first sent away beta version

[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 that are defined in chapter \ref{chapter:simulation}.
Both methods are used to investigate selected diffusion processes of some of the defect configurations.
While the quantum-mechanical description yields results that excellently compare to experimental findings, shortcomings of the classical potential approach are identified.
These shortcomings are further investigated and the basis for a workaround, as proposed later on in the classical MD simulation chapter, is discussed.

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}[tp]
\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}[tp]
\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}[tp]
\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}[tp]
\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}[tp]
\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
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 \\
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}[tp]
\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 the 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}[tp]
\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}[tp]
\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}[tp]
\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}[tp]
\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}[tp]
\begin{center}
\begin{minipage}{8cm}
\begin{center}
\includegraphics[width=6cm]{c_pd_vasp/bc_2333.eps}\\
\vspace*{0.2cm}
\hrule
\vspace*{0.2cm}
\includegraphics[width=6cm]{c_100_mig_vasp/im_spin_diff.eps}
\vspace*{0.2cm}
\framebox{
 \footnotesize
 \begin{minipage}[t]{7.5cm}
  \begin{minipage}[t]{1.4cm}
  {\color{red}Si}\\
  {\tiny sp$^3$}\\[0.8cm]
  \underline{${\color{black}\uparrow}$}
  \underline{${\color{black}\uparrow}$}
  \underline{${\color{black}\uparrow}$}
  \underline{${\color{red}\uparrow}$}\\
  sp$^3$
  \end{minipage}
  \begin{minipage}[t]{1.6cm}
  \begin{center}
  {\color{red}M}{\color{blue}O}\\[0.8cm]
  \underline{${\color{blue}\uparrow}{\color{white}\downarrow}$}\\
  $\sigma_{\text{ab}}$\\[0.5cm]
  \underline{${\color{red}\uparrow}{\color{blue}\downarrow}$}\\
  $\sigma_{\text{b}}$
  \end{center}
  \end{minipage}
  \begin{minipage}[t]{1.2cm}
  \begin{center}
  {\color{blue}C}\\
  {\tiny sp}\\[0.2cm]
  \underline{${\color{white}\uparrow\uparrow}$}
  \underline{${\color{white}\uparrow\uparrow}$}\\
  2p\\[0.4cm]
  \underline{${\color{blue}\uparrow}{\color{blue}\downarrow}$}
  \underline{${\color{blue}\uparrow}{\color{blue}\downarrow}$}\\
  sp
  \end{center}
  \end{minipage}
  \begin{minipage}[t]{1.6cm}
  \begin{center}
  {\color{blue}M}{\color{green}O}\\[0.8cm]
  \underline{${\color{blue}\uparrow}{\color{white}\downarrow}$}\\
  $\sigma_{\text{ab}}$\\[0.5cm]
  \underline{${\color{green}\uparrow}{\color{blue}\downarrow}$}\\
  $\sigma_{\text{b}}$
  \end{center}
  \end{minipage}
  \begin{minipage}[t]{1.4cm}
  \begin{flushright}
  {\color{green}Si}\\
  {\tiny sp$^3$}\\[0.8cm]
  \underline{${\color{green}\uparrow}$}
  \underline{${\color{black}\uparrow}$}
  \underline{${\color{black}\uparrow}$}
  \underline{${\color{black}\uparrow}$}\\
  sp$^3$
  \end{flushright}
  \end{minipage}
 \end{minipage}
}
\end{center}
\end{minipage}
\begin{minipage}{7cm}
\includegraphics[width=7cm]{c_pd_vasp/bc_2333_ksl.ps}
\end{minipage}
\end{center}
\caption[Structure, charge density isosurface, molecular orbital diagram and Kohn-Sham level diagram of the bond-centered interstitial configuration.]{Structure, charge density, molecular orbital diagram 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 in between 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.
In fact, the net magnetization of two electrons is already suggested by simple molecular orbital theory considerations with respect to the bonding of the C atom.
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.

\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}[tp]
\begin{center}
%
\begin{minipage}{15cm}
\centering
\framebox{\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}\\[0.5cm]
%
\begin{minipage}{15cm}
\centering
\framebox{\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}\\[0.5cm]
%
\begin{minipage}{15cm}
\centering
\framebox{\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}[tp]
\begin{center}
\includegraphics[width=0.7\textwidth]{im_00-1_nosym_sp_fullct_thesis_vasp_s.ps}
\end{center}
\caption[Migration barrier and structures of the \hkl<0 0 -1> DB (left) to BC (right) transition.]{Migration barrier and structures of the \hkl<0 0 -1> DB (left) to BC (right) transition. Bonds of the C atom are illustrated by blue lines.}
\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}[tp]
\begin{center}
\includegraphics[width=0.7\textwidth]{00-1_0-10_vasp_s.ps}
\end{center}
\caption[Migration barrier and structures of the \hkl<0 0 -1> DB (left) to the \hkl<0 -1 0> DB (right) transition.]{Migration barrier and structures of the \hkl[0 0 -1] DB (left) to the \hkl[0 -1 0] DB (right) transition. Bonds of the C atom are illustrated by blue lines.}
% todo read above caption! enable [] hkls in short caption
\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}[tp]
\begin{center}
\includegraphics[width=0.7\textwidth]{00-1_ip0-10_nosym_sp_fullct_vasp_s.ps}
\end{center}
\caption[Reorientation barrier and structures of the \hkl<0 0 -1> DB (left) to the \hkl<0 -1 0> DB (right) transition in place.]{Reorientation barrier and structures of the \hkl<0 0 -1> DB (left) to the \hkl<0 -1 0> DB (right) transition in place. Bonds of the carbon atoms are illustrated by blue lines.}
\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}[tp]
\begin{center}
\includegraphics[width=0.7\textwidth]{vasp_mig/110_mig_vasp.ps}
%\begin{picture}(0,0)(140,0)
%\includegraphics[width=2.2cm]{vasp_mig/00-1_b.eps}
%\end{picture}
%\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.
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 DB interstitial.
Fig. \ref{fig:defects:110_mig_vasp} shows the migration barrier of the \hkl<1 1 0> C-Si DB to the BC, \hkl<0 0 -1> and \hkl<0 -1 0> (in place) transition.
Indeed less than \unit[0.7]{eV} are necessary to turn a \hkl<0 -1 0>- to a \hkl<1 1 0>-type C-Si DB interstitial.
This transition is carried out in place, i.e. the Si DB pair is not changed and both, the Si and C atom share the initial lattice site.
Thus, this transition does not contribute to long-range diffusion.
Once the C atom resides in the \hkl<1 1 0> DB interstitial configuration it can migrate into the BC configuration requiring approximately \unit[0.95]{eV} of activation energy, which is only slightly higher than the activation energy needed for the \hkl<0 0 -1> to \hkl<0 -1 0> pathway as shown in Fig. \ref{fig:defects:00-1_0-10_mig}.
As already known from the migration of the \hkl<0 0 -1> to the BC configuration discussed in Fig. \ref{fig:defects:00-1_001_mig}, another \unit[0.25]{eV} are needed to turn back from the BC to a \hkl<1 0 0>-type interstitial.
However, due to the fact that this migration consists of three single transitions with the second one having an activation energy slightly higher than observed for the direct transition, this sequence of paths is considered very unlikely to occur.
The migration barrier of the \hkl<1 1 0> to \hkl<0 0 -1> transition, in which the C atom is changing its Si partner and, thus, moving to the neighbored lattice site, corresponds to approximately \unit[1.35]{eV}.
During this transition the C atom is escaping the \hkl(1 1 0) plane approaching the final configuration on a curved path.
This barrier is much higher than the ones found previously, which again make this transition very unlikely to occur.
For this reason, the assumption that C diffusion and reorientation is achieved by transitions of the type presented in Fig. \ref{fig:defects:00-1_0-10_mig} is reinforced.

%As mentioned earlier the procedure to obtain the migration barriers differs from the usually applied procedure in two ways.
%Firstly constraints to move along the displacement direction are applied on all atoms instead of solely constraining the diffusing atom.
%Secondly the constrainted directions are not kept constant to the initial displacement direction.
%Instead they are updated for every displacement step.
%These modifications to the usual procedure are applied to avoid abrupt changes in structure and free energy on the one hand and to make sure the expected final configuration is reached on the other hand.
%Due to applying updated constraints on all atoms the obtained migration barriers and pathes might be overestimated and misguided.
%To reinforce the applicability of the employed technique the obtained activation energies and migration pathes for the \hkl<0 0 -1> to \hkl<0 -1 0> transition are compared to two further migration calculations, which do not update the constrainted direction and which only apply updated constraints on three selected atoms, that is the diffusing C atom and the Si dumbbell pair in the initial and final configuration.
%Results are presented in figure \ref{fig:defects:00-1_0-10_cmp}.
%\begin{figure}[tp]
%\begin{center}
%\includegraphics[width=13cm]{vasp_mig/00-1_0-10_nosym_sp_cmp.ps}
%\end{center}
%\caption[Comparison of three different techniques for obtaining migration barriers and pathways applied to the \hkl<0 0 -1> to \hkl<0 -1 0> C-Si dumbbell transition.]{Comparison of three different techniques for obtaining migration barriers and pathways applied to the \hkl<0 0 -1> to \hkl<0 -1 0> C-Si dumbbell transition.}
%\label{fig:defects:00-1_0-10_cmp}
%\end{figure}
%The method without updating the constraints but still applying them to all atoms shows a delayed crossing of the saddle point.
%This is understandable since the update results in a more aggressive advance towards the final configuration.
%In any case the barrier obtained is slightly higher, which means that it does not constitute an energetically more favorable pathway.
%The method in which the constraints are only applied to the diffusing C atom and two Si atoms, ... 

% todo if there is plenty of time ... reinvestigate above stuff

\subsection{Migration described by classical potential calculations}
\label{subsection:defects:mig_classical}

