Grow hin Film
Paul-
-Henri
Haume
esser
To my father
First published 2016 in Great Britain and the United States by ISTE Press Ltd and Elsevier Ltd
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Acknowledgments
AsascientistinCEA-Leti,Ihavebeenfortunateenoughtobeconfronted withavarietyofstimulatingindustrialchallengesandacademicquestionsin thesamefieldofexpertise:theprecipitationofmetalsfromsolutions,either byelectrochemicalorchemicalprocesses.Ihavetriedtoputinthisbookthe essenceofwhatIhavelearntinthisfield.
Ihavelearntalotfrommycolleagues.Someofthemcontributedinthis work:MarylineCordeau,CélineJayet,MurielChaupin,SégolèneOlivier, AnneRoule,ThierryMourier,SylvainMaîtrejean,OlivierPolletandXavier Avalehaveallbeofgreathelpinthedevelopmentoftheelectroplatingand electrolessprocesses.IwillnotforgetCatherineSantini,mycolleagueinCPE Lyon,whohasintroducedmetothefascinatingworldofionicliquidsand whocontinuesbringingnewideasintheresearchonnanoparticle(NP) synthesis.IamalsogratefultoVincentJousseaume,FrédéricGaillard,Eric ChaînetandDanielLincotforsharingwithmetheirdeepscientific knowledge.
Ihavelearntalotfrommystudents.Ithankthemall:SandrineDaSilva, TifennDecorps,KarimSidiAliCherif,MarianaAmuntencei,Julien Cuzzocrea,PhilippeArquillière,IngaHelgadottirandWalidDarwich. Withoutthem,lifewouldhavebeenmuchlessenjoyableandthisbookwould nothavebeenpossible.
Finally,Iwouldliketothankmyfamily:mychildren,Oscar,Léonardand Mélisande,andmywife,Sandrine.Ithankthemfortheirpatienceandsupport asdeeplyasIlovethem.
Introduction
Masteringtheartofmetalworkwasadecisivemomentformankind severalthousandyearsago.Sincethemiddleofthelastcentury,another decisivetechnologicalbreakthroughchangedourworld.Microcomputing enteredourhomesmorethan30yearsago.Wehavebeencarrying microcomputersinourpocketssincetheearly2000s.Theyarenowinvading oureverydaylifeinavarietyof“connected”items.Thisspectacularevolution mirrorsthetremendousprogressintheminiaturizationofintegratedcircuits (ICs).Thesedevicespackaneverincreasingnumberofcomponentsinan eversmallervolume.Asaresult,thecircuitryneededtointerconnectthese componentsbecomesfinerandmorecomplexastechnologyprogresses.The metallicconductorshavenowsub-micronicsizes.Atthesametime,new objectsandapplicationsareemerginginwhichmetalisusedasnanoparticles (NPs).Asaresult, wemustnowmastertheartofmetalworkatthemicroscale andevenatthenanoscale.
Sinceancienttimes,metalhasbeenshapedintothedesiredobjectusinga top-down approach.Arawpieceofmetaliscut,hammeredormoldedintothe finalobject.TofabricatetheverysmallmetallicstructuresinICs,itis preferabletousea bottom-up approachinwhichmetalis grown toitsfinal shape.Thisisusuallydoneby precipitating themetalfromasolution containingsuitablechemicalprecursors.Forthisreason,theartof metalsworkatsmallscaleswouldbemoreofanalchemist’sroleratherthana blacksmith’s.
Thepurposeofthisbookistoexplainthewaysinwhichthisprecipitation reactioncanbecontrolledtoproducethedesiredobject.
Thenature,thesizeandtheshapeofthelatterisdictatedbyitsexactuse. ConductionlinesinICsarequitelong(uptoseveralcentimeters)butvery narrow(downtoseveraltensofnanometers).MetallicNPsaremetallic clustersbelow 10nm indiameter.InChapter1,bothtypesofmetallicobjects arepresented.TheprogressofICtechnologyisbrieflydescribed.Tosustain thepaceofcircuitminiaturization,Cuhasbeenintroducedasaconduction metalinICs.Thevariousprocessesinvolvedinthefabricationofadvanced Cuinterconnectstructuresandtheassociatedchallengesarereviewedfor conventionalintegrationaswellasmorerecentthree-dimensionalstacking strategies.InthelastsectionofChapter1,theneedformetallicNPsis formalized,andthedifficultiesassociatedwiththeelaborationand stabilizationofthesenano-objectsarediscussed.
