HYBRID FINITE-ELEMENT/FINITE-VOLUME METHOD FOR INTEGRATED HYDRODYNAMIC THERMAL CAPILLARY ANALYSIS OF TURBULENCE AND HEAT TRANSFER OF CZOCHRALSKI CRYSTAL GROWTH OF SILICON
A. Lipchin Massachusetts Institute of Technology, Department of Chemical Engineering 77 Massachusetts Avenue, Cambridge, Massachusetts 02139, USA Room 66-250. Phone (617) 253-6547, fax (617) 258-5042, e-mail
[email protected] R.A. Brown Massachusetts Institute of Technology, Department of Chemical Engineering 77 Massachusetts Avenue, Cambridge, Massachusetts 02139, USA Room 1-206. Phone (617) 253-5726, fax (617) 253-8549, e-mail
[email protected]
srf
ABSTRACT
Modeling turbulent convection in the melt is the most dicult part of the simulation of the transport processes in the growth of large-diameter silicon crystals by the Czochralski (CZ) process. We report the application of hybrid nite-element/ nitevolume methods for the integrated modeling of turbulence in the melt and heat transfer by conduction, convection and radiation throughout the CZ system. Turbulence calculations are based on a low Reynolds number model proposed by Jones and Launder. The integrated hybrid method is based on coupled solution of energy transport in every phase in the CZ system by an integrated nite element analysis with the velocity eld in the melt and turbulence variables interpolated from a nite volume solution of the coupled convection and energy transport problem in the melt alone. Sample calculations make clear the advantages of the hybrid method.
t t
1 INTRODUCTION
The Czochralski process is the most used method for growing large diameter silicon crystals which are used as substrates in microelectronic devices. Requirements in device manufacturing for higher quality and larger diameter silicon crystals lead to increasing demands in the performance and producing of Czochralski (CZ) systems. Melt motion, dopant transport and heat transfer a ect the crystal quality through the distribution of impurities, such as oxygen, boron and carbon, and the formation of crystallographic defects in the material. Integrated modeling and simulation of heat and mass transfer in CZ system is necessary to establish quantitative relationships between growth conditions and crystal quality. If quantitatively accurate, integrated modeling, coupled with models of microdefect formation, will be useful for systematic design and control of the CZ process. The Czochralski system is show schematically in Fig.1. The silicon melt is held in a quartz crucible placed in a graphite holder. Heat is supplied by radiative heat ux from a graphite resistance heater. The silicon crystal is pulled from the melt at
NOMENCLATURE
Cp K T Tm V g k p r u
z
coecient of thermal expansion of the melt dissipation rate of turbulent energy surface emissivity dimensionless temperature, T=Tm molecular dynamic viscosity turbulent dynamic viscosity molecular kinimatic viscosity turbulent kinimatic viscosity density stress tensor rotation rate
Heat capacity Thermal conductivity temperature melting temperature crystal pulling rate acceleration due to gravity turbulent kinetic energy pressure radial cylindrical coordinate melt velocity vertical cylindrical coordinate
1
and Brown (1993). Simulation of this convection in the context of an IHTCM and its analysis is only feasible using turbulence models, such as k ? models used by Kinney and Brown (1992) and by Zhang et al. (1996). Modeling convection in CZ systems is especially challenging because of the high anisotropy of the motion caused by the variety of driving forces and the variations of their magnitudes throughout the ow. This leads to a
ow with regions of intense convection connected to parts of the melts with relatively weak convection. Accurate prediction of such motions is dicult for twoequation engineering models of turbulence built on scalar measures of turbulent intensity. In this paper, we review the performance of three k ? models for analysis of convection in a prototype CZ system | commonly called Czochralski bulk ow | to highlight the di erences in the predictions of these models. The models are (a) the standard k ? model with wall functions along solid surfaces (WF; Henkes et al.,1991), (b) a two-layer k ? model including a one-equation model for the
ow near solid boundaries (TL; Iacovides et al., 1996), and (c) the low-Reynolds number k ? model of Jones and Launder (JL; Henkes et al.,1991). The two-layer model consists of the standard k ? model for the core and Wolfshtein's one-equation model for the near-wall regions. The one-equation model incorporates an equation for the kinetic energy k, but approximates the dissipation and the turbulent viscosity t from prescribed length scales. This approach was used successfully for rotating and curved ows by Iacovides et al. (1996) and for turbulent convection by other researchers. Of the three formulations, the JL model has been selected for use in the IHTCM analysis described here. The IHTCM analysis is based on a novel numerical method in which di erent computational schemes are applied to solve the energy equation throughout the whole system and the turbulent motion equations in the melt. A nite element (FE) discretization and Newton method are used to compute the temperature in the CZ system as previously done by us in other applications. We have not described this analysis here, but direct the reader to other references (Sackinger et al., 1989; Bornside et al., 1990). A nite volume (FV) discretization and sequential iterations are applied for velocity and turbulence quantities in the melt. The FE-simulation of the thermal-capillary analysis and the FV-simulation for melt convection are iterated sequentially until convergence. The governing equations and the turbulence models used in this analysis are presented in Section 2. Performance of the turbulence models for the prototype model of convection in a CZ system are compared in Section 3. The hybrid FE/FV numerical method is presented in Section 4. Sample results of global simulation for the prototype Czochralski system are reported in Section 5.