\begin{figure}[tp]
\begin{center}
\includegraphics[width=0.7\textwidth]{bc_00-1_albe_s.ps}
%\includegraphics[width=13cm]{bc_00-1.ps}\\[5.6cm]
%\begin{pspicture}(0,0)(0,0)
%\psframe[linecolor=red,fillstyle=none](-7,2.7)(7.2,6)
%\end{pspicture}
%\begin{picture}(0,0)(140,-100)
%\includegraphics[width=2.4cm]{albe_mig/bc_00-1_red_00.eps}
%\end{picture}
%\begin{picture}(0,0)(10,-100)
%\includegraphics[width=2.4cm]{albe_mig/bc_00-1_red_01.eps}
%\end{picture}
%\begin{picture}(0,0)(-120,-100)
%\includegraphics[width=2.4cm]{albe_mig/bc_00-1_red_02.eps}
%\end{picture}
%\begin{picture}(0,0)(25,-80)
%\includegraphics[width=2.5cm]{110_arrow.eps}
%\end{picture}
%\begin{picture}(0,0)(215,-100)
%\includegraphics[height=2.2cm]{001_arrow.eps}
%\end{picture}\\
%\begin{pspicture}(0,0)(0,0)
%\psframe[linecolor=blue,fillstyle=none](-7,-0.5)(7.2,2.8)
%\end{pspicture}
%\begin{picture}(0,0)(160,-10)
%\includegraphics[width=2.2cm]{albe_mig/bc_00-1_01.eps}
%\end{picture}
%\begin{picture}(0,0)(100,-10)
%\includegraphics[width=2.2cm]{albe_mig/bc_00-1_02.eps}
%\end{picture}
%\begin{picture}(0,0)(10,-10)
%\includegraphics[width=2.2cm]{albe_mig/bc_00-1_03.eps}
%\end{picture}
%\begin{picture}(0,0)(-120,-10)
%\includegraphics[width=2.2cm]{albe_mig/bc_00-1_04.eps}
%\end{picture}
%\begin{picture}(0,0)(25,10)
%\includegraphics[width=2.5cm]{100_arrow.eps}
%\end{picture}
%\begin{picture}(0,0)(215,-10)
%\includegraphics[height=2.2cm]{010_arrow.eps}
%\end{picture}
\end{center}
\caption[Migration barrier and structures of the \ci{} BC to \hkl<0 0 -1> DB transition using the classical EA potential.]{Migration barrier and structures of the \ci{} BC to \hkl[0 0 -1] DB transition using the classical EA potential. Two migration pathways are obtained for different time constants of the Berendsen thermostat. The lowest activation energy is \unit[2.2]{eV}.}
\label{fig:defects:cp_bc_00-1_mig}
% red: ./visualize -w 640 -h 480 -d saves/c_in_si_mig_bc_00-1_s20 -nll -0.56 -0.56 -0.7 -fur 0.2 0.2 0.0 -c 0.75 -1.25 -0.25 -L -0.25 -0.25 -0.25 -r 0.6 -B 0.1
% blue: ./visualize -w 640 -h 480 -d saves/c_in_si_mig_bc_00-1_s20_tr100/ -nll -0.56 -0.56 -0.7 -fur 0.2 0.2 0.0 -c 0.0 -0.25 1.0 -L 0.0 -0.25 -0.25 -r 0.6 -B 0.1
\end{figure}
Fig. \ref{fig:defects:cp_bc_00-1_mig} shows the evolution of structure and energy along the \ci{} BC to \hkl<0 0 -1> DB transition.
Since the \ci{} BC configuration is unstable relaxing into the \hkl<1 1 0> DB configuration within this potential, the low kinetic energy state is used as a starting configuration.
Two different pathways are obtained for different time constants of the Berendse
n thermostat.
With a time constant of \unit[1]{fs} the C atom resides in the \hkl(1 1 0) plane
 resulting in a migration barrier of \unit[2.4]{eV}.
However, weaker coupling to the heat bath realized by an increase of the time constant to \unit[100]{fs} enables the C atom to move out of the \hkl(1 1 0) plane already at the beginning, which is accompanied by a reduction in energy, approaching the final configuration on a curved path.
The energy barrier of this path is \unit[0.2]{eV} lower in energy than the direct migration within the \hkl(1 1 0) plane.
However, the investigated pathways cover an activation energy approximately twice as high as the one obtained by quantum-mechanical calculations.
If the entire transition of the \hkl<0 0 -1> into the \hkl<0 0 1> configuration is considered a two step process passing the intermediate BC configuration, an additional activation energy of \unit[0.5]{eV} is necessary to escape the BC towards the \hkl<0 0 1> configuration.
Assuming equal preexponential factors for both diffusion steps, the total probability of diffusion is given by $\exp\left((2.2\,\text{eV}+0.5\,\text{eV})/k_{\text{B}}T\right)$.
Thus, the activation energy should be located within the range of \unit[2.2-2.7]{eV}.

\begin{figure}[tp]
\begin{center}
\includegraphics[width=0.7\textwidth]{00-1_0-10_albe_s.ps}
\end{center}
\caption{Migration barrier and structures of the \ci{} \hkl<0 0 -1> to \hkl<0 -1 0>  DB transition using the classical EA potential.}
% 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}[tp]
\begin{center}
\includegraphics[width=0.7\textwidth]{00-1_ip0-10.ps}
\end{center}
\caption{Reorientation barrier of the \ci{} \hkl<0 0 -1> to \hkl<0 -1 0> DB transition in place using the classical EA potential.}
\label{fig:defects:cp_00-1_ip0-10_mig}
\end{figure}
Figures \ref{fig:defects:cp_00-1_0-10_mig} and \ref{fig:defects:cp_00-1_ip0-10_mig} show the migration barriers of the \ci{} \hkl<0 0 -1> to \hkl<0 -1 0> DB transition.
In the first case, the transition involves a change in the lattice site of the C atom whereas in the second case, a reorientation at the same lattice site takes place.
In the first case, the pathways for the two different time cosntants look similar.
A local minimum exists in between two peaks of the graph.
The corresponding configuration, which is illustrated for the results obtained for a time constant of \unit[1]{fs}, looks similar to the \ci{} \hkl<1 1 0> configuration.
Indeed, this configuration is obtained by relaxation simulations without constraints of configurations near the minimum.
Activation energies of roughly \unit[2.8]{eV} and \unit[2.7]{eV} are needed for migration.

The \ci{} \hkl<1 1 0> configuration seems to play a decisive role in all migration pathways in the classical potential calculations.
As mentioned above, the starting configuration of the first migration path, i.e. the BC configuration, is fixed to be a transition point but in fact is unstable.
Further relaxation of the BC configuration results in the \ci{} \hkl<1 1 0> configuration.
Even the last two pathways show configurations almost identical to the \ci{} \hkl<1 1 0> configuration, which constitute local minima within the pathways.
Thus, migration pathways involving the \ci{} \hkl<1 1 0> DB configuration as a starting or final configuration are further investigated.
\begin{figure}[tp]
\begin{center}
\includegraphics[width=0.7\textwidth]{110_mig.ps}
\end{center}
\caption[Migration barriers of the \ci{} \hkl<1 1 0> DB to BC (blue), \hkl<0 0 -1> (green) and \hkl<0 -1 0> (in place, red) transition.]{Migration barriers of the \ci{} \hkl<1 1 0> DB to BC (blue), \hkl<0 0 -1> (green) and \hkl<0 -1 0> (in place, red) transition. Solid lines show results for a time constant of \unit[1]{fs} and dashed lines correspond to simulations employing a time constant of \unit[100]{fs}.}
\label{fig:defects:110_mig}
\end{figure}
Fig. \ref{fig:defects:110_mig} shows migration barriers of the \ci{} \hkl<1 1 0> DB to \hkl<0 0 -1>, \hkl<0 -1 0> (in place) and BC configuration.
As expected there is no maximum for the transition into the BC configuration.
As mentioned earlier the BC configuration itself constitutes a saddle point configuration relaxing into the energetically more favorable \hkl<1 1 0> DB configuration.
An activation energy of \unit[2.2]{eV} is necessary to reorientate the \hkl<0 0 -1> into the \hkl<1 1 0> DB configuration, which is \unit[1.3]{eV} higher in energy.
Residing in this state another \unit[0.90]{eV} is enough to make the C atom form a \hkl<0 0 -1> DB configuration with the Si atom of the neighbored lattice site.
In contrast to quantum-mechanical calculations, in which the direct transition is the energetically most favorable transition and the transition composed of the intermmediate migration steps is very unlikely to occur, the just presented pathway is much more conceivable in classical potential simulations, since the energetically most favorable transition found so far is likewise composed of two migration steps with activation energies of \unit[2.2]{eV} and \unit[0.5]{eV}, for which the intermediate state is the BC configuration, which is unstable.
Thus the just proposed migration path, which involves the \hkl<1 1 0> interstitial configuration, becomes even more probable than the initially porposed path, which involves the BC configuration that is, in fact, unstable.
Due to these findings, the respective path is proposed to constitute the diffusion-describing path.
The evolution of structure and configurational energy is displayed again in Fig. \ref{fig:defects:involve110}.
\begin{figure}[tp]
\begin{center}
\includegraphics[width=0.7\textwidth]{00-1_110_0-10_mig_albe.ps}
\end{center}
\caption[Migration barrier and structures of the \ci{} \hkl<0 0 -1> (left) to the \hkl<0 -1 0> DB (right) transition involving the \hkl<1 1 0> DB (center) configuration.]{Migration barrier and structures of the \ci{} \hkl[0 0 -1] (left) to the \hkl[0 -1 0] DB (right) transition involving the \hkl[1 1 0] DB (center) configuration. Migration simulations are performed utilizing time constants of \unit[1]{fs} (solid line) and \unit[100]{fs} (dashed line) for the Berendsen thermostat.}
\label{fig:defects:involve110}
\end{figure}
Approximately \unit[2.2]{eV} are needed to turn the \ci{} \hkl[0 0 -1] into the \hkl[1 1 0] DB located at the neighbored lattice site in \hkl[1 1 -1] direction.
Another barrier of \unit[0.90]{eV} exists for the rotation into the \ci{} \hkl[0 -1 0] DB configuration for the path obtained with a time constant of \unit[100]{fs} for the Berendsen thermostat.
Roughly the same amount would be necessary to excite the C$_{\text{i}}$ \hkl[1 1 0] DB to the BC configuration (\unit[0.40]{eV}) and a successive migration into the \hkl[0 0 1] DB configuration (\unit[0.50]{eV}) as displayed in Fig. \ref{fig:defects:110_mig} and Fig. \ref{fig:defects:cp_bc_00-1_mig}.
The former diffusion process, however, would more nicely agree with the ab initio path, since the migration is accompanied by a rotation of the DB orientation.
By considering a two step process and assuming equal preexponential factors for both diffusion steps, the probability of the total diffusion event is given by $\exp(\frac{\unit[2.24]{eV}+\unit[0.90]{eV}}{k_{\text{B}}T})$, which corresponds to a single diffusion barrier that is 3.5 times higher than the barrier obtained by {em ab initio} calculations.

\subsection{Conclusions}

Although classical potential simulations reproduce the same order in energy of the \ci{} \hkl<1 0 0> and \hkl<1 1 0> DB interstitial configurations as obtained by more accurate quantum-mechanical calculations, the obtained migration pathways and resulting activation energies differ to a great extent.
On the one hand, the most favorable pathways differ.
However, the pathway, which is considered most probable in the classical potential treatment, exhibits the same starting and final configuration of the DB structure as well as the change in orientation during migration as obtained by quantum-mechanical calculations.
On the other hand, the activation energy obtained by classical potential simulations is tremendously overestimated by a factor of 2.4 to 3.5.
The overestimated barrier is due to the short range character of the potential, which drops the interaction to zero within the first and next neighbor distance.
Since the total binding energy is accommodated within a short distance, which according to the universal energy relation would usually correspond to a much larger distance, unphysical high forces between two neighbored atoms arise.
This is explained in more detail in a previous study \cite{mattoni2007}.
Thus, atomic diffusion is wrongly described in the classical potential approach.
The probability of already rare diffusion events is further decreased for this reason.
However, agglomeration of C and diffusion of Si self-interstitials are an important part of the proposed SiC precipitation mechanism.
Thus, a serious limitation that has to be taken into account for appropriately modeling the C/Si system using the otherwise quite promising EA potential is revealed.
Possible workarounds are discussed in more detail in section \ref{section:md:limit}.

\section{Combination of point defects and related diffusion processes}

The study proceeds with a structural and energetic investigation of pairs of the ground-state and, thus, most probable defect configurations that are believed to be fundamental in the Si to SiC conversion.
Investigations are restricted to quantum-mechanical calculations.
\begin{figure}[tp]
% ./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
\begin{center}
\subfigure[]{\label{fig:defects:combos_ci}\includegraphics[width=0.3\textwidth]{combos_ci_col.eps}}
\hspace{0.5cm}
\subfigure[]{\label{fig:defects:combos_si}\includegraphics[width=0.3\textwidth]{combos.eps}}
\end{center}
\caption{Position of the initial \ci{} \hkl[0 0 -1] DB (I) (a) and of the lattice site chosen for the initial \si{} \hkl<1 1 0> DB (\si) (b). Lattice sites for the second defect used for investigating defect pairs are numbered from 1 to 5. For black/red/blue numbers, one/two/four possible atom(s) exist for the second defect to create equivalent defect combinations.}
\label{fig:defects:combos}
\end{figure}
Fig.~\ref{fig:defects:combos} schematically displays the initial \ci{} \hkl[0 0 -1] DB structure (Fig.~\ref{fig:defects:combos_ci}) as well as the lattice site chosen for the initial \si{} \hkl<1 1 0> DB (Fig.~\ref{fig:defects:combos_si}) and various positions for the second defect (1-5) that are used for investigating defect pairs.
The color of the number denotes the amount of possible atoms for the second defect resulting in equivalent configurations.
Binding energies of the defect pair are determined by equation \ref{eq:basics:e_bind}.
Next to formation and binding energies, migration barriers are investigated, which allow to draw conclusions on the probability of the formation of such defect complexes by thermally activated diffusion processes.

