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\begin{document}

\title{First-principles and empirical potential simulation study of intrinsic
       and carbon-related defects in silicon}

\titlerunning{First-principles and empirical potential simulation study
              of intrinsic and carbon-related defects in silicon}

\author{%
 F. Zirkelbach\textsuperscript{\Ast,\textsf{\bfseries 1}},
 B. Stritzker\textsuperscript{\textsf{\bfseries 1}},
 K. Nordlund\textsuperscript{\textsf{\bfseries 2}},
 W. G. Schmidt\textsuperscript{\textsf{\bfseries 3}},
 E. Rauls\textsuperscript{\textsf{\bfseries 3}},
 J. K. N. Lindner\textsuperscript{\textsf{\bfseries 3}}
}

\authorrunning{F. Zirkelbach et al.}

\mail{e-mail
  \textsf{frank.zirkelbach@physik.uni-augsburg.de}
}

\institute{%
  \textsuperscript{1}\, Experimentalphysik IV, Universit\"at Augsburg,
                        86135 Augsburg, Germany\\
  \textsuperscript{2}\, Department of Physics, University of Helsinki,
                        00014 Helsinki, Finland\\
  \textsuperscript{3}\, Department Physik, Universit\"at Paderborn,
                        33095 Paderborn, Germany}

\received{XXXX, revised XXXX, accepted XXXX}

\published{XXXX}

\keywords{Silicon, carbon, silicon carbide, defect formation, defect migration,
          density functional theory, empirical potential, molecular dynamics.}
%\pacs{61.72.J-,61.72.Yx,61.72.uj,66.30.J-,79.20.Rf,31.15.A-}

\abstract{%
Results of atomistic simulations aimed at understanding precipitation of the highly attractive wide band gap semiconductor material silicon carbide in silicon are presented.
The study involves a systematic investigation of intrinsic and carbon-related defects as well as defect combinations and defect migration by both, quantum-mechanical first-principles as well as empirical potential methods.
Comparing formation and activation energies, ground-state structures of defects and defect combinations as well as energetically favorable agglomeration of defects are predicted.
Moreover, the highly accurate ab initio calculations unveil limitations of the analytical method based on a Tersoff-like bond order potential.
A work-around is proposed in order to subsequently apply the highly efficient technique on large structures not accessible by first-principles methods.
The outcome of both types of simulation provides a basic microscopic understanding of defect formation and structural evolution particularly at non-equilibrium conditions strongly deviated from the ground state as commonly found in SiC growth processes.
A possible precipitation mechanism, which conforms well to experimental findings clarifying contradictory views present in the literature is outlined.
}

\maketitle

\section{Introduction}

Silicon carbide (SiC) is a promising material for high-temperature, high-power and high-frequency electronic and optoelectronic devices, which can operate under extreme conditions \cite{edgar92,morkoc94,wesch96,capano97,park98}.
Ion beam synthesis (IBS) consisting of high-dose carbon implantation into crystalline silicon (c-Si) and subsequent or in situ annealing is a promising technique to fabricate nano-sized precipitates and thin films of the favorable cubic SiC (3C-SiC) polytype topotactically aligned to and embedded in the silicon host \cite{borders71,lindner99,lindner01,lindner02}.
However, the process of formation of SiC precipitates in Si during C implantation is not yet fully understood and controversial ideas exist in the literature.
Based on experimental high resolution transmission electron microscopy (HREM) studies \cite{werner96,werner97,eichhorn99,lindner99_2,koegler03} it is assumed that incorporated C atoms form C-Si dimers (dumbbells) on regular Si lattice sites.
The highly mobile C interstitials agglomerate into large clusters followed by the formation of incoherent 3C-SiC nanocrystallites once a critical size of the cluster is reached.
In contrast, a couple of other studies \cite{strane94,nejim95,guedj98} suggest initial coherent SiC formation by agglomeration of substitutional instead of interstitial C followed by the loss of coherency once the increasing strain energy surpasses the interfacial energy of the incoherent 3C-SiC precipitate and the c-Si substrate.

