For investigating the \si{} structures a Si atom is inserted or removed according to Fig. \ref{fig:basics:ins_pos} of section \ref{section:basics:defects}.
The formation energies of \si{} configurations are listed in Table \ref{tab:defects:si_self} for both methods used in this work as well as results obtained by other {\em ab initio} studies \cite{al-mushadani03,leung99}.
+\bibpunct{}{}{,}{n}{}{}
\begin{table}[tp]
\begin{center}
\begin{tabular}{l c c c c c}
\multicolumn{6}{c}{Other {\em ab initio} studies} \\
Ref. \cite{al-mushadani03} & 3.40 & 3.45 & - & - & 3.53 \\
Ref. \cite{leung99} & 3.31 & 3.31 & 3.43 & - & - \\
-% todo cite without []
\hline
\hline
\end{tabular}
\caption[Formation energies of Si self-interstitials in crystalline Si determined by classical potential MD and DFT calculations.]{Formation energies of Si self-interstitials in crystalline Si determined by classical potential MD and DFT calculations. The formation energies are given in eV. T denotes the tetrahedral and H the hexagonal interstitial configuration. V corresponds to the vacancy configuration. Dumbbell configurations are abbreviated by DB. Formation energies for unstable configurations are marked by an asterisk and determined by using the low kinetic energy configuration shortly before the relaxation into the more favorable configuration starts.}
\label{tab:defects:si_self}
\end{table}
+\bibpunct{[}{]}{,}{n}{}{}
\begin{figure}[tp]
\begin{center}
\begin{flushleft}
In fact, the same type of interstitial arises using random insertions.
In addition, variations exist, in which the displacement is only along two \hkl<1 0 0> axes ($E_\text{f}=3.8\,\text{eV}$) or along a single \hkl<1 0 0> axes ($E_\text{f}=3.6\,\text{eV}$) successively approximating the tetdrahedral configuration and formation energy.
The existence of these local minima located near the tetrahedral configuration seems to be an artifact of the analytical potential without physical authenticity revealing fundamental problems of analytical potential models for describing defect structures.
-% todo - energy barrier of what ?!?!
-However, the energy barrier is small.
+However, the energy barrier required for a transition into the tetrahedral configuration is small.
\begin{figure}[tp]
\begin{center}
\includegraphics[width=0.7\textwidth]{nhex_tet.ps}
There is no magnetization density.
An acceptor level arises at approximately $E_v+0.35\,\text{eV}$ while a band gap of about \unit[0.75]{eV} can be estimated from the Kohn-Sham level diagram for plain Si.
However, strictly speaking, the Kohn-Sham levels and orbitals do not have a direct physical meaning and, thus, these values have to be taken with care.
-% todo band gap problem
+% todo - band gap problem, skip it?
\subsection{Bond-centered interstitial configuration}
\label{subsection:bc}
\begin{center}
\includegraphics[width=0.7\textwidth]{im_00-1_nosym_sp_fullct_thesis_vasp_s.ps}
\end{center}
-\caption[Migration barrier and structures of the \hkl<0 0 -1> DB (left) to BC (right) transition.]{Migration barrier and structures of the \hkl<0 0 -1> DB (left) to BC (right) transition. Bonds of the C atom are illustrated by blue lines.}
+\caption[Migration barrier and structures of the {\hkl[0 0 -1]} DB to BC transition.]{Migration barrier and structures of the \hkl<0 0 -1> DB (left) to BC (right) transition. Bonds of the C atom are illustrated by blue lines.}
\label{fig:defects:00-1_001_mig}
\end{figure}
In Fig. \ref{fig:defects:00-1_001_mig} results of the \hkl<0 0 -1> to \hkl<0 0 1> migration fully described by the migration of the \hkl<0 0 -1> to the BC configuration is displayed.
\begin{center}
\includegraphics[width=0.7\textwidth]{00-1_0-10_vasp_s.ps}
\end{center}
-\caption[Migration barrier and structures of the \hkl<0 0 -1> DB (left) to the \hkl<0 -1 0> DB (right) transition.]{Migration barrier and structures of the \hkl[0 0 -1] DB (left) to the \hkl[0 -1 0] DB (right) transition. Bonds of the C atom are illustrated by blue lines.}
-% todo read above caption! enable [] hkls in short caption
+\caption[Migration barrier and structures of the {\hkl[0 0 -1]} DB to the {\hkl[0 -1 0]} DB transition.]{Migration barrier and structures of the \hkl[0 0 -1] DB (left) to the \hkl[0 -1 0] DB (right) transition. Bonds of the C atom are illustrated by blue lines.}
\label{fig:defects:00-1_0-10_mig}
\end{figure}
Fig. \ref{fig:defects:00-1_0-10_mig} shows the migration barrier and structures of the \ci{} \hkl<0 0 -1> to \hkl<0 -1 0> DB transition.
