A Bayesian Approach to Diameter Estimation in the Diameter Control System of Silicon Single Crystal Growth

Abstract

In the diameter control system of silicon single crystal growth, the variation of the diameter of the aperture (i.e., a halo with high brightness, which appears at the junction of a solid crystal and a liquid solution) is consistent with the change in the diameter of the growing crystal. Therefore, the diameter of the aperture can be used as a control variable for adjusting the casting speed and temperature so that the grown silicon single crystal approximates to a perfect cylinder. It is obvious that the measured diameter of the current aperture plays an important role in the diameter control system of silicon single crystal growth. In fact, the obtained aperture image from a charge-coupled device camera is a halo of ellipses instead of circles. To estimate the diameter (or radius) of the elliptical aperture, we propose a Bayesian approach, in which a Bayesian model is derived to define a posterior distribution for the unknown parameters. This distribution is too complicated for analytical extraction of moments to sample directly. An efficient computational algorithm based on a Markov chain Monte Carlo method is derived to estimate the posterior distribution and draw samples from the distribution. Comparing with the classical Hough transform-based algorithm and the direct least-squares fitting method, the proposed algorithm has higher estimation accuracy. Some simulated and experimental examples are presented to illustrate the algorithm's effectiveness.