Calibration and Traceability

1. Theory

• Traceability: A measurement result is traceable when it can be related to a reference through a documented, unbroken chain of calibrations, each contributing to the uncertainty (NIST glossary, VIM).
• Uncertainty growth: Every link adds terms to the budget, so uncertainty grows down the chain from national standards to production gauges (Morehouse).
• Reference materials: Certified artifacts such as NIST SRM 2059 carry the chain to the fab floor (SRM 2059 certificate).
• Decision rules: Test uncertainty ratio, false-accept probability and guard bands decide whether a calibrated instrument is declared in tolerance (Keysight; Mensor).

2. Working Principle

• Calibrate: Measure a certified standard and fit a calibration function or scale factor (NIST e-Handbook; SRM 2059 certificate).
• Propagate: Combine the certificate uncertainty with repeatability, resolution and environment terms (Morehouse; GUM page).
• Decide: Compare the reading with an acceptance limit set by the tolerance and the guard band (Keysight).
• Repeat: Recalibrate at intervals, and use check standards at the end-points and middle of the calibration interval to detect drift (NIST e-Handbook, 2.3.7).

3. Applications

• Dimensional metrology: Pitch and linewidth standards for optical, CD-SEM and AFM scales (SRM 2059 certificate).
• Electrical and pressure gauges: Pressure calibrator verification (Mensor).
• Quality systems: ISO/IEC 17025 accreditation and Z540.3 false-accept control (Keysight; Transcat).

Traceability Chain

Name	Formula	Related Formulae	Variables/Constants
Traceability chain	$u_{c,n}^2=u_{c,n-1}^2+{\sum }_i{u}_{i,n}^2$	$U_n=k{u}_{c,n},u_{c,n}>u_{c,n-1}$	$\begin{array}{r}{u}_{c,n}:combinedstandarduncertaintyatlevel\\ nofthechain\end{array}$ $\begin{array}{r}{u}_{i,n}:contributionsaddedatthatlevel\\ (stability,repeatability,resolution,\\ temperature)\end{array}$ $\begin{array}{r}k:coveragefactor;NISTglossary:\\ traceabilityrelatesaresulttoa\\ referencethroughadocumentedunbroken\\ chainofcalibrations,eachcontributing\\ totheuncertainty;Morehouse:eachlink\\ addsdoubtandnoneremovesit\end{array}$ $\begin{array}{r}(myreading:anuncertaintysmallerthan\\ thereferenc{e}^{\prime }sisasignthebudgetis\\ incomplete)\end{array}$
Inherited reference uncertainty	$u_{ref}=\frac{U_{ref}}{k_{ref}}$	$k_{ref}=2\Rightarrow \approx 95\text{per cent}$	$\begin{array}{r}{U}_{ref},k_{ref}:expandeduncertaintyand\\ coveragefactoronthecalibration\\ certificate;Morehouseexample:NISTU=\\ 0.001percentwithk=2gives0.0005per\\ cent,plusstability0.0004,repeatability\\ 0.0003,resolution0.0002andtemperature\\ 0.0002percent,combined0.000762per\\ centandU=0.001523percent\end{array}$ $\begin{array}{r}(computed:squarerootof0.58times1\times 10^{-6}\\ gives0.000762percent,1.5timesthe\\ referenceU)\end{array}$
Certified photomask linewidth	$w=w_{cert}\pm U,U<26\mathrm{nm}\left(k=2\right)$	$U_{C1}<1.455\ln x+4\mathrm{nm}$	$w_{cert}:certifiedwidth$ $\begin{array}{r}x:linepositionin\mu \mathrm{m};SRM2059\\ certificate:certifiedwiththeNISTUV\\ transmissionmicroscopeand\\ interferometricstage,sub-2\mu \mathrm{m}features\\ combinedwithAFM,traceabletothemetre\\ throughtheNISTLinescaleInterferometer\\ viaSRM2800,validindefinitelyif\\ undamaged,notrecommendedforSEMuse;\\ exampleA1nominal0.250\mu \mathrm{m},measured0.172\\ \mu \mathrm{m},U=13.6\mathrm{nm};pitchC91.000\mu \mathrm{m},U=2.8\\ \mathrm{nm}\end{array}$ $\begin{array}{r}(computed:theboundgives7.4,10.7and\\ 12.0\mathrm{nm}at10,100and250\mu \mathrm{m},abovethe\\ listed4.3,7.5and11.0\mathrm{nm})\end{array}$

