The barrier is smaller than \unit[0.2]{eV}.
Hence, these artifacts have a negligible influence in finite temperature simulations.
-The bond-centered configuration is unstable and, thus, is not listed.
+The bond-centered (BC) configuration is unstable and, thus, is not listed.
The \si{} \hkl<1 0 0> DB constitutes the most unfavorable configuration for both, the EA and {\textsc vasp} calculations.
+The quantum-mechanical treatment of the \si{} \hkl<1 0 0> DB demands for spin polarized calculations.
+The same applies for the vacancy.
+In the \si{} \hkl<1 0 0> DB configuration the net spin up density is localized in two caps at each of the two DB atoms perpendicularly aligned to the bonds to the other two Si atoms respectively.
+For the vacancy the net spin up electron density is localized in caps at the four surrounding Si atoms directed towards the vacant site.
+No other intrinsic defect configuration, within the ones that are mentioned, is affected by spin polarization.
In the case of the classical potential simulations bonds between atoms are displayed if there is an interaction according to the potential model, i.e. if the distance of two atoms is within the cut-off radius $S_{ij}$ 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.
The same applies to the bonds between the interstitial and the upper two atoms in the \si{} \hkl<1 1 0> DB configuration.
A more detailed description of the chemical bonding is achieved through quantum-mechanical calculations by investigating the accumulation of negative charge between the nuclei.
-%\clearpage{}
-%\cleardoublepage{}
+
+
\section{Carbon point defects in silicon}
+\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}.
The relaxed configurations are visualized in Fig. \ref{fig:defects:c_conf}.
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 \si{}, a \ci{} \hkl<1 1 0> DB configuration exists.
-For the EA potential the formation energy is situated in the same order as found by quantum-mechanical results.
+It constitutes the second most favorable configuration, reproduced by both methods.
Similar structures arise in both types of simulations.
The Si and C atom share a regular Si lattice site aligned along the \hkl<1 1 0> direction.
The C atom is slightly displaced towards the next nearest Si atom located in the opposite direction with respect to the site-sharing Si atom and even forms a bond with this atom.
-The \ci{} \hkl<1 1 0> DB structure is energetically followed by the bond-centered configuration.
-However, even though EA based results yield the same difference in energy with respect to the \hkl<1 1 0> defect as DFT does, the bond-centered configuration is found to be a unstable within the EA description.
-The bond-centered configuration relaxes into the \ci{} \hkl<1 1 0> DB configuration.
-This, like in the hexagonal case, is also true for the unmodified Tersoff potential and the given relaxation conditions.
+The \ci{} \hkl<1 1 0> DB structure is energetically followed by the BC configuration.
+However, even though EA based results yield the same difference in energy with respect to the \hkl<1 1 0> defect as DFT does, the BC configuration is found to be a unstable within the EA description.
+The BC configuration descends into the \ci{} \hkl<1 1 0> DB configuration.
+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}.
-In another {\em ab inito} study Capaz et al. \cite{capaz94} determined this configuration as 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.
-In calculations performed in this work the bond-centered 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, which is discussed in section \ref{subsection:100mig}.
+In another {\em ab inito} study, Capaz~et~al. \cite{capaz94} in turn found 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 neglection 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}.
+
+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.
+Both methods (EA and DFT) predict the \ci{} \hkl<1 0 0> DB configuration to be most stable.
+Also the remaining defects and their energetical order are described fairly well.
+It is thus concluded that, so far, modeling of the SiC precipitation by the EA potential might lead to trustable results.
\subsection[C \hkl<1 0 0> dumbbell interstitial configuration]{\boldmath C \hkl<1 0 0> dumbbell interstitial configuration}
\label{subsection:100db}
\caption[Charge density isosurface and Kohn-Sham levels of the \ci{} \hkl<1 0 0> DB structure obtained by {\textsc vasp} calculations.]{Charge density isosurface and Kohn-Sham levels of the \ci{} \hkl<1 0 0> DB structure obtained by {\textsc vasp} calculations. Yellow and grey spheres correspond to Si and C atoms. The blue surface is the charge density isosurface. In the energy level diagram red and green lines and dots mark occupied and unoccupied states.}
\label{img:defects:charge_den_and_ksl}
\end{figure}
-The Si atom numbered '1' and the C atom compose the DB structure.
