\chapter{Point defects in silicon}
+Given the conversion mechnism of SiC in crystalline silicon introduced in \ref{section:assumed_prec} the understanding of carbon and silicon interstitial point defects in c-Si is of great interest.
+Both types of defects are examined in the following both by classical potential as well as density functional theory calculations.
+
+In case of the classical potential calculations a simulation volume of nine silicon lattice constants in each direction is used.
+Calculations are performed in an isothermal-isobaric NPT ensemble.
+Coupling to the heat bath is achieved by the Berendsen thermostat with a time constant of 100 fs.
+The temperature is set to zero Kelvin.
+Pressure is controlled by a Berendsen barostat again using a time constant of 100 fs and a bulk modulus of 100 GPa for silicon.
+To exclude surface effects periodic boundary conditions are applied.
+
+Due to the restrictions in computer time three silicon lattice constants in each direction are considered sufficiently large enough for DFT calculations.
+The ions are relaxed by a conjugate gradient method.
+The cell volume and shape is allowed to change using the pressure control algorithm of Parinello and Rahman \cite{}.
+Periodic boundary conditions in each direction are applied.
+
+\begin{figure}
+\begin{center}
+\includegraphics[width=10cm]{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}$\bullet$}) and \hkl<1 1 0> dumbbell ({\color{magenta}$\bullet$}) interstitial configurations.}
+\label{fig:defects:ins_pos}
+\end{figure}
+
+The interstitial atom positions are displayed in Fig. \ref{fig:defects:ins_pos}.
+In seperated simulation runs the silicon or carbon atom is inserted at the tetrahedral $(0,0,0)$ ({\color{red}$\bullet$}), the hexagonal $(-1/8,-1/8,1/8)$ ({\color{green}$\bullet$}), the nearly \hkl<1 0 0> dumbbell $(-1/4,-1/4,-1/8)$ ({\color{yellow}$\bullet$}) and the nearly \hkl<1 1 0> dumbbell $(-1/8,-1/8,-1/4)$ ({\color{magenta}$\bullet$}) interstitial position.
+For the dumbbell configurations the nearest silicon atom is displaced by $(0,0,-1/8)$ and $(-1/8,-1/8,0)$ respectively of the unit cell length to avoid to high forces.
+A vacancy or a substitutional atom is realized by removing one silicon atom and switching the type of one silicon atom respectively.
+
+From an energetic point of view the free energy of formation $E_{\text{f}}$ is suitable for the characterization of defect structures.
+For defect configurations consisting of a single atom species the formation energy is defined as
+\begin{equation}
+E_{\text{f}}=\left(E_{\text{coh}}^{\text{defect}}
+ -E_{\text{coh}}^{\text{defect-free}}\right)N
+\end{equation}
+where $N$ and $E_{\text{coh}}^{\text{defect}}$ are the number of atoms and the cohesive energy per atom in the defect configuration and $E_{\text{coh}}^{\text{defect-free}}$ is the cohesive energy per atom of the defect-free structure.
+Evtl Paper mit Ef rauskramen lenen schreiben ...
+Defects consisting of two or more atom species ...
+
\section{Silicon self-interstitials}
\section{Carbon related point defects}
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