Dielectric Function Modeling

Spectroscopic ellipsometry (SE) is commonly used to measure the optical constants of thin films and bulk materials. The optical constants vary with wavelength, which is referred to as dispersion. Rather than independently determine the optical constants at each wavelength, it is convenient to use an equation to describe their dispersion. A dispersion equation simplifies the description of the optical constants and improves the efficiency of data analysis. We begin this chapter by describing the optical constants, optical resonance, and the Kramers-Kronig relations. Different absorption phenomena are also briefly described. Many dispersion equations relate an optical resonance or absorption in terms of the complex dielectric function. Multiple resonance and absorption features can be summed to describe the overall dielectric function for the material. Finally, we review the common dispersion equations used for photovoltaic materials. The Cauchy and Sellmeier equations are used to describe transparent materials. The Lorentz, Harmonic, and Gaussian equations describe a resonant absorption. The Tauc-Lorentz and Cody-Lorentz were developed for amorphous semiconductors with dispersion features necessary to describe the optical functions near the bandgap energy. Additional dispersion equations are designed to describe the critical points in semiconductor band structure. We conclude this review with a description of polynomials, splines, and basis-splines, which are used to empirically match the optical functions of many materials.