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1、,FEMAG晶体生长模拟软件-How to extend Cz modeling techniques to FZ growth?,Modeling of FZ growth,FEMAGSoft 2013,Global temperature field(left),melt flow(right),and alternating magnetic field(bottom),Quasi-steady simulation of the Floating Zone(FZ)growth of a 100 mm silicon crystal(1mm/min pull rate),Modeling
2、 of FZ growth(contd),Turbulent viscosity is low and the melt flow can be computed by means of a laminar model.,FEMAGSoft 2013,Modeling of FZ growth(contd),Induction heating,FEMAGSoft 2013,Modeling of FZ growth(contd),Induction Heating in FZ semi-conductor growth,FEMAGSoft 2013,J current densityJsour
3、ce imposed by external source Jeddy induced by time-dependent magnetic field,inductor,susceptor,Modeling of FZ growth(contd),Slottedinductor,Top view Section S-S,S,S,Jsource,N:number of slits,FEMAGSoft 2013,Modeling of FZ growth(contd),Numerical results,Non-slotted inductor,Slotted inductor,z,r,susc
4、eptor,inductor,z,r,susceptor,inductor,FEMAGSoft 2013,Modeling of FZ growth(contd),Real part of magnetic flux,FEMAGSoft 2013,Imaginary part of magnetic flux,Modeling of FZ growth(contd),Difficulties,FEMAGSoft 2013,melt-atmosphere interface shape(magnetic pressure),open melting front(thin fluid film),
5、tangential stress exerted onto the melt free surface:-generally undesired resulting shear flow-potentially useful effect to control the flow.,Induction Heating in FZ semi-conductor growth,Modeling of FZ growth(contd),FEMAGSoft 2013,Induction Heating in FZ semi-conductor growth,B magnetic inductionm0
6、 magnetic permeability of vacuums electric conductivityw angular frequency,Dissipated power:Force density:,Heat flux2)Normal stress3)Tangential stress,Alternating magnetic field effects:,Inductor,Susceptor,Modeling of FZ growth(contd),Development of a mathematical model of the electromagnetic field
7、distribution in planar and axisymmetric configurations,Hypothesis:low value of the magnetic skin depth,FEMAGSoft 2013,Model based on using:-a matched asymptotic expansion technique to approximate the electromagnetic field inside the conductors-a Finite Element numerical representation of the electro
8、magnetic field outside the conductors.,Modeling of FZ growth(contd),Mean dissipated power:,Mean body force density:,Equivalent normal heat flux qneq:,Equivalent surfacestress teq:,FEMAGSoft 2006,Equivalent magnetic stresses and heat flux,Modeling of FZ growth(contd),Flow and temperature calculations
9、 are performed with a 200 mm diameter crystal.The melt viscosity is set to 5 times the actual silicon viscosity to obtain steady results.,RePolycrystal=7460ReCrystal=3730Pe=261Gr=2.7x107Ma=6116,Floating Zone Silicon GrowthSimulation,FEMAGSoft 2013,Modeling of FZ growth(contd),Temperature field and i
10、solines of the norm of the magnetic flux function.,FEMAG-FZ quasi-steadysimulation of the growthof a 200 mm silicon crystal,FEMAGSoft 2013,Modeling of FZ growth(contd),With equivalent magnetic tangential stress.,Without equivalent magnetic tangential stress.,Temperature field(left)and Stokes stream
11、function(right)in the melt,FEMAGSoft 2013,6.Modeling of FZ growth(contd),FEMAG-FZ quasi-steady simulation of the growth of a 200 mm silicon crystal,FEMAGSoft 2013,(right)Temperature field and isolines of the norm of the magnetic flux function.(bottom)Stream function isolines in the melt,Modeling of
12、FZ growth(contd),Model validation,FEMAGSoft 2013,Modeling of FZ growth(contd),Crystal radius:51 mmFeed rotation rate:15 RPMCrystal rotation rate:(a)5 RPM,(b)10 RPM,(c)15 RPMMarangoni coefficient:1.0 10-4 N/mK,(a),(b),(c),Good correspondence between predicted and experimental results,Effect of crysta
13、lrotation rateon the melt flow(FZ growth),FEMAGSoft 2013,With the courtesy of IKZ,Berlin,Modeling of FZ growth(contd),FEMAGSoft 2013,Calculation of point defects in a growing FZ crystal,Modeling of FZ growth(contd),FEMAGSoft 2013,Global temperature field(left),melt flow(right),and alternating magnet
14、ic field(bottom),Quasi-steady simulation of the Floating Zone(FZ)growth of a 100 mm silicon crystal(1mm/min pull rate),Modeling of FZ growth(contd),FEMAGSoft 2013,Growth of a 100 mm silicon crystal(1mm/min pull rate),Predicted defect delta-(CI-CV)distribution by means of a quasi-steady simulation,Mo
15、deling of FZ growth(contd),FEMAGSoft 2013,Second example,Modeling of FZ growth(contd),FEMAGSoft 2013,Rs=5.1 cm,Rf=4.7 cm,Ws=10 rpm,Wf=-15 rpmvpul=3.4 mm/min,Temperature field,Modeling of FZ growth(contd),FEMAGSoft 2013,Rs=5.1 cm,Rf=4.7 cm,Ws=10 rpm,Wf=-15 rpmvpul=3.4 mm/min,Streamlines,Modeling of F
16、Z growth(contd),FEMAGSoft 2013,Rs=5.1 cm,Rf=4.7 cm,Ws=10 rpm,Wf=-15 rpmvpul=3.4 mm/min,Difference of interstitial and vacancy concentrations(CI-CV),Modeling of FZ growth(contd),FEMAGSoft 2013,Rs=5.1 cm,Rf=4.7 cm,Ws=10 rpm,Wf=-15 rpmvpul=3.4 mm/min,Difference of interstitial and vacancy concentration
17、s(CI-CV)(detail),Modeling of FZ growth(contd),FEMAGSoft 2013,von Mises invariant:global view and detail,Ratio of the von Mises invariant over the CRSS,Modeling of FZ growth(contd),FEMAGSoft 2013,Calculation of thermal stresses in a growing FZ crystal without convection,Modeling of FZ growth(contd),F
