Spectroscopic Ellipsometer

1. Theory

• Definition: A spectroscopic ellipsometer measures the change in polarization of light reflected obliquely from a sample, expressed as $\Psi$ and $\Delta$ through $\rho =r_p/r_s=\tan \Psi e^{i\Delta }$, over a range of wavelengths (HORIBA; Woollam).
• Model-based: Thickness and optical constants are not measured directly; they come from fitting a layer model to the data (HORIBA; Woollam).
• Sensitivity: Because $\Delta$ is a phase, it responds to films down to sub-monolayer thickness (Woollam).
• Dispersion models: Cauchy, Sellmeier, Lorentz, Tauc-Lorentz and Cody-Lorentz functions describe how n and k vary with wavelength (Woollam).

2. Working Principle

• Optics: A light source, polarizer, the sample at oblique incidence, an analyzer and a spectrometer; a rotating element or phase modulator encodes the polarization (Woollam; HORIBA).
• Angle: Measurements near the Brewster angle of the substrate, about 70 degrees for silicon, give the largest sensitivity (background knowledge; computed).
• Regression: Model parameters are adjusted to minimize the mean squared error between measured and modelled spectra (Woollam).
• Extensions: Mueller-matrix and depolarization measurements, and in-situ monitoring of growth (HORIBA).

3. Applications

• Gate and passivation films: Thickness of oxides, nitrides and high-k layers (Woollam; HORIBA).
• Optical constants: n and k of new materials for device and lithography design (HORIBA).
• Composition and crystallinity: Alloy fraction, band gap and amorphous or crystalline phase (HORIBA).
• Porosity: Low-k porosity with ellipsometric porosimetry, linked to the Wire Capacitance and Low-k Dielectrics page (background knowledge).

Polarization and Reflection

Name	Formula	Related Formulae	Variables/Constants
Ellipsometric ratio	$\rho =\frac{r_p}{r_s}=\tan \Psi e^{i\Delta }$	$I_s=\sin 2\Psi \sin \Delta ,I_c=\sin 2\Psi \cos \Delta ,I_c^{\prime }=\cos 2\Psi$	$\rho :complexreflectanceratio$ $\begin{array}{r}{r}_p,r_s:Fresnelreflectioncoefficients\\ forlightpolarizedparalleland\\ perpendiculartotheplaneofincidence\end{array}$ $\Psi :amplitude-ratioangle$ $\begin{array}{r}\Delta :phasedifference,I_s,I_c,I_c^{\prime }:\\ phase-modulatedellipsometersignals;\\ HORIBA:theseequations,I_{s}with{I}_{c}giving\\ \Delta over0to360degreesand{I}_{s}with{I}_c^{\prime }\\ giving\Psi over0to90degrees;Woollam:\Psi \\ and\Delta asamplituderatioandphase\\ difference,rotating-analyzer,\\ rotating-polarizer,rotating-compensator\\ andphase-modulationconfigurations\end{array}$ $\begin{array}{r}(computed:baresiliconat633\mathrm{nm}and70\\ degrees,N=3.882-0.019i,gives\Psi =\\ 10.57and\Delta =179.2degrees)\end{array}$
Fresnel coefficients and Snell’s law	$r_p=\frac{N_1\cos {\theta }_0-N_0\cos {\theta }_1}{N_1\cos {\theta }_0+N_0\cos {\theta }_1},r_s=\frac{N_0\cos {\theta }_0-N_1\cos {\theta }_1}{N_0\cos {\theta }_0+N_1\cos {\theta }_1}$	$N_0\sin {\theta }_0=N_1\sin {\theta }_1,\tan {\theta }_B=\frac{n_1}{n_0}$	$\begin{array}{r}N=n-ik:complexrefractiveindex(n\\ index,kextinctioncoefficient)\end{array}$ ${\theta }_0,{\theta }_1:incidenceandrefractionangles$ $\begin{array}{r}{\theta }_B:Brewsterangle;Woollam:Snel{l}^{\prime }slaw\\ andFresnelcoefficientsasitsEqs.7and\\ 8(formulasnotlegibleinthecopyread),\\ complexindexanddielectricfunction;\\ HORIBA:N=n-ik;formulas(background\\ knowledge,Fujiwara)\end{array}$ $\begin{array}{r}(computed:{\theta }_B=75.6degreesforsiliconat\\ 633\mathrm{nm},whichiswhyanglesof65to75\\ degreesareused)\end{array}$
Pseudo-dielectric function	$⟨\epsilon ⟩={\sin }^2{\theta }_0\left[1+{\tan }^2{\theta }_0{\left(\frac{1-\rho }{1+\rho }\right)}^2\right]$	$\epsilon =N^2={\left(n-ik\right)}^2={\epsilon }_1-i{\epsilon }_2$	$\begin{array}{r}⟨\epsilon ⟩:pseudo-dielectricfunction,exactfor\\ abare,smoothsubstrate\end{array}$ $\begin{array}{r}{\epsilon }_1,{\epsilon }_2:realandimaginarydielectric\\ function;Woollam:pseudooptical\\ constantsforbulksamples(itsEq.12)\\ andtheindex-dielectricrelation;form\\ (backgroundknowledge,Fujiwara)\end{array}$ $\begin{array}{r}(computed:N=3.882-0.019igives\epsilon =\\ 15.07-0.148i)\end{array}$

