\begin{center}
\includegraphics[width=12cm]{c_sub_si110.ps}
\end{center}
-\caption{Binding energy of combinations of a substitutional C and a Si \hkl<1 1 0> dumbbell self-interstitial with respect to the separation distance.}
+\caption[Binding energy of combinations of a substitutional C and a Si \hkl<1 1 0> dumbbell self-interstitial with respect to the separation distance.]{Binding energy of combinations of a substitutional C and a Si \hkl<1 1 0> dumbbell self-interstitial with respect to the separation distance. The binding energy of the defect pair is well approximated by a Lennard-Jones 6-12 potential, which is used for curve fitting.}
\label{fig:defects:csub_si110}
\end{figure}
According to the formation energies none of the investigated structures is energetically preferred over the C-Si \hkl<1 0 0> dumbbell interstitial, which exhibits a formation energy of 3.88 eV.
Thus, the C-Si \hkl<1 0 0> dumbbell structure remains the ground state configuration of a C interstitial in c-Si with a constant number of Si atoms.
{\color{blue}
-However the binding energy quickly drops to zero with respect of the distance indicating a possibly low interaction capture radius of the defect pair.
+However the binding energy quickly drops to zero with respect to the distance, which is reinforced by the Lennard-Jones fit estimating almost zero interaction energy already at 0.6 nm.
+This indicates a possibly low interaction capture radius of the defect pair.
Highly energetic collisions in the IBS process might result in separations of these defects exceeding the capture radius.
For this reason situations most likely occur in which the configuration of substitutional C can be considered without a nearby interacting Si self-interstitial and, thus, unable to form a thermodynamically more stable C-Si \hkl<1 0 0> dumbbell configuration.
}
+\label{section:defects:noneq_process_01}
The energetically most favorable configuration of the combined structures is the one with the substitutional C atom located next to the \hkl<1 1 0> interstitial along the \hkl<1 1 0> direction (configuration \RM{1}).
Compressive stress along the \hkl<1 1 0> direction originating from the Si \hkl<1 1 0> self-intesrtitial is partially compensated by tensile stress resulting from substitutional C occupying the neighboured Si lattice site.
While first results support the proposed precipitation model the latter suggest the formation of silicon carbide by succesive creation of substitutional carbon instead of the agglomeration of C-Si dumbbell interstitials followed by an abrupt transition.
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}
{\color{blue}
In addition, there are experimental findings, which might be exploited to reinforce the non-validity of the proposed precipitation model.
\end{equation}
which results in a saturation value of 93 \%.
Obviously the decrease in temperature accelerates the saturation and inhibits further formation of substitutional carbon.
+\label{subsubsection:md:ep}
Conclusions drawn from investigations of the quality evolution correlate well with the findings of the radial distribution results.
\begin{figure}[!ht]
In this case, the Si and the C atom are bound to a central Si atom.
To summarize, the amorphous phase remains though sharper peaks in the radial distributions at distances expected for a-SiC are observed indicating a slight acceleration of the dynamics due to elevated temperatures.
+\subsubsection{Conclusions concerning the usage of increased temperatures}
+
Regarding the outcome of both, high and low concentration simulations at increased temperatures, encouraging conclusions can be drawn.
With the disappearance of the peaks at the respective cut-off radii one limitation of the short range potential seems to be accomplished.
In addition, sharper peaks in the radial distributions lead to the assumption of expeditious structural formation.
The increase in temperature leads to the occupation of new defect states, which is particularly evident but not limited to the low carbon concentration simulations.
+
+{\color{blue}
The question remains whether these states are only occupied due to the additional supply of kinetic energy and, thus, have to be considered unnatural for temperatures applied in IBS or whether the increase in temperature indeed enables infrequent transitions to occur faster, thus, leading to the intended acceleration of the dynamics and weakening of the unphysical quirks inherent to the potential.
-In the first case these occupied states would be expected to be higher in energy than the states occupied at low temperatures.
-Since substitutional C without the presence of a Si self-interstitial is energetically more favorable than the lowest defect structure obtained without removing a Si atom, that is the \hkl<1 0 0> dumbbell interstitial, and the migration of Si self-interstitials towards the sample surface can be assumed for real life experiments \cite{}, this approach is accepted as an accelerated way of approximatively describing the structural evolution.
-{\color{red}Todo: C sub and Si self-int is energetically less favorable! Maybe fast migration of Si (mentioned in another Todo)? If true, we have to care about Si removal in simulations? In any way these findings suggest a different prec model.}
+As already pointed out in section~\ref{section:defects:noneq_process_01} on page~\pageref{section:defects:noneq_process_01} and section~\ref{section:defects:noneq_process_02} on page~\pageref{section:defects:noneq_process_02} IBS is a nonequilibrium process, which might result in the formation of the thermodynamically less stable substitutional carbon and Si self-interstitital configuration.
