Optical CD Scatterometer (OCD)

1. Theory

• Definition: An optical critical-dimension (OCD) scatterometer measures the reflectance, ellipsometric or Mueller-matrix spectrum of a periodic test grating and infers its profile, width, height, sidewall angle, corner rounding and film thicknesses, by fitting a rigorous electromagnetic model (Bodermann et al.; Heider et al.).
• Zeroth-order diffraction: For pitches below the wavelength only the specular order propagates, yet its spectrum depends on the full cross section (LibreTexts grating equation; computed; Bodermann et al.).
• Inverse problem: The profile is found by minimizing the misfit between measured and modelled spectra; the problem is ill-posed and may not have a unique solution (Bodermann et al.).
• Sensitivity: Different wavelength ranges carry information about different parameters, so the measured spectral range limits what can be resolved (Heider et al.).

2. Working Principle

• Optics: Broadband normal-incidence reflectometry or spectroscopic and Mueller ellipsometry from the DUV to the IR (Heider et al.; Bodermann et al.; Spectroscopic Ellipsometer page).
• Modelling: RCWA, FEM or FDTD solvers compute spectra for parameterized profiles (Bodermann et al.).
• Fitting: Library lookup or real-time regression finds the best-fit profile, with uncertainties from the covariance matrix (NAG e04ycc; library against regression my reading).
• Hybridization: Data from several methods are fitted jointly to reduce parameter correlation (Bodermann et al.).

3. Applications

• Lithography and etch control: Resist and etched line CD, height and sidewall angle (Bodermann et al.).
• Trench and 3-D structures: Deep trench depth and CD at several heights (Heider et al.).
• Reference metrology: Comparison with AFM, CD-SEM and GISAXS on calibrated standards (Bodermann et al.).

Diffraction and Modelling

Name	Formula	Related Formulae	Variables/Constants
Grating equation	$\sin {\theta }_m=\sin {\theta }_i+\frac{m\lambda }{P}$	$\text{zeroth order only if}P<\frac{\lambda }{1+\sin {\theta }_i}$	${\theta }_m:angleoforderm$ ${\theta }_i:incidenceangle$ $m:diffractionorder$ $\lambda :wavelength$ $\begin{array}{r}P:pitch;LibreTexts:n\lambda =d(\sin \alpha +\sin \beta ),written\\ herewiththeoppositesignforthe\\ diffractedangle;zero-ordercondition\\ computedfrom|\sin {\theta }_m|\le 1;Heideretal.:1.7\mu \mathrm{m}\\ pitch,190to1000\mathrm{nm}broadbandplusIRto\\ 15000\mathrm{nm};Bodermannetal.:100\mathrm{nm}pitch\\ standard,DUVat266\mathrm{nm}and193to210\mathrm{nm}\end{array}$ $\begin{array}{r}(computed:P=100\mathrm{nm}at193or266\mathrm{nm}\\ givesonlythezerothorder;P=1.7\mu \mathrm{m}\\ gives17propagatingordersat190\mathrm{nm}and\\ thefirstorderat36.0{}^{\circ }at1000\mathrm{nm})\end{array}$
Rigorous forward model	${\mathbf{S}}_{model}=f\left(CD,h,SWA,\dots ;\lambda ,{\theta }_i,\varphi \right)$	$N_{orders}=2M+1(\text{RCWA truncation})$	$\begin{array}{r}\mathbf{S}:modelledreflectance,ellipsometricor\\ Muellerspectra\end{array}$ $CD:criticaldimension$ $h:height$ $SWA:sidewallangle$ $\varphi :azimuth$ $\begin{array}{r}M:highestretainedFourierorder;\\ Bodermannetal.:RCWA,FDTD,FEMand\\ waveguidesolversfromMaxwel{l}^{\prime }s\\ equations;Civilettietal.:RCWAfields\\ andpermittivityexpandedinFourier\\ seriestruncatedtoafinitenumberof\\ modes\end{array}$ $\begin{array}{r}(myreading:libraryprecomputationtrades\\ accuracyforspeed,regressiontrades\\ speedforaccuracy)\end{array}$
Inverse problem and fit quality	${\chi }^2={\sum }_k\frac{{\left[S_{meas}\left({\lambda }_k\right)-S_{model}\left({\lambda }_k;\mathbf{p}\right)\right]}^2}{{\sigma }_k^2}$	$\mathrm{Cov}\left(\mathbf{p}\right)\approx {\left({\mathbf{J}}^T\mathbf{WJ}\right)}^{-1}$	$\mathbf{p}:profileparameters$ ${\sigma }_k:measurementnoise$ $\mathbf{J}:sensitivity(Jacobian)matrix$ $\begin{array}{r}\mathbf{W}:weightmatrix;NAGe04ycc:C={\sigma }^2(J^{T}J{)}^{-1},\\ theweightedformhere(computed);\\ Bodermannetal.:severelyill-posed\\ inverseproblem,solutioningeneralnot\\ unique\end{array}$ $\begin{array}{r}(myreading:stronglycorrelated\\ parametersshowaslargeoff-diagonal\\ covarianceandinflateeachuncertainty)\end{array}$

