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Crack patterns in thin films and X-ray optics thermal deformations

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Crack patterns in thin films and X-ray optics thermal deformations
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Kravchenko, Grygoriy A
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Residual stress
Fracture
Cracks interaction
Mo/Si multilayers
Thermal distortions
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ABSTRACT: Thin films and multilayers are widely used in many applications, ranging from X-ray optics to microelectronic devices. In service, the X-ray optics elements are exposed to the X-ray beam, which heats up the structure resulting in the thermal deformations, and consequently in distortions of the reflective surface. In addition, the excessive heating may activate interdiffusion in the multilayers coatings and result in degradation of their reflective performance and even film cracking. Therefore, analysis of the thermally-induced deformations and stresses in the X-ray optical elements is important. The presented work is organized in two major parts. The first part examines formation of the peculiar periodic crack patterns observed in the thermally loaded Mo/Si multilayers. Film stress evolution during thermal cycling of the multilayers on Si substrate is analyzed.Results of the high-speed microscopic observations of crack propagation in the annealed Mo/Si multilayers are presented. The observations provide experimental evidence of the mechanism underlying formation of the periodic crack patterns. In the second part, thermal deformations and the resulting surface curvature changes in the X-ray optics elements are analyzed. Finite element modeling is used to assess the potential to thermally control curvature in the X-ray mirrors consisting of the Mo/Si multilayers on a Si substrate. Influence of heating due to the X-ray beam irradiation on thermal deformations in the X-ray mirror bonded to a thick substrate is analyzed in-depth. The detailed consideration includes analysis of the thermal and structural mechanics simulations. Based on simulations of different model configurations, influence of structural composition on thermal distortions of the optics elements is addressed.Results of this analysis can be used to mitigate distortions of the X-ray optics caused by the X-ray beam and provide basis for further studies of thermally controlling surface curvature in the optical elements.
Thesis:
Thesis (M.S.M.E.)--University of South Florida, 2008.
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by Grygoriy A. Kravchenko.
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CrackPatternsinThinFilmsandX-rayOpticsThermalDeformations by GrygoriyA.Kravchenko Athesissubmittedinpartialfulllment oftherequirementsforthedegreeof MasterofMechanicalEngineering DepartmentofMechanicalEngineering CollegeofEngineering UniversityofSouthFlorida MajorProfessor:AlexA.Volinsky,Ph.D. AutarK.Kaw,Ph.D. NathanCrane,Ph.D. DateofApproval: November7,2008 Keywords:residualstress,fracture,cracksinteraction,Mo/Simultilayers,thermaldistortions, curvaturecontrol,X-rayoptics c Copyright2008,GrygoriyA.Kravchenko

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Acknowledgments Firstandforemost,IwouldliketothankmyadvisorAlexVolinskyforinvitingmetotheUniversity ofSouthFloridaandprovidingmewithgreatsupport,encouragementandassistancetoaccomplish thiswork.IamthankfultoAlexforgivingmefreedomtochooseandconductresearchfroma broadspectrumofinterestingproblemsheproposed. ItismypleasuretoacknowledgehospitalityofProf.DirkMeyerandallmembersofhisgroup fromtheInstituteofStructuralPhysicsattheTechnicalUniversityinDresden,whoprovidedexcellentworkingandlivingconditionsduringstayofourgroupthere.IamalsoindebtedtoDirk MeyerforsharingsomeinterestingideasontheX-rayopticselements,whichwereelaboratedin thecourseofthiswork. IappreciatehelpofDr.StefanBraunfromtheFraunhoferInstituteforMaterialandBeam TechnologyinDresdenforprovidingtheMo/Sisamplesandthestressdata.IamgratefultoRobert ShieldsandJamesRachwalfortheirsupportintheexperimentalwork. ThisworkwassupportedbytheNationalScienceFoundationthroughGrantCMMI0600266.

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TableofContents ListofTables iii ListofFigures iv Abstract vii ChapterOneIntroduction1 1.1Fractureandcrackpatternsinthinlms1 1.2ThermaldeformationsinX-rayoptics8 ChapterTwoCrackPatternsintheAnnealedMo/SiMultilayers11 2.1Residuallmstress11 2.1.1Curvaturemethod11 2.1.2ResidualstressesintheMo/Simultilayers13 2.2Microscopyanalysisofthecrackpatterns16 2.2.1High-speedcamerasetupandsamples16 2.2.2Aposteriorianalysisofcracks18 2.2.3High-speedphotographyresults24 2.3ConclusionsforChapterTwo31 ChapterThreeThermalDeformationofX-rayOptics32 3.1Wafercurvatureduetothethrough-thicknesstemperaturegradient32 3.1.1Modeldescription33 3.1.2Finiteelementmodel34 3.1.3Results36 3.2ThermaldeformationsofX-raymirrorduetobeamirradiation39 3.2.1Modeldescription39 3.2.2Geometryandmaterialproperties39 3.2.3Computationoftheheattransfercoefcient42 3.2.4Boundaryconditions43 3.2.5Finiteelementmodel44 3.2.6Thermomechanicalsimulationresults45 3.3ConclusionsforChapterThree53 ChapterFourSummaryandFutureOutlook54 References 56 i

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Appendices 61 AppendixA:Surfacecurvature62 AppendixB:Finiteelementmodelsconvergencestudy68 ii

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ListofTables Table1.MaterialpropertiesofX-raymirrorcomponentsat27 C34 Table2.ModelcongurationsoftheX-rayopticselement40 Table3.MechanicalpropertiesofmaterialsoftheX-rayopticalelementat27 C41 Table4.ThermalpropertiesofmaterialsoftheX-rayopticalelementat27 C42 Table5.Constantsandexpressionsusedforcalculationoftheaverageheattransfercoefcient43 Table6.Unatnessfordifferentmodelcongurations52 iii

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ListofFigures Figure1.Delaminationofalmfromasubstrate2 Figure2.Steady-statechannelcrackingduetoatensilelmstress3 Figure3.Arrayofchannelingcracksduetoatensilelmstress4 Figure4.Illustrationofchannelcrackingaccompaniedbydelamination5 Figure5.Interfacedebondingandcrackkinking5 Figure6.ApplicationofG obelmirrorinX-raydiffraction8 Figure7.Mo/Simultilayerstackwiththicknes/curvaturevariationdepositedonSisubstrate9 Figure8.BurntraceseenonthebulkSisinglecrystalmonochromator,DeutcscheSynchrotron9 Figure9.G obelMirrorbondedtoathicksubstrate10 Figure10.Substratecurvaturecausedbystressinthelm11 Figure11.StressevolutionintheMo/SimultilayersonSisubstrateduringthermalcycling14 Figure12.High-speedcameraexperimentalsetup17 Figure13.Treeleaf-likebranchingchannelingcracks19 Figure14.Burstchannelingcrackstransformingintothesinusoidalandcrescentpatterns19 Figure15.Channelingcrackstransformingintoasinusoidalfollowedbyacrescentform20 Figure16.Square-sinusoidalortheChineseWalltypechannelingcracksaccompaniedby delamination21 Figure17.Sinusoidalchannelingcracksaccompaniedbydelamination21 Figure18.Sawtooth-typechannelingcrackfollowingastraightpathaccompaniedbydelamination22 Figure19.Sinusoidalchannelingcracksfollowingacurvedpathaccompaniedbydelamination23 iv

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Figure20.SpiralchannelcrackingintheannealedMo/Silmaccompaniedbydelamination23 Figure21.BurstchannelingcrackstransformingintoasinusoidalintheannealedMo/Si lm24 Figure22.Propagationofastraightchannelingcrackaccompaniedbydelaminationatselectedinstantsoftime25 Figure23.Propagationofwavychannelingcracksatdifferentinstantsoftime.Oneofthe channelsturnsintoaChinesewall-typepattern.26 Figure24.Selectedframesshowinggrowthoftheburst-likechannelingcracksfollowedby formationoftheperiodiccrackpatterns27 Figure25.Selectedframesshowingformationofthesinusoidalandcrescent-likecracks29 Figure26.Selectedframesshowingformationofcrescent-typecracks30 Figure27.GeometryofX-raymirrorindicatingstructuralcompositionwithfrontreectiveandbacksides33 Figure28.FiniteelementmodelofX-raymirrorillustratingtheappliedtemperatureloading35 Figure29.TotalradialstrainsintheX-raymirrorduetothethrough-thicknesstemperature gradient36 Figure30.Out-of-planedisplacementsoftheX-raymirrorduetothethrough-thickness temperaturegradient37 Figure31.Slopeandcurvatureofthewaferreectivesurfaceasafunctionoftheradial position38 Figure32.GeometryofanX-rayopticalelementsubjectedtotheheatuxfromanX-ray source40 Figure33.Averageheattransfercoefcientasafunctionoftemperatureforthreedifferent platesurfaceorientations43 Figure34.Meshofthequarter-symmetricniteelementmodeloftheX-rayopticalelement45 Figure35.ThermalsimulationresultsforModel1Siwaferonglass46 Figure36.CalculatedaverageheattransfercoefcientModel146 Figure37.ThermalsimulationresultsforModel2SiwaferonSi47 Figure38.ThermalsimulationresultsforSimonochromatorModel547 Figure39.Deformedshaperesultsfordifferentmodelcongurations48 v

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Figure40.Gaussiancurvatureofsurfacefordifferentcongurations49 Figure41.ComparisonofGaussiancurvaturealongtheX-raybeamlinefordifferentmodel congurations50 Figure42.Unatnessofsurfacefordifferentmodelcongurations51 Figure43.ComparisonofsurfaceunatnessalongtheX-raybeamlinefordifferentmodel congurations52 Figure44.Contourplotof z = x 2 + y 2 on 100 100 gridpoints66 Figure45.Curvatureoffunction z = x 2 + y 2 on 100 100 gridpoints67 Figure46.Accuracyofnumericallycomputedcurvatureson 100 100 gridpointsfor y =0 67 Figure47.FiniteelementmodelusedformeshconvergencestudyoftheX-raymirrorsubjectedtothethrough-thicknesstemperaturegradient68 Figure48.TotalradialstrainsintheX-raymirrorduetothethrough-thicknesstemperature gradientcalculatedusingthenemesh68 Figure49.Out-of-planedisplacementsoftheX-raymirrorduetothethrough-thickness temperaturegradientcalculatedusingthenemesh69 Figure50.FiniteelementmodelmeshesusedforconvergencestudyModel469 Figure51.CalculatedaverageheattransfercoefcientModel470 Figure52.CalculatedtemperaturedistributionModel470 Figure53.Calculatedthrough-thicknesstemperaturegradientModel471 Figure54.Calculatedout-of-planedisplacements u z Model471 vi

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CrackPatternsinThinFilmsandX-rayOpticsThermalDeformations GrygoriyA.Kravchenko ABSTRACT Thinlmsandmultilayersarewidelyusedinmanyapplications,rangingfromX-rayopticstomicroelectronicdevices.Inservice,theX-rayopticselementsareexposedtotheX-raybeam,which heatsupthestructureresultinginthethermaldeformations,andconsequentlyindistortionsofthe reectivesurface.Inaddition,theexcessiveheatingmayactivateinterdiffusioninthemultilayers coatingsandresultindegradationoftheirreectiveperformanceandevenlmcracking.Therefore,analysisofthethermally-induceddeformationsandstressesintheX-rayopticalelementsis important. Thepresentedworkisorganizedintwomajorparts.Therstpartexaminesformationofthe peculiarperiodiccrackpatternsobservedinthethermallyloadedMo/Simultilayers.Filmstress evolutionduringthermalcyclingofthemultilayersonSisubstrateisanalyzed.Resultsofthe high-speedmicroscopicobservationsofcrackpropagationintheannealedMo/Simultilayersare presented.Theobservationsprovideexperimentalevidenceofthemechanismunderlyingformation oftheperiodiccrackpatterns. Inthesecondpart,thermaldeformationsandtheresultingsurfacecurvaturechangesinthe X-rayopticselementsareanalyzed.FiniteelementmodelingisusedtoassessthepotentialtothermallycontrolcurvatureintheX-raymirrorsconsistingoftheMo/SimultilayersonaSisubstrate. InuenceofheatingduetotheX-raybeamirradiationonthermaldeformationsintheX-raymirror bondedtoathicksubstrateisanalyzedin-depth.Thedetailedconsiderationincludesanalysisofthe thermalandstructuralmechanicssimulations.Basedonsimulationsofdifferentmodelcongurations,inuenceofstructuralcompositiononthermaldistortionsoftheopticselementsisaddressed. ResultsofthisanalysiscanbeusedtomitigatedistortionsoftheX-rayopticscausedbytheX-ray vii

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beamandprovidebasisforfurtherstudiesofthermallycontrollingsurfacecurvatureintheoptical elements. viii

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ChapterOne Introduction 1.1Fractureandcrackpatternsinthinlms Thinlmsarewidelyusedinvariousapplications,suchasmicroelectronicdevicesandmicroelectro-mechanicalsystemsMEMS;theyserveasprotectiveandthermalbarriercoatings.Thin lmmultilayersareusedasreectivecoatingsinopticalapplications.Oftenlmdepositionprocessesand/orserviceconditionsleadtohighlevelsofstress,whichmaycauselmfractureand resultinfailureofthedevice.Theindustryhasbeenfacingmanyreliabilityissuesconnectedto fractureofthinlmsduringseveralpastdecades.Difcultiesintheanalysisinvolvecomplicated natureofmechanismsunderlyingevolutionofthelmstressesandfractureprocesses,complex geometryandpoorlyknownmaterialsproperties.Althoughtherehasbeenagreatprogressinresearchofthinlmsfracture,reectedbythousandsofscienticpublications,predictionofthin lmsreliabilitystillremainsachallengingtask. Typically,thinlmsaredepositedonsubstratesatelevatedtemperaturesbychemicalvapor deposition,physicalvapordeposition,electro-platingandothermethods.Whencooled,thermal expansionmismatchbetweenthelmandthesubstrateresultsinthemismatchstrains,andconsequentlyinthelmstress.Inaddition,thedepositionprocessesmayresultinhighintrinsicstressesin thelm.Ontopofthat,thermalloadinginservice,suchasthermalcycling,mayresultinsignicant redistributionofthestressescausedbyplasticowinmetalliclms,changeofthemicrostructure, interdiffusionprocesses[DN88]. Failureinthinlmsmayoccurindifferentmodesdependingontheloadingconditions,material propertiesandgeometry.Undertensilestressesthinlmstypicallyfailthroughcrackinganddelamination.Compressivestressesmayresultinthinlmbucklingdelaminationfollowedbycracking 1

