Tool Matching and Total Measurement Uncertainty (TMU)

1. Theory

• Accuracy against a reference: A measurement system under test (MSUT) is compared with a reference measurement system (RMS), such as CD-AFM, on the same artifacts (IBM TMU patent).
• Errors in both variables: Because both systems have noise, a Mandel (Deming) regression is used instead of ordinary least squares (IBM TMU patent; SimplyAgree).
• TMU: Total measurement uncertainty is the scatter about the best-fit line once the reference’s own uncertainty is removed (IBM TMU patent).
• Matching: Tool matching precision and fleet measurement precision combine precision, offsets, slope differences and nonlinearity across tools (IBM fleet patent).

2. Working Principle

• Artifacts: Measure a set of features spanning the process window, for example a focus-exposure matrix, on the MSUT and the RMS (IBM TMU patent).
• Regression: Fit slope and offset with the error-variance ratio set from the two precisions (IBM TMU patent; SimplyAgree).
• TMU and nonlinearity: Compute net residual error, subtract the reference uncertainty and compare the residuals with precision (IBM TMU patent).
• Fleet: Compare each tool with a benchmark tool and pool the terms into FMP (IBM fleet patent).

3. Applications

• CD metrology: CD-SEM and scatterometry accuracy against reference measurements (IBM TMU patent).
• Fleet matching: Keeping many CD-SEM or overlay tools interchangeable for production sampling (IBM fleet patent).
• Model referencing: Scatterometry, whose accuracy is known only after verification to a reference (Thiel et al.).
• Gauge qualification: Extends Gauge R&R, which does not test accuracy, to a reference comparison (Gauge R&R page).

Reference Regression

Name	Formula	Related Formulae	Variables/Constants
Mandel regression slope	$\hat{\beta }=\frac{S_{yy}-\lambda S_{xx}+\sqrt{{\left(S_{yy}-\lambda S_{xx}\right)}^2+4\lambda S_{xy}^2}}{2S_{xy}}$	$\lambda \to \infty :\hat{\beta }=\frac{S_{xy}}{S_{xx}},\lambda \to 0:\hat{\beta }=\frac{S_{yy}}{S_{xy}}$	$x:measurementsystemundertest(MSUT)$ $y:referencemeasurementsystem(RMS)$ $\begin{array}{r}S:sumsofsquaresaboutthemeans;IBM\\ TMUpatent:slopeasthekeymetric,\\ limitingformsaswritten\end{array}$ $\begin{array}{r}(myreading:theslopesayshowtheMSUT\\ scalemustbestretchedtomatchthe\\ reference)\end{array}$
Error-variance ratio	$\lambda =\frac{{\sigma }_y^2}{{\sigma }_x^2}=\frac{U_{RMS}^2}{U_{MSUT}^2}$	${\lambda }_{SimplyAgree}=\frac{{\sigma }_{ϵ,x}^2}{{\sigma }_{\delta ,y}^2}$	$U_{RMS}:referenceprecision$ $\begin{array}{r}{U}_{MSUT}:correctedprecisionorTMUofthe\\ MSUT;IBM:x=MSUT,y=RMS,OLSvalid\\ onlyiftheMSUTprecisionismuchbetter\\ thantheRMSprecision;SimplyAgree:\lambda as\\ x-errorovery-errorvariance,default1,\\ assumedconstant,estimablefrom\\ replicates\end{array}$ $\begin{array}{r}(myreading:thetwoconventionsare\\ reciprocal,soaratiocopiedbetween\\ toolsmustbeinverted)\end{array}$
Corrected precision and offset	${\sigma }_{corr}=\hat{\beta }{\sigma }_x,\Delta =\bar{y}-\bar{x}$	$\frac{{\hat{\beta }}_{OLS}}{\beta }\approx \frac{{\sigma }_{true}^2}{{\sigma }_{true}^2+{\sigma }_x^2}$	${\sigma }_x:MSUTprecision(onesigma)$ $\begin{array}{r}\Delta :offset;IBM:slopecorrectsprecision\\ totheRMSscale;MetricGate:attenuation\\ ofanordinaryleast-squaresslopebythe\\ reliabilityratio\end{array}$ $\begin{array}{r}(assumed:5\mathrm{nm}spreadoftrueCDs,1.5\mathrm{nm}\\ MSUTprecision;computed:OLSslopebiased\\ lowbyafactor0.917)\end{array}$

