Resolución numérica de las ecuaciones de la magnetohidrodinámica en el proceso Czochralski para la obtención de cristales semiconductores

Abstract

The objective of this work is to introduce and numerically solve the mathematical model for the bahavior of liquid semiconductors in the Czochralski process to obtain semiconductor crystals under the influence of a magnetic field. Such model is framed within the Magnetohydrodynamics equations. The numerical solution of this mathematical model presents important difficulties due to the involvement of phenomena from Fluid Mechanics, Electromagnetism and Heat Transfer. In order to numerically solve this model a stabilized finite element approximation based on tha algebraic version of a subgrid scale model in used. This approach makes possible to satisfy the free divergence condition over the velocity of the fluid together with the free divergence condition over the magnetic field and also to avoid numerical oscillations. The numerical scheme used in this work is based on the scheme proposed in (1). This numerical scheme is applied to the numerical benchmark proposed by Bückle (2). Such example is the most common numerical benchmark for the Czochralski process. The cases analysed in this work only differ from those proposed by Bückle in the fact that a magnetic field has been added. The use of magnetic fields in the Czochralski process is headed to diminish and suppress the perturbations inthe liquid which can originate defects in the semiconductor cristal.

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