Periodic boundary conditions in each direction are applied.
All point defects are calculated for the neutral charge state.
-\begin{figure}[h]
+\begin{figure}[th]
\begin{center}
\includegraphics[width=9cm]{unit_cell_e.eps}
\end{center}
Quantum-mechanical total-energy calculations are an invalueable tool to investigate the energetic and structural properties of point defects since they are experimentally difficult to assess.
The formation energies of some of the silicon self-interstitial configurations are listed in table \ref{tab:defects:si_self} for both methods used in this work as well as results obtained by former studies \cite{leung99}.
-\begin{table}[h]
+\begin{table}[th]
\begin{center}
\begin{tabular}{l c c c c c}
\hline
\label{tab:defects:si_self}
\end{table}
The final configurations obtained after relaxation are presented in figure \ref{fig:defects:conf}.
-\begin{figure}[h]
+\begin{figure}[t!h!]
\begin{center}
%\hrule
%\vspace*{0.2cm}
The formation energy of 3.96 eV for this type of interstitial is equal to the result for the hexagonal one in the original work \cite{albe_sic_pot}.
Obviously the authors did not carefully check the relaxed results assuming a hexagonal configuration.
In figure \ref{fig:defects:kin_si_hex} the relaxation process is shown on the basis of the kinetic energy plot.
-\begin{figure}[h]
+\begin{figure}[th]
\begin{center}
\includegraphics[width=10cm]{e_kin_si_hex.ps}
\end{center}
The same type of interstitial arises using random insertions.
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 (DAS MAL DURCHRECHNEN).
+However, the energy barrier is small.
+{\color{red}Todo: Check!}
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 same applies to the bonds between the interstitial and the upper two atoms in the \hkl<1 1 0> dumbbell configuration.
A more detailed description of the chemical bonding is achieved by quantum-mechanical calculations by investigating the accumulation of negative charge between the nuclei.
-Todo: Plot the electron density for these types of defect to derive conclusions of existing bonds ...
+{\color{red}Todo: Plot the electron density for these types of defect to derive conclusions of existing bonds?}
\section{Carbon related point defects}
The type of reservoir of the carbon impurity to determine the formation energy of the defect was chosen to be SiC.
This is consistent with the methods used in the articles \cite{tersoff90,dal_pino93}, which the results are compared to in the following.
Hence, the chemical potential of silicon and carbon is determined by the cohesive energy of silicon and silicon carbide.
-\begin{table}[h]
+\begin{table}[th]
\begin{center}
\begin{tabular}{l c c c c c c}
\hline
\caption[Formation energies of carbon point defects in crystalline silicon determined by classical potential molecular dynamics and density functional calculations.]{Formation energies of carbon point defects in crystalline silicon determined by classical potential molecular dynamics and density functional calculations. The formation energies are given in eV. T denotes the tetrahedral, H the hexagonal, B the bond-centered and S the substitutional interstitial configuration. The dumbbell configurations are abbreviated by DB. Formation energies for unstable configurations are marked by an asterisk and determined by using the low kinetic energy configuration shortly before the relaxation into the more favorable configuration starts.}
\label{tab:defects:c_ints}
\end{table}
-\begin{figure}[h]
+\begin{figure}[th]
\begin{center}
\begin{flushleft}
\begin{minipage}{4cm}