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ofthelm.Inaddition,lmfailuremayresultinfractureofthesubstratethroughcrackpenetration acrossthelm/substrateinterfaceorduetothecrackkinking[HS92b],[Suo03]. Simultaneousoccurrenceofthedifferentfailuremodesmayresultintheirinteractionandsignicantlycomplicateanalysisoftheproblem.Furthercomplicationsarisewhenastructureissubjectedtothermalshockingordynamicimpulseloadingconditions.Crackpropagationunderthese circumstancesisnotwellunderstood. Thedelaminationofalmfromasubstrate[EDH88]duetoauniformtensilestressisillustrated inFigure1.AccordingtothestrainenergyreleaserateSERRcriteria,thefractureoccurswhenthe SERRoftheinterfacecrack G i duetotheloadingequalsthecriticalvalue )]TJ/F22 7.9701 Tf 6.818 -1.637 Td [(i foragivenmaterials combination: G i =)]TJ/F22 7.9701 Tf 18.333 -1.636 Td [(i Thiscriticalvalueisalsocalledtheinterfacefracturetoughnessanditadditionallydependsonthe ratiobetweenthenormalandshearstressesaheadofthepropagatingcrack[MS65],whichisdened throughthemodemixity [RSW90]. Figure1.Delaminationofalmfromasubstrate Atypicalfailuremodeforabrittlethinlmundertensionisthefracturebylmcracking,where thecrackpathformsachannel[EDH88].IllustrationofthiscaseisprovidedinFigure2.Thecrack frontadvancesinthelmperpendiculartotheinterfacewithoutpenetrationsothatthesubstrate remainsintact.UtilizingtheSERRcriterion,thelmfractureoccurswhenthecrackdrivingforce 2

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equalsthefracturetoughnessofthelm: G f =)]TJ/F22 7.9701 Tf 18.333 -1.777 Td [(f Thedepictedproblemisthree-dimensional,howeverforthestraightcracksexceedingafew timesthelmthickness,thesteady-statesolutionoftheproblemcanbeobtainedintheplane-strain formulation[Gec79]. Figure2.Steady-statechannelcrackingduetoatensilelmstress Analysisoftheproblemshowsthattherelativelythicksubstrateconstrainsthecrackpropagation,asopposedtoacaseofafree-standinglmundertensileloading.Thecrackdrivingforce increasesforincreasinglycompliantsubstrates,sinceacompliantsubstrateprovideslessconstraint onthecrack[Beu92].Theplasticyieldinginthesubstratealsofacilitatesthechannelingcrack growthbyprovidinglessconstraintonthecrack[BK96].Afractionofresultsdoneintheeldmay alsobefoundin[HE89],[SH90],[YSE92],[HS92a],[HHL95]. Growthofmultiplechannelingcracksacrossthelmprovideaneffectivewayofreleasing energyinatensilestrainedlmonasubstrateFigure3.Incaseofthestraightchannelingcracks, theformedarrayofcrackswasfoundapproximatelyperiodic.Thecracksspacingdependsonthe elasticmismatchbetweenthelmandthesubstrate,magnitudeofthetensilestressinthelmand itsfracturetoughness[Tho90],[F.91],[TOG92],[SSF00]. 3

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Figure3.Arrayofchannelingcracksduetoatensilelmstress Inpracticalapplications,suchasMEMSandmicroelectronicdevicesonthinwafers,thesubstrateisofacomparablethicknesswiththelmsuchthatstressinthelmisallowedtorelax duetothesubstratedeformation.Analysisoftheproblemshowedthatathinnersubstratereduces thecrackdrivingforce[Vla03].Finiteelementmodelingwasutilizedtostudythechanneling cracksinthinlmsandmultilayersincludingeffectsofplasticity[AB02],creepoftheunderlayermaterial[LHP03],thermalcyclingloading[BE01],[BA03],inuenceofthesubstratethickness[HPHS03],periodicinterconnectstructures[AJB02],andbufferlayers[TMV05]. Stressconcentrationattherootofachannelingcrackmaycausedecohesionalongthelmsubstrateinterface[HE89].Dependingonfracturetoughnessofthesubstrateandtheinterfaceas wellasontheelasticmismatchbetweenthebondedmaterials,almcrackmaypenetrateinto thesubstrate[YSE92].FilmdecohesionaroundtherootofachannelingcrackFigure4relaxes constraintofthesubstrateincreasingthecrackdrivingforce.Caseofasymmetriclmdelamination accompanyingachannelinglmcrackwasstudiedin[MPH07].Theauthorsanalyzeddelamination conditionsasafunctionofelasticmismatch,lmstressandtheinterfacefracturetoughnessusinga plane-strainFEMmodel. However,asseenfromtheillustrationFigure4,thedelaminationaroundthecrackrootmay notnecessarilybesymmetric.Intheasymmetriccongurationthelmstressdistributionaheadof apropagatingcrackwillstronglydependonthedelaminationfrontgeometryandontherelative 4

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Figure4.Illustrationofchannelcrackingaccompaniedbydelamination positionofthecrack.Consequently,thecrackdeviatesfromthedirectpathattractedbyhigher intensitiesinthestresseld. DeviationofacrackfromitsoriginalpathisillustratedinFigure5.Combinationoftheexternal normalandshearcomponentsactingonthecrackarerepresentedbythemode-Iandmode-IIstress intensityfactors K 1 and K 2 ,respectively.Thenonzeroshearingcomponentforcesthecrackto kink.Note,thatthekinkisassumedtobeinnitesimallysmallmuchsmallerthanthecrack. Stress-andenergy-basedfracturemechanicscriteriaforpredictionofthecrackkinkingdirection maybefoundinthefollowingpapers:[ES63],[Sih73],[TP81],[Wu78],[TP82],[CR80],[HH89], [HBE91],[YX92],[Kan94]. Figure5.Interfacedebondingandcrackkinking Predictionofthecrackpathundergeneralloadingconditionsatthecracktipisamuchmore involvedtaskthanthoseformulatedunderthesteady-stateconditions.Sincethecrackpathisnot knownapriori,iterativeapproachmustbeundertakenforsolutionoftheproblem.Numerical 5

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methodsweresuccessfullyappliedformanyproblems,suchaspredictionofasymmetricgrowthof delaminationbetweentwobondedwafers[Tur04]andmodelingofevolvingcrackpatternsinthin lms[LHP03],[SC03],tonameafew.Fracturemechanics-basedanalysisofasingleandmultiple channelcracksinthinelasticlms[XH00]wasusedtopredictthecrackpathtendency.Itwasalso shownthatamode-Icrackgrowth K II =0 alongaspiralmayexistinabiaxiallystressedlmin casetherearecurvedawstostartthecracking.However,tothebestknowledgeoftheauthor,no publishedattemptshavebeenmadesofartomodelinteractionofapropagatinglmcrackwiththe advancingdelaminationfront. Whilepatternscontainingstraightcracksareverycommontolmsinresidualtension,combinationofdifferenteffectssuchastemperaturegradients,externallyappliedstress,inuenceof boundariesandasymmetrymayproduce,underfavorableconditions,somepeculiarcrackpatterns. Firstnotionofhelix-likecrackpatternsinresidualtensioninPyrexglassplateswaspublished in[Arg59].Oscillatingandbrunchedcracksalongalargetemperaturegradientinaglassplate rapidlyimmersedintowaterwerereportedin[YS97].Basedontheexperimentalobservations, theauthorsconcludedthattheoscillationarisesfromthetwocompetingmechanisms:deviationof thecrackfromthestraightpathtoreleasemorestrainenergy,andrestorationforcewhichdrives thecrackawayfromthelateraledgestowardstheplatecenterlinewiththelargesttensilestresses. Fromtheexperimentsondryingprecipitatesonthesubstrates[NLJR02]itwasfoundthattheadvancingdelaminationfrontforcestherunningtunnellmcracktoturninwards.Theauthorswere abletomodelthespiralcrackingwithaspring-blockmodel[LN00].Experimentalobservationsof thesimultaneouslyrunningoutwardsspiral,sinusoidal,saw-toothandcrescent-likecrackpatterns indryingsol-gelsilicatethinlmsonglassandsteelsubstrateswerereportedin[SW03].Basedon theexperimentaldata,itwassuggestedthattheobservedcurvingcrackpathswereattributableto lmdelamination.Observationsofspiralcrackpathsandotherinterestingpatternscanbefoundin otherpapers[DHJSC94],[CC95],[Gar90]. Recently,interestingsinusoidalandspiralcrackpatternswereobservedintheMo/SimultilayersdepositedontheSisubstrate.Thesamplesweresubjectedtothe3-pointbending,annealedathightemperatureofabout500 Candslowlycooledinavacuumchamber,followed 6

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bytheirmicroscopicobservations[MLL + 04],[VMM].Similartoobservationsofotherauthors e.g.,see[NLJR02],[SW03],thethrough-thicknesscrackswereaccompaniedbydebondingof theadjacentareas.Basedontheexperimentalndingstheauthorssuggestedthatacombination ofbiaxiallmstress,temperatureandtheexternallyappliedstresspossiblywithasymmetriclm debondingcausestheseperiodiccrackpatterns. TheworkpresentedinChapterTwoisthecontinuationofthestudyofthecrackingbehaviourin theannealedMo/Silmaimingatndingtheexactrootcauseoftheobservedcrackpatterns.This wasaccomplishedbyanalyzingthestressstateinthelmasafunctionoftemperaturesection2.1 andfurtherresultsofmicroscopicobservationssection2.2.Inparticular,crackspropagationand crackpatternsevolutioncapturedusingthehigh-speedphotographyarepresentedanddiscussed. Itisbelievedthatthepresentexperimentalworkwouldcontributetoabetterunderstandingofthe crackpatternsformationinthinlmsandinspiredevelopmentofthetheoreticalandnumerical modelscapableofaccuratelypredictingtheobservedcrackingphenomena. 7

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1.2ThermaldeformationsinX-rayoptics MultilayerX-rayopticsareusedintheextremeultra-violetlithographyEUVL,X-raydiffractometersXRD,X-rayreectometersXRR,synchrotronsourcesandotherX-raydevices[DBH + 00]. Nowadays,themostwidespreadopticsforXRD/XRRsystemsaretheso-calledG obelmirrors [HOH + 05].TheG obelmirrorsconvertX-raybeamcomingfromanX-raysourceintoanintense parallelorfocusedmonochromaticbeam.Figure6showsaprincipleschemeoftheGoebelmirror congurationinaX-raydiffractometer. Figure6.ApplicationofG obelmirrorinX-raydiffraction Themirrorconsistsofastackoftypically50-200alternatingnanometer-thicklayersmadefrom twodifferentmaterialswithacontinuouslyvaryingcurvatureandthicknessFigure7.Toachieve highbeamreectingperformanceitisnecessarytodepositultra-precisemultilayerstackontoan ultra-preciseprepolishedsubstrateofparabolicform[Bra02].Commonmaterialcombinationsused inreectiveX-rayopticsareMo/Si,W/B 4 C,W/Si,Ni/Candothers.IncaseofmultilayersconsistingofalternatingMo/Sinanolayers,heatingabove110 Ccausesinterfacialchemicalreactionthat signicantlydegradesthemirrorperformance[Boe01],[BMP + 03]. InthesynchrotrondiffractionopticstypicallytheSibulksinglecrystalisusedtoselectaparticularX-raywavelengthwhentheincomingbeamdiffractsfromthesinglecrystal.Inthistype ofcongurationlargeamountofheatisgeneratedinthecrystal,whichhastobeproperlycooled, typicallywithwaterorliquidnitrogen[BJD + 05],otherwiseitmaymeltinthehighpowersynchrotronbeam.SuchacaseisshowninFigure8,wheretheburntraceisseenonthesurfaceofaSi 8

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Figure7.Mo/Simultilayerstackwiththicknes/curvaturevariationdepositedonSisubstrate monochromatorfromexcessiveheatingbythesynchrotronbeam[Sta].Besidesthat,reductionof thebeamqualityiscausedbythermaldistortionsofthecrystallatticethatleadstoabroadeningof therockingcurveandreducesthepeakintensity[BFKM00],[Beg97],[Kho91]. Figure8.BurntraceseenonthebulkSisinglecrystalmonochromator,DeutcscheSynchrotron Although,thereectivityopticsofthein-houseX-raysystemsareusuallynotexposedtohigh incomingbeamenergies,evenlowtemperaturechangesmayleadtodeviationsfromtheirinitial ultra-precisegeometry.Thisresultsinlossofthebeamconditioningquality,asmentionedabove. Geometricaldistortionsduetotemperaturechangesareassociatedwiththethermalexpansionmismatchofdifferentmaterials.SinceSisubstrateandthemultilayerstackshowninFigure7have differentcoefcientsofthermalexpansion,uponheating/coolingthestructureinevitablydeforms. Iftemperaturechangeissmallontheorderof10 C,thedeformationissmalltooanddoes notcausesubstantialreductionofbeamconditioning,sincethicknessoftheSisubstrateistypically 2000timeslargerthanthicknessofthemultilayerstack.However,thewaferwithdepositedlayers is,inturn,bondedtoathickersubstrateFigure9toincreaseitsstructuralstabilityandimprove 9

