Light-Matter Interaction for Optical Inspection

1. Theory

• Definition: Optical inspection and metrology read how light is reflected, absorbed and scattered by a wafer; the complex index $\tilde{n}=n+ik$ sets reflectance and absorption, and small defects scatter light in proportion to $d^6/{\lambda }^4$ (Almeida; Chew and Jiao, Lecture 23; Hattori).
• Reflection: At normal incidence $R=[(n-1{)}^2+k^2]/[(n+1{)}^2+k^2]$, and at oblique incidence TE and TM reflect differently, with TM reflection vanishing at the Brewster angle (Almeida; Chew and Jiao, Lecture 17).
• Absorption: Intensity decays as $e^{-\alpha z}$ with $\alpha =4\pi k/\lambda$; silicon absorbs within nanometres in the deep UV but over about a hundred nanometres at 400 $\mathrm{nm}$ (Almeida; Polyanskiy; Nagakubo et al.).
• Scattering regimes: Particles much smaller than the wavelength, size parameter below 1, follow Rayleigh scattering; larger ones need Mie theory (NASA page; Chew and Jiao, Lecture 23).
• Background: Surface roughness scatters as the Bennett-Porteus total integrated scatter, producing haze that hides small particles (Eckhardt Optics; Hattori).

2. Working Principle

• Illumination: CW or pulsed lasers from 532 to 266 $\mathrm{nm}$ illuminate the wafer, at normal or oblique incidence (Hattori).
• Collection: Scattered light is collected away from the specular beam, and its intensity is calibrated against polystyrene latex spheres (Hattori).
• Discrimination: Particles are separated from haze and crystal-originated pits by signal size, incidence geometry and polarization behaviour (Hattori; NASA page).
• Films: Interference in transparent films modulates the reflected and scattered light, which reflectometry and ellipsometry use to measure thickness (Almeida).

3. Applications

• Unpatterned wafer inspection: Particle and haze maps (Hattori; Laser-Scattering Particle Inspection page).
• Patterned wafer inspection: Choice of wavelength and polarization for defect contrast (Patterned-Wafer Optical Inspection page; Chew and Jiao, Lecture 17).
• Thin-film metrology: Thickness from interference fringes (Almeida).
• Surface quality: Roughness from total integrated scatter (Eckhardt Optics).

Reflection and Absorption

Name	Formula	Related Formulae	Variables/Constants
Normal-incidence reflectance	$R={\left|\frac{\tilde{n}-1}{\tilde{n}+1}\right|}^2=\frac{{\left(n-1\right)}^2+k^2}{{\left(n+1\right)}^2+k^2}$	$\tilde{n}=n+ik$	$\tilde{n}:complexindex$ $\begin{array}{r}k:extinctioncoefficient;Almeida:this\\ form;Polyanskiy:Sin=1.010,k=2.909\\ at206.6\mathrm{nm};Nagakuboetal.:n=5.52,k\\ =0.268at401.6\mathrm{nm}\end{array}$ $\begin{array}{r}(computed:R=0.68at206.6\mathrm{nm}and0.48\\ at401.6\mathrm{nm}forbareSi)\end{array}$
Absorption and penetration depth	$\alpha =\frac{4\pi k}{\lambda },{\delta }_p=\frac{1}{\alpha }$	$I\left(z\right)=I_0\left(1-R\right)e^{-\alpha z}$	$\alpha :absorptioncoefficient$ $\begin{array}{r}{\delta }_p:1/eintensitydepth;Almeidaand\\ Polyanskiy:\alpha =4\pi k/\lambda ;Almeida:Bee{r}^{\prime }s\\ lawwithreflectionlosses;Nagakuboet\\ al.:measured83800\mathrm{c}{\mathrm{m}}^{-1}at401.6\mathrm{nm}\end{array}$ $\begin{array}{r}(computed:Si{\delta }_p=5.7\mathrm{nm}at206.6\mathrm{nm}and\\ 119\mathrm{nm}at401.6\mathrm{nm};4\pi k/\lambda fromNagakub{o}^{\prime }s\\ kgives83800\mathrm{c}{\mathrm{m}}^{-1},matchingtheir\\ measurement)\end{array}$
Oblique incidence and polarization	$R_{TE}=\frac{{\beta }_{1z}/{\mu }_1-{\beta }_{2z}/{\mu }_2}{{\beta }_{1z}/{\mu }_1+{\beta }_{2z}/{\mu }_2}$	$\sin {\theta }_B=\sqrt{\frac{{\epsilon }_2}{{\epsilon }_1+{\epsilon }_2}}\iff \tan {\theta }_B=\frac{n_2}{n_1}$	${\beta }_{iz}:normalwavenumbercomponents$ $\begin{array}{r}{\theta }_B:Brewsteranglewhere{R}_{TM}vanishes;Chew\\ andJiao,Lecture17,Eqs.2.13,2.15and\\ 3.7;tangentform(computedfromEq.3.7\\ fornon-magneticmedia)\end{array}$ $\begin{array}{r}(assumed:SiO2n=1.46;computed:{\theta }_B=\\ 55.6degreesfromair)\end{array}$

