Application of a Nonlinear Model Reduction Method to Rapid Thermal Processing (RTP) Reactors

If-024

Copyright © 19% IFAC 13th Triennial World Congress. San Fran!;isw. USA

Application of a Nonlinear Model Reduction Method to Rapid Thermal Processing (RTP) Reactors * H. Alillg, J.L. Ebert, A.Emami-Naeini, R.L. Kosut In tegrated. Systems Inc., 3260 Jay Street, Santa. Clara, CA 95054

Abstl'act: Processes described by partial differential equations (PPEs) such as thermal modelH of RTP rea.ctors are frequently discretized into a high-order lumped model (or the purpose of simula.tio~ . The advantage of qsing a lumped model lies in t he fact. that. it ~pond 8 with " system of ordinary differential equa.tions (ODEs) . •e high order of the lumped model can still be prohibitive for pra.cticaJ applications l:iuch as control design which ma.kes model reduction necessary. We have applied t he "method of snapshots" (also called: PrQper Orthogo-

nal Decomposition method) for thf'! model reduction of ,a: generic RTP simulation model The snapshot method is based on principal CO;p1POnent anal)'sis and projection of the ODEs on a lower-dimension aJ space of basis fun ctio ns. We describe how this method a.pplies to the RTP model and demonstrate how the model order can be reduced by a factor 4 wit hout essential loss of accuracy over," wide operating range .

Keywords:

nonlinear mod el!'!, model reduction, singular value

decom positio n.

1

Introduction

In conventional batch furnace processing, multiple wafer. (H)-lOO) are closely stacked into a quartz holder and th e entire batch is placed in an electric furnacc. Si nce the wafer spacing is typically very small (compared t o th e wafer di· ameter) the wafer stack is essentially heated from the edge and ramp rates are limited by the radial heat transfer through t he stack. If the stack is heated too rapidly t.h e radial ternperatur~ gradients can cause excessive thermal stresses and result in plastic deformation (warp and slip) of the wafers. This processing rate limitation will become increasingly severe with the advent of larger wafe rs (e.g ., 300 mm). In contrast to batch furnace techniques, in a typi cal RTP system a single wafer is heated across

its surface by radiation from an a rray of t ungs ten hal ogen lamps. Wi t h proper control of the distribution of incident r::..ci ia nt fl ux one can achieve ramp rates of 50°C/ s or more while m ai ntaining good temperature uniformity. From a manufactur-

ing point of view ! RTP fits naturally into current cluster-tool concepts that promise fabrication lines that a re more Hexible a nd much less expensive as compared to present billion dollar state-ofthe-art fab'. RTP is an essential technology for single-wafer processing. RTP 's viability has been demonstrated fo r process steps such as silicidation, R'l'CVD , and annealing [1] . It has also been proposed as an efficient way to dean wafers.

le

Precise temperature control is critical to this promising technology, In these systems, m a ny heaters affect the temperature at each location where it is measured . Control that explicitly accounts for the influe nce of each heat source on each t emperature sensor is -needed for acceptable performance. With such strong coupling1 it is difficu It to obtain acceptable control of the temperature profile using single loop conventional controllers com mon ly used in industrial applications. Moreover, since previous approaches relied heavily on precise calibration, slight changes in ch amber design or wafer geometry can require substantia l an d time-consuming effo rts in control redesign. The necessity for meeting extremely high performance

"This work is supported by the Advanced Research Projects Agency (ARPA) under Contract No. NOOO 14-94-C.0187

713

specifications requires that the control system be optimal with respect to the specific process being controlled. As a result, model-based multivariable control system design is a must.. Physical models of RTP processing reactors are typically described by partial differential equations (PDEs). Solving these model equations is computationally extremely expensi ve. The standard way of finding approximate :"!olutions is by discretizing the spatial model :"!tatf~s int.o many lumped elements, then iterating an ordinary differential equation (ODE) type of solution strategy. As a consequence of the discretization these models oftent.1mes contain many states tha.t are almost linearly dependent. This suggests that lower order models can be found which approximate the model behavior quit.e well. Our goal is to apply nonlinear model reduction and control design methods for the purpose of finding optimal open-loop input sequences for the tracking of a given desired temperature profile. These input sequences can then be augmented with a feedback controller to minimize the effects of modeling errors and disturbances. In this paper we will describe this procedure and show results of applicat.ion to a model of a generic RTP processing system. We will first discuss the generic RTF model in the next section, then describe a model reduction method bas~d on snapl;hotl; of the state, and the non linear feedback-linearization control design method. The results are then validated on bolh reduced-order and Ihe full-order models, in open and closed loop.

