Periodic boundary conditions in each direction are applied.
All point defects are calculated for the neutral charge state.
-\begin{figure}[h]
+\begin{figure}[th]
\begin{center}
\includegraphics[width=9cm]{unit_cell_e.eps}
\end{center}
Quantum-mechanical total-energy calculations are an invalueable tool to investigate the energetic and structural properties of point defects since they are experimentally difficult to assess.
The formation energies of some of the silicon self-interstitial configurations are listed in table \ref{tab:defects:si_self} for both methods used in this work as well as results obtained by former studies \cite{leung99}.
-\begin{table}[h]
+\begin{table}[th]
\begin{center}
\begin{tabular}{l c c c c c}
\hline
\label{tab:defects:si_self}
\end{table}
The final configurations obtained after relaxation are presented in figure \ref{fig:defects:conf}.
-\begin{figure}[h]
+\begin{figure}[t!h!]
\begin{center}
%\hrule
%\vspace*{0.2cm}
The formation energy of 3.96 eV for this type of interstitial is equal to the result for the hexagonal one in the original work \cite{albe_sic_pot}.
Obviously the authors did not carefully check the relaxed results assuming a hexagonal configuration.
In figure \ref{fig:defects:kin_si_hex} the relaxation process is shown on the basis of the kinetic energy plot.
-\begin{figure}[h]
+\begin{figure}[th]
\begin{center}
\includegraphics[width=10cm]{e_kin_si_hex.ps}
\end{center}
In addition, variations exist in which the displacement is only along two \hkl<1 0 0> axes ($E_\text{f}=3.8\text{ eV}$) or along a single \hkl<1 0 0> axes ($E_\text{f}=3.6\text{ eV}$) successively approximating the tetdrahedral configuration and formation energy.
The existence of these local minima located near the tetrahedral configuration seems to be an artifact of the analytical potential without physical authenticity revealing basic problems of analytical potential models for describing defect structures.
However, the energy barrier is small.
-Todo: Check!
+{\color{red}Todo: Check!}
Hence these artifacts should have a negligent influence in finite temperature simulations.
The bond-centered configuration is unstable and the \hkl<1 0 0> dumbbell interstitial is the most unfavorable configuration for both, the Erhard/Albe and VASP calculations.
The same applies to the bonds between the interstitial and the upper two atoms in the \hkl<1 1 0> dumbbell configuration.
A more detailed description of the chemical bonding is achieved by quantum-mechanical calculations by investigating the accumulation of negative charge between the nuclei.
-Todo: Plot the electron density for these types of defect to derive conclusions of existing bonds?
+{\color{red}Todo: Plot the electron density for these types of defect to derive conclusions of existing bonds?}
\section{Carbon related point defects}
The type of reservoir of the carbon impurity to determine the formation energy of the defect was chosen to be SiC.
This is consistent with the methods used in the articles \cite{tersoff90,dal_pino93}, which the results are compared to in the following.
Hence, the chemical potential of silicon and carbon is determined by the cohesive energy of silicon and silicon carbide.
-\begin{table}[h]
+\begin{table}[th]
\begin{center}
\begin{tabular}{l c c c c c c}
\hline
\caption[Formation energies of carbon point defects in crystalline silicon determined by classical potential molecular dynamics and density functional calculations.]{Formation energies of carbon point defects in crystalline silicon determined by classical potential molecular dynamics and density functional calculations. The formation energies are given in eV. T denotes the tetrahedral, H the hexagonal, B the bond-centered and S the substitutional interstitial configuration. The dumbbell configurations are abbreviated by DB. Formation energies for unstable configurations are marked by an asterisk and determined by using the low kinetic energy configuration shortly before the relaxation into the more favorable configuration starts.}
\label{tab:defects:c_ints}
\end{table}
-\begin{figure}[h]
+\begin{figure}[th]
\begin{center}
\begin{flushleft}
\begin{minipage}{4cm}
Figure \ref{fig:defects:100db_cmp} schematically shows the \hkl<1 0 0> dumbbell structure and table \ref{tab:defects:100db_cmp} lists the details of the atomic displacements, distances and bond angles obtained by analytical potential and quantum-mechanical calculations.
