% ------ Albe potential ---------
For the classical potential calculations, defect structures were modeled in a supercell of nine Si lattice constants in each direction consisting of 5832 Si atoms.
Reproducing the SiC precipitation was attempted by the successive insertion of 6000 C atoms (the number necessary to form a 3C-SiC precipitate with a radius of $\approx 3.1$ nm) into the Si host, which has a size of 31 Si unit cells in each direction consisting of 238328 Si atoms.
-At constant temperature 10 atoms are inserted at a time.
-Three different regions within the total simulation volume are considered for a statistically distributed insertion of the C atoms: $V_1$ corresponding to the total simulation volume, $V_2$ corresponding to the size of the precipitate and $V_3$, whih holds the necessary amount of Si atoms of the precipitate.
+At constant temperature 10 atoms were inserted at a time.
+Three different regions within the total simulation volume were considered for a statistically distributed insertion of the C atoms: $V_1$ corresponding to the total simulation volume, $V_2$ corresponding to the size of the precipitate and $V_3$, whih holds the necessary amount of Si atoms of the precipitate.
+After C insertion the simulation has been continued for \unit[100]{ps} and cooled down to \unit[20]{$^{\circ}$C} afterwards.
A Tersoff-like bond order potential by Erhart and Albe (EA)\cite{albe_sic_pot} has been utilized, which accounts for nearest neighbor interactions only realized by a cut-off function dropping the interaction to zero in between the first and second next neighbor distance.
The potential was used as is, i.e. without any repulsive potential extension at short interatomic distances.
Constant pressure simulations are realized by the Berendsen barostat\cite{berendsen84} using a time constant of \unit[100]{fs} and a bulk modulus of \unit[100]{GPa} for Si.
Instead the tetrahedral defect configuration is favored.
This limitation is assumed to arise due to the cut-off.
In the tetrahedral configuration the second neighbors are only slightly more distant than the first neighbors, which creates the particular problem.
-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 terahedral configuration.
+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{Defect combinations}
+\subsection{Formation energies of C$_{\text{i}}$ and C$_{\text{s}}$-Si$_{\text{i}}$}
-\subsection{Mobility}
+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 next 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.
-A measure for the mobility of the interstitial carbon is the activation energy for the migration path from one stable position to another.
-The stable defect geometries have been discussed in the previous subsection.
-In the following the migration of the most stable configuration, i.e. C$_{\text{i}}$, from one site of the Si host lattice to a neighboring site has been investigated by both, EA and DFT calculations utilizing the constraint conjugate gradient relaxation technique (CRT)\cite{kaukonen98}.
-Three migration pathways are investigated.
-The starting configuration for all pathways was the \hkl[0 0 -1] dumbbell interstitial configuration.
-In path~1 and 2 the final configuration is a \hkl[0 0 1] and \hkl[0 -1 0] dumbbell interstitial respectively, located at the next neighbored Si lattice site displaced by $\frac{a_{\text{Si}}}{4}\hkl[1 1 -1]$, where $a_{\text{Si}}$ is the Si lattice constant.
-In path~1 the C atom resides in the \hkl(1 1 0) plane crossing the BC configuration whereas in path~2 the C atom moves out of the \hkl(1 1 0) plane.
-Path 3 ends in a \hkl[0 -1 0] configuration at the initial lattice site and, for this reason, corresponds to a reorientation of the dumbbell, a process not contributing to long range diffusion.
+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$_{\text{i}}$.
+This is particularly important since the energy of formation of C$_{\text{s}}$ is drastically underestimated by the EA potential.
+A possible occurence of C$_{\text{s}}$ could then be attributed to a lower energy of formation of the C$_{\text{s}}$-Si$_{\text{i}}$ combination due to the low formation energy of C$_{\text{s}}$, which obviously is wrong.
+Since quantum-mechanical calculation reveal the Si$_{\text{i}}$ \hkl<1 1 0> DB as the ground state configuration of Si$_{\text{i}}$ in Si it is assumed to provide the energetically most favorable configuration in combination with C$_{\text{s}}$.
+Empirical potentials, however, predict Si$_{\text{i}}$ T to be the energetically most favorable configuration.
+Thus, investigations of the relative energies of formation of defect pairs need to include combinations of C$_{\text{s}}$ with Si$_{\text{i}}$ T.
+Results of VASP and EA calculations are summarized in Table~\ref{tab:defect_combos}.
