Defect Detection Fundamentals

1. Theory

• Definition: Defect inspection compares an image of the wafer with a reference and flags locations where the difference exceeds a threshold set above the noise (Takeuchi patent; Barnes et al.).
• Reference choice: Neighbouring dies or neighbouring cells within a die serve as references; cell comparison is more sensitive because the patterns compared are closer (Takeuchi patent).
• Noise and threshold: Process variation and detector noise set the noise in the difference image, and Barnes et al. count pixels above $4\sigma$ (Barnes et al.).
• Detection trade-off: Lowering the threshold raises capture of defects of interest but also nuisance events; tools are tuned on a capture rate against nuisance rate curve (diagnostic patent).
• Physics of signal: Wavelength, angle, polarization and focus change the defect signal, as described on the Light-Matter Interaction page (Barnes et al.; Light-Matter Interaction page).

2. Working Principle

• Scan and align: Swaths of dies are imaged and aligned to sub-pixel accuracy (Takeuchi patent).
• Compare: Gray-level difference images are formed between the test die and its neighbours (Takeuchi patent).
• Arbitrate: Double detection assigns the defect to the die that differs from both neighbours (Takeuchi patent).
• Classify and sample: Events are binned into defects of interest and nuisance, and a sample is sent for SEM review, preferably near the classifier boundary (diagnostic patent; E-Beam Inspection and Defect Review page).

3. Applications

• Patterned wafer inspection: Random particles and pattern defects in production (Zhu et al.; Takeuchi patent).
• Unpatterned wafer inspection: Particle and COP classification (Hattori).
• Recipe optimization: Selecting angle, polarization and focus per defect type (Barnes et al.).

Comparison and Thresholding

Name	Formula	Related Formulae	Variables/Constants
Die-to-die difference image	$\Delta I\left(x,y\right)=I_{test}\left(x,y\right)-I_{ref}\left(x,y\right)$	$\left|\Delta I\right|>T\Rightarrow \text{defect candidate}$	$\begin{array}{r}{I}_{test},I_{ref}:alignedgraylevelsofthetest\\ andneighbouringdie\end{array}$ $\begin{array}{r}T:threshold;Takeuchipatent:gray-level\\ differenceimageandfixedthreshold,die\\ thresholdabovecellthreshold;Barneset\\ al.:normalizeddie-to-defectdifference\\ images\end{array}$ $\begin{array}{r}(myreading:cell-to-cellcomparisonuses\\ anearbyrepeatinsidethedie,soless\\ processvariationentersthedifference)\end{array}$
Double detection	$\left|I_B-I_A\right|>T\wedge \left|I_B-I_C\right|>T\Rightarrow B\text{defective}$	$P_{fa,double}\approx P_{fa}^2(\text{independent noise})$	$\begin{array}{r}A,B,C:threeadjacentdies;Takeuchi\\ patent:adieisdefectivewhenit\\ disagreeswithneighboursonbothsides;\\ independenceassumption(myreading,since\\ B^{\prime }sownnoiseiscommontoboth\\ differences)\end{array}$ $\begin{array}{r}(computed:P_{fa}=3.2\times 10^{-5}perpixelat4\sigma \\ becomesabout1\times 10^{-9}ifindependent)\end{array}$
Threshold and false-alarm rate	$P_{fa}=Q\left(\frac{T}{{\sigma }_n}\right)=\frac{1}{2}\mathrm{erfc}\left(\frac{T}{\sqrt{2}{\sigma }_n}\right)$	$N_{false}=N_{pix}{P}_{fa}$	${\sigma }_n:difference-imagenoise$ $\begin{array}{r}Q:Gaussiantail;Barnesetal.:4\sigma pixel\\ threshold,noise3\sigma =2percentofmean;\\ Gaussiantail(computed;myreadingofa\\ Gaussiannoisemodel)\end{array}$ $\begin{array}{r}(assumed:1\times 10^{10}pixels;computed:P_{fa}=\\ 1.35\times 10^{-3},3.2\times 10^{-5}and2.9\times 10^{-7}at3,4and\\ 5\sigma ,so4\sigma stillgivesabout3.2\times 10^{5}false\\ pixels,fallingtoabout10with\\ independentdoubledetection)\end{array}$

