\label{chapter:defects}
Regarding the supposed conversion mechanisms of SiC in c-Si as introduced in section \ref{section:assumed_prec} the understanding of C and Si interstitial point defects in c-Si is of fundamental interest.
-During implantation defects such as vacancies (V), substitutional C (C$_{\text{s}}$), interstitial C (C$_{\text{i}}$) and Si self-interstitials (Si$_{\text{i}}$) are created, which are believed to play a decisive role in the precipitation process.
-In the following, these defects are systematically examined by computationally efficient, classical potential as well as highly accurate DFT calculations with the parameters and simulation conditions as outlined in chapter \ref{chapter:simulation}.
+During implantation, defects such as vacancies (V), substitutional C (C$_{\text{s}}$), interstitial C (C$_{\text{i}}$) and Si self-interstitials (Si$_{\text{i}}$) are created, which are believed to play a decisive role in the precipitation process.
+In the following, these defects are systematically examined by computationally efficient, classical potential as well as highly accurate DFT calculations with the parameters and simulation conditions that are defined in chapter \ref{chapter:simulation}.
Both methods are used to investigate selected diffusion processes of some of the defect configurations.
While the quantum-mechanical description yields results that excellently compare to experimental findings, shortcomings of the classical potential approach are identified.
These shortcomings are further investigated and the basis for a workaround, as proposed later on in the classical MD simulation chapter, is discussed.
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.
However, the energy barrier is small.
-\begin{figure}[ht]
+\begin{figure}[!ht]
\begin{center}
\includegraphics[width=0.7\textwidth]{nhex_tet.ps}
\end{center}
The same applies to the bonds between the interstitial and the upper two atoms in the \si{} \hkl<1 1 0> DB configuration.
A more detailed description of the chemical bonding is achieved through quantum-mechanical calculations by investigating the accumulation of negative charge between the nuclei.
-
-
+%\clearpage{}
\section{Carbon point defects in silicon}
% todo smaller images, therefore add mo image
\clearpage{}
-\cleardoublepage{}
% todo migration of \si{}!
\begin{figure}[ht]
\begin{center}
-\includegraphics[width=13cm]{vasp_mig/00-1_0-10_nosym_sp_fullct.ps}\\[1.6cm]
-\begin{picture}(0,0)(140,0)
-\includegraphics[width=2.5cm]{vasp_mig/00-1_a.eps}
-\end{picture}
-\begin{picture}(0,0)(20,0)
-\includegraphics[width=2.5cm]{vasp_mig/00-1_0-10_sp.eps}
-\end{picture}
-\begin{picture}(0,0)(-120,0)
-\includegraphics[width=2.5cm]{vasp_mig/0-10.eps}
-\end{picture}
-\begin{picture}(0,0)(25,20)
-\includegraphics[width=2.5cm]{100_arrow.eps}
-\end{picture}
-\begin{picture}(0,0)(200,0)
-\includegraphics[height=2.2cm]{001_arrow.eps}
-\end{picture}
+\includegraphics[width=0.7\textwidth]{00-1_0-10_vasp_s.ps}
+%\includegraphics[width=13cm]{vasp_mig/00-1_0-10_nosym_sp_fullct.ps}\\[1.6cm]
+%\begin{picture}(0,0)(140,0)
+%\includegraphics[width=2.5cm]{vasp_mig/00-1_a.eps}
+%\end{picture}
+%\begin{picture}(0,0)(20,0)
+%\includegraphics[width=2.5cm]{vasp_mig/00-1_0-10_sp.eps}
+%\end{picture}
+%\begin{picture}(0,0)(-120,0)
+%\includegraphics[width=2.5cm]{vasp_mig/0-10.eps}
+%\end{picture}
+%\begin{picture}(0,0)(25,20)
+%\includegraphics[width=2.5cm]{100_arrow.eps}
+%\end{picture}
+%\begin{picture}(0,0)(200,0)
+%\includegraphics[height=2.2cm]{001_arrow.eps}
+%\end{picture}
\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.}
+\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. {\color{red} Prototype design, adjust related figures!}}
+% todo read above caption! enable [] hkls in short caption
\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{figure}[ht]
\begin{center}
-\includegraphics[width=13cm]{vasp_mig/110_mig_vasp.ps}
+\includegraphics[width=0.7\textwidth]{vasp_mig/110_mig_vasp.ps}
%\begin{picture}(0,0)(140,0)
%\includegraphics[width=2.2cm]{vasp_mig/00-1_b.eps}
%\end{picture}
\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.
-% HIER WEITER
This is investigated in the following in order to find possible migration pathways that have an activation energy lower than the ones found up to now.
-The next energetically favorable defect configuration is the \hkl<1 1 0> C-Si dumbbell interstitial.
-Figure \ref{fig:defects:110_mig_vasp} shows the migration barrier of the \hkl<1 1 0> C-Si dumbbell to the bond-centered, \hkl<0 0 -1> and \hkl<0 -1 0> (in place) transition.
-Indeed less than 0.7 eV are necessary to turn a \hkl<0 -1 0>- to a \hkl<1 1 0>-type C-Si dumbbell interstitial.
-This transition is carried out in place, that is the Si dumbbell pair is not changed and both, the Si and C atom share the initial lattice site.
+The next energetically favorable defect configuration is the \hkl<1 1 0> C-Si DB interstitial.
+Fig. \ref{fig:defects:110_mig_vasp} shows the migration barrier of the \hkl<1 1 0> C-Si DB to the BC, \hkl<0 0 -1> and \hkl<0 -1 0> (in place) transition.
+Indeed less than \unit[0.7]{eV} are necessary to turn a \hkl<0 -1 0>- to a \hkl<1 1 0>-type C-Si DB interstitial.
+This transition is carried out in place, i.e. the Si DB pair is not changed and both, the Si and C atom share the initial lattice site.
Thus, this transition does not contribute to long-range diffusion.
-Once the C atom resides in the \hkl<1 1 0> interstitial configuration it can migrate into the bond-centered configuration by employing approximately 0.95 eV of activation energy, which is only slightly higher than the activation energy needed for the \hkl<0 0 -1> to \hkl<0 -1 0> pathway shown in figure \ref{fig:defects:00-1_0-10_mig}.
-As already known from the migration of the \hkl<0 0 -1> to the bond-centered configuration as discussed in figure \ref{fig:defects:00-1_001_mig} another 0.25 eV are needed to turn back from the bond-centered to a \hkl<1 0 0>-type interstitial.
-However, due to the fact that this migration consists of three single transitions with the second one having an activation energy slightly higher than observed for the direct transition it is considered very unlikely to occur.
-The migration barrier of the \hkl<1 1 0> to \hkl<0 0 -1> transition, in which the C atom is changing its Si partner and, thus, moving to the neighbored lattice site is approximately 1.35 eV.
+Once the C atom resides in the \hkl<1 1 0> DB interstitial configuration it can migrate into the BC configuration requiring approximately \unit[0.95]{eV} of activation energy, which is only slightly higher than the activation energy needed for the \hkl<0 0 -1> to \hkl<0 -1 0> pathway as shown in Fig. \ref{fig:defects:00-1_0-10_mig}.
+As already known from the migration of the \hkl<0 0 -1> to the BC configuration discussed in Fig. \ref{fig:defects:00-1_001_mig}, another \unit[0.25]{eV} are needed to turn back from the BC to a \hkl<1 0 0>-type interstitial.
+However, due to the fact that this migration consists of three single transitions with the second one having an activation energy slightly higher than observed for the direct transition, this sequence of paths is considered very unlikely to occur.
+The migration barrier of the \hkl<1 1 0> to \hkl<0 0 -1> transition, in which the C atom is changing its Si partner and, thus, moving to the neighbored lattice site, corresponds to approximately \unit[1.35]{eV}.
During this transition the C atom is escaping the \hkl(1 1 0) plane approaching the final configuration on a curved path.
This barrier is much higher than the ones found previously, which again make this transition very unlikely to occur.
-For this reason the assumption that C diffusion and reorientation is achieved by transitions of the type presented in figure \ref{fig:defects:00-1_0-10_mig} is reinforced.
-
-As mentioned earlier the procedure to obtain the migration barriers differs from the usually applied procedure in two ways.
-Firstly constraints to move along the displacement direction are applied on all atoms instead of solely constraining the diffusing atom.
-Secondly the constrainted directions are not kept constant to the initial displacement direction.
-Instead they are updated for every displacement step.
-These modifications to the usual procedure are applied to avoid abrupt changes in structure and free energy on the one hand and to make sure the expected final configuration is reached on the other hand.
-Due to applying updated constraints on all atoms the obtained migration barriers and pathes might be overestimated and misguided.
-To reinforce the applicability of the employed technique the obtained activation energies and migration pathes for the \hkl<0 0 -1> to \hkl<0 -1 0> transition are compared to two further migration calculations, which do not update the constrainted direction and which only apply updated constraints on three selected atoms, that is the diffusing C atom and the Si dumbbell pair in the initial and final configuration.
-Results are presented in figure \ref{fig:defects:00-1_0-10_cmp}.
-\begin{figure}[ht]
-\begin{center}
-\includegraphics[width=13cm]{vasp_mig/00-1_0-10_nosym_sp_cmp.ps}
-\end{center}
-\caption[Comparison of three different techniques for obtaining migration barriers and pathways applied to the \hkl<0 0 -1> to \hkl<0 -1 0> C-Si dumbbell transition.]{Comparison of three different techniques for obtaining migration barriers and pathways applied to the \hkl<0 0 -1> to \hkl<0 -1 0> C-Si dumbbell transition.}
-\label{fig:defects:00-1_0-10_cmp}
-\end{figure}
-The method without updating the constraints but still applying them to all atoms shows a delayed crossing of the saddle point.
-This is understandable since the update results in a more aggressive advance towards the final configuration.
-In any case the barrier obtained is slightly higher, which means that it does not constitute an energetically more favorable pathway.
-The method in which the constraints are only applied to the diffusing C atom and two Si atoms, ... {\color{red}Todo: does not work!} ...
-
-\subsection{Migration barriers obtained by classical potential calculations}
+For this reason, the assumption that C diffusion and reorientation is achieved by transitions of the type presented in Fig. \ref{fig:defects:00-1_0-10_mig} is reinforced.
+
+%As mentioned earlier the procedure to obtain the migration barriers differs from the usually applied procedure in two ways.
+%Firstly constraints to move along the displacement direction are applied on all atoms instead of solely constraining the diffusing atom.
+%Secondly the constrainted directions are not kept constant to the initial displacement direction.
+%Instead they are updated for every displacement step.
+%These modifications to the usual procedure are applied to avoid abrupt changes in structure and free energy on the one hand and to make sure the expected final configuration is reached on the other hand.
+%Due to applying updated constraints on all atoms the obtained migration barriers and pathes might be overestimated and misguided.
+%To reinforce the applicability of the employed technique the obtained activation energies and migration pathes for the \hkl<0 0 -1> to \hkl<0 -1 0> transition are compared to two further migration calculations, which do not update the constrainted direction and which only apply updated constraints on three selected atoms, that is the diffusing C atom and the Si dumbbell pair in the initial and final configuration.
+%Results are presented in figure \ref{fig:defects:00-1_0-10_cmp}.
+%\begin{figure}[ht]
+%\begin{center}
+%\includegraphics[width=13cm]{vasp_mig/00-1_0-10_nosym_sp_cmp.ps}
+%\end{center}
+%\caption[Comparison of three different techniques for obtaining migration barriers and pathways applied to the \hkl<0 0 -1> to \hkl<0 -1 0> C-Si dumbbell transition.]{Comparison of three different techniques for obtaining migration barriers and pathways applied to the \hkl<0 0 -1> to \hkl<0 -1 0> C-Si dumbbell transition.}
+%\label{fig:defects:00-1_0-10_cmp}
+%\end{figure}
+%The method without updating the constraints but still applying them to all atoms shows a delayed crossing of the saddle point.
+%This is understandable since the update results in a more aggressive advance towards the final configuration.
+%In any case the barrier obtained is slightly higher, which means that it does not constitute an energetically more favorable pathway.
+%The method in which the constraints are only applied to the diffusing C atom and two Si atoms, ...
+
+% todo if there is plenty of time ... reinvestigate above stuff
+
+\subsection{Migration described by classical potential calculations}
\label{subsection:defects:mig_classical}
-The same method for obtaining migration barriers and the same suggested pathways are applied to calculations employing the classical EA potential.
-Since the evaluation of the classical potential and force is less computationally intensive higher amounts of steps can be used.
-The time constant $\tau$ for the Berendsen thermostat is set to 1 fs in order to have direct velocity scaling and with the temperature set to zero Kelvin perform a steepest descent minimazation to drive the system into a local minimum.
-However, in some cases a time constant of 100 fs resuls in lower barriers and, thus, is shown whenever appropriate.
