Czochralski Growth of Oxide Crystals: Numerical Simulation and Experiments

A mathematical model that explores the basic transport phenomena in a Czochralski process, their interaction and influence on the growth of high quality oxide crystals is presented. Rare earth garnets YAG and Nd-doped YAG are considered as representative oxide materials for the purpose of modeling and numerical simulation. The model proposed is evolutionary in time, and axisymmetric in space. All three velocity components enter the calculations, and in this respect, the model is posed in 2 1 2 dimensions. The computational domain consists of the melt in the crucible, the crystal growing out of it, the seed rod, the gas phase and the enclosure. The conservation equations for mass, momentum and energy for all the three phases are jointly represented with the provision to account for abrupt changes in transport properties across the zonal boundaries. The zonal boundaries between two phases, for example, the phase-change interface move under the influence of flow and temperature fields and in turn can affect the transport behavior in the adjoining regions. The interfaces are followed by assigning a fixed grid line to them, and the interface movement is calculated using auxiliary equations that govern the interface dynamics. Grids are regenerated with every new location of the interface using a numerically derived coordinate transformation. Assuming that the solid and liquid phases are separated by a sharp interface, the location and shape of the solidification front is obtained by the local energy balance between the heat fluxes in the crystal and melt along with the release of latent heat of fusion. The gas-melt interface is obtained by a force balance among the pressure forces, viscous forces, surface tension forces and centrifugal forces. The force balance accounts for thermocapillary (Marangoni) flow at the free surface. In addition, gas phase convection is taken to originate from the temperature difference between the melt surface and the enclosure, and not because of the surface velocity of the melt. No-slip boundary conditions are used along all the solid surfaces. The crucible wall is assumed to be at a uniform temperature, while the temperature of the heat shield enclosing the crystal decreases linearly along the height until it reaches a low