If present, symmetries in reciprocal space may further reduce the number of calculations.
For supercells, i.e.\ repeating unit cells that contain several primitive cells, restricting the sampling of the Brillouin zone (BZ) to the $\Gamma$ point can yield quite accurate results.
In fact, with respect to BZ sampling, calculating wave functions of a supercell containing $n$ primitive cells for only one $\vec{k}$ point is equivalent to the scenario of a single primitive cell and the summation over $n$ points in $\vec{k}$ space.
-In general, finer $\vec{k}$ point meshes better account for the periodicity of a system, which in some cases, however, might be fictitious anyway.
+In general, finer $\vec{k}$-point meshes better account for the periodicity of a system, which in some cases, however, might be fictitious anyway.
\subsection{Structural relaxation and Hellmann-Feynman theorem}
\caption{Kinetic energy plot of the relaxation process of the hexagonal silicon self-interstitial defect simulation using the EA potential.}
\label{fig:defects:kin_si_hex}
\end{figure}
-To exclude failures in the implementation of the potential or the MD code itself the hexagonal defect structure was double-checked with the \textsc{parcas} MD code~\cite{parcas_md}.
+To exclude failures in the implementation of the potential or the MD code itself, the hexagonal defect structure was double-checked with the \textsc{parcas} MD code~\cite{parcas_md}.
The respective relaxation energetics are likewise plotted and look similar to the energetics obtained by \textsc{posic}.
In fact, the same type of interstitial arises using random insertions.
In addition, variations exist, in which the displacement is only along two \hkl<1 0 0> axes ($E_\text{f}=3.8\,\text{eV}$) or along a single \hkl<1 0 0> axes ($E_\text{f}=3.6\,\text{eV}$) successively approximating the tetrahedral configuration and formation energy.
A net amount of five Si-Si and one Si-C bond are additionally formed during transition.
An activation energy of \unit[0.6]{eV} necessary to overcome the migration barrier is found.
This energy is low enough to constitute a feasible mechanism in SiC precipitation.
-To reverse this process \unit[5.4]{eV} are needed, which make this mechanism very improbable.
+To reverse this process, \unit[5.4]{eV} are needed, which make this mechanism very improbable.
%
The migration path is best described by the reverse process.
Starting at \unit[100]{\%}, energy is needed to break the bonds of Si atom 1 to its neighbored Si atoms as well as the bond of the C atom to Si atom number 5.
One possibility is to simply skip the force contributions containing the derivatives of the cut-off function, which was successfully applied to reproduce the brittle propagation of fracture in SiC at zero temperature~\cite{mattoni2007}.
Another one is to use variable cut-off values scaled by the system volume, which properly describes thermomechanical properties of 3C-SiC~\cite{tang95} but might be rather ineffective for the challenge inherent to this study.
-To conclude the obstacle needed to get passed is twofold.
+To conclude, the obstacle needed to get passed is twofold.
The sharp cut-off of the employed bond order model potential introduces overestimated high forces between next neighbored atoms enhancing the problem of slow phase space propagation immanent to MD simulations.
This obstacle could be referred to as {\em potential enhanced slow phase space propagation}.
Due to this, pushing the time scale to the limits of computational resources or applying one of the above mentioned accelerated dynamics methods exclusively will not be sufficient enough.
Next to the employment of longer time scales and a maximum temperature a few more changes are applied.
In the following simulations, the system volume, the amount of C atoms inserted and the shape of the insertion volume are modified from the values used in first MD simulations.
-To speed up the simulation the initial simulation volume is reduced to 21 Si unit cells in each direction and 5500 inserted C atoms in either the whole volume or in a sphere with a radius of 3 nm corresponding to the size of a precipitate consisting of 5500 C atoms.
+To speed up the simulation, the initial simulation volume is reduced to 21 Si unit cells in each direction and 5500 inserted C atoms in either the whole volume or in a sphere with a radius of 3 nm corresponding to the size of a precipitate consisting of 5500 C atoms.
The \unit[100]{ps} sequence after C insertion intended for structural evolution is exchanged by a \unit[10]{ns} sequence, which is hoped to result in the occurrence of infrequent processes and a subsequent phase transition.
The return to lower temperatures is considered separately.
A more detailed investigation showed the formation of a preceding $(2\times 1)$ and $(5\times 2)$ pattern within the exposure to the Si containing gas~\cite{yoshinobu90,fuyuki93}.
The $(3\times 2)$ superstructure contains approximately 1.7 monolayers of Si atoms, crystallizing into 3C-SiC with a smooth and mirror-like surface after C$_2$H$_6$ is inserted accompanied by a reconstruction of the surface into the initial C terminated $(2\times 2)$ pattern.
A minimal growth rate of 2.3 monolayers per cycle exceeding the value of 1.7 is due to physically adsorbed Si atoms not contributing to the superstructure.
-To realize single monolayer growth precise control of the gas supply to form the $(2\times 1)$ structure is required.
+To realize single monolayer growth, precise control of the gas supply to form the $(2\times 1)$ structure is required.
However, accurate layer-by-layer growth is achieved under certain conditions, which facilitate the spontaneous desorption of an additional layer of one atom species by supply of the other species~\cite{hara93}.
Homoepitaxial growth of the 6H polytype has been realized on off-oriented substrates utilizing simultaneous supply of the source gases~\cite{tanaka94}.
Depending on the gas flow ratio either 3C island formation or step flow growth of the 6H polytype occurs, which is explained by a model including aspects of enhanced surface mobilities of adatoms on a $(3\times 3)$ reconstructed surface.
However, these layers show an extremely poor interface to the Si top layer governed by a high density of SiC precipitates, which are not affected in the C redistribution during annealing and, thus, responsible for the rough interface.
Hence, to obtain sharp interfaces and single-crystalline SiC layers temperatures between \unit[400]{$^{\circ}$C} and \unit[600]{$^{\circ}$C} have to be used.
Indeed, reasonable results were obtained at \unit[500]{$^{\circ}$C}~\cite{lindner98} and even better interfaces were observed for \unit[450]{$^{\circ}$C}~\cite{lindner99_2}.
-To further improve the interface quality and crystallinity a two-temperature implantation technique was developed~\cite{lindner99}.
+To further improve the interface quality and crystallinity, a two-temperature implantation technique was developed~\cite{lindner99}.
To form a narrow, box-like density profile of oriented SiC nanocrystals, \unit[93]{\%} of the total dose of \unit[$8.5\cdot 10^{17}$]{cm$^{-2}$} is implanted at \unit[500]{$^{\circ}$C}.
The remaining dose is implanted at \unit[250]{$^{\circ}$C}, which leads to the formation of amorphous zones above and below the SiC precipitate layer and the destruction of SiC nanocrystals within these zones.
After annealing for \unit[10]{h} at \unit[1250]{$^{\circ}$C} a homogeneous, stoichiometric SiC layer with sharp interfaces is formed.
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]
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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