-\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.
-
-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.
-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.
-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.
-Regardless of the rather small correction of \unit[0.3]{eV} due to the spin, the difference we found is much smaller (\unit[0.94]{eV}), which would nicely compare to experimentally observed migration barriers of \unit[0.70-0.87]{eV}\cite{lindner06,tipping87,song90}.
-However, since the BC configuration constitutes a real local minimum another barrier exists which is about \unit[1.2]{eV} in height.
-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 to a C$_{\text{i}}$ \hkl[0 -1 0] DB located at the neighbored Si lattice site in \hkl[1 1 -1] direction.
-Calculations in this work reinforce this path by an additional improvement of the quantitative conformance of the barrier height (\unit[0.90]{eV}) to experimental values.
-A more detailed description can be found in a previous study\cite{zirkelbach10a}.
-
-Next to the C$_{\text{i}}$ BC configuration the vacancy and Si$_{\text{i}}$ \hkl<1 0 0> DB have to be treated by taking into account the spin of the electrons.
-For the vacancy the net spin up electron density is localized in caps at the four surrounding Si atoms directed towards the vacant site.
-In the Si$_{\text{i}}$ \hkl<1 0 0> DB configuration the net spin up density is localized in two caps at each of the two DB atoms perpendicularly aligned to the bonds to the other two Si atoms respectively.
-No other configuration, within the ones that are mentioned, is affected.
-
-Concerning the mobility of the ground state Si$_{\text{i}}$, we found an activation energy of \unit[0.67]{eV} for the transition of the Si$_{\text{i}}$ \hkl[0 1 -1] to \hkl[1 1 0] DB located at the neighbored Si lattice site in \hkl[1 1 -1] direction.
-Further investigations revealed a barrier of \unit[0.94]{eV} for the Si$_{\text{i}}$ \hkl[1 1 0] DB to Si$_{\text{i}}$ H, \unit[0.53]{eV} for the Si$_{\text{i}}$ \hkl[1 1 0] DB to Si$_{\text{i}}$ T and \unit[0.35]{eV} for the Si$_{\text{i}}$ H to Si$_{\text{i}}$ T transition.
-%Obtained values are of the same order of magnitude than values derived from other ab initio studies\cite{bloechl93,sahli05}.
-These are of the same order of magnitude than values derived from other ab initio studies\cite{bloechl93,sahli05}.
-
-\subsection{Pairs of C$_{\text{i}}$}
-
-C$_{\text{i}}$ pairs of the \hkl<1 0 0> type have been investigated in the first part.
-Fig.~\ref{fig:combos_ci} schematically displays the initial C$_{\text{i}}$ \hkl[0 0 -1] DB structure and various positions for the second defect (1-5) that have been used for investigating defect pairs.
-Table~\ref{table:dc_c-c} summarizes resulting binding energies for the combination with a second C-Si \hkl<1 0 0> DB obtained for different orientations at positions 1 to 5.
-\begin{figure}
-\subfloat[]{\label{fig:combos_ci}\includegraphics[width=0.45\columnwidth]{combos_ci.eps}}
-\hspace{0.1cm}
-\subfloat[]{\label{fig:combos_si}\includegraphics[width=0.45\columnwidth]{combos.eps}}
-\caption{Position of the initial C$_{\text{i}}$ \hkl[0 0 -1] DB (I) (Fig.~\ref{fig:combos_ci}) and of the lattice site chosen for the initial Si$_{\text{i}}$ \hkl<1 1 0> DB (Si$_{\text{i}}$) (Fig.~\ref{fig:combos_si}). Lattice sites for the second defect used for investigating defect pairs are numbered from 1 to 5.}
-\label{fig:combos}
-\end{figure}
-\begin{table}
-\begin{tabular}{l c c c c c c }
- & 1 & 2 & 3 & 4 & 5 & R \\
-\hline
- \hkl[0 0 -1] & -0.08 & -1.15 & -0.08 & 0.04 & -1.66 & -0.19\\
- \hkl[0 0 1] & 0.34 & 0.004 & -2.05 & 0.26 & -1.53 & -0.19\\
- \hkl[0 -1 0] & -2.39 & -0.17 & -0.10 & -0.27 & -1.88 & -0.05\\
- \hkl[0 1 0] & -2.25 & -1.90 & -2.25 & -0.12 & -1.38 & -0.06\\
- \hkl[-1 0 0] & -2.39 & -0.36 & -2.25 & -0.12 & -1.88 & -0.05\\
- \hkl[1 0 0] & -2.25 & -2.16 & -0.10 & -0.27 & -1.38 & -0.06\\
-\end{tabular}
-\caption{Binding energies in eV of C$_{\text{i}}$ \hkl<1 0 0>-type defect pairs. Equivalent configurations exhibit equal energies. Column 1 lists the orientation of the second defect, which is combined with the initial C$_{\text{i}}$ \hkl[0 0 -1] DB. The position index of the second defect is given in the first row 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 defect separation distance ($\approx \unit[1.3]{nm}$) due to periodic boundary conditions.}
-\label{table:dc_c-c}
-\end{table}
-Most of the obtained configurations result in binding energies well below zero indicating a preferable agglomeration of this type of defects.
