Multiplexed MPC for mutli zone thermal processing in semiconductor manufacturing

Work on applications of model predictive control MPC as feedback controller for bake plate temperature control has been done experimentally in many papers.. In this thesis, we have desig

Multiplexed MPC for Multi-Zone Thermal Processing in Semiconductor

Manufacturing

Andreas

A THESIS SUBMITTED FOR THE DEGREE OF MASTER OF ENGINEERING DEPARTMENT OF ELECTRICAL AND COMPUTER

ENGINEERING NATIONAL UNIVERSITY OF SINGAPORE

2008

Acknowledgements

I would like to express my gratitude to my supervisor, associate professor Ho Weng Khuen for his guidance through my M.Eng study Without his gracious encouragement and generous guidance, I would not be able to finish my work His unwavering confidence and patience have aided me tremendously I would like to extend special thanks to associate professor Ling Keck Voon His wealth of knowledge and accurate foresight have greatly impressed and benefited me I am indebted to him for his care and advice

I would also like to express my thanks to my friends and colleagues, Mrs Wu Bing Fang, Ms Nandar Lyn, Mr Yan Han, Mr Feng Yong, Mr Chen Ming and many others in the advanced control technology lab who have helped me a lot during my study

I would like to acknowledge the National University of Singapore and AUN SEED-Net for providing research facilities and financial support

Finally, I want to thank my parents, without their support, I could never achieve this goal I want to dedicate this thesis to my brother and sister and hope that they will enjoy it

i

Contents Acknowledgements i

Contents ii

List of Figures iv

Summary vi

1 Introduction 1

1.1 Motivations 1

1.2 Contributions 5

1.3 Organization 6

2 Bake Plate Thermal Modeling 8

3 Controller Design 14

3.1 Introduction to Model Predictive Control (MPC) 14

3.2 Synchronized Model Predictive Control (SMPC) 16

3.2.1 SMPC Model Formulation 16

3.2.2 Prediction Model 17

3.2.3 Optimization Problem without Constraints 19

3.2.4 Constraints 20

ii

Contents iii

3.2.5 Optimization Problem with Constraints 23

3.2.6 Infinite Horizon 23

3.2.7 SMPC Design for 3-zones Bake Plate 27

3.3 Multiplexed Model Predictive Control (MMPC) 28

3.3.1 Problem Formulation 32

3.3.2 MMPC Design for 3-zones Bake Plate 37

3.4 Kalman Filter 38

4 Experimental Result 42

4.1 Experimental Setup 42

4.2 System Identification 44

4.3 Result and Discussion 49

4.3.1 Tuning MPC Parameters 51

4.3.2 White Noise 53

5 Conclusions and Recommendations 56

5.1 Conclusions 56

5.2 Recommendations for Further Study 57

Bibliography 59

Appendix 64

A Derivation of the equivalent LQ Problem of SMPC 64

B Derivation of the Stabilizing Terminal Weight for MMPC 66

List of Figures

1-1 Close-loop experimental result using SMPC and MMPC controller for

un-constrained case 4

2-1 Diagram of bake plate (a) top view; (b) side view 9

3-1 Basic structure of MPC 16

3-2 Flowchart of SMPC controller design 29

3-3 Pattern of inputs update for traditional or synchronized MPC (dashed line) and for multiplexed MPC (solid line) 31

3-4 Flowchart of MMPC controller design 39

4-1 Top view photograph of multizone bake plate 43

4-2 Side view photograph of multizone bake plate 43

4-3 Experimental setup diagram 44

4-4 Step response of bake plate 46

4-5 Comparison of simulation and experimental result for bake plate model From top to bottom, step input applied at zone-1, zone-2, and zone-3 47

iv

List of Figures v

4-6 Diagram of close-loop experiment 484-7 Experimental result of SMPC and MMPC for constrained case 524-8 Experimental result of MMPC with different input weight r 544-9 Experimental result of SMPC and MMPC when states taken directly fromdistorted measurements 55

Summary

Photolithography process is regarded as the center and the most important process in semiconductor manufacturing due to its strong influence on cost and performance of a microchip In the photolithography sequences, the most important variable to be controlled is critical dimension (CD) which is the minimum feature size dimension One

of major source of CD variation is the thermal processing in lithography, such as exposure bake (PEB) and post-apply bake Thermal processing of semiconductor wafers

post-is commonly performed by placement of the wafer on a heated plate for a given period of time A general requirement for these systems is the ability to reject the load disturbance induced by placement of a cold wafer on the bake plate Sluggish response can cause difficulties with, for example, repeatability of the manufacturing process if the recovery time of the temperature disturbance is longer than the baking time of the wafer and the next wafer comes before the temperature recovers