Decision Rules and Risk

Name	Formula	Related Formulae	Variables/Constants
Test uncertainty ratio	$TUR=\frac{T_U-T_L}{2U_{95}}$	$TUR\ge 4\iff U_{95}\le 0.25T$	$T_U,T_L:upperandlowertolerancelimits$ $T:half-widthofasymmetrictolerance$ $\begin{array}{r}{U}_{95}:expandeduncertaintyofthewhole\\ calibrationprocess;Transcat:Z540.3\\ 5.3(b)2percentfalse-acceptlimitwith\\ 4:1TURwheretheprobabilitycannotbe\\ estimated;TURform(myreading);Mensor:\\ TARcomparesaccuracyspecifications,TUR\\ thefullprocessuncertainty,aTARof\\ 10:1wasoncethenorm,4:1isnow\\ acceptedandZ540-1requiresthe\\ standard{s}^{\prime }collectiveuncertaintynotto\\ exceed25percentofthetolerance\end{array}$ $\begin{array}{r}(myreading:TURisthestrictertest\\ becauseitincludesmorethanthe\\ calibrator)\end{array}$
False-accept probability	$P_{FA}={\iint }_{\left(|e|>T,|y|\le A\right)}{p}_0\left(e\right)p_m\left(y-e\right)dedy$	$P_{FA}\le 2\text{per cent}$	$e:trueerroroftheunitundertest$ $y:measurederror$ $p_0:priordistributionofuniterrors$ $p_m:measurementdistribution$ $\begin{array}{r}A:acceptancelimit;Keysight:Gaussian\\ jointdensity,Z540.3limitof2percent;\\ worst-casefalseaccept2.281percentat\\ TUR4:1(in-toleranceprobability64.65\\ percent),2.968percentat3:1and5.420\\ percentat1.5:1withoutaguardband\end{array}$ $\begin{array}{r}(myreading:4:1aloneslightlyexceeds\\ the2percentlimitintheworstcase)\end{array}$
Managed guard band	$A=T-M{U}_{95}$	$M_2=1.04-e^{0.38\ln TUR-0.54}$	$A:acceptancelimit$ $\begin{array}{r}M:fractionof{U}_{95}removed;Keysight\\ empiricalfit:M=5.32,15.36and35.89\\ percentatTUR4,3and1.5,falseaccept\\ generallybetween1and2percent;afull\\ U_{95}guardbandcutsfalseacceptbelow0.15\\ percentbutraisesfalserejects;Mensor\\ example:0.1\mathrm{kPa}specification,0.04\mathrm{kPa}\\ uncertainty,readingsmuststaybelow0.06\\ \mathrm{kPa}\end{array}$ $\begin{array}{r}(computed:thefitgives5.31,15.53and\\ 36.02percentatTUR4,3and1.5,and\\ becomesnegativeaboveTUR4.6,meaningno\\ guardband)\end{array}$

