Phase field model for precipitates in crystals

Abstract

Oxygen precipitate caused by oxygen supersaturation is the most common and important defects in Czochralski (CZ) silicon. The presence of oxygen precipitate in silicon wafer has both harmful and beneficial effects on the microelectronic device production. Oxygen precipitates are useful for gathering metallic contaminants away from the device regions and for increasing the mechanical strength of the wafer [Borghesi, 1995], but they also can destroy the electrical and mechanical characteristics of the semiconductor and microelectronic devices [Abe, 1985; Kolbesen, 1985]. The understanding of the mechanism of the formation and growth of the oxygen precipitates in CZ silicon is a key to improve the quality of silicon wafer. The goal of this thesis is to provide a full understanding of the growth of an isolated oxygen precipitate in CZ silicon and its morphological evolution by means of phase-field method, and to gain the insight of the morphological transition of the oxygen precipitate and the distribution of oxygen, vacancy, and self-interstitial around the single oxygen precipitate. The traditional approach to simulate multiphase system is the sharp interface model. Sharp interface model requires tracking the interface between phases, which make the simulation much difficult and complicate. Phase-field model offers an alternative approach for predicting mesoscale morphological and microstructure evolution of inhomogeneous multiphase system. The most significant computational advantage of a phase-field model is that explicit tracking of the interface is unnecessary. In this thesis, the phase-field model is applied to simulate the evolution of oxygen precipitates in CZ silicon. A phase-field model for a two-component inhomogeneous system was first derived to set up the framework of phase-field method and a dynamically adaptive finite element method also was built to specifically solve phase-field equations. This model was used to investigate the effects of interfacial and elastic properties on the growth of a single precipitate, coarsening of two precipitates, and competitive growth of multiple precipitates. For an isolated precipitate growth, both elastic energy and interfacial energy affect the precipitate morphological evolution.

(cont.) Numerical results show the shape of the precipitate is determined by the relative contributions of elastic energy and interfacial energy, the degree of elastic anisotropy, and the degree of interfacial anisotropy. A dimensionless length scale LS3 was defined to represent the relative contributions of the interfacial energy and elastic energy. For large LS3 (LS3 > 5), the anisotropic elasticity plays a dominant role and precipitate evolves to held the elastic anisotropy even if the interfacial anisotropy is very strong. However, if LS3 ~1 or elasticity is isotropic, the strong anisotropy ([epsilon]4 =/> 0.05 ) of the interface will be the dominant factor to determine the precipitate shape. The growth rate of an isolated precipitate follows the diffusion-controlled power law. The elasticity significantly decreases the precipitate growth rate, while the anisotropy of the interface does not. Coarsening of two precipitates was also explored with different interfacial and elastic properties. The results also show that both elasticity and interfacial anisotropy enhance the coarsening rate. For competitive growth of multiple precipitates, a gap was found to be developed between the precipitates because of the precipitate screening, but this gap could be destroyed by increasing the interfacial energy or introducing elastic energy. Based on the framework of the previous phase-field model, another phase-field model coupling CALPHAD thermodynamic assessment was developed to simulate the growth of the oxygen precipitate in CZ isilicon. An asymptotic analysis was performed to understand the phase-field model at the sharp interface limit and all physical principles of the solid precipitate growth problem were recovered. a Cristobalite and amorphous oxygen precipitates were calculated at different orientations and temperatures. Disk-like shape, square, ellipse, a slightly deformed sphere are reproduced for oxygen precipitates, which agrees with the experimental observations very well. In addition, the growth rates of amorphous precipitates and a cristobalite precipitates at different temperatures show that at high temperature 1100 °C, amorphous precipitate has the largest growth rate, while at low temperature 900 °C, a cristobalite precipitate grows faster.

(cont.) This qualitatively explained why different polymorphs and shapes of the oxygen precipitate were observed in experiments at different annealing temperatures.