+The Si atom labeled '1' and the C atom compose the DB structure.
They share the lattice site which is indicated by the dashed red circle.
They are displaced from the regular lattice site by length $a$ and $b$ respectively.
The atoms no longer have four tetrahedral bonds to the Si atoms located on the alternating opposite edges of the cube.
However, due to the increased electronegativity of the C atom the electron density is attracted by and, thus, localized around the C atom.
In the same figure the Kohn-Sham levels are shown.
There is no magnetization density.
-An acceptor level arises at approximately $E_v+0.35\,\text{eV}$ while a band gap of about \unit[0.75]{eV} can be estimated from the Kohn-Sham level diagram for plain silicon.
-However, these values have to be ...
+An acceptor level arises at approximately $E_v+0.35\,\text{eV}$ while a band gap of about \unit[0.75]{eV} can be estimated from the Kohn-Sham level diagram for plain Si.
+However, strictly speaking, the Kohn-Sham levels and orbitals do not have a direct physical meaning and, thus, these values have to be taken with care.
+% todo band gap problem
\subsection{Bond-centered interstitial configuration}
\label{subsection:bc}
\caption[Structure, charge density isosurface and Kohn-Sham level diagram of the bond-centered interstitial configuration.]{Structure, charge density isosurface and Kohn-Sham level diagram of the bond-centered interstitial configuration. Gray, green and blue surfaces mark the charge density of spin up, spin down and the resulting spin up electrons in the charge density isosurface, in which the carbon atom is represented by a red sphere. In the energy level diagram red and green lines mark occupied and unoccupied states.}
\label{img:defects:bc_conf}
\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 EA potential simulations which reveal this configuration to be unstable relaxing into the \hkl<1 1 0> configuration.
+In the BC insterstitial configuration the interstitial atom is located inbetween two next neighbored Si atoms forming linear bonds.
+In a previous study this configuration was found to constitute an intermediate saddle point configuration determining the migration barrier of one possibe migration path of a \ci{} \hkl<1 0 0> DB configuration into an equivalent one \cite{capaz94}.
+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 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}.
-After slightly displacing the carbon 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 resulting structures relax back into the bond-centered configuration.
-As we will see in later migration simulations the same would happen to structures where the carbon atom is displaced along the migration direction, which approximately is the \hkl<1 1 0> direction.
-These relaxations indicate that the bond-cenetered configuration is a real local minimum instead of an assumed saddle point configuration.
-Figure \ref{img:defects:bc_conf} shows the structure, the charge density isosurface and the Kohn-Sham levels of the bond-centered configuration.
-The linear bonds of the carbon atom to the two silicon atoms indicate the $sp$ hybridization of the carbon atom.
-Two electrons participate to the linear $\sigma$ bonds with the silicon neighbours.
+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> interstitital 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.
+Fig. \ref{img:defects:bc_conf} shows the structure, charge density isosurface and Kohn-Sham levels of the BC configuration.
+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 figure \ref{img:defects:bc_conf}.
-The blue torus, reinforcing the assumption of the p orbital, illustrates the resulting spin up electron density.
+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.
+% todo smaller images, therefore add mo image
\clearpage{}
\cleardoublepage{}
-\section{Migration of the carbon interstitials}
-\label{subsection:100mig}
+% todo migration of \si{}!
-In the following the problem of interstitial carbon migration in silicon is considered.
-Since the carbon \hkl<1 0 0> dumbbell interstitial is the most probable hence most important configuration the migration simulations focus on this defect.
+\section{Migration of the carbon interstitial}
+\label{subsection:100mig}
+A measure for the mobility of interstitial C is the activation energy necessary for the migration from one stable position to another.
+The stable defect geometries have been discussed in the previous subsection.
+In the following, the problem of interstitial C migration in Si is considered.