-
\begin{figure}[ht]
\begin{center}
-\includegraphics[width=13cm]{bc_00-1.ps}\\[5.6cm]
-\begin{pspicture}(0,0)(0,0)
-\psframe[linecolor=red,fillstyle=none](-7,2.7)(7.2,6)
-\end{pspicture}
-\begin{picture}(0,0)(140,-100)
-\includegraphics[width=2.4cm]{albe_mig/bc_00-1_red_00.eps}
-\end{picture}
-\begin{picture}(0,0)(10,-100)
-\includegraphics[width=2.4cm]{albe_mig/bc_00-1_red_01.eps}
-\end{picture}
-\begin{picture}(0,0)(-120,-100)
-\includegraphics[width=2.4cm]{albe_mig/bc_00-1_red_02.eps}
-\end{picture}
-\begin{picture}(0,0)(25,-80)
-\includegraphics[width=2.5cm]{110_arrow.eps}
-\end{picture}
-\begin{picture}(0,0)(215,-100)
-\includegraphics[height=2.2cm]{001_arrow.eps}
-\end{picture}\\
-\begin{pspicture}(0,0)(0,0)
-\psframe[linecolor=blue,fillstyle=none](-7,-0.5)(7.2,2.8)
-\end{pspicture}
-\begin{picture}(0,0)(160,-10)
-\includegraphics[width=2.2cm]{albe_mig/bc_00-1_01.eps}
-\end{picture}
-\begin{picture}(0,0)(100,-10)
-\includegraphics[width=2.2cm]{albe_mig/bc_00-1_02.eps}
-\end{picture}
-\begin{picture}(0,0)(10,-10)
-\includegraphics[width=2.2cm]{albe_mig/bc_00-1_03.eps}
-\end{picture}
-\begin{picture}(0,0)(-120,-10)
-\includegraphics[width=2.2cm]{albe_mig/bc_00-1_04.eps}
-\end{picture}
-\begin{picture}(0,0)(25,10)
-\includegraphics[width=2.5cm]{100_arrow.eps}
-\end{picture}
-\begin{picture}(0,0)(215,-10)
-\includegraphics[height=2.2cm]{010_arrow.eps}
-\end{picture}
+\includegraphics[width=0.7\textwidth]{bc_00-1_albe_s.ps}
+%\includegraphics[width=13cm]{bc_00-1.ps}\\[5.6cm]
+%\begin{pspicture}(0,0)(0,0)
+%\psframe[linecolor=red,fillstyle=none](-7,2.7)(7.2,6)
+%\end{pspicture}
+%\begin{picture}(0,0)(140,-100)
+%\includegraphics[width=2.4cm]{albe_mig/bc_00-1_red_00.eps}
+%\end{picture}
+%\begin{picture}(0,0)(10,-100)
+%\includegraphics[width=2.4cm]{albe_mig/bc_00-1_red_01.eps}
+%\end{picture}
+%\begin{picture}(0,0)(-120,-100)
+%\includegraphics[width=2.4cm]{albe_mig/bc_00-1_red_02.eps}
+%\end{picture}
+%\begin{picture}(0,0)(25,-80)
+%\includegraphics[width=2.5cm]{110_arrow.eps}
+%\end{picture}
+%\begin{picture}(0,0)(215,-100)
+%\includegraphics[height=2.2cm]{001_arrow.eps}
+%\end{picture}\\
+%\begin{pspicture}(0,0)(0,0)
+%\psframe[linecolor=blue,fillstyle=none](-7,-0.5)(7.2,2.8)
+%\end{pspicture}
+%\begin{picture}(0,0)(160,-10)
+%\includegraphics[width=2.2cm]{albe_mig/bc_00-1_01.eps}
+%\end{picture}
+%\begin{picture}(0,0)(100,-10)
+%\includegraphics[width=2.2cm]{albe_mig/bc_00-1_02.eps}
+%\end{picture}
+%\begin{picture}(0,0)(10,-10)
+%\includegraphics[width=2.2cm]{albe_mig/bc_00-1_03.eps}
+%\end{picture}
+%\begin{picture}(0,0)(-120,-10)
+%\includegraphics[width=2.2cm]{albe_mig/bc_00-1_04.eps}
+%\end{picture}
+%\begin{picture}(0,0)(25,10)
+%\includegraphics[width=2.5cm]{100_arrow.eps}
+%\end{picture}
+%\begin{picture}(0,0)(215,-10)
+%\includegraphics[height=2.2cm]{010_arrow.eps}
+%\end{picture}
\end{center}
-\caption{Migration barrier and structures of the bond-centered to \hkl<0 0 -1> dumbbell transition using the classical EA potential.}
+\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}. {\color{red} Prototype design, adjust related figures!}}
\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}
-Figure \ref{fig:defects:cp_bc_00-1_mig} shows the migration barrier and corresponding structures of the bond-centered to \hkl<0 0 -1> dumbbell transition.
-Since the bond-centered configuration is unstable relaxing into the \hkl<1 1 0> C-Si dumbbell interstitial configuration within this potential the low kinetic energy state is used as a starting configuration.
-Depending on the time constant activation energies of 2.4 eV and 2.2 eV respectively are obtained.
-The migration path obtained by simulations with a time constant of 1 fs remains in the \hkl(1 1 0) plane.
-Using 100 fs as a time constant the C atom breaks out of the \hkl(1 1 0) plane already at the beginning of the migration accompanied by a reduction in energy.
-The energy barrier of this path is 0.2 eV lower in energy than the direct migration within the \hkl(1 1 0) plane.
+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
+ resulting in a migration barrier of \unit[2.4]{eV}.
+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.
-For the entire transition of the \hkl<0 0 -1> into the \hkl<0 0 1> configuration by passing the bond-centered configuration an additional activation energy of 0.5 eV is necessary to escape from the bond-centered and reach 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{figure}[ht]
\begin{center}
\includegraphics[height=2.2cm]{001_arrow.eps}
\end{picture}
\end{center}
-\caption{Migration barrier and structures of the \hkl<0 0 -1> to \hkl<0 -1 0> C-Si dumbbell 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{figure}[ht]
\begin{center}
-\includegraphics[width=13cm]{00-1_ip0-10.ps}
+\includegraphics[width=0.7\textwidth]{00-1_ip0-10.ps}
\end{center}
-\caption{Migration barrier of the \hkl<0 0 -1> to \hkl<0 -1 0> C-Si dumbbell 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}
-Figure \ref{fig:defects:cp_00-1_0-10_mig} and \ref{fig:defects:cp_00-1_ip0-10_mig} show the migration barriers of \hkl<0 0 -1> to \hkl<0 -1 0> C-Si dumbbell transition, with a transition of the C atom to the neighbored lattice site in the first case and a reorientation within the same lattice site in the latter case.
-Both pathways look similar.
+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 inbetween two peaks of the graph.
-The corresponding configuration, which is illustrated for the migration simulation with a time constant of 1 fs, looks similar to the \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 2.8 eV and 2.7 eV respectively are needed for migration.
+Activation energies of roughly \unit[2.8]{eV} and \unit[2.7]{eV} are needed for migration.
-The \hkl<1 1 0> configuration seems to play a decisive role in all migration pathways.
-In the first migration path it is the configuration resulting from further relaxation of the rather unstable bond-centered configuration, which is fixed to be a transition point in the migration calculations.
-The last two pathways show configurations almost identical to the \hkl<1 1 0> configuration, which constitute a local minimum within the pathway.
-Thus, migration pathways with the \hkl<1 1 0> C-Si dumbbell interstitial configuration as a starting or final configuration are further investigated.
+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.
\begin{figure}[ht]
\begin{center}
-\includegraphics[width=13cm]{110_mig.ps}
+\includegraphics[width=0.7\textwidth]{110_mig.ps}
\end{center}
-\caption[Migration barriers of the \hkl<1 1 0> dumbbell to bond-centered (blue), \hkl<0 0 -1> (green) and \hkl<0 -1 0> (in place, red) C-Si dumbbell transition.]{Migration barriers of the \hkl<1 1 0> dumbbell to bond-centered (blue), \hkl<0 0 -1> (green) and \hkl<0 -1 0> (in place, red) C-Si dumbbell transition. Solid lines show results for a time constant of 1 fs and dashed lines correspond to simulations employing a time constant of 100 fs.}
+\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}.}
\label{fig:defects:110_mig}
\end{figure}
-Figure \ref{fig:defects:110_mig} shows migration barriers of the C-Si \hkl<1 1 0> dumbbell to \hkl<0 0 -1>, \hkl<0 -1 0> (in place) and bond-centered configuration.
-As expected there is no maximum for the transition into the bond-centered configuration.
-As mentioned earlier the bond-centered configuration itself constitutes a saddle point configuration relaxing into the energetically more favorable \hkl<1 1 0> configuration.
-An activation energy of 2.2 eV is necessary to reorientate the \hkl<0 0 -1> dumbbell configuration into the \hkl<1 1 0> configuration, which is 1.3 eV higher in energy.
-Residing in this state another 0.9 eV is enough to make the C atom form a \hkl<0 0 -1> dumbbell 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 supposable in classical potential simulations, since the energetically most favorable transition found so far is also composed of two migration steps with activation energies of 2.2 eV and 0.5 eV, for which the intermediate state is the bond-centered configuration, which is unstable.
-Thus the just proposed migration path involving the \hkl<1 1 0> interstitial configuration becomes even more probable than path 1 involving the unstable bond-centered configuration.
-
-Although classical potential simulations reproduce the order in energy of the \hkl<1 0 0> and \hkl<1 1 0> C-Si dumbbell interstitial configurations as obtained by more accurate quantum-mechanical calculations the obtained migration pathways and resulting activation energies differ to a great extent.
-On the one hand the most favorable pathways differ.
-On the other hand the activation energies obtained by classical potential simulations are tremendously overestimated by a factor of almost 2.4.
+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.
+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.
+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}[ht]
+\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 were 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.
+Another barrier of \unit[0.90]{eV} exists for the rotation into the \ci{} \hkl[0 -1 0] DB configuration for the path obtained with a time constant of \unit[100]{fs} for the Berendsen thermostat.
+Roughly the same amount would be necessary to excite the C$_{\text{i}}$ \hkl[1 1 0] DB to the BC configuration (\unit[0.40]{eV}) and a successive migration into the \hkl[0 0 1] DB configuration (\unit[0.50]{eV}) as displayed in Fig. \ref{fig:defects:110_mig} and Fig. \ref{fig:defects:cp_bc_00-1_mig}.
+The former diffusion process, however, would more nicely agree with the ab initio path, since the migration is accompanied by a rotation of the DB orientation.
+By considering a two step process and assuming equal preexponential factors for both diffusion steps, the probability of the total diffusion event is given by $\exp(\frac{\unit[2.24]{eV}+\unit[0.90]{eV}}{k_{\text{B}}T})$, which corresponds to a single diffusion barrier that is 3.5 times higher than the barrier obtained by {em ab initio} calculations.
+
+\subsection{Conclusions}
+
+Although classical potential simulations reproduce the same order in energy of the \ci{} \hkl<1 0 0> and \hkl<1 1 0> DB interstitial configurations as obtained by more accurate quantum-mechanical calculations, the obtained migration pathways and resulting activation energies differ to a great extent.
+On the one hand, the most favorable pathways differ.
+On the other hand, the activation energies obtained by classical potential simulations are tremendously overestimated by a factor of 2.4 to 3.5.
Thus, atomic diffusion is wrongly described in the classical potential approach.
The probability of already rare diffusion events is further decreased for this reason.
-Since agglomeration of C and diffusion of Si self-interstitials are an important part of the proposed SiC precipitation mechanism a problem arises, which is formulated and discussed in more detail in section \ref{subsection:md:limit}.
+However, agglomeration of C and diffusion of Si self-interstitials are an important part of the proposed SiC precipitation mechanism.
+Thus, a serious limitation that has to be taken into account for appropriately modeling the C/Si system using the otherwise quite promising EA potential is revealed.
+Possible workarounds are discussed in more detail in section \ref{subsection:md:limit}.
\clearpage{}
-\cleardoublepage{}
\section{Combination of point defects}
-The structural and energetic properties of combinations of point defects are examined in the following.
-Investigations are restricted to quantum-mechanical calculations for two reasons.
-First of all, as mentioned in the last section, they are far more accurate.
-Secondly, the restrictions in size and simulation time for this type of calculation due to limited computational resources, necessitate to map the complex precipitation mechanism to a more compact and simplified modelling.
-The investigations of defect combinations approached in the following are still feasible within the available computational power and allow to draw conclusions on some important ongoing mechanisms during SiC precipitation.
-
-\subsection[Combinations with a C-Si \hkl<1 0 0>-type interstitial]{\boldmath Combinations with a C-Si \hkl<1 0 0>-type interstitial}
-\label{subsection:defects:c-si_comb}
-
-This section focuses on combinations of the \hkl<0 0 -1> dumbbell interstitial with a second defect.
-The second defect is either another \hkl<1 0 0>-type interstitial occupying different orientations, a vacany or a substitutional carbon atom.
-Several distances of the two defects are examined.
-
-\begin{figure}[ht]
+The study proceeds with a structural and energetic investigation of pairs of the ground-state and, thus, most probable defect configurations that are believed to be fundamental in the Si to SiC conversion.
+Investigations are restricted to quantum-mechanical calculations.
+\begin{figure}[t]
\begin{center}
-\begin{minipage}{7.5cm}
-\includegraphics[width=7cm]{comb_pos.eps}
-% ./visualize_contcar -w 640 -h 480 -d results/.../CONTCAR -nll -0.20 -0.20 -0.6 -fur 1.2 1.2 0.6 -c 0.5 -1.5 0.3 -L 0.5 0 0 -r 0.6 -m 3.0 0.0 0.0 0.0 3.0 0.0 0.0 0.0 3.0 -A -1 2.465
-\end{minipage}
-\begin{minipage}{6.0cm}
-\underline{Positions given in $a_{\text{Si}}$}\\[0.3cm]
-Initial interstitial I: $\frac{1}{4}\hkl<1 1 1>$\\
-Relative silicon neighbor positions:
-\begin{enumerate}
- \item $\frac{1}{4}\hkl<1 1 -1>$, $\frac{1}{4}\hkl<-1 -1 -1>$
- \item $\frac{1}{2}\hkl<1 0 1>$, $\frac{1}{2}\hkl<0 1 -1>$,\\[0.2cm]
- $\frac{1}{2}\hkl<0 -1 -1>$, $\frac{1}{2}\hkl<-1 0 -1>$
- \item $\frac{1}{4}\hkl<1 -1 1>$, $\frac{1}{4}\hkl<-1 1 1>$
- \item $\frac{1}{4}\hkl<-1 1 -3>$, $\frac{1}{4}\hkl<1 -1 -3>$
- \item $\frac{1}{2}\hkl<-1 -1 0>$, $\frac{1}{2}\hkl<1 1 0>$
-\end{enumerate}
-\end{minipage}\\
-\begin{picture}(0,0)(190,20)
-\includegraphics[width=2.3cm]{100_arrow.eps}
-\end{picture}
-\begin{picture}(0,0)(220,0)
-\includegraphics[height=2.2cm]{001_arrow.eps}
-\end{picture}
+\subfigure[]{\label{fig:defects:combos_ci}\includegraphics[width=0.3\textwidth]{combos_ci.eps}}
+\hspace{0.5cm}
+\subfigure[]{\label{fig:defects:combos_si}\includegraphics[width=0.3\textwidth]{combos.eps}}
\end{center}
-\caption[\hkl<0 0 -1> dumbbell interstitial defect and positions of next neighbored silicon atoms used for the second defect.]{\hkl<0 0 -1> dumbbell interstitial defect and positions of next neighbored silicon atoms used for the second defect. Two possibilities exist for red numbered atoms and four possibilities exist for blue numbered atoms.}
-\label{fig:defects:pos_of_comb}
+\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.}
+\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.
+Binding energies of the defect pair are determined by equation \ref{eq:basics:e_bind}.
+Next to formation and binding energies, migration barriers are investigated, which allow to draw conclusions on the probability of the formation of such defect complexes by thermally activated diffusion processes.
