View source for Refractive index and extinction coefficient of thin film materials

{{Short description|Material science measurement method}}
{{Multiple issues|
{{Orphan|date=December 2023}}
{{Notability|date=August 2023}}
}}

A. R. Forouhi and I. Bloomer deduced dispersion equations for the [[Refractive index|refractive index, ''n'', and extinction coefficient, ''k'']], which were published in 1986<ref name=FB1986>{{cite journal|last=Forouhi|first=A.R.|author2=Bloomer, I. |title=Optical Dispersion Relations for Amorphous Semiconductors and Amorphous Dielectrics|journal=Physical Review B|date=1986|volume=34|issue=10|pages=7018–7026|doi=10.1103/physrevb.34.7018|pmid=9939354|bibcode = 1986PhRvB..34.7018F }}</ref> and 1988.<ref name=FB1988>{{cite journal|last=Forouhi|first=A.R.|author2=Bloomer, I. |title=Optical Properties of Crystalline Semiconductors and Dielectrics|journal=Physical Review B|date=1988|volume=38|issue=3|pages=1865–1874|doi=10.1103/physrevb.38.1865|pmid=9946471 |bibcode = 1988PhRvB..38.1865F }}</ref> The 1986 publication relates to amorphous materials, while the 1988 publication relates to crystalline. Subsequently, in 1991, their work was included as a chapter in ''The Handbook of Optical Constants''.<ref name=OpticalHandbook>{{cite book|last=Forouhi|first=A.R.|title=Handbook of Optical Constants II|date=1991|publisher=Academic Press|page=Chapter 7|author2=Bloomer, I. |editor=Palik, E.D.}}</ref> The [[Forouhi–Bloomer model|Forouhi–Bloomer dispersion equations]] describe how photons of varying energies interact with thin films. When used with a spectroscopic [[reflectometry]] tool, the Forouhi–Bloomer dispersion equations specify ''n'' and ''k'' for amorphous and crystalline materials as a function of photon energy ''E''. Values of ''n'' and ''k'' as a function of photon energy, ''E'', are referred to as the spectra of ''n'' and ''k'', which can also be expressed as functions of the wavelength of light, ''λ'', since ''E'' = ''hc''/''λ''. The symbol ''h'' is the [[Planck constant]] and ''c'', the speed of light in vacuum. Together, ''n'' and ''k'' are often referred to as the "optical constants" of a material (though they are not constants since their values depend on photon energy).

The derivation of the Forouhi–Bloomer dispersion equations is based on obtaining an expression for ''k'' as a function of photon energy, symbolically written as ''k''(''E''), starting from first principles quantum mechanics and solid state physics. An expression for ''n'' as a function of photon energy, symbolically written as ''n''(''E''), is then determined from the expression for ''k''(''E'') in accordance to the [[Kramers–Kronig relations]]<ref name=Roman>{{cite book|last=Roman|first=P.|title=Advanced Quantum Theory|url=https://archive.org/details/advancedquantumt0000roma|url-access=registration|date=1965|publisher=Addison-Wesley}}</ref> which states that ''n''(''E'') is the [[Hilbert transform]] of ''k''(''E'').

The Forouhi–Bloomer dispersion equations for ''n''(''E'') and ''k''(''E'') of amorphous materials are given as:
: <math> k(E) = \frac{A(E-E_\text{g})^2}{E^2-BE+C} \ </math>
: <math> n(E) = n(\infty)+\frac{(B_0 E + C_0 )}{E^2-BE+C} \ </math>

The five parameters ''A'', ''B'', ''C'', ''E''<sub>g</sub>, and ''n''(∞) each have physical significance.<ref name=FB1986 /><ref name=OpticalHandbook /> ''E''<sub>g</sub> is the optical energy band gap of the material. ''A'', ''B'', and ''C'' depend on the band structure of the material. They are positive constants such that 4''C'' − ''B''<sup>2</sup> > 0. Finally, ''n''(∞), a constant greater than unity, represents the value of ''n'' at ''E'' = ∞. The parameters ''B''<sub>0</sub> and ''C''<sub>0</sub> in the equation for ''n''(''E'') are not independent parameters, but depend on ''A'', ''B'', ''C'', and ''E''<sub>g</sub>. They are given by:
: <math> B_0 = \frac{A}{Q} \ \left (\frac{-B^2}{2} \ + E_gB - {E_g}^2 + C \right) </math>
: <math> C_0 = \frac{A}{Q} \ \left [({E_g}^2 + C) \frac{B}{2} \ - 2E_g C \right] </math>
where
: <math> Q = \frac{1}{2} \ (4C - B^2 )^{\frac{1}{2}} </math>

Thus, for amorphous materials, a total of five parameters are sufficient to fully describe the dependence of both ''n'' and ''k'' on photon energy, ''E''.

