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

Abstract

We report the results of a two-dimensional reference model for the formation of facets on the left and the 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 size grains [Boukellal et al., J. Cryst. Growth 522, 37 (2019)]. 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.

Interplay between facetting and confinement in directional solidification of silicon: A direct comparison between experiments and phase-field simulations

“…We only give the main lines of our TIPM model here; more details can be found elsewhere [17]. In order to take into account the dependence of the surface energy on the crystal orientation, the anisotropy function a s ( n) must be defined, where n is the unit vector normal to the solid-liquid interface.…”

“…Both functions are characteristic of the material under study. In the case of pure silicon, based on symmetry considerations and on experimental results, we recently proposed an analytical form for both functions [16,17]. In the phase-field model, the singularities of these anisotropy functions have to be smoothed out to avoid numerical divergences.…”

“…where α is the conical angle between the normal to the interface n and the unit vector n 111 normal to a {111} facet, and where is a parameter much smaller than one that is used to avoid a sharp cusp at α = 0 [26]. For the reasonable choice = 10 −2 , the amplitudes δ 2.5 and δ 0 1.75 were obtained by fitting phase-field equilibrium shapes [17] to their experimental counterparts [15]. For the kinetic attachment,…”

“…Since then, this model has proved successful, notably in providing quantitative support to theoretical results [14]. In our previous work, a comparison of TIPM simulations with experimental results for silicon free growth [15] and for directional solidification [6] recently allowed to quantitatively parametrize anisotropy functions for both the surface energy and the kinetic attachment coefficient [16,17]. In the case of silicon directional solidification, the quantitativeness of our numerical code was recently assessed by comparing two-dimensional (2D) simulations with an analytical model [17].…”

“…In our previous work, a comparison of TIPM simulations with experimental results for silicon free growth [15] and for directional solidification [6] recently allowed to quantitatively parametrize anisotropy functions for both the surface energy and the kinetic attachment coefficient [16,17]. In the case of silicon directional solidification, the quantitativeness of our numerical code was recently assessed by comparing two-dimensional (2D) simulations with an analytical model [17]. In the present study, the 3D version of the phase-field code is implemented to conform to the thin layer geometry of the experimental setup.…”