\begin{document}
-% ref mod: no capital letters (by editors)
\title{Combined ab initio and classical potential simulation study on the silicon carbide precipitation in silicon}
\author{F. Zirkelbach}
\author{B. Stritzker}
% --------------------------------------------------------------------------------
\section{Introduction}
-% TOOD: redo complete intro!
-
The wide band gap semiconductor silicon carbide (SiC) is well known for its outstanding physical and chemical properties.
The high breakdown field, saturated electron drift velocity and thermal conductivity in conjunction with the unique thermal and mechanical stability as well as radiation hardness makes SiC a suitable material for high-temperature, high-frequency and high-power devices operational in harsh and radiation-hard environments\cite{edgar92,morkoc94,wesch96,capano97,park98}.
Different modifications of SiC exist, which solely differ in the one-dimensional stacking sequence of identical, close-packed SiC bilayers\cite{fischer90}.
Utilized and enhanced, ion beam synthesis (IBS) has become a promising method to form thin SiC layers of high quality and exclusively of the 3C polytype embedded in and epitactically aligned to the Si host featuring a sharp interface\cite{lindner99,lindner01,lindner02}.
However, only little is known about the SiC conversion in C implanted Si.
-High resolution transmission electron microscopy (HREM) studies\cite{werner96,werner97,eichhorn99,lindner99_2,koegler03} suggest the formation of C-Si dimers (dumbbells) on regular Si lattice sites, which agglomerate into large clusters indicated by dark contrasts and otherwise undisturbed Si lattice fringes in HREM.
-A topotactic transformation into a 3C-SiC precipitate occurs once a critical radius of 2 nm to 4 nm is reached, which is manifested by the disappearance of the dark contrasts in favor of Moir\'e patterns due to the lattice mismatch of \unit[20]{\%} of the 3C-SiC precipitate and c-Si.
+\begin{figure}
+\begin{center}
+\subfigure[]{\label{fig:hrem:c-si}\includegraphics[width=0.48\columnwidth]{tem_c-si-db.eps}}
+\subfigure[]{\label{fig:hrem:sic}\includegraphics[width=0.48\columnwidth]{tem_3c-sic.eps}}
+\end{center}
+\caption{High resolution transmission electron microscopy (HREM) micrographs\cite{lindner99_2} of agglomerates of C-Si dimers showing dark contrasts and otherwise undisturbed Si lattice fringes (a) and equally sized Moir\'e patterns indicating 3C-SiC precipitates (b).}
+\label{fig:hrem}
+\end{figure}
+High resolution transmission electron microscopy (HREM) studies\cite{werner96,werner97,eichhorn99,lindner99_2,koegler03} suggest the formation of C-Si dimers (dumbbells) on regular Si lattice sites, which agglomerate into large clusters indicated by dark contrasts and otherwise undisturbed Si lattice fringes in HREM, as can be seen in Fig.~\ref{fig:hrem:c-si}.
+A topotactic transformation into a 3C-SiC precipitate occurs once a critical radius of 2 nm to 4 nm is reached, which is manifested by the disappearance of the dark contrasts in favor of Moir\'e patterns (Fig.~\ref{fig:hrem:sic}) due to the lattice mismatch of \unit[20]{\%} of the 3C-SiC precipitate and c-Si.
The insignificantly lower Si density of SiC ($\approx \unit[4]{\%}$) compared to c-Si results in the emission of only a few excess Si atoms.
In contrast, investigations of strained Si$_{1-y}$C$_y$/Si heterostructures formed by MBE\cite{strane94,guedj98}, which incidentally involve the formation of SiC nanocrystallites, suggest an initial coherent precipitation by agglomeration of substitutional instead of interstitial C.
Coherency is lost once the increasing strain energy of the stretched SiC structure surpasses the interfacial energy of the incoherent 3C-SiC precipitate and the Si substrate.
These two different mechanisms of precipitation might be attributed to the respective method of fabrication.
While in CVD and MBE surface effects need to be taken into account, SiC formation during IBS takes place in the bulk of the Si crystal.
However, in another IBS study Nejim et~al.\cite{nejim95} propose a topotactic transformation that is likewise based on the formation of substitutional C.