Theprecipitationofmetalisthecoreprocessinthebottom-upfabrication ofsmallmetallicobjects.Itisaspecialcaseofphasetransformation.Because suchphasetransformationsareofparamountpracticalandtechnological importance,theyhavebeenextensivelystudied.InChapter2,thegeneral conceptsassociatedwithphasetransformationareintroduced.Averysimple thermodynamicanalysisoftheproblemshowsthatitgenerallystartswiththe nucleationofextremelysmalldomainsofthenewphaseintheoldphase. Hence,controllingthisnucleationisthemostefficientstrategytooptimize bottom-upprocesses,especiallywhenextremelysmallsizesaretargeted.For thisreason,theso-called classicalnucleationtheory isdiscussedindetailin thischapter.Controllingthegrowth,aggregationandcoalescenceofthese nucleiisalsomandatory,asisillustratedbypracticalexamplesinthe subsequentchapters.Therefore,abrieftheoreticaldescriptionofthesesteps ofphasetransformationisalsoproposedinChapter2.
InChapter3,thesegeneralconceptsaredevelopedfurtherinthemore specificcaseofchemicalandelectrochemicalprecipitationsofmetals.In particular,theelectrochemicalmechanismsassociatedwithCuelectroplating arediscussed.Similarly,importantaspectsoftheelaborationandstabilization ofmetallicNPsinsolutionarebrieflyexposed.Hence,bothChapters2and3 shouldbeaniceintroductionforthestudentorthescientisttoquestionsof nucleation,growth,aggregationandcoalescence,especiallywhenchemical precipitationreactionsareinvolved.
Chapters4–6arededicatedtothedescriptionanddiscussionofvery specificprocessesforthefabricationofsmallmetallicobjects.InChapter4, thecontrolledgrowthofCuinelectroplatingprocessesisshowntoallowfor theso-calledsuperconformalfillofholesandtrenches.Thisparticularregime isusedtofabricatethemostaggressiveinterconnectstructures.InChapter5,
twoexamplesofthinfilmdepositionby(electro)chemicalreactionsare described.ThesearethinCuseedlayersandmetallicbarriersalsoneededin thefabricationoftheinterconnectstructures.Finally,averyoriginaland promisingapproachtofabricatemetallicNPsisdevelopedinChapter6.This verysimpleprocessusesanewclassofsolvents,theionicliquids(ILs).
Theseexampleshavebeenselectedbasedontheirrelevanceregardingthe technologicalchallengesmentionedinChapter1.Theyareallaimedat overcomingcurrentlimitationsofexistingprocesses.Assuch,theyprovide up-to-dateinformationforthetechnologistinterestedinthesequestions. Throughoutthesechapters,itisalsoshownhowtheconceptsintroducedin Chapters2and3canbe(atleastqualitatively)usedtoguidetheoptimization ofthesenewprocesses.Assuch,theyshouldbeexemplarytoresearchers involvedinsimilardevelopments.
TheFabricationofMicroandNanostructures
AsFeynmanfirststatedinhisvisionarylecturein1959[FEY92],thereis plentyofroomatthebottom.Indeed,thesourceofmostofourcurrent technologicalprogressesisourabilitytomanipulateandshapematterata verysmallsize,downtothenanometerscale.Inthischapter,itisnotour purposetoextensivelycovertheprogressesinmicrofabricationorthe emergenceofnanoscience.Rather,weshallillustratecurrentchallengesin bothfieldsthroughselectedexamples:thefabricationofadvanced interconnectstructuresinmicroelectronicdevicesandtheelaborationof metallicnanoparticles(NPs).
1.1.Thefabricationofadvancedinterconnectstructuresin microelectronicdevices
Oneofthemostprominentdiscoveriesduringthe20thCenturywasthe transistorinventedbyBardeenandBrattainattheBellLaboratoriesin1947 [BAR48].Lesspraised,butnotlessimportant,isthefirstdemonstrationin themid-1950sofanintegratedcircuitonasiliconsubstrate[KIL64].Indeed, thiswasthestartingpointofastillon-goingtechnologicalracetowardmore integrated,thusmorecomplex,circuits.Astheygetsmallerand“smarter”, thesedevicesarenolongerrestrictedtodeskcomputers,buthavebeen introducedinavarietyofmobileapplications(laptops,phones,tablets,GPS systems,etc.),cars,televisions,etc.Withtheadventofthe“Internetof Things”,theyaremeanttobringbrains(oratleastcommunicationskills)toa muchvasternumberofobjectsinoureverydaylife.
