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BOSTON UNIVERSITYCOLLEGEOFENGINEERING
DISSERTATION
MODELING AND SIMULA TION OF TRANSIENT ENHANCEDDIFFUSION BASED ON INTERA CTIONS OF POINT AND EXTENDED
DEFECTS
by
ALP H. GENCERB.S.,Bogazici University, 1993M.S.,BostonUniversity, 1995
Submittedin partialfulfillment of therequirementsfor thedegreeof
Doctorof Philosophy1999
ACKNOWLEDGEMENTS
FirstandforemostI would like to expressmy thanksto my professor, Prof.ScottDunham,whoseknowledgemadethis thesispossiblein the first place. His deepknowledgein the field of processanddevicemodeling,andhiscuriosityto learnmorehasalwayskepthim andmeat thecuttingedge.Without his forwardlooking vision andstrive for perfectionthis pieceof work would not beasgoodasI liked it to be. I am alsograteful to him to allow me to spendtime with my family andworkremotelywhenmy life tooka turnwith thebirth of my daughter.
I amalsothankful to Prof.TheodoreMoustakasfor passingon his deepknowledgeon semicon-ductorphysicsto me,for encouragingmeto join BU, andsupportingmewith ateachingassistantshipin my first year, for giving valuablesuggestionsasa memberof my proposalanddissertationcom-mittee. I alsowould like to expressmy thanksto Prof. DanielCole, for servingon my dissertationcommitteeandproviding a long list of suggestionsandcorrections.Finally, I am alsothankful toProf.Karl Ludwig for bringingaphysicist’saspectto this work.
During theyearsI spentat BU’s ProcessModelingLab, I metseveralpeople.It wasa greatfunworkingwith all of them.In particular, I would like to thankAnu Agarwal for herencouragingmetojoin thegroup,SoumyaBanerjeeandMitra Navi for beinggreatfriends,andSrinivasanChakravarthiandBrendonMurphy for working with meall the time. I alsowant to expressmy specialthankstoFredWittel, who not only thoughtmehow to bea greatscientistandhelpedmein surviving my firstyearwith thegroup,but alsoshowedmehow to beagooddad.
I alsowould like to expressmy thanksto my parentsfor providing me theeducationthatmadethis thesisa reality. I ammorethanthankful to my parents-in-law, who providedconstantsupportinevery mannerthey couldandput methroughthis work. Without their support,I would not find thecourage,powerandopportunityto completemy Ph.D.work.
Finally, I would like to expressmy love to my wife Aysenandmy daughterAylin. Without them,I certainlywouldn’t bethepersonwho I amtoday.
ii
MODELING AND SIMULA TION OF TRANSIENT ENHANCEDDIFFUSION BASED ON INTERA CTIONS OF POINT AND EXTENDED
DEFECTS
(OrderNo. )ALP H. GENCER
BostonUniversity, CollegeOf Engineering,1999Major Professor:ScottT. Dunham,Professorof ElectricalEngineering
ABSTRACT
As device sizesin VLSI technologyget smaller, the importanceof predictive processmodelingincreases.Oneof the biggestchallengesin predictive processmodelingtoday lies in modelingofTransientEnhancedDiffusion (TED). TED is the greatlyenhanceddiffusion of dopantsin siliconseenduringannealingof thedamagecreatedby theion implantationof thedopants.As onemovestosmallerthermalbudgets,TED is often theprimarysourceof diffusionandthusdeterminesthefinaljunctiondepth.
It is known that TED is causedby the excessinterstitial concentrationthat persistsdue to ionimplantation.But how thisexcessinterstitialconcentrationevolvesovertimeandaffectsthediffusionof dopantsremainsunclear. Ourwork attemptsto understandthephysicalprocessesoccurringduringion implant annealing,expressthemin a mathematicalmodel,integratethis model into a diffusionequationsolverandquantitatively matchtheexperimentalobservations.
To this end, we have developeda solid physicalmodel (KPM) for the evolution of extendeddefects(
�311� defectsanddislocationloops)which areobserved underTED conditions. We have
alsodevelopeddifferentversionsof KPM thathave applicabilityunderdifferentcircumstances,andhavedifferentlevelsof computationalefficiency. We believe thattherangeof modelsdevelopedwillgive theusertheability to makea compromisebetweenaccuracy andcomputationaltime.
We have appliedKPM to�311� defectsthat areobserved undernon-amorphizingimplant con-
ditions andwe wereable to get a goodagreement.We have thenusedthis model to predictTEDbehavior basedon marker layer experimentsandwe found a goodmatch. To extendthe model todislocationloops,weassumedthatdislocationloopsform by unfaultingof
�311� defectsasobserved
experimentally. Weaccountedfor this transformationin ourmodelandwewereableto obtainagoodmatchto theexperimentaldatawithoutany modificationsin the
�311� defectmodel.
Our work alsoinvolvedin developinga computersoftwarethat is capableof solving themodelsthatwehavepostulated.To thisend,wehavedevelopedDOPDEES,aone-dimensionalmulti purposepartial differentialequationinitial valuesolver. To enablefastertechnologytransfer, we have alsodevelopedProcessModeling Modules(PMM) which consistsa set of scriptsthat encapsulatethemodelsthatwehavedevelopedin a readyto useform.
iii
Contents
1 UnderstandingTED 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Experimentaldataon TED . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2.1 Earlywork onTED . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2.2 Separatingtheenhancementfrom thedamagedose . . . . . . . . . . . . . . 4
1.2.3 Thesourceof theinterstitials. . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2.4 Interactionsinvolving dislocationloopsandboron-interstitialclusters . . . . 6
2 Modelsof Importance for TED 7
2.1 Coupleddiffusionof dopantsandpoint defects . . . . . . . . . . . . . . . . . . . . 7
2.1.1 Fermidiffusionmodel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.1.2 Pair diffusionwith asinglepointdefect . . . . . . . . . . . . . . . . . . . . 7
2.1.3 Pair diffusionwith bothpointdefects . . . . . . . . . . . . . . . . . . . . . 9
2.1.4 Pair diffusionwith asinglepointdefect,includingFermilevel effects . . . . 10
2.1.5 Pair diffusionwith bothpointdefects,includingFermilevel effects . . . . . 12
2.1.6 Fivestreammodel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.2 Initial conditions:Damagecreatedby ion implantation . . . . . . . . . . . . . . . . 15
2.2.1 Non-amorphizingimplants . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2.2 Amorphizingimplants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.3 Otherparameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.3.1 Pointdefectproperties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.3.2 Surfacerecombination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3 Modeling ExtendedDefectswith Kinetic Precipitation Model 21
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.2 Full Kinetic PrecipitationModel (FKPM) . . . . . . . . . . . . . . . . . . . . . . . 21
3.3 ReducedKinetic PrecipitationModels(RKPM) . . . . . . . . . . . . . . . . . . . . 23
iv
3.3.1 3-momentmodel(3KPM) . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.3.2 2-momentmodel(2KPM) . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.4 AnalyticalKinetic PrecipitationModel (AKPM) . . . . . . . . . . . . . . . . . . . . 26
3.5 Simplesolid-solubilitymodel(SSS) . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4 Modeling�311� Defectsand TED 32
4.1 Evolutionof�311� DefectProfile . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.2 Predictionof TED . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.3 Comparisonof models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.4 Effectof “ � n” factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
5 Models for DislocationLoops 45
5.1 ExtendingKPM to dislocationloops . . . . . . . . . . . . . . . . . . . . . . . . . . 45
5.2 Analytical loopmodel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
5.3 SimulatingTED . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
5.4 Modelingheterogeneousnucleation . . . . . . . . . . . . . . . . . . . . . . . . . . 51
5.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
6 SoftwareDevelopmentEfforts 54
6.1 DOPDEES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
6.2 ProcessModelingModules(PMM) . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
6.3 Simulationof dopantdiffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
7 Summary and Future Dir ections 62
7.1 Modelingandexperiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
7.2 Softwaredevelopment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
A Calculation of the Kinetic Precipitation Rate 64
A.1 Sphericaldefects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
A.2 Disc-shapeddefects: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
A.3�311� defects:. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
B BU Parameter Set 68
v
List of Figures
1.1 Time dependenceof TED. Implantationof 5 � 1013 – 2 � 1014cm� 2 Si at 200keV,anddiffusionat 800� C. Datafrom Packan[39]. . . . . . . . . . . . . . . . . . . . . 3
1.2 Temperaturedependenceof TED. Implantationof 1 � 1014cm� 2 Si at 200keV, anddiffusionat 800� Cand850� C. Datafrom Packan[39]. . . . . . . . . . . . . . . . . . 3
1.3 Energy dependenceof TED. Implantationof 1 � 1012cm� 2 Si at 10–200keV, anddiffusionat 800–1000� C. Datafrom Packan[39]. . . . . . . . . . . . . . . . . . . . 4
1.4 Dosedependenceof TED. Implantationof 1 � 1012 – 1 � 1014cm� 2 Si at 200keV,anddiffusionat 800–1000� C. Datafrom Packan[39]. . . . . . . . . . . . . . . . . . 5
2.1 Comparisonof UT-Marlowe andTRIMCSR simulationsof a 40keV 5 � 1013cm� 2
Si implantat 7o tilt and45o rotation. . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.2 Total andnet damagecreatedby a 40keV 5 � 1013cm� 2 Si implant. Monte Carlosimulationswith TRIMCSR. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.3 Damagecreatedby a 40keV 5 � 1013cm� 2 Si implant beforeandafter very shortannealing.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.4 Net damage(I � V) dosedivided by implant dose(plus factor) for a seriesof non-amorphizing40keV Si implants. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.5 TRIMCSRsimulationof totalandnetimplantdamagefor a50keV 1 � 1015cm� 2 Siimplant.Amorphizationthresholdhasbeenassumedto be10%. . . . . . . . . . . . 19
3.1 Growth anddissolutionof precipitatesby attachmentandemissionof soluteatoms. . 22
3.2 Distribution of�311� defectdensitiesover defectsizesand bestfit to log-normal
distribution. z2 � 0 � 8 hasbeenusedin all fits. Datafrom PanandTu [41]. . . . . . . 25
3.3 Changeof λn versusdefectsizefor�311� defectsanddislocationloops. Thecalcu-
lationcanbefoundin AppendixA. . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.4 Thecorrespondingγi asa functionof m1 for a0 � 3 � 855,a1 � 15� 9, a2 � � 1 � 4. Theparametersfor γi areK0 � 3, K1 � 14� 5 andK2 � 366. . . . . . . . . . . . . . . . . 29
3.5 Thecorrespondingγi asafunctionof m1 for a0 � 1, a1 � 4 � 7, a2 � 0. Theparametersfor γi areK0 � 8, K1 � 0 � 14andK2 � 8. . . . . . . . . . . . . . . . . . . . . . . . . 29
3.6 Exampleof γi asdefinedby Eq.3.27.In thisparticularplot K0 � 4, K1 � 2 andK2 � 7havebeenused. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
vi
3.7 Comparisonof 2KPM andAKPM underidenticalconditionsandequivalentparame-tersasgivenin Fig. 3.4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4.1 Evolution of densityof interstitialsin�311� defects(m1) andcomparisonto model.
Datafor 5 � 1013cm� 2 Si implantat 40keV with annealat 815� C. . . . . . . . . . . 33
4.2 Evolution of theaveragelengthof�311� defectsandcomparisonto model.Datafor
5 � 1013cm� 2 Si implantat 40keV with annealat 815� C. . . . . . . . . . . . . . . . 33
4.3 Evolution of density of�311� defects(m0) and comparisonto model. Data for
5 � 1013cm� 2 Si implantat 40keV with annealat 815� C. . . . . . . . . . . . . . . . 34
4.4 The�311� defectprofile for variousannealingtimesat 815� C. . . . . . . . . . . . . 34
4.5 C n andCss versus�311� defectsizefor annealsat815� C. . . . . . . . . . . . . . . . 35
4.6 Normalizeddistributionof�311� defectsizesfor varioustimesfor annealsat 815� C.
Theverticalline representstheminimumobservabledefectsize(20A). . . . . . . . 36
4.7 Evolution of densityof interstitialsin�311� defects(m1) andcomparisonto model.
Datafor 5 � 1013cm� 2 Si implantsat 40keV with annealsat 815� C and670� C. . . . 36
4.8 Predictionof dosedependenceof interstitialsin�311� defectsduringinitial stagesof
TED. Datafrom Poateet al. [46] for 40keV Si implantsand1hr annealat 670� C. . . 37
4.9 Energy dependenceof TED. Total broadeningof a deepB marker profile due toimplantationof 1 � 1014cm� 2 29Si with annealingat800� C for 60min. . . . . . . . 38
4.10 Timedependenceof TED. Totalbroadeningof aB markerprofiledueto implantationof 1 � 1014cm� 2 29Si at 80and200keV with annealingat 800� C. . . . . . . . . . . 38
4.11 Dosedependenceof TED. Totalbroadeningof aB markerprofiledueto 29Si implan-tationat200keV with annealingat800� C for 45min. . . . . . . . . . . . . . . . . 39
4.12 Comparisonof 1, 2 and3-momentmodelsfor evolutionof�311� defects.Both2 and
3-momentmodelscapturetheexponentialdecayof interstitialsin�311� defects. . . 40
4.13 Evolutionof densityof interstitialsin�311� defects(m1) andcomparisonto analytical
model(AKPM) and3-momentmodel(3KPM). . . . . . . . . . . . . . . . . . . . . 40
4.14 Evolutionof theaveragelengthof�311� defectsandcomparisonto analyticalmodel
(AKPM). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.15 Comparisonof 1, 2 and3-momentmodelsfor TED predictions.The modelsdiffersignificantlyonly at shorttimebehavior, which this datacannotdistinguish. . . . . . 42
4.16 Comparisonof 1 and2-momentmodelsfor shorttimediffusivity enhancements.Datafrom Chao[6] for a5 � 1013cm� 2 50keV implantwith annealsat 750� C. . . . . . . 42
4.17 Comparisonof predictionsfor the dosedependenceof TED using“ � 1” and“ � n”models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
5.1�311� defectanddislocationloop energy asa functionof defectsize. Theloopsare
morestableat largersizes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
5.2�311� defectanddislocationloopC n asa functionof defectsize. . . . . . . . . . . . 46
vii
5.3 Evolutionof densityof interstitialsin extendeddefects(m1) andcomparisonto model.Datafrom Panet al. [42] for 1 � 1016cm� 2 Si implantat 50keV. . . . . . . . . . . . 47
5.4 Evolutionof averagedefectsize(m1 m0) for 1 � 1016cm� 2 Si implantsat50keV andcomparisonto datafrom Panet al. [42] . . . . . . . . . . . . . . . . . . . . . . . . 47
5.5 Evolutionof densityof interstitialsin dislocationloops(m1) andcomparisontomodel.Datafor 1 � 1015cm� 2 Si implantat 50keV. . . . . . . . . . . . . . . . . . . . . . . 48
5.6 The γ1 function for dislocationloopsat asderived from ∆Gexcn for dislocationloops
anda log-normalclosureassumption. . . . . . . . . . . . . . . . . . . . . . . . . . 49
5.7 The γ1 function for dislocationloopsat 1000� Cfor the AKPM. For m1 � ncrit, γ1 isidenticalto that for
�311� defects,exceptfor a multiplier to accountfor differences
in solid solubility. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
5.8 Evolutionof totalnumberof interstitialsstoredin dislocationloopsand�311� defects
for theAKPM. Datafor 1 � 1016cm� 2 Si implantat 50keV with annealat850� C. . . 50
5.9 Evolutionof averagedefectsizefor 1 � 1016cm� 2 Si implantsat 50keV from AKPM. 51
5.10 Predictionof TEDusingFKPM for amorphizingandnon-amorphizingimplants.Datafor 50keV Si implantswith annealingat 750� C. . . . . . . . . . . . . . . . . . . . . 52
5.11 Distributionof C n for heterogeneousnucleation.NotethatC n will bedifferentfor thepureloopdistributionbasedonspatiallocation. . . . . . . . . . . . . . . . . . . . . 53
6.1 A structurein DOPDEESconsistsof regions,which eachhave a numberof fields(solutionvariables)anda grid structure.Eachfield is associatedwith anequationofthetype∂ f ∂t � ∑opi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
6.2 DOPDEEShandlesPDEsby discretizingthemusingfinite differencesandfeedingtheminto astandardODE solver. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
6.3 Modelsimplementedin PMM andtheir interdependencies.Modulescaninherit pa-rametersfrom lower level modulesanddo consistency checks. . . . . . . . . . . . . 58
6.4 DOPDEESinput script for thesolvingtheequationsystemfor co-diffusionof boronandarsenic. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
6.5 DOPDEESinput script for thesolvingthesameequationsystem,but with theFermimodelimplementedusingaPMM script. . . . . . . . . . . . . . . . . . . . . . . . . 61
6.6 Theinitial dopantprofilesandfinal profilesaftera30min annealat 900� C accordingto simulationusingtheFermimodel. . . . . . . . . . . . . . . . . . . . . . . . . . . 61
A.1 Divisionof thecapturecross-sectionof a�311� defectinto spheresof equalarea. . . 66
viii
Chapter 1
UnderstandingTED
1.1 Intr oduction
Oneof the most importantchallengesin VLSI technologytoday is to shrink device sizesto theirlimits, sinceboth the speedandthe yield (fraction of deviceswhich work) increaseasdevicesgetsmaller. However, as devices get smallerand junctionsget shallower, somephenomenabegin toget moreandmoreimportantandthe validity of currentprocesssimulatorsbecomesquestionable.Predictive processsimulationplaysa major rolesin today’s TechnologyCAD (TCAD), andhence,weneedimprovementsin currentprocesssimulators.
As onetries to fabricateshallower junctions,onemovesto smallerthermalbudgetsduring pro-cessingwith thehopeof having lessdiffusionof dopants.But, onecomponentof thediffusion,calledTransientEnhancedDiffusion(TED), remains.TED is thegreatlyenhanceddiffusionof dopantsinsiliconseenduringannealingof thedamagecreatedby theion implantationof thedopants.It is oftenthecasethat for today’s processingparametersTED is theonly significantcomponentof diffusion.Thus,theneedfor accuratemodelsthatcanaccountfor thebehavior seenduringTED emerges.
Ourwork aimsatobtainingaphysicalmodelthatcanbeappliedto thebehavior of dopantsunderTED conditions.AlthoughtherearenumerousarticlesonexperimentalobservationsconcerningTEDin theliterature[2, 9, 29, 30,31,36,40], little fundamentalmodelinghasbeendone[13, 19,34,45].Our work hasthefollowing uniquefeatures:� Our work relieson a physicalmodelof diffusion and interactionof dopantatomsandpoint
defects,ratherthanonanempiricalapproach.� Thework is basedon theevolutionof thesizedistributionof extendeddefectsduringTED.� It alsoincludesdeactivationof dopantsandincorporationof point defectsin dopantclustersorprecipitates.
Having accountedfor theprimaryphysicalprocessesthatareoccurringduringTED, ourgoalwasto arrive at a meaningful,andyet computationallyefficient model that can be incorporatedinto adiffusionequationsolver. It is noteworthy to mentionthata physicalmodelcanbeincorporatedintoany diffusionequationsolver, sinceit is independentof thepropertiesof thesolver, like thenumberof dimensions.
1
1.2 Experimental data on TED
1.2.1 Early work on TED
Early work on TED concentratedon experimentalobservations,measuringtheextentof TED. Afterthe inventionof rapid thermalannealing(RTA), first observationsof TED werereported:even forveryshorttimesof annealingtherewasaconsiderableamountof dopantdiffusion.