a rate of several centimeters per hour. After an initial transient, the crystal is maintained at constant diameter by changing the pulling rate and the heater power. The crystal and crucible are rotated in opposite directions to stabilize the melt motion near melt/crystal interface and to provide axial symmetry in the delivering of heat to the crucible.
Growth chamber Si crystal
Heat shield Heater
Si melt
Silica crucible
Crucible holder
Figure 1: Schematic view of Czochralski silicon crystal growing system.
Analysis of heat transfer in the CZ systems must account for heat transfer by conduction in all phases, radiation between all surfaces, crystal growth, latent heat release, and convective heat transport in the melt, which is turbulent in the large CZ system of interest in commercial silicon productions (Kinney and Brown, 1993). Heat transfer a ects the shape of the melt/crystal solidi cation interface, which in turn a ects the shape of the melt/gas meniscus, which is set by a balance of gravity, surface tension and hydrodynamic forces. CZ growth is a meniscus-de ned crystal growth process; models that account for this aspect are termed thermal-capillary analysis (Brown, 1988). Our goal has been to develop an Integrated Hydrodynamic Thermal-Capillary Model (IHTCM) of large-scale CZ systems. A major complexity of this analysis is description of the convection in the melt, which is driven by buoyancy, surface-tension di erences and by rotation of the crystal and crucible. Although direct measurements of the ow has not been possible there is substantial evidence that it is turbulent; see references in Kinney
2
excepting the melt/crystal interface, where solidi cation heat is released. At the exposed surfaces conductive uxes are balanced by radiative uxes. Radiative heat exchange is calculated in di usive-gray surface approximation; the ambient temperature is the function of the point on a surface. On the free melt surface the capillary pressure under the curved boundary is balanced by hydrodynamic pressure and normal viscous stress. The wetting angles at melt/crystal/gas and melt/crucible/gas point are prescribed as material properties. The meniscus shape and the solidi cation interface are computed as free surfaces in the nite element algorithm. In the quasi-steady-state approximation the power to the heater is computed from the constraint of xed crystal radius. Details of the nite element algorithm used to solve the thermal-capillary problem are presented in (Sackinger et al., 1989) and (Bornside et al., 1990). The IHTCM includes analysis of turbulent convection in the melt using the k ? model. The convection is assumed to be axisymmetric and steady. Boundary conditions on k and are that both are zero on solid surfaces and have no normal gradients at the melt free surface. Thermocapillary convection is not included in our analysis, because it has been shown to be insigni cant for the intense turbulent convection seen in these systems.
