Enabling the investigation of the evolution of structure on the atomic scale, molecular dynamics (MD) simulations are chosen for modeling the behavior and precipitation of C introduced into an initially crystalline Si environment.
To be able to model systems with a large amount of atoms computational efficient classical potentials to describe the interaction of the atoms are most often used in MD studies.
For reasons of flexibility in executing this non-standard task and in order to be able to use a novel interaction potential~\cite{albe_sic_pot} an appropriate MD code called \textsc{posic}\footnote{\textsc{posic} is an abbreviation for {\bf p}recipitation {\bf o}f {\bf SiC}} including a library collecting respective MD subroutines was developed from scratch\footnote{Source code: http://www.physik.uni-augsburg.de/\~{}zirkelfr/posic}.
-The basic ideas of MD in general and the adopted techniques as implemented in \textsc{posic} in particular are outlined in section \ref{section:md}, while the functional form and derivative of the employed classical potential is presented in appendix \ref{app:d_tersoff}.
-An overview of the most important tools within the MD package is given in appendix \ref{app:code}.
+The basic ideas of MD in general and the adopted techniques as implemented in \textsc{posic} in particular are outlined in section~\ref{section:md}, while the functional form and derivative of the employed classical potential is presented in appendix~\ref{app:d_tersoff}.
+An overview of the most important tools within the MD package is given in appendix~\ref{app:code}.
Although classical potentials are often most successful and at the same time computationally efficient in calculating some physical properties of a particular system, not all of its properties might be described correctly due to the lack of quantum-mechanical effects.
Thus, in order to obtain more accurate results quantum-mechanical calculations from first principles based on density functional theory (DFT) were performed.
The Vienna {\em ab initio} simulation package (\textsc{vasp})~\cite{kresse96} is used for this purpose.
-The relevant basics of DFT are described in section \ref{section:dft} while an overview of utilities mainly used to create input or parse output data of \textsc{vasp} is given in appendix \ref{app:code}.
+The relevant basics of DFT are described in section~\ref{section:dft} while an overview of utilities mainly used to create input or parse output data of \textsc{vasp} is given in appendix~\ref{app:code}.
The gain in accuracy achieved by this method, however, is accompanied by an increase in computational effort constraining the simulated system to be much smaller in size.
Thus, investigations based on DFT are restricted to single defects or combinations of two defects in a rather small Si supercell, their structural relaxation as well as some selected diffusion processes.
-Next to the structure, defects can be characterized by the defect formation energy, a scalar indicating the costs necessary for the formation of the defect, which is explained in section \ref{section:basics:defects}.
-The method used to investigate migration pathways to identify the prevalent diffusion mechanism is introduced in section \ref{section:basics:migration} and modifications to the \textsc{vasp} code implementing this method are presented in appendix \ref{app:patch_vasp}.
+Next to the structure, defects can be characterized by the defect formation energy, a scalar indicating the costs necessary for the formation of the defect, which is explained in section~\ref{section:basics:defects}.
+The method used to investigate migration pathways to identify the prevalent diffusion mechanism is introduced in section~\ref{section:basics:migration} and modifications to the \textsc{vasp} code implementing this method are presented in appendix~\ref{app:patch_vasp}.
\section{Molecular dynamics simulations}
\label{section:md}
Three ingredients are required for a MD simulation:
\begin{enumerate}
\item A model for the interaction between system constituents is needed.
- Interaction potentials and their accuracy for describing certain systems of elements will be outlined in section \ref{subsection:interact_pot}.
+ Interaction potentials and their accuracy for describing certain systems of elements will be outlined in section~\ref{subsection:interact_pot}.
\item An integrator is needed, which propagates the particle positions and velocities from time $t$ to $t+\delta t$, realized by a finite difference scheme which moves trajectories discretely in time.
- This is explained in section \ref{subsection:integrate_algo}.
+ This is explained in section~\ref{subsection:integrate_algo}.
\item A statistical ensemble has to be chosen, which allows certain thermodynamic quantities to be controlled or to stay constant.
- This is discussed in section \ref{subsection:statistical_ensembles}.
+ This is discussed in section~\ref{subsection:statistical_ensembles}.
\end{enumerate}
These ingredients will be outlined in the following.
The discussion is restricted to methods employed within this study.
\begin{equation}
F^i = - \nabla_{{\bf r}_i} E \textrm{ .}
\end{equation}
-Details of the Tersoff potential derivative are presented in appendix \ref{app:d_tersoff}.
+Details of the Tersoff potential derivative are presented in appendix~\ref{app:d_tersoff}.
\subsubsection{Improved analytical bond order potential}
For this reason, Erhart and Albe provide a reparametrization of the Tersoff potential based on three independently fitted parameter sets for the Si-Si, C-C and Si-C interaction~\cite{albe_sic_pot}.
The functional form is similar to the one proposed by Tersoff.
-Differences in the energy functional and the force evaluation routine are pointed out in appendix \ref{app:d_tersoff}.
+Differences in the energy functional and the force evaluation routine are pointed out in appendix~\ref{app:d_tersoff}.
Concerning Si the elastic properties of the diamond phase as well as the structure and energetics of the dimer are reproduced very well.
The new parameter set for the C-C interaction yields improved dimer properties while at the same time delivers a description of the bulk phase similar to the Tersoff potential.
The potential succeeds in the description of the low as well as high coordinated structures.
Structures of maximum configurational energy do not necessarily constitute saddle point configurations, i.e.\ the method does not guarantee to find the true minimum energy path.
Whether a saddle point configuration and, thus, the minimum energy path is obtained by the CRT method, needs to be verified by calculating the respective vibrational modes.
-Modifications used to add the CRT feature to the \textsc{vasp} code and a short instruction on how to use it can be found in appendix \ref{app:patch_vasp}.