+
+\subsection[Pairs of \ci{} \hkl<1 0 0>-type interstitials]{\boldmath Pairs of \ci{} \hkl<1 0 0>-type interstitials}
+\label{subsection:defects:c-si_comb}
+
+\ci{} pairs of the \hkl<1 0 0>-type are investigated in the first part.
\begin{table}[ht]
\begin{center}
\begin{tabular}{l c c c c c c}
\hline
& 1 & 2 & 3 & 4 & 5 & R\\
\hline
- \hkl<0 0 -1> & {\color{red}-0.08} & -1.15 & {\color{red}-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> & {\color{orange}-2.39} & -0.17 & {\color{green}-0.10} & {\color{blue}-0.27} & {\color{magenta}-1.88} & {\color{gray}-0.05}\\
- \hkl<0 1 0> & {\color{cyan}-2.25} & -1.90 & {\color{cyan}-2.25} & {\color{purple}-0.12} & {\color{violet}-1.38} & {\color{yellow}-0.06}\\
- \hkl<-1 0 0> & {\color{orange}-2.39} & -0.36 & {\color{cyan}-2.25} & {\color{purple}-0.12} & {\color{magenta}-1.88} & {\color{gray}-0.05}\\
- \hkl<1 0 0> & {\color{cyan}-2.25} & -2.16 & {\color{green}-0.10} & {\color{blue}-0.27} & {\color{violet}-1.38} & {\color{yellow}-0.06}\\
- \hline
- C substitutional (C$_{\text{S}}$) & 0.26 & -0.51 & -0.93 & -0.15 & 0.49 & -0.05\\
- Vacancy & -5.39 ($\rightarrow$ C$_{\text{S}}$) & -0.59 & -3.14 & -0.54 & -0.50 & -0.31\\
+ \hkl[0 0 -1] & {\color{red}-0.08} & -1.15 & {\color{red}-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] & {\color{orange}-2.39} & -0.17 & {\color{green}-0.10} & {\color{blue}-0.27} & {\color{magenta}-1.88} & {\color{gray}-0.05}\\
+ \hkl[0 1 0] & {\color{cyan}-2.25} & -1.90 & {\color{cyan}-2.25} & {\color{purple}-0.12} & {\color{violet}-1.38} & {\color{yellow}-0.06}\\
+ \hkl[-1 0 0] & {\color{orange}-2.39} & -0.36 & {\color{cyan}-2.25} & {\color{purple}-0.12} & {\color{magenta}-1.88} & {\color{gray}-0.05}\\
+ \hkl[1 0 0] & {\color{cyan}-2.25} & -2.16 & {\color{green}-0.10} & {\color{blue}-0.27} & {\color{violet}-1.38} & {\color{yellow}-0.06}\\
+% \hline
+% C substitutional (C$_{\text{S}}$) & 0.26 & -0.51 & -0.93 & -0.15 & 0.49 & -0.05\\
+% Vacancy & -5.39 ($\rightarrow$ C$_{\text{S}}$) & -0.59 & -3.14 & -0.54 & -0.50 & -0.31\\
\hline
\hline
\end{tabular}
\end{center}
-\caption[Energetic results of defect combinations.]{Energetic results of defect combinations. The given energies in eV are defined by equation \eqref{eq:defects:e_of_comb}. Equivalent configurations are marked by identical colors. The first column lists the types of the second defect combined with the initial \hkl<0 0 -1> dumbbell interstitial. The position index of the second defect is given in the first row according to figure \ref{fig:defects:pos_of_comb}. R is the position located at $\frac{a_{\text{Si}}}{2}\hkl<3 2 3>$ relative to the initial defect, which is the maximum realizable distance due to periodic boundary conditions.}
+\caption[Binding energies in eV of \ci{} \hkl<1 0 0>-type defect pairs.]{Binding energies in eV of \ci{} \hkl<1 0 0>-type defect pairs. The given energies in eV are defined by equation \eqref{eq:basics:e_bind}. Equivalent configurations are marked by identical colors. The first column lists the types of the second defect combined with the initial \ci \hkl[0 0 -1] DB interstitial. The position index of the second defect is given in the first row according to Fig.~\ref{fig:defects:combos_ci}. R is the position located at $\frac{a_{\text{Si}}}{2}\hkl[3 2 3]$ relative to the initial defect, which is the maximum realizable distance due to periodic boundary conditions.}
\label{tab:defects:e_of_comb}
\end{table}
-Figure \ref{fig:defects:pos_of_comb} shows the initial \hkl<0 0 -1> dumbbell interstitial defect and the positions of next neighbored silicon atoms used for the second defect.
-Table \ref{tab:defects:e_of_comb} summarizes energetic results obtained after relaxation of the defect combinations.
-The energy of interest $E_{\text{b}}$ is defined to be
-\begin{equation}
-E_{\text{b}}=
-E_{\text{f}}^{\text{defect combination}}-
-E_{\text{f}}^{\text{C \hkl<0 0 -1> dumbbell}}-
-E_{\text{f}}^{\text{2nd defect}}
-\label{eq:defects:e_of_comb}
-\end{equation}
-with $E_{\text{f}}^{\text{defect combination}}$ being the formation energy of the defect combination, $E_{\text{f}}^{\text{C \hkl<0 0 -1> dumbbell}}$ being the formation energy of the C \hkl<0 0 -1> dumbbell interstitial defect and $E_{\text{f}}^{\text{2nd defect}}$ being the formation energy of the second defect.
-For defects far away from each other the formation energy of the defect combination should approximately become the sum of the formation energies of the individual defects without an interaction resulting in $E_{\text{b}}=0$.
-Thus, $E_{\text{b}}$ can be best thought of a binding energy, which is required to bring the defects to infinite separation.
-In fact, a \hkl<0 0 -1> dumbbell interstitial created at position R with a distance of $\frac{a_{\text{Si}}}{2}\hkl<3 2 3>$ ($\approx 12.8$ \AA) from the initial one results in an energy as low as -0.19 eV.
-There is still a low interaction which is due to the equal orientation of the defects.
-By changing the orientation of the second dumbbell interstitial to the \hkl<0 -1 0>-type the interaction is even more reduced resulting in an energy of $E_{\text{b}}=-0.05\text{ eV}$ for a distance, which is the maximum that can be realized due to periodic boundary conditions.
-The energies obtained in the R column of table \ref{eq:defects:e_of_comb} are used as a reference to identify, whether less distanced defects of the same type are favorable or unfavorable compared to the far-off located defect.
-Configurations wih energies greater than zero or the reference value are energetically unfavorable and expose a repulsive interaction.
-These configurations are unlikely to arise or to persist for non-zero temperatures.
-Energies below zero and below the reference value indicate configurations favored compared to configurations in which these point defects are separated far away from each other.
-
-Investigating the first part of table \ref{tab:defects:e_of_comb}, namely the combinations with another \hkl<1 0 0>-type interstitial, most of the combinations result in energies below zero.
-Surprisingly the most favorable configurations are the ones with the second defect created at the very next silicon neighbor and a change in orientation compared to the initial one.
-This leads to the conclusion that an agglomeration of C-Si dumbbell interstitials as proposed by the precipitation model introduced in section \ref{section:assumed_prec} is indeed an energetically favored configuration of the system.
-The reason for nearby interstitials being favored compared to isolated ones is most probably the reduction of strain energy enabled by combination in contrast to the strain energy created by two individual defects.
+Table~\ref{tab:defects:e_of_comb} summarizes resulting binding energies for the combination with a second \ci{} \hkl<1 0 0> DB obtained for different orientations at positions 1 to 5 after structural relaxation.
+Most of the obtained configurations result in binding energies well below zero indicating a preferable agglomeration of this type of the defects.
+For increasing distances of the defect pair, the binding energy approaches to zero as it is expected for non-interacting isolated defects.
+%
+In fact, a \ci{} \hkl[0 0 -1] DB interstitial created at position R separated by a distance of $\frac{a_{\text{Si}}}{2}\hkl<3 2 3>$ ($\approx$\unit[12.8]{\AA}) from the initial one results in an energy as low as \unit[-0.19]{eV}.
+There is still a low interaction remaining, which is due to the equal orientation of the defects.
+By changing the orientation of the second DB interstitial to the \hkl<0 -1 0>-type, the interaction is even more reduced resulting in an energy of \unit[-0.05]{eV} for a distance, which is the maximum that can be realized due to periodic boundary conditions.
+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.
+
\begin{figure}[ht]
\begin{center}
-\begin{minipage}[t]{7cm}
-a) \underline{$E_{\text{b}}=-2.25\text{ eV}$}
-\begin{center}
-\includegraphics[width=6cm]{00-1dc/2-25.eps}
-\end{center}
-\end{minipage}
-\begin{minipage}[t]{7cm}
-b) \underline{$E_{\text{b}}=-2.39\text{ eV}$}
-\begin{center}
-\includegraphics[width=6cm]{00-1dc/2-39.eps}
+\subfigure[\underline{$E_{\text{b}}=-2.25\,\text{eV}$}]{\label{fig:defects:225}\includegraphics[width=0.3\textwidth]{00-1dc/2-25.eps}}
+\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}
-\end{minipage}
-\end{center}
-\caption{Relaxed structures of defect complexes obtained by creating a) \hkl<1 0 0> and b) \hkl<0 -1 0> dumbbels at position 1.}
+\caption{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}
-Figure \ref{fig:defects:comb_db_01} shows the structure of these two configurations.
-The displayed configurations are realized by creating a \hkl<1 0 0> (a)) and \hkl<0 -1 0> (b)) dumbbell at position 1.
-Structure \ref{fig:defects:comb_db_01} b) is the energetically most favorable configuration.
-After relaxation the initial configuration is still evident.
-As expected by the initialization conditions the two carbon atoms form a bond.
-This bond has a length of 1.38 \AA{} close to the nex neighbor distance in diamond or graphite, which is approximately 1.54 \AA.
-The minimum of binding energy observed for this configuration suggests prefered C clustering as a competing mechnism to the C-Si dumbbell interstitial agglomeration inevitable for the SiC precipitation.
-{\color{red}Todo: Activation energies to obtain separated C confs FAILED (again?) - could be added in the combined defect migration chapter and mentioned here, too!}
-However, for the second most favorable configuration, presented in figure \ref{fig:defects:comb_db_01} a), the amount of possibilities for this configuration is twice as high.
-In this configuration the initial Si (I) and C (I) dumbbell atoms are displaced along \hkl<1 0 0> and \hkl<-1 0 0> in such a way that the Si atom is forming tetrahedral bonds with two silicon and two carbon atoms.
-The carbon and silicon atom constituting the second defect are as well displaced in such a way, that the carbon atom forms tetrahedral bonds with four silicon neighbors, a configuration expected in silicon carbide.
-The two carbon atoms spaced by 2.70 \AA{} do not form a bond but anyhow reside in a shorter distance as expected in silicon carbide.
-The Si atom numbered 2 is pushed towards the carbon atom, which results in the breaking of the bond to atom 4.
-The breaking of the $\sigma$ bond is indeed confirmed by investigating the charge density isosurface of this configuration.
-{\color{red}Todo: Is this conf really benificial for SiC prec?}
+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}.
+In the present study, a further relaxation of this defect structure is observed.
+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}.
+The corresponding defect structure is displayed in Fig.~\ref{fig:defects:225}.
+In this configuration the initial Si and C DB atoms are displaced along \hkl[1 0 0] and \hkl[-1 0 0] in such a way that the Si atom is forming tetrahedral bonds with two Si and two C atoms.
+The C and Si atom constituting the second defect are as well displaced in such a way, that the C atom forms tetrahedral bonds with four Si neighbors, a configuration expected in SiC.
+The two carbon atoms, which are spaced by \unit[2.70]{\AA}, do not form a bond but anyhow reside in a shorter distance than expected in SiC.
+Si atom number 2 is pushed towards the C atom, which results in the breaking of the bond to Si atom number 4.
+Breaking of the $\sigma$ bond is indeed confirmed by investigating the charge density isosurface of this configuration.
+
+Apart from that, a more favorable configuration is 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 \ci{} DBs in Si.
+The atomic arrangement is shown in Fig.~\ref{fig:defects:239}.
+The initial configuration is still evident in the relaxed configuration.
+The two \ci{} 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}.
+This bond has a length of \unit[1.38]{\AA} close to the next neighbor distance in diamond or graphite, which is approximately \unit[1.54]{\AA}.
+The minimum of the binding energy observed for this configuration suggests prefered C clustering as a competing mechnism to the \ci{} DB interstitial agglomeration inevitable for the SiC precipitation.
+However, the second most favorable configuration ($E_{\text{f}}=-2.25\,\text{eV}$) is represented four times, i.e. two times more often than the ground-state configuration, within the systematically investigated configuration space.
+Thus, particularly at high temepratures that cause an increase of the entropic contribution, this structure constitutes a serious opponent to the ground state.
+In fact, following results on migration simulations will reinforce the assumption of a low probability for C clustering by thermally activated processes.
\begin{figure}[ht]
\begin{center}
-\begin{minipage}[t]{5cm}
-a) \underline{$E_{\text{b}}=-2.16\text{ eV}$}
-\begin{center}
-\includegraphics[width=4.8cm]{00-1dc/2-16.eps}
+\subfigure[\underline{$E_{\text{b}}=-2.16\,\text{eV}$}]{\label{fig:defects:216}\includegraphics[width=0.25\textwidth]{00-1dc/2-16.eps}}
+\hspace{0.2cm}
+\subfigure[\underline{$E_{\text{b}}=-1.90\,\text{eV}$}]{\label{fig:defects:190}\includegraphics[width=0.25\textwidth]{00-1dc/1-90.eps}}
+\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}
-\end{minipage}
-\begin{minipage}[t]{5cm}
-b) \underline{$E_{\text{b}}=-1.90\text{ eV}$}
-\begin{center}
-\includegraphics[width=4.8cm]{00-1dc/1-90.eps}
-\end{center}
-\end{minipage}
-\begin{minipage}[t]{5cm}
-c) \underline{$E_{\text{b}}=-2.05\text{ eV}$}
-\begin{center}
-\includegraphics[width=4.8cm]{00-1dc/2-05.eps}
-\end{center}
-\end{minipage}
-\end{center}
-\caption{Relaxed structures of defect complexes obtained by creating a a) \hkl<1 0 0> and b) \hkl<0 1 0> dumbbell at position 2 and a c) \hkl<0 0 1> dumbbel at position 3.}
+\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.}
\label{fig:defects:comb_db_02}
\end{figure}
-Figure \ref{fig:defects:comb_db_02} shows the next three most energetically favorable configurations.
-The relaxed configuration obtained by creating a second \hkl<1 0 0> dumbbell at position 2 is shown in figure \ref{fig:defects:comb_db_02} a).
-A binding energy of -2.16 eV is observed.
-After relaxation the second dumbbell is aligned along \hkl<1 1 0>.
-The bond of the silicon atoms 1 and 2 does not persist.
-Instead the silicon atom forms a bond with the initial carbon interstitial and the second carbon atom forms a bond with silicon atom 1 forming four bonds in total.
-The carbon atoms are spaced by 3.14 \AA, which is very close to the expected C-C next neighbor distance of 3.08 \AA{} in silicon carbide.
