraywindowinFig. 2-3A movestothehighergatevoltagerange.AsIon=Ioffdecreasesandthegatevoltageatwhichtheon-stateisdenedincreases,theaveragecarriervelocityinitiallyincreases,whichleadstoasmallerintrinsicdelay,becausetheloweringofthechannelconductionbandedgeresultsinalargeraverageelectronkineticenergy.Asthegatevoltageatwhichtheon-stateisdenedfurtherincreasesandtheon-oratiofurtherdecreases,thebottomoftheconductionbandhitsthedrainFermilevelandthedraininjectionisconsiderablyincreasedasshowninFig. 2-9 ,whichsignicantlyincreasesthepopulationofthekstates.Asaresult,theaveragecarriervelocitydecreases,whichresultsinalargerintrinsicdelay. Figure 2-8 alsoshowsthattheintrinsicdelayataspeciedon-oratiodependsonthestrainapplied.Forexample,atanon-oratioofIon=Ioff=102,theintrinsicdelayof(16;0)CNTFETwith2%oftensilestrainisroughly1.8timeslargerthanthatofunstrained(16;0)CNTFET.ThestraineectofintrinsicdelayonCNTFETscanbeunderstoodbytherolesofband-structure-limitedvelocityandSchottkybarrierheight.Figure 2-10 showsthe1stand2ndlowestsubbandsof(16;0)CNTs,wherethesolidlineisforanunstrainedCNTandthedashedlineisfora2%strainedCNT,andthebottomofconductionbandsarealigned.Theband-structure-limitedvelocity,v=1 dk,isdecreasedwith2%tensilestrainbecausetheslopeoftheE-kdecreasesanditiseasiertooccupythebottomofthe2ndsubbandduetothedecreasedgapbetweenthe1standthe2ndsubband.ThedecreaseofthevelocitymakestheCNTFETslower.Furthermore,thelargerSchottkybarrierheightduetothe2%ofstrainresultsinlargerquantumreectionatthechannel/draininterface,whichlowerstheaveragecarriervelocityand 32

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2-8D andtheintrinsicdelayofthe(17;0)CNTFETwith4otorsionalstrainisabout4/5ofthatoftheunstrained(17;0)CNTFETatIon=Ioff=102.However,a5ooftorsionalstrainona(16;0)CNTFETdegradesthespeedofthedeviceandincreasestheintrinsicdelaybyafactorof1.35atIon=Ioff=102.Generally,asmallerdelayataspeciedon-oratiocanbeachievedbyapplyingastrainthatleadstoadecreaseofbandgap,whichincreasestheband-structure-limitedvelocityandreducesquantumreectionatthechannel/drainjunction.Atthesametime,thereductionofthebandgapleadstoanincreaseoftheminimumleakagecurrent,andtherefore,adecreaseofthemaximumachievableon-oratioasshowninFig. 2-8 33

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B C Typeofappliedstrains.A)Tensileuniaxialstrain.B)Compressiveuniaxialstrain.C)Torsionalstrain. B RealtomodespaceapproachforstrainedCNT.A)Partof2-D(n;0)zigzagnanotubelatticeinrealspace.B)Uncoupled1-Dmodespacelattice. 34

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B C D Transfercharacteristicswithdierentstrains.A)Uniaxialstrainon(16;0)CNT.B)Uniaxialstrainon(17;0)CNT.C)Torsionalstrainon(16;0)CNT.D)Torsionalstrainon(17;0)CNT. B Bandgapvs.strain.A)Uniaxialstrain.B)Torsionalstrain. 35

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B Minimumleakagecurrentvs.strain.A)Uniaxialstrain.B)Torsionalstrain. B Oncurrentvs.strain.A)Uniaxialstrain.B)Torsionalstrain. 36

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B Onstateof2%strained(16;0)CNTFETcomparedtotheunstraineddevice.A)Conductionbandprolealongthechannelposition.B)Energy-resolvedcurrentspectrum. 37

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B C D Intrisicdelayvs.Ion=Ioffunderdierentstrain.A)Uniaxialstrainon(16;0)CNT.B)Uniaxialstrainon(17;0)CNT.C)Torsionalstrainon(16;0)CNT.D)Torsionalstrainon(17;0)CNT. 38

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Conductionbandprolealongthechannelpositionfor2%strained(17;0)CNTFETatahighgateoverdrive. Figure2-10. Firstandsecondlowestsubbandof(16;0)CNTsunderunstrained(solidline)and2%strained(dashedline)condition 39

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Averticalpartialgatecarbonnanotube(CNT)eld-eecttransistor(FET),whichisamenabletotheverticalCNTgrowthprocessandoersthepotentialforaparallelCNTarraychannel,issimulatedusingaself-consistentatomisticapproach.Weshowthattheunderlapbetweenthegateandthebottomelectrode(requiredforisolationbetweenelectrodes)isadvantageousfortransistoroperationbecauseitsuppressesambipolarconduction.AverticalCNTFETwithagatelengththatcoversonly1/6ofthechannellengthhasamuchsmallerminimumleakagecurrentthanonewithoutunderlap,whilemaintainingcomparableon-current.Bothn-typeandp-typetransistoroperationswithbalancedperformancemetricscanbeachievedonasinglepartialgateFETbyusingproperbiasschemes.Evenwithagateunderlap,itisdemonstratedthatincreasingtheCNTdiameterstillleadstoasimultaneousincreaseofon-currentandminimumleakagecurrent.Alongwithapartialgate,thesimulatedtransistorfeaturesasignicantamountofairbetweenthesurfaceofthechannelCNTandthegateinsulator,asiscausedbytheverticalCNTgrowthprocess.Filingthisporewithahigh-insulatorisshowntohavethepotentialtodecreasetheon-current,duetoelectrostaticphenomenaatthesource-channelcontact. 41 { 43 ]haveledtothesuccessfulimplementationofatwo-terminalresistiveSWNTdevice[ 44 ]withinaverticallyalignedporousanodicalumina(PAA)template[ 45 ].TheSWNTscanbeplacedwithinpredenedlocationsinthePAAtemplateusing 40

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46 47 ],andgoodelectricalcontactsbetweentheSWNTsandmetalhavealsobeendemonstrated.Couplingthisverticaltwo-terminalPAA-baseddevicewithasidegatecanpotentiallyleadtoaverticalcoaxiallygatedCNTFETwithidealgateelectrostaticcontrol[ 48 49 ],oraverticalparallelarrayCNTFETwithlargeon-currentandsignicantlyreducedparasiticeects[ 50 ]. Theeectofnon-uniformgategeometryalongthechanneldirectiononthehorizontalFETswaspreviouslyexploredexperimentally[ 51 ]byusingatrench,andtheoretically[ 52 ]intermsofelectrostaticengineeringviavaryinggateoxidethicknessalongthechannel.However,astudyofthedeviceperformancefortheverticalpartialgatedCNTFEThasnotyetbeenperformed.WhilesomeofthephysicalpropertiesoftheverticalCNTFETmaybesharedbythehorizontalCNTFETpreviouslystudied,amorecarefulapproachfortheverticalCNTFETisnecessarytoaccentuatethekeydierencesbetweenthetwodevices.AcarefulstudyoftheverticalCNTFETmayhelpdevelopcentraldesignrulesrequiredfortheimplementationofasuccessfulmanufacturingprocess.Inthisstudy,wesimulatedaverticalpartialgateSchottkybarrier(SB)CNTFETusingself-consistentatomisticsimulations.Thesimulationresultsindicatethattheunderlaprequiredforisolationbetweenthegateandthebottomelectrodesisadvantageousfortransistoroperation,asitsuppressestheambipolarconduction.Signicantincreaseoftheunderlaplengthhasonlyasmalleectontheon-currentandsubthresholdswing,butsignicantlylowerstheminimumleakagecurrent.Variationsofthepartialgatelength,therefore,shouldnotleadtoananotransistorvariabilityproblem.Bothn-typeandp-typetransistoroperationswithbalancedperformancemetricscanbeachievedonasinglepartialgateFETbyusingproperbiasschemes,whichisadvantageousforCMOSelectronicsapplications.ItisdemonstratedthatincreasingthediameterofthechannelCNTwillincreaseboththeon-currentandtheminimumleakagecurrent.Thisisattributedtoasmallerbandgap,andthussmallerSchottkybarrierheight.Oneofthefeaturesofthetwo-terminalPAASWNTarrayisaconsiderableamountofairbetweenthesurfaceofthe 41

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3.2.1DeviceStructure 3-1 .Thenominaldevicehasa120nm-long(17;0)CNTasthechannelmaterial,whichisoverlappedbythegatefor60nmfromthesource-channelcontact.TwoinsulatorsarepresentbetweenthegateandtheCNT-2nmofPAAand10nmofAl2O3,bothofwhichhaveanapproximatedielectricconstantof10.A2nm-radiusairpore("r=1)istreatedbetweenthegateoxideandthechannelCNT.Boththechannellengthandairpore/gateoxideradiusarescaleddownbyafactorof10withrespecttoatypicalPAAprocesstofacilitateatomisticsimulations.Scalingdownthedeviceispreferredforecientgatecontrolandfastswitching.ThequalitativeconclusionsondevicephysicsanddesignrulesshouldapplytolargerverticalCNTFETs.Themetalsource/drainisdirectlyattachedtotheCNTchannel,andtheSchottkybarrierheightbetweensource/drainandchannelistreatedasbeingequaltohalfthebandgapofthechannelmaterial(Bn=Eg=2)[ 13 53 ].ThemetalcontactFermilevelliesinthemiddleoftheCNTbandgap.TheCNTisassumedtobeatthecenteroftheairpore.Inpractice,theCNTmaybeplacednearthePAAwall.However,thequalitativeconclusionsoftheworkremainthesameregardlessoftheexactpositionoftheCNTintheairpore. 42

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TheretardedGreen'sfunctionoftheCNTchannelisgivenby whereHisTBHamiltonianmatrixoftheCNTchannelwithapzorbitalbasisset,1and2areself-energiesofthemetalsourceanddraincontacts,respectively. Thechargedensitycanbecomputedas wheresgn(E)isthesignfunction,EF1;F2isthesource(drain)Fermilevel,andD1;2(E;x)isthelocaldensityofstatesduetothesource(drain)contact,whichiscomputedbytheNEGFmethod.Thechargeneutralitylevel,EN(x),isatthemiddleofbandgapbecausetheconductionbandandthevalencebandoftheCNTaresymmetric. Thesource-draincurrentiscalculatedas hZ1Trace1Gr2Gr+(f1f2)dE where1;2=i1;2+1;2arebroadeningfunctionsofthesource(drain)contacts,f1;2areequilibriumFermifunctionsofsource(drain)contacts,andthefactorof4comesfromspinandvalleydegeneraciesoftwo[ 11 ].Thegateleakagecurrentisneglectedforsimplicity.Theatomisticsimulationapproachusedinthisstudyhasbeencarefullyvalidatedbyexperimentalmeasurements[ 54 ],anditshowsaverygoodquantitativeagreement. 43

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whereU*ristheelectronpotentialenergywhichdeterminesthediagonalentryofthepotentialenergymatrixinEqn.( 3{1 ),"isthepermittivity,andQ*risthechargedensity.Becausetheelectriceldonlyvariesin2Ddimensionsofrandzaxesforthesimulatedcoaxialdevicestructure,thesimulatedregionissimplychosenfromacrosssectionofthedevicestructureasshowninFig. 3-1A .ThePoissonequationisnumericallysolvedusingtheniteelementmethod(FEM)becauseitnaturallylendsitselftotreatingcomplexdevicegeometriesaswellasboundariesbetweendierentdielectricmaterials. 3.3.1TransferCharacteristics 3-2A showsplotsofID-VGatVD=0:4V.ComparedtotheID-VGofafullgateCNTSBFET,whichhassymmetricambipolarcharacteristics,verticalpartialgateCNTSBFETsshownearlyunipolarcharacteristics.Figure 3-2B and 3-2C showtheconductionandvalencebandprolesalongthechannelpositionforVG=0:9VandVG=0:6V,respectively.AtVG=0:9V,theelectrostaticpotentialofthechannelatthesourceendissucientlycontrolledbythegateelectrodeandtheSchottkybarrierthinenoughforholestotunnelthoughthebarrier.However,atVG=0:6V,thechannelpotentialatthedrainendisnoteectivelycontrolledbythegateelectrode,andhenceelectronsinthedrainencounteraverythickSchottkybarrierandhardlyowthroughthechannelduetosignicantgateunderlapatthedrainend.ThispartialgateeectisadvantageousfortransistoroperationbecauseitsuppressestheambipolarconductionoftypicalfullgateCNTSBFETs,thusresultinginsmallerminimumcurrentandalargermaximumon/ocurrentratio. 44

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3-3A showstheon-stateofp-typeoperationatVG=VD=VDD,whereVDDisthepowersupplyvoltage.Similarly,forn-typeoperation,thedrainvoltagemustbepositive,withapositivegatevoltagerequiredforoperationintheon-state.ThisbiasingalsocausesareductionintheSchottkybarrierbetweensourceandchannel,inthiscaseenablingelectronconductionthroughthechannel.Figure 3-3B showstheon-stateofn-typeoperationatVG=VD=VDD.TheinsetsshowninFig. 3-3A and 3-3B areschematicdrawingsofthebiasschemesforp-typeandn-typeoperation,respectively.Inbothcasesthekeyparameterthatdeterminesthecurrentowthroughthechannelisthesource/channelSchottkybarrier,asthepartialgatecanonlymodulatetheelectrostaticpotentialatthisend.Restated,thegatecontroloverthechannelelectrostaticpotentialonthedrainsideissopoorthattheSchottkybarrierbetweenthedrainandchannelcannotbeeectivelyreducedwiththepartialgate.TheSchottkybarrierbetweenthedrainandchannelalwaysremainsthick,regardlessofgatevoltage.ThisisdemonstratedinFig. 3-2 3-4A showstheIDversusVDcharacteristicsofapartialgateCNTSBFETwithgatevoltagesofVG=0:5;0:6,and0:7V.ForafullgateCNTSBFETitisknownthatsource-draincurrentislinearlyincreasedasdrainvoltageisincreased,providedthatthegatevoltageissucientlyhigh.Thisisnotthecaseforapartialgate 45

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52 ].Figure 3-4B showsthebandprolealongthechannelpositionforVD=0:1V(thesolidline)andVD=0:4V.Astheapplieddrainvoltageincreases,thebarrieratthedrainendisreduced,whichenablescarrierstobecollectedbythedrain.Thesource-draincurrentexponentiallyincreaseswithVDbecausethereectionprobabilityofthesource-injectedcarriersbythedrainbarrierexponentiallydecreasesasVDincreases.Asthedrainvoltagefurtherincreases,thedrain-sideSchottkybarriervanishesandsaturationcurrentisachieved(thepotentialproleatthesourceendremainsunchangedbecausetheelectrostaticpotentialnearthesourceistotallycontrolledbythegatevoltage). 3-5A showstheID-VGcurveofpartialgateFETsforgatelengthsLG=20;60;and120nmatVD=0:4V.AllotherparametersareequaltothoseofthenominaldevicedescribedinFig. 3-1B .TheresultsshowthattheambipolarcharacteristicoffullgateCNTSBFETsisfurthersuppressedasthegatelengthisreducedbecausetheunderlaplengthhasadirecteectontheSchottkybarrierbetweendrainandchannel.AtVG=0:6V,thesource-draincurrentof60nmand20nmgatelengthsis4and5ordersofmagnitudesmallerthanthatofa120nmgatelength,respectively.Itcanalsobeshownthatthepartialgategeometryhighlyaectstheminimumleakagecurrent,suchthatamuchsmallerminimumleakagecurrentcanbeobtainedbyusingashortergatelength.Theminimumleakagecurrentassociatedwitha20nmgatelengthissmallerthanthatofa120nmgatelengthbyanorderof2.Figure 46