\begin{center}
\includegraphics[width=0.7\textwidth]{00-1_ip0-10_nosym_sp_fullct_vasp_s.ps}
\end{center}
-\caption[Reorientation barrier and structures of the \hkl<0 0 -1> DB (left) to the \hkl<0 -1 0> DB (right) transition in place.]{Reorientation barrier and structures of the \hkl<0 0 -1> DB (left) to the \hkl<0 -1 0> DB (right) transition in place. Bonds of the carbon atoms are illustrated by blue lines.}
+\caption[Reorientation barrier and structures of the {\hkl[0 0 -1]} DB to the {\hkl[0 -1 0]} DB transition in place.]{Reorientation barrier and structures of the \hkl[0 0 -1] DB (left) to the \hkl[0 -1 0] DB (right) transition in place. Bonds of the carbon atoms are illustrated by blue lines.}
\label{fig:defects:00-1_0-10_ip_mig}
\end{figure}
The third migration path, in which the DB is changing its orientation, is shown in Fig. \ref{fig:defects:00-1_0-10_ip_mig}.
%\includegraphics[width=2.2cm]{vasp_mig/0-10_b.eps}
%\end{picture}
\end{center}
-\caption{Migration barriers of the \hkl<1 1 0> DB to BC (blue), \hkl<0 0 -1> (green) and \hkl<0 -1 0> (in place, red) C-Si DB transition.}
+\caption[{Migration barriers of the \hkl[1 1 0] DB to BC, \hkl[0 0 -1] and \hkl[0 -1 0] (in place) C-Si DB transition.}]{Migration barriers of the \hkl[1 1 0] DB to BC (blue), \hkl[0 0 -1] (green) and \hkl[0 -1 0] (in place, red) C-Si DB transition.}
\label{fig:defects:110_mig_vasp}
\end{figure}
Further migration pathways, in particular those occupying other defect configurations than the \hkl<1 0 0>-type either as a transition state or a final or starting configuration, are totally conceivable.
%\includegraphics[height=2.2cm]{010_arrow.eps}
%\end{picture}
\end{center}
-\caption[Migration barrier and structures of the \ci{} BC to \hkl<0 0 -1> DB transition using the classical EA potential.]{Migration barrier and structures of the \ci{} BC to \hkl[0 0 -1] DB transition using the classical EA potential. Two migration pathways are obtained for different time constants of the Berendsen thermostat. The lowest activation energy is \unit[2.2]{eV}.}
+\caption[Migration barrier and structures of the \ci{} BC to {\hkl[0 0 -1]} DB transition using the classical EA potential.]{Migration barrier and structures of the \ci{} BC to \hkl[0 0 -1] DB transition using the classical EA potential. Two migration pathways are obtained for different time constants of the Berendsen thermostat. The lowest activation energy is \unit[2.2]{eV}.}
\label{fig:defects:cp_bc_00-1_mig}
% red: ./visualize -w 640 -h 480 -d saves/c_in_si_mig_bc_00-1_s20 -nll -0.56 -0.56 -0.7 -fur 0.2 0.2 0.0 -c 0.75 -1.25 -0.25 -L -0.25 -0.25 -0.25 -r 0.6 -B 0.1
% blue: ./visualize -w 640 -h 480 -d saves/c_in_si_mig_bc_00-1_s20_tr100/ -nll -0.56 -0.56 -0.7 -fur 0.2 0.2 0.0 -c 0.0 -0.25 1.0 -L 0.0 -0.25 -0.25 -r 0.6 -B 0.1
\end{figure}
-Fig. \ref{fig:defects:cp_bc_00-1_mig} shows the evolution of structure and energy along the \ci{} BC to \hkl<0 0 -1> DB transition.
-Since the \ci{} BC configuration is unstable relaxing into the \hkl<1 1 0> DB configuration within this potential, the low kinetic energy state is used as a starting configuration.
+Fig. \ref{fig:defects:cp_bc_00-1_mig} shows the evolution of structure and energy along the \ci{} BC to \hkl[0 0 -1] DB transition.
+Since the \ci{} BC configuration is unstable relaxing into the \hkl[1 1 0] DB configuration within this potential, the low kinetic energy state is used as a starting configuration.
Two different pathways are obtained for different time constants of the Berendse
n thermostat.
With a time constant of \unit[1]{fs} the C atom resides in the \hkl(1 1 0) plane
However, weaker coupling to the heat bath realized by an increase of the time constant to \unit[100]{fs} enables the C atom to move out of the \hkl(1 1 0) plane already at the beginning, which is accompanied by a reduction in energy, approaching the final configuration on a curved path.
The energy barrier of this path is \unit[0.2]{eV} lower in energy than the direct migration within the \hkl(1 1 0) plane.
