Solving this controversy and understanding the effective underlying processes will enable significant technological progress in 3C-SiC thin film formation driving the superior polytype for potential applications in high-performance electronic device production\cite{wesch96}.\r
\r
Atomistic simulations offer a powerful tool of investigation providing detailed insight not accessible by experiment.\r
-A lot of theoretical work has been done on intrinsic point defects in Si\cite{bar-yam84,bar-yam84_2,car84,batra87,bloechl93,tang97,leung99,colombo02,goedecker02,al-mushadani03,posselt08,ma10}, threshold displacement energies in Si\cite{mazzarolo01,holmstroem08} important in ion implantation, C defects and defect reactions in Si\cite{tersoff90,dal_pino93,capaz94,burnard93,leary97,capaz98,zhu98,mattoni2002,park02,jones04}, the SiC/Si interface\cite{chirita97,kitabatake93,cicero02,pizzagalli03} and defects in SiC\cite{bockstedte03,rauls03a,gao04,posselt06,gao07}.\r
+A lot of theoretical work has been done on intrinsic point defects in Si\cite{bar-yam84,bar-yam84_2,car84,batra87,bloechl93,tang97,leung99,colombo02,goedecker02,al-mushadani03,hobler05,posselt08,ma10}, threshold displacement energies in Si\cite{mazzarolo01,holmstroem08} important in ion implantation, C defects and defect reactions in Si\cite{tersoff90,dal_pino93,capaz94,burnard93,leary97,capaz98,zhu98,mattoni2002,park02,jones04}, the SiC/Si interface\cite{chirita97,kitabatake93,cicero02,pizzagalli03} and defects in SiC\cite{bockstedte03,rauls03a,gao04,posselt06,gao07}.\r
However, none of the mentioned studies consistently investigates entirely the relevant defect structures and reactions concentrated on the specific problem of 3C-SiC formation in C implanted Si.\r
% but mattoni2002 actually did a lot. maybe this should be mentioned!\r
In fact, in a combined analytical potential molecular dynamics and ab initio study\cite{mattoni2002} the interaction of substitutional C with Si self-interstitials and C interstitials is evaluated.\r
Throughout this work an energy cut-off of \unit[300]{eV} was used to expand the wave functions into the plane-wave basis.\r
Sampling of the Brillouin zone was restricted to the $\Gamma$-point.\r
The defect structures and the migration paths were modelled in cubic supercells containing $216\pm2$ Si atoms.\r
-The ions and cell shape are allowed to change in order to realize a constant pressure simulation.\r
+The ions and cell shape were allowed to change in order to realize a constant pressure simulation.\r
+Ionic relaxation was realized by the conjugate gradient algorithm.\r
Spin polarization has been fully accounted for.\r
\r
Migration and recombination pathways have been ivestigated utilizing the constraint conjugate gradient relaxation technique (CRT)\cite{kaukonen98}.\r
Instead, Capaz et al.\cite{capaz94}, investigating migration pathways of the C$_{\text{i}}$ $\langle 1 0 0 \rangle$ DB, find this defect to be \unit[2.1]{eV} lower in energy than the bond-centered (BC) configuration, which is claimed to constitute a saddle point configuration in the migration path within the $(1 1 0)$ plane and, thus, interpreted as the barrier of migration for the respective path.\r
However, the present study indicates a local minimum state for the BC defect if spin polarized calculations are performed resulting in a net magnetization of two electrons localized in a torus around the C atom.\r
Another DFT calculation without fully accounting for the electron spin results in the smearing of a single electron over two non-degenerate Kohn-Sham states and an increase of the total energy by \unit[0.3]{eV} for the BC configuration.\r