\subsection[Pairs of \ci{} \hkl<1 0 0>-type interstitials]{\boldmath Pairs of \ci{} \hkl<1 0 0>-type interstitials}
\label{subsection:defects:c-si_comb}

\ci{} pairs of the \hkl<1 0 0>-type are investigated in the first part.
\begin{table}[tp]
\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[Binding energies in eV of \ci{} \hkl<1 0 0>-type defect pairs.]{Binding energies in eV of \ci{} \hkl<1 0 0>-type defect pairs. The given energies in eV are defined by equation \eqref{eq:basics:e_bind}. Equivalent configurations are marked by identical colors. The first column lists the types of the second defect combined with the initial \ci \hkl[0 0 -1] DB interstitial. The position index of the second defect is given in the first row according to Fig.~\ref{fig:defects:combos_ci}. R is the position located at $\frac{a_{\text{Si}}}{2}\hkl[3 2 3]$ relative to the initial defect, which is the maximum realizable distance due to periodic boundary conditions.}
\label{tab:defects:e_of_comb}
\end{table}
Table~\ref{tab:defects:e_of_comb} summarizes resulting binding energies for the combination with a second \ci{} \hkl<1 0 0> DB obtained for different orientations at positions 1 to 5 after structural relaxation.
Most of the obtained configurations result in binding energies well below zero indicating a preferable agglomeration of this type of the defects.
For increasing distances of the defect pair, the binding energy approaches to zero as it is expected for non-interacting isolated defects.
%
In fact, a \ci{} \hkl[0 0 -1] DB interstitial created at position R separated by a distance of $\frac{a_{\text{Si}}}{2}\hkl<3 2 3>$ ($\approx$\unit[12.8]{\AA}) from the initial one results in an energy as low as \unit[-0.19]{eV}.
There is still a low interaction remaining, which is due to the equal orientation of the defects.
By changing the orientation of the second DB interstitial to the \hkl<0 -1 0>-type, the interaction is even more reduced resulting in an energy of \unit[-0.05]{eV} for a distance, which is the maximum that can be realized due to periodic boundary conditions.
Energetically favorable and unfavorable configurations can be explained by stress compensation and increase respectively based on the resulting net strain of the respective configuration of the defect combination.
Antiparallel orientations of the second defect, i.e. \hkl[0 0 1] for positions located below the \hkl(0 0 1) plane with respect to the initial one (positions 1, 2 and 4) form the energetically most unfavorable configurations.
In contrast, the parallel and particularly the twisted orientations constitute energetically favorable configurations, in which a vast reduction of strain is enabled by combination of these defects.

\begin{figure}[tp]
\begin{center}
\subfigure[\underline{$E_{\text{b}}=-2.25\,\text{eV}$}]{\label{fig:defects:225}\includegraphics[width=0.3\textwidth]{00-1dc/2-25.eps}}
\hspace{0.5cm}
\subfigure[\underline{$E_{\text{b}}=-2.39\,\text{eV}$}]{\label{fig:defects:239}\includegraphics[width=0.3\textwidth]{00-1dc/2-39.eps}}
\end{center}
\caption{Relaxed structures of defect combinations obtained by creating \hkl[1 0 0] (a) and \hkl[0 -1 0] (b) DBs at position 1.}
\label{fig:defects:comb_db_01}
\end{figure}
Mattoni~et~al. \cite{mattoni2002} predict the ground-state configuration of \ci{} \hkl<1 0 0>-type defect pairs for a \hkl[1 0 0] or equivalently a \hkl[0 1 0] defect created at position 1 with both defects basically maintaining the as-isolated DB structure, resulting in a binding energy of \unit[-2.1]{eV}.
In the present study, a further relaxation of this defect structure is observed.
The C atom of the second and the Si atom of the initial DB move towards each other forming a bond, which results in a somewhat lower binding energy of \unit[-2.25]{eV}.
The corresponding defect structure is displayed in Fig.~\ref{fig:defects:225}.
In this configuration the initial Si and C DB atoms are displaced along \hkl[1 0 0] and \hkl[-1 0 0] in such a way that the Si atom is forming tetrahedral bonds with two Si and two C atoms.
The C and Si atom constituting the second defect are as well displaced in such a way, that the C atom forms tetrahedral bonds with four Si neighbors, a configuration expected in SiC.
The two carbon atoms, which are spaced by \unit[2.70]{\AA}, do not form a bond but anyhow reside in a shorter distance than expected in SiC.
Si atom number 2 is pushed towards the C atom, which results in the breaking of the bond to Si atom number 4.
Breaking of the $\sigma$ bond is indeed confirmed by investigating the charge density isosurface of this configuration.
 
Apart from that, a more favorable configuration is found for the combination with a \hkl[0 -1 0] and \hkl[-1 0 0] DB respectively, which is assumed to constitute the actual ground-state configuration of two \ci{} DBs in Si.
The atomic arrangement is shown in Fig.~\ref{fig:defects:239}.
The initial configuration is still evident in the relaxed configuration.
The two \ci{} atoms form a strong C-C bond, which is responsible for the large gain in energy resulting in a binding energy of \unit[-2.39]{eV}.
This bond has a length of \unit[1.38]{\AA} close to the next neighbor distance in diamond or graphite, which is approximately \unit[1.54]{\AA}.
The minimum of the binding energy observed for this configuration suggests prefered C clustering as a competing mechnism to the \ci{} DB interstitial agglomeration inevitable for the SiC precipitation.
However, the second most favorable configuration ($E_{\text{f}}=-2.25\,\text{eV}$) is represented four times, i.e. two times more often than the ground-state configuration, within the systematically investigated configuration space.
Thus, particularly at high temepratures that cause an increase of the entropic contribution, this structure constitutes a serious opponent to the ground state.
In fact, following results on migration simulations will reinforce the assumption of a low probability for C clustering by thermally activated processes.

\begin{figure}[tp]
\begin{center}
\subfigure[\underline{$E_{\text{b}}=-2.16\,\text{eV}$}]{\label{fig:defects:216}\includegraphics[width=0.25\textwidth]{00-1dc/2-16.eps}}
\hspace{0.2cm}
\subfigure[\underline{$E_{\text{b}}=-1.90\,\text{eV}$}]{\label{fig:defects:190}\includegraphics[width=0.25\textwidth]{00-1dc/1-90.eps}}
\hspace{0.2cm}
\subfigure[\underline{$E_{\text{b}}=-2.05\,\text{eV}$}]{\label{fig:defects:205}\includegraphics[width=0.25\textwidth]{00-1dc/2-05.eps}}
\end{center}
\caption{Relaxed structures of defect combinations obtained by creating \hkl[1 0 0] (a) and \hkl[0 1 0] (b) DBs at position 2 and a \hkl[0 0 1] (c) DB at position 3.}
\label{fig:defects:comb_db_02}
\end{figure}
Fig.~\ref{fig:defects:comb_db_02} shows the next three energetically favorable configurations.
The relaxed configuration obtained by creating a \hkl[1 0 0] DB at position 2 is shown in Fig. \ref{fig:defects:216}.
A binding energy of \unit[-2.16]{eV} is observed.
After relaxation, the second DB is aligned along \hkl[1 1 0].
The bond of Si atoms 1 and 2 does not persist.
Instead, the Si atom forms a bond with the initial \ci{} and the second C atom forms a bond with Si atom 1 forming four bonds in total.
The C atoms are spaced by \unit[3.14]{\AA}, which is very close to the expected C-C next neighbor distance of \unit[3.08]{\AA} in SiC.
Figure \ref{fig:defects:205} displays the results of a \hkl[0 0 1] DB inserted at position 3.
The binding energy is \unit[-2.05]{eV}.
Both DBs are tilted along the same direction remaining parallely aligned and the second DB is pushed downwards in such a way, that the four DB atoms form a rhomboid.
Both C atoms form tetrahedral bonds to four Si atoms.
However, Si atom number 1 and number 3, which are bound to the second \ci{} atom are also bound to the initial C atom.
These four atoms of the rhomboid reside in a plane and, thus, do not match the situation in SiC.
The Carbon atoms have a distance of \unit[2.75]{\AA}.
In Fig. \ref{fig:defects:190} the relaxed structure of a \hkl[0 1 0] DB constructed at position 2 is displayed.
An energy of \unit[-1.90]{eV} is observed.
The initial DB and especially the C atom is pushed towards the Si atom of the second DB forming an additional fourth bond.
Si atom number 1 is pulled towards the C atoms of the DBs accompanied by the disappearance of its bond to Si number 5 as well as the bond of Si number 5 to its neighbored Si atom in \hkl[1 1 -1] direction.
The C atom of the second DB forms threefold coordinated bonds to its Si neighbors.
A distance of \unit[2.80]{\AA} is observed for the two C atoms.
Again, the two C atoms and its two interconnecting Si atoms form a rhomboid.
C-C distances of \unit[2.70-2.80]{\AA} seem to be characteristic for such configurations, in which the C atoms and the two interconnecting Si atoms reside in a plane.

Configurations obtained by adding a \ci{} \hkl<1 0 0> DB at position 4 are characterized by minimal changes from their initial creation condition during relaxation.
There is a low interaction of the DBs, which seem to exist independent of each other.
This, on the one hand, becomes evident by investigating the final structure, in which both of the DBs essentially retain the structure expected for a single DB and, on the other hand, is supported by the observed binding energies, which vary only slightly around zero.
This low interaction is due to the large distance and a missing direct connection by bonds along a chain in the crystallographic \hkl<1 1 0> direction.
Both, C and Si atoms of the DBs form threefold coordinated bonds to their neighbors.
The energetically most unfavorable configuration ($E_{\text{b}}=0.26\,\text{eV}$) is obtained for the \ci{} \hkl[0 0 1] DB, which is oppositely orientated with respect to the initial one.
A DB taking the same orientation as the initial one is less unfavorble ($E_{\text{b}}=0.04\,\text{eV}$).
Both configurations are unfavorable compared to far-off, isolated DBs.
Nonparallel orientations, i.e. the \hkl[0 1 0], \hkl[0 -1 0] and its equivalents, result in binding energies of \unit[-0.12]{eV} and \unit[-0.27]{eV}, thus, constituting energetically favorable configurations.
The reduction of strain energy is higher in the second case, where the C atom of the second DB is placed in the direction pointing away from the initial C atom.

\begin{figure}[tp]
\begin{center}
\subfigure[\underline{$E_{\text{b}}=-1.53\,\text{eV}$}]{\label{fig:defects:153}\includegraphics[width=0.25\textwidth]{00-1dc/1-53.eps}}
\hspace{0.7cm}
\subfigure[\underline{$E_{\text{b}}=-1.66\,\text{eV}$}]{\label{fig:defects:166}\includegraphics[width=0.25\textwidth]{00-1dc/1-66.eps}}\\
\subfigure[\underline{$E_{\text{b}}=-1.88\,\text{eV}$}]{\label{fig:defects:188}\includegraphics[width=0.25\textwidth]{00-1dc/1-88.eps}}
\hspace{0.7cm}
\subfigure[\underline{$E_{\text{b}}=-1.38\,\text{eV}$}]{\label{fig:defects:138}\includegraphics[width=0.25\textwidth]{00-1dc/1-38.eps}}
\end{center}
\caption{Relaxed structures of defect combinations obtained by creating \hkl[0 0 1] (a), \hkl[0 0 -1] (b), \hkl[0 -1 0] (c) and \hkl[1 0 0] (d) DBs at position 5.}
\label{fig:defects:comb_db_03}
\end{figure}
Energetically beneficial configurations of defect combinations are observed for interstititals of all orientations placed at position 5, a position two bonds away from the initial interstitial along the \hkl[1 1 0] direction.
Relaxed structures of these combinations are displayed in Fig. \ref{fig:defects:comb_db_03}.
Fig. \ref{fig:defects:153} and \ref{fig:defects:166} show the relaxed structures of \hkl[0 0 1] and \hkl[0 0 -1] DBs.
The upper DB atoms are pushed towards each other forming fourfold coordinated bonds.
While the displacements of the Si atoms in case (b) are symmetric to the \hkl(1 1 0) plane, in case (a) the Si atom of the initial DB is pushed a little further in the direction of the C atom of the second DB than the C atom is pushed towards the Si atom.
The bottom atoms of the DBs remain in threefold coordination.
The symmetric configuration is energetically more favorable ($E_{\text{b}}=-1.66\,\text{eV}$) since the displacements of the atoms is less than in the antiparallel case ($E_{\text{b}}=-1.53\,\text{eV}$).
In Fig. \ref{fig:defects:188} and \ref{fig:defects:138} the non-parallel orientations, namely the \hkl[0 -1 0] and \hkl[1 0 0] DBs, are shown.
Binding energies of \unit[-1.88]{eV} and \unit[-1.38]{eV} are obtained for the relaxed structures.
In both cases the Si atom of the initial interstitial is pulled towards the near by atom of the second DB.
Both atoms form fourfold coordinated bonds to their neighbors.
In case (c) it is the C and in case (d) the Si atom of the second interstitial, which forms the additional bond with the Si atom of the initial interstitial.
The respective atom of the second DB, the \ci{} atom of the initial DB and the two interconnecting Si atoms again reside in a plane.
As observed before, a typical C-C distance of \unit[2.79]{\AA} is, thus, observed for case (c).
In both configurations, the far-off atom of the second DB resides in threefold coordination.

