勾形磁场下提拉法生产单晶硅的数值模拟

©1994-2010ChinaAcademicJournalElectronicPublishingHouse.Allrightsreserved.(1.,100022;2.,100088):,,,,,,:;;:O78:A:10002985X(2004)0220217206NumericalSimulationinCzochralskiSiMeltunderaCuspMagneticFieldYUHui2ping1,SUIYun2kang1,ZHANGFeng2yi2,WANGXue2feng2(1.CollegeofMechanicalEngineering&ApplyingElectronicTechnique,BeijingUniversityofTechnology,Beijing100022,China;2.GeneralResearchInstituteforNonferrousMetals,Beijing100088,China)(Received10September2003,accepted25December2003)Abstract:Theequationsusedtocalculatethecuspmagneticfieldstrengthinthecrystalsilicongrowthweregiven.NumericalcalculationwascarriedouttoinvestigatethemeltconvectionandoxygentransportationinaCzochralskigrowthsystemwithandwithoutacuspmagneticfield.ThecuspmagneticfiledstrengthandLorentzforceataverticalcrosssectionwerecalculatedandthestreamfunction,velocityataverticalcrosssection,oxygenconcentrationwerealsocalculated.Numericalresultshowsthatthemeltflowisdramaticallysuppressedandtheoxygenconcentrationatthegrowthinterfacedecreaseswiththeapplicationofthecuspmagneticfield.Themechanismofthereductionsoftheconvectionandtheconcentrationofoxygenwasanalyzed.Keywords:Czochralskimethod;cuspmagneticfield;crystalsilicon1,,,,,,,,,[124],,OzoeIwamoto[5],Series[6],,,:2003209210;:2003212225:(JQ01005200372):(19732),,,E2mail:[e-mail]©1994-2010ChinaAcademicJournalElectronicPublishingHouse.Allrightsreserved.[7],Organ[8],,,,,,Series[6]Hirata[9],,,,,,,21,Rc,H,c,,Rx(RxRc),x,,,Hc,Rcl:;;Tx;Tc;,TxTc;;;,[9];[10];Bousinesq,3r3=r/Rc,x3=x/Rc,u3=Rcu/,v3=Rcv/,w3=Rcw/,p3=R2c(p+Rcgx)/2,T3=(T-Tx)/(Tc-Tx),C3=C/c0,b3=b/b0,j3=Rcj/(b0),3=/(b0),Pr=v/a,Sc=v/D,Gr=2g(Tc-Tm)R3c/2,Ha=/b0Rc,c0=3.991023exp(-2.0104/Tm)atoms/cm3[1],Gr;Pr;Sc;Ha;r3x3;u3v3w3;T3;C3;p3;b3;j3;3;b02.1(1),1,3,3,s355x3(u33)+1r355r3(r3v33)=55x33535x3+1r355r3r33535r3+s3(1)1Table1Exchangecoefficientsandsourcesinthedimensionlessgoverningequations33s3Continuityequation100Momentumequationsu31-5p35x3+GrT3-Ha2j3b3rv31-5p35r3-v3r32+w32r3+Ha2j3b3xw31-w3r32-v3w3r3+Ha2(j3xb3r-j3rb3x)EnergyequationT31/Pr0OxygenequationC31/Sc0:55x3(j3x)+1r355r3(r3j3r)=0(2)81233©1994-2010ChinaAcademicJournalElectronicPublishingHouse.Allrightsreserved.