fixed stupid bug + introduction ...

[lectures/latex.git] / posic / publications / emrs2012.tex

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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{\Ast,\textsf{\bfseries 1}},
 K. Nordlund\textsuperscript{\Ast,\textsf{\bfseries 2}},
 W. G. Schmidt\textsuperscript{\Ast,\textsf{\bfseries 3}},
 E. Rauls\textsuperscript{\Ast,\textsf{\bfseries 3}},
 J. K. N. Lindner\textsuperscript{\Ast,\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}, threshold displacement energies in Si \cite{mazzarolo01,holmstroem08} important in ion implantation, C defects and defect reactions in Si \cite{tersoff90,dal_pino93,capaz94,burnard93,leary97,capaz98,zhu98,mattoni2002,park02,jones04}, the SiC/Si interface \cite{chirita97,kitabatake93,cicero02,pizzagalli03} and defects in SiC \cite{bockstedte03,rauls03a,gao04,posselt06,gao07}.
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, a highly accurate first-principles treatment is utilized to systematically investigate relevant intrinsic as well as carbon related defect structures and defect mobilities in silicon.
These findings are compared to empirical potential results and completed by molecular dynamics simulations  to draw conclusions on the SiC precipitation mechanism in Si.

\section{Methodology}

The first-principles DFT calculations were performed with the plane-wave based Vienna ab initio simulation package (VASP)\cite{kresse96}.
The Kohn-Sham equations were solved using the generalized-gradient exchange-correlation (XC) functional approximation proposed by Perdew and Wang\cite{perdew86,perdew92}.
The electron-ion interaction was described by norm-conserving ultra-soft pseudopotentials\cite{hamann79} as implemented in VASP\cite{vanderbilt90}.
Throughout this work an energy cut-off of \unit[300]{eV} was used to expand the wave functions into the plane-wave basis.
To reduce the computational effort sampling of the Brillouin zone was restricted to the $\Gamma$-point, which has been shown to yield reliable results\cite{dal_pino93}.
The defect structures and the migration paths were modelled in cubic supercells with a side length of \unit[1.6]{nm} containing $216$ Si atoms.
Formation energies and structures are reasonably converged with respect to the system size.
The ions and cell shape were allowed to change in order to realize a constant pressure simulation.
The observed changes in volume were less than \unit[0.2]{\%} of the volume indicating a rather low dependence of the results on the ensemble choice.
Ionic relaxation was realized by the conjugate gradient algorithm.
Spin polarization has been fully accounted for.

Migration and recombination pathways have been investigated utilizing the constraint conjugate gradient relaxation technique (CRT)\cite{kaukonen98}.
While not guaranteed to find the true minimum energy path, the method turns out to identify reasonable pathways for the investigated structures.
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.
%In the same way defect formation energies are determined in the article\cite{dal_pino93} used for comparison.
This corresponds to the definition utilized in another study on C defects in Si\cite{dal_pino93} that we compare our results to.
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 show binding energies below zero while non-interacting isolated defects result in a binding energy of zero.
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.

\section{Results}

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.
First of all, structure and energetics of separated defects are presented.
The investigations proceed with pairs of the ground state and, thus, most probable defect configurations that are believed to be fundamental in the Si to SiC conversion.

\subsection{Separated defects in silicon}
\label{subsection:sep_def}
% we need both: Si self-int & C int ground state configuration (for combos)

