Instability of the bouyancy driven convection in Si melts during Czochralski crystal growth

Abstract

In this paper we study conditions under which the bouyancy driven convection in Czochralski systems will be unstable. The variable parameters are: the crucible radius R c, the crystal radius R D, the melt depth H, and the temperature difference between the crucible wall and the solid-liquid interface λ T, respectively. In order to initiate the instability of the flow we impose a small temperature disturbance over the axisymmetric temperature distribution at the crucible wall. 36 cases are simulated numerically in order to get enough material to make some inferences about this instability. It is shown that (1) the buoyancy driven flow is highly unstable for all nearly real growing conditions, (2) the instability of the flow depends on all varied parameters, but stronger on the melt depth than on the other parameters, and (3) the characteristics length L in the Grashof number (Gr) must be a function of the crystal radius ( R D), the crucible radius ( R c), and the melt depth H. So far as we know, it is for the first time that such a numerical study has been done for a Czochralski system. Nevertheless, the similar problem relating to the large-scale circulations in the atmosphere exists in the meteorology. The laboratory counterpart of this problem is known as the “sloping” convection in a rotating fluid, which deals with the dependence of the type of the flow on rotation rate Ω in an annulus filled with a water-glycerol solution and subjected to a horizontal temperature gradient [see R. Hide and P.J. Mason, Advan. Phys. 24 (1975)].