Mpirical Modeling of Heating Element Power for the Czochralski Crystallization Process

Abstract

The Czochralski (CZ) crystallization process is used to produce monocrystalline silicon. Monocrystalline silicon is used in solar cell wafers and in computers and electronics. The CZ process is a batch process, where multicrystalline silicon is melted in a crucible and later solidifies on a monocrystalline seed crystal. The crucible is heated using a heating element where the power is manipulated using a triode for alternating current (TRIAC). As the electric resistance of the heating element increases by increased temperature, there are significant dynamics from the TRIAC input signal (control system output) to the actual (measured) heating element power. The present paper focuses on empirical modeling of these dynamics. The modeling is based on a dataset logged from a real-life CZ process. Initially the dataset is preprocessed by detrending and handling outliers. Next, linear ARX, ARMAX, and output error (OE) models are identified. As the linear models do not fully explain the process’ behavior, nonlinear system identification is applied. The Hammerstein-Wiener (HW) model structure is chosen. The final model identified is a Hammerstein model, i.e. a HW model with nonlinearity at the input, but not at the output. This model has only one more identified parameter than the linear OE model, but still improves the optimization criterion (mean squared ballistic simulation errors) by a factor of six. As there is no nonlinearity at the output, the dynamics from the prediction error to the model output are linear, which allows a noise model to be added. Comparison of a Hammerstein model with noise model and the linear ARMAX model, both optimized for mean squared one-step-ahead prediction errors, shows that this optimization criterion is 42% lower for the Hammerstein model. Minimizing the number of parameters to be identified has been an important consideration throughout the modeling work.