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ELECTRONICMATERIALSSCIENCE不得转载谢谢合作LWM 不得转载谢谢合作LWM ELECTRONICMATERIALSSCIENCEEugeneA.IreneUniversityofNorthCarolinaChapelHill,NorthCarolinaAJohnWiley&Sons,Inc.,Publication不得转载谢谢合作LWM Copyright©2005byJohnWiley&Sons,Inc.Allrightsreserved.PublishedbyJohnWiley&Sons,Inc.,Hoboken,NewJersey.PublishedsimultaneouslyinCanada.Nopartofthispublicationmaybereproduced,storedinaretrievalsystem,ortransmittedinanyformorbyanymeans,electronic,mechanical,photocopying,recording,scanning,orotherwise,exceptaspermittedunderSection107or108ofthe1976UnitedStatesCopyrightAct,withouteitherthepriorwrittenpermissionofthePublisher,orauthorizationthroughpaymentoftheappropriateper-copyfeetotheCopyrightClearanceCenter,Inc.,222RosewoodDrive,Danvers,MA01923,978-750-8400,fax978-646-8600,oronthewebatwww.copyright.com.RequeststothePublisherforpermissionshouldbeaddressedtothePermissionsDepartment,JohnWiley&Sons,Inc.,111RiverStreet,Hoboken,NJ07030,(201)748-6011,fax(201)748-6008.LimitofLiability/DisclaimerofWarranty:Whilethepublisherandauthorhaveusedtheirbesteffortsinpreparingthisbook,theymakenorepresentationsorwarrantieswithrespecttotheaccuracyorcompletenessofthecontentsofthisbookandspecificallydisclaimanyimpliedwar-rantiesofmerchantabilityorfitnessforaparticularpurpose.Nowarrantymaybecreatedorextendedbysalesrepresentativesorwrittensalesmaterials.Theadviceandstrategiescontainedhereinmaynotbesuitableforyoursituation.Youshouldconsultwithaprofessionalwhereappropriate.Neitherthepublishernorauthorshallbeliableforanylossofprofitoranyothercommercialdamages,includingbutnotlimitedtospecial,incidental,consequential,orotherdamages.ForgeneralinformationonourotherproductsandservicespleasecontactourCustomerCareDepartmentwithintheU.S.at877-762-2974,outsidetheU.S.at317-572-3993orfax317-572-4002.Wileyalsopublishesitsbooksinavarietyofelectronicformats.Somecontentthatappearsinprint,however,maynotbeavailableinelectronicformat.LibraryofCongressCataloging-in-PublicationData:Irene,EugeneA.Electronicmaterialsscience/EugeneA.Irene.p.cm.Includesbibliographicalreferencesandindex.ISBN0-471-69597-1(cloth)1.Electronics—Materials.2.Electronicapparatusandappliances—Materials.I.Title.TK7871.I742005621.381—dc222004016686PrintedintheUnitedStatesofAmerica.10987654321不得转载谢谢合作LWM CONTENTSPrefacexi1IntroductiontoElectronicMaterialsScience11.1Introduction/11.2StructureandDiffraction/31.3Defects/41.4Diffusion/51.5PhaseEquilibria/51.6MechanicalProperties/61.7ElectronicStructure/61.8ElectronicPropertiesandDevices/71.9ElectronicMaterialsScience/82StructureofSolids92.1Introduction/92.2Order/102.3TheLattice/122.4CrystalStructure/162.5Notation/172.5.1NamingPlanes/172.5.2LatticeDirections/192.6LatticeGeometry/212.6.1PlanarSpacingFormulas/212.6.2ClosePacking/222.7TheWigner-SeitzCell/24不得转载谢谢合作LWMv viCONTENTS2.8CrystalStructures/252.8.1StructuresforElements/252.8.2StructuresforCompounds/262.8.3SolidSolutions/28RelatedReading/29Exercises/293Diffraction313.1Introduction/313.2PhaseDifferenceandBragg’sLaw/333.3TheScatteringProblem/373.3.1CoherentScatteringfromanElectron/383.3.2CoherentScatteringfromanAtom/403.3.3CoherentScatteringfromaUnitCell/403.3.4StructureFactorCalculations/433.4ReciprocalSpace,RESP/453.4.1WhyReciprocalSpace?/453.4.2DefinitionofRESP/463.4.3DefinitionofReciprocalLatticeVector/483.4.4TheEwaldConstruction/503.5DiffractionTechniques/533.5.1RotatingCrystalMethod/533.5.2PowderMethod/533.5.3LaueMethod/553.6WaveVectorRepresentation/55RelatedReading/58Exercises/584DefectsinSolids614.1Introduction/614.2WhyDoDefectsForm?/624.2.1ReviewofSomeThermodynamicsIdeas/624.3PointDefects/664.4TheStatisticsofPointDefects/674.5LineDefects—Dislocations/714.5.1EdgeDislocations/734.5.2ScrewDislocations/744.5.3Burger’sVectorandtheBurgerCircuit/764.5.4DislocationMotion/77不得转载谢谢合作LWM CONTENTSvii4.6PlanarDefects/774.6.1GrainBoundaries/774.6.2TwinBoundaries/784.7Three-DimensionalDefects/79RelatedReading/79Exercises/805DiffusioninSolids815.1IntroductiontoDiffusionEquations/815.2AtomisticTheoryofDiffusion:Fick’sLawsandaTheoryfortheDiffussionConstructD/835.3RandomWalkProblem/875.3.1RandomWalkCalculations/895.3.2RelationofDtoRandomWalk/895.3.3Self-DiffusionVacancyMechanisminaFCCCrystal/905.3.4ActivationEnergyforDiffusion/915.4OtherMassTransportMechanisms/915.4.1PermeabilityversusDiffusion/915.4.2ConvectionversusDiffusion/945.5MathematicsofDiffusion/945.5.1SteadyStateDiffusion—Fick’sFirstLaw/955.5.2Non–SteadyStateDiffusion—Fick’sSecondLaw/97RelatedReading/108Exercises/1086PhaseEquilibria1116.1Introduction/1116.2TheGibbsPhaseRule/1116.2.1Definitions/1116.2.2EquilibriumAmongPhases—ThePhaseRule/1136.2.3ApplicationsofthePhaseRule/1156.2.4ConstructionofPhaseDiagrams:TheoryandExperiment/1166.2.5TheTieLinePrinciple/1206.2.6TheLeverRule/1216.2.7ExamplesofPhaseEquilibria/1256.3NucleationandGrowthofPhases/1306.3.1ThermodynamicsofPhaseTransformations/1306.3.2Nucleation/133RelatedReading/137Exercises/138不得转载谢谢合作LWM viiiCONTENTS7MechanicalPropertiesofSolids—Elasticity1397.1Introduction/1397.2ElasticityRelationships/1417.2.1TrueversusEngineeringStrain/1437.2.2NatureofElasticityandYoung’sModulus/1447.3AnAnalysisofStressbytheEquationofMotion/1477.4Hooke’sLawforPureDilatationandPureShear/1507.5Poisson’sRatio/1517.6RelationshipsAmongE,e,andv/1517.7RelationshipsAmongE,G,andn/1537.8ResolvingtheNormalForces/156RelatedReading/157Exercises/1588MechanicalPropertiesofSolids—Plasticity1618.1Introduction/1618.2PlasticityObservations/1618.3RoleofDislocations/1638.4DeformationofNoncrystallineMaterials/1758.4.1ThermalBehaviorofAmorphousSolids/1758.4.2Time-DependentDeformationofAmorphousMaterials/1778.4.3ModelsforNetworkSolids/1798.4.4Elastomers/183RelatedReading/186Exercises/1869ElectronicStructureofSolids1879.1Introduction/1879.2Waves,Electrons,andtheWaveFunction/1879.2.1RepresentationofWaves/1879.2.2MatterWaves/1899.2.3Superposition/1909.2.4ElectronWaves/1959.3QuantumMechanics/1969.3.1Normalization/1979.3.2DispersionofElectronWavesandtheSE/1979.3.3ClassicalandQMWaveEquations/1999.3.4SolutionstotheSE/200不得转载谢谢合作LWM CONTENTSix9.4ElectronEnergyBandRepresentations/2159.4.1ParallelBandPicture/2159.4.2kSpaceRepresentations/2169.4.3BrillouinZones/2199.5RealEnergyBandStructures/2219.6OtherAspectsofElectronEnergyBandStructure/224RelatedReading/226Exercises/22710ElectronicPropertiesofMaterials22910.1Introduction/22910.2OccupationofElectronicStates/23010.2.1DensityofStatesFunction,DOS/23010.2.2TheFermi-DiracDistributionFunction/23210.2.3OccupancyofElectronicStates/23510.3PositionoftheFermiEnergy/23610.4ElectronicPropertiesofMetals:ConductionandSuperconductivity/24010.4.1FreeElectronTheoryforElectricalConduction/24010.4.2QuantumTheoryofElectronicConduction/24410.4.3Superconductivity/24710.5Semiconductors/25310.5.1IntrinsicSemiconductors/25310.5.2ExtrinsicSemiconductors/25710.5.3SemiconductorMeasurements/26110.6ElectricalBehaviorofOrganicMaterials/264RelatedReading/266Exercises/26611JunctionsandDevicesandtheNanoscale26911.1Introduction/26911.2Junctions/27011.2.1Metal–MetalJunctions/27011.2.2Metal–SemiconductorJunctions/27111.2.3Semiconductor–SemiconductorPNJunctions/27411.3SelectedDevices/27511.3.1PassiveDevices/27611.3.2ActiveDevices/279不得转载谢谢合作LWM xCONTENTS11.4NanostructuresandNanodevices/29011.4.1HeterojunctionNanostructures/29011.4.22-Dand3-DNanostructures/293RelatedReading/294Exercises/295Index297不得转载谢谢合作LWM PREFACEStartinginthe1960sthefieldofmaterialssciencehasundergonesignificantchanges,fromafieldderivedlargelyfromwell-establisheddisciplinesofmetallurgyandceramicstoafieldthatincludesmicroelectronics,polymers,biomaterials,andnanotechnology.Thestringentmaterialsrequirements,suchasextremepurity,perfectcrystallinityanddefect-freematerialsforthemicroelectronicsrevolutioninthe1960s,weretheprimemovers.Majordevelopmentsinothertechnologicallysignificantfields,suchaspolymers,optics,high-strengthmaterialsthatcanwithstandhostileenvironmentsforspaceandatmos-phericflight,prostheticsanddentalmaterials,andsuperconductivity,havealongwithmicroelectronicschangedmaterialssciencefromaprimarilymetallurgicalfieldtoabroaddisciplinethatincludesever-growingnumbersofclassesofmaterialsandsubdisciplines.Thisbookisatextbookthatambitiouslyendeavorstopresentthefundamentalsofthemodernbroadfieldofmaterialsscience,electronicsmaterialsscience,andtodosoasafirstcourseinmaterialsscienceaimedatgraduatestudentswhohavenothadapreviousintroductorycourseinmaterialsscience.Thebook’scontentsderivefromcoursenotesthatIhaveusedinteachingthisfirstcourseformorethan20yearsatUNC.Theinitialchallengeinteachingaonesemesterfirstcourseinthisbroaddisciplineofelectronicsmaterialsscienceistheselectionoftopicsthatprovidesufficientfundamen-talstofacilitatefurtheradvancedstudy,eitherformallywithadvancedcoursesorviaselfstudyduringthecourseofperformingadvanceddegreeresearch.Itisthemainintentofthisbooktoprovidefundamentalintellectual“tools”forelectronicmaterialssciencethatcanbedevelopedthroughfurtherstudyandresearch.Thebookisspecificallydirectedtomaterialsscientistswhowillfocusonelectronicsandopticalmaterialsscience,althoughwithanemphasisonfundamentals,thematerialselectedhasbenefitedpolymerandbiomaterialsscientistsaswell,enablingawidevarietyofmaterialsscience,chem-istry,andphysicsstudentstopursuediversefieldsandqualifyforavarietyofadvancedcourses.Withsuchabroadintentvirtuallyallofmaterialssciencewouldberelevant,sincemodernelectronicsmaterialsincludemanydiversematerials,morphologies,andstructures.However,therewasaself-limitingmechanism,namelyitallhadtofitintoonesemester.Consequentlyfundamentalsarestressedanddescriptivematerialislimited.Thenextchallengefortheinstructoristoconsiderthelevelofstudents.Inmaterialssciencecurriculatypicallyfoundinengineeringschools,afirstcourseinmaterialsscienceisusuallyrequiredbeforetheendofthesecondundergraduateyear,soastoprovidethebasisformorespecializedandadvancedjuniorandseniorlevelundergraduatecoursesinthevariousareasofmaterialsscience.Thusmostintroductory(firstcourse)materialssciencetextsarewrittenforfirstorsecondyearengineeringstudents,andthereforeassumemeagermathematicalexperience,andonlyelementarychemistryandphysics.In不得转载谢谢合作LWMxi xiiPREFACEthesetextsprinciplesareoftenintroducedusingformulasthatarenotderived,followedbydescriptivematerialandexamplestoreinforcetheideasandprovidepracticewithproblemsolving.Therearenumeroushigh-qualitytextsavailableatthislevel.OvertheyearsIhaveusedanumberofthemeitherasprimarytextsand/orasreferencemateri-alsforthematerialssciencecoursesthatIteachatUNC.However,theleveloftheavail-ableintroductorytextsistoolowforafirstcourseinmaterialsscienceofferedtograduatestudentsandtochemistryandphysicsundergraduatesintheirsenioryear.Fortheunder-graduatesatUNCwherethereisnomaterialssciencedepartment,thefirstmaterialssciencecoursewaspartofanAppliedSciencesCurriculumwithMaterialsScience(elec-tronicmaterialsandpolymers)asatrack.Forthechemistryandgraduatestudentswhowilldograduatelevelresearchinmaterialsscience,thereareonlyfewadvancedmateri-alscoursesavailableatUNC.Thusthefirstmaterialssciencecourseofferedtothesestu-dentsmustnotonlybeatahigherlevel,itmustalsomorecompletelyequipthestudentsforadvancedcoursesandindependentstudyintheirrespectiveresearchinterests.ThistexthasbeenwrittenfromthenotesthatIhavegeneratedovertheyearsofteachingthishigherlevel,butintroductorymaterialssciencecourseatUNC.Thenoteswereusedtosupplementandraisetheleveloftheavailableintroductorytexts.Chapters1through11arecoveredintheirentiretyinasinglesemestercourseatUNC.Theresultisafastpacedcoursewithadearthofdescriptivematerial.InthiscourseIassumethatthestudentshavehadatleasttwosemestersofcalculus,generalchemistry,elementarybutcalculus-basedphysics,andtheequivalenceoftwosemestersofphysicalchemistry,whichincludesthermodynamicsandquantummechanics.Mostofthestu-dentstakingthecoursehavehadsignificantlymorepreparationthanassumed.WiththeseassumptionsIamabletomovemorequicklythroughthematerial.Alsothereisnottheusualinitialtreatmentofchemicalbonding,sinceitisassumedthatstudentshavealreadyhadatleasttwochemistrycoursesthatcoveratomicandmolecularstructureandchem-icalbondingandchemicalreactions.Derivationsofimportantformulasusuallyomittedinafirstmaterialscourseareincludedwhereitisfeltthatthederivationisinstructive,andnotsimplyamathematicalexercise.Nonetheless,thisauthorbelievesthatitisnec-essarytohavethestudentreachacomfortlevelwithsomemorephysicalandmathe-maticalareassothattheycanreadoriginalpaperswithouttrepidation.Theearlyintroductionofreciprocalspaceisconsideredessentialtounderstanddiffractionasastructuraltool,andalsoelectronbandtheory(askspace)andmuchofsolidstatephysics.Reciprocalspaceisthenaturalcoordinatespace.Themathematicalnatureofdiffusionisintroducedtopresentthe“flavor”ofthefield.ElectronenergybandsaretreatedfromtheKronig-Penneymodel,andnotsimplyassumedtoexistfromsemanticarguments,asisdonefortypicalsecond-yeartexts.Theareaofdefects,phaseequilibria,andmechan-icalpropertiesaretreatedsimilarlytointroductorymaterialssciencetextswiththeaddi-tionofsomeimportantderivationssothatastudentscangleananappreciationoftheoriginoftheformulasaswellasthemethodologyusedinvariousfieldsofmaterialsscience.Iamgratefultoallmystudents,pastandpresent,foralltheirhelpwiththistextbook.Itwastheirquestionsandenduringcuriositythathaveoftendrivenmetoseekbetter,clearerexplanations.Overtheyearsmygraduatestudentshavemadeperceptive(andusuallytactful)commentsaboutmycoursepointingoutbothstrongandweakareas.DuringthewritingandeditingofthisbookmyPh.D.graduatestudents(N.Suvorova,C.Lopez,R.Shrestha,andD.Yang)andpostdoctoral(Dr.LeYan)havereadandcom-mentedonthemanydrafts.Ihavetriedtomakethechangesandcorrectionsthattheysuggested,butIassumeresponsibilityfortheremaininguncleardiscussionsanderrors.不得转载谢谢合作LWM PREFACExiiiIamgratefultomycolleaguesatIBM(ThomasJ.WatsonResearchLaboratory)whereIspentmyfirstprofessional10yearsinscience,andwhereIwasabletolearnelectron-icsmaterialssciencefromleadingscientists,andtothepeopleatWileyforhavingconfi-denceinmethroughthepublishingprocess.Finally,Iamgratefultomyfamily(mywifeMaryAnn,andMichaelandChristina)whoenduredmylonghoursofworkovermanyyearsthatledtothisbook,aswellasallmyotherscientificendeavors.不得转载谢谢合作LWM 不得转载谢谢合作LWM 1INTRODUCTIONTOELECTRONICMATERIALSSCIENCE1.1INTRODUCTIONMaterialssciencecanbethoughtofasacombinationofthesciencesofchemistryandphysicswithinabackdropofengineering.Chemistryhelpstodefinethesyntheticpath-ways,andprovidesthechemicalmakeupofamaterial,aswellasitsmolecularstructure.Physicsprovidesanunderstandingoftheordering(orlackthereof)ofatomsand/moleculesandelectronicstructure,andphysicsalsoprovidesthebasicprinciplesthatenableadescriptionofmaterialsproperties.Thecombinedinformationprovidedbyphysicsandchemistryaboutamaterialleadstothedeterminationandcorrelationofmaterialspropertieswiththeprocessusedtopreparethematerial,andwiththemateri-alsstructureandmorphology.Thepropertiesoncedeterminedandunderstoodareexploitedthroughjudiciousengineering.Inasenseengineeringbringsfocustotheprop-ertiesthatmaterialspossess,andtothematerialitselfifsuitableapplicationsarefound.Evidencefortheleadershipofengineeringiswitnessedbythemanynationalgoalsthatpervadethenationalresearchfundingagenciessuchasnanotechnology,biotechnology,andmicroelectronics.Ineachofthesefieldstheadvantagesofcertainmaterialsproper-tiesareextolled.Thegoalsineverycaseincludethepreparationofnewmaterialswithenhancedpropertiesforparticularengineeringobjectives.Materialsscienceasweknowittodayfindsitsoriginsintraditionalmetallurgyandmetallurgicalengineeringdepartments.Consequentlymanyuniversitymaterialssciencecurriculaandtextbooksinuseinthesecurriculaareheavilyweightedtowardtraditionaltopicsrelatedtometallurgy.Moremodernareasarerelegatedtowardspecialtopicscoursesandtextbookscoveringselectedareas.Thistextisaimedtowardelectronicmate-rialssciencewheretheengineeringobjectiveisbettermaterialsformicroelectronicsandphotonics.ElectronicMaterialsScience,byEugeneA.IreneISBN0-471-69597-1Copyright©2005JohnWiley&Sons,Inc.不得转载谢谢合作LWM1 2INTRODUCTIONTOELECTRONICMATERIALSSCIENCEWhiletherehasbeengrowinginterestandunderstandinginelectronicmaterialsforcenturies,therewasamajorrevolutioninelectronicsthatbeganinthelate1940swiththeinventionofthetransistorbyBardeen,Brattain,andShockley.Thisinventionirreversiblychangedtheentireelectronicsarena.Essentiallybeforethistimeallactiveelectroniccircuitscomponentsweremadeofcloselyspacedsimilarmetalelements(electron-emittingfilaments,grids,electrodes)containedwithinaglassvacuumenvelope,so-calledvacuumtubes.Thesedevicescouldswitchcurrents,provideamplificationandrectification,andalongwithpassivecomponentsenabletheconstructionofradios,tele-visions,andevenanaloganddigitalcomputers.Abouttheearlyelectronicdevicesbasedonvacuumtubes,itisamusingtorecallthattheseearlyelectronicmarvelswerealllargerthantoday’sversions.Nonewerelargerthantheearly(1960s)analoganddigitalcom-putersthatusedvacuumtubes,andthatfilledlargeroomsandevenentirebuildings,buthadlesscomputingpowerthatthelaptopwithwhichthistextiswritten.Then,aftertheinventionofthetransistor,iswasmorethan10yearsbeforetheideasaboutthesolidstatedevicescouldbetrulyfeltwiththeimplementationofreliablediscretetransistorsreplacingvacuumtubesontheelectronicsmarket,andinallkindsofconsumerdevices.Duringthisperiodofincubationfrominventiontowidespreadapplications,thereweresomewhatdormantareasofscienceandengineeringthatbecameveryactiveandmademajoradvancesthatwerespurredonbythepotentialmarketsforthenewsolidstatedevices.Firstitwasrealizedthatsinglecrystalsofsemiconductorelectronicmaterialshadtobemadeinlargequantitiesratherthaninlaboratorysizesandwithcrystallineperfectionandchemicalpurityneverbeforeimaginedinmanufacturing.Thenthenotionofelectronicbandstructurethatderivedfromtheearliestdaysofquantummechanicshadtobemodernizedandunderstoodforthenewsolidstateelectronicmaterials.Fromthenewresultsofelectronicenergybandstructure,dopingcouldbeunderstood,andtheroleofcrystallographicdefectsbecamecentraltoelectronicsmaterials.Latticediffusionofdopantsintocrystalsdevelopedgreatlyinthisera.Itwasalsorealizedthatthenewclassofelectronicdeviceswouldrequirethejoiningofdifferentsolidstatematerialssuchasmetalswithsemiconductorswithinsulatorsineverypermutation.Thustherewasrenewedinterestinphaseequilibria,notonlytounderstandtheimportantmetallurigi-caltransformationsthatgovernsteelandotheralloysbut,withemphasisonalloysbetweenelectronicallydissimilarmaterialsandwithhomogeneityranges,soastounder-standatomicvacanciesandcorrelatecrystallatticevacancieswithresultingelectronicproperties.Alongwithalltheseadvancesinunderstandingandpracticeofthesolidstatesincetheinventionofthetransistor,anotherinventioncametotheforethatalsorevo-lutionizedthewaywelive.Thisinventionistheintegratedcircuit(IC).Theintegratedcircuitenablestheconfiguringofsolidstateelectronicmaterialsinordertofabricatedevicessuchastransistorsandrectifiersonthesurfaceofsemiconductors,andtolinkthemalltogethertomakeacompleteelectronicsystemorsubsystemtobefurtherlinked.TheIChaspavedthewayforallthemodernelectronicdevicesespeciallythedigitaldevicesthatperformlogicandmemory.Inadditiontoenablingtheefficientmanufac-tureofmultiplesolidstatedevices,theICpavedthewayforanothermajorrevolution,namelynanotechnologyornanoscience.TheveryheartoftheIC,asitisimplementedwithplanartechnology,enablesthedownwardsizescalingtopresentdevicedimensionsinthenanoscalerange.Theareasofelectronicmaterialsscienceandmicroelectronicsareclearlytheforerunnersofnanotechnology,andmanyofthetechniquesdevelopedforICsarefullyintegratedintomodernnanotechnology.Thustheareasofelectronicsmate-rials/microelectronicsandnanotechnologyareintimatelyrelatedinthatitisclearthatmicroelectronicsisthepredecessorofnanotechnology,andthatadvancesinnanotech-不得转载谢谢合作LWM 1.2STRUCTUREANDDIFFRACTION3nologywillundoubtedlyimpactmicroelectronics.Asmicroelectronicstookholdofallthedevicesweuse,theareaofopticaldevicesorphotonicsalsodevelopedusingthesolidstateideasaboutmaterialsaswellastheabilitytointegrateopticalandelectronicdevicesonachip.Thestudyofelectronicmaterialssciencemustthenincludethefactorsthatenableamaterialtobepreparedandunderstood,anditspropertiesdeterminedandoptimizedfordefinedapplications,inparticular,electronicsand/orphotonicsapplications.ThesetypicalfactorsselectedforstudycomprisethenamesofChapters2through11:Struc-ture,Diffraction,Defects,PhaseEquilibria,Diffusion,MechanicalProperties(twochap-ters),ElectronicStructure,ElectronicProperties,andDevices.Manyofthesetopicsandchaptershavethesamenamesonefindsintraditionalmaterialssciencetexts,andthatisnoaccident.Itisclearthatafoundationintraditionalmaterialsscienceisimplicitinelec-tronicsmaterialsscience.Thedifferenceisinemphasis,sinceasapracticalmatteronetextoronecoursecannotdoitall.Inthefollowingparagraphsthereasonsarediscussedwhytheseheadingsarechosenforastudyofelectronicsmaterialsscience,andtheempha-sisisexplained.1.2STRUCTUREANDDIFFRACTIONMaterialsscienceisoftendescribedasbeingcomprisedofstructure-propertyrelation-ships.Inthiscontextstructurerefersnotonlytothearrangementofthebasicbuildingblocks,orlong-rangeorderingbutalsotothechemicalstructureorshort-rangeorder-ing.ThismorecompletenotionoforderingisdiscussedearlyinChapter2ofthistextwiththeappropriatenomenclature,andthisthemeisrevisitedmanytimesthroughoutthebook.Differentstructurescanrepresentbothdifferentchemicalbondinganddiffer-entarrangementsofatomsand/ormolecules,andpossiblyevendifferentstatesofaggre-gation(roughness,largegrained,etc.).Allthesestructuralaspectscanleadtodifferentproperties,includingelectronicandopticalproperties.Itisimportanttouseaconsistentnomenclaturetoidentifytheuniquestructuralfeaturessothatmaterialsscientistscommunicateinastandardlanguage.ThesetopicsarediscussedinChapter2onthestructureofsolids.InChapter3ondiffractionwestudythedeterminationofcrystalstructure.Thebasicideathatunderliesthisimportantfamilyoftechniques,diffractiontechniques,istheprin-cipleofsuperposition.Itwillbeseeninthetextthatmuchofthefundamentalsofmate-rialssciencecanbeunderstoodbyreferringtoafewthebasictenetsofchemistryandphysics.Amongthetenetsthatarecontinuallyrevisitedisthesuperpositionprinciplethatisusedfordiffraction,mechanicalproperties,andelectronicstructure(withthefirstreviewofthistenetinChapter3andagainmorethoroughlyinChapter9).Forexample,thenatureofawavefunctionthatisusedtodescribeanelectroncanbeunderstoodbyconsideringthewavefunctiontobemadeupofmanywavesinacomplexblend,namelythenotionofmodulation.LaterinChapter3theconceptofreciprocalspaceisintroduced.Theideafollowsfromthenotionthatitisimportantinsciencetooperateinthecoordinatespacemostappropriatetothesystem.Itisfoundthatforcrystalstructureobtainedbydiffraction,reciprocaldistancescorrelatethestructurewithdiffractionexperiments.Fromastudyofstructureanddiffractiononemaygleantheerroneousideathatonly,oratleastmostly,crystallinematerialsareimportantinmaterialsscienceandelectronicmaterialsscience.Thisisfarfromthetruth,butitisanaturaltendencythatfollowsfrom不得转载谢谢合作LWM 4INTRODUCTIONTOELECTRONICMATERIALSSCIENCEpayingcloseandearlyattentiontoonlyperfectcrystals.Infactalargefractionofusefulmaterialsinallfieldsarenotcrystallineatall(e.g.,thedielectricsusedinmicroelectronicICs),andanotherlargefractionispartiallycrystalline(alloysusedforcontactsinmicro-electronics)oratleastdefectiveintheircrystallinenature.However,thenonperfectlycrys-tallinematerialsaremoredifficulttodescribeuniversallyandsimply.Thatistosay,eachmaterialmustbedescribedusinganumberofstructuralaspectswherecrystallinitymaybeoneoftheimportantaspects.However,asisusualinscience,theidealstateistheeasiesttodescribethoroughly,andthisisthereasonwhyvirtuallyallstudiesofmateri-alssciencecommencewithadiscussionofidealorperfectcrystals.AlsoelectronicstructurethatisdiscussedinChapter9onelectronicstructureisimportantfordeterminingmanypropertiesparticularlyelectricalproperties.ItwillbeseeninChapter9thatthestructureofthematerialwillgreatlyinfluencetheelectronicstructureandinturntheelectronicandopticalproperties.1.3DEFECTSTodispelthemisleadingattentiontoperfectcrystals,inChapter4ondefectsinsolidswelookatdifferentkindsofdefects.Thedefinitionsforseveralofthemorecommonmaterialdefectsarediscussed.Ithasbeenfoundoverandoverthatsimplestructuraldefectssuchassubstitutionalandinterstitialdefectscanalterelectricalpropertiesandmasstransportviadiffusionbyordersofmagnitude,whileatthesametimehardlyaffectthemeltingpointorthethermalconductivityforamaterial.Furthermorelinedefectsareimplicatedasthemainfactorintheplasticdeformationofcrystallinematerials.Thenotionofgrainboundariesastheboundariesinbetweensinglecrystalgrainsisalsoimplicatedinthemechanicalpropertiesofmaterialsandinelectronicpropertiesofpoly-crystallinesemiconductors.Thusboththestructureanditslevelofperfectionprovideabackdropfromwhichthebehaviorandpropertiesofamaterialareunderstood,partic-ularly,electronicmaterials.AlsoinChapter4anotherfundamentaltenetofmaterialsscienceisintroducedandusedliberallyinfollowingchapters.ThistenetistheBoltzmanndistributionfromwhichbothequilibriumthermodynamicsandactivationenergies,orenergybarriers,forprocessescanbeunderstood.Thisconceptisintroducedbyconsideringasimpletwoallowedstateproblem,andassessinghowtwoenergeticallydistinctstatesseparatedbyadifferenceinenergy,DE,canbepopulated.Theresultisafamiliarexponential-DE/kTtermeoftenreferredtoastheBoltzmannfactor.However,inthefieldofchemicalkineticsanArrheniusfactorwiththesameformastheBoltzmannfactorisoftendis-cussedinrelationtothevelocityofchemicalreactions,buttheArrheniusfactorisoftenintroducedwithoutadequatediscussionaboutitsorigin,oratbestasanempiricalresult.Theimportanceofthisideaissuchthatitisintroducedanddiscussedearlyinthetext.Furthermorethelawsofthermodynamicsderivefromtheaverageorstatisticalnatureofatomsorcompoundsthatcompriseamaterial.Thisstatisticalnotioniscrucialtowardtheunderstandingtheaveragepropertiesofamacroscopicpieceofamaterialthatcontainsalargenumberofatomsand/ormolecules.Suchthermodynamicsproper-tiesincludethephaseofthematerial,thevaporpressure,anddecompositiontempera-ture.Ontheotherhand,quantummechanicsmayberequiredtounderstandthepropertiesthatdependonthespecificinteractionsofatomsand/ormoleculeswithinamaterialsuchastheabsorptionoremissionoflightandtheelectronicandthermalconductivity.不得转载谢谢合作LWM 1.5PHASEEQUILIBRIA51.4DIFFUSIONInvirtuallyallsolidstatereactionsandtransformation,mattermoves;thatis,atomsand/ormoleculesaretransportedtoandfromthereactionsite.Ofteninthesolidstatethatmotionisbyarandomprocess,andsuchrandomprocessesaretermeddiffusiveprocesses.EarlyinChapter5ondiffusioninsolidstheformforavarietyofdiffusionequationsarecompared,anditisobservedthatseeminglyunrelatedphenomenaaregov-ernedbyequationswiththesameform,namelythereisafluxinresponsetoaforce.Thatflux(withunitsofamount/area·time)canbematter,heat,charge,energy,andsoon.EventhefamousSchroedingerequationofquantummechanics(seeChapter9)hastheformofadiffusionequation.AlthoughonlymassdiffusioniscoveredinChapter5,heattransport,forexample,involvesthesolutionofsimilarequations.Inthefieldofmassdiffusionmanytreatmentsdealpurelywiththeunderlyingphysicsthatenablerandommattertransport,whileotherapproachesdealexclusivelywiththemathematicsofsolvingthedifferentialdiffusionequations.InChapter5bothareasareaddressed.Inadditionanotherfundamentaltenetinmaterialsscienceisintroduced,namelytherandomwalkproblem.Whileappliedstrictlytodiffusioninthischapter,therandomwalkproblemyieldsinsightintohowrandomprocessescanyieldsimpleunder-standableresultspreciselybecauseoftheassumedrandomnessofthesystem.Thisisapowerfulideathathelpshonetheintuitionofamaterialsscientistwhomustoftendealwithseeminglyunsolvableproblemsinvolvingrandomnessandcomplexity.Inthefieldofelectronicmaterialsdiffusionplaysacentralrolethatincludesthetransportofdopants,otherpointdefects(vacanciesandimpurities,andelectroniccarrierdiffusioninelectronicandopticaldevices.1.5PHASEEQUILIBRIATraditionalintroductorymaterialssciencetextsusuallycoverthetopicofphaseequilib-riaadequatelyforunderstandingelectronicmaterials.Themainreasonisbasedonthefactthatmostintroductorymaterialssciencetextsemphasizemetallurgicalmaterials,namelymetalsandalloys,eventhoughthesetextshaveoftenbeenmodernizedwiththeadditionofpolymersandelectronicmaterials.Metallurgydealsextensivelywithmixedcompositionalloyssuchassteel.Anunderstandingofsteelandotherimportantalloysrequiresadetailedknowledgeofthephasediagramforthesystem,inordertoknowunderwhatconditionstoexpectcertainalloyphasesandthecompositionofthephases.However,oftentimesadvancedphysicsandchemistrycoursesspendlittletimeonthistopic,andwhilesomeformsofphaseequilibriumarecoveredinundergraduatechem-istrycourses,solidstatephasediagramsareoftenbarelymentioned.Itisclear,however,thatmoderntrendsinmaterialsscienceandelectronicmaterialsscienceincludecomplexmaterialsthatcanhaveseveralphasesandwidehomogeneity(stoichiometry)ranges.Includedinthekindsofelectronicandphotonicmaterialsinwhichphaseequilibriaareimportantaremodernbinarysemiconductorsthatareusedextensivelyforbothelec-tronicandopticaldevices,ceramicsuperconductors,alloysuperconductors,magneticalloys,highdielectricconstantinsulators,andpolymerblends.InChapter6onphaseequilibriaweprovidesimplederivationsoftheGibbsphaseruleandtheleverruleandoutlinestheproceduretoestimatephasediagramsfromknownthermodynamicdata.Allmaterialsscientistsdealwiththeformationofphasesfromsomeprimalstate,andhenceoftentheinitialstageofphaseformation,nucleation不得转载谢谢合作LWM 6INTRODUCTIONTOELECTRONICMATERIALSSCIENCEbecomesimportantindeterminingfinalproductmorphologies.Forthisreasonnucle-ationisaddedinthechapter.Anunderstandingofnucleationphenomenaisalsoimpor-tanttotheunderstandingoftheprocessesthatareusedtopreparethethinfilmsusedformostmodernelectronicandopticaldevices.1.6MECHANICALPROPERTIESInthefirstofthetwochaptersonmechanicalpropertiestheemphasisisthedevelop-mentofthebasicideasandtheresultingrelationshipsamongtheelasticconstants.InChapter7ontheelasticitypropertyofsolids,theseconstantsareusedtodescribethebehaviorofmaterialsthatdeformelastically,whichmeansthatasforcesareapplied,thematerialdeforms,butthematerialreturnstoitsoriginalstateastheforcesareremoved.Mostmaterialsexhibitthisbehaviorwhensmallforcesareappliedforshortperiodsoftime.Thereismoreinterestwhenlargerforcesareappliedthatleaveamaterialperma-nentlydeformedorevencausesfractureofthematerial,sincedeformationandfailurerelatetheusefulnessofamaterialforfabricatingproductssuchascars,bridges,andhomes.However,aswasthecaseforstructure,firstthesimpleridealcaseofelasticityisconsideredandthenconsiderationisgiventoamorecomplicatedbehaviorcalledplas-ticity.InChapter8ontheplasticitypropertyofsolidstheunderlyingideasarepresentedforpermanentdeformationorplasticity.Theimplicationofdislocationsfortheplasticdeformationofcrystallinematerialsisdiscussedandcreepisbrieflydiscussed.Inthischapterthedeformationofnoncrystallinematerialssuchaspolymersisdiscussed,andseveralmodelsthatareusedtointerpretthemechanicalresponseofthesekindsofmate-rialsaredeveloped.Inmicroelectronicsandphotonicsmanyofthedevicesareconstructedbylayeringfilmsofdissimilarmaterials.Thereforedifferencesinthermalexpansionaswellaschem-icalincompatibilitiesattheinterfacescanleadtoperformanceandreliabilityissuesforthedevices.Furthermoremanyoftheextremestructuralfeaturesandextremelysmallsizesoffeaturesofthemoderndevicescanexacerbatethemechanicalissuesthatmayexistforplanarandlargerdevices.Inadditiontheapplicationsofforcesonacrystallatticecanaltertheatomicspacingandthereforeaffecttheelectronicnature,meaningtheelectronicenergybandstructure,ofamaterial.Afullanalysisofthesecomplicatedstructuralandelectronicissuesisbeyondthescopeofthistext,butafirst-ordertreat-mentoftheimportantrelationshipspropertiesisessentialsothatadvancedstudyandappreciationoftheimplicationsofmechanicalpropertiescanbeaccomplished.Manymodernmicroelectronicsproductssuchascomputerchipsarefabricatedfromthinfilmsofdissimilarmaterials.Also,oncethelayeredstructuresareformed,theprod-uctsgothroughvarioustemperaturecyclesaspartofthefurtherprocessing.Thesestruc-turesarepronetothedevelopmentofstressesthatcanleadtodevicefailureandtoshorterusefullifetimes.Consequentlythemechanicalissuesofthermalexpansion,stresses,anddefectformationthatarecrucialtofurtherstudyofelectronicmaterialreli-abilityarecoveredinthesetwochapters.1.7ELECTRONICSTRUCTUREInChapter9onelectronicstructureweconsideranotheraspectofthestructureofmate-rials,namelytheelectronicstructure.Thebasicideasrelatingtoelectronicstructure不得转载谢谢合作LWM 1.8ELECTRONICPROPERTIESANDDEVICES7includeaconsiderationofthearrangementofatomsandmoleculesaswasintroducedinChapters2and3plustheadditionofaconsiderationoftheinteractionsoftheatomsormoleculesintheirvariousstructuralmotifs.Theinteractionsamongatomsandmoleculesishandledusingquantummechanics.Quantummechanicsenableschemiststoestimate,ifnotcalculate,thestructureofmanyimportantmoleculesusingtheSchrödingerequation.Similarlyquantummechanicsenablesthecalculationoftheallowedanddisallowedenergiesfortheelectronsinanarrayofatomsormoleculesincondensedphases,suchasliquidsorsolids.Theallowedenergiesarecalledenergybands,andthedisallowedenergiesarecalledtheforbiddenenergygaps(FEG)orsimplybandgaps.Anold(1931)butusefulmodelforthecalculationofelectronicenergybandstruc-tureforsolidsispresented,theKronig-Penney(KP)model.DespiteitssimplicitytheKPmodelcontainsmanyoftheimportantphysicalideasthatareusedinmoremodernmodels,butwithoutdifficultmathematics.ConsequentlytheKPmodelisusefulasavehicletounderstandtheoriginofallowedelectronicenergybandsandgaps,buttheKPmodeldoesnotenablequantitativeestimationsofenergybands.Nonetheless,manyimportantconclusionscanbemaderegardingtheelectronicstructureofmaterialsusingtheKPmodel.Associatedwiththeenergybandstructureisanextensivenomenclatureandrepresentationlanguage,andthislanguageisintroducedtodescribeelectronenergybandstructure.Inthischapterthereisheavyrelianceonthestructuralideasandrecip-rocalspacethatwereintroducedinChapters2and3.Itisclearthatfundamentaltounderstandingelectronicandopticalpropertiesofsolidsandthedevicesistheelectronicenergybandstructure;thusChapters10and11makeheavyuseoftheideasdevelopedinthischapter.Furthermoremodernideasaboutnanotechnologythatincludequantumwellstructures,quantumdots,andothersmallintricatestructuresareunderstoodintermsoftheenergybandstructureandthecom-parisonsthataremadetolargerdevices.1.8ELECTRONICPROPERTIESANDDEVICESInChapter10onelectronicpropertieswemakeheavyuseoftheresultsfromChapter9,inparticular,theelectronicenergybandstructure,andaddstothisdevelopmenttheuseofthestatisticsforelectrons,namelyFermistatistics.Anestimateismadeaboutthenumberofelectronicstatesformaterials,theso-calleddensityofstates(DOS)iscalcu-lated.Fromtheenergybandstructure,thedensityofstates(DOS),andtheprobabilityforoccupancy,theFermi-Diracdistributionfunction,theelectronicarrangementforsolidsisdeduced.Fromthisarrangementtheelectronicnatureofthematerialsisrevealed,andresultingpropertiesareunderstood.Thedifferentkindsofelectronicmaterialsarealsodiscussed:conductors,semiconductors,superconductors,andnon-conductors.Electronicconductionistreatedbothclassicallyandintermsofquantummechanicalideas.ForsuperconductionthepopularBCStheoryisintroduced.LastlyinChapter10theelectronicnatureoforganicmaterialsisintroduced,andsincemanyoftheorganicmaterialsinuseareamorphous,theelectronicnatureofamorphousmate-rialsisdiscussed.Inthefinalchapter,Chapter11onjunctions,devices,andthenanoscale,wereachapointwherewecandistilltheideasdevelopedinChapters9and10thatarefundamentaltodesigningandunderstandingelectronicandopticaldevices.Virtuallyallmodernelectronicandopticaldevicesusethejunctionsofmaterials.ThusinChapter11wecommencewithjunctionsandtheelectronicsimplicationsofjoiningdissimilarmate-rials.Fromjunctions,passivedevicesthatdonotchangeflowingcurrentsorapplied不得转载谢谢合作LWM 8INTRODUCTIONTOELECTRONICMATERIALSSCIENCEpotentialscanbeconstructedsuchasthermocouplesandsolidstaterefrigerators.Then,usingvariousjunctions,thischapterintroduceselectronicdevicesthatareimportantintoday’smicroelectronictechnologysuchasdiodes,solarcells,transistors,andthedevicesthatcomprisecomputerchips.Thebasisideasaboutopticaldevicesareintroducedwithexamples.Thelastsectiondealswithnanotechnologyandthekindsofdevicesthatwillemergefromongoingresearchinfabricatingnanoscalestructuresfrommaterials.1.9ELECTRONICMATERIALSSCIENCEModernscienceandtechnologyrequireshighlytrainedmaterialsscientistswhocanfunc-tionindiverseareassuchasmetallurgy,biology,ceramics,electronics,andoptics,tonameseveralfields.Itisclearthattherearemanycommonalitiesinthefields.Forexample,forallsolidstatematerials,structurewithallitsimplicationsisimportant.Forbiology,mol-ecularstructureismoreimportantthaniselectronicenergybandstructureatthisjunc-tureindevelopment.Thatisnottosaythatwiththedevelopmentofbiomaterialsandnanotechnologythefuturewillbringbio-inspiredelectronicandopticaldevices.Formanyfieldsstructuraldefectsareimportantasaremechanicalproperties.Forthefieldsofelectronicsandoptics,electronicstructureandpropertiesarefundamentaltounder-standtheresultingdevices.However,defectsandmechanicalinteractionsarealsocrucial.Thustopicsinthistextwerechosenmoreasamatterofpracticality,inthattoadequatelycoverallareasofimportancetoelectronicmaterialswouldresultinanimpracticallylargetext.Carefulchoiceshadtobemadeinselectingthemostgermanematerialforelec-tronicmaterialsscience.不得转载谢谢合作LWM 2STRUCTUREOFSOLIDS2.1INTRODUCTIONAsthestudyofmaterialsprogressesinsuccessivechapters,theimportanceofstructureindictatingmanyofthematerialspropertieswillbecomeclearer.Knowledgeofstruc-turealongwithchemicalcompositioncomprisesthemostfundamentalpropertiesknownaboutmaterials,andbothkindsofpropertiesarerequiredtocompletethecharacteri-zationofamaterial.Achemistasamolecularscientisttypicallyfocusesattentionontheatomiccompositionandmolecularstructureofthechemicalormoleculeunderstudy.Molecularstructurereferstothearrangementoftheatomsinaparticularmolecule.Inadditiontocomposition,amaterialsscientistmustnotonlyknowstructureatthemol-ecularlevelbutalsoathigherlevelssuchasthearrangementofmolecules,namelywhetherthemoleculesareordered(ornot)onscaleslargerthanthemolecularsize.Thisissobecauseagivenmaterialwithaspecifiedcompositioncan,andoftendoes,exhibitwidelydifferingpropertiesthatarerelatedtothestructure.Asimpleexampleofthisiswater,H2O,asshowninFigure2.1.Water,inthesolid,liquid,andgaseousstatespos-sessesdifferentstructures,widelydifferentproperties,butthesamechemicalcomposi-tion.Figure2.1adisplaysthestructureofamoleculeofwaterwhileFigure2.1bdisplayswhatstate(solid,liquid,vapor)andstructureofwaterexistatvariouspressuresandtem-peratures.Itispossibletohavebothavariationofthemolecularstructure(therela-tionshipoftheH’sandO’s)andavariationinthearrangementofthewatermolecules(therelationshipoftheH2Ounits).Figure2.1billustratesseveralsolidphasesofwater(X1,1C,and1H)thatexistathighpressuresandlowtemperatures.Whilethisexampleseemssimpleenough,itisnot.Inthisexamplethedifferencesbetweendifferentstatesofmatterwerecompared,therebyexaggeratingthestructuraldissimilarities.However,wecouldhavechosentodiscussonlysolidH2O,anditsdifferentstructuresascanbeElectronicMaterialsScience,byEugeneA.IreneISBN0-471-69597-1Copyright©2005JohnWiley&Sons,Inc.不得转载谢谢合作LWM9 10STRUCTUREOFSOLIDSa)Ho104OHb)9Liquid10)XI1C1H106Solid103VaporPressure(Pa1100200300400500600Temperature(K)Figure2.1(a)Awatermolecule;(b)phasediagramofwatershowingthreesolidphases.obtainedbypreparingiceunderdifferentconditions.Onefindsmanypropertiesthatdifferwithstructure,buttherearealsosomepropertiesthatdonotdependstronglyonstructure.Animportantobjectiveofmaterialsscienceistounderstandstructure-propertyrelationships,namelywhysuchcorrelationsmayormaynotobtain.Inthiswayamaterialsengineercanrationalizematerialspropertiesanddesignmaterialswithoptimumpropertiesforaspecificapplication.TheH2Oexampleindicatesthefactthatmorethanoneleveloforderingisimpor-tant.Ontheatomicscalethechemicalbondingbetweenatomsisthesame,ornearlythesame,formanystructurallydifferentformsofH2O.Thischemicalbondinglevelofstruc-tureistermedshort-rangeorder,orlocalstructure,asopposedtothelong-rangeorder-ingoftheH2Omoleculesinicecrystals.Short-rangeorderisintimatelyrelatedtochemicalbondingandhencedictatesstoichiometry.Long-rangeorderreferstothearrangementofthechemicalbuildingblocksthatmaybemolecularoratomic.Thischapterendeavorstofirstdescribeorder,thenstructureandthenomenclatureusedtoindicatethekindofstructureforsolids.Manyimportantmaterialsdonotpossessorder;hencewemustalsoconsiderthekindsofdisorderedmaterials.Theimplicationsofstruc-tureareincludedinalltheremainingchaptersofthistextmostlyexplicitly,butalsoimplicitly.2.2ORDERThereareseveral,sometimesconfusingtermsrelatedtoorderthatrequireimmediateattention.Inthediscussionabovethenotionsoflong-andshort-rangeorderwereintro-不得转载谢谢合作LWM 2.2ORDER11a)b)c)Figure2.2(a)AnSiO4tetrahedron;(b)anorderedarraySiO4tetrahedra;(c)adisorderedarrayofSiO4tetrahedra.duced.TheseconceptsaswellasafewotherrelatedconceptsarefurtherillustratedusingFigure2.2forsolidsilicondioxide,SiO2.FirstweseethatinFigure2.2a,whichrepre-sentsabuildingblocktetrahedronforSiO2,theratioofsiliconatoms(shadedcircles)to1oxygenatoms(opencircles)inthethree-dimensional(3-D)representationis–forasingle4isolatedtetrahedralstructuralunit.The3-DbondinginSiO2istetrahedral,whichmeansthatsurroundingeachSitherearefourO’slocatedattheapicesofatetrahedronandwiththetetrahedralO–Si–Oangleof109°54¢.TheseSiO4buildingblocksarethenassem-bledtocreatethe3-DsolidSiO2material.ThisassemblyisseeninFigures2.2bandcwheretheO’sattheapicesofthetetrahedronsbridgetoadjacentSi’syieldinganoverallstoichiometryofSi/2O’sorSiO2.TheindividualSiO4tetrahedraeachcomposedofSiatomstetrahedrallysurroundedbyO’shaveconsiderablelocalorshort-rangeorder,andtheycomprisethebasicbuildingblocksofSiO2.However,thetetrahedrathatarejoined不得转载谢谢合作LWM 12STRUCTUREOFSOLIDSthroughthebridgingOsattheapicesofthetetrahedracanexistinarangeofangles;thatistosay,therecanbeawidedistributionofSi–O–Siangles.Ifthedistributionisverynarrow,thenthetetrahedraareallarrangedinanorderlyfashionandthematerialhaslong-rangeaswellasshort-rangeorderasshowninFigure2.2b.Withtheadditionoflong-rangeorderthematerialiscalled“crystalline,”andthepossiblecrystalstructureswillbediscussedlater.IfthedistributionofSi–O–Sianglesiswide,thenthetetrahedraarearrangedhaphazardlyalthougheachtetrahedronisthesameasallothers.Thismate-rialwithshort-rangebutnotlong-rangeorderiscallednoncrystalline,orglassy.ThisisseeninFigure2.2cwherenoapparentrepeatisseenintheframeshown.Anotherpos-sibilityisthatthereareregionsofcrystallineorder,buteachregionisunalignedwithanadjacentregionthatisalsocrystalline.Thiskindofmaterialiscalledpolycrystallineorapolycrystallineaggregate.Eachgrainofthepolycrystallineaggregateisitselfasinglecrystal.Last,thematerialmayhaveneithershort-norlong-rangeorder,andthismate-rialiscalledamorphous.Combinationsofthesetypesarealsopossibleinthatamater-ialmaybepartcrystallineandpartamorphous.Whilethedistinctionsmadewiththesedefinitionsforcrystalline,noncrystalline,glassy,andamorphousareconsistent,itisoftenfoundthatthetermsnoncrystallineandamorphousrefertomaterialswithnolong-rangeorderandthetermsareusedinterchangeably,andglassyisusedtodescribeamorphousornoncrystallineoxideglasses.Inthistextwewillusethetermamorphoustodescribematerialswithoutlong-rangeorder.Theunifyingthemeforcrystallinematerialsisthelong-rangeorderingthatcanbethoughtofasauniformtranslationofabasicbuildingblock.Inthiswayoneimaginesthatanentiremacroscopicpieceofamaterialisbuiltsimplybydiscretetranslationsofabasicbuildingunitthroughthreedimensions.Wereturntothispointbelow.2.3THELATTICEThelanguageusedfordescribingcrystalstructureshelpsoneunderstandthedifferencesamongthevarietyofpossiblestructures.Thislanguagecommenceswiththemathemat-icalnotionofapointlattice.Figure2.3ashowsalatticetobeanarrayofpointsinspacesoarrangedthateachpointhasidenticalsurroundings.Thesmallestunit,orunitcell,canbeobtainedbyconstructingplanesthroughpoints,andthelinesresultingfromtheintersectionoftheplanesatlatticepointsdefinetheunitcell.Figure2.3ashowsaunitcellindarkeroutlineanddefinedbythecellparametersa,b,candangles(notshown)a,b,gcalledlatticeparameters.TheanglesaredefinedusingFigure2.3bwhereaistheanglebetweenvectorsaandb,btheanglebetweenaandc,andgistheanglebetweenbandc.Itshouldbenoticedthattheunitcellsodefinedembodiesthesymmetryoftheentirelattice.Theentirelatticecanbegeneratedbysimplytranslatingtheunitcellby|a|intheadirection,by|b|inthebdirection,andby|c|inthecdirection.Thustranslationbecomesanimportantoperationinunderstandingthelong-rangeorderingrepresentedbythelattice.Thequestionastohowmanydifferentkindsofunitcellsarenecessarytofillallspacebytranslationandhowtoaccomplishthisforallpossiblesymmetriesisasolvedmath-ematicalquestionforwhichwehereinacceptthesolutionwithoutproof.ThelatticesthataccomplishthistaskarecalledBravaislattices,andthereare14suchBravaislattices,asshowninFigure2.4.These14Bravaislatticesareorganizedinto7crystalsystemsaccordingtothebasicsymmetrythatthelatticepossesses:cubic,tetrahedral,hexagonal(ortrigonal),不得转载谢谢合作LWM 2.3THELATTICE13a)b)bcaFigure2.3(a)Mathematicalpointlatticewithaunitcelloutlined;(b)unitcellwithlatticevectorsindicated.CubicTetrahdralOrthorhombicHexagonalRhombohedralTriclinicMonoclinicFigure2.4Sevencrystalsystemsand14Bravaislattices.不得转载谢谢合作LWM 14STRUCTUREOFSOLIDSTable2.1SevencrystalsystemsintermsoflatticeparametersCrystalSystemUnitCellVectorsUnitCellAnglesCubica=b=ca=b=g=90°Tetragonala=bπca=b=g=90°Orthorhombicaπbπca=b=g=90°Hexagonal(trigonal)a=bπca=b=90°,g=120°Monoclinicaπbπca=b=90°πgTriclinicaπbπcaπbπgπ90°Rhombohedrala=b=ca=b=gπ90°Table2.2SevencrystalsystemsintermsofminimumdistinguishingsymmetryelementsCrystalSystemMinimumSymmetryElementsCubic4ThreefoldrotationaxesTetragonal1FourfoldrotationaxisOrthorhombic3TwofoldrotationaxesHexagonal(trigonal)1SixfoldrotationaxisMonoclinic1TwofoldrotationaxisTriclinicNoneRhombohedral1Threefoldrotationaxisorthorhombic,rhombohedral,monoclinic,andtriclinic.Someofthesesystemscanhavedifferentlattices:simpleorprimitive(P),bodycentered(BC),andfacecentered(FC).Laterwewilldiscussface-centeredcubicstructures(abbreviatedFCC),andbody-centeredcubicstructures(BCC),amongothers.Table2.1summarizesthedescriptionoftheunitcellsforthesevencrystalsystems.Inadditionthereareotherbasicsymmetryoperationsthatdistinguisheachofthesevencrystalsystems.Symmetryoperationsbringalatticepointintocoincidence.Table2.2showsdistinguishingorminimumsymmetryelementsthatdefineanddistinguisheachofthesevencrystalsystems.Ofcourse,themorehighlysymmetricalsystemscontainthesymmetryelementsofthelowersymmetryones.Differenttypesoflatticesthatmakeupthe14Bravaislatticescanbeobtainedbyseveralfundamentaltranslationsofaprimitivelatticepositionbytheunitcellparame-ter(s)orfractionsthereof.Ifwestartononecornerofaprimitiveunitcellandassignthispositionthecoordinates0,0,0,thenothermajortranslationsare111Bodycentered:0,0,0Æ,,222111111Facecentered:0,0,0Æ0,,,,0,,022222211Basecentered:0,0,0Æ,,022Thefractionscorrespondtofractionsofthea,b,clatticeparametersforthespecificcrystalsystem.Figure2.5showsthemajortranslationsforacubicunitcell.Figure2.5a不得转载谢谢合作LWM 2.3THELATTICE15a)b)c)Figure2.5(a)Primitive,(b)body-centered,and(c)face-centeredcubicunitcellsshowinguniquelatticepositions(largefilledcircles).foraprimitivecellshowsthatanyapexisindistinguishablefromtheothers.Thusthereisonlyoneuniquepositionthatisreproducedbyatranslationofa,andthispositionisdenotedbythecoordinates0,0,0.Figure2.5bforaBCCshowsthattherearethesamecornerpositionssummarizedby0,0,0,butthereisalsoauniquecentercellpositionlabeled–1,–1,–1thatcannotbegeneratedstartingfrom0,0,0andatranslationofa.For222theFCCFigure2.5cshowsthattherearethecornerpositionssummarizedby0,0,0andtherearesixfacepositions.However,onlythreeofthefacepositionsneedtobespeci-fied,sincetheothersareobtainedfromatranslationbythelatticeparametera.Thesemajortranslationsareusefulforgeneratingtheuniquepositionsinalatticestructure.Themajortranslationswillbeusedinthefollowingchapterwhenweconsiderthescat-teringofradiationfromtheuniquelatticepositionsofcrystalsandthedifferentphasesproducedtherefrom(i.e.,diffraction).Itshouldbenotedthatinmanyinstancesthemag-nitudesoflatticeparametersareindicatedwithazerosubscriptasa0.Thenumberoflatticepointsforaunitcell,N,iscalculatedbycountingthepointsthatboundandareinteriortothecellandthenconsideringthesharingofpointsbyadjacentcells.Forexample,theeightlatticepointsatthecellcornersintheunitcellinFigure2.3aareeachsharedbyeightadjacentcells(Nc),thepointsonthefaceofacellbytwocells(Nf),andofcourse,theinteriorpoints(Ni)belongsolelytothecellinques-tion.Hencethefollowingrelationshipsummarizesthis:不得转载谢谢合作LWM 16STRUCTUREOFSOLIDSNNfcNN=++i(2.1)28Aprimitiveunitcellisdefinedasacellthatcontainsonelatticepoint.Thedensityoflatticepointsmaybecalculatedbyconsideringthenumberoflatticepointsforthecelldividedbytheunitcellvolume.If,asweshowlater,atomsormoleculesareassociatedwithlatticepoints,thenthetheoreticaldensityofamaterialcanbeobtained.For3example,foracubicunitcellofdimensiona0,thevolumeofthecellisa0.Ifthecellcon-tainsNatomsofthetypewithamolecularweightofM(g/mole),thenthedensityisgivenasNM()number◊()gmoler=3(2.2)N()numberpermoleorAvogadro’snumber◊a00ThistheoreticaldensityissometimescalledtheX-raydensity,anditcanbecomparedwithmeasureddensity.Thedifferenceisameasureoftheperfectionofamaterial.AswillbecoveredinChapter3,whenoneusesX-raydiffractiontomeasureatomicposi-tions,anaveragepositionorstructureismeasured.Localvacantpositionsandsparseimpuritiesareignored.ThustheX-raydensityisbasedontheoverallstructurewithoutimperfections.2.4CRYSTALSTRUCTUREThemathematicallatticesdisplayedinFigure2.4serveasthestartingpointforunder-standingcrystalstructures.Theyprovidethesmallestnumberofallowedsymmetriesintermsofeasilyimaginedunitcellsthatarenecessarytofillandthusdefineallspace.Buttheinformationfromthesemathematicallatticesisinsufficienttodescriberealcrystalstructures.Whatislackingiscalledthe“basis.”Thebasisistheatoms,ormolecules,thatcomprisetherealmaterialandthatareinsomefixedrelationtothelatticepoints.Forexample,thesimplestcaseisthemonatomicelementalsolidwhereoneatomresidesexactlyateachlatticepointofoneoftheBravaislattices.TheBravaislatticethenbecomesthecrystalstructure.Thecrystalstructuresfortheelementsareofthistype.However,morecomplicatedmaterialssuchastheSiO2,aswasdiscussedabove,havethebasicbuildingblockssuchasSiO4tetrahedraassociatedwiththelatticepoints;morecomplexmaterialssuchasproteinsandDNAhavemoreelaboratebuildingblocksasso-ciatedwith(notnecessarilyat)thelatticepointsoftheBravaislattices,therebyyieldingacrystalstructure.ThusthedefiningrelationshipforcrystalstructureisCrystalstructure=+PointlatticeBasis(2.3)Figure2.6illustratestheformulaabove.Thestructureshownattherightismadeupofanarrayofthebuildingblocks.Thebuildingblocksarethetrianglesofthreeopencircles(e.g.,tomodelatriangularmolecule)andthearrayattheleftisaBravaislattice(primitivecubic).Thereisthesamerelationshipbetweeneachbuildingblockandthearrayorlattice.Formostelementsthebasisisunity,whichmeansthereisliterallyanatomatthelatticepoints.Ifastructureisconsideredwherethebasisisknowntobeatomsormolecules,thentheidealdensitycanbecalculated.Dependingonthelatticetype,thenumberofatoms不得转载谢谢合作LWM 2.5NOTATION17+=Figure2.6Latticeplusbasisyieldsacrystalstructure.ormolecules,N,inthestructurecanbecalculatedfromformula(2.1)forN.Withknowl-edgeoftheidentityofthebasiselementsormolecules,theatomicormolecularweights,MW,areknown.FromNandMWforeachspeciespresentintheunitcell,themassofatomsinthecelliscalculated.Now,iftheunitcellparametersareknown,thenthevolumeofthecellisalsocalculated.Thusthemassdividedbythevolumefortheunitcellyieldsthedensityforthestructure.Aswasdiscussedabove,thedensitycalculatedinthiswayisconsideredtobeideal,sinceitassumesthatalltheunitcellsareasperfectastheoneusedforthecalculation.Laterweconsiderthatdefectscanoccurandaltertheideality.Forexample,supposethatoneinonehundredlatticesitesarevacant.Thiswillaltertheactualdensityby1%.Similarlythepresenceofimpurities,eitherassubstitutesforatomsorinadditionto,willaltertheidealdensity.Thereforethedifferencesbetweenidealandrealdensitiescansignalandquantifythepresenceofsomesortoflatticeimperfection.2.5NOTATIONAsone’sunderstandingofstructuredeepens,itbecomesincreasinglyimportanttobeabletodiscussspecificplanesanddirectionsinthevariouscrystallinematerials.Thisimportancederivesfromthefactthatthechemicalbondingthat,beingdirectional,isoftendifferentindifferentdirectionsandondifferentplanesinacrystalstructure.Thusitisnotsurprisingthatmanymaterialpropertiesaredifferentindifferentbondingdirec-tions.Someoftheseproperties,alongwithcrystallographicdifferences,willbediscussedinlaterchapters.Inordertodealwithdirectionaldifferences,amethodologytonamedirectionsandplanesincrystallinematerialsisincommonuse.2.5.1NamingPlanesTheacceptedsystemfornamingplanesistheMillerindexnotation.NamingplanesislinkedwithfindingtheintersectionsoftheplaneswiththebasiclatticevectorsthatdefinethefundamentalBravaislatticeforastructure.However,simplyusingintersectionsissometimescumbersomebecauseinterestingplanesareoftenparalleltooneormoreoftheunitcelllatticevectors.Inthiscasetheintersectionisatinfinity,andeitherthewordorinfinitysymbol•needstobecarriedalonginthenomenclature.Inordertoobviatethissituation,thereciprocalsoftheinterceptsaretakensothat1/•becomes0.Frac-tionsobtainedaftertakingthereciprocaloftheinterceptsarecleared,andtheresultingsetsofusuallythreewholenumbers(anexceptiontothreeiscoveredbelow)areplaced不得转载谢谢合作LWM 18STRUCTUREOFSOLIDSa)cabb)cabc)cabFigure2.7Examplesofplanesshowinginterceptsonlatticevectors.inbetweenroundedbrackets()indicativeofspecificplanes.Figure2.7illustratestheMillerindexsystem.Figure2.7ashowsthea,b,caxeswiththelargerdiagonalplaneinterceptsofa=1,b=1,andc=•(i.e.,theplaneisparalleltothecaxis).Thereciprocalsare1/1,1/1,and1/•,whichyieldthe(110)plane.Thesmallplanehasinterceptsofa=1/3,b=1/2,andc=1.Thecorrespondingreciprocalsare3,2,1,sotheplaneisthe(321)plane.InFigure2.7bthelargerplanehasinterceptsof1,1,1,sotheplaneis(111).Thesmallerplanehasinterceptsof1/3,1/2,2/3.Thereciprocalsare3,2,3/2and,uponclearingfractions,becomesthe(643)plane.Figure2.7cshowstheshadowedplanewithinterceptsof•,1/2,•,whichyieldsthe(020)plane.Wecanimaginetheplanesperpendiculartoandbisect-ingtheshaded(020)plane.Theseplaneswouldbetheeitherthe(200)planeorthe(002)plane.Thesethreeplanescompriseafamilyofplanesdenotedby{200}.SimilarlyinFigure2.7ctheplanesthatboundthefigureare{100},namelythefamilyof(100)planes.Thehexagonalsystemoftenusesanadditionalindex,meaningfourindexesratherthanthree,as(h,k,i,l).Thenewindexiissymmetricallyrelatedtothefirsttwoas-i=h+k.Becausethisfourthindexisnotunique,itissometimesomittedorreplacedbyaperiodas(hk.l)toindicatehexagonalsymmetry.BecausetheMillerindexesareobtainedfromthereciprocalsoftheintercepts,theplaneswiththesmallestintercepts(relativetoalatticeparameter)havethelargestMiller不得转载谢谢合作LWM 2.5NOTATION19abFigure2.8Two-dimensionalcubiclatticeshowingdifferentlowandhighMillerindexplanes.indexes.Lowindexplanesarethemostcommononesfoundinnature,andhencetheinterceptsbeingfractionalinterceptscorrespondwiththelatticeparameters.Figure2.8showsa2-Dprojectionofthelowandhighindexplanes.Therearethreesetsofplanesshown:(11),(12),and(17).Noticethatthelowindexplanesalsocontainthegreatestnumberoflatticepointsperunitlengthin2-D(areain3-D).Theseplaneswiththehighestatom/moleculeconcentrationalsopossessthehighestbonddensityandthuselec-trondensity.Thereforeallthosepropertiesthatcorrelatewithatom,bond,and/orelec-trondensityaredeterminedbythelowindexplanesofthematerial.Itiseasytoseewhywhendescribingthepropertiesofacrystalitisimportanttoalsospecifythedirectioninwhichthepropertywasmeasuredandtheappropriateplaneinvolved.2.5.2LatticeDirectionsInordertonameadirection,onemustfirstconstructalineparalleltothedirectiontobenamed,butthatintersectstheoriginofthelatticevectors.Thenatanypointontheconstructedlineaperpendicularisdroppedtoeachlatticevector.Theinterceptstothelatticevectorsclearedoffractionsarethedirectionindexes.AnexampleisshowninFigure2.9a.Thelinelisdrawnfromtheoriginparalleltothelinewhosedirectionistobedetermined(?).Theinterceptsonthea,b,caxesarenotedtobea=1,b=1,c=1/2.Theinterceptsareclearedoffractionsyieldingthedirection[221].Anyinterceptscon-sistentwithbeingparalleltothedirectioninquestionwillwork.Figure2.9bshowsacubicunitcellwithlowindexplanesanddirections.Noticethatthedirectionsinsquarebracketsareperpendiculartotheplanesinroundedbracketswiththesameindices.不得转载谢谢合作LWM 20STRUCTUREOFSOLIDSa)c?labb)cc[111](111)aabb[110]c(100)a[100]bFigure2.9Proceduretoindexadirection:(a)Thedirectiondesiredtobeindexed(?)withaline(1)drawnparalleltothedesiredlineandwithintercepts;(b)severaldirectionswithassociatedplanes.Directionindexesareenclosedinsquarebrackets[].Afamilyofdirections,suchas[111],[111¯],[11¯1],and[1¯11],where1¯indicatesanegativevaluefortheindex,canbeindi-catedusingangularbracketsas<111>.ItisbothinterestingandusefultorealizethatfororthogonalBravaislattices(a=b=g=90°)the[100],[110],and[111]areperpen-dicularto(100),(110),and(111),respectively.Table2.3summarizesthekindsofbrack-etsthatareconventionallyusedtoindicateplanes,directionsandfamiliesofeach.不得转载谢谢合作LWM 2.6LATTICEGEOMETRY21Table2.3NomenclatureforplanesanddirectionsPlane()Familyofplanes{}Direction[]Familyofdirections·Òcc/laa/h3aa1b/k2adhklbFigure2.10Plane(shaded)withanglesandfractionalintercepts.2.6LATTICEGEOMETRY2.6.1PlanarSpacingFormulasFromFigure2.10itisseenthattheperpendiculardistancefromtheorigin,(000)ofthecoordinatesystemtotheplaneshown,islabeleddhkl.distheperpendiculartotheplane.WiththeplanarMillerindexesof(hkl),thefractionalinterceptsthat(hkl)makeswiththecoordinatesystemarea/h,b/k,c/lfortheaxesa,b,c.Rememberthataisthefulllatticevectorlength.Assumeunitlengtha=1,then1/histhefractionalintercept.Fromthisfigurewecandefinethefollowinganglesa:a1betweendanda,a2betweendandb,a3betweendandcThedirectioncosinesareobtained:dddcosaaa123===,cos,cos(2.4)ahklbcForacubicsystem|a|=|b|=|c|=a0andd=|d|,2222dhkl()++222cosaaa1++=cos2cos31=2(2.5)a0不得转载谢谢合作LWM 22STRUCTUREOFSOLIDSTable2.4PlanarspacingformulasforthesevencrystalsystemsCrystalSystemPlaneSpacingFormulas2221hkl++Cubic=22da02221hk+lTetragonal=+222dac002221hklOrthorhombic=++2222dabc00022214Êhh++kkˆlHexagonal2=ÁË2˜¯+2d3ac00222221()hkl++sinaa++2()hkklh+l()cos-cosaRhombohedral=2223da()13-+cosaa2cos0222211Êhksinaal2hlcosˆMonoclinic=++Á-˜22Ë222¯dsinaa0b0c0ac0011222Triclinic=+()AhBk+Cl+222Dhk+Ekl+Fhl22dV2221/2V=a0b0c0(1-cosa1-cosa2-cosa3+2cosa1cosa2cosa3)222222222A=b0c0sina1;B=a0c0sina2;C=a0b0sina322D=a0b0c0(cosa1cosa2-cosa3);E=a0b0c0(cosa2cosa3-cosa1)2F=a0b0c0(cosa3cosa1-cosa2)orinthemorecommonform:2221hkl++=(2.6)22da0FromTable2.4wecanseethatthealgebraicmethodusedtocalculatetheinterplanarspacingformulasrapidlybecomestediousforthecrystalsystemswithlowersymmetry.InChapter3,whenwecoverreciprocalspacethatcomprisesthenaturalcoordinatesforthelinkbetweenstructureanddiffraction,wewillseethatthisprocedureissimplerinreciprocalspace.2.6.2ClosePackingInordertoobtainafirst-ordernotionofclosepacking,considertheprimitive,body-centeredandface-centeredcubiccellsinFigure2.4.Ateachlatticepositionimagineasphere(atomormoleculeasthebasis)ofradiusR.Forsimplicity,considerallthespherestobeequivalent.Nowimagineeachofthecelldimensionsshrinkinguniformly(butnottheradiusofthespheres)untilthespheresjusttouch.Itisclearthattheshortestcon-nectingdimensionsarethemostimportant,andalongthesedirectionsthesphereswilltouchfirst.FortheprimitivecubicstructureshowninFigure2.11a,eachoftheeightspheres(twoshown)touchsixnearestneighbors,andnosphereisuntouched.Atetra-不得转载谢谢合作LWM 2.6LATTICEGEOMETRY23a)ao}a=2Rob)4Ra=o3c)4Ra=o2Figure2.11Closepackingdirectionsfor(a)PC,(b)BCC,and(c)FCCcubicunitcellswherethecloselypackeddirectionisindicatedbythetouchingofatoms(shaded).Therelationshipbetweenthelatticepara-metera0andtheatomicradiusRisalsogiven.hedralshapedholeisformedatthecenterofthiscloselypackedstructurethathasanedgelengthorlatticeparameterofa0=2R,whereRistheradiusofthespheres.FortheBCC,however,thesphereatthecenterofthecellcontactstheeightcornerspheres,yield-ingacellbodydiagonallengthof4RasisillustratedinFigure2.11b.Thiscloselypackedstructureyieldsalatticeparameterofa0=4R/3,andthecornerspheresdonottouch.Ateachofthesixcubicfacesanoctahedralholeexists.TheoctahedronshapefortheinterstitialholesinthelatticeiscompletedwiththeinclusionofadjacentBCCcells.FortheFCCshowninFigure2.11c,closepackingisachievedwhentheface-centeredspherestouchthecornerspheres.Again,thecornerspheresdonocontacteachother.Thefacediagonalis4Randthecelldimensionsarea0=4R/.2Interestinglytherearetheoctahedralholesatthecenterofeachcell,andinadditionatthecelledgestherearetetrahedralinterstitialsitesformed.Tetrahedralsiteshavefourspheresandoctahedralsiteshavesix.Theexistenceandsizeoftheseintersticesareimportantbecausetransportofspeciescantakeplacethroughtheintersticesandforeignspeciescanoccupyinterstices.Theseideaswillbedevelopedfurtherinfollowingchapters.Anotheridearelatedtotheconceptofpackingisthepossibilityofpackingspheresinonelayeruponanother.AsshowninFigure2.12,thefirstandbottomlayerofspheres(dashed)thataretouchingiscalledtheAlayer.ThenextclosedpackedlayerisimaginedtoformbysimplyallowingthespheresforthesecondorBlayertofallintothetroughsmadebythreeAlayerclosepackedspheres.Nowtoformthethirdlayer,therearetwopossibilities.IfthepossibilitythatthethirdlayerformsindirectcorrespondencetotheAlayer,thenthisthirdlayerisalsonamedanAlayer(anotherAlayer).TheclosepackingoflayersfollowstheorderABABA....Thisformhashexagonalsymmetryandis不得转载谢谢合作LWM 24STRUCTUREOFSOLIDSCLayerBLayerALayerAnotherALayerFigure2.12Closepackinginlayerswhereatomsareassumedtobespherical.ThebottomAlayer(dashed)iscoveredwithBlayeratomsintroughsintheAlayer.TheClayerislikewiseformedbutintwodifferentways.consequentlycalledhexagonalclosepacked(HCP).Alternatively,ifthethirdlayerformsintheotherposition,whichregistersneitherwiththeAorBlayers,itformsaClayerwiththeorderABCABC....ThispackingisalsoclosepackingandpossessesFCCsymmetry,soitistermedaccordingly.AtomsofbothHCPandFCCclosepackingareshowninFigure2.12.2.7THEWIGNER-SEITZCELLUpuntilnowwehavechosentheunitcellboundariessomewhatintuitivelybyextract-ingaportionofthelargerlattice.Itisreasonabletoexpectthatbythismethodthecare-fullychosenpieceswillreproducetheentirelatticebytranslationandthereforefulfilltheunitcelldefinition.Thereareothermethodstoselecttheunitcellthatkeeptherequire-mentsthesame,namelythattheunitcellmustcontainthesymmetryofthelatticeandfillallspacebytranslation.Itisparticularlyusefulinsomeapplicationstochooseacellthatisprimitive,acellthatcontainsasinglelatticepoint.Onewaytodothisiswithasquare2-DlatticeasdepictedinFigure2.13.Asthefigureshows,onestartsatanylatticepointinthe2-Darrayanddrawslatticevectorsemanatingfromthestartingpointtofirstnearestneighbors(solidarrows).Thebisectors(dashedlines)ofthesevectorsarecon-structedandextended.Theareaincludedwithinthebisectorsformsanewunitprimi-tivecell(shaded)andiscalledaWigner-Seitzcellafterthescientistswhomadeuseofthiskindofcell.In3-Dthelatticevectorbisectorsareplanesthatencloseavolumesur-roundingthechoseninitiallatticepoint.WewillrevisitthisconstructioninChapter3afterreciprocalspaceisintroduced,andwewillproduceasimilarunitcellinreciprocal不得转载谢谢合作LWM 2.8CRYSTALSTRUCTURES25Figure2.13SchematicoftheWigner-Seitzcellformation.A2-Dlatticeisshownwithvectorsdrawntonearestneighborsandnextnearestneighbors.Thesevectorsarebisectedtoformtwo(shaded)primitiveunitcells.spacethatiscalledaBrillouinzone.TheBrillouinzonehasaspecialsignificanceinelec-tronbandtheory,aswillbecomeapparentinChapter9andsubsequentchapters.2.8CRYSTALSTRUCTURES2.8.1StructuresforElementsAswasmentionedabove,foracrystalstructuretoform,bothlatticeandbasisarerequired.Wefirstconsiderastructurecomposedofatomsofthesameelementatalllatticesites.Alltheelementsfitthisexampleexceptforuranium,wherethestableroomtemperaturecrystalstructureisbase-centeredorthorhombicbutwithtwoatomsnotat,butnear,eachlatticeposition.Manyelementsarecubic,eitherBCCorFCC.Table2.5givessomeexamples.ForthemostpartthesekindsofstructuresareeasilyvisualizedbysimplyimaginingaBravaispointlatticeandplacingatomsatthelatticepointsorinsomefewcasesnearthesepoints.InFigure2.4thetoprowgivesexamplesofPC,BCC,andFCCmetalswherethebasisistheshownlatticepoints.Figure2.14ashowsthediamondcubic(DC)latticeofcarbon(C).ThisissimilarinformtotheFCC(opencircles)butincludesfourextraCatomsat–1–1–1,–3–3–3,–3–3–1,and–1–1–3thatareshownasblackcircles444444444444forcontrast.Severalimportantsemiconductors,SiandGe,havethisdiamondcubicstructure.