半導(dǎo)體材料與技術(shù)課件：chapter2-5（第一章）

Chapter2

ElectricalandThermalConductioninSolid2.1Classicaltheory:TheDrudemodel(德魯特模型)2.2Temperaturedependenceofresistivity:idealpuremetals(電阻對時間的依賴性：理想純金屬)2.3Matthiessen’sandNordheim’srules(馬西森和諾德海姆定則)2.4Resistivityofmixturesandporousmaterials(混合物和孔洞材料的電阻率)2.5TheHalleffectandHalldevices(霍爾效應(yīng)和霍爾器件)2.6Thinmetalfilms(金屬薄膜)2.7Thermalconduction(熱傳導(dǎo))2.8Electricalconductivityofnonmetals(非金屬的電導(dǎo))

FromPrinciplesofelectronicMaterialsDevices,SOKasap(McGraw-Hill,2005)Content?Electricalconductioninvolvesthemotionofchargesinamaterialundertheinfluenceofanappliedfield.?Amaterialcangenerallybeclassifiedasaconductorifitcontainsalargenumberof‘free’ormobilechargecarriers.?Inmetals,thevalenceelectronsthatarefreetomovewithinthemetalarecalledasconductionelectrons.?Objectivesofelectricalconduction:conductionelectrons;accelerationoffreechargecarriers;driftvelocity(漂移速度);electroncollisions(碰撞)

