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Sampling Approaches to Metrology in SemiconductorManufacturing

Tyrone Vincent1 and Broc Stirton2, Kameshwar Poolla3

1Colorado School of Mines, Golden CO2GLOBALFOUNDRIES, Austin TX

3University of California, Berkeley CA

(IMPACT) Metrology Sampling 1 / 30

Approach

Outline

1 Approach

2 Modeling and Identification

3 Experimental Results

4 Conclusion


Approach

Context

the progression for Advanced Process Control (APC) places increaseddemand on metrology

lot level variation↘

wafer level variation↘

within-wafer variation


Approach

Goal

decrease the number of measurements needed to determine keyfeatures on wafers


Approach

Approach

develop a correlation model of metrology measurements for aparticular process

conduct an a priori analysis to determine the optimally informativesites that should be measured by minimizing an expected predictionerror at the unmeasured sites.

use a subset of measurement sites together with the correlation modelto predict measurements at sites which are not measured


Approach

Prediction Process

IdentificationSet

m sitesmeasured

Prediction Set

q sites measuredm− q sitespredicted


Modeling and Identification

Outline

1 Approach



4 Conclusion



Sources of Variation

Lgate(f, d, k) = L0 + La(k) + Lb(f, d, k) + Lc(f, k)

+Ld(d, k) + Le(d) + Lf (f, d, k)

pos

Lgate

(a)

(b)

(c)

(d)

(e)

(f)

(c)

(d)

(e)

(f)

(c)

(d)

(e)

(f)

(c)

(d)

(e)

(f)

(c)

(d)

(e)

(f)

(c)

(d)

(e)

(f)

(c)

(d)

(e)

(f)“random” variationlayout dependent variation

field level systematicvariation and bias

wafer level systematicvariation and bias



Metrology Model 1

wafer sequence

−1 −0.5 0 0.5 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

spatial variation

yk(p) = ȳ(p) +∑

i Ci(p)xi,k + nk(p)

parameters

yk(p) - measurements at position p on wafer kCi(p) - variation basis functionxi,k - scaling factor for variation function ink(p) - measurement noise



Identification Process 1

vectorized model

yk =[yk(p1) yk(p2) · · · yk(pm)

]Txk =

[x1,k x2,k · · · xn,k

]Tnk =

[nk(p1) nk(p2) · · · nk(pm)

]Tyk = ȳ + Cxk + nk

case 1: xk iid random sequence

identification process: C and covariance of xk identified using PrincipleComponent Analysis



Metrology Model 2

wafer sequence

−1 −0.5 0 0.5 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

spatial variation

0 5 10 15 20 25 30−2

−1

0

1

2

3

4

5

Wafer #

Mea

sure

men

t

time variation

yk = ȳ+∑

i Ci(p)xi,k +nkx1,k+1...xn,k+1

= Ax1,k...xn,k

+wk



Identification Process 2

Case 2: xk time correlated

xk+1 = Axk + wk

yk = ȳ + Cxk + nk

parameters A, C and covariance of wk identified using CanonicalCorrelation Analysis



Prediction

measured data

yMk = ȳM + CMxk + n

Mk

unmeasured values

zUk = ȳU + CUxk

common variable xk can beestimated from yMk and used topredict zUk

- M- U



Performance Prediction

models include uncertainty in process variation and measurement error

performance prediction possible

SU := tr Cov[ẑUk − zUk

]



Measurement Sequencing

greedy algorithm for minimizing prediction error

1 Set M = ∅, U = {1, . . . ,m}2 For each element i ∈ U, calculate SU−{i}.3 Select the element j for which trSU−{j} is minimum.4 Remove j from U and add it to M.5 If |M| < q goto to step 2, otherwise quit.



Implementation

model requires data - how do we get it?

re-modeling may be needed - how do we check?



Implementation Process 1

initialdata set︷︸︸︷

predictedfeatures

{measuredfeatures

{

lower rate measurementsto monitor performance





predictedfeatures

{measuredfeatures

{modelexpires





predictedfeatures

{measuredfeatures

{modelexpires

data set torenew model︷︸︸︷





predictedfeatures

{measuredfeatures

{modelexpires


modelexpires





predictedfeatures

{measuredfeatures

{modelexpires



Experimental Results

Outline

1 Approach



4 Conclusion



Will this work?

performance is process specific! best case:

process variation is “low order”process variation is “stationary”

validation requires real process data



Real Process Data use for Verification

poly-gate critical dimension (CD) process data fromGLOBALFOUNDRIES Fab1

1789 wafers with common litho-etch equipment sequence

Same feature measured on 18 die.