Several geometries have been calculated to be stable for individual intrinsic and C related defects in Si.
Fig.~\ref{fig:sep_def} shows the obtained structures while the corresponding energies of formation are summarized and compared to values from literature in Table~\ref{table:sep_eof}.
\begin{figure}
\begin{minipage}[t]{0.32\columnwidth}
\underline{Si$_{\text{i}}$ \hkl<1 1 0> DB}\\
\includegraphics[width=\columnwidth]{si110.eps}
\end{minipage}
\begin{minipage}[t]{0.32\columnwidth}
\underline{Si$_{\text{i}}$ hexagonal}\\
\includegraphics[width=\columnwidth]{sihex.eps}
\end{minipage}
\begin{minipage}[t]{0.32\columnwidth}
\underline{Si$_{\text{i}}$ tetrahedral}\\
\includegraphics[width=\columnwidth]{sitet.eps}
\end{minipage}\\
\begin{minipage}[t]{0.32\columnwidth}
\underline{Si$_{\text{i}}$ \hkl<1 0 0> DB}\\
\includegraphics[width=\columnwidth]{si100.eps}
\end{minipage}
\begin{minipage}[t]{0.32\columnwidth}
\underline{Vacancy}\\
\includegraphics[width=\columnwidth]{sivac.eps}
\end{minipage}
\begin{minipage}[t]{0.32\columnwidth}
\underline{C$_{\text{s}}$}\\
\includegraphics[width=\columnwidth]{csub.eps}
\end{minipage}\\
\begin{minipage}[t]{0.32\columnwidth}
\underline{C$_{\text{i}}$ \hkl<1 0 0> DB}\\
\includegraphics[width=\columnwidth]{c100.eps}
\end{minipage}
\begin{minipage}[t]{0.32\columnwidth}
\underline{C$_{\text{i}}$ \hkl<1 1 0> DB}\\
\includegraphics[width=\columnwidth]{c110.eps}
\end{minipage}
\begin{minipage}[t]{0.32\columnwidth}
\underline{C$_{\text{i}}$ bond-centered}\\
\includegraphics[width=\columnwidth]{cbc.eps}
\end{minipage}
\caption{Configurations of silicon and carbon point defects in silicon. Silicon and carbon atoms are illustrated by yellow and gray spheres respectively. Bonds are drawn whenever considered appropriate to ease identifying defect structures for the reader. Dumbbell configurations are abbreviated by DB.}
\label{fig:sep_def}
\end{figure}
\begin{table*}
\begin{tabular}{l c c c c c c c c c}
 & 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
 Present study & 3.39 & 3.42 & 3.77 & 4.41 & 3.63 & 1.95 & 3.72 & 4.16 & 4.66 \\
 \multicolumn{10}{c}{Other ab initio studies} \\
 Ref.\cite{al-mushadani03} & 3.40 & 3.45 & - & - & 3.53 & - & - & - & - \\
 Ref.\cite{leung99} & 3.31 & 3.31 & 3.43 & - & - & - & - & - & - \\
 Ref.\cite{dal_pino93,capaz94} & - & - & - & - & - & 1.89\cite{dal_pino93} & x & - & x+2.1\cite{capaz94}
\end{tabular}
\caption{Formation energies of silicon and carbon point defects in crystalline silicon given in eV. T denotes the tetrahedral, H the hexagonal and BC the bond-centered interstitial configuration. V corresponds to the vacancy configuration. Dumbbell configurations are abbreviated by DB.}
\label{table:sep_eof}
\end{table*}
Results obtained by the present study compare well with results from literature\cite{leung99,al-mushadani03,dal_pino93,capaz94}.
Regarding intrinsic defects in Si, the \hkl<1 1 0> self-interstitial dumbbell (Si$_{\text{i}}$ \hkl<1 1 0> DB) is found to be the ground state configuration closely followed by the hexagonal and tetrahedral configuration, which is consensus for Si$_{\text{i}}$\cite{leung99,al-mushadani03}.
In the case of a C impurity, next to the C$_{\text{s}}$ configuration, in which a C atom occupies an already vacant Si lattice site, the C \hkl<1 0 0> interstitial dumbbell (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{burnard93,leary97,dal_pino93,capaz94,jones04} and experimental\cite{watkins76,song90} investigations, which all predict this configuration to be the ground state.
%However, to our best knowledge, no energy of formation for this type of defect based on first-principles calculations has yet been explicitly stated in literature.
However, to our best knowledge, no energy of formation for this type of defect based on first-principles calculations is available.

