Inelastic mean free path from reflectivity of slow electrons

The inelastic mean free path (IMFP) of electrons is derived using a new approach based on the low-energy electron reflectivity from ultrathin films. The thickness-dependent quantum size oscillations as a function of electron energy observed in the reflectivity of slow electrons are modeled using an absorbing Fabry-Pérot interferometer consisting of vacuum, film, and substrate. The absorbing properties of the film are represented by the imaginary part of the complex refractive index associated with the IMFP which determines the amplitude of the electron reflectivity oscillations. Using this formalism for an Fe film on W(110), the IMFP in Fe is found in the energy range from 4 to 18 eV above the vacuum level. In contrast to the common notion, the IMFP in Fe is shown to have a very weak energy dependence at low energy. The results are in good agreement with independent IMFP measurements found in thickness-dependent photoemission experiments.

Figure 1

Tungsten band structure along the [110] direction above the vacuum level. For details see text.

Figure 2

(a) Reflectivity vs energy measured in LEEM. (b) Fe band structure along the [110] direction above the vacuum level derived from LEEM (circles). The solid line is a two-band model fit to the experimental data. Zero denotes the vacuum level.

Figure 3

(a) Reflectivity vs energy for 6 ML Fe derived from experiment (dots) and theory (lines) according to Eq. (1). (b) Difference between reflectivity and zero line. For details see text. The line codes in (b) are the same as in (a). (c) Ratio of calculated oscillations amplitude, assuming energy-independent IMFP, to amplitude determined from experiment. Triangles, circles, and squares are from 5, 6, and 7 ML data, respectively. Zero denotes the vacuum level.

Figure 4

(a) Inelastic mean free path vs energy calculated from Fe reflectivity curves. The black full line is a fit to the mean values. (b) Reflectivity vs energy for 6 ML Fe derived from the experiment (dots) and the model (solid) with $L$ values derived from the fit in (a).

Figure 5

Example spectra showing the tungsten $4f_{7/2}$ core level as a function of Fe thickness at a photon energy of 46 eV. The bottom inset shows a LEEM image of the submonolayer Fe/W(110) with dark contrast corresponding to the Fe monolayer. The top inset shows the exponential dependence of the W signal vs Fe thickness. The intensity of each peak is found by fitting a Doniach-Šunjić function (Ref. 16) as shown by solid lines along with the data.

Figure 6

IMFP determined by reflectivity (black dots) and photoemission (red squares) experiments. Green circles, triangle, and star are from Refs. 17, 18, 19, respectively, blue diamonds Ref. 24.