Strictly, these supercells become the unit cells, which, by a periodic sequence, compose the bulk material that is actually investigated by this approach.
Thus, importance need to be attached to the construction of the supercell.
Three basic types of supercells to compose the initial Si bulk lattice, which can be scaled by integers in the different directions, are considered.
-The basis vectors of the supercells are shown in Fig. \ref{fig:simulation:sc}.
+The basis vectors of the supercells are shown in Fig.~\ref{fig:simulation:sc}.
\begin{figure}[t]
\begin{center}
\subfigure[]{\label{fig:simulation:sc1}\includegraphics[width=0.3\textwidth]{sc_type0.eps}}
\caption{Basis vectors of three basic types of supercells used to create the initial Si bulk lattice.}
\label{fig:simulation:sc}
\end{figure}
-Type 1 (Fig. \ref{fig:simulation:sc1}) constitutes the primitive cell.
+Type 1 (Fig.~\ref{fig:simulation:sc1}) constitutes the primitive cell.
The basis is face-centered cubic and is given by $x_1=(0.5,0.5,0)$, $x_2=(0,0.5,0.5)$ and $x_3=(0.5,0,0.5)$.
Two atoms, one at $(0,0,0)$ and the other at $(0.25,0.25,0.25)$ with respect to the basis, generate the Si diamond primitive cell.
-Type 2 (Fig. \ref{fig:simulation:sc2}) covers two primitive cells with 4 atoms.
+Type 2 (Fig.~\ref{fig:simulation:sc2}) covers two primitive cells with 4 atoms.
The basis is given by $x_1=(0.5,-0.5,0)$, $x_2=(0.5,0.5,0)$ and $x_3=(0,0,1)$.
-Type 3 (Fig. \ref{fig:simulation:sc3}) contains 4 primitive cells with 8 atoms and corresponds to the unit cell shown in Fig. \ref{fig:sic:unit_cell}.
+Type 3 (Fig.~\ref{fig:simulation:sc3}) contains 4 primitive cells with 8 atoms and corresponds to the unit cell shown in Fig.~\ref{fig:sic:unit_cell}.
The basis is simple cubic.
In the following an overview of the different simulation procedures and respective parameters is presented.
\end{figure}
To estimate a critical size the formation energies of several intrinsic defects in Si with respect to the system size are calculated.
An energy cut-off of \unit[250]{eV} and a $4\times4\times4$ Monkhorst-Pack $k$-point mesh \cite{monkhorst76} is used.
-The results are displayed in Fig. \ref{fig:simulation:ef_ss}.
+The results are displayed in Fig.~\ref{fig:simulation:ef_ss}.
The formation energies converge fast with respect to the system size.
Thus, investigating supercells containing more than 56 primitive cells or $112\pm1$ atoms should be reasonably accurate.
\caption{Lattice constants of 3C-SiC with respect to the cut-off energy used for the plane-wave basis set.}
\label{fig:simulation:lc_ce}
\end{figure}
-Fig. \ref{fig:simulation:lc_ce} shows the respective lattice constants of the relaxed 3C-SiC structure with respect to the cut-off energy.
+Fig.~\ref{fig:simulation:lc_ce} shows the respective lattice constants of the relaxed 3C-SiC structure with respect to the cut-off energy.
As can be seen, convergence is reached already for low energies.
Obviously, an energy cut-off of \unit[300]{eV}, although the minimum acceptable, is sufficient for the plane-wave expansion.
\caption{Evolution of the total energy of 3C-SiC in the $NVE$ ensemble for two different initial temperatures.}
\label{fig:simulation:verlet_e}
\end{figure}
-The evolution of the total energy is displayed in Fig. \ref{fig:simulation:verlet_e}.
+The evolution of the total energy is displayed in Fig.~\ref{fig:simulation:verlet_e}.
Almost no shift in energy is observable for the simulation at \unit[0]{$^{\circ}$C}.
Even for \unit[1000]{$^{\circ}$C} the shift is as small as \unit[0.04]{eV}, which is a quite acceptable error for $10^5$ integration steps.
Thus, using a time step of \unit[100]{ps} is considered small enough.
\caption[Radial distribution of a 3C-SiC precipitate embedded in c-Si at $20\,^{\circ}\mathrm{C}$.]{Radial distribution of a 3C-SiC precipitate embedded in c-Si at \unit[20]{$^{\circ}$C}. The Si-Si radial distribution of plain c-Si is plotted for comparison. Green arrows mark bumps in the Si-Si distribution of the precipitate configuration, which do not exist in plain c-Si.}
\label{fig:simulation:pc_sic-prec}
\end{figure}
-Fig. \ref{fig:simulation:pc_sic-prec} shows the radial distribution of the obtained precipitate configuration.
+Fig.~\ref{fig:simulation:pc_sic-prec} shows the radial distribution of the obtained precipitate configuration.
The Si-Si radial distribution for both, plain c-Si and the precipitate configuration show a maximum at a distance of \unit[0.235]{nm}, which is the distance of next neighbored Si atoms in c-Si.
Although no significant change of the lattice constant of the surrounding c-Si matrix was assumed, surprisingly, there is no change at all within observational accuracy.
Looking closer at higher order Si-Si peaks might even allow the guess of a slight increase of the lattice constant compared to the plain c-Si structure.
%\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}
-%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}.
+%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.
%Thus, it is only the c-Si surrounding undergoing a structural phase transition, which is very well reflected by the difference observed for the two Si-Si distributions.
%This is surprising since the melting transition of plain c-Si for the same heating conditions is expected at temperatures around \unit[3125]{K}, as will be discussed later in section \ref{subsection:md:tval}.
%\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.
+%Fig.~\ref{fig:simulation:sic_melt} shows cross section images of the atomic structures at different times and temperatures.
%As can be seen from the image at \unit[520]{ps} 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 at \unit[720]{ps}.
%As predicted from the radial distribution data the precipitate itself indeed remains stable.
%
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