© 2002 WIT Press, Ashurst Lodge, Southampton, SO40 7AA, UK. All rights reserved. Web: www.witpress.com Email [email protected] Paper from: Advanced Computational Methods in Heat Transfer, B Sunden and CA Brebbia (Editors). ISBN 1-85312-906-2

Numerical analysis of conjugate heat transfer in a melt-cruciblemodel of Czochralski systems: combined Coriolis and MHD effects on time-dependent3D melt

E.M. Smirnov, A.G. Abramov, N.G. Ivanov, A.B. Korsakov, D.K. Zajtsev Department of Aerodynamics, ,%PetersburgState Technical Universiq, Russia

Abstract

3D unsteady convection of low-Prandtl-number liquid (Pr = 0.015) in a model Czochralski crystal growth system is numerically simulated on the base of conjugate heat transfer formulation. Computations using the full Navier-Stokes equations with no turbulence model have been carried out for the of 5x10’. Effects of crucible rotation and stationary axial magnetic field on the convection are investigated.

1Introduction

The majority of monocrystalline silicon wafers used m electronics are produced by the Czochralski (CZ) method with the crystal pulled vertically from the melt. The Si-melt convection determines directly the heat transfer through the melt and the crystallisation rate at the melt-crystal interface (e.g. Muller [l]).Natural convection is combined with the effects induced by crystal rotation and by rotation of the crucible containing the melt. Axisymmetric models for the analyses of melt convection and heat transfer has been in wide use since the end of the 80’s. However, to predict flow instabilities and melt turbulence, only 3D time-dependent calculations of CZ melt flow may be applied. A number of attempts have been reported on this subject (see Basu et al. [2] and references therein). © 2002 WIT Press, Ashurst Lodge, Southampton, SO40 7AA, UK. All rights reserved. Web: www.witpress.com Email [email protected] Paper from: Advanced Computational Methods in Heat Transfer, B Sunden and CA Brebbia (Editors). ISBN 1-85312-906-2

As judged from the literature, previous computations of 3D unsteady phenomena during the CZ crystal growth were based on segregated formulations for melt convection. The segregated formulations require the use of proper approximations defining thermal boundary conditions at the melt-crucible interface. As a rule, an axisymmetric distribution of wall temperature is assumed. Solutions that are closer to the reality can be obtained considering a conjugate heat transfer problem. Setting such a problem may start from the suggestion that unsteady phenomena attributed to instability of the melt are concentrated in the hot zone including the melt, the silica crucible and the graphite support. To control the melt convection and heat transfer during the industrial CZ process the application of steady magnetic fields seems to be rather attractive. This is a motivation for performing both experimental and numerical studies of magnetohydrodynamic (MHD) effects on melt convection (e.g. Grabner et al. [3]). Sure, MHD effects add to the complexity of the problem. A great deal of effort has been put into numerical simulation of magnetoconvection for CZ configurations, but most of the contributions are restricted to the axisymmetric formulation only. However, the magnetic fields used in practice are usually not strong enough to suppress 3D instabilities in the melt completely (e.g. Vizman et al. [4]). The present work is aimed at getting an experience in numerical simulation of 3D unsteady magnetoconvection in a model CZ configuration on the base of conjugate heat transfer formulation.

2 Computational model

The domain geometry and boundary conditions used for the present heat transfer calculations are typical for CZ crystal growth (Figure la). A low-Prandtl-number liquid (melt) is placed in a thin-wall crucible of radius R, made of material with a comparatively low thermal conductivity. We assume h,A, = 0.08, where h, and h, are the crucible and the melt thermal conductivities respectively. The crucible wall thickness, 6, is taken constant. A support of radius Rsupand height Hsupis also included into the computational domain. The ratio of the support thermal conductivity, hsup,to the melt conductivity is taken as 0.7. A concave melt- crystal interface of radius R, occupies the central part of the melt upper boundary. At Y > R, the melt free surface is flat, while for R, 5 r 5 R, the typical meniscus geometry is assumed. The vertical distance from the lowest bottom point to the flat free surface, H, is taken as the length scale. The crucible and crystal rotate about the vertical axis with the angular velocities Q and Cl,, respectively. To nondimensionalise heat fluxes, the reference heat flux, qreb is introduced, given by the following expression: qref= h,31(p2gP3H6),where p, is the melt density, g is the gravitational acceleration, p is the coefficient of thermal expansion. Consequently, the reference temperature difference, AT, may be taken as the ratio q,,,Hlh,,,. As one can see below, this choice of reference © 2002 WIT Press, Ashurst Lodge, Southampton, SO40 7AA, UK. All rights reserved. Web: www.witpress.com Email [email protected] Paper from: Advanced Computational Methods in Heat Transfer, B Sunden and CA Brebbia (Editors). ISBN 1-85312-906-2 Advmcmi CompututiodMethods in HatTrumf?v 109 values ensures that the normalised temperature, 0 = (T-TJIAT, is of the order of unity, here T, is the temperature of the melt-crystal interface. The buoyancy velocity, V, = (gj3ATH)”’, is considered as the velocity scale. The ratio HIV, serves as the time scale.

