1. Chapter 4a Stochastic 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
Introduction to Stochastic Modeling
Spatial Correlation Modeling
Monte Carlo Simulation

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. Outline
Process Variation
Introduction to Stochastic Modeling
Spatial Correlation Modeling
Monte Carlo Simulation

11. R.V. X can take value from its domain randomly
Review of Probability
R.V. X can take value from its domain randomly
Domain can be continuous/discrete, finite/infinite
PDF vs. CDF x dx
f (x) 1 x
F (x)

12. Review of Probability Mean and Variance Normal Distribution σx μx
f (x)

13. Multivariate Distribution
Similar definition can be extended for multivariate cases
Joint PDF (JPDF), Covariance
Becomes much more complicated
Correlation MATTERS!!

14. Correlation Coefficient
Normalized covariance:
Always lies between -1 and 1
Correlation of 1  x ~ y, -1 

15. Principal Component Analysis
PAC helps to compress and classify data
It reduces the dimensionality of a data set (sample) by finding a new set of variables
The new set has a smaller number of variables
The new set nonetheless retains most of the original information.
By information we mean the variation present in the sample, given by the correlations between the original variables. The new variables, called principal components (PCs), are uncorrelated, and are ordered by the fraction of the total information each retains from high to low.

16. A sample of n observations in the 2-D space x = (x1, x2)
Geometric Interpretation of Principal Components
A sample of n observations in the 2-D space x = (x1, x2)
Goal: to account for the variation in a sample
in as few variables as possible, to some accuracy

17. The 1st PC z1 is a minimum distance fit to a line in x space
Geometric Interpretation of Principal Components
The 1st PC z1 is a minimum distance fit to a line in x space
The 2nd PC z2 is a minimum distance fit to a line in the plane
perpendicular to the 1st PC
PCs are a series of linear least squares fits to a sample,
each orthogonal to all the previous.

18. Independent Component Analysis
Independent Component Analysis (ICA) vs. Principal Component Analysis (PCA)
Similarity
Feature extraction
Dimension reduction
Difference
PCA uses up to second order moment of the data to produce uncorrelated components
ICA strives to generate components as independent as possible
PCA can only be used to Gaussian distribution, while ICA can be applied to arbitrary distribution.

19. Introduction to Process Variation Introduction to Stochastic Modeling
Outline
Introduction to Process Variation
Introduction to Stochastic Modeling
Spatial Correlation Modeling
Robust Extraction of Valid Spatial Correlation Function
Robust Extraction of Valid Spatial Correlation Matrix
Monte Carlo Simulation

20. 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]

21. 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

22. 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

23. 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

24. Introduction to Stochastic Modeling Spatial Correlation Modeling
Outline
Process Variation
Introduction to Stochastic Modeling
Spatial Correlation Modeling
Robust Extraction of Valid Spatial Correlation Function
Robust Extraction of Valid Spatial Correlation Matrix
Monte Carlo Simulation

25. 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

26. 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

27. 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

28. 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

29. Introduction to Stochastic Modeling Spatial Correlation Modeling
Outline
Process Variation
Introduction to Stochastic Modeling
Spatial Correlation Modeling
Robust Extraction of Valid Spatial Correlation Function
Robust Extraction of Valid Spatial Correlation Matrix
Monte Carlo Simulation

30. 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

31. 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

32. 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

33. Outline
Process Variation
Introduction to Stochastic Modeling
Spatial Correlation Modeling
Monte Carlo Simulation

34. Monte Carlo Simulation
Problem Formulation
Given a set of random variables X=(X1, X2, … Xn)T and a function of X, Y=f(X), estimate the distribution of the Y
Method
Generate N samples of X=(X1, X2, … Xn)T
For each sample of X, calculate the correspondent sample of Y=f(X)
Obtain the distribution of Y from the samples of Y

35. Advantage and Disadvantage of MC simulation
Pro:
Accurate
Error→0 when N→∞
Flexible
Works for any arbitrary distribution of X
Works for any arbitrary function of f
Simple
Easy to implement
Usually used as golden case in statistical analysis
Con:
Not efficient
Need large N to obtain high accuracy
Need to run large number of iterations
Not suitable for statistical optimization

36. Example
Given X1 and X2 are independent standard Gaussian RVs, estimate the distribution of max(X1, X2)

37. Quasi Monte Carlo Simulation
Basic idea
Use deterministic samples instead of pure random samples
Select deterministic samples to cover the whole sample space evenly

38. Discrepancy measures how evenly the samples are in the sample place
Definition
N is total number of samples, A(B, P) is the number of points in bounding box B, λs(B) is the volume of B
Discrepancy measures how evenly the samples are in the sample place

39. Low Discrepancy Sequence
Sample sequence with low discrepancy
Low discrepancy array generation algorithms
Faure sequence
Neiderreiter sequence
Sobol sequence
Halton Sequence

40. Example: Halton Sequence
Basic idea
Choose a prime number as base (let's say 2)
Write natural number sequence 1, 2, 3, ... in base
Reverse the digits, including the decimal sign
Convert back to base 10:
1 = 1.0 => 0.1 = 1/2
2 = 10.0 => 0.01 = 1/4
3 = 11.0 => 0.11 = 3/4
4 = => = 1/8
5 = => = 5/8
6 = => = 3/8
7 = => = 7/8
High dimensional array
Use different base for different dimension
Example 2-d array, X-base 2, y-base 3
1 => x=1/2 y=1/3
2 => x=1/4 y=2/3
3 => x=3/4 y=1/9
4 => x=1/8 y=4/9
5 => x=5/8 y=7/9
6 => x=3/8 y=2/9
7 => x=7/8 7=5/9

41. Advantage and Disadvantage of QMC Simulation
Efficient
Use fewer sample than random Monte Carlo simulation
Disadvantage
Only works in low dimension cases
Very slow when number of random variations become large
Not very common in statistical analysis

42. Comparison of MC and QMC
QMC converges faster than MC

43. Reference
Sani R. Nassif, “Design for Variability in DSM Technologies”. ISQED 2000:
L. I. Smith. “A Tutorial on Principal Components Analysis”. Cornell University, USA, 2002.
Jinjun Xiong, Vladimir Zolotov, Lei He, "Robust Extraction of Spatial Correlation," ( Best Paper Award ) IEEE/ACM International Symposium on Physical Design , 2006.
Singhee, A., Rutenbar, R. “From Finance to Flip Flops: A study of Fast Quasi-Monte Carlo Methods from Computational Finance Applied to Statistical Circuit Analysis.”