Calibration Functions

Name	Formula	Related Formulae	Variables/Constants
Inverse calibration curve	$X^{\prime }=\frac{Y^{\prime }-\hat{a}}{\hat{b}}$	$u^2=\frac{s_{Y^{\prime }}^2+s_{\hat{a}}^2}{{\hat{b}}^2}+{\left(\frac{Y^{\prime }-\hat{a}}{{\hat{b}}^2}\right)}^2{s}_{\hat{b}}^2+\frac{2\left(Y^{\prime }-\hat{a}\right)}{{\hat{b}}^3}{s}_{\hat{a}\hat{b}}$	$Y^{\prime }:instrumentreading$ $\hat{a},\hat{b}:fittedinterceptandslopeofY=a+bX$ $\begin{array}{r}s:standarddeviationsandcovarianceof\\ thefit;NISTe-Handbook:propagation\\ formulaandexamplefita=0.2357623,b=\\ 0.9870377,s_a^2=5.905067\times 10^{-4},s_b^2=\\ 1.183759\times 10^{-5},s_{ab}=-7.649453\times 10^{-5}\end{array}$ $\begin{array}{r}(assumed:Y^{\prime }=5.0,s_{Y^{\prime }}=0.01;computed:\\ X^{\prime }=4.8268,fitpart0.0115,total\\ 0.0153)\end{array}$
Scale-factor calibration	$s=\frac{P_{cert}}{P_{meas}},w=s{w}_{meas}$	$\frac{u_w}{w}=\sqrt{{\left(\frac{u_P}{P}\right)}^2+{\left(\frac{u_{meas}}{w_{meas}}\right)}^2}$	$\begin{array}{r}{P}_{cert},P_{meas}:certifiedandmeasuredpitchof\\ astandard\end{array}$ $\begin{array}{r}w:calibratedwidth;SRM2059pitchC9:\\ 1.000\mu \mathrm{m}withU=2.8\mathrm{nm};uncorrelated\\ relative-uncertaintypropagation(GUM\\ page)\end{array}$ $\begin{array}{r}(computed:pitchrelativestandard\\ uncertainty0.14percent,whichis0.07\\ \mathrm{nm}ona50\mathrm{nm}line)\end{array}$
Thermal length correction	$\Delta L=L\alpha \Delta T$	$L_{20}=L\left[1-\alpha \left(T-20\right)\right]$	$\alpha :expansioncoefficientofthestandard$ $T:temperaturein{}^{\circ }C$ $\begin{array}{r}{L}_{20}:lengthatthe20{}^{\circ }Creference\\ temperature;CorningHPFSdatasheet:\\ fusedsilica0.52ppmperkelvinfrom5to\\ 35{}^{\circ }C;20{}^{\circ }Creferencetemperature(my\\ reading)\end{array}$ $\begin{array}{r}(assumed:1Koffsetandafused-silica\\ substratewith0.5ppmperkelvin;\\ computed:0.125\mathrm{nm}over250\mu \mathrm{m}ofa\\ fused-silicamask,smallagainstUof11\\ \mathrm{nm})\end{array}$

Important Questions

Ques: What is metrological traceability and why must uncertainty grow down the chain?

Traceability is a property of a result that can be related to a reference through a documented, unbroken chain of calibrations, each contributing to the measurement uncertainty (NIST glossary). Because every level adds terms in quadrature, a lower level can never have smaller uncertainty than its reference (Morehouse). Morehouse’s example starts from a NIST expanded uncertainty of 0.001 per cent (k = 2), adds stability, repeatability, resolution and temperature terms, and reaches U = 0.0015 per cent, 1.5 times the reference (computed). Secondary force laboratories land at 0.01 to 0.05 per cent and production floors at 0.5 per cent or more. The same article quotes primary deadweight machines at 0.0016 to 0.0025 per cent, so its example reference of 0.001 per cent is better than the primary tier it describes; the higher range is more likely realistic (my reading). Reference: NIST Computer Security Resource Center glossary, ‘metrological traceability’, quoting NIST IR 8323r1 and the VIM, 3rd ed. (fetch-tool summary); Morehouse Instrument Company, ‘Force Measurement Traceability: Why Every Link in the Chain must Carry more Doubt than the One Above it’ (fetch-tool summary).

Ques: Is a 4:1 test uncertainty ratio enough?

