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## Robust Extraction of Spatial Correlation Jinjun Xiong, Vladimir Zolotov, Lei He

Dee, Lewis, Host has reference to this Academic Journal, PHwiki organized this Journal Robust Extraction of Spatial Correlation Jinjun Xiong, Vladimir Zolotov, Lei He EE, University of Cali as long as nia, Los Angeles IBM T.J. Watson Research Center, Yorktown Heights Acknowledgements to Dr. Ch in addition to u Visweswariah Sponsors: NSF, UC MICRO, Actel Process Variations in Nanometer Manufacturing R in addition to om fluctuations in process conditions changes physical properties of parameters on a chip What you design what you get Huge impact on design optimization in addition to signoff Timing analysis (timing yield) affected by 20% [Orshansky, DAC02] Leakage power analysis (power yield) affected by 25% [Rao, DAC04] Circuit tuning: 20% area difference, 17% power difference [Choi, DAC04], [Mani DAC05] R in addition to om dopants Oxide thickness Process Variation Classification Systematic vs r in addition to om variation Systematic variation has a clear trend/pattern (deterministic variation [Nassif, ISQED00]) Possible to correct (e.g., OPC, dummy fill) R in addition to om 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 Intra-die variation

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Spatial 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] Leff highly correlated Leff almost independent Leff slightly correlated A Missing Link Previous statistical analysis/optimization work modeled spatial correlation as a correlation matrix known a priori [Chang ICCAD 03, Su LPED 03, Rao DAC04, Choi DAC 04, Zhang DATE05, Mani DAC05, Guthaus ICCAD 05] Process variation has to be characterized from silicon measurement Measurement has inevitable noises Measured correlation matrix may not be valid (positive semidefinite) Missing link: technique to extract a valid spatial correlation model Correlate with silicon measurement Easy to use as long as both analysis in addition to design optimization Agenda Motivations Process Variation Modeling Robust Extraction of Valid Spatial Correlation Function Robust Extraction of Valid Spatial Correlation Matrix Conclusion

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 [Orshansky TCAD02, Cain SPIE03] Fr models the r in addition to om variation with zero mean ZD2D,rnd: inter-chip r in addition to om variation Xg ZWID,rnd: within-chip spatial variation Xs with spatial correlation Xr: Residual uncorrelated r in addition to om variation How to extract Fr focus of this work Simply averaging across dies will not work Assume variation is Gaussian [Le DAC04] Process Variation Characterization via Correlation Matrix Characterized by variance of individual component + a positive semidefinite spatial correlation matrix as long as M points of interests In practice, superpose fixed grids on a chip in addition to 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 cant be too close (accuracy) Measurement is expensive cant af as long as d measurement as long as all points Overall variance Global variance Spatial variance R in addition to om variance Spatial correlation matrix Process Variation Characterization via Correlation Function A more flexible model is through a correlation function If variation follows a homogeneous in addition to isotropic r in addition to om (HIR) field spatial correlation described by a valid correlation function (v) Dependent on their distance only Independent of directions in addition to absolute locations Correlation matrices generated from (v) are always positive semidefinite Suitable as long as a matured manufacturing process Spatial covariance 1 1 1 d1 d1 d1 2 3 Overall process correlation

Overall Process Correlation without Measurement Noise Uncorrelated r in addition to om part Intra-chip spatially correlated part Inter-chip globally correlated part 1 0 Distance Correlation Distance Overall Process Correlation v(0)=1 perfect correlation, same device Overall process correlation Die-scale Silicon Measurement [Doh et al., SISPAD 05] Samsung 130nm CMOS technology 4×5 test modules, with each module containing 40 patterns of ring oscillators 16 patterns of NMOS/PMOS Model spatial correlation as a first-order decreasing polynomial function Correlation between measured NMOS saturation current Measurement error prevails Wafer-scale Silicon Measurement [Friedberg et al., ISQED 05] UC Berkeley Micro-fabrication Labs 130nm technology 23 die/wafer, 308 module/die, 3 patterns/module Die size: 28x22mm2 Average measurements as long as critical dimension Model spatial correlation as a decreasing PWL function

