\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}.
+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]
\caption[Relaxed Si self-interstitial defect configurations obtained by classical potential calculations.]{Relaxed Si self-interstitial defect configurations obtained by classical potential calculations. Si atoms and bonds are illustrated by yellow spheres and blue lines. Bonds of the defect atoms are drawn in red color.}
\label{fig:defects:conf}
\end{figure}
-The final configurations obtained after relaxation are presented in Fig. \ref{fig:defects:conf}.
+The final configurations obtained after relaxation are presented in Fig.~\ref{fig:defects:conf}.
The displayed structures are the results of the classical potential simulations.
There are differences between the various results of the quantum-mechanical calculations but the consensus view is that the \hkl<1 1 0> dumbbell (DB) followed by the hexagonal and tetrahedral defect is the lowest in energy.
The \si{} atom then begins to slowly move towards an energetically more favorable position very close to the tetrahedral one but slightly displaced along the three coordinate axes.
The formation energy of \unit[3.96]{eV} for this type of interstitial is equal to the result for the hexagonal one in the original work \cite{albe_sic_pot}.
Obviously, the authors did not carefully check the relaxed results assuming a hexagonal configuration.
-In Fig. \ref{fig:defects:kin_si_hex} the relaxation process is shown on the basis of the kinetic energy plot.
+In Fig.~\ref{fig:defects:kin_si_hex} the relaxation process is shown on the basis of the kinetic energy plot.
\begin{figure}[tp]
\begin{center}
\includegraphics[width=0.7\textwidth]{e_kin_si_hex.ps}
\caption{Migration barrier of the tetrahedral Si self-interstitial slightly displaced along all three coordinate axes into the exact tetrahedral configuration using classical potential calculations.}
\label{fig:defects:nhex_tet_mig}
\end{figure}
-This is exemplified in Fig. \ref{fig:defects:nhex_tet_mig}, which shows the change in configurational energy during the migration of the interstitial displaced along all three coordinate axes into the tetrahedral configuration.
+This is exemplified in Fig.~\ref{fig:defects:nhex_tet_mig}, which shows the change in configurational energy during the migration of the interstitial displaced along all three coordinate axes into the tetrahedral configuration.
The barrier is smaller than \unit[0.2]{eV}.
Hence, these artifacts have a negligible influence in finite temperature simulations.
\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}.
+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}.
+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.
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.
-For comparison, the obtained structures for both methods are visualized in Fig. \ref{fig:defects:100db_vis_cmp}.
+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}
\hline
\end{tabular}\\[0.5cm]
\end{center}
-\caption[Atomic displacements, distances and bond angles of the \ci{} \hkl<1 0 0> DB structure obtained by \textsc{posic} and {\textsc vasp} calculations.]{Atomic displacements, distances and bond angles of the \ci{} \hkl<1 0 0> DB structure obtained by {\textsc posic} and {\textsc vasp} calculations. The displacements and distances are given in nm and the angles are given in degrees. Displacements, distances and angles are schematically displayed in Fig. \ref{fig:defects:100db_cmp}. In addition, the equilibrium lattice constant for crystalline Si is listed.}
+\caption[Atomic displacements, distances and bond angles of the \ci{} \hkl<1 0 0> DB structure obtained by \textsc{posic} and {\textsc vasp} calculations.]{Atomic displacements, distances and bond angles of the \ci{} \hkl<1 0 0> DB structure obtained by {\textsc posic} and {\textsc vasp} calculations. The displacements and distances are given in nm and the angles are given in degrees. Displacements, distances and angles are schematically displayed in Fig.~\ref{fig:defects:100db_cmp}. In addition, the equilibrium lattice constant for crystalline Si is listed.}
\label{tab:defects:100db_cmp}
\end{table}%
\begin{figure}[tp]
The other two bonds are bonds to the two Si edge atoms located in the opposite direction of the DB atom.
The distance of the two DB atoms is almost the same for both types of calculations.
However, in the case of the \textsc{vasp} calculation, the DB structure is pushed upwards compared to the results using the EA potential.
-This is easily identified by comparing the values for $a$ and $b$ and the two structures in Fig. \ref{fig:defects:100db_vis_cmp}.
