Strictly, these supercells become the unit cells, which, by a periodic sequence, compose the bulk material that is actually investigated by this approach.
Thus, importance need to be attached to the construction of the supercell.
Three basic types of supercells to compose the initial Si bulk lattice, which can be scaled by integers in the different directions, are considered.
-The basis vectors of the supercells are shown in Fig. \ref{fig:simulation:sc}.
+The basis vectors of the supercells are shown in Fig.~\ref{fig:simulation:sc}.
\begin{figure}[t]
\begin{center}
\subfigure[]{\label{fig:simulation:sc1}\includegraphics[width=0.3\textwidth]{sc_type0.eps}}
\caption{Basis vectors of three basic types of supercells used to create the initial Si bulk lattice.}
\label{fig:simulation:sc}
\end{figure}
-Type 1 (Fig. \ref{fig:simulation:sc1}) constitutes the primitive cell.
+Type 1 (Fig.~\ref{fig:simulation:sc1}) constitutes the primitive cell.
The basis is face-centered cubic and is given by $x_1=(0.5,0.5,0)$, $x_2=(0,0.5,0.5)$ and $x_3=(0.5,0,0.5)$.
Two atoms, one at $(0,0,0)$ and the other at $(0.25,0.25,0.25)$ with respect to the basis, generate the Si diamond primitive cell.
-Type 2 (Fig. \ref{fig:simulation:sc2}) covers two primitive cells with 4 atoms.
+Type 2 (Fig.~\ref{fig:simulation:sc2}) covers two primitive cells with 4 atoms.
The basis is given by $x_1=(0.5,-0.5,0)$, $x_2=(0.5,0.5,0)$ and $x_3=(0,0,1)$.
-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}.
+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.
\section{DFT calculations}
\label{section:simulation:dft_calc}
-The first-principles DFT calculations are performed with the plane-wave-based Vienna {\em ab initio} simulation package ({\textsc vasp}) \cite{kresse96}.
+The first-principles DFT calculations are performed with the plane-wave-based Vienna {\em ab initio} simulation package (\textsc{vasp}) \cite{kresse96}.
The Kohn-Sham equations are solved using the GGA utilizing the exchange-correlation functional proposed by Perdew and Wang (GGA-PW91) \cite{perdew86,perdew92}.
-The electron-ion interaction is described by norm-conserving ultra-soft pseudopotentials as implemented in {\textsc vasp} \cite{vanderbilt90}.
+The electron-ion interaction is described by norm-conserving ultra-soft pseudopotentials as implemented in \textsc{vasp} \cite{vanderbilt90}.
An energy cut-off of \unit[300]{eV} is used to expand the wave functions into the plane-wave basis.
Sampling of the Brillouin zone is restricted to the $\Gamma$ point.
Spin polarization has been fully accounted for.
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}.
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.
+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}.
% todo - point defects are calculated for the neutral charge state.
\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.
-The results are displayed in Fig. \ref{fig:simulation:ef_ss}.
+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.
\caption{Lattice constants of 3C-SiC with respect to the cut-off energy used for the plane-wave basis set.}
\label{fig:simulation:lc_ce}
\end{figure}
-Fig. \ref{fig:simulation:lc_ce} shows the respective lattice constants of the relaxed 3C-SiC structure with respect to the cut-off energy.
+Fig.~\ref{fig:simulation:lc_ce} shows the respective lattice constants of the relaxed 3C-SiC structure with respect to the cut-off energy.
As can be seen, convergence is reached already for low energies.
Obviously, an energy cut-off of \unit[300]{eV}, although the minimum acceptable, is sufficient for the plane-wave expansion.
\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.
-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}.
+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.
+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]
\begin{center}
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.
+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.
Thus, the PAW \& LDA combination is not pursued.
Since the lattice constant and cohesive energy of 3C-SiC calculated by the PAW method using the GGA are not improved compared to the ultra-soft pseudopotential calculations using the same XC functional, this concept is likewise stopped.
To conclude, the combination of ultra-soft pseudopotentials and the GGA XC functional are considered the optimal choice for the present study.
The quality of the integration algorithm and the occupied time step is determined by the ability to conserve the total energy.
Therefore, 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}.