Particle Scattering

Name	Formula	Related Formulae	Variables/Constants
Rayleigh scattering cross section	${\Sigma }_s=\frac{8\pi a^2}{3}{\left(\frac{{\epsilon }_s-\epsilon }{{\epsilon }_s+2\epsilon }\right)}^2{\left(\beta a\right)}^4$	${\Sigma }_s\propto \frac{d^6}{{\lambda }^4}$	$a:particleradius,d=2a,\beta =2\pi /\lambda$ $\begin{array}{r}{\epsilon }_s:particlepermittivity;ChewandJiao,\\ Lecture23,Eq.1.21,validforamuch\\ smallerthan\lambda ;Hattori:d^6/{\lambda }^{4}intensity\end{array}$ $\begin{array}{r}(assumed:PSLindex1.59;computed:the\\ permittivityfactoris0.114;halvingd\\ cutssignal64times,halving\lambda raisesit\\ 16times)\end{array}$
Size parameter and Mie regime	$x=\frac{2\pi a}{\lambda }$	$x<1\text{Rayleigh},{\Sigma }_s\to \pi a^2\text{for large particles}$	$\begin{array}{r}x:sizeparameter;NASApage:Rayleigh\\ belowx=1,Mieabovewithweaker\\ wavelengthdependenceanddepolarization;\\ ChewandJiao:full-waveMiesolutionfor\\ largerparticles,crosssectiontendingto\\ \pi a^2\end{array}$ $\begin{array}{r}(computed:at266\mathrm{nm},x=0.35fora30\mathrm{nm}\\ particleand1.18for100\mathrm{nm};x=1at85\\ \mathrm{nm}diameter)\end{array}$
Equal-signal particle size	$d_2=d_1{\left(\frac{{\lambda }_2}{{\lambda }_1}\right)}^{2/3}$	${\Sigma }_s\left(d_1,{\lambda }_1\right)={\Sigma }_s\left(d_2,{\lambda }_2\right)$	$\begin{array}{r}fromthe{d}^6/{\lambda }^{4}scaling(computed);Hattori:\\ 30\mathrm{nm}PSLdetectedat266\mathrm{nm},commercial\\ toolsabout48to60\mathrm{nm},sourcesfrom532\\ to266\mathrm{nm}\end{array}$ $\begin{array}{r}(computed:a60\mathrm{nm}particleat532\mathrm{nm}\\ scattersasmuchasa37.8\mathrm{nm}particleat\\ 266\mathrm{nm})\end{array}$