+This is easily identified by comparing the values for $a$ and $b$ and the two structures in Fig.~\ref{fig:defects:100db_vis_cmp}.
Thus, the angles of bonds of the Si DB atom ($\theta_1$ and $\theta_2$) are closer to $120^{\circ}$ signifying the predominance of $sp^2$ hybridization.
On the other hand, the C atom forms an almost collinear bond ($\theta_3$) with the two Si edge atoms implying the predominance of $sp$ bonding.
-This is supported by the image of the charge density isosurface in Fig. \ref{img:defects:charge_den_and_ksl}.
+This is supported by the image of the charge density isosurface in Fig.~\ref{img:defects:charge_den_and_ksl}.
The two lower Si atoms are $sp^3$ hybridized and form $\sigma$ bonds to the Si DB atom.
The same is true for the upper two Si atoms and the C DB atom.
In addition the DB atoms form $\pi$ bonds.
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.
-Fig. \ref{img:defects:bc_conf} shows the structure, charge density isosurface and Kohn-Sham levels of the BC configuration.
+Fig.~\ref{img:defects:bc_conf} shows the structure, charge density isosurface and Kohn-Sham levels of the BC configuration.
In fact, the net magnetization of two electrons is already suggested by simple molecular orbital theory considerations with respect to the bonding of the C atom.
The linear bonds of the C atom to the two Si atoms indicate the $sp$ hybridization of the C atom.
Two electrons participate to the linear $\sigma$ bonds with the Si neighbors.
The other two electrons constitute the $2p^2$ orbitals resulting in a net magnetization.
-This is supported by the charge density isosurface and the Kohn-Sham levels in Fig. \ref{img:defects:bc_conf}.
+This is supported by the charge density isosurface and the Kohn-Sham levels in Fig.~\ref{img:defects:bc_conf}.
The blue torus, which reinforces the assumption of the $p$ orbital, illustrates the resulting spin up electron density.
In addition, the energy level diagram shows a net amount of two spin up electrons.
\caption{Conceivable migration pathways among two \ci{} \hkl<1 0 0> DB configurations.}
\label{img:defects:c_mig_path}
\end{figure}
-Three different migration paths are accounted in this work, which are displayed in Fig. \ref{img:defects:c_mig_path}.
+Three different migration paths are accounted in this work, which are displayed in Fig.~\ref{img:defects:c_mig_path}.
The first migration investigated is a transition of a \hkl<0 0 -1> into a \hkl<0 0 1> DB interstitial configuration.
During this migration the C atom is changing its Si DB partner.
The new partner is the one located at $a_{\text{Si}}/4 \hkl<1 1 -1>$ relative to the initial one, where $a_{\text{Si}}$ is the Si lattice constant.
\caption[Migration barrier and structures of the {\hkl[0 0 -1]} DB to BC transition.]{Migration barrier and structures of the \hkl<0 0 -1> DB (left) to BC (right) transition. Bonds of the C atom are illustrated by blue lines.}
\label{fig:defects:00-1_001_mig}
\end{figure}
-In Fig. \ref{fig:defects:00-1_001_mig} results of the \hkl<0 0 -1> to \hkl<0 0 1> migration fully described by the migration of the \hkl<0 0 -1> to the BC configuration is displayed.
+In Fig.~\ref{fig:defects:00-1_001_mig} results of the \hkl<0 0 -1> to \hkl<0 0 1> migration fully described by the migration of the \hkl<0 0 -1> to the BC configuration is displayed.
To reach the BC configuration, which is \unit[0.94]{eV} higher in energy than the \hkl<0 0 -1> DB configuration, an energy barrier of approximately \unit[1.2]{eV} given by the saddle point structure at a displacement of \unit[60]{\%} has to be passed.
This amount of energy is needed to break the bond of the C atom to the Si atom at the bottom left.
In a second process \unit[0.25]{eV} of energy are needed for the system to revert into a \hkl<1 0 0> configuration.
