Process Corners and Statistical Models

1. Theory

• Definition: Process corner and statistical models describe how device parameters vary between dies, wafers and lots (global variation) and between neighbouring devices (local mismatch), so circuits can be designed to work across manufacturing spread (Saha; Watts et al.).
• Corners: Fixed model sets such as typical, fast and slow for each device type bound the expected range; foundries add resistor, capacitor, leakage and bipolar corners (SkyWater documentation).
• Statistical models: Each parameter is the nominal value plus random global and local deviations, sampled in Monte Carlo simulation (Saha; Watts et al.).
• Correlations: Physical process steps move many model parameters together, so realistic statistics are built on a few independent factors found by principal component analysis (Power et al.).

2. Working Principle

• Separating scales: Global variation shifts all devices on a die alike and is extracted from electrical-test data across wafers; local variation follows Pelgrom’s area law (Saha; Mismatch and Variability page).
• Independent factors: PCA rotates correlated parameters into uncorrelated components, each tied to a process variable such as oxide thickness or implant dose (Power et al.).
• Worst cases: Placing components, not individual parameters, at their limits and scoring circuit-relevant figures selects realistic corners (Power et al.).
• Propagation: Monte Carlo sampling or first-order sensitivity sums carry parameter spread to circuit performance (Power et al.; background knowledge for the sensitivity form).

3. Applications

• Design sign-off: Simulating digital timing and analog specifications at all corners, temperatures and supplies (SkyWater documentation; background knowledge).
• Yield estimation: Monte Carlo of offsets, gains and delays to predict the fraction of working dies (Saha; Power et al.).
• Analog matching: Monte Carlo mismatch at fixed corners for comparators, references and data converters (Saha; Mismatch and Variability page).
• Process control: Linking model-parameter spread to process fluctuations (Power et al.).

Corners

Name	Formula	Related Formulae	Variables/Constants
Process corners	$\text{TT, FF, SS, FS, SF}:(\text{pMOS speed},\text{nMOS speed})\text{in the SkyWater convention}$	$\text{R/C corners: tt, hl, lh, ll, hh}(\text{resistor},\text{capacitor})$	$\begin{array}{r}T,F,S:typical,fastandslowdevice\\ sets;SkyWater130nmdocumentation:tt,\\ ff,ss,fs(fastpmos,slownmos)andsf\\ (slowpmos,fastnmos)forFETsandPNPs,\\ thetwo-lettercodegivingpMOSfirst,\\ parasiticRCcornerstt,hlandlh,linear\\ R/Ccornerstt,lh,hl,llandhh,\\ cell-transistorcorners,wafer,leakage\\ (averageworst-case)andNPNt,fands\\ sections;cornersarecombinedwith\\ temperatureandsupply(background\\ knowledge;TemperatureScalingpage)\end{array}$ $(nocomputedvalue)$
Probability of simultaneous worst cases	$P\left(\text{all }k\text{ independent parameters beyond }3\sigma \right)={\left[1-\Phi \left(3\right)\right]}^k$	$1-\Phi \left(3\right)=1.35\times 10^{-3},1-\Phi \left(2.5\right)=6.2\times 10^{-3}$	$\Phi :standardnormalcumulativedistribution$ $\begin{array}{r}k:numberofparametersplacedattheir\\ limitstogether(backgroundknowledge);\\ Poweretal.:atraditional\pm 2.5\sigma \\ per-parameterworstcasefoundoverly\\ pessimisticcomparedwithcorrelatedworst\\ cases\end{array}$ $\begin{array}{r}(computed:1.35\times 10^{-3}foroneparameter,\\ 1.8\times 10^{-6}fortwoand4.5\times 10^{-15}forfive\\ independentparametersat+3\sigma )\end{array}$

Global and Local Variation

Name	Formula	Related Formulae	Variables/Constants
Global plus local statistical model	$P_i=P_{i0}+\Delta P_{i,mismatch}+\Delta P_{i,global}$	$\Delta P_{i,mismatch}\sim N\left(0,1\right){\sigma }_i,{\sigma }_{V_{T0}}\approx \frac{A_{VT}}{\sqrt{WL}},{\sigma }_{tot}=\sqrt{{\sigma }_{global}^2+{\sigma }_{local}^2}$	$P_i:modelparameter$ $P_{i0}:nominalvaluefromagoldendie$ $\Delta P_{i,mismatch}:local(intra-die)randomshift$ $\Delta P_{i,global}:global(die,waferandlot)shift$ ${\sigma }_i:standarddeviation$ $\begin{array}{r}{A}_{VT}:Pelgromcoefficient(quadraturesum,\\ backgroundknowledge);Saha:theseforms,\\ alocalsetVT0,TOX,U0,K1,XW,XLanda\\ globalsetaddingRDSW,overlapand\\ junctioncapacitances,global\\ distributionsfromelectrical-testdata,\\ TCADdatafornewtechnologies,local\\ variabilityincreasinglyimportantbelow\\ 90\mathrm{nm},andbothaMonteCarlomodeland\\ fixedSS/FFcornerswithMonteCarlo\\ mismatchatthecorners;Pelgromlawon\\ theMismatchandVariabilitypage\end{array}$ $\begin{array}{r}(computed:anassumed{A}_{VT}=2\mathrm{mV}\mu \mathrm{m}gives\\ 28.3\mathrm{mV}fora0.1by0.05\mu \mathrm{m}device;with\\ anassumed20\mathrm{mV}globalsigmathetotalis\\ 34.7\mathrm{mV})\end{array}$