+Indeed 3C-SiC is metastable and if not fabricated by IBS requires strong deviation from equilibrium and/or low temperatures to stabilize the 3C polytype \cite{}.
+In IBS highly energetic C atoms are able to generate vacant sites, which in turn can be occupied by highly mobile C atoms.
+This is found to be favorable in the absence of the Si self-interstitial, which turned out to have a low interaction capture radius with a substitutional C atom very likely preventing the recombination into thermodynamically stable C-Si dumbbell interstitials for appropriate separations of the defect pair.
+Results gained in this chapter show preferential occupation of Si lattice sites by substitutional C enabled by increased temperatures supporting the assumptions drawn from the defect studies of the last chapter.
+
+Thus, employing increased temperatures is not exclusively usefull to accelerate the dynamics approximatively describing the structural evolution.
+Moreover it can be considered a necessary condition to deviate the system out of equilibrium enabling the formation of 3C-SiC obviously realized by a successive agglomeration of substitutional C.
+}
\subsection{Valuation of a practicable temperature limit}
\label{subsection:md:tval}
-\begin{figure}[!ht]
-\begin{center}
-\includegraphics[width=12cm]{fe_and_t.ps}
-\end{center}
-\caption{Free energy and temperature evolution of plain silicon at temperatures in the region around the melting transition.}
-\label{fig:md:fe_and_t}
-\end{figure}
The assumed applicability of increased temperature simulations as discussed above and the remaining absence of either agglomeration of substitutional C in low concentration simulations or amorphous to crystalline transition in high concentration simulations suggests to further increase the system temperature.
So far, the highest temperature applied corresponds to 95 \% of the absolute silicon melting temperature, which is 2450 K and specific to the Erhart/Albe potential.
However, melting is not predicted to occur instantly after exceeding the melting point due to additionally required transition enthalpy and hysteresis behaviour.
According to this study temperatures of 100 \% and 120 \% of the silicon melting point could be used.
However, defects, which are introduced due to the insertion of C atoms are known to lower the transition point.
Indeed simulations show melting transitions already at the melting point whenever C is inserted.
-Thus, a system temperature of 95 \% of the silicon melting point is used in the following.
+Thus, the system temperature of 95 \% of the silicon melting point is considered the maximum limit.
+\begin{figure}[!t]
+\begin{center}
+\includegraphics[width=12cm]{fe_and_t.ps}
+\end{center}
+\caption{Free energy and temperature evolution of plain silicon at temperatures in the region around the melting transition.}
+\label{fig:md:fe_and_t}
+\end{figure}
\subsection{Long time scale simulations at maximum temperature}
-HERE: Quality evolution showed that without cooling it could have increased ... mention that, while at temperatures already simulated, the time time scale is extended! ...
+As discussed in section~\ref{subsection:md:limit} and~\ref{subsection:md:inct} a further increase of the system temperature might help to overcome limitations of the short range potential and accelerate the dynamics involved in structural evolution.
+Furthermore these results indicate that increased temperatures are necessary to drive the system out of equilibrium enabling conditions needed for the formation of a metastable cubic polytype of SiC.
-As discussed in section \ref{subsection:md:limit} and \ref{subsection:md:inct} a further increase of the system temperature might help to overcome limitations of the short range potential and accelerate the dynamics involved in structural evolution.
A maximum temperature to avoid melting is determined in section \ref{subsection:md:tval} to be 120 \% of the Si melting point but due to defects lowering the transition point a maximum temperature of 95 \% of the Si melting temperature is considered usefull.
This value is almost equal to the temperature of $2050\,^{\circ}\mathrm{C}$ already used in former simulations.
-Thus, this approach reduces to the application of longer time scales.
-Super!
+Since the maximum temperature is reached the approach is reduced to the application of longer time scales.
+This is considered usefull since the estimated evolution of quality in the absence of the cooling down sequence in figure~\ref{fig:md:tot_si-c_q} predicts an increase in quality and, thus, structural evolution is liekyl to occur if the simulation is proceeded at maximum temperature.
-Next to a longer time scale of simulating at maximum temperature a few more changes are applied.
-In the following simulations the system volume, the amount of C atoms inserted and the shape of the insertion volume are modified from the values used in the first MD simulations.
+Next to the employment of longer time scales and a maximum temperature a few more changes are applied.
+In the following simulations the system volume, the amount of C atoms inserted and the shape of the insertion volume are modified from the values used in first MD simulations.
