Energy Dissipation of Fast Electrons in Polymethylmethacrylate: Toward a Universal Curve for Electron-Beam Attenuation in Solids between $\sim 0\mathrm{eV}$ and Relativistic Energies

Spectroscopy of correlated electron pairs was employed to investigate the energy dissipation process, as well as the transport and the emission of low-energy electrons on a polymethylmethacrylate surface, providing secondary electron spectra causally related to the energy loss of the primary. Two groups are identified in the cascade of slow electrons, corresponding to different stages in the energy dissipation process. The characteristic lengths for attenuation due to collective excitations and momentum relaxation are quantified for both groups and are found to be distinctly different: ${\lambda }_1=(12\pm 2)\AA$ and ${\lambda }_2=(62\pm 11)\AA$. The results strongly contradict the commonly employed model of exponential attenuation with the electron inelastic mean free path as characteristic length, but they essentially agree with a theory used for decades in astrophysics and neutron transport, albeit with characteristic lengths expressed in units of angstroms rather than light-years.

Figure 1

(a) Double-differential secondary electron-electron energy-loss coincidence spectrum (SE2ELCS) for $E_0=500\mathrm{eV}$ electrons striking a PMMA surface. The red curve indicates the maximum energy in vacuum for an emitted electron created by an energy loss $\mathrm{\Delta }E$ of the primary. (b) Schematic illustration of the electronic structure. (c) Differential inverse inelastic mean free path (DIIMFP) for PMMA for energies of 11, 173, 500, and 1000 eV above the vacuum level [47]. The maximum energy loss for 11 eV electrons (above vacuum) is seen to be 15.5 eV, since no allowed states exist at energies below $E_{\mathrm{vac}}-\chi$. (d) SE spectra obtained by integrating the data in (a) over the indicated ranges of $\mathrm{\Delta }E$ [see the arrows in (a)].

Figure 2

(a) Red data points show the singles energy-loss spectra (acquired during the coincidence run) for a primary energy of $E_0=1000\mathrm{eV}$. (b) Corresponding coincidence spectra, obtained by integrating the double-differential data over $E_2$. Black curves are a fit of these data to a linear combination of multiple self-convolutions of the DIIMFP, shown by the filled colored curves. (c) Reduced partial intensities ${\gamma }_n=C_n/C_{n=1}$, where the quantities $C_n$ are the areas under the filled curves for the spectra shown in (a) and (b). The green line represents the identity ${\gamma }_n=n$.

Figure 3

(a) Experimental attenuation curves of the SE yield as a function of depth, ${\gamma }_n^{\mathrm{coi}}/(n\times {\gamma }_n^{\mathrm{sing}})(⟨z⟩)$, for primary energies of 173, 500, and 1000 eV. The red solid curves represent a fit to a double exponential function [Eq. (1)]. The inset shows the MC results for the average depth $⟨z_n⟩$ at which, on average, $n$ inelastic collisions of the primaries occur. (b) MC simulation of the results shown in (a) for 1000 eV (see text). The data in (a) and (b) were offset by multiplication to improve the distinguishability of individual curves.

Figure 4

(a) Trajectories of electrons emitted isotropically at a depth of 50 Å and reaching the surface without energy loss. Left (blue): Single scattering albedo $c\approx 1$. Right (red): $c\ll 1$. (b) Electron inelastic mean free path (IMFP, black [54]), transport mean free path (TrMFP, red), and effective attenuation length [EAL, blue, Eq. (2)]. The (magenta) circles are the results of MC model calculations for the EAL, and the (cyan) triangles are earlier experimental data for the IMFP [22]. The green diamonds represent the results for ${\lambda }_{1,2}$ derived from the data in Fig. 3. The blue shaded region represents the uncertainty in the EAL when the TrMFP is increased/decreased by a factor of 3.