Design and optimization of superconducting CUSP electromagnetic field structure based on a COMSOL-GMDH-MOSO hybrid strategy

As the semiconductor industry gains momentum, there is a significant focus on producing high-quality silicon wafers with larger diameters. However, the preparation of major-diameter, high-quality silicon single-crystal has encountered challenges such as intense convection inside the melt and high oxygen impurity concentration at the interface between the fluid and solid phases. To tackle these problems, the external superconducting CUSP electromagnetic field structure was designed to optimize the crystal growth parameters. A hybrid strategy of the multiphysics field coupling method, Group Method of Data Handling (GMDH) type neural network identification, and the Multiple Objective Snake Optimizer (MOSO) algorithm was proposed to optimize the superconducting CUSP electromagnetic field structure. A two-dimensional axisymmetric model was established for 18-inch silicon single-crystal preparation using the Magnet system Applied Czochralski (FCZ) method. The amount of longitudinal plies of superconducting CUSP electromagnetic field coils, coil spacing, and the thickness of magnetic shields were selected as design variables, and the radial electromagnetic induction intensity at the graphite crucible inner wall and the superconducting magnet volume were used as dual objective functions. The required samples for GMDH training were provided by COMSOL Multiphysics (COMSOL) numerical calculations, and polynomial mathematical equations were established with the radial electromagnetic induction intensity at the graphite crucible inner wall and the superconducting magnet volume as the dual objective functions. Then the Pareto optimal solution for the electromagnetic field structure design variables was obtained using the dual objective function optimized by MOSO. With the hybrid strategy, numerical simulations and theoretical analysis of the crystal preparation process parameters under the effect of the superconducting electromagnetic field with the optimal structure were carried out. The results displayed that the added superconducting CUSP electromagnetic field reduced the crystal radial and axial thermal gradients and greatly suppressed the melt convection. As a result, the heat distribution inside the hot fused silicon was more even, the thermal gradient at the solid-liquid interface was reduced by 68.4%, and the oxygen impurity content was reduced by 82.1%, which effectively improved the crystal quality. The thermal field simulation results verified that the proposed hybrid strategy provided a new numerical calculation method for obtaining the optimal superconducting electromagnetic field structure.