[lectures/latex.git] / posic / thesis / defects.tex

\chapter{Point defects in silicon}
\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}.
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.

However, the implantation of highly energetic C atoms results in a multiplicity of possible defect configurations.
Next to individual Si$_{\text{i}}$, C$_{\text{i}}$, V and C$_{\text{s}}$ defects, combinations of these defects and their interaction are considered important for the problem under study.
Thus, the study proceeds examining pairs of most probable defect configurations and related diffusion processes exclusively by first-principles methods.
These systems can still be described by the highly accurate but computationally costly method.
Respective results allow to draw conclusions concerning the SiC precipitation in Si.

\section{Silicon self-interstitials}

For investigating the \si{} structures a Si atom is inserted or removed according to Fig. \ref{fig:basics:ins_pos} of section \ref{section:basics:defects}.
The formation energies of \si{} configurations are listed in Table \ref{tab:defects:si_self} for both methods used in this work as well as results obtained by other {\em ab initio} studies \cite{al-mushadani03,leung99}.
\begin{table}[t]
\begin{center}
\begin{tabular}{l c c c c c}
\hline
\hline
 & \hkl<1 1 0> DB & H & T & \hkl<1 0 0> DB & V \\
\hline
\multicolumn{6}{c}{Present study} \\
{\textsc vasp} & 3.39 & 3.42 & 3.77 & 4.41 & 3.63 \\
{\textsc posic} & 4.39 & 4.48$^*$ & 3.40 & 5.42 & 3.13 \\
\multicolumn{6}{c}{Other {\em ab initio} studies} \\
Ref. \cite{al-mushadani03} & 3.40 & 3.45 & - & - & 3.53 \\
Ref. \cite{leung99} & 3.31 & 3.31 & 3.43 & - & - \\
% todo cite without []
\hline
\hline
\end{tabular}
\end{center}
\caption[Formation energies of Si self-interstitials in crystalline Si determined by classical potential MD and DFT calculations.]{Formation energies of Si self-interstitials in crystalline Si determined by classical potential MD and DFT calculations. The formation energies are given in eV. T denotes the tetrahedral and H the hexagonal interstitial configuration. V corresponds to the vacancy configuration. Dumbbell configurations are abbreviated by DB. Formation energies for unstable configurations are marked by an asterisk and determined by using the low kinetic energy configuration shortly before the relaxation into the more favorable configuration starts.}
\label{tab:defects:si_self}
\end{table}
\begin{figure}[t]
\begin{center}
\begin{flushleft}
\begin{minipage}{5cm}
\underline{Tetrahedral}\\
$E_{\text{f}}=3.40\,\text{eV}$\\
\includegraphics[width=4.0cm]{si_pd_albe/tet.eps}
\end{minipage}
\begin{minipage}{10cm}
\underline{Hexagonal}\\[0.1cm]
\begin{minipage}{4cm}
$E_{\text{f}}^*=4.48\,\text{eV}$\\
\includegraphics[width=4.0cm]{si_pd_albe/hex_a.eps}
\end{minipage}
\begin{minipage}{0.8cm}
\begin{center}
$\Rightarrow$
\end{center}
\end{minipage}
\begin{minipage}{4cm}
$E_{\text{f}}=3.96\,\text{eV}$\\
\includegraphics[width=4.0cm]{si_pd_albe/hex.eps}
\end{minipage}
\end{minipage}\\[0.2cm]
\begin{minipage}{5cm}
\underline{\hkl<1 0 0> dumbbell}\\
$E_{\text{f}}=5.42\,\text{eV}$\\
\includegraphics[width=4.0cm]{si_pd_albe/100.eps}
\end{minipage}
\begin{minipage}{5cm}
\underline{\hkl<1 1 0> dumbbell}\\
$E_{\text{f}}=4.39\,\text{eV}$\\
\includegraphics[width=4.0cm]{si_pd_albe/110.eps}
\end{minipage}
\begin{minipage}{5cm}
\underline{Vacancy}\\
$E_{\text{f}}=3.13\,\text{eV}$\\
\includegraphics[width=4.0cm]{si_pd_albe/vac.eps}
\end{minipage}
\end{flushleft}
%\hrule
\end{center}
\caption[Relaxed Si self-interstitial defect configurations obtained by classical potential calculations.]{Relaxed Si self-interstitial defect configurations obtained by classical potential calculations. The Si atoms and the bonds (only for the interstitial atom) are illustrated by yellow spheres and blue lines.}
\label{fig:defects:conf}
\end{figure}
The final configurations obtained after relaxation are presented in Fig. \ref{fig:defects:conf}.
The displayed structures are the results of the classical potential simulations.

There are differences between the various results of the quantum-mechanical calculations but the consensus view is that the \hkl<1 1 0> dumbbell (DB) followed by the hexagonal and tetrahedral defect is the lowest in energy.
This is nicely reproduced by the DFT calculations performed in this work.

It has turned out to be very difficult to capture the results of quantum-mechanical calculations in analytical potential models.
Among the established analytical potentials only the EDIP \cite{bazant97,justo98} and Stillinger-Weber \cite{stillinger85} potential reproduce the correct order in energy of the defects.
However, these potenitals show shortcomings concerning the description of other physical properties and are unable to describe the C-C and C-Si interaction.
In fact the EA potential calculations favor the tetrahedral defect configuration.
This limitation is assumed to arise due to the cut-off.
In the tetrahedral configuration the second neighbors are only slightly more distant than the first neighbors, which creates the particular problem.
Indeed, an increase of the cut-off results in increased values of the formation energies \cite{albe_sic_pot}, which is most significant for the tetrahedral configuration.
The same issue has already been discussed by Tersoff \cite{tersoff90} with regard to the description of the tetrahedral C defect using his potential.
While not completely rendering impossible further, more challenging empirical potential studies on large systems, the artifact has to be taken into account in the investigations of defect combinations later on in this chapter.

