Crystal radius and temperature regulation in Czochralski crystallization process

In this paper we explore modelling and regulation techniques for temperature and crystal growth control in an industrially important process of Czochralski crystal growth. First, a geometrical model for both crystal growth and radius evolution dynamics is developed and a controller is designed based on an input-output linearization to regulate and track a desired reference crystal radius. The pulling based regulated crystal radius control provides a time-varying shape (geometry) evolution of the underlying time-varying partial differential equation describing the crystal temperature dynamics. A low dimensional time-varying parabolic PDE model obtained by Galerkin method is used for regulation of the triple point temperature (melt-solid-encapsulant intersection). The coupling among finite-dimensional crystal radius and triple point temperature regulation is given by the perturbation in the crystal growth parameters and it is induced by possible solid-melt interface temperature fluctuations. We provide a unified regulation framework for such a complex coupled system of crystal growth and temperature regulation in order to improve operational and economic features associated with the Czochralski crystal growth process.