However, the investigated pathways cover an activation energy approximately twice as high as the one obtained by quantum-mechanical calculations.
-If the entire transition of the \hkl<0 0 -1> into the \hkl<0 0 1> configuration is considered a two step process passing the intermediate BC configuration, an additional activation energy of \unit[0.5]{eV} is necessary to escape the BC towards the \hkl<0 0 1> configuration.
+If the entire transition of the \hkl[0 0 -1] into the \hkl[0 0 1] configuration is considered a two step process passing the intermediate BC configuration, an additional activation energy of \unit[0.5]{eV} is necessary to escape the BC towards the \hkl[0 0 1] configuration.
Assuming equal preexponential factors for both diffusion steps, the total probability of diffusion is given by $\exp\left((2.2\,\text{eV}+0.5\,\text{eV})/k_{\text{B}}T\right)$.
Thus, the activation energy should be located within the range of \unit[2.2-2.7]{eV}.
\begin{center}
\includegraphics[width=0.7\textwidth]{00-1_0-10_albe_s.ps}
\end{center}
-\caption{Migration barrier and structures of the \ci{} \hkl<0 0 -1> to \hkl<0 -1 0> DB transition using the classical EA potential.}
+\caption{Migration barrier and structures of the \ci{} \hkl[0 0 -1] to \hkl[0 -1 0] DB transition using the classical EA potential.}
% red: ./visualize -w 640 -h 480 -d saves/c_in_si_mig_00-1_0-10_s20 -nll -0.56 -0.56 -0.8 -fur 0.3 0.2 0 -c -0.125 -1.7 0.7 -L -0.125 -0.25 -0.25 -r 0.6 -B 0.1
\label{fig:defects:cp_00-1_0-10_mig}
\end{figure}
\begin{center}
\includegraphics[width=0.7\textwidth]{00-1_ip0-10.ps}
\end{center}
-\caption{Reorientation barrier of the \ci{} \hkl<0 0 -1> to \hkl<0 -1 0> DB transition in place using the classical EA potential.}
+\caption{Reorientation barrier of the \ci{} \hkl[0 0 -1] to \hkl[0 -1 0] DB transition in place using the classical EA potential.}
\label{fig:defects:cp_00-1_ip0-10_mig}
\end{figure}
-Figures \ref{fig:defects:cp_00-1_0-10_mig} and \ref{fig:defects:cp_00-1_ip0-10_mig} show the migration barriers of the \ci{} \hkl<0 0 -1> to \hkl<0 -1 0> DB transition.
+Figures \ref{fig:defects:cp_00-1_0-10_mig} and \ref{fig:defects:cp_00-1_ip0-10_mig} show the migration barriers of the \ci{} \hkl[0 0 -1] to \hkl[0 -1 0] DB transition.
In the first case, the transition involves a change in the lattice site of the C atom whereas in the second case, a reorientation at the same lattice site takes place.
In the first case, the pathways for the two different time cosntants look similar.
A local minimum exists in between two peaks of the graph.
-The corresponding configuration, which is illustrated for the results obtained for a time constant of \unit[1]{fs}, looks similar to the \ci{} \hkl<1 1 0> configuration.
+The corresponding configuration, which is illustrated for the results obtained for a time constant of \unit[1]{fs}, looks similar to the \ci{} \hkl[1 1 0] configuration.
Indeed, this configuration is obtained by relaxation simulations without constraints of configurations near the minimum.
Activation energies of roughly \unit[2.8]{eV} and \unit[2.7]{eV} are needed for migration.
-The \ci{} \hkl<1 1 0> configuration seems to play a decisive role in all migration pathways in the classical potential calculations.
+The \ci{} \hkl[1 1 0] configuration seems to play a decisive role in all migration pathways in the classical potential calculations.
As mentioned above, the starting configuration of the first migration path, i.e. the BC configuration, is fixed to be a transition point but in fact is unstable.
-Further relaxation of the BC configuration results in the \ci{} \hkl<1 1 0> configuration.
-Even the last two pathways show configurations almost identical to the \ci{} \hkl<1 1 0> configuration, which constitute local minima within the pathways.
-Thus, migration pathways involving the \ci{} \hkl<1 1 0> DB configuration as a starting or final configuration are further investigated.
+Further relaxation of the BC configuration results in the \ci{} \hkl[1 1 0] configuration.
+Even the last two pathways show configurations almost identical to the \ci{} \hkl[1 1 0] configuration, which constitute local minima within the pathways.
