Performance Analysis of EWMA Controllers Subject to

Metrology Delay

報告者：碩研工管二甲 蔡依潾

原著： Ming-Feng Wu中華民國 2008年 8 月

Main point PrefacePreface

Questions of this study

EWMA controller◦ Single EWMA controller

Literature Reviews

Metrology delay

The Single EWMA Controller subject to Metrology delay

Example◦ Analysis

conclusion

PrefacePreface

半導體產業需要很高的投資成本

先進製程控制量測延遲是自然存在的問題

這篇研究主要分成二個部份

◦1.Exponentrally weighted moving average（ EWMA） 控制器存在量測延遲下做探討

◦2.利用虛擬量測系統 (virtual metrology system， VM) 來探討量測延遲的問題

Questions of this study1.Is the investment in advanced metrology justified？

2.How do we retune the controller parameters

if the metrology delay is changed?

3.Can virtual metrology be used?

4.Do the above guidelines apply in case of variable delays？

EWMA EWMA Controller

The Single EWMA Controller(1/6)

Model:

is the observed process output at the run.

is the process input at the run.

is the intercept parameter.

is the slope parameter.

is the process disturbance.

tY

1tX

t

ttt XY 1

The Single EWMA Controller(2/6)

The Single EWMA Controller(3/6)

The Single EWMA Controller(4/6)

The Single EWMA Controller(5/6)

調整製程

: is the target of the process

: the estimate of

: the discount factor,

b

aTX t

t

)(

11 1 tttt abXYa

10

T

b

調整截距

11 )( tt aTY

The Single EWMA Controller(6/6)

穩定條件

必要條件

0b

lim ( ) is bounded.t tVar Y

TYE tt )(lim

Literature Reviews

ProcessI-O model

EWMA feedback control scheme

Without metrology delay With metrology delay

MISO(SISO) system

With no linear drift

Ingolfsson & Sachs (1993)Tseng et al. (2003)

With a linear drift

Butler & Stefani (1994)Chen & Guo (2001)

Tseng et al. (2002) Su & Hsu (2004)

Tseng et al. (2007)

MIMOsystem

With no linear drifts

Tseng et al. (2002) Good & Qin (2006)

With linear drifts

Del Castillo & Rajagopal (2002, 2003)

Tseng et al. (2007)Lee et al. (2008)

Metrology delay

741 32 85 6Production Line

d=0 1 2 3 4 5 6 7 8

d=1 1 2 3 4 5 6 7

d=2 1 2 3 4 5 6

d=3 1 2 3 4 5

Lot-to-Lot metrology delay

The Single EWMA Controller subject to Metrology delay(1/2)

Model:

is the observed process output at the run.

is the process input at the run.

is the initial bias of process.

is the process gain.

is the disturbance input.

tY

1tX

1t t tY x

t

Process predicted model

is the model offset parameter.

is the model gain parameter.

Disturbance is estimated to be

tt bXaY ˆ

a

b

111 ˆ)1()(ˆ tdtdtt bXaY

b

aTX t

t̂

delay

The Single EWMA Controller subject to Metrology delay(2/2)

Model mismatch Bias correction

Time-correction noise reduce

1r 1s

Formula of Process Output

1 ( 1)

01 ( 1)

1 (1 )( )

1 (1 ) ( )

d

t t t td

z zY a W

z z

11t tz Y Y

/ b

2

1 1

1, 1

0, [2, 1]

( 1) , [ 2, 2 2]

, [2 3, )

i i d

i i d

i

i dp

r s r i d d

rp sp i d

00

( )t

it i

i

p z a

0

ti

t i ti

W p z

Backshift operator

( 公式1 ）

Bias subject to Metrology delay

0Let ( )t tB a

1 ( 1)

01 ( 1)

1 (1 )( )

1 (1 ) ( )

d

t td

z zY a

z z

1

1 1

1, [1, ]

( 1)(1 )1 , [ 1, 2 1]

1, [2 2, )

t d

t

t t d

t d

r s rB t d d

rrB sB t d

( 公式2 ）

tt pppB 10

Proof 1-1:◦ Give

◦ From 公式 2 when

1

10

22 dt

Proof (1/2) TYE tt )(lim

Proof 1-2:◦ For any ， We can find some value of

◦ So that

From 公式 2 when

1 10

0tB

12 dt

Proof (2/2) TYE tt )(lim

The first property of bias subject to Metrology delay

For an overestimated process gain (i.e., )

◦ is a monotonic decreasing sequence.

For an underestimated process gain (i.e., )

◦ There exists some values of larger than which is oscillatory.

1

tB

1

tB

Proof

The following can be easily derived from公式 1 ：

Proof 2:

lim ( ) is bounded.t tVar Y

The second property of bias subject to Metrology delay

In case of , for all t>d

Then

Therefore

In case of and

The optimal value of is equal to one

1 21 )10(

),,(),,( 21 dBdB tt

),,(),,( 21 dSSEdSSE

1 )10(

),1,(),,(min dSSEdSSE

The effects of time delay don the optimalλand SSE(1/2)

We can found that the optimal value ofλdoes not change much with different run numbers.