+Modifications used to add the CRT feature to the \textsc{vasp} code and a short instruction on how to use it can be found in appendix~\ref{app:patch_vasp}.
% todo - advantages of pw basis concenring hf forces
\end{equation}
By this means values of 2.973 nm and 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:md:sic_prec}.
+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:md:sic_prec}.
\begin{table}[!ht]
\begin{center}
\begin{tabular}{l c c c c}
\caption[Radial distribution of a 3C-SiC precipitate embeeded in c-Si at $20\,^{\circ}\mathrm{C}$.]{Radial distribution of a 3C-SiC precipitate embeeded in c-Si at $20\,^{\circ}\mathrm{C}$. 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:md:pc_sic-prec}
\end{figure}
-Figure \ref{fig:md:pc_sic-prec} shows the radial distribution of the obtained precipitate configuration.
+Figure~\ref{fig:md: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 0.235 nm, which is the distance of next neighboured 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.
\frac{N^{\text{3C-SiC}}_{\text{Si}}}{4/a_{\text{3C-SiC of precipitate configuration}}}}
{\frac{N^{\text{total}}_{\text{Si}}}{8/a_{\text{plain c-Si}}}}
\end{equation}
-with the notation used in table \ref{table:md:sic_prec}.
+with the notation used in table~\ref{table:md:sic_prec}.
The lattice constant of plain c-Si at $20\,^{\circ}\mathrm{C}$ can be determined more accurately by the side lengthes of the simulation box of an equlibrated 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 $a_{\text{c-Si of precipitate configuration}}$ since peaks in the radial distribution match the ones of plain c-Si.
In the following the 3C-SiC/c-Si interface is described in further detail.
One important size analyzing the interface is the interfacial energy.
It is determined exactly in the same way than the formation energy as described in equation \eqref{eq:defects:ef2}.
-Using the notation of table \ref{table:md:sic_prec} and assuming that the system is composed out of $N^{\text{3C-SiC}}_{\text{C}}$ C atoms forming the SiC compound plus the remaining Si atoms, the energy is given by
+Using the notation of table~\ref{table:md:sic_prec} and assuming that the system is composed out of $N^{\text{3C-SiC}}_{\text{C}}$ C atoms forming the SiC compound plus the remaining Si atoms, the energy is given by
\begin{equation}
E_{\text{f}}=E-
N^{\text{3C-SiC}}_{\text{C}} \mu_{\text{SiC}}-
\caption{Free energy and temperature evolution of a constructed 3C-SiC precipitate embedded in c-Si at temperatures above the Si melting point.}
\label{fig:md:fe_and_t_sic}
\end{figure}
-Figure \ref{fig:md:fe_and_t_sic} shows the free energy and temperature evolution.
+Figure~\ref{fig:md:fe_and_t_sic} shows the free energy and temperature evolution.
The sudden increase of the free energy indicates possible melting occuring around 2840 K.
\begin{figure}[!ht]
\begin{center}
\caption{Radial distribution of the constructed 3C-SiC precipitate embedded in c-Si at temperatures below and above the Si melting transition point.}
\label{fig:md:pc_500-fin}
\end{figure}
-Investigating the radial distribution function shown in figure \ref{fig:md:pc_500-fin}, which shows configurations below and above the temperature of the estimated transition, indeed supports the assumption of melting gained by the free energy plot.
+Investigating the radial distribution function shown in figure~\ref{fig:md:pc_500-fin}, which shows configurations below and above the temperature of the estimated transition, indeed supports the assumption of melting gained by the free energy plot.
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 is expected at temperatures around 3125 K, as discussed in section \ref{subsection:md:tval}.
+This is surprising since the melting transition of plain c-Si is expected at temperatures around 3125 K, as discussed in section~\ref{subsection:md:tval}.
Obviously the precipitate lowers the transition point of the surrounding c-Si matrix.
This is indeed verified by visualizing the atomic data.
% ./visualize -w 640 -h 480 -d saves/sic_prec_120Tm_cnt1 -nll -11.56 -0.56 -11.56 -fur 11.56 0.56 11.56 -c -0.2 -24.0 0.6 -L 0 0 0.2 -r 0.6 -B 0.1
\caption{Cross section image of atomic data gained by annealing simulations of the constructed 3C-SiC precipitate in c-Si at 200 ps (top left), 520 ps (top right) and 720 ps (bottom).}
\label{fig:md:sic_melt}
\end{figure}
-Figure \ref{fig:md:sic_melt} shows cross section images of the atomic structures at different times and temperatures.
+Figure~\ref{fig:md:sic_melt} shows cross section images of the atomic structures at different times and temperatures.
As can be seen from the image at 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 720 ps.
As predicted from the radial distribution data the precipitate itself remains stable.
0, & r_{ij} > S_{ij}
\end{array} \right.
\end{equation}
-with $\theta_{ijk}$ being the bond angle between bonds $ij$ and $ik$ as shown in Figure \ref{img:tersoff_angle}.\\
+with $\theta_{ijk}$ being the bond angle between bonds $ij$ and $ik$ as shown in Figure~\ref{img:tersoff_angle}.\\
\\
For a three body potential, if $V_{ij}$ is not equal to $V_{ji}$, the derivative is of the form
\begin{equation}
\chapter{Point defects in silicon}
\label{chapter:defects}
-Regarding the supposed conversion mechanisms of SiC in c-Si as introduced in section \ref{section:assumed_prec} the understanding of C and Si interstitial point defects in c-Si is of fundamental interest.
+Regarding the supposed conversion mechanisms of SiC in c-Si as introduced in section~\ref{section:assumed_prec} the understanding of C and Si interstitial point defects in c-Si is of fundamental interest.