The interaction of \ci{} \hkl<1 0 0> DBs is investigated along the \hkl[1 1 0] bond chain assuming a possible reorientation of the DB atom at each position to minimize its configurational energy.
Therefore, the binding energies of the energetically most favorable configurations with the second DB located along the \hkl[1 1 0] direction and resulting C-C distances of the relaxed structures are summarized in Table~\ref{tab:defects:comb_db110}.
\begin{table}[tp]
\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 energies $E_{\text{b}}$, C-C distance and types of energetically most favorable \ci{} \hkl<1 0 0>-type defect pairs separated along the \hkl[1 1 0] bond chain.}
\label{tab:defects:comb_db110}
\end{table}
%
\begin{figure}[tp]
\begin{center}
\includegraphics[width=0.7\textwidth]{db_along_110_cc_n.ps}
\end{center}
\caption[Minimum binding energy of DB combinations separated along \hkl<1 1 0> with respect to the C-C distance.]{Minimum binding energy of dumbbell combinations separated along \hkl[1 1 0] with respect to the C-C distance. The blue line is a guide for the eye and the green curve corresponds to the most suitable fit function consisting of all but the first data point.}
\label{fig:defects:comb_db110}
\end{figure}
The binding energy of these configurations with respect to the C-C distance is plotted in Fig.~\ref{fig:defects:comb_db110}.
The interaction is found to be proportional to the reciprocal cube of the C-C distance for extended separations of the \ci{} DBs and saturates for the smallest possible separation, i.e. the ground-state configuration.
The ground-state configuration was ignored in the fitting process.
Not considering the previously mentioned elevated barriers for migration, an attractive interaction between the \ci{} \hkl<1 0 0> DB defects indeed is detected with a capture radius that clearly exceeds \unit[1]{nm}.
The interpolated graph suggests the disappearance of attractive interaction forces, which are proportional to the slope of the graph, in between the two lowest separation distances of the defects.
This finding, in turn, supports the previously established assumption of C agglomeration and absence of C clustering.

%\subsection{Diffusion processes among configurations of \ci{} pairs}

To draw further conclusions on the probability of C clustering, transitions into the ground-state configuration are investigated.
Based on the lowest energy migration path of a single \ci{} \hkl<1 0 0> DB, the configuration, in which the second \ci{} DB is oriented along \hkl[0 1 0] at position 2 is assumed to constitute an ideal starting point for a transition into the ground state.
In addition, the starting configuration exhibits a low binding energy (\unit[-1.90]{eV}) and is, thus, very likely to occur.
However, a smooth transition path is not found.
Intermediate configurations within the investigated turbulent pathway identify barrier heights of more than \unit[4]{eV} resulting in a low probability for the transition.
The high activation energy is attributed to the stability of such a low energy configuration, in which the C atom of the second DB is located close to the initial DB.
Due to an effective stress compensation realized in the respective low energy configuration, which will necessarily be lost during migration, a high energy configuration needs to get passed, which is responsible for the high barrier.
Low barriers are only identified for transitions starting from energetically less favorable configurations, e.g. the configuration of a \hkl[-1 0 0] DB located at position 2 (\unit[-0.36]{eV}).
Starting from this configuration, an activation energy of only \unit[1.2]{eV} is necessary for the transition into the ground state configuration.
The corresponding migration energies and atomic configurations are displayed in Fig.~\ref{fig:036-239}.
\begin{figure}[tp]
\begin{center}
\includegraphics[width=0.7\textwidth]{036-239.ps}
\end{center}
\caption{Migration barrier and structures of the transition of a C$_{\text{i}}$ \hkl[-1 0 0] DB at position 2 (left) into a C$_{\text{i}}$ \hkl[0 -1 0] DB at position 1 (right). An activation energy of \unit[1.2]{eV} is observed.}
\label{fig:036-239}
\end{figure}
Since thermally activated C clustering is, thus, only possible by traversing energetically unfavored configurations, extensive C clustering is not expected.
Furthermore, the migration barrier of \unit[1.2]{eV} is still higher than the activation energy of \unit[0.9]{eV} observed for a single C$_{\text{i}}$ \hkl<1 0 0> DB in c-Si.
The migration barrier of a C$_{\text{i}}$ DB in a complex system is assumed to approximate the barrier of a DB in a separated system with increasing defect separation.
Accordingly, lower migration barriers are expected for pathways resulting in larger separations of the C$_{\text{i}}$ DBs.
% acknowledged by 188-225 (reverse order) calc
However, if the increase of separation is accompanied by an increase in binding energy, this difference is needed in addition to the activation energy for the respective migration process.
Configurations, which exhibit both, a low binding energy as well as afferent transitions with low activation energies are, thus, most probable C$_{\text{i}}$ complex structures.
On the other hand, if elevated temperatures enable migrations with huge activation energies, comparably small differences in configurational energy can be neglected resulting in an almost equal occupation of such configurations.
In both cases the configuration yielding a binding energy of \unit[-2.25]{eV} is promising.
First of all, it constitutes the second most energetically favorable structure.
Secondly, a migration path with a barrier as low as \unit[0.47]{eV} exists starting from a configuration of largely separated defects exhibiting a low binding energy (\unit[-1.88]{eV}).
The migration barrier and corresponding structures are shown in Fig.~\ref{fig:188-225}.
\begin{figure}[tp]
\begin{center}
\includegraphics[width=0.7\textwidth]{188-225.ps}
\end{center}
\caption{Migration barrier and structures of the transition of a C$_{\text{i}}$ \hkl[0 -1 0] DB at position 5 (left) into a C$_{\text{i}}$ \hkl[1 0 0] DB at position 1 (right). An activation energy of \unit[0.47]{eV} is observed.}
\label{fig:188-225}
\end{figure}
Finally, as already mentioned above, this type of defect pair is represented two times more often than the ground-state configuration.
The latter is considered very important at high temperatures, accompanied by an increase in the entropic contribution to structure formation.
As a result, C defect agglomeration indeed is expected, but only a low probability is assumed for C-C clustering by thermally activated processes with regard to the considered process time in IBS.

\subsection[Combinations of the \ci{} \hkl<1 0 0> and \cs{} type]{\boldmath Combinations of the \ci{} \hkl<1 0 0> and \cs{} type}
\label{subsection:defects:c-csub}

\begin{table}[tp]
\begin{center}
\begin{tabular}{c c c c c c}
\hline
\hline
1 & 2 & 3 & 4 & 5 & R \\
\hline
0.26$^a$/-1.28$^b$ & -0.51 & -0.93$^A$/-0.95$^B$ & -0.15 & 0.49
 & -0.05\\
\hline
\hline
\end{tabular}
\end{center}
\caption{Binding energies of combinations of the \ci{} \hkl[0 0 -1] defect with a \cs{} atom located at positions 1 to 5 according to Fig.~\ref{fig:defects:combos_ci}. R corresponds to the position located at $\frac{a_{\text{Si}}}{2}\hkl[3 2 3]$ relative to the initial defect position, which is the maximum realizable distance due to periodic boundary conditions.}
\label{tab:defects:c-s}
\end{table}
%\begin{figure}[tp]
%\begin{center}
%\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}[tp]
%\begin{center}
%\begin{minipage}[t]{7cm}
%a) \underline{Pos: 2, $E_{\text{b}}=-0.51\text{ eV}$}
%\begin{center}
%\includegraphics[width=6cm]{00-1dc/0-51.eps}
%\end{center}
%\end{minipage}
%\begin{minipage}[t]{7cm}
%b) \underline{Pos: 4, $E_{\text{b}}=-0.15\text{ eV}$}
%\begin{center}
%\includegraphics[width=6cm]{00-1dc/0-15.eps}
%\end{center}
%\end{minipage}
%\end{center}
%\caption{Relaxed structures of defect complexes obtained by creating a carbon substitutional at position 2 (a)) and 4 (b)).}
%\label{fig:defects:comb_db_05}
%\end{figure}
%
Table~\ref{tab:defects:c-s} lists the energetic results of \cs{} combinations with the initial \ci{} \hkl[0 0 -1] DB.
For \cs{} located at position 1 and 3, the configurations $\alpha$ and A correspond to the naive relaxation of the structure by substituting the Si atom by a C atom in the initial \ci{} \hkl[0 0 -1] DB structure at positions 1 and 3 respectively.
However, small displacements of the involved atoms near the defect result in different stable structures labeled $\beta$ and B respectively.
Fig.~\ref{fig:093-095} and \ref{fig:026-128} show structures A, B and $\alpha$, $\beta$ together with the barrier of migration for the A to B and $\alpha$ to $\beta$ transition respectively.

% A B
%./visualize_contcar -w 640 -h 480 -d results/c_00-1_c3_csub_B -nll -0.20 -0.4 -0.1 -fur 0.9 0.6 0.9 -c 0.5 -1.5 0.375 -L 0.5 0 0.3 -r 0.6 -A -1 2.465
\begin{figure}[tp]
\begin{center}
\includegraphics[width=0.7\textwidth]{093-095.ps}
\end{center}
\caption{Migration barrier and structures of the transition of the initial \ci{} \hkl[0 0 -1] DB and C$_{\text{s}}$ at position 3 (left) into a configuration of a twofold coordinated Si$_{\text{i}}$ located in between two C$_{\text{s}}$ atoms occupying the lattice sites of the initial DB and position 3 (right). An activation energy of \unit[0.44]{eV} is observed.}
\label{fig:093-095}
\end{figure}
Configuration A consists of a C$_{\text{i}}$ \hkl[0 0 -1] DB with threefold coordinated Si and C DB atoms slightly disturbed by the C$_{\text{s}}$ at position 3, facing the Si DB atom as a neighbor.
By a single bond switch, i.e. the breaking of a Si-Si in favor of a Si-C bond, configuration B is obtained, which shows a twofold coordinated Si atom located in between two substitutional C atoms residing on regular Si lattice sites.
This configuration has been identified and described by spectroscopic experimental techniques \cite{song90_2} as well as theoretical studies \cite{leary97,capaz98}.
Configuration B is found to constitute the energetically slightly more favorable configuration.
However, the gain in energy due to the significantly lower energy of a Si-C compared to a Si-Si bond turns out to be smaller than expected due to a large compensation by introduced strain as a result of the Si interstitial structure.
Present results show a difference in energy of states A and B, which exactly matches the experimental value of \unit[0.02]{eV} \cite{song90_2}, reinforcing qualitatively correct results of previous theoretical studies on these structures.
% mattoni: A favored by 0.4 eV - NO, it is indeed B (reinforce Song and Capaz)!
%
% AB transition
The migration barrier is identified to be \unit[0.44]{eV}, almost three times higher than the experimental value of \unit[0.16]{eV} \cite{song90_2} estimated for the neutral charge state transition in p- and n-type Si.
Keeping in mind the formidable agreement of the energy difference with experiment, the overestimated activation energy is quite unexpected.
Obviously, either the CRT algorithm fails to seize the actual saddle point structure or the influence of dopants has exceptional effect in the experimentally covered diffusion process being responsible for the low migration barrier.
% not satisfactory!