withlatticevibrations(晶格振動),crystaldefects,impurities(雜質(zhì))etc.?Thermalconductioninsolid2.1Classicaltheory:theDrudemodelTheelectriccurrentdensityJisdefinedas:Driftvelocityinthexdirection(averageoverNelectrons):漂移速度Driftofelectronsinaconductorinthepresenceofanappliedelectricfield.2.1Classicaltheory:theDrudemodelThenumberofelectronsperunitvolumen:Electronsdriftwithanaveragevelocityvdxinthexdirection.(Existheelectricfield.)(a)Aconductionelectronintheelectrongasmovesaboutrandomlyinametal(withameanspeedu)beingfrequentlyandrandomlyscatteredbythermalvibrationsoftheatoms.Intheabsenceofanappliedfieldthereisnonetdriftinanydirection.(b)Inthepresenceofanappliedfield,Ex,thereisanetdriftalongthex-direction.Thisnetdriftalongtheforceofthefieldissuperimposed(疊加)ontherandommotionoftheelectron.Aftermanyscatteringeventstheelectronhasbeendisplacedbyanetdistance,Δx,fromitsinitialpositiontowardthepositiveterminalvxi:thevelocityinthexdirectionoftheelectroniuxi:thevelocityaftercollision(initialvelocity)Ex;appliedfieldinthexdirectionme:themassofanelectronti:thelastcollisiontime(relaxationtime(弛豫時間))Velocitygainedinthex-directionattimetfromtheelectricfield(Ex)forthreeelectrons.TherewillbeNelectronstoconsiderinthemetal.Driftvelocityvdx(averagevelocityforallsuchelectronsalongx):Supposethatτisthemeanfreetime(ormeantimebetweencollisions):Driftmobility(漂移遷移率)μd:whereOhm’slaw:I=V/RwhereσisconductivitySummationoperator(求和符號)Example(SupposeeachCuatomdonatesoneelectron.)Example(SupposeeachCuatomdonatesoneelectron.)Example(driftvelocityandmeanspeed):Whatistheappliedelectricfieldthatwillimposeadriftvelocityequalto0.1percentofthemeanspeedu(~106m/s)ofconductionelectronsincopper?WhatisthecorrespondingcurrentdensitythroughaCuwireofadiameterof1mm?Electricfield:Currentdensity:Acurrentthrougha1mm-diametercopperwire:Whenanelectricfieldisappliedtoaconductor,forallpracticalpurposes,themeanspeedisunaffected.2.2Temperaturedependenceofresistivity:idealpuremetalsSincethescatteringcrosssectionalareaisS,inthevolumeSltheremustbeatleastonescatterer,Ns(Suτ)=1.NS:thenumberofscatteringcentersperunitvolume.meanfreepathWhereuisthemeanspeedScatteringofanelectronfromthethermalvibrationsoftheatoms.Theelectrontravelsameandistancel=uτbetweencollisions.Themeanfreetimeisgivenas:Anatomcoversacross-sectionalareaπa2withthevibrationamplitudea.Theaveragekineticenergyoftheoscillationsisgivenas:Whereωistheoscillationfrequency.C:constantA:temperatureindependentconstant???Example(temperaturedependenceofresistivitiy):whatisthepercentagechangeintheresistanceofapuremetalwirefromSaskatchewan’ssummer(20°C)towinter(-30°C),neglectingthechangesinthedimensionsofthewire?Example(driftmobilityandresistivityduetolatticevibrations):Giventhatthemeanspeedofconductionelectronsincopperis1.5x106m/sandthefrequencyofvibrationofthecopperatomsatroomtemperatureisabout4x1012S-1,estimatethedriftmobilityofelectronsandtheconductivityofcopper.Thedensityofcopperis8.96g/cm3andtheatomicmassMatis62.56g/mol.?Example(driftmobilityandresistivityduetolatticevibrations):Giventhatthemeanspeedofconductionelectronsincopperis1.5x106m/sandthefrequencyofvibrationofthecopperatomsatroomtemperatureisabout4x1012S-1,estimatethedriftmobilityofelectronsandtheconductivityofcopper.Thedensityofcopperis8.96g/cm3andtheatomicmassMatis62.56g/mol.2.3Matthiessen’sandNordheim’srules2.3.1Matthiessen’sruleandthetemperaturecoefficientofresistivity(α)Thetheoryofconductionthatconsidersscatteringfromlatticevibrationsonlyworkswellwithpuremetals.Inametalalloy(合金),anelectroncanbescatteredbytheimpurityatomsduetounexpectedchangeinthepotentialenergyPEbecauseofalocaldistortion(畸變).StrainedregionbyimpurityexertsascatteringforceF=-d(PE)/dxIfweassumethetwoscatteringmechanismsareindependent.Wenoweffectivelyhavetwotypesofmeanfreetimes:τTfromthermalvibrationonlyandτIfromcollisionswithimpurities.Thenetprobabilityofscattering1/τisgivenas:Twodifferenttypesofscatteringprocessesinvolvingscatteringfromimpuritiesaloneandthermalvibrationsalone.Thedriftmobility:Theeffective(oroverall)resistivityρ(Matthiessen’srule):Consideringotherscatteringeffects(dislocations(位錯),grainboundaries(晶界)

andothercrystaldefects(缺陷)),theeffectiveresistivityofametalmaybewrittenas:WhereρRistheresidualresistivity.Theresidualresistivityshowsverylittletemperaturedependence.WhereAandBaretemperatureindependentconstants.Thetemperaturecoefficientα0isdefinedas:Whereρ0istheresistivityatthereferencetemperatureT0,usually273K(or293K),andδρ=ρ-ρ0,isthechangeintheresistivityduetoasmallincreaseintemperatureδT=T-T0.Whenα0isconstantoveratemperaturerangeT0toT:Example:temperaturecoefficientIf??IfFrequently,theresistivityversustemperaturebehaviorofpuremetalscanbeempiricallyrepresentedbyapowerlaw:n:thecharacteristicindexρ=AT+Bisoversimplified.Asthetemperaturedecreases,typicallybelow~100Kformanymetals,theresistivitybecomesρ=DT5+ρR,whereDisaconstant.Tinmeltsat505Kwhereasnickelandirongothroughamagnetictonon-magnetic(Curie)transformationsatabout627Kand1043Krespectively.Thetheoreticalbehavior(ρ~T)isshownforreference.[FromMetalsHandbook]Theresistivityofvariousmetalsasafunctionoftemperatureabove0°C.Aboveabout100K,ρ∝TAtlowtemperatures,ρ∝T5AtthelowesttemperaturesρapproachestheresidualresistivityρR.Theinsetshowstheρvs.Tbehaviourbelow100Konalinearplot(ρRistoosmallonthisscale).TheresistivityofCufromlowesttohighesttemperatures(nearmeltingtemperature,1358K)onalog-logplot.Typicaltemperaturedependenceoftheresistivityofannealedandcoldworked(deformed)coppercontainingvariousamountofNiinatomicpercentage(dataadaptedfromJ.O.Linde,Ann.Pkysik,5,219(1932)).Example(Matthiessen’srule–Cualloys)2.3.2SolidsolutionsandNordheim’sruleThetemperature-independentimpuritycontributionρI