+Since the \ci{} \hkl<1 0 0> DB is the most probable, hence, most important configuration, the migration of this defect atom from one site of the Si host lattice to a neighboring site is in the focus of investigation.
\begin{figure}[ht]
\begin{center}
\begin{minipage}{15cm}
\end{minipage}
\end{minipage}
\end{center}
-\caption{Migration pathways of the carbon \hkl<1 0 0> interstitial dumbbell in silicon.}
+\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 shown in figure \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> dumbbell interstitial configuration.
-During this migration the carbon atom is changing its silicon dumbbell partner.
-The new partner is the one located at $\frac{a}{4}\hkl<1 1 -1>$ relative to the initial one.
-Two of the three bonds to the next neighboured silicon atoms are preserved while the breaking of the third bond and the accompanying formation of a new bond is observed.
-The carbon atom resides in the \hkl(1 1 0) plane.
-This transition involves an intermediate bond-centerd configuration.
-Results discussed in \ref{subsection:bc} indicate, that the bond-ceneterd configuration is a real local minimum.
-Thus, the \hkl<0 0 -1> to \hkl<0 0 1> migration can be thought of a two-step mechanism in which the intermediate bond-cenetered configuration constitutes a metastable configuration.
-Due to symmetry it is enough to consider the transition from the bond-centered to the \hkl<1 0 0> configuration or vice versa.
-In the second path, the carbon atom is changing its silicon partner atom as in path one.
-However, the trajectory of the carbon atom is no longer proceeding in the \hkl(1 1 0) plane.
-The orientation of the new dumbbell configuration is transformed from \hkl<0 0 -1> to \hkl<0 -1 0>.
-Again one bond is broken while another one is formed.
+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.
+Two of the three bonds to the next neighbored Si atoms are preserved while the breaking of the third bond and the accompanying formation of a new bond is observed.
+The C atom resides in the \hkl(1 1 0) plane.
+This transition involves the intermediate BC configuration.
+However, results discussed in the previous section indicate that the BC configuration is a real local minimum.
+Thus, the \hkl<0 0 -1> to \hkl<0 0 1> migration can be thought of a two-step mechanism in which the intermediate BC configuration constitutes a metastable configuration.
+Due to symmetry it is enough to consider the transition from the BC to the \hkl<1 0 0> configuration or vice versa.
+In the second path, the C atom is changing its Si partner atom as in path one.
+However, the trajectory of the C atom is no longer proceeding in the \hkl(1 1 0) plane.
+The orientation of the new DB configuration is transformed from \hkl<0 0 -1> to \hkl<0 -1 0>.
+Again, one bond is broken while another one is formed.
As a last migration path, the defect is only changing its orientation.
-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.
-
-\subsection{Migration barriers obtained by quantum-mechanical calculations}
+Thus, this path is not responsible for long-range migration.
+The Si DB partner remains the same.
+The bond to the face-centered Si 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.
-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.
+\subsection{Migration paths obtained by first-principles calculations}
\begin{figure}[ht]
\begin{center}
\includegraphics[height=2.2cm]{001_arrow.eps}
\end{picture}
\end{center}
-\caption[Migration barrier and structures of the \hkl<0 0 -1> dumbbell (left) to bond-centered (right) transition.]{Migration barrier and structures of the \hkl<0 0 -1> dumbbell (left) to bond-centered (right) transition. Bonds of the carbon atoms are illustrated by blue lines.}
+\caption[Migration barrier and structures of the \hkl<0 0 -1> DB (left) to BC (right) 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 figure \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> dumbbell to the bond-ceneterd configuration is displayed.
-To reach the bond-centered configuration, which is 0.94 eV higher in energy than the \hkl<0 0 -1> dumbbell configuration, an energy barrier of approximately 1.2 eV, given by the saddle point structure at a displacement of 60 \%, has to be passed.
-This amount of energy is needed to break the bond of the carbon atom to the silicon atom at the bottom left.
-In a second process 0.25 eV of energy are needed for the system to revert into a \hkl<1 0 0> configuration.