Detection Performance

Name	Formula	Related Formulae	Variables/Constants
Capture rate and nuisance rate	$CR=\frac{N_{DOI,detected}}{N_{DOI,total}}$	$NR=\frac{N_{nuisance}}{N_{reported}}$	$\begin{array}{r}DOI:defectofinterest;diagnostic\\ patent:capturerateascorrectly\\ classifiedDOIsoverallDOIs,nuisance\\ rateestimatedfromposteriorsintheDOI\\ bin,caprateagainstnuisanceratein\\ placeofanROC\end{array}$ $\begin{array}{r}(assumed:90of100DOIsfoundamong300\\ reportedevents;computed:CR=90per\\ centandNR=70percent)\end{array}$
Classification accuracy	$Ac{c}_k=\frac{N_{k,correct}}{N_k}$	$\text{particle and COP accuracy}53\text{to}93\%$	$\begin{array}{r}Ac{c}_k:per-classaccuracy;diagnostic\\ patent:per-classaccuracy;Hattori:tools\\ with100,80and80\mathrm{nm}sensitivity\\ classifyingparticlesandCOPsat84/85,\\ 84/84and53/93percent\end{array}$ $\begin{array}{r}(myreading:thethirdtooltrades\\ particleaccuracyforCOPaccuracy,asa\\ cutlineshiftwould)\end{array}$
Inspection pixel count	$N_{pix}=\frac{\pi {\left(D/2\right)}^2}{p^2}$	$t_{scan}=\frac{N_{pix}}{R_{pix}}$	$D:waferdiameter$ $p:pixelsizeatthewafer$ $\begin{array}{r}{R}_{pix}:pixelrate;geometry(computed);\\ E-BeamInspectionandDefectReviewpage\\ fore-beampixelrates\end{array}$ $\begin{array}{r}(computed:a300\mathrm{mm}waferis70686\mathrm{m}{\mathrm{m}}^2,\\ 1.77\times 10^{14}pixelsat20\mathrm{nm},2.8\times 10^{13}at50\mathrm{nm}\\ and7.1\times 10^{12}at100\mathrm{nm})\end{array}$

Signal Enhancement and Defect Types

Name	Formula	Related Formulae	Variables/Constants
Defect metric	$S={\sum }_{\Delta I_i>4\sigma }\Delta I_i$	$A_{defect}=N\left(\Delta I_i>4\sigma \right)$	$S:integrateddefectintensity$ $\begin{array}{r}{A}_{defect}:areainpixels;Barnesetal.:this\\ metricoverregionsaboveaminimumarea\end{array}$ $\begin{array}{r}(computed:Barne{s}^{\prime }Jdefectrisesfrom\\ 0.50to6.03a.u.,12times,andfrom8to\\ 42pixels,5.25times,withvolumetric\\ data)\end{array}$
Volumetric and angular gain	$\frac{S_{3D}}{S_{2D}}=7.5\text{to}14.9$	$S\left(\theta ,\varphi ,\text{pol}\right)\text{optimized per defect}$	$\begin{array}{r}Barnesetal.:through-focusimagingfrom\\ about-1to+1\mu \mathrm{m},bridgesbestseenat\\ particularpolarandazimuthalanglesand\\ polarization,rotatingwithbridge\\ direction,gainsatshorterwavelength;\\ Light-MatterInteractionpageforthe\\ scatteringphysics\end{array}$ $\begin{array}{r}(myreading:adefectgainof7.5atfixed\\ noiseallowsthethresholdfactortorise\\ from4toabout30sigma-equivalent\\ margin,orsmallerdefectstopass)\end{array}$
Random against systematic defects	$\text{random: particles}\text{systematic: pattern-dependent}$	$\text{die-to-die misses defects repeated in every die}$	$\begin{array}{r}Zhuetal.:randomparticlesand\\ systematicdefectshardertoidentifyat\\ andbeyond10\mathrm{nm};die-to-diecomparison\\ (Takeuchipatent)\end{array}$ $\begin{array}{r}(myreading:adefectprintedidentically\\ inalldiescancelsinadie-to-die\\ difference,whichiswhydie-to-database\\ methodsexist)\end{array}$

Important Questions

Ques: How does die-to-die inspection decide which die holds the defect?

By comparing each die with both neighbours. The Takeuchi patent forms a gray-level difference image between corresponding pixels of adjacent dies and flags pixels whose difference exceeds a threshold; because a single difference cannot tell which die is wrong, it uses double detection, calling a die defective only when it disagrees with the dies on both sides. The die-comparison threshold is set higher than the cell-comparison threshold, since cell comparison uses closer patterns and is more sensitive, while colour variation between dies causes false defects. A defect repeated identically in every die cancels in such a difference (my reading), and Zhu et al. stress that systematic defects, as well as random particles, become harder to find at and beyond 10 $\mathrm{nm}$. Reference: N. Takeuchi (Tokyo Seimitsu), ‘Pattern inspection method and apparatus’, US Patent 7,366,343 (2008), Justia: die-to-die comparison of corresponding patterns, gray-level difference image, defect candidate where the difference exceeds a threshold, die-comparison threshold set higher than cell-comparison, double detection judging a die defective when it disagrees with neighbours on both sides, false defects from colour variation, cell comparison more sensitive, frames hundreds of pixels wide (fetch-tool summary); J. Zhu, J. Liu, T. Xu, S. Yuan, Z. Zhang, H. Jiang, H. Gu, R. Zhou and S. Liu, ‘Optical wafer defect inspection at the 10 nm technology node and beyond’, Int. J. Extreme Manuf. 4, 032001 (2022), abstract only: identifying, locating and classifying random particles and systematic defects harder at and beyond 10 nm, review of detectability evaluation, inspection systems and post-processing, deep learning and computational imaging (fetch-tool summary).

