Temperature distribution reconstruction in Czochralski crystal growth process

Abstract

A mechanical geometric crystal growth model is developed to describe the crystal length and radius evolution. The crystal radius regulation is achieved by feedback linearization and accounts for parametric uncertainty in the crystal growth rate. The associated parabolic partial differential equation (PDE) model of heat conduction is considered over the timevarying crystal domain and coupled with crystal growth dynamics. An appropriately defined infinite-dimensional representation of the thermal evolution is derived considering slow time-varying process effects. The computational framework of the Galerkin's method is used for parabolic PDE order reduction and observer synthesis for temperature distribution reconstruction over the entire crystal domain. It is shown that the proposed observer can be utilized to reconstruct temperature distribution from boundary temperature measurements. The developed observer is implemented on the finite-element model of the process and demonstrates that despite parametric and geometric uncertainties present in the model, the temperature distribution is reconstructed with the high accuracy.

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

“…Heater input power P law, the heat balance, and the thermodynamic equilibrium condition at the three-phase line, with v p and melt temperature T mel as inputs and r cry and meniscus height h men as outputs. Abdollahi et al [8,9] constructed a physical model to represent the relationship between P , v p , r cry , and T mel , considering the radiative heat transfer from the heater and the conductive heat transfer in the crystal. Rahmanpour et al [10] developed a simple model with fewer states and parameters than the above models, with P and v p as inputs and r cry and T mel as outputs.…”

Low‐order optimal regulation of parabolic PDEs with time‐dependent domain

“…It is assumed that the evolution of the domain Ω( t ) is smooth and known a priori , as it can be easily measured in many chemical and material process systems. In the example of the model of industrial CZ semiconductor crystal growth, there are robust control practices in achieving a desired crystal shape by manipulating the pulling rate of the crystal from the melt and other control inputs (see the work of Abdollahi et al and series of studies by Gevelber et al for more details). Hence, the PDE domain evolution is considered independent of the thermal field and the mentioned assumption is valid in this case.…”

“…In (19), k = 6.25 is the dimensionless process parameter and represents the domain velocity which is the derivative of the height function L ( t ) with respect to time. A simplified radius control strategy arising from geometric model provides the domain evolution in terms of L ( t ) and R ( t ) as shown in Figure , see the work of Abdollahi et al for more details.…”

Backstepping output-feedback control of moving boundary parabolic PDEs

“…Considering such problems as infinite-dimensional systems, direct state estimation is not possible in a large number of cases, because analytic expressions for the two-parameter semigroups describing the nonautonomous behaviour of the system can not be found. As an early-lumping approach, Galerkin's method is used for an eigenfunctions-based observer design by Abdollahi et al (2013) for the boundary control of a 2D heat equation with time-dependent spatial domain.…”