Refractive index and extinction coefficient of thin film materials information

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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^[1] and 1988.^[2] 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”.^[3] 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 the 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^[4] 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:

${\displaystyle k(E)={\frac {A(E-E_{g})^{2}}{E^{2}-BE+C}}\ }$

${\displaystyle n(E)=n(\infty )+{\frac {(B_{0}E+C_{0})}{E^{2}-BE+C}}\ }$

The five parameters A, B, C, E_g, and n(∞) each have physical significance.^[1][3] E_g 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 E_g. They are given by:

${\displaystyle B_{0}={\frac {A}{Q}}\ \left({\frac {-B^{2}}{2}}\ +E_{g}B-{E_{g}}^{2}+C\right)}$

${\displaystyle C_{0}={\frac {A}{Q}}\ \left[({E_{g}}^{2}+C){\frac {B}{2}}\ -2E_{g}C\right]}$

where

${\displaystyle Q={\frac {1}{2}}\ (4C-B^{2})^{\frac {1}{2}}}$

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:

${\displaystyle k(E)=\sum _{i=1}^{q}\left[{\frac {A_{i}(E-E_{g_{i}})^{2}}{E^{2}-B_{i}E+C_{i}}}\right]}$

${\displaystyle n(E)=n(\infty )+\sum _{i=1}^{q}\left[{\frac {B_{0_{i}}E+C_{0_{i}}}{E^{2}-B_{i}E+C_{i}}}\right]}$

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_g, as well as its own values of B₀ and C₀. Analogous to the amorphous case, the terms all have physical significance.^[2][3]