Simulation of large-scale silicon melt flow in magnetic Czochralski growth

Abstract

We have modeled the silicon melt flow and temperature as well as the applied and induced magnetic fields in a large-scale, cylindrically symmetric Czochralski crystal growth system. The validity of a cylindrically symmetric model is evaluated by comparing melt flow simulations to previously published three-dimensional simulations and experimental measurements in identical setups. The applied magnetic field, due to a given external magnet configuration and its current distribution, is first solved. Subsequently, the coupled magnetohydrodynamical system, i.e., the Navier–Stokes, heat and induction equations, is solved in the melt region. The mathematical model is discretized by the finite element method. The numerical methods used are explained in this article. The applied external magnetic fields we have considered are time-independent and axisymmetric. The magnetohydrodynamical problem is time-dependent, and the velocity and the induced magnetic field have azimuthal components. The velocity distributions and temperature time series for various cusp-type fields are compared. It is found that a magnetic field of the order of 25 mT strongly damps the temperature oscillations in the melt outside about half the crucible radius. A higher field of about 100 mT stabilizes the flow in most of the melt.