Use of mathematical expansions to model crystal growth from the melt under the effect of magnetic fields

High-quality silicon crystals provide the basis of many industrial technological advances, including computers and telecommunication devices. The increasing size and extremely high quality requirements of silicon wafers have made furnace design and crystal manufacturing a very challenging task. Numerical simulations have become an essential and powerful tool to overcome the difficulties of the experimental approach with a view to understanding the crystal growth process but also to finding an appropriate path to optimize the crystal pulling conditions in industry.

This thesis deals with the use of alternating and steady transverse magnetic fields in silicon growth from the melt. The use of magnetic fields represents a powerful tool to damp out turbulence and control the melt flow. This technique can also be used to heat the system. We focus on the numerical modeling of (i) induction heating in the Floating Zone process and of (ii) melt convection under the effect of transverse magnetic fields in the Czochralski process. For each of these topics, our work is subdivided in two parts : firstly mathematical modeling, based on asymptotic or Fourier expansions, and secondly numerical implementation and simulation of the considered processes.

First, a theoretical and numerical model of the alternating magnetic field distribution (as generated by induction heating) has been developed by means of an asymptotic expansion technique. Moreover, a new methodology has been developed to calculate the thermal and mechanical effects of alternating magnetic fields on the liquid conductor flow, leading to accurate expressions for the equivalent magnetic heat flux and surface stresses in the 2D and 3D cases. Second, investigation of the effect of a transverse magnetic field on the melt flow in semi-conductor crystal growth has been performed by the simplified FLET method ("Fourier Limited Expansion Technique".)