For comparison, the obtained structures for both methods visualized out of the atomic position data are presented in figure \ref{fig:defects:100db_vis_cmp}.
-\begin{figure}[h]
+\begin{figure}[th]
\begin{center}
\includegraphics[width=12cm]{100-c-si-db_cmp.eps}
\end{center}
\label{fig:defects:100db_cmp}
\end{figure}
%
-\begin{table}[h]
+\begin{table}[th]
\begin{center}
Displacements\\
\begin{tabular}{l c c c c c c c c c}
\caption[Atomic displacements, distances and bond angles of the \hkl<1 0 0> dumbbell structure obtained by the Erhard/Albe potential and VASP calculations.]{Atomic displacements, distances and bond angles of the \hkl<1 0 0> dumbbell structure obtained by the Erhard/Albe potential and VASP calculations. The displacements and distances are given in nm and the angles are given in degrees. Displacements, distances and angles are schematically displayed in figure \ref{fig:defects:100db_cmp}. In addition, the equilibrium lattice constant for crystalline silicon is listed.}
\label{tab:defects:100db_cmp}
\end{table}
-\begin{figure}[h]
+\begin{figure}[t!h!]
\begin{center}
\begin{minipage}{6cm}
\begin{center}
\caption{Comparison of the visualized \hkl<1 0 0> dumbbel structures obtained by Erhard/Albe potential and VASP calculations.}
\label{fig:defects:100db_vis_cmp}
\end{figure}
-\begin{figure}[h]
+\begin{figure}[th]
\begin{center}
\includegraphics[height=10cm]{c_pd_vasp/eden.eps}
\includegraphics[height=12cm]{c_pd_vasp/100_2333_ksl.ps}
\subsection{Bond-centered interstitial configuration}
\label{subsection:bc}
-\begin{figure}[h]
+\begin{figure}[t!h!]
\begin{center}
\begin{minipage}{8cm}
\includegraphics[width=8cm]{c_pd_vasp/bc_2333.eps}\\
In the following the problem of interstitial carbon migration in silicon is considered.
Since the carbon \hkl<1 0 0> dumbbell interstitial is the most probable hence most important configuration the migration simulations focus on this defect.
-\begin{figure}[h]
+\begin{figure}[t!h!]
\begin{center}
\begin{minipage}{15cm}
\underline{\hkl<0 0 -1> $\rightarrow$ \hkl<0 0 1>}\\
Thus, it is not responsible for long-range migration.
The silicon dumbbell partner remains the same.
The bond to the face-centered silicon atom at the bottom of the unit cell breaks and a new one is formed to the face-centered atom at the forefront of the unit cell.
-Todo: \hkl<1 1 0> to \hkl<1 0 0> and bond-centerd configuration (in progress).
-Todo: \hkl<1 1 0> to \hkl<0 -1 0> (rotation of the DB, in progress).
-Todo: Comparison with classical potential simulations or explanation to only focus on ab initio calculations.
+{\color{red}Todo: Comparison with classical potential simulations or explanation to only focus on ab initio calculations.}
Since the starting and final structure, which are both local minima of the potential energy surface, are known, the aim is to find the minimum energy path from one local minimum to the other one.
One method to find a minimum energy path is to move the diffusing atom stepwise from the starting to the final position and only allow relaxation in the plane perpendicular to the direction of the vector connecting its starting and final position.
Relaxation of each individual atom is only allowed in the plane perpendicular to the last individual displacement vector.
The modifications used to add this feature to the VASP code and a short instruction on how to use it can be found in appendix \ref{app:patch_vasp}.
Due to these constraints obtained activation energies can effectively be higher.
-Todo: To refine the migration barrier one has to find the saddle point structure and recalculate the free energy of this configuration with a reduced set of constraints.