不得转载谢谢合作LWM 26STRUCTUREOFSOLIDSTable2.5SelectedelementsandcompoundswithcrystalstructuresandlatticeparametersLatticeParametersSubstanceStructurea,b,c,a,b,g(nm,°)AlFCC0.5311AsRhomb0.4132,57.1°AgFCC0.4086AuFCC0.4079BTet0.880,0.505CDC0.3567Hex(graphite)0.2461,0.6708CuFCC0.3615GaOrtho0.4523,0.4524,0.7661GeDC0.5658InTet0.4598,0.4947Fea,BCC0.2867b,FCC0.3647g,BCC0.2932PbFCC0.4950NiFCC0.3524POrtho0.3314,0.4377,0.1048PtFCC0.3924SiDC0.5431Sna,DC0.6489b,Tet0.5832CsClPC0.356GaAsDC/zincblende0.5653NaClFCC0.5639SiO2a,Hex0.490,0.5392.8.2StructuresforCompoundsChemicalcompoundsarecharacterizedasbeingbothchemicallyandphysicallydistinctfromtheiratomicconstituents.Thischaracteristicisattributedtothefactthatthewaychemicalbondingalterstheelectrondensitydistributionisdependentonboththeelec-tronsandthenucleiinvolved.Thuseachcompoundisadefiniteanduniquecombina-tionofatomsthatisalsouniqueinnature.Thespecificarrangementofatomsmustbeuniquesincetheatomicandmolecularforcesarerelatedtotheparticulararrangement.Infactitisprobablethatthecrystalstructuresforallcompoundsaredistinct.Whilethisintuitivereasoningsuggestsconsiderabletruth,thewaythecrystalstructurefillsspaceisdescribedwellwithintheframeworkalreadydeveloped.Whendealingwithchemicalcompoundstherearesomerulestokeepinmind.First,alltheatomsinagivencom-poundcrystalstructuremusthavethesamesymmetry.Second,whendescribingacrystalstructureforacompound,alltherepeatedtranslationsmustbeginandendonthesameatom.Whilecompoundspresentagreatdiversitytocrystallographers,theideasalreadydevelopedplusthetworulesgivenabovegreatlysimplifythearea.Weconsiderafewcommoncasesfrommyriadpossibilities.不得转载谢谢合作LWM 2.8CRYSTALSTRUCTURES27a)b)c)d)Figure2.14Severalcrystalstructures:(a)Diamondcubic;(b)NaClstructure;(c)CsClstructure;(d)ZnSstructure.Inorganicsolidcompoundstypicallyformsimplestructures.Forexample,solidNaClpossessesFCCstructure(notethatelementalNaatroomtemperatureisBCCandClisagas).Whatdoesthismean?Inordertovisualizethisstructure,itiswellfirsttoimaginetheappropriatepointlattice,namelytheFCCBravaislattice.Thenapplytherulesstatedabove.Figure2.14bshowstheNaClstructuretobetwointerpenetratingFCCpointlattices.ThebasisforoneisNaatthelatticepoints,whileClisthebasisfortheother.Itissometimesdifficulttodeterminetherepeatingcellbecausetherearetoofewatomsinview.ThiscanbeillustratedwiththeCsClstructureshowninFigure2.14c.InFigure2.14cthetoppanelshowsthatCsClhasaBCCstructurewithClatthecenter,butthisisnotcorrectfrombothastructuralandchemicalpointofview.SincesucharepeatingunitcannotfaithfullyreproducethecorrectstoichiometryofCsClataratioof1/1,amoreextendedviewoftheCsClgeometryisshowninthelowerpanel.Thestructurehereisprimitivecubicwithtwointerpenetratingprimitivecubiclattices,thefirstwithCsasthebasisandtheotherwithCl.NotonlyaretheNaClandCsClstructuresinterest-inginthemselves,buttherearenumerousotherchemicalcompoundsthathavethesamestructures.KCl,CaSe,andPbTehavetheNaClstructure,whileCsBrandNiAlaswell不得转载谢谢合作LWM 28STRUCTUREOFSOLIDSasmanyimportantorderedalloys(discussedbelow)displaytheCsClstructure.InFigure2.14donceagainisshownaFCCstructurethatappearssimilartotheDCdiscussedaboveandshowninFigure2.14aforC.However,Figure2.14dgivestheso-calledZnSstructureinwhichtwoFCClatticesinterpenetrate,oneforZnandtheotherforS.Itshouldnowbeclearthatcompoundstructurescanbedifficulttovisualizefromasmallsamplingsize.Table2.5listssomecommonlyusedmaterialsinelementalandcompoundformwithstructuresandlatticeparameters.2.8.3SolidSolutionsInadditiontopureelementsthathaveadefinitecrystalstructureandchemicalcom-poundsthataredistinguishedbyhavingbothadefinitestoichiometryandadefinitecrystalstructure,thereareotherkindsofcrystallinematerialsforwhichknowledgeofstructureallowsonetounderstandthematerialspropertiesandreactions.Solidsolutionsareprominentinthiscategoryofmaterials,andthesolidsolutionisdefinedasaphasewithvariablestoichiometry(seeChapter6onphaseequilibria).Structurallythesolidsolutionretainsthecrystalstructureofthehostorsolvent,andtheotherminorcon-stituentsorsolutesaddeithersubstitutionallyorinterstitiallytothehostlattice.IntheformercaseasdepictedinFigure2.15asomeofthehostspecies(opencircles)areran-domlyreplacedbysolutespecies(filledcircles)atthelatticesites.Foraninterstitialsolidsolutionthesolutespeciesarepositionedintheintersticesofthehostlatticeratherthanatthelatticesites.Thearrangementofsolutespeciesistypicallyrandomonthehostlattice.However,undercertainconditionsforsubstitutionalsolidsolutionsandforspe-cificmaterials,thesoluteatomstakeregularpositionsinthehostlattice.AsecondlatticearisesasshowninFigure2.15b(theorderingofthesolute-filledcircles)duetothisaddi-tionalordering.Thissituationistermedasuperlatticebecauseaneworderingisaddedtothealreadyexistingorder,whichinthiscaseexistsinthehostlattice.Forsubstitu-tionalsolutions,thehostandguestatomsneedtobesimilarinsizeandchargeforanysubstantivesolubility(CuNisolutionsasmentionedbelowdisplay100%solubilityduetothesimilaritiesoftheatoms).Forinterstitialsolutions,theguesttypicallyneedstobesufficientlysmalltofitintheinterstices.Anotableexampleofaninterstitialsolidsolu-tionissteel,whichisessentiallyaninterstitialsolutionofCatomsintheintersticesintheFelattice.Later,inChapter11,wewillseeotherkindsofsuperlatticesthatarecom-posedofrepetitivethinfilmlayers.Thisorderingyieldsunusualquantummechanicalbehavior.Solidsolutionsarealsoreferredtoasalloys.Thestoichiometryforsolidsolutionsoralloyscanvarysometimestoallproportions.Forexample,CuandNicanmixinallpro-portions,butthequestionarisesastowhatstructureresults,CuorNi?InfactbothCuandNihavethesamestructure,FCC.Inadditionbothelementshavenearlythesamelatticeconstantof0.36nmforCuand0.35nmforNiandsimilaratomicradii(0.25nm)andelectronicstructure.Thusintuitionsuggeststhattheseelementsarecontinuouslymis-cibleinallproportions.Inthecaseofinterstitialalloys,typicallythesoluteissmallerthanthehostsothatthehostintersticescanhousesolutewithoutconsiderabledistor-tiontothehostlattice.OnetechnologicallysignificantexampleissteelcomposedofCinFe,asmentionedabove.Cwitharadiusofabout0.15nmsomehowfitsintotheocta-hedralholesinBCCFe(a-Fe)thathasadiameterofabout0.072nmintheFelattice.DuetothesizeofCrelativetothesizeoftheoctahedralhole,thesolubilityofCissmall,andthisresultsinconsiderablelocaldistortionoftheFelattice.Itshouldbe不得转载谢谢合作LWM EXERCISES29a)b)Figure2.15Alloyformationwithguestatoms(filledcircles)takingpositiononthehoststructure(opencircles):(a)Randomdistributionofguestatoms;(b)ordereddistributionformingasuperlattice.rememberedthathoweverconvenientintuitionhereisatpredictingtheresults,allsoluteatomsaddenergytothesystembydistortingtheoriginallattice,evenwhereatomicsizesareclose.Thesolubilitylimitinsuchanadditionprocesscan,inprinciple,beobtainedbydetermininghowmuchadditionalstrainthehostlatticecanwithstandbeforebreak-ingdownorrejectingfurtherforeignspecies.RELATEDREADINGC.R.Barrett,W.D.Nix,andA.S.Tetelman.1973.ThePrinciplesofEngineeringMaterials.Pren-ticeHall,EnglewoodCliffs,NJ.Areadableelementarytextforafirstcourseinmaterialsscience.E.D.Cullity.1956.ElementsofX-rayDiffraction.AddisonWesley,Reading,MA.Alleditionsofthisbookcontainvoluminousstructureinformationinreadabletextform,andinappendixestheX-raydiffractiontechniquesarediscussedatlength.P.A.ThorntonandV.J.Colangelo.1985.FundamentalofEngineeringMaterials.PrenticeHall,EnglewoodCliffs,NJ.Areadableelementarytextforafirstcourseinmaterialsscience.EXERCISES1.(a)Inthecubicstructurecalculatetheanglesbetweenthefollowingplanes:(100)and(100)thereare2;(100)and(110)thereare2;(110)and(110)thereare3(111)and(100)thereare1;(111)and(110)thereare2;(111)and(111)thereare3.Hint:Theanglebetweenplanesistheanglebetweenplanenormals.(b)Illustratetheanglesbetweenthe(100)and(110)withsketches.2.(a)Forcubicsymmetrydrawthe(111)boundedby[110].(b)Identifyeachofthethreeboundingdirections.(c)Showmathematicallythat[111]isperpendiculartoeachdirectionin(b).不得转载谢谢合作LWM 30STRUCTUREOFSOLIDS3.Showusingsketchesormaththatthe[111]isperpendicularto(111)inacubicsystembutnototherwise.4.Sketchandshadethefollowingplanes:(111),(110),(100),(220),(330),(113),and(115).MakeastatementontherelationshipbetweentheMillerindexesandthedis-tancefromtheoriginfortheplanessketched.5.ForFCCandBCCmetalscalculatethelatticeparameterintermsoftheatomicradiioftheatoms.Sketchthestructuresandindicatetheclosepackeddirections.6.ForCuwithacubicstructure,MW=63.54g/mol,a0=0.3168nmanddensity=38.92g/cm(a)DeterminewhetherCuisPC,BCC,orFCC.(b)CalculatetheatomicradiusforCu.7.ForBCCFewithatomicradiusis0.1238nm:(a)Calculatea0.(b)CalculatetherforaperfectcrystalofFe.(c)LookupavalueforpFeandcomparewiththecalculatedvalueanddiscussreasonsforthedifference.8.Calculatethe%volumechangeforasubstanceassumingconstantatomicradius:(a)IfamaterialweretochangefromPCtoBCCuponheating(b)andthenfromBCCtoFCCifheatedfurther.29.Calculatetheatomicareadensity(#/areainnm)onthe(111),(110),and(100)planesforanyFCCandBCCelementthatyouchoose.10.Makecarefulsketchesofanoctahedral(sixnearestneighbors)andatetrahedral(fournearestneighbors)siteintheFCC.11.ShowthatlowMillerindexplaneshaveahigherlatticepointdensitythathighindexplanes.12.DeterminethespecificBravaislatticeforacubicmetal(atomicwt=192g/mol,a0=30.3839nm)withadensityof22.5g/cm.13.Discussbrieflythedifferencebetweenshort-andlong-rangeorderinanymaterialyouchoose(SiO2,etc.).Usealabeledsketch.14.ForaBCCwitha0=0.1nm,calculatetheatomicradiusassumingclosepacking.15.Giventhedensity,D,foranFCCelementwithaknownatomicweight,calculatetheBragganglefor(111).16.Explain(briefly)thestructuraldifference(s)betweenacompoundandanalloy.17.WhataretheMillerindexesfortheintercepts:a=2,b=1,andc=0.5?不得转载谢谢合作LWM 3DIFFRACTION3.1INTRODUCTIONThestudyofdiffractioninmaterialsscienceisfundamentalandimportantforseveralreasons.Firstandforemostisthedeterminationofthestructureofsolids.Thelatticetypeandthepreciselocationsofatomsormoleculesrelativetothelattice,thebasis,areaccessedusingdiffractiontechniques.Alargenumberofstudiesofstructureandprop-ertieshaveattestedtoaclearlinkbetweenstructureandproperties.Thustheestablish-mentofsturcture-propertyrelationshipshasbecomethedefiningfeatureofthefieldofmaterialsscience.Ofspecialimportancetoelectronicmaterialsscientistsisthestudyofelectronictrans-port,whichisessentiallythemannerinwhichelectronsinteractwiththesolid(i.e.,theinteractionofelectronswiththeatoms/molecules).Theinteractionoftheelectronicwavefunctionsoftheatoms/moleculesinthelatticegivesrisetotheelectronicenergybandstructure(thesubjectofChapter9),andthusestablishestheelectronictransport.Themoderndescriptionofelectronicbandstructureusesterminologyderivedfromthestruc-tureofsolids.Hencetheknowledgeofstructureloomsfundamentallyimportant.Alsorelatedtoelectronicpropertiesisthesubjectofdefects.Aswillbecoveredinlaterchap-ters(particularlyChapter4),defectsincrystallinesolidsaredefinedrelativetoaperfectdefect-freestructure.Hencetheidealorperfectstructureneedstobeestablished,inordertothenestablishclearnotionsaboutdefects.TheX-raydiffractiontechniquesdiscussedheretypicallyignoredefectsandyieldtheidealstructure.Thereare,however,diffractiontechniquesthatcandeterminecrystallographicabnormalities.Furthermoredefectsdom-inatetheelectronicpropertiesofsemiconductors.Themechanicalpropertiesofsolidsdependonchemicalbonding,whichincrystallinesolidsishighlyoriented.Thusthemechanicalpropertiesbecomefirstorderfunctionsofstructure.Finally,surfacesinelec-ElectronicMaterialsScience,byEugeneA.IreneISBN0-471-69597-1Copyright©2005JohnWiley&Sons,Inc.不得转载谢谢合作LWM31 32DIFFRACTIONtronicsdevicesareoftremendoussignificance,andthestructureofsurfaces,whileoftenrelatedtounderlyingbulkstructure,isoftendifferentfrombulkstructure.Henceanunderstandingofthemanypropertiesthatdependonstructureofthesurfacerequiresthedeterminationofsurfacestructure.Allthestructuralinformationcrucialtovirtuallyallthesubfieldsofmaterialsscienceareaccessibleusingdiffractiontechniques.Diffractionmeasurestheeffectsonamplitudeandphaseoftheinteractionofmono-chromaticelectromagneticradiation(emr)andmatterwaveswithasolid.Historically,aswellasscientifically,thediffractionofXraysisofgreatsignificance.ThewavelengthofXraysrangesfromfractionsofanmtoafullnm,whichisoftheorderofthesizeofatomicandinterplanarspacingsintypicalcrystalstructures.Thus,aswewillshortlysee,theX-raywavelengthrangeisidealformaximizingthephasechangesthatoccurwhentheemrisscatteredfromatomsinalattice.Atthispointitisusefultocalculatetheapproximatewavelengthforphotons,electrons,andneutrons.ForphotonstheformulaishcEh==n(3.1)l-10Usingdimensionsforlofabout0.1nmor1Å(1¥10m),inthisformula,wecalcu-8lateEforaphoton(wherec=1¥10m/s)asfollows:-34866310.¥◊Js◊310¥ms-15E==¥198910.J-10110m¥-19Theconversion1eV=1.602¥10JisthenusedtoconvertEtoeVsothatE=12415.7eV.SolvingforlinÅyieldstheapproximateresult:31-l=124010.V¥(3.2)-31Forparticleswithmasssuchaselectronswhereme=9.11¥10kg,thedeBroglierela-tionshipformatterwavesisusedasfollows:hhl==pmv122eVKineticenergy==eVmv,v=(3.3)2mthenh150l==forVinvolts2eVVmm6Soelectronsat1¥10Vyieldsawavelength,lª0.01Å.Likewise,forneutronswhere-27mn=1.68¥10kg,12hEm==v,l(3.4)22mE不得转载谢谢合作LWM 3.2PHASEDIFFERENCEANDBRAGG’SLAW33atE=kT,l=1Å,Tª300-400K.Thuselectronsintheappropriateenergyrange,andthermalneutronsalsoyieldausefulrangeofwavelengthsfordiffraction.Theproductionofhigh-intensitymonochromaticXraysisawell-developedtechnol-ogy.Suitableavailablewavelengthswithhigh-wavelengthprecisionareusedtorenderXraysnearlyidealfordiffractionfromsolids.However,thediffractionofotherkindsofwavesinadditiontoXraysisimportant,andamongthesearematterwavesfromelec-tronsandneutrons.ParticlediffractiondependsontheenergyderivedfromwavelengthsofparticlesthroughthedeBroglierelationship,asillustratedabove.Themaindifferencebetweenparticleandnonparticleemrdiffractionistheefficiencyofthediffraction,whichistypicallygreaterforparticles.Butotherthanthis,anunderstandingofthefundamentalaspectsofX-raydiffractionunderliesanunderstandingofalltheotherdiffractionphe-nomena.Therefore,inthefollowingtreatmentofdiffraction,X-raydiffractionisempha-sizedasawaytodiscoveringfundamentalideas,andlaterotherdiffractionresultswillbediscussedintermsoftheobtainedresults.Beforebeginningtheserioustreatmentofhowtheemrinteractionwithalatticerevealsstructuralinformation,itisusefultobrieflylookatdiffractionresultsfromvarioustechniquesillustratedinFigure3.1.Thearrayoflines,arcs,andspotsonapho-tographicplatearetypicalofexperimentalresultsofdiffractionfromcrystallinemate-rials.Figure3.1ashowsso-calledpowderdiffractionresultstakenfromcrystallinematerialgroundtoafinepowderbeforediffraction.Figure3.1bisdiffractiontakenfromasinglecrystalwherethedefinitearrangementofspotsisindicativeofthecrystalstruc-tureofthematerial.Figure3.1cshowstwoelectrondiffractionmicrographswheretheleft-handimageisforapolycrystallinematerialdisplayingringswhiletherightsideshowsbothsinglecrystalspotsandpolycrystallinerings.Therawdiffractiondatainthisfigurebespeaksofanunderlyingorder.Inthischapterthephysicsthatgivesrisetotheresultisexplored,asthiswillnaturallyyieldtheabilitytointerprettheresultsintermsofunder-lyingcrystalstructures.Apowerfulstructuraltool.3.2PHASEDIFFERENCEANDBRAGG’SLAWFigure3.2showsthreerays(A,B,andC)ofmonochromaticradiation(Efield)travel-ingtotherightandinphaseatplane11¢wheretheraysoriginate.RaysAandBtravelthesamedistanceandareinphaseateverypointequidistantfromthesourceatplane11¢becausetheraysareparallel.Forexample,atplane22¢theraysAandBareinphase.RayC,whichisalsotravelingparalleltoAandB,isinphaseat22¢.However,rayCtravelsagreaterdistancetoplane33¢.Atplane33¢thephaseofCisdifferentfromthephasesat22¢.Thislong-windedexampleisusedsimplytoshowthatmonochromaticwavesproducedinphaseatthesourcewillremaininphaseforanequaldistanceoftravel,buteachraymaynotremaininphaseifwavesareaddedorcomparedatdifferentdistancesoftravel.Letusassumethateachrayhasthesameelectricfield(E)field(E=EA=EB=EC)andconsideradetectorplacedat22¢.Thedetector“sees”theresul-tantelectricfielddisturbance(thesuperpositionofthewaves).Sincethethreewavestrav-eledexactlythesamepath,theyallarriveat22¢inphaseandtheelectricfieldsaddto3Ebecausethepathdifferenceis0.However,ifforwaveCweputitsdetectorat33¢whererayCis180°outofphasewithAorB,butleavethedetectorsforAandBat22¢,thetotalEfieldwillbe2E-1E=1E.Wewillseethatwhenemrofsuitablewavelengths(i.e.,forl’softhesizeofthelattices)isscatteredfromcrystalstructures,thepathand,consequently,phasedifferenceswillcauseEfielddifferencesthatleadtointensity不得转载谢谢合作LWM 34DIFFRACTIONa)b)c)Figure3.1Variousdiffractionpatterns:(a)PowderX-raydiffraction;(b)LauebackreflectionX-raydiffrac-tion;(c)transmissionelectronmicroscopydiffractionofpolycrystallinematerial(left)andpolycrystallinemate-rialonasinglecrystalsubstrate.123ABC1'1¢2'2¢3'3¢Figure3.2ThreewavesA,B,Cinphaseattheorigin11¢and22¢.WaveCdisplaysdifferentphasefordif-ferentdistancetraveledto33¢.不得转载谢谢合作LWM 3.2PHASEDIFFERENCEANDBRAGG’SLAW35differencesthatareactuallymeasured.Theprocessofinteractionofemrwithacrystalstructureneedspreliminaryconsideration.TheelectricfieldoscillationsofaperiodicX-rayemrinteractwiththeelectroncloudsoftheatomsinalatticeofatomsorthecrystalstructure.Ingeneral,afractionoftheemrfieldisabsorbedandpartisre-emitted.Thatpartthatisre-emittedunalteredwithrespecttotheinitialphaseandwavelengthistheonlypartfromwhichdiffractionwilltakeplace.Thisso-calledunmodifiedemrretainsphasecoherencywiththeincidentemr.Theunmodifiedcomponentofthescatteredradi-ationresemblesthereflectionoflightfromamirror,andforthisreasondiffractionisoftenspokenaboutas“reflection”ofXrays.However,despitethecommonterminologythatX-raydiffractionisreflection,thescatteringofXraysfrommaterialsisdecidedlydifferentfromlightreflection.ThecoherentlyscatteredcomponentoftheincidentXraysisasmallfractionofthetotalX-rayintensity,typicallylessthan1%,comparedtotrueopticalreflectionwherenearly100%canbeobtainedusinggoodmirrors.Thecoherentlyscatteredemrhasnophasechange,whereasreflectioninvolvesaphasechange.Forreflec-tion,theangleofincidenceisthesameastheangleofreflection,whereasfordiffraction,specificanglesarederived.Furthermorereflectionisasurfaceeffectwhereasdiffractionisabulkeffect.So,althoughcommonparlancemayrelatereflectionanddiffraction,thebasicphysicsisdecidedlydifferent.Figure3.3showstheinteractionofX-rayemrwiththreeequidistantrows(planesbutseenendonasarowofatoms)ofatoms(filledcircles)whereeachplaneisseparatedbydistancex.Rays1,2,and3areincidentatqand,afterscatteringatatomsatO,E,andB,propagatetothedetectoratthefarrightthatcollectsallthescatteredrays.First,weconsidertwoparallelrays1and3incidentatq(asmeasuredfromtheplanesurface)ontothetopandbottomplanes,respectively.FrompointOonthetopplaneaperpendicularisdrawntotheincidentbeams2atDand3atA.Thisissimilarlydonetoexitingrays2¢and3¢atFandC,respectively.NowtheperpendicularlineODAformsanincidentrayfrontuptowhichthethreeincidentraysareinphaseandhavetraveledthesamedis-tancefromthesource.Likewise,lineOFCformsafrontinthescatteredrays,totherightofwhichtheraystravelthesamedistancetothedetector.Inbetweenthesetwofronts(ODAandOFC)theraystraveldifferentdistancesthatcanresultinphasedifferences.NN111¢qq22OOqq2¢xxDDFFccEE3¢33AAxxCCBBaaFigure3.3Threewavesimpingingonthreeparallelequidistantplanes.不得转载谢谢合作LWM 36DIFFRACTIONConsideringrays1and3,weseethatbeforescattering,ray3travelsadistanceABmorethanray1,andthatafterscattering,ray3travelsadistanceBCmorethanray1.Sothetotalpathdifference,d,betweenrays1and3isAB+BC.disthepathdifferencebetweenincidentandscatteredrays.Ifthispathdifferenceisanintegralnumber,n,ofwave-lengths,l,ornl,thenrays1and3thatbecome1¢and3¢,respectively,areexactlyinphaseattheexitwavefrontplaneatangleq.UnderthisconditioneachsegmentABandBCisequalto2¥sin(q).ThissimpleconstructionyieldstheusualformforBragg’slaw:ndl=2sin()q(3.5)wheredistheinterplanarspacing(2xinthefigureforrays1and3)betweenthetopandbottomplanes.UndertheconditionsshowninFigure3.3,noticethatforscatteringfromthetoplayer,surfacescattering,atequalincidentandscatteredanglesthereisnopathdifferencebetweenincidentandscatteredwaves.Consequentlythereisnodiffractionfromasur-faceundertheseconditions.However,unliketheincidentradiation,itisnotrequiredtomeasurethescatteredemratthesameangleq.Inprinciple,thedetectorcanbeplacedtoreceivescatteredradiationatanyangle.Forexample,letusassumethatinFigure3.3,disnotequaltonlbutisl/2.Thentheintensityinthescatteredwavefrontisidenti-cally0,sincetheelectricfieldsare180°outofphaseandaddto0.Now,ifthedetectorismovedtodifferentscatteringangles,thendifferentintensitiesareobtainedduetothedifferentpaths.Maximainintensitywillappearsharplyatcertainangles,andthesearetermedBraggangles.Inactualdiffractionexperimentsmanyraysareconsideredfrommanyscatteringcenters(atoms),sothesharpnessofthein-phasescatteringatspecificanglesisgreat.Toobtaindiffractionfromasurface,theemrisbroughttothesurfaceatnormalincidence,asshowninFigure3.4.Inthiscasealltheincidentraysareinphaseatthesurfaceatoms.Thedetectorscanshemisphericallyabovethesurfaceforbackscat-teredradiation,andthemaximaarenoted.TheanglesatwhichmaximaareobtainedareusedintheBraggrelationshiptodetermineinterplanardistancesandindexes.Ifemrormatterwavesthatdonotpenetratethesurfaceatomlayerareused,suchasthematterwavesfromlow-energyelectrons,thentheresultingdiffractionisrepresentativeofthesurfacestructure.Thistechniqueiscalledlow-energyelectrondiffractionorLEED.Fortypicaldiffractionproblemsitisoftenconvenienttodefinetheincidentangle,q,fromthenormaltothetopplane,andthenthediffractionangle,theanglebetweentheincidentanddiffractedbeam,isreportedas2q,whereqiscalledtheBraggangle.IncidentIncidentDiffracteDiffractedPathDifferencePathDifferenceFigure3.4Incidentwavesatnormalincidencetosurfaceatom,diffractedatvariousangles.不得转载谢谢合作LWM 3.3THESCATTERINGPROBLEM37ConsidertheBraggequation(3.5)andacubiclatticewherefromTable2.42221()hkl++=(3.6)22da02Then,aftersquaringtheBraggequationandsubstitutingtheformulaford,thefollow-ingformulaisobtainedandusedtointerpretmeasuredscatteringangles:2222l()hkl++2sin()q=(3.7)24a0where2qismeasured,andlisknown.FromthemeasuredBragganglesthekindofunitcell(asdeterminedbythehklvaluesfortheplanesdiffracting)andsizecanbedeter-mined.However,inordertoobtainthespecificpositionofscatteringsites(i.e.,theatomicpositions),moreinformationisrequired.Thisinformationiscontainedintheintensitiesofthescatteredradiation.Toseehowatompositionsaffectscatteredintensities,wereturntoFigure3.3andfocusonthemiddlerowofatomsthatisexactlyhalfwaybetweenthetopandbottomrows.Wesettheconditionsofdistancesandanglessothatthepathdifferenceforrays1and3scatteredfromthetopandbottomplanesyielda0pathdifferenceinthescat-teredwaveswhenABC=nl.Thissituationshouldyieldamaximuminthescatteredintensityifweconsideronlythetopandbottomplanesandrays11¢and33¢.However,wehavethepresenceofthemiddleplaneinFigure3.3toconsider,anditispositionedatexactlyhalfthedistancefromthetopandbottomplanes.Thepathdifferenceofthe1rayscatteredfromthetopplanetothemiddleplaneisexactly–thattothebottomplane2sothatfromasimilartrianglesargumentthefollowingisobtained:1DE+=EF()AB+BC(3.8)2Thusthecoherentcomponentoftheemrscatteredfromthemiddleplaneis180°outofphasefromthescatteredradiationofboththetopandbottomplanes.Withtheradia-tionfromthetopandbottomplanesarrangedtobeexactlyinphaseatq,theadditionofthemiddleplanereducesthescatteredradiation.Theadditionofthenewplaneinbetweenthetopandbottomchangestheoriginallysetupmaximumatqtosomelesserintensity.Ifthiswerethebody-centeredatominaBCCorface-centeredatomsintheFCC,onecanimagineadrasticalterationforcertainscatteringanglesrelativetoasimplecubicstructure.Thisresultwillbemadeclearerwhenweconsiderhowtoquantifythescatteredintensitiesandtherebydetermineatompositions,butitshouldbeevidentherethatthespecificatompositionsgreatlyalterthescatteredintensities.Forthisreasonatompositionscanbeconverselydeterminedfromananalysisoftheintensities.Beforewecon-tinuewithsolvingthisproblem,weneedtotakeastepbackandconsiderthedifferentscatteringintensitiesfromanelectron,fromanatom,andthenfromanarrayofatoms.3.3THESCATTERINGPROBLEMWefirstconsiderscatteringfromagasandthenfromatomsonacrystallattice.Allgasparticles(atomsand/ormolecules)orscatterersinthegasareatrandompositionsatany不得转载谢谢合作LWM 38DIFFRACTIONinstant,andsothephasesofthescatteredemratthedetectorarealsorandom.Theinten-sityatadetectorisgivenasI=EE*(3.9)whereEistheelectricfieldoftheemr.ForNscatterersinthegas2IN=E(3.10)inthecaseofapuregasofonekindofatomormoleculewherethescatteringfromeachparticleisthesame.However,ifthephasesatthedetectorarecoherent,thenEforNscatterersisNEandtheintensityatthedetectorisgivenas222IN=()EE=N(3.11)20Thehugedifference,incoherence,ofcourse,becomesevidentwhenNisoforder~10.Butevenasmallamountofscatteredradiation(itisaninefficientprocess)ismadesignificantbycoherency.Thisfactrendersdiffractionanimportantandmeasurablephenomenon.3.3.1CoherentScatteringfromanElectronTheX-rayemrpropagatesfrom-xto+xandinteractswithanelectronatO,asshowninFigure3.5.Thescatteredintensity,I,ismeasuredatapointintheXZplane,P,adis-tanceOP(OP=r)fromtheelectron,andatanangle2qdeterminedfromthedirectionoftheincidentradiation,I0,from-xtowardxtothedirectionOP,orsimplyitistheanglebetweenthetransmittedandscatteredradiation.bistheanglebetweenXandPdirectionsandis90°inthisexample,sincePwaschosentobeintheXZplane.Theoscil-latingelectricfieldoftheemrintheYZplanewithrandompolarizationinducesanoscil-lationintheelectronwhichinturnre-emitstheemr.ThecoherentscatteredportionofthiseventisgovernedbytheThompsonequation:+ZPa2qO-X+Xb+YFigure3.5Electromagneticradiationtravellingfrom-XtoX,stikesandelectronatO.Theradiationismea-suredatPintheXZplane.不得转载谢谢合作LWM 3.3THESCATTERINGPROBLEM392IKsin()a0I=(3.12)2r2whereKisaconstantthatisproportionalto1/mwheremhereistheelectronmass,andaistheanglebetweenthedirectionofvibrationoftheelectricfield,E,oftheincident-302emr,andthescatteringdirection,OPinFigure3.5.WithKª7.94¥10m,Ip/I0ª7.94-26¥10at1cmintheforwarddirection.Accordingtothisrelationship,foremrpropa-gatinginthexdirection,I=0ata=0(intheZdirection).Foratransversewave,char-acteristicofanXray,theEfielddirectionisorthogonaltothepropagationdirection,whichinFigure3.5istheYZplane.Thusthereis0scatteredintensityintheplanenormaltothepropagationdirection.Themaximumscatteredintensity,wheresin(a)=1at90°and270°,isintheforwardandbackwardscatteringdirections,+OXand-OX.-ForI0intheOXdirection(comingfrom-X)andencounteringaneatO,wecancal-culateIatPintheXZplane.(NotethatforsimplicitythepointPisintheXZplane.Inrealitythedetectorcanbeanywhere.)ConsideranunpolarizedI0thathasrandomEoscillatingintheYZplanebutpropagatinginthe+Xdirection.Thisbeamcanberesolvedintoyandzcomponentsas222EEE=+yz2212EEE=+(3.13)yz21III==oyoz02WecanwriteK2KII==sinbbI,for=90∞pyoy2oy2rr(3.14)K2II==sinaa,90∞-2qpzoz2rwiththeidentitysin(a-b)=sin(a)cos(b)-cos(a)sin(b)andwitha=90°andb=2q,22sin(a)=cos(2q).UsingtherelationshipaboveforI,weobtain2KÈ12+cosq˘IIII=+=(3.15)ppypz02Í2˙rÎ˚ThisistheThompsonequationforscatteringfromanelectron.Theimportantmessageisthattheintensitymeasuredatanypositionisafunctionofqsimplyduetothenatureofthescatteringphysicsoftransverseelectromagneticradiationwaves.Alsoscat-teringintheforwardandreversedirectionsisstrongestandweakestat90°tothedirec-tionofpropagation.AsaresulttheintensitiesatspecificBragganglesmustbecorrectedforthisqdependencebeforetheintensitychangesduetoatomicpositioncanbedetermined.ItisalsopossibletohaveincoherentorComptonscatteringfromanelectron.Thensomeoftheenergyoftheincidentphotonwillconverttokineticenergyfortheelectron.Theincreaseinlfortheincidentphotonwilldependonthescatteringangle:不得转载谢谢合作LWM 40DIFFRACTION2Dl=00486.sinq(3.16)Withrandomphase,comptonmodifiedradiationisproduced,sothereisnodiffraction.Incoherentscatteringisusuallyaproblemforlooselyboundelectrons(typicallyoccur-ringforatomsoflowatomicnumber,Z,e.g.,C),whichformabackgroundtocoher-entlyscatteredintensities.3.3.2CoherentScatteringfromanAtomScatteringfromanatomiseasilyunderstoodbyconsideringthatanatomhasanumberofelectronsassociatedwithitandeachatomwillscatteremr,asdiscussedabove.Theradiationcoherentlyscatteredfromthenucleusisverysmallbecause,asmentionedabove,2Kµ1/mandissmallfortherelativelylargemassofthenucleuscomparedtotheelec-tronmass.AnatomicscatteringfactorfisdefinedasIatomf=(3.17)IelectronEquation(3.17)comparestheatomicscatteredintensitytotheintensityscatteredfromasingleelectron.fisgivenbytheatomicnumber(Z)forscatteringinthedirectionofpropagationoftheemr,sinceZdeterminesthenumberofelectrons,andinthatdirec-tionnophasedifferencesexistfortheemrscatteredfromtheelectronsofanatom.InFigure3.6thisplaneislabeled11¢.However,atallotherangles(e.g.,plane22¢)fisreducedfromZaccordingtotheangleq(assinq)andthewavelength,l,bythefactorsin(q)/l.Shorterwavelengthsexacerbatethephasedifferencesatanyangle.Tablesofatomicscatteringfactorsfortheelementsareavailable,andFigure3.7isaplotoftheatomicscatteringfactorforCuandAl.3.3.3CoherentScatteringfromaUnitCellScatteringfromaunitcellisastraightforwardproblematthispoint.Allweneedtodoistokeeptrackofthescatteringfromtheatomsintheunitcellattheuniqueposi-tionsinthecell,andthephasedifferencesthatoccurduetotherelationshipofoneatominthecelltotheothers.Weformulatetheproblem,again,withthehelpofFigure3.3wheremonochromaticX-rayradiation(rays1,2,3)isincidentatqonthreeparallel2A12¢2qB1¢Figure3.6X-rayscatteringfromanatomthathasseveralelectrons.Theincidentradiationtravelsfromtheleft,scattersfromtwoelectrons(smallfilledcircles),andismeasuredatplanes11¢and22¢.不得转载谢谢合作LWM 3.3THESCATTERINGPROBLEM4130CuZ=2920fAlZ=131000.5sinqlFigure3.7AtomicscatteringfactorsfforCuandAlasafunctionoftheratioofangletowavelength.planeswithatomsatpositionslabeledO,E,andB.Coherentlyscatteredradiationisalsomeasuredatqandshownas1¢,2¢,and3¢.The2-Dunitcelldimensionsareaandc.Firstwecalculatethepathdifferencebetween1and3asd13.Thepathdifferenced13isgiveninFigure3.3asABCandwillresultinconstructiveinterferenceaccordingtoBragg’slawwhendq==ABC2dsin()=l(3.18)13001Noticethatthedistancebetweenthebottomandtopplanesistheclatticeparameterandsocanbeindicatedasd001.NowconsidertheeffectoftheplanewithatomEthatishalfwaybetweenOandBhasonthissituation.EisxawayfromOandxawayfromB.IfonenoticesthesimilartrianglesinFigure3.3(ODEsimilartoOABandOEFsimilartoOBC),thenweobtainthefollowingEFOEDEOEDE==,=(3.19)BCOBABOBBCWhenusedinABC,theequationaboveyieldsOEOEDEF==2BCABC(3.20)OBOBWithABC=lasgivenabove,weobtainÈ˘OEÍx˙dl12===DEFÍ˙l(3.21)OBÍÊcˆ˙ÎËl¯˚不得转载谢谢合作LWM 42DIFFRACTIONNowitisusefultoconvertthecalculatedpathdifferencestoangularphasedifferences.Thisisdonesimply,usingtherelationshipthatonewavelength(l)traverses360°.Fromtheconversion360°=2pradians,thephasedifferencefisthengivenasÊdˆf=2p(3.22)Ël¯whered/listhefractionof2pradiansyieldingtheangularphasedifference.ThephasedifferencebetweenOandEisthenreadilycalculated:Êd12ˆÊlxˆf=22pp=(3.23)12Ël¯Ëc¯In3-D,letu=x/a,v=y/b,andw=z/c,whichyieldsfp=2lw(3.24)12111wheretheu,v,warethecoordinatesoftheatoms(recallthemajortranslations–,–,–,222...,fromChapter2).Thiscanbegeneralizedforthehklreflections,thatis,diffractionfromthehklplane(hkl):fp=+2()hukvlw+(3.25)Thisrelationshipforthephasedifferencefisthephasedifferenceforapointwiththepositionsx/a,y/b,z/c,orasabove,u,v,w,andtheoriginfortheh,k,lreflection.Wenowcanproceedtoquantifythephasedifference,f,foranyoccupiedatomicposi-tionbyanyatomwithscatteringability,f.Tocompletethisproblem,weneedtoaddtogetherallthef’sforalltheuniqueatomicpositionsinagivenunitcell.Wereturntotheuseoftheuniquelatticepositionsfromwhichscatteringwillalsobedifferentforthedifferentuniquepositions.Oneefficientwaytoaccomplishthetaskofsimultaneouslyaddingtogethertwoquan-tities,amplitudeandphase,istousecomplexarithmetic.Theaddedfeaturefortheuseofcomplexnotationistheabilitytouseexponentials,andthiswillbeshownbelow.Wefirstconsiderthatperiodicwavescanhavedifferentphases(faandfb)andamplitudes(aandb),asisshownfortwowavesinFigure3.8.NotethatthewavesneedtohavetheabffabFigure3.8Twowaves,aandb,eachwithdifferentamplitudeandphase.不得转载谢谢合作LWM 3.3THESCATTERINGPROBLEM43iYabfafbROXFigure3.9WavesaandbfromFigure3.8canberepresentedascomplexnumberswithdifferentprojec-tionsontherealRandimaginaryiaxes.samewavelengths.Figure3.9showsthesetwowavesdisplayedonthecomplexplaneasvectorsaandbwithalengthanddirection.ForwaveainFigure3.9,theprojectionontherealaxisisOXandontheimaginaryaxisisOY.ThecomponentsaregivenasOX=acos()ffandOY=iasin()(3.26)aaandtheresultantisaicos()ff+asin()aaxItcanbereadilyshownthatforthesubstitutionofpowerseriesforcos,sin,andewecanobtainfortheresultantwave:aeifa=acos()ff+iasin()(3.27)aaifwhichderivesfromtheidentitye=cosf+isinf.Usingthisexpression,wenowsubstituteforaandffromthecalculationsabove:if2pihukvlw()++ae=fe(3.28)Foraunitcellweneedtoconsiderscatteringfromallthedifferentatomicsites.Whendone,thisyieldstheso-calledstructurefactor,F,asthesumoverallthensites:Ff=Âe2pihukvlw()nnn++(3.29)hklnnTheintensityiscalculatedintheusualwayfromtheelectricfieldasI=EE*orFF*3.3.4StructureFactorCalculationsThesimplestproblemtoaddressisaprimitivecubic.ThefirststepinthecalculationofFistodeterminetheuniqueatomcoordinates.Foraprimitivecubic,(u,v,w)=(0,0,0).不得转载谢谢合作LWM 44DIFFRACTIONIfweassumeonlyonekindofatom(withthereforeonekindoff),thenthestructurefactorabovefromequation(3.29)isgivenas2000pih()◊+◊+◊kl0Ffhkl==Ânefe=1(3.30)n22forallvaluesof(h,k,l).ThenwecanproceedtocalculatetheintensityI=F=f,whichisthesameforallhkl,all(hkl)planesdiffractandwithequalintensity.11Tothisprimitivecubiccellweaddabasecenteratomatthecoordinates(–,–,0).Thus,2211forthiscasetherearetwouvw’sat0,0,0and–,–,0.ForthissituationFfromequation22(3.29)givenas2000ppih()◊+◊+◊kl21i()21h+2kFf=+efehklppihk()+ihk()+=+ffe=+fe()1(3.31)Tointerpretthisresult,wesimplyusetheresultnipne=-()1(3.32)(evenpi)(oddpi)Thisrelationteachesthatfornevene=1andfornodde=-1.22Returningtotheproblem,wehaveforthesum(h+k)even,F=2fandF=4f,butfor(h+k)oddF=f(1-1)=0.Thevalueforldoesnotmatterbecauseitwasalwaysmultipliedby0andremovedfromF.ItisusefultomakeatableofallowedplaneindexeswiththeselectionrulesdevelopedfromF(thiswayonlyplaneswithh+kevenwillyieldanintensityandallintensitieswillbethesame).AllotherplanesyieldI=0.Inordertoachieveh+keven,handkneedtobeeitherevenoroddbutnotmixed:11x,22x,33x,44x,55x,35x,24x,butnot12x,23x,34x.Herexisanyintegerand,ofcourse,0isneutral.111NowweconsideraBCCwithuvwof0,0,0and–,–,–,andthisyields22220ppii2121212()h++klFf=+efepihkl()++=+fe[]1(3.33)ThenwededucethatF=2fforh+k+leven,andF=0forh+k+lodd.Thisresultdoesnotallowthe(100)todiffract.Recallthatthiswastheexampleweconsideredwhenweputaplaneinbetweenadiffractingsetofplanesatd/2.ThelastexampleillustratedhereistheFCCwhereuniqueatompositionsareatthe111111following(u,v,w):(000),(––0),(–0–),(0––).ThisyieldsFwithfourterms:22222220ppii2()h22+ki2p()h22+li2p()k22+lFf=+efe+fe+fepppihk()+ihl()+ikl()+=+fe[]1+e+e(3.34)22Ifhklarealloddoralleven,thenthesumofanytwoisevenandF=4fandF=16(i.e.,thesumoftwooddortwoevenintegersisaneveninteger).Ifhklaremixed,thentherearetwooddortwoeven.Whentakeninpairstheywillyieldthreesums,twoofwhichwillbebetweenanevenandanodd.Theresultwillbetwooddsumsandoneeven.Thustwo-1’saddedtotwo+1’swillyieldF=0as不得转载谢谢合作LWM 3.4RECIPROCALSPACE,RESP45Table3.1Resultsofstructurefactorcalculations222h+k+lPCBCCFCCDC1(100)———2(110)(110)——3(111)—(111)(111)4(200)(200)(200)—5(210)———6(211)(211)——7————8(220)(220)(220)(220)9(300),(221)———10(310)(310)——11(311)—(311)(311)evenpppiiioddoddFfe=+[]1++ee=+f[]11110-+-=(3.35)Table3.1summarizesthestructurefactorcalculationsforthePC,BCC,FCC,andDCcubicstructuresforlow-indexplanes.3.4RECIPROCALSPACE,RESPItisalwaysadvantageousinsciencetodefinecoordinatesthatmatchcloselythesystembeingstudied.Forexample,whenstudyingasystemwithsphericalsymmetry,theapplic-ableformulasarealmostalwayssimplifiedwiththeuseofsphericalcoordinatesratherthanCartesiancoordinates.Familiarproblemsinclassicalandquantummechanicsinvolveatransformofcoordinatestoreducethealgebraandarriveatthesimplestfor-mulas.Reciprocaldistancesaretheappropriateunitstousetoconnectthediffractionphysicsdiscussedabovewiththeexperimentalresults(theexperimentaldiffractionpat-ternsshowninFigure3.1).3.4.1WhyReciprocalSpace?FromthediscussionofcrystalstructuresinChapter2,wearrivedattheconclusionthataspecificstructurehaslatticeparametersassociatedwiththeunitcellsizeandshape,andinterplanarspacings,orso-calleddspacings.UptothispointinChapter3wehavedis-cussedhowemrandmatterwaves,inparticular,Xrays,interactwiththearrangementofatomstoyielddiffractioneventsasreinforcedintensitiesatspecificangularlocations,andwithspecificintensitiesrelativetotheincidentradiation.Thefundamentalrelation-shipunitingplanarindexes,planarspacings,andthemeasurablediffractionanglesisBragg’slaw.Whenrearrangedappropriately,thisrelationshiprevealsthenaturalcoordi-nates,whichallowthesimplestinterpretationofadiffractionexperiment.Specifically,itisseenthatthemeasuredBragganglesareproportionalto1/d,sorewritingBragg’slaw(equation3.5),weobtainthefollowingformulaintermsofreciprocaldspacings:1sin()ql=◊n(3.36)hkl2dhkl不得转载谢谢合作LWM 46DIFFRACTIONFurthermorefromequation(3.6)foracubicstructure,1/disproportionalto1/aas22212nl()hkl++sin()qhkl=◊(3.37)2a0ThusthemeasuredBraggangles,q,arefoundtobedirectlyrelatedtoreciprocaldis-tancesinthelattice.Whileforthecaseofsimplestructuresthisfactmaynotseemsoimpressive,forcomplexstructuresandmanyotherideasinsolidstatephysics,notablyelectronicenergybandstructure(tobediscussedinChapter9),reciprocalspaceratherthanrealspaceprovidesthemostsuitablecoordinateswithwhichtodescribeinterac-tionsofemrandelectronswithacrystallinematerial.Thiscasewillbedemonstratedthroughoutthistext,andthereasonsarethosesimplystatedabove:thatthesuperposi-tionofwavesdependsonthereciprocalsizeoftheunitcell.Whetherthewavesaremass-lesswaves,suchasX-rayphotonsormatterwavesderivedfromparticleswithmasssuchaselectrons,islargelyimmaterialtodescribingthediffractioneffectsthatoftendomi-natetheinformationcarryingabilityoftheinteraction.Whilereciprocalspace(RESP)istremendouslyimportantfortheunderstandingandsimplificationofmyriadproblemsofphysics,thisnewspaceisnotintuitiveasisrealspace.HencemostpeoplestudyingRESPforthefirsttimeandevenformanytimesthere-aftercannotreally“feel”thenewcoordinatesystem.Whilesomewhatuncomfortable,thisshouldnotresultindespair.Muchistobegainedfromsimplyrememberingthedef-initionsandlearninghowtomanipulateandcalculateusingtheideasofRESP.Asfamil-iarityincreases,sowillonesabilityto“feel”thisconcept.Forthoseofuswhohavespentagoodlyfractionoftheirlifetimestudyingvariousaspectsofmaterialsphysics,thisdis-comfortwithnewideasiscommonplace.However,itistobetoleratedwithpatience,andtherealizationthatwhenhugeamountsofinformationarevirtuallyperfectlyunder-stood,simplybychoosingthenaturalcoordinates,thenthepowerandrelevanceoftheideasandtheimpactonscienceisalsounderstood.3.4.2DefinitionofRESPFirstareciprocallattice,REL,isdefinedbyreciprocallatticevectorsa*,b*,andc*using()bc¥()ca¥()ab¥a*,=b*,=c*=(3.38)abc◊¥()bca◊¥()cab◊¥()Figure3.10ashowsthatthemagnitudeofthevector(b¥c)istheareaoftheplane(shaded)definedbytherealspacevectorsbandc,andthedirectionisnormaltothebcplane.a·(b¥c)isthevolumeofthesoliddefinedbya,b,andc.Sotheareaofthebaseofa3-Dfiguredividedintothevolumeyieldstheheight.Thisheightisperpendiculartothebasalareaandhasamagnitudeof1/a*andforacubicsystem:abcbcacab***^^^and,and,and,Thesolidfiguredefinedbya,b,andchasabasalarea(b¥c),andthevolumea·(b¥c)canbethoughtofasaunitcellboundedontopandbottombyplaneswithindexes(hkl)andwitharea(b¥c).Thedistancebetweentheplanesistheinterplanarspacing,dhkl,不得转载谢谢合作LWM 3.4RECIPROCALSPACE,RESP47a)a*cbb)a*(100)dq(100)cab˚–1c)Ac*(hkl)r*a*O˚–1Ab*˚–1AFigure3.10Reciprocalspace(RESP)representations:(a)Definitionofa*;(b)a*inanon-orthogonalsystem;(c)reciprocallatticevectorr*inRESP.whichistheheightofthe3-Dfigureandistheperpendiculardistancefromthebottomtothetopplane.Now,aswasdeterminedabove,thisheighthasthedirectionofa*,11a*==(3.39)adcosq100Fororthogonalvectorsa,b,andc,theheightalsohasthemagnitudeofaor1/a*.ThisgeometryisclarifiedfurtherbyconsideringatricliniccellinFigure3.10bwherethetopplaneisthe(100)plane,andd100or1/a*isshownatanangleqtoa.Thevolumeofthis不得转载谢谢合作LWM 48DIFFRACTIONtricliniccellistheareaofthebase(b¥c),dottedbytheheightwhichisd100or|a|cosqor1/|a*|.Someotherrelationshipsare1aa*,=◊*a=1d100butabc*^andthusab**◊=0,ac◊=0Sincecos90°=0,11cb*,==*dd0010103.4.3DefinitionofaReciprocalLatticeVectorItisusefultodefineareciprocallatticevector,r*,asfollows:rabc*=++hkl***(3.40)hklr*isdrawnfromtheoriginofRESPtothehklpoint,whichisaplaneinrealspace(hkl)asisshowninFigure3.10c.WiththehelpofFigure3.11itcanbeshownthat1rr*^()hkland*=(3.41)dhklFromFigure3.11theplaneABCisdefinedinrealspacewithindexes(hkl).Sothefol-lowingrelationshipsobtainamongthedefiningvectors:OAAB+=OBandOAAC+=OC(3.42)Ifr*is^toplaneABC,thenr*isalso^toallthelinesintheplane,AB,AC,andsoon.WecanexpressthevariousvectorsintermsoffractionalinterceptsasbacaAB=-=-OBOAandAC=-=-OCOA(3.43)khlhCc/ldNOb/kBa/hAFigure3.11NormalNtoplaneABCwithfractionalintercepts.不得转载谢谢合作LWM 3.4RECIPROCALSPACE,RESP49Thenwecanformthescalar(dot)productsofr*withlinesintheplane,inordertotesttheorthogonality:Êbaˆra**◊=++AB()hkb*lc*◊-=-++=1100(3.44)Ëkh¯Êcaˆra**◊=++AC()hkb*lc*◊-=-++=1100(3.45)Ëlh¯Thusr*is^totwolinesintheABCplane;hencer*mustbe^totheplaneABCor(hkl).WithdhklasthedistancefromtheoriginOtotheplaneABC,weobservethatdistheprojectionofa/hontheABCplanenormalN.Wehavethefollowingrelationships:ÊaˆÊarˆÊ*ˆ1d=◊=N◊=(3.46)hklËhh¯Ë¯Ërr*¯*Fromtheseequalitieswecanseeclearlytheimportanceofr*asthevectorfromtheoriginofreciprocalspacetoanhklpointwithmagnitude1/d;henceitisrelatedtotheinter-planarspacings.Furthermorethehklpointisaplaneinrealspace.Anexampleforthecubicsystemisthedistanceformulaequation(3.6):2221()hkl++=(3.6)22da0ItwasderivedinChapter2usingdirectioncosines.Althoughthismethodissatisfactoryforthecubicsystem,itbecomesunwieldyformorecomplicatedcrystalsystems.Since|r*|=1/d,wecanusetheexpressionrrrr*◊=***cosq(3.47)andsubstituteequation(3.40):rabc****=++hkl(3.40)Theresultis222222()hklhklhklabcabcabc*********++◊++()=++2222222()hkl++1=++()hkla*==(3.48)22ad0ifa=b=candthusa*=b*=c*,andforacubicsystema*=1/a.Thustheuseofrec-iprocalspacesimplifiesthederivationofthedspacingformulas.Fortheothercrystalsystemstheaxesmaybeunequalandnotorthogonal.Thedirectionsforthevariousr*expresstherelativeplanarangles,astheanglesbetweenr*.Withthisknowledgeonecanderiveinterplanarangleformulas.Forexample,forplanes(h1k1l1)and(h2k2l2)theinterplanaranglecanbecalculatedasfollows:不得转载谢谢合作LWM 50DIFFRACTIONrrrr****◊=cos()q(3.49)1212whereqistheanglebetweentheplanes.Weknowthatrabc*=++hkl***andrabc*=++hkl***11112222andfromabovethata*·a*=1,a*·b*=0,anda*·c*=0.Themagnitudeforthevectorr*222r*=++hkl(3.50)Theinterplanarangleformulaforthecubicsystemiswrittenrr**◊hh++kkll12121212cosq==(3.51)222222rr**◊hklhkl1++112++2212ItisusefultoconstructaRELinRESP.Fromthedefinitionsaboveseveralpointsareusefultoguidetheconstruction:1.ThesymmetryoftheRELmustbethesameasthatofthereallattice.2.PlanesinrealspacearepointsinRESP.3.DistancesinRESParereciprocaloftherealspacedistancesfororthogonalsystems.4.r*hklisthevectorfromtheoriginofRESPtotheplane(pointinRESP)andequals1/dhkl.WecommencetheconstructionbyarbitrarilyassigningtheoriginofRESPasOinFigure3.12.Wealsorestricttheconstructiontoacubicreallatticewitha0=2Å.Thusthe-1axesinFigure3.12areinunitsofÅ.Forthe(100),therealspacedistanceis2Åwhile-1a*=0.5Å.Thisdistanceisthesameforthe(010)and(001)inacubiclattice.ThethreeplanesareindicatedaspointsontheRELinRESP.Notethatforthe(200)inrealspacethedistanceishalfthatofthe(100)or1Åfromthedspacingformulaaboveand-11/dis1Å,andthe(200),(020),and(002)planesareshown.Forthe(300)plane,-1d=0.67Åand1/d=1.5Å,andtwooftheseplanesareshown.Thecalculationsaresum-2marizedinTable3.2usingthepreviousformulafor1/dfromTable2.4andrecallingequa-tion(3.41)that|rhkl*|=1/dhkl.FromtheseplanesitiseasilyseenthatastheMillerindexesincrease,theplanes(pointsinRESP)increaseindistancefromOforRESP.Thisisexactlytheoppositeofrealspace.AnumberofotherplanesarealsoshownintheREL.Insummary,theRELhasthesamesymmetryastherealspacelattice,butthedistancesarereversed.3.4.3TheEwaldConstructionTheRESPandRELideaspresentedaboveenabletherepresentationofdiffractioncon-ditionstobesimplified.Anaddedpieceofsimplificationtotheconceptsandimplica-tionsoftheRESPisanelegantgeometricalrepresentationofdiffractionbyEwald.Thesignificanceofthisconstructionisthatitgoesbeyondsimplyhelpingoneunderstandorsummarizediffraction.Itsgeometricrepresentationenablesonetoconstructandderive不得转载谢谢合作LWM 3.4RECIPROCALSPACE,RESP51–1A˚(003)(002)(112)(222)(001)(111)(011)(100)(200)(300)O–1A˚0.51.01.5(010)(110)(210)(020)(120)(220)–1A˚Figure3.12Amapofreciprocalspace(RESP)showingselectedplanesaspointsinRESP.Table3.2CubicREL,a0=0.2nmhkl1/d2nm-2|r*|=1/dnm-1(Å-1)(100)255(0.5)(200)10010(1)(300)22515(1.5)(400)40020(2.0)(110)507.07(0.7)(220)20014.1(1.4)(330)45021.2(2.1)(440)80028.3(2.8)(210)12511.2(1.1)(111)758.7(0.9)(112)15012.2(1.2)(222)30017.3(1.7)alltheexperimentaldiffractionmethodsdirectlyandsimplyintermsofageometricconstruction.Essentiallythisconstructionconsidersthegeometricalconsequencesofemrormatterwavesincidentwithmagnitude1/landdirectionS0(aunitvector)astheincidentvector不得转载谢谢合作LWM 52DIFFRACTIONS0/l,interactingwithafixedREL.Figure3.13showstheEwaldconstructionusinga2-DarrayofRELpoints.ThesepointsinRESPareofcourseplanesinrealspace.WecanconsideranyRELpointastheoriginfortheconstructionandlabelthispointO(attheleftoftheREL).Foranywavelengthl,weconstructacircle(in2-D)withradius1/l,thattouchesO(in3-Dthisconstructionyieldsasphere).WiththecenterofthecirclelabeledC,welabelthelineCO,asthevectorS0/lpointingfromCtoO,meaningitorig-inatesatCandterminatesonO.AnydirectioncanbechosenforS0.Inreality,andasdiscussedbelow,thisdirectionisexperimentallydeterminedbythedirectionoftheinci-dentemrbeamfromthemachinethatproducesthebeamofXraysthatimpingeonasample.FromOdrawavectorOPtoanyotherpointontheRELthatliesexactlyonthecircle.NoticethatOPtopointPhklisaRELvector,r*hkl(bydefinition).Thediffractedbeam,S/lcompletestheconstructionwhendrawnfromCtoP.ItshouldbenotedthatS/lhasthesamemagnitudeasS0/l,sincebothareradiiofthesamecircle,butthesevectorshavedifferentdirections.Nowwewillseewhatinformationissummarizedbythisconstruction.Fromthisconstructionthereareseveralrelationshipsthatareusefultowritedown:SS-OP==r*0(3.52)hkll1r*sin()q=2(3.53)1lRecallfromequation(3.46)thatr*hkl=1/dhkl.Substitutingthisintoequation(3.53),weobtainBragg’slaw:l=2dsin()q(3.5)WithpointPhklontheperimeterofthecirclewithradius1/l,Bragg’slawissatisfied.Thismeansthatthe(hkl)planerepresentedbyPhkldiffracts,yieldingadiffractedPhklr*hklS/lq-S0qOS0/lCFigure3.13Ewaldconstruction(circle)in2-DRESP.不得转载谢谢合作LWM 3.5DIFFRACTIONTECHNIQUES53beamat2qrelativetotheincidentbeam.Ingeneral,allthepointsthatlieontheEwaldcirclewilldiffractatdifferentanglesrelativetotheincidentbeam.Becausethiscon-structioncanbemade,withthesevererestrictionthatanr*existsonthecircle,lisappro-priatetotheRELandsowilleffectdiffractionfromcertainRELpointsattheappropriatedirection(s).Itmaybethecasethatwithaparticularchoiceofdirectionandwavelengthfortheincidentbeam,therearenoRELpointsthatlieonthecircle.ThepointsPhklanddirections2qareidentifiedintheconstructionafterthevectorS0/lisidentified.TheEwaldconstructionin2-Dyieldsadiffractioncircle,andforausualdif-fractionproblemin3-D,adiffractionsphereisobtainedthatisusuallycalledanEwaldsphere.3.5DIFFRACTIONTECHNIQUESThegreatpoweroftheRESPrepresentationalongwiththeEwaldconstructionismanifestinthesimplicitywithwhichalltheexperimentaldiffractiontechniquesarerepresented.TheexperimentalmethodsaredirectlydistinguishedbyusingtheRESPrep-resentationandconsideringthedifferentwaysRELpoints,namelytheplanesinrealspace,arebroughtontothesurfaceoftheEwaldsphere(i.e.,broughtintodiffractingconditions).AmongtheparametersthataffecttheEwaldspherearethewavelength,l,whichdeterminestheradiusofthesphere1/l,andthedirectionoftheincidentradia-tionormoregenerallytheorientationoftheEwaldsphererelativetotheREL.BelowonlyseveralprominentX-raydiffractiontechniquesarepresentedfromthepointofviewofhoweachtechniquemanipulatestheRESPandEwaldspheretoenablediffraction.Noattemptismadetotreattheexperimentaldetails.3.5.1RotatingCrystalMethodThismethodisusedinanumberofmanifestationsfordeterminingentirecrystalstruc-tures,inparticular,theshapeandsizeofunitcellsandatompositions.ThismethodalsonicelyillustratesthepointmadeaboveabouttherelationshipofthesizeandorientationoftheEwaldsphererelativetotheREL.ThetechniqueusesmonochromaticXraysandrequiresasinglecrystalofmaterial.AsshowninFigure3.14thecrystalinthefigure,asrepresentedbyitsREL,isbeingrotatedaboutanaxis.PriortorotationonepointoftheRELtouchesthesurfaceoftheEwaldsphere.WhentheRELrotatesotherplanes(pointsinRESP)eventuallytouchthesurfaceoftheEwaldsphereandgivesrisetodiffractionspotswithdiffractionvectorsS/lonthesurfaceofacone.ItistypicaltorotatearoundseveralaxestoproduceamapofRELanddeducesymmetryandstructurefromthesym-metryrelationshipsamongthespotsandrelativeintensities.3.5.2PowderMethodThepowdermethodalsousesmonochromaticXrays,butinthismethodthesampleisfinelyground.Imaginethatonestartswithasinglecrystal,andasthisisgroundtoafinerandfinerpowder,thecrystalsgetsmallerandsmaller.Inapinchoffinepowderonehasalargenumberofpossiblecrystalorientations(notreallyalltheorientations,whichwouldbeideal,butahugenumberinafinepowder).HencealltheseorientationsoftheRELofthematerialcanbesimultaneouslypaintedbytheincidentX-raybeamofseveralmmindiameterwithouttheneedforrotation.Thisexperimentalsituation,不得转载谢谢合作LWM 54DIFFRACTIONFigure3.14Rotatingcrystalmethodinwhichamaterialsreciprocallattice(REL)isrotatedsothatRELpointsintersecttheEwaldsphere(shaded)ofradius1/l.2/lS/l0OFigure3.15PowdermethodwhereanEwaldsphere(shaded)ofradius1/lisrotatedaboutOtoproducediffractionfromvariousoritentationsofcrystalsinthepowder.showninFigure3.15,canbethoughtofashavingtheRELfixedandS0/lrotatingabouttheoriginO.Thiswayanewsphereofradius2/lisgenerated,calledthelimitingsphere,thatcontainsallthepossiblespheresofreflectionandthusdefinesallpossiblehkl’sforltodiffract.Thismethodproduceswhatarecalledconesofdiffraction.ThereasonthatitproducesconesisthatnotonlyarealltheBragganglesqproducedbythefinelygroundcrystallitesintheincidentemrbeamatonce,butdifferentrotationsabouttheincidentbeamfarealsopresentincrystallitesateachq.Thisresultsinaconeofdiffraction.Ifafilmstripisplacedtointercepttheconesandrecordthediffractedradiation,arcs,calledDebyearcs,areseenonthefilm.Figure3.1ashowsarepresentationofafilmstripusedinonekindofpowderdiffractiontechniquewiththeDebyearcs.Fromthepositionsofthearcs,hklindexesareobtainedaswellasdspacings,fromwhichtheidentityofthe不得转载谢谢合作LWM 3.6WAVEVECTORREPRESENTATION552/lSWLO1/lpFigure3.16LauemethodwhereradiationwithwavelengthfromtheshortestlimitlSWLtotheadsorptionedgeoftheemulsionlpdefinethoseEwaldspheresintheshadedregionthatyielddiffraction.powdercanbededucedbycomparisonwithavailablelibrariesofpowderpatterns.Thecirclesare,inreality,holesinthefilmthroughwhichanincidentX-raybeamentersandleavestheapparatus.3.5.3LaueMethodIntheLauemethod“white”radiation(manyl’s)isusedwithsinglecrystalsamples.Thismethodistypicallyusedtodeterminetheorientationofaspecificfaceofacrystalrela-tivetotheincidentradiation.Aswewillseemanytimesinmaterialsscience,manyofthepropertiesofmaterialsaredeterminedbycrystallography.Thusitisimportanttoknowwhichcrystallographicorientationonwhichsinglecrystalpropertyisdetermined.Eachofthemanyl’sthatareusedinthetechniquedefinesasphereinwhichthereexistsafamilyofEwaldsphereshavingradiifrom1/lp,wherelpisthewavelengththatisabsorbedbythephotographicemulsionusedtorecordthediffraction(theKedgeforAgassumingthataAgphotographicemulsionisusedtorecordthediffraction)to1/lSWLfortheshortestwavelengthintheX-rayspectrum(theshortwavelengthlimit,SWL),andthissituationisdepictedinFigure3.16.Inessence,thesphereswithradiiintheshadedregionrepresentpotentialdiffractionspheres.TheavailablewavelengthsselectplanesfromtheRELwithwhichtodiffractforagivenorientationoftheRELatO.Fromthediffractionspotpattern,theorientationoftheRELisdeduced.Thismethodistypicallyusedtoobtainthecrystalorientation.Forexample,aSicrystalbouleispulledfromthemeltyieldingasausage-shapedSisinglecrystal.Itisnecessarytoslicethebouleintopre-ciselyorientedSiwafersformicroelectronicsprocessing.Afterafirstcutismade,theLauepatternfromtheflatboulesurfaceisusedtodeterminetheprecisesawingangleforalltheslices.Figure3.1bshowstheresultsoftheLauemethodusingasinglecrystal.TypicallyanexperiencedexperimentercanaccuratelyguessthecrystalorientationsimplyfromthesymmetryoftheRELonthefilm.3.6WAVEVECTORREPRESENTATIONThewavevector,orkspace,representationofreciprocalspaceisimportantforelectronenergybandstructures(tobeintroducedinChapter9).However,theRESPdevelopment不得转载谢谢合作LWM 56DIFFRACTIONGk'qqCkOFigure3.17kspacerepresentationofRESPwhereG,k,andk¢areanalogoustor*,S0/l,andS/l,respectively.presentedabovesetsthestageforthisrepresentation.kspaceisobtainedbysimplyexpandingRESPbythefactor2p.Togeneratethisspace,wedefineanewvector,k,calledthewavevector,21pʈkS==remember,S(3.54)llË0¯Nextwedefine,foranorthogonalsystem,222pppa*,*,===bc*(3.55)abcInRESPr*=ha*+kb*+lc*,butnowwiththeREL’sexpandedby2p,thenewRELvectorislabeledG,thediffractionvectorinkspace.ThesituationanalogoustotheEwalddiffractionsphereisshowninkspaceinFigure3.17wherekkG¢-=(3.56)DkG=(3.57)Squaringbothsidesyields222k¢=++k2kGG(3.58)2Theconditionfordiffractioniswhen2kG+G=0,whichmeansthatk=k¢andthatGliesatapointoftheREL.Thesolutionofthisquadraticisk=±G/2.ThusthewavevectorthatbisectsaRELvectorGwillbediffractedwitha*=(2p/a)x,wherexisaunitvector.Then,foralineararrayofregularlyspacedlatticepointsintheREL,thedif-fractionconditionisgivenas不得转载谢谢合作LWM 3.6WAVEVECTORREPRESENTATION57G2G2G1G1G1G1G2G2Figure3.18BrillouinzonesinkspacedefinedbythebisectorsoftheGivectors.Ê2pˆGa==n*nx(3.59)Ëa¯npk=±,n=12,,...(3.60)aElectronspropagatinginacrystallatticeoftenhavedeBrogliewavelengthscommen-suratewiththesubnanometricandnanometriccrystaldimensions.Thussomeoftheelectronwavesinamaterialwiththeappropriatewavelengthsanddirectionsinrecipro-calspacewilldiffract.Theelectronsofcertainenergyanddirectionthatdiffractyieldintensitiesoutsideofthecrystal,anditisasiftheseelectronsarepreventedfromprop-agating.Sothereareenergyregionsanddirectionsinwhichelectronscanpropagate,calledallowed“energybands”andenergyregionswhereelectronscannotpropagate.Theelectronsthatcannotpropagateareeffectivelydiffractedoutofthecrystal,andtheassociatedenergiesgiverisetoenergy“bandgaps.”Thebandgapregionsofelectronenergyaresaidtohavenoallowedenergystatestoandfromwhichelectronscanflow.(MorewillbesaidaboutallowedanddisallowedelectronenergiesinChapter9.)Thekspacerepresentationcanbeusedtorepresenttheseallowedzones,calledBrillouinzonesafterthescientistwhomadethisideapopular.TheconstructionofBrillouinzonesissimilartotheconstructionofaWigner-Seitzunitcell(seeFigure2.13),anditis不得转载谢谢合作LWM 58DIFFRACTIONdepictedinFigure3.18.Firsta2-Dcrystallatticeisrepresentedbythe2-DRELpoints.Anoriginischosenthatisoften,butnotnecessarily,aRELpoint.FromthispointGvectorsaredrawnfirsttothenearestneighborplanesandlabeledG1,andthentosuc-ceedingplanes,orRELpoints,andlabeledG2,G3,andsoon.Now,fromthediscussionabove,diffractiontakesplaceatG/2.SowebisectG1/2andextendthebisectorssothattheyintersectandfromanenclosedregion(shadedregion)aroundtheorigin.ThisregioniscalledthefirstBrillouinzone,andanelectroncanpropagateuptotheregion’sedgewhereuponitisdiffracted.NoticethatthisregionisalsoaprimitivecellcontainingonelatticepointinRESP.ThefirstBrillouinzoneinRESPisdirectlycomparabletotheWigner-Seitzprimitivecellinrealspace.SucceedingBrillouinzonesareobtainedbybisectingsucceedingGvectorsandconsideringtheareaenclosedlesstheareaoftheprecedingBrillouinzone.Interestingshapesareobtainedfor3-DRELs,andthisimpor-tantideawillberevisitedwhenthesubjectofelectronbandstructureiscoveredinChapter9.RELATEDREADINGE.D.Cullity.1956.ElementsofX-rayDiffraction.AddisonWesley,Reading,MA.Alleditionsofthisbookcontainvoluminousstructureinformationinreadabletextform,andinappendixestheX-raydiffractiontechniquesarediscussedatlength.D.McKieandC.McKie.1986.EssentialofCrystallography.BlackwellScientificPublications,Cambridge,MA.Athoroughtreatmentofcrystallographyandofdiffractionanddiffractiontechniques.J.M.Schultz.1992.DiffractionforMaterialsScientists.PrenticeHall,EnglewoodCliffs,NJ.Amoreadvancedtreatmentofdiffractiontheoryandpractice.EXERCISES1.ForAlpredictthediffractionangles(2q)forthefirstthreeplanesforl=0.1542nm.2.ForFCCCuwithatomicradiusof0.2552nm:(a)Calculatea0(b)GivenXraysofl=0.152nm,andmeasured2qvaluesfortwoCusamplesof42°46¢and41°55¢.Whichsampleispure?3.FromthefollowinginformationdeterminewhetherthestructureisBCCorFCC.Xrayl=0.171nm,2qvaluesare60°00¢and70°34¢.4.Constructa2-Drealspacelatticewithanassumeda0andthatshowsfiveplanes.ThenconstructtheRELshowingthesamefiveplanesandr*.Discussdifferencesindirectionsandspacingsinrealandreciprocalspaces,andrelateyourresponsewithChapter2,exercise4.5.YouareassignedtobuildanX-raymonochromatortocovertheenergyrangeof54about10–10eV.Designit,suggestingmaterials(justify)andgeometry(sketch).Emphasizeprinciples.6.Provethatthe(111)diffractsfortheFCCbutnottheBCC.不得转载谢谢合作LWM EXERCISES597.Forasimpleorprimitivecubic(PC),structurecalculatetheindexesforthe10lowestindexplanesthatdiffract.8.CalculatethefirstfivelowestindexplanesthatdiffractfortheBCCandFCC.DiscusswhytheplanesaredifferentforthePC,BCC,andFCC.9.Calculatethefivelowestindexplanesforthediamondcubic,DC,structure.Thisstructurehas8atomsperunitcelllocatedat:0001/21/201/201/201/21/21/41/41/43/43/41/43/41/43/41/43/43/4andsketchtheDCunitcell.10.FortheFCCmetalwitha0=0.51603nm(Ce):(a)Makelistofthediffractingplanes.(b)CalculatethediffractionanglesforCuKaradiationl=0.15418nm.11.UsingasketchofanEwaldconstructionfora2-DsquarelatticeinRESPandwithawavelengthlthattouchesatleastonepointoftheREL,explainwhatexperimentalchangesyouwouldneedtomaketogetanotherpointthatyouchoosetodiffract.12.Determine(calculate)whetherthe(111)diffractsinamonatomicBCCsolid.Thenshowhowthiscalculationwouldbemodifiedifthesolidwerenotmonatomic.13.DiscusswhatstructuralinformationyoucanandcannotobtainfromBragg’slaw.14.FromChapter2,Exercise16,whereyoudiscussedthestructuraldifference(s)betweenacompoundandanalloy,nowexplainthedifferencesyouwouldexpectfromdiffraction.不得转载谢谢合作LWM 不得转载谢谢合作LWM 4DEFECTSINSOLIDS4.1INTRODUCTIONInthediscussionsinChapters2and3theemphasiswasonperfectcrystallinesolids.AttheoutsetofChapter2abriefandcursorycomparisonofcrystallineandnoncrystallinesolidswaspresented.Asthestudyofmaterialsscienceproceedsthroughoutthistext,morereferencestoanddiscussionsaboutbothglassyandamorphousnoncrystallinesolidswillbemade.Noncrystallinesolidsofferliterallyalimitlessvarietyofstructures,andmanyofthemaredependentonspecificpreparationconditions.Ontheotherhand,crystallinesolids,inparticular,singlecrystals,arewelldescribedusingthe14Bravaislat-tices.Thusitisusuallyconsideredusefulinthestudyofmaterialstocommencewiththestructurallysimplestmaterials—singlecrystals—andthenproceedtowardgreaterstruc-turalcomplexity.Beforegettingtononcrystallinematerials,anotherideaiscrucialtoenableunderstandingofthestructuralnatureofbothrealcrystallineandnoncrystallinematerials.Thisideaisaboutdefects,thatdefectsoccurinallrealmaterialstosomedegreeandcanbeevokedtoexplainthedifferencebetweencrystallineandnoncrystallinesolids(whereinthelattercasethedefectlevelisveryhigh).Inadditiondefectsareinherenttomanyphysicalandchemicalpropertiesofsolids.Amongtheimportantpropertiesthatarelargelycontrolledbydefects,anddiscussedinthistext,arethefollowing:•Resistivity,r,throughcurrentcarrierscatteringfromdefects.•Conductivity,s,insemiconductorsduetosubstitutionallatticedefects.•Deformationandstrengthduetodislocations.•Nucleationofphasesandsitespecificchemicalreactionsatsurfacedefects.•Diffusionviadefectmechanisms.ElectronicMaterialsScience,byEugeneA.IreneISBN0-471-69597-1Copyright©2005JohnWiley&Sons,Inc.不得转载谢谢合作LWM61 62DEFECTSINSOLIDSTherearemanykindsofdefects.Onewaytosystematizethestudyofdefectsisbyacategorizationaccordingtothedimensionofthedefect:•0-DPointdefects:impurities,vacancies,interstitials.•1-DLinedefects:dislocations.•2-DPlanardefects:interfacialdefectssuchassurfaces,grainboundaries,andphaseboundaries.