+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/recrystallisation 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 amorphization or recrystallization probability is computed as detailed below and another random number between $0$ and $1$ decides whether there is amorphization or recrystallization 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 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 amorphization probability and a term taking into account the crystalline neighbourhood which is needed for epitaxial recrystallization.
+