\section{Methodology}
The plane-wave based Vienna ab initio simulation package (VASP) \cite{kresse96} is used for the first-principles calculations based on density functional theory (DFT).
-Exchange and correlation is taken into account by the generalized-gradient approximation as proposed by Perdew and Wang \cite{perdew86,perdew92}.
+Exchange and correlation is taken into account by the generalized-gradient approximation \cite{perdew86,perdew92}.
Norm-conserving ultra-soft pseudopotentials \cite{hamann79} as implemented in VASP \cite{vanderbilt90} are used to describe the electron-ion interaction.
A kinetic energy cut-off of \unit[300]{eV} is employed.
Defect structures and migration paths were modelled in cubic supercells with a side length of \unit[1.6]{nm} containing $216$ Si atoms.
These structures are large enough to restrict sampling of the Brillouin zone to the $\Gamma$-point and formation energies and structures are reasonably converged.
The ions and cell shape are allowed to change in order to realize a constant pressure simulation realized by the conjugate gradient algorithm.
-Spin polarization has been fully accounted for.
+Spin polarization is fully accounted for.
Migration and recombination pathways are investigated utilizing the constraint conjugate gradient relaxation technique (CRT) \cite{kaukonen98}.
The defect formation energy $E-N_{\text{Si}}\mu_{\text{Si}}-N_{\text{C}}\mu_{\text{C}}$ is defined by choosing SiC as a particle reservoir for the C impurity, i.e. the chemical potentials are determined by the cohesive energies of a perfect Si and SiC supercell after ionic relaxation.
The velocity Verlet algorithm \cite{verlet67} and a fixed time step of \unit[1]{fs} is used to integrate the equations motion.
Structural relaxation of defect structures is treated by the same algorithms at zero temperature.
-\section{Results}
+\section{Defect configurations in silicon}
-\subsection{Carbon and silicon defect configurations}
-
-Several geometries have been calculated to be stable for individual intrinsic and C related defects in Si.
-Fig.~\ref{fig:sep_def} shows the obtained structures while the corresponding energies of formation are summarized and compared to values from literature in Table~\ref{table:sep_eof}.
+Table~\ref{tab:defects} summarizes the formation energies of relevant defect structures for the EA and DFT calculations, which are shown in Figs.~\ref{fig_intrinsic_def} and \ref{fig:carbon_def}.
+\begin{table*}
+\centering
+\begin{tabular}{l c c c c c c c c c}
+\hline
+$E_{\text{f}}$ [eV] & Si$_{\text{i}}$ \hkl<1 1 0> DB & Si$_{\text{i}}$ H & Si$_{\text{i}}$ T & Si$_{\text{i}}$ \hkl<1 0 0> DB & V & C$_{\text{s}}$ & C$_{\text{i}}$ \hkl<1 0 0> DB & C$_{\text{i}}$ \hkl<1 1 0> DB & C$_{\text{i}}$ BC \\
+\hline
+VASP & 3.39 & 3.42 & 3.77 & 4.41 & 3.63 & 1.95 & 3.72 & 4.16 & 4.66 \\
+Erhart/Albe & 4.39 & 4.48$^*$ & 3.40 & 5.42 & 3.13 & 0.75 & 3.88 & 5.18 & 5.59$^*$ \\
+\hline
+\end{tabular}
+\caption{Formation energies of C and Si point defects in c-Si given in eV. T denotes the tetrahedral, H the hexagonal and BC the bond-centered interstitial configuration. V corresponds to the vacancy. Subscript i and s indicates the interstitial and substitutional configuration. Dumbbell configurations are abbreviated by DB. Formation energies of unstable configurations are marked by an asterisk.}
+\label{tab:defects}
+\end{table*}
\begin{figure}
-\begin{minipage}[t]{0.48\columnwidth}
+\centering
+\begin{minipage}[t]{0.43\columnwidth}
+\centering