+{\color{red}Todo: To refine the migration barrier one has to find the saddle point structure and recalculate the free energy of this configuration with a reduced set of constraints.}
-\begin{figure}[h]
+\begin{figure}[t!h!]
\begin{center}
\includegraphics[width=13cm]{im_00-1_nosym_sp_fullct_thesis.ps}\\[1.5cm]
\begin{picture}(0,0)(150,0)
This amount of energy is needed to break the bond of the carbon atom to the silicon atom at the bottom left.
In a second process 0.25 eV of energy are needed for the system to revert into a \hkl<1 0 0> configuration.
-\begin{figure}[h]
+\begin{figure}[t!h!]
\begin{center}
\includegraphics[width=13cm]{vasp_mig/00-1_0-10_nosym_sp_fullct.ps}\\[1.6cm]
\begin{picture}(0,0)(140,0)
Figure \ref{fig:defects:00-1_0-10_mig} shows the migration barrier and structures of the \hkl<0 0 -1> to \hkl<0 -1 0> dumbbell transition.
The resulting migration barrier of approximately 0.9 eV is very close to the experimentally obtained values of 0.73 \cite{song90} and 0.87 eV \cite{tipping87}.
-\begin{figure}[h]
+\begin{figure}[t!h!]
\begin{center}
\includegraphics[width=13cm]{vasp_mig/00-1_ip0-10_nosym_sp_fullct.ps}\\[1.8cm]
\begin{picture}(0,0)(140,0)
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{figure}[th]
\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]
\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}
-\begin{table}[h]
+\begin{table}[th]
\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.
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
+The energy of interest $E_{\text{b}}$ is defined to be
\begin{equation}
-E=
+E_{\text{b}}=
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.
+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, 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 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.
-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.
+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{figure}[t!h!]
\begin{center}
\begin{minipage}[t]{7cm}
-a) \underline{$E=-2.25\text{ eV}$}
+a) \underline{$E_{\text{b}}=-2.25\text{ eV}$}
\begin{center}
\includegraphics[width=6cm]{00-1dc/2-25.eps}
\end{center}
\end{minipage}
\begin{minipage}[t]{7cm}
-b) \underline{$E=-2.39\text{ eV}$}
+b) \underline{$E_{\text{b}}=-2.39\text{ eV}$}
\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.}
+\caption{Relaxed structures of defect complexes obtained by creating a) \hkl<1 0 0> and b) \hkl<0 -1 0> dumbbels at position 1.}
\label{fig:defects:comb_db_01}
\end{figure}
Figure \ref{fig:defects:comb_db_01} shows the structure of these two configurations.
+The displayed configurations are realized by creating a \hkl<1 0 0> (a)) and \hkl<0 -1 0> (b)) dumbbell at position 1.
+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.
+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.
+{\color{red}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 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.
+{\color{red}Todo: Is this conf really benificial for SiC prec?}
+
+\begin{figure}[t!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.
+Figure \ref{fig:defects:comb_db_02} c) displays the results of another \hkl<0 0 1> dumbbell inserted at position 3.
+The binding energy is -2.05 eV.
+Both dumbbells are tilted along the same direction remaining parallely aligned and the second dumbbell is pushed downwards in such a way, that the four dumbbell atoms form a rhomboid.
+Both carbon atoms form tetrahedral bonds to four silicon atoms.
+However, silicon atom 1 and 3, which are bond to the second carbon dumbbell interstitial are also bond to the initial carbon atom.
+These four atoms of the rhomboid reside in a plane and, thus, do not match the situation in silicon carbide.
+The carbon atoms have a distance of 2.75 \AA.
+In figure \ref{fig:defects:comb_db_02} b) a second \hkl<0 1 0> dumbbell is constructed at position 2.
+An energy of -1.90 eV is observed.
+The initial dumbbell and especially the carbon atom is pushed towards the silicon atom of the second dumbbell forming an additional fourth bond.
+Silicon atom number 1 is pulled towards the carbon atoms of the dumbbells accompanied by the disappearance of its bond to silicon number 5 as well as the bond of silicon number 5 to its next neighboured silicon atom in \hkl<1 1 -1> direction.
