TIME-DEPENDENT SIMULATION OF CZOCHRALSKI SILICON CRYSTAL-GROWTH

We have developed a detailed mathematical model and numerical simulati on tools based on the streamline up-wind/Petrov-Galerkin (SUPG) finite element formulation for the Czochralski silicon crystal growth. In th is paper we consider the mathematical modeling and numerical simulatio n of the time-dependent melt flow and temperature held in a rotational ly symmetric crystal growth environment. Heat inside the Czochralski f urnace is transferred by conduction, convection and radiation, Radiati ng surfaces an assumed to be opaque, diffuse and gray. Hence the radia tive heat exchange can be modeled with a non-local boundary condition on the radiating part of the surface. The position of the crystal-melt interface is solved by the enthalpy method. The melt flow is assumed to be laminar and governed by the cylindrically symmetric and incompre ssible Navier-Stokes equations coupled with the calculation of tempera ture.