\section{Combination of point defects}
+The structural and energetic properties of combinations of point defects are investigated in the following.
+The focus is on combinations of the \hkl<0 0 -1> dumbbell interstitial with a second defect.
+The second defect is either another \hkl<1 0 0>-type interstitial occupying different orientations, a vacany or a substitutional carbon atom.
+Several distances of the two defects are examined.
+Investigations are restricted to quantum-mechanical calculations.
\begin{figure}[h]
\begin{center}
\begin{minipage}{7.5cm}
\end{minipage}
\begin{minipage}{6.0cm}
\underline{Positions given in $a_{\text{Si}}$}\\[0.3cm]
-Initial interstitial: $\frac{1}{4}\hkl<1 1 1>$\\
+Initial interstitial I: $\frac{1}{4}\hkl<1 1 1>$\\
Relative silicon neighbour positions:
\begin{enumerate}
- \item $\frac{1}{4}\hkl<1 1 -1>$, $\frac{1}{4}\hkl<-1 -1 -1>$ ()
+ \item $\frac{1}{4}\hkl<1 1 -1>$, $\frac{1}{4}\hkl<-1 -1 -1>$
\item $\frac{1}{2}\hkl<1 0 1>$, $\frac{1}{2}\hkl<0 1 -1>$,\\[0.2cm]
$\frac{1}{2}\hkl<0 -1 -1>$, $\frac{1}{2}\hkl<-1 0 -1>$
\item $\frac{1}{4}\hkl<1 -1 1>$, $\frac{1}{4}\hkl<-1 1 1>$
\caption[\hkl<0 0 -1> dumbbell interstitial defect and positions of next neighboured silicon atoms used for the second defect.]{\hkl<0 0 -1> dumbbell interstitial defect and positions of next neighboured silicon atoms used for the second defect. Two possibilities exist for red numbered atoms and four possibilities exist for blue numbered atoms.}
\label{fig:defects:pos_of_comb}
\end{figure}
-The structural and energetic properties of combinations of point defects are investigated in the following.
-The focus is on combinations of the \hkl<0 0 -1> dumbbell interstitial with a second defect.
-The second defect is either another \hkl<1 0 0>-type interstitial occupying different orientations, a vacany or a substitutional carbon atom.
-Several distances of the two defects are examined.
-Investigations are restricted to quantum-mechanical calculations.
-Figure \ref{fig:defects:pos_of_comb} shows the initial \hkl<0 0 -1> dumbbell interstitial defect and the positions of the next neighboured silicon atoms used for the second defect.
+\begin{table}[h]
+\begin{center}
+\begin{tabular}{l c c c c c}
+\hline
+\hline
+ & 1 & 2 & 3 & 4 & 5 \\
+ \hline
+ \hkl<0 0 -1> & {\color{red}-0.08} & -1.15 & {\color{red}-0.08} & 0.04 & -1.66\\
+ \hkl<0 0 1> & 0.34 & 0.004 & -2.05 & 0.26 & -1.53\\
+ \hkl<0 -1 0> & {\color{orange}-2.39} & -2.16 & {\color{green}-0.10} & {\color{blue}-0.27} & {\color{magenta}-1.88}\\
+ \hkl<0 1 0> & {\color{cyan}-2.25} & -0.36 & {\color{cyan}-2.25} & {\color{purple}-0.12} & {\color{violet}-1.38}\\
+ \hkl<-1 0 0> & {\color{orange}-2.39} & -1.90 & {\color{cyan}-2.25} & {\color{purple}-0.12} & {\color{magenta}-1.88}\\
+ \hkl<1 0 0> & {\color{cyan}-2.25} & -0.17 & {\color{green}-0.10} & {\color{blue}-0.27} & {\color{violet}-1.38} \\
+ \hline
+ C substitutional (C$_{\text{S}}$) & 0.26 & -0.51 & -0.93 & -0.15 & 0.49 \\
+ Vacancy & -5.39 ($\rightarrow$ C$_{\text{S}}$) & -0.59 & -3.14 & -0.54 & -0.50 \\
+\hline
+\hline
+\end{tabular}
+\end{center}
+\caption[Energetic results of defect combinations.]{Energetic results of defect combinations. The given energies in eV are defined by equation \eqref{eq:defects:e_of_comb}. Equivalent configurations are marked by identical colors. The first column lists the types of the second defect combined with the initial \hkl<0 0 -1> dumbbell interstitial. The position index of the second defect is given in the first row according to figure \ref{fig:defects:pos_of_comb}.}
+\label{tab:defects:e_of_comb}
+\end{table}
+Figure \ref{fig:defects:pos_of_comb} shows the initial \hkl<0 0 -1> dumbbell interstitial defect and the positions of next neighboured silicon atoms used for the second defect.
