\chapter{Point defects in silicon}
+\label{chapter:defects}
Given the conversion mechnism of SiC in crystalline silicon introduced in section \ref{section:assumed_prec} the understanding of carbon and silicon interstitial point defects in c-Si is of great interest.
Both types of defects are examined in the following both by classical potential as well as density functional theory calculations.
Periodic boundary conditions in each direction are applied.
All point defects are calculated for the neutral charge state.
-\begin{figure}[th]
-\begin{center}
-\includegraphics[width=9cm]{unit_cell_e.eps}
-\end{center}
-\caption[Insertion positions for the tetrahedral ({\color{red}$\bullet$}), hexagonal ({\color{green}$\bullet$}), \hkl<1 0 0> dumbbell ({\color{yellow}$\bullet$}), \hkl<1 1 0> dumbbell ({\color{magenta}$\bullet$}) and bond-centered ({\color{cyan}$\bullet$}) interstitial configuration.]{Insertion positions for the tetrahedral ({\color{red}$\bullet$}), hexagonal ({\color{green}$\bullet$}), \hkl<1 0 0> dumbbell ({\color{yellow}$\bullet$}), \hkl<1 1 0> dumbbell ({\color{magenta}$\bullet$}) and bond-centered ({\color{cyan}$\bullet$}) interstitial configuration. The black dots ({\color{black}$\bullet$}) correspond to the silicon atoms and the blue lines ({\color{blue}-}) indicate the covalent bonds of the perfect c-Si structure.}
-\label{fig:defects:ins_pos}
-\end{figure}
-
-The interstitial atom positions are displayed in figure \ref{fig:defects:ins_pos}.
-In seperated simulation runs the silicon or carbon atom is inserted at the
-\begin{itemize}
- \item tetrahedral, $\vec{r}=(0,0,0)$, ({\color{red}$\bullet$})
- \item hexagonal, $\vec{r}=(-1/8,-1/8,1/8)$, ({\color{green}$\bullet$})
- \item nearly \hkl<1 0 0> dumbbell, $\vec{r}=(-1/4,-1/4,-1/8)$, ({\color{yellow}$\bullet$})
- \item nearly \hkl<1 1 0> dumbbell, $\vec{r}=(-1/8,-1/8,-1/4)$, ({\color{magenta}$\bullet$})
- \item bond-centered, $\vec{r}=(-1/8,-1/8,-3/8)$, ({\color{cyan}$\bullet$})
-\end{itemize}
-interstitial position.
-For the dumbbell configurations the nearest silicon atom is displaced by $(0,0,-1/8)$ and $(-1/8,-1/8,0)$ respectively of the unit cell length to avoid too high forces.
-A vacancy or a substitutional atom is realized by removing one silicon atom and switching the type of one silicon atom respectively.
-
-From an energetic point of view the free energy of formation $E_{\text{f}}$ is suitable for the characterization of defect structures.
-For defect configurations consisting of a single atom species the formation energy is defined as
-\begin{equation}
-E_{\text{f}}=\left(E_{\text{coh}}^{\text{defect}}
- -E_{\text{coh}}^{\text{defect-free}}\right)N
-\label{eq:defects:ef1}
-\end{equation}
-where $N$ and $E_{\text{coh}}^{\text{defect}}$ are the number of atoms and the cohesive energy per atom in the defect configuration and $E_{\text{coh}}^{\text{defect-free}}$ is the cohesive energy per atom of the defect-free structure.
-The formation energy of defects consisting of two or more atom species is defined as
-\begin{equation}
-E_{\text{f}}=E-\sum_i N_i\mu_i
-\label{eq:defects:ef2}
-\end{equation}
-where $E$ is the free energy of the interstitial system and $N_i$ and $\mu_i$ are the amount of atoms and the chemical potential of species $i$.
-The chemical potential is determined by the cohesive energy of the structure of the specific type in equilibrium at zero Kelvin.
-For a defect configuration of a single atom species equation \eqref{eq:defects:ef2} is equivalent to equation \eqref{eq:defects:ef1}.
-
\section{Silicon self-interstitials}
Point defects in silicon have been extensively studied, both experimentally and theoretically \cite{fahey89,leung99}.
\hline
& T & H & \hkl<1 0 0> DB & \hkl<1 1 0> DB & V \\
\hline
- Erhard/Albe MD & 3.40 & 4.48$^*$ & 5.42 & 4.39 & 3.13 \\
+ Erhart/Albe MD & 3.40 & 4.48$^*$ & 5.42 & 4.39 & 3.13 \\
VASP & 3.77 & 3.42 & 4.41 & 3.39 & 3.63 \\
LDA \cite{leung99} & 3.43 & 3.31 & - & 3.31 & - \\
GGA \cite{leung99} & 4.07 & 3.80 & - & 3.84 & - \\
It has turned out to be very difficult to capture the results of quantum-mechanical calculations in analytical potential models.
Among the established analytical potentials only the EDIP \cite{bazant97,justo98} and Stillinger-Weber \cite{stillinger85} potential reproduce the correct order in energy of the defects.
However, these potenitals show shortcomings concerning the description of other physical properties and are unable to describe the C-C and C-Si interaction.
-In fact the Erhard/Albe potential calculations favor the tetrahedral defect configuration.
+In fact the Erhart/Albe potential calculations favor the tetrahedral defect configuration.
The hexagonal configuration is not stable opposed to results of the authors of the potential \cite{albe_sic_pot}.
In the first two pico seconds while kinetic energy is decoupled from the system the Si interstitial seems to condense at the hexagonal site.
The formation energy of 4.48 eV is determined by this low kinetic energy configuration shortly before the relaxation process starts.
\begin{center}
\includegraphics[width=10cm]{e_kin_si_hex.ps}
\end{center}
-\caption{Kinetic energy plot of the relaxation process of the hexagonal silicon self-interstitial defect simulation using the Erhard/Albe classical potential.}
+\caption{Kinetic energy plot of the relaxation process of the hexagonal silicon self-interstitial defect simulation using the Erhart/Albe classical potential.}
\label{fig:defects:kin_si_hex}
\end{figure}
To exclude failures in the implementation of the potential or the MD code itself the hexagonal defect structure was double-checked with the PARCAS MD code \cite{parcas_md}.
In addition, variations exist in which the displacement is only along two \hkl<1 0 0> axes ($E_\text{f}=3.8\text{ eV}$) or along a single \hkl<1 0 0> axes ($E_\text{f}=3.6\text{ eV}$) successively approximating the tetdrahedral configuration and formation energy.
The existence of these local minima located near the tetrahedral configuration seems to be an artifact of the analytical potential without physical authenticity revealing basic problems of analytical potential models for describing defect structures.
However, the energy barrier is small.