% a b
\begin{figure}[tp]
\begin{center}
\includegraphics[width=0.7\textwidth]{comb_mig_026-128_vasp.ps}
\end{center}
\caption{Migration barrier and structures of the transition of the initial C$_{\text{i}}$ \hkl[0 0 -1] DB and C$_{\text{s}}$ at position 1 (left) into a C-C \hkl[1 0 0] DB occupying the lattice site at position 1 (right). An activation energy of \unit[0.1]{eV} is observed.}
\label{fig:026-128}
\end{figure}
Configuration $\alpha$ is similar to configuration A, except that the C$_{\text{s}}$ atom at position 1 is facing the C DB atom as a neighbor resulting in the formation of a strong C-C bond and a much more noticeable perturbation of the DB structure.
Nevertheless, the C and Si DB atoms remain threefold coordinated.
Although the C-C bond exhibiting a distance of \unit[0.15]{nm} close to the distance expected in diamond or graphite should lead to a huge gain in energy, a repulsive interaction with a binding energy of \unit[0.26]{eV} is observed due to compressive strain of the Si DB atom and its top neighbors (\unit[0.230]{nm}/\unit[0.236]{nm}) along with additional tensile strain of the C$_{\text{s}}$ and its three neighboring Si atoms (\unit[0.198-0.209]{nm}/\unit[0.189]{nm}).
Again a single bond switch, i.e. the breaking of the bond of the Si atom bound to the fourfold coordinated C$_{\text{s}}$ atom and the formation of a double bond between the two C atoms, results in configuration b.
The two C atoms form a \hkl[1 0 0] DB sharing the initial C$_{\text{s}}$ lattice site while the initial Si DB atom occupies its previously regular lattice site.
The transition is accompanied by a large gain in energy as can be seen in Fig.~\ref{fig:026-128}, making it the ground-state configuration of a C$_{\text{s}}$ and C$_{\text{i}}$ DB in Si yet \unit[0.33]{eV} lower in energy than configuration B.
This finding is in good agreement with a combined ab initio and experimental study of Liu et~al.~\cite{liu02}, who first proposed this structure as the ground state identifying an energy difference compared to configuration B of \unit[0.2]{eV}.
% mattoni: A favored by 0.2 eV - NO! (again, missing spin polarization?)
A net magnetization of two spin up electrons, which are equally localized as in the Si$_{\text{i}}$ \hkl<1 0 0> DB structure is observed.
In fact, these two configurations are very similar and are qualitatively different from the C$_{\text{i}}$ \hkl<1 0 0> DB that does not show magnetization but a nearly collinear bond of the C DB atom to its two neighbored Si atoms while the Si DB atom approximates \unit[120]{$^{\circ}$} angles in between its bonds.
Configurations $\alpha$, A and B are not affected by spin polarization and show zero magnetization.
Mattoni et~al.~\cite{mattoni2002}, in contrast, find configuration $\beta$ less favorable than configuration A by \unit[0.2]{eV}.
Next to differences in the XC functional and plane-wave energy cut-off, this discrepancy might be attributed to the neglect of spin polarization in their calculations, which -- as has been shown for the C$_{\text{i}}$ BC configuration -- results in an increase of configurational energy.
Indeed, investigating the migration path from configurations $\alpha$ to $\beta$ and, in doing so, reusing the wave functions of the previous migration step the final structure, i.e. configuration $\beta$, is obtained with zero magnetization and an increase in configurational energy by \unit[0.2]{eV}.
Obviously a different energy minimum of the electronic system is obtained indicating hysteresis behavior.
However, since the total energy is lower for the magnetic result it is believed to constitute the real, i.e. global, minimum with respect to electronic minimization.
%
% a b transition
A low activation energy of \unit[0.1]{eV} is observed for the a$\rightarrow$b transition.
Thus, configuration a is very unlikely to occur in favor of configuration b.

% repulsive along 110
A repulsive interaction is observed for C$_{\text{s}}$ at lattice sites along \hkl[1 1 0], i.e. positions 1 (configuration a) and 5.
This is due to tensile strain originating from both, the C$_{\text{i}}$ DB and the C$_{\text{s}}$ atom residing within the \hkl[1 1 0] bond chain.
This finding agrees well with results by Mattoni et~al.~\cite{mattoni2002}.
% all other investigated results: attractive interaction. stress compensation.
In contrast, all other investigated configurations show attractive interactions.
The most favorable configuration is found for C$_{\text{s}}$ at position 3, which corresponds to the lattice site of one of the upper neighbored Si atoms of the DB structure that is compressively strained along \hkl[1 -1 0] and \hkl[0 0 1] by the C-Si DB.
The substitution with C allows for most effective compensation of strain.
This structure is followed by C$_{\text{s}}$ located at position 2, the lattice site of one of the neighbor atoms below the two Si atoms that are bound to the C$_{\text{i}}$ DB atom.
As mentioned earlier, these two lower Si atoms indeed experience tensile strain along the \hkl[1 1 0] bond chain, however, additional compressive strain along \hkl[0 0 1] exists.
The latter is partially compensated by the C$_{\text{s}}$ atom.
Yet less of compensation is realized if C$_{\text{s}}$ is located at position 4 due to a larger separation although both bottom Si atoms of the DB structure are indirectly affected, i.e. each of them is connected by another Si atom to the C atom enabling the reduction of strain along \hkl[0 0 1].
\begin{figure}[tp]
\begin{center}
\subfigure[\underline{$E_{\text{b}}=-0.51\,\text{eV}$}]{\label{fig:defects:051}\includegraphics[width=0.25\textwidth]{00-1dc/0-51.eps}}
\hspace{0.2cm}
\subfigure[\underline{$E_{\text{b}}=-0.15\,\text{eV}$}]{\label{fig:defects:015}\includegraphics[width=0.25\textwidth]{00-1dc/0-15.eps}}
\hspace{0.2cm}
\subfigure[\underline{$E_{\text{b}}=0.49\,\text{eV}$}]{\label{fig:defects:049}\includegraphics[width=0.25\textwidth]{00-1dc/0-49.eps}}
\end{center}
\caption{Relaxed structures of defect combinations obtained by creating \cs{} at positions 2 (a), 4 (b) and 5 (c) in the \ci{} \hkl[0 0 -1] DB configuration.}
\label{fig_defects:245csub}
\end{figure}
Fig.~\ref{fig_defects:245csub} lists the remaining configurations and binding energies of the relaxed structures obtained by creating a \cs{} at positions 2, 4 and 5 in the \ci{} \hkl[0 0 -1] DB configuration.
% todo explain some configurations, source: old text some lines below

% c agglomeration vs c clustering ... migs to b conf
% 2 more migs: 051 -> 128 and 026! forgot why ... probably it's about probability of C clustering
Obviously, agglomeration of C$_{\text{i}}$ and C$_{\text{s}}$ is energetically favorable except for separations along one of the \hkl<1 1 0> directions.
The energetically most favorable configuration (configuration $\beta$) forms a strong but compressively strained C-C bond with a separation distance of \unit[0.142]{nm} sharing a Si lattice site.
Again, conclusions concerning the probability of formation are drawn by investigating respective migration paths.
Since C$_{\text{s}}$ is unlikely to exhibit a low activation energy for migration the focus is on C$_{\text{i}}$.
Pathways starting from the next most favored configuration, i.e. \cs{} located at position 2, into configuration $\alpha$ and $\beta$ are investigated, which show activation energies above \unit[2.2]{eV} and \unit[2.5]{eV}.
The respective barriers and structures are displayed in Fig.~\ref{fig:051-xxx}.
For the transition into configuration $\beta$, as before, the non-magnetic configuration is obtained.
If not forced by the CRT algorithm, the structures beyond \perc{50} and below \perc{90} displacement of the transition approaching configuration $\alpha$ would settle into configuration $\beta$.
\begin{figure}[tp]
\begin{center}
\includegraphics[width=0.7\textwidth]{comb_mig_051-xxx_conf.ps}
\end{center}
\caption{Migration barrier and structures of the transition of a configuration equivalent to the one of the initial \hkl<0 0 -1> \ci{} DB with \cs{} located at position 2 into the $\alpha$ and $\beta$ configurations.}
\label{fig:051-xxx}
\end{figure}
Although lower than the barriers for obtaining the ground state of two C$_{\text{i}}$ defects, the activation energies are yet considered too high.
For the same reasons as in the last subsection, structures other than the ground-state configuration are, thus, assumed to arise more likely due to much lower activation energies necessary for their formation and still comparatively low binding energies.

% old c_int - c_substitutional stuff

%Figures \ref{fig:defects:comb_db_04} and \ref{fig:defects:comb_db_05} show relaxed structures of substitutional carbon in combination with the \hkl<0 0 -1> dumbbell for several positions.
%In figure \ref{fig:defects:comb_db_04} positions 1 (a)), 3 (b)) and 5 (c)) are displayed.
%A substituted carbon atom at position 5 results in an energetically extremely unfavorable configuration.
%Both carbon atoms, the substitutional and the dumbbell atom, pull silicon atom number 1 towards their own location regarding the \hkl<1 1 0> direction.
%Due to this a large amount of tensile strain energy is needed, which explains the high positive value of 0.49 eV.
%The lowest binding energy is observed for a substitutional carbon atom inserted at position 3.
%The substitutional carbon atom is located above the dumbbell substituting a silicon atom which would usually be bound to and displaced along \hkl<0 0 1> and \hkl<1 1 0> by the silicon dumbbell atom.
%In contrast to the previous configuration strain compensation occurs resulting in a binding energy as low as -0.93 eV.
%Substitutional carbon at position 2 and 4, visualized in figure \ref{fig:defects:comb_db_05}, is located below the initial dumbbell.
%Silicon atom number 1, which is bound to the interstitial carbon atom is displaced along \hkl<0 0 -1> and \hkl<-1 -1 0>.
%In case a) only the first displacement is compensated by the substitutional carbon atom.
%This results in a somewhat higher binding energy of -0.51 eV.
%The binding energy gets even higher in case b) ($E_{\text{b}}=-0.15\text{ eV}$), in which the substitutional carbon is located further away from the initial dumbbell.
%In both cases, silicon atom number 1 is displaced in such a way, that the bond to silicon atom number 5 vanishes.
%In case of \ref{fig:defects:comb_db_04} a) the carbon atoms form a bond with a distance of 1.5 \AA, which is close to the C-C distance expected in diamond or graphit.
%Both carbon atoms are highly attracted by each other resulting in large displacements and high strain energy in the surrounding.
%A binding energy of 0.26 eV is observed.
%Substitutional carbon at positions 2, 3 and 4 are the energetically most favorable configurations and constitute promising starting points for SiC precipitation.
%On the one hand, C-C distances around 3.1 \AA{} exist for substitution positions 2 and 3, which are close to the C-C distance expected in silicon carbide.
%On the other hand stretched silicon carbide is obtained by the transition of the silicon dumbbell atom into a silicon self-interstitial located somewhere in the silicon host matrix and the transition of the carbon dumbbell atom into another substitutional atom occupying the dumbbell lattice site.


\subsection[Combinations of a \ci{} \hkl<1 0 0> DB and vacancy]{\boldmath Combinations of a \ci{} \hkl<1 0 0> DB and a vacancy}
\label{subsection:defects:c-v}

In the last section, configurations of a C$_{\text{i}}$ DB with C$_{\text{s}}$ occupying a vacant site have been investigated.
Additionally, configurations might arise in IBS, in which the impinging C atom creates a vacant site near a C$_{\text{i}}$ DB, but does not occupy it.
These structures are investigated in the following.

