Analytical study of modified Czochralski crystal growth problem

F. Mokhtari^1,2 - A. Bouabdallah² - M. Zizi² - S. Hanchi³ - A. Alemany⁴

¹ Unit of Development of Silicon Technology, 2 Bd Frantz Fanon, BP 399, Alger Gare, Algeria
² LTSE Laboratory, University of Science and Technology USTHB, BP 32 El-Alia, BabEzzouar, Algeria
³ UER Mécanique EMP, BP 17 Bordj ElBahri, Algeria
⁴ Laboratoire EPM, CNRS, Grenoble, France

Abstract
An analytical study of Czochralski crystal growth problem is presented, therefore, we propose a model based on a three-dimensional axisymmetric approach using the Galerkin method for solving the system of equations of heat and momentum in a modified crystal growth process geometry as cylindrical-spherical. To facilitate the procedure of resolution related to the considered problem, we impose some approximation; the molten silicon is assumed to be a viscous, Newtonian and incompressible fluid satisfying the Boussinesq assumption. The thermophysical properties of the fluid are constant except for the density variation in the buoyancy force term. The flow is symmetric in the axial direction. Fixed temperatures are imposed at the walls of the melt crucible and the crystal melt interface. Thus we determine analytically the expressions of temperature and velocity field in the melt, and discuss the temperature properties for different values of the Grashof number. Figs 4, Refs 10.