During implantation, defects such as vacancies (V), substitutional C (C$_{\text{s}}$), interstitial C (C$_{\text{i}}$) and Si self-interstitials (Si$_{\text{i}}$) are created, which are believed to play a decisive role in the precipitation process.
In the following, these defects are systematically examined by computationally efficient, classical potential as well as highly accurate DFT calculations with the parameters and simulation conditions that are defined in chapter~\ref{chapter:simulation}.
Both methods are used to investigate selected diffusion processes of some of the defect configurations.
\section{Silicon self-interstitials}
-For investigating the \si{} structures a Si atom is inserted or removed according to Fig.~\ref{fig:basics:ins_pos} of section \ref{section:basics:defects}.
-The formation energies of \si{} configurations are listed in Table \ref{tab:defects:si_self} for both methods used in this work as well as results obtained by other {\em ab initio} studies~\cite{al-mushadani03,leung99}.
+For investigating the \si{} structures a Si atom is inserted or removed according to Fig.~\ref{fig:basics:ins_pos} of section~\ref{section:basics:defects}.
+The formation energies of \si{} configurations are listed in Table~\ref{tab:defects:si_self} for both methods used in this work as well as results obtained by other {\em ab initio} studies~\cite{al-mushadani03,leung99}.
\bibpunct{}{}{,}{n}{}{}
\begin{table}[tp]
\begin{center}
\subsection{Defect structures in a nutshell}
-For investigating the \ci{} structures a C atom is inserted or removed according to Fig.~\ref{fig:basics:ins_pos} of section \ref{section:basics:defects}.
-Formation energies of the most common C point defects in crystalline Si are summarized in Table \ref{tab:defects:c_ints}.
+For investigating the \ci{} structures a C atom is inserted or removed according to Fig.~\ref{fig:basics:ins_pos} of section~\ref{section:basics:defects}.
+Formation energies of the most common C point defects in crystalline Si are summarized in Table~\ref{tab:defects:c_ints}.
The relaxed configurations are visualized in Fig.~\ref{fig:defects:c_conf}.
Again, the displayed structures are the results obtained by the classical potential calculations.
The type of reservoir of the C impurity to determine the formation energy of the defect is chosen to be SiC.
This is consistent with the methods used in the articles~\cite{tersoff90,dal_pino93}, which the results are compared to in the following.
-Hence, the chemical potential of Si and C is determined by the cohesive energy of Si and SiC as discussed in section \ref{section:basics:defects}.
+Hence, the chemical potential of Si and C is determined by the cohesive energy of Si and SiC as discussed in section~\ref{section:basics:defects}.
\begin{table}[tp]
\begin{center}
\begin{tabular}{l c c c c c c}
Due to the high formation energy of the BC defect resulting in a low probability of occurrence of this defect, the wrong description is not posing a serious limitation of the EA potential.
Tersoff indeed predicts a metastable BC configuration.
However, it is not in the correct order and lower in energy than the \ci{} \hkl<1 1 0> DB.
-Quantum-mechanical results of this configuration are discussed in more detail in section \ref{subsection:bc}.
+Quantum-mechanical results of this configuration are discussed in more detail in section~\ref{subsection:bc}.
In another {\em ab initio} study, Capaz~et~al.~\cite{capaz94} in turn found the BC configuration to be an intermediate saddle point structure of a possible migration path, which is \unit[2.1]{eV} higher than the \ci{} \hkl<1 0 0> DB structure.
This is assumed to be due to the neglecting of the electron spin in these calculations.
Another \textsc{vasp} calculation without fully accounting for the electron spin results in the smearing of a single electron over two non-degenerate states for the BC configuration.
This problem is resolved by spin polarized calculations resulting in a net spin of one accompanied by a reduction of the total energy by \unit[0.3]{eV} and the transformation into a metastable local minimum configuration.
It is worth to note that all other listed configurations are not affected by spin polarization.
However, in calculations performed in this work, which fully account for the spin of the electrons, the BC configuration in fact is a real local minimum and an energy barrier is needed to reach this configuration starting from the \ci{} \hkl<1 0 0> DB configuration.
-This is discussed in more detail in section \ref{subsection:100mig}.
+This is discussed in more detail in section~\ref{subsection:100mig}.
To conclude, discrepancies between the results from classical potential calculations and those obtained from first principles are observed.
Within the classical potentials EA outperforms Tersoff and is, therefore, used for further studies.
As the \ci{} \hkl<1 0 0> DB constitutes the ground-state configuration of a C atom incorporated into otherwise perfect c-Si it is the most probable and, hence, one of the most important interstitial configurations of C in Si.
The structure was initially suspected by IR local vibrational mode absorption~\cite{bean70} and finally verified by electron paramagnetic resonance (EPR)~\cite{watkins76} studies on irradiated Si substrates at low temperatures.
-Fig.~\ref{fig:defects:100db_cmp} schematically shows the \ci{} \hkl<1 0 0> DB structure and Table \ref{tab:defects:100db_cmp} lists the details of the atomic displacements, distances and bond angles obtained by classical potential and quantum-mechanical calculations.
+Fig.~\ref{fig:defects:100db_cmp} schematically shows the \ci{} \hkl<1 0 0> DB structure and Table~\ref{tab:defects:100db_cmp} lists the details of the atomic displacements, distances and bond angles obtained by classical potential and quantum-mechanical calculations.