-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.
-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.
-Antiparallel orientations of the second defect, i.e. \hkl[0 0 1] for positions located below the \hkl(0 0 1) plane with respect to the initial one (positions 1, 2 and 4) form the energetically most unfavorable configurations.
-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.
-
-Mattoni et al.\cite{mattoni2002} predict the ground state configuration for a \hkl[1 0 0] or equivalently a \hkl[0 1 0] defect created at position 1 with both defects basically maintaining the as-isolated DB structure, resulting in a binding energy of \unit[-2.1]{eV}.
-In this work we observed a further relaxation of this defect structure.
-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}.
-Apart from that, we found a more favorable configuration 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.
-The atomic arrangement is shown in the bottom right of Fig.~\ref{fig:036-239}.
-The two C$_{\text{i}}$ 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}.
-
-Investigating migration barriers allows to predict the probability of formation of defect complexes by thermally activated diffusion processes.
-% ground state configuration, C cluster
-Based on the lowest energy migration path of a single C$_{\text{i}}$ DB the configuration, in which the second C$_{\text{i}}$ DB is oriented along \hkl[0 1 0] at position 2 is assumed to constitute an ideal starting point for a transition into the ground state.
-In addition, the starting configuration exhibits a low binding energy (\unit[-1.90]{eV}) and is, thus, very likely to occur.
-However, a barrier height of more than \unit[4]{eV} was detected resulting in a low probability for the transition.
-The high activation energy is attributed to the stability of such a low energy configuration, in which the C atom of the second DB is located close to the initial DB.
-Low barriers have only been identified for transitions starting from energetically less favorable configurations, e.g. the configuration of a \hkl[-1 0 0] DB located at position 2 (\unit[-0.36]{eV}).
-Starting from this configuration, an activation energy of only \unit[1.2]{eV} is necessary for the transition into the ground state configuration.
-The corresponding migration energies and atomic configurations are displayed in Fig.~\ref{fig:036-239}.
-\begin{figure}
-\includegraphics[width=\columnwidth]{036-239.ps}
-\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.}
-\label{fig:036-239}
-\end{figure}
-% strange mig from -190 -> -2.39 (barrier > 4 eV)
-% C-C migration -> idea:
-% mig from low energy confs has extremely high barrier!
-% low barrier only from energetically less/unfavorable confs (?)! <- prove!
-% => low probability of C-C clustering ?!?
-%
-% should possibly be transfered to discussion section
-Since thermally activated C clustering is, thus, only possible by traversing energetically unfavored configurations, extensive C clustering is not expected.
-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.
-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.
-Accordingly, lower migration barriers are expected for pathways resulting in larger separations of the C$_{\text{i}}$ DBs.
-% acknowledged by 188-225 (reverse order) calc
-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.
-Configurations, which exhibit both, a low binding energy as well as afferent transitions with low activation energies are, thus, most probable C$_{\text{i}}$ complex structures.
-On the other hand, if elevated temperatures enable migrations with huge activation energies, comparably small differences in configurational energy can be neglected resulting in an almost equal occupation of such configurations.
-In both cases the configuration yielding a binding energy of \unit[-2.25]{eV} is promising.
-First of all, it constitutes the second most energetically favorable structure.
-Secondly, a migration path with a barrier as low as \unit[0.47]{eV} exists starting from a configuration of largely separated defects exhibiting a low binding energy (\unit[-1.88]{eV}).
-The migration barrier and corresponding structures are shown in Fig.~\ref{fig:188-225}.