The hexagonal configuration is not stable opposed to results of the authors of the potential \cite{albe_sic_pot}.
In the first two pico seconds, while kinetic energy is decoupled from the system, the \si{} seems to condense at the hexagonal site.
The formation energy of \unit[4.48]{eV} is determined by this low kinetic energy configuration shortly before the relaxation process starts.
The \si{} atom then begins to slowly move towards an energetically more favorable position very close to the tetrahedral one but slightly displaced along the three coordinate axes.
The formation energy of \unit[3.96]{eV} for this type of interstitial is equal to the result for the hexagonal one in the original work \cite{albe_sic_pot}.
Obviously the authors did not carefully check the relaxed results assuming a hexagonal configuration.
In Fig. \ref{fig:defects:kin_si_hex} the relaxation process is shown on the basis of the kinetic energy plot.
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.7\textwidth]{e_kin_si_hex.ps}
\end{center}
\caption{Kinetic energy plot of the relaxation process of the hexagonal silicon self-interstitial defect simulation using the EA potential.}
\label{fig:defects:kin_si_hex}
\end{figure}
To exclude failures in the implementation of the potential or the MD code itself the hexagonal defect structure was double-checked with the {\textsc parcas} MD code \cite{parcas_md}.
The respective relaxation energetics are likewise plotted and look similar to the energetics obtained by {\textsc posic}.
In fact, the same type of interstitial arises using random insertions.
In addition, variations exist, in which the displacement is only along two \hkl<1 0 0> axes ($E_\text{f}=3.8\,\text{eV}$) or along a single \hkl<1 0 0> axes ($E_\text{f}=3.6\,\text{eV}$) successively approximating the tetdrahedral configuration and formation energy.
The existence of these local minima located near the tetrahedral configuration seems to be an artifact of the analytical potential without physical authenticity revealing fundamental problems of analytical potential models for describing defect structures.
However, the energy barrier is small.
\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.7\textwidth]{nhex_tet.ps}
\end{center}
\caption{Migration barrier of the tetrahedral Si self-interstitial slightly displaced along all three coordinate axes into the exact tetrahedral configuration using classical potential calculations.}
\label{fig:defects:nhex_tet_mig}
\end{figure}
This is exemplified in Fig. \ref{fig:defects:nhex_tet_mig}, which shows the change in configurational energy during the migration of the interstitial displaced along all three coordinate axes into the tetrahedral configuration.
The barrier is smaller than \unit[0.2]{eV}.
Hence, these artifacts have a negligible influence in finite temperature simulations.

The bond-centered (BC) configuration is unstable and, thus, is not listed.
The \si{} \hkl<1 0 0> DB constitutes the most unfavorable configuration for both, the EA and {\textsc vasp} calculations.
The quantum-mechanical treatment of the \si{} \hkl<1 0 0> DB demands for spin polarized calculations.
The same applies for the vacancy.
In the \si{} \hkl<1 0 0> DB configuration the net spin up density is localized in two caps at each of the two DB atoms perpendicularly aligned to the bonds to the other two Si atoms respectively.
For the vacancy the net spin up electron density is localized in caps at the four surrounding Si atoms directed towards the vacant site.
No other intrinsic defect configuration, within the ones that are mentioned, is affected by spin polarization.

In the case of the classical potential simulations bonds between atoms are displayed if there is an interaction according to the potential model, i.e. if the distance of two atoms is within the cut-off radius $S_{ij}$ introduced in equation \eqref{eq:basics:fc}.
For the tetrahedral and the slightly displaced configurations four bonds to the atoms located in the center of the planes of the unit cell exist in addition to the four tetrahedral bonds.
The length of these bonds are, however, close to the cut-off range and thus are weak interactions not constituting actual chemical bonds.
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.




\section{Carbon point defects in silicon}

\subsection{Defect structures in a nutshell}

For investigating the \ci{} structures a C atom is inserted or removed according to Fig. \ref{fig:basics:ins_pos} of section \ref{section:basics:defects}.
Formation energies of the most common C point defects in crystalline Si are summarized in Table \ref{tab:defects:c_ints}.
The relaxed configurations are visualized in Fig. \ref{fig:defects:c_conf}.
Again, the displayed structures are the results obtained by the classical potential calculations.
The type of reservoir of the C impurity to determine the formation energy of the defect is chosen to be SiC.
This is consistent with the methods used in the articles \cite{tersoff90,dal_pino93}, which the results are compared to in the following.
Hence, the chemical potential of Si and C is determined by the cohesive energy of Si and SiC as discussed in section \ref{section:basics:defects}.
\begin{table}[t]
\begin{center}
\begin{tabular}{l c c c c c c}
\hline
\hline
 & T & H & \hkl<1 0 0> DB & \hkl<1 1 0> DB & S & BC \\
\hline
\multicolumn{6}{c}{Present study} \\
 {\textsc posic} & 6.09 & 9.05$^*$ & 3.88 & 5.18 & 0.75 & 5.59$^*$ \\
 {\textsc vasp} & Unstable & Unstable & 3.72 & 4.16 & 1.95 & 4.66 \\
\multicolumn{6}{c}{Other studies} \\
 Tersoff \cite{tersoff90} & 3.8 & 6.7 & 4.6 & 5.9 & 1.6 & 5.3 \\
 {\em Ab initio} \cite{dal_pino93,capaz94} & - & - & x & - & 1.89 & x+2.1 \\
\hline
\hline
\end{tabular}
\end{center}
\caption[Formation energies of C point defects in c-Si determined by classical potential MD and DFT calculations.]{Formation energies of C point defects in c-Si determined by classical potential MD and DFT calculations. The formation energies are given in eV. T denotes the tetrahedral, H the hexagonal and BC the bond-centered interstitial configuration. S corresponds to the substitutional interstitial configuration. The dumbbell configurations are abbreviated by DB.  Formation energies for unstable configurations are marked by an asterisk and are determined by using the low kinetic energy configuration shortly before the relaxation into the more favorable configuration starts.}
\label{tab:defects:c_ints}
\end{table}
\begin{figure}[t]
\begin{center}
\begin{flushleft}
\begin{minipage}{4cm}
\underline{Hexagonal}\\
$E_{\text{f}}^*=9.05\,\text{eV}$\\
\includegraphics[width=4.0cm]{c_pd_albe/hex.eps}
\end{minipage}
\begin{minipage}{0.8cm}
\begin{center}
$\Rightarrow$
\end{center}
\end{minipage}
\begin{minipage}{4cm}
\underline{\hkl<1 0 0>}\\
$E_{\text{f}}=3.88\,\text{eV}$\\
\includegraphics[width=4.0cm]{c_pd_albe/100.eps}
\end{minipage}
\begin{minipage}{0.5cm}
\hfill
\end{minipage}
\begin{minipage}{5cm}
\underline{Tetrahedral}\\
$E_{\text{f}}=6.09\,\text{eV}$\\
\includegraphics[width=4.0cm]{c_pd_albe/tet.eps}
\end{minipage}\\[0.2cm]
\begin{minipage}{4cm}
\underline{Bond-centered}\\
$E_{\text{f}}^*=5.59\,\text{eV}$\\
\includegraphics[width=4.0cm]{c_pd_albe/bc.eps}
\end{minipage}
\begin{minipage}{0.8cm}
\begin{center}
$\Rightarrow$
\end{center}
\end{minipage}
\begin{minipage}{4cm}
\underline{\hkl<1 1 0> dumbbell}\\
$E_{\text{f}}=5.18\,\text{eV}$\\
\includegraphics[width=4.0cm]{c_pd_albe/110.eps}
\end{minipage}
\begin{minipage}{0.5cm}
\hfill
\end{minipage}
\begin{minipage}{5cm}
\underline{Substitutional}\\
$E_{\text{f}}=0.75\,\text{eV}$\\
\includegraphics[width=4.0cm]{c_pd_albe/sub.eps}
\end{minipage}
\end{flushleft}
\end{center}
\caption[Relaxed C point defect configurations obtained by classical potential calculations.]{Relaxed C point defect configurations obtained by classical potential calculations. The Si/C atoms and the bonds (only for the interstitial atom) are illustrated by yellow/grey spheres and blue lines.}
\label{fig:defects:c_conf}
\end{figure}