Correlated Statistics

Name	Formula	Related Formulae	Variables/Constants
Principal components of correlated parameters	$P_{\alpha }={\sum }_{j=1}^m{a}_{\alpha j}{X}_j,m\ll n$	$X_j\sim N\left(0,1\right)\text{independent},a_{\alpha j}=\mathrm{corr}\left(P_{\alpha },X_j\right)$	$\begin{array}{r}{P}_{\alpha }:normalizedparameter(zeromean,unit\\ variance)\end{array}$ $X_j:independentprincipalcomponents$ $a_{\alpha j}:loadings$ $m:retainedcomponents$ $\begin{array}{r}n:originalparameters;Poweretal.:75\\ correlatedparametersfromthree\\ geometriesreducedtom=7components\\ explainingalmost81percentofthe\\ variance,VARIMAXrotation,components\\ identifiedasoxidethickness,length\\ reduction,thresholdimplant,P-well\\ implant,widthreduction,n-channel\\ flat-bandvoltageandjunctiondepth,from\\ over800measuredparametersets(about\\ 700afterremovingoutliersbeyond4\sigma )\end{array}$ $(nocomputedvalue)$
Correlated worst-case search	$F\left(P\right)={\sum }_k\left(B_1\frac{I_{DS,k}}{I_{DS,nom}}+B_2\frac{g_{ds,k}}{g_{ds,nom}}+B_3\frac{g_{m,k}}{g_{m,nom}}\right)$	$X_j=\pm 2.5\Rightarrow 2^m\text{corner candidates}$	$F(P):figureofmeritsummedoverbiaspointsk$ $\begin{array}{r}{B}_1,B_2,B_3:weights;Poweretal.:B_1=B_2\\ =B_3=1and10biaspoints(nominalF=\\ 30),eachofthe7componentsat\pm 2.5\\ giving{2}^7=128candidates,themaximum\\ andminimumF(about50and17)chosenas\\ worst-casesets,whichmatchedmeasured\\ distributions\end{array}$ $(nocomputedvalue)$
Monte Carlo and sensitivity (gradient) analysis	${\sigma }^2\left(e\right)\approx {\sum }_{j=1}^m{\left(\frac{\partial e}{\partial X_j}\right)}^2{\sigma }^2\left(X_j\right)$	$\frac{\Delta \sigma }{\sigma }\approx \frac{1}{\sqrt{2N}}(\text{standard error of a Monte Carlo sigma})$	$\begin{array}{r}e:circuitperformance,\partial e/\partial X_j:\\ sensitivitytocomponentj\end{array}$ $\begin{array}{r}N:numberofMonteCarlosamples\\ (standard-errorform,background\\ knowledge);Poweretal.:correlatedMonte\\ Carlowith100simulationspredicteda221\\ \mu \mathrm{A}standarddeviationagainst240\mu \mathrm{A}\\ measured,uncorrelatedparameters190\mu \mathrm{A},\\ andgradientanalysiswithm+1=8\\ simulations219\mu \mathrm{A},assuminguncorrelated\\ factorsandanearlylinearresponse;\\ Wattsetal.:MonteCarlodistributions\\ andfixedanduser-definablecornerswith\\ PSP\end{array}$ $\begin{array}{r}(computed:100samplesgivea7percent\\ standarderroronsigma,consistentwith\\ the8percentgapbetween221and240\mu \mathrm{A};\\ theuncorrelatedresultis21percent\\ low;200samplesareneededfor5per\\ cent)\end{array}$

Important Questions

Ques: What do TT, FF, SS, FS and SF mean, and which letter refers to which device?

They are typical, fast and slow combinations of the n- and p-channel transistors, but the letter order is a convention. In the SkyWater 130 nm documentation the first letter is the pMOS and the second the nMOS: fs is fast pMOS with slow nMOS, sf is slow pMOS with fast nMOS. Many other foundries and textbooks put the nMOS first, so that FS means fast nMOS and slow pMOS (background knowledge). The same label can therefore mean opposite skews in two kits, which matters for ratioed logic, SRAM write margins and inverter switching points. The SkyWater kit also defines resistor and capacitor corners such as hl (high resistance, low capacitance) and lh, leakage corners, and fast, typical and slow NPN sets. Always check the kit’s definition before interpreting a skewed corner (my reading). Reference: SkyWater 130 nm PDK documentation, ‘Spectre simulation’ page (docs/sim/spectre.rst, s8phirs_10r models; the page does not name SkyWater) (fetch-tool summary).

