Oscillations of the longitudinal solutal profile in Bridgman growth of doped crystals

Numerical simulation is used in order to explain some unexpected oscillations of the longitudinal solutal profile detected in Bridgman growth of doped crystals. The modeling is performed for the Bridgman growth of doped semiconductors and fluoride crystals. From numerical analysis it is found that these oscillations appear in the conditions of a quasi-diffusive or purely diffusive transport of the solute in the melt, when a sharp variation of the solute concentration exists near the solidification front. In this case, the flow mixing effect is negligible, and the solute rehomogenization near the interface occurs slowly giving rise to some fluctuations of the concentration in the longitudinal direction of the crystal. The apparition of these oscillations is also conditioned by a supplementary condition related to the characteristic time of the flow. The amplitude and the period of these oscillations are analyzed as function on the diffusion coefficient and the growth rate. The numerical results show that the oscillation amplitude becomes huge at smaller values of the diffusion coefficient or higher growth rates. A critical growth rate can be also carried from a simpler formula proposed in the paper. Finally, optimal growth conditions are proposed in order to avoid the concentration fluctuations and to improve the crystal's chemical homogeneity.