\subsubsection{Improved analytical bond order potential}
Although the Tersoff potential is one of the most widely used potentials, there are some shortcomings.
-Describing the Si-Si interaction Tersoff was unable to find a single parameter set to describe well both, bulk and surface properties.
+Describing the Si-Si interaction, Tersoff was unable to find a single parameter set to describe well both, bulk and surface properties.
Due to this, and since the first approach labeled T1~\cite{tersoff_si1} turned out to be unstable~\cite{dodson87}, two further parametrizations exist, T2~\cite{tersoff_si2} and T3~\cite{tersoff_si3}.
While T2 describes well surface properties, T3 yields improved elastic constants and should be used for describing bulk properties.
However, T3, which is used in the Si/C potential, suffers from an underestimation of the dimer binding energy.
+\mathcal{O}(\delta t^3)
\text{ .}
\end{equation}
-Since the forces for the new positions are required to update the velocity the determination of the forces has to be carried out within the integration algorithm.
+Since the forces for the new positions are required to update the velocity, the determination of the forces has to be carried out within the integration algorithm.
\subsection{Statistical ensembles}
\label{subsection:statistical_ensembles}
The substitutional or vacancy defect is realized by replacing or removing one atom contained in the supercell.
Interstitial defects are created by adding an atom at positions located in the space between regular lattice sites.
Once the intuitively created defect structure is generated structural relaxation methods will yield the respective local minimum configuration.
-Since the supercell approach applies periodic boundary conditions enough bulk material surrounding the defect is required to exclude possible interaction of the defect with its periodic image.
+Since the supercell approach applies periodic boundary conditions, enough bulk material surrounding the defect is required to exclude possible interaction of the defect with its periodic image.
\begin{figure}[t]
\begin{center}
Obviously, agglomeration of C$_{\text{i}}$ and C$_{\text{s}}$ is energetically favorable except for separations along one of the \hkl<1 1 0> directions.
The energetically most favorable configuration (configuration $\beta$) forms a strong but compressively strained C-C bond with a separation distance of \unit[0.142]{nm} sharing a Si lattice site.
Again, conclusions concerning the probability of formation are drawn by investigating respective migration paths.
-Since C$_{\text{s}}$ is unlikely to exhibit a low activation energy for migration the focus is on C$_{\text{i}}$.
+Since C$_{\text{s}}$ is unlikely to exhibit a low activation energy for migration, the focus is on C$_{\text{i}}$.
Pathways starting from the next most favored configuration, i.e.\ \cs{} located at position 2, into configuration $\alpha$ and $\beta$ are investigated, which show activation energies above \unit[2.2]{eV} and \unit[2.5]{eV}.
The respective barriers and structures are displayed in Fig.~\ref{fig:051-xxx}.
For the transition into configuration $\beta$, as before, the non-magnetic configuration is obtained.
Scaling the absolute temperatures accordingly results in maximum temperatures of \unit[1460--2260]{$^{\circ}$C}.
Since melting already occurs shortly below the melting point of the potential (\unit[2450]{K})~\cite{albe_sic_pot} due to the presence of defects, temperatures ranging from \unit[450--2050]{$^{\circ}$C} are used.
The simulation sequence and other parameters except for the system temperature remain unchanged as in section~\ref{section:initial_sims}.
-Since there is no significant difference among the $V_2$ and $V_3$ simulations only the $V_1$ and $V_2$ simulations are carried on and referred to as low C and high C concentration simulations.
+Since there is no significant difference among the $V_2$ and $V_3$ simulations, only the $V_1$ and $V_2$ simulations are carried on and referred to as low C and high C concentration simulations.
A simple quality value $Q$ is introduced, which helps to estimate the progress of structural evolution.
In bulk 3C-SiC every C atom has four next neighbored Si atoms and every Si atom four next neighbored C atoms.
A maximum temperature to avoid melting is determined in section~\ref{section:md:tval} to be 120 \% of the Si melting point but due to defects lowering the transition point a maximum temperature of 95 \% of the Si melting temperature is considered useful.
This value is almost equal to the temperature of $2050\,^{\circ}\mathrm{C}$ already used in former simulations.
-Since the maximum temperature is reached the approach is reduced to the application of longer time scales.
+Since the maximum temperature is reached, the approach is reduced to the application of longer time scales.
This is considered useful since the estimated evolution of quality in the absence of the cooling down sequence in figure~\ref{fig:md:tot_si-c_q} predicts an increase in quality and, thus, structural evolution is likely to occur if the simulation is proceeded at maximum temperature.
Next to the employment of longer time scales and a maximum temperature a few more changes are applied.
\label{eq:simulation:constr_sic_03}
\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.
+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}.
\begin{table}[t]
\begin{center}
The same lattice constant is assumed for the c-Si surrounding in the precipitate configuration $a_{\text{c-Si prec}}$ since peaks in the radial distribution match the ones of plain c-Si.
Using $a_{\text{3C-SiC prec}}=4.34\,\text{\AA}$ as observed from the radial distribution finally results in an increase of the initial volume by \unit[0.12]{\%}.
However, each side length and the total volume of the simulation box is increased by \unit[0.20]{\%} and \unit[0.61]{\%} respectively compared to plain c-Si at \unit[20]{$^{\circ}$C}.
-Since the c-Si surrounding resides in an uncompressed state the excess increase must be attributed to relaxation of strain with the strain resulting from either the compressed precipitate or the 3C-SiC/c-Si interface region.
+Since the c-Si surrounding resides in an uncompressed state, the excess increase must be attributed to relaxation of strain with the strain resulting from either the compressed precipitate or the 3C-SiC/c-Si interface region.
This also explains the possibly identified slight increase of the c-Si lattice constant in the surrounding as mentioned earlier.
As the pressure is set to zero the free energy is minimized with respect to the volume enabled by the Berendsen barostat algorithm.
Apparently the minimized structure with respect to the volume is a configuration of a small compressively stressed precipitate and a large amount of slightly stretched c-Si in the surrounding.
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