Dopant transport during semiconductor crystal growth with magnetically damped buoyant convection

This paper presents an asymptotic model for the unsteady transport of a dopant during the growth of a semiconductor crystal from a melt with an externally applied magnetic field. The melt is divided into (1) mass-diffusion boundary layers where convective and diffusive mass transfer are comparable, and (2) a core region where diffusion is negligible, so that the concentration of each fluid particle is constant. A Lagrangian description of motion is used to track each fluid particle during its transits across the core between diffusion layers. The dopant distribution in each layer depends on the concentrations of all fluid particles which are entering this layer. The dopant distribution is very non-uniform throughout the melt and is far from the instantaneous steady state at every stage during crystal growth. The transient asymptotic model predicts the dopant distribution in the entire crystal.