Chapter 4 b Process Variation Modeling f x

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

Outline • Introduction to Process Variation • Spatial Correlation Modeling

As Good as Models Tell… Our device and interconnect models have assumed: • n n n 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 • n n 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 • n n n Channel length (gate length) Channel width Thickness of dioxide (tox) Threshold (Vth) … Operation factors • n n Power supply fluctuation Temperature Coupling … Material physics: device fatigue • n n n Electron migration hot electron effects …

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

Process Variation Trend • Keep increasing as technology scales down n n 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) • • Environmental 15 % Device fatigue 10% Device characteristics (Vt, Tox) 5% Extreme case: guard band [-40%, 55%] n -10% ~ +25% 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 • n n Normal distribution Uniform distribution

Types of Process Variation Systematic vs. Random variation • n n Systematic variation has a clear trend/pattern (deterministic variation [Nassif, ISQED 00]) – 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 • n n 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 n n Robust Extraction of Valid Spatial Correlation Function Robust Extraction of Valid Spatial Correlation Matrix

Process Variation Exhibits Spatial Correlation of device parameters depends on spatial locations • n The closer devices the higher probability they are similar Impact of spatial correlation • n n Considering vs not considering 30% difference in timing [Chang ICCAD 03] Spatial variation is very important: 40~65% of total variation [Nassif, ISQED 00]

Modeling of Process Variation f 0 is the mean value with the systematic variation considered • n n h 0: nominal value without process variation ZD 2 D, 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 • n n ZD 2 D, 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 • n • n n 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 correlation matrix Spatial variance Random variance

Process Variation Characterization via Correlation Function A more flexible model is through a correlation function • n n 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 d 1 2 Spatial covariance d 1 1 3 d 1 1 1 Overall process correlation

Outline • Process Variation • Spatial Correlation Modeling n n 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 inconsistency Extract: global variance G 2, spatial variance S 2, random variance R 2, and spatial correlation function (v) Such that: G 2, S 2, R 2 capture the underlying variation model, and (v) is always valid M measurement sites 1 i 1 2 …M k N sample chips fk, i: measurement at chip k and location i Global variance Spatial variance Random variance Valid spatial correlation function How to design test circuits and place them are not addressed in this work

Extraction Individual Variation Components Variance of the overall chip variation • Unbiased Sample Variance • Variance of the global variation • Spatial covariance We obtain the product of spatial variance S 2 and spatial correlation function (v) • n n Need to separately extract S 2 and (v) has to be a valid spatial correlation function

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 n • 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 n n n •

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

Outline • Process Variation • Spatial Correlation Modeling n n 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 n M measurement sites 1 i 1 2 fk, i: measurement at chip k and location i …M k N sample chips Valid correlation matrix

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 • n

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 n

Reference n n 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.