Not always. Mensor notes that Z540-1 requires the collective uncertainty of the standards not to exceed 25 per cent of the tolerance, the 4:1 rule that replaced the older 10:1 norm. Z540.3, however, limits false-accept probability to 2 per cent, and Keysight’s analysis gives a worst-case false accept of 2.281 per cent at TUR 4:1, at an in-tolerance probability of 64.65 per cent. The two criteria therefore disagree slightly: 4:1 alone can exceed 2 per cent. The probability-based criterion governs, because Z540.3 makes the 2 per cent limit primary and allows 4:1 only where the probability cannot be estimated (Transcat); a small guard band then restores compliance (Keysight; my reading). In practice the worst case is rare, since most fleets have in-tolerance probabilities well above 65 per cent (my reading). Reference: Mensor, ‘How Accurate Should My Calibrator Be?’, Mensor blog (fetch-tool summary); Keysight Technologies, ‘A Guard-Band Strategy for Managing False-Accept Risk’, white paper 5991-1267EN (2014) (fetch-tool summary); Transcat, ‘ISO-17025:2017 vs. ANSI Z540.3’ FAQ: Z540.3 section 5.3(b), false-accept probability shall not exceed 2 per cent, and where this cannot be estimated the test uncertainty ratio shall be at least 4:1 (fetch-tool summary).

Ques: How does a managed guard band set the acceptance limit?

It moves the acceptance limit inside the tolerance by a fraction of the expanded uncertainty, $A=T-M{U}_{95}$ (Keysight). Keysight’s empirical multiplier, $M=1.04-e^{0.38\ln TUR-0.54}$, gives 5.31, 15.53 and 36.02 per cent at TUR 4, 3 and 1.5 (computed), close to its tabulated 5.32, 15.36 and 35.89 per cent, and holds false accept between 1 and 2 per cent. The fit turns negative above TUR 4.6 (computed), meaning no guard band is needed. A full $U_{95}$ guard band cuts false accept below 0.15 per cent but rejects many good units. Mensor’s example uses a 0.1 $\mathrm{kPa}$ specification and 0.04 $\mathrm{kPa}$ uncertainty, so readings must stay below 0.06 $\mathrm{kPa}$, which it calls a 60 per cent guard band; it is really 40 per cent of the tolerance removed (computed). Reference: Keysight Technologies, ‘A Guard-Band Strategy for Managing False-Accept Risk’, white paper 5991-1267EN (2014) (fetch-tool summary); Mensor, ‘How Accurate Should My Calibrator Be?’, Mensor blog (fetch-tool summary).

Ques: How does SRM 2059 carry length traceability to a photomask CD tool?

SRM 2059 is a photomask linewidth standard with isolated lines, spaces and pitch patterns certified with the NIST ultraviolet transmission microscope, whose scale comes from SRM 2800 and the NIST Linescale Interferometer, so results are traceable to the metre (SRM 2059 certificate). Features below about 2 $\mu \mathrm{m}$ combine optical and AFM values. Expanded uncertainties are below 26 $\mathrm{nm}$ with k = 2, for example 13.6 $\mathrm{nm}$ on line A1 and 2.8 $\mathrm{nm}$ on pitch C9. Certified widths differ strongly from nominal, A1 being 0.172 $\mu \mathrm{m}$ against 0.250 $\mu \mathrm{m}$, so nominal values cannot be used as a calibration (my reading). The certificate does not recommend SEM use. A 1 K temperature offset changes a 250 $\mu \mathrm{m}$ pitch by only 0.125 $\mathrm{nm}$ if the substrate is fused silica, 0.52 ppm per kelvin in Corning’s data (assumed substrate; computed). Reference: NIST, Certificate of Analysis, Standard Reference Material 2059, Photomask Linewidth Standard (archived certificate) (fetch-tool summary); Corning, ‘HPFS Wafer’ data sheet: fused silica expansion coefficient 0.52 ppm/K from 5 to 35 degC (fetch-tool summary).