-{\color{red}Todo: Check!}
+\begin{figure}[th]
+\begin{center}
+\includegraphics[width=12cm]{nhex_tet.ps}
+\end{center}
+\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 figure \ref{fig:defects:nhex_tet_mig}, which shows the change in potential energy during the migration of the interstitial displaced along all three coordinate axes into the tetrahedral configuration.
+The technique used to obtain the migration data is explained in a later section (\ref{subsection:100mig}).
+The barrier is less than 0.2 eV.
Hence these artifacts should have a negligent influence in finite temperature simulations.
-The bond-centered configuration is unstable and the \hkl<1 0 0> dumbbell interstitial is the most unfavorable configuration for both, the Erhard/Albe and VASP calculations.
+The bond-centered configuration is unstable and the \hkl<1 0 0> dumbbell interstitial is the most unfavorable configuration for both, the Erhart/Albe and VASP calculations.
In the case of the classical potential simulations bonds between atoms are displayed if there is an interaction according to the potential model, that is if the distance of two atoms is within the cutoff region $S_{ij}$ introduced in equation \eqref{eq:basics:fc}.
For the tetrahedral and the slightly displaced configurations four bonds to the atoms located in the center of the planes of the unit cell exist in addition to the four tetrahedral bonds.
\hline
& T & H & \hkl<1 0 0> DB & \hkl<1 1 0> DB & S & B \\
\hline
- Erhard/Albe MD & 6.09 & 9.05$^*$ & 3.88 & 5.18 & 0.75 & 5.59$^*$ \\
+ Erhart/Albe MD & 6.09 & 9.05$^*$ & 3.88 & 5.18 & 0.75 & 5.59$^*$ \\
%VASP & unstable & unstable & 3.15 & 3.60 & 1.39 & 4.10 \\
VASP & unstable & unstable & 3.72 & 4.16 & 1.95 & 4.66 \\
Tersoff \cite{tersoff90} & 3.8 & 6.7 & 4.6 & 5.9 & 1.6 & 5.3 \\
It is simply given by the atomic density of pure silicon, which is $5\times 10^{22}\text{ cm}^{-3}$.
Tersoff \cite{tersoff90} and Dal Pino et al. \cite{dal_pino93} pointed out that by combining this prefactor with the calculated values for the energy of formation ranging from 1.6 to 1.89 eV an excellent agreement with the experimental solubility data within the entire temeprature range of the experiment is obtained.
This reinterpretation of the solubility data, first proposed by Tersoff and later on reinforced by Dal Pino et al. is in good agreement with the results of the quantum-mechanical calculations performed in this work.
-Unfortunately the Erhard/Albe potential undervalues the formation energy roughly by a factor of two.
+Unfortunately the Erhart/Albe potential undervalues the formation energy roughly by a factor of two.
Except for Tersoff's tedrahedral configuration results the \hkl<1 0 0> dumbbell is the energetically most favorable interstital configuration.
The low energy of formation for the tetrahedral interstitial in the case of the Tersoff potential is believed to be an artifact of the abrupt cutoff set to 2.5 \AA{} (see ref. 11 and 13 in \cite{tersoff90}) and the real formation energy is, thus, supposed to be located between 3 and 10 eV.
Thus, this configuration is of great importance and discussed in more detail in section \ref{subsection:100db}.
The highest energy is observed for the hexagonal interstitial configuration using classical potentials.
-Quantum-mechanical calculations reveal this configuration to be unstable, which is also reproduced by the Erhard/Albe potential.
+Quantum-mechanical calculations reveal this configuration to be unstable, which is also reproduced by the Erhart/Albe potential.
In both cases a relaxation towards the \hkl<1 0 0> dumbbell configuration is observed.
In fact the stability of the hexagonal interstitial could not be reproduced in simulations performed in this work using the unmodifed Tersoff potential parameters.
Unfortunately, apart from the modified parameters, no more conditions specifying the relaxation process are given in Tersoff's study on carbon point defects in silicon \cite{tersoff90}.
The tetrahedral is the second most unfavorable interstitial configuration using classical potentials and keeping in mind the abrupt cutoff effect in the case of the Tersoff potential as discussed earlier.
Again, quantum-mechanical results reveal this configuration unstable.
-The fact that the tetrahedral and hexagonal configurations are the two most unstable configurations in classical potential calculations and, thus, are less likely to arise in MD simulations acts in concert with the fact that these configurations are found to be unstable in the more accurate quantum-mechanical calculations.
+The fact that the tetrahedral and hexagonal configurations are the two most unstable configurations in classical potential calculations and, thus, are less likely to arise in MD simulations acts in concert with the fact that these configurations are found to be unstable in the more accurate quantum-mechanical description.
Just as for the Si self-interstitial a carbon \hkl<1 1 0> dumbbell configuration exists.
-For the Erhard/Albe potential the formation energy is situated in the same order as found by quantum-mechanical results.
+For the Erhart/Albe potential the formation energy is situated in the same order as found by quantum-mechanical results.
Similar structures arise in both types of simulations with the silicon and carbon atom sharing a silicon lattice site aligned along \hkl<1 1 0> where the carbon atom is localized slightly closer to the next nearest silicon atom located in the opposite direction to the site-sharing silicon atom even forming a bond to the next but one silicon atom in this direction.
-The bond-centered configuration is unstable for the Erhard/Albe potential.
+The bond-centered configuration is unstable for the Erhart/Albe potential.
The system moves into the \hkl<1 1 0> interstitial configuration.
This, like in the hexagonal case, is also true for the unmodified Tersoff potential and the given relaxation conditions.
Quantum-mechanical results of this configuration are discussed in more detail in section \ref{subsection:bc}.