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Figure9.G obelMirrorbondedtoathicksubstrate handlingwhenthemirrorismountedinadevice.Whenheatedorcooled,deformationofthis structuremaybecomeconsiderablesincethicknessoftheSiwaferandthesubstrateiscomparable. GeometricaldistortionsoftheX-rayopticsmayalsoariseduetothetemperaturegradients existinginthestructure.Fortheplate-likestructuresthicknessismuchlessthantheotherdimensions,inuenceofthethrough-thicknesstemperaturegradientonthesurfacecurvaturewouldbe muchhigherthaninuenceofthein-planetemperaturegradients.Thisbehaviourmaybeutilizedto controlsurfacecurvatureoftheopticalelementsbyapplyingnecessaryheating/coolingconditions. Anothersourceofgeometricaldistortionsthatshouldbenotedherearisesfromhighresidual stressesduetothenatureoflmdepositionprocesses[DN88].Thisissueisaddressedinsection2.1.2inagreaterdetail. Inviewoftheabove,discussioninChapterThreewillbeconcentratedontheanalysisofdeformedstateoftheX-raymirrorcausedbythermalloading.Potentialofthermalcurvaturecontrol inX-rayopticsisaddressedinsection3.1.Thisisaccomplishedbyinvestigatinginuenceofthe through-thicknesstemperaturegradientoncurvatureofthemirrorsurfaceusingtheniteelement modeling.ProblemofthermaldistortionsinX-rayopticsexposedtotheX-raybeamirradiationis discussedinsection3.2.NumericalanalysiscontainsresultsofthermalandmechanicalniteelementsimulationsofdifferentX-raymirrorcongurationsonathicksubstrate.Impactofboundary conditions,materialpropertiesandgeometryisassessed.Waysofreducingmirrordistortionsare proposedbasedonthesestudies. 10

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ChapterTwo CrackPatternsintheAnnealedMo/SiMultilayers 2.1Residuallmstress 2.1.1Curvaturemethod Curvaturemeasurementisthetraditionalexperimentaltechniquefordeterminingstressinthinlms depositedonsubstrates.ThemethodisbasedontheobservationmadebyStoney[Sto09]thatstress inthelmstrainsthesubstratesoasitbendsFigure10. Figure10.Substratecurvaturecausedbystressinthelm ThebiaxialstressinthelmcanbecalculatedfromthesubstratecurvatureusingtheStoney equation: = k M s h 2 s 6 h f ; 11

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where k isthesubstratecurvature, h s and h f arethesubstrateandthelmthicknesses.Thebiaxial modulusofthesubstrate M s isgivenby M s = E s 1 )]TJ/F21 10.9091 Tf 10.909 0 Td [( s ; where E s and s aretheelasticmodulusandthePoisson'sratioofthesubstratematerial,respectively. Note,thattheStoneyformulainequationdoesnotcontainmaterialpropertiesofthelm, onlythoseofthesubstrate.Effectsoflmthicknessonsubstratecurvatureinbimaterialswasrst analyzedindetailin[Tim25].Theanalysisshowsthatthethinlmapproximationgiveserrorof 15%forthelm/substratethicknessratioof h f =h s = 0.05incasetheelasticmismatchbetweenthe materialsisneglected[FS04]. Thenatureoflmstressesfallsintotwomajorcategories:growthorintrinsicstressesand inducedorextrinsicstresses.Thegrowthstressesdependontheconditionsoflmdepositionprocessesandareconnectedtovariouscomplexphysicalphenomenaoccurringinthelmmaterialas wellasatthematerialsinterfaces[DN88].Theextrinsiclmstressesinsemiconductorapplications aretypicallycausedbythetemperaturechangebetweenthelmdepositionprocessesandthein serviceconditions,sincethelmandthesubstratematerialshavedifferentcoefcientsofthermal expansion.Thentheelasticthermalmismatchstraininalmwithrespecttothesubstrateis th = s )]TJ/F21 10.9091 Tf 10.909 0 Td [( f T; where s and f arethesubstrateandlmcoefcientsoflinearthermalexpansion, T isthe temperaturechange.Thecorrespondingmismatchstressis th = th M f ; 12

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where M f isthebiaxialmodulusofthelmmaterial.Rearrangingequationsand,thecoefcientofthermalexpansionofthelmmaterialcanbeestimatedby f = s )]TJ/F21 10.9091 Tf 33.375 7.38 Td [( T M f T )]TJ/F21 10.9091 Tf 10.909 0 Td [(T ref ; where T isthelmstressatthetemperature T withrespecttothereferencestress-freetemperature T ref SimilarlytotheStoneyformulaforstress,theelasticmismatchstraininalm m withrespect tothesubstratecanalsobedeterminedfromthecurvaturemeasurements: m = k M s h 2 s 6 M f h f : Note,thatthephysicaloriginofthemismatchstrainisimmaterial. 2.1.2ResidualstressesintheMo/Simultilayers Thissectionpresentsresultsofthecurvaturestressmeasurementsduringthermalcyclingofthe Mo/SimultilayersonSisubstrate.TheMo/SimultilayersweredepositedonaSisubstrate usingmagnetronDCsputterdepositionattheFraunhoferInstituteforMaterialandBeamTechnologyinDresden,Germany[Boe01].ThesputteredMo/Silmconsistedof60alternatinglayers withthecorrespondingthicknessesof2.7and4.2nmeach,producingatotalthicknessof353.4nm. TheSisubstratewas525 mthickwiththediameterof100mm.Itshouldbenoted,thatthepresentedstressdatawascalculatedbasedonthethinlmapproximationandthesmalldeformation assumption. StressevolutionintheMo/Simultilayersstackasafunctionoftemperatureispresentedin Figure11.Threeheatingcycleswiththepeaktemperatureof500 Cwereappliedtothesystem,as showninthegure.IntheinitialstateatroomtemperaturethetotalstressintheMo/Simultilayers furtherreferredasthelmstressiscompressiveandequalsabout-360MPa.Therstheating cyclecontainsdifferentstagesofthelmstressevolution.Withoutconsideringthemechanisms underlyingkineticsofthelmstressevolution,onecouldallocatethefollowingfourstages: 13

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Figure11.StressevolutionintheMo/SimultilayersonSisubstrateduringthermalcycling.Courtesy ofS.Braun2008,FraunhoferInstituteforMaterialandBeamTechnology,Dresden,Germany 23
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denseh-MoSi 2 -phaseconsumesMoandSimaterialfromthemultilayers.Thisdensication processresultsinvolumecontractionofthemultilayers,andconsequentlyinarapidincrease ofthetensilelmstress. Initialstageofcoolingfrom500 Crevealsthatthecrystallizationprocesscontinuesuntilthe temperaturefallsbelow475 C.Thisshortportioncanalsobecharacterizedbyarapidincreaseof tensilestressintheformingh-MoSi 2 -phaseduetothereasonsdiscussedabove.Thesubsequent coolingresultsinalinearincreaseofthelmstressuptoabout800MPacausedbytheelastic thermalmismatchstrainbetweentheSisubstrateandthemultilayers. Heatingduringthesecondthermalcyclefollowsthecurvepathofthecoolingstageoftherst cycleuptothetemperatureof475 C,wherethestressisseentoincreaserapidly,whichsimilarto therstcycle.Formationoftheh-MoSi 2 -phaseandthethermalelasticmismatchstrainarethetwo competingmechanismsinuencingstressinthemultilayersinoppositeways,asdescribedabove. Whentemperatureequals475 Cthetwocompetingmechanismsequilibrateeachother.Therate ofthecrystallizationprocessdependsontheamountofMoandSiphasesleftinthemultilayers, andconsequentlyitdecreasesintimeassumingconstanttemperatureconditions. SincemuchoftheMoandSimaterialswereconsumedinthecrystallizationprocessoccurredin therstcycle,theincreaseofthelmstressisnowmuchsmallercomparedtothatoftherstone, althoughtheannealingtimeswereequal.Thecoolingportionofthecurveislinearapartfromthe temperaturescloseto500 Cwherethecrystallizationprocessstillplaysasignicantroleinthelm stressevolution.Atroomtemperaturethestressinthemultilayersstackreachesthevalueofabout 1000MPa.Thethirdcycleisverysimilartothesecondoneintermsofthestressvs.temperature behaviour.Thesmallincreaseinthetensilelmstresscausedbyformationoftheh-MoSi 2 -phase indicatesthatalmostallMoandSielementswereconsumedinthecrystallizationprocess.Thelm stressreaches1100MPawhenthestructureiscooledtotheroomtemperature. Basedontheconclusionthatchangeofthelmstressbelow475 Cisgovernedbythethermal expansionmismatch,thecoefcientofthermalexpansionoftheMo/Simultilayerscanbeestimated. 15

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Considerthefollowingvaluesofthelmandsubstratematerialproperties[ME]: s =2 : 6 10 )]TJ/F19 7.9701 Tf 6.586 0 Td [(6 K )]TJ/F19 7.9701 Tf 6.587 0 Td [(1 ;E f =250 GPa ; f =0 : 3 ;M f =357 GPa : Now,forthematerialconstantsgivenaboveandthestressvaluesmeasuredatthecorresponding temperaturesof23 Cand400 CseeFigure11: C =1100 MPa ; C =640 MPa ; thecoefcientofthermalexpansionoftheMo/Simultilayeraftertherstcycleiscalculatedfrom : f =5 : 2 10 )]TJ/F19 7.9701 Tf 6.254 -2.812 Td [(6 K )]TJ/F19 7.9701 Tf 6.587 0 Td [(1 : Fromtheparallelrunofthestressvs.temperaturecurvesforthesecondandthirdthermalcycles below475 CitisreasonabletoassumethattheCTEoftheMo/Silmdoesnotchangesignicantly andmaybetaken,ascalculatedabove. 2.2Microscopyanalysisofthecrackpatterns 2.2.1High-speedcamerasetupandsamples TheMo/SimultilayersusedinthisstudyweremanufacturedbythemagnetronDCsputterdepositionattheFraunhoferInstituteforMaterialandBeamTechnologyinDresden,Germany.Thestack consistedof40MoandSialternatinglayerswith2.7and4.2nminthickness,respectively.The multilayersweredepositedontoa525 mthick-Sisubstratewithadiameterof100mm.The twotestedwafershaddesignationsPS221andPS227,accordingtothemanufacturer. Annealingwasperformedusingahightemperatureovencapableofheatingupto1100 C. Priortoannealing,theinvestigatedsampleswerecutoutfromthewaferintopiecesabout2 2mm 2 largeusingadiamondscriber.Thecuttingprocedureinduceddefectsandawsontheedgesofthe -Siwafer.Thecutoutsampleswereplacedontoathickceramicplateandannealedintheoven at500 Cfor20minutes.Afterannealingthesamplesbatchwastakenoutofthehotovenand 16

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exposedtotheambientenvironmentwithtemperatureofabout23 C.Thesampleswereallowedto cooldownslowlyunderthenaturalairconvectionwhilerestingontheceramicplate1minutes. Forcrackgrowthobservationsthesampleswereplacedontoasteelstageofthemicroscopecovered witha1mmthickceramicplate.Thethinceramicplatewasusedtopreventimmediatesample coolingduringopticalobservations.Temperatureofthesampleswasnotmeasuredduringtesting. Figure12showstheexperimentalsetupusedtoinvestigatethecrackingbehaviourintheannealedMo/Simultilayers.Torecordthecrackpropagationintimeahigh-speeddigitalcamerawas attachedtoalong-rangeopticalmicroscope,asshowninthegure. Figure12.High-speedcameraexperimentalsetup Onesetofhigh-speedphotographyresultspresentedinthisstudywasobtainedusingadigital consumercameraPhilipsSPC/1300NCcapableofrecording90framespersecondfpswiththe resolutionofupto320 240pixels.Inthecourseofexperimentsitbecameevident,thatevenfor theslowerpropagatingcrackstherateof90fpswasinsufcienttocaptureimportantstagesofthe crackgrowth.Basedonthisexperience,thehigh-speedphotographyresultsatalaterstageofthe workwereobtainedusingahigh-endprofessionalfast-motioncameraPhotronFASTCAM-Ultima 1024.Propagationofthelmcracksusingthisequipmentwasrecordedatarateof1000fpswith theresolutionof512 512pixels. 17

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2.2.2Aposteriorianalysisofcracks ThesectionpresentsresultsofopticalobservationsofthecrackpropagationintheannealedMo/Si multilayersusingthehigh-speedphotography.Inmostcasescoolingofthetestedsamplesledto formationandgrowthofcracksintheMo/Silm.However,afewsamplesdidnotrevealany crackingevenwhencooledtotheroomtemperature,althoughallthesampleswerecutoutfromthe samePS221wafer.Althoughnotveriedindepth,absenceofacriticalstarterdefectintheMo/Si lmmayexplainthisrarebehaviour.Thisargumentcanbesupportedbythefactthatnocracks areusuallyobservedintheMo/Siwafersannealedat500 C.NocrackswereobservedintheSi substrate. Opticalobservationsshowedthatthecoolingrateofthesamplescorrelatedwiththespeedof crackspropagationintheannealedMo/Silm.Onlytheslowpropagatingcracksrevealedformationofthepeculiarcrackpatterns.Obviously,thehighestcoolingratesareachievedwhenahot sampleisplacedontothemetalstageofthemicroscopeundertheroomtemperatureconditions.The fasterpropagatingcrackswereobservedimmediatelyafterasamplewasbroughtincontactwiththe microscopestage.Asthesamplecoolsdownthecoolingratedecreases.Accordingly,thecracks wereobservedtopropagateslowerafterthesampletemperaturedecreased.Atthispointitisalso importanttonote,thatthelimitedeldofviewofthemicroscopepreventedfromsimultaneously observingthewholesurfaceofasample. Inmostcasesthefastpropagatingcracksprecededtheslowgrowingcracks.However,insome rarecasesthissuccessionwasobservedtobebroken.Anexampleofatypicalpatternformedby thefastgrowingcracksispresentedinFigure13.Inbothimagesthecrackpatternhasatreeleaflikegeometry,whichwasformedbythecracksbranchingandlooping.Thedarkerportionsofthe imagescorrespondtothelmdelaminatedareasthatlieout-of-planewithrespecttotheoriginal horizontalplaneofthesample.ThemicroscopeimageshowninFigure13acontainsboththe treeleaf-likecracksandthecrackshavingasquare-sinusoidalgeometryintheheavilydelaminated areaseenintheupperrightportion.TherightpartofFigure13bshowsacompletelydelaminated partofthelmwhichisdisplacedwithrespecttothepresumablynon-delaminatedportionseenon theleftside.Theblackareaattheupperleftcornerofthisimagerepresentsapieceofabroken 18