=(j3xj3rj3)T=-535x3-w3b3r,-535r3+w3b3x,u3b3r-v3b3xT(3)2.22.2.1,:u3=v3=0,w3=r3Rex;u3=v3=0,w3=r3Rec,,u3=5v35x3=5w35x3=05u35r3=v3=w3=0,Rex=R2cxv,Rec=R2ccv2.2.2,5T35x3=0;Tc,,T3=1;Tm,T3=0;T=Tm+(Tc-Tm)r-RxRc-Rx,T3=r3-1-,=RxRc;5T35r3=02.2.3,,,0;,5C35x3=0;5C35r3=0;C3=exp(2.0104(1/Tm-1/Tc)),,C3=exp(2.0104(1/Tm-1/Tc))1.031,I,,,1Fig.1Sketchofthesiliconmeltunderacuspmagneticfield2Fig.2Calculationofthemagneticfieldstrengthofanode1,2,Rcl,IBiot2Savart,21:9122:©1994-2010ChinaAcademicJournalElectronicPublishingHouse.Allrightsreserved.=I4dI2r12r312(4)B:1;I:;:;r12:12,dI2:,r12r1211XOY,r1O1XOY,Z11Z,O1OX,O2OX2(Rclcos,Rclsin,0),1(r1cos,r1sin,z1),dl2(-Rclsind,Rclcosd,0),r12=(r1cos-Rclcos,r1sin-Rclsin,z1)dl2r12=ijk-RclsindRclcosd0r1cos-Rclcosr1sin-Rclsinz1(5):Bx1=I420Rclz1cosr312d,By1=I420Rclz1sinr312d,Bz1=I420Rcl[Rcl-r1cos(-)r312d(6)Bx1,By1,r1,1(1r1),:Br1=cosBx1+sinBy1,B1=-sinBx1+cosBy1(7)(6)(7):Br1=I420Rclz1cos(-)r312d,B1=0,Bz1=I420Rcl[Rcl-r1cos(-)]r312d(8)r12=[r21+R2cl-2r1Rclcos(-)+z21]1/21,1,2Hc,ZZ1,ZZ2=2Hc-ZZ1,(8),,1:Br=I420RclZZ1cos(-)r21+R2cl-2r1Rclcos(-)+ZZ21]3/2+Rcl(2Hc-ZZ1)cos(-)[r21+R2cl-2r1Rclcos(-)+(2Hc-ZZ1)2]3/2dBz=I420Rcl[Rcl-r1cos(-)]r21+R2cl-2r1Rclcos(-)+ZZ21]3/2-Rcl[Rcl-r1cos(-)][r21+R2cl-2r1Rclcos(-)+(2Hc-ZZ1)2]3/2d(9),,,,,,SIMPLE[10]4,H=0.03m,Rc=0.03m,Rx=0.012m,c=-7r/min,x=24r/min,=2.610-5m2s-1,=1.410-4K-1,Cp=1000Jkg-1K-1,=2.4103kgm-3,=710-4kgm-1s-1,=2.910-7m2s-1,Tm=1683K,k=64Wm-1K-1,D=510-8m2s-1,=410-7H/m,=1.29106-1m-1,Pr=0.011,Gr=107,Rex=7800,Rec=-2250,Sc02233©1994-2010ChinaAcademicJournalElectronicPublishingHouse.Allrightsreserved.=5.8,Ha=50,b0=388Oe31.253.(a);(b)Fig.3Contoursofstreamfunction(a)without(b)withmagneticfield4.(a);(b)Fig.4Velocityataverticalcrosssection(a)without(b)withmagneticfield5Fig.5Oxygenconcentrationatthegrowthinterface6(a)(b)Fig.6Numericalresultsataverticalcrosssection(a)magneticfieldstrength;(b)Lorentzforce3(a)3(a),,,,,,,,3(a)/,,,,,,,,3(b),13.01,-0.27783.28,-12.56,,,4(a)4(b),,,,5,,,,,,,()();,,1222:©1994-2010ChinaAcademicJournalElectronicPublishingHouse.Allrightsreserved.(b),,,,,6(a),/,/,,(6(a))(6(b)),,,,4,,,,5,:(1)Gr107,;(2),,;(3),,[1]HirataH.Three2dimensionalNumericalAnalysisofEffectsofCuspMagneticFieldontheFlows,OxygenTransportandHeatTranferinaCzochralskiSiliconMelt[J].J.CrystalGrowth,1992,125,1812207.[2]JosephMortonL,NancyMa,DavidF,BlissGeorgeBryantG.DopantSegregationduringLiquid2encapsulatedCzochralskiCrystalGrowthinaSteadyAxialMagneticField[J].J.CrystalGrowth,2002,242:471.[3]Jong2SeonKim,Tai2yongLee.Time2dependentSimulationoftheGrowthofLargeSiliconCrystalsbytheCzochralskiTechniqueUsingaTurbulentModelforMeltConvection[J].J.CrystalGrowth,1997,180:450.[4]