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showstheconductionandthevalencebandprolesalongthechannelpositionfordierentgatelengthsatVG=0:6V.Eventhoughthepartialgatehasasevereeectontheminimumleakagecurrent,ithasaweakeectontheon-currentofthedevice.ThisisbecausethegatevoltagehasgoodcontrolovertheSchottkybarrierbetweenthesourceandthechannel,whichcompletelydeterminesthecarrierinjectionfromthesource.AtVG=0:6V,thedierenceofsource-draincurrentsbetweenLG=60nmand120nmislessthan5%.Thesubthresholdswingisalsoalmostinvarianttothegatelength. 3-6A showstheresultingIDVGcurve.Theresultsareinterestinginthattheon-currentforthedevicelackinganairporeissmallerthanthatofthenominaldevicestructure.ThisresultisunderstoodviacarefulobservationoftheSchottkybarrierbetweenthesourceandchannel.Figure 3-6B comparestheelectriceldpatternforbothcasesatabiasofVD=0:4VandVG=0:6V.Itdemonstratesthattheredirectionoftheelectriceldcausedbytheair/insulatorinterfaceactuallyreducesthebarrierbetweenthesourceandchannel.Nearthesourcecontact,theelectriceldismoreheavilycontrolledbythesourcepotential.Sincethedielectricofairissmallerthanthatoftheinsulator,theairporeessentiallymakesitmoredicultforthesourcetomodulatetheelectriceldnearthesourcecontact.Theelectriceldmodulationcausedbythesourceincreasesthesource/channelSchottkybarrier,andthussuppressingthismodulationdecreasesthebarrier.AtVG=0:6V,thesource-draincurrentisreducedby34%whentheairporeislledwithAl2O3.However,theo-currentisnotaectedbyllingthepore,sincetheSchottkybarrieratdrainsideissothickthatthedecrementishardtorecognize.AtVG=0:6V,thecurrentofbothcasesdoesnotshowanyappreciabledierence. 47

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3-7 .AsthediameteroftheCNTincreases,sodoestheon-currentandminimumleakagecurrent.Usually,aCNTwithalargerdiameterhasasmallerbandgap(thisistrueforthechiralindexrangesofCNTssimulatedinthisstudy,althoughitisinvalidforverysmalldiametertubesduetocurvatureeects).Therefore,alargerdiameterCNTchannelhassmallerSchottkybarrierheightsatthesourceanddraincontacts,whichresultsinincreasesinbothon-currentandminimumleakagecurrent.Atalargepositivegatevoltage,theelectronleakagecurrentislimitedbythethickSchottkybarrieratthedrainendwithabarrierheightofBn=Eg=2,whichdecreasesasthediameterincreases.Ingeneral,alargeron-currentandasmallerminimumleakagecurrentaredesiredfortransistors,sotheeectofCNTdiametercausesadeviceperformancetrade-o.Thesource-draincurrentofa(26;0)CNTFETisafactorof33largerthanthatofa(17;0)CNTFETatVG=0:4V.Atthesametime,theminimumleakagecurrentofa(26;0)CNTFETisontheorderof2timeslargerthanthatofa(17;0)CNTFET. 48

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49

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B Verticalpartial-gateCNTFET.A)Crosssectionofthemodeledgeometry.B)Deviceparametersofthenominaldevice. 50

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B C Suppressionofambipolarconduction.A)ID-VGcharacteristicsatVD=0:4V.B)BandprolealongthechannelpositionatVG=0:9V.C)BandprolealongthechannelpositionatVG=0:6V. 51

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B Deviceoperationsofverticalpartial-gateCNTFET.A)P-typeoperationwithVG=VD=VDD.B)N-typeoperationwithVG=VD=+VDD. B Outputcharacteristicsofpartial-gateCNTFET.A)ID-VDcurvewithdierentgatevoltages.B)BandprolealongthechannelpositionatVD=0:4Vand0:1VwithVG=0:5V. 52

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B Eectofgatelengthinpartial-gateCNTFET.A)ID-VGcurveswithdierentgatelengths.B)Bandprolealongthechannelpositionfordierentgatelengths. 53

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B Eectofairporeinpartial-gateCNTFET.A)ID-VGcurveswithandwithoutairpore.B)Electriceldcontournearthesourcecontactwith(left)andwithout(right)airpore. Figure3-7. EectofCNTdiameterinpartial-gateCNTFET. 54

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Theeectofphononscatteringontheintrinsicdelayandcut-ofrequencyofSchottkybarrierCNTFETsisexamined.Carriersaremostlyscatteredbyopticalandzoneboundaryphononsbeyondthebeginningofthechannel.WeshowthatthescatteringhasasmalldirecteectontheDCon-currentoftheCNTFET,butitresultsinsignicantdecreaseofintrinsiccut-ofrequencyandincreaseofintrinsicdelay. 55 { 57 ].Theoreticalcalculationsofthecut-ofrequencyanddelaytimeofCNTFETshavebeenbasedontheassumptionofballistictransport,whichpredictsTHzoperation[ 50 58 59 ].Little,however,isknownabouthowscatteringaectsthespeedandhighfrequencyperformanceofCNTFETs.TheunderstandingisessentialforassessingtheperformancepotentialofCNTFETsforelectronicsapplications. TheeectofscatteringontheDCcurrentofmetallicandsemiconductingCNTshasbeenpreviouslyexamined[ 60 ].Thedominantscatteringmechanisminahigh-qualityCNTisphononscattering.Atlowbiases,acousticphonon(AP)scatteringwithameanfreepath(mfp)1misdominant,andathighbiases,opticalphonon(OP)andzoneboundary(ZB)phononscatteringwithamfpof10nmismostimportant.ACNTFETwithachannellengthseveraltimeslongerthanphononscatteringmfpcanstilldeliveranearballisticDCon-current[ 61 ].Inthiswork,theeectofphononscatteringontheintrinsicdelayandcut-ofrequencyofCNTFETsisexamined.Weshowthatalthoughitsdirecteectonon-currentandtransconductanceissmall,phononscatteringresultsinsignicant(i)pile-upofchargeinthechannelandincreaseoftheintrinsicgatecapacitance,(ii)randomwalksofcarriersinthechannelanddecreaseoftheaverage 55

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61 62 ]usingtheperturbationpotentialcomputedbyMahan[ 63 ].AnatomisticdescriptionofthenanotubeusingatightbindingHamiltonianwithapzorbitalbasissetisapplied.Theatomistictreatmentiscomputationallyexpensiveinrealspace,butsignicantsavingofcomputationalcostcanbeachievedbythemodespaceapproach. Wecomputetheintrinsicdelayandcut-ofrequencyofthedeviceattheballisticlimitandinthepresenceofphononscattering(assumingzeroparasiticcapacitance).ThedesignofCNTFETsinexperimentstodatehasnotbeenoptimizedforhigh-frequencyperformance,whichislimitedbyparasiticcapacitancesbetweenelectrodes.Therecent 56

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50 56 58 59 ], 2gm 2gm@VG 2@ID wheregmisthetransconductance,Cgistheintrinsicgatecapacitance,IDisthesource-draincurrent,Qch=qRLch0Ne(x)dxisthetotalchargeintheCNTchannel,Ne(x)istheelectrondensityasafunctionofthechannelposition,andLchisthechannellength.Thederivative,@ID=@Qch,inEqn.( 4{1 )isobtainedbyrunningthedetailedDCsimulationsandcomputingtheratioofthecurrentvariationtothechannelchargevariationattwoslightlydierentgatevoltagesandVD=0:5V. Theintrinsicdelay,whichcharacterizeshowfastatransistorintrinsicallyswitches,isanimportantperformancemetricfordigitalelectronicsapplications.Theintrinsicdelay,=(QonQoff)=Ion,iscomputedbyrunningDCsimulationsattheo-state(VG=0,VD=0:5V)andtheon-state(VD=VG=0:5V),whereQonandQoarethetotalchargeinthechannelaton-stateando-state,respectively,andIonistheon-current. 4.3.1OnCurrent 4-1A showsthesimulatedIDvs.VGcharacteristicsforVD=0:5Vattheballisticlimit(thedashedline)andinthepresenceofscattering(thesolidline).Inthepresenceofphononscattering,theCNTFETdeliversanearballisticDCon-current(85%oftheballisticvalue)andthetransconductance,gm=@ID/@VGisalsocloseto 57

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61 62 ].AnelectroninjectedfromthesourcecanbeacceleratedbytheelectriceldandobtainakineticenergylargerthantheOP/ZBphononenergy.ItcanthenemitanOP/ZBphononandgetbackscattered,butbecausetheOP/ZBphononenergyinCNTislarge(>160meV),thebackscatteredcarrierfacesamuchthickerandhigherbarrier,andhaslittlechancetotunnelbacktothesource.Theresultisthattheelectronrattlesaroundinthechannelandnallyexitstothedrain.OPandZBphononscatteringhasasmalldirecteectontheDCon-currentoftheCNTFETundermodestgatebiases. Figure 4-1B plotstheon-currentversusthechannellength.Inthepresenceofphononscattering,theCNTFETdeliversabout95%oftheballisticon-currentatachannellengthof20nmandabout80%at200nm.Theballisticon-currentisnearlyindependentofthechannellength,andtheon-currentinthepresenceofphononscatteringslightlydecreasesasthechannellengthincreases,mostlyduetotheincreasinglyimportantroleofnearelasticacousticandradialbreathingmode(RBM)phononscattering.OPandZBscatteringwithashortmfphasasmalldirecteectontheon-current.TheCNTFETdeliversaDCon-current80%oftheballisticvalueforachannellengthupto200nm. 4-2A plotstheaverageelectronvelocityaton-stateasafunctionofthechannelposition,whichiscomputedas(x)=Ion/[qNe(x)],whereNe(x)istheelectrondensityaton-state.Attheballisticlimit,theelectronvelocityincreasesatthebeginningofthechannelandthenremainsnearlyconstantatabout6107cm/s.Becausetherstsubbandbendsatthebeginningofthechannelandthenremainsnearlyat,electronsonlypopulatethe+kstateswithhighkineticenergiesbeyondthebeginningofthechannel,whichhavealarge 58

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Figure 4-2B plotstheelectrondensityNe(x)aton-stateversusthechannelposition.Theelectrondensityattheballisticlimitremainsnearlyconstantinthechannelbecausethesubbandproleisnearlyatduetogoodgateelectrostaticcontrol.Itreachespeakvaluesatthemetal/CNTcontactsduetometalinducedgapstates(MIGS)attheendsofthechannel.Phononscatteringresultsinasignicantincreaseoftheelectrondensityinthechannelbecausecarriersarescatteredbyphononsandrattlearoundinthechannelregion. 4-3A showstheintrinsiccut-ofrequencyversusthechannellengthattheballisticlimit(thecircles)andinthepresenceofphononscattering(thecrosses).ThedashedlineisattingoftheballisticresultbyfT=110GHzm=Lch,andthesolidlineisattingofthescatteringresultbyfT=40GHzm=Lch.AlthoughthetransistordeliversanearballisticDCon-current(>80%),thecut-ofrequencyinthepresenceofphononscatteringisonlyabout40%oftheballisticvalueforachannellengthofLch=50-200nm.Thereasonisthatrandomwalksofelectronsduetophononscatteringresultinpile-upofchargeinthechannelandsignicantincreaseoftheintrinsicgatecapacitance,althoughitseectontheDCcurrentandtransconductanceissmall.Aschannellengthdecreasesfrom50nmto20nm[ 64 ],theratiooffTinthepresenceofphononscatteringtothatattheballisticlimitincreasestoabout54%duetoreducedOP/ZBscatteringandquasi-ballistictransport.Inapreviouswork[ 50 ],weshowedthatthefTofaballisticCNTFETisabout1.5timeslargerthanthatofaballisticSiMOSFETatthesamechannellengthdueto 59

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64 ],weexpectscatteringdecreasestheintrinsicfTofaSiMOSFETbyafactorof0.4-0.5. Figure 4-3B plotstheintrinsicdelayoftheCNTFETvs.thechannel.Thedashedlineisalinearttingtotheballisticresultby=Lch1:71ps=m.TheintrinsicdelayattheballisticlimitscaleslinearlywiththechannellengthforLch>20nm.Inthepresenceofphononscattering,theintrinsicdelaysignicantlyincreasesbecausephononscatteringlowerstheaveragecarriervelocityinthechannel.TheintrinsicdelayinthepresenceofphononscatteringatLch=200nmisabout210%largerthanthatattheballisticlimit.Whenthechannellengthdecreasestoavalueclosethephononscatteringmfp,thespeeddegradationbyscatteringislesssevere.Atachannellengthof20nm,theintrinsicdelaywithscatteringisonlyabout40%largerthanthatattheballisticlimit. 15 ].WeshowthatthedirecteectofphononscatteringontheDCsource-draincurrentandtransconductanceissmall.Theindirecteectofphononscatteringthroughself-consistentelectrostaticsisalsosmallbecauseofthegoodgateelectrostaticcontrol[ 61 ].Thetransistordeliversanearballisticon-current.Ontheotherhand,theintrinsicgatecapacitanceCg,whichistheseriescombinationofthegateinsulatorcapacitanceCinsandtheCNTquantum(semiconductor)capacitanceCQ,isclosetoCQatthequantumcapacitancelimit(CinsCQ)[ 15 ].PhononscatteringresultsinasignicantincreaseofCQ,whichleadstoanincreaseofCganddecreaseoftheintrinsiccut-ofrequency,fT=gm=(2Cg). WealsoexaminedthecasewhenthegateinsulatoristhickandthetransistoroperatesclosetotheconventionalMOSFETlimit(CinsCQsothatCgCins)[ 15 ].Incontrast,phononscatteringdoesnotleadtoanincreaseoftheintrinsicgatecapacitanceattheMOSFETlimit.It,however,cansignicantlylowertheDCsource-draincurrent 60

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61 ].Theintrinsiccut-ofrequencystilldecreases,butduetothedecreaseoftheDCtransconductance,insteadoftheincreaseoftheintrinsicgatecapacitance. 61

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B Eectofphononscatteringonthecurrentwithdierentgatevoltagesordierentchannellengths.A)IDvs.VGatVD=0:5Vattheballisticlimit(thedashedline)andinthepresenceofphononscattering(thesolidline).Thechannellengthis100nm.B)Oncurrentvs.channellengthatballisticlimit(dashedline)andinthepresenceofphononscattering(solidline).Theinsetsketchestherstsubbandproleattheonstate. B Eectofphononscatteringonaveragecarriervelocityandelectrondensity.A)Averagecarriervelocityvs.channelpositionatVG=VD=0:5Vattheballisticlimit(dashedlines)andinthepresenceofphononscattering(solidlines).B)Electrondensityvs.channelpositionatVG=VD=0:5Vattheballisticlimit(dashedlines)andinthepresenceofphononscattering(solidlines). 62

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B Eectofphononscatteringoncutofrequencyandintrinsicdelay.A)Cutofrequencyvs.channellengthatonstate.ThecirclesarenumericallycomputedfTattheballisticlimitandthedashedlineisattingcurveoffT=110GHzm=Lch.ThecrossesarenumericallycomputedfTinthepresenceofphononscatteringandthesolidlineisattingcurveoffT=40GHzm=Lch.B)Intrinsicdelayvs.channellengthattheballisticlimit(thecircles)andinthepresenceofphononscattering(thecrosses).Thedashedlineisalinearttingoftheballisticresultby=Lch1:71ps=m. 63