Resulting binding energies of a C$_{\text{i}}$ DB and a nearby vacancy are listed in the second row of Table~\ref{tab:defects:c-v}.
\begin{table}[tp]
\begin{center}
\begin{tabular}{c c c c c c}
\hline
\hline
1 & 2 & 3 & 4 & 5 & R \\
\hline
-5.39 ($\rightarrow$ C$_{\text{S}}$) & -0.59 & -3.14 & -0.54 & -0.50 & -0.31\\
\hline
\hline
\end{tabular}
\end{center}
\caption{Binding energies of combinations of the \ci{} \hkl[0 0 -1] defect with a vacancy located at positions 1 to 5 according to Fig.~\ref{fig:defects:combos_ci}. R corresponds to the position located at $\frac{a_{\text{Si}}}{2}\hkl[3 2 3]$ relative to the initial defect position, which is the maximum realizable distance due to periodic boundary conditions.}
\label{tab:defects:c-v}
\end{table}
\begin{figure}[tp]
\begin{center}
\subfigure[\underline{$E_{\text{b}}=-0.59\,\text{eV}$}]{\label{fig:defects:059}\includegraphics[width=0.25\textwidth]{00-1dc/0-59.eps}}
\hspace{0.7cm}
\subfigure[\underline{$E_{\text{b}}=-3.14\,\text{eV}$}]{\label{fig:defects:314}\includegraphics[width=0.25\textwidth]{00-1dc/3-14.eps}}\\
\subfigure[\underline{$E_{\text{b}}=-0.54\,\text{eV}$}]{\label{fig:defects:054}\includegraphics[width=0.25\textwidth]{00-1dc/0-54.eps}}
\hspace{0.7cm}
\subfigure[\underline{$E_{\text{b}}=-0.50\,\text{eV}$}]{\label{fig:defects:050}\includegraphics[width=0.25\textwidth]{00-1dc/0-50.eps}}
\end{center}
\caption{Relaxed structures of defect combinations obtained by creating a vacancy at positions 2 (a), 3 (b), 4 (c) and 5 (d).}
\label{fig:defects:comb_db_06}
\end{figure}
Figure \ref{fig:defects:comb_db_06} shows the associated configurations.
All investigated structures are preferred compared to isolated, largely separated defects.
In contrast to C$_{\text{s}}$ this is also valid for positions along \hkl[1 1 0] resulting in an entirely attractive interaction between defects of these types.
Even for the largest possible distance (R) achieved in the calculations of the periodic supercell a binding energy as low as \unit[-0.31]{eV} is observed.
The creation of a vacancy at position 1 results in a configuration of substitutional C on a Si lattice site and no other remaining defects.
The \ci{} DB atom moves to position 1 where the vacancy is created and the \si{} DB atom recaptures the DB lattice site.
With a binding energy of \unit[-5.39]{eV}, this is the energetically most favorable configuration observed.
A great amount of strain energy is reduced by removing the Si atom at position 3, which is illustrated in Fig.~\ref{fig:defects:314}.
The DB structure shifts towards the position of the vacancy, which replaces the Si atom usually bound to and at the same time strained by the \si{} DB atom.
Due to the displacement into the \hkl[1 -1 0] direction the bond of the DB Si atom to the Si atom on the top left breaks and instead forms a bond to the Si atom located in \hkl[1 -1 1] direction, which is not shown in Fig.~\ref{fig:defects:314}.
A binding energy of \unit[-3.14]{eV} is obtained for this structure composing another energetically favorable configuration.
A vacancy ctreated at position 2 enables the relaxation of Si atom number 1 mainly in \hkl[0 0 -1] direction.
The bond to Si atom number 5 breaks.
Hence, the \si{} DB atom is not only displaced along \hkl[0 0 -1] but also and to a greater extent in \hkl[1 1 0] direction.
The C atom is slightly displaced in \hkl[0 1 -1] direction.
A binding energy of \unit[-0.59]{eV} indicates the occurrence of much less strain reduction compared to that in the latter configuration.
Evidently this is due to a smaller displacement of Si atom 1, which would be directly bound to the replaced Si atom at position 2.
In the case of a vacancy created at position 4, even a slightly higher binding energy of \unit[-0.54]{eV} is observed, while the Si atom at the bottom left, which is bound to the \ci{} DB atom, is vastly displaced along \hkl[1 0 -1].
However the displacement of the C atom along \hkl[0 0 -1] is less compared to the one in the previous configuration.
Although expected due to the symmetric initial configuration, Si atom number 1 is not displaced correspondingly and also the \si DB atom is displaced to a greater extent in \hkl[-1 0 0] than in \hkl[0 -1 0] direction.
The symmetric configuration is, thus, assumed to constitute a local maximum, which is driven into the present state by the conjugate gradient method used for relaxation.
Fig.~\ref{fig:defects:050} shows the relaxed structure of a vacancy created at position 5.
The Si DB atom is largely displaced along \hkl[1 1 0] and somewhat less along \hkl[0 0 -1], which corresponds to the direction towards the vacancy.
The \si DB atom approaches Si atom number 1.
Indeed, a non-zero charge density is observed in between these two atoms exhibiting a cylinder-like shape superposed with the charge density known from the DB itself.
Strain reduced by this huge displacement is partially absorbed by tensile strain on Si atom number 1 originating from attractive forces of the C atom and the vacancy.
A binding energy of \unit[-0.50]{eV} is observed.

The migration pathways of configuration \ref{fig:defects:314} and \ref{fig:defects:059} into the ground-state configuration, i.e. the \cs{} configuration, are shown in Fig.~\ref{fig:314-539} and \ref{fig:059-539} respectively.
\begin{figure}[tp]
\begin{center}
\includegraphics[width=0.7\textwidth]{314-539.ps}
\end{center}
\caption{Migration barrier and structures of the transition of the initial C$_{\text{i}}$ \hkl[0 0 -1] DB and a V created at position 3 (left) into a C$_{\text{s}}$ configuration (right). An activation energy of \unit[0.1]{eV} is observed.}
\label{fig:314-539}
\end{figure}
\begin{figure}[tp]
\begin{center}
\includegraphics[width=0.7\textwidth]{059-539.ps}
\end{center}
\caption{Migration barrier and structures of the transition of the initial C$_{\text{i}}$ \hkl[0 0 -1] DB and a V created at position 2 (left) into a C$_{\text{s}}$ configuration (right). An activation energy of \unit[0.6]{eV} is observed.}
\label{fig:059-539}
\end{figure}
Activation energies as low as \unit[0.1]{eV} and \unit[0.6]{eV} are observed.
In the first case the Si and C atom of the DB move towards the vacant and initial DB lattice site respectively.
In total three Si-Si and one more Si-C bond is formed during transition.
The activation energy of \unit[0.1]{eV} is needed to tilt the DB structure.
Once this barrier is overcome, the C atom forms a bond to the top left Si atom and the \si{} atom capturing the vacant site is forming new tetrahedral bonds to its neighbored Si atoms.
These new bonds and the relaxation into the \cs{} configuration are responsible for the gain in configurational energy.
For the reverse process approximately \unit[2.4]{eV} are needed, which is 24 times higher than the forward process.
In the second case the lowest barrier is found for the migration of Si number 1, which is substituted by the C$_{\text{i}}$ atom, towards the vacant site.
A net amount of five Si-Si and one Si-C bond are additionally formed during transition.
An activation energy of \unit[0.6]{eV} necessary to overcome the migration barrier is found.
This energy is low enough to constitute a feasible mechanism in SiC precipitation.
To reverse this process \unit[5.4]{eV} are needed, which make this mechanism very unprobable.
%
The migration path is best described by the reverse process.
Starting at \unit[100]{\%}, energy is needed to break the bonds of Si atom 1 to its neighbored Si atoms as well as the bond of the C atom to Si atom number 5.
At \unit[50]{\%} displacement, these bonds are broken.
Due to this and due to the formation of new bonds, e.g. the bond of Si atom number 1 to Si atom number 5, a less steep increase of configurational energy is observed.
In a last step, the just recently formed bond of Si atom number 1 to Si atom number 5 is broken up again as well as the bond of the initial Si DB atom and its Si neighbor in \hkl[-1 -1 -1] direction, which explains the repeated boost in energy.
Finally, the system gains some configurational energy by relaxation into the configuration corresponding to \unit[0]{\%} displacement.
%
The direct migration of the C$_{\text{i}}$ atom onto the vacant lattice site results in a somewhat higher barrier of \unit[1.0]{eV}.
In both cases, the formation of additional bonds is responsible for the vast gain in energy rendering almost impossible the reverse processes.

In summary, pairs of C$_{\text{i}}$ DBs and vacancies, like no other before, show highly attractive interactions for all investigated combinations independent of orientation and separation direction of the defects.
Furthermore, small activation energies, even for transitions into the ground state exist.
If the vacancy is created at position 1 the system will end up in a configuration of C$_{\text{s}}$ anyways.
Based on these results, a high probability for the formation of C$_{\text{s}}$ must be concluded.

\subsection{Combinations of \si{} and \cs}
\label{subsection:si-cs}

So far the C-Si \hkl<1 0 0> DB interstitial was found to be the energetically most favorable configuration.
In fact substitutional C exhibits a configuration more than \unit[3]{eV} lower with respect to the formation energy.
However, the configuration does not account for the accompanying Si self-interstitial that is generated once a C atom occupies the site of a Si atom.
With regard to the IBS process, in which highly energetic C atoms enter the Si target being able to kick out Si atoms from their lattice sites, such configurations are absolutely conceivable and a significant influence on the precipitation process might be attributed to them.
Thus, combinations of \cs{} and an additional \si{} are examined in the following.
The ground-state of a single \si{} was found to be the \si{} \hkl<1 1 0> DB configuration.
For the follwoing study the same type of self-interstitial is assumed to provide the energetically most favorable configuration in combination with \cs.

\begin{table}[tp]
\begin{center}
\begin{tabular}{l c c c c c c}
\hline
\hline
 & \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} & \RM{6} & \RM{6} & \RM{2} & \RM{8} & \RM{5} \\
3 & \RM{3} & \RM{1} & \RM{3} & \RM{1} & \RM{1} & \RM{3} \\
4 & \RM{4} & \RM{7} & \RM{9} & \RM{10} & \RM{10} & \RM{9} \\
5 & \RM{5} & \RM{8} & \RM{6} & \RM{2} & \RM{6} & \RM{2} \\
\hline
\hline
\end{tabular}
\end{center}
\caption{Equivalent configurations labeled \RM{1}-\RM{10} of \hkl<1 1 0>-type Si$_{\text{i}}$ DBs created at position I and C$_{\text{s}}$ created at positions 1 to 5 according to Fig.~\ref{fig:defects:combos_si}. The respective orientation of the Si$_{\text{i}}$ DB is given in the first row.}
\label{tab:defects:comb_csub_si110}
\end{table}
\begin{table}[tp]
\begin{center}
\begin{tabular}{l c c c c c c c c c c}
\hline
\hline
 & \RM{1} & \RM{2} & \RM{3} & \RM{4} & \RM{5} & \RM{6} & \RM{7} & \RM{8} & \RM{9} & \RM{10} \\
\hline
$E_{\text{f}}$ [eV]& 4.37 & 5.26 & 5.57 & 5.37 & 5.12 & 5.10 & 5.32 & 5.28 & 5.39 & 5.32 \\
$E_{\text{b}}$ [eV] & -0.97 & -0.08 & 0.22 & -0.02 & -0.23 & -0.25 & -0.02 & -0.06 & 0.05 & -0.03 \\
$r$ [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 energies $E_{\text{f}}$, binding energies $E_{\text{b}}$ and C$_{\text{s}}$-Si$_{\text{i}}$ separation distances of configurations combining C$_{\text{s}}$ and Si$_{\text{i}}$ as defined in Table~\ref{tab:defects:comb_csub_si110}.}
\label{tab:defects:comb_csub_si110_energy}
\end{table}
Table~\ref{tab:defects:comb_csub_si110} classifies equivalent configurations of \hkl<1 1 0>-type Si$_{\text{i}}$ DBs created at position I and C$_{\text{s}}$ created at positions 1 to 5 according to Fig.~\ref{fig:defects:combos_si}.
Corresponding formation as well as binding energies and the separation distances of the C$_{\text{s}}$ atom and the Si$_{\text{i}}$ DB lattice site are listed in Table~\ref{tab:defects:comb_csub_si110_energy}.
In total, ten different configurations exist within the investigated range.
Configuration \RM{1} constitutes the energetically most favorable structure exhibiting a formation energy of \unit[4.37]{eV}.
Obviously, the configuration of a Si$_{\text{i}}$ \hkl[1 1 0] DB and a neighbored C$_{\text{s}}$ atom along the bond chain, which has the same direction as the alignment of the DB, enables the largest possible reduction of strain.
%
The relaxed structure is displayed in the bottom right of Fig.~\ref{fig:162-097}.
Compressive strain originating from the Si$_{\text{i}}$ is compensated by tensile strain inherent to the C$_{\text{s}}$ configuration.
The Si$_{\text{i}}$ DB atoms are displaced towards the lattice site occupied by the C$_{\text{s}}$ atom in such a way that the Si$_{\text{i}}$ DB atom closest to the C atom does no longer form bonds to its top Si neighbors, but to the next neighbored Si atom along \hkl[1 1 0].
%
In the same way the energetically most unfavorable configuration can be explained, which is configuration \RM{3}.
The \cs{} is located next to the lattice site shared by the \si{} \hkl[1 1 0] DB in \hkl[1 -1 1] direction.
Thus, the compressive stress along \hkl[1 1 0] of the \si{} \hkl[1 1 0] DB is not compensated but intensified by the tensile stress of the \cs{} atom, which is no longer loacted along the direction of stress.

