Optimization of the crystal surface temperature distribution was performed in order to control point defects in a single crystal grown via the Czochralski process. In the optimization problem, an optimal solution was sought which minimized the number of point defects while the radial uniformity of the temperature was maximized. In order to solve the optimization problem within the equality constraints imposed by the partial differential equations, the variational principle was used. Based upon the calculus of variations and the method of Lagrange multipliers, the Euler–Lagrange equations were derived and solved using a finite difference method. In order to handle inequality constraints, a penalty function method was applied. An optimum distribution of the

surface temperature was expected to aid the design of thermal shield configurations and heater/cooler positions.

Optimization of Surface Temperature Distribution for Control of Point Defects in the Silicon Single Crystal. H.S.Woo, J.H.Jeong, I.S.Kang: Journal of Crystal Growth, 2003, 247[3-4], 320-32