Measurement Channels and Sensitivity

Name	Formula	Related Formulae	Variables/Constants
Polarization measurements	${\mathbf{S}}_{out}=\mathbf{M}{\mathbf{S}}_{in}$	$\mathbf{M}={\mathbf{M}}_0+\text{residuum}(\text{depolarization})$	$\mathbf{S}:Stokesvectors$ $\mathbf{M}:4by4Muellermatrix$ $\begin{array}{r}{\mathbf{M}}_0:best-fitnon-depolarizing\\ Mueller-Jonesmatrix;Bodermannetal.:\\ Muellerellipsometry190to2500\mathrm{nm},\\ decompositioninto{M}_{0}plusresiduum,sand\\ psensitiveEUVscatterometry;\\ ellipsometrypage:\tan \Psi e^{i\Delta }=r_p/r_s;Trippe:\\ S^{\prime }=MSwitha4by4Muellermatrixableto\\ describedepolarization\end{array}$ $\begin{array}{r}(myreading:off-diagonalMuellerelements\\ vanishforsymmetricgratingsatzero\\ azimuth,sotheyflagasymmetrysuchas\\ sidewall-angledifferences)\end{array}$
Spectral range and sensitivity	$\frac{\partial S}{\partial p_j}\left(\lambda \right)(\text{sensitivity spectrum})$	${\lambda }_{UV}\text{for small CD},{\lambda }_{IR}\text{for depth}$	$\begin{array}{r}\partial S/\partial p_j:sensitivitytoparameterj;\\ Heideretal.:full190to1000\mathrm{nm}range\\ matchingcross-sectionSEM,limited375to\\ 750\mathrm{nm}range(350to750inacaption)\\ failingondeeptrenches,Rssensitivity\\ 190to450\mathrm{nm},Rp800to1000\mathrm{nm},\\ sensitivitycollapsingto190to280\mathrm{nm}as\\ CDshrinks,SWAderivedfromCDanddepth\\ (seetheQ/A)\end{array}$ $\begin{array}{r}(computed:thelimitedrangeremoves190\\ to375\mathrm{nm},23percentofthebroadband\\ spanandmostoftheUVsensitivity)\end{array}$
Hybrid metrology	$\hat{p}=\frac{{\sum }_i{p}_i/{\sigma }_i^2}{{\sum }_{i}1/{\sigma }_i^2},{\sigma }_{\hat{p}}={\left({\sum }_i\frac{1}{{\sigma }_i^2}\right)}^{-1/2}$	$\text{joint fit of several data sets}$	$\hat{p}:combinedestimate$ $\begin{array}{r}{\sigma }_i:individualuncertainty;Kirchner:\\ weightsinverselyproportionaltothe\\ squareofeachuncertainty,combined\\ uncertainty(computed);Bodermannetal.:\\ P100standardbyscatterometry,MMEand\\ combinedfit,CD32.3(1.8),33.0(1.5),\\ 32.5(1.1)\mathrm{nm},h105.7(2.3),105.8(2.2),\\ 105.7(1.6)\mathrm{nm},SWA91.4,90.9,91.2{}^{\circ },\\ topcornerrounding14.9(3.0),13.8\\ (1.9),14.0(1.1)\mathrm{nm}(seetheQ/A)\end{array}$ $\begin{array}{r}(computed:simpleweightingreproduces\\ 1.15and1.59\mathrm{nm}forCDandhbutgives\\ 1.6\mathrm{nm}fortoproundingagainstthejoint\\ fi{t}^{\prime }s1.1)\end{array}$

