\begin{center}
\includegraphics[width=9cm]{unit_cell_e.eps}
\end{center}
-\caption{Insertion positions for the tetrahedral ({\color{red}$\bullet$}), hexagonal ({\color{green}$\bullet$}), \hkl<1 0 0> dumbbell ({\color{yellow}$\bullet$}), \hkl<1 1 0> dumbbell ({\color{magenta}$\bullet$}) and bond-centered ({\color{cyan}$\bullet$}) interstitial configuration.}
+\caption[Insertion positions for the tetrahedral ({\color{red}$\bullet$}), hexagonal ({\color{green}$\bullet$}), \hkl<1 0 0> dumbbell ({\color{yellow}$\bullet$}), \hkl<1 1 0> dumbbell ({\color{magenta}$\bullet$}) and bond-centered ({\color{cyan}$\bullet$}) interstitial configuration.]{Insertion positions for the tetrahedral ({\color{red}$\bullet$}), hexagonal ({\color{green}$\bullet$}), \hkl<1 0 0> dumbbell ({\color{yellow}$\bullet$}), \hkl<1 1 0> dumbbell ({\color{magenta}$\bullet$}) and bond-centered ({\color{cyan}$\bullet$}) interstitial configuration. The black dots ({\color{black}$\bullet$}) correspond to the silicon atoms and the blue lines ({\color{blue}-}) indicate the covalent bonds of the perfect c-Si structure.}
\label{fig:defects:ins_pos}
\end{figure}
\caption[Formation energies of silicon self-interstitials in crystalline silicon determined by classical potential molecular dynamics and density functional calculations.]{Formation energies of silicon self-interstitials 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 V the vacancy interstitial configuration. The dumbbell configurations are abbreviated by DB.}
\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{center}
+\hrule
+\vspace*{0.2cm}
+\begin{flushleft}
+\begin{minipage}{5cm}
+\underline{\hkl<1 1 0> dumbbell}\\
+$E_{\text{f}}=3.39\text{ eV}$\\
+\includegraphics[width=4.0cm]{si_pd_vasp/110_2333.eps}
+\end{minipage}
+\begin{minipage}{5cm}
+\underline{Hexagonal}\\
+$E_{\text{f}}=3.42\text{ eV}$\\
+\includegraphics[width=4.0cm]{si_pd_vasp/hex_2333.eps}
+\end{minipage}
+\begin{minipage}{5cm}
+\underline{Tetrahedral}\\
+$E_{\text{f}}=3.77\text{ eV}$\\
+\includegraphics[width=4.0cm]{si_pd_vasp/tet_2333.eps}
+\end{minipage}\\[0.2cm]
+\begin{minipage}{5cm}
+\underline{\hkl<1 0 0> dumbbell}\\
+$E_{\text{f}}=4.41\text{ eV}$\\
+\includegraphics[width=4.0cm]{si_pd_vasp/100_2333.eps}
+\end{minipage}
+\begin{minipage}{5cm}
+\underline{Vacancy}\\
+$E_{\text{f}}=3.63\text{ eV}$\\
+\includegraphics[width=4.0cm]{si_pd_vasp/vac_2333.eps}
+\end{minipage}
+\begin{minipage}{5cm}
+\begin{center}
+VASP\\
+calculations\\
+\end{center}
+\end{minipage}
+\end{flushleft}
+\vspace*{0.2cm}
+\hrule
+\begin{flushleft}
+\begin{minipage}{5cm}
+\underline{\hkl<1 1 0> dumbbell}\\
+$E_{\text{f}}=4.39\text{ eV}$\\
+\includegraphics[width=4.0cm]{si_pd_albe/110.eps}
+\end{minipage}
+\begin{minipage}{5cm}
+\underline{Hexagonal}\\
+$E_{\text{f}}=3.96\text{ eV}$\\
+\includegraphics[width=4.0cm]{si_pd_albe/hex.eps}
+\end{minipage}
+\begin{minipage}{5cm}
+\underline{Tetrahedral}\\
+$E_{\text{f}}=3.40\text{ eV}$\\
+\includegraphics[width=4.0cm]{si_pd_albe/tet.eps}
+\end{minipage}\\[0.2cm]
+\begin{minipage}{5cm}
+\underline{\hkl<1 0 0> dumbbell}\\
+$E_{\text{f}}=5.42\text{ eV}$\\
+\includegraphics[width=4.0cm]{si_pd_albe/100.eps}
+\end{minipage}
+\begin{minipage}{5cm}
+\underline{Vacancy}\\
+$E_{\text{f}}=3.13\text{ eV}$\\
+\includegraphics[width=4.0cm]{si_pd_albe/vac.eps}
+\end{minipage}
+\begin{minipage}{5cm}
+\begin{center}
+Erhard/Albe potential\\
+calculations\\
+\end{center}
+\end{minipage}
+\end{flushleft}
+\hrule
+\end{center}
+\caption[Relaxed silicon self-interstitial defect configurations.]{Relaxed silicon self-interstitial defect configurations. The silicon atoms and the bonds (only for the interstitial atom) are illustrated by yellow spheres and blue lines.}
+\label{fig:defects:conf}
+\end{figure}
There are differences between the various results of the quantum-mechanical calculations but the consesus view is that the \hkl<1 1 0> dumbbell followed by the hexagonal and tetrahedral defect is the lowest in energy.
This is nicely reproduced by the DFT calculations performed in this work.
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{}.
Obviously the authors did not carefully check the relaxed results assuming a hexagonal configuration.
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.
+However, the energy barrier is small (DAS MAL DURCHRECHNEN).
+Hence these artefacts 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 bond-centered configuration is unstable for both, the Erhard/Albe and VASP calculations.
+In the case of the classical potential simulations bonds between atoms are displayed if there is an interaction according to the potential model, that is if the distance of two atoms is within the cut-off region $S_{ij}$ introduced in equation \eqref{eq:basics:fc}.
+For the tetrahedral and the slightly displaced configurations four bonds to the atoms located in the center of the planes of the unit cell exist in addition to the four tetrahedral bonds.
+The length of these bonds are, however, close to the cut-off range and thus are weak interactions not constituting actual chemical bonds.
+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 ...
\section{Carbon related point defects}
+
+
\section[Migration of the carbon \hkl<1 0 0> interstitial]{\boldmath Migration of the carbon \hkl<1 0 0> interstitial}
\section{Combination of point defects}
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