& & & & \multicolumn{3}{c}{Atom 2} & \multicolumn{3}{c}{Atom 3} \\
& $a$ & $b$ & $|a|+|b|$ & $\Delta x$ & $\Delta y$ & $\Delta z$ & $\Delta x$ & $\Delta y$ & $\Delta z$ \\
\hline
-Erhard/Albe & 0.084 & -0.091 & 0.175 & -0.015 & -0.015 & -0.031 & -0.014 & 0.014 & 0.020 \\
+Erhart/Albe & 0.084 & -0.091 & 0.175 & -0.015 & -0.015 & -0.031 & -0.014 & 0.014 & 0.020 \\
VASP & 0.109 & -0.065 & 0.174 & -0.011 & -0.011 & -0.024 & -0.014 & 0.014 & 0.025 \\
\hline
\hline
\hline
& $r(1C)$ & $r(2C)$ & $r(3C)$ & $r(12)$ & $r(13)$ & $r(34)$ & $r(23)$ & $r(25)$ & $a_{\text{Si}}^{\text{equi}}$\\
\hline
-Erhard/Albe & 0.175 & 0.329 & 0.186 & 0.226 & 0.300 & 0.343 & 0.423 & 0.425 & 0.543 \\
+Erhart/Albe & 0.175 & 0.329 & 0.186 & 0.226 & 0.300 & 0.343 & 0.423 & 0.425 & 0.543 \\
VASP & 0.174 & 0.341 & 0.182 & 0.229 & 0.286 & 0.347 & 0.422 & 0.417 & 0.548 \\
\hline
\hline
\hline
& $\theta_1$ & $\theta_2$ & $\theta_3$ & $\theta_4$ \\
\hline
-Erhard/Albe & 140.2 & 109.9 & 134.4 & 112.8 \\
+Erhart/Albe & 140.2 & 109.9 & 134.4 & 112.8 \\
VASP & 130.7 & 114.4 & 146.0 & 107.0 \\
\hline
\hline
\end{tabular}\\[0.5cm]
\end{center}
-\caption[Atomic displacements, distances and bond angles of the \hkl<1 0 0> dumbbell structure obtained by the Erhard/Albe potential and VASP calculations.]{Atomic displacements, distances and bond angles of the \hkl<1 0 0> dumbbell structure obtained by the Erhard/Albe potential and 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 figure \ref{fig:defects:100db_cmp}. In addition, the equilibrium lattice constant for crystalline silicon is listed.}
+\caption[Atomic displacements, distances and bond angles of the \hkl<1 0 0> dumbbell structure obtained by the Erhart/Albe potential and VASP calculations.]{Atomic displacements, distances and bond angles of the \hkl<1 0 0> dumbbell structure obtained by the Erhart/Albe potential and 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 figure \ref{fig:defects:100db_cmp}. In addition, the equilibrium lattice constant for crystalline silicon is listed.}
\label{tab:defects:100db_cmp}
\end{table}
\begin{figure}[t!h!]
\begin{center}
\begin{minipage}{6cm}
\begin{center}
-\underline{Erhard/Albe}
+\underline{Erhart/Albe}
\includegraphics[width=5cm]{c_pd_albe/100_cmp.eps}
\end{center}
\end{minipage}
\end{center}
\end{minipage}
\end{center}
-\caption{Comparison of the visualized \hkl<1 0 0> dumbbel structures obtained by Erhard/Albe potential and VASP calculations.}
+\caption{Comparison of the visualized \hkl<1 0 0> dumbbel structures obtained by Erhart/Albe potential and VASP calculations.}
\label{fig:defects:100db_vis_cmp}
\end{figure}
\begin{figure}[th]
One bond is formed to the other dumbbell atom.
The other two bonds are bonds to the two silicon edge atoms located in the opposite direction of the dumbbell atom.
The distance of the two dumbbell atoms is almost the same for both types of calculations.
-However, in the case of the VASP calculation, the dumbbell structure is pushed upwards compared to the Erhard/Albe results.
+However, in the case of the VASP calculation, the dumbbell structure is pushed upwards compared to the Erhart/Albe results.
This is easily identified by comparing the values for $a$ and $b$ and the two structures in figure \ref{fig:defects:100db_vis_cmp}.
Thus, the angles of bonds of the silicon dumbbell atom ($\theta_1$ and $\theta_2$) are closer to $120^{\circ}$ signifying the predominance of $sp^2$ hybridization.
On the other hand, the carbon atom forms an almost collinear bond ($\theta_3$) with the two silicon edge atoms implying the predominance of $sp$ bonding.
\end{figure}
In the bond-centerd insterstitial configuration the interstitial atom is located inbetween two next neighboured silicon atoms forming linear bonds.
In former studies this configuration is found to be an intermediate saddle point configuration determining the migration barrier of one possibe migration path of a \hkl<1 0 0> dumbbel configuration into an equivalent one \cite{capaz94}.
-This is in agreement with results of the Erhard/Albe potential simulations which reveal this configuration to be unstable relaxing into the \hkl<1 1 0> configuration.
+This is in agreement with results of the Erhart/Albe potential simulations which reveal this configuration to be unstable relaxing into the \hkl<1 1 0> configuration.
However, this fact could not be reproduced by spin polarized VASP calculations performed in this work.
Present results suggest this configuration to be a real local minimum.
In fact, an additional barrier has to be passed to reach this configuration starting from the \hkl<1 0 0> interstitital configuration, which is investigated in section \ref{subsection:100mig}.
The blue torus, reinforcing 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.
-\section[Migration of the carbon \hkl<1 0 0> interstitial]{\boldmath Migration of the carbon \hkl<1 0 0> interstitial}
+\section{Migration of the carbon interstitials}
\label{subsection:100mig}
In the following the problem of interstitial carbon migration in silicon is considered.
Thus, it is not responsible for long-range migration.
The silicon dumbbell partner remains the same.
The bond to the face-centered silicon atom at the bottom of the unit cell breaks and a new one is formed to the face-centered atom at the forefront of the unit cell.
-{\color{red}Todo: Comparison with classical potential simulations or explanation to only focus on ab initio calculations.}
-
-Since the starting and final structure, which are both local minima of the potential energy surface, are known, the aim is to find the minimum energy path from one local minimum to the other one.
-One method to find a minimum energy path is to move the diffusing atom stepwise from the starting to the final position and only allow relaxation in the plane perpendicular to the direction of the vector connecting its starting and final position.
-No constraints are applied to the remaining atoms in order to allow relaxation of the surrounding lattice.
-To prevent the remaining lattice to migrate according to the displacement of the defect an atom far away from the defect region is fixed in all three coordinate directions.
-However, it turned out, that this method tremendously failed applying it to the present migration pathways and structures.
-Abrupt changes in structure and free energy occured among relaxed structures of two successive displacement steps.
-For some structures even the expected final configurations were never obtained.
-Thus, the method mentioned above was adjusted adding further constraints in order to obtain smooth transitions, either in energy as well as structure is concerned.
-In this new method all atoms are stepwise displaced towards their final positions.
-Relaxation of each individual atom is only allowed in the plane perpendicular to the last individual displacement vector.
-The modifications used to add this feature to the VASP code and a short instruction on how to use it can be found in appendix \ref{app:patch_vasp}.
-Due to these constraints obtained activation energies can effectively be higher.
-{\color{red}Todo: To refine the migration barrier one has to find the saddle point structure and recalculate the free energy of this configuration with a reduced set of constraints.}
+
+\subsection{Migration barriers obtained by quantum-mechanical calculations}
+
+In the following migration barriers are investigated using quantum-mechanical calculations.
+The amount of simulated atoms is the same as for the investigation of the point defect structures.
+Due to the time necessary for computing only ten displacement steps are used.
\begin{figure}[t!h!]
\begin{center}
Thus, this pathway is more likely to be composed of two consecutive steps of the second path.
Since the activation energy of the first and last migration path is much greater than the experimental value, the second path is identified to be responsible as a migration path for the most likely carbon interstitial in silicon explaining both, annealing and reorientation experiments.
-The activation energy of roughly 0.9 eV nicely compares to experimental values.
+The activation energy of roughly 0.9 eV nicely compares to experimental values reinforcing the correct identification of the C-Si dumbbell diffusion mechanism.