Work on applications of model predictive control (MPC) as feedback controller for bake plate temperature control has been done experimentally in many papers In a recent work, a variant of MPC called Multiplexed MPC, or MMPC, which claimed to have the potential for faster disturbance recovery response over the conventional MPC

vi

Summary vii

was proposed One characteristic of MPC is online optimization Since optimization is conducted every sampling time, therefore computational power is likely an issue All MPC theory to date and as far as we know the implementation, assume that all the control inputs are updated at the same instant or we called synchronized MPC (SMPC) In contrast, MMPC updates only one control input at a time This will lead to suboptimal control signals However, with reduced computational time, MMPC can use shorter update period, and updating all inputs one after another consecutively in the same period with SMPC

In this thesis, we have designed MMPC feedback controller for bake plate temperature control and conduct the experiment to show the improvement from standard MPC controller Since the model is important for MPC controller to work properly, we have conducted bake plate physical modeling and system identification The computational advantage of MMPC becomes even more significant when constraints are considered and with increasing number of zones and control horizon

Chapter 1

Introduction

Semiconductor manufacturing has greatly affected the world due to the wide application

of semiconductor devices The industry development can basically be resembled by the

so called integrated circuit (IC) scaling The number of transistors on a single IC doubles

in every two years according to Moore’s law (Hamilton, 2003) Critical dimension (CD)

of patterns is currently reduced below 100nm A more stringent demand on the CD ation is imposed By the year 2010, a CD control requirement of 4.7nm is expected for45nm technology node (International Technology Roadmap for Semiconductors, 2005 ).The industry has moved through several lithography generations to achieve smaller fea-ture sizes However, technology transition is expensive and time consuming To reducethe cost a better way is to extend the life cycle of current lithography generation The

vari-1

Chapter 1 Introduction 2

challenge is to maintain CD variation within specifications while pushing feature size toits absolute minimum achievable value One solution is the introduction of advancedequipment and process control (Moynes, 2006; Miyagi et al., 2006)

According to Franssila (2004), Microfabrication processes consist of four basic ations which are high-temperature processes, thin-film deposition processes, patterning,layer transfer and bonding Photolithography, a process which include some of these ba-sic processes, is regarded as the center and the most important process due to its stronginfluence on cost and performance of a microchip In the photolithography sequences,the most important variable to be controlled is critical dimension (CD) which is the min-imum feature size dimension CD is perhaps the single variable with the most impact ondevice speed and performance (Tay et al., 2004; Edgar, 2000) The CD is significantlyaffected by several variables (Kim et al., 2004) Exposure was regarded as an importantsource for CD variation (Postnikov et al., 2003), and the errors may originate from ex-posure dose, grid size and illumination condition Another major source of CD variation

oper-is the thermal processing in lithography, such as post-exposure bake (PEB) (Li, 2001;Cain et al., 2005), and post-apply bake (Raptis, 2001)

Thermal processing of semiconductor wafers is commonly performed by placement

of the wafer on a heated plate for a given period of time The heated plate is of largethermal mass relative to the wafer and is held at a constant temperature by a feedbackcontroller that adjusts the resistive heater power in response to a temperature sensorembedded in the plate near the surface The plate is designed with multiple radial zone

Chapter 1 Introduction 3

configurations The wafer may be placed in direct contact or on proximity pins Processesthat utilize this thermal approach include photoresist processing, chemical vapor deposi-tion and rapid thermal annealing, and span a large temperature range (Campbell, 1996;Schaper et al., 1994)

A general requirement for these systems is the ability to reject the load disturbanceinduced by placement of a cold wafer on the bake plate Figure 1-1 shows the closed-looptemperature response of a bake plate used for photoresist processing when a 200mmwafer at a room temperature was placed on the bake plate Initially the temperaturedropped and then recovered because of closed-loop control In manufacturing, wafersare processed in quick successions, one after another Sluggish response can cause dif-ficulties with, for example, repeatability of the manufacturing process if the recoverytime of the temperature disturbance is longer than the baking time of the wafer andthe next wafer comes before the temperature recovers When this happens, there isnot only wafer-to-wafer non-repeatability in temperature processing trajectory, but alsoplate-to-plate non-repeatability as the feedback controllers generally do not respond thesame If the processing temperature is not critical, then this type of response is accept-able However, for some processes such as chemically amplified photoresist processing

of the post-exposure bake step, temperature control is critical (Sturtevant et al., 1993 ;Pawlowski, 1997; ElAwady et al., 1999)

