Gray-box modeling of 300 mm diameter Czochralski single-crystal Si production process

Abstract

More than 95% of 300 mm diameter single-crystal silicon ingots, the raw material for semiconductors, are produced by the Czochralski process. The demand for improving yield, throughput, and control performance has been increasing. The present study developed a gray-box model that can predict controlled variables from manipulated variables with higher accuracy than the conventional first-principle model (Zheng et al., 2018), aiming at realizing model predictive control of the Czochralski process. The proposed gray-box model used a statistical model to predict the temperature gradient of the crystal at the solid–liquid interface Gcry, which was constant in the first-principle model. The crystal length and the melt temperature are used as the input variables to predict Gcry. The prediction accuracy of the proposed gray-box model was compared with that of the first-principle model using real process data obtained during the production of four silicon ingots. The results demonstrated that the proposed model reduced the root mean square errors of the crystal radius, the crystal growth rate, and the heater temperature by 94.1%, 62.7%, and 70.6% on average, respectively.