Physically-based, lumped-parameter models for the prediction of oxygen concentration during Czochralski growth of silicon crystals

Lumped-parameter models are derived from boundary layer and other physical arguments to describe oxygen concentration levels during the Czochralski (CZ) growth of silicon. These models are assessed against predictions from a detailed, high-fidelity 2D-3D numerical simulation of the entire CZ puller, whose solutions are realistic but require intense computational effort. Comparisons of predictions show that the lumped-parameter model captures the correct trends of melt oxygen levels influenced by melt height, crucible rotation, and crystal rotation. A simple fitting of coefficients provides reasonably good quantitative predictions by the lumped-parameter model, and its near-instantaneous computations make it an interesting candidate for real-time growth optimization and control. Possible model improvements and extensions are discussed.