Wheeler, A.A.
(1982)
The effect upon Czochralski growth of periodic modulation of the growth rate.
*Journal of Crystal Growth*, 56 (1), 67-76.
(doi:10.1016/0022-0248(82)90013-6).

## Abstract

In this paper we describe the results of a mathematical treatment of the Czochralski growth system when the growth rate is periodically modulated. Such a modulation simulates the effect of fluctuations in temperature which occur in the melt from which the crystal is grown. In this treatment we consider the system in the limits Sc? ?, R? ?, ?? 0, ??0, where Sc is the Schmidt number, R is the Reynolds number of the liquid phase, ? the Prandtl number, and ? the ratio of the solution diffusivities in the liquid and solid phases. Here we describe the effect of the modulation upon the solute, temperature and velocity fields. The velocity filed is perturbed by an amount O(Sc-1/2) from the steady-state flow induced by the rotation of the crystal, while the temperature field in the liquid exhibits a three-region structure. (Here the symbol O(x) means of order x. We use it here in its mathematical sense. That is if y = O(x), then there exists a constant k independent of y and x such that -kx <y < kx for all x.) A solute boundary layer of thickness O((Sc)-1/2) forms adjacent to the crystal interface in the liquid. When the crystal does not melt back the effect of solute diffusion in the solid is confined to a perturbation O(?). However if the crystal melts back the effect of solute diffusion in the crystal becomes important, producing a time dependent solute boundary layer in the crystal.

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