Thin Films

Name	Formula	Related Formulae	Variables/Constants
Thin-film three-phase model	$r_{012}=\frac{r_{01}+r_{12}{e}^{-i2\beta }}{1+r_{01}{r}_{12}{e}^{-i2\beta }}$	$\beta =\frac{2\pi d}{\lambda }{N}_1\cos {\theta }_1$	$\begin{array}{r}{r}_{01},r_{12}:Fresnelcoefficientsofthe\\ ambient-filmandfilm-substrateinterfaces\\ (pors)\end{array}$ $\beta :filmphasethickness$ $d:filmthickness$ $\begin{array}{r}\lambda :wavelength;Woollam:phasethickness\\ (itsEq.9),interferenceofmultiple\\ waves,\Delta sensitivetosub-monolayerfilms;\\ formulas(backgroundknowledge,Fujiwara)\end{array}$ $\begin{array}{r}(computedwithn=1.457SiO2onSiat633\\ \mathrm{nm},70degrees:eachnanometrechanges\Delta \\ byabout2.9degreesbut\Psi byonly0.014;\\ 10\mathrm{nm}gives\Psi =11.48,\Delta =152.0,and100\\ \mathrm{nm}gives41.04,79.8)\end{array}$
Interference period	$d_{period}=\frac{\lambda }{2\sqrt{n_1^2-{\sin }^2{\theta }_0}}$	$\text{solutions repeat every }{d}_{period}\text{at a single wavelength}$	$\begin{array}{r}{d}_{period}:filmthicknessforonefullcycleof\\ \Psi and\Delta ;relationfrom\beta =\pi (background\\ knowledge);Woollam:asinglewavelength\\ givesonlytwodatavaluesandtherefore\\ atmosttwounknowns,spectroscopicdata\\ givehundredsofpoints,anexamplefit\\ withaglobalminimumat749\mathrm{nm}anda\\ localminimumnear0.45\mu \mathrm{m}\end{array}$ $\begin{array}{r}(computed:284\mathrm{nm}forSiO2at633\mathrm{nm}and\\ 70degrees)\end{array}$
Penetration depth	${\delta }_p=\frac{\lambda }{4\pi k}$	$\alpha =\frac{4\pi k}{\lambda },I=I_0{e}^{-\alpha z}$	${\delta }_p:depthatwhichintensityfallsto1/e$ $\alpha :absorptioncoefficient$ $\begin{array}{r}z:depth;Woollam:extinctioncoefficient\\ andabsorptioncoefficient(itsEq.5),\\ Bee{r}^{\prime }slaw(Eq.6),metalsmeasurableup\\ totypicallyabout100\mathrm{nm};forms\\ (backgroundknowledge)\end{array}$ $\begin{array}{r}(computed:k=1at500\mathrm{nm}gives{\delta }_p=40\\ \mathrm{nm})\end{array}$