\underline{Si$_{\text{i}}$ \hkl<1 1 0> DB}\\
-\includegraphics[width=\columnwidth]{si110_bonds.eps}
+\includegraphics[width=0.9\columnwidth]{si110_bonds.eps}
\end{minipage}
-\begin{minipage}[t]{0.48\columnwidth}
+\begin{minipage}[t]{0.43\columnwidth}
+\centering
\underline{Si$_{\text{i}}$ hexagonal}\\
-\includegraphics[width=\columnwidth]{sihex_bonds.eps}
+\includegraphics[width=0.9\columnwidth]{sihex_bonds.eps}
\end{minipage}\\
-\begin{minipage}[t]{0.48\columnwidth}
+\begin{minipage}[t]{0.43\columnwidth}
+\centering
\underline{Si$_{\text{i}}$ tetrahedral}\\
-\includegraphics[width=\columnwidth]{sitet_bonds.eps}
+\includegraphics[width=0.9\columnwidth]{sitet_bonds.eps}
\end{minipage}
-\begin{minipage}[t]{0.48\columnwidth}
+\begin{minipage}[t]{0.43\columnwidth}
+\centering
\underline{Si$_{\text{i}}$ \hkl<1 0 0> DB}\\
-\includegraphics[width=\columnwidth]{si100_bonds.eps}
+\includegraphics[width=0.9\columnwidth]{si100_bonds.eps}
\end{minipage}
\caption{Configurations of intrinsic silicon point defects. Dumbbell configurations are abbreviated by DB.}
\label{fig:intrinsic_def}
\end{figure}
\begin{figure}
-\begin{minipage}[t]{0.48\columnwidth}
+\centering
+\begin{minipage}[t]{0.43\columnwidth}
+\centering
\underline{C$_{\text{s}}$}
-\includegraphics[width=\columnwidth]{csub_bonds.eps}
+\includegraphics[width=0.9\columnwidth]{csub_bonds.eps}
\end{minipage}
-\begin{minipage}[t]{0.48\columnwidth}
+\begin{minipage}[t]{0.43\columnwidth}
+\centering
\underline{C$_{\text{i}}$ \hkl<1 0 0> DB}\\
-\includegraphics[width=\columnwidth]{c100_bonds.eps}
+\includegraphics[width=0.9\columnwidth]{c100_bonds.eps}
\end{minipage}\\
-\begin{minipage}[t]{0.48\columnwidth}
+\begin{minipage}[t]{0.43\columnwidth}
+\centering
\underline{C$_{\text{i}}$ \hkl<1 1 0> DB}\\
-\includegraphics[width=\columnwidth]{c110_bonds.eps}
+\includegraphics[width=0.9\columnwidth]{c110_bonds.eps}
\end{minipage}
-\begin{minipage}[t]{0.48\columnwidth}
+\begin{minipage}[t]{0.43\columnwidth}
+\centering
\underline{C$_{\text{i}}$ bond-centered}\\
-\includegraphics[width=\columnwidth]{cbc_bonds.eps}
+\includegraphics[width=0.9\columnwidth]{cbc_bonds.eps}
\end{minipage}
\caption{Configurations of carbon point defects in silicon. Silicon and carbon atoms are illustrated by yellow and gray spheres respectively. Dumbbell configurations are abbreviated by DB.}
\label{fig:carbon_def}
\end{figure}
-\begin{table*}
-\begin{tabular}{l c c c c c c c c c}
- & Si$_{\text{i}}$ \hkl<1 1 0> DB & Si$_{\text{i}}$ H & Si$_{\text{i}}$ T & Si$_{\text{i}}$ \hkl<1 0 0> DB & V & C$_{\text{s}}$ & C$_{\text{i}}$ \hkl<1 0 0> DB & C$_{\text{i}}$ \hkl<1 1 0> DB & C$_{\text{i}}$ BC \\
-\hline
- Present study & 3.39 & 3.42 & 3.77 & 4.41 & 3.63 & 1.95 & 3.72 & 4.16 & 4.66 \\
- \multicolumn{10}{c}{Other ab initio studies} \\
- Ref.\cite{al-mushadani03} & 3.40 & 3.45 & - & - & 3.53 & - & - & - & - \\
- Ref.\cite{leung99} & 3.31 & 3.31 & 3.43 & - & - & - & - & - & - \\
- Ref.\cite{dal_pino93,capaz94} & - & - & - & - & - & 1.89\cite{dal_pino93} & x & - & x+2.1\cite{capaz94}
-\end{tabular}
-\caption{Formation energies of silicon and carbon point defects in crystalline silicon given in eV. T denotes the tetrahedral, H the hexagonal and BC the bond-centered interstitial configuration. V corresponds to the vacancy configuration. Dumbbell configurations are abbreviated by DB.}
-\label{table:sep_eof}
-\end{table*}
-Results obtained by the present study compare well with results from literature\cite{leung99,al-mushadani03,dal_pino93,capaz94}.