•3-DBulkdefects:voids,cracks,pores.Thesekindsofdefectsaredefinedanddiscussedbelowwithparticularattentionto0-Dpointdefectsand1-Dlinedefects.Butbeforewebeginthediscussion,weneedtolaytheimportantphysicalgroundwork.4.2WHYDODEFECTSFORM?Foranydefecttoformofthetypementionedabove,achemicalbondneedstobebroken.Thus,defectsraisethefreeenergyofaperfectcrystal,andtheymaynotformsponta-neously.Ifoneconsidersthatundernearequilibriumconditionsthesolidwillattainthelowestenergyconfiguration,thatis,theequilibriumcrystalstructureforcrystallinemate-rialsoranetworkstructurefornoncrystallinematerials,thenwhydovirtuallyallmate-rialsdisplayasubstantialnumberofdefects,particularlypointdefects?Oftenformostsolidsthenumberofdefectsislargeenoughtomeasurablyalter(raise)theenthalpyofthesolid.Otherobservationsthatexplainthenatureofdefectsincludethefollowing:solidsthatdisplaylowdefectconcentrationsaredifficulttopreserve,virtuallyanypro-cessingofperfectcrystalswillincreasethenumberofdefects,differentkindsofdefectsareassociatedwithdifferentproperties,underagivenprocesssomekindsofdefectswillincreasewhileotherwilldisappear.Howarealltheseobservationsreconciled?Agoodapproachtothisquestionistoturntothefieldofthermodynamics.Sobeforeproceed-ingtotheheartofthedefectsissues,wewillbrieflyreviewseveralrelevantprinciplesofthermodynamics.However,althoughthethermodynamicsofthedifferentkindsofdefectscanyieldanunderstandingofdefectsincrystallinesolidsthatcanleadtoaquan-titativedescriptionofmanyimportantproperties,theareaofdefectsinsolidsremainsanopenandfertileareaofmaterialsresearch.4.2.1ReviewofSomeThermodynamics4.2.1.1FirstLawofThermodynamicsSeveralrelationshipswedevelopherewillbeusedlater.Wefirstconsiderthesimplesystemofanidealgasatconstanttemperature(dT=0)thatundergoesanisothermalexpansionfromV1toV2,whereV2>V1.WecandependontheFirstLawforthissituation,whichassertstheconservationofenergy.ItiswrittenintermsofthechangeoftheinternalenergyEforasystem:dE=+=dqdw0(4.1)whereqisheatinto(oroutof)thesystemandwisworkdoneonorbythesystem.Thechangeinwork(donebythesystem)isdw=-pdV(4.2)ext不得转载谢谢合作LWM 4.2WHYDODEFECTSFORM?63foraconstantexternalpressure,pext.WorkisthenthepartoftheenergyEthatiscon-certedtoatask(onedoeswork).Strictlyspeaking,workisdefinedintermsofforce,F,anddisplacementdxaswd=◊ÚFx(4.3)Equations(4.2)and(4.3)areconcordant,sincep=F/AwithAthearea.However,thereisanothercomponentofE,namelytheheat,q,thatisnotfocusedtoperformatask.Heatisthecomponentoftheenergyfoundintherandommotionoftheatoms/mole-culesinamaterial.SoqisthatpartofEthatcannotbefullyharnessedtodirectlydowork.Apartofqisthenirrecoverable.Itshouldalsobeunderstoodthatthetempera-tureTisanindicatorofthedirectionofflowofheat.Heatflowsfromhightolowtem-perature.ThescaleforTisarbitraryandchosenforconvenience.NowassumingtheapplicabilityoftheidealgaslawnRTp=(4.4)VandusingthedifferentialformofFirstLawaswrittenabove,weobtainforareversibleprocessnRTdVdq=(4.5)revVUponintegrationofequation(4.5)fromstate1to2,theresulttobeusedbelowisÊV2ˆDqrev=nRTlnÁ˜(4.6)ËV¯14.2.1.2SecondLawofThermodynamicsTheSecondLawdealswiththeavailabilityofenergyonceasystemabsorbs(oremits)energy.Itisobservedthatalltheenergywithinasystemisnotlateravailable.Thisideawasmentionedaboveinrelationtotheheat,q.Thethermodynamicstatefunctionthatquantifiestheunavailablepartoftheenergyiscalledtheentropy,andissymbolizedbyS.TheuseofSandmoreimportantlyforachangeDSordScanyieldinformationaboutthedirectionofreactionsinwhichenergyisexchanged.WecommencewithafewstatementsaboutDS:DDDSSS=+>0(4.7)totsyssurforaspontaneousprocesswherethesubscriptstot,sys,andsurrefertototal,system,andsurroundings,respectively.Theinfinitesimalchange(asindicatedusingthesubscriptrevforreversible)inentropyforanisolatedsystemcanbeobtainedusingdqrevdS=(4.8)sysTToobtaindSsur,oneneedstoconsiderthespecificprocessconditions.Forexample,atconstantp,DH=qP.ThusdStotiswrittenas不得转载谢谢合作LWM 64DEFECTSINSOLIDSÊdHˆdS=-dS(4.9)totsysËT¯wherethe-dHindicatesthatenthalpyflowsoutofthesystemtothesurroundings.Thepointofreferenceistakeninthesystem.ThenthisexpressioncanbesimplifiedtoTdS=-TdSdH(4.10)totsysAndnowbydefiningdG=-TdStotandremovingthed’s,weobtaintheformulathatdefinesG,theGibbsfreeenergy:GHT=-S(4.11)ItshouldbenotedthatthisnewthermodynamicstatefunctioncalledtheGibbsfreeenergyisnothingmore(orless)thanthenegativeofthesumoftheentropyforthesystemandsurroundingsatconstantpressure.SinceitiswrittenintermsofHratherthanq,itisquiteusefulbecausevaluesforHarerelativelyavailableinTables.Weneedausefulexpressionforourpresentpurposes.Soweintegrateequation(4.8),using(4.5),toobtainDSsys:2dqrevÊV2ˆDSsys==Ú1nRlnÁ˘¯(4.12)TV1ThechangeinGibbsfreeenergy,DGisoftenusedtodeterminethespontaneityofachemicalprocessatconstantpressure(i.e.,inanopenvesselonearth):DDDGHT=-S(4.13)wheretheenthalpyH=E+pVisrelatedtotheinternalenergyandTistheabsolutetemperature.TheHelmholtzfreeenergy,A=E-TS,isusedsimilarlyforthelesscommonconstantvolumeprocesses.ForDG,theprocessisspontaneouswithDGbeingnegative(orTDSpositive),whichcanbeobtainedreadilywithnegativeDHandpositiveDSsys.4.2.1.3NotionofStateWewanttoapplythethermodynamicreasoningdevelopedabovetotheproblemofdefects,inparticular,theentropyofdefects.Soitisimportanttohaveaclearideaofthemeaningofathermodynamicstate.Itispartofourexperiencethattheexpansionofanidealgas(oranygas)fromV1toV2,aswasdiscussedabove,isspontaneous.ThatthisoccursspontaneouslyissummarizedbytheconditionthatatDH=0,DStot>0forthecaseofV2>V1,asisgivenbyequations(4.12)and(4.9).WecanvisualizetheexpansionastwoequalvolumechambersAandBseparatedbyavalve,asshowninFigure4.1.Whenthevalveisopened,thegasinitiallyinchamberAwillfillalltheavailablevolume(atareductioninp).WeletV1=AandV2=A+B,andthenV1willchangetoV2.ThereverseprocessofthegasparticlesreturningintoV1isnotobservedsolongasthenumberofgasparticlesislarge.Whilewehavenodiffi-cultyinassertingourexperiencethatthisisindeedthecase,wemayhavesomedifficultyinexplainingwhythisisso.Sinceeachgasparticlevelocityvectorisequallylikelyeveninthereversedirection,itisobviousthatonlysinglegasparticleisequallylikelytobeinAorB,ifVA=VB.IfVBislargerthanVA,thenwecanintuitivelyconcludethatouroneparticleofinterest不得转载谢谢合作LWM 4.2WHYDODEFECTSFORM?65ABFigure4.1TwoequalvolumesAandBseparatedbyavalve.willmorelikelybefoundinthelargervolume.HoweverwecannotexcludeitfrombeinginVA,especiallyifbothvolumeswerecloseinsize.Thusourexperienceandintuitionindicatethattheprobabilityoffindingtheparticleisrelatedtothesizeofthecontainer.Onecanexpressthisexperiencemoredefinitivelyintermsofallowedparticlestates.Simply,inthelargervolumetherearemoreavailableandallowedparticlestates.Inoursimpleunrestrictedexample,wecanconsideranallowedstateastheminimumvolumenecessarytocontainagasparticle,anditisanavailablestateifunoccupied.Obviouslytherearemoreofthesevolumestatesinthelargervolume.Extendingthisideaofvolumestatestomanyparticles,weconsiderthatiftherearemorethanonegasparticlethatwewishtofollowthenthereisagreaterprobabilityfortheparticlesbeinginthedifferentvolumesthaninthesamevolume.Thisisso,becausethereissimplymorevolumeavail-able,sinceV2>V1.Furthermoreaparticleresidinginonevolumelowerstheprobabil-ityforthesecondparticletobeinthatvolume,becausethepresenceofthefirstparticlereducesthenumberofavailablestatesforthesecondparticle.ThenthenumberofstatesinV2isgreaterthanthenumberinV1.Thustherearemoreplaces(volumestates)inV2forparticlesthaninV1.Iftheprobabilityforeachstateasdefinedisequivalent,thentheprobabilityishigherforparticlestooccupyV2thanV1.4.2.1.4BoltzmannRelationshipItisimportanttorememberthatDStotisameasureofthechaoticcomponentoftheenergyinasystemandalsoameasureoftherandomnessofasystem.Whennoheatcanflow(DT=0)asystemcanstillhaverandomness(relativetoapreviousstateortoanothersystem).Previouslyweconsideredordered(crystalline)andrelativelydisorderedpiecesofmaterial.Boltzmannrecognizedthatamaterialhasaconfigurationalentropyororderingregardlessoftheflowofheat.TheBoltzmannrelationshipendeavorstoquantifytherandomnessbyrelatingthermodynamicalandstatisticalvariablesanditcanbewrittenforachangeasDWSk=ln(4.14)totInthisequationWistheratiooftheprobabilityofthefinaltoinitialstatesandgivenaswfW=(4.15)wiwherewfisrelatedtothenumberofwaysofformingthefinalstateandwiisthenumberofwaysofformingtheinitialstate.Stotreferstothetotalentropy,whichisthesumofSsys+Ssur.IftheTwerethesameforthesystemandsurroundings,thennoqwouldflow不得转载谢谢合作LWM 66DEFECTSINSOLIDStothesurroundingsanddSsur=0.WetakethisisothermalcaseandapplytheBoltzmannrelationshiptothesystemdiscussedabovecomprisedofV1andV2andsomenumberofgasparticles.Whenthevalveisopenedtheparticlesarrangethemselvesinthelargervolumewithmoreavailablestates.TherearemorewaystoarrangetheparticlesintheV2statebyvirtueofthemoreavailablestates.ThenWisgreaterthanunity,andDStotispositive(DSsys=+andDSsur=0).OnecanseefromtheGibbsfreeenergyrelationshipthatforallothervariablesbeingthesame,thehigherentropyassociatedwiththesystemwithhigherprobabilityyieldsalowerfreeenergy(DSappearsas-TDS),andathighertemperaturestheentropyisevenmoreeffectiveatreducingDG.Wenowhavesometoolswithwhichtoattempttounderstandwhysomekindsofdefectsalwaysexist.4.3POINTDEFECTSThreedifferentkindsofcommonpointdefectsareshowninFigure4.2:vacancies,inter-stitials,andsubstitionals.Vacanciesandsubstitutionalpointdefectsrefertothenormallatticepositions.Thesubstitutionaldefectisadifferentatomonalatticesitenormallyoccupiedbyanormallatticeatom.Aninterstitialpointdefectisanatomintheinter-sticesinbetweennormallatticesites,anditcanbeaself-interstitial,namelytheinter-stitialisanormallatticeatomdisplacedfromitsnormalposition,oritcanbeaforeignatom.Thesedefectsoccurinnumbersdictatedbytheamountofenergynecessarytoproducethedefect.Belowitwillbearguedthattheconfigurationalentropyattainedbycreatingadefectlatticehelpstooffsettheenthalpyofformationofthedefects.TheSubstitutionalVacancyInterstitialFigure4.22-Dsquarelatticedisplayingseveralpointdefects:Substitutionalimpurity,interstitialimpurity,andvacancy.Vacancydiffusionisalsoshown(arrow).不得转载谢谢合作LWM 4.4THESTATISTICSOFPOINTDEFECTS67enthalpyofformationforaninterstitialoftenisastrongfunctionofthesizeandchargeoftheinterstitialtobeformed.Thereisalsothepossibilityofformingchargedpointdefectsinioniccrystals,andtheseareknownasFrenkelandSchottkydefects.TheFrenkeldefectisaninterstitialofachargedatomthat,whenformed,createstworegionsofdifferentpolarity.Itisoftenreferredtoasaninterstitialpairdefect.TheSchottkydefectisalsoapairdefect,butitistheabsenceofbothions.Overallchargeneutralitymustbemaintainedfortheformationofchargeddefects.Oftenthemotionofatomsinasolidtakesplacebyatomsmigratingintovacanciesorintersticesthatwillleaveavacantpositionbehindforanotheratomtomigrateinto.Self-diffusionviavacanciesisshowninFigure4.2withthearrowtoindicatethepossibledirectionofmigrationofanatomthatleavesbehindavacancy.Theatommovesonewayandthevacancytheopposite.ThediffusionviapointdefectswillbetreatedagaininChapter5ondiffusion.4.4THESTATISTICSOFPOINTDEFECTSWecomparetheentropyforacrystalthathassomenumberofdefectlatticesites(eitherinterstitialsorvacancies)tothatforaperfectsinglecrystalusingthetoolsdevelopedabove.Interstitialsaresimplyanatomresidinginbetweenlatticesitessuchasinthetetra-hedraloroctahedralintersticesdiscussedearlier,andvacanciesaresimplyatomsmissingfromlatticesites.Forexample,inordertoformaself-interstitialoravacancyfromaperfectlattice,somebondsneedtobebrokenandpossiblyrearrangementoccurs.Thusenergyisrequiredforeachdefectaddedtothelattice.Furthermorethereisonlyonepos-sibleconfigurationforaperfectsinglecrystal,andforasimplemonatomicsoliditiswithallthelatticepointsoccupiedorassociatedwithatoms.Thereforewi=1foraperfectcrystalastheinitialcondition.Thenfromequation(4.15)W=wfandDS=klnwffromequation(4.14).Nowweneedtocalculatewfforamaterialthathassomenumberofdefectspresent.Itisclearthatthenumberofdefectswilldeterminethenumberofwaysthedefectscanbearrangedonthelattice.AlsoDSwillincreasewiththenumberofwaystoarrangethesystem.TheinfluenceofSontheresultingfreeenergyisoffsetorcounter-balancedbytheenergyrequiredtocreatethedefects.Forinterstitialsandvacancies,respectively,wecanwriteDDEnE=◊andDDEnE=◊(4.16)iivvwherethen’sarethenumbersofthedefectsandtheE’sarethecreation/formationener-giesperdefect.BecauseofthesmallvolumechangesassociatedwithsolidmaterialsEandHareoftenusedinterchangeablyforsolids.Thusequation(4.16)teachesthatDEorDHincreaseslinearlywiththenumberofdefects.Inthiscasewherethereisenergyrequiredtoformthedefects,thisenergyneedstoflowtothesystemandthusDSsurπ0.ThenwecanusetheexpressionforDG,whichsumsthesurroundingsandsystementropies.WearenowreadytocalculateDSforanynumberofdefects.Firstwecalcu-latewf,thenWandDS,andfinallythenumberofpointdefectsatanytemperature.Weconsiderthatthedefectisavacancyofalatticesite,althoughthesamekindofargumentisappropriateforanytypeoflatticedefect.ForonevacancyonalatticeofNsites:wf=N.Forthenextvacancy:wf=N-1,becauseonlyN-1sitesremainavailable.Nowtogeneralize,wecanwritewN==forthefirst,1wN-=forthesecond,wN-2forthethird,etc.fff不得转载谢谢合作LWM 68DEFECTSINSOLIDSWealsoincludetheassumptionthatallthesitesandallthevacanciesarerespectivelyindistinguishable.Thisgivesrisetoadenominator1,2,3,toaccountfortheindistin-guishabledefectsandyieldsforwf:NN()-12()N-...w=(4.17)f123◊◊...Weproceedtogeneralizefornvvacancies.TheN-1termforthesecondvacancyisfor-mulatedbysubtractingfromthenumberoflatticesitesNthenumberofvacanciesremaining,(nv-1)or(2-1),tobeaddedtothelattice;thatis,N-1asobtainedfromN-(nv-1),yieldingN-(2-1).ThisresultcanbegeneralizedasNN()-123()N-()N-...()Nn-+1vw=(4.18)f134◊◊...nvForexample,fornv=5,NN()-1234()N-()N-()N-w=(4.19)f5!whereN-4comesfromN-(5-1)withnv=5.ThenumeratordoesnotgoallthewaytoN!.Itstops,fornvatN-nv+1.ToforcethenumeratortobeN!,wemultiplybothnumeratoranddenominatorby(N-nv)!ThisyieldsN!w=(4.20)fnNn!!()-vvNowwithDS=kln(wf/1)weobtainÊwfˆÊN!ˆDDGEk=-Tln=-nEkTvvDlnÁ˜(4.21)Ë1!¯ËnNn()-!¯vvEquilibriumiswhenDGisaminimumor∂(DG)/∂nv=0:∂()DG∂()ln!NnN---ln!ln()n!vv==0DEk-T(4.22)v∂n∂nvvWeproceedtoevaluatethederivativeofeachtermontheright-handsideofthisexpression.Firstterm:∂()ln!N=0(4.23)∂nvsinceNisnotafunctionofnv.不得转载谢谢合作LWM 4.4THESTATISTICSOFPOINTDEFECTS69Secondterm:∂lnn!dXln!v=ªlnnfromtherelationshiplnX(4.24)v∂ndXvThirdterm:∂ln()Nn-!∂[]()Nn-ln()NnNn---()vvvv=(4.25)∂n∂nvvwhichisobtainedusingSterling’sformula:ln!XXXX=-ln(4.26)Upondifferentiationthefirsttermontherightsideyieldstwoterms:∂[]()Nn-ln()Nn-()Nn-vvv=---ln()Nn(4.27)v∂n()Nn-vvForthesecondtermontherightof(4.26)itis∂[]--()Nnv=1(4.28)∂nvUponcombiningtheresultsweobtainNn-DEvv---ln()Nn++1lnn=-(4.29)vvNn-kTvRearrangingandsimplifyingyieldsDEvlnnN--ln()n+-=-11(4.30)vvkTItfurtheryieldsÈnv˘DEvln=-ÍÎNn-v˙˚kT(4.31)andforNn>=nve-()DEkvTvNwherenv/Nistheconcentrationofvacancies(i.e.,thenumberofvacanciesnvdividedbyN,thetotalnumberofsites).Asimilardevelopmentforinterstitialswouldyieldasimilarfinalresultwhereonlythesubscriptswouldbedifferent.Thusitisconcludedthatforany(positive)valuefortheenergyrequiredtoproducethepointdefect,highertemper-atureswillproducemorepointdefects.ThisisattributabletothegreaterstabilityofthesystemderivedfromtheTSterminthefreeenergyequation(4.13).不得转载谢谢合作LWM 70DEFECTSINSOLIDSThederivationaboveisspecifictothecaseofcreatingdefectsandhowthesystemconfigurationortheconfigurationalentropyimpactsthecalculationofthenumberofdefectsexistentatanyT.Thisnotionofpointdefectcreationcanalsobereadilyenvi-sionedasaso-calledtwo-stateproblem.Weconsidertwoallowedstates,1and2,andthatthechangeintheoccupationofoneofthestates,dn,isproportionaltotheenergydifferencesbetweenthestates,dE.Inparticular,themoreenergyittakestopopulatethehigherenergy(state2),thesmalleristhechanceforthechangeofstatetooccur,thenweusethenegativesign,-dE,andwecanwritethissituationfortherelativepopulationofastateasdnµ-dE(4.32)nIfweconvertthedtodandintegratetheresultingdiffererntialequationfromstate1tostate2usingaconstantCtochangetheproportionalitytoanequality,wehave22dnÚÚ=-CdE(4.33)n11Weobtaintheresultasfollows:lnnnC-=ln-DE21andwithC=1/kT,n1-DEkT=e(4.34)n2Thisisessentiallythesameresultasequation(4.32)forthecreationofvacancies.Theprocessiscalledatemperature-activatedprocessinwhichn2isthenumberoflatticesitesthat,whenvacated,producen1vacancies.AvacancyinthismaterialrequiringDEvforproductionfromanoccupiedsiteyieldsnv-DEkvT=Ae(4.35)NwhereAisrepresentativeofaspecificapplication.Equation(4.35)isaformreminiscentoftheso-calledArrheniusactivationenergythatisusedtoobtainorcharacterizerateprocesses.Nowweseethatthisexponentialformforenergyistheresultofconsideringthestatisticsofpopulatingvariousstatesthat-E/kTrequiredifferentenergies.Theexponentialexpressioneiscommonplaceinthephy-sicalsciencesandisoftenreferredtoastheBoltzmannfactor.Letusturntotheimplicationsofthecalculationsmadeabove.Formetalstheenthalpynecessarytocreateavacancyis1toseveraleV.Fromequation(4.31)itcanbedemon-stratedthatatanyTthereexistsafinitenumberofvacanciesinanycrystal.Thismeansthatthevacanciesformspontaneouslytominimizethefreeenergyofthesystem.ThissituationisrepresentedinFigure4.3,whereitisshownthatasthenumberofvacanciesincreases,theenergyassociatedwiththebondbreakingincreaseslinearly.Incontrast,不得转载谢谢合作LWM 4.5LINEDEFECTS—DISLOCATIONS71atTnEvnVEnergyDG-TSDFigure4.3Thermodynamicfunction(E,G,S)variationwiththenumberofvacanciesataconstanttemperature.theenergycomponentassociatedwiththeentropydecreasesrapidly.SincethefreeenergychangeDGisthesumofthesetwocomponents,apointofinflectionmustoccur.Atthispointtheequilibriumnumberofvacanciesatthespecifiedtemperatureisobtained(thedottedline).4.5LINEDEFECTS—DISLOCATIONSCrystallinesolidsthathavebeensubjectedtotensileforcesoftendisplaygroupsofpar-allellines,calledsliplines.Thesliplinesoccurastheappliedtensionexceedsathresh-oldforthematerial,butbeforefracture.Figure4.4ashowsatensileforceFTappliedperpendicularlytothefaceofasinglecrystalsample.Thetensileforce’sdirectionstretchesthematerialparalleltothedirectionoftheappliedforce,andthisresultsintheelongatedsampleshowninFigure4.4b.Figure4.4bincludesaclose-upsketchofsliplines,orbands.Thesemacroscopiclineshaveamicroscopicorigin,whichthediagramofFigure4.4bhelpselucidate.Notethattheappliedforceisresolvedintoforcecomponentsthatexistonanyplaneinthecrystal.Itisusualtoresolvetheappliedforceintonormalandparallelforcecomponentsonaspecificplane.WithFTastheappliedforcecomponentnormaltothetopplaneofthesample,andFPtheforceresolvedontheplanesparalleltotheslipbands,FPiswrittenasFF=cosq(4.36)PTTheangleqbetweenthenormaltothetopplaneandthenormaltotheslipplaneasisshowninthefigure.ThusFTandFParebothtensileforcecomponents,withtheformerperpendiculartothetopplaneandthelattertotheslipplane.Itisalsopossibletoresolve不得转载谢谢合作LWM 72DEFECTSINSOLIDSFTFPqFTfFTa)b)Figure4.4(a)Applicationoftensileforcestoacrystal;(b)riseofslipbandsduetotheappliedforces,whichcanberesolvedontospecificplanesanddirections.theappliedforceFTintheslipplane(ratherthanperpendiculartoit)inanydirectionintheslipplane.ThiscomponentofforceiscalledtheshearforcecomponentFSandisgivenasFF=cosf(4.37)STwithfbeingtheanglebetweenFTandaparticulardirectionintheplane.FS,theshearcomponentoftheappliedforce,istheforcethatappearstobealikelycandidatetogiverisetothesliplines,sincethisforceexistsinthedirectionoftheplanardisplacement,namelytheslip.Actuallyitismoreappropriatetodiscussslipintermsoftheshearstress,t,whichistheresolvedshearforce,FS,perunitareaonaplane.InChapter7,twillbeshowntobeamaximumwhenbothqandfare45°totheappliedforce.However,sliplinesarenotalwaysfoundonlyat45°totheappliedforces.Thisstronglysuggeststhatthespecificcrystallographyinfluencesthesliplineformation.Infactitwasdiscoveredthatcertainplanes,calledslipplanes,andcertaindirectionsintheplanes,calledslipdirections,werealmostalwaysimplicatedintheslipphenomena.Thecombinationofslipplaneanddirectioniscalledaslipsystem,andthiscombinationisspecificwithinthecrystalstructures.Table8.1displaystheslipsystemsforcubicstructuresthatwillbeusedlaterforelucidatingmechanicalproperties.Theslipdirectionisfoundtohaveclosepackingintheslipplane,whichisaplanealreadyofhigh(usuallyhighest)atomdensity.Fromthesefactsapictureofthephysicsofslipemerges.Considera3-Dlatticeforamonatomicsolidwhereeachatomisheldtothenextbychemicalbondsthatarerep-resentedmechanicallybysprings(seea2-DrepresentationinFigure7.2wheremechan-icalpropertiesarediscussed).Furtherweimaginethatthespringswiththehighest不得转载谢谢合作LWM 4.5LINEDEFECTS—DISLOCATIONS73tensionarethosetotheclosestneighbors(representingthestrongestbonds).Withtheapplicationofanexternalforcetothesolid,thespringswilldistend.Withmorespringsofhighesttensionontheplaneswithmoreatomsandinthedirectionsofdensestpacking,itistheseplanesanddirectionsthatfirst“feel”orbearthedistension.Anotherwaytoviewthisisthattheshortesttightestspringswillbeartheappliedforcebeforethelongerandlooseronesdoso.Whenasufficientlylargeshearforceexistsonaslipplane,oneplanewillyieldintheslipdirectionrelativetotheother.Theatomisticresultofthismotion,andtheresultingdeformation,iscalledadislocation.Theresultofmotionofoneplanerelativetoanotheryieldsadislocationoflessthananatomicspacinginthedirectionoftherelativemotion.Whenexactlyafulllatticedisplacementisachieved,registrywillagainbeachievedacrossthedislocationline.Thusthemaximumdisregistryisatahalfplanarspacing.Whenalargenumberofplanesparticipateallwithfractionallatticespacingdeformation,thenthesumisamacroscopicdisplacementthatresultsinthestep-liketotaldeformationshowninFigure4.4.Itisobserved(andwewilldiscusslaterinChapters7and8)thattheactualshearforcethatisrequiredforthedislocationtypeofdeformationisaboutG/1000,whereGisthebulkmodulus(definedanddiscussedinChapter7).Evenwithoutproperback-grounddefinitions,itisinterestingtocomparethismeasuredvalueofforceofaroundG/1000tocreateadislocationwiththetheoreticalforcesrequiredfordeformationofamaterialwithoutdefects,astheseareoftheorderofG/30orafactorofabout30larger.Thedifferenceliesinthepresence,multiplication,andmovementofdislocationsinacrystallinesolid.WereservethisdiscussiontoChapter8,whichisdevotedtomechani-calproperties.Herewedefinemorepreciselywhatadislocationisandthekindsofdis-locations.However,beforeweproceedtothatsubject,weshouldreflectontheimportantfactthatitisonlybytheinputofconsiderableenergyorwork(forcethroughadis-placement,seeequation4.3)thatdislocationsareformed.Thisisincontrasttothepointdefects,suchasvacancies,thatformspontaneously.Itisconcludedthattheenergyinputrequiredtoformalinedefectis5to10timeslargerandtheconfigurationalentropygainedwiththeformationoflinedefectsissmall.Thuslinedefectsarelesslikelytoformspontaneously.4.5.1EdgeDislocationsImagineablockofcrystallinematerial,likethatinshowninFigure4.5a,thatiscuthalfwaywithaknifehavingacuttingdirectionnormaltoonefaceofthesolid.Thecutismadeinbetweentwoplanesofatoms,therebyslicingthroughthebonds.One-halfoftheslicedcubeisheldinplace(theleftsideinFigure4.5a),andtheotherside(therightside)iscompressed,asshownbythearrow,keepingtherearedgestogether.Atsomepointinthecompressionoftherighthalfanextraplaneiscreated.Theadditionalplaneinthecompressedhalfwillappearasadisplacementofone-halftheatomicspacinginthedirectionoftheappliedcompressiveforce.Withthetopandbottomhalvesreattached(i.e.,thebondsmadeacrossthecutorshearplane),amaximumdisplacementofhalfanatomicspacingoccurs,andtoeithersidethereislessdisplacementlaterallywherethedistortedbondsrelax.Figure4.5bshowsaviewofthisresultnormaltothedirectionofthedisplacement(i.e.,onedge)andwiththecompressionsymmetricfrombothsides.Theresultisthesameasifanextraplaneofatomswereinsertedinto(only)thebottomhalfofthecrystal.Thiskindofdislocationiscalledanedgedislocationanditisdefinedbythefactthatthedislocationline(theblackfilledcircleatthecenterofthe^,the不得转载谢谢合作LWM 74DEFECTSINSOLIDSa)b)DislocationLinebFigure4.5(a)Compressiveforce(arrow)exertedonpartofacrystalleadingtodisplacementofplanes;(b)formationofanedgedislocation.Thedislocationlineisnormaltothedirectionoftheforce,andtheBurgersvectorbindicatesthemagnitudeanddirectionofthedislocation.symboltoindicateanedgedislocation)isperpendiculartothedirectionoftheshear.Thedislocationlinerunsinandoutoftheplaneofthepaperundertheextrahalfplane.Thiskindofdislocationissymbolizedby:or,fortheextrahalfplaneinthebottomortoppartofthecrystal,respectively.Itiseasytoimaginethatifsomehow^weretoapproach,thedislocationsinthetopandbottomofthecrystalwouldannihilate,andperfectregistrywouldberestored.4.5.2ScrewDislocationsToformtheedgedislocation,wecompressedtherighthalfofthecrystalrelativetotheleftinFigure4.5a.Theforceswereappliednormaltothedislocationline.Ifwe,instead,hadsimplylaterallypushedthetophalfofacutcrystal(aswascutinthecaseabove)toonesideasisshowninFigure4.6a,theresultwouldbecalledascrewdislocation.Thistypeofdislocationresultswhentheshearingforceisappliedinthesamedirection(parallel)asthedislocationline.Theouterpartofthesolidisdeformedmorethanthe不得转载谢谢合作LWM 4.5LINEDEFECTS—DISLOCATIONS75a)b)DislocationLinebFigure4.6(a)Shearforce(arrow)appliedtopartofacrystalandresultingdisplacementofplanesparal-leltotheforce,calledascrewdislocation;(b)displacementintersectingasurfacewithBurgersvectorbpar-alleltothedislocationline.dislocationcore.Aspiralappearsaroundthepartofthecrystalthatisnotcutnordeformed,asintheviewshowninFigure4.6b(alsodisplayedinlargerviewinFigure8.4a).Inthisfigurethedislocationlineisnormaltothecrystalsurfacebutparalleltothedirectionoftheappliedforcethatcausedthedislocation.Thepureedgeorscrewdislocationsarethentheresultofspecificdirectionalforcesappliedrelativetotheslipsysteminagivenlattice.Itisunlikelythatsuchforceswilloccurpreciselyinsuchawaytoproduceonlyanedgeoronlyascrewdislocation,exceptinourimaginations.Thus,inreality,acombinationofedgeandscrewcharacteristicswillbedisplayedinmostdislocationsasthedislocationlinesnakesthroughthedeformedsolid.However,ashortsegmentofadislocationmightbeaccuratelydescribedasapureedgeorscrewtypedislocation.Whatisnowneededisaquantitativedescriptorfordislocations.ThisdescriptoriscalledtheBurgersvectorbecauseitdescribesboththemagnitudeanddirectionofadis-locationline.不得转载谢谢合作LWM 76DEFECTSINSOLIDS4.5.3Burger’sVectorandtheBurgerCircuitForbothedgeandscrewtypedislocations,thedislocationisdefinedquantitativelybyavectorbcalled“Burger’s”vector.Asavectoritexpressesboththemagnitudeanddirec-tionofthedislocation.NotethatitislabeledinbothFigures4.5and4.6.(InChapter8bwillbeusedtocalculatetheenergyofadislocation.)ThemagnitudeanddirectionofbisobtainedfromaBurgercircuit,whichisnowdescribed.Figure4.7illustratestheBurgercircuitforbothedge(Figure4.7a)andscrewdislo-cations(Figure4.7b).ToformaBurgercircuit,onefirstselectsastartingplaceintheundeformedregionofthecrystal(labeledSinFigure4.7).Thenoneproceedsfromlatticepositiontolatticepositionaroundthedislocationinaclockwisemanner,asindicatedbythearrow.Theverticalandhorizontallatticepositionsnecessarytogohalfwayaroundthedislocationareseparatelytabulated.Fromthehalfwaypositionareturntothestart-ingpositionisattemptedusingthetabulatednumberofverticalandhorizontaljumpstogettothehalfwaypoint.Ifadislocationexistswithinthecircuit,itwillnotbepossi-bletoreturntothestartingpositionusingthesetabulatedsteps.Thisis,ofcourse,duea)1234567851423324b15S7654321b)123456751423342765432151SbFigure4.7Burger’scircuitsfor(a)edgedislocationand(b)screwdislocation.ThestartpointSforthecircuit,theclockwisedirection,andtheBurgervectorsbareshown.不得转载谢谢合作LWM 4.6PLANARDEFECTS77totheexistenceofadislocation,whichisadisplacement.Nevertheless,thetabulatedjumpsareexecutedtoleaveadisplacementfromthestartingposition.Thenthelastpieceofthecircuitorthedifferencenecessarytoreturntothestartingpointisnoted,andthissegmentisbinbothmagnitudeanddirection.InFigure4.7aforanedgedislocationwearbitrarilystartinthelowerleftsideofthecrystal.Wecancountanarbitrarynumberoflatticepositions,say,5towardthetopand8acrosstotherightinaclockwisemanner.Thenumberoflatticepositionsisdeterminedsoastoencirclethedislocation.Havingaccomplishedthisandbeinghalfwayaroundtheedgedislocation,weproceedtoreturnby5jumpsdownwardfromthehalfwaypointandthen8backtothelefttocompletethecircuit.However,8jumpswouldpassthestartposition.Thenweformavectorbpointingfrom8to7tocompletethecircuit.TheBurgervectoristhislastsegmentnecessarytocompletethecircuitbacktoS.InFigure4.7bweproceedinthesamemannerforascrewdislocation.WecommencetheBurgercircuitinthelowerleftandproceedclockwiseupward,first5jumpsandthen7totherighttothehalfwaypoint.Thenwereturnwith5jumpsdownandthen7totheleft.However,thesefinal7totheleftareonaplanebehindtheplaneonwhichwestarted.Toindicatethisonthefigure,italicsareusedforthejumpnumbers.TheseventhjumptotheleftwouldbeonepositiondirectlybehindthestartpositionS.So,tocompletethecircuit,weneedajumpperpendiculartotheplaneofthepapertothestartposition,andthisvectorformsbforthescrewdislocationdepicted.4.5.4DislocationMotionThemotionofalinedefectordislocationisanimportantconceptindefiningthemechanicalpropertiesofmaterials,inparticular,plasticity.(ThiscontextwillbediscussedinChapter8.)Hereweconsiderthelow-resistance(low-energy)pathbywhichdisloca-tionsareobservedtomove.Wemightimaginethesimultaneousbondbreakingofallbondsaroundtheextrahalf-planethatformsanedgedislocation.Thentheextraplanecanberemovedandtransportedtotheedgeofthecrystalandreattached.Thisistan-tamounttomajorsurgeryinvolvingsimultaneousmultiple-bondbreaking.Itisnot,however,Nature’spathofchoice.Anedgedislocationcanactuallymoverelativelyeffort-lesslyinalattice(asinplasticdeformation,whichisdiscussedinChapter8,Figure8.2).Theatomsneedonlybedisplacedasmallamountfromtheirequilibriumpositionasashearstresstisapplied.Thiswayatomsoneithersideoftheshearplanefirstgooutofalignmentandthenintoalignmentwithatomsthatwerenotinalignmentpriortothedeformation.Onlyasmallnumberofatomsthenneedsbondingrearrangements,andtheextrahalf-planemovesonelatticepositionatatimeasthestresscontinues.Finallytheextrahalf-planemovestothesurface,whereitresultsinanatomicstep.Themotionofmanydislocationsinthatplanewillincreasethesizeofthestep.4.6PLANARDEFECTS4.6.1GrainBoundariesInpreviouschapterscrystalline,polycrystalline,andamorphousmaterialsweredis-cussed.Polycrystallinematerialsaremadeupofsinglecrystalsthatarebondedtogetherandhavelittleornocrystallographicrelationshiptooneanother.Thebondingregioniscalledagrainboundary.Grainboundariesareclassifiedaccordingtothemagnitudeofthemisorientationthatoccursattheboundaries.Figure4.8comparesaperfectcrystal不得转载谢谢合作LWM 78DEFECTSINSOLIDSqa)b)Figure4.8(a)Aperfectcrystaland(b)alow-anglegrainboundarythatresemblesanarrayofedgedislocations.(atomsarerepresentedassquares)withonethathasalow-anglegrainboundary.ImaginethatFigure4.8aisaperfectcrystalwithatomsateachsquare.Alltheatomsareinreg-istryintheperfectcrystal.InFigure4.8bthelefthalfofthecrystalistiltedrelativetotherighthalf,andtobondthehalvestogether,anotherrowofatomsisnecessary;itisdepictedasshadedsquares.Eachsideiscalledasinglecrystal,sinceeachisaperfectalbeitsmallcrystal.Theangleqisameasureofthemisregistrywithsmallanglesresult-ingin“low-anglegrainboundaries”andlargeanglesresultinginwhatarecalled“high-anglegrainboundaries.”Forlow-anglegrainboundariesasshowninFigure4.8b,theextrarowofatomsresultsinanedgedislocation.Onecanimaginethegrainboundaryextendedovermanymoreatomsthanshownintheillustrationandatthegrainbound-aryanarrayofedgedislocations.Theenergyofthemisorientationvarieswithq.Italwayscostsenergytoproduceasurface,andtheproductionofsmall-grainedmaterialwithlargesurface-to-volumeratiosisnotasstableaslargergrainedmaterials.Thereforethetendencyisforthegrainsizetoincreasewhenthecircumstancespermit.Onesimplewaytoachievegraingrowth,andhencereducethegrainboundaryarea,istoannealatelevatedtemperatures.Thehightemperatureenhancesatomicmobility(aswillbedis-cussedinChapter5),whichenablesatomsto“find”andsettleintolowerenergyconfig-urations.Itisalsopossibletohavelarge-anglegrainboundarieswherethemisregistryangleqiscomparativelylarge.Insuchcasesthegrainboundaryregionismorecomplexandcannotbecharacterizedassimplyasanarrayofedgedislocations.Infactdisconti-nuitiescanarise.4.6.2TwinBoundariesAtwinboundary(sometimescalledaninterfaceboundary)iswhereinagivensinglecrystal(ofteninagrainofapolycrystallineaggregate)onepartofthecrystalisamirror不得转载谢谢合作LWM RELATEDREADING79MirrorPlaneFigure4.9Atwinboundarywiththemirrorsymmetryplaneshown.imageoftheotheradjacentpart.Figure4.9showsthetwinplaneorboundaryasamirrorplane.Thesedefectsareoftentheresultofappliedstresstoanalreadyformedgrainstructure.4.7THREE-DIMENSIONALDEFECTSAmong3-Ddefectsaremacroscopicirregularitiessuchasvoidedpockets,cracks,andporestructurewhoseindividualnaturesareevidentfromthenames.Virtuallyallrealmaterialshavethesekindsofdefects,andsomereferencetoeachwillbegiventhrough-outthistextasappropriate.Thefactthatlittlespaceisdevotedto3-Ddefectsshouldnotdiminishtheirimportance.Forexample,cracksandcrackpropagationinglassesdominatethefracturemechanism.Poresdominatematerialstransportinmanymateri-als.Voidscandominatestrength.Indeed,becausethedefinitionsarealmostself-evidentandthestructurescomplex,fewunifyinggeneralizationscanbemade.RELATEDREADINGC.R.Barrett,W.D.Nix,andA.S.Tetelman.1973.ThePrinciplesofEngineeringMaterials.PrenticeHall,EnglewoodCliffs,NJ.Areadableelementarytextforafirstcourseinmaterialsscience.D.Hull.1975.IntroductiontoDislocations.PergamonPress,NewYork.Areadableandwell-illustratedtreatmentofdislocations.P.A.ThorntonandV.J.Colangelo.1985.FundamentalofEngineeringMaterials.PrenticeHall,EnglewoodCliffs,NJ.Areadableelementarytextforafirstcourseinmaterialsscience.不得转载谢谢合作LWM 80DEFECTSINSOLIDSEXERCISES31.TheroomtemperaturedensityforCuanFCCmetalisgivenas8.94g/cm.At1000°C3thedensitywasfoundtobe8.92g/cm.Calculatehowmanysitesarevacantat1000°Cassumingthatthedensitychangesareduetovacancies.32.FCCAlhasa0=0.4050nm.Themeasureddensityis2.698g/cm.CalculatethenumberofvacanciesintheAl.3.Ametalhasanenthalpyforformationforavacancyof2eV.Calculatethenumberofvacanciesthatcanbequenchedat500°C,andthencalculatethetemperatureincreasenecessarytoincreasethenumberofvacanciesbyafactorof10fromthe500°Cvalue.4.Showwithsketcheswhathappenswhenanedgedislocationinthetophalfofacrystalmeetsanedgedislocationinthebottomhalf:+^.5.SketchtherelationshipoftheappliedforceandthedirectionoftheBurgersvectorforbothedgeandscrewdislocationsinalattice.6.Discusswhypointdefectsformspontaneouslybutlinedefectsdonot.不得转载谢谢合作LWM 5DIFFUSIONINSOLIDS5.1INTRODUCTIONTODIFFUSIONEQUATIONSDiffusionproblemsanddiffusion-likeproblemspervadescienceandengineering.Basi-callytheseproblemsconsiderafluxorflowinresponsetoaspatialgradientcalledaforce,namelythedrivingforcefortheflow.Someexamplesintermsofaonedimen-sionalgradientare-kdTJ=Fourier’slawofheatflowhdx-sdVJ=Ohm’slawedx(5.1)-cdPJ=Poiseuille’slawmdx-DdCJ=Fick’sfirstlawDdxFourier’slawisforaflowofheatJhinresponseto(andproportionalto)thetempera-turegradientdT/dxwithaheattransfercoefficientkbeingtheconstantofproportion-alitybetweenthefluxanddrivingforce.LikewiseOhm’slawisforaflowofelectronsJe(anelectriccurrentflux)thatisproportionaltothepotentialgradientdV/dxwithsbeingtheconductivity.ThenextexampleisPoiseuilleflow,withJmbeingamassfluxinresponsetoapressuregradient.Thesubjectofthischapterisanothermassfluxcalleddiffusionwithitsflux,JD,thatisproportionaltoaconcentrationgradient.Intheseequa-tionsthenegativesignindicatesafluxdownthegradient,andofcourse,allthefluxesElectronicMaterialsScience,byEugeneA.IreneISBN0-471-69597-1Copyright©2005JohnWiley&Sons,Inc.不得转载谢谢合作LWM81 82DIFFUSIONINSOLIDScanbewrittenin3-D.Alloftheselawshavemuchincommoninthattheyignoreatomicstructureandassumeacontinuumofmatter,andallareintuitive,andthereforeagreewithexperience.Forexample,weexpectanelectricalcurrenttoflowinresponsetoapotentialappliedtoaconductor,andforwantofmoreinformation,areasonableguessisthattherelationshipislinear.Oftensuchlawsthatcorrespondtoexperiencearetermed“phenomenological”or“thermodynamic”laws.Also,likethelawsofthermodynamics,theselawsapplytolargenumbersofatoms/moleculesorotherobjects,andthuslikether-modynamicsareunderstoodonastatisticalbasis.Thatis,thelawsdescribestatisticalbehaviorofalargenumberofobjects.Theselawsandthephenomenagovernedcanoperateindividuallyand/orsimultane-ously.Inthelattercasethereexistscoupling.Forexample,ifthereexistsaspatialtem-peraturegradient,dT/dx,notonlywillheatflowbutmassmayflowduetoa“thermalmigration”crossterm.Possiblywithmassandheatflows,additionalspeciescanmigrateduetogradientsotherthanthechemicalgradientofthespeciesinquestion.Atheoret-icalunderstandingofsimultaneousfluxeswasdevelopedbyOnsagercommencingwiththeprincipleofmicroscopicreversibility.Thefluxoftheithspeciescanbegivenasthesumofallthecontributionstothisflow:JLii=ÂkXk(5.2)ForseveralfluxeswecanwriteJLXLXLX=++1111122133JLXLXLX=++(5.3)2211222233JLXLXLX=++3311322333whereX’sarethedrivingforces.Forexample,wecanassumethatJ1isaheatflux.ThentheprimarydrivingforceisX1whichisdT/dx,andL11=k,theheatflowcoefficient.Theotherfluxesmaybediffusionalfluxesofeachofthevariouscomponents.Forexample,J2canhaveX2=dC2/dx,andJ3canhaveX3=dC3/dx.Theinterestingmajorcontribu-tionofOnsageristhatthereisarelationshipamongtheL’s:LL=(5.4)ikkiHerewedonotconsiderthisimportantfieldofnonequilibriumthermodynamicsfurther.However,theusefulmessageisthattherearemanyimportantphysicalphenomenathataregovernedbysimilarformulas.InthischapterwespecificallyconsideronlysolutionstotheFickiandiffusionalformulas,butwecanrecognizethesimilaritiesinmathemati-calformofotherphenomenaandconsequentlythesimilaritiesintheformofresultingsolutions.Alargenumberofapplicationsexistforthediffusionequationsfrommassflowtoalloftheformulasabovefordifferentfluxes(heat,charge,etc.).Evenquantummechanics,whichmakesmuchuseoftheSchrödingerequation(seeChapter9)andwhichisconstructedofafirsttemporalderivativeandsecondspatialderivatives,hastheformofFick’ssecondlaw.Eachoftheselawshassimilarmathematicalform.Buttheynotonlygovernvastlydifferentphysics,theboundaryconditionsforthesolutionsaredif-ferent.Hencethesolutionsaredifferent.Nevertheless,therearesimilaritiesinthefun-damentalphysicsthatunderlythephenomenologicalequations,mainlythatfluxesareinresponsetoforces.不得转载谢谢合作LWM 5.2ATOMISTICTHEORYOFDIFFUSION83Inthischapterthreeaspectsofthesubjectofmassflowbydiffusioninsolidsareconsidered:physicalmodelsandideas,mathematicalaspects,andapplicationssuchasnucleationandphasechanges.5.2ATOMISTICTHEORYOFDIFFUSION:FICK’SLAWSANDATHEORYFORTHEDIFFUSSIONCONSTRUCTDConsidertwoadjacentplanesatXandX+DXshowninFigure5.1a,thatareadistanceDXapart.(Laterwewillconsiderthisseparationtobealatticespacingorlatticepara-meter,a0.)ThereareotherparallelplanessomeDXapartandothersfartheraway,butwenowonlyconsiderthesetwo.WeproceedbydefiningthenumberdensityofatomsN=#/areaandsettheinitialnumberdensitiesasNxontheplaneXandNx+DXontheplaneX+DX.OnlyjumpsofatomstotheleftorrightinunitsofDX(ora0)areper-mitted(1-D).Thismeansthattheatomsmustjumptoanemptystateontheadjacentplane.Weassumethatsufficientemptystatesareavailable,andajumpdoesnotdependonapreviousjumptoleaveanemptystate.AjumpfrequencyGisdefinedbyunits#jumps/sec·atom.ThenthenumberofatomsjumpingfromplaneXinthetimeintervaldtisNt◊◊Gd[]()#atomsarea◊◊()#jumpssecatom◊()timexWithonlyleftorrightjumpsofdistanceDX,thenumberofjumpstotherightfromXtoX+DXwillbegivenas1#R=◊NtGd◊(5.5)x2JRJLa)XXX+DJLJRb)X-DXXXX+DFigure5.1(a)Fluxestoandfromtwoadjacentplanesforsteadystatediffusion;(b)fluxestotherightandleftfromandtoacenterplanefornon–steadystatediffusion.不得转载谢谢合作LWM 84DIFFUSIONINSOLIDSAsimilarargumentappliesfortheatomsjumpingfromplaneX+DXtoplaneX,ortotheleft:1#LN=◊Gd◊t(5.6)xx+D2Theotherhalfofthejumpsoccur(toaplanetotheleftofXandrightofX+DX)withequalfrequency,buttheresultofthosejumpsdonotappearonthetwoplanesthatarepresentlybeingconsidered.Forourpurposetheotherhalfofthejumpsareunobserved.Noticeinequations(5.5)and(5.6)thatforatoms,planesandjumpsareallthesame;thenumberofjumpsinagiventimeintervalisonlyafunctionofthenumberofatomsperplane.Thenetflux,J,isobtainedfromthedifferenceinthefluxestotherightandleft,andgivenas1JN=-G()N(5.7)xx+Dx2Tomakeequation(5.7)conformtotheusualformforFick’slaws,weproceedtoconvertthenumberofatomsonaplane(N)toconcentrations(C).WithconcentrationasthenumberpervolumeorC=#/V,itfollowsthatC·DX=N,thenumberperarea.Thenweobtain1JC=◊GD()X-◊CXD(5.8)xx+Dx2DXisintroducedintothenumeratoranddenominatorofequation(5.8)bymultiplyingbyDX/DX,toobtain12ÏDC¸JX=◊◊GDÌ˝(5.9)2ÓDX˛Convertingtosmallchanges(Dtod)andsettingDX=a0,thelatticespacing,obtainsthefollowingformula:12ÏdC¸Ja=G0Ì˝(5.10)2ÓdX˛12SinceD=G2a0,wehavewhatlookslikeFick’sfirstlaw:-DdcJ=(5.11)dxDisdefinedabove,andtheminussignisaddedtoindicatethefluxdownthegradient.1In3-Dwithofthefluxalongeachcoordinate,3Ê1ˆ2Da=G(5.12)Ë6¯0不得转载谢谢合作LWM 5.2ATOMISTICTHEORYOFDIFFUSION85ThesimplewayweusedtoobtainFick’sfirstlawrevealsthedynamicsunderlyingnetdiffusiveflux.ItisbasicallyrandomjumpsalongwithanimbalanceinN’sonplanesXandX+DX.Inotherwords,theplanewithmoreN’swillhavethegreatestnumberofjumpsifotherpotentialvariables,X(ora0)andG,areequal.WecansummarizethephysicsofFickiandiffusionasthatprocessofmassflowthatdependsontheconcen-trationgradientandrandomness.Strictlyspeaking,wecouldhaveusedthethermody-namicchemicalactivityinsteadofconcentration.However,thesearenearlyequivalentinsimplecases.If,ontheotherhand,thejumpsweresomehowbiased,thenthenetfluxcalculatedusingequation(5.7)wouldbeinerror.Laterwewillbrieflydiscussconvec-tionalmassflowwheretherandomnessassumptionislifted.ToobtainafeelforthemagnitudeofG,itisinstructivetoconsidersomerealnumbers.-62FordiffusionofcarboninaFe,Disabout10cm/sat900°C(thisisaratherlargeD-82fordiffusioninsolids).Wecanassumethata0isabout1Åor10cm.FromG=2D/a0,-6-1610-1G=2·10/10or10s.ThusCchangespositionorjumps10billiontimespersecond!13-13Considerthatanatomicvibrationisabout10s.Sothereare10vibrationsforevery-82jump.FormanymetalsDisaround10cm/snearthemeltingpoint,andthisyieldsG8-15ofabout10s.Thusonlyonein10vibrationsresultsinajump.Sowhilejumpfre-quenciesappeartobehighnumbers,theactualjumpingofatomstoadjacentsitesisaninfrequentoccurrenceontheatomicscale.Fick’sfirstlawisusefulwhenconcentrationsareestablishedatallpointsinasystem.However,itisoftenthecasethattheC’sareafunctionoftimeinaprocess.Forexample,atthebeginningofthediffusionofNiintopureCu,thereisnoNiintheCu;theNiconcentrationinCubuildsupovertime.Thesametreatmentasdoneabovecanbepursuedtofindthetimeevolutionofconcentrationonagivenplane(C(t)onX).Con-siderFigure5.1bwherethreeplanesareconsidered.ThenumberarrivingatXfromthetwoadjacentplanesisGGNNN=+(5.13)Arrivexx-+DDxx22ThenumberleavingXisNN=G(5.14)leavingxThetotalchangeisobtainedfromthedifference.SincehalfofthejumpsarefromXtotheleftandhalfaretotheright,thenetchangeatXisgivenasdNGx=-[]()NNNN+-()(5.15)xx-+DDxxxxdt2Heretheleft-handterminthesquarebracketsisforthechangefromtheleft(L),andtheright-handtermisforthechangefromtheright(R).However,ifLπR,thenasteadystatedoesnotobtain,anddC/dTπ0,andN(orC)isafunctionoftime.NowconvertN’stoC’sasbefore,usingdNdCNCx=◊DDand=◊x(5.16)dtdt不得转载谢谢合作LWM 86DIFFUSIONINSOLIDSThefollowingresultisobtained:dCG◊=DxxD[]()CCCC-+-()(5.17)xx-+DDxxxxdt2TheexpressioninsquarebracketsisessentiallyDContheXplane.Toputthisexpres-sioninappropriateform,wedividethroughbyDxandthenmultiplytheright-handsidebyDx/DxtoobtaindCG2()DCG2dÊDCˆ=Dx=Dx(5.18)dt22Dx2dxËDx¯Uponconvertingequation(5.18)toderivatives,wehaveFick’ssecondlaw:2dCG2dC=Dx(5.19)2dt2dx12withD=G2a0forDx=a0,asbefore.WhilethemethodtoobtainFick’ssecondlawisappealinginthatitfollowsthemethodtoobtaintheFick’efirstlaw,itisusefultotakeastepbackandconsiderwhatishappening.InlookingagainatFigure5.1b,weseetwofluxproblemsratherthanoneaswasconsideredinFigure5.1aforFick’sfirstlaw.Thatis,weseeoneproblemyieldinganetflux,say,JLbyatplanesX-DXandXandasecondproblemyieldinganetfluxJRatplanesX+DXandX.SincetheplanesX-DXandX+DXcanarbitrarilyhavedifferentN’s(againassumingequalG’sandrandomjumps),thenthetwonetfluxesarenotequal,JLπJRingeneral.ForthisreasonthenumberofatomsonXperunitarea,Nx,willbeafunctionoftime,dNx/dt,andspecificallydNx/dtwillbedeterminedbythechangeinJor(JR-JL)ordJforsmallchanges.AlsoweknowthatdNdC=dx(5.20)dtdtbecauseC=N/VandthevolumeV=Adx.Puttingthistogether,wehavedNdC==dx-dJdtdtthen2dCddÊ-DdCˆdC=-J=-=D(5.21)dtdxdxËdx¯dx2whichisFick’ssecondlawagain.TheformulaabovefordN/dthasanegativesignbecausethefluxeswithaminussignfromtheX-DXandX+DXplanesyieldanetgainor+dN/dt.ThemessagefromthisprocedureisthatthechangeinCwitht,ordC/dt,isgivenbythechangeinthefluxwithxordJ/dx.不得转载谢谢合作LWM 5.3RANDOMWALKPROBLEM875.3RANDOMWALKPROBLEMFromtheprecedingdiscussions,weknowthatdiffusionisthetimeintegralofalltherandomjumps.Nowimaginethejumpstobeoccurringatarapidstatisticalrate(aswasestimatedabove),andeveryonceinawhileasnapshotoftheatomsistakeninordertodeterminetheirrelativepositions.Thediffusiondistanceisthefinalplaceofanobservedatomafteraspecifiedtime.Withtherandomjumpsinvirtuallyalldirectionsandwithahugenumberofjumps,isseemsimpossibletoanalyticallydeterminethedisplacementofatoms.Theproblemseemsintractableatfirst.However,theveryrandomnessandlargenumbersenablesasolutionwithfar-rangingimplications.Theproblemisaclassicalphysicsproblemcalledtherandomwalkproblem.Itisusualforarandomwalkproblem,asappliedtodiffusion,toincludealargenumberofrandomjumps,witheachjumpbeingunbiasedbythepreviousjumps,andtoaimatcomputinghowfarfromthestartpositionaparticularatomcanbeexpectedtobeafteratimeinterval.Figure5.2showsacompositeofanumberofsnapshotsforaparticularatomwhereriarethevectorsthatcorrespondtoeachoftheijumpsthatthisatommakes.Thereisadefinedstart,S,andfinish,F,point.ThefinaldisplacementfromStoF,designatedbythevectorRncanbeobtainedbyvectoraddition:nRrrrni=+++=123...Âr(5.22)i=1InordertofindthemagnitudeofthevectorRn,thedotorscalarproductofRnwithitself(q=0°)isdetermined:2R=◊RRnnn=◊+◊++◊rrrr...rr11121nrrrr◊+◊++◊...rr21222nrr◊+...rr◊++◊...rr31333n...rr◊nn2(5.23)=RcosqnFRnSriFigure5.2RandomJumpsfromthestartpositionStothefinishFaregivenbythevectorsri.Theresul-tantofthejumpsisRn.不得转载谢谢合作LWM 88DIFFUSIONINSOLIDSThisarrayofproductscanbesimplifiedandwrittenasaseriesofsums,wherethefirstisasumofdiagonaltermsnÂrrii◊(5.24)iandthesecondconsistsofsemidiagonaltermsasthesumsofri·ri+1andri+1·ri.Thesemidiagonaltermsareequalandcanbecombined,andtherearen-1ofthesesemi-diagonalterms:n-12rr◊(5.25)Âii+1i=1Thenextandsucceedingtermsaredevelopedasabove:n-22rr◊+...(5.26)Âii+2i=1Insummary,thisyieldsnn-1nj-22R=+r2rr◊(5.27)niÂÂÂiij+i==11j=1iThiscanbeputintomoreusableformasnn-1nj-22R=+r2rrcosq(5.28)niÂÂÂiij++iij,i==11j=1iwhereqistheanglebetweentheadjacentvectors(ofcourse,q=0whenthei’sarethesameandthusthecosinetermisunityforthefirsttermontherightofequation5.28).Themanipulationsabovehavemadenorestrictionsupontherandomnessofsuccessivejumps,thejumpdistances,theangles,oranythingelse.Thefirstassumptionistocon-sidercrystallinematerialsand,inparticular,cubiccrystalswhereeachjumpdistanceis2identical(i.e.,allr’sarethesame)thenRnisgivenasn-1nj-222Rnni=+rr2ÂÂcosq,i+jj=1i=1n-1nj-2Ê2ˆ=+nrÁ1ÂÂcosqiij,+˜(5.29)Ën¯j=1i=12wheretheidentityofther’sisdropped.ThisequationgivesRnforoneparticleafternjumps.Theaveragevalueisobtainedfrommanyparticleseachwithnjumps.Thevalue2ofnrremainsunchanged,butthedoublesumtermsareaveragedwiththeresult:Ê2n-1nj-ˆ22Rnni=+rÁ1ÂÂcosq,i+j˜(5.30)Ënj=1i=1¯不得转载谢谢合作LWM 5.3RANDOMWALKPROBLEM89Nowcomestheimportantpart.Ifwesimplyassumethatthereisrandomnessinthejumpssothateachjumpdirectionisindependentoftheprecedingjumpandalljumpsareequallyprobablenegativeandpositivejumps,thenthe+and-valuesofcosinetermswilladdto0,andthefinalresultisobtained:22Rn=rorn(5.31)2R=nrnFromequation(5.31)welearnthattherootmeansquaredisplacementgivenontheleftsideoftheequationisproportionalton.Foramorphousmaterialsthejumpdistanceisirregular,andthustheaveragevaluecanbeusedforrrsothat22Rn=r(5.32)n5.3.1RandomWalkCalculations-8Startwithsomeunitcellwhereatomicdistancesareoftheorderof10cmor1Å-10-62(10m).ConsideradiffusivityofaboutD=10cm/sthatapproximatelycorresponds-5toCinaFeat900°C,aswasusedabove.Now,equation(5.31)yieldsRn=10mfor110-12-10secondusingn=10s(seesection5.2whereP=2D/a0)andr=10m.In3hoursor414-3about10seconds,nisabout10yieldingRn=10mastherandomwalkdistance.However,ifweconsiderthestraightlinedistanceobtainedbyplacingallther’sendto-1010-144end,orthetotaldistancetraveled,weobtain(10m·10s·10s)10morabout6miles-3comparedtothenetdistanceof10m.5.3.2RelationofDtoRandomWalkWiththerandomwalkresultintermsofthelatticeparameter,222Rn==rna(5.33)n012UsingtheresultsinSection5.2whereG=#/t·atom=n/t,wehaveD=2a0Gand1212ÊÊ1ˆ2ˆDt==aGtanornain3-D(5.34)2020ËË6¯0¯211or2Dt=na0perdirection.In3-Dmultiplyby,becausethereare33jumpsinanyonedirection.FinallyweobtainanexpressionforDintermsoffundamentalparameters:226Dt==naR(5.35)0nandÊ1ˆ2Da=G(5.36)Ë6¯0不得转载谢谢合作LWM 90DIFFUSIONINSOLIDS113322}aoFigure5.3FCClatticeshowing12nearestneighborsforanatomatthecenterofplane1.5.3.3Self-DiffusionVacancyMechanisminaFCCCrystalInFigure5.3weseetwojoinedFCCunitcells(withoutallthepositionsoccupiedforclarity)wherethenearestneighborsandplanesareidentified(planeslabeled1,2,3).Weconsidertheself-diffusionproblemwhereatracer(anisotope)atomisusedtomonitorthemigration.Gistakenastheaveragenumberofjumpspersecondperatom,asabove.Withn1asthenumberoftraceratomsonplane1thenn1·G·dtisthenumberoftracerthatwilljumpfromplane1inatimeintervaldt.G·dtisproportionaltothenumberofnearestneighborsitesmultipliedbytheprobabilityofnearestneighborsitesbeingvacant,pv,andtheprobabilitythatatraceratomwilljumptoaparticularvacantsiteindt(i.e.,w·dtwherewisthefrequencyforthevibration).AllofthisissummarizedasG◊dwdtpt=12◊◊◊(5.37)vwherethenumber12comesfromthenumberofnearestneighborsforanFCC,asseeninFigure5.3.Forthiscalculationconsidertheatomatthecenterofplane1,andnotethatthereare12nearestneighborswith4eachonplanes1,2,and3.Forafluxfromplane1toplane2,J12only4of12providepotentialnearestneighborsites,soJn=◊◊◊4pw(5.38)121v212Likewiseforthereverseflux,Jn=◊◊◊4pw(5.39)212v121Inapuremetalwheretheadjacentsitesareindistinguishable,ww=(5.40)2112pp=(5.41)vv12andn1=x·C1,whichisthenumberperarea=C1·a0/2.Asaresulta0J=◊◊◊◊-4pw()CC(5.42)v122不得转载谢谢合作LWM 5.4OTHERMASSTRANSPORTMECHANISMS91andwith(C1-C2)=-(a0/2)dC/dX,weobtain2dCJap=-◊◊◊w(5.43)0vdXpN==#ofvacancies(5.44)vv2thenD=a0·Nv·w.Onceagain,wehaveDintermsoffundamentalparameters.For2thecaseofinterstitialdiffusionasimilardevelopmentyieldsD=g·a0·w,wheregisageometricalconstant.5.3.4ActivationEnergyforDiffusionExperimentallyitisfoundthatDfollowsanArrheniusexpressionas-EkDTDDe=(5.45)0Toseethatthisfunctionalityisobservedfordiffusionandformanyphysicalandchem-icalprocesses,consider,again,atwo-statesituationaswastreatedinChapter4forthepredictionofthenumberofvacanciesinaperfectcrystal.Thepresentsituationinvolvesthemigrationofanatomfromonepositiontoavacantposition.ForthisitrequiresanenergyinputofED.(Thisexamplecanalsobetreatedasavacancymovingintheoppositedirection.)Asisusualinvirtuallyallprocesses,foradiffusingspeciestomovetoanewsite(whethersubstitutionalorinterstitial),someenergyinputisrequired(forbondbreak-ing,forsqueezingthroughintersticesandoversaddlepoints,etc.).Thusthissituationcanbecastasthetwo-stateprobleminChapter4(e.g.,seethedevelopmentofequation4.34)whereaBoltzmann-likeenergyfactordeterminestheoccupancyofthestatesbasedontheenergydifferencesamongstates.5.4OTHERMASSTRANSPORTMECHANISMS5.4.1PermeabilityversusDiffusionFigure5.4aillustratesthecommonsituationoftwointerfaceswithsolidstatediffusionoccurringbetweentheinterfaces.Atinterface1,say,agassolidinterface,althoughthisisnotnecessary,agasdissolvesinthesolid,meaningthegascrossesboundary1anddis-solves.Followingthisinterfacereaction,thegasdissolvedinthesoliddiffusesbyonemechanismoranothertointerface2whereitexitsintovacuum.Ifwetagaparticulargasspeciesandfollowitfromthegasonthelefttothevacuumontherightandmeasureitsflux,wecallthisnetobservedfluxforthethreeseriesprocessesapermeationflux,andtheoverallprocesspermeation.Forsuchaseriesprocess,theslowestofthethreestepsintheseriesprocessdeterminesthepermeationrate.Itmaybediffusionoroneoftheinterfacereactionsat1or2(solubilizationordesolubolization,respectively).Theper-meationfluxorrateisoftentheexperimentallydeterminedrate.Assuchitisoftentheratethatiserroneouslyascribedtodiffusion,particularly,whereitisbelievedthatdiffu-sionisoccurringsomewhereinthesystem.Oneuseful,butnotexclusive,testtodeter-minewhetherthemeasuredfluxisreallyadiffusionflux,andnotlimitedbysomeother不得转载谢谢合作LWM 92DIFFUSIONINSOLIDSPP'P'JDC1C2a)12O2SiO2SiPo2J=JJ2D=JJ=kCJ3=kreactC2J1C1C2Lb)12Figure5.4(a)Threeseriesprocessesshowingdissolutionatplane1,diffusionfromplane1toplane2,andremovalatplane2;(b)anactualexampleofthreeprocessesinseriesfortheoxidationofSiinO2.process,istodetermineifthemeasuredfluxisproportionalto1/L,whereListhepathlengthinthesolid,namelythedistancebetweeninterfaces1and2.ThistestisobtainedfromFick’sfirstlawwherethefluxJisproportionaltoDC/DxandDxisthepathlength.Ifthetestispositive,thefluxislikelyadiffusiveflux.Insummary,permeabilityincludesthereactionsat1and2anddiffusioninthecaseabove,andingeneral,alltheprocessesfrominputtooutput.Permeabilityisameasureofthetotaltransportofmatterthroughasystem.Astraightforwardexampleofthreeprocessesinseries,whereoneisdiffusion,isthethermaloxidationofsiliconinoxygengas.Inthisprocessacleanedsiliconsurface,typ-icallyasinglecrystalsurface,isexposedtooxygenattemperaturesabove500°C.AlmostinstantlyaSiO2filmforms,andfromthispointintimeforwardtheoxidegrows.Tofollowthefilmformationkinetics,oneobservestheincreaseinthethicknessLofSiO2asafunc-tionoftime,dL/dt.AsshowninFigure5.4b,thefirstprocessintheseriesprocessschemeisthedissolutionofO2intheSiO2atthegas-solidO2–SiO2interface.Thisprocessistyp-icallyfastandresultsinaconcentrationofO2intheSiO2ofC1attheouterinterface.ThefluxcorrespondingtothisprocessisJkp=(5.46)1soloxThisprocessisessentiallyHenry’slaw.ThedissolutionofO2inSiO2isproportionaltothepressureofO2withtheHenrylawconstantksol.ThesecondprocessisthediffusionofO2throughtheSiO2ofthicknessLtotheSiO2–Siinterface,resultinginasmallerconcentrationofO2,C2.InthesteadystatethefluxcorrespondingtothisprocessisgivenasDdCD()CC-oxox12J==(5.47)2dLL不得转载谢谢合作LWM 5.4OTHERMASSTRANSPORTMECHANISMS93ThethirdprocessintheseriesschemeisthereactionoftheO2atconcentrationC2withSiattheinnerSiO2–Siinterface.ThisprocesscontinuallyremovesO2andpreventsaccu-mulation.Thefluxcorrespondingtothisprocessisexpressedbyafirst-orderreactionbetweenO2withSi:Jk=C(5.48)32reactAfirst-orderrateexpressionwithkreactthefirst-orderrateconstantwaschoseninitiallyforsimplicitytodescribethereactionoftheO2arrivingattheSisurfacetoreactwithaconstantconcentrationofSiatthecrystalsurface.Laterresearchconfirmedtheassump-tionoffirstorderinO2.ThisthreefluxschemecanbesimplifiedbyconsideringthatJ1,agasphaseflux,ismuchfasterthantheothertwofluxesthatoccurinthesolidstate.Withprocessesinseriestheslowestprocessdeterminestheobservedrate.Therefore,becauseJ1ismuchfasterthantheotherfluxes,weneednotconsideritfurther.FluxesJ2andJ3mustbeequalafteraninitialtransient.Toseethis,considertheconsequencesofusingtheformulasaboveforJ2andJ3.Ifthefluxeswerenotequal,theneitherJ2orJ3wouldbethelargerflux.IfJ2islarger,thenthesupplyfluxofoxidanttotheSiO2–Siinterface(thatdependsonthedifferenceC1-C2)wouldincreaseC2attheinterface.TheincreaseinC2wouldinturnincreaseJ3,whichwouldreduceC2andregulatetheprocess.Likewise,ifJ3werelargerthanJ2,thenJ3wouldreduceC2,andinsodoing,increasethedifferenceC1-C2andtherebyincreaseJ2,againregulatingtheseriessteps.Thusthisself-regulatingsetofseriesfluxesmustbeequal,andhenceasteadystateobtainsaftersomeinitialtransientduringwhichthefluxesequalize.NowwithJ2=J3,C2canbeobtainedasfollows:D()CC-ox12kC=(5.49)react2LandDCox1C2=(5.50)kLD+reactoxArateexpressioncanbeformedusingthisvalueforC2intherelationship:JJJ===WdLdtk=C(5.51)23react2withC2substitutedfromabove,andWistheconversionfrommolesofoxygengasinJ222-3tomolesofoxygeninsolidSiO2,andWhasthevalueofabout2.3¥10cm.Thisvalue3isobtainedfromthedensityofSiO2of2.2g/cmdividedby60g/molforthemolecularweightofSiO2,allmultipliedbyAvagadro’snumber.Arateequationcanbewrittenwiththeseresultsasfollows:dLkDCreactox1W=(5.52)dtkLD+reactoxUponintegrationthisyieldsalinear-parabolicdependenceofthicknessLontimeofoxidationtasfollows:tA+=+constL2BL(5.53)不得转载谢谢合作LWM 94DIFFUSIONINSOLIDS2P’1PFigure5.5Pressuredifferenceacrossapipecausesconvectivegasflowfromplane1toplane2.Themotionofagasparticle(filledcircle)inthepipeisbiasedbytheconvectiveflow(arrows).whereA=W/2DoxC1andB=W/kreactC1.AandBarecalledtheparabolicandlinearrateconstants,respectively.DiffusionthroughDoxentersintotheparabolicconstantwhilethelinearconstantdependsupontheinterfacereactionthroughkreact.Thislinear-parabolicformulahasbeenverifiedforthethermaloxidationofsilicon.Thisrealexampleillus-trateshowmasstransport,intermsofdiffusion,couplesintothepermeationproblem,anditillustratesthatoftentheoverallpermeationprocessisobservedinwhichdiffusionisonlyapartoftheprocess.5.4.2ConvectionversusDiffusionFigure5.5showsacylindricalsystem(apipe)withtwoboundaryplanes1and2.Dif-fusioncanbethoughtofasoccurringfromeitherplaneasarandomjumpingfromanoccupiedstate,say,onplane1,withequalprobabilityinalldirectionsandobservedasafluxwhenthespeciesarrivesatanunoccupiedstateonplane2.ThediffusionalfluxiswrittenasJ12,andasdiscussedabove,itdependsonN1andN2or,better,onthecon-centrationgradientfrom1to2.Atlowtotalgaspressurestherandomnessrequiredfordiffusionaltransportwillbemaintained.However,wecanalsoimaginealargepressuredropDp(=p-p¢)fromplanes1to2,andhightotalpressuresofaspecies.Thepressuredropfromplanes1to2givesrisetoaconcertedflowofgasasisindicatedbythearrowsinFigure5.5.Thenwecanimaginethataparticleonplane1isrepeatedlybumpedbytheparticlesfallingdowntheDp/Dxgradient(whereDxisthedistancefrom1to2)fromtheleftat1toward2.Repeatedcollisionswithotherparticlesmovinginthisdirectionwilleventuallyimpartavelocitycomponenttotheoriginallyrandomlymovingparticlethatundergoescollisions.Thusthemotionoftheparticlewillnolongerberandom,butratherstronglyinfluencedbythesurroundingflowingparticles.Itisasifawindwereblowingfrom1to2thatcarriesrandomlymovingparticlesalong.Thiskindofmotioniscalledconvectivetransport.ConvectivetransportoccursathighpandlargeDp,whilediffusionaltransportoccursatlowpandDpsoastomaintainrandomness.5.5MATHEMATICSOFDIFFUSIONInthissectionwesolveFick’slawsforseveralimportantcases.ThisisincontrasttotheapproachtakenabovewhereFick’slawswereobtainedfromfirstprinciples.Bothapproachesareimportant,sincebytheatomisticapproachwecanunderstandthenature不得转载谢谢合作LWM 5.5MATHEMATICSOFDIFFUSION95ofdiffusionandbythephenomenologicalapproachwecanunderstandthedynamicalprocessofdiffusion.5.5.1SteadyStateDiffusion—Fick’sFirstLawThephenomenologicaldiffusionfluxequationshownatthebeginningofthischapterasequation(5.1),knownasFick’sfirstlaw,canbeapplieddirectlytosituationswherethereexistsasteadystateconditioninconcentration.Thismeansthatateverypointinthemediumunderstudy,theconcentrationsarenotchangingintime.Thisconditiondoesnotmeanthattheconcentration(orbetterchemicalactivity)isthesameateverypoint.Indeed,ifthatwerethecase,thenthedrivingforcefordiffusion,theconcentrationgradient,wouldbeabsent(dC/dx=0).ThesteadystateisexpressedasdC=0(5.54)dtwiththexaxisparalleltothegradientintheconcentration,afluxJisproducedinresponsetotheforce,dC/dx:-dC.Jµ(5.55)dxItisintuitivethatthefluxisdowntheconcentrationgradientinthedirectionofdecreas-ingC;henceanegativesignisappropriatetoindicatethis.Withaconstantofpropor-tionality,aphenomenologicalcoefficient,D,weobtainFick’sfirstlaw:-DdCJ=(5.56)dxItisusefultoexaminetheunitsfordiffusionproblems:2-Dd()displacementtimeC2J()massdisplacement◊time=4dx()massdisplacementInusualunitsformass,displacement,andtime,22Dd()cms◊CJ()molcms=-4dx()molcmItisfoundthatthediffusioncoefficient,D,oftencalledthediffusivityvarieswithdirectioninacrystallattice.ThisisentirelysensiblebecausethederivationsaboveofFick’slawsarebasedontherandomjumpingfromandtoatomicpositionsorinterstices,bothofwhichhavecrystallographicbias.ThusDisatensorofrank2(ithasthreevariations)correspondingtothethreefoldcoordinatesystem:D00D00aa00Dand00D(5.57)ab00D00Dac不得转载谢谢合作LWM 96DIFFUSIONINSOLIDSforcubicandorthorhombicsystems,respectively.Theappropriatefluxequationsare-DdC-DdC-DdC111213J=++xdxdydz-DdC-Ddc-DdC212223J=++(5.58)ydxdydz-DdC-DdC-DdC313233J=++zdxdydzHowever,inmostcasesanaverageDisusedinasinglefluxequation.AnExampleofFick’sFirstLaw—SteadyStateDiffusion.ThecarburizingofFeprovidesthatproducessteelisaclassicexampleofasteadystatesituation.Anironpipewithradiusr,wallthicknessdr,andlengthlisshowninFigure5.6.Tocarburizethepipetomakesteel,onecanflowCH4(acarburizingorcarboncontaininggas)downthepipewiththepipeattemperatureTandwithagasthatcanreactwithand/orcarrytheCawayfromtheoutersurfaceofthepipe.TheCfromtheCH4willcommencetodiffuseintotheFe.WhentheconcentrationofC,[C],inthepipedoesnotchangewithtimeasgivenbyequation(5.54),asteadystateinCisgivenasd[]C=0(5.59)dtThentheamountofC,qdividedbythetimeisconstant,q/t,isconstant.ThissteadystateconditionoccursafteratransientwhenCbuildsupfromaninitialzeroconcen-tration.ThefluxiswrittenasqJ=(5.60)CAt◊WithareaofasectionofpipewithlengthlgivenasA=2prl,thefluxisqJ=(5.61)C2prlt◊drr}lFigure5.6Ironpipeoflengthlradiusrandwallthicknessdrcaburizedbyflowingacarboncontaininggasdowntheinsideoftheheatedpipe.不得转载谢谢合作LWM 5.5MATHEMATICSOFDIFFUSION97Understeadystate,byFick’sfirstlaw,wehaveq-DdC=(5.62)2p◊rltdrThiscanbereadilysolvedforqas-◊D22pprlt◊dC-◊DltdCq==(5.63)drdrlnExperimentsaredonewhereqismeasured(thetotalamountofC)aswellasCasafunc-tionofrwithtandlknown.OnewaytoobtainCasafunctionofdepthristoliterallyshavethepipeonalathe,keepingtrackofthedepthintothepipewallandchemicallyanalyzetheshavingsforC.AplotofCversuslnryieldsastraightlinewithslope=-q/D·2pltfromwhichDcanbereadilyextracted.ThisanalysisassumesasteadystateandthatDisnotafunctionofconcentration.EarlierinSection5.4.1theexampleofpermeabilitydiscussedwastheoxidationofSiwithgaseousO2whereasteadystateapproximationamongtheseriesprocessesledtotheequation2(5.53)tA+=+constLBLwhereA=W/2DoxC1andB=W/kreactC1.ThisproblemissimilartotheFecarburizationproblemaboveinthatthereisasupplyofdiffusantononesideofasolidandamech-anismtoremovediffusantontheothersideofthesolid,althoughthesemethodsaredif-ferentinthetwoproblems.Theseriesofthreeprocessescanleadtoasteadystate,andFick’sfirstlawcanapplytothediffusionstep.ForlargeSiO2filmthicknessesthepara-22bolictermwithLwillbemuchlargerthanthelinearterm,L>>L.ConsequentlytheoverallSioxidationprocesswillbedominatedbythediffusiontermthatistheslowprocessforlargethicknessesandlongtimes.Theresultisthatthicknessandtimearerelatedbyaparabolicrelationshipandtheinterfacereactionscanbeignored.Ontheotherhand,forsmallSiO2filmthicknessesthelineartermcandominate.ThisexampleofSioxidationisanimportantoneinmicroelectronics,sinceSiistypicallyoxidizedtoformanelectronicallyusefulsurfacetobuildcomputerchips.5.5.2Non–SteadyStateDiffusion—Fick’sSecondLawManyimportantproblemsindiffusioncannotbesolvedusingthesteadystateassump-tionthatd[C]/dt=0.Forexample,intheprecedingproblemonthecarburizingofFe,onecouldinquireaboutthedistributionofcarbonbeforesteadystateisachievedatthebeginningoftheexperiment,whentheconcentrationofCischangingintime.Thusanexpressionford[C]/dtisrequired.WeobtainedtheresultusingFick’sfirstlawandmassbalance,namelybykeepingtrackofarrivalsanddeparturesonagivenreferenceplane.ThisresultwasFick’ssecondlawequation(5.21),whichisoftentermedacontinuityequation:2dCdC=D(5.21)2dtdx不得转载谢谢合作LWM 98DIFFUSIONINSOLIDSWewillillustratebelowthatthesolutionstothiskindofequationaredependentontheboundaryconditions,ortheconditionsofthediffusionexperiment.5.5.2.1SolutionstoFick’sSecondLawWewillconsiderseveralusefulsolutionstoFick’ssecondlawthataredistinguishedbyinitialandboundaryconditions.Aswasdoneabove,inallsolutionsweconsiderDtobeaconstantandnotafunctionofconcentration.Weusethefirst,“thinfilmsolution”tocomputethediffusionofafixedamountofmaterialintoanother(orthesame)material.Next,ratherthanstartingwithafixedamountofdiffusant,weconsideraninfinitesourceofdiffusant,atleastforshortdiffusiontimes.Last,weconsiderthecasewheresurfaceconcentrationisheldtozerooranyfixedvalue.Thesurfaceconcentrationcanbeheldtozerowiththeuseofareactantthatreactswithdiffusantcomingtothesurfaceandtherebyremovesitfromtheproblem,aswasdonefortheFecarburizationandtheSioxidationcasesabove.Forsimplicity,onthesolutionswewillatfirstassumethattheinitialconcentrationofdiffusantiszero.Laterwereturntoconsidermorecomplexsolutionswherethisisnotthecase.Thesesolutionsareallshorttimesolutions.Theshorttimesolutionsareimportantinmicroelectronicswhere,forexample,adopantisdesiredinalocalizedregion,andalsofordeterminingD’sforvariousspecies.Thefinalsolutionweconsiderisageneralformforalongtimesolution.Thelimitofthelongtimesolutionsiscompletehomogenization,wheredC/dx=0.Thelongtimesolutionsareparticularlyimportantinmetallurgywhereuniformdistributionsofselectedalloycomponentsaredesired.Therearemanypossibletechnologicallyandscientificallyimportantdiffusionsitua-tionsrepresentedbyFick’ssecondlaw,eachofwhichhasdifferentboundaryconditionsandconsequentlydifferentmathematicalsolutions.Thetreatmentinthissectionisintendedtocoveronlythemoreinterestingcasesandprovideabriefoutlineofthemath-ematicalmethods.Formoreexamplesthereaderisencouragedtoconsultthereadingmaterialscitedattheendofthischapter.ThinFilmSolution.ForthisanalysisFigure5.7showsathinfilmofonematerialcoatedontotheendofabarofanotherorthesamematerial.Ifthematerialsarethesame,theprocessistermedself-diffusion.Thetotalamountofdiffusant,a,isfixedatthenumberofatomsoriginallyinthedepositedfilm.Thesupplyofdiffusantbecomesdepletedasthediffusionprocessescontinues.Alargeliteratureonthesubjectofself-diffusioninmetalsexistsforthethinfilmsolution.InourexampleweconsideraradiotracerofametalA,A*,coatedononesurfaceofabarofpureA,andwithasecondbarofpuremetalAweldedtothefreesurfaceofthecoating.ThisisnothingmorethanathinfilmofA*sandwichedinbetweentwobarsofpureA.ThisdiffusioncoupleisheatedatconstanttemperatureTforatimet.Aftersomediffusiontime,thedistributionoftheconcentrationA*,CA*(x),isconsideredaGaussiandistributionwhosetailsofA*AA**AProfileofA*ProfileofA*Figure5.7BarofmetalAcoatedwithathinfilmofanisotopeofA,A*.HeatingthebarproducesaGaussiandiffusionprofile.