Work on applications of model predictive control (MPC) as feedback controller forbake plate temperature control can be found in (Ho et al., 2000; Lee et al., 2002) In

Chapter 1 Introduction 4

Figure 1-1: Close-loop experimental result using SMPC and MMPC controller for constrained case

un-Chapter 1 Introduction 5

addition, a linear quadratic gaussian (LQG) controller has been applied to a the-art 49-zone bake plate (Schaper el al., 1999) LQG and MPC are optimal controlstrategies In a recent work, a variant of MPC called Multiplexed MPC, or MMPC,which claimed to have the potential for faster disturbance recovery response over theconventional MPC was proposed (Ling et al., 2005; Ling et al., 2006) In this paper,

state-of-we report the successful application of MMPC to improve the temperature recoveryperformance of a multi-zone bake plate Figure 1-1 shows the improvement of MMPCover the standard MPC

In this thesis, both conventional MPC or synchronized MPC (SMPC) and multiplexedMPC (MMPC) controllers were designed for bake plate application These feedbackcontrollers will be used to maintain bake plate temperatures at set point 90oC Theemphasis will be put on how MMPC performs compare to SMPC for disturbance rejec-tion Observation was made in the presence of disturbance caused by wafer placement

on top of the bake plate at set point 90oC This study has major contribution as thefirst experimental application of MMPC and support previous studies and simulation ofMMPC (Ling et al., 2005; Ling et al., 2006; Ling et al., 2008) The scope of this thesiscovering bake plate modeling, SMPC and MMPC controllers design and its application

in real experiment

In the early part of this thesis, physical model of bake plate without wafer will be

Chapter 1 Introduction 6

derived using heat transfer law Furthermore, system identification is conducted to getthe true model for our specific bake plate Using open loop experiment, we can observethe step response of the bake plate Therefore, we can obtain model estimation byfitting experiment data into the structure of physical model we have derived Fromexperimental result, we have found that MMPC outperforms SMPC in term of recoverytime after wafer with room temperature is dropped on top of the plate However, we alsofound that MMPC is not as robust to white noise as SMPC In the experiment, kalmanfilter was used to obtain the true states

This thesis is organized as follow, Chapter 2 discuss plant modeling In this chapter,

a theoretical model of bake plate is constructed using heat transfer law In Chapter 3,standard formulation of SMPC and MMPC problems is given for both finite and infi-nite horizon, constrained and unconstrained In Chapter 4, the experimental setup isexplained in details To verify the accuracy of theoritical model, open loop system iden-tification experiment is conducted In this experiment, step input is given to one of thezones for every zone, then the step response result of open loop experiment and theoreticalmodel simulation will be compared The second part of this Chapter presents close-loopexperimental result of the designed controller for bake plate temperatures control appli-cation with some discussion about the tuning Finally Chapter 5 gives conclusions andrecommendations for future work Appendix A derives equivalent linear quadratic (LQ)

Chapter 1 Introduction 7

problem for SMPC to give fair basis for comparison with MMPC Appendix B derivesstabilizing terminal weight for infinite horizon MMPC

Chapter 2

Bake Plate Thermal Modeling

The plant used in this project is a multi-zone bake plate which comprises of an aluminiumplate with installed heaters at the bottom of the plate Every heater is connected to inputpower so that it can heats up the plate according to the power given The bake plate

as shown in Figure 2-1 can be divided into multiple zones where each zone has its ownseparate heaters and every zone is powered separately Between each zone there is 1mmair gap to reduce the effect of heat transfer between zones A physical model of an m-zonebake plate has been derived in (Ho et al., 2007) based on heat transfer laws Because ofthe good heat conduction of metal, the temperature within each zone of bake plate isassumed to be sufficiently uniform Thus a distributed lumped model can satisfactorilydescribe the plant characteristics Heat transfer due to radiation can be safely neglectedsince its effect is small compared to conduction and convection at the temperature range

of interest Given the energy balance and heat transfer law, the bake plate can be modeled

8

Chapter 2 Bake Plate Thermal Modeling 9

Figure 2-1: Diagram of bake plate (a) top view; (b) side view

Chapter 2 Bake Plate Thermal Modeling 10

Ci = heat capacity of the ith zone (J/K)

Ti(t) = The ith zone temperature above ambient (K)

pi = heater power to zone i (W )

ri = thermal resistance between zone i and surrounding air (K/W )

r(i−1)i = thermal resistance between zone i − 1 and zone i; r(i−1)i =∞ for i = 1 (K/W )

ri(i+1) = thermal resistance between zone i and zone i + 1; ri(i+1)=∞ for i = m (K/W )

