\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}

\section{Combination of point defects}

