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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.〔〔 == 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 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 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, as well as metallic compounds. The formalism for crystalline materials was found to also apply to polymers, 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 Tauc-Lorentz, can be found in the literature. Two well-known models—Cauchy and 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.〔〔〔〔〔〔〔〔〔〔〔〔〔〔〔〔 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 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, Eg, and ''n(∞)'' that constitute ''n(λ)'' and ''k(λ)'', along with film thickness, ''t'', by utilizing a nonlinear least squares regression analysis fitting procedure. The fitting procedure entails an iterative improvement of the values of A, B, C, Eg, ''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)''. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Refractive index and extinction coefficient of thin film materials」の詳細全文を読む スポンサード リンク
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