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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
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Approach
Outline
1 Approach
2 Modeling and Identification
3 Experimental Results
4 Conclusion
(IMPACT) Metrology Sampling 2 / 30
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Approach
Context
the progression for Advanced Process Control (APC) places increaseddemand on metrology
lot level variation↘
wafer level variation↘
within-wafer variation
(IMPACT) Metrology Sampling 3 / 30
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Approach
Goal
decrease the number of measurements needed to determine keyfeatures on wafers
(IMPACT) Metrology Sampling 4 / 30
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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
(IMPACT) Metrology Sampling 5 / 30
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Approach
Prediction Process
IdentificationSet
m sitesmeasured
Prediction Set
q sites measuredm− q sitespredicted
(IMPACT) Metrology Sampling 6 / 30
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Modeling and Identification
Outline
1 Approach
2 Modeling and Identification
3 Experimental Results
4 Conclusion
(IMPACT) Metrology Sampling 7 / 30
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Modeling and Identification
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
(IMPACT) Metrology Sampling 8 / 30
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Modeling and Identification
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
(IMPACT) Metrology Sampling 9 / 30
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Modeling and Identification
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
(IMPACT) Metrology Sampling 10 / 30
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Modeling and Identification
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
(IMPACT) Metrology Sampling 11 / 30
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Modeling and Identification
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
(IMPACT) Metrology Sampling 12 / 30
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Modeling and Identification
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
(IMPACT) Metrology Sampling 13 / 30
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Modeling and Identification
Performance Prediction
models include uncertainty in process variation and measurement error
performance prediction possible
SU := tr Cov[ẑUk − zUk
]
(IMPACT) Metrology Sampling 14 / 30
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Modeling and Identification
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.
(IMPACT) Metrology Sampling 15 / 30
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Modeling and Identification
Implementation
model requires data - how do we get it?
re-modeling may be needed - how do we check?
(IMPACT) Metrology Sampling 16 / 30
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Modeling and Identification
Implementation Process 1
initialdata set︷ ︸︸ ︷
predictedfeatures
{measuredfeatures
{
lower rate measurementsto monitor performance
(IMPACT) Metrology Sampling 17 / 30
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Modeling and Identification
Implementation Process 1
initialdata set︷ ︸︸ ︷
predictedfeatures
{measuredfeatures
{modelexpires
(IMPACT) Metrology Sampling 17 / 30
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Modeling and Identification
Implementation Process 1
initialdata set︷ ︸︸ ︷
predictedfeatures
{measuredfeatures
{modelexpires
data set torenew model︷ ︸︸ ︷
(IMPACT) Metrology Sampling 17 / 30
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Modeling and Identification
Implementation Process 1
initialdata set︷ ︸︸ ︷
predictedfeatures
{measuredfeatures
{modelexpires
data set torenew model︷ ︸︸ ︷
modelexpires
(IMPACT) Metrology Sampling 17 / 30
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Modeling and Identification
Implementation Process 1
initialdata set︷ ︸︸ ︷
predictedfeatures
{measuredfeatures
{modelexpires
data set torenew model︷ ︸︸ ︷
modelexpires
data set torenew model︷ ︸︸ ︷
(IMPACT) Metrology Sampling 17 / 30
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Modeling and Identification
Implementation Process 2
initialdata set︷ ︸︸ ︷
predictedfeatures
{measuredfeatures
{modelexpires
data set torenew model︷ ︸︸ ︷
(IMPACT) Metrology Sampling 18 / 30
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Experimental Results
Outline
1 Approach
2 Modeling and Identification
3 Experimental Results
4 Conclusion
(IMPACT) Metrology Sampling 19 / 30
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Experimental Results
Will this work?
performance is process specific! best case:
process variation is “low order”process variation is “stationary”
validation requires real process data
(IMPACT) Metrology Sampling 20 / 30
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Experimental Results
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.
(IMPACT) Metrology Sampling 21 / 30
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Experimental Results
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
(IMPACT) Metrology Sampling 22 / 30
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Experimental Results
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.
(IMPACT) Metrology Sampling 23 / 30
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Experimental Results
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]
PCA C matrix, column 2
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]
PCA C matrix, column 3
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]
PCA C matrix, column 4
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
(IMPACT) Metrology Sampling 24 / 30
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Experimental Results
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
(IMPACT) Metrology Sampling 25 / 30
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Experimental Results
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
Number of measurements per wafer
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
(IMPACT) Metrology Sampling 26 / 30
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Experimental Results
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
Number of measurements per wafer
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
(IMPACT) Metrology Sampling 27 / 30
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Experimental Results
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
Number of measurements per wafer
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
(IMPACT) Metrology Sampling 28 / 30
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Conclusions
Outline
1 Approach
2 Modeling and Identification
3 Experimental Results
4 Conclusion
(IMPACT) Metrology Sampling 29 / 30
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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.
(IMPACT) Metrology Sampling 30 / 30
ApproachModeling and IdentificationExperimental ResultsConclusion
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