The theoretical description performed in this work is improved compared to a former study \cite{capaz94}, which underestimates the experimental value by 35 \%.
In addition the bond-ceneterd configuration, for which spin polarized calculations are necessary, is found to be a real local minimum instead of a saddle point configuration.
+\begin{figure}[th!]
+\begin{center}
+\includegraphics[width=13cm]{vasp_mig/110_mig_vasp.ps}
+%\begin{picture}(0,0)(140,0)
+%\includegraphics[width=2.2cm]{vasp_mig/00-1_b.eps}
+%\end{picture}
+%\begin{picture}(0,0)(20,0)
+%\includegraphics[width=2.2cm]{vasp_mig/00-1_ip0-10_sp.eps}
+%\end{picture}
+%\begin{picture}(0,0)(-120,0)
+%\includegraphics[width=2.2cm]{vasp_mig/0-10_b.eps}
+%\end{picture}
+\end{center}
+\caption{Migration barriers of the \hkl<1 1 0> dumbbell to bond-centered (blue), \hkl<0 0 -1> (green) and \hkl<0 -1 0> (in place, red) C-Si dumbbell transition.}
+\label{fig:defects:110_mig_vasp}
+\end{figure}
+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 dumbbell interstitial.
+Figure \ref{fig:defects:110_mig_vasp} shows the migration barrier of the \hkl<1 1 0> C-Si dumbbell to the bond-centered, \hkl<0 0 -1> and \hkl<0 -1 0> (in place) transition.
+Indeed less than 0.7 eV are necessary to turn a \hkl<0 -1 0>- to a \hkl<1 1 0>-type C-Si dumbbell interstitial.
+This transition is carried out in place, that is the Si dumbbell 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> interstitial configuration it can migrate into the bond-centered configuration by employing approximately 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 shown in figure \ref{fig:defects:00-1_0-10_mig}.
+As already known from the migration of the \hkl<0 0 -1> to the bond-centered configuration as discussed in figure \ref{fig:defects:00-1_001_mig} another 0.25 eV are needed to turn back from the bond-centered 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 it 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 neighboured lattice site is approximately 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 figure \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.
+Secondly the constrainted directions are not kept constant to the initial displacement direction.
+Instead they are updated for every displacement step.
+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}.
+\begin{figure}[th!]
+\begin{center}
+\includegraphics[width=13cm]{vasp_mig/00-1_0-10_nosym_sp_cmp.ps}
+\end{center}
+\caption[Comparison of three different techniques for obtaining migration barriers and pathways applied to the \hkl<0 0 -1> to \hkl<0 -1 0> C-Si dumbbell transition.]{Comparison of three different techniques for obtaining migration barriers and pathways applied to the \hkl<0 0 -1> to \hkl<0 -1 0> C-Si dumbbell transition.}
+\label{fig:defects:00-1_0-10_cmp}
+\end{figure}
+The method without updating the constraints but still applying them to all atoms shows a delayed crossing of the saddle point.
+This is understandable since the update results in a more aggressive advance towards the final configuration.
+In any case the barrier obtained is slightly higher, which means that it does not constitute an energetically more favorable pathway.
+The method in which the constraints are only applied to the diffusing C atom and two Si atoms, ... {\color{red}Todo: does not work!} ...
+
+\subsection{Migration barriers obtained by classical potential calculations}
+\label{subsection:defects:mig_classical}
+
+The same method for obtaining migration barriers and the same suggested pathways are applied to calculations employing the classical Erhart/Albe potential.
+Since the evaluation of the classical potential and force is less computationally intensive higher amounts of steps can be used.
+The time constant $\tau$ for the Berendsen thermostat is set to 1 fs in order to have direct velocity scaling and with the temperature set to zero Kelvin perform a steepest descent minimazation to drive the system into a local minimum.
+However, in some cases a time constant of 100 fs resuls in lower barriers and, thus, is shown whenever appropriate.
+
+\begin{figure}[th!]
+\begin{center}
+\includegraphics[width=13cm]{bc_00-1.ps}\\[5.6cm]
+\begin{pspicture}(0,0)(0,0)
+\psframe[linecolor=red,fillstyle=none](-7,2.7)(7.2,6)
+\end{pspicture}
+\begin{picture}(0,0)(140,-100)
+\includegraphics[width=2.4cm]{albe_mig/bc_00-1_red_00.eps}
+\end{picture}
+\begin{picture}(0,0)(10,-100)
+\includegraphics[width=2.4cm]{albe_mig/bc_00-1_red_01.eps}
+\end{picture}
+\begin{picture}(0,0)(-120,-100)
+\includegraphics[width=2.4cm]{albe_mig/bc_00-1_red_02.eps}
+\end{picture}
+\begin{picture}(0,0)(25,-80)
+\includegraphics[width=2.5cm]{110_arrow.eps}
+\end{picture}
+\begin{picture}(0,0)(215,-100)
+\includegraphics[height=2.2cm]{001_arrow.eps}
+\end{picture}\\
+\begin{pspicture}(0,0)(0,0)
+\psframe[linecolor=blue,fillstyle=none](-7,-0.5)(7.2,2.8)
+\end{pspicture}
+\begin{picture}(0,0)(160,-10)
+\includegraphics[width=2.2cm]{albe_mig/bc_00-1_01.eps}
+\end{picture}
+\begin{picture}(0,0)(100,-10)
+\includegraphics[width=2.2cm]{albe_mig/bc_00-1_02.eps}
+\end{picture}
+\begin{picture}(0,0)(10,-10)
+\includegraphics[width=2.2cm]{albe_mig/bc_00-1_03.eps}
+\end{picture}
+\begin{picture}(0,0)(-120,-10)
+\includegraphics[width=2.2cm]{albe_mig/bc_00-1_04.eps}
+\end{picture}
+\begin{picture}(0,0)(25,10)
+\includegraphics[width=2.5cm]{100_arrow.eps}
+\end{picture}
+\begin{picture}(0,0)(215,-10)
+\includegraphics[height=2.2cm]{010_arrow.eps}
+\end{picture}
+\end{center}
+\caption{Migration barrier and structures of the bond-centered to \hkl<0 0 -1> dumbbell transition using the classical Erhart/Albe potential.}
+\label{fig:defects:cp_bc_00-1_mig}
+% 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}
+Figure \ref{fig:defects:cp_bc_00-1_mig} shows the migration barrier and corresponding structures of the bond-centered to \hkl<0 0 -1> dumbbell transition.
+Since the bond-centered configuration is unstable relaxing into the \hkl<1 1 0> C-Si dumbbell interstitial configuration within this potential the low kinetic energy state is used as a starting configuration.
+Depending on the time constant activation energies of 2.4 eV and 2.2 eV respectively are obtained.
+The migration path obtained by simulations with a time constant of 1 fs remains in the \hkl(1 1 0) plane.