increaseswiththeconcentrationofsoluteatoms.Thismeansthatasthealloyconcentrationincreases,theresistivityρincreasesandbecomeslesstemperaturedependentasρIoverwhelmsρT,leadingtoα<<1/273.Forexample:Nichrome(80%ofNiand20%ofCr)hasaresistivity,thatincreasesalmost16timescomparedtothatofpureNi.Thealloy(Nichrome)hasaverylowvalueofα.Example(Cu-Nisystem)(a)PhasediagramoftheCu-Nialloysystem.Abovetheliquiduslineonlytheliquidphaseexists.IntheL+Sregion,theliquid(L)andsolid(S)phasescoexistwhereasbelowthesolidusline,onlythesolidphase(asolidsolution)exists.(b)TheresistivityoftheCu-NialloyasafunctionofNicontent(at.%)atroomtemperature.[fromMetalsHandbook-10thEditionandConstitutionofBinaryAlloys-Anisomorphousbinaryalloysystem(onephasefcc).-Solidsolutionphaseexistsinthewholecompositionrange.-Themaximumofρisataround50%ofNi.AnimportantsemiempiricalequationthatcanbeusedtopredicttheresistivityofanalloyisNordheim’srulewhichrelatestheimpurityresistivitypItotheatomicfractionXofsoluteatomsinasolidsolution,asfollows:WhereCistheconstanttermedtheNordheimcoefficient.Fordilutesolutions,Nordheim’srulepredictsthelinearbehavior,thatis,ρI=CXforX<<1.thelinearbehavioragreeswellwithexperimentalobservations.TheresistivityofanalloyofcompositionXis:Whereρmatrix=ρT+ρIistheresistivityofthematrixduetoscatteringfromthermalvibrationandfromotherdefects,intheabsenceofalloyingelements.-Cu3AuandCuAuareintermediatealloyingphases.-Nointermediatephasesformedafterquenching.-Nordheim’sruleonlyappliestosolidsolutionthataresingle-phasesolids.EpositionatroomtemperatureinCu-Aualloys.Thequenchedsample(dashedcurve)isobtainedbyquenchingtheliquidandhastheCuandAuatomsrandomlymixed.TheresistivityobeystheNordheimrule.Ontheotherhand,whenthequenchedsampleisannealedortheliquidslowlycooled(solidcurve),certaincompositions(Cu3AuandCuAu)resultinanorderedcrystallinestructureinwhichCuandAuatomsarepositionedinanorderedfashioninthecrystalandthescatteringeffectisreduced.Example(resistivityofCu-Aualloys)Example(Nordheim’srule):Predicttheresistivityofthealloy90wt%Au–10wt%Cu.IfwistheweightfractionofCu(w=0.1),andifMAuandMCuaretheatomicmassesofAuandCu,theatomicfractionXofCuisgivenby:GiventhatρAu=22.8nΩmandC=450nΩm,Note:Thisvalueisonly0.5%differentfromtheexperimentalvalue.Example(resistivityduetoimpurities):Themeanspeedofconductionelectronsincopperisabout1.5x106m/s.Itsroomtemperatureresistivityis17nΩm,andtheatomicconcentrationNatinthecrystalis8.5x1022cm-3.Supposethatweadd1at%Autoformasolidsolution.Whatistheresistivityofthealloy,theeffectivemeanfreepath,andthemeanfreepathduetocollisionwithAuatomsonly?TheNordheimcoefficientCofAuinCuis5500nΩm.WithC=0.01,theoverallresistivityis:Sincetheeffectivemeanfreepathl=uτandtheeffectivedriftmobilityμd=eτ/me,theexpressionfortheconductivitybecomes:?

