The most common empirical potentials for covalent systems are the Stillinger-Weber\cite{stillinger85} (SW), Brenner\cite{brenner90}, Tersoff\cite{tersoff_si3} and environment-dependent interatomic (EDIP)\cite{bazant96,bazant97,justo98} potential.\r
Until recently\cite{lucas10}, a parametrization to describe the C-Si multicomponent system within the mentioned interaction models did only exist for the Tersoff\cite{tersoff_m} and related potentials.\r
Whether such potentials are appropriate for the description of the physical problem has, however, to be verified first by applying classical and quantum-mechanical methods to relevant processes that can be treated by both methods.\r
-By combination of empirical potential molecular dynamics (MD) and density functional theory (DFT) calculations the SW turned out to be best suited for simulations of dislocation nucleation processes\cite{godet03} and threshold displacement energy calculations\cite{holmstroem08} important in ion implantation.\r
-Also the Tersoff potential yields a qualitative agreement concerning the interaction of Si self-interstitials and substitutional C\cite{mattoni02}.\r
+For instance, by combination of empirical potential molecular dynamics (MD) and density functional theory (DFT) calculations, SW turned out to be best suited for simulations of dislocation nucleation processes\cite{godet03} and threshold displacement energy calculations\cite{holmstroem08} important in ion implantation, while the Tersoff potential yielded a qualitative agreement for the interaction of Si self-interstitials with substitutional C\cite{mattoni02}.\r
An extensive comparison\cite{balamane92} concludes that each potential has its strengths and limitations and none of them is clearly superior to others.\r
Despite their shortcomings these potentials are assumed to be reliable for large-scale simulations\cite{balamane92,huang95,godet03} on specific problems under investigation providing insight into phenomena that are otherwise not accessible by experimental or first principles methods.\r
Remaining shortcomings have frequently been resolved by modifying the interaction\cite{tang95,mattoni2007} or extending it\cite{devanathan98_2} with data gained from ab initio calculations\cite{nordlund97}.\r
\r
\subsection{Carbon interstitials in various geometries}\r
\r
-Table~\ref{tab:defects} summarizes the formation energies of the interstitial configurations for the Erhart/Albe and VASP calculations performed in this work as well as further results from literature.\r
+Table~\ref{tab:defects} summarizes the formation energies of defect structures for the Erhart/Albe and VASP calculations performed in this work as well as further results from literature.\r
The formation energies are defined in the same way as in the articles used for comparison\cite{tersoff90,dal_pino93} chosing SiC as a reservoir for the carbon impurity.\r
Relaxed geometries are displayed in Fig.~\ref{fig:defects}.\r
Astonishingly there is only little literature present to compare with.\r
-\begin{table}[th]\r
+\begin{table*}\r
+\begin{ruledtabular}\r
\begin{tabular}{l c c c c c c}\r
-\hline\r
-\hline\r
+%\hline\r
+%\hline\r
& T & H & \hkl<1 0 0> DB & \hkl<1 1 0> DB & S & B \\\r
\hline\r
Erhart/Albe & 6.09 & 9.05$^*$ & 3.88 & 5.18 & 0.75 & 5.59$^*$ \\\r
VASP & unstable & unstable & 3.72 & 4.16 & 1.95 & 4.66 \\\r
Tersoff\cite{tersoff90} & 3.8 & 6.7 & 4.6 & 5.9 & 1.6 & 5.3 \\\r
- ab initio & - & - & x & - & 1.89 \cite{dal_pino93} & x+2.1 \cite{capaz94} \\\r
- % there is no more data!\r
-\hline\r
-\hline\r
+ ab initio\cite{dal_pino93,capaz94} & - & - & x & - & 1.89 \cite{dal_pino93} & x+2.1 \cite{capaz94} \\\r
+ % there is no more ab initio data!\r
+%\hline\r
+%\hline\r
\end{tabular}\r
-\caption{Formation energies of carbon point defects in crystalline silicon determined by classical potential and ab initio methods. 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.}\r
+\end{ruledtabular}\r
+\caption{Formation energies of carbon point defects in crystalline silicon determined by classical potential and ab initio methods. 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 obtained by classical potential MD are marked by an asterisk and determined by using the low kinetic energy configuration shortly before the relaxation into the more favorable configuration starts.}\r
\label{tab:defects}\r
-\end{table}\r
+\end{table*}\r
\begin{figure}\r
\begin{minipage}[t]{0.32\columnwidth}\r
\underline{Tetrahedral}\\\r
\label{fig:defects}\r
\end{figure} \r
\r
-Discrepancies are observed between the results from classical potential calculations and those obtained from first principles.\r
+Substitutional carbon (C$_{\text{sub}}$) in silicon, which is in fact not an interstitial defect, is found to be the lowest configuration with regard to energy for all potential models.\r
+VASP calculations performed in this work are in good agreement with results obtained by classical potential simulations by Tersoff\cite{tersoff90} and ab initio calculations done by Dal Pino et~al\cite{dal_pino93}.\r
+However, the Erhart/Albe potential dramatically underestimtes the C$_{\text{sub}}$ formation energy, which is a definite drawback of the potential.\r
+\r
+Except for the Tersoff potential the \hkl<1 0 0> dumbbell is the energetically most favorable interstital configuration, in which the C and Si dumbbell atoms share a Si lattice site.\r
+This finding is in agreement with several theoretical\cite{burnard93,leary97,dal_pino93,capaz94} and experimental\cite{watkins76,song90} investigations.\r
+Tersoff as well, considers the \hkl<1 0 0> to be the ground state configuration and believes an artifact due to the abrupt C-Si cut-off used in the potential to be responsible for the small value of the tetrahedral formation energy\cite{tersoff90}.\r
+A qualitative difference is observed investigating the dumbbell structures.\r
+While the C-Si distance of the dumbbell atoms is almost equal for both methods, the vertical position of the dumbbell inside the tetrahedra spanned by the four next neighboured Si atoms differs significantly.\r
+The dumbbell based on the Erhart/Albe potential is almost centered around the regular Si lattice site as can be seen in Fig. \ref{fig:defects} whereas for VASP calculations it is translated upwards with the C atom forming an almost collinear bond to the two Si atoms of the top face of the tetrahedra and the bond angle of the Si dumbbell atom to the two bottom face Si atoms approaching \unit[120]{$^\circ$}.\r
+This indicates predominant sp and sp$^2$ hybridization for the C and Si dumbbell atom respectively.\r
+Obviously the classical potential is not able to reproduce the clearly quantum-mechanical character of bonding.\r
+% empirical potential adjusts according to minimum of angular function, no QM!\r
+\r
+\r
+% pick up again later, that this is why erhart/albe is more promising for the specific problem under investigation\r
+\r
+\r
+\r
\r
While the Erhart/Albe potential predicts ... as stable, DFT does not. ...(further comparisons, trend "too high/low" E-formation,...)... \r
\r
+To conclude, discrepancies are observed between the results from classical potential calculations and those obtained from first principles.\r
Nevertheless, both methods predict the \hkl<1 0 0> dumbbell configuration to be most stable.\r
\r
-\r
\subsection{Mobility}\r
A measure for the mobility of the interstitial carbon is the activation energy for the migration path from one stable \r
position to another. The stable defect geometries have been discussed in the previous subsection. We now investigate \r
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