+\begin{table}
+\begin{ruledtabular}
+\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 & - \\
+ Erhart/Albe & 3.88 & 4.93 & 5.25$^{\text{a}}$/5.08$^{\text{b}}$/4.43$^{\text{c}}$
+\end{tabular}
+\end{ruledtabular}
+\caption{Formation energies of defect configurations of a single C impurity in otherwise perfect c-Si determined by classical potential and ab initio methods. The formation energies are given in electron volt. T denotes the tetrahedral and the subscripts i and s indicate the interstitial and substitutional configuration. Superscripts a, b and c denote configurations of C$_{\text{s}}$ located at the first, second and third next neighbored lattice site with respect to the Si$_{\text{i}}$ atom.}
+\label{tab:defect_combos}
+\end{table}
+Obviously the EA potential properly describes the relative energies of formation.
+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 enrgies 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 next neighbored C$_{\text{s}}$, which is the most favored configuration of C$_{\text{s}}$ and Si$_{\text{i}}$ according to quantum-mechanical caluclations\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.
+This is attributed to an effective reduction in strain enabled by the respective combination.
+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}.
+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 next 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.
+A C$_{\text{i}}$ \hkl[0 0 -1] DB turns into the \hkl[0 0 1] configuration at the next neighbored lattice site.
+The transition involves the C$_{\text{i}}$ BC configuration, which, however, was found to be unstable relaxing into the C$_{\text{i}}$ \hkl<1 1 0> DB configuration.
+If the migration is considered to occur within a single step the kinetic energy of \unit[2.2]{eV} is enough to turn the \hkl<1 0 0> DB into the BC and back into a \hkl<1 0 0> DB configuration.
+If, on the other hand, a two step process is assumed the BC configuration will most probably relax into the C$_{\text{i}}$ \hkl<1 1 0> DB configuration resulting in different relative energies of the intermediate state and the saddle point.
+For the latter case a migration path, which involves a C$_{\text{i}}$ \hkl<1 1 0> DB configuration is proposed and displayed in Fig.~\ref{fig:mig}.
+\begin{figure}
+\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.}
+\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 next 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}.
+The former diffusion process, however, would more nicely agree to 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.
+
+\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, 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}
+\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.}
+\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.
+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 next 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}.
+
+For high C concentrations the defect concentration is likewise increased and a considerable amount of damamge is introduced in the insertion volume.
+A subsequent superposition of defects generates new displacement arrangements for the C-C as well as Si-C pair distances, which become hard to categorize and trace and obviously lead to a broader distribution.
+Short range order indeed is observed, i.e. the large amount of strong next neighbored C-C bonds at \unit[0.15]{nm} as expected in graphite or diamond and Si-C bonds at \unit[0.19]{nm} as expected in SiC, but only hardly visible is the long range order.
+This indicates the formation of an amorphous SiC-like phase.
+In fact resulting Si-C and C-C radial distribution functions compare quite well with these obtained by cascade amorphized and melt-quenched amorphous SiC using a modifed Tersoff potential\cite{gao02}.
+
+In both cases, i.e. low and high C concentrations, the formation of 3C-SiC fails to appear.
+With respect to the precipitation model the formation of C$_{\text{i}}$ \hkl<1 0 0> DBs indeed occurs for low C concentrations.
+However, sufficient defect agglomeration is not observed.
+For high C concentrations a rearrangment of the amorphous SiC structure, which is not expected at prevailing temperatures, and a transition into 3C-SiC is not observed either.
+On closer inspection two reasons for describing this obstacle become evident.
+
+First of all there is the time scale problem inherent to MD in general.
+To minimize the integration error the discretized time step must be chosen smaller than the reciprocal of the fastest vibrational mode resulting in a time step of \unit[1]{fs} for the current problem under study.
+Limitations in computer power result in a slow propgation in phase space.
+Several local minima exist, which are separated by large energy barriers.
+Due to the low probability of escaping such a local minimum a single transition event corresponds to a multiple of vibrational periods.
+Long-term evolution such as a phase transformation and defect diffusion, in turn, are made up of a multiple of these infrequent transition events.
+Thus, time scales to observe long-term evolution are not accessible by traditional MD.
+New accelerated methods have been developed to bypass the time scale problem retaining proper thermodynamic sampling\cite{voter97,voter97_2,voter98,sorensen2000,wu99}.
+
+However, the applied potential comes up with an additional limitation already mentioned in the introductory part.
+The cut-off function of the short range potential limits the interaction to next neighbors, which results in overestimated and unphysical high forces between next neighbor atoms.
+This behavior, as observed and discussed for the Tersoff potential\cite{tang95,mattoni2007}, is supported by the overestimated activation energies necessary for C diffusion as investigated in section \ref{subsection:cmob}.
+Indeed it is not only the strong C-C bond which is hard to break inhibiting C diffusion and further rearrengements in the case of the high C concentration simulations.
