\end{equation}
where $E$ is the free energy of the interstitial system and $N_i$ and $\mu_i$ are the amount of atoms and the chemical potential of species $i$.
The chemical potential is determined by the cohesive energy of the structure of the specific type in equilibrium at zero Kelvin.
-For a defect configuration of a single atom species equation \ref{eq:defects:ef2} is equivalent to equation \ref{eq:defects:ef1}.
+For a defect configuration of a single atom species equation \eqref{eq:defects:ef2} is equivalent to equation \eqref{eq:defects:ef1}.
\section{Silicon self-interstitials}
To exclude failures in the implementation of the potential or the MD code itself the hexagonal defect structure was double-checked with the PARCAS MD code \cite{}.
The same type of interstitial arises using random insertions.
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 artefact of the analytical potential without physical authenticity revealing basic problems of analytical potential models for describing defect structures.
+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 (DAS MAL DURCHRECHNEN).
-Hence these artefacts should have a negligent influence in finite temperature simulations.
+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.
As the \hkl<1 0 0> dumbbell interstitial is the lowest configuration in energy it is the most probable hence important interstitial configuration of carbon in silicon.
It was first identified by infra-red (IR) spectroscopy \cite{bean70} and later on by electron paramagnetic resonance (EPR) spectroscopy \cite{watkins76}.
-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 displacements obtained by analytical potential and quantum-mechanical calculations.
+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{center}
-\includegraphics[width=10cm]{100-c-si-db_cmp.eps}
+\includegraphics[width=12cm]{100-c-si-db_cmp.eps}
\end{center}
+\caption[Sketch of the \hkl<1 0 0> dumbbell structure.]{Sketch of the \hkl<1 0 0> dumbbell structure. Atomic displacements, distances and bond angles are listed in table \ref{tab:defects:100db_cmp}.}
\label{fig:defects:100db_cmp}
-\caption[Sketch of the \hkl<1 0 0> dumbbell structure.]{Sketch of the \hkl<1 0 0> dumbbell structure. Atomic displacements and distances are listed in table \ref{tab:defects:100db_cmp}.}
\end{figure}
%
\begin{table}[h]
\begin{center}
+Displacements\\
\begin{tabular}{l c c c c c c c c c}
\hline
\hline
& $a$ & $b$ & $|a|+|b|$ & $\Delta x$ & $\Delta y$ & $\Delta z$ & $\Delta x$ & $\Delta y$ & $\Delta z$ \\
\hline
Erhard/Albe & 0.084 & -0.091 & 0.175 & -0.015 & -0.015 & -0.031 & -0.014 & 0.014 & 0.020 \\
-VASP & & & & & & & & & \\
+VASP & 0.109 & -0.065 & 0.174 & -0.011 & -0.011 & -0.024 & -0.014 & 0.014 & 0.025 \\
\hline
\hline
-\end{tabular}
+\end{tabular}\\[0.5cm]
+\end{center}
+\begin{center}
+Distances\\
+\begin{tabular}{l c c c c c c c c r}
+\hline
+\hline
+ & $r(1C)$ & $r(2C)$ & $r(3C)$ & $r(12)$ & $r(13)$ & $r(34)$ & $r(23)$ & $r(25)$ & $a_{\text{Si}}^{\text{equi}}$\\
+\hline
+Erhard/Albe & 0.175 & 0.329 & 0.186 & 0.226 & 0.300 & 0.343 & 0.423 & 0.425 & 0.543 \\
+VASP & 0.174 & 0.341 & 0.182 & 0.229 & 0.286 & 0.347 & 0.422 & 0.417 & 0.548 \\
+\hline
+\hline
+\end{tabular}\\[0.5cm]
\end{center}
\begin{center}
-\begin{tabular}{l c c c c c c c c}
+Angles\\
+\begin{tabular}{l c c c c }
\hline
\hline
- & $r(1C)$ & $r(2C)$ & $r(3C)$ & $r(12)$ & $r(13)$ & $r(34)$ & $r(23)$ & $r(25)$\\
+ & $\theta_1$ & $\theta_2$ & $\theta_3$ & $\theta_4$ \\
\hline
-Erhard/Albe & & & & & & & \\
-VASP & & & & & & & \\
+Erhard/Albe & 140.2 & 109.9 & 134.4 & 112.8 \\
+VASP & 130.7 & 114.4 & 146.0 & 107.0 \\
\hline
\hline
\end{tabular}\\[0.5cm]
\end{center}
+\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}