-%The formation of substitutional C, however, is accompanied by Si self-interstitial atoms that previously occupied the lattice sites and concurrently by a reduction of volume due to the lower lattice constant of SiC compared to Si.
The formation of substitutional C, however, is accompanied by Si self-interstitial atoms that previously occupied the lattice sites and a concurrent reduction of volume due to the lower lattice constant of SiC compared to Si.
Both processes are believed to compensate one another.
All these potentials are short range potentials employing a cut-off function, which drops the atomic interaction to zero in between the first and second nearest neighbor distance.
In a combined ab initio and empirical potential study it was shown that the Tersoff potential properly describes binding energies of combinations of C defects in Si\cite{mattoni2002}.
However, investigations of brittleness in covalent materials\cite{mattoni2007} identified the short range character of these potentials to be responsible for overestimated forces necessary to snap the bond of two neighbored atoms.
-%In a previous study\cite{zirkelbach10a} we approved explicitly the influence on the migration barrier for C diffusion in Si.
-In a previous study\cite{zirkelbach10a} we determined the influence on the migration barrier for C diffusion in Si.
+In a previous study\cite{zirkelbach10} we determined the influence on the migration barrier for C diffusion in Si.
Using the Erhart/Albe (EA) potential\cite{albe_sic_pot}, an overestimated barrier height compared to ab initio calculations and experiment is obtained.
A proper description of C diffusion, however, is crucial for the problem under study.
The formation energy $E-N_{\text{Si}}\mu_{\text{Si}}-N_{\text{C}}\mu_{\text{C}}$ of a defect configuration 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.
Migration and recombination pathways have been investigated utilizing the constraint conjugate gradient relaxation technique\cite{kaukonen98}.
+Time constants of \unit[1]{fs}, which corresponds to direct velocity scaling, and \unit[100]{fs}, which results in weaker coupling to the heat bath allowing the diffusing atoms to take different pathways, were used for the Berendsen thermostat for structural relaxation within the migration calculations.
\section{Results}
The C interstitial defect with the lowest energy of formation has been found to be the C-Si \hkl<1 0 0> interstitial dumbbell (C$_{\text{i}}$ \hkl<1 0 0> DB), which, thus, constitutes the ground state of an additional C impurity in otherwise perfect c-Si.
This finding is in agreement with several theoretical\cite{burnard93,leary97,dal_pino93,capaz94} and experimental\cite{watkins76,song90} investigations.
Astonishingly EA and DFT predict almost equal formation energies.
-There are, however, geometric differences with regard to the DB position within the tetrahedron spanned by the four neighbored Si atoms, as already reported in a previous study\cite{zirkelbach10a}.
+There are, however, geometric differences with regard to the DB position within the tetrahedron spanned by the four neighbored Si atoms, as already reported in a previous study\cite{zirkelbach10}.
Since the energetic description is considered more important than the structural description, minor discrepancies of the latter are assumed non-problematic.
The second most favorable configuration is the C$_{\text{i}}$ \hkl<1 1 0> DB followed by the C$_{\text{i}}$ bond-centered (BC) configuration.
For both configurations EA overestimates the energy of formation by approximately \unit[1]{eV} compared to DFT.
Thus, nearly the same difference in energy has been observed for these configurations in both methods.
However, we have found the BC configuration to constitute a saddle point within the EA description relaxing into the \hkl<1 1 0> configuration.
Due to the high formation energy of the BC defect resulting in a low probability of occurrence of this defect, the wrong description is not posing a serious limitation of the EA potential.
-A more detailed discussion of C defects in Si modeled by EA and DFT including further defect configurations are presented in a previous study\cite{zirkelbach10a}.
+A more detailed discussion of C defects in Si modeled by EA and DFT including further defect configurations can be found in our recently published article\cite{zirkelbach10}.
Regarding intrinsic defects in Si, both methods predict energies of formation that are within the same order of magnitude.
However discrepancies exist.
Indeed, an increase of the cut-off results in increased values of the formation energies\cite{albe_sic_pot}, which is most significant for the tetrahedral configuration.
The same issue has already been discussed by Tersoff\cite{tersoff90} with regard to the description of the tetrahedral C defect using his potential.