Accuracy and Practice

Name	Formula	Related Formulae	Variables/Constants
Comparison and reference values	${\Delta }_{CD}=C{D}_{OCD}-C{D}_{ref}$	$U\left(C{D}_{ref}\right)<1\mathrm{nm}$	$\begin{array}{r}{\Delta }_{CD}:offsetagainstareferencemethod;\\ Bodermannetal.:DUVscatterometry25.9\\ (1.4)and50.0(4.4)\mathrm{nm},combinedwithMME\\ 26.4(0.9)and50.5(1.6),GISAXS25.1and\\ 48.2\mathrm{nm},referencestandardU(CD)below1\\ \mathrm{nm}characterizedbyscatterometry,MME,CD\\ andtiltAFM,CD-SEMandGISAXS,CD-AFM\\ gapat5\mathrm{nm}orbelow\end{array}$ $\begin{array}{r}(computed:combinedOCDandGISAXSdiffer\\ by1.3\mathrm{nm},1.4timesthe0.9\mathrm{nm}OCD\\ uncertainty)\end{array}$
Short-wavelength scatterometry	$\lambda \approx 5\text{to}6\mathrm{nm},{\alpha }_i=30^{\circ }(\text{EUV})$	$\lambda =0.12\text{to}0.73\mathrm{nm},{\alpha }_i\approx 1^{\circ }(\text{X-ray})$	$\begin{array}{r}{\alpha }_i:grazingangle;Bodermannetal.:SX700\\ softX-rayreflectometer0.7to25\mathrm{nm},EUV\\ exampleat5to6\mathrm{nm}and30{}^{\circ },much\\ reducedfootprintbuthigheruncertainty\\ andS/Nissues,GISAXSat0.12to0.73\mathrm{nm}\\ andabout1{}^{\circ };CD-SAXSpage:transmission\\ alternative\end{array}$ $\begin{array}{r}(computed:at5.5\mathrm{nm}a100\mathrm{nm}pitch\\ supportsabout36propagatingorders,so\\ EUVscatterometryisfarfromthe\\ zero-orderregime)\end{array}$
Target and throughput constraints	$A_{target}\gtrsim {\left(\text{spot size}\right)}^2$	$N_{periods}=\frac{L_{target}}{P}$	$A_{target}:gratingpadarea$ $L_{target}:padwidth$ $\begin{array}{r}{N}_{periods}:periodsacrossthepad;targetat\\ leastthespotsize(myreading);\\ Bodermannetal.:EUVreducedfootprint;\\ Heideretal.:measurementontrench\\ arrayswith1.7\mu \mathrm{m}pitch\end{array}$ $\begin{array}{r}(assumed:40\mu \mathrm{m}pad;computed:400periods\\ ofa100\mathrm{nm}grating)\end{array}$

Important Questions

Ques: How can a specular reflection measure a 30 nm line?

Because its spectrum depends on the whole grating cross section. The grating equation $\sin {\theta }_m=\sin {\theta }_i+m\lambda /P$ shows that for a 100 $\mathrm{nm}$ pitch at 193 or 266 $\mathrm{nm}$ only the zeroth order propagates (LibreTexts grating equation; computed). That order’s amplitude and polarization still change with width, height, sidewall angle and underlying films, so a rigorous solver such as RCWA, which truncates its Fourier expansion to a finite number of modes (Civiletti et al.), FEM or FDTD predicts spectra for each candidate profile (Bodermann et al.). Fitting the measured spectrum by minimizing ${\chi }^2$ gives the parameters, with uncertainties from the covariance $({\mathbf{J}}^T\mathbf{WJ}{)}^{-1}$ (NAG e04ycc gives the unweighted form; computed). Bodermann et al. measured a nominal 35 $\mathrm{nm}$ line on a 100 $\mathrm{nm}$ pitch as 32.5 +/- 1.1 $\mathrm{nm}$ by combined scatterometry and Mueller ellipsometry. Reference: Chemistry LibreTexts, ‘Grating Equation’: n lambda = d(sin alpha + sin beta), angles counter-clockwise from the normal, the sine magnitude at most 1, n = 0 as the mirror order (fetch-tool summary); B. J. Civiletti, A. Lakhtakia and P. B. Monk, ‘Analysis of the Rigorous Coupled Wave Approach for s-Polarized Light in Gratings’, arXiv:1904.12822: fields and permittivity expanded in Fourier series truncated to a finite number of modes, with truncation error (fetch-tool summary); NAG C Library, e04ycc ‘Covariance matrix for nonlinear least squares’: C = sigma^2 (J^T J)^-1 with sigma^2 = F/(m - n), diagonal elements as parameter variances (fetch-tool summary); B. Bodermann, A. Diener, S. Heidenreich, F. Scholze, V. Soltwisch and M. Wurm, ‘OCD Metrology for Advanced Lithography’, FCMN 2017, Monterey, slides hosted by NIST (fetch-tool summary).

