실리콘 단결정 표면온도 분포의 최적화

고영덕, 정자훈, 강인석

포항공과대학교 화학공학과/지능자동화 연구센터

Optimization of Surface Temperature Profile of a Silicon Single Crystal

Y. Koh, J. H. Jeong, I. S. Kang

Dept. of Chem. Eng. and ARC, POSTECH

1. Introduction

For the past nearly 30 years, the distributed parameter systems, which are described by partial differential equations (PDEs), have occupied an important place in control and system theories. The importance of the distributed parameter systems has grown owing to the expansion of engineering systems that are mainly governed by PDEs.  The optimization problem of distributed parameter system may be defined in various ways, and in some cases it can be formulated as the minimization of an objective function defined on the spatial domain through the solution of a boundary value problem.

In the present work, we are interested in the system where the basic form of the model is known but the model contains unknown functions that must be determined by minimizing the objective function with or without constraints. This problem is called simply the inverse problem. As a sample problem, we consider an inverse heat conduction problem (IHCP) for the crystal part in the Czochralski process. We want to find analytically the optimal temperature profile at the surface of the crystal, which results in the optimal  temperature distribution inside the crystal for production of high quality single crystals.

2. Problem Statement

We consider the steady heat conduction problem in the crystal (Fig. 1). The governing equation and the boundary conditions are given as

Governing eq'n :

[eqn: bold TRIANGLED ^2 T``=``0``]	(1)

B.C.s :

[eqn: T``=``f``(z`)``]	at the crystal surface,

[eqn: T``=``1``]	at the crystal-melt interface,

[eqn: T``=``0``]	at the top of the crystal,

symmetry condition        at the axis of symmetry.

As shown in Fig. 1, we consider both the 2-D rectangular geometry and the axisymmetric cylindrical geometry. The solutions of two problems are represented in the forms of series solutions as

[eqn: z`]                                      [eqn: z`]

[eqn: l`]      [eqn: T=0`]                                   [eqn: l`]       [eqn: T=0`]

[eqn: {partial T} over {partial x}=0`]  [eqn: bold triangled ^2 T=0`]  [eqn: `T=f`(z)`]     [eqn: {partial T} over {partial r}=0`]  [eqn: bold triangled ^2 T=0`]   [eqn: T=f`(z)`]

[eqn: 0`]    [eqn: T=1`]   [eqn: 1`]     [eqn: x`]               [eqn: 0`]      [eqn: T=1`]   [eqn: 1`]       [eqn: r`]

[eqn: phi`]

(a)                               (b)

Fig. 1. Steady heat conduction in a single crystal : (a) rectangular geometry ;

(b) cylindrical geometry.

[eqn: T=1-{z} over {l} ~+~ Sum from n=1 to inf ``A sub n``cosh`({{n pi `x} over l})``sin`({n pi `z} over {l})]           (2a)

for  2-D rectangular geometry,

[eqn: T=1-{z} over {l}~+~Sum from n=1 to inf``A sub n ``I sub 0`({{n pi `r`} over l})``sin({n pi `z} over {l})]              (2b)

for axisymmetric cylindrical geometry,

where the coefficients are determined by the surface temperature condition, i.e. [eqn: f(z)].

Now we want to find [eqn: f``(z`)] (or [eqn: A sub n``]) that optimizes the temperature distribution inside the domain for each case. In other words, [eqn: f``(z`)`] is determined by minimizing the objective function in the space ([eqn: W `sub{2} sup{m}`` [`0`,`l`]``]).  Hereinafter, we consider only the 2-D problem due to the space limitation, although the results for the cylindrical geometry are also available. In the present study, the objective function is defined as

[eqn: J sub1``=``integral from 0 to l integral from 0 to 1`` LEFT ({partial T} over {partial x} RIGHT ) ^2 dx``dz~+~alpha `integral from 0 to 1 LEFT ( {partial T} over {partial z} RIGHT ) sub z=0 dx``] [eqn: (alpha >0)]      (3)

where [eqn: alpha] is a positive parameter determined by the operating conditions. Each term in the objective function has some physical meaning. Minimization of the first term represents the improved uniformity of the x-directional  temperature distribution, which is known to be essential in production of wafers of high quality. The second term has negative value and its absolute value is proportional to the heat removal from the crystal-melt interface. Thus minimization of the second term means the maximization of production.