Total Measurement Uncertainty

Name	Formula	Related Formulae	Variables/Constants
Net residual error	$D=\sqrt{\frac{\sum d_i^2}{N-2}}$	$D_M=\sqrt{\frac{{\lambda }^2+{\hat{\beta }}^2}{{\left(\lambda +{\hat{\beta }}^2\right)}^2}{D}^2}$	$d_i:residualsfromthefit$ $N:numberofartifacts$ $\begin{array}{r}{D}_M:Mandelnetresidualerror;IBMTMU\\ patentequations11and12\end{array}$ $\begin{array}{r}(myreading:D_{M}collectseverythingthe\\ straight-linefitcannotexplain)\end{array}$
Total measurement uncertainty	$TMU=\sqrt{D_M^2-U_{RMS}^2}$	$TM{U}_{OLS}=\sqrt{D^2-U_{RMS}^2}$	$\begin{array}{r}IBM:TMUisthenetresidualerrorwithout\\ thereferenceuncertainty,nofactorof3\\ intheseformulas,precisionquotedasone\\ sigma;resistFEMexampleagainstanAFM\\ reference:CD-SEMAcorrectedprecision\\ 1.5\mathrm{nm},TMU20.3\mathrm{nm},nonlinearity100;\\ CD-SEMB1.8\mathrm{nm},26.1\mathrm{nm},137\end{array}$ $\begin{array}{r}(computed:TMUis13.5and14.5timesthe\\ precisionfortoolsAandB)\end{array}$
Nonlinearity	$NL=\frac{D_M^2}{Var\left(input\right)}$	$Var\left(input\right)={\sigma }_y^2+{\hat{\beta }}^2{\sigma }_x^2$	$\begin{array}{r}NL:ratioofresidualvariancetothe\\ varianceexpectedfromprecisionalone;\\ IBMequations13and14\end{array}$ $\begin{array}{r}(myreading:avalueof100meansthe\\ residualsareabout10timeslargerthan\\ precisionexplains,soprofileormaterial\\ effectsdominate)\end{array}$

Tool and Fleet Matching

Name	Formula	Related Formulae	Variables/Constants
Tool matching precision	$TMP=3\sqrt{{\hat{\beta }}^2{\sigma }_{MSUT}^2+{\left(\Delta -{\Delta }_{BMS}\right)}^2+{\left(SIS-SI{S}_{BMS}\right)}^2+{\sigma }_{NL}^2}$	$TM{P}_2=3\sqrt{{\hat{\beta }}_2^2{\sigma }_{MSUT}^2+\frac{{\Delta }_2^2}{4}+\frac{SI{S}_2^2}{4}+\frac{{\sigma }_{NL,2}^2}{4}}$	$BMS:benchmarkmeasurementsystem$ $\begin{array}{r}SIS:slope-inducedshift,{\sigma }_{NL}^2=Mandel\\ residualvarianceminusBMSandMSUT\\ variances;IBMfleetpatent:TMPandthe\\ two-toolspecialcase,thresholduser\\ selected\end{array}$ $\begin{array}{r}(assumed:twotools,slope1,precision\\ 0.3\mathrm{nm},offset0.4\mathrm{nm},noslope\\ difference,nonlinearity0.2\mathrm{nm};computed:\\ TMP=1.12\mathrm{nm})\end{array}$
Slope-induced shift	$SIS=\upsilon W\left(1-{\hat{\beta }}_{MSUT}\right)$	$\upsilon \le 1$	$W:processwindowsizeordatarange$ $\begin{array}{r}\upsilon :user-selectedfraction;IBMfleet\\ patent:definitionaswritten\end{array}$ $\begin{array}{r}(assumed:\upsilon =0.5,10\mathrm{nm}window,slope\\ 0.95;computed:0.25\mathrm{nm}shiftatthe\\ windowedge)\end{array}$
Fleet measurement precision	$FMP=3\sqrt{V_{pp}+V_{po}+V_{ps}+V_{pn}}$	$V_{pp}=\frac{1}{N}\sum {\hat{\beta }}_i^2{\sigma }_i^2,V_{po}=\frac{1}{N}\sum {\left({\Delta }_i-{\Delta }_{BMS}\right)}^2$	$N:toolsinthefleet$ $\begin{array}{r}{V}_{ps},V_{pn}:pooledslope-shiftand\\ nonlinearityvariances;IBMfleetpatent:\\ pooledterms,FMPthresholdsetfromfleet\\ experience,referencetothe1999ITRS\\ precisionspecificationof1.8\mathrm{nm}\end{array}$ $\begin{array}{r}(assumed:precisions0.30,0.35and0.40\\ \mathrm{nm},offsets0,0.5and-0.3\mathrm{nm},\\ nonlinearity0.2\mathrm{nm};computed:FMP=1.58\\ \mathrm{nm}against1.06\mathrm{nm}fromprecisionalone)\end{array}$

Important Questions

Ques: How can a tool with 1.5 nm precision have a TMU of 20 nm?