1.1.1. Ultralarge-scaleintegration
Sincethemid-1950s,thisevolutionhasdemandedtointegratemoreand moretransistorsperchip[MOO65].Todoso,thesizeofthetransistorshas progressivelybeenreduced,aswellasthesizeofallothersurrounding structuressuchasinterconnects.Bythemid-1960s,theindustrymovedfrom small-tomedium-scaleintegration(lessthan1,000transistorsperchip),then tolarge-scaleintegration(103 to 105 transistors)intheearly1970s.Since 1983,theso-calledverylargescaleintegrationschemehasbeenadopted,in whichthenumberoftransistorsexceeds 105 .Morerecently,theterm ultralarge-scaleintegrationhasappearedforcircuitscontainingmorethan 106 transistors.
Asaresult,thedensityoftransistorshasincreasedexponentiallyoverthe years,aspredictedsincetheearly1960sbyMoore[MOO65].Accordingto Moore’slaw,thedensityoftransistorsdoublesevery18months(Figure1.1 (left)).
Figure1.1. Evolutionof(left)thenumberoftransistorsperchipand (right)CPUclockfrequencyovertheyears
Untiltheendofthe1990s,thistrendwasonlysustainedbythe miniaturizationofthetransistorsandinterconnects,withoutanysignificant modificationoftheirstructure.Inthisperiod,theprogresswasmeasuredby therampupofthedevice’sclockfrequency,whichincreasedfrom 0.5MHz to 3GHz (Figure1.1(right)).Indeed,asthesizeofthetransistorswas reduced,sowastheircharacteristicresponsetime.However,inthemeantime, theresponsetimeoftheinterconnectstructureswasincreasingdueto capacitivecoupling.Soonenough,thedelaytimeassociatedwiththe interconnectsbecamelimiting.Currently,thedelaytimeassociatedwith interconnectstructuresexceedstheresponsetimeoftransistorsby 3 ordersof magnitude(Figure1.2).
Figure1.2. Evolutionofdelaysassociatedwithtransistorsand interconnectsastechnologyprogresses.Adaptedfrom[YEA13]
AsshowninFigure1.3,typicalinterconnectstructuresareformedbymetal linesisolatedbyadielectricmaterial[LE13].Severallevelsoftheselinesare neededtoformalltherequiredinterconnectionsinacircuit.Eachofthese “metallevels”isconnectedtothelevelsaboveandbelowbycontactholes called vias.
Electrically,thelinesareresistors(R).Adjacent,parallelmetalliclinesalso formcapacitors(C ,seeFigure1.3).Initsmostsimplisticdescription,sucha circuithasacut-offfrequencygivenby:
Inotherwords,thiscircuitisexpectednottobeabletopropagatesignals whosefrequencyexceeds fRC .Toincreasethiscut-offfrequency,itisthus necessarytoeitherreduce R or C.Inthelate1990s,chipmanufacturersdid both.
1.1.2. Thedamascenearchitecture
Untilthen,themetalanddielectricmaterialusedtofabricateinterconnect structureswereAlandSiO2 ,respectively.Toreduce R and C ,thesematerials
wereabandonedandreplacedbyCuandso-called low-K dielectricmaterials. ThelatterareSiO2 derivatives,incorporatingapolarchemicalspeciessuchas methylgroupstodecreasetheirrelativepermittivity[LE13].Morerecently, porousvariantshavebeenintroduced:theincorporationofnanosizedporesin thematerialallowedfurtherreductionofthe K valuedownto2.2[GRI01].
Figure1.3. Typicallocalarrangementofamicrochip,includinglocal, intermediateandglobalinterconnectsarrangedinseveral“metallevels” connectedtoeachotherbyvias
1.1.2.1.
Cuasaconductingmetal
CuwasselectedasareplacementmetalforAlbecauseofitslower resistivity(ρCu =1.67 μΩ·cm),butalsobecauseofitsbetterresistanceto electromigration(EM)ascomparedtothemoreconductiveAgandAu(see section5.2.1formoredetailsaboutEM).