Figure 1: (a) The solution domain (b) Multiblock computational grid.

Considering as a prototype the EKZ-1300 CZ furnace used for 100 mm Si- crystal growth (e.g. Evstratov et al. [.5]), basic geometrical ratios can be taken as RJH = 1.7,RJH = 0.6, RJH = 0.5, WH = 0.08, R,,,/H = 2.0, Hsup/H=1S. Three-dimensional melt convection is computed on the base of the full Navier-Stokes equation with no turbulence model. The governing equations accounting for the effects of a stationary magnetic field on the melt motion are written under the assumption of small magnetic , Re, = pBoHVb<< 1, where pB is the magnetic permeability, o is the electric conductivity of the melt. With this conventional approximation the back influence of the melt motion on the magnetic field is neglected. In the present work the MHD effects on the melt motion are studied for the case of a pure vertical magnetic field, with constant axial component of the magnetic induction vector, B. With given geometry and thermal boundary conditions, the behaviour of the melt is determined by tie Pmndtl number, Pr= vla (v is the kinematic viscosity,a is the thermal diffusivity), tie Rayleigh number, Ra = gpATH’/(va), the , Ro = Vb/RcH,the Stuart number, N = oB2H/(pmVb),and the angular velocity ratio, QJQ. Using the Boussinesq’s approach for incorporation of the gravity buoyancy effects, a vector formulation of the momentum equation can be written in a reference rotating together with the crucible as follows:

---vp*+*AV-@egdV -~(eaxV)+N[-V@+VxB]xB. - dt &G Ro © 2002 WIT Press, Ashurst Lodge, Southampton, SO40 7AA, UK. All rights reserved. Web: www.witpress.com Email [email protected] Paper from: Advanced Computational Methods in Heat Transfer, B Sunden and CA Brebbia (Editors). ISBN 1-85312-906-2

Here p* is the reduced pressure, Q is the electric potential. Via the term depending on the normalised temperature, 0, eqn (1) is coupled with the energy equation (not written). Zero relative velocity conditions on the crucible wall are imposed, while on the rotating disk the velocity distribution is given by V,,= -(Qs + Q)Y.Zero shear stresses are imposed on the free surface. Heat flux boundary distributions are adopted to be uniform. The support is heated from the periphery with non- dimensionalised flux q,,= ~.OX~O-~.Heat flux leaving the free surface, g,, is twice g,,. This ratio of heat fluxes was chosen to get Q,,= 1.l& where Qhis the heat rate supplied to the support, and Q, is the heat rate leaving the free surface. At these conditions, the heat rate supplied to the melt-crystal interface is one order less than the heat rate passing through the melt, that is typical for the growth process. The Poisson equation governing the electrical potential distribution, A@ = B.(V x V), is solved on the assumption that all boundaries are electrically insulating. Computations in the framework of the above formulation have been carried out using a well-validated in-house code (named SINF). This advanced 3D Navier-Stokes solver is based on the second-order finite-volume spatial discretisation using the cell-centred variable arrangement and body-fitted block- structured grids. The discretisation of the time derivatives is done with a three- time-level, second-order implicit scheme. The artificial-compressibility method is applied at each time step. A QUICK-type upwind scheme is employed to compute convective fluxes. The code is in wide use for solving both fundamental and industrial problems, in particular, it has been used for simulation of 3D turbulent convection during industrial-scale CZ Si-crystal growth (Evstratov et al. [5]). Additional details for the solver can be found elsewhere [6]. The computations were performed with Ra= 5x105, Pr = 0.015. The domain occupied by the melt was covered by a three-block grid consisting of 79,872 cells, with nodes clustered to the wall and to the free surface. Six additional blocks of 29,952 cells covered the crucible and the support. The grid is shown in Figure lb. As a result of preliminary computations the non- dimensional time step of0.25 has been chosen. 3 Crucible rotation effects