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Thescalingbehaviorsofcarbon-nanotube-enabledverticalorganiceld-eecttransistors(OFET)arecomprehensivelyexaminedbyusingtwo-dimensionalself-consistentdevicesimulations.Becauseoftheuniqueandcomplexverticalstructure,devicesimulationisneededtoclarifythedevicephysicsanditsoperation.WendthattheverticalOFETshowscleartransistorswitchingbehaviorallowingordersofmagnitudemodulationofthesource-draincurrenteveninthepresenceofelectrostaticscreeningbythesourceelectrode.Thechannellengthshouldbecarefullyengineeredduetothetrade-obetweendevicecharacteristicsinthesubthresholdandabove-thresholdregions.Theactivelayershouldhaveasmallpermittivityforreducingtheelectrostaticshortchanneleect.Reducingtheeectivegateoxidethicknessbyeitherathinneroxideorahigh-gateinsulatorresultsinasimilarimprovementofperformancefortheverticalOFET. 65 ]arewidelyusedfordisplaydevicessuchasliquid-crystaldisplay(LCD).ThemainstreamofactivelayersforTFTsisinorganicmateriallikeamorphoussilicon(a-Si),buttheresearchesfororganicTFTshavebeenalsoveryactiveformorethantwodecadestoachievelargeareaorexibledisplays,low-temperatureandlow-costprocessing[ 66 { 69 ].ThebasicgeometricalshapeofTFThasbeenahorizontalstructure,inwhichthesource-draincurrentowsinthedirectionparalleltothechannel-gateoxideinterface.TheperformanceofanorganicTFTislimitedbyitslowmobility(1081cm2=Vs),atleastordersofmagnitudelowercomparedtocrystallinematerials[ 66 ].ThedeviceperformanceofsuchaplanarorganicTFTcanbeboostedupbypatterningthesourceandthedraincloseforashortchannellength,buttherequirementsforhighresolutionlithographyosettheadvantageofcheapfabricationcostoeredbytheorganicTFTtechnology. 64

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70 71 ],inwhichthegate,source,activelayeranddrainareverticallystacked.TheCNT-enabledverticalorganiceld-eecttransistor(OFET)isschematicallyshowninFig. 5-1A .Inthisconstruction,thenanotubes(depositedbythemethodsreportedinref.[ 72 ])formadilutesourceelectrodelayeronagatedielectricandunderlyinggatingelectrode.Theactivelayerisathinorganicsemiconductordepositedontothenanotubes,andnallyadrainelectrodeisdepositedontotheorganicactivelayer.Thenanotubelayeriswellabovethepercolationthresholdtoactasthesourceelectrode[ 73 74 ],andatthesametime,itisdiluteenoughtoallowthegateeldtopenetratethroughformodulatingtheorganicchannellayer.ComparedtothehorizontalOFET,asignicantadvantageisthatthechannellength,whichisdeterminedbythethicknessoftheactivelayer,canbemadealmostarbitrarilythinwithouttheneedforcostlyhigh-resolutionelectrodepatterning.Thiscomplexdevicestructuremakesthescalinglawsandoptimizationschemesofhorizontaleld-eecttransistors(FETs)invalid.Therefore,devicemodelingandnumericalsimulationareneededtoelucidatedevicephysicsandscalingprinciples,whichareindispensablefordeviceoptimization. Inthisstudy,acomprehensivecomputationalstudyonthescalingbehaviorsofCNT-enabledverticalOFETsispresented.Adrift-diusiontransportequationfortheorganicmaterialchannelisself-consistentlysolvedwithaPoissonelectrostaticequationforthetwo-dimensionalcrosssectionoftheverticalOFET.TheresultsshowthattheverticalOFEToperatesinadierentwaytotheconventionalhorizontalOFET.Thescalingcharacteristicsofchannellengthandgateoxideareexplored.Therolesofthemobilityandpermittivityoftheactivelayeronthedeviceoperationareinvestigatedaswell. 65

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5-1A .Thegate,source,channel,anddrainareverticallydeposited.Adilutepercolativecarbonnanotubenetworkisrequiredforallowingthegateelectriceldtopenetrateintoorganicchannelregion.Inordertosimplifythemodelingandperformthenumericalsimulation,thefollowingassumptionsaremade:(i)percolatingCNTnetworkissparse,sothatasinglemetallicCNTisstudiedforeachregion,and(ii)a2Dcross-section,whichisperpendiculartoCNTlengthdirection,issimulatedandtheplaneisuniformalongtheCNTaxis.Figure 5-1B showsthecross-sectionofthesimulateddevice.Thenominaldeviceparametersareasfollows.Thethicknessoftheactiveorganiclayer,whichisthechannellength,is240nm.Thechannelorganicsemiconductorhasamobilityof=105cm2=Vsandadielectricconstantofch=6.Arelativelythick200-nmSiO2withadielectricconstantofox=3:9isusedasthegateinsulator.ApowersupplyvoltageofVDD=5Visused.AsinglemetallicCNTwiththediameterofdCNT=2nmisusedasthesource,andthereisonlyoneCNTwithinaneighboringdistanceof1000nminthehorizontaldirection.Theaboveparametersarenominal,andtheyarevariedtoexplorevariousissues. ThecharacteristicsofCNT-enabledverticalOFETsaresimulatedbysolvingthe2Ddrift-diusionequationself-consistentlywithPoissonequation[ 75 76 ].Weexaminethep-typeunipolarconductionatsteadystate.Therecombination-generation(RG)rateissettozeroduetotheoperationattheunipolartransportregime.ThetransportandPoissonequationsareexpressedasfollows. @t=1 66

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"0p; whereJpisholecurrentdensity,pisholedensity,pisholemobility,Visvoltage,Dpisholediusioncoecient,rand0arerelativedielectricconstantandvacuumpermittivity,respectively.The2Ddrift-diusionequationissolvedusingScharfetter-Gummeldiscritization[ 77 ].The2DPoissonequationissolvedusingthenitevolumemethod.Anon-linearPoissonschemeisutilizedtoimprovetheconvergenceoftheself-consistentloopbetweentheelectrostaticandcarriertransportequations. ASchottkybarriercanformbetweenthemetallicCNTcontactandtheactivelayer.Importantdierences,however,existbetweentheCNT-organicsemiconductorcontactsandtheconventionalmetal-semiconductorcontacts.First,duetothefullypassivatedCNTsurfaceandlackofcovalentbondingbetweentheCNTandtheactivelayer,thedensityofmetal-inducedgapstates[ 78 ]andtheeectofFermilevelpiningissignicantlyreduced.Second,theelectriceldatthesurfaceofthenanotubeissignicantlyenhancedduetoitsnanometer-scaleradius,whichcouldenhancecarrierinjectionfromthemetallicCNTcontacttotheactivelayer.Third,theworkfunctionoftheCNT,whichisthedierencebetweentheFermilevelandthevacuumlevel,canbemodulatedbythegatevoltageduetothelowquasi-1Ddensity-of-states(DOS),whichcouldresultinmodulationoftheSchottkybarrierheightbythegatevoltage[ 71 ].TheswitchingcharacteristicsoftheverticalOFETcanstemfromcombinedeectsofmodulatingtheSchottkybarrierandmodulatingtheactivelayer.Forthemodeleddevice,asimpleestimation,however,showsthatthechannelresistanceismuchlargerthanthecontactresistanceduetothelowmobility(105cm2=Vs)andconsiderablechannellength(240nm)oftheorganicchannel.Gatemodulationofthechannelresistanceshoulddominate.AnOhmiccontactisassumedinthisstudyforsimplicity,byxingthechargedensityattheCNT-activelayerinterfaceastheboundaryconditionforEq. 5{1 ,atthevaluewhentheFermilevelis 67

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5-2A showsthecross-sectionalpotentialenergycontouratVG=0andVD=5V.Ifthegatevoltageisapplied,thepotentialnearthechannel-oxideinterfaceiscontrolledbythegate.TheinsetofFig.2aisthepotentialenergycontournearthesourceelectrodeatVG=10VandVD=5V,showingthatthepotentialaroundtheCNTismodulatedbythegate.MoredetailedbarrierheightcontrolunderthesebiasconditionscanbeseeninFig. 5-2B .Becausethedeviceisoperatedinthep-typeregion,theHOMO(EH)proleoftheactivelayerneedstobecarefullyobserved.Figure 5-2B showsvacuumenergylevelshiftedbyanionizationenergyoftheactivelayeralongtheverticaldirectionatthecrosssectiontakenthroughthecenteroftheCNT(x=0),whichshowsHOMOlevelinthechannelregion(y>2nm)andtheoriginalelectronpotentialenergy(Em)ofCNTwithoutgatebiasinthesourceregion(0
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Thetransfercharacteristicsareexaminednext.Figure 5-3A showsthetransfercharacteristicsatVD=5and2:5V.Theresultsindicatethepossibilitytomodulatethesource-draincurrentbyordersofmagnitudeinaverticalOFET,consistentwithgateelectrostaticsexploredinFigure 5-2 .Thetransconductanceisgm=8:5pS=m2andthesubthresholdswingisS=357mV/decatVD=5V.IfthedrainvoltageisreducedtoVD=2:5V,adecreaseofthethresholdvoltageisobserved.Forexample,VG=9:72and5:89VarenecessaryforacommonocurrentofID=1010A/mforVD=5and2:5V,respectively,whichtranslatestothedrain-inducedbarrierlowering(DIBL)of1532mV/V.ThelargevalueofDIBLisduetothegeometricstructureofthemodeledverticalFET.Thechannellengthisonly240nmwhereasagateoxidethicknessisrelativelythick(200nm).Whilethechannellength,whichisthethicknessoftheorganiclayer,canbemadealmostarbitrarilyshortwithoutanexpensivehigh-resolutionlithographyprocess,itshouldbenoticedthatareducedorganiclmthicknesscouldresultinalargeDIBLandsevereelectrostaticshortchanneleects. WenextshowtheoutputcharacteristicsatVG=0,5,and10VinFig. 5-4A .Itisobservedthatthisdevicedoesnotshowcurrentsaturationwithinthesimulatedbiasrange(uptoVD=-30V).Figure4bplotsholedensityalongthechannelpositionforVD=-2to-10Vsteppedby-2V,showingthattheholedensitymonotonicallyincreasesasthemagnitudeofthedrainbiasincreases.Incomparison,foranelectrostaticallywelldesignedhorizontalMOSFET,thechargedensityatthebeginningofthechannelisxedbythegatevoltageandisnearlyindependentofthedrainvoltage,whichresultsinthesaturationofthesource-draincurrentathighdrainvoltages.Thecontinuousincreaseofthechannelchargedensitywiththedrainbias,however,resultsinnocurrentsaturationinthemodeledverticalOFET.TheresultisconsistentwiththelargeDIBLobservedin 69

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5-3 ,andthequalitativetrendofnosource-draincurrentsaturationagreeswiththeexperimentaldata[ 71 ]. TheeectofchannellengthscalingonthedevicecharacteristicsisexploredinFig. 5-5 .Ashorterchannellengthisbenecialintermsoftransconductancebecauseitreducesthechannelresistanceinthediusivetransportregime.Ashortchannellength,however,increasesthesubthresholdswingatthesametime,becauseashorterchannellengthincreasestheelectrostaticcouplingbetweenthedrainandtheorganicchannel,andtherefore,leadingtomoresevereelectrostaticshortchanneleect.Whenthechannellengthisscaleddownfrom240to180nm,Sisincreasedby29%.Acarefuldesignofthechannellength,therefore,isneededtoachieveatrade-obetweentheabove-thresholdcharacteristicsandthesubthresholdcharacteristics. Wenextstudytheeectofthemobilityofchannelorganicmaterial.Themobilityoftheorganicactivelayercanvarybyordersofmagnitude,dependingonthechoiceofthechannelmaterial.Figure 5-6A showsID-VGcharacteristicsforthedierentmobilityvaluesfrom107to104cm2=Vs.Themobilityhasadirecteectonthesource-draincurrent,andthesource-draincurrentislinearlyincreasedwithmobility.Thetransconductorcegmisproportionaltothevalueofthemobility.Themobility,however,doesnothaveanyeectonthesubthresholdswing,whichisS=357mV/decregardlessofdierentvaluesofmobility.Intermsofmaximizingtheon-currentandtransconductance,largechannelmobilityisdesired. Thechoiceofdierentorganicmaterialsalsoaectsthedielectricconstantoftheactivelayer.Figure 5-7A showsID-VGcharacteristicsforthedierentdielectricconstantoftheactivelayer.Alargerdielectricconstantoftheorganiclayerenhancestheelectrostaticcouplingbetweenthedrainandthechannel,withoutaectingtheelectrostaticcouplingbetweenthegateandthechannel.Theelectrostaticshortchanneleectbecomesmorepronouncedandthesubthresholdswingincreases.AsshowninFig. 5-7B ,strongerelectrostaticcouplingbetweenthedrainandthechannelresultsinalowerpotential 70

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Athinnergateoxideorevenahigh-gateinsulatorcanbeused,andwenextexaminethescalingbehaviorofthegateinsulator.Figure 5-8A plotsthetransfercharacteristicsfortwodierentgatedielectrics.Alargergatedielectricconstantimprovesthegatecontroloverthechannelinbothsubthresholdandabove-thresholdbiasranges,whichcanbeshownbysmallerS(Fig. 5-8A )andlargergm(insetofFig. 5-8A ).Ahigh-gateinsulator,therefore,ispreferredfortheverticalOFET.Inordertoinvestigatethegeneralscalingbehaviorsofthegateoxideeect,thesubthresholdswingandthetransconductanceareplottedasafunctionoftheinverseeectivegateoxidethickness,ox=tox,byvaryingeitherthegateoxidethicknessorthegateoxidedielectricconstant,asshowninFigure 5-8B .Astheeectivegateoxidethicknessdecreasesbyeitherreducingtheoxidethicknessorusingahigh-gateinsulator,thesubthresholdswingdecreasesandtransconductanceincreases,leadingtoabettertransistorperformance.Inaddition,thetwocurvescloselymatchinboththemainpanelandtheinsetofFig. 5-8B ,whichindicatesthattheperformancegainachievedbyusingahigh-gateinsulatorcanbealmostequivalentlyachievedbyusingathinnergateoxidewiththesameeectivegateinsulatorthicknessfortheverticalOFET. 71

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72

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B A)SchematicstructureofaverticalorganicFET.Gate,source,activelayer,anddrainareverticallystackedup.Poroussourceelectrodeofpercolatingcarbonnanotubenetworkallowsgateelectriceldtopenetratetothechannelregion.B)Cross-sectionofthesimulateddevicestructure. B A)Cross-sectionalpotentialenergycontouratVG=0andVD=5V.TheinsetispotentialenergyatVG=10VandVD=5VaroundtheCNT(shownbywhitecircle).B)Vacuumenergylevelshiftedbyanionizationenergyoftheactivelayeralongthechannelatx=0underVG=0and10V(VD=5V).ThisshowsHOMOlevel(EH)intheactivelayerregion(y>2nm)andtheoriginalelectronpotentialenergy(Em)ofCNTwithoutgatebiasinthesourceelectroderegion(0
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B A)TransfercharacteristicsforVD=2:5and5Vinaalongscale(linearplotsareshownintheinset).B)Vacuumenergylevelshiftedbyanionizationenergyoftheactivelayeralongthechannelatx=0underVG=0V. B A)OutputcharacteristicsforVD=0,5,and10V.Nocurrentsaturationisobservedwithinthesimulatedbiasrange(uptoVD=30V).B)HoledensityinthechannelforVD=2and10V. 74

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B Channellengthscaling:ID-VGcurvesfordierentchannellengthinA)alinearandB)alogscale. B Eectofchannelmobility:A)ID-VGcurveswithdierentmobilityoftheactivelayer(theunitisgivenincm2=Vs).B)Transconductancevs.channelmobilityplot.TheinsetshowsSvs.mobility. 75