\cs{} occupying an already vacant Si lattice site, which is in fact not an interstitial defect, is found to be the lowest configuration in energy for all potential models.
An experiemntal value of the formation energy of \cs{} was determined by a fit to solubility data yielding a concentration of $3.5 \times 10^{24} \exp{(-2.3\,\text{eV}/k_{\text{B}}T)} \text{ cm}^{-3}$ \cite{bean71}.
However, there is no particular reason for treating the prefactor as a free parameter in the fit to the experimental data.
It is simply given by the atomic density of pure silicon, which is $5\times 10^{22}\text{ cm}^{-3}$.
Tersoff \cite{tersoff90} and Dal Pino et al. \cite{dal_pino93} pointed out that by combining this prefactor with the calculated values for the energy of formation ranging from \unit[1.6-1.89]{eV} an excellent agreement with the experimental solubility data within the entire temeprature range of the experiment is obtained.
This reinterpretation of the solubility data, first proposed by Tersoff and later on reinforced by Dal~Pino~et~al. is in good agreement with the results of the quantum-mechanical calculations performed in this work.
Unfortunately the EA potential undervalues the formation energy roughly by a factor of two, which is a definite drawback of the potential.

Except for Tersoff's results for the tedrahedral configuration, the \ci{} \hkl<1 0 0> DB is the energetically most favorable interstital configuration.
As mentioned above, the low energy of formation for the tetrahedral interstitial in the case of the Tersoff potential is believed to be an artifact of the abrupt cut-off set to \unit[2.5]{\AA} (see Ref. 11 and 13 in \cite{tersoff90}) and the real formation energy is, thus, supposed to be located between \unit[3-10]{eV}.
Keeping these considerations in mind, the \ci{} \hkl<1 0 0> DB is the most favorable interstitial configuration for all interaction models.
This finding is in agreement with several theoretical\cite{burnard93,leary97,dal_pino93,capaz94,jones04} and experimental\cite{watkins76,song90} investigations, which all predict this configuration to be the ground state.
However, no energy of formation for this type of defect based on first-principles calculations has yet been explicitly stated in literature.
The defect is frequently generated in the classical potential simulation runs, in which C is inserted at random positions in the c-Si matrix.
In quantum-mechanical simulations the unstable tetrahedral and hexagonal configurations undergo a relaxation into the \ci{} \hkl<1 0 0> DB configuration.
Thus, this configuration is of great importance and discussed in more detail in section \ref{subsection:100db}.
It should be noted that EA and DFT predict almost equal formation energies.

The highest energy is observed for the hexagonal interstitial configuration using classical potentials.
Quantum-mechanical calculations reveal this configuration to be unstable, which is also reproduced by the EA potential.
In both cases a relaxation towards the \ci{} \hkl<1 0 0> DB configuration is observed.
Opposed to results of the first-principles calculations, Tersoff finds this configuration to be stable \cite{tersoff90}.
In fact, the stability of the hexagonal interstitial could not be reproduced in simulations performed in this work using the unmodifed Tersoff potential parameters.
Unfortunately, apart from the modified parameters, no more conditions specifying the relaxation process are given in Tersoff's study on C point defects in Si.

The tetrahedral is the second most unfavorable interstitial configuration using classical potentials if the abrupt cut-off effect of the Tersoff potential is taken into account.
Again, quantum-mechanical results reveal this configuration to be unstable.
The fact that the tetrahedral and hexagonal configurations are the two most unstable configurations in classical potential calculations and, thus, are less likely to arise in MD simulations, acts in concert with the fact that these configurations are found to be unstable in the more accurate quantum-mechanical description.

Just as for \si{}, a \ci{} \hkl<1 1 0> DB configuration exists.
It constitutes the second most favorable configuration, reproduced by both methods.
Similar structures arise in both types of simulations.
The Si and C atom share a regular Si lattice site aligned along the \hkl<1 1 0> direction.
The C atom is slightly displaced towards the next nearest Si atom located in the opposite direction with respect to the site-sharing Si atom and even forms a bond with this atom.

