Coupled convection, segregation, and thermal stress modeling of low and high pressure Czochralski crystal growth

Czochralski (Cz) method is a dominant single crystal growth technology for microelectronics applications. The demand for large diameter, low defect density, and uniform single crystals has motivated extensive research on Cz Si growth as well as high pressure liquid-encapsulated Czochralski (HPLEC) growth of III-V compound crystals, e.g., GaAs and InP. The transport phenomena of Cz growth is quite complex, particularly under the industrial growth conditions. The relationship between the process parameters and material properties is further complicated by convective flows of the gas if a high pressure condition is to be maintained for the growth. Two important factors that greatly influence the quality of the crystals, are: (a) impurity and dopant distributions and (b) thermal stresses in the crystal.; A comprehensive model which incorporates all of the major physical mechanisms of HPLEC growth, has been developed. For numerical simulation, a novel scheme of combined finite volume (FVM) and finite element (FEM) methods has been devised for thermal-mechanical calculations, that uses multizone adaptive grid generation (MAGG) technique for both FVM and FEM modules. By combining the FVM for thermal transport modeling and FEM for solid stress calculations, valuable experiences in both fields have been employed, and a reliable and robust predictive tool for a large class of problems has been developed. This requires minimum effort and cost in both software development and computing environment and shows a great promise. It makes the investigation of coupled thermal convection and stress phenomena much easier to perform. A two time-scale, mass conserving scheme has also been developed to perform macro-segregation calculations.; Both Cz and HPLEC (high pressure liquid-encapsulant Czochralski) processes have been investigated. It is found that both melt and gas convective flows have significant influence on stress distribution in the crystal. It is shown that pure conduction-based models can not make accurate predictions of stresses in as-grown crystals. Use of a heat transfer coefficient to account for gas convection as many investigators have done in the past, is therefore not sufficient. Both melt and gas convection must be accounted in all future models if more accurate flow, temperature and stress calculations are desired. The predicted stress distributions agree qualitatively with experimental results. For macro-segregation analysis, it is found that the dopant distribution is controlled by the melt flow pattern.