全球半导体晶体生长建模著名商业软件FEMAG-横向磁解读

1、Cz Si growth under a TMFLIG.k ll：mxrtAasAMt全球半导体晶体生长建模著名商业软件FEMAGGlobal Simulation of Czochralski SiliconGrowth under the Effectof a Transverse Magnetic FieldCz Si growth under a TMF (confd)MHD boundary layers: Hartmann layersThe Hartmann layers develop along the surfaces where the normalcomponent o

2、f the magnetic field is non-negligible An order of magnitude of the thickness of a Hartmann layer is = LHa1(L = Rs or RJLIG.k ll：mxrtAasAMtCz Si growth under a TMFTypically SH= 0.05 - 0.08 mm in industrial furnaces.yGg JugvaruaMMtCz Si growth under a TMF (confd)FEMAGTransverse magnetic fields: FLET

3、methodHypothesis: B =BQCI= BQ(COS0er- sin OeQObjective: global, quasi-steady or lime-dependentcalculations at a reasonable costMethod: Fourier decomposition of all fields(velocity, pressure, temperature)Principal issue: which and how many modes ?0113Fourier Limited Expansion Technique (FLET)The fiel

4、d variables are expanded as Fourier series in the azimuthal (0)direction:M0= 9+l.os(2i+l)0) + 0+I)”sin(2i+1)0) i=0MT = T +为(T(2,)rcos(2i6) + 72 sin(2i)i=lA/v = v。+ 工(V(N)CCOS(2诃)+v)sir)(2i0)MP = p + Z (cos(2i0) + pQJ”sin(2i0)t=lf =+ 力(f(2l)rcos(2ti?) + f)sin(2M).j=lFEMAGSaOMIJ IFEMAGGgJUUvaruaMMtCz

5、Si growth under a TMF (confd)yGJUUmruatAMtCz Si growth under a TMF (confd)Fourier Limited Expansion Technique (FLET)while the different mode coefficients:0+lk+l)sr(2i)T(2iUv(2i)cv(20p(2r)e卩”are 2D Finite Element functions of the meridional coordinates (n z) andwhile the number of modes M is as small

6、 as possible without loss ofaccuracy (spectral convergence).This results in a system whose size is that of the 2D system multiplied by the number of Fourier modes considered and . hencein a dramatic system size reduction.EMAGS嗓0 202 II、fj；Cz Si growth under a TMF (confd)Radiation transferCoupling be

7、tween 3D and 2D axisymmetric heat transfer acrossa radiative enclosureMain modeling hypothesis: lhe viewed and hidden parts arc calculated as axisymmetric or. equivalently, each surface of the enclosure is viewed as axisymmetric fromthe other surfacesThis hypothesis is satisfactory because: generall

8、y 3D components arc rotating with respect to the 2D environment 3D component mostly view 2D components because of the presence of heatshieldsyGJUCiRxructtAMCz Si growth under a TMF (confd)Flow and global heat transfer in a silicon Cz puller under theeffect of a TMF (quasi-steadysimulation)Growth of

9、a 300 mm diameter Sicrystal under the effect of a 0.5 TTMF.Top: top view of the velocitymagnitude and streamlines.Bottom: velocity field magnitude andcross section showing a sharpHartmann layer along the melt-crucibleinterface.Cz Si growth under a TMF (confd)yGJUC . iRxructtAMIRXHACtMMtCz Si growth

10、under a TMF (confd)/s.FEMAG只;m Cz Si growth under a TMF (confd)Q Soft _Flow and global heat transfer in a silicon Cz pullerunder the effect of a TMFGrowth of a 300 mmdiameter Si crystalunder the effect of a0.5 T TMF.In a typical TMFconfigurationextremely thinHartmann layersof 50-80 mindevelopFEMASl3

11、IRXHACtMMtCz Si growth under a TMF (confd)Cz Si growth under a TMF (confd)Growth of a 300 mm diameter Si crystal under the effect of a 0.5 TTMF.Hartmann boundary layers along the melt-cryrstal and rnelt cruciblc interfaces andassociated boundary layer meshes.Strong Hartmann backflows develop Cz Si g

12、rowth under a TMF(confd)Czochralski growth of a silicon crystalunder a 500 mT horizontal magnetic fieldGlobal simulation the growth of 300 mm and 400 mm crystals$JUUvaructMMzruauMCz Si growth under a TMF (confd) k.ti：Left: melt surfaceRight: meridional cross-sections parallel and perpendicular to th

