Multi-objective optimization of Magnetic Czochralski crystal growth process using surrogate modeling and evolutionary algorithm

The optimization of the Magnetic Czochralski (MCZ) crystal growth process is crucial for producing high-quality single-crystal silicon but is challenged by the complex interplay of multiple physical phenomena and computationally expensive simulations. Achieving both high crystal quality and production efficiency often involves navigating conflicting objectives. To address this challenge, this study proposes and implements a multi-objective optimization framework based on surrogate modeling. The framework integrates high-fidelity Computational Fluid Dynamics (CFD) simulations, ensemble learning surrogate models, and the Non-dominated Sorting Genetic Algorithm II. The primary objectives were to minimize the solid–liquid interface deflection (|δ|) and maximize the crystal pulling velocity (V_pull), while satisfying the v/G Voronkov criterion. Eight key process and geometric parameters were investigated. Accurate surrogate models (R²'0.90) were successfully developed, serving as efficient proxies for the CFD simulations. Model interpretation using SHapley Additive exPlanations and Response Surface Analysis revealed that while deflection is predominantly driven by the pulling velocity, the v/G ratio is governed by a more complex, multi-parameter interaction, highlighting the limitations of single-parameter tuning. The optimization successfully generated a 3D Pareto front of optimal solutions, offering a range of actionable process recipes from conservative (low-deflection) to aggressive (high-V_pull) that significantly outperform the initial base cases. This work provides not only a set of optimized process parameters but also demonstrates a robust methodology for navigating the complex design space of the MCZ process. The findings offer a valuable data-driven tool for process engineers to make informed decisions that balance the competing demands of crystal quality and production efficiency.