For crystalline materials which have multiple peaks in their ''n'' and ''k'' spectra, the Forouhi–Bloomer dispersion equations can be extended as follows:
: <math> k(E) = \sum_{i=1}^q \left [\frac{A_i(E - E_{g_i})^2}{E^2-B_iE+C_i} \right] </math>
: <math> n(E) = n(\infty)+\sum_{i=1}^q \left [\frac{B_{0_i}E+C_{0_i}}{E^2-B_iE+C_i} \right] </math>

The number of terms in each sum, ''q'', is equal to the number of peaks in the ''n'' and ''k'' spectra of the material. Every term in the sum has its own values of the parameters ''A'', ''B'', ''C'', ''E''<sub>g</sub>, as well as its own values of ''B''<sub>0</sub> and ''C''<sub>0</sub>. Analogous to the amorphous case, the terms all have physical significance.<ref name=FB1988 /><ref name=OpticalHandbook />

== Characterizing thin films ==
The refractive index (''n'') and extinction coefficient (''k'') are related to the interaction between a material and incident light, and are associated with refraction and absorption (respectively). They can be considered as the "fingerprint of the material". Thin film material coatings on various substrates provide important functionalities for the [[Microfabrication|microfabrication industry]], and the ''n'', ''k'', as well as the thickness, ''t'', of these thin film constituents must be measured and controlled to allow for repeatable [[Semiconductor device fabrication|manufacturing]].

