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.
Bibliographical noteFunding Information:
The authors would like to acknowledge partial support of this work from the University of Minnesota and the CZ3003 project, funded partly by the German Federal Ministry for Economic Affairs and Energy (BMWi) under contract number 0324357A. The authors would like to thank Dr. Thomas Jung for significant, original contributions. We also wish to thank an anonymous reviewer, whose comments substantially improved the paper.
© 2021 Elsevier B.V.
- A1. Computer simulation
- A1. Convection
- A2. Czochralski method
- A2. Single Crystal Growth
- B2. Semiconducting silicon