The \ci{} \hkl<1 1 0> DB structure is energetically followed by the BC configuration.
However, even though EA based results yield the same difference in energy with respect to the \hkl<1 1 0> defect as DFT does, the BC configuration is found to be a unstable within the EA description.
The BC configuration descends into the \ci{} \hkl<1 1 0> DB configuration.
Due to the high formation energy of the BC defect resulting in a low probability of occurrence of this defect, the wrong description is not posing a serious limitation of the EA potential.
Tersoff indeed predicts a metastable BC configuration.
However,  it is not in the correct order and lower in energy than the \ci{} \hkl<1 1 0> DB.
Quantum-mechanical results of this configuration are discussed in more detail in section \ref{subsection:bc}.
In another {\em ab inito} study, Capaz~et~al. \cite{capaz94} in turn found BC configuration to be an intermediate saddle point structure of a possible migration path, which is \unit[2.1]{eV} higher than the \ci{} \hkl<1 0 0> DB structure.
This is assumed to be due to the neglection of the electron spin in these calculations.
Another {\textsc vasp} calculation without fully accounting for the electron spin results in the smearing of a single electron over two non-degenerate states for the BC configuration.
This problem is resolved by spin polarized calculations resulting in a net spin of one accompanied by a reduction of the total energy by \unit[0.3]{eV} and the transformation into a metastable local minimum configuration.
It is worth to note that all other listed configurations are not affected by spin polarization.
However, in calculations performed in this work, which fully account for the spin of the electrons, the BC configuration in fact is a real local minimum and an energy barrier is needed to reach this configuration starting from the \ci{} \hkl<1 0 0> DB configuration.
This is discussed in more detail in section \ref{subsection:100mig}.

To conclude, discrepancies between the results from classical potential calculations and those obtained from first principles are observed.
Within the classical potentials EA outperforms Tersoff and is, therefore, used for further studies.
Both methods (EA and DFT) predict the \ci{} \hkl<1 0 0> DB configuration to be most stable.
Also the remaining defects and their energetical order are described fairly well.
It is thus concluded that, so far, modeling of the SiC precipitation by the EA potential might lead to trustable results.

\subsection[C \hkl<1 0 0> dumbbell interstitial configuration]{\boldmath C \hkl<1 0 0> dumbbell interstitial configuration}
\label{subsection:100db}

As the \ci{} \hkl<1 0 0> DB constitutes the ground-state configuration of a C atom incorporated into otherwise perfect c-Si it is the most probable and, hence, one of the most important interstitial configurations of C in Si.
The structure was initially suspected by IR local vibrational mode absorption \cite{bean70} and finally verified by electron paramegnetic resonance (EPR) \cite{watkins76} studies on irradiated Si substrates at low temperatures.

Fig. \ref{fig:defects:100db_cmp} schematically shows the \ci{} \hkl<1 0 0> DB structure and Table \ref{tab:defects:100db_cmp} lists the details of the atomic displacements, distances and bond angles obtained by classical potential and quantum-mechanical calculations.
For comparison, the obtained structures for both methods are visualized in Fig. \ref{fig:defects:100db_vis_cmp}.
\begin{figure}[ht]
\begin{center}
\includegraphics[width=12cm]{100-c-si-db_cmp.eps}
\end{center}
\caption[Sketch of the \ci{} \hkl<1 0 0> dumbbell structure.]{Sketch of the \ci{} \hkl<1 0 0> dumbbell structure. Atomic displacements, distances and bond angles are listed in Table \ref{tab:defects:100db_cmp}.}
\label{fig:defects:100db_cmp}
\end{figure}
\begin{table}[ht]
\begin{center}
Displacements\\
\begin{tabular}{l c c c c c c c c c}
\hline
\hline
 & & & & \multicolumn{3}{c}{Atom 2} & \multicolumn{3}{c}{Atom 3} \\
 & $a$ & $b$ & $|a|+|b|$ & $\Delta x$ & $\Delta y$ & $\Delta z$ & $\Delta x$ & $\Delta y$ & $\Delta z$ \\
\hline
{\textsc posic} & 0.084 & -0.091 & 0.175 & -0.015 & -0.015 & -0.031 & -0.014 & 0.014 & 0.020 \\
{\textsc vasp} & 0.109 & -0.065 & 0.174 & -0.011 & -0.011 & -0.024 & -0.014 & 0.014 & 0.025 \\
\hline
\hline
\end{tabular}\\[0.5cm]
\end{center}
\begin{center}
Distances\\
\begin{tabular}{l c c c c c c c c r}
\hline
\hline
 & $r(1C)$ & $r(2C)$ & $r(3C)$ & $r(12)$ & $r(13)$ & $r(34)$ & $r(23)$ & $r(25)$ & $a_{\text{Si}}^{\text{equi}}$\\
\hline
{\textsc posic} & 0.175 & 0.329 & 0.186 & 0.226 & 0.300 & 0.343 & 0.423 & 0.425 & 0.543 \\
{\textsc vasp} & 0.174 & 0.341 & 0.182 & 0.229 & 0.286 & 0.347 & 0.422 & 0.417 & 0.548 \\
\hline
\hline
\end{tabular}\\[0.5cm]
\end{center}
\begin{center}
Angles\\
\begin{tabular}{l c c c c }
\hline
\hline
 & $\theta_1$ & $\theta_2$ & $\theta_3$ & $\theta_4$ \\
\hline
{\textsc posic} & 140.2 & 109.9 & 134.4 & 112.8 \\
{\textsc vasp} & 130.7 & 114.4 & 146.0 & 107.0 \\
\hline
\hline
\end{tabular}\\[0.5cm]
\end{center}
\caption[Atomic displacements, distances and bond angles of the \ci{} \hkl<1 0 0> DB structure obtained by {\textsc posic} and {\textsc vasp} calculations.]{Atomic displacements, distances and bond angles of the \ci{} \hkl<1 0 0> DB structure obtained by {\textsc posic} and {\textsc vasp} calculations. The displacements and distances are given in nm and the angles are given in degrees. Displacements, distances and angles are schematically displayed in Fig. \ref{fig:defects:100db_cmp}. In addition, the equilibrium lattice constant for crystalline Si is listed.}
\label{tab:defects:100db_cmp}
\end{table}
\begin{figure}[ht]
\begin{center}
\begin{minipage}{6cm}
\begin{center}
\underline{\textsc posic}
\includegraphics[width=5cm]{c_pd_albe/100_cmp.eps}
\end{center}
\end{minipage}
\begin{minipage}{6cm}
\begin{center}
\underline{\textsc vasp}
\includegraphics[width=5cm]{c_pd_vasp/100_cmp.eps}
\end{center}
\end{minipage}
\end{center}
\caption{Comparison of the \ci{} \hkl<1 0 0> DB structures obtained by {\textsc posic} and {\textsc vasp} calculations.}
\label{fig:defects:100db_vis_cmp}
\end{figure}
\begin{figure}[ht]
\begin{center}
\includegraphics[height=10cm]{c_pd_vasp/eden.eps}
\includegraphics[height=12cm]{c_pd_vasp/100_2333_ksl.ps}
\end{center}
\caption[Charge density isosurface and Kohn-Sham levels of the \ci{} \hkl<1 0 0> DB structure obtained by {\textsc vasp} calculations.]{Charge density isosurface and Kohn-Sham levels of the \ci{} \hkl<1 0 0> DB structure obtained by {\textsc vasp} calculations. Yellow and grey spheres correspond to Si and C atoms. The blue surface is the charge density isosurface. In the energy level diagram red and green lines and dots mark occupied and unoccupied states.}
\label{img:defects:charge_den_and_ksl}
\end{figure}
The Si atom labeled '1' and the C atom compose the DB structure.
They share the lattice site which is indicated by the dashed red circle.
They are displaced from the regular lattice site by length $a$ and $b$ respectively.
The atoms no longer have four tetrahedral bonds to the Si atoms located on the alternating opposite edges of the cube.
Instead, each of the DB atoms forms threefold coordinated bonds, which are located in a plane.
One bond is formed to the other DB atom.
The other two bonds are bonds to the two Si edge atoms located in the opposite direction of the DB atom.
The distance of the two DB atoms is almost the same for both types of calculations.
However, in the case of the {\textsc vasp} calculation, the DB structure is pushed upwards compared to the results using the EA potential.
This is easily identified by comparing the values for $a$ and $b$ and the two structures in Fig. \ref{fig:defects:100db_vis_cmp}.
Thus, the angles of bonds of the Si DB atom ($\theta_1$ and $\theta_2$) are closer to $120^{\circ}$ signifying the predominance of $sp^2$ hybridization.
On the other hand, the C atom forms an almost collinear bond ($\theta_3$) with the two Si edge atoms implying the predominance of $sp$ bonding.
This is supported by the image of the charge density isosurface in Fig. \ref{img:defects:charge_den_and_ksl}.
The two lower Si atoms are $sp^3$ hybridized and form $\sigma$ bonds to the Si DB atom.
The same is true for the upper two Si atoms and the C DB atom.
In addition the DB atoms form $\pi$ bonds.
However, due to the increased electronegativity of the C atom the electron density is attracted by and, thus, localized around the C atom.
In the same figure the Kohn-Sham levels are shown.
There is no magnetization density.
An acceptor level arises at approximately $E_v+0.35\,\text{eV}$ while a band gap of about \unit[0.75]{eV} can be estimated from the Kohn-Sham level diagram for plain Si.
However, strictly speaking, the Kohn-Sham levels and orbitals do not have a direct physical meaning and, thus, these values have to be taken with care.
% todo band gap problem

