The final configurations obtained after relaxation are presented in figure \ref{fig:defects:conf}.
\begin{figure}[h]
\begin{center}
-\hrule
-\vspace*{0.2cm}
+%\hrule
+%\vspace*{0.2cm}
+%\begin{flushleft}
+%\begin{minipage}{5cm}
+%\underline{\hkl<1 1 0> dumbbell}\\
+%$E_{\text{f}}=3.39\text{ eV}$\\
+%\includegraphics[width=3.0cm]{si_pd_vasp/110_2333.eps}
+%\end{minipage}
+%\begin{minipage}{5cm}
+%\underline{Hexagonal}\\
+%$E_{\text{f}}=3.42\text{ eV}$\\
+%\includegraphics[width=3.0cm]{si_pd_vasp/hex_2333.eps}
+%\end{minipage}
+%\begin{minipage}{5cm}
+%\underline{Tetrahedral}\\
+%$E_{\text{f}}=3.77\text{ eV}$\\
+%\includegraphics[width=3.0cm]{si_pd_vasp/tet_2333.eps}
+%\end{minipage}\\[0.2cm]
+%\begin{minipage}{5cm}
+%\underline{\hkl<1 0 0> dumbbell}\\
+%$E_{\text{f}}=4.41\text{ eV}$\\
+%\includegraphics[width=3.0cm]{si_pd_vasp/100_2333.eps}
+%\end{minipage}
+%\begin{minipage}{5cm}
+%\underline{Vacancy}\\
+%$E_{\text{f}}=3.63\text{ eV}$\\
+%\includegraphics[width=3.0cm]{si_pd_vasp/vac_2333.eps}
+%\end{minipage}
+%\begin{minipage}{5cm}
+%\begin{center}
+%VASP\\
+%calculations\\
+%\end{center}
+%\end{minipage}
+%\end{flushleft}
+%\vspace*{0.2cm}
+%\hrule
\begin{flushleft}
\begin{minipage}{5cm}
-\underline{\hkl<1 1 0> dumbbell}\\
-$E_{\text{f}}=3.39\text{ eV}$\\
-\includegraphics[width=3.0cm]{si_pd_vasp/110_2333.eps}
-\end{minipage}
-\begin{minipage}{5cm}
-\underline{Hexagonal}\\
-$E_{\text{f}}=3.42\text{ eV}$\\
-\includegraphics[width=3.0cm]{si_pd_vasp/hex_2333.eps}
-\end{minipage}
-\begin{minipage}{5cm}
\underline{Tetrahedral}\\
-$E_{\text{f}}=3.77\text{ eV}$\\
-\includegraphics[width=3.0cm]{si_pd_vasp/tet_2333.eps}
-\end{minipage}\\[0.2cm]
-\begin{minipage}{5cm}
-\underline{\hkl<1 0 0> dumbbell}\\
-$E_{\text{f}}=4.41\text{ eV}$\\
-\includegraphics[width=3.0cm]{si_pd_vasp/100_2333.eps}
+$E_{\text{f}}=3.40\text{ eV}$\\
+\includegraphics[width=4.0cm]{si_pd_albe/tet.eps}
\end{minipage}
-\begin{minipage}{5cm}
-\underline{Vacancy}\\
-$E_{\text{f}}=3.63\text{ eV}$\\
-\includegraphics[width=3.0cm]{si_pd_vasp/vac_2333.eps}
+\begin{minipage}{10cm}
+\underline{Hexagonal}\\[0.1cm]
+\begin{minipage}{4cm}
+$E_{\text{f}}^*=4.48\text{ eV}$\\
+\includegraphics[width=4.0cm]{si_pd_albe/hex_a.eps}
\end{minipage}
-\begin{minipage}{5cm}
+\begin{minipage}{0.8cm}
\begin{center}
-VASP\\
-calculations\\
+$\Rightarrow$
\end{center}
\end{minipage}
-\end{flushleft}
-\vspace*{0.2cm}
-\hrule
-\begin{flushleft}
-\begin{minipage}{5cm}
-\underline{\hkl<1 1 0> dumbbell}\\
-$E_{\text{f}}=4.39\text{ eV}$\\
-\includegraphics[width=3.0cm]{si_pd_albe/110.eps}
-\end{minipage}
-\begin{minipage}{5cm}
-\underline{Hexagonal}\\
+\begin{minipage}{4cm}
$E_{\text{f}}=3.96\text{ eV}$\\
-\includegraphics[width=3.0cm]{si_pd_albe/hex.eps}