Mostof theearlypapers[2, 36, 49] dealwith dopantimplantsinto silicon. A (high)doseof boron,phosphorusor arsenicwasimplantedinto silicon,andafteranRTA stepthefinal profilewasmeasuredto determinetheextentof TED. Angeluccietal. observedthatboronandphosphorusshow TED to alargeextent,but arsenicshows little TED, andantimony shows almostno TED [3, 2]. To show this,Angelucciformeduniform layersof thedopants,patternedthesample,createda damageby siliconimplantationandannealedat varioustimesandtemperatures.For B andP, theSi-implantedregionsshoweda largeincreasein junctiondepth,whereasfor As andSb,thedifferencewasminimal.
The enhancement/retardationof diffusion during oxidation,nitridation andoxynitridation indi-catesthatboronandphosphorusdiffuseprimarily via pairingwith interstitialsandantimony andar-senicdiffuseprimarily via pairingwith vacancies[18]. It hasbeenobservedthatboronandphospho-rus show enhanceddiffusion during oxidation,andduring oxidationinterstitial-typestackingfaultsgrow. Therefore,oxidationmustbe injecting interstitialsandboronandphosphorusmusthave aninterstitialdominateddiffusionmechanism.Similarly, antimony diffusionis enhancedduringnitrida-tion, but retardedduringoxidation.Thereforeantimony mustbediffusingprimarily via vacancies.
Combiningthe knowledgeon primary diffusionmechanismsof individual dopantswith the ob-servationson TED for differentdopants,we mayconcludethatTED is relatedto interstitialassisteddiffusion. Indeed,afteranion implantation,thereis a high super-saturationof interstitialsdueto thedamageof theimplant,andthus,this resultis quitelogical.
If welookatthetimebehavior of TED,wecanobservethattheenhancementis nearlyindependentof the ion-implantdamagefor initial times(Fig. 1.1); andafter someperiod(durationof TED) theenhancementgoesaway [2, 39], suchthatwe areleft with normaldiffusionwhich is many ordersofmagnitudesmaller. This meansthat for early stagesof TED the excessinterstitial concentrationisapproximatelyfixed,andaftersometime it dropsto its equilibriumvalue.
Anotherseeminglyanomalousobservationis that theamountof TED is largerat lower tempera-tures[36, 39], asdemonstratedin Fig. 1.2. This canbeexplainedin thefollowing fashion:Althoughthediffusionof dopantsis fasterat highertemperatures,thedurationof TED lastsshorterat highertemperatures,sothattheoverall junctionmovementgetssmallerasthetemperatureis increased.
Hence,thefollowing conclusionsarein order:� TED is causedby excessinterstitialconcentrationthatpersistsafterion implantation.� Theexcessinterstitial concentrationremainsapproximatelyfixedduringTED, andthendropsto its equilibriumvalue.� At higher temperaturesthe excessinterstitial concentrationdisappearsmore rapidly, i.e. thedurationof TED is shorter.
All of theabovequalitativeobservationswill helpusto build ourmodel.
2
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Time (min)
800oC 850oC
(D
t)1/
2 (n
m)
�
� � � �
�
� � � � �
Figure1.2: Temperaturedependenceof TED. Implantationof 1 � 1014cm� 2 Si at 200keV, anddif-fusionat 800� Cand850� C. Datafrom Packan[39].
3
|�0� |�
20|�
40|�
60|�
80|�
100� |�120� |�
140� |�160� |�
180� |�200� |�
220�|�0
|�10
|�20
|�30
|�40
|�50
|�60
|�70
|�80
|�90
|�100
|�110 |� |� |� |� |� |� |� |� |� |� |� |�
|� |� |� |� |� |� |� |� |� |� |� |�
Energy (keV)
(D
t)1/
2 (n
m)
! !! ! !
" "" " "
# ## # #
Figure1.3: Energy dependenceof TED. Implantationof 1 � 1012cm� 2 Si at 10–200keV, anddiffu-sionat 800–1000� C. Datafrom Packan[39].
1.2.2 Separatingthe enhancementfr om the damagedose
In orderto gainmoreinsightinto themechanismof TED,experimentsdevotedto this topichavebeendone[30, 31,40, 44]. Sincein theexperimentsdiscussedin theprevioussectionit wasimpossibletoseparatetheenhancementfrom thedamagedose,researchersdesignedexperimentsfor this purpose.They put a tracerdoseof the dopant(usuallyboronor phosphorus)deepinto silicon. The damageis usuallycreatedby implantingsilicon into thesamesample.Thus,thedamagedoseis controlledindependentlyfrom the dopantdose. This enablesus to investigateregionswherethe materialisintrinsic,andthedopantis far from its solidsolubility limit.
Sinceamorphizationandsolid phaseepitaxialregrowth is a complex processitself, experimentsaimedat avoiding additionalunknownsusedimplantdosesbelow theamorphizationthreshold.Theamorphizationof silicon with silicon implantsoccursusuallyat dosesabove 2 � 1014cm� 2, whichalsodependson theimplantenergy.
Theexperimentsconfirmtheresultsmentionedin theabovesubsectionsandaddsomemoreinter-estingaspectsto thepicture.Firstof all, theamountof TED increaseswith increasingsilicon implantenergy [39] (Fig. 1.3. This is a bit surprising,since,althoughthe amountof total damagegener-atedincreaseswith implantenergy, thenet interstitialexcessis nearlyindependentfrom theimplantenergy, only the positionof thepeakof damagedependingon implantenergy. Looking at the timebehavior atdifferentimplantenergies,wecanconcludethattheinitial enhancementis aboutthesamein all cases,only thedurationof TED is longerfor deeperimplants[39].
Thereasonbehindthis observationis very simpleif oneassumestheprimarysink for theexcess
4
$%&
|' |' |� |' |' |' |' |' |' |' |' |� |' |' |' |' |' |' |' |' |�|�0
|�10
|�20
|�30
|�40
|�50
|�60
|�70
|�80
|�90
|�100
|�110 |( |( |� |( |( |( |( |( |( |( |( |� |( |( |( |( |( |( |( |( |�
|� |� |� |� |� |� |� |� |� |� |� |� 800oC 900oC 1000oC
1012 1013 1014
Dose (cm) -2)
(D
t)1/
2 (n
m)
* * * * * *
+ + + + +, , , , ,
Figure1.4: Dosedependenceof TED. Implantationof 1 � 1012 – 1 � 1014cm� 2 Si at 200keV, anddiffusionat 800–1000� C. Datafrom Packan[39].
interstitialsto bethesurface:Thefartherawaythedamagefrom thesurfaceis, theharderit will betoget rid of it. Although interstitial-vacancy (I/V) recombinationmight play a role in beinga sink forinterstitials,otherexperimentsindicatethatit is fast,suchthatproductof concentrationof interstitialsandconcentrationof vacancies(CICV) dropsto its equilibriumvaluein theveryearlystagesof TED,which leavesusthesurfaceastheprimarysink.
Oneother interestingobservation is that the dependenceof TED on the silicon implant doseisnon-linear[40] (Fig. 1.4). Evenfor silicon implantdosesaslow as1 � 1012cm� 2 substantialdopantmovementis observed. This behavior is one of the biggestchallengesin TED and still no fullysatisfactoryexplanationhasbeenfound.
Anotherobservation is the fact that increasingthe dopantimplant dosedecreasesTED [31]. Itmight be expectedthat the dopants,too, are “using up” interstitials. It hasbeenobserved that aconstantratio of the marker layer doseto the damagedosegivesan almostconstantdiffusion en-hancement.
Hence,thefollowing conclusionsarein order:� Higherenergy damageimplantsresultin moreprofilemovement.� Surfaceis theprimarysink for excessinterstitials.� TED increasessub-linearlywith increasingdamagedose.� TED decreaseswith increasingdopantmaker layerdose.
5
1.2.3 The sourceof the interstitials
As mentionedin section1.2.1,theexcessinterstitialconcentrationremainsapproximatelyfixeddur-ing TED, andthendropsto its equilibriumvalue. This tells us that theremustbea mechanismthatstoresthe interstitialscreatedby the implantdamageandthenreleasesthemduringTED, actingasa “source”of interstitials. In fact, if sucha meta-stablestatefor interstitialsdidn’t exist, they wouldrapidlydiffuseto thesurfaceandTED wouldbeover in averyshorttime.
Recentexperimentsrevealedthe actualsourceof the interstitialsduring TED [10, 17]. Eagle-shamet al. createdthe damageby implantingSi into Si andthenannealedthe samplesat varioustemperatures.They thenperformedplan-view andcrosssectionalTransmissionElectronMicroscopy(TEM) on thesamples.They clearlysaw thedefectsthatstoreexcessinterstitials.
Thesedefectsare the so-called“rod-like” or “�311� ” defects. The atomic structureof
�311�
defectshasbeenonly recentlyresolved[50]. It is believedthatinterstitialsform chainsalong � 110-directionandthesechainscometogetherto form a
�311� plane.Thisdefectcangetvery long(about
1µm) in the � 110- direction,henceis giventhename“rod-like” defects.
Thefactthatthetimeneededfor dissolutionof�311� defectsis equalto thedurationof TED [17]
is anexcellentindicatorthat�311� defectsarethesourceof theinterstitialsduringTED.
1.2.4 Interactions involving dislocation loopsand boron-interstitial clusters
Studiesconcentratingon formationandevolution of dislocationloopsgeneratedby Si implantationhave beenpublished[7, 35, 52]. It hasbeenobserved that the growth rateof dislocationloops isconstantin time, as it would be predictedby a bulk-diffusion mechanism.A quantitative model,which predictstheloopevolutioncorrectly, hasalsobeendeveloped[7].
Experimentsby the samegroupalsoreport interactionsbetween�311� defectsanddislocation
loops[33]. Resultsclearlyshow thatthedistributionof trappedinterstitialsbetween�311� defectsand
dislocationloopsis dependenton the implantdose.Interestingly, theamountof trappedinterstitialsin�311� defectsseemsto saturatearound2 � 1013cm� 2 at 700� C.
Implantationof boroninto silicon at very low dosesandenergiesproducesno visible�311� de-
fects,but substantialTED is observed [57]. The thresholdfor�311� formation is estimatedto be
5 � 1012cm� 2 for silicon implantsand1 � 1014cm� 2 for boronimplants.Thequestionwhetherbe-low thesedosestiny interstitialclustersexist or not remainsunanswered.
Recentobservationsby Cowernet al. [9, 10], suggestthat in systemswhereB is presentin largedoses,excessinterstitialshelp boronatomsto form boronclustersandarethemselvesincorporatedinto theseclusters(so-calledboroninterstitialclusters,BICs),therebyreducingthenumberof mobileboronatoms.AlthoughtheBICsarenotvisibleevenwith highresolutionTEM, thediffusionprofilesindicatethatboronis becomingimmobilewhereit is presentat highdoses.
6
Chapter 2
Modelsof Importance for TED
2.1 Coupleddiffusion of dopantsand point defects
SinceTED is animmediateresultof enhancementof dopantdiffusiondueto presenceof pointdefects(mainly interstitials) in excessof their equilibrium values(i.e. existenceof a point defectsuper-saturation),thefirst thingwewill investigateis theeffectof point defectsondopantdiffusion. In thissection,we presentvariousdiffusionmodelsanddiscusstheir applicabilityto TED. In thefollowingequations,B representsour dopant,I representsinterstitialsandV representsvacancies.C denotestheconcentrationandD denotesthediffusivity of a certainspeciesindicatedby thesubscript.
2.1.1 Fermi diffusion model
In the mostsimplistic view, dopantdiffusion canbe describedby the “Fermi” diffusion model, inwhich theeffectof point defectson dopantdiffusionis completelyignored:
∂CB
∂t �/.∇ 0 DB .∇CB (2.1)
whereDB is a functionof theFermilevel:
DB � D0B � D 1B 2 p ni 3 � D �B 2 n ni 3 (2.2)
This model is only valid underequilibrium conditionswhenboth interstitialsandvacanciesattheir equilibrium levels. Obviously, TED is a total deviation from this assumption,sowe can’t usethis modelunderTED conditions.However, diffusivity of dopantsis determinedusingthe“Fermi”model,so theparametersin themoresophisticateddiffusionmodelsshouldrelateto DB asdefinedabove.
2.1.2 Pair diffusion with a singlepoint defect
Dopantsareknown to bediffusingvia pairingwith pointdefects[18]. Wewantto build up thetheorythat leadsto thepair diffusionof dopants.For this purposewe startby assumingthedopantdiffuses
7
only via pair formationwith interstitials:
B � I 4 BI (2.3)
In thischemicalreaction,theforwardratewill beproportionalto theCBCI product,andthereverseratewill beproportionalto CBI . Thenetforwardreactionratecanbewrittenas:
kBI 2 KBICBCI � CBI 3 (2.4)
whereKBI is theequilibriumconstantandhasunitsof cm3 (inverseconcentration).In equilibrium,CBI will beequalto KBICBCI. Thesystemresultsin thefollowing continuityequations:
∂CB
∂t � � kBI 2 KBICBCI � CBI 3∂CI
∂t � .∇ 0 DI .∇CI � kBI 2 KBICBCI � CBI 3 (2.5)
∂CBI
∂t � .∇ 0 DBI .∇CBI � kBI 2 KBICBCI � CBI 3If weassumethatpairingis fast(thatis kBI is very large),thenthepairingreactionwill alwaysbe
in equilibrium,andwecanreducethecontinuityequationsto two by eliminatingthetermsassociatedwith this reaction:
∂ 2 CB � CBI 3∂t � .∇ 0 DBI .∇CBI (2.6)
∂ 2 CI � CBI 3∂t � .∇ 0 DI .∇CI � .∇ 0 DBI .∇CBI
with CBI � KBICBCI. NotethatCB � CBI � CTB , thetotal concentrationof our dopantin silicon. This
is thevaluereportedby SecondaryIon MassSpectrometry(SIMS).SimilarlyCI � CBI � CTI , thetotal
interstitialconcentration.If weassumethatCBI 5 CB andCBI 5 CI, thenCB 6 CTB andCI 6 CT
I . Thisapproximationswill bevalid underequilibriumandoxidationenhanceddiffusion(OED) conditions.WecantheneliminateCBI from theaboveequations:
∂CB
∂t � .∇ 0 DBI .∇ 2 KBICBCI 3 (2.7)
∂CI
∂t � .∇ 0 DI .∇CI � .∇ 0 DBI .∇ 2 KBICBCI 3which canbemodifiedinto:
∂CB
∂t � .∇ 0 DBIKBIC I .∇ 2 CBCI
C I 3 (2.8)
∂CI
∂t � .∇ 0 DI .∇CI � .∇ 0 DBIKBIC I .∇ 2 CBCI
C I 3If CI � C I , thenthediffusivity of thedopantshouldbeequalto DB in the“Fermi” model. Thus
wecansaythatDB � DBIKBIC I .
∂CB
∂t � .∇ 0 DB
C I .∇ 2 CBCI 3 (2.9)
∂CI
∂t � .∇ 0 DI .∇CI � .∇ 0 DB
C I .∇ 2 CBCI 38
It is evidentthatthedopantwill diffusenotonly becausethereis agradientin CB, but alsobecausethereis agradientin CI andviceversa.
2.1.3 Pair diffusion with both point defects
If weassumethatthedopantcanpair bothwith vacanciesandinterstitials,wecanwrite two possiblegroupsof reactions.In eachcasethe rightmostcolumngivesthe net forward rateof the reactions.First,wehavedopant/pointdefectpairingreactions:
B � I 4 BI kBI 2 KBICBCI � CBI 3B � V 4 BV kBV 2 KBVCBCV � CBV 3 (2.10)
And thenwehave thedirectandindirectrecombinationreactions:
I � V 4 /0 kR 2 CICV � C I C V 3BI � V 4 B kR 2 CBICV � KBICBC I C V 3BV � I 4 B kR 2 CBVCI � KBVCBC I C V 3
BI � BV 4 2B kR 2 CBICBV � KBIKBVC2BC I C V 3 (2.11)
Note that the rateconstantfor eachof the recombinationreactionsis assumedto be kR, whichis a simplification. In fact, if we assumediffusion limited processes,the reactionratesfor the fourreactionsshouldbe4πa 2 DI � DV 3 , 4πa 2 DBI � DV 3 , 4πa 2 DI � DBV 3 and4πa 2 DBI � DBV 3 , respectively.For thepurposesof this reactionset,we canassumethatDBI � DI andDBV � DV . This is not a badassumptionsincebothpoint defectsandpairsarefastdiffusers.This assumptionmakesall the fourratesequal.
If we assumethatpairing is fast,sothatCBI � KBICBCI andCBV � KBVCBCV, we maycombinetherecombinationreactionsinto asingle,effectiveone.Notethatnoneof therecombinationreactionswill effect total dopantconcentration(CT
B ):
kR 2 CICV � C I C V 37� kR 2 CICV � C I C V 3kR 2 CBICV � KBICBC I C V 37� kRKBICB 2 CICV � C I C V 3
kR 2 CBVCI � KBVCBC I C V 37� kRKBVCB 2 CICV � C I C V 3kR 2 CBICBV � KBIKBVC2
BC I C V 37� kRKBIKBVC2B 2 CICV � C I C V 3 (2.12)
which gives:keff
R � kR 2 1 � KBICB 3 2 1 � KBVCB 3 (2.13)
Thecontinuityequationsreduceto:
∂ 2 CB � CBI � CBV 3∂t � .∇ 0 DBI .∇CBI � .∇ 0 DBV .∇CBV
∂ 2 CI � CBI 3∂t � .∇ 0 DI .∇CI � .∇ 0 DBI .∇CBI � keff
R 2 CICV � C I C V 3 (2.14)
∂ 2 CV � CBV 3∂t � .∇ 0 DV .∇CV � .∇ 0 DBV .∇CBV � keff
R 2 CICV � C I C V 3with CBI � KBICBCI andCBV � KBVCBCV. Wecandenotethediffusivity of thedopantatomdueto theinterstitialcy mechanismunderequilibriumconditions(CI � C I ) by DI
B. Similarly, diffusivity of the
9
dopantdueto thevacancy mechanismwhenCV � CV 8 canbedenotedby DVB. NotethatDI
B � DVB �
DB, the total diffusivity of thedopantunderequilibriumconditions.This givesthe following setofcontinuityequations:
∂CTB
∂t � .∇ 0 DIB .∇ 9
CBCI
C I : � .∇ 0 DVB .∇ 9
CBCV
C V :∂CT
I
∂t � .∇ 0 DI .∇CI � .∇ 0 DIB .∇ 9
CBCI
C I : � keffR 2 CICV � C I C V 3 (2.15)
∂CTV
∂t � .∇ 0 DV .∇CV � .∇ 0 DVB .∇ 9
CBCV
C V : � keffR 2 CICV � C I C V 3
2.1.4 Pair diffusion with a singlepoint defect,including Fermi level effects
Theprecedingsetof of equationsignoredthedependenceof dopantdiffusivity on Fermi level. Thisdependencestemsfrom thefactthatthenumberof point defectsavailablefor diffusionchangeswiththe Fermi level. We assumethat interstitialscanbe neutral,positively or negatively charged. Weassumethat the charging reactionsaremuchfasterthanpairing reactions,sincethey areelectronicreactionsandthe mobility of electronsis muchlarger thanthe mobility of dopants.Therefore,theelectronicreactionsarealwayscloseto equilibrium:
e�;� e1 4 /0 np � n2i
I0 � e� 4 I � CI < � KI < CI0 2 n ni 3I0 � e1 4 I 1 CI = � KI = CI0 2 p ni 3 (2.16)
Eachof thesecharged interstitialscanpair with dopants. Again, we assumethat pairing is inequilibrium:
B � � I0 4 BI0 CBI0 � KBI0CBCI0
B � � I 1 4 BI 1 CBI = � KBI = CB 2 KI = CI0 2 p ni 3>3B � � I � 4 BI � CBI < � KBI < CB 2 KI < CI0 2 n ni 3>3 (2.17)
Thenotationabove maybeconfusing,but is laid out asfollows: CBI0 is theconcentrationof theresultingpair from thereactionof anionizeddopantandI0. If thedopantis anacceptor, it will carrya netnegativecharge in its ionizedstate,makingBI0 negatively charged. If thedopantis a donor, itwill carryanetpositivechargein its ionizedstate,makingBI0 positively charged.
