Efficient IC Statistical Modeling and Extraction Using a Bayesian Inference Framework Li Yu, Ibrahim (Abe) Elfadel1, and Duane Boning Electrical Engineering and Computer Science Microsystems Technology Laboratories, MIT 1Masdar Institute of Science and Technology Device Variation & Statistical Compact Modeling Issues How to build valid statistical models enabling robust circuit design? How to efficiently extract transistor parameters ? 2 How to predict circuit performance using mixtures of on-chip test measurements? What is the lower bound for the number of IV measurements to fit a model and how select those measurements? Traditional Methods: System Identification Approaches ๏ฎ System identification approaches: try to determine a mathematical relation between input and output without going into the physical details of the system ๏ฎ Parameter extraction through least-square optimization, ๐บ ๐ = ๐๐๐๐๐๐๐ ๐ − ๐(๐, ๐) ๐ ๏ฎ Statistical modeling/moment estimation through Backward Propagation of Variance (BPV), ๐๐ญ๐ ๐ ๐ต ๐ ๐๐ญ๐ = ๐=๐( ) ๐๐๐ ๐ ๐๐๐ ๏ฎ Performance modeling through response surface modeling (RSM), ๐บ ๐ = ๐๐๐๐๐๐๐ถ ๐ − ๐ถ โ ๐ ๐ 3 Limitations in System Identification Approaches ๏ฎ Challenges with solution uniqueness/expressiveness ๏ท We wish to extract parameters with physical meaning ๏ท We need to preserve physical correlation among parameters ๏ท We need to limit the parameter values to a specific domain ๏ฎ Challenges from over-fitting ๏ท We wish to model the system with limited measurements/samples ๏ท Some methods require uncorrelated measurements/variables ๏ท Need to perform feature selection ๏ฎ Challenges with noisy data ๏ท Equal weight is assigned to different measurements ๏ท We wish to identify data with systematic offset or measurement noise 4 Our Approach to IC Statistical Modeling and Extraction ๏ฎ MIT Virtual Source (MVS) Compact MOSFET๏ฎ Bayesian Extraction Method with Model (Antoniadis et al.) Very Limited Data • • • Ultra-compact model: small number of physically based parameters Applicable to nano-scale MOS devices Fit model from small number of early device and monitor measurements Statistical IC Modeling and Extraction Methods ๏ฎ Statistical Formulations • • Variation model and extraction using Back-Propagation of Variance (BPV) Projection onto physical subspace spanned by MVS model (in contrast to PCA projections) ๏ฎ Applications • • • Early technology evaluation and trends Efficient circuit performance evaluation Efficient statistical library cell timing characterization Shaloo Rakheja; Dimitri Antoniadis (2013), "MVS 1.0.1 Nanotransistor Model (Silicon),“ http://nanohub.org/resources/19684. 