However, even configuration \RM{1} is energetically less favorable than the \hkl<1 0 0> C$_{\text{i}}$ DB, which, thus, remains the ground state of a C atom introduced into otherwise perfect c-Si.
The transition involving the latter two configurations is shown in Fig.~\ref{fig:162-097}.
\begin{figure}[tp]
\begin{center}
\includegraphics[width=0.7\textwidth]{162-097.ps}
\end{center}
\caption{Migration barrier and structures of the transition of a \hkl[1 1 0] Si$_{\text{i}}$ DB next to C$_{\text{s}}$ (right) into the C$_{\text{i}}$ \hkl[0 0 -1] DB configuration (left). An activation energy of \unit[0.12]{eV} and \unit[0.77]{eV} for the reverse process is observed.}
\label{fig:162-097}
\end{figure}
An activation energy as low as \unit[0.12]{eV} is necessary for the migration into the ground-state configuration.
Accordingly, the C$_{\text{i}}$ \hkl<1 0 0> DB configuration is assumed to occur more likely.
However, only \unit[0.77]{eV} are needed for the reverse process, i.e. the formation of C$_{\text{s}}$ and a Si$_{\text{i}}$ DB out of the ground state.
Due to the low activation energy this process must be considered to be activated without much effort either thermally or by introduced energy of the implantation process.

\begin{figure}[tp]
\begin{center}
\includegraphics[width=0.7\textwidth]{c_sub_si110.ps}
\end{center}
\caption{Binding energies of combinations of a C$_{\text{s}}$ and a Si$_{\text{i}}$ DB with respect to the separation distance. The interaction strength of the defect pairs are well approximated by a Lennard-Jones 6-12 potential, which is used for curve fitting.}
\label{fig:dc_si-s}
\end{figure}
Fig.~\ref{fig:dc_si-s} shows the binding energies of pairs of C$_{\text{s}}$ and a Si$_{\text{i}}$ \hkl<1 1 0> DB with respect to the separation distance.
The interaction of the defects is well approximated by a Lennard-Jones (LJ) 6-12 potential, which is used for curve fitting.
Unable to model possible positive values of the binding energy, i.e. unfavorable configurations, located to the right of the minimum, the LJ fit should rather be thought of as a guide for the eye describing the decrease of the interaction strength, i.e. the absolute value of the binding energy, with increasing separation distance.
The binding energy quickly drops to zero.
The LJ fit estimates almost zero interaction already at \unit[0.6]{nm}.
 indicating a low interaction capture radius of the defect pair.
%As can be seen, the interaction strength, i.e. the absolute value of the binding energy, quickly drops to zero with increasing separation distance.
%Almost zero interaction may be assumed already at distances about \unit[0.5-0.6]{nm}, indicating a low interaction capture radius of the defect pair.
In IBS, highly energetic collisions are assumed to easily produce configurations of defects exhibiting separation distances exceeding the capture radius.
For this reason C$_{\text{s}}$ without a Si$_{\text{i}}$ DB located within the immediate proximity, which is, thus, unable to form the thermodynamically stable C$_{\text{i}}$ \hkl<1 0 0> DB, constitutes a most likely configuration to be found in IBS.
In particular in IBS, which constitutes a system driven far from equilibrium, respective defect configurations might exist that do not combine into the ground-state configuration.
Thus, the existence of C$_{\text{s}}$ is very likely.
\label{section:defects:noneq_process_01}


% the ab initio md, where to put

Similar to what was previously mentioned, configurations of C$_{\text{s}}$ and a Si$_{\text{i}}$ DB might be particularly important at higher temperatures due to the low activation energy necessary for its formation.
At higher temperatures, the contribution of entropy to structural formation increases, which might result in a spatial separation even for defects located within the capture radius.
Indeed, an {em ab initio} MD run at \unit[900]{$^{\circ}$C} starting from configuration \RM{1}, which -- based on the above findings -- is assumed to recombine into the ground state configuration, results in a separation of the C$_{\text{s}}$ and Si$_{\text{i}}$ DB by more than 4 neighbor distances realized in a repeated migration mechanism of annihilating and arising Si$_{\text{i}}$  DBs.
The atomic configurations for two different points in time are shown in Fig.~\ref{fig:defects:md}.
\begin{figure}[tp]
\begin{center}
\begin{minipage}{0.40\textwidth}
\includegraphics[width=\columnwidth]{md_vasp_01.eps}
\end{minipage}
\hspace{1cm}
\begin{minipage}{0.40\textwidth}
\includegraphics[width=\columnwidth]{md_vasp_02.eps}
\end{minipage}\\
\begin{minipage}{0.40\textwidth}
\begin{center}
$t=\unit[2230]{fs}$
\end{center}
\end{minipage}
\hspace{1cm}
\begin{minipage}{0.40\textwidth}
\begin{center}
$t=\unit[2900]{fs}$
\end{center}
\end{minipage}
\end{center}
\caption{Atomic configurations of an ab initio molecular dynamics run at \unit[900]{$^{\circ}$C} starting from a configuration of C$_{\text{s}}$ located next to a Si$_{\text{i}}$ \hkl[1 1 0] DB (atoms 1 and 2). Equal atoms are marked by equal numbers. Bonds are drawn for substantial atoms only.}
\label{fig:defects:md}
\end{figure}
Si atoms 1 and 2, which form the initial DB, occupy Si lattice sites in the final configuration while Si atom 3 is transferred from a regular lattice site into the interstitial lattice.
These results support the above assumptions of an increased entropic contribution to structural formation involving C$_{\text{s}}$ to a greater extent.

% link to migration of \si{}!
The possibility for separated configurations of \cs{} and \si{} becomes even more likely if one of the constituents exhibits a low barrier of migration.
In this case, the \si{} is assumed to constitute the mobile defect compared to the stable \cs{} atom.
Thus, migration paths of \si{} are investigated in the following excursus.
Acoording to Fig.~\ref{fig:defects:si_mig1}, an activation energy of \unit[0.67]{eV} is necessary for the transition of the \si{} \hkl[0 -1 1] to \hkl[1 1 0] DB located at the neighbored Si lattice site in \hkl[1 1 -1] direction.
\begin{figure}[tp]
\begin{center}
\includegraphics[width=0.7\textwidth]{si_110_110_mig_02_conf.ps}
\end{center}
\caption[Migration barrier and structures of the \si{} \hkl<1 1 0> DB.]{Migration barrier and structures of the \si{} \hkl[0 -1 1] DB (left) to the \hkl[1 1 0] DB (right) transition. Bonds are illustrated by blue lines.}
% todo read above caption! enable [] hkls in short caption
\label{fig:defects:si_mig1}
\end{figure}
The barrier, which is even lower than the one for \ci{}, indeed indicates highly mobile \si.
In fact, a similar transition is expected if the \si{} atom, which does not change the lattice site during transition, is located next to a \cs{} atom.
Due to the low barrier the initial separation of the \cs{} and \si{} atom are very likely to occur.
Further investigations revealed transition barriers of \unit[0.94]{eV} for the Si$_{\text{i}}$ \hkl[1 1 0] DB to the hexagonal Si$_{\text{i}}$, \unit[0.53]{eV} for the Si$_{\text{i}}$ \hkl[1 1 0] DB to the tetrahedral Si$_{\text{i}}$ and \unit[0.35]{eV} for the hexagonal Si$_{\text{i}}$ to the tetrahedral Si$_{\text{i}}$ configuration.
The respective configurational energies are shown in Fig.~\ref{fig:defects:si_mig2}.
\begin{figure}[tp]
\begin{center}
\includegraphics[width=0.7\textwidth]{si_mig_rest.ps}
\end{center}
\caption{Migration barrier of the \si{} \hkl[1 1 0] DB into the hexagonal (H) and tetrahedral (T) configuration as well as the hexagonal \si{} to tetrahedral \si{} transition.}
% todo read above caption! enable [] hkls in short caption
\label{fig:defects:si_mig2}
\end{figure}
The obtained activation energies are of the same order of magnitude than values derived from other ab initio studies \cite{bloechl93,sahli05}.
The low barriers indeed enable configurations of further separated \cs{} and \si{} atoms by the highly mobile \si{} atom departing from the \cs{} defect as observed in the previously discussed MD simulation.

% kept for nostalgical reason!

%\section{Migration in systems of combined defects}

%\begin{figure}[tp]
%\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}[tp]
%\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}

\section{Applicability: Competition of \ci{} and \cs-\si{}}
\label{section:ea_app}

As has been shown, the energetically most favorable configuration of \cs{} and \si{} is obtained for \cs{} located at the neighbored lattice site along the \hkl<1 1 0> bond chain of a Si$_{\text{i}}$ \hkl<1 1 0> DB.
However, the energy of formation is slightly higher than that of the C$_{\text{i}}$ \hkl<1 0 0> DB, which constitutes the ground state for a C impurity introduced into otherwise perfect c-Si.

For a possible clarification of the controversial views on the participation of C$_{\text{s}}$ in the precipitation mechanism by classical potential simulations, test calculations need to ensure the proper description of the relative formation energies of combined structures of C$_{\text{s}}$ and Si$_{\text{i}}$ compared to C$_{\text{i}}$.
This is particularly important since the energy of formation of C$_{\text{s}}$ is drastically underestimated by the EA potential.
A possible occurrence of C$_{\text{s}}$ could then be attributed to a lower energy of formation of the C$_{\text{s}}$-Si$_{\text{i}}$ combination due to the low formation energy of C$_{\text{s}}$, which is obviously wrong.