?Theeffectivemeanfreepath,l=8.8nm.Repeatthecalculationusingρmatrix=17x10-9Ωm:?Themeanfreepath,lCu=37nm.FromMatthiessen’srule:2.4Resistivityofmixturesandporousmaterials2.4.1HeterogeneousmixtureTheeffectiveresistivityofamaterialhavingalayeredstructure.(a)Alongadirectionperpendicular(垂直)

tothelayers.(b)Alongadirectionparallel(水平)totheplaneofthelayers.(c)Materialwithadispersedphase(分散的相)inacontinuousmatrix.-Series(串聯(lián))-Parallel(并聯(lián))-Dispersed(分散的)Twodistinctphasesα,βTwodistinctphasesα,βinSeries(串聯(lián))Seriesruleofmixture–Resistivity-mixtureruleTheeffectiveresistanceReff:WhereLα(orLβ)isthetotallength(thickness)oftheα-phase(orβ-phase)layers,andthetotallengthL=Lα+Lβ.χαandχβarethevolumefractions.Fromχα=Lα/Landχβ=Lβ/L:Parallelruleofmixture–conductivity-mixtureruleTwodistinctphasesα,βinParallel(并聯(lián))

????Althoughtheretworulesrefertospecialcases,forarandommixtureofphaseαandβ,wewouldnotexpecteitherequationtoapplyrigorously.Whentheresistivitiesoftworandomlymixedphasesarenotmarkedlydifferent,theseriesmixturerulecanbeappliedatleastapproximately.Dispersedspheresρdandσdwithavolumefractionχinacontinuousmatrixwithρcandσc(ReynoldsandHough):Whereσistheoverallconductivity.Spheresarerandomlydispersed.Iftheresistivityofonephaseisappreciably(相當(dāng)大的)differentthantheother,let’sseetwosemiempricalrules.Therearetwospecialcasesunderdispersedspheresinacontinuousmatrix.Case1:ifρd>10ρc:Whereχdisthevolumefractionofthedispersedphased.Case2:ifρd<ρc/10:Example(theresistivity-mixturerule):consideratwo-phasealloyconsistingofphaseαandphaseβrandomlymixed.Thesolidconsistsofarandommixtureoftwotypesofresistivitiesραofαandρβofβ.WecandividethesolidintoabundleofNparallelfibersoflengthLandcross-sectionalareaA/N.Inthisfiber(infinitesimallythin),theαandβphasesareinseries,soifχα=Vα/Visthevolumefractionofphaseα,andχβisthatofβ,thenthetotallengthofallαregionspresentinthefiberisχαL.(a)Atwophasesolid.(b)Athinfibercutoutfromthesolid.cThetworesistancesareinseries,sothefiberresistanceis:TheresistanceofthesolidismadeupofNfibersinparallel:Theeffectiveresistivity:Theseriesrulefailswhentheresistivitiesofthetwophasesarevastlydifferent.Example(acomponentwithdispersedairpores):whatistheeffectiveresistivityof95/5(95%Cu-5%Sn)bronze(青銅),whichismadefrompowderedmetalcontainingporesat15%(volumepercent,vol%)?Theresistivityof95/5bronzeis1x10-7Ωm.Poresareinfinitelyresistive(ρd=∞)thanthebronzematrix,sothatρd>>10ρc:Example(combinedNordheimandmixturerules):Brass(黃銅)isanalloycomposedofCuandZn.Considerabrasscomponentmadefromsintering90at%Cuand10at%Znbrasspowder.Thecomponentcontainsdispersedairporesat15vol%.TheNordheimcoefficientCofZninCuis300nΩm.Predicttheeffectiveresistivityofthisbrasscomponent,iftheresistivityofpureCuis16nΩmatroomtemperature.Theresistivityofthebrassalloy:Theeffectiveresistivityofthecomponent:2.4.2Two-phasealloy(Ag-Ni)resistivityandelectricalcontacts-Nordheim’srulecanbeusedinthecompositionranges–0-X1