While not completely rendering impossible further, more challenging empirical potential studies on large systems, the artifact has to be taken into account in the following investigations of defect combinations.
-%This artifact does not necessarily render impossible further challenging empirical potential studies on large systems.
-%However, it has to be taken into account in the following investigations of defect combinations.
\subsection{Formation energies of C$_{\text{i}}$ and C$_{\text{s}}$-Si$_{\text{i}}$}
-As has been shown in a previous study\cite{zirkelbach10b}, the energetically most favorable configuration of C$_{\text{s}}$ and Si$_{\text{i}}$ is obtained for C$_{\text{s}}$ located at the neighbored lattice site along the \hkl<1 1 0> bond chain of a Si$_{\text{i}}$ \hkl<1 1 0> DB.
+As has been shown in a previous study\cite{zirkelbach11a}, the energetically most favorable configuration of C$_{\text{s}}$ and Si$_{\text{i}}$ is obtained for C$_{\text{s}}$ located at the neighbored lattice site along the \hkl<1 1 0> bond chain of a Si$_{\text{i}}$ \hkl<1 1 0> DB.
However, the energy of formation is slightly higher than that of the C$_{\text{i}}$ \hkl<1 0 0> DB, which constitutes the ground state for a C impurity introduced into otherwise perfect c-Si.
For a possible clarification of the controversial views on the participation of C$_{\text{s}}$ in the precipitation mechanism by classical potential simulations, test calculations need to ensure the proper description of the relative formation energies of combined structures of C$_{\text{s}}$ and Si$_{\text{i}}$ compared to C$_{\text{i}}$.
\begin{tabular}{l c c c}
& C$_{\text{i}}$ \hkl<1 0 0> & C$_{\text{s}}$ \& Si$_{\text{i}}$ \hkl<1 1 0> & C$_{\text{s}}$ \& Si$_{\text{i}}$ T\\
\hline
- VASP & 3.72 & 4.37 & - \\
+ VASP & 3.72 & 4.37 & 4.17$^{\text{a}}$/4.99$^{\text{b}}$/4.96$^{\text{c}}$ \\
Erhart/Albe & 3.88 & 4.93 & 5.25$^{\text{a}}$/5.08$^{\text{b}}$/4.43$^{\text{c}}$
\end{tabular}
\end{ruledtabular}
Combined structures of C$_{\text{s}}$ and Si$_{\text{i}}$ T are energetically less favorable than the ground state C$_{\text{i}}$ \hkl<1 0 0> DB configuration.
With increasing separation distance the energies of formation decrease.
However, even for non-interacting defects, the energy of formation, which is then given by the sum of the formation energies of the separated defects (\unit[4.15]{eV}) is still higher than that of the C$_{\text{i}}$ \hkl<1 0 0> DB.
-Unexpectedly, the structure of a Si$_{\text{i}}$ \hkl<1 1 0> DB and a neighbored C$_{\text{s}}$, which is the most favored configuration of C$_{\text{s}}$ and Si$_{\text{i}}$ according to quantum-mechanical calculations\cite{zirkelbach10b}, likewise constitutes an energetically favorable configuration within the EA description, which is even preferred over the two least separated configurations of C$_{\text{s}}$ and Si$_{\text{i}}$ T.
+Unexpectedly, the structure of a Si$_{\text{i}}$ \hkl<1 1 0> DB and a neighbored C$_{\text{s}}$, which is the most favored configuration of C$_{\text{s}}$ and Si$_{\text{i}}$ according to quantum-mechanical calculations\cite{zirkelbach11a}, likewise constitutes an energetically favorable configuration within the EA description, which is even preferred over the two least separated configurations of C$_{\text{s}}$ and Si$_{\text{i}}$ T.
This is attributed to an effective reduction in strain enabled by the respective combination.
+Quantum-mechanical results reveal a more favorable energy of fomation for the C$_{\text{s}}$ and Si$_{\text{i}}$ T (a) configuration.
+However, this configuration is unstable involving a structural transition into the C$_{\text{i}}$ \hkl<1 1 0> interstitial, thus, not maintaining the tetrahedral Si nor the substitutional C defect.