不得转载谢谢合作LWM 5.5MATHEMATICSOFDIFFUSION99penetrateintothebarsshowninFigure5.7(ononesideofthesandwich)asthedashedcurve.InFigure5.7thecenterofthetracerthinfilmistheoriginfortheproblematx=0.AftermoretimethematerialspreadsintothebarandthepeakinCA*atx=0,CA*(0)falls(theGaussianflattens),butsincetheamountofmaterialisconstant,theareaundertheCA*(x)Gaussianisconstant.Next,forthisexperiment,weneedtosetuptheboundaryconditionsforthesolution.TheboundaryconditionsaretheallimportantfactorsincorrectlysolvingFick’ssecondlawequationforaspecificcase.Firsttheinitialandboundaryconditionsforthespecificsituationmustbecastinanalyticalform:1.Beforethediffusionexperimentstarts,att=0,theconcentrationofdiffusant,A*,is0withinthebarofA,soCA*=0exceptatx=0.2.Atx=0,thereisnogradientsincethereisauniformlayerofA*,sodCA*/dx=0.3.AstheexperimentensuesandafterashorttimetheconcentrationofA*remainsatzeroawayfromthesurfaceofthebar.Inotherwords,thetimefortheexperi-mentsisveryshortrelativetothetimeneededforhomogenizationtooccur,oreventooshortforA*toreachtheendofthebar.ThusCA*(•,t)=0.4.Theamountofdiffusingmaterial,A*,isfixed.Thereisnosourcebeyondthenumberofatomsoriginallydepositeduponthebar,andthisisexpressedas•C=a(5.64)ÚA*-•ThesolutionofFick’ssecondlawforthissituationisaGaussianoftheforma-xD24tCxt(),=e(5.65)pDtThissolution(showntobethecorrectsolutionbelow)canbeusedtofindDbytakingthe2logarithmofbothsidesandthendifferentiatingwithrespecttoxtoyieldthefollowing:dC()ln-1A*=(5.66)2dx4Dt2whichcanbevisualizedasaplotoflnCA*versusx.TheslopeoftheresultingstraightlineyieldsD.Ifintheexampleabovearadiotracerdepositedononeendofthebarisofthesameordifferentmaterial,firsttherewillbeafixedamountofdiffusant(a),andthenallofthediffusantwilltravelinthedirectionx>0with0definedastheplaneofthesurfaceofthebar.IntheexampledepictedinFigure5.7thereisnopossibilityofsolidstatedif-fusioninthex<0directionbecausethereisnosolidonthatsideofthedeposit.However,supposethatafterdepositionofthematerialontothebar,anotherbarisweldedsothatasymmetricalsandwichiscreatedwithdiffusantinthemiddleofthetwoequalbars,aswasdiscussedabove.Inthiscasehalfofa(orA*)diffusesinx>0andhalfinx<0direction.Thesolutionforeithersymmetricalhalfofthediffusioncoupleisa-xD24tCxt(),=e(5.67)2pDt不得转载谢谢合作LWM 100DIFFUSIONINSOLIDSWithonlyhalfofthecoupleitisasiftheotherhalfofthediffusingmaterialisreflectedtothepresenthalf,effectivelydoublingtheamountofdiffusant,andconsequentlythe2inthedenominatordisappears.MathematicalInterlude.Equation(5.67)(orequation5.65)wasstatedtobethesolutionforthethinfilmcase.However,inordertoshowthatC(x,t)aboveisasolu-tiontoFick’ssecondlaw,weneedtodosomemath.Inparticular,weneedtodothefollowing:1.Differentiateoncewithrespecttotandtwicewithrespecttox.ThenthesetwotermsshoulddifferbyDasgivenbyFick’ssecondlaw.2.Testthelimits:(a)at|x|>0,Capproacheszero,CÆ0,astÆ0;(b)x=0,CÆ•astÆ0.ThefirstderivativeofC(x,t)inequation(5.67)is2dCa-12--xD224txxD4tapDÊ1ˆ-32=()pDt◊◊+◊◊ee-()pDtdt24Dt22Ë2¯2xD224txD4taxeapDe=◊-◊(5.68)212328Dt()pDt4()pDtNowthederivativeofC(x,t)istakentwicewithrespecttox:-xD24tdCa12-xD24t2x-axe=()pDt◊◊=e◊(5.69)12dx244DtDt()pDt2-xD24tdC-axe-x-xD24t-a=◊◊+◊e21212dx42Dt()pDtDt4DtDt()p2-xD24t-axe-a-xD24t=◊-◊e(5.70)2212128Dt()pDt4DtDt()pNowacomparisonofequations(5.68)and(5.70)showsthatthedifferencebetweenthese22twofinalexpressionsfordC/dtanddC/dxisD.Whilethisagreementisnecessary,itisnotsufficient.Inadditiontheconsistencyofinitialconditionswiththesolutionalsoneedstobetested.Firstatx>0,CÆ0astÆ0.SimplyinsertingtheseinthesolutionC(x,t)aboveyieldstheindeterminateform:C=•/0.Toproceed,wemakethefollowingsubstitutions:2-BtaxAeA=,B==,thenC(5.71)122()4BD4DtWenexttakethederivativeofnumeratoranddenominator,andapplythelimitsaccord-ingtoL’Hospital’srule.Thisisasfollows:-32At0=(5.72)Bte•不得转载谢谢合作LWM 5.5MATHEMATICSOFDIFFUSION101whichremainsindeterminate.Tocontinue,wemakesomefurthersubstitutions:B22-2z==,thentBz(5.73)tand-z2AeAzC==(5.74)22-2zBzBeTakingthederivativeyields1(5.75)2zzewhichyields0inthelimitaszÆ•(tÆ0).Thisistheappropriatelimit.Alsox=0,forcÆ•astÆ0needstobechecked.ThisisdoneasÊaˆlimCxt(),astÆ=0limÁ2˜=•(5.76)Ë2()pDt¯asisrequired.Thusthesolutionischeckedasappropriateandcorrect.Semi-infiniteSolidSolution.Figure5.8ashowsakindofproblemaddressedwiththesemi-infinitesolidsolution.MaterialAisjoinedtomaterialB;eachispureanddevoidoftheotheratthebeginningoftheexperiment(laterwewillrelaxthisrequirement).Onewaytosolvethisproblemistothinkofeachsideofthediffusioncoupleasmadea)ABb)CBDa0aiX0c)CDa0X0Figure5.8(a)BarofmetalAjoinedtometalBat0;(b)thinslabsimaginedateachsideofthecoupleforwhichthethinfilmsolutioncanbeused;(c)theGaussianineachslabduringdiffusion.不得转载谢谢合作LWM 102DIFFUSIONINSOLIDSupofismallslicesofwidthDaandeachoftheseslicesareaiawayfromtheoriginasisshowninFigure5.8b.AsAdecreasesonsideA,Bincreases,andonsideBasBdecreases,Aincreases.WecommencetheanalysisbyconsideringeventsonsideBwherex>0.OnecanimaginetwoidenticalbarsjoinedwhereoneispureCuandtheotherispureNi.LaterwecanconsidermorecomplexsituationssuchasapureNibarjoinedtoaCubarinitiallywith10%Ni.Asforthethinfilmcaseabove,thefirststepistowritetheinitialandboundaryconditionsinanalyticalform:1.Atthestartofthediffusionexperiment,thereisnomixingofAandB.CB=0forx<0att=0andCA=0forx>0att=0.2.Initiallyatt=0,CB=CiBforx>0andCA=CiAforx<0att=0.UsingFigure5.8bconsiderthataregioninBiscomposedofnthinslices,eachofwidthDaandcross-sectionalarea1,andeachsliceisatx-ai.Theuseofx-aifortheposi-tionisusefulbecause,whenx=ai,eachoftheGaussiansiscenteredontheslabinques-tion.Thiswillbeclearerbelow.AttheoutsetonesliceinBhasCiB·V((amount/vol)·vol)=CiB·Da·1ofsolute.Foreachslicethethinfilmsolutiondiscussedaboveapplies;thatis,ineachsliceaGaussianobtains(eachslicehasafixedamountofmaterial)andchangeswithtimeintheslicewhereequation(5.67)isagaingiven:a-xD24tCxt(),=e(5.77)2pDtFigure5.8cshowsthissituationwherethesolutionwouldbethesuperpositionoftheGaussians:in=Cxa2DtiB--()i4Cxt(),ª◊ÂDaei(5.78)2pDtiThissumcanbeconvertedintoanintegralbytakingthelimitofalargenumberofverythinslices:nÆ•,DaiÆ0,yielding•C--()24iBxaiDtCxt(),=Úeda(5.79)2pDt0Nowsubstituting(x-a)/2Dt=hwecanrewritetheintegralasxDt2C-h2Cxt(),=iBÚedh(5.80)p-•whereda=-2Dtdh(5.81)withtheminussignreversingthelimits.Ata=0,xh=(5.82)2Dt不得转载谢谢合作LWM 5.5MATHEMATICSOFDIFFUSION103Table5.1Errorfunctionvalueszerf(z)zerf(z)zerf(z)000.550.56331.30.93400.250.02820.600.60391.40.95230.050.05640.650.64201.50.96610.100.11250.700.67881.60.97630.150.16800.750.71121.70.98380.200.22270.800.74211.80.98910.250.27630.850.77071.90.99280.300.32860.900.79702.00.99530.350.37940.950.82092.20.99810.400.42841.00.84272.40.99930.450.47551.10.88022.60.99980.500.52051.20.91032.80.9999andata=•,h=-•.Thisintegraliscalledanerrorfunction,andthevaluesareshowninTable5.1:z2-h2erf()ze=Údh(5.83)p0ThefinalsolutioninreadilyusableformisCxiBÈʈ˘Cxt(),=+1erf(5.84)2ÎÍË2Dt¯˚˙atx=0,C(0,t)=CiB/2.Thisformisobtainedfromthesplitintegrals:0xDt2CʈiBCxt(),=ÁÚerf()zz+Úerf()˜(5.85)2˯•0whereerf(•)=1,erf(-z)=-erf(z),erf(0)=0.ThissolutionpinstheconcentrationofBattheinterfacetohalfoftheinitialconcentrationofB,orCiB/2.ThecasethatwesolvedwasontheBsideoftheABcouple(x>0),andthesolutionabovetracksthedecreaseinBontheBside.Nowusingtheresultsobtained,wecanexploretheAsideandthechangeinB(increase)onthatside.TheAsideisx<0.Forthispurposeweslightlymodifyequation(5.84),thex>0solutionforBinB(decreases):CxiÈʈ˘Cxt(),=+1erf(5.86)2ÎÍË2Dt¯˚˙Atx=0,c(0,t)=Ci/2;atx=•,c(•,t)=CiandCireferstotheinitialconcentrationvalueforB.Usingthisformula,wesubstitutex<0or-xandobtainCxiÈʈ˘Cxt(),=-1erf(5.87)2ÎÍË2Dt¯˚˙不得转载谢谢合作LWM 104DIFFUSIONINSOLIDSsinceerf(-z)=-erf(z).Theterminsquarebrackets[]inequation(5.87)iscalledtheerrorfunctioncompliment,erfc,anditisoftenseparatelytabulated.NowletusconsiderseveralotherimportantsolutionswherethesurfaceorboundaryisheldateitheraconstantC(0,t)=CSwiththeinitialzeroconcentrationofdiffusantinthemedium,Ci=0,andthenwhereCiπ0wherethereisalreadysomesoluteinitiallyinthesolid.ThesemorecomplicatedcasesarebestsolvedwiththeuseoftheLaplacetransform.MathematicalInterlude.TheLaplacetransformLofafunctionf(t)forpositivevaluesoftisdefinedas•-ptLp()=Úeftdt()(5.88)0wherepissufficientlylargetoforcetheintegraltoconverge.Forexample,iff(t)=1,then•-pt1Lp()==Úedt(5.89)p0usingthedefiniteintegral:•-ax1Úedx=(5.90)a0OtherSolutionsforSemi-infiniteSolids.Consideracasewherethesurfaceconcentrationisfixedattheoutsetofthediffusionexperiment.Wesolveequation(5.21):2dCdC=D(5.21)2dtdxusingLaplacetransforms.ForC=CSatx=0andt>0,andtheinitialconditionsthat-ptC=0atx>0andt=0,wemultiplybothsidesbyeandintegratefrom0to•withrespecttot:••2-ptdC-ptdCedtDe-=0(5.91)ÚÚ2dtdx00Ifweinterchangetheorderofintegrationanddifferentiationandassumethat•-ptCSCC¢==edt(5.92)ÚSp022ThenthesecondtermontheleftisgivenasdC¢/dx,andthefirsttermontheleftcanbeintegratedbypartstoyieldpC¢asÚÚudv=-uvvdu(5.93)不得转载谢谢合作LWM 5.5MATHEMATICSOFDIFFUSION105and••-ptdC-pt•-ptÚÚedt=[]Ce0+=pCedtpC¢(5.94)dt00ThenFick’ssecondlawistransformedintoanordinarydifferentialequation:2dC¢D=pC¢(5.95)2dxwiththesolution:CS-pxDC¢=e(5.96)pForequation(5.96)C¢remainsfiniteasxgoesto•.FromatableofLaplacetransformsthefunctionwhosetransformisgivenbythisexpressionforC¢isxCxtC(),=erfc(5.97)S2DtForthecasewhereCSisagainconstantbutCiπ0,bythesamemethodsthefollowingsolutionisobtained:CxtC(),-xS=erf(5.98)CC-2DtiSNowforthecaseofCi=0aswastreatedabove,thesameerfcformulaisobtainedforanyCS.ForthecaseofCS=0,thereisoutdiffusionofthebackgrounddiffusant,Ci,withtime.ItisgivenasxCxtC(),=erf(5.99)i2DtSinceerfz=0forz=0anderfz=1forz=•,atanypositionwithinthesolid,x,C(x,t)isdecreasingastincreases.5.5.2.2LongTimeSolution—HomogenizationThelastcasediscussedishomogenization,oralongtimesolution.AnotherusualwaytosolveadifferentialequationoftheformofFick’ssecondlawistoseparatevariables.Anysolutionofthisequationmustinvolvebothxandt,sowecanwriteC(x,t)ingeneralformasCxtFxGt(),=()()(5.100)IfwethenusethisinFick’ssecondlaw(equation5.21),weobtain21dGDdF=(5.101)2GdtFdx不得转载谢谢合作LWM 106DIFFUSIONINSOLIDSwhichseparatesthevariables.Bothsidesmustbeequaltothesameconstant,whichfor2conveniencewecandenoteas-lD.Nowwecanwriteandsolvetheseparatedifferen-tialequations:1dG2=-lDGdt(5.102)-l2DtSolution:Ge=2DdF2=-lD2Fdx(5.103)Solution:sinFAxBx=+()llcosCombiningtheseindividualsolutions,weobtain-l2DtCxte(),=+()AsinllxBcosx(5.104)whereAandBareconstants.Amoregeneralsolutionwouldbeasumofsolutions:•-l2Cxt(),=+ÂemDt()AsinllxBcosx(5.105)mmmmm=1whereallconstantsaredeterminedbytheinitialandboundaryconditions.Inordertousethissolution,asetofspecificinitialandboundaryconditionsneedtobespecifiedandfromwhichtheconstants(Am,Bm)areobtained.Wedonottreatthisfurtherexcepttopointoutthatthesolutionappearstobethesuperpositionoftheperiodicwavesofconcentrations.Forexample,tounderstandthediffusionintoaslabofmaterial,wecanimagineitasapeakinconcentrationatonesidethatdecreasestoanearlyhomogeneousstateintimeforafixedamountofdiffusantorbyholdingthesurfaceatafixedconcentration.5.5.2.3DiffusionLengthItisoftenusefulinthinkingaboutsolidstatediffusionproblemstobeabletoestimatehowfaranatomwilltravelinacertaintimeorhowlongwillittaketodiffuseacertaindistance.ThatproblemisstraightforwardtosolveifweconsiderthechangeinshapeofaGaussianwithtime.WiththehelpofFigure5.9thatshowstheevolutionofaGaussianovertimewhereÚadx=constant,wecalculatethedistancebetweentheplanedefinedatx=0,C=CmaxandtheplanewhereC=1/e(Cmax).Thedistancefromx=0tothisplaneiscalledthediffusionlength.WestartfromtheexpressionaCt()0,=(5.106)122()pDtThisisthesolutiontoequation(5.67)attheplanewherex=0.Clearly,Cdecreasesasf(1/t),butfromconservationofmatterÚCdx=constant.Thedistancetothe1/e(Cmax)planeincreasesast.Atx=0,aCt()0,=(5.106)122()pDt不得转载谢谢合作LWM 5.5MATHEMATICSOFDIFFUSION107CCmax1/eCmax-XXtt012Figure5.9EvolutionofaGaussianprofileovertime(solidtodashedprofiles).Aconcentrationpointontheoriginalprofilemovefromt1tot2inthetimedifference.and1ln[]Ct()02,=--lnapln()ln()Dt(5.107)2Atx,2ÊaˆÊ-xˆCxt(),=expÁ˜(5.108)Ë24pDt¯ËDt¯Thusweobtain21xln[]Cxt(),=--lnapln2()ln()Dt-(5.109)24DtTheconditionsforthesolutionareÊ1ˆatxC==01,,andatxC=1(5.110)Ëe¯Thenusingrelations(5.110)andsubtractingequation(5.109)fromequation(5.107),weobtain2xln11-ln()e=(5.111)4Dtand2x01+=(5.112)4Dtwhichyieldsthefinalresult:xD=2t(5.113)不得转载谢谢合作LWM 108DIFFUSIONINSOLIDSThisformulateachesthatthedistancetraveledisproportionaltot.So,ifwewanttomeasure(usingchemicalanalysis)thedistanceBtraveledintoAatvarioustimes,wecanplotthedataasxversust.Ifdiffusionisthemechanismformasstransport,astraightlinewillresultandyieldDastheslope.RELATEDREADINGR.J.BorgandG.J.Dienes.1988.AnIntroductiontoSolidStateDiffusion.AcademicPress,SanDiego.SimilartoShewmon’sclassicbutwithmanymoderntopicsandexamples.R.Ghez.2001.DiffusionPhenomena.KluwerAcademic,Dordrecht.MoreadvancedthattheShewmonbookandtheBrogandDienesbookbutreadableandwithexcellentinsightsintomoderndiffusionproblemsinscienceandtechnology.J.Crank.1975.TheMathematicsofDiffusion.ClarendonPress,Oxford,England.Atreasuretroveofsolveddiffusionproblemswithalltherelevantmathematics.P.G.Shewmon.1963.DiffusioninSolids.McGraw-Hill,NewYork.Aclassicbookindiffusionformaterialsscientists.EXERCISES1.CisdiffusedintoatubeofFeattherateof3.6g/100h.Thetubehasani.d.=0.86cm,ano.d.=1.11cm,andalengthof10cm.ThevariationofCwithradiusisgivenbelow.CalculateDanddetermineifDisdependentonconcentration.r(cm)wt%C0.5530.280.5400.460.5270.650.5160.820.4911.090.4791.200.4661.320.4491.422.AradioactiveCuthinfilmwasdepositedontoapureCubar.Afteranisothermalannealfor20hours,theradioactiveCuwasdeterminedatvariousdepthsintheCubar.DetermineD.-2Activity(counts/min-mg)Averagedistance(10cm)500014000225003150045005153.Foradiffusionof10radiotracerCuatomsintoCuat800°Ctoadiffusionlength-52xDof10cm,usingD0(Cu)=0.16cm/sandED=2.07eVcalculatethediffusioncoefficientDandtimet.4.YoufindthatthediffusioncoefficientDofboronisafactorof10greaterthanthatofAsat1150°Candthatthesolidsolubility(assumeequaltothesurfaceconcen-不得转载谢谢合作LWM EXERCISES109tration)ofAsisafactorof10higherthanthatofboron:DB=10DAS;C0(As)=10C0(B).Forthesamediffusiontime,t:(a)Whichspecies,boronorAs,hasthelargestdiffusionlength?2(b)Whichhasthelargestnumberofatoms/cmdiffusedintoSi?(c)Whatwouldleadtothegreatestchangeindiffusionlength:a20%changeintemperature,time,orsurfaceconcentration?5.Showthata-xD24tCxt(),=e2pDtisthesolutiontoFick’ssecondlawforthethinfilmproblem.-46.ConsideranalloywithauniformconcentrationofXof0.25wt%(D=1.6¥10m/sat950°C).Ifthealloyisthentreatedat950°Cwithagasthatbringstheconcentra-tionofXto1.2wt%,calculatehowlongitwouldtaketoachieve0.8wt%AT0.5mmbelowthesurfaceofthealloy.7.PfromadepositwasdiffusedintoSiat1000°Cfor2hours,resultinginajunctiondepthof0.537microns.EstimateD.8.(a)AtomsofAghavemigratedadistancexintimetinasolidasgivenbelow.AssumethatthemigrationwasbyFickiandiffusion,estimateD.(b)Usingyourcalculationsfrom(a),justifythatdiffusionwasagoodassumptionforthemechanism.x(cm)t(s)0020.0021060.2109.(a)AthinplasticmembraneisusedtoseparateHfromacarriergasstream.The3concentrationofHononesideofthemembraneisconstantat0.025mol/m3whileontheothersideitisalsoconstantat0.0025mol/m.Themembraneis-62100mmthick.GiventhatthefluxofHis2.25¥10mol/m-s,calculateDforHfromtheseconditions.(b)SupposethatinitiallytheconcentrationofHwaszerointhemembranebutbuilt3upto0.0025mol/cmat10mmfromthesurfacein1minute,calculateD.(c)Explainhowyoucantestwhethertheprocessintheplasticmembraneswasadiffusionprocessorpermeation.10.ASicrystalisputincontactwithInvapor,inordertodiffuseInintoSisothatat-3thedepthof10cmtheconcentrationofInishalfofthesurfaceconcentration.For-122DIn=8¥10cm/sat1600K,howlongmustyouheatSiincontactwithInvaportoperformthisdiffusion.10-811.Anatommakes10randomjumpsof10cmeachandeachwithidenticalproba-bility.Determinehowfarfromthestartingpointtheatomtravels.12.MetalAisdiffusedintometalBat1300°C(BalsodiffusesinA).D0andEDforA2inBis8cm/sand80kcal/mol,respectively.CalculateDforAinBat1300°C,and不得转载谢谢合作LWM 110DIFFUSIONINSOLIDSthenfindhowlongitwilltakefortheconcentrationofAinBat0.001cmfromthejunctiontobehalfofthesurfaceconcentrationofA.13.Anisotopeofametalwascoatedontoabarofthesamemetalthatwasinitiallyfreeoftheisotope.Afterheatingfor1hourat1000°C,theisotopewasfoundtobeat-5-3-6arelativeconcentrationof4¥10at10cmbelowthesurface,and2.3¥10at-38¥10cm.CalculateD.不得转载谢谢合作LWM 6PHASEEQUILIBRIA6.1INTRODUCTIONPhaseequilibriadealswiththeexistenceofphases,namelywhatphasesexistwhenequilibriumoccurs.Foramaterialsscientistaknowledgeofphaseequilibriaprovidesaroadmaptothechemicalsystem(s),theingredients,thestatesofmatterwithwhichoneisdealing.Itisparticularlyimportantforworkonanewmaterialormaterialssystemtofirstperusewhatisknownaboutthephaseequilibriuminthesystemunderstudy.6.2THEGIBBSPHASERULE6.2.1DefinitionsTheGibbsphaserule(orjustphaserule)establishestherelationshipamongphasesandcomponentsandintensivevariables.Asisusualandcrucialtounderstandingthermo-dynamics,wecommencewithasetofdefinitions.Anintensivevariableisavariablethatdoesnotrelyonthetotalamountofmaterialpresentsuchaspressure,temperature,energypermole,enthalpypermole,accelerationduetogravity,compositionfraction,andrefractiveindex.Thisiscontrastedwithexten-sivevariablesthatdodependonthetotalamountofmaterialsuchas:mass,weight,volume.ThenumberofintensivevariablesisdesignatedhereinasF.Fissometimescalledthenumberofdegreesoffreedom,sinceitsnumericalvalueisthenumberofintensivevariablesthatneedtobespecifiedtodescribeaparticularequilibriumsituation.Aphase,P,isahomogeneous,physicallydistinct,andmechanicallyseparableportionofmater-ialwithacompositionandstructure.Acomponent,C,isaspecifickindofatomorElectronicMaterialsScience,byEugeneA.IreneISBN0-471-69597-1Copyright©2005JohnWiley&Sons,Inc.不得转载谢谢合作LWM111 112PHASEEQUILIBRIAmolecule.TheGibbsphaseruletobederivedbelowestablishesarelationshipamongF,C,P,andotherintensivevariables(typicallytemperatureTandpressurep).ThedefinitionsforPandCbreakdownforsmallquantities,namelyonanatomisticscale,asdootherthermodynamicvariables.Asthermodynamicvariables,PandCapplytolargenumbersofatomsormolecules.Strictlyspeaking,itisimpropertousethephaserulewhendescribingatomicsizedcollectionsandevennanometersizes.Indeed,mostofequilibriumthermodynamicsisnotreliableindealingwithsmallsystems.However,asafirstapproximationitisuseful,andoftendone,inmaterialssciencetocommencedescrib-ingtheexistentphasesusingtheanticipatedthermodynamicvaluesforsmallsystemsbasedonthelargesystemvalues.Aphasediagramisaplotoftheintensivevariables,suchasfreeenergy,pressure(p),temperature(T),andmolefraction(X),andthisplotisamapofthephasesthatexistforthematerialssystem.Atequilibriumthemoststablephasehasthelowestfreeenergy.Figure6.1isaphasediagramforthesinglecomponent(C=1),water.First,itisseenthatundertheexcursionofthepandTintensivevariableswaterdisplaysthreedistinctphases:solid,liquid,andgas.RecallthatinChapter2,Figure2.1bisalsoaphasediagramforwaterbutoveramuchmoreextensivepandTrange.Inthisfigurewaterwasshowntohaveavarietyofdifferentphases,butallinthesamesolidstateofmatterandthisistermed“polymorphism,”whichreferstodifferentphasesofthesamecompositionandusuallyinthesamestateofmatter.InFigure2.1beachofthesolidcrystallinewaterphasesdepictedhasadistinctcrystalstructure.Figure6.1isfocusedonthatregionofthewaterp,Tphasediagramatornearp=1atmthatisofinteresttopeopleonearth.Noticefirstthatthelinesonthediagramareessentiallytwophaseboundariesatwhichtwophasescoexistatthep,Tconditionsspecifiedbytheexactpositionontheline.Thereisonepointonthep,TdiagramlabeledAwherethreephasescoexist.Needlesstosay,thispointiscalledatriplepoint.ThispointisalsocalledaninvariantpointbecausetherePressure(atm)Liquid1SolidVaporA0100Temperature(C)oFigure6.1Pressure–temperaturephasediagramforwater.不得转载谢谢合作LWM 6.2THEGIBBSPHASERULE113isonlyonesetofconditionsatwhichthethreephasescoexist,p=0.006atm,T=0.0075°C.NextweturnattentiontoanotherC=1phasediagraminFigure6.2forsulfur.Onceagain,allthreestatesofmatterarerepresentedandthesolidsulfurhastwophases,rhombicandmonoclinicsulfur.TherearetwotriplepointsatAandB,andatpressureshigherthanDthemonoclinicphasenolongerexistsatleastasfarasisshownfortheconditionsonthisdiagram.LastwelookattheFep,TphasediagraminFigure6.3andseethreedistinctpolymorphsofsolidFeandthreetriplepointsatA,B,andC.Soevenwithsinglecomponentmaterialsadiversephasebehaviorcanberealizedatearthambientconditions.Beforeproceedingtomorecomplicatedmulticomponent(C>1)phasediagrams,wereturntotheGibbsphaseruleandderiveit.Thephasediagramsusedinthefollowingdiscussionsareonlyapproximationsforinstructionaluse,andaccuratediagramsbasedoncurrentdataareavailableincompila-tionsandintheoriginalliterature.6.2.2EquilibriumAmongPhases—ThePhaseRuleTheGibbsphaserulerelatesF,P,andCandenablesthedeterminationoftheminimumvalueofFforequilibrium.Thederivationcommenceswiththeconsiderationofaseriesofphases,P(1)...PasshowninFigure6.4.EachphasehascomponentsC(1)...C.Imagineaflaskcontainingoilandwaterthatseparatesintotwoimmisciblephases.Nowmixinseveralcomponentseachofwhichhassomesolubilityinbothoilandwater.Ifthetwophasesareincontactsothatallthegradientsarereducedtozero,thenequilib-riuminT,pandchemicalpotential,m,willobtain.Indeed,foreachcomponent,C,the1042D101Monoclinic10-2RhombicLiquidPressure(atm)-410CBA-610VaporO50100150200Temperature(°C)Figure6.2Pressure–temperaturephasediagramforsulfur.不得转载谢谢合作LWM 114PHASEEQUILIBRIA104SolidLiquid1Fe(BCC)d10-4aFe(BCC)gFe(FCC)CPressure(atm)Vapor10-8B10-12A500100015002000Temperature(°C)Figure6.3Pressure–temperaturephasediagramforiron.P(1)P(2)P(3)...PC(1)C(1)C(1)C(1)C(2)C(2)C(2)C(2)............CCCCFigure6.4PphaseseachwithCcomponentsinequilibrium.chemicalpotentialwillbethesameamongthePphasesinwhichthecomponentappears.Atequilibriummm()12=()==...m()P111mm()12=()==...m()P222(6.1)MMMmm()12=()==...m()PCCCforeachcomponent1,2,...,C.NowFigure6.4depictsPphasesinintimatecontactwitheachphasehavingeachoftheCcomponents.ItshouldbeunderstoodthatineachofthephasestheCcomponentsdonothavetobeinthesameconcentrationasinthe不得转载谢谢合作LWM 6.2THEGIBBSPHASERULE115otherphases.Indeed,thecomponentswilladjustinthephasesaccordingtotheirindi-vidualsolubilities.Itisrequiredthatforequilibriumthechemicalpotentialforeachcom-ponentbethesameineachphase.Forouroilandwatercasetherearetwophases,eachwithanequilibriumfractionoftheCcomponents.Ingeneral,foreachphasethemolefractions,Xi,sumto1andaregivenasSX=1(6.2)iIneachphasebecausethesumofthemolefractionsisunityonlyC-1compositionalvariablesarerequired(thelastisknownbydifference)inaphase.ThusthetotalnumberofcompositionalvariablesforallPphasesisPC()-1(6.3)ForeachC,equilibriumobtainsatthephaseboundaries.IfweassumethatallPphasesareinintimatecontactsothateachCcanequilibrateamongallPphases,thenthereareP-1equilibria,oroneforeachCateachboundary.HencethenumberofequilibriumrelationsisCP()-1(6.4)whichisP-1equilibriaforeachC.IfwenowincludethecommonsituationofTandp(notethatweusePforphasesandpforpressure,forclarity)asvariablesthatneedtobespecified,thetotalnumberofintensivevariablesisgivenbythenumberofvariablesminusthenumberofrelationsamongthevariables:FPC=-()12+-[]CP()-1(6.5)F=-+PCP2-+CPC(6.6)FCP=-+2(6.7)If,asisoftenthecase,weestablishtheequilibriaonthesurfaceoftheearthinourlab-oratory,thepressureisfixed(atnear1atm).Thenthe2forTandpisreducedto1foronlyT(pisspecified).Likewise,ifwedoworkonanotherplanetwherethereisadif-ferentgravity,and/orperhapsworkaroundastrongmagneticfieldorwherevermoreintensivevariablesrequirespecification,thenthenumber2willincreaseappropriately.Sincewewillconsiderthatthephaseequilibriaareestablishedonearthinanormallaboratory,weusuallyneedonlytospecifyT.HencetheformofthephaseruleformcommonlyusedisFCP=-+1(6.8)6.2.3ApplicationsofthePhaseRuleWenowconsiderthepressure(p)versusTphasediagramforH2OshowninFigure6.1inwhichpisavariableandthususeF=C-P+2.WeseethatC=1,sincewehaveonlyH2O.InthisfigurethelinesrepresentonedegreeoffreedomF=1.ThelinesindicatetwophasesthatcoexistatthespecificpointonalinesoP=2.ThusF=1-2+2=1.Anotherwaytoviewthisisthattostayonaline,oneneedstospecifyeitherTorp,and不得转载谢谢合作LWM 116PHASEEQUILIBRIAtheotherisdetermined.However,offthelineintheareasbetweenlinesthereisacon-tinuousrangeofTandpevenwhenoneisspecified.InthesesinglephaseareasC=1andP=1,andthusF=2.Noticetheintersectionoflinesatthepointp=0.006atmandT=0.0075°C.Aswaspreviouslymentioned,thispoint(A)iscalledaninvariantpoint,sinceitexistsasapointinintensivevariablespaceandisthereforecompletelyspecified.Unlikealine,novariationispossiblethatcanmaintainthecoexistencedescribed,namelythecoexistenceofthreephasesatapoint.Theinvariantpointhasnodegreesoffreedom:F=0,andisthereforecompletelyspecified.ForFigure6.2theonlydifferenceisthattherearetwoinvariantpointsatAandBwherethreephasescoexistandF=0.Alsothisdiagramteachesthatifonestartstoheattherhombicphaseatanyp<50atm,rhombicSdoesn’tmelt!Rather,itconvertstothemonoclinicphasethatdoesmeltatTgreaterthanabout130°Catpgreaterthanabout-510atmbutatlowerpressurestherhombicphasesublimes.Thephasediagramalsosug-geststhatifonewantstopreparethemonoclinicphaseat1atm,oneroutecouldbetoheattheSuntilitmelts,cooltothemonoclinicphase,andthenquenchrapidlyinliquidnitrogen(95K)inordertopreventatomsofSfromrearrangingfurther.Figure6.3forFedisplaysthreedifferentsolidphasesforFeandthreeinvariantpoints(A,B,C).Thediagramshowsthatstartingat1atmand25°C,FeisinasinglephaseregionaFe(F=2),whichuponheatingtoabout910°CisontheequilibriumlinethatseparatesaFefromgFe(F=1);convertstogFe,whichinturnconvertstodFeatabout1394°C;andthenmeltsatabout1538°C.Onceagain,thesehightemperaturephasesofFecanbeobtainedatlaboratoryambientbyquenching.6.2.4ConstructionofPhaseDiagrams:TheoryandExperiment6.2.4.1TheorySincephasediagramsrepresentequilibriumconditions,itispossiblebyequilibriumthermodynamicsreasoningtoobtainformulasfromwhichtocalculate,oratleastestimate,equilibriumphasediagrams.Firstconsiderthattherearethreeconditionsfortrueequilibrium.Asystematequilibriummustbeatthermal,chemical,andmechanicalequilibria.Fortwophases,aandb,atequilibriumwhereaandbarecomposedoftwosubstances(purecompoundsorelements),meaningAandBareineachphase,thefollowingconditionsmustobtainatequilibrium:1.Ta=Tb.2.ma(A)=mb(A)andma(B)=mb(B).3.dwa-b=0.Conditions1and2arereasonablyobvious,butcondition3requiresthatreversibleworknotbedonewhenacomponentchangesbetweenthetwophases.Forexample,ifagasandsolidphasesareindynamicequilibriumwheregasischangingtosolid,andviceversa,thisinterchange,ifatequilibrium,takesplacewithoutpdVworkinthegasorsolidoranyotherkindofwork.Considerthephaseequilibriumbetweenphasesofonecomponentsuchaswaterinsolid,liquid,orvaporordiamondandgraphite.ThevariationofGwithpatdT=0isexploredstartingwithanexpressionforG,thenH,thenEasG=-HTS,,H=+EpVE=+qw(6.9)dG=--dHTdSSdT(6.10)不得转载谢谢合作LWM 6.2THEGIBBSPHASERULE117dG=++--dEpdVVdpTdSSdT(6.11)dG=+++--dqdwpdVVdpTdSSdT(6.12)Nowwithdwindicatingworkdonebythesystemdw=-pdVanddq/T=dS,thendGbecomesdG=-SdTVdp+(6.13)WithdT=0asabove,dGVdp=(6.14)Dependingonthephasesexisting,thenatureofthisequationcantakedifferentforms.Forexample,foramoleofanidealgaswherepV=RT,dG=RTdlnp()(6.15)whichisobtainedfromdp/p.However,forarelativelyincompressiblesolidwithavolumeVS,dGVdp=(6.16)SNowtoreturntoourtwophasesaandb,wewritedG=dG(6.17)abdG=-SdTVdp+=dG=-SdTVdp+(6.18)aaabbbFromwhichuponrearranging,wehaveDDSdT=Vdp(6.19)whereDS=Sb-SaandDV=Vb-Va.Atequilibrium,withDG=0=DH-TDS,DS=DH/Tandcombining,weobtainÊDHˆdT=DVdp(6.20)ËT¯ThendpÊ1ˆDH=(6.21)dTËDV¯TwhichiscalledtheClapeyronequation.Ifthechangeinphaseisfromsolidtoliquid,thenDHcorrespondstomeltingorfusion,andDVcorrespondstodifferenceinthespe-cificvolumesfortheliquidandsolidphases.不得转载谢谢合作LWM 118PHASEEQUILIBRIAConsiderasolidphaseinequilibriumwithavaporphasewherethespecificvolumeofthevaporismuchlargerthatthatforthesolid(DV=Vv).ThentheClapeyronequationcanbewrittenasdpÊ1ˆDHvap=Á˜(6.22)dTËV¯TvIfweassumethatthevaporbehaveslikeanidealgas,thenweobtaindpÊpˆDHvap=(6.23)dTËRT¯TFromequation(6.23)weobtainDHvapÊ1ˆdlnp()=d(6.24)RËT¯Usingthisrelationship,onecanthendefinethesolid–vapororliquid–vaporboundaryinap–Tphasediagram.Forthesolid–liquidboundarythefollowingequationisapplicable:dpÊ1ˆDHfus=Á˜(6.25)dTËDV¯TSLForsolid–solidequilibriaasimilarformisusedwiththeappropriatevolumeandDH.ForamoreprecisecalculationthechangesintheHvalueswouldneedtobeappropri-atelycorrectedusingheatcapacity(CP)values.Therefore,ifthethermodynamicproper-tiesareknown,thenphasediagramscanbecalculatedoratleastapproximated.6.2.4.2ExperimentOnetraditionalmethodfordeterminingphasediagramsistomeasuresolidificationtemperaturesduringtheslowcoolingofmoltenmixturesofvariouscompositions.Figure6.5showstheresultfromavarietyofstartingcompositionsofCuandNi.ForpureCu(farleft)andpure(Ni)thehorizontalplateauseeninthecoolingcurve(T=constant)plateaucorrespondstothephasetransitionCuorNiL()()ÆCuorNiS()()(6.26)TheTatwhichthephasetransitionoccursisthemeltingpoint,whichforapurecrys-tallinematerialisawell-definedTatafixedp.Thetimedurationfortheplateau,orwidth,correspondstothetimenecessaryforthecompletionofthephasetransition.ThemoreCupresent,thelongerittakes.However,duringtheentiretimetheTremainscon-stant,sincethelatentheatoffusion,DHf,isreleasedduringsolidification.Oncethenewphaseisformed,theDHfassociatedwiththetransitionisnolongeravailable.Thuscoolingofthesolidnowcontinuesbutatadifferentratethantheliquidbecauseofthedifferingheatcapacities(Cp).ExactlythesamephysicalpictureobtainsforpureNi,thoughtheplateauoccursatadifferentT.However,forallthemixedcompositions,theregionofthecoolingcurvesbetweenthebeginningandendofthemorehorizontalregionsisnotlevel.Theslopeisalsodifferentinthethreedifferentregionsofthecurve:不得转载谢谢合作LWM 6.2THEGIBBSPHASERULE11915001400C)°13001200LiquidTemperature(1100Solid1000100%Cu80%50%20%0%(100%Ni)TimeFigure6.5Temperature–timedataforthecoolingofvariouscompositionsofCuNialloys.beforephasechange,duringphasechange,andafterphasechange.Inthephasechangeregion,themixturehasbeenobservedtobecomposedoftwophases:liquidandsolid.Figure6.6showsthedatafromFigure6.5replottedwiththeinformationintermsofTandcomposition.Inanycompositionthatissubjectedtocooling,thestartofthetwophaseregionfromthemeltiscalledthe“liquidus”temperaturewhiletheendofthistwo-phaseregion,wherecompletesolidificationoccurs,iscalledthe“solidus”temperature.Thesetemperaturesformtheboundaryforthetwo-phaseregionwherebothsolidandliquidcoexistandvarywiththestartingcompositions.Theupperboundaryofthetwo-phaseregioniscalledtheliquiduslineandthelowerthesolidusline.Diffractiontechniquesareusefulindeterminingboththesolidcrystallinephasesthatexistandthestructureofthephases.Forexample,aseriesofmixturesofCuandNicanbemade,melted,andcooled.X-raypowderdiffractioncanbeusedtoidentifythepres-enceofcrystallinephases.Diffractioncanalsobeusedatelevatedtemperaturetodeter-minesolidtosolidphasetransitionsaswasdiscussedaboveforSandFe.Electronmicroscopy,particularlytransmissionelectronmicroscopy(TEM)whereelectrondif-fractionisobtained,isalsoroutinelyusedtoidentifythepresenceofsolidstatephases.Metallurgistsroutinelyuseopticalmicroscopyinthereflectionmodetoobservethepresenceofdifferentphasesthathavedifferentreflectivities.Samplesthatcontaingrainsofdifferentphasesarecarefullypolishedtoremoveroughnessfromaffectingthereflec-tivity.Experiencedobserverscanevenidentifythephasesinwell-knownmaterialssystemsusingopticalmicroscopy.Wenowexploreinmoredetailthewealthofinformationcontainedwithinphasediagrams.不得转载谢谢合作LWM 120PHASEEQUILIBRIA1500Liquid1400C)°1300LiquidusSolidus1200Liquid+SolidSolutionTemperature(1100SolidSolution1000100%Cu80%50%20%0%(100%Ni)Weight%CuFigure6.6Temperature–composition(wt%)phasediagramfortheCu–NiSystem.6.2.5TheTieLinePrincipleAtielineisanisothermallinethroughatwo-phaseregion.Tielinesareshownasthehorizontaldashedlinesinthetwo-phaseregionoftheCu–NiTversuscompositionphasediagraminFigure6.7.Onanytielinethecompositionsofthetwophasesinequilibrium(liquidandsolidsolution)aregivenbytheintersectionsofthetielinewiththesolidusCa(orCa¢)andliquidusCL(orC¢L)lines.Figure6.7illustratesthisforanoverallcom-positionCoof60%Cubyweight,whichexistsalongtheverticaldashedlinethatinter-sectsbothtielinesat60%Cu.Considerthatthis60%CualloywithNiisat1250°CasindicatedbyColocatedonthetoptieline.Thistielineintersectstheliquidusatabout70%Cu.Thismeansthattheliquidphaseinequilibriumwiththeasolidsolutionisabout70%Cuand30%Ni(CL).TheintersectionofthistielinewiththesoludusandCSyieldsthecompositionforthesolidphasealloy,a,ofabout37%Cuand63%Ni.IfthisCoiscooledfrom1250°Cto1200°C,Coremainsthesame(theoverallcom-positionofthesampleisnotmodifiedsimplybycooling!),butCLevolvestoC¢LandCStoCS¢,asindicatedbythearrowsandanewtieline(thebottomdashedline)canbedrawn.Thisnewtielineat1200°CconnectsCL¢andCS¢.Whiletheoverallcompositionhasnotchanged,thecompositionoftheliquidphasehaschangedfromabout70%Cutoabout82%Cu,andthesolidaphasehaschangedfromabout37%toabout58%Cu.SoboththeliquidandaphasehaveincreasedinCu.Thismaysoundabitstrangeatfirst,butwewillseebelowthatthisisnotamistake.Insummary,inthetwo-phaseregionofaT-compositionphasediagram,atielinecanbedrawnatanytemperaturethroughthecompositionandtemperaturedesired.This不得转载谢谢合作LWM 6.2THEGIBBSPHASERULE12115001400LiquidLiquidusa°C)L+1300e(CSolidusLCCaC’oL1200C’aTemperaturTieLine1100SolidSolution1000100%Cu80%60%50%20%0%Weight%Cu(100%Ni)Figure6.7Temperature–compositionphasediagramfortheCu–Nisystem.Tielines(dashedhorizontallines)areshownforcoolinga60%Cualloyfrom1250°Cto1200°C.horizontallineextendstotheendsofthetwo-phaseregion.ThecompositionoftheliquidphaseisgivenasCLandthesolidphaseasCa,namelytheintersectionsofthetielinewiththeliquidusandsoludus,respectively.AttheTofthistielineCLisinequilib-riumwithCa.Ascoolingtakesplace,anotherlineparalleltothefirstwilldisplaydif-ferentintersectionswiththeliquidusandsolidus,indicatingdifferentliquidandsolidphasecompositionsbutwiththesameoverallcompositionCo.Onanytielinetheweightfraction(orcompositionfraction)ofeachphasecanbeobtainedinasimpleway,andthiswillbeconsiderednext.6.2.6TheLeverRuleBeforetheleverruleisderived,itisusefultostatewhatitisandthenthegoalofthederivationwillbeclearattheoutset.Essentiallytheleverrulequantifiestheinformationabouttheamountofeachofthephasespresent,aswasderivedfromthetielineprinci-ple.InFigure6.8,whichisaslightlymodifiedphasediagramfromFigure6.7,itisseenthatthesegmentofeithertielinetotheleftofCoislabeledSaandthesegmenttotherightislabeledSL.Wewillshowbelowthatthesesegmentlengths,SaandSL,arerespec-tivelyproportionaltotheamountoftheaandLphasespresentwhichisastatementoftheleverrule.Wecommencethederivationwiththedefinitions.Letfa=weightfractionofaandfL=weightfractionofL不得转载谢谢合作LWM 122PHASEEQUILIBRIA15001400LiquidLiquidusaCoL+°C)SS1300aLe(C}SolidusL}CCaC’oL}peratur1200}C’SaSLTema1100SolidSolution1000100%Cu80%60%50%20%0%(100%Ni)Weight%CuFigure6.8Temperature–compositionphasediagramfortheCu–Nisystemshowingtielinesandillustrat-ingtheleverruleusingtielinesegments(S’s).Sincetheweightfractionsofthecomponentsaddto1,wecanwritetherelationshipff+=1(6.27)aLTheoverallcompositionCoisgivenbythesumofthefractionofamountsofthetwophaseseachmultipliedbythecompositionasCf=+=+-CfCfCCf()1(6.28)oLaaLLaaaCCCCCf=+-(6.29)oLaaLaThensolvingforthefractions,weobtainCC-CC-oLaof=andf=or()1-f(6.30)aLaC-CCC-aLaLNoticethatSa(=Co-CL)isthepartofthetielinefarthestfroma,thatSL(=Ca-Co)isthepartfarthestfromL,andthatthedenominatorCa-CListhetotallengthofthetieline.Thenthef’sobtainedabovearefractionsofthetieline,thefractionofaispro-portionaltothesegmentofthetielineontheoppositesideofCo,andlikewiseforthe不得转载谢谢合作LWM 6.2THEGIBBSPHASERULE123Lphase.Phasediagramsareoftendisplayedinweightfractionsespeciallyinthemate-rialsengineeringcommunity.Theleverrulehasthusfarbeendefinedintermsofweightpercent.Theleverrulecanalsobeexpressedsimilarlyintermsofmolefractions.Con-siderthatFigure6.8isplottedintermsofthemolefractionofcopper(XCu)ratherthanweightfractionorpercent.Ofcourse,sinceweightfractionandmolefractionforagivenalloyarenotingeneralnumericallythesame,thefigurewouldbescaleddifferently.Ignor-ingthatfactforthepresent,wecanstartbydefiningthetotalnumberofmolesoftheliquidandalloyphasesasnt:nnn=+(6.31)tLawherenaisthenumberofmolesofthealloyphaseandnListhenumberofmolesoftheliquidphase.Eachphase(aandL)hasbothCuandNi.ThetotalamountofCu,ntXCu,canbeobtainedbysummingtheCuinbothphasesasnX=+nXnX(6.32)tCuaCuLCuAlsotheCuinaisnaXCu(a),andtheCuintheLphaseisnLXCu(L).ThusnX=nX()a+nX()L(6.33)tCuaCuLCuEquatingtheexpressionsforntXCu,weobtainnX+=nXnX()a+nX()L(6.34)aaCuLCuCuLCuNowrearranging,weobtainnX()-X()a=nXLX()()-(6.35)aCuCuLCuCuWiththeaidofFigure6.9,whichisthesamephasediagramasFigures6.7and6.8exceptitisintermsofmolefraction,weobservethattheexpressioninparenthesisontheleftisthelengthofthelinesegmenttotherightofXCu(SL),andthatontherightisthelengthofthelinesegmenttotheleftofXCu(Sa).Usingtheselinesegments,wecanwritenS()=nS()(6.36)aaLLandnSµ(6.37)aaThismeansthattheamountofthesolidsolutionphase,a,isproportionaltothelinesegmentontheoppositesideofthecompositionofthetwo-phasemixture(XCu).LikewisenSµ(6.38)LLSoweseethattheamountoftheliquidphase,L,isproportionaltothelinesegmentontheoppositesideofthecompositionofthetwo-phasemixture(XCu).ItshouldbenotedthatFigure6.9isnearlythesameasFigure6.8intermsofthescalesfortheweightand不得转载谢谢合作LWM 124PHASEEQUILIBRIA15001400LiquidLiquidusaL+∞C)SS1300aLe(X(L)Cu}}SolidusXCuX(a)Cu1200Temperatur1100SolidSolution100010.80.60.50.20(pureCu)(pureNi)MoleFractionCuFigure6.9Temperature–compositionphasediagramfortheCu–Nisystemusingmolefractions.molefractions.However,aswasmentionedabove,weightfractionandmolefractionarenotingeneralthesame.Theformulaforweightfraction(WF)isWeightiWF=(6.39)iS()allweightsandformolefraction(X)isMassMWiiX=(6.40)iSMassMWiiThusforweightsandmassesingramsthefractionswilldifferbythemolecularweightsfortheconstituents.ForCuandNithemolecular(atomic)weightsareabout58.7g/moland63.5g/mol,respectively.ForourchosenvaluefortheweightfractionfortheCu/Nisystemof0.60,themolefractionofCuisabout0.62.Socoincidentallythemassandmolefractionsarecloseinvaluetoeachother,andthedifferenceignoredforthepurposeofFigure6.9.Insummary,thetielineprincipleenablesthedeterminationofthecompositionofthephases,andtheleverruleenablesthedeterminationofamountsofeachphase.Thereexistawidevarietyofphasediagrams.Ratherthanattempttosurveyalargenumberofcaseshere,theemphasiswillbeongeneralprinciplesthatapplytomanykindsofphasediagrams.Thetopicsaddressedbelowrelatingtorealphasediagramsincludecomplete不得转载谢谢合作LWM 6.2THEGIBBSPHASERULE125solidsolubility,partialsolubility,compoundformation,homogeneityrange,andternaryandpseudobinaryphasediagrams.6.2.7ExamplesofPhaseEquilibriaTheequilibriumphasediagramforCu–NishowninFigure6.7illustratesthatthereiscompletemiscibilityinthesolidphasefortheCu–Nialloy.Thismeansthatanycom-positionofCuinNi,andviceversa,canbepreparedunderambientconditions.RecallfromChapter2thatanalloyassumesthecrystalstructureofthehostmaterial.Theques-tionthatarisesis,Whichconstituentisthehostfora50–50alloy?Whilethisseemstobeunanswerable,natureobviatesthequestion.ItisfoundthatbothCuandNiareFCCwithlatticeparametersof0.3615nmand0.3524nm,respectively.FurthermoreCuandNiaresidebysideinthePeriodicTableoftheElements,andthusbothhavesimilarelectronicstructure(differingbyoneelectron)andmasses.ConsequentlyitisunderstandablethatCuandNicaninterchangeforeachotherontheFCClatticeandresultincompletemiscibility.SimilarlyFigure6.10displaysthecompletesolidmiscibil-ityphasediagramforGe–Si.Theseelementsalsohavethesamecrystalstructure(diamondcubic)andsimilarlatticeconstants(0.5658and0.5431nmforGeandSi,respectively)andareadjacentinthesamegroupinthePeriodicTable.WhileSi’selec-2tronicstructurehad2electronsinitsouterMshellas2pandGehas2electronsinits2outerNshellas3p,theelectronicenergybandstructuresaresimilar.Thuscompletemiscibilityisanticipated.ThecompletesolidsolubilityideaspresentedabovecanbecontrastedwithcompletesolidimmiscibilityasshownfortheBi–CdphasediagramshowninFigure6.11.Bi1400LiquidLiquidus1300SolidusC)°1200aL+e(1100Temperatur1000SolidSolution900100%Ge80%50%20%0%Weight%Ge(100%Si)Figure6.10Temperature–compositionphasediagramfortheGe–Sisystem.不得转载谢谢合作LWM 126PHASEEQUILIBRIA400C)°e(Liquid300200TemperaturL+BiL+Cd100EutecticBi+Cd100%Bi80%50%20%0%(100%Cd)Weight%BiFigure6.11Temperature–compositionphasediagramfortheBi–Cdsystem.(rhombohedral)andCd(hexagonal)havedifferentcrystalstructures,widelydifferentlatticeparameters(Bi:0.4736nmanda=57.14°;Cd:0.2979and0.5617nm),anddif-ferentelectronicstructures,andtheyarenotadjacent(horizontallyorvertically)inthePeriodicTable.Thusalargedegreeofimmiscibilityisanticipated.Alsointhisphasediagramthereisapointwherethreephases(L,Bi,andCd)areinequilibrium,atripleorinvariantpoint.Inthiskindofphasediagramthisinvariantpointiscalledtheeutec-tic,anditisthelowesttemperatureatwhichsolidificationoccurs.IfwepreparealiquidofBiandCdattheeutecticcompositionandthenslowlycooltheliquid,atexactlytheeutectictemperaturetheliquidwillconverttoBiandCd,sothethreephasescoexistattheeutectictemperature.ThisreactioncanbeexpressedasLÆ+SolidSolid(6.41)12whereinthiscasethesolidsareBiandCd.Fortwocomponentsandthreecoexistingphasesthephaserulepredictsnodegreesoffreedom,aninvariantpoint:FCP=-+12()atconstantP==-+=310(6.42)Itisalsopossibletohaveotherinvariantpoints.Ifasolidphase,a,convertsuponcoolingtotwoothersolidphases,anotherinvariantpoint(F=0)wouldexist.ThisreactionisgivenasabgÆ+(6.43)andiscalledaeutectoidreaction.Theinvariantpointiscalledaeutectoid.Alsothefol-lowingreactionsinvolvingthreephasesyieldinvariantpoints:不得转载谢谢合作LWM 6.2THEGIBBSPHASERULE127ab+ÆLperitectic(6.44)abg+Æperitectoid(6.45)LLÆ+amonotectic(6.46)12Whileimmiscibilityiscommonplace,totalorneartotalimmiscibilityisrare,andlimitedsolubilityismoreoftenobserved.Figure6.12showsalimitedsolubilityphasediagramforthePb–Snsystem.Onthe100%Pbandthe100%Snsidesofthephasediagramlimitedmiscibilityisseen.OnthePbsideofthediagram,thesolidsolution(alloy)thatisformediscalledtheaphaseinwhichthecrystalstructureisthatofPbwithSnatomstakingsubstitutionalsitesonthePbstructure.OntheSnsideofthephasediagram,thealloyformediscalledthebphase,andisbasicallythecrystalstructureofSnwithsomePbatomsonsubstitutionalsites.PbhastheFCCstructureandSnthetetragonalstructure.PbislargerthanSn,soonecanexpectlessPbinSnthanSninPb,andthisisobservedwiththeaphaseextendingtoabout19%SninPbwhilethebphaseextendstolessthan6%PbinSn.Inbetweenthesetwoalloyphasesthereisimmiscibil-ityandamixtureoftheimmiscibleaandbphasesisfound.Theratioofthecomposi-tionandamountofeachofthesephasescanbedeterminedatanyTuptotheeutectic(about183°C)usingthetielineprincipleandleverruleaswasdescribedabove.AnexampleofhowthequantitativeinformationisextractedfromthephasediagramsisillustratedwiththeuseoftheSn–PbPhasediagramshowninFigure6.12.Wecom-menceat250°Cfor80/20=Pb/Sn(seethedashedlineinFigure6.12withthefilledcircle),andfollowthechangeswhenthesystemiscooledto200°C.At250°CtheapproximatecompositionsareCa=87%PbandCL=67%Pb,ascanbeobtainedfromatielineat250°C(solidlineat250°C).Nowapplytheleverruletodeterminetheamounts:400C)°CoLiquid300a+L200ab+LTemperature(b100ab+100%Pb80%50%20%0%(100%Sn)Weight%PbFigure6.12Temperature–compositionphasediagramforthePb–Snsystem.不得转载谢谢合作LWM 128PHASEEQUILIBRIACC-L0f=aCC-LawithCL-Ca=20andCL-C0=67-80,thenf=65wt%aaCC-8087-0af===35wt%LL2020Nowat200°CtheapproximatecompositionsareCC==83wt%%andL45wtPba4580-f==92wt%aa4583-8083-fL==8wt%L4583-Thus,astheTislowered,thecompositionsofbothaandLmovetowardenrichmentofSn,thecomponentthatmeltsatlowerT.Startingwith1gofalloyat250°C,wehavewtPb=+=◊+◊=fCfC06508703506708.....gaaLLFor1gofalloyat200°C,wtPb=◊+◊=09208300804508.....gThusthePb,whichis80%byweight,remainsat80%.Thusfarwehaveconsideredcaseswheretheconstituentshaveeitherformedornotformedanalloyinthesolidphase.Nowwediscussthecasewhere,insteadofforminganalloyorsolidsolution,anewchemicalcompoundisformed.Itshouldberecalledthatthismeansthatastructurethatisdifferentfromeitherofthestartingcomponentsispro-duced.Figure6.13fortheMg–Nisystemshowscompoundformationinadditiontothefeaturesthatwehavealreadyseen.Averticallineonaphasediagramisacompoundwherethewidthofthelineisameasureofthecompound’sstoichiometryandiscalledthehomogeneityrange.Forexample,ageometriclinewithzerowidthisindicativeofperfectstoichiometry.However,manycompoundshavefinitehomogeneityranges,asisseennotonlyforintermetalliccompoundsasshowninFigure6.13butalsoformanyothermaterials.SolidsolutionsoccuratboththeMgandNisidesofthephasediagramyieldingtheaandballoyphases,respectively.Atabout0.46and0.17weightfractionofMg,thecompoundsMg2NiandMgNi2form,respectively.Thesecompoundsaredenotedbyver-ticallinesonthephasediagram.Atwo-phaseregionexistsinbetweenthesecompoundsindicativeoftheirimmiscibility.Twoeutecticsareseennear0.77and0.10weightfrac-tionMg,asareindicatedbytheLÆsolidphase1+solidphase2reactionoccurringattheeutecticT’sfortheseinvariantpoints.ThecompoundMgNi2meltsabove1000°Cto不得转载谢谢合作LWM 6.2THEGIBBSPHASERULE1291500Liquid°C)CongruentMelting1000L+bIncongruentMeltingL+MgNi2MgNi2MgNiL+MgNi2500L+a2Temperature(MgNi+2baMgNi+MgNiba+MgNi22210.80.50.20(pureMg)(pureNi)WeightFractionMgFigure6.13Temperature–compositionphasediagramfortheMg–Nisystem.liquidofthesamecomposition.Thisiscalledcongruentmelting.However,thecom-poundMg2Nimeltsabove700°C.HoweverthecompositionoftheliquidhasmoreMgthanisindicatedbythestoichiometryofMg2Ni,andasolidphaseMgNi2isalsopro-duced.Atielinedrawnabovethemeltinthetwo-phaseregionindicatestheresultingLandsolidphasecompositions.Thiskindofmeltingwherethemeltedcompounddoesnotexistinthemeltiscalledincongruentmelting.Whiletheverticallinesthatareindicativeofcompoundsappearnarrow,acharac-teristicofstoichiometriccompounds,inrealitythereisalwayssomewidthtothelines.Thelinewidthrepresentsarangeofstoichiometry,albeitsometimesverysmall,inwhichthecompoundexists.Itispossiblethatsomeimportantmaterialspropertiescanchangesignificantlyacrossthewidthofthehomogeneityrange.ThisfactisillustratedinFigure6.14forGaAs,animportantelectronicandopticalmaterial.Notethatthecom-poundGaAsforms,andthereisnoalloyformationofGaAswitheitherGaorAs.Con-sequentlybothsidesofGaAsdisplaytwophaseregions;GaAsmeltscongruently.TheinterestingnuanceisthewidthoftheverticallineusedtoindicatethecompoundGaAs.Thelineshowsmeasurablewidthatanytemperatureandbelowabout800°Cthewidthisabout0.02atomicfraction,andthiswidthismainlyontheGarichside.ThismeansthataGarichGaAscompoundisformed.Above1100°Cthewidthismaximumandabout0.15atomicfractionandnearlysymmetricalaboutthe0.50atomicfraction.TheGarichsideofthestoichiometryyieldsanN-typematerial,whichmeansthatthedom-inantelectricalchargecarriersareelectrons.AlsothisN-typematerialisseveralordersofmagnitudemoreconductivethanamorestoichiometricmaterial.MorewillbesaidlateraboutsemiconductorsinChapters9through11.Sufficeforthepresentdiscussionthatthehomogeneityrangeisaveryimportantmaterialsparametertoknowsomethingabout.Theprecedingdiscussionhasdealtexclusivelywithsinglecomponentandbinaryphasediagrams.However,phaseequilibriainsystemsthathavethreecomponentsare不得转载谢谢合作LWM 130PHASEEQUILIBRIA1500LiquidC)°1000HomogeneityL+GaAse(RangeL+GaAs500TemperaturGaAs+AsGa+GaAs(pureGa)0.250.50.75(pureAs)AtomicFractionAsFigure6.14Temperature–compositionphasediagramfortheGa–Assystem.alsoimportant.Consider,forexample,ceramicmaterialsthatareusuallycombinationsofvariousoxides,andspecificallythephaseequilibriainthealumosilicatematerialssystemthatcontainAl,Si,andO,andthatcanberepresentedonaternaryphasediagram.Figure6.15ashowsaternarydiagramfortheAl,Si,andOcomponentsonwhichthecompositionmapofexistentphasescanbedisplayed.Inthiskindofphasediagraminvolvingthreeelements,manydifferentstructures(phases),compounds,andsolutionscanexist.Fortheuninitiatedthetriangularrepresentationcanbecomplexanddifficulttocomprehend.Furthermorethetwo-dimensionalternarydiagramisforasingletemperature,andevenmorecomplicated,thethree-dimensionaldiagramisrequiredtodisplaythetemperatureinformation.Whileathree-dimensionaldiagramiscomplete,suchadiagramtypicallydisplaysmoreinformationthanisrequired.Anexperimental-istusuallystartstopreparethedesiredmaterialsin,say,theAl,Si,OsystembycombiningvariousproportionsofpowdersofAl2O3andSiO2thatarereadilyavailableandeasytohandle.ThephasediagraminFigure6.15acanbesimplifiedbyconsideringAl2O3andSiO2asthestartingcomponents.OnFigure6.15athissituationisindicatedbythedashedlinethatconnectsthesetwobinarycompounds.Inbetweentheseendpointsarethephaseequilibriathancanexistforthislimitedpartoftheentireternarydiagram.Thislineiscalledapseudobinarycutoftheternarydiagram.Figure6.15bisapseudobinarycutoftheAl,Si,OternarydiagramwithAl2O3andSiO2thestartingcomponents.Thepseudobinarydiagramhasallthenormalfeaturesthatwerepreviouslydiscussed.6.3NUCLEATIONANDGROWTHOFPHASES6.3.1ThermodynamicsofPhaseTransformationsAnimportantapplicationofthethermodynamicsofphaseequilibriaistodeterminethelowestenergypathwayfornewphasestoappear.Oneaspectofthisgeneralissueofphasetransformationisnucleation.Nucleationistheprocessbywhichthefirstvestigeofanew不得转载谢谢合作LWM 6.3NUCLEATIONANDGROWTHOFPHASES131a)O100.10.2MoleFractionO0.550.MoleFractionSi0.20.10Si1Al00.10.20.51MoleFractionAlb)2500LiquidC)°2000L+MulliteL+AlO23L+MulliteL+SiO21500AlO23Temperature(+SiO+Mullite2MulliteMullite(solidsolution)100010.80.50.20(pureSiO)2(pureAlO)23WeightFractionAlO23Figure6.15(a)TriangulardiagramusedtorepresenttheternaryAl–Si–Osystemshowingthelinecon-nectingSiO2withAl2O3;(b)detailswithinthepseudobinarycutfrom(a).phaseappears.Thesmallestpartsofthenewphasearecallednucleiandarethecollectedbuildingblocks(atomsormolecules)ofthenewphase.Wefirstconsiderthethermodynamicsforthetransformationoftheaphasetothebphase:abÆ(6.47)Theveryinitialappearanceofbphaseiscallednucleationthatresultsinnucleiwithadistributionofsizes.Belowwewilldiscussthecriticalnucleisizeastheminimumsize不得转载谢谢合作LWM 132PHASEEQUILIBRIAforastablenucleus.Onceastablenucleusforms,thenewlyemergentphasecangrowandresultinthecoalescenceofthenuclei.Forgrowthofthenewphasematerial,transporttothestablenucleiiscrucial.Considerthetwophases,aandb,inequilibriumattemperatureTE.ConsiderthataismorestableatT>TEandbatTTE).ThenDGvbecomesmorepositive,andthetransformationaswritten(aÆb)islessfavorable.If,ontheotherhand,thereactionisendothermic,DEvispositive,thetransformationismorefavorableathigherT.Thedirectionofthetransformationisobtainedfromthisthermodynamictreatment,butatthispointnothingislearnedaboutkinetics(thepathway)ofthetransformation.Tothisendthespecificthermodynamicsrelatingtotheformationofthenewphaseneedstobeunderstood,andthisiscalledthethermodynamicsofnucleation.不得转载谢谢合作LWM 6.3NUCLEATIONANDGROWTHOFPHASES1336.3.2NucleationFigure6.16showsthenucleiofbthatformfromtheabove-mentionedphasetransfor-mationinthepreviouslyhomogeneousaphase.Tocalculatethenumberofnuclei,n*,weagainconsiderthetwo-stateproblem,andthefactthatitwillcostenergytoproducethebphasenuclei.TheresultistheBoltzmannfactorintermsoftheenergeticsofthespecificcaseathand:ÊDG*ˆnN*=-exp(6.54)ËRT¯whereNisthenumberofsitesavailablefornucleiformationandDG*istheenergythatmustbesuppliedfornucleiformation(i.e.,thebarrierforformation).WereturntotheproblemofcalculatingDG(andthenDG*)fortheaÆbphasetrans-formationbyfirstassumingtheformationofsmallsphericalparticlesofbintheaphase.Laterthisassumptionaboutthenucleishapeisjustified.Asthebphaseformsintheaphase,aninterfaceiscreatedbetweentheaphaseandthebnuclei.Thisinterfacehasanassociatedsurfaceenergy,g,inunitsofenergyperareaproduced.Figure6.17illus-tratestheoriginofthissurfaceenergy.Ifwefocusononeatominthedepictedtwo-dimensionalsolid,wenoticethattheatomisbondedtofouradjacentatoms.Forthepurposeoftabulatingtheenergyrequiredtoformanewsurface,thebulkatomsfirstneedtobefreedbybreakingthefourbondsforeachatomandthentransportingthefreedatom(s)toformthenewsurfacewherethreebondsreform.Fromanenergyaccountingpointofview,itcostsfourunitsofbondenergytofreeanatomandthreeunitsarereturneduponsurfacebondformation.Thusthereisanetexpenditureofoneunitofbondenergytoformaoneatomsurfaceinthissimplemodel.Theformationofarealsurfaceis,ofcourse,morecomplicatedwithstructuralrearrange-ments.Nevertheless,thissimplemodeldemonstratesthatthetotalamountofenergyneededtoformasurfaceisalwayspositiveandproportionaltothenumberofatomsonthenewsurface,hencetothesurfaceareaproduced.RememberthatuptonowwehaveabFigure6.16Anaphasewithsphericalnucleiofbphase.不得转载谢谢合作LWM 134PHASEEQUILIBRIAFigure6.17Bulkatomswithfourboundsandsurfaceatomswiththreebonds.beenconsideringenergypervolumeandnotperareawhencalculatingthermodynamicproperties.Wereturntoconsidertheformationofasphericalnucleusofbwithradiusr,andcalculatethetotalenergyforthisprocess,DGtot,usingthefollowingrelationship:DG=+VolumefreeenergychangeSurfaceenergychange(6.55)totForasphericalnucleusthevolumefreeenergytermisthechangeinthechemicalfreeenergyassociatedwiththetransformation,DGV.However,aswasmentionedabove,thisenergyiscalculatedinunitsofenergypervolume.Inordertoaddthevolumefreeenergytothesurfaceterm,weneedtoremovethegeometricalpartbymultiplyingbythevolumeforthesphericalnucleus,toyieldÊ43ˆprG◊D(6.56)Ë3¯vAswesawabove,wecanadjusttheconditionsandeventhechemistrysuchthatthisvolumetermwillbefavorablemeaning,DGvisnegative.Also,aswasdiscussedabove,thesurfaceenergyterm,g,isenergyperarea.Sobeforeaddingtothevolumeterm,itmustbemultipliedbythesphericalnucleusarea,toyield2()4pgr◊(6.57)不得转载谢谢合作LWM 6.3NUCLEATIONANDGROWTHOFPHASES135Thistermisalwayspositivebecauseitalwaystakesenergytocreateasurface.Itisnownoticedthatthesphereexhibitsthesmallestsurfacetovolumeratio.Hencewithgposi-tive,thesphericalshapednucleiwillprovidethesmallestenergyrequirement,andhencebethefavoredshape.Wecanaddthevolumeandsurfaceenergytermstoobtainthetotalenergy:3Ê4prˆ2DDGtot=Á˜◊+Grv()4pg◊(6.58)Ë3¯Thechangesinthetwoenergytermsontheright-handsideofequation(6.58)dependonnucleisize,asshowninFigure6.18.Aswasdiscussedaboveforthevolumeterm,theconditionscanbeadjustedsothatthistermisnegativeanddecreaseswithincreasingr.Thus,withthesurfacetermpositiveandgrowingmorepositivewithincreasingr,itisinferredthatthesumofthesetermshasaninflectionpointwhichislabeledasr*inFigure6.18.Forsmallnucleisizeswhererr*,theneg-ativevolumetermdominatesandpullsthetransformationprocesstotheright.ThisfigureteachesthatapositiveDGtotresultsforsmallnuclei(rr*).Thismeansthatthesmallnucleiarethermodynamicallyunsta-blerelativetolargernuclei.Atr*thebarrier,DG*,isobtained,andr*,itiscalculatedbelow.Beforeproceedingfurtherwiththisdevelopment,itisusefultotrytoimaginewhatisoccurringandwhy.Itisstraightforwardtoimagineatomsormoleculesmovingaroundandcollidingwithsomeofthecollisionsofaphasespeciesconvertingtobphase,sincewehavealreadymadethisprobablebyadjustingtheconditionstoyieldanegativeDGv.Theinitialresultisthatasmallnumberofatfirstsmallnucleiofbform.Ifthesenucleiaresmallerthanthecriticalsize(r100%)andyetreturntotheiroriginalshape.ThermalexpansionisalsoreadilyunderstoodbyreferencetothePEcurvesinFigure7.4.Figure7.4ashowshorizontallinesinthePEcurvethatrepresentdifferenttempera-tures.Thecenterofeachlineisdifferentduetotheasymmetryinthepotentialcurve.Thecenterpointrepresentsthemeannearestneighbordistance.Forhighertemperaturesthemeanseparationincreasestolargerinteratomicseparations.Themagnitudeofthischangeiscalledthethermalexpansion.Thecoefficient,a,governsthermalexpansionandisdefinedasDla=()∞C(7.12)lOfcourse,weexpectadifferentawheneveradifferentpotentialobtains,asislikelyfordifferentcrystallographicdirections.Typicallyanaveragevalueisreported.Table7.2alsodisplayssomevaluesofaforseveralmaterials.Notethataforrubberandnylonarelargecomparedtoaforinorganicmaterials;SiO2hasoneofthesmallestavalues.Inter-estingly,whenfilmsofrubberandSiO2aredepositedonasubstrateofSiatsomeele-vatedtemperature,say100°C,nostressesoccurduetothermalexpansion.However,whenthesampleiscooledfromthedepositiontemperaturetoroomtemperature,boththeSisubstrateandthefilm(rubberorSiO2)contract.Therubberwillcontractalmost100timesmorethanSi,andSiO2willcontract10timeslessthanSi.ForthecaseofarubberfilmonSi,thelargecontractionoftherubberandtherelativelysmallcontractionoftheSicauseacompressivestresstodevelopneartheSisurfaceandatensilestresstodevelopintherubberfilm.ForthecaseofanSiO2filmonSi,theoppositeoccurs.Namelythe不得转载谢谢合作LWM 7.3ANANALYSISOFSTRESSBYTHEEQUATIONOFMOTION147a)b)Figure7.5(a)Asolidofdifferentmateriallayers;(b)thesamebaras(a)butafterachangeintempera-tureshowingstrainresultingfromdifferentthermalexpansioncoefficients.smallaforSiO2andtherelativelylargeaforSicauseacompressivestressintheoxidefilmandtensilestressintheSisurface.Thesestressescandeformthesubstrate.Figure7.5illustratesthecaseforafilmwithlowerathanthatofthesubstrate.InFigure7.5athefilm-substratesampleisundeformedatthedepositiontemperature,andinFigure7.5bthesamesampleisobservedaftercoolingtoroomtemperature.Thisfiguregreatlyexaggeratesthedeformationimpartedbyathinfilmtoathicksubstrate.However,formaterialsofequalthicknesswithdissimilara’s,significantbendingcanoccurandthiseffectwasusedinolderthermalswitches.7.3ANANALYSISOFSTRESSBYTHEEQUATIONOFMOTIONInthissectionwedevelopamethodfortheanalysisoftheequationofmotionusingthesegmentindicatedbyx,x+dxinFigure7.1.Wesupposethatthelongitudinalwavecom-pressesthesolid,andthatthiscompressedsegmenttravelsdownthesolidbarasthewaveadvances.OuranalysiscommenceswiththebasicNewtonianformulaforforce2dux()Fa==mrAdx◊(7.13)2dt2wheretheunitsfortherighthandexpressionare(mass/volume·area·dx·distance/timeandyieldsmass·acceleration.Usingthisrelationshipandrememberingthats=Ee,s=F/Aandde=du(x)/dx(equations7.2and7.6),wecanrewritethestressformulasasfollows:F=+[]ss()xdx-()xA◊(7.14)不得转载谢谢合作LWM 148MECHANICALPROPERTIESOFSOLIDS—ELASTICITYwhereFisthenetforceactingonthesegment.Nowwewritethechangeinstressindif-ferentialform:∂s◊=+dxss()xdx-()x(7.15)∂xSubstitutingfors,s=Ee,weobtain2∂s∂e()E∂∂{}Eduxdx◊[]()ux()===◊E(7.16)2∂x∂xdx∂xThen2∂ux()F=◊E◊◊Adx(7.17)2∂xEquatingthetworesultsabove(equations7.13and7.17)forFyields22dux()∂ux()rAdx◊=◊E◊◊Adx(7.18)22dt∂xEquation(7.18)canberearrangedtoyield22∂ux()rdux()-◊=0(7.19)22∂xEdtwhichisa1-Dwaveequation.Asolutiontothisequationisanexponentialoftheformikxt()-wuCe=◊(7.20)wherek=2p/l(recallkspacefromChapter3),wisthefrequency,andCistheampli-tude.Also,sincethemomentumpisgivenashp=(7.21)lwhichisthedeBroglierelationship(seeChapter9),wecanwritekas2ppk=(7.22)hToshowthatu(x)aboveisasolution,itisnecessarytodifferentiatetwicewithrespecttobothtimeandposition:duikxt()-w=ikCedxdu2(7.23)22ikxt()-w=ikCe2dx不得转载谢谢合作LWM 7.3ANANALYSISOFSTRESSBYTHEEQUATIONOFMOTION149andduikxt()-w=-iCewdt2(7.24)du22ikxt()-w=iCwe2dtCombined,theyyield22ikxt()-wwrikxt()--+kCewCe=0(7.25)EForthisequationtohold,thefollowingrelationshipsmustalsohold:1222EÈE˘w==k,w(7.26)rÍÎr˙˚soweobtaintheinterestingresultw=vk(7.27)sTherelationshipabove,showninFigure7.6,iscalledadispersionrelationshipbecauseonevariableistheenergyandvsisthespeedofsoundinthemedium.Thevstermhas2theunitsofvelocity.Forexample,Ecanbeinunitsofdynes/cmwithadyneaunitof2322forceasg-cm/sanddensity,r,inunitsofg/cm,andsoforE/rtheunitsarecm/sor2v.TogetasenseofthemagnitudeofE,recallthatatypicalvelocityofsoundisabout5335¥10cm/sandrvariesfromabout2to8g/cm,soweuse5g/cm.Thesereasonable122valuesgiveanorderofmagnitudeapproximationforEofabout10dynes/cm.wDwDkkFigure7.6Frequencywversuslatticevectorkcurveforasolid.不得转载谢谢合作LWM 150MECHANICALPROPERTIESOFSOLIDS—ELASTICITYssssssFigure7.7Deformationofacuberesultingfromnormalstresses(s)appliedtothecube’sfaces.txqh(p/2)-gtFigure7.8Deformationofacuberesultingfromshearstresses(t)appliedtothecube’sfaces.Thereareexperimentaltechniquesavailableinwhichthespeedofpropagationofacousticwavescanbemeasuredformaterials.Withavalueofvsmeasuredexperimen-tallyforaparticularacousticwavelength,andavalueforthedensityofthematerial,r,avaluefortheYoung’smodulus,E,canbeobtained.7.4HOOKE’SLAWFORPUREDILATATIONANDPURESHEARWefirsttreatpuredilatation,atermthatmeansthechangeinshapeduetosomeforce.Whiledilatationisusuallythoughtofasanexpansion,itcanbeboth+and-.Apuredilatationisshowninthreedimensions(3-D)inFigure7.7.Notethatthecubeisuni-formlydeformedbytheapplicationofequalforcestoitsfacessothats/Aisthesameoneachface.In3-Dthefractionalchangeinvolume,DV/V,isthestrain.ThusBVDs=(7.28)VwhereBisthebulkmodulus.Pureshearproducesnochangeinvolume,onlyachangeinshape,asshowninFigure7.8.Ameasureoftheshearstrain,g,isgivenbytheangleq.FromFigure7.8,tanq=x/h,andforsmallq,tanq=q.Thusg=x/h,andtheshearstress,tµg,isgivenas不得转载谢谢合作LWM 7.6RELATIONSHIPSAMONGE,e,ANDn151s}Dy}DxsFigure7.9Lateraldeformation(DxandDy)ofasolid(horizontalarrows)resultingfromappliednormalstresses(s).tg=G(7.29)whereGistheshearmodulus.7.5POISSON’SRATIOAnaxialforceperarea,s,appliedtothetopandbottomofasolid(intheydirection),asshowninFigure7.9,causesstrains(deformations)laterally(inthexandzdirections),asindicatedbythearrowsforthexdirectioninthefigure.Theratiooftheinducedtrans-versestrain(inx)relativetotheaxialstrain(y)isthePoissonratioexn=-(7.30)eyFormanymetalstherangeforthePoissonratioisn0.25–0.35and0.18forSiO2,andsinceitisaratiooflikevalues,itisunitless.Intuitiontellsus,anditiscommonlyobserved,thatforatensileaxialstraininy,apositivenresultsthatisindicativeofaneg-ativestraininx.However,inrarematerialsandforcertaindirectionsinthesematerials,anegativenisobserved.Theseunusualmaterialsarecalledauxeticmaterials.7.6RELATIONSHIPSAMONGE,e,ANDnTherelationship(s)amongthevariousmechanicalpropertieshelpuscalculateunknownpropertiesoratleastestimateunknownproperties.TherelationshipsamongE,e,andnarereadilyobtainedbyHooke’slawandsuperposition.Superpositionallowssmall(elastic)deformationsfromdifferentappliedstresses(forcesperarea)tobeadded,aswasdiscussedabove.Wecommencebyconsideringtheorthogonalx,y,zcoordinatesandtheapplicationofastressinthexdirection,sx,thatcausesstrainsexandalsolateral不得转载谢谢合作LWM 152MECHANICALPROPERTIESOFSOLIDS—ELASTICITYstrainsiny,eyandz,ez.Usingequation(7.30),weobtainthethreeorthogonalstrains:sx-nsyx-nszxe==,,ee=(7.31)xyzEEESimilarly,forstressesappliedinthey,syandz,szdirections,thefollowingrelationshipsareobtained:s-ns-nsyxyzye==,,ee=(7.32)yxzEEEandsz-nsxz-nsyze==,,ee=(7.33)zxyEEEUsingsuperposition,weproceedtocombinethedeformationsineachofthex,y,zdirec-tions;thatis,wecombineex,ey,andez.Considerthatastressappliedinthexdirectionnotonlycausesadeformationinxbutalsoinyandz,accordingtothePoissonratio,andthesignofthelateraldeformationsisoppositetothatintheapplieddirection.Theresultofsuperposition,assumingthatnisisotropic,issnsnsxyze=--(7.34)xEEEsnsnsyxze=--(7.35)yEEEsnsnszxye=--(7.36)zEEETheassumptionthatnisisotropicisnotentirelytrueingeneral,butitgreatlysimplifiesthealgebra.Becauseformostmaterialsthereisasmallrangeofn,theapproximationwillnotcauseseriouserror.Tofurthersimplifythealgebraandthefinalresult,wecon-siderahydrostaticappliedpressure,p.RecallthatstressisF/A,orpressure.Ahydrosta-ticpressureisanequalpressureinalldirectionssothatsx=sy=sz=s=p.Fromthisweobtainthefollowingsummaryformulas:sse=-2nEEÈE˘(7.37)s=eÎÍ12-n˚˙ThisresultisessentiallyacorrectiontoHooke’slawasitwasexpressedaboveinequa-tion(7.2):s=Ee.Thefactthatlateraldeformationsoccur,andaresummarizedbythePoissonratioenablesthecorrectiontobemadebytallyingthelateraldeformationsusingthesuperpositionprinciple.Aswasmentionedabove,nvaluesaround0.25areexpected不得转载谢谢合作LWM 7.7RELATIONSHIPSAMONGE,G,ANDn153formostmaterials(metalsbeinghigherandionicsolidslower).Thisleadstoacorrec-tionbyafactorofabout2as11=ª2(7.38)12-n105-.7.7RELATIONSHIPSAMONGE,G,ANDnRefertoFigure7.8,whichdisplaysacubedeformedtoanobliqueparallelepiped,andHooke’slaw(equation7.2),whichweusedtoderiveaformulafortheshearstresstintermsoftheshearstraing(equation7.29)ast=Gg,whereGistheshearmodulus.Thisformulaisgeneralizedforthex,y,zcoordinatesastxytyztzxgxy===,,gyzgzx(7.39)GGGAtthisjuncturewehavethreeconstantsforamaterialthatdescribeelasticity,E,G,andn.Belowwewillshowthattheseelasticconstantsarerelatedas:EG=(7.40)21()+nAssumingthattheapproximateveracityofequation(7.40)aswewilljustifybelow,wecandeducetheapproximationEGª(7.41)25.ThisrelationshipenablesanestimationofGfromameasuredvalueofE.AlsoforE>Gitfollowsthatsheardeformationsarelargerthannormaldeformationsduetoanappliedstress.Ifthenoneconsidershowmuchdeformationamaterialcanwithstandbeforethematerialfails(fractures)orpermanentlydeforms(plasticdeformation),itcanbereasonedthatthematerialwillmorelikelyfailorpermanentlydeformasaresultofshearstrains.WereturntothisimportantinsightinChapter8.ToderivetherelationshipsamongE,G,andn,weconsidertheprismaticelementofacubeformedbyadiagonalplanethatbisectsthecubeinFigure7.8toyieldthepris-maticelementseenontherightinFigure7.10.Thep/4deformstobintheleft-handprismofFigure7.10,andbisgivenas()pgpg2-b==-(7.42)242Forthisvalueforbtobeobtained,acubewithanglesoriginallyatp/2mustdeformunderstresstop/2-gasshowninFigure7.8.Sotheprismaticelementwithanglesofp/4deformsasshownattheleftsideofFigure7.10top/4-g/2.NowusingtheidentitytanAB-tantan()AB-=(7.43)1+tanABtan不得转载谢谢合作LWM 154MECHANICALPROPERTIESOFSOLIDS—ELASTICITY1xp/41-nexb11+exFigure7.10Prismaticelementofacube(leftpanel)comparedtoadeformedelement(rightpanel).whereAandBareanyangles,wecanobtainanexpressionfortanbwithsubstitutionsp/4=Aandg/2=B:tanpg42-tan12-tangtanb==(7.44)142+tanpgtan12+tangThisexpressioncanbesimplifiedbyconsideringthattan45°=1andthat,forelasticdeformations,g/2issmall.Thenthetanofasmallargumentisapproximately12-()g2-gtanb==(7.45)12+()g2+gForthedeformedelementontherightsideofFigure7.10,weobtaintherelation1-nextanb=(7.46)1+exWecanequatethetworelationshipsfortanb(equations7.45and7.46)andobtain()21-ge()+=+()21gn()-e(7.47)xxThisequationyieldsthefollowingvalueforg:en()1+xg=(7.48)112+-en[]()xNextwemakesomeapproximationstoobtainaphysicallymeaningfulandsimpleresult.Sincenwasapproximatedtobeabout0.25,andforsmallelasticdeformationsexis0.1orless,ex·(1-n)/2<0.04.Becausetheproductofexand(1-n)/2issmallrelativeto1,weobtaingasge=+()1n(7.49)xRecallthatg=t/Gandex=sx/Esubstitutioninequation(7.49)yieldst()1+nsx=+enx()1=(7.50)GE不得转载谢谢合作LWM 7.7RELATIONSHIPSAMONGE,G,ANDn155Then,solvingforG,wehaveÊEˆtmaxG=(7.51)Ë1+n¯sxwheretmaxisthemaximumshearstress.Belowitwillbeshownthatthemaximumshearstressisobtainedat45°fromtheapplicationofanormalstresstoasolid.WiththeresultaboveforGandthefollowingrelationshipstobedevelopedbelow(seeequation7.60),weobtainFFst==∞,at45xmaxAA2t1(7.52)maxthen=s2xFromthisweproceedtothedesiredrelationshipamongE,G,andn:EG=(7.53)21()+nTofullyunderstandthederivationabove,itisnecessarytoresolvetherelevantforcesandstresses,thatwillyieldtherelationshipsusedabove.Figure7.11showsappliedforcestoabarofsolid.Theforcesareonthesameaxis,FAPandFAP¢,andareappliedtoplanesAqandA,respectively.Theseplanesrepresentarbitrarycutstothebarofmaterial.BothFAPandFAP¢areperpendiculartoplaneA,butbecauseplaneAqisatangleqtoplaneA,FAPisnotperpendiculartoplaneAq.Theappliedforcescanberesolvedinanydesireddirection.Thetwocomponentsofanystressarethenormalcomponent,s,whichisper-pendiculartoanyplaneinquestion,say,theAqplane,andtheshearcomponent,t,thatisintheAqplane.ThesecomponentsofstressarenowcalculatedfortheAqplanefromthegivenappliedstresses.ThenormalcomponentofforceontheAqplaneisFNandisgivenasFF=cosq(7.54)NAPAFNqqFF’APApfAf’FPF’PFigure7.11Appliednormalforces(FAP)toplaneA.TheforcesareresolvedonplaneAqintermsofthenormal(FN)andinplaneorshearforcecomponents(FP).