2 GOVERNING EQUATIONS
The Reynolds average governing equations for turbulent energy transport and convection in the melt are Cpu rT
= r((K + Tt )rT );
(1)
and
u ru = ?rp + r + g (T ? Tm )ez ; (2) @ui @uj + ): (3) r u; i;j = ( + t )( @x j @xi In the k ? turbulence model the turbulent viscosity t is expressed in terms of kinetic energy k and dissipation functions
as
k2 t t = = f C ;
(4)
where k and are computed from transport equations u rk = r(( + t )rk ) + k Pk + Gk ? + D;
(5)
u r = r(( + t )r) + e C1 f1 (Pk + C3 Gk ) ? C2 f2 ) + E; k
(6)
Pk
@ui @uj @ui + ) ; = t( @x j @xi @xj
Gk
= ? Tt @T : @z
(7)
3 NUMERICAL METHOD
The numerical analysis begins with FE discretization for the whole CZ system. FE discretization has advantages for computational domains with complex geometry because FE method can be easily implemented for unstructured grids. Turbulent heat transfer in the melt is incorporated by including the velocity eld and turbulent viscosity computed from the previous FV iteration. The thermal-capillary model is discretized by Galerkin's method, which leads to large set of nonlinear algebraic equations because of di use grey-body radiation and the unknown interface shapes. These equations are solved eciently by Newton's method. The equations of motion in the melt are discretized by the FV method and solved sequentially equation by equation for set melt shape and temperature boundary conditions. Each complete iteration of this FE/FV coupled algorithm consists of two steps: (i) computation of temperature eld in the entire CZ system and the melt shape by the FE method for xed melt velocity, turbulent viscosity and normal component viscous stress on the free surface; (ii) calculation of the velocity, pressure, temperature, k, and in the melt by the FV method for xed melt shape and surface temperature distribution taken from (i). In the rst step the FE mesh in the melt and adjacent regions is calculated using an elliptic mesh generator, simultaneously with solution of the thermal-capillary model. In (ii) the FV mesh is constructed from the FE mesh. In the FE/FV algorithm the
and each variable has its usual meaning. The constants in the k ? model are taken as C = 0:09, k = 1:00, e = 1:30, C1 = 1:30, C2 = 1:92, T = 0:9, and C3 = tanh(juv =uh j), where uv and uh are the vertical and horizontal components of velocity, respectively. The damping functions (f1 , f2 , f ) and the terms (D, E ) model the transition from the viscous sublayer to fully developed turbulence in the bulk ow. The forms for these terms used in the low-Reynolds-number model of Jones and Launder (Henkes et al.,1991) give good predictions for convective boundary layers and convection in enclosures; see Nobile (1993). For this model f1 = 1:0; f2 = 1 ? 0:3 exp f = exp
?2:5
?
?Ret2
1 + Ret=50 ; p k2 Ret = ; D = ?2(r k )2 ; E = 2t (
@ 2 ui 2 ): @xj @xk
;
(8) (9) (10) (11)
The IHTCM is formulated using the quasi-steady-state crystal growth approximation (Derby and Brown, 1986). Energy
uxes are assumed to be continuous at all internal interfaces,
3
temperature in the melt is calculated in both steps. Excellent agreement between these two temperature distributions proves the consistency of the sequential calculations. The FV computer code uses collocated variable arrangements and was developed from the code and ideas of Ferziger and Peric (1996). Details of the FE algorithm are discussed by Bornside (1990).
6 7
4 COMPARISON OF TURBULENCE MODELS
8 7 6 5 4 3 2 1
2 3 4 5
1
T = Tm
8
The performance of the three turbulence models is made for the Czochralski bulk ow prototype problem. This approximation sets the free melt surface and melt/crystal interface as planar and simpli es the thermal boundary conditions on the melt to be for a vertical cylindrical crucible with the xed constant temperature Tm +T along boundaries, where Tm = 1683 K is the melting temperature, see Fig.2. The crucible height is equal to its radius. The ambient temperature for radiation is assumed constant along the free surface and equal to Tm . We discussed details of the numerical scheme and showed ow structures and distributions of turbulence quantities for the three turbulence models in (Lipchin and Brown (1998)). Here we presented only
ow structure for TL model in Fig.2 and distributions of the vertical velocity at the mid-height of the crucible in Fig.3 for comparison with the results reported by Zhang et al. (1996) for pure buoyancy ow. In agreement with these results velocity discrepancy for di erent turbulence models is small. The differences computed for turbulence quantities (k, , t ) are much larger. With increasing rotation rates of crucible and crystal the
