Chapter 4a Stochastic Modeling μx σx x f (x) Prof. Lei He Electrical Engineering Department University of California, Los Angeles URL: eda.ee.ucla.edu Email: lhe@ee.ucla.edu

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

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? …

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.

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

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

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

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)

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

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

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

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.

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

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.

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.

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

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 Global variance Overall variance Spatial variance Random variance Spatial 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 d1 Spatial covariance 2 d1 1 1 3 Overall process correlation d1 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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 = 100.0 => 0.001 = 1/8 5 = 101.0 => 0.101 = 5/8 6 = 110.0 => 0.011 = 3/8 7 = 111.0 => 0.111 = 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

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

Comparison of MC and QMC QMC converges faster than MC

Reference Sani R. Nassif, “Design for Variability in DSM Technologies”. ISQED 2000: 451-454 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.”