Modelling of the bulk flow and transport processes in a Czochralski crystal puller with a magnetic field

The control of heat and mass transport from the melt to the crystal during crystal growth may be facilitated by the application of a magnetic field. Since semiconductor materials in the liquid phase have the properties of a liquid metal, the interaction between the melt and magnetic field suppresses any time-dependent heat transport and small-scale turbulence and decreases the magnitude of the melt velocities. The use of matched asymptotic expansions provides analytical solutions for the bulk flow in a Czochralski crystal puller with an axial magnetic field. Treating the crystal as a slight electrical conductor alters the radial and axial flows driven by centrifugal pumping. The motion due to buoyancy and thermocapillarity are found by considering the temperature as a known function and solving the non-linear heat equation by a finite-difference scheme for different field strengths and melt depths. The mass transport for the dopants in this system is considered as a time-dependent process and requires a mixture of Lagrangian and Eulerian descriptions to keep track of the concentration at each point of space and time.