To speed up the simulation the initial simulation volume is reduced to 21 Si unit cells in each direction and 5500 inserted C atoms in either the whole volume or in a sphere with a radius of 3 nm corresponding to the size of a precipitate consisting of 5500 C atoms.
-The 100 ps sequence after C insertion intended for structural evolution is exchanged by a 10 ns sequence, which is hoped to result in the occurence of infrequent processes.
+The 100 ps sequence after C insertion intended for structural evolution is exchanged by a 10 ns sequence, which is hoped to result in the occurence of infrequent processes and a subsequent phase transition.
The return to lower temperatures is considered seperately.
-\begin{figure}[!ht]
+\begin{figure}[!t]
\begin{center}
-\includegraphics[width=12cm]{fe_100.ps}
-\includegraphics[width=12cm]{q_100.ps}
+\includegraphics[width=12cm]{c_in_si_95_v1_si-c.ps}\\
+\includegraphics[width=12cm]{c_in_si_95_v1_c-c.ps}
\end{center}
-\caption[Evolution of the free energy and quality of a simulation at 100 \% of the Si melting temperature.]{Evolution of the free energy and quality of a simulation at 100 \% of the Si melting temperature. Matt colored parts of the graphs represent the C insertion sequence.}
-\label{fig:md:exceed100}
+\caption{Si-C (top) and C-C (bottom) radial distribution for low concentration simulations at 95 \% of the potential's Si melting point at different points in time of the simulation.}
+\label{fig:md:95_long_time_v1}
\end{figure}
-\begin{figure}[!ht]
+\begin{figure}[!t]
\begin{center}
-\includegraphics[width=12cm]{fe_120.ps}
-\includegraphics[width=12cm]{q_120.ps}
+\includegraphics[width=12cm]{c_in_si_95_v2.ps}
\end{center}
-\caption[Evolution of the free energy and quality of a simulation at 120 \% of the Si melting temperature.]{Evolution of the free energy and quality of a simulation at 120 \% of the Si melting temperature. Matt colored parts of the graphs represent the C insertion sequence.}
-\label{fig:md:exceed120}
+\caption{Si-C and C-C radial distribution for high concentration simulations at 95 \% of the potential's Si melting point at different points in time of the simulation.}
+\label{fig:md:95_long_time_v2}
\end{figure}
-Figure \ref{fig:md:exceed100} and \ref{fig:md:exceed120} show the evolution of the free energy per atom and the quality at 100 \% and 120 \% of the Si melting temperature.
-
-{\color{red}Todo: Melting occurs, show and explain it and that it's due to the defects created.}
-{\color{red}Todo: Due to melting, after insertion, simulation is continued NVE, so melting hopefully will not occur, before it will be cooled down later on.}
+Figure \ref{fig:md:95_long_time_v1} shows the evolution in time of the radial distribution for Si-C and C-C pairs for a low C concentration simulation.
+Differences are observed for both types of atom pairs indeed indicating proceeding structural changes even well beyond 100 ps of simulation time.
+Peaks attributed to the existence of substitutional C increase and become more distinct.
+However, no increase of the amount of total C-C pairs within the observed region can be identified.
+Carbon, whether substitutional or as a dumbbell does not agglomerate within the simulated period of time.
+
+Figure \ref{fig:md:95_long_time_v2} shows the evolution in time of the radial distribution for Si-C and C-C pairs for a high C concentration simulation.
+There are only small changes identifiable.
+Explain more ...
+
+For both, high and low concentration simulations the radial distribution converges as can be seen by the nearly identical graphs for the last two points in time.
+Changes exist ... bridge to results after cooling down to 20 degree C.
-{\color{red}Todo: In additions simulations at 95 \% of the Si melting temperature are started again for longer times.}
+{\color{red}Todo: Cooling down to 20 degree C and compare.}
+
+{\color{red}Todo: Remember NVE simulations (prevent melting).}
\subsection{Further accelerated dynamics approaches}
-{\color{red}Todo: self-guided MD!}
+Since longer time scales are not sufficient \ldots
+
+{\color{red}Todo: self-guided MD?}
{\color{red}Todo: other approaches?}
-{\color{red}
-Todo: ART MD?
-Also, how about forcing a migration of a $V_2$ configuration to a constructed prec configuration, detrmine the maximum saddle point and let the simulation run.
+{\color{red}Todo: ART MD?\\
+How about forcing a migration of a $V_2$ configuration to a constructed prec configuration, detrmine the maximum saddle point and let the simulation run?
}
+\section{Conclusions concerning the SiC conversion mechanism}
+
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