A Semianalytical Thermal Stress Model for the Czochralski Growth of Type III-V Compounds

In this paper we describe a semianalytical approach to computing the temperature and thermal stress inside a III-V compound grown with the Czochralski technique. An analysis of the growing conditions indicates that the crystal growth occurs on the conductive time scale. A perturbation method for the temperature field is developed for an arbitrary crystal profile using the Biot number as a (small) expansion parameter. The zeroth order solution is one-dimensional in the axial direction. Explicit solutions are obtained for a cylindrical and a conical crystal. Under typical growth conditions, a parabolic temperature profile in the radial direction is shown to arise naturally as the first order correction. As a result, the thermal stress is obtained explicitly and its magnitude is shown to depend on the zeroth order temperature and Biot number. Both the axial temperature gradient and crystal profile are shown to be important for controlling thermal stress and defect density. Some issues relevant to growth conditions are also discussed.