hot lamps at wavelengths shorter than approximately 41lm, but are opaque to radiation at longer wavelengths. The silicon wafer and guard ring are heated by this short wavelength lamp radiation. As the wafer and guard ring temperatures increase they begin to emit radiation which in turn heats the quartz showerhead and window. Radiation is the dominant mode of heat transfer in the system; however, transfer by conduction through the solid elements (wafer, window, showerhead, etc.) and conduction through (and convection to) the gas between the various elements is not negligible, especially at lower wafer temperatures (e.g. 700 K). A physical model of this non-linear system was constructed which predicts the dynamic temperature response of the 116 nodes in the discretized system illustrated in Figure 2. Details of this physical model are described in Ebert et.a!. [3] . The model computes the radiative transfer between the 250 free surfaces which bound the 116 solid elements using a two-band radiative property model in which the quartz elements are assumed to be completely transparent at wavelengths shorter than 4JLill and are assumed to be completely opaque at longer wavelength:"!. The. conduction into each node is computed for over 100 conduction paths (or branches) including numerous conduction paths between the wafer and showerhead. COIlvection boundary conditions are added to approximate losses to gases that flow through the system. The resulting model consists of 116 states corresponding to the 116 node temperatures.

3 2

The Method of Snapshots

The Generic RTP Model

A generic rapid thermal processing (RTP) syst.em Wa.<; proposed by Jensen [61 for this study of non-linear model reduction an cont.rol design techniques. and is illustrated in Figure 1 . While no system exists with these exact dimensions or elements. several commercial systems do have similar features, and models of such systems are similar to this system. This system consists of five independently powered lamps that form axisymmetric rings at radii 1'"1 through T5. The walls of the chamber are highly reflective (95%) and water-cooled. A thick (6.35 mm) quart" window and thinner (1 mm) quartz showcrhead transmit radiation from the

We will briefly describe a nonlinear model reduction method which we refer to as the Method of Snapshots. The general formulation of this method is described in [7, 8].

Consider the generic (116 states) model equations:

RTP

high-order

Here,

714

'Ve collect a set of observed state vectors ,,("lex) = .p(x, t n ) in a matrix X which has a singular value decomposition (2) Let us a.'lsume that only the first k singular values in E have. a significant value, and that the rest have a negligible magnitude. We may then approximate the state by projecting the differential equation (1) on the space spanned by the matrix U l which is composed of the first k columns of () . The projection of (1) on the space spanned by Ut is done by replacing 1; with lh z and taking the inner product. wit.h t.hf' columns of U l This leads to

,4,(UtZ)4

+ C + RIL)) =

0

(3)

input sequence consists of the sum of four independent PRBS sequences, chosen in such a manner that each of those sequences covers one of the four dominant. time scales in the process (lamp, wafer, showerhead and window temperature). The fluctuation of the state around the nominal trajectory is then described by the difference of the perturbed sequence (Figure 6) and the nominal one (Figure 5), which is our choice for the matrix X , The fluctuations of the wafer and window temperatures are depicted in Figures 7 and 8, respectively. The singular values of X are displayed in Figure g. The point where the singular values drop below 0.1 % point is approximately for M :::: 27 .

"

'.

•

•

"

' ..,

"

"

•

•

.'~

or, equivalently:

~.

-

T,,,_

d_

..-J

.~u

'. The solution in the original coordinates is then given by ~(t) = Ut :(/) This method can be applied to general setl; of differential equat.ions (ODEs/PDEs, linear/nonlinear). Although there is no guarantee that a satisfactory low-order approximat.ion can be obtained, this method seems to work well on high-order spatially disc.retized RTP models. This is due 1.0 the fact that there are many dependencies in the discretized model that can fairly well be described within a linear subspace of the state space.

4

'-

'''-

'''-

Figure 1: Schematic of the axisymmetric generic RTP system

Results :0'1' 'l"i"0

The first. step in the method is to perform model simulations resulting in an appropriate state matrix X There are several possibilities; since we are interested in optimal control around a nominal open-loop optimal temperature trajectory, we used the variations around the nominal trajectory as the basis for constructing X . For this purpose, we first ran the nominal optimal model simulation hy exciting the model with t.he. opt.imal input sequence 100 times in succession (Figure 3, only the first 500 seconds are displayed). Next, we added a sequence of independent. Pseudo-Random Binary Sequences (PRBS) to the same opt.imal input sequence and used that as input for another simulation run (Figure 4). Each component of the

'.., '"

y.., - :

:

:

1!l7'1iMi>G<-IIOIItr

"""<'h.di",, po",-,

(hfonch •• )

!

U i!GI

.

,. Jlldk.. ..rq-f..,.. _ _ _

-L" ...... -,: -.. :.. "-';:. " ......

forrd.ti~lI'IIIlSf••

Figure 2: Discretized geometry for reduction of t.he governing PDEs We performed model validation on the nominal sequence first, for orders M = ]0,15,20,25,30) 35 and 40, The RMS error between the full-order and reduced order models is shown in Figure 10

715

J

Figure 3: The Optimal Input. Sequence

]&{··, Vbi.',~;···}l4l"WK·jJF\.,1 !ir:ttJ\·.t1\",F-'tr:s,:~ ~ :h , t)<·K··.t3L'Kj.N

Figure 6: The PRBS-perturbed Nominal Traject.ory

J

!k.rR2f5Ku}Q[• • rSd,j"filij.J i ~:t±52s"tKjt~"-~1 Figure 4: The Optimal Input. Sequence with PRBS Added

Figure 7: Fluctuations in Wafer Temperature around the Nominal Trajectory due to PRBS

J

J

Figure .1: The Optima.l Nominal Trajectory

Figure 8: Fluctuations in Window Temperature around the Nominal Trajectory due to PRBS

716

__

"" ......,........ ...

