+\caption{Radial distribution function for C-C and Si-Si (Fig.~\ref{fig:450:a}) as well as Si-C (Fig.~\ref{fig:450:b}) pairs for C inserted at \unit[450]{$^{\circ}$C}. In the latter case the resulting C-Si distances for a C$_{\text{i}}$ \hkl<1 0 0> DB are given additionally and the Si-C cut-off distance is marked by an arrow. Insets in Fig.~\ref{fig:450:a} show magnified regions of the respective distribution functions.}
+\label{fig:450}
+\end{figure}
+There is no significant difference between C insertion into $V_2$ and $V_3$.
+Thus, in the following, the focus is on low ($V_1$) and high ($V_2$, $V_3$) C concentration simulations only.
+
+In the low C concentration simulation the number of C-C bonds is small, as can be seen in the upper part of Fig.~\ref{fig:450:a}.
+On average, there are only 0.2 C atoms per Si unit cell.
+By comparing the Si-C peaks of the low concentration simulation with the resulting Si-C distances of a C$_{\text{i}}$ \hkl<1 0 0> DB in Fig.~\ref{fig:450:b} it becomes evident that the structure is clearly dominated by this kind of defect.
+One exceptional peak at \unit[0.26]{nm} (marked with an arrow in Fig.~\ref{fig:450:b}) exists, which is due to the Si-C cut-off, at which the interaction is pushed to zero.
+Investigating the C-C peak at \unit[0.31]{nm}, which is also available for low C concentrations as can be seen in the upper inset of Fig.~\ref{fig:450:a}, reveals a structure of two concatenated, differently oriented C$_{\text{i}}$ \hkl<1 0 0> DBs to be responsible for this distance.
+Additionally, in the inset of the bottom part of Fig.\ref{fig:450:a} the Si-Si radial distribution shows non-zero values at distances around \unit[0.3]{nm}, which, again, is due to the DB structure stretching two neighbored Si atoms.
+This is accompanied by a reduction of the number of bonds at regular Si distances of c-Si.
+A more detailed description of the resulting C-Si distances in the C$_{\text{i}}$ \hkl<1 0 0> DB configuration and the influence of the defect on the structure is available in a previous study\cite{zirkelbach09}.
+
+For high C concentrations, the defect concentration is likewise increased and a considerable amount of damage is introduced in the insertion volume.
+A subsequent superposition of defects generates new displacement arrangements for the C-C as well as 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 hardly visible is the long range order.
+This indicates the formation of an amorphous SiC-like phase.
+In fact, 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.
+On closer inspection two reasons for describing this obstacle become evident.
+
+First of all, there is the time scale problem inherent to MD in general.
+To minimize the integration error the discretized time step must be chosen smaller than the reciprocal of the fastest vibrational mode resulting in a time step of \unit[1]{fs} for the investigated materials system.
+Limitations in computer power result in a slow propagation in phase space.
+Several local minima exist, which are separated by large energy barriers.
+Due to the low probability of escaping such a local minimum, a single transition event corresponds to a multiple of vibrational periods.
+Long-term evolution, such as a phase transformation and defect diffusion, in turn, are made up of a multiple of these infrequent transition events.
+Thus, time scales to observe long-term evolution are not accessible by traditional MD.
+New accelerated methods have been developed to bypass the time scale problem retaining proper thermodynamic sampling\cite{voter97,voter97_2,voter98,sorensen2000,wu99}.
+
+However, the applied potential comes up with an additional limitation, as previously mentioned in the introduction.
+The cut-off function of the short range potential limits the interaction to nearest neighbors, which results in overestimated and unphysical high forces between neighbored atoms.
+This behavior, as observed and discussed for the Tersoff potential\cite{tang95,mattoni2007}, is supported by the overestimated activation energies necessary for C diffusion as investigated in section \ref{subsection:cmob}.
+Indeed, it is not only the strong, hard to break C-C bond inhibiting C diffusion and further rearrangements in the case of the high C concentration simulations.
+This is also true for the low concentration simulations dominated by the occurrence of C$_{\text{i}}$ \hkl<1 0 0> DBs spread over the whole simulation volume, which are unable to agglomerate due to the high migration barrier.
+
+\subsection{Increased temperature simulations}
+
+Due to the problem of slow phase space propagation, which is enhanced by the employed potential, pushing the time scale to the limits of computational resources or applying one of the above mentioned accelerated dynamics methods exclusively might not be sufficient.
+Instead, higher temperatures are utilized to compensate 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 (2450 K)\cite{albe_sic_pot} due to the presence of defects, a maximum temperature of \unit[2050]{$^{\circ}$C} is used.
+
+Fig.~\ref{fig:tot} shows the resulting radial distribution functions for various temperatures.