This section describes results obtained at computations of cases with no magnetic field action. To simulate the effects of crucible rotation two runs have been performed. For the first run (Ro = 2, QK2, = -0.25), the computations were started from some initial velocity and temperature fields. After a transient period of more than 300 time units, a statically-developed self-oscillating regme has been obtained. For the second run (Ro = 10, = -0.05), the computations were started from an instantaneous field which had been obtained for the first case. A transient period before the achievement of the new statically-developed © 2002 WIT Press, Ashurst Lodge, Southampton, SO40 7AA, UK. All rights reserved. Web: www.witpress.com Email [email protected] Paper from: Advanced Computational Methods in Heat Transfer, B Sunden and CA Brebbia (Editors). ISBN 1-85312-906-2 Advmcmi Compututiod Methods in HatTrumf?v 1 1 1 self-oscillating regime was more than 150 time units. The length of the samples computed after passing the transient period was about 300 time units.

Figure 2: Snapshots of (left) velocity patterns and (right) temperature contours in a vertical section; (a,d) Ro = 10, N = 0, (b,e) Ro = 2, N = 1, (Q) Ro=2,N=l.

4 e> Q Figure 3: Snapshots of (top) velocity patterns and (bottom) temperature contours in the plane Z*= 0.25; (a,d) Ro = 10, N = 0, (b,e) Ro = 2, N = 0, (qf) Ro=2,N= 1. © 2002 WIT Press, Ashurst Lodge, Southampton, SO40 7AA, UK. All rights reserved. Web: www.witpress.com Email [email protected] Paper from: Advanced Computational Methods in Heat Transfer, B Sunden and CA Brebbia (Editors). ISBN 1-85312-906-2

Figures 2a,b show instantaneous velocity fields in a vertical section for both the cases computed. The global circulation undergoes a noticeable transformation due to increase in the Rossby number. For Ro = 10, the flow is rather intensive, in particular, a toroidal vortex with maximum velocity of about 0.1V, is observed on the melt periphery. For Ro = 2, the motion is strongly suppressed by rotation due to the influence of the Coriolis force. For both the cases, a number of vortices is observed in the under-crystal region. Note, that comparison of instantaneous and time-averaged velocity fields indicates that fluctuations of melt velocity are comparable or even higher than the averaged local velocity, especially in the under-crystal region. Snapshots of velocity patterns given in the rotating reference for a horizontal plane of z* = Z,-Z = 0.25 demonstrate strong transformations of flow structure with the change of the Rossby number (Figures 3a,b). A number of vortices arise on the periphery of the melt at Ro = 10 (Figure 3a). The basic mechanism responsible for occurrence of multiple-vortex flow structure on the melt periphery is the so-called baroclinic instability (e.g. Nakamura et al. [7]). With decrease in the Rossby number, the periphery vortices shift to smaller radii (Figure 3b). From Figures 3a,b the above-mentioned suppression of melt convection, attributed to increase in the Coriolis force, is also evident. Nevertheless, the melt behaviour in the under-crystal region remains essentially three-dimensional. The reasons for “survivability” of strong 3D unsteady effects in the under-crystal region are not clear yet (Evstratov et al. [S]). Close attention should be paid to the temperature field transformation due to influence of rotation because thermal state of the melt in the under-crystal region determines the quality of a growing single crystal. Instantaneous normalised temperature distributions over a vertical section, obtained for different Ro, are compared in Figures 2d,e (the extended domain is shown). In case of Ro = 10, the role of convection is well pronounced on the periphery of the melt (Figure 2d), while in the under-crystal region isotherms are nearly horizontal for both the regimes. Note, that in the solid part of the domain the temperature field is almost axysimmetric, a slight deviation fi-om the axial symmetry takes place in the vicinity of the melt-crucible interface only. Figures 3d,e present instantaneous melt temperature distributions over the horizontal plane of z“= 0.25. A strong spatial non-uniformity of temperature field is observed. The melt temperature fluctuations at two monitoring points rotating together with the crucible are shown in Figure 4. Only the time behaviour related to the statically-developed self-oscillating regime is presented for the cases with no magnetic field. One point, P1, is placed in the under-crystal region (r= 0.1, z“= 0.1) and the other one, P2, is positioned on the melt periphery (r= 0.87, z*= 0.35). It is evident that the temperature evolution is chaotic in time for both the regimes. The decrease in the Rossby number results in some reduction of the amplitude of fluctuations. The related plots of power spectral density (PSD) are presented in Figure 5. Several leading frequencies can be extracted for both the regimes. The PSD © 2002 WIT Press, Ashurst Lodge, Southampton, SO40 7AA, UK. All rights reserved. Web: www.witpress.com Email [email protected] Paper from: Advanced Computational Methods in Heat Transfer, B Sunden and CA Brebbia (Editors). ISBN 1-85312-906-2 Advmcmi CompututiodMcthods in HatTrumf?v 1 13 starts to drop at a normalised frequency,J; of about 0.1. It seems that the flow is only in the beginning of a turbulent state, since the Kolmogorov proportionality of f -j13 does not occur even for the case ofRo = 10, with more intensive fluctuations.