不得转载谢谢合作LWM 156MECHANICALPROPERTIESOFSOLIDS—ELASTICITYThisforceisconvertedtothenormalstressbydividingbytheareaAq:AAq=(7.55)cosqWiththenormalforceandtheplanearea,thenormalstresssisgivenas2FcosqAPs=(7.56)ATheforceintheAqplane,theshearforce,isFP:FF=cosf(7.57)PAPThisforceisconvertedtotheshearstress,t,bydividingbyAqasabove:FcoscosfqAPt==sfqAPcoscos(7.58)ATheformulasforsandtinresponsetotheappliedforceillustrateSchmid’slaw.TheshearforcecanhavearangeofdirectionsintheAqplanegivenanotherforcevector,asdrawninFigure7.11,whereFP¢andtheresolutionofFAPtodifferentdirectionsrequiredifferentf’sandthushavedifferentvalues.Noticethatthenormalforceisamaximumwhenq=0:FAPss==max(7.59)AWhenq=0,thenf=90°andt=0.Theshearforceisamaximumwhenf=q=45°(aswasmentionedabove),whichisgivenasFFcos45cos45APAPtt==max=(7.60)AA27.8RESOLVINGTHENORMALFORCESIncontinuingthemethodaboveforresolvingforces,theshearstressonanyplaneinasolidcanbeobtainedfromknowledgeofthenormalstresses,whichareshowninFigure7.12.Inthisfiguretheshadedplaneistheoneinquestion,anditsareaAqcanbefoundfromtherelationshipAAyxAq==(7.61)cosqqsinUsingthefactthats=F/A,wecanwritetheexpressionfortheshearstresstresultingfromsyandthenfromsxtoobtainthefollowingforcerelationships:FF=∞-cos()90qq=Fsin(7.62)tyyy不得转载谢谢合作LWM RELATEDREADING157syt90-qoAyqqsxo90-qAAxqFigure7.12Appliednormalstresses(s)areresolvedontheAqplaneintermsoftheshearstress(t).andFF=cosq(7.63)txxTheforcesinFigure7.12canbesummedatequilibriumasFFF--=0(7.64)xytThisexpressioncanberewrittenmakingsubstitutionsofsAforeachF,assqAAAsincosq--sqcossinqt=0(7.65)xyqqqSolvingequation(7.65)fortyieldsthefollowing:tssqq=-()sincos(7.66)xyBytheidentity,sin22qqq=◊sincos(7.67)andweobtainthefinalresult:()ss-xyt=sin2q(7.68)2RELATEDREADINGC.R.Barrett,W.D.Nix,andA.S.Tetelman1973.ThePrinciplesofEngineeringMaterials.Pren-ticeHall,EaglewoodCliffs,NJ.Areadableelementarytextforafirstcourseinmaterialsscience.J.M.GereandS.P.Timoshenko1984.MechanicsofMaterials.Brooks/ColeEngineering,Monterey,CA.Anauthoritativetreatmentofmechanicsofmaterials,startingfromthebasicsofelasticityandplasticityandincludingmanyimportantpracticalexamples.P.A.ThorntonandV.J.Colangelo.1985.FundamentalofEngineeringMaterials.PrenticeHall,EaglewoodCliffs,NJ.Areadableelementarytextforafirstcourseinmaterialsscience.不得转载谢谢合作LWM 158MECHANICALPROPERTIESOFSOLIDS—ELASTICITYEXERCISES1.AsphericalAlpressurevesselhas18ini.d.and0.25inwallthickness.Theultimate33stressis24¥10psiandyieldstressintensionis16¥10psi.Thetankmusthaveasafetyfactorof2.1withrespecttoultimatestressand1.5withrespecttoyieldstress.Whatisthemaximumallowablepressureinthetank?2.Aloadof1000lbissuspendedfromeachoftwoidenticallysizedwiresof0.25≤diam-66eter.Onewireissteel(E=30¥10psi)andtheotherisAl(E=10.5¥10psi).Deter-minetheaxialengineeringstrainineach,anddiscusstheatomisticdifferencesinthematerialsthatleadtothisresult.3.FromthetablebelowcalculatethePoissonratioforeachmaterialanddiscusswhythevaluesaresimilarornot(dependingonyourresults).MaterialEG66(psi◊10)(psi◊10)Carbonsteel3012Alloysteel3012Castiron156Al104Brass146Cu1764.AnAlbarandanalloysteelbar(seetableabove)areeachsubjectedto24,000psitensilestress.Calculatethelateralstrainsforbothbars.Discussyouranswer.5.Foracubiccrystalcalculatetheforceinthe[110]foraforceof184Ninthe[100].6.AppliedtothecubicmaterialshownbelowisaforceFonthe(100)plane.Calculatetheshearstressesinthe(110)planeinthetwodirectionsshown.Aretheshearstressesequal?F(100)(110)不得转载谢谢合作LWM EXERCISES1597.ExplainwhyHooke’slawrequiresacorrectionbyPoisson’sratio.IncludeinyourdiscussionhowthePoissonratiowouldeffecttheelasticdeformationduetoanappliedforce.8.ExplainwhyEfordifferentmaterialsisingeneraldifferent.不得转载谢谢合作LWM 不得转载谢谢合作LWM 8MECHANICALPROPERTIESOFSOLIDS—PLASTICITY8.1INTRODUCTIONPlasticdeformationreferstononrecoverablestrainscaused,forexample,bystressesthatexceedtheelasticlimitforaparticularmaterial.ElasticbehaviorwasillustratedinFigure7.3a.ThenonrecoverablestressstrainbehaviorisrepresentedtotherightofthedashedlineinFigure7.3a.Itiscommonpracticetodeformapaperclipbyasmallamountinordertocliptwosheetsofpaper.Thepaperclipwithstandsthissmalldeformation,anditsoriginalshapereturnsonceitisremovedfromthepapers.However,whenwebendthepaperclipsignificantlyforalargegroupofpapers,itremainsinthedeformedposi-tion.Thisdeformationisanexampleofaplasticdeformation.Inthefirstpartofthischapterwereviewobservationsaboutplasticdeformation.Thenwecomparetheseobser-vationswithsimpletheory,andobservationsaremadeabouttheoriginsofplasticdefor-mation.Incrystallinematerialsdislocationsarethemainfactorinplasticdeformation.Creepisanotherimportantmechanismforplasticdeformationandwillbebrieflydis-cussed.Fromcrystallinematerialsweproceedtoatreatmentofpolymericmaterials,andmodelsusedtodescribethedeformationofthesekindsofmaterials.8.2PLASTICITYOBSERVATIONSWhenabarofsinglecrystalmaterialisstressedanddeformed,characteristicobserva-tionscanbemaderelativetoplasticdeformation.Thestressesandresultingdeforma-tionsareshownschematicallyinFigure8.1.InFigure8.1a,abarofsinglecrystalmaterialisputintensionviastress(s)appliedperpendiculartotheshadedcrystalfaces,asshownbythearrows.Anadditionalshadedplaneisshowninsidethebaratsomearbi-ElectronicMaterialsScience,byEugeneA.IreneISBN0-471-69597-1Copyright©2005JohnWiley&Sons,Inc.不得转载谢谢合作LWM161 162MECHANICALPROPERTIESOFSOLIDS—PLASTICITYa)sb)sssc)d)Figure8.1(a)Tensilestresssappliedtoabarofsolid;(b)elongationofthebarduetostress;(c)macro-scopicdeformationduetoelongation;(d)slipbands.traryangle.AswesawinChapter7,theappliedforcescanberesolvedasnormalandshearforcesonthisoranyotherplaneinthesolid.Figure8.1bshowsthedeformation(elongation)ofthecrystalundertensilestress.Ifthedeformationislarge,therewillbeasmallrecoverablecomponent(elastic)andanonrecoverablecomponenttothedefor-mation,theso-calledplasticdeformation.Closeexaminationofthecrystaleitherduringorafteranonrecoverabledeformationrevealsthattheoutsidesurfacebecomesstepped,asshowninFigure8.1c.Thestepsleadtostriationsatspecificanglesrelativetotheappli-cationoftheappliedstress.Theanglescorrespondtospecificplanesthathaveslippedlaterally(relativetos),causingthesteps.Thelateralmotionoftheplanesmustbedueprimarilytoshearforcesresolvedontheplanesthatmove,andthoseplanesarecalledslipplanes.RecallalsofromChapter7thattheshearstressesaremoreefficaciousatpro-ducingstrainowingtothesmallerGrelativetoE.Specifically,wefoundinChapter7thatEGª(7.41)25.FromthisresultwecanexpressthestrainsintermsofEasg=2.5t/E(fromt=Gg)whilee=s/E(froms=Ee).Thustheshearstrainscaneasilybelargerthannormalstrains.不得转载谢谢合作LWM 8.3ROLEOFDISLOCATIONS163Ofcourse,eachplanethatslipsmovesonlybyatomicdimensions.Thus,toobservetheslip,oneisinrealityobservingthelateralmotionofmanyplanes.Afterdeformation,furtherobservationsoftheoutsidesurfacerevealsthestepaccumulationsastheslipbandsdepictedinFigure8.1d.RecallthatthisscenariowasbrieflydiscussedinChapter4,usingFigure4.4wheredislocationswereimplicatedintheformationofthestriationsorslipbands.Naturally,edgeandscrewdislocationsariseasaresultofthemotionofoneplanerelativetoanother.Thereforeitisonlyasmallstretchoftheimaginationtorelatedislocationmotiontoplasticdeformationincrystals.However,whileitisunavoid-abletoimplicatedislocationsincrystalplasticdeformation,thedetailsarefarlessobvious.Forthedislocationstobecausativeofmacroscopicdeformation,theremustbemanydislocationseachwithBurger’svectorbofatomicdimensionthat,whenadded,yieldsamacroscopicdeformation.Thedislocationmustbeabletomovetothecrystalsurface,soastobeadditive,andthenumberofdislocationsmustbevariable.Forexample,onecandeformasolidshortoffracture.Ifdislocationsarecausative,thenthenumbermustbeproportionaltothemagnitudeofthedeformation.Onthislatterpointithasbeenobservedinmanystudiesthatasdeformationincreases,sodoesthedisloca-tiondensity.Inthischapterthedetailsabouthowdislocationscanmoveandmultiplyincrystalsarediscussed.Bothamorphousandcrystallinesolidshavebeenobservedtoplasticallydeformbymeansotherthandislocations,andthisisdiscussedlaterinthischapter.Themechani-calpropertiesofpolymericsolidsareofgreatindustrialinterest,andtheypresentinter-estingscientificcases.Forexample,weknowthatarubberbandcanbestretched(deformed)tohundredsofpercentofitsoriginallength,yetitcanelasticallyrelaxtoitsinitialshape.Thisiscalledelastomericbehavior,andwillbediscussed.Alsomanysolidscanexhibitatime-dependentelasticandplasticresponsetoanappliedstressorstrain,andthiskindofbehaviorisalsodiscussed.Differentsimplemodelsforthemechanicalbehaviorofsolidsaredeveloped.8.3ROLEOFDISLOCATIONSFirstweaddresstheissueofhowdislocationsmovethroughacrystallinesolid,andthiscanbeunderstoodwiththehelpofFigure8.2.Figure8.2ashowsanedgedislocationatsomedistancefromthesurface.Therowofatomsformingtheextrahalfplaneatthetopofthecrystal(indicatedbythesymbol)isshownnottobebondedtotheatomsbelowitintheundisturbedpartofthecrystal.However,withtheshearstress(t)appliedasindicatedbythearrows,thetoppartofthecrystaldistortsleft,andthebottomparttotheright.Thenetresultofthisdistortionisthatatomsthatweredistantfromoneanotherarenowpushedcloser,andsomebondingoccursthateffectivelymovestheextrahalfplanetotheleft,asisindicatedbythesymbolinFigure8.2b,nowonespacingleftofitsoriginalposition.Theendresultofcontinualstressisthattheextrahalfplaneintersectsthesurfacewhereitobviouslycannotmovefurther.Theaccumulationoftheseplanesleadstothesteppedstructurediscussedabove,andamacroscopicdistortionviathesumofmanysuchmovementsofdislocations.Thismechanismforthemotionofadislocationisanenergyefficientmethod.Ratherthanthemotionofanentirehalfplanethedistancetothesurface,thebond-breakingandbond-makingmechanismisakintothemotionofaperiodicdisturbancethatisarelativelyefficientpathwayfortransport.Whileallpathwaysarepossible,thepathwaysthatareactuallyobservedarethosethataremostefficient,meaningthosewiththehighestprobability.不得转载谢谢合作LWM 164MECHANICALPROPERTIESOFSOLIDS—PLASTICITYta)ttb)tc)Figure8.2(a)Shearstresstappliedtoasolidthathasanedgedislocation;(b)thedistortioncausesmotionofthedislocationviabondbreakingandmaking;(c)thedislocationeventuallymovestothesurface.Table8.1Slipsystemsforface(FCC)andbody(BCC)centeredanddiamondcubic(DC)latticesCubicLatticeSlipPlaneSlipDirectionBurgersVectorFCC{111}·110Òa/2·110ÒBCC{110}and{112}·111Òa/2·111ÒDC{111}·110Òa/2·110ÒInthediscussionaboveitwasmentionedthatsliplinesformasaresultoftheresolvedforcesontocertainplanesthatarepronetoslip.Aplaneandthepreferreddirectionsforsliparecalledaslipsystem.Table8.1includesslipsysteminformationforcommoncubiccrystalsystemsaswellasBurger’svectorforthestabledislocationsinthecrystal.Noticethattheslipplanesarealllowindexplanes.RecallfromChapter2thatlowindexplanesareplaneswiththehighestdensityofatoms.Furthermorethepreferredslipdirectionsarethedirectionsofclosestapproach(seealsoChapter2onclosepacking).IfweagainconsidertheballandspringmodelthatwasusedinChapter7(Figure7.2),theslipsystemsarerationalized.UsingFigure7.2thatissketchedforasingleplane,imagine不得转载谢谢合作LWM 8.3ROLEOFDISLOCATIONS165otherplanesanddirectionsalsoconnectedbysprings(thechemicalbonds),butthatthespringsforotherthannearestneighborsarelooserspringsrepresentativeofweakerbondingthanfornearestneighbors.Thenimaginegrippingthesolidandpullingitsoastoputthesolidintensionbytryingtopullitapart.Thespringsresistanddistendsoastoreducetheappliedforce.Thetightestspringsaretheonesthatbeartheappliedforce.Nowreturningtothedislocationissue,itcanberealizedthattheshortestbondingdis-tancesonthedenselypackedplanesarethosethat“feel”theappliedstressanddistortasaresultofthestress.Consequentlythemostdenseplanesanddirectionscomprisethepredominantslipsystem.Thisisnottosaythatotherplanesanddirectionswillnotpermitslip.However,onceagain,themostprobablepathisthatdictatedbytheslipsystem,andonaverage,itisthepathwaymostobserved.Previouslyitwasmentionedthatdislocationsareimplicatedintheplasticdeforma-tionofcrystals.Ifthisistruethenacrystaldevoidofdislocationswillresistplasticdefor-mation.Theargumentmadehereisausefuloneforbothunderstandingandquantifyingthedegreetowhichdislocationsaffectthestrengthofmaterials.Wecommencethisargu-mentbyfirstconsideringasimplemodelforthestrengthofmaterials.WeconsiderinFigure8.3atworowsofanundistortedcrystallinesolidandthesamerowsinFigure8.3bwithanappliedshearstresstothetopandbottomrows,asindicatedbythearrows.InFigure8.3bthedistortioncausedbythestressismeasuredasx(withg=x/a)andcanadvancetob/2,butthenfromb/2forwardthedistortionisactuallydecreasingasthetopandbottomrowscomebacktoregistry.Giventhatasxincreases,a)b}a{b)xt}tc)txb/2bFigure8.3(a)Twoplanesinregistry;(b)disregistryasastresstisapplied;(c)periodicstressastheplanesmovepastoneanother.不得转载谢谢合作LWM 166MECHANICALPROPERTIESOFSOLIDS—PLASTICITYgincreasesandthustlikewiseincreases,wecanplotthevariationoftwithx.Thisperi-odicvariationisshowninFigure8.3c.ThisperiodicvariationcanbewrittenasÊxˆtt=◊sin2p(8.1)max˯bThisformulashowsthattvariessinusoidallywiththefractionx/bandthefraction2p·(x/b)isthex/bfractionof360°.Ifthestrainissmall,wecanusethefactthatthesineofasmallargumentcanbeapproximatedbytheargument,sothatthefollowingapproximationcanbeused:xttp=◊2(8.2)maxbWitht=Ggandg=x/aandwithaªb,wecanfurtherapproximatetasGtª(8.3)max2p7ortmaxisapproximately0.1G.Gvaluesformaterials(metals)typicallylieinthe10psirange,sothemaximumshearstressachievablebyamaterial(maximumstrength)isabout610psi.However,typicallyobservedstrengthsforcrystallinematerialsrangefrom3to5-3-5ordersofmagnitudelessthanthemaximum,ortmaxisabout(10to10)·G.Closescrutinyoftheseresultsisrevealing.Itisobservedthatfordislocationfreemetalsthevalueoftmaxisabout0.1to0.01G,whichisclosetotheapproximatecalculationmadeabove,consideringalltheassumptionsmadeandthesimplicityofthemodelforstrength.Furthermore,inmaterialswheredislocationscannotreadilymovesuchasceramicsandfornoncrystallinematerialswheredislocationsdonotexist,theactualandtheoreticalstrengthsarealsoreasonablycloseinmagnitude.Thereforeitisquiteclearthatformetalsdislocationsplayaveryimportantroleindeformationandstrength.Tocompleteourunderstandingabouttheimplicationofdislocationsinplasticdefor-mation,itisusefultomakefurthercalculationsaboutdislocations,particularlytheenergyforadislocationandtheenergytomoveadislocation.Itisintuitivethattheenergyofadislocationmustbeproportionaltothedeformationassociatedwiththedislocation.RecallthattheBurgervector,b,isameasureofthedeformationofadislocation,andthustheenergyofadislocationisproportionaltoitsb.WiththeuseofFigure8.4,whichdisplaysascrewdislocation,wecanobtainanexpressionforthestrain,g.Thedisloca-tionshowninFigure8.4aintheannuluswithwidthdrisunrolledinFigure8.4b,where2pristhecircumferenceoftheannulusandbistheBurger’svector.Theshearstrainexpressionthatwewillneedbelowisgivenasbg=(8.4)2prToassesswhetherdislocationwillform,itisnecessarytocalculatetheelasticenergyforadislocation.Inparticular,sinceadislocationisaline,weneedtocalculatetheenergyperlengthdesignatedbyG.Wecommencewithacalculationoftheenergypervolumeelementinwhichthedislocationresides.Weusethefactthatenergyisgivenbytheinte-gralofforcetimesdisplacement(recallequation4.3writtenforwork)asEnergy=◊ÚFdx(8.5)不得转载谢谢合作LWM 8.3ROLEOFDISLOCATIONS167a)drb)rqbl{l{{2rpFigure8.4(a)Screwdislocationintersectingasurface;(b)the“unrolled”annuluscontainingthedislocation.Thenenergypervolumecanbeobtainedusingt=Gg,andthefactthatvolume(V)isarea(A)timesheightx:EnergyÚÚÚF◊dxtg◊◊AdxGd◊◊x===(8.6)VolumeVAx◊xFromequation(8.4)forgandFigure8.4bandfromthefactthatdg=dx/x,weobtainfortheenergypervolumethefollowingformula:EnergyÚGd◊◊gx1Gb2==◊◊=ÚGdGggg2=ʈ(8.7)Volumex22Ë2pr¯Nowweconvertthisexpressiontoenergybymultiplyingbyvolumeandintegratingandthendividingbythelengthlofadislocationtoobtaintheenergy/length,G,as2EnergyEnergyVolumeGbʈ2prldrG==ÚÚ◊=◊(8.8)LengthVolumel22Ëpr¯lThisintegrationiscarriedoutoverthelengthofthedislocationfrom0tor.However,weareusingelastictheoryfortheexpressions,andelastictheorydoesnotholdforlargedeformationssuchasthoseatthecoreofthedislocation(nearr=0)wherethedefor-mationislargest.Thereforethecoredislocationenergy(Ecore)mustbeseparatelycalcu-latedandthenaddedtotheresultfromequation(8.8),22rGÊbbˆGrG=ÚÁ˜drE+=coreln+Ecore(8.9)rc22Ëppr¯4rcwiththelimitsofintegrationcommencingwherethedislocationcoredeformationcanbeapproximatedbyelastictheoryformulas(rc)andextendstotheendofthedisloca-tionr.Thereareseveralapproximationsthatcanbemadetosimplifythisexpression.Firstwedonothereattempttoreproducecalculationsoftheenergyforatypicaldislo-cationcore.However,estimationsofthisenergyrevealthatthecoreenergyistypically不得转载谢谢合作LWM 168MECHANICALPROPERTIESOFSOLIDS—PLASTICITYlessthan10%ofG.Thus,forthepurposeshere,weneglectthissmallcontributiontoG.Formacroscopicdislocationsoftheorderof1cm(r=1cm)andfordislocationscores-62withapproximatesizeof10cm,andEcore=0,GªGb.ThisfinalapproximationisimportantbecauseitshowsthefinaldesiredresultthatGisproportionaltothesquareofBurger’svector:2Gµb(8.10)Beforeproceedingwiththeuseofthisexpressiontoapproximatedislocationenergies,weillustratetheuseofthisexpressiontotestthestabilityofdislocations.Specifically,thisrelationshipforGcanbeevaluatedforvariousvaluesofthesquareofBurger’svectorofdislocations.Wecommencewithabriefreviewofvectoralgebra.Forthevectors,Vijk=++abcandVijk=++abc(8.11)11112222ThevectorsumisgivenasVV+=+()abbbcci++()j++()k(8.12)12121212ThevectordotproductcanbewrittenasVV◊={}aa++bbcccosq(8.13)12121212IfV1=V2,thedotproductisgivenas222VV◊=++abc(8.14)11111TheformofaBurgervectoris,ofcourse,thatofanyvector.Thereforewecanusevectoralgebratopredictwhathappenswhendislocationsmeetandcancombine.Forexample,forthefollowingdislocations,addingwecanwrite,usingvectoralgebra,11111+Æ111100(8.15)22Furthermoredislocationreactionscanbetested.Fortheexampleabovewecancalcu-lateGforeachofthethreedislocationsandcomparetheresults,andthenmakeajudg-mentwhetherthereactionwilltendtoproceedtotherightasitiswrittenortheopposite.Forthe1·111Òdislocation,G=a2()111=3/4a2,andforthe1·11¯1¯ÒG=3/4a2,and244422theright-handsideofreaction(8.15)forthea·100ÒyieldsG=a.Thustheleft-handside22energyis1.5a,andtheright-handsideis1a.Reaction(8.15)canclearlyproceedtotherighttowardlowerenergy.Wereturntoacalculationoftheenergyofadislocationaccordingtoequation(8.9)fortheenergyperunitlength,G.Formanymaterials,especiallymetals,avalueof74242Gª10psior7¥10dynes/cm(1psiª7¥10dynes/cm).Wecanassumevaluesforthelengthofdislocationsfrom,say,0.2nmtobe1to10nmandbeyond.Thisyieldsatotal-12-11-10energyof1.6¥10,1.6¥10,and1.6¥10dynes-cmforthe0.2,1,and10nmdis--12locationlengths,respectively.ConvertingtoeV(1eVª1.6¥10dyne-cm),weobtain1eVforthesmallestto100eVforthelargestofthethreedislocationlengths.Recallthat不得转载谢谢合作LWM 8.3ROLEOFDISLOCATIONS169inChapter4itwasindicatedthatpointdefectssuchasvacanciesandinterstitialsrequireabout1eVforformation,andsomerelyatroomtemperature(ª0.025eV)afinitenumberofsuchdefectswillexist.However,exceptforthesmallestdislocations,theenergyrequiredtoformdislocationsisatleastanorderofmagnitudelargerthantheenergyrequiredforpointdefects.Thuswiththenumberofdislocationsbeingexponentiallyrelatedtothenegativeoftheenergyrequired,wecansafelypredictthattheformationofdislocationsisnotspontaneousatroomtemperature.Thismeansthatappliedforcesarenecessarytoproducedislocations.Aswasdiscussedabove,ithasbeenobservedthatunderappliedforcesdislocationsareproducedthatmoveandcanmultiply.RecallfromChapter4thatanimportantfactorfortheproductionofpointdefectswastheconfigurationalentropyderivedfromplacingdisorderintoaperfectlattice.Linedefects,however,contributefarlessconfigurationalentropy(i.e.,disorder)becausethedislocationisrestrictedinthenumberofwaysaparticulardislocationcanbearrangedinthesolid.Thustheentropyofdisordergaineddoesnothelptooffsetthelargerenergyneededtoproduceadislocation.Dislocationsarenotformedspontaneouslybecausetheyrequireanexternalstress.Themotionofdislocationsisalsonecessaryforplasticdeformation.Thedislocationsonceproducedareabletomovetothesurfaceofthesolidandtherebyaccumulate,add,andultimatelyyieldamacroscopicplasticdeformation.Experimentshaveshownthattheenergyrequiredtomoveadislocationvarieswiththekindofsolid.Aswesaw,themostefficientkindofdislocationmotionrequiresbondbreakingandbondmaking.Thusitisconsistenttoconsiderthebondinginordertoobtaininsightsintodislocationmotion.Forexample,weknowthatmetalstypicallyhaveweakerbonding,comparedtoionicand/orcovalentsolids.Thisisconsistentwithobservationsthatindicatethatthestressnecessarytomovedislocations(smove)isgreaterforionicandcovalentsolidsthanformetals,andthiscanbequantitativelysummarizedasfollows:2410psi(formetals)£≥s10psi(forcovalentorionicsolids)moveTheenergyrequiredcanbecalculatedfromtheseobservedstressvaluesandthencom-paredtotheenergyavailableatroomtemperature(ª0.025eV).Forthiscalculationweneedtoestimatetheareaoverwhichadislocationisoperative.Forthesizesofdisloca--12-142tionlinesoffrom1to10nmwecanestimateareaof10toabout10cm.TheforcecanbecalculatedusingtheseareaestimatesfromF=s·AandthentheenergyfromF·dx,wherexisthemagnitudeofthedistortionorthemagnitudeoftheBurgervector.4Weestimatethistobeabout0.1nm.Withtheconversionsofpsitodynes(1psi=7¥10222dynes/cm)whereadyneis(g·cm)/s·cm),andthenconvertingtoeV(1eV=1.6¥-1210dynes-cm)forcomparison,weobtaintheresultsforthetwoareasinTable8.2.Noteinthetablefromtheenergiesinthetoprowformetalsatroomtemperature(0.025eV)Table8.2EnergytomovedislocationsAppliedStressEnergy(eV)Energy(eV)(psi)area=10-12cm2area=10-14cm21023¥10-23¥10-410433¥10-2不得转载谢谢合作LWM 170MECHANICALPROPERTIESOFSOLIDS—PLASTICITYTable8.3Approximations-1TheoreticalstrengthG/2p10G-3-5Actualstrength(withdislocations)(10–10)GEnergyofadislocation≥10eVEnergytomoveadislocation<1eVthatthedislocationsinmetalscanmoveatroomtemperature.Forothermaterials,dis-locationmotionwilldependonspecificstressesandtemperature,andcanbeexpectedintheweakerbondedmaterials.BeforeproceedingfartheritisusefultosummarizethemainideasdevelopedthusfarusingtheapproximationsmadeaboveanddisplayedinTable8.3.FromthefirsttworowsofTable8.3arethedislocationsimplicatedinplasticdeformationandstrengthofmate-rials.Theyapplytocrystallinematerialsinwhichdislocationscanbedefined.Thethirdrowindicatesthatthedislocationenergyisrelativelyhighandthatdislocationsdonotformspontaneouslyinmostmaterialsandrequireexternalstresstobeproduced.Oncedislocationsareproduced,thefourthrowindicatesthatthedislocationscanmoveeasilyviaaperiodicmotionofbondbreakingandbondmaking.Alltheconclusionsareappro-priateformetalswherethebondingisweakeranddislocationscanmoveeasily.Othercrystallinematerialscanalsoexhibitplasticdeformationviadislocationsbutthespecificsofthematerialsneedcarefulscrutiny.Thelastmajorpointisthatthedislocationdensitymustincreaseduringtheapplica-tionofprogressivelyincreasingstressinordertoaccountforincreasingplasticdefor-mationwithstress.Therearemanywaysthatthiscanoccur,andhereweintroducejustafew.Firstandeasiesttoimagineistheexpansionofadislocationlineorloop,asshowninFigure8.5aandb.DislocationdensityrDisgivenbyaproductofthenumberofdis-locations,n,witheachmultipliedbyitslength,l,anddividedbythevolume,V,consid-eredasÂinliirD=(8.16)VWhileitisintuitivethatanincreasednumberofdislocationsinavolumewillincreasethedislocationdensity,itisbelessobviousthattheincreaseinlengthofanyonedislo-cationinthevolumewithoutincreasingthenumberwillincreasethedislocationdensity.ThelinedislocationdepictedinFigure8.5acanbestretchedbytheapplicationofstressestothesolidfromthesolidlinetothedottedline.Thedislocationlengthlthenincreases,andaccordingtoequation(8.16),sodoesthedislocationdensity.Likewisethestretch-ingofthedislocationloopshowninFigure8.5bincreasestheperimeteroftheloop,andthuscausesanincreaseindislocationdensity.Forthedislocationloopwecancalculatethestressnecessaryforexpansionandto2increaserDbyrecallingthattheenergyperlengthisgivenbyG=Gb.Fortheloopthisenergyperlengthismultipliedbythelength2prtogivethetotalenergy.Fortheexpan-2sionfromrtor+drtheextraenergyneededisthengivenas2pGb·dr.Alsoforexpan-siontheenergyorworkofexpansionisgivenbyF·drwheretheforceperlengthcanbegivenbyt·bortheforceperareamultipliedbythelengthofthedislocation.Thisforceperlengthisinturnmultipliedbythelengthof2prtoyieldfortheworkavalue2t·b2pr·dr.Basicallythereisworknecessarytoexpandtheloop(2pGb·dr)andthereis不得转载谢谢合作LWM 8.3ROLEOFDISLOCATIONS171a)BAb)Figure8.5(a)AdislocationlinepinnedatAandBinasolidandstretchedviastress;(b)adislocationloopthatisstretchedviastress.workofresistanceorlinetension(t·b2pr·dr).Thissituationcanbeimaginedasthestretchingofarubberband.Weneedtoovercomethelinetensionoftherubberbandtoexpandit.Atequilibriumthesetwoworktermsneedtobeequal.SotheresultforttoachieveequilibriumisgivenasGbt=(8.17)randtheloopdislocationdensitywillincreasewhent>Gb/r(i.e.,theloopwillexpand).ForthelinedislocationshowninFigure8.5athatispinnedatpositionsAandB,the2linetensionTistheresistance.Theenergyforadislocation(G=Gb)fromthepinningattwopositionsis2G=tbl(8.18)atequilibrium,andhencetheconditionforthelineexpansionis2222GGbGbtt>>oror(8.19)blbllInadditiontoincreasingrDbydislocationexpansion,therearemechanismsthatcanincreasethenumberofdislocations.Thebest-knownmechanismistheFrank-Read不得转载谢谢合作LWM 172MECHANICALPROPERTIESOFSOLIDS—PLASTICITYa)ABqqlTTb)ABFigure8.6(a)Adislocationonaplane,pinnedatAandBandstressed;(b)theevolutionofthestresseddislocationyieldinganewdislocation.source.ThismechanismcanbeunderstoodwiththeuseofFigure8.6.Figure8.6ashowsadislocationontheshadedplanethatispinnedatpointsAandBwithlengthl.Ashearstresstisappliedinsuchawayastostretchthedislocationinthedirectionshownbythearrowaswasdiscussedabove.SincethedislocationispinnedatAandB,itcannotmove,butitcanexpand.ThedislocationhasalinetensionGthatistheresistancetotheexpansioncausedbyt.Thestretchingandresistanceforcescanbeequatedaswasdoneabovetoyieldtqbl=2Gsin(8.20)不得转载谢谢合作LWM 8.3ROLEOFDISLOCATIONS173wherebisBurger’svector.Atq=90°G=Gmax,andthefollowingformulaisobtained:2222GGbGbt===(8.21)blbllAstheappliedstressincreasesandexceedst,thedislocationexpands,asshownbythesuccessiveimagesinFigure8.6b.Noticethatinthetopimageofthedislocation,thedirectionofthedislocationisindicatedbythehalfplanesymbol,.Asthedislocationexpands,itbowsout,asshowninthesecondpanel,andthedirectionofthehalfplanesbecomesagainshown.Ultimately,asthedislocationexpandsfurther,theright-andleft-handbowedsectionstouch,asshowninthefourthpanelofFigure8.6b.Atthispointintheevolutionoftheexpandingdislocationwherethedislocationpartstouch,thedis-locationhalfplanesannihilate,asshownbythedasheddislocationsymbol,.AnewdislocationlineformsandtheloopthathasformedstaysinplaceasshowninthebottompanelofFigure8.6b.Insummary,theoriginalpinneddislocation,whenstretched,liter-allypunchesoutdislocationloopsandreconstitutesitself.ThissourceofdislocationsundershearstressesiscalledtheFrank-Readsourceandisobservedinmanycrystalsystems,suchasSiwhereseriesofconcentricloopssurroundingadislocationlinehavebeenreported.Itisalsopossibleforplasticdeformationtooccurwithoutdislocations.Onesuchmechanismiscalledcreep,andwediscussnowaspecifickindofcreepcalledNabarro-HerringcreepusingFigure8.7.Figure8.7pictoriallysummarizesactualSioxidationPolySiGrainsSiOSubstrate2a)SiO2ttSiOSubstrate2b)SiO2SiOSubstrate2c)Figure8.7(a)PolycrystallineSigrainsonasubstrate;(b)oxidationcausesSiO2formationonthefreesurfaceandingrainboundariescausingstressthatleadstoSimigrationfromthegrainboundaries;(c)theintergranularoxidationcausesthicknessfluctuationsintheoxide.不得转载谢谢合作LWM 174MECHANICALPROPERTIESOFSOLIDS—PLASTICITYexperimentsperformedsomeyearsago.SpecificallypolycrystallineSi(polySi)grainsaresubjectedtoO2gasatelevatedtemperatures.Beforeproceeding,itiswelltoknowthattheoxidationofsinglecrystalSiinO2resultsintheoxidationofSitoformSiO2.TheoxidationofthefreesurfaceofasinglecrystalofSiproducesauniformfilmofSiO2.FurtheracomparisonofthemolarvolumesforSi(VM,Si)withthatofSiO2(VM,SiO2)yieldsavaluefortheratioof(VM,Si)/(VM,SiO2)=2.2.ThismeansthatforeveryatomofSicon-vertedtoamoleculeofSiO2,thenewsolidSiO2producedhas2.2timethevolumeofSithatitreplaces.InmetallurgythismolarvolumeratioofproducttoreactantisoftenreferredtoasthePilling-Bedworthratio.OxideswithlargeratiossuchasthatforSiO2andSiusuallyresultintheoxideflakingoff(alsocalledscaling)asitisformed.Thisfailureinthecoatingisduetothefactthatthestressesdevelopedwiththemolarvolumeexpansionexceedthemechanicalstrengthofthecoating.However,itisobservedthattheSiO2filmdoesnotfailwhenSiisoxidized.Thefactthatscalingdoesnotoccurisattributedtotheviscousflowoftheamorphousoxideproduced,andtheflowrelievesthestressesthatcanaccumulatefromtothelargemolarvolumeexpansionthatoccursduringSioxidation(viscousflowisdiscussedbelowandreferstothemotionofgroupsofatomsormolecules).Returningtothecreepproblem,weseeinFigure8.7aaregulararrangementofgrainsofSionaSiO2substrate,whoseregularityisshownonlyforillus-trationpurposes.Thesubstratetakesnopartinthescenarioandismerelyarelativelyinertsupport.WhenthispolycrystallineSiisexposedtoO2atelevatedtemperatures,theSiisthermodynamicallydriventoreactwithO2toformSiO2,andattemperaturesbelowabout1200°Ctheoxidewillbeamorphous.NowthedifferencebetweentheoxidationofsinglecrystalSi,andthepolycrystallineSishowninthefigureisduetothegrainbound-aries.ForsinglecrystalSionlythefreesurfaceofSioxidizes,butforpolycrystallineSi,boththefreesurfaceandgrainboundaries(whichareboundarieswithdisorder)canoxidize.InthegeometryshowninFigure8.7thegrainboundarieswilloxidizemoreslowlythanthefreesurface,becauseO2needstopenetratetheboundaries,butwillneverthelessoxidize.Therelativelylargevolumeofoxide(2.2timesthevolumeofSi)thatformsonthefreesurfacecanreadilyexpandinthefreedirection,provideditcanflow(laterwedealwithviscositiesandtheabilitytoflow).Athightemperaturetheoxideviscosityissufficientlysmallthattheoxidecanflowinthefreedirection.However,theoxidethatisconfinedinthegrainboundariescannotflowasreadilyasthatonthefreesurface.Consequentlyalateralstressdevelopsduetothelackoffreevolumethatisnec-essarytoaccommodatetherelativelylargevolumeofSiO2formingwiththeSi.Whenthestressinthegrainboundariesduetooxidationreachessufficientlyhighvalues,theflowofSiand/orSiO2occurstorelievethestress.Inthismaterialssystem,aflowofSiisobservedfromthegrainboundaryregiontowardthetopofthefreesideofthegrains,asshownbythedashedarrowsinthegrainsinFigure8.7b.SuchaflowofSiatomsreducestheintergranularstressbycreatingvolumefortheformingSiO2.AtthesametimemoreSiatthefreesurfacecanoxidizewithminimalaccumulationofstress.ThisflowofSi(orwhateverelse)inresponsetoalateralstressiscalledNabarro-Herringcreep.Oftenratherthantalkabouttheflowofatoms,materialsscientistsrefertotheflowofvacancies(volume)intheoppositedirection.Insummary,intheplasticdeformationofcrystallinematerials,dislocationsareclearlyimplicatedformetalsandmayalsobecrucialforothermaterials.Dislocationscanaccu-mulateviamotion,andtheymultiplyunderstresstoproduceplasticdeformation.Othermechanismsforplasticdeformationsuchascreeparealsopossible.不得转载谢谢合作LWM 8.4DEFORMATIONOFNONCRYSTALLINEMATERIALS1758.4DEFORMATIONOFNONCRYSTALLINEMATERIALSNoncrystallineoramorphousmaterialscanplasticallydeformbutwithoutdislocations.Inamorphousmaterialsdislocationscannotbedefinedbecausethisclassofmaterialsdoesnothavelongrangeorder.Manyamorphousmaterialsofimportanceareeitherso-callednetworksolidsorpolymers,andthisdichotomyprovidesareferenceframefromwhichtodiscussthedeformationofamorphousmaterials.ThenetworksolidsincludeinorganicceramicssuchasSiO2aswasdiscussedinChapter2andillustratedinFigure2.2asacontinuousnetworkofSiO4tetrahedra.Networksolidstypicallyhaveshortrangeorderdictatedbychemicalbondingthatdefinesthebasicbuildingblocks,andtheseblocksareconnectedtoformnetworksthatextendthroughoutthematerial.Polymersaresimilaruptothispoint,butinadditiontohavinganetwork,thenetworkshaveanisotropicbonds.Inonedirectionthechemicalbondingisstrongindefiningthebuildingblocksandcombinationofbuildingblocks,thepolymerbackbone,inaone-dimensionalstrand.Thesestrandsorchainsarethenbondedtogether,ofteninatwistedandtangledarray.Itisthisinterchainbondingthatdiscriminatesapolymerfromanetworksolid.Typicallyitismuchweakerthanthebackbonebondingyieldingananisotropy,andinsomecasesthechainscanbeextensivelytangled.Ifapolymerwithextensivelytangledchainsisstretchedsoastostraightenthetangles,thenitisthesameasanetworksolid.Thus,inordertounderstandthemechanicalpropertiesofamorphousmaterials,wecommencewithadiscussionofnetworkamorphoussolids,andthenaddthefeatureoftangledchainsandnotethedifference.8.4.1ThermalBehaviorofAmorphousSolidsInordertoobtainacoherentpictureofamorphoussolids,itisusefultocommencewiththethermalbehaviorofthesolids.Figure8.8showsthetypicalchangeinthemolarvolumeofasolidwithtemperature.Thebehaviorisbestunderstoodstartingfromthemoltenstateathightemperature,andatfirstslowlycooling.AsingletrajectoryisseenintheliquidstateandisindicatedbythesolidlineabovethetemperatureTm.However,atthemeltingtemperature,Tm,theliquidconvertstoitscrystallinestate,C.Noticethattheslopesoftheliquidandsolidlinesaredifferentandareindicativeofdifferentthermalexpansioncoefficientsforliquidandsolidforms.Usuallythisliquidtocrystallinesolidtransformationyieldsthemostcompactformforthematerial,andhencethesmallestmolarvolume.Forsmallmoleculesandatomicsolidsthisisthepathwaythatismostoftenobserved.Forexample,toobtainthepathwayleadingtoanamorphoussolidformetals,extremelyrapidcoolingmustbeperformed.Onemethodofproducingamor-phousmetalsinvolvesactuallysettingoffanexplosionintheliquidthatsplattersthemoltenmetalontoliquidnitrogencooled(77K)plates.Thisso-calledsplatcoolinginvolvestheveryrapidcoolingofthemetalbeforetheatomscanattaincrystallineorder,andthemaintenanceofthelowtemperaturetopreventoratleastreduceatomicmigra-tion.However,forlargernetworksofatomsandmolecules,coolingevenatnominalratescanyieldamorphoussolids,becausetherequiredrearrangementhasfewpathways(entropy,S)andrequiresconsiderableenergy(enthalpy,H).ItisseeninFigure8.8thattwoamorphoussolids,AandA¢,areproducedbycoolingtheliquidatratestoofasttoachievethecrystallinestateCforthesolid.Infactaseriesofamorphoussolidscanbeproduced,eachsomewhatdifferentinmolarvolumewithonlytwoofthissetshowninthefigure.WhilethesolidsA,A¢,andCarecomposedofthesamebuildingblocks,theyhavedifferentstructuresduetothefinalarrangementofthebuildingblocks.Thebottom-不得转载谢谢合作LWM 176MECHANICALPROPERTIESOFSOLIDS—PLASTICITYLiquidSolidAA’MolarVolumeCTT’ggTfTmTemperatureFigure8.8Molarvolumeversustemperatureforasolidformedatdifferentcoolingratesfromtheliquid,andresultingindifferentstructures.Melttemperature,Tm,fictivetemperature,Tf,andglasstransitiontem-peraturesTgindicated.mostsolid,thecrystallineone,C,hasmoreorderwhilethetopmostoneAhastheleastorder.Thedifferentstructuresareusuallyidentifiedbythetemperatureatwhichthestructureingoingfromliquidtosolidnolongerchanges.ThistemperatureiscalledtheglasstransitiontemperatureandlabeledasTg.NoticethatAandA¢havedifferentTg’s.ThedifferentTg’srepresentdifferentarrangementsofthebuildingblocksforthenetworksolidsand/orpolymers,anddifferentstructuresareobtainedusingdifferentpreparationprocedures.Inthecaseabovedifferentcoolingrateswereusedtoproducethedifferentsolidstatestructures.AlsoinFigure8.8isanextrapolatedtemperature,Tf,thatiscalledthefictivetemperature.TfisthetemperatureatwhichtheextrapolationofthesolidandliquidvolumeversusTlinesmeet.Tfrepresentsastructurethatwouldexistatthejunc-ture.Assuchitisnotanactualstructurebutdoesservetodifferentiatedifferentamor-phousstructuresofthesamematerial.Tfwasreferredtointheolderliteratureonglassesbutishardlyusedanymore.Sometimestheamorphousstructuralformforanetworksolidisreferredtoasanundercooledliquid.ThemeasurementofthecharacteristictemperatureTgisnowpossiblebyavarietyofprecise,commerciallyavailabletechniques;mostmeasurethermalpropertiesofthemate-rialasafunctionofT.Asapracticalmatter,severaloftheimportanttemperaturesasso-ciatedwithamorphousnetworksolids,mostlyoxideglasses,arerelatedtotheviscosityoftheglassortheviscosityrange.Wewilldefineviscositymorethoroughlybelow,buttheviscosity,h,isessentiallyameasureoftheresistanceofamaterialtoshearforces.Theunitsforharepoise,whichisg·cm/s.Amaterialwithalowerhwillflowmoreeasilyunderanappliedforce,whileamaterialwithahighviscosityresistsfloworflowsmoreslowlyunderanappliedforce.Becauseamorphoussolidsaresometimesreferredtoas不得转载谢谢合作LWM 8.4DEFORMATIONOFNONCRYSTALLINEMATERIALS177Table8.4ImportantviscosityrangesforglassesRangeIdentityViscosityRange(poise)13Glasstransitiontemperature,Tg1013Annealingrange~1046Workingrange10–104Softeningormeltingrange<10LoadingUnloadingStressStressTimeTimeElasticainStrainStrTimeTimeFigure8.9Leftsideshowsappliedstress-time(loading)toasolidandtheanelasticstrainresponse;rightsideshowsthestressunloadingandtheanelasticstrainresponse.undercooledliquids,itisnaturaltoalsorefertotheviscosityofasolid.Table8.4summarizestheimportantviscosityrangesforglasses.13Thetransitiontemperatureisat10poise,whichisneartheso-calledannealingrange.Thisistherangeusedbyglassblowerstoannealawaystressesinthefinishedwork.Ofcourse,whenactuallyshapingtheglass,amuchlowerviscosityisrequired,andthisiscalledtheworkingrange.Evenintheworkingrangetheglassblowerdoesn’twanttheglasstobesorunnyasnottobeworkable,asitwouldbeatlowerviscosityinthesoft-eningormeltingrangewheretheglassappearstobeliquid.8.4.2Time-DependentDeformationofAmorphousMaterialsUptonowwehaveignoredthetimedependenceofadeformation,orthetimeelapsedbetweentheapplicationofastressandtheresultantstrain.Forpurelyelastic(totallyrecoverable)deformation,thetimedifferencebetweenstressandstrainiszero.Thephe-nomenonwherethereisafullyrecoverable,buttime-dependentrecovery,iscalledanelas-ticity.AnelasticitycanbeunderstoodbyreferringtotheloadingandunloadingcurvesshowninFigure8.9.Thesecurvesshowtheapplicationofstress(loading)inthetopleft不得转载谢谢合作LWM 178MECHANICALPROPERTIESOFSOLIDS—PLASTICITYpanel,andtheremovaloftheappliedstress(unloading)inthetoprightpanel;inthebottomtwopanelsthecorrespondingstrainsaredisplayedwithparticularattentiontothetimephase.NoteinFigure8.9thatforanelasticity,theapplicationofastepfunc-tionshapedappliedstressyieldsatime-dependentstrainthatultimatelyreachesamaximum,butcomparedwithapurelyelasticresponse(dashed),themaximumistimedelayed.Whenthestressisremoved(unloading),onceagaintheanelasticresponseisatimedelayed,butwithfullrecovery.Thedifferencebetweenpurelyelasticbehaviorandanelasticbehaviorliesinthetimelagbetweenstressandstrain.Fortheplasticdeformationofcrystallinematerials,dislocationsand/orcreepmech-anismswerefoundtobeoperative.Forplasticdeformationofnoncrystallinematerials,dislocationscannotbeinvoked.Ratherfornoncrystallinematerialsplasticdeformationischaracterizedbyviscoelasticbehavior.Viscoelasticbehavior,orviscoelasticity,issome-whatanalogoustoanelasticityinthatthereisatimelagbetweenstressandstrain.However,viscoelasticdeformationisnotfullyrecoverableasisanelasticity,andtheresultisplasticdeformation.Specifically,viscoelasticbehaviorischaracterizedbytheshear.stresstbeingproportionaltothestrainrategastgandµ=thg˙(8.22)whereh,theconstantofproportionality,iscalledtheviscosity.Aswasmentionedabove,theunitsforharepoise,whichisg·cm/s,andasisseenfromthedefiningformulaabove,22thisunitcomesfromthestress(dynes/cmwhereadyneisg·cm/s)dividedbythestrain.-1.rateg(s)yieldsg·cm/sorthepoise.Aplotoftversusgshouldyieldastraightlinewithaslopeh,andthisiscommonlyreferredtoasidealNewtonianbehaviorforthemater-ial.NewtonianfluidbehaviorisshowninFigure8.10asthesolidline.Thedashedlineshowsnonidealviscoelasticbehaviorwherethereisadeviationatlargerstrainrates.Specifically,thedeviationathigherstrainratesindicatesthatlowershearstressesarerequiredfornon-NewtonianbehaviortocausegivenshearratesascomparedtoNewtonianliquids.Anotherwaytolookatitisthatlowervaluesoftheviscosity(theslope)occurathigherstressesorhighershearrates.Thisisreferredtoasshear-thinningtNewtonianLiquidShearThinningShearStress,.StrainRate,gFigure8.10ShearstressversusstrainforaNewtonianliquid(solidline)andanon-idealNewtonianliguid(dashedline).不得转载谢谢合作LWM 8.4DEFORMATIONOFNONCRYSTALLINEMATERIALS179Atyxhyx{Figure8.11Twoplaneswithrelativemotioncausedbystresstimmersedinaliquidofviscosityh.behavior,anditprovidesausefulpropertyforfluids.Forexample,houseceilingpaintismadesomewhatthickerthanwallpainttopreventdripping.However,inadditionchem-icalsareaddedsothatasthepaintisspread(sheared)withabrushorroller(forceisapplied),andthepaintthinssoastocoverandbespreadable.Thesubstancesaddedarecalledthixotropicagents,andtheshear-thinningfluidiscalledathixotrope.Alsosomeliquidrocketfuelsarethixotropes.Thefuelsareessentiallygelledsolidsatnormalpres-suressoastoreducehazardsduetoleaks,butwhenpumpedatelevatedpressure(stressed)thefuelflowsreadily.Itisusefultoconsidertheoriginofviscosityfromaphenomenologicalviewpoint.Figure8.11showstwoequalarea(A)platesimmersedinafluid.Ausefulwaytoimaginethisistoconsiderthetwometalplatesinacontainerofhoney.NowapplyaforceFtothetopplate,asisshownbythearrow,whilekeepingthebottomplateisstationary.TheforceappliedperareayieldstywiththecoordinatesshowninFigure8.11.Asthetopplateismovedtotherightbytheappliedforce,thehoneynearestthetopplatewillalsomovetotheright.Thefartherfromthetopplateonemoves,thelessvelocitywillbeimpartedtothefluid,andthisisindicatedbytheprogressivelysmallerarrowsinbetweentheplates.Basicallythereisavelocitygradientforvythatdependsonthexpositionorvy(x).Thisgradientisapproximatedasvy(x)/x.Whenvy=dy/dtandg=dy/x,sothegradientisexpressedasvy1==g˙y˙(8.23)xx.Fromequation(8.22),tyµgwiththeproportionalityconstanth,sothevelocitygradi-entisalsoproportionaltotheshearstress,withtheviscositydeterminingthesteepnessormagnitudeofthegradient.Nextweseehowthemechanicalmakeupofnetworksolidsmodelsisconstructedusingelasticityandviscoelasticitytosimulateparticularsolids.8.4.3ModelsforNetworkSolidsModelscanbeformulatedusingthelawsgoverningelasticityandviscoelasticity,sincemostrealmaterialshavecharacteristicsofbothbehaviors.Modelparametersareessen-tiallyelasticandviscoelasticconstants.Theseconstantsenableamechanicaldescriptionofthematerial,andthuscomprisethemechanicalpropertiesofthematerial.Thetwomainmodelelementsrepresentelasticityandviscoelasticity.Forapurelyelasticresponseaspringisused.Recallthatwepreviouslyconsideredasolidtobecom-不得转载谢谢合作LWM 180MECHANICALPROPERTIESOFSOLIDS—PLASTICITYLoadUnloada)StressTimeStrainTimeb)StrainTimeFigure8.12(a)Springand(b)dashpotwithstrainresponseafterloading.posedofspringsaschemicalbonds.Figure8.12ashowsaspringandtheloadingandunloadingresponseofaspring.Noticethatthedeformationofthespring,thestrain,isinphasewiththeappliedstress.Forapurelyviscousresponseadashpotisused.Adashpotcanbeenvisionedasacylindricalcontainerwithoneendopen,andinsertedintheopenendofthecontainerisafittedpiston.Thecontainerhasaviscousfluidsuchashoneyonbothsidesofthepiston.AdashpotsketchandtheloadingandunloadingresponsesforthedashpotareshowninFigure8.12b.Noticethatthedashpotdeforma-tionlagstheappliedstress,andthatwhenthedashpotisunloaded,itremainsatthelaststrainstate.Thereisnoforcetoreturnthedashpottoitsoriginalposition.Acombina-tionofspring(s)anddashpot(s)canbeusedinvariousarrangementstosimulatethemechanicalbehaviorofawidevarietyofamorphoussolids.ThefirstarrangementtoconsiderisasolidmodeledasaspringanddashpotinseriesasshowninFigure8.13.AsolidthatbehavesapproximatelylikeaspringanddashpotinseriesiscalledaMaxwellsolid,andmanyinorganicnetworksolidsdisplaythisbehav-iortosomedegree;oneexampleisSiO2.Totheconnectedelementsastresssisappliedsoastocauseatotaldeformationofe.Boththespringanddashpotdeformtoe1ande2,respectively.Withtheapplicationofthedeformingstress,thespringinstantaneouslydistendstoe1.Thenthedashpotbeginstodistendultimatelyreachinge2.Whenthestressisremoved(unload)thespringinstantaneouslycontractstoitsoriginallength(elasticresponse).However,thedashpotremainsdistendedbecausethereisnodrivingforceto不得转载谢谢合作LWM 8.4DEFORMATIONOFNONCRYSTALLINEMATERIALS181LoadUnloade1StresseTime2ee}12{Straine}1TimeFigure8.13Maxwellmodelforasolidwithaspringanddashpotinseriesandwithloadingandunloading.recompressit.Withthebasicformulasforelasticityandviscoelasticitydiscussedabove.s=Ee1ande2=s/h,thefollowingisobtainedforthetotaldeformatione:eee=+(8.24)12Takingthetimederivative,oneobtainsss˙˙e=+(8.25)Eh.Foraconstantdeformation,e=constant,e=0,andweobtainthefollowing:dsE-=s(8.26)dthThesolutionforthisdifferentialequationis-ttss=oe(8.27)where.1/th=E/ThenextarrangementincomplexityistoarrangethespringanddashpotinparallelasshowninFigure8.14.ThiskindofsolidiscalledaVoigtsolid,anditisobservedforsomepolymerswherechainstraighteningoccursfirstasastressisapplied.TheVoigtmodelintroducesatime-dependentelasticity.IntheMaxwellsolidtheappliedstresswasinstantaneouslytakenupbythespring,andthedashpotdidnotrespondattheinstant不得转载谢谢合作LWM 182MECHANICALPROPERTIESOFSOLIDS—PLASTICITYLoadUnloadsStressTimeCreepRelaxationsStrainTimeFigure8.14Voigtmodelforasolidwithaspringanddashpotinparallelandwithloadingandunloading.thattheforcewasapplied.However,theVoigtsolidbehavesdifferently.Inthismodelthespringcannotinstantaneouslydistendbecauseitisinparallelwiththerelativelynon-responsivedashpot.Infacttheslowlyrespondingdashpotcontrolsorlimitstherateofdeformationanddictatesthatdeformationoccursasafunctionoftime.Whenthesolidisunloaded,thespringisdistended,anditstoresenergythatisthenavailabletocom-pressthedashpot.Theappliedstresscanbewrittenasthesumoftheelasticandviscouscomponents:ss=+s(8.28)elasticviscous.ThensubstitutingEeandhefortheelasticandviscousstresses,respectively,weobtainseh=+Ee˙(8.29)Noticethatforthepurposeofsimplificationwehaveequatedthestrainseandg.Nowdividebyhandrearrangetoobtainadifferentialequationasfollows:Es˙e+-=e0(8.30)hhRearrangingobtainsÊEsˆdde+-te=0(8.31)Ëhh¯Thisexpressionresultsinawell-knownintegralform:deÚ=-Údt(8.32)()aeb+不得转载谢谢合作LWM 8.4DEFORMATIONOFNONCRYSTALLINEMATERIALS183Thenequation(8.31)isreadilyintegratedtoyield1ln()aeb+=-+tC(8.33)aTheconstantofintegrationCcanbeevaluatedatt=0,e=0whichyieldsC=(1/a)lnb.Wheninsertedintoequation(8.33),ityieldsthefollowingresult:11ln()aeb+=-+tlnb(8.34)aaEquation(8.34)issimplifiedtothefollowing:1Êaˆlne+1=-t(8.35)aËb¯Thisexpressionhasa=E/handb=-s/h,andthusab=-E/s.Itsexponentialformisobtainedass-()Ethe=-()1e(8.36)EItisusefultoexplorethestrainresponseusingthisfinalformula.Astimeincreases,thetexponentialpartdecreases,sinceitisessentially1/e.Thus,astincreases,eapproachess/E,thepurelyelasticresponse.Asthepolymerchainsbecomestretched(i.e.,thetanglesarestraightened),thematerialsbecomesmoreelasticinmechanicalbehavior.Forlargervaluesoftheviscosity(e.g.,formoreviscouspolymers),theexponentialisessentially1/h1/e,andasthistermincreases,edecreases.Consequentlythecreepisslowerthaninamoreviscoussystem.TounderstandhowtheMaxwellandVoigtmodelsareusedtosimulatearealsolid,imagineapolymericsolidwithtangledchains.Asstressisapplied,thesoliddeformsslowlyinresponsetothestressstraighteningoutthechains.Oncethechainsaretaut,thesolidbeginstobehavelikeaMaxwellsolidwithinitiallyelasticbehaviorbutfollowedbyacreepresponse.ThesimulationforaVoigtsolidisdepictedinFigure8.14withaspringanddashpotparallelnetwork.Thetemperaturedependencecanalsobemodeledwithacombinationofmodels.ABurgersolidisaseriescombinationofMaxwellandVoigtmodelsasisshowninFigure8.15.Thespringatthetopsimulateslow-temperaturebondstretching,andthebottomdashpotsimulateshigh-temperatureplasticdeformationaboveTm.TheVoigtmodelsimulateschainstraighteningthatoccursaboveTg.Thebestwaytousethesemodelsistofirstobtaintime-dependentstressandstraindataforthesolidsampleinquestion.Temperature-dependentdataareideal.Thentheelementsareaddedinparallelorseriestosimulatethebehavior.Last,dataandmodelarecompared,possiblyusingregressionanalysis,andthevaluesforthemodelpara-metersareobtainedandverifiedindependently.8.4.4ElastomersSomepolymersexhibitelasticbehaviorforverylargedeformations.Thesesolidsarecalledelastomers,andthisbehaviorwasdiscussedinChapter7andillustratedinFigure不得转载谢谢合作LWM 184MECHANICALPROPERTIESOFSOLIDS—PLASTICITYsLoadUnloadE1StressEhTime22s/E}1}s/E2Strainh3}s/E1}sh(-)/tt321t1Timet2sFigure8.15Burgermodelforasolidwithaspringanddashpotinbothseriesandparallelandwithloadingandunloading.7.3b.Figure7.3bshowsthattheelastomerdisplaysmorestiffnessafteralargedeforma-tion,whicheventuallyleadstofracture.Theunusualbehaviordisplayedbyelastomersisattributedtolongtangledchainsinthematerial.Theextendedelasticregionisduetothelongchainsuntanglingandstraighteningasthematerialisdeformed.Afterthechainsarestraightened,moreconventionalnetworksolidelasticbehaviorisobservedwiththerapidlyrisingstress–strainbehaviorandfinallyfracture.Theuniquefeatureofelastomersisthatafterthestraightening,thechainsre-tangleasthestressisremoved.Thisbehav-iorisunderstoodbyrecallingtheconfigurationalentropynotionintroducedinChapter4thatderivesfromtheBoltzmannrelationship:DWSk=ln(4.14)totWistheratiooftheprobabilityofthefinaltoinitialstatesas:wfW=(4.15)wiwherewfisrelatedtothenumberofwaysofformingthefinalstateandwiisthenumberofwaysofformingtheinitialstate.Stotreferstothetotalentropy,meaningthesumofSsys+Ssur.IftheTwerethesameforthesystemandsurroundings,thennoqwouldflowtothesurroundingsanddSsur=0.Sowewritetheconfigurationalentropyofthesystem(thesolid)asÊwfˆDDSSkksys===configlnWlnÁ˜(8.37)Ëw¯i不得转载谢谢合作LWM 8.4DEFORMATIONOFNONCRYSTALLINEMATERIALS185ABFigure8.16SeveralpossibleconfigurationsforapolymerchainbetweentwopointsAandB.Nowweconsiderthenumberofwaysofarranginglongchainsinasolid.Figure8.16showsthreechainsofapproximatelyequallengthconnectedtothesameAandBendpoints.Thisis,ofcourse,threewaysofarrangingchainsfromliterallyaninfinitesetofways,andinrealitytherearebondingandstericconstraintsthatlimitthenumberofways.Because,therearealargenumberofwaystoarrangethechains(largewf),com-paredtoonewaytoarrangeastraightenedchainbetweentwopoints(wi),itiseasytoseethattheconfigurationalentropywillfavorthetangledarrangementbyalargemeasure.TheworkchangedwfromextendingthechainsdxusingaforceFisgivenasdw=◊Fdx(8.38)UsingtheGibbsfreeenergyrelationship,G=H(orEforsolids)-TS,andremember-ingthatGisthenon-pVworkinthesystem,wecanalsowritethefollowing:dwÊ∂GˆÊ∂EˆÊ∂SˆF===-T(8.39)dxË∂x¯T,PË∂x¯T,PË∂x¯T,PIftheelongationdoesnotproduceasignificantchangeinbonding,then∂E/∂xª0,andthechangeinentropywithelongationisgivenbytheBoltzmannrelationship:Ê∂SˆwfF=-T=-Tkln(8.40)Ë∂x¯T,PwiNotethatforstraighterchainswfissmallerwhilefortangledchainswfislarger.Asthechainsarestretched,theratiowf/wi<1.Forthisreasonthelnterminequation(8.40)isnegativeandFpositive.HenceitispredictedthatmoreFisneededathighertempera-tures,meaningtheYoung’smodulusincreaseswithT,whichisexactlywhatweobserved不得转载谢谢合作LWM 186MECHANICALPROPERTIESOFSOLIDS—PLASTICITYhere.Thereforethismodelofchainstraighteningyieldsaconsistentpictureofelastomericbehavior.RELATINGREADINGC.R.Barrett,W.D.Nix,andA.S.Tetelman.1973.ThePrinciplesofEngineeringMaterials.Pren-ticeHall,EaglewoodCliffs,NJ.Areadableelementarytextforafirstcourseinmaterialsscience.J.M.GereandS.P.Timoshenko.1984.MechanicsofMaterials.Brooks/ColeEngineering,Monterey,CA.Anauthoritativetreatmentofmechanicsofmaterials,startingfromthebasicsofelasticityandplasticityandincludingmanyimportantpracticalexamples.P.A.ThorntonandV.J.Colangelo.1985.FundamentalofEngineeringMaterials.PrenticeHall,EaglewoodCliffs,NJ.Areadableelementarytextforafirstcourseinmaterialsscience.EXERCISES1.DiscussthefactthatfortheBCandFCCtheminimumdeformationoccursforslipsystems.2.Showwhythetheoreticalstrengthofamaterialisseldomrealized.3.Explainwhyrubbertendstocrystallizewhenitisstretched.4.HowcanyoudistinguishexperimentallybetweenmechanismsofNabarro-Herringcreep,anddislocationgenerationandmotiontoexplainplasticdeformationinamaterial.5.FromtheMaxwellmodelpredicttheeffectofahigherviscosityonthestressresponseduringloadingwithaconstantstrain.LikewisepredicttheeffectofahigherYoung’smodulus.6.Dothesameanalysisasinproblem5,butfortheVoigtmodel.不得转载谢谢合作LWM 9ELECTRONICSTRUCTUREOFSOLIDS9.1INTRODUCTIONTheelectronicstructureofsolidmaterialsisfundamentaltounderstandingvirtuallyallthepropertiesofmaterials,includingthearrangementofatomsandmolecules,thermo-dynamicproperties,mechanicalproperties,andelectronicproperties.Itistheelectronicpropertiesthatarethefocusofthischapterandthenexttwochapters,butbeforethesepropertiescanbediscussedinChapter10anddevicesinChapter11,abasicunder-standingoftheelectronicstructureisrequired.Thischaptercommenceswithwavemechanics,includingelectronparticle-waveduality,andthenprovidesaquantummechanicaltreatmentofelectronsinperiodicstruc-tures.Thisdiscussionleadstotheelectronicenergybandstructuremodelthatisusedinmuchofthisandthefollowingchapters.9.2WAVES,ELECTRONS,ANDTHEWAVEFUNCTION9.2.1RepresentationofWavesFigure9.1showsthesinefunctionforsimpleharmonicmotion(SHM).Att=0thewavetravelinginthisfigureisrepresentedasy=fxAx()=sin(9.1)ThisrepresentationcanbemodifiedtoyieldSHMatanytimetforawavetravelingwithvelocityv:ElectronicMaterialsScience,byEugeneA.IreneISBN0-471-69597-1Copyright©2005JohnWiley&Sons,Inc.不得转载谢谢合作LWM187 188ELECTRONICSTRUCTUREOFSOLIDSyxx’Figure9.1Aperiodicwavetravelinginthexdirectionandobservedatx¢.y=fxvt()-=-Asin(xvt)(9.2)Inequation(9.2)theproductvtsubtractedfromxexpressesthefactthatanobserveratx’inFigure9.1“sees”thewavemovingpastfromtheleft.Henceattthewaveampli-tudethatwaspreviouslyatthet=0valueisnowatthevtvalue,andtheamplitudenowatxisgivenbythevaluesin(x-vt),indicatingthatthewaveistravelingfromlefttoright.Thedisplacementintermsofangularmeasureinfractionsofawavelength,2p,isgivenas(x/l)2p.Sincek=2p/l,weobtainY=Asin()xvt-=-Asin()kkxvt(9.3)Bytherelationshipsv=lnandw=2pn,andthenv=lw/2p=w/k,thefollowingrepre-sentationisobtained:Y=Aksin()xt-w(9.4)Ofcourse,forawavetravelingbothleftandright,thewaveisasfollows:YY=-Aksin()xtAkww++sin()xtor=±Aksin()xtw(9.5)Complexnumbers(seethediscussionaboutcomplexnumbersinChapter3)arealsousedtorepresentharmonicwaves.Foronethingtheyareeasilymanipulatedmathe-matically(differentiated,integrated,etc.).Acomplexnumberzisdefinedaszai=+b(9.6)whereaistherealpart,Re(z),andbtheimaginarypart,Im(z).OntheimaginarynumberaxesshowninFigure9.2,weseethatzprojectedontherealaxisiszcosq,andontheimaginaryaxis,itiszsinq.Thenthesetermsareaddedtofindtheresultantzas不得转载谢谢合作LWM 9.2WAVES,ELECTRONS,ANDTHEWAVEFUNCTION189ImzqReFigure9.2Imaginarynumberzisrepresentedbyprojectionsontoreal(Re)andimaginary(Im)coordinates.zz=+()cosqqisin(9.7)Then,bytheEulerformula,wehaveiq(9.8)ei=+cosqqsinItfollowsthatanexponentialrepresentationisappropriateforyfromequation(9.4):()ikxt()-w(9.9)Y=AeIncludingbothleftandrighttravelingwavesandnowusingYtodenotethecompletewave(ratherthany),wehaveixkk--ixitww-itYY=+()AeBeeand=ye(9.10)Laterwewillbeinterestedinthederivatives∂Y=-iwY(9.11)∂tand2dYYixkk--ixitw∂2=-()AikeBikeeand=-kY(9.12)2dx∂x9.2.2MatterWavesInthissectionwerelatepurewaveswithmatter.Wecommencewithareviewofsomephysicsformulas,andthenpiecethepicturetogether.ForpurewavesthetotalenergycarriedbyawaveisgivenasEh=u(9.13)HereuisthefrequencyandhisPlank’sconstant.Foraparticlethekineticenergyisgivenas不得转载谢谢合作LWM 190ELECTRONICSTRUCTUREOFSOLIDS2121pEmkin===vpv(9.14)222mSincephereisthemomentum,p=mv,andforanelectron,p=mev.Also,formatter,thetotalenergyisgivenas2Em==cpc(9.15)Thespeedoflightcisgivenbyc=ul,wherelisthewavelength.Since,accordingtodeBroglie,matterandenergyareunified,thetotalenergiesmustbeequal,soequations(9.13)and(9.15)yieldtherelationshiphpu=c(9.16)Substitutingc=ulintoequation(9.16)yieldshpuu=l(9.17)Nowthefollowingrelationshipisobtained:h1ll=µand(9.18)pmThisremarkableresult,knownasthedeBroglierelationshipteachesthatbothparticlesandwaveshavewavelengthsandthatthewavelengthofaparticleisinverselypropor-tionaltotheparticle’smass.Interestingly,oncethisresultbecameaccepted,thatthereisadualityofwavesandmatter,themechanicsformallyusedforwavescouldbeappliedtoparticlesofmatteraswell.Beforepursuingthemechanicsassociatedwithduality,wewillperformsomecalcu-lationstounderstandhowthedeBroglierelationshipoperatesforelectrons.Electronic-31-19restmassisme=9.1¥10kg.Ifweassume1eVelectronenergyor1.6¥10J,with1/25v=(2E/m)fromequation(9.14),weobtainanelectronvelocityv=5.9¥10m/s.-25Thisyieldsamomentum,mv=5.4¥10kg-m/s.Then,usingequation(9.18),wehave-34-25-9l=h/p=6.63¥10J-s/5.4¥10kg-m/s=1.2¥10m.Thevalueforthewavelengthisobtainedasl=1.2nm,or1eVelectronenergyyieldsadeBrogliewavelengthofabout1nm.RecallthatinChapter3similarcalculationsweredoneforneutrons,toachievel=0.1nmforanenergycorrespondingtoabout400K(calledthermalneutrons).Atthispointthefoundationhasbeenputinplacetotreatelectronsaswavesandemploywavemechanics.Attheheartofthatpossibilityofusingwavemechanicstoexploretheelectronicstructureofmatteristheissueofwhatmathematicalfunctionsaretobeusedforelectronswithinthewavemechanicsmachinerythatembodytheproper-tiesofmatterinawave-likefunction.Thesefunctionsarecalledwavefunctions.Beforeweproceedtolearnwhattheyare,wewillreviewtheconceptofsuperposition(alsodiscussedinChapter3).Superpositionisahelpfulconceptforwavefunctions.9.2.3SuperpositionAswewillseebelow,andhavealreadyseeninChapter3onthesubjectofdiffraction,itisimportanttoconsiderseveralwavestravelinginthesamedirectionand/orimping-不得转载谢谢合作LWM 9.2WAVES,ELECTRONS,ANDTHEWAVEFUNCTION191Eo2Esino2a2Eo}a2aEo1Esina}o11a}i}Ecoso1a1Ecoso2a2Figure9.3SuperpositionofvectorsE01andE02toyieldresultantEo.ingatthesamepoint.Inmanycasesthecombinedeffectofmultiplewavescanbeobtainedusingthesuperpositionprinciple.Fortwowaves,1and2,thenetdisplacementatthepointofintersectionofthetwowaves,yisobtainedasyyy=+(9.19)12Forwavesofthesamefrequency,superpositionreferstotheadditionofwaveswhiletrackingbothwaveamplitudeandphase.Thisistheideabehinddiffractionwherewavesofthesamefrequencyarescatteredtoyieldinformationaboutthesymmetryandposi-tionofstructuralelements(seeChapter3).Vectorsareusedtotracktwoquantities.Intheadditionoftwowaves,1and2,wecommencebyassigningamplitudesE01andE02tothewavesandphasesa1anda2.Figure9.3showstheadditionofthesetwovectorstoyieldtheresultantvectorE0.Thisfigureshowsthattheycomponentsoftheresultantarethetwosinecomponents,E01sina1andE02sina2,andthatthexcomponentsoftheresultantarethetwocosinecomponents,E01cosa1andE02cosa2.Thesecomponentsareseparatelyaddedandthensquaredtoyieldthesquareoftheresultantasgivenbythefollowing:222EE=+()sinaaEsin++()EcosaaEcos(9.20)0011022011022Theneachofthesquaredtermsisexpandedtoyield2222222EE=++sinaaEsin2EEsinasinaa+Ecos001102201021201122(9.21)++EEcosaa2Ecoscosa02101021222Bytheidentitysinz+cosz=1,thefollowingexpressionisobtained:不得转载谢谢合作LWM 192ELECTRONICSTRUCTUREOFSOLIDS222EEEEE0=++010220102()sinaa1sin2+cosaa1cos2(9.22)Bytheidentitycos(a1-a2)=sina1sina2+cosa1cosa2,thefinalresultisobtainedfortheadditionoftwowaves:222EEEEE0=++010220102cos()aa1-2(9.23)Byanalogywiththetwo-waveresult,wecangeneralizeforNwavesasfollows:N2N2NNN2È˘È˘2EE0=ÍÂÂÂ0iisinaa˙+ÍE0iicos˙=+E0i2ÂÂE00ijEcos()aji-a(9.24)Îi===1˚Îi1˚i1j>ii=1ThesumofNwavesofidenticalfrequencyisawaveofthesamefrequencywithanewamplitudeandphase.Thisresultcanbeusedtocomparediffractionimageswherethephasesarecoherentwiththosewherethephasesarenot.Ifthea’sarerandom,then(a1-a2)isalsorandom.AsintherandomwalkproblemofChapter5,thesumovertherandomcosinetermswillbe0.Ifwethenconsiderthattheamplitudesarenearlythesame,wecanwrite22EE=N(9.25)001However,ifthephasesareequal,then(a1-a2)is0andthecosineis1.HenceweobtainNNNN222È˘22EE0=+ÂÂ0i2ÂÂE00ijEE=Í0i˙=NE01(9.26)i==1j>ii1Î1˚Thiscanbeverifiedforthecaseof2waves(N=2)withequalamplitudeswherethefollowingisobtained:NNN222222EE0=+ÂÂ0i22ÂE00ijEEEE=+=0ii20N0i(9.27)i==1ji>i1Theresultinequation(9.26)wasdiscussedinChapter3fordiffractionwherethephasesarecoherent,andequation(9.25)appliestoincoherentscattering.ThedifferenceinthecoherentandincoherentphasesisbetweenNandthesquareofN,whereNisalargenumber.Thuscoherentphasediffractionyieldsalargesignalrelativetothebackground.Anotherusefulresultisforwavestravelingindifferentdirections.Again,forsimilaramplitudeswecanwriteforsuperpositionEEEE=+=sin()kkxt+ww+-Esin()xt(9.28)1200Notethatselectionofsinefunctionsisarbitrary.Bytheidentity11sinab+=sin2sin()a+bacos()-b(9.29)22witha=kx+wt,b=kx-wt,weobtainthefinalresultasEE=2sin()()kxcoswt(9.30)0不得转载谢谢合作LWM 9.2WAVES,ELECTRONS,ANDTHEWAVEFUNCTION193102sin(kx)cos(kx)1090180270360kxFigure9.4Plotofstandingwavefromsuperpositionwithnodesatkx=ml/4.ThisisastandingwaveasshowninFigure9.4.Forkx=2px/landwt=2pnt=2vpvt/l=kx,thenodesareatx=ml/4,wheremis0,1,2,...,andwhereeitherthesinorcosiszero.Thesuperpositioncaseforourneedsaheadisthatofwavesofnearlythesameampli-tudebutwithdifferingfrequencywandwavenumberk.Superpositionofthesewaves(inthiscasecosinewavesarearbitrarilychosen)yieldsEEEE=+=cos()kx-wwt+-Ecos()kxt(9.31)12011022Bytheidentity11cosab+=cos2cos()a+bacos()-b(9.32)22weobtainÈkk12+ww12+˘Èkk12-ww12-˘EE=20cosxt-cosxt-(9.33)ÎÍ22˚˙ÎÍ22˚˙Aftermakingthesubstitutionskk+ww+kk-ww-12121212khh=,,w=andk1=,w1=(9.34)2222weobtainthesimplifiedresult:EE=-20cos[]kxhhwwtcos[]kx11-t(9.35)Weseethatwh>wl,sothefirsttermcontainsahigh-frequency(sumoffrequencies;seeequation9.34)waveandthesecondcosinetermbeingthedifferenceinfrequenciesisalow-frequencyterm.Figure9.5adisplaysthiskindofcompositewavewhere,asabove,不得转载谢谢合作LWM 194ELECTRONICSTRUCTUREOFSOLIDS10cos(10Z)cos(1Z)1090180270360450540630720a)Z10cos(1000Z)cos(Z)1090180270360450540630720b)Z10[cos(1000Z)cos(Z)]*1090180270360450540630720c)ZFigure9.5Threedifferentmodulatedwavesappearingmore“particle-like”from(a)through(c).*indicatesthefunctionraisedtothe100power.