CD-SEM

1. Theory

• Definition: A CD-SEM is a scanning electron microscope specialized for automated top-down measurement of linewidths, spaces, hole diameters and edge roughness on wafers, at low landing energies, 300 $\mathrm{eV}$ in Verduin et al.’s measurements (Verduin et al.).
• Edge detection: CDs come from edges found in line-scan profiles by threshold or model-based algorithms (Verduin et al.; Mack and Bunday).
• Roughness: Line-edge and linewidth roughness are three standard deviations of edge positions and widths along a line; image noise biases them upward (Villarrubia and Bunday; Fractilia).
• Beam effects: The electron dose shrinks resist and adds shot noise, so dose must be minimized and corrected for (Verduin et al.).

2. Working Principle

• Low-voltage column: Low landing energy, 300 $\mathrm{eV}$ in Verduin et al., with dose kept low because it damages resist (Verduin et al.; Mack and Bunday).
• Analysis: Line-scan profiles are averaged along the line and edges located; many lines are measured for roughness statistics (Verduin et al.).
• Noise correction: The random noise contribution is estimated and subtracted from roughness and PSD metrics (Fractilia; Villarrubia and Bunday).

3. Applications

• Roughness: LER, LWR and local CD uniformity for EUV and ArF processes (Fractilia).
• Overlay and edge placement: Contributions to edge-placement error (Fractilia).
• Resist development: Dose and shrinkage studies (Verduin et al.).

Imaging and Edge Detection

Name	Formula	Related Formulae	Variables/Constants
Dose and current	$D=\frac{It}{A}$	$1e^{-}/\mathrm{n}{\mathrm{m}}^2=16\mu C/c{\mathrm{m}}^2$	$D:electrondoseperarea$ $I:beamcurrent$ $\begin{array}{r}t:dwelltimeoverareaA;definition\\ (computed);Verduinetal.:imagingat300\\ \mathrm{eV}and10\mathrm{pA}correspondingto1e^{-}/\mathrm{n}{\mathrm{m}}^{2}or\\ 16\mu C/c{\mathrm{m}}^2,severeresistshrinkageata\\ normaldoseof10e^{-}/\mathrm{n}{\mathrm{m}}^2(160\mu C/c{\mathrm{m}}^2),\\ lowest-doseimagingonaHitachiCG4000\\ CD-SEM\end{array}$ $\begin{array}{r}(computed:q/1n{m}^2=1.602\times 10^{-19}C/1\times 10^{-14}c{m}^2\\ =16.0\mu C/c{\mathrm{m}}^2;10\mathrm{pA}delivers62400\\ electronspermillisecond)\end{array}$
Edge detection	$x_{edge}:I\left(x\right)=I_{min}+f\left(I_{max}-I_{min}\right)(\text{threshold, }f\approx 0.5)$	$x_{edge}=\arg {\max }_x\left|\frac{dI}{dx}\right|(\text{maximum derivative}),CD=x_{right}-x_{left}$	$\begin{array}{r}{x}_{edge}:edgepositionfromtheline-scan\\ profileI(x)\end{array}$ $f:thresholdfraction$ $\begin{array}{r}CD:criticaldimension;MackandBunday:\\ thresholdbetweenminimumandmaximumgrey\\ levelsandmodel-baseddetection;\\ maximum-derivativeform(myreading);\\ Verduinetal.:integratingthelineimage\\ alongtheline,fittingtheintegrated\\ profilewithamodelcontainingtheedge\\ position,findingedgesateachdepth,\\ biasdependingonresistthickness,LER\\ decreasingwithimagingenergy\end{array}$ $\begin{array}{r}(computed:a1\mathrm{nm}shiftofeachedgein\\ oppositedirectionschangestheCDby2\\ \mathrm{nm})\end{array}$
Shot noise	$SNR\approx \sqrt{N\delta }(\text{secondary-electron counting})$	$N=D{A}_{px}$	$SNR:signal-to-noiseratioofapixel$ $N:electronsperpixel$ $\delta :secondaryyield$ $\begin{array}{r}{A}_{px}:pixelarea;MackandBunday:few\\ detectedsecondariesperpixelandhigh\\ shotnoise;PoissonSNR(computed);\\ Verduinetal.:LERincreasingatlower\\ dosefromshotnoise,someLERreduction\\ athigherdose,SEMimagessimulatedwith\\ PoissonnoiseandPSDanalysis\end{array}$ $\begin{array}{r}(assumed:\delta =1,1\mathrm{nm}pixels;computed:1\\ e^{-}/\mathrm{n}{\mathrm{m}}^{2}givesoneelectronperpixel,SNR\\ about1;10e^{-}/\mathrm{n}{\mathrm{m}}^{2}givesabout3.2)\end{array}$