-Figure \ref{fig:defects:comb_db_02} c) displays the results of another \hkl<0 0 1> dumbbell inserted at position 3.
-The binding energy is -2.05 eV.
-Both dumbbells are tilted along the same direction remaining parallely aligned and the second dumbbell is pushed downwards in such a way, that the four dumbbell atoms form a rhomboid.
-Both carbon atoms form tetrahedral bonds to four silicon atoms.
-However, silicon atom 1 and 3, which are bound to the second carbon dumbbell interstitial are also bound to the initial carbon atom.
-These four atoms of the rhomboid reside in a plane and, thus, do not match the situation in silicon carbide.
-The carbon atoms have a distance of 2.75 \AA.
-In figure \ref{fig:defects:comb_db_02} b) a second \hkl<0 1 0> dumbbell is constructed at position 2.
-An energy of -1.90 eV is observed.
-The initial dumbbell and especially the carbon atom is pushed towards the silicon atom of the second dumbbell forming an additional fourth bond.
-Silicon atom number 1 is pulled towards the carbon atoms of the dumbbells accompanied by the disappearance of its bond to silicon number 5 as well as the bond of silicon number 5 to its next neighbored silicon atom in \hkl<1 1 -1> direction.
-The carbon atom of the second dumbbell forms threefold coordinated bonds to its silicon neighbors.
-A distance of 2.80 \AA{} is observed for the two carbon atoms.
-Again, the two carbon atoms and its two interconnecting silicon atoms form a rhomboid.
-C-C distances of 2.70 to 2.80 \AA{} seem to be characteristic for such configurations, in which the carbon atoms and the two interconnecting silicon atoms reside in a plane.
-
-Configurations obtained by adding a second dumbbell interstitial at position 4 are characterized by minimal changes from their initial creation condition during relaxation.
-There is a low interaction of the dumbbells, which seem to exist independent of each other.
-This, on the one hand, becomes evident by investigating the final structure, in which both of the dumbbells essentially retain the structure expected for a single dumbbell and on the other hand is supported by the observed binding energies which vary only slightly around zero.
-This low interaction is due to the larger distance and a missing direct connection by bonds along a crystallographic direction.
-Both carbon and silicon atoms of the dumbbells form threefold coordinated bonds to their next neighbors.
-The energetically most unfavorable configuration ($E_{\text{b}}=0.26\text{ eV}$) is obtained for the \hkl<0 0 1> interstitial oppositely orientated to the initial one.
-A dumbbell taking the same orientation as the initial one is less unfavorble ($E_{\text{b}}=0.04\text{ eV}$).
-Both configurations are unfavorable compared to far-off isolated dumbbells.
-Nonparallel orientations, that is the \hkl<0 1 0>, \hkl<0 -1 0> and its equivalents, result in binding energies of -0.12 eV and -0.27 eV, thus, constituting energetically favorable configurations.
-The reduction of strain energy is higher in the second case where the carbon atom of the second dumbbell is placed in the direction pointing away from the initial carbon atom.
+Fig.~\ref{fig:defects:comb_db_02} shows the next three energetically favorable configurations.
+The relaxed configuration obtained by creating a \hkl[1 0 0] DB at position 2 is shown in Fig. \ref{fig:defects:216}.
+A binding energy of \unit[-2.16]{eV} is observed.
+After relaxation, the second DB is aligned along \hkl[1 1 0].
+The bond of Si atoms 1 and 2 does not persist.
+Instead, the Si atom forms a bond with the initial \ci{} and the second C atom forms a bond with Si atom 1 forming four bonds in total.
+The C atoms are spaced by \unit[3.14]{\AA}, which is very close to the expected C-C next neighbor distance of \unit[3.08]{\AA} in SiC.
+Figure \ref{fig:defects:205} displays the results of a \hkl[0 0 1] DB inserted at position 3.
+The binding energy is \unit[-2.05]{eV}.
+Both DBs are tilted along the same direction remaining parallely aligned and the second DB is pushed downwards in such a way, that the four DB atoms form a rhomboid.
+Both C atoms form tetrahedral bonds to four Si atoms.
+However, Si atom number 1 and number 3, which are bound to the second \ci{} atom are also bound to the initial C atom.
+These four atoms of the rhomboid reside in a plane and, thus, do not match the situation in SiC.
+The Carbon atoms have a distance of \unit[2.75]{\AA}.
+In Fig. \ref{fig:defects:190} the relaxed structure of a \hkl[0 1 0] DB constructed at position 2 is displayed.
+An energy of \unit[-1.90]{eV} is observed.
+The initial DB and especially the C atom is pushed towards the Si atom of the second DB forming an additional fourth bond.
+Si atom number 1 is pulled towards the C atoms of the DBs accompanied by the disappearance of its bond to Si number 5 as well as the bond of Si number 5 to its neighbored Si atom in \hkl[1 1 -1] direction.
+The C atom of the second DB forms threefold coordinated bonds to its Si neighbors.
+A distance of \unit[2.80]{\AA} is observed for the two C atoms.
+Again, the two C atoms and its two interconnecting Si atoms form a rhomboid.
+C-C distances of \unit[2.70-2.80]{\AA} seem to be characteristic for such configurations, in which the C atoms and the two interconnecting Si atoms reside in a plane.
+
+Configurations obtained by adding a \ci{} \hkl<1 0 0> DB at position 4 are characterized by minimal changes from their initial creation condition during relaxation.
+There is a low interaction of the DBs, which seem to exist independent of each other.
+This, on the one hand, becomes evident by investigating the final structure, in which both of the DBs essentially retain the structure expected for a single DB and, on the other hand, is supported by the observed binding energies, which vary only slightly around zero.
+This low interaction is due to the large distance and a missing direct connection by bonds along a chain in the crystallographic \hkl<1 1 0> direction.
+Both, C and Si atoms of the DBs form threefold coordinated bonds to their neighbors.
+The energetically most unfavorable configuration ($E_{\text{b}}=0.26\,\text{eV}$) is obtained for the \ci{} \hkl[0 0 1] DB, which is oppositely orientated with respect to the initial one.
+A DB taking the same orientation as the initial one is less unfavorble ($E_{\text{b}}=0.04\,\text{eV}$).
+Both configurations are unfavorable compared to far-off, isolated DBs.
+Nonparallel orientations, i.e. the \hkl[0 1 0], \hkl[0 -1 0] and its equivalents, result in binding energies of \unit[-0.12]{eV} and \unit[-0.27]{eV}, thus, constituting energetically favorable configurations.
+The reduction of strain energy is higher in the second case, where the C atom of the second DB is placed in the direction pointing away from the initial C atom.
\begin{figure}[ht]
\begin{center}
-\begin{minipage}[t]{7cm}
-a) \underline{$E_{\text{b}}=-1.53\text{ eV}$}
-\begin{center}
-\includegraphics[width=6.0cm]{00-1dc/1-53.eps}
-\end{center}
-\end{minipage}
-\begin{minipage}[t]{7cm}
-b) \underline{$E_{\text{b}}=-1.66\text{ eV}$}
-\begin{center}
-\includegraphics[width=6.0cm]{00-1dc/1-66.eps}
-\end{center}
-\end{minipage}\\[0.2cm]
-\begin{minipage}[t]{7cm}
-c) \underline{$E_{\text{b}}=-1.88\text{ eV}$}
-\begin{center}
-\includegraphics[width=6.0cm]{00-1dc/1-88.eps}
+\subfigure[\underline{$E_{\text{b}}=-1.53\,\text{eV}$}]{\label{fig:defects:153}\includegraphics[width=0.25\textwidth]{00-1dc/1-53.eps}}
+\hspace{0.7cm}
+\subfigure[\underline{$E_{\text{b}}=-1.66\,\text{eV}$}]{\label{fig:defects:166}\includegraphics[width=0.25\textwidth]{00-1dc/1-66.eps}}\\
+\subfigure[\underline{$E_{\text{b}}=-1.88\,\text{eV}$}]{\label{fig:defects:188}\includegraphics[width=0.25\textwidth]{00-1dc/1-88.eps}}
+\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}
-\end{minipage}
-\begin{minipage}[t]{7cm}
-d) \underline{$E_{\text{b}}=-1.38\text{ eV}$}
-\begin{center}
-\includegraphics[width=6.0cm]{00-1dc/1-38.eps}
-\end{center}
-\end{minipage}
-\end{center}
-\caption{Relaxed structures of defect complexes obtained by creating a a) \hkl<0 0 1>, a b) \hkl<0 0 -1>, a c) \hkl<0 -1 0> and a d) \hkl<1 0 0> dumbbell at position 5.}
+\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.}
\label{fig:defects:comb_db_03}
\end{figure}
-Energetically beneficial configurations of defect complexes are observed for second interstititals of all orientations placed at position 5, a position two bonds away from the initial interstitial along the \hkl<1 1 0> direction.
-Relaxed structures of these complexes are displayed in figure \ref{fig:defects:comb_db_03}.
-Figure \ref{fig:defects:comb_db_03} a) and b) show the relaxed structures of \hkl<0 0 1> and \hkl<0 0 -1> dumbbells.
-The upper dumbbell atoms are pushed towards each other forming fourfold coordinated bonds.
-While the displacements of the silicon atoms in case b) are symmetric to the \hkl(1 1 0) plane, in case a) the silicon atom of the initial dumbbel is pushed a little further in the direction of the carbon atom of the second dumbbell than the carbon atom is pushed towards the silicon atom.
-The bottom atoms of the dumbbells remain in threefold coordination.
-The symmetric configuration is energetically more favorable ($E_{\text{b}}=-1.66\text{ eV}$) since the displacements of the atoms is less than in the antiparallel case ($E_{\text{b}}=-1.53\text{ eV}$).
-In figure \ref{fig:defects:comb_db_03} c) and d) the nonparallel orientations, namely the \hkl<0 -1 0> and \hkl<1 0 0> dumbbells are shown.
-Binding energies of -1.88 eV and -1.38 eV are obtained for the relaxed structures.
-In both cases the silicon atom of the initial interstitial is pulled towards the near by atom of the second dumbbell so that both atoms form fourfold coordinated bonds to their next neighbors.
-In case c) it is the carbon and in case d) the silicon atom of the second interstitial which forms the additional bond with the silicon atom of the initial interstitial.
-The atom of the second dumbbell, the carbon atom of the initial dumbbell and the two interconnecting silicon atoms again reside in a plane.
-A typical C-C distance of 2.79 \AA{} is, thus, observed for case c).
-The far-off atom of the second dumbbell resides in threefold coordination.
-
-Assuming that it is possible for the system to minimize free energy by an in place reorientation of the dumbbell at any position the minimum energy orientation of dumbbells along the \hkl<1 1 0> direction and the resulting C-C distance is shown in table \ref{tab:defects:comb_db110}.
+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.
+Relaxed structures of these combinations are displayed in Fig. \ref{fig:defects:comb_db_03}.
+Fig. \ref{fig:defects:153} and \ref{fig:defects:166} show the relaxed structures of \hkl[0 0 1] and \hkl[0 0 -1] DBs.
+The upper DB atoms are pushed towards each other forming fourfold coordinated bonds.
+While the displacements of the Si atoms in case (b) are symmetric to the \hkl(1 1 0) plane, in case (a) the Si atom of the initial DB is pushed a little further in the direction of the C atom of the second DB than the C atom is pushed towards the Si atom.
+The bottom atoms of the DBs remain in threefold coordination.
+The symmetric configuration is energetically more favorable ($E_{\text{b}}=-1.66\,\text{eV}$) since the displacements of the atoms is less than in the antiparallel case ($E_{\text{b}}=-1.53\,\text{eV}$).
+In Fig. \ref{fig:defects:188} and \ref{fig:defects:138} the non-parallel orientations, namely the \hkl[0 -1 0] and \hkl[1 0 0] DBs, are shown.
+Binding energies of \unit[-1.88]{eV} and \unit[-1.38]{eV} are obtained for the relaxed structures.
+In both cases the Si atom of the initial interstitial is pulled towards the near by atom of the second DB.
+Both atoms form fourfold coordinated bonds to their neighbors.
+In case (c) it is the C and in case (d) the Si atom of the second interstitial, which forms the additional bond with the Si atom of the initial interstitial.
+The respective atom of the second DB, the \ci{} atom of the initial DB and the two interconnecting Si atoms again reside in a plane.
+As observed before, a typical C-C distance of \unit[2.79]{\AA} is, thus, observed for case (c).
+In both configurations, the far-off atom of the second DB resides in threefold coordination.
+
+The interaction of \ci{} \hkl<1 0 0> DBs is investigated along the \hkl[1 1 0] bond chain assuming a possible reorientation of the DB atom at each position to minimize its configurational energy.
+Therefore, the binding energies of the energetically most favorable configurations with the second DB located along the \hkl[1 1 0] direction and resulting C-C distances of the relaxed structures are summarized in Table~\ref{tab:defects:comb_db110}.
\begin{table}[ht]
\begin{center}
\begin{tabular}{l c c c c c c}
\hline
$E_{\text{b}}$ [eV] & -2.39 & -1.88 & -0.59 & -0.31 & -0.24 & -0.21 \\
C-C distance [\AA] & 1.4 & 4.6 & 6.5 & 8.6 & 10.5 & 10.8 \\
-Type & \hkl<-1 0 0> & \hkl<1 0 0> & \hkl<1 0 0> & \hkl<1 0 0> & \hkl<1 0 0> & \hkl<1 0 0>, \hkl<0 -1 0>\\
+Type & \hkl[-1 0 0] & \hkl[1 0 0] & \hkl[1 0 0] & \hkl[1 0 0] & \hkl[1 0 0] & \hkl[1 0 0], \hkl[0 -1 0]\\
\hline
\hline
\end{tabular}
\end{center}
-\caption{Binding energy and type of the minimum energy configuration of an additional dumbbell with respect to the separation distance in bonds along the \hkl<1 1 0> direction and the C-C distance.}
+\caption{Binding energies $E_{\text{b}}$, C-C distance and types of energetically most favorable \ci{} \hkl<1 0 0>-type defect pairs separated along the \hkl[1 1 0] bond chain.}
\label{tab:defects:comb_db110}
\end{table}
+%
\begin{figure}[ht]
\begin{center}
-\includegraphics[width=12.5cm]{db_along_110.ps}\\
-\includegraphics[width=12.5cm]{db_along_110_cc.ps}
+\includegraphics[width=0.7\textwidth]{db_along_110_cc_n.ps}
\end{center}
-\caption{Minimum binding energy of dumbbell combinations with respect to the separation distance in bonds along \hkl<1 1 0> and C-C distance.}
+\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}
-Figure \ref{fig:defects:comb_db110} shows the corresponding plot of the data including a cubic spline interplation and a suitable fitting curve.
