Optimization of the Crystal Surface Temperature Distribution in the Single-Crystal Growth Process by the Czochralski Method

Abstract

The optimization of the crystal surface temperature distribution is performed for single-crystal growth in the Czochralski process. In the optimization problem, we seek an optimal solution in the sense that the index of crystalline defects is minimized while the single-crystal growth rate is maximized. In the objective function, the von Mises stress is considered a driving force that induces crystalline defects. In order to solve the optimization problem with the equality constraints given by the governing partial differential equations, the variational method is used. Based on the calculus of variations and the method of Lagrange multiplier, the Euler–Lagrange equations are derived in the form of coupled partial differential equations. They are solved by using the finite-difference method and the iterative numerical scheme proposed in this work. In order to handle inequality constraints, the penalty function method is applied. The optimal distributions of the crystal surface temperature obtained in this work may provide an insight into the optimal design of thermal surroundings, such as thermal shield configurations and heater/cooler positions.