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 $(\unit[0.70-0.87]{eV})$\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
+However, since the BC configuration constitutes a real local minimum another barrier exists which is about \unit[1.2]{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$_{\text{i}}$ \hkl[0 0 -1] DB migrates into a C$_{\text{i}}$ \hkl[0 -1 0] DB located at the next neighbored Si lattice site in \hkl[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
A more detailed description can be found in a previous study\cite{zirkelbach10a}.\r
In contrast, all other investigated configurations show attractive interactions.\r
The most favorable configuration is found for C$_{\text{s}}$ at position 3, which corresponds to the lattice site of one of the upper next neighbored Si atoms of the DB structure that is compressively strained along \hkl[1 -1 0] and \hkl[0 0 1] by the C-Si DB.\r
The substitution with C allows for most effective compensation of strain.\r
-This structure is followed by C$_{\text{s}}$ located at position 2, the next neighbour atom below the two Si atoms bound to the C$_{\text{i}}$ DB atom.\r
+This structure is followed by C$_{\text{s}}$ located at position 2, the next neighbor atom below the two Si atoms bound to the C$_{\text{i}}$ DB atom.\r
As mentioned earlier these two lower Si atoms indeed experience tensile strain along the \hkl[1 1 0] bond chain, however, additional compressive strain along \hkl[0 0 1] exists.\r
The latter is partially compensated by the C$_{\text{s}}$ atom.\r
Yet less of compensation is realized if C$_{\text{s}}$ is located at position 4 due to a larger separation although both bottom Si atoms of the DB structure are indirectly affected, i.e. each of them is connected by another Si atom to the C atom enabling the reduction of strain along \hkl[0 0 1].\r
The eneregtically most favorable configuration (configuration b) forms a strong but compressively strained C-C bond with a separation distance of \unit[0.142]{nm} sharing a Si lattice site.\r
Again, conclusions concerning the probability of formation are drawn by investigating migration paths.\r
Since C$_{\text{s}}$ is unlikely to exhibit a low activation energy for migration the focus is on C$_{\text{i}}$.\r
-Pathways starting from the two next most favored configurations were investigated, all of them showing activation energies above \unit[2.?-2.?]{eV}.\r
-Although lower than the barriers for obtaining the ground state of two C$_{\text{i}}$ defects the activation energy is yet considered too high.\r
+Pathways starting from the two next most favored configurations were investigated, all of them showing activation energies above \unit[2.2-3.5]{eV}.\r
+Although lower than the barriers for obtaining the ground state of two C$_{\text{i}}$ defects the activation energies are yet considered too high.\r
For the same reasons as in the last subsection, structures other than the ground state configuration are, thus, assumed to arise more likely due to much lower activation energies necessary for their formation and still comparatively low binding energies.\r
\r
\subsection{C$_{\text{i}}$ next to V}\r
Even for the largest possible distance (R) achieved in the calculations of the periodic supercell a binding energy as low as \unit[-0.31]{eV} is observed.\r
The ground state configuration is obtained for a V at position 1.\r
The C atom of the DB moves towards the vacant site forming a stable C$_{\text{s}}$ configuration resulting in the release of a huge amount of energy.\r
-The second most favored configuration is accomplished for a V located at position 3 due to the reduction of compressive strain of the Si DB atom and its two upper Si neighbours present in the C$_{\text{i}}$ DB configuration.\r
