Use of x-ray scattering functions in Kramers-Kronig analysis of reflectance

Abstract

Kramers-Kronig analysis is commonly used to estimate the optical properties of new materials. The analysis typically uses data from far infrared through near ultraviolet (say, 40--40,000 cm$^{-1}$ or 5 mev--5 eV) and uses extrapolations outside the measured range. Most high-frequency extrapolations use a power law, 1/$\omega^n$, transitioning to $1/\omega^{4}$ at a considerably higher frequency and continuing this free-carrier extension to infinity. The mid-range power law is adjusted to match the slope of the data and to give pleasing curves, but the choice of power (usually between 0.5 and 3) is arbitrary. Instead of an arbitrary power law, it is is better to use X-ray atomic scattering functions such as those presented by Henke and co-workers. These basically treat the solid as a linear combinations of its atomic constituents and, knowing the chemical formula and the density, allow the computation of dielectric function, reflectivity, and other optical functions. The "Henke reflectivity" can be used over photon energies of 10 eV--34 keV, after which a $1/\omega^{4}$ continuation is perfectly fine. The bridge between experimental data and the Henke reflectivity as well as two corrections that needed to be made to the latter are discussed.