The use of magnetic fields during the growth of semiconductor crystals from the melt in a Czochralski (CZ) crystal puller has shown promise in controlling the heat and mass transport to the growth interface. The magnetic field suppresses turbulence and thermal convection in the melt in which large thermal gradients are present, thus improving the quality of the crystal. In this paper, analytical solutions are presented for the isothermal melt motion and electric current density driven by the differential rotation of the crystal and crucible about their common vertical axis. There is an applied, non-uniform, axisymmetric magnetic field with only radial and axial components which are independent of the azimuthal coordinate. The melt motion with a uniform axial magnetic field represents a singular limit of the flow considered here: as the radial magnetic field component goes to zero, the radial and axial (meridional) velocity components decrease in magnitude by a factor of M-1, where M is the large Hartmann number. The uniform axial field is a singular limit because the centrifugal acceleration due to the azimuthal velocity is exactly perpendicular to the magnetic field. Since the radial isothermal motion near the growth interface controls the radial distributions of dopants and impurities in the crystals, a non-uniform axisymmetric magnetic field is better than the uniform axial field. In addition, the axisymmetric field avoids the detrimental deviations from axisymmetric heat and mass transport associated with a uniform transverse (horizontal) magnetic field.
Two classes of shaped fields are considered, with only one class leading to the presence of the large meridional flow driven by differential rotation. The small electrical conductivity of the crystal plays an important role in determining the behaviour of the melt's angular velocity, which is constant along each magnetic field line. Results for two simple field configurations are presented in order to illustrate the effect of the field configuration on the nature of the meridional circulation and the potential for flow tailoring with the shaped field.