Inertial effects in the rotationally driven melt motion during the Czochralski growth of silicon crystals with a strong axial magnetic field

This article treats the melt motion driven by the rotations of the crystal and crucible about their common vertical axis during the Czochralski growth of silicon crystals with a strong, uniform, vertical magnetic field produced by a solenoid around the crystal growth furnace. Since molten silicon is an excellent electrical conductor, the interaction parameter N and the Hartmann number H a are both large for typical magnetic field strengths, so that composite singular perturbation techniques for N ≫ 1 and H a ≫ 1 are appropriate. An inertialess solution, which also assumed that N ≫ H a^3/2, was presented in a previous paper. In the inertialess solution, the largest gradient of the azimuthal velocity υ_θ and the largest secondary flow with radial and axial velocities υ_r and υ_z both occur inside an interior layer with an O(H a^-1/2) radial thickness at the vertical cylinder directly beneath the periphery of the crystal, where the growth interface meets the free surface. For all current experimental studies, the assumption that N ≫ H a^3/2 is not satisfied. The appropriate assumption is that N = O(H a^3/2), and inertial effects are not negligible inside the interior layer. An intersection region, which is formed by the intersection of the interior layer and a Hartmann layer with an O(H a^-1) axial thickness adjacent to the crystal-melt interface and free surface, is intrinsically coupled to the interior layer. This article treats inertial effects in the interior layer and intersection region for N = O(H a^3/2) and H a ≫ 1. Non-linear governing equations were derived and solved numerically. A fourth-order Adams - Bashforth - Moulton predictor-corrector method was used to solve the transport equations for the primary azimuthal velocity and for the secondary-flow vorticity. Poisson equations, which govern the stream functions for both the secondary flow and the electric current density, were solved using a matrix diagonalization technique. The effects of inertia on the melt motion are discussed. This type of study provides for a fuller understanding of the melt motion, without which defect-free crystals will be difficult to grow on a consistent basis.