\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}[h]
+\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.}
+\label{fig:defects:ins_pos}
+\end{figure}
+
+The interstitial atom positions are displayed in figure \ref{fig:defects:ins_pos}.
+In seperated simulation runs the silicon or carbon atom is inserted at the
+\begin{itemize}
+ \item tetrahedral, $\vec{p}=(0,0,0)$, ({\color{red}$\bullet$})
+ \item hexagonal, $\vec{p}=(-1/8,-1/8,1/8)$, ({\color{green}$\bullet$})
+ \item nearly \hkl<1 0 0> dumbbell, $\vec{p}=(-1/4,-1/4,-1/8)$, ({\color{yellow}$\bullet$})
+ \item nearly \hkl<1 1 0> dumbbell, $\vec{p}=(-1/8,-1/8,-1/4)$, ({\color{magenta}$\bullet$})
+ \item bond-centered, $\vec{p}=(-1/8,-1/8,-3/8)$, ({\color{cyan}$\bullet$})
+\end{itemize}
+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 too 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
+\label{eq:defects:ef1}
+\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.
+The formation energy of defects consisting of two or more atom species is defined as
+\begin{equation}
+E_{\text{f}}=E-N_1\mu_1-N_2\mu_2 - \ldots
+\label{eq:defects:ef2}
+\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 species equation \ref{eq:defects:ef2} is equivalent to equation \ref{eq:defects:ef1}.
+
\section{Silicon self-interstitials}
+
+
\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}
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