The Forouhi–Bloomer dispersion equations for ''n'' and ''k'' were originally expected to apply to semiconductors and dielectrics, whether in amorphous, polycrystalline, or crystalline states. However, they have been shown to describe the ''n'' and ''k'' spectra of transparent conductors,<ref name=Torkaman>{{cite journal|last=Torkaman|first=N.M.|author2=Ganjkhanlou, Y. |author3=Kazemzad, M. |author4=Dabaghi, H.H. |author5= Keyanpour-Rad, M. |title=Crystallographic Parameters and Electro-Optical Constants in ITO Thin Films|journal=Materials Characterization|date=2010|volume=61|issue=3|pages=362–370 |doi=10.1016/j.matchar.2009.12.020}}</ref> as well as metallic compounds.<ref name=Lakhdar>{{cite journal|last=Lakhdar|first=M.H.|author2=Ouni, B. |author3=Amlouk, M. |title=Thickness Effect on the Structural and Optical Constants of Stibnite Thin Films Prepared by Sulfidation Annealing of Antimony Films|journal=Optik|date=2014|volume=125 |issue=10 |pages=2295–2301 |doi=10.1016/j.ijleo.2013.10.114 |bibcode=2014Optik.125.2295L }}</ref><ref name=Khanbashi>{{cite journal|last=Al-Khanbashi|first=H.A.|author2=Shirbeeny, W. |author3=Al-Ghamdi, A.A. |author4=Bronstein, L.M. |author5= Mahmoud, W.E. |title=Spectroscopic Ellipsometry of Zn1−xCuxO Thin Films Based on a Modified Sol–Gel Dip-Coating Technique|journal=Spectrochimica Acta Part A: Molecular and Biomolecular Spectroscopy|date=2014|volume=118|pages=800–805 |doi=10.1016/j.saa.2013.09.085|pmid=24157332|bibcode = 2014AcSpA.118..800A }}</ref><ref name=Nakamura>{{cite journal|last=Nakamura|first=T.|author2=Moriyama, T. |author3=Nabatova-Gabain, N. |author4= Adachi, S. |title=Emission Decay Rate of a Light Emitter on Thin Metal Films|journal=Japanese Journal of Applied Physics|date=2014|volume=53|issue=4|page=5201|doi=10.7567/jjap.53.045201 |bibcode = 2014JaJAP..53d5201N |s2cid=120461637 }}</ref><ref name=Winkler>{{cite journal|last=Winkler|first=M.T.|author2=Wang, W. |author3=Gunawan, O. |author4=Hovel, H.J. |author5=Todorova, T.K. |author6= Mitzi, D.B. |title=Optical Designs that Improve the Efficiency of Cu2ZnSn(S,Se)4 Solar Cells|journal=Energy & Environmental Science|date=2014|volume=7|issue=3|pages=1029–1036 |doi=10.1039/c3ee42541j}}</ref><ref name=Miao>{{cite journal|last=Miao|first=L.|author2=Su, L.F. |author3=Tanemura, S. |author4=Fisher, C.A.J. |author5=Zhao, L.L. |author6=Liang, Q. |author7= Xu, G. |title=Cost-Effective Nanoporous SiO2–TiO2 Coatings on Glass Substrates with Antireflective and Self-Cleaning Properties|journal=Applied Energy|date=2013|volume=112|pages=1198–1205 |doi=10.1016/j.apenergy.2013.03.043}}</ref><ref name=Zhang>{{cite journal|last=Zhang|first=F. |author2=Zhang, R.J. |author3=Zhang, D.X. |author4=Wang, Z.Y. |author5=Xu, J.P. |author6=Zheng, Y.X. |author7=Chen, L.Y. |author8=Huang, R.Z. |author9=Sun, Y. |author10=Chen, X. |authorlink11=Meng Xiang-Jin |author11=Meng, X.J. |author12=Dai, N.|title=Temperature-Dependent Optical Properties of Titanium Oxide Thin Films Studied by Spectroscopic Ellipsometry|journal=Applied Physics Express|date=2013|volume=6|issue=12 |page=121101|doi=10.7567/apex.6.121101|bibcode = 2013APExp...6l1101Z |s2cid=94319396 }}</ref><ref name=Sheng>{{cite journal|last=Sheng-Hong|first=Y.|author2=Sen, C. |author3=Ning, Y. |author4= Yue-Li, Z. |title=Optical Study of Sol-gel Processed Nd-doped BiFeO3 Multiferroic Films by Spectroscopic Ellipsometry|journal=Ferroelectrics|date=2013|volume=454|issue=1|pages=78–83 |doi=10.1080/00150193.2013.842802|bibcode=2013Fer...454...78S |s2cid=122654864 }}</ref><ref name=Balakrishnan>{{cite journal|last=Balakrishnan|first=G.|author2=Sundari, S.T. |author3=Kuppusami, P. |author4=Chandra, P.M. |author5=Srinivasan, M.P. |author6=Mohandas, E. |author7=Ganesan, V. |author8= Sastikumar, D. |title=A Study of Microstructural and Optical Properties of Nanocrystalline Ceria Thin Films Prepared by Pulsed Laser Deposition|journal=Thin Solid Films|date=2011|volume=519|issue=8|pages=2520–2526 |doi=10.1016/j.tsf.2010.12.013|bibcode = 2011TSF...519.2520B }}</ref><ref name=Cheng>{{cite journal|last=Cheng|first=K.W.|author2=Huang, C.M. |author3=Pan, G.T. |author4=Chang, W.S. |author5=Lee, T.C. |author6= Yang, T.C.K. |title=Effect of Sb on the growth and photoelectrochemical response of AgIn5S8 Film Electrodes Created by Solution Growth Technique|journal=Chemical Engineering Science|date=2010|volume=65|issue=1|pages=74–79 |doi=10.1016/j.ces.2009.02.002}}</ref><ref name=Das>{{cite journal|last=Das|first=N.S.|author2=Ghosh, P.K. |author3=Mitra, M.K. |author4= Chattopadhyay, K.K. |title=Effect of Film Thickness on the Energy Band Gap of Nanocrystalline CdS Thin Films Analyzed by Spectroscopic Ellipsometry|journal=Physica E: Low-dimensional Systems and Nanostructures|date=2010|volume=42|issue=8|pages=2097–2102 |doi=10.1016/j.physe.2010.03.035|bibcode = 2010PhyE...42.2097D }}</ref> The formalism for crystalline materials was found to also apply to polymers,<ref name=Xiong>{{cite journal|last=Xiong|first=K.|author2=Hou, L. |author3=Wang, P. |author4=Xia, Y. |author5=Chen, D. |author6= Xiao, B. |title=Phosphor-Doping Enhanced Efficiency in Bilayer Organic Solar Cells due to Longer Exciton Diffusion Length|journal=Journal of Luminescence|date=2014|volume=151|pages=193–196 |doi=10.1016/j.jlumin.2014.02.016|bibcode = 2014JLum..151..193X }}</ref><ref name=Huynh>{{cite journal|last=Huynh|first=T.P.|author2=Pietrzyk-Le, A. |author3=Chandra-Bikram, K.C. |author4=Noworyta, K.R. |author5=Sobczak, J.W. |author6=Sharma, P.S. |author7=D'Souza, F. |author8= Kutner, W. |title=Electrochemically Synthesized Molecularly Imprinted Polymer of Thiophene Derivatives for Flow-Injection Analysis Determination of Adenosine-5′-Triphosphate (ATP)|journal=Biosensors and Bioelectronics|date=2013|volume=41|pages=634–641 |doi=10.1016/j.bios.2012.09.038|pmid=23131778}}</ref><ref name=Zhu>{{cite journal|last=Zhu|first=D.|author2=Shen, W. |author3=Ye, H. |author4=Liu, X. |author5= Zhen, H. |title=Determination of the Optical Constants of Polymer Light-Emitting Diode Films from Single Reflection Measurements|journal=Journal of Physics D: Applied Physics|date=2008|volume=41|issue=23|series=23|page=235104 |doi=10.1088/0022-3727/41/23/235104|bibcode = 2008JPhD...41w5104Z |s2cid=123065042 }}</ref> which consist of long chains of molecules that do not form a crystallographic structure in the classical sense.