\subsection{Bond-centered interstitial configuration}
\label{subsection:bc}

\begin{figure}[ht]
\begin{center}
\begin{minipage}{8cm}
\includegraphics[width=8cm]{c_pd_vasp/bc_2333.eps}\\
\hrule
\vspace*{0.2cm}
\includegraphics[width=8cm]{c_100_mig_vasp/im_spin_diff.eps}
\end{minipage}
\begin{minipage}{7cm}
\includegraphics[width=7cm]{c_pd_vasp/bc_2333_ksl.ps}
\end{minipage}
\end{center}
\caption[Structure, charge density isosurface and Kohn-Sham level diagram of the bond-centered interstitial configuration.]{Structure, charge density isosurface and Kohn-Sham level diagram of the bond-centered interstitial configuration. Gray, green and blue surfaces mark the charge density of spin up, spin down and the resulting spin up electrons in the charge density isosurface, in which the carbon atom is represented by a red sphere. In the energy level diagram red and green lines mark occupied and unoccupied states.}
\label{img:defects:bc_conf}
\end{figure}
In the BC insterstitial configuration the interstitial atom is located inbetween two next neighbored Si atoms forming linear bonds.
In a previous study this configuration was found to constitute an intermediate saddle point configuration determining the migration barrier of one possibe migration path of a \ci{} \hkl<1 0 0> DB configuration into an equivalent one \cite{capaz94}.
This is in agreement with results of the EA potential simulations, which reveal this configuration to be unstable relaxing into the \ci{} \hkl<1 1 0> configuration.
However, this fact could not be reproduced by spin polarized {\textsc vasp} calculations performed in this work.
Present results suggest this configuration to correspond to a real local minimum.
In fact, an additional barrier has to be passed to reach this configuration starting from the \ci{} \hkl<1 0 0> interstitital configuration, which is investigated in section \ref{subsection:100mig}.
After slightly displacing the C atom along the \hkl[1 0 0] (equivalent to a displacement along \hkl[0 1 0]), \hkl[0 0 1], \hkl[0 0 -1] and \hkl[1 -1 0] direction the distorted structures relax back into the BC configuration.
As will be shown in subsequent migration simulations the same would happen to structures where the C atom is displaced along the migration direction, which approximately is the \hkl[1 1 0] direction.
These relaxations indicate that the BC configuration is a real local minimum instead of an assumed saddle point configuration.
Fig. \ref{img:defects:bc_conf} shows the structure, charge density isosurface and Kohn-Sham levels of the BC configuration.
The linear bonds of the C atom to the two Si atoms indicate the $sp$ hybridization of the C atom.
Two electrons participate to the linear $\sigma$ bonds with the Si neighbors.
The other two electrons constitute the $2p^2$ orbitals resulting in a net magnetization.
This is supported by the charge density isosurface and the Kohn-Sham levels in Fig. \ref{img:defects:bc_conf}.
The blue torus, which reinforces the assumption of the $p$ orbital, illustrates the resulting spin up electron density.
In addition, the energy level diagram shows a net amount of two spin up electrons.
% todo smaller images, therefore add mo image

\clearpage{}
\cleardoublepage{}

% todo migration of \si{}!

