# Lineal Energy of Proton in Silicon by a Microdosimetry Simulation
## 3 Results and Discussions  
### Effect of SV Thickness on \(y\) Distribution  
[PERSON] et al. [11] showed that smaller feature sizes have smaller SEE thresholds. In this study, six SV thicknesses were simulated to mimic different feature sizes. Figure 2 shows the effect of different SV thickness on the lineal energy (\(y\)) distribution. In the 10,000 nm case (green line), no secondary particle has \(y\) > 10 MeV-cm\({}^{2}\)/mg. The \(\varepsilon\) for most secondary particles (more than 99.999%) were less than 2.33 MeV, corresponding to 1 MeV-cm\({}^{2}\)/mg. Let us consider the secondaries with \(y\) > 10 MeV-cm\({}^{2}\)/mg for other SV thickness. Their differential fluences decreases as the SV thickness decreases from 1000 nm to 10 nm, and this trend can be explained as follows. The \(y\) value of a particle is usually maximized when the SV thickness is close to the particle range, owing to a pronounced nature named Bragg peak. The Bragg peak describes that the heavy charged particles have little energy loss as they enter the material and then peak before the end of their path. In Figure 3, the cumulative distribution function of kinetic energy of secondary particles generated by 200 MeV proton irradiated on silicon is plotted. In this figure, more than 80% of secondary particles with atomic number (Z) larger than 2 have the kinetic energy lower than 10 MeV and 99% lower than 20 MeV. The continuous slowing down approximation (CSDA) ranges of these particles are about few thousand nm. During the slowing down process, the stopping power of particles increases as they loss their kinetic energy. The stopping power reach maximum at the end of the particle's range. When the SV size close to the particle range, it can cover the maximum stopping power. Since the particle range is approximately 1 \(\upmu\)m, the \(y\) values for the case of SV thickness = 1000 nm is larger than that for SV = 100 nm, and then 10 nm. On the other hand, when the thickness decreases to 1 nm, the differential fluences increases significantly, and the reason will be discussed later using  
Figure 2: Lineal energy spectra of different sensitive volumes in silicon irradiated by a 200 MeV proton beam. a CROSSER and STOPPER theory [18]. As the boundary of the CROSSER and STOPPER will be visible only at lower LET, hence this theory does not apply in this discussion where \(y>10\) MeV-cm\({}^{2}\)/mg. For the case of \(y<10\) MeV-cm\({}^{2}\)/mg, Figure 2 shows that the differential fluences decrease first when thickness decreased from 10,000 nm to 1000 nm, and then 100 nm. However, it increases when SV thickness continues to decrease from 100 nm to 10 nm and 1 nm. To understand the above-mentioned trends, additional SV thicknesses of 20, 40 and 50 nm are simulated respectively. Figure 4 plots the spectra in log scale, which can present the events with smaller \(y\). In these spectra, \"shoulders\" can be seen for SV thickness from 10-100 nm, and the \"shoulder\" left shift with the SV thickness. The shoulder of 10 nm case can be found in Figure 2 which is around \(y=2\)-3 MeV-cm\({}^{2}\)/mg. The presence of this \"shoulder\" can be understood from the CROSSER and STOPPER theory [18] as elaborated below. When particles travel through different SVs, they can end up into either CROSSERs or STOPPERs. A CROSSER refers to particle with high enough energy that can go through the  
Figure 4: Lineal energy spectra of different sensitive volumes in silicon irradiated by a 200 MeV proton beam (log \(y\) scale). Figure 3: Cumulative distribution function of kinetic energy of secondary particles generated by 200 protons irradiated on silicon. entire SV, and a STOPPER refers to particle that stop inside SV because of its insufficient energy [18]. In Figures 2 and 4, the events on the left side of shoulder are contributed from both the CROSSER and STOOPER, but the right side can only be from the CROSSER.