Other dispersion models that can be used to derive ''n'' and ''k'', such as the [[Tauc–Lorentz model]], can be found in the literature.<ref name=Laidani>{{cite journal|last=Laidani|first=N.|author2=Bartali, R. |author3=Gottardi, G. |author4=Anderle, M. |author5= Cheyssac, P. |title=Optical Absorption Parameters of Amorphous Carbon Films from Forouhi–Bloomer and Tauc–Lorentz Models: A Comparative Study|journal=Journal of Physics: Condensed Matter|date=2008|volume=20|issue=1|page=15216 |doi=10.1088/0953-8984/20/01/015216|bibcode = 2008JPCM...20a5216L |citeseerx=10.1.1.369.5532|s2cid=7359667 }}</ref><ref name=Easwarakhanthan>{{cite journal|last=Easwarakhanthan|first=T.|author2=Beyssen, D. |author3=Brizoual, L.L. |author4= Alnot, P. |title=Forouhi–Bloomer and Tauc–Lorentz Optical Dispersions Applied Using Spectroscopic Ellipsometry to Plasma-Deposited Fluorocarbon Films|journal=Journal of Applied Physics|date=2007|volume=101|issue=7|pages=073102–073102–7|doi=10.1063/1.2719271|bibcode = 2007JAP...101g3102E }}</ref> Two well-known models—[[Cauchy's equation|Cauchy]] and [[Sellmeier equation|Sellmeier]]—provide empirical expressions for ''n'' valid over a limited measurement range, and are only useful for non-absorbing films where ''k''=0. Consequently, the Forouhi–Bloomer formulation has been used for measuring thin films in various applications.<ref name=Torkaman /><ref name=Lakhdar /><ref name=Khanbashi /><ref name=Nakamura /><ref name=Winkler /><ref name=Miao /><ref name=Zhang /><ref name=Sheng /><ref name=Balakrishnan /><ref name=Cheng /><ref name=Das /><ref name=Xiong /><ref name=Huynh /><ref name=Zhu /><ref name=Laidani /><ref name=Easwarakhanthan />

In the following discussions, all variables of photon energy, ''E'', will be described in terms of wavelength of light, ''λ'', since experimentally variables involving thin films are typically measured over a spectrum of wavelengths. The ''n'' and ''k'' spectra of a thin film cannot be measured directly, but must be determined indirectly from measurable quantities that depend on them. Spectroscopic reflectance, ''R''(''λ''), is one such measurable quantity. Another, is spectroscopic transmittance, ''T''(''λ''), applicable when the substrate is transparent. Spectroscopic reflectance of a thin film on a substrate represents the ratio of the intensity of light reflected from the sample to the intensity of incident light, measured over a range of wavelengths, whereas spectroscopic transmittance, ''T''(''λ''), represents the ratio of the intensity of light transmitted through the sample to the intensity of incident light, measured over a range of wavelengths; typically, there will also be a reflected signal, ''R''(''λ''), accompanying ''T''(''λ'').

