Due to restrictions by the {\textsc vasp} code, {\em ab initio} MD could only be performed at constant volume.
In MD simulations the equations of motion are integrated by a fourth order predictor corrector algorithm for a timestep of \unit[1]{fs}.
-% todo
-% All point defects are calculated for the neutral charge state.
+% todo - point defects are calculated for the neutral charge state.
Most of the parameter settings, as determined above, constitute a tradeoff regarding the tasks that need to be addressed.
These parameters include the size of the supercell, cut-off energy and $k$ point mesh.
\subsection{Energy cut-off}
To determine an appropriate cut-off energy of the plane-wave basis set a $2\times2\times2$ supercell of type 3 containing $32$ Si and $32$ C atoms in the 3C-SiC structure is equilibrated for different cut-off energies in the LDA.
-% todo
-% mention that results are within lda
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.7\textwidth]{sic_32pc_gamma_cutoff_lc.ps}
A nice agreement with experimental results is achieved.
Clearly, a competent parameter set is found, which is capabale of describing the C/Si system by {\em ab initio} calculations.
-% todo
-% rewrite dft chapter
-% ref for experimental values!
+% todo - ref for experimental values!
\section{Classical potential MD}
+\label{section:classpotmd}
The classical potential MD method is much less computationally costly compared to the highly accurate quantum-mechanical method.
Thus, the method is capable of performing structural optimizations on large systems and MD calulations may be used to model a system over long time scales.
There are basically no free parameters, which could be set by the user and the properties of the potential and its parameters are well known and have been extensively tested by the authors \cite{albe_sic_pot}.
Therefore, test calculations are restricted to the time step used in the Verlet algorithm to integrate the equations of motion.
Nevertheless, a further and rather uncommon test is carried out to roughly estimate the capabilities of the EA potential regarding the description of 3C-SiC precipitation in c-Si.
-% todo
-% rather a first investigation than a test
\subsection{Time step}
Below, a spherical 3C-SiC precipitate enclosed in a c-Si surrounding is investigated by means of MD.
On the one hand, these investigations are meant to draw conclusions on the capabilities of the potential for modeling the respective tasks in the C/Si system.
-Since, on the other hand, properties of the 3C-SiC precipitate, the surrounding and the interface can be obtained, the calculations could be considered to constitute a first investigation rather than a test of the capabilities of the potential.
+Since, on the other hand, properties of the 3C-SiC precipitate, its surrounding as well as the interface can be obtained, the calculations could be considered to constitute a first investigation rather than a test of the capabilities of the potential.
+
+\subsubsection{Interfacial energy}
To construct a spherical and topotactically aligned 3C-SiC precipitate in c-Si, the approach illustrated in the following is applied.
A total simulation volume $V$ consisting of 21 unit cells of c-Si in each direction is created.
The amount of C atoms together with the observed lattice constant of the precipitate leads to a precipitate radius of \unit[29.93]{\AA}.
Thus, the interface tension, given by the energy of the interface devided by the surface area of the precipitate is \unit[20.15]{eV/nm$^2$} or \unit[$3.23\times 10^{-4}$]{J/cm$^2$}.
This value perfectly fits within the eperimentally estimated range of \unit[$2-8\times10^{-4}$]{J/cm$^2$} \cite{taylor93}.
-Thus, the EA potential is considered an appropriate choice for the current study properly describing the energetics of interfaces.
+Thus, the EA potential is considered an appropriate choice for the current study concerning the accurate description of the energetics of interfaces.
+Furthermore, since the calculated interfacial energy is located in the lower part of the experimental range, the obtained interface structure might resemble an authentic configuration of an energetically favorable interface structure of a 3C-SiC precipitate in c-Si.
+
+% todo - nice to reproduce this value!
-% todo
-% nice to reproduce this value!
+\subsubsection{Stability of the precipitate}
+
+To investigate the stability of the precipiate, which is assumed to be stable even at temperatures above the Si melting temperature, the configuration is heated up beyond the critical point, at which the Si melting transition occurs.
+For this, the transition point of c-Si needs to be evaluated first.
+According to the authors of the potential, the Si melting point is \degk{2450}.
+However, melting is not predicted to occur instantly after exceeding the melting point due to the additionally required transition enthalpy and hysteresis behaviour.
+To determine the transition point, c-Si is heated up using a heating rate of \unit[1]{$^{\circ}$C/ps}.
+\begin{figure}[tp]
+\begin{center}
+\includegraphics[width=0.7\textwidth]{fe_and_t.ps}
+\end{center}
+\caption{Total energy and temperature evolution of c-Si at temperatures in the region around the melting transition.}
+\label{fig:simulation:fe_and_t}
+\end{figure}
+Fig.~\ref{fig:simulation:fe_and_t} shows the total energy and temperature evolution in the region around the transition temperature.
+Indeed, a transition and the accompanied critical behaviour of the total energy is first observed at approximately \degk{3125}, which corresponds to \unit[128]{\%} of the Si melting temperature.
+The difference in total energy is \unit[0.58]{eV} per atom corresponding to \unit[55.7]{kJ/mole}, which compares quite well to the Si enthalpy of melting of \unit[50.2]{kJ/mole}.
