Regarding the supposed conversion mechanisms of SiC in c-Si as introduced in section \ref{section:assumed_prec} the understanding of C and Si interstitial point defects in c-Si is of fundamental interest.
During implantation, defects such as vacancies (V), substitutional C (C$_{\text{s}}$), interstitial C (C$_{\text{i}}$) and Si self-interstitials (Si$_{\text{i}}$) are created, which are believed to play a decisive role in the precipitation process.
-In the following, these defects are systematically examined by computationally efficient, classical potential as well as highly accurate DFT calculations with the parameters and simulation conditions that are defined in chapter \ref{chapter:simulation}.
+In the following, these defects are systematically examined by computationally efficient, classical potential as well as highly accurate DFT calculations with the parameters and simulation conditions that are defined in chapter~\ref{chapter:simulation}.
Both methods are used to investigate selected diffusion processes of some of the defect configurations.
While the quantum-mechanical description yields results that excellently compare to experimental findings, shortcomings of the classical potential approach are identified.
These shortcomings are further investigated and the basis for a workaround, as proposed later on in the classical MD simulation chapter, is discussed.
\begin{minipage}{5cm}
\underline{Tetrahedral}\\
$E_{\text{f}}=3.40\,\text{eV}$\\
-\includegraphics[width=4.0cm]{si_pd_albe/tet.eps}
+\includegraphics[width=4.0cm]{si_pd_albe/tet_bonds.eps}
\end{minipage}
\begin{minipage}{10cm}
\underline{Hexagonal}\\[0.1cm]
\begin{minipage}{4cm}
$E_{\text{f}}^*=4.48\,\text{eV}$\\
-\includegraphics[width=4.0cm]{si_pd_albe/hex_a.eps}
+\includegraphics[width=4.0cm]{si_pd_albe/hex_a_bonds.eps}
\end{minipage}
\begin{minipage}{0.8cm}
\begin{center}
\end{minipage}
\begin{minipage}{4cm}
$E_{\text{f}}=3.96\,\text{eV}$\\
-\includegraphics[width=4.0cm]{si_pd_albe/hex.eps}
+\includegraphics[width=4.0cm]{si_pd_albe/hex_bonds.eps}
\end{minipage}
\end{minipage}\\[0.2cm]
\begin{minipage}{5cm}
\underline{\hkl<1 0 0> dumbbell}\\
$E_{\text{f}}=5.42\,\text{eV}$\\
-\includegraphics[width=4.0cm]{si_pd_albe/100.eps}
+\includegraphics[width=4.0cm]{si_pd_albe/100_bonds.eps}
\end{minipage}
\begin{minipage}{5cm}
\underline{\hkl<1 1 0> dumbbell}\\
$E_{\text{f}}=4.39\,\text{eV}$\\
-\includegraphics[width=4.0cm]{si_pd_albe/110.eps}
+\includegraphics[width=4.0cm]{si_pd_albe/110_bonds.eps}
\end{minipage}
\begin{minipage}{5cm}
\underline{Vacancy}\\
\end{flushleft}
%\hrule
\end{center}
-\caption[Relaxed Si self-interstitial defect configurations obtained by classical potential calculations.]{Relaxed Si self-interstitial defect configurations obtained by classical potential calculations. The Si atoms and the bonds (only for the interstitial atom) are illustrated by yellow spheres and blue lines.}
+\caption[Relaxed Si self-interstitial defect configurations obtained by classical potential calculations.]{Relaxed Si self-interstitial defect configurations obtained by classical potential calculations. Si atoms and bonds are illustrated by yellow spheres and blue lines. Bonds of the defect atoms are drawn in red color.}
\label{fig:defects:conf}
\end{figure}
The final configurations obtained after relaxation are presented in Fig. \ref{fig:defects:conf}.