Instead, Capaz et al.\cite{capaz94}, investigating migration pathways of the C$_{\text{i}}$ \hkl<1 0 0> DB, find this defect to be \unit[2.1]{eV} lower in energy than the bond-centered (BC) configuration.
The BC configuration is claimed to constitute the saddle point within the C$_{\text{i}}$ \hkl[0 0 -1] DB migration path residing in the \hkl(1 1 0) plane and, thus, interpreted as the barrier of migration for the respective path.
However, the present study indicates a local minimum state for the BC defect if spin polarized calculations are performed resulting in a net magnetization of two electrons localized in a torus around the C atom.
Another DFT calculation without fully accounting for the electron spin results in the smearing of a single electron over two non-degenerate Kohn-Sham states and an increase of the total energy by \unit[0.3]{eV} for the BC configuration.
Regardless of the rather small correction of \unit[0.3]{eV} due to the spin, the difference we found is much smaller (\unit[0.94]{eV}), which would nicely compare to experimentally observed migration barriers of \unit[0.70-0.87]{eV}\cite{lindner06,tipping87,song90}.
However, since the BC configuration constitutes a real local minimum another barrier exists which is about \unit[1.2]{eV} in height.
Indeed Capaz et al. propose another path and find it to be the lowest in energy\cite{capaz94}, 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 (\unit[0.90]{eV}) to experimental values.
A more detailed description can be found in a previous study\cite{zirkelbach10a}.

Next to the C$_{\text{i}}$ BC configuration the vacancy and Si$_{\text{i}}$ \hkl<1 0 0> DB have to be treated by taking into account the spin of the electrons.
For the vacancy the net spin up electron density is localized in caps at the four surrounding Si atoms directed towards the vacant site.
In the  Si$_{\text{i}}$ \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.
No other configuration, within the ones that are mentioned, is affected.

Concerning the mobility of the ground state Si$_{\text{i}}$, we found an activation energy of \unit[0.67]{eV} for the transition of the Si$_{\text{i}}$ \hkl[0 1 -1] to \hkl[1 1 0] DB located at the neighbored Si lattice site in \hkl[1 1 -1] direction.
Further investigations revealed a barrier of \unit[0.94]{eV} for the Si$_{\text{i}}$ \hkl[1 1 0] DB to Si$_{\text{i}}$ H, \unit[0.53]{eV} for the Si$_{\text{i}}$ \hkl[1 1 0] DB to Si$_{\text{i}}$ T and \unit[0.35]{eV} for the Si$_{\text{i}}$ H to Si$_{\text{i}}$ T transition.
%Obtained values are of the same order of magnitude than values derived from other ab initio studies\cite{bloechl93,sahli05}.
These are of the same order of magnitude than values derived from other ab initio studies\cite{bloechl93,sahli05}.

\subsection{Pairs of C$_{\text{i}}$}

C$_{\text{i}}$ pairs of the \hkl<1 0 0> type have been investigated in the first part.
Fig.~\ref{fig:combos_ci} schematically displays the initial C$_{\text{i}}$ \hkl[0 0 -1] DB structure and various positions for the second defect (1-5) that have been used for investigating defect pairs.
Table~\ref{table:dc_c-c} summarizes resulting binding energies for the combination with a second C-Si \hkl<1 0 0> DB obtained for different orientations at positions 1 to 5.
\begin{figure}
\subfloat[]{\label{fig:combos_ci}\includegraphics[width=0.45\columnwidth]{combos_ci.eps}}
\hspace{0.1cm}
\subfloat[]{\label{fig:combos_si}\includegraphics[width=0.45\columnwidth]{combos.eps}}
\caption{Position of the initial C$_{\text{i}}$ \hkl[0 0 -1] DB (I) (Fig.~\ref{fig:combos_ci}) and of the lattice site chosen for the initial Si$_{\text{i}}$ \hkl<1 1 0> DB (Si$_{\text{i}}$) (Fig.~\ref{fig:combos_si}). Lattice sites for the second defect used for investigating defect pairs are numbered from 1 to 5.} 
\label{fig:combos}
\end{figure}
\begin{table}
\begin{tabular}{l c c c c c c }
 & 1 & 2 & 3 & 4 & 5 & R \\
\hline
 \hkl[0 0 -1] & -0.08 & -1.15 & -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] & -2.39 & -0.17 & -0.10 & -0.27 & -1.88 & -0.05\\
 \hkl[0 1 0] & -2.25 & -1.90 & -2.25 & -0.12 & -1.38 & -0.06\\
 \hkl[-1 0 0] & -2.39 & -0.36 & -2.25 & -0.12 & -1.88 & -0.05\\
 \hkl[1 0 0] & -2.25 & -2.16 & -0.10 & -0.27 & -1.38 & -0.06\\
\end{tabular}
\caption{Binding energies in eV of C$_{\text{i}}$ \hkl<1 0 0>-type defect pairs. Equivalent configurations exhibit equal energies. Column 1 lists the orientation of the second defect, which is combined with the initial C$_{\text{i}}$ \hkl[0 0 -1] DB. The position index of the second defect is given in the first row according to Fig.~\ref{fig:combos}. 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 defect separation distance ($\approx \unit[1.3]{nm}$) due to periodic boundary conditions.}
\label{table:dc_c-c}
\end{table}
Most of the obtained configurations result in binding energies well below zero indicating a preferable agglomeration of this type of defects.
For increasing distances of the defect pair the binding energy approaches to zero (R in Table~\ref{table:dc_c-c}) as it is expected for non-interacting isolated defects.
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.