Roughness

Name	Formula	Related Formulae	Variables/Constants
Line-edge and linewidth roughness	$LER=3{\sigma }_{x_{edge}},LWR=3{\sigma }_{CD}$	${\sigma }_{CD}^2=2{\sigma }_{edge}^2\left(1-\rho \right)$	$\begin{array}{r}LER,LWR:line-edgeandlinewidth\\ roughnessasthreestandarddeviations\end{array}$ ${\sigma }_{edge}:standarddeviationofedgepositions$ $\begin{array}{r}\rho :correlationcoefficientbetweenleft\\ andrightedges;VillarrubiaandBunday:\\ LWRusuallyestimatedasthreestandard\\ deviationsofthelinewidth;correlation\\ relation(computedfromwidthasthe\\ differenceoftwoedges)\end{array}$ $\begin{array}{r}(computed:uncorrelatededgesgiveLWR=\\ \sqrt{2}LER;\rho =0.5givesLWR=LER)\end{array}$
Noise bias	${\sigma }_{meas}^2={\sigma }_{true}^2+{\sigma }_{noise}^2$	${\sigma }_{true}=\sqrt{{\sigma }_{meas}^2-{\sigma }_{noise}^2}$	${\sigma }_{meas}:measuredroughness$ ${\sigma }_{true}:trueroughness$ $\begin{array}{r}{\sigma }_{noise}:contributionofimagenoisetoedge\\ detection;VillarrubiaandBunday:image\\ noiseaddsapositivenonrandomcomponent,\\ abiascomparabletotheroughnessitself\\ forroadmap-smoothsamples,removableby\\ simplealgorithmchanges;Fractilia:\\ randomSEMnoisemeasuredand\\ statisticallysubtracted,givingunbiased\\ LWR,LER,PSD(0)andcorrelationlength,\\ systematicSEMerrorsandfielddistortion\\ alsoremoved;quadratureformfor\\ independentnoise(computed);Mack:image\\ noiseproduceswhitenoise\end{array}$ $\begin{array}{r}(assumed:{\sigma }_{meas}=2.0\mathrm{nm},{\sigma }_{noise}=1.5\mathrm{nm};\\ computed:{\sigma }_{true}=1.32\mathrm{nm},sothebiased\\ valueis51percenttoohigh)\end{array}$
Roughness power spectrum	$PSD\left(f\right)=\frac{PSD\left(0\right)}{1+{\left(2\pi f\xi \right)}^{2H+1}}$	${\sigma }^2={\int }_{-\infty }^{\infty }PSD\left(f\right)df$	$\begin{array}{r}PSD(f):powerspectraldensityofedge\\ deviationsagainstspatialfrequencyf,\\ PSD(0):low-frequencylevel\end{array}$ $\xi :correlationlength$ $H:roughness(Hurst)exponent$ $\begin{array}{r}{\sigma }^2:totalvariance;Cutleretal.andMack:\\ PSD(0),correlationlength,slope2H+1\\ andvarianceastheareaunderthePSD;\\ theexplicitform(myreading);Fractilia:\\ PSD(0)andcorrelationlengthamongthe\\ unbiasedmetrics;Verduinetal.:PSD\\ analysisofsimulatedimages;whitenoise\\ addsaflatfloortothePSD\end{array}$ $\begin{array}{r}(computed:awhite-noisefloorisconstant\\ inf,soitisseparatedfromthe\\ roughnessPSDathighfrequencies)\end{array}$

