Next to individual Si$_{\text{i}}$, C$_{\text{i}}$, V and C$_{\text{s}}$ defects, combinations of these defects and their interaction are considered important for the problem under study.\r
In the following the structure and energetics of separated defects are presented.\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
-Fig.~\ref{fig:combos} schematically displays the positions for the initial interstitial defect (Si$_{\text{i}}$/C$_{\text{i}}$) and the neighbured defect (1-5) used for investigating defect pairs.\r
-\begin{figure}\r
-\includegraphics[width=0.5\columnwidth]{cs.eps}\r
-\caption{Positions for the initial defect (Si$_{\text{i}}$/C$_{\text{i}}$) and the neighboured defect (1-5) used for investigating defect pairs.} \r
-\label{fig:combos}\r
-\end{figure}\r
\r
\subsection{Separated defects in silicon}\r
% we need both: Si self-int & C int ground state configuration (for combos)\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
-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 neighboured Si lattice site in \hkl[1 1 -1] direction.\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
\r
For the latter two the net spin up electron density is localized in caps at the four surrounding Si atoms directed towards the vacant site and in two caps at each of the two DB atoms perpendicularly aligned to the bonds to the other two Si atoms respectively.\r
No other configuration, within the ones that are mentioned, is affected.\r
\r
-Concerning the mobility of the ground state Si$_{\text{i}}$, an activation energy shortly below \unit[0.7]{eV} was found for the migration of a Si$_{\text{i}}$ \hkl[0 1 -1] into a \hkl[1 1 0] DB configuration located at the next neighboured Si lattice site in \hkl[1 1 -1] direction.\r
+Concerning the mobility of the ground state Si$_{\text{i}}$, an activation energy shortly below \unit[0.7]{eV} was found for the migration of a Si$_{\text{i}}$ \hkl[0 1 -1] into a \hkl[1 1 0] DB configuration located at the next neighbored Si lattice site in \hkl[1 1 -1] direction.\r
% look for values in literature for neutraly charged Si_i diffusion\r
\r
\subsection{Pairs of C$_{\text{i}}$}\r
\r
C$_{\text{i}}$ pairs of the \hkl<1 0 0>-type have been considered in the first part.\r
-Table~\ref{table:dc_c-c} summarizes the binding energies obtained for configurations, in which an initial C$_{\text{i}}$ \hkl[0 0 -1] DB located at position Si$_{\text{i}}$/C$_{\text{i}}$ is combined with a defect of the same type occupying various orientations at positions 1 to 5 (see Fig.~\ref{fig:combos}).\r
+Fig.~\ref{fig:combos_ci} schematically displays the position of the initial C$_{\text{i}}$ \hkl[0 0 -1] DB and the various positions for the second defect (1-5) used for investigating the defect pairs.\r
+Table~\ref{table:dc_c-c} summarizes the binding energies for the combination with a second C-Si \hkl<1 0 0> DB obtained for different orientations.\r
+\begin{figure}\r
+%\begin{minipage}{0.49\columnwidth}\r
+\subfigure[]{\label{fig:combos_ci}\includegraphics[width=0.45\columnwidth]{combos_ci.eps}}\r
+\hspace{0.1cm}\r
+\subfigure[]{\label{fig:combos_si}\includegraphics[width=0.45\columnwidth]{combos.eps}}\r
+\caption{Positions of the initial C$_{\text{i}}$ \hkl[0 0 -1] DB (I) (Fig.~\ref{fig:combos_ci}), the lattice site chosen for the initial Si$_{\text{i}}$ \hkl<1 1 0> DB (Si$_{\text{i}}$) occupying various orientations (Fig.~\ref{fig:combos_si}) and neighbored positions (1-5) for the second defect used for investigating defect pairs.} \r
+\label{fig:combos}\r
+\end{figure}\r
\begin{table}\r
\begin{ruledtabular}\r
\begin{tabular}{l c c c c c c }\r
\end{table}\r
Most of the obtained configurations result in binding energies well below zero indicating a preferable agglomeration of these type of defects.\r
For increasing distances of the defect pair the binding energy approaches to zero (R in Table~\ref{table:dc_c-c}) as it is expected for non-interacting isolated defects.\r