Ques: How does the threshold set the false-alarm rate?

Through the tail of the noise distribution. Barnes et al. flag pixels above $4\sigma$ in difference images whose simulated noise has $3\sigma$ equal to 2 per cent of the mean reference intensity. For Gaussian noise the per-pixel false-alarm probability is $Q(T/\sigma )$: $1.35\times 10^{-3}$, $3.2\times 10^{-5}$ and $2.9\times 10^{-7}$ at 3, 4 and 5 $\sigma$ (computed). Even at $4\sigma$, an assumed $10^{10}$ pixels would give about $3.2\times 10^5$ false pixels (computed). Requiring disagreement with both neighbours, as in the Takeuchi patent, would cut that to about 10 if the two differences were independent (computed), though they share the test die’s own noise, so the real gain is smaller (my reading). Barnes et al. also require a minimum area. Reference: B. Barnes, M. Sohn, F. Goasmat, H. Zhou, A. Vladar, A. Arceo and R. Silver (NIST), ‘Scatterfield Microscopy, Review of Techniques that Push the Limit of Optical Defect Metrology’: die-to-defect difference images after normalization, simulated noise 3 sigma = 2 per cent of mean reference intensity, defect metric as summed intensity of pixels above 4 sigma, 193 and 450 nm, gains at shorter wavelengths, bridge detection depending on polar and azimuthal angle and polarization, through-focus volumetric data raising sensitivity 7.5 to 14.9 times, J defect area 8 to 42 pixels and intensity 0.50 to 6.03 (fetch-tool summary); N. Takeuchi (Tokyo Seimitsu), ‘Pattern inspection method and apparatus’, US Patent 7,366,343 (2008), Justia: die-to-die comparison of corresponding patterns, gray-level difference image, defect candidate where the difference exceeds a threshold, die-comparison threshold set higher than cell-comparison, double detection judging a die defective when it disagrees with neighbours on both sides, false defects from colour variation, cell comparison more sensitive, frames hundreds of pixels wide (fetch-tool summary).

Ques: What are capture rate and nuisance rate, and how are they traded?

They measure how many real defects are found and how many reported events are noise. The diagnostic patent defines capture rate as correctly classified defects of interest over all defects of interest, estimates nuisance rate from the posterior probabilities of nuisance among events in the DOI bin, and notes that a curve of capture rate against nuisance rate can replace an ROC, so a user reads the nuisance rate expected for a desired capture rate. Finding 90 of 100 DOIs among 300 reported events is 90 per cent capture with 70 per cent nuisance (computed). Hattori’s three commercial tools classified particles and COPs at 84/85, 84/84 and 53/93 per cent, showing the same trade-off between classes (my reading). Review samples are best taken near the classifier boundary (diagnostic patent). Reference: US Patent Application 2018/0197714, ‘Diagnostic Methods for the Classifiers and the Defects Captured by Optical Tools’ (ipiq.io; assignee not shown): capture rate as correctly classified DOIs over all DOIs, nuisance rate from posteriors of nuisance in the DOI bin, per-class accuracy, two classes DOI and nuisance, ROC or cap rate against nuisance rate to choose a cutline, SEM review sampling near the classifier boundary (fetch-tool summary); T. Hattori (Sony), ‘Challenges of Smaller Particle Detection on Both Bulk-Silicon and SOI Wafers’, NIST presentation (2005): particle and COP classification accuracies of 84/85, 84/84 and 53/93 per cent for three commercial tools of 100, 80 and 80 nm sensitivity, haze masking small particles (fetch-tool summary).

Ques: How can defect signal be raised above the noise?