Precision measures only repeatability, while TMU measures agreement with a reference. In IBM’s focus-exposure study against an AFM reference, CD-SEM A had a corrected precision of 1.5 $\mathrm{nm}$ but a TMU of 20.3 $\mathrm{nm}$, and CD-SEM B 1.8 $\mathrm{nm}$ and 26.1 $\mathrm{nm}$ (IBM TMU patent), so TMU is 13.5 to 14.5 times the precision (computed). TMU is the Mandel net residual error with the reference uncertainty removed, $\sqrt{D_M^2-U_{RMS}^2}$. The nonlinearity values of 100 and 137 show the residuals are about ten times what precision explains (my reading). The likely cause is that resist profile changes across the focus-exposure matrix alter the SEM edge signal differently from the AFM tip, so a constant offset cannot correct it (my reading). A Gauge R&R study would rate both tools as good (Gauge R&R page). Reference: C. N. Archie, G. W. Banke Jr. and M. J. Sendelbach, ‘Assessment and optimization for metrology instrument including uncertainty of total measurement uncertainty’, US patent 7,286,247 B2, IBM (filed 2005, priority 2002) (fetch-tool summary); Gauge Repeatability and Reproducibility (Gauge R&R) page of this vault.

Ques: Is TMU a one-sigma or three-sigma quantity?

The sources differ. IBM’s TMU patent writes $TMU=\sqrt{D_M^2-U_{RMS}^2}$ with no factor of 3 and quotes precision as one sigma. The companion fleet patent multiplies its tool matching and fleet measurement precision by 3, and the 2010 ITRS table quoted CD tool uncertainty at 3 sigma (IBM fleet patent; Thiel et al.). A TMU reported next to an FMP may therefore be on a different basis, differing by a factor of 3. For the IBM TMU equations as written, the one-sigma reading is more likely correct, because TMU is built from residual standard deviations and one-sigma precisions (my reading). Any comparison should state the multiplier. On a one-sigma basis IBM’s 20.3 $\mathrm{nm}$ TMU would be 61 $\mathrm{nm}$ at three sigma (computed). Reference: C. N. Archie, G. W. Banke Jr. and M. J. Sendelbach, ‘Assessment and optimization for metrology instrument including uncertainty of total measurement uncertainty’, US patent 7,286,247 B2, IBM (filed 2005, priority 2002) (fetch-tool summary); C. N. Archie, G. W. Banke Jr. and E. P. Solecky, ‘Determining root cause of matching problem and/or fleet measurement precision problem for measurement system’, US patent 7,532,999 B2, IBM (filed 2005) (fetch-tool summary); B. Thiel, A. Cepler, A. Diebold and R. Matyi, ‘Advances in CD-Metrology (CD-SAXS, Mueller Matrix based Scatterometry, and SEM)’, CNSE Albany, about 2011, hosted by NIST: 2010 ITRS Table MET3 CD tool uncertainty at 3 sigma with P/T = 0.2, scatterometry accuracy known only after verification to reference (fetch-tool summary).

Ques: Why use Mandel regression instead of ordinary least squares?

Because both instruments have noise. Ordinary least squares assumes an error-free x, and noise in x biases the slope low by the reliability ratio ${\sigma }_{true}^2/({\sigma }_{true}^2+{\sigma }_x^2)$ (MetricGate). For an assumed 5 $\mathrm{nm}$ spread of true CDs and 1.5 $\mathrm{nm}$ MSUT precision, that factor is 0.917 (computed). IBM’s slope uses $\lambda ={\sigma }_y^2/{\sigma }_x^2$ with x as the MSUT and y as the reference, and tends to the OLS slope $S_{xy}/S_{xx}$ as $\lambda$ grows. The SimplyAgree Deming routine defines its ratio the other way, x-error over y-error variance, with a default of 1. The two conventions are reciprocal, so a ratio of 4 in one is 0.25 in the other, and copying a value between packages unchanged gives the wrong slope (my reading). Both assume the ratio is constant over the range. Reference: C. N. Archie, G. W. Banke Jr. and M. J. Sendelbach, ‘Assessment and optimization for metrology instrument including uncertainty of total measurement uncertainty’, US patent 7,286,247 B2, IBM (filed 2005, priority 2002) (fetch-tool summary); SimplyAgree R package vignette, ‘Errors-in-Variables Regression’ (Deming regression) (fetch-tool summary); MetricGate, ‘Measurement Error in Predictors Biases Your Slopes’ (2026): reliability ratio lambda = sigma_X^2/(sigma_X^2 + sigma_u^2), expected OLS slope about lambda times the true slope (fetch-tool summary).

