Critical-Dimension Small-Angle X-Ray Scattering (CD-SAXS)

1. Theory

• Definition: Critical-dimension small-angle X-ray scattering (CD-SAXS) transmits a hard X-ray beam through a wafer and records diffraction from periodic line gratings, giving pitch, width, height, sidewall angle and roughness averaged over many lines (Hu et al.; Wu et al.; Sunday et al.).
• Fourier relation: Scattered intensity is the squared Fourier transform of the electron-density profile, damped by a Debye-Waller roughness factor (Sunday et al.).
• Reciprocal space: A grating of pitch P produces orders spaced $2\pi /P$ in $q_x$; their intensity along $q_z$ encodes the cross-section shape (Hu et al.).
• Rotation: Tilting the sample sweeps the orders through $q_z$, revealing height and sidewall angle (Hu et al.; Sunday et al.).

2. Working Principle

• Source: Hard X-rays above about 15 $\mathrm{keV}$ from a synchrotron or high-brightness laboratory source pass through the silicon wafer (Sunday et al.; Wu et al.).
• Detection: A 2-D area detector several metres downstream records the diffraction orders (Hu et al.).
• Tilt series: Images at many rotation angles, for example -20 to 20 ${}^{\circ }$ or over 45 ${}^{\circ }$ in 1 ${}^{\circ }$ steps, build the $q_x$-$q_z$ map (Hu et al.; Sunday et al.).
• Modelling: Stacked-trapezoid models are fitted, with uncertainties from Markov-chain Monte Carlo (Sunday et al.).

3. Applications

• Lithography focus and dose control: Width, height and sidewall angle across focus series (Wu et al.).
• Dense patterns: Sub-50 $\mathrm{nm}$ pitch gratings, fins and self-aligned multiple patterning (Sunday et al.).
• Reference metrology: Validation of CD-SEM and optical scatterometry with TEM cross sections (Sunday et al.; Hu et al.).
• Line roughness: Averaged sidewall and top-surface roughness (Wu et al.).

Scattering Geometry

Name	Formula	Related Formulae	Variables/Constants
Scattering vector	$q=\frac{4\pi \sin \theta }{\lambda },\mathbf{q}=\mathbf{k}-{\mathbf{k}}^{\prime }$	$q=\frac{2\pi }{d},k=\frac{2\pi }{\lambda }$	$q:momentumtransfer$ $\theta :halfthescatteringangle$ $\begin{array}{r}\lambda :wavelength,\mathbf{k},{\mathbf{k}}^{\prime }:incidentand\\ scatteredwavevectors\end{array}$ $\begin{array}{r}d:real-spaceperiod;Sundayetal.:q=\\ 2\pi /d,k=2\pi /\lambda ,q=k-k^{\prime };Huetal.:\lambda =\\ 0.729\mathring{\mathrm{A}},detectorat700cmwith78.75\mu \mathrm{m}\\ pixels;standardform(background\\ knowledge,Als-NielsenandMcMorrow)\end{array}$ $(computed:0.729\mathring{\mathrm{A}}is17.0\mathrm{keV})$
Grating pitch from peak spacing	$q_{x,n}=\frac{2\pi n}{P}$	$\Delta \left(2\theta \right)\approx \frac{\lambda }{P}$	$q_{x,n}:positionofdiffractionordern$ $\begin{array}{r}P:pitch;Huetal.:verticalridges\\ spacedby2\pi /D,2000\mathring{\mathrm{A}}modelrepeat;Wuet\\ al.:periodicityalong{q}_{x}at{q}_z=0,pitch\\ 237.1+/-0.5and330.5+/-0.5\mathrm{nm},\\ sub-nanometreprecision;Sundayetal.:\\ 128\mathrm{nm}periodicitywithnominal32\mathrm{nm}\\ pitch\end{array}$ $\begin{array}{r}(computed:P=32\mathrm{nm}gives\Delta q=0.196\mathrm{n}{\mathrm{m}}^{-1}\\ and2.28\mathrm{mrad},15.9\mathrm{mm}or202pixelsat7\\ \mathrm{m};330.5+/-0.5\mathrm{nm}is0.15percent\\ precision)\end{array}$
Intensity model	$I\left(\mathbf{q}\right)={\left|F\left(\mathbf{q}\right)\right|}^2{e}^{-q^2{\sigma }_{DW}^2}$	$F\left(\mathbf{q}\right)=\int \rho \left(\mathbf{r}\right)e^{-i\mathbf{q}\cdot \mathbf{r}}dV$	$F:formfactor$ $\rho :electron-densitycontrast$ $\begin{array}{r}{\sigma }_{DW}:Debye-Wallerroughness;Sundayetal.:\\ theserelations,six-trapezoidstack\\ model,MCMCuncertainties;Wuetal.:\\ amplitude,Debye-Wallerandroughness\\ terms,3to5\mathrm{nm}RMSroughnessfor\\ high-LERresist;Huetal.:contoursas\\ theformfactorofanisolatedtrapezoid\end{array}$ $\begin{array}{r}(computedforP=32\mathrm{nm}:\sigma =3\mathrm{nm}keeps71\\ percentoforder1and25percentof\\ order2;\sigma =1\mathrm{nm}keeps38percentof\\ order5)\end{array}$