5 Outline ๏ฎ Introduction ๏ฎ Foundation ๏ท MIT Virtual Source (MVS) Transistor Model: A Physical Solution ๏ฎ Performance Estimation with Mixed & Small Size Samples ๏ท Physical Subspace Projection ๏ท Maximum a Posteriori (MAP) Estimation ๏ฎ Compact Model Parameter Extraction ๏ท Bayesian Inference: Learning Precision and Prior at Different Biases ๏ท Optimal Sampling of Transistor Measurements ๏ฎ Statistical Library Characterization ๏ท Novel Delay/Slew Model ๏ท Exploring Sparsity in Library Input Space ๏ฎ Summary/Acknowledgement 6 MIT Virtual Source (MVS) Model ๏ฎ Virtual source velocity ๐ฃ๐ฅ0 ๏ฎ Continuity function ๐น๐ links linear and saturation region ๐น๐ = ๐๐๐ ๐๐๐ ๐๐ก (1+(๐๐๐ ๐๐๐ ๐๐ก )๐ฝ )1 ๐ฝ ๏ฎ Inversion charge density ๐๐๐ฅ0 ๐๐๐ฅ0 = ๐ถ๐๐๐ฃ ๐๐๐ก ln(1 + exp ๐๐บ๐ −(๐๐ −๐ผ๐๐ก ๐น๐ ) ๐๐๐ก ) ๐ผ๐ท ๐ = ๐๐๐ฅ0 ๐ฃ๐ฅ0 ๐น๐ ๏ฎ Function ๐น๐ is a smoothing function Parameters ๐๐๐ (V) ๐0 Description Strong inversion threshold voltage Sub-threshold swing factor ๐ฟ (๐๐/๐) Drain-induced barrier lowering ๐ฃ๐ฅ0 (๐๐/๐ ) Virtual source carrier velocity ๐ (๐๐2 /๐ โ ๐ ) ๐ ๐ 0 (๐โ๐ โ ๐๐) Low-field mobility Series resistance per side Shaloo Rakheja; Dimitri Antoniadis (2013), "MVS 1.0.1 Nanotransistor Model (Silicon),“ http://nanohub.org/resources/19684. 7 ๏ฎ Dynamic model partitions channel charge; includes parasitic caps: ๐ถ๐๐ , ๐ถ๐๐ , ๐ถ๐๐ฃ Outline ๏ฎ Introduction ๏ฎ Foundation ๏ท MIT Virtual Source (MVS) Transistor Model: A Physical Solution ๏ฎ Performance Estimation with Mixed & Small Size Samples ๏ท Physical Subspace Projection ๏ท Maximum a Posteriori (MAP) Estimation ๏ฎ Compact Model Parameter Extraction ๏ท Bayesian Inference: Learning Precision and Prior at Different Biases ๏ท Optimal Sampling of Transistor Measurements ๏ฎ Statistical Library Characterization ๏ท Novel Delay/Slew Model ๏ท Exploring Sparsity in Library Input Space ๏ฎ Summary/Acknowledgement 8 Challenge: Statistical Performance Modeling with Mixed and Limited Data ๏ฎ Goal: approximate circuit performance as function of process variations Performance modeling Process domain Performance domain ๏ฎ Response surface modeling (RSM) is widely used ๏ท Performance ๐ โ๐ = ๏ท ๏ท ๏ท ๏ท ๐ โ๐ฟ : โ๐ฟ: ๐๐ โ๐ฟ : ๐ถ๐๐ : ๐ ๐ผ๐๐ โ ๐๐ โ๐ = ๐1 โ๐ , ๐2 โ๐ , … , ๐๐ โ๐ โ ๐ผ๐1 ๐ผ๐2 … ๐ผ๐๐ target performance of interest (e.g. frequency of a digital circuit) vector of random variables to model process variations basis functions (e.g., linear or quadratic polynomials) model coefficients ๏ฎ Requires large sample size in order to solve (i.e., sample size K > M model parameters) ๏ฎ High dimensionality of โ๐ฟ poses challenges in applying RSM ๏ท Principal component analysis (PCA) or related approaches often required 9 X. Li et al., “Projection-based performance modeling for inter/intra-die variations,” ICCAD, 2005. Proposed Method – Physical Subspace Projection and Maximum a Posteriori Estimation Traditional Method RSM PCA Performance measurements Principal components Target performance Proposed Method Physical subspace projection MAP Process Shift calibration Device-array measurements Physical subspace projection Performance domain: target performance of interest RO measurements Performance domain: measurements from test structures 10 Probability map of physical subspace: {๐๐ก๐ , ๐๐ก๐ , ๐ฃ๐ฅ๐ , ๐ฃ๐ฅ๐ } Physical Subspace Projection: Intuition ๏ฎ MVS model simplifies the complex process to capture a target transistor model from measurement by ๏ท Limited number of parameters ๏ท Most parameters