Topography of laminar and oscillatory thermal and thermocapillary convection in a Czochralski process

Transcription

1 Topography of laminar and oscillatory thermal and thermocapillary convection in a Czochralski process J. A. Szymczyk*, J. Aleksic**, P. Ch. Zielke** *Department of Thermo Fluid Dynamics and Turbo Machines, Fachhochschule Stralsund (University of Applied Sciences), Germany ** Stralsund Institute for Energy and Environment e.v., Germany Abstract The Czochralski process is the most common method for growing large semiconductor mono-crystals. All experiments and measurements for the explanation and optimisation of the process were made on experimental equipment simulating the process at room temperature. In melt thermocapillary, thermal and forced convection are induced according to thermal, kinematical and geometrical boundary conditions. Thermocapillary convection is a boundary driven convection caused by a temperature gradient on the fluid surface. Because of the friction the impulse of the flow is transmitted to deeper fluid layers and the convection is spread through the whole fluid. 1 Introduction The application of semiconductor mono-crystals and their oxides has enormously increased during recent decades. Last year s production of silicon mono-crystals was t and it is growing exponentially. About 80% of them are produced with the Czochralski method. The quality of the mono-crystals mainly depends on the interface and transport phenomenon occurring in the melt during the production process. One of the possible causes for fluctuations of the growth rate is the instationary convection in the melt, generated by the thermal and kinematic boundary conditions. To avoid any kind of disturbing factors on the flow and in this way on the crystal quality, it is necessary to investigate the process and determine all influences on the convection. On the other hand, considering the production s costs of final mono-crystal products (like solar cells or micro chips) the main part of 37% are the crystallisation costs. It is of great

2 importance to optimise this process and to reduce these costs. Most investigations of the Czochralski process are numerical, e.g. with the use of finite elements method like the group of Tanahashi. It is used for two-dimensional flow simulation of the crystal growth at high Reynolds No. with the help of the one-dimensional Burger s equation [1], for solving the Navier-Stokes equation for incompressible fluids [2], for analysis of viscous-elastic flow under high share rate [3], for three-dimensional simulation of an transient Marangoni convection [4] etc. Therefore it is necessary to simulate the process experientially with parameters commensurably to the real crystal growth process. 2 The experimental equipment and optical measurement systems The experimental equipment was designed in a way which best matches the real Czochralski-process [5-10]. The experimental equipment allows the setting of four thermal and two kinematical boundary conditions. There is also the possibility of using diverse stick/cubicle geometries, fluids with different Prandtl No. and to influence the thermocapillary convection additionally. Thermo - chromic liquid crystals (TLC) were put in the experimental fluid. These little particles change colour with temperature (highlight the temperature field), and act as tracers for the velocity measurements. Two optical measurement systems were used simultaneously. For quantitative determination of the temperature fields we developed a new method based on the selective light reflection from the TLCs depending on the temperature called Particle Image Thermometry (PIT)[5], [7]. To determine the velocity fields we used Particle Image Velocimetry (PIV). 3 Topography of the flow Due to the dominantly influencing variables resp. dominantly convection kind the measuring cubicle can be divided into four areas (Fig. 1): 1. the thermocapillary area (free surface area with the boundary layer and a dominating tangential thermocapillary force), 2. the gravitation area (the cold jet area with a dominating gravitation force), 3. the thermal area (the hot wall area with the boundary layer and a dominating thermal force) and 4. the friction area (the interior part of the cubicle with a dominating friction force). The convection roll builds up in the friction area as a consequence of the influence of the first three areas (Fig. 2).

3 Figure 1: Areas in the cubicle with the boundary layers considering the dominating forces Heat re moval Heat removal Heat supply Figure 2: Flow topography with the acting forces

4 Due to the radial temperature gradient dt/dr on the fluid surface (area 1) between the wall and the middle of the cubicle arises a tangential tension, which causes the fluid to move towards the increasing interface tension from the edge to the middle of the cubicle. A swift, cold, downward flow (called jet) forms in the middle of the cubicle (area 2) and is reinforced trough the thermal convection. Due to the hot wall (area 3) a thermal lifting force appears and move the particles in this area to the top of the cubicle. 4 Presentation of the results The topography of the flow and temperature fields are explained fo r different Prandtl (Pr), Reynolds (Re), Marangoni (Mg), Bond (Bo) and Froude (Fr) Numbers and aspect ratios. For small Prandtl Numbers the oscillatory Marangoni convection at the fluid surface is determined and analysed. During the process of crystal growth the aspect ratio constantly changes and becomes smaller. It is necessary that the stability of the flow be maintained during this change. The flow topography and the temperature distribution were investigated for a 80 mm crucible and 30 mm crystal diameter and constant Prandtl No (Fig. 3A). During the whole experiment, the temperature and the velocity fields stay stable and axial symmetry remains unchanged due to the very small Reynolds No. With a decreasing aspect ratio, the Marangoni No. also decreases. It suggests minor boundary driven convection and heat transfer, which show also the less extended warm areas in these configurations (Fig. 3 B, D, F). The thermocapillary convection rolls are well formed and spread trough the whole fluid (Fig. 3 C, E, G). The cold jet forms and appears for all aspect ratios. In all three configurations no secondary temperature gradient or convection roll occurred. 80 mm 29.5 C H 19.5 C 30 mm 20.0 C A Pr = 8227

5 B H=72mm, A = 0.9, Mg=591, Re=0.07 C D H=60mm, A = 0.75, Mg=492, Re=0.06 E F H=40mm, A= 0.5, Mg=382, Re=0.04 G Figure 3: Boundary conditions (A) with the appendant temperature fields in C (B, D, F) and streamlines (C, E, G) and the dimensionless numbers (Pr, Mg, Re) The boundary layer form for the thermocapillary convection (delta) was determined graphically with the help of horizontal velocity profiles. As expected, the boundary layer becomes proportionally thinner for decreasing aspect ratios with a thickness of about half of the fluid height H (Fig. 4). This value is very high compared to the fluid height, which explains the stable and laminar fluid flow.

