Mathematical modeling of the dissolution process of silicon into germanium melt

Abstract

Numerical simulations were carried out to study the thermosolutal and flow structures observed in the dissolution experiments of silicon into a germanium melt. The dissolution experiments utilized a material configuration similar to that used in the Liquid Phase Diffusion (LPD) and Melt-Replenishment Czochralski (Cz) crystal growth systems. In the present model, the computational domain was assumed axisymmetric. Governing equations of the liquid phase (Si-Ge mixture), namely the equations of conservation of mass, momentum balance, energy balance, and solute (species) transport balance were solved using the Stabilized Finite Element Methods (ST-GLS for fluid flow, SUPG for heat and solute transport). Measured concentration profiles and dissolution height from the samples processed with and without the application of magnetic field show that the amount of silicon transported into the melt is slightly higher in the samples processed under magnetic field, and there is a difference in dissolution interface shape indicating a change in the flow structure during the dissolution process. The present mathematical model predicts this difference in the flow structure. In the absence of magnetic field, a flat stable interface is observed. In the presence of an applied field, however, the dissolution interface remains flat in the center but curves back into the source material near the edge of the wall. This indicates a far higher dissolution rate at the edge of the silicon source.