Ques: Why does the spectral range matter so much?

Because each parameter is sensitive only in certain wavelength bands. Heider et al. found that UV reflectance, Rs from 190 to 450 $\mathrm{nm}$, and near-IR Rp from 800 to 1000 $\mathrm{nm}$ carried trench depth and CD information; restricting data to 375 to 750 $\mathrm{nm}$ (350 to 750 in one caption) gave results that matched cross-section SEM poorly, especially for trench depth. Dropping 190 to 375 $\mathrm{nm}$ removes 23 per cent of the broadband span (computed) but nearly all the UV sensitivity. As CD shrinks, their simulations show sensitivity narrowing to 190 to 280 $\mathrm{nm}$, so smaller features need deeper UV. Bodermann et al. extend to 190 to 2500 $\mathrm{nm}$ Mueller ellipsometry, an extension of the Spectroscopic Ellipsometer page’s method, and down to EUV and X-ray wavelengths for the same reason. Reference: F. Heider, J. Roberts, J. Huang, J. Lam and R. Forouhi, ‘Effects of measured spectral range on accuracy and repeatability of OCD analysis’, Solid State Technology 56(5) (2013) (fetch-tool summary); B. Bodermann, A. Diener, S. Heidenreich, F. Scholze, V. Soltwisch and M. Wurm, ‘OCD Metrology for Advanced Lithography’, FCMN 2017, Monterey, slides hosted by NIST (fetch-tool summary); Spectroscopic Ellipsometer page of this vault.

Ques: Why is OCD an ill-posed inverse problem?

Because different profiles can produce nearly the same spectrum. Bodermann et al. describe the inverse problem as severely ill-posed with a solution that is not unique in general. Parameters such as height and sidewall angle, or CD and corner rounding, can trade off against each other, appearing as large off-diagonal terms in the covariance matrix (my reading). Remedies include fixing well-known parameters, adding sensitivity through more wavelengths, angles or polarizations, and combining methods. Their 25.9 +/- 1.4 $\mathrm{nm}$ CD and 50.0 +/- 4.4 $\mathrm{nm}$ height from DUV scatterometry tightened to 26.4 +/- 0.9 and 50.5 +/- 1.6 $\mathrm{nm}$ with Mueller ellipsometry added; the height uncertainty fell by a factor of 2.75 (computed). Reference: B. Bodermann, A. Diener, S. Heidenreich, F. Scholze, V. Soltwisch and M. Wurm, ‘OCD Metrology for Advanced Lithography’, FCMN 2017, Monterey, slides hosted by NIST (fetch-tool summary); NAG C Library, e04ycc ‘Covariance matrix for nonlinear least squares’: C = sigma^2 (J^T J)^-1 with sigma^2 = F/(m - n), diagonal elements as parameter variances (fetch-tool summary).

Ques: Is the combined CD 32.5 nm or does simple averaging give 32.7 nm?

Bodermann et al. report 32.3 +/- 1.8 $\mathrm{nm}$ by scatterometry, 33.0 +/- 1.5 $\mathrm{nm}$ by Mueller ellipsometry, which relates output to input Stokes vectors through a 4 by 4 matrix (Trippe), and 32.5 +/- 1.1 $\mathrm{nm}$ combined. Inverse-variance weighting, which Kirchner gives as the optimal choice, of the two gives 32.7 +/- 1.15 $\mathrm{nm}$ (computed), the same uncertainty but a slightly different value; for height it reproduces 105.75 +/- 1.6 $\mathrm{nm}$ almost exactly (computed). The difference arises because their combination is a joint fit of both data sets, which also adjusts correlated parameters, rather than averaging finished results (my reading). The joint fit is more likely correct, since it uses the full information; this is clearest for top corner rounding, where simple weighting gives 1.6 $\mathrm{nm}$ uncertainty but the joint fit 1.1 $\mathrm{nm}$ (computed against reported). Reference: B. Bodermann, A. Diener, S. Heidenreich, F. Scholze, V. Soltwisch and M. Wurm, ‘OCD Metrology for Advanced Lithography’, FCMN 2017, Monterey, slides hosted by NIST (fetch-tool summary); S. Trippe, ‘Polarization and Polarimetry: A Review’, arXiv:1401.1911: S’ = M S with a 4 by 4 Mueller matrix, which can describe depolarization (fetch-tool summary); J. Kirchner, ‘Data Analysis Toolkit 12: Weighted averages and their uncertainties’, UC Berkeley, 2006: optimal weights inversely proportional to the square of each uncertainty (fetch-tool summary).

