在稳态动态磁场作用下大尺寸晶体直拉生长的硅熔体流动模拟使解读.docx
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在稳态动态磁场作用下大尺寸晶体直拉生长的硅熔体流动模拟使解读.docx
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在稳态动态磁场作用下大尺寸晶体直拉生长的硅熔体流动模拟使解读
JournalofCrystalGrowth230(200192–99
NumericalinvestigationofsiliconmeltflowinlargediameterCZ-crystalgrowthundertheinfluenceofsteadyanddynamic
magneticfields
J.Virbulisa,*,Th.Wetzelb,A.Muiznieksb,B.Hannaa,E.Dornbergera,
E.Tomziga,A.M.uhlbauerb
W.v.Ammona
a
WackerSiltronicAG,P.O.Box1140,Burghausen,D-84479,Germany
b
InstituteforElectroheat,UniversityofHannover,Wilhelm-Buschstr.4,D-30167,Hannover,Germany
Abstract
TurbulentsiliconmeltflowsarestudiedinlargediameterCzochralskicruciblesundertheinfluenceofalternating,steadyandcombinedmagneticfields.Theinvestigationsarebasedontheexperimentallyverifiedtwo-dimensionalaxisymmetricmathematicalmodels.Theinfluenceofsteady,alternatingandcombinedmagneticfieldsontheflowpatternandtemperaturefieldisinvestigated.Globalheattransferandmeltflowcalculationsarecoupledandtheinfluenceofmeltconvectionontheinterfaceshapeisstudiedandcomparedwithexperimentaldata.r2001ElsevierScienceB.V.Allrightsreserved.
PACS:
47.27.i;47.65.+a
Keywords:
A1.Fluidflows;A1.Heattransfer;A1.Magneticfields;A2.Czochralskimethod;B2.Semiconductingsilicon
1.Introduction
Theconversiontolargesilicon(Sisinglecrystalsof300mmdiameterrequireslargerbatchsizeswhichgeneratesturbulentmeltconvectionwithGrashofnumberupto1010.Severaleffortsofthecrystalgrowthindustryarededicatedtocontrolthemeltflowandthetemperaturedis-tribution.Inparticular,itsfocusisonthecorrectpredictionoftheinterfaceshapeandtherelatedpointdefectdistributioninthecrystal,ofthe
oxygentransport,ofthecrucibleoverheatingandofthedislocationfreegrowth.Besidesconven-tionalmeans,electromagneticsteady(DCanddynamic(ACfieldsoffernewpossibilitiestomeetthecontinuouslyincreasingdemandsforcrystalquality,yieldimprovementandcostreduction.Anumericalsimulationhelpsininvestigatingthewidefieldofpossibleprocessconditions,tosavealotofexperimentalcostsforlargediametercrystalgrowthandtoreducethetimetomarket.
Globalheattransfercalculations[1–3]providegoodagreementwithtemperaturemeasurementsincrystalandinsulation[4]andareestablishedasastandardtoolforCzochralski(CZprocessdevelopment.
*Correspondingauthor.Tel.:
+49-8677-83-4227;fax:
+49-8677-83-7303.
E-mailaddress:
janis.virbulis@(J.Virbulis.0022-0248/01/$-seefrontmatterr2001ElsevierScienceB.V.Allrightsreserved.PII:
S0022-0248(0101321-5
SeveralnumericalstudiesaredevotedtomeltflowinCZcrucibles.Mostofthemaremadefortwo-dimensional(2Didealizedcylindricalgeo-metrieswithsmallcruciblesizesandsimplifiedthermalboundaryconditions.Insomecases,mag-neticfieldeffectshavebeenstudied,e.g.Ref.[5].Three-dimensional(3Ddirectnumericalsimula-tionwithoutturbulencemodelsinidealizedgeometriesshownon-symmetrictime-dependentflowbehaviorinsmall[6]andmedium(Gr=108[7]crucibles.Globalheattransfercoupledwith2DmeltflowinlargecruciblesusinglowRenumberk2emodelshasbeencalculated[8,9]withouttheinfluenceofmagneticfields.
Inthiswork,wepresentnumericalinvestiga-tionsofturbulentSi-meltflowsinlargediameterCZcruciblesundertheinfluenceofAC,DCandcombinedmagneticfields.Theinvestigationsarebasedontheexperimentallyverifiedsystemofaxi-symmetricmathematicalmodels,wherethemeltconvectionandtheglobalheattransferarecoupled.Theinfluenceofmeltconvectionontheinterfaceshapeandthermalgradientsinthecrystalisstudiedandcomparedwithexperimentaldata.2.Numericalmodels
2.1.Couplingbetweenglobalheattransfer
andmeltconvection
GlobalheattransferinthewholeaxisymmetricCZsystemissimulatedwiththefiniteelementcodeFEMAG,describedindetailinRef.[1].Heattransferismodeledusingquadraticfiniteelementsinsolidsandviewfactorsinradiativeenclosures.Conductiveandconvectiveheattransferinargongasisneglectedduetolowpressure.Thegrowthprocessisassumedtobequasi-steadyandthereleaseoflatentheatoffusionatthecrystallizationinterfaceisproportionaltotheimposedgrowthrate.