+In either case, no configuration more favorable than the C$_{\text{i}}$ \hkl<1 0 0> DB has been found.
Thus, a proper description with respect to the relative energies of formation is assumed for the EA potential.
\subsection{Carbon mobility}
\label{subsection:cmob}
To accurately model the SiC precipitation, which involves the agglomeration of C, a proper description of the migration process of the C impurity is required.
-As shown in a previous study\cite{zirkelbach10a}, quantum-mechanical results properly describe the C$_{\text{i}}$ \hkl<1 0 0> DB diffusion resulting in a migration barrier height of \unit[0.90]{eV}, excellently matching experimental values of \unit[0.70-0.87]{eV}\cite{lindner06,tipping87,song90} and, for this reason, reinforcing the respective migration path as already proposed by Capaz et~al.\cite{capaz94}.
+As shown in a previous study\cite{zirkelbach10}, quantum-mechanical results properly describe the C$_{\text{i}}$ \hkl<1 0 0> DB diffusion resulting in a migration barrier height of \unit[0.90]{eV}, excellently matching experimental values of \unit[0.70-0.87]{eV}\cite{lindner06,tipping87,song90} and, for this reason, reinforcing the respective migration path as already proposed by Capaz et~al.\cite{capaz94}.
During transition a C$_{\text{i}}$ \hkl[0 0 -1] DB migrates towards a C$_{\text{i}}$ \hkl[0 -1 0] DB located at the neighbored lattice site in \hkl[1 1 -1] direction.
However, it turned out that the description fails if the EA potential is used, which overestimates the migration barrier (\unit[2.2]{eV}) by a factor of 2.4.
In addition a different diffusion path is found to exhibit the lowest migration barrier.
\begin{center}
\includegraphics[width=\columnwidth]{110mig.ps}
\end{center}
-\caption{Migration barrier and structures of the \hkl[0 0 -1] DB (left) to the \hkl[0 -1 0] DB (right) transition involving the \hkl[1 1 0] DB (center) configuration.}
+\caption{Migration barrier and structures of the \hkl[0 0 -1] DB (left) to the \hkl[0 -1 0] DB (right) transition involving the \hkl[1 1 0] DB (center) configuration. 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:mig}
\end{figure}
Approximately \unit[2.24]{eV} are needed to turn the C$_{\text{i}}$ \hkl[0 0 -1] DB into the C$_{\text{i}}$ \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 C$_{\text{i}}$ \hkl[0 -1 0] DB configuration.
-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 our previous study\cite{zirkelbach10a}.
+Another barrier of \unit[0.90]{eV} exists for the rotation into the C$_{\text{i}}$ \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 our previous study\cite{zirkelbach10}.
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 ab initio calculations.
Accordingly, the effective barrier of migration of C$_{\text{i}}$ is overestimated by a factor of 2.4 to 3.5 compared to the highly accurate quantum-mechanical methods.
-This constitutes a serious limitation that has to be taken into account for modeling the C-Si system using the EA potential.
+This constitutes a serious limitation that has to be taken into account for modeling the C-Si system using the otherwise quite promising EA potential.
\subsection{Molecular dynamics simulations}
Fig.~\ref{fig:450} shows the radial distribution functions of simulations, in which C was inserted at \unit[450]{$^{\circ}$C}, an operative and efficient temperature in IBS\cite{lindner99}, for all three insertion volumes.
\begin{figure}
\begin{center}
-\includegraphics[width=\columnwidth]{../img/sic_prec_450_si-si_c-c.ps}\\
-\includegraphics[width=\columnwidth]{../img/sic_prec_450_si-c.ps}
+\subfigure[]{\label{fig:450:a}
+\includegraphics[width=\columnwidth]{sic_prec_450_si-si_c-c.ps}
+}
+\subfigure[]{\label{fig:450:b}
+\includegraphics[width=\columnwidth]{sic_prec_450_si-c.ps}
+}
\end{center}
-\caption{Radial distribution function for C-C and Si-Si (top) as well as Si-C (bottom) pairs for C inserted at \unit[450]{$^{\circ}$C}. In the latter case the resulting C-Si distances for a C$_{\text{i}}$ \hkl<1 0 0> DB are given additionally.}
+\caption{Radial distribution function for C-C and Si-Si (Fig.~\ref{fig:450:a}) as well as Si-C (Fig.~\ref{fig:450:b}) pairs for C inserted at \unit[450]{$^{\circ}$C}. In the latter case the resulting C-Si distances for a C$_{\text{i}}$ \hkl<1 0 0> DB are given additionally and the Si-C cut-off distance is marked by an arrow. Insets in Fig.~\ref{fig:450:a} show magnified regions of the respective distribution functions.}
\label{fig:450}
\end{figure}
There is no significant difference between C insertion into $V_2$ and $V_3$.
