Influence of the three dimensionality of the HF electromagnetic field on resistivity variations in Si single crystals during FZ growth

Abstract

Three-dimensional numerical modelling is carried out to analyse the floating zone crystal growth with the needle-eye technique used for the production of high-quality silicon single crystals with large diameters (⩾100mm). Since the pancake inductor has only one turn, the EM field and the distribution of heat sources and EM forces are only roughly axisymmetric. The non-symmetry together with crystal rotation reflects itself on the hydrodynamic, thermal and dopant concentration fields in the molten zone and causes variations of resistivity in the grown single crystal, which are known as the so-called rotational striations. The non-symmetric high-frequency electromagnetic field of the pancake inductor is calculated by boundary element method. The obtained non-symmetric power distribution on the free melt surface and the corresponding EM forces are used for the coupled calculation of the 3D steady-state hydrodynamic and temperature fields in the molten zone on a body fitted structured 3D grid by a commercial program package with control volume approach. The buoyancy, Marangoni and EM forces are considered. The afterwards calculated corresponding 3D dopant concentration field is used to derive the variations of resistivity in a longitudinal cut of the grown crystal. The results are compared with experimental measurements (photo-scanning method) and with results of 2D transient flow calculations. Rotational striations are found in both 3D-calculated and experimental resistivity distributions and show a qualitative agreement. A Fourier analysis for the resistivity variations is performed and the observed differences are explained by modelling limitations.