\label{fig:simulation:sc}
\end{figure}
Type 1 (Fig. \ref{fig:simulation:sc1}) constitutes the primitive cell.
-The basis is face-centered cubic (fcc) and is given by $x_1=(0.5,0.5,0)$, $x_2=(0,0.5,0.5)$ and $x_3=(0.5,0,0.5)$.
+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.
The basis is given by $x_1=(0.5,-0.5,0)$, $x_2=(0.5,0.5,0)$ and $x_3=(0,0,1)$.
\section{DFT calculations}
\label{section:simulation:dft_calc}
-The first-principles DFT calculations are performed with the plane-wave-based Vienna {\em ab initio} simulation package ({\textsc vasp}) \cite{kresse96}.
+The first-principles DFT calculations are performed with the plane-wave-based Vienna {\em ab initio} simulation package (\textsc{vasp}) \cite{kresse96}.
The Kohn-Sham equations are solved using the GGA utilizing the exchange-correlation functional proposed by Perdew and Wang (GGA-PW91) \cite{perdew86,perdew92}.
-The electron-ion interaction is described by norm-conserving ultra-soft pseudopotentials as implemented in {\textsc vasp} \cite{vanderbilt90}.
+The electron-ion interaction is described by norm-conserving ultra-soft pseudopotentials as implemented in \textsc{vasp} \cite{vanderbilt90}.
An energy cut-off of \unit[300]{eV} is used to expand the wave functions into the plane-wave basis.
Sampling of the Brillouin zone is restricted to the $\Gamma$ point.
Spin polarization has been fully accounted for.
The conjugate gradient algorithm is used for ionic relaxation.
Migration paths are determined by the modified version of the CRT method as explained in section \ref{section:basics:migration}.
The cell volume and shape is allowed to change using the pressure control algorithm of Parrinello and Rahman \cite{parrinello81} in order to realize constant pressure simulations.
-Due to restrictions by the {\textsc vasp} code, {\em ab initio} MD could only be performed at constant volume.
+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 time step of \unit[1]{fs}.
% todo - point defects are calculated for the neutral charge state.
\subsection{Potential and exchange-correlation functional}
To find the most suitable combination of potential and XC functional for the C/Si system a $2\times2\times2$ supercell of type 3 of Si and C, both in the diamond structure, as well as 3C-SiC is equilibrated for different combinations of the available potentials and XC functionals.
-To exclude a possibly corrupting influence of the other parameters highly accurate calculations are performed, i.e. an energy cut-off of \unit[650]{eV} and a $6\times6\times6$ Monkhorst-Pack $k$-point mesh is used.
-Next to the ultra-soft pseudopotentials \cite{vanderbilt90} {\textsc vasp} offers the projector augmented-wave method (PAW) \cite{bloechl94} to describe the ion-electron interaction.
-The two XC functionals included in the test are of the LDA \cite{ceperley80,perdew81} and GGA \cite{perdew86,perdew92} type as implemented in {\textsc vasp}.
+To exclude a possibly corrupting influence of the other parameters highly accurate calculations are performed, i.e.\ an energy cut-off of \unit[650]{eV} and a $6\times6\times6$ Monkhorst-Pack $k$-point mesh is used.
+Next to the ultra-soft pseudopotentials \cite{vanderbilt90} \textsc{vasp} offers the projector augmented-wave method (PAW) \cite{bloechl94} to describe the ion-electron interaction.
+The two XC functionals included in the test are of the LDA \cite{ceperley80,perdew81} and GGA \cite{perdew86,perdew92} type as implemented in \textsc{vasp}.
\begin{table}[t]
\begin{center}
Table \ref{table:simulation:potxc} shows the lattice constants and cohesive energies obtained for the fully relaxed structures with respect to the utilized potential and XC functional.
As expected, cohesive energies are poorly reproduced by the LDA whereas the equilibrium lattice constants are in good agreement.
Using GGA together with the ultra-soft pseudopotential yields improved lattice constants and, more importantly, a very nice agreement of the cohesive energies to the experimental data.
-The 3C-SiC calculations employing the PAW method in conjunction with the LDA suffers from the general problem inherent to LDA, i.e. overestimated binding energies.
+The 3C-SiC calculations employing the PAW method in conjunction with the LDA suffers from the general problem inherent to LDA, i.e.\ overestimated binding energies.
Thus, the PAW \& LDA combination is not pursued.
Since the lattice constant and cohesive energy of 3C-SiC calculated by the PAW method using the GGA are not improved compared to the ultra-soft pseudopotential calculations using the same XC functional, this concept is likewise stopped.
To conclude, the combination of ultra-soft pseudopotentials and the GGA XC functional are considered the optimal choice for the present study.
The quality of the integration algorithm and the occupied time step is determined by the ability to conserve the total energy.
Therefore, simulations of a $9\times9\times9$ 3C-SiC unit cell containing 5832 atoms in total are carried out in the $NVE$ ensemble.
-The calculations are performed for \unit[100]{ps} corresponding to $10^5$ integration steps and two different initial temperatures are considered, i.e. \unit[0]{$^{\circ}$C} and \unit[1000]{$^{\circ}$C}.
+The calculations are performed for \unit[100]{ps} corresponding to $10^5$ integration steps and two different initial temperatures are considered, i.e.\ \unit[0]{$^{\circ}$C} and \unit[1000]{$^{\circ}$C}.
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.7\textwidth]{verlet_e.ps}
An interfacial energy of \unit[2267.28]{eV} is obtained.
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 divided 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 experimentally estimated range of \unit[$2-8\times10^{-4}$]{J/cm$^2$} \cite{taylor93}.
+This value perfectly fits within the experimentally 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 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.
%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.
%The precipitate structure is rapidly heated up to \unit[2050]{$^{\circ}$C} with a heating rate of approximately \unit[75]{$^{\circ}$C/ps}.
-%From that point on the heating rate is reduced to \unit[1]{$^{\circ}$C/ps} and heating is continued upto \unit[120]{\%} of the Si melting temperature of the potential, i.e. \unit[2940]{K}.
+%From that point on the heating rate is reduced to \unit[1]{$^{\circ}$C/ps} and heating is continued upto \unit[120]{\%} of the Si melting temperature of the potential, i.e.\ \unit[2940]{K}.
%\begin{figure}[t]
%\begin{center}
%\includegraphics[width=0.7\textwidth]{fe_and_t_sic.ps}
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