+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.
\begin{figure}[ht]
\begin{center}
\includegraphics[height=2.2cm]{001_arrow.eps}
\end{picture}
\end{center}
-\caption[Migration barrier and structures of the \hkl<0 0 -1> dumbbell (left) to the \hkl<0 -1 0> dumbbell (right) transition.]{Migration barrier and structures of the \hkl<0 0 -1> dumbbell (left) to the \hkl<0 -1 0> dumbbell (right) transition. Bonds of the carbon atoms are illustrated by blue lines.}
+\caption[Migration barrier and structures of the \hkl<0 0 -1> DB (left) to the \hkl<0 -1 0> DB (right) 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}
-Figure \ref{fig:defects:00-1_0-10_mig} shows the migration barrier and structures of the \hkl<0 0 -1> to \hkl<0 -1 0> dumbbell transition.
-The resulting migration barrier of approximately 0.9 eV is very close to the experimentally obtained values of 0.73 \cite{song90} and 0.87 eV \cite{tipping87}.
+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}[ht]
\begin{center}
\includegraphics[height=2.2cm]{001_arrow.eps}
\end{picture}
\end{center}
-\caption[Migration barrier and structures of the \hkl<0 0 -1> dumbbell (left) to the \hkl<0 -1 0> dumbbell (right) transition in place.]{Migration barrier and structures of the \hkl<0 0 -1> dumbbell (left) to the \hkl<0 -1 0> dumbbell (right) transition in place. Bonds of the carbon atoms are illustrated by blue lines.}
+\caption[Reorientation barrier and structures of the \hkl<0 0 -1> DB (left) to the \hkl<0 -1 0> DB (right) 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 dumbbell is changing its orientation is shown in figure \ref{fig:defects:00-1_0-10_ip_mig}.
-An energy barrier of roughly 1.2 eV is observed.
-Experimentally measured activation energies for reorientation range from 0.77 eV to 0.88 eV \cite{watkins76,song90}.
+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.
-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 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.
+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 C interstitial in Si explaining both, annealing and reorientation experiments.
+The activation energy of roughly \unit[0.9]{eV} nicely compares to experimental values reinforcing the correct identification of the C-Si DB diffusion mechanism.
+Slightly increased values compared to experiment might be due to the tightend constraints applied in the modified CRT approach.
+Nevertheless, the theoretical description performed in this work is improved compared to a former study \cite{capaz94}, which underestimates the experimental value by \unit[35]{\%}.
+In addition, it is finally shown that the BC configuration, for which spin polarized calculations are necessary, constitutes a real local minimum instead of a saddle point configuration due to the presence of restoring forces for displacements in migration direction.
\begin{figure}[ht]
\begin{center}
%\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.}
+\caption{Migration barriers of the \hkl<1 1 0> DB to BC (blue), \hkl<0 0 -1> (green) and \hkl<0 -1 0> (in place, red) C-Si DB 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.
+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.
+% HIER WEITER
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.
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.
+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 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.
\caption{Migration barrier of the \hkl<0 0 -1> to \hkl<0 -1 0> C-Si dumbbell transition in place using the classical EA 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.
+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 neighbored 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.
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.
+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 neighbored 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.