Background and Films

Name	Formula	Related Formulae	Variables/Constants
Surface roughness scatter (haze)	$TIS=R_0\left[1-\exp \left(-{\left(\frac{4\pi R_q\cos {\theta }_i}{\lambda }\right)}^2\right)\right]$	$TIS\approx R_0{\left(\frac{4\pi R_q\cos {\theta }_i}{\lambda }\right)}^2(R_q\ll \lambda )$	$R_q:rmsroughness$ $R_0:specularreflectance$ $\begin{array}{r}{\theta }_i:incidenceangle;EckhardtOptics:\\ Bennett-Porteusform,morescatterat\\ short\lambda andnormalincidence;Hattori:\\ hazefrommicroroughnessmaskssmall\\ particles;small-roughnessexpansion\\ (computed)\end{array}$ $\begin{array}{r}(computed:R_q=0.1\mathrm{nm}atnormalincidence\\ givesTIS/R_0=2.2\times 10^{-5}at266\mathrm{nm}and5.6\times 10^{-6}\\ at532\mathrm{nm})\end{array}$
Particle-to-haze contrast	$\frac{{\Sigma }_s}{TIS}\propto \frac{{\lambda }^{-4}}{{\lambda }^{-2}}={\lambda }^{-2}$	$\text{oblique incidence lowers }\cos {\theta }_i\text{ in TIS}$	$\begin{array}{r}ratiooftheRayleighandsmall-roughness\\ scalings(computed;myreadingthatthis\\ setsthegainofshorterwavelengthsover\\ aroughbackground);EckhardtOptics:less\\ scatteratgrazingincidence\end{array}$ $\begin{array}{r}(computed:goingfrom532to266\mathrm{nm}raises\\ particlesignal16timesandhaze4times,\\ a4timescontrastgain)\end{array}$
Thin-film interference	$2nd\cos \theta =\left(m+\frac{1}{2}\right)\lambda (\text{reflection minimum})$	$d_{AR}=\frac{\lambda }{4n}$	$n,d:filmindexandthickness$ $\begin{array}{r}m:order;Almeida:pathdifferenceandthe\\ quarter-waveconditionforafilmoflower\\ indexthanthesubstrate\end{array}$ $\begin{array}{r}(assumed:SiO2n=1.46;computed:d_{AR}=91\\ \mathrm{nm}at532\mathrm{nm})\end{array}$

Important Questions

Ques: Why does a shorter laser wavelength detect smaller particles?

Because Rayleigh scattering grows as ${\lambda }^{-4}$. Chew and Jiao give the cross section of a small sphere as $(8\pi a^2/3)[({\epsilon }_s-\epsilon )/({\epsilon }_s+2\epsilon ){]}^2(\beta a{)}^4$, so it scales as $d^6/{\lambda }^4$, as Hattori also states. Halving the diameter cuts the signal 64 times, while halving the wavelength raises it 16 times (computed). Equal signal therefore needs $d_2=d_1({\lambda }_2/{\lambda }_1{)}^{2/3}$: a 60 $\mathrm{nm}$ particle at 532 $\mathrm{nm}$ scatters like a 37.8 $\mathrm{nm}$ one at 266 $\mathrm{nm}$ (computed). Hattori calls shorter wavelength the most effective route to sensitivity and reports 30 $\mathrm{nm}$ PSL spheres detected on bulk silicon with a 266 $\mathrm{nm}$ CW laser, against about 48 to 60 $\mathrm{nm}$ for the commercial tools he surveyed (see also the Laser-Scattering Particle Inspection page). Reference: W. C. Chew and D. Jiao, ECE 604 Lecture 23, ‘Rayleigh Scattering’ and ‘Mie Scattering’, Purdue University (29 November 2018): induced dipole p = 4 pi eps (eps_s - eps)/(eps_s + 2 eps) a^3 E_i, scattering cross section Sigma_s = (8 pi a^2/3)((eps_s - eps)/(eps_s + 2 eps))^2 (beta a)^4 growing as the fourth power of frequency, valid for particles much smaller than the wavelength, full-wave Mie solution needed for larger particles, cross section tending to pi a^2 (fetch-tool summary); T. Hattori (Sony), ‘Challenges of Smaller Particle Detection on Both Bulk-Silicon and SOI Wafers’, NIST presentation (March 2005): Rayleigh intensity scaling as d^6/lambda^4, shorter wavelength the most effective route to sensitivity, haze from microroughness, COPs, microscratches and organics masking small particles, PSL reference spheres, 30 nm PSL on bulk Si and 40 nm on SOI with a 266 nm, 100 mW CW laser, commercial limits about 48, 50 and 60 nm, sources at 532, 488, 405, 355 and 266 nm, particle and COP classification 53 to 93 per cent (fetch-tool summary); Laser-Scattering Particle Inspection page of this vault.