Limitations of Previous Work Both modeled spatial correlation as monotonically decreasing functions (i.e., first-order polynomial or PWL) Devices close by are more likely correlated than those far away But not all monotonically decreasing functions are valid For example, (v)=-v2+1 is monotonically decreasing on [0,21/2] When d1=31/32, d2=1/2, d3=1/2, it results in a non-positive definite matrix Smallest eigen-value is -0.0303 Theoretic Foundation from R in addition to om Field Theory Theorem: a necessary in addition to sufficient condition as long as the function (v) to be a valid spatial correlation function [Yaglom, 1957] For a HIR field, (v) is valid iff it can be represented in the as long as m of where J0(t) is the Bessel function of order zero () is a real nondecreasing function such that as long as some non-negative p For example: We cannot show whether decreasing polynomial or PWL functions belong to this valid function category but there are many that we can Agenda Motivations Process Variation Modeling Robust Extraction of Valid Spatial Correlation Function Robust = immune to measurement noise Robust Extraction of Valid Spatial Correlation Matrix Conclusion

Robust Extraction of Spatial Correlation Function Given: noisy measurement data as long as the parameter of interest with possible inconsistency Extract: global variance G2, spatial variance S2, r in addition to om variance R2, in addition to spatial correlation function (v) Such that: G2, S2, R2 capture the underlying variation model, in addition to (v) is always valid N sample chips M measurement sites 1 1 M Global variance Spatial variance R in addition to om variance Valid spatial correlation function 2 fk,i: measurement at chip k in addition to location i i k How to design test circuits in addition to 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 in addition to spatial correlation function (v) Need to separately extract S2 in addition to (v) (v) has to be a valid spatial correlation function Unbiased Sample Variance [Hogg in addition to Craig, 95] Robust Extraction of Spatial Correlation Solved by as long as ming 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 as long as the purpose of modeling One such a function family is given by [Bras in addition to Iturbe, 1985] K is the modified Bessel function of the second kind is the gamma function Real numbers b in addition to s are two parameters as long as the function family More tractable enumerate all possible values as long as b in addition to s

Robust Extraction of Spatial Correlation Re as long as mulate another constrained non-linear optimization problem Different choices of b in addition to s different shapes of the function each function is a valid spatial correlation function Experimental Setup based on Monte Carlo Model Monte Carlo model = different variation amount (inter-chip vs spatial vs r in addition to om) + different measurement noise levels Easy to model various variation scenarios Impossible to obtain from real measurement Confidence in applying our technique to real wafer data Our extraction is accurate in addition to robust Results on Extraction Accuracy More measurement data (Chip x site ) more accurate extraction More expensive Guidance in choosing minimum measurements with desired confidence level

Agenda Motivations Process Variation Modeling Robust Extraction of Valid Spatial Correlation Function Robust Extraction of Valid Spatial Correlation Matrix Conclusion 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 as long as EVD N sample chips M measurement sites 1 1 M Valid correlation matrix 2 i k fk,i: measurement at chip k in addition to 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

Robust Extraction of Correlation Matrix Find a closest correlation matrix to the measured matrix A Convex optimization problem [Higham 02, Boyd 05] Solved via an alternative projection algorithm [Higham 02] Details in the paper Results on Correlation Matrix Extraction A is the measured spatial correlation matrix is the extracted spatial correlation matrix is the smallest eigenvalue of the matrix Original matrix A is not positive, as is negative Extracted matrix is always valid, as is always positive Conclusion in addition to Future Work Robust extraction of statistical characteristics of process parameters is crucial In order to achieve the benefits provided by SSTA in addition to robust circuit optimization Developed two novel techniques to robustly extract process variation from noisy measurements Extraction of spatial correlation matrix + spatial correlation function Validity is guaranteed with minimum error Provided theoretical foundations to support the techniques Future work Apply this technique to real wafer data Use the model as long as robust mixed signal circuit tuning with consideration of correlated process variations

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