Energy distribution of elastically scattered electrons from double layer samples

Abstract

We present a theoretical description of the spectra of electrons elastically scattered from thin double layered Au–C samples. The analysis is based on the Monte Carlo simulation of the recoil and Doppler effects in reflection and transmission geometries of the scattering at a fixed angle of 44.3$°$ and a primary energy of 40 keV. The relativistic correction is taken into account. Besides the experimentally measurable energy distributions the simulations give many partial distributions separately, depending on the number of elastic scatterings (single, and multiple scatterings of different types). Furthermore, we present detailed analytical calculations for the main parameters of the single scattering, taking into account both the ideal scattering geometry, i.e. infinitesimally small angular range, and the effect of the real, finite angular range used in the measurements. We show our results for intensity ratios, peak shifts and broadenings for four cases of measurement geometries and layer thicknesses. While in the peak intensity ratios of gold and carbon for transmission geometries were found to be in good agreement with the results of the single scattering model, especially large deviations were obtained in reflection geometries. The separation of the peaks, depending on the geometry and the thickness, generally smaller, and the peak width generally larger than it can be expected from the nominal values of the primary energy, scattering angle, and mean kinetic energy of the atoms. We also show that the peaks are asymmetric even for the case of the single scattering due to the finite solid angle. Finally, we present a qualitative comparison with the experimental data. We find our resulting energy distribution of elastically scattered electrons to be in good agreement with recent measurements.

Introduction

In recent times the recoil energies of scattered electrons for atoms with large mass differences can be well resolved by using an energetic electron beam in the range of a few keV [1], [2], [3], [4], [5], [6] to a few tens of keV, and with large scattering angles in the measurements [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18]. This technique is called as Electron Rutherford Backscattering Spectroscopy (ERBS), which relies on the quasi-elastic electron-atom scattering. In this case, we take advantage of the fact that the energy of the elastically scattered electrons is shifted from the primary values, due to the momentum transfer between the primary electron and the target atoms (recoil effect), and thereby the peak, due to electrons scattered elastically, splits into component peaks, which can be associated with the electrons scattered mainly from different target atoms of the sample, respectively. Furthermore, the thermal motion of the scattering atoms causes broadening in the primary electron energy distribution, usually referred to as Doppler broadening. So, from the accurate determination of the full width at half maximum (FWHM) of the peaks, the average kinetic energy of the atoms in a solid can be determined. Moreover, from the accurate peak shape analysis we can determine the Compton profile [19], [20] or we can prognosticate different fine interaction processes, such as, final state interactions.

Just recently this technique was recommended as an alternative technique of the determination of the hydrogen content at surfaces [2], [3]. These investigations may require such an analysis to improve the understanding of the processes that involve the presence of H atoms at surfaces in polymers, carbon based hard coatings, or new H storage materials. This procedure has also been used to check surface degradation in several polymers [5]. Filippi and Calliari extended this strategy of the quantification of H at more complex surfaces, containing other atoms like O [6].

Following the same principles but using higher electron kinetic energies extend the information depth during the investigations. Besides the fundamental research interest, applying electrons with relativistic energies ($E>10$ keV), can highlight important technical applications. For example, from the measurement of the relative peak intensities we can estimate the thickness of the layer from where the signal originates. So far, many experiments have been done also in the relativistic electron energy range. Vos and Went [12], [14] proposed to study surface atomic composition and vibrational properties. They also used various kinds of specimen, such as bulk elements [10], [11], overlayer/substrate systems [10], [11], [12], [13], alloy and composite bulks [11], [14], to measure the recoil energies, the Doppler broadening of elastic peaks, and especially the peak intensity ratio between different atoms. However, according to our best knowledge, the detailed theoretical analysis, taking into accounts the effect of the different types of scattering on the measurable quantity, and thereby the test of the validity of the single scattering approach, which is used mostly in the interpretation of the experimental data, is still missing.

In this work, accurate Monte Carlo simulations are presented for double layer samples in order to simulate the spectrum of elastically scattered electrons having 40 keV primary energy. The analysis was based on the Monte Carlo simulations of the recoil and Doppler effects in reflection and transmission geometries of the scattering at a fixed scattering angle of 44.3$°$. The relativistic correction was also taken into account. Besides the experimentally measurable energy distributions we present many partial distributions separately, depending on the number of elastic scatterings (single, and multiple scatterings of different types). Furthermore, we present detailed analytical calculations for the main parameters of the single scattering taking into account both the ideal scattering geometry, i.e. infinitesimally small angular range, and the effect of the real, finite angular range used in the measurements. We show our results for intensity ratios, peak shifts and broadenings in four cases of measurement geometries and layer thicknesses. The effects of the multiple and mixed scatterings on the parameters of the elastic peak is also investigated. Finally, we make an attempt to compare our Monte Carlo results with the experimental results measured by Vos and Went [16]. Here we would like to note that some of the input data of our Monte Carlo calculation may differ slightly from the data realized in the given experiments (see, for example, the real solid angle during the measurement or the accurate knowledge of the thickness of the layers).