N=20 N=2004,1 d

The effects of time delay don the optimalλand SSE(2/2)

0 1 2 3 4 50

0.2

0.4

0.6

0.8

1

Delay

λO

PT

ξ ≦ 1.0

ξ =2.0 ξ =1.8 ξ =1.6

ξ =1.4

ξ =1.2

0 1 2 3 4 50

2

4

6

8

10

12

14

16

18

20

Delay

SS

E = 1.4 = 1.0

= 1.6 = 1.2

= 0.6

= 0.4

The effects of time delay d on the optimalλfor different noise to initial bias

ratios

0 1 2 3 4 50

0.2

0.4

0.6

0.8

1

Delay

OP

T

= 0.4

= 0.6

= 0.8

= 1.0

= 1.2 = 1.4 = 1.6 = 1.8 = 2.0

0 1 2 3 4 50

0.2

0.4

0.6

0.8

1

Delay O

PT

= 0.4

= 0.6

= 0.8

= 1.0

= 1.2 = 1.4 = 1.6 = 1.8 = 2.0

0.1 0.2 2

2( )a

A ratio of metrology noise to the magnitude of error initial bias estimate

The effects of time delay d on the optimalSSE for different noise to initial bias

ratios

0 1 2 3 4 50

2

4

6

8

10

12

14

16

18

20

Delay

SS

E

= 1.4 = 1.0

= 1.6 = 1.2

= 0.6

= 0.4

0 1 2 3 4 50

2

4

6

8

10

12

14

16

18

20

Delay

SS

E = 1.4 = 1.0 = 1.6

= 1.2

= 0.6 = 0.4

0.1 0.2

Time-correlated noise reduction

1

1

Consider that follows a ARMA(1,1) time series model with a metrology noise,

that is,

1 .

1

Note that when 1 the process disturbance becomes a non-s

t

t t t

z

z

tationary process

disturbance IMA(1,1); when 1 the process disturbance is stationary process

disturbance.

1 ( 1) 1

1 ( 1) 1

1 (1 ) 1( , , , , )

1 (1 ) ( ) 1

d

t t t td

z z zY d W

z z z

Non-stationary process disturbance IMA(1,1)

0 1 2 3 4 5 6 7 8 9 100

0.2

0.4

0.6

0.8

1

λO

PT

Delay

ξ =0.4 ξ =0.6 ξ =1.0 ξ =1.4 ξ =1.8

0 1 2 3 4 5 6 7 8 9 100

0.2

0.4

0.6

0.8

1

Delay

o

pt = 1

= 0.4

= 0.6

= 0.8

= 1.2

= 1.4

= 1.6 = 1.80 1 2 3 4 5 6 7 8 9 10

0

0.2

0.4

0.6

0.8

1

Delay

o

pt

1

= 1.2

= 1.4

= 1.6

= 1.8

0 1 2 3 4 5 6 7 8 9 100

2

4

6

8

10

12

14

16

18

20

Delay

AM

SE

= 0.4

= 0.6

= 1

= 1.4 = 1.6

0 1 2 3 4 5 6 7 8 9 100

2

4

6

8

10

12

14

16

18

20

Delay

AM

SE

= 0.4 = 0.6

= 1

= 1.4 = 1.6

0 1 2 3 4 5 6 7 8 9 100

2

4

6

8

10

12

14

16

18

20

DelayA

MS

E

= 0.4 = 0.6 = 1 = 1.4 = 1.6

0 0.5 1

2 0

Stationary process disturbance ARMA(1,1)

0 1 2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

= 1.5

= 0.5

= 1

Delay

opt

0 1 2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

= 1.5

= 0.5

= 1

Delay

opt

0 1 2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

= 1.5 = 0.5 = 1

Delay

opt

0 1 2 3 4 5 6 7 8 9 101

1.5

2

2.5

3

3.5

Delay

AM

SE

= 1 = 0.5 = 1.5

0 1 2 3 4 5 6 7 8 9 101

1.5

2

2.5

3

3.5

Delay

AM

SE

= 1

= 0.5 = 1.5

0 1 2 3 4 5 6 7 8 9 101

1.5

2

2.5

3

3.5

Delay

AM

SE

= 1.5

= 0.5

= 1

0.8, 0 0.8, 1 0.8, 0.5

2 0

Example

Tungsten CVD process

沈澱速率 溫度 partial pressure of hydrogen d=3

)4200ln(

5.0

8800

)10*2ln(

2

1

80

T

C

C

C

assume

]ln[1

)ln( 2210 HCT

CCRw wR

T2H

ttt z

z

IMANoise

1

1

1

5.01

)1,1(：

Analysis(1/3)

Analysis(2/3)

We find H2 pressure is unchanged.

Due to the fact that C2 is so much smaller than C1

Analysis(3/3)

Optimal AMSE of the tungsten CVD process at difference metrology delay

conclusion1.Is the investment in advanced metrology justified？

2.How do we retune the controller parameters if the metrology delay is changed?

心得心得這篇報告主要探討 SEWMA有 delay的問題◦探討有 delay狀況下， 的影響◦探探討有 delay狀況下，分別探討穩定與不穩定干擾之情況控制器如何調整

我覺得新的發展方向可以針對 DEWMA做 delay的探討