-Regarding intrinsic defects in Si, the \hkl<1 1 0> self-interstitial dumbbell (Si$_{\text{i}}$ \hkl<1 1 0> DB) is found to be the ground state configuration closely followed by the hexagonal and tetrahedral configuration, which is consensus for Si$_{\text{i}}$\cite{leung99,al-mushadani03}.
-In the case of a C impurity, next to the C$_{\text{s}}$ configuration, in which a C atom occupies an already vacant Si lattice site, the C \hkl<1 0 0> interstitial dumbbell (C$_{\text{i}}$ \hkl<1 0 0> DB) constitutes the energetically most favorable configuration, in which the C and Si dumbbell atoms share a regular Si lattice site.
-This finding is in agreement with several theoretical\cite{burnard93,leary97,dal_pino93,capaz94,jones04} and experimental\cite{watkins76,song90} investigations, which all predict this configuration to be the ground state.
-%However, to our best knowledge, no energy of formation for this type of defect based on first-principles calculations has yet been explicitly stated in literature.
-However, to our best knowledge, no energy of formation for this type of defect based on first-principles calculations is available.
+
+Regarding intrinsic defects in Si, classical potential and {\em ab initio} methods predict energies of formation that are within the same order of magnitude.
+The EA potential does not reproduce the correct ground state, i.e. the interstitial Si (Si$_{\text{i}}$) \hkl<1 1 0> dumbbell (DB), which is consensus for Si$_{\text{i}}$ \cite{leung99,al-mushadani03}.
+Instead, the tetrahedral configuration is favored, a limitation assumed to arise due to the sharp cut-off as has already been discussed by Tersoff \cite{tersoff90}.
+
+In the case of C impurities, although discrepancies exist, classical potential and first-principles methods depict the correct order of the formation energies.
+Next to the substitutional C (C$_{\text{s}}$) configuration, which is not an interstitial configuration since the C atom occupies an already vacant Si lattice site, the interstitial C (C$_{\text{i}}$) \hkl<1 0 0> DB constitutes the energetically most favorable configuration, in which the C and Si dumbbell atoms share a regular Si lattice site.
+This finding is in agreement with several theoretical \cite{dal_pino93,capaz94,burnard93,leary97,jones04} and experimental \cite{watkins76,song90} investigations, which all predict this configuration to be the ground state.
+It is worth to note that the bond-centered (BC) configuration constitutes a real local minimum in spin polarized calculations in contrast to results \cite{capaz94} without spin predicting a saddle point configuration as well as to the empirical description, which shows a relaxation into the C$_{\text{i}}$ \hkl<1 0 0> DB ground-state configuration.
+
+\section{Mobility of the carbon defect}
Instead, Capaz et al.\cite{capaz94}, investigating migration pathways of the C$_{\text{i}}$ \hkl<1 0 0> DB, find this defect to be \unit[2.1]{eV} lower in energy than the bond-centered (BC) configuration.
The BC configuration is claimed to constitute the saddle point within the C$_{\text{i}}$ \hkl[0 0 -1] DB migration path residing in the \hkl(1 1 0) plane and, thus, interpreted as the barrier of migration for the respective path.
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