Elsevier

Journal of Process Control

Volume 21, Issue 10, December 2011, Pages 1361-1369
Journal of Process Control

Optimal control of convection–diffusion process with time-varying spatial domain: Czochralski crystal growth

https://doi.org/10.1016/j.jprocont.2011.07.017Get rights and content

Abstract

This paper considers the optimal control of convection–diffusion systems modeled by parabolic partial differential equations (PDEs) with time-dependent spatial domains for application to the crystal temperature regulation problem in the Czochralski (CZ) crystal growth process. The parabolic PDE model describing the temperature dynamics in the crystal region arising from the first principles continuum mechanics is defined on the time-varying spatial domain. The dynamics of the domain boundary evolution, which is determined by the mechanical subsystem pulling the crystal from the melt, are described by an ordinary differential equation for rigid body mechanics and unidirectionally coupled to the convection–diffusion process described by the PDE system. The representation of the PDE as an evolution system on an appropriate infinite-dimensional space is developed and the analytic expression and properties of the associated two-parameter semigroup generated by the nonautonomous operator are provided. The LQR control synthesis in terms of the two-parameter semigroup is considered. The optimal control problem setup for the PDE coupled with the finite-dimensional subsystem is presented and numerical results demonstrate the regulation of the two-dimensional crystal temperature distribution in the time-varying spatial domain.

Highlights

► Boundary control of parabolic PDEs. ►Surface diffusion does not lead to diffusion limitations.►Optimal control of crystal growth process.

Introduction

A large number of industrial systems exhibit time-varying features in which certain parameters of the system change over the course of the process. The methods employed in the formation and treatment of materials may result in, for example, chemical reactions, phase transitions, deformations or a combination of these behaviours, and therefore introduce complexities in model-based controller design. The Czochralski (CZ) crystal growth process, utilized for the production of semiconductor materials for the microelectronics industry, is a prime example in which a time-dependent feature of the system is the change in material domain and is the motivating example behind our study.

In the CZ crystal growth process, large boules of single crystals, typically Si, GaAs, InP, and CdTe, are formed in a thermal environment, whereby a crystal seed is slowly drawn from a pool of melt by a mechanical pulling arm. The material growth by solidification at the crystal–melt interface is affected by variations in the thermal fields of the ambient and melt temperatures, as well as the rate of pulling. These conditions are significant factors which contribute to the overall product quality where the objective of the batch processing strategy is to yield high-purity, defect and dislocation free crystals with constant diameter. The latter specification is vulnerable to fluctuations in heat transfer caused by turbulent convection in the melt environment, and also to longer term disturbances in the ambient temperature and changes in the melt level.

The complexity in modelling the dynamics of the CZ crystal growth process is reflected in the numerous works dedicated to the analyses of the multi-physics system which include studies of the transport phenomena associated with the crystal temperature, crystal–melt interface, melt dynamics, and crystal pull rate (see [6], [7], [8], [9]). A more complete survey of the modelling and dynamical analyses of the process is contained in the review articles by Brown [6] and Lan [16], which also describe the usage and challenges in the design and implementation of active control methodologies for single crystal growth. For example, the maintenance of the crystal shape is a subject of considerable interest. Several controlled growth methods are based on models which incorporate the relationships between the crystal, ambient and melt temperatures, and have led to proposed strategies in which diameter control is achieved via combinations of crucible heater, bottom heater and crystal pull rate actuation (see [7], [8], [9]).

Another important control problem which has garnered less attention is the regulation of the crystal temperature distribution during the process which is important in counteracting the fluctuations in the rate at which the crystal cools which can cause large thermoelastic stresses leading to micro-defect and dislocation generation within the grown crystal [23], [25]. Therefore, the complex transport phenomena models of the crystal temperature dynamics are developed by utilizing mass and energy balance relations which yield parabolic partial differential equations (PDEs) defined on time-varying spatial domains [9]. In the process control field there are several works which consider various model representations and control strategies for parabolic PDEs with time-varying spatial domains along with different control objectives which include the temperature stabilization problem using robust control methods [3], the boundary stabilization by manipulation of the temperature field [13], and the inverse Stefan problem in which the boundary evolution is known a priori [11]. Another approach considers the optimal stabilization of the temperature distribution of a material, for example in annealing type processes, by varying the spatial domain in which the domain motion is described by a finite-dimensional mechanical subsystem (see [26], [27]).