To solve this controversy and in order to understand the effective underlying processes on a microscopic level atomistic simulations are performed.
% ????
A lot of theoretical work has been done on intrinsic point defects in Si \cite{bar-yam84,bar-yam84_2,car84,batra87,bloechl93,tang97,leung99,colombo02,goedecker02,al-mushadani03,hobler05,sahli05,posselt08,ma10} and C defects and defect reactions in Si \cite{tersoff90,dal_pino93,capaz94,burnard93,leary97,capaz98,zhu98,mattoni2002,park02,jones04}.
However, none of the mentioned studies consistently investigates entirely the relevant defect structures and reactions concentrated on the specific problem of 3C-SiC formation in C implanted Si.
% ????

In the present study, an accurate first-principles treatment is utilized to systematically investigate relevant intrinsic as well as carbon related defect structures and defect mobilities in silicon, which allow to draw conclusions on the mechanism of SiC precipitation in Si.
These findings are compared to empirical potential results, which, by taking into account the drawbacks of the less accurate though computationally efficient method enabling molecular dynamics (MD) simulations of large structures, support and complete previous findings on SiC precipitation based on the quantum-mechanical treatment.

\section{Methodology}

The plane-wave based Vienna ab initio simulation package (VASP) \cite{kresse96} is used for the first-principles calculations based on density functional theory (DFT).
Exchange and correlation is taken into account by the generalized-gradient approximation \cite{perdew86,perdew92}.
Norm-conserving ultra-soft pseudopotentials \cite{hamann79} as implemented in VASP \cite{vanderbilt90} are used to describe the electron-ion interaction.
A kinetic energy cut-off of \unit[300]{eV} is employed.
Defect structures and migration paths were modelled in cubic supercells with a side length of \unit[1.6]{nm} containing $216$ Si atoms.
These structures are large enough to restrict sampling of the Brillouin zone to the $\Gamma$-point and formation energies and structures are reasonably converged.
The ions and cell shape are allowed to change in order to realize a constant pressure simulation realized by the conjugate gradient algorithm.
Spin polarization is fully accounted for.

Migration and recombination pathways are investigated utilizing the constraint conjugate gradient relaxation technique (CRT) \cite{kaukonen98}.
The defect formation energy $E-N_{\text{Si}}\mu_{\text{Si}}-N_{\text{C}}\mu_{\text{C}}$ is defined by choosing SiC as a particle reservoir for the C impurity, i.e. the chemical potentials are determined by the cohesive energies of a perfect Si and SiC supercell after ionic relaxation.
The binding energy of a defect pair is given by the difference of the formation energy of the complex and the sum of the two separated defect configurations.
Accordingly, energetically favorable configurations result in binding energies below zero while unfavorable configurations show positive values for the binding energy.
The interaction strength, i.e. the absolute value of the binding energy, approaches zero for increasingly non-interacting isolated defects.

Within the empirical approach, defect structures are modeled in a supercell of nine Si lattice constants in each direction consisting of 5832 Si atoms.
Reproducing SiC precipitation is attempted by successive insertion of 6000 C atoms to form a minimal 3C-SiC precipitate with a radius of about \unit[3.1]{nm} within the Si host consisting of 31 unit cells (238328 atoms) in each direction.
At constant temperature 10 atoms are inserted at a time.
Three different regions inside the total simulation volume are considered for a statistically distributed insertion of C atoms.
$V_1$ corresponds to the total simulation volume, $V_2$ to the size of the precipitate and $V_3$ holds the necessary amount of Si atoms of the precipitate.
After C insertion, the simulation is continued for \unit[100]{ps} and cooled down to \unit[20]{$^{\circ}$C} afterwards.
A Tersoff-like bond order potential by Erhart and Albe (EA) \cite{albe_sic_pot} has been utilized, which accounts for nearest neighbor interactions realized by a cut-off function dropping the interaction to zero in between the first and second nearest neighbor distance.
The Berendsen barostat and thermostat \cite{berendsen84} with a time constant of \unit[100]{fs} enables the isothermal-isobaric ensemble.
The velocity Verlet algorithm \cite{verlet67} and a fixed time step of \unit[1]{fs} is used to integrate the equations motion.
Structural relaxation of defect structures is treated by the same algorithms at zero temperature.