13、e magnetic fieldBottom:temperature fieldGrowth of a 300 mm crystal under a 500 mT TMF k.ti：zruauMCz Si growth under a TMF (confd)Left: melt surfaceRight: meridional cross-sections parallel and perpendicular to the magnetic fieldBottom:temperature fieldTop:velocityfieldOI3Top:velocityfieldzruauMCz Si

14、 growth under a TMF (confd)Growth of a 400 mm crystal under a 500 mT TMFCz Si growth under a TMF (confd)Czochralski growth of a silicon crystalunder a 3000 or 5000 Ghorizontal magnetic fieldGlobal simulation of the growth of 300 mm and 400 mm crystalsCz Si growth under a TMF (confd)Geometry Issued f

15、rom following publications : K Takano & al Mat Sc AFng . 873 (?000) 30-35 Y. Shicaisho& al . JCG 229 (2001) 17-21Parameters tor the reference simulation :Crystal Diameter = 0 4mCruobte Diameter 0.9 (mJCrystal Rotation ra:e -10 (RPMCruciate Rotation rate “5 (RPM)Total charge = 400 kgPull Rate

16、 0.45 (mm. mmHorizontal Magnetc Field 3000 Gauss (in x direction)Geometry and operatingconditionsGraphiteSiliconInsulatorQuartz.52 m78申y G $ MVCz Si growth under a TMF (confd)Gas flow: noDescription of the simulationsIDCrucible CrystalRotation diameter rateType of Magnetic magneticstrength fieldTota

17、lVerticalcharge position of theTMFCrystal Ar gas lengthflowmmGaussPeg(mJmm15400horizontal3000400/科500no gas25300horizontal3000400/+/ 500no gas35400horizontal5000400/*500no gas45400ovoid+/ 3000400K1-1.2+/-500no gas55400ovoidW-3000400h2=1.37+/ 500no gas65400horizo nlal3000400/+/ 400no gas 75400horizon

18、tal3000400/500with gas85400400/W-500no gas90.2400horizontal3000400/500no gas y G $ MVCz Si growth under a TMF (confd)y G $ MVCz Si growth under a TMF (confd)Gas flow: noTemperature field (K): top left: parallel cross section; bottom left:perpendicular cross section; top right: top viewCasel17101720T

19、emporature )6901700Cry stal diameter: 400 mm Pullingrate: 0.45 mm/min Cniciblerotation rate: 5 RPM Magneticfield type: horizontal Magneticfield strength: 3(X)0 GraruaMMiCz Si growth under a TMF (confd)Velocity field (m.s *): top left: parallel cross section; bottom left:perpendicular cross sectionFE

20、MAG：SoftIK il：FEMAGIK il：Magnetic field type: horizontalMagnetic field strength: 3000 GGas How: noCz Si growth under a TMF (confd)Cry stal diameter: 400 mmPulling rate: 0.45 mni/minCnicible rotation rate: 5 RPM9.X&4.23IOOe*24ConcentrationO104-242s24raruaMMiCz Si growth under a TMF (confd)Oxygen

21、concentration (m ): top left: parallel cross section in melt;bottom left: perpendicular cross section in melt; lop right: crystalCry stal diameter: 400 mm Pulling rate:0.45 mni/min Crucible rotation rate: 5 RPM Magnetic field type: horizontal Magnetic field strength:3000 G Gas How: noCz Si growth un

22、der a TMF (confd)Gas flow: noGas flow: noCz Si growth under a TMF (confd)Temperature field (K): top left: parallel cross section;bottom left: perpendicular cross sectionMagnetic field type: horizontalMagnetic field strength: 3000 GTemperature field (K): top left: parallel cross section; bottom left:

23、perpendicular cross sectionCry stal diameter: 300 mmPulling rare: 0.45 mni/minCnicible rotation rate: 5 RPMMagnetic field type: horizontalMagnetic field strength: 3000 G、K tU0113Cry stal diameter: 40() mmPulling rare: 0.45 mni/minCnicible rotation rate: 5 RPM0113MFIACCMMCz Si growth under a TMF (con

24、fd)Gas flow: noMFIACCMMCz Si growth under a TMF (confd)Gas flow: noVelocity field (m.s *): top left: parallel cross section; bottom left:perpendicular cross sectionCz Si growth under a TMF (confd)FEMAG Soft. K fl：Magnetic field type: horizontalMagnetic field strength: 3000 GGas flow: noFEMAG Soft00.

