**3. Monte Carlo Method for Two-Dimensional Imaging of Ion Track Overlapping Effect**

We have developed the following Monte Carlo algorism for the present study. The target square for the Monte Carlo calculation consisted of 1000 × 1000 grid of cells. Its dimension was defined to be 100 nm × 100 nm throughout this work; a single impact on the target square was equivalent to the ion fluence of 10<sup>10</sup> cm−<sup>2</sup> . The target was randomly bombarded with energetic ions until the ion fluence reached the given value, Φ.

Numbers of ion impacts for all cells were examined after every single bombardment to visualize the two-dimensional impact map and to calculate the fraction of ion overlapping area, *A*(Φ,*r*) for all the number of the impacts by the ion-track, *r*, to compare the results by using the Poisson distribution function. It is worth noting here that the analysis by using the Poisson distribution function ignores temporal decay and spatial distribution of an effect of an impact. Hence, these conditions can also be included in the Monte Carlo calculations.

All the Monte Carlo calculations were carried out with the homemade code written in LabVIEW 2019 (32 bit) for Windows.
