-\begin{figure}[!ht]
-\begin{center}
-\includegraphics[width=12cm]{pc_0.ps}
-\end{center}
-\caption[Radial distribution of a 3C-SiC precipitate embeeded in c-Si at $20\,^{\circ}\mathrm{C}$.]{Radial distribution of a 3C-SiC precipitate embeeded in c-Si at $20\,^{\circ}\mathrm{C}$. The Si-Si radial distribution of plain c-Si is plotted for comparison. Green arrows mark bumps in the Si-Si distribution of the precipitate configuration, which do not exist in plain c-Si.}
-\label{fig:md:pc_sic-prec}
-\end{figure}
-Figure \ref{fig:md:pc_sic-prec} shows the radial distribution of the obtained precipitate configuration.
-The Si-Si radial distribution for both, plain c-Si and the precipitate configuration show a maximum at a distance of 0.235 nm, which is the distance of next neighboured Si atoms in c-Si.
-Although no significant change of the lattice constant of the surrounding c-Si matrix was assumed, surprisingly there is no change at all within observational accuracy.
-Looking closer at higher order Si-Si peaks might even allow the guess of a slight increase of the lattice constant compared to the plain c-Si structure.
-A new Si-Si peak arises at 0.307 nm, which is identical to the peak of the C-C distribution around that value.
-It corresponds to second next neighbours in 3C-SiC, which applies for Si as well as C pairs.
-The bumps of the Si-Si distribution at higher distances marked by the green arrows can be explained in the same manner.
-They correspond to the fourth and sixth next neighbour distance in 3C-SiC.
-It is easily identifiable how these C-C peaks, which imply Si pairs at same distances inside the precipitate, contribute to the bumps observed in the Si-Si distribution.
-The Si-Si and C-C peak at 0.307 nm enables the determination of the lattic constant of the embedded 3C-SiC precipitate.
-A lattice constant of 4.34 \AA{} compared to 4.36 \AA{} for bulk 3C-SiC is obtained.
-This is in accordance with the peak of Si-C pairs at a distance of 0.188 nm.
-Thus, the precipitate structure is slightly compressed compared to the bulk phase.
-This is a quite surprising result since due to the finite size of the c-Si surrounding a non-negligible impact of the precipitate on the materializing c-Si lattice constant especially near the precipitate could be assumed.
-However, it seems that the size of the c-Si host matrix is chosen large enough to even find the precipitate in a compressed state.
-
-The absence of a compression of the c-Si surrounding is due to the possibility of the system to change its volume.
-Otherwise the increase of the lattice constant of the precipitate of roughly 4.31 \AA{} in the beginning up to 4.34 \AA{} in the relaxed precipitate configuration could not take place without an accompanying reduction of the lattice constant of the c-Si surrounding.
-If the total volume is assumed to be the sum of the volumes that are composed of Si atoms forming the c-Si surrounding and Si atoms involved forming the precipitate the expected increase can be calculated by
-\begin{equation}
- \frac{V}{V_0}=
- \frac{\frac{N^{\text{c-Si}}_{\text{Si}}}{8/a_{\text{c-Si of precipitate configuration}}}+
- \frac{N^{\text{3C-SiC}}_{\text{Si}}}{4/a_{\text{3C-SiC of precipitate configuration}}}}
- {\frac{N^{\text{total}}_{\text{Si}}}{8/a_{\text{plain c-Si}}}}
-\end{equation}
-with the notation used in table \ref{table:md:sic_prec}.
-The lattice constant of plain c-Si at $20\,^{\circ}\mathrm{C}$ can be determined more accurately by the side lengthes of the simulation box of an equlibrated structure instead of using the radial distribution data.
-By this a value of $a_{\text{plain c-Si}}=5.439\text{ \AA}$ is obtained.
-The same lattice constant is assumed for the c-Si surrounding in the precipitate configuration $a_{\text{c-Si of precipitate configuration}}$ since peaks in the radial distribution match the ones of plain c-Si.
-Using $a_{\text{3C-SiC of precipitate configuration}}=4.34\text{ \AA}$ as observed from the radial distribution finally results in an increase of the initial volume by 0.12 \%.
-However, each side length and the total volume of the simulation box is increased by 0.20 \% and 0.61 \% respectively compared to plain c-Si at $20\,^{\circ}\mathrm{C}$.
-Since the c-Si surrounding resides in an uncompressed state the excess increase must be attributed to relaxation of strain with the strain resulting from either the compressed precipitate or the 3C-SiC/c-Si interface region.
-This also explains the possibly identified slight increase of the c-Si lattice constant in the surrounding as mentioned earlier.
-As the pressure is set to zero the free energy is minimized with respect to the volume enabled by the Berendsen barostat algorithm.
-Apparently the minimized structure with respect to the volume is a configuration of a small compressively stressed precipitate and a large amount of slightly stretched c-Si in the surrounding.
-
-In the following the 3C-SiC/c-Si interface is described in further detail.