25、004Cry stal diameter: 300 mmPulling rare: 0.45 mni/minCnicible rotation rate: 5 RPMMFIACCMMCz Si growth under a TMF (confd)Gas flow: noMagnetic field type: horizontalMagnetic field strength: 3000 GVelocity field (m.s *): top left: parallel cross section; bottom left:perpendicular cross sectionCry st

26、al diameter: 400 mmPulling rare: 0.45 mni/minCnicible rotation rate: 5 RPMCz Si growth under a TMF (confd)Oxygen concentration (m3): top left: parallel cross section in melt;bottom left: perpendicular cross section in melt; top right: crystalConcentrationO8e+23124l,00e249.00et23Cry stal diameter: 30

27、0 mm Pullingrate: 0.45 mm/min Cruciblerotation rate: 5 RPM Magneticfield type: horizontal Magneticfield strength: 3000 G Gas flow:noConcentranonO 5e+23Cz Si growth under a TMF (confd)Cz Si growth under a TMF (confd)Oxygen concentration (m3): top left: parallel cross section in melt;bottom left: perp

28、endicular cross section in melt; top right: crystalConcentrationOle24Case 11.00e*249.00&23Cry stal dianwtcr: 400 mm Pullingrate: 0.45 mm/min Cruciblerotation rate: 5 RPM Magneticfield type: horizontal Magneticfield strength: 3000 G Gas Flow:noQgZJ mg $223/S.FEMAG怎爲Cz Si growth under a TMF (confd

29、)BBBBSoft _Temperature field (K): top left: parallel cross section; bottom left:perpendicular cross section; top right: top viewTomperoturo16901700Cry stal dianwter: 4(X) mm Pulling rate:0.45 mrn/min Cnicible rotation rate: 5RPM Magnetic field type: horizontalMagnetic field strength: 5000 G Gasflow:

30、 no/S.FEMAG怎爲Cz Si growth under a TMF (confd)BBBBSoft _Temperature field (K): top left: parallel cross section;bottom left: perpendicular cross section; top right: top viewTemperatureCase 316801720Temperature170017201700 ,1710Cry stal dianwtcr: 4(X) mm Pulling rate:0.45 mrn/min Cnicible rotation rat

31、e: 5RPM Magnetic field type: horizontalMagnetic field strength: 3000 G Gasflow: noy G 1 KIVCz Si growth under a TMF (confd)Magnetic field type: horizontalMagnetic Held strength: 3000 GGas flow: noVelocity field (m.s *): top left: parallel cross section; bottom left:perpendicular cross sectionCz Si g

32、rowth under a TMF (confd)Velocity field (m.s *): top left: parallel cross section; bottom left:perpendicular cross sectionCry staldianwtcr: 400 mmPulling rare: 0.45 mni/minCrucible rotation rate: 5 RPMMagnetic field type: horizontalMagnetic Held strength: 5000 (；Gas flow: noMogrwtucloCry staldianwtc

33、r: 400 mmPulling rare: 0.45 mni/minCrucible rotation rate: 5 RPMCry stal dianwter: 400 mmPulling rate: 0.45 mm/minCrucible rotation rate: 5 RPMMagnetic field type: horizontalMagnetic field strength: 5000 GGas flow: no/S.FEMAG土瓷Cz Si growth under a TMF (confd)yassBSoft _Cry stal dianwter: 400 mmPulli

34、ng rate: 0.45 mm/minCrucible rotation rate: 5 RPMMagnetic field type: horizontalMagnetic field strength: 5000 GGas flow: noOxygen concentration (m3): top left: parallel cross section in melt;bottom left: perpendicular cross section in melt; top right: crystalCry stal dianwtcr: 400 mm Pulling rate:0.

35、45 mm/min Crucible rotation rate: 5RPM Magnetic field type: horizontalMagnetic field strength: 5000 G Gas flow:noCz Si growth under a TMF (confd)Oxygen concentration (m3): top left: parallel cross section in melt;bottom left: perpendicular cross section in melt; top right: crystalCase 3Case 3Concen

36、tratlonO1G*242GI24ConcentKrtonO:nQ oenConcen tratlonO1G*242GI24iW G K ft：/:SFEMAG总二;Cz Si growth under a TMF (confd)Q. Soft _Ovoid magnetic field9, top left: sketch of the magnetic field: bottomleft: side view: right: top viewConcentrationO2+24Cz Si growth under a TMF (confd)Oxygen concentration (m3