Profile Reconstruction

Name	Formula	Related Formulae	Variables/Constants
Trapezoid form factor	$F\left(q_x,q_z\right)={\int }_0^{H}w\left(z\right)\mathrm{sinc}\left(\frac{q_{x}w\left(z\right)}{2}\right)e^{i{q}_{z}z}dz$	$w\left(z\right)=w_0-2z\tan \beta$	$H:lineheight,w(z):widthatheightz$ $w_0:bottomwidth$ $\begin{array}{r}\beta :sidewallanglefromvertical;trapezoid\\ integral(backgroundknowledge,\\ Als-NielsenandMcMorrow);Huetal.:\\ major-maximalinestiltingwithsidewall\\ angle;Wuetal.:linewidthsmeasurable\\ from10to300\mathrm{nm}\end{array}$ $\begin{array}{r}(myreading:anonzero\beta tiltsthe\\ intensitystreaksinthe{q}_x-q_{z}mapby\beta ,\\ whichishowlinearfitsofthemaxima\\ givetheangle)\end{array}$
Height and profile from sample rotation	$q_z=q_x\tan \omega$	$\Delta q_z=\frac{2\pi }{H}$	$\omega :samplerotationaboutthelineaxis$ $\begin{array}{r}\Delta q_z:fringeperiodalong{q}_z;Huetal.:\\ rotationfrom-20to20{}^{\circ }inabout1{}^{\circ }\\ steps,heightabout318\mathrm{nm}from2\pi /\Delta q_z;\\ Sundayetal.:variable-angletransmission\\ over45{}^{\circ }in1{}^{\circ }steps,121imagesat10\\ s(about30minutes),widthbelow100\mathrm{nm}\\ andheightabove400\mathrm{nm}\end{array}$ $\begin{array}{r}(computed:H=318\mathrm{nm}gives\Delta q_z=0.0198\\ \mathrm{n}{\mathrm{m}}^{-1};fora400\mathrm{nm}lineatP=32\mathrm{nm},one\\ fringeneeds4.6{}^{\circ }ofrotation)\end{array}$
Sidewall angle and width results	${\beta }_A=5.7\pm 0.2^{\circ }(\text{maxima fits}),\approx 5.0^{\circ }(\text{intensity fit})$	$w_{p0}=148,w_{p2}=153\pm 1,w_{p4}=160\pm 1\mathrm{nm}$	$\begin{array}{r}Huetal.:sampleA5.7+/-0.2{}^{\circ }from\\ linearfitsofmajormaximaandabout5.0\\ {}^{\circ }fromsecond-orderintensity,sampleB\\ about3.0{}^{\circ },nominal150\mathrm{nm}lines,\\ qualitativeagreementwith30{}^{\circ }tiltedSEM\\ crosssections;Wuetal.:IBMfocus\\ seriesat330.5\mathrm{nm}pitchwithheights550,\\ 605+/-10and460+/-10\mathrm{nm}andsidewall\\ angles2+/-0.5to5.6+/-0.5{}^{\circ }(seethe\\ Q/A)\end{array}$ $\begin{array}{r}(computed:widthgrows12\mathrm{nm}andsidewall\\ angleby3.6{}^{\circ }across0.4\mu \mathrm{m}ofdefocus)\end{array}$