Thus, in the following, the focus is on low ($V_1$) and high ($V_2$, $V_3$) C concentration simulations only.
-In the low C concentration simulation the number of C-C bonds is small.
+In the low C concentration simulation the number of C-C bonds is small, as can be seen in the upper part of Fig.~\ref{fig:450:a}.
On average, there are only 0.2 C atoms per Si unit cell.
-By comparing the Si-C peaks of the low concentration simulation with the resulting Si-C distances of a C$_{\text{i}}$ \hkl<1 0 0> DB it becomes evident that the structure is clearly dominated by this kind of defect.
-One exceptional peak exists, which is due to the Si-C cut-off, at which the interaction is pushed to zero.
-Investigating the C-C peak at \unit[0.31]{nm}, which is also available for low C concentrations as can be seen in the inset, reveals a structure of two concatenated, differently oriented C$_{\text{i}}$ \hkl<1 0 0> DBs to be responsible for this distance.
-Additionally, the Si-Si radial distribution shows non-zero values at distances around \unit[0.3]{nm}, which, again, is due to the DB structure stretching two neighbored Si atoms.
+By comparing the Si-C peaks of the low concentration simulation with the resulting Si-C distances of a C$_{\text{i}}$ \hkl<1 0 0> DB in Fig.~\ref{fig:450:b} it becomes evident that the structure is clearly dominated by this kind of defect.
+One exceptional peak at \unit[0.26]{nm} (marked with an arrow in Fig.~\ref{fig:450:b}) exists, which is due to the Si-C cut-off, at which the interaction is pushed to zero.
+Investigating the C-C peak at \unit[0.31]{nm}, which is also available for low C concentrations as can be seen in the upper inset of Fig.~\ref{fig:450:a}, reveals a structure of two concatenated, differently oriented C$_{\text{i}}$ \hkl<1 0 0> DBs to be responsible for this distance.
+Additionally, in the inset of the bottom part of Fig.\ref{fig:450:a} the Si-Si radial distribution shows non-zero values at distances around \unit[0.3]{nm}, which, again, is due to the DB structure stretching two neighbored Si atoms.
This is accompanied by a reduction of the number of bonds at regular Si distances of c-Si.
A more detailed description of the resulting C-Si distances in the C$_{\text{i}}$ \hkl<1 0 0> DB configuration and the influence of the defect on the structure is available in a previous study\cite{zirkelbach09}.
Fig.~\ref{fig:tot} shows the resulting radial distribution functions for various temperatures.