eO.10 a---7

0.m 0 ox

l!."11 11.06

m MM

l/22 0 22 e n 20

0 IR 0.18

0 16 11.16

0 14 U.14 " 12 0 411 80 120 l60 200 4 l Figure 4: Temperature fluctuations at (top) point P1 and (bottom) point P2; (a,d) Ro = 10,N = 0, @,e)Ro = 2, N = 0, (c,Q Ro = 2. N = 1.

IE-l 1E-l

I€-2 IE-2

IE-3 1E-3

IE-4 1 E-4

IE-5 I E-5

IE-2 0. I 1E-2 0.1 f 4 b) f

E,IEfO IE-l

IE-2

1E-3

1 E-4

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1E-6 IE-2 0.1 4 J Q J Figure 5:Power spectral density of temperature fluctuations at (top) point P1 and (bottom) point P2; (a,d) Ro = 10, N = 0, @,e) Ro = 2, N = 0, (c$) Ro=2,N= 1. © 2002 WIT Press, Ashurst Lodge, Southampton, SO40 7AA, UK. All rights reserved. Web: www.witpress.com Email [email protected] Paper from: Advanced Computational Methods in Heat Transfer, B Sunden and CA Brebbia (Editors). ISBN 1-85312-906-2

028

0.24

0.20

"'pmm,, , , syy~ 0.12 0.0 0.50.0 1.51.0 O S 2.5 Figure 6: Time-averaged temperature distribution along the melt-crucible interface: (solid) Ro = 10, N = 0, (dashed) Ro = 2, N = 0, (dotted- dashed) Ro = 2, N = 1.

Figure 6 compares the computer-generated time-averaged temperature distributions along the melt-crucible interface. Increase in the Coriolis force action leads to significant increase in a characteristic temperature difference of the melt, the latter is treated as the difference between the maximum temperature of the crucible wall and the melting temperature, T,. For Ro = 10, the temperature distribution over the crucible wall is more uniform, as compared with the case of Ro = 2.

4 Magnetic field induced changes of melt convection

Numerical solution of the conjugate heat transfer problem accounting for the magnetic field action has been obtained setting the Stuart number equal to 1, the Rossby number equal to 2, and the angular velocity ratio QQ = -0.25. Computations were started from an instantaneous field being in hand for the correspondent case with no magnetic field. It has been established that the transient period that is required for achieving the statistically-developed regime is much longer as compared with the case of no magnetic field. It has been estimated as about 500 time units (see Figures 4@. The length of the sample computed after passing the transient period was about 300 time units. Significant changes in 3D behaviour of the melt due to imposing the magnetic field are observed. A comparison of Figures 2b and 2c shows that the influence of the magnetic field leads to much more regular melt motion. It is remarkable that the application of the magnetic field results in formation of a toroidal motion directed upward in the vicinity of the axis (Figure 2c). Convective heat transfer by this motion affects the temperature distribution in the under-crystal region. In Figure 2f one can see that temperature isolines are deflected in this region considerably. © 2002 WIT Press, Ashurst Lodge, Southampton, SO40 7AA, UK. All rights reserved. Web: www.witpress.com Email [email protected] Paper from: Advanced Computational Methods in Heat Transfer, B Sunden and CA Brebbia (Editors). ISBN 1-85312-906-2 Advmcmi CompututiodMethods in HatTrumf?v 1 15