-One important size analyzing the interface is the interfacial energy.
-It is determined exactly in the same way than the formation energy as described in equation \eqref{eq:defects:ef2}.
-Using the notation of table \ref{table:md:sic_prec} and assuming that the system is composed out of $N^{\text{3C-SiC}}_{\text{C}}$ C atoms forming the SiC compound plus the remaining Si atoms, the energy is given by
-\begin{equation}
- E_{\text{f}}=E-
- N^{\text{3C-SiC}}_{\text{C}} \mu_{\text{SiC}}-
- \left(N^{\text{total}}_{\text{Si}}-N^{\text{3C-SiC}}_{\text{C}}\right)
- \mu_{\text{Si}} \text{ ,}
-\label{eq:md:ife}
-\end{equation}
-with $E$ being the free energy of the precipitate configuration at zero temperature.
-An interfacial energy of 2267.28 eV is obtained.
-The amount of C atoms together with the observed lattice constant of the precipitate leads to a precipitate radius of 29.93 \AA.
-Thus, the interface tension, given by the energy of the interface devided by the surface area of the precipitate is $20.15\,\frac{\text{eV}}{\text{nm}^2}$ or $3.23\times 10^{-4}\,\frac{\text{J}}{\text{cm}^2}$.
-This is located inside the eperimentally estimated range of $2-8\times 10^{-4}\,\frac{\text{J}}{\text{cm}^2}$ \cite{taylor93}.
-
-Since the precipitate configuration is artificially constructed the resulting interface does not necessarily correspond to the energetically most favorable configuration or to the configuration that is expected for an actually grown precipitate.
-Thus annealing steps are appended to the gained structure in order to allow for a rearrangement of the atoms of the interface.
-The precipitate structure is rapidly heated up to $2050\,^{\circ}\mathrm{C}$ with a heating rate of approximately $75\,^{\circ}\mathrm{C}/\text{ps}$.
-From that point on the heating rate is reduced to $1\,^{\circ}\mathrm{C}/\text{ps}$ and heating is continued to 120 \% of the Si melting temperature, that is 2940 K.
-\begin{figure}[!ht]
-\begin{center}
-\includegraphics[width=12cm]{fe_and_t_sic.ps}
-\end{center}
-\caption{Free energy and temperature evolution of a constructed 3C-SiC precipitate embedded in c-Si at temperatures above the Si melting point.}
-\label{fig:md:fe_and_t_sic}
-\end{figure}
-Figure \ref{fig:md:fe_and_t_sic} shows the free energy and temperature evolution.
-The sudden increase of the free energy indicates possible melting occuring around 2840 K.
-\begin{figure}[!ht]
-\begin{center}
-\includegraphics[width=12cm]{pc_500-fin.ps}
-\end{center}
-\caption{Radial distribution of the constructed 3C-SiC precipitate embedded in c-Si at temperatures below and above the Si melting transition point.}
-\label{fig:md:pc_500-fin}
-\end{figure}
-Investigating the radial distribution function shown in figure \ref{fig:md:pc_500-fin}, which shows configurations below and above the temperature of the estimated transition, indeed supports the assumption of melting gained by the free energy plot.
-However the precipitate itself is not involved, as can be seen from the Si-C and C-C distribution, which essentially stays the same for both temperatures.
-Thus, it is only the c-Si surrounding undergoing a structural phase transition, which is very well reflected by the difference observed for the two Si-Si distributions.
-This is surprising since the melting transition of plain c-Si is expected at temperatures around 3125 K, as discussed in the last section.
-Obviously the precipitate lowers the transition point of the surrounding c-Si matrix.
-For the rearrangement simulations temperatures well below the transition point should be used since it is very unlikely to recrystallize the molten Si surrounding properly when cooling down.
-To play safe the precipitate configuration at 100 \% of the Si melting temperature is chosen and cooled down to $20\,^{\circ}\mathrm{C}$ with a cooling rate of $1\,^{\circ}\mathrm{C}/\text{ps}$.
-{\color{blue}Todo: Wait for results and then compare structure (PC) and interface energy, maybe a energetically more favorable configuration arises.}
-{\color{red}Todo: Mention the fact, that the precipitate is stable for eleveated temperatures, even for temperatures where the Si matrix is melting.}
-{\color{red}Todo: Si starts to melt at the interface, show pictures and explain, it is due to the defective interface region.}
-
-\subsection{Simulations at temperatures around the silicon melting point}
-
-As discussed in section \ref{subsection:md:limit} and \ref{subsection:md:inct} a further increase of the system temperature might help to overcome limitations of the short range potential and accelerate the dynamics involved in structural evolution.
-A maximum temperature to avoid melting was determined in section \ref{subsection:md:tval}, which is 120 \% of the Si melting point.
-In the following simulations the system volume, the amount of C atoms inserted and the shape of the insertion volume are modified from the values used in the first MD simulations to now match the conditions given in the simulations of the self-constructed precipitate configuration for reasons of comparability.