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B Eectofchanneldielectric:A)ID-VGcurvesfordierentpermittivityofactivelayer.B)Vacuumenergylevelshiftedbyanionizationenergyoftheactivelayeralongthechannelatx=0underVG=10V. B Eectofgateoxide:A)ID-VGcurvesforgatedielectricconstantof4and16inalogscaleandalinearscale(inset).B)Sandgm(inset)areplottedasafunctionofinverseeectivegateoxidethickness:toxisscaleddownfrom200to50nm,andoxisvariedfrom4to20. 76

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Atomistic3Dsimulationstudyoftheperformanceofgraphenenanoribbon(GNR)Schottkybarrier(SB)FETsispresentedbymeansoftheself-consistentsolutionofthePoissonandSchrodingerequationswithinthenon-equilibriumGreen'sfunction(NEGF)formalism.Theimpactofnon-idealitiesondeviceperformancehasbeeninvestigated,takingintoaccountthepresenceofsinglevacancy,edgeroughnessandionizedimpuritiesalongthechannel. 1 ],andfromtherehugeeorthasbeendirectedtounderstandthephysicalpropertiesofthenewmaterialandtoexploititspotentialsinelectronicapplicationstocomeafterMoore'slawandITRSrequirements[ 25 79 ].Carbonatomsnotonlycanbecombinedintheformoftubesofnanoscaledimensions,butcanalsobearrangedinastabletwo-dimensionalgraphenesheet[ 3 80 81 ].Electronsingraphenebehaveasmasslessfermionsandtravelthroughthelatticewithlongmeanfreepath,asshownbythehighmobility[ 3 80 81 ].Grapheneisazerogapmaterial,withlineardispersionincorrespondenceoftheFermienergy,whichmakesitparticularlyunsuitablefortransistorapplications.However,energygapcanbeinducedbymeansoflateralconnement[ 8 ],realizedforexamplebyetchingthegraphenesheetinnarrowstripes,so-calledgraphenenanoribbons(GNRs). TheoreticalworkshaveshownthatGNRshaveenergygapwhichisinverselyproportionaltotheirwidth[ 82 83 ],andduetotheirreduceddimensions,edgestatesplayanimportantrole,deningnon-nullenergygap,forallribbonwidths[ 10 84 ].GNReld-eecttransistors(GNRFETs)havebeenfabricatedveryrecently[ 7 85 77

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].GNRFETsdemonstratedexperimentallytodatearerealizedbyconnectingthechanneltometalswithSchottkycontacts[ 7 8 ],thereforeobtainingaSchottkybarrierFET(SBFET).Becausefabricationtechniquesareattheveryrststeps,simulationscanrepresentanimportanttooltoevaluatedeviceperformance.Semiclassicaltop-of-the-barriersimulationshavebeenperformed[ 87 88 ],whilequantumsimulationsbasedonatight-bindingapproachhavefollowed[ 89 { 92 ]inordertoassessdevicepotential.However,duetotheembryonicstageofthisneweldofresearch,manyissuesstillremainunsolved.State-of-the-artetchingtechniquesareforinstancefarfromatomisticresolutionsothatedgeroughnesscanplayanimportantroleondeviceperformance[ 93 94 ].Inaddition,defectsorionizedimpuritiescanrepresentelasticscatteringcenters,whichcangreatlydegradetheexpectedfully-ballisticbehavior. Inthisstudy,GNRSBFETisnumericallystudied.Theapproachisbasedontheself-consistentsolutionofthethree-dimensional(3D)PoissonandSchrodingerequationswithinthenon-equilibriumGreen'sfunction(NEGF)formalism[ 95 ],bymeansofarealspacepztight-bindingHamiltonian,inwhichenergyrelaxationattheGNRedgesisconsidered.Dierenttypesofnon-idealitieshavebeeninvestigated.Inparticular,wehavestudiedtheeectofasinglevacancydefect,anionizedimpurityinthechannel,andedgeroughnessonthedeviceperformance.Vacanciesandedgeroughnesscangreatlyaectdeviceelectricalperformance,morethanionizedimpuritiesactuallydo. 95 ]self-consistentlywith3DPoissonequation[ 89 { 92 ].Atight-bindingHamiltonianwithanatomisticpzorbitalbasissetisusedtodescribeatomisticdetailsoftheGNRchannel.Coherenttransportisassumed.SimulateddevicestructureisdepictedinFig. 6-1 .ThesourceandthedrainaremetalswithSchottkybarrierheightofBn=Bp=Eg=2.Double-gategeometryisusedthrough1.5nmSiO2gateoxide(=3.9).Foranidealdevicesimulation,perfectlypatterned15nm-longN

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83 ]armchair-edgeGNR(A-GNR)isusedasachannelmaterial,whichhasawidthof1:35nmandabandgapof0:6eV.Edgebondrelaxationistreatedaccordingtoab-initiocalculation,andatight-bindingparameteroft0=2.7eVisused[ 10 ].PowersupplyvoltageisVDD=0.5V.Roomtemperature(T=300K)operationisassumed. Non-idealitiesaretreatedasfollows.LatticevacanciesoredgeroughnessareconsideredasatomisticdefectsofthechannelGNR,wheretheexistenceofcarriersisessentiallyprohibited.Theseatomisticvacancyoredgeroughnesscanbeimplementedbybreakingthenearestbonds(t0=0)inthedevicechannelHamiltonianmatrixoftheperfectlatticeaccordingtothegeometryofthedefectivelattice.Forsimplicity,itisassumedthattopologicalstructureofGNRisnotaectedbythedefect,whichmayprovideaperturbationtothequantitativeresults,butthequalitativeconclusionsofthisstudywillnotbechanged.Anionizedimpurityistreatedasanexternalxedcharge,whichcanplayanimportantroleforelectrostaticpotentialofthedevice.Inotherwords,intheself-consistentiterativeloopbetweenthetransportequationandthePoissonequation,theinputchargeintothePoissonequationalwaysincludesaxedexternalchargeaswellastheoutputchargefromtheSchrodingerequation. 6.3.1IdealStructure 6-2A depictsthetransfercharacteristics.SBFETshowsthetypicalambipolarbehavior.OcurrentIoff=107AisselectedandonstateisdenedatVon=Voff+VDD.ThentheoperatingvoltagerangesareshownbythegraywindowsinFig. 6-2A .Throughthegateworkfunctiontuning,VoffcanbeshiftedtoVG=0V(VD=VDD),andthetransfercharacteristicsaftertheworkfunctionengineeringareshowninFig. 6-2B 6-3 showsthetransfercharacteristicsinthelinearandinthelogarithmicscale,fordierentpositionsofadefect.Alldefectsareplacedinthemiddle 79

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6-3 ,thedefectnearthesourcehasthelargesteect.Ascomparedtotheidealdevice,thedefectresultsin46%smallerIon.ThisisbecausethecarriertransportinthedeviceistotallycontrolledbytheSchottkybarrieratthesourceend. ThedetailsoftheIonreductioncanbeexplainedbythereducedquantumtransmissionandself-consistentelectrostaticeect.ForanSBFETwithadefectnearthesource,thickerSBisinduced(Fig. 6-4A )duetotheelectronaccumulation,andquantumtransmissionisreduced(Fig. 6-4B )attheonstate,whichresultinasmallerIon.WhenadefectislocatedathalfwayalongthechannelornearthedrainofanSBFET,accumulatedelectronsliftsuppotentialbarrierandreducestheenergywindowofelectroninjectionfromthesourcetochannel,whichresultsinreducedcurrentwithalatticevacancy.Nextweshowthatthetransfercharacteristicisalsoverysensitivetothepositionalongthechannelwidthdirection.Thepositionofadefectvariesfromthecentertothenear-edgeasshownintheinsetofFig. 6-5A .ForbothdevicesithasthelargesteectontheIonwhenitislocatedatthepositionmarkedin-between.BecauseN=12A-GNRhasthelargesteectivecouplingstrengthatthatposition[ 39 ],thedevicehasseverelyreducedtransmission(Fig. 6-5B )andhencethesmallestoncurrent(Fig. 6-5A ).Incomparison,itonlyhassmalleectswhenthedefectisatthecenterorneartheedgeduetotherelativelysmalleectivecouplingstrength. 6-6 ,whichisobtainedbyremovingcarbonatomsfrombothedgesinthesameprobability.Ingeneral,theocurrentsareincreaseddueto 80

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94 ].Figure 6-7A clearlyshowsthelocaldensityofstatesintheband-gapregionatostate.Ontheotherhand,theoncurrentsaregenerallydecreasedduetothereducedquantumtransport[ 94 ].Eventhoughthegapstatesnearthebeginningofthechannelmayfacilitatequantumtransport,overallquantumtransmissionisreducedbythecarriertransportthroughtheimperfect-edgeGNRasshowninFig. 6-7B .ForthestructureofFig. 6-6 ,Ioffisincreasedbyafactorof7,andIonisreducedby40%. InordertoinvestigatethegeneralbehaviorofGNRFETswithedgeroughness,randomlygenerated100samplesaresimulated.Figure 6-8 isahistogramofIonforSBFETsinthepresenceofedgeroughness,wherecarbonatomsarerandomlyaddedintoorremovedfromtheedgesofGNRwithprobabilityP=0.05.TheresultshowsthatIonisgenerallydecreasedbyedgeroughness,andthemeanvalueis25%smallerthantheidealone.Inaddition,theperformancevariationcanbeverylargefromdevicetodevice,whichiscausedbythedierentatomisticdetailsofeachirregular-edgeGNR. 96 ].ItislocatedinthemiddleoftheGNRwidthatdierentpositionsalongthetransportdirection.Figure 6-9 showsID-VGcurvesinthepresenceofanionizedimpurity.Ithasthelargesteectwith20%largerIonwhenlocatednearthesourcebecauseoftheseverelyreducedSchottkybarrieratsourceend(Fig. 6-10A ),whichisakeyfactortodeterminethecarriertransportintunnelingdevices.Ifanimpurityislocatedfarfromthesource,thealterationofSchottkybarrierissignicantlyreduced,anditonlyhasasmalleectontheIon. Next,wesimulated100casesatrandomlydistributedpositionsmaintainingthedistancebetweenLiionandGNRsurfacetoexploreitsgeneraleectontheIon.Figure 6-11 isahistogramofIoninthepresenceofapositiveionizedimpurity,whichshows 81

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SofarwefocusedonapositiveionizedimpurityneartheGNRsurface.Inordertoinvestigatetheeectbyanegativechargeimpurity,anexternalimpurityofanelectronisplacedat0.5nmawayfromGNRsurface.TheelectronislocatedinthemiddleoftheGNRwidthatdierentpositionsalongthetransportdirection.Anelectronimpurityincreasestheself-consistentelectrostaticpotentialasshowninFig. 6-12B .Therefore,theoncurrentisdecreasedby33-47%inthepresenceofelectronimpurity. 82

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Simulateddevicestructure. B TransfercharacteristicsA)beforeandB)afterwork-functionengineering. 83

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Transfercharacteristicsinthepresenceofasinglelatticevacancyalongthetransportdirectioninalogscale(leftaxis)andinalinearscale(rightaxis). B A)ConductionbandprolealongthechannelpositioninthepresenceofalatticevacancyattheONstate.B)Energy-resolvedcurrentspectruminthepresenceofavacancynearthesource. 84

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B A)Transfercharacteristicsinthepresenceofasinglelatticevacancyalongthechannelwidthdirection.B)Energy-resolvedcurrentspectrum. Figure6-6. AtomisticcongurationofasimulatedGNRchannelinthepresenceofedgeroughness. B A)TransfercharacteristicsandB)LDOSattheOFFstate(VG=0VandVD=VDD)withtheGNRchannelshowninFig. 6-6 .ThesolidlineshowsthebandproleoftheidealGNRFET. 85

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HistogramofIoninthepresenceofedgeroughnessofGNRbyaddingorremovingcarbonatomswithprobabilityP=0.05.One-hundredsamplesarerandomlygeneratedandsimulated.Themeanis6.36A,themedianis6.31A,andthestandarddeviationis2A. Figure6-9. Transfercharacteristicsinthepresenceofapositiveionizedimpurityalongthetransportdirectioninalogscale(leftaxis)andinalinearscale(rightaxis). 86

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B A)ConductionbandprolealongthechannelpositioninthepresenceofapositiveionizedimpurityattheONstate.B)Energy-resolvedcurrentspectruminthepresenceofapositiveionizedimpuritynearthesource. Figure6-11. HistogramofIoninthepresenceofanionizedimpuritywith+0:4qat1.84AawayfromtheGNRsurface. 87

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B A)Transfercharacteristicsinthepresenceofachargeimpuritywithqat0.5nmawayfromtheGNRsurface.B)ConductionbandprolealongthetransportpositionattheONstate. 88

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7-1A isanexampleofarmchair-edgeGNR(A-GNR)structure,andFig. 7-1B showsitsbandstructure.IfHpassivationisassumedontheedgestoterminateitsdanglingbonds,thedispersionrelationofA-GNRcomesdirectlyfromthequantumconnementeect.Inotherwords,thesubbandscanbeobtainedatthequantizedkvaluesingraphenebandstructure.Inreality,however,otherchemicalspeciescanbeattachedontheedgesofGNR,andempiricalinputparametersforarbitrarysystemsarenotstraightforward.Therefore,inthisstudy,wepresentaframeworkofdevicesimulationsfornon-uniformGNRsbyusingdensityfunctionaltheory(DFT)calculationwithstandardNEGFformalism. 7-2 )usingSIESTApackage[ 97 ],wherestandardKohn-Shamself-consistentdensity-functionalmethodinthelocaldensityapproximation(LDA)isused,andthebasissetislinearcombinationofatomisticorbitals 89

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7-2B and 7-2C showvariationsofbandstructurefromH-terminatedA-GNR(Fig. 7-2A ),whicharehardtocaptureinTBapproachwithoutcarefullyextractedempiricalparameters.ComparedtotheH-terminatedA-GNR,terminatingtheedgeswithOHgroupsorFatomsshowsqualitativelysimilarE-k.Thereis,however,acleareectonreducingthespacingbetweenthe1standthe2ndvalencesubbands.The1standthe2ndvalencesubbandvalleysaredegenerateinOH-terminatedGNRwhilethereisanenergydierenceof120meVbetweenthemintheF-terminatedGNR. InordertoassesstheperformancelimitsoftheGNReld-eecttransistors(FETs)withdierentedgeterminations,asemiclassicaltop-of-the-barrier(TOB)transistormodel,whichtakesthebandstructuresofGNRsasaninput,canbeused[ 15 ].Themodelassessestheperformancelimitsbyassuming(i)thechannelisballistic(noscattering),and(ii)thetransistorcontactsareideal(withaperfecttransmission).Thesemiclassicalmodeldoesnottreatquantumtunneling.Comparisontothedetailedquantumsimulationsbasedonthenon-equilibriumGreen'sfunctionformalismdemonstratedthatthemodelisvalidfortheMOSFETstructureifthechannellengthislargerthan10nm.Attopofthebarrier,thestatesinthe+kbranchesofE-karelledaccordingtothesourceFermilevelwhilethestatesinthekbranchesarelledaccordingtothedrainFermilevel(Fig. 7-3 ).Asimple2Delectrostaticmodelisusedtocomputethesurfacepotentialatthetopofthebarrierself-consistentlywithcarrierstatistics.Two-dimensionalshortchanneleectscanbemodeledbyusingnon-zeroCSandCDvalues.Theconclusionsoftheedgechemistryeects,however,remainthesameforarangeofphysicalCSandCDvalues.We,therefore,setCS=CD=0forsimplicity.ApowersupplyofVDD=0:6Visused,andtheocurrentforalldevicesissettoIoff=1nAforafaircomparisonoftheon-state 90