Beam Effects and Uncertainty

Name	Formula	Related Formulae	Variables/Constants
Resist shrinkage	$CD\left(n\right)=C{D}_0-\Delta C{D}_{\infty }\left(1-e^{-n/n_0}\right)$	$C{D}_0=CD\left(n\right)+\Delta C{D}_{\infty }\left(1-e^{-n/n_0}\right)(\text{back-extrapolation})$	$CD(n):CDafternframesordoseunits$ $\begin{array}{r}C{D}_0:unexposedCD,\Delta C{D}_{\infty }:saturated\\ shrinkage\end{array}$ $\begin{array}{r}{n}_0:characteristicdose;saturating\\ exponentialform(myreading);Verduinet\\ al.:severeshrinkagewithinnormal\\ acquisitiondoseandinthefirstfew\\ frames,recommendingminimum-dose\\ measurement;Fractilia:lowerdose\\ possiblewithunbiasedanalysis,reducing\\ shrinkage\end{array}$ $\begin{array}{r}(assumed:\Delta C{D}_{\infty }=3\mathrm{nm},n_0=2frames;\\ computed:1.18\mathrm{nm}lostafterthefirst\\ frameand2.6\mathrm{nm}afterfour)\end{array}$
Precision and total measurement uncertainty	$P=3{\sigma }_{repeat}$	$TMU=\sqrt{P^2+u_{match}^2+u_{linearity}^2+\dots }$	$P:precision$ $\begin{array}{r}{\sigma }_{repeat}:standarddeviationofrepeated\\ measurements\end{array}$ $\begin{array}{r}TMU:totalmeasurementuncertaintyagainst\\ areference\end{array}$ $\begin{array}{r}u:uncertaintycomponentsfromtool\\ matching,linearityandothereffects;\\ Thieletal.:ITRSCDtooluncertainty\\ quotedat3sigma;TMUpage;quadrature\\ combination(myreading);Fractilia:LWR,\\ LCDUandEPEmakingupmorethanhalfof\\ theerrorbudgetatcurrentnodes\end{array}$ $\begin{array}{r}(assumed:{\sigma }_{repeat}=0.5\mathrm{nm},u_{match}=0.3\mathrm{nm},u_{lin}=\\ 0.4\mathrm{nm};computed:P=1.5\mathrm{nm}anda1-sigma\\ combinedvalueof0.71\mathrm{nm})\end{array}$
Low-voltage imaging	$E_0\approx 0.3\text{to}1\mathrm{keV}$	$R_{KO}\left(1\mathrm{keV},\text{resist}\right)\text{of order tens of nanometres}$	$\begin{array}{r}{E}_0:landingenergy;lowenergieslimit\\ chargingandpenetrationintoresist(my\\ reading;SEMpagefortherangeformula);\\ Verduinetal.:300\mathrm{eV}imaging,\\ simulationsat20\mathrm{kV}witha10.5\mathrm{nm}spot\end{array}$ $\begin{array}{r}(computed:onsilicontheKanaya-Okayama\\ rangeis32\mathrm{nm}at1\mathrm{keV}fromtheSEMpage)\end{array}$

Important Questions

Ques: How does a CD-SEM differ from an ordinary SEM?

It is automated, top-down and low-voltage. A CD-SEM finds a target on a wafer, scans it at landing energies near a few hundred electronvolts, and measures linewidths from line-scan profiles; Verduin et al. imaged resist at 300 $\mathrm{eV}$ and 10 $\mathrm{pA}$ on a Hitachi CG4000 at the lowest available dose. Low energy should limit charging of resist and oxides and confine the beam to the top tens of nanometres (my reading); on silicon the range is 32 $\mathrm{nm}$ at 1 $\mathrm{keV}$ (SEM page, computed). The tool reports critical dimensions, not images, and its precision must be a fraction of a nanometre. Because the beam itself changes resist and adds noise, dose and algorithm choices matter more than in imaging SEM. Reference: T. Verduin, M. D. Hermans, S. R. Lokhorst, C. W. Hagen and P. Kruit, ‘Quantum Noise Effects in e-Beam Lithography and Metrology’, TU Delft presentation hosted by NIST (fetch-tool summary); Scanning Electron Microscope (SEM) page of this vault.