-To quantify, the initial simulation volume now consists of 21 Si unit cells in each direction and 5500 C atoms are inserted in either the whole volume or in a sphere with a radius of 3 nm.
-Since the investigated temperatures exceed the Si melting point the initial Si bulk material is heated up slowly by $1\,^{\circ}\mathrm{C}/\text{ps}$ starting from $1650\,^{\circ}\mathrm{C}$ before the C insertion sequence is started.
-The 100 ps sequence at the respective temperature intended for the structural evolution is exchanged by a 10 ns sequence, which will hopefully result in the occurence of infrequent processes.
-The return to lower temperatures is considered seperately.
-
-\begin{figure}[!ht]
+\subsection{Conclusions concerning the usage of increased temperatures}
+
+Regarding the outcome of both, high and low C concentration simulations at increased temperatures, encouraging conclusions can be drawn.
+With the disappearance of the peaks at the respective cut-off radii, one limitation of the short range potential seems to be accomplished.
+In addition, sharper peaks in the radial distribution functions lead to the assumption of expeditious structural formation.
+The increase in temperature leads to the occupation of new defect states, which is particularly evident but not limited to the low C concentration simulations.
+
+The question remains, whether these states are only occupied due to the additional supply of kinetic energy and, thus, have to be considered unnatural for temperatures applied in IBS or whether the increase in temperature indeed enables infrequent transitions to occur faster, thus, leading to the intended acceleration of the dynamics and weakening of the unphysical quirks inherent to the potential.
+As already pointed out in section~\ref{section:defects:noneq_process_01} and section~\ref{section:defects:noneq_process_02}, IBS is a non-equilibrium process, which might result in the formation of the thermodynamically less stable \cs{} and \si{} configuration.
+Indeed, 3C-SiC is metastable and if not fabricated by IBS requires strong deviation from equilibrium and low temperatures to stabilize the 3C polytype.
+In IBS, highly energetic C atoms are able to generate vacant sites, which in turn can be occupied by highly mobile \ci{} atoms.
+This is in fact found to be favorable in the absence of the \si{}, which turned out to have a low interaction capture radius with the \cs{} atom and very likely prevents the recombination into a thermodynamically stable \ci{} DB for appropriate separations of the defect pair.
+Results gained in this chapter show preferential occupation of Si lattice sites by \cs{} enabled by increased temperatures supporting the assumptions drawn from the defect studies of the last chapter.
+
+Moreover, the cut-off effect as detailed in section~\ref{section:md:limit} is particularly significant for configurations that are deviated to a great extent from their equilibrium structures.
+Thus, for instance, it is not surprising that short range potentials show overestimated melting temperatures while properties of structures that are only slightly deviated from equilibrium are well described.
+Due to this, increased temperatures are considered exceptionally necessary for modeling non-equilibrium processes and structures such as IBS and 3C-SiC.
+
+Thus, it is concluded that increased temperatures are not exclusively useful to accelerate the dynamics approximatively describing the structural evolution.
+Moreover, it can be considered a necessary condition to deviate the system out of equilibrium enabling the formation of 3C-SiC, which is obviously realized by a successive agglomeration of \cs{}.
+
+\ifnum1=0
+
+\section{Long time scale simulations at maximum temperature}
+
+As discussed in section~\ref{section:md:limit} and~\ref{section:md:inct}, a further increase of the system temperature might help to overcome limitations of the short range potential and accelerate the dynamics involved in structural evolution.
+Furthermore, these results indicate that increased temperatures are necessary to drive the system out of equilibrium enabling conditions needed for the formation of a metastable cubic polytype of SiC.
+
+A maximum temperature to avoid melting is determined in section~\ref{section:md:tval} to be 120 \% of the Si melting point but due to defects lowering the transition point a maximum temperature of 95 \% of the Si melting temperature is considered useful.
+This value is almost equal to the temperature of $2050\,^{\circ}\mathrm{C}$ already used in former simulations.
+Since the maximum temperature is reached, the approach is reduced to the application of longer time scales.
+This is considered useful since the estimated evolution of quality in the absence of the cooling down sequence in figure~\ref{fig:md:tot_si-c_q} predicts an increase in quality and, thus, structural evolution is likely to occur if the simulation is proceeded at maximum temperature.
+
+Next to the employment of longer time scales and a maximum temperature, a few more changes are applied.
+In the following simulations, the system volume, the amount of C atoms inserted and the shape of the insertion volume are modified from the values used in first MD simulations.
+To speed up the simulation, the initial simulation volume is reduced to 21 Si unit cells in each direction and 5500 inserted C atoms in either the whole volume or in a sphere with a radius of 3 nm corresponding to the size of a precipitate consisting of 5500 C atoms.
+The \unit[100]{ps} sequence after C insertion intended for structural evolution is exchanged by a \unit[10]{ns} sequence, which is hoped to result in the occurrence of infrequent processes and a subsequent phase transition.
+The return to lower temperatures is considered separately.
+
+\begin{figure}[tp]