37、): top left: parallel cross section in melt;bottom left: perpendicular cross section in melt; top right: crystal9.0Ge*23100e-24Cry stal diameter: 400 mm Pullingrare: 0.45 mni/min Cruciblerotation rate: 5 RPM Magneticfield type: horizontal Magneticfield strength: 3000 G Gas flow:no氐、KII：Cz Si growth

38、under a TMF (confd)R -1.0 m, Z-1.2 m, Radius 0.3 mCUrrent，nt6nSity2 06+06A】Magnetic MagNtudeCz Si growth under a TMF (confd)Ovoid magnetic field9, top left: sketch of the magnetic field; bottom left:side view; right: top viewCz Si growth under a TMF (confd)Temperature field (K): top left: parallel c

39、ross section; bottom left:perpendicular cross section; top right: top viewR -1.0 m, Z-1.2 (m, Radius 0.3 mJyGiKti：Cz Si growth under a TMF (confd)Cry stal diameter: 40() mm Pulling rate: 0.45mni/min Crucible rotation rate: 5 RPMMagnetic field type: ovoid Magnetic fieldstrength: 3000 G Gas flow: no16

40、80Case 41680T令mp令roture 1690lemperature17001710_ iRxrtAanCz Si growth under a TMF (confd)1IU匸Temperature field (K): top left: parallel cross section; bottom left:perpendicular cross section; top right: top viewCry staldianwtcr: 4(X) mm Pulling rate:0.45 mm/min Cnicihle rotation rate: 5RPM Magnetic f

41、ield type: horizontalMagnetic field strength: 3000 G Gasflow: noMagnetic field strength: 3000 G16801710Temperature 16901700Cry stal dianwter: 400 mmPulling rate: 0.45 mm/minCnicihle rotation rate: 5 RPMMagnetic field type: ovoidiRxrtAanCz Si growth under a TMF (confd)Cz Si growth under a TMF (confd)

42、Velocity field (m.s *): top left: parallel cross section; bottom left:perpendicular cross sectionGas flow: noCz Si growth under a TMF (confd)Cry stal dianwter: 400 mmPulling rare: 0.45 mni/minCnicible rotalion rate: 5 RPMMagnetic field type: ovoidMagnetic field strength: +3000GGas flow: noConcentrah

43、onOle*24I 00e24Velocity field (m.s *): top left: parallel cross section; bottom left:perpendicular cross sectionCz Si growth under a TMF (confd)Oxygen concentration (m *): top left: parallel cross section in melt;bottom left: perpendicular cross section in melt; top right: crystalMagnetic field type

44、: horizontalMagnetic field strength: 3(X)0 GGas flow: noCry stal dianwtcr: 400 mmPulling rare: 0.45 mni/minCnicible rotalion rate: 5 RPMCase 49 00e242*245e 23Cz Si growth under a TMF (confd)Ovoid magnetic field (lifted), top left: sketch of the magnetic field;bottom left: side view; right: top viewR

45、 -1.0 m, Z-1.37 m, Radius - 0.3 mCurrent Intensity - 2.0e*06 AXl-Mli夕Wc 1kll：varuauM2. Cz Si growth under a TMF (confd)Cn sta! dianwtcr: 40() mmPulling rate: 0.45 mm/minCnicible rotation rate: 5 RPMMagnetic field type: ovoid (lifted)Magnetic field strength: +3000GGas flow: no80Ovoid magnetic field (

46、lifted): top left: sketch of the magnetic field;bottom left: side view; right: top viewCz Si growth under a TMF (confd)R -1.0 m, Z-1.37 m), Radius - 0.3 mCurrent Intensity 2.0306 AWc 1kll：varuauMWc 1kll：varuauM2. Cz Si growth under a TMF (confd)Cn sta! dianwtcr: 40() mmPulling rate: 0.45 mm/minCnici

47、ble rotation rate: 5 RPMMagnetic field type: ovoid (lifted)Magnetic field strength: +3000GGas flow: no80Temperature field (K): top left: parallel cross section; bottom left:perpendicular cross section; top right: top view1700Temperature 1690 17001710vaflACtfAMtCz Si growth under a TMF (confd)Tempera

48、ture field (K): top left: parallel cross section; bottom left:perpendicular cross section; lop right: top viewCry stal dianwtcr: 400 mm Pulling rate:0.45 mnx/min Cnicible rotation rate: 5RPM Magnetic field type: ovoid Magneticfield strength:土3000 G Gas flow: noCz Si growth under a TMF (confd)Velocity field (m.s!): top left: parallel cross section; bo