\section{Migration of the carbon interstitial}
\label{subsection:100mig}

A measure for the mobility of interstitial C is the activation energy necessary for the migration from one stable position to another.
The stable defect geometries have been discussed in the previous subsection.
In the following, the problem of interstitial C migration in Si is considered.
Since the \ci{} \hkl<1 0 0> DB is the most probable, hence, most important configuration, the migration of this defect atom from one site of the Si host lattice to a neighboring site is in the focus of investigation.
\begin{figure}[ht]
\begin{center}
\begin{minipage}{15cm}
\underline{\hkl<0 0 -1> $\rightarrow$ \hkl<0 0 1>}\\
\begin{minipage}{4.5cm}
\includegraphics[width=4.5cm]{c_pd_vasp/100_2333.eps}
\end{minipage}
\begin{minipage}{0.5cm}
$\rightarrow$
\end{minipage}
\begin{minipage}{4.5cm}
\includegraphics[width=4.5cm]{c_pd_vasp/bc_2333.eps}
\end{minipage}
\begin{minipage}{0.5cm}
$\rightarrow$
\end{minipage}
\begin{minipage}{4.5cm}
\includegraphics[width=4.5cm]{c_pd_vasp/100_next_2333.eps}
\end{minipage}
\end{minipage}\\
\begin{minipage}{15cm}
\underline{\hkl<0 0 -1> $\rightarrow$ \hkl<0 -1 0>}\\
\begin{minipage}{4.5cm}
\includegraphics[width=4.5cm]{c_pd_vasp/100_2333.eps}
\end{minipage}
\begin{minipage}{0.5cm}
$\rightarrow$
\end{minipage}
\begin{minipage}{4.5cm}
\includegraphics[width=4.5cm]{c_pd_vasp/00-1-0-10_2333.eps}
\end{minipage}
\begin{minipage}{0.5cm}
$\rightarrow$
\end{minipage}
\begin{minipage}{4.5cm}
\includegraphics[width=4.5cm]{c_pd_vasp/0-10_2333.eps}
\end{minipage}
\end{minipage}\\
\begin{minipage}{15cm}
\underline{\hkl<0 0 -1> $\rightarrow$ \hkl<0 -1 0> (in place)}\\
\begin{minipage}{4.5cm}
\includegraphics[width=4.5cm]{c_pd_vasp/100_2333.eps}
\end{minipage}
\begin{minipage}{0.5cm}
$\rightarrow$
\end{minipage}
\begin{minipage}{4.5cm}
\includegraphics[width=4.5cm]{c_pd_vasp/00-1_ip0-10_2333.eps}
\end{minipage}
\begin{minipage}{0.5cm}
$\rightarrow$
\end{minipage}
\begin{minipage}{4.5cm}
\includegraphics[width=4.5cm]{c_pd_vasp/0-10_ip_2333.eps}
\end{minipage}
\end{minipage}
\end{center}
\caption{Conceivable migration pathways among two \ci{} \hkl<1 0 0> DB configurations.}
\label{img:defects:c_mig_path}
\end{figure}
Three different migration paths are accounted in this work, which are displayed in Fig. \ref{img:defects:c_mig_path}.
The first migration investigated is a transition of a \hkl<0 0 -1> into a \hkl<0 0 1> DB interstitial configuration.
During this migration the C atom is changing its Si DB partner.
The new partner is the one located at $a_{\text{Si}}/4 \hkl<1 1 -1>$ relative to the initial one, where $a_{\text{Si}}$ is the Si lattice constant.
Two of the three bonds to the next neighbored Si atoms are preserved while the breaking of the third bond and the accompanying formation of a new bond is observed.
The C atom resides in the \hkl(1 1 0) plane.
This transition involves the intermediate BC configuration.
However, results discussed in the previous section indicate that the BC configuration is a real local minimum.
Thus, the \hkl<0 0 -1> to \hkl<0 0 1> migration can be thought of a two-step mechanism in which the intermediate BC configuration constitutes a metastable configuration.
Due to symmetry it is enough to consider the transition from the BC to the \hkl<1 0 0> configuration or vice versa.
In the second path, the C atom is changing its Si partner atom as in path one.
However, the trajectory of the C atom is no longer proceeding in the \hkl(1 1 0) plane.
The orientation of the new DB configuration is transformed from \hkl<0 0 -1> to \hkl<0 -1 0>.
Again, one bond is broken while another one is formed.
As a last migration path, the defect is only changing its orientation.
Thus, this path is not responsible for long-range migration.
The Si DB partner remains the same.
The bond to the face-centered Si atom at the bottom of the unit cell breaks and a new one is formed to the face-centered atom at the forefront of the unit cell.

\subsection{Migration paths obtained by first-principles calculations}

\begin{figure}[ht]
\begin{center}
\includegraphics[width=13cm]{im_00-1_nosym_sp_fullct_thesis.ps}\\[1.5cm]
\begin{picture}(0,0)(150,0)
\includegraphics[width=2.5cm]{vasp_mig/00-1.eps}
\end{picture}
\begin{picture}(0,0)(-10,0)
\includegraphics[width=2.5cm]{vasp_mig/bc_00-1_sp.eps}
\end{picture}
\begin{picture}(0,0)(-120,0)
\includegraphics[width=2.5cm]{vasp_mig/bc.eps}
\end{picture}
\begin{picture}(0,0)(25,20)
\includegraphics[width=2.5cm]{110_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 BC (right) transition.]{Migration barrier and structures of the \hkl<0 0 -1> DB (left) to BC (right) transition. Bonds of the C atom are illustrated by blue lines.}
\label{fig:defects:00-1_001_mig}
\end{figure}
In Fig. \ref{fig:defects:00-1_001_mig} results of the \hkl<0 0 -1> to \hkl<0 0 1> migration fully described by the migration of the \hkl<0 0 -1> to the BC configuration is displayed.
To reach the BC configuration, which is \unit[0.94]{eV} higher in energy than the \hkl<0 0 -1> DB configuration, an energy barrier of approximately \unit[1.2]{eV} given by the saddle point structure at a displacement of \unit[60]{\%} has to be passed.
This amount of energy is needed to break the bond of the C atom to the Si atom at the bottom left.
In a second process \unit[0.25]{eV} of energy are needed for the system to revert into a \hkl<1 0 0> configuration.