\caption[Migration barrier and structures of the {\hkl[0 0 -1]} DB to the {\hkl[0 -1 0]} DB transition.]{Migration barrier and structures of the \hkl[0 0 -1] DB (left) to the \hkl[0 -1 0] DB (right) transition. Bonds of the C atom are illustrated by blue lines.}
\label{fig:defects:00-1_0-10_mig}
\end{figure}
-Fig. \ref{fig:defects:00-1_0-10_mig} shows the migration barrier and structures of the \ci{} \hkl<0 0 -1> to \hkl<0 -1 0> DB transition.
+Fig.~\ref{fig:defects:00-1_0-10_mig} shows the migration barrier and structures of the \ci{} \hkl<0 0 -1> to \hkl<0 -1 0> DB transition.
The resulting migration barrier of approximately \unit[0.9]{eV} is very close to the experimentally obtained values of \unit[0.70]{eV} \cite{lindner06}, \unit[0.73]{eV} \cite{song90} and \unit[0.87]{eV} \cite{tipping87}.
\begin{figure}[tp]
\caption[Reorientation barrier and structures of the {\hkl[0 0 -1]} DB to the {\hkl[0 -1 0]} DB transition in place.]{Reorientation barrier and structures of the \hkl[0 0 -1] DB (left) to the \hkl[0 -1 0] DB (right) transition in place. Bonds of the carbon atoms are illustrated by blue lines.}
\label{fig:defects:00-1_0-10_ip_mig}
\end{figure}
-The third migration path, in which the DB is changing its orientation, is shown in Fig. \ref{fig:defects:00-1_0-10_ip_mig}.
+The third migration path, in which the DB is changing its orientation, is shown in Fig.~\ref{fig:defects:00-1_0-10_ip_mig}.
An energy barrier of roughly \unit[1.2]{eV} is observed.
Experimentally measured activation energies for reorientation range from \unit[0.77]{eV} to \unit[0.88]{eV} \cite{watkins76,song90}.
Thus, this pathway is more likely to be composed of two consecutive steps of the second path.
Further migration pathways, in particular those occupying other defect configurations than the \hkl<1 0 0>-type either as a transition state or a final or starting configuration, are totally conceivable.
This is investigated in the following in order to find possible migration pathways that have an activation energy lower than the ones found up to now.
The next energetically favorable defect configuration is the \hkl<1 1 0> C-Si DB interstitial.
-Fig. \ref{fig:defects:110_mig_vasp} shows the migration barrier of the \hkl<1 1 0> C-Si DB to the BC, \hkl<0 0 -1> and \hkl<0 -1 0> (in place) transition.
+Fig.~\ref{fig:defects:110_mig_vasp} shows the migration barrier of the \hkl<1 1 0> C-Si DB to the BC, \hkl<0 0 -1> and \hkl<0 -1 0> (in place) transition.
Indeed less than \unit[0.7]{eV} are necessary to turn a \hkl<0 -1 0>- to a \hkl<1 1 0>-type C-Si DB interstitial.
This transition is carried out in place, i.e.\ the Si DB pair is not changed and both, the Si and C atom share the initial lattice site.
Thus, this transition does not contribute to long-range diffusion.
-Once the C atom resides in the \hkl<1 1 0> DB interstitial configuration it can migrate into the BC configuration requiring approximately \unit[0.95]{eV} of activation energy, which is only slightly higher than the activation energy needed for the \hkl<0 0 -1> to \hkl<0 -1 0> pathway as shown in Fig. \ref{fig:defects:00-1_0-10_mig}.
-As already known from the migration of the \hkl<0 0 -1> to the BC configuration discussed in Fig. \ref{fig:defects:00-1_001_mig}, another \unit[0.25]{eV} are needed to turn back from the BC to a \hkl<1 0 0>-type interstitial.
+Once the C atom resides in the \hkl<1 1 0> DB interstitial configuration it can migrate into the BC configuration requiring approximately \unit[0.95]{eV} of activation energy, which is only slightly higher than the activation energy needed for the \hkl<0 0 -1> to \hkl<0 -1 0> pathway as shown in Fig.~\ref{fig:defects:00-1_0-10_mig}.
+As already known from the migration of the \hkl<0 0 -1> to the BC configuration discussed in Fig.~\ref{fig:defects:00-1_001_mig}, another \unit[0.25]{eV} are needed to turn back from the BC to a \hkl<1 0 0>-type interstitial.