Ques: When does the d-to-the-sixth rule stop working?

When the particle is no longer much smaller than the wavelength. The NASA page places Rayleigh scattering below a size parameter $x=2\pi a/\lambda$ of 1, with Mie scattering above it showing much weaker wavelength dependence and depolarizing the light. Chew and Jiao state that their result assumes particles much smaller than the wavelength and that a full-wave Mie solution is needed beyond that, with the cross section tending to $\pi a^2$ rather than growing without limit. At 266 $\mathrm{nm}$, a 30 $\mathrm{nm}$ particle has $x$ = 0.35 but a 100 $\mathrm{nm}$ particle has 1.18, and $x$ = 1 falls at 85 $\mathrm{nm}$ diameter (computed). Sizing larger particles from a $d^6$ calibration would therefore under-estimate them, and PSL-equivalent sizes are only nominal there (my reading). Reference: NASA Goddard, ‘Rayleigh Scattering and Mie Scattering’ lecture page (author not named): Rayleigh scattering for size parameter x below 1 with very strong wavelength dependence, Mie scattering for larger particles with much weaker wavelength dependence, Mie scattering depolarizing light (fetch-tool summary); W. C. Chew and D. Jiao, ECE 604 Lecture 23, ‘Rayleigh Scattering’ and ‘Mie Scattering’, Purdue University (29 November 2018): induced dipole p = 4 pi eps (eps_s - eps)/(eps_s + 2 eps) a^3 E_i, scattering cross section Sigma_s = (8 pi a^2/3)((eps_s - eps)/(eps_s + 2 eps))^2 (beta a)^4 growing as the fourth power of frequency, valid for particles much smaller than the wavelength, full-wave Mie solution needed for larger particles, cross section tending to pi a^2 (fetch-tool summary).

Ques: Why is haze the limit for small-particle detection?

Because roughness scatter forms a background that small particles must exceed. Hattori lists microroughness, crystal-originated pits, micro-scratches and organic films as sources of variable haze that mask small particles. Eckhardt Optics gives the Bennett-Porteus total integrated scatter, $R_0[1-\exp (-(4\pi R_q\cos {\theta }_i/\lambda {)}^2)]$, which for smooth wafers becomes $R_0(4\pi R_q\cos {\theta }_i/\lambda {)}^2$ (computed). For 0.1 $\mathrm{nm}$ roughness at normal incidence that is $2.2\times 10^{-5}{R}_0$ at 266 $\mathrm{nm}$ and $5.6\times 10^{-6}{R}_0$ at 532 $\mathrm{nm}$ (computed). Haze grows as ${\lambda }^{-2}$ and particle signal as ${\lambda }^{-4}$, so moving from 532 to 266 $\mathrm{nm}$ improves particle-to-haze contrast four times (computed; my reading). The same source notes that oblique incidence scatters less. Hattori’s 30 $\mathrm{nm}$ limit on bulk silicon against 40 $\mathrm{nm}$ on SOI fits this picture, the film stack adding background (my reading). Reference: T. Hattori (Sony), ‘Challenges of Smaller Particle Detection on Both Bulk-Silicon and SOI Wafers’, NIST presentation (March 2005): Rayleigh intensity scaling as d^6/lambda^4, shorter wavelength the most effective route to sensitivity, haze from microroughness, COPs, microscratches and organics masking small particles, PSL reference spheres, 30 nm PSL on bulk Si and 40 nm on SOI with a 266 nm, 100 mW CW laser, commercial limits about 48, 50 and 60 nm, sources at 532, 488, 405, 355 and 266 nm, particle and COP classification 53 to 93 per cent (fetch-tool summary); Eckhardt Optics, ‘Optical Scattering and Surface Roughness’ web resource: Bennett-Porteus TIS = R0 [1 - exp(-(4 pi Rq cos theta_i/lambda)^2)], scatter proportional to reflectance and higher at short wavelengths and normal incidence, superpolished surfaces Rq 0.1 to 0.2 nm (fetch-tool summary).

