Linear and Nonlinear State Estimation in the Czochralski Process*

The Czochralski process is the only method used commercially for production of monocrystalline silicon for semiconductor and solar cell applications. This paper explores the use of mathematical modeling as an aid in estimation of system state variables in the standard Czochralski process. A state-space model of the process is presented, describing the dynamics of the crystal radius and meniscus height with crystal radius as measured output. For the purpose of estimating the actual crystal radius during growth, three types of state estimators are developed based on the state-space model; the Kalman filter, the extended Kalman filter and the unscented Kalman filter. It is found that the latter two provide highly accurate state estimates with excellent noise suppression.