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delaminatedlmwhichwasdisplacedbyalargedistanceasaresultofitssuddendelaminationand cracking. a b Figure13.Treeleaf-likebranchingchannelingcracks Thefastpropagatingcrackswereobservedtoformagrass-likepatterngeometrypresentedin Figure14.Thisoblongpatternwasformedasaresultofgrowthandbranchingofthechanneling cracksaccompaniedbytheirintersectionandtermination.Inthiscasearadialstructureofthecrack patterncanberecognized.Thegurealsoshowsthatthecracksfromthegrass-likepatternturn Figure14.Burstchannelingcrackstransformingintothesinusoidalandcrescentpatterns intothesinusoidal-andcrescent-likeshapes.Onemayalsoobserve,thatthelmisdelaminatedto alargepart. AnotherexampleofthefastchannelingparallelcracksthatturnsimultaneouslyintothesinusoidalandcrescenttypesofcracksisshowninFigure15.Theparallelstraightcracksemanated 19

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fromamothercrackbybranching.Itisinterestingtonote,thattheamplitudeofthecrescentcracks increasesastheypropagate.Asintheguresabove,thelmdelaminationisclearlyseen. Figure15.Channelingcrackstransformingintoasinusoidalfollowedbyacrescentform ExperimentalstudiesofcrackingbehaviourintheannealedMo/Silmdiscoveredanewtype ofacrackpatternthatwasnotpreviouslyreportedintheliterature,tothebestknowledgeofthe author.Thiscrackpatternmaybebestdescribedasasquare-sinusoidalortheChineseWall pattern,whichispresentedinFigure16.Itcanbeobserved,thattheperiodandtheamplitudeof thepatternremainedconstantasthecrackpropagated.Figure16bshowsthatafterthepathof thesquare-sinusoidalcrackwasdisturbedbythebrancheddaughtercrack,themothercrackcould restoreitsinitialperiodicsquare-sinusoidalform.Onecanalsoobservethesmalldelaminatedparts aroundtheconsideredcrack.InthelowerpartofFigure16bonemayalsonoticepresenceofa straightchannelingcrackterminatingatthesquare-sinusoidalcrack. ResultsofopticalobservationspresentedsofardonotgivetheanswerwhysuchpeculiarperiodicalcrackpatternsformintheannealedMo/Silm.Remarkablyenough,discoveryofthe mechanismexplainingtheobservedcrackbehaviourwasmadebychanceduringadjustingthemicroscopefocus.Itwasnoticed,thatasinusoidal-likecrackwascloselysurroundedbyanareaof thelmlyingoutofthefocusplaneoftheinitiallyadjustedmicroscope.Thisareacanonlybe attributedtoadelaminatedareaofthelm. Figures17aand17bpresentthesinusoidal-likelmcracksaccompaniedbydelamination. Inbothimagesthedelaminationcanberecognizedasaslightlyraisedareaaroundthecrackrelative 20

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a b Figure16.Square-sinusoidalortheChineseWalltypechannelingcracksaccompaniedbydelamination a b Figure17.Sinusoidalchannelingcracksaccompaniedbydelamination 21

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totherestofthesamplesurface.Thedelaminationfollowsandcompletelyenclosesthechanneling crack,whichterminatedbyloopingonitself.Sincestressinthelmistensileseesection2.1.2 suchcloseddelaminationcannottakeplacealoneinthelm,asopposedtothecaseofbucklingdelaminationundercompressivelmstresses[HHE00],[MCL02].Inthepresentcase,thechanneling crackreleasesthelmedgesnecessaryforthedelaminationtotakeplace. Figure18.Sawtooth-typechannelingcrackfollowingastraightpathaccompaniedbydelamination Duringcrackpropagationthedirection,amplitudeandperiodofthepatternmaysuddenly change,asalsoseenfromthemicroscopeimagesFigure17.However,theundisturbedcrack propagationundertheuniformconditionsresultsinaconstantgeometryofthecrackpattern,as presentedinFigure18showingasawtooth-likecrackpattern.Onecanalsoobservethatthedelaminationfrontrepeatsthemajordirectionofthecrackpropagationandremainsataconstantdistance fromthecrackcenterline. Acloseinteractionbetweenachannelingcrackandlmdelaminationproducingtheperiodic crackpatternscanbeevidencedfromthemicroscopeimagesinFigure19.Itisinterestingtonote, thatdirectionofthecrackpropagationchangesassoonasitreachesthedelaminationfront,although theperiodicalpatternispertained. SpiralcrackpatternswerealsofoundintheannealedMo/Silm.Twoexamplesrepresenting suchcracksareshowninFigures20aand20b.Closetotheimaginarycenterthechannelcracks showsomesimilaritywithanArchimedesspiral,whichismoredistinctintherightimage.Afterpropagatingapproximatelythreespiralcyclesthecrackformedacrescentpatternfollowedby 22

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a b Figure19.Sinusoidalchannelingcracksfollowingacurvedpathaccompaniedbydelamination a b Figure20.SpiralchannelcrackingintheannealedMo/Silmaccompaniedbydelamination 23

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terminationandloopingonitself.Asinthecasesofsinusoidal-likecracks,thedelaminatedarea ispresentaroundthespiralcracks.Itshouldbenoted,thatthespiralcrackswererarelyobserved intherapidlycooledannealedMo/Silm.Onemaysuggest,thatthesepatternsformundermuch slowercoolingrateswhichprovideuniformthermalloadingconditions[MLL + 04],[VMM]. Figure21.BurstchannelingcrackstransformingintoasinusoidalintheannealedMo/Silm ExperimentalobservationsshowedthattheperiodiccrackpatternsintheannealedMo/Silm canalsobetriggeredbyimpulsedynamicalloading.Figure21presentsthecrescent-typecrack patternformedasaresultofthedroptest.Thepresentcaserevealsamuchlargerratioofthe amplitudetotheperiodlengthofthecrescentpatterncomparedwiththepurelythermallyloaded samples,shownabove.However,formationofthecrackpatternscausedbytheimpulsedynamic loadingwasnotpursuedinthecurrentwork. 2.2.3High-speedphotographyresults Sofarthediscussionwasconcentratedonthepostmortemmicroscopeimagesoftheformedcrack patterns.Basedontheseobservationsitwasconcludedthatinteractionofachannelingcrackwith thelmdelaminationfrontleadstoformationofthepeculiarperiodicpatterns.Toshedlighton theprocessofthecrackpatternsformationresultsofhigh-speedphotographyarepresentedand discussedinthefollowingsection. 24

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Figure22.Propagationofastraightchannelingcrackaccompaniedbydelaminationatselected instantsoftime Figure22presentsselectedframescapturedduringpropagationofastraightchannelingcrack intheannealedMo/Silm.ThecrackingeventwasrecordedusingthePhilipsdigitalcameraata rateof90fps.Theembossdigitallterwasappliedtotheoriginallyrecordedframes.Itisseen thatasthechannelrunsthroughthelmitisaccompaniedbyasymmetriclmdelaminationatall stagesofthecrackgrowth.Theimagecomparisonshowsthatthedelaminationfrontcontinuesto advanceintimeandismuchslowerthanthelmcrack.Theaveragespeedofthechannelingcrack isapproximately0.4mm/s,whichisrelativelyslowwhencomparedtothecracksformingtheleaforgrass-likepatternsseebelow.Unfortunately,magnicationandresolutionoftheusedoptical equipmentdoesnotallowtounambiguouslydeterminewhetherthedelaminationrunsaheadofthe 25

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channelingcrackorviceversa.However,itshouldbenotedthattheframesf-40andf-55indicate thatthechannelingcrackrunsbehindtheadvancingdelamination. Figure23.Propagationofwavychannelingcracksatdifferentinstantsoftime.Oneofthechannels turnsintoaChinesewall-typepattern. Figure23showssimultaneouspropagationoftwoparallelwavychannelingcracks.Itcanbe seenthatbothcracksmakea90 leftturnfollowedbyformationoftheChineseWallpatternincase oftheleftcrack.Propagationoftheleftwavychannelisdisturbedframef-60byascratchonthe samplesurfaceleftbythediamondscriber,whichcanbeseenatthebottomoftheimages.Obviously,thisdeviatesthechannelingcrackfromitsoriginaldirection.Althoughtheaccompanying delaminationishardlyseeninthepresentedimages,acloseobservationrevealsitspresence.The initiallysmalldelaminatedareasaroundthechannelingcrackscontinuetogrowcausingdelaminationofthegrosslmareas. Formationoftheperiodicalcrackpatternsfromtheburstinggrass-likepatterncracks,discussed above,wascapturedusingthefast-motionPhotroncamera.Thesampleswerepreparedfromthe PS227wafer.Figure24presentsdifferentstagesofthecrackpatternformationrecordedatarate of1000fps.Comparingthetworstframesframe-1546andframe-1545theaveragecrackpropagationspeedcanbeestimatedatleast200mm/s.Thethirdframeframe-1544revealsthatafter 0.002sgrowthofthecracksslowsdownsubstantially.Thefastcrackgrowthwasalwaysobserved 26

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Figure24.Selectedframesshowinggrowthoftheburst-likechannelingcracksfollowedbyformationoftheperiodiccrackpatterns 27

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tobeaccompaniedbylargedisplacementofthedustparticlesinitiallyrestingonthelmsurface. Thisyingparticleseventsuggeststhatthefastgrowthofthechannelingcracksisalsoaccompaniedbythelmdelamination,whichsuddenlycreatestheout-of-planedisplacementofthelm. However,duetotheshortperiodoftimewhenthesecrackspropagateandlimitedopticsresolution, delaminationduringthefastcrackgrowthwasnotdirectlyobserved.After0.007sframe-1539the delaminationisclearlyseentospreadaroundthealmostmotionlesschannelcracks.Thefollowing framesshowthattheslowcrackpropagationphasebeginswhereformationoftheperiodiccrack patternsisaccompaniedbylmdelamination. Figure25presentsframesshowingformationoftheperiodiccrackpatternsselectedevery 0.005s.Atallstagesthechannelingcracksinteractwiththeadvancingdelaminationfront,although itisimpossibletodeterminewhetherthecracksrunaheadorbehindthedelaminationjudgingby thesemicroscopeimages.Itcanbeobservedthatthedelaminationgrowthisfasterinthevicinity ofthechannelingcracksasthelatterprovidethenecessaryfreelmedges.However,itisseenthat afterdelaminationisemittedbyachannelingcracktheformeralsocontinuestogrow,althoughata slowerrate. Theinteractionbetweenachannelingcrackandtheadvancingdelaminationfrontisthekey pointintheformationoftheobservedperiodiccrackpatterns.Intheimmediatevicinitybehindthe delaminationfrontthestoredstrainenergyduetothetensilelmstressisnotcompletelyreleased. Becauseofconstraintsprovidedbythesubstrate,thestrainenergyreleaseratenecessarytocracka lmattachedtoasubstrateismuchhigherthanthestrainenergyreleaseratenecessarytocrackthe unattachedlm.Basedontheopticalobservationsitmaybesuggested,thatduringtheslowcrack growthphasethestrainenergystoredinthelmislowerthanthatnecessaryforachannelingcrack tooccurinthenon-delaminatedlm.Ontheotherhand,thestillstoredstrainenergyinthelm behindthedelaminationfrontisenoughforthelmcrackingtooccur.Thisexplanationissupported bythefactthatthechannelingcrackalwayspropagatesinclosevicinityofthedelaminationfront. Formationofthecrescent-likecrackspresentedinFigure26illustratespropagationofthechannelingcracksalongtheadvancingdelaminationfront.Intherstimageframe-2911thelong channelingcrackontherighthasjustgrewbehindthedelaminationfrontandstopped,althoughit 28

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Figure25.Selectedframesshowingformationofthesinusoidalandcrescent-likecracks 29

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Figure26.Selectedframesshowingformationofcrescent-typecracks couldcontinuegrowingalongtheavailabledelaminationseenintheimage.After0.008sframe2903thecrackchangeditspathintothedirectionoftheself-emitteddelamination.Onemaynotice thatasmallerchannelingcrackseeninthelowerrightpartoftheimagegrowsalongthedelaminationfronttoo.Inthethirdandfourthimagesframe-2860andframe-2795bothchannelingcracks growbehindthedelaminationfrontsimultaneouslystimulatingdebondingofthelm.Thelater stagesofthecrackpatternformationessentiallyrepeatthedescribedcrackingbehaviour. Itisknownfromthefracturemechanics,thatacrackpropagatesinthedirectionsoastomaximizethestrainenergyreleaserate.Appliedtotheconsideredcase,itmaybesuggested,thata 30

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channelingcrackchangesitspropagationdirectionwhenitismoreadvantageoustoturninthe directionoftheself-emitteddelamination,intermsofthestrainenergyreleaseratemaximization. 2.3ConclusionsforChapterTwo TheMo/Silmmustbeannealedupto500 Cinordertoformtheh-MoSi 2 .Asaresultofthis phasetransformationandthethermalmismatchbetweenthelmandthesubstrate,hightensile lmstressariseswhenthestructureiscooled.Atacertaintemperature,thestrainenergystored inthesystemisreleasedbylmcrackinganddelamination.Itwasobservedthatthecoolingrate inuencesthespeedatwhichthelmcrackspropagate.Thedecreasingcoolingrateresultedinthe slowerpropagatingcracks.Theperiodiccrackpatternswereobservedtoformbythequasi-statically propagatingcracks. Microscopicobservationsusingthehigh-speedcamerarevealedthatthepeculiarcrackpatterns formasaresultoftheinteractionbetweenthepropagatingchannelingcracksandtheadvancing delaminationfront.Periodicityofthepatternscanbeexplainedbyahigherspeedofchanneling cracksrelativetothedelaminationandbytheperiodicalprocesswhenachannelingcrackturns intothedirectionoftheself-emitteddelaminationoccurringatcertainfavorableconditions.These successivecrackturnsproducetheobservedperiodiccrackpatterns.However,aquantitativedescriptionoftheconditions,whichforceachannelingcracktoturnwouldhelptobetterunderstand theobservedbehaviour. 31