The measurable quantities, ''R''(''λ'') and ''T''(''λ'') depend not only on ''n''(''λ'') and ''k''(''λ'') of the film, but also on film thickness, ''t'', and ''n''(''λ'') and ''k''(''λ'') of the substrate. For a silicon substrate, the ''n''(''λ'') and ''k''(''λ'') values are known and are taken as a given input. The challenge of characterizing thin films involves extracting ''t'', ''n''(''λ'') and ''k''(''λ'') of the film from the measurement of ''R''(''λ'') and/or ''T''(''λ''). This can be achieved by combining the Forouhi–Bloomer dispersion equations for ''n''(''λ'') and ''k''(''λ'') with the [[Fresnel equations]] for the reflection and transmission of light at an interface<ref name=Heavens2>{{cite book|last=Heavens|first=O.S.|title=Optical Properties of Thin Solid Films|url=https://archive.org/details/opticalpropertie0000heav|url-access=registration|date=1965|publisher=Dover|location=New York}}</ref> to obtain theoretical, physically valid, expressions for reflectance and transmittance. In so doing, the challenge is reduced to extracting the five parameters ''A'', ''B'', ''C'', ''E''<sub>g</sub>, and ''n''(∞) that constitute ''n''(''λ'') and ''k''(''λ''), along with film thickness, ''t'', by using a nonlinear least squares regression analysis<ref name=Levenberg>{{cite journal|last=Levenberg|first=K.|title=A Method for the Solution of Certain Non-Linear Problems in Least Squares|journal=Quarterly of Applied Mathematics|date=1944|volume=2|issue=2|page=164|doi=10.1090/qam/10666|doi-access=free}}</ref><ref name=Marquardt>{{cite journal|last=Marquardt|first=D.W.|title=An Algorithm for Least-Squares Estimation of Nonlinear Parameters|journal=Journal of the Society for Industrial and Applied Mathematics|date=1963|volume=11|issue=2|series=2|pages=431–441|doi=10.1137/0111030|hdl=10338.dmlcz/104299|hdl-access=free}}</ref> fitting procedure. The fitting procedure entails an iterative improvement of the values of ''A'', ''B'', ''C'', ''E''<sub>g</sub>, ''n''(∞), ''t'', in order to reduce the sum of the squares of the errors between the theoretical ''R''(''λ'') or theoretical ''T''(''λ'') and the measured spectrum of ''R''(''λ'') or ''T''(''λ'').
 
Besides spectroscopic reflectance and transmittance, spectroscopic [[ellipsometry]] can also be used in an analogous way to characterize thin films and determine ''t'', ''n''(''λ'') and ''k''(''λ'').

== Measurement examples ==
The following examples show the versatility of using the Forouhi–Bloomer dispersion equations to characterize thin films using a tool based on near-normal incident spectroscopic reflectance. Near-normal spectroscopic transmittance is also used when the substrate is transparent. The ''n''(''λ'') and ''k''(''λ'') spectra of each film are obtained along with film thickness, over a wide range of wavelengths from deep ultraviolet to near infrared wavelengths (190–1000&nbsp;nm).

In the following examples, the notation for theoretical and measured reflectance in the spectral plots is expressed as "R-theor" and "R-meas", respectively.

Below are schematics depicting the thin film measurement process:

[[File:Thin Film Analysis Process.gif|center|800px|Thin film characterization involves determining the film's thickness (''t'') plus its refractive index (''n'') and extinction coefficient (''k'') over as wide a wavelength range as possible, preferably covering ultra-violet through near infra-red wavelengths (190–1000&nbsp;nm). By measuring near-normal incident reflectance (''R'') of the film (from 190 to 1000&nbsp;nm), and analyzing ''R'' using the Forouhi–Bloomer dispersion equations, the film can be completely characterized.]]

The Forouhi–Bloomer dispersion equations in combination with [[Rigorous coupled-wave analysis|Rigorous Coupled-Wave Analysis (RCWA)]] have also been used to obtain detailed profile information (depth, CD, sidewall angle) of trench structures. In order to extract structure information, polarized broadband reflectance data, [[Polarization (waves)|''Rs'' and ''Rp'']], must be collected over a large wavelength range from a periodic structure (grating), and then analyzed with a model that incorporates Forouhi–Bloomer dispersion equations and RCWA. Inputs into the model include grating pitch and ''n'' and ''k'' spectra of all materials within the structure, while outputs can include Depth, CDs at multiple locations, and even sidewall angle. The ''n'' and ''k'' spectra of such materials can be obtained in accordance with the methodology described in this section for thin film measurements.