Mattoni et al.\cite{mattoni2002} predict the ground state configuration 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 this work we observed a further relaxation of this defect structure.
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}.
Apart from that, we found a more favorable configuration 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 C$_{\text{i}}$ DBs in Si.
The atomic arrangement is shown in the bottom right of Fig.~\ref{fig:036-239}.
The two C$_{\text{i}}$ 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}.

Investigating migration barriers allows to predict the probability of formation of defect complexes by thermally activated diffusion processes.
% ground state configuration, C cluster
Based on the lowest energy migration path of a single C$_{\text{i}}$ DB the configuration, in which the second C$_{\text{i}}$ DB is oriented along \hkl[0 1 0] at position 2 is assumed to constitute an ideal starting point for a transition into the ground state.
In addition, the starting configuration exhibits a low binding energy (\unit[-1.90]{eV}) and is, thus, very likely to occur.
However, a barrier height of more than \unit[4]{eV} was detected resulting in a low probability for the transition.
The high activation energy is attributed to the stability of such a low energy configuration, in which the C atom of the second DB is located close to the initial DB.
Low barriers have only been 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}
\includegraphics[width=\columnwidth]{036-239.ps}
\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}
%  strange mig from -190 -> -2.39 (barrier > 4 eV)
% C-C migration -> idea:
%  mig from low energy confs has extremely high barrier!
%  low barrier only from energetically less/unfavorable confs (?)! <- prove!
%  => low probability of C-C clustering ?!?
%
% should possibly be transfered to discussion section
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}
\includegraphics[width=\columnwidth]{188-225.ps}
\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, this type of defect pair is represented four times (two times more often than the ground state configuration) within the systematically investigated configuration space.
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.
% alternatively: ... considered period of time (of the IBS process).
%
% ?!?
% look for precapture mechanism (local minimum in energy curve)
% also: plot energy all confs with respect to C-C distance
%       maybe a pathway exists traversing low energy confs ?!?

% point out that configurations along 110 were extended up to the 6th NN in that direction
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{table:dc_110}.
\begin{table}
\begin{tabular}{l c c c c c c }
 & 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 [nm] & 0.14 & 0.46 & 0.65 & 0.86 & 1.05 & 1.08 
\end{tabular}
\caption{Binding energies $E_{\text{b}}$ and C-C distance of energetically most favorable C$_{\text{i}}$ \hkl<1 0 0>-type defect pairs separated along the \hkl[1 1 0] bond chain.}
\label{table:dc_110}
\end{table}
The binding energy of these configurations with respect to the C-C distance is plotted in Fig.~\ref{fig:dc_110}.
\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 interaction is found to be proportional to the reciprocal cube of the C-C distance for extended separations of the C$_{\text{i}}$ and saturates for the smallest possible separation, i.e. the ground state configuration.
Not considering the previously mentioned elevated barriers for migration an attractive interaction between the C$_{\text{i}}$ 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.

\begin{table}
\begin{tabular}{l c c c c c c }
 & 1 & 2 & 3 & 4 & 5 & R \\
\hline
C$_{\text{s}}$ & 0.26$^a$/-1.28$^b$ & -0.51 & -0.93$^A$/-0.95$^B$ & -0.15 & 0.49 & -0.05\\
V & -5.39 ($\rightarrow$ C$_{\text{S}}$) & -0.59 & -3.14 & -0.54 & -0.50 & -0.31
\end{tabular}
\caption{Binding energies of combinations of the C$_{\text{i}}$ \hkl[0 0 -1] defect with a substitutional C or vacancy located at positions 1 to 5 according to Fig.~\ref{fig: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{table:dc_c-sv}
\end{table}