Notethedopantis anacceptor(carryinganegativecharge)it is veryunlikely thatit will pairwitha negatively chargedinterstitial dueto repulsion,makingCBI < � 0. A similar statementis true fordonors.But we keeptheCBI < termto make theanalysismoregeneralandassumethatKBI < � 0 foracceptorsandthatKBI = � 0 for donors.
Notethatdueto thechargeof ions,thecontinuityequationwill notonly haveadiffusioncompo-nent( .Jdiff), but alsoa drift component( .Jdrift). Thedrift termsarisebecausethechargedspeciescanalsomovebecauseof forcesof theelectricfield, createdby thegradientof electronconcentrationinthesubstrate.Thetotalflux of pointdefectsandpairswill beequalto thesumof fluxesof eachchargestate:
10
∂CTB
∂t � � .∇ 0 1 1
∑i ? � 1
2 .JdiffBIi � .Jdrift
BI i 3 (2.18)
∂CTI
∂t � � .∇ 0 1 1
∑i ? � 1
2 .JdiffI i � .Jdrift
I i 3 � .∇ 0 1 1
∑i ? � 1
2 .JdiffBIi � .Jdrift
BI i 3Next wedothegenericderivationof .Jdiff
BI i� .Jdrift
BI i. Assumethatthedopantis anacceptor, andhence
carriesa chargeof � q. If thedopantwasa donor, it would carrya chargeof � q. This will make thecharge on a BIi pair 2 i � z3 q, wherez � � 1 if thedopantis a donorandz � � 1 if thedopantis anacceptor. Then,we canwrite thediffusiontermas:.Jdiff
BI i � � DBI i .∇CBIi� � DBI i .∇ 2 KBIiCB 2 KI iCI0 2 p ni 3 i 3>3� � DBI i KBI i KI i .∇ 2 CBCI0 2 p ni 3 i 3� � DiB C I0 @ 2 p ni 3 i .∇ 2 CBCI0 3 � CBCI0 .∇ 2 p ni 3 i A� � DiB C I0 @ 2 p ni 3 i .∇ 2 CBCI0 3 � CBCI0i 2 p ni 3 i � 1 .∇ 2 p ni 3 A (2.19)
And thedrift termbecomes: .JdriftBI i � µBI i 2 i � z3 q .B CBI i� DBI i 2 q kT 3 2 i � z3 .B CBIi (2.20)
whereµ � D kT is the mobility accordingto the Einsteinrelationshipand .B is the electric fieldvector. The electricfield canbe calculatedfrom the gradientof the potential,which in turn canbefoundfrom thelocalcarrierconcentration,givenby aBoltzmanndistribution(Ψ denotestheintrinsicpotential):
n � ni exp
9Ψ
kT q : (2.21).B � � .∇Ψ� � .∇ 2 kT qln 2 n ni 3>3� .∇ 2 kT qln 2 p ni 3>3 (2.22).JdriftBIi � DBIi 2 i � 13 CBIi .∇ ln 2 p ni 3� DBIi 2 i � 13 2 KBIiCB 2 KI iCI0 2 p ni 3 i 3 2 ni DCE3F.∇ 2 CE ni 3� DBIi KBI i KI i 2 i � 13 CBCI0 2 p ni 3 i � 1 .∇ 2 p ni 3� Di
B C I0 2 i � 13 CBCI0 2 p ni 3 i � 1 .∇ 2 p ni 3 (2.23)
11
Adding thediffusionanddrift termsweget:.JBIi � � DiB C I0 @ 2 p ni 3 i .∇ 2 CBCI0 3 � CBCI0 2 p ni 3 i � 1 .∇ 2 p ni 3 A� � DiB C I0 2 p ni 3 i @ .∇ 2 CBCI0 3 � CBCI0 .∇ ln 2 p ni 3 A (2.24)
Summingoverall thechargestates,wewill get:.JBI � 1 1
∑i ? � 1
.JBI i � � 2 D0B � D 1B 2 p ni 3 � D �B 2 n ni 3G3
C I0 @ .∇ 2 CBCI0 3 � CBCI0 .∇ ln 2 p ni 3 A (2.25)
For donors,weneedto replaceln 2 p ni 3 with ln 2 n ni 3 . It mayreadilybeobservedthatunderequilib-rium conditions(CI0 � C I0), this reducesto “Fermi model” (Eq.2.1).
If we carry out a similar analysisfor .JI i we canseethat the term with the gradientof 2 p ni 3 isbeingcanceledby thedrift term:.Jdiff
I i � � DI i .∇CI i� � DI i .∇ 2 KI iCI0 2 p ni 3 i 3� � DI i KI i .∇ 2 CI0 2 p ni 3 i 3� � DI i KI i @ 2 p ni 3 i .∇CI0 � CI0 .∇ 2 p ni 3 i A� � DI i KI i @ 2 p ni 3 i .∇CI0 � CI0i 2 p ni 3 i � 1 .∇ 2 p ni 3 A (2.26).JdriftI i � DI i 2 q kT 3 i .B CI i� DI i iCI i .∇ ln 2 p ni 3� DI i i 2 KI iCI0 2 p ni 3 i 3 2 ni p3 .∇ 2 p ni 3� DI i KI i iCI0 2 p ni 3 i � 1 .∇ 2 p ni 3 (2.27).JI i � .Jdiff
I i � .JdriftI i � � DI i KI i 2 p ni 3 i .∇CI0 (2.28).JI � 1 1
∑i ? � 1
.JI i � � DI0 2 1 � KI = 2 p ni 3 � K �I 2 n ni 3>3 .∇CI0 (2.29)
Thelastequationwasobtainedby thesimplifying assumptionthatinterstitialsat differentchargestatesdiffuseequallyfast,that is DI0 � DI = � DI < . Thereis no experimentalevidenceto show thatthey diffuseat differentspeeds.
2.1.5 Pair diffusion with both point defects,including Fermi level effects
We will merelystatethe resultshere. Derivation is analogousto the previous sections. The onlypoint to payattentionhereis thatthefractionalinterstitialcy componentof diffusionmaychangewithFermi level. For exampleit is known that phosphorusdiffusesprimarily via interstitialswhenit isintrinsic, but it diffusesprimarily via vacancieswhenpresentin large concentrations[11]. We canhandlethis situationby having separatecomponentslike DI =
B , which standfor “dif fusivity of borondueto thepositively chargedinterstitials.” Wemaywrite:
12
DI0B � DV0
B � D0B
DI =B � DV =
B � D 1B (2.30)
DI0B � DI =
B
D0B � D 1B � f intr
I
where f intrI is the fractionalinterstitialcy componentof diffusionfor intrinsic boron. That is, f intr
I isthe fraction of diffusivity causedby the interstitialsfor intrinsic conditions. Of course,thesethreeequationsarenot enoughto determinethe four unknowns DI0
B , DV0
B , DI =B andDV =
B , so, anotheras-sumptionmustbemadebasedon theexperimentaldatafor thedopant.For example,we canassumethatborondiffusesalwaysthroughinterstitialcy mechanismat all Fermilevels.Wecanthendefine:
DIB � DI0
B � DI =B
9pni : � DI <
B
9nni :
DVB � DV0
B � DV =B
9pni : � DV <
B
9nni : (2.31)
This resultsin thefollowing systemof equations:
∂CTB
∂t � � .∇ .JBI � .∇ .JBV
∂CTI
∂t � � .∇ .JI � .∇ .JBI � keffR 2 CI0CV0 � C I0C V0 3 (2.32)
∂CTV
∂t � � .∇ .JV � .∇ .JBV � keffR 2 CI0CV0 � C I0C V0 3
where .JBI � � DIB C I0 @ .∇ 2 CBCI0 3 � CBCI0 .∇ ln 2 p ni 3 A.JBV � � DVB C V0 @ .∇ 2 CBCV0 3 � CBCV0 .∇ ln 2 p ni 3 A.JI � � DI0χI .∇CI0 (2.33).JV � � DV0χV .∇CV0
keffR � kR 2 χI � πICB 3 2 χV � πVCB 3
with
χI � 1 � KI = 9 pni : � KI < 9 n
ni :χV � 1 � KV = 9 p
ni : � KV < 9 nni : (2.34)
πI � KBI0 � KBI = KI = 9 pni : � KBI < KI < 9 n
ni :πV � KBV0 � KBV = KV = 9 p
ni : � KBV < KV < 9 nni :
13
If CBI 5 CB andCBV 5 CB, onecanassumethatCTB � CB andwrite:
CTI � 2 χI � πICB 3 CI0
CTV � 2 χV � πVCB 3 CV0 (2.35)
A noteabouttherecombinationratekeffR : Actually therateis toohighasgivenhere,becauseit also
includesrecombinationtermsthatactuallywould not bepresent.For example,anV � andI � wouldbe very unwilling to recombinebecausethey would repeleachother. If the dopantis an acceptor,we would have 4 suchpairs: I � andV � ; I 1 andV 1 , BI0 andV � ; BV0 and I � (Note that BI0 isactuallynegatively charged). For anacceptor, BI � andBV � areunlikely to exists,sowe don’t haveto considerthem.
Thus we have only 4 extra pairs out of the 25 pairs that can recombinewith eachother. Theerrorof includingthese4 pairsin theexpressionfor recombinationis oftennegligible, andmakestheequationsmuchsimpler. If moreaccuracy is desired,thefollowing mustbesubtractedfrom thekeff
Rabove:
kR CKI � KV � 2 n ni 3 2 � KI = KV = 2 p ni 3 2 � KBI0KV � 2 n ni 3 � KBV0KI � 2 n ni 3IH (2.36)
2.1.6 Fivestreammodel
If we don’t make theassumptionthatpairingis fast,we shouldkeepall five variables.We cancarryouta similaranalysisto eliminatethecharging reactions.Theresultingsystemcanbeexpressedas:
∂CB
∂t � � RBI � RBV � RBI 1 V � RBV 1 I � 2RBI 1 BV
∂CBI
∂t � � .∇ .JBI � RBI � RBI 1 V � RBI 1 BV
∂CBV
∂t � � .∇ .JBV � RBV � RBV 1 I � RBI 1 BV (2.37)
∂CTI
∂t � � .∇ .JI � RBI � RI 1 V
∂CTV
∂t � � .∇ .JV � RBV � RI 1 V
where .JBI � � DIB
C I0 J .∇ 9CBI
πI : � CBI
πI.∇ ln
9pni :LK.JBV � � DV
B
C V0 J .∇ 9CBV
πV : � CBV
πV.∇ ln
9pni :LK
RBI � kBI CCBCI � χI
πICBI H
RBV � kBV CCBCV � χI
πICBV H (2.38)
RBI 1 V � kR 2 CBICV � C I0C V0πIχVCB 314
RBV 1 I � kR 2 CBVCI � C I0C V0πVχICB 3RBV 1 BI � kR 2 CBVCBI � C I0C V0πIπVC2
B 3andothervariableshavesamemeaningsasin theprevioussection.
2.2 Initial conditions: Damagecreatedby ion implantation
To developmodelsfor TED,weneedto haveagoodunderstandingof thedamagetheion implantationprocesscreates.Sincethereis no accurateway of measuringthedamageprofile experimentally, wehaveto rely onMonte-Carlosimulationsfor calculationof theimplantdamage.In thiswork, weusedtheMonte-Carloion implantsimulatorsTRIMCSR[4, 53] andUT-Marlowe[51].
Thedamagecreationprocesscanbesummarizedasfollows: As anion with high kinetic energyentersthe silicon substrate,it undergoesa seriesof collisionswith a numberof silicon atoms. Thekinetic energy of the ion is high enoughto displacetheseatomsfrom their original sites,leaving avacancy behind.Thesesecondaryionsalsocollide with othersilicon atoms,etc.,creatinga collisioncascade.Oneimplantedion caneasilycreatethousandsof Frenkel pairs(interstitial-vacancy pairs),dependingon the energy andmassof the implantedion. If the createddamageis high enough,thesubstratewill becomeamorphous.
2.2.1 Non-amorphizing implants
We first want to investigatenon-amorphizingimplants. In thesetypesof implants,the doseof theimplantedion is small enoughthat the createddamagedoesn’t reachthe amorphizationthreshold.Thus,thesubstratemaintainsit crystallinenature,althoughthereis extensive damagein thecrystalstructure.
TRIMCSRassumesthatthestructureis amorphousto begin with. Therefore,it ignoresany chan-nelingthatmayoccurdueto thecrystallinenatureof thesubstrate.On theotherhand,UT-Marlowetakesthe crystalstructureof the substrateinto accountandcanpredict the tilt androtationdepen-denceof thetail of theprofile. Thedifferencebetweenthetwo simulatorsis obviousasrepresentedinFigure2.1.Eventhoughtheimplanthasbeenperformedat7o tilt and45o rotationto minimizechan-neling,still substantialchannelingoccurs.Therefore,UT-Marlowewill beourchoiceof simulatorfornon-amorphizingimplants.
Looking at the damagecreatedby the ion implantationprocess,we canseethat the numberofFrenkel pairsgeneratedis muchhigher than the numberof implantedions (Fig. 2.2). In fact, theinterstitial andvacancy curvesarealmostindistinguishablefrom eachother, but thereis a vacancy-rich regionnearthesurfaceandaninterstitial-richregiondeeperin thesubstrate.Thisstemsfrom thefactthattheimplantationdrivessomesiliconatomsdeeperinto thesubstrate.
We know that even during the implantationprocessthe Frenkel pairswill recombine.After anextremelysmallthermalbudget(1msat600� C), wewouldexpectthatmostFrenkel pairswouldhaverecombinedbecausethesystemis sofarawayfrom equilibrium.Thiswould leaveuswith anet“ � 1”damage,whereeachincomingion displacesonesilicon atom,suchthatthenetI � V doseis equaltotheimplantdose.
15
TRIMCSR UT-Marlowe
|�0� |�
500� |�1000
|�1500
|�2000
|�2500
|�3000
|�3500
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Depth (A)O1015
1016
1017
1018
1019
Con
cent
ratio
n (c
mP -3 )
Figure2.1: Comparisonof UT-Marlowe andTRIMCSR simulationsof a 40keV 5 � 1013cm� 2 Siimplantat7o tilt and45o rotation.
Total Interstitials Total Vacancies Net Interstitials Net Vacancies
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0.02� |�0.04� |�
0.06� |�0.08� |�
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|� |N |N |N |N |N |N |N |N |� |N |N |N |N |N |N |N |N |� |N |N |N |N |N |N |N |N |� |N |N |N |N |N |N |N |N |� |N |N |N |N |N |N |N |N |� |N |N |N |N |N |N |N |N |�
1016
1017
1018
1019
1020
1021
1022
Depth (Q µm)
Con
cent
ratio
n (c
mP -3 )
Figure2.2: Total andnetdamagecreatedby a 40keV 5 � 1013cm� 2 Si implant. MonteCarlosimu-lationswith TRIMCSR.
16
Implanted ions Total interstitials Total vacancies Recombined I-V
|�0� |�
500� |�1000
|�1500
|�2000
|�2500
|�3000
|�3500
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|M |M |M |M |M |M |M |M |� |M
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|� |N |N |N |N |N |N |N |N |� |N |N |N |N |N |N |N |N |� |N |N |N |N |N |N |N |N |� |N |N |N |N |N |N |N |N |� |N |N |N |N |N |N |N |N |� |N |N |N
Depth (A)O1016
1017
1018
1019
1020
1021
Con
cent
ratio
n (c
mP -3 )
Figure2.3: Damagecreatedby a40keV 5 � 1013cm� 2 Si implantbeforeandafterveryshortanneal-ing.
However, I/V recombinationis not the only processoccurringduring the post implant phase.Interstitialsandandvacanciescanalsodiffuseandrecombineat thesurface. Ab-initio calculationsshow thatthediffusivity of vacanciesis higherthanthediffusivity of interstitials,particularlyat lowertemperatures.Thuswe would expect the vacanciesin the vacancy-rich region nearthe surfacetorecombineat thesurfacemorereadilythanthey wouldrecombinewith aninterstitial.To demonstratethis effect, we have performedsimulationsassumingthatDV R DI andbothsurfacerecombinationandI/V recombinationarediffusionlimited processes.Figure2.3shows thatwe indeedgeta regionnearthesurfacewherethenet I � V concentrationis higherthanpredictedby a “ � 1” approach,andthenetI � V doseis largerthan“ � 1”.
This effect is, however, dosedependent.If we have a relatively high dose,thereis a higherprob-ability thata vacancy will first find aninterstitialandrecombinewith it. If we have a relatively lowdose,thereis a higherchancethat thevacancy will hit thesurfacebeforefinding anotherinterstitial.Therefore,we would expectthenet I � V doseafter recombinationto be increasingwith decreasingimplantdose.We have run a seriesof simulationsto affirm this perception,andindeedwe seethatthe“plus factor” canbeashighas10 (Fig. 2.4).
Notethattherearealsootherphenomenaoccurringduringtheimplantationprocess,suchasamor-phouspocket formation,whichmight increasethemagnitudeof the“ � n” effect. Formationof amor-phouspocketsconfinestheinterstitialsandvacanciesto arelativelysmallregionin space.If avacancymanagesto escapetheamorphouspocket, it hasa fairly highchanceof hitting thesurfaceratherthananotheramorphouspocket. We believe that we have taken this effect into accountto someextendby assumingDI � 0 in our simulations,but clearly atomisticsimulationsarerequiredto accurately
17
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lus
fact
or S
1012 1013 10� 14
Dose (cm) -3)
TT
T T T T TFigure 2.4: Net damage(I � V) dosedivided by implant dose(plus factor) for a seriesof non-amorphizing40keV Si implants.
estimatetheseeffects.
2.2.2 Amorphizing implants
Thepictureis slightlydifferentfor amorphizingimplants.If thesubstrateis amorphizedupto acertaindepth(theamorphous/crystallineinterface),this portionwill regrow epitaxially, giving a defectfreeregion aroundthepeakof the implant,andleaving anexcessinterstitial profile only nearthe tail oftheprofile (Fig. 2.5).This interstitialrich region is alsocalledtheendof range(EOR)regionandthisis wherethedislocationloopsform.
Note that the net damagedoseis much lessthan the implant dose,sincemuchof the damagegoesaway duringsolid phaseepitaxialregrowth. Therefore,TED from anamorphizingimplantcanactuallybe lessthanfrom a non-amorphizingimplant. However, dislocationloops,which areverystable,have a negative impacton thedevice characteristics:They arein themiddleof thedepletionregion,andthereforeincreasetheleakagecurrentof thejunction.
Anotherimportantdifferencefromthenon-amorphizingimplantsis thattheamorphizationthresh-old hasnouniquevalue.Amorphizationthresholdis thetotal vacancy concentrationabovewhich thesubstrateis assumedto beamorphized.It is usuallyexpressedasapercentageof thesiliconatomcon-centration(5 � 1022cm� 3). Studieshave shown that this thresholddependson implant temperatureandimplantdoserate,sincebothof thesefactorsaffect thedamageaccumulationprocess[47].
Moreover, thenetI � V doseis averystrongfunctionof theamorphous/crystalline(a/c)interface.
