+Fig.~\ref{fig:md:pc_si-c} displays the Si-C radial distribution function for all three insertion volumes together with the Si-C bonds as observed in a \ci{} \hkl<1 0 0> DB configuration.
+The first peak observed for all insertion volumes is at approximately \unit[0.186]{nm}.
+This corresponds quite well to the expected next neighbor distance of \unit[0.189]{nm} for Si and C atoms in 3C-SiC.
+By comparing the resulting Si-C bonds of a \ci{} \hkl<1 0 0> DB with the C-Si distances of the low concentration simulation, it is evident that the resulting structure of the $V_1$ simulation is clearly dominated by this type of defect.
+This is not surprising, since the \ci{} \hkl<1 0 0> DB is found to be the ground-state defect of a C interstitial in c-Si and, for the low concentration simulations, a C interstitial is expected in every fifth Si unit cell only, thus, excluding defect superposition phenomena.
+The peak distance at \unit[0.186]{nm} and the bump at \unit[0.175]{nm} corresponds to the distance $r(3C)$ and $r(1C)$ as listed in Table~\ref{tab:defects:100db_cmp} and visualized in Fig.~\ref{fig:defects:100db_cmp}.
+In addition, it can be easily identified that the \ci{} \hkl<1 0 0> DB configuration contributes to the peaks at about \unit[0.335]{nm}, \unit[0.386]{nm}, \unit[0.434]{nm}, \unit[0.469]{nm} and \unit[0.546]{nm} observed in the $V_1$ simulation.
+Not only the peak locations but also the peak widths and heights become comprehensible.
+The distinct peak at \unit[0.26]{nm}, which exactly matches the cut-off radius of the Si-C interaction, is again a potential artifact.
+
+For high C concentrations, i.e. the $V_2$ and $V_3$ simulation corresponding to a C density of about 8 atoms per c-Si unit cell, the defect concentration is likewise increased and a considerable amount of damage is introduced in the insertion volume.
+The consequential superposition of these defects and the high amounts of damage generate new displacement arrangements for the C-C as well as for the Si-C pair distances, which become hard to categorize and trace and obviously lead to a broader distribution.
+Short range order indeed is observed, i.e. the large amount of strong neighbored C-C bonds at \unit[0.15]{nm} as expected in graphite or diamond and Si-C bonds at \unit[0.19]{nm} as expected in SiC, but only hardly visible is the long range order.
+This indicates the formation of an amorphous SiC-like phase.
+In fact the resulting Si-C and C-C radial distribution functions compare quite well with these obtained by cascade amorphized and melt-quenched amorphous SiC using a modified Tersoff potential \cite{gao02}.
+
+In both cases, i.e. low and high C concentrations, the formation of 3C-SiC fails to appear.
+With respect to the precipitation model, the formation of C$_{\text{i}}$ \hkl<1 0 0> DBs indeed occurs for low C concentrations.
+However, sufficient defect agglomeration is not observed.
+For high C concentrations, a rearrangement of the amorphous SiC structure, which is not expected at prevailing temperatures, and a transition into 3C-SiC is not observed either.
+
+\section{Limitations of conventional MD and short range potentials}
+\label{section:md:limit}
+
+Results of the last section indicate possible limitations of the MD method regarding the task addressed in this study.
+Low C concentration simulations do not reproduce the agglomeration of C$_{\text{i}}$ \hkl<1 0 0> DBs.
+High concentration simulations result in the formation of an amorphous SiC-like phase, which is unexpected since IBS experiments show crystalline 3C-SiC precipitates at prevailing temperatures.
+Keeping in mind the results
+On closer inspection, however, two reasons for describing this obstacle become evident, which are discussed in the following.
+
+The first reason is a general problem of MD simulations in conjunction with limitations in computer power, which results in a slow and restricted propagation in phase space.
+In molecular systems, characteristic motions take place over a wide range of time scales.
+Vibrations of the covalent bond take place on the order of \unit[10$^{-14}$]{s}, of which the thermodynamic and kinetic properties are well described by MD simulations.
