Spectral Reflectometer

1. Theory

• Definition: A spectral reflectometer shines light, usually at normal incidence, on a film and records reflectance against wavelength; interference between reflections from the top and bottom of the film creates fringes whose spacing and amplitude give thickness and, with a model, optical constants (Agilent; EPFL).
• Fringe spacing: The number of fringes in a wavenumber span is proportional to the optical thickness $nd$ (Agilent; background knowledge).
• Model fitting: For thin films or multilayers, the full spectrum is fitted with the Airy reflectance model (background knowledge, Fujiwara).
• Relation to ellipsometry: Reflectometry uses intensity only, with no phase, so it is simpler and faster but less sensitive to very thin films (background knowledge; Spectroscopic Ellipsometer page).

2. Working Principle

• Optics: A broadband source, a fibre or microscope reflection probe, and a spectrometer (PhysicsOpenLab; EPFL).
• Calibration: Spectra are normalized to a reference of known reflectance, such as bare silicon, and checked against traceable standards (EPFL; background knowledge).
• Analysis: Fringe counting for thick transparent films, regression for thin films and stacks (Agilent; background knowledge).
• Mapping: Motorized stages measure many points across a wafer (EPFL).

3. Applications

• Dielectric and resist thickness: Oxide, nitride and photoresist films after deposition or spin coating (EPFL; background knowledge).
• CMP and etch monitoring: Remaining film thickness, including in-situ endpoint (background knowledge).
• Coatings: Polymer, hard and anti-reflection coatings (Agilent; PhysicsOpenLab).
• Thin metals and semiconductors: Thin metal films and semiconductor layers with known n,k (EPFL).

Thin-Film Interference

Name	Formula	Related Formulae	Variables/Constants
Single-film reflectance	$R={\left|\frac{r_{01}+r_{12}{e}^{-i2\beta }}{1+r_{01}{r}_{12}{e}^{-i2\beta }}\right|}^2$	$\beta =\frac{2\pi nd\cos {\theta }_1}{\lambda },r_{ij}=\frac{n_i-n_j}{n_i+n_j}(\text{normal incidence})$	$R:reflectance$ $r_{01},r_{12}:interfacereflectioncoefficients$ $\beta :phasethickness$ $n:filmindex$ $d:thickness$ ${\theta }_1:angleinthefilm$ $\begin{array}{r}\lambda :wavelength;Airyform(background\\ knowledge,Fujiwara;samemodelasthe\\ SpectroscopicEllipsometerpage);Agilent:\\ interferenceofthetworeflections\\ dependingontheiropticalpath\\ difference;PhysicsOpenLab:angleusually\\ 0degrees\end{array}$ $\begin{array}{r}(computed:baresiliconwithn=3.88\\ reflects34.8percentatnormal\\ incidence)\end{array}$
Interference extrema	$2nd\cos {\theta }_1=m\lambda \text{or}\left(m+\frac{1}{2}\right)\lambda$	$d_{QW}=\frac{\lambda }{4n}$	$m:interferenceorder$ $\begin{array}{r}{d}_{QW}:quarter-wavethickness;which\\ conditiongivesamaximumdependson\\ whetherthefilmindexliesbetweenthe\\ ambientandsubstrateindices(background\\ knowledge)\end{array}$ $\begin{array}{r}(computed:aquarter-waveSiO2film,n=\\ 1.46,at633\mathrm{nm}is108\mathrm{nm})\end{array}$
Thickness from fringe counting	$d=\frac{\Delta m}{2\Delta \tilde{\nu }\sqrt{n^2-{\sin }^2\theta }}$	$d=\frac{{\lambda }_1{\lambda }_2}{2n\left({\lambda }_1-{\lambda }_2\right)}(\text{adjacent maxima, normal incidence})$	$\Delta m:numberoffringesintheregion$ $\Delta \tilde{\nu }:wavenumberspan({\tilde{\nu }}_1-{\tilde{\nu }}_2)$ $\begin{array}{r}\theta :angleofincidence;Agilent:the\\ variablesd,m,n,\theta andthewavenumber\\ region,apolymercoatingwithn=1.51at\\ 7degreesgiving4.95\mu \mathrm{m}from16fringes\\ between420and765\mathrm{nm}(or756\mathrm{nm}ina\\ caption;seetheQ/A);PhysicsOpenLab:\\ thicknessfromthenumberofmaxima\\ betweentwowavelengthswithknownindex;\\ formulas(backgroundknowledge,checked\\ againsttheAgilentexample)\end{array}$ $\begin{array}{r}(computed:\Delta \tilde{\nu }=10738\mathrm{c}{\mathrm{m}}^{-1}for420to765\\ \mathrm{nm}givesd=4.95\mu \mathrm{m})\end{array}$