are directly inferable from IV curve ๏ฎ A gap still exists with traditional RSM approaches ๏ท How to translate backwards from measurements in the performance domain (e.g., RO frequency) to MVS domain? ๏ท Typically very small number of measurements are monitored for post-silicon validation purposes and therefore a low dimensional subspace is preferable ๏ท Hard to weight mixed measurements from different types of measurement structures, i.e., device & circuit structures (RO’s) ๏ฎ Idea: find the best (most likely) physical MVS model parameters that explain past and new measurement data 11 (1) Physical Subspace Projection ๏ฎ The purpose of physical subspace projection is to transfer mixture of measurements into a unique probability space spanned by MVS parameter space ๐ฟ ๏ฎ Assumption: ๐ฟ are the subspace variables satisfying a multivariate Gaussian distribution ๐~๐ฉ ๐๐ , ๐ , ๏ท The initial value of ๐ equals the covariance of subspace variables under within-die variation ๏ท Key advantage: subspace variables can be correlated ๏ฎ We choose a conjugate Gaussian prior for ๐๐ ~๐ฉ ๐0 , Σ0 ๏ท ๐0 are the nominal value for the subspace variables ๏ท Σ0 are the variance of subspace variables under die-to-die variation ๏ฎ The probability of observing data point ๐น๐ (๐๐ ) in ๐th group associated with subspace distribution ๐๐๐ ๐น๐ ๐๐ ๐๐ , ๐ : ๏ท 12 (2) Maximum a Posteriori Estimation ๏ฎ Posterior distribution after observing all data is ๐๐ ๐ ๐ญ, ๐๐ฟ ๐ฝ = ๐๐ ๐(๐๐ฟ ) โ ๐๐ ๐(๐ญ๐ |๐๐ฟ , ๐ฝ) โ…โ ๐๐ ๐(๐ญ๐ |๐๐ฟ , ๐ฝ) ๏ฎ Our goal is to find ๐๐ฟ that maximizes ๐ฅ๐ง ๐๐ ๐ ๐ญ, ๐๐ฟ ๐ฝ = ๐ฅ๐ง ๐ฟ ๐๐ ๐ ๐ญ, ๐๐ฟ ๐ฝ ๐ ๐๐ฟ ๏ฎ However, ๐ is a unknown parameter since it is a local parameter for particular die ๏ฎ Initial value of ๐ is the covariance of subspace variables under only intradie variation ๏ฎ Use an expectation maximization (EM) algorithm to find covariance ๐ Initiate ๐ ๐ ๐๐๐ = ๐ ๐๐๐ค ๐(๐|๐น, ๐ ๐๐๐ ) = ๐(๐, ๐น|๐ ๐๐๐ ) ๐(๐น|๐ ๐๐๐ ) โ ๐, ๐ ๐๐๐ = ๐ ๐|๐น, ๐ ๐๐๐ ๐๐๐(๐, ๐น|๐)๐๐ฅ ๐ ๐๐๐ค = ๐๐๐๐๐๐ฅโ(๐, ๐ ๐๐๐ ) ๐ ๐๐๐ − ๐ ๐๐๐ค < ๐ Yes stop 13 No (3) Prediction of Performance using MVS Model ๏ฎ Once ๐ฝ and ๐๐ฟ are obtained, we can estimate the mean and standard deviation of target performance ๐๐ and ๐๐ using the MVS model and Spice simulations ๏ฎ However, there may be mismatch between the nominal performance values of post-layout simulations and the measurements: typical shift of 15% or less ๏ง Shifts in the corresponding performance distributions are due to modeling and extraction inaccuracy ๏ฎ Therefore a very small sample size (~5) can be used to calibrate the difference between measurements and MVS model prediction ๏ฎ This step is referred to as process shift calibration 14 Experimental Results ๏ฎ ๏ฎ ๏ฎ ๏ฎ Measurements from on-chip test structures Designed in 28-nm bulk CMOS process Measured from 3186 dies in 27 wafers Test sites and structures: ๏ท Device-array ๏ท RO-array 15 Validation (I): Compared with Naïve Approach ๏ฎ Naïve approach: Model the output (RO frequency) as function of mean values from measurement of a single type of device DUT (e.g., individual NMOS, PMOS devices) ๏ฎ Our approach: Capture correlations across multiple DUT types (use a mixture group of measurements) Performance modeling Process domain Performance domain Relative error compared to average measurement data 6 5 groups(#1,2,3,4,5) 3 groups(#1,2,3) single group(#1) single group(#2) single group(#3) Relative Error(%) 5 4 Prediction of NOR RO frequency using mean of single group of transistor DUT measurements 3.25x samples reduction Prediction of NOR RO frequency using mean of mixture group of transistor DUT measurements 3 ~10x samples reduction 2 1 1 2 3 4 Sample sizes 16 Relative prediction error for DUT 6 (NOR RO) versus replicate samples per die (sample size) using different DUT group means to fit models. Prediction of NOR RO frequency using mean of mixture group of both transistor and RO DUT measurements (proposed method) Validation (II): Compared with PCA+RSM Method ๏ฎ Two methods are compared with our proposed method ๏ท Standard least square regression (LSR) after PCA (blue circles) ๏ท Advanced least angle regression (LAR) utilizing sparsity after PCA (black triangles) ๏ท Proposed method: physical subspace projection with MAP (red squares) 27 folds …… Cross validation on 27 wafers (3186 die) 17 Relative prediction error for DUT 6 group (NOR RO) versus number of training dies Comparison for Priors with Different Prediction Error 18 Outline ๏ฎ Introduction ๏ฎ Foundation ๏ท MIT Virtual Source (MVS) Transistor Model: A Physical Solution ๏ฎ Performance Estimation with Mixed & Small Size Samples ๏ท Physical Subspace Projection ๏ท Maximum a Posteriori (MAP) Estimation ๏ฎ Compact Model Parameter Extraction ๏ท Bayesian Inference: Learning Precision and Prior at Different Biases ๏ท Optimal Sampling of Transistor Measurements ๏ฎ Statistical Library Characterization ๏ท Novel Delay/Slew Model ๏ท Exploring Sparsity in Library Input Space ๏ฎ Summary/Acknowledgement 19 Previous Optimization Flow for I-V Parameter Extraction in the MVS Model Automatically adjust ๐๐กโ0 to achieve selfconsistency in ๐๐๐ฅ๐ Sub I-V Measurement All I-V Measurement ๏ฎ Parameters are divided into two groups: Least square optimization for subparameters set ๐, ๐ฟ … ๏ท ๐ท๐๐๐ for the sub-threshold region: ๐ฝ๐๐ , ๐๐, ๐น ๏ท ๐ท๐๐๐๐๐ for the abovethreshold region: ๐, ๐น๐๐ , ๐๐๐ Least square optimization for aboveparameters set ๐ฃ๐ฅ๐ , ๐ … No Converge ๏ฎ Optimized through the least-square error function : ๏ท ๐บ ๐ท๐๐๐ = ๐ ๐ ๏ท ๐บ ๐ท๐๐๐๐๐ = ๐ ๐=๐{ln ๐ ๐ ๐น๐ − ln๐ ๐ฝ๐ , ๐ท๐๐๐๐๐ , ๐ท๐๐๐ } 2 ๐ ๐=๐{๐น๐ − ๐ ๐ฝ๐ , ๐ท๐๐๐๐๐ , ๐ท๐๐๐ }2 ๏ท Substantial number of measurements for each device (~20 measurements) needed in traditional extraction of MVS model 20 L. Yu et al., “An ultra-compact virtual source FET model for deeply-scaled devices: Parameter extraction and validation for standard cell libraries and digital circuits,” ASPDAC 2013. New Approach: Bayesian Inference and Past/Prior Information to Enable MVS Model with Limited New Data ๏ฎ Problems in traditional Least Square Estimation (LSE) objective function ๏ท Equal weights: (1) Systematic offset (2) Noisy region ๏ท Numerical solution Id (A/micron) Modeling error (1) Initial value (2) Searching boundary Vg (V) ๏ฎ Novel objective function utilizing Bayesian Inference 21 L. Yu et al., “Remembrance of Transistors Past: Compact Model Parameter Extraction Using Incomplete New Measurements and a Bayesian Framework,” DAC 2014. Id (A/micron) ๏ท Good consistency of MVS model parameters across various nodes ๏ท Different weight according to data “uncertainty” ๏ท A proper prior distribution learned from historical data Measurement error Vg (V) ๏ฎ Modeling errors: MVS model error + measurement error MVS model error 0.5 0.5 Idlin Idsat 0.4 ๏ท Inability of MVS to capture certain physical effects ๏ท E.g. gate tunneling effects Idlin Idsat 0.4 ๏ณ(log10Id) ๏ฎ The learning of precision ๐ฝln๐น๐ −1 and ๐ฝ๐น๐ −1 is a key step in this work at Different Biases ๏ณ(log10Id) Learning Precision ๐ฝ๐น๐ −1 0.3 0.2 0.1 0.3 0.2 0.1 0 0 0.5 0 0 1 0.5 Vg(V) 1 Vg(V) -5 10−๐ × x๐๐ ๏ฎ Measurement errors: ๏ท Noise or other inaccuracies in current measurements ๏ท Important in low ๐ฝ๐ region 22 1.5 1.5 11 0.5 0.5 00 11 0.8 0.8 ๐ฝ๐ ๐ฝ 0.8 0.8 22 0.6 0.6 0.6 0.6 0.4 0.4 0.4 0.4 Vg (V) 11 0.2 0.2 00 0.2 0.2 ๐ฝ๐ ๐ฝ Vd (V) Extraction of average uncertainty ๐ฝ๐น๐ −1 Learning a Prior Distribution ๏ฎ Assuming each current measurement is independent, the likelihood function ๐๐๐ ๐ญ ๐๐ท๐๐๐ , ๐ฝln๐น๐ is ๏ท ๐๐๐ ๐ญ ๐๐๐ ๐ข๐ , ๐ฝln๐น๐ = ๐ ๐=1 ๐๐๐ ๐น๐ ๐๐๐ ๐ข๐ , ๐ฝln๐น๐ ๏ฎ According to Bayes’ rule, we have ๏ท ๐๐๐ ๐๐๐ ๐ข๐ ๐ญ ∝ ๐๐๐(๐๐๐ ๐ข๐ ) โ ๐ ๐=1 ๐๐๐ ๐น๐ ๐๐๐ ๐ข๐ , ๐ฝln๐น๐ ๏ฎ Prior ๐๐0 , Σ๐ 0 provides information before new measurements about expected transistor I-V curves 23 Mean and standard deviation of the transistor IV Mean and standard deviation of the transistor IV curve learned from historical transistor data (no curve learned from historical transistor data (with particular technology information ). particular technology information ). Sequential Bayesian Learning from Prior and I-V Measurement (sub-threshold parameters: DIBL & SS) Id (A/micron) ๐๐ = 0.05๐, ๐๐ = 0.05๐ Vg (V) 24 ๐๐ = 0.9๐, ๐๐ = 0.05๐ ๐๐ = 0.05๐, ๐๐ = 0.3๐ ๐๐ = 0.9๐, ๐๐ = 0.3๐ Maximum a Posteriori Estimation ๏ฎ Goal: argmax ๐๐๐ ๐๐๐ ๐ข๐ ๐ญ and argmax ln๐๐๐ ๐๐๐๐๐๐ฃ๐ ๐ญ ๐ท๐๐๐ ๐ท๐๐๐๐๐ ๏ฎ Equivalent to minimization with new error function 1 ๏ท ๐บ ๐ท๐๐๐ = 2 ๐๐๐ ๐ข๐ − ๐๐0 25 ๐ ๐ Σ๐ 0 −1 ๐๐๐ ๐ข๐ − ๐๐0 + ๐ ๐ ๐=๐ ๐ฝln๐น๐ {ln ๐น๐ − Example I: Early Technology Evaluation (14nm - 28nm) Technology 2 Technology 1 ๐ฝ๐ ๐ฝ 26 ๐ฝ๐ ๐ฝ ๐ฝ๐ ๐ฝ Technology 3 ๐ฝ๐ ๐ฝ ๐ฝ๐ ๐ฝ ๐ฝ๐ ๐ฝ ๐ฝ๐ ๐ฝ ๐ฝ๐ ๐ฝ ๐ฝ๐ ๐ฝ ๐ฝ๐ ๐ฝ ๐ฝ๐ ๐ฝ ๐ฝ๐ ๐ฝ Technology 4 Parameter Consistency with Number of Measurements 20 15 10 5 0 0 % error vs. baseline LSE Bayesian 10 5 0 0 20 Measurements 40 LSE Bayesian 40 20 10 0 20 Measurements 40 ๐ฟ 70 LSE Bayesian 60 50 40 30 20 10 0 ๏ท Average extraction error for 50 transistors (variation & noise) ๏ท Baseline: parameters extracted from measurements with 20๐๐ ๐๐๐ intervals (~100 measurements total) 30 0 40 ๐๐ก0 15 27 20 Measurements 50 0 20 Measurements 40 ๐0 15 % error vs. baseline