+The second most favored configuration is accomplished for a V located at position 3 due to the reduction of compressive strain of the Si DB atom and its two upper Si neighbors present in the C$_{\text{i}}$ DB configuration.\r
This configuration is follwed by the structure, in which a vacant site is created at position 2.\r
Similar to the observations for C$_{\text{s}}$ in the last subsection a reduction of strain along \hkl[0 0 1] is enabled by this configuration.\r
Relaxed structures of the latter two defect combinations are shown in the bottom left of Fig.~\ref{fig:314-539} and \ref{fig:059-539} respectively together with their energetics during transition into the ground state.\r
$E_{\text{b}}$ [eV] & -0.97 & -0.08 & 0.22 & -0.02 & -0.23 & -0.25 & -0.02 & -0.06 & 0.05 & -0.03 \\\r
$r$ [nm] & 0.292 & 0.394 & 0.241 & 0.453 & 0.407 & 0.408 & 0.452 & 0.392 & 0.456 & 0.453\\\r
\end{tabular}\r
-\caption{Formation energies $E_{\text{f}}$, binding energies $E_{\text{b}}$ and C$_{\text{s}}$-Si$_{\text{i}}$ separation distances of the combinational C$_{\text{s}}$ and Si$_{\text{i}}$ configurations as defined in table \ref{table:dc_si-s}. Energies are given in eV while the separation is given in nm.}\r
+\caption{Formation energies $E_{\text{f}}$, binding energies $E_{\text{b}}$ and C$_{\text{s}}$-Si$_{\text{i}}$ separation distances of the combinational C$_{\text{s}}$ and Si$_{\text{i}}$ configurations as defined in Table~\ref{table:dc_si-s}. Energies are given in eV while the separation is given in nm.}\r
\label{table:dc_si-s_e}\r
\end{ruledtabular}\r
\end{table*}\r
Table~\ref{table:dc_si-s} classifies equivalent configurations of \hkl<1 1 0>-type Si$_{\text{i}}$ DBs created at position I and C$_{\text{s}}$ created at positions 1 to 5 according to Fig.~\ref{fig:combos_si}.\r
-Corresponding formation as well as binding energies and the C$_{\text{s}}$-Si$_{\text{i}}$ distances are listed in Table~\ref{table:dc_si-s_e}.\r
+Corresponding formation as well as binding energies and the separation distances of the C$_{\text{s}}$ atom and the Si$_{\text{i}}$ DB lattice site are listed in Table~\ref{table:dc_si-s_e}.\r
+In total ten different configurations exist within the investigated range.\r
+Configuration \RM{1} constitutes the energetically most favorable structure exhibiting a formation energy of \unit[4.37]{eV}.\r
+Obviously the configuration of a \hkl[1 1 0] Si$_{\text{i}}$ DB and a next neighbored C$_{\text{s}}$ in the same direction as the alignment of the DB, as displayed in the bottom right of Fig.~\ref{fig:162-097}, enables the largest possible reduction of strain.\r
+The Si$_{\text{i}}$ DB atoms are displaced towards the lattice site occupied by the C$_{\text{s}}$ atom in such a way that the Si DB atom closest to the C atom does no longer form bonds to its top Si neighbors but to the second next neighbored Si atom along \hkl[1 1 0].\r
+However, this configuration is energetically less favorable than the \hkl<1 0 0> C$_{\text{i}}$ DB, which, thus, remains the ground state of a C atom introduced into otherwise perfect c-Si.\r
+The transition involving the latter two configurations is shown in Fig.~\ref{fig:dc_si-s}.\r
+\begin{figure}\r
+\includegraphics[width=\columnwidth]{162-097.ps}\r
+\caption{Migration barrier and structures of the transition of a \hkl[1 1 0] Si$_{\text{i}}$ DB next to C$_{\text{s}}$ (right) into the C$_{\text{i}}$ \hkl[0 0 -1] DB configuration (left). An activation energy of \unit[0.12]{eV} is observed.}\r
+\label{fig:162-097}\r
+\end{figure}\r
+An activation energy as low as \unit[0.12]{eV} is necessary for the migration into the ground state configuration.\r
+Thus, the C$_{\text{i}}$ \hkl<1 0 0> DB configuration is assumed to occur more likely.\r
+However, only \unit[0.77]{eV} are needed for the reverse process, i.e. the formation of C$_{\text{s}}$ and a Si$_{\text{i}}$ DB out of the ground state.\r
