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A.R. Forouhi and I. Bloomer deduced dispersion equations for the refractive index, n, and extinction coefficient, k, which were published in 1986 and 1988. 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”. The 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 wavelength

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  • Refractive index and extinction coefficient of thin film materials
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  • A.R. Forouhi and I. Bloomer deduced dispersion equations for the refractive index, n, and extinction coefficient, k, which were published in 1986 and 1988. 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”. The 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 wavelength
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  • A.R. Forouhi and I. Bloomer deduced dispersion equations for the refractive index, n, and extinction coefficient, k, which were published in 1986 and 1988. 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”. The 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 wavelength of light, λ, since E = hc/λ. The symbol h represents Planck’s 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 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: The five parameters A, B, C, Eg, and n(∞) each have physical significance. Eg 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 4C-B2 > 0. Finally, n(∞), a constant greater than unity, represents the value of n at E = ∞. The parameters B0 and C0 in the equation for n(E) are not independent parameters, but depend on A, B, C, and Eg. They are given by: where 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: 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, Eg, as well as its own values of B0 and C0. Analogous to the amorphous case, the terms all have physical significance.
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  • n
  • k
  • Complex Trench Structure Measured Rs and Rp
  • Complex Trench Structure Schematic
  • Optical Properties
  • Reflectance
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  • 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 SiO2 and Si3N4 films was utilized. With the n and k spectra of these films at hand, and utilizing Rigorous Coupled Wave Analysis , film thicknesses, various depths inside the trench, and CDs are then determined.
  • Ex. 3: Reflectance and transmittance spectra from 190nm - 1000nm of ITO deposited on the glass substrate described above, plus the n and k spectra of the ITO film. ITO thickness of 133nm and its n and k spectra were simultaneously determined by fitting measured reflectance and transmittance spectra to theoretical expressions of these quantities using the Forouhi-Bloomer equations.
  • Ex. 2: Reflectance spectra collected over the 190nm-1000nm wavelength range for a photoresist film on silicon substrate, plus the n and k spectra of the photoresist. The film thickness was found to be 498nm. The thickness and the n and k spectra of the photoresist were all simultaneously determined.
  • Ex. 1: Amorphous materials typically exhibit one broad maximum in their n and k 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 and k spectra. This is demonstrated for the case of amorphous silicon progressing to poly-silicon and further progressing to crystalline silicon.
  • Ex. 3: Reflectance and transmittance spectra from 190nm - 1000nm 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 in the deep UV wavelength range is of the order of k = 3x10−4, and this small non-zero value is consistent with T = 0 in the deep UV.
  • Ex. 1: Reflectance spectra collected over 190nm-1000nm wavelength range for an amorphous silicon film on an oxidized silicon substrate plus the n and k spectra of the a-Si film. The film thickness was found to be 1147nm. The thicknesses of a-Si and SiO2 films, plus the n and k spectra of the a-Si were all simultaneously determined. The n and k spectra of the SiO2 film was held fixed.
  • Ex. 4: Multi-spectral analysis was used to analyze the reflectance spectra of a Ge40Se60 film deposited on two different substrates: both silicon and oxidized-silicon substrates. The measurements yielded a single n and k spectra of Ge40Se60. A thickness of 33.6 nm for Ge40Se60 on the oxidized silicon substrate was found, while a thickness of 34.5 nm of Ge40Se60 on the silicon substrate was found. In addition the thickness of the oxide layer was determined to be 166nm.
  • Ex. 5: Measured Rs and Rp reflectance collected on the Complex Trench Structure.
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  • Comparison_of_k.png
  • Comparison_of_n.png
  • Complex Trench Structure Measured Rs and Rp.png
  • Complex_Trench_Structure_Image.png
  • Ex. 2 - Amorphous Silicon on Oxided RR.png
  • Ex. 2 - Amorphous Silicon on Oxided nk.png
  • Ex. 5 - 193nm Photoresist RR.png
  • Ex. 6 - Glass RRTT.png
  • Ex. 6 - Glass nk.png
  • Ex. 6 - ITO nk.png
  • Ex. 6 - ITO on Glass RRTT.png
  • Ex. 8 - Multi-Spectral Analysis of Ge40Se60 RR.png
  • Ex. 8 - Multi-Spectral Analysis of Ge40Se60 nk.png
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