-Regardless of the rather small correction due to the spin, the difference we found is much smaller (\unit[0.9]{eV}), which would nicely compare to experimental findings $(\unit[0.73-0.87]{eV})$\cite{tipping87,song90} for the migration barrier.\r
+Regardless of the rather small correction due to the spin, the difference we found is much smaller (\unit[0.9]{eV}), which would nicely compare to experimental findings $(\unit[0.70-0.87]{eV})$\cite{lindner06,tipping87,song90} for the migration barrier.\r
However, since the BC configuration constitutes a real local minimum another barrier exists which is about \unit[1.2]{eV} ($\unit[0.9]{eV}+\unit[0.3]{eV}$) in height.\r
Indeed Capaz et al. propose another path and find it to be the lowest in energy\cite{capaz94}, in which a C$_{\text{i}}$ $\langle 0 0 -1\rangle$ DB migrates into a C$_{\text{i}}$ $\langle 0 -1 0\rangle$ DB located at the next neighboured Si lattice site in $[1 1 -1]$ direction.\r
Calculations in this work reinforce this path by an additional improvement of the quantitative conformance of the barrier height (\unit[0.9]{eV}) to experimental values.\r
% also: plot energy all confs with respect to C-C distance\r
% maybe a pathway exists traversing low energy confs ?!?\r
\r
+% point out that configurations along 110 were extended up to the 6th NN in that direction\r
The binding energies of the energetically most favorable configurations with the seocnd DB located along the $\langle 1 1 0\rangle$ direction and resulting C-C distances of the relaxed structures are summarized in Table~\ref{table:dc_110}.\r
\begin{table}\r
\begin{ruledtabular}\r
The binding energy of these configurations with respect to the C-C distance is plotted in Fig.~\ref{fig:dc_110}\r
\begin{figure}\r
\includegraphics[width=\columnwidth]{db_along_110_cc_n.ps}\r
-\caption{Minimum binding energy of dumbbell combinations separated along $\langle 1 1 0\rangle$ with respect to the C-C distance.}\r
+\caption{Minimum binding energy of dumbbell combinations separated along $\langle 1 1 0\rangle$ with respect to the C-C distance. The blue line is a guide for the eye and the green curve corresponds to the most suitable fit function consisting of all but the first data point.}\r
\label{fig:dc_110}\r
\end{figure}\r
+The interaction is found to be proportional to the reciprocal cube of the C-C distance for extended separeations of the C$_{\text{i}}$ and saturates for the smallest possible separation, i.e. the ground state configuration.\r
\r
-\r
-\r
-\subsection{C$_I$ next to C$_{\text{s}}$}\r
+\subsection{C$_{\text{i}}$ next to C$_{\text{s}}$}\r
\r
% c_i and c_s, capaz98, mattoni2002 (restricted to 110 -110 bond chain)\r
+\begin{table}\r
+\begin{ruledtabular}\r
+\begin{tabular}{l c c c c c c }\r
+ & 1 & 2 & 3 & 4 & 5 & R \\\r
+\hline\r
+C$_{\text{s}}$ & 0.26 & -0.51 & -0.93 & -0.15 & 0.49 & -0.05\\\r
+Vacancy & -5.39 ($\rightarrow$ C$_{\text{S}}$) & -0.59 & -3.14 & -0.54 & -0.50 & -0.31\r
+\end{tabular}\r
+\end{ruledtabular}\r
+\caption{Binding energies of combinations of the C$_{\text{i}}$ $[0 0 -1]$ defect with a substitutional C or vacancy located at positions 1 to 5 according to Fig.~\ref{fig:combos}. R corresponds to the position located at $\frac{a_{\text{Si}}}{2} \langle3 2 3 \rangle$ relative to the initial defect position, which is the maximum realizable distance due to periodic boundary conditions.}\r
+\label{table:dc_c-sv}\r
+\end{table}\r
+Table~\ref{table:dc_c-sv} lists the binding energies of C$_{\text{s}}$ next to the C$_{\text{i}}$ $[0 0 -1]$ DB.\r
\r
\r
-\subsection{C$_I$ next to V}\r
+\subsection{C$_{\text{i}}$ next to V}\r
\r
\subsection{C$_{\text{s}}$ next to Si$_{\text{i}}$}\r
\r
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