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 \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.
+All point defects are calculated for the neutral charge state.
+
+\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-\sum_i N_i\mu_i
+\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 atom species equation \ref{eq:defects:ef2} is equivalent to equation \ref{eq:defects:ef1}.
+
 \section{Silicon self-interstitials}
 
+Point defects in silicon have been extensively studied, both experimentally and theoretically \cite{fahey89,leung99}.
+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{center}
+\begin{tabular}{l c c c c c}
+\hline
+\hline
+ & T & H & \hkl<1 0 0> DB & \hkl<1 1 0> DB & V \\
+\hline
+ Erhard/Albe MD & 3.40 & unstable & 5.42 & 4.39 & 3.13 \\
+ VASP & 3.77 & 3.42 & 4.41 & 3.39 & 3.63 \\
+ LDA \cite{leung99} & 3.43 & 3.31 & - & 3.31 & - \\
+ GGA \cite{leung99} & 4.07 & 3.80 & - & 3.84 & - \\
+\hline
+\hline
+\end{tabular}
+\end{center}
+\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}
+
+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.
+
+It has turned out to be very difficult to capture the results of quantum-mechanical calculations in analytical potential models.
+Among the established analytical potentials only the EDIP \cite{} and Stillinger-Weber \cite{} potential reproduce the correct order in energy of the defects.
+However, these potenitals show shortcomings concerning the description of other physical properties and are unable to describe the C-C and C-Si interaction.
+In fact the Erhard/Albe potential calculations favor the tetrahedral defect configuration.
+The hexagonal configuration is not stable opposed to results of the authors of the potential \cite{}.
+The Si interstitial atom moves towards an energetically more favorable position very close to the tetrahedral one but slightly displaced along the three coordinate axes.
+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 bond-centered configuration is unstable for both, the Erhard/Albe and VASP calculations.
+
+
 \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}