However,theintroductionofCuhasnotbeenstraightforward.Indeed,this metalwasnotcompatiblewiththeprocessflowusedtofabricateametallevel withAl.ThelatterisshowninFigure1.4(left).Inafirststep,Alisdeposited onthesubstrate(whichalreadyhasametallevelorcontactplugstothe transistorsifthisisthefirstmetallevel).Theconnectionstructures(linesor vias)arethenpatternedintothemetal.Inthisstep,theexcessmetalisetched toleaveonlythefinalstructures.Then,thedielectricmaterialisdeposited.A chemicalmechanicalpolishing(CMP)processisfinallyappliedtoremovethe excessinsulatingmaterialandtoplanarizethemetallevel.
TheFabricationofMicro-andNanostructures5
Itturnsoutthatthereisnoindustrialprocesscapableofproperlyetching Cu.Forthisreason,analternativeprocessflowwasdevised(Figure1.4 (right))[AND98].Inthissequence,alayerofdielectricmaterialisdeposited first,inwhichtheinterconnectstructuresareetched.Then,themetalfillsthe trenches(lines)orholes(vias)andtheexcessmetalisfinallypolishedaway. Thisarchitecturehasbeencalled damascene,afteranancienttechniqueused injewelleryinwhichgoldisinterlacedintoironorsteel[HES07].
Aldeposition
Dielectricdeposition
Dielectricdeposition
Dielectricpatterning
Figure1.4. Comparisonbetween(left)theconventionalintegrationof Aland(right)thedamascenearchitecture
Thesuccessofthisapproach,whichwascrucialtofurtherimprovethe performancesofmicroelectronicdevices,dependedononecondition:a processwasrequiredwhichwascapableoffillingthetrenchesandholes withoutleavinganyvoid.ThisprocesswasintroducedbyresearchersatIBM inthelate1990s[AND98]:itisCuelectroplating.
1.1.2.2. Cuelectroplating,seedlayerandbarrierdeposition
Conceptually,Cuelectroplatingisrathersimple.Itconsistsof electrolyticallydepositingcopperontothesubstratefromanelectrolyte1, essentiallycomposedofCusulfateandsulfuricacid.However,itwasfound thatbyincorporatingappropriateadditivesinthiselectrolyteaspecific depositionregimetakesplace,inwhichdepositionissignificantlyfaster insidethefeatures(Figure1.5).Thisparticularregimeisreferredtoas
1Theprinciplesofmetalelectroplatingarediscussedinsection3.1.
superconformal deposition[AND98].Thissuperconformalregimeistheonly onecapableofreliablyfillingtheinterconnectfeaturesandcanbeconsidered asthecornerstoneofthedamasceneapproach.Itsdescriptionisthemain topicofsection4.2.
Figure1.5. Schematicrepresentationofthea)superconformal;b) subconformaldepositionregimes;c)earlyexampleofsuperconformal filladaptedfrom[AND98]
ToinitiateCuelectroplating,aconductingsurfaceisneeded.Therefore,a thinmetalliclayerisdepositedfirstonthesubstrate.Usually,aCucoatingis used,depositedbyphysicalvapordeposition(PVD)[NIS07].ThisthinCu lineriscalledthe seedlayer .Theelaborationofthisthinmetalliclayerwithan electroplatingprocessalternativetoPVDisdiscussedinsection5.1.
Atthisstage,thefinaldifficultyremainswiththeuseofCu.Indeed,this metalisknowntoeasilydiffuseintoothermaterialssuchasSioritsoxides. Also,Cuisacontaminantthatcreatesdeeplevelsintheelectronicstructure ofSi,compromisingtheoperationoftransistors[IST02].Inlarger concentrations,CucanevenformcompoundswithSiandcompletelydestroy thedevices[NEW82].Forthesereasons,itismandatorytoconfineCuwithin theinterconnectstructures.Thisiswhyspecificbarrierlayershavebeen addedinthestructurestopreventCudiffusion.Thefirstoneisdeposited beforethemetal(i.e.beforetheCuseedlayer),andisusuallyabilayerof TaNandTa.ThisbarrierisdepositedbyPVD,usuallyinthesameequipment
c)
TheFabricationofMicro-andNanostructures7
asfortheseedlayer2.Thesecondbarrierisdepositedoncethemetallevelis formed,afterthepolishingstep.Thisupperbarrierisusuallyformedbya dielectricmaterial(typicallyaSinitrideand/orcarbide)depositedby plasma-enhancedchemicalvapordeposition[LE13].Recently,ametallic coatinghasbeenproposedtoadvantageouslyreplacethislayer.Thiswillbe thesubjectofsection5.2.