+Using 100 fs as a time constant the C atom breaks out of the \hkl(1 1 0) plane already at the beginning of the migration accompanied by a reduction in energy.
+The energy barrier of this path is 0.2 eV lower in energy than the direct migration within the \hkl(1 1 0) plane.
+However, the investigated pathways cover an activation energy approximately twice as high as the one obtained by quantum-mechanical calculations.
+For the entire transition of the \hkl<0 0 -1> into the \hkl<0 0 1> configuration by passing the bond-centered configuration an additional activation energy of 0.5 eV is necessary to escape from the bond-centered and reach the \hkl<0 0 1> configuration.
+
+\begin{figure}[th!]
+\begin{center}
+\includegraphics[width=13cm]{00-1_0-10.ps}\\[2.4cm]
+\begin{pspicture}(0,0)(0,0)
+\psframe[linecolor=red,fillstyle=none](-6,-0.5)(7.2,2.8)
+\end{pspicture}
+\begin{picture}(0,0)(130,-10)
+\includegraphics[width=2.2cm]{albe_mig/00-1_0-10_red_00.eps}
+\end{picture}
+\begin{picture}(0,0)(0,-10)
+\includegraphics[width=2.2cm]{albe_mig/00-1_0-10_red_min.eps}
+\end{picture}
+\begin{picture}(0,0)(-120,-10)
+\includegraphics[width=2.2cm]{albe_mig/00-1_0-10_red_03.eps}
+\end{picture}
+\begin{picture}(0,0)(25,10)
+\includegraphics[width=2.5cm]{100_arrow.eps}
+\end{picture}
+\begin{picture}(0,0)(185,-10)
+\includegraphics[height=2.2cm]{001_arrow.eps}
+\end{picture}
+\end{center}
+\caption{Migration barrier and structures of the \hkl<0 0 -1> to \hkl<0 -1 0> C-Si dumbbell transition using the classical Erhart/Albe potential.}
+% red: ./visualize -w 640 -h 480 -d saves/c_in_si_mig_00-1_0-10_s20 -nll -0.56 -0.56 -0.8 -fur 0.3 0.2 0 -c -0.125 -1.7 0.7 -L -0.125 -0.25 -0.25 -r 0.6 -B 0.1
+\label{fig:defects:cp_00-1_0-10_mig}
+\end{figure}
+\begin{figure}[th!]
+\begin{center}
+\includegraphics[width=13cm]{00-1_ip0-10.ps}
+\end{center}
+\caption{Migration barrier of the \hkl<0 0 -1> to \hkl<0 -1 0> C-Si dumbbell transition in place using the classical Erhart/Albe potential.}
+\label{fig:defects:cp_00-1_ip0-10_mig}
+\end{figure}
+Figure \ref{fig:defects:cp_00-1_0-10_mig} and \ref{fig:defects:cp_00-1_ip0-10_mig} show the migration barriers of \hkl<0 0 -1> to \hkl<0 -1 0> C-Si dumbbell transition, with a transition of the C atom to the neighboured lattice site in the first case and a reorientation within the same lattice site in the latter case.
+Both pathways look similar.
+A local minimum exists inbetween two peaks of the graph.
+The corresponding configuration, which is illustrated for the migration simulation with a time constant of 1 fs, looks similar to the \hkl<1 1 0> configuration.
+Indeed, this configuration is obtained by relaxation simulations without constraints of configurations near the minimum.
+Activation energies of roughly 2.8 eV and 2.7 eV respectively are needed for migration.
+
+The \hkl<1 1 0> configuration seems to play a decisive role in all migration pathways.
+In the first migration path it is the configuration resulting from further relaxation of the rather unstable bond-centered configuration, which is fixed to be a transition point in the migration calculations.
+The last two pathways show configurations almost identical to the \hkl<1 1 0> configuration, which constitute a local minimum within the pathway.
+Thus, migration pathways with the \hkl<1 1 0> C-Si dumbbell interstitial configuration as a starting or final configuration are further investigated.
+\begin{figure}[!ht]
+\begin{center}
+\includegraphics[width=13cm]{110_mig.ps}
+\end{center}
+\caption[Migration barriers of the \hkl<1 1 0> dumbbell to bond-centered (blue), \hkl<0 0 -1> (green) and \hkl<0 -1 0> (in place, red) C-Si dumbbell transition.]{Migration barriers of the \hkl<1 1 0> dumbbell to bond-centered (blue), \hkl<0 0 -1> (green) and \hkl<0 -1 0> (in place, red) C-Si dumbbell transition. Solid lines show results for a time constant of 1 fs and dashed lines correspond to simulations employing a time constant of 100 fs.}
+\label{fig:defects:110_mig}
+\end{figure}
+Figure \ref{fig:defects:110_mig} shows migration barriers of the C-Si \hkl<1 1 0> dumbbell to \hkl<0 0 -1>, \hkl<0 -1 0> (in place) and bond-centered configuration.
+As expected there is no maximum for the transition into the bond-centered configuration.
+As mentioned earlier the bond-centered configuration itself constitutes a saddle point configuration relaxing into the energetically more favorable \hkl<1 1 0> configuration.
+An activation energy of 2.2 eV is necessary to reorientate the \hkl<0 0 -1> dumbbell configuration into the \hkl<1 1 0> configuration, which is 1.3 eV higher in energy.
+Residing in this state another 0.9 eV is enough to make the C atom form a \hkl<0 0 -1> dumbbell configuration with the Si atom of the neighboured lattice site.
+In contrast to quantum-mechanical calculations, in which the direct transition is the energetically most favorable transition and the transition composed of the intermmediate migration steps is very unlikely to occur the just presented pathway is much more supposable in classical potential simulations, since the energetically most favorable transition found so far is also composed of two migration steps with activation energies of 2.2 eV and 0.5 eV, for which the intermediate state is the bond-centered configuration, which is unstable.
+Thus the just proposed migration path involving the \hkl<1 1 0> interstitial configuration becomes even more probable than path 1 involving the unstable bond-centered configuration.
+
+Although classical potential simulations reproduce the order in energy of the \hkl<1 0 0> and \hkl<1 1 0> C-Si dumbbell interstitial configurations as obtained by more accurate quantum-mechanical calculations the obtained migration pathways and resulting activation energies differ to a great extent.
+On the one hand the most favorable pathways differ.
+On the other hand the activation energies obtained by classical potential simulations are tremendously overestimated by a factor of almost 2.4.
+Thus, atomic diffusion is wrongly described in the classical potential approach.
+The probability of already rare diffusion events is further decreased for this reason.
+Since agglomeration of C and diffusion of Si self-interstitials are an important part of the proposed SiC precipitation mechanism a problem arises, which is formulated and discussed in more detail in section \ref{subsection:md:limit}.
+
\section{Combination of point defects}
-The structural and energetic properties of combinations of point defects are investigated in the following.
