A thermal elastic model for constrained crystal growth with facets
type: Item
instance of: scholarly article
title: A thermal elastic model for constrained crystal growth with facets
zbMATH Open document ID: 1266.74013
author: C. Sean Bohun
author: Jinbiao Wu
author: Huaxiong Huang
published in: Journal of Engineering Mathematics
publication date: 2011-09-27T00:00:00Z
review text: 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…
Mathematics Subject Classification ID: 74E15
Mathematics Subject Classification ID: 74F05
Mathematics Subject Classification ID: 74H10
Mathematics Subject Classification ID: 74S20
zbMATH DE Number: 5950610
zbMATH Keywords: geometric anisotropy
zbMATH Keywords: asymptotic expansion
zbMATH Keywords: facet formation
zbMATH Keywords: finite-difference method
zbMATH Keywords: thermal stress
reviewed by: Ivan A. Parinov
MaRDI profile type: Publication
full work available at URL: s10665-010-9378-8
OpenAlex ID: W2164903432
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DOI: 10.1007/S10665-010-9378-8
description: scientific article; zbMATH DE number 5950610