We examine a cylindrically symmetric, transient and thermally coupled melt flow in Czochralski crystal growth process. We consider the melt only, i.e., we neclect the thermal environment surrounding the melt (furnace, heaters, crystal and crucible).

The next figure shows the surface of the silicon melt with the growing crystal being raised slowly from the melt. The thin tube on top of the crystal is the seed crystal, which is dipped into the melt to initiate the growth process.

Our mathematical model consists of the coupled Navier-Stokes and heat equations. The melt is considered incompressible, and the Boussinesq approximation is used. On the crucible wall, we take Dirichlet boundary conditions for the velocity and temperature. We assume that on the top boundary of the melt there is a crystal at a constant temperature rotating in the opposite direction to the rotation of the crucible. At the melt-gas interface between the crystal and crucible wall, we set idealized radiation boundary condition for the temperature.

The equations are discretized by a stabilized finite element method in space and implicit Euler in time. The nonlinearities in the Navier-Stokes equation and in the radiation boundary condition are solved with a Newton type iteration.

Transient Cylindrically Symmetric Simulation of the Silicon Melt Flow

The following figures show the temperature field and the velocity field from a transient cylindrically symmetric simulation of the melt flow. The flow has entered a developed state which seems to be quasiperiodic, with a period of about 20 seconds. The material parameters, boundary conditions and dimensions of the model correspond to a real industrial growth environment.

Here are a couple of animations from a simulation with the same physics, but with a bit different geometry

• Animation of the velocity field (mpg ~4 MB)
• Animation of the temperature field (mpg ~2 MB)

See also the simulation of the melt flow using adaptive mesh refinement..

3D Simulation of the Silicon Melt Flow

It is known, that the melt flow is actually three-dimensional, and it might also be mildly turbulent. We have also made some simulations of the melt flow in 3D.

The figures below are from a simulation with density reduced by a factor of ten. The figures show the temperature and velocity fields at surface of the crucible. With this reduced value of density the flow patterns did approach steady state cylindrically symmetric configuration after transient more complicated patterns. However, as the figures shows, even in this simulation the (slight) break of symmetry is evident. It is more than likely that this break of symmetry will be much more pronounced as we approach the realistic parameters.

Read More about Czochralski Crystal Growth.

[1] P.A. Sackinger, R.A. Brown, J.J. Brown, International Journal for Numerical Methods in Fluids, VOL. 9, 453-492 (1989).