\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}
\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.}
\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.
The resulting migration barrier of approximately \unit[0.9]{eV} is very close to the experimentally obtained values of \unit[0.70]{eV} \cite{lindner06}, \unit[0.73]{eV} \cite{song90} and \unit[0.87]{eV} \cite{tipping87}.

\begin{figure}[ht]
\begin{center}
\includegraphics[width=13cm]{vasp_mig/00-1_ip0-10_nosym_sp_fullct.ps}\\[1.8cm]
\begin{picture}(0,0)(140,0)
\includegraphics[width=2.2cm]{vasp_mig/00-1_b.eps}
\end{picture}
\begin{picture}(0,0)(20,0)
\includegraphics[width=2.2cm]{vasp_mig/00-1_ip0-10_sp.eps}
\end{picture}
\begin{picture}(0,0)(-120,0)
\includegraphics[width=2.2cm]{vasp_mig/0-10_b.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[Reorientation barrier and structures of the \hkl<0 0 -1> DB (left) to the \hkl<0 -1 0> DB (right) transition in place.]{Reorientation barrier and structures of the \hkl<0 0 -1> DB (left) to the \hkl<0 -1 0> DB (right) transition in place. Bonds of the carbon atoms are illustrated by blue lines.}
\label{fig:defects:00-1_0-10_ip_mig}
\end{figure}
The third migration path, in which the DB is changing its orientation, is shown in Fig. \ref{fig:defects:00-1_0-10_ip_mig}.
An energy barrier of roughly \unit[1.2]{eV} is observed.
Experimentally measured activation energies for reorientation range from \unit[0.77]{eV} to \unit[0.88]{eV} \cite{watkins76,song90}.
Thus, this pathway is more likely to be composed of two consecutive steps of the second path.

Since the activation energy of the first and last migration path is much greater than the experimental value, the second path is identified to be responsible as a migration path for the most likely C interstitial in Si explaining both, annealing and reorientation experiments.
The activation energy of roughly \unit[0.9]{eV} nicely compares to experimental values reinforcing the correct identification of the C-Si DB diffusion mechanism.
Slightly increased values compared to experiment might be due to the tightend constraints applied in the modified CRT approach.
Nevertheless, the theoretical description performed in this work is improved compared to a former study \cite{capaz94}, which underestimates the experimental value by \unit[35]{\%}.
In addition, it is finally shown that the BC configuration, for which spin polarized calculations are necessary, constitutes a real local minimum instead of a saddle point configuration due to the presence of restoring forces for displacements in migration direction.

\begin{figure}[ht]
\begin{center}
\includegraphics[width=13cm]{vasp_mig/110_mig_vasp.ps}
%\begin{picture}(0,0)(140,0)
%\includegraphics[width=2.2cm]{vasp_mig/00-1_b.eps}
%\end{picture}
%\begin{picture}(0,0)(20,0)
%\includegraphics[width=2.2cm]{vasp_mig/00-1_ip0-10_sp.eps}
%\end{picture}
%\begin{picture}(0,0)(-120,0)
%\includegraphics[width=2.2cm]{vasp_mig/0-10_b.eps}
%\end{picture}
\end{center}
\caption{Migration barriers of the \hkl<1 1 0> DB to BC (blue), \hkl<0 0 -1> (green) and \hkl<0 -1 0> (in place, red) C-Si DB transition.}
\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.
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.
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}
\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}
\end{center}
\caption{Migration barrier and structures of the bond-centered to \hkl<0 0 -1> dumbbell transition using the classical EA potential.}
\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.
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.

\begin{figure}[ht]
\begin{center}
\includegraphics[width=13cm]{00-1_0-10.ps}\\[2.4cm]
\begin{pspicture}(0,0)(0,0)
\psframe[linecolor=red,fillstyle=none](-6,-0.5)(7.2,2.8)
\end{pspicture}
\begin{picture}(0,0)(130,-10)
\includegraphics[width=2.2cm]{albe_mig/00-1_0-10_red_00.eps}
\end{picture}
\begin{picture}(0,0)(0,-10)
\includegraphics[width=2.2cm]{albe_mig/00-1_0-10_red_min.eps}
\end{picture}
\begin{picture}(0,0)(-120,-10)
\includegraphics[width=2.2cm]{albe_mig/00-1_0-10_red_03.eps}
\end{picture}
\begin{picture}(0,0)(25,10)
\includegraphics[width=2.5cm]{100_arrow.eps}
\end{picture}
\begin{picture}(0,0)(185,-10)
\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.}
% 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}
\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.}
\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.
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.
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.

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.
\begin{figure}[ht]
\begin{center}
\includegraphics[width=13cm]{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.}
\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.
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}.

\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]
\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}
\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}
\end{figure}
\begin{table}[ht]
\begin{center}
\begin{tabular}{l c c c c c c}
\hline
\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\\
\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.}
\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.
\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}
\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.}
\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?}

\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}
\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.}
\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.

\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}
\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.}
\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}.
\begin{table}[ht]
\begin{center}
\begin{tabular}{l c c c c c c}
\hline
\hline
 & 1 & 2 & 3 & 4 & 5 & 6\\
\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>\\
\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.}
\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}
\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.}
\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.

\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}
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.

\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}
\end{center}
\end{minipage}
\begin{minipage}[t]{7cm}
b) \underline{Pos: 3, $E_{\text{b}}=-3.14\text{ eV}$}
\begin{center}
\includegraphics[width=6.0cm]{00-1dc/3-14.eps}
\end{center}
\end{minipage}\\[0.2cm]
\begin{minipage}[t]{7cm}
c) \underline{Pos: 4, $E_{\text{b}}=-0.54\text{ eV}$}
\begin{center}
\includegraphics[width=6.0cm]{00-1dc/0-54.eps}
\end{center}
\end{minipage}
\begin{minipage}[t]{7cm}
d) \underline{Pos: 5, $E_{\text{b}}=-0.50\text{ eV}$}
\begin{center}
\includegraphics[width=6.0cm]{00-1dc/0-50.eps}
\end{center}
\end{minipage}
\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}
\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.

