Optimizing the Temperature Profile during Sublimation Growth of SiC Single Crystals:  Control of Heating Power, Frequency, and Coil Position

We use a numerical optimization method to determine the control parameters frequency, power, and coil position for the radio frequency (RF) induction heating of the growth apparatus during sublimation growth of SiC single crystals via physical vapor transport (PVT) (also called the modified Lely method). The control parameters are determined to minimize a functional, tuning the radial temperature gradient on the single-crystal surface as well as the vertical temperature gradient between SiC source and seed, both being crucial for high-quality growth. The optimization is subject to constraints with respect to a required temperature difference between source and seed, a required temperature range at the seed, and an upper bound for the temperature in the entire apparatus. The numerical computations use a stationary mathematical model for the heat transport, including heat conduction, radiation, and RF heating to solve the forward problem, and a Nelder−Mead method for optimization. In agreement with published experimental results, a minimal radial temperature gradient is found to coincide with a minimal temperature at the single-crystal surface. A maximal temperature gradient between source and seed is found to coincide with a low coil position.