+Thus, migration pathways involving the \ci{} \hkl[1 1 0] DB configuration as a starting or final configuration are further investigated.
\begin{figure}[tp]
\begin{center}
\includegraphics[width=0.7\textwidth]{110_mig.ps}
\end{center}
-\caption[Migration barriers of the \ci{} \hkl<1 1 0> DB to BC (blue), \hkl<0 0 -1> (green) and \hkl<0 -1 0> (in place, red) transition.]{Migration barriers of the \ci{} \hkl<1 1 0> DB to BC (blue), \hkl<0 0 -1> (green) and \hkl<0 -1 0> (in place, red) transition. Solid lines show results for a time constant of \unit[1]{fs} and dashed lines correspond to simulations employing a time constant of \unit[100]{fs}.}
+\caption[{Migration barriers of the \ci{} \hkl[1 1 0] DB to BC, \hkl[0 0 -1] and \hkl[0 -1 0] (in place) transition.}]{Migration barriers of the \ci{} \hkl[1 1 0] DB to BC (blue), \hkl[0 0 -1] (green) and \hkl[0 -1 0] (in place, red) transition. Solid lines show results for a time constant of \unit[1]{fs} and dashed lines correspond to simulations employing a time constant of \unit[100]{fs}.}
\label{fig:defects:110_mig}
\end{figure}
-Fig. \ref{fig:defects:110_mig} shows migration barriers of the \ci{} \hkl<1 1 0> DB to \hkl<0 0 -1>, \hkl<0 -1 0> (in place) and BC configuration.
+Fig. \ref{fig:defects:110_mig} shows migration barriers of the \ci{} \hkl[1 1 0] DB to \hkl[0 0 -1], \hkl[0 -1 0] (in place) and BC configuration.
As expected there is no maximum for the transition into the BC configuration.
-As mentioned earlier the BC configuration itself constitutes a saddle point configuration relaxing into the energetically more favorable \hkl<1 1 0> DB configuration.
-An activation energy of \unit[2.2]{eV} is necessary to reorientate the \hkl<0 0 -1> into the \hkl<1 1 0> DB configuration, which is \unit[1.3]{eV} higher in energy.
-Residing in this state another \unit[0.90]{eV} is enough to make the C atom form a \hkl<0 0 -1> DB configuration with the Si atom of the neighbored lattice site.
+As mentioned earlier, the BC configuration itself constitutes a saddle point configuration relaxing into the energetically more favorable \hkl[1 1 0] DB configuration.
+An activation energy of \unit[2.2]{eV} is necessary to reorientate the \hkl[0 0 -1] into the \hkl[1 1 0] DB configuration, which is \unit[1.3]{eV} higher in energy.
+Residing in this state another \unit[0.90]{eV} is enough to make the C atom form a \hkl[0 0 -1] DB configuration with the Si atom of the neighbored lattice site.
In contrast to quantum-mechanical calculations, in which the direct transition is the energetically most favorable transition and the transition composed of the intermmediate migration steps is very unlikely to occur, the just presented pathway is much more conceivable in classical potential simulations, since the energetically most favorable transition found so far is likewise composed of two migration steps with activation energies of \unit[2.2]{eV} and \unit[0.5]{eV}, for which the intermediate state is the BC configuration, which is unstable.
-Thus the just proposed migration path, which involves the \hkl<1 1 0> interstitial configuration, becomes even more probable than the initially porposed path, which involves the BC configuration that is, in fact, unstable.
+Thus the just proposed migration path, which involves the \hkl[1 1 0] interstitial configuration, becomes even more probable than the initially porposed path, which involves the BC configuration that is, in fact, unstable.
Due to these findings, the respective path is proposed to constitute the diffusion-describing path.
The evolution of structure and configurational energy is displayed again in Fig. \ref{fig:defects:involve110}.
\begin{figure}[tp]
\begin{center}
\includegraphics[width=0.7\textwidth]{00-1_110_0-10_mig_albe.ps}
\end{center}
-\caption[Migration barrier and structures of the \ci{} \hkl<0 0 -1> (left) to the \hkl<0 -1 0> DB (right) transition involving the \hkl<1 1 0> DB (center) configuration.]{Migration barrier and structures of the \ci{} \hkl[0 0 -1] (left) to the \hkl[0 -1 0] DB (right) transition involving the \hkl[1 1 0] DB (center) configuration. Migration simulations are performed utilizing time constants of \unit[1]{fs} (solid line) and \unit[100]{fs} (dashed line) for the Berendsen thermostat.}
+\caption[Migration barrier and structures of the \ci{} {\hkl[0 0 -1]} to the {\hkl[0 -1 0]} DB transition involving the {\hkl[1 1 0]} DB configuration.]{Migration barrier and structures of the \ci{} \hkl[0 0 -1] (left) to the \hkl[0 -1 0] DB (right) transition involving the \hkl[1 1 0] DB (center) configuration. Migration simulations are performed utilizing time constants of \unit[1]{fs} (solid line) and \unit[100]{fs} (dashed line) for the Berendsen thermostat.}
\label{fig:defects:involve110}
\end{figure}
Approximately \unit[2.2]{eV} are needed to turn the \ci{} \hkl[0 0 -1] into the \hkl[1 1 0] DB located at the neighbored lattice site in \hkl[1 1 -1] direction.