18
Total interstitials Total vacancies
Net I-V
|�0� |�
200� |�400� |�
600� |�800� |�
1000|�
1200|�
1400|�
1600|�
1800
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|� |N |N |N |N |N |N |N |N |� |N |N |N |N |N |N |N |N |� |N |N |N |N |N |N |N |N |� |N |N |N |N |N |N |N |N |� |N |N |N |N |N |N |N |N |� |N |N |N |N |N |N |N |N |� |N
Depth (A)O
amorphous crystalline
1017
1018
1019
1020
1021
1022
1023
Con
cent
ratio
n (c
mP -3 )
Figure2.5: TRIMCSR simulationof total andnet implant damagefor a 50keV 1 � 1015cm� 2 Siimplant.Amorphizationthresholdhasbeenassumedto be10%.
Under conditionsof Fig. 2.5, a 30A changein the a/c interfacewould changethe net I � V doseby 25%. Even if thea/c interfaceis measuredexperimentally, theendof rangedamagestill cannotbe determinedaccuratelybecauseof the sensitivity of the net I � V doseon the locationof the a/cinterface. Therefore,it is important to have anothermeasure,suchas the numberof interstitialstrappedin dislocationloops,to modelthenetdamageunderamorphizingconditions.
2.3 Other parameters
2.3.1 Point defectproperties
Oneof themostimportantparametersfor TED is point defectproperties,especiallythoseof intersti-tials. We tried to useasaccuratevaluesaspossible,but sinceno directmeasurementof point defectpropertiescanbeperformed,all parametervaluesweobtainin oursimulationshaveto bere-calibratedwhena differentsetof point defectpropertiesis used.In this section,we investigatetherelationshipbetweenpoint defectpropertiesandmodelingresults.
The most importantparameteris the self diffusion coefficient via interstitials (DIC I product).The value of this productdirectly effects simulationresults. More precisely, DIC I and U Dt (thejunctionmovementduring TED) areinverselyproportionalto eachother. Luckily, theDIC I canberelatively well determinedfrom experiments.However, themostreliableexperimentsfor determiningDIC I aremetaldiffusionexperiments,whichareusuallyperformedat highertemperaturesthanTED
19
conditions.Therefore,a smallerror in theactivationenergy of DIC I mayleadto a big error in TEDpredictions.
The valuesof diffusivity of interstitials(DI) andequilibrium concentrationof interstitials(C I )individually don’t affect TED resultsdirectly, aslong astheir productis constant.The durationofTED is controlledby theflux of interstitialstowardsthesurface,which is directlyproportionalto theratio of solid solubility of interstitial (Css) to C I . Thus,aslong astheDIC I productandCss C I ratiois constant,theresultswill remainunchanged.
The vacancy parametersdon’t affect the simulationresultsboth for TED and extendeddefectevolution,exceptfor the“+n” effect describedin section2.2.1.Evenfor thateffect, theactualvalueof DV is irrelevant;theonly thing thatis importantis DV R DI.
In summary, the primarily relevant point defectpropertiesareDIC I andCss C I . In this thesis,point defectvaluesfrom metaldiffusionareused[5, 58] consistently. I/V recombinationis notasig-nificantprocessfor TED (exceptfor thefirst µs); thereforewe canassumefor simplificationthatit isdiffusionlimited. Wehavealsoassumedthatinterstitialprecipitation(formationof extendeddefects)is alsodiffusionlimited. This assumptionis necessaryif we assumefastsurfacerecombination(seenext section),in orderto ensureasteadyinterstitialflux towardsthesurface.
2.3.2 Surfacerecombination
UnderTED conditions,the surfaceis the main sink for interstitials. Databy Eagleshamet al. [17]suggesta surfacerecombinationratethat is decreasingwith time, sincetherateat which interstitialsareconsumedis decreasingapproximatelyexponentiallywith time. Otherexperimentalobservationsalsoindicatethat a fasteffective interstitial surfaceregrowth exists for small thermalbudgets,witha muchslower onefor longertimesandhighertemperatures[15]. Thus,ratherthanusinga simplemodel with constantsurfacerecombinationrate, we usedthe film segregation model of AgarwalandDunham[1] to accountfor surfaceregrowth. In thismodel,theprimarymechanismby which thesurfaceconsumesinterstitialsis notsurfacerecombination,but segregationof interstitialsto theoxidelayer.
However, after performinga seriesof simulations,we find that the surfacehasno strongeffecton the results,aslong asit hasa fastrecombinationrate. The main reasonbehindthe exponentialdecayin Eaglesham’s datais Ostwald ripeningprocess.We find that usinga fast,constantsurfacerecombinationmodelgivesalmostexactly thesameresultsasthefilm segregationmodel.Therefore,weswitchedto aconstantsurfacerecombinationmodelwhereweassumedthattheratioof thesurfacerecombinationvelocityandinterstitialdiffusivity is V 0 � 1A � 1.
20
Chapter 3
Modeling ExtendedDefectswith KineticPrecipitation Model
3.1 Intr oduction
As mentionedin section1.2.3,TransmissionElectronMicroscopy (TEM) observations[10, 17] showthat
�311� defects(also known as “rod-like defects”) form, grow and eventually dissolve during
annealingof ion implantedsamples.Initially, thereis ahugedriving forceto form�311� defects,such
that, for sub-amorphizingsilicon implants,almostthe whole net excessinterstitial doseaggregatesinto theseextendeddefectswithin a very short period of time (less than 5s at 815� C) [17]. Asannealingcontinues,thesedefectsundergo Ostwald ripening: Their averagesizeincreasesandtheirnumberdecreases.The total numberof interstitialsboundto thesedefectsdecreasesapproximatelyexponentially, andeventually, all
�311� defectsdissolveanddisappear. It hasbeenreportedthat the
time scalefor their dissolutionis aboutthesameasthetime scaleof TED [10]. Theseobservationsstronglysuggestthat
�311� defectsplay a centralrole underTED conditions.If we couldmodelthe
evolutionof thesedefectscorrectly, weshouldbeableto predictTED.
In general,nucleationandgrowth processesplay a critical role in a largerangeof materialspro-cessingsystems.Classicalmodelingapproachesdivide suchprocessesinto two discretesteps,withnucleationandgrowth being modeledusing fundamentallydifferentassumptions,eachvalid onlyunderidealizedconditions.Thus,althoughtheseapproachesarevery usefulfor understandingqual-itative behavior, they areunsuitablein many casesfor thedevelopmentof quantitative models,par-ticularly undercomplex annealingconditions(e.g.,multi-stepanneals).Noting thepower of moderncomputersto solve complex systemsof coupleddifferentialequations,we have developeda unifiedapproachto modelingof nucleationandgrowth processeswhichextendsnucleationtheoryto includethebehavior of supercriticalaswell assub-criticalaggregates.
3.2 Full Kinetic Precipitation Model (FKPM)
The major challengein modelingthe evolution of precipitatesandextendeddefectsis the fact thatdifferentsizeddefectshave very differentproperties.TheFull Kinetic PrecipitationModel (FKPM)[14] treatsprecipitatesof differentsizesasindependentspecies( fn) andaccountsfor theirkineticsby
21
I 1 I n
n+1n2
A
AA
Figure3.1: Growth anddissolutionof precipitatesby attachmentandemissionof soluteatoms.
consideringtheattachmentandemissionof soluteatoms(Fig 3.1).
Thedriving forcefor precipitationis theminimizationof thefreeenergy of thesystem,wherethefreeenergy of asizen extendeddefectis givenby:
∆Gn � � nkT lnCA
Css� ∆Gexc
n (3.1)
Here,CA denotesthesoluteconcentration,Css is thesolid solubility and∆Gexcn is theexcesssurface
andstrainenergy of a sizen extendeddefect.We usuallyassumethat∆Gexcn hasa polynomialform,
sinceany functioncanbeapproximatedby a polynomial:
∆Gexcn � a0nβ0 � a1nβ1 � a2nβ2 (3.2)
with β2 � β1 � β0 � 1. The largestexponent(β0) controlsthe asymptoticbehavior for large sizes.Theoreticalcalculations[43] for thefunctionalform of ∆Gexc
n show thatit is reasonableto assumethatβ0 � 0 � 5 since
�311� defectsareplanardefects.Weusedβ1 � � 0 � 2 W β1 � � 1 ascorrectionsfor small
sizebehavior. a0, a1 anda2 remainastemperatureindependentfitting parameters.
The main reactionin the systemis the attachmentand emissionof soluteatomsto and fromprecipitates. If In denotesthe net growth rate from size n to n � 1, we may write the followingequation:
In �/X Dλn 2 CA fn � C n fn1 1 3 for n V 2Dλ1 Y C2
A � C 1 f2 Z for n � 1(3.3)
Note that I1 is differentfrom otherterms,becauseit representsthe ratefor formationof thedefectsby reactionof two interstitials.
Thegrowth rateof precipitatesis written in theform Dλn, whereλn incorporateseffectsof bothdiffusionto theprecipitate/siliconinterfaceandthereactionat theinterface.λn is calculatedbasedonsolvingthesteady-statediffusionequationin theneighborhoodof a precipitate,takingits shapeintoaccount(seeAppendixA). C n representsthe interstitial concentrationat which therewould be nochangein freeenergy for a precipitategrowing from sizen to sizen � 1. C n canbefoundby setting∆Gn � ∆Gn1 1 andsolvingfor CA.
C n � Cssexp
9∆Gexc
n1 1 � ∆Gexcn
kT : (3.4)
Theevolution of thesizedistribution fn is givenby thedifferencebetweenthenet rateat whichdefectsgrow from sizen � 1 to n (In � 1) andthenetrateof growth from sizen to n � 1 (In). Sincethe
22
fundamentalgrowth processis the incorporationof a soluteatom,the total changein CA includesatermfrom eachgrowth reaction,giving asumover In:
∂ fn∂t � In � 1 � In (3.5)
∂CA
∂t � � 2I1 � ∞
∑n? 2
In (3.6)
3.3 ReducedKinetic Precipitation Models (RKPM)
TheFull Kinetic PrecipitationModeladdsanextradimension,namelyprecipitatesize,to theproblembeingsolved. Even if onelimits thenumberof precipitatesizesthatwill besolved for, thenumberof variablesis still very largefor efficient solutionof theequationsystem.If thesystemhasmultiplespatialdimensions,thenumberof solutionvariablesbecomesprohibitively large.
To minimizethecomputationalbudget,wehavedevelopedamoreefficientversionof thismodel,basedonthework of ClejanandDunham[8]. Insteadof calculatingall the fn, onecalculatesonly thelowestmomentsof thedistribution(mi � ∑∞
n? 2ni fn, wherei � 0 W 1 W 2 WG�>�>� ). This transformsthesystemof equationsto thefollowing set:
∂mi
∂t � 2i I1 � ∞
∑n? 2 [ 2 n � 13 i � ni \ In (3.7)
Note that thesumsover the In canall bewritten in termsof sumsover fn, nfn, etc. Hence,theycanbecalculatedfrom themomentsif momentsareusedto describethedistribution. This reducesthesystemof equationsto besolvedto:
∂mi
∂t � DA [ 2iλ1C2A � m0CAγ 1i � m0Cssγ �i \
γ 1i � ∞
∑n? 2 [ 2 n � 13 i � ni \ λn fn (3.8)
γ �i � λ1C 1 f2 � ∞
∑n? 2 [ ni � 2 n � 13 i \ λn � 1C n � 1 fn
whereC n � C n Css and fn � fn m0.
Sinceno finite numberof momentscanfully describea full distribution, we needa closureas-sumption,which is anassumptionabouttheform of thedistribution, fn � f 2 n W zi 3 . Thezi areparame-tersof thedistributionwhichcanbedeterminedfrom themoments.Thenumberof momentsweneedto keeptrackof equalsthenumberof parametersin thedistribution function.This resultsin differentversionsof RKPM:
3.3.1 3-momentmodel (3KPM)
If nothingis known aboutthedistribution of extendeddefectsover sizespace,it is logical to useanenergy minimizing closureassumption[8]. Theenergy minimizing closureassumptionassumesthat
23
thedistribution will be theonethatminimizesthe free energy, given the moments.It canbe foundfrom constrainedminimizationof the freeenergy (∆Gn). This resultsin the following (normalized)distributionwith threeparameters:
fn � z0exp Y � ∆Gexcn kT � z1n � z2n2 Z (3.9)
Theresultingsystemis a3-momentsystem(3KPM):
∂m0
∂t � D [ λ1C2A � m0Cssγ �0 \
∂m1
∂t � D [ 2λ1C2A � m0CAγ 11 � m0Cssγ �1 \ (3.10)
∂m2
∂t � D [ 4λ1C2A � m0CAγ 12 � m0Cssγ �2 \
∂CA
∂t � � ∂m1
∂t
with
γ �0 � λ1C 1 f2
γ 11 � ∞
∑n? 2
λn fn
γ �1 � λ1C 1 f2 � ∞
∑n? 2
λn � 1C n � 1 fn (3.11)
γ 12 � ∞
∑n? 2
2 2n � 13 λn fn
γ �2 � λ1C 1 f2 � ∞
∑n? 2
2 2n � 13 λn � 1C n � 1 fn
For the3-momentmodel(3KPM), thedistribution function is not readily integrable.Therefore,we can’t analyticallyreplacea functionof theparametersof thedistribution functionwith a functionof the moments,which are our solution variables. Instead,we have to solve the following set ofnon-linear, coupledequationsatevery timestepandgrid point:
1 � ∞
∑n? 2
z0exp Y � ∆Gexcn kT � z1n � z2n2 Z
m1 � ∞
∑n? 2
nz0exp Y � ∆Gexcn kT � z1n � z2n2 Z (3.12)
m2 � ∞
∑n? 2
n2z0exp Y � ∆Gexcn kT � z1n � z2n2 Z
To make thesimulationcomputationallyefficient,we solve theabovenon-linearsetof equationsfor arangeof m1 andm2 valuesandcalculatetheγi for thesevalues.Thentheγi arestoredin alookup
24
] 30s^ 60s_ 120s
|�0� |�
200� |�400� |�
600� |�800� |�
1000|�
1200|�
1400
|�0
|�100
|�200
|�300
|�400
|�500 |� |� |� |� |� |� |� |�
|� |� |� |� |� |�
Defect size (atoms)`
Def
ects
Den
sity
(10a8
cm-2
)
bb
bb b b b bc
c c c c c c c cd d d d d d d d d d dFigure3.2: Distribution of
�311� defectdensitiesover defectsizesandbestfit to log-normaldistri-
bution. z2 � 0 � 8 hasbeenusedin all fits. Datafrom PanandTu [41].
tableandinterpolationis usedto find valuesof γi for valuesof mi thathavenotbeentabulated.ClejanandDunhamhave appliedthis systemto dopantdeactivationandshowed that theuseof a momentbasedapproachdoesn’t refrainusfrom capturingthephysicsof thesystem[8].
3.3.2 2-momentmodel (2KPM)
For�311� defectsanddislocationloops,thesizedistributionhasbeenmeasuredexperimentally[41].
Theresultssuggestthatthedistribution is roughlylog-normal:
fn � z0exp 2 � ln 2 n z1 3 2 z2 3 (3.13)
Whenweanalyzethedata,wefind thatz2 appearsto beindependentof annealingtime. Thedistribu-tionscanbeapproximatedby log-normaldistributionswith z2 � 0 � 8 (Fig. 3.2). Thus,we canreducethe numberof parameters(hencenumberof moments)to 2. The resultingsystemis a 2-momentsystem(2KPM):
∂m0
∂t � I1 � D [ λ1C2A � m0Cssγ �0 \
∂m1
∂t � 2I1 � Dm0 [CAγ 11 � Cssγ �1 \ (3.14)
∂CA
∂t � � ∂m1
∂t
25
{311} defects Approximation Dislocation loops
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|M |M |M |M
|M |M |M |M |�1000 |� |( |( |( |( |( |( |( |( |� |( |( |( |( |( |( |( |( |� |( |( |( |( |( |( |( |( |� |( |( |( |( |( |( |( |( |� |( |( |( |( |( |( |( |( |�
|� |N |N |N |N |N |N |N |N |� |N |N |N |N |N |N |N |N |�
Defect size (n)100 101 102 103 10� 4 105
λn
(Ang
stro
ms)
Figure3.3: Changeof λn versusdefectsizefor�311� defectsanddislocationloops.Thecalculation
canbefoundin AppendixA.
with
γ �0 � λ1C 1 f2
γ 11 � ∞
∑n? 2
λn fn
γ �1 � ∞
∑n? 2
λnC n fn1 1 (3.15)
Note that for the 2KPM, the integral of the distribution function canbe found analytically, andthereforetheparametersof thedistributioncanbecalculatedfromthemomentsbymeansof analyticalfunctions.Thiseliminatestheneedfor thelookuptableandinterpolation,giving speedandrobustnessadvantages.However, still theγi needto becalculatedby sums,which is aniterativeprocess.
3.4 Analytical Kinetic Precipitation Model (AKPM)
RKPM, althoughcomputationallymoreefficient than the FKPM, canstill be very costly for largesimulationsin multiple dimensions.We have developedanevenmoreefficient versionstartingwith2KPM.
First of all, we notethat for�311� defects,thekinetic precipitationrateλn is a weakfunctionof
theprecipitatesize(ascomparedto dislocationloops)andconvergesto a constantastheprecipitate
26
sizeincreases(Fig. 3.3). Therefore,it would be reasonableto replaceλn by a constantvalueλ forall sizes.This valuecanbefoundapproximatelyfrom a weightedsumof λn. Therefore,our systemreducesto:
∂m0
∂t � I1 � Dλ [C2A � m0Cssγ0
\∂m1
∂t � 2I1 � Dλm0 CCA � Cssγ1 H (3.16)
∂CA
∂t � � ∂m1
∂t
with
γ0 � C 1 f2
γ1 � ∞
∑n? 2
C n fn1 1 (3.17)
Note that in theabove equations,fn is givenby thedistribution functionwe assume,andcanbedeterminedfully if m1 is given. For example,if thedistribution function is a geometricdistributionfunction( fn � z0zn
1), thenthedistributioncanbedeterminedby solvingthefollowing setof equations:
1 � ∞
∑n? 2
z0zn1
m1 � ∞
∑n? 2
nz0zn1 (3.18)
Thesolutionof this systemgives:
z1 � m1 � 2m1 � 1
z0 � 1 � z1
z21
(3.19)
Hencethenormalizeddistributioncanbewritten in termsof m1:
fn � 1m1 � 1
9m1 � 2m1 � 1 : n � 2
(3.20)
Sincewe alsoassumea functionalform for C n, theγi areuniquelydefinedif m1 is known. so,infact,theγi arefunctionsof m1:
γ0 � γ0 2 m1 3γ1 � γ1 2 m1 3 (3.21)
27
Looking at thefunctionalform of C n and fn, wemaywrite thefollowing limits for thesefunctions:
limm1 e 2
γ0 � C 1lim
m1 e ∞γ0 � 0 (3.22)
limm1 e 2
γ1 � 0
limm1 e ∞
γ1 � 1
Thus,for every ∆Gexcn fn pair, we canfind a correspondingγ0 2 m1 3 , γ1 2 m1 3 pair; andinsteadof using
theparametersof ∆Gexcn asourfitting parameters,wecanusecorrespondingparametersof γ0 2 m1 3 and
γ1 2 m1 3 asfitting parameters.