Practical Limits

Name	Formula	Related Formulae	Variables/Constants
Transmission through the wafer	$T=\exp \left[-\left(\frac{\mu }{\rho }\right)\rho \frac{d}{\cos \varphi }\right]$	$E>13\text{to}15\mathrm{keV}(\text{Si transparent})$	$\begin{array}{r}T:transmission,\mu /\rho :massattenuation\\ coefficient\end{array}$ $d:waferthickness$ $\begin{array}{r}\varphi :tiltangle;Huetal.:pathd/\cos \varphi ,d\\ about1\mathrm{mm},about20percenttransmission\\ at0.729\mathring{\mathrm{A}};Wuetal.:Sitransparent\\ above13\mathrm{keV};Sundayetal.:hardX-rays\\ above15\mathrm{keV},idealabove20\mathrm{keV};Si\mu /\rho \\ of10.34,4.464and1.436\mathrm{c}{\mathrm{m}}^2/gat15,20\\ and30\mathrm{keV}(backgroundknowledge,NIST\\ XCOM)\end{array}$ $\begin{array}{r}(computed:775\mu \mathrm{m}Sitransmits15,45and\\ 77percentat15,20and30\mathrm{keV};1\mathrm{mm}at\\ 17\mathrm{keV}transmits18percent,matching\\ H{u}^{\prime }s20percent)\end{array}$
Source flux and measurement time	$t\propto \frac{N_{required}}{\Phi {\eta }_{scatter}}$	${\Phi }_{lab}\approx 2.2\times 10^6\text{photons/s};{\Phi }_{needed}=10^8\text{to}10^{10}$	$\Phi :incidentflux$ $\begin{array}{r}{\eta }_{scatter}:fractionscatteredintothepeaks;\\ Sundayetal.:labsource2.2\times 10^{6}photons/s\\ and300\mu \mathrm{m}beam,ideal1\times 10^{8}to1\times 10^{10}\\ photons/sfor10s,spotbelow100\mu \mathrm{m},\\ exposures0.1to20s;Wuetal.:\\ synchrotronspotabout40by40\mu \mathrm{m},1to5\\ spersample;Huetal.:beam120by180\mu \mathrm{m}\\ (seetheQ/A)\end{array}$ $\begin{array}{r}(computed:thelabfluxis45to4500\\ timesbelowtheideal)\end{array}$
Roughness: SAXS against SEM	${\sigma }_{SAXS}\text{includes sidewall and top surface}$	${\sigma }_{LER,SEM}\text{edge only}$	$\begin{array}{r}Wuetal.:CD-SAXSroughnessincludesboth\\ sidewallandtopsurface,differentfrom\\ SEMLER,3to5\mathrm{nm}RMSforhigh-LER\\ resist,a5\mathrm{nm}randomdeviationonone\\ sample;Huetal.:LERasapossiblecause\\ offasterintensitydecay;Sundayetal.:\\ validationagainstcross-sectionTEMwith\\ top-downSEMforcontext\end{array}$ $\begin{array}{r}(computed:a40\mu \mathrm{m}spotat32\mathrm{nm}pitch\\ covers1250lines,5cmoflinelength,\\ whileSEMsamplesmicrometresofafew\\ lines)\end{array}$

Important Questions

Ques: How does CD-SAXS measure the pitch of a grating?

From the spacing of diffraction orders in reciprocal space. The scattering vector is $q=4\pi \sin \theta /\lambda$, and a grating of pitch P gives orders at $q_x=2\pi n/P$ (Hu et al.; Sunday et al.; background knowledge, Als-Nielsen and McMorrow). For Hu’s 0.729 $\mathring{\mathrm{A}}$ beam, 17.0 $\mathrm{keV}$ (computed), a 32 $\mathrm{nm}$ pitch produces orders 0.196 $\mathrm{n}{\mathrm{m}}^{-1}$ apart, 2.28 $\mathrm{mrad}$ in angle and 15.9 $\mathrm{mm}$ apart on a detector 7 $\mathrm{m}$ away, about 202 pixels (computed). Wu et al. report pitches of 237.1 +/- 0.5 and 330.5 +/- 0.5 $\mathrm{nm}$, 0.15 per cent precision (computed), and sub-nanometre precision generally. Because pitch comes from peak positions alone, it is independent of the shape model, unlike width and sidewall angle. Reference: T. Hu, R. L. Jones, W.-L. Wu, E. K. Lin, Q. Lin and J. Quintana, ‘Small Angle X-ray Scattering Metrology for Sidewall Angle and Cross Section of Nanometer Scale Gratings’, IBM Research Report RC23206 (2004) (fetch-tool summary); D. F. Sunday, D. Windover, W. Wu and R. J. Kline, ‘Current Status of CDSAXS: Is it Fab-Ready?’, NIST MSED slides (fetch-tool summary; partly garbled); J. Als-Nielsen and D. McMorrow, Elements of Modern X-ray Physics, 2nd ed., Wiley 2011, and NIST XCOM attenuation data (background knowledge; not opened); W.-L. Wu, R. L. Jones, T. Hu, C. L. Soles and E. K. Lin, ‘Small Angle X-ray Scattering Metrology for Sidewall Angle and Cross Section of Nanometer Scale Line Gratings’, ULSI conference slides, Richardson TX (2005) (fetch-tool summary).