Motivated by the process complexity and the lack of optimal control realization for the CZ crystal growth process, in this work we provide a model development for the parabolic PDE on the time-varying spatial domain, and consider the optimal control formulation for the CZ crystal temperature regulation problem. As previously mentioned, it is of interest to control the rate at which the crystal cools in order to prevent material defects and dislocation generation and it is also of interest to stabilize the rate of pulling around some desired value. Therefore, the optimal regulation of the temperature distribution in the time-varying spatial domain around a pre-specified nominal distribution is required. The system is characterized by the unidirectional coupling of the domain motion, which is determined by the mechanical pulling which draws the crystal from the melt with dynamics described by a second order ODE, to the parabolic PDE system which describes the transient temperature distribution in the grown crystal. The controller synthesis for the crystal temperature regulation problem is considered from the perspective of infinite-dimensional systems theory whereby the PDE is represented as an evolutionary equation on an appropriately defined function space with nonautonomous operator which generates a two-parameter semigroup. In this form, the control problem is considered using linear-quadratic optimal control theory for nonautonomous infinite-dimensional systems. To address the issue of practical realization, the finite-dimensional system representation of the PDE system is obtained, and we consider the simultaneous control problem of the crystal temperature regulation and the stabilization of the domain motion around a nominal steady state value. A low order controller for the crystal temperature regulation problem is proposed and numerical results are provided including the comparison of the optimal controller to conventional proportional controllers.

This paper is organized as follows: in Section 2 the boundary evolution due to the mechanical pulling arm is described in terms of a second order ODE and the crystal temperature dynamics are described by a parabolic PDE defined on the time-varying spatial domain. In Section 3, the PDE is represented as an abstract evolution equation on an infinite-dimensional space with nonautonomous parabolic operator which generates a two-parameter semigroup. This representation enables the use of time-varying infinite-dimensional systems theory to pose the time-varying optimal control problem in Section 4. In Section 5, the finite-dimensional system representation of the PDE is determined and augmented with the mechanical pulling arm subsystem, to facilitate the numerical implementation of the control problem for the temperature regulation and domain motion. The optimal control synthesis of the augmented system is presented and numerical results are provided in Section 6. Finally, Section 7 concludes the paper with a summary of results.

Section snippets

Model description

The crystal region is considered as an axisymmetric and time-varying spatial domain with unit radius R = 1 and length l(t). The spatial domain motion is due to the crystal pull rate v(t) which determines the growth in the crystal at the boundary z = l(t), where l(t) > 0 is the crystal length, see Fig. 1. The boundary evolution is determined by a mechanical actuator pulling the crystal from the melt. In practice the crystal pull rate is slow, and we approximate the dynamics of the mechanical subsystem

Infinite-dimensional system representation

The optimal control formulation proposed in the subsequent section requires the representation of the PDE in Eq. (2) as an evolution system on some appropriate Banach space. In order to handle the time-dependence of the spatial domain, the following function space description provides a suitable framework such that the representation of the PDE in Eq. (2) can be handled using standard infinite-dimensional systems theory.

Controller design

In this section, we consider the optimal control problem for the PDE system given by Eq. (2). Although in several works the boundary control problem for nonautonomous systems of parabolic type has been explored within different frameworks (see for example [1], [2]), in this work, we consider the approach to the boundary control formulation which is proposed in [22].

Numerical implementation

This section provides an overview of the numerical approach utilized to simulate the closed loop PDE system in Eqs. (2), (3). A more thorough treatment of the Galerkin method pertaining to the variational form of the problem and existence and uniqueness of solutions is omitted (see for example [12], [19], [14]).

Simulation results

In this section, the numerical simulation of the crystal temperature regulation problem in the presence of the time-varying spatial domain is provided. We consider the situation in which a perturbation has occurred in the crystal temperature distribution which arises, for example, from fluctuations in the melt environment. It is of interest to optimally stabilize the crystal temperature around the nominal steady state distribution of x(r, z, t) = 0 throughout the crystal region which is

Summary

In this work, we considered the optimal control of the CZ crystal growth and temperature regulation problem. The convection–diffusion parabolic PDE process model of the crystal temperature dynamics defined on the time-varying spatial domain was derived from first principles continuum mechanics. The domain evolution was described by a second order ODE model for the mechanical pulling arm subsystem which is unidirectionally coupled to the crystal temperature dynamics. The representation of the

References (27)

  • BensoussanA. et al.

    Representation and Control of Infinite Dimensional Systems

    (2007)
  • BrownR.

    Theory of transport processes in single crystal growth from the melt

    AIChE J.

    (1988)
  • DerbyJ.J. et al.

    Finite-element methods for analysis of the dynamics and control of czochralski crystal growth

    J. Sci. Comput.

    (1987)
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