Furthermore these investigations might establish the prediction of conditions necessary for the simulation of the precipitation process.
To construct a spherical and topotactically aligned 3C-SiC precipitate in c-Si, the approach illustrated in the following is applied.
-A total simulation volume $V$ consisting of 21 unit cells of c-Si in each direction is used.
+A total simulation volume $V$ consisting of 21 unit cells of c-Si in each direction is created.
To obtain a minimal and stable precipitate 5500 carbon atoms are considered necessary.
+This corresponds to a spherical 3C-SiC precipitate with a radius of approximately 3 nm.
The initial precipitate configuration is constructed in two steps.
In the first step the surrounding silicon matrix is created.
This is realized by just skipping the generation of silicon atoms inside a sphere of radius $x$, which is the first unknown variable.
\hline
& C in 3C-SiC & Si in 3C-SiC & Si in c-Si surrounding & total amount of Si\\
\hline
-Expected & 5500 & 5500 & 68588 & 74088\\
Obtained & 5495 & 5486 & 68591 & 74077\\
-Difference & 5 & 14 & -3 & 11\\
+Expected & 5500 & 5500 & 68588 & 74088\\
+Difference & -5 & -14 & 3 & -11\\
+Notation & $N^{\text{3C-SiC}}_{\text{C}}$ & $N^{\text{3C-SiC}}_{\text{Si}}$
+ & $N^{\text{c-Si}}_{\text{Si}}$ & $N^{\text{total}}_{\text{Si}}$ \\
\hline
\hline
\end{tabular}
After the initial configuration is constructed some of the atoms located at the 3C-SiC/c-Si interface show small distances, which results in high repulsive forces acting on the atoms.
Thus, the system is equilibrated using strong coupling to the heat bath, which is set to be $20\,^{\circ}\mathrm{C}$.
-Once the main part of th excess energy is carried out previous settings for the Berendsen thermostat are restored and the system is relaxed for another 10 ps.
+Once the main part of the excess energy is carried out previous settings for the Berendsen thermostat are restored and the system is relaxed for another 10 ps.
+
+\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}.
-PC and energy of that one.
-Now let's see, whether annealing will lead to some energetically more favorable configurations.
+Since interface region is constructed and not neccesarily corresponds to the energetically most favorable layout we will now try hard to improve this ...
+Let's see, whether annealing will lead to some energetically more favorable configurations.
-Estimate surface energy ...
\subsection{Simulations at temperatures exceeding the silicon melting point}
LL Cool J is hot as hell!
+
+A different simulation volume and refined amount as well as shape of insertion volume for the C atoms, to stay compareable to the results gained in the latter section, is used throughout all following simulations.
+
+\subsection{Todo}
+
+{\color{red}TODO: self-guided MD!}
+
+{\color{red}TODO: other approaches!}
+
+{\color{red}TODO: ART MD?}
+
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