The basis is simple cubic.
In the following, an overview of the different simulation procedures and respective parameters is presented.
-These procedures and parameters differ depending on whether classical potentials or {\em ab initio} methods are used and on what is going to be investigated.
+These procedures and parameters differ depending on whether classical potentials or {\em ab initio} methods are used as well as on the object of investigation.
\section{DFT calculations}
\label{section:simulation:dft_calc}
% 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.
+These parameters include the size of the supercell, cut-off energy and $\vec{k}$-point mesh.
The choice of these parameters is considered to reflect a reasonable treatment with respect to both, computational efficiency and accuracy, as will be shown in the next sections.
Furthermore, criteria concerning the choice of the potential and the exchange-correlation (XC) functional are being outlined.
Finally, the utilized parameter set is tested by comparing the calculated values of the cohesive energy and the lattice constant to experimental data.
\caption{Defect formation energies of several defects in c-Si with respect to the size of the supercell.}
\label{fig:simulation:ef_ss}
\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.
+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 $\vec{k}$-point mesh~\cite{monkhorst76} is used.
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.
The calculation is usually two times faster and half of the storage needed for the wave functions can be saved since $c_{i,q}=c_{i,-q}^*$, where the $c_{i,q}$ are the Fourier coefficients of the wave function.
As discussed in section~\ref{subsection:basics:bzs}, this does not pose a severe limitation if the supercell is large enough.
Indeed, it was shown~\cite{dal_pino93} that already for calculations involving only 32 atoms, energy values obtained by sampling the $\Gamma$ point differ by less than \unit[0.02]{eV} from calculations using the Baldereschi point~\cite{baldereschi73}, which constitutes a mean-value point in the BZ.
-Thus, the calculations of the present study on supercells containing $108$ primitive cells can be considered sufficiently converged with respect to the $k$-point mesh.
+Thus, the calculations of the present study on supercells containing $108$ primitive cells can be considered sufficiently converged with respect to the $\vec{k}$-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.
+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.
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.7\textwidth]{sic_32pc_gamma_cutoff_lc.ps}
\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.
+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 $\vec{k}$-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]
However, in some cases a time constant of \unit[100]{fs} turned out to result in lower barriers.
Defect structures as well as the simulations modeling the SiC precipitation are performed in the isothermal-isobaric $NpT$ ensemble.
-In addition to the bond order formalism the EA potential provides a set of parameters to describe the interaction in the C/Si system, as discussed in section~\ref{subsection:interact_pot}.
-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}.
+In addition to the bond order formalism, the EA potential provides a set of parameters to describe the interaction in the C/Si system, as discussed in section~\ref{subsection:interact_pot}.
+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.
\subsection{Time step}
-The quality of the integration algorithm and the occupied time step is determined by the ability to conserve the total energy.
+The quality of the integration algorithm and the occupied time step of \unit[1]{fs} is determined by the ability to conserve the total energy.
Therefor, 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}.
\begin{figure}[t]
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.
+Thus, using a time step of \unit[1]{fs} is considered small enough.
\subsection{3C-SiC precipitate in c-Si}
\label{section:simulation:prec}
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.
-To obtain a minimal and stable precipitate 5500 carbon atoms are considered necessary according to experimental results as discussed in section~\ref{subsection:ibs} and~\ref{section:assumed_prec}.
+To obtain a minimal and stable precipitate, 5500 carbon atoms are considered necessary according to experimental results as discussed in section~\ref{subsection:ibs} and~\ref{section:assumed_prec}.
This corresponds to a spherical 3C-SiC precipitate with a radius of approximately \unit[3]{nm}.
The initial precipitate configuration is constructed in two steps.
In the first step the surrounding Si matrix is created.
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