Given in the rotating reference, the velocity and temperature distributions over the horizontal section of Z' = 0.25 (Figures 3qf)show that the under-crystal flow affected by the magnetic field can be characterised as an 0-shape structure. A comparison of flow fields obtained for successive instants allows to conclude that this structure is in a slow precession motion with respect to the rotating observer. For the case considered, Figures 4c,f illustrate the time history of the melt- temperature at two monitoring points, P1 and P2. It should be stressed, that in contrast to the data given for the cases with no magnetic field (Figure 4, left), the results for the whole time interval computed are presented (including the transient period of more than 500 time units). It is interesting, that the high- frequency components of the oscillations are suppressed during a relatively short time period, directly after the magnetic field application. Then the flow evolves slowly into a quasi-periodic regme. The residual oscillations visible for point P2 are attributed to the above-mentioned horizontal precession of the 0-shaped structure, but not to melt fluctuations in the vertical direction. The correspondent spectra shown in Figures 5qf confirm that the flow subjected to the magnetic field action may be treated as one close to quasi- periodic. Two dominating frequencies are seen in these spectra, the first peak is at fK 0.03, and the second one is at fE 0.2. The application of the magnetic field leads to further increase in the temperature difference inside the melt, that is demonstrated,in Figure 6. It is obvious that this increase is due to damping of global melt convection, and to suppression of turbulence contribution to heat transfer.

5 Conclusions

3D unsteady low-Prandtl-number liquid flow and heat transfer m the hot zone of a typical Czochralski crystal growth system have been simulated on the base of conjugate heat transfer formulation. For the Rayleigh number of 5x10' computations have been carried out on the base of the full Navier-Stokes equations with no turbulence model. Computations carried out for two values of the Rossby number with no magnetic field action have shown that fluctuations of melt velocity are comparable or even higher than the averaged local velocity, especially in the under-crystal region. The size and radial position of vortices arising on the melt periphery due to the baroclinic instability significantly change with variations of crucible rotation. To suppress melt fluctuations in the most unstable under- crystal zone one would have to use a higher cnxible rotation rate than that considered in the present contribution. Increase in the Coriolis force action leads to significant increase in a characteristic temperature difference of the melt, mostly due to suppression of the turbulence contribution to heat transfer. The transient period that is required for achieving the statistically- developed regime of the convection subjected to the Lorentz force action can be © 2002 WIT Press, Ashurst Lodge, Southampton, SO40 7AA, UK. All rights reserved. Web: www.witpress.com Email [email protected] Paper from: Advanced Computational Methods in Heat Transfer, B Sunden and CA Brebbia (Editors). ISBN 1-85312-906-2 much longer as compared with the case of no magnetic field. Under the conditions considered the axial magnetic field, characterized by the Stuart number close to unity, has produced a strong stabilizing action on the melt convection. Over the major part of the melt convection became practically axisymmetric. In the under-crystal region a toroidal vortex developed, with motion directed upward in the vicinity of the axis. This vortex produces a considerable effect on temperature isoline patterns. For horizontal planes the vortex can be treated as an 0-shape structure that is in a slow precession motion with respect to the rotating observer. The residual oscillations observed in the melt can be mostly attributed to this precession. Application of the magnetic field leads to further increase in the temperature difference across the melt.

Acknowledgements

The study has been supported in part by the Russian Foundation of Basic Research, grant 01-02-16697.

References

[l] Muller, G. Convection and inhomogeneities in crystal growth from the melt, Crystals, vo1.12, ed. C.J.M. Rooijmanns, Springer-Verlag: Berlin, 1988. [2] Basu, B., Enger, S., Breuer, M. & Durst, F., Three-dimensional simulation of flow and thermal field m a Czochralski melt using a block-structured finite- volume method. J. Crystal Growth,219, p.p. 123-143,2000. [3] Grabner, O., Muller, G., Virbulis, J., Tomzig, E. & v.Ammon, W., Effects of various magnetic field configurations in Czochralski silicon melts. MicroelectronicEngineering, 56,p.p. 83-88,2001. [4] Vizman, D., Friedrich, J. & Muller, G., Comparison of the predictions from 3D numerical simulation with temperature distributions measured in Si Czochralski melts under the influence of different magnetic fields. J. Crystal Growth, 230, P.P. 73-80, 2001. [5]Evstratov, I. Yu., Kalaev, V.V., Zhmakin, AI., Makarov, YLN., Abramov, A.G., Ivanov, N.G., Smirnov, E.M., Dornberger, E., Virbulis, J., Tomzig, E., & v.Ammon, W., Modeling analysis of unsteady three-dimensional turbulent melt flow during Czochralski growth of Si crystals J. Crystal Growth, 230, P.P. 22-29,2001. [6] Smirnov, E.M., Solving the fidl Navier-Stokes equations for very-long-duct flows using the artificial compressibility method. CD-ROM Proc. of the ECCOMAS 2000 Conf,Barcelona, Spain, 17p., 2000. [7] Nakamura, S., Eguchi, M., Azami, T. & Hibiya, T., Thermal waves of a nonaxisymmetric flow in a Czochralski-type silicon melt. J. Crystal Growth, 207, P.P. 55-61, 1999.