+To avoid discretization errors, the integration time step needs to be chosen smaller than the fastest vibrational frequency in the system.
+On the other hand, infrequent processes, such as conformational changes, reorganization processes during film growth, defect diffusion and phase transitions are processes undergoing long-term evolution in the range of microseconds.
+This is due to the existence of several local minima in the free energy surface separated by large energy barriers compared to the kinetic energy of the particles, i.e. the system temperature.
+Thus, the average time of a transition from one potential basin to another corresponds to a great deal of vibrational periods, which in turn determine the integration time step.
+Hence, time scales covering the necessary amount of infrequent events to observe long-term evolution are not accessible by traditional MD simulations, which are limited to the order of nanoseconds.
+New methods have been developed to bypass the time scale problem.
+The most famous approaches are hyperdynamics (HMD) \cite{voter97,voter97_2}, parallel replica dynamics \cite{voter98}, temperature accelerated dynamics (TAD) \cite{sorensen2000} and self-guided dynamics (SGMD) \cite{wu99}, which accelerate phase space propagation while retaining proper thermodynamic sampling.
+
+In addition to the time scale limitation, problems attributed to the short range potential exist.
+The sharp cut-off function, which limits the interacting ions to the next neighbored atoms by gradually pushing the interaction force and energy to zero between the first and second next neighbor distance, is responsible for overestimated and unphysical high forces of next neighbored atoms \cite{tang95,mattoni2007}.
+This is supported by the overestimated activation energies necessary for C diffusion as investigated in section \ref{subsection:defects:mig_classical}.
+Indeed, it is not only the strong C-C bond, which is hard to break, inhibiting C diffusion and further rearrangements.
+This is also true for the low concentration simulations dominated by the occurrence of C-Si DBs spread over the whole simulation volume.
+The bonds of these C-Si pairs are also affected by the cut-off artifact preventing C diffusion and agglomeration of the DBs.
+This can be seen from the almost horizontal progress of the total energy graph in the continuation step of Fig.~\ref{fig:md:energy_450}, even for the low concentration simulation.
+These unphysical effects inherent to this type of model potentials are solely attributed to their short range character.
+While cohesive and formational energies are often well described, these effects increase for non-equilibrium structures and dynamics.
+However, since valuable insights into various physical properties can be gained using this potentials, modifications mainly affecting the cut-off were designed.
+One possibility is to simply skip the force contributions containing the derivatives of the cut-off function, which was successfully applied to reproduce the brittle propagation of fracture in SiC at zero temperature \cite{mattoni2007}.
+Another one is to use variable cut-off values scaled by the system volume, which properly describes thermomechanical properties of 3C-SiC \cite{tang95} but might be rather ineffective for the challenge inherent to this study.
+
+To conclude the obstacle needed to get passed is twofold.
+The sharp cut-off of the employed bond order model potential introduces overestimated high forces between next neighbored atoms enhancing the problem of slow phase space propagation immanent to MD simulations.
+This obstacle could be referred to as {\em potential enhanced slow phase space propagation}.
+Due to this, pushing the time scale to the limits of computational resources or applying one of the above mentioned accelerated dynamics methods exclusively will not be sufficient enough.
+
+Instead, the approach followed in this study, is the use of higher temperatures as exploited in TAD to find transition pathways of one local energy minimum to another one more quickly.
+Since merely increasing the temperature leads to different equilibrium kinetics than valid at low temperatures, TAD introduces basin-constrained MD allowing only those transitions that should occur at the original temperature and a properly advancing system clock \cite{sorensen2000}.
+The TAD corrections are not applied in coming up simulations.
+This is justified by two reasons.
+First of all, a compensation of the overestimated bond strengths due to the short range potential is expected.
+Secondly, there is no conflict applying higher temperatures without the TAD corrections, since crystalline 3C-SiC is also observed for higher temperatures than \unit[450]{$^{\circ}$C} in IBS \cite{nejim95,lindner01}.
+It is therefore expected that the kinetics affecting the 3C-SiC precipitation are not much different at higher temperatures aside from the fact that it is occurring much more faster.
