Silicon single crystal produced by the Czochralski (CZ) process provides a majority of silicon substrates for the fabrication of microelectronic devices. A single crystal is being pulled from silicon melt and there are several complex features of fluid dynamics and heat transfer involved in the system. Complete transport phenomenon involved in this system is briefly summarized by Brown¹. Various driving forces influence the local flow structure of melt which in tern affects the transport of heat and mass transfer in the melt. Local hot spot in crucible increases corrosion rate of crucible and non-uniformities in flow affect the crystal quality.

During Czochralski (Cz) crystal growth process, various forces interact at different time scales which creates instability in flow. These driving forces are buoyancy, surface tension, Coriolis and centrifugal force. Convection in large crucible is turbulent in nature and accurate prediction of small-scale fluctuation is essential for capturing local flow and temperature profile. Considerable work has been done numerically and experimentally to study the flow structure of melt and its effect on crystal quality^2-3. Temperature difference in crucible manifests natural convection and marangoni stresses at melt-gas interface. Below crystal, Ekman layer form due to viscous sub layer. Due to crystal-crucible rotation, a strong Taylor-Proudman cell forms at the bottom of the crystal. Natural convection creates one additional vortex near crucible wall region away from the crystal. At high Rayleigh number (>10⁸), flow is unstable and non-symmetric. Rotational speed of crystal and crucible, marangoni stresses play an important role in local flow structure. These local flow structures affect the macro and micro defects in crystal. Numerous efforts have been carried in order to investigate the nature of these structures. However, due to nature of complexities involved in flow and numerical limitations, more efforts are needed to resolve different instabilities associated within the system.

In this work, two dimensional axisymmetric and complete three dimensional models were developed for melt flow inside the crucible in order to study the effect of rotational flow and marangoni flow. Flow is turbulent or transition to turbulent in nature, hence using k-e model with Renormalization Group Theory (RNG) and low Reynolds number turbulence model was used model the fluctuating component of flow. Resultant model equations, i.e., equations describing the conservation of mass, energy, momentum and turbulence closure terms were implemented in FLUENT 6.2. The temperature profile along the boundaries of the crucible and crystal were implemented from global simulations of literature.

Simulations were carried out at different crystal-crucible rotational speed and in presence and absence of marangoni stresses. Vortex formed near the crucible wall decreases with increase in crucible speed. Unsteady buoyancy plume was observed below crystal whose magnitude is depends on rotational speed and marangoni stresses generated at melt-gas interface. In co-current rotation, anti-clockwise cell is predominant under crystal (as shown in Figure 1a), while counter rotation, Taylor-Proudman cell is predominant under crystal (see Figure 1b). These local flow structures affect the temperature profile in melt considerably and higher temperature below crystal was observed with co-counter rotation of crystal and crucible. Simulated results will be helpful for understanding the effect of local flow macro and micro defect in crystal.

References:

1. Brown R. A., Theory of Transport Processes in Single Crystal Growth from the Melt. AIChE J, 1988, 34(6), 881.

2. Nakanishi H., Watanabe M. and Terashima K., Dependence of Si Melt Flow in a Crucible on Surface Tension Variation in the Czochralski Process, Jou. of Crystal Growth, 236, (2002), 523-328.

3. Lipchin A., R. A. Brown, Hybrid finite-volume/finite-element simulation of heat transfer and melt turbulence in Czochralski crystal growth of silicon, Journal of Crystal Growth 216 (2000) 192-203.