1102 Etienne Brodu and Emmanuel Bouzy


Figure 6. Depth resolution dependence with sample thickness at 25 keV by on-axis transmission Kikuchi diffraction in scanning electron microscope for cubic Si. The values are the same as in Figure 5, only displayed differently.


methodology presented in Methodology for the Determina- tion of the Depth Resolution section is presented in Figure 5 for cubic silicon. The dependence with sample thickness could be studied thanks to the thickness gradient produced during the FIB thinning of the Si lamella (Fig. 3). In the low incident energy range, large thicknesses could not be tested because of the disappearing of diffraction spots from pat- terns, spots on which we rely for the determination of the depth resolution and sample thickness (see Methodology for the Determination of the Depth Resolution section). For example, at 10 keV on Si, a thickness of about 100nm is the upper limit that could be tested, in agreementwith a previous study on the thickness and energy dependence of the absorption of diffraction spots (Brodu et al., 2017). A larger range of thickness could be tested at 30 keV. The uncertainty displayed in Figure 5 for the depth resolution almost fully results from the difficulty to determine when the Kikuchi diffraction associated to the top crystal is no more present in patterns. Similarly, the uncertainty for the local sample thickness results from the determination of the first emer- gence of spots. According to the results presented in Figures 5 and 6, the depth resolution seems largely inde- pendent from the sample thickness in the range 70–200 nm. The absence of dependence with sample thickness of the depth resolution is good news because the sample thickness is often unknown in everyday TKD experiments. On the other hand, the depth resolution presents a clear dependence with incident energy in the range 10–30 keV, with a close to linear dependence, as discussed in the next section. This dependence opens the door to the tuning of the depth reso- lution via the choice of incident energy, although it might be difficult to take advantage of this dependence because of the simultaneous, and opposite effect on the lateral resolution.


Towards a Physical Model for the Depth Resolution


Diffraction Pattern Evolution Along a Straight Line Crossing the Twin Boundary section showed that the Kikuchi dif- fraction produced by a crystal disappears progressively from


diffraction patterns when another crystal of increasing thickness is added underneath it. In particular, when the bottom crystal thickness reaches the depth resolution, the Kikuchi diffraction produced by the top crystal is no more visible in patterns.Wecan expect this behavior to result from the subsequent interaction of the Kikuchi diffraction pro- duced by the top crystal along the travel inside the bottom crystal. These interactions are of two main types: (1) inco- herent interaction with plasmon and phonon (other electron–electron interactions can be neglected) and (2) elastic interaction. In the case of Bragg diffraction, the con- tribution of the elastic interaction to the depletion of the wave produced by the top crystal along the travel inside the bottom crystal is clearly visible in Figure 7, with the presence of spots resulting from double diffraction. All three electron– matter interactions involved here (elastic, plasmon, phonon) have in common a close to linear dependence with energy of their mean free paths (Hall & Hirsch, 1965; Crewe et al., 1970; Williams & Barry Carter, 1996; Mayol & Salvat, 1997; Tanuma et al., 2011; Shinotsuka et al., 2015). According to this dependence of the mean free paths, if we imagine that a specific number of interactions is necessary to make the Kikuchi diffraction associated to the top crystal disappear from patterns, then it could be expected that the depth resolution measured would also show a close to linear dependence with energy. In addition, in this scenario, the depth resolution should be independent of the total sample thickness. This latter prediction was confirmed in Depth Resolution Dependence With Sample Thickness and Inci- dent Energy section. As for the former prediction, Figure 5 shows that the mean free path for plasmon (Shinotsuka et al., 2015) and elastic scattering (Mayol & Salvat, 1997), as a


function of incident energy multiplied by a constant number, fits the experimental results of depth resolution. Yamamoto (1977) also identified a linear dependence of the depth resolution in electron channeling patterns in the SEM on Al and Cr. The meaning of the multiplication constant is here straightforward: this constant consists in the average number of collisions per electron occurring along the travel path corresponding to the depth resolution; 1.5 plasmon and 2.6 elastic interactions here for cubic Si. Values of the mean free path for phonon scattering could not be found for Si in the right energy range, hence, it is not plotted. Since the energy dependence of all three interactions have similar dependen- cies with incident energy, it is not possible to discriminate between the respective responsibilities of these interactions from this study (i.e., to attribute the depth resolution to one interaction over another). At low energy (≤15 keV), the values seem to drift apart from the model, but this might very well be explained by the beam broadening. Indeed, it is expected that the beam broadening at low energy would lead to an overestimation of the depth resolution, as illustrated in Figure 8 via Monte Carlo simulations [CASINO (Drouin et al., 2007)]. Although Figure 8 is mostly intended for illustration, it is accurate in terms of geometry and electron trajectory. From it, one can tell that the beam broadening could very well induce a bias in the determination of the


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