+The carbon atom of the second dumbbell forms threefold coordinated bonds to its silicon neighbours.
+A distance of 2.80 \AA{} is observed for the two carbon atoms.
+Again, the two carbon atoms and its two interconnecting silicon atoms form a rhomboid.
+C-C distances of 2.70 to 2.80 \AA{} seem to be characteristic for such configurations, in which the carbon atoms and the two interconnecting silicon atoms reside in a plane.
+
+Configurations obtained by adding a second dumbbell interstitial at position 4 are characterized by minimal changes from their initial creation condition during relaxation.
+There is a low interaction of the dumbbells, which seem to exist independent of each other.
+This, on the one hand, becomes evident by investigating the final structure, in which both of the dumbbells essentially retain the structure expected for a single dumbbell and on the other hand is supported by the observed binding energies which vary only slightly around zero.
+This low interaction is due to the larger distance and a missing direct connection by bonds along a crystallographic direction.
+Both carbon and silicon atoms of the dumbbells form threefold coordinated bonds to their next neighbours.
+The energetically most unfavorable configuration ($E_{\text{b}}=0.26\text{ eV}$) is obtained for the \hkl<0 0 1> interstitial oppositely orientated to the initial one.
+A dumbbell taking the same orientation as the initial one is less unfavorble ($E_{\text{b}}=0.04\text{ eV}$).
+Both configurations are unfavorable compared to far-off isolated dumbbells.
+Nonparallel orientations, that is the \hkl<0 1 0>, \hkl<0 -1 0> and its equivalents, result in binding energies of -0.12 eV and -0.27 eV, thus, constituting energetically favorable configurations.
+The reduction of strain energy is higher in the second case where the carbon atom of the second dumbbell is placed in the direction pointing away from the initial carbon atom.
+
+\begin{figure}[t!h!]
+\begin{center}
+\begin{minipage}[t]{7cm}
+a) \underline{$E_{\text{b}}=-1.53\text{ eV}$}
+\begin{center}
+\includegraphics[width=6.0cm]{00-1dc/1-53.eps}
+\end{center}
+\end{minipage}
+\begin{minipage}[t]{7cm}
+b) \underline{$E_{\text{b}}=-1.66\text{ eV}$}
+\begin{center}
+\includegraphics[width=6.0cm]{00-1dc/1-66.eps}
+\end{center}
+\end{minipage}\\[0.2cm]
+\begin{minipage}[t]{7cm}
+c) \underline{$E_{\text{b}}=-1.88\text{ eV}$}
+\begin{center}
+\includegraphics[width=6.0cm]{00-1dc/1-88.eps}
+\end{center}
+\end{minipage}
+\begin{minipage}[t]{7cm}
+d) \underline{$E_{\text{b}}=-1.38\text{ eV}$}
+\begin{center}
+\includegraphics[width=6.0cm]{00-1dc/1-38.eps}
+\end{center}
+\end{minipage}
+\end{center}
+\caption{Relaxed structures of defect complexes obtained by creating a a) \hkl<0 0 1>, a b) \hkl<0 0 -1>, a c) \hkl<0 -1 0> and a d) \hkl<1 0 0> dumbbell at position 5.}
+\label{fig:defects:comb_db_05}
+\end{figure}
+Energetically beneficial configurations of defect complexes are observed for second interstititals of all orientations placed at position 5, a position two bonds away from the initial interstitial along the \hkl<1 1 0> direction.
+Relaxed structures of these complexes are displayed in figure \ref{fig:defects:comb_db_05}.
+Figure \ref{fig:defects:comb_db_05} a) and b) show the relaxed structures of \hkl<0 0 1> and \hkl<0 0 -1> dumbbells.
+The upper dumbbell atoms are pushed towards each other forming fourfold coordinated bonds.
+While the displacements of the silicon atoms in case b) are symmetric to the \hkl(1 1 0) plane, in case a) the silicon atom of the initial dumbbel is pushed a little further in the direction of the carbon atom of the second dumbbell than the carbon atom is pushed towards the silicon atom.
