Diffusion-controlled Solid State Reactions: in Alloys, Thin-Films, and Nanosystems

491 Pages·2010·6.8 MB·English

by  Andriy M. Gusak

AndriyM.Gusak
Diffusion-controlled
SolidStateReactions
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Andriy M. Gusak
Diffusion-controlled Solid State Reactions
in Alloys, Thin Films, and Nano Systems
With the collaboration of
Yuriy A. Lyashenko
Semen V. Kornienko
Mykola O. Pasichnyy
Aram S. Shirinyan
Tatyana V. Zaporozhets
TheAuthor AllbookspublishedbyWiley-VCHare
carefullyproduced.Nevertheless,authors,
Prof.AndriyM.Gusak editors,andpublisherdonotwarrantthe
CherkasyStateUniversity informationcontainedinthesebooks,
DepartmentofTheoreticalPhysics includingthisbook,tobefreeoferrors.
Cherkasy,Ukraine Readersareadvisedtokeepinmindthat
[email protected] statements,data,illustrations,procedural
detailsorotheritemsmayinadvertentlybe
inaccurate.
ConsultantEditor
Dr.habil.Ju¨rnW.P.Schmelzer LibraryofCongressCardNo.:appliedfor
PhysicsDepartment
UniversityofRostock BritishLibraryCataloguing-in-Publication
Rostock,Germany Data
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publicationintheDeutscheNationalbibli-
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ISBN:978-3-527-40884-9
V
Foreword
IhaveknownProf.AndriyM.Gusaksince1991,andIvisitedhiminthecampus
of Cherkassy National University twice. He has visited our campus at UCLA
seven times and during each visit he has spent from one to four months in
my research group to work together on diffusion-controlledsolid-state reactions.
It is a topic relevant to processing and reliability of modern microelectronic
devices, especially thin film-based devices. For example, metal–silicon reactions
are crucial in forming millions of electrical contacts and gates on a piece of Si
chip. Another example is Pb-free solder joint reactions in flip-chip technology,
which are the most important processing steps in the packaging of consumer
electronic products. As we approach the end of Moore’s law of miniaturization,
a change of paradigm is being developedin order to focuson more applications
of the existing CMOS technology besides pushing harder and harder to achieve
smaller and smaller devices. When nanotechnology becomes mature and finds
applications, diffusion-controlledsolid-state reactions in nanoscale materials will
be of major concern from the point of view of stability and reliability of the
nanodevices.
Prof.Gusakisnotonlyanoutstandingphysicist,butalsoasuperbmathemati-
cianinapplyingmathematicalanalysistonewphysicalfindings.Hehaspublished
many highly cited papers, especially in kinetics of solid-state reactions. He has
made significant contributions to reaction kinetics in bulk materials, thin films,
andnanoscalemicrostructures.Hehascoveredkineticprocessesfromnucleation,
growth, to ripening. On reliability issues, he has applied the concept of irre-
versible processesto electromigration,thermomigration,and stressmigration in
interconnectandelectronicpackagingtechnology.Ihavebenefitedtremendously
from working with him on the reliability issues, and I appreciate his deep in-
sight and subtle understanding of some of the very salient nature of solid-state
reactions.
In this book, Prof. Gusak and his former students have made a systematic
presentationofthefollowingtopics:
(i) Theinitialstagesofreactivereactions,includingnucleationandlateralgrowth
of nucleated islands. The nucleation of different phases is considered as a
coupled process in open systems, under conditions of external fluxes and
gradientsofchemicalpotential(Chapters3,4,and5).
Diffusion-ControlledSolidStateReactions.AndriyM.Gusak
Copyright2010WILEY-VCHVerlagGmbH&Co.KGaA,Weinheim
ISBN:978-3-527-40884-9
VI Foreword
(ii) Other flux-driven processes in open systems leading to specific morphol-
ogyevolution – flux-driveripeningofintermetallicscallopsduringreaction
betweenmoltensolderandmetal(copperornickel),flux-drivenlateralgrain
growth in thin films during deposition, flux-driven bifurcations of Kirk-
endall planes and the related problem of dynamic stability or instability of
these planes, electromigration-driven grain rotation, and electromigration
andthermomigration-drivenevolutionoftwo-phasealloys(Chapter6).
(iii) Voidformation,growth,shrinkage,andmigrationduringreactivediffusion
innanoparticlesandduringelectromigration(Chapter7).
(iv) Phase growth and competition of growth under stressing by direct electric
current(Chapter8).
(v) Interdiffusionandreactivediffusion(includingthenucleationstageandthe
formationandgrowthoftwo-phasezones)intheternarysystems(Chapters
9and10).
(vi) Specialemphasisondiffusionandphasetransformationsinnanosystemsis
made in Chapters 2 and 13 (although other chapters, especially Chapter 7,
alsoincludesomeanalysisofnanosystems).