To demonstratethis transformation,weuseanexample.Assumethatthesizedistribution is againgeometricalasgiven by Eq. 3.20. For simplicity of calculation,assumethat C n is alsogiven in ageometricfashion:
C n � abn � 1 � 1 (3.23)
ThisgivesC 1 � 1 � a and f2 � 1 2 m1 � 13 , resultingin:
γ0 � 1 � am1 � 1
(3.24)
For γ1, wehave to do thefollowing summation:
γ1 � ∞
∑n? 2
2 abn � 1 � 13 1m1 � 1
9m1 � 2m1 � 1 : n � 1� m1 � 2
m1 � 1 J abm1 2 1 � b3 � 1 � b
� 1K (3.25)
Similarly, startingwith the polynomial form of ∆Gexcn asgiven by Eq. 3.2 and the log-normal
distribution of the defectsizesasgiven in Eq. 3.13,we calculatethe correspondingγ0 2 m1 3 , γ1 2 m1 3pairsnumerically. Figures3.4and3.5show thesetof calculatedγi for a givensetof coefficientsforthepolynomialof ∆Gexc
n . Ourcalculationsshow thattheresultscanbeapproximatedby thefollowingfunctionsof m1:
γ0 2 m1 3f� K1
m1 � 1(3.26)
γ1 2 m1 3f� m1 � 2m1 � K0
91 � 2 K0 � 23 K2
m1 � K0 :To get a feel for thesefunctions,we have plotted them in Figure 3.6, in addition to Figures
3.4 and 3.5. The parameterK0 controlsthe x-scale,parameterK1 definesthe overall scaleof γ0
andtheparameterK2 definesthemagnitudeof thepeakfor γ1. Usingthesethreeparameters,we cancoverawide rangeof curveshapes.
Sincethe γi calculatedfrom 2KPM and determinedby the analytical functionsof AKPM areapproximatelyequal,we would expect that both modelswould give the samesimulationresults.
28
gh
|' |' |' |' |' |' |' |' |� |' |' |' |' |' |' |' |' |� |' |' |' |' |' |' |' |' |�|�0
|�10
|�20
|�30
|�40
|�50
|�60
|�70
|�80
|�90
|�100 |( |( |( |( |( |( |( |( |� |( |( |( |( |( |( |( |( |� |( |( |( |( |( |( |( |( |�
|� |� |� |� |� |� |� |� |� |� |�
Average sizei
γ1 (From 2KPM) γ1 (AKPM fit) γ0 (From 2KPM) γ0 (AKPM fit)
101 102 103
γ
jj
j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j jk k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k kFigure3.4: The correspondingγi asa function of m1 for a0 � 3 � 855, a1 � 15� 9, a2 � � 1 � 4. Theparametersfor γi areK0 � 3, K1 � 14� 5 andK2 � 366.
|' |' |' |' |' |' |' |' |� |' |' |' |' |' |' |' |' |� |' |' |' |' |' |' |' |' |�|�0.0
|�0.5
|�1.0
|�1.5|�2.0
|�2.5 |( |( |( |( |( |( |( |( |� |( |( |( |( |( |( |( |( |� |( |( |( |( |( |( |( |( |�
|� |� |� |� |� |�
Average sizei
γ1 (From 2KPM) γ1 (AKPM fit) γ0 (From 2KPM) γ0 (AKPM fit)
101 102 103
γ
ll
l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l lm m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m
Figure3.5: Thecorrespondingγi asa functionof m1 for a0 � 1, a1 � 4 � 7, a2 � 0. Theparametersforγi areK0 � 8, K1 � 0 � 14andK2 � 8.
29
|' |' |' |' |' |' |' |' |� |' |' |' |' |' |' |' |' |� |' |' |' |' |' |' |' |' |�|�0.0
|�0.5
|�1.0
|�1.5
|�2.0
|�2.5 |( |( |( |( |( |( |( |( |� |( |( |( |( |( |( |( |( |� |( |( |( |( |( |( |( |( |�
|� |� |� |� |� |�
Average sizei
γ0
γ1
101 102 103
γ
Figure3.6: Exampleof γi asdefinedby Eq. 3.27. In this particularplot K0 � 4, K1 � 2 andK2 � 7havebeenused.
2KPM AKPM
|�0� |�
20|�
40|�
60|�
80|�
100� |�120
|�140
|�160� |�
180� |�200�|�0
|�1
|�2
|�3|�4
|�5 |� |� |� |� |� |� |� |� |� |� |�
|� |� |� |� |� |�
Time (s)
m1
(101
3 cm
-3)
Figure3.7: Comparisonof 2KPM andAKPM underidenticalconditionsandequivalentparametersasgivenin Fig. 3.4.
30
Indeed,whenwetestbothmodelsunderthesameconditions,theresultsarealmostindistinguishable(Fig 3.7). To obtainFig 3.7,we have usedtheparametersof Fig. 3.4,with an initial 5 � 1013cm� 2
interstitialdose.
It is alsopossibleto find C n, andhence∆Gexcn , if theparametersof theγi (Ki) andthedistribution
function is given,althoughthis procedureis lessstraightforward. Sinceγ0 is theproductof C 1 andf2, knowing γ0, wecaneasilyfind C 1. Ontheotherhand,γ1 is only dependentonC 2 throughC ∞. So,wecanfind C n, by solvinga largesetof linearequationsthatis definedby:
γ1 2 m1 3n� nmax
∑n? 2
C n fn 2 m1 3 (3.27)
atdifferentm1. OnceC n is determined,∆Gexcn canbefoundby solvingtheexpressionfor C n (Eq.3.4)
for ∆Gexcn , whichgivesa recurrencerelation:
∆Gexcn1 1 � ∆Gexc
n � kT lnC n (3.28)
Obviously, ∆Gexc1 mustbesetasa referencepoint.
3.5 Simplesolid-solubility model (SSS)
All of the above modelsaccountfor the fact that the characteristicsof the precipitateschangewithchangingaveragesize, and thereforecan be consideredas “sophisticated”models. The simplestmodelfor precipitationis to assumeaconstantsolidsolubility cut-off. Thisamountsto assumingthatall soluteatomsabovethesolidsolubility level will form precipitatesinstantaneously. Wehaveaddeda kinetic ratefor this systemandwill useit for comparisonswith KPM modelsdiscussedin thetext.Therateequationsfor this modelcanbeformulatedasfollows:
∂m1
∂t � Dλm1 2 CA � Css3 � X Dλ 2 CA � Css3 2 for CA - Css
0 for CA o Css(3.29)
∂CA
∂t � � ∂m1
∂t
The first term in the above equationis for growth/dissolutionof the precipitates,whereasthesecondtermis for nucleationof precipitateswhenCA - Css. Sincethismodelkeepstrackof only onemoment(m1), we canclassifyit asa 1-momentmodel.This modelis similar to themodelproposedby Rafferty [48], in thesensethatbothmodelscapturethedynamicsof clusterformation,but useaconstantsolid solubility.
31
Chapter 4
Modeling p 311q Defectsand TED
In this chapter, wewill applythemodelsthatweredevelopedin thepreviouschapterto�311� defect
evolutiondataby Eagleshametal. [17] to obtainparametervaluesfor thesemodels.Wewill thenusetheseparametersandmodelsto predictTED andcomparethepredictionsto boronmarker layerdataby Packan[39] andChao[6].
4.1 Evolution of r 311s DefectProfile
Eagleshamet al. [17] implanted5 � 1013cm� 2 Si into silicon at 40keV andannealedthe samplesat varioustemperatures.They measuredthe resulting
�311� defectdistributions using TEM and
reportedthemomentsof thisdistribution. However, they wereunableto observedefectssmallerthana cut-off of 20A andassumedthat all
�311� defectshave a constantwidth of 50A. An elongated
defectshouldn’t have its width largerthanits length,soweassumedw � min 2 l W 50A 3 (seeAppendixA). Wefit ourdatato momentsthatwecalculatethesamewayEagleshametal. did, but wewill showthattherealmomentsmightbedifferentfrom those.
First,weappliedthe3-momentmodel(3KPM) to Eaglesham’sdata.Weoptimizedourparameters(a0, a1, a2 andCss, with only Css beingtemperaturedependent)to matchthedatarepresented.It canbe seenthat we wereable to obtaina goodagreementbetweenthe dataandthe model (Figs. 4.1,4.2 and4.3). We show model resultsboth for the full distribution (actual)and the portion of thedistribution with sizegreaterthan20A (observable). Note that the averagesizeof the observabledefects(l - 20A) seemsto diverge from the averagesizeof all defects,and we will explain thisbehavior later.
To gainabetterunderstandingof thesystemlet usinvestigatehow thedepthprofilechangesovertime (Fig. 4.4). It canbe observed that the
�311� defectpile is beingconsumedfrom the sidethat
is closerto the surface,andthis processmovesthe profile deeperinto the substratewith time. Asa matterof fact, theapproximatelyexponentialdecayof thenumberof interstitialsin
�311� defects
arisesfrom thecombinationof theOstwaldripeningbehavior andtheshift of the�311� defectfrontier
away from thesurface.
Fig. 4.5 shows a plot of C n versusdefectlengthfor this system.It canbe observed thatC n hasa peakaroundn � 7, which would be the leaststableclustersize. The fact that small clustersare
32
t Data Model (Observable) Model (Actual)
|�0� |�
20|�
40|�
60|�
80|�
100� |�120� |�
140� |�160� |�
180� |�200�|� |M |M |M
|M |M |M |M |M |� |M
|M |M |M |M |M |M |M |�
|M |M |M |M |M |M |M
|M |� |� |� |� |� |� |� |� |� |� |� |�
|� |N |N |N |N |N |N |N |N |� |N |N |N |N |N |N |N |N |� |N |N |N |N |N |N |N |N |�
Time (s)u1011
1012
1013
1014
Den
sity
of I
nter
stiti
als
in D
efec
ts (
cm
v-2
) w w w w w w w w w wFigure4.1: Evolutionof densityof interstitialsin
�311� defects(m1) andcomparisonto model.Data
for 5 � 1013cm� 2 Si implantat 40keV with annealat 815� C.
x Data Model (Observable) Model (Actual)
|�0� |�
20|�
40|�
60|�
80|�
100� |�120
|�140
|�160� |�
180� |�200�|�0
|�20
|�40
|�60
|�80
|�100
|�120
|�140|�160
|�180
|�200
|�220
|� |� |� |� |� |� |� |� |� |� |�
|� |� |� |� |� |� |� |� |� |� |� |�
Time (s)
Ave
rage
leng
th o
f {31
1} D
efec
ts (
A)
y y y yy y y y y y
Figure4.2: Evolution of the averagelengthof�311� defectsandcomparisonto model. Data for
5 � 1013cm� 2 Si implantat 40keV with annealat815� C.
33
z Data Model (Observable) Model (Actual)
|�0� |�
20|�
40|�
60|�
80|�
100� |�120� |�
140� |�160� |�
180� |�200�|M |M |M |M |� |M
|M |M |M |M |M |M |M |�
|M |M |M |M |M |M |M |M
|� |M |M |M
|M |M |M |M |M |� |� |� |� |� |� |� |� |� |� |� |�
|N |N |N |N |� |N |N |N |N |N |N |N |N |� |N |N |N |N |N |N |N |N |� |N |N |N |N |N |N |N |N |�
Time (s)u109
1010
1011
1012
Den
sity
of {
311}
Def
ects
(cm
-2) { { { { { { { { { {
Figure 4.3: Evolution of density of�311� defects(m0) and comparisonto model. Data for
5 � 1013cm� 2 Si implantat 40keV with annealat815� C.
|�0.00� |�
0.05� |�0.10| |�
0.15| |�0.20� |�
0.25�|� |M |M |M |M |M |M|M |M |� |M |M |M |M |M
|M |M |M |� |M |M |M |M|M |M |M |M |� |M |M |M
|M |M |M |M |M |� |M |� |� |� |� |� |�
|� |N |N |N |N |N |N |N |N |� |N |N |N |N |N |N |N |N |� |N |N |N |N |N |N |N |N |� |N |N |N |N |N |N |N |N |� |N1s
20s
60s
120s
200s
1015
1016
1017
1018
1019
Depth (Q µm)
Con
cent
ratio
n of
Inte
rstit
ials
in D
efec
ts (
cm
P
-3)
Figure4.4: The�311� defectprofile for variousannealingtimesat 815� C.
34
|�1| |' |' |' |' |' |' |' |' |�
10|' |' |' |' |' |' |' |' |�
100� |' |' |' |' |' |' |' |' |�1000
|� |M |M |M
|M |M |M |M |M |� |M
|M |M |M |M |M |M |M |�
|M |M |M |M |M |M |M
|M |� |� |( |( |( |( |( |( |( |( |� |( |( |( |( |( |( |( |( |� |( |( |( |( |( |( |( |( |�
|� |N |N |N |N |N |N |N |N |� |N |N |N |N |N |N |N |N |� |N |N |N |N |N |N |N |N |�
Defect size (n)
Cn*
Css
1012
1013
1014
1015
Equ
ilibr
ium
con
cent
ratio
n of
inte
rstit
ials
(cm
}-3
)
Figure4.5:C n andCss versus�311� defectsizefor annealsat 815� C.
stablegivesrise to the very rapid formationof�311� defects,whereasthecontinuouslydecreasing
C n for l - 10A givesriseto theongoingOstwald ripeningin thesystem.Theleveling off aftersizesl - 300A stopsthegrowth of
�311� defectsaroundl � 150A, assuggestedby thedata.Notethatthe
exactshapeof C n dependson theform assumed,but thegeneralslopeis necessaryin orderto matchtheexperimentalresults.
Theshapeof thedistributionoverthesizes(Fig.4.6)alsotellsusthatthereis anunstableregionofclustersizes,suchthatthedistributionsaredouble-peaked.Thisshapeof thedefectsizedistributionisanimmediateresultof theenergy-minimizingclosureassumption.ThefactthatEagleshametal. can-not observe clusterssmallerthan20A resultsin anoverestimationof theaveragesize(seeFig. 4.2),aswell asanunderestimationof m0, thenumberof clusters(Fig.4.3).Theclustersin thefirst peakarecompletelyinvisible to them. Luckily, theseclustersdon’t contributesignificantlyto m1. Note thatfor a log-normalclosureassumption,thereis nosecondpeak,suchthattheaveragesizeof observableclustersandall clustersis thesame.
We characterizedthe parametersof the systemfor different temperatures(specifically, 815� C,738� C, 705� C and670� C), takingonly Css astemperaturedependent.Two of thesefits areshown inFig. 4.7.
We thenappliedthesamesystemandparametersto thedosedependenceof interstitialsin�311�
defects. Poateet al. [46] reportedthat above a thresholddosethe numberof interstitialsin�311�
defectsat theinitial stagesof TED increaseslinearlywith implantdose,but with aslopeof about1 � 4.Our modelwasableto accuratelypredictthis behavior (Fig. 4.8),boththeslopeaswell asthedosebelow whichno
�311� defectsform.
35
|' |' |' |' |' |�10
|' |' |' |' |' |' |' |' |�100� |' |'|� |M |M |M |M |M |M |M |M |� |M
|M |M |M |M |M |M |M |� |M |M |M |M |M |M |M
|M |� |M |M |M |M |M |M |M |M |� |M
|M |M |M |M |M |M |M |� |M |M |M |M |M |M |M
|M |� |M |M |M |M |M |M |M |M |� |( |( |( |( |( |� |( |( |( |( |( |( |( |( |� |( |(
|� |N |N |N |N |N |N |N |N |� |N |N |N |N |N |N |N |N |� |N |N |N |N |N |N |N |N |� |N |N |N |N |N |N |N |N |� |N |N |N |N |N |N |N |N |� |N |N |N |N |N |N |N |N |� |N |N |N |N |N |N |N |N |�
Average Length (A)~5s
45s
200s�10-7
10-6
10-5
10-4
10-3
10-2
10-1
100
Nor
mal
ized
Con
cent
ratio
ns o
f Def
ects
(f
n)
Figure4.6: Normalizeddistributionof�311� defectsizesfor varioustimesfor annealsat815� C. The
verticalline representstheminimumobservabledefectsize(20A).
��|' |' |' |' |' |' |' |� |' |' |' |' |' |' |' |' |� |' |' |' |' |' |' |' |' |� |' |' |' |' |' |' |' |' |� |' |' |' |' |' |' |' |' |�|� |M |M |M
|M |M |M |M |M |� |M|M |M |M |M |M |M |M |�
|M |M |M |M |M |M |M |M|� |M |( |( |( |( |( |( |( |� |( |( |( |( |( |( |( |( |� |( |( |( |( |( |( |( |( |� |( |( |( |( |( |( |( |( |� |( |( |( |( |( |( |( |( |�
|� |N |N |N |N |N |N |N |N |� |N |N |N |N |N |N |N |N |� |N |N |N |N |N |N |N |N |� |N
Time (s)
Data (815oC) Model (815oC) Data (670oC) Model (670oC)
101 102 103 10� 4 1051011
1012
1013
1014
Den
sity
of I
nter
stiti
als
in D
efec
ts (
cm
v
-2) � � � � � � � � � �
� � � � ��
Figure4.7: Evolutionof densityof interstitialsin�311� defects(m1) andcomparisonto model.Data
for 5 � 1013cm� 2 Si implantsat40keV with annealsat 815� C and670� C.
36
� Data Model
|�0.0� |�
0.2� |�0.4� |�
0.6� |�0.8� |�
1.0� |�1.2� |�
1.4� |�1.6�|�0.0
|�0.5
|�1.0
|�1.5
|�2.0 |� |� |� |� |� |� |� |� |�
|�
|�
|�
|�
|�40keV Si implant
Implant Dose (10� 14 cm-2)
{31
1} In
ters
titia
l Dos
e (1
0
v14
cm
-2)
� � � �� �
�
Figure4.8: Predictionof dosedependenceof interstitialsin�311� defectsduring initial stagesof
TED. Datafrom Poateet al. [46] for 40keV Si implantsand1hr annealat 670� C.
4.2 Prediction of TED
We usedthe parametersobtainedin the previous sectionto predictTED behavior asa function ofimplantandannealingconditions.Forcomparison,weuseddatafromPackan[39], whousedimplantsof 29Si into silicon at energiesrangingfrom 10keV to 200keV anddosesrangingfrom 1 � 1012 to2 � 1014cm� 2. TED wasmeasuredby observingprofilemovementof adeepboronmarker layer. Nofurtherfitting wasdoneat thisstageandall parameterswerefrom work citedpreviously[1, 5] or frommatchingdataby Eagleshamet al.
Theresultsshow thatthemodelpredictionsdoagoodjob in matchingthetimeandenergy depen-dence(Figs.4.9and4.10). Themodelnot only givestheright magnitudefor theobservedenhance-ments,but alsopredictslengthof time TED lasts. In addition,it alsocorrectlypredictsthathigherdosesor energiesmake TED last longer, but give similar enhancementsfor shorttimes(Fig. 4.10).Thepredicteddosedependencematchesmostof thedata(Fig.4.11),with substantialdifferencesonlyfor thelowestimplantdose.Pleasenotethat in thesesimulationsa “+1” damagedosewasusedandtheeffectof usingaplusvaluethatis dosedependentwill beinvestigatedlater.
4.3 Comparisonof models
In this section,we will comparefour extendeddefectmodels:3-momentmodel(3KPM), which wasdemonstratedin the previous section,2-momentmodel (2KPM), the simplesolid solubility model
37
� Data Model
|�0� |�
20|�
40|�
60|�
80|�
100� |�120� |�
140� |�160� |�
180� |�200� |�
220�|�0
|�10
|�20
|�30
|�40
|�50
|�60
|�70
|�80
|�90
|�100
|�110
|�120
|� |� |� |� |� |� |� |� |� |� |� |�
|� |� |� |� |� |� |� |� |� |� |� |� |�
Energy (keV)
(D
t)� 1/2
(nm
)
Temp = 800oC , Dose = 1014 cm-2 , Time = 60min
� �� � �
Figure4.9: Energy dependenceof TED. Total broadeningof a deepB marker profile dueto implan-tationof 1 � 1014cm� 2 29Si with annealingat 800� C for 60min.
