\begin{center}
\begin{minipage}{7.5cm}
\includegraphics[width=7cm]{comb_pos.eps}
+% ./visualize_contcar -w 640 -h 480 -d results/.../CONTCAR -nll -0.20 -0.20 -0.6 -fur 1.2 1.2 0.6 -c 0.5 -1.5 0.3 -L 0.5 0 0 -r 0.6 -m 3.0 0.0 0.0 0.0 3.0 0.0 0.0 0.0 3.0 -A -1 2.465
\end{minipage}
\begin{minipage}{6.0cm}
\underline{Positions given in $a_{\text{Si}}$}\\[0.3cm]
\end{figure}
\begin{table}[h]
\begin{center}
-\begin{tabular}{l c c c c c}
+\begin{tabular}{l c c c c c c}
\hline
\hline
- & 1 & 2 & 3 & 4 & 5 \\
+ & 1 & 2 & 3 & 4 & 5 & R\\
\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} \\
+ \hkl<0 0 -1> & {\color{red}-0.08} & -1.15 & {\color{red}-0.08} & 0.04 & -1.66 & -0.19\\
+ \hkl<0 0 1> & 0.34 & 0.004 & -2.05 & 0.26 & -1.53 & -0.19\\
+ \hkl<0 -1 0> & {\color{orange}-2.39} & -0.17 & {\color{green}-0.10} & {\color{blue}-0.27} & {\color{magenta}-1.88} & {\color{gray}-0.05}\\
+ \hkl<0 1 0> & {\color{cyan}-2.25} & -1.90 & {\color{cyan}-2.25} & {\color{purple}-0.12} & {\color{violet}-1.38} & {\color{yellow}-0.06}\\
+ \hkl<-1 0 0> & {\color{orange}-2.39} & -0.36 & {\color{cyan}-2.25} & {\color{purple}-0.12} & {\color{magenta}-1.88} & {\color{gray}-0.05}\\
+ \hkl<1 0 0> & {\color{cyan}-2.25} & -2.16 & {\color{green}-0.10} & {\color{blue}-0.27} & {\color{violet}-1.38} & {\color{yellow}-0.06}\\
\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 \\
+ C substitutional (C$_{\text{S}}$) & 0.26 & -0.51 & -0.93 & -0.15 & 0.49 & -0.05\\
+ Vacancy & -5.39 ($\rightarrow$ C$_{\text{S}}$) & -0.59 & -3.14 & -0.54 & -0.50 & -0.31\\
\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}.}
+\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}. R is the position located at $\frac{a_{\text{Si}}}{2}\hkl<3 2 3>$ relative to the initial defect, which is the maximum realizable distance due to periodic boundary conditions.}
\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.
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 without an interaction resulting in $E_{\text{b}}=0$.
Thus, $E_{\text{b}}$ can be best thought of a binding energy, which is required to bring the defects to infinite separation.
-In fact, further \hkl<0 0 -1> dumbbell interstitials 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) relative to the initial one result in energies as low as -0.19 eV and -0.12 eV.
+In fact, a \hkl<0 0 -1> dumbbell interstitial created at position R with a distance of $\frac{a_{\text{Si}}}{2}\hkl<3 2 3>$ ($\approx 12.8$ \AA) from the initial one results in an energy as low as -0.19 eV.
There is still a low interaction which is due to the equal orientation of the defects.
-By changing the orientation of the second dumbbell interstitial to the \hkl<0 -1 0>-type the interaction is even mor reduced, which results in an energy of $E_{\text{b}}=...\text{ eV}$ for a distance of $\frac{a_{\text{Si}}}{2}\hkl<2 3 2>$, which is the maximum that can be reached due to periodic boundary conditions.
-Configurations wih energies greater than zero are energetically unfavorable and expose a repulsive interaction.
+By changing the orientation of the second dumbbell interstitial to the \hkl<0 -1 0>-type the interaction is even mor reduced resulting in an energy of $E_{\text{b}}=-0.05\text{ eV}$ for a distance, which is the maximum that can be realized due to periodic boundary conditions.
+The energies obtained in the R column of table \ref{eq:defects:e_of_comb} are used as a reference to identify, whether less distanced defects of the same type are favorable or unfavorable compared to the far-off located defect.
+Configurations wih energies greater than zero or the reference value 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.
+Energies below zero and below the reference value 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.
Structure \ref{fig:defects:comb_db_01} b) is the energetically most favorable configuration.
After relaxation the initial configuration is still evident.
As expected by the initialization conditions the two carbon atoms form a bond.
-This bond has a length of 1.38 \AA close to the nex neighbour distance in diamond or graphite, which is approximately 1.54 \AA.