Ques: What is the uncertainty of a value read from a linear calibration curve?

It combines the reading noise with the uncertainty of the fitted line. Inverting $Y=a+bX$ gives $X^{\prime }=(Y^{\prime }-\hat{a})/\hat{b}$, and the NIST e-Handbook propagates variances and the intercept-slope covariance through the partial derivatives. With its example fit (a = 0.2358, b = 0.9870) and an assumed reading of 5.0, $X^{\prime }$ is 4.8268 and the fit alone contributes a standard uncertainty of 0.0115 (computed). The negative covariance of -$7.65\times 10^{-5}$ removes most of the intercept and slope terms, and ignoring it would overstate the uncertainty (computed). Adding an assumed reading standard deviation of 0.01 gives 0.0153 total (computed). The fit term grows away from the centre of the calibration data, so standards should bracket the working range, and the NIST e-Handbook controls the end-points and middle with check standards (my reading; NIST e-Handbook, 2.3.7). Reference: NIST/SEMATECH e-Handbook of Statistical Methods, section 2.3.6.7.3, ‘Comparison of check standard analysis and propagation of error’ (fetch-tool summary); GUM Uncertainty Budgets page of this vault; NIST/SEMATECH e-Handbook of Statistical Methods, section 2.3.7, ‘Instrument control for linear calibration’: check standards controlling the end-points and the middle of the calibration interval to detect drift (fetch-tool summary).

References

1. NIST Computer Security Resource Center, “metrological traceability,” glossary entry citing NIST IR 8323r1 and JCGM 200 (VIM, 3rd ed.). Available: https://csrc.nist.gov/glossary/term/traceability_metrological
2. Morehouse Instrument Company, “Force Measurement Traceability: Why Every Link in the Chain must Carry more Doubt than the One Above it.” Available: https://mhforce.com/force-measurement-traceability/
3. Mensor, “How Accurate Should My Calibrator Be?” Available: https://www.mensor.com/en-us/blog_how_accurate_should_my_calibrator_be_en_um.WIKA
4. Keysight Technologies, “A Guard-Band Strategy for Managing False-Accept Risk,” white paper 5991-1267EN, 2014. Available: https://xdevs.com/doc/HP_Agilent_Keysight/A%20Guard-Band%20Strategy%20for%20Managing%20False-Accept%20Risk%20-%20White%20Paper%205991-1267EN%20c20140530%20%5B10%5D.pdf
5. Transcat, “ISO-17025:2017 vs. ANSI Z540.3,” FAQ. Available: https://www.transcat.com/iso17025-vs-ansiz540
6. Corning, “HPFS Wafer” data sheet. Available: https://www.corning.com/media/worldwide/csm/documents/Semiconductor_Technologies_Solutions/Corning_HPFSWafer_DataSheet.pdf
7. NIST/SEMATECH, e-Handbook of Statistical Methods, section 2.3.7, “Instrument control for linear calibration.” Available: https://itl.nist.gov/div898/handbook/mpc/section3/mpc37.htm
8. National Institute of Standards and Technology, “Certificate of Analysis, Standard Reference Material 2059, Photomask Linewidth Standard.” Available: https://tsapps.nist.gov/srmext/certificates/archives/2059.pdf
9. NIST/SEMATECH, “2.3.6.7.3. Comparison of check standard analysis and propagation of error,” e-Handbook of Statistical Methods. Available: https://itl.nist.gov/div898/handbook/mpc/section3/mpc3673.htm
10. GUM Uncertainty Budgets page of this vault.

Related pages in this vault: Physical Constants, GUM Uncertainty Budgets, Gauge Repeatability and Reproducibility (Gauge R&R), CD-SEM, Atomic Force Microscope (AFM), Optical Microscope, Vacuum Gauges, Pyrometers and Wafer Temperature Sensors, Statistical Process Control (SPC), Tool Matching and Total Measurement Uncertainty (TMU).