Assuming that ambient temperature is constant then at steady state pi(t) = pi(∞) and

Ti(t) = Ti(∞), with ˙Ti(∞) = 0, Eq 2.1 can be formulated as

Chapter 2 Bake Plate Thermal Modeling 11

Because the baking process is not conducted at room temperature but at set point 90oC,therefore it is easier if the variables used are relative temperatures with respect to thesteady state temperatures rather than absolute temperatures Defining new variables

Chapter 2 Bake Plate Thermal Modeling 12

Chapter 2 Bake Plate Thermal Modeling 13

Given the continuous-time model of Eq 2.11, a discrete-time model, with discretizationinterval of h seconds, suitable for digital control design can be obtained as

Chapter 3

Controller Design

Model predictive control (MPC) is a class of control algorithms which make explicit use

of a model of the process to obtain the control signal by minimizing an objective function.The model is used to predict the process output at future time instant (horizon) Know-ing these process output, a control sequence can be calculated to minimize the designedobjective function For each instant, this process is repeated and horizon is displaced to-ward the future However, only the first control signal of the sequences is applied at eachstep, this is known as receding strategy These three components are the main part ofMPC Acronym MPC denotes all types of predictive control laws, for which many otherabbreviations exist such as GPC (Generalized Predictive Control), DMC (Dynamic Ma-trix Control), MAC (Model Algorithmic Control), PFC (Predictive Functional Control),

14

Chapter 3 Controller Design 15

EPSAC (Extended Prediction Self Adaptive Control) and EHAC (Extended HorizonAdaptive Control) (Camacho and Bordons, 2004; Roberts, 2000) These various MPCalgorithms only differ among themselves in the model used and cost function to be min-imized One of the most attractive features of MPC is that it can handle multivariablesystem naturally and can also handle input, output constraints explicitly by includingthem into problem formulation

As is logical, however, MPC also has its drawbacks One of these is that although theresulting control law is easy to implement and requires little computation, its derivation

is more complex than that of classical PID controllers The computation has to be carriedout at every sampling time When constraints are considered, the amount of computationrequired is even higher Although this, with the computing power available today, is not

an essential problem, one should bear in mind that many industrial process controlcomputers are not at their best regarding their computing power Another drawback isthe need for an appropriate model of the process to be available The design algorithm isbased on prior knowledge of the model and is independent of it, but it is obvious that thebenefit obtained will be affected by the discrepancies existing between the real processand the model used However as long as the model is good enough for the purpose, onedoes not need to model all the physics, chemistry and internal behaviour of the process

to get reliable model The basic structure of MPC is depicted in Figure 3-1 All MPCtheory to date and as far as we know the implementation, assume that all the controlinputs are updated at the same instant (Maciejowski, 2002) Therefore, from this point

Chapter 3 Controller Design 16

Figure 3-1: Basic structure of MPConward, this type of MPC will be identified as synchronized MPC (SMPC)

Chapter 3 Controller Design 17

Chapter 3 Controller Design 18

or in general i-step ahead prediction of future output at current time instant k is

Chapter 3 Controller Design 19

We can set SMPC cost function as

There-J = (Yk− Wk)TQ(Yk− Wk) + ∆UkTR∆Uk (3.3)

= (Φ + G∆Uk− Wk)TQ(Φ + G∆Uk− Wk) + ∆UkTR∆Uk

= ∆UkT(GTQG + R)∆Uk+ 2∆UkTGTQ(Φxk− Wk) + const

Minimizing Eq 3.3 with respect to ∆Uk, we get a linear feedback control law:

∆Uk = K1Wk+ K2xk

Chapter 3 Controller Design 20

umin ≤ uk+i ≤ umax i = 0, 1,· · · , Nu − 1

∆umin ≤ ∆uk+i ≤ ∆umax i = 0, 1,· · · , Nu − 1

ymin ≤ yk+i ≤ ymax i = 1, 2,· · · , N2

We can arrange this equation to make it in standard form Ω∆ ˆU ≤ ω

Chapter 3 Controller Design 21

For input increment constraints, we can formulate it as

−∆uk+i ≤ −∆umin

∆ˆuk+i ≤ ∆umax

max

¸T

Chapter 3 Controller Design 22

For output constraints, the formulation is

Chapter 3 Controller Design 23

outputs, the number of inequalities will be 2(2mN u + pN 2)