\subsection{C$_{\text{i}}$ next to C$_{\text{s}}$}

The first row of Table~\ref{table:dc_c-sv} lists the binding energies of C$_{\text{s}}$ next to the C$_{\text{i}}$ \hkl[0 0 -1] DB.
For C$_{\text{s}}$ located at position 1 and 3 the configurations a and A correspond to the naive relaxation of the structure by substituting the Si atom by a C atom in the initial C$_{\text{i}}$ \hkl[0 0 -1] DB structure at positions 1 and 3 respectively.
However, small displacements of the involved atoms near the defect result in different stable structures labeled b and B respectively.
Fig.~\ref{fig:093-095} and \ref{fig:026-128} show structures A, B and a, b together with the barrier of migration for the A to B and a to b transition respectively. 

% A B
%./visualize_contcar -w 640 -h 480 -d results/c_00-1_c3_csub_B -nll -0.20 -0.4 -0.1 -fur 0.9 0.6 0.9 -c 0.5 -1.5 0.375 -L 0.5 0 0.3 -r 0.6 -A -1 2.465
\begin{figure}
\includegraphics[width=\columnwidth]{093-095.ps}
\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 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 was 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}
\includegraphics[width=\columnwidth]{026-128.ps}
\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 a is similar to configuration A, except that the C$_{\text{s}}$ atom at position 1 is facing the C DB atom as a neighbor resulting in the formation of a strong C-C bond and a much more noticeable perturbation of the DB structure.
Nevertheless, the C and Si DB atoms remain threefold coordinated.
Although the C-C bond exhibiting a distance of \unit[0.15]{nm} close to the distance expected in diamond or graphite should lead to a huge gain in energy, a repulsive interaction with a binding energy of \unit[0.26]{eV} is observed due to compressive strain of the Si DB atom and its top neighbors (\unit[0.230]{nm}/\unit[0.236]{nm}) along with additional tensile strain of the C$_{\text{s}}$ and its three neighboring Si atoms (\unit[0.198-0.209]{nm}/\unit[0.189]{nm}).
Again a single bond switch, i.e. the breaking of the bond of the Si atom bound to the fourfold coordinated C$_{\text{s}}$ atom and the formation of a double bond between the two C atoms, results in configuration b.
The two C atoms form a \hkl[1 0 0] DB sharing the initial C$_{\text{s}}$ lattice site while the initial Si DB atom occupies its previously regular lattice site.
The transition is accompanied by a large gain in energy as can be seen in Fig.~\ref{fig:026-128}, making it the ground state configuration of a C$_{\text{s}}$ and C$_{\text{i}}$ DB in Si yet \unit[0.33]{eV} lower in energy than configuration B.
This finding is in good agreement with a combined ab initio and experimental study of Liu et~al.\cite{liu02}, who first proposed this structure as the ground state identifying an energy difference compared to configuration B of \unit[0.2]{eV}.
% mattoni: A favored by 0.2 eV - NO! (again, missing spin polarization?)
A net magnetization of two spin up electrons, which are equally localized as in the Si$_{\text{i}}$ \hkl<1 0 0> DB structure is observed.
In fact, these two configurations are very similar and are qualitatively different from the C$_{\text{i}}$ \hkl<1 0 0> DB that does not show magnetization but a nearly collinear bond of the C DB atom to its two neighbored Si atoms while the Si DB atom approximates \unit[120]{$^{\circ}$} angles in between its bonds.
Configurations a, A and B are not affected by spin polarization and show zero magnetization.
Mattoni et~al.\cite{mattoni2002}, in contrast, find configuration b less favorable than configuration A by \unit[0.2]{eV}.
Next to differences in the XC functional and plane-wave energy cut-off this discrepancy might be attributed to the neglect of spin polarization in their calculations, which -- as has been shown for the C$_{\text{i}}$ BC configuration -- results in an increase of configurational energy.
Indeed, investigating the migration path from configurations a to b and, in doing so, reusing the wave functions of the previous migration step the final structure, i.e. configuration b, was 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].

