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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. (en)
- Модель Форухи — Блумер — дисперсионные уравнения для среды с поглощением выведенные А. Р. Форухи и И. Блумер для комплексного показателя преломления n +ik, которые были опубликованы в 1986 и 1988 годах. Публикация 1986 г. относится к аморфным материалам, а публикация 1988 г. — к кристаллическим. Впоследствии, в 1991 году, их работа была включена в качестве главы в «Справочник оптических констант». Дисперсионные уравнения Форухи — Блумер описывают, как фотоны различной энергии взаимодействуют с тонкими плёнками. При использовании в спектроскопической дисперсионные уравнения Форухи — Блумер позволяют определять n (коэффициент преломления) и k (коэффициент поглощения) для аморфных и кристаллических материалов как функции энергии фотона E. Значения n(E) и k(E) называются спектрами n и k, которые также могут выражаться в зависимости от длины волны света λ, поскольку E = hc/λ, где h - постоянная Планка, а c — скорость света в вакууме. Вместе n и k часто называют «оптическими константами» материала (хотя они не являются константами, поскольку их значения зависят от энергии фотонов). (ru)
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- Ex. 2: Reflectance spectra collected over the 190–1000nm 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. (en)
- Ex. 3: Reflectance and transmittance spectra in 190–1000nm range 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. (en)
- Ex. 1: Reflectance spectra collected over 190–1000nm 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. (en)
- 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. (en)
- 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. (en)
- Ex. 5: Measured Rs and Rp reflectance collected on the Complex Trench Structure. (en)
- 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 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. (en)
- 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. (en)
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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 wavelengt (en)
- Модель Форухи — Блумер — дисперсионные уравнения для среды с поглощением выведенные А. Р. Форухи и И. Блумер для комплексного показателя преломления n +ik, которые были опубликованы в 1986 и 1988 годах. Публикация 1986 г. относится к аморфным материалам, а публикация 1988 г. — к кристаллическим. Впоследствии, в 1991 году, их работа была включена в качестве главы в «Справочник оптических констант». Дисперсионные уравнения Форухи — Блумер описывают, как фотоны различной энергии взаимодействуют с тонкими плёнками. При использовании в спектроскопической дисперсионные уравнения Форухи — Блумер позволяют определять n (коэффициент преломления) и k (коэффициент поглощения) для аморфных и кристаллических материалов как функции энергии фотона E. Значения n(E) и k(E) называются спектрами n и k, кото (ru)
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