# Lineal Energy of Proton in Silicon by a Microdosimetry Simulation
## 1 Introduction
[PERSON] used a set of heavy ion with specific kinetic energy to derive the single event cross sections and predict the single event upset rate of a 0.25 \(\upmu\)m CMOS SRAM well [14]. As the feature size reduced to a few nanometres, this \"single-value assumption\" may no longer valid. [PERSON] et al. found that in a 90 nm CMOS irradiated with heavy ions, although the effective LET is the same, the single event cross sections can have more than three order of magnitude differences [15]. In other words, LET distributions can be different in various feature sizes of the devices under an exact irradiation condition. With today nano-meter microelectronic devices, the variability of LET from a given radiation can be very large, and this leads to the necessity to use the concept of lineal energy distribution to predict SEE in a given device. In fact, LET is commonly used in the cm/mm scale for deterministic events. For microelectronics, microdosimetry is more appropriate to describe the stochastic events. Down to nanometre scale, many researchers used realistic track structure to predict local effect of different radiations [16; 17; 18; 19]. This track structure approach will be practical in the simulation, but it is still difficult to compare with measurements. In addition, these track structure codes can only simulate interactions in limited materials and semiconductor materials is yet to be included. Furthermore, the track structure will only be meaningful when detailed geometry of a specific microelectronic device is given. On the other hand, microdosimetry is easier to be modelled for radiation effect prediction [20]. We have used this microdosimetry approach to investigate the equivalence of neutrons and protons in SEEs testing [21]. ICRU report 16 considered that the local energy density and individual event size will be more relevant than LET for microscopic structures [6]. These concepts of local energy density and individual event size have been clarified in the ICRU report 36 for the introduction of microdosimetry [22]. The basic concepts of microdosimetry are developed by [PERSON] and co-workers [23; 24], and they used two stochastic quantities, specific energy (\(z\)) and lineal energy (\(y\)) to replace dose and LET. Lineal energy (\(y\)) of microdosimetry is a way to estimate radiation quality/impact microscopically [22], and it is defined as,  
\[y=\frac{\epsilon}{\bar{l}}=\frac{\sum_{i}\epsilon_{i}}{\bar{l}} \tag{2}\]  
where the parameters in the equation is defined below. When a radiation particle enters a target, it will interact with the target material along its track in the material. Energy deposit \(\epsilon_{i}\) is defined as the energy deposited in \(i^{th}\) \"Single\" interaction. The sum of the \(\epsilon_{i}\) in the target within one track is defined as the energy imparted (\(\epsilon\)). Mean chord length (\(\bar{l}\)) is the mean length of randomly oriented chords in a given volume [22]. In large size target, the number of interactions will be large, and the distribution of \(y\) will converge to an expected value, which is LET. However, at the nanometric scale, the number of interactions is small, and only very few interaction events occur in a radiation track. Thus, \(y\) would have a wide distribution. In this case, providing discrete values of LET will be meaningless because different \(y\) values would have different contributions to SEEs and the relationship is non-linear. Monte Carlo simulation is commonly used in microdosimetry studies [25; 26], and it relies on many physical models among which nuclear interaction physics is the most important for microdosimetry study on SEE. However, a Monte Carlo simulation with detailed nuclear interaction is very time and computation resources consuming. When computing power was expensive in the past, secondary particles generation and their transport in silicon could not be included in the simulation. For example, [PERSON] et al.