Finally,theconventionalmetallizationsequencecomprisingbarrierand seeddeposition,Cuelectroplating,polishingandencapsulationisdepictedin Figure1.6.
Figure1.6. Schematicrepresentationofthemetallizationsequenceof hollowtrenchesandviasincluding(1)barrierdeposition,(2)seedlayer deposition,(3)superconformalCufill,(4)CMPand(5)encapsulation
1.1.3. 3Dintegration
Theco-integrationoflow-KdielectricmaterialsandCuisstillusedtoday intheindustry.Becauseofthisarchitecturalmodification,thedevice 2Thisisneededtoavoidairexposureofthebarrier,whichwouldresultintheoxidation oftheTa-basedbarrier,seesection5.1.4.
manufacturershavebeenabletopursueMoore’slawinrecentyears[LE13]. However,ascharacteristicdimensionsareshrinkingdowntoafew(tensof) nanometers,physicallimitsarebeingapproached.
Forinstance,thewidthoftheCulinesisreducedto 40nm inthemost recenttechnologies[ITR13].Thisdimensionbecomescomparablewiththe meanfreepathofelectronsinCu,whichisabout 50nm at 293K [HAN02]. Thismeansthatelectronsarenolongermovinginaninfinitemedium;the probabilitythattheycollidewithawallbecomessignificant.Thisphenomenon causesanincreaseintheresistivityoftheCulines.Insuchnarrowlines,the resistivityofCucanexceed 3μΩ·cm [HAU06].
Becauseoftheselimitations,pursuingMoore’slawbyminiaturization onlyisbecomingmoreandmoredifficultandwillevenbeimpossibleinthe foreseeablefuture.Forthisreason,chipmanufacturersaredevisingan alternativeapproach,referredtoasthree-dimensional3Dintegration [LED08].ThisapproachisschematicallyexplainedinFigure1.7.The conceptisquitesimple:severallayersoftransistors(oranyotherdevice)can bestackedtoincreasetheirnumberperunitsurfacearea.Inaddition,this significantlyshortensthe“longdistance”interconnectsinchips,resultingin improvedperformances.Thisstrategyisconceptuallysimpleandelegant,but raisestechnologicalchallenges,suchasdrillingandmetallizingdeepcontact holestoconnectadjacentlayers.Thesestructures,calledthroughsiliconvias (TSVs),arekeyenablersforthe3Dintegration[KAT10].
ThemetallizationofTSVsproceedsthroughasimilarsequenceasfor damasceneinterconnects(Figure1.6).However,allthedepositionprocesses, includingbarrier,seedlayerandCuelectroplatingneedtoberevisedto complywiththedepthofTSVs,aswillbediscussedinsections4.3and5.1.
1.2.ElaborationofmetallicNPs
Todaynanosciencesareamajorfieldofresearch.Indeed,objectsatthe nanoscale(whosesizerangesfrom 1 to 10nm)areatthecrossroadsbetween themoleculesmanipulatedbychemistsandthetechnologicaldevices fabricatedbytechnologists(Figure1.8).Thesedifferentscientificfieldsare nowconvergingintheexplorationofthisfascinating“nanoworld”[BAL05].
Nano-objectspossesspropertiesbetweenthebulkmaterialandthe molecule.Forinstance,metallicNPsofferopticalpropertiesthatcannotbe obtainedfrombulkmetals.Thesepropertieshavebeenusedsinceantiquity, likeforinstanceinthefamousLycurguscupwhosecolorchangesfromgreen
TheFabricationofMicro-andNanostructures9 toreddependingontheincidenceoflight(Figure1.9).Thisdichroicbehavior isduetothepresenceofgoldNPsintheglass.