\hspace{0.5cm}
\subfigure[]{\label{fig:defects:combos_si}\includegraphics[width=0.3\textwidth]{combos.eps}}
\end{center}
-\caption{Position of the initial \ci{} \hkl[0 0 -1] DB (I) (a) and of the lattice site chosen for the initial \si{} \hkl<1 1 0> DB (\si) (b). Lattice sites for the second defect used for investigating defect pairs are numbered from 1 to 5. For black/red/blue numbers, one/two/four possible atom(s) exist for the second defect to create equivalent defect combinations.}
+\caption[Position of the initial \ci{} {\hkl[0 0 -1]} DB and of the lattice site chosen for the initial \si{} \hkl<1 1 0> DB.]{Position of the initial \ci{} \hkl[0 0 -1] DB (I) (a) and of the lattice site chosen for the initial \si{} \hkl<1 1 0> DB (\si) (b). Lattice sites for the second defect used for investigating defect pairs are numbered from 1 to 5. For black/red/blue numbers, one/two/four possible atom(s) exist for the second defect to create equivalent defect combinations.}
\label{fig:defects:combos}
\end{figure}
Fig.~\ref{fig:defects:combos} schematically displays the initial \ci{} \hkl[0 0 -1] DB structure (Fig.~\ref{fig:defects:combos_ci}) as well as the lattice site chosen for the initial \si{} \hkl<1 1 0> DB (Fig.~\ref{fig:defects:combos_si}) and various positions for the second defect (1-5) that are used for investigating defect pairs.
\hspace{0.5cm}
\subfigure[\underline{$E_{\text{b}}=-2.39\,\text{eV}$}]{\label{fig:defects:239}\includegraphics[width=0.3\textwidth]{00-1dc/2-39.eps}}
\end{center}
-\caption{Relaxed structures of defect combinations obtained by creating \hkl[1 0 0] (a) and \hkl[0 -1 0] (b) DBs at position 1.}
+\caption[Relaxed structures of defect combinations obtained by creating {\hkl[1 0 0]} and {\hkl[0 -1 0]} DBs at position 1.]{Relaxed structures of defect combinations obtained by creating \hkl[1 0 0] (a) and \hkl[0 -1 0] (b) DBs at position 1.}
\label{fig:defects:comb_db_01}
\end{figure}
Mattoni~et~al. \cite{mattoni2002} predict the ground-state configuration of \ci{} \hkl<1 0 0>-type defect pairs 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}.
\hspace{0.2cm}
\subfigure[\underline{$E_{\text{b}}=-2.05\,\text{eV}$}]{\label{fig:defects:205}\includegraphics[width=0.25\textwidth]{00-1dc/2-05.eps}}
\end{center}
-\caption{Relaxed structures of defect combinations obtained by creating \hkl[1 0 0] (a) and \hkl[0 1 0] (b) DBs at position 2 and a \hkl[0 0 1] (c) DB at position 3.}
+\caption[Relaxed structures of defect combinations obtained by creating {\hkl[1 0 0]} and {\hkl[0 1 0]} DBs at position 2 and a {\hkl[0 0 1]} DB at position 3.]{Relaxed structures of defect combinations obtained by creating \hkl[1 0 0] (a) and \hkl[0 1 0] (b) DBs at position 2 and a \hkl[0 0 1] (c) DB at position 3.}
\label{fig:defects:comb_db_02}
\end{figure}
Fig.~\ref{fig:defects:comb_db_02} shows the next three energetically favorable configurations.
\hspace{0.7cm}
\subfigure[\underline{$E_{\text{b}}=-1.38\,\text{eV}$}]{\label{fig:defects:138}\includegraphics[width=0.25\textwidth]{00-1dc/1-38.eps}}
\end{center}
-\caption{Relaxed structures of defect combinations obtained by creating \hkl[0 0 1] (a), \hkl[0 0 -1] (b), \hkl[0 -1 0] (c) and \hkl[1 0 0] (d) DBs at position 5.}
+\caption[Relaxed structures of defect combinations obtained by creating {\hkl[0 0 1]}, {\hkl[0 0 -1]}, {\hkl[0 -1 0]} and {\hkl[1 0 0]} DBs at position 5.]{Relaxed structures of defect combinations obtained by creating \hkl[0 0 1] (a), \hkl[0 0 -1] (b), \hkl[0 -1 0] (c) and \hkl[1 0 0] (d) DBs at position 5.}
\label{fig:defects:comb_db_03}
\end{figure}
Energetically beneficial configurations of defect combinations are observed for interstititals of all orientations placed at position 5, a position two bonds away from the initial interstitial along the \hkl[1 1 0] direction.