For comparison, the obtained structures for both methods are visualized in Fig.~\ref{fig:defects:100db_vis_cmp}.
\begin{figure}[tp]
\begin{center}
\includegraphics[width=12cm]{100-c-si-db_cmp.eps}
\end{center}
-\caption[Sketch of the \ci{} \hkl<1 0 0> dumbbell structure.]{Sketch of the \ci{} \hkl<1 0 0> dumbbell structure. Atomic displacements, distances and bond angles are listed in Table \ref{tab:defects:100db_cmp}.}
+\caption[Sketch of the \ci{} \hkl<1 0 0> dumbbell structure.]{Sketch of the \ci{} \hkl<1 0 0> dumbbell structure. Atomic displacements, distances and bond angles are listed in Table~\ref{tab:defects:100db_cmp}.}
\label{fig:defects:100db_cmp}
\end{figure}%
\begin{table}[tp]
This is in agreement with results of the EA potential simulations, which reveal this configuration to be unstable relaxing into the \ci{} \hkl<1 1 0> configuration.
However, this fact could not be reproduced by spin polarized \textsc{vasp} calculations performed in this work.
Present results suggest this configuration to correspond to a real local minimum.
-In fact, an additional barrier has to be passed to reach this configuration starting from the \ci{} \hkl<1 0 0> interstitial configuration, which is investigated in section \ref{subsection:100mig}.
+In fact, an additional barrier has to be passed to reach this configuration starting from the \ci{} \hkl<1 0 0> interstitial configuration, which is investigated in section~\ref{subsection:100mig}.
After slightly displacing the C atom along the \hkl[1 0 0] (equivalent to a displacement along \hkl[0 1 0]), \hkl[0 0 1], \hkl[0 0 -1] and \hkl[1 -1 0] direction the distorted structures relax back into the BC configuration.
As will be shown in subsequent migration simulations the same would happen to structures where the C atom is displaced along the migration direction, which approximately is the \hkl[1 1 0] direction.
These relaxations indicate that the BC configuration is a real local minimum instead of an assumed saddle point configuration.
%These modifications to the usual procedure are applied to avoid abrupt changes in structure and free energy on the one hand and to make sure the expected final configuration is reached on the other hand.
%Due to applying updated constraints on all atoms the obtained migration barriers and pathes might be overestimated and misguided.
%To reinforce the applicability of the employed technique the obtained activation energies and migration pathes for the \hkl<0 0 -1> to \hkl<0 -1 0> transition are compared to two further migration calculations, which do not update the constrainted direction and which only apply updated constraints on three selected atoms, that is the diffusing C atom and the Si dumbbell pair in the initial and final configuration.
-%Results are presented in figure \ref{fig:defects:00-1_0-10_cmp}.
+%Results are presented in figure~\ref{fig:defects:00-1_0-10_cmp}.
%\begin{figure}[tp]
%\begin{center}
%\includegraphics[width=13cm]{vasp_mig/00-1_0-10_nosym_sp_cmp.ps}
\caption{Reorientation barrier of the \ci{} \hkl[0 0 -1] to \hkl[0 -1 0] DB transition in place using the classical EA potential.}
\label{fig:defects:cp_00-1_ip0-10_mig}
\end{figure}
-Figures \ref{fig:defects:cp_00-1_0-10_mig} and \ref{fig:defects:cp_00-1_ip0-10_mig} show the migration barriers of the \ci{} \hkl[0 0 -1] to \hkl[0 -1 0] DB transition.
+Figures~\ref{fig:defects:cp_00-1_0-10_mig} and~\ref{fig:defects:cp_00-1_ip0-10_mig} show the migration barriers of the \ci{} \hkl[0 0 -1] to \hkl[0 -1 0] DB transition.
In the first case, the transition involves a change in the lattice site of the C atom whereas in the second case, a reorientation at the same lattice site takes place.
In the first case, the pathways for the two different time constants look similar.
A local minimum exists in between two peaks of the graph.
The probability of already rare diffusion events is further decreased for this reason.
However, agglomeration of C and diffusion of Si self-interstitials are an important part of the proposed SiC precipitation mechanism.
Thus, a serious limitation that has to be taken into account for appropriately modeling the C/Si system using the otherwise quite promising EA potential is revealed.
-Possible workarounds are discussed in more detail in section \ref{section:md:limit}.
+Possible workarounds are discussed in more detail in section~\ref{section:md:limit}.
\section{Combination of point defects and related diffusion processes}
\end{figure}
Fig.~\ref{fig:defects:combos} schematically displays the initial \ci{} \hkl[0 0 -1] DB structure (Fig.~\ref{fig:defects:combos_ci}) as well as the lattice site chosen for the initial \si{} \hkl<1 1 0> DB (Fig.~\ref{fig:defects:combos_si}) and various positions for the second defect (1--5) that are used for investigating defect pairs.
The color of the number denotes the amount of possible atoms for the second defect resulting in equivalent configurations.
-Binding energies of the defect pair are determined by equation \ref{eq:basics:e_bind}.
+Binding energies of the defect pair are determined by equation~\ref{eq:basics:e_bind}.
Next to formation and binding energies, migration barriers are investigated, which allow to draw conclusions on the probability of the formation of such defect complexes by thermally activated diffusion processes.
\subsection[Pairs of \ci{} \hkl<1 0 0>-type interstitials]{\boldmath Pairs of \ci{} \hkl<1 0 0>-type interstitials}
The bond of Si atoms 1 and 2 does not persist.
Instead, the Si atom forms a bond with the initial \ci{} and the second C atom forms a bond with Si atom 1 forming four bonds in total.
The C atoms are spaced by \unit[3.14]{\AA}, which is very close to the expected C-C next neighbor distance of \unit[3.08]{\AA} in SiC.
-Figure \ref{fig:defects:205} displays the results of a \hkl[0 0 1] DB inserted at position 3.
+Figure~\ref{fig:defects:205} displays the results of a \hkl[0 0 1] DB inserted at position 3.
The binding energy is \unit[-2.05]{eV}.