Longglobalinterconnects
Figure1.7. Schematicrepresentationof3Dintegrationascomparedto theconventional2Dlayout.Thelonginterconnectlinesin2Dconventional assemblyarereplacedbyTSVsinthe3Dintegration.Foracolorversionof thisfigure,seewww.iste.co.uk/haumesser/metals.zip
Figure1.8. Theconvergenceofchemistryand technologyinthe“nanoworld”
TheLycurguscup,fourth century–BritishMuseum,London
Moreover,theseuniquecharacteristicsaretunablewiththesizeandshape ofthesenano-objects.Forinstance,theircolormayvaryastheirdiameter decreases(thesamefortheirfluorescence,seeFigure1.10).Thisisbecause theirphysicalpropertiesevolvewiththeirsize,eitherbyclassicalorquantum effects.Inaddition,NPsusuallyexhibitenhancedorunexpectedchemical reactivitybecausetheypossessasignificantamountofsurfaceatoms.The fractionofsurfaceatomsiscalled dispersion andscaleswiththeinverse diameteroftheNP[ROD06].Below 3nm,surfaceatomstypicallybecome themainconstituentsofNPs(Figure1.11).Becausesurfaceandedgeatoms havefewerneighborsascomparedtotheinneratoms,theyaremoreproneto interactwithothersurroundingspecies.Thisabilitymaybeusedtoenhance ormodifythereactivityofthelatter,whichisthebasisofthecatalytic propertiesofmetallicNPs[FRE11].Itisalsoresponsibleforthe chemisorptionofcompoundsonmetallicNPs,whichisusedtostabilize colloidalsuspensions(seesection3.2.2).
Today,theversatilepropertiesofmetallicNPsareusedinavarietyof applications,suchasmedicalimaging(usingluminescentormagnetic properties),pigments,(bio)sensors,energyconversionandstorage,and catalysis[GOE10].
Figure1.9.
Figure1.10. FluorescenceofCdSe-CdScoreshellNPswitha diameterof1.7nm(blue)upto6nm(red).Foracolorversionofthis figure,seewww.iste.co.uk/haumesser/metals.zip
Figure1.11. Evolutionofthedispersionwith sizeforicosahedralRu-NPs
Inalltheseapplications,thesizeoftheNPsmustbeaccuratelycontrolled tocomplywiththedesiredproperties.Thismeansthatnotonlythe averagesizeneededforagivenapplicationisrespected,butalso thatsizedistributionisasnarrowaspossibletoensurehomogeneous performances.
TherearetwomainapproachestoelaborateNPs.Inthefirstone,called top-down,thenanostructuresarecarvedoutofthebulkmaterial.Thismaybe achievedeitherbyextensivegrinding,orusingmoresophisticatedapproaches involvinglithographypatterning.Theformerispronetocontaminationand agglomerationoftheNPs.Besides,NPsbelow 50nm areonlymarginally reachable.Withthelattertechniques,majordrawbacksaretheircostandthe wasteofremovedmaterial.
Hence,apreferredstrategyisthe bottom-up elaborationofNPsby combiningatoms.Thiscanbeachievedeitherfromthegasortheliquidphase atamoderatetolowcost.Inmostcases,theseprocessesyieldquitepureNPs. However,thecontrolofsizeandthetendencytoagglomerationremain challenging.TheelaborationandstabilizationofmetallicNPsfromliquid phaseprocessesisfurtherdescribedinsection3.2.Aspecificandinnovative approachtothisquestionistheobjectofChapter6.
1.3.Conclusions
Thefabricationofmetallicobjectsatthesmallsize,eitherthinfilmsor NPs,isofparamountimportanceforcurrentandfuturetechnological applications.Thiselaborationispreferentiallycarriedoutusingbottom-up approaches,inwhichthestructuresareassembledfromindividualatoms. Mostoftheseprocessesmustbecontrolledatthenanometerscale.Thisis mandatoryeithertoobtainNPswithpredictablesize,orformthinfilmswith optimizedstructureandproperties.
AswillbeshownChapter2,thetransformationofisolatedatomsinafluid phase3 intobulkmetalusuallyproceedsthroughthenucleationofsmall clustersthatgrowandcoalesceintothedensemetal.Thisisthecaseforthe precipitationofametalfromasolutioncontainingsuitableprecursors,which isthetransformationinvolvedineitherthe(electro)chemicaldepositionof metallicthinfilmsorthechemicalsynthesisofmetallicNPs.Fortunately, thereexistmanywaysinwhichthisnucleation–growth–coalescencesequence canbemodifiedbytuningexperimentalconditions.Thereasonswhyare explainedinChapters2and3.Inthesubsequentchapters,theseconceptswill beappliedtoprocessesoftechnologicalinterestinthefabricationof interconnectstructures(Chapters4and5)ortothesynthesisofmetallicNPs withaccuratesizecontrol(Chapter6).