-The funtion found most suitable for curve fitting is $f(x)=a/x^3$ comprising the single fit parameter $a$.
-Thus, far-off located dumbbells show an interaction proportional to the reciprocal cube of the distance and the amount of bonds along \hkl<1 1 0> respectively.
-This behavior is no longer valid for the immediate vicinity revealed by the saturating binding energy of a second dumbbell at position 1, which is ignored in the fitting procedure.
-
+The binding energy of these configurations with respect to the C-C distance is plotted in Fig.~\ref{fig:defects:comb_db110}.
+The interaction is found to be proportional to the reciprocal cube of the C-C distance for extended separations of the \ci{} DBs and saturates for the smallest possible separation, i.e. the ground-state configuration.
+The ground-state configuration was ignored in the fitting process.
+Not considering the previously mentioned elevated barriers for migration, an attractive interaction between the \ci{} \hkl<1 0 0> DB defects indeed is detected with a capture radius that clearly exceeds \unit[1]{nm}.
+The interpolated graph suggests the disappearance of attractive interaction forces, which are proportional to the slope of the graph, in between the two lowest separation distances of the defects.
+This finding, in turn, supports the previously established assumption of C agglomeration and absence of C clustering.
+
+\subsection{Diffusion processes among configurations of \ci{} pairs}
+
+Based on the lowest energy migration path of a single \ci{} \hkl<1 0 0> DB, the configuration, in which the second \ci{} 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} is 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 are only 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}[ht]
\begin{center}
-\begin{minipage}[t]{5cm}
-a) \underline{Pos: 1, $E_{\text{b}}=0.26\text{ eV}$}
-\begin{center}
-\includegraphics[width=4.8cm]{00-1dc/0-26.eps}
+\includegraphics[width=0.7\textwidth]{036-239.ps}
\end{center}
-\end{minipage}
-\begin{minipage}[t]{5cm}
-b) \underline{Pos: 3, $E_{\text{b}}=-0.93\text{ eV}$}
+\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}
+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}.
+\begin{figure}[ht]
\begin{center}
-\includegraphics[width=4.8cm]{00-1dc/0-93.eps}
+\includegraphics[width=0.7\textwidth]{188-225.ps}
\end{center}
-\end{minipage}
-\begin{minipage}[t]{5cm}
-c) \underline{Pos: 5, $E_{\text{b}}=0.49\text{ eV}$}
+\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.}
+\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.
+The latter is considered very important at high temperatures, accompanied by an increase in the entropic contribution to structure formation.
+As a result, C defect agglomeration indeed is expected, but only a low probability is assumed for C-C clustering by thermally activated processes with regard to the considered process time in IBS.
+
+\subsection[Combinations of the \ci{} \hkl<1 0 0> and \cs{} type]{\boldmath Combinations of the \ci{} \hkl<1 0 0> and \cs{} type}
+\label{subsection:defects:c-csub}
+
+\begin{table}[ht]
\begin{center}
-\includegraphics[width=4.8cm]{00-1dc/0-49.eps}
-\end{center}
-\end{minipage}
+\begin{tabular}{c c c c c c}
+\hline
+\hline
+1 & 2 & 3 & 4 & 5 & R \\
+\hline
+0.26$^a$/-1.28$^b$ & -0.51 & -0.93$^A$/-0.95$^B$ & -0.15 & 0.49
+ & -0.05\\
+\hline
+\hline
+\end{tabular}
\end{center}
-\caption{Relaxed structures of defect complexes obtained by creating a carbon substitutional at position 1 (a)), 3 (b)) and 5 (c)).}
-\label{fig:defects:comb_db_04}
-\end{figure}
+\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.}
+\label{tab:defects:c-s}
+\end{table}
+%\begin{figure}[ht]
+%\begin{center}
+%\begin{minipage}[t]{5cm}
+%a) \underline{Pos: 1, $E_{\text{b}}=0.26\text{ eV}$}
+%\begin{center}
+%\includegraphics[width=4.8cm]{00-1dc/0-26.eps}
+%\end{center}
+%\end{minipage}
+%\begin{minipage}[t]{5cm}
+%b) \underline{Pos: 3, $E_{\text{b}}=-0.93\text{ eV}$}
+%\begin{center}
+%\includegraphics[width=4.8cm]{00-1dc/0-93.eps}
+%\end{center}
+%\end{minipage}
+%\begin{minipage}[t]{5cm}
+%c) \underline{Pos: 5, $E_{\text{b}}=0.49\text{ eV}$}
+%\begin{center}
+%\includegraphics[width=4.8cm]{00-1dc/0-49.eps}
+%\end{center}
+%\end{minipage}
+%\end{center}
+%\caption{Relaxed structures of defect complexes obtained by creating a carbon substitutional at position 1 (a)), 3 (b)) and 5 (c)).}
+%\label{fig:defects:comb_db_04}
+%\end{figure}
+%\begin{figure}[ht]
+%\begin{center}
+%\begin{minipage}[t]{7cm}
+%a) \underline{Pos: 2, $E_{\text{b}}=-0.51\text{ eV}$}
+%\begin{center}
+%\includegraphics[width=6cm]{00-1dc/0-51.eps}
+%\end{center}
+%\end{minipage}
+%\begin{minipage}[t]{7cm}
+%b) \underline{Pos: 4, $E_{\text{b}}=-0.15\text{ eV}$}
+%\begin{center}
+%\includegraphics[width=6cm]{00-1dc/0-15.eps}
+%\end{center}
+%\end{minipage}
+%\end{center}
+%\caption{Relaxed structures of defect complexes obtained by creating a carbon substitutional at position 2 (a)) and 4 (b)).}
+%\label{fig:defects:comb_db_05}
+%\end{figure}
+%
+Table~\ref{tab:defects:c-s} lists the energetic results of \cs{} combinations with the initial \ci{} \hkl[0 0 -1] DB.
+For \cs{} located at position 1 and 3, the configurations a and A correspond to the naive relaxation of the structure by substituting the Si atom by a C atom in the initial \ci{} \hkl[0 0 -1] DB structure at positions 1 and 3 respectively.
+However, small displacements of the involved atoms near the defect result in different stable structures labeled b and B respectively.
+Fig.~\ref{fig:093-095} and \ref{fig:026-128} show structures A, B and a, b together with the barrier of migration for the A to B and a to b transition respectively.
+
+% A B
+%./visualize_contcar -w 640 -h 480 -d results/c_00-1_c3_csub_B -nll -0.20 -0.4 -0.1 -fur 0.9 0.6 0.9 -c 0.5 -1.5 0.375 -L 0.5 0 0.3 -r 0.6 -A -1 2.465
\begin{figure}[ht]
\begin{center}
-\begin{minipage}[t]{7cm}
-a) \underline{Pos: 2, $E_{\text{b}}=-0.51\text{ eV}$}
-\begin{center}
-\includegraphics[width=6cm]{00-1dc/0-51.eps}
+\includegraphics[width=0.7\textwidth]{093-095.ps}
\end{center}
-\end{minipage}
-\begin{minipage}[t]{7cm}
-b) \underline{Pos: 4, $E_{\text{b}}=-0.15\text{ eV}$}
+\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.}
+\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.
+By a single bond switch, i.e. the breaking of a Si-Si in favor of a Si-C bond, configuration B is obtained, which shows a twofold coordinated Si atom located in between two substitutional C atoms residing on regular Si lattice sites.
+This configuration has been identified and described by spectroscopic experimental techniques \cite{song90_2} as well as theoretical studies \cite{leary97,capaz98}.
+Configuration B is found to constitute the energetically slightly more favorable configuration.
+However, the gain in energy due to the significantly lower energy of a Si-C compared to a Si-Si bond turns out to be smaller than expected due to a large compensation by introduced strain as a result of the Si interstitial structure.
+Present results show a difference in energy of states A and B, which exactly matches the experimental value of \unit[0.02]{eV} \cite{song90_2}, reinforcing qualitatively correct results of previous theoretical studies on these structures.
+% mattoni: A favored by 0.4 eV - NO, it is indeed B (reinforce Song and Capaz)!
+%
+% AB transition
+The migration barrier ss identified to be \unit[0.44]{eV}, almost three times higher than the experimental value of \unit[0.16]{eV} \cite{song90_2} estimated for the neutral charge state transition in p- and n-type Si.
+Keeping in mind the formidable agreement of the energy difference with experiment, the overestimated activation energy is quite unexpected.
+Obviously, either the CRT algorithm fails to seize the actual saddle point structure or the influence of dopants has exceptional effect in the experimentally covered diffusion process being responsible for the low migration barrier.
+% not satisfactory!
+
+% a b
+\begin{figure}[ht]
\begin{center}
-\includegraphics[width=6cm]{00-1dc/0-15.eps}
+\includegraphics[width=0.7\textwidth]{026-128.ps}
\end{center}
-\end{minipage}
-\end{center}
-\caption{Relaxed structures of defect complexes obtained by creating a carbon substitutional at position 2 (a)) and 4 (b)).}
-\label{fig:defects:comb_db_05}
+\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.}
+\label{fig:026-128}
\end{figure}
-The second part of table \ref{tab:defects:e_of_comb} lists the energetic results of substitutional carbon and vacancy combinations with the initial \hkl<0 0 -1> dumbbell.
-Figures \ref{fig:defects:comb_db_04} and \ref{fig:defects:comb_db_05} show relaxed structures of substitutional carbon in combination with the \hkl<0 0 -1> dumbbell for several positions.
-In figure \ref{fig:defects:comb_db_04} positions 1 (a)), 3 (b)) and 5 (c)) are displayed.
-A substituted carbon atom at position 5 results in an energetically extremely unfavorable configuration.
-Both carbon atoms, the substitutional and the dumbbell atom, pull silicon atom number 1 towards their own location regarding the \hkl<1 1 0> direction.
-Due to this a large amount of tensile strain energy is needed, which explains the high positive value of 0.49 eV.
-The lowest binding energy is observed for a substitutional carbon atom inserted at position 3.
-The substitutional carbon atom is located above the dumbbell substituting a silicon atom which would usually be bound to and displaced along \hkl<0 0 1> and \hkl<1 1 0> by the silicon dumbbell atom.
-In contrast to the previous configuration strain compensation occurs resulting in a binding energy as low as -0.93 eV.
-Substitutional carbon at position 2 and 4, visualized in figure \ref{fig:defects:comb_db_05}, is located below the initial dumbbell.
-Silicon atom number 1, which is bound to the interstitial carbon atom is displaced along \hkl<0 0 -1> and \hkl<-1 -1 0>.
-In case a) only the first displacement is compensated by the substitutional carbon atom.
-This results in a somewhat higher binding energy of -0.51 eV.
-The binding energy gets even higher in case b) ($E_{\text{b}}=-0.15\text{ eV}$), in which the substitutional carbon is located further away from the initial dumbbell.
-In both cases, silicon atom number 1 is displaced in such a way, that the bond to silicon atom number 5 vanishes.
-In case of \ref{fig:defects:comb_db_04} a) the carbon atoms form a bond with a distance of 1.5 \AA, which is close to the C-C distance expected in diamond or graphit.
-Both carbon atoms are highly attracted by each other resulting in large displacements and high strain energy in the surrounding.
-A binding energy of 0.26 eV is observed.
-Substitutional carbon at positions 2, 3 and 4 are the energetically most favorable configurations and constitute promising starting points for SiC precipitation.
-On the one hand, C-C distances around 3.1 \AA{} exist for substitution positions 2 and 3, which are close to the C-C distance expected in silicon carbide.
-On the other hand stretched silicon carbide is obtained by the transition of the silicon dumbbell atom into a silicon self-interstitial located somewhere in the silicon host matrix and the transition of the carbon dumbbell atom into another substitutional atom occupying the dumbbell lattice site.
-
+Configuration a 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.
+Nevertheless, the C and Si DB atoms remain threefold coordinated.
+Although the C-C bond exhibiting a distance of \unit[0.15]{nm} close to the distance expected in diamond or graphite should lead to a huge gain in energy, a repulsive interaction with a binding energy of \unit[0.26]{eV} is observed due to compressive strain of the Si DB atom and its top neighbors (\unit[0.230]{nm}/\unit[0.236]{nm}) along with additional tensile strain of the C$_{\text{s}}$ and its three neighboring Si atoms (\unit[0.198-0.209]{nm}/\unit[0.189]{nm}).
+Again a single bond switch, i.e. the breaking of the bond of the Si atom bound to the fourfold coordinated C$_{\text{s}}$ atom and the formation of a double bond between the two C atoms, results in configuration b.
+The two C atoms form a \hkl[1 0 0] DB sharing the initial C$_{\text{s}}$ lattice site while the initial Si DB atom occupies its previously regular lattice site.
+The transition is accompanied by a large gain in energy as can be seen in Fig.~\ref{fig:026-128}, making it the ground-state configuration of a C$_{\text{s}}$ and C$_{\text{i}}$ DB in Si yet \unit[0.33]{eV} lower in energy than configuration B.
+This finding is in good agreement with a combined ab initio and experimental study of Liu et~al.~\cite{liu02}, who first proposed this structure as the ground state identifying an energy difference compared to configuration B of \unit[0.2]{eV}.
+% mattoni: A favored by 0.2 eV - NO! (again, missing spin polarization?)
+A net magnetization of two spin up electrons, which are equally localized as in the Si$_{\text{i}}$ \hkl<1 0 0> DB structure is observed.
+In fact, these two configurations are very similar and are qualitatively different from the C$_{\text{i}}$ \hkl<1 0 0> DB that does not show magnetization but a nearly collinear bond of the C DB atom to its two neighbored Si atoms while the Si DB atom approximates \unit[120]{$^{\circ}$} angles in between its bonds.
+Configurations a, A and B are not affected by spin polarization and show zero magnetization.
+Mattoni et~al.~\cite{mattoni2002}, in contrast, find configuration b less favorable than configuration A by \unit[0.2]{eV}.
+Next to differences in the XC functional and plane-wave energy cut-off, this discrepancy might be attributed to the neglect of spin polarization in their calculations, which -- as has been shown for the C$_{\text{i}}$ BC configuration -- results in an increase of configurational energy.
+Indeed, investigating the migration path from configurations a to b and, in doing so, reusing the wave functions of the previous migration step the final structure, i.e. configuration b, is obtained with zero magnetization and an increase in configurational energy by \unit[0.2]{eV}.
+Obviously a different energy minimum of the electronic system is obtained indicating hysteresis behavior.
+However, since the total energy is lower for the magnetic result it is believed to constitute the real, i.e. global, minimum with respect to electronic minimization.
+%
+% a b transition
+A low activation energy of \unit[0.1]{eV} is observed for the a$\rightarrow$b transition.