Dispersion Models and Fitting

Name	Formula	Related Formulae	Variables/Constants
Cauchy and Sellmeier dispersion	$n\left(\lambda \right)=A+\frac{B}{{\lambda }^2}+\frac{C}{{\lambda }^4}$	${\epsilon }_1\left(\lambda \right)=1+{\sum }_j\frac{A_j{\lambda }^2}{{\lambda }^2-{\lambda }_j^2}$	$A,B,C:Cauchycoefficients$ $\begin{array}{r}{A}_j,{\lambda }_j:Sellmeieramplitudesandresonance\\ wavelengths;Woollam:Cauchy(Eq.13)not\\ constrainedbyKramers-Kronigconsistency\\ andabletogiveunphysicaldispersion,\\ Sellmeier(Eq.14)enforcingit;formulas\\ (backgroundknowledge)\end{array}$ $\begin{array}{r}(assumed:A=1.45,B=0.0036\mu {\mathrm{m}}^2;\\ computed:n(633\mathrm{nm})=1.459)\end{array}$
Lorentz and Tauc-Lorentz oscillators	$\epsilon \left(E\right)={\epsilon }_{1,offset}+\frac{AB{E}_c}{E_c^2-E^2-iBE}$	${\epsilon }_2^{TL}\left(E\right)=\frac{A{E}_{0}C{\left(E-E_g\right)}^2}{{\left(E^2-E_0^2\right)}^2+C^2{E}^2}\frac{1}{E}(E>E_g),{\epsilon }_2^{TL}=0(E\le E_g)$	$A:amplitude$ $BandC:broadening$ $E_c,E_0:centreenergy$ $E_g:bandgap$ $\begin{array}{r}{\epsilon }_{1,offset}:offset;Woollam:Lorentzoscillator\\ parameters(Eq.15),Tauc-Lorentzand\\ Cody-Lorentzincludingthebandgap;\\ HORIBA:bandgap,compositionand\\ crystallinityamongoutputs;Tauc-Lorentz\\ formafterJellisonandModine(background\\ knowledge)\end{array}$ $\begin{array}{r}(computed:E=1240/\lambda gives1.96\mathrm{eV}at633\\ \mathrm{nm})\end{array}$
Regression and roughness	$MSE=\frac{1}{2M-P}{\sum }_{j=1}^M\left[{\left(\frac{{\Psi }_j^{mod}-{\Psi }_j^{exp}}{{\sigma }_{\Psi ,j}}\right)}^2+{\left(\frac{{\Delta }_j^{mod}-{\Delta }_j^{exp}}{{\sigma }_{\Delta ,j}}\right)}^2\right]$	${\sum }_i{f}_i\frac{{\epsilon }_i-\epsilon }{{\epsilon }_i+2\epsilon }=0(\text{Bruggeman EMA})$	$MSE:meansquarederror$ $M:numberofwavelengths$ $P:numberoffitparameters$ $\sigma :measurementuncertainties$ $\begin{array}{r}{f}_i,{\epsilon }_i:volumefractionsanddielectric\\ functionsofmixedconstituents;Woollam:\\ MSEtocomparecurves,unknownsnot\\ exceedingtheinformationinthedata;\\ HORIBA:roughness,interfaces,gradingand\\ depolarizationfromthemodel;MSEform\\ (onecommondefinition)andEMA\\ (backgroundknowledge)\end{array}$ $\begin{array}{r}(computed:200wavelengthsand4\\ parametersgive2M-P=396degreesof\\ freedom)\end{array}$

Important Questions

Ques: What do Psi and Delta mean?

They describe how reflection changes the polarization of light. Light polarized between the p and s planes is split by the sample into two components with different reflection coefficients; $\tan \Psi$ is the ratio of their amplitudes and $\Delta$ the difference of their phase shifts, so $\rho =r_p/r_s=\tan \Psi e^{i\Delta }$ (HORIBA; Woollam). For bare silicon at 633 $\mathrm{nm}$ and 70 degrees, $\Psi =10.57$ and $\Delta =179.2$ degrees (computed). Because they are ratios, they are insensitive to source intensity fluctuations. Phase-modulated instruments measure $\sin 2\Psi \sin \Delta$, $\sin 2\Psi \cos \Delta$ and $\cos 2\Psi$, which together fix $\Delta$ over the full 0 to 360 degrees and $\Psi$ over 0 to 90 degrees (HORIBA). Sign conventions for $\Delta$ differ between instruments (background knowledge). Reference: HORIBA Scientific, ‘Spectroscopic Ellipsometry’ technology page (fetch-tool summary); J. A. Woollam Co., ‘Ellipsometry Tutorial’ (Gaertner ellipsometer handout copy, Purdue University; equations lost in extraction) (fetch-tool summary).