+\includegraphics[width=4.0cm]{si_pd_albe/hex.eps}
\end{minipage}
-\begin{minipage}{5cm}
-\underline{Tetrahedral}\\
-$E_{\text{f}}=3.40\text{ eV}$\\
-\includegraphics[width=3.0cm]{si_pd_albe/tet.eps}
\end{minipage}\\[0.2cm]
\begin{minipage}{5cm}
\underline{\hkl<1 0 0> dumbbell}\\
$E_{\text{f}}=5.42\text{ eV}$\\
-\includegraphics[width=3.0cm]{si_pd_albe/100.eps}
+\includegraphics[width=4.0cm]{si_pd_albe/100.eps}
\end{minipage}
\begin{minipage}{5cm}
-\underline{Vacancy}\\
-$E_{\text{f}}=3.13\text{ eV}$\\
-\includegraphics[width=3.0cm]{si_pd_albe/vac.eps}
+\underline{\hkl<1 1 0> dumbbell}\\
+$E_{\text{f}}=4.39\text{ eV}$\\
+\includegraphics[width=4.0cm]{si_pd_albe/110.eps}
\end{minipage}
\begin{minipage}{5cm}
-\begin{center}
-Erhard/Albe potential\\
-calculations\\
-\end{center}
+\underline{Vacancy}\\
+$E_{\text{f}}=3.13\text{ eV}$\\
+\includegraphics[width=4.0cm]{si_pd_albe/vac.eps}
\end{minipage}
\end{flushleft}
-\hrule
+%\hrule
\end{center}
-\caption[Relaxed silicon self-interstitial defect configurations.]{Relaxed silicon self-interstitial defect configurations. The silicon atoms and the bonds (only for the interstitial atom) are illustrated by yellow spheres and blue lines.}
+\caption[Relaxed silicon self-interstitial defect configurations obtained by classical potential calculations.]{Relaxed silicon self-interstitial defect configurations obtained by classical potential calculations. The silicon atoms and the bonds (only for the interstitial atom) are illustrated by yellow spheres and blue lines.}
\label{fig:defects:conf}
\end{figure}
Carbon is a common and technologically important impurity in silicon.
Concentrations as high as $10^{18}\text{ cm}^{-3}$ occur in Czochralski-grown silicon samples.
It is well established that carbon and other isovalent impurities prefer to dissolve substitutionally in silicon.
-However, radiation damage can generate carbon interstitials \cite{watkins76} which enough mobility at room temeprature to migrate and form defect complexes.
+However, radiation damage can generate carbon interstitials \cite{watkins76} which have enough mobility at room temeprature to migrate and form defect complexes.
Formation energies of the most common carbon point defects in crystalline silicon are summarized in table \ref{tab:defects:c_ints}.
The type of reservoir of the carbon impurity to determine the formation energy of the defect was chosen to be SiC.
-This is consistent with the methods used in the articles \cite{tersoff90,dal_pino93} which the results are compared to in the following.
+This is consistent with the methods used in the articles \cite{tersoff90,dal_pino93}, which the results are compared to in the following.