Both DBs are tilted along the same direction remaining aligned in parallel and the second DB is pushed downwards in such a way, that the four DB atoms form a rhomboid.
Both C atoms form tetrahedral bonds to four Si atoms.
\end{figure}
Energetically beneficial configurations of defect combinations are observed for interstitials of all orientations placed at position 5, a position two bonds away from the initial interstitial along the \hkl[1 1 0] direction.
Relaxed structures of these combinations are displayed in Fig.~\ref{fig:defects:comb_db_03}.
-Fig.~\ref{fig:defects:153} and \ref{fig:defects:166} show the relaxed structures of \hkl[0 0 1] and \hkl[0 0 -1] DBs.
+Fig.~\ref{fig:defects:153} and~\ref{fig:defects:166} show the relaxed structures of \hkl[0 0 1] and \hkl[0 0 -1] DBs.
The upper DB atoms are pushed towards each other forming fourfold coordinated bonds.
While the displacements of the Si atoms in case (b) are symmetric to the \hkl(1 1 0) plane, in case (a) the Si atom of the initial DB is pushed a little further in the direction of the C atom of the second DB than the C atom is pushed towards the Si atom.
The bottom atoms of the DBs remain in threefold coordination.
The symmetric configuration is energetically more favorable ($E_{\text{b}}=-1.66\,\text{eV}$) since the displacements of the atoms is less than in the antiparallel case ($E_{\text{b}}=-1.53\,\text{eV}$).
-In Fig.~\ref{fig:defects:188} and \ref{fig:defects:138} the non-parallel orientations, namely the \hkl[0 -1 0] and \hkl[1 0 0] DBs, are shown.
+In Fig.~\ref{fig:defects:188} and~\ref{fig:defects:138} the non-parallel orientations, namely the \hkl[0 -1 0] and \hkl[1 0 0] DBs, are shown.
Binding energies of \unit[-1.88]{eV} and \unit[-1.38]{eV} are obtained for the relaxed structures.
In both cases the Si atom of the initial interstitial is pulled towards the near by atom of the second DB.
Both atoms form fourfold coordinated bonds to their neighbors.
Table~\ref{tab:defects:c-s} lists the energetic results of \cs{} combinations with the initial \ci{} \hkl[0 0 -1] DB.
For \cs{} located at position 1 and 3, the configurations $\alpha$ and A correspond to the naive relaxation of the structure by substituting the Si atom by a C atom in the initial \ci{} \hkl[0 0 -1] DB structure at positions 1 and 3 respectively.
However, small displacements of the involved atoms near the defect result in different stable structures labeled $\beta$ and B respectively.
-Fig.~\ref{fig:093-095} and \ref{fig:026-128} show structures A, B and $\alpha$, $\beta$ together with the barrier of migration for the A to B and $\alpha$ to $\beta$ transition respectively.
+Fig.~\ref{fig:093-095} and~\ref{fig:026-128} show structures A, B and $\alpha$, $\beta$ together with the barrier of migration for the A to B and $\alpha$ to $\beta$ transition respectively.
% A B
%./visualize_contcar -w 640 -h 480 -d results/c_00-1_c3_csub_B -nll -0.20 -0.4 -0.1 -fur 0.9 0.6 0.9 -c 0.5 -1.5 0.375 -L 0.5 0 0.3 -r 0.6 -A -1 2.465
% old c_int - c_substitutional stuff
-%Figures \ref{fig:defects:comb_db_04} and \ref{fig:defects:comb_db_05} show relaxed structures of substitutional carbon in combination with the \hkl<0 0 -1> dumbbell for several positions.
-%In figure \ref{fig:defects:comb_db_04} positions 1 (a)), 3 (b)) and 5 (c)) are displayed.
+%Figures~\ref{fig:defects:comb_db_04} and~\ref{fig:defects:comb_db_05} show relaxed structures of substitutional carbon in combination with the \hkl<0 0 -1> dumbbell for several positions.
+%In figure~\ref{fig:defects:comb_db_04} positions 1 (a)), 3 (b)) and 5 (c)) are displayed.
%A substituted carbon atom at position 5 results in an energetically extremely unfavorable configuration.
%Both carbon atoms, the substitutional and the dumbbell atom, pull silicon atom number 1 towards their own location regarding the \hkl<1 1 0> direction.
%Due to this a large amount of tensile strain energy is needed, which explains the high positive value of 0.49 eV.
%The lowest binding energy is observed for a substitutional carbon atom inserted at position 3.
%The substitutional carbon atom is located above the dumbbell substituting a silicon atom which would usually be bound to and displaced along \hkl<0 0 1> and \hkl<1 1 0> by the silicon dumbbell atom.
%In contrast to the previous configuration strain compensation occurs resulting in a binding energy as low as -0.93 eV.
-%Substitutional carbon at position 2 and 4, visualized in figure \ref{fig:defects:comb_db_05}, is located below the initial dumbbell.
+%Substitutional carbon at position 2 and 4, visualized in figure~\ref{fig:defects:comb_db_05}, is located below the initial dumbbell.
%Silicon atom number 1, which is bound to the interstitial carbon atom is displaced along \hkl<0 0 -1> and \hkl<-1 -1 0>.
%In case a) only the first displacement is compensated by the substitutional carbon atom.
%This results in a somewhat higher binding energy of -0.51 eV.
%The binding energy gets even higher in case b) ($E_{\text{b}}=-0.15\text{ eV}$), in which the substitutional carbon is located further away from the initial dumbbell.
%In both cases, silicon atom number 1 is displaced in such a way, that the bond to silicon atom number 5 vanishes.
-%In case of \ref{fig:defects:comb_db_04} a) the carbon atoms form a bond with a distance of 1.5 \AA, which is close to the C-C distance expected in diamond or graphit.