Ques: Why is ellipsometry so sensitive to very thin films?

Because a thin film mainly shifts the phase, which $\Delta$ measures directly. For SiO2 on silicon at 633 $\mathrm{nm}$ and 70 degrees, each nanometre of oxide changes $\Delta$ by about 2.9 degrees but $\Psi$ by only 0.014 degrees (computed with the three-phase model). Since $\Delta$ can be measured to a few hundredths of a degree, thickness resolution well below 0.1 $\mathrm{nm}$ follows (background knowledge), consistent with Woollam’s statement that $\Delta$ is sensitive to sub-monolayer films. Sensitivity is highest near the substrate’s Brewster angle, 75.6 degrees for silicon (computed), where $r_p$ is smallest. For very thin films, however, thickness and refractive index are strongly correlated, so the index is usually fixed and only thickness fitted (background knowledge, Fujiwara). Reference: J. A. Woollam Co., ‘Ellipsometry Tutorial’ (Gaertner ellipsometer handout copy, Purdue University; equations lost in extraction) (fetch-tool summary); H. Fujiwara, Spectroscopic Ellipsometry: Principles and Applications, Wiley 2007 (background knowledge; not opened).

Ques: Why is spectroscopic data needed rather than a single wavelength?

Because one wavelength gives only two numbers. Woollam notes a single-wavelength ellipsometer measures $\Psi$ and $\Delta$ only, so it can find at most two unknowns, and the solution repeats every interference period, 284 $\mathrm{nm}$ for SiO2 at 633 $\mathrm{nm}$ and 70 degrees (computed). A thicker film can therefore match several thicknesses. Spectroscopic data supply hundreds of points for a few unknowns; Woollam’s example fit finds the global minimum at 749 $\mathrm{nm}$ but also a local minimum near 0.45 $\mu \mathrm{m}$, showing that the starting guess still matters. More wavelengths also allow the dispersion of n and k to be fitted, and multiple angles add independent information. The number of unknowns must never exceed the information in the data. Reference: J. A. Woollam Co., ‘Ellipsometry Tutorial’ (Gaertner ellipsometer handout copy, Purdue University; equations lost in extraction) (fetch-tool summary).

Ques: Which dispersion model should be used?

One that matches the material and respects physics. The Cauchy formula $A+B/{\lambda }^2+C/{\lambda }^4$ works for transparent films such as SiO2, with assumed coefficients 1.45 and 0.0036 $\mu {\mathrm{m}}^2$ giving 1.459 at 633 $\mathrm{nm}$ (computed), but Woollam warns that it is not Kramers-Kronig consistent and can produce unphysical dispersion, especially near absorption. The Sellmeier form enforces that consistency. Absorbing materials need oscillators: Lorentz terms with amplitude, broadening and centre energy, and for amorphous semiconductors and high-k dielectrics the Tauc-Lorentz or Cody-Lorentz models, which include a band gap below which absorption vanishes (Woollam; Tauc-Lorentz form after Jellison and Modine, background knowledge). The fitted gap is then a model parameter, not a direct measurement. Reference: J. A. Woollam Co., ‘Ellipsometry Tutorial’ (Gaertner ellipsometer handout copy, Purdue University; equations lost in extraction) (fetch-tool summary); H. Fujiwara, Spectroscopic Ellipsometry: Principles and Applications, Wiley 2007 (background knowledge; not opened).

Ques: What thickness range can ellipsometry measure, a few microns or tens of microns?