Below are schematics depicting the measurement process for trench structures. Examples of trench measurements then follow.

[[File:OCD Analysis Flow New.PNG|center|800px|The key properties of trench structures are trench depth, critical dimensions, plus profile (or sidewall angle). The term "critical dimension" is usually abbreviated as "CD". CDs signify widths of the trench at various levels inside the trench – top, middle, and bottom of trench.

These key properties can be determined by measuring polarized reflectance ''Rs'' and ''Rp'' over as wide a wavelength range as possible, preferably covering ultra-violet through near infra-red wavelengths (190–1000&nbsp;nm).

Analysis of the measured ''Rs'' and ''Rp'', based on the Forouhi–Bloomer dispersion equations (to characterize any films within the trench structure), combined with Rigorous Coupled Wave Analysis (RCWA) (to determine the geometry), will provide trench depth, CDs and profiles.
]]

=== Example 1: Amorphous silicon on oxidized silicon substrate (a-Si/SiO<sub>2</sub>/Si-Sub) ===
{{multiple image
| footer    = Ex. 1: Reflectance spectra collected over 190–1000nm wavelength range for an amorphous silicon film (a-Si) on an oxidized silicon substrate (SiO<sub>2</sub>/Si-Sub) plus the ''n(λ)'' and ''k(λ)'' spectra of the a-Si film. The film thickness was found to be 1147nm. The thicknesses of a-Si and SiO<sub>2</sub> films, plus the ''n(λ)'' and ''k(λ)'' spectra of the a-Si were all simultaneously determined. The ''n(λ)'' and ''k(λ)'' spectra of the SiO<sub>2</sub> film was held fixed.
| width     = 250

| image1    = Ex. 2 - Amorphous Silicon on Oxided RR.png
| alt1      = Reflectance

| image2    = Ex. 2 - Amorphous Silicon on Oxided nk.png
| alt2      = Optical properties
}}
{{multiple image
| footer    = Ex. 1: Amorphous materials typically exhibit one broad maximum in their ''n(λ)'' and ''k(λ)'' spectra. As a material transitions from the amorphous state to the fully crystalline state, the broad maximum sharpens up and other sharp peaks start to appear in the ''n(λ)'' and ''k(λ)'' spectra. This is demonstrated for the case of amorphous silicon progressing to poly-silicon and further progressing to crystalline silicon. 
| width     = 250

| image1    = Comparison_of_n.png
| alt1      = n

| image2    = Comparison_of_k.png
| alt2      = k
}}
Example 1 shows one broad maximum in the ''n(λ)'' and ''k(λ)'' spectra of the a-Si film, as is expected for amorphous materials. As a material transitions toward crystallinity, the broad maximum gives way to several sharper peaks in its ''n(λ)'' and ''k(λ)'' spectra, as demonstrated in the graphics.

When the measurement involves two or more films in a stack of films, the theoretical expression for reflectance must be expanded to include the ''n''(''λ'') and ''k''(''λ'') spectra, plus thickness, ''t'', of each film. However, the regression may not converge to unique values of the parameters, due to the non-linear nature of the expression for reflectance. So it is helpful to eliminate some of the unknowns. For example, the ''n''(''λ'') and ''k''(''λ'') spectra of one or more of the films may be known from the literature or previous measurements, and held fixed (not allowed to vary) during the regression. To obtain the results shown in Example 1, the ''n''(''λ'') and ''k''(''λ'') spectra of the SiO<sub>2</sub> layer was fixed, and the other parameters, ''n''(''λ'') and ''k''(''λ'') of a-Si, plus thicknesses of both a-Si and SiO<sub>2</sub> were allowed to vary.

=== Example 2: 248 nm photoresist on silicon substrate (PR/Si-Sub) ===
{{multiple image
| footer    = Ex. 2: Reflectance spectra collected over the 190–1000&nbsp;nm wavelength range for a photoresist film on silicon substrate, plus the ''n''(''λ'') and ''k''(''λ'') spectra of the photoresist. The film thickness was found to be 498&nbsp;nm. The thickness and the ''n''(''λ'') and ''k(λ)'' spectra of the photoresist were all simultaneously determined.
| width     = 250

| image1    = Ex. 5 - 193nm Photoresist RR.png
| alt1      = Reflectance

| image2    = 248_nm_Photoresist_n_and_k_Dispersion.png
| alt2      = Optical properties
}}

Polymers such as [[photoresist]] consist of long chains of molecules which do not form a crystallographic structure in the classic sense. However, their ''n''(''λ'') and ''k''(''λ'') spectra exhibit several sharp peaks rather than a broad maximum expected for non-crystalline materials. Thus, the measurement results for a polymer are based on the Forouhi–Bloomer formulation for crystalline materials. Most of the structure in the ''n''(''λ'') and ''k''(''λ'') spectra occurs in the deep UV wavelength range and thus to properly characterize a film of this nature, it is necessary that the measured reflectance data in the deep UV range is accurate.