Since quantum-mechanical calculations reveal the Si$_{\text{i}}$ \hkl<1 1 0> DB as the ground-state configuration of Si$_{\text{i}}$ in Si, it was assumed to provide the energetically most favorable configuration in combination with C$_{\text{s}}$ in the calculations carried out in section \ref{subsection:si-cs}.
Empirical potentials, however, predict Si$_{\text{i}}$ T to be the energetically most favorable configuration.
Thus, investigations of the relative energies of formation of defect pairs need to include combinations of C$_{\text{s}}$ with Si$_{\text{i}}$ T.
Results of {\em ab initio} and classical potential calculations are summarized in Table~\ref{tab:defect_combos}.
\begin{table}[tp]
\begin{center}
\begin{tabular}{l c c c}
\hline
\hline
 & C$_{\text{i}}$ \hkl<1 0 0> & C$_{\text{s}}$ \& Si$_{\text{i}}$ \hkl<1 1 0> &  C$_{\text{s}}$ \& Si$_{\text{i}}$ T\\
\hline
 {\textsc vasp} & 3.72 & 4.37 & 4.17$^{\text{a}}$/4.99$^{\text{b}}$/4.96$^{\text{c}}$ \\
 {\textsc posic} & 3.88 & 4.93 & 5.25$^{\text{a}}$/5.08$^{\text{b}}$/4.43$^{\text{c}}$\\
\hline
\hline
\end{tabular}
\end{center}
\caption{Formation energies of defect configurations of a single C impurity in otherwise perfect c-Si determined by classical potential and {\em ab initio} methods. The formation energies are given in eV. T denotes the tetrahedral and the subscripts i and s indicate the interstitial and substitutional configuration. Superscripts a, b and c denote configurations of C$_{\text{s}}$ located at the first, second and third nearest neighbored lattice site with respect to the Si$_{\text{i}}$ atom.}
\label{tab:defect_combos}
\end{table}
Obviously the EA potential properly describes the relative energies of formation.
Combined structures of C$_{\text{s}}$ and Si$_{\text{i}}$ T are energetically less favorable than the ground state C$_{\text{i}}$ \hkl<1 0 0> DB configuration.
With increasing separation distance the energies of formation decrease.
However, even for non-interacting defects, the energy of formation, which is then given by the sum of the formation energies of the separated defects (\unit[4.15]{eV}) is still higher than that of the C$_{\text{i}}$ \hkl<1 0 0> DB.
Unexpectedly, the structure of a Si$_{\text{i}}$ \hkl<1 1 0> DB and a neighbored C$_{\text{s}}$, which is the most favored configuration of a C$_{\text{s}}$ and Si$_{\text{i}}$ DB according to quantum-mechanical calculations, likewise constitutes an energetically favorable configuration within the EA description, which is even preferred over the two least separated configurations of C$_{\text{s}}$ and Si$_{\text{i}}$ T.
This is attributed to an effective reduction in strain enabled by the respective combination.
Quantum-mechanical results reveal a more favorable energy of fomation for the C$_{\text{s}}$ and Si$_{\text{i}}$ T (a) configuration.
However, this configuration is unstable involving a structural transition into the C$_{\text{i}}$ \hkl<1 1 0> DB interstitial, thus, not maintaining the tetrahedral Si nor the \cs{} defect.

Thus, the underestimated energy of formation of C$_{\text{s}}$ within the EA calculation does not pose a serious limitation in the present context.
Since C is introduced into a perfect Si crystal and the number of particles is conserved in simulation, the creation of C$_{\text{s}}$ is accompanied by the creation of Si$_{\text{i}}$, which is energetically less favorable than the ground state, i.e. the C$_{\text{i}}$ \hkl<1 0 0> DB configuration, for both, the EA and {\em ab initio} treatment.
In either case, no configuration more favorable than the C$_{\text{i}}$ \hkl<1 0 0> DB has been found.
Thus, a proper description with respect to the relative energies of formation is assumed for the EA potential.

\section{Conclusions concerning the SiC conversion mechanism}

\ifnum1=0

Obtained results for separated point defects in Si are in good agreement to previous theoretical work on this subject, both for intrinsic defects \cite{leung99,al-mushadani03} as well as for C point defects \cite{dal_pino93,capaz94}.
The ground-state configurations of these defects, i.e. the Si$_{\text{i}}$ \hkl<1 1 0> and C$_{\text{i}}$ \hkl<1 0 0> DB, are reproduced and compare well to previous findings of theoretical investigations on Si$_{\text{i}}$ \cite{leung99,al-mushadani03} as well as theoretical \cite{dal_pino93,capaz94,burnard93,leary97,jones04} and experimental \cite{watkins76,song90} studies on C$_{\text{i}}$.
A quantitatively improved activation energy of \unit[0.9]{eV} for a qualitatively equal migration path based on studies by Capaz et.~al.~\cite{capaz94} to experimental values \cite{song90,lindner06,tipping87} ranging from \unit[0.70-0.87]{eV} reinforce their derived mechanism of diffusion for C$_{\text{i}}$ in Si
However, it turns out that the BC configuration is not a saddle point configuration as proposed by Capaz et~al.~\cite{capaz94} but constitutes a real local minimum if the electron spin is properly accounted for.
A net magnetization of two electrons, which is already clear by simple molecular orbital theory considerations on the bonding of the $sp$ hybridized C atom, is settled.
By investigating the charge density isosurface it turns out that the two resulting spin up electrons are localized in a torus around the C atom.
With an activation energy of \unit[0.9]{eV} the C$_{\text{i}}$ carbon interstitial can be expected to be highly mobile at prevailing temperatures in the process under investigation, i.e. IBS.
Since the \ci{} \hkl<1 0 0> DB is the ground-state configuration and highly mobile, possible migration of these DBs to form defect agglomerates, as demanded by the model introduced in section \ref{section:assumed_prec}, is considered possible.

Unfortunately the description of the same processes fails if classical potential methods are used.
Already the geometry of the most stable DB configuration differs considerably from that obtained by first-principles calculations.
The classical approach is unable to reproduce the correct character of bonding due to the deficiency of quantum-mechanical effects in the potential.
Nevertheless, both methods predict the same type of interstitial as the ground-state configuration and also the order in energy of the remaining defects is reproduced fairly well.
From this, a description of defect structures by classical potentials looks promising.
%
However, focussing on the description of diffusion processes the situation changes completely.
Qualitative and quantitative differences exist.
First of all, a different pathway is suggested as the lowest energy path, which again might be attributed to the absence of quantum-mechanical effects in the classical interaction model.
Secondly, the activation energy is overestimated by a factor of 2.4 to 3.5 compared to the more accurate quantum-mechanical methods and experimental findings.
This is attributed to the sharp cut-off of the short range potential.
As already pointed out in a previous study \cite{mattoni2007}, the short cut-off is responsible for overestimated and unphysical high forces of next neighbor atoms.
The overestimated migration barrier, however, affects the diffusion behavior of the C interstitials.
By this artifact, the mobility of the C atoms is tremendously decreased resulting in an inaccurate description or even absence of the DB agglomeration as proposed by one of the precipitation models.

Quantum-mechanical investigations of two \ci{} of the \hkl<1 0 0>-type and varying separations and orientations state an attractive interaction between these interstitials.
Obtained results for the most part compare well with results gained in previous studies \cite{leary97,capaz98,mattoni2002,liu02} and show an astonishingly good agreement with experiment \cite{song90}.
%
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> bond chain 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 \ci{} DB agglomeration.
%
The ground state configuration is found to consist of a C-C bond, which is responsible for the vast gain in energy.
However, based on investigations of possible migration pathways, these structures are less likely to arise than structures, in which both C atoms are interconnected by another Si atom, which is due to high activation energies of the respective pathways or alternative pathways featuring less high activation energies, which, however, involve intermediate unfavorable configurations.
Thus, agglomeration of C$_{\text{i}}$ is expected while the formation of C-C bonds is assumed to fail to appear by thermally activated diffusion processes.

In contrast, C$_{\text{i}}$ and vacancies are found to efficiently react with each other exhibiting activation energies as low as \unit[0.1]{eV} and \unit[0.6]{eV} resulting in stable C$_{\text{s}}$ configurations.
In addition, a highly attractive interaction exhibiting a large capture radius, effective independent of the orientation and the direction of separation of the defects, is observed.
Accordingly, the formation of C$_{\text{s}}$ is very likely to occur.
Comparatively high energies necessary for the reverse process reveal this configuration to be extremely stable.
Thus, C interstitials and vacancies located close together are assumed to end up in such a configuration, in which the C atom is tetrahedrally coordinated and bound to four Si atoms as expected in SiC.

Investigating configurations of C$_{\text{s}}$ and Si$_{\text{i}}$, formation energies higher than that of the C$_{\text{i}}$ \hkl<1 0 0> DB are obtained keeping up previously derived assumptions concerning the ground state of C$_{\text{i}}$ in otherwise perfect Si.
However, a small capture radius is identified for the respective interaction that might prevent the recombination of defects exceeding a separation of \unit[0.6]{nm} into the ground state configuration.
In addition, a rather small activation energy of \unit[0.77]{eV} allows for the formation of a C$_{\text{s}}$-Si$_{\text{i}}$ pair originating from the C$_{\text{i}}$ \hkl<1 0 0> DB structure by thermally activated processes.
Thus, elevated temperatures might lead to thermodynamically unstable configurations of C$_{\text{s}}$ and a remaining Si atom in the near interstitial lattice, which is supported by the result of an {\em ab initio} molecular dynamics run.
%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.

\fi

% todo - sync with conclusion chapter

These findings allow to draw conclusions on the mechanisms involved in the process of SiC conversion in Si.
Agglomeration of C$_{\text{i}}$ is energetically favored and enabled by a low activation energy for migration.
Although ion implantation is a process far from thermodynamic equilibrium, which might result in phases not described by the Si/C phase diagram, i.e. a C phase in Si, high activation energies are believed to be responsible for a low probability of the formation of C-C clusters.

In the context of the initially stated controversy present in the precipitation model, these findings suggest an increased participation of C$_{\text{s}}$ already in the initial stage due to its high probability of incidence.
In addition, thermally activated, C$_{\text{i}}$ might turn into C$_{\text{s}}$.
The associated emission of Si$_{\text{i}}$ serves two needs: as a vehicle for other C$_{\text{s}}$ atoms and as a supply of Si atoms needed elsewhere to form the SiC structure.
As for the vehicle, Si$_{\text{i}}$ is believed to react with C$_{\text{s}}$ turning it into highly mobile C$_{\text{i}}$ again, allowing for the rearrangement of the C atom.
The rearrangement is crucial to end up in a configuration of C atoms only occupying substitutionally the lattice sites of one of the two fcc lattices that build up the diamond lattice.
On the other hand, the conversion of some region of Si into SiC by \cs{} is accompanied by a reduction of the volume since SiC exhibits a \unit[20]{\%} smaller lattice constant than Si.
The reduction in volume is compensated by excess Si$_{\text{i}}$ serving as building blocks for the surrounding Si host or a further formation of SiC.

To conclude, precipitation occurs by successive agglomeration of C$_{\text{s}}$.
However, the agglomeration and rearrangement of C$_{\text{s}}$ is only possible by mobile C$_{\text{i}}$, which has to be present at the same time.
Accordingly, the process is governed by both, C$_{\text{s}}$ accompanied by Si$_{\text{i}}$ as well as C$_{\text{i}}$.
It is worth to mention that there is no contradiction to results of the HREM studies \cite{werner96,werner97,eichhorn99,lindner99_2,koegler03}.
Regions showing dark contrasts in an otherwise undisturbed Si lattice are attributed to C atoms in the interstitial lattice.
However, there is no particular reason for the C species to reside in the interstitial lattice.
Contrasts are also assumed for Si$_{\text{i}}$.
Once precipitation occurs, regions of dark contrasts disappear in favor of Moir\'e patterns indicating 3C-SiC in c-Si due to the mismatch in the lattice constant.
Until then, however, these regions are either composed of stretched coherent SiC and interstitials or of already contracted incoherent SiC surrounded by Si and interstitials, where the latter is too small to be detected in HREM.
In both cases Si$_{\text{i}}$ might be attributed a third role, which is the partial compensation of tensile strain that is present either in the stretched SiC or at the interface of the contracted SiC and the Si host.

Furthermore, the experimentally observed alignment of the \hkl(h k l) planes of the precipitate and the substrate is satisfied by the mechanism of successive positioning of C$_{\text{s}}$.
In contrast, there is no obvious reason for the topotactic orientation of an agglomerate consisting exclusively of C-Si dimers, which would necessarily involve a much more profound change in structure for the transition into SiC.

%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}

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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.

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