Ques: Why are corners that put every parameter at 3 sigma too pessimistic?

Because independent parameters rarely reach their extremes together. If one parameter exceeds +3$\sigma$ with probability $1.35\times 10^{-3}$, two independent ones do so together with $1.8\times 10^{-6}$, and five with $4.5\times 10^{-15}$ (computed), far rarer than any yield target. Real parameters are correlated, so the realistic worst case lies along the directions in which process steps move them together. Power et al. found the traditional $\pm 2.5\sigma$ per-parameter approach overly pessimistic, while worst cases built from 7 principal components at $\pm 2.5$, ranked by a current, conductance and transconductance figure of merit, matched measured distributions. Over-pessimistic corners waste area and power through over-design (background knowledge). Fixed corners remain useful for digital timing, where the performance measures move monotonically with speed. Reference: J. A. Power, B. Donnellan, A. Mathewson and W. A. Lane, ‘Relating Statistical MOSFET Model Parameter Variabilities to IC Manufacturing Process Fluctuations Enabling Realistic Worst Case Design’, IEEE Transactions on Semiconductor Manufacturing 7(3), 306-318 (1994) (some equations garbled in the extract) (fetch-tool summary).

Ques: How does principal component analysis simplify statistical models?

By replacing many correlated parameters with a few independent factors. Power et al. measured over 800 parameter sets, keeping about 700 after removing outliers beyond 4$\sigma$, with 75 parameters from three geometries. PCA of the correlation matrix showed that 7 components explain almost 81 per cent of the variance, and after VARIMAX rotation the components could be identified with process variables: gate oxide thickness, channel-length reduction, threshold-adjust implant, P-well implant, channel-width reduction, n-channel flat-band voltage and junction depth. Each normalized parameter is then $P_{\alpha }=\sum a_{\alpha j}{X}_j$ with independent $X_j$, so sampling 7 numbers generates fully correlated model cards. The 19 per cent of variance left out is the price, which they correct for each parameter’s retained variance. Reference: J. A. Power, B. Donnellan, A. Mathewson and W. A. Lane, ‘Relating Statistical MOSFET Model Parameter Variabilities to IC Manufacturing Process Fluctuations Enabling Realistic Worst Case Design’, IEEE Transactions on Semiconductor Manufacturing 7(3), 306-318 (1994) (some equations garbled in the extract) (fetch-tool summary).

Ques: How many Monte Carlo runs are needed?

More than intuition suggests for tight estimates. The standard error of an estimated standard deviation is about $\sigma /\sqrt{2N}$ (background knowledge), so 100 runs give about 7 per cent and 5 per cent needs 200 runs (computed). Power et al. predicted a drain-current standard deviation of 221 $\mu \mathrm{A}$ from 100 correlated Monte Carlo runs against 240 $\mu \mathrm{A}$ measured, an 8 per cent gap consistent with that sampling error (computed). Ignoring correlations gave 190 $\mu \mathrm{A}$, 21 per cent low (computed), a model error that more runs would not fix. Estimating tails, such as a 6$\sigma$ SRAM failure rate, needs far more samples or importance sampling (background knowledge). Their gradient analysis needed only 8 simulations for 219 $\mu \mathrm{A}$, an efficient alternative when the response is close to linear. Reference: J. A. Power, B. Donnellan, A. Mathewson and W. A. Lane, ‘Relating Statistical MOSFET Model Parameter Variabilities to IC Manufacturing Process Fluctuations Enabling Realistic Worst Case Design’, IEEE Transactions on Semiconductor Manufacturing 7(3), 306-318 (1994) (some equations garbled in the extract) (fetch-tool summary).

Ques: How are global and local variation combined in a model card?

As separate random terms added to the nominal parameters. Saha writes each parameter as $P_i=P_{i0}+\Delta P_{i,mismatch}+\Delta P_{i,global}$, with the nominal value from a golden die, mismatch drawn from Pelgrom’s law, and global spread from electrical-test data or, for new technologies, TCAD. The local set is VT0, TOX, U0, K1, XW and XL, and the global set adds RDSW and the overlap and junction capacitances. For an assumed $A_{VT}$ of 2 $\mathrm{mV}\mu \mathrm{m}$, a 0.1 by 0.05 $\mu \mathrm{m}$ device has 28.3 $\mathrm{mV}$ of local threshold spread, which with an assumed 20 $\mathrm{mV}$ of global spread totals 34.7 $\mathrm{mV}$ (computed). Saha also describes running Monte Carlo mismatch on top of fixed SS and FF corners, which bounds global spread while still sampling local mismatch. Reference: S. K. Saha, ‘Process Variability Modeling for VLSI Circuit Simulation’, Technical Proceedings of NSTI-Nanotech 2011, Vol. 2, 751-755 (2011) (equations partly garbled in the extract) (fetch-tool summary); Mismatch and Variability page of this vault.

