Das Stefan-Problem bei der Kristallzucht nach Czochralski

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Kopetsch, H. (1987)
  • Publisher: Kernforschungsanlage J├╝lich, Verlag
  • Subject:
    arxiv: Condensed Matter::Superconductivity

The shape of the crystal-melt interface in Czochralski crystal growth may strongly influence the quality of the grown crystal. Thus a numerical algorithm has been developed which allows us to study the dynamics of this interface subject to various growth conditions. Especially the hydrodynamics in the melt is taken into account. Mathematically, a moving boundary problem (Stefan problem) has to be solved along with the flow and temperature field in melt and crystal which is treated by the method of boundary fitted coordinates. The time-dependent domains of the field equations are mapped onto time-independent rectangles via algebraic transformations. These imply a constant crystal radius which is desired in the real growth process. Hence the pulling rate cannot be choosen arbitrarily but has to beadjusted at every time step. On the new computational domains the field equations are solved numerically by vectorizing finite difference methods. A MAC-scheme is employed for the hydrodynamics. With this technique a few hours of the growth of a silicon crystal were simulated during which time the melt level in the crucible drops significantly. Although the process para meters are kept constant the flow field exhibits fluctuations which affect the growth rate and the deflection of thecrystal-melt interface. This is likely to impair the crystal quality. For the first time an interface instability which is observed in oxid crystal growth and induced through crystal rotation could be simulated numerically. The amplitude of the interface flips from deeply convex into the melt to strongly concave with increasing crystal rotation rate. Maximum concavity corresponds to the point where the vortex induced through crystal rotation has become strong enough to extend from beneath the crystal to the free melt surface. The interface inversion is accompanied by vigorous oscillations of the pull rate which indicate unstable process conditions. Qualitative and partially quantitative agreement with experiments is found.
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