+Thus, configuration a is very unlikely to occur in favor of configuration b.
+
+% repulsive along 110
+A repulsive interaction is observed for C$_{\text{s}}$ at lattice sites along \hkl[1 1 0], i.e. positions 1 (configuration a) and 5.
+This is due to tensile strain originating from both, the C$_{\text{i}}$ DB and the C$_{\text{s}}$ atom residing within the \hkl[1 1 0] bond chain.
+This finding agrees well with results by Mattoni et~al.~\cite{mattoni2002}.
+% all other investigated results: attractive interaction. stress compensation.
+In contrast, all other investigated configurations show attractive interactions.
+The most favorable configuration is found for C$_{\text{s}}$ at position 3, which corresponds to the lattice site of one of the upper 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.
+The substitution with C allows for most effective compensation of strain.
+This structure is followed by C$_{\text{s}}$ located at position 2, the lattice site of one of the neighbor atoms below the two Si atoms that are bound to the C$_{\text{i}}$ DB atom.
+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.
+The latter is partially compensated by the C$_{\text{s}}$ atom.
+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].
\begin{figure}[ht]
\begin{center}
-\begin{minipage}[t]{7cm}
-a) \underline{Pos: 2, $E_{\text{b}}=-0.59\text{ eV}$}
-\begin{center}
-\includegraphics[width=6.0cm]{00-1dc/0-59.eps}
+\subfigure[\underline{$E_{\text{b}}=-0.51\,\text{eV}$}]{\label{fig:defects:051}\includegraphics[width=0.25\textwidth]{00-1dc/0-51.eps}}
+\hspace{0.2cm}
+\subfigure[\underline{$E_{\text{b}}=-0.15\,\text{eV}$}]{\label{fig:defects:015}\includegraphics[width=0.25\textwidth]{00-1dc/0-15.eps}}
+\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}
-\end{minipage}
-\begin{minipage}[t]{7cm}
-b) \underline{Pos: 3, $E_{\text{b}}=-3.14\text{ eV}$}
+\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.}
+\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.
+
+% c agglomeration vs c clustering ... migs to b conf
+% 2 more migs: 051 -> 128 and 026! forgot why ... probably it's about probability of C clustering
+Obviously agglomeration of C$_{\text{i}}$ and C$_{\text{s}}$ is energetically favorable except for separations along one of the \hkl<1 1 0> directions.
+The energetically 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.
+Again, conclusions concerning the probability of formation are drawn by investigating migration paths.
+Since C$_{\text{s}}$ is unlikely to exhibit a low activation energy for migration the focus is on C$_{\text{i}}$.
+Pathways starting from the two next most favored configurations were investigated, which show activation energies above \unit[2.2]{eV} and \unit[3.5]{eV} respectively.
+Although lower than the barriers for obtaining the ground state of two C$_{\text{i}}$ defects, the activation energies are yet considered too high.
+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.
+
+% old c_int - c_substitutional stuff
+
+%Figures \ref{fig:defects:comb_db_04} and \ref{fig:defects:comb_db_05} show relaxed structures of substitutional carbon in combination with the \hkl<0 0 -1> dumbbell for several positions.
+%In figure \ref{fig:defects:comb_db_04} positions 1 (a)), 3 (b)) and 5 (c)) are displayed.
+%A substituted carbon atom at position 5 results in an energetically extremely unfavorable configuration.
+%Both carbon atoms, the substitutional and the dumbbell atom, pull silicon atom number 1 towards their own location regarding the \hkl<1 1 0> direction.
+%Due to this a large amount of tensile strain energy is needed, which explains the high positive value of 0.49 eV.
+%The lowest binding energy is observed for a substitutional carbon atom inserted at position 3.
+%The substitutional carbon atom is located above the dumbbell substituting a silicon atom which would usually be bound to and displaced along \hkl<0 0 1> and \hkl<1 1 0> by the silicon dumbbell atom.
+%In contrast to the previous configuration strain compensation occurs resulting in a binding energy as low as -0.93 eV.
+%Substitutional carbon at position 2 and 4, visualized in figure \ref{fig:defects:comb_db_05}, is located below the initial dumbbell.
+%Silicon atom number 1, which is bound to the interstitial carbon atom is displaced along \hkl<0 0 -1> and \hkl<-1 -1 0>.
+%In case a) only the first displacement is compensated by the substitutional carbon atom.
+%This results in a somewhat higher binding energy of -0.51 eV.
+%The binding energy gets even higher in case b) ($E_{\text{b}}=-0.15\text{ eV}$), in which the substitutional carbon is located further away from the initial dumbbell.
+%In both cases, silicon atom number 1 is displaced in such a way, that the bond to silicon atom number 5 vanishes.
+%In case of \ref{fig:defects:comb_db_04} a) the carbon atoms form a bond with a distance of 1.5 \AA, which is close to the C-C distance expected in diamond or graphit.
+%Both carbon atoms are highly attracted by each other resulting in large displacements and high strain energy in the surrounding.
+%A binding energy of 0.26 eV is observed.
+%Substitutional carbon at positions 2, 3 and 4 are the energetically most favorable configurations and constitute promising starting points for SiC precipitation.
+%On the one hand, C-C distances around 3.1 \AA{} exist for substitution positions 2 and 3, which are close to the C-C distance expected in silicon carbide.
+%On the other hand stretched silicon carbide is obtained by the transition of the silicon dumbbell atom into a silicon self-interstitial located somewhere in the silicon host matrix and the transition of the carbon dumbbell atom into another substitutional atom occupying the dumbbell lattice site.
+
+
+\subsection[Combinations of a \ci{} \hkl<1 0 0> DB and vacancy]{\boldmath Combinations of a \ci{} \hkl<1 0 0> DB and a vacancy}
+\label{subsection:defects:c-v}
+
+In the last section, configurations of a C$_{\text{i}}$ DB with C$_{\text{s}}$ occupying a vacant site have been investigated.
+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.
+These structures are investigated in the following.
+Resulting binding energies of a C$_{\text{i}}$ DB and a nearby vacancy are listed in the second row of Table~\ref{tab:defects:c-v}.
+\begin{table}[ht]
\begin{center}
-\includegraphics[width=6.0cm]{00-1dc/3-14.eps}
+\begin{tabular}{c c c c c c}
+\hline
+\hline
+1 & 2 & 3 & 4 & 5 & R \\
+\hline
+-5.39 ($\rightarrow$ C$_{\text{S}}$) & -0.59 & -3.14 & -0.54 & -0.50 & -0.31\\
+\hline
+\hline
+\end{tabular}
\end{center}
-\end{minipage}\\[0.2cm]
-\begin{minipage}[t]{7cm}
-c) \underline{Pos: 4, $E_{\text{b}}=-0.54\text{ eV}$}
+\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.}
+\label{tab:defects:c-v}
+\end{table}
+\begin{figure}[ht]
\begin{center}
-\includegraphics[width=6.0cm]{00-1dc/0-54.eps}
+\subfigure[\underline{$E_{\text{b}}=-0.59\,\text{eV}$}]{\label{fig:defects:059}\includegraphics[width=0.25\textwidth]{00-1dc/0-59.eps}}
+\hspace{0.7cm}
+\subfigure[\underline{$E_{\text{b}}=-3.14\,\text{eV}$}]{\label{fig:defects:314}\includegraphics[width=0.25\textwidth]{00-1dc/3-14.eps}}\\
+\subfigure[\underline{$E_{\text{b}}=-0.54\,\text{eV}$}]{\label{fig:defects:054}\includegraphics[width=0.25\textwidth]{00-1dc/0-54.eps}}
+\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}
-\end{minipage}
-\begin{minipage}[t]{7cm}
-d) \underline{Pos: 5, $E_{\text{b}}=-0.50\text{ eV}$}
+\caption{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.
+All investigated structures are preferred compared to isolated, largely separated defects.
+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.
+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.
+The creation of a vacancy at position 1 results in a configuration of substitutional C on a Si lattice site and no other remaining defects.
+The \ci{} DB atom moves to position 1 where the vacancy is created and the \si{} DB atom recaptures the DB lattice site.
+With a binding energy of \unit[-5.39]{eV}, this is the energetically most favorable configuration observed.
+A great amount of strain energy is reduced by removing the Si atom at position 3, which is illustrated in Fig.~\ref{fig:defects:314}.
+The DB structure shifts towards the position of the vacancy, which replaces the Si atom usually bound to and at the same time strained by the \si{} DB atom.
+Due to the displacement into the \hkl[1 -1 0] direction the bond of the DB Si atom to the Si atom on the top left breaks and instead forms a bond to the Si atom located in \hkl[1 -1 1] direction, which is not shown in Fig.~\ref{fig:defects:314}.
+A binding energy of \unit[-3.14]{eV} is obtained for this structure composing another energetically favorable configuration.
+A vacancy ctreated at position 2 enables the relaxation of Si atom number 1 mainly in \hkl[0 0 -1] direction.
+The bond to Si atom number 5 breaks.
+Hence, the \si{} DB atom is not only displaced along \hkl[0 0 -1] but also and to a greater extent in \hkl[1 1 0] direction.
+The C atom is slightly displaced in \hkl[0 1 -1] direction.
+A binding energy of \unit[-0.59]{eV} indicates the occurrence of much less strain reduction compared to that in the latter configuration.
+Evidently this is due to a smaller displacement of Si atom 1, which would be directly bound to the replaced Si atom at position 2.
+In the case of a vacancy created at position 4, even a slightly higher binding energy of \unit[-0.54]{eV} is observed, while the Si atom at the bottom left, which is bound to the \ci{} DB atom, is vastly displaced along \hkl[1 0 -1].
+However the displacement of the C atom along \hkl[0 0 -1] is less compared to the one in the previous configuration.
+Although expected due to the symmetric initial configuration, Si atom number 1 is not displaced correspondingly and also the \si DB atom is displaced to a greater extent in \hkl[-1 0 0] than in \hkl[0 -1 0] direction.
+The symmetric configuration is, thus, assumed to constitute a local maximum, which is driven into the present state by the conjugate gradient method used for relaxation.
+Fig.~\ref{fig:defects:050} shows the relaxed structure of a vacancy created at position 5.
+The Si DB atom is largely displaced along \hkl[1 1 0] and somewhat less along \hkl[0 0 -1], which corresponds to the direction towards the vacancy.
+The \si DB atom approaches Si atom number 1.
+Indeed, a non-zero charge density is observed inbetween these two atoms exhibiting a cylinder-like shape superposed with the charge density known from the DB itself.
+Strain reduced by this huge displacement is partially absorbed by tensile strain on Si atom number 1 originating from attractive forces of the C atom and the vacancy.
+A binding energy of \unit[-0.50]{eV} is observed.
+
+The migration pathways of configuration \ref{fig:defects:314} and \ref{fig:defects:059} into the ground-state configuration, i.e. the \cs{} configuration, are shown in Fig.~\ref{fig:314-539} and \ref{fig:059-539} respectively.
+\begin{figure}[ht]
\begin{center}
-\includegraphics[width=6.0cm]{00-1dc/0-50.eps}
+\includegraphics[width=0.7\textwidth]{314-539.ps}
\end{center}
-\end{minipage}
+\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.}
+\label{fig:314-539}
+\end{figure}
+\begin{figure}[ht]
+\begin{center}
+\includegraphics[width=0.7\textwidth]{059-539.ps}
\end{center}
-\caption{Relaxed structures of defect complexes obtained by creating vacancies at positions 2 (a)), 3 (b)), 4 (c)) and 5 (d)).}
-\label{fig:defects:comb_db_06}
+\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.}
+\label{fig:059-539}
\end{figure}
-Figure \ref{fig:defects:comb_db_06} displays relaxed structures of vacancies in combination with the \hkl<0 0 -1> dumbbell interstital.
-The creation of a vacancy at position 1 results in a configuration of substitutional carbon on a silicon lattice site and no other remaining defects.
-The carbon dumbbell atom moves to position 1 where the vacancy is created and the silicon dumbbell atom recaptures the dumbbell lattice site.
-With a binding energy of -5.39 eV, this is the energetically most favorable configuration observed.
-A great amount of strain energy is reduced by removing the silicon atom at position 3, which is illustrated in figure \ref{fig:defects:comb_db_06} b).
-The dumbbell structure shifts towards the position of the vacancy which replaces the silicon atom usually bound to and at the same time strained by the silicon dumbbell atom.
-Due to the displacement into the \hkl<1 -1 0> direction the bond of the dumbbell silicon atom to the silicon atom on the top left breaks and instead forms a bond to the silicon atom located in \hkl<1 -1 1> direction which is not shown in the figure.
-A binding energy of -3.14 eV is obtained for this structure composing another energetically favorable configuration.
-A vacancy ctreated at position 2 enables a relaxation of the silicon atom number 1 mainly in \hkl<0 0 -1> direction.
-The bond to silicon atom number 5 breaks.
-Hence, the silicon dumbbell atom is not only displaced along \hkl<0 0 -1> but also and to a greater extent in \hkl<1 1 0> direction.
-The carbon atom is slightly displaced in \hkl<0 1 -1> direction.
-A binding energy of -0.59 eV indicates the occurrence of much less strain reduction compared to that in the latter configuration.
-Evidently this is due to a smaller displacement of silicon atom number 1, which would be directly bound to the replaced silicon atom at position 2.
-In the case of a vacancy created at position 4, even a slightly higher binding energy of -0.54 eV is observed, while the silicon atom at the bottom left, which is bound to the carbon dumbbell atom, is vastly displaced along \hkl<1 0 -1>.
-However the displacement of the carbon atom along \hkl<0 0 -1> is less than it is in the preceding configuration.
-Although expected due to the symmetric initial configuration silicon atom number 1 is not displaced correspondingly and also the silicon dumbbell atom is displaced to a greater extent in \hkl<-1 0 0> than in \hkl<0 -1 0> direction.
-The symmetric configuration is, thus, assumed to constitute a local maximum, which is driven into the present state by the conjugate gradient method used for relaxation.
-Figure \ref{fig:defects:comb_db_06} d) shows the relaxed structure of a vacancy created at position 5.
-The silicon dumbbell atom is largely displaced along \hkl<1 1 0> and somewaht less along \hkl<0 0 -1>, which corresponds to the direction towards the vacancy.
-The silicon dumbbell atom approaches silicon number 1.
-Indeed a non-zero charge density is observed inbetween these two atoms exhibiting a cylinder-like shape superposed with the charge density known from the dumbbell itself.
-Strain reduced by this huge displacement is partially absorbed by tensile strain on silicon atom number 1 originating from attractive forces of the carbon atom and the vacancy.
-A binding energy of -0.50 eV is observed.