18、EMAGSoft 2013,Effect of a heat shield:temperature field,No convection,Rs=5.1 cm,Rf=4.7 cm,vpul=3.4 mm/min,a)Without heat shieldb)With a heat shield,Modeling of FZ growth(contd),FEMAGSoft 2013,Effect of a heat shield:von Mises stress,a),b),growth orientation,Modeling of FZ growth(contd),FEMAGSoft 201
19、3,a),b),growth orientation,Effect of a heat shield:von Mises stress,Modeling of FZ growth(contd),Typical FEMAG-FZ global unstructured mesh for heat transfer and induction heating,FEMAGSoft 2013,Modeling of FZ growth(contd),FEMAGSoft 2013,FEMAG-FZ time-dependent simulation of the growth of a silicon
20、crystalUse of an equivalent thermal conductivity,Modeling of FZ growth(contd),Free interface constraining loci(secondary mesh)in FZ growth,FEMAGSoft 2013,Modeling of FZ growth(contd),FEMAGSoft 2013,Inverse modeling in FZ growthmuch more difficult problem than in Cz growthcan lead to misleading inter
21、pretations of the simulation results since completely inverse models result in the calculation of the melt volume and hence parametric studies are difficult to interpretwith classical simplified models,the open melting front(OMF)is imposed and the melting front is either imposed or calculated(as an
22、isotherm),Modeling of FZ growth(contd),FEMAGSoft 2013,Open Melting Front after extraction of the single crystal,Modeling of FZ growth(contd),FEMAGSoft 2013,Main issue:modeling of the Open Melting Front(OMF),Physical problem:the flow of the molten silicon along the OMF and the angle at which the melt
23、-gas interface detaches from the OMF require accurate modeling in view of their direct impact on the radiation transfer to the OMF and on the melt-gas interface shape,Numerical problem:the coupled solution of a problem with 4 interfaces(solidification front,melting front,melt-gas interface,and OMF)a
24、nd 3 tri-junctions represents a difficult problem of computational geometry.,Modeling of FZ growth(contd),FEMAGSoft 2013,Other key issues:,Species transport(dopant and impurities):the problem is similar to species transport in Cz growth,but much more difficult since almost no turbulent mixing is pre
25、sent in FZ growth,Oscillations of the crystal and/or feed-rod rotation rates:this technique is often used to better mix the melt and can be simulated by use of a quasi-dynamic model,3D effects:non-axisymmetric effects are generated(i)by the inductor shape and possibly(ii)by the use of non-aligned cr
26、ystal and feed-rod rotation axes,Modeling of FZ growth(contd),FEMAGSoft 2013,Investigation of ACRT technique,Modeling of FZ growth(contd),Investigation of ACRT technique,FEMAGSoft 2013,Modeling of FZ growth(contd),FEMAGSoft 2013,Quasi-steady simulation:global temperature field,Quasi-steady simulatio
27、n:local temperature field,Modeling of FZ growth(contd),FEMAGSoft 2013,Quasi-steady simulation:global temperature field,Quasi-steady simulation:local temperature field and meridional velocity vectors,Modeling of FZ growth(contd),FEMAGSoft 2013,Quasi-dynamic simulations,Temperature field and meridiona
28、l velocity,Modeling of FZ growth(contd),FEMAGSoft 2013,Quasi-dynamic simulations,Azimuthal velocity,Modeling of FZ growth(contd),FEMAGSoft 2013,Top right:temperature field and meridional velocityBottom left:azimuthal velocity,Quasi-dynamic simulations,Modeling of FZ growth(contd),FEMAGSoft 2013,Defi
29、nition of an average flow for species transport inverse simulation,Average flow:quasi-dynamic results are further time-averaged in order to provide mean velocity,viscosity,and heat and species diffusivity fields,Inverse simulations:time-averaged fields are used in quasi-steady or inverse dynamic sim
30、ulations in order to predict species transport in the melt and incorporation into the crystal,Ultimate goal:to predict the resistivity distribution in the crystal.,Modeling of FZ growth(contd),FEMAGSoft 2013,Quasi-steady and quasi-dynamic“ACRT”simulation:local temperature field and average meridiona
31、l velocity vectors,Quasi-steady simulation:local temperature field and meridional velocity vectors,The average flow is weaker than the quasi-steady flow,Modeling of FZ growth(contd),FEMAGSoft 2013,Quasi-steady results,Radial velocity,Azimuthal velocity,Axial velocity,Average quasi-dynamic results,Mo
32、deling of FZ growth(contd),FEMAGSoft 2013,Boron concentration prediction,Calculations from the initial quasi-steady simulation,Calculations from the averaged quasi-dynamic simulation,Modeling of FZ growth(contd),FEMAGSoft 2013,Conclusion,Physics of the flow:due to the very low molten silicon viscosi
33、ty,the feed-rod almost slips onto the melt surfaceSo alternating the feed-rod rotation sense only affects a thin boundary layer along the fusion interface and a limited region around the axis,Mixing efficiency:with feed alternate rotations the ACRT technique has limited efficiency to mix the silicon melt,Possible solutions:selection of improved parameters or use of non-aligned crystal and feed-rod rotation axes in order to generate efficient 3D mixing.,Modeling of FZ growth(contd),
