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}.
+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.
In MD simulations the equations of motion are integrated by a fourth order predictor corrector algorithm for a time step of \unit[1]{fs}.
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.
+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.
\caption[Equilibrium lattice constants and cohesive energies of fully relaxed structures of Si, C (diamond) and 3C-SiC for different potentials and XC functionals.]{Equilibrium lattice constants and cohesive energies of fully relaxed structures of Si, C (diamond) and 3C-SiC for different potentials (ultra-soft PP and PAW) and XC functionals (LDA and GGA). Deviations of the respective values from experimental values are given. Values are in good (green), fair (orange) and poor (red) agreement.}
\label{table:simulation:potxc}
\end{table}
-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.
+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.
\caption{Equilibrium lattice constants and cohesive energies of Si, C (diamond) and 3C-SiC using the entire parameter set as determined in the beginning of this section.}
\label{table:simulation:paramf}
\end{table}
-Table \ref{table:simulation:paramf} shows the respective results and deviations from experiment.
+Table~\ref{table:simulation:paramf} shows the respective results and deviations from experiment.
A nice agreement with experimental results is achieved.
Clearly, a competent parameter set is found, which is capable of describing the C/Si system by {\em ab initio} calculations.
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}.
+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.
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.
\end{equation}
By this means values of \unit[2.973]{nm} and \unit[4.309]{\AA} are obtained for the initial precipitate radius and lattice constant of 3C-SiC.
Since the generation of atoms is a discrete process with regard to the size of the volume the expected amounts of atoms are not obtained.
-However, by applying these values the final configuration varies only slightly from the expected one by five carbon and eleven silicon atoms, as can be seen in Table \ref{table:simulation:sic_prec}.
+However, by applying these values the final configuration varies only slightly from the expected one by five carbon and eleven silicon atoms, as can be seen in Table~\ref{table:simulation:sic_prec}.
\begin{table}[t]
\begin{center}
\begin{tabular}{l c c c c}
\frac{N^{\text{3C-SiC}}_{\text{Si}}}{4/a_{\text{3C-SiC prec}}}}
{\frac{N^{\text{total}}_{\text{Si}}}{8/a_{\text{plain c-Si}}}}
\end{equation}
-with the notation used in Table \ref{table:simulation:sic_prec}.
+with the notation used in Table~\ref{table:simulation:sic_prec}.
Here, $a_{\text{c-Si prec}}$ denotes the lattice constant of the surrounding crystalline Si and $a_{\text{3C-SiC prec}}$ is the lattice constant of the precipitate.
The lattice constant of plain c-Si at \unit[20]{$^{\circ}$C} can be determined more accurately by the side lengths of the simulation box of an equilibrated structure instead of using the radial distribution data.
By this, a value of $a_{\text{plain c-Si}}=5.439\,\text{\AA}$ is obtained.
To finally draw some conclusions concerning the capabilities of the potential, the 3C-SiC/c-Si interface is now addressed.
One important size analyzing the interface is the interfacial energy.
A good estimate of the interfacial energy should be obtained by utilizing the formula for determining the defect formation energy as described in equation \eqref{eq:basics:ef2}.
-Using the notation of Table \ref{table:simulation:sic_prec} and assuming that the system is composed out of $N^{\text{3C-SiC}}_{\text{C}}$ C atoms forming the SiC compound plus the remaining Si atoms, the energy is given by
+Using the notation of Table~\ref{table:simulation:sic_prec} and assuming that the system is composed out of $N^{\text{3C-SiC}}_{\text{C}}$ C atoms forming the SiC compound plus the remaining Si atoms, the energy is given by
\begin{equation}
E_{\text{f}}=E-
N^{\text{3C-SiC}}_{\text{C}} E_{\text{coh}}^{\text{SiC}}-
%\caption{Total energy and temperature evolution of a 3C-SiC precipitate embedded in c-Si at temperatures above the Si melting point.}
%\label{fig:simulation:fe_and_t_sic}
%\end{figure}
-%Figure \ref{fig:simulation:fe_and_t_sic} shows the total energy and temperature evolution.
+%Figure~\ref{fig:simulation:fe_and_t_sic} shows the total energy and temperature evolution.
%The sudden increase of the total energy indicates possible melting occuring around \unit[2840]{K}.
%\begin{figure}[ht]
%\begin{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}
-%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}.
+%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}.
%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
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