(vii) Sincereactionscanoccuratvariouswaysandproceedindifferentdirections,
thereareunsolved(ingeneralcase)fundamentalproblemsofchoicebetween
theevolutionpathsinclosedandinopensystems.Thesetopicsareintensively
discussedinChapters11and12.
I highly recommend this book to students and to experienced researchers
working in the field of materials science, nonequilibrium thermodynamics, and
nucleationtheory.Iamsureitwillbecomeasourceofnewideasandinspiration
foranybodydealingwiththesetopics.
LosAngeles,November2009 King-NingTu
http://www.seas.ucla.edu/eThinFilm/
VII
Contents
Editor’sPreface XVII
ListofContributors XIX
1 Introduction 1
AndriyM.Gusak
2 NonequilibriumVacanciesandDiffusion-ControlledProcessesat
Nanolevel 11
AndriyM.Gusak
2.1 Introduction 11
2.2 BeyondDarken’sApproximation 12
2.3 TheModelforRegularChainsofIdealVacanciesSinks/Sources 17
2.4 DescriptionofInterdiffusioninAlloysatRandomPowerofDistributed
VacancySinks 20
2.5 LinearPhaseGrowthandNonequilibriumVacancies 22
2.6 IntermetallicLayerGrowthatImposedCurrentandNonequilibrium
VacanciesDampingEffect 25
2.7 PossibleRoleofNonequilibriumVacanciesinSpinodal
Decomposition 26
2.8 NanoshellCollapse 29
2.9 TheRoleofNonequilibriumVacanciesinDiffusionCoarsening 32
2.10 Conclusions 34
References 34
3 DiffusivePhaseCompetition:Fundamentals 37
AndriyM.Gusak
3.1 Introduction 37
3.2 StandardModelandtheAnomalyProblem 37
3.3 CriteriaofPhaseGrowthandSuppression:Approximationof
UnlimitedNucleation 45
3.4 IncubationTime 47
3.5 ShouldWeRelyUpontheIngenuityofNature?NucleationProblems
andMeta-Quasi-EquilibriumConcept 49
Diffusion-ControlledSolidStateReactions.AndriyM.Gusak
Copyright2010WILEY-VCHVerlagGmbH&Co.KGaA,Weinheim
ISBN:978-3-527-40884-9
VIII Contents
3.6 SuppressionofanIntermediatePhasebySolidSolutions 52
3.6.1 UnlimitedNucleation 53
3.6.2 FiniteRateofNucleiFormation 54
3.7 PhaseCompetitioninaModelofDividedCouple 55
References 59
4 NucleationinaConcentrationGradient 61
AndriyM.Gusak
4.1 Introduction 61
4.2 NucleationinNonhomogeneousSystems:GeneralApproach 63
4.3 ThermodynamicsofthePolymorphicModeofNucleationina
ConcentrationGradient 65
4.3.1 HomogeneousNucleation:GeneralRelations 65
4.3.2 SphericalNuclei 66
4.3.3 EllipsoidalNuclei 68
4.3.4 MCSimulationsoftheShapeoftheNucleus 70
4.3.5 StressEffects 71
4.4 ThermodynamicsoftheTransversalModeofNucleationina
ConcentrationGradient 74
4.4.1 HomogeneousNucleation:GeneralRelations 74
4.5 ThermodynamicsoftheLongitudinalModeofNucleationina
ConcentrationGradient 79
4.6 NucleationinSystemswithLimitedMetastableSolubility 81
4.6.1 NucleationofaLineCompoundattheInterfaceDuring
Interdiffusion 82
4.6.2 NucleationinbetweenDiluteSolutions 86
4.6.3 NucleationinbetweenTwoGrowingIntermediatePhaseLayers 86
4.6.4 NucleationinbetweenaGrowingIntermediatePhaseandaDilute
Solution 88
4.6.5 ApplicationtoParticularSystems 91
4.7 Conclusions 95
References 97
5 ModelingoftheInitialStagesofReactiveDiffusion 99
MykolaO.PasichnyyandAndriyM.Gusak
5.1 Introduction 99
5.2 FirstPhaseNucleationDelayinAl–CoThinFilms 100
5.2.1 TheProblemofNucleationinaConcentrationGradientField 101
5.2.2 BasicModel 102
5.2.3 TransversalMode 105
5.2.4 PolymorphicMode 107
5.2.4.1 PolymorphicModewithoutShapeOptimization 108
5.2.4.2 PolymorphicModewithShapeOptimization 109
5.2.5 DiscussionandConclusions 110
Contents IX
5.3 KineticsofLateralGrowthofIntermediatePhaseIslandsattheInitial