不得转载谢谢合作LWM 9.2WAVES,ELECTRONS,ANDTHEWAVEFUNCTION195oneperiodicfunctionaltersormodulatestheother.InFigure9.5aweseetwofrequen-cies.Theamplitudeofthehigh-frequencycomponentischangingatalower-frequency.Ineffectthereisalow-frequencyenvelopethatcontainsthehigh-frequencycomponent.Thevelocitiesforthetwocomponentsare,ingeneral,different.Thehigh-frequencycomponentofvelocityiscalledthephasevelocity,vp,andthisvelocitycanbecalculatedstartingfromthefollowingrelationship:wv==nl(9.36)kwherewistheangularfrequency,w=2pn.Thenweapplyequation(9.36)tohighandlowfrequenciesresultingfromtheadditionandsubtraction,respectively,ofsimilarmagnitudecomponents.Wereturntoequation(9.34)andobtainforthehigh-frequencycomponentofthewaveenvelopeww+w12vp=@(9.37)kkk+12Bysimilarreasoning,wecanobtainthelowervelocityfortheenvelopecalledthegroupvelocity,vg:ww-dw12vg=@(9.38)kk-dk12Theresultobtainedhereofaddingtogetherorsuperimposingmultiplewaves,asgivenbyequation(9.35),canalsobeusedtodescribethephenomenonofaddingintelligencewaves(speechormusicorimages)toacarrierwavethatcanbetransmittedaselectro-magneticwavesandreceivedbyaradioorTV,forexample.Itisusefultoconsiderhowthismodulationisaccomplished,inthatitisrelatedtodefiningwavefunctions.Forspeech,acousticwavesmustfirstbetransformedtovaryingelectricalsignalsusingatransducer.Forspeechormusic,acommonlyusedtransducerisamicrophonethattrans-formsthelongitudinalacousticwavestovaryingelectricsignals.Thentheserelativelyslowlyvaryingelectricwavescanbesuperimposedwithrelativelyrapidlyvaryingradiofrequency(RF)waves.TheRFwavesarebettersuitedtolong-rangetransmission.Theresultingmodulatedcompositewavehasaslowlyvaryingamplitudethatisthefrequencyofthespeechandformsanenvelopethatcontainstherapidlyvaryingcarrierwave.Aradioistunedtothecarrierfrequency,andadetectorcircuitisusedthatisnotsensitivetothehigh-frequencycarrierbutratherdetectsthechangingamplitudeofthecarrier,whichistheintelligence.Thislowerfrequencywaveisthenfedtoanothertransducer(speaker)whereitischangedbackintoaudiblespeech.Figure9.5bshowstwo-wavemodulationwherethehighfrequencycarrierisamplitudemodulated(AM)byalowerfrequencywave.OfcoursethewavesresultingfromspeechwouldnotbeperiodicandwouldbemuchmorecomplexthanthisexampleshowninFigure9.5b.Themodulatedwavepropagatesatthegroupvelocity,andthusthisvelocityismoreimportantbecauseenergyisalsotransmittedbythewaveatvg.9.2.4ElectronWavesTheideasandexamplesdevelopedaboveaboutthesuperpositionofwavesarenowextendedbeyondtheadditionofafewwavestothesuperpositionofmanywaveswithdisparatefrequenciesandamplitudes.Figure9.5aandbcanbeusedtocomparewhathappensasaresultofmodulationwheretwowavesthataresuperimposeddifferwidely不得转载谢谢合作LWM 196ELECTRONICSTRUCTUREOFSOLIDSinfrequency.InFigure9.5athefrequencyofboththehigh-andlow-frequencycompo-nentcanbediscerned.However,inFigure9.5bitisdifficulttodiscernthefrequencyofthehigh-frequencycomponent.Ifweconsiderthesuperpositionofmanywaveswithdisparatefrequencies,thepossibilitiesareendless,butonerelevantexampleisshowninFigure9.5c.Noticeinthiscasethatwidenodesareproducedthatseparatepulsesthathaveaverycomplexwaveform.Itisvirtuallyimpossibletodiscernfrequencyforthehigh-frequencycomponent.Evenmorecomplexmodulationscanproducepulsesthathavestilllargerseparations,andalsoisolatedpulses.Itisnowarguedthatthesewidelysepa-ratedpulseshavesomepropertiessimilartoparticles.Inthesecomplexwaveformpulses,itisnotpossibletodeterminethefrequency,hencetheenergy,inthepulse.Asthepulsenarrows,however,itbecomeseasiertodetermineitsposition.Thisisthebasicideacon-tainedintheHeisenberguncertaintyprinciplethatappliestomatter.ItisexpressedashDDpx◊≥(9.39)2whereDpandDxareuncertaintiesinthesimultaneousdeterminationofpositionand2momentum(recallthatmomentumandkineticenergyarerelatedasE=p/2m)ofapar-ticle.Thisexpressionindicatesthatitisnotpossibletopreciselydeterminetheenergyandpositionofasmallparticle.Tomeasurethepositionusing,forexample,photons,thepositionoftheelectronmustbeaccuratelydetectedbyshortwavelengthphotonssuchasgorXrays.Anyinteractionoftheprobephotonwiththeelectrontobemea-suredcanaltertheelectronmomentum(orenergy)andhenceresultinuncertainty.Thuscomplexmodulationyieldsawavepacketthathaspropertiessimilartothoseofasmallparticleinthateachparticlehasareadilydefinedposition.ForthestandingwaveshowninFigure9.4,thefrequencyisreadilydiscernablebutnotitsposition.Thereforeonecantakeanintellectualleapandexpressaparticle,say,anelectron,asacomplexwaveresult-ingfromthemodulationorsuperpositionofmanysimplerwaves.Theresultantdescrip-tionoftheparticleiscalleda“wavefunction,”andthesymbolYisusuallyused.Oncehavingmadethisscientifichypothesis,onecanusetheavailablemachinery/mechanicsandexplorethevariousconclusions.Thisis,ofcourse,thefieldofquantummechanics,whichhasallowedahugeandever-growingnumberofcorrectconclusions.Allofthe“miracles”ofquantummechanicscommencewiththewavefunctionasanappropriatedescriptoroftheelectronand/orotherinterestingparticles,withtheappropriateequa-tionstosolveforenergyorotherinterestingproperties.WecansummarizethissectionbyrealizingthatthewavefunctionY,whichisusedtorepresentaparticleinquantummechanics,derivesfromthefactthatcomplexwavesresultingfromsuperpositionsharesomeimportantbehavior.Thisnotionisattheheartof“duality”andprovidesthebasisforquantummechanicsBelowwewilladopttheuseofYanduseappropriateformulastoperformavarietyofcalculationsaimedatelucidatingtheelectronicstructureofsolids.9.3QUANTUMMECHANICSInthissectionweadoptthewavefunctiondescriptionoftheelectron,anddevelopsomebasicideasabouttheSchrödingerequation(SE).ThenwepresentseveralsolutionsfortheSEthatareusefulfortheunderstandingelectronicstructure.Theobjectiveistoarriveatasimplemodelforasolid,theso-calledKronig-Penney(KP)model.Despiteitssim-plicityandassumptions,theKPmodelcontainsmostoftheessentialfeaturesrelating不得转载谢谢合作LWM 9.3QUANTUMMECHANICS197totheoriginofelectronicstructureofsolidswithoutthepotentiallycomplicatedmath-ematicsnecessaryformodelsthatprovideprecisecalculations.9.3.1NormalizationRecalltheexponentialformforYexpressedinequation(9.10):-itwY=ye(9.10)whereikx-ikxy=Ae+Be(9.40)Clearly,Yiscomplex.Sincetheintroductionofthatrepresentation,wehavelearnedthatthewavefunctionYcanbeusedtorepresentelectronsthatarereal.ThisdilemmacanbeovercomeusingapostulatefortheprobabilityP(x,t)offindinganelectroninsomerangeofx(in1-D),say,fromxtox+dxasPxt(),*=YYdxin1-D(9.41)andPd=YY*Vin-D3(9.42)Thispostulaterenderstheprobabilityreal,andYY*(thesuperscript*referstothecomplexconjugate)canbeinterpretedasameasureoftheelectrondensityforYasthewavefunctionrepresentinganelectron.FurthermoreitfollowsthatthesumofalltheprobabilitiesPoffindinganelectronatallpointsmustequalunity.Thiscanbeexpressedby•ÚYY*dV=1(9.43)-•TheintegralequationaboveiscalledthenormalizationconditionforY.Fortheform-iwtiwtforY,thecomplexconjugateforeiseandhenceYY*=yy*.Forthecasewhere2yrepresentsastandingwave,yisrealandyy*=y.9.3.2DispersionofElectronWavesandtheSEStartingfromtherelationshipinSection9.2.2,equations(9.13)through(9.18),wecanwrite,usingk=2p/landS=h/2p,hp===hkmve(9.44)lThekineticenergyisgivenas22212hkpKE===mv(9.45)222mmeeThetotalenergyisthesumofkineticandpotentialenergies(PE).Theresultingdisper-sionrelationshipcanbeexpressedasfollows:不得转载谢谢合作LWM 198ELECTRONICSTRUCTUREOFSOLIDS222hkpEK=+===+=+EPEhwwhVV(9.46)22mmeewhereVisthepotential.Thisyieldsforw,2hkVw=+(9.47)2mheTheserelationshipswillbeusedbelow.Assumethatasolutionforawaveequationisawavefunctionoftheformabove:ikx--ikxitwY=()Ae+Bee(9.48)Thenthederivativeswithrespecttotimeandxintroducedaboveareasfollows:∂Y-itw=-iiwwY=-ye(9.11)∂tand2∂Y2=-kY(9.12)2∂xSolvingthesederivativesforYandequatingtheresultsyields22-∂kYY∂=-w2(9.49)it∂∂xUsingthedispersionrelationshipforwfromaboveobtains222-∂kYY∂ÏhkV¸=-2Ì+˝(9.50)it∂∂xÓ2meh˛Nowdivideequation(9.50)byk2andmultiplybyh-toobtain22-∂hhYY∂ÏV¸=-2Ì+2˝(9.51)it∂∂xÓ2mek˛AftermultiplyingthebracketedtermsbythesecondderivativeofYandthensubstitut-ingequation(9.12)aboveinthesecondterminbrackets,wehave22-∂hhYY∂=-+VY(9.52)2it∂2me∂xThisequationcanbefurthersimplifiedbyseparatingthespatialandtemporalpartsofY.Toaccomplishtheseparation,weusethetimederivativeofYfromequation(9.11),keepinginmindthatyisonlyafunctionofposition.Theresultis22dYYd-wit2=2e(9.53)dxdxNowwecansubstitutetheseexpressionsintoequation(9.52)aboveandobtain不得转载谢谢合作LWM 9.3QUANTUMMECHANICS1992dy--itww22mieeitm-itw22e+-()iewy-Ve()y=0(9.54)dxhhiwtAftermultiplyingthroughbyeandrecognizingthatE=Sw,weobtainin1-D,2dy2me+-()EVy=0(9.55)22dxhThisequationisthetypicalformforthetime-independentSE,andtheonethatwillbeusedinthefollowingsections.9.3.3ClassicalandQMWaveEquationsRecallfromChapter7,equation(7.19),thattheclassicalwaveequationcontainsasecondspatialandsecondtemporalderivative.Thiswaveequationiseasilyobtainedfromequa-tion(9.3)usingtheearlierrepresentationofawave:y=fxvt()±=±Asin()xvt(9.56)Considerawavey=f(x¢),wherex¢=x±vtwithxthedistancetraveledandvtheveloc-ity.Thiswavemaybeaharmonicwavewherethefunctionistrigonometric,asshowninequation(9.56).Thechainruleisusednexttodevelopformulaswithpositionxandtimetasvariables.Thefirstderivativesare∂x¢∂x=1and=±v(9.57)∂x∂tWiththesederivatives,andbythechainrule,wecanobtainthefollowingformulas:22∂y∂f∂x¢∂f∂y∂∂f∂f==◊1and==(9.58)22∂x∂x¢∂x∂x¢∂x∂x¢∂x¢x¢22∂y∂f∂x¢∂f∂y∂Ê∂fˆ∂x¢∂f2(9.59)==◊±vand=◊±v◊=v∂t∂x¢∂t∂x¢∂t2∂x¢Ë∂x¢¯∂t∂x¢2Noticethesimilarityofthesecondspatialandtemporalderivatives(rightsidesofequa-tions9.58and9.59).Thenequatingthesederivativesyields222∂y1∂y21∂y=andin-D3—=y(9.60)22222∂xv∂tv∂tThisisthesameclassicalwaveequationweusedinChapter7asequation(7.19).However,aswasshownabove,theSEhasadifferentform:22-∂hhYY∂=-+VY(9.52)2it∂2me∂xTheSEhasafirstderivativewithrespecttotime.ThereasonforthedifferentformfortheSElieswiththeformforthedispersionequation(9.46),whichderivesfromwave-particleduality.Consequently,whentheproperdispersionformulathatincludesduality不得转载谢谢合作LWM 200ELECTRONICSTRUCTUREOFSOLIDSisusedintheformulation,theSElooksmorelikeadiffusionequation(recallFick’ssecondlawfromChapter5,equations5.19and5.21)thanaclassicalwaveequation.InthefollowingsectiontheSEisappliedtoarriveattheelectronicstructureforsolids.9.3.4SolutionstotheSEUltimatelyinthissectionwewillsolvethetime-independentSE(equation9.55)forasolid.Ourobjectivewillbetofindtheunperturbedallowedelectronicenergylevelsinthesolid.Theseallowedenergylevelscomprisetheelectronicstructureofthematerial.Thetime-dependentsolutiontotheSEisnotusefulherebecauseweareinterestedintheequilibriumortimeinvariantelectronicstructure.However,todeterminethetimeevo-lutionofeventsthatcandisturbtheequilibriumstructuresuchasvariousspectroscopiesthatuseaperturbingradiation,forexample,acompletesolutionisrequired.Beforepro-ceedingdirectlytothetaskofsolvingthetime-independentSEforasolid,wetakeaslightdetourandapplytheSEtoseveralsimplerproblems,namelyfreeandboundelec-trons.ThesesolutionscanyieldinsightintoboththesolutionmethodoftheSEandtheanticipatedresults.Inadditionitshouldberecognizedthatonecanobtainprecisesolu-tionsbyusingrealisticinputstotheSE.Thisispossibleinsomecases,andsolutionshavebeenfoundthatareincloseagreementwithreality.However,themathematicalcom-plexity,evenforverysimplesystems,canbedistracting.Thusthestrategyweusehereistomodelsystemsthatdonotgivepreciseresultsbutratherphysicallycorrectandmean-ingfulresultsbywayofsimplemathematics.ThiswillleadustothesolutionoftheSEforthecomplicatedcaseofasolidcomposedofmanyatoms,eventhoughwedonotobtainaccuratenumericalresults.9.3.4.1FreeElectronSolutiontotheSEAswasalludedtoabove,itissometimesuseful,andalmostalwayssimpler,toperformcalculationsonmodelsystemsasopposedtorealsystems.Inthisspiritweconsidertotallyfreeelectrons,eventhoughwerealizethatelectronsinsolidsarenevercompletelyfreeofthepotentialthatexistsinthesolid.However,someelectronsinsomesolids(goodconductors,e.g.,CuandAu)behaveasiftheyarefree,ifnotentirelyfree.Thus,usingthemodeloftotallyfreeelectronscanyieldsomeinsightintohowthesekindofelectronsingoodconductorsbehave.ThetimeindependentSEwrittenin1-Dis2dy2me+-()EVy=0(9.55)22dxhStartingfromequation(9.55),thecondition(s)thatrelatetoentirelyfreeelectronscanbeimposed.Essentiallytheconditionforanentirelyfreeelectronisthatbindingpoten-tialfortheelectronbezero:V=0.FromthisconditiontheSEbecomes2dy2me(9.61)+=Ey022dxhThesolutionforthisSEcanbewrittendownimmediatelywhenwerealizethatthistime-independentSEisvirtuallyidenticaltotheclassicalwaveequationwhenthetimedepen-denceisremoved.Thus,fromapurelymathematicalpointofview,wearedealingwiththeundampedvibratingstringproblemofclassicalphysics.Inthisspecificcasethestringisfreetovibrate,sinceitisnotheldorconstrainedinanyway.Thesolutionforthiskind不得转载谢谢合作LWM 9.3QUANTUMMECHANICS201ofvibrationcanbeimmediatelyrationalized.Alltypesofvibrationsarepossible,andwecanwritethissolutionin1-Dasixaa-ixy()xA=+eBe(9.62)whereAandBareconstantstobedeterminedandaisgivenasa=k,atwhichequa-tion(9.62)isthesameasequation(9.40).Thiscanbefoundbyconsideringthatthetime-independentSEtobesolvedisadifferentialequationoftheform2dfx()a◊+bfx()=0(9.63)2dxThisequationcanbesolvedtoyieldixaa-ixfx()=+AeBe(9.64)1/2witha=(b/a).FromtheSEabovetheratioofthecoefficientsaandbisb/a=2mE/-h2.RecallingthatE=p2/2m,p=h/l,andh-=h/2p,weobtainaasfollows:ee222mEmppp22ppeea======k(9.65)22hh2mhhleNowconsideringthepropagationofthewaveinonedirection,weobtainthefinalresult:ikxy()xA=e(9.66)Fromequation(9.65)fora,wherewefounda=k,weobtainfortheenergy2h2E=k(9.67)2me2Thisresultisrevealingofthebehavioroffreeelectrons.FirstweseethatEk,whichindicatesthatallenergiesarepossible.Thusthereexistsacontinuumofallowedelectronstatescorrespondingtovaluesofk.Second,theallowedenergiesarearrangedinaparabolicband,andthisisshowninFigure9.6.Thisparabolicbandisknownasthefreeelectronband.E-kkFigure9.6Prabolicelectronenergybandforfreeelectrons.不得转载谢谢合作LWM 202ELECTRONICSTRUCTUREOFSOLIDS9.3.4.2StronglyandWeaklyBoundElectronSolutiontotheSEWecanmodelthesituationforstronglyboundelectronsbyassumingthattheelectronsareinapotentialwellofinfinitedepth.Thisistheso-calledparticleinaboxformulationinwhichthewallsofthe1-Dboxareinfinitelyhigh.Underthisassumptionofinfinitelyhighbarriers,thereiszeroprobabilityfortheelectrontobeoutsidethebox.Wefirstsolvethisproblem,andthenwerelaxthestrictrequirementontheinfinitelyhighwallstofinitewallsandresolvetheweaklyboundelectronproblem.Wecommencewiththe1-DtimeindependentSEasbefore:2dy2me(9.55)+-()EVy=022dxhbutinthiscaseVπ0.Thisproblemandthedifferentialequationaboveareanalogoustotheclassicvibratingstringproblemthatweusedforthefreeelectrons,butratherthanthestringbeingunconstrainedasaboveforthefreeelectronproblem,nowthestringistightly(infinitelytightly!)heldattwopositionsalongitslength.Thus,whenthestringispluckedinbetweenthepinningpoints,itwillvibrateandsustainthosevibrationsthatdonotdestructivelyinterfere.Furthermore,becauseoftheinfinitelytightpinning,thevibrationscannotpropagatebeyondthepinnedpoints.Withinthepinnedpointsthewavescanpropagatetotheleftandright.Asforthecaseabovethegeneralsolutiontothedifferentialequationisasitwasabove:ixaa-ixy()xA=+eBe(9.62)Thebarriers(pinningpoints)areshowninFigure9.7.Nowthisgeneralsolution,equa-tion(9.62),needstobetailoredtobeinconformitywiththespecificconditionsoftheproblem,theso-calledboundaryconditions.WithreferencetoFigure9.7theboundaryconditionsare(1)at0andl(thepinningpoints),V=•;(2)y=0atx£0,andx≥l.Theseboundaryconditionsmeanthatthevibrationsdonotexisttotherightorleftofthebarriersbutdoexistintheregion0tol,soweadjustthegeneralsolutionaccord-ingly.Thiskindofproblemiscalledaboundaryvalueproblem.Thefunctionstobesolvedarecalledeigenfunctions,andtheresultantvaluesthatsolvetheproblemarecalledeigenvalues.Theconditiony=0atx=0inequation(9.62)yieldsthefollowing:iiaa00-Ae+=Be0(9.68)Thesolutionforequation(9.68)isthatA=-B.Theconditiony=0atx=linequation(9.62)yieldsthefollowing:V=•V=•0lFigure9.71-Dpotentialwelloflengthlandwithinfinitepotentialbarriers.不得转载谢谢合作LWM 9.3QUANTUMMECHANICS203iiaa11-Ae+=Be0(9.69)Tosimplifytheanalysisofresult(9.69),wecanmakeuseofanEulerrelationship:1iirr-sinr=-()ee(9.70)2iForthecaseofequation(9.69),r=alandA=-B,weobtainilaa-ilAe()-e==20Aisinal(9.71)Thisequationwillholdtrueifandonlyif(iff)al=np,wherenisaninteger,n=0,1,2,3....Thereforewecanexpressthisconditionasnp2mEea==(9.72)2lhandsolvefortheenergy:222hnpE=◊(9.73)22melTheimplicationsofthissolutionarequiteimportant.Unlikethefreeelectron(V=0)solution(equation9.67)whereallenergieswereallowed,nowasgivenbyequa-tion(9.73)forthetightlyboundelectrons,onlycertainvaluesarepermitted,specifically2thosevaluesofEwherenisaninteger.Thustheenergyisquantizedinunitsofn.Thequantizationoftheenergyisaresultoftheboundaryconditionsimposedonthesolu-tionoftheSE.Returningtothepluckedstringanalogyusedabove,weconsiderthatthestringisheldinfinitelytightlyattwopoints,andpluckedinbetweenthepinningpoints.Soweaskwhatwavelengthsforthevibrationsarepermittedthatdonotsufferfromdestructiveinterference.ItisclearfromFigure9.8a(forn=1,2,and3)thatthewavelengthsthatsurvivearethosethat“fit”integrallyinthelengthofthestring(thelengthofthebox,l).Theseallowedwavesconstructivelyinterfereandleadtostandingwaves.Alltheothersdestructivelyinterfereanddieout.OfparticularinterestisFigure9.8b,whichshows2yy*,andforrealythisisy.Theprobabilityisthatforn=1theparticleisnearthecenterofthebox;forn=2andhigher,theparticlehasmultiplepointsofhighproba-bility.Thisisadecidedlynonclassicalresult.Foraclassicalparticleinaboxtheparticlewouldhaveuniformprobabilityeverywhereinthebox.Interestinglywecanextrapolatetohighvaluesofn(e.g.,n=100)sothatthenumberofpeaksarehigh,butasnincreases,moreuniformprobabilityresults.Forhighenergies(highn)theparticlethenbehavesclassically.Itisfurtherusefultoconsidertheallowedenergiesasafunctionofnasshown2inFigure9.9a.Recallthatequation(9.73)showsthatasnincreases,Eincreasesasn,andtheenergylevelsaremoreandmoreseparated.Thisresultisdifferentfromwhatisexpectedforatomsboundinmoleculesthathaveabindingpotential.Forthecaseofboundatoms,asnincreases,theenergylevelsbecomemorecloselyspaceduptotheion-izationlevelwhereanelectronhasenoughenergytoescapefromtheatom.ThiscaseisshowninFigure9.9bandistheresultofthefactthattheformforthebindingpotentialVisdecidedlydifferent.Usuallythebindingenergyisreferencedtotheenergyneededforanelectrontoberemovedfromtheatom,andthislevelissetas0,sothebinding不得转载谢谢合作LWM 204ELECTRONICSTRUCTUREOFSOLIDSa)n=3yn=2n=10lb)n=3yy*n=2n=10lFigure9.8(a)Threesolutions(n=1,2,and3)fortheboundelectronproblem;(b)probabilitiesforthethreesolutionsin(a).energiesarenegativeaswasshowninChapter7,Figure7.4.SinceinFigure9.9a,V=•,whileinFigure9.9b,Visproportionaltothereciprocaloftheseparationthatis2characteristicofaCoulombicorMorsepotentialthatresultsinE1/n,andhencethecloserElevelsareathighern.Wecannowmodifythispictureslightlyandreleaseourinfinitelystronggriponthestring,inordertopermitsomeofthevibrationsto“leak”fromtheboundregiontotheadjacentregionsofthestring.ThismeansthatwelifttherestrictionthatV=•andsetVatsomefinitevalueandresolvetheSE.Forthisproblemweconsideranelectronwavepropagatingin1-Dfromlefttoright(+xdirection)asisshowninFigure9.10.Figure9.10alsoshowsthatthewaveistravelingintwodifferentregionswithrespecttothebindingpotential.Inregion1theelectronisfreeandV=0,andinregion2theelectronexperiencesafinitebindingpotential,V=Vfiinit.Thefinitebindingpotentialisgreaterthanthekineticenergyoftheelectronwave,Vfiinit>Ekin,butitisnotinfinite.Theoriginx=0issetattheborderofregions1(x<0)and2(x>0).WeneedtowriteseparateSE’sforthetworegionsandsolvetheSE’ssothatthesolutionsmatchatx=0.TheSEforregion1isthesameastheSEaboveforV=0andisasfollows:不得转载谢谢合作LWM 9.3QUANTUMMECHANICS205n=3Energyn=2n=1a)nfi•fi•n=3n=2-Energyn=1b)Figure9.9(a)Energylevelscorrespondingtoequation(9.73)withenergyspacingincreasingwithn;(b)energylevelsformolecularpotentialwithenergyspacingdecreasingwithn.VVfiniteRegion1Region2V=0V=Vfinite0x0Figure9.10Electronintworegions:Region1isthefreeelectronregion,andregion2hasafinitebindingpotential.2dy2me(9.74)+=Ey0(Region1)22dxhAndforregion2whereV=VfiinittheSEisasfollows:2dy2m+-e()EVy=0(Region2)(9.75)22finitedxh不得转载谢谢合作LWM 206ELECTRONICSTRUCTUREOFSOLIDSWehavealreadysolvedtheseSE’sabove,sothesolutionscanbewrittenimmediatelyasfollows:y()xA=+eBixaa11e-ix(Region1)1(9.76)where2mEea=(9.77)12hixaa22-ixy2()xC=+eDe(Region2)(9.78)where2mEV()-efinitea=(9.79)22hIf|Vfiinit|>|E|,thena2willbeimaginary.Thecomplicationsarisingfromthiscanbeavoidedbymakingthefollowingsubstitution:ba=i(9.80)2Withthissubstitution,bcanbewrittenasfollows:2mV()-Eefiniteb=(9.81)2hForthesolutioninregion2,weobtainthefollowing:bbxx-y2()xC=+eDe(9.83)ItisnowinstructivetodeterminetheconstantsA,B,C,andDforwhichthedetailsoftheproblemneedtobeconsulted.Weconsiderthebehaviorofy2(x)asxapproaches•andimposethecontinuityconditionsthaty1(x)=y2(x)attheborderx=0,andthatthederivativesdy1/dxanddy2/dxarealsoequalatx=0.AsxÆ•,weobtainthefollowing:b•y2()xC=+e0(9.84)However,thisresultisnotphysical,becauseaswasgivenbyequation(9.43),theprobabilityoverallspacemustbe1:•ÚYY*dV=1(9.43)-•ThewaytoensurethisconditionisforC=0,andtheny2(x)becomes-bxy2()xD=e(9.85)Fortheconditiony1(x)=y2(x)atx=0,weobtainixaab11--ixxAe+=BeDe(9.86)不得转载谢谢合作LWM 9.3QUANTUMMECHANICS207whichatx=0becomesABD+=(9.87)Fortheconditiondy1/dxanddy2/dxatx=0,weobtainAiaabeixaab11-=Bie--ix-Dex11(9.88)whichatx=0becomesAiaab11==Bi-D(9.89)Fromtheresultsin(9.89)and(9.87),AandBintermsofDcanbeobtainedasfollows:DiÊbˆDiÊbˆA=+Á1˜andB=-1(9.90)2Ëa¯2Ë2¯1Forourpresentpurposesthemostinterestingresultisthesolutionaboveinequation-bx(9.85)wherey2(x)=Deandwherey2(x)isshownversusxinFigure9.11ainregiona)VVfiniteRegion1-yy-xb1Region2-=De20x0b)n=3yn=2n=10lFigure9.11(a)SolutionstotheSEintworegionsfromFigure9.10asy1andy2;(b)othersolutionswithdashedportionindicatingleakingintoadjacentregions.不得转载谢谢合作LWM 208ELECTRONICSTRUCTUREOFSOLIDS2.Figure9.11ashowstheshapeofthewavefunctiony2inregion2tobeanexponen-tiallydecayingwavefunction.Thusthewavefunctionpenetratesintoregion2,butitdecaysrapidly.Thispenetrationofthewavefunctionintothebarrieriscalledquantummechanicaltunneling,andthetunnelingresultsfromchangingthebarrierheightfrominfinitelyhightofinite.LikewiseFigure9.11bshowstheresultforotherwavefunc-tionscorrespondingtodifferentn’saswasshowninFigure9.8abovefortheinfinitebarrier.Tunnelingisanimportantresultofquantummechanicsbecauseitshowsthatitisprobableforanelectrontoexistinaregionthatisclassicallydisallowed.Oneconse-quenceoftunnelingisthatifanelectronicdetectorisplacedclosetothebarrierinregion2,theelectronisdetected(butweakly),andthisistheunderlyingideaforscanningtun-nelingmicroscopythathasbecomeanimportantsurfacesciencetool.Aswillbedis-cussedinChapter11,thereareanumberofelectronicdevicesthatoperatebasedontunneling.Alternatively,asmoderndeviceshavegottensmallerinthenanotechnologydomain,electrodesindevicesaremorecloselyspacedandtunnelingcancauseexces-sivelyhighleakagecurrentsbetweenthecloseelectrodesthatdisruptnormaldeviceoperation.Somedeviceresearchersseetunnelingasasizelimitationcriterionforelec-tronicdevices.9.3.4.3PeriodicSolidSolutiontotheSE–Kronig-PenneyModelTheSEsolutionsdiscussedaboveyieldinsightsaboutthesolutionforasolidmaterial.Weproceedbyfirstconsideringthatthereisanatomicbindingpotentialforelectronsthatischaracteristicoftheiratomtype.RecallFigure7.4,whichshowedtheshapeforanatomicpotential.Wenowimagineanarrayofatoms,inparticular,anorderedarrayforasinglecrystalofaparticularmaterial.Associatedwitheachatominthecrystalisanatomicpotentialthatcharacterizestheatom’sinteractionwithadjacentatoms.Thearrayofatomicpotentialsisthisalsoorderedwiththeperiodicityofthecrystallattice.AnexampleofonekindoforderedpotentialisshowninFigure9.12awheretherealcrystalpotentialisperiodicandofacomplexshape.TheshapeshownisthatofanessentiallyCoulombicpotential,whichdependsonthereciprocaloftheseparationaswasmentionedpreviously.Thusthebindingisstrongerclosertotheatomiccore.Differentatomsofcoursedisplaydifferentshapedpotentials,andthesolutionoftheSEforonesuchpotentialwillrequiremodificationforanotherkindofatom.Formaterialswithdifferentkindsofatoms,differentstructuresandmorphologies,thesituationbecomesevenmorecomplicated.Yetinthosecasesinwhichpotentialsareavailable,precisecalculationsoftheelectronicstructurehavebeenmade.ThepointisthatproceduresarepresentlyknowntosolvecomplicatedSE’s.Theseproceduresenableprecisecalculationsthataremathematicallyintenseandbeyondthescopeofthistext.Nevertheless,itisimportanttounderstandtheimplicationsofthecalculationsandtogainafeelforelectronicstructure,whetherobtainedfromexperimentortheory.Tothisendasimplemodelforasolidthatyieldsphysicallycor-rect,thoughinaccurate,numericalresultsisuseful.Thisideawassuccessfullypursuedintheearly1930sandiscalledtheKronig-Penneymodel.Thismodelwillbeoutlinedbelow.TheKronig-Penney(KP)modelcommenceswithasimplificationofthepotentialforaperiodicsolid.Ratherthanattemptingtoapproximatecomplexshapedpotentials,theKPmodelusestheconjoiningoftworegionsandtherepeatingoftheseregionswithcrystalperiodicity.Figure9.12bshowstheKPperiodicpotentialasmadeupofapoten-tialfreeregion(V=0):region1withawidthaastheseparationbetweenatoms(similartothelatticeparameter),andregion2havingafinitebindingpotential(Vfiinit)andwidth不得转载谢谢合作LWM 9.3QUANTUMMECHANICS209a)VxSurfaceb)VVfinitebRegion1Region2-b0aa+bFigure9.12(a)1-Dperiodicbindingpotentialforasolid;(b)Kronig-Penneypotentialforaperiodicsolid.b(similartothebarrierwidth).Althoughthissimplepotentialcannotyieldcorrectnumericalresults,aswewillseebelowwhentheresultsfromtheKPmodelarecomparedwithrealelectronicstructures,theKPmodelwillbefoundtorevealthecorrectoverallphysics,butwithoutsomedetails.FurthermoretheKPmodelisamenabletoalgebraicanalyticalsolution,andfollowsdirectlyfromthesolutionsaboveforthefreeandboundelectrons.TheproceduretoobtainasolutionoftheSEusingtheKPmodelisessentiallythesameprocedureaswasusedaboveforthefinitepotential.First,forregions1and2inFigure9.12b,thetworelevantSE’sarewritten不得转载谢谢合作LWM 210ELECTRONICSTRUCTUREOFSOLIDS2dy2me+=Ey0(Region1)(9.91)22dxh2dy2me+-()EVy=0(Region2)(9.92)22finitedxhAsbeforetheenergiesforregions1and2,whichareequations(9.77)and(9.79),respec-tively,areexpressedas22mEmV()-Eeefiniteab==and(9.93)22hhThedifferencewiththeKPmodelisthatthepotentialisperiodicthroughoutthecrystal,andthesolutionforthetworegionsmustthereforebeasimultaneoussolutionextend-ingovertheentirecrystal.Solutionsforthiskindofproblemexist,andtheyaregivenbytheso-calledBlochtheoremtoyield(in1-D)theformikxy()xu=()xe(9.94)Inthisexpressionu(x)isaperiodicfunctionwiththesameperiodasthebarriers(a+b).NoticethatfromourpreviousdiscussionofwavesinSection9.2,thissolutionisessen-tiallyamodulatedwavewhereu(x)isthemodulatingfunction.ThesewavesareoftenreferredtoasBlochwaves.Sinceourdiscussionwillbeonlyin1-D,itshouldbeunder-stoodthattheperiodicityisdifferentineachcrystaldirection.Thusthesolutionin3-Disformorecomplicated.TheBlochtheoremisusedtoprovidethecorrectsolutionforourperiodicpotentialproblem.TheappropriatederivativesofyaretakenandtheSEisreformattedwiththesederivatives.Thefirstderivativeofthesolutiony(x)inequation(9.94)yieldstwoterms:dxy()ikxikxdux()=uxike()+e(9.95)dxdxThesecondderivativeofequation(9.94)yieldsfourterms:22dxy()22ikxikxdux()ikxdux()ikxdux()=uxike()+ike+e+ike(9.96)22dxdxdxdxWiththetermscollected,weobtain22dxy()ikxÊdux()dux()2ˆ=eÁ+2ik-kux()˜(9.97)2Ë2¯dxdxdxThesederivativesarenowsubstitutedinregions1and2oftheSE,andthefollowingsubstitutionsarealsomadefromequation(9.93):2222hhabE=-andVEfinite=(9.98)22mmee不得转载谢谢合作LWM 9.3QUANTUMMECHANICS211Forregion1,theSEis2ikxÊdux()dux()22ˆeÁ+20ik-kux()˜+ay()x=(9.99)Ë2¯dxdxikxWesubstituteiny(x)=u(x)e,andthefinalequationforregion1becomes2dux()dux()22+20ik--()kua()x=(Region1)(9.100)2dxdxForregion2,theSEis2ikxÊdux()dux()22ˆeÁ+20ik-kux()˜-by()x=(9.101)Ë2¯dxdxikxAftersubstitutingy(x)=u(x)e,aswasdoneforregion1above,weobtainforregion2,2dux()dux()22+20ik-+()kub()x=(Region2)(9.102)2dxdxRegions1and2SE’scanbeseentohavethesameformasthedifferentialequationforadampedvibrationinclassicalphysics.Ifu(x)isdefinedasthedisplacementforthevibration,thentheformisasfollows:2dux()dux()+A+Bux()=0(9.103)2dxdxThisequationhasageneralsolutionoftheform:2--Ax2ixxxixAux()=+e()CeDewherex=-B(9.104)4222Forregions1and2,A=2ik;forregion1,B=-(k-a);andforregion2,B=-(k-2b).Puttingthistogetherobtainsthefollowingsolutions:--ikxixaaixux()=+e()CeDe(Region1)(9.105)--ikxixbbixux()=+E()AeBe(Region2)(9.106)A,B,C,andDareconstantsthatneedtobedeterminedbythespecificconditionsoftheproblem,namelythatthesolutionsandtheirderivativesarecontinuousatthebound-ary(x=0)andatallperiods(x=a+b).ThuswehavefourequationsinthefourunknownsA,B,C,andD.Atx=0thesolutionsgivenbyequations(9.105)and(9.106)andthederivativesareequated(subscripts1and2areusedtoindicatetheregions)toyieldthefirsttwoequa-tionsasfollows:Foruxux12()=()atx=+0:CDAB=+(9.107)不得转载谢谢合作LWM 212ELECTRONICSTRUCTUREOFSOLIDSdux()dux()12For=atx=0:dxdx(9.108)Ci()aabb-k+--Di()k=-Ai()k+--Bi()kAlsotwoequationscanbeobtainedfortheperiodicboundaryconditionx=a+b.TheseequationsaremosteasilyobtainedbyusingFigure9.12bandrealizingthatu1(x)atamustequalu2(x)at-b,andlikewiseforthederivatives,andthisrequirementyieldsthefollowingtwoequations:Forux()at()xaux==()at()x=-b:12(9.109)()iiaa-ka()--iika()ikib-bb()ikib+Ce+=+DeAeBedxu()dux()12Forat()xa==at()xb=-:dxdx(9.110)ia()aab-k-+ia()k--ib()kib()b+kCi()aa-ke-+Di()ke=-Ai()bbke-+Bi()keNowtherearefourindependentequationsinthefourunknownsA,B,C,andD,namelyequations(9.107)through(9.110).AsimultaneoussolutionisobtainedbyformingadeterminantofthecoefficientsC,D,A,B,andsettingthedeterminantequaltozero.Thedeterminanthastheform:1111aabb--kkkk()+--()+ia()aabb-k-+ia()k--ib()kib()+keeeeia()aa-k-+ia()k--ib()bbkib()+k()aabb-ke-+()ke()-ke-+()keTheresultaftersettingthedeterminantequalto0,considerablealgebraandapplyingEuler’sformulasisasfollows:22ba-sinh()baba◊sin()+cosh()babak◊cos()=+cos()()ab(9.111)2abThisresultcanbefurthersimplifiedusingseveralphysicallyrelevantassumptions.ThefirstistoassumethatVfiiniteislargecomparedwiththekineticenergyfortheelectron,sothatfromequation(9.93)theexpressionforbabovebecomes222mVmVbefiniteefinitebb==andb(9.112)22hh1/21/2222Also,sincebVfiinieandaE,thenb>aandb>>a.Thusacanberemovedfromthefirstterminequation(9.111).Nowweassumethattheelectronbindingpotentialdropsoffsharplyattheatomiccores.Effectivelythisassumesanarrowbarrier.Thus,asbÆ0,bbbecomessmall.Thecoshofasmallargumentis1andthesinhofasmallargu-mentistheargument.IncludingalltheseassertionssimplifiestheKPresultsinequation(9.111)tothefollowing:不得转载谢谢合作LWM 9.3QUANTUMMECHANICS2132b()baabaak◊sin()+cos()=cos()a(9.113)2abUponsubstitution,usingequation(9.112),thisbecomesmbVefinite2a◊sin()aaaak+cos()=cos()a(9.114)hItistypicaltodefineatermPasfollows:ambVefiniteP=(9.115)2hThefinalformfromtheKPanalysisisthenP◊sin()aaaak+cos()=cos()a(9.116)aaThefinalsimplifiedform,equation(9.116),iscalledtheKPformula.Thisequationisofgreatinterestbecausefromitimportantrevelationsaboutelectronicstructureofsolidscanbedirectlymade,aswewillseebelow.First,weshouldobservethattheright-handsideoftheKPformulaisacosinefunctionwiththeargument(ka).Theonlyper-missiblevaluesfortheleft-handsideofthefinalformthenmustliebetween1and-1.Thefirsttermontheleftisessentiallysin(aa)/aa,andPisacompositeconstantdefinedaboveinequation(9.115).Thesin(aa)/aafunctioniscalledasincfunction,anditischar-acterizedasaperiodicfunctionwithdecreasingamplitude,asshowninFigure9.13a.Recallfromequation(9.93)thataisanexpressionoftheenergyfortheelectron.Thustheleft-handsideofthefinalKPformuladefinesalltheenergiesforelectrons.Whenalltheseenergiesareboundbytheright-handsideofequation(9.116),theallowedenergiesliebetweenvaluesof+1and-1fortheleft-handsideofequation(9.116).Theseallowedenergyregionsarecalledallowedenergybands.Figure9.13bshowsaplotoftheleft-handsideoftheKPformulaversusaa(orenergy)fortwovaluesofP;onevalueofPis2.5timestheotherforcomparison.Alsoplottedarethehorizontallinesatvalues+1and-1thatformtheboundariesforallowedelectronenergysolutions,asdictatedbytheright-handsideoftheKPformula(equation9.116).Thescalesareconvenientlychosen,anddonotreflectrealnumbers.However,theshapeofthesolutionintermsofallowedenergybandsisevident.TheKPformulaplottedinFigure9.13bshowsseveralimportantfeatures.FromthezeroofenergyatthecentertheKPfunction,thereisasharprise.Thenthefunctiondecreasestraversingthe+1to-1regioninitsdecent;itrepeatswithever-decreasingamplitudeuntilallthevaluesliebetween+1and-1.ThevaluesofenergythattheKPfunction(fKP)takesintheregion+1≥fKP≥-1aretheallowedenergies.Thisregioniscalledanallowedenergyband.ItisalsoseeninFigure9.13bthattheplotcorrespond-ingtohigherP(dashedline)traversesthe+1,-1regionmoresteeplythenthelowerPplot(solidline).ThismeansthatforhigherP(andV)thereisanarrowerrangeofallowedenergies.Notethatbothbeforeandafteranallowedbandthereisaregionofenergiesthatliesaboveorbelowtheallowedregion.Theseregionsareobviouslynotallowedener-gies,sotheyarecalledenergybandgaps.Theelectronicstructureisthereforecomposedofallowedenergybandsseparatedbydisallowedenergygaps,ormoresimplyofbandsandgaps.不得转载谢谢合作LWM 214ELECTRONICSTRUCTUREOFSOLIDSa)10.5sin(Z)/Z00.5108090072054036018001803605407209001080Zb)Psin(Z)/Z+cos(Z)+1–1ZFigure9.13(a)Plotofsinc(Z)functionwhereZisanangleq;(b)plotofleftsideofKPformulaequation(9.116)wheretheargumentZisaa.不得转载谢谢合作LWM 9.4ELECTRONENERGYBANDREPRESENTATIONS2159.4ELECTRONENERGYBANDREPRESENTATIONS9.4.1ParallelBandPictureTheallowedbandsanddisallowedgapsderivedfromtheKPmodelcanbedisplayedinaparallelbandpicture,asshowninFigure9.14.Inthispictureallinformationonhowanenergybandcanvarywithdirectionislost.Nevertheless,thesimplepicturecon-tributestoourunderstandingofhowlowerelectronenergybandsaredominatedbytheinteratomicpotential.Consequentlythesebandsarenarrowandwidelyseparated.Incontrast,thebandsthathousethehigherenergyelectronsarewiderandmorecloselyseparated.Exactlyhowthesebandsareoccupieddeterminestheoverallelectronicprop-ertiesofthematerial.WewillreturntothisimportantpointofoccupancyinChapter10.Fornowweconsiderthatforanelectrontomove(i.e.,forelectronicconduction),quantummechanicsrequiresthatelectronsbeinallowedstatesandthattherebeemptyallowedquantumstatesaccessibleinenergyfortheelectronstomoveinto.Anotherwaytoexpressthisisthatfilledandemptyallowedstatesarerequired.TheKPmodelyieldstheallowedstatesforagivenpotential.Theoreticallythesebandsofstatesextendtoinfi-niteenergy,sothereareinfinitestatesavailable.However,theatomicnumberforanatomdictateshowmanyelectronsareapportionedtoeachatominthesolid.Thusthereareafinitenumberofelectronsthatoccupytheallowedenergybandsstartingfromthelowestelectronenergiestothehighestlyingbands.Thisleavestheoutermostbandssometimesfilledandsometimesnotfilled.Thelastbandtohaveanyoccupyingelectronsiscalledthevalenceband(VB).AftertheVBthereisagapandthenanotherbandthatisempty.Thisnextemptybandiscalledtheconductionband(CB).ThreeimportantsituationsariseandaredepictedinFigure9.15.Figure9.15ashowsthesituationwithacompletelyfilledVBandemptyCB,andarelativelylargebandgap.Figure9.15bshowsthesamesituationasFigure9.15aexceptthatthebandgapisrelativelynarrow.Figure9.15cshowapartiallyfilledVBthatalsoEnergyFigure9.14Parallelenergybandrepresentationwithallowedelectronenergybands(shaded)separatedbydisallowedenergygaps.不得转载谢谢合作LWM 216ELECTRONICSTRUCTUREOFSOLIDSEnergya)b)c)Figure9.15Valenceband(lower)andconductionbandwithvariousamountsofelectrons(shaded)for(a)awidebandgapmaterialsuchasaninsulator,(b)anarrowbandgapmaterialsuchasasemiconductor,and(c)ametallicmaterial.hasemptystatesintheVB.ManyvariationsarepossibleinapartiallyfilledVB.Start-ingwithFigure9.15c,notethattheVBhasbothfilledandunfilledstatesincloseenergyproximity.Consequentlytheapplicationofasmallpotentialenableselectronfloworcon-duction.Thekindsofmaterialsthathavethisenergybandstructurearegoodconduc-torsandincludemetals.Bycontrast,Figure9.15ashowsnoemptystatesintheVBandanenergeticallyfarawaybutemptyCB.BecausetheCBisenergeticallyunavailable,eventheapplicationofstrongpotentialwillnotenableelectronicconduction.Sothiskindofmaterialsisanonconductororaninsulator.InbetweenthesetwokindsofbehaviorisFigure9.15bwithafilledVBandemptyCB,buttheCBissignificantlycloserinenergytotheVBthaninFigure9.15a.Thereforetherearemethodstoenableconductionsuchastheapplicationofareasonableexternalpotential,andthiskindofmaterialiscalledasemiconductor.InChapter10thesedifferentkindsofmaterialswillbediscussedatsomelength,andtheenergiesofthegapsthatdistinguishthematerialswillbequantified.However,inordertogetasemiconductortoconduct,aspecialprocesscalleddopingisusuallyrequired.Thisprocessactuallyaddseitherfilledoremptystatestothebandgap,therebynarrowingthegapandenablingconduction.Beforediscussingthepracticalsideofelec-tronicstructure,itisusefultofirstconsidertheenergybandrepresentationsandtheirimplications.9.4.2kSpaceRepresentationsInequation(9.115)wesawthatwhenP=0,theelectronsarefree(V=0).TheKPformulaforthiscaseisgivenbysubstitutingP=0intoequation(9.116).Thisyieldscos()aak=cos()a(9.117)Fromthiswededucethata=k.Recallequation(9.65)forfreeelectrons:22haE=(9.65)2meWitha=kthisyieldsequation(9.67):22hkE=(9.67)2me不得转载谢谢合作LWM 9.4ELECTRONENERGYBANDREPRESENTATIONS217ThisequationshowstheparabolicdependenceofEwithk,aswaspreviouslydiscussedandillustratedinFigure9.6.Fromequation(9.117),whichistheKPformulaforP=0,andfurtherfromtheperiodicityofthecosinefunction,thefollowingisalsotruein1-D:cos()apak=cos()xxak=+cos()an2(9.118)Equatingthecosineargumentsandsubstitutingforafromtheformulaabove,andthensolvingforE,weobtainapakan=+x2(9.119)Thus2222hhapÊ2nˆE==k+(9.120)22mmËxa¯eeThisisaformulaforafamilyofparabolas,eachcenteredat2np/a,wherenis0,1,2,...,orevenmultiplesofp/aandpartofthisfamilyisshowninFigure9.16.Noticethattheparabolasintersectatallmultiplesofp/a.RecallfromtheKPformulathattheleft-handside,cos(ka),cannotexceedvaluesof±1.Valuesoftheright-handsideoftheKPformulathatexceedthislimitrepresentdisallowedelectronenergystates.Thiswilloccurwhentheargumentofcos(kxa),kxa=nporkx=np/a,wheren=±0,1,2,...Thusexactlywheretheparabolascross,theenergyatthecrossingisdisallowed.Inspectroscopythisiscalledanoncrossingrule.Inordertorepresentthedisallowedcrossings,atthepointsofthecrossingstheparabolasaredistortedtopreventcrossing,andtheresultisshowninFigure9.17fortheparabolacenteredat0.Thisrepresentationofallowedanddisal-lowedenergiesiscalledtheextendedzonescheme.Allthesameinformationcanbecom-pressedintothefirstallowedenergyzone,±p/abytranslatingthepiecesoftheextendedzoneschemebyn2p/a,wheren=1,2,3...,andtheresultisshownalongwiththeextendedzoneschemeinFigure9.18wherethearrowsindicatethetranslations.Theregioninbetweentheverticaldashedlinesrepresentstheresultofthetranslationsofthepartsoftheextendedzonescheme(asindicatedbythearrows)intothefirstallowedenergyzone,andthisrepresentationiscalledthereducedzonescheme.Ek-4p/a-3p/a-p2/a-p/a0p/a2/ap3p/a4p/aFigure9.16Freeelectronparabolicbandsintersectingatnp/a,wherenis0,1,2....不得转载谢谢合作LWM 218ELECTRONICSTRUCTUREOFSOLIDSE-kk-3p/a-2p/a-p/a0p/a2/ap3p/aFigure9.17Onefreeelectronparabolicenergybandshowingdistortionsnearthedisallowedenergiesatnp/a,wherenisaninteger.E-kk-3p/a-2p/a-p/a0p/a2/ap3p/aFigure9.18Reducedzoneschemewheresegmentsofthefreeelectronparabola’saretranslated(arrows)intothefirstBrillouinzone.Theusefulfeatureofthereducedzonescheme,asshowninFigure9.19,isthattheallowedelectronenergybandsanddisallowedgapsarereadilyseenbyscanningverti-callyintherepresentation.Totherightinthefiguretheallowedenergybandsareshownincrosshatchandseparatedbybandgaps.Theparabolicshapeofthefreeelectronbands不得转载谢谢合作LWM 9.4ELECTRONENERGYBANDREPRESENTATIONS219Ek-2p/a-p/a0p/a2/apFigure9.19Reducedzonerepresentationofelectronenergybandstructurewithallowedenergybandsindicatedbyshadedareasattheright.isseennearthemidpartofthezone.Thealteredshapeisnearthebandedgeswheretheforbiddenenergiesoccur.9.4.3BrillouinZonesAttheendofChapter3theWigner-Seitzunitcellinkspacewasintroduced,andthereciprocalspacewascalledaBrillouinzone.Inthischapteranduptothissectionwehavedroppedthevector(bold)notationforkthatwasappropriatelyusedinChapter3.Thisisbecausethusfarinthischapterwehavemainlybeeninterestedinthemagnitudeofkandnotitsdirection.Nowweagaintakeupthevectordesignationforkbecausethedirectionaswellasthemagnitudearebeingconsidered.UsingFigure3.18,wecanseethatwavespropagatinginacrystalthathavewavelengthsoftheorderofthesizeoftheunitcellcandiffractincertainappropriatedirectionswherethediffractioncondi-tionsaremet.Namelythewavelengthorintegralmultiplesofthewavelengthfitinaspacebetweenscatteringatomsormolecules.ThisiscalledtheBraggcondition.Specif-icallythediffractionconditionsaresatisfiedusingequation(3.59)whenthekspacevectorG=na*.Witha*=2p/aforanorthogonalsystemandrecallingthatintermsofk,dif-fractionoccursatG/2,thediffractionconditionintermsofkwasfoundtobek=±np/a,wheren=1,2,...(seeequation3.60).RecallthatFigure3.18gavethekspacerepre-sentationofdiffractionwherethefirstandsecondBrillouinzonesarewithinG1/2andG2/2,respectively,forGdrawntothenearestandnextnearestneighbors.ThisfigurefromChapter3isslightlymodifiedtoincludethediffractionconditionforthefirstBrillouinzoneatp/aanddisplayedasFigure9.20.HereinChapter9wehaverevisiteddiffractionfromanothervantagepoint.Specifi-callywefoundthatforfreeelectronsinacubiclattice,theallowedelectronenergiesextendinkspacefrom0tonp/a,atwhichpositionthereexistsagapintheallowedelec-不得转载谢谢合作LWM 220ELECTRONICSTRUCTUREOFSOLIDSp/ap/aGp/a2G2p/aG1G1G1Gp/a1p/aG2G2p/ap/aFigure9.20First(shaded)andsecond(cross-hatched)Brillouinzonesin2-D.tronenergies.Thisgapreferstothoseenergiesforelectronwavesthatarenotpermittedtopropagateinthecrystal.Ineffecttheseenergies(withcorrespondingwavelengths)arediffractedoutofthecrystal.ThisnotiongivesrisetoanotherfullyconsistentapproachtoobtainelectronenergybandstructurecalledtheZimanapproachafteritsproponent.UsingFigure9.20andalittleimagination,itiseasytoconstructa3-DBrillouinzonerepresentationforasimple(primitive)cubicbyaddingtheothertwodimensionsandthisisrepresentedinFigure9.21awiththeboundariesindicated.Itismoredifficulttocon-structtheBrillouinzonesformorecomplicatedstructures,buttheprocedureisthesame.Figure9.21band9.21cshowthefirstBrillouinzonesfortheBCCandFCClattices.AlsoshowninFigure9.21band9.21caresomeusefultermsymbolsthathavebeenadopted.InthesefiguresitisseenthattheoriginofkspaceisdesignatedbythesymbolG.FromthefigurethefollowingdirectionsfromtheoriginGinthedirectionindicatedarelabeledasfollows:GÆ[100]isnamedDGÆ[110]isnamedSGÆ[111]isnamedLTheendpointsforthezonearealsolabeled.TheendpointforthefirstzoneintheDdirec-tionisHforBCCandXforFCC;theendpointforthefirstzoneintheSdirectionisNforBCCandKforFCC;theendpointforthefirstzoneintheLdirectionisPforBCCandLforFCC.Thesesymbolsareusedtodenoteabandina2-Drepresentationtobeintroducedbelow.不得转载谢谢合作LWM 9.5REALENERGYBANDSTRUCTURES221a)kzp/a-p/a-p/ap/akxp/a-p/akyb)kzLPGDHkySkNxc)kzKSkyGDXkxLLFigure9.213-DBrillouinzonesfor(a)PC,(b)BCC,and(c)FCCstructureswithtermsymbolsforimpor-tantdirections.9.5REALENERGYBANDSTRUCTURESFigure9.22ashowstheenergybandstructureforthemetalAlinseveraloftheimpor-tantcrystallographicdirectionsinitsFCCstructure.AmetalsuchasAlischosenfirstbecauseitwouldmorecloselyapproximateaV=0material.Itiseasilynoticedthatmanyofthebandsandpartsofthebandsareparabolicinshape,therebygivingcredencetotheearlierassertionabouttheV=0approximation.Itshouldalsobenoticedthatseveralofthebandsarenotparabolic,anindicationthatinrealityVπ0evenforagoodcon-ductor,althoughtheapproximationisusefulforthismaterial.IntheGÆXdirection不得转载谢谢合作LWM 222ELECTRONICSTRUCTUREOFSOLIDS2'324.02'1522.025'320.03'18.0116.0Al5'14.0312.010.012'8.0316.04'14.0ENERGY(eV)2'2.00.0–2.01–4.0GDXZWQLLGSX(a)EnergybandstructureforAl.(AdaptedfromHandbookoftheBandStructureofElementalSolids,D.A.Papaconstantopoulos,PlenumPress,1986)1510a-SiO250–5Energy(eV)–10–15–20GGKHAMLA(b)Energybandstructurefora-quartz.(AdaptedfromPhysicalReviewB,R.B.Laughlin,J.D.Joannopoulos,andD.J.Chadi,vol.20,p.5228,1979)Figure9.22Electronenergybandstructurefor(a)Al,(b)a-SiO2,(c)Si,and(d)GaAS.不得转载谢谢合作LWM 9.5REALENERGYBANDSTRUCTURES223Si15.02'3153.0111.025'–1.03'(eV)4–3.01GY–5.01–7.0ENER11–8.02'–11.–13.GDXWQLLGSX(c)TheenergybandstructureforSi.(AdaptedfromHandbookoftheBandStructureofElementalSolids,D.A.Papaconstantopoulos,PlenumPress,1986)76GaAsL3G155G1543X32L1G1K1XG111G150E(eV)–1G15KL2'3–2–3X5K1–4LGXKGk(d)EnergybandstructureforGaAs.(AdaptedfromPhysicalReview,M.I.CohenandT.K.Bergstresser,vol.141,p.789,1966)Figure9.22Continuedthelowerparabolicbandandtheupperbandshowagapinbetweenthebands.Thusinthisparticulardirectionanelectronnearthetopofthelowerbandisprecludedfromattainingtheupperbandstatesunlesstheelectronatthetopofthelowerbandreceivesenergygreaterthanthegapbetweenthebands.However,iftheelectronchangesdirection,thentherearedirectionsinwhichthereisnogap.InfactforAlatanyenergyonthediagramthereisalwaysadirectionthatcanbefoundinwhichthereisno不得转载谢谢合作LWM 224ELECTRONICSTRUCTUREOFSOLIDSenergygap.Theabilityorprocessbywhichanelectroncanprobabilistically“find”anappropriatedirectioniscalledpercolation.Theideaofpercolationisderivedfromwaterpercolatingupthroughsandfromaspring.Thewaterfindspathsaroundthegrainsofsand.IncontrasttoafreeelectronkindofelectronenergybandstructureistheenergybandstructureforSiO2thatisshowninFigure9.22b.Noticethatwithagoodimaginationonecanseesomeparabolicregionsinthebandstructure.However,therearelargeenergygaps,andtherearenopercolationdirectionsthatcouldenableanelectrontopercolatetothehigherenergylevelswithoutalargeinputofenergy.InbetweentheseextremecasesforelectronenergybandstructurearethebandstructuresforSiandGaAsshowninFigure9.22cand9.22d,respectively.Forbothoftheseso-calledsemiconductors,somebandregionsareparabolic.LikeSiO2bothofthesematerialshavebandgapsinalldirec-tions.However,theminimumgapforSiO2isaround9eV,andtheminimumgapsforSiandGaAsareabout1.1eVand1.4eV,respectively.AswedelveintosemiconductingmaterialsmoredeeplyinChapter10,wewillseethatevena1eVgapislargerelativetotheenergyavailableatroomtemperature(kTatroomTª0.025eV).Thustheelectronsaren’tvery“free”tomigrateinthesesemiconductingmaterialsastheydointhemetals.Infact,unlessthesenearly1eVgapmaterialsaredopedtoaddappropriateenergylevels,theyarereasonablygoodelectricalinsulators.InobservingthebandstructureofSiandGaAs,inparticularthegaps,thegapforGaAsofabout1.4eV,isobservedtobeaverticaltransitionfromabandthatisconcavedownwardtoonethatisconcaveupward.Thistransitiontakesplacewithoutachangeink(thehorizontalaxisinFigure9.22)andiscalledanopticaltransition(orak=0transition),sincenoneoftheenergyinthetransitionislosttotheGaAslattice.Incon-trast,theminimumgapinSiof1.1eVtakesplacefromGÆXwithachangeink,anonverticaltransition.ThismeansthatmomentumistransferredtotheSilattice.Themainimplicationofthisbandfeature,namelyk=0orkπ0transition,isthatthosematerialswithk=0bandgapsaremoreusefulforopticaldeviceswheretheopticaltran-sition,k=0,hasahigherprobabilitytoabsorbandemitaphoton.ForSi,forexample,theminimumgapat1.1eVisanonopticaltransition,butthereareopticaltransitionsinSiat3.4and4.3eVresultinginpeaksintheopticalabsorptionatthoseenergies.9.6OTHERASPECTSOFELECTRONENERGYBANDSTRUCTUREReturningagaintothesimplestcaseforfreeelectronenergybandsinasimplecubicsolid,theparabolicshapewasthemaincharacteristicofthebandsin1-D.Wecannowimaginethe1-DparabolainFigure9.6rotatedaroundtheenergyaxistosweepoutafunnel-like3-Dparabolicfigure,asshowninFigure9.23a.Horizontalcutsinthisfigurearecirclesthatrepresentdifferentenergies.InFigure9.23bthecircularcutsarestackedontopofthefirstBrillouinzonewithsmallerenergiesbeingsmallercircles.NowforagivenBrillouinzonesize,thelargestenergycirclethatjustfitsinsidethezoneisshownattheborderoftheBrillouinzone.Largerenergiesareallowedinthezone,butonlynearthecornersoftheBrillouinzone.Thisisillustratedbytheoutermostcirclethathasfourofitsarcswithinthefirstzone,andfourarcspenetratingbeyondthezoneboundaryandintothesecondzone.Alsothehigherenergiesinthefirstzonearefoundinthecornersofthezonewheretherearefewerelectronstatesforthesehigherenergies.WewillreturntothispointinChapter10whenweconsiderthedensityofallowedelectronstateswherewewillfindadecreaseinthedensityofstatesforhigherenergies.不得转载谢谢合作LWM 9.6OTHERASPECTSOFELECTRONENERGYBANDSTRUCTURE225a)E-ky-kxkxkyb)p/ap/ap/ap/ap/ap/ap/ap/aFigure9.23(a)Freeelectronbandsin3-D;(b)projectionofverticalcutsinthebandsin(a).Thelasttopicforthischapterisadiscussionofelectronmass.Itwillbeshownbelowthatthemassforanelectronisnotaconstantinamaterial,butrathermassisdifferentdependingonwhichbandtheelectroninquestionresides.Thustheterm“effectivemass,”oftenwrittenasm*,isusedtodenotethisquantummechanicalmass.Recalltheformuladevelopedearlierforthegroupvelocityforawavepacket,vg:dwv=(9.38)gdkSubstituting2pn=wyieldsd()2pnv=(9.121)gdkSubstitutionE=hn,weobtaindE()21p()hdEv==(9.122)gdkkhdThestrategyistomakeananalogytotherelationshipF=maandsolveformusingourdevelopmentabove.Theaccelerationaisgivenasdv2g11dÊdEˆdEdka===(9.123)dthhdtËdk¯dk2dt不得转载谢谢合作LWM 226ELECTRONICSTRUCTUREOFSOLIDSWecanfindanexpressionfordk/dtusingp=hk:dpkddkp1d==handthus(9.124)dtdtdthdtThenusingp=mv,wecanwritefora,2211dEdpdEdmva==(9.125)2222hhdkdtdkdtWithsubstitutionofF=ma=mdv/dt,thefinalrelationshipisnowwritten21dEa=F(9.126)22hdkThismeansthatthecoefficientofFinthisformulais1/morintermsoftheeffectivemass,m*:2-12ÊdEˆm*=hÁ˜(9.127)Ë2¯dkCalculusteachesthatthesecondderivativeofafunctionE(k)yieldsthecurvatureofthefunction.Thusm*isproportionaltothereciprocalofthecurvatureorm*1/cur-vatureofaparticularband.Furthermoretheradiusofcurvatureisgivenastherecip-rocalofthecurvature.Theresultisthattheeffectivemassisthenproportionaltotheradiusofcurvatureofanenergyband,theE(k)versuskcurve.FornowwecanapplythisideatothefreeelectronbandsseeninFigure9.19.Nearthebottomofthelowestenergyband,thebandiscurved.Thusithasalargecurvatureatthisposition,andhenceasmalleffectivemass.Alsonearthezoneedgethereissimilarlylargecurvature.However,inbetweenthesecurvedregionsthecurvatureisquitesmall,yieldingalargeeffectivemass.Thusm*isrelativelylargeinthecenterofthebandbetween0andp/acomparedtoadjacentendregionsofthebandthathavehighercurvatureandthusasmallerm*.Interestinglythecurvaturealsochangesfromconcaveup(positivevalue)near0toconcavedownnearp/a(negativevalue).Theregionofthebandthatisconcaveupwardiscalledanelectronband,andtheconcavedownwardbandregioniscalledaholeband.Theregionsofthebandsthatcorrespondtolargenegativeeffectivemassesarecalledheavyholebands(forlargepositiveeffectivemasses,heavyelectronbands),andthosecorrespondingtosmallnegativeeffectivemassesarecalledlightholebands(smallpositiveeffectivemasses,lightelectronbands).Themassescanthenbelistedasme*forelectronsandmh*forholes.Theotherbandsalsoshowbothkindsofcurva-tureaswellasdifferentcurvatures.Inthenextchapterholeswillbedefinedmorecare-fully,anditwillbecomeclearerhowtheconceptofelectronsandholesascarriersofcurrentiscentraltounderstandingelectronicpropertiesofmaterialsandelectronicdevices.RELATEDREADINGD.A.Davies.1978.WavesAtomsandSolids.Longman,London.Averywell-writtentextcoveringmanyofthetopicsinChapters9,10,and11withgoodinsights.不得转载谢谢合作LWM EXERCISES227R.E.Hummel.1992.ElectronicPropertiesofMaterials.Springer-Verlag,NewYork.Thistextpro-videswell-writtencoverageofthematerialinChapters9,10,and11attheappropriatelevel.Theauthorhasusedthisbookasatextfortheelectronicmaterialspartofthematerialssciencecourse.J.P.McKelvey.1993.SolidStatePhysicsforEngineeringandMaterialsScience.Krieger.AhigherleveltextthanHummel,andalsowellwritten,readable,butforthetopicscoveredismorecomplete.M.A.Omar.1993.ElementarySolidStatePhysics.AddisonWesley,Reading,MA.AtextthatcoversmanyofthetopicsinChapters9,10,and11andwithmanymoretopicsnotcoveredinthepresenttext.Areadabletextinthesubject.EXERCISES1.(a)Sketchamodulatedwaveform(cosa·cosb)wherethefrequencyofthehigh-requencycomponentis100¥thelow-frequencycomponent.(b)Showthegroupandphasevelocitiesonthesketch.(c)Explainhowthiskindofwaveformcanbeusedtodescribeanelectronthathasthecharacteristicsofaparticle.2.Calculatetheenergyforanelectronandforphotonswithwavelengthsof0.1nm,1nm,and10nm.3.ShowthattheformfortheSchrödingerequationderivesfromthedualityprincipleofdeBroglie.4.Fortwoadjacentregions1and2wheretheelectronisfreeandbound,respectively,sketchthepropagationofmatterwavesfromthefreeelectronsidetotheboundside(fromregion1to2).Discusswhathappenstothewavefunctionwhenthebindingpotentialandthelengthofsidesareseparatelyincreased.5.StartingwiththeKPformula,showhowtheKPmodelgivesrisetotheparallelenergybandpictureformaterialsbyidentifyingtheatomisticparametersthatdeterminebandwidthsandseparations.6.(a)Startingfromasquare2-DlatticeinRESP,constructthefirsttwoBrillouinzones.(b)Usingyourconstructionexplainhowenergystates(intermsofboththeenergiesandnumber)varywithdifferentdirectionsinthefirstBrillouinzone.Fromthisresultsketchadensityofstatesversusenergycurve.7.UsingtheSibanddiagram(Figure9.22c),pointoutholeandelectronbands,andforoneofeachkindofband,traceoutanddiscussthevariationintheeffectivemassfortheparticleinkspace.8.UsingtheSibanddiagram(Figure9.22c)answerthefollowing:(a)Identifyoneholebandwithlarge(heavyholeband)andanotherwithrelativelysmaller(lightholeband)m*.Includethebasisforyourselection.h(b)LocatetheFermilevel.(c)Howwouldthisdiagramchangeifthismaterialbecameamorphous.(d)Explainwhymanyofthebandshaveaparabolicshape.不得转载谢谢合作LWM 不得转载谢谢合作LWM 10ELECTRONICPROPERTIESOFMATERIALS10.1INTRODUCTIONInthischapterwedealwiththoseaspectsofelectronicstructurethataffectandunder-lietheelectronicpropertiesofconductors,nonconductorsorinsulators,andsemicon-ductors.WestartwiththeelectronicenergybandstructureofChapter9thatdefinestheallowedelectronicstates.Becausebandstructurealoneisnotsufficienttodefineelec-tronicproperties,weneedtolookathowtheallowedstatesarefilled.ThiswasbrieflymentionedinChapter9whenwenotedthatthedispositionofthelastbandsfilleddeter-minestoalargeextenttheresultingproperties.Thustheoccupationoftheallowedstatesiswherewestartourfocusinthechapter.Wewillcalculatethenumberofallowedstates,theso-calleddensityofstates(DOS)function.Inadditionwewillassesstheprobabilitythatagivenallowedstateisfilled.Since,forelectronsandespeciallyelectronswithlowenergies,theBoltzmannprobabilityfunctionisnotappropriate,anewdistributionfunc-tionwillbeused.ThisfunctioniscalledtheFermi-Diracdistributionfunction.Thisfunc-tionisreadilyrationalized,anditleadstoadefinitionoftheFermienergylevelthat1essentiallyistheelectronenergyattheprobability,P=–forthestatetobeoccupied.At2absolutezerotemperatureallstatesbelowtheFermienergy,EF,arefilledandallstatesaboveEFareempty.Oncethebasicsmentionedaboveareestablished,thenelectronicconductioncanbeconsidered.Thiswedofrombothaclassicalperspective,soastoobtaintheintuitivefundamentalrelationships,andamorecorrectquantummechanicalperspective,whereweshowthatonlyelectronsnearEFparticipate.Inadditionwewillconsidertheimpor-tanttopicofsuperconductivityinordertogainsomebasicideasaboutthismodernsubject.Freeelectronconductionisthesimplesttounderstandandthusprovidesthestartingpointforelectronicproperties.ThisdiscussionisfollowedbyalookatsemiconductorpropertiesandthemainideasthatunderliemostmodernelectronicElectronicMaterialsScience,byEugeneA.IreneISBN0-471-69597-1Copyright©2005JohnWiley&Sons,Inc.不得转载谢谢合作LWM229 230ELECTRONICPROPERTIESOFMATERIALSdevices.DevicesandothermodernareasofelectronicmaterialssciencearereservedforChapter11.10.2OCCUPATIONOFELECTRONICSTATESTheoccupancyofallowedelectronicstatesN(E)dependsonthenumberofallowedstates,theso-calleddensityofstates(DOS),multipliedbytheprobabilitythatastatewillbeoccupied,theso-calledFermi-DiracdistributionfunctionF(E).Allthisismultipliedby2,sinceeachquantumstatecanhavetwoelectronsofoppositespin.WenowproceedtodeveloptheappropriaterelationshipsforDOS,F(E),andN(E).10.2.1DensityofStatesFunction,DOSRecallfromChapter9theenergyformulafortheboundelectronproblem:22hp2E=◊n(9.73)22mleThisformulateachesthattheelectronenergyinawelloflengthlisquantizedinunits2ofn(orn)wherenisanypositiveinteger.Thissetofintegersdefinestheallowedstates.Inthreedimensionsnisgivenas2222nnnn=++(10.1)xyzTobettervisualizethedevelopment,itishelpfultodefineasphericalstatespacethatcontainsallnintheorthogonalcoordinatesystemdefinedbythestatesnx,ny,andnz.ThisstatespaceisillustratedinFigure10.1,whichshowsthatanyintegerncanbeobtainedbyavectorfromtheorigintotheintegern.Withnasapositiveinteger,werestrictourinteresttothefirstoctantofthissphericalspacewhereallnarepositiveintegers.nzn-ny–nxnxny-nzFigure10.1Statespacedefinedin3-Dbyintegersnx,ny,andnz.不得转载谢谢合作LWM 10.2OCCUPATIONOFELECTRONICSTATES231TheobjectiveistocalculatethenumberofstatesforanenergyEthatislessthansomemaximumenergyinthesystem,En.Westartwiththenumberofstates,h,inthefirstoctantofpositiven’s.hisgivenbythevolumeofaspherewithradiusn,butonlyinthefirstoctantas1Ê43ˆhp=◊n(10.2)8Ë3¯Usingequation(9.73),werelateEtonandsolveforntoobtainthefollowingexpression:212Ï2mle¸n=Ì22En˝(10.3)Óhp˛Thisexpressionfornissubstitutedintoequation(10.2)forhtoobtaintheresult232pÏ2mle¸32h=Ì22˝E(10.4)6Óhp˛TheDOS,whichisthenumberofstatesperunitenergy,isobtainedasthederivativeofhwithrespecttoenergy,abbreviatedasZ(E),andgivenas33232dhlÏ2me¸12VÏ2me¸12ZE()==2Ì2˝E=2Ì2˝E(10.5)dE4ppÓhh˛4Ó˛3whereV=l.TheDOSfunctionisplottedinFigure10.2asthesolidlineonZ(E)versus1/2E.ItisseenthatthedensityofelectronstatesperunitenergyincreasesasE.ThedashedlineathigherEwillbeconsideredlater.TheexplanationforthedecreaseinZ(E)athigherelectronenergieslieswithafactdiscussedneartheendofChapter9,wherewelearnedthattherearefewerelectronstatesathigherenergiesneartheBrillouinzoneedges(recallFigure9.23b).ThustheDOSfunctiondecreasesathigherenergiesbecauseoftherestric-tionsoftheallottedareaineachBrillouinzone.Z(E)EFigure10.2DensityofstatesfunctionZ(E)asafunctionofenergy.SolidlineforparabolicincreaseinZ(E),anddashedlineindicatesthatZ(E)decreasesforhigherenergiesintheBrillouinzone.不得转载谢谢合作LWM 232ELECTRONICPROPERTIESOFMATERIALS10.2.2TheFermi-DiracDistributionFunctionTheFermi-Diracdistributionfunction,F(E),yieldstheprobabilityforanallowedstatetobeoccupiedbyanelectron.Forlargenon-quantummechanicalparticlessuchasmol-ecules,dust,billiardballs,andtrucks,theprobabilityforoccupancyofaparticularenergystateisgivenbytheBoltzmanndistribution,P(E),andtheprobabilityisanexpo-nentialfunctionoftheenergyaswasdiscussedinChapter4(e.g.,seethedevelopmentofequation4.34):-EkTPE()=()conste(10.6)Itisimportantnowtonoticethattherearenorestrictionsonhowmanyparticlescanoccupyanygivenenergystate,andthatstatesofhigherenergyaremoresparselypopu-lated.Itisthisprobablisticnotionthatdrivesvarioussystemstoultimatelyattainthelowestenergystatepossible.AlsoasdiscussedinChapter4,P(E)derivesfromtheassumedproportionalityofthechangeinpopulation,dn,ofnstatesasbeingpropor-tionaltoEasdnµdE(10.7)nForelectronswemustconsiderthequantummechanicalselectionrulesthatdonotpermitelectronstohaveidenticalenergydescriptionsintermsofquantumnumbers.Essentiallyweneedtoemploythereasoningthatatmosttwoelectronscanoccupyaquantumstateandthattheseelectronsdifferonlyinspin.Wepresenttheresultbelowintheformofanew,moreappropriatedistributionfunction,theFermi-Diracdistributionfunction,F(E),anddefineitintermsofa“new”energytermtheFermienergy,EF:1FE()=(10.8)1+e()EEkT-FOnesimplewaytoarriveatthisFermi-Dirac(FD)distributionfunctionistoconsidertwoallowedstates,1and2,asshowninFigure10.3.Aswasstatedabove,weneedtokeepinmindthePauliexclusionprinciplethatpermitstwoelectronsperstate,andtheprincipleofdetailedbalancingthatequatesforwardandreversekineticallysimpleprocesses.Akineticallysimpleprocessisaprocessofasinglestepasitiswritten.Thismeansthatthewrittenstepisnotacomposite,composedofperhapsmanysteps.Alsoitisassumedthattheprobabilityoftheithstatebeingoccupiedisfi=f(Ei),meaningtheprobabilityisafunctionoftheenergy.Forexample,theprobabilityislowerforahigherenergystatetobeoccupiedthanforalowerenergystate.1E1DE2E2Figure10.3TwoallowedelectronstatesseparatedbyDE.不得转载谢谢合作LWM 10.2OCCUPATIONOFELECTRONICSTATES233-Considerthefollowingelectron(e)interactionsbetweenthetwoallowedenergystates--inFigure10.3:aneinstate1goestostate2andaneinstate2goestostate1.Thisisillustratedbythedouble-endedarrowinthefigure.Thefollowingenergybalanceobtainsforthetransitions:EEEE()1221-()=()-()(10.9)Foranytransitiontooccur,itisrequiredthattheinitialstatebeoccupiedandthefinalstateempty,andthatelectronsdonotoccupythesamestate.Theprobabilityforstate1beingoccupiedisf1andforstate2beingemptyis1-f2.Theprobabilityforbotheventstooccuratonceisobtainedfromtheproductofprobabilities:Pf=◊-()1f(10.10)1212Fortheinverseprocess,thatis,state1emptyandstate2filled,wecanthenwritePf=◊-()1f(10.11)2121DetailedbalancerequiresthatP=Pinverse.Thusweobtainffff◊-()11=◊-()(10.12)1221Intermsofratioswecanrewritethisresultasff12=(10.13)11-f-f12Thisequationinquotientformcanbeputintomoreusefuldifferenceformbytakingthelogarithmsofbothsidesasfollows:ff12G=lnandG=ln(10.14)1211-f-f12ReferringtoFigure10.3,weseeachangeinenergyDEbetweenthetwostatesandtheprobabilitiesforoccupancyofonestateoranotherisafunctionoftheenergydif-ference.ThisiswrittenDGµ-DE.Inusingthedifferentialform,wecanconverttheenergytosmallchangesandobtaindG=-LdE,whereListheconstantofpropor-tionality.IntegrationwillyieldtheprobabilityG1.ThustheintegralofdGisevaluatedfromprobability0toG1.Theenergyintegralisevaluatedfromtheenergypositionwhere1theprobabilitycorrespondstoG1=0toE.G1=0whenf1=–,andthisenergyisdefined2astheFermienergyEF.Withtheseintegrationlimitsincluded,theintegraltobeevalu-atedisG1EdG=-LdE(10.15)ÚÚ10EFTheresultis不得转载谢谢合作LWM 234ELECTRONICPROPERTIESOFMATERIALSf1GLEE=-()=ln(10.16)1F1-f1Rearrangementofthisexpressionintermsoff1,whichistheprobabilitythatastateisoccupied,andnowrenamedasF(E)yieldsthefollowingresult:1FE()=(10.8)1+e()EEkT-FwhereListheconstantofintegrationandhasbeenexperimentallydeterminedtobeL=1/kT.Figure10.4displaysaplotoftheFermi-DiracdistributionfunctionF(E)versusenergyatabsolutezero(solidline),andatsometemperatureaboveabsolutezero(dashedline).ThestepfunctionappearanceofF(E)atabsolutezeroindicatesthattheprobabil-ityisunitythatanelectronwillbeatorbelowtheFermienergy,EF.ThismeansthatalltheelectronsarefoundatorbelowEF,andthatEFrepresentsthehighestenergyofanyelectroninthematerial.However,atanytemperatureabove0KthereisatailextendingaboveEF.InFigure10.4thetailisexaggeratedandistypicallyabout5kTwherekTisabout0.025eVat300K.WithEFvaluesformetalsofabout5eV,thetailrepresentsonlyasmallpartoftheelectronconcentration.Nevertheless,atT>0KafewelectronshaveafiniteprobabilitytoinhabitstatesthathaveenergiesgreaterthanEF.Becausethisoccurs,thereisafiniteprobabilitythatsomestatesbelowEFareunoccupied.Theseemptystatesinthevalencebandarecalledholes,andelectronsinthevalencebandcanmoveintotheholesunderanelectricfield.Todistinguishthiselectronmotionfromthatintheconductionband,itisusualtodescribethemotionoftheholesinthedirectionoppo-sitetoelectronsinthevalenceband,andthusspeakofholemotionindescribingcon-ductioninthevalenceband.ItisusefultoalsoconsiderthemathematicalformofF(E)inequation(10.8)above,andthenprobevariousTandEregions.Fromequation(10.8)forF(E),forEEFandTapproaching0K,theexponentialinthedenom-inatorgrowslargesothatF(E)goesto0asisalsodepictedinFigure10.4aboveEFand1T=0KF(E)12T>0K0EEFFigure10.4Fermi-Diracdistributionfunction,F(E)at0K(solidline)andhigherT(dashedline).TheFermi1levelEFatF(E)=–.2不得转载谢谢合作LWM 10.2OCCUPATIONOFELECTRONICSTATES235forthesolidline(T=0K).ForlargeE,theexponentialterminthedenominatordom--E/kTinates,andF(E)ªe,whichistheBoltzmanndistribution.Soatlargeelectronener-giestheBoltzmanndistributioncanbeusedtodescribeelectronprobabilities.10.2.3OccupancyofElectronicStatesWiththedensityofstatesZ(E)andtheFermi-DiracdistributionfunctionF(E)obtained,wecancalculatethenumberofoccupiedelectronstatesN(E)inamaterial:NE()=◊2ZEFE()◊()(10.17)Figure10.5adisplaystheshapeofN(E)fromtheproductaboveforparabolicZ(E)asshowninthesolidlineofFigure10.2.However,aswasdiscussedinChapter9(seeFigure9.23),therearefewerstatesneartheBrillouinzoneedges.ThusZ(E)isnotincreasinginaparabolicmannerwithenergy.Rather,asEincreases,Z(E)fallsrapidlytowardthezoneboundary,andthisisillustratedbythedashedlineforZ(E)inFigure10.2.Withthiscorrectfall-offinZ(E)intheproduct,themorecorrectshapeforN(E)isthatappear-inginFigure10.5b,whereamaximumisevidentnearthemidbandregion.Thisfactwillbeimportantlaterwhenelectronicconductionisdiscussed.ThechangeinthenumberofoccupiedstatesperenergyintervaldNoisgivenasN(E)dE.TheintegrationofN(E)dEfrom0energytotheFermilevelEFatT=0K(F(E)=1)willthenyieldthenumberofelectronsinallowedstatesasa)N(E)EEFb)N(E)EEFFigure10.5NumberofoccupiedelectronstatesN(E)=2Z(E)F(E).(a)ParabolicZ(E);(b)falloffinZ(E)athigherenergies.