Crystal Growth and Solute Trapping
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CRYSTAL GROWTH AND SOLUTE TRAPPING
MICHAEL J. AZIZ* Division of Applied Sciences,
Harvard University,
Cambridge,
MA 02138
ABSTRACT A simple model for solute trapping during rapid solidification is presented in terms of a single unknown parameter, the interfacial diffusivity Di. A transition from equilibrium segregation to complete solute trapping occurs over roughly an order of magnitude in growth speed, as the interface speed surpasses the maximum speed with which solute atoms can diffuse across the interface to remain ahead of the growing crystal. This diffusive speed is given by Di/X, where X is the interatomic spacing, and is typically of the order 10 meters per second. Comparison is made with experiment. The steady-state speed of a planar interface is predicted by calculating the free energy dissipated by irreversible processes at the interface and equating it to the available driving free energy. A solute drag term and an intrinsic interfacial mobility term are included in the dissipation calculations. Steady-state solutions are presented for Bi-doped Si during pulsed laser annealing.
INTRODUCTION Laser annealing experiments have reached a crystal growth regime where deviations from local equilibrium are obvious and interface motion is no longer heat-flow limited. These experiments allow us the opportunity to study the interface kinetics in high-mobility systems for the first time. In this paper I describe a simple model for the kinetics of the fundamental atomic processes occurring at the interface during rapid solidification of binary alloys, one of which, of course, is doped silicon. The result is a pair of "interface response functions" which predict (a) how much impurity should be incorporated into the solid, and (b) how fast the interface should move; in terms of the local conditions at the interface, namely temperature and composition. The experiments (1,2] that have inspired the modeling efforts have shown that the chemical potential of the minor component of a binary alloy often increases as the partition coefficient k approaches unity during rapid solidification. A number of plausible models have been proposed [3-9] to explain how the host atoms persuade the impurity atoms to increase their chemical potential and join the crystal. The basis for the model described in this paper is shown in Fig. 1, which illustrates the fundamental difference between crystallization and interdiffusion according to the model for collision-limited growth of pure systems with simple interatomic potentials, which has been developed by Turnbull and coworkers [10]. A crystallization event consists of an atom shifting its position a small distance to move from a potential well in the liquid structure to a well on the crystal lattice as the solid arrows indicate. Since no bonds must be broken, the activation barrier for this reaction ought to be quite small; in the Turnbull collision-limited growth model the barrier is zero. A diffusive jump, on the other hand, is a different atomic process in this system. Position shi
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