However, due to the fact that this migration consists of three single transitions with the second one having an activation energy slightly higher than observed for the direct transition, this sequence of paths is considered very unlikely to occur.
The migration barrier of the \hkl<1 1 0> to \hkl<0 0 -1> transition, in which the C atom is changing its Si partner and, thus, moving to the neighbored lattice site, corresponds to approximately \unit[1.35]{eV}.
During this transition the C atom is escaping the \hkl(1 1 0) plane approaching the final configuration on a curved path.
This barrier is much higher than the ones found previously, which again make this transition very unlikely to occur.
-For this reason, the assumption that C diffusion and reorientation is achieved by transitions of the type presented in Fig. \ref{fig:defects:00-1_0-10_mig} is reinforced.
+For this reason, the assumption that C diffusion and reorientation is achieved by transitions of the type presented in Fig.~\ref{fig:defects:00-1_0-10_mig} is reinforced.
%As mentioned earlier the procedure to obtain the migration barriers differs from the usually applied procedure in two ways.
%Firstly constraints to move along the displacement direction are applied on all atoms instead of solely constraining the diffusing atom.
% red: ./visualize -w 640 -h 480 -d saves/c_in_si_mig_bc_00-1_s20 -nll -0.56 -0.56 -0.7 -fur 0.2 0.2 0.0 -c 0.75 -1.25 -0.25 -L -0.25 -0.25 -0.25 -r 0.6 -B 0.1
% blue: ./visualize -w 640 -h 480 -d saves/c_in_si_mig_bc_00-1_s20_tr100/ -nll -0.56 -0.56 -0.7 -fur 0.2 0.2 0.0 -c 0.0 -0.25 1.0 -L 0.0 -0.25 -0.25 -r 0.6 -B 0.1
\end{figure}
-Fig. \ref{fig:defects:cp_bc_00-1_mig} shows the evolution of structure and energy along the \ci{} BC to \hkl[0 0 -1] DB transition.
+Fig.~\ref{fig:defects:cp_bc_00-1_mig} shows the evolution of structure and energy along the \ci{} BC to \hkl[0 0 -1] DB transition.
Since the \ci{} BC configuration is unstable relaxing into the \hkl[1 1 0] DB configuration within this potential, the low kinetic energy state is used as a starting configuration.
Two different pathways are obtained for different time constants of the Berendsen thermostat.
With a time constant of \unit[1]{fs} the C atom resides in the \hkl(1 1 0) plane
\caption[{Migration barriers of the \ci{} \hkl[1 1 0] DB to BC, \hkl[0 0 -1] and \hkl[0 -1 0] (in place) transition.}]{Migration barriers of the \ci{} \hkl[1 1 0] DB to BC (blue), \hkl[0 0 -1] (green) and \hkl[0 -1 0] (in place, red) transition. Solid lines show results for a time constant of \unit[1]{fs} and dashed lines correspond to simulations employing a time constant of \unit[100]{fs}.}
\label{fig:defects:110_mig}
\end{figure}
-Fig. \ref{fig:defects:110_mig} shows migration barriers of the \ci{} \hkl[1 1 0] DB to \hkl[0 0 -1], \hkl[0 -1 0] (in place) and BC configuration.
+Fig.~\ref{fig:defects:110_mig} shows migration barriers of the \ci{} \hkl[1 1 0] DB to \hkl[0 0 -1], \hkl[0 -1 0] (in place) and BC configuration.
As expected there is no maximum for the transition into the BC configuration.
As mentioned earlier, the BC configuration itself constitutes a saddle point configuration relaxing into the energetically more favorable \hkl[1 1 0] DB configuration.
An activation energy of \unit[2.2]{eV} is necessary to reorientate the \hkl[0 0 -1] into the \hkl[1 1 0] DB configuration, which is \unit[1.3]{eV} higher in energy.
In contrast to quantum-mechanical calculations, in which the direct transition is the energetically most favorable transition and the transition composed of the intermediate migration steps is very unlikely to occur, the just presented pathway is much more conceivable in classical potential simulations, since the energetically most favorable transition found so far is likewise composed of two migration steps with activation energies of \unit[2.2]{eV} and \unit[0.5]{eV}, for which the intermediate state is the BC configuration, which is unstable.