Ques: How do sample rotations reveal height and sidewall angle?

By sweeping each order through $q_z$. At rotation $\omega$, an order at $q_x$ samples $q_z=q_x\tan \omega$, so a tilt series maps the form factor along $q_z$ (background knowledge, Als-Nielsen and McMorrow). Fringes spaced $2\pi /H$ give height: Hu et al. found about 318 $\mathrm{nm}$, a period of 0.0198 $\mathrm{n}{\mathrm{m}}^{-1}$ (computed). For a 400 $\mathrm{nm}$ line at 32 $\mathrm{nm}$ pitch, one fringe needs about 4.6 ${}^{\circ }$ of rotation (computed), so Sunday’s 45 ${}^{\circ }$ range in 1 ${}^{\circ }$ steps samples many fringes. A sloped sidewall tilts the intensity streaks, and Hu et al. obtained sidewall angles from linear fits of the major maxima, rotating from -20 to 20 ${}^{\circ }$. Sunday et al. fit six stacked trapezoids with Monte Carlo uncertainty estimates. Reference: T. Hu, R. L. Jones, W.-L. Wu, E. K. Lin, Q. Lin and J. Quintana, ‘Small Angle X-ray Scattering Metrology for Sidewall Angle and Cross Section of Nanometer Scale Gratings’, IBM Research Report RC23206 (2004) (fetch-tool summary); D. F. Sunday, D. Windover, W. Wu and R. J. Kline, ‘Current Status of CDSAXS: Is it Fab-Ready?’, NIST MSED slides (fetch-tool summary; partly garbled); J. Als-Nielsen and D. McMorrow, Elements of Modern X-ray Physics, 2nd ed., Wiley 2011, and NIST XCOM attenuation data (background knowledge; not opened).

Ques: Is the sidewall angle of Hu’s sample A 5.7 or 5.0 degrees?

Hu et al. report 5.7 +/- 0.2 ${}^{\circ }$ from linear fits of the major maxima and about 5.0 ${}^{\circ }$ from fitting the second-order peak intensity. The maxima-tilt method depends only on the direction of the streaks and is less sensitive to roughness and intensity calibration, so 5.7 ${}^{\circ }$ is more likely, with the intensity fit pulled lower by roughness damping that the simple model omits (my reading). Wu’s later deck shows similar scatter: angles of 2 +/- 0.5 ${}^{\circ }$ and 5.6 +/- 0.5 ${}^{\circ }$ across a focus series, with uncertainties stated inconsistently. A 3 $\mathrm{nm}$ Debye-Waller roughness already reduces order 2 at 32 $\mathrm{nm}$ pitch to 25 per cent (computed), so intensity-based angles need roughness in the model. Hu et al. found qualitative agreement with tilted SEM cross sections. Reference: T. Hu, R. L. Jones, W.-L. Wu, E. K. Lin, Q. Lin and J. Quintana, ‘Small Angle X-ray Scattering Metrology for Sidewall Angle and Cross Section of Nanometer Scale Gratings’, IBM Research Report RC23206 (2004) (fetch-tool summary); W.-L. Wu, R. L. Jones, T. Hu, C. L. Soles and E. K. Lin, ‘Small Angle X-ray Scattering Metrology for Sidewall Angle and Cross Section of Nanometer Scale Line Gratings’, ULSI conference slides, Richardson TX (2005) (fetch-tool summary).

Ques: Why do synchrotron and laboratory CD-SAXS times differ so much?

