Energy dependence of the energy loss function parametrization of indium in the Drude–Lindhard model

Abstract

The energy loss function (ELF) is a key parameter for the calculations of energy losses undergone by electrons in matter. It is often the only input in the calculations performed within the models based on the semi-classical dielectric response theory. Its perfect knowledge is thus of primordial importance. To evaluate the ELF, it is usual to consider as a model an expansion in Drude-Lindhard (DL) type oscillators with fixed values of the strength, width, energy and dispersion for the various oscillators. However, for materials that are characterized by a single sharp oscillator as aluminium or indium, it has been shown [Phys. Rev. B 46 (1992) 2486] that the damping parameter that corresponds in the DL model to the width of the oscillator increases for decreasing incident electron energy. To emphasize this effect, we perform in this work systematic calculation of the ELF for an indium target and for incident electron energies between 200 and 2000 eV. The ELF is determined by comparing REELS (reflection electron energy loss spectroscopy) experimental inelastic electron scattering cross sections with cross sections calculated within the semi-classical dielectric response model, which is implemented in the QUEELS-ε(k,ω)-REELS software (Quantitative analysis of Electron Energy Losses at Surfaces) [Surf. Interface Anal. 36 (2004) 824]. We also perform measurements and calculations for varying incident and exit angles of the electron, namely for angles to the surface normal between 15 o and 75 o , to check the validity of our results for all geometries. Our results show that the damping parameter of indium in the DL model for electron energy of 200 eV is three times larger than for energy of 2000 eV.

Local inverse inelastic mean free path at interface

“…The Al/In interface has been chosen for its academic interest rather than its practical importance. Indeed each of the two materials exhibits only one plasmon peak in its ELF 13,16 . Figure 1 shows the local‐DIIMFP for a 500‐eV electron travelling from Al to In and from In to Al at various distances from the interface crossing point (with the interface crossing angle °).…”

“…Indeed each of the two materials exhibits only one plasmon peak in its ELF. 13,16 Figure 1 shows the local-DIIMFP μ 1!2 ðE, α, ℏω, rÞ for a 500-eV electron travelling from Al to In and from…”

“… and are the strength, width, energy and dispersion of the i th oscillator, respectively, and the step function is included to describe the effect of the energy band gap present in semiconductors and insulators. For the materials studied in this work, the parameters in the expansion are taken from Pauly et al (Al, Si), 13 (In), 16 (Au) 17 and (SiO 2 ) 18 . Once the ELF of a material is known, it is straightforward to obtain from Kramers–Krönig transformation 12 the real part of the reciprocal of the complex dielectric function, Re , as well as the real and imaginary parts of the dielectric function, and , used in Equation ().…”

Extracting the dielectric function from high‐energy REELS measurements

“…However, the width of this loss function did depend on the incoming electron energy, whereas the dielectric function of a material should describe REELS measurements taken at any energy. Pauly suggested that this could be explained by assuming that Γ depends on q , as the ranges of q values contributing to the spectrum (see the integration limits in Eqn ) changes with incoming electron energy . Calliari et al .…”