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Chapter 4b Process Variation Modeling
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Transcript of Chapter 4b Process Variation Modeling
Chapter 4b
Process Variation Modeling
Prof. Lei HeElectrical Engineering Department
University of California, Los Angeles
URL: eda.ee.ucla.eduEmail: [email protected]μx
σx
x
f (x)
Outline
Introduction to Process Variation
Spatial Correlation Modeling
As Good as Models Tell…
Our device and interconnect models have assumed:
Delays are deterministic values Everything works at the extreme case (corner case) Useful timing information is obtained
However, anything is as good as models tell Is delay really deterministic? Do all gates work at their extreme case at the same
time? Is the predicated info correlated to the measurement? …
Factors Affecting Device Delay
Manufacturing factors Channel length (gate length) Channel width Thickness of dioxide (tox) Threshold (Vth) …
Operation factors Power supply fluctuation Temperature Coupling …
Material physics: device fatigue Electron migration hot electron effects …
Lithography Manufacturing Process
As technology scales, all kinds of sources of variations Critical dimension (CD) control for minimum feature size Doping density Masking
Process Variation Trend
Keep increasing as technology scales down Absolute process variations do not scale well Relative process variations keep increasing
0.00%
15.00%
30.00%
45.00%
1997 1999 2002 2005 2006
Leff Tox Vth
Variation Impact on Delay
Extreme case: guard band [-40%, 55%] In reality, even higher as more sources of variation
Can be both pessimistic and optimistic
Sources of Variation Impact on Delay
Interconnect wiring(width/thickness/Inter layer dielectric thickness)
-10% ~ +25%
Environmental 15 %
Device fatigue 10%
Device characteristics(Vt, Tox)
5%
Variation-aware Delay Modeling
Delay can be modeled as a random variable (R.V.)
R.V. follows certain probability distribution
Some typical distributions Normal distribution Uniform distribution
Types of Process Variation Systematic vs. Random variation Systematic variation has a clear trend/pattern (deterministic variation
[Nassif, ISQED00])– Possible to correct (e.g., OPC, dummy fill)
Random variation is a stochastic phenomenon without clear patterns– Statistical nature statistical treatment of design
Inter-die vs intra-die variation Inter-die variation: same devices at different dies are manufactured
differently Intra-die (spatial) variation: same devices at different locations of the
same die are manufactured differently
Outline
Introduction to Process Variation
Spatial Correlation Modeling Robust Extraction of Valid Spatial Correlation Function Robust Extraction of Valid Spatial Correlation Matrix
Process Variation Exhibits Spatial Correlation
Correlation of device parameters depends on spatial locations
The closer devices the higher probability they are similar
Impact of spatial correlation Considering vs not considering 30% difference in timing
[Chang ICCAD03] Spatial variation is very important: 40~65% of total variation
[Nassif, ISQED00]
Modeling of Process Variation
f0 is the mean value with the systematic variation considered h0: nominal value without process variation ZD2D,sys: die-to-die systematic variation (e.g., depend on locations at wafers) ZWID,sys: within-die systematic variation (e.g., depend on layout patterns at
dies) Extracted by averaging measurements across many chips
Fr models the random variation with zero mean ZD2D,rnd: inter-chip random variation Xg
ZWID,rnd: within-chip spatial variation Xs with spatial correlation ρ Xr: Residual uncorrelated random variation
How to extract Fr key problem Simply averaging across dies will not work Assume variation is Gaussian
Process Variation Characterization via Correlation Matrix
Characterized by variance of individual component + a positive semidefinite spatial correlation matrix for M points of interests
In practice, superpose fixed grids on a chip and assume no spatial variation within a grid
Require a technique to extract a valid spatial correlation matrix Useful as most existing SSTA approaches assumed such a valid
matrix