Ques: How are edges and linewidths extracted?

From the brightness profile across the line. Mack and Bunday place edges at a threshold between the minimum and maximum grey levels of a filtered linescan, or invert a calibrated linescan model; the steepest slope is a further option (my reading). Verduin et al. integrate the line image along its length, fit the integrated profile with a model containing the edge position, and locate edges at each depth. The CD is the distance between left and right edges, so a 1 $\mathrm{nm}$ shift of each edge in opposite directions changes it by 2 $\mathrm{nm}$ (computed). Different algorithms give different absolute CDs on the same profile, so offsets are calibrated against reference metrology (my reading); Verduin et al. find the bias depends on resist thickness. Reference: T. Verduin, M. D. Hermans, S. R. Lokhorst, C. W. Hagen and P. Kruit, ‘Quantum Noise Effects in e-Beam Lithography and Metrology’, TU Delft presentation hosted by NIST (fetch-tool summary); C. A. Mack and B. D. Bunday, ‘Using the Analytical Linescan Model for SEM Metrology’, Proc. SPIE (Metrology, Inspection, and Process Control XXXI), 2017: threshold between minimum and maximum grey levels on a filtered linescan, model-based edge detection by inverting a calibrated linescan model, resist shrinkage limiting dose, small numbers of detected secondaries giving high shot noise (fetch-tool summary).

Ques: How are LER and LWR related?

Through the correlation between the two edges. Each is usually reported as three standard deviations, of edge position for line-edge roughness and of width for linewidth roughness; Villarrubia and Bunday note LWR is usually estimated as three standard deviations of the linewidth. Since width is the difference of two edge positions, ${\sigma }_{CD}^2=2{\sigma }_{edge}^2(1-\rho )$ (computed). Uncorrelated edges give LWR equal to $\sqrt{2}$ times LER, while a correlation of 0.5 makes them equal (computed). Positive correlation, edges wiggling together, means the line meanders but keeps its width; negative correlation means width varies more. For device performance, LWR affects transistor variability and LER affects line placement, so both are reported. Reference: J. S. Villarrubia and B. Bunday, ‘Unbiased Estimation of Linewidth Roughness’, Proc. SPIE 5752, 480 (2005) (abstract) (fetch-tool summary); C. Cutler et al. (with C. Mack), ‘Roughness power spectral density as a function of resist parameters and its impact through process’, Proc. SPIE 10587, 1058707, 2018: PSD slope 2H + 1 with H the roughness exponent, variance as the area under the curve, PSD(0) proportional to variance, 3 sigma LER and LWR (fetch-tool summary).

Ques: Why is measured roughness biased, and how is it corrected?

Because image noise moves the detected edges even on a perfect line. Villarrubia and Bunday showed that noise adds a positive, nonrandom component to the three-sigma metric, comparable to the roughness itself for samples as smooth as the roadmap requires. Since noise and roughness add as variances, ${\sigma }_{meas}^2={\sigma }_{true}^2+{\sigma }_{noise}^2$ for independent white noise, as Mack describes it (computed); with an assumed 2.0 $\mathrm{nm}$ measured and 1.5 $\mathrm{nm}$ noise, the true value is 1.32 $\mathrm{nm}$, so the biased number is 51 per cent too high (computed). Fractilia’s MetroLER measures the random noise in each image and subtracts it statistically, reporting unbiased LWR, LER, PSD(0) and correlation length, and also removes systematic SEM errors and field distortion. Reference: J. S. Villarrubia and B. Bunday, ‘Unbiased Estimation of Linewidth Roughness’, Proc. SPIE 5752, 480 (2005) (abstract) (fetch-tool summary); Fractilia, ‘You can’t fix what you don’t measure accurately’, MetroLER product page (fetch-tool summary); C. A. Mack, ‘More on Stochastics and the Phenomenon of Line-Edge Roughness’, ICPST, Chiba, 2017: PSD as variance per frequency increment, variance as the area under the PSD, PSD(0), correlation length and roughness exponent, 3 sigma roughness, image noise producing white noise and biased roughness (fetch-tool summary).