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7-4 showsoutputcharacteristicsoftheOH-terminatedGNRFET.Thep-typeoperationofOH-terminateddeviceisworsethann-typeconduction,whichmaybecomparedtothewell-balancedp-typeandn-typeoperationsofH-terminateddevcie,becauseofitsrelativelysmallerband-limitedvelocityofthezonecentervalleysinthevalencesubbandsasshowninFig. 7-2B Figure 7-5A isaunitcellofH-terminatedN=21A-GNRwithOimpurityontheedgeforabandstructurecalculation.OneedgeOatominevery8.52Ahasalargeeect,andtheelectronicpropertiesneartheFermilevelischangedalot.Figure 7-5B showsthedispersionrelationanddensity-of-states(DOS)neartheFermilevel.ThedashedlineispartialorprojectedDOS(PDOS)onthecarbonbackbone,anditisextendedby0:2eVaboveEF.OxygenatomattachedontheedgemakesH-terminatedA-GNRintoap-typesemiconductor.ThestabilityofO-attachedA-GNRwithHterminationisexaminednext.Forthis,bindingenergybetweentheoxygenandthehydrogenatomsiscalculatedby whereERtotisthetotalenergyoftheO-attachedstructure(asshownintherightofFig. 7-6 )andELtotisthetotalenergyofO-detachedone(asintheleftofFig. 7-6 ).Thecalculatedbindingenergyis8:02eVandthenegativevaluemeansthattheO-attached 91

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Wealsoinvestigatedanotherstructureforn-typedopingeect.ThestructureisshowninFig. 7-7A ,whereNatomsubstitutesCatominH-terminatedN=12A-GNRinevery8.52A.DOSandPDOSonthecarbonatomsarealsoplottedbythesolidandthedashedline,respectively(Fig. 7-7B ).PDOSofthecarbonbackboneisextendedby250meVbelowEF,andtheGNRisclearlyn-doped.Nitrogenatomhasonemoreelectronthancarbon,andNsubstitutionprovidesadditionalelectrontotheGNR.Therefore,n-typeelectronicpropertycanbeachieved.Inordertoexaminethestabilityofthisstructure,totalenergiesfordierentstructuresinFig. 7-8 arecompared.ThetotalenergyofFig. 7-8D isminimum,whichindicatesthatthesystemcanloweritsenergybyformingabondbetweennitrogenandGNR.Sofarab-initiocalculationhasbeenextensivelyusedtodescribeelectronicpropertiesofperiodicallyrepeatedstructure.Inthefollowingsection,wepresentadevicesimulationforanon-periodicstructureusingDFT-NEGFmethod. 7-9 ,whereoneOatomisattachedtotheedge.Eveninthepresenceofoneimpurity,simpletight-bindingparametersinpzorbitalbasissetcannotexactlydescribeatomisticdetailsofGNR.However,Hamiltonianandoverlapmatricescanbeconstructedfromab-initiocalculationbypartitioningthewholematrices.Dashed-lineboxesinFig. 7-9 arenecessaryblocksforDFTcalculation.OneisforuniformGNRstructure,andtheresttwoareneededforthe 92

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Becausethebasissetisnonorthogonal,retardedGreen'sfunctioniswrittenas whereHisHamiltonianmatrix,Sisoverlapmatrix,Uisself-consistentelectrostaticpotential,1and2areself-energiesatthesourceandthedraincontacts,respectively.Figure 7-10 showssimulationresultsunderequilibriumbiascondition.OneOatomattachedtotheedgeofA-GNRcanhavealargeeectonthetransmissionofdevice(Fig. 7-10A ),whichindicatesitspotentialapplicationlikesensors.Figure 7-10B islocaldensity-of-states(LDOS),andgapstatesareinducedbytheattachedOatom. 93

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B A)Quasi-1Dgraphenenanoribbon(GNR)structure.B)GNRsubbands,whichcomesfromthequantumconnementalongthewidthdirection. 94

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B C Eectofedgeatoms:apartofGNRanditsbandstructurewithA)H,B)OH,andC)Ftermination. Figure7-3. Top-of-the-barrierballistictransistormodel. 95

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OutputcharacteristicsofOH-terminatedN=12armchair-edgeGNRFET. B P-typedopingeect:A)OatomisattachedontheedgeofH-terminatedN=21A-GNR.B)It'sbandstructureanddensity-of-states(DOS)andprojectedDOS(PDOS)onCatoms. 96

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StabilityofO-attachedA-GNR.BondingbetweenOandCmakesithavelowerenergy. B N-typedopingeect:A)NreplacesCatominH-terminatedN=21A-GNR.B)It'sbandstructureanddensity-of-states(DOS)andprojectedDOS(PDOS)onCatoms. 97

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B C D StabilityofN-substitutionalA-GNR.BondsbetweenNandCmakeithavelowerenergy. Figure7-9. GNRwithoneimpurityatom.Blocksfortheinputofab-initiocalculationareshown.0and1correspondtoon-siteblocksofuniformGNRandonewithimpurity,respectively,and00isforcouplingofuniformGNRand01and10arecouplingsbetweentheuniformGNRandonewithimpurity. 98

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B A)TransmissionandB)localdensity-of-states(LDOS)forthestructureshowninFig. 7-9 99

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100

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ThesecondsubjectisdevicesimulationsofGNRFETs.EventhoughGNRisalsographene-basedquasi-1DnanostructurelikeCNT,thedierencesinshape,boundarycondition,andexistenceofedgesanddanglingbondsmakeitoperateinadierentway.Atomistic3DsimulationstudyoftheperformanceofGNRSBFETsispresented.Theimpactsofnon-idealitiesondeviceperformancehavebeeninvestigated.Inthepresenceofasinglelatticevacancy,IoncanbereducedduetoseverelyaectedSchottkybarrierthicknessofthetunnelingdevice.EdgeroughnessofGNRresultsinlargerocurrentandsmalleroncurrent,ingeneral,andthevariabilityofdeviceperformanceisverylargebecauseofthetotallydierentatomisticcongurationofGNRinsuchsmallchanneldevices.Inthepresenceofapositiveionizedimpurity,Ioncanbeincreasedby20%,andanegativechargeimpurityalwayslargelydisturbsthecarriertransportofGNRFETsduetothelocallyincreasedelectrostaticpotential.TheedgesofGNR,whichdonotexistinCNT,canbeadvantagesordisadvantages.Ifanappropriatecontrolbydierentedgechemicalspeciesispossible,itwouldbedenitelypositive.Inthiswork,edge-chemistryengineeringofGNRisstudiedbydensity-functionaltheory(DFT).Totallynewelectronicbandstructureisobtainedbydierentedge-terminationatoms.Inaddition,onlyafractionofimpurityatomcanalsomuchaectonthematerialpropertiesofGNR.Inordertoperformdevicesimulationsofnon-uniformGNRdevices,multiscalesimulationschemecanbeusedinnon-equilibriumGreen'sfunction(NEGF)formalismanddensity-functionalmethod. 101

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102

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A.Javey,J.Guo,D.B.Farmer,Q.Wang,E.Yenilmez,R.G.Gordon,M.Lundstrom,andH.J.Dai,\Self-alignedballisticmoleculartransistorsandelectricallyparallelnanotubearrays,"NanoLett.,vol.4,pp.1319{1322,2004. [55] J.AppenzellerandD.J.Frank,\Frequencydependentcharacterizationoftransportpropertiesincarbonnanotubetransistors,"Appl.Phys.Lett.,vol.84,pp.1771{1773,2004. [56] P.J.Burke,\Acperformanceofnanoelectronics:towardsaballisticthznanotubetransistor,"Solid-StateElectronics,vol.48,pp.1981{1986,Oct./Nov.2004. [57] S.Rosenblatt,H.Lin,V.Sazonova,S.Tiwari,andP.L.McEuen,\Mixingat50ghzusingasingle-walledcarbonnanotubetransistor,"Appl.Phys.Lett.,vol.87,p.153111,2005. [58] K.AlamandR.Lake,\Performanceof2nmgatelengthcarbonnanotubeeld-eecttransistorswithsource/drainunderlaps,"Appl.Phys.Lett.,vol.87,p.073104,2005. [59] L.C.Castro,D.L.John,D.L.Pulfrey,M.Pourfath,A.Gehring,andK.H.,\Methodforpredictingftforcarbonnanotubefets,"IEEETrans.Nanotech.,vol.4,pp.699{704,2005. [60] A.Javey,J.Guo,M.Paulsson,Q.Wang,D.Mann,M.Lundstrom,andH.J.Dai,\High-eldquasiballistictransportinshortcarbonnanotubes,"Phys.Rev.Lett.,vol.92,p.106804,2004. [61] J.Guo,\Aquantum-mechanicaltreatmentofphononscatteringincarbonnanotubetransistors,"J.Appl.Phys.,vol.98,p.063519,2005. [62] S.Koswatta,S.Hasan,M.Lundstrom,M.P.Anantram,andD.E.Nikonov,\Ballisticityofnanotubeeld-eecttransistors:Roleofphononenergyandgatebias,"Appl.Phys.Lett.,vol.89,p.023125,2006. [63] G.D.Mahan,\Electron-opticalphononinteractionincarbonnanotubes,"J.Appl.Phys.,vol.68,p.125409,2003. [64] R.V.Seidel,A.P.Graham,J.Kretz,B.Rajasekharan,G.S.Duesberg,M.Liebau,E.Unger,F.Kreupl,andW.Hoenlein,\Sub-20nmshortchannelcarbonnanotubetransistors,"NanoLett.,vol.5,pp.147{150,2005. [65] C.R.KaganandP.Andry,Thin-lmtransistors.NewYork:MarcelDekker,2003. [66] C.D.DimitrakopoulosandD.J.Mascaro,\Organicthin-lmtransistors:Areviewofrecentadvances,"IBMJ.ResearchandDevelopment,vol.45,pp.11{27,2001. [67] M.Muccini,\Abrightfuturefororganiceld-eecttransistors,"NatureMaterials,vol.5,pp.605{613,2006. 107

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J.H.Burroughes,D.D.C.Bradley,A.R.Brown,R.N.Marks,K.Mackay,R.H.Friend,P.L.Burns,andA.B.Holmes,\Light-emitting-diodesbasedonconjugatedpolymers,"Nature,vol.347,pp.539{541,1990. [69] Q.B.Pei,G.Yu,C.Zhang,Y.Yang,andA.J.Heeger,\Polymerlight-emittingelectrochemical-cells,"Science,vol.269,pp.1086{1088,1995. [70] M.LipingandY.Yang,\Uniquearchitectureandconceptforhigh-performanceorganictransistors,"Appl.Phys.Lett.,vol.85,pp.5084{5086,1995. [71] B.Liu,M.A.McCarthy,Y.Yoon,D.Y.Kim,Z.Wu,F.So,P.H.Holloway,J.R.Reynolds,J.Guo,andA.G.Rinzler,\Carbon-nanotube-enabledverticaleldeectandlight-emittingtransistors,"Adv.Mater.,vol.20,pp.3605{3609,2008. [72] Z.C.Wu,Z.H.Chen,X.Du,J.M.Logan,J.Sippel,M.Nikolou,K.Kamaras,J.R.Reynolds,D.B.Tanner,A.F.Hebard,andA.G.Rinzler,\Transparent,conductivecarbonnanotubelms,"Science,vol.305,pp.1273{1276,2004. [73] D.H.Zhang,K.Ryu,X.L.Liu,E.Polikarpov,J.Ly,M.E.Tompson,andC.W.Zhou,\Transparent,conductive,andexiblecarbonnanotubelmsandtheirapplicationinorganiclight-emittingdiodes,"NanoLett.,vol.6,pp.1880{1886,2006. [74] D.L.R.B.K.L.B.Hu,G.GrunerandJ.Cech,\Patternabletransparentcarbonnanotubelmsforelectrochromicdevices,"J.Appl.Phys.,vol.101,p.016102,2007. [75] M.A.Alam,A.Dodabalapur,,andM.R.Pinto,\Atwo-dimensionalsimulationoforganictransistors,"IEEETrans.ElectronDevices,vol.44,pp.1332{1337,1997. [76] S.Selberherr,A.Schutz,andH.W.Potzl,\Minimos-atwo-dimensionalmos-transistoranalyzer,"IEEETrans.ElectronDevices,vol.27,pp.1540{1550,1980. [77] D.L.ScharfetterandH.K.Gummel,\Large-signalanalysisofasiliconreaddiodeoscillator,"IEEETrans.ElectronDevices,vol.16,pp.64{77,Jan.1969. [78] J.Terso,\Schottky-barrierheightsandthecontinuumofgapstates,"Phys.Rev.Lett.,vol.52,pp.465{468,1984. [79] A.Javey,J.Guo,Q.Wang,M.Lundstrom,andH.J.Dai,\Ballisticcarbonnanotubeeld-eecttransistors,"Nature,vol.424,pp.654{657,Aug.2003. [80] C.Berger,Z.M.Song,T.B.Li,X.B.Li,A.Y.Ogbazghi,R.Feng,Z.T.Dai,A.N.Marchenkov,E.H.Conrad,P.N.First,andW.A.deHeer,\Ultrathinepitaxialgraphite:2delectrongaspropertiesandaroutetowardgraphene-basednanoelectronics,"J.Phys.Chem.B,vol.108,no.52,pp.19912{19916,2004. [81] Y.B.Zhang,Y.W.Tan,H.L.Stormer,andP.Kim,\Experimentalobservationofthequantumhalleectandberry'sphaseingraphene,"Nature,vol.438,pp.201{204,Nov.2005. 108

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V.Barone,O.Hod,andG.E.Scuseria,\Electronicstructureandstabilityofsemiconductinggraphenenanoribbons,"NanoLett.,vol.6,pp.2748{2754,Dec.2006. [83] K.Nakada,M.Fujita,G.Dresselhaus,andM.S.Dresselhaus,\Edgestateingrapheneribbons:Nanometersizeeectandedgeshapedependence,"Phys.Rev.B,vol.54,pp.17954{17961,Dec.1996. [84] C.T.White,J.W.Li,D.Gunlycke,andJ.W.Mintmire,\Hiddenone-electroninteractionsincarbonnanotubesrevealedingraphenenanostrips,"NanoLett.,vol.7,no.3,pp.825{830,2007. [85] X.L.Li,X.R.Wang,L.Zhang,S.W.Lee,andH.J.Dai,\Chemicallyderived,ultrasmoothgraphenenanoribbonsemiconductors,"Science,vol.319,pp.1229{1232,Feb.2008. [86] M.C.Lemme,T.J.Echtermeyer,M.Baus,andH.Kurz,\Agrapheneeld-eectdevice,"IEEEElectronDeviceLetters,vol.28,pp.282{284,Apr.2007. [87] G.C.Liang,N.Neophytou,D.E.Nikonov,andM.S.Lundstrom,\Performanceprojectionsforballisticgraphenenanoribboneld-eecttransistors,"IEEETrans.ElectronDevices,vol.54,pp.677{682,Apr.2007. [88] Y.Ouyang,Y.Yoon,J.K.Fodor,andJ.Guo,\Comparisonofperformancelimitsforcarbonnanoribbonandcarbonnanotubetransistors,"Appl.Phys.Lett.,vol.89,p.203107,Nov.2006. [89] X.Guan,M.Zhang,Q.Liu,andZ.Yu,\Simulationinvestigationofdouble-gatecnr-mosfetswithfullyself-consistentnegfandtbmethod,"inIEDMTech.Dig.,(Washington,DC),pp.761{764,2007. [90] G.C.Liang,N.Neophytou,M.S.Lundstrom,andD.E.Nikonov,\Ballisticgraphenenanoribbonmetal-oxide-semiconductoreld-eecttransistors:Afullreal-spacequantumtransportsimulation,"J.Appl.Phys.,vol.102,p.054307,Sep.2007. [91] Y.Ouyang,Y.Yoon,andJ.Guo,\Scalingbehaviorsofgraphenenanoribbonfets:Athree-dimensionalquantumsimulationstudy,"IEEETrans.ElectronDevices,vol.54,pp.2223{2231,Sep.2007. [92] G.FioriandI.Giuseppe,\Simulationofgraphenenanoribboneldeecttransistors,"IEEEElectronDeviceLetters,vol.28,pp.760{762,Aug.2007. [93] D.Basu,M.J.Gilbert,L.F.Register,S.K.Banerjee,andA.H.MacDonald,\Eectofedgeroughnessonelectronictransportingraphenenanoribbonchannelmetal-oxide-semiconductoreld-eecttransistors,"Appl.Phys.Lett.,vol.92,p.042114,Jan.2008. [94] Y.YoonandJ.Guo,\Eectofedgeroughnessingraphenenanoribbontransistors,"Appl.Phys.Lett.,vol.91,p.073103,Aug.2007. 109

