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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 {\em 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
% shorten
% \cite{edgar92,morkoc94,wesch96,capano97,park98}.
\cite{edgar92,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
% shorten
% \cite{borders71,lindner99,lindner01,lindner02}.
\cite{borders71,lindner01}.
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 
% shorten
% \cite{werner96,werner97,eichhorn99,lindner99_2,koegler03}
\cite{werner96,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,serre95} 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
% shorten
% \cite{bar-yam84,bar-yam84_2,car84,batra87,bloechl93,tang97,leung99,colombo02,goedecker02,al-mushadani03,hobler05,sahli05,posselt08,ma10} 
\cite{bar-yam84,car84,bloechl93,tang97,leung99,al-mushadani03,hobler05,sahli05,posselt08,ma10} 
and C defects and defect reactions in Si 
% shorten
%\cite{tersoff90,dal_pino93,capaz94,burnard93,leary97,capaz98,zhu98,mattoni2002,park02}.
\cite{tersoff90,dal_pino93,capaz94,burnard93,leary97,capaz98,mattoni2002}.
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 {\em 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 on statistically distributed positions at a time.
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=0.8\columnwidth]{si110_bonds.eps}
\end{minipage}
\begin{minipage}[t]{0.43\columnwidth}
\centering
\underline{Si$_{\text{i}}$ hexagonal}\\
\includegraphics[width=0.8\columnwidth]{sihex_bonds.eps}
\end{minipage}\\
\begin{minipage}[t]{0.43\columnwidth}
\centering
\underline{Si$_{\text{i}}$ tetrahedral}\\
\includegraphics[width=0.8\columnwidth]{sitet_bonds.eps}
\end{minipage}
\begin{minipage}[t]{0.43\columnwidth}
\centering
\underline{Si$_{\text{i}}$ \hkl<1 0 0> DB}\\
\includegraphics[width=0.8\columnwidth]{si100_bonds.eps}
\end{minipage}
\caption{Configurations of intrinsic Si 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=0.8\columnwidth]{csub_bonds.eps}
\end{minipage}
\begin{minipage}[t]{0.43\columnwidth}
\centering
\underline{C$_{\text{i}}$ \hkl<1 0 0> DB}\\
\includegraphics[width=0.8\columnwidth]{c100_bonds.eps}
\end{minipage}\\
\begin{minipage}[t]{0.43\columnwidth}
\centering
\underline{C$_{\text{i}}$ \hkl<1 1 0> DB}\\
\includegraphics[width=0.8\columnwidth]{c110_bonds.eps}
\end{minipage}
\begin{minipage}[t]{0.43\columnwidth}
\centering
\underline{C$_{\text{i}}$ bond-centered}\\
\includegraphics[width=0.8\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} 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 empirical 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 indicates a high probability for the the formation of C$_{\text{s}}$ and a Si$_{\text{i}}$ DB out of the ground state, which 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 {\em 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}

Results of the MD simulations at \unit[450]{$^{\circ}$C}, an operative and efficient temperature in IBS \cite{lindner01}, indicate the formation of C$_{\text{i}}$ \hkl<1 0 0> DBs if C is inserted into the total simulation volume.
However, no agglomeration is observed within the simulated time, which was increased up to several nanoseconds.
To overcome the drastically overestimated migration barriers of the C defect, which hamper C agglomeration, the simulation temperature is successively increased up to ßunit[2050]{$^{\circ}$C}.
Fig.~\ref{fig:tot} shows the resulting radial distribution function of Si-C bonds for various elevated temperatures.
\begin{figure}
\includegraphics[width=\columnwidth]{tot_pc_thesis.ps}
\caption{Radial distribution function for Si-C pairs for C insertion at various elevated temperatures. Si-C distances of a single C$_{\text{s}}$ defect configuration are plotted.}
\label{fig:tot}
\end{figure}
A transformation from a structure dominated by C$_{\text{i}}$ into a C$_{\text{s}}$ dominated structure with increasing temperature can clearly be observed if compared with the radial distribution of C$_{\text{s}}$ in c-Si.
Thus, the C$_{\text{s}}$ defect and, thus, stretched coherent structures of SiC, must be considered to play an important role in the IBS at elevated temperatures.
This, in fact, is in agreement with experimental findings of annealing experiments \cite{nejim95,strane94,serre95} and also with the previous DFT results, which suggest C$_{\text{s}}$ to be involved at higher temperatures and in conditions out of thermodynamic equilibrium.

\section{Summary and discussion}

Although investigations of defect combinations show the agglomeration of C$_{\text{i}}$ DBs to be energetically most favorable, configurations that may arise during IBS were presented, their dynamics indicating an C$_{\text{s}}$ to play an important role particularly at high temperatures.
This is supported by the empirical MD results, which show an increased participation of C$_{\text{s}}$ at increased temperatures that allow the system to deviate from the ground state.

Based on these findings, it is concluded that in IBS at elevated temperatures, the conversion into SiC takes place by an initial agglomeration of C$_{\text{s}}$ into coherent, tensily strained structures of SiC followed by precipitation into incoherent SiC structures once a critical radius is reached.
Rearrangement of stable C$_{\text{s}}$ is enabled by excess Si$_{\text{i}}$, which not only acts as a vehicle for C but also as a supply of Si atoms needed elsewhere to form the SiC structure and to reduce possible strain at the interface of coherent SiC precipitates and the Si host.

It is worth to point out that 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.

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

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