\begin{minipage}{6.0cm}
\underline{Positions given in $a_{\text{Si}}$}\\[0.3cm]
Initial interstitial I: $\frac{1}{4}\hkl<1 1 1>$\\
-Relative silicon neighbour positions:
+Relative silicon neighbor positions:
\begin{enumerate}
\item $\frac{1}{4}\hkl<1 1 -1>$, $\frac{1}{4}\hkl<-1 -1 -1>$
\item $\frac{1}{2}\hkl<1 0 1>$, $\frac{1}{2}\hkl<0 1 -1>$,\\[0.2cm]
\includegraphics[height=2.2cm]{001_arrow.eps}
\end{picture}
\end{center}
-\caption[\hkl<0 0 -1> dumbbell interstitial defect and positions of next neighboured silicon atoms used for the second defect.]{\hkl<0 0 -1> dumbbell interstitial defect and positions of next neighboured silicon atoms used for the second defect. Two possibilities exist for red numbered atoms and four possibilities exist for blue numbered atoms.}
+\caption[\hkl<0 0 -1> dumbbell interstitial defect and positions of next neighbored silicon atoms used for the second defect.]{\hkl<0 0 -1> dumbbell interstitial defect and positions of next neighbored silicon atoms used for the second defect. Two possibilities exist for red numbered atoms and four possibilities exist for blue numbered atoms.}
\label{fig:defects:pos_of_comb}
\end{figure}
\begin{table}[ht]
\caption[Energetic results of defect combinations.]{Energetic results of defect combinations. The given energies in eV are defined by equation \eqref{eq:defects:e_of_comb}. Equivalent configurations are marked by identical colors. The first column lists the types of the second defect combined with the initial \hkl<0 0 -1> dumbbell interstitial. The position index of the second defect is given in the first row according to figure \ref{fig:defects:pos_of_comb}. R is the position located at $\frac{a_{\text{Si}}}{2}\hkl<3 2 3>$ relative to the initial defect, which is the maximum realizable distance due to periodic boundary conditions.}
\label{tab:defects:e_of_comb}
\end{table}
-Figure \ref{fig:defects:pos_of_comb} shows the initial \hkl<0 0 -1> dumbbell interstitial defect and the positions of next neighboured silicon atoms used for the second defect.
+Figure \ref{fig:defects:pos_of_comb} shows the initial \hkl<0 0 -1> dumbbell interstitial defect and the positions of next neighbored silicon atoms used for the second defect.
Table \ref{tab:defects:e_of_comb} summarizes energetic results obtained after relaxation of the defect combinations.
The energy of interest $E_{\text{b}}$ is defined to be
\begin{equation}
Energies below zero and below the reference value indicate configurations favored compared to configurations in which these point defects are separated far away from each other.
Investigating the first part of table \ref{tab:defects:e_of_comb}, namely the combinations with another \hkl<1 0 0>-type interstitial, most of the combinations result in energies below zero.
-Surprisingly the most favorable configurations are the ones with the second defect created at the very next silicon neighbour and a change in orientation compared to the initial one.
+Surprisingly the most favorable configurations are the ones with the second defect created at the very next silicon neighbor and a change in orientation compared to the initial one.
This leads to the conclusion that an agglomeration of C-Si dumbbell interstitials as proposed by the precipitation model introduced in section \ref{section:assumed_prec} is indeed an energetically favored configuration of the system.
The reason for nearby interstitials being favored compared to isolated ones is most probably the reduction of strain energy enabled by combination in contrast to the strain energy created by two individual defects.
\begin{figure}[ht]
Structure \ref{fig:defects:comb_db_01} b) is the energetically most favorable configuration.
After relaxation the initial configuration is still evident.
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.
+This bond has a length of 1.38 \AA{} close to the nex neighbor 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 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.
+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 neighbors, a configuration expected in silicon carbide.
The two carbon atoms spaced by 2.70 \AA{} do not form a bond but anyhow reside in a shorter distance as expected in silicon carbide.
The Si atom numbered 2 is pushed towards the carbon atom, which results in the breaking of the bond to atom 4.
The breaking of the $\sigma$ bond is indeed confirmed by investigating the charge density isosurface of this configuration.
After relaxation the second dumbbell is aligned along \hkl<1 1 0>.
The bond of the silicon atoms 1 and 2 does not persist.
Instead the silicon atom forms a bond with the initial carbon interstitial and the second carbon atom forms a bond with silicon atom 1 forming four bonds in total.
-The carbon atoms are spaced by 3.14 \AA, which is very close to the expected C-C next neighbour distance of 3.08 \AA{} in silicon carbide.
+The carbon atoms are spaced by 3.14 \AA, which is very close to the expected C-C next neighbor distance of 3.08 \AA{} in silicon carbide.
Figure \ref{fig:defects:comb_db_02} c) displays the results of another \hkl<0 0 1> dumbbell inserted at position 3.
