\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}

\section{Combination of point defects}