The optimization problem with constraints can be formulated as

min J = ∆UkT(GTQG + R)∆Uk+ 2∆UkTGTQ(Φxk− Wk) + const

to make use of infinite horizon by setting N u = N 2 = N = ∞ in cost function

It has been known that making the horizon infinite in predictive control will lead toguaranteed stability (Bitmead et al., 1990) However, problem arises when constraintsare involved because it is impossible to solve optimization problems with infinite variable

to be solved Muske and Rawlings (1993a, 1993b, 1995) have made some works to solve

Chapter 3 Controller Design 24

this problem The idea is to re-parameterize the predictive control problem with infinitehorizon in term of finite number of parameters, so optimization can still be performed.One realization of infinite horizon is to compose it in two separated modes Mode 1 issimilar with previous finite horizon problem, while mode 2 use fixed linear control law

to solve optimization problem beyond the horizon To illustrate how it works, regulatorproblem will be used as example

The cost function for regulator problem is

J =

∞

X

i=0

(kxk+i+1k2Q+k∆uk+ik2R)

We can separate it into two parts where J = J1 + J2

k+N

¸T

Chapter 3 Controller Design 25

J1 = (Ψxk+ Θ∆Uk)TQ(Ψxk+ Θ∆Uk) + ∆UkTR∆Uk

For mode 2, we can define control law as

∆uk+i =−Kxk+i f or i≥ N

xk+i+N +1 = (A− BK) xk+i+N = Φxk+i+N = Φi+1xk+N f or i≥ 0

∆uk+i+N = −Kxk+i+N =−KΦixk+N

Chapter 3 Controller Design 26

J2 = (ΨNxk+ ΘN∆Uk)TP (ΨNxk+ ΘN∆Uk)

Combining J1 and J2, the cost function for infinite horizon MPC is

J = ∆UkTSu∆Uk+ ∆UkTSxxk+ const

Chapter 3 Controller Design 27

that beyond N the constraints are always satisfied

We will design infinite horizon SMPC controller for the bake plate model From ouroriginal state space model

Chapter 3 Controller Design 28

derived so far Figure 3-2 shows step by step how to design SMPC controller Predictionmodel matrix and constraints matrix are composed according to Eq 3.5 and Eq 3.4

We can put the infinite horizon weight or end point weighting P , obtained from LQRsolution, as the last part of state weighting matrix Q and solve the optimization usingquadratic programming (QP) For unconstrained case, SMPC solution is linear controllaw

As we know, one characteristic of MPC is online optimization Since optimization isconducted every sampling time, therefore computational power is likely an issue especially

in a system with lack of resources If there is not enough time to compute the controlsignal before next sampling instant due to complexity of the problem then the controllerwill crash since there is not enough memory available to start new optimization whileprevious optimization still running Or even worse, it will give wrong control signals sincethe plant demands inputs from controller Therefore, it is important to find a way toreduce the computational time needed by MPC controller to calculate the control signals.All MPC methods to date, require all input channels to be updated simultaneously.The way of traditional MPC became very big burden with the increase of the inputnumber Theory states that computational complexity including time requirement tend

to vary as O(m3), where m is the number of control inputs Multiplexed model predictivecontrol (MMPC) tries to exploit this weakness of traditional MPC by updating only one

Chapter 3 Controller Design 29

Figure 3-2: Flowchart of SMPC controller design

Chapter 3 Controller Design 30

channel at a time instead of all channels Off course, this will lead to suboptimal controlsignals However, with reduced computational time, MMPC can use shorter updateperiod, and updating all channels one after another consecutively in the same periodwith traditional MPC This is why MMPC called multiplexed MPC Here, we assumethat fresh measurement of plant states are available at reduced sampling period In manycases, it is better to response faster albeit suboptimal than optimal but very late Oneexample is in disturbance rejection case Figure 3-3 shows the pattern of input update inthe MMPC scheme with m = 3 compare to conventional MPC which updates all inputsimultaneously or we called synchronized MPC (SMPC)

MMPC scheme which updates all inputs "not" simultaneously really suit industrialpractice since complex plant usually has large number of input to control and limitedcommunication channel between controller and actuators so that it is impossible to up-date all control input simultaneously One should note that there are many possiblevariation of MMPC scheme regarding its input pattern update Sometimes, it is moreuseful to update one subset instead of only one input, or not to update all the inputsbut decide in real time which one is more important to be updated more frequent Thisvariation of MMPC resembles statistical process control (SPC), which is used widely inmanufacturing processes (Box and Luceno, 1997)

Chapter 3 Controller Design 31

Figure 3-3: Pattern of inputs update for traditional or synchronized MPC (dashed line)and for multiplexed MPC (solid line)