Ques: How deep does inspection light probe into silicon?

From a few nanometres to over a hundred, depending on wavelength. Almeida gives $\alpha =4\pi k/\lambda$ and Beer’s-law decay. The refractiveindex.info record cited by Polyanskiy lists $k$ = 2.909 at 206.6 $\mathrm{nm}$, so the 1/e depth is 5.7 $\mathrm{nm}$ (computed). Nagakubo et al. measured $n$ = 5.52 and $k$ = 0.268 at 401.6 $\mathrm{nm}$ with $\alpha$ = 83800 $\mathrm{c}{\mathrm{m}}^{-1}$, a depth of 119 $\mathrm{nm}$ (computed); recomputing $\alpha$ from their $k$ gives the same 83800 $\mathrm{c}{\mathrm{m}}^{-1}$ (computed), confirming the relation. Bare silicon reflects 68 per cent at 206.6 $\mathrm{nm}$ and 48 per cent at 401.6 $\mathrm{nm}$ (computed from Almeida’s formula). Deep-UV light therefore senses only the top few nanometres, while visible light can reach buried features under thin films (my reading). Reference: R. Almeida, Optical and Photonic Glasses, Lecture 14, ‘Refraction of Light in Absorbing Materials’, International Materials Institute, Lehigh University (2005): R = |(n* - 1)/(n* + 1)|^2 = ((n - 1)^2 + k^2)/((n + 1)^2 + k^2), alpha = 4 pi k/lambda, T = (1 - R)^2 exp(-alpha x), antireflection condition 2nd cos theta = (m + 1/2) lambda giving d = lambda/(4n) (fetch-tool summary); M. N. Polyanskiy, ‘Refractiveindex.info database of optical constants’, Scientific Data 11, 94 (2024): example record Si (Aspnes) with n = 1.010, k = 2.909 at 206.6 nm and n = 1.083, k = 2.982 at 210.1 nm, alpha = 4 pi k/lambda (fetch-tool summary); A. Nagakubo, H. Ogi and M. Hirao, ‘Refractive index and extinction coefficient of Si at 400 nm between 10 and 300 K’, Jpn. J. Appl. Phys. 54, 128001 (2015), Osaka repository: n = 5.5215 and kappa = 0.2679 at 401.6 nm and 295 K, alpha = 83800 per cm (fetch-tool summary).

Ques: How do polarization and thin films change what an optical tool sees?