+
+The precipitate structure is heated up using the same heating rate.
+As can be seen in Fig.~\ref{fig:simulation:sic_melt}, which shows a cross-sectional image of the configuration at different temperatures, the precipitate is stable whereas melting of the surrounding Si host matrix starting at the interface region is observed.
+\begin{figure}[tp]
+\begin{center}
+\subfigure[]{\label{fig:simulation:sic_melt1}\includegraphics[width=7cm]{sic_prec/melt_01.eps}}
+\subfigure[]{\label{fig:simulation:sic_melt2}\includegraphics[width=7cm]{sic_prec/melt_02.eps}}
+\subfigure[]{\label{fig:simulation:sic_melt3}\includegraphics[width=7cm]{sic_prec/melt_03.eps}}
+\end{center}
+\caption{Cross section image of the 3C-SiC precipitate in c-Si at temperatures before (a), at the onset of (b) and after (c) the Si melting transition.}
+\label{fig:simulation:sic_melt}
+\end{figure}
+This is verified by the radial distribution function shown in Fig.~\ref{fig:simulation:pc_500-fin}, which displays configurations before and after the Si transition occurs.
+\begin{figure}[tp]
+\begin{center}
+\includegraphics[width=0.7\textwidth]{pc_500-fin.ps}
+\end{center}
+\caption{Radial distribution of a 3C-SiC precipitate embedded in c-Si at temperatures below and above the Si melting transition point.}
+\label{fig:simulation:pc_500-fin}
+\end{figure}
+The precipitate itself is not involved in the transition, as indicated by the Si-C and C-C distribution, which essentially stays the same.
+It is only the c-Si surrounding undergoing a structural phase transition, which is very well reflected by the difference observed for the respective Si-Si distributions.
+The temperature of transition is determined to be around \degk{2840}.
+This is surprising since the melting transition of c-Si for the same heating conditions is expected at temperatures around \degk{3125} as discussed above.
+Obviously, the precipitate lowers the transition point of the surrounding c-Si matrix.
+This is indeed verified by the cross-sectional images of the configurations shown in Fig.~\ref{fig:simulation:sic_melt}.
+Melting of the Si surrounding in fact starts in the defective interface region of the 3C-SiC precipitate and the c-Si surrounding propagating outwards until the whole Si matrix is affected.
+
+\subsubsection{Concluding remarks}
+
+To conclude, the obtained results, particularly the accurate value of the interface energy, the Si enthalpy of melting as well as the stability of the 3C-SiC structure give quite a good feeling concerning the applicability of the potential.
%Since the precipitate configuration is artificially constructed, the resulting interface does not necessarily correspond to the energetically most favorable configuration or to the configuration that is expected for an actually grown precipitate.
%Thus, annealing steps are appended to the gained structure in order to allow for a rearrangement of the atoms of the interface.
%\includegraphics[width=0.7\textwidth]{pc_500-fin.ps}
%\end{center}
%\caption{Radial distribution of a 3C-SiC precipitate embedded in c-Si at temperatures below and above the Si melting transition point.}
-%%\label{fig:simulation:pc_500-fin}
+%\label{fig:simulation:pc_500-fin}
%\end{figure}
%Investigating the radial distribution function shown in figure \ref{fig:simulation:pc_500-fin}, which shows configurations below and above the temperature of the estimated transition, indeed supports the assumption of melting gained by the total energy plot in Fig. \ref{fig:simulation:fe_and_t_sic}.
%However, the precipitate itself is not involved, as can be seen from the Si-C and C-C distribution, which essentially stays the same for both temperatures.
%Obviously the precipitate lowers the transition point of the surrounding c-Si matrix.
%This is indeed verified by visualizing the atomic data.
%% ./visualize -w 640 -h 480 -d saves/sic_prec_120Tm_cnt1 -nll -11.56 -0.56 -11.56 -fur 11.56 0.56 11.56 -c -0.2 -24.0 0.6 -L 0 0 0.2 -r 0.6 -B 0.1
-%\begin{figure}[t]
-%%\begin{center}
+%\begin{figure}[tp]
+%\begin{center}
%\begin{minipage}{7cm}
%\includegraphics[width=7cm,draft=false]{sic_prec/melt_01.eps}
%\end{minipage}
%\includegraphics[width=7cm,draft=false]{sic_prec/melt_03.eps}
%\end{minipage}
%\end{center}
-%\caption{Cross section image of atomic data gained by annealing simulations of the constructed 3C-SiC precipitate in c-Si at \unit[200]{ps} (top left), \unit[520]{ps} (top right) and \unit[720]{ps} (bottom).}
+%\caption{Cross section image of the precipitate configuration gained by annealing simulations of the constructed 3C-SiC precipitate in c-Si at \unit[200]{ps} (top left), \unit[520]{ps} (top right) and \unit[720]{ps} (bottom).}
%\label{fig:simulation:sic_melt}
%\end{figure}
%Fig. \ref{fig:simulation:sic_melt} shows cross section images of the atomic structures at different times and temperatures.
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