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}
This is realized by just skipping the generation of Si atoms inside a sphere of radius $x$, which is the first unknown variable.
The Si lattice constant $a_{\text{Si}}$ of the surrounding c-Si matrix is assumed to not alter dramatically and, thus, is used for the initial lattice creation.
In a second step 3C-SiC is created inside the empty sphere of radius $x$.
-The lattice constant $y$, the second unknown variable, is chosen in such a way, that the necessary amount of carbon is generated and that the total amount of silicon atoms corresponds to the usual amount contained in the simulation volume.
+The lattice constant $y$, the second unknown variable, is chosen in such a way that the necessary amount of carbon is generated and that the total amount of silicon atoms corresponds to the usual amount contained in the simulation volume.
This is entirely described by the equation
\begin{equation}
\frac{8}{a_{\text{Si}}^3}(
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