+Due to the low activation energy this process must be considered to be activated without much effort either thermally or by introduced energy of the implantation process.\r
\r
\begin{figure}\r
\includegraphics[width=\columnwidth]{c_sub_si110.ps}\r
\caption{Binding energies of combinations of a C$_{\text{s}}$ and a Si$_{\text{i}}$ DB with respect to the separation distance. The binding energies of the defect pairs are well approximated by a Lennard-Jones 6-12 potential, which is used for curve fitting.}\r
\label{fig:dc_si-s}\r
\end{figure}\r
-\r
-Non-zero temperature, entropy, spatial separation of these defects possible, indeed observed in ab initio MD run.\r
+Fig.~\ref{fig:dc_si-s} shows the binding energies of pairs of C$_{\text{s}}$ and a Si$_{\text{i}}$ \hkl<1 1 0> DB with respect to the separation distance.\r
+The interaction of the defects is well approximated by a Lennard-Jones 6-12 potential, which was used for curve fitting.\r
+The binding energy quickly drops to zero with the fit estimating almost zero interaction at \unit[0.6]{nm}.\r
+This indicates a low interaction capture radius of the defect pair.\r
+In IBS highly energetic collisions are assumed to easily produce configurations of these defects with separation distances exceeding the capture radius.\r
+For this reason C$_{\text{s}}$ without a nearby interacting Si$_{\text{i}}$ DB, which are, thus, unable to form the thermodynamically stable C$_{\text{i}}$ \hkl<1 0 0> DB constitutes a most likely configuration to be found in IBS.\r
+\r
+As mentioned above, configurations of C$_{\text{s}}$ and Si$_{\text{i}}$ DBs might be especially important at higher temperatures due to the low activation energy necessary for its formation.\r
+At higher temperatures the contribution of entropy to structural formation increases, which might result in a spatial separation even for defects located within the capture radius.\r
+Indeed an ab initio molecular dynamics run at \unit[900]{$^{\circ}$C} starting from configuration \RM{1}, which -- based on the above findings -- is assumed to recombine into the ground state configuration, results in a separation the C$_{\text{s}}$ and Si$_{\text{i}}$ DB by more than 4 next neighbor distances realized in a repeated migration mechnism of annihilating and arising Si DBs.\r
+The atomic configurations for two different points in time are shown in Fig.~\ref{fig:md}.\r
+Si atoms 1 and 2, which form the initial DB, occupy usual Si lattice sites in the final configuration while atom 3 occupies an interstitial site.\r
+\begin{figure}\r
+\begin{minipage}{0.49\columnwidth}\r
+\includegraphics[width=\columnwidth]{md01.eps}\r
+\end{minipage}\r
+\begin{minipage}{0.49\columnwidth}\r
+\includegraphics[width=\columnwidth]{md02.eps}\\\r
+\end{minipage}\\\r
+\begin{minipage}{0.49\columnwidth}\r
+\begin{center}\r
+$t=\unit[2230]{fs}$\r
+\end{center}\r
+\end{minipage}\r
+\begin{minipage}{0.49\columnwidth}\r
+\begin{center}\r
+$t=\unit[2900]{fs}$\r
+\end{center}\r
+\end{minipage}\r
+\caption{Atomic configurations of an ab initio molecular dynamics run at \unit[900]{$^{\circ}$C} starting from a configuration of C$_{\text{s}}$ located next to a Si$_{\text{i}}$ DB (atoms 1 and 2). Equal atoms are marked by equal numbers. Blue lines correpsond to bonds, which are drawn for substantial atoms.}\r
+\label{fig:md}\r
+\end{figure}\r
\r
\section{Discussion}\r
\r
+The ground state configuration of a C atom in otherwise perfect c-Si is the C$_{\text{i}}$ \hkl<1 0 0> DB.\r
Our calculations show that point defects which unavoidably are present after ion implantation significantly influence the mobility of implanted carbon \r
in the silicon crystal.\r
A large capture radius has been found for... \r
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