+%In case of~\ref{fig:defects:comb_db_04} a) the carbon atoms form a bond with a distance of 1.5 \AA, which is close to the C-C distance expected in diamond or graphit.
%Both carbon atoms are highly attracted by each other resulting in large displacements and high strain energy in the surrounding.
%A binding energy of 0.26 eV is observed.
%Substitutional carbon at positions 2, 3 and 4 are the energetically most favorable configurations and constitute promising starting points for SiC precipitation.
\caption[Relaxed structures of defect combinations obtained by creating a vacancy at positions 2, 3, 4 and 5.]{Relaxed structures of defect combinations obtained by creating a vacancy at positions 2 (a), 3 (b), 4 (c) and 5 (d).}
\label{fig:defects:comb_db_06}
\end{figure}
-Figure \ref{fig:defects:comb_db_06} shows the associated configurations.
+Figure~\ref{fig:defects:comb_db_06} shows the associated configurations.
All investigated structures are preferred compared to isolated, largely separated defects.
In contrast to C$_{\text{s}}$ this is also valid for positions along \hkl[1 1 0] resulting in an entirely attractive interaction between defects of these types.
Even for the largest possible distance (R) achieved in the calculations of the periodic supercell a binding energy as low as \unit[-0.31]{eV} is observed.
Strain reduced by this huge displacement is partially absorbed by tensile strain on Si atom number 1 originating from attractive forces of the C atom and the vacancy.
A binding energy of \unit[-0.50]{eV} is observed.
-The migration pathways of configuration \ref{fig:defects:314} and \ref{fig:defects:059} into the ground-state configuration, i.e.\ the \cs{} configuration, are shown in Fig.~\ref{fig:314-539} and \ref{fig:059-539} respectively.
+The migration pathways of configuration~\ref{fig:defects:314} and~\ref{fig:defects:059} into the ground-state configuration, i.e.\ the \cs{} configuration, are shown in Fig.~\ref{fig:314-539} and~\ref{fig:059-539} respectively.
\begin{figure}[tp]
\begin{center}
\includegraphics[width=0.7\textwidth]{314-539.ps}
This is particularly important since the energy of formation of C$_{\text{s}}$ is drastically underestimated by the EA potential.
A possible occurrence of C$_{\text{s}}$ could then be attributed to a lower energy of formation of the C$_{\text{s}}$-Si$_{\text{i}}$ combination due to the low formation energy of C$_{\text{s}}$, which is obviously wrong.
-Since quantum-mechanical calculations reveal the Si$_{\text{i}}$ \hkl<1 1 0> DB as the ground-state configuration of Si$_{\text{i}}$ in Si, it was assumed to provide the energetically most favorable configuration in combination with C$_{\text{s}}$ in the calculations carried out in section \ref{subsection:si-cs}.
+Since quantum-mechanical calculations reveal the Si$_{\text{i}}$ \hkl<1 1 0> DB as the ground-state configuration of Si$_{\text{i}}$ in Si, it was assumed to provide the energetically most favorable configuration in combination with C$_{\text{s}}$ in the calculations carried out in section~\ref{subsection:si-cs}.
Empirical potentials, however, predict Si$_{\text{i}}$ T to be the energetically most favorable configuration.
Thus, investigations of the relative energies of formation of defect pairs need to include combinations of C$_{\text{s}}$ with Si$_{\text{i}}$ T.
Results of {\em ab initio} and classical potential calculations are summarized in Table~\ref{tab:defect_combos}.
A net magnetization of two electrons, which is already clear by simple molecular orbital theory considerations on the bonding of the $sp$ hybridized C atom, is settled.
By investigating the charge density isosurface it turns out that the two resulting spin up electrons are localized in a torus around the C atom.
With an activation energy of \unit[0.9]{eV} the C$_{\text{i}}$ carbon interstitial can be expected to be highly mobile at prevailing temperatures in the process under investigation, i.e.\ IBS.
-Since the \ci{} \hkl<1 0 0> DB is the ground-state configuration and highly mobile, possible migration of these DBs to form defect agglomerates, as demanded by the model introduced in section \ref{section:assumed_prec}, is considered possible.
+Since the \ci{} \hkl<1 0 0> DB is the ground-state configuration and highly mobile, possible migration of these DBs to form defect agglomerates, as demanded by the model introduced in section~\ref{section:assumed_prec}, is considered possible.
Unfortunately the description of the same processes fails if classical potential methods are used.
Already the geometry of the most stable DB configuration differs considerably from that obtained by first-principles calculations.
The intention of this work is to contribute to the understanding of C in Si by means of atomistic simulations targeted on the task to elucidate the SiC conversion mechanism in silicon.
The outline of this work is as follows:
-In chapter \ref{chapter:sic_rev} a review of the Si/C compound is given, including the very central discussion on two controversial precipitation mechanisms present in literature in section \ref{section:assumed_prec}.
-Chapter \ref{chapter:basics} introduces some basics and internals of the utilized atomistic simulations as well as special methods of application.
-Details of the simulation and associated test calculations are presented in chapter \ref{chapter:simulation}.
-In chapter \ref{chapter:defects} results of investigations of single defect configurations, structures of combinations of two individual defects as well as some selected diffusion pathways in silicon are shown.
+In chapter~\ref{chapter:sic_rev} a review of the Si/C compound is given, including the very central discussion on two controversial precipitation mechanisms present in literature in section~\ref{section:assumed_prec}.
+Chapter~\ref{chapter:basics} introduces some basics and internals of the utilized atomistic simulations as well as special methods of application.
+Details of the simulation and associated test calculations are presented in chapter~\ref{chapter:simulation}.
+In chapter~\ref{chapter:defects} results of investigations of single defect configurations, structures of combinations of two individual defects as well as some selected diffusion pathways in silicon are shown.
