Melt motion in a Czochralski crystal puller with an axial magnetic field: motion due to buoyancy and thermocapillarity

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Abstract
In the Czochralski process a single crystal is grown from liquid in a crucible. An axial magnetic field suppresses turbulence in the melt and thus reduces the density of microdefects in the crystal. This paper treats the melt motion due to buoyancy and thermocapillarity. The magnitude of this motion decreases roughly like B−2, as the magnetic field strength B is increased. The separate circulations due to buoyancy and thermocapillarity are roughly equal at an early stage of growing a crystal. However the circulation due to thermocapillarity is nearly independent of the melt depth, while that due to buoyancy is proportional to the square of the depth. Therefore as the crystal grows and the melt depth decreases, thermocapillarity becomes progressively more dominant. When the heat flux into the melt is used to define the characteristic temperature difference and velocity, the stream functions are rather insensitive to changes in the thermal boundary conditions at the free surface and at the crucible bottom, provided the overall heat balance of the system is correctly estimated. This is fortunate because there is considerable uncertainty about these boundary conditions. The exception to this insensitivity is that the melt motion due to thermocapillarity is sensitive to changes in the amount of heat lost through the part of the free surface adjacent to the crystal.