Calibration and Limits

Name	Formula	Related Formulae	Variables/Constants
Normalization	$R_s=\frac{I_s-I_{dark}}{I_{ref}-I_{dark}}{R}_{ref}$	$R_{ref}=0.348(\text{bare Si at normal incidence})$	$R_s:samplereflectance$ $I_s,I_{ref},I_{dark}:sample,referenceanddarkspectra$ $\begin{array}{r}{R}_{ref}:knownreferencereflectance;method\\ (backgroundknowledge);EPFL:\\ NIST-traceablecalibrationstandardof\\ 1000\mathrm{nm}SiO2onSi,materiallibrary\\ extendablewithimportedn,kfiles\end{array}$ $\begin{array}{r}(computed:asamplegiving80percentof\\ thesiliconreferencesignalreflects27.9\\ percent)\end{array}$
Thick-film limit	$\Delta {\lambda }_{fringe}=\frac{{\lambda }^2}{2nd}$	$\Delta {\lambda }_{fringe}\gtrsim \text{several}\times \delta {\lambda }_{spec}$	$\Delta {\lambda }_{fringe}:fringespacing$ $\begin{array}{r}\delta {\lambda }_{spec}:spectrometerresolution;relation\\ (backgroundknowledge);EPFL:F54range\\ about20\mathrm{nm}toabout40\mu \mathrm{m}initstableand\\ 20\mathrm{nm}to45\mu \mathrm{m}initsintroduction(seethe\\ Q/A),380to1050\mathrm{nm};Agilent:2\mathrm{nm}\\ bandwidthand1\mathrm{nm}dataintervalover400\\ to800\mathrm{nm}\end{array}$ $\begin{array}{r}(computed:a40\mu \mathrm{m}SiO2filmhasfringes\\ 8.6\mathrm{nm}apartat1000\mathrm{nm}butonly1.4\mathrm{nm}\\ apartat400\mathrm{nm})\end{array}$
Thin-film limit and model fitting	$d\ll \frac{{\lambda }_{min}}{4n}\Rightarrow \text{no full fringe; fit amplitude of }R\left(\lambda \right)$	${\chi }^2={\sum }_j{\left[R_j^{mod}\left(d,n\right)-R_j^{exp}\right]}^2$	${\lambda }_{min}:shortestwavelength$ $\begin{array}{r}{\chi }^2:fitresidual;approach(background\\ knowledge);EPFL:minimumabout20\mathrm{nm},\\ material-andobjective-dependent,\\ accuracy2\mathrm{nm}or0.2percent,precision\\ 0.2\mathrm{nm},stability0.05\mathrm{nm}\end{array}$ $\begin{array}{r}(computed:at380\mathrm{nm}aquarterwavein\\ SiO2is65\mathrm{nm},soa20\mathrm{nm}filmshowsless\\ thanathirdofafringe)\end{array}$

Instrument Practice

Name	Formula	Related Formulae	Variables/Constants
Spot size and mapping	$D_{spot}=\frac{D_{ap}}{M}$	$t_{map}\approx t_0+N_{pts}{t}_{pt}$	$D_{spot}:measuredspot$ $D_{ap}:fieldaperture$ $M:objectivemagnification$ $t_{map}:maptime$ $\begin{array}{r}{N}_{pts}:numberofpoints;EPFL:250\mu \mathrm{m}aperture\\ giving250\mu \mathrm{m}at5xand17\mu \mathrm{m}at15x,500\mu \mathrm{m}\\ aperturegiving500and33\mu \mathrm{m},r-theta\\ stagewith5pointsin5\mathrm{s},25in25\mathrm{s}and\\ 56in29\mathrm{s},wafersto200\mathrm{mm};relations\\ (myreadingofthelistedvalues)\end{array}$ $\begin{array}{r}(computed:250/15=16.7\mu \mathrm{m},matching17\mu \mathrm{m};\\ the56-pointtimeimpliesabout0.5\mathrm{s}per\\ pointatfullspeed)\end{array}$
Measured examples	$d_{PVC}\approx 10\mu \mathrm{m},d_{sheet}=70\mu \mathrm{m}(\text{nominal }100),d_{AR}\approx 8\mu \mathrm{m}$	$\frac{d_{meas}}{d_{nom}}=\frac{n_{assumed}}{n_{true}}(\text{index error only})$	$\begin{array}{r}PhysicsOpenLab:foodfilm,plasticsheet,\\ LCDanti-reflectioncoating,interference\\ filterbelow1\mu \mathrm{m},micanear10\mu \mathrm{m}andsoap\\ filmsof2to8\mu \mathrm{m}lastingover10\mathrm{s},using\\ areflectionprobe,lightsourceand\\ spectrometer;index-errorrelation\\ (backgroundknowledge)\end{array}$ $\begin{array}{r}(computed:a30percentlowresultwould\\ needtheindextobeoverestimatedby43\\ percent,soanominal-thicknessor\\ fringe-samplingcauseismorelikely)\end{array}$

Important Questions

Ques: How does a spectral reflectometer find film thickness?