\begin{center}
\includegraphics[width=0.7\textwidth]{db_along_110_cc_n.ps}
\end{center}
-\caption[Minimum binding energy of DB combinations separated along \hkl<1 1 0> with respect to the C-C distance.]{Minimum binding energy of dumbbell combinations separated along \hkl[1 1 0] 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.}
+\caption[Minimum binding energy of DB combinations separated along {\hkl[1 1 0]} with respect to the C-C distance.]{Minimum binding energy of dumbbell combinations separated along \hkl[1 1 0] 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.}
\label{fig:defects:comb_db110}
\end{figure}
The binding energy of these configurations with respect to the C-C distance is plotted in Fig.~\ref{fig:defects:comb_db110}.
\begin{center}
\includegraphics[width=0.7\textwidth]{036-239.ps}
\end{center}
-\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.}
+\caption[Migration barrier and structures of the transition of a C$_{\text{i}}$ {\hkl[-1 0 0]} DB at position 2 into a C$_{\text{i}}$ {\hkl[0 -1 0]} DB at position 1.]{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}
Since thermally activated C clustering is, thus, only possible by traversing energetically unfavored configurations, extensive C clustering is not expected.
\begin{center}
\includegraphics[width=0.7\textwidth]{188-225.ps}
\end{center}
-\caption{Migration barrier and structures of the transition of a C$_{\text{i}}$ \hkl[0 -1 0] DB at position 5 (left) into a C$_{\text{i}}$ \hkl[1 0 0] DB at position 1 (right). An activation energy of \unit[0.47]{eV} is observed.}
+\caption[Migration barrier and structures of the transition of a C$_{\text{i}}$ {\hkl[0 -1 0]} DB at position 5 into a C$_{\text{i}}$ {\hkl[1 0 0]} DB at position 1.]{Migration barrier and structures of the transition of a C$_{\text{i}}$ \hkl[0 -1 0] DB at position 5 (left) into a C$_{\text{i}}$ \hkl[1 0 0] DB at position 1 (right). An activation energy of \unit[0.47]{eV} is observed.}
\label{fig:188-225}
\end{figure}
Finally, as already mentioned above, this type of defect pair is represented two times more often than the ground-state configuration.
\hline
\end{tabular}
\end{center}
-\caption{Binding energies of combinations of the \ci{} \hkl[0 0 -1] defect with a \cs{} atom located at positions 1 to 5 according to Fig.~\ref{fig:defects: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.}
+\caption[Binding energies of combinations of the \ci{} {\hkl[0 0 -1]} defect with a \cs{} atom located at positions 1 to 5 according to Fig.~\ref{fig:defects:combos_ci}.]{Binding energies of combinations of the \ci{} \hkl[0 0 -1] defect with a \cs{} atom located at positions 1 to 5 according to Fig.~\ref{fig:defects: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.}
\label{tab:defects:c-s}
\end{table}
%\begin{figure}[tp]
\begin{center}
\includegraphics[width=0.7\textwidth]{093-095.ps}
\end{center}
-\caption{Migration barrier and structures of the transition of the initial \ci{} \hkl[0 0 -1] DB and C$_{\text{s}}$ at position 3 (left) into a configuration of a twofold coordinated Si$_{\text{i}}$ located in between two C$_{\text{s}}$ atoms occupying the lattice sites of the initial DB and position 3 (right). An activation energy of \unit[0.44]{eV} is observed.}
+\caption[Migration barrier and structures of the transition of the initial \ci{} {\hkl[0 0 -1]} DB and C$_{\text{s}}$ at position 3 into a configuration of a twofold coordinated Si$_{\text{i}}$ located in between two C$_{\text{s}}$ atoms occupying the lattice sites of the initial DB and position 3.]{Migration barrier and structures of the transition of the initial \ci{} \hkl[0 0 -1] DB and C$_{\text{s}}$ at position 3 (left) into a configuration of a twofold coordinated Si$_{\text{i}}$ located in between two C$_{\text{s}}$ atoms occupying the lattice sites of the initial DB and position 3 (right). An activation energy of \unit[0.44]{eV} is observed.}
\label{fig:093-095}
\end{figure}
Configuration A consists of a C$_{\text{i}}$ \hkl[0 0 -1] DB with threefold coordinated Si and C DB atoms slightly disturbed by the C$_{\text{s}}$ at position 3, facing the Si DB atom as a neighbor.