+The bottom atoms of the dumbbells remain in threefold coordination.
+The symmetric configuration is energetically more favorable ($E_{\text{b}}=-1.66\text{ eV}$) since the displacements of the atoms is less than in the antiparallel case ($E_{\text{b}}=-1.53\text{ eV}$).
+In figure \ref{fig:defects:comb_db_05} c) and d) the nonparallel orientations, namely the \hkl<0 -1 0> and \hkl<1 0 0> dumbbells are shown.
+Binding energies of -1.88 eV and -1.38 eV are obtained for the relaxed structures.
+In both cases the silicon atom of the initial interstitial is pulled towards the near by atom of the second dumbbell so that both atoms form fourfold coordinated bonds to their next neighbours.
+In case c) it is the carbon and in case d) the silicon atom of the second interstitial which forms the additional bond with the silicon atom of the initial interstitial.
+The atom of the second dumbbell, the carbon atom of the initial dumbbell and the two interconnecting silicon atoms again reside in a plane.
+A typical C-C distance of 2.79 \AA{} is, thus, observed for case c).
+The far-off atom of the second dumbbell resides in threefold coordination.
+
+Assuming that it is possible for the system to minimize free energy by an in place reorientation of the dumbbell at any position the minimum energy orientation of dumbbells along the \hkl<1 1 0> direction and the resulting C-C distance is shown in table \ref{tab:defects:comb_db110}.
+\begin{table}[t!h!]
+\begin{center}
+\begin{tabular}{l c c c c c c}
+\hline
+\hline
+ & 1 & 2 & 3 & 4 & 5 & 6\\
+\hline
+$E_{\text{b}}$ [eV] & -2.39 & -1.88 & -0.59 & -0.31 & -0.24 & -0.21 \\
+C-C distance [\AA] & 1.4 & 4.6 & 6.5 & 8.6 & 10.5 & 10.8 \\
+Type & \hkl<-1 0 0> & \hkl<1 0 0> & \hkl<1 0 0> & \hkl<1 0 0> & \hkl<1 0 0> & \hkl<1 0 0>, \hkl<0 -1 0>\\
+\hline
+\hline
+\end{tabular}
+\end{center}
+\caption{Binding energy and type of the minimum energy configuration of an additional dumbbell with respect to the separation distance in bonds along the \hkl<1 1 0> direction and the C-C distance.}
+\label{tab:defects:comb_db110}
+\end{table}
+\begin{figure}[t!h!]
+\begin{center}
+\includegraphics[width=12.5cm]{db_along_110.ps}\\
+\includegraphics[width=12.5cm]{db_along_110_cc.ps}
+\end{center}
+\caption{Minimum binding energy of dumbbell combinations with respect to the separation distance in bonds along \hkl<1 1 0> and C-C distance.}
+\label{fig:defects:comb_db110}
+\end{figure}
+Figure \ref{fig:defects:comb_db110} shows the corresponding plot of the data including a cubic spline interplation and a suitable fitting curve.
+The funtion found most suitable for curve fitting is $f(x)=a/x^3$ comprising the single fit parameter $a$.
+Thus, far-off located dumbbells show an interaction proportional to the reciprocal cube of the distance and the amount of bonds along \hkl<1 1 0> respectively.
+This behavior is no longer valid for the immediate vicinity revealed by the saturating binding energy of a second dumbbell at position 1, which was ignored in the fitting procedure.
-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 ...
+{\color{red}Todo: DB mig along 110?}
-Minimum E (reorientation) per distance
+{\color{red}Todo: Si int and C sub ...}
-Todo: Si int and C sub ...
-Todo: Model of kick-out and kick-in mechnism?
+{\color{red}Todo: Model of kick-out and kick-in mechnism?}
-Todo: Jahn-Teller distortion (vacancy) $\rightarrow$ actually three possibilities! :(
+{\color{red}Todo: Jahn-Teller distortion (vacancy) $\rightarrow$ actually three possibilities! :(}
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