Ques: How do tool matching precision and fleet measurement precision combine terms?

Both are 3 times a root sum of squares (IBM fleet patent). Tool matching precision adds slope-weighted precision, the offset from a benchmark tool, the slope-induced shift $\upsilon W(1-\hat{\beta })$ and nonlinearity. For two assumed tools with 0.3 $\mathrm{nm}$ precision, 0.4 $\mathrm{nm}$ offset and 0.2 $\mathrm{nm}$ nonlinearity, the two-tool form gives 1.12 $\mathrm{nm}$ (computed). Fleet measurement precision pools the same terms over N tools; three assumed tools with 0.30 to 0.40 $\mathrm{nm}$ precision and offsets up to 0.5 $\mathrm{nm}$ give 1.58 $\mathrm{nm}$, against 1.06 $\mathrm{nm}$ from precision alone (computed). Offsets therefore add about half again to the fleet figure, which is why offset correction is usually the first matching step (my reading). The patent cites the 1999 ITRS precision specification of 1.8 $\mathrm{nm}$ as a benchmark. Reference: C. N. Archie, G. W. Banke Jr. and E. P. Solecky, ‘Determining root cause of matching problem and/or fleet measurement precision problem for measurement system’, US patent 7,532,999 B2, IBM (filed 2005) (fetch-tool summary).

Ques: Why does TMU become uncertain when it approaches the reference uncertainty?

Because it is a difference of squares. With $TMU=\sqrt{D_M^2-U_{RMS}^2}$, an assumed net residual of 2.0 $\mathrm{nm}$ and reference uncertainty of 1.8 $\mathrm{nm}$ give a TMU of 0.87 $\mathrm{nm}$, but a 10 per cent error in the reference uncertainty moves it from 1.17 to 0.28 $\mathrm{nm}$ (computed). IBM’s patent addresses this with techniques for an accurate TMU estimate when TMU is close to or less than the reference uncertainty. The practical consequence is that a reference must be considerably better than the tool it qualifies, or the TMU tells little (my reading). For CD-SEM the AFM reference itself carries tip-shape uncertainty, so its own TMU should be established first, for instance against cross-section data (my reading). Reference: C. N. Archie, G. W. Banke Jr. and M. J. Sendelbach, ‘Assessment and optimization for metrology instrument including uncertainty of total measurement uncertainty’, US patent 7,286,247 B2, IBM (filed 2005, priority 2002) (fetch-tool summary).

References

1. C. N. Archie, G. W. Banke Jr. and M. J. Sendelbach, “Assessment and optimization for metrology instrument including uncertainty of total measurement uncertainty,” US Patent 7,286,247 B2, International Business Machines Corp., 2007. Available: https://patents.google.com/patent/US7286247
2. Gauge Repeatability and Reproducibility (Gauge R&R) page of this vault.
3. C. N. Archie, G. W. Banke Jr. and E. P. Solecky, “Determining root cause of matching problem and/or fleet measurement precision problem for measurement system,” US Patent 7,532,999 B2, International Business Machines Corp., 2009. Available: https://patents.google.com/patent/US7532999
4. MetricGate, “Measurement Error in Predictors Biases Your Slopes,” June 2026. Available: https://metricgate.com/blogs/measurement-error-in-regressors-attenuation/
5. B. Thiel, A. Cepler, A. Diebold and R. Matyi, “Advances in CD-Metrology (CD-SAXS, Mueller Matrix based Scatterometry, and SEM),” College of Nanoscale Science and Engineering, University at Albany, about 2011. Available: https://www.nist.gov/document/thiel2011pdf
6. SimplyAgree R package, “Errors-in-Variables Regression,” vignette. Available: https://cran.nics.utk.edu/cran/web/packages/SimplyAgree/vignettes/Deming.html

Related pages in this vault: Gauge Repeatability and Reproducibility (Gauge R&R), GUM Uncertainty Budgets, Calibration and Traceability, Statistical Process Control (SPC), CD-SEM, Atomic Force Microscope (AFM), Optical CD Scatterometer (OCD), Overlay Metrology Tool.