Modelling Plastic Stress Relaxation in Shaped Sapphire Crystal Growth

Thermally induced stresses in growing shaped sapphire crystals are modelled using transient finite element simulations. Boundary conditions and mass changes are fixed on an expanding remeshed and updated grid. The actual stresses are obtained taking into account plastic relaxation through dislocation glide along basal, prismatic and pyramidal slip planes. For this purpose, phenomenological creep laws available for this material are implemented in the frame of thermally activated plasticity. The model is calibrated by comparison with directional mechanical tests, and validation is performed by growth simulations and dislocation density measurements on as-grown crystals.

A two-dimensional, quasi-steady-state, thermal-capillary model is developed for a micro-pulling-down (μ-PD) system to study limitations to steady growth of sapphire. The model incorporates mass, energy, and momentum conservation equations, and also accounts for the physics of the melt meniscus, the solidification front, and the crystal radius. Limit points with respect to pull rate are found under higher-gradient thermal conditions but are shown to unfold with changes in die heating and ambient temperature. Limit points related to crystal size and capillary effects are also found with respect to static head (melt height); however, classical criteria of capillary instability are shown to be invalid. Thus, a more fundamental understanding is obtained for μ-PD operating limits, their origins, and their possible avoidance.


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