-The focus is on combinations of the \hkl<0 0 -1> dumbbell interstitial with a second defect.
+The structural and energetic properties of combinations of point defects are examined in the following.
+Investigations are restricted to quantum-mechanical calculations for two reasons.
+First of all, as mentioned in the last section, they are far more accurate.
+Secondly, the restrictions in size and simulation time for this type of calculation due to limited computational resources, necessitate to map the complex precipitation mechanism to a more compact and simplified modelling.
+The investigations of defect combinations approached in the following are still feasible within the available computational power and allow to draw conclusions on some important ongoing mechanisms during SiC precipitation.
+
+\subsection[Combinations with a C-Si \hkl<1 0 0>-type interstitial]{\boldmath Combinations with a C-Si \hkl<1 0 0>-type interstitial}
+\label{subsection:defects:c-si_comb}
+
+This section focuses on combinations of the \hkl<0 0 -1> dumbbell interstitial with a second defect.
The second defect is either another \hkl<1 0 0>-type interstitial occupying different orientations, a vacany or a substitutional carbon atom.
Several distances of the two defects are examined.
-Investigations are restricted to quantum-mechanical calculations.
\begin{figure}[th]
\begin{center}
As expected by the initialization conditions the two carbon atoms form a bond.
This bond has a length of 1.38 \AA{} close to the nex neighbour distance in diamond or graphite, which is approximately 1.54 \AA.
The minimum of binding energy observed for this configuration suggests prefered C clustering as a competing mechnism to the C-Si dumbbell interstitial agglomeration inevitable for the SiC precipitation.
-{\color{red}Todo: Activation energy to obtain a configuration of separated C atoms again or vice versa to obtain this configuration from separated C confs?}
+{\color{red}Todo: Activation energies to obtain separated C confs FAILED (again?) - could be added in the combined defect migration chapter and mentioned here, too!}
However, for the second most favorable configuration, presented in figure \ref{fig:defects:comb_db_01} a), the amount of possibilities for this configuration is twice as high.
In this configuration the initial Si (I) and C (I) dumbbell atoms are displaced along \hkl<1 0 0> and \hkl<-1 0 0> in such a way that the Si atom is forming tetrahedral bonds with two silicon and two carbon atoms.
The carbon and silicon atom constituting the second defect are as well displaced in such a way, that the carbon atom forms tetrahedral bonds with four silicon neighbours, a configuration expected in silicon carbide.
A binding energy of -0.50 eV is observed.
{\color{red}Todo: Jahn-Teller distortion (vacancy) $\rightarrow$ actually three possibilities. Due to the initial defect, symmetries are broken. The system should have relaxed into the minumum energy configuration!?}
-{\color{blue}Todo: Si int + vac and C sub/int ...?
-Investigation of vacancy, Si and C interstitital.
-As for the ground state of the single Si self-int, a 110 is also assumed as the lowest possibility in combination with other defects (which is a cruel assumption)!
+\subsection{Combinations of Si self-interstitials and substitutional carbon}
+
+So far the C-Si \hkl<1 0 0> interstitial was found to be the energetically most favorable configuration.
+In fact substitutional C exhibits a configuration more than 3 eV lower in formation energy, however, the configuration does not account for the accompanying Si self-interstitial that is generated once a C atom occupies the site of a Si atom.
+With regard to the IBS process, in which highly energetic C atoms enter the Si target being able to kick out Si atoms from their lattice sites, such configurations are absolutely conceivable and a significant role for the precipitation process might be attributed to them.
+Thus, combinations of substitutional C and an additional Si self-interstitial are examined in the following.
+The ground state of a single Si self-interstitial was found to be the Si \hkl<1 1 0> self-interstitial configuration.
+For the follwoing study the same type of self-interstitial is assumed to provide the energetically most favorable configuration in combination with substitutional C.
+
+\begin{table}[!ht]
+\begin{center}
+\begin{tabular}{l c c c c c c}
+\hline
+\hline
+C$_{\text{sub}}$ & \hkl<1 1 0> & \hkl<-1 1 0> & \hkl<0 1 1> & \hkl<0 -1 1> &
+ \hkl<1 0 1> & \hkl<-1 0 1> \\
+\hline
+1 & \RM{1} & \RM{3} & \RM{3} & \RM{1} & \RM{3} & \RM{1} \\
+2 & \RM{2} & A & A & \RM{2} & C & \RM{5} \\
+3 & \RM{3} & \RM{1} & \RM{3} & \RM{1} & \RM{1} & \RM{3} \\
+4 & \RM{4} & B & D & E & E & D \\
+5 & \RM{5} & C & A & \RM{2} & A & \RM{2} \\
+\hline
+\hline
+\end{tabular}
+\end{center}
+\caption{Equivalent configurations of \hkl<1 1 0>-type Si self-interstitials created at position I of figure \ref{fig:defects:pos_of_comb} and substitutional C created at positions 1 to 5.}
+\label{tab:defects:comb_csub_si110}
+\end{table}
+\begin{table}[!ht]
+\begin{center}
+\begin{tabular}{l c c c c c c c c c c}
+\hline
+\hline
+Conf & \RM{1} & \RM{2} & \RM{3} & \RM{4} & \RM{5} & A & B & C & D & E \\
+\hline
+$E_{\text{f}}$ [eV]& 4.37 & 5.26 & 5.57 & 5.37 & 5.12 & 5.10 & 5.32 & 5.28 & 5.39 & 5.32 \\
+$E_{\text{b}}$ [eV] & -0.97 & -0.08 & 0.22 & -0.02 & -0.23 & -0.25 & -0.02 & -0.06 & 0.05 & -0.03 \\
+$r$ [nm] & 0.292 & 0.394 & 0.241 & 0.453 & 0.407 & 0.408 & 0.452 & 0.392 & 0.456 & 0.453\\
+\hline
+\hline
+\end{tabular}
+\end{center}
+\caption{Formation $E_{\text{f}}$ and binding $E_{\text{b}}$ energies in eV of the combinational substitutional C and Si self-interstitial configurations as defined in table \ref{tab:defects:comb_csub_si110}.}
+\label{tab:defects:comb_csub_si110_energy}
+\end{table}
+Table \ref{tab:defects:comb_csub_si110} shows equivalent configurations of \hkl<1 1 0>-type Si self-interstitials and substitutional C.
+The notation of figure \ref{fig:defects:pos_of_comb} is used with the six possible Si self-interstitials created at the usual C-Si dumbbell position.
+Substitutional C is created at positions 1 to 5.
+Resulting formation and binding energies of the relaxed structures are listed in table \ref{tab:defects:comb_csub_si110_energy}.
+In addition the separation distance of the ssubstitutional C atom and the Si \hkl<1 1 0> dumbbell interstitial, which is defined to reside at $\frac{a_{\text{Si}}}{4} \hkl<1 1 1>$ is given.
+In total 10 different configurations exist within the investigated range.
+
+\begin{figure}[th!]