\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> \\
\hline
1 & \RM{1} & \RM{3} & \RM{3} & \RM{1} & \RM{3} & \RM{1} \\
2 & \RM{2} & A & A & \RM{2} & C & \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} \\
\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.}
\label{tab:defects:comb_csub_si110}
\end{table}
\begin{table}[ht]
\begin{center}
\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 \\
\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 \\
$r$ [nm] & 0.292 & 0.394 & 0.241 & 0.453 & 0.407 & 0.408 & 0.452 & 0.392 & 0.456 & 0.453\\
\hline
\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}.}
\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.

\begin{figure}[ht]
\begin{center}
\includegraphics[width=12cm]{c_sub_si110.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}
\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.

{\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.
}
\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.}

{\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.
\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}
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.

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.

\clearpage{}
\cleardoublepage{}

\section{Conclusions concerning the SiC conversion mechanism}

The ground state configuration of a carbon interstitial in crystalline siliocn is found to be the C-Si \hkl<1 0 0> dumbbell interstitial configuration, in which the threefold coordinated carbon and silicon atom share a usual silicon lattice site.
This supports the assumption of C-Si \hkl<1 0 0>-type dumbbel interstitial formation in the first steps of the IBS process as proposed by the precipitation model introduced in section \ref{section:assumed_prec}.

Migration simulations reveal this carbon interstitial to be mobile at prevailing implantation temperatures requireing an activation energy of approximately 0.9 eV for migration as well as reorientation processes.
This enables possible migration of the defects to form defect agglomerates as demanded by the model.
Unfortunately classical potential simulations show tremendously overestimated migration barriers indicating a possible failure of the necessary agglomeration of such defects.

Investigations of two carbon interstitials of the \hkl<1 0 0>-type and varying separations and orientations state an attractive interaction between these interstitials.
Depending on orientation, energetically favorable configurations are found in which these two interstitials are located close together instead of the occurernce of largely separated and isolated defects.
This is due to strain compensation enabled by the combination of such defects in certain orientations.
For dumbbells oriented along the \hkl<1 1 0> direction and the assumption that there is the possibility of free orientation, an interaction energy proportional to the reciprocal cube of the distance in the far field regime is found.
These findings support the assumption of the C-Si dumbbell agglomeration proposed by the precipitation model.

Next to the C-Si \hkl<1 0 0> dumbbell interstitial configuration, in which the C atom is sharing a Si lattice site with the corresponding Si atom the C atom could occupy the site of the Si atom, which in turn forms a Si self-interstitial.
Combinations of substitutional C and a \hkl<1 1 0> Si self-interstitial, which is the ground state configuration for a Si self-interstitial and, thus, assumed to be the energetically most favorable configuration for combined structures, show formation energies 0.5 eV to 1.5 eV greater than that of the C-Si \hkl<1 0 0> interstitial configuration, which remains the energetically most favorable configuration.
However, the binding energy of substitutional C and the Si self-interstitial quickly drops to zero already for short separations indicating a low interaction capture radius.
Thus, due to missing attractive interaction forces driving the system to form C-Si \hkl<1 0 0> dumbbell interstitial complexes substitutional C, while thermodynamically not stable, constitutes a most likely configuration occuring in IBS, a process far from equlibrium.

Due to the low interaction capture radius substitutional C can be treated independently of the existence of separated Si self-interstitials.
This should be also true for combinations of C-Si interstitials next to a vacancy and a further separated Si self-interstitial excluded from treatment, which again is a conveivable configuration in IBS.
By combination of a \hkl<1 0 0> dumbbell with a vacancy in the absence of the Si self-interstitial it is found that the configuration of substitutional carbon occupying the vacant site is the energetically most favorable configuration.
Low migration barriers are necessary to obtain this configuration and in contrast comparatively high activation energies necessary for the reverse process.
Thus, carbon interstitials and vacancies located close together are assumed to end up in such a configuration in which the carbon atom is tetrahedrally coordinated and bound to four silicon atoms as expected in silicon carbide.

While first results support the proposed precipitation model the latter suggest the formation of silicon carbide by succesive creation of substitutional carbon instead of the agglomeration of C-Si dumbbell interstitials followed by an abrupt transition.
Prevailing conditions in the IBS process at elevated temperatures and the fact that IBS is a nonequilibrium process reinforce the possibility of formation of substitutional C instead of the thermodynamically stable C-Si dumbbell interstitials predicted by simulations at zero Kelvin.
\label{section:defects:noneq_process_02}

{\color{blue}
In addition, there are experimental findings, which might be exploited to reinforce the non-validity of the proposed precipitation model.
High resolution TEM shows equal orientation of \hkl(h k l) planes of the c-Si host matrix and the 3C-SiC precipitate.
Formation of 3C-SiC realized by successive formation of substitutional C, in which the atoms belonging to one of the two fcc lattices are substituted by C atoms perfectly conserves the \hkl(h k l) planes of the initial c-Si diamond lattice.
Silicon self-interstitials consecutively created to the same degree are able to diffuse into the c-Si host one after another.
Investigated combinations of C interstitials, however, result in distorted configurations, in which C atoms, which at some point will form SiC, are no longer aligned to the host.
It is easily understandable that the mismatch in alignement will increase with increasing defect density.
In addition, the amount of Si self-interstitials equal to the amount of agglomerated C atoms would be released all of a sudden probably not being able to diffuse into the c-Si host matrix without damaging the Si surrounding or the precipitate itself.
In addition, IBS results in the formation of the cubic polytype of SiC only.
As this result conforms well with the model of precipitation by substitutional C there is no obvious reason why hexagonal polytypes should not be able to form or an equal alignement would be mandatory assuming the model of precipitation by C-Si dumbbell agglomeration.
}

{\color{red}Todo: C mobility higher than Si mobility? -> substitutional C is more likely to arise, since it migrates 'faster' to vacant sites?}