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ChapterThree ThermalDeformationofX-rayOptics 3.1Wafercurvatureduetothethrough-thicknesstemperaturegradient X-raymirrorstypicallyhaveaparabolicreectivesurfacealongthebeamapplicationlineandzero curvatureintheperpendiculardirectionrefertosection1.2.Theaveragecurvatureoftheparabolic surfaceisabout1/10m )]TJ/F19 7.9701 Tf 6.587 0 Td [(1 [Bra]andcannotbechangedafterproduction.Forsuchxedsystems differentmeasuresareundertakentomitigateanyparasitemirrorsurfacedistortionse.g.,thermal deformationscausedbyheating,whichreducethereectedbeamquality.Someaspectsconnected tothisproblemareaddressedlaterinsection3.2.Atthispoint,theoppositeproblemformulation presentsagreatinterest:whataretheconditionstoachievethenecessaryvariablecurvatureof themirrorsurfacewhichcouldadditionallybevaryingintime?Thisapproachhaslongbeenutilizedinadaptiveopticssuchasimagestabilizationsystemsinconsumerphotoequipmentorimage correctionduetoatmosphericandotherdistortionsinpowerfulastronomytelescopes. IncontextoftheX-raymirrorsonecouldimaginenumerouswaystocontrolcurvatureofthe reectivesurface.Probably,themostconventionalonewouldbebendingofthemirror.Thiscanbe achievedbydirectlyapplyingmechanicalloadstothestructure,byutilizationofthepiezoelectric effect,etc.Amoreunconventionalway,atleastfromtheauthor'spointofview,isutilizationof thermalloads.Itisobvious,thatmentionedaboveparasitethermaldistortionscouldalsobeutilized tocontrolshapeoftheX-rayopticsandthuscurvatureofthereectivesurface.Thermalloadscan beapplied,forinstance,byattachingheatingorcoolingelementstothemirror.Toidentifythe necessarythermalloading,effectoftemperaturedistributiononthermaldeformationsinthemirror needstobeanalyzedrst. 32

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ThissectionintendstoprovideinsightforthermallyinducingthetargetcurvatureinaSiwafer withadepositedreectivemultilayer.Forthesakeofsimplicity,thediscussionisnarrowedtoaxisymmetriccaseofuniformwaferdeformations.Giventheactualgeometry,materialsandtemperatureconstraints,resultsofthisstudyhelptoidentifythetemperatureloadingnecessarytoproduce therequiredsurfacecurvatureoftheX-rayopticalelement. 3.1.1Modeldescription ConsideraxisymmetricwafergeometryshowninFigure27.Forconvenience,cylindricalcoordinate systemisdenedatthecentreofthewaferwithdirections ;;z .Thewaferis20mmindiameter anditiscomposedofa525 mthickSisubstrate,aswellasthefront-andbacksidelayers.The waferfrontsiderepresentsX-rayreectivecoatingconsistingof40xMo/Sialternatinglayerswith auniformthicknessof2.7/4.2nmeach,asshownontheblowout.Thebacksideofthewaferis sputteredwitha3 mthicktungstenlm.Thechoiceinfavourofthetungstenlmisbasedonits abilitytobedepositedeitherwithcompressiveortensilestresses[Wat08],enablingonetomagnify orcompensatewafercurvaturewhichmaybecausedbythefrontsidelayers.Thewaferistakento beinitiallyat,aswellasstressfree. Figure27.GeometryofX-raymirrorindicatingstructuralcompositionwithfrontreectiveand backsides MaterialpropertiesofthemirrorconstituentsarelistedinTable1.Thematerialsareassumed tobeisotropicelastic.ThePoisson'sratioandthecoefcientofthermalexpansionfortheMo/Si multilayerwereestimatedusingthecompositemixturerules[AZP84]andareinputintermsofthe effectivevalues. 33

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Table1.MaterialpropertiesofX-raymirrorcomponentsat27 C MaterialYoung'smodulus,Poisson'sratio,CTE, E ,GPa ,10 )]TJ/F19 7.9701 Tf 6.587 0 Td [(6 1/K Silicon 1 149.00.282.6 Tungsten 1 400.00.284.4 40xMo/Si170.0 2 0.303.5 1 materialdatatakenfrom[cI] 2 Measuredusingtheindentationtest[Vol] Analysisofthemirrorgeometryandthethermalexpansionmismatchbetweenthematerialssuggeststhatunderuniformheatingconditionsthemidplaneofthestructurewilltakeaconcaveform. Thiscanreadilybeshown,sincethicknessofthetungstenlayerisaboutoneorderofmagnitude higherthanthatoftheMo/SimultilayertungstenhasahigherCTEtoo.However,forthesputtered Mo/Simultilayertheupperlimitofoperationtemperatureisabout100 C[Boe01],[BMP + 03].Exceedingthistemperatureleadstodegradationofthemultilayersintermsofitsabilitiestoreectthe X-raysduetoareactionbetweentheindividuallayers.Preliminarysimulationsshowedthatthe targetradiusofcurvatureof10mcannotbeachievedbyauniformheatingoftheX-raymirrorin thegiventemperaturerange. Now,letusexplorethenonuniformtemperatureloadingofthestructure.Forsimplicity,assume thetemperaturegradientisconstant.Inthiscasethetemperaturevarieslinearlywiththewafer thicknessandisuniforminotherdirections.Undertheseconditions,acorrespondinguniform straineldisgeneratedresultinginthecurveddeformationshape.Asaresultofsymmetryand translationalinvariancethedeformedshapeofthemidplaneisspherical,providedthedeformation issmall.Itisnowtobeanswered,whethersuchthermalloadingisabletoproducetherequired radiusofcurvature,mentionedabove,intheprescribedtemperaturerange.Thenextsectionfocuses onanalysisofthisproblemusingtheniteelementmodeling. 3.1.2Finiteelementmodel CommerciallyavailableniteelementpackageANSYS[Ans]wasusedtoanalyzethermaldeformationsoftheX-raymirror.Takingadvantageoftheaxisymmetricmirrorgeometry,acorresponding axisymmetricmodelwascreated.Themodeldimensionsandtheniteelementmeshtogetherwith 34

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theboundaryconditionsareshowninFigure28.Asindicated,themodelincludestheMo/Simultilayeronthetopandthetungstenlmonthebottomsurfaces,respectively.Themodelwasmeshed withthe8-nodedtwo-dimensionalPLANE183elementswiththeaxisymmetrickeyoptionturned on.TheMo/Simultilayerwasmodeledwithonelayerofelementsinthicknessdirectionwiththe effectivematerialproperties,asmentionedabove.Thissimplicationdoesnotsignicantlyinuencethedeformedshapebutdrasticallyreducesnumberofelementsinthemodel.Thetungsten lmwasalsomodeledviaonlayerofelementsinthethicknessdirection.Materialpropertiesused inthemodelarelistedinTable1. Figure28.FiniteelementmodelofX-raymirrorillustratingtheappliedtemperatureloading Themodelwasallowedtodeformfreelybyconstrainingtheaxisofsymmetrywiththesymmetryboundaryconditions.Inaddition,thebottomnodeonthesymmetryaxiswasconstrainedto preventtranslationalmovementintheverticaldirection. Figure28alsoshowsapplicationofheatingandcoolingloadstothebottomandtopsurfacesof themodel.Thisdesignationshouldratherbetreatedconditionallyinthesensethatthetemperature atthebottomsurface T bot = 42 Cishigherthanthetemperatureatthetopsurface T top = 23 C toproducethevaryingtemperaturedistributioninthethicknessdirection.Infact,thethroughthicknesstemperaturegradient dT=dz wasapplieddirectlytothemodel,asshownattheleftportion ofthegure. ConvergencestudyseeAppendixBhasshownthattheusedniteelementmeshisadequateto accuratelyreproducethermaldeformationsofthemodel. 35

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3.1.3Results TheexpandedaxisymmetricsimulationplotFigure29showsdistributionofradialstrains intheX-raymirror.Itcanbeobserved,thattheuniformthrough-thicknesstemperaturegradient producesthecorrespondinguniformstraineldwithalineardistributionacrossthethickness,as expectedforsmalldeformations.Highertemperatureatthebottomsurfaceresultsinlargerelongationofthematerialbelowtheneutralsurfaceofbendingwhichproducestheconcavespherical shapeofthemirror.Note,thatthefree-edgesingularities[Bog71]arenotreproducedintheplot whichwasachievedbyexcludingelementsatthecorrespondinglocations. Figure29.TotalradialstrainsintheX-raymirrorduetothethrough-thicknesstemperaturegradient Figure30illustratestheout-of-planedisplacementeldoftheX-raymirror.Duetotheaxisymmetry,thecircumferentialcurvature k andtheinteractiontermthetwist k arezero.Consequently,theonlycurvaturetermassociatedwiththeradialdirectionisnonzero: k = k: Asseenfromthedisplacementsplot,thebowdoesnotexceed5 mforthe20mmlargewaferwith totalthicknessofabout528 munderthe19.1 Ctemperaturegradient.Thisresultcanbeutilized tocalculatethewafercurvaturepresentedbelow.Notethatbecausethesolutionislinearelastic,the calculatedresultcanbescaledwiththevalueofthetemperaturegradient. 36

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Figure30.Out-of-planedisplacementsoftheX-raymirrorduetothethrough-thicknesstemperature gradient Theout-of-planedisplacement u z canbewrittenintermsoftheradialposition anditscurvature k ,asfollows: u z = 1 2 k 2 : Fromequationonecandeterminetheuniformcurvatureas: k = 2 u z 2 =9 : 8 10 )]TJ/F19 7.9701 Tf 6.586 0 Td [(5 mm )]TJ/F19 7.9701 Tf 6.586 0 Td [(1 whichisthesoughtcurvatureof1/10m )]TJ/F19 7.9701 Tf 6.586 0 Td [(1 ,asstatedabove. Theresultfromequationcanreadilybeveriedbydirectlycalculatingthesecondderivative withrespecttotheradialposition,asknownfromtheanalyticgeometry.Again,assumingthe curvatureissmall,equationoftheneutrallineofbending[PAK + 85]is k = d 2 u z d : Tondslopeandcurvatureofthewafersurface,therstandsecondnumericalderivativeswere appliedtothedisplacementeld u z .TheresultsforaSisubstrateandtheX-raymirrorconsisting 37

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oftheMo/SiandWlayersareshowningures31aand31b.Theslopevarieslinearlywith thedistancefromthewafercentre.ThetwolinesrepresentingtheslopeforSiwaferandX-ray mirrorFigure31aalmostcoincideontheplotsuggestingthedeformedshapeofbothstructures isessentiallythesame.AnalogousbehaviourcanbeobservedonthecurvaturesplotFigure31b. aslope bcurvature Figure31.Slopeandcurvatureofthewaferreectivesurfaceasafunctionoftheradialposition Itisseen,thatthecurvatureisindependentonthepositionforthemostpartofthewaferand theresultfromequationisreproduced.However,tworemarksshouldbemade.Splash-like behaviourontherightportionofthecurverepresentingcurvatureoftheX-raymirrorisattributed tosingularbehaviourofthestraineldatintersectionofthefreeedgeswiththeinterfacebetween Mo/SistackandSi.TheSisubstratedoesnothavematerialinterfacesandthecurvatureplotin thiscaseisat.Theabruptfalloffseenattheleft-andtherightmostpartsofthecurvesshould bedisregarded.Theauthorassumesthatthiserroneousbehaviouriscausedbyapplicationofthe centraldifferenceschemefortheedgepointsinsteadoftheforwardandbackwardones.Thisissue needstobeclariedwiththeANSYSdevelopmentteam. EssentiallythesameresultfortheSisubstrateandtheX-raymirrorintermsoftheslopeand curvaturevaluesshowsthattheinuenceoftherelativelythinlayersisinsignicant.Inotherwords, foragivengeometryandmaterialsset,inuenceofthetemperaturegradientismuchhigherthanthe inuenceofthethermalmismatchbetweenthematerials.Theelasticmismatchinthiscasecanbe neglectedatallbecauseitsimpactisevenmuchsmallerthanthermalmismatchinuence[Tim25]. 38

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3.2ThermaldeformationsofX-raymirrorduetobeamirradiation 3.2.1Modeldescription BeamirradiationofG obelmirrorsandSimonochromatorsusuallytakesplacealonganarrowstripe ofthereectivesurfaceseeFigures6and8.Inaddition,themirroriscomposedofdifferent materialsthatalsoresultsinacomplicatedmodelgeometry.Alongwiththenonhomogeneous thermalboundaryconditionsandnonlinearconvectiveandradiationheattransfer,itisvirtually impossibletoconstructanadequateanalyticalsolutionabletoaccuratelypredictthermomechanical behaviourofthemirror.Tocircumventtheseproblems,theniteelementmodelingwasutilizedin thecourseofthisworkwiththesimulationstrategydescribedbelow. ThecommerciallyavailableniteelementpackageANSYS[Ans]wasutilizedforcomputation ofthermaldeformationsintheX-raymirror.Theanalysiswascarriedoutintwosequentialsteps: thermalandstructuralmechanicssimulations.Inthethermalanalysisnaturalcoolingtoairconvectionandheattransferbyradiationfromthemodelsurfacewereassumedastheonlyboundary conditionsappliedtothemodel.Theassumptionofnaturalairconvectionisvalidsincenospecial devicessuchasairfans,radiatorsorsimilarareusuallyutilized.TheStefan-Boltzmannlawforheat owbyradiationwasusedinthethermalanalysis.However,inviewofasmalltemperaturedifferencebetweentheX-raymirrorwiththeambientandthelowabsolutetemperaturerangeoverall,the heatowbyradiationissmallcomparedwiththatofconvection.Bearingthisinmind,noradiation reectingsurfaceinteractingwiththeX-raymirrorweremodeledtosimplifytheanalysis. Thecalculatedtemperaturedistributionfromthethermalanalysishasbeenusedasinputfor thestructuralmechanicsmodelwhichwasassumedtodeformfreely.Thesequentialuncoupled thermal-mechanicalniteelementsimulationwasutilizedastheinuenceofthestructuralmechanicsvariablesonthethermalstateofthemirrorwasassumedtobenegligiblysmall. 3.2.2Geometryandmaterialproperties ModelgeometryoftheX-rayopticalelementispresentedinFigure32.The0.525mmthickSi waferisbondedwithoutslippingtoa5mmthicksubstrate.ThetopsurfaceoftheSiwafercontains 39