+\begin{figure}
+\begin{center}
+\subfigure[]{\label{fig:tot:si-c}
+\includegraphics[width=\columnwidth]{../img/tot_pc_thesis.ps}
+}
+\subfigure[]{\label{fig:tot:si-si}
+\includegraphics[width=\columnwidth]{../img/tot_pc3_thesis.ps}
+}
+\subfigure[]{\label{fig:tot:c-c}
+\includegraphics[width=\columnwidth]{../img/tot_pc2_thesis.ps}
+}
+\end{center}
+\caption{Radial distribution function for Si-C (Fig.~\ref{fig:tot:si-c}), Si-Si (Fig.~\ref{fig:tot:si-si}) and C-C (Fig.~\ref{fig:tot:c-c}) pairs for the C insertion into $V_1$ at elevated temperatures. For the Si-C distribution resulting Si-C distances of a C$_{\text{s}}$ configuration are plotted. In the C-C distribution dashed arrows mark C-C distances occurring from C$_{\text{i}}$ \hkl<1 0 0> DB combinations, solid arrows mark C-C distances of pure C$_{\text{s}}$ combinations and the dashed line marks C-C distances of a C$_{\text{i}}$ and C$_{\text{s}}$ combination.}
+\label{fig:tot}
+\end{figure}
+In Fig.~\ref{fig:tot:si-c}, the first noticeable and promising change observed for the Si-C bonds is the successive decline of the artificial peak at the cut-off distance with increasing temperature.
+Obviously, sufficient kinetic energy is provided to affected atoms that are enabled to escape the cut-off region.
+Additionally, a more important structural change was observed, which is illustrated in the two shaded areas in Fig.~\ref{fig:tot:si-c}.
+Obviously, the structure obtained at \unit[450]{$^{\circ}$C}, which was found to be dominated by C$_{\text{i}}$, transforms into a C$_{\text{s}}$ dominated structure with increasing temperature.
+Comparing the radial distribution at \unit[2050]{$^{\circ}$C} to the resulting bonds of C$_{\text{s}}$ in c-Si excludes all possibility of doubt.
+
+The phase transformation is accompanied by an arising Si-Si peak at \unit[0.325]{nm} in Fig.~\ref{fig:tot:si-si}, which 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 \unit[0.308]{nm} the existing SiC structures embedded in the c-Si host are stretched.
+
+According to the C-C radial distribution displayed in Fig.~\ref{fig:tot:c-c}, agglomeration of C fails to appear even for elevated temperatures, as can be seen on the total amount of C pairs within the investigated separation range, which does not change significantly.
+However, a small decrease in the amount of neighbored C pairs can be observed with increasing temperature.
+This high temperature behavior is promising since breaking of these diamond- and graphite-like bonds is mandatory for the formation of 3C-SiC.
+Obviously, acceleration of the dynamics occurred by supplying additional kinetic energy.
+A slight shift towards higher distances can be observed for the maximum located shortly above \unit[0.3]{nm}.
+Arrows with dashed lines mark C-C distances resulting from C$_{\text{i}}$ \hkl<1 0 0> DB combinations while arrows with solid lines mark distances arising from combinations of C$_{\text{s}}$.
+The continuous dashed line corresponds to the distance of C$_{\text{s}}$ and a neighbored C$_{\text{i}}$ DB.
+Obviously, the shift of the peak is caused by the advancing transformation of the C$_{\text{i}}$ DB into the C$_{\text{s}}$ defect.
+Quite high g(r) values are obtained for distances in between the continuous dashed line and the first arrow with a solid line.
+For the most part, these structures can be identified as configurations of C$_{\text{s}}$ with either another C atom that basically occupies a Si lattice site but is displaced by a Si interstitial residing in the very next surrounding or a C atom that nearly occupies a Si lattice site forming a defect other than the \hkl<1 0 0>-type with the Si atom.
+Again, this is a quite promising result since the C atoms are taking the appropriate coordination as expected in 3C-SiC.
+
+Fig.~\ref{fig:v2} displays the radial distribution for high C concentrations.
+\begin{figure}
+\begin{center}
+\subfigure[]{\label{fig:v2:si-c}
+\includegraphics[width=\columnwidth]{../img/12_pc_thesis.ps}
+}
+\subfigure[]{\label{fig:v2:c-c}
+\includegraphics[width=\columnwidth]{../img/12_pc_c_thesis.ps}
+}
+\end{center}
+\caption{Radial distribution function for Si-C (Fig.~\ref{fig:v2:si-c}) and C-C (Fig.~\ref{fig:v2:c-c}) pairs for the C insertion into $V_2$ at elevated temperatures. Arrows mark the respective cut-off distances.}
+\label{fig:v2}
+\end{figure}