Modeling of Particle Engulfment during the Growth of Crystalline Silicon for Solar Cells

Abstract

A major challenge for the growth of multi-crystalline silicon is the formation of carbide and nitride precipitates in the melt that are engulfed by the solidification front to form inclusions. These lower cell efficiency and can lead to wafer breakage and sawing defects. Minimizing the number of these engulfed particles will promote lower cost and higher quality silicon and will advance progress in commercial solar cell production. To better understand the physical mechanisms responsible for such inclusions during crystal growth, we have developed finite-element, moving-boundary analyses to assess particle dynamics during engulfment via solidification fronts. Two-dimensional, steady-state and dynamic models are developed using the Galerkin finite element method and elliptic mesh generation techniques in an arbitrary Eulerian-Lagrangian (ALE) implementation. This numerical approach allows for an accurate representation of forces and dynamics previously inaccessible by approaches using analytical approximations. We reinterpret the significance of premelting via the definition of an unambiguous critical velocity for engulfment from steady-state analysis and bifurcation theory. Parametric studies are then performed to uncover the dependence of critical growth velocity upon some important physical properties. We also explore the complicated transient behaviors due to oscillating crystal growth conditions as well as the nonlinear nature related with temperature gradients and solute effects in the system. When compared with results for the SiC-Si system measured during ParSiWal experiments conducted by our collaborators, our model predicts a more realistic scaling of critical velocity with particle size than that predicted by prior theories. However, the engulfment growth velocity observed in the subsequent experiment onboard the TEXUS sounding rocket mission turned out to be unexpectedly higher. To explain this model discrepancy, a macroscopic model is developed in order to account for the natural convection in the terrestrial experiments. We demonstrate that the convective flows are able to keep most small particles suspended in the melt, so that the observed critical velocities and their variance are enhanced in the experiments conducted on earth. According to simulation results, some solutions, which are applicable in photovoltaic industry, to the inclusion problem are also discussed and studied.