% c agglomeration vs c clustering ... migs to b conf
% 2 more migs: 051 -> 128 and 026! forgot why ... probably it's about probability of C clustering
Obviously agglomeration of C$_{\text{i}}$ and C$_{\text{s}}$ is energetically favorable except for separations along one of the \hkl<1 1 0> directions.
The energetically most favorable configuration (configuration b) forms a strong but compressively strained C-C bond with a separation distance of \unit[0.142]{nm} sharing a Si lattice site.
Again, conclusions concerning the probability of formation are drawn by investigating migration paths.
Since C$_{\text{s}}$ is unlikely to exhibit a low activation energy for migration the focus is on C$_{\text{i}}$.
Pathways starting from the two next most favored configurations were investigated, which show activation energies above \unit[2.2]{eV} and \unit[3.5]{eV} respectively.
Although lower than the barriers for obtaining the ground state of two C$_{\text{i}}$ defects the activation energies are yet considered too high.
For the same reasons as in the last subsection, structures other than the ground state configuration are, thus, assumed to arise more likely due to much lower activation energies necessary for their formation and still comparatively low binding energies.

\subsection{C$_{\text{i}}$ next to V}

In the last subsection 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.
Resulting binding energies of a C$_{\text{i}}$ DB and a nearby vacancy are listed in the second row of Table~\ref{table:dc_c-sv}.
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 ground state configuration is obtained for a V at position 1.
The C atom of the DB 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 at position 3 due to the reduction of compressive strain of the Si DB atom and its two upper Si neighbors present in the C$_{\text{i}}$ DB configuration.
This configuration is followed by the structure, in which a vacant site is created at position 2.
Similar to the observations for C$_{\text{s}}$ in the last subsection a reduction of strain along \hkl[0 0 1] is enabled by this configuration.
Relaxed structures of the latter two defect combinations are shown in the bottom left of Fig.~\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 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}
\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 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.
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.
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 Vs, 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.
Based on these results, a high probability for the formation of C$_{\text{s}}$ must be concluded.

\subsection{C$_{\text{s}}$ next to Si$_{\text{i}}$}

As shown in section~\ref{subsection:sep_def}, C$_{\text{s}}$ exhibits the lowest energy of formation.
Considering a perfect Si crystal and conservation of particles, however, the occupation of a Si lattice site by a slowed down implanted C atom is necessarily accompanied by the formation of a Si self-interstitial.
There are good reasons for the existence of regions exhibiting such configurations with regard to the IBS process.
Highly energetic C atoms are able to 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.
%Thus, configurations of C$_{\text{s}}$ and Si self-interstitials are investigated in the following.
Provided that the first C atom, which created the V and Si$_{\text{i}}$ pair has enough kinetic energy to escape the affected region, the C$_{\text{s}}$-Si$_{\text{i}}$ pair can be described as a separated defect complex.
The Si$_{\text{i}}$ \hkl<1 1 0> DB, which was found to exhibit the lowest energy of formation within the investigated self-interstitial configurations, is assumed to provide the energetically most favorable configuration in combination with C$_{\text{s}}$.

\begin{table}
\begin{tabular}{l c c c c c c}
 & \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} \\
\end{tabular}
\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:combos_si}. The respective orientation of the Si$_{\text{i}}$ DB is given in the first row.}
\label{table:dc_si-s}
\end{table}
\begin{table*}
\begin{tabular}{l c c c c c c c c c c}
 & \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\\
\end{tabular}
\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{table:dc_si-s}.}
\label{table:dc_si-s_e}
\end{table*}
Table~\ref{table:dc_si-s} 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: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{table:dc_si-s_e}.
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].

However, the configuration 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}
\includegraphics[width=\columnwidth]{162-097.ps}
\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}
%\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}
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 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.

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 ab initio molecular dynamics 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:md}.
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.
\begin{figure}
\begin{minipage}{0.49\columnwidth}
\includegraphics[width=\columnwidth]{md01.eps}
\end{minipage}
\begin{minipage}{0.49\columnwidth}
\includegraphics[width=\columnwidth]{md02.eps}\\
\end{minipage}\\
\begin{minipage}{0.49\columnwidth}
\begin{center}
$t=\unit[2230]{fs}$
\end{center}
\end{minipage}
\begin{minipage}{0.49\columnwidth}
\begin{center}
$t=\unit[2900]{fs}$
\end{center}
\end{minipage}
\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:md}
\end{figure}

\section{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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