\label{chapter:md}
The molecular dynamics (MD) technique is used to gain insight into the behavior of C existing in different concentrations in c-Si on the microscopic level at finite temperatures.
-The simulations are restricted to classical potential simulations utilizing the analytical EA bond order potential as described in section \ref{subsection:interact_pot}.
-Parameters are chosen according to the discussion in section \ref{section:classpotmd}.
+The simulations are restricted to classical potential simulations utilizing the analytical EA bond order potential as described in section~\ref{subsection:interact_pot}.
+Parameters are chosen according to the discussion in section~\ref{section:classpotmd}.
At the beginning, simulations are performed, which try to mimic the conditions during IBS.
Results reveal limitations of the employed potential and MD in general.
$V_2$ approximately corresponds to the volume of a minimal 3C-SiC precipitate.
$V_3$ is approximately the volume containing the amount of Si atoms necessary to form such a precipitate, which is slightly smaller than $V_2$ due to the slightly lower Si density of 3C-SiC compared to c-Si.
The two latter insertion volumes are considered since no diffusion of C atoms is expected within the simulated period of time at prevalent temperatures.
-This is due to the overestimated activation energy for the diffusion of a \ci \hkl<1 0 0> DB, as pointed out in section \ref{subsection:defects:mig_classical}.
+This is due to the overestimated activation energy for the diffusion of a \ci \hkl<1 0 0> DB, as pointed out in section~\ref{subsection:defects:mig_classical}.
For rectangularly shaped precipitates with side length $L$ the amount of C atoms in 3C-SiC and Si atoms in c-Si is given by
\begin{equation}
N_{\text{C}}^{\text{3C-SiC}} =4 \left( \frac{L}{a_{\text{SiC}}}\right)^3
N_{\text{Si}}^{\text{c-Si}} =8 \left( \frac{L}{a_{\text{Si}}}\right)^3 \text{ .}
\label{eq:md:n_prec}
\end{equation}
-Table \ref{table:md:ins_vols} summarizes the side length of each of the three different insertion volumes determined by equations \eqref{eq:md:n_prec} and the resulting C concentrations inside these volumes.
+Table~\ref{table:md:ins_vols} summarizes the side length of each of the three different insertion volumes determined by equations \eqref{eq:md:n_prec} and the resulting C concentrations inside these volumes.
Looking at the C concentrations, simulations can be distinguished in simulations occupying low ($V_1$) and high ($V_2$, $V_3$) concentrations of C.
\begin{table}[tp]
\begin{center}
In addition to the time scale limitation, problems attributed to the short range potential exist.
The sharp cut-off function, which limits the interacting ions to the next neighbored atoms by gradually pushing the interaction force and energy to zero between the first and second next neighbor distance, is responsible for overestimated and unphysical high forces of next neighbored atoms~\cite{tang95,mattoni2007}.
-This is supported by the overestimated activation energies necessary for C diffusion as investigated in section \ref{subsection:defects:mig_classical}.
+This is supported by the overestimated activation energies necessary for C diffusion as investigated in section~\ref{subsection:defects:mig_classical}.
Indeed, it is not only the strong C-C bond, which is hard to break, inhibiting C diffusion and further rearrangements.
This is also true for the low concentration simulations dominated by the occurrence of C-Si DBs spread over the whole simulation volume.
The bonds of these C-Si pairs are also affected by the cut-off artifact preventing C diffusion and agglomeration of the DBs.
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 potential (\unit[2450]{K})~\cite{albe_sic_pot} due to the presence of defects, temperatures ranging from \unit[450--2050]{$^{\circ}$C} are used.
-The simulation sequence and other parameters except for the system temperature remain unchanged as in section \ref{section:initial_sims}.
+The simulation sequence and other parameters except for the system temperature remain unchanged as in section~\ref{section:initial_sims}.
Since there is no significant difference among the $V_2$ and $V_3$ simulations only the $V_1$ and $V_2$ simulations are carried on and referred to as low C and high C concentration simulations.
A simple quality value $Q$ is introduced, which helps to estimate the progress of structural evolution.
\includegraphics[width=0.7\textwidth]{tot_pc_thesis.ps}\\
\includegraphics[width=0.7\textwidth]{tot_ba.ps}
\end{center}
-\caption[Si-C radial distribution and evolution of quality $Q$ for the low concentration simulations at different elevated temperatures.]{Si-C radial distribution and evolution of quality $Q$ according to equation \ref{eq:md:qdef} for the low concentration simulations at different elevated temperatures. All structures are cooled down to \degc{20}. The gray line shows resulting Si-C bonds in a configuration of \cs{} in c-Si (C$_\text{sub}$) at zero temperature. Arrows in the quality plot mark the end of C insertion and the start of the cooling down step. A fit function according to equation \eqref{eq:md:fit} shows the estimated evolution of quality in the absence of the cooling down sequence.}
+\caption[Si-C radial distribution and evolution of quality $Q$ for the low concentration simulations at different elevated temperatures.]{Si-C radial distribution and evolution of quality $Q$ according to equation~\ref{eq:md:qdef} for the low concentration simulations at different elevated temperatures. All structures are cooled down to \degc{20}. The gray line shows resulting Si-C bonds in a configuration of \cs{} in c-Si (C$_\text{sub}$) at zero temperature. Arrows in the quality plot mark the end of C insertion and the start of the cooling down step. A fit function according to equation \eqref{eq:md:fit} shows the estimated evolution of quality in the absence of the cooling down sequence.}
\label{fig:md:tot_si-c_q}
\end{figure}
Fig.~\ref{fig:md:tot_si-c_q} shows the radial distribution of Si-C bonds for different temperatures and the corresponding evolution of quality $Q$ as defined above for the low concentration simulation.
In addition, structures form that result in distances residing in between the ones obtained from combinations of mixed defect types and the ones obtained by \cs{} configurations, as can be seen by quite high $g(r)$ values in between the continuous dashed line and the first arrow with a solid line.
For the most part, these structures can be identified as configurations of \cs{} with either another C atom that basically occupies a Si lattice site but is displaced by a \si{} atom residing in the very next surrounding or a C atom that nearly occupies a Si lattice site forming a defect other than the \hkl<1 0 0>-type with the Si atom.
Again, this is a quite promising result since the C atoms are taking the appropriate coordination as expected in 3C-SiC.
-%However, this is contrary to the initial precipitation model proposed in section \ref{section:assumed_prec}, which assumes that the transformation into 3C-SiC takes place in a very last step once enough C-Si DBs agglomerated.
+%However, this is contrary to the initial precipitation model proposed in section~\ref{section:assumed_prec}, which assumes that the transformation into 3C-SiC takes place in a very last step once enough C-Si DBs agglomerated.
To summarize, results of low concentration simulations show a phase transition in conjunction with an increase in temperature.
The \ci{} \hkl<1 0 0> DB dominated structure turns into a structure characterized by the occurrence of an increasing amount of \cs{} with respect to temperature.
As discussed in section~\ref{section:md:limit} and~\ref{section: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.
-A maximum temperature to avoid melting is determined in section \ref{section: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 useful.
+A maximum temperature to avoid melting is determined in section~\ref{section: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 useful.
This value is almost equal to the temperature of $2050\,^{\circ}\mathrm{C}$ already used in former simulations.
Since the maximum temperature is reached the approach is reduced to the application of longer time scales.
This is considered useful 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 likely to occur if the simulation is proceeded at maximum temperature.
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