Hence, the chemical potential of silicon and carbon is determined by the cohesive energy of silicon and silicon carbide.
\begin{table}[h]
\begin{center}
& T & H & \hkl<1 0 0> DB & \hkl<1 1 0> DB & S & B \\
\hline
Erhard/Albe MD & 6.09 & 9.05$^*$ & 3.88 & 5.18 & 0.75 & 5.59$^*$ \\
- VASP (C [dia] reservoir) & unstable & unstable & 3.15 & 3.60 & 1.39 & 4.10 \\
- VASP (SiC reservoir) & unstable & unstable & 3.72 & 4.16 & 1.95 & 4.66 \\
+ %VASP & unstable & unstable & 3.15 & 3.60 & 1.39 & 4.10 \\
+ VASP & unstable & unstable & 3.72 & 4.16 & 1.95 & 4.66 \\
Tersoff \cite{tersoff90} & 3.8 & 6.7 & 4.6 & 5.9 & 1.6 & 5.3 \\
ab initio & - & - & x & - & 1.89 \cite{dal_pino93} & x+2.1 \cite{capaz94} \\
\hline
\end{table}
Substitutional carbon in silicon is found to be the lowest configuration in energy for all potential models.
-
-
+An experiemntal value of the formation energy of substitutional carbon was determined by a fit to solubility data yielding a concentration of $3.5 \times 10^{24} \exp{(-2.3\text{ eV}/k_{\text{B}}T)} \text{ cm}^{-3}$ \cite{bean71}.
+However, there is no particular reason for treating the prefactor as a free parameter in the fit to the experimental data.
+It is simply given by the atomic density of pure silicon, which is $5\times 10^{22}\text{ cm}^{-3}$.
+Tersoff \cite{tersoff90} and Dal Pino et. al. \cite{dal_pino93} pointed out that by combining this prefactor with the calculated values for the energy of formation ranging from 1.6 to 1.89 eV an excellent agreement with the experimental solubility data within the entire temeprature range of the experiment is obtained.
+This reinterpretation of the solubility data, first proposed by Tersoff and later on reinforced by Dal Pino et. al. is in good agreement with the results of the quantum-mechanical calculations performed in this work.
Except for Tersoff's tedrahedral configuration results the \hkl<1 0 0> dumbbell is the energetically most favorable interstital configuration.
-The low energy of formation for the tetrahedral interstitial in the case of the Tersoff potential is believed to be an artifact of the short cutoff (see ref. 13 in \cite{tersoff90}) and the real formation energy is supposed to be located between 3 and 10 eV.
-The formation energy for substitutional carbon is about 3 eV lower than the \hkl<1 0 0> dumbbell for both classical potentials.
+The low energy of formation for the tetrahedral interstitial in the case of the Tersoff potential is believed to be an artifact of the abrupt cutoff set to 2.5 \AA (see ref. 11 and 13 in \cite{tersoff90}) and the real formation energy is, thus, supposed to be located between 3 and 10 eV.
+Keeping these considerations in mind, the \hkl<1 0 0> dumbbell is the most favorable interstitial configuration for all interaction models.
+In addition to the theoretical results compared to in table \ref{tab:defects:c_ints} there is experimental evidence of the existence of this configuration \cite{watkins76}.
+It is frequently generated in the classical potential simulation runs in which carbon is inserted at random positions in the c-Si matrix.
+In quantum-mechanical simulations the unstable tetrahedral and hexagonal configurations undergo a relaxation into the \hkl<1 0 0> dumbbell configuration.
+Thus, this configuration is of great importance and discussed in more detail in section \ref{subsection:100db}.
The highest energy is observed for the hexagonal interstitial configuration using classical potentials.
Quantum-mechanical calculations reveal this configuration to be unstable, which is also reproduced by the Erhard/Albe potential.
In both cases a relaxation towards the \hkl<1 0 0> dumbbell configuration is observed.
+The tetrahedral is the second most unfavorable interstitial configuration using classical potentials and keeping in mind the abrupt cutoff effect in the case of the Tersoff potential as discussed earlier.
+Again, quantum-mechanical results reveal this configuration unstable.
+The fact that the tetrahedral and hexagonal configurations are the two most unstable configurations in classical potential calculations and, thus, are less likely to arise in MD simulations acts in concert with the fact that these configurations are found to be unstable in the more accurate quantum-mechanical calculations.
+
+Bond-centered ...
\subsection[\hkl<1 0 0> dumbbell interstitial configuration]{\boldmath\hkl<1 0 0> dumbbell interstitial configuration}
+\label{subsection:100db}
\subsection{Bond-centered interstitial configuration}
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