The optimization problem defined above can be solved analytically without great difficulties. However, [eqn: f``(z`)`] that minimizes the objective function turns out to be unstable in the sense that the solution is fluctuating along the z-direction. So we need regularization, which is a procedure that adds factors to reduce fluctuations in the unknown function such as [eqn: f``(z`)`]. As a result, a new objective function is reconstructed as following.

[eqn: J sub 2`` = ``J sub 1 ~+~J sub regul``]                                       (4)

Regularization term [eqn: J sub regul``] is based on the norm defined in the space[eqn: W ``sub{2} sup{m}`` [0`,`l``]``]. In the present work, the second order regularization

[eqn: J_{regul} = beta int_0^l [f`''(z)]^2 dz~~~(beta>0)]	(5)

is adopted to reduce fluctuations of [eqn: f`(z)``] in[eqn: Left[``0``,`l`Right]``].

Now the question is that what degree of regularization should be used or how the suitable regularization parameter is determined. In the present work, the parameter has been determined as the minimal value of [eqn: beta`] with which the resulting [eqn: f(z)] satisfies a constraint. The constraint has been imposed by consideration of thermal stress in the crystal and the feasible solution set has been defined as

[eqn: \{ f(z) ~such~that ~max vert f~'(z) vert le ~certain~value \}].                    (6)

3. Results and Discussion

The analytical solution to the above optimization problem (2-D case) is given as

[eqn: f`(z)=1-{z} over{l} + Sum from n=1 to inf A sub n cosh({n pi}over{l} ) sin( {n pi z} over {l})].                       (7)

The optimal surface temperature profiles are shown in Fig. 2 for the case of [eqn: l=4]. As we can see in the figure, the fluctuations of [eqn: f``(z`)``] are reduced as the regularization parameter is increased. It can also be noticed that the absolute value of the temperature gradient at the bottom corner decreases with the increase of the regularization parameter. Physically, the sudden change of temperature gradient near the bottom results in extremely high thermal stress and it is detrimental to the quality of single crystal. Therefore,

[eqn: max `| f~'(z)|] should not be larger than a certain value.

Fig. 2. Temperature profile at x=1 ([eqn: alpha=0.2`]) in 2-D case

The temperature distribution in the crystal is shown in Fig. 3 when the regularized optimal temperature profile is imposed at the crystal surface. In Fig. 4, how to determine the regularization parameter is illustrated. From the figure we can also see that the value of [eqn: max left|f``'(z`)right|] is reduced at the cost of increase of [eqn: min. `J``]. When the maximum permissible value of [eqn: max left|f``'(z`)right|] is given, the minimum value of [eqn: beta``] can be found.

Finally it should be mentioned that the results from the present work can play an important role in design of thermal shields in Czochralski process. The thermal shields should be designed in a way to obtain the crystal surface temperature profile which is as close as possible to the optimal profile.

Acknowledgment

This work has been supported by the grant from the Automation Research Center (ARC, POSTECH), which is funded by the Korea Science and Engineering Foundation (KOSEF).

References

1. J. V. Beck, Ben Blackwell & Charles R. St. Clair, Jr, Inverse Heat Conduction, John Wiley & Sons, 1985

2. H. S. Carslaw & J. C. Jaeger, Conduction of Heat in Solids, Oxford University Press, 1959

3. S. Omatu & J. H. Seinfeld, Distributed Parameter Systems : theory and applications, Oxford University Press, 1989

Fig. 3. Temperature distribution                Fig. 4. Selection of regularization

([eqn: alpha=0.2`,`beta=0.001`])                          parameter([eqn: alpha=0.2`])