Ques: How does the electron beam change what it measures?

It shrinks resist. Verduin et al. report severe shrinkage within a normal acquisition dose of 10 $e^{-}/\mathrm{n}{\mathrm{m}}^2$, 160 $\mu C/c{\mathrm{m}}^2$, and in the very first few frames of imaging, and recommend measuring at the minimum dose. If shrinkage follows an exponential approach to a saturated loss (my reading), with an assumed 3 $\mathrm{nm}$ saturation and a two-frame constant, 1.18 $\mathrm{nm}$ is lost after the first frame and 2.6 $\mathrm{nm}$ after four (computed). Back-extrapolating the CD to zero dose recovers the unexposed value. The conversion is fixed: one electron per square nanometre is 16 $\mu C/c{\mathrm{m}}^2$ (computed). Fractilia argues that unbiased analysis allows lower dose at the same accuracy, reducing shrinkage. Reference: T. Verduin, M. D. Hermans, S. R. Lokhorst, C. W. Hagen and P. Kruit, ‘Quantum Noise Effects in e-Beam Lithography and Metrology’, TU Delft presentation hosted by NIST (fetch-tool summary); Fractilia, ‘You can’t fix what you don’t measure accurately’, MetroLER product page (fetch-tool summary).

Ques: Why does roughness depend on imaging dose?

Because shot noise grows at low dose. With Poisson statistics, a pixel receiving N electrons and producing $N\delta$ secondaries has SNR about $\sqrt{N\delta }$ (computed); Mack and Bunday stress that few detected secondaries per pixel mean high shot noise. At Verduin et al.’s lowest dose of 1 $e^{-}/\mathrm{n}{\mathrm{m}}^2$ and 1 $\mathrm{nm}$ pixels, each pixel receives about one electron, an SNR near 1, rising only to about 3.2 at 10 $e^{-}/\mathrm{n}{\mathrm{m}}^2$ (computed with an assumed yield of 1). Verduin et al. found that LER increases at lower dose from shot noise, with some reduction at higher dose, and simulated images with Poisson noise to analyse the power spectrum. The trade-off is direct: higher dose lowers noise but shrinks resist, so noise subtraction and averaging along long lines are essential. Reference: T. Verduin, M. D. Hermans, S. R. Lokhorst, C. W. Hagen and P. Kruit, ‘Quantum Noise Effects in e-Beam Lithography and Metrology’, TU Delft presentation hosted by NIST (fetch-tool summary); Fractilia, ‘You can’t fix what you don’t measure accurately’, MetroLER product page (fetch-tool summary); C. A. Mack and B. D. Bunday, ‘Using the Analytical Linescan Model for SEM Metrology’, Proc. SPIE (Metrology, Inspection, and Process Control XXXI), 2017: threshold between minimum and maximum grey levels on a filtered linescan, model-based edge detection by inverting a calibrated linescan model, resist shrinkage limiting dose, small numbers of detected secondaries giving high shot noise (fetch-tool summary).

Ques: How is CD-SEM performance specified?