+\begin{center}
+\includegraphics[width=12cm]{c_sub_si110.ps}
+\end{center}
+\caption[Binding energy of combinations of a substitutional C and a Si \hkl<1 1 0> dumbbell self-interstitial with respect to the separation distance.]{Binding energy of combinations of a substitutional C and a Si \hkl<1 1 0> dumbbell self-interstitial with respect to the separation distance. The binding energy of the defect pair is well approximated by a Lennard-Jones 6-12 potential, which is used for curve fitting.}
+\label{fig:defects:csub_si110}
+\end{figure}
+According to the formation energies none of the investigated structures is energetically preferred over the C-Si \hkl<1 0 0> dumbbell interstitial, which exhibits a formation energy of 3.88 eV.
+Further separated defects are assumed to approximate the sum of the formation energies of the isolated single defects.
+This is affirmed by the plot of the binding energies with respect to the separation distance in figure \ref{fig:defects:csub_si110} approximating zero with increasing distance.
+Thus, the C-Si \hkl<1 0 0> dumbbell structure remains the ground state configuration of a C interstitial in c-Si with a constant number of Si atoms.
+
+{\color{blue}
+However the binding energy quickly drops to zero with respect to the distance, which is reinforced by the Lennard-Jones fit estimating almost zero interaction energy already at 0.6 nm.
+This indicates a possibly low interaction capture radius of the defect pair.
+Highly energetic collisions in the IBS process might result in separations of these defects exceeding the capture radius.
+For this reason situations most likely occur in which the configuration of substitutional C can be considered without a nearby interacting Si self-interstitial and, thus, unable to form a thermodynamically more stable C-Si \hkl<1 0 0> dumbbell configuration.
}
+\label{section:defects:noneq_process_01}
+
+The energetically most favorable configuration of the combined structures is the one with the substitutional C atom located next to the \hkl<1 1 0> interstitial along the \hkl<1 1 0> direction (configuration \RM{1}).
+Compressive stress along the \hkl<1 1 0> direction originating from the Si \hkl<1 1 0> self-intesrtitial is partially compensated by tensile stress resulting from substitutional C occupying the neighboured Si lattice site.
+In the same way the energetically most unfavorable configuration can be explained, which is configuration \RM{3}.
+The substitutional C is located next to the lattice site shared by the \hkl<1 1 0> Si self-interstitial along the \hkl<1 -1 0> direction.
+Thus, the compressive stress along \hkl<1 1 0> of the Si \hkl<1 1 0> interstitial is not compensated but intensified by the tensile stress of the substitutional C atom, which is no longer loacted along the direction of stress.
+
+{\color{red}Todo: Erhart/Albe calc for most and less favorable configuration!}
+
+{\color{red}Todo: Mig of C-Si DB conf to or from C sub + Si 110 in progress.}
+
+{\color{red}Todo: Mig of Si DB located next to a C sub (also by MD!).}
\section{Migration in systems of combined defects}
-During carbon implantation into crystalline silicon the energetic carbon atoms may kick out silicon atoms from their lattice sites.
-A vacancy accompanied by a silicon self-interstitial is generated.
-The silicon self-interstitial may migrate to the surface or recombine with other vacancies.
-Once a vacancy and a carbon interstitial defect exist the energetically most favorable configuration is the configuration of a substitutional carbon atom, that is the carbon atom occupying the vacant site.
+As already pointed out in the previous section energetic carbon atoms may kick out silicon atoms from their lattice sites during carbon implantation into crystalline silicon.
+However configurations might arise in which C atoms do not already occupy the vacant site but instead form a C interstitial next to the vacancy.
+These combinations have been investigated shortly before in the very end of section \ref{subsection:defects:c-si_comb}.
+In the absence of the Si self-interstitial the energetically most favorable configuration is the configuration of a substitutional carbon atom, that is the carbon atom occupying the vacant site.
In addition, it is a conceivable configuration the system might experience during the silicon carbide precipitation process.
Energies needed to overcome the migration barrier of the transformation into this configuration enable predictions concerning the feasibility of a silicon carbide conversion mechanism derived from these microscopic processes.
-This is especially important for the case, in which the vacancy is created at position 3, as discussed in the last section and figure \ref{fig:defects:comb_db_06} b).
+This is especially important for the case, in which the vacancy is created at position 3, as displayed in figure \ref{fig:defects:comb_db_06} b).
Due to the low binding energy this configuration might constitute a trap, which it is hard to escape from.
However, migration simulations show that only a low amount of energy is necessary to transform the system into the energetically most favorable configuration.
\begin{figure}[!t!h]
\label{fig:defects:comb_mig_02}
\end{figure}
Figure \ref{fig:defects:comb_mig_01} and \ref{fig:defects:comb_mig_02} show the migration barriers and structures for transitions of the vacancy-interstitial configurations examined in figure \ref{fig:defects:comb_db_06} a) and b) into a configuration of substitutional carbon.
+If the vacancy is created at position 1 the system will end up in a configuration of substitutional C anyways.
-In the first case the focus is on the migration of silicon atom number 1 towards the vacant site created at position 2, while the carbon atom substitutes the site of the migrating silicon atom.
+In the first case the focus is on the migration of silicon atom number 1 towards the vacant site created at position 2 while the carbon atom substitutes the site of the migrating silicon atom.
An energy of 0.6 eV necessary to overcome the migration barrier is found.
This energy is low enough to constitute a feasible mechanism in SiC precipitation.
To reverse this process 5.4 eV are needed, which make this mechanism very unprobable.
Due to this and due to the formation of new bonds, that is the bond of silicon atom number 1 to silicon atom number 5 and the bond of the carbon atom to its siliocn neighbour in the bottom left, a less steep increase of free energy is observed.
At a displacement of approximately 30 \% the bond of silicon atom number 1 to the just recently created siliocn atom is broken up again, which explains the repeated boost in energy.
Finally the system gains energy relaxing into the configuration of zero displacement.
+{\color{red}Todo: Direct migration of C in progress.}
Due to the low binding energy observed, the configuration of the vacancy created at position 3 is assumed to be stable against transition.
However, a relatively simple migration path exists, which intuitively seems to be a low energy process.
In fact, migration simulations yield a barrier as low as 0.1 eV.
This energy is needed to tilt the dumbbell as the displayed structure at 30 \% displacement shows.
Once this barrier is overcome, the carbon atom forms a bond to the top left silicon atom and the interstitial silicon atom capturing the vacant site is forming new tetrahedral bonds to its neighboured silicon atoms.
-These new bonds and the relaxation into the substitutional carbon configuration are responsible for the gain free energy.
-For the reverse process approximately 2.4 eV are nedded, which is 24 times higher than the forward process.
+These new bonds and the relaxation into the substitutional carbon configuration are responsible for the gain in free energy.
+For the reverse process approximately 2.4 eV are needed, which is 24 times higher than the forward process.
