Quantitativeness of phase-field simulations for directional solidification of faceted silicon monograins in thin samples.

We report the results of a two-dimensional reference model for the formation of facets on the left and right side of a silicon monograin that is solidified by pulling a thin sample in a constant temperature gradient. Anisotropy functions of both the surface energy and the kinetic attachment coefficient are adapted from a recent model for free growth of silicon micrometer-sized grains [Boukellal et al., J. Cryst. Growth 522, 37 (2019)0022-024810.1016/j.jcrysgro.2019.06.005.]. More precise estimates of the physical parameters entering these functions are obtained by reanalyzing available experimental results. We show that the reference model leads to a differential equation for the shape of the solid-liquid interface. The numerical solutions of this equation give a reference law Λ(V_{f}) relating the facet length Λ to the facet normal velocity V_{f}. In parallel, phase-field simulations of the reference model are performed for two growth orientations, [001] and [011]. Facet lengths Λ obtained from simulations at different facet velocities are first extrapolated to the limit of vanishing interface width. This extrapolation is made possible by constructing a master curve common to the whole range of V_{f} values considered. The extrapolated Λ values are then compared with the ones predicted by the Λ(V_{f}) reference law. Both sets give comparable values, with an accuracy of a few percent, which confirms that the phase-field model can give quantitative results for faceted solidification of silicon.