\section{Defect configurations in silicon}

Table~\ref{tab:defects} summarizes the formation energies of relevant defect structures for the EA and DFT calculations, which are shown in Figs.~\ref{fig_intrinsic_def} and \ref{fig:carbon_def}.
\begin{table*}
\centering
\begin{tabular}{l c c c c c c c c c}
\hline
$E_{\text{f}}$ [eV] & Si$_{\text{i}}$ \hkl<1 1 0> DB & Si$_{\text{i}}$ H & Si$_{\text{i}}$ T & Si$_{\text{i}}$ \hkl<1 0 0> DB & V & C$_{\text{s}}$ & C$_{\text{i}}$ \hkl<1 0 0> DB & C$_{\text{i}}$ \hkl<1 1 0> DB & C$_{\text{i}}$ BC \\
\hline
VASP & 3.39 & 3.42 & 3.77 & 4.41 & 3.63 & 1.95 & 3.72 & 4.16 & 4.66 \\
Erhart/Albe & 4.39 & 4.48$^*$ & 3.40 & 5.42 & 3.13 & 0.75 & 3.88 & 5.18 & 5.59$^*$ \\
\hline
\end{tabular}
\caption{Formation energies of C and Si point defects in c-Si given in eV. T denotes the tetrahedral, H the hexagonal and BC the bond-centered interstitial configuration. V corresponds to the vacancy. Subscript i and s indicates the interstitial and substitutional configuration. Dumbbell configurations are abbreviated by DB. Formation energies of unstable configurations are marked by an asterisk.}
\label{tab:defects}
\end{table*}
\begin{figure}
\centering
\begin{minipage}[t]{0.43\columnwidth}
\centering
\underline{Si$_{\text{i}}$ \hkl<1 1 0> DB}\\
\includegraphics[width=0.9\columnwidth]{si110_bonds.eps}
\end{minipage}
\begin{minipage}[t]{0.43\columnwidth}
\centering
\underline{Si$_{\text{i}}$ hexagonal}\\
\includegraphics[width=0.9\columnwidth]{sihex_bonds.eps}
\end{minipage}\\
\begin{minipage}[t]{0.43\columnwidth}
\centering
\underline{Si$_{\text{i}}$ tetrahedral}\\
\includegraphics[width=0.9\columnwidth]{sitet_bonds.eps}
\end{minipage}
\begin{minipage}[t]{0.43\columnwidth}
\centering
\underline{Si$_{\text{i}}$ \hkl<1 0 0> DB}\\
\includegraphics[width=0.9\columnwidth]{si100_bonds.eps}
\end{minipage}
\caption{Configurations of intrinsic silicon point defects. Dumbbell configurations are abbreviated by DB.}
\label{fig:intrinsic_def}
\end{figure}
\begin{figure}
\centering
\begin{minipage}[t]{0.43\columnwidth}
\centering
\underline{C$_{\text{s}}$}
\includegraphics[width=0.9\columnwidth]{csub_bonds.eps}
\end{minipage}
\begin{minipage}[t]{0.43\columnwidth}
\centering
\underline{C$_{\text{i}}$ \hkl<1 0 0> DB}\\
\includegraphics[width=0.9\columnwidth]{c100_bonds.eps}
\end{minipage}\\
\begin{minipage}[t]{0.43\columnwidth}
\centering
\underline{C$_{\text{i}}$ \hkl<1 1 0> DB}\\
\includegraphics[width=0.9\columnwidth]{c110_bonds.eps}
\end{minipage}
\begin{minipage}[t]{0.43\columnwidth}
\centering
\underline{C$_{\text{i}}$ bond-centered}\\
\includegraphics[width=0.9\columnwidth]{cbc_bonds.eps}
\end{minipage}
\caption{Configurations of carbon point defects in silicon. Silicon and carbon atoms are illustrated by yellow and gray spheres respectively. Dumbbell configurations are abbreviated by DB.}
\label{fig:carbon_def}
\end{figure}

Regarding intrinsic defects in Si, classical potential and {\em ab initio} methods predict energies of formation that are within the same order of magnitude.
The EA potential does not reproduce the correct ground state, i.e. the interstitial Si (Si$_{\text{i}}$) \hkl<1 1 0> dumbbell (DB), which is consensus for Si$_{\text{i}}$ \cite{leung99,al-mushadani03}.
Instead, the tetrahedral configuration is favored, a limitation assumed to arise due to the sharp cut-off as has already been discussed by Tersoff \cite{tersoff90}.

