\chapter{Silicon carbide precipitation simulations}
+\label{chapter:md}
The molecular dynamics (MD) technique is used to gain insight into the behavior of carbon existing in different concentrations in crystalline silicon on the microscopic level at finite temperatures.
Both, quantum-mechanical and classical potential molecular dynamics simulations are performed.
Molecular dynamics simulations of a single, two and ten carbon atoms in $3\times 3\times 3$ unit cells of crytsalline silicon are performed.
+{\color{red}Todo: ... in progress ...}
+
\section{Classical potential MD simulations}
In contrast to the quantum-mechanical MD simulations the developed classical potential MD code is able to do constant pressure simulations using the Berendsen barostat.
\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.}
-{\color{red}Todo: In additions simulations at 95 \% of the Si melting temperature are started again for longer times.}
+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.
+This finding complies with the predicted increase of quality evolution as explained earlier.
+More and more C forms tetrahedral bonds to four Si neighbours occupying vacant Si sites.
+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 visible by the unchanging area beneath the graphs.
+
+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.
+A slight increase of the Si-C peak at approximately 0.36 nm attributed to the distance of substitutional C and the next but one Si atom along \hkl<1 1 0> is observed.
+In the same time the C-C peak at approximately 0.32 nm corresponding to the distance of two C atoms interconnected by a Si atom along \hkl<1 1 0> slightly decreases.
+Obviously the system preferes a slight increase of isolated substitutional C at the expense of incoherent C-Si-C precipitate configurations, which at a first glance actually appear as promising configurations in the precipitation event.
+On second thoughts however, this process of splitting a C atom out of this structure is considered necessary in order to allow for the rearrangement of C atoms on substitutional lattice sites on the one hand and for C diffusion otherwise, which is needed to end up in a structure, in which one of the two fcc sublattices is composed out of carbon only.
+
+For both, high and low concentration simulations the radial distribution converges as can be seen by the nearly identical graphs of the two most advanced configurations.
+Changes exist ... bridge to results after cooling down to 20 degree C.
+
+{\color{red}Todo: Cooling down to $20\,^{\circ}\mathrm{C}$ by $1\,^{\circ}\mathrm{C/s}$ in progress.}
+
+{\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, determine the saddle point configuration and continue the simulation from this configuration?
}
+\section{Conclusions concerning the SiC conversion mechanism}
+
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