By precision, matching and total measurement uncertainty. Precision is three times the standard deviation of repeated measurements of the same site; total uncertainty also combines tool-to-tool matching, linearity against a reference and other components in quadrature (my reading; Thiel et al. quote ITRS CD tool uncertainty at 3 sigma; TMU page). With an assumed 0.5 $\mathrm{nm}$ repeatability, 0.3 $\mathrm{nm}$ matching and 0.4 $\mathrm{nm}$ linearity, precision is 1.5 $\mathrm{nm}$ and the combined one-sigma value 0.71 $\mathrm{nm}$ (computed). Accuracy also depends on edge-detection bias, which Verduin et al. find varies with resist height, and on the roughness bias described by Villarrubia and Bunday. Fractilia notes that LWR, local CD uniformity and edge-placement error make up more than half of the error budget at current nodes, so roughness metrology now matters as much as mean CD. Reference: T. Verduin, M. D. Hermans, S. R. Lokhorst, C. W. Hagen and P. Kruit, ‘Quantum Noise Effects in e-Beam Lithography and Metrology’, TU Delft presentation hosted by NIST (fetch-tool summary); J. S. Villarrubia and B. Bunday, ‘Unbiased Estimation of Linewidth Roughness’, Proc. SPIE 5752, 480 (2005) (abstract) (fetch-tool summary); Fractilia, ‘You can’t fix what you don’t measure accurately’, MetroLER product page (fetch-tool summary); B. Thiel, A. Cepler, A. Diebold and R. Matyi, ‘Advances in CD-Metrology’, CNSE Albany, about 2011: 2010 ITRS CD tool uncertainty at 3 sigma (fetch-tool summary); Tool Matching and Total Measurement Uncertainty (TMU) page of this vault.

References

1. T. Verduin, M. D. Hermans, S. R. Lokhorst, C. W. Hagen and P. Kruit, “Quantum Noise Effects in e-Beam Lithography and Metrology,” TU Delft presentation hosted by NIST. Available: https://www.nist.gov/document/026pdf
2. Scanning Electron Microscope (SEM), page in this vault.
3. C. A. Mack and B. D. Bunday, “Using the analytical linescan model for SEM metrology,” Proc. SPIE, Metrology, Inspection, and Process Control for Microlithography XXXI, 2017. Available: https://www.fractilia.com/s/SPIE2017AnalyticalLinescanModel.pdf
4. C. A. Mack, “More on stochastics and the phenomenon of line-edge roughness,” 34th International Photopolymer Science and Technology Conference, Chiba, 2017. Available: https://www.fractilia.com/s/Morestochasticsandthephenomenonofline-edgeroughness_Mack.pdf
5. C. Cutler et al., “Roughness power spectral density as a function of resist parameters and its impact through process,” Proc. SPIE 10587, 1058707, 2018. Available: https://www.fractilia.com/s/SPIE2018_Roughnesspowerspectraldensityasafunctionofresistparameters.pdf
6. B. Thiel, A. Cepler, A. Diebold and R. Matyi, “Advances in CD-Metrology (CD-SAXS, Mueller Matrix based Scatterometry, and SEM),” CNSE, University at Albany, about 2011. Available: https://www.nist.gov/document/thiel2011pdf
7. Tool Matching and Total Measurement Uncertainty (TMU) page of this vault.
8. J. S. Villarrubia and B. Bunday, “Unbiased Estimation of Linewidth Roughness,” Proc. SPIE, vol. 5752, pp. 480-488, 2005. Available: https://nist.gov/publications/unbiased-estimation-linewidth-roughness
9. Fractilia, “You can’t fix what you don’t measure accurately,” MetroLER product page. Available: https://fractilia.com/299609321176

Related pages in this vault: Scanning Electron Microscope (SEM), Atomic Force Microscope (AFM), Critical-Dimension Small-Angle X-Ray Scattering (CD-SAXS), Overlay Metrology Tool, Optical CD Scatterometer (OCD), Optical Microscope, Transmission Electron Microscope (TEM and STEM), Focused Ion Beam (FIB), EBIC and EBAC, E-Beam Inspection and Defect Review, Gauge Repeatability and Reproducibility (Gauge R&R), Calibration and Traceability, Statistical Process Control (SPC), Tool Matching and Total Measurement Uncertainty (TMU), High-NA DUV Immersion Scanner, Photoresist and Process Window, Resolution Enhancement and Multiple Patterning, Electron-Matter Interaction, Electron Optics and the SEM Column.