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E.R.Mucciolo,A.H.C.Neto,andC.H.Lewenkoph,\Conductancequantizationandtransportgapindisorderedgraphenenanoribbons,"cond-mat/0806.3777. [96] S.Hong,Y.Yoon,andJ.Guo,\Metal-semicondutorjunctionofgraphenenanoribbons,"Appl.Phys.Lett.,vol.92,p.083107,Feb.2008. [97] J.M.Soler,E.Artacho,J.D.Gale,A.Garcia,J.Junquera,P.Ordejon,andD.Sanchez-Portal,\Thesiestamethodforabinitioorder-nmaterialssimulation,"J.Phys.:Condens.Matter,vol.14,pp.2745{2779,2002. 110

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YoungKiYoonreceivedtheB.E.inmaterialscienceandengineeringfromKoreaUniversity,Seoul,Koreain1999.HealsoreceivedtheM.S.andPh.D.inelectricalandcomputerengineeringfromUniversityofFlorida(UF)in2005and2008,respectively.HeworkedforProf.JingGuoinComputationalNanotechnologyGroupatUFfrom2005to2008.DuringhisPh.D.work,heparticipatedinmorethan14journalpapersand10conferencepresentations.Hisresearchinterestsincludethephysincs,modeling,andsimulationofnanoscaledevices. 111


Permanent Link: http://ufdc.ufl.edu/UFE0022887/00001

Material Information

Title: Carbon Nanotube and Graphene Device Modeling and Simulation
Physical Description: 1 online resource (111 p.)
Language: english
Creator: Yoon, Young
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2008

Subjects

Subjects / Keywords: carbon, cnt, device, gnr, graphene, nanoribbon, nanotube, negf, quantum, simulation, transport
Electrical and Computer Engineering -- Dissertations, Academic -- UF
Genre: 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

Notes

Abstract: The performance of the semiconductors has been improved and the price has gone down for decades. It has been continuously scaled down in size year by year, and now it encounters the fundamental scaling limit. We, therefore, should prepare a new era beyond the conventional semiconductor technologies. One of the most promising devices is possible by carbon nanotube (CNT) or graphene nanoribbon (GNR) in terms of its excellent charge transport properties. Their fundamental material properties and device physics are totally different to those of the conventional devices. In this nano-regime, more sophisticated device modeling and simulation are really needed to elucidate nano-device operation and to save our resources from errors. The numerical simulation works in this dissertation will provide novel view points on the emerging devices. In this dissertation, CNT and GNR devices are numerically studied. The first part of this work is on CNT devices, and a common structure of CNT device has CNT channel, metal source and drain contacts, and gate electrode. We investigate the strain, geometry, and scattering effects on the device performance of CNT field-effect transistors (FETs). It is shown that even a small amount of strain can result in a large effect on the performance of CNTFETs due to the variation of the bandgap and band-structure-limited velocity. A type of strain which produces a larger bandgap results in increased Schottky barrier (SB) height and decreased band-structure-limited velocity, and hence a smaller minimum leakage current, smaller on current, larger maximum achievable on-off ratio, and larger intrinsic delay. We also examine geometry effect of partial gate CNTFETs. In the growth process of vertical CNT, underlap between the gate and the bottom electrode is advantageous for transistor operation because it suppresses ambipolar conduction of SBFETs. Both n-type and p-type transistor operations with balanced performance metrics can be achieved on a single partial gate FET by using proper bias schemes. The effect of phonon scattering on the intrinsic delay and cut-off frequency of Schottky barrier CNTFETs is also examined. Carriers are mostly scattered by optical and zone boundary phonons beyond the beginning of the channel. The scattering has a small direct effect on the DC on current of the CNTFET, but it results in significant decrease of intrinsic cut-off frequency and increase of intrinsic delay. Semiconducting CNT is useful for the channel in CNTFETs, whereas metallic CNT can be used as an electrode. If a porous CNT film is used as a source electrode, vertical thin-film transistors (TFTs) can be constructed. Vertical organic FET (OFET) shows clear transistor switching behavior allowing orders of magnitude modulation of the source-drain current even in the presence of electrostatic screening by the source electrode. The channel length should be carefully engineered due to the trade-off between device characteristics in the subthreshold and above-threshold regions. The second subject is device simulations of GNRFETs. Even though GNR is also graphene-based quasi-1D nanostructure like CNT, the differences in shape, boundary condition, and existence of edges and dangling bonds make it operate in a different way. Atomistic 3D simulation study of the performance of GNR SBFETs is presented. The impacts of non-idealities on device performance have been investigated. The edges of GNR, which do not exist in CNT, can be advantages or disadvantages. If an appropriate control by different edge atoms is possible, it would be definitely positive. Totally new electronic band structure is obtained by different edge-termination atoms. In addition, only a fraction of impurity atom can also much affect on the material properties of GNR. In order to perform device simulations of non-uniform GNR devices, multiscale simulation scheme can be used in non-equilibrium Green's function (NEGF) formalism and density-functional method.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: 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 Young Yoon.
Thesis: Thesis (Ph.D.)--University of Florida, 2008.
Local: Adviser: Guo, Jing.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2009-06-30

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2008
System ID: UFE0022887:00001

Permanent Link: http://ufdc.ufl.edu/UFE0022887/00001

Material Information

Title: Carbon Nanotube and Graphene Device Modeling and Simulation
Physical Description: 1 online resource (111 p.)
Language: english
Creator: Yoon, Young
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2008

Subjects

Subjects / Keywords: carbon, cnt, device, gnr, graphene, nanoribbon, nanotube, negf, quantum, simulation, transport
Electrical and Computer Engineering -- Dissertations, Academic -- UF
Genre: 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

Notes

Abstract: The performance of the semiconductors has been improved and the price has gone down for decades. It has been continuously scaled down in size year by year, and now it encounters the fundamental scaling limit. We, therefore, should prepare a new era beyond the conventional semiconductor technologies. One of the most promising devices is possible by carbon nanotube (CNT) or graphene nanoribbon (GNR) in terms of its excellent charge transport properties. Their fundamental material properties and device physics are totally different to those of the conventional devices. In this nano-regime, more sophisticated device modeling and simulation are really needed to elucidate nano-device operation and to save our resources from errors. The numerical simulation works in this dissertation will provide novel view points on the emerging devices. In this dissertation, CNT and GNR devices are numerically studied. The first part of this work is on CNT devices, and a common structure of CNT device has CNT channel, metal source and drain contacts, and gate electrode. We investigate the strain, geometry, and scattering effects on the device performance of CNT field-effect transistors (FETs). It is shown that even a small amount of strain can result in a large effect on the performance of CNTFETs due to the variation of the bandgap and band-structure-limited velocity. A type of strain which produces a larger bandgap results in increased Schottky barrier (SB) height and decreased band-structure-limited velocity, and hence a smaller minimum leakage current, smaller on current, larger maximum achievable on-off ratio, and larger intrinsic delay. We also examine geometry effect of partial gate CNTFETs. In the growth process of vertical CNT, underlap between the gate and the bottom electrode is advantageous for transistor operation because it suppresses ambipolar conduction of SBFETs. Both n-type and p-type transistor operations with balanced performance metrics can be achieved on a single partial gate FET by using proper bias schemes. The effect of phonon scattering on the intrinsic delay and cut-off frequency of Schottky barrier CNTFETs is also examined. Carriers are mostly scattered by optical and zone boundary phonons beyond the beginning of the channel. The scattering has a small direct effect on the DC on current of the CNTFET, but it results in significant decrease of intrinsic cut-off frequency and increase of intrinsic delay. Semiconducting CNT is useful for the channel in CNTFETs, whereas metallic CNT can be used as an electrode. If a porous CNT film is used as a source electrode, vertical thin-film transistors (TFTs) can be constructed. Vertical organic FET (OFET) shows clear transistor switching behavior allowing orders of magnitude modulation of the source-drain current even in the presence of electrostatic screening by the source electrode. The channel length should be carefully engineered due to the trade-off between device characteristics in the subthreshold and above-threshold regions. The second subject is device simulations of GNRFETs. Even though GNR is also graphene-based quasi-1D nanostructure like CNT, the differences in shape, boundary condition, and existence of edges and dangling bonds make it operate in a different way. Atomistic 3D simulation study of the performance of GNR SBFETs is presented. The impacts of non-idealities on device performance have been investigated. The edges of GNR, which do not exist in CNT, can be advantages or disadvantages. If an appropriate control by different edge atoms is possible, it would be definitely positive. Totally new electronic band structure is obtained by different edge-termination atoms. In addition, only a fraction of impurity atom can also much affect on the material properties of GNR. In order to perform device simulations of non-uniform GNR devices, multiscale simulation scheme can be used in non-equilibrium Green's function (NEGF) formalism and density-functional method.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: 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 Young Yoon.
Thesis: Thesis (Ph.D.)--University of Florida, 2008.
Local: Adviser: Guo, Jing.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2009-06-30

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2008
System ID: UFE0022887:00001


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Firstandforemost,Imustexpressmysinceregratitudetomyadvisor,Prof.JingGuo,forhisguidanceandencouragement.HisbeliefinmyabilityhasmademeachievethegoalsIdreamed.Thisworkwouldnothavebeenpossiblewithouthissupportandinspiration.Ideeplythankhimforhisinvaluableadviceandnumerousdiscussionswehad.Ialsowouldliketothankallthemembersofmysupervisorycommittee(Prof.ScottThompson,Prof.AndrewG.Rinzler,Prof.AntUral,andProf.SusanB.Sinnott)fortheirinterestsandhelpfulsuggestionsonmyresearch.Onaccountoftheirdeepknowledgeandkeeninsightonelectrondevices,thisworkcouldbeasmuchawlessasitis.Thanksarealsonecessaryformycolleagues(YijianOuyang,PeiZhao,JyotsnaChauhan,SeokminHong,andJamesFodor)inComputationalNanotechnologyLaboratoryfortheirfriendshipandsupport.IalsothankProf.GainlucaFioriandProf.GiuseppeIannacconeatUniversityofPisa,Italy,Prof.HongjieDaiandXinranWangatStanfordUniversity,andProf.KartikMohanramandMihirChoudhuryatRiceUniversityforcollaborationsandhelpfuldiscussions.Ithanktomyparentsandparents-in-lawwithallmyheartfortheirendlesssupportandloveduringthelongjourney.Finally,Iwouldliketoexpressmygreatestthankstomywife,AramJeon,forhersupportwithlove.ShealwayshasbeenstandingbytoencouragemeininnumerablewaysthroughoutmyPh.D.study.TheloveIfeelformyfamilygoesfarbeyondthewordsIcanexpress. 4

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page ACKNOWLEDGMENTS ................................. 4 LISTOFTABLES ..................................... 7 LISTOFFIGURES .................................... 8 ABSTRACT ........................................ 13 CHAPTER 1INTRODUCTION .................................. 16 1.1Overview .................................... 16 1.2CarbonNanotube(CNT)vs.GrapheneNanoribbon(GNR) ........ 16 1.3NanoelectronicsSimulation ........................... 18 1.3.1ReviewofNEGFFormalism ...................... 18 1.3.2SelfconsistentSimulationScheme ................... 19 2ANALYSISOFSTRAINEFFECTSINBALLISTICCARBONNANOTUBEFETS ......................................... 22 2.1Introduction ................................... 22 2.2Approach .................................... 24 2.3Results ...................................... 27 2.3.1TransferCharacteristics ......................... 27 2.3.2MinimumCurrent,Imin 29 2.3.3OnCurrent,Ion 30 2.3.4IntrinsicDelay .............................. 31 2.4ConclusionsandDiscussions .......................... 33 3ACOMPUTATIONALSTUDYOFVERTICALPARTIALGATECARBONNANOTUBEFETS ................................. 40 3.1Introduction ................................... 40 3.2Approach .................................... 42 3.2.1DeviceStructure ............................. 42 3.2.2QuantumTransport ........................... 42 3.2.3Electrostatics .............................. 44 3.3Results ...................................... 44 3.3.1TransferCharacteristics ......................... 44 3.3.2DeviceOperation ............................ 45 3.3.3OutputCharacteristics ......................... 45 3.3.4EectofGateLength .......................... 46 3.3.5EectofAirPore ............................ 47 3.3.6EectofCNTDiameter ........................ 48 5

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.................................... 48 4EFFECTSOFSCATTERINGONRFPERFORMANCEOFCARBONNANOTUBEFETS ......................................... 55 4.1Introduction ................................... 55 4.2Approach .................................... 56 4.3Results ...................................... 57 4.3.1OnCurrent ............................... 57 4.3.2ChargeandVelocity ........................... 58 4.3.3IntrinsicCutoFrequencyandDelay ................. 59 4.4DiscussionsandConclusions .......................... 60 5ACOMPUTATIONALSTUDYOFCARBON-NANOTUBE-ENABLEDVERTICALORGANICFIELD-EFFECTTRANSISTORS ................... 64 5.1Introduction ................................... 64 5.2Approach .................................... 65 5.3Results ...................................... 68 5.4DiscussionsandConclusions .......................... 71 6EFFECTOFNON-IDEALITIESINGRAPHENENANORIBBONFETS ... 77 6.1Introduction ................................... 77 6.2Approach .................................... 78 6.3Results ...................................... 79 6.3.1IdealStructure .............................. 79 6.3.2AtomisticVacancy ........................... 79 6.3.3EdgeRoughness ............................. 80 6.3.4IonizedImpurity ............................. 81 6.4Conclusion .................................... 82 7FRAMEWORKOFDEVICESIMULATIONSFORNON-UNIFORMGRAPHENENANORIBBONS ................................... 89 7.1Introduction ................................... 89 7.2DFT-TOBModel ................................ 89 7.3DopingEectinGrapheneNanoribbonbyEdge-ChemistryEngineering .. 91 7.4DFT-NEGFMethod .............................. 92 7.5Conclusion .................................... 93 8CONCLUSIONANDFUTUREWORK ...................... 100 8.1Conclusion .................................... 100 8.2SuggestionforFutureWorks .......................... 102 REFERENCES ....................................... 103 BIOGRAPHICALSKETCH ................................ 111 6

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Table page 1-1Carbonnanotube(CNT)vs.graphenenanoribbon(GNR) ............ 21 7