\begin{figure}
\begin{center}
-\includegraphics[width=\columnwidth]{../img/tot_pc_thesis.ps}\\
-\includegraphics[width=\columnwidth]{../img/tot_pc3_thesis.ps}\\
-\includegraphics[width=\columnwidth]{../img/tot_pc2_thesis.ps}
+\subfigure[]{\label{fig:tot:si-c}
+\includegraphics[width=\columnwidth]{tot_pc_thesis.ps}
+}
+\subfigure[]{\label{fig:tot:si-si}
+\includegraphics[width=\columnwidth]{tot_pc3_thesis.ps}
+}
+\subfigure[]{\label{fig:tot:c-c}
+\includegraphics[width=\columnwidth]{tot_pc2_thesis.ps}
+}
\end{center}
-\caption{Radial distribution function for Si-C (top), Si-Si (center) and C-C (bottom) pairs for the C insertion into $V_1$ at elevated temperatures. For the Si-C distribution resulting Si-C distances of a C$_{\text{s}}$ configuration are plotted. In the C-C distribution dashed arrows mark C-C distances occurring from C$_{\text{i}}$ \hkl<1 0 0> DB combinations, solid arrows mark C-C distances of pure C$_{\text{s}}$ combinations and the dashed line marks C-C distances of a C$_{\text{i}}$ and C$_{\text{s}}$ combination.}
+\caption{Radial distribution function for Si-C (Fig.~\ref{fig:tot:si-c}), Si-Si (Fig.~\ref{fig:tot:si-si}) and C-C (Fig.~\ref{fig:tot:c-c}) pairs for the C insertion into $V_1$ at elevated temperatures. For the Si-C distribution resulting Si-C distances of a C$_{\text{s}}$ configuration are plotted. In the C-C distribution dashed arrows mark C-C distances occurring from C$_{\text{i}}$ \hkl<1 0 0> DB combinations, solid arrows mark C-C distances of pure C$_{\text{s}}$ combinations and the dashed line marks C-C distances of a C$_{\text{i}}$ and C$_{\text{s}}$ combination.}
\label{fig:tot}
\end{figure}
-The first noticeable and promising change observed for the Si-C bonds is the successive decline of the artificial peak at the cut-off distance with increasing temperature.
+In Fig.~\ref{fig:tot:si-c}, the first noticeable and promising change observed for the Si-C bonds is the successive decline of the artificial peak at the cut-off distance with increasing temperature.
Obviously, sufficient kinetic energy is provided to affected atoms that are enabled to escape the cut-off region.
-Additionally, a more important structural change was observed, which is illustrated in the two shaded areas of the graph.
+Additionally, a more important structural change was observed, which is illustrated in the two shaded areas in Fig.~\ref{fig:tot:si-c}.
Obviously, the structure obtained at \unit[450]{$^{\circ}$C}, which was found to be dominated by C$_{\text{i}}$, transforms into a C$_{\text{s}}$ dominated structure with increasing temperature.
Comparing the radial distribution at \unit[2050]{$^{\circ}$C} to the resulting bonds of C$_{\text{s}}$ in c-Si excludes all possibility of doubt.
-The phase transformation is accompanied by an arising Si-Si peak at \unit[0.325]{nm}, which corresponds to the distance of next neighbored Si atoms along the \hkl<1 1 0> bond chain with C$_{\text{s}}$ in between.
+The phase transformation is accompanied by an arising Si-Si peak at \unit[0.325]{nm} in Fig.~\ref{fig:tot:si-si}, which corresponds to the distance of next neighbored Si atoms along the \hkl<1 1 0> bond chain with C$_{\text{s}}$ in between.
Since the expected distance of these Si pairs in 3C-SiC is \unit[0.308]{nm} the existing SiC structures embedded in the c-Si host are stretched.
-According to the C-C radial distribution, agglomeration of C fails to appear even for elevated temperatures, as can be seen on the total amount of C pairs within the investigated separation range, which does not change significantly.
+According to the C-C radial distribution displayed in Fig.~\ref{fig:tot:c-c}, agglomeration of C fails to appear even for elevated temperatures, as can be seen on the total amount of C pairs within the investigated separation range, which does not change significantly.
However, a small decrease in the amount of neighbored C pairs can be observed with increasing temperature.
This high temperature behavior is promising since breaking of these diamond- and graphite-like bonds is mandatory for the formation of 3C-SiC.
Obviously, acceleration of the dynamics occurred by supplying additional kinetic energy.
Fig.~\ref{fig:v2} displays the radial distribution for high C concentrations.