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40xMo/Simultilayerstackwiththeoveralluniformthicknessofabout240nmanda3 mthick tungstenlmonitsbacksideseetheblowoutinFigure9.Thein-planedimensionsofthemirror are60 20mm mm.TheX-raybeamexposedareahasdimensionsof60 1mm mmandis locatedatthecenterofthereectivesurface,asillustratedinthegure. Figure32.GeometryofanX-rayopticalelementsubjectedtotheheatuxfromanX-raysource CompositionoftheX-rayopticalelementcanbechangedbyvaryingmaterialand/orthickness ofthesubstrate,bondingofadditionalstructures,etc.Asdiscussedinsection3.1,themajorfactors inuencinggeometrydistortionsoftheX-rayopticalelementunderthermalloadingarerelative thicknessofitsconstituents,thermalexpansionmismatchandpresenceofthethrough-thickness temperaturegradient.ToanalyzeinuenceofthesefactorsfortheconsideredhereX-rayoptical element,fourdifferentmodelcongurationswereanalyzed,aspresentedinTable2. Table2.ModelcongurationsoftheX-rayopticselement ModelvariationGeometrycongurationHeatux,W/m 2 Model1Siwafer 1 onglasssubstrate1 10 4 Model2Siwafer 1 onSisubstrate1 10 4 Model3Siwafer 1 oninvarsubstrate1 10 4 Model4Siwafer 1 onbothsidesofglasssubstrate1 10 4 Model5Simonochromator 2 1.33 10 6 1 contains40xMo/Si=60.7/4.2nm[Bra] 2 DeutscheElektronenSynchrotron[Sta] 40

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Model1consistsoftheX-raymirrorbondedtoaglasssubstrate.CombinationoftheX-raymirrorbondedtoaSisubstrateisdesignatedasModel2.Inuenceofthethermalexpansionmismatch inthismodelisverysmallduetotherelativelythinMo/SiandWlms. Table3.MechanicalpropertiesofmaterialsoftheX-rayopticalelementat27 C MaterialYoung'smodulus,Poisson'sratio,CTE, E ,GPa ,10 )]TJ/F19 7.9701 Tf 6.586 0 Td [(6 1/K Silicon149.00.282.6 Glass70.00.170.5 Invar148.00.301.3 Molybdenum330.00.305.4 Mo/Sistack170.0 1 0.303.5 1 Measuredusingtheindentationtest[Vol] InModel3theX-raymirrorisbondedtoinvarsubstrate.AsseenfromtheTable3,thethermal mismatchbetweenSiandinvarissmallerthanthatbetweenSiandglass.Owingtothisfact,invaris thematerialofchoiceforsubstratesofthemodernX-rayoptics,suchasG obelmirrors[Bra].This congurationservesasthereference. Model4presentsastructurecomposedoftheglasssubstratebondedbetweentheX-raymirror containstheMo/SiandWlmsandablanketSiwafer.Thicknessofthelatterisequal0.525mm. NeglectingthesmallrelativethicknessoftheMo/SiandWlms,itcanbesaidthatthestructureis essentiallysymmetricalwithrespecttoitsthicknessdirection. Allmodels1aresubjecttoaheatuxfromalowpowerX-raysourceof1 10 4 W/m 2 .The Model5representsabulkSimonochromatorexposedtoahighpowerX-raysynchrotronsourceof 800WutilizedattheDeutscheElektronenSynchrotronDESYsite[Sta].Itwasassumedthatonly 10%ofthebeampowerislostduetothematerialheating. ThermalpropertiesofthematerialsusedinsimulationsarepresentedinTable4.Duetothe smallrelativethicknessoftheMo/SiandWlmscomparedtotheSiwaferandratherclosethermal properties,theselayerswerenotexplicitlymodeledinthethermalsimulationsandwereincludedin theSiwafer.Materialinterfacesweremodeledasideal. Asseenfromthetable,theglassmaterialhasthelowestthermalconductivityat27 Ctemperature,whereasSithermalconductivityisabout100timeshigher. 41

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Table4.ThermalpropertiesofmaterialsoftheX-rayopticalelementat27 C MaterialDensity,Thermalconductivity,Specicheat,Emissivity, ,kg/m 3 k ,W/m-K c p ,J/kg-K e Silicon2330130.07140.6 Glass22001.37500.9 Invar805010.25151.0 materialdatatakenfrom[cI] 3.2.3Computationoftheheattransfercoefcient Thenatureoftheconvectionheatowheavilydependsontheowconditionsnearthesurface, anditisnonlinear.Initssimplestform,theconvectionheatowcanbeaccountedforbyutilizing theheattransfercoefcient.UsingtheNewton'slawofcoolingforconvectionheatow,theheat transferbetweenamovinguidandasurfacecanbedetermined[LL06]: Q c = h c A s T s )]TJ/F21 10.9091 Tf 10.909 0 Td [(T a ; where h c istheaverageconvectiveheattransfercoefcient; A s isthecross-sectionalareaforheatowthroughthesurface; T s isthetemperatureofthesurface; T a isthetemperatureoftheambient. Thenaturalcoolingconditionstoairconvectionareassumedwiththeambienttemperatureof Theprocedureforestimationoftheaverageheattransfercoefcient h c followstherecommendationsdescribedin[Bla00].Assumingthenaturalcoolingconditionstoairconvectionwiththe ambienttemperatureof27 C,theconstantsandexpressionsnecessaryforcomputationof h c are presentedinTable5.Itmustbenotedthattheaverageheattransfercoefcientdependsonthe surfaceorientation.Forsimplicity,positionoftheX-raymirrorreectivesurfacewasassumed horizontallocatedsideup. Figure33showstheplotoftheaverageheattransfercoefcient h c asafunctionoftemperature differencebetweenthesurfacewithnaturalconvectiontoairandtheambienttemperature.The curvesrepresentthethreeorientationsofthemirrorsurfaceswithgeometrydimensionsshownin 42

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Table5.Constantsandexpressionsusedforcalculationoftheaverageheattransfercoefcient CongurationCharacteristiclength,Constant,Heattransfercoefcient, L ch C 0 C h c ,W/m 2 -K Verticalplate H 1.51 C 0 TL ch 0 : 25 Horizontalplate WL= [2 W + L ] heatedsideup1.38 C 0 TL ch 0 : 25 heatedsidedown0.69 C 0 TL ch 0 : 25 Figure32.Accordingtothecalculations,theverticalwallandthehorizontalsidedownhavethe highestandthelowest h c ,respectively.Thesaturation-likebehaviourofthe h c T curvessuggests Figure33.Averageheattransfercoefcientasafunctionoftemperatureforthreedifferentplate surfaceorientations thatactivetypesofcoolingwithhigherheattransfercoefciente.g.,forcedairow,watercooling, etc.needtobeappliedforamoreeffectiveheatremoval,ifnecessary. 3.2.4Boundaryconditions HeatgeneratedintheopticalelementduetotheX-raybeamirradiationisappliedatthecorrespondingareaFigure32asaheatuxintothesystem,whichcausesincreaseintemperature.Theheat removalismodeledbythesurfaceconvectiontoair,asdiscussedinsection3.2.3,andbytheradiationheatowbasedontheStefan-Boltzmannlaw[LL06].Thislawpostulatesthattheradiation heatowbetweenasurfaceanditssurroundingsisgovernedbythehighlynonlinearequationwith 43

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respecttothetemperature: Q r = AF T )]TJ/F21 10.9091 Tf 5 -8.836 Td [(T 4 1 )]TJ/F21 10.9091 Tf 10.909 0 Td [(T 4 2 ; where =5 : 67 10 )]TJ/F19 7.9701 Tf 6.586 0 Td [(8 W/m 2 -K 4 istheStefan-Boltzmannconstant; A istheeffectiveareaoftheemittingsurface; F T istheexchangeradiationfactor.Inabsenceofotherreectingsurfaces F T = e ; T 1 istheabsolutetemperaturesoftheemittingsurface; T 2 istheabsolutetemperaturesoftheambient. Analysisoftheequation13suggeststhattheheattransferbyradiationiscomparablysmall forlowabsolutetemperaturesandlowtemperaturedifferencesbetweenaradiatingsurfaceandits surroundings.ThiscaseappliestotheX-rayreectiveopticsexposedtoalowX-raybeampower density.LargecontributionoftheheatowbyradiationmaybeanticipatedfortheX-raydiffractive opticssuchasmonochromators,sincetheirtemperaturemayriseupto500 Candmore. 3.2.5Finiteelementmodel Figure34showsmeshoftheniteelementmodelusedinsimulations.Owingtothetwoplanes ofsymmetry xz and yz ,aquarter-symmetricmodelwasused,whichresultedinreductionof themodelsizebyafactoroffour.Forthermalsimulationsthemodelwasmeshedwiththethreedimensionaleight-nodedsolidelementsSOLID70[Ans].Theheattransfercoefcient h c wasevaluatedatdifferentialtemperaturebetweenthesurfaceandtheambientbysettingthekeyoption KEYOPT = 3.OverallnumberofSOLID70elementsintheniteelementmeshis60 20 10 = 1200. Three-dimensionalthermalsurfaceeffectelementsSURF152wereoverlaidatthecorresponding areasofthemodelforgeneratingheatuxduetotheX-raybeamexposureandradiationheat exchangewiththeambient. Thestructuralmechanicssimulationutilizedthesamemodeldiscretizationasthethermalanalysis.However,differenttypesofelementswereused.TheSiwaferconsistingoftheMo/Simultilayer andWlmweremodeledusingthree-dimensional8-nodedsolid-like-shellelementsSOLSH190. 44

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Figure34.Meshofthequarter-symmetricniteelementmodeloftheX-rayopticalelement Thismaterialscompositionwasincludedinthemodelasasinglelayerofelementsinthethickness direction,whichallowedtodrasticallyreducethemodelsize.Thesubstratewasmeshedwiththe three-dimensional8-nodedSOLID185elements. ConvergencestudyhasshownthattheusedniteelementmeshisadequatetoaccuratelyreproducethermaldeformationofthemodelseeAppendixB. 3.2.6Thermomechanicalsimulationresults Asmentionedaboveseesection3.2.2,theMo/SiandWlmconstituentswerenotexplicitly modeledinthermalsimulations.Duetothisreasonitismoreconvenientandappropriatetodiscuss thecorrespondingthermalresultsreferringtotheblanketSiwaferinsteadoftheX-raymirror. Steady-stateresultsofthermalsimulationsarepresentedinthetextbelow.Figures35aand35b showexpandedhalf-symmetricplotsoftemperaturedistributionandthethrough-thicknesstemperaturegradientinModel1.Asseenfromthegures,theregionofthehighesttemperatureofabout 314.5 KislocatedinthevicinityoftheX-raybeamexposedarea.Thecoldestpointsliefarthermostfromthebeamirradiatedareawiththetemperatureofabout312.2 K.Thehighesttemperature gradientarisesintheglasssubstrateattheintersectionsofitsouteredgeswiththeSisubstrate/glass interface.Overall,thetemperaturegradientintheSiismuchlowerthaninglass,whichcanbe explainedbythehigherthermalconductivityoftheformer. 45

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aTemperaturedistribution bThrough-thicknesstemperaturegradient Figure35.ThermalsimulationresultsforModel1Siwaferonglass Simulationplotoftheaverageheattransfercoefcient h c shownfortheSiwaferofModel1 ispresentedinFigure36.Thisresultexactlycorrespondstothepreviouslycalculatedvaluesof h c T seeFigure33giventhetemperaturedistributionshowninFigure35a. Figure36.CalculatedaverageheattransfercoefcientModel1 SiwaferonSisubstratecombinationModel2revealsadifferentbehaviourfromModel1 intermsofthetemperaturedistributioninthestructure.Thetemperatureisessentiallyuniform Figure37awithavalueofapproximately313.7 K,whichliesbetweentheminimumandmaximumtemperaturevaluesofModel1.Simulationresultsshowaverysmalltemperaturegradient inthestructureFigure37bwhich,asalreadystatedabove,canbeexplainedbythehighthermal conductivityofSi.Thehighesttemperaturegradientisobservedinthebeamexposedarea. SimulationresultsshowedthatModel3SiwaferoninvarandModel4Siwafer/glass/Si waferdemonstratedsimilartemperaturedistributionaswascalculatedforModel1.Theseresults arenotpresented. TheSimonochromatorisexposedtoa133timeshigherheatuxthantheX-raymirror.TemperaturedistributioninthiscaseispresentedinFigure38a.Obviously,thenaturalconvection toairfailstoeffectivelyremovesuchhighamountofheat,whichleadstothetemperatureriseto 46

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aTemperaturedistribution bThrough-thicknesstemperaturegradient Figure37.ThermalsimulationresultsforModel2SiwaferonSi about500 C.Thehottestpointslieinthevicinityofthebeamexposedareawhereas,thecoldest onesarelocatedattheside-walledges,asexpected.Thethrough-thicknesstemperaturegradient Figure38breachesmaximumvaluesofabout7600K/malongthebeamheatedarea. aTemperaturedistribution bThrough-thicknesstemperaturegradient Figure38.ThermalsimulationresultsforSimonochromatorModel5 Inthestructuralmechanicssimulations,deformationoftheX-rayopticalelementwasmodeled aslinearelastic.Itwasassumedthatthestructureisfreefromstressesandstrainsattheambient initialtemperature.Inwhatfollows,simulationresultsdemonstratedeformationbehaviourofthe opticalelementcorrespondingtothecalculatedtemperaturedistributionspresentedabove. Figures39adshowequallyscaledmagnicationfactor2000deformedshapesofthe Models1.Thedashedlinesshowthemodelcontouroutlinesintheundeformedstate.This side-by-sidecomparisonrevealsthattheX-raymirroronglasssubstrateModel1andtheX-ray mirroronSisubstrateModel2havethehighestandthesmallestgeometrydistortionsamong thefourcases.Similarly,itcanbeobserved,thattheX-raymirroroninvarsubstrateModel3 getsmoredistorted,thanthe[X-raymirror/glasssubstrate/Siwafer]structureModel4.Here, thegeometrydistortionisunderstoodasdeviationfromtheoriginalshape.SinceModel2has 47