Average wafer

−1.5 −1 −0.5 0 0.5 1 1.5−1.5

−1

−0.5

0

0.5

1

1.5

[7]

[10]

[6]

[8]

[11]

[9]

Average wafer

X position

[5]

[1]

[12]

[4]

[2]

[13]

[3]

Y p

ositi

on

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8



PCA variance contribution

0 5 10 150

0.01

0.02

0.03

0.04

0.05

(2σ)2

State index

Var

ianc

e C

ontr

ibut

ion

PCA state dimension

0 2 4 6 8 10 120

0.1

0.2

0.3

0.4

0.5

0.6

n=1

n=2

n=3

n=7n=13

Number of measurements per wafer

Ave

rage

pre

dict

ion

erro

r (u

nmea

sure

d si

tes

only

)A plot of average prediction er-ror vs. # of sites measured, fordifferent model dimensions, n.



First four basis elements

−1.5 −1 −0.5 0 0.5 1−1.5

−1

−0.5

0

0.5

1

1.5

[7]

[10]

[6]

[11]

[8]

[9]

[5]

PCA C matrix, column 1

X position

[1]

[12]

[4]

[2]

[13]

[3]

Y p

ositi

on

−0.3

−0.295

−0.29

−0.285

−0.28

−0.275

−0.27

−0.265

−0.26

−1.5 −1 −0.5 0 0.5 1−1.5

−1

−0.5

0

0.5

1

1.5

[7]

[10]

[6]

[8]

[11]

[9]

[5]


X position

[1]

[12]

[4]

[2]

[13]

[3]

Y p

ositi

on

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

−1.5 −1 −0.5 0 0.5 1−1.5

−1

−0.5

0

0.5

1

1.5

[7]

[10]

[6]

[8]

[11]

[9]

[5]


X position

[1]

[12]

[4]

[2]

[13]

[3]

Y p

ositi

on

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

−1.5 −1 −0.5 0 0.5 1−1.5

−1

−0.5

0

0.5

1

1.5

[7]

[10]

[6]

[8]

[11]

[9]

[5]


X position

[1]

[12]

[4]

[2]

[13]

[3]

Y p

ositi

on

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4



Data Splits for Validation

Full Data Set for Litho i/Etch j

Split: wafer Ns

Identification Set:N wafers

m sites measured

Prediction Set:M wafers

q sites measuredm− q sites predicted



Prediction Error vs. Model Window

0 2 4 6 8 10 120

0.1

0.2

0.3

0.4

0.5

N=50

N=200N=899


Ave

rage

pre

dict

ion

erro

r (u

nmea

sure

d si

tes

only

)

Prediction error with M = 899

Required size of model set ∼ 200 wafers



Prediction Error vs. Prediction Window

0 2 4 6 8 10 120

0.1

0.2

0.3

0.4

0.5

M=50

M=1598


Ave

rage

pre

dict

ion

erro

r (u

nmea

sure

d si

tes

only

)

Prediction Error with N = 200.

Performance degrades gracefully with prediction window



Time correlation

0 1 2 3 4 50

0.1

0.2PCA M=50

CCA M=50

0 1 2 3 4 50

0.1

0.2PCA M=400

CCA M=400


Ave

rage

pre

dict

ion

erro

r (u

nmea

sure

d si

tes

only

)

No improvement with more complex model, except for small numberof measurements per wafer


Conclusions

Outline

1 Approach



4 Conclusion


Conclusions

Conclusions

Intelligent strategies can reduce the expense of metrology withoutcompromising effectiveness

Reduced sampling realized through models which predict data atunmeasured sites

The real challenge: SPC and fault detection

Why do we use metrology? To tell us if something goes wrong [alsoclosed-loop process control]

What are optimal sampling strategies that don’t compromise SPC orfault detection?

Metrics: time to detect a process drift or failure, false alarm rates, etc

Supported in part by NSF grant ECS 01-34132, Berkeley IMPACT program and DOE Programs DE-ZDO-2-30628-07 and

DE-FG36-08GO88100.


ApproachModeling and IdentificationExperimental ResultsConclusion

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