+This is also true for the low concentration simulations dominated by the occurrence of C$_{\text{i}}$ \hkl<1 1 0> DBs spread over the whole simulation volume, which are unable to agglomerate due to the high migration barrier.
+
+\subsection{Increased temperature simulations}
+
+Due to the potential enhanced problem of slow phase space propagation, pushing the time scale to the limits of computational ressources or applying one of the above mentioned accelerated dynamics methods exclusively might not be sufficient.
+Instead higher temperatures are utilized to compensate overestimated diffusion barriers.
+These are overestimated by a factor of 2.4 to 3.5.
+Scaling the absolute temperatures accordingly results in maximum temperatures of \unit[1460-2260]{$^{\circ}$C}.
+Since melting already occurs shortly below the melting point of the potetnial (2450 K) due to the defects, a maximum temperature of \unit[2050]{$^{\circ}$C} is used.
+Fig.~\ref{fig:tot} shows the resulting bonds for various temperatures.
\begin{figure}
\begin{center}
-\includegraphics[width=\columnwidth]{path2_vasp_s.ps}
+\includegraphics[width=\columnwidth]{../img/tot_pc_thesis.ps}\\
+\includegraphics[width=\columnwidth]{../img/tot_pc3_thesis.ps}\\
+\includegraphics[width=\columnwidth]{../img/tot_pc2_thesis.ps}
\end{center}
-\caption{Migration barrier and structures of the \hkl[0 0 -1] dumbbell (left) to the \hkl[0 -1 0] dumbbell (right) transition as obtained by first-principles methods. The activation energy of \unit[0.9]{eV} agrees well with experimental findings of \unit[0.70]{eV}\cite{lindner06}, \unit[0.73]{eV}\cite{song90} and \unit[0.87]{eV}\cite{tipping87}.}
-\label{fig:vasp_mig}
-\end{figure}
-The lowest energy path (path~2) as detected by the first-principles approach is illustrated in Fig.~\ref{fig:vasp_mig}, in which the \hkl[0 0 -1] dumbbell migrates towards the next neighbored Si atom escaping the $(1 1 0)$ plane forming a \hkl[0 -1 0] dumbbell.
-The activation energy of \unit[0.9]{eV} excellently agrees with experimental findings ranging from \unit[0.70]{eV} to \unit[0.87]{eV}\cite{lindner06,song90,tipping87}.
+\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. In the latter case dashed arrows mark C-C distances occuring 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}
+Obviously a phase transition occurs ... WEITER
+Barfoo ...
\begin{figure}
\begin{center}
-\includegraphics[width=\columnwidth]{path1_albe_s.ps}
+\includegraphics[width=\columnwidth]{../img/12_pc_thesis.ps}\\
+\includegraphics[width=\columnwidth]{../img/12_pc_c_thesis.ps}
\end{center}
-\caption{Migration barrier and structures of the bond-centered (left) to \hkl[0 0 -1] dumbbell (right) transition utilizing the classical potential method. Two different pathways are obtained for different time constants of the Berendsen thermostat. The lowest activation energy is \unit[2.2]{eV}.}
-\label{fig:albe_mig}
+\caption{Radial distribution function for Si-C (top) and C-C (bottom) pairs for the C insertion into $V_2$ at elevated temperatures.}
+\label{fig:v2}
\end{figure}
-Calculations based on the EA potential yield a different picture.
-Fig.~\ref{fig:albe_mig} shows the evolution of structure and energy along the lowest energy migration path (path~1) based on the EA potential.
-Due to symmetry it is sufficient to merely consider the migration from the BC to the C$_{\text{i}}$ configuration.
-Two different pathways are obtained for different time constants of the Berendsen thermostat.
-With a time constant of \unit[1]{fs} the C atom resides in the \hkl(1 1 0) plane resulting in a migration barrier of \unit[2.4]{eV}.
-However, weaker coupling to the heat bath realized by an increase of the time constant to \unit[100]{fs} enables the C atom to move out of the \hkl(1 1 0) plane already at the beginning, which is accompanied by a reduction in energy, approaching the final configuration on a curved path.
-The energy barrier of this path is \unit[0.2]{eV} lower in energy than the direct migration within the \hkl(1 1 0) plane.
-It should be noted that the BC configuration is actually not a local minimum configuration in EA based calculations since a relaxation into the \hkl<1 1 0> dumbbell configuration occurs.
-However, investigating further migration pathways involving the \hkl<1 1 0> interstitial did not yield lower migration barriers.
-Thus, the activation energy should at least amount to \unit[2.2]{eV}.
\section{Discussion}
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