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aModel1 bModel2 cModel3 dModel4 Figure39.Deformedshaperesultsfordifferentmodelcongurations negligibleinuenceofthethermalexpansionmismatchbetween,asdiscussedabove,theonlyway geometrydistortionsmayoccurhereisviathethrough-thicknesstemperaturegradient.However,by comparingtheinitialmodelcontouroutlinedashedlineswiththedeformedshapeFigure39bit isevidentthatthestructureexperiencesonlylinearvolumetricexpansionwithoutchangeinshape. Thus,thethrough-thicknesstemperaturegradientFigure35bistoosmalltocauseanyobservable mirrordistortions.Now,recallingthatthethrough-thicknesstemperaturegradientforModel1 Figure35bandinmuchthesamewayforModels3and4isclosetotheresultsofModel2away frommaterialscorners,itcanbeinferredthatimpactofthetemperaturegradientongeometry distortionsisnegligiblysmall.Therefore,thethermalexpansionmismatchbetweentheSiwafer andthethicksubstrateistheonlyrelevantsourceoftheopticalelementdistortionsfortheobtained temperatureelds.Thedeformedshaperesultsagreeverywellwiththisassumption.SinceSi/glass combinationhasahigherthermalexpansionmismatchthanSi/invar,theformerwouldhavehigher geometrydistortionsgiventheequaltemperaturedifference.Intheessentiallysymmetriccaseof the[X-raymirror/glass/Siwafer]structureModel4thethermalexpansionmismatchbetweenSi 48

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andglassiscompensated,whichleadstocomparativelysmallgeometrydistortionsoftheoptical element. IntheabovetexttheX-raymirrorgeometrydistortionsweredescribedandmutuallycompared basedonthequalitativevisualobservationsusingthesimulationsplots.Now,letusturnourattentiontothequantitativedescriptionofthemirrordeformationbehaviour. Geometricaldistortionsofthestructureresultincurvatureofthereectivemirrorsurface.In ordertocomputesurfacecurvatures,thecorrespondingsimulationresultswereimportedfromthe ANSYSmodeldatabasefollowedbythecomputationstepsdescribedinAppendixA. ThecontourplotsinFigures40adpresentGaussiancurvatureoftheX-rayreective surfaceforModels1,respectively.Sincethemodelisquarter-symmetric,onlyone-fourthofthe surfaceisshown.Itcanbeobserved,thatthecentralareaofthemirrorhasauniformcurvaturefor aModel1 bModel2 cModel3 dModel4 Figure40.Gaussiancurvatureofsurfacefordifferentcongurations allsimulatedmodels.Inthevicinityoftheouteredgesthecurvatureofthesurfaceisnonuniform changingitssignandvalue.Thisbehaviourisattributedtothewellknownedgesingularityeffect [Bog71],whichisconnectedtotheelasticmismatchbetweenthebondedmaterials.Itcouldbe mentioned,thattheedgesingularitycouldhavebeenbettercapturedbytheniteelementmodel 49

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eitherbydecreasingelementsizenearthematerialcornersorbyutilizationofdifferentelement types,howeveritwasnotintendedinthepresentwork. Comparingtheuniformlycurvedpartsofthesurface,theconclusionsdrawnfromthedeformed shapeplotscanbeconrmed.Particularly,thevaluesforModel2X-raymirroronSisubstrateare closetozeroindicatingthatthesurfaceremainsat.Model1X-raymirrorontheglasssubstrate hasthehighestcurvature.ThecompensatedX-raymirroronglassModel4performsbetterthan X-raymirrorontheinvarsubstrate,althoughthelattercombinationofmaterialshasalowerthermal expansionmismatch. TheactualareaofinterestofthemirrorwhichreectstheincomingX-raysisonly1mmwide seeFigure32.Curvatureofthisareaisthereforecriticalforthebeamconditioning.Toillustrate curvatureofthebeamexposedarea,thecurvesshowninFigure41wereextractedalongthesymmetryplanexz.Consideringtheuniformportionofthecurvesawayfromfreeedges,theGaussian curvatureofModel1isabout8timeshigherthanforModel3.Model4,inturn,hasthecurvature ofmorethanorderofmagnitudesmaller,thanModel3.ThecurvatureofModel2cannotbedistinguishedfromzeroontheplot.Itisnecessarytonotethattherightportionofthecurvesclosetothe edgedepictsthedeformationsingularityatthecornersofdissimilarbondedmaterials,asmentioned earlier. Figure41.ComparisonofGaussiancurvaturealongtheX-raybeamlinefordifferentmodelcongurations 50

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Thesurfacecurvaturesinlongitudinalandtransversedirectionstothebeamapplicationline areequalbecauseofnegligibleeffectoftemperaturegradientsinthestructureandtheassumed materialsisotropy. ItisusefultocomparesurfacecurvatureoftheX-raymirrormodelsusingthesurfaceunatness parameterseeequationinsectionA.ResultsofthesurfaceunatnessareillustratedinFigures42ad.ComparisonshowstheirfullcorrespondencetotheGaussiancurvaturebehaviour discussedabove.Theguresalsoshowtheintegralvalueofthesurfaceunatness U surf = Z S udS; incaseoneneedstocomparetheoveralldeviationofsurfacefromatness. aModel1 bModel2 cModel3 dModel4 Figure42.Unatnessofsurfacefordifferentmodelcongurations Figure43illustratesthesurfaceunatnesscurvesextractedinthelongitudinaldirectionofthe mirroralongthebeamapplicationline.Consideringonlytheuniformpartsofthecurvesawayfrom thesurfaceedges,theunatnessparameter u line isabout3and10timeshigherforreectivesurfaces ofModel1thanforModels3and4,respectively.UnatnessofModel2iszero.Itisimportantto 51

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notethattherightportionoftheplotstartingapproximatelyfrom0.025mdemonstratesnonuniform surfacecurvatureduetothementionedaboveedgesingularityandshouldbedisregarded. Figure43.ComparisonofsurfaceunatnessalongtheX-raybeamlinefordifferentmodelcongurations Table6summarizesresultsillustratedinFigures42and43.Thevaluesofunatnessalongthe beamline u line correspondtotheuniformpartofthecurves.Incaseswhenthewholesurfacearea isutilizedtoreecttheX-raybeam,itmaybenecessarytocomparethecorrespondingintegral parameter.Thevaluesofunatnessoverthesurfacearealsolistedinthetable.Note,thatduetothe freeedgesingularity,cautionmustbetakenwheninterpretingthesevaluesespeciallyforModel4. Table6.Unatnessfordifferentmodelcongurations ModelVariationUnatnessalongbeamline,Unatnessoversurface, u line ,1/m 2 U surf ,m 2 /m 2 Model15.88 10 )]TJ/F19 7.9701 Tf 6.587 0 Td [(3 1.55 10 )]TJ/F19 7.9701 Tf 6.586 0 Td [(6 Model25.95 10 )]TJ/F19 7.9701 Tf 6.587 0 Td [(6 1.77 10 )]TJ/F19 7.9701 Tf 6.586 0 Td [(9 Model32.05 10 )]TJ/F19 7.9701 Tf 6.587 0 Td [(3 5.51 10 )]TJ/F19 7.9701 Tf 6.586 0 Td [(7 Model45.78 10 )]TJ/F19 7.9701 Tf 6.587 0 Td [(4 2.10 10 )]TJ/F19 7.9701 Tf 6.586 0 Td [(6 52

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3.3ConclusionsforChapterThree Analysisofthermally-inducedSiwaferdeformationsshowedthatasphericalcurvaturewiththe targetradiusof10mcanbeachievedbyapplicationofaconstantthrough-thicknesstemperature gradientofabout20 Ctoa525 mthickwafer.Takingtheroomtemperatureasthelowesttemperatureofoperation,thefoundtemperaturegradientlieswithintheprescribedtemperaturerange withtheupperlimit,whichis100 C.Thisindicatesgoodpotentialforachievingthenecessary curvatureinX-rayopticsapplicationsusingthethermalloading.However,toachievetherequired parabolicsurfacecurvature,furtherniteelementstudiesinvolvingoptimizationproceduresneed tobeconducted. DeformationsofX-rayopticsbondedtothethicksubstrates,duetotheX-raybeamirradiation, wereanalyzedwiththehelpofsequentialthermalandstructuralniteelementsimulations.Results ofthermalsimulationsshowedthatalowpowerX-raybeamwithintensityof1W/cm 2 heatsupthe opticalelementunderthenaturalsurfaceconvectiontoairconditionsuptoabout15 Cwiththeresultingalmostuniformtemperaturedistributioninthestructure.Consequently,itwasshownthatthe thermalexpansionmismatchbetweentheSiwaferandthethicksubstrateisthemajorinuencing factorcausingthermaldeformationsoftheopticalelement.Resultsoftheniteelementsimulationsandthesurfacecurvatureanalysisrevealedthat,thebestperformanceintermsofthelowest geometrydistortionsoftheX-raymirrorwereachievedwhenthethermalmismatchiscompensated. ThermalanalysisofaSimonochromatorexposedtoahighenergyX-raybeamshowedthatamore efcientheatremovalthanthenaturalconvectiontoairisnecessarytoavoiditsoverheating. 53

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ChapterFour SummaryandFutureOutlook Thisworkwasorganizedintwoparts.TherstpartChapterTwodiscussedexperimentalobservationsofcrackspropagationandformationofperiodiccrackpatternsintheannealedMo/Si multilayerlmdepositedontheSisubstrate.InthesecondpartChapterThree,analysisofthermaldeformationsintheX-rayopticselementswascarriedoutusingtheniteelementmodeling. InChapterTwo,resultsofstressmeasurementsduringthermalcyclinghaveshownthathigh tensilestressesariseasaresultofh-MoSi 2 -phaseformationwhentheMo/Simultilayerisannealed andcooleddowntotheroomtemperature.Microscopicobservationshaveshownthatthesetensile stressesleadtocrackinganddelaminationofthelmfromtheSisubstrateformingtheperiodic crackpatterns.Thecrackspropagationhasbeenrecordedusingahigh-speedcameraandanalyzed. Theseobservationsrevealedthatinteractionofapropagatingcrackandthelmdelaminationcauses suchpeculiarpatterns.Basedontheseobservations,aqualitativedescriptionhasbeengiventhat explainstheconditions,underwhichthephenomenonoccurs. TheconductedexperimentalobservationsoncrackingintheannealedMo/Simultilayershas demonstratedthatthecrackpatternscaneasilybereproducedinasimpleexperiment.However, recordingofthecrackingeventisachallengingtask.Animportantopenquestionstilltobedenitivelyanswerediswhetherthechannelingcrackspropagatebehindorbeforethedelaminationfront. Thisresultcouldbeusedforfurtherinterpretationandbuildingofanumericalmodeltoaccurately capturethecrackingbehaviour.Ahigherimageresolutionofthemicroscopeandthecameracould providethenecessaryexperimentalevidence.Theanalysisoftheresultscanalsobeimprovedby conductingthetemperatureandlmstressmeasurementsduringthemicroscopyobservations. InChapterThree,theniteelementmodelingwasusedtoanalyzedeformationbehaviourof theX-rayopticselementsduetothethermalloadingcausedbytheX-rayirradiation.Here,ithas beenshownthatathrough-thicknesstemperaturegradientcouldbeaneffectivewayofcontrolling 54

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curvatureofaSiwafercoatedwiththeMo/Sireectivemultilayers.Thisinitialassessmentneedsto befurtherelaboratedtoincludetheoptimizationprocedureforndingtheoptimumthermalloading conditionsandthestructuralcompositioninordertoproducethedesiredwafersurfacecurvatures. Inaddition,ithastobeveriedifthefoundoptimumthermalloadingisrealistictobeachievedby thereadilyavailablemeanssupportedbytheexperimentalevidence. ThermomechanicalanalysisofdeformationbehaviourintheX-raymirrorsbondedtothethick substratescanbeusedtominimizegeometrydistortionsoftheopticsreectivesurface.Different modelsrepresentingvariousstructuralcompositionsandusedmaterialshavebeenanalyzed.In caseofthelowpowerX-raybeamirradiation,thestudyhasshownthattheopticsdeformationsare primarilycausedbythethermalexpansionmismatchbetweenthethicksubstrateandtheSiwafer. ThethermalsimulationresultshaveindicatedthatahighpowerdensityX-raysynchrotronsource wouldresultinexcessiveoverheatingoftheSimonochromatorunlessitiseffectivelycooled. RenementoftheseanalyzescouldbeachievedbycomparingthesimulationresultswithexperimentalmeasurementsoftheX-raymirrortemperatureanddeformations.Knowledgeoftheheat owcomingintothesystemduetotheX-raybeamexposurecouldalsocontributetotheaccuracy ofthemodel. 55

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References [AB02]J.M.AmbricoandM.RBegley.Theroleofinitialawsizeandplasticityinadjacent layersinchannelingcrackgrowthinthinlms. ThinSolidFilms ,419:144,2002. [AJB02]J.M.Ambrico,E.E.Jones,andM.RBegley.Crackinginthinmulti-layerswith nite-widthandperiodicarchitectures. Int.J.SolidsStruct. ,39:1443,2002. [Ans]AnsysInc. DocumentationforANSYS .Version11.0. [Arg59]A.S.Argon.Surfacecracksonglass.In ProceedingsoftheRoyalSocietyofLondon volume250,pages472,1959. [AZP84]N.A.Alfutov,P.A.Zinoviev,andB.G.Popov. Analysisofmultilayercompositeplates andshells .Moscow,Mashinostroenie,1984.inRussian. [BA03]M.RBegleyandJ.M.Ambrico.Channelcrackingduringthermalcyclingofthinlm multi-layers. Int.J.Fracture ,119:325,2003. [BE01]M.RBegleyandA.G.Evans.Progressivecrackingofamulti-layersystemupon thermalcycling. J.Appl.Mech. ,68:513,2001. [Beg97]H.R.Beguiristain. Thermaldistortioneffectsonbeamlineopticaldesignforhighux synchrotronradiation .PhDthesis,UniversityofCaliforniaBerkley,1997. [Beu92]J.L.Beuth.Crackingofthinbondedlmsinresidualtension. Int.J.SolidsStruct. 29:1657,1992. [BFKM00]D.H.Bilderback,A.K.Freund,G.S.Knapp,andD.M.Mills.Thehistoricaldevelopmentofcryogenicallycooledmonochromatorsforthird-generationsynchrotron radiationsources. J.SynchrotronRadiation ,7:53,2000. [BJD + 05]D.Bhattacharyya,S.N.Jha,N.C.Das,V.Verma,S.G.Markandeya,andA.K.Ghosh. Simulationoftemperaturedistributionbyniteelementanalysisondifferentcomponentsoftheexafsbeamlineatindus-iisynchrotronsource. Sadhana ,30:735, 2005. [BK96]J.L.BeuthandN.W.Klingbeil.Crackingofthinlmsbondedtoelastic-plasticsubstrates. J.Mech.Phys.Solids ,44:1411,1996. [Bla00]G.RBlackwell. Theelectronicpackaginghandbook .CRCPressLLC,2000. [BMP + 03]T.Boettger,D.C.Meyer,P.Pauer,S.Braun,M.Moss,H.Mai,andE.Beyer.Thermalstabilityofmo/simultilayerswithboroncarbideinterlayers. ThinSolidFilms 444:165,2003. 56