The binding energy is -2.05 eV.
Both dumbbells are tilted along the same direction remaining parallely aligned and the second dumbbell is pushed downwards in such a way, that the four dumbbell atoms form a rhomboid.
In figure \ref{fig:defects:comb_db_02} b) a second \hkl<0 1 0> dumbbell is constructed at position 2.
An energy of -1.90 eV is observed.
The initial dumbbell and especially the carbon atom is pushed towards the silicon atom of the second dumbbell forming an additional fourth bond.
-Silicon atom number 1 is pulled towards the carbon atoms of the dumbbells accompanied by the disappearance of its bond to silicon number 5 as well as the bond of silicon number 5 to its next neighboured silicon atom in \hkl<1 1 -1> direction.
-The carbon atom of the second dumbbell forms threefold coordinated bonds to its silicon neighbours.
+Silicon atom number 1 is pulled towards the carbon atoms of the dumbbells accompanied by the disappearance of its bond to silicon number 5 as well as the bond of silicon number 5 to its next neighbored silicon atom in \hkl<1 1 -1> direction.
+The carbon atom of the second dumbbell forms threefold coordinated bonds to its silicon neighbors.
A distance of 2.80 \AA{} is observed for the two carbon atoms.
Again, the two carbon atoms and its two interconnecting silicon atoms form a rhomboid.
C-C distances of 2.70 to 2.80 \AA{} seem to be characteristic for such configurations, in which the carbon atoms and the two interconnecting silicon atoms reside in a plane.
There is a low interaction of the dumbbells, which seem to exist independent of each other.
This, on the one hand, becomes evident by investigating the final structure, in which both of the dumbbells essentially retain the structure expected for a single dumbbell and on the other hand is supported by the observed binding energies which vary only slightly around zero.
This low interaction is due to the larger distance and a missing direct connection by bonds along a crystallographic direction.
-Both carbon and silicon atoms of the dumbbells form threefold coordinated bonds to their next neighbours.
+Both carbon and silicon atoms of the dumbbells form threefold coordinated bonds to their next neighbors.
The energetically most unfavorable configuration ($E_{\text{b}}=0.26\text{ eV}$) is obtained for the \hkl<0 0 1> interstitial oppositely orientated to the initial one.
A dumbbell taking the same orientation as the initial one is less unfavorble ($E_{\text{b}}=0.04\text{ eV}$).
Both configurations are unfavorable compared to far-off isolated dumbbells.
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 figure \ref{fig:defects:comb_db_03} c) and d) the nonparallel orientations, namely the \hkl<0 -1 0> and \hkl<1 0 0> dumbbells are shown.
Binding energies of -1.88 eV and -1.38 eV are obtained for the relaxed structures.
-In both cases the silicon atom of the initial interstitial is pulled towards the near by atom of the second dumbbell so that both atoms form fourfold coordinated bonds to their next neighbours.
+In both cases the silicon atom of the initial interstitial is pulled towards the near by atom of the second dumbbell so that both atoms form fourfold coordinated bonds to their next neighbors.
In case c) it is the carbon and in case d) the silicon atom of the second interstitial which forms the additional bond with the silicon atom of the initial interstitial.
The atom of the second dumbbell, the carbon atom of the initial dumbbell and the two interconnecting silicon atoms again reside in a plane.
A typical C-C distance of 2.79 \AA{} is, thus, observed for case c).
\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.
+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 neighbored 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.
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.
The migration path is best described by the reverse process.
-Starting at 100 \% energy is needed to break the bonds of silicon atom 1 to its neighboured silicon atoms and that of the carbon atom to silicon atom number 5.
+Starting at 100 \% energy is needed to break the bonds of silicon atom 1 to its neighbored silicon atoms and that of the carbon atom to silicon atom number 5.
At a displacement of 60 \% these bonds are broken.
-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.
+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 neighbor 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.}
The migration path and the corresponding differences in free energy are displayed in figure \ref{fig:defects:comb_mig_02}.
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.
+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 neighbored silicon atoms.
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.
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