3D mathematical model system for melt hydrodynamics in the silicon single crystal FZ-growth process with rotating magnetic field

K. Lācis¹ - A. Muiżnieks^1,2 - G. Ratnieks³

¹ Department of Physics, University of Latvia, 8 Zellu str., LV-1002, Riga, Latvia
² Institute for Electrothermal Processes, 4 Wilhelm-Busch-Str., 30167 Hannover, Germany
³ Siltronic AG, 24 Johannes-Hess Str., 84489 Burghausen, Germany

Abstract
A system of three-dimensional numerical models is described to analyse the melt hydrodynamics in the floating zone crystal growth by the needle-eye technique under a rotating magnetic field for the production of high quality silicon single crystals of large diameters ( 100... 200 mm ). Since the pancake inductor has only one turn, the high frequency (HF) electromagnetic (EM) field and the distribution of heat sources and EM forces on the melt free surface have distinct asymmetric features. This asymmetry together with the displacement of the crystal and feed rod axis and crystal rotation manifests itself as three dimensional 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. Additionally, the rotating magnetic field can be used to influence the melt hydrodynamics and to reduce the flow asymmetry. In the present 3D model system, the shape of the molten zone is obtained from symmetric FZ shape calculations. The asymmetric HF EM field is calculated by the 3D boundary element method. The low-frequency rotating magnetic field and a corresponding force density distribution in the melt are calculated by the 3D finite element method. The obtained asymmetric HF field power distribution on the free melt surface, the corresponding HF EM forces and force density of the rotating magnetic field are used for the coupled calculation of 3D steady-state hydrodynamic and temperature fields in the molten zone on a body fitted structured 3D grid by a commercial program package with a control volume approach. Beside the EM forces, also the buoyancy and Marangoni forces are considered. After HD calculations a corresponding 3D dopant concentration field is calculated and used to derive the variations resistivity in the grown crystal. The capability of the system of models is illustrated by a calculation example of a realistic FZ system. Tables 1, Figs 9, Refs 15.