But correlation matrix based on grids may be still too complex Spatial resolution is limited points can’t be too close (accuracy) Measurement is expensive can’t afford measurement for all points
2 2 2 2F G S R
1,
1,
1
1
M
M
Overall varianceGlobal variance
Spatial variance
Random varianceSpatial correlation matrix
Process Variation Characterization via Correlation Function
A more flexible model is through a correlation function If variation follows a homogeneous and isotropic random (HIR) field spatial correlation described by a valid correlation function (v) – Dependent on their distance only– Independent of directions and absolute locations – Correlation matrices generated from (v) are always positive
semidefinite Suitable for a matured manufacturing process
2 2cov( , ) ( )i j G SF F v Spatial covariance
1
1
1
d1
d1
d1
2
3
2 2
2 2 2
cov( , ) ( )i j G Sv
i j G S R
F F v
Overall process correlation
Outline
Process Variation
Spatial Correlation Modeling Robust Extraction of Valid Spatial Correlation Function Robust Extraction of Valid Spatial Correlation Matrix
Robust Extraction of Spatial Correlation Function
Given: noisy measurement data for the parameter of interest with possible inconsistencyExtract: global variance G
2, spatial variance S2, random
variance R2, and spatial correlation function (v)
Such that: G2, S
2, R2 capture the underlying variation model,
and (v) is always valid
N sample chips
M measurement sites
1
1
M
2 2 2 2F G S R
Global variance
Spatial variance
Random variance
( )v Valid spatial correlation function
2
…
fk,i: measurement at chip k and location i
i
k
How to design test circuits and place them are not addressed in this work
Variance of the overall chip variation
Variance of the global variation
Spatial covariance
We obtain the product of spatial variance S2 and spatial
correlation function (v)
Need to separately extract S2 and (v)
(v) has to be a valid spatial correlation function
Extraction Individual Variation Components
Unbiased Sample Variance
Robust Extraction of Spatial Correlation
Solved by forming a constrained non-linear optimization problem
Difficult to solve impossible to enumerate all possible valid functions
In practice, we can narrow (v) down to a subset of functions Versatile enough for the purpose of modeling
One such a function family is given by
K is the modified Bessel function of the second kind is the gamma function Real numbers b and s are two parameters for the function family
More tractable enumerate all possible values for b and s
11
1( ) 2 ( ) ( 1)2
s
s
b vv K b v s
2
2 2
, ( )min : ( ) cov( )s
s gv
v v
Robust Extraction of Spatial Correlation
Reformulate another constrained non-linear optimization problem
2
212 1 2
1, ,
min : 2 ( ) ( 1) cov( )2s
s
s s gb s
b vK b v s v
2 2. . : s fcs t
Different choices of b and s different shapes of the function each function is a valid spatial correlation function
Outline
Process Variation
Spatial Correlation Modeling Robust Extraction of Valid Spatial Correlation Function Robust Extraction of Valid Spatial Correlation Matrix
Robust Extraction of Spatial Correlation Matrix
Given: noisy measurement data at M number of points on a chip
Extract: the valid correlation matrix that is always positive semidefinite
Useful when spatial correlation cannot be modeled as a HIR field
Spatial correlation function does not exist SSTA based on PCA requires to be valid for EVD
N sample chips
M measurement sites
1
1
M
1
1
1
0
1
M
M
Valid correlation matrix
2
…i
k
fk,i: measurement at chip k and location i
Extract Correlation Matrix from Measurement
Spatial covariance between two locations
Variance of measurement at each location
Measured spatial correlation
Assemble all ij into one measured spatial correlation matrix A But A may not be a valid because of inevitable
measurement noise
ijA
Robust Extraction of Correlation Matrix
Find a closest correlation matrix to the measured matrix A Convex optimization problem Solved via an alternative projection algorithm
Details in the paper
Reference
Sani R. Nassif, “Design for Variability in DSM Technologies”. ISQED 2000: 451-454
Jinjun Xiong, Vladimir Zolotov, Lei He, "Robust Extraction of Spatial Correlation," ( Best Paper Award ) IEEE/ACM International Symposium on Physical Design , 2006.