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Appendices 61

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AppendixA:Surfacecurvature DeformationofX-rayopticscausesgeometrychangesofthereectivesurface.Dependingongeometry,materialproperties,structuralcomposition,boundaryconditionsetc.,theresultingsurface distortionsmayexceedtheprescribedtolerancevalues.Inaddition,thenalsurfacegeometrymay takecomplicatedshapesprohibitingsimpleanalysisofthedistortedstateonthereectiveperformance. Differentialgeometryisapowerfultoolfortheanalysisofsurfacegeometryproperties.However,asoftimeofwriting,thecurrentversionoftheniteelementprogramANSYS,utilizedduring thiswork,doesnotcontainpostprocessingproceduresforcomputationofsurfaceproperties,such ascurvatures.InthecourseofthisworktheauthorhasimplementedthenecessarynumericalproceduresinaseparatePython-based[VR08]programcodeforpostprocessingoftheoutputsimulation results. ThissectionbrieyreviewsmaterialofdifferentialgeometryappliedtoanalysisoftheX-ray mirrordeformations,discussedinsection3.2.Thereviewisfollowedbyoutliningsomeaspectsof numericalimplementation.Thereafter,theprogramcodevericationiscarriedoutbycomparing numericallycomputedsurfacecurvatureswiththeanalyticalresults. Asurfacecanbedenedinparametricformwhenthesurfacecoordinatesareexpressedas functionsoftwoindependentvariables u and v [I.74]: x = u;v ;y = u;v ;z = u;v : Itisassumedthattheabovefunctionsaresingle-valued,continuousandhavecontinuousderivatives uptothesecondorderatsomedomainofchangeof u;v Besidesthecoordinatedescriptionintermsof x;y;z values,onecandescribeasurfaceby deningavariableradius-vector r u;v goingfromsomexedpoint O toapoint M atthesurface. Partialderivativesofthisradius-vectorwithrespecttotheparameters u;v givetangentvectorsto thecoordinatelines r 0 u ; r 0 v 62

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AppendixAContinued Aunitnormalvectortothesurfacecanthenbewrittenintermsofthepartialderivativesofthe radius-vector,asfollows: m = r 0 u r 0 v j r 0 u r 0 v j : Inexpandedformthevectorcrossproductis r 0 u r 0 v = @y @u @z @v )]TJ/F21 10.9091 Tf 12.45 7.38 Td [(@z @u @y @v + @z @u @x @v )]TJ/F21 10.9091 Tf 12.109 7.38 Td [(@x @u @z @v + @x @u @y @v )]TJ/F21 10.9091 Tf 12.357 7.38 Td [(@y @u @x @v : Coefcientsoftherstfundamentalformaregivenby E u;v = r 00 uu = @x @u 2 + @y @u 2 + @z @u 2 ; F u;v = r 00 uv = @x @u @x @v + @y @u @y @v + @z @u @z @v ; G u;v = r 00 uv = @x @v 2 + @y @v 2 + @z @v 2 : Forcaseswhen F =0 ,thecoordinatelines u = C 1 and v = C 2 areorthogonal. Inaddition,itcanbeshownthat j r 0 u r 0 v j = EG )]TJ/F21 10.9091 Tf 10.909 0 Td [(F 2 : Coefcientsofthesecondfundamentalformaregivenbyexpressions L = r 00 uu m ;N = r 00 vv m ;M = r 00 uv m : 63

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AppendixAContinued Or,byrecallingtheexpressionsandonecanrewritetheequationsas L = r 00 uu r 0 u r 0 v p EG )]TJ/F21 10.9091 Tf 10.909 0 Td [(F 2 ;M = r 00 uv r 0 u r 0 v p EG )]TJ/F21 10.9091 Tf 10.909 0 Td [(F 2 ; N = r 00 vv r 0 u r 0 v p EG )]TJ/F21 10.9091 Tf 10.909 0 Td [(F 2 : Inthecaseofexplicitlygivensurface z = f x;y x and y playaroleofparametersthatresults inthefollowingexpressionsfortheradius-vectorconstituentsanditsderivatives: r x;y;z ; r 0 x ; 0 ;p ; r 0 y ; 1 ;q ; r 00 xx ; 0 ;r ; r 00 xy ; 0 ;s ; r 00 yy ; 0 ;t ; where p = @f @x ;q = @f @y ;r = @ 2 f @x 2 ;t = @ 2 f @y 2 ;s = @ 2 f @x@y : Applyingformulasandonearrivesatthefollowingexpressionsforcoefcientsofthe twofundamentalforms: E =1+ p 2 ;F = pq;G =1+ q 2 ; L = r p 1+ p 2 + q 2 ;M = s p 1+ p 2 + q 2 ;N = t p 1+ p 2 + q 2 : MeanandGaussiancurvaturesofasurfacecanbeobtainedfromthefollowingexpressions: H = EN )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 FM + GL 2 EG )]TJ/F21 10.9091 Tf 10.909 0 Td [(F 2 ;K = LN )]TJ/F21 10.9091 Tf 10.909 0 Td [(M 2 EG )]TJ/F21 10.9091 Tf 10.909 0 Td [(F 2 : 64

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AppendixAContinued Principalcurvatures P min and P max arethenfoundfrom P min = H + p H 2 )]TJ/F21 10.9091 Tf 10.909 0 Td [(K;P max = H )]TJ/F26 10.9091 Tf 10.909 10.822 Td [(p H 2 )]TJ/F21 10.9091 Tf 10.91 0 Td [(K: Deviationfromatnessisausefulwayofmeasuringthelocalunatness,whichcanbepresented inthefollowingform: u = P 2 min + P 2 max : Considera2-dimensionalarrayoffunctionvalues z denedonameshgrid x;y .Thebase parameters p;q;r;t;s seeequationsareobtainedbyapplyingnumericalgradient[Oli06]to thefunction z .Forexample,therstderivativeswithrespectto x and y directionsare 0 B @ p q 1 C A = r z = 0 B @ z x z y 1 C A : Analogously,thesecondderivativesarecalculatedbyapplyingthenumericalgradienttothepreviouslyfoundvalues p and q .Afterallthebaseparameters p;q;r;t;s arefound,computationofthe fundamentalformsequationAandthesurfacepropertiesisstraightforward. Asanexampleforvericationoftheimplementednumericalprocedure,consideraparabolaof revolution z = x 2 + y 2 .Thefunctionwasdiscretizedonagridof 100 100 pointsinthefollowing interval x;y = )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 ; 2; )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 ; 2 : Figure44showscontourplotofthediscretizedfunctiononthe xy -plane.Gradationsofgraycolor representthecorrespondingvaluesof z 65

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AppendixAContinued Figure44.Contourplotof z = x 2 + y 2 on 100 100 gridpoints Applyinganalysisproceduredescribedaboveonecanndanalyticalexpressionsforthemean andGaussiancurvatures,asfollows H = 2+4 x 2 +4 y 2 +4 x 2 +4 y 2 p 1+4 x 2 +4 y 2 ; K = 4 +4 x 2 +4 y 2 +4 x 2 +4 y 2 : NumericallycalculatedvaluesofthesurfacecurvaturesarepresentedinFigures45aand45b. Asexpected,themaximumvaluesarereachedattheaxesoriginwithvaluesof2and4forthemean andtheGaussiancurvatures,respectively.Notethatthecurvaturefunctionsretaintheaxisymmetry oftheexaminedfunction z x;y Side-by-sidecomparisonoftheresultscalculatedanalyticallyusingequationsandwith thenumericaldataextractedalongthe y =0 axisisshowninFigure46.Excellentagreementcan beobserved. Itcanbeconcludedthattheimplementednumericalcodeiscapableofaccuratelycalculating thegeometricalsurfacepropertiesprovidedtheinputdatahasadequatediscretization. 66

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AppendixAContinued aMeancurvature bGaussiancurvature Figure45.Curvatureoffunction z = x 2 + y 2 on 100 100 gridpoints Figure46.Accuracyofnumericallycomputedcurvatureson 100 100 gridpointsfor y =0 67

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AppendixB:Finiteelementmodelsconvergencestudy ConvergencestudyoftheniteelementmodelrepresentingtheX-raymirrorsubjectedtothe through-thicknesstemperaturegradientwasperformedusinganerniteelementmesh,presented inFigure47.Sizeoftheelementsintheradialandthethrough-thicknessdirectionsrefertoFigure28wastaken2.0and1.5timessmallerthanforthecoarsemodel.Forthemeshrepresenting theMo/Siandtungstenlayers,theelementssizeinthicknessdirectionwasnotvaried. Figure47.FiniteelementmodelusedformeshconvergencestudyoftheX-raymirrorsubjectedto thethrough-thicknesstemperaturegradient Figures48and49showdistributionofthetotalradialstrainsandtheout-of-planedisplacements intheX-raymirrorsubjectedtothethrough-thicknesstemperaturegradientcalculatedusingthe nerniteelementmodel. Figure48.TotalradialstrainsintheX-raymirrorduetothethrough-thicknesstemperaturegradient calculatedusingthenemesh ConvergencestudywasalsoperformedforthemodelrepresentingtheX-raymirrorexposed toanX-raybeam,describedinsection3.2.5.Comparisonofthecoarseandneniteelement 68

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AppendixBContinued Figure49.Out-of-planedisplacementsoftheX-raymirrorduetothethrough-thicknesstemperature gradientcalculatedusingthenemesh meshesusedfortheconvergencestudyisshowninFigure50.Forthein-planedirection,sizeof theelementsofthenemeshFigure50bis4timessmallercomparedtothethecoarsemesh Figure50a,whichwasusedthroughouttheworkseesection3.2.Thenumberofelementsin thethicknessdirectionintheglasssubstratewasalsoincreasedbyafactorof1.25. acoarsemodel bnemodel Figure50.FiniteelementmodelmeshesusedforconvergencestudyModel4 ComparisonofthecalculatedaverageheattransfercoefcientFigures51aand51bshows thatthecoarseandthenemodelsgiveessentiallythesameresult. 69

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AppendixBContinued acoarsemodel bnemodel Figure51.CalculatedaverageheattransfercoefcientModel4 Resultsofthetemperatureandthethrough-thicknessgradientdistributionscalculatedusingthe coarseandthenemodelsarepresentedinFigures52and53.Comparisonoftheguresshowsthat thecoarseandthenemodelsproducealmostidenticalresults.Slightdifferences,however,can beobservedforthetemperaturegradientresultsattheintersectionsofthefreeedgesandmaterials interfaces,asseenfromtheFigures53aand53b. acoarsemodel bnemodel Figure52.CalculatedtemperaturedistributionModel4 Simulationplotsoftheout-of-planedisplacements u z calculatedusingthecoarseFigure54a andtheneFigure54bmodelsshow,thatthereareonlyminordifferencesbetweenthem.Finally,itcanalsobeconcluded,thattheniteelementmeshofthecoarsemodelisfullysufcientto reproducethethermaldeformationsoftheX-raymirror. 70

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AppendixBContinued acoarsemodel bnemodel Figure53.Calculatedthrough-thicknesstemperaturegradientModel4 acoarsemodel bnemodel Figure54.Calculatedout-of-planedisplacements u z Model4 71


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TJ145 (Online)
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Kravchenko, Grygoriy A.
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Crack patterns in thin films and X-ray optics thermal deformations
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by Grygoriy A. Kravchenko.
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[Tampa, Fla] :
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2008.
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Title from PDF of title page.
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Thesis (M.S.M.E.)--University of South Florida, 2008.
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Includes bibliographical references.
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Text (Electronic thesis) in PDF format.
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ABSTRACT: Thin films and multilayers are widely used in many applications, ranging from X-ray optics to microelectronic devices. In service, the X-ray optics elements are exposed to the X-ray beam, which heats up the structure resulting in the thermal deformations, and consequently in distortions of the reflective surface. In addition, the excessive heating may activate interdiffusion in the multilayers coatings and result in degradation of their reflective performance and even film cracking. Therefore, analysis of the thermally-induced deformations and stresses in the X-ray optical elements is important. The presented work is organized in two major parts. The first part examines formation of the peculiar periodic crack patterns observed in the thermally loaded Mo/Si multilayers. Film stress evolution during thermal cycling of the multilayers on Si substrate is analyzed.Results of the high-speed microscopic observations of crack propagation in the annealed Mo/Si multilayers are presented. The observations provide experimental evidence of the mechanism underlying formation of the periodic crack patterns. In the second part, thermal deformations and the resulting surface curvature changes in the X-ray optics elements are analyzed. Finite element modeling is used to assess the potential to thermally control curvature in the X-ray mirrors consisting of the Mo/Si multilayers on a Si substrate. Influence of heating due to the X-ray beam irradiation on thermal deformations in the X-ray mirror bonded to a thick substrate is analyzed in-depth. The detailed consideration includes analysis of the thermal and structural mechanics simulations. Based on simulations of different model configurations, influence of structural composition on thermal distortions of the optics elements is addressed.Results of this analysis can be used to mitigate distortions of the X-ray optics caused by the X-ray beam and provide basis for further studies of thermally controlling surface curvature in the optical elements.
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Advisor: Alex A. Volinsky, Ph.D.
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Residual stress
Fracture
Cracks interaction
Mo/Si multilayers
Thermal distortions
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Dissertations, Academic
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