� Data (80keV) Model (80keV)� Data (200keV) Model (200keV)
|�0� |�
10|�
20|�
30|�
40|�
50|�
60|�
70
|�0
|�10
|�20
|�30
|�40
|�50
|�60
|�70
|�80|�90
|�100
|�110
|�120
|�130 |� |� |� |� |� |� |� |�
|� |� |� |� |� |� |� |� |� |� |� |� |� |�
Time (min)
(D
t)� 1/2
(nm
)
Temp=800� oC , Dose = 1014 cm-2�� � � � � �
�
� � � � � �
Figure4.10:Timedependenceof TED. Totalbroadeningof aB markerprofiledueto implantationof1 � 1014cm� 2 29Si at 80 and200keV with annealingat800� C.
38
� Data Model
|� |� |� |� |� |� |� |� |� |� |� |� |� |� |� |� |� |� |� |� |�|�0
|�10
|�20
|�30
|�40
|�50
|�60
|�70
|�80
|�90
|�100
|�110
|�120 |� |� |� |� |� |� |� |� |� |� |� |� |� |� |� |� |� |� |� |� |�
|� |� |� |� |� |� |� |� |� |� |� |� |�
1012 1013 1014
Dose (cm) -2)
(Dt)
1/2
(nm
)
Temp=800� oC , Energy = 200keV , Time = 45min
� � � � � �
Figure4.11:Dosedependenceof TED. Total broadeningof a B marker profiledueto 29Si implanta-tion at 200keV with annealingat 800� C for 45min.
(SSS)andtheanalyticalmodel(AKPM). Wefirst review theassumptionsof thesefour models:� 1-momentmodel: In thismodelweassumethatsoluteatomsabovesolidsolubility precipitateto extendeddefectswith a diffusion limited rate. This modeldoesnot accountfor Ostwaldripening.� 2-moment model: We uselog-normaldistribution of extendeddefectswith fixed z2. Thevariableswehavearem0 andm1.� 3-momentmodel: Weuseenergy minimizingclosureassumptionwith threemoments.� Analytical model: We useanalyticalfunctionsfor the reactionratesthat dependon averagesize.
Figure4.12shows thecomparisonof 1-, 2- and3-momentmodelsto the�311� dissolutiondata
from Eagleshamet al. [17]. It can be observed that both the 2 and3-momentmodelsaccuratelycapturethe observed behavior. However, the 1-momentmodelcannotaccountfor the exponentialdecayin thenumberof interstitialsstoredin
�311� defects,sinceit neglectsOstwaldripeningprocess,
andpredictsanapproximatelylineardecayin thenumberof interstitialsstoredin�311� defects.The
exponentialdecayrateis a resultof decreasingsuper-saturationof interstitialswith time,which is, inturn,a resultof theripeningof defects.
WhenweapplyAKPM to thesamesetof data,wefind thatit alsodescribestheOstwaldripeningprocessequallywell, althoughthecomputationtime is muchsmaller(Fig. 4.13andFig. 4.14). This
39
� Data 3-moment model 2-moment model 1-moment model
|�0� |�
20|�
40|�
60|�
80|�
100� |�120� |�
140� |�160� |�
180� |�200�|� |M |M |M
|M |M |M |M |M |� |M
|M |M |M |M |M |M |M |�
|M |M |M |M |M |M |M
|M |� |� |� |� |� |� |� |� |� |� |� |�
|� |N |N |N |N |N |N |N |N |� |N |N |N |N |N |N |N |N |� |N |N |N |N |N |N |N |N |�
Time (s)u1011
1012
1013
1014
Den
sity
of I
nter
stiti
als
in D
efec
ts (
cm
v-2
) � � � � � � � � � �Figure4.12: Comparisonof 1, 2 and3-momentmodelsfor evolution of
�311� defects.Both 2 and
3-momentmodelscapturetheexponentialdecayof interstitialsin�311� defects.�
Data (Eaglesham) Analytical model 3-moment model
|�0� |�
20|�
40|�
60|�
80|�
100� |�120� |�
140� |�160� |�
180� |�200�|M |M |M |M |M |M |M |�
|M |M |M |M |M|M |M |M |� |M
|M |M |M |M |M |M |M|� |� |� |� |� |� |� |� |� |� |� |�
|N |N |N |N |N |N |N |� |N |N |N |N |N |N |N |N |� |N |N |N |N |N |N |N |N |�
Time (s)u1012
1013
1014
Den
sity
of I
nter
stiti
als
in D
efec
ts (
cm
v
-2) � � � � � � � � � �
Figure4.13: Evolution of densityof interstitialsin�311� defects(m1) andcomparisonto analytical
model(AKPM) and3-momentmodel(3KPM).
40
� Data Analytical model
|�0� |�
20|�
40|�
60|�
80|�
100� |�120� |�
140� |�160� |�
180� |�200�|�0
|�20
|�40
|�60
|�80
|�100
|�120
|�140
|�160
|�180
|�200 |� |� |� |� |� |� |� |� |� |� |�
|� |� |� |� |� |� |� |� |� |� |�
Time (s)u
Ave
rage
leng
th o
f {31
1} D
efec
ts (
A)
� � � �� � � � � �
Figure4.14: Evolution of the averagelengthof�311� defectsandcomparisonto analyticalmodel
(AKPM).
makesthe AKPM modelof choicefor simulationof�311� defectsin large systems.The fact that
AKPM doesn’t requireany proprietaryoperatorsenableseasyintegrationto many PDEsolvers.
If we look at TED datafrom Packan[39], thedifferencesbetweenthe threemodelsis relativelysmall(Fig. 4.15).Onecanreadilyobservethatthefinal amountof TED predictedby all threemodelsis approximatelythesame.This is dueto the fact that thefinal amountof TED is not dependentonhow the interstitial super-saturationevolvesover time, but ratheron effectivenessof the surfaceinconsuminginterstitials.Only atshorttimesdoesthereseemto beasignificantdifferencebetweenthemodels.If we look at shorttime behavior moreclosely, thedifferencebetweenthemodelsbecomesmoreevident (Fig. 4.16). The1-momentmodelpredictsa constantenhancementof diffusivity untilTED is over, whereasthe2-momentmodelaccountsfor thereductionof diffusivity enhancement—or interstitialsuper-saturation— duringTED. Again,this is aneffectcausedby theOstwaldripeningprocess.
4.4 Effect of “ � n” factor
Let’sinvestigatethedifferencesbetweenusinga“ � 1” modelfor initial damage,andthemorerigorous“ � n” modeldescribedin section2.2.1usingAKPM. Sincethe“ � n” modelpredictsa plusvalueashigh as10 (Fig. 2.4), we would expectthat the TED predictionsat lower doseswould bewould besubstantiallyincreased.However, simulationsshow only a marginal increase(Fig. 4.17).Thereasonbehindthiscanbeunderstoodby investigatingthedepthprofileof thedamage(Fig.2.3).Althoughthe
41
� Data (Packan) 3-moment model 2-moment model 1-moment model
|�0� |�
5� |�10
|�15
|�20
|�25
|�30� |�
35
|�0
|�1000
|�2000
|�3000
|�4000
|�5000
|�6000
|�7000
|�8000 |� |� |� |� |� |� |� |�
|� |� |� |� |� |� |� |� |�
Time (min)
Dt (
nm 2 )
850oC, Si, 200keV
5e13 cm-2
1e14 cm-2
¡ ¡ ¡ ¡¢¢ ¢ ¢ ¢
Figure 4.15: Comparisonof 1, 2 and 3-momentmodelsfor TED predictions. The modelsdiffersignificantlyonly at shorttimebehavior, which thisdatacannotdistinguish.
£ Data (Chao) AKPM model SSS model
|� |' |' |' |' |' |' |' |' |� |' |' |' |' |' |' |' |' |� |' |' |' |' |' |' |' |' |�|M |M |M |M |M |M|M |�1000
|M |M |M |M|M |M |M |M |�10000
|M |M |M |� |( |( |( |( |( |( |( |( |� |( |( |( |( |( |( |( |( |� |( |( |( |( |( |( |( |( |�
|N |N |N |N |N |N |N |� |N |N |N |N |N |N |N |N |� |N |N |N
Time (s)101 102 103 104
<D
>/D
*
750oC, Si, 50keV, 5e13cm-2¤ ¤ ¤ ¤
¤Figure4.16: Comparisonof 1 and2-momentmodelsfor shorttime diffusivity enhancements.Datafrom Chao[6]for a5 � 1013cm� 2 50keV implantwith annealsat 750� C.
42
¥ Data "+n" model "+1" model
|�0� |�
1| |�2� |�
3¦ |�4� |�
5� |�6� |�
7§ |�8� |�
9|�
10
|�0
|�10
|�20
|�30
|�40
|�50
|�60
|�70
|�80
|�90
|�100
|�110 |� |� |� |� |� |� |� |� |� |� |�
|� |� |� |� |� |� |� |� |� |� |� |�
Dose (10) 13 cm-2)
(D
t)1/
2 (n
m) © © © © © ©
Figure4.17:Comparisonof predictionsfor thedosedependenceof TED using“ � 1” and“ � n” mod-els.
lowerimplantdosesindeedresultin ahigherplusvalue,muchof thisextrainterstitialconcentrationisnearthesurface,suchthatit is dissolvedveryquickly. Therefore,theneteffectof thisexcessdamageis minimal.
However, to fully understandtheeffects,a morerigorous“ � n” modelhasto bedevelopedbasedon atomisticsimulations,which takesdiffusion of interstitialsandvacanciesduring annealingintoaccountandalsoconsidersfactorssuchasamorphouspocket formation.
4.5 Summary
In summary, we have developeda rangeof KPM modelsandtestedthemfor validity andaccuracyunderdifferentconditions.For theevolution of
�311� defectsandfor TED behavior thedifference
betweenthe2 and3-momentmodels(2KPM, 3KPM) andtheanalyticalmodel(AKPM) is negligible.Theorderingof modelswith respectto their computationalefficiency is asfollows (from mosttimeconsumingto leasttime consuming):3KPM, 2KPM, AKPM, SSS.For the rangeof availabledata,AKPM seemsto bethebestcompromisebetweencomputationalefficiency andaccuracy.
Thesimplesolid solubility model(SSS)is not ableto accountfor theOstwald ripeningprocess,andthusis unsuitablefor modelingof extendeddefectkineticsandshort-timeTED behavior. How-ever, it doesa goodjob in predictingthefinal amountof TED. But thekineticsof
�311� defectsat
the initial stagesof TED is importantif we have temperatureramps.Dependingon thespeedof thetemperatureramp,wemayendupwith differentamountsof clusteredinterstitials.TheSSSmodelis
43
not capableof makingthis distinctionsinceit doesn’t accountfor Ostwald ripening. Anotherpointwherethe
�311� defectkinetics is importantis whenwe have a competingprocess,suchasboron
interstitial cluster(BIC) formation. In sucha case,both�311� defectsandBICs will competein
capturingthe interstitials,andthekineticsbecomeimportant.With respectto all of these,we won’trecommendusingSSSmodelfor modelingextendeddefects.
44
Chapter 5
Models for DislocationLoops
5.1 Extending KPM to dislocation loops
Our first approachto modelingof dislocationloopswasto extendtheKPM modelto includedislo-cationloopsin additionto
�311� defects.We did this by assumingthat therearetwo populationof
extendeddefectswhich caninteractwith eachother. We assumedthat for smallersizesit wasener-geticallymorefavorableto stayasa
�311� defect,but above a certainsizeit wasmorefavorableto
transforminto a dislocationloop (Fig 5.2). Our simulationsgavea cross-over aroundn � 2200.Thetransferratefrom
�311� defectsinto dislocationloopscanbeexpressedas:
DI
b2 ª f « 311¬n � f loop
n exp ®� ∆G« 311¬n � ∆Gloop
n
kT ¯±° (5.1)
whereb is a “capturedistance.” We found a valueof 20µm for b. Thus the transferfrom�311�
defectsinto dislocationloopsis a ratherslow process.
Pan et al. [42] implanted1 � 1016cm� 2 Si into silicon at 50keV andannealedthe samplesat850� C and1000� C. They measuredtheresultingdislocationloopdistributionsusingTEM. Wemod-eledtheir datausinga similar approachto
�311� defects. The parametersfor
�311� defectswere
from previouswork [25]. It canbearguedthatCss for loopsshouldbejustC I , sinceaninfinite sizeperfectloop is nothingbut an extra planein silicon. For partial loopsCss shouldbeslightly higher,but smallchangesin Css hadno significanteffect in our simulationresults.Initial damageconditionswereobtainedasdescribedin Section2.2.2,wheretheamorphizationthresholdhasbeenadjustedtogetthecorrectdoseof interstitialsin loops.
Our resultsshow thatwewereableto correctlymodeltheevolutionof thesystemandtransforma-tion of
�311� defectsinto dislocationloops(Fig. 5.3). We werealsoableto get thecorrectOstwald
ripeningbehavior (Fig.5.4).Similarmatchesfor datafrom Lui etal. [35] thatincludedlongerannealswhich led to substantialloopdissolution(Fig. 5.5).
The relatively slow dissolutionrateof dislocationloopsstemsfrom the factsthat they cangrowvery largeandCss for loopsis closeto C I . This resultsin C n for loopsapproachingC I , so that theysustainonly a small super-saturationof interstitials. Sincetheseloopsaredeepin thesubstrateandsustainonly aminimal super-saturation,theflux to thesurfaceis smallandthusdissolvethey slowly.
45
{311}s Dislocation Loops
|' |' |' |' |' |' |' |' |� |' |' |' |' |' |' |' |' |� |' |' |' |' |' |' |' |' |� |' |' |' |' |' |' |' |' |� |' |' |' |' |' |' |' |' |�|�0
|�1
|�2
|�3
|�4
|�5
|�6 |( |( |( |( |( |( |( |( |� |( |( |( |( |( |( |( |( |� |( |( |( |( |( |( |( |( |� |( |( |( |( |( |( |( |( |� |( |( |( |( |( |( |( |( |�
|� |� |� |� |� |� |�
Size²
Def
ect e
nerg
y (e
V/a
tom
)
³
101 102 103 104 105
Figure5.1:�311� defectanddislocationloopenergy asa functionof defectsize.Theloopsaremore
stableat largersizes.
{311}s Dislocation Loops
|' |' |' |' |' |' |' |' |� |' |' |' |' |' |' |' |' |� |' |' |' |' |' |' |' |' |� |' |' |' |' |' |' |' |' |� |' |' |' |' |' |' |' |' |�|� |M |M |M |M |M |M |M |M |� |M |M |M|M |M |M |M |M |� |M |M |M |M |M |M |M |M |� |M
|M |M |M |M |M |M |M |� |M |M |M |M |M |M |M |M |�|M |M |M |M |M |M |M |M |� |M |M |M |M |M |M |M |M
|� |M |M |M |M |M |M |M |M |� |( |( |( |( |( |( |( |( |� |( |( |( |( |( |( |( |( |� |( |( |( |( |( |( |( |( |� |( |( |( |( |( |( |( |( |� |( |( |( |( |( |( |( |( |�
|� |N |N |N |N |N |N |N |N |� |N |N |N |N |N |N |N |N |� |N |N |N |N |N |N |N |N |� |N |N |N |N |N |N |N |N |� |N |N |N |N |N |N |N |N |� |N |N |N |N |N |N |N |N |� |N |N |N |N |N |N |N |N |� |N |N |N |N |N |N |N |N |�
Size² CI
*
101 102 103 104 105108
109
1010
1011
1012
1013
1014
1015
1016
C
´ n* (c
m-3
)
Figure5.2:�311� defectanddislocationloopC n asa functionof defectsize.
46
µ¶ {311} defects Dislocation loops
|� |' |' |' |' |' |' |' |' |� |' |' |' |' |' |' |' |' |� |' |' |' |' |' |' |' |' |� |' |' |' |' |' |' |' |' |� |' |' |' |' |' |' |' |' |� |' |' |' |' |' |' |' |' |�|� |M
|M |M |M |M |M
|M |M |� |M
|M |M |M |M |M
|M |M |� |� |( |( |( |( |( |( |( |( |� |( |( |( |( |( |( |( |( |� |( |( |( |( |( |( |( |( |� |( |( |( |( |( |( |( |( |� |( |( |( |( |( |( |( |( |� |( |( |( |( |( |( |( |( |�
|� |N |N |N |N |N |N |N |N |� |N |N |N |N |N |N |N |N |�
Time (s)u
Data (1000oC) Data (850oC)
10-1 100 101 102 103 10� 4 1051013
1014
1015
Inte
rstit
ials
bou
nd to
Def
ects
(cm
·-2
) ¸ ¸ ¸ ¸ ¸ ¸ ¸ ¹ ¹ ¹ ¹ ¹ ¹ ¹
Figure5.3: Evolution of densityof interstitialsin extendeddefects(m1) andcomparisonto model.Datafrom Panetal. [42] for 1 � 1016cm� 2 Si implantat 50keV.º»
{311} defects Dislocation loops
|� |' |' |' |' |' |' |' |' |� |' |' |' |' |' |' |' |' |� |' |' |' |' |' |' |' |' |� |' |' |' |' |' |' |' |' |� |' |' |' |' |' |' |' |' |� |' |' |' |' |' |' |' |' |�|M |M |M |M |M |M |M|�1000
|M |M |M |M |M|M |M |M |�10000
|M |M |M |M |M|M |M |M |�100000 |� |( |( |( |( |( |( |( |( |� |( |( |( |( |( |( |( |( |� |( |( |( |( |( |( |( |( |� |( |( |( |( |( |( |( |( |� |( |( |( |( |( |( |( |( |� |( |( |( |( |( |( |( |( |�
|N |N |N |N |N |N |N |� |N |N |N |N |N |N |N |N |� |N |N |N |N |N |N |N |N |�
Time (s)
Ave
rage
Siz
e ³ Data (1000oC) Data (850oC)
10-1 100 101 102 103 10� 4 105
¼ ¼ ¼ ¼ ¼ ¼ ¼½ ½ ½ ½ ½ ½ ½
Figure5.4: Evolution of averagedefectsize(m1 m0) for 1 � 1016cm� 2 Si implantsat 50keV andcomparisonto datafrom Panetal. [42]
47
¾¿ {311} defects Dislocation loops
|�1| |' |' |' |' |' |' |' |' |�
10|' |' |' |' |' |' |' |' |�
100� |' |' |' |' |' |' |' |' |�1000
|' |' |' |' |' |' |' |' |�10000� |' |' |' |' |' |' |' |' |�
100000
|�0
|�1
|�2
|�3
|�4
|�5
|�6
|�7 |� |( |( |( |( |( |( |( |( |� |( |( |( |( |( |( |( |( |� |( |( |( |( |( |( |( |( |� |( |( |( |( |( |( |( |( |� |( |( |( |( |( |( |( |( |�
|� |� |� |� |� |� |� |�
Time (s)u
Data (1000oC) Data (800oC)
Inte
rstit
ials
bou
nd to
Def
ects
(1014
cm
-2)
ÀÀ À À
Á Á Á Á Á
Figure5.5: Evolution of densityof interstitialsin dislocationloops(m1) andcomparisonto model.Datafor 1 � 1015cm� 2 Si implantat 50keV.