From interference fringes in the reflectance spectrum. Light reflected from the top surface and from the film-substrate interface adds constructively or destructively depending on their optical path difference, so reflectance oscillates with wavelength (Agilent). Extrema occur where $2nd\cos {\theta }_1$ equals a whole or half-integer number of wavelengths, and the full spectrum follows the Airy formula (background knowledge). For a transparent film of known index, the number of fringes in a wavenumber span gives thickness directly; for thinner films or stacks, the spectrum is fitted with a layer model. Spectra are normalized to a known reference, often bare silicon at 34.8 per cent (computed). The F54 at EPFL covers 380 to 1050 $\mathrm{nm}$ and measures dielectrics, semiconductors and thin metals. Reference: A. R. Hind and L. Chomette, ‘The determination of thin film thickness using reflectance spectroscopy’, Agilent Technologies application note, 2011 (thickness formula not legible; variables listed) (fetch-tool summary); EPFL Center of MicroNanoTechnology (CMi), ‘Filmetrics F54’ equipment page (fetch-tool summary); H. Fujiwara, Spectroscopic Ellipsometry: Principles and Applications, Wiley 2007 (background knowledge; not opened).

Ques: Did Agilent’s example count fringes between 420 and 765 nm or 420 and 756 nm?

The note gives both: its results text says 16 fringes between 420 and 765 $\mathrm{nm}$, while a figure caption says 420 to 756 $\mathrm{nm}$. Using $d=\Delta m/[2\Delta \tilde{\nu }\sqrt{n^2-{\sin }^2\theta }]$ with its index of 1.51 and 7 degree incidence, 420 to 765 $\mathrm{nm}$ spans 10738 $\mathrm{c}{\mathrm{m}}^{-1}$ and gives 4.95 $\mu \mathrm{m}$, exactly the thickness the note reports, whereas 756 $\mathrm{nm}$ would give 5.02 $\mu \mathrm{m}$ (computed). So 765 $\mathrm{nm}$ is the more likely correct value and the caption the typo (my reading). The example also shows how fringe counting depends on the index: a 1 per cent index error gives a 1 per cent thickness error. The measurement used a 2 $\mathrm{nm}$ bandwidth, 1 $\mathrm{nm}$ data interval and 400 to 800 $\mathrm{nm}$ range. Reference: A. R. Hind and L. Chomette, ‘The determination of thin film thickness using reflectance spectroscopy’, Agilent Technologies application note, 2011 (thickness formula not legible; variables listed) (fetch-tool summary).

Ques: What sets the thinnest film a reflectometer can measure?

The absence of fringes. When the film is much thinner than a quarter wave, the spectrum shows no full oscillation and thickness must be fitted from the reflectance level and slope. At the 380 $\mathrm{nm}$ short limit of the F54, a quarter wave in SiO2 is 65 $\mathrm{nm}$, so a 20 $\mathrm{nm}$ film shows less than a third of a fringe (computed). EPFL quotes a minimum of about 20 $\mathrm{nm}$, material- and objective-dependent, with an accuracy of 2 $\mathrm{nm}$ or 0.2 per cent and precision of 0.2 $\mathrm{nm}$. Below this, thickness and index cannot be separated from intensity alone, and ellipsometry, which also measures phase, is preferred (background knowledge; Spectroscopic Ellipsometer page). Deep-ultraviolet systems push the limit lower by using shorter wavelengths. Reference: EPFL Center of MicroNanoTechnology (CMi), ‘Filmetrics F54’ equipment page (fetch-tool summary); Spectroscopic Ellipsometer page of this vault.

Ques: Is the F54 upper limit 40 or 45 microns?

EPFL’s page gives about 40 $\mu \mathrm{m}$ in its specification table and 45 $\mu \mathrm{m}$ in its introduction, both described as material- and objective-dependent. The limit comes from spectral resolution: fringe spacing is ${\lambda }^2/(2nd)$, so a 40 $\mu \mathrm{m}$ SiO2 film has fringes 8.6 $\mathrm{nm}$ apart at 1000 $\mathrm{nm}$ but only 1.4 $\mathrm{nm}$ apart at 400 $\mathrm{nm}$ (computed), and fringes must span several spectrometer resolution elements to be counted. The specification table is the more likely authoritative figure, with 45 $\mu \mathrm{m}$ possible for favourable materials (my reading). Thicker films, for example in MEMS, are better measured with near-infrared reflectometers or FTIR, which use longer wavelengths and wider fringes (background knowledge). Reference: EPFL Center of MicroNanoTechnology (CMi), ‘Filmetrics F54’ equipment page (fetch-tool summary); A. R. Hind and L. Chomette, ‘The determination of thin film thickness using reflectance spectroscopy’, Agilent Technologies application note, 2011 (thickness formula not legible; variables listed) (fetch-tool summary).

