Type 3 (Fig.~\ref{fig:simulation:sc3}) contains 4 primitive cells with 8 atoms and corresponds to the unit cell shown in Fig.~\ref{fig:sic:unit_cell}.
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.
+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 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.
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.
+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]
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.
+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.
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}.
-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}.
+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.
\subsection{Time step}
-The quality of the integration algorithm and the occupied time step is determined by the ability to conserve the total energy.
+The quality of the integration algorithm and the occupied time step of \unit[1]{fs} is determined by the ability to conserve the total energy.
Therefor, simulations of a $9\times9\times9$ 3C-SiC unit cell containing 5832 atoms in total are carried out in the $NVE$ ensemble.
The calculations are performed for \unit[100]{ps} corresponding to $10^5$ integration steps and two different initial temperatures are considered, i.e.\ \unit[0]{$^{\circ}$C} and \unit[1000]{$^{\circ}$C}.
\begin{figure}[t]
The evolution of the total energy is displayed in Fig.~\ref{fig:simulation:verlet_e}.
Almost no shift in energy is observable for the simulation at \unit[0]{$^{\circ}$C}.
Even for \unit[1000]{$^{\circ}$C} the shift is as small as \unit[0.04]{eV}, which is a quite acceptable error for $10^5$ integration steps.
-Thus, using a time step of \unit[100]{ps} is considered small enough.
+Thus, using a time step of \unit[1]{fs} is considered small enough.
\subsection{3C-SiC precipitate in c-Si}
\label{section:simulation:prec}
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.
+In the first step, the surrounding Si matrix is created.
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 $x$.
\text{ .}
\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.
-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}.
+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}.
\begin{table}[t]
\begin{center}
\begin{tabular}{l c c c c}
\end{center}
\end{table}
-After the initial configuration is constructed some of the atoms located at the 3C-SiC/c-Si interface show small distances, which results in high repulsive forces acting on the atoms.
+After the initial configuration is constructed, some of the atoms located at the 3C-SiC/c-Si interface show small distances, which results in high repulsive forces acting on the atoms.
Thus, the system is equilibrated using strong coupling to the heat bath, which is set to be \unit[20]{$^{\circ}$C}.
-Once the main part of the excess energy is carried out previous settings for the Berendsen thermostat are restored and the system is relaxed for another \unit[10]{ps}.
+Once the main part of the excess energy is carried out, previous settings for the Berendsen thermostat are restored and the system is relaxed for another \unit[10]{ps}.
\begin{figure}[t]
\begin{center}
A lattice constant of \unit[4.34]{\AA} compared to \unit[4.36]{\AA} for bulk 3C-SiC is obtained.
This is in accordance with the peak of Si-C pairs at a distance of \unit[0.188]{nm}.
Thus, the precipitate structure is slightly compressed compared to the bulk phase.
-This is a quite surprising result since due to the finite size of the c-Si surrounding a non-negligible impact of the precipitate on the materializing c-Si lattice constant especially near the precipitate could be assumed.
+This is a quite surprising result since due to the finite size of the c-Si surrounding, a non-negligible impact of the precipitate on the materializing c-Si lattice constant especially near the precipitate could be assumed.
However, it seems that the size of the c-Si host matrix is chosen large enough to even find the precipitate in a compressed state.
The absence of a compression of the c-Si surrounding is due to the possibility of the system to change its volume.
-Otherwise the increase of the lattice constant of the precipitate of roughly \unit[4.31]{\AA} in the beginning up to \unit[4.34]{\AA} in the relaxed precipitate configuration could not take place without an accompanying reduction of the lattice constant of the c-Si surrounding.
+Otherwise, the increase of the lattice constant of the precipitate of roughly \unit[4.31]{\AA} in the beginning up to \unit[4.34]{\AA} in the relaxed precipitate configuration could not take place without an accompanying reduction of the lattice constant of the c-Si surrounding.
If the total volume is assumed to be the sum of the volumes that are composed of Si atoms forming the c-Si surrounding and Si atoms involved forming the precipitate, the expected increase can be calculated by
\begin{equation}
\frac{V}{V_0}=
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.
+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.
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.
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