The Czochralski crystal growth system with a periodic crystal growth rate and no back-melting

Abstract

By considering the Czochralski crystal growth for a periodic crystal growth rate a solution procedure is developed for the hydrodynamic, temperature, and solute fields in the limit Sc → ∞, Re → ∞, σ → 0, ∆ → 0, where Sc is the Schmidt number, Re is the Reynolds number, σ the Prandtl number of the liquid phase and ∆ = D S / D L , D S and D L being the solute diffusivities in the solid and liquid phases respectively. With this process, solutions are developed as double perturbation series in SC -½ and ∆ , with the method of matched asymptotic expansions used to overcome any singular behaviour. The effect upon the hydrodynamic field of a periodic crystal growth rate is shown to be confined to a perturbation O ( Sc -½ ). In the liquid solute field a thin boundary layer forms within the viscous boundary layer; the structure of the solute field in this region is demonstrated. The solute field in the solid phase is also considered. For the temperature field a three-layer structure is revealed in the liquid phase, directly due to the fluctuating growth rate. The temperature field in the solid phase is also considered. This work lays the foundations for considering the role of solute diffusion in the solid when the crystal is allowed to melt back.