Thus the just proposed migration path, which involves the \hkl[1 1 0] interstitial configuration, becomes even more probable than the initially proposed path, which involves the BC configuration that is, in fact, unstable.
Due to these findings, the respective path is proposed to constitute the diffusion-describing path.
-The evolution of structure and configurational energy is displayed again in Fig. \ref{fig:defects:involve110}.
+The evolution of structure and configurational energy is displayed again in Fig.~\ref{fig:defects:involve110}.
\begin{figure}[tp]
\begin{center}
\includegraphics[width=0.7\textwidth]{00-1_110_0-10_mig_albe.ps}
\end{figure}
Approximately \unit[2.2]{eV} are needed to turn the \ci{} \hkl[0 0 -1] into the \hkl[1 1 0] DB located at the neighbored lattice site in \hkl[1 1 -1] direction.
Another barrier of \unit[0.90]{eV} exists for the rotation into the \ci{} \hkl[0 -1 0] DB configuration for the path obtained with a time constant of \unit[100]{fs} for the Berendsen thermostat.
-Roughly the same amount would be necessary to excite the C$_{\text{i}}$ \hkl[1 1 0] DB to the BC configuration (\unit[0.40]{eV}) and a successive migration into the \hkl[0 0 1] DB configuration (\unit[0.50]{eV}) as displayed in Fig. \ref{fig:defects:110_mig} and Fig. \ref{fig:defects:cp_bc_00-1_mig}.
+Roughly the same amount would be necessary to excite the C$_{\text{i}}$ \hkl[1 1 0] DB to the BC configuration (\unit[0.40]{eV}) and a successive migration into the \hkl[0 0 1] DB configuration (\unit[0.50]{eV}) as displayed in Fig.~\ref{fig:defects:110_mig} and Fig.~\ref{fig:defects:cp_bc_00-1_mig}.
The former diffusion process, however, would more nicely agree with the {\em ab initio} path, since the migration is accompanied by a rotation of the DB orientation.
By considering a two step process and assuming equal preexponential factors for both diffusion steps, the probability of the total diffusion event is given by $\exp(\frac{\unit[2.24]{eV}+\unit[0.90]{eV}}{k_{\text{B}}T})$, which corresponds to a single diffusion barrier that is 3.5 times higher than the barrier obtained by {\em ab initio} calculations.
\label{fig:defects:comb_db_02}
\end{figure}
Fig.~\ref{fig:defects:comb_db_02} shows the next three energetically favorable configurations.
-The relaxed configuration obtained by creating a \hkl[1 0 0] DB at position 2 is shown in Fig. \ref{fig:defects:216}.
+The relaxed configuration obtained by creating a \hkl[1 0 0] DB at position 2 is shown in Fig.~\ref{fig:defects:216}.
A binding energy of \unit[-2.16]{eV} is observed.
After relaxation, the second DB is aligned along \hkl[1 1 0].
The bond of Si atoms 1 and 2 does not persist.
However, Si atom number 1 and number 3, which are bound to the second \ci{} atom are also bound to the initial C atom.
These four atoms of the rhomboid reside in a plane and, thus, do not match the situation in SiC.
The C atoms have a distance of \unit[2.75]{\AA}.
-In Fig. \ref{fig:defects:190} the relaxed structure of a \hkl[0 1 0] DB constructed at position 2 is displayed.
+In Fig.~\ref{fig:defects:190} the relaxed structure of a \hkl[0 1 0] DB constructed at position 2 is displayed.
An energy of \unit[-1.90]{eV} is observed.
The initial DB and especially the C atom is pushed towards the Si atom of the second DB forming an additional fourth bond.
Si atom number 1 is pulled towards the C atoms of the DBs accompanied by the disappearance of its bond to Si number 5 as well as the bond of Si number 5 to its neighbored Si atom in \hkl[1 1 -1] direction.
\label{fig:defects:comb_db_03}
\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.
+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.
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.
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.
\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.
\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.
%\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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