3Thefluidphasemaybeamelted,butalsoagaseousorliquid,solution.
PhaseTransition:Nucleation,Growth, AggregationandCoalescence
Inthischapter,thegeneralquestionofphasetransformationisdiscussed fromaquitefundamentalstandpoint.Thenotionof nucleation isintroduced. Thisconceptiscentraltounderstandandcontrolthesolidificationprocesses describedinChapters5and6forthefabricationofthinfilmsornanoparticles. Thesubsequentstepsofphasetransformation,growth,aggregationand coalescencearealsodiscussed.Alargepartofthisdiscussionisinspiredfrom therecentbookbyKashchiev[KAS00].Here,onlytheessentialconcepts andequationsareretainedandupdatedwhenevernecessary.
2.1.Whatistheeasiestpathforaphasetransition?
Phasetransitionsareprofoundtransformationsofmatter.Likeother naturalprocesses,theymustfollowtheeasiestpath.Therefore,itisagood ideatofigureoutwhatthispathmightbe.Insuchasituation,thermodynamic considerationsareusuallyextremelyuseful.Inthissection,basic thermodynamicnotionswillberecalled.Then,theywillbeusedtoidentify thedrivingforceforphasetransition.Resistancesagainsttransformationwill beconsideredaswell.Finally,ageneraldiscussionofthewholeprocesswill beproposed.
2.1.1. Gibbsenergyofahomogeneousphase
Aphase α canbethermodynamicallycharacterizedbyitsentropy Sα , volume Vα andcomposition.Thelatterisdescribedbythequantities nα,i of
thevariouschemicalspeciesthatconstitutethephase.Itsenergy Uα isrelated tothesevariablesthrough[GUG85]:
where T and P arethetemperatureandpressure,respectively,and
isthe chemicalpotential ofspecies i inphase α.However,itismorepractical tousetheso-called Gibbsenergy Gα asacharacteristicfunctionofphase α. Thisfunctionisdefinedby:
Since Uα ishomogeneousoffirstdegree,itispossibletouseEuler’s theoremtoobtain:
2.1.2. Thedrivingforce:supersaturation
Letusconsiderthetransformationofahomogeneousphaseintoanother homogeneousphase.Letusagreetocalltheinitialphase“old”andthefinal phase“new”.Thetransformationoftheoldphaseintothenewphaseis possibleonlyif:
Letuslimitourselvestothecasewherethistransformationconcerns a singlespecies,andletusagreetocalltheunitbuildingblockofthenewphase (whichcanbeanatom,amolecule,etc.)a monomer .Ifthetransformation
PhaseTransition:Nucleation,Growth,AggregationandCoalescence15
involves M monomers,theGibbsenergiescanberelatedtotheirchemical potentialsintheoldandnewphases:
Bydefinition, Δμ iscalledsupersaturation.Itcorrespondstothefavorable balanceinGibbsenergy,whichdrivesthephasetransformation.The larger(themorenegative)thisdifference,themorefavorablewillbethe phasetransformation.
2.1.3. Theresistanceforce:theenergeticbarrier
Nowcomesthequestionoftheactualpathfollowedbythesystemto transformtheoldphaseintothenewphase.Thistransitionnecessarily proceedsbysomecontinuousevolutionofthesystemfromtheinitial(old phase)tothefinalstate(newphase).Inotherwords,thesystemhastogo throughasuccessionofintermediatestates.Generally,thesestateshappento belessstablethantheoldphase,sothat μinter >μold (or Ginter >Gold ,see Figure2.1).Inotherwords,theoldphasecorrespondstoa localminimumof theGibbsenergy (thetrueminimumcorrespondingtothenewphase).For thisreason,theoldphaseissaidtobeina metastablestate.
Thesystemneedstoovercomeanenergeticbarrierinordertotransform theoldphaseintothenewphase,andthisbarrierplaysacentralrolein thephasetransformation.
