A thermal elastic model for constrained crystal growth with facets
The paper presents a model predicting the facet formation for constrained crystal growth such as the Czochralski process. Under the assumptions of weak lateral heat flux and anisotropic effects, the perturbation solutions are derived for temperature and related thermal stress. The temperature inside the crystal is governed by the heat equation, whereas the domain is varying with time as the crystal being pulled out of the melt. Weak lateral heat flux is characterized by a small Biot number which allows one to expand the temperature into an asymptotic series and reduce a 3D problem to a series of 2D ones. Due to the assumed weak geometric anisotropic effects, these 2D problems are converted into a series of 1D (axisymmetric) problems. The authors also neglect the melt flow effect and assume that the axial heat flux from the melt at the crystal melt interface does not vary in the cross-sectional (radial and circumferential) directions. The thermal field is obtained by solving a quasi-steady equation, and a perturbation solution for temperature is presented. It is assumed that the ratio of the growth rates is determined by the crystal-lattice structure unless the constraint for directional growth is active. Using the growth model, the authors predict the facet formation and shape evolution during the entire growth process, starting from an arbitrary seed shape in any crystallographic orientation. Then, an expression for the thermal stress is determined by using isotropic elastic constants. Similar to the temperature problem, the expansions for thermal stresses are obtained, and an explicit solution is derived based on 1D temperature solutions. The stresses due to lateral temperature variation are obtained in the approximation of plane strain state. The thermal elastic stress is used as an indicator for predicting the formation of grown-in defects (dislocations) inside the crystal. Computational results are presented for idealized growth conditions in crystal with cubic symmetry, where the growth angle is fixed and an equilibrium lateral crystal shape is ssumed. The numerical results show that the facet formation greatly affects the thermal stress distribution, and defects are more likely to form in faceted crystals with certain seed orientations due to high stress levels while other orientations may be favored because of reduced stress levels.
scientific article; zbMATH DE number 5950610
scientific article