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Figure page 1-1Latticestructureofgraphene ............................ 20 1-2Bandstructureofgraphene,whichhasalinearE-krelationneartheFermipoint. 20 1-3TwospecialkindsofGNRs.A)N=10armchair-edgeGNR.B)N=7zigzag-edgeGNR. ......................................... 20 1-4BandstructureofsemiconductingCNTandGNR.A)(13,0)CNT.B)N=12armchair-edgeGNR. ................................. 21 1-5Densityofstatesof(13;0)CNT(solidline)andN=12A-GNR(dashedline). 21 2-1Typeofappliedstrains.A)Tensileuniaxialstrain.B)Compressiveuniaxialstrain.C)Torsionalstrain. .............................. 34 2-2RealtomodespaceapproachforstrainedCNT.A)Partof2-D(n;0)zigzagnanotubelatticeinrealspace.B)Uncoupled1-Dmodespacelattice. ...... 34 2-3Transfercharacteristicswithdierentstrains.A)Uniaxialstrainon(16;0)CNT.B)Uniaxialstrainon(17;0)CNT.C)Torsionalstrainon(16;0)CNT.D)Torsionalstrainon(17;0)CNT. ................................ 35 2-4Bandgapvs.strain.A)Uniaxialstrain.B)Torsionalstrain. ........... 35 2-5Minimumleakagecurrentvs.strain.A)Uniaxialstrain.B)Torsionalstrain. .. 36 2-6Oncurrentvs.strain.A)Uniaxialstrain.B)Torsionalstrain. .......... 36 2-7Onstateof2%strained(16;0)CNTFETcomparedtotheunstraineddevice.A)Conductionbandprolealongthechannelposition.B)Energy-resolvedcurrentspectrum. ....................................... 37 2-8Intrisicdelayvs.Ion=Ioffunderdierentstrain.A)Uniaxialstrainon(16;0)CNT.B)Uniaxialstrainon(17;0)CNT.C)Torsionalstrainon(16;0)CNT.D)Torsionalstrainon(17;0)CNT. ......................... 38 2-9Conductionbandprolealongthechannelpositionfor2%strained(17;0)CNTFETatahighgateoverdrive. ............................... 39 2-10Firstandsecondlowestsubbandof(16;0)CNTsunderunstrained(solidline)and2%strained(dashedline)condition ...................... 39 3-1Verticalpartial-gateCNTFET.A)Crosssectionofthemodeledgeometry.B)Deviceparametersofthenominaldevice. ...................... 50 8

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..................... 51 3-3Deviceoperationsofverticalpartial-gateCNTFET.A)P-typeoperationwithVG=VD=VDD.B)N-typeoperationwithVG=VD=+VDD. ......... 52 3-4Outputcharacteristicsofpartial-gateCNTFET.A)ID-VDcurvewithdierentgatevoltages.B)BandprolealongthechannelpositionatVD=0:4Vand0:1VwithVG=0:5V. .............................. 52 3-5Eectofgatelengthinpartial-gateCNTFET.A)ID-VGcurveswithdierentgatelengths.B)Bandprolealongthechannelpositionfordierentgatelengths. 53 3-6Eectofairporeinpartial-gateCNTFET.A)ID-VGcurveswithandwithoutairpore.B)Electriceldcontournearthesourcecontactwith(left)andwithout(right)airpore. .................................... 54 3-7EectofCNTdiameterinpartial-gateCNTFET. ................. 54 4-1Eectofphononscatteringonthecurrentwithdierentgatevoltagesordierentchannellengths.A)IDvs.VGatVD=0:5Vattheballisticlimit(thedashedline)andinthepresenceofphononscattering(thesolidline).Thechannellengthis100nm.B)Oncurrentvs.channellengthatballisticlimit(dashedline)andinthepresenceofphononscattering(solidline).Theinsetsketchestherstsubbandproleattheonstate. ........................... 62 4-2Eectofphononscatteringonaveragecarriervelocityandelectrondensity.A)Averagecarriervelocityvs.channelpositionatVG=VD=0:5Vattheballisticlimit(dashedlines)andinthepresenceofphononscattering(solidlines).B)Electrondensityvs.channelpositionatVG=VD=0:5Vattheballisticlimit(dashedlines)andinthepresenceofphononscattering(solidlines). ....... 62 4-3Eectofphononscatteringoncutofrequencyandintrinsicdelay.A)Cutofrequencyvs.channellengthatonstate.ThecirclesarenumericallycomputedfTattheballisticlimitandthedashedlineisattingcurveoffT=110GHzm=Lch.ThecrossesarenumericallycomputedfTinthepresenceofphononscatteringandthesolidlineisattingcurveoffT=40GHzm=Lch.B)Intrinsicdelayvs.channellengthattheballisticlimit(thecircles)andinthepresenceofphononscattering(thecrosses).Thedashedlineisalinearttingoftheballisticresultby=Lch1:71ps=m. .................. 63 5-1A)SchematicstructureofaverticalorganicFET.Gate,source,activelayer,anddrainareverticallystackedup.Poroussourceelectrodeofpercolatingcarbonnanotubenetworkallowsgateelectriceldtopenetratetothechannelregion.B)Cross-sectionofthesimulateddevicestructure. ................ 73 9

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........................................ 73 5-3A)TransfercharacteristicsforVD=2:5and5Vinaalongscale(linearplotsareshownintheinset).B)Vacuumenergylevelshiftedbyanionizationenergyoftheactivelayeralongthechannelatx=0underVG=0V. ...... 74 5-4A)OutputcharacteristicsforVD=0,5,and10V.Nocurrentsaturationisobservedwithinthesimulatedbiasrange(uptoVD=30V).B)HoledensityinthechannelforVD=2and10V. .......................... 74 5-5Channellengthscaling:ID-VGcurvesfordierentchannellengthinA)alinearandB)alogscale. .................................. 75 5-6Eectofchannelmobility:A)ID-VGcurveswithdierentmobilityoftheactivelayer(theunitisgivenincm2=Vs).B)Transconductancevs.channelmobilityplot.TheinsetshowsSvs.mobility. ........................ 75 5-7Eectofchanneldielectric:A)ID-VGcurvesfordierentpermittivityofactivelayer.B)Vacuumenergylevelshiftedbyanionizationenergyoftheactivelayeralongthechannelatx=0underVG=10V. .................... 76 5-8Eectofgateoxide:A)ID-VGcurvesforgatedielectricconstantof4and16inalogscaleandalinearscale(inset).B)Sandgm(inset)areplottedasafunctionofinverseeectivegateoxidethickness:toxisscaleddownfrom200to50nm,andoxisvariedfrom4to20. ........................ 76 6-1Simulateddevicestructure. ............................. 83 6-2TransfercharacteristicsA)beforeandB)afterwork-functionengineering. .... 83 6-3Transfercharacteristicsinthepresenceofasinglelatticevacancyalongthetransportdirectioninalogscale(leftaxis)andinalinearscale(rightaxis). ........ 84 6-4A)ConductionbandprolealongthechannelpositioninthepresenceofalatticevacancyattheONstate.B)Energy-resolvedcurrentspectruminthepresenceofavacancynearthesource. ............................ 84 6-5A)Transfercharacteristicsinthepresenceofasinglelatticevacancyalongthechannelwidthdirection.B)Energy-resolvedcurrentspectrum. .......... 85 6-6AtomisticcongurationofasimulatedGNRchannelinthepresenceofedgeroughness. ....................................... 85 10

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6-6 .ThesolidlineshowsthebandproleoftheidealGNRFET. ............................ 85 6-8HistogramofIoninthepresenceofedgeroughnessofGNRbyaddingorremovingcarbonatomswithprobabilityP=0.05.One-hundredsamplesarerandomlygeneratedandsimulated.Themeanis6.36A,themedianis6.31A,andthestandarddeviationis2A. ............................. 86 6-9Transfercharacteristicsinthepresenceofapositiveionizedimpurityalongthetransportdirectioninalogscale(leftaxis)andinalinearscale(rightaxis). .. 86 6-10A)ConductionbandprolealongthechannelpositioninthepresenceofapositiveionizedimpurityattheONstate.B)Energy-resolvedcurrentspectruminthepresenceofapositiveionizedimpuritynearthesource. .............. 87 6-11HistogramofIoninthepresenceofanionizedimpuritywith+0:4qat1.84AawayfromtheGNRsurface. ............................. 87 6-12A)Transfercharacteristicsinthepresenceofachargeimpuritywithqat0.5nmawayfromtheGNRsurface.B)ConductionbandprolealongthetransportpositionattheONstate. ............................... 88 7-1A)Quasi-1Dgraphenenanoribbon(GNR)structure.B)GNRsubbands,whichcomesfromthequantumconnementalongthewidthdirection. ......... 94 7-2Eectofedgeatoms:apartofGNRanditsbandstructurewithA)H,B)OH,andC)Ftermination. ................................ 95 7-3Top-of-the-barrierballistictransistormodel. .................... 95 7-4OutputcharacteristicsofOH-terminatedN=12armchair-edgeGNRFET. ... 96 7-5P-typedopingeect:A)OatomisattachedontheedgeofH-terminatedN=21A-GNR.B)It'sbandstructureanddensity-of-states(DOS)andprojectedDOS(PDOS)onCatoms. .............................. 96 7-6StabilityofO-attachedA-GNR.BondingbetweenOandCmakesithavelowerenergy. ......................................... 97 7-7N-typedopingeect:A)NreplacesCatominH-terminatedN=21A-GNR.B)It'sbandstructureanddensity-of-states(DOS)andprojectedDOS(PDOS)onCatoms. ...................................... 97 7-8StabilityofN-substitutionalA-GNR.BondsbetweenNandCmakeithavelowerenergy. ......................................... 98 11

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......... 98 7-10A)TransmissionandB)localdensity-of-states(LDOS)forthestructureshowninFig. 7-9 ...................................... 99 12

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Theperformanceofthesemiconductorshasbeenimprovedandthepricehasgonedownfordecades.Ithasbeencontinuouslyscaleddowninsizeyearbyyear,andnowitencountersthefundamentalscalinglimit.We,therefore,shouldprepareanewerabeyondtheconventionalsemiconductortechnologies.Oneofthemostpromisingdevicesispossiblebycarbonnanotube(CNT)orgraphenenanoribbon(GNR)intermsofitsexcellentchargetransportproperties.Theirfundamentalmaterialpropertiesanddevicephysicsaretotallydierenttothoseoftheconventionaldevices.Inthisnano-regime,moresophisticateddevicemodelingandsimulationarereallyneededtoelucidatenano-deviceoperationandtosaveourresourcesfromerrors.Thenumericalsimulationworksinthisdissertationwillprovidenovelviewpointsontheemergingdevices. Inthisdissertation,CNTandGNRdevicesarenumericallystudied.TherstpartofthisworkisonCNTdevices,andacommonstructureofCNTdevicehasCNTchannel,metalsourceanddraincontacts,andgateelectrode.Weinvestigatethestrain,geometry,andscatteringeectsonthedeviceperformanceofCNTeld-eecttransistors(FETs).ItisshownthatevenasmallamountofstraincanresultinalargeeectontheperformanceofCNTFETsduetothevariationofthebandgapandband-structure-limitedvelocity.AtypeofstrainwhichproducesalargerbandgapresultsinincreasedSchottkybarrier(SB)heightanddecreasedband-structure-limitedvelocity,andhenceasmallerminimumleakagecurrent,smalleroncurrent,largermaximumachievableIon/Ioff,and 13

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ThesecondsubjectisdevicesimulationsofGNRFETs.EventhoughGNRisalsographene-basedquasi-1DnanostructurelikeCNT,thedierencesinshape,boundarycondition,andexistenceofedgesanddanglingbondsmakeitoperateinadierentway.Atomistic3DsimulationstudyoftheperformanceofGNRSBFETsispresented.Theimpactsofnon-idealitiesondeviceperformancehavebeeninvestigated.TheedgesofGNR,whichdonotexistinCNT,canbeadvantagesordisadvantages.Ifanappropriatecontrolbydierentedgeatomsispossible,itwouldbedenitelypositive.Totallynewelectronicbandstructureisobtainedbydierentedge-terminationatoms.Inaddition,onlyafractionofimpurityatomcanalsomuchaectonthematerialpropertiesofGNR.Inordertoperformdevicesimulationsofnon-uniformGNRdevices,multiscale 14

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15

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1 ],anumberofactiveresearcheshavebeenachievedtoexplorethefundamentalmaterialpropertiesofCNTsanditspossibleapplications[ 2 ].Recently,inaddition,grapheneandgraphenenanoribbon(GNR)havebeenstrongtopicsofresearchessincethegraphenemonolayerwasexperimentallyobtainedforthersttimein2004[ 3 ].CNTsandGNRsarestrongcandidatesforfuturenanoelectronicapplicationsduetotheirexcellentelectriccharacteristicsasfollows.Thegraphene-basednanostructureshaveextremelyhighcarriermobilities[ 3 4 ].Thebandstructuresaresymmetricanddirect,whichisusefulforoptoelectronicapplications.Theycanbemetallicorsemiconductingdependingonthestructures.Thereisnodanglingbondsonthesurface,andtheyareamenabletothedepositionofhigh-gateinsulator.SignicanteortshavebeendevotedtotheapplicationsofCNTsandGNRsforthelastdecade.ThersttypeofCNTandGNReld-eecttransistors(FETs)weredemonstratedin1998and2007,respectively[ 5 { 8 ].EventhoughagreatdealofeorthasbeenmadetounderstandthedevicephysicsofnanodevicesbasedonCNTsorGNRs,manyoftransistoroperationsarestillunclearundervariousconditions. Inthisstudy,devicesimulationsareperformedtounderstandthefundamentaldeviceoperationofCNTFETsorGNRFETsandtoelucidatethekeyfactorcontrollingthedeviceperformance.Ournumericalsimulationwillalsohelpexperimentalistssavetheirintensiveeortsinthefabricationbysharingtheunderstandingonnanodeviceoperations. 16

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Grapheneisanatomisticallythin2DmonolayerofcarbonatomsasshowninFig. 1-1 .Bandstructureofgraphenecanbecalculatedbyatight-binding(TB)modelinapzorbitalbasissetpercarbonatombyconsideringitsnearestneighborsonly[ 9 ].First,Hamiltonianmatrixcanbewrittenas wheret3eVistight-bindingparameterbetweencarbonatomsand~a1and~a2arebasisvectorsofgraphene.Then,thebandstructureofgraphenecanbeobtainedas whichcanbesimpliedbyTaylorexpansionneartheFermipointas wherevFc=300isFermivelocity.ThebandstructureofgrapheneneartheFermipointisshowninFig. 1-2 ,wherethereisnobandgapbetweentheconductionbandandthevalenceband.Puregrapheneitself,therefore,isnotsuitablefornanoelectronicapplications.Foragraphene-basedmaterial,quantumconnementcangiverisetoabandgap,whichwillbeshowninthefollowingtwocases:CNTandGNR. CNTandGNRarequasi-1Dnanostructuresderivedfrom2Dgraphene.CNTisconceptuallyarolled-upgraphene,andachiralvector~c=n~a1+m~a2describesitsuniquestructurealongthecircumferentialdirectionofthetube.TheperiodicboundaryconditionofCNTischaracterizedby~k~c=2q,whereqisaquantumnumber.Duetothe 17

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GNRisanarrowstripofgraphene,whichcanbeobtainedbystate-of-the-artpatterningtechniquelikee-beamlithography.ThestructureofGNRcanbedenedwithavector~c=n~a1 1-3 showstwospecialcasesofGNRs:(i)armchair-edgeGNR(A-GNR)and(ii)zigzag-edgeGNR(Z-GNR).Accordingtothesimpletight-bindingmethod,Z-GNRisalwaysmetallic,andA-GNRcanbemetallicorsemiconductingdependingonstructure(itismetallicifn+1isamultipleof3,andsemiconductingotherwise).IncomparisontoCNT,GNRhasinnite-wallboundarycondition.Table 1-1 isabriefsummaryofthecomparisonofCNTandGNR. BandstructuresofsemiconductingCNTandGNRarecomparedinFig. 1-4 ,whichshowsthesamebandgapforboth(13;0)CNTandN=12A-GNRunderthesimpletight-bindingmethod.However,CNTandGNRwiththesamebandgapshowroughlyafactorof2dierentdensity-of-states(DOS)ascomparedinFig. 1-5 ,whichiscausedbythebanddegeneracyofCNT.Eventhoughthesimpletight-bindingmethodpredictsthesamebandgapfortheabovesemiconductingCNTandGNR,thebandgapofNA-GNRhasadeviationfromEgof(n+1;0)CNTduetotheedgebondrelaxation[ 10 ].Furthermore,thebandgapofZ-GNRisalsoslightlyopenedandithasabandgapowingtothespin-polarizededgestates[ 10 ]. 1.3.1ReviewofNEGFFormalism 11 12 ].ThevalidityofNEGFformalismhasalreadybeenshownfromthepreviousstudiesinvariousareas[ 13 { 17 ].Inthissection,briefsummaryofNEGFmethodisgiven. 18