+Moreover, the interest of this study is focused on structural evolution of a system far from equilibrium instead of equilibrium properties which rely upon proper phase space sampling.
+On the other hand, during implantation, the actual temperature inside the implantation volume is definitely higher than the experimentally determined temperature tapped from the surface of the sample.
+
+\section{Increased temperature simulations}
+\label{section:md:inct}
+
+Due to the limitations of short range potentials and conventional MD as discussed above, elevated temperatures are used in the following.
+Increased temperatures are expected to compensate the overestimated diffusion barriers.
+These are overestimated by a factor of 2.4 to 3.5.
+Scaling the absolute temperatures accordingly results in maximum temperatures of \unit[1460-2260]{$^{\circ}$C}.
+Since melting already occurs shortly below the melting point of the potential (\unit[2450]{K}) \cite{albe_sic_pot} due to the presence of defects, temperatures ranging from \unit[450-2050]{$^{\circ}$C} are used.
+The simulation sequence and other parameters except for the system temperature remain unchanged as in section \ref{section:initial_sims}.
+Since there is no significant difference among the $V_2$ and $V_3$ simulations only the $V_1$ and $V_2$ simulations are carried on and referred to as low C and high C concentration simulations.
+
+A simple quality value $Q$ is introduced, which helps to estimate the progress of structural evolution.
+In bulk 3C-SiC every C atom has four next neighbored Si atoms and every Si atom four next neighbored C atoms.
+The quality could be determined by counting the amount of atoms, which form bonds to four atoms of the other species.
+However, the aim of the simulation is to reproduce the formation of a 3C-SiC precipitate embedded in c-Si.
+The amount of Si atoms and, thus, the amount of Si atoms remaining in the c-Si diamond lattice is much higher than the amount of inserted C atoms.
+Thus, counting the atoms, which exhibit proper coordination, is limited to the C atoms.
+The quality value is defined to be
+\begin{equation}
+Q = \frac{\text{Amount of C atoms with 4 next neighbored Si atoms}}
+ {\text{Total amount of C atoms}} \text{ .}
+\label{eq:md:qdef}
+\end{equation}
+By this, bulk 3C-SiC will still result in $Q=1$ and precipitates will also reach values close to one.
+However, since the quality value does not account for bond lengths, bond angles, crystallinity or the stacking sequence, high values of $Q$ not necessarily correspond to structures close to 3C-SiC.
+Structures that look promising due to high quality values need to be further investigated by other means.
+
+\subsection{Low C concentration simulations}
+
+\begin{figure}[tp]
+\begin{center}
+\includegraphics[width=0.7\textwidth]{tot_pc_thesis.ps}\\
+\includegraphics[width=0.7\textwidth]{tot_ba.ps}
+\end{center}
+\caption[Si-C radial distribution and evolution of quality $Q$ for the low concentration simulations at different elevated temperatures.]{Si-C radial distribution and evolution of quality $Q$ according to equation \ref{eq:md:qdef} for the low concentration simulations at different elevated temperatures. All structures are cooled down to \degc{20}. The gray line shows resulting Si-C bonds in a configuration of \cs{} in c-Si (C$_\text{sub}$) at zero temperature. Arrows in the quality plot mark the end of C insertion and the start of the cooling down step. A fit function according to equation \eqref{eq:md:fit} shows the estimated evolution of quality in the absence of the cooling down sequence.}
+\label{fig:md:tot_si-c_q}
+\end{figure}
+Fig.~\ref{fig:md:tot_si-c_q} shows the radial distribution of Si-C bonds for different temperatures and the corresponding evolution of quality $Q$ as defined above for the low concentration simulation.
+The first noticeable and promising change in the Si-C radial distribution is the successive decline of the artificial peak at the Si-C cut-off distance with increasing temperature up to the point of disappearance at temperatures above \degc{1650}.
+Obviously, sufficient kinetic energy is provided to affected atoms that are enabled to escape the cut-off region.