5.2 Analytical loop model
Although themodelpresentedin theprevioussectiongivesgoodpredictionsof loop evolution, it iscomputationallyexpensive. Therefore,we seekto find ananalyticalmodelfor loop evolution basedon theanalyticalmodelfor
�311� defects(AKPM). Theapproachwetake is similar: Weassumethat�
311� defectstransforminto dislocationloopsby an unfaulting reactiononcethey reacha criticalsize.Wecannotethefollowing pointsfor thefunctionsthatareinvolvedin AKPM:� Loopshavealowersolidsolubility than
�311� defects,typically aroundC I . Therefore,wehave
to setthesolid solubility of the�311� defects/dislocationloop distribution to Cloop
ss . Sincetheγi areinverselyproportionalto thesolid solubility they have to bemultiplied by thedifferencefor smallersizes.� To ensurelim
n ∞γ1 � 1, weneeda functionalchangeafterthe
�311� /loopcrossoverpoint (ncrit).� Theγi have to becontinuousat the
�311� /loopcrossoverpoint (ncrit).
To understandwhattypeof functionthisrestrictionsgive,weproceedasearlier:Weuse∆Gexcn for
dislocationloops(which is identicalto ∆Gexcn for
�311� defectsatsizessmallerthanncrit), anda log-
normaldistribution function;andfind thecorrespondingγi aswe did for�311� defectsby Eq. 3.17.
Theresultsareshown in Fig 5.6,togetherwith analyticalfunctionsthatcapturethebehavior of γ1.
48
Ã
|' |' |' |' |' |' |' |' |� |' |' |' |' |' |' |' |' |� |' |' |' |' |' |' |' |' |� |' |' |' |' |' |' |' |' |� |' |' |' |' |' |' |' |' |� |' |' |' |' |' |' |' |' |�|� |M |M |M |M |M |M |M |M |�
|M |M |M |M |M |M |M |M |� |M
|M |M |M |M |M |M |M |� |M |M
|M |M |M |M |M |M |� |M |M |M |M
|M |M |M |M |� |M |( |( |( |( |( |( |( |( |� |( |( |( |( |( |( |( |( |� |( |( |( |( |( |( |( |( |� |( |( |( |( |( |( |( |( |� |( |( |( |( |( |( |( |( |� |( |( |( |( |( |( |( |( |�
|� |N |N |N |N |N |N |N |N |� |N |N |N |N |N |N |N |N |� |N |N |N |N |N |N |N |N |� |N |N |N |N |N |N |N |N |� |N |N |N |N |N |N |N |N |� |N
Defect SizeÄOnly {311}s {311}s and loops
γ1 (From 2KPM) γ1 (AKPM fit)
101 102 103 104 105 106100
101
102
103
104
105
γ1 Å
Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å ÅFigure5.6: Theγ1 functionfor dislocationloopsat asderivedfrom ∆Gexc
n for dislocationloopsandalog-normalclosureassumption.
We find asetof γi functionsto beusedwith AKPM by extendingthesetof functionsweusedfor�311� defects:
γ0 � ÆÇÇÇÈ ÇÇÇÉC « 311¬
ss
Cloopss
K1
m1 � 1for m1 � ncrit
K3
91
m1 � 1 : αfor m1 - ncrit
(5.2)
γ1 � ÆÇÇÇÈ ÇÇÇÉC « 311¬
ss
Cloopss
m1 � 2m1 � K0
91 � 2 K0 � 23 K2
m1 � K0 : for m1 � ncrit
1 � K4
9K0 � 2
m1 � K0 : αfor m1 - ncrit
(5.3)
whereK3 andK4 arechosensuchthatcontinuityof γ0 andγ1 areensured(Fig. 5.7). In our simula-tions,we have useda cross-over point of ncrit � 1000andleft α asa fitting parameters.Pleasenotethatncrit shouldnot be interpretedas“the point where
�311� defectto dislocationloop transforma-
tion happens,” but rather“the sizebelow which no�311� defectto dislocationloop transformation
happens.” Thetransformationwill happenatany sizegreaterthanncrit.
Weagaincompareourmodelagainstdatafrom Panetal.[42]. As onecanreadilyobserve,agoodmatchcanbefound(Fig.5.8andFig. 5.9).Sincethismodelcan’t distinguishbetweentheinterstitialscontainedin dislocationloopsand the interstitialscontainedin
�311� defects,we can’t plot them
separately. However, to obtainan ideaon how the�311� defectswould evolve, we canturn off the
49
|' |' |' |' |' |' |' |' |� |' |' |' |' |' |' |' |' |� |' |' |' |' |' |' |' |' |� |' |' |' |' |' |' |' |' |� |' |' |' |' |' |' |' |' |�|� |M |M |M
|M |M |M |M |M |� |M
|M |M |M |M |M |M |M |�
|M |M |M |M |M |M |M |M
|� |M |( |( |( |( |( |( |( |( |� |( |( |( |( |( |( |( |( |� |( |( |( |( |( |( |( |( |� |( |( |( |( |( |( |( |( |� |( |( |( |( |( |( |( |( |�
|� |N |N |N |N |N |N |N |N |� |N |N |N |N |N |N |N |N |� |N |N |N |N |N |N |N |N |� |N
Defect SizeÄ
Only {311}s {311}s and loops
101 102 103 104 105100
101
102
103
γ1
Figure 5.7: The γ1 function for dislocationloops at 1000� Cfor the AKPM. For m1 � ncrit, γ1 isidenticalto thatfor
�311� defects,exceptfor amultiplier to accountfor differencesin solidsolubility.
Ê Loop model {311} model
|� |' |' |' |' |' |' |' |' |� |' |' |' |' |' |' |' |' |� |' |' |' |' |' |' |' |' |� |' |' |' |' |' |' |' |' |�|�0
|�1
|�2
|�3|�4
|�5 |� |( |( |( |( |( |( |( |( |� |( |( |( |( |( |( |( |( |� |( |( |( |( |( |( |( |( |� |( |( |( |( |( |( |( |( |�
|� |� |� |� |� |�
Time (s)
Data (850oC)
101 102 103 104 105
Inte
rstit
ials
bou
nd to
Def
ects
(1014
cm
-2) Ë Ë Ë Ë Ë Ë Ë
Figure5.8: Evolutionof totalnumberof interstitialsstoredin dislocationloopsand�311� defectsfor
theAKPM. Datafor 1 � 1016cm� 2 Si implantat 50keV with annealat 850� C.
50
ÌÍ Model
|� |' |' |' |' |' |' |' |' |� |' |' |' |' |' |' |' |' |� |' |' |' |' |' |' |' |' |� |' |' |' |' |' |' |' |' |� |' |' |' |' |' |' |' |' |�|�100
|M |M |M |M |M
|M |M |M |�1000
|M |M |M |M |M
|M |M |M |�10000
|M |M |M |� |( |( |( |( |( |( |( |( |� |( |( |( |( |( |( |( |( |� |( |( |( |( |( |( |( |( |� |( |( |( |( |( |( |( |( |� |( |( |( |( |( |( |( |( |�
|� |N |N |N |N |N |N |N |N |� |N |N |N |N |N |N |N |N |� |N |N |N
Time (s)u
Ave
rage
Siz
e (a
tom
s)
Data (1000oC) Data (850oC)
100 101 102 103 10� 4 105
Î Î Î Î Î Î ÎÏ Ï Ï Ï Ï Ï Ï
Figure5.9: Evolutionof averagedefectsizefor 1 � 1016cm� 2 Si implantsat 50keV from AKPM.
loop portionof themodelandplot�311� defectsin theabsenceof loops(Fig. 5.8). Themodelalso
givesa goodfit to theOstwald ripeningprocessof dislocationloops(Fig. 5.9).
5.3 Simulating TED
We have appliedour loop modelto TED conditionsto predictdopantmovementof a buriedmarkerlayerdueto anamorphizingimplant. To this end,we have useddatafrom Chao[6] andHuang[32].Chaoalso had datafor non-amorphizingimplants,and for thoseimplantsthe modelpredictsthatdislocationloopswill never form in agreementwith experiment.
Our resultsshow thatwe wereableto predictTED behavior over a largerangeof implantdoses(Fig. 5.10). Although the fact that a 1 � 1015cm� 2 implant causeslessTED thana 1 � 1014cm� 2
implantmayseemsa little paradoxicalin thefirst look, it is very reasonablewhenwe considerthatmostof theimplantdamageannealsoutby epitaxialregrowth for amorphizingimplants.
5.4 Modeling heterogeneousnucleation
Themodelwedemonstratedin theprecedingsectionsassumedthatthesolemechanismof dislocationloop formationwasthroughtransformationfrom
�311� defects,andthat the
�311� defectsgrow to
a certainsize (ncrit), after which it becomesenergetically favorableto transforminto a dislocationloop. Moreover, weassumedthatthenucleationof
�311� defects,andhenceformationof dislocation
51
|�0� |�
10|�
20|�
30|�
40|�
50|�
60|�
70|�
80|�
90� |�100� |�
110| |�120� |�
130¦|�0
|�10
|�20
|�30
|�40
|�50
|�60
|�70
|�80
|�90
|�100
|�110 |� |� |� |� |� |� |� |� |� |� |� |� |� |�
|� |� |� |� |� |� |� |� |� |� |� |�
Time (min)
1016 cm-2 (Huang) 1015 cm-2 (Chao) 1014 cm-2
1013 cm-2
(D
t)1/
2 (n
m) Ð Ð ÐÑ Ñ ÑÒ Ò
Ò
Ó ÓFigure5.10: Predictionof TED usingFKPM for amorphizingandnon-amorphizingimplants.Datafor 50keV Si implantswith annealingat 750Ô C.
loopsis ahomogeneousprocess.Thatis, in thewafer, thereis nopreferredsitefor Õ 311Ö defect/loopformationandthesedefectswill form anywherewhenthereis enoughdriving force to form thesedefects.
However, experimentalobservationsby Pan et al. revealedthat after annealingan amorphizingimplant,thereexistsalsoloopsthatarein sizemuchsmallerncrit [41]. Althoughour previousmodeldoesn’t rule out this possibility, it is very unlikely that a small Õ 311Ö defectwill transforminto adislocationloopsinceit is energeticallyunfavorable.Panet al. concludedthattheseloopsmusthavenucleateddirectly from thematrix.
But why do we observe loopsthatnucleatedirectly without Õ 311Ö defectsif we have an amor-phizingimplant,andobserveonly Õ 311Ö defectsif wehaveanon-amorphizingimplant?Theanswermay be heterogeneousnucleationwith the amorphous/ crystallineinterfaceactingasa nucleationsitefor dislocationloops.Therefore,wecanhavedislocationloopsthataremuchsmallerthan Õ 311Ödefects,sincethey have formedby nucleationassistedby theamorphous/crystallineinterface.
We now seekways to extendour model to include this capability. Oneway of accomplishingthis goalis to have two separatedistributionsof extendeddefects.Thefirst distributionwould bethedistribution for Õ 311Ö defectsanddislocationloopsthatwereformedby transformationfrom Õ 311Ödefects,which is thedistribution we consideredin previoussection.Theseconddistribution wouldbefor dislocationloopsthat formedheterogeneouslyat theamorphous/crystallineinterface.We candealwith the heterogeneousnatureof the seconddistribution by lowering the formationenergy oftheseextendeddefectsaroundthe amorphous/crystallineinterface. A simpleway of achieving thisgoal is depictedin Fig 5.11. Sincewe can’t seesmallsizedefects,thedeterminationof how ∆Gexc
n ,
52
{311}/loop distribution Loops at a/c interface Loops elsewhere
|× |× |× |× |× |× |× |× |Ø |× |× |× |× |× |× |× |× |Ø |× |× |× |× |× |× |× |× |Ø |× |× |× |× |× |× |× |× |Ø |× |× |× |× |× |× |× |× |Ø |× |× |× |× |× |× |× |× |Ø|Ù |Ú |Ú |Ú |Ú |Ú |Ú |Ú |Ú |Ù
|Ú |Ú |Ú |Ú |Ú |Ú |Ú |Ú |Ù |Ú |Ú
|Ú |Ú |Ú |Ú |Ú |Ú |Ù |Ú |Ú |Ú |Ú |Ú |Ú
|Ú |Ú |Ù |Ú |Ú |Ú |Ú |Ú |Ú |Ú |Ú |Ù
|Ú |Ú |Ú |Ú |Ú |Ú |Ú |Ú |Ù |Û |Û |Û |Û |Û |Û |Û |Û |Ü |Û |Û |Û |Û |Û |Û |Û |Û |Ü |Û |Û |Û |Û |Û |Û |Û |Û |Ü |Û |Û |Û |Û |Û |Û |Û |Û |Ü |Û |Û |Û |Û |Û |Û |Û |Û |Ü |Û |Û |Û |Û |Û |Û |Û |Û |Ü
|Ý |Þ |Þ |Þ |Þ |Þ |Þ |Þ |Þ |Ý |Þ |Þ |Þ |Þ |Þ |Þ |Þ |Þ |Ý |Þ |Þ |Þ |Þ |Þ |Þ |Þ |Þ |Ý |Þ |Þ |Þ |Þ |Þ |Þ |Þ |Þ |Ý |Þ |Þ |Þ |Þ |Þ |Þ |Þ |Þ |Ý |Þ |Þ |Þ |Þ |Þ |Þ |Þ |Þ |Ý
Defect sizeß101 102 103 104 105 106100
101
102
103
104
105
106
C
à n*
Figure5.11: Distribution of C án for heterogeneousnucleation.Note thatC án will bedifferentfor thepureloopdistributionbasedon spatiallocation.
andhenceC án lookslikeat smallsizeswill have to dependheavily on atomisticcalculations.
Oneissuehereis the“thickness”of theheterogeneousnucleationregion. An initial approximationcouldbetheroughnessof a/c interface.However, theregionswherethis nucleationeffectsthepointdefectconcentrationscangrow to themaximumloopsize.
5.5 Summary
In summary, wewereableto modeltheformationandevolutionof dislocationloopsby extendingourmodelfor Õ 311Ö defectsto a systemwith two distributionsandaccountingfor the transferbetweenÕ 311Ö defectsanddislocationloops. We alsodevelopeda computationallyefficient versionof thismodel,basedonAKPM.
The loop modelextendsour TED predictioncapabilityto amorphizingdoses.We wereabletoobtaingoodTED predictionsfor a largerangeof implantdoses.
53
Chapter 6
Software DevelopmentEfforts
In orderto carryout thesimulationsdescribedin theprecedingchapters,we havedevelopedamulti-purposepartialdifferentialequationsolver, DOPDEES,whichis capableof solvinginitial valueprob-lemsin onespatialdimensionusinga finite differencesmethod. The systemof partial differentialequationsis specifiedusinga “dial-an-operator”paradigm,and the programusesTcl (a re-usableandextendiblecommandlanguagedevelopedat UC Berkeley [38, 54]) asa front end. DOPDEEShascommandsfor structuregenerationandmodification,aswell asfeaturesthatmake simulationoftime-varyingparameters(e.g. temperatureramps)possible.Thesolver is fastandsuitablefor useasaprocesssimulatorandmodeltestbedin 1-D.
ProcessModelingModules(PMM) is aframework for platform-independentscriptizationof com-monlyencounteredmodelsin theprocessmodelingcommunity. PMM freestheuserfrom theburdenof specifyingall equationsandparametersfor commonlyknown models,but still retainsthepowertospecifyarbitraryequationsandmodels.It alsoprovidesa framework for transferringmodelsdirectlyfrom the university to the industry, without the needfor a vendorto implementthosemodelsin acommercialprocesssimulator.
6.1 DOPDEES
Theshrinkingdimensionsof VLSI devicesleadsto morecomplex phenomenabecomingimportantduringfabrication.This requiresa flexible environmentfor developingprocessmodels.Thesystemshouldnot restrict the userto predefinedmodels,but rathershouldenableimplementationof anysystemof continuumequations.DOPDEESis suchaneffort.
DOPDEESis a multi-purposePDEinitial valuesolver in onespatialdimensionthatusesa “dial-an-operator”paradigmfor specificationof theequationsystem[37]. Theprimaryaim of thecodeisrapiddevelopmentof continuummodelsandit maybeusedfor avarietyof systems.It is intendedtobeeasyto useandeasyto extendwith new operators.
DOPDEEScanbeusedin any systemwheretheuserwantsto obtainsolutionsfor coupledpartialdifferentialequations(PDEs). Theprogramwasoriginally written to solve diffusion/reactionprob-lemsandthis is why it comeswith operatorssuitedfor suchsystems.However, if the operatorsorfunctionstheuserneedsarenot included,it is relatively easyto addthem. Anotherpositive featureis that theuseris ableto selecttheintegrationengine,asdifferentsystemsbenefitfrom differentnu-
54
Field
Structure
Region Grid point
1 1Equation
Operator
EvaluableFunctionFigure6.1: A structurein DOPDEESconsistsof regions,whicheachhaveanumberof fields(solutionvariables)andagrid structure.Eachfield is associatedwith anequationof thetype∂ f â ∂t ã ∑opi .
mericalalgorithmswhich associatedtrade-offs in speedandstability. Therearealsocommandsforgrid generationandresultextraction. Theuserinterfaceof DOPDEESis in Tcl, which providesforpowerful input scripting.
DOPDEESsolvesa given setof partial differentialequations.The (onedimensional)spaceisassumedto bedivided into chunkscalledregions, andit is assumedthatdifferentequationsneedtobesolvedin differentregions(Fig 6.1).Thesolutionvariablesarecalledfields, andit is assumedthatfor all fieldsequationsof thetype∂ f â ∂t ãåä>ä>ä exist thatdescribethePDEs.A field specifiedin oneregion doesn’t exist in others,unlessexplicitly specified.Actually, theonly communicationbetweenregionshappensthroughboundarytransfers.
The right handsidesof the partial differential equationsare specifiedas a sum of operators:∂ f â ∂t ã ∑opi This approachis calleddial-an-operator, sincethe usercanchoosethe operatorsontheright handside.Theoperatorscanmakeuseof functions, which canbedefinedin termsof fieldsanddefinedparametersaswell asotherfunctions.
DOPDEESactuallydiscretizestheuser-specifiedpartialdifferentialequationsandthensolvestheresultingsetof ordinarydifferentialequations(ODE)usingastandardODEsolver, or solver“engine”(Fig. 6.2). This considerablyreducesthecodedevelopmenttime, while theuseof well-known, bug-freecodeleadsto stability in theprogram.
DOPDEESsupportstime-varying parameters,which enablesthe userto simulatesystemslike
55
Discretization
System of PDEs
System of ODEs
ODE solverlsode rkc dvode
Field
Structure
Region
1 1Equation
Operator
EvaluableFunction
Figure6.2: DOPDEEShandlesPDEsby discretizingthemusingfinite differencesandfeedingtheminto astandardODEsolver.
56
temperatureramps.The parametersandfields arehandledalike usingthe samefunctions,the onlydifferencebeingthataparameteris asinglevaluedquantityin a region,whereasfieldshavedifferentvaluesateachspatialgrid point. Whenfunctionsaredefined,theresultingquantityis singlevaluedifall argumentsaresinglevalued,otherwiseit is assumedto bemulti-valued.This eliminatestheneedfor writing separatefunctionsfor parametersandfields,but still leadsto efficientexecution.
6.2 ProcessModeling Modules (PMM)
The conceptof ProcessModelingModules(PMM) developedfrom our realizationthat therewasagapto befilled betweenaprocessmodelingsoftwarelikeSUPREMandapartialdifferentialequation(PDE) solver like DOPDEESor Alamode. Processsimulatorsare not flexible enoughto providechoiceovertheequationsto besolvedfor; they arehard-codedandusercontrolis limited to afixedsetof modelsandparametervalues.On theotherhand,whenusingageneralPDEsolver, it is necessaryto specifyevery singleequationandparameterin all input decks,evenif someof theequationsandparametersareconsideredto bewell-known. In additionto theoverheadinvolved,thismayalsoleadto inconsistenciesbetweenusers,or groupsof users.