Ques: How accurate is OCD against reference methods?

Within about a nanometre on well-characterized samples. Bodermann et al. compared DUV scatterometry, Mueller ellipsometry and GISAXS on the same structure: combined OCD gave 26.4 +/- 0.9 $\mathrm{nm}$ CD and GISAXS 25.1 $\mathrm{nm}$, a 1.3 $\mathrm{nm}$ difference, 1.4 times the OCD uncertainty (computed), and heights of 50.5 and 48.2 $\mathrm{nm}$. Their reference standard targets U(CD) below 1 $\mathrm{nm}$, characterized by scatterometry, Mueller ellipsometry, CD and tilt AFM, CD-SEM and GISAXS. Heider et al. found good matches with cross-section SEM only when the full spectral range was used. Agreement depends on whether each method measures the same CD definition, top, middle or bottom width, which must be specified (my reading). Reference: B. Bodermann, A. Diener, S. Heidenreich, F. Scholze, V. Soltwisch and M. Wurm, ‘OCD Metrology for Advanced Lithography’, FCMN 2017, Monterey, slides hosted by NIST (fetch-tool summary); F. Heider, J. Roberts, J. Huang, J. Lam and R. Forouhi, ‘Effects of measured spectral range on accuracy and repeatability of OCD analysis’, Solid State Technology 56(5) (2013) (fetch-tool summary).

References

1. Chemistry LibreTexts, “Grating Equation.” Available: https://chem.libretexts.org/Bookshelves/Analytical_Chemistry/Supplemental_Modules_(Analytical_Chemistry)/Instrumentation_and_Analysis/Atomic_Emission_Spectroscopy_(AES)/05_Spectrometers/02_Grating_Equation
2. B. J. Civiletti, A. Lakhtakia and P. B. Monk, “Analysis of the rigorous coupled wave approach for s-polarized light in gratings,” arXiv:1904.12822. Available: https://arxiv.org/abs/1904.12822
3. NAG Ltd., “nag_opt_lsq_covariance (e04ycc): covariance matrix for nonlinear least squares,” NAG C Library, Mark 26. Available: https://support.nag.com/numeric/cl/nagdoc_cl26.0/pdf/e04/e04ycc.pdf
4. S. Trippe, “Polarization and polarimetry: a review,” J. Korean Astron. Soc.; arXiv:1401.1911. Available: https://arxiv.org/abs/1401.1911
5. J. Kirchner, “Data Analysis Toolkit 12: Weighted averages and their uncertainties,” UC Berkeley, 2006. Available: https://seismo.berkeley.edu/~kirchner/Toolkits/Toolkit_12.pdf
6. B. Bodermann, A. Diener, S. Heidenreich, F. Scholze, V. Soltwisch and M. Wurm, “OCD Metrology for Advanced Lithography,” presented at FCMN 2017, Monterey, 2017. Available: https://www.nist.gov/document/bodermannpdf
7. F. Heider, J. Roberts, J. Huang, J. Lam and R. Forouhi, “Effects of measured spectral range on accuracy and repeatability of OCD analysis,” Solid State Technology, vol. 56, no. 5, 2013. Available: https://sst.semiconductor-digest.com/issue/?id=27157
8. Spectroscopic Ellipsometer page of this vault.

Related pages in this vault: Critical-Dimension Small-Angle X-Ray Scattering (CD-SAXS), Spectroscopic Ellipsometer, CD-SEM, Overlay Metrology Tool, Atomic Force Microscope (AFM), Spectral Reflectometer, Scanning Electron Microscope (SEM), Fin Field Effect Transistor (FinFET), Patterned-Wafer Optical Inspection, Stylus Profiler, Optical Emission Spectroscopy Endpoint (OES), Laser Interferometric Endpoint, Gauge Repeatability and Reproducibility (Gauge R&R), 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, Light-Matter Interaction for Optical Inspection.