Numerical modelling for the diameter increase of silicon crystals grown with the pedestal method

•The pedestal method for silicon crystal growth was modelled numerically.•High-frequency inductor was optimized via gradient descent method.•Middle-frequency inductor was used to increase the depth of the melting front.•Crystal diameters from 20 mm to 100 mm were considered. The pedestal method is one of crucible-free crystal growth methods, that has been less researched than the well-known floating zone (FZ) method. However, the pedestal method may be a cost-effective alternative to FZ, if large diameter feed rods are available. The investigated system contains two electromagnetic inductors: high-frequency inductor for pedestal top surface melting and middle-frequency inductor for pedestal side heating. The present work describes recent advances in numerical modelling of heat transfer and phase boundaries in axially symmetrical approximation, neglecting the melt flow. The shape of high-frequency inductor was optimized with the algorithm of gradient descent. As the risk of melt freezing is substantial during both the seeding and the cylindrical phase, the distance between the centres of melting and crystallization interfaces was used as a target function. Optimal inductor geometry and system geometrical parameters were obtained for systems with crystal diameters from 20 mm to 100 mm, thus sketching a pathway for crystal diameter increase in industrial growth system prototypes.