The Czochralski crystal growth system with a periodic crystal growth rate and back-melting

Abstract

This paper considers the effect upon the Czochralski crystal growth process of modulating the crystal growth rate periodically, by imposing upon a constant mean growth rate a harmonic component. The case is considered when the amplitude of the harmonic component is sufficiently large that the crystal melts during part of the periodic cycle. The model of the Czochralski system discussed in the preceding paper is adopted. The system is considered in the realistic limit of Sc → ∞, σ → 0 ∆ → 0, where Sc = v / D L is the Schmidt number, σ= v / K L is the Prandtl number, and ∆ = D S / D L is the ratio of the solute diffusivities in the liquid and solid phases, v being the kinematic viscosity of the liquid, and K L the thermal diffusivity of the liquid. When the crystal melts back, large solute gradients are formed in the solid phase. It is due to the presence of these that the diffusion of solute in the solid becomes important, being responsible for the formation of a time-dependent solute boundary layer adjacent to the interface in the crystal. Four distinct periods throughout the cycle are identified in which this boundary layer has different structures. The results of numerical calculations arising from this work are presented.