-{\color{red}Todo: Jahn-Teller distortion (vacancy) $\rightarrow$ actually three possibilities. Due to the initial defect, symmetries are broken. The system should have relaxed into the minumum energy configuration!?}
-
-\subsection{Combinations of Si self-interstitials and substitutional carbon}
-
-So far the C-Si \hkl<1 0 0> interstitial was found to be the energetically most favorable configuration.
-In fact substitutional C exhibits a configuration more than 3 eV lower in formation energy, however, the configuration does not account for the accompanying Si self-interstitial that is generated once a C atom occupies the site of a Si atom.
-With regard to the IBS process, in which highly energetic C atoms enter the Si target being able to kick out Si atoms from their lattice sites, such configurations are absolutely conceivable and a significant role for the precipitation process might be attributed to them.
-Thus, combinations of substitutional C and an additional Si self-interstitial are examined in the following.
-The ground state of a single Si self-interstitial was found to be the Si \hkl<1 1 0> self-interstitial configuration.
-For the follwoing study the same type of self-interstitial is assumed to provide the energetically most favorable configuration in combination with substitutional C.
+Activation energies as low as \unit[0.1]{eV} and \unit[0.6]{eV} are observed.
+In the first case the Si and C atom of the DB move towards the vacant and initial DB lattice site respectively.
+In total three Si-Si and one more Si-C bond is formed during transition.
+The activation energy of \unit[0.1]{eV} is needed to tilt the DB structure.
+Once this barrier is overcome, the C atom forms a bond to the top left Si atom and the \si{} atom capturing the vacant site is forming new tetrahedral bonds to its neighbored Si atoms.
+These new bonds and the relaxation into the \cs{} configuration are responsible for the gain in configurational energy.
+For the reverse process approximately \unit[2.4]{eV} are needed, which is 24 times higher than the forward process.
+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.
+A net amount of five Si-Si and one Si-C bond are additionally formed during transition.
+An activation energy of \unit[0.6]{eV} necessary to overcome the migration barrier is found.
+This energy is low enough to constitute a feasible mechanism in SiC precipitation.
+To reverse this process \unit[5.4]{eV} are needed, which make this mechanism very unprobable.
+%
+The migration path is best described by the reverse process.
+Starting at \unit[100]{\%}, energy is needed to break the bonds of Si atom 1 to its neighbored Si atoms as well as the bond of the C atom to Si atom number 5.
+At \unit[50]{\%} displacement, these bonds are broken.
+Due to this and due to the formation of new bonds, e.g. the bond of Si atom number 1 to Si atom number 5, a less steep increase of configurational energy is observed.
+In a last step, the just recently formed bond of Si atom number 1 to Si atom number 5 is broken up again as well as the bond of the initial Si DB atom and its Si neighbor in \hkl[-1 -1 -1] direction, which explains the repeated boost in energy.
+Finally, the system gains some configurational energy by relaxation into the configuration corresponding to \unit[0]{\%} displacement.
+%
+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}.
+In both cases, the formation of additional bonds is responsible for the vast gain in energy rendering almost impossible the reverse processes.
+
+In summary, pairs of C$_{\text{i}}$ DBs and vacancies, like no other before, show highly attractive interactions for all investigated combinations independent of orientation and separation direction of the defects.
+Furthermore, small activation energies, even for transitions into the ground state exist.
+If the vacancy is created at position 1 the system will end up in a configuration of C$_{\text{s}}$ anyways.
+Based on these results, a high probability for the formation of C$_{\text{s}}$ must be concluded.
+
+%\clearpage{}
+
+\subsection{Combinations of \si{} and \cs}
+\label{subsection:si-cs}
+
+So far the C-Si \hkl<1 0 0> DB interstitial was found to be the energetically most favorable configuration.
+In fact substitutional C exhibits a configuration more than \unit[3]{eV} lower with respect to the formation energy.
+However, the configuration does not account for the accompanying Si self-interstitial that is generated once a C atom occupies the site of a Si atom.
+With regard to the IBS process, in which highly energetic C atoms enter the Si target being able to kick out Si atoms from their lattice sites, such configurations are absolutely conceivable and a significant influence on the precipitation process might be attributed to them.
+Thus, combinations of \cs{} and an additional \si{} are examined in the following.
+The ground-state of a single \si{} was found to be the \si{} \hkl<1 1 0> DB configuration.
+For the follwoing study the same type of self-interstitial is assumed to provide the energetically most favorable configuration in combination with \cs.
\begin{table}[ht]
\begin{center}
\begin{tabular}{l c c c c c c}
\hline
\hline
-C$_{\text{sub}}$ & \hkl<1 1 0> & \hkl<-1 1 0> & \hkl<0 1 1> & \hkl<0 -1 1> &
- \hkl<1 0 1> & \hkl<-1 0 1> \\
+ & \hkl[1 1 0] & \hkl[-1 1 0] & \hkl[0 1 1] & \hkl[0 -1 1] &
+ \hkl[1 0 1] & \hkl[-1 0 1] \\
\hline
1 & \RM{1} & \RM{3} & \RM{3} & \RM{1} & \RM{3} & \RM{1} \\
-2 & \RM{2} & A & A & \RM{2} & C & \RM{5} \\
+2 & \RM{2} & \RM{6} & \RM{6} & \RM{2} & \RM{8} & \RM{5} \\
3 & \RM{3} & \RM{1} & \RM{3} & \RM{1} & \RM{1} & \RM{3} \\
-4 & \RM{4} & B & D & E & E & D \\
-5 & \RM{5} & C & A & \RM{2} & A & \RM{2} \\
+4 & \RM{4} & \RM{7} & \RM{9} & \RM{10} & \RM{10} & \RM{9} \\
+5 & \RM{5} & \RM{8} & \RM{6} & \RM{2} & \RM{6} & \RM{2} \\
\hline
\hline
\end{tabular}
\end{center}
-\caption{Equivalent configurations of \hkl<1 1 0>-type Si self-interstitials created at position I of figure \ref{fig:defects:pos_of_comb} and substitutional C created at positions 1 to 5.}
+\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.}
\label{tab:defects:comb_csub_si110}
\end{table}
\begin{table}[ht]
\begin{tabular}{l c c c c c c c c c c}
\hline
\hline
-Conf & \RM{1} & \RM{2} & \RM{3} & \RM{4} & \RM{5} & A & B & C & D & E \\
+ & \RM{1} & \RM{2} & \RM{3} & \RM{4} & \RM{5} & \RM{6} & \RM{7} & \RM{8} & \RM{9} & \RM{10} \\
\hline
$E_{\text{f}}$ [eV]& 4.37 & 5.26 & 5.57 & 5.37 & 5.12 & 5.10 & 5.32 & 5.28 & 5.39 & 5.32 \\
$E_{\text{b}}$ [eV] & -0.97 & -0.08 & 0.22 & -0.02 & -0.23 & -0.25 & -0.02 & -0.06 & 0.05 & -0.03 \\
\hline
\end{tabular}
\end{center}
-\caption{Formation $E_{\text{f}}$ and binding $E_{\text{b}}$ energies in eV of the combinational substitutional C and Si self-interstitial configurations as defined in table \ref{tab:defects:comb_csub_si110}.}
+\caption{Formation energies $E_{\text{f}}$, binding energies $E_{\text{b}}$ and C$_{\text{s}}$-Si$_{\text{i}}$ separation distances of configurations combining C$_{\text{s}}$ and Si$_{\text{i}}$ as defined in Table~\ref{tab:defects:comb_csub_si110}.}
\label{tab:defects:comb_csub_si110_energy}
\end{table}
-Table \ref{tab:defects:comb_csub_si110} shows equivalent configurations of \hkl<1 1 0>-type Si self-interstitials and substitutional C.
-The notation of figure \ref{fig:defects:pos_of_comb} is used with the six possible Si self-interstitials created at the usual C-Si dumbbell position.
-Substitutional C is created at positions 1 to 5.
-Resulting formation and binding energies of the relaxed structures are listed in table \ref{tab:defects:comb_csub_si110_energy}.
-In addition the separation distance of the ssubstitutional C atom and the Si \hkl<1 1 0> dumbbell interstitial, which is defined to reside at $\frac{a_{\text{Si}}}{4} \hkl<1 1 1>$ is given.
-In total 10 different configurations exist within the investigated range.
+Table~\ref{tab:defects:comb_csub_si110} 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:defects:combos_si}.
+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{tab:defects:comb_csub_si110_energy}.
+In total, ten different configurations exist within the investigated range.
+Configuration \RM{1} constitutes the energetically most favorable structure exhibiting a formation energy of \unit[4.37]{eV}.
+Obviously, the configuration of a Si$_{\text{i}}$ \hkl[1 1 0] DB and a neighbored C$_{\text{s}}$ atom along the bond chain, which has the same direction as the alignment of the DB, enables the largest possible reduction of strain.
+%
+The relaxed structure is displayed in the bottom right of Fig.~\ref{fig:162-097}.
+Compressive strain originating from the Si$_{\text{i}}$ is compensated by tensile strain inherent to the C$_{\text{s}}$ configuration.
+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$_{\text{i}}$ DB atom closest to the C atom does no longer form bonds to its top Si neighbors, but to the next neighbored Si atom along \hkl[1 1 0].
+%
+In the same way the energetically most unfavorable configuration can be explained, which is configuration \RM{3}.
+The \cs{} is located next to the lattice site shared by the \si{} \hkl[1 1 0] DB in \hkl[1 -1 1] direction.
+Thus, the compressive stress along \hkl[1 1 0] of the \si{} \hkl[1 1 0] DB is not compensated but intensified by the tensile stress of the \cs{} atom, which is no longer loacted along the direction of stress.
+However, even configuration \RM{1} 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.
+The transition involving the latter two configurations is shown in Fig.~\ref{fig:162-097}.
\begin{figure}[ht]
\begin{center}
-\includegraphics[width=12cm]{c_sub_si110.ps}
+\includegraphics[width=0.7\textwidth]{162-097.ps}
\end{center}
-\caption[Binding energy of combinations of a substitutional C and a Si \hkl<1 1 0> dumbbell self-interstitial with respect to the separation distance.]{Binding energy of combinations of a substitutional C and a Si \hkl<1 1 0> dumbbell self-interstitial with respect to the separation distance. The binding energy of the defect pair is well approximated by a Lennard-Jones 6-12 potential, which is used for curve fitting.}
-\label{fig:defects:csub_si110}
+\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.}
+\label{fig:162-097}
\end{figure}
-According to the formation energies none of the investigated structures is energetically preferred over the C-Si \hkl<1 0 0> dumbbell interstitial, which exhibits a formation energy of 3.88 eV.
-Further separated defects are assumed to approximate the sum of the formation energies of the isolated single defects.
-This is affirmed by the plot of the binding energies with respect to the separation distance in figure \ref{fig:defects:csub_si110} approximating zero with increasing distance.
-Thus, the C-Si \hkl<1 0 0> dumbbell structure remains the ground state configuration of a C interstitial in c-Si with a constant number of Si atoms.
+An activation energy as low as \unit[0.12]{eV} is necessary for the migration into the ground-state configuration.
+Accordingly, the C$_{\text{i}}$ \hkl<1 0 0> DB configuration is assumed to occur more likely.
+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.
+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.
-{\color{blue}
-However the binding energy quickly drops to zero with respect to the distance, which is reinforced by the Lennard-Jones fit estimating almost zero interaction energy already at 0.6 nm.
-This indicates a possibly low interaction capture radius of the defect pair.
-Highly energetic collisions in the IBS process might result in separations of these defects exceeding the capture radius.
-For this reason situations most likely occur in which the configuration of substitutional C can be considered without a nearby interacting Si self-interstitial and, thus, unable to form a thermodynamically more stable C-Si \hkl<1 0 0> dumbbell configuration.
-}
+\begin{figure}[ht]
+\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.}
+\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.
+The interaction of the defects is well approximated by a Lennard-Jones (LJ) 6-12 potential, which is used for curve fitting.
+Unable to model possible positive values of the binding energy, i.e. unfavorable configurations, located to the right of the minimum, the LJ fit should rather be thought of as a guide for the eye describing the decrease of the interaction strength, i.e. the absolute value of the binding energy, with increasing separation distance.
+The binding energy quickly drops to zero.
+The LJ fit estimates almost zero interaction already at \unit[0.6]{nm}.
+ indicating a low interaction capture radius of the defect pair.
+%As can be seen, the interaction strength, i.e. the absolute value of the binding energy, quickly drops to zero with increasing separation distance.
+%Almost zero interaction may be assumed already at distances about \unit[0.5-0.6]{nm}, indicating a low interaction capture radius of the defect pair.
+In IBS, highly energetic collisions are assumed to easily produce configurations of defects exhibiting separation distances exceeding the capture radius.
+For this reason C$_{\text{s}}$ without a Si$_{\text{i}}$ DB located within the immediate proximity, which is, 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.
+In particular in IBS, which constitutes a system driven far from equilibrium, respective defect configurations might exist that do not combine into the ground-state configuration.
+Thus, the existence of C$_{\text{s}}$ is very likely.
\label{section:defects:noneq_process_01}
-The energetically most favorable configuration of the combined structures is the one with the substitutional C atom located next to the \hkl<1 1 0> interstitial along the \hkl<1 1 0> direction (configuration \RM{1}).
-Compressive stress along the \hkl<1 1 0> direction originating from the Si \hkl<1 1 0> self-intesrtitial is partially compensated by tensile stress resulting from substitutional C occupying the neighbored Si lattice site.
-In the same way the energetically most unfavorable configuration can be explained, which is configuration \RM{3}.
-The substitutional C is located next to the lattice site shared by the \hkl<1 1 0> Si self-interstitial along the \hkl<1 -1 0> direction.
-Thus, the compressive stress along \hkl<1 1 0> of the Si \hkl<1 1 0> interstitial is not compensated but intensified by the tensile stress of the substitutional C atom, which is no longer loacted along the direction of stress.
-
-{\color{red}Todo: EA calc for most and less favorable configuration!}
-{\color{red}Todo: Mig of C-Si DB conf to or from C sub + Si 110 in progress.}
+% the ab initio md, where to put
-{\color{red}Todo: Mig of Si DB located next to a C sub (also by MD!).}
-
-\clearpage{}
-\cleardoublepage{}
-
-\section{Migration in systems of combined defects}
-
-As already pointed out in the previous section energetic carbon atoms may kick out silicon atoms from their lattice sites during carbon implantation into crystalline silicon.
-However configurations might arise in which C atoms do not already occupy the vacant site but instead form a C interstitial next to the vacancy.
-These combinations have been investigated shortly before in the very end of section \ref{subsection:defects:c-si_comb}.
-In the absence of the Si self-interstitial the energetically most favorable configuration is the configuration of a substitutional carbon atom, that is the carbon atom occupying the vacant site.
