\chapter{Details of simulation parameters and test calculations}
\label{chapter:simulation}
-All calculations are carried out utilizing the supercell approach, which means that the simulation cell contains a multiple of unti cells and periodic boundary conditions are imposed on the boundaries of that simulation cell.
+All calculations are carried out utilizing the supercell approach, which means that the simulation cell contains a multiple of unit cells and periodic boundary conditions are imposed on the boundaries of that simulation cell.
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.
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 electronic ground state is calculated by an interative Davidson scheme \cite{davidson75} until the difference in total energy of two subsequent iterations is below \unit[$10^{-4}$]{eV}.
+The electronic ground state is calculated by an iterative Davidson scheme \cite{davidson75} until the difference in total energy of two subsequent iterations is below \unit[$10^{-4}$]{eV}.
Defect structures and the migration paths have been modeled in cubic supercells of type 3 containing 216 Si atoms.
-The conjugate gradiant algorithm is used for ionic relaxation.
+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 Parinello and Rahman \cite{parrinello81} in order to realize constant pressure simulations.
+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.
-In MD simulations the equations of motion are integrated by a fourth order predictor corrector algorithm for a timestep of \unit[1]{fs}.
+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.
\hline
\end{tabular}
\end{center}
-\caption[Equilibrium lattice constants and cohesive energies of fully relaxed structures of Si, C (diamond) and 3C-SiC for different potentials and XC functionals.]{Equilibrium lattice constants and cohesive energies of fully relaxed structures of Si, C (diamond) and 3C-SiC for different potentials (ultr-soft PP and PAW) and XC functionals (LDA and GGA). Deviations of the respective values from experimental values are given. Values are in good (green), fair (orange) and poor (red) agreement.}
+\caption[Equilibrium lattice constants and cohesive energies of fully relaxed structures of Si, C (diamond) and 3C-SiC for different potentials and XC functionals.]{Equilibrium lattice constants and cohesive energies of fully relaxed structures of Si, C (diamond) and 3C-SiC for different potentials (ultra-soft PP and PAW) and XC functionals (LDA and GGA). Deviations of the respective values from experimental values are given. Values are in good (green), fair (orange) and poor (red) agreement.}
\label{table:simulation:potxc}
\end{table}
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.
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 ultr-soft pseudopotentials and the GGA XC functional are considered the optimal choice for the present study.
+To conclude, the combination of ultra-soft pseudopotentials and the GGA XC functional are considered the optimal choice for the present study.
\subsection{Lattice constants and cohesive energies}
\end{table}
Table \ref{table:simulation:paramf} shows the respective results and deviations from experiment.
A nice agreement with experimental results is achieved.
-Clearly, a competent parameter set is found, which is capabale of describing the C/Si system by {\em ab initio} calculations.
+Clearly, a competent parameter set is found, which is capable of describing the C/Si system by {\em ab initio} calculations.
% todo - ref for experimental values!
\label{section:classpotmd}
The classical potential MD method is much less computationally costly compared to the highly accurate quantum-mechanical method.
-Thus, the method is capable of performing structural optimizations on large systems and MD calulations may be used to model a system over long time scales.
+Thus, the method is capable of performing structural optimizations on large systems and MD calculations may be used to model a system over long time scales.
Defect structures are modeled in a cubic supercell (type 3) of nine Si lattice constants in each direction containing 5832 Si atoms.
Reproducing the SiC precipitation was attempted in cubic c-Si supercells, which have a size up to 31 Si unit cells in each direction consisting of 238328 Si atoms.
A Tersoff-like bond order potential by Erhart and Albe (EA) \cite{albe_sic_pot} is used to describe the atomic interaction.
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.
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 minimazation driving the system into a local minimum, if the temperature is set to zero Kelvin.
+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.
\label{fig:simulation:pc_sic-prec}
\end{figure}
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 neighboured Si atoms in c-Si.
+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.
A new Si-Si peak arises at \unit[0.307]{nm}, which is identical to the peak of the C-C distribution around that value.
The bumps of the Si-Si distribution at higher distances marked by the green arrows can be explained in the same manner.
They correspond to the fourth and sixth next neighbor distance in 3C-SiC.
It is easily identifiable how these C-C peaks, which imply Si pairs at same distances inside the precipitate, contribute to the bumps observed in the Si-Si distribution.
-The Si-Si and C-C peak at \unit[0.307]{nm} enables the determination of the lattic constant of the embedded 3C-SiC precipitate.
+The Si-Si and C-C peak at \unit[0.307]{nm} enables the determination of the lattice constant of the embedded 3C-SiC precipitate.
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.
\end{equation}
with the notation used in Table \ref{table:simulation:sic_prec}.
Here, $a_{\text{c-Si prec}}$ denotes the lattice constant of the surrounding crystalline Si and $a_{\text{3C-SiC prec}}$ is the lattice constant of the precipitate.
-The lattice constant of plain c-Si at \unit[20]{$^{\circ}$C} can be determined more accurately by the side lengthes of the simulation box of an equilibrated structure instead of using the radial distribution data.
+The lattice constant of plain c-Si at \unit[20]{$^{\circ}$C} can be determined more accurately by the side lengths of the simulation box of an equilibrated structure instead of using the radial distribution data.
By this, a value of $a_{\text{plain c-Si}}=5.439\,\text{\AA}$ is obtained.
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]{\%}.
where $E$ is the total energy of the precipitate configuration at zero temperature.
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 devided 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 eperimentally estimated range of \unit[$2-8\times10^{-4}$]{J/cm$^2$} \cite{taylor93}.
+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}.
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.
\subsubsection{Stability of the precipitate}
-To investigate the stability of the precipiate, which is assumed to be stable even at temperatures above the Si melting temperature, the configuration is heated up beyond the critical point, at which the Si melting transition occurs.
+To investigate the stability of the precipitate, which is assumed to be stable even at temperatures above the Si melting temperature, the configuration is heated up beyond the critical point, at which the Si melting transition occurs.
For this, the transition point of c-Si needs to be evaluated first.
According to the authors of the potential, the Si melting point is \degk{2450}.
-However, melting is not predicted to occur instantly after exceeding the melting point due to the additionally required transition enthalpy and hysteresis behaviour.
+However, melting is not predicted to occur instantly after exceeding the melting point due to the additionally required transition enthalpy and hysteresis behavior.
To determine the transition point, c-Si is heated up using a heating rate of \unit[1]{$^{\circ}$C/ps}.
\begin{figure}[tp]
\begin{center}
\label{fig:simulation:fe_and_t}
\end{figure}
Fig.~\ref{fig:simulation:fe_and_t} shows the total energy and temperature evolution in the region around the transition temperature.
-Indeed, a transition and the accompanied critical behaviour of the total energy is first observed at approximately \degk{3125}, which corresponds to \unit[128]{\%} of the Si melting temperature.
+Indeed, a transition and the accompanied critical behavior of the total energy is first observed at approximately \degk{3125}, which corresponds to \unit[128]{\%} of the Si melting temperature.
The difference in total energy is \unit[0.58]{eV} per atom corresponding to \unit[55.7]{kJ/mole}, which compares quite well to the Si enthalpy of melting of \unit[50.2]{kJ/mole}.
The precipitate structure is heated up using the same heating rate.
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