\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 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 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}.
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 electronic ground state is calculated by an iterative Davidson scheme \cite{davidson75} until the difference in total energy of two subsequent iterations is below \unit[$10^{-4}$]{eV}.
+The electronic ground state is calculated by an iterative Davidson scheme~\cite{davidson75} until the difference in total energy of two subsequent iterations is below \unit[$10^{-4}$]{eV}.
Defect structures and the migration paths have been modeled in cubic supercells of type 3 containing 216 Si atoms.
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.
+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.
In MD simulations the equations of motion are integrated by a fourth order predictor corrector algorithm for a time step of \unit[1]{fs}.
\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.
+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 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.
Throughout this work sampling of the BZ is restricted to the $\Gamma$ point.
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.
+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.
\subsection{Energy cut-off}
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}.
+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}
Thus, the method is capable of performing structural optimizations on large systems and MD calculations may be used to model a system over long time scales.
Defect structures are modeled in a cubic supercell (type 3) of nine Si lattice constants in each direction containing 5832 Si atoms.
Reproducing the SiC precipitation was attempted in cubic c-Si supercells, which have a size up to 31 Si unit cells in each direction consisting of 238328 Si atoms.
-A Tersoff-like bond order potential by Erhart and Albe (EA) \cite{albe_sic_pot} is used to describe the atomic interaction.
-Constant pressure simulations are realized by the Berendsen barostat \cite{berendsen84} using a time constant of \unit[100]{fs} and a bulk modulus of \unit[100]{GPa} for Si.
-The temperature is controlled by the Berendsen thermostat \cite{berendsen84} with a time constant of \unit[100]{fs}.
-Integration of the equations of motion is realized by the velocity Verlet algorithm \cite{verlet67} using a fixed time step of \unit[1]{fs}.
+A Tersoff-like bond order potential by Erhart and Albe (EA)~\cite{albe_sic_pot} is used to describe the atomic interaction.
+Constant pressure simulations are realized by the Berendsen barostat~\cite{berendsen84} using a time constant of \unit[100]{fs} and a bulk modulus of \unit[100]{GPa} for Si.
+The temperature is controlled by the Berendsen thermostat~\cite{berendsen84} with a time constant of \unit[100]{fs}.
+Integration of the equations of motion is realized by the velocity Verlet algorithm~\cite{verlet67} using a fixed time step of \unit[1]{fs}.
For structural relaxation of defect structures the same algorithm is utilized with the temperature set to zero Kelvin.
This also applies for the relaxation of structures within the CRT calculations to find migration pathways.
In the latter case the time constant of the Berendsen thermostat is set to \unit[1]{fs} in order to achieve direct velocity scaling, which corresponds to a steepest descent minimization driving the system into a local minimum, if the temperature is set to zero Kelvin.
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}.
+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.
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.
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