LSE Bayesian % error vs. baseline 25 ๐ % error vs. baseline % error vs. baseline ๐ฃ๐ฅ0 LSE Bayesian 10 5 0 0 20 Measurements 40 Statistical Extraction using Measurement Results (28nm) ๐๐ก ๐๐ก ๐ฟ ๐0 ๐ฃ๐ฅ๐ ๐ ๐0 ๐ฃ๐ฅ๐ ๐ฟ ๐ 1 -0.104 0.363 0.339 0.328 -0.104 1 -0.001 -0.311 0.075 0.363 -0.001 1 0.616 0.529 0.339 -0.311 0.616 1 0.819 0.328 0.075 0.529 0.819 1 Allows and captures correlation between MVS model parameters 28 Outline ๏ฎ Introduction ๏ฎ Foundation ๏ท MIT Virtual Source (MVS) Transistor Model: A Physical Solution ๏ท Statistical MVS Model through Backward Propagation of Variance ๏ฎ Performance Estimation with Very Small Sample Size ๏ท Physical Subspace Projection ๏ท Maximum a Posteriori (MAP) Estimation ๏ฎ Compact Model Parameter Extraction ๏ท Bayesian Inference: Learning Precision and Prior at Different Biases ๏ท Optimal Sampling of Transistor Measurements ๏ฎ Statistical Library Characterization ๏ท Novel Delay/Slew Model ๏ท Exploring Sparsity in Library Input Space ๏ฎ Summary/Acknowledgement 29 Problem Definition: Statistical Library Timing Characterization ๏ท Traditional approach by lookup table with both ๐ and ๐ generated from Monte Carlo Simulation Supply voltage: ๐๐๐ No impact No impact ๏ท Interpolation is needed given a group of input combinations ๏ท Recent work explores the sparsity in MC sampling space (process space) ๏ท We propose a novel delay/slew model with four parameters combined with Bayesian inference exploiting sparsity in input vector space Driving strength: ๐ผ๐๐๐ Input slew: ๐๐๐ Output load: ๐ถ๐๐๐๐ ๏ท ๐ก๐,๐−โ = ๐๐ (๐๐๐ +๐′) (๐ถ๐๐๐ ๐ผ๐๐๐,๐๐๐๐ + ๐ถ๐๐๐๐ + ๐ผ โ ๐ ๐๐๐ค) ๏ท ๐ก๐,โ−๐ = ๐๐ (๐๐๐ +๐′) (๐ถ๐๐๐ ๐ผ๐๐๐,๐๐๐๐ + ๐ถ๐๐๐๐ + ๐ผ โ ๐ ๐๐๐ค) ๐ ๐๐๐ ๐ผ ๐๐๐ = ๐๐ , ๐ =๐ +๐ผ ๐ =๐ , ๐ = ๐๐ ๐๐ ๐๐ ๐๐ ๐๐ 2 2 30 ๏ท ๐ผ๐๐๐,๐๐๐๐ = 2 = ๐(๐๐กโ , ๐๐๐๐ก๐, ๐๐, ๐ฃ๐ฅ๐ , ๐๐ข, … ) Model Validation Tech Cell ๐๐ ๐ถ๐๐๐ (๐๐น) ๐′(V) ๐ผ error A INV 0.389 0.951 -0.266 0.0922 1.56% A NAND2 0.372 1.328 -0.209 0.0342 1.98% A NOR2 0.356 1.186 -0.241 0.102 B INV 0.416 1.046 -0.287 0.1029 1.50% B NAND2 0.403 1.471 -0.228 0.0339 2.05% B NOR2 0.374 1.276 -0.253 0.1041 1.12% C INV -0.272 0.1069 1.84% C NAND2 0.383 1.120 -0.258 0.050 C NOR2 0.368 1.225 -0.264 0.1170 1.47% 31 0.389 0.978 0.91% 1.94% delay slew delay slew Bayesian Inference Characterization Flow 32 Validation Validation input spread Nominal characterization Vdd (V) 1 0.9 0.8 0.7 1.5 6 -11 4 1 x 10 -15 0.5 0 0 Input Slew (s) 15 x 10 2 Output Cap (F) ๐(๐๐ ) 50 Proposed Model+Baysian Inference Lookup Table Statistical characterization ๐(๐๐ ) 11 3.5 Proposed Model+Baysian Inference Lookup Table x 10 Lookup Table Proposed Model+Baysian Inference Ideal 3 10 5 17X reduction 2.5 30 Frenquency Prediction Error(%) Prediction Error(%) 40 20 20X reduction 2 1.5 1 10 0.5 0 1 33 2 3 5 10 20 Training Samples 50 100 0 1 2 3 5 10 20 30 50 Training Samples 100 0 0.8 1 1.2 1.4 1.6 Delay (s) 1.8 2 2.2 2.4 -11 x 10 Acknowledgments ๏ฎ Funding and support provided in part through the MIT/Masdar Institute Cooperative Program, and in part through collaboration with PDF Solutions, Inc. 34