변분법 응용: 초크랄스키 공정에서 단결정 표면 온도 분포의 최적화

정 자훈, 강 인석

포항 공과 대학교 화학공학과 및 지능자동화 연구센터

Application of calculus of variations

to optimization of single crystal surface temperature distribution

in the Czochralski process

J.H. Jeong, I.S. Kang

Dept. of Chem. Eng. and Automation Res. Center, Postech

INTRODUCTION

In the Czochralski(CZ) process, the quality of single crystal is determined by crystalline defects, thermoelastic stresses and so on.  Since the microscopic phenomena inside single crystal are strongly influenced by thermal conditions, the crystal thermal history has been considered as the most important key word for the control of the quality of single crystal [1]. It is the crystal thermal history that we are concerned with in the present work. The general theme is pursued by performing studies on the optimization of the crystal surface temperature distribution.

Many researches have aimed at the optimal design of thermal surroundings to improve the quality of single crystal. Although the increases of capabilities of numerical analysis have stimulated rigorous simulations to suggest proper operating conditions, most numerical works for the optimal design have been accomplished based on the simulations case by case. Thus, in the present work, we want to obtain the optimal crystal surface temperature in a systematic way.

The area of optimization and control of distributed parameter systems described by the partial differential equations is a highly active and rapidly growing area in applied mathematics and engineering. In the optimal control within the lumped parameter systems, described by ordinary differential equations, the solution methods based on calculus of variations have already been popular [2]. But in case of the distributed parameter systems, only few practical problems have been studies, although there are well-developed theories for the optimal control of systems governed by partial differential equations [3].

Calculus of variations has been developed as a general technique for the optimization problems such as the isoperimetric problem, but most researches have been concentrated on derivation of the different expressions of the principles by inferences [4]. Surprisingly enough, there have been few attempts to apply calculus of variations to the optimization problems in science and engineering.

The first goal of this work is to propose a better methodology for optimization of the spatial distribution of a certain variable such as the crystal surface temperature distribution in CZ process. We formulate a simple optimization problem by both considering the thermal stress distribution inside the crystal and the crystal growth rate. Under the assumption that the radial component of the thermal gradient is the dominant factor to induce the thermal stress, the problem can be formulated in terms of temperature only. Based on calculus of variations and the method of Lagrange multiplier, we derive and solve Euler-Lagrange equations. We solve the same problem by using the conjugate direction method, which is among the techniques of nonlinear programming [5]. Then we compare the histories of convergence during iterations between two methods. The variational method adopted in this work can be extended to more realistic problems relevant to CZ process such as the thermal stress problem and the point defect problem. In the thermal stress problem, the thermal stress is directly involved in the formulation and a multiplier tensor is used. In the point defect dynamics problem, the nonlinear state equations add some extra-complexity.

METHODS OF ANALYSIS

Calculus of variations provides necessary conditions for the optimization problem, which can be summarized by followings, equivalent to each other.

[eqn: {rm Min.} I``(u) ~~LRARROW~~ delta I``(u^* )=0~],	(1)

where [eqn: u^* ``] represents the optimal solution. In this work, the optimization problem with the equality constraints of partial differential equations is considered, given by

[eqn: {rm Min.}``I``(u) ``=`` int_{Omega} ``F {rm d}{Omega}``], s.t. [eqn: L(u) = 0 ~{rm i n}`` Omega``],	(2)

where [eqn: L``] is the partial differential operator. The boundary values on the essential boundary [eqn: Gamma_e``] are fixed but they are free on the natural boundary [eqn: Gamma_n``]. One of the general methods to handle the equality constraint is the method of Lagrange multipliers, which provides the augmented objective function as

[eqn: I_a ``=`` I ``+`` I_u ``], where [eqn: I ``= int_{Omega} ``F {rm d}{Omega}``], [eqn: I_u ``= int_{Omega} ``p``L(u) {rm d}{Omega}`` equiv`` <p, L (u)> ``],	(4)

where [eqn: p``] is Lagrange multiplier and we suppose that the integrand of [eqn: F``] is given by [eqn: F``=``F``(u, nabla u``)``]. When the boundary shape is fixed, the variation of [eqn: delta I``] is given by

[eqn: delta I ``= < {partial F}over{partial u} - nabla cdot {rm bold F}_1 , delta u ``> + B_n ({rm bold F}_1 , delta u``)``]	(5)

where[eqn: {rm bold F}_1 equiv {partial F}over{partial {nabla u}}``], [eqn: B_n ({rm bold F}_1 , delta u``) ``equiv int_{Gamma} {rm bold n} cdot {rm bold F}_1 ``delta u {rm d} Gamma ``].

In the case that [eqn: L``] is given as a linear operator, the variation of [eqn: delta I_u``] is given by

[eqn: delta I_u = <delta p, L(u)> + <p, L(delta u)> = <delta p, L(u)> + <delta u, L^* (p)> + J_L (delta u, p)``],	(6)

where [eqn: L^*``]is the adjoint operator of [eqn: L``] and [eqn: J_L ``] is the conjunct corresponding to [eqn: L``]. By using (1), (4), (5), (6), we can obtain Euler-Lagrange equations, given by

[eqn: L(u``)=0``], [eqn: L^* (p) ``= -{partial F}over{partial u} + nabla cdot {rm bold F}_1 ``].	(7)

Also the boundary conditions can be obtained from

[eqn: B_n ({rm bold F}_1 , delta u``)+ J_L (delta u, p)``=``0``], [eqn: delta u``=0`` {rm on} ``Gamma_e``], [eqn: delta u`` != 0`` {rm on} ``Gamma_n``].	(8)