Because of incident flux. Wu et al. used a synchrotron spot of about 40 by 40 $\mu \mathrm{m}$ with 1 to 5 s per sample, whereas Sunday et al. report a laboratory source of about $2.2\times 10^6$ $photons/s$ in a 300 $\mu \mathrm{m}$ beam, needing 121 images of 10 s, about 30 minutes, per scan. Their ideal fab source needs $1\times 10^8$ to $1\times 10^{10}$ $photons/s$ for a 10 s measurement, 45 to 4500 times more flux (computed), and a spot below 100 $\mu \mathrm{m}$ to fit test pads. The wafer itself also absorbs: a 775 $\mu \mathrm{m}$ silicon wafer transmits 15, 45 and 77 per cent at 15, 20 and 30 $\mathrm{keV}$ (computed from background attenuation data, NIST XCOM), and Hu’s 1 $\mathrm{mm}$ at 17 $\mathrm{keV}$, 18 per cent, matches their 20 per cent (computed). Reference: W.-L. Wu, R. L. Jones, T. Hu, C. L. Soles and E. K. Lin, ‘Small Angle X-ray Scattering Metrology for Sidewall Angle and Cross Section of Nanometer Scale Line Gratings’, ULSI conference slides, Richardson TX (2005) (fetch-tool summary); D. F. Sunday, D. Windover, W. Wu and R. J. Kline, ‘Current Status of CDSAXS: Is it Fab-Ready?’, NIST MSED slides (fetch-tool summary; partly garbled); J. Als-Nielsen and D. McMorrow, Elements of Modern X-ray Physics, 2nd ed., Wiley 2011, and NIST XCOM attenuation data (background knowledge; not opened); T. Hu, R. L. Jones, W.-L. Wu, E. K. Lin, Q. Lin and J. Quintana, ‘Small Angle X-ray Scattering Metrology for Sidewall Angle and Cross Section of Nanometer Scale Gratings’, IBM Research Report RC23206 (2004) (fetch-tool summary).

Ques: How does CD-SAXS roughness compare with SEM line-edge roughness?

They measure different things. Wu et al. state that CD-SAXS roughness includes both sidewall and top surface, unlike SEM line-edge roughness, which follows only the edge; they report 3 to 5 $\mathrm{nm}$ RMS for a high-LER resist. In SAXS, roughness enters through the Debye-Waller factor $e^{-q^2{\sigma }^2}$ (Sunday et al.), suppressing higher orders: at 32 $\mathrm{nm}$ pitch, 1 $\mathrm{nm}$ roughness leaves 38 per cent of order 5 (computed), so rough lines show few orders. SAXS averages over every line in the beam, about 1250 lines and 5 cm of total edge length for a 40 $\mu \mathrm{m}$ spot at 32 $\mathrm{nm}$ pitch (computed), while SEM samples short lengths of a few lines (my reading). Hu et al. note LER as one cause of faster-than-expected intensity decay. Reference: W.-L. Wu, R. L. Jones, T. Hu, C. L. Soles and E. K. Lin, ‘Small Angle X-ray Scattering Metrology for Sidewall Angle and Cross Section of Nanometer Scale Line Gratings’, ULSI conference slides, Richardson TX (2005) (fetch-tool summary); D. F. Sunday, D. Windover, W. Wu and R. J. Kline, ‘Current Status of CDSAXS: Is it Fab-Ready?’, NIST MSED slides (fetch-tool summary; partly garbled); T. Hu, R. L. Jones, W.-L. Wu, E. K. Lin, Q. Lin and J. Quintana, ‘Small Angle X-ray Scattering Metrology for Sidewall Angle and Cross Section of Nanometer Scale Gratings’, IBM Research Report RC23206 (2004) (fetch-tool summary).

References

1. T. Hu, R. L. Jones, W.-L. Wu, E. K. Lin, Q. Lin and J. Quintana, “Small Angle X-ray Scattering Metrology for Sidewall Angle and Cross Section of Nanometer Scale Gratings,” IBM Research Report RC23206, 2004. Available: https://dominoweb.draco.res.ibm.com/reports/rc23206.pdf
2. D. F. Sunday, D. Windover, W. Wu and R. J. Kline, “Current Status of CDSAXS: Is it Fab-Ready?,” NIST Materials Science and Engineering Division slides. Available: https://www.nist.gov/document/klinepdf
3. J. Als-Nielsen and D. McMorrow, Elements of Modern X-ray Physics, 2nd ed., Wiley, 2011; NIST XCOM photon cross-section database (cited from background knowledge).
4. W.-L. Wu, R. L. Jones, T. Hu, C. L. Soles and E. K. Lin, “Small Angle X-ray Scattering Metrology for Sidewall Angle and Cross Section of Nanometer Scale Line Gratings,” ULSI conference, Richardson, TX, 2005. Available: https://www.nist.gov/document/wu2005pdf

Related pages in this vault: X-Ray Reflectometer (XRR), CD-SEM, X-Ray Diffractometer (XRD and HRXRD), Optical CD Scatterometer (OCD), Spectroscopic Ellipsometer, Transmission Electron Microscope (TEM and STEM), X-Ray Topography, Scanning Electron Microscope (SEM), Rutherford Backscattering Spectrometer (RBS).