-In addition, it is a conceivable configuration the system might experience during the silicon carbide precipitation process.
-Energies needed to overcome the migration barrier of the transformation into this configuration enable predictions concerning the feasibility of a silicon carbide conversion mechanism derived from these microscopic processes.
-This is especially important for the case, in which the vacancy is created at position 3, as displayed in figure \ref{fig:defects:comb_db_06} b).
-Due to the low binding energy this configuration might constitute a trap, which it is hard to escape from.
-However, migration simulations show that only a low amount of energy is necessary to transform the system into the energetically most favorable configuration.
+Similar to what was previously mentioned, configurations of C$_{\text{s}}$ and a Si$_{\text{i}}$ DB might be particularly important at higher temperatures due to the low activation energy necessary for its formation.
+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.
+Indeed, an {em ab initio} MD 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 of the C$_{\text{s}}$ and Si$_{\text{i}}$ DB by more than 4 neighbor distances realized in a repeated migration mechanism of annihilating and arising Si$_{\text{i}}$ DBs.
+The atomic configurations for two different points in time are shown in Fig.~\ref{fig:defects:md}.
\begin{figure}[ht]
\begin{center}
-\includegraphics[width=13cm]{vasp_mig/comb_mig_3-2_vac_fullct.ps}\\[2.0cm]
-\begin{picture}(0,0)(170,0)
-\includegraphics[width=3cm]{vasp_mig/comb_2-1_init.eps}
-\end{picture}
-\begin{picture}(0,0)(80,0)
-\includegraphics[width=3cm]{vasp_mig/comb_2-1_seq_03.eps}
-\end{picture}
-\begin{picture}(0,0)(-10,0)
-\includegraphics[width=3cm]{vasp_mig/comb_2-1_seq_06.eps}
-\end{picture}
-\begin{picture}(0,0)(-120,0)
-\includegraphics[width=3cm]{vasp_mig/comb_2-1_final.eps}
-\end{picture}
-\begin{picture}(0,0)(25,20)
-\includegraphics[width=2.5cm]{100_arrow.eps}
-\end{picture}
-\begin{picture}(0,0)(230,0)
-\includegraphics[height=2.2cm]{001_arrow.eps}
-\end{picture}
+\begin{minipage}{0.40\textwidth}
+\includegraphics[width=\columnwidth]{md01.eps}
+\end{minipage}
+\hspace{1cm}
+\begin{minipage}{0.40\textwidth}
+\includegraphics[width=\columnwidth]{md02.eps}\\
+\end{minipage}\\
+\begin{minipage}{0.40\textwidth}
+\begin{center}
+$t=\unit[2230]{fs}$
\end{center}
-\caption{Transition of the configuration of the C-Si dumbbell interstitial in combination with a vacancy created at position 2 into the configuration of substitutional carbon.}
-\label{fig:defects:comb_mig_01}
-\end{figure}
-\begin{figure}[ht]
+\end{minipage}
+\hspace{1cm}
+\begin{minipage}{0.40\textwidth}
\begin{center}
-\includegraphics[width=13cm]{vasp_mig/comb_mig_4-2_vac_fullct.ps}\\[1.0cm]
-\begin{picture}(0,0)(150,0)
-\includegraphics[width=2cm]{vasp_mig/comb_3-1_init.eps}
-\end{picture}
-\begin{picture}(0,0)(60,0)
-\includegraphics[width=2cm]{vasp_mig/comb_3-1_seq_03.eps}
-\end{picture}
-\begin{picture}(0,0)(-45,0)
-\includegraphics[width=2cm]{vasp_mig/comb_3-1_seq_07.eps}
-\end{picture}
-\begin{picture}(0,0)(-130,0)
-\includegraphics[width=2cm]{vasp_mig/comb_3-1_final.eps}
-\end{picture}
-\begin{picture}(0,0)(25,20)
-\includegraphics[width=2.5cm]{100_arrow.eps}
-\end{picture}
-\begin{picture}(0,0)(230,0)
-\includegraphics[height=2.2cm]{001_arrow.eps}
-\end{picture}
+$t=\unit[2900]{fs}$
+\end{center}
+\end{minipage}
\end{center}
-\caption{Transition of the configuration of the C-Si dumbbell interstitial in combination with a vacancy created at position 3 into the configuration of substitutional carbon.}
-\label{fig:defects:comb_mig_02}
+\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.}
+\label{fig:defects:md}
\end{figure}
-Figure \ref{fig:defects:comb_mig_01} and \ref{fig:defects:comb_mig_02} show the migration barriers and structures for transitions of the vacancy-interstitial configurations examined in figure \ref{fig:defects:comb_db_06} a) and b) into a configuration of substitutional carbon.
-If the vacancy is created at position 1 the system will end up in a configuration of substitutional C anyways.
+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.
-In the first case the focus is on the migration of silicon atom number 1 towards the vacant site created at position 2 while the carbon atom substitutes the site of the migrating silicon atom.
-An energy of 0.6 eV necessary to overcome the migration barrier is found.
-This energy is low enough to constitute a feasible mechanism in SiC precipitation.
-To reverse this process 5.4 eV are needed, which make this mechanism very unprobable.
-The migration path is best described by the reverse process.
-Starting at 100 \% energy is needed to break the bonds of silicon atom 1 to its neighbored silicon atoms and that of the carbon atom to silicon atom number 5.
-At a displacement of 60 \% these bonds are broken.
-Due to this and due to the formation of new bonds, that is the bond of silicon atom number 1 to silicon atom number 5 and the bond of the carbon atom to its siliocn neighbor in the bottom left, a less steep increase of free energy is observed.
-At a displacement of approximately 30 \% the bond of silicon atom number 1 to the just recently created siliocn atom is broken up again, which explains the repeated boost in energy.
-Finally the system gains energy relaxing into the configuration of zero displacement.
-{\color{red}Todo: Direct migration of C in progress.}
-
-Due to the low binding energy observed, the configuration of the vacancy created at position 3 is assumed to be stable against transition.
-However, a relatively simple migration path exists, which intuitively seems to be a low energy process.
-The migration path and the corresponding differences in free energy are displayed in figure \ref{fig:defects:comb_mig_02}.
-In fact, migration simulations yield a barrier as low as 0.1 eV.
-This energy is needed to tilt the dumbbell as the displayed structure at 30 \% displacement shows.
-Once this barrier is overcome, the carbon atom forms a bond to the top left silicon atom and the interstitial silicon atom capturing the vacant site is forming new tetrahedral bonds to its neighbored silicon atoms.
-These new bonds and the relaxation into the substitutional carbon configuration are responsible for the gain in free energy.
-For the reverse process approximately 2.4 eV are needed, which is 24 times higher than the forward process.
-Thus, substitutional carbon is assumed to be stable in contrast to the C-Si dumbbell interstitial located next to a vacancy.
+% kept for nostalgical reason!
+
+%\section{Migration in systems of combined defects}
+
+%\begin{figure}[ht]
+%\begin{center}
+%\includegraphics[width=13cm]{vasp_mig/comb_mig_3-2_vac_fullct.ps}\\[2.0cm]
+%\begin{picture}(0,0)(170,0)
+%\includegraphics[width=3cm]{vasp_mig/comb_2-1_init.eps}
+%\end{picture}
+%\begin{picture}(0,0)(80,0)
+%\includegraphics[width=3cm]{vasp_mig/comb_2-1_seq_03.eps}
+%\end{picture}
+%\begin{picture}(0,0)(-10,0)
+%\includegraphics[width=3cm]{vasp_mig/comb_2-1_seq_06.eps}
+%\end{picture}
+%\begin{picture}(0,0)(-120,0)
+%\includegraphics[width=3cm]{vasp_mig/comb_2-1_final.eps}
+%\end{picture}
+%\begin{picture}(0,0)(25,20)
+%\includegraphics[width=2.5cm]{100_arrow.eps}
+%\end{picture}
+%\begin{picture}(0,0)(230,0)
+%\includegraphics[height=2.2cm]{001_arrow.eps}
+%\end{picture}
+%\end{center}
+%\caption{Transition of the configuration of the C-Si dumbbell interstitial in combination with a vacancy created at position 2 into the configuration of substitutional carbon.}
+%\label{fig:defects:comb_mig_01}
+%\end{figure}
+%\begin{figure}[ht]
+%\begin{center}
+%\includegraphics[width=13cm]{vasp_mig/comb_mig_4-2_vac_fullct.ps}\\[1.0cm]
+%\begin{picture}(0,0)(150,0)
+%\includegraphics[width=2cm]{vasp_mig/comb_3-1_init.eps}
+%\end{picture}
+%\begin{picture}(0,0)(60,0)
+%\includegraphics[width=2cm]{vasp_mig/comb_3-1_seq_03.eps}
+%\end{picture}
+%\begin{picture}(0,0)(-45,0)
+%\includegraphics[width=2cm]{vasp_mig/comb_3-1_seq_07.eps}
+%\end{picture}
+%\begin{picture}(0,0)(-130,0)
+%\includegraphics[width=2cm]{vasp_mig/comb_3-1_final.eps}
+%\end{picture}
+%\begin{picture}(0,0)(25,20)
+%\includegraphics[width=2.5cm]{100_arrow.eps}
+%\end{picture}
+%\begin{picture}(0,0)(230,0)
+%\includegraphics[height=2.2cm]{001_arrow.eps}
+%\end{picture}
+%\end{center}
+%\caption{Transition of the configuration of the C-Si dumbbell interstitial in combination with a vacancy created at position 3 into the configuration of substitutional carbon.}
+%\label{fig:defects:comb_mig_02}
+%\end{figure}
\clearpage{}
-\cleardoublepage{}
+
+\section{Applicability: Competition of \ci{} and \cs-\si{}}
+\label{section:ea_app}
+
+As has been shown, the energetically most favorable configuration of \cs{} and \si{} is obtained for \cs{} located at the neighbored lattice site along the \hkl<1 1 0> bond chain of a Si$_{\text{i}}$ \hkl<1 1 0> DB.
+However, the energy of formation is slightly higher than that of the C$_{\text{i}}$ \hkl<1 0 0> DB, which constitutes the ground state for a C impurity introduced into otherwise perfect c-Si.
+
+For a possible clarification of the controversial views on the participation of C$_{\text{s}}$ in the precipitation mechanism by classical potential simulations, test calculations need to ensure the proper description of the relative formation energies of combined structures of C$_{\text{s}}$ and Si$_{\text{i}}$ compared to C$_{\text{i}}$.
+This is particularly important since the energy of formation of C$_{\text{s}}$ is drastically underestimated by the EA potential.
+A possible occurrence of C$_{\text{s}}$ could then be attributed to a lower energy of formation of the C$_{\text{s}}$-Si$_{\text{i}}$ combination due to the low formation energy of C$_{\text{s}}$, which is obviously wrong.
+
+Since quantum-mechanical calculations reveal the Si$_{\text{i}}$ \hkl<1 1 0> DB as the ground-state configuration of Si$_{\text{i}}$ in Si, it was assumed to provide the energetically most favorable configuration in combination with C$_{\text{s}}$ in the calculations carried out in section \ref{subsection:cs-si}.
+Empirical potentials, however, predict Si$_{\text{i}}$ T to be the energetically most favorable configuration.
+Thus, investigations of the relative energies of formation of defect pairs need to include combinations of C$_{\text{s}}$ with Si$_{\text{i}}$ T.
+Results of {\em ab initio} and classical potential calculations are summarized in Table~\ref{tab:defect_combos}.
+\begin{table}[t]
+\begin{center}
+\begin{tabular}{l c c c}
+\hline
+\hline
+ & C$_{\text{i}}$ \hkl<1 0 0> & C$_{\text{s}}$ \& Si$_{\text{i}}$ \hkl<1 1 0> & C$_{\text{s}}$ \& Si$_{\text{i}}$ T\\
+\hline
+ {\textsc vasp} & 3.72 & 4.37 & 4.17$^{\text{a}}$/4.99$^{\text{b}}$/4.96$^{\text{c}}$ \\
+ {\textsc posic} & 3.88 & 4.93 & 5.25$^{\text{a}}$/5.08$^{\text{b}}$/4.43$^{\text{c}}$\\
+\hline
+\hline
+\end{tabular}
+\end{center}
+\caption{Formation energies of defect configurations of a single C impurity in otherwise perfect c-Si determined by classical potential and {\em ab initio} methods. The formation energies are given in eV. T denotes the tetrahedral and the subscripts i and s indicate the interstitial and substitutional configuration. Superscripts a, b and c denote configurations of C$_{\text{s}}$ located at the first, second and third nearest neighbored lattice site with respect to the Si$_{\text{i}}$ atom.}
+\label{tab:defect_combos}
+\end{table}
+Obviously the EA potential properly describes the relative energies of formation.
+Combined structures of C$_{\text{s}}$ and Si$_{\text{i}}$ T are energetically less favorable than the ground state C$_{\text{i}}$ \hkl<1 0 0> DB configuration.
+With increasing separation distance the energies of formation decrease.
+However, even for non-interacting defects, the energy of formation, which is then given by the sum of the formation energies of the separated defects (\unit[4.15]{eV}) is still higher than that of the C$_{\text{i}}$ \hkl<1 0 0> DB.
+Unexpectedly, the structure of a Si$_{\text{i}}$ \hkl<1 1 0> DB and a neighbored C$_{\text{s}}$, which is the most favored configuration of a C$_{\text{s}}$ and Si$_{\text{i}}$ DB according to quantum-mechanical calculations, likewise constitutes an energetically favorable configuration within the EA description, which is even preferred over the two least separated configurations of C$_{\text{s}}$ and Si$_{\text{i}}$ T.
+This is attributed to an effective reduction in strain enabled by the respective combination.
+Quantum-mechanical results reveal a more favorable energy of fomation for the C$_{\text{s}}$ and Si$_{\text{i}}$ T (a) configuration.
+However, this configuration is unstable involving a structural transition into the C$_{\text{i}}$ \hkl<1 1 0> DB interstitial, thus, not maintaining the tetrahedral Si nor the \cs{} defect.
+
+Thus, the underestimated energy of formation of C$_{\text{s}}$ within the EA calculation does not pose a serious limitation in the present context.
+Since C is introduced into a perfect Si crystal and the number of particles is conserved in simulation, the creation of C$_{\text{s}}$ is accompanied by the creation of Si$_{\text{i}}$, which is energetically less favorable than the ground state, i.e. the C$_{\text{i}}$ \hkl<1 0 0> DB configuration, for both, the EA and {\em ab initio} treatment.
+In either case, no configuration more favorable than the C$_{\text{i}}$ \hkl<1 0 0> DB has been found.
+Thus, a proper description with respect to the relative energies of formation is assumed for the EA potential.
\section{Conclusions concerning the SiC conversion mechanism}
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