不得转载谢谢合作LWM 236ELECTRONICPROPERTIESOFMATERIALSEEFFEFNN=()EdEZ=22()()EFEdEZ=()EdE(10.18)oÚÚ00Ú0Then,withF(E)=1,wesubstituteforZ(E)usingequation(10.5)toobtain:32VmÏ2e¸12ZE()=2Ì2˝E(10.5)4pÓh˛IntegratingoverEyieldsthefollowing:32VmÏ2e¸32No=2Ì2˝EF(10.19)3pÓh˛IfweletNVbethenumberofelectronspervolume,No/V,andrearrangesolvingforEF,thefollowingisobtained:2223hENFV=()3p(10.20)2meThisinterestingresultteachesthattheFermilevelisafunctionofthenumberofelec-tronsandthevolumeoftheunitcellthatdependsonthecrystalstructure.10.3POSITIONOFTHEFERMIENERGYTheFermienergyEF,alsoreferredtoastheFermilevel,isthehighestenergyoccupiedstateatT=0Kforanelectron,andveryclosetothatforT>0K,whichiswithinabout0.1eVatroomtemperature.AsapracticalmatterEFrepresentsthehighestenergyelectronsinanequilibriumsolidnotwithstandingtheFermitail.Thereforeanyexcita-tionsofelectronsimposed,forexample,byanexternalelectricfield,orevenopticalexci-tation,willpotentiallylifttheequilibriumenergylevelstohigherenergies.Clearly,knowledgeofthestartinglevel,EF,isrelevanttounderstandingtheexcitation.Aswillbeseenbelow,knowledgeofthepositionofEFisinfactcrucialtounderstandvirtuallyallelectronicpropertiesanddevices.Thereforewenowaddresstheproblemoftheloca-tionofEFinamaterial.Wecommencewithasemanticargumentbasedonwhatwealreadyknowabouttheenergybandstructure,andthisargumentwillenableaverycloseestimationofthepositionofEFthatwillsufficeformostproblems.ThenamorepreciseanalyticformulationoftheproblemispresentedwherethepositionofEFiscalculated.WereturntoFigure9.15.Recallthatitdisplaystheparallelbandpictureforthreecases:Figure9.15aisforafullvalencebandwithawidegapandanemptyconductionband,Figure9.15bshowsafullvalencebandwithanarrowgapandanemptyconduc-tionband,andFigure9.15cshowsapartiallyfullvalenceband.Ineachcase,withsomeadditionallogic,wecanlabelthepositionoftheFermilevelfromthedefinitionsabove.Wecommencewiththecaseofapartiallyfilledvalenceband,andmodifyitslightlywiththeadditionoftheFermilevelEFasshowninFigure10.6a.InFigure10.6athemostenergeticelectronsforapartiallyfilledvalencebandarethoseatthetopofthefilledlevelsinthevalenceband.Adottedlineisdrawn,andthisenergylevelislabeledasEF不得转载谢谢合作LWM 10.3POSITIONOFTHEFERMIENERGY237EnergyEEFConductionBandFConductionBandEiConductionBandEDEFValenceBandValenceBandValenceBanda)b)c)Figure10.6Parallelelectronenergybandsfor(a)ametal,(b)asemiconductor(orinsulator),and(c)adopedsemiconductor.thehighestlevelforfilledstates.Ofcourse,wekeepinmindthecaveatgivenabove,thatthisisstrictlytrueonlyatT=0Kbutclosetotrueforroomtemperature.Figure10.6bdisplaysafullvalenceband,agap,andthenanemptyconductionband.Dependingonthesizeofthegap,thiscasecanrepresenteitherasemiconductor(narrowgap)oraninsulator(widegap).TorationalizethepositionforEFinthissituation,we1recalltheprobabilisticdefinitionfortheFermienergyastheprobabilityof–foranelec-21trontogotothehigherallowedlevel.Theprobabilitywillbe–halfwayinenergybetween2thefilledvalencebandandtheemptyconductionband.Thusthedottedlineisinthemiddleofthegap.Thisisaninterestingcase:whiletheFermilevelisnearthemiddleofthegap,therearenoallowedstatesatthatposition.Figure10.6cisthesamecaseasFigure10.6bexceptthatthereisanaddedlevelofstatesrepresentedbyadashedlineslabeledEDneartheconductionband.Thislevelistypicallycalledadopinglevel.Itresultsfromaddingimpurityatomsthatyieldanearlymonoenergeticlevelofstates.Iftheimpuritystatesaddedarefilledwithelectrons,theleveliscalledadonorlevelandlabeledED,asshowninFigure10.6c.Ifthestatesareempty,theyarecalledacceptorstatesbecausetheycanaccommodateoracceptelectronsfromthematerial;thesestatesarelabeledEAandnotshown.Dopingwillbediscussedinmoredetaillaterinthischapter,butfornowweneedonlyadmittoitspossibility,andtrytolocateEF.InFigure10.6cthedonorlevelisveryclosetotheconductionband.Infactthedistanceinenergyisoftheorderof1or2kT(0.025–0.050eV).ThustheenergydistancetotheconductionbandisaboutkT.Thismeansthatatroomtemperaturethefilleddonorstatescanionizetheelectronstotheconductionband(recalltheearliertwo--E/kTstateproblemandequation(10.6)inwhichthenumberintheupperstateisnearlye,whereEisthesmallenergydistancebetweenEDandECB).Inthiscasethentheproba-1bility–point,EF,isinbetweentheEDlevelandtheECB.Thisisindicatedbythedotted2EFinthefigure.InadditionnearmidgapisanotherdottedlinelabeledEi.ThisiscalledtheintrinsicFermilevel,anditindicateswhereEFwillbeifthematerialwerenotdoped,thatis,ifthematerialexhibitsintrinsicpropertiesratherthanpropertiesduetodoping(extrinsicproperties).Morewillbesaidaboutthislater.不得转载谢谢合作LWM 238ELECTRONICPROPERTIESOFMATERIALSThus,bythedefinitionofEF,itspositionontheenergybanddiagramcanbededuced.Inrealitythisexerciseisonlyapproximatelycorrect,butitiscloseenoughformostdis-cussionsaboutelectronicbehavior.Itisalsoworthwhiletoperformamoreaccuratecalculationtouncovertheassumptionsimplicitlymadeabove,andtherebydeepenunder-standingofthisenergylevel.Whatwewillfindbelowisthatifthevalenceandconduc-tionbandsdonothavethesameshapeorsymmetry,thenacorrectionisrequiredtothesymmetryargumentmadeabove.Recallthatpreviouslyourdiscussionofshapeorcur-vatureofelectronenergybandsinChapter9revolvedaroundtheconceptofeffectivemass.Effectivemasswillariseagaininourdiscussionhereandinthesamecontext.TocalculatethepositionofEFinanenergygap,itisnecessarytofirstrealizethatasanelectronmovesfromafilledvalencebandtoanemptyconductionband,aholeisleftbehindinthevalenceband.Wecalculatethenumberofelectronsintheconductionbandnandthenequatentothecalculatednumberofholesinthevalencebandp.Theequa-tionisthensolvedforEF,whichappearsintheequation.Forelectronsintheconduc-tionband,nisobtainedfromtheequationforN(E),equation(10.18),asEC2nNEeZEFEd=()=Ú()()E(10.21)EC1EC1andEC2arethebottomandtopoftheconductionband,respectively.Thisexpres-sioncanbesimplifiedbyrealizingthatatT>0,E>EFintheconductionbandthenE-EF>>kT.ThusintheconductionbandF(E)canbeapproximatedbytheBoltzmanndistributionP(E):()=1--()EEFkTFEgoestoe(10.22)1+e()EE-FkT3Forthecasewherel=1,thenV=l=1.SubstitutingforVinequation(10.5)obtainsthefollowingformforZ(E):32pÈ8me˘12ZE()=E(10.23)24ÎÍh˚˙TheassumptionaboutV=1willdropoutbelowwhentheresultisequatedwiththatforholesinthesamevolume.Theintegrationlimitsaresetforbothconvenienceandcon-sistency.Forelectrons,wesettheenergyatthetopofthevalencebandto0.Thenthelowerintegrationlimitfortheconductionbandbecomes0+EgorEg,andtheupperissetconvenientlyat•.Ofcourse,theconductionbanddoesnotextendtoinfinity,buttheexponentiallydecreasingfunctionswilltakecareofthis.Theinfinitylimitsimplifiesthemath,aswillbeseenbelow.WithEgasthebottomoftheconductionband,anyenergyEintheconductionbandisnowchangedtoE-Egforconsistency.Withequa-tion(10.22)forF(E)andequation(10.23)forZ(E),theintegralfornorN(E)inequa-tion(10.21)canbereformulatedas•32nNE=()=pÈ8me˘()EEe-12--()EEkTFdE(10.24)Ú2g2ÎÍh˚˙EgToperformtheintegration,wenotefirstthat不得转载谢谢合作LWM 10.3POSITIONOFTHEFERMIENERGY239EE-FÈEE-ggFEE-˘(10.25)-=-Í+˙kTÎkTkT˚Ifweletx=(E-Eg)/kT,thenthefirsttermabovebecomes-xanddE=kTdx.Thisyields,fortheintegral,•32pÈ8me˘3212-xnNE=()=kTxedx(10.26)Ú22ÎÍh˚˙EgTheintegral,nowinastandarddefiniteintegralform,canbereadilysolvedusingthefollowing:•1212-xpÚxedx=(10.27)20Theresultforelectronsintheconductionbandis321È8pmkTe˘()EEkFg-TnNE=()=e(10.28)24ÎÍh˚˙Wecanproceedtocalculatethenumberofholespleftinthevalenceband.FromtheFermi-Diracdistributionfunctionweknowtheprobabilityforelectrons.Tofindtheholes(usingthesubscript“h”todenoteholes),ortheabsenceofelectronsinthevalenceband,weuseFh(E)=1-F(E)andobtain:1e()EEkT-FFE()=-1=(10.29)h1+e()EEkT-F1()EEkT-F+eForE0K,foreveryelectronaboveEFaholeisleftbelowEF.Thusaholeisanoccurrenceinthevalencebandofmaterials.Onceaholeiscreatedinthevalencebandofasemiconductor,electricalconductioncantakeplacenotonlyintheconductionbandbythemotionofelectronsthroughtheemptyallowedstatesintheconductionband,butalsoviaelectronsmovinginthenowvacatedstatesinthevalenceband,theholes.Soitisalwayselectronmotion,butinthevalencebanditiscalledholemotion(oppositeindirectiontoelectronmotion)todistinguishitfromelectronmotionintheconductionband.Also,becausethemotioninthedifferentbandsisviaelec-tronswithdifferentenergiesandindifferentenvironments,theeaseofelectronmotioncanbequitedifferent,andholesusuallyhaveslowermotion.Basicallyholemotionissimilartovacancymotion,whichweencounteredpreviouslyasNabarro-Herringcreep.Theideaofholesandvacanciesmovingprovidesaconvenientanddescriptiveterminology.10.4ELECTRONICPROPERTIESOFMETALS:CONDUCTIONANDSUPERCONDUCTIVITY10.4.1FreeElectronTheoryforElectricalConductionTheclassicaltheoryofelectricalconductioninmetalsiscalledtheDrudetheory.Itisbasedontheideathatmanyelectronsinmetalsarenearlyfreeandthereforecanmigrate不得转载谢谢合作LWM 10.4ELECTRONICPROPERTIESOFMETALS:CONDUCTIONANDSUPERCONDUCTIVITY241easilywithamodestappliedpotentialV.ThemotionofelectronsperunittimeiscalledelectroncurrentI,andthefluxofelectronsJeisthecurrentperarea,I/A.TheelectricfieldEisgivenasthepotentialperdistanceL,E=V/L.Werecallthefluxequations(5.1),fromwhichOhm’slawisfortheelectronfluxbeingproportionaltotheappliedelectricfieldasJEµ(10.37)eTheconstantofproportionalityistheconductivitysasJE=s(10.37)eTheelectricfieldcanbeexpressedasthegradientinpotential,aswasdoneinChapter5.Theconductivitysisthereciprocaloftheresistivityras1s=(10.38)rOhm’slawisoftenexpressedasVI=(10.39)RWhereRistheresistanceandisexpressedintermsofrasrLR=(10.40)AWhereLandAarethelengthandcross-sectionalareaoftheconductor.Recallthattheunitforflux,andinparticular,electronflux,isnumberofelectrons/A·t.ThusJecanalsobeexpressedasJEv==◊◊sNe(10.41)edwhereNisthenumberofcharges/volume,vdisthedriftvelocityofelectrons,andeistheunitelectroniccharge.Thensolvingequation(10.41)fortheconductivity,wehavethefollowingequationfors:Neveds==Neeem(10.42)Ewheremeistheelectrondriftvelocityperelectricfield,calledtheelectronmobility,andisanimportantdevicequantitythatdeterminesoperationspeedofmanyelectronicdevices.Inthefreeelectrontheory,thefreeelectronsareconsideredtobeinrandommotionandthusprovidingnonetcurrent.However,ifanelectricfieldisimposed,theresultingelectromotiveforceeEcausesaconcertedmotionoftheelectrons.Theelectro-motiveforcecanbewrittenasequaltotheclassicalNewtonianforceontheelectrons,F=maas不得转载谢谢合作LWM 242ELECTRONICPROPERTIESOFMATERIALSdvmma==-eE(10.43)dtThenegativesignindicatesthatthedirectionoftheelectronmotionandtheelectricfieldareopposite.ThisequationcanbeintegratedtoyieldmetvE=(10.44)Equation(10.44)impliesthatforaconstantappliedfieldE,thevelocityofanelectronandtheelectronfluxincreasesindefinitelywithtime.Rather,whatisobservedisthatforagivenmaterialaconstantcurrentorelectronfluxisobtainedwiththeapplicationofaconstantfield.Thereforeequation(10.44)embodiesanonphysicalresult,andthemodelisobviouslyinneedofcorrection.Wecancorrectthedifferentialequationabovebyimposingaviscousforceontheelectronswiththeformmv/t,wheretisthecharacteris-tictimeforthedecayoftheelectronvelocitywhentheelectricfieldisremoved;tiscalledarelaxationtime.Theviscousforceisderivedfromthefactthataselectronsmovethroughthesolid,theycollidewithnuclei,andthesecollisionstendtoreducetheveloc-ityintheforwarddirection.thasunitsoftimeandisthetimeinbetweencollisions.Ifthetimeisshort,thenalargercorrectiontermisobtained.Thiscorrectiontoequation(10.44)yieldsdvmvm=-eE-(10.45)dttNowsupposethatanelectricfieldEisappliedtoestablishsomeelectronvelocityandthenEisturnedoff(E=0).Imposingtheseconditionsonequation(10.45),weobtainthedifferentialequationfortherelaxationofthevelocityasfollows:ddvt=-(10.46)vtThisintegratestothefollowingexpressionforvelocity:vtdddvtÚÚ=-(10.47)v0v0tTheresultis-ttvv=e(10.48)d0Equation(10.48)indicatesthatwhenthefieldisremoved,thevelocityexponentiallydecaysovertimewithashapedeterminedbytherelaxationtimet,asisshowninFigure10.7a.Alsotheviscousdrag-correcteddifferentialequation(10.45)canbeexploredwhenthevelocityreachesafinalvelocityvfintheappliedfield.Thisvelocityincreaseswhenthefieldisapplied,butitcannotincreaseindefinitely(asinequation10.44,theuncor-rectedform).Sothevelocitylevelsoffatsomefinalvelocityvf.Theendpointisevidentwhenthevelocitynolongerchanges,oratdv/dt=0:不得转载谢谢合作LWM 10.4ELECTRONICPROPERTIESOFMETALS:CONDUCTIONANDSUPERCONDUCTIVITY243a)vdtimeb)vfvdtimeFigure10.7Electrondriftvelocityvdversustime(a)decreasesexponentiallywhentheelectricfieldisremovedand(b)reachesafinalvelocityvfunderconstantfield.mveEf==eEvandt(10.49)ftmThechangeinvelocityresultingfromtheapplicationoftheappliedfielduntilthevelocitynolongerchangesisillustratedinFigure10.7b.Forconsistencytheunits2forvfcanbechecked:ehasunitsofamps·time,Ehasunitsof(mass·length)/(amps·3time·length),andthasunitsoftime.Theresultisthatvfcorrectlyhasunitsofdis-tance/time.Nowwecanpullallthistogetherandobtaintheclassicalexpressionfortheconduc-tivitys,sothatwecancompareitlaterwiththequantummechanicalresult.Fromequa-tion(10.41)forJeaboveweobtainthestartingformulaforsassEv=◊◊Ne(10.50)dThenwesubstituteforvffromequation(10.49)andsolvefors:2Nest=(10.51)mThustheclassicalresultillustratesthedependenceoftheconductivityonthenumberofelectronsNandtherelaxationtimeinbetweencollisions,t.Thefreeelectrontheoryprovidesgoodcorrespondencewiththeobservationsformetals.However,thistheoryprovidesnounderstandingaboutinsulatorandsemicon-ductormaterialsthatalsohavelargenumbersofvalenceelectrons.Thusamorecom-pletetheoryisneededthatexplainsallkindsofmaterials’electronicbehavior,andthequantumtheorydoesthisverywell.不得转载谢谢合作LWM 244ELECTRONICPROPERTIESOFMATERIALS10.4.2QuantumTheoryofElectronicConductionThemainthrustofthequantumtheoryisthatunlikethefreeelectrontheory,alltheelectronsinamaterialarenotequalwithrespecttotheconductionprocess.RecallfromelectronbandtheoryinChapter9thatallowedenergystatesarebasedonthechemicalbondingthatdeterminesthestrengthofthebindingpotentialsandtheordering(shortandlongrange)ofthematerial.Theseallowedquantumstatesarefilledwiththeavail-ableelectronsfortheparticularmaterial.Whenalltheavailableelectronsareinplace,thehighestfilledlevelformetalscloselydeterminesthepositionoftheFermilevel,EF.FormaterialswithacompletelyfilledvalencebandEFisnearmidgap.Theelectronsareundergoingrandommotionswithrandomvelocitiesintheabsenceofanelectricfield.Theserandomvelocitiesgiverisetorandommomentaaswell.Thereforemomentumorkspacecanbeusefulininterpretingthesequenceofeventsthatoccurwhenanelectricfieldisappliedtotheelectronsthatoriginallyhaverandommomenta.Figure10.8adis-plays2-DkspaceforamaterialwithaFermilevelthatisthesameinalldirections.RecallfromChapter9thattheelectronenergyisproportionaltokandforfreeelectronsisgivenbytheformula2h2E=k(9.67)2mea)kzEFk2k1-kxkxk4k3-kzb)E+-kzE(withE)FE(original)F-kxkx-kzFigure10.8kspacein2-DforacubicmaterialshowingtheFermienergyEFfor(a)withoutanappliedfieldand(b)withanappliedfieldE.不得转载谢谢合作LWM 10.4ELECTRONICPROPERTIESOFMETALS:CONDUCTIONANDSUPERCONDUCTIVITY245InFigure10.8aanyelectronenergywhereE£EFcanbeidentifiedbyavectortotheappropriatekvaluewithinthecircledefinedbyEF.WhenanelectricfieldEisimposedinthedirectionshownbythearrowinFigure10.8b,thetotalrandomnessoftheelec-tronvelocitiesislost.SomeelectronstravelingoppositetothedirectionofthefieldEshowninFigure10.8breceivemaximumincreaseinenergyfromtheelectricfield,whilethoseelectronstravelingintheoppositedirectionarereducedinenergybythefield.ThisisindicatedbytheformationofanewFermicircleshownastheshadedcircleinFigure10.8b.ThoseelectronsintheshadedcrescentontheleftofFigure10.8breceiveenergyfromtheelectricfield,whilethoseelectronsthatwereintheunshadedcrescentontherightarereducedinenergybyvirtueoftheappliedelectricfield.Itistheelectronsintheshadedcrescentthatcontributetoelectronicconduction;theelectronsintheoverlapregionofthetwoFermicirclesdonotcontributetoelectronicconduction.Thisisunder-stoodbyconsideringthatalltheelectronswithkvaluesintheoverlapregionhaveacor-respondingelectronwiththeoppositek.Themomentaforalltheelectronsinanydirectionintheoverlapregioncompensate,andtheeffectscancel.However,electronswithenergiesandmomentaintheshadedcrescenttotheleftdonothavecompensatingmomenta,andconsequentlythereisanetelectronmomentumtotheleftforFigure10.8bforthegivenelectricfielddirection.Thus,fromaquantummechanicalpointofview,notallelectronsparticipateinelec-tronicconduction.Rather,itisonlythoseelectronsthathaveenergiesnearEFthatpar-ticipate.Thisisthefirstdeviationfromthefreeelectrontheory.ThemomentumdiagramsshowninFigure10.8aand10.8bcanreadilybeconvertedtovelocitiesfortheelectronsusingequation(9.44).TheresultisshowninFigure10.9a.Thusthevelocitiescorre-spondingtotheFermivelocityaretheuncompensatedvelocitiesthatareincludedinelec-tronicconduction,drivenbytheappliedelectricfield.Figure10.9bshowstheparabolicshapeforthenumberofoccupiedstatesN(E)thatobtainforpurelyfreeelectrons,aswasdiscussedearlier.InadditionthisfigureshowsEFandthatwithanappliedelectricfieldthereisarangeofenergies(andvelocities)nearEF(vF)thatparticipateinelectronicconduction.Theshadedbarinthefigureshowsthelargedensityofoccupiedelectronicstatesinthesameregionastheparticipatingelectronvelocities.Wecanputtheseideastogether,startingwithequation(10.41)fortheelectronflux:Jv=◊◊Ne(10.52)ewhereclassicallyvwasthedriftvelocityvd.NowwesubstitutevFforvandforNwesubstituteN(EF)dE,sincewearenowconcernedwithN(E)nearEF,toobtainJv=eNEdE()(10.53)eFFSincebyequation(9.67),Eisrelatedtok,theexpressionforJecanbewrittenasdEJv=eNE()dk(10.54)eFFdkThen,usingequation(9.67)forE(k),wecanwritethefollowing:2dEh=k(10.55)dmke不得转载谢谢合作LWM 246ELECTRONICPROPERTIESOFMATERIALSa)E+-vzv(withE)Fv(original)F-vxvx-vzb)N(E)E+EFDEEFFigure10.9(a)2-DvelocityspaceforacubicmaterialwithanappliedelectricfieldE;(b)thedensityofoccupiedelectronstatesN(E).Thestatesaffectedbytheelectricfieldareshaded.Usingthisexpression,andrecallingthatp=hk(equation9.44),wecanwrite22dEhpvhmeF===hv(10.56)FdmmkhheeThenwesubstitutethisresultintheexpressionforJeabovetoobtain2Jv=heNE()dk(10.57)eFFNowanexpressionfordkisneededwecanfinditasfollows:StartingfromF=d(mv)/dt,weapplythefactthattheforceF=eE:dm()vpkddF====heE(10.58)dtdtdtThenfromequation(10.58),weobtaindkaseEEedk==dtt(10.59)hh不得转载谢谢合作LWM 10.4ELECTRONICPROPERTIESOFMETALS:CONDUCTIONANDSUPERCONDUCTIVITY247whereasbeforethetimeintervalt(substitutedfordt)isthetimebetweencollisions.Sub-stitutingthisexpressionfordkintoequation(10.57)forJeyields22Jv=eNE()Et(10.60)eFFThisexpressionisforthe1-Dcaseinthexdirection.In3-DforasphericalFermisurface,212vv()x=(10.61)FF3Sothe3-DexpressionforJbecomes122Jv=eNE()Et(10.62)eFF3Fromthiswefinds=J/Eandobtainthequantummechanicalsthatcanbecomparedwiththeclassicalresult:Je122st==veNE()(10.63)FFE3Thisquantummechanicalresultinequation(10.63)canbecomparedwiththeclassicalresultinequation(10.51).ThequantummechanicalJedependsonvFandnotmerelyonanyv,andalsoonthenumberofelectronsattheFermienergylevelbutnotalltheelectrons.Thequantummechanicaltreatmenthasabroaderapplicability.Forexample,formetalsthatarematerialswithpartiallyfilledvalencebands,werecallFigure10.5b,whichshowsthattherearethemostelectronsnearthemiddleofaband.Thusmetalsshouldbethematerialswiththehighestelectronicconductivity.Forinsulatorsandsemicon-ductors,thereisafilledvalencebandandthenagap.WiththeFermilevelinthegap,thenumberofoccupiedstatesneartheFermilevelissmall,andweshouldanticipatelowconductivity.10.4.3SuperconductivityAswasnotedabove,normalelectricalconductivityischaracterizedbyelectronsexperi-encingaresistancewhenelectroniccurrentflowsthroughamaterial.However,undersomecircumstances,typicallylowtemperature,somematerials(27elements,manyalloysandcompounds)exhibitzeroresistancetocurrentflow.Asaresultofzeroresistancethecurrentthatflowsinasuperconductordoesnotdecayintime,andassuchiscalledasupercurrent.Itisobviousthatsuperconductorswiththelosslesscurrentscanrevolu-tionizemanyareasofelectronics,electricaltransmission,andmagnetics.Consequentlytherehasbeengreatinterestinsuperconductivematerials,especiallysincethediscoveryofsuperconductorsthatexistatrelativelyhightemperatures.Thesematerialswillbedis-cussedbelow,butbeforethatwefocusonthebasicsandconsiderthephenomenonofzeroelectricalresistancethatiscalledsuperconductivity.Figure10.10ashowsanidealresistivityversustemperaturecharacteristicforasuper-conductor.AttemperatureT=Tcthereisatransitionfromnormalbehaviorwhereresis-tivitydecreaseswithdecreasingT.ThenormalbehavioratT>Tcisattributedtoreduced不得转载谢谢合作LWM 248ELECTRONICPROPERTIESOFMATERIALSa)rTcT(K)b)H@0KcNormalHcSuperconductingT(K)Tcc)TypeIrTypeIIHc1HcHc2HFigure10.10(a)ResistivityversusTforanidealsuperconductor;(b)criticalmagneticfieldversusTforasuperconductor;(c)resistivityversusmagneticfieldfortypeIandIIsuperconductors.scatteringofelectronsastheatomsinthelatticevibratelessatlowerT.AtTc,thecrit-icalsuperconductingtransitiontemperature,theelectricalresistancedisappears.ThisfiguredisplaysidealbehaviorwhilematerialswithdefectsandimpuritiesmightdisplayalesssharptransitionatTcand/oralesssteepapproachtozeroresistivity.Table10.1showsalistingofsomecommonsuperconductorswiththeirassociatedTc.ItisseeninthetablethatallthemetalandalloysuperconductorshavelowTc’sthatwillrequirecontinualandextensivecoolingforthematerialtooperateinthesupercon-ductingstate.TherequirementforcoolingtoTnearabsolutezeroreducesthetechno-logicalimportanceofsuperconductorsbecausesuchcoolingisexpensiveandcumbersome.However,aremarkablediscoverymadebyBednorzandMuellerin1986hasstartedarenewalofinterest.Theseworkersdiscoveredthatcertainoxidecompounds不得转载谢谢合作LWM 10.4ELECTRONICPROPERTIESOFMETALS:CONDUCTIONANDSUPERCONDUCTIVITY249Table10.1SomesuperconductingmaterialsMaterialTc(K)W0.01Hg4.15Al1.2Nb3Ge23.1LaBaCuO40YBa2Cu3O792(twoofmanynowdiscoveredarelistedabove)exhibitsuperconductivityatconsiderablyhighertemperaturesthanmetalsandalloys.Thisdiscoverymayleadthewayforappli-cationswithouttheexpensivecoolingapparatusrequiredforthelow-temperaturesuper-conductormaterials.Researchwiththenewhigh-temperaturesuperconductormaterialsthataretypicallycomplexoxidesispresentlyanactiveandfertileareaofelectronicmate-rialsresearch.Intheearlydaysofsuperconductionresearch,itwasfoundthattheapplicationofamagneticfieldcoulddestroysuperconduction.ForAlthecriticalfieldatwhichsuper-conductivityisdestroyedisaround100GaussandforHgaround400Gauss.Forreasonsnotfullyunderstood,thecriticalfieldislowerforthehigh-temperatureoxidesupercon-ductors,andhencethisisalimitationonthesematerialsatthistime.Themagneticfieldthatdestroyssuperconductivityisafunctionoftemperature.ThetemperatureatwhichthecriticalmagneticfieldHcexiststhatnegatesthesupercurrentisplottedinFigure10.10b.ItisseenthatatT=0K,thematerialcanwithstandthehighestmagneticfieldbeforeitlosessuperconductivity,butathighertemperatures,alowermagneticfieldwillforcethematerialbackintothenormalstate.Acriticalmagneticfieldcancomenotonlyfromanexternallyappliedfieldbutalsofromthesupercurrentitself.Whenamaterialisbroughtintothesuperconductingstatethesupercurrent(oranyflowingcurrent)givesrisetoamagneticfield.Ifthecurrentissufficientlyhigh,thepro-ducedmagneticfieldmayexceedHcandsuperconductivitywillbedestroyed.Also,asinanyconductors,theflowingcurrentinducesamagneticfieldwithamagneticfieldinten-sityH.Themagneticfieldcausesacurrentthatopposesthesupercurrentandeventu-allycausesthematerialtorevertfromthesuperconductingstatebacktothenormalstate.Figure10.10cdisplayscriticalmagneticfieldbehaviorfortwotypesofsuperconductors,referredtoastypeIandtypeIIsuperconductors.TypeIsuperconductorsarealsocalledsoftsuperconductors,andtheyarecharacterizedbyasharponsetofsuperconductionatHc(solidline).TypeIIsuperconductors,oftenreferredtoashardsuperconductors,displayalesspronouncedonsetofsuperconduction.TypeIIsuperconductorsarechar-acterizedbytwovaluesforthecriticalmagneticfieldHc1andHc2.Inbetweenthereareregionsinthematerialthatremainsuperconductingandregionsthatarenormal.ThemixedsuperconductingandnormalregioninbetweenHc1andHc2isreferredtoasthevortexstate.TypeIIsuperconductorsareusuallythechoiceforfabricatingsupercon-ductingmagnets,sincethesematerialscanwithstandhighermagneticfieldsbeforerevert-ingtothenormalstate.AswasshownaboveusingFigure10.10a,themostinterestingcharacteristicofthesuperconductingstateisthatrgoesto0atTc.However,asapracticalmatter,itisnoteasytoknowwhetherrisactuallyatormerelyveryclosetozeroresistance—thatis,whenthemeterisreallyatorverynearzero.Thus,inordertodefinitivelydeterminethe不得转载谢谢合作LWM 250ELECTRONICPROPERTIESOFMATERIALSsuperconductingstate,ameasurementofralonemaybeinsufficient,oratleastbyitselfuntrustworthy.AmoredefinitiveassessmentcanbemadeusingtheMeisnereffect.Aswasmentionedabovewhenasuperconductorisinthesuperconductingstate,thesupercurrentcanflow.Ifatthesametimeamagneticfieldisimposedonthesuperconductorinthesuperconductingstate,theresultingsupercurrentwillcauseamagneticfieldtooccurthatopposesanappliedmagneticfield,andexpelsthemagneticfluxBduetotheappliedfieldH.ThetotalexpulsionofthemagneticfluxthatoccursinsuperconductorsiscalledtheMeisnereffect.ThemagneticfluxisgivenasBH=+m()M(10.64)0-7wheremoisthepermeabilityoffreespace(4p¥10Hz/m).Miscalledthemagnetiza-tion,anditrepresentsthemagnetizationthatoccurswithinthematerial.BecauseofasupercurrentwithinasuperconductorB=0,andthusH=-M.TheratioofM/Hiscalledthemagneticsusceptibilityc,andthemagnitudec=-1forasuperconductoris-5consideredperfectdiamagnetism.Formetalscª-10.Thecombinationofthetwomeasurementsabove,namelyrgoingtozeroandtheexpulsionofmagneticflux,pro-videsconvincingevidencethatamaterialistrulyasuperconductor.Aquantummechanicaltheorywasproposedtoexplainsuperconductivity,in1957,byBardeen,Cooper,andSchrieffer,thuscalledBCStheory.BCStheoryexplainsvirtu-allyallthesuperconductorphenomenarelatedtothelow-temperaturesuperconductors,namelythemetalsandalloys,butthereissomedisputeaboutitsapplicabilitytoallaspectsofthehigh-temperaturesuperconductors,theoxidesuperconductors.Neverthe-less,BCStheoryprovidesimportantinsightsintosuperconductivity,andthusabriefcon-ceptualoutlinewillbegivenherein.AkeyaspectofBCStheoryisthattheparticipatingelectrons,thesuperelectrons,pairupintoso-calledCooperpairs.Figure10.11ashowsa2-DFermicirclewithtwoelectronsaandbneartheFermisurface.BecauseofCoulombicforces,electronsrepeleachother.However,forelectronsaandbtherepulsionshouldbeminimumbecausetheyaremaximallyscreenedfromeachotherbytheelectronsinbetween(intheshadedregion),andtheyarethegreatestdistanceapartinkspacehavingkand-kvectorsneartheFermisurface.InBCStheoryitisfoundthatwiththerepulsionsataminimumandwithspinsopposite,twoelectronscanattracteachotherwithasmallattractivepoten-tialV.ManypairscanformatorverynearEF.AswehavepreviouslyseeninChapter9,thisbindingpotentialcangiverisetoanenergybandstructureandagap,thesuper-conductinggapillustratedinFigure10.11b.FromthedetailsofBCStheorythissuper-conductinggapEscgisgivenasEe=8hw-2NEV()F(10.65)scgDwherewDistheDebyefrequencyandfoundtovarywiththereciprocalofmassas1wµ(10.66)DM-1/2SinceEscgisproportionaltowD,thegapisalsoproportionaltoM,andthisisknownastheisotopeeffect.Forexample,ifaheavierisotopeissubstitutedforalighterone,thenthegapwilldecreaseandTcwillalsodecrease.TheisotopeeffecthasalsobeenusedalongwiththeMeisnereffecttoconfirmsuperconductivitybehavior.Escgmeasuredvalues不得转载谢谢合作LWM 10.4ELECTRONICPROPERTIESOFMETALS:CONDUCTIONANDSUPERCONDUCTIVITY251a)kzEFelectronak-kxkx-kelectronb-kzb)N(E)GapEEFc)AtomicCore+--ee12+Figure10.11(a)2-DkspaceforasuperconductorshowingelectronsaandbnearEFwithoppositek;(b)densityofoccupiedelectronstatesforasuperconductorshowingthesuperconductinggap;(c)phononmechanismforformingCooperpairs.-4areabout10eVorintheinfraredrange(ª1.2m).ItisalsointerestingthatasEscgincreasesbecauseoflargerelectronbindingenergies,Tcrises.Theoxidehigh-temperaturesuperconductorsallhavelargepotentialsandareinsulatorsatroomtem-perature,yettheyareexcellenthighTcsuperconductors.ThemechanismfortheoriginoftheelectronpairingintheBCStheoryisattributedtoaphononeffect,namelyaquantizedatomicvibration.ThisphononeffectcanbesomewhatsimplisticallyvisualizedusingFigure10.11c.Considertwoatomiccoreposi--tionsindicatedbytheshadedcircleswith+charges.Anelectron(e1)travelingfromthe-righttolefthasapathbetweenthetwocores.Therapidlymovingelectrone1causesadisplacementofpositivecores,asisindicatedbythelightershadedcores.Anotherelec--trone2travelingintheoppositedirectionbutataninstantlater,soasnottobeaffected-bye1,“sees”adifferentenvironmentwithrespecttothecores.Infacttheslowdisplace-mentofthecorescannotrelaxbacktotheequilibriumpositionbeforethearrivalof--e2.Thegreatercenterofpositivechargeduetothedisplacementofthecorescausese2--toaccelerate.Thustheelectronse1ande2arethoughtofasboundbythevibrationofthecores,thephonon.不得转载谢谢合作LWM 252ELECTRONICPROPERTIESOFMATERIALSa)sc1sc2sc1sc2EscgThinInsulatorb)IVEscgFigure10.12Josephsontunneling(a)betweentwoidenticalsuperconductorsand(b)whenafieldisappliedtunnelingofCooperpairscanoccuryielding(c)asharpcurrentrise.Amongtheinterestingapplicationsforsuperconductorsissuperconductingtunnel-ing,ortheso-calledJosephsontunneling.Tunnelinginsuperconductorsoccurssimilarlytotunnelinginnormalmaterials,andthisisillustratedinFigure10.12a.Theleftpanelshowstwonearlyidenticalsuperconductorsseparatedbyathininsulatingbarrier,typi-callylessthan2nmthick.IfapotentialisappliedequivalenttoormorethanthegapEscg,thentunnelingwilltakeplacefromoccupiedstatesinonesuperconductortoemptystatesintheotherasisindicatedintheright-handpanel.ThecurrentIversusvoltageVcharacteristicisseeninFigure10.12b.Thesharpriseisinthepicosecondtimerange,whichmakesdeviceswitchingveryfastwithsuperconductors.Whatisdifferentinsuper-conductorsisthatforlowcurrents,Cooperpairsmigrateacrossthetunnelbarrier,andconsequentlytheinsulatorbecomesasuperconductorwherecurrentcanflowwithoutanaccompanyingvoltage.Forlargercurrents,theCooperpairscanradiateenergyastheydroptolowerenergystatesasfollows:2eV=hw(10.67)12whereV12isthedcvoltageacrossthejunction.Thusanapplieddcvoltageproduceselec-tromagneticradiationtypicallyatmicrowavefrequencies.Thiseffectisduetothefactthatacrossthebarriertheincidentwavefunctionreceivesaphaseshift.ThisphaseshiftcanbeexpressedintermsofthecurrentfluxJasfollows:JJ=sinf(10.68)21WithavoltageVappliedthephaseshiftisincreased:JJ=+sin()ffD(10.69)21ThephaseshiftduetotheappliedpotentialDfcanbeexpressedasfollows:不得转载谢谢合作LWM 10.5SEMICONDUCTORS253EteVtDf==(10.70)hhwheretheenergyisexpressedaseV.ForthetwoelectronsofaCooperpair,thisphaseismultipliedby2andreinsertedintothefluxequationtoyieldÊeVtˆJJ=+sinf2(10.71)21Ëh¯Thusanapplieddcpotentialyieldsatime-dependentcurrent.ThefactorEt/hcanbeexpressedintermsofafrequencyasfollows:Ethtn2eV===wtt(10.72)hhhTheresultantfrequencyisgivenbyn=484V,whichisintheGHzrange.10.5SEMICONDUCTORSSemiconductorsarethosematerialsthathaveafilledvalencebandfollowedinenergybyabandgap,andthenanemptyconductionband,asisshowninFigure10.13a.ThebandgapislargerelativetokT,butnotverylargeandtypicallylessthan2eV.Forexample,Si,themostutilizedsemiconductorintheworld,hasabandgapof1.1eVwhileGehasagapof0.7eVandGaAs,animportantcompoundsemiconductor,hasagapof1.4eV.Forcomparison,SiO2,acommoninsulatorusedinmicroelectronicshasabandgapofabout9eV,anddiamond,anothergoodinsulator,hasagapofabout5.5eV.BelowwewillusetheFermi-DiracdistributionfunctionF(E)andthedensityofstatesfunctionZ(E)tocalculatethenumberofelectronsintheconductionbandforsemiconductors93andcomparethatnumberwithinsulators.ForSi,wewillfindabout10electrons/cmintheconductionbandatroomtemperature,andforinsulatorswithwidergaps,severalordersofmagnitudefewerelectronsarepresent.However,fortypicaldeviceoperation,suchasthedevicesinadesktopPC,aboutthreeordersofmagnitudemoreelectroniccarriersareneededfordeviceoperationthanavailableinthepureSisemiconductor.Thepuresemiconductorexhibitsintrinsicelectronicpropertiesdictatedbythenumberofelectroniccarriersavailable,andthusthiskindofsemiconductoriscalledanintrinsicsemiconductor.Intrinsicsemiconductorsarerarelyusefulbecauseofthelownumberofcurrentcarriers.Thusthesemiconductorsrequireadditionalsubstancesthatcanalterthenumberofcarriers.Thesesubstancesarecalleddopants,andtheprocessinvolvediscalleddoping.Theresultisasemiconductorwhoseelectronicpropertiesaredominatedbythedopants,andthiskindofsemiconductoriscalledanextrinsicsemiconductor.Inthefollowingsectionswewillexplorebothintrinsicandextrinsicsemiconductors.10.5.1IntrinsicSemiconductorsApuresemiconductorsuchasSi,whichhasabandgapofabout1.1eV,isshowninFigure10.13a.AswasdiscussedaboveinSection10.3,thisintrinsicsemiconductoraswellasotherintrinsicsemiconductorshaveEFclosetothecenterofthegap,andbecause不得转载谢谢合作LWM 254ELECTRONICPROPERTIESOFMATERIALSa)EnergyConductionBandEgEFValenceBandb)EnergyCBElectronsEgEFHolesVB01EF(E)Figure10.13(a)Parallelbandschemeforanintrinsicsemiconductor;(b)bandstructurewiththeFermi-Diracfunctionindicatingholesinthevalencebandandelectronsintheconductionband.thisisanenergygap,thereisnodensityofallowedstatesinthegap.Thustheonlywaytoeffectelectronicconductionistosomehowenableelectronstoaccesstheallowedemptystatesintheconductionband.Amongthemanywaystodothisaretopromotevalenceband(VB)electronstotheconductionband(CB)viaaphotonprocess,and/orviaathermalprocess,and/orbyaddingallowedelectronstates.Thislatternotionofaddingstatesiscalleddopingandwillbediscussedbelowforextrinsicsemiconductors.Thephotonprocessispossiblebutimpracticalfordeviceoperation,exceptfordevicesthatareusedfordetectingphotons.Sincetheusualelectronicdevicesareoperatedatroomtemperature,wewillfullychar-acterizeintrinsicsemiconductorsatroomtemperature,andthenexplorechangesthatmayoccuratothertemperatures.AtroomtemperaturewerecallthattheFermi-Dirac(FD)distribution(equation10.8)yieldsbandtailingtoenergiesaboveEF.Figure10.13bshowsthebandstructureasinFigure10.13abutsuperimposedistheFDdistributionfunctionforroomtemperature.NotethatwhenlowinenergyintheVB,FD=1.Like-wise,whenhighinenergyintheCB,FD=0.TheseregionsoftheVBandCBarefilledandempty,respectively.However,intheregionoftheupperbandedgefortheVBandlowerbandedgefortheCB,thesituationisdifferentfromthe0Kpictureanddifferentfromregionsinthebulkofthebandsdescribedabove.Specifically,theverytopoftheVBhasemptystates,holes,andthebottomoftheCBhasoccupiedelectronicstates.不得转载谢谢合作LWM 10.5SEMICONDUCTORS255TheseregionsareshadedinFigure10.13binthecircledregions.Aswasdiscussedabove,basedonthetailofFDdistributionfunction,althoughthereareelectronsandholesavailableforconduction,thenumberofavailablecarriersistoolowforpracticaldevices.Belowwewillcalculatethisnumber,andexaminethetemperaturedependence.Itisimportanttonotethatforintrinsicsemiconductors,thenumberofelectronsandholesareequal,meaningthatforeveryelectronintheCBthereisaholeintheVB.InordertocalculatethenumberofelectronsNeintheCB(whichisequaltothenumberofholesintheVB)foranintrinsicsemiconductor,weneedintegrateN(E)overtheCB.RecallthatNE()=2ZEFE()()(10.17)Alsorecallthat32VÏ2me¸121ZE()=Ì˝EFand()E=(10.5and10.8)4p2Óh2˛1+e()EE-FkTF(E)canbesimplifiedbyrealizingthatE-EFforthisproblemisECB-EFandisabout0.5eV(forSiwithEg=1.1eV).kTatroomTisabout0.025eV.Thustheexponentialterminthedenominatoris()EEkCB-FT050025..ee=>>1(10.73)ThereforeF(E)issimplifiedtoaBoltzmann-likeformasfollows:--()EEkCBFTFEe()=(10.74)TheintegrationlimitsaresetforconveniencewiththezeroofenergyatthebottomofCBandintegrateto•.Onceagain,thisupperlimitisconvenientyetfictitous,butsinceF(E)goesto0,theconveniencedoesnotimposeinaccuracy.Theintegraltoevaluateisasfollows:32•VmÏ2¸eEe12◊--()EEkTFdE(10.75)Ú2Ì2˝02pÓh˛Thisintegralcanbeputintoaformthatcanbereadilyintegrated.First,noticethatEFisaconstant.Thenbygroupingconstantsandenergyvariables,weobtainforthenumberofelectronsintheCB(Ne)thefollowingintegral:32VmÏ2¸•N=eeEEkFT12◊e-EkTdE(10.76)e2p2ÌÓh2˝˛Ú0Theformoftheintegralinequation(10.76)canberecognizedasthedefiniteintegral:•1p12-nxÚxedx=(10.77)02nnwheren=1/kTandx=E.Thisyieldsthefollowingresult:不得转载谢谢合作LWM 256ELECTRONICPROPERTIESOFMATERIALS3232N=VÏ2me¸eEkFFT◊◊kT()pkT12=VÏ2mkTe¸eEkT(10.78)e2Ì2˝Ì2˝2pÓhh˛24Óp˛ThisexpressionforNecanberewrittenforconveniencewithseveralsubstitutions.ThenumberofelectronsintheCBpervolumeisn=Ne/V,EF=-Eg/2,andformetheeffec-tiveelectronmassratiototherestmass,m*e/mo,isusedtoobtainthefinalresult:323212mm32Ê*ˆÊm*ˆn=Ïo¸eTe32Ekg2T48210.21mK--332eTe32-Ekg2TÌ2˝Á˜=¥()Á˜4Óph˛Ëmo¯Ëmo¯(10.79)3215--332Êm*ˆ32-Ek2Tnc=¥48210.()mKeTegÁ˜Ëmo¯Inequation(10.79)therearetwotemperature-dependentterms:anexponentialtermandapre-exponentialterm,withtheformerbeingdominantandyieldinganexponen-tialincreaseinthenumberofelectronsnfoundintheCBofanintrinsicsemiconduc-163tor.At25°Cthereareabout10electrons/minSiwithagapof1.1eV,assumingthat3283theeffectivemassrationisnear1.In1mofSithereare10Siatoms/m.Thusonly112in10SiatomscontributeanelectrontotheCB.ConsequentlySiisagoodinsulatoratroomtemperature.ThesameformulacanbeusedtocalculatethenumberofholesintheVBwithonlythesubstitutionoftheeffectivemassforholesm*forthatforhelectrons.Theconductivityforanintrinsicsemiconductorshouldincludetheconductivityofbothelectronsandholesasfollows:3221--332Êm*ˆ32-Ek2Tsmm=+=¥nepe48210.()mKeTemmeg(10.80)ehÁ˜()eh+Ëmo¯forthecasewherem*e=m*;otherwise,anotherratiotermisneeded.hAswaspreviouslydiscussedformetals,theelectronmobilitygenerallydecreasesastemperatureincreasesbecausetheatomicvibrationsincreasewiththetemperature,andthensodoestheelectronscattering.Consequentlytheconductivityformetalsdecreaseswithtemperature.Theoppositeistrueforsemiconductors.Itisseeninequation(10.79)abovethatnincreasesexponentiallywithtemperature,sincetheexponentialterm3/2hasTinthedenominatorofthenegativeexponentandTalsoappearsinthepre-exponential.Theconductivity,whichisafunctionofn(andp),mustthenalsodisplayanexponentialincreasewithtemperature;thisisshownschematicallyinFigure10.14alongwithn.Thesemetalandsemiconductorconductivityvaluesneedtobekeptinper-spective.Infacttheydifferbysome8to10ordersofmagnitudeatroomtemperature8-2withgoodmetalsnear10S/mandsemiconductorsnear10S/m,dependingstronglyonthesizeofthebandgap.Conductioninmetalscandecreasebyanorderofmagnitudeortwowhenheatedtomorethanseveralhundreddegrees,andsemiconductorconduc-tionincreasesbythesameamount.However,thesematerialswillremainfarapartintheabsolutevaluesoftheconductivity.Thenumberoffreeelectronsinmetalsis28–29-3around10m,whileforintrinsicsemiconductorsthenumberofcarriersisaround15–16-310m.ForheavilydopedSi,forexample,thenumberofelectronscangoashighas21-310m.Thusthenumberofavailablecarriersformetalsistypicallyconsiderablymore6than10thenumberinsemiconductors.不得转载谢谢合作LWM 10.5SEMICONDUCTORS257lnslnnTFigure10.14Logarithmicplotoftheconductivitysandelectronconcentrationnforanintrinsicsemicon-ductorversustemperature.10.5.2ExtrinsicSemiconductorsExtrinsicsemiconductorsderivetheirelectronicpropertieslargelyfromtheadditionofimpurityatomscalleddopantsthateffectprofoundchangesintheelectronicbehaviorofthesemiconductor,andyetarepresentinlowconcentrations,typicallylessthan0.1%.Usuallydopingtakesplaceininorganicsemiconductorssimplybytheadditionofatomswithdifferentelectronicconfigurations.Theseatomconfigurationsmustsubstitute(fit)foratomsonthesemiconductorlattice.However,othermethodsofdopingarepossible,suchastheionimplantationofchemicallyinertmoietiesthatcreatelocalizeddamageinthesemiconductor,andcreateelectronicstatesthatactasdopants.Essentiallydopantscreatelocalizedelectronicstatesinthebandgapofthesemicon-ductor.PreviouslyinChapter9,whenelectronicenergybandswerediscussed,wesawthatdelocalizedorextendedstatesarisefromtheintermixingofwavefunctionsfromall3theatomsinasolid.Forexample,forSiwithadensityof2.33g/cmandanatomicweight223of28g/mol,thereareabout5¥10atoms/cm,andtheatomsaretenthsofanmapart.22–233Thuswavefunctionsoforderof10mixforSiofonecmtoformtheSielectronbands,theso-calledextendedallowedelectronstatesforthematerial.Theresistivityof5-6ultrapureSiisoftheorderofr=10W-cm,comparedwithCuwherer=10W-cm.Themaximumphosphorus(P)dopantthatcanbeaddedtodopeSiwithanexcessofelectrons,calledN-typedoping,isthesolubilitylimitforPinSi,whichisabout20-310cm.Thisislessthan1%maximumconcentrationofPinSiorlessthanonePin-3100Si’s,andityieldsaresistivityforSiofaboutr=10W-cm.Forso-calledmetaloxidesemiconductorfieldeffecttransistors(MOSFET’s),thedopingintheactivedeviceregion15–16-3ofSiisabout10cmP’sinSi,andthusmuchlowerthanthesolubilitylimit,yield-7ingaresistivityofaboutr=5W-cm.AtthisdopinglevelthereisonePforevery10Siatoms,andthedopantatomsareabout50nmapart.Thesearenormaldopingrangesthatrepresentlowconcentrationsrelativetothenumberofsemiconductoratoms.Con-sequentlythedopantatomsarefarapartanddonothaveoverlappingwavefunctions,sotheydonotformbands.Rather,thedopantstypicallygiverisetonewlocalizedelectronstates.Figure10.15showsthedistortioncreatedifalargerandasmalleratomareinsertedinanotherwiseuniformlattice.First,thelocalorderisaltered,inthatanumberofsur-roundingatomshavealteredcoordinates.ThedistortiondoesnotextendthroughouttheSianddampsoutafterseverallatticespacings.Second,thebindingpotentialinaregionsurroundingthesubtitutionalatomisaltered.FromtheKronig-Penney(KP)modeldis-cussedinChapter9,weknowthata,bandthepotentialbarrierV(orPintheKPmodel)不得转载谢谢合作LWM 258ELECTRONICPROPERTIESOFMATERIALSSubstitutionalSubstitutionalFigure10.15Substitutionalimpuritiesona2-Dlattice.aremodifiedinthevicinityofthesubstitutionaldopantatom.Thesemodificationsundoubtedlyleadtodifferentallowedelectronstates.However,becauseofthesparsenessofthedopants,thenewstatesarelocalizedinthatthewavefunctionsdonotoverlap.Also,ifweconsidertheelectronconfigurationofSi,whichliesingroupIVofthePeri-22odicTable,theouterelectronshell,theMshell,hasaspelectronconfigurationthat3hybridizesinsolidSitosp,andthisleadstotetrahedralcovalentbondingofeachSitofournearestneighborSiatoms.TwocommondopantsforSiareP,whichdopesSiwithexcesselectronstocreateN-typeSi,andB,whichdopesSiwithexcessholestocreateP-typeSi.PisingroupVofthePeriodicTablewithonemoreelectronintheMshellwith23anspelectronicconfiguration;BisingroupIIIwithonelessouterelectronthanSiin21itsLshellwithanspconfiguration.IfweimaginePandBsubstitutingonthetetrahe-3dralbondedspSilattice,thenrelativetoaSiatom,PhasanextraelectronandBhasonetoofewelectrons.TheresultisthatPproducesnewelectronstatesthatarefilledwithelectronsandwithinafewhundredthsofaneVbelowtheCBofSi,whileBsubstitutedontheSilatticeproducesnewelectronstatesthatareinitiallyemptyandthatarehun-dredthsofaneVabovetheSiVB.ThesenewstatesareshowninFigure10.16.InFigure10.16atheleveloffilledelectronstatesneartheCBislabeledED,wheretheDsignifiesthateachofthefilledstatesatthisfilledlevelcandonateitselectrontotheCB,sotheDsubscriptsignifiesthatthislevelisadonorlevel.InFigure10.16bthelevelofemptyaccep-torstatesisnearthevalencebandwiththesubscriptA.Table10.2showsseveraldopantsthatcanbeusedtodopeSiandGe.Itisclearthatthelevelsofalloweddonorandaccep-torstatesareclosetotheCBandVB,respectively.Withroomtemperatureyieldingabout0.025eVenergy,mostallofthedonor-occupiedstateswillhavesufficientkineticenergyavailabletotheelectronstoraisetheenergytothatofthenearbyCB.ThusmostofthedonorstatesareionizedandyieldtheirelectronstotheCB.LikewisetheemptyacceptorlevelarecloseenoughinenergyfromthefilledVBthatelectronsfromthetopoftheVBcanoccupytheemptyacceptorstatesandleaveholesbehindintheVB.Itisusefultocomparetheconductivityforanintrinsicandextrinsicsemiconductor.Recalltheformuladescribesconductivity:不得转载谢谢合作LWM 10.5SEMICONDUCTORS259EnergyEFConductionBandConductionBandEDEiEFEAValenceBandValenceBanda)b)Figure10.16Parallelbandschemefor(a)anN-typeand(b)aP-typesemiconductorshowingthedonor(ED)andacceptor(EA)levels,aswellastheFermilevels(EF)andtheintrinsicFermilevels(Ei).Table10.2CommondopantsforSiandGeandthelevelsproducedSiLevelGeLevelDopant(eV)(eV)DonorsP0.0450.012(belowCB)Sb0.0390.0096As0.0490.0127AcceptorsB0.0450.0104(aboveVB)In0.1600.0112Ga0.0650.0108sm=Ne(10.42)ccwhereNeisthenumberofcarriers(N(EF))andmcthecarriermobility.Weneedtofirstcomparethenumberofcarriers.AswasindicatedaboveforintrinsicSiatroomT,there10320areabout10electrons/cmwhilefordopedSithatnumbercanrisetoabout103electrons(orholes)/cm.ThusatroomTwewouldanticipateahigherconductivityfordopedSi.InFigure10.14wesawthatthetemperaturedependenceofanintrinsicsemiconduc-torexponentiallyincreasesincarrierconcentrationwithariseinT.TheTdependenceforanextrinsicsemiconductorismorecomplicatedwithatypicaldependence,showninFigure10.17.AtverylowtemperatureswhenkTislessthantheionizationenergyforthedopantstates(lessthantheenergydifferencebetweenthedopantlevelandnearestbandedge),thecarrierconcentrationisthesameasforanintrinsicsemiconductor.Thisregionissometimesreferredtoasthe“freeze-out”regionwherethethermalenergyisinsufficienttoionizeallofthedopants.DependingonT,somefractionofthedopantscancontributecarriers.ThisisfollowedbyarelativelyTindependentcarrierconcentra-tionwhereallthepossiblecarriersaregeneratedbythedopant,andtheseextrinsiccar-riersgreatlyexceedtheintrinsicnumberofcarriers.However,atstillhigherT,thenumber不得转载谢谢合作LWM 260ELECTRONICPROPERTIESOFMATERIALSnIntrinsicnDIntrinsic+freezeoutExtrinsicTFigure10.17TemperaturedependenceofthecarrierconcentrationforanN-typesemiconductor.ConductionBandEnergyEDEFEiEAValenceBandTFigure10.18TemperaturedependenceoftheFermilevelsforbothN-andP-typesemiconductors.TheextrinsicFermilevels(EF)movetowardtheintrinsicFermilevel(Ei).ofintrinsiccarriersproducedexceedstheextrinsicnumber,andthetotalnumberrisesexponentiallywithT.Asthecarriersgainenergy,moreandmoreofboththeintrinsicandextrinsiccar-rierswillentertheconductionband.IntheN-typeSiwithEF,shownbetweentheCBedgeandEDinFigure10.16a,asTincreases,EFnecessarilydecreasesinenergyaselec-tronsarefirstexhaustedfromthedonorlevel.UltimatelyEFdropstoEiwhenallthedonorelectronsareexhaustedandthematerialisdominatedbyintrinsiccarriers.Thephenomenonissimilarforholes,inthattheEFstartsoutatlowTinbetweentheVBedgeandEAshowninFigure10.16b.AsTrisesmore,theelectronsareabletoreachthedonorlevelraisingEF.Ultimately,asTcontinuestorise,thenumberofholesproducedwilloutnumbertheextrinsiccarriers,andEFwillrisetotheintrinsiclevelEi..ThecasesforbothN-andP-typeSi(orothersemiconductors)areshowninFigure10.18.CarriermobilitygenerallydecreaseswithTinsemiconductorsforthesamereasonasformetals,namelyelectronscatteringfromtheatoms.Wecannowproceedtorelateallthisinformationaboutcarriersintermsoftheirquantityandmobilitytodeterminehow不得转载谢谢合作LWM 10.5SEMICONDUCTORS261sHighDopingLowDoping300T(K)Figure10.19Temperaturedependenceoftheconductivity(s)forheavilyandlightlydopedsemiconductors.theconductivityvarieswithT,andthisisshowninFigure10.19forbothlightlyandheavilydopedsemiconductors.Asisexpected,theheavilydopedsemiconductordisplaysahigherconductivity.Asthetemperatureincreaseswellbeyondroomtemperature,bothsamplesdisplayhigherconductivity,whichapproachesintrinsicconductivityastheintrinsiccarriersdominateconduction(seeFigure10.17).BothsamplesarelimitedinconductivitybydecreasedmobilityathigherT,andthisisnoticeableatlowerTwhereasmalldipduetoscatteringisseenasTrises.10.5.3SemiconductorMeasurementsItisusefultoconsidertheprinciplesthatunderlieseveralimportantsemiconductormea-surements,inthatthemeasurementsareusefulinthemselvesbutfurthersupportandconfirmthemodelsusedheretodescribeelectronicconductioninmaterials.Inthissectionthreemeasurementsareconsidered.ThefirstistheHalleffectmeasurementthatusesthesimultaneousapplicationofelectricandmagneticfields,andcanaccessthemajoritycarriertype(electronorhole)andthemajoritycarrierconcentration.Thesecondmeasurementisthedeterminationoftheeffectivemassusingelectroncyclotronresonance,andthethirdisthefour-pointprobemeasurementofresistivity.FortheHallmeasurementwerefertoFigure10.20whereanelectricfield,Ex,isappliedtoanN-typesemiconductorsolidinthe+xdirection.Thisfieldgivesrisetoacurrentflux-Jx.Withthisfieldappliedandcurrentflowing,amagneticfieldissimulta-neouslyappliedinthezdirectionasisshowninthisfigureasBz.Thismagneticfieldcausesadeflectionintheelectriccurrentthatcanbepredictedbytheso-calledright-handrule.WiththethumboftherighthandpointedinthedirectionofEx,pointtheindexfingeroftherighthandinthedirectionofBz.Themiddlefingeroftherighthandnowpointsinthedirectionofthedeflection,whichisthe+ydirection.Thisdeflectionoftheelectronflowinthe+ydirectiongivesrisetoanewelectricfieldcalledtheHallfieldalsoin+y.TheforcefromthisHallfieldisgivenasFE=e(10.81)HyThisforceisinequilibriumwiththeLorentzforce,FL,thatresultsfromBz,andisgivenas不得转载谢谢合作LWM 262ELECTRONICPROPERTIESOFMATERIALSBzz+--yEx-xx-zyFigure10.20Hallmeasurementgeometrywithelectricfieldapplied(Ex)andmagneticfield(Bz)causingapotentialiny,theHallpotential.FvB=e(10.82)L-xzwherev-xistheelectronvelocityinthe-xdirectionofthecurrent.EquilibriumrequiresthatFH+FL=0,andthisyieldstheHallfieldEyasEv=-B(10.83)yx-zThisequationalongwithequation(10.52)canbeusedtoexpressthecurrentflow:-NEyJv==Ne(10.84)--xxBezEquation(10.84)canbesolvedforthecarrierconcentrationNasfollows:JB-xzN=-(10.85)eEyApositiveNisindicativeofelectronsservingasmajoritycarriers.WiththecurrentJandamagneticfieldapplied,andthereforeknown,thepotentialthatyieldsthefieldEyismeasuredandNiscalculated.UsuallyaHallcoefficientRHisreportedandgivenas1R=-(10.86)HeNThesignofRHindicatesthecarriertype,wherenegative(-)isforelectronsandpositive(+)forholes.Theeffectivemassforelectronscanbemeasuredusingtheelectroncyclotronreso-nanceeffect,asisshowninFigure10.21a.Aslabofsemiconductorisplacedinamag-neticfieldandthefieldcausesaprecessionoftheelectroninacircularorspiralmotionasshowninthisfigure.Theangularfrequencyoftheprecessionwcisgivenasfollows:eBwc=(10.87)m*不得转载谢谢合作LWM 10.5SEMICONDUCTORS263Forafreeelectronthefrequencyiswcn==28.B()GHzforinkiloGaussB(10.88)c2pThusforB=1kGaussnc=2.8GHz,whichisinthemicrowaverange.ExperimentallyoneappliesamagneticfieldinthekiloGaussrange,andparalleltothemagneticfieldanRFfieldinthemicrowaveregionisscannedinfrequency.Whentheprecessionfrequencyisreached,resonanceoccursandenergyfromtheRFfieldisabsorbed.Tomeasurethefrequency,theRFabsorptionversusappliedRFfrequencyismeasured,asillustratedinFigure10.21.FromncandB,m*iscalculated.ThismeasurementistypicallyperformedatlowTnearliquidHeorN2temperaturestoreducecollisionsduringprecession.Thefour-pointprobemethodisoftenusedtomeasuresorr.Thismethodavoidstheproblemofhavingveryhighresistancecontactsthatovershadowthemeasurementofthematerialproperties.Theseso-calledSchottkycontacts,orjunctions,willbeaddressedinChapter11.Figure10.22showsfoursharpprobesA,B,C,Dthataretyp-icallyequallyspaced(e.g.,attypicallyabout1mm).AcurrentIfromacurrentsourceisappliedtoflowbetweenAandD.ThevoltageVacrossBandCismeasured.ByOhm’slaw,therelationshipobtainslIlVI===RIr(10.89)AsAwhereRtheresistanceisexpressedintermsoftheresistivityrorconductivitysandthelengthlandareaA.ForthespecificgeometryshowninFigure10.22,l=drthedistance2betweenprobes,andAis2pr,orthesurfaceareaofahemispherethroughwhichthecurrentpasses.Thentheformulaaboveindifferentialformbecomesa)Bb)awwcFigure10.21(a)MagneticfieldBcausesprecessionofelectrons;(b)microwaveabsorptionpeakatprecessionfrequency.不得转载谢谢合作LWM 264ELECTRONICPROPERTIESOFMATERIALSBCADS1S2S3rdrFigure10.22Four-pointprobe(A,B,C,D)geometrywithequalseparations(S).IdV=dr(10.90)22psrToobtainthevoltagedropfromBtoC,weintegrateasfollows:CSS12+IdV=dr(10.91)ÚÚBS212psrTheintegrationofequation(10.91)yieldstheresultIÊ11ˆVVCB-=Á-˜(10.92)2psËSSS+¯112Thisformulaisuseful,buttobemoreaccurate,thevoltagedropsatAandDmustbetakenintoaccountbecausethecurrentpassesthroughthosecontacts.Thecorrectedformulaisasfollows:IÊ1111ˆVVCB-=Á+--˜(10.93)2psËSSSSSS++¯131223ForallthespacingequaltoS,weobtainthefollowingresultfors:1s=(10.94)2pSVBC10.6ELECTRICALBEHAVIOROFORGANICMATERIALSThebasisfortheelectronicpropertiesofmaterialshasbeentheelectronicenergybandstructure,whichisbasedontheperiodicstructureandtheconsequentperiodicpoten-tialforcrystallinematerials.Itisnowworthwhiletoreconsiderthelongandshortrangeorder,aswasdiscussedinChapter2.ApoignantexampleofthedistinctionbetweenlongandshortrangeorderingisSiO2.RecallthatSiO2iscomposedofSiO4tetrahedra,as3showninFigure2.2a,thathavespbondingwithinatetrahedron,andthetetrahedraarejoinedviathebridgingO’sateachapexofthetetrahedra.Ifthetetrahedraarearrangedinaperiodicmanner,thenSiO2willhavelongrangeorderandbedeemedtobecrys-talline,asshowninFigure2.2b.Alternatively,thetetrahedracanbearrangedwith不得转载谢谢合作LWM 10.6ELECTRICALBEHAVIOROFORGANICMATERIALS265Z(E)EVBEmEFEmECBEFigure10.23Electronenergybandstructureintermsofthedensityofelectronstates,Z(E),versuselectronenergy.Valenceandconductionbandedgesareshownwithbandtailingandlocalizedstatesinthebandgap.randomanglesandthusthematerialwillbeamorphous,asshowninFigure2.2c.Thelackoflongrangeorder,butthepresenceofthesamechemicalbondingorshortrangeorder,givesrisetoamorecomplicatedelectronenergybandstructure.Ofcourse,theprecisenatureoftheenergybandstructurewilldependonthespecificmaterial,andthespecificnatureoftheorderingorlackthereof.However,Figure10.23showsatypicalelectronicenergybandstructureforanamorphousmaterialthatcontainsmanyofthefeaturesofimportance.DuetothechemicalbondingandtheshortrangeorderingthereisaVBandCBwithextendedstates,asindicatedbytheenergyatthebandedges,EVBandECB.Incrystallinematerials,thedensityofstatesgoestozeroforboththeVBandCB.However,becauseofthelongrangedisorderthisdoesnotoccur.Instead,thereareallowedstatespenetratingintotheenergygap,andthisiscalled“bandtailing.”InbetweenEVBandEmandinbetweenECBandEmarelocalizedstatesduetothelackoflongrangeorder.Theuseofthe“m”subscriptreferstothemobilityoftheselocalizedstates,whichappearssimilartothatofthedelocalizedstatesbut,becausetheyarelocal-ized,electrontransportisviahoppingfromstatetostate.Thisconductionmechanismtypicallydisplaysasignificantlylowermobility,ascomparedwithelectrontransportthroughthequasi-continuousstatesintheallowedenergybandsincrystallinematerials.Thus,thedifferenceinenergybetweenthetwoEmenergiesindicatesamobilitygap.Inthemobilitygapthereisalsothepossibilityoflocalizedstatesduetodefects,so-calleddefectstates.Inmanycasesthestateshaveapeakandamid-gappeak,asindicatedinFigure10.23.TheexampleshowninFigure10.23isonlyshowntoindicatethekindsofstatesthatcanoccur,andindeedforwhichexperimentalevidenceexists.Aspecificmate-rialwithaspecificlevelofdisorderanddefectscangreatlyalterthelevelandpositionsofthelocalizedstates.不得转载谢谢合作LWM 266ELECTRONICPROPERTIESOFMATERIALSRELATEDREADINGD.A.Davies.WavesAtomsandSolids.Longman,London.Awell-writtentextcoveringmanyofthetopicsinChapters9,10,and11withgoodinsights.R.E.Hummel.ElectronicPropertiesofMaterials.Springer-Verlag,NewYork.Thistextprovidesawell-writtencoverageofthematerialinChapters9,10,and11attheappropriatelevel.Theauthorhasusedthisbookasatextfortheelectronicmaterialspartofthematerialssciencecourse.J.P.McKelvey.SolidStatePhysicsforEngineeringandMaterialsScience.Krieger.1993.HigherlevelthanHummel,well-written,readable,andforthetopicscoveredmorecomplete.M.A.Omar.ElementarySolidStatePhysics.AddisonWesley,Reading,MA.AtextthatcoversmanyofthetopicsinChapters9,10,and11andalsomanymoretopicsnotcoveredinthepresenttext.Areadabletextonthesubject.EXERCISES1.Calculateforasemiconductor,withEg=0.1,1and10eV,thetemperatureatwhichthereisa10%probabilitythattheelectrontobeintheCB.2.Explainwhyahalf-filledvalencebandyieldsamaterialwiththehighestconductiv-3ity.Inyourresponseexplainwhytheconductivityisnothigherfor,say,a–filledVB.43.Explainthedifferencebetweenelectronandholeconduction.4.Whyareelectronsandholesinequalnumbersinanintrinsicsemiconductor.Howcanthisbalancebecomealtered.5.Calculatethenumberofelectronsintheconductionbandofasemiconductorwithabandgapof0.1,and1and10eVatroomT.Whatassumptionshaveyoumadeaboutthemassoftheelectrons.6.Repeatproblem5for500K.Discussthedifferencesintheresultsfromexercises5and6.7.DiscusswhentheclassicalandQMtheoriesforelectricalconductionyieldthesameanddifferentresults.8.Explainwhydopant-producedstatesarelocalizedstateswhiletheallowedstatesintheVBorCBareextendedordelocalizedstates.9.Explainhowasubstitutionalimpuritycanactasadopant.10.ExplainhowaCooperpairformsandleadstoasuperconductingbandgap.11.ExplainhowaCooperpaircanleadtosuperconductivity.12.Explainwhyasapracticalmatteritisinsufficienttomeasureonlyrwhendeter-miningTc.Whatothermeasurementscanbedonetoconfirmsuperconductivity.13.Explainwhyamorphousmaterialsexhibitanenergybandstructuresimilartocrys-tallinematerialsinsomefeaturesbutdifferentinotherfeatures.Listthesimilaritiesanddifferences.14.Explainhoppingconductioninamorphousmaterials.不得转载谢谢合作LWM EXERCISES26715.(a)UsingsketchesofvelocityspaceforelectronsthatincludetheFermivelocity,explainhowelectronicconductionoccursinthepresenceofanelectricfield.(b)Usingthesketchesin(a)discussthedifference(s)fromtheclassicalmodel(Drude)forconduction.16.(a)Sketchtheparallelbandstructureforap-typeextrinsicsemiconductorshowingboththeintrinsicFermilevel(Ei)andtherealFermilevel(EF).(b)OnthissketchshowtheevolutionofbothFermilevelswithincreasingtemperatureanddiscussbrieflywhythesechange(s)ifanyareoccurring.(c)Brieflydiscussthedifference(s)inthetemperaturebehaviorofsemiconductorandmetalconductivity.17.CalculatetheprobabilityforanallowedelectronstatetobeoccupiedbelowtheFermienergyat0K.-18.Discusstwowaystopromoteaneintotheconductionbandofaninsulator.19.Discusswhatinformationismissingfromtheparallelenergybandpicture.不得转载谢谢合作LWM 不得转载谢谢合作LWM 11JUNCTIONSANDDEVICESANDTHENANOSCALE11.1INTRODUCTIONTheworkinthefieldofelectronicdeviceshascoveredvastterritory.ConsequentlytheintentinthischapteristousethebasicideasofelectronicstructureandpropertiesfromChapters9and10andprovideafirstlevelofunderstandingabouthowtheselected,importantdevicesoperate.Amongthesewillberectifiers,solarcells,andtransistors,includingthosekindsoftransistorsthatpresentlydominatecomputertechnology.Amajorityoftheseimportantdevicesoperatebasedonformingjunctionsbetweenmate-rials.Thereforeitisusefultofirstconsidertheelectronicconsequencesofjoiningmate-rialstoformjunctions.Itwillbeseenthatthedevicesdiscussedinthischapteroperateontheprinciplesdevelopedforjunctions.Electronicdeviceshaveevolvedconsiderablysincethevacuumtubesuptothe1960s,throughtransistorsinthe1970s,tochipsthatcontainhugenumbersofdevicesthatareintegratedtoacommonpurposesuchasmemoryorlogicincomputers.Allthisevolu-tionhasoccurredthroughthedownsizingofdevicesizethathasbeendrivenlargelybytheneedforfastermorecapablecomputers.Thetechnologyhasspilledoutfromcom-puterstootherareassuchasbiotechnology,medicine,sensors,communications,andphotonics.Nowwestandonanotherthresholdinsizethatisatthescaleofmoleculesandsmallgroupsofatoms,thenanoscale.Recentmaterialsdiscoveriesofnanoscalematerialssuchasbuckyballs(C60),carbon(andother)nanotubes,quantumdots,andquantumwireshaveledresearcherstothinkaboutnewdeviceswherethesmallsizedic-tatesthedevicecharacteristics.Thenanoscaleareahasjustbegun,sotheoutflowofcon-sumerproductsismodestatthistime.However,foranelectronicsmaterialsscientistitisatimeforconsideringhowthenanoscaledevicewillimpactelectronicstechnology.Tothisendanintroductorysectiononnanotechnologyandnanodevicesisincluded.ElectronicMaterialsScience,byEugeneA.IreneISBN0-471-69597-1Copyright©2005JohnWiley&Sons,Inc.不得转载谢谢合作LWM269 270JUNCTIONSANDDEVICESANDTHENANOSCALE11.2JUNCTIONSInthissectionthecharacteristicsofmetal–metal,metal–semiconductor,andsemicon-ductorjunctionsarediscussed.Thefocusisonthebehavioroftheelectronswhentwomaterialsarebroughttogether,inparticular,weconsidertheelectronsinthevalenceandconductionbandsthatdeterminetheelectronicproperties.11.2.1Metal–MetalJunctionsTounderstandjunctionsandtheeventsthatoccurwhenmaterialsarebroughttogether,wecommencewiththeparallelbandpictureasshowninFigure11.1afortwometals.Inthispicturethemetalsdisplayapartiallyfilledvalenceband,withEFshowingtheapprox-imatepositionofthehighestenergyelectrons.OntheleftisonemetalM1andontherightanothermetalM2,andeachhasadifferentEF.TheenergyrequiredtomoveanelectronfromEFtoinfinitedistancefromthemetaliscalledtheworkfunctionfM,andalsocalledtheionizationenergy.Thezeroofenergyatinfinitedistanceisoftentimescalledthevacuumlevel.Thustheelectronsboundinamaterialareatnegativeenergieswithrespecttothevacuumlevel.Whenthesetwometalsarejoined,electronsfromthemetalwiththehigherEF,metalM2,flowtoM1,whichhasalowerEF.Electronsendeavortoachievethelowestallowedenergies.ThusmetalM2thatwasneutralbeforejoiningtometalM1,aswasM1beforejoining,nowbecomescharged.M1,whichgainselectrons,a)VacuumLevelf2f1electronEflowF2EF1M2M1b)VacuumLevelEFMM12Figure11.1(a)TwoseparateddifferentmetalswithFermilevels(EF)andworkfunctions(f)indicated;(b)thesamemetalsasin(a)butafterjoiningandatequilibrium.不得转载谢谢合作LWM 11.2JUNCTIONS271becomesnegativewhileM2,whichloseselectrons,becomespositivelycharged.LikewiseEF1forM1riseswhileEF2forM2drops.Whenalltheelectronsflowfromhighertolowerenergy,equilibriumwillresultwheretheEF’swilllevelatEF1=EF2=EFasisshowninFigure11.1b.ThedifferenceinpotentialgeneratedbetweenM1andM2asaresultoftheequilibrationoftheFermilevelsiscalledthecontactpotentialf1–2andisgivenasfff=-(11.1)12-12BecausethecontactpotentialresultsinnodifferenceinFermilevels,itisdifficulttomeasureexperimentallyasapotentialdifferenceinanexternalcircuit.Alsothechargeonthemetalscannotcreateanelectricfieldinthemetals.However,onewaytomeasurethecontactpotentialistofirstbringthemetalsincontact,soastoenablechargeexchange,andthenseparatethebarsasshowninFigure11.2.Thenwiththeseparatedbarsheldinvacuum,anelectronbeamisprovidedtomovethroughthegapinbetweentheseparatedmetals.Thecontactpotentialf1–2willcausetheelectronbeamtodeviateinproportiontothemagnitudeoffM1–M2.Anothermorepracticalmethod,calledtheKelvinmethod,istobringtwodifferentmetalsincontactandthenseparatethem,butwithcloseproximity.Oneofthemetalsisshapedasasharptipandhasaknownworkfunction.Thecontactwillenablechargeexchange,andthenwhenthemetalsareseparated,themetalwiththesharptipismechan-icallyvibrated.Thechargemotioncausesanaccurrentthatcanbereducedtozerobytheapplicationofanexternaldcpotentialthatisexactlyequaltothecontactpotential.Then,withavalueforthecontactpotentialobtainedfromthecurrentnulling,andwithonemetalworkfunctionknown,theothermetalworkfunctioncanbecalculatedusingequation(11.1).11.2.2Metal–SemiconductorJunctionsTheessentialdifferencebetweenmetal–metalandmetal–semiconductorjunctionsisthatthesemiconductor(andalsoforaninsulator)unlikeametalcansupportaninternalelec-tricfieldasaresultofextrachargebeingpresent.Theregioninthesemiconductorwherethefieldisgeneratediscalledthespacechargeregion,andtheevolutionofthespacechargeregioncanbeunderstoodasshowninFigure11.3.Figure11.3agivesaparallelbandpictureofametalontheleftandaN-typesemiconductorontheright.Thesemi-conductorhasaworkfunctionfSdefinedasthedistanceinenergyfromEFtothevacuumlevelasforametal.InthiscaseforanN-typesemiconductor,EFisbetweenthedonorlevelandtheconductionband(labeledasEFS).Thedistancebetweentheconduction-+-+-+M-+M1-+2-+-+-+ElectronbeamFigure11.2Twodifferentmetalspreviouslyincontactandequilibrated,andnowseparatedandwithadirectedelectronbeam.不得转载谢谢合作LWM 272JUNCTIONSANDDEVICESANDTHENANOSCALEa)VacuumLevel++fcCBS+++electronEfMFSflowEDEFMMSVBSpaceChargeb)SurfaceRegionPotentialBarrierS-MBarrierM-S{}f-fMSf-cM{EFMSFigure11.3(a)Aseparatedmetal(M)andN-typesemiconductor(S)withwithFermilevels(EF),workfunc-tions(f),electronaffinity(c),bandedges(EvbandEcb),anddonorlevel(ED)withEFMfS.Therearethreeothercasestoconsider:fMfSandfMfSandfMfSshowstheelectronenergybandsinthespacea)VacuumLevelfelectronEMFMflowfSCBEFSEMDSVBb)SpaceChargeSurfaceRegionPotential{BarrierM-S{EFMSFigure11.4(a)Separatedmetal(M)andN-typesemiconductor(S)withFermilevels(EF),workfunctions(f),bandedges(EvbandEcb),anddonorlevel(ED)withEFM>EFS;(b)MandSaftercontactandequilibrationwithbandbendingandthedevelopmentofaspacechargeregion(cross-hatched)andasurfacepotential.