In the case of C impurities, although discrepancies exist, classical potential and first-principles methods depict the correct order of the formation energies.
Next to the substitutional C (C$_{\text{s}}$) configuration, which is not an interstitial configuration since the C atom occupies an already vacant Si lattice site, the interstitial C (C$_{\text{i}}$) \hkl<1 0 0> DB constitutes the energetically most favorable configuration, in which the C and Si dumbbell atoms share a regular Si lattice site.
This finding is in agreement with several theoretical \cite{dal_pino93,capaz94,burnard93,leary97,jones04} and experimental \cite{watkins76,song90} investigations, which all predict this configuration to be the ground state.
It is worth to note that the bond-centered (BC) configuration constitutes a real local minimum in spin polarized calculations in contrast to results \cite{capaz94} without spin predicting a saddle point configuration as well as to the empirical description, which shows a relaxation into the C$_{\text{i}}$ \hkl<1 0 0> DB ground-state configuration.

\section{Mobility of the carbon defect}

In the following, the migration barriers of the ground-state C defect are investigated by both, first-principles as well as the empircal method.
The migration pathways are shown in Figs.\ref{fig:vasp_mig} and \ref{fig:albe_mig} respectively.

\begin{figure}
\begin{center}
\includegraphics[width=\columnwidth]{path2_vasp_s.ps}
\end{center}
\caption{Migration barrier and structures of the C$_{\text{i}}$ \hkl[0 0 -1] DB (left) to the \hkl[0 -1 0] DB (right) transition as obtained by first principles methods.}
\label{fig:vasp_mig}
\end{figure} 
\begin{figure}
\begin{center}
\includegraphics[width=\columnwidth]{110mig.ps}
\end{center}
\caption{Migration barrier and structures of the C$_{\text{i}}$ \hkl[0 0 -1] DB (left) to the hkl[0 -1 0] DB (right) transition involving the \hkl[1 1 0] DB (center) configuration within EA description. Migration simulations were performed utilizing time constants of \unit[1]{fs} (solid line) and \unit[100]{fs} (dashed line) for the Berendsen thermostat.}
\label{fig:albe_mig}
\end{figure}

In qualitative agreement with the results of Capaz~et~al.\  \cite{capaz94}, the lowest migration barrier of the ground-state C$_{\text{i}}$ defect within the quantum-mechanical treatment is found for the path, in which a C$_{\text{i}}$ \hkl[0 0 -1] DB migrates to a C$_{\text{i}}$ \hkl[0 -1 0] DB located at the neighbored Si lattice site in \hkl[1 1 -1] direction.
Calculations in this work reinforce this path by an additional improvement of the quantitative conformance of the barrier height of \unit[0.90]{eV} to experimental values (\unit[0.70-0.87]{eV}) \cite{lindner06,tipping87,song90}.

In contrast, the empirical approach does not reproduce the same path.
Related to the instability of the BC configuration \cite{zirkelbach11}, a pathway involving the C$_{\text{i}}$ \hkl<1 1 0> DB as an intermediate configuration must be considered most plausible.
Considering a two step diffusion process and assuming equal preexponential factors, an total effective migration barrier 3.5 times higher than the one obtained by first-principles methods is obtained.
A more detailed description can be found in previous studies \cite{zirkelbach10,zirkelbach11}.