andX2-100%B.-MixturerulebetweenX1andX2.(a)Thephasediagramforabinary,eutecticformingalloy.(b)Theresistivityvscompositionforthebinaryalloy.Whenweapplyamagneticfieldinaperpendiculardirectiontoanappliedelectricfield(whichisdrivingtheelectriccurrent),wefindthereisatransverseelectricfieldinthesamplethatisperpendiculartothedirectionofboththeappliedelectricfieldExandthemagneticfieldBzbecauseofLorentzforce：2.5TheHalleffectandHalldevicesIllustrationoftheHalleffect.Thez-directionisoutfromtheplaneofpaper.Theexternallyappliedmagneticfieldisalongthez-direction.AmovingchargeexperiencesaLorentzforceinamagneticfield.(a)Apositivechargemovinginthexdirectionexperiencesaforcedownwards.(b)Anegativechargemovinginthe-xdirectionalsoexperiencesaforcedownwards.Lorentzforce:WhereqisthechargeTheaccumulationofelectronsnearthebottomresultsinaninternalelectricfieldEH(Hallfield).ElectronaccumulationcontinuesuntiltheincreaseinEHissufficienttostopthefurtheraccumulationofelectrons.Whenthishappened,themagnetic-fieldforceevdBzthatpushestheelectronsdownjustbalancetheforceeEHthatpreventsfurtheraccumulation.Inthesteadystate:FromJx=envdx:HallcoefficientRH:Formetals:Note:Fromσ=enμd?μd

=σ/(en)?HallmobilityμH

=|σRH|Example(Hall-effectWattmeter)WattmeterbasedontheHalleffect.LoadvoltageandloadcurrenthaveLassubscript.CdenotesthecurrentcoilsforsettingupamagneticfieldthroughtheHalleffectsample(semiconductor)VH=wEH=wRHJxBz∝IxBz∝VLILWisthethickness.Example(Hallmobility):TheHallcoefficientandconductivityofcopperat300Khavebeenmeasuredtobe-0.55x10-10m3A-1s-1and5.9x107

Ω-1m-1,respectively.Calculatethedriftmobilityofelectronsincopper.FromμH=|σRH|Example(conductionelectronconcentrationincopper)Sincetheconcentrationofcopperatomsis8.5x1028m-3,theaveragenumberofelectronscontributedperatomis(1.15x1029)/(8.5x1028)=Thinmetalfilms(a)GrainboundariescausescatteringoftheelectronandthereforeaddtotheresistivitybyMatthiessen'srule.(b)Foraverygrainysolid,theelectronisscatteredfromgrainboundarytograinboundaryandthemeanfreepathisapproximatelyequaltothemeangraindiameter.PolycrystallinefilmsandgrainboundaryscatteringThemeanfreepathl:λ:meanfreepathinthesinglecrystald:grainsize.Fromρcrystal∝1/λandρ∝1/l:Mayadas-Shatkezformula:WhereRisaparameter,whichisbetween0.24to0.40forcopperForexample:thepredictedρ/ρcrystal

≈1.20foraCufilm,ifR=0.3andd≈3λ=120nm(sincethebulkcrystalλ≈40nm).SurfacescatteringConductioninthinfilmsmaybecontrolledbyscatteringfromthesurfaces.DisthefilmthicknessFromamorerigorouscalculation(Fuchs-Sondheimerequation):Thevalueofpisdependentonthepreparationconductionandmicrostructure.p=0.9-1formostepitaxialthinfilms,unlessverythin(D<<λ).a)ρfilmoftheCupolycrystallinefilmsvs.reciprocalmeangrainsize(diameter),1/d.FilmthicknessD=250nm-900nmdoesnotaffecttheresistivity.Thestraightlineisρfilm=17.8nm+(595nΩmnm)(1/d),(b)ρfilmoftheCuthinpolycrystallinefilmsvs.filmthicknessD.Inthiscase,annealing(heattreating)thefilmstoreducethepolycrystallinitydoesnotsignificantlyaffecttheresistivitybecauseρ