\begin{center}
\includegraphics[width=0.7\textwidth]{comb_mig_026-128_vasp.ps}
\end{center}
-\caption{Migration barrier and structures of the transition of the initial C$_{\text{i}}$ \hkl[0 0 -1] DB and C$_{\text{s}}$ at position 1 (left) into a C-C \hkl[1 0 0] DB occupying the lattice site at position 1 (right). An activation energy of \unit[0.1]{eV} is observed.}
+\caption[Migration barrier and structures of the transition of the initial C$_{\text{i}}$ {\hkl[0 0 -1]} DB and C$_{\text{s}}$ at position 1 into a C-C {\hkl[1 0 0]} DB occupying the lattice site at position 1.]{Migration barrier and structures of the transition of the initial C$_{\text{i}}$ \hkl[0 0 -1] DB and C$_{\text{s}}$ at position 1 (left) into a C-C \hkl[1 0 0] DB occupying the lattice site at position 1 (right). An activation energy of \unit[0.1]{eV} is observed.}
\label{fig:026-128}
\end{figure}
Configuration $\alpha$ is similar to configuration A, except that the C$_{\text{s}}$ atom at position 1 is facing the C DB atom as a neighbor resulting in the formation of a strong C-C bond and a much more noticeable perturbation of the DB structure.
\hspace{0.2cm}
\subfigure[\underline{$E_{\text{b}}=0.49\,\text{eV}$}]{\label{fig:defects:049}\includegraphics[width=0.25\textwidth]{00-1dc/0-49.eps}}
\end{center}
-\caption{Relaxed structures of defect combinations obtained by creating \cs{} at positions 2 (a), 4 (b) and 5 (c) in the \ci{} \hkl[0 0 -1] DB configuration.}
+\caption[Relaxed structures of defect combinations obtained by creating \cs{} at positions 2, 4 and 5 in the \ci{} {\hkl[0 0 -1]} DB configuration.]{Relaxed structures of defect combinations obtained by creating \cs{} at positions 2 (a), 4 (b) and 5 (c) in the \ci{} \hkl[0 0 -1] DB configuration.}
\label{fig_defects:245csub}
\end{figure}
Fig.~\ref{fig_defects:245csub} lists the remaining configurations and binding energies of the relaxed structures obtained by creating a \cs{} at positions 2, 4 and 5 in the \ci{} \hkl[0 0 -1] DB configuration.
\hline
\end{tabular}
\end{center}
-\caption{Binding energies of combinations of the \ci{} \hkl[0 0 -1] defect with a vacancy located at positions 1 to 5 according to Fig.~\ref{fig:defects: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.}
+\caption[Binding energies of combinations of the \ci{} {\hkl[0 0 -1]} defect with a vacancy located at positions 1 to 5 according to Fig.~\ref{fig:defects:combos_ci}.]{Binding energies of combinations of the \ci{} \hkl[0 0 -1] defect with a vacancy located at positions 1 to 5 according to Fig.~\ref{fig:defects: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.}
\label{tab:defects:c-v}
\end{table}
\begin{figure}[tp]
\hspace{0.7cm}
\subfigure[\underline{$E_{\text{b}}=-0.50\,\text{eV}$}]{\label{fig:defects:050}\includegraphics[width=0.25\textwidth]{00-1dc/0-50.eps}}
\end{center}
-\caption{Relaxed structures of defect combinations obtained by creating a vacancy at positions 2 (a), 3 (b), 4 (c) and 5 (d).}
+\caption[Relaxed structures of defect combinations obtained by creating a vacancy at positions 2, 3, 4 and 5.]{Relaxed structures of defect combinations obtained by creating a vacancy at positions 2 (a), 3 (b), 4 (c) and 5 (d).}
\label{fig:defects:comb_db_06}
\end{figure}
Figure \ref{fig:defects:comb_db_06} shows the associated configurations.
\begin{center}
\includegraphics[width=0.7\textwidth]{314-539.ps}
\end{center}
-\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.}
+\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 into a C$_{\text{s}}$ configuration.]{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.}
\label{fig:314-539}
\end{figure}
\begin{figure}[tp]
\begin{center}
\includegraphics[width=0.7\textwidth]{059-539.ps}
\end{center}
-\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.}
+\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 into a C$_{\text{s}}$ configuration.]{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.}
\label{fig:059-539}
\end{figure}
Activation energies as low as \unit[0.1]{eV} and \unit[0.6]{eV} are observed.
\hline
\end{tabular}
\end{center}
-\caption{Equivalent configurations labeled \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:defects:combos_si}. The respective orientation of the Si$_{\text{i}}$ DB is given in the first row.}
+\caption[Equivalent configurations labeled \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:defects:combos_si}.]{Equivalent configurations labeled \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:defects:combos_si}. The respective orientation of the Si$_{\text{i}}$ DB is given in the first row.}
\label{tab:defects:comb_csub_si110}
\end{table}
\begin{table}[tp]
\begin{center}
\includegraphics[width=0.7\textwidth]{162-097.ps}
\end{center}
-\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} and \unit[0.77]{eV} for the reverse process is observed.}
+\caption[Migration barrier and structures of the transition of a {\hkl[1 1 0]} Si$_{\text{i}}$ DB next to C$_{\text{s}}$ into the C$_{\text{i}}$ {\hkl[0 0 -1]} DB configuration.]{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} and \unit[0.77]{eV} for the reverse process is observed.}
\label{fig:162-097}
\end{figure}
An activation energy as low as \unit[0.12]{eV} is necessary for the migration into the ground-state configuration.