+Additionally, a more important structural change is observed, which is illustrated in the two shaded areas in Fig.~\ref{fig:md:tot_si-c_q}.
+%
+In the gray shaded region a decrease of the peak at \unit[0.186]{nm} and of the bump at \distn{0.175} accompanied by an increase of the peak at \distn{0.197} with increasing temperature is visible.
+Similarly, the peaks at \distn{0.335} and \distn{0.386} shrink in contrast to a new peak forming at \distn{0.372} as can be seen in the yellow shaded region.
+Obviously, the structure obtained from the \degc{450} simulations, which is dominated by the existence of \ci{} \hkl<1 0 0> DBs, transforms into a different structure with increasing simulation temperature.
+Investigations of the atomic data reveal \cs{} to be responsible for the new Si-C bonds.
+The peak at \distn{0.197} corresponds to the distance of a \cs{} atom and its next neighbored Si atoms.
+The one at \distn{0.372} constitutes the distance of a \cs{} atom to the second next Si neighbor along a \hkl<1 1 0> direction.
+Comparing the radial distribution for the Si-C bonds at \degc{2050} to the resulting Si-C bonds in a configuration of a \cs{} atom in c-Si excludes all possibility of doubt.
+The resulting bonds perfectly match and, thus, explain the peaks observed for the increased temperature simulations.
+To conclude, by increasing the simulation temperature, the structure characterized by the \ci{} \hkl<1 0 0> DB structure transforms into a structure dominated by \cs{}.
+
+This is likewise reflected in the quality values obtained for different temperatures.
+While simulations at \degc{450} exhibit \perc{10} of fourfold coordinated C, simulations at \degc{2050} exceed the \perc{80} range.
+Since \cs{} has four nearest neighbored Si atoms and is the preferential type of defect in elevated temperature simulations, the increase of the quality values become evident.
+The quality values at a fixed temperature increase with simulation time.
+After the end of the insertion sequence marked by the first arrow, the quality is increasing and a saturation behavior, yet before the cooling process starts, can be expected.
+The evolution of the quality value of the simulation at \degc{2050} inside the range, in which the simulation is continued at constant temperature for \unit[100]{fs}, is well approximated by the simple fit function
+\begin{equation}
+f(t)=a-\frac{b}{t} \text{ ,}
+\label{eq:md:fit}
+\end{equation}
+which results in a saturation value of \perc{93}.
+Obviously, the decrease in temperature accelerates the saturation and inhibits further formation of \cs{}.
+\label{subsubsection:md:ep}
+Conclusions drawn from investigations of the quality evolution correlate well with findings of the radial distribution results.
+
+\begin{figure}[tp]
+\begin{center}
+\includegraphics[width=0.7\textwidth]{tot_pc2_thesis.ps}\\
+\includegraphics[width=0.7\textwidth]{tot_pc3_thesis.ps}
+\end{center}
+\caption[C-C and Si-Si radial distribution for the low concentration simulations at different elevated temperatures.]{C-C and Si-Si radial distribution for the low concentration simulations at different elevated temperatures. All structures are cooled down to $20\,^{\circ}\mathrm{C}$. Arrows with dashed lines mark C-C distances of \hkl<1 0 0> DB combinations and those with solid lines mark C-C distances of combinations of substitutional C. The dashed line corresponds to the distance of a substitutional C with a next neighbored \hkl<1 0 0> DB.}
+\label{fig:md:tot_c-c_si-si}
+\end{figure}
+The formation of \cs{} also affects the Si-Si radial distribution displayed in the lower part of Fig.~\ref{fig:md:tot_c-c_si-si}.
+Investigating the atomic structure indeed shows that the peak arising at \distn{0.325} with increasing temperature is due to two Si atoms that form direct bonds to the \cs{} atom.
+The peak corresponds to the distance of next neighbored Si atoms along the \hkl<1 1 0> bond chain with C$_{\text{s}}$ in between.
+Since the expected distance of these Si pairs in 3C-SiC is \distn{0.308}, the existing SiC structures embedded in the c-Si host are stretched.