Thus, substitutional carbon is assumed to be stable in contrast to the C-Si dumbbell interstitial located next to a vacancy.
-
-{\color{red}Todo: DB mig along 110 (at the starting of this section)?}
-
-{\color{red}Todo: Migration of Si int + vac and C sub/int ...?}
-
-{\color{red}Todo: Model of kick-out and kick-in mechnism?}
\section{Conclusions concerning the SiC conversion mechanism}
-The ground state configuration of a carbon interstitial in crystalline siliocn is found to be the C-Si \hkl<1 0 0> dumbbell interstitial configuration.
-The threefold coordinated carbon and silicon atom share a usual silicon lattice site.
-Migration simulations reveal the carbon interstitial to be mobile at prevailing implantation temperatures requireing an activation energy of approximately 0.9 eV for migration as well as reorientation processes.
+The ground state configuration of a carbon interstitial in crystalline siliocn is found to be the C-Si \hkl<1 0 0> dumbbell interstitial configuration, in which the threefold coordinated carbon and silicon atom share a usual silicon lattice site.
+This supports the assumption of C-Si \hkl<1 0 0>-type dumbbel interstitial formation in the first steps of the IBS process as proposed by the precipitation model introduced in section \ref{section:assumed_prec}.
+
+Migration simulations reveal this carbon interstitial to be mobile at prevailing implantation temperatures requireing an activation energy of approximately 0.9 eV for migration as well as reorientation processes.
+This enables possible migration of the defects to form defect agglomerates as demanded by the model.
+Unfortunately classical potential simulations show tremendously overestimated migration barriers indicating a possible failure of the necessary agglomeration of such defects.
Investigations of two carbon interstitials of the \hkl<1 0 0>-type and varying separations and orientations state an attractive interaction between these interstitials.
Depending on orientation, energetically favorable configurations are found in which these two interstitials are located close together instead of the occurernce of largely separated and isolated defects.
This is due to strain compensation enabled by the combination of such defects in certain orientations.
For dumbbells oriented along the \hkl<1 1 0> direction and the assumption that there is the possibility of free orientation, an interaction energy proportional to the reciprocal cube of the distance in the far field regime is found.
-These findings support the assumption of the C-Si dumbbell agglomeration proposed by the precipitation model introduced in section \ref{section:assumed_prec}.
+These findings support the assumption of the C-Si dumbbell agglomeration proposed by the precipitation model.
-By combination of the \hkl<1 0 0> dumbbell with a vacancy it is found that the configuration of substitutional carbon arising by the carbon interstitial atom occupying the vacant site is the energetically most favorable configuration.
+Next to the C-Si \hkl<1 0 0> dumbbell interstitial configuration, in which the C atom is sharing a Si lattice site with the corresponding Si atom the C atom could occupy the site of the Si atom, which in turn forms a Si self-interstitial.
+Combinations of substitutional C and a \hkl<1 1 0> Si self-interstitial, which is the ground state configuration for a Si self-interstitial and, thus, assumed to be the energetically most favorable configuration for combined structures, show formation energies 0.5 eV to 1.5 eV greater than that of the C-Si \hkl<1 0 0> interstitial configuration, which remains the energetically most favorable configuration.
+However, the binding energy of substitutional C and the Si self-interstitial quickly drops to zero already for short separations indicating a low interaction capture radius.
+Thus, due to missing attractive interaction forces driving the system to form C-Si \hkl<1 0 0> dumbbell interstitial complexes substitutional C, while thermodynamically not stable, constitutes a most likely configuration occuring in IBS, a process far from equlibrium.
+
+Due to the low interaction capture radius substitutional C can be treated independently of the existence of separated Si self-interstitials.
+This should be also true for combinations of C-Si interstitials next to a vacancy and a further separated Si self-interstitial excluded from treatment, which again is a conveivable configuration in IBS.
+By combination of a \hkl<1 0 0> dumbbell with a vacancy in the absence of the Si self-interstitial it is found that the configuration of substitutional carbon occupying the vacant site is the energetically most favorable configuration.
Low migration barriers are necessary to obtain this configuration and in contrast comparatively high activation energies necessary for the reverse process.
Thus, carbon interstitials and vacancies located close together are assumed to end up in such a configuration in which the carbon atom is tetrahedrally coordinated and bound to four silicon atoms as expected in silicon carbide.
-In contrast to the above, this would suggest a silicon carbide precipitation by succesive creation of substitutional carbon instead of the agglomeration of C-Si dumbbell interstitials followed by an abrupt precipitation.
-{\color{red}Todo:
-C atoms in 100 DB tightend in 110 direction, which is important for stress compensation in some combined constellations.
-}
-{\color{red}Todo:
-Most of the combinations should only be thought of intermediate configurations, which need to be transformed in the later SiC precipitation process.
-}
-{\color{red}Todo:
-Better structure, better language, better methodology!
-}
-{\color{red}Todo:
-Fit of lennard-jones an other rep + attr potentials in 110 interaction data!
+While first results support the proposed precipitation model the latter suggest the formation of silicon carbide by succesive creation of substitutional carbon instead of the agglomeration of C-Si dumbbell interstitials followed by an abrupt transition.
+Prevailing conditions in the IBS process at elevated temperatures and the fact that IBS is a nonequilibrium process reinforce the possibility of formation of substitutional C instead of the thermodynamically stable C-Si dumbbell interstitials predicted by simulations at zero Kelvin.
+\label{section:defects:noneq_process_02}
+
+{\color{blue}
+In addition, there are experimental findings, which might be exploited to reinforce the non-validity of the proposed precipitation model.
+High resolution TEM shows equal orientation of \hkl(h k l) planes of the c-Si host matrix and the 3C-SiC precipitate.
+Formation of 3C-SiC realized by successive formation of substitutional C, in which the atoms belonging to one of the two fcc lattices are substituted by C atoms perfectly conserves the \hkl(h k l) planes of the initial c-Si diamond lattice.
+Silicon self-interstitials consecutively created to the same degree are able to diffuse into the c-Si host one after another.
+Investigated combinations of C interstitials, however, result in distorted configurations, in which C atoms, which at some point will form SiC, are no longer aligned to the host.
+It is easily understandable that the mismatch in alignement will increase with increasing defect density.
+In addition, the amount of Si self-interstitials equal to the amount of agglomerated C atoms would be released all of a sudden probably not being able to diffuse into the c-Si host matrix without damaging the Si surrounding or the precipitate itself.
+In addition, IBS results in the formation of the cubic polytype of SiC only.
+As this result conforms well with the model of precipitation by substitutional C there is no obvious reason why hexagonal polytypes should not be able to form or an equal alignement would be mandatory assuming the model of precipitation by C-Si dumbbell agglomeration.
}
+{\color{red}Todo: C mobility higher than Si mobility? -> substitutional C is more likely to arise, since it migrates 'faster' to vacant sites?}
+
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