These allow to draw conclusions with respect to the SiC precipitation mechanism in Si.
-More complex systems aiming to model the transformation of C incorporated in bulk Si into a SiC nucleus are examined in chapter \ref{chapter:md}.
-Finally, a summary and concluding remarks are given in chapter \ref{chapter:summary}.
+More complex systems aiming to model the transformation of C incorporated in bulk Si into a SiC nucleus are examined in chapter~\ref{chapter:md}.
+Finally, a summary and concluding remarks are given in chapter~\ref{chapter:summary}.
\label{chapter:md}
The molecular dynamics (MD) technique is used to gain insight into the behavior of C existing in different concentrations in c-Si on the microscopic level at finite temperatures.
-The simulations are restricted to classical potential simulations utilizing the analytical EA bond order potential as described in section \ref{subsection:interact_pot}.
-Parameters are chosen according to the discussion in section \ref{section:classpotmd}.
+The simulations are restricted to classical potential simulations utilizing the analytical EA bond order potential as described in section~\ref{subsection:interact_pot}.
+Parameters are chosen according to the discussion in section~\ref{section:classpotmd}.
At the beginning, simulations are performed, which try to mimic the conditions during IBS.
Results reveal limitations of the employed potential and MD in general.
$V_2$ approximately corresponds to the volume of a minimal 3C-SiC precipitate.
$V_3$ is approximately the volume containing the amount of Si atoms necessary to form such a precipitate, which is slightly smaller than $V_2$ due to the slightly lower Si density of 3C-SiC compared to c-Si.
The two latter insertion volumes are considered since no diffusion of C atoms is expected within the simulated period of time at prevalent temperatures.
-This is due to the overestimated activation energy for the diffusion of a \ci \hkl<1 0 0> DB, as pointed out in section \ref{subsection:defects:mig_classical}.
+This is due to the overestimated activation energy for the diffusion of a \ci \hkl<1 0 0> DB, as pointed out in section~\ref{subsection:defects:mig_classical}.
For rectangularly shaped precipitates with side length $L$ the amount of C atoms in 3C-SiC and Si atoms in c-Si is given by
\begin{equation}
N_{\text{C}}^{\text{3C-SiC}} =4 \left( \frac{L}{a_{\text{SiC}}}\right)^3
N_{\text{Si}}^{\text{c-Si}} =8 \left( \frac{L}{a_{\text{Si}}}\right)^3 \text{ .}
\label{eq:md:n_prec}
\end{equation}
-Table \ref{table:md:ins_vols} summarizes the side length of each of the three different insertion volumes determined by equations \eqref{eq:md:n_prec} and the resulting C concentrations inside these volumes.
+Table~\ref{table:md:ins_vols} summarizes the side length of each of the three different insertion volumes determined by equations \eqref{eq:md:n_prec} and the resulting C concentrations inside these volumes.
Looking at the C concentrations, simulations can be distinguished in simulations occupying low ($V_1$) and high ($V_2$, $V_3$) concentrations of C.
\begin{table}[tp]
\begin{center}
In addition to the time scale limitation, problems attributed to the short range potential exist.
The sharp cut-off function, which limits the interacting ions to the next neighbored atoms by gradually pushing the interaction force and energy to zero between the first and second next neighbor distance, is responsible for overestimated and unphysical high forces of next neighbored atoms~\cite{tang95,mattoni2007}.
-This is supported by the overestimated activation energies necessary for C diffusion as investigated in section \ref{subsection:defects:mig_classical}.
+This is supported by the overestimated activation energies necessary for C diffusion as investigated in section~\ref{subsection:defects:mig_classical}.
Indeed, it is not only the strong C-C bond, which is hard to break, inhibiting C diffusion and further rearrangements.
This is also true for the low concentration simulations dominated by the occurrence of C-Si DBs spread over the whole simulation volume.
The bonds of these C-Si pairs are also affected by the cut-off artifact preventing C diffusion and agglomeration of the DBs.
These are overestimated by a factor of 2.4 to 3.5.
Scaling the absolute temperatures accordingly results in maximum temperatures of \unit[1460--2260]{$^{\circ}$C}.
Since melting already occurs shortly below the melting point of the potential (\unit[2450]{K})~\cite{albe_sic_pot} due to the presence of defects, temperatures ranging from \unit[450--2050]{$^{\circ}$C} are used.
-The simulation sequence and other parameters except for the system temperature remain unchanged as in section \ref{section:initial_sims}.
+The simulation sequence and other parameters except for the system temperature remain unchanged as in section~\ref{section:initial_sims}.
Since there is no significant difference among the $V_2$ and $V_3$ simulations only the $V_1$ and $V_2$ simulations are carried on and referred to as low C and high C concentration simulations.
A simple quality value $Q$ is introduced, which helps to estimate the progress of structural evolution.
\includegraphics[width=0.7\textwidth]{tot_pc_thesis.ps}\\
\includegraphics[width=0.7\textwidth]{tot_ba.ps}
\end{center}
-\caption[Si-C radial distribution and evolution of quality $Q$ for the low concentration simulations at different elevated temperatures.]{Si-C radial distribution and evolution of quality $Q$ according to equation \ref{eq:md:qdef} for the low concentration simulations at different elevated temperatures. All structures are cooled down to \degc{20}. The gray line shows resulting Si-C bonds in a configuration of \cs{} in c-Si (C$_\text{sub}$) at zero temperature. Arrows in the quality plot mark the end of C insertion and the start of the cooling down step. A fit function according to equation \eqref{eq:md:fit} shows the estimated evolution of quality in the absence of the cooling down sequence.}
+\caption[Si-C radial distribution and evolution of quality $Q$ for the low concentration simulations at different elevated temperatures.]{Si-C radial distribution and evolution of quality $Q$ according to equation~\ref{eq:md:qdef} for the low concentration simulations at different elevated temperatures. All structures are cooled down to \degc{20}. The gray line shows resulting Si-C bonds in a configuration of \cs{} in c-Si (C$_\text{sub}$) at zero temperature. Arrows in the quality plot mark the end of C insertion and the start of the cooling down step. A fit function according to equation \eqref{eq:md:fit} shows the estimated evolution of quality in the absence of the cooling down sequence.}
\label{fig:md:tot_si-c_q}
\end{figure}
Fig.~\ref{fig:md:tot_si-c_q} shows the radial distribution of Si-C bonds for different temperatures and the corresponding evolution of quality $Q$ as defined above for the low concentration simulation.
