1. Chapter 4b Process Variation Modelingμx σx x
f (x)
Prof. Lei He
Electrical Engineering Department
University of California, Los Angeles
URL: eda.ee.ucla.edu

2. Outline
Introduction to Process Variation
Spatial Correlation Modeling

3. Our device and interconnect models have assumed:
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? …

4. 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
Similar sets of factors affect interconnect delays as well.

5. Lithography Manufacturing Process
As technology scales, all kinds of sources of variations
Critical dimension (CD) control for minimum feature size
Doping density
Masking

6. Process Variation Trend
Keep increasing as technology scales down
Absolute process variations do not scale well
Relative process variations keep increasing

7. Variation Impact on Delay
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%
Extreme case: guard band [-40%, 55%]
In reality, even higher as more sources of variation
Can be both pessimistic and optimistic

8. 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

9. 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

10. Introduction to Process Variation
Outline
Introduction to Process Variation
Spatial Correlation Modeling
Robust Extraction of Valid Spatial Correlation Function
Robust Extraction of Valid Spatial Correlation Matrix

11. 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]

12. 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

13. 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
Global variance
Overall variance
Spatial variance
Random variance
Spatial correlation matrix

14. 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 d1
Spatial covariance 2 d1 1 1 3
Overall process correlation d1 1

15. Spatial Correlation Modeling
Outline
Process Variation
Spatial Correlation Modeling
Robust Extraction of Valid Spatial Correlation Function
Robust Extraction of Valid Spatial Correlation Matrix

16. Robust Extraction of Spatial Correlation Function
Given: noisy measurement data for the parameter of interest with possible inconsistency
Extract: global variance G2, spatial variance S2, random variance R2, and spatial correlation function (v)
Such that: G2, S2, R2 capture the underlying variation model, and (v) is always valid
M measurement sites 1 2
fk,i: measurement at chip k and location i
Global variance
Spatial variance i … M
Random variance 1 k
Valid spatial correlation function
N sample chips
How to design test circuits and place them are not addressed in this work

17. Extraction Individual Variation Components
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
Unbiased Sample Variance

18. 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

19. Robust Extraction of Spatial Correlation
Reformulate another constrained non-linear optimization problem
Different choices of b and s  different shapes of the function each function is a valid spatial correlation function

20. Spatial Correlation Modeling
Outline
Process Variation
Spatial Correlation Modeling
Robust Extraction of Valid Spatial Correlation Function
Robust Extraction of Valid Spatial Correlation Matrix

21. 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
M measurement sites 1 2
fk,i: measurement at chip k and location i i … 1 M k
Valid correlation matrix
N sample chips

22. 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

23. 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

24. Reference
Sani R. Nassif, “Design for Variability in DSM Technologies”. ISQED 2000:
Jinjun Xiong, Vladimir Zolotov, Lei He, "Robust Extraction of Spatial Correlation," ( Best Paper Award ) IEEE/ACM International Symposium on Physical Design , 2006.