Optimization of surface temperature distribution for control of point defects in the silicon single crystal

Abstract

Optimization of crystal surface temperature distribution is performed for the control of point defects in a silicon single crystal grown by the Czochralski process. In the optimization problem, we seek an optimal solution that minimizes the level of point defects while the radial uniformity of temperature is maximized. In order to solve the optimization problem with the equality constraints described by partial differential equations, the variational principle is used. Based on the calculus of variations and the method of Lagrange multiplier, the Euler-Lagrange equations are derived and solved by the finite difference method. In order to handle inequality constraints, the penalty function method is also applied. The optimal distribution of the crystal surface temperature is expected to provide an insight for the design of thermal surroundings such as the thermal shield configuration and the heater/cooler position. (C) 2002 Elsevier Science B.V. All rights reserved.