\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 \\
+ Erhard/Albe MD & 3.40 & 4.48$^*$ & 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
\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.}
+\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. Formation energies for unstable configurations are marked by an asterisk and determined by using the low kinetic energy configuration shortly before the relaxation into the more favorable configuration starts.}
\label{tab:defects:si_self}
\end{table}
The final configurations obtained after relaxation are presented in figure \ref{fig:defects:conf}.
\begin{minipage}{5cm}
\underline{\hkl<1 1 0> dumbbell}\\
$E_{\text{f}}=3.39\text{ eV}$\\
-\includegraphics[width=4.0cm]{si_pd_vasp/110_2333.eps}
+\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=4.0cm]{si_pd_vasp/hex_2333.eps}
+\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=4.0cm]{si_pd_vasp/tet_2333.eps}
+\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=4.0cm]{si_pd_vasp/100_2333.eps}
+\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=4.0cm]{si_pd_vasp/vac_2333.eps}
+\includegraphics[width=3.0cm]{si_pd_vasp/vac_2333.eps}
\end{minipage}
\begin{minipage}{5cm}
\begin{center}
\begin{minipage}{5cm}
\underline{\hkl<1 1 0> dumbbell}\\
$E_{\text{f}}=4.39\text{ eV}$\\
-\includegraphics[width=4.0cm]{si_pd_albe/110.eps}
+\includegraphics[width=3.0cm]{si_pd_albe/110.eps}
\end{minipage}
\begin{minipage}{5cm}
\underline{Hexagonal}\\
$E_{\text{f}}=3.96\text{ eV}$\\
-\includegraphics[width=4.0cm]{si_pd_albe/hex.eps}
+\includegraphics[width=3.0cm]{si_pd_albe/hex.eps}
\end{minipage}
\begin{minipage}{5cm}
\underline{Tetrahedral}\\
$E_{\text{f}}=3.40\text{ eV}$\\
-\includegraphics[width=4.0cm]{si_pd_albe/tet.eps}
+\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=4.0cm]{si_pd_albe/100.eps}
+\includegraphics[width=3.0cm]{si_pd_albe/100.eps}
\end{minipage}
\begin{minipage}{5cm}
\underline{Vacancy}\\
$E_{\text{f}}=3.13\text{ eV}$\\
-\includegraphics[width=4.0cm]{si_pd_albe/vac.eps}
+\includegraphics[width=3.0cm]{si_pd_albe/vac.eps}
\end{minipage}
\begin{minipage}{5cm}
\begin{center}
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{}.
+The hexagonal configuration is not stable opposed to results of the authors of the potential \cite{albe_sic_pot}.
+In the first two pico seconds while kinetic energy is decoupled from the system the Si interstitial seems to condense at the hexagonal site.
+The formation energy of 4.48 eV is determined by this low kinetic energy configuration shortly before the relaxation process starts.
+The Si interstitial atom then begins to slowly move 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{albe_sic_pot}.
Obviously the authors did not carefully check the relaxed results assuming a hexagonal configuration.
+In figure \ref{fig:defects:kin_si_hex} the relaxation process is shown on the basis of the kinetic energy plot.
+\begin{figure}[h]
+\begin{center}
+\includegraphics[width=12cm]{e_kin_si_hex.ps}
+\end{center}
+\caption{Kinetic energy plot of the relaxation process of the hexagonal silicon self-interstitial defect simulation using the Erhard/Albe classical potential.}
+\label{fig:defects:kin_si_hex}
+\end{figure}
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 same type of interstitial arises using random insertions.
In addition, variations exist in which the displacement is only along two \hkl<1 0 0> axes ($E_\text{f}=3.8\text{ eV}$) or along a single \hkl<1 0 0> axes ($E_\text{f}=3.6\text{ eV}$) successively approximating the tetdrahedral configuration and formation energy.
The bond-centered configuration is unstable and the \hkl<1 0 0> dumbbell interstitial is the most unfavorable configuration for both, the Erhard/Albe and VASP calculations.
-In the case of the classical potential simulations bonds between atoms are displayed if there is an interaction according to the potential model, that is if the distance of two atoms is within the cut-off region $S_{ij}$ introduced in equation \eqref{eq:basics:fc}.
+In the case of the classical potential simulations bonds between atoms are displayed if there is an interaction according to the potential model, that is if the distance of two atoms is within the cutoff region $S_{ij}$ introduced in equation \eqref{eq:basics:fc}.
For the tetrahedral and the slightly displaced configurations four bonds to the atoms located in the center of the planes of the unit cell exist in addition to the four tetrahedral bonds.
-The length of these bonds are, however, close to the cut-off range and thus are weak interactions not constituting actual chemical bonds.
+The length of these bonds are, however, close to the cutoff range and thus are weak interactions not constituting actual chemical bonds.
The same applies to the bonds between the interstitial and the upper two atoms in the \hkl<1 1 0> dumbbell configuration.
A more detailed description of the chemical bonding is achieved by quantum-mechanical calculations by investigating the accumulation of negative charge between the nuclei.
\section{Carbon related point defects}
+Formation energies of the most common carbon point defects in crystalline silicon are summarized in table \ref{tab:defects:c_ints}.
+\begin{table}[h]
+\begin{center}
+\begin{tabular}{l c c c c c c}
+\hline
+\hline
+ & T & H & \hkl<1 0 0> DB & \hkl<1 1 0> DB & S & B \\
+\hline
+ Erhard/Albe MD & 5.41 & 8.37$^*$ & 3.21 & 4.50 & 0.07 & 4.91$^*$ \\
+ VASP & unstable & unstable & 3.15 & 3.60 & 1.39 & 4.10 \\
+ 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
+\hline
+\end{tabular}
+\end{center}
+\caption[Formation energies of carbon point defects in crystalline silicon determined by classical potential molecular dynamics and density functional calculations.]{Formation energies of carbon point defects 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 S the substitutional interstitial configuration. The dumbbell configurations are abbreviated by DB. Formation energies for unstable configurations are marked by an asterisk and determined by using the low kinetic energy configuration shortly before the relaxation into the more favorable configuration starts.}
+\label{tab:defects:c_ints}
+\end{table}
+Except for Tersoff's tedrahedral configuration results the \hkl<1 0 0> dumbbell is the energetically most favorable configuration for all types of interaction models.
+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.
+
\section[Migration of the carbon \hkl<1 0 0> interstitial]{\boldmath Migration of the carbon \hkl<1 0 0> interstitial}
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