6 - H [mm] delta (H = 72 mm) delta (H = 60 mm) delta (H = 40 mm) R [mm] Figure 4: Boundary layer forms for the thermocapillary convection for different fluid heights H Figure 5 shows the vertical velocity profiles v for the same configurations and for the heights of the centre of the convection rolls. The profiles are symmetrical for all aspect ratios. The values of velocity are extremely large in the middle of the cubicle marking the cold jet, with a thickness of approximately 20 mm for all aspect ratios. The jet velocity increases with increasing aspect ratio. The centres of convection rolls are all 1/4 cubical diameter (20 mm) far from the symmetrical axis. 0,1 0,05 v [mm/s] ,05-0,1 v (H = 72 mm) v (H = 60 mm) v (H = 40 mm) -0,15-0,2-0,25 R [mm] Figure 5: Vertical velocity profiles for the height of the convection roll centre for different fluid heights H

7 5 Summary and Conclusions The topography of the flow and temperature fields are explained with the help of whole-field optical measuring systems and thermo -chromic liquid crystals (TLCs) for different Prandtl, Reynolds, Marangoni, Bond and Froude Numbers and aspect ratios. For small Prandtl Numbers the oscillatory Marangoni convection at the fluid surface is detected. The presented results have shown the effect of changing aspect ratios on the quantitatively determined temperature and velocity fields and boundary layers. The aspect ratio has no influence on the flow topography, cold jet thickness or on the distance between the convection roll centres. The fluid becomes negligibly warmer for bigger aspect ratios. The boundary layers are stable with a thickness of about half fluid height. The aspect ratio has generally no influence on the melt flow topography in a Czochralski simulated crystal grow. In experiments with small Prandtl No. an oscillatory flow occurred. The oscillation frequency is very complicated and dependent on the Prandtl No., temperature difference on the fluid surface and on the rotation rate of the cubicle and crystal. The cause for the oscillation as well as the phase behaviour of velocity and temperature oscillations have to be analysed. Acknowledgement The authors would like to thank the Federal Ministry of Education and Research, for promoting the project, under number: The second author wishes to express his thanks to the University of Rostock and the Stralsund Institute for Energy and Environment e.v., Germany, for the financial support. References [1] Ikeda, H.; Tanahashi, T.: New Hybrid-Streamline-Upwind Finite-Element method for a Dual Space (Verification for Two-dimensional Advection- Diffusion Equation), JSME International Journal, Series B, Vol. 39, No. 4, 1996 [2] Makihara, T.; Shibata, E.; Tanahashi, T.: Application of the GSMAC-CIP method to Incompressible Navier-Stokes Equations at High Reynolds Numbers, IJCFD Vol. 12, pp , 1999 [3] Fuijeda, T.; Tanahashi, T.: Finite Element Analysis of Visco-elastic Flow Under high Shear Rate Using the GSMAC-method, IJCFD Vol. 13, pp , 2000 [4] Kohno, H.; Tanahashi, T.: Three-dimensional Numerical Simulation of Unsteady Marangoni Convection in the CZ method Using GSMAC-FEM, ICES 2K: August 2000, Los Angeles, U.S.A, 2000 [5] Aleksic, J., Szymczyk, J. A., Leder, A., Kowalewski T. A.: Experimental Investigations on Thermal, Thermocapillary and Forced Convection in

8 Czochralski Crystal Growth Configuration, CMEM 2001, 4-6 Jun 2001, Alicante, Spain, Computational Methods and Experimental Measurements X, pp , ISBN , WIT Press, 2001 [6] Aleksic, J., Zielke, P., Szymc zyk, J. A., Delgado, A.: Diagnose des Geschwindigkeits- und Temperaturfeldes in einer Czochralski Simulation, 8. Fachtagung Lasermethoden in der Strömungsmesstechnik neuere Entwicklungen und Anwendungen GALA 2000 Tagungsband, 37, ISBN , Shaker Verlag 2000 [7] Aleksic, J., Szymczyk, J. A., Leder, A.: Interaction of the thermocapillary, thermal and forced convection in the Czochralski-configuration of the silicon crystal growth, Developments in Theoretical and Applied Mechanics, Volume XXI, XXI Southeastern Conference on Theoretical and Applied Mechanics, May 19-21, 2002, Orlando, Florida, USA, pp , ISBN , Rivercross Publishing, Inc. Orlando, 2002 [8] Szymczyk, J. A., Aleksic, J., Zielke P.: Wärmetransport in einer Zweiphasenströmung einer Czochralski- Konfiguration, Heat Transfer and Renewable Sources of Energy 2002, Miedzyzdroje, Poland , pp , ISBN , 2002 [9] Aleksic, J., Zielke, P., Szymczyk, J. A.: Temperature and Flow Visualization in a Simulation of the Czochralski Process using TLCs, VIM- 01, International Symposium on Visualization and Imaging in Transport Phenomena, May 05-10, 2002, Antalya, Turkey, submitted for publication [10] Aleksic, J., Szymczyk, J. A., Leder A.,: Anwendung optischer Ganzfeldverfahren zur Strömungsdiagnose einer Zweiphasenströmung, 10. Fachtagung Lasermethoden in der Strömungsmsstechnik neuere Entwicklungen und Anwendungen, GALA 2002 Tagungsband, 15, ISBN , Universität Rostock 2002