不得转载谢谢合作LWM 274JUNCTIONSANDDEVICESANDTHENANOSCALEa)VacuumLevelfCBSfMEelectron++Aflow+++EFSEFMVBMSb)SpaceChargeSurfaceRegionPotential{EFElectronBarrierElectronBarrierM-S{}S-MMSFigure11.5(a)Separatedmetal(M)andP-typesemiconductor(S)withFermilevels(EF),workfunctions(f),bandedges(EvbandEcb),andacceptorlevel(EA)withEFMEFS;(b)MandSaftercontactandequili-brationwithbandbendingandthedevelopmentofaspacechargeregion(cross-hatched)andasurfacepotential.side,themajoritycarriersonthePsidearealsoreduced.ThustheregioninbetweenthePandNsemiconductors,thejunctionregion,isdevoidornearlydevoidofcarriers.Thisregioniscalledthedepletionregion.ElectronsfromtheN-typesidehavealargebarriertoovercome,inordertotraversethebarrier,asdoholesfromtheP-typeside.Thusitwouldbedifficultforcurrentflowineitherdirection.AtfirstglanceaPNjunctiondoesn’tseemveryuseful,butactuallythiskindofjunctionisattheheartofthepresent-daymicroelectronicsindustry,andthePNjunctionappearsinmanydevices.However,torenderthisjunctionuseful,typicallyanexternalpotentialisappliedthatiscalleda“bias.”Figure11.8ashowsthesamePNjunctionasinFigure11.7butwithaforwardbias,thatis,apositive(+)potentialontheP-typeandanegative(-)potentialontheN-type.Theeffectofthebiasistopushthemajoritycarrierstogether,reducethedepletionwidth,andreducethebarriers.TheoppositeoccurswithreversebiasasisshowninFigure11.8b.Aswewillseebelowfordevices,thebiasgivesrisetorectifierdevicesandwhencom-binedwithanotherPNjunctionresultsinatransistor.11.3SELECTEDDEVICESSimplyexpressed,electronicdevicesaredevicesthatdosomethingusefulusingelectricalcurrentorpotential.Somedevicesperformausefultasksimplybytheoutputofausefulpotential.Thermocouplesthataremetal–metaljunctionsfallintothiscategoryofpassive不得转载谢谢合作LWM 276JUNCTIONSANDDEVICESANDTHENANOSCALEa)VacuumLevelCBfelectron++fSCBSflow+++EFSEAEDEFSVBP-SVBN-Sb)CBEFVB{DepletionWidthFigure11.7(a)SeparatedP-andN-typesemiconductorswithFermilevels(EF),workfunctions(f),bandedges(EvbandEcb),anddopinglevel(EAandED);(b)P-SandN-Saftercontactandequilibrationwiththedevelopmentofaspacechargeregion(cross-hatched),thatis,depletedofcarriers.devices.Aswillbedescribedbelow,thermocouplestakeadvantageofthetemperaturedependenceofthecontactpotential.Thermoelectricdevicesareanothertypeofpassivedevice,thatusesacurrenttoperformcooling,inwhichcasethedeviceembodiesather-moelectricrefrigerator.Ontheotherhand,thereareactivedevicesthatswitchacurrent,calledtransistors,andthereareactivedevicesthatenhanceapotential,calledamplifiers,andthatlimitcurrenttoflowinonedirection,so-calledrectifiers.Furthertherearedevicesthatconvertoneformofenergyintoanother.Forexample,adevicethatcon-vertsphotonenergyintoanelectricalcurrentiscalledaphotocell.Allofthesolidstateversionsofelectronicdevicesbothpassiveandactivetakeadvantageofmaterialsprop-erties.Alimitedsampleofactiveandpassivedevicesisincludedinthefollowingsec-tions,inordertodemonstratethecleverwaysthattechnologymakesuseofthesolidstateproperties.11.3.1PassiveDevicesAthermocoupleconsistsofmetal–metaljunctions.Aswasdiscussedabove,whentwometalsarejoinedacontactpotentialdevelops.Thecontactpotentialchangeswithtem-perature;thatis,thecontactpotentialdisplaystemperaturedependencethatisamani-festationoftheSeebeckeffect.Essentially,ifabarofmetal(orsemiconductor)initiallyatthermalequilibriumisheatedatoneend,theelectronsinthehotendwillattaina不得转载谢谢合作LWM 11.3SELECTEDDEVICES277a)CBVBb)CBVBFigure11.8(a)JoinedP-andN-typesemiconductorswithforwardbias;(b)joinedP-andN-typesemi-conductorswithreversebias.higherkineticenergyandhighervelocitythanthoseinthecoolend.Consequentlyelec-tronswillflowfromthehottocoldendsonaverage,causingaSeebeckpotentialVStodevelopthatopposeselectronflow.Thesizeofthepotentialwillvarywithtemperatureandmaterial,andtheconstantofproportionalitybetweenthechangeintemperatureDTandthechangeincontactpotentialDViscalledtheSeebeckcoefficientforagivenmaterial.Toproduceameasurableoutput,twojunctionsbetweendifferentmetalsAandBareformedasshowninFigure11.9.Thethermocouplejunctionisindicatedbyabeadthatisoftentheresultofspotweldingwiresoftheappropriatemetals.OnejunctionisheatedtoTh,andtheotherisheldatafixedtemperatureforreference,Tref.Attheheatedjunc-tioneachmetalwillexhibitadifferentSeebeckpotential.Thedifferenceinpotentialbetweenthehotandcoldjunctionisameasureoftherelativejunctiontemperature,providedthatthejunctionscanbecalibratedagainstaknowntemperaturestandard.Tablesofpotentialvaluesversustemperatureexistformanycommonmetalsandalloysfromwhichthermocouplescanbefabricatedbyspot-weldingwirestogether.Asetoftwojunctionsiscalledathermocouple,andiscommonlyusedtomeasuretemperature.ThetypicaloutputforthermocouplesisinthemVrangeandiseasytomeasureaccu-rately.Onewaytouseathermocoupleistoplaceonejunctionintoanovenforwhichthetemperatureisdesired,andtheotherisimmersedinaconstanttemperaturebathsuchasicewatertoprovideastablereferencepotential.Modernthermocouplesgeneratethepotentialoftheicebathfordifferentcombinationofmetalsbysimplyusing不得转载谢谢合作LWM 278JUNCTIONSANDDEVICESANDTHENANOSCALEThTrefMetalBMetalAVFigure11.9ThermocoupleformedwithtwodissimilarmetalsAandBwithtwoABjunctions(·).Onejunc-tionisatahighertemperature(Th)thantheother(Tref).MMLRP-Type+holeflow-electronflowFigure11.10Peltiereffectillustratedusingtwometal(MLandMR)junctionstoaP-typesemiconductor,withcurrentflowinthesemiconductorandexternalcircuitindicated.aprecisionhigh-impedancevoltagesource.Thusonlyonerealmetal–metaljunctionisnecessary.TheSeebeckeffectyieldsanelectronflowasaresultofatemperaturedifferenceorgradient.Theinverseofthatistoprovideanexternalcurrentsourcefromapowersupplytosustainacurrentthatpusheselectronsfromthecoldtothehotendofabarofmetal.Thiswaycoolingoftheheatedendofthebarwilloccur.AlsothepotentialcouldbereversedsothatthecurrentflowsinadditiontotheSeebeckcurrenttoproduceacceler-atedheatingofthecoolendofthebar.TheheatingandcoolingassociatedwithcurrentflowiscalledtheThompsoneffect.Whenajunctionisinvolvedtheheatingorcoolingeffectcanbemuchlarger,andiscalledthePeltiereffect.Figure11.10illustratesthePeltiereffectwithmetal–semiconductorjunctions.ThisfigureshowsaP-typesemiconductorwherethemajoritycurrentcarriersareholesandtherearetwometalsemiconductorjunc-tionsMRandML.Intheexternalcircuitthecurrentiscarriedbyelectronswithaflowasshowninthefigure,andthatderivesfromthedirectionofthedcpowersupply.Forcurrenttoflow,theremustbeachangeincarriersnearthecontactelectrodesandarecombinationofelectronsfromthemetalwithholesfromthesemiconductor.Electronsatthepowersupplypotentialareinjectedintothesemiconductorattherightsidecontact,andtheseelectronscombinewithholesandlosetheirenergytothesemiconductor.Thusthesemiconductorgetshotterneartheright-handcontactMR.Attheleft-handcontactMLelectronsandholesrecombine,andtheenergyoftheholesisremovedbythecurrentflowintheexternalcircuit.Hencecoolingoccursattheleftsidecontact.ThusthePeltiereffectgivesrisetotheconstructionofathermoelectricrefrigeratorand/orheater.不得转载谢谢合作LWM 11.3SELECTEDDEVICES27911.3.2ActiveDevicesDevicesthatcontrolandadjustcurrentinacircuitarehereintermed“active”devices.Aswillbediscussedbelow,thedeviceschosenfordiscussionareamongthemostcom-monlyfounddevices,andtheyareallbasedonjunctionsfortheiroperation.11.3.2.1RectifiersRectifiersaredevicesthatpermitcurrenttoflowpredominantlyinonedirection.Inthiswayanaccurrentcanbetransformedintoacurrentthatflowsinonedirectionoradccurrent.Typicalsolidstaterectifierscanbedesignedusingmetal–semiconductorandsemiconductor–semiconductorPNjunctions.Manyrectifiersaretwo-terminaldeviceswithoneterminalatthePandtheotherontheNtypesemiconductororonthemetal,andareconsequentlycalleddiodes.Formetal–semiconductorjunctionswereferbacktoFigure11.3,andthefactthatthecurrentineitherdirectionisgivenbyexponentialequation(11.2).Startingfromequa-tion(11.2),wecanwriteanexpressionforthecurrentfromMtoSasfollows:--()ffMSkTIA=e(11.3)MSInequation(11.3)weignoredanydifferencebetweenfSandc.Likewisewecanwriteforthereversedirection,--()ffSMkTIA=e(11.4)SMThusthenetcurrentInet=ISM-IMS=0.Thissituationismarkedlychangedwiththeapplicationofabiasvoltage.Firstweapplyaforwardbiastothemetal–semiconductorjunctionshowninFigure11.3,whichlowersthebarrierfromStoM.SincethisisanN-typesemiconductor,foraforwardbiasweapplyanegativepotentialtothesemi-conductor,andthecurrentISMismodifiedas---()ffSMeVkTIA=e(11.5)SMThenetcurrentisnolonger0andcanbewrittenasthedifferencebetweenforwardandreversecurrents:eVkTII=-()e1(11.6)net0whereI0iscalledthesaturationcurrent.Thisformulaisoftenreferredtoastherectifierformula.ThusitisseenthatforthebiaspotentialV=0,Inet=0asshownbefore.For+V,InetrisesexponentiallywithVandfor-V,InetisessentiallyI0orthesaturationcurrent.TheseconditionsareshownforanidealrectifierinFigure11.11a.SimilarlyaPNjunctioncanalsobeusedforrectification.AswasshowninFigures11.7and11.8,thedepletionregionprovidesabarrier,andforwardandreversebiascanalterthebarrierheight.Alsowecananticipatetheuseofarectifierformulasuchasequa-tion(11.6)developedabove.However,inaPNjunctionwehavetokeeptrackofbothmajorityandminoritycarriersinboththePandNsidesofthejunction.IfaforwardbiasVisappliedtothePNjunctionasshowninFigure11.8awithapositive(+)poten-tialonPandanegative(-)onN,themajoritycarriersthatwereinequilibriumontheirrespectivesidesarepushedintothejunctionareaandcombinewiththeiropposite,whileminoritycarriersoneachsideofthedepletionregionofthejunctionwillriseinanexpo-不得转载谢谢合作LWM 280JUNCTIONSANDDEVICESANDTHENANOSCALEa)II0-V+Vb)I-V+VAvalancheRegionFigure11.11(a)Idealdiodecurrent(I)versusappliedvoltage(V)characteristicshowingcurrentinonedirection;(b)amorerealisticdiodecharacteristicshowingsomereversecurrentandavalanchebreakdowninthereversecurrentregion.eV/kTnentialwayasafunctionofV.TheholesontheNsiderisetopNandelectronsoneV/kTthePsiderisetonPe.Thesearenonequilibriumvaluesandaregreaterthanthevaluesinthebulkofthesemiconductoroneitherside.Consequentlytheseexcessminoritycar-rierswilldiffuseintothebulk.RecallfromChapter5ondiffusionthattheminoritycarriersdiffusewithacharacteristiclengthL,asisgivenbyaslightlymodifiedequation(5.113):1212LD=()ttandLD=()(11.7)pppnnnWhereratherthanadiffusiontimetbeingused,trepresentsthelifetimefortheminor-itycarrier.TheexcessofminoritycarriersisthedifferenceinthenumberwithandwithoutanappliedpotentialV,andthevaluesareexpressedas不得转载谢谢合作LWM 11.3SELECTEDDEVICES281eVkTeVkTppp=-andnn=-en(11.8)excessNNexcessPPThehole(minoritycarrier)quantitiesatadistancexintheNandPsidesofthejunc-tioncanbeexpressedaseVkT-xLeVkT-xLpx()=-()eNNpe()pN+pandnx()=-()nePneP()nP+n(11.9)ThecurrentdensityJforholesandelectronscanbecalculatedfromadiffusionformula(equation5.1or5.11)asdpeDpPneVkTdneDpnpeVkTJe=-D=()eJ-11and=-eD=()e-(11.10)pPnndxLdxLPnThisyieldsanetcurrentfluxintheformoftherectifierformula:ÏeDpPneDpnp¸eVkTJnet=+Ì˝()e-1(11.11)ÓLPLn˛FinallythesaturationcurrentisgivenasÏeDpPneDnnp¸J0=+Ì˝(11.12)ÓLPLn˛Figure11.11bdisplaysmorerealisticrectifierbehaviorwithaninitiallyslightlychang-ingcurrentinthereversebiasregime.Afteralargereversebiasasuddenandlargeincreaseinthereversebiascurrentisobserved.Inthishigh-currentregionelectronsareacceleratedbytheappliedpotential.Therapidlymovingelectronscausemoreelectronstogainenergyandcontributetothereversecurrent.Thisissometimescalledimpactionization,anditreferstotheionizationcausedbyelectronimpact.Ultimatelythiscas-cadingeffect,calledanavalanche,createshighcurrent.Inthisregionthehighcurrentscancausepermanentdamagetothematerial,andthustoabreakdown.Theavalancheeffectthatleadstobreakdowninlightlydopedsemiconductorscanbetoleratedandren-deredindefinitelyusefulinheavilydopedsemiconductors.ThiskindofrectifieriscalledaZenerdiodeorZenerrectifier.Therectifiercanbeusedasavoltagereferencebecauseitbreaksdownatprecisevoltagesthatdependonthedoping.TherectifierformulaforPNjunctionsassumesthatthetransferofchargeisviadif-fusionofcarriers,electrons,andholes.However,forveryheavilydopedPandNsemi-conductorsthejunctionformedwillhaveathindepletionwidthoftheorderofafewnanometers.Withhighappliedbiaspotentials,theoccupiedelectronbandscanbecomealignedwithemptybandsontheothersideofthejunction.Inthiseventandwithathinbarrierordepletionregion,electrontunnelingcanoccur.Theresultingrectifieriscalledatunnelrectifierordiode,andsometimesanEsakidiodeafteritsinventor.Theopera-tionofatunneldiodecanbeunderstoodwiththeuseofFigure11.12.Figure11.2aillus-tratesthebandstructurewithnobiasappliedtoaheavilyP-andN-dopedjunction.Theheavydopingcreatesanarrowdepletionwidththatactsasatunnelingbarrier.Withnoappliedbias,thereisnocurrentflowandthisisthepointontheI-versus-Vplotcorre-spondingtoa)inFigure11.12e.EFisindicatedbythedottedlineineachframeofFigure11.12.Notethatfilledelectronstatesarenotinenergyproximitytoemptystatesacross不得转载谢谢合作LWM 282JUNCTIONSANDDEVICESANDTHENANOSCALEa)PNb)PNtunnelingEFEFV=0V=-c)PNd)PNelectronflowtunnelingEEFFholeflowV=+V=++tunnelingreducede)Id)c)tunnelingceased-V+Vb)normaldiodea)Figure11.12HeavilydopedPNjunctionwithdifferentbiasstatesillustratingtunneldiodeoperation:(a)Noappliedbiasandnotunneling;appliedreverse(b)orforward(c)biasyieldsfilledstatesaboveemptystatesandtunnelingoccurs;(d)tunnelingceasesforalargerforwardbias;(e)theI-Vcharacteristicforatunneldiodewiththebiasconditionsindicatedandwithanidealdiodecharacteristic.thenarrowdepletionzone.However,wheneitheraforward(Figure11.12c)orreverse(Figure11.12b)biasisapplied,occupiedstatesalignwithemptystatesandwiththenarrowbarrier,soelectrontunnelingcanoccurinthedirectionindicatedbythearrowsinFigure11.12band11.12c.ThetunnelingcurrentisindicatedontheI-versus-Vcurve(Figure11.12e)aspointsbandcforreverseandforwardbias,respectively.Asmoreforwardbiasisapplied,thebandsreachoptimumalignment,whichischaracterizedbyamaximuminthetunnelingcurrent.Afterthemaximum,thebandsbegintobemis-alignedwithincreasingforwardbias.Consequentlythetunnelingdecreases,andthecurrentdecreases.ThisdownwardslopeoftheI-versus-Vcharacteristiciscalledaneg-不得转载谢谢合作LWM 11.3SELECTEDDEVICES283ativeresistanceregion,sinceanapplicationofOhm’slawtothisregionrequiresa-R.Finally,withtheapplicationofalargeforwardbiasasshowninFigure11.12d(orreversebiasbutnotshowninFigure11.12),thefilledstatesarenolongeralignedinenergywithemptystates,andthustheonlymodeforcurrentflowisdiffusionofcarriersaswithanormal(nontunneling)PNjunction.ConsequentlytheI-versus-VcharacteristicreturnstothenormaldiodecaseasshownbypointdinFigure11.12e.Althoughnotcoveredhere,thenegativeresistancecharacteristicsofthetunneldiodecanbeexploitedinoscillatorandamplifiercircuits.11.3.2.2PhotocellsAnotherimportantkindofPNjunctiondiodeisaphotodiode.Figure11.13ashowsthecrosssectionofaPNjunctionwithathinPregionsoastopermitlighttopenetratetothedepletionregion.Thephotonswithenergygreaterthanthebandgapcancreateelectron-holepairsbypromotingvalencebandelectronstotheconductionbandleavinganequalnumberofholesbehind.Theelectron-holepairscreatedawayfromthedepletionwidthareoflittleconsequenceandeventuallymerelyre-combine.However,asisdepictedinFigure11.13b,theelectron-holepairscreatedinthedepletionwidthareofgreatinterestbecausetheelectronscanfall“downhill”intotheCBoftheN-typesemiconductorwhiletheholesflow“uphill”totheP-typesideofthejunction.Thiswaycurrentiscreatedandphotonenergyiseffectivelytransducedintoelectricalenergy.Tooptimizeaphotodiode,itisbesttooperateinthereversebiasregion.Inthiscasethedownhillshapeoftheenergybandsforbothelectronsandholesmaximizesthecurrent.Alsotheonlycurrentflowinginreversebiasisthesaturationcurrent.Aswasmentionedabove,thethicknessofthetoplayerneedstobeminimizedtoallowmaximumphotonfluxtothedepletionregion,andtheareaneedstobemaximizedtoharvestasa)hnDepletionPPRegionNb)PNCBe-’sEFVBh+’s{DepletionRegionFigure11.13(a)PhotocellPNjunctionwithathinP-typeregiontoallowlightpenetration;(b)thejunctiondepletionregionwherephoto-createdelectron-holepairsareseparated.不得转载谢谢合作LWM 284JUNCTIONSANDDEVICESANDTHENANOSCALEmuchofthelightaspossible.Thedepletionregioniswheretheusefulelectron-holepairsarecreatedbythephotons.Thisregioncanbemaximizedbyusinglightlydopedsemi-conductorsandevenbytheuseofanintrinsicsemiconductorthatisinsulatinginbetweenPandNregions.SuchadiodeiscalledaPINdiode.Thecurrentproducedcanbefedtoaresistorwhereapotentialcanbemeasured.11.3.2.3TransistorsTransistorsarealsocomprisedofjunctions.Typicallytransistorsaresolidstatedevicesthatcancontroltheflowofcurrent.AgoodanalogyisawatervalveasshowninFigure11.14.Watercurrentflowsintothevalvebodyandout,asisindicatedbythearrows,andthecontrolofthewatercurrentisbymeansofavalvethatcanturntheflowonorofforatvariousflowrates.ThesimplewatervalveshowninFigure11.14isathree-terminaldevice:in,out,andcontrol.Atransistorisalsoathree-(atleast)terminaldevicethatcontrolstheelectricalcurrentinandout(twoterminals)bymeansofathirdterminal.Herewediscussthreetypesoftransistors:thebipolartransistorthatusestwocarriertypesforoperation,electronsandholes;themetaloxidesemiconductorfieldeffecttransistor(MOSFET)thatusesminoritycarriersandispresentlytheheartofthecomputerindustry;andanovelorganictransistorthatappearstooperatelikeaMOSFETbutactuallyisdifferentandoperatesusingmajoritycarriers.Thisorganictransistoriscalledathinfilmtransistor,TFT.BipolarTransistor.Figure11.15ashowsabipolartransistorasconsistingoftwoPNjunctions.Specifically,thisisaNPNtransistorconsistingoftwoPNjunctionswithacommonPregionsandwichedinbetweentwoNregions.ItisalsopossibletofabricateaPNPtransistorwithtwoPregionsandaNregionsandwichedinbetween.Thethreeregionsinthiskindoftransistorarecontactedandthethreecontactsarecalledtheemitter(e),thebase(b),andthecollector(c).AsbeforeforPNjunctions,thedepletionregionsareshownascrosshatchedareas.Figure11.15bshowstheenergybandstructureforthiskindofdevicewithnoappliedbias.Therearetwodepletionzonesatthetwojunctions,andthesedepletionzonescausetheelectronenergybandstorisefromtheNtoPregions.Theserisesimpedetheflowofelectronicchargeintothedevicefromemittertobaseandfromcollectortobase.Bymeansofjudiciouslyapplyingexternalbiastothejunctions,variouskindsofdeviceoperationcanbeobtained.Forexample,acommonapplicationistoamplifyanelectricalsignal.Wecanimagineanaudiosignal(music)beingtransducedbyamicrophonetoelectricalsignalsofthesamefrequencydistribution.Theelectricalsignals,electrons,areplacedontotheemitterofourNPNbipolaramplifier.TheNPNamplifierissetupwithaforwardbiasede–bjunctionandareversebiasedc–bjunctionasisshowninFigure11.15c.Theincomingcurrentislargelydroppedasapotentialacrosstheresistanceofthee–bjunction.However,aftertheslightlyuphilljourneyofelectronsinjectedintothee–bjunctiondepletionregion,those“lucky”electronsthatdriftnearthecenterofthejunctionwill“feel”thelargedownhillControlInOutFigure11.14Watervalveasanexampleofathree-terminaldevice:in,out,andcontrol.不得转载谢谢合作LWM 11.3SELECTEDDEVICES285a)ebcNPNb)CBVBc)CBVB}}ForwardReversee-bBiasc-bBiasFigure11.15(a)NPNbipolartransistorshowingtwoPNjunctionsandthethreeterminals:emitter(e),base(b),andcollector(c);(b)theenergybandstructureofanunbiasedNPNtransistorshowingthedepletionregions;(c)theNPNtransistorwithforward-biasedemitter-baseterminalsandreverse–biasedcollector-baseterminals.potentialcreatedbyvirtueofthereversebiasatthec–bjunction.Thisisastronglyforwardbiastoanelectronenteringfromthebsideofthec–bjunction.This“lucky”electrongainsenergy,anditsenergyisamplified,andifdesired,amplifiedtotheextentnecessarytodrivespeakersthattransducetheelectricalsignaltoanacousticsignal.Ifwevarythereversebiasonthec–bjunction,wecanvarythevolumeofthefinalacousticsignal.Theemitterregionisusuallyheavilydopedtoenhancetheoperationofthisamplifier,therebyincreasingthenumberofcarriersdriftingintothebaseregion.Alsothebaseismadethinsoastoimprovethechancesofaninjectedelectroninreachingthedownwardpotentialtopropelittothecollector.Thegainofanamplifierisameasureofthesizeoftheoutputsignalpotentialrelativetoitsinputpotential.Thereareotherconfigurationsofamplifiersthatcanincreasetheoutputsignalandprovideregulation.不得转载谢谢合作LWM 286JUNCTIONSANDDEVICESANDTHENANOSCALETheyareallbasedonthetwo-junctionbiasideaspresentedabove.Inadditionitisstraightforwardtoconsiderthatthebasepotentialcanpreventemitter-injectedcarriersfromreachingthecollector.Forexample,asmallpositiveoranegativepotentialonthebasecanreducethepossibilityofelectronsdriftingtothecollector.Thusthebasepotentialcanturnthedeviceonbyallowingcurrenttoflowtothecollector,oroffbynotallowingcurrentflow.Thisisanexampleofabipolarswitch.Thiskindofswitchcanbefabricatedwithathinbaseregionandcanthusoperateveryfast,andincertaincircumstancesthisswitchisusedforfastlogicoperationsincomputers.Oldergenerationsofmainframecomputersusedbipolarswitchesforitsmostcompetentandfastestswitchingoperations.MorerecentlymuchofthemorecostlybipolarcircuitryhasbeenreplacedbylesscostlyconfigurationsofMOSFETdevices,asdiscussedbelow.Itshouldbenoticedthatabipolardeviceusesbothelectronsandholesforoperation.MOSFET.TheMOSFETisattheheartofpresent-daymicroelectronics,andthisdeviceisbasedonsiliconasthesemiconductormaterial.ThemainreasonforthischoiceofSiasthemainsemiconductormaterialusedinmicroelectronicshastodowiththeabilitytoreducesurfaceandinterfaceelectronicstatestoacceptablelevelsviafilmsofSiO2,whichalsoservesasadielectricinthedevices.Intrinsicinterfaceelectronicstatesarisefromtheterminationofacrystalatitssurface(moreonthistopicisbeyondthescopeofthistext).ThetypicalMOSFETisshownasFigure11.16a,whichspecificallydisplaysanN-channelMOSFET.AN-channelMOSFETisconstructedonP-typeSi.TheMOSFETdeviceisusuallyathree-terminaldevicewiththeterminalsnamedSource(S),Drain(D),andGate(G).Abacksidecontactissometimesused,butitisnotnecessaryfordeviceoperation.TheMOSFETismostoftenusedasaswitchwheretheStoDcurrentisswitchedonandoffusingthepotentialatGtoeffecttheswitching.ThechannelregionseparatestheN-typeSandD,andthisistheregionwherecarriersflowfromStoDtoturnonthedevice.NoticethattheN-typedopinginSandDis+labelednwherethesuperscript+indicatesheavydoping.InitiallytheN-typeSandDareseparatedbyaP-typeregion.Thus,ifoneinjectsamajoritycarrier,anelectron,fromSintothechannelforthepurposeofeffectingconduction,theelectronwilleventuallyre-combinewiththeholesintheP-typechannelregionbeforetheelectroneverreachesD.Ifanegativepotentialisappliedtothegate,-VG,themajoritycarriersthatareholesintheSisubstratewillbeattractedtothechannel.Thisconditioniscalledaccumula-tion,anditreferstomajoritycarriersthatwillaccumulateinthechannel.AccumulationfurtherpreventsS–Dcurrent,andthusmaintainstheoffstatefortheMOSFET.Now,astheVGismade,morepositivethemajoritycarrierholesarerepelledfromthechannel.Whenthenumberofholesinthechannelbecomeslessthantheequilibriumnumber,thechannelissaidtobedepleted.Theconditioninthechannelgoesfromaccumulationwith-VGtodepletionwithamorepositiveVG.AsVGgetsyetmorepositive,theminoritycarrierelectronsintheP-typeSiwilloutnumbertheholesthatarebeingdepleted.Theresultingconditioniscalledinversionbecausethemajoritycarriertypeinthechannel(andonlyinthechannel)hasbeenchangedfromholestoelectrons;namelythecarriertypehasbeeninvertedbyvirtueofchangingVG.TheimportantaspectofinversionisthatnowelectronsfromSinjectedintotheinvertedchannelcanmakeittoDandthusturnonthedevice.TheI-versus-Vcharac-teristicisshowninFigure11.16b,anditisseenthatfortheS–Dcurrent,ISDiszeroasVGincreasesbutthenrisessteeplywhenVGissufficientlypositiveinthisN-channeldevicetoeffectinversionandturn-onofthedevice.AVGvalueatwhichISDissufficientlyhightoindicateunambiguousturn-oniscalledthethresholdvoltageandlabeledasVT.The不得转载谢谢合作LWM 11.3SELECTEDDEVICES287a)GateGateSourceSourceDrainDrainMetalMetalGateDielectricGateDielectricn+n+n+n+ChannelRegionP-typeSiliconP-typeSiliconElectronsElectronsholesholesBackSideContactb)ISD0VVGTFigure11.16(a)N-Channelmetaloxidesemiconductorfieldeffecttransistor(MOSFET)withthreetermi-nals—source,gate,anddrain—andwithheavilydopedsourceanddrainregions;(b)thesourcetodraincurrent(ISD)versusgatevoltage(VG)characteristicfortheMOSFET,showingonandoffstatesatthethresh-oldvoltage(VT).onandoffstatesareallthatarenecessarytoperformcomputermemoryoperationsbasedonboolean1and0.AMOSFETcanbeconfiguredforcomputermemoryappli-cation,asshownschematicallyinFigure11.17.Inthisfigure,Disconnectedtoacapac-itorandthentoground.Bytheoperationofthegatethedevicecanturnon,asdiscussedabove.Withthedeviceon,ISDwillflowandchargethestoragecapacitor.Thedeviceisturnedonusingtheconnectiontotheso-calledwordline.Currentflowsfromthebitlinetochargethestoragecapacitor.Whetherthecapacitoralreadyhasastoredcharge(astoredbinarybit1)ornot(abinarybit0)canbesensedusingthewordlineagainandthebitline.Ifthewordlineturnsthedeviceon,thencurrentwillflowifthecapacitorisatbinary0.But,ifthestoragecapacitorisalreadycharged,nocurrentwillflowfromthebitline.Thiswaybooleanlogiccanbereadandwrittentomemory.Thiskindofmemoryiscalleddynamicrandomaccessmemory(DRAM),anditmustberefreshedbythepowersupply.Ofcourse,foraP-channelMOSFETdevicethatisfabricatedonN-typeSi,theturn-onwillbeatnegativevaluesofVT.SimilarlyaP-channelDRAMcanbefabricated.Both不得转载谢谢合作LWM 288JUNCTIONSANDDEVICESANDTHENANOSCALEWordLineGSDCapacitorBitLineGroundFigure11.17MOSFETconnectedtowordandbitlinesthatoperateandaccesscharge(information)storedonthecapacitor.P-ChannelPowerSupplySGDVinVoutDGSGroundN-ChannelFigure11.18P-ChannelandN-channelMOSFETsconnectedasavoltageinverter.TheMOSFETsoperateinacomplimentarymannerandthecircuitiscalledacomplimentaryMOSFET,orCMOS.N-channelandP-channeldevicesareusedformicroelectronicsapplications.TheN-channelMOSFETispreferredbecausethemobilityofelectronsinSiishigher,andthusthisdeviceisfasterthanthecorrespondingP-channelMOSFET.However,acommonmemorydeviceiscomposedofbothN-channelandP-channelMOSFETsconnectedtogethersothattheG’sandD’sareconnectedasshowninFigure11.18.Thisconfigu-rationwithbothkindsofMOSFETsiscalledacomplementaryMOSFETorCMOS.Withoutgoingintospecificoperatingconditions,thisdeviceoperatesbyplacingapoten-tialatVinandobservingeitheranonoroffstateatVout.WithVinsufficientlynegativetheP-channeldeviceisonandwhensufficientlypositivetheN-channelturnson.Whenoneison,theotherisoff.SowhentheP-channelisonwithVinnegativeandtheN-channelisoff,theoutputVoutgoestowardthepowersupplyvoltage,whichisusuallysomesmallpositivevoltage(e.g.,+severalvolts).WhentheP-channelisoffwithapos-itiveVinbuttheN-channelison,Voutgoesto0(groundpotential).Thus,alowVinyieldsahighVout,andahighVinyieldsalowVout.Thiscircuitiscalledaninverter,anditisusedasthebasisforstaticrandomaccessmemory(SRAM).SRAMdoesnotrequire不得转载谢谢合作LWM 11.3SELECTEDDEVICES289refreshinglikeDRAM.TherealadvantageofCMOSisthatitdrawsverylittlecurrent,andonlywhenthedeviceswitchesfromonestatetoanother.Thisiscrucialforhighdevicedensitymicroelectronicschipsthatcanbedamagedfromthegreatheatcreatedbyhighcurrentflow.OrganicTransistors.Atthepresenttimevirtuallyallimportantelectronicdevicesaremadeusinginorganicmaterialsastheactivecomponents.However,inthesedevicesoftenanorganicconstituentwillbeusedforonepurposeoranother.Alsodisplaydevicesoftenuseavarietyoforganicmaterialsforkeycomponents.Thereareamanygoodreasonsfortheuseoforganicmaterialstosubstituteforinorganicmaterialsinelectronicdevices.Onereasonisthecostoflargeareadevices,anotherisflexibility,andanotheriscompatibilitywithbiologicalsystems.Thustherehasbeenconsiderableactivityinbothresearchanddevelopmentaimedatorganicelectronics.Oneinterestingdevelopmentthathasledtoconsiderableresearchistheabilitytoprepareanorganicfilm-baseddevicethatoperateslikeaMOSFET.Thisdeviceistypicallynamedathinfilmtransistor(TFT),sinceitdoesnotoperateinexactlythesamewayasaMOSFET.Figure11.19showsatypicalTFT,butmanyconfigurationshavebeenfabricated.Asemiconductorpolymerfilmisdepositedontoastructurethathasaninsulatordepositedontoaconductingsubstrate.Theinsulatorhasmetalcontactsdepositeduponitpriortotheorganicfilmdeposition.TheorganicsemiconductorcanbeN-orP-type,althoughmoststableorganicsemiconductorsareP-type.Purelyasamatterofconvenience,thesubstrateconductorisusuallyalowresistivitySiwafer,andaSiO2filmisgrownontheSiwaferthatservesasthegateinsulator.Theconveniencestemsfromtheavailabilityofhigh-qualitySiwafersandtheeaseofgrowthoftheexcellentinsulatorSiO2,butanysuitableinsulatorandconductingsubstratecanbeused.Thedeviceoperationisstraightforward.ApotentialisappliedtoGinsuchawaytoattractthemajoritycarrierstotheorganicfilm–insulatorinterface.ThiswayahighlyconductingchanneliscreatedinbetweentheconductingmetalSandDcontacts,therebyenablingtheturn-onofISD.TheessentialdifferenceinoperationoftheTFTascomparedwiththeMOSFETisthattheTFTusesonlymajoritycarriers.TheTFTtakesadvantageofthelowconductivityoforganicsemiconductorsthatisduetothelocalizedstateconductionratherthanextendedstates.SemiconductingPolymerSDInsulatorMetalContactsConductorGFigure11.19Thinfilmtransistor(TFT)configurationusingandorganicsemiconductorfilmwiththreeter-minalsS,G,andD.不得转载谢谢合作LWM 290JUNCTIONSANDDEVICESANDTHENANOSCALEThislowconductivitycanbegreatlyenhancedbyconcentratingthecarriersinachannel.ConsiderableimprovementsinTFTswillbenecessarytorendertheuseoforganicmaterialsinmicroelectronics.11.4NANOSTRUCTURESANDNANODEVICESSizeisthedeterminingfactorintheemergingscienceofnanostructuresandnanodevices.Aspresentlypracticedinthemostadvancedcomputerchips,thesmallestfeaturesizesareontheorderof50nm,andthemostcriticalthinfilmthickness,thegateoxideinMOSFETsislessthan5nm.Futuredevicestodayindevelopmentwillhaveevensmallerdimensions.Therefore,inatleastthefieldofmicroelectronics,nanostructuresandnanodevicesarepartofreality.Alreadytheadvancesinultra-thinfilmdepositionandlithographythathavetakenplaceinthefieldofmicroelectronicshaveenabledthecre-ationofultra-thinfilmsinthenanometerrange,andnanometersizestructuresforuseinotherfields.Forexample,nanometersizemotors,pumps,andothermachinesarebeingfabricatedbyadvancedlithographytechniques.Thus,whilethefieldofnanostructuresandnanodeviceshasbegunwithelectronicsmaterialsanddevices,presentlymanyotherengineeringobjectivesinmedicine,optics,mechanics,sensors,andsoon,areadvancingintothenanotechnologyarena.Inkeepingwiththethemeofthistext,namelyelectronicmaterials,thenanostructureandnanodevicediscussionbelowislimitedtoelectronicandoptoelectronicapplications.However,isshouldbeunderstoodthatvirtuallyallmodernscienceandengineeringfieldswillsoonhavetheirnamestartwiththeprefixnano-.11.4.1HeterojunctionNanostructuresManyoftheearliestandnowmostdevelopednanostructuresandnanoelectronicdevicesarefabricatedwithjunctions,aswasthecaseforconventionalelectronicdevicesdiscussedabove.Thedifferencewiththeso-callednanostructureswithnano-sizeddimensionsisthattheycontainarelativelysmallnumberofatoms.Forapieceofmaterialwithlargenumbersofatomsinaperiodicarray,theresultsofChapter9yieldedelectronenergybandsthatcontainedquasi-continuousallowedenergylevels.However,forasingleatomtheboundelectronstatesforasystemofsizelisgivenbyequation(9.73)asfollows:222hnpE=◊(9.73)22melThis1-Dformulayieldsdiscreteenergylevelsfortheintegernandforsmalll.Thusforelectronsconfinedtothenanometerthickfilmsandinthedirectionnormaltothefilmsurface,theelectronswillexperiencediscreteallowedlevels.Thiskindofstructureiscalledaquantumwell,sincethediscretequantumlevelsresultfromtheconfinementoftheelectronstolinthedirectionnormaltotheplaneofthefilm.Specifically,aquantumwellisfabricatedbysandwichinganultra-thinfilm(<10nm)materialthathasarelativelysmallbandgapinbetweenlayersofmaterialwithalargergap,asshowninFigure11.20aforthesemiconductorGaAssandwichedinbetweeninsulatingGaAlAslayers.Tomaintainthehighestmaterialquality,eachofthelayersshouldbesinglecrystalline.Thatmeansthateachofthelayersrequiresdepositioninthesinglecrystallinemorphologyonadifferentsinglecrystalmaterial.Thedepositionofafilmwithasinglecrystalmorphology,andwithsomecrystallographicrelationshiptothesinglecrystallinelayer不得转载谢谢合作LWM 11.4NANOSTRUCTURESANDNANODEVICES291a)GaAlAsGaAsGaAlAsb)VacuumLevelf1CBf2EF1CBVBEF2VBc)VacuumLevelDE}cEF}DEvFigure11.20(a)Quantumwellstructuremadefromthinfilmlayers;(b)separatedwideandnarrowgapsemiconductors;(c)whenthesemiconductorsin(b)arejoined,aquantumwellisformedwithbandoffsetsindicatedDEcandDEv.beneath,iscalledepitaxy.Allthesuccessfulquantumwellstructuresreportedinthelit-eratureareepitaxiallayers.Inpractice,itisdifficulttofindmaterialssystemsthatarecompatible,thatis,thatdonotreactwitheachother,orareotherwisealteredbycontact,andthathavelatticesmatchingcloselyenoughtofabricatethefilmsandwichstructureshowninFigure11.20a.Significantlatticeparametermismatchandstructuraldissimilaritybetweenthenarrowandwidebandgapmaterialsusedforthequantumwellwillatbestyieldepitaxywithlargenumbersofinterfacialdefects,ornoepitaxyatall.RecallfromChapter7thatifmaterialshaveadifferentthermalexpansioncoefficient,astresscandevelopwithachangeintemperature.Manythinfilmepitaxyprocessesinvolvetemperaturecyclesthatimposemechanicalstressesontheinterfacesandleadtodefectformationandreducedquality.Thereforeitisanenormousmaterialssciencechallengetoidentifysuitablemate-rialssystemsforobtainingthedesiredquantumeffectsanddevices.Wherelatticemisfit不得转载谢谢合作LWM 292JUNCTIONSANDDEVICESANDTHENANOSCALEorhighstressesleadtointerfacialdefects,theresultingquantumwelldeviceswillnotbeoperable.OneofthemostimportantmaterialssystemsforfabricatingquantumwellsisGaAsasthenarrowgapmaterial(Eg=1.4eV,ao=0.565nm)andGaxAlyAswithalargerbandgap(AlAs=2.4eV,ao=0.566nm).SotheGaAlAsalloyhasagapsomewhereinbetween1.4and2.4,dependingonthevaluesforxandy.Thejunctionresultingfromthesetwodissimilarmaterialsiscalledaheterojunction,andthistermindicatesdiffer-entmaterialsacrossthejunction.Laterforopticaldevicefabricationitwillbeusefultousex=0.7,y=0.3,sincethisalloyhasthelargestGaAlAsgap(2eV)andcaneffectquantumconfinementwhileretainingadirectgapbandstructurethatisusefulforopticaldevices,aswasmentionedinChapter9,Section9.5.Figure11.20ashowsaquantumwellformedusingGaAsasthenarrowgapmaterial(Eg=1.4eV)andGaxAlyAs(2eV).Figure11.20bshowsseparatedsemiconductorscorrespondingtonarrowandwidegapmaterials.WhenthesematerialsarejoinedasinFigure11.20a,theresultisshowninFigure11.20cwherethedissimilarbandgapsleadtothebandoffsets,DEc(barrierforelectrons)andDEv(barrierforholes).TheoffsetsresultwhentheFermilevelsequilibrate,aswasdiscussedearlierinChapter11.IftheoriginalvaluesofEFforthematerialsareclose,thenthereislittlebandbanding,andtheidealquantumwellstructuredepictedcanbeapproximated.Theactualenergylevelsaredeterminedbythesizeofthewell(l)andtheoffsets(thestrengthoftheconfinement).Thequantumwellstructureisfabricatedfromthreelayersinasandwichstructure.Modernfilmmakingtechniquessuchasmolecularbeamepitaxy(MBE)usuallyuseatomicbeamstoformelementalorcompoundlayersonasubstrate.Differentlayerscanbeproducedwithdifferentatomsorthesameatomsindifferingproportions(alloys),andthelayerscanberepeated.Ineffectdifferentalternatingnanometerthicklayerscanbealternatedvirtuallyindefinitely.Suchanarrayofrepeatingquantumwellsiscalledasuperlattice,andthesuperlatticestructurecanbeusedinmanydeviceapplications.MBEsystemsareoperatedinultra-highvacuumsystemsandarethereforelarge,complicated,expensive,andtime-consumingtokeepinoperation.However,avarietyofhigh-qualityMBEsystemsarecommerciallyavailableandextensivelyusedinthescientificandengi-neeringareasofelectronicmaterials.Oneimportantapplicationofthesuperlatticeistoenhancetheperformanceofpho-todetectors.RecallSection11.3.2.2aboveandFigure11.13whichshowsthataphoto-cellisaPNjunctioninwhichanincidentphotoncreateselectronholepairsthatareseparatedacrossthedepletionwidthofthejunction.Thecurrentflowderivesfromthecarriersthatareessentiallyproducedbyabsorptionoftheincidentlight.Aphotodetec-toristhesamekindofdevice,butitisusedtosenselightandmeasureitsintensity,ratherthenproduceausablecurrentorpotential.Figure11.21ashowsasuperlatticestructurecomprisedofmultiplequantumwellsthatwereshowninFigure11.20c.Figure11.21bshowsthemultiplequantumwellstructureinsertedinbetweenaPandNsemiconduc-tor,andwithanexternalelectricfieldapplied.Fromthisfigureitisseenthatphoto-inducedcarriersgeneratedasaresultoflightincidentonthedevicecangainsufficientenergyfromtheappliedfieldtoovercomethebarriers(DEcthebarrierforelectronsandDEvthebarrierforholes)betweenquantumwells.Theacceleratedcarrierscancreateadditionalcarriersbyimpactionization.Impactionizationistheprocessbywhichcar-riersareproducedfromtheenergeticcollisionsofalreadyproducedcarriers.Thiscreatesanavalancheeffectthatenhancestheoriginalsignalfromtheoriginalphotoproducedcarriers.Thephotocellenhancedbytheimpactionizationviathesuperlatticeiscalledanavalanchephotocell(orphotodiode).Byjudiciouschoiceofthematerialsthatcom-不得转载谢谢合作LWM 11.4NANOSTRUCTURESANDNANODEVICES293a)b)-eP}DEc}DEv+hNFigure11.21(a)Multiplequantumwellsformingasuperlattice;(b)thesuperlatticein(a)withanappliedforwardbias.Thebandoffsetsanddirectionforelectronandholemotionareindicated.prisethesuperlatticejunctions,thebandoffsetvalues,DEcandDEv,canbeengineeredsothatonecarrieramplifiesviaavalancheatagreaterratethantheothertherebyimprov-ingthesignaltonoiseratioofthedevice.Also,withthepropervaluesoftheappliedfieldandsufficientlythinquantum(<10nm)inthesuperlattice,electrontunnelingcanoccurfromwelltowellandsuchadeviceiscalledatunnelingphotocell(orphotodiode).Thetunnelingprobabilityforholesistypicallylessthanthatofelectronssothatholesaccumulateorbecomelocalizedrelativetoelectrons.Theoveralleffectisthatelectronsareseparatedfromandtransportedingreaternumbersthanholes,andthegainofthedevice,namelythenumberofelectronsproducedasaresultofthephotonicprocess,canbeenhancedwiththeuseofthequantumwellnanostructure.Thequantumwellandsuperlatticenanostructuresdiscussedaboveareexamplesoftheintegrationofopticalandelectricaldevicesintoonestructure,andtheycompriseonekindofoptoelectronics.Solidstatelasersandopticalswitchescanalsobefabricatedbyusingsimilarsuperlatticeheterojunctionsthatarealsooptoelectronicdevices.11.4.22-Dand3-DNanostructuresThedevicesbasedonquantumwellsandsuperlatticesdiscussedabovearetheresultofquantumeffectsinonedirection(linequation9.73).Itisalsoimportantandusefultoachieve2-Dand3-Dquantumeffectsthatleadtoquantumwiresanddots,respectively.FurtherquantizationcanbeobtainedbyconfiguringtheGaAlAstosurroundGaAssoastoproduceabarofGaAsthatextendsinonedirection,asshowninFigure11.22a,andcalledaquantumwireoracubeofGaAsthatissurroundedonalldirectionsbythewiderbandgapGaAlAs,asisshowninFigure11.22candcalledaquantumdot.Theadditionalquantumconfinementyieldquantizationintheotherdirectionsandnew不得转载谢谢合作LWM 294JUNCTIONSANDDEVICESANDTHENANOSCALEa)GaAsGaAlAsb)GaAsGaAlAsFigure11.22(a)Quantumwireresultingfrom2-Dconfinementand(b)quantumdotresultingfrom3-Dconfinement.electronicstatessimilartothequantumwell.Devicesutilizing2-Dand3-Dquantizationarenowbeingresearched.Forexample,arraysofquantumdotsareenvisionedforcom-putermemorywherechargecarrierscanmovefromonedottoanotherviatunnelingwhenasuitablepotentialpulseisapplied.Alsothe3-Dquantizationcanleadtowave-lengthselectivityforopticaldevices.Whileseveralelectronics-basednanostructuresanddeviceshavebeenpresentedhere,therearemanyothernanostructuresreceivingatten-tion.Forexample,nano-sizedmotorcomponentsarebeingfabricatedthatcandovarioustaskssuchaspumpingminuteamountsoffluid.Suchadevicehaspotentialmedicalapplicationstoadministermedicinesinresponsetoanano-sizedsensorthatdetectsdefi-ciency.Thepumpandsensorcanbeintegratedwithsuitablenano-electronicsandimplantedinthepatient.Whilethissoundslikesciencefiction,itisrapidlybecomingreality.However,withalloftheemergingnanoscienceandnanotechnology,significantmaterialschallengeslieaheadinfindingsuitablematerialsandprocessestoproducetheintricatenanostructures,andsuitablemeasurementsthatoperateatthenanoscale.RELATEDREADINGD.A.Davies.1978.WavesAtomsandSolids.Longman,London.Awell-writtentextcoveringmanyofthetopicsinChapters9,10,and11withgoodinsights.S.Dimitrijev.2000.UnderstandingSemiconductorDevices.OxfordUniversityPress,Oxford.Thistextisatthenextlevelofunderstandingforjunctiondevices.Itisreadableandwellillustrated.不得转载谢谢合作LWM EXERCISES295R.E.Hummel.1992.ElectronicPropertiesofMaterials.Springer-Verlag,NewYork.Thistextpro-videswell-writtencoverageofthematerialinChapters9,10,and11attheappropriatelevel.Theauthorhasusedthisbookasatextfortheelectronicmaterialspartofthematerialssciencecourse.J.P.McKelvey.1993.SolidStatePhysicsforEngineeringandMaterialsScience.Krieger,HigherlevelthanHummel,well-written,readable,andforthetopicscoveredmorecomplete.M.A.Omar.1993.ElementarySolidStatePhysics.AddisonWesley,Reading,MA.AtextthatcoversmanyofthetopicsinChapters9,10,and11andcalsomanymoretopicsnotcoveredinthepresenttext.Areadabletextinthesubject.EXERCISES1.Explainthesequenceofeventsthatoccurwhentwodifferentmetalsarebroughtintocontact.2.Repeatexercise1forametalandforintrinsicN-typeandP-typesemiconductors.3.Calculatethecontactpotentialandthedirectionofbandbending(sketch)forajunc-tionwithametalworkfunctionof4eVandasemiconductorworkfunctionof3eV.4.Explainwhyitisdifficulttomeasurethecontactpotentialandhowitcanbemeasured.5.Usingparallelenergybandsketches,explainhowtoformanohmiccontactwithametalandanN-andP-typesemiconductor.6.Explainhowathermocoupleandthermoelectricrefrigeratorworks.7.Show,usingparallelbandenergyandI-Vsketches,howaPNdiodeworkswithandwithoutanappliedbias.8.ExplainhowaZenerdiodeworksandwhatitcanbeusedfor.9.Explainhowaphotocelloperatesandhowtooptimizethedevice.10.ExplainhowbipolarandMOSFETtransistorswork.11.Intermsofquantummechanicalimplicationsonelectronicdevices,discussmakingdimensionsofasolidsmallerintothenmrange.12.Explainhowaphotocellcanbeenhancedbyasuperlatticestructure.不得转载谢谢合作LWM 不得转载谢谢合作LWM INDEXNote:Pagenumbersfollowedbyfrefertofigures,pagenumbersfollowedbytrefertotables.Acceptorstates,237AtomsAccumulationcondition,286coherentscatteringfrom,40Activeelectronicdevices,276,279–290elasticdisplacementof,144photocells,283–284numberdensityof,83rectifiers,279–283Auxeticmaterials,151transistors,284–290Avalancheeffect,281Allowedenergybands/levels/states,57,200,201,213.SeealsoExtendedallowedBandgaps,7,57electronstates“Bandtailing,”265electroninteractionsbetween,233Basis,16–17Allowedquantumstates,65,244,229BCCcells,closepackingdirectionsfor,23.filledandempty,215SeealsoBody-centeredcubicentriesAlloys,28BCS(Bardeen,Cooper,andSchrieffer)theory,understanding,5250–251Al-Si-Osystem,131felectronpairingin,251Aluminum(Al),energybandstructurefor,222fBias,275Amorphousmaterials/solids,12.SeealsoBi-Cdsystem,temperature-compositionphaseNoncrystallinematerialsdiagramfor,126fthermalbehaviorof,175–177Bindingenergies,203–204time-dependentdeformationof,177–179Bipolarswitch,286Anelasticity,177–178Bipolartransistors,284–286Angularphasedifferences,42Bismuth.SeeBi-CdsystemAnisotropicbonds,175Blochtheorem,210Arrheniusactivationenergy,70Blochwaves,210Arrheniusfactor,4Body-centeredcubic(BCC)lattices,slipArsenic.SeeGa-Assystem;Galliumarsenidesystemsfor,164t.SeealsoBCCcells(GaAs)Body-centeredcubicstructures,14,Atomicbindingpotential,20815fAtomicscatteringfactor,40Boltzmanndistribution,4,235Atomistictheoryofdiffusion,83–86Boltzmannfactor,4,70,133Atompositions,effectonscatteredintensities,Boltzmannrelationship,65–6637Bondenergy,133ElectronicMaterialsScience,byEugeneA.IreneISBN0-471-69597-1Copyright©2005JohnWiley&Sons,Inc.不得转载谢谢合作LWM297 298INDEXBonding.SeealsoChemicalbondingConductivity,241.SeealsoConductionanisotropic,175formulafor,258–259dislocationmotionand,169,170Conesofdiffraction,54Boundaryvalueproblem,202Configurationalentropy,184Boundelectronproblem,solutionsto,202–208Congruentmelting,129Braggangles,36–37,46Constantofintegration,183Braggcondition,219Contactpotential,271,276Bragg’slaw,33–37,45,52,53Continuityequation,97–98Bravaislattices,12–14,16Convection,versusdiffusion,94Brillouinzones,25,57,58,219–221,224Convectivetransport,94Brittlesolids,142Cooperpairs,250,252,253Bulkatoms,133,134fCopper.SeeCu-NientriesBulkmodulus,73Coredislocationenergy,167–168Burgercircuit,76–77Coulombicpotential,208Burgersolid,183,184fCracks,79Burger’svector,75,76–77,166,168Creep,161,173–174Criticalnucleisize,131–132Cadmium.SeeBi-CdsystemCrystallinematerials,3–4.SeealsoCrystalCarbondiffusion,85structuresCarrierconcentration,259deformationof,161–162Carriermobility,260jumpdistancein,88–89Cesiumchloride(CsCl)structure,27namingdirectionsandplanesin,17–21Chainrule,199plasticdeformationof,174,178Chemicalbonding,144.SeealsoBondingCrystallineorder,12Chemicalcompounds.SeeCompoundsCrystallographicorientation,55Chlorine.SeeCesiumchloride(CsCl)Crystallography,influencesonsliplinestructure;Sodiumchloride(NaCl)formation,72structureCrystalstructures,3,16–17,25–29Clapeyronequation,117–118forcompounds,26Classicalwaveequations,199–200determining,53Closepackingconcept,22–24Crystalsystems,14Coherentphasediffraction,192planarspacingformulasfor,22tCoherentscatteringCube,deformationof,150ffromanatom,40CuNialloys,temperature-timedataforfromanelectron,38–40cooling,119ffromaunitcell,40–43Cu-Nisystem,temperature-compositionphaseComplementaryMOSFET(CMOS),288–289.diagramsfor,121f,122f,124fSeealsoMetaloxidesemiconductorfieldCurrentdensity,281effecttransistors(MOSFETs)Complexarithmetic,42–43deBroglierelationship,32,148,190Complexnumbers,188Debyearcs,54Components,equilibriumand,113–115Debyefrequency,250CompoundsDefectsformationof,128–129incrystallinesolids,31structuresfor,26–28kindsof,62Comptonscattering,39material,4Conduction.SeealsoConductivity;Electronicpropertiescontrolledby,61conductionDefectstates,265freeelectrontheoryfor,240–243Deformationinmetals,240–247ofnoncrystallinematerials,175–186Conductionband(CB),215,216time-dependent,177–179electronsin,238–239Degreesoffreedom,111,116insemiconductors,254–256DG,calculating,132,133–135不得转载谢谢合作LWM INDEX299DG*,calculating,135–136Dislocationreactions,testing,168Densityofstates(DOS)function,7,229,Dispersionrelationship,149230–231,253Distanceformulaequation,49Depletionregion,275,284Disturbances,short-orlong-wavelength,139Devices.SeeElectronicdevices;NanodevicesDonorlevel,237Diamondcubic(DC)lattices,slipsystemsfor,Dopants,253164tDoping,216,254Diffraction,3–4,31–59.SeealsoDiffractionlevelof,237techniquesinsemiconductors,257phasedifferenceandBragg’slaw,33–37Drudetheory,240reciprocalspaceand,45–53Ductility,142scatteringand,37–45Dynamicrandomaccessmemory(DRAM),wavevectorrepresentationand,55–58287Diffractiontechniques,53–55phasedeterminationand,119E.SeeYoung’smodulus(E)Diffusion,5,81–110Edgedislocations,73–74activationenergyfor,91Burgercircuitfor,76–77atomistictheoryof,83–86Effectivemass,225,226,238masstransportmechanismsand,91–94forholesandelectrons,240mathematicsof,94–108Eigenfunctions,202non-steadystate,97–108Eigenvalues,202randomwalkproblemand,87–91Elasticconstants,6,179–181steadystate,95–97relationshipsamong,153–156versusconvection,94Elasticdeformation,144versuspermeability,91–94Elasticity,139–159Diffusionalflux,94Hooke’slawand,150–151Diffusioncoefficient,95natureof,144–147Diffusionconstruct(D),83–86normalforceresolutionand,156–157relationtorandomwalk,89Poisson’sratioand,151Diffusiondistance,87relationshipsof,141–147Diffusionequations,81–83stressanalysisand,147–150Diffusionlength,106–108Elasticlimit,141,143Diffusionproblems,unitsfor,95Elastomericbehavior,163Dilatation.SeePuredilatationElastomers,142,146,183–186Directionindexes,19–20Electricalconduction,freeelectrontheoryfor,Directions240–243lattice,19–21Electromagneticradiation(emr),32.Seealsonomenclaturefor,21tEmrdiffractionDiscreteness,144interactionwithcrystalstructure,35Dislocation(s),71–77Electromagneticwave,139Burger’svector/Burgercircuitand,76–77Electronaffinity,272defined,73Electronbands,226.SeealsoElectronenergyedge,73–74bandrepresentationselasticenergyfor,166–167structureof,2,264–265energyrequiredtomove,169tElectroncurrent,241increasingthenumberof,171–173Electrondiffractionmicrographs,33,34fmotionof,77,169–170Electrondriftvelocity,243froleof,163–174Electronenergybandrepresentations,215–221,screw,74–75222f.SeealsoElectronbandsstabilityof,168Electronflux,242Dislocationdensity,170Electron-holepairs,283Dislocationlengths,168–169,170Electronicconduction,215,229Dislocationloop,170–171quantumtheoryof,244–247不得转载谢谢合作LWM 300INDEXElectronicdevices,7–8,275–290Entropy,63–64active,279–290Epitaxy,291evolutionof,269e.SeeStrain(e)passive,276–278Equationofmotion,147–150Electronicmaterialsscience,1–8Equilibrium,conditionsfor,116defectsand,4Errorfunctioncomplement(erfc),104,105diffusionand,5Errorfunctionvalues,103tmechanicalpropertiesand,6Euler’sformulas,203,212phaseequilibriaand,5–6Eutectictemperature,126structureanddiffractionin,3–4Eutectoidreaction,126studyof,3Ewaldconstruction,50–53Electronicproperties,7–8,229–267Ewaldspheres,53,55electricalbehavioroforganicmaterials,Exothermicreaction,132264–265Extendedallowedelectronstates,257Fermienergyposition,236–240Extendedzonescheme,217ofmetals,240–253Extensivevariables,111occupationofelectronicstates,230–236Extrinsicsemiconductors,253,257–261semiconductors,253–264Tdependenceof,259–261Electronicstructure,6–7,8,187–227.SeealsoElectronenergybandrepresentationsFace-centeredcubic(FCC)structures,14,15f.aspectsof,224–226SeealsoFCCentriesquantummechanicsand,196–214Face-centeredlattices,slipsystemsfor,164trealenergybandstructures,221–224FCCcells,closepackingdirectionsfor,23waves,electrons,andwavefunction,187–196FCCcrystal,self-diffusionvacancymechanismElectronictransport,31in,90–91Electronmass,225–226Fecarburization,96–97.SeealsoIron(Fe)Electronmobility,241,256Fermi-Diracdistributionfunction,7,229,230,Electrons232–235,253,254coherentscatteringfrom,38–40plotof,234fdeBroglierelationshipand,190Fermienergy(level),229,232energiesfor,213positionof,236–240wavelengthfor,32–33Fermistatistics,7Electrontunneling,282Fickiandiffusion/flow,physicsof,81,85Electronvelocity,242–243Fick’sfirstlaw,84–86,92,95–97Electronwaves,195–196Fick’ssecondlaw,86,97–108dispersionof,197–199solutionsto,98–106Elements,structuresfor,25–26Fictivetemperature,176Emrdiffraction,particleandnonparticle,Filmformationkinetics,9233.SeealsoElectromagneticradiationFilms,layered,6(emr)Finitebindingpotential,204Endothermicreaction,132Fluxequations,96EnergyFluxes,81–83availabilityof,63–64Forbiddenenergygaps(FEG),7ofadislocation,166–169Force(F),141.SeealsoNewtonianforceEnergybandgaps,213.SeealsoEnergygapformula;Normalforces;Shearforce;Energybandstructures,7Tensileforcesaspectsof,224–226appliedtosolids,139real,221–224Forcerelationships,156–157Energygap,positionofFermienergyin,238.Fourier’slaw,81SeealsoEnergybandgapsFour-pointprobemethod,263–264Energyintegral,233–234Fractionalintercepts,48Engineeringstrain,143Frank-Readsource,171–173Enthalpy,62,64Freeelectronband,201ofdefectformation,66–67Freeelectronequation,216不得转载谢谢合作LWM INDEX301Freeelectronsolution,200–201Idealgaslaw,63Freeelectrontheory,240–243IdealNewtonianbehavior,178“Freeze-out”region,259Immiscibility,complete/total,125–126,127Frenkeldefects,67Impactionization,281,292Frequency,vibration,90Incoherentscattering,39–40,192Incongruentmelting,129G.SeeGibbsfreeenergy(G);ShearmodulusIndexplanes,lowandhigh,19(G)Inorganicsolidcompounds,27G(diffractionvectorinkspace),56Integratedcircuit(IC),2Ga-Assystem,temperature-compositionphaseIntensivevariables,111–112diagramfor,130fplotof,112–113Galliumarsenide(GaAs)Interfaceboundary,78–79bandgapof,253Interplanarangleformulas,49–50energybandstructurefor,223f,224Interplanarspacing,46–47,49Gas,scatteringfrom,37–38Interstitialdiffusion,91Gas-solidinterface,91Interstitialpairdefects,67Gaussiandiffusionprofile,98–99Interstitialpointdefects,66evolutionof,107fInterstitials,statisticsof,67Germanium(Ge)Interstitialsolutions,28bandgapof,253IntrinsicFermilevel,237dopantsfor,259tIntrinsicsemiconductors,253–257Ge-Sisystem,temperature-compositionphaseconductivityfor,256diagramfor,125fInvariantpoints,116,126Gibbsfreeenergy(G),64,132,185.SeealsoInversioncondition,286DGentriesIonizationenergy,270Gibbsfreeenergyrelationship,66Iron(Fe).SeealsoFecarburizationGibbsphaserule,111–130phasesfor,116applicationsof,115–116pressure-temperaturephasediagramfor,114fleverruleand,121–125Isothermalexpansion,62–63tielineprincipleand,120–121Isotopeeffect,250–251Glasses,viscosityrangesfor,177tGlasstransitiontemperature,176–177Josephsontunneling,252Grainboundaries,77–78,174Jumpdistance,88Groupvelocity,195Jumpfrequencies,83–84,85Junctions,7–8,270–275.SeealsoHalleffectmeasurement,261–263Heterojunctionnanostructures;Metal-Hardsuperconductors,249metaljunctions;Metal-semiconductorHarmonicwaves,187–188junctions;PNjunctions;SchottkyHeatcapacityvalues,118contacts(junctions);Semiconductor-Heavyholebands,226semiconductorPNjunctionsHeisenberguncertaintyprinciple,196Helmholtzfreeenergy,64Kelvinmethod,271Henry’slaw,92k=0transition,224Heterogeneousnucleation,137Kineticenergy,197Heterojunctionnanostructures,290–293KPformula,213,216,217.SeealsoKronig-“High-anglegrainboundaries,”78Penney(KP)modelhklindexes,54Kronig-Penney(KP)model,7,196–197,Holeband,226257–258.SeealsoKPformula;Holemotion,240SE-Kronig-PenneymodelHoles,valenceband,234,238,239kspace,55–58,219,220,244Homogeneityrange,128,129representationsof,216–219Homogenization,98,99,105–106Hooke’slaw,141,150–151Laplacetransforms,104,105Hydrostaticpressure,152Lateraldeformations,152不得转载谢谢合作LWM 302INDEXLatticedirections,19–21dislocationsin,166Latticegeometry,21–24electronicpropertiesof,240–253Latticeparameters,12Metal-semiconductorjunctions,271–274,279Latticepoints,15–16Mg-Nisystem,temperature-compositionphaseLattices,12–16diagramfor,129fLatticevector,reciprocal,48–50Microelectronics,290Lauediffractiontechnique,55Microscopicreversibility,82Layeredstructures,6Millerindexnotation,17–19Lead.SeePb-SnsystemMiscibility,complete,125Leverrule,121–125Models,ofnetworksolids,179–183L’Hospital’srule,100–101Molarvolumeexpansion,174Lightholebands,226Molecularbeamepitaxy(MBE),292Limitedsolubilityphasediagram,127Molecularstructure,9Linearrateconstant,94Molecularweights,17Linedefects,4,71–77,169Molefractions,115,123“Liquidus”temperature,119formulafor,124tielineand,120–121Morsepotential,144Longchains,arranginginsolids,185–186MOSFETdevices,286–289.SeealsoMetalLongitudinalwave,139oxidesemiconductorfieldeffectLong-rangeorder,10–12,264–265transistors(MOSFETs);TransistorsLongtimesolution,Fick’ssecondlaw,Motion,equationof,147–150105–106Motorcomponents,nano-sized,294“Low-anglegrainboundaries,”78Multiplewaves,superimposing,195–196Low-energyelectrondiffraction(LEED),36Lowindexplanes,164Nabarro-Herringcreep,173,240Nanodevices,290–294Macroscopicdeformation,163Nanoscalematerials,269Magnesium.SeeMg-NisystemNanostructuresMagneticfield,superconductivitydestructionheterojunction,290–293by,2492-Dand3-D,293–294Massdiffusion,5Nanotechnology,2–3Massflux(flow),81–83N-channelMOSFET,286,287f,288Masstransportmechanisms,91–94Negativeresistanceregion,282–283Materialdefects,4Netflux,84–85Materials,theoreticaldensityof,16.SeealsoNetworksolids,175Electronicproperties;Organicmaterialsmodelsof,179–183Materialsscience,changesin,xiNeutrons,wavelengthfor,32Matterwaves,189–190Newtonianforceformula,147Maximumshearstress,155,166Nickel.SeeCu-Nientries;Mg-NisystemMaxwellsolid,180,181–182,183Noncrossingrule,217Mechanicalproperties,6Noncrystallinematerials,deformationof,relationship(s)among,151–153175–186.SeealsoAmorphousMeisnereffect,250materials/solidsMeltingpoint,118Noncrystallinesolids,61Metallurgy,1,5Non-Newtonianbehavior,178Metal-metaljunctions,270–271,276–277Non-steadystatediffusion,83f,97–108MetaloxidesemiconductorfieldeffectNonverticaltransition,224transistors(MOSFETs),284.SeealsoNormalcomponentofstress,155–156ComplementaryMOSFET(CMOS);Normalizationcondition,197MOSFETdevicesforY,197dopingin,257Normalforces,resolving,156–157MetalsNormalstresses,156–157amorphous,175Notation,fordirectionsandplanes,17–21conductivityfor,256N-typedoping,257,286不得转载谢谢合作LWM INDEX303N-typematerial,129Phasetransformations,thermodynamicsof,N-typesemiconductors,261,271–273,130–132279Phasetransitions,118v.SeePoisson’sratio(v)diffractiontechniquesand,119Nucleation,5–6,130–137Phasevelocity,195activationenergyfor,136Phenomenological(thermodynamic)laws,82Nuclei,ripeningof,136–137Phononeffect,251Phonons,140–141Occupiedelectronstates,calculatingthePhotocells,276,283–284numberof,235–236Photodetectors,292“Ohmic”contacts,274Photodiodes,293Ohm’slaw,81,241optimizing,2831-Dlinedefects,62Photons,wavelengthfor,321-Dstrain,143Pilling-Bedworthratio,1741-Dwaveequation,148PINdiode,284Opticalmicroscopy,119Planardefects,77–79Opticaltransition,224Planarspacingformulas,21–22Order,long-andshort-range,10–11Planes,naming,17–19,21tOrganicmaterials,electricalbehaviorof,Plasticdeformation,161–163264–265dislocationsin,165Organictransistors,289–290Plasticity,6,161–186Orthogonalstrains,152dislocationsand,163–174Oxidecompounds,superconductivityof,noncrystallinematerialdeformationand,248–249175–186observationsconcerning,161–163Pairdefects,67PNjunctions,279Parabolicrateconstant,94rectifierformulafor,281Parallelbandpicture,215–216semiconductor-semiconductor,274–275Partiallyfilledvalenceband,236–237Pointdefects,66–67,169“Particleinabox”formulation,202statisticsof,67–71Particlestates,65Pointlattices,12–16Passivedevices,276–278Poiseuilleflow,81Pathdifference,36Poisson’sratio(v),151Pauliexclusionprinciple,232relationshiptoEande,151–153Pb-Snsystem,temperature-compositionphaserelationshiptoEandG,153–156diagramfor,127fPolycrystallinematerials,12,77P-channelMOSFETdevice,287–288Polymericsolids,mechanicalpropertiesof,163Peltiereffect,278Polymers,175Percolationprocess,224Polymorphism,112Periodicboundarycondition,212Porestructure,79Periodicsolidsolution,208–214Potentialenergy(PE)curves,145–146Permeability,versusdiffusion,91–94Powderdiffraction,33,34f,53–55Permeationflux(rate),91–92Primitive(P)lattices,14,15fPhasediagrams,112–113Primitivecubicstructure,closepackingconstructing,116–119directionsfor,22–23informationextractionfrom,127–128Prismaticelement,153–154Phasedifferences,33–37,42Y.SeeWavefunction(Y)Phaseequilibria,2,5–6,111–138P-typesemiconductors,273–274,278examplesof,125–130Puredilatation,Hooke’slawfor,150–151Gibbsphaseruleand,111–130Pureshear,Hooke’slawfor,150–151nucleationandphasegrowth,130–137Phaserule.SeeGibbsphaseruleQuantumdot,293Phases,growthof,130–137Quantummechanical(QM)waveequations,Phaseshift,252–253199–200不得转载谢谢合作LWM 304INDEXQuantummechanicaltunneling,208Semiconductors,216,224,253–264.SeealsoQuantummechanics(QM),7,196–214SemiconductormeasurementsQuantumtheoryofelectronicconduction,conductivityfor,256244–247extrinsic,257–261Quantumwell,290–291intrinsic,253–257structureof,292Semiconductor-semiconductorPNjunctions,274–275Radiofrequency(RF)waves,195Semi-infinitesolidsolution,Fick’ssecondlaw,Randomwalkproblem,87–91101–104Realenergybandstructures,221–224Shear.SeePureshearReciprocallatticevector,48–50.SeealsoRELShearcomponentofstress,155–156entriesSheardeformations,153Reciprocalspace.SeeRESP(reciprocalspace)Shearforce,onaslipplane,73Rectification,273Shearforcecomponent,72Rectifiers,276,279–283Shearmodulus(G),relationshiptoEandv,Reducedzonescheme,217–219153–156“Reflection”ofXrays,35Shearstrains,162REL(reciprocallattice).SeealsoReciprocalShearstress,72,164flatticevectorShear-thinningbehavior,178–179constructinginRESP,50Short-rangeordering,10–12,264–265cubic,51tShortwavelengths,139RELpoints,2-Darrayof,52s.SeeStress(s)RESP(reciprocalspace),xii,3,45–53Silicon(Si).SeealsoAl-Si-Osystem;Sidefined,46–48oxidationReversebias,275,281,283bandgapof,253Rotatingcrystaldiffractiontechnique,53dopantsfor,258,259tenergybandstructurefor,223f,224Saturationcurrent,279,281,283thermaloxidationinoxygengas,Scalar(dot)products,4992–93Scanningtunnelingmicroscopy,208Silicondioxide(SiO2)Scattering,37–45bandgapof,253fromanatom,40energybandstructurefor,224fromanelectron,38–40long-andshort-rangeorderingin,fromaunitcell,40–43264–265Schmid’slaw,156orderin,11–12Schottkycontacts(junctions),263,274Simpleharmonicmotion(SHM),187Schottkydefects,67Sincfunction,213,214fSchrödingerequation(SE),5,196.SeealsoSioxidation,97.SeealsoSilicon(Si)SE-Kronig-Penneymodelexperimentsin,173–174alternativeformfor,199–200Slip,physicsof,72–73electronwavedispersionand,197–199Sliplines(bands),71,164energybandsand,7Slipplanes,162–163freeelectronsolutionto,200–201Slipsystems,72,164–165strongly/weaklyboundelectronsolutionto,forlattices,164t202–208Sodiumchloride(NaCl)structure,27Screwdislocations,74–75,166,167fSoftsuperconductors,249Burgercircuitfor,76–77Soliddefects,61–80Seebeckeffect,276–277,278formationof,62–66SE-Kronig-Penneymodel,periodicsolidlinedefects,71–77solutionto,208–214.SeealsoSchrödingerplanardefects,77–79equation(SE)pointdefects,66–71Self-diffusionprocess,98three-dimensional,79Self-diffusionvacancymechanism,90–91Solidificationtemperatures,118Semiconductormeasurements,261–264Solid-liquidboundary,118不得转载谢谢合作LWM INDEX305Solids.SeealsoAmorphoussolids;Diffusion;structureof,292Elasticity;Electronicstructure;Plasticity;Superpositionprinciple,3,146,151,152,Soliddefects190–195crystalstructureof,16–17,25–29Surface,energyneededtoform,133.Seealsoforcesappliedto,139Surfaceenergy(g)latticegeometry,21–24Surfaceconcentration,98orderin,10–12Surfaceenergy(g),134–135pointlatticesin,12–16Surfacescattering,36stress-strainbehaviorfor,141–142Symmetryoperations,14structureof,9–30Wigner-Seitzcell,24–25Taylorexpansion,144Solid-solidequilibria,118Temperature(T)Solidsolutions,28–29extrinsicsemiconductorsand,259–261“Solidus”temperature,119superconductorsand,248–249tielineand,120–121Temperature-activatedprocess,70Solubilitylimit,29Tensileforces,71–72Solutespecies,28Tensilestress,162Spacechargeregion,271,273Theoreticaldensity,16Spacingformulas,planar,21–22Thermalbehavior,ofamorphoussolids,Splatcooling,175175–177Staticrandomaccessmemory(SRAM),Thermalexpansion,6,146–147288–289coefficientsfor,146tStatistics,ofpointdefects,67–71Thermaloxidation,ofsiliconinoxygengas,Steadystatediffusion,83f,95–9792–93Sterling’sformula,69Thermocouples,275–276,277–278Stoichiometriccompounds,129ThermodynamicsStrain(e),141Boltzmannrelationshipand,65–66relationshiptoEandv,151–153conceptofstatein,64–65trueversusengineering,143FirstLawof,62–63Strainresponse,183ofnucleation,132Stress(s),141ofphasetransformations,130–132componentsof,155–156SecondLawof,63–64formulasfor,147–148Thermodynamicstate,64Strongly/weaklyboundelectronsolution,Thermodynamicvariables,112202–208Thinfilmepitaxyprocesses,291Structuraldefects,4Thinfilmsolution,Fick’ssecondlaw,98–100Structurefactor,43Thinfilmtransistor(TFT),284,289–290calculationswith,43–45Thixotropicagents,179Structure-propertyrelationships,3–4,10,31Thompsoneffect,278Substitutionalpointdefects,66Thompsonequation,38–39Substitutionalsolutions,28Three-componentsystems,129–130Sulfur(S),pressure-temperaturephasediagram3-Dbonding,11for,113f3-DBrillouinzone,220,221fSuperconductingstate,determining,2503-Dbulkdefects,62Superconductivity,229Three-dimensionaldefects,79inmetals,247–2533-Dlatticevector,24Superconductors3-Dnanostructures,293–294applicationsfor,252Threefluxscheme,92–93metalandalloy,248–249Tielineprinciple,120–121resistivityversusTfor,247–248Time-dependentdeformation,ofamorphoustunnelingin,252materials,177–179Supercooling,132Tin.SeePb-SnsystemSupercurrent,247Totalentropy,65,184Superlattice,28Toughness,142不得转载谢谢合作LWM 306INDEXTracer(isotope)atom,90Viscosityranges,forglasses,177tTransducer,195Viscousforce,242Transistors,284–290.SeealsoMOSFETVoidedpockets,79(metaloxidesemiconductorfieldeffectVoigtsolid,181,182,183transistor)devicesbipolar,284–286Waterinventionof,2phasesof,9–10organic,289–290pressure-temperaturephasediagramfor,Transmissionelectronmicroscopy(TEM),119112fTruestrain,143Waveequations,classicalandQM,199–200Tunneling,quantummechanical,208Waveformpulses,complex,196Tunnelingcurrent,282Wavefunction(Y),190,196,198Tunnelingphotocell,293normalizationconditionfor,197Tunnelrectifier(diode),281Wavemechanics,190Twinboundaries,78–79Wavepacket,groupvelocityfor,225Two-dimensionalnucleation,137Waves,representationof,187–1892-Dkspace,244Wavevectorrepresentation,55–582-Dlatticevector,24Weightfraction(WF),1232-Dnanostructures,293–294formulafor,1242-Dplanardefects,62Wettingbehavior,137Two-stateproblem,70,91“White”radiation,55TypeI/IIsuperconductors,249Wigner-Seitzcell,24–25,219Work,73,144Undercooledliquid,176,177Workfunctiosn,270,271Unitcell,coherentscatteringfrom,40–43Unmodifiedemr,35X-raydensity,16X-raydiffraction,31–33VacanciesXraysenthalpynecessarytocreate,70high-intensitymonochromatic,33statisticsof,67–69wavelengthof,32Vacancypointdefects,66Valenceband(VB),215,216Young’smodulus(E),141,145–147insemiconductors,254–256relationshiptoeandv,151–153Vectordotproduct,168relationshiptoGandv,153–156Velocitygradient,179valuesof,146Viscoelasticconstants,179–181Viscoelasticdeformation,178Zenerdiode(rectifier),281Viscosity,176–1770-Dpointdefects,62originof,179Zimanapproach,220不得转载谢谢合作LWM