SyMO: A Hybrid Approach for Multi-Objective Optimization of Crystal Growth Processes

Abstract

Crystal growth, particularly silicon, is pivotal in the semiconductor industry. It serves as the foundation for electronic devices, solar cells, and various advanced technologies. The Czochralski method is a prominent technique for producing large single silicon crystals, well-known for its complexity due to the precise control required over temperature gradients, interface dynamics, and impurity incorporation— all critical factors for growing uniform, high-quality crystals. This paper proposes a hybrid SyMO (Symbolic regression Multi-objective Optimization) framework that combines Computational Fluid Dynamics (CFD), machine learning, and mathematical optimization techniques to investigate the effects of various process parameters, furnace geometries, and radiation shield material properties on key crystal quality metrics. The data set created from axisymmetric CFD simulations is used to fit symbolic regression models to effectively capture complex nonlinear relationships, ensuring accurate interface deflection and $v/G$ ratio predictions. The SR equations are integrated into a multi-objective optimization model that simultaneously optimizes crystal quality and process efficiency. The obtained results are validated through additional CFD simulations to confirm the accuracy of the solution. It is demonstrated that the SyMO successfully generalizes the critical dependencies across various parameters and provides robust, high-quality solutions.