APPLICATION TO CRYSTAL GROWTH PROBLEM

The governing equation for the temperature field in the crystal is given from the heat transfer equation. From the characteristics in CZ process, we can obtain the simple governing equation in the form of Laplace equation. That is, the state equation is given by [eqn: L``(u``)=nabla^2 u``], where [eqn: u``] is denoted by the dimensionless temperature. As mentioned previously, we formulate the optimization problem by considering the thermal stress and the crystal growth rate under the assumption that the radial component of the thermal gradient is the dominant factor to induce the thermal stress inside the single crystal. The objective function is

[eqn: {rm Min.} ``I``(u) = int_{Omega} F {rm d} Omega + alpha G(u)],	(9)

where [eqn: alpha``] is the positive weighting factor. [eqn: F``] and [eqn: G``(u``)``] are given by

[eqn: F ``= left( {partial u}over{partial r} right)^2 ``], [eqn: G``(u``)= int_{Gamma_1} -{rm bold n} cdot nabla u ``{rm d}Gamma``],	(10)

where [eqn: Gamma_1 ``] corresponds to the crystal-melt interface and[eqn: {rm bold n}``]denotes the outward unit normal from the interface to the melt phase.

From (7), we can obtain the Euler-Lagrange equations, which are expressed in the cylindrical coordinates as

[eqn: {1}over{r} {partial}over{partial r} left( r {partial u}over{partial r} right) + {partial^2 u}over{partial z^2} = 0``], [eqn: {1}over{r} {partial}over{partial r} left( r {partial u}over{partial r} right) + {partial^2 u}over{partial z^2} = {2}over{r} {partial}over{partial r} left(r {partial u}over{partial r} right) ``]	(11)

By using (8), we can obtain the boundary conditions to the PDEs in (11)

[eqn: u=1, ``p = alpha``] on [eqn: z=0``],  [eqn: 2{partial u}over{partial r}={partial p}over{partial r}, ``p = 0 ``] on [eqn: r=1``],	(12)

[eqn: u=0, ``p = 0``] on [eqn: z=l``],  [eqn: {partial u}over{partial r}=0, ``{partial p}over{partial r} = 0``] on r=1.	(13)

In order to solve the PDEs with boundary conditions, we use the finite difference method and the alternative directional iterative method. Fig. 1 shows the typical results of the optimal distributions of [eqn: u``], [eqn: p``] in the case of [eqn: alpha``=0.1``]. Even though the problem about [eqn: p``] has a singularity at [eqn: (r,z)=(1,0)] due to nonzero value of [eqn: alpha``], there is no difficulty to get fully converged solution by iterations. From Fig. 1, we can see that weighting of [eqn: alpha``] on crystal growth rate results in steep temperature gradient near the crystal-melt interface and thus surface temperature gradually increases due to the term of the minimization of radial thermal gradient in the objective function. By using the penalty function method, we can reflect technical opinion to avoid the optimal solution below a certain critical value.

COMPARISON WITH OTHER METHOD

In this work, the optimization problem is formulated with the temperature only, then analytical solution for the optimal distribution is available. Koh [6] provided the exact solution in the series form as a part of his thesis. Based on his solution, the distribution of crystal surface temperature can be represented in the series form with unknown coefficients. In this work, we use the finite truncation of the infinite series, which means the projection of the infinite dimensional function space to a finite dimensional vector space. As a search method, we select the conjugate-direction method, because the gradient-based methods are known to be not efficient without the accurate expression for gradient. For predicting the optimal step size, we use the golden section method.  Until termination criteria are satisfied, repeated are solving differential equations, evaluating the objective function value by numerical integration, finding the search direction and optimal step size and updating the unknown coefficients.

The histories of convergence during iterations in the variational method and the search method are compared. Fig. 2(a) shows that both errors of [eqn: u``] and [eqn: p``] decrease when the variational method is used. In Fig. 2(b), two errors for change of step size and of objective function value per each iteration are shown. From this figure, we can see that iterations cannot make converged solution and strong fluctuations of numerical error occur near a certain tolerance in the case of search method

CONCLUSION

An efficient numerical algorithm based on the calculus of variations is proposed for the optimization problem in distributed parameter systems, at least in the view point of solution convergency. Since the realistic design variables are the configurations of thermal shield, the heater/cooler positions, the optimal distributions of the crystal surface temperature obtained in this work can provide an insight for the optimal design. Furthermore, the methodology adopted in this work can be applied to a much wider class of the problems.

ACKNOWLEDGEMENT

The authors wish to acknowledge the financial support by the grant from the Korea Science and Engineering Foundation (KOSEF) through the Automation Research Center (ARC) at the Pohang University of Science and Technology (POSTECH).

REFERENCES

1. F. Dupret and N. Van Den Bogaert, in: "Handbook of Crystal Growth", vol. 2, Ed. D.T.J. Hurle, 875 (1994)

2. D.E. Kirk, □Optimal control theory: an introduction□, Prentice-Hall Inc. (1970).

3. J.L. Lions, □Optimal control of systems governed by partial differential equations□, Springer-Verlag (1971).

4. R. Weinstock, "Calculus of variations: with applications to physics and engineering", Dover Publications Inc. (1974).

5. M. Avriel, "Nonlinear programming: analysis and methods", Prentice-Hall Inc. (1976).

6. Y. Koh, M.S. thesis, POSTECH, S. Korea (1998).

Fig. .Distributions of the

optimal solutions [eqn: u``], [eqn: p``]

when [eqn: alpha``=0.1``]

Fig. . Histories of convergence during iterations when [eqn: alpha``=0.1``]

(a) variation method

(b) search method