\section{Defect combinations}

The implantation of highly energetic C atoms results in a multiplicity of possible point defects and respective combinations.
Thus, defect combinations of an initial C$_{\text{i}}$ DB and further types of defects created at certain neighbor positions have been investigated \cite{zirkelbach11} exclusively by DFT calculations.
Some of the most important results are presented in the following.

\begin{figure}
\includegraphics[width=\columnwidth]{db_along_110_cc_n.ps}
\caption{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:dc_110}
\end{figure}

The agglomeration of interstitial C is found to be energetically favorable.
As can be seen in Fig.~\ref{fig:dc_110}, which shows the binding energies of DB combinations separated along the \hkl[1 1 0] chain, a capture radius clearly exceeding \unit[1]{nm} is observed.
The interaction is proportional to the reciprocal cube of the C-C distance for extended separations of the defects.
However, the interpolated graph shows the disappearance of attractive forces corresponding to the slope of the graph in between the two lowest separation distances, which clearly indicates a preferable C agglomeration but the absence of C clustering.

In IBS, configurations may arise, in which the impinging C atom creates a vacant site (V) near a C$_{\text{i}}$ DB, but does not occupy it.
All these structures were found to be energetically preferred compared to isolated largely separated defects \cite{zirkelbach11} showing an entirely attractive interaction between defects of these types.
The ground-state configuration is obtained for a V located right next to the C atom of the DB.
The C atom moves towards the vacant site forming a stable C$_{\text{s}}$ configuration resulting in the release of a huge amount of energy.
The second most favorable configuration is accomplished for a V located right next to the Si atom of the DB structure.
This is due to the reduction of compressive strain of the Si DB atom and its two upper Si neighbors present in the isolated C$_{\text{i}}$ DB configuration.
This configuration is followed by the structure, in which the V is created at one of the neighbored lattice site below one of the Si atoms that are bound to the C atom of the initial DB.
Relaxed structures of the latter two defect combinations are shown in the bottom left of Figs.~\ref{fig:314-539} and \ref{fig:059-539} respectively together with their energetics during transition into the ground state.
\begin{figure}
\includegraphics[width=\columnwidth]{314-539.ps}
\caption{Migration barrier and structures of the transition of the initial C$_{\text{i}}$ \hkl[0 0 -1] DB and a V created right next to the Si atom of the initial DB (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}
\includegraphics[width=\columnwidth]{059-539.ps}
\caption{Migration barrier and structures of the transition of the initial C$_{\text{i}}$ \hkl[0 0 -1] DB and a V created next to one of the Si atoms that is bound to the C atom of the initial DB (left) into a C$_{\text{s}}$ configuration (right). An activation energy of \unit[0.6]{eV} is observed.}
\label{fig:059-539}
\end{figure}
These transitions exhibit activation energies as low as \unit[0.1]{eV} and \unit[0.6]{eV}.
In the first case the Si and C atom of the DB move towards the vacant and initial DB lattice site respectively.
In the second case Si number 1, which is substituted by the C$_{\text{i}}$ atom, migrates towards the vacant site.
In both cases, the formation of additional bonds is responsible for the vast gain in energy rendering almost impossible the reverse processes.
Considering the small activation energies, a high probability for the formation of stable C$_{\text{s}}$ must be concluded.