The figure shows a measurement example of a photoresist (polymer) material used for 248&nbsp;nm micro-lithography. Six terms were used in the Forouhi–Bloomer equations for crystalline materials to fit the data and achieve the results.

=== Example 3: Indium tin oxide on glass substrate (ITO/Glass-Sub) ===
{{multiple image
| footer    = Ex. 3: Reflectance and transmittance spectra in 190–1000nm range for an uncoated glass substrate. Note that ''T'' = 0 for the glass substrate in the DUV, indicating absorption in this range of the spectrum. It is found that the value of ''k''(''λ'') in the deep UV wavelength range is of the order of ''k'' = {{val|3|e=−4}}, and this small non-zero value is consistent with ''T'' = 0 in the deep UV.
| width     = 250

| image1    = Ex. 6 - Glass RRTT.png
| alt1      = Reflectance

| image2    = Ex. 6 - Glass nk.png
| alt2      = Optical properties
}}
{{multiple image
| footer    = Ex. 3: Reflectance and transmittance spectra in 190–1000nm range of ITO deposited on the glass substrate described above, plus the ''n''(''λ'') and ''k''(''λ'') spectra of the ITO film. ITO thickness of 133nm and its ''n''(''λ'') and ''k''(''λ'') spectra were simultaneously determined by fitting measured reflectance and transmittance spectra to theoretical expressions of these quantities using the Forouhi–Bloomer equations.
| width     = 250

| image1    = Ex. 6 - ITO on Glass RRTT.png
| alt1      = Reflectance

| image2    = Ex. 6 - ITO nk.png
| alt2      = Optical properties
}}
[[Indium tin oxide]] (ITO) is a conducting material with the unusual property that it is transparent, so it is widely used in the flat panel display industry. Reflectance and transmittance measurements of the uncoated glass substrate were needed in order to determine the previously unknown ''n''(''λ'') and ''k''(''λ'') spectra of the glass. The reflectance and transmittance of ITO deposited on the same glass substrate were then measured simultaneously, and analyzed using the Forouhi–Bloomer equations.

As expected, the ''k''(''λ'') spectrum of ITO is zero in the visible wavelength range, since ITO is transparent. The behavior of the ''k''(''λ'') spectrum of ITO in the near-infrared (NIR) and infrared (IR) wavelength ranges resembles that of a metal: non-zero in the NIR range of 750–1000&nbsp;nm (difficult to discern in the graphics since its values are very small) and reaching a maximum value in the IR range (''λ''&nbsp;>&nbsp;1000&nbsp;nm). The average ''k'' value of the ITO film in the NIR and IR range is 0.05.

=== Example 4: Multi-spectral analysis of germanium (40%)–selenium (60%) thin films ===
{{multiple image
| footer    = Ex. 4: Multi-spectral analysis was used to analyze the reflectance spectra of a Ge<sub>40</sub>Se<sub>60</sub> film deposited on two different substrates: both silicon and oxidized-silicon substrates. The measurements yielded a single ''n''(''λ'') and ''k''(''λ'') spectra of Ge<sub>40</sub>Se<sub>60</sub>. A thickness of 33.6&nbsp;nm for Ge<sub>40</sub>Se<sub>60</sub> on the oxidized silicon substrate was found, while a thickness of 34.5&nbsp;nm of Ge<sub>40</sub>Se<sub>60</sub> on the silicon substrate was found. In addition the thickness of the oxide layer was determined to be 166&nbsp;nm.
| width     = 250

| image1    = Ex. 8 - Multi-Spectral Analysis of Ge40Se60 RR.png
| alt1      = Optical properties

| image2    = Ex. 8 - Multi-Spectral Analysis of Ge40Se60 nk.png
| alt2      = Optical properties
}}
When dealing with complex films, in some instances the parameters cannot be resolved uniquely. To constrain the solution to a set of unique values, a technique involving multi-spectral analysis can be used. In the simplest case, this entails depositing the film on two different substrates and then simultaneously analyzing the results using the Forouhi–Bloomer dispersion equations.