Ques: How is a reflectometer calibrated?

Against a reference of known reflectance and a thickness standard. Each sample spectrum is divided by a reference spectrum after dark subtraction and multiplied by the reference’s known reflectance; with bare silicon as reference, a sample giving 80 per cent of its signal reflects 27.9 per cent (background knowledge; computed). Errors in the reference reflectance or in the dark level shift the whole spectrum and bias thin-film fits, which rely on absolute reflectance. Thickness accuracy is then verified with a traceable film; EPFL uses a NIST-traceable 1000 $\mathrm{nm}$ SiO2 on Si standard. Optical constants come from a material library, which EPFL notes can be extended by importing n,k files, so the result is only as good as the assumed dispersion. Reference: EPFL Center of MicroNanoTechnology (CMi), ‘Filmetrics F54’ equipment page (fetch-tool summary); H. Fujiwara, Spectroscopic Ellipsometry: Principles and Applications, Wiley 2007 (background knowledge; not opened).

Ques: What spatial resolution and mapping speed are possible?

Set by the microscope optics and stage. On the EPFL F54, a 250 $\mu \mathrm{m}$ field aperture gives a 250 $\mu \mathrm{m}$ spot with a 5x objective and 17 $\mu \mathrm{m}$ with 15x, consistent with the aperture divided by magnification, 16.7 $\mu \mathrm{m}$ (computed), while a 500 $\mu \mathrm{m}$ aperture gives 500 and 33 $\mu \mathrm{m}$. An r-theta stage measures 5 points in 5 $\mathrm{s}$, 25 in 25 $\mathrm{s}$ and 56 in 29 $\mathrm{s}$ on wafers up to 200 $\mathrm{mm}$, so the incremental time at full speed is about 0.5 $\mathrm{s}$ per point (my reading of the listed values). Smaller spots allow measurement on patterned test pads, but they collect less light and need the pad to be larger than the spot. Mapping reveals deposition non-uniformity across the wafer. Reference: EPFL Center of MicroNanoTechnology (CMi), ‘Filmetrics F54’ equipment page (fetch-tool summary).

Ques: How reliable are simple home-built reflectometers?

Good to the order of magnitude, but sensitive to the assumed index and to sampling. PhysicsOpenLab built one from a reflection probe, light source and spectrometer and found about 10 $\mu \mathrm{m}$ for a food film of nominal 10 $\mu \mathrm{m}$, about 8 $\mu \mathrm{m}$ for an LCD anti-reflection coating, about 10 $\mu \mathrm{m}$ for mica, 2 to 8 $\mu \mathrm{m}$ for soap films and below 1 $\mu \mathrm{m}$ for an interference filter. A plastic sheet of nominal 0.1 $\mathrm{mm}$ read 70 $\mu \mathrm{m}$, 30 per cent low. Since thickness scales inversely with the assumed index, that error would need the index overestimated by 43 per cent (computed), which is implausible, so an incorrect nominal value or undersampled fringes are the more likely causes (my reading). Commercial tools add calibration, dispersion libraries and fitting. Reference: PhysicsOpenLab, ‘Thin Film Reflectometry’ (equation as image) (fetch-tool summary).

References

1. A. R. Hind and L. Chomette, “The determination of thin film thickness using reflectance spectroscopy,” Agilent Technologies application note, 2011. Available: https://www.agilent.com/cs/library/applications/uv90.pdf
2. EPFL Center of MicroNanoTechnology (CMi), “Filmetrics F54,” equipment page. Available: https://www.epfl.ch/research/facilities/cmi/equipment/metrology/filmetrics-f54/
3. H. Fujiwara, Spectroscopic Ellipsometry: Principles and Applications. Chichester: Wiley, 2007 (cited from background knowledge).
4. Spectroscopic Ellipsometer, page in this vault.
5. PhysicsOpenLab, “Thin Film Reflectometry.” Available: https://physicsopenlab.org/?p=16808

Related pages in this vault: Spectroscopic Ellipsometer, FTIR Spectrometer, UV-Vis Spectrophotometer, Optical CD Scatterometer (OCD), X-Ray Reflectometer (XRR), White-Light Interferometer, Optical Microscope, Terahertz Time-Domain Spectrometer, Photoreflectance Spectrometer, Stylus Profiler, Picosecond Ultrasonic Film Metrology, Optical Emission Spectroscopy Endpoint (OES), Laser Interferometric Endpoint, Thermal Oxidation Furnace, CMP Polisher, Light-Matter Interaction for Optical Inspection, Spin Coater and Resist Track.