-Energetically favorable and unfavorable configurations can be explained by stress compensation and increase respectively, which is due to the resulting net strain of the respective configuration of the defect combination.\r
+Energetically favorable and unfavorable configurations can be explained by stress compensation and increase respectively based on the resulting net strain of the respective configuration of the defect combination.\r
Antiparallel orientations of the second defect (\hkl[0 0 1]) at positions located below the \hkl(0 0 1) plane with respect to the initial one (positions 1, 2 and 4) show the energetically most unfavorable configurations.\r
In contrast, the parallel and particularly the twisted orientations constitute energetically favorable configurations, in which a vast reduction of strain is enabled by combination of these defects.\r
\r
In this work we found a further relaxation of this defect structure.\r
The C atom of the second and the Si atom of the initial DB move towards each other forming a bond, which results in a somewhat lower binding energy of \unit[-2.25]{eV}.\r
Furthermore a more favorable configuration was found for the combination with a \hkl[0 -1 0] and \hkl[-1 0 0] DB respectively, which is assumed to constitute the actual ground state configuration of two C$_{\text{i}}$ DBs in Si.\r
+The atomic arrangement is shown in the bottom right of Fig.~\ref{fig:036-239}.\r
The two C atoms form a strong C-C bond, which is responsible for the large gain in energy resulting in a binding energy of \unit[-2.39]{eV}.\r
\r
Investigating migration barriers enables to predict the probability of formation of the thermodynamic ground state defect complex by thermally activated diffusion processes.\r
High activation energies are necessary for the migration of low energy configurations, in which the C atom of the second DB is located in the vicinity of the initial DB.\r
-The transition of the configuration, in which the second DB oriented along \hkl[0 1 0] type at position 2 (\unit[-1.90]{eV}) into a \hkl[0 1 0] DB at position 1 (\unit[-2.39]{eV}) for instance, revealed a barrier height of more than \unit[4]{eV}.\r
+The transition of the configuration, in which the second DB oriented along \hkl[0 1 0] at position 2 (\unit[-1.90]{eV}) into a \hkl[0 1 0] DB at position 1 (\unit[-2.39]{eV}) for instance, revealed a barrier height of more than \unit[4]{eV}.\r
Low barriers do only exist from energetically less favorable configurations, e.g. the configuration of the \hkl[-1 0 0] DB located at position 2 (\unit[-0.36]{eV}).\r
-Starting from this onfiguration, an activation energy of only \unit[1.2]{eV} is necessary for the transition into the ground state configuration.\r
+Starting from this configuration, an activation energy of only \unit[1.2]{eV} is necessary for the transition into the ground state configuration.\r
+The corresponding migration energies and atomic configurations are displayed in Fig.~\ref{fig:036-239}.\r
+\begin{figure}\r
+\includegraphics[width=\columnwidth]{036-239.eps}\r
+\caption{Migration barrier and structures of the transition of a C$_{\text{i}}$ \hkl[-1 0 0] DB at position 2 (left) into a C$_{\text{i}}$ \hkl[0 -1 0] DB at position 1 (right). An activation energy of \unit[1.2]{eV} is observed.}\r
+\label{fig:036-239}\r
+\end{figure}\r
% strange mig from -190 -> -2.39 (barrier > 4 eV)\r
% C-C migration -> idea:\r
% mig from low energy confs has extremely high barrier!\r
%\r
% should possibly be transfered to discussion section\r
Since thermally activated C clustering is, thus, only possible by traversing energetically unfavored configurations, mass C clustering is not expected.\r
-Furthermore, the migration barrier is still higher than the activation energy observed for a single C$_{\text{i}}$ \hkl<1 0 0> DB in c-Si.\r
-The migration barrier of a C$_{\text{i}}$ DB in a complex system is assumed to approximate the barrier in a separated system with increasing defect separation distance.\r
-Thus, lower migration barriers are expected for separating C$_{\text{i}}$ DBs.\r
-% calculate?!?\r
