### Abstract

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.

Original language | English (US) |
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Pages (from-to) | 267-289 |

Number of pages | 23 |

Journal | Zeitschrift fur Angewandte Mathematik und Physik |

Volume | 51 |

Issue number | 2 |

DOIs | |

State | Published - Mar 2000 |

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### Keywords

- Asymptotic analysis
- Inertial effects
- Magnetic Czochralski crystal growth
- Rotationally driven flow

### ASJC Scopus subject areas

- Mathematics(all)
- Physics and Astronomy(all)
- Applied Mathematics

### Cite this

*Zeitschrift fur Angewandte Mathematik und Physik*,

*51*(2), 267-289. https://doi.org/10.1007/s000330050198