StageofReactiveDiffusion 112
5.3.1 ProblemFormulation 112
5.3.2 PhysicalModel 114
5.3.3 NumericalResults 116
5.3.4 AnalyticalSolutionfortheSteadyState 117
5.3.5 AsymptoticThicknessofanIsland 118
5.3.6 Estimates 119
5.3.7 Conclusions 121
5.4 MC-SchemeofReactiveDiffusion 121
5.4.1 FormulationoftheProblem 121
5.4.2 TheModel 122
5.4.3 NucleationofPhaseA B attheInterfaceA–A B 124
2 1 1 2
5.4.4 CompetitiveNucleationofPhasesA B andA B attheInterface
1 2 2 1
A–B 129
5.4.5 LateralCompetition 131
5.4.6 Conclusions 131
References 132
FurtherReading 133
6 Flux-DrivenMorphologyEvolution 135
AndriyM.Gusak
6.1 Introduction 135
6.2 GrainGrowthandRipening:Fundamentals 136
6.2.1 MainApproximationsoftheLSWApproach 136
6.2.2 TraditionalApproachestotheDescriptionofGrainGrowth 138
6.3 AlternativeDerivationoftheAsymptoticSolutionoftheLSW
Theory 139
6.4 Flux-DrivenRipeningatReactiveDiffusion 142
6.4.1 ExperimentalResults 143
6.4.2 BasicApproximations 144
6.4.3 BasicEquations 145
6.5 Flux-DrivenGrainGrowthinThinFilmsduringDeposition 148
6.5.1 ‘‘MushroomEffect’’ontheSurfaceofaPairofGrains:Deterministic
Approach 150
6.5.2 AnalysisofFlux-DrivenGrainGrowth 151
6.5.3 StochasticApproach 154
6.5.4 MonteCarloSimulationofFlux-DrivenGrainGrowth 155
6.5.5 LateralGrainGrowthinAluminumNanofilmduringDeposition 156
6.5.5.1 Hillert’sModel 160
6.5.5.2 ModelsLeadingtoaRayleighDistribution 161
6.5.5.3 PairInteractionModel(DiNunzio) 161
6.6 Flux-InducedInstabilityandBifurcationsofKirkendallPlanes 163
6.6.1 KirkendallEffectandVelocityCurve 164
6.6.2 StableandUnstableK-Planes 165
X Contents
6.6.3 ExperimentalResults 166
6.6.4 GeneralInstabilityCriterion 168
6.6.5 EstimationofMarkers’DistributionsNeartheVirtualK-Plane 169
6.6.6 SpatialDistributionofMarkers 170
6.6.7 PossibleAlternativetotheMultilayerMethod 171
6.7 Electromigration-InducedGrainRotationinAnisotropicConducting
BetaTin 173
6.8 ThermomigrationinEutecticTwo-PhaseStructures 178
6.8.1 ThermomigrationInducedBackStressinTwo-PhaseMixtures 183
6.8.2 Thermomigration-DrivenKirkendallEffectinBinaryMixtures 184
6.8.3 StochasticTendenciesinThermomigration 185
References 186
7 NanovoidEvolution 189
TatyanaV.ZaporozhetsandAndriyM.Gusak
7.1 Introduction 189
7.2 KineticAnalysisoftheInstabilityofHollowNanoparticles 191
7.2.1 Introduction 191
7.2.2 MechanismofNanoshellShrinkage 192
7.2.3 ModelsofNanovoidShrinkage 194
7.2.3.1 Model1:ShrinkageofPureElementNanoshells 195
7.2.3.2 Model2:ShrinkageofaBinaryCompoundNanoshellwithSteady
StateApproximationforBothVacanciesandBSpecies 197
7.2.3.3 Model3:SteadyStateandNon–SteadyStateVacanciesfor
ComponentB 200
7.2.3.4 Model4:Non–SteadyStateVacanciesandAtoms 204
7.2.4 SegregationofPureBattheInternalSurface 205
7.2.5 KineticMonteCarloSimulationofShrinkageofaNanoshell 206
7.2.5.1 Model1MC:PureB-ShellinVacuum 207
7.2.5.2 Model2MC:OrderedIMCNanoshellinVacuum 208
7.2.6 InfluenceofVacancySegregationonNanoshellShrinkage 208
7.2.7 Summary 215
7.3 FormationofCompoundHollowNanoshells 216
7.3.1 Introduction 216
7.3.2 ModelofNanoshellFormation 216
7.3.3 SimplifiedAnalysisoftheCompetitionBetween‘‘Kirkendall-Driven’’
and‘‘Curvature-Driven’’Effects 218
7.3.4 RigorousKineticAnalysis 220
7.3.5 ResultsandDiscussion 225
7.3.6 Summary 228
7.4 HollowNanoshellFormationandCollapseinOneRun:Modelfora
SolidSolution 229
7.4.1 Introduction 229
7.4.2 Shrinkage 229