-\caption[Atomic displacements and distances of the \hkl<1 0 0> dumbbell structure obtained by the Erhard/Albe potential and VASP calculations.]{Atomic displacements and distances 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 schematically displayed in figure \ref{fig:defects:100db_cmp}.}
\end{table}
+\begin{figure}[h]
+\begin{center}
+\begin{minipage}{6cm}
+\begin{center}
+\underline{Erhard/Albe}
+\includegraphics[width=5cm]{c_pd_albe/100_cmp.eps}
+\end{center}
+\end{minipage}
+\begin{minipage}{6cm}
+\begin{center}
+\underline{VASP}
+\includegraphics[width=5cm]{c_pd_vasp/100_cmp.eps}
+\end{center}
+\end{minipage}
+\end{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{center}
+\includegraphics[height=10cm]{c_pd_vasp/eden.eps}
+\includegraphics[height=12cm]{c_pd_vasp/100_2333_ksl.ps}
+\end{center}
+\caption[Charge density isosurface and Kohn-Sham levels of the C \hkl<1 0 0> dumbbell structure obtained by VASP calculations.]{Charge density isosurface and Kohn-Sham levels of the C \hkl<1 0 0> dumbbell structure obtained by VASP calculations. Yellow and grey spheres correspond to silicon and carbon atoms. The blue surface is the charge density isosurface. In the energy level diagram red and green lines and dots mark occupied and unoccupied states.}
+\label{img:defects:charge_den_and_ksl}
+\end{figure}
+The silicon atom numbered '1' and the C atom compose the dumbbell structure.
+They share the lattice site which is indicated by the dashed red circle and which they are displaced from by length $a$ and $b$ respectively.
+The atoms no longer have four tetrahedral bonds to the silicon atoms located on the alternating opposite edges of the cube.
+Instead, each of the dumbbell atoms forms threefold coordinated bonds, which are located in a plane.
+One bond is formed to the other dumbbell atom.
+The other two bonds are bonds to the two silicon edge atoms located in the opposite direction of the dumbbell atom.
+The distance of the two dumbbell atoms is almost the same for both types of calculations.
+However, in the case of the VASP calculation, the dumbbell structure is pushed upwards compared to the Erhard/Albe results.
+This is easily identified by comparing the values for $a$ and $b$ and the two structures in figure \ref{fig:defects:100db_vis_cmp}.
+Thus, the angles of bonds of the silicon dumbbell atom ($\theta_1$ and $\theta_2$) are closer to $120^{\circ}$ signifying the predominance of $sp^2$ hybridization.
+On the other hand, the carbon atom forms an almost collinear bond ($\theta_3$) with the two silicon edge atoms implying the predominance of $p$ and $sp$ bonding.
+This is supported by the image of the charge density isosurface in figure \ref{img:defects:charge_den_and_ksl}.
+In the same figure the Kohn-Sham levels are shown.
+There is no magnetization density.
+An acceptor level arises resulting in a band gap of 0.35 eV compared to 0.75 eV as obtained for plain silicon.
\subsection{Bond-centered interstitial configuration}
\label{subsection:bc}
+In the bond-centerd insterstitial configuration the interstitial atom is located inbetween two next neighboured silicon atoms.
+In former studies this configuration is found to be an intermediate saddle point configuration determining the migration barrier of one possibe migration path of a \hkl<1 0 0> dumbbel configuration into another one \cite{capaz94}.
+Hier ist es aber ein echtes Minimum.
+Eine 'weitere' Barriere muss ueberschritten werden um dahin zu kommen.
+Genaueres in section \ref{subsection:100mig}.
+Die Konfiguration besitzt ein magnetisches Moment.
+Bild der spin-ladungen.
+
+
\section[Migration of the carbon \hkl<1 0 0> interstitial]{\boldmath Migration of the carbon \hkl<1 0 0> interstitial}
\label{subsection:100mig}
+In the following the problem of interstitial carbon migration in silicon is considered.
+
\section{Combination of point defects}
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