\begin{figure}
\begin{center}
-\includegraphics[width=\columnwidth]{../img/12_pc_thesis.ps}\\
-\includegraphics[width=\columnwidth]{../img/12_pc_c_thesis.ps}
+\subfigure[]{\label{fig:v2:si-c}
+\includegraphics[width=\columnwidth]{12_pc_thesis.ps}
+}
+\subfigure[]{\label{fig:v2:c-c}
+\includegraphics[width=\columnwidth]{12_pc_c_thesis.ps}
+}
\end{center}
-\caption{Radial distribution function for Si-C (top) and C-C (bottom) pairs for the C insertion into $V_2$ at elevated temperatures.}
+\caption{Radial distribution function for Si-C (Fig.~\ref{fig:v2:si-c}) and C-C (Fig.~\ref{fig:v2:c-c}) pairs for the C insertion into $V_2$ at elevated temperatures. Arrows mark the respective cut-off distances.}
\label{fig:v2}
\end{figure}
+\begin{figure}
+\begin{center}
+\includegraphics[width=\columnwidth]{2050.eps}
+\end{center}
+\caption{Cross section along the \hkl(1 -1 0) plane of the atomic structure of the high concentration simulation for a C insertion temperature of \unit[2050]{$^{\circ}$C}.}
+\label{fig:v2as}
+\end{figure}
+A cross-section along the \hkl(1 -1 0) plane of the atomic structure for a C insertion temperature of \unit[2050]{$^{\circ}$C} is shown in Fig.~\ref{fig:v2as}.
The amorphous SiC-like phase remains.
No significant change in structure is observed.
However, the decrease of the cut-off artifact and slightly sharper peaks observed with increasing temperature, in turn, indicate a slight acceleration of the dynamics realized by the supply of kinetic energy.
However, it is not sufficient to enable the amorphous to crystalline transition.
In contrast, even though bonds of neighbored C atoms could be partially dissolved in the system exhibiting low C concentrations, the amount of neighbored C pairs even increased in the latter case.
-Moreover, the C-C peak at \unit[0.252]{nm}, which gets slightly more distinct, equals the second nearest neighbor distance in diamond and indeed is made up by a structure of two C atoms interconnected by a third C atom.
+Moreover, the C-C peak at \unit[0.252]{nm} in Fig.~\ref{fig:v2:c-c}, which gets slightly more distinct, equals the second nearest neighbor distance in diamond and indeed is made up by a structure of two C atoms interconnected by a third C atom.
Obviously, processes that appear to be non-conducive are likewise accelerated in a system, in which high amounts of C are incorporated within a short period of time, which is accompanied by a concurrent introduction of accumulating, for the reason of time non-degradable damage.
% non-degradable, non-regenerative, non-recoverable
Thus, for these systems even larger time scales, which are not accessible within traditional MD, must be assumed for an amorphous to crystalline transition or structural evolution in general.
% maybe put description of bonds in here ...
Nevertheless, some results likewise indicate the acceleration of other processes that, again, involve C$_{\text{s}}$.
-The increasingly pronounced Si-C peak at \unit[0.35]{nm} corresponds to the distance of a C and a Si atom interconnected by another Si atom.
-Additionally, the C-C peak at \unit[0.31]{nm} corresponds to the distance of two C atoms bound to a central Si atom.
+The increasingly pronounced Si-C peak at \unit[0.35]{nm} in Fig.~\ref{fig:v2:si-c} corresponds to the distance of a C and a Si atom interconnected by another Si atom.
+Additionally, the C-C peak at \unit[0.31]{nm} in Fig.~\ref{fig:v2:c-c} corresponds to the distance of two C atoms bound to a central Si atom.
For both structures the C atom appears to reside on a substitutional rather than an interstitial lattice site.
However, huge amounts of damage hamper identification.
The alignment of the investigated structures to the c-Si host is lost in many cases, which suggests the necessity of much more time for structural evolution to maintain the topotactic orientation of the precipitate.
-\section{Summary and discussion}
+\section{Discussion and Summary}
Investigations are targeted at the initially stated controversy of SiC precipitation, i.e. whether precipitation occurs abruptly after enough C$_{\text{i}}$ agglomerated or after a successive agglomeration of C$_{\text{s}}$ on usual Si lattice sites (and Si$_{\text{i}}$) followed by a contraction into incoherent SiC.
-Results of a previous ab initio study on defects and defect combinations in C implanted Si\cite{zirkelbach10b} suggest C$_{\text{s}}$ to play a decisive role in the precipitation of SiC in Si.
+Results of a previous ab initio study on defects and defect combinations in C implanted Si\cite{zirkelbach11a} suggest C$_{\text{s}}$ to play a decisive role in the precipitation of SiC in Si.