+This bond has a length of 1.38 \AA{} close to the nex neighbour distance in diamond or graphite, which is approximately 1.54 \AA.
The minimum of binding energy observed for this configuration suggests prefered C clustering as a competing mechnism to the C-Si dumbbell interstitial agglomeration inevitable for the SiC precipitation.
Todo: Activation energy to obtain a configuration of separated C atoms again or vice versa to obtain this configuration from separated C confs?
However, for the second most favorable configuration, presented in figure \ref{fig:defects:comb_db_01} a), the amount of possibilities for this configuration is twice as high.
-In this configuration the two carbon atoms are spaced by 2.70 \AA.
-The initial Si (I) and C (I) dumbbell atoms are displaced along \hkl<1 0 0> and \hkl<-1 0 0> in such a way that the Si atom is forming tetrahedral bonds with two silicon and two carbon atoms.
+In this configuration the initial Si (I) and C (I) dumbbell atoms are displaced along \hkl<1 0 0> and \hkl<-1 0 0> in such a way that the Si atom is forming tetrahedral bonds with two silicon and two carbon atoms.
The carbon and silicon atom constituting the second defect are as well displaced in such a way, that the carbon atom forms tetrahedral bonds with four silicon neighbours, a configuration expected in silicon carbide.
+The two carbon atoms spaced by 2.70 \AA{} do not form a bond but anyhow reside in a shorter distance as expected in silicon carbide.
The Si atom numbered 2 is pushed towards the carbon atom, which results in the breaking of the bond to atom 4.
The breaking of the $\sigma$ bond is indeed confirmed by investigating the charge density isosurface of this configuration.
-Todo: But is this configuration beneficial for SiC prec?
+Todo: Is this conf really benificial for SiC prec?
-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.
+\begin{figure}[h]
+\begin{center}
+\begin{minipage}[t]{5cm}
+a) \underline{$E_{\text{b}}=-2.16\text{ eV}$}
+\begin{center}
+\includegraphics[width=4.8cm]{00-1dc/2-16.eps}
+\end{center}
+\end{minipage}
+\begin{minipage}[t]{5cm}
+b) \underline{$E_{\text{b}}=-1.90\text{ eV}$}
+\begin{center}
+\includegraphics[width=4.8cm]{00-1dc/1-90.eps}
+\end{center}
+\end{minipage}
+\begin{minipage}[t]{5cm}
+c) \underline{$E_{\text{b}}=-2.05\text{ eV}$}
+\begin{center}
+\includegraphics[width=4.8cm]{00-1dc/2-05.eps}
+\end{center}
+\end{minipage}
+\end{center}
+\caption{Relaxed structures of defect complexes obtained by creating a a) \hkl<1 0 0> and b) \hkl<0 1 0> dumbbell at position 2 and a c) \hkl<0 0 1> dumbbel at position 3.}
+\label{fig:defects:comb_db_02}
+\end{figure}
+Figure \ref{fig:defects:comb_db_02} shows the next three most energetically favorable configurations.
+The relaxed configuration obtained by creating a second \hkl<1 0 0> dumbbell at position 2 is shown in figure \ref{fig:defects:comb_db_02} a).
+A binding energy of -2.16 eV is observed.
+After relaxation the second dumbbell is aligned along \hkl<1 1 0>.
+The bond of the silicon atoms 1 and 2 does not persist.
+Instead the silicon atom forms a bond with the initial carbon interstitial and the second carbon atom forms a bond with silicon atom 1 forming four bonds in total.
+The carbon atoms are spaced by 3.14 \AA, which is very close to the expected C-C next neighbour distance of 3.08 \AA{} in silicon carbide.
-Explanation of results of defects created along <110>.
--1.90 ...
+-2.05 ... both C atoms correctly coordinated, however (check C-C distance, too close?) wrong coordination of the C-Si-C bonds which reside in a plane ... all the 4 participating atoms reside in a plane ...
+
+-1.90 ... again, one C atom bound to 4 Si atoms but the second one only bond to three Si atoms. However, C-C slighlty higher.
--2.16 ...
+The 2.7 diatnce characteristic for configurations with one C atom bound to 4 silicon and.
+Different energies due to slightly varying constellation.
the more far-off ones:
--0.27 and -0.12 ...
+-0.27 and -0.12 ... carbon is threefold coordinated.
+initial structures evident, seem to be independent of each other ...
but better ...
-1.88 and -1.38 ...
+-1.38: Si (I) moves towards 2nd Si int in 110 direction, such that both Si 4fold coordinated and C remain 3fold ...
+-1.88: threefold coordinated c atoms but all participating Si atoms fourfold coordinated ...
+
+Explanation of results of defects created along <110>.
Minimum E (reorientation) per distance
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