ow structure becomes more complex and the velocity discrepancies for the three models become larger. However, the di erences in the computed values of the heat uxes thorough boundaries, the maximal values of the stream function and turbulent viscosity decrease with increasing rotation rates. The di erences between the maximal values of t for the JL/WF models and JL/TL models are 29% and 45% for pure buoyancy ow, 10% and 15% for crys = 20 rpm, cruc = ?8 rpm, and 6% and 5% for crys = 100 rpm, cruc = ?20 rpm. In general the predictions of the JL and WF models are closer to each other than to the predictions of the TL model. For the wall function approach the choice of distance of the rst computational point from the wall (the point where the wall function boundary conditions applied) is crucial and not well de ned. The value should be selected to satisfy two contradicting requirements. First, the distance should be large enough so that most of the rst layer nodes are outside of viscous sublayer. Also, the layer should be small enough that the velocity remains essentially parallel to the wall. It is dicult or impossible to satisfy both these constraints. Moreover, the mesh used with the WF model is inadequate for computation of the oxygen distribution in the melt because the very steep boundary di usive boundary layers adjacent to the crucible and melt/crystal interface are unresolved. Of the three
T = Tm + ∆T
2.5E+03 2.2E+03 1.9E+03 1.6E+03 1.2E+03 9.2E+02 6.0E+02 2.9E+02
a
T = Tm 8 7
6 4
5
2
1
3
5
8 7 6 5 4 3 2 1
1.1E+02 9.7E+01 8.2E+01 6.6E+01 5.0E+01 3.5E+01 1.9E+01 3.3E+00
4
1
3 2
T = Tm + ∆T
b
Figure 2: Stream function (scaled with Rcruc ) of turbulent flow , srf : , Rcruc , and for TL model T Rcrys . Results are shown for (a) pure buoyancy-driven flow and (b) flow with rotation ( crys , cruc ) 6 , and P r g T Rcruc 3 = Cp =K with Gr : .
0 01
= 50 K = 0 05 = 200 mm = 80 mm
= 20 rpm = ?8 rpm = 6 10 = = =
k ? models, only the low-Reynolds number model (JL) gives
consistent results in the presence of ow separation along the boundaries and approaches qualitatively the laminar ow as the intensity of the convection is decreases. We use the JL model in the analysis of the IHTCM reported below.
4
Silica crucible
3.0
Si crystal
JL
Si melt
TZ 1.0 4
-2.0
6
7
7
-5.0
0.25
0.5
0.75
5 4
3 1
Growth chamber
a 0
4
Heater
-4.0
3
Crucible holder
-3.0
2
8
-1.0
1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3
6
0.0
-6.0
5 8 7 6 5 4 3 2 1
4
WF
5
8
vertical velocity × 10
Heat shield
5
2.0
a
b
1
r/R cruc Figure 4: (a) mesh and (b) temperature Czochralski system.
distribution for prototype
3.0
JL 2.0
WF
3
vertical velocity × 10
The turbulent ow and temperature eld in the melt is shown in Fig. 5. Two secondary ows are clearly seen: a buoyancydriven vortex near the crucible wall and a Taylor-Proudman recirculation under the rotating crystal. The temperature eld appears very di usive, with smooth isotherms that are not convected by the ow. This smoothness is a result of the intense turbulence in the bulk ow, as shown by the contours of t in Fig. 5 b; the maximum value of t is 1097 along the free surface.
TZ
1.0
0.0
-1.0
-2.0
6 CONCLUSIONS
b -3.0
0
0.25
0.5
0.75
The large di erences between the predictions of di erent turbulence models appeals strongly to the direct comparison with experiments. Unfortunately, there are no direct measurement of melt convection in large-scale CZ systems. We believe that the evaluation of turbulence models is possible only in the framework of integrated modeling of CZ process and comparison with indirect measures of convection such as the shape of melt/crystal interface, the oxygen distribution in the crystal, the heat power required for growing a prescribed crystal radius. In the absence of these comparison, the ability of turbulence model to reproduce qualitative features of the ow points strongly to the desirability of the low-Reynolds number model of JL over the conventional k ? model and the two layer model tested here; see Lipchin and Brown (1998). The new hybrid FE/FV method for solution of the IHTCM shows great promise as a robust numerical method for coupling the turbulence calculations with accurate solution of the thermal-capillary model of the CZ system.
