Optical properties of silicon and germanium determined by high-precision analysis of reflection electron energy loss spectroscopy spectra

We present a detailed analysis and comparison of four models describing the extension of the electron-energy loss function from the optical limit of q\ensuremath{\rightarrow}0 into the (q,\ensuremath{\omega}) plane to obtain the bulk and surface terms of differential inverse inelastic mean free paths. We found that the best model that describes accurately and times efficiently the calculation of the energy loss function of free-electron-like materials is the combination of the Penn algorithm [Phys. Rev. B 35, 482 (1987)] with the Ritchie-Howie method [Philos. Mag. 36, 463 (1977)]. Applying this model in our reverse Monte Carlo method, we determined, with high-precision, electron-energy loss functions of silicon and germanium based on the theoretical analysis of the high-energy resolution reflected electron energy loss spectroscopy (REELS) spectra, measured at 3, 4, and 5 keV incident electron energies. The refractive index $n$, the extinction coefficient $k$, and the complex dielectric function ($\ensuremath{\varepsilon}={\ensuremath{\varepsilon}}_{1}+i{\ensuremath{\varepsilon}}_{2}$) were calculated from the obtained energy loss function in a wide energy loss range of 0--200 eV. The accuracy of the obtained results is justified with various sum rules. We found that the calculated optical data of Si and Ge fulfill the sum rules with an average accuracy of 0.11% or even better. Therefore, the use of these optical data in materials science and surface analysis is highly recommended for further applications.