In addition, it is instructive to investigate combinations of C$_{\text{s}}$ and Si$_{\text{i}}$, which can be created in IBS by highly energetic C atoms that kick out a Si atom from its lattice site, resulting in a Si self-interstitial accompanied by a vacant site, which might get occupied by another C atom that lost almost all of its kinetic energy.
Provided that the first C atom has enough kinetic energy to escape the affected region, the remaining C$_{\text{s}}$-Si$_{\text{i}}$ pair can be described as a separated defect complex.
Considering the energetically most favorable Si$_{\text{i}}$ defect, i.e. the Si$_{\text{i}}$ \hkl<1 1 0> DB, the most favorable combination is found for C$_{\text{s}}$ located right next to that DB enabling the largest possible reduction of strain.
The configuration and the transition into the ground-state configuration, i.e. the C$_{\text{i}}$ hkl<1 0 0> DB is displayed in Fig.~\ref{fig:162-097}
\begin{figure}
\includegraphics[width=\columnwidth]{162-097.ps}
\caption{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).}
\label{fig:162-097}
\end{figure}
Due to the low barrier of \unit[0.12]{eV}, the C$_{\text{i}}$ \hkl<1 0 0> DB configuration is very likely to occur.
However, the barrier of only \unit[0.77]{eV} for the reverse process indictaes a high probability for the the formation of C$_{\text{s}}$ and a Si$_{\text{i}}$ DB out of the ground state, wich must be considered to be activated without much effort either thermally or by introduced energy of the implantation process.
\begin{figure}
\includegraphics[width=\columnwidth]{c_sub_si110.ps}
%\includegraphics[width=\columnwidth]{c_sub_si110_data.ps}
%\caption{Binding energies of combinations of a C$_{\text{s}}$ and a Si$_{\text{i}}$ DB with respect to the separation distance.}
\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}
Furthermore, the interaction strength, i.e. the absolute value of the binding energy, quickly drops to zero with increasing separation distance as can be seen in Fig.~\ref{fig:dc_si-s}.
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 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 LJ fit estimates almost zero interaction already at \unit[0.5-0.6]{nm}, indicating a low interaction capture radius of the defect pair.
In IBS separations exceeding this capture radius are easily produced.
For these reasons, it must be concluded that configurations of C$_{\text{s}}$ and Si$_{\text{i}}$ instead of the thermodynamically stable C$_{\text{i}}$ \hkl<1 0 0> DB play a decisive role in IBS, a process far from equilibrium.
Indeed, in a previous study, an ab initio molecular dynamics run at \unit[900]{$^{\circ}$C} results in a separation of the C$_{\text{s}}$ and Si$_{\text{i}}$ DB located right next to each other \cite{zirkelbach11}.

To summarize, these obtained  results suggest an increased participation of C$_{\text{s}}$ already in the initial stages of precipitation under IBS conditions.

\section{Large scale empirical potential MD results}

\section{Summary and discussion}

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, have been 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.

The investigation of defect pairs indicated a general trend of defect agglomeration mainly driven by the potential of strain reduction.
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}.
For configurations involving two C impurities the ground state configurations have been found to consist of C-C bonds, which are 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 Vs were 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, we observed a highly attractive interaction exhibiting a large capture radius, effective independent of the orientation and the direction of separation of the defects.
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.

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 were obtained keeping up previously derived assumptions concerning the ground state of C$_{\text{i}}$ in otherwise perfect Si.
However, a small capture radius was 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 configurations of C$_{\text{s}}$ and a remaining Si atom in the near interstitial lattice, which is supported by the result of the molecular dynamics run.

% add somewhere: nearly same energies of C_i -> Si_i + C_s, Si_i mig and C_i mig

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.
% TODO: add SiC structure info to intro
On the other hand, the conversion of some region of Si into SiC by substitutional C 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.

We conclude that 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.

In addition, 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.

\section{Summary}

In summary, C and Si point defects in Si, combinations of these defects and diffusion processes within such configurations have been investigated.
We have shown that C interstitials in Si tend to agglomerate, which is mainly driven by a reduction of strain.
Investigations of migration pathways, however, allow to conclude that C clustering is hindered due to high activation energies of the respective diffusion processes.
A highly attractive interaction and a large capture radius has been identified for the C$_{\text{i}}$ \hkl<1 0 0> DB and the vacancy indicating a high probability for the formation of C$_{\text{s}}$.
In contrast, a rapidly decreasing interaction with respect to the separation distance has been identified for C$_{\text{s}}$ and a Si$_{\text{i}}$ \hkl<1 1 0> DB resulting in a low probability of defects exhibiting respective separations to transform into the C$_{\text{i}}$ \hkl<1 0 0> DB, which constitutes the ground state configuration for a C atom introduced into otherwise perfect Si. 
%Based on these findings conclusions on basic processes involved in the SiC precipitation in bulk Si are drawn.
Obviously, the precipitation process is governed by the formation of C$_{\text{s}}$ already in the initial stages.
Agglomeration and rearrangement of C$_{\text{s}}$, however, is only possible by mobile C$_{\text{i}}$, which, thus, needs to be present at the same time.
Si$_{\text{i}}$ constitutes the vehicle for the rearrangement of C$_{\text{s}}$.

\section*{Acknowledgment}
We gratefully acknowledge financial support by the Bayerische Forschungsstiftung (Grant No. DPA-61/05) and the Deutsche Forschungsgemeinschaft (Grant No. DFG SCHM 1361/11).

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