+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]
\begin{center}
\includegraphics[width=0.7\textwidth]{verlet_e.ps}
\caption{Evolution of the total energy of 3C-SiC in the $NVE$ ensemble for two different initial temperatures.}
\label{fig:simulation:verlet_e}
\end{figure}
-The evolution of the total energy is displayed in Fig. \ref{fig:simulation:verlet_e}.
+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.
\caption[Radial distribution of a 3C-SiC precipitate embedded in c-Si at $20\,^{\circ}\mathrm{C}$.]{Radial distribution of a 3C-SiC precipitate embedded in c-Si at \unit[20]{$^{\circ}$C}. The Si-Si radial distribution of plain c-Si is plotted for comparison. Green arrows mark bumps in the Si-Si distribution of the precipitate configuration, which do not exist in plain c-Si.}
\label{fig:simulation:pc_sic-prec}
\end{figure}
-Fig. \ref{fig:simulation:pc_sic-prec} shows the radial distribution of the obtained precipitate configuration.
+Fig.~\ref{fig:simulation:pc_sic-prec} shows the radial distribution of the obtained precipitate configuration.
The Si-Si radial distribution for both, plain c-Si and the precipitate configuration show a maximum at a distance of \unit[0.235]{nm}, which is the distance of next neighbored Si atoms in c-Si.
Although no significant change of the lattice constant of the surrounding c-Si matrix was assumed, surprisingly, there is no change at all within observational accuracy.
Looking closer at higher order Si-Si peaks might even allow the guess of a slight increase of the lattice constant compared to the plain c-Si structure.
An interfacial energy of \unit[2267.28]{eV} is obtained.
The amount of C atoms together with the observed lattice constant of the precipitate leads to a precipitate radius of \unit[29.93]{\AA}.
Thus, the interface tension, given by the energy of the interface divided by the surface area of the precipitate is \unit[20.15]{eV/nm$^2$} or \unit[$3.23\times 10^{-4}$]{J/cm$^2$}.
-This value perfectly fits within the experimentally estimated range of \unit[$2-8\times10^{-4}$]{J/cm$^2$} \cite{taylor93}.
+This value perfectly fits within the experimentally estimated range of \unit[2--8$\times10^{-4}$]{J/cm$^2$} \cite{taylor93}.
Thus, the EA potential is considered an appropriate choice for the current study concerning the accurate description of the energetics of interfaces.
Furthermore, since the calculated interfacial energy is located in the lower part of the experimental range, the obtained interface structure might resemble an authentic configuration of an energetically favorable interface structure of a 3C-SiC precipitate in c-Si.
%Since the precipitate configuration is artificially constructed, the resulting interface does not necessarily correspond to the energetically most favorable configuration or to the configuration that is expected for an actually grown precipitate.
%Thus, annealing steps are appended to the gained structure in order to allow for a rearrangement of the atoms of the interface.
%The precipitate structure is rapidly heated up to \unit[2050]{$^{\circ}$C} with a heating rate of approximately \unit[75]{$^{\circ}$C/ps}.
-%From that point on the heating rate is reduced to \unit[1]{$^{\circ}$C/ps} and heating is continued upto \unit[120]{\%} of the Si melting temperature of the potential, i.e. \unit[2940]{K}.
+%From that point on the heating rate is reduced to \unit[1]{$^{\circ}$C/ps} and heating is continued upto \unit[120]{\%} of the Si melting temperature of the potential, i.e.\ \unit[2940]{K}.
%\begin{figure}[t]
%\begin{center}
%\includegraphics[width=0.7\textwidth]{fe_and_t_sic.ps}
%\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}.
%\caption{Cross section image of the precipitate configuration gained by annealing simulations of the constructed 3C-SiC precipitate in c-Si at \unit[200]{ps} (top left), \unit[520]{ps} (top right) and \unit[720]{ps} (bottom).}
%\label{fig:simulation:sic_melt}
%\end{figure}
-%Fig. \ref{fig:simulation:sic_melt} shows cross section images of the atomic structures at different times and temperatures.
+%Fig.~\ref{fig:simulation:sic_melt} shows cross section images of the atomic structures at different times and temperatures.
%As can be seen from the image at \unit[520]{ps} melting of the Si surrounding in fact starts in the defective interface region of the 3C-SiC precipitate and the c-Si surrounding propagating outwards until the whole Si matrix is affected at \unit[720]{ps}.
%As predicted from the radial distribution data the precipitate itself indeed remains stable.
%
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