Thus,oneform of softwareprovidesalmostno flexibility , while theotherrequiresthesameworkto bedoneoverandover, oftenleadingto errors.Weattackthisdilemmaby writing reusablemodulesthat canbe incorporatedinto input decksof PDE solvers. This approachretainsthe flexibility of aPDEsolver, while providing reasonabledefaultsfor modelsandparameters.
Severalstandardmodelshavebeenimplementedin PMM, aswell asmany modelsthathavebeendevelopedat BostonUniversity. Fig 6.3 shows availablemodelsaswell astheir interdependencies.PMM (ProcessModeling Modules)usesa hierarchicalmodeldescription,suchthatmodelscanin-cludeothermodels,andensuresinter-operabilityof modelsby consistency checks.Currently, PMMincludesacollectionof modulesfrequentlyencounteredin VLSI processmodeling:æ Dopantdiffusion,simpleandcoupledwith point defects.æ Pointdefectdiffusion,generationandrecombination.æ Dopant-dopantpairing.æ Two-streamdiffusionin polysilicon.æ Clusteringof point defectsandprecipitationof dopants.
PMM modeldescriptionsareindependentof thePDEsolverused,thusproviding amechanismfortransferof modelsbetweensimulationpackages.In additionto DOPDEES,PMM hasbeeninterfacedwith theAlamodesolver [55], which is partof theSUPREMOO7system.We have alsopublisheda white paperdescribingthe commonscriptingplatform [56]. The TCAD softwarevendorAvant!Corp. shows an effort to integratePMM with their new processsimulator. We believe that the useof thePMM conceptwill speedmodeldevelopmentandfacilitatetransferof thosemodelsbetweengroupsof users.Themostdirect impactwould berapid transferof modelsfrom universityresearchgroupsto industry, without theneedto wait for implementationby commercialsoftwarevendors.
57
poly
five-stream
full-cpl
fermi
carrierconc
params-B
params-P
params-As
params-IVpoint-defect
kpm-AsV
kpm-BIC
kpm-loop
kpm-311
params-AsV
params-BIC
params-311
params-poly
oxide-I params-oxide
Figure6.3: Modelsimplementedin PMM andtheir interdependencies.Modulescaninherit parame-tersfrom lower level modulesanddoconsistency checks.
6.3 Simulation of dopant diffusion
As anexamplewe will considerdiffusionof two dopants,boronandarsenic,in silicon. The diffu-sivity of bothboronandarsenicaredependenton thelocal Fermilevel, andthereforethey canaffecteachothersdiffusionthroughthenetdopantconcentration.At high concentrations,arsenicis knowto becomeimmobileandelectricallyinactive. We will considera simplemodelwheretheeffect ofpoint defectinteractionsis not taken into account(theFermimodel)andthedeactivationof arsenicis assumedto happenabruptlyabove thesolid solubility. More rigorousmodelsexistsbothfor diffu-sionanddeactivationandhavebeenimplementedin DOPDEES/PMMframework. Underconditionswherethepointdefectconcentrationsarecloseto theirequilibriumvalues,thismodelshouldprovidereasonableresults.
Thesetof PDEsin our systemcanbeformulatedasfollows:
∂CB
∂tã ç∇DB è ç∇CB é CB ç∇ ln ê nâ ni ëGì (6.1)
∂CAs
∂tã ç∇DAs è ç∇Cact
As í CactAs ç∇ ln ê nâ ni ëGì (6.2)
CactAs ã min ê CAs î Cssë (6.3)
nni
ã ï CactAs é CB
2ni ð í ï CactAs é CB
2ni ð 2 í 1 (6.4)
58
The diffusivities of boronandarsenicareassumedto dependon nâ ni in a standardmanner. Inaddition to the silicon region, we also have an oxide region on top of the structureand boron isassumedto segregatepreferentiallyto theoxide. Figure6.4 shows thescriptusedin DOPDEESformodeldefinition. Note that theoperatorsneedto specifyto which field (i.e., equation)they belongto. UsingPMM greatlysimplifiestheinputfile sinceit callsthepre-definedFermimodule(Fig. 6.5).Notethattheusercanstill specifyanarbitrarysetof PDEs,for this casePDEsfor boronsegregationto anddiffusion in the oxide, whenusingPMM modules.This ensuresmaximumflexibility whilesimplifying theinput. Figure6.6shows theinitial andfinal dopingprofilesin thestructure.
59
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60
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As B Initial As Initial B
|Ø-0.05
|Ø0.00
Y |Ø0.05
Y |Ø0.10
|Ø0.15
|Ø0.20
|Ø0.25
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|Ú |Ú |Ú |Ú |Ú |Ú |Ú |Ù |Ü |Ü |Ü |Ü |Ü |Ü |Ü
|Þ |Þ |Þ |Þ |Þ |Þ |Ý |Þ |Þ |Þ |Þ |Þ |Þ |Þ |Þ |Ý |Þ |Þ |Þ |Þ |Þ |Þ |Þ |Þ |Ý |Þ |Þ |Þ |Þ |Þ |Þ |Þ |Þ |Ý |Þ |Þ |Þ |Þ |Þ |Þ |Þ |Þ |Ý |Þ |Þ |Þ |Þ |Þ |Þ |Þ |Þ |Ý
1016
1017
1018
1019
1020
1021
Depth (Z µm)
Con
cent
ratio
n (c
m
[
-3)
Figure6.6: Theinitial dopantprofilesandfinal profilesaftera 30min annealat 900Ô C accordingtosimulationusingtheFermimodel.
61
Chapter 7
Summary and Futur e Dir ections
7.1 Modeling and experiments
Our aim in this work wasto understandthephysicalprocessesoccurringunderTransientEnhancedDiffusion(TED) conditions,expressthemin amathematicalmodel,integratethis modelinto adiffu-sionequationsolverandquantitatively matchtheexperimentalobservations.
To this end, we have developeda solid physicalmodel (KPM) for the evolution of extendeddefects( Õ 311Ö defectsanddislocationloops)which areobserved underTED conditions. We havealsodevelopeddifferentversionsof KPM thathave applicabilityunderdifferentcircumstances,andhave anassociatedrangeof computationalefficiency. The full setof modelsdevelopedin this workgivestheusertheability to makeappropriatetrade-offs betweenaccuracy andcomputationaltime.
We have appliedKPM to Õ 311Ö defectsthat areobserved undernon-amorphizingimplant con-ditions andwe wereable to get a goodagreementwith the availableexperimentaldata. We havethenusedthismodelto predictTED behavior basedonmarker layerexperimentsandwe foundgoodagreementfor themagnitudeof TED, aswell asthetimeenergy anddosedependence.To extendthemodelto dislocationloops,we assumedthatdislocationloopsform by unfaultingof Õ 311Ö defectsasobservedexperimentally. We accountedfor this transformationin our modelandwe wereabletoobtaina goodmatchto theexperimentaldatawithout any modificationsin the Õ 311Ö defectmodel.Wealsoindicatedhow thesamemodelcouldbeusedfor heterogeneousnucleation.
Note that we have only dealtwith silicon implantsandmarker layer experimentsin this work.A naturalextensionof our work is to extendit to dopantimplants,which areusedfor makingrealdevices. One of the phenomenathat becomesimportantunder theseconditionsis the decreasedsolid solubility of boron,which hasbeenattributedto formationof boron-interstitialclusters(BICs).ModelingBICs with a comprehensive, yet computationallyefficient modelremainsstill a challengeat this point,althoughseveralresearchershavemadesubstantialprogressin this field in recentyears.
Oneimportantfact is that theTED predictionsstronglydependon point defectproperties,espe-cially theDIC áI product.All theresultsobtainedin thiswork wouldbedifferentif weusedadifferentvalueof DIC áI . Although work hasbeendoneto measurepoint defectpropertiesat highertemper-atures,the extrapolationto lower (TED) temperaturesis errorprone. Obtainingaccuratevaluesforpoint defectparameterseitherthroughexperimentsor throughab-initio calculationsis essential.
62
7.2 Softwaredevelopment
Our work alsoinvolvedin developinga computersoftwarethat is capableof solvingthemodelsthatwe have postulated.To this end,we have developedDOPDEES,a one-dimensionalmulti-purposepartial differentialequationinitial valuesolver. To enablefastertechnologytransfer, we have alsodevelopedProcessModeling Modules(PMM) which consistsa set of scriptsthat encapsulatethemodelsthatwehavedevelopedin a readyto useform.
AlthoughDOPDEES/PMMframework is goodfor modeldevelopment/testing,it haslimited ap-plicability in a productionenvironment,primarily due to its 1-D nature. One commercialTCADsoftwarevendor(Avant! Corp.) hasdevelopeda similar framework (TaurusPMEI) thatcanbeusedin 1, 2 and3 dimensionalproblems.The modelsthatwe implementedundertheDOPDEES/PMMframework will beportedto TaurusPMEI framework for wider usewithin theindustry.
63
Appendix A
Calculation of the Kinetic Precipitation Rate
In thisappendix,wederivethekineticprecipitationratefor variousprecipitateshapes.To thisend,wesolvethesteadystatediffusionfield aroundaprecipitate,calculatingtheflux attheinterfacefollowingthetechniqueusedby Dunham[12]. Westartwith thefollowing definitions:
D: Diffusivity of soluteatoms.
k: Surfacereactionrate(k \ 1 ã 0 if thereactionis diffusionlimited).
An: Surfaceareaof asizen precipitate.
rn: Radiusof thesphere.
Rn: Radiusof thedisc.
b: Reactiondistance(onelatticespacing).
Cintn : Concentrationat thesurfaceof asizen precipitate.
C án: Concentrationthatwouldbein equilibriumwith asizen precipitate.
C: C é C∞.
cs: Concentrationat thesurfaceof asphere.
Our basicequationis flux in to thesurface ã amountof materialconsumed:
AnDdCdr
]]]]r ^ surface
ã Ank ê Cintn é C án ë (A.1)
We will executethefollowing stepsfor eachprecipitateshape:
1. Write down C ê r ë asa functionof cs (sumoverspheresfor usingr ratherthanb).
2. Write down Cintn asa functionof cs (setr ã r int in thepreviousstep).
3. Take thederivativeof C ê r ë with respectto r andevaluateat theinterface.
4. Solve for cs from step1 andsubstituteinto above to find Reff.
5. Write down λn.
64
A.1 Spherical defects
1.
C ê r ë ã csrn í br
(A.2)
2.Cint
n ã cs (A.3)
3.dCdr
]]]]r ^ rn _ b
ã é cs
rn í b(A.4)
4.dCdr
]]]]r ^ rn _ b
ã é Cintn
rn í b` Reff ã rn í b (A.5)
5.
Dλn ã An
Reff
D í k \ 1ã DAn
Reff í Dk \ 1(A.6)
If thesystemis diffusionlimited this reducesto:
λn ã An
Reff (A.7)
If not wehave
Λ ã Dk
` λn ã An
Reff í Λ(A.8)
For thesphere,An ã 4π ê rn í bë 2, hence:
λn ã 4π ê rn í bë 2ê rn í bë í Λaã 4π ê rn í bë (A.9)
A.2 Disc-shapeddefects:
1.C ê r ë ã csln ê 8Rn â r ë (A.10)
2.Cint
n ã csln ê 8Rn â bë (A.11)
3.dCdr
]]]]r ^ b
ã é cs
b(A.12)
4.dCdr
]]]]r ^ b
ã é Cintn
bln ê 8Rn â bë ` Reff ã bln ê 8Rn â bë (A.13)
65
b
{311} defect
[113]
[332]
[110]
l
w r
FigureA.1: Divisionof thecapturecross-sectionof a Õ 311Ö defectinto spheresof equalarea.
5.
λn ã An
Reff í Λaã An
Reff (A.14)
For thedisc(torus)An ã 4π2Rnb, soweget:
λnaã 4π2Rn
ln ê 8Rn â bë (A.15)
A.3b311c defects:
Weassumethat Õ 311Ö defectsareplanarwith arectangularshapeandgrow primarily in theirlength.[50]Thecapturecross-sectionof a Õ 311Ö defectis assumedto becylindrical regionsof radiusb with half-spheresateachend(Fig.A.1), whereb is thesilicon latticespacing.In orderto find thediffusionfieldarounda Õ 311Ö defect,wedividedthis region into spheresof radiusb whichhavea total surfaceareaequalto thatof thesumof thetwo cylindersandfour half-spheres.Wethenaddedupthesteady-statediffusion fields of thesespheresto find the concentrationat a distancer along the mid-line of thewidth of a Õ 311Ö defect:
1.
C ê r ë ã 2N d 2
∑i ^ 1
csber2 ígf i w
N h 2 í 2N d 2
∑i ^ 1
csbe ê l í r ë 2 ígf i wN h 2ã 2 i wd 2
0
csbjr2 í y2
Nw
dy í 2 i wd 2
0
csbj ê l í r ë 2 í y2
Nw
dyã 4 i wd 2
0
csjr2 í y2
dy í 4 i wd 2
0
csj ê l í r ë 2 í y2dyã 4cs kl ln mn w
2 í er2 í ê w2 ë 2r
op í ln mn w2 í e ê l í r ë 2 í ê w2 ë 2
l í r
oprqs66
(A.16)
2.
Cintn ã 4cs t ln ï w
2 í eb2 í ê w2 ë 2 ð é ln ê bë í ln ï w
2 í e ê l í bë 2 í ê w2 ë 2 ð é ln ê l í bë�u (A.17)
3.
dCdr
]]]]r ^ b
ã é 2csw kl 1
be
b2 í ê w2 ë 2 í 1ê l í bë e ê l í bë 2 í ê w2 ë 2 qs(A.18)
4.
Reff ã v ln è w2 í e
b2 í ê w2 ë 2 ì é ln ê bë í ln è w2 í e ê l í bë 2 í ê w2 ë 2 ì é ln ê l í bë�w
w2
kl 1
be
b2 í ê w2 ë 2 í 1ê l í bë e ê l í bë 2 í ê w2 ë 2 qsã t ln ï 1 í e
1 í ê 2bw ë 2 ð é ln ê 2b
w ë í ln ï 1 í e1 í ê 2 x l _ by
w ë 2 ð é ln ê 2 x l _ byw ë ukl 1
be
1 í ê 2bw ë 2 í 1ê l í bë e 1 í ê 2 x l _ by
w ë 2 qs(A.19)
5.
λn ã An
Reff í Λaã An
Reff (A.20)
For the Õ 311Ö defectAn ã 4πbw í 8πb2, soweget:
λnaã 4πb ê w í 2bë
Reff (A.21)
67
Appendix B
BU Parameter Set
In this sectionwewantto list all theparametersusedin this thesis. !�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�! úõùøù�!�! ú������óò ñøñ ù���;Iò� �� ÿ�' ù���;õþ� õò ñ�,V9���� 7 :óú 7 ÷ø÷ � �óÿ�; % �6����;Iò�5 ,���5�A)����G % �(�z{2}|Iòóÿ��óò� �� �(�~�<øþ{2 ò;õþ .�M6;T��(�����6� ñ! >� 2 ô�� -/ & AB24�JA)����G�D����D�A)N$A)N3�4��"3�FE�E��� �( ÷�*(��� �(�. !�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!8� � þP���SLþGñóò ô@��óñ ñ��9øô+���Iò$�69�S�� þ���+øÿ��� ����óòóÿ��óò ô�� þGñóò 7 :óú 7 ÷ø÷ � ����% �6����;Iò32 � � þP� þGñ ôLÿ��óôQ����;��69�S ô@��óñLñ��9øô+���Iò�� ÿ��óÿ�S8+��ST2 7 òõô����� Iñ���9 ����óòóÿGñóò����óÿ�< ò89�� þøÿ�;��õô�@Iôøô({�XD�Dõòóÿ�'��-2�<øþ{2 ò;õþ�D��)�� �(�D�(���� !�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�% ô������� ÿ��óòóÿIô����õô���� ÿ�� ÿ � �(�����6�$� � � üEþøÿ�� ô��øò������ ��!"��� �õôøô����óò;ñ)(������ ÿ�'���9 � �(�����6� %�� � ü ò*(���(���+-,Gü�(�����6��� � �6�3/1!�324"�"�"�"�"�56�(�����6��(��6E�� ü ò*(�� ü�(�����6� %�� �6��O�E�O"32�A6E�A)������G� M ÿIô���� ÿGñ���� ò� õò�� ô���� ÿJ�� ÿ��óòóÿIô����õô���� ÿ�� ÿ � �(�����6�;ÿ ? � üF9õþøÿ��P(�����; ü4����@ òóÿ�� þGñ�"324�øò�A)N��324N������ ��!"���üF9õþøÿ��P(���+ ò��.óöø÷õùIúøû$AB24�����% �I� ÿ��V� þ����� ÿ��óòóÿIô����õô���� ÿJ9����8� ��� M(�����6�$�6�I� ÿýüEþøÿ�� ô$A ò�C�� ��!"�87 ��(��óÿIôJ�)@����'Iò ñ(�����6�$�óÿ òõô ? ��õò9�9 ? úKA(�����6�$�óÿ òõô ? ��õò9�9 ?�% ñ$A(�����6�$�óÿ òõô ? ��õò9�9 ?�> !�A !�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�! ú�����6�>òõôIò� ñ�9������� ñóòóÿ����-2 !�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�!�� �� ��6; ñ�� þ�<��� �� ô�S$��9 % ñ(�����6�$� ñøñ ?�% ñ ü4����@ òóÿ�� þGñ��324���øò������32FE�G�� ��!"�87 �69�9õþGñ�� & � ô�SJ��9 % ñ(�����6� 7 �69�9 ? � ?�% ñ ü4����@ òóÿ�� þGñ�A�ET24����� ��� ñ!�A6�(�����6� 7 �69�9 ? � ?�% ñ ü4����@ òóÿ�� þGñJA)�8ET24����� ��� ñ!�A6�(�����6� 7 �69�9 ? ( ?�% ñ��(�����6� 7 �69�9 ? ; ?�% ñ��8� ����� ô���� ÿ��� $� ÿIôIò� ñ ô�� ô��)�� ���S������(�� ÿ òóÿIô(�����6� � � ? � ?�% ñ��324"(�����6� � � ? � ?�% ñ��324�(�����6� � � ? ( ?�% ñ�!�A)����� M6���Iò� õò & �óÿIô(�����6� � � ? ; ?�% ñ�!�A)����� M6���Iò� õò & �óÿIô68
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CURRICULUM V ITAE
EDUCATIONæ Ph.D. in Electrical Engineering 1999BostonUniversity, Departmentof ElectricalandComputerEngineering,Boston,MA.æ Master of Sciencein Electrical Engineering 1995BostonUniversity, Departmentof Electrical,ComputerandSystemsEngineering,Boston,MA.GPA: 4.00/4.00.æ Bachelorof Sciencein Electrical Engineering 1993Bogazici University, Departmentof ElectricalandElectronicsEngineering,Istanbul, Turkey.GPA: 3.92/4.00(High Honors).æ Bachelorof Sciencein Physics 1993Bogazici University, Departmentof Physics,Istanbul, Turkey. GPA: 3.92/4.00(High Honors).æ Abitur 1989GermanHigh School,Istanbul, Turkey.
EXPERIENCEæ Avant! Corporation, Lincoln, MA 1998–presentApplicationsEngineerat theTCAD BusinessUnit.æ BostonUniversity, Boston,MA 1994–1998ResearchAssistantfor Assoc.Prof. ScottDunhamin the Departmentof ElectricalandCom-puterEngineering.æ BostonUniversity, Boston,MA 1993–1994Teachingassistantfor Prof. T.D. Moustakasin the Departmentof Electrical and ComputerEngineering.æ KondaResearch Inc., Istanbul, Turkey 1991–1993IT consultantandapplicationdeveloper(part-time).æ SiemensAG, Munich,Germany 1991SummerInternat theDepartmentof CorporateResearchandDevelopment.
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