\begin{minipage}{4cm}
\underline{Hexagonal}\\
$E_{\text{f}}^*=9.05\,\text{eV}$\\
-\includegraphics[width=4.0cm]{c_pd_albe/hex.eps}
+\includegraphics[width=4.0cm]{c_pd_albe/hex_bonds.eps}
\end{minipage}
\begin{minipage}{0.8cm}
\begin{center}
\begin{minipage}{4cm}
\underline{\hkl<1 0 0>}\\
$E_{\text{f}}=3.88\,\text{eV}$\\
-\includegraphics[width=4.0cm]{c_pd_albe/100.eps}
+\includegraphics[width=4.0cm]{c_pd_albe/100_bonds.eps}
\end{minipage}
\begin{minipage}{0.5cm}
\hfill
\begin{minipage}{5cm}
\underline{Tetrahedral}\\
$E_{\text{f}}=6.09\,\text{eV}$\\
-\includegraphics[width=4.0cm]{c_pd_albe/tet.eps}
+\includegraphics[width=4.0cm]{c_pd_albe/tet_bonds.eps}
\end{minipage}\\[0.2cm]
\begin{minipage}{4cm}
\underline{Bond-centered}\\
$E_{\text{f}}^*=5.59\,\text{eV}$\\
-\includegraphics[width=4.0cm]{c_pd_albe/bc.eps}
+\includegraphics[width=4.0cm]{c_pd_albe/bc_bonds.eps}
\end{minipage}
\begin{minipage}{0.8cm}
\begin{center}
\begin{minipage}{4cm}
\underline{\hkl<1 1 0> dumbbell}\\
$E_{\text{f}}=5.18\,\text{eV}$\\
-\includegraphics[width=4.0cm]{c_pd_albe/110.eps}
+\includegraphics[width=4.0cm]{c_pd_albe/110_bonds.eps}
\end{minipage}
\begin{minipage}{0.5cm}
\hfill
\begin{minipage}{5cm}
\underline{Substitutional}\\
$E_{\text{f}}=0.75\,\text{eV}$\\
-\includegraphics[width=4.0cm]{c_pd_albe/sub.eps}
+\includegraphics[width=4.0cm]{c_pd_albe/sub_bonds.eps}
\end{minipage}
\end{flushleft}
\end{center}
-\caption[Relaxed C point defect configurations obtained by classical potential calculations.]{Relaxed C point defect configurations obtained by classical potential calculations. The Si/C atoms and the bonds (only for the interstitial atom) are illustrated by yellow/gray spheres and blue lines.}
+\caption[Relaxed C point defect configurations obtained by classical potential calculations.]{Relaxed C point defect configurations obtained by classical potential calculations. Si/C atoms and bonds are illustrated by yellow/gray spheres and blue lines. Bonds of the defect atoms are drawn in red color.}
\label{fig:defects:c_conf}
\end{figure}
\includegraphics[width=7cm]{c_pd_vasp/bc_2333_ksl.ps}
\end{minipage}
\end{center}
-\caption[Structure, charge density isosurface, molecular orbital diagram and Kohn-Sham level diagram of the bond-centered interstitial configuration.]{Structure, charge density, molecular orbital diagram isosurface and Kohn-Sham level diagram of the bond-centered interstitial configuration. Gray, green and blue surfaces mark the charge density of spin up, spin down and the resulting spin up electrons in the charge density isosurface, in which the carbon atom is represented by a red sphere. In the energy level diagram red and green lines mark occupied and unoccupied states.}
+\caption[Structure, charge density isosurface, molecular orbital diagram and Kohn-Sham level diagram of the bond-centered interstitial configuration.]{Structure, charge density isosurface, molecular orbital diagram and Kohn-Sham level diagram of the bond-centered interstitial configuration. Gray, green and blue surfaces mark the charge density of spin up, spin down and the resulting spin up electrons in the charge density isosurface, in which the carbon atom is represented by a red sphere. In the energy level diagram red and green lines mark occupied and unoccupied states.}
\label{img:defects:bc_conf}
\end{figure}
In the BC interstitial configuration the interstitial atom is located in between two next neighbored Si atoms forming linear bonds.
Approximately \unit[2.2]{eV} are needed to turn the \ci{} \hkl[0 0 -1] into the \hkl[1 1 0] DB located at the neighbored lattice site in \hkl[1 1 -1] direction.
Another barrier of \unit[0.90]{eV} exists for the rotation into the \ci{} \hkl[0 -1 0] DB configuration for the path obtained with a time constant of \unit[100]{fs} for the Berendsen thermostat.
Roughly the same amount would be necessary to excite the C$_{\text{i}}$ \hkl[1 1 0] DB to the BC configuration (\unit[0.40]{eV}) and a successive migration into the \hkl[0 0 1] DB configuration (\unit[0.50]{eV}) as displayed in Fig. \ref{fig:defects:110_mig} and Fig. \ref{fig:defects:cp_bc_00-1_mig}.
-The former diffusion process, however, would more nicely agree with the ab initio path, since the migration is accompanied by a rotation of the DB orientation.
-By considering a two step process and assuming equal preexponential factors for both diffusion steps, the probability of the total diffusion event is given by $\exp(\frac{\unit[2.24]{eV}+\unit[0.90]{eV}}{k_{\text{B}}T})$, which corresponds to a single diffusion barrier that is 3.5 times higher than the barrier obtained by {em ab initio} calculations.
+The former diffusion process, however, would more nicely agree with the {\em ab initio} path, since the migration is accompanied by a rotation of the DB orientation.
+By considering a two step process and assuming equal preexponential factors for both diffusion steps, the probability of the total diffusion event is given by $\exp(\frac{\unit[2.24]{eV}+\unit[0.90]{eV}}{k_{\text{B}}T})$, which corresponds to a single diffusion barrier that is 3.5 times higher than the barrier obtained by {\em ab initio} calculations.
\subsection{Conclusions}
Both C atoms form tetrahedral bonds to four Si atoms.