1
r/R cruc Figure 3: The vertical velocity (scaled with =Rcruc ) at the midheight of the crucible (a) pure buoyancy flow and (b) rotating flow;
crys = 20 rpm, cruc = ?8 rpm.
5 INTEGRATED FE/FV SOLUTION OF IHTCM
The FE/FV algorithm was applied to solution of IHTCM for the prototype Czochralski con guration shown in Fig.4. The prototype system is for the growth of 104 mm radius crystal from a 220 mm radius crucible; the crystal and crucible rotation rates are cruc = ?8:9 rpm and crys = 15 rpm.
5
34, No. 6, pp. 881-911.
Stream function 8 4
1 2 3
7 6 5 4 3 2
1
8 7 6 5 4 3 2 1
123.1 107.4 91.6 75.9 60.2 44.5 28.7 13.0
8 7 6 5 4 3 2 1
975.2 853.3 731.4 609.5 487.6 365.7 243.8 121.9
[3] Derby, J.J., and Brown, R.A., 1986, \Thermal-Capillary Analysis of Czochralski and Liquid Encapsulated Czochralski Crystal Growth," J. Crystal Growth, Vol. 74, pp. 605624. [4] Ferziger, J.H. and Peric, M., 1996, \Computational Methods for Fluid Dynamics,", Springer. [5] Iacovides, H., Launder, B.E., and Li, H.-Y., 1996, \The Computation of Flow Development through Stationary and Rotating U-Ducts of Strong Curvature," Int. J. Heat and Fluid Flow, Vol. 17, pp. 22-33.
a Turbulent viscosity
8
6
5
7
4
3 2 1
[6] Henkes, R.A.W.M., Van Der Vlugt, F.F., and Hoogendoorn, C.J., 1991, \Natural-Convection Flow in a Square Cavity Calculated with Low-Reynolds-Number Turbulence Models," Int. J. Heat Mass Transfer, Vol. 34, No.2, pp. 377-388. [7] Kinney, T.A., and Brown, R.A., 1993, \Application of Turbulence Modeling to the Integrated Hydrodynamic Thermal-Capillary Model of Czochralski Crystal Growth of Silicon," Journal of Crystal Growth, Vol. 132 pp. 551-574.
b
3
6
8 7
4
5
2
1
Temperature 8 7 6 5 4 3 2 1
[8] Lipchin, A and Brown, R.A., 1998, \Comparison of the Three Turbulence Models for Simulation of the Melt Convection in Czochralski Crystal Growth of Silicon", J. Crystal Growth, submitted.
1.014 1.012 1.010 1.008 1.007 1.005 1.003 1.001
[9] Nobile, E., 1993, \Comparison of turbulence models for side-heated cavities," Turbulent Natural convection in enclosures. A computational and experimental benchmark study (R.A.W.M. Henkes and C.J. Hoogendoorn, Edrs., EETI, Paris), pp. 214-233.
1
c
[10] Sackinger, P.A, Brown, R.A., and Derby, J.J., 1989, \A Finite Element Method for Analysis of Fluid Flow, Heat Transfer and free Interfaces in Czochralski Crystal Growth Growth," International Journal for Numerical Methods in Fluids, Vol. 9, pp. 453-492.
Figure 5: (a) stream function (scaled with Rcruc ) of turbulent flow, (b) turbulent viscosity t (scaled with ), and (c) temperature field in the melt. The Grashof number defined in terms of the crucible inner radius and the maximum temperature difference in the melt is 9. Gr
= 5 10
[11] Zhang, T., Ladeinde, F., Zhang, H., and Prasad, V., 1996, \Comparison of Turbulence Models for Natural Convection in Enclosures: Applications to Crystal Growth Processes," Proceedings, 31st ASME National Heat Transfer Conference, Part.1, Houston, TX, HTD-Vol.323, pp. 17-26.
References
[1] Bornside D.E., Kinney, T.A., Brown, R.A., and Kim, G., 1990, \Finite Element/Newton Method for the Analysis of Czochralski Crystal Growth with Di use-Grey Radiative Heat Transfer," International Journal for Numerical Methods in Engineering, Vol. 30, pp.133-154. [2] Brown, R.A., 1988, \Theory of Transport Processes in Single Crystal Growth from the Melt," AIChE Journal, Vol.
6