For example, the single measurement of reflectance in 190–1000&nbsp;nm range of Ge<sub>40</sub>Se<sub>60</sub>/Si does not provide unique ''n''(''λ'') and ''k''(''λ'') spectra of the film. However, this problem can be solved by depositing the same Ge<sub>40</sub>Se<sub>60</sub> film on another substrate, in this case oxidized silicon, and then simultaneously analyzing the measured reflectance data to determine:
* Thickness of the Ge<sub>40</sub>Se<sub>60</sub>/Si film on the silicon substrate as 34.5&nbsp;nm,
* Thickness of the Ge<sub>40</sub>Se<sub>60</sub>/Si film on the oxidized silicon substrate as 33.6&nbsp;nm,
* Thickness of SiO<sub>2</sub> (with ''n'' and ''k'' spectra of SiO<sub>2</sub> held fixed), and
* ''n'' and ''k'' spectra, in 190–1000&nbsp;nm range, of Ge<sub>40</sub>Se<sub>60</sub>/Si.

=== Example 5: Complex trench structure ===
{{multiple image
| footer    = Ex. 5: A trench structure consisting of various films and complex profile. The Poly-Si film was measured on a blanket area of the sample and its ''n'' and ''k'' spectra were determined based on the Forouhi–Bloomer dispersion equations. A fixed table of values for the ''n'' and ''k'' spectra of the SiO<sub>2</sub> and Si<sub>3</sub>N<sub>4</sub> films was used. With the ''n'' and ''k'' spectra of these films at hand, and using Rigorous Coupled Wave Analysis (RCWA), film thicknesses, various depths (heights) inside the trench, and CDs are then determined.
| width     = 300

| image1    = Complex_Trench_Structure_Image.png 
| alt1      = Complex Trench Structure Schematic
}}
{{multiple image
| footer    = Ex. 5: Measured ''Rs'' and ''Rp'' reflectance collected on the Complex Trench Structure.
| width     = 300

| image1    = Complex Trench Structure Measured Rs and Rp.png
| alt1      = Complex Trench Structure Measured Rs and Rp
}}

The trench structure depicted in the adjacent diagram repeats itself in 160&nbsp;nm intervals, that is, it has a given pitch of 160&nbsp;nm. The trench is composed of the following materials:
{{columns-list|colwidth=30em|
* M1: Si<sub>3</sub>N<sub>4</sub>
* M2: Poly-Si
* M3: Sidewall Oxide (SiO<sub>2</sub>)
* M4: SiO<sub>2</sub>
* M5: Si<sub>3</sub>N<sub>4</sub>
* M6: SiO<sub>2</sub>
* M7: Si Substrate
* M8: Air
}}
Accurate ''n'' and ''k'' values of these materials are necessary in order to analyze the structure. Often a blanket area on the trench sample with the film of interest is present for the measurement. In this example, the reflectance spectrum of the poly-silicon was measured on a blanket area containing the poly-silicon, from which its ''n'' and ''k'' spectra were determined in accordance with the methodology described in this article that uses the Forouhi–Bloomer dispersion equations. Fixed tables of ''n'' and ''k'' values were used for the SiO<sub>2</sub> and Si<sub>3</sub>N<sub>4</sub> films.

Combining the ''n'' and ''k'' spectra of the films with [[Rigorous coupled-wave analysis|Rigorous Coupled-Wave Analysis (RCWA)]] the following critical parameters were determined (with measured results as well):

{| class="wikitable" style="text-align: center;"
|-
! !! Measured Parameter !! Results
|-
|1|| Si Depth || 27.4&nbsp;nm
|-
|2|| CD @ Top of Si || 26.4&nbsp;nm
|-
|3|| SiO<sub>2</sub> Liner Width || 40.2&nbsp;nm
|-
|4|| Si<sub>3</sub>N<sub>4</sub> Height || 28&nbsp;nm
|-
|3|| Poly-Si Width || 92.6&nbsp;nm
|-
|3|| Poly-Si Height || 85.6&nbsp;nm
|}

== References ==
{{reflist}}

[[Category:Light]]
[[Category:Thin films]]
[[Category:Semiconductors]]
[[Category:Refraction]]
[[Category:Metrology]]