However, Si atom number 1 and number 3, which are bound to the second \ci{} atom are also bound to the initial C atom.
These four atoms of the rhomboid reside in a plane and, thus, do not match the situation in SiC.
-The Carbon atoms have a distance of \unit[2.75]{\AA}.
+The C atoms have a distance of \unit[2.75]{\AA}.
In Fig. \ref{fig:defects:190} the relaxed structure of a \hkl[0 1 0] DB constructed at position 2 is displayed.
An energy of \unit[-1.90]{eV} is observed.
The initial DB and especially the C atom is pushed towards the Si atom of the second DB forming an additional fourth bond.
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}.
+This finding is in good agreement with a combined {\em 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.
Similar to what was previously mentioned, configurations of C$_{\text{s}}$ and a Si$_{\text{i}}$ DB might be particularly important at higher temperatures due to the low activation energy necessary for its formation.
At higher temperatures, the contribution of entropy to structural formation increases, which might result in a spatial separation even for defects located within the capture radius.
-Indeed, an {em ab initio} MD run at \unit[900]{$^{\circ}$C} starting from configuration \RM{1}, which -- based on the above findings -- is assumed to recombine into the ground state configuration, results in a separation of the C$_{\text{s}}$ and Si$_{\text{i}}$ DB by more than 4 neighbor distances realized in a repeated migration mechanism of annihilating and arising Si$_{\text{i}}$ DBs.
+Indeed, an {\em ab initio} MD run at \unit[900]{$^{\circ}$C} starting from configuration \RM{1}, which -- based on the above findings -- is assumed to recombine into the ground state configuration, results in a separation of the C$_{\text{s}}$ and Si$_{\text{i}}$ DB by more than 4 neighbor distances realized in a repeated migration mechanism of annihilating and arising Si$_{\text{i}}$ DBs.
The atomic configurations for two different points in time are shown in Fig.~\ref{fig:defects:md}.
\begin{figure}[tp]
\begin{center}
\end{center}
\end{minipage}
\end{center}
-\caption[Atomic configurations of an ab initio molecular dynamics run at {\unit[900]{$^{\circ}$C}} starting from a configuration of C$_{\text{s}}$ located next to a Si$_{\text{i}}$ {\hkl[1 1 0]} DB.]{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. For substantial atoms, bonds are drawn in red color.}
+\caption[Atomic configurations of an {\em 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.]{Atomic configurations of an {\em 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. For substantial atoms, bonds are drawn in red color.}
\label{fig:defects:md}
\end{figure}
Si atoms 1 and 2, which form the initial DB, occupy Si lattice sites in the final configuration while Si atom 3 is transferred from a regular lattice site into the interstitial lattice.
\caption[Migration barrier of the \si{} {\hkl[1 1 0]} DB into the hexagonal and tetrahedral configuration as well as the hexagonal \si{} to tetrahedral \si{} transition.]{Migration barrier of the \si{} \hkl[1 1 0] DB into the hexagonal (H) and tetrahedral (T) configuration as well as the hexagonal \si{} to tetrahedral \si{} transition.}
\label{fig:defects:si_mig2}
\end{figure}
-The obtained activation energies are of the same order of magnitude than values derived from other ab initio studies \cite{bloechl93,sahli05}.
+The obtained activation energies are of the same order of magnitude than values derived from other {\em ab initio} studies \cite{bloechl93,sahli05}.
The low barriers indeed enable configurations of further separated \cs{} and \si{} atoms by the highly mobile \si{} atom departing from the \cs{} defect as observed in the previously discussed MD simulation.
% kept for nostalgical reason!
% todo - sync with conclusion chapter
-These findings allow to draw conclusions on the mechanisms involved in the process of SiC conversion in Si, which is elaborated in more detail within the comprehensive description in chapter~\ref{chapter:summary}.
+These findings allow to draw conclusions on the mechanisms involved in the process of SiC conversion in Si.
+% which is elaborated in more detail within the comprehensive description in chapter~\ref{chapter:summary}.
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.
Furthermore, the experimentally observed alignment of the \hkl(h k l) planes of the precipitate and the substrate is satisfied by the mechanism of successive positioning of C$_{\text{s}}$.
In contrast, there is no obvious reason for the topotactic orientation of an agglomerate consisting exclusively of C-Si dimers, which would necessarily involve a much more profound change in structure for the transition into SiC.
+Conclusions on the SiC precipitation mechanism in Si, which additionally include and consider results of the molecular dynamics investigations presented in the following, are elaborated in more detail within the comprehensive description in chapter~\ref{chapter:summary}.
+
%Prevailing conditions in the IBS process at elevated temperatures and the fact that IBS is a nonequilibrium process reinforce the possibility of formation of substitutional C instead of the thermodynamically stable C-Si dumbbell interstitials predicted by simulations at zero Kelvin.
\label{section:defects:noneq_process_02}
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