Three-dimensional numerical analyses of the effects of a cusp magnetic field on the flows, oxygen transport and heat transfer in a Czochralski silicon melt

Abstract

The effects of a cusp magnetic field on the flows, oxygen transport and heat transfer in an actual Czochralski silicon melt are quantitatively analyzed using boundary conditions based on experimental measurements and a numerical method involving the direct solution of the Navier-Stokes and Maxwell equations for three-dimensional unsteady flows. The unstable and instantaneously asymmetric flow components and forced comvections emanating from the rotating crucible bottom in the absence of a magnetic field are effectively suppressed when a cusp magnetic field is applied. On the other hand, the rotational flows caused by crucible rotation become uniform and the circumferential velocity increases upon application of a cusp magnetic field. As a result of these rotating melt flows together with crystal rotation in the opposite direction, strong stable forced convections are produced that move towards the centre directly below the growth interface. Due to these changes in the melt flows, the concentration of oxygen at the growth interface is decreased and the radial uniformity is improved; temperature fluctuations within the melt are significantly reduced; and the radial temperature gradient is increased, whereas the temperature gradient decreases with respect to depth. These results regarding temperature and oxygen concentration are all in good agreement with experimental observations.