\begin{center}
\includegraphics[width=0.7\textwidth]{c_sub_si110.ps}
\end{center}
-\caption{Binding energies of combinations of a C$_{\text{s}}$ and a Si$_{\text{i}}$ DB with respect to the separation distance. The interaction strength of the defect pairs are well approximated by a Lennard-Jones 6-12 potential, which is used for curve fitting.}
+\caption[Binding energies of combinations of a C$_{\text{s}}$ and a Si$_{\text{i}}$ DB with respect to the separation distance.]{Binding energies of combinations of a C$_{\text{s}}$ and a Si$_{\text{i}}$ DB with respect to the separation distance. The interaction strength of the defect pairs are well approximated by a Lennard-Jones 6-12 potential, which is used for curve fitting.}
\label{fig:dc_si-s}
\end{figure}
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.
\begin{figure}[tp]
\begin{center}
\begin{minipage}{0.40\textwidth}
-\includegraphics[width=\columnwidth]{md_vasp_01.eps}
+\includegraphics[width=\columnwidth]{md01_bonds.eps}
\end{minipage}
\hspace{1cm}
\begin{minipage}{0.40\textwidth}
-\includegraphics[width=\columnwidth]{md_vasp_02.eps}
+\includegraphics[width=\columnwidth]{md02_bonds.eps}
\end{minipage}\\
\begin{minipage}{0.40\textwidth}
\begin{center}
\end{center}
\end{minipage}
\end{center}
-\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$_{\text{i}}$ \hkl[1 1 0] DB (atoms 1 and 2). Equal atoms are marked by equal numbers. Bonds are drawn for substantial atoms only.}
+\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$_{\text{i}}$ {\hkl[1 1 0]} DB.]{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$_{\text{i}}$ \hkl[1 1 0] DB (atoms 1 and 2). Equal atoms are marked by equal numbers. For substantial atoms, bonds are drawn in red color.}
\label{fig:defects:md}
\end{figure}
Si atoms 1 and 2, which form the initial DB, occupy Si lattice sites in the final configuration while Si atom 3 is transferred from a regular lattice site into the interstitial lattice.
These results support the above assumptions of an increased entropic contribution to structural formation involving C$_{\text{s}}$ to a greater extent.
-% link to migration of \si{}!
-% todo - make it a subsection
+\section{Mobility of the silicon self-interstitial}
+
The possibility for separated configurations of \cs{} and \si{} becomes even more likely if one of the constituents exhibits a low barrier of migration.
In this case, the \si{} is assumed to constitute the mobile defect compared to the stable \cs{} atom.
Thus, migration paths of \si{} are investigated in the following excursus.
\begin{center}
\includegraphics[width=0.7\textwidth]{si_110_110_mig_02_conf.ps}
\end{center}
-\caption[Migration barrier and structures of the \si{} \hkl<1 1 0> DB.]{Migration barrier and structures of the \si{} \hkl[0 -1 1] DB (left) to the \hkl[1 1 0] DB (right) transition. Bonds are illustrated by blue lines.}
-% todo read above caption! enable [] hkls in short caption
+\caption[Migration barrier and structures of the \si{} {\hkl[0 -1 1]} DB to the {\hkl[1 1 0]} DB transition.]{Migration barrier and structures of the \si{} \hkl[0 -1 1] DB (left) to the \hkl[1 1 0] DB (right) transition. Bonds are illustrated by blue lines.}
\label{fig:defects:si_mig1}
\end{figure}
The barrier, which is even lower than the one for \ci{}, indeed indicates highly mobile \si.
\begin{center}
\includegraphics[width=0.7\textwidth]{si_mig_rest.ps}
\end{center}
-\caption{Migration barrier of the \si{} \hkl[1 1 0] DB into the hexagonal (H) and tetrahedral (T) configuration as well as the hexagonal \si{} to tetrahedral \si{} transition.}
-% todo read above caption! enable [] hkls in short caption
+\caption[Migration barrier of the \si{} {\hkl[1 1 0]} DB into the hexagonal and tetrahedral configuration as well as the hexagonal \si{} to tetrahedral \si{} transition.]{Migration barrier of the \si{} \hkl[1 1 0] DB into the hexagonal (H) and tetrahedral (T) configuration as well as the hexagonal \si{} to tetrahedral \si{} transition.}
\label{fig:defects:si_mig2}
\end{figure}
The obtained activation energies are of the same order of magnitude than values derived from other ab initio studies \cite{bloechl93,sahli05}.
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