By choosing wavelength, angle, polarization and focus. Barnes et al. found that narrow bridges are detected best at particular polar and azimuthal angles, that the best angles rotate with the bridge direction, that polarization matters most for 8 $\mathrm{nm}$ bridges, and that shorter wavelengths give further gains. Collecting images through focus from about -1 to +1 $\mu \mathrm{m}$ and analysing them as volumes raised sensitivity 7.5 to 14.9 times; for one defect the integrated signal rose from 0.50 to 6.03, 12 times, and its area from 8 to 42 pixels, 5.25 times (computed). The Light-Matter Interaction page explains why shorter wavelengths help, through the $d^6/{\lambda }^4$ scattering law, and why polarization changes the background. Reference: B. Barnes, M. Sohn, F. Goasmat, H. Zhou, A. Vladar, A. Arceo and R. Silver (NIST), ‘Scatterfield Microscopy, Review of Techniques that Push the Limit of Optical Defect Metrology’: die-to-defect difference images after normalization, simulated noise 3 sigma = 2 per cent of mean reference intensity, defect metric as summed intensity of pixels above 4 sigma, 193 and 450 nm, gains at shorter wavelengths, bridge detection depending on polar and azimuthal angle and polarization, through-focus volumetric data raising sensitivity 7.5 to 14.9 times, J defect area 8 to 42 pixels and intensity 0.50 to 6.03 (fetch-tool summary); Light-Matter Interaction for Optical Inspection page of this vault.

Ques: Why can optical inspection not simply use nanometre pixels everywhere?

Because the wafer is enormous in pixels. A 300 $\mathrm{mm}$ wafer is 70686 $\mathrm{m}{\mathrm{m}}^2$, which is $1.77\times 10^{14}$ pixels at 20 $\mathrm{nm}$, $2.8\times 10^{13}$ at 50 $\mathrm{nm}$ and $7.1\times 10^{12}$ at 100 $\mathrm{nm}$ (computed). Each pixel is a false-alarm opportunity, so smaller pixels need higher thresholds or better arbitration (computed from the false-alarm relation; my reading). Optical tools therefore scan whole wafers at pixels larger than the defects and rely on scattering contrast, while e-beam tools image smaller areas at higher resolution and review sampled locations (E-Beam Inspection and Defect Review page). Zhu et al. note that inspection at and beyond 10 $\mathrm{nm}$ increasingly combines optics with computational imaging and deep learning in post-processing. Reference: E-Beam Inspection and Defect Review page of this vault; J. Zhu, J. Liu, T. Xu, S. Yuan, Z. Zhang, H. Jiang, H. Gu, R. Zhou and S. Liu, ‘Optical wafer defect inspection at the 10 nm technology node and beyond’, Int. J. Extreme Manuf. 4, 032001 (2022), abstract only: identifying, locating and classifying random particles and systematic defects harder at and beyond 10 nm, review of detectability evaluation, inspection systems and post-processing, deep learning and computational imaging (fetch-tool summary).

References

1. N. Takeuchi, “Pattern inspection method and apparatus,” US Patent 7,366,343, Tokyo Seimitsu Co., 2008. Available: https://patents.justia.com/patent/7366343
2. B. Barnes, M. Sohn, F. Goasmat, H. Zhou, A. Vladar, A. Arceo and R. Silver, “Scatterfield microscopy, review of techniques that push the limit of optical defect metrology,” NIST presentation. Available: https://www.nist.gov/document/silverfinalpdf
3. “Diagnostic methods for the classifiers and the defects captured by optical tools,” US Patent Application 2018/0197714. Available: https://ipiq.io/patents/5407294
4. T. Hattori, “Challenges of smaller particle detection on both bulk-silicon and SOI wafers,” Sony Corporation, NIST presentation, 2005. Available: https://www.nist.gov/document/hattori2005pdf
5. J. Zhu, J. Liu, T. Xu, S. Yuan, Z. Zhang, H. Jiang, H. Gu, R. Zhou and S. Liu, “Optical wafer defect inspection at the 10 nm technology node and beyond,” International Journal of Extreme Manufacturing, vol. 4, 032001, 2022 (abstract read). Available: https://www.sciopen.com/article/10.1088/2631-7990/ac64d7
6. Light-Matter Interaction for Optical Inspection page of this vault.
7. E-Beam Inspection and Defect Review page of this vault.

Related pages in this vault: Patterned-Wafer Optical Inspection, Laser-Scattering Particle Inspection, E-Beam Inspection and Defect Review, Light-Matter Interaction for Optical Inspection, Statistical Process Control (SPC), Yield Models and Critical Area, Defect Sources and Classification, EUV Scanner, Nanoimprint Lithography Tool, Wafer-Map Spatial Signature Analysis, Defect Inspection Workflow and Sampling Plans, Automatic Defect Classification and Machine Learning, Defect-to-Yield Correlation and Kill Ratio.