To support previous assumptions MD simulations, which are capable of modeling the necessary amount of atoms, i.e. the precipitate and the surrounding c-Si structure, have been employed in the current study.
-In a previous comparative study\cite{zirkelbach10a} we have shown that the utilized empirical potential fails to describe some selected processes.
+In a previous comparative study\cite{zirkelbach10} we have shown that the utilized empirical potential fails to describe some selected processes.
Thus, limitations of the employed potential have been further investigated and taken into account in the present study.
We focussed on two major shortcomings: the overestimated activation energy and the improper description of intrinsic and C point defects in Si.
Overestimated forces between nearest neighbor atoms that are expected for short range potentials\cite{mattoni2007} have been confirmed to influence the C$_{\text{i}}$ diffusion.
-The migration barrier was estimated to be larger by a factor of 2.4 to 3.5 compared to highly accurate quantum-mechanical calculations\cite{zirkelbach10a}.
+The migration barrier was estimated to be larger by a factor of 2.4 to 3.5 compared to highly accurate quantum-mechanical calculations\cite{zirkelbach10}.
Concerning point defects, the drastically underestimated formation energy of C$_{\text{s}}$ and deficiency in the description of the Si$_{\text{i}}$ ground state necessitated further investigations on structures that are considered important for the problem under study.
It turned out that the EA potential still favors a C$_{\text{i}}$ \hkl<1 0 0> DB over a C$_{\text{s}}$-Si$_{\text{i}}$ configuration, which, thus, does not constitute any limitation for the simulations aiming to resolve the present controversy of the proposed SiC precipitation models.
However, we observed a phase transition of the C$_{\text{i}}$-dominated into a clearly C$_{\text{s}}$-dominated structure.
The amount of substitutionally occupied C atoms increases with increasing temperature.
Entropic contributions are assumed to be responsible for these structures at elevated temperatures that deviate from the ground state at 0 K.
-Indeed, in a previous ab initio MD simulation\cite{zirkelbach10b} performed at \unit[900]{$^{\circ}$C} we observed the departing of a Si$_{\text{i}}$ \hkl<1 1 0> DB located next to a C$_{\text{s}}$ atom instead of a recombination into the ground state configuration, i.e. a C$_{\text{i}}$ \hkl<1 0 0> DB.
+Indeed, in a previous ab initio MD simulation\cite{zirkelbach11a} performed at \unit[900]{$^{\circ}$C} we observed the departing of a Si$_{\text{i}}$ \hkl<1 1 0> DB located next to a C$_{\text{s}}$ atom instead of a recombination into the ground state configuration, i.e. a C$_{\text{i}}$ \hkl<1 0 0> DB.
% postannealing less efficient than hot implantation
Experimental studies revealed increased implantation temperatures to be more efficient than postannealing methods for the formation of topotactically aligned precipitates\cite{eichhorn02}.
The latter has been directly identified in the present simulation study, i.e. structures of two C$_{\text{s}}$ atoms and Si$_{\text{i}}$ located in the vicinity.
It is, thus, concluded that precipitation occurs by successive agglomeration of C$_{\text{s}}$ as already proposed by Nejim et~al.\cite{nejim95}.
-This agrees well with a previous ab initio study on defects in C implanted Si\cite{zirkelbach10b}, which showed C$_{\text{s}}$ to occur in all probability.
+This agrees well with a previous ab initio study on defects in C implanted Si\cite{zirkelbach11a}, which showed C$_{\text{s}}$ to occur in all probability.
However, agglomeration and rearrangement is enabled by mobile C$_{\text{i}}$, which has to be present at the same time and is formed by recombination of C$_{\text{s}}$ and Si$_{\text{i}}$.
In contrast to assumptions of an abrupt precipitation of an agglomerate of C$_{\text{i}}$\cite{werner96,werner97,eichhorn99,lindner99_2,koegler03}, however, structural evolution is believed to occur by a successive occupation of usual Si lattice sites with substitutional C.
This mechanism satisfies the experimentally observed alignment of the \hkl(h k l) planes of the precipitate and the substrate, whereas 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.
% --------------------------------- references -------------------
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\end{document}
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