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whereHistheHamiltonianmatrixofthechannel,1and2areself-energiesofthesourceanddraincontacts,respectively. Thechargedensitycanbecomputedas wheresgn(E)isthesignfunction,EF1;F2isthesource(drain)Fermilevel,andD1;2(E;x)isthelocaldensity-of-statesduetothesource(drain)contact,whichiscomputedbytheNEGFmethod.Thechargeneutralitylevel,EN(x),isatthemiddleofbandgapbecausetheconductionbandandthevalencebandoftheCNTorGNRaresymmetric. Thesource-draincurrentiscalculatedas hZ1Trace1Gr2Gr+(f1f2)dE perspinpervalley,where1;2=i1;2+1;2arebroadeningfunctionsofthesource(drain)contacts,f1;2areequilibriumFermifunctionsofsource(drain)contacts[ 11 ]. 19

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Latticestructureofgraphene Figure1-2. Bandstructureofgraphene,whichhasalinearE-krelationneartheFermipoint. B TwospecialkindsofGNRs.A)N=10armchair-edgeGNR.B)N=7zigzag-edgeGNR. 20

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B BandstructureofsemiconductingCNTandGNR.A)(13,0)CNT.B)N=12armchair-edgeGNR. Figure1-5. Densityofstatesof(13;0)CNT(solidline)andN=12A-GNR(dashedline). Table1-1: Carbonnanotube(CNT)vs.graphenenanoribbon(GNR) CNTGNR DenitionRolled-upgraphenePatternedgrapheneShapeTubeRibbonVector~c=n~a1+m~a2~c=n~a1 21

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Theeectofuniaxialandtorsionalstrainontheperformanceofballisticcarbonnanotube(CNT)Schottkybarrier(SB)eld-eecttransistors(FETs)isexaminedbyself-consistentlysolvingthePoissonequationandtheSchrodingerequationusingthenon-equilibriumGreen'sfunction(NEGF)formalism.AmodespaceapproachcanbeusedtoreducethecomputationalcostofatomisticsimulationsforstrainedCNTsbyordersofmagnitude.Itisshownthatevenasmallamountofuniaxial(<2%)ortorsional(<5o)straincanresultinalargeeectontheperformanceoftheCNTFETsduetothevariationofthebandgapandband-structure-limitedvelocity.SemiconductingCNTchannelswithdierentchiralitiesareinuencedindrasticallydierentwaysbyacertainappliedstrain,whichisdeterminedbya(n-m)mod3rule.Ingeneral,atypeofstrainwhichproducesalargerbandgapresultsinincreasedSchottkybarrierheightanddecreasedband-structure-limitedvelocity,andhenceasmallerminimumleakagecurrent,smalleron-current,largermaximumachievableIon=Ioff,andlargerintrinsicdelay.TheothertypeofstrainthatreducesthebandgapresultsintheoppositeeectonthedeviceperformancemetricsofCNTFETs. 18 19 ].Strainalsoplaysanimportantroleintheelectricalpropertiesofcarbonnanotubes(CNTs),andhasbeenasubjectofstrongresearchinterest.Thepioneeringexperimentalworks[ 20 { 23 ]showedthattheconductanceofaCNTcanbevariedbyordersofmagnitudebyapplyingalocalstrainusingaSTMtip,andthephenomenonwassubsequentlyinvestigatedbytheoreticalcalculations[ 24 ].WiththerecentadvanceonCNTdevices[ 25 26 ],uniaxialortorsionalstraincanbeappliedthroughoutthewholeCNTchannelasapotentialapproachtoimprovethedeviceperformance.ACNTdeviceunderuniaxialstrainhasbeen 22

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27 ].Theresultscanbeimportantfortheapplicationssuchasexibleelectronics[ 28 ]andnanoscalepressuresensors.ACNTelectromechanicaldeviceundertorsionalstrainhasbeenexaminedbypressingapedalthatisattachedtoasuspendedCNTchannel[ 29 ].InadditiontothecontrolledwaysofappliedstrainstotheCNTchannel,straincanalsobeunintentionallyinducedinthedevicefabricationprocess.PrevioustheoreticalstudieshavebeenfocusingonlocaldeformationofCNTsortheeectsofstrainatthemateriallevel[ 9 24 30 { 32 ],andthestraineectsonthedeviceperformanceareunclear.Withthesignicantadvanceonexperimentaltechniquesforapplyinguniaxialandtorsionalstrains,itisimportanttoexplorehowstrainaectstheperformanceofCNTFETsandwhetheritispossibletousestrainforimprovingtheperformanceofCNTFETs. Inthiswork,theeectsofuniaxialandtorsionalstrainsonthecharacteristicsofballisticCNTSchottkybarrier(SB)FETsareexaminedusingthenon-equilibriumGreen'sfunction(NEGF)formalism.AmodespaceapproachpreviouslydevelopedforunstrainedCNTs[ 16 ]canbeextendedtoaCNTchannelunderuniformuniaxialortorsionalstrain,whichreducesthecomputationalcostoftheatomisticquantumsimulationbyordersofmagnitude.Weshowthatevenasmalluniaxial(<2%)ortorsional(<5o)straincanresultinalargeeectonthedeviceperformancemetrics,suchastheon-current,theminimumleakagecurrent,andtheintrinsicdelay,duetothevariationofthebandgapandband-structure-limitedvelocity.SemiconductingCNTchannelswithdierentCNTchiralitiesrespondindrasticallydierentwaystothesametypeofstrain.Forexample,thetensilestrainona(17;0)CNTcausesalargeron-currentandasmallerintrinsicdelay,butthesamestrainona(16;0)CNTresultsintheoppositeeect.TheresultsindicatetheimportanteectsofuniaxialandtorsionalstrainonthecharacteristicsofCNTFETsandthepossibilitytoimprovetheperformanceofCNTFETsbyapplyingacarefullydesignedtypeofstrain.Atthesametime,carefultrade-obetweendierentdeviceperformancemetricsmustbetakencareof. 23

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2-1A and 2-1B showuniaxialtensileandcompressivestrain,respectively,andtheuniaxialstrainisdescribedbythepercentageofthelengthchange.Figure 2-1C showstorsionalstrainandrepresentsthetorsionalstrainbydegree.Withintherangeofstrainweconsidered(<2%),CNTsshowalinearresponsetotheuniaxialstrain[ 33 ],andhencestrainenergypercarbonatomisassumedtobeequalforallatoms.Severetorsionmayoccuratteningontubes[ 34 ],which,however,isneglectedinthisstudybecauseourrangeoftorsionalstrain(<5o)ismuchsmallerthanthepredictedcriticalangle(70oand60ofor40nm-long(16;0)and(17;0)CNT,respectively)thatchangestheatomisticconguration.Apowersupplyvoltageof0.4Visused.Themetalsource/drainisdirectlyattachedtotheCNTchannel,andtheSchottkybarrierheightbetweenthesource/drainandthechannelisBn=Eg=2,whereEgisthebandgapofchannelCNT.ThemetalcontactFermilevelliesinthemiddleoftheCNTbandgap.Thegateleakagecurrentisneglectedforsimplicity. TheDCcharacteristicsofballisticCNTFETsaresimulatedbysolvingtheSchrodingerequationusingthenon-equilibriumGreen'sfunction(NEGF)formalismself-consistentlywiththePoissonequation.Atightbinding(TB)Hamiltonianwithapzorbitalbasissetisusedtodescribeanatomisticphysicalobservationofthechannel.Theatomistictreatmentiscomputationallyexpensiveinrealspace,butsignicantsavingofcomputationalcostcanbeachievedbythemodespaceapproach[ 16 ]. Figure 2-2A showsapartof(n;0)CNTlatticeinrealspace.t1,t2,andt3arebindingparametersbetweenthenearestneighborcarbonatoms;ti=t0(r0=ri)2[ 35 ],wheresubscriptioni=1,2,and3,t0isbindingparameterbetweencarbonatomsofunstrainedCNT,r0isbondinglengthbetweencarbonatomsofunstrainedCNT,andriisbonding 24

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whererandrareoriginalradiusanditsvariation,landlareoriginallengthanditsvariationofstrainedCNT,"cisthecircumferentialstrainand"tisthetransversestrain[ 36 ].=20isused.Thetight-bindingapproximationisusedtodescribetheinteractionbetweencarbonatomsandonlythenearestneighborcouplingisconsidered.ForsemiconductingCNTswithadiameterlargerthan1nm,itisknownthatorbitalcalculationresultsagreewithfourorbitalresultswithinasmallrangeofstrain[ 9 ],andfourorbitalTBresultsagreewithabinitioandexperimentalresults[ 37 ].ACNTcanbeconsideredasa1-Dstructurecomposedofsuccessivetwotypesofringsalongthelengthdirectionwithncarbonatomsineachring,andHamiltonianmatrixfortheCNTchannelisconstructedas 25

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arecouplingmatricesbetweenadjacentrings,and+2isconjugatetransposeof2.ThesizeoftheHamiltonianmatrixissquareofthetotalnumberofcarbonatomsintheCNTchannel,whichresultsinanexpensivecomputationalcost.ByusingtheplanewaveswiththewavevectorssatisfyingtheperiodicboundaryconditionasthenewbasissetinthecircumferentialdirectionoftheCNT,thebasistransformationonthe(n;0)nanotubedecouplesaproblemintonuncoupled1-DmodespacelatticeasshowninFig. 2-2B .Foreachmode,thetwotypesofcouplingbetweentheadjacentmodespacelatticepointsbecomet1and n; whereq=1;2;...;n.Becausethephasefactorhasnoeectonchargedensityandcurrent,itcanbeignoredandb2q=t2+t3ei2q nisusedinstead.Generally,onlyafewmodesarerelevanttothecarriertransport,whichreducesthecomputationalcostfurther. TheretardedGreen'sfunctionofstrainedCNTchannelisgivenas whereHisHamiltonianmatrixofthestrainedCNT,1and2areself-energiesofthemetalsourceanddraincontacts,respectively.Theelectroncorrelationfunctionover 26

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wheref1;2areequilibriumFermifunctions,and1;2=i1;2+1;2arebroadeningfunctionsofsourceanddraincontacts,respectively.NEGFtransportequationsaresolvediterativelywiththePoissonequationforaself-consistentsolution.Thesource-draincurrentiscalculatedas hZ1Trace1Gr2Gr+(f1f2)dE; wherethefactorof4comesfromspindegeneracyandvalleydegeneracyof2[ 11 ]. WecomputetheintrinsicdelayofthedevicetoseethestraineectonthespeedofCNTFETs.Intrinsicdelayisanimportantperformancemetricfordigitalapplicationsandcanbecalculatedas=(QonQoff)/Ion,whereIonissource-draincurrentaton-state,QonandQoffarethetotalchargeinthechannelaton-stateando-state,respectively.ThetotalchargeinthechannelcanbeobtainedbyQch=qRLch0Ne(x)dx,whereqiselectronchargemagnitude,Ne(x)isthenumberofelectronsasafunctionofthechannelposition,andLchisthechannellength. 2.3.1TransferCharacteristics 2-3 plotsIDvs.VGcharacteristicswithdierentuniaxialstrainon(a)a(16;0)CNTand(b)a(17;0)CNTchannel,andwithdierenttorsionalstrainon(c)a(16;0)CNTand(d)a(17;0)CNTchannel.TheeectofeitheruniaxialortorsionalstrainonthecurrentofCNTFETsissevereevenwithsmallrangeofstrain.Byapplying2%uniaxialstrainor5ooftorsionalstrainontheCNTchannel,theID-VGcharacteristicschangesignicantlyevenonalogscaleplot. ThechangeofID-VGcharacteristicsofCNTFETsunderuniaxialstraincanbeunderstoodbytherelationshipbetweenbandgapandstrain.Figure 2-4A showsbandgap 27

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9 ].Thebandgapof(17;0)CNT(thedashedlineinthegure)with2%tensilestrainisabout2/3ofunstrainedone,whichresultsinsmallerSchottkybarrierheightofthedeviceandlargerdraincurrent.Ontheotherhand,thebandgapof(16;0)CNT(thesolidlineinthegure)with2%tensilestrainisabout30%largerthanthatofunstrained(16;0)CNT,whichresultsinlargerSchottkybarrierheightandreducedcurrent.Becausethedevicecharacteristicsstronglydependonthebandgapofthechannelmaterial,theIDvs.VGcharacteristicschangesignicantlyinthepresenceofasmallstrain.TheeectoftorsionalstrainonthecurrentofCNTFETscanbedescribedasthesamewayasthatofuniaxialstrain.Figure 2-4B showsbandgapvs.torsionalstrain.Thesolidlineisfora(16;0)CNTandthedashedlineisfora(17;0)CNT.A5ooftorsionalstrainona(16;0)CNTincreasesthebandgapoftheCNTby17%,whichresultsinlargerSchottkybarrierheightandsmallercurrent,whilethebandgapofthe(17;0)CNTwith5otorsionalstrainis4/5oftheunstrainedone,whichresultsinsmallerSchottkybarrierheightandlargercurrent. ThemaindierencebetweentheeectofuniaxialandtorsionalstrainonCNTisthefreedomofbandgapchange.Forexample,uniaxialstrainonaCNTmayeitherincreaseordecreasethebandgapbyapplyingeithertensileorcompressivestrain.Torsionalstrain,however,onlyincreasesthebandgapofthe(16;0)CNT,whereasitonlydecreasesthebandgapofthe(17;0)CNT,regardlessofwhetherthetorsionalangleispositiveornegative.Asaresult,applyinguniaxialstraincaneitherincreaseordecreasethecurrent,butapplyingtorsionalstraincanonlyvarythecurrentinoneway,asshowninFig. 2-3 .Ingeneral,thebehaviorofbandgapchangebystrainsisdeterminedbythe(n-m)mod3rule[ 9 ];n=3q+1(qisaninteger)groupCNTsbehaveasa(16;0)CNTandn=3q+2groupCNTsarelikea(17;0)CNT[ 32 ]. 28

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2-5 showsIminvs.(a)uniaxialstrainand(b)torsionalstrain.Iministheminimumsource-draincurrentdeliveredbyastrainedCNTFET.Thesolidlinesarefor(16;0)CNTFETsandthedashedlinesarefor(17;0)CNTFETs.IminisachievedatVG=VD/2sincethesimulatedCNTFETisanambipolardevicewithBn=Eg=2andazeroatbandvoltageisassumed.ThesubsetofFig. 2-5A showsthebandgapprolealongthechannelpositionofunstrained(solidline)and2%strained(dashedline)(16;0)CNTFET.Iminof2%strained(16;0)CNTFETis1/75oftheunstrainedCNTFEToneduetotheincreaseofthebandgap,andIminof(16;0)CNTFETwith2%compressivestrainisroughly90timeslargerthanthatoftheunstrainedCNTFETduetothedecreaseofthebandgap. WenextperformasimpleestimationforthechangeofIminaftertheapplicationofstrain.ThechangeofImincanbeestimatedasafactorofexpEg 38 ], h