In addition, structures form that result in distances residing in between the ones obtained from combinations of mixed defect types and the ones obtained by \cs{} configurations, as can be seen by quite high $g(r)$ values in between the continuous dashed line and the first arrow with a solid line.
For the most part, these structures can be identified as configurations of \cs{} with either another C atom that basically occupies a Si lattice site but is displaced by a \si{} atom residing in the very next surrounding or a C atom that nearly occupies a Si lattice site forming a defect other than the \hkl<1 0 0>-type with the Si atom.
Again, this is a quite promising result since the C atoms are taking the appropriate coordination as expected in 3C-SiC.
-%However, this is contrary to the initial precipitation model proposed in section \ref{section:assumed_prec}, which assumes that the transformation into 3C-SiC takes place in a very last step once enough C-Si DBs agglomerated.
+%However, this is contrary to the initial precipitation model proposed in section~\ref{section:assumed_prec}, which assumes that the transformation into 3C-SiC takes place in a very last step once enough C-Si DBs agglomerated.
To summarize, results of low concentration simulations show a phase transition in conjunction with an increase in temperature.
The \ci{} \hkl<1 0 0> DB dominated structure turns into a structure characterized by the occurrence of an increasing amount of \cs{} with respect to temperature.
As discussed in section~\ref{section:md:limit} and~\ref{section:md:inct} a further increase of the system temperature might help to overcome limitations of the short range potential and accelerate the dynamics involved in structural evolution.
Furthermore, these results indicate that increased temperatures are necessary to drive the system out of equilibrium enabling conditions needed for the formation of a metastable cubic polytype of SiC.
-A maximum temperature to avoid melting is determined in section \ref{section:md:tval} to be 120 \% of the Si melting point but due to defects lowering the transition point a maximum temperature of 95 \% of the Si melting temperature is considered useful.
+A maximum temperature to avoid melting is determined in section~\ref{section:md:tval} to be 120 \% of the Si melting point but due to defects lowering the transition point a maximum temperature of 95 \% of the Si melting temperature is considered useful.
This value is almost equal to the temperature of $2050\,^{\circ}\mathrm{C}$ already used in former simulations.
Since the maximum temperature is reached the approach is reduced to the application of longer time scales.
This is considered useful since the estimated evolution of quality in the absence of the cooling down sequence in figure~\ref{fig:md:tot_si-c_q} predicts an increase in quality and, thus, structural evolution is likely to occur if the simulation is proceeded at maximum temperature.
Defect structures and the migration paths have been modeled in cubic supercells of type 3 containing 216 Si atoms.
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}.
+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.
In MD simulations the equations of motion are integrated by a fourth order predictor corrector algorithm for a time step of \unit[1]{fs}.
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.
\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.
+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.
\caption{Equilibrium lattice constants and cohesive energies of Si, C (diamond) and 3C-SiC using the entire parameter set as determined in the beginning of this section.}
\label{table:simulation:paramf}
\end{table}
-Table \ref{table:simulation:paramf} shows the respective results and deviations from experiment.
+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 capable of describing the C/Si system by {\em ab initio} calculations.
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}.
+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.
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.
\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}.
+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}
\frac{N^{\text{3C-SiC}}_{\text{Si}}}{4/a_{\text{3C-SiC prec}}}}
{\frac{N^{\text{total}}_{\text{Si}}}{8/a_{\text{plain c-Si}}}}
\end{equation}
-with the notation used in Table \ref{table:simulation:sic_prec}.
+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 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.
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.
A good estimate of the interfacial energy should be obtained by utilizing the formula for determining the defect formation energy as described in equation \eqref{eq:basics:ef2}.
-Using the notation of Table \ref{table:simulation:sic_prec} and assuming that the system is composed out of $N^{\text{3C-SiC}}_{\text{C}}$ C atoms forming the SiC compound plus the remaining Si atoms, the energy is given by
+Using the notation of Table~\ref{table:simulation:sic_prec} and assuming that the system is composed out of $N^{\text{3C-SiC}}_{\text{C}}$ C atoms forming the SiC compound plus the remaining Si atoms, the energy is given by
\begin{equation}
E_{\text{f}}=E-
N^{\text{3C-SiC}}_{\text{C}} E_{\text{coh}}^{\text{SiC}}-
%\caption{Total energy and temperature evolution of a 3C-SiC precipitate embedded in c-Si at temperatures above the Si melting point.}
%\label{fig:simulation:fe_and_t_sic}
%\end{figure}
-%Figure \ref{fig:simulation:fe_and_t_sic} shows the total energy and temperature evolution.
+%Figure~\ref{fig:simulation:fe_and_t_sic} shows the total energy and temperature evolution.
%The sudden increase of the total energy indicates possible melting occuring around \unit[2840]{K}.
%\begin{figure}[ht]
%\begin{center}
%\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}.
+%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}.
%Obviously the precipitate lowers the transition point of the surrounding c-Si matrix.
%This is indeed verified by visualizing the atomic data.
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