Radial segregation induced by natural convection and melt/solid interface shape in vertical bridgman growth

Abstract

The influence of natural convection in the melt and the shape of the melt/solid interface on radial dopant segregation are analyzed for a prototype vertical Bridgman crystal growth system by finite element methods that solve simultaneously for the velocity field in the melt, the shape of the solidification isotherm, and the temperature distribution in both phases. Results are presented for crystal and melt with thermophysical properties similar to those of gallium-doped germanium in Bridgman configurations with melt below (thermally destabilizing) and above (stabilizing) the crystal. Steady axisymmetric flows are classified according to Rayleigh number and are either nearly equal to the growth velocity, have a weak cellular structures or have large amplitude cells. The flows in the two Bridgman configurations are driven by different temperature gradients and are in opposite directions. Finite element calculations for the transport of a dilute dopant by these flow fields reveal radial segregation levels as large as 60% of the mean concentration. Radial segregation is most severe at an intermediate value of Rayleigh number above which the dopant distribution along the interface levels as the intensity of the flow increases. The complexity of the concentration field coupled with calculations of the effective segregation coefficient show the coarseness of the usual diffusion-layer approximation for describing the dopant distribution adjacent to the crystal. The length of the insulated zone and the ratio of thermal conductivities between melt and crystal are identified as additional critical parameters for setting the degree of radial segregation.