1104 Etienne Brodu and Emmanuel Bouzy


Figure 9. Depth resolution dependence with incident energy by on-axis transmission Kikuchi diffraction in scanning electron microscope for cubic Si. The thickness of single Si crystal for which Bragg diffraction spots disappear completely from patterns is added for comparison (Brodu et al., 2017). The elastic and plasmon mean free paths are taken from Mayol & Salvat (1997) and Shinotsuka et al. (2015), respectively, and the mean absorp- tion coefficient is obtained from Reimer (1997: 309).


might be possible to have a clue on the scattering process responsible for the absorption leading to the depth resolu- tion. The inelastic and elastic mean free path at 30 keV are, respectively, 35, 25, and 19nm (Shinotsuka et al., 2015) and 24, 7.9, and 3.5nm (Mayol&Salvat, 1997) for Al, Cu, and Pt. While the depth resolution reported by Sneddon et al. divi- ded by the inelastic mean free path is far from constant with Z (2.3, 1.3, and 0.7), the depth resolution divided by the elastic mean free path is close to constant (3.4, 4.2, and 3.7). It suggest that the electron–nucleus interaction drives the absorption responsible for the depth resolution. Hence, elastic and quasi-elastic (phonon) scattering would be the major contribution, over plasmon scattering, to the depth resolution of TKD. To go further, it is well known that the damping of a


diffracted wave of intensity I along its travel path t,on average (i.e., independently from the crystallographic direc- tion), can be described by the decreasing exponential I(t)= I0 exp(−µ0 t) (Ichimiya & Lehmpfuhl, 1978; Spence & Zuo, 1992: 78). Note that the exact same dependence can be obtained by using the previous formalism with the mean free path. Indeed, the probability for a given number of inter- actions to occur along a travel path of variable length, as given by Poisson’s law, is of the same form. This exponential dependence with thickness of the damping observed in Figure 4 could not be confirmed here because profile analysis was unsuccessful (see Methodology for the Determination of


the Depth Resolution section). Still, this behavior is highly probable. It is possible to verify that the depth resolution measured and the mean absorption coefficient reported in Reimer (1997: 309) are in line: at 30 keV, a depth resolution of 65nm was measured (Fig. 5), which gives I(t)/I0=exp (−µ0 t)=0.07, for t=65nm and µ0 (E)=0.012 ×100/ E=0.041 at 30 keV (Reimer, 1997: 309). A remaining con- trast of 7% is likely to have made the diffraction contrast drop below the threshold for detection, which means that we can make a direct use of the mean absorption coefficient to evaluate the contribution of a layer at a given depth. A first example of the calculation is as follows: in the case of a division of the depth resolution at 30 keV in five layers of 65/5=13nm each, if the bottom layer produces a contrast of 100% (as the most intense contribution on the pattern, it is used as the reference), then the first layer right above it produces a relative contrast of 58%, the next 34%, then 20% and finally the last 12%. If instead the depth resolution reported in Figure 5 is divided into two layers, then the bottom one produces again a contrast of 100% and the one above it produces a relative contrast of 27%. While this cal- culation was done at 30 keV, these percentages are valid for any incident energy. Alternatively, it is possible to calculate the thickness of the surface layer producing a given percen- tage of the total diffraction contrast: at 30 keV, a layer of 17nm produces 50% of the contrast, 34nm produces 75%, and 56nm produces 90%. Finally, we can try to gather Bragg and Kikuchi dif-


fraction inside a common theoretical framework. Like any diffracted wave, Bragg spots are absorbed along their travel inside a crystal. A previous study reported the sample thickness for which Bragg spots in Si disappear from patterns as a function of incident energy (Brodu et al., 2017). At that specific sample thickness (see Fig. 9), the spots were con- sidered absorbed, and the coherence with the incident beam was lost. Unlike Kikuchi diffraction, spots could not “restart” at this point because the coherence with the incident beamis necessary. Kikuchi diffraction on the other hand can still produce diffraction even though the sample is several times thicker than the characteristic length for absorption, because the coherence with the incident beam is not necessary. Unlike spots, Kikuchi diffraction beneficiates from its own in-depth localized sources (Zaefferer, 2007; Winkelmann, 2010). This is the reason why very thick samples still produce Kikuchi diffraction by transmission while spots have long disappeared from patterns, as well as why there is a depth resolution with Kikuchi diffraction, that we can make use of for orientation mapping, while there is none with spots. Another difference lies in the absorption length: it was determined that an average of 7.8 plasmon collisions were necessary for the Bragg diffraction spots to disappear from patterns produced by a single Si crystal (Brodu et al., 2017) (the corresponding number of phonon collisions could not be determined), while it is about 1.5 for Kikuchi diffraction according to the present study. We attribute this difference to the fact that Bragg spots are the direct reflections of the incident beam, while Kikuchi diffraction is not as direct and


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