+Table \ref{tab:defects:e_of_comb} summarizes energetic results obtained after relaxation of the defect combinations.
+The energy of interest $E$ is defined to be
+\begin{equation}
+E=
+E_{\text{f}}^{\text{defect combination}}-
+E_{\text{f}}^{\text{C \hkl<0 0 -1> dumbbell}}-
+E_{\text{f}}^{\text{2nd defect}}
+\label{eq:defects:e_of_comb}
+\end{equation}
+with $E_{\text{f}}^{\text{defect combination}}$ being the formation energy of the defect combination, $E_{\text{f}}^{\text{C \hkl<0 0 -1> dumbbell}}$ being the formation energy of the C \hkl<0 0 -1> dumbbell interstitial defect and $E_{\text{f}}^{\text{2nd defect}}$ being the formation energy of the second defect.
+For defects far away from each other the formation energy of the defect combination should approximately become the sum of the formation energies of the individual defects with zero interaction resulting in $E=0$.
+In fact, for another \hkl<0 0 -1> dumbbell interstitial created at position $\frac{a_{\text{Si}}}{2}\hkl<3 2 3>$ ($\approx 10.2$ \AA) and $\frac{a_{\text{Si}}}{2}\hkl<2 3 2>$ ($\approx 12.8$ \AA, maximum distance due to periodic boundary conditions) relative to the initial one an energy of -0.19 eV and ... is obtained.
+Configurations wih energies greater than zero are energetically unfavorable and expose a repulsive interaction.
+These configurations are unlikely to arise or to persist for non-zero temperatures.
+Energies below zero indicate configurations favored compared to configurations in which these point defects are separated far away from each other.
+
+Investigating the first part of table \ref{tab:defects:e_of_comb}, namely the combinations with another \hkl<1 0 0>-type interstitial, most of the combinations result in energies below zero.
+Surprisingly the most favorable configurations are the ones with the second defect created at the very next silicon neighbour and a change in orientation compared to the initial one.
+This leads to the conclusion that an agglomeration of C-Si dumbbell interstitials as proposed by the precipitation model introduced in section\ref{section:assumed_prec} is indeed an energetically favored configuration of the system.
+The reason for nearby interstitials being favored compared to isolated ones is most probably the reduction of strain energy enabled by combination in contrast to the strain energy created by two individual defects.
+\begin{figure}[h]
+\begin{center}
+\begin{minipage}[t]{7cm}
+a) \underline{$E=-2.25\text{ eV}$}
+\begin{center}
+\includegraphics[width=6cm]{00-1dc/2-25.eps}
+\end{center}
+\end{minipage}
+\begin{minipage}[t]{7cm}
+b) \underline{$E=-2.39\text{ eV}$}
+\begin{center}
+\includegraphics[width=6cm]{00-1dc/2-39.eps}
+\end{center}
+\end{minipage}
+\end{center}
+\caption{Relaxed structure of defect complexes consisting of two \hkl<1 0 0>-type dumbbell interstitial defects.}
+\label{fig:defects:comb_db_01}
+\end{figure}
+Figure \ref{fig:defects:comb_db_01} shows the structure of these two configurations.
+
+Structure b) is the energetically most favorable configuration.
+The two carbon atoms form a bond to each other.
+This suggests an unwanted C clustering in SiC production.
+However, for the second most favorable configuration, presented in figure a), the amount of possibilitie for this configuration is twice as high.
+Investigating C-Si and C-C bond lengths ...
+
+001 at pos 2 looks as if there is no interaction.
+There is an interaction but in the same time strain is reduced due to the opposing orientations of the defects, which leads to this low energy value.
+
+Explanation of results of defects created along <110>.
+
+-1.90 ...
+
+-2.16 ...
+
+the more far-off ones:
+-0.27 and -0.12 ...
+but better ...
+-1.88 and -1.38 ...
+Minimum E (reorientation) per distance
+Todo: Si int and C sub ...
+Todo: Model of kick-out and kick-in mechnism?
+Todo: Jahn-Teller distortion (vacancy) $\rightarrow$ actually three possibilities! :(
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