Ques: What does the sensitivity (gradient) method offer over Monte Carlo?

Speed, at the cost of linearity. With independent factors, the variance of a performance is approximately $\sum (\partial e/\partial X_j{)}^2{\sigma }^2(X_j)$, which needs only one simulation per factor plus the nominal, $m+1$ in all (background knowledge for the general form). Power et al. used 8 simulations for 7 components and obtained a 219 $\mu \mathrm{A}$ standard deviation, against 221 $\mu \mathrm{A}$ from 100 correlated Monte Carlo runs and 240 $\mu \mathrm{A}$ measured. The method assumes uncorrelated factors and a nearly linear response, so it fails for strongly non-linear performances, such as comparator offset near a threshold or yield near a specification edge, where Monte Carlo is required. The sensitivities also show which process factor dominates each performance, guiding design and process control (background knowledge). Reference: J. A. Power, B. Donnellan, A. Mathewson and W. A. Lane, ‘Relating Statistical MOSFET Model Parameter Variabilities to IC Manufacturing Process Fluctuations Enabling Realistic Worst Case Design’, IEEE Transactions on Semiconductor Manufacturing 7(3), 306-318 (1994) (some equations garbled in the extract) (fetch-tool summary).

Ques: Are statistical models in a design kit always reliable?

Not necessarily; their qualification must be checked. The SkyWater 130 nm documentation states that a Monte Carlo section covering process and mismatch variation exists, but that process Monte Carlo is not supported and its models are not qualified, so designers should rely on the corners there; for Monte Carlo, all corner sections must be excluded and the mc section included. Watts et al. describe Monte Carlo distributions and both fixed and user-definable corners for PSP, built by varying model parameters to represent random and systematic sources. Mixing corner and Monte Carlo sections double-counts global variation (background knowledge). A kit’s statistical models are only as good as the electrical-test data behind them, which is thin for new processes, as Saha notes. Reference: SkyWater 130 nm PDK documentation, ‘Spectre simulation’ page (docs/sim/spectre.rst, s8phirs_10r models; the page does not name SkyWater) (fetch-tool summary); J. S. Watts, Y. M. Lee and J.-E. Park, ‘Modeling MOSFET Process Variation using PSP’, Technical Proceedings of the 2007 NSTI Nanotechnology Conference, Vol. 3, 554-557 (2007) (abstract only) (fetch-tool summary); S. K. Saha, ‘Process Variability Modeling for VLSI Circuit Simulation’, Technical Proceedings of NSTI-Nanotech 2011, Vol. 2, 751-755 (2011) (equations partly garbled in the extract) (fetch-tool summary).

References

1. SkyWater 130 nm PDK documentation, “Spectre simulation” (docs/sim/spectre.rst, master-old branch). Available: https://foss-eda-tools.googlesource.com/skywater-pdk-docs/+/refs/heads/master-old/docs/sim/spectre.rst
2. J. A. Power, B. Donnellan, A. Mathewson and W. A. Lane, “Relating Statistical MOSFET Model Parameter Variabilities to IC Manufacturing Process Fluctuations Enabling Realistic Worst Case Design,” IEEE Transactions on Semiconductor Manufacturing, vol. 7, no. 3, pp. 306-318, Aug. 1994. Available: https://mural.maynoothuniversity.ie/id/eprint/10471/1/00311334.pdf
3. S. K. Saha, “Process Variability Modeling for VLSI Circuit Simulation,” in Technical Proceedings of NSTI-Nanotech 2011, vol. 2, pp. 751-755, 2011. Available: https://briefs.techconnect.org/wp-content/volumes/Nanotech2011v2/pdf/563.pdf
4. J. S. Watts, Y. M. Lee and J.-E. Park, “Modeling MOSFET Process Variation using PSP,” in Technical Proceedings of the 2007 NSTI Nanotechnology Conference, vol. 3, pp. 554-557, 2007. Available: https://briefs.techconnect.org/papers/modeling-mosfet-process-variation-using-psp/

Related pages in this vault: Mismatch and Variability, Binning and Geometry Scaling, Temperature Scaling, Static Random Access Memory (SRAM), Surface-Potential Models (PSP and HiSIM), Parameter Extraction Methods, Model Quality and Benchmark Tests, GUM Uncertainty Budgets, Gauge Repeatability and Reproducibility (Gauge R&R), Statistical Process Control (SPC).