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→0 into the (q,ω) 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 ($\varepsilon ={\varepsilon }_1+i{\varepsilon }_2$) 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.

Figure 1

Comparison of the optical ELF of Ge between the measured data [63, 67, 68] (squares) with the fitting by Drude-type oscillators.

Figure 2

Surface plots of the energy loss function of Ge as a function of momentum transfer and energy loss, calculated by (a) the simple Ritchie-Howie method, (b) the Ritchie-Howie method, (c) the FPA method, and (d) the SPA method.

Figure 3

Comparison of the ELFs of Ge at given momentum transfers as a function of the electron energy loss: (a) $0.1{\AA }^{-1}$, (b) $0.5{\AA }^{-1}$, (c) $1.0{\AA }^{-1}$, and (d) $1.5{\AA }^{-1}$. The ELFs are calculated from the optical ELF by using four different extension algorithms. Blue line: simple Ritchie-Howie method; green line: Ritchie-Howie method; red line: FPA method; purple line: SPA method.

Figure 4

Comparison of the ELFs of Ge at given energy losses as a function of the momentum transfer: (a) 5 eV, (b) 16 eV, (c) 30 eV, and (d) 50 eV. The ELFs are calculated from the optical ELF by using four different extension algorithms. Blue line: simple Ritchie-Howie method; green line: Ritchie-Howie method; red line: FPA method; purple line: SPA method.

Figure 5

(a) Comparison of simulated REELS spectra of Ge based on the simple Ritchie-Howie method (blue), the Ritchie-Howie method (green), and the present FPA-Ritchie-Howie method (red) with the experimental REELS spectrum (black) measured at the incident energy of 5000 eV. (b) Comparison of the relative errors between the experimental data and the simulated REELS spectra.

Figure 6

(a) The final simulated REELS spectra (dash lines) of silicon at 3000, 4000, and 5000 eV, in comparison with experimental results (solid lines). (b) The final ELFs, $\mathrm{Im}\left[-1/\phantom{-1\varepsilon \left(\omega \right)}\varepsilon \left(\omega \right)\right]$, obtained from the REELS spectra.

Figure 7

Comparison of the ELFs between the present data and other sources [63, 68] for Si.

Figure 8

Comparison of the (a) optical constants and (b) dielectric functions between the present data and other sources [63, 68] for Si.

Figure 9

(a) The final simulated REELS spectra (dash lines) of germanium at 3000, 4000, and 5000 eV, in comparison with experimental results (solid lines). (b) The final ELFs, $\mathrm{Im}\left[-1/\phantom{-1\varepsilon \left(\omega \right)}\varepsilon \left(\omega \right)\right]$, obtained from the REELS spectra.

Figure 10

Comparison on the ELFs between the present data and other sources [44, 63, 67, 68] for Ge.

Figure 11

Comparison of the (a) optical constants and (b) dielectric functions between the present data and other sources [44, 68] for Ge.

Figure 12

Inertial sum rule and dc-conductivity sum rule checks of Si in comparison with other source [63].

Figure 13

Inertial sum rule and dc-conductivity sum rule checks of Ge in comparison with other source [44].