They change both the specular background and the defect signal. Chew and Jiao give separate Fresnel coefficients for TE and TM waves and show that TM reflection vanishes at the Brewster angle, $\sin {\theta }_B=\sqrt{{\epsilon }_2/({\epsilon }_1+{\epsilon }_2)}$, which is $\tan {\theta }_B=n_2/n_1$, 55.6 degrees for an assumed oxide of index 1.46 (computed). The NASA page notes that small particles scatter most strongly perpendicular to the polarization and that Mie scattering depolarizes light. Almeida gives the thin-film reflection minimum, $2nd\cos \theta =(m+\frac{1}{2})\lambda$, so a 91 $\mathrm{nm}$ oxide is anti-reflecting at 532 $\mathrm{nm}$ (computed). Film stacks can therefore boost or null the field at a defect, and inspection recipes tune wavelength and polarization per layer (my reading; Patterned-Wafer Optical Inspection page). Reference: W. C. Chew and D. Jiao, ECE 604 Lecture 17, Purdue University (30 October 2018): Fresnel coefficients R_TE = (beta1z/mu1 - beta2z/mu2)/(beta1z/mu1 + beta2z/mu2) and R_TM with eps, Brewster condition sin theta1 = sqrt(eps2/(eps1 + eps2)) with theta1 + theta2 = pi/2, critical angle sin^-1(n2/n1), evanescent wave beyond it (fetch-tool summary); NASA Goddard, ‘Rayleigh Scattering and Mie Scattering’ lecture page (author not named): Rayleigh scattering for size parameter x below 1 with very strong wavelength dependence, Mie scattering for larger particles with much weaker wavelength dependence, Mie scattering depolarizing light (fetch-tool summary); R. Almeida, Optical and Photonic Glasses, Lecture 14, ‘Refraction of Light in Absorbing Materials’, International Materials Institute, Lehigh University (2005): R = |(n* - 1)/(n* + 1)|^2 = ((n - 1)^2 + k^2)/((n + 1)^2 + k^2), alpha = 4 pi k/lambda, T = (1 - R)^2 exp(-alpha x), antireflection condition 2nd cos theta = (m + 1/2) lambda giving d = lambda/(4n) (fetch-tool summary); Patterned-Wafer Optical Inspection page of this vault.

References

1. W. C. Chew and D. Jiao, ECE 604 Lecture 23, “Rayleigh scattering” and “Mie scattering,” Purdue University, 2018. Available: https://engineering.purdue.edu/wcchew/ece604f18/latex%20lecture%20notes/LectureNotes23.pdf
2. W. C. Chew and D. Jiao, ECE 604 Lecture 17, Purdue University, 2018. Available: https://engineering.purdue.edu/wcchew/ece604f18/latex%20lecture%20notes/LectureNotes17.pdf
3. R. Almeida, Optical and Photonic Glasses, Lecture 14, “Optical properties continued: refraction of light in absorbing materials,” International Materials Institute, Lehigh University, 2005. Available: https://www.lehigh.edu/IMI/teched/OPG/lecture14.pdf
4. NASA Goddard Space Flight Center, “Rayleigh scattering and Mie scattering,” lecture page. Available: https://acd-ext.gsfc.nasa.gov/anonftp/acd/daac_ozone/Lecture4/Text/Lecture_4/raymie.html
5. 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
6. Eckhardt Optics, “Optical scattering and surface roughness.” Available: https://www.eckop.com/resources/scatterometer-resources/optical-scattering-versus-surface-roughness/
7. A. Nagakubo, H. Ogi and M. Hirao, “Refractive index and extinction coefficient of Si at 400 nm between 10 and 300 K,” Japanese Journal of Applied Physics, vol. 54, 128001, 2015. Available: https://ir.library.osaka-u.ac.jp/repo/ouka/all/84532/JpnJApplPhys_54_12_128001.pdf
8. M. N. Polyanskiy, “Refractiveindex.info database of optical constants,” Scientific Data, vol. 11, 94, 2024. Available: https://www.nature.com/articles/s41597-023-02898-2.pdf
9. Laser-Scattering Particle Inspection page of this vault.
10. Patterned-Wafer Optical Inspection page of this vault.

Related pages in this vault: Laser-Scattering Particle Inspection, Patterned-Wafer Optical Inspection, Spectroscopic Ellipsometer, Spectral Reflectometer, Optical CD Scatterometer (OCD), High-NA DUV Immersion Scanner, Electron-Matter Interaction, Defect Detection Fundamentals, Spin Coater and Resist Track.