% additional stuff\r
\usepackage{miller}\r
\r
+% roman numbers\r
+\newcommand{\RM}[1]{\MakeUppercase{\romannumeral #1{}}}\r
+\r
\begin{document}\r
\r
%\title{Mobility of Carbon in Silicon -- a first principles study}\r
The investigations proceed with pairs of the ground state and, thus, most probable defect configurations that are believed to be fundamental in the Si to SiC transition.\r
\r
\subsection{Separated defects in silicon}\r
+\label{subsection:sep_def}\r
% we need both: Si self-int & C int ground state configuration (for combos)\r
\r
Several geometries have been calculated to be stable for individual intrinsic and C related defects in Si.\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 $(\unit[0.70-0.87]{eV})$\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 the last subsection configurations of a C$_{\text{i}}$ DB with C$_{\text{s}}$ occupying a vacant site created by the implantation process have been investigated.\r
Additionally, configurations might arise in IBS, in which the impinging C atom creates a vacant site near a C$_{\text{i}}$ DB but does not occupy it.\r
Resulting binding energies of a C$_{\text{i}}$ DB with a nearby vacancy are listed in the second row of Table~\ref{table:dc_c-sv}.\r
-Obviously all investigated structures are prefered compared to isolated largely separated defects of this type.\r
+All investigated structures are prefered compared to isolated largely separated defects.\r
+In contrast to C$_{\text{s}}$ this is also valid for positions along \hkl[1 1 0] resulting in an entirely attractive interaction between defects of these types.\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.\r
-Figure\r
-\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
+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
+\begin{figure}\r
+\includegraphics[width=\columnwidth]{314-539.ps}\r
+\caption{Migration barrier and structures of the transition of the initial C$_{\text{i}}$ \hkl[0 0 -1] DB and a V created at position 3 (left) into a C$_{\text{s}}$ configuration (right). An activation energy of \unit[0.1]{eV} is observed.}\r
+\label{fig:314-539}\r
+\end{figure}\r
+\begin{figure}\r
+\includegraphics[width=\columnwidth]{059-539.ps}\r
+\caption{Migration barrier and structures of the transition of the initial C$_{\text{i}}$ \hkl[0 0 -1] DB and a V created at position 2 (left) into a C$_{\text{s}}$ configuration (right). An activation energy of \unit[0.6]{eV} is observed.}\r
+\label{fig:059-539}\r
+\end{figure}\r
+Activation energies as low as \unit[0.1]{eV} and \unit[0.6]{eV} are observed.\r
+In the first case the Si and C atom of the DB move towards the vacant and initial DB lattice site respectively.\r
+In total three Si-Si and one more Si-C bond is formed during the transition.\r
+In the second case the lowest barrier is found for the migration of Si number 1 , which is substituted by the C$_{\text{i}}$ atom, towards the vacant site.\r
+A net amount of five Si-Si and one Si-C bond are additionally formed during the transition.\r
+The direct migration of the C$_{\text{i}}$ atom onto the vacant lattice site results in a somewhat higher barrier of \unit[1.0]{eV}.\r
+In both cases, the formation of additional bonds is responsible for the vast gain in energy rendering almost impossible the reverse processes.\r
+\r
+In summary, pairs of C$_{\text{i}}$ DBs and Vs, like no other before, show highly attractive interactions for all investigated combinations indpendent of orientation and separation direction of the defects.\r
+Furthermore, small activation energies, even for transitions into the ground state exist.\r
+Based on these results, a high probability for the formation of C$_{\text{s}}$ must be concluded.\r
\r
\subsection{C$_{\text{s}}$ next to Si$_{\text{i}}$}\r
\r
+As shown in section~\ref{subsection:sep_def} C$_{\text{s}}$ exhibits the lowest energy of formation.\r
+Considering a perfect Si crystal and conservation of particles, however, the occupation of a Si lattice site by a slowed down implanted C atom is necessarily accompanied by the formation of a Si self-interstitial.\r
+There are good reasons for the existence of regions exhibiting such configurations with regard to the IBS process.\r
+Highly energetic C atoms are able to kick out a Si atom from its lattice site, resulting in a Si self-interstitial accompanied by a vacant site, which might get occupied by another C atom, which lost almost all of its kinetic energy.\r
+Thus, configurations of C$_{\text{s}}$ and Si self-interstitials are investigated in the following.\r
+The Si$_{\text{i}}$ \hkl<1 1 0> DB, which was found to exhibit the lowest energy of formation within the investigated self-interstitial configurations, is assumed to provide the energetically most favorable configuration in combination with C$_{\text{s}}$.\r
+\r
+\begin{table}\r
+\begin{ruledtabular}\r
+\begin{tabular}{l c c c c c c}\r
+ & \hkl[1 1 0] & \hkl[-1 1 0] & \hkl[0 1 1] & \hkl[0 -1 1] &\r
+ \hkl[1 0 1] & \hkl[-1 0 1] \\\r
+\hline\r
+1 & \RM{1} & \RM{3} & \RM{3} & \RM{1} & \RM{3} & \RM{1} \\\r
+2 & \RM{2} & \RM{6} & \RM{6} & \RM{2} & \RM{8} & \RM{5} \\\r
+3 & \RM{3} & \RM{1} & \RM{3} & \RM{1} & \RM{1} & \RM{3} \\\r
+4 & \RM{4} & \RM{7} & \RM{9} & \RM{10} & \RM{10} & \RM{9} \\\r
+5 & \RM{5} & \RM{8} & \RM{6} & \RM{2} & \RM{6} & \RM{2} \\\r
+\end{tabular}\r
+\caption{Equivalent configurations labeld \RM{1}-\RM{10} 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}. The respective orientation of the Si$_{\text{i}}$ DB is given in the first row.}\r
+\label{table:dc_si-s}\r
+\end{ruledtabular}\r
+\end{table}\r
+\begin{table*}\r
+\begin{ruledtabular}\r
+\begin{tabular}{l c c c c c c c c c c}\r
+ & \RM{1} & \RM{2} & \RM{3} & \RM{4} & \RM{5} & \RM{6} & \RM{7} & \RM{8} & \RM{9} & \RM{10} \\\r
+\hline\r
+$E_{\text{f}}$ [eV]& 4.37 & 5.26 & 5.57 & 5.37 & 5.12 & 5.10 & 5.32 & 5.28 & 5.39 & 5.32 \\\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
+\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 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
+The configurations of C$_{\text{s}}$ and Si$_{\text{i}}$ DBs might be especially important at higher temperatures accompanied by an increase of the entropic contribution.\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
+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 considered to produce configurations of these defects with separation distances exceeding the capture radius.\r
+\r
Non-zero temperature, entropy, spatial separation of these defects possible, indeed observed in ab initio MD run.\r
\r
\section{Discussion}\r