Fromthesesimpleconsiderations,anextremelyimportantconclusioncan bedrawn: thetransformationishighlyunlikelytoinvolvetheMspeciesat once.Indeed,theassociatedbarrierwouldbe M (μinter μold ).By comparison,foranalternativepathwayinvolvingasmallnumberofspecies n<<M ,theassociatedenergeticbarrierisonly n (μinter μold ).If n is reallysmallascomparedto M (severalordersofmagnitudeinmostpractical cases),thelatterbarrierisbyfareasiertoovercome.Forthisreason, first-ordertransformationsusuallyproceedby nucleation (formationofsmall aggregatesor clusters ofthenewphase),whichsubsequently grow to progressivelytransformtheoldphaseintothenewphase.
Figure2.1. Schematicrepresentationofthetransitionpath fromtheoldtothenewphase
Bydefinition, nucleation istheprocessbywhichextremelysmall(usually nanometric)aggregates(or clusters)ofthenewphaseareformed,which irreversiblygrowintomacroscopicdomainsofthenewphase.These clustersmayforminthebulkoftheoldphase(homogeneousnucleation), butalsoattheinterfacewithasubstratephase(heterogeneousnucleation).
Consequently,toproperlydescribethephasetransition,itisnecessaryto determinethecharacteristicsoftheseclusters.Again,thisisequivalentto findingthe easiest path(withsmallestenergeticbarrier)betweentheoldand thenewphases.Thisispreciselythepurposeofthevarioustheoriesof nucleation.
2.1.4. Nucleus,subnucleusandsupernucleus
Indeed,thegranulartheoriesofnucleationallaimatdeterminingthework neededtoformaclusterofagivensize(expressedherebythenumber n of monomerscontainedinthecluster).Thisworkcorrespondstothedifference inGibbsenergybetweentheinitialstate(homogeneousoldphase)andthe
PhaseTransition:Nucleation,Growth,AggregationandCoalescence17
intermediatestate(aclusterof n monomersembeddedintheoldphase,see Figure2.1): W (n)= Ginter (n) Gold
Remarkablyenough,mostofthenucleationtheoriesleadtoageneral expressionofthisworkasthebalancebetweensupersaturationandenergetic penalties[KAS00]:
ThefirsttermcorrespondstothegaininGibbsenergyassociatedwiththe formationofthe n-sizedclusterinthesupersaturatedoldphase.Thesecond termgathersallcontributionsresistingthenucleation(i.e.allphenomena associatedwiththeformationoftheclusterthatcauseanincreaseinthe system’senergy).
Onthebasisofthisdescription,theenergeticbarriertobeovercomeforthe phasetransformationisclearlythemaximumvalueofthiswork.Thisvalue, noted W ∗ ,correspondstotheformationofaclusterwithasize n∗ ofthenew phase.Thisspecificsizeiscalled critical,andsuchaclusterisreferredtoasa nucleus (Figure2.1).Indeed,anysmallercluster(n<n∗ )willspontaneously redissolve(backtothelocalenergyminimum,i.e.theoldphase),whereasany largercluster(n>n∗ )willgrowtoformmoreofthenewphase.
Thereisacriticalsize n∗ (correspondingtothenucleus)belowwhich clustersarenon-viableandredissolve(n<n∗ , subnuclei),andabove whichclusters(n>n∗ , supernuclei)cangrow(Figure2.1).Thiscritical sizeisdeterminedbytheconditionofmaximum W (n):
(n)
Inthiscase,equation[2.8]becomes:
W ∗ iscalled workofnucleation.
2.1.5. Phasetransition:astep-by-steptransformation
Fromtheconsiderationsabove,nucleationhasbeenidentifiedasthe preferredprocesstotransformtheoldphaseintothenewphase.However, nucleationisjustastart.Thenucleiareusuallyverysmall,andinmanycases alotofmonomersremainintheoldphase.Therefore,asignificantportionof thetransformationimpliesthegrowthofsupernuclei.Astheirsizeand numberincrease,theystarttocoalesce,untilallmonomersareconsumed. ThesedifferentstagesareschematicallydepictedinFigure2.2.Ourgoalin thefollowingsectionsistostudyeachoftheseprocessesaswellastheir interplay.
2.2.Nucleation
Oldphase Nucleation Growth Coalescence Newphase Aggregation
Theconceptofnucleationhasbeenintroducedintheprevioussectionon thebasisofenergeticconsiderations.Here,weshalldescribethis phenomenonintermsofelementaryprocesses.Wepurposelyrestrict ourselvestotheso-called classicalnucleationtheory,whichissufficientto
Figure2.2. Thedifferentstagesofphasetransformation