MeltconvectionissimulatedwiththeprogrampackageCFD-ACE[10].Thecouplingbetweenglobalthermalandmeltflowsimulationsisrealizedbyexchangeoftheheatfluxesatthemeltboundaries.ThefirstglobalsimulationiscarriedoutusingtheeffectiveheattransportcoefficientincludingconductiveandconvectivecontributionsinthemeltwithFEMAG.ThecalculatedheatfluxesalongthecrucibleandmeltfreesurfacesareprovidedasthermalboundaryconditionsforthemeltflowsimulationwithCFD-ACE.MeltflowsimulationusestheinterfaceshapecalculatedinFEMAG.Afterflowsimulation,temperaturesonmeltboundariesinCFD-ACEandFEMAGarecomparedandnormalheatfluxesonthemeltboundariesinFEMAGareadjustedtoreachthesamecrucibletemperatureasinCFD-ACE.Atthecrystallizationinterface,heatfluxesfromthemeltsideinFEMAGareadjustedtogetthesamedistributionasinCFD-ACE.Newheatfluxesalongthecrucibleandmeltfreesurfacesaswellasthenewinterfaceshapeareusedtoperformthenextmeltflowcalculation.Typically,fiveglobaliterationsarerequiredtoreachthefinalsolution.2.2.Meltconvectionanddirectdampingofturbulencebyasteadymagneticfield
Buoyancy,centrifugal,MarangoniandLorentzforcesareconsideredintheconvectionsimulationwithCFD-ACE.AlowReynoldsnumberk2eturbulencemodelisusedbecauseofthinvelocityandconcentrationboundarylayers.ACandDCmagneticfielddistributionsarecalculatedwithFZHDEM[11,12].SincetheeffectsofACandDCfieldsaredifferent,twoseparateLorentzforcetermsareused.ForACfields,meanvaluesoftheLorentzforceareapplied,whicharecalculatedasaproductofalternatingfieldandinthemeltinducedalternatingcurrent.Theseforcesdependonthefrequencyandonthegeometryofcoilsandmelt,butareindependentofthemeltvelocities.ForDCfields,Lorentzforcedependsonflowvelocity.Anadditionalequationofelectricpoten-tial,whichiscausedbyinteractionofsteadymagneticfieldandmeltvelocity,issolvedtogetherwiththehydrodynamicequations.2Daxisym-metricmodelsareusedforthemeltflowinaCZcruciblebecausetheyareagoodcompromisebetweenefficiencyandaccuracyforengineeringapplications.
Inthestandardmodel,theLorentzforcesuppressesthemeanvelocityand,hencelessturbu-lenceisgenerated.However,velocityfluctuations,
J.Virbulisetal./JournalofCrystalGrowth230(200192–9993
modeledbyusingtheturbulentenergyk,arealsodirectlydampedbythemagneticfield.Hence,weimprovethestandardmodelbyaddinganegativetermforturbulencegenerationPBtothekequation
PB¼@cBkkB2;ð1Þwherekistheelectricconductivity.AusefulandprecisevalueofcBhastobeverifiedbycompar-isonstoexperimentaldata.ModelsofmeltconvectionanddirectdampingofturbulencebysteadymagneticfieldarereportedinmoredetailinRef.[12].
2.3.Additionalturbulencegeneration
Oneofthemainproblemsfor2Daxisymmetricmodelistheartificialsymmetryline.Belowtheinterface,themeltflowisdirectedtowardsthecenterandthendownwardsalongthesymmetryaxis.Duetothemassconservation,theflowvelocityneartheaxisisveryhigh(seeFig.3a.Inreality,thesymmetrylinedoesnotexistandthepositionofthisjetcanfluctuateabouttheaxis,generatinganadditionalmixinginthisregion.Inmetallurgicalapplications,similarproblemsaresolvedbyapplyingadditionalturbulenceintheareasbetweentwovorticeswherehighvelocitiesexist[13].WeproposeamodeltakingintoaccountthiseffectinCZcrucibles.Anadditionallowfrequencyturbulencewillbegeneratedinthepresenceofajetonthecenterlineaccordingtotheequation
kLF¼u2=2:
ð2ÞTherefore,weintroduceanadditionalgenerationterminkequationproportionaltothesquareofvelocityinameridionalplane
PLF¼cLFðu2rþu2
z
Þ:
ð3Þ
ThevalueoftheconstantcLFcanbefoundbycomparingthesimulationresultswithexperimen-taldata.ThismodelisverifiedanddiscussedinSection3.2.3.Verificationofmodels
3.1.Couplingbetweenglobalheattransfer
andmeltconvection
Fig.1showsthetemperaturesandFig.2showstheheatfluxesalongthecrucible,freemeltsurfaceandcrystallizationinterface,calculatedfora200mmcrystalanda24incrucible.Thenumericalgridwith12,300cells,usedforthemeltflowcalcu-lations,isfineenoughtoreachagridindependentsolution.Thesolidlinesaretheresultswithoutconvection,theboldsolidlinesarethefinalresultswithconvectionandthedashedlinesresultafterthefirstconvectioncalculation.Thecrucibletemperatureisstronglyinfluencedby
convection,Fig.2.Heatfluxatthecruciblewall,thefreemeltsurfaceandthecrystallizationinterface.Solidline:
withoutconvection,boldsolidline:
withconvection,dashedline:
afterfirstcoupling
iteration.
Fig.1.Temperatureatthecruciblewallandthefreemeltsurface.Solidline:
withoutconvection,boldsolidline:
withconvection,dashedline:
afterfirstcouplingiteration.
J.Virbulisetal./JournalofCrystalGrowth230(200192–9994
ofcourse,dependingontheinitialguessoftheeffectiveheattransfercoefficientinthemelt,usedinglobalsimulation.Buttheheatfluxinourtestcaseischangedatthecruciblebylessthan10%andatthefreesurfacebylessthan5%,ifconvectionisconsidered.Theheatfluxatthecrystallizationinterfaceischangedsignificantly,and,asaresult,theinterfaceshapeischanged,aswell.
Afterthefirstconvectionsimulation,thetem-peraturediffersfromthefinalsolutionbylessthan2K.T
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- 关 键 词:
- 稳态 动态 磁场 作用 尺寸 晶体 生长 硅熔体 流动 模拟 解读