The sources differ. Woollam says ellipsometry is typically used from sub-nanometre films to a few microns, with difficulty resolving oscillations above several tens of microns, while HORIBA quotes a few angstroms to several tens of microns. Both agree on the lower limit. The upper limit depends on spectral resolution: the interference period in wavenumber shrinks as thickness grows, and when it approaches the spectrometer bandwidth, the oscillations wash out (background knowledge). The more likely reading is that routine work covers up to a few microns, while tens of microns are possible with high-resolution or infrared instruments and transparent films (my reading). For absorbing films the limit is set instead by penetration depth; Woollam gives about 100 $\mathrm{nm}$ for metals. Reference: J. A. Woollam Co., ‘Ellipsometry Tutorial’ (Gaertner ellipsometer handout copy, Purdue University; equations lost in extraction) (fetch-tool summary); HORIBA Scientific, ‘Spectroscopic Ellipsometry’ technology page (fetch-tool summary); H. Fujiwara, Spectroscopic Ellipsometry: Principles and Applications, Wiley 2007 (background knowledge; not opened).

Ques: How are roughness and interfaces handled?

With effective-medium layers in the model. A rough surface or a mixed interface is represented as a layer whose dielectric function is a mixture of its constituents, usually calculated with the Bruggeman effective-medium approximation $\sum f_i({\epsilon }_i-\epsilon )/({\epsilon }_i+2\epsilon )=0$, for example 50 per cent film and 50 per cent void (background knowledge, Fujiwara). HORIBA lists surface roughness, interface thickness, grading and depolarization among the properties obtainable from the model. Adding layers adds parameters, so each must lower the mean squared error meaningfully without creating correlations; with 200 wavelengths and 4 parameters, the fit has 396 degrees of freedom (computed). Atomic force microscopy is often used to confirm the fitted roughness. Reference: HORIBA Scientific, ‘Spectroscopic Ellipsometry’ technology page (fetch-tool summary); H. Fujiwara, Spectroscopic Ellipsometry: Principles and Applications, Wiley 2007 (background knowledge; not opened).

Ques: How do absorbing films and metals limit the measurement?

Light must reach the bottom of the film and return. Intensity decays as $e^{-\alpha z}$ with $\alpha =4\pi k/\lambda$, giving a penetration depth $\lambda /(4\pi k)$: 40 $\mathrm{nm}$ for k = 1 at 500 $\mathrm{nm}$ (computed). Woollam gives about 100 $\mathrm{nm}$ as the typical maximum for metals, beyond which the film behaves like a bulk substrate and only its optical constants can be found. Semiconductors are transparent below their band gap, so thick layers are measured in the infrared or near-gap region, and thin layers in the ultraviolet where absorption is strong (background knowledge). For bulk samples the measured data convert directly to a pseudo-dielectric function; silicon at 633 $\mathrm{nm}$ gives $\epsilon =15.07-0.148i$ (computed). Reference: J. A. Woollam Co., ‘Ellipsometry Tutorial’ (Gaertner ellipsometer handout copy, Purdue University; equations lost in extraction) (fetch-tool summary); H. Fujiwara, Spectroscopic Ellipsometry: Principles and Applications, Wiley 2007 (background knowledge; not opened).

References

1. HORIBA Scientific, “Spectroscopic Ellipsometry,” technology page. Available: https://www.horiba.com/usa/scientific/technologies/spectroscopic-ellipsometry/spectroscopic-ellipsometry/
2. J. A. Woollam Co., “Ellipsometry Tutorial,” reproduced in a Gaertner ellipsometer handout, Purdue University Research Instrumentation Center. Available: https://www.chem.purdue.edu/ric/docs/Gaertner-Ellipsometer-Handout-2---JAWoollamCo-Ellipsometry-Tutorial.pdf
3. H. Fujiwara, Spectroscopic Ellipsometry: Principles and Applications. Chichester: Wiley, 2007 (cited from background knowledge).

Related pages in this vault: Spectral Reflectometer, Terahertz Time-Domain Spectrometer, Wire Capacitance and Low-k Dielectrics, Photoreflectance Spectrometer, X-Ray Reflectometer (XRR), UV-Vis Spectrophotometer, FTIR Spectrometer, Optical CD Scatterometer (OCD), MOS Capacitor, Picosecond Ultrasonic Film Metrology, Quartz Crystal Microbalance (QCM), Laser Interferometric Endpoint, GUM Uncertainty Budgets, Thermal Oxidation Furnace, Rapid Thermal Oxidation System, Light-Matter Interaction for Optical Inspection, LPCVD Furnace, PECVD Reactor, Atomic Layer Deposition (ALD) Reactor, Wet Etch Bench.