+For the Monte Carlo simulation the target is divided into cells with a cube length of $3 \, nm$.
+Each cell is either in a crystalline or amorphous state and stores the local carbon concentration.
+The simulation starts with a complete crystalline target and zero carbon concentration.
+
+The simulation algorithm consists of three parts.
+In a first amorphisation/re-crystallisation step random numbers are computed to specify the volume at position $\vec{r}$ in which a collision occurs.
+Two uniformly distributed random numbers $x$ and $y$ are generated to determine the lateral position of $\vec{r}$.
+Using the rejection method a random number $z$ specifying the depth coordinate of $\vec{r}$ is distributed according to the nuclear stopping power profile which, as will be seen below, is identical to the number of collisions caused by the ions per depth.
+The local amorphisation or recrystallisation probability is computed as detailed below and another random number between $0$ and $1$ decides whether there is amorphisation or recrystallisation or the state of that volume is unchanged.
+This step is repeated for the mean number of steps of cells in which collisions are caused by one ion, gained from {\em TRIM} [12] collision data.
+In a second step, the ion gets incorporated in the target at randomly chosen coordinates with the depth coordinate being distributed according to the {\em TRIM} implantation profile.
+In a last step the carbon diffusion, controlled by two simulation parameters $d_v$ and $d_r$, as well as sputtering, controlled by the parameter $n$ are treated.
+Every $d_v$ simulation steps, a fraction $d_r$ of the amount of carbon in crystalline volumes gets transfered to an amorphous neighbour in order to allow for a reduction of the supersaturation of carbon in crystalline volumes.
+Every $n$ steps a crystalline, carbon-free layer is inserted at the bottom of the cell array while the first layer is removed, where $n$ results from a RBS derived [5] sputter rate.
+
+In order to calculate the amorphisation probability, three factors have to be taken into account corresponding to our model.
+In the simulation, each of these mechanisms contributes to a local amorphisation probability of the cell at $\vec{r}$.
+The strength of each mechanism is controlled by simulation parameters.
+The local amorphisation probability at volume $\vec{r}$ is calculated by
+\begin{equation}
+p_{c \rightarrow a}(\vec{r}) = p_b + p_c c_C(\vec{r}) + \sum_{\textrm{amorphous neighbours}} \frac{p_s c_C(\vec{r'})}{(r-r')^2} \textrm{ .}
+\end{equation}
+
+The normal (ballistic) amorphisation is controlled by $p_b$ and is set constant.
+This choice is justified by analysing {\em TRIM} collision data that show identical depth profiles for the number of collisions per depth and the nuclear stopping power.
+Thus, on average an ion is loosing a constant energy per collision.
+The carbon induced amorphisation is proportional to the local amount of carbon $c_C(\vec{r})$ and controlled by weight factor $p_c$.
+The stress enhanced amorphisation is weighted by $p_s$.
+The forces originating from the amorphous volumes $\vec{r'}$ in the vicinity of $\vec{r}$ are assumed to be proportional to the amount of carbon $c_C(\vec{r'})$ in the neighbour cell.
+The sum is limited to volumes located in the same layer because of stress relaxation towards the surface. Since the stress amplitude is decreasing with the square of the distance $r-r'$, a cutoff radius is used in the simulation.
+If an amorphous volume is hit by collisions, a recrystallisation probability is given by
+\begin{equation}
+p_{a \rightarrow c}(\vec r) = (1 - p_{c \rightarrow a}(\vec r)) \Big(1 - \frac{\sum_{direct \, neighbours} \delta (\vec{r'})}{6} \Big) \, \textrm{,}
+\end{equation}
+\[
+\delta (\vec r) = \left\{
+\begin{array}{ll}
+ 1 & \textrm{if the cell at position $\vec r$ is amorphous} \\
+ 0 & \textrm{otherwise} \\
+\end{array}
+\right.
+\]
+which is basically $1$ minus the amorphisation probability and a term taking into account the crystalline neighbourhood which is needed for epitaxial recrystallisation.
+