晶體生長工藝之區(qū)熔法

1、,FEMAGSoft 2013,晶體生長工藝之區(qū)熔法,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 of FZ growth (contd),Turbulent

2、 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 density Jsource imposed by external sour

3、ce 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,susceptor,inductor,z,r,susc

4、eptor,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),tangential stress e

5、xerted 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 induction m0 magnetic per

6、meability of vacuum s electric conductivity w angular frequency,Dissipated power: Force density:,Heat flux 2) Normal stress 3) Tangential stress,Alternating magnetic field effects :,Inductor,Susceptor,Modeling of FZ growth (contd), Development of a mathematical model of the electromagnetic field dis

7、tribution 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 elec

8、tromagnetic 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 calc

9、ulations 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 = 7460 ReCrystal = 3730 Pe = 261 Gr = 2.7x107 Ma = 6116,Floating Zone Silicon Growth Simulation,FEMAGSoft 2013,Modeling of FZ growth (contd

10、),Temperature field and isolines of the norm of the magnetic flux function.,FEMAG-FZ quasi-steady simulation of the growth of a 200 mm silicon crystal,FEMAGSoft 2013,Modeling of FZ growth (contd),With equivalent magnetic tangential stress.,Without equivalent magnetic tangential stress.,Temperature f

11、ield (left) and Stokes stream 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 funct

12、ion isolines in the melt,Modeling of FZ growth (contd),Model validation,FEMAGSoft 2013,Modeling of FZ growth (contd),Crystal radius: 51 mm Feed rotation rate: 15 RPM Crystal rotation rate: (a) 5 RPM, (b) 10 RPM, (c) 15 RPM Marangoni coefficient: 1.0 10-4 N/mK,(a),(b),(c),Good correspondence between

13、predicted and experimental results,Effect of crystal rotation rate on 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 temper

14、ature 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 of FZ growth (contd),FEMAGSoft 2013,Growth of a 100 mm silicon crystal (1mm/min pull rate),Predicted defect

15、 delta -(CI-CV) distribution by means of a quasi-steady simulation,Modeling 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 rpm vpul = 3.4 mm/min,Temperature field,Modeling of FZ growth (contd),FEMAGSoft

16、2013,Rs = 5.1 cm, Rf = 4.7 cm, Ws = 10 rpm, Wf = -15 rpm vpul = 3.4 mm/min,Streamlines,Modeling of FZ growth (contd),FEMAGSoft 2013,Rs = 5.1 cm, Rf = 4.7 cm, Ws = 10 rpm, Wf = -15 rpm vpul = 3.4 mm/min,Difference of interstitial and vacancy concentrations (CI - CV ),Modeling of FZ growth (contd),FEM

17、AGSoft 2013,Rs = 5.1 cm, Rf = 4.7 cm, Ws = 10 rpm, Wf = -15 rpm vpul = 3.4 mm/min,Difference of interstitial and vacancy concentrations (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

18、 of FZ growth (contd),FEMAGSoft 2013,Calculation of thermal stresses in a growing FZ crystal without convection,Modeling of FZ growth (contd),FEMAGSoft 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 shie

19、ld,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 2013,a),b), growth orientation,Effect of a heat shield: von Mises stress,Modeling of FZ growth (contd),Typical FEMAG-FZ global unstructured mesh

20、 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 crystal Use of an equivalent thermal conductivity,Modeling of FZ growth (contd),Free interface constraining loci (secondary mesh) in FZ g

21、rowth,FEMAGSoft 2013,Modeling of FZ growth (contd),FEMAGSoft 2013,Inverse modeling in FZ growth much more difficult problem than in Cz growth can lead to misleading interpretations of the simulation results since completely inverse models result in the calculation of the melt volume and hence parame

22、tric studies are difficult to interpret with classical simplified models, the open melting front (OMF) is imposed and the melting front is either imposed or calculated (as an isotherm),Modeling of FZ growth (contd),FEMAGSoft 2013,Open Melting Front after extraction of the single crystal,Modeling of

23、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-gas interface detaches from the OMF require accurate modeling in view of their direct impact on the radiation transfer

24、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) and 3 tri-junctions represents a difficult problem of computational geometry.,Modeling of FZ growth (contd),FEMAGSo

25、ft 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 present in FZ growth,Oscillations of the crystal and/or feed-rod rotation rates: this technique is often used to

26、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 crystal and feed-rod rotation axes,Modeling of FZ growth (contd),FEMAGSoft 2013,Investigation of ACRT tech

27、nique,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 simulation: local temperature field,Modeling of FZ growth (contd),FEMAGSoft 2013,Quasi-steady simulation: glo

28、bal 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 meridional velocity,Modeling of FZ growth (contd),FEMAGSoft 2013,Quasi-dynamic simulations,Azimuthal vel

29、ocity,Modeling of FZ growth (contd),FEMAGSoft 2013,Top right: temperature field and meridional velocity Bottom left: azimuthal velocity,Quasi-dynamic simulations,Modeling of FZ growth (contd),FEMAGSoft 2013,Definition of an average flow for species transport inverse simulation,Average flow: quasi-dy

30、namic 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 simulations 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 meridional velocity vectors,Quasi-steady simulation: local temperature fiel