-However, low binding energies ... and the difference needs to be overcome too.\r
-It is bound to precapture state and only \r
-However if the activation energy is $>>$ than the difference in energy of the two configurations both states are equally occupied.\r
-And at increased temperatures that enable such diffusion processes the entropy comes into play.\r
-A promising configuration ... -2.25, and the amoun tof equivalent configurations is twice as high.\r
+Furthermore, the migration barrier of \unit[1.2]{eV} is still higher than the activation energy of \unit[0.9]{eV} observed for a single C$_{\text{i}}$ \hkl<1 0 0> DB in c-Si.\r
+The migration barrier of a C$_{\text{i}}$ DB in a complex system is assumed to approximate the barrier of a DB in a separated system with increasing defect separation.\r
+Thus, lower migration barriers are expected for pathways resulting in larger separations of the C$_{\text{i}}$ DBs.\r
+% calculate?!? ... hopefully acknowledged by 188-225 calc\r
+However, if the increase of separation is accompanied by an increase in binding energy, this difference is needed in addition to the activation energy for the respective migration process.\r
+Configurations, which exhibit both, a low binding energy as well as targeting transitions with low activation energies are, thus, most probable C$_{\text{i}}$ complex structures.\r
+On the other hand, if elevated temperatures enable migrations with huge activation energies, the comparably small differences in configurational energy can be neglected resulting in an almost equal occupation of these configurations.\r
+In both cases the configuration yielding a binding energy of \unit[-2.25]{eV} is promising since it constitutes the second most energetically favorable structure, exhibits a low migration barrier of transition starting from more separated defect structures and\r
+% 188 - 225 transition in progress\r
+is represented four times (two times more often than the ground state configuration) within the investigated configuration space.\r
+The latter is considered very important for high temperatures, which is accompanied by an increase in the entropic contribution to structure formation.\r
Thus, C agglomeration indeed is expected but only a low probability is assumed for C clustering by thermally activated processes with regard to the considered period of time.\r
% ?!?\r
% look for precapture mechnism (local minimum in energy curve)\r
C-C distance [nm] & 0.14 & 0.46 & 0.65 & 0.86 & 1.05 & 1.08 \r
\end{tabular}\r
\end{ruledtabular}\r
-\caption{Binding energies $E_{\text{b}}$ and C-C distance of energetically most favorable C$_{\text{i}}$ \hkl<1 0 0>-type defect pairs separated along bonds in \hkl[1 1 0] direction.}\r
+\caption{Binding energies $E_{\text{b}}$ and C-C distance of energetically most favorable C$_{\text{i}}$ \hkl<1 0 0>-type defect pairs separated along bonds in the \hkl[1 1 0] direction.}\r
\label{table:dc_110}\r
\end{table}\r
The binding energy of these configurations with respect to the C-C distance is plotted in Fig.~\ref{fig:dc_110}\r
V & -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$_{\text{i}}$ \hkl[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 $\frac{a_{\text{Si}}}{2}\hkl[3 2 3]$ relative to the initial defect position, which is the maximum realizable distance due to periodic boundary conditions.}\r
+\caption{Binding energies of combinations of the C$_{\text{i}}$ \hkl[0 0 -1] defect with a substitutional C or vacancy located at positions 1 to 5 according to Fig.~\ref{fig:combos_ci}. R corresponds to the position located at $\frac{a_{\text{Si}}}{2}\hkl[3 2 3]$ 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
\r
Spin polarization for C-C Int resulting spin up electrons located as in the case of the Si 100 int.\r
% mattoni: A favored by 0.2 eV - NO! (again, missing spin polarization?)\r
\r
+% mig a-b\r
+% 2 more migs: 051 -> 128 and 026! forgot why ...\r
\r
\subsection{C$_{\text{i}}$ next to V}\r
\r
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