filmiscontrolledmainlybysurfacescattering.From(a)Microelec.Engin.and(b)Appl.Surf.Sci.2.7ThermalconductionMetalsarebothgoodelectricalandgoodthermalconductors.Freeconductionelectronsinametalplayanimportantroleinheatconduction.Whenametalpieceisheatedatoneend,theamplitudeoftheatomicvibrationandthustheaveragekineticenergyoftheelectronsintheregionincreases.Electronsgainenergyfromenergeticatomicvibrationswhenthetwocollide.Byvirtueoftheirincreasedrandommotion,theseenergeticelectronsthentransfertheextraenergytothecolderregionsbycollidingwiththeatomicvibrationsthere.Thus,electronsactas“energycarriers”Note:Innonmetals,thethermalconductionisduetolatticevibrations.Thermalconductioninametalinvolvestransferringenergyfromthehotregiontothecoldregionbyconductionelectrons.Moreenergeticelectrons(shownwithlongervelocityvectors)fromthehotterregionsarriveatcoolerregionsandcollidetherewithlatticevibrations

andtransfertheirenergy.Lengthsofarrowedlinesonatomsrepresentthemagnitudesofatomicvibrations.Thethermalconductivitymeasurestheabilityofheattransportationthroughthemedium.δT/δx:thetemperaturegradientA:thecross-sectionalareaThesign“-”:indicatestheheatformhotendtocoldend.(Fourier’slaw)κ:thermalconductivity(Fourier’slaw)Heatflowinametalrodheatedatoneend.Considertherateofheatflow,dQ/dt,acrossathinsectionδxoftherod.TherateofheatflowisproportionaltothetemperaturegradientδT/δxandthecrosssectionalareaA.Inmetals,electronsparticipateintheprocessofchargeandheattransport,whicharecharacterizedbyσ(electricalconductivity)andk,respectively.Therefore,itisnosurprisingtofindthatthetwocoefficientsarerelatedbytheWiedemann-Franz-Lorenzlaw.Wiedemann-Franz-Lorenzlaw:WhereCWFL=π2k2/3e2=2.44x10-8WΩK-2istheLorenznumber(ortheWiedemann-Franz-Lorenzcoefficent).ExperimentsshowthattheWiedemann-Franz-Lorenzlawisreasonablyobeyedatclosetoroomtemperatureandabove.Thermalconductivity,κvs.electricalconductivityσforvariousmetals(elementsandalloys)at20°C.ThesolidlinerepresentstheWFLlawwithCWFL≈2.44×108WΩK-2.Thermalconductivityvs.temperaturefortwopuremetals(CuandAl)andtwoalloys(brassandAl-14%Mg).DataextractedfromThermophysicalPropertiesofMatter?Nonmetalsdonothavefreeelectrons.Theenergytransferinvolveslatticevibration.Thespringscouplethevibrationstoneighbouringatomsandthusallowthelargeamplitudevibrationstopropagate,asavibrationalwave,tothecoolerregionsofthecrystal.?Thestrongerthecoupling,thegreaterwillbethethermalconductivity,forexampleκ≈1000W/mKindiamond.?Thecoefficientofheattransferdependsnotonlyontheefficiencyofcoupling,andhenceonthenatureofinteratomicbonding,butalsoonthepropagationinthecrystal.Conductionofheatininsulatorsinvolvesthegenerationandpropagationofatomicvibrationsthroughthebondsthatcoupletheatoms.(Anintuitivefigure.)Example(thermalconductivity):A95%Cu-5%Snbronzebearingmadeofpowderedmetalcontains15vol%porosity.Calculateitsthermalconductivi