Aperiodic stepwise growth model for the velocity and orientation dependence of solute trapping
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M. J. Aziza) Solid State Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831 (Received 30 January 1987; accepted 8 April 1987) An atomistic model for the dependence on interface orientation and velocity v of the solute partition coefficient k during rapid solidification is developed in detail. Starting with a simple stepwise growth model, the simple continuous growth model result is obtained for k(v) when the growth steps are assumed to pass at random intervals rather than periodically. The model is applied to rapid solidification of silicon. Crystal growth at all orientations is assumed to occur by the rapid lateral passage of (111) steps at speeds determined by the interface velocity and orientation. Solute escape is parametrized by a diffusion coefficient at the edge of the moving step and a diffusion coefficient at the terrace, far from the step edge. The model results in an excellent fit to data for the velocity and orientation dependence of k of Bi in Si.
I. INTRODUCTION When solidification of binary liquid alloys occurs at interface velocities v on the order of or greater than the diffusive velocity of the solute atoms at the liquid/solid interface, nonequilibrium fractions k of the solute in the liquid at the interface are incorporated into the solid. This phenomenon is known as solute trapping. On the one hand, the continuous growth model1 (CGM) fits the k(v) data2 quite well. In this model, solute-solvent redistribution across the interface and the simultaneous advance of the interface are treated as separate processes that occur under microscopically steady-state conditions. As originally derived, however, the CGM does not provide a detailed atomistic description of the process of solute trapping. On the other hand, the stepwise growth model3 (SGM) provides a detailed atomistic picture of the solute trapping process: crystallization of a monolayer, occuring by the rapid lateral passage of a growth step, and solute-solvent redistribution alternate in time. However, in its simplest form the SGM yields a form for k(v) with too strong a velocity dependence to fit the data. Using pulsed laser melting techniques, Aziz and White4 recently measured the orientation dependence of k at constant v and accounted for their results with an extension of the CGM. They assumed, as did Pfeiffer,5 that growth occurs by the periodic rapid lateral passage of {111} steps separated by an interval determined by the velocity and orientation of the interface, as shown in Fig. 1. They also assumed that all of the solute escape occurs at the edge of the moving step and that there is no escape once the solute atom is incorporated into the ter' Permanent address: Division of Applied Sciences, Harvard University, Cambridge, Massachusetts 02138.
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J. Mater. Res. 2 (4), Jul/Aug 1987
http://journals.cambridge.org
race. Their model recovers the CGM velocity dependence and accounts fairly well for the observed sharp rise in A: as the {111} orientation is approached. However, the model predicts that &— 1 at { i l l } w
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