::0:;:

0.(1Il1

I

'Ill

t'jl!11!'!Hm~~~'Hm

!lj~

n!w!!nmnnnm!1 .; ..

Ill!

!n!;,,~~lrH~1m~mm

!I\l

'...... ~ '-,

iHIIIIIIIIIIIHU~" 'll :"i,':················

Ilil

,....,r

! ~!!!!!!! !!!! 11! !lll!!!!! I

I

I

ti111H1!!mm1nmn

I

.. +

!'!

."

::E

!~!mmH!!lH1~mmH

Figure 9: Singular Values of the State Matrix (Normalized)

......

"~~

for each of the four groups of temperatures. Order 30 seems to be a good choice. One of the most important measures is the wafer temperature which is off by only one third of a degree. We have used other validation datascts using steps, ramps and ramps with varying slopes as input to further validate the 30th order model. Also an independent PRBS-perturbed sequence was used (not displayed). The results are quite comparable to that of validation on the nominal sequence, and are given in Table I:

____

Wafer Shower Lamp Window

a.,

a Function of Model

Steps 0.290 1.543 0.007 0.664

Ramps 0.290 1.543 0.007 0.664

Var Ramps 0.291 1.505 0.007 0.679

Table 1: RMS validation results for the 30-th order model

Although it has been shown that the state dimension can be reduced significantly) this does not imply that the reduced-order model can be simulated much faster than the original high-order one. In particular, the original model has a tridiagonal conductivity matrix Ae which is transformed to a full matrix for the reduced-order model. Also, the term uT A, (Ut Z)4 cannot be computed efficiently by expanding the power term in products the individual components of z. Therefore one needs to go back to the full-order ,tate term U, z first which adds some high-dimensional computations back in. We are still investigating how we can take better advantage of the reduced-order model.

5

Figure 10: RMS Error Order

PRBS 0.316 1.549 0.018 0.7:l3

Conclnsions

The POD method is a simple method for approximating a set of PDEs by a set of ODEs. The basis of the method is formed by principal componentanalysis. For applications in the areas of pattern recognition and fluid mechanics where oftentimes a very high dimension is required to represent the snapshots, this method may therefore be infeasible for that type of application. For nonlinear processes, it is not guaranteed that the fluctuations of the state vector can adequately be projected on a lower-dimensional linear space. "'hile there is no guarantee) we have been

717

ab le to app ly this method successfully to the RTP model. The POD method was able to reduce the requi red state dimension of the Generic RTP Model siglli ficant.ly. How this low-dimcnsionality can be exploil,ed for the purposes of fa."ter simulation or low-order controll er design is still a subject that needs to be studied .

[8] L. Sirovich: TUrbulence and the Dynami cs of Coherent Structures parts I, Il and Ill, Quarterly of Applied Mathematics, Vo l. XLV , 1987, no. 3, pp. 561-571 , pp . 573-582, pp. 583-590.

[91 A.K. Bangia, F . Batcho, I .G. Kevrekidis: Flows in 2-D Comp lex Ceometries: Bifurcation Studies with "Alternative" Global Spectral Methods , (Princeton University) , in preparation.

Acknowledgements This is joint work with MIT and Princeton University.

In particular, we wish to thank 1. G .

Kev rekidis (Princeton) and K. Jensen (MIT) for th e helpful d iscussions that we had with them .

References 11) Texas Instruments TechnJcal Journal, Vo1.9, No . 5, September-October, 1992.

[21 F. L. Degertekin, J. Pie, Y.J. Lee, B.T. KhuriYakub, K. C. Saraswat: In-Situ Temperature Monito111t..'l m RTF by Aco'Ustical Techniques, MRS Sympos ium, San Francisco, April 1993.

[31 J.L. I%ert , A. Emami-Naeni, R .L. Kosut : Th erma l modeling of rapid thermal proce:lsing sy.sf.ems, submitted to 3rd International Ra.pid

Thermal Processing Conference, Amsterdam. Au g 30-Sep I , 1995.

[4] H. Gay, W.H. Il.ay: Application of Singular Value Method. {or Jdentification and Model Based Ccmtro / of Distributed Parameter Systems, IFAC SYlllpo.i um on Model-Based Control, Atlanta, June 1988.

[51 H. Gay, W.H. Ray: Identificati on and Contml of Linea r Distribll.tf'd Paramet er Systems through the use of Expel'~mentally Delermined Singu lar Functi ons, IFAC Symposium on Control of Distributed Parameter Systems, Los Angeles, CA, July 1986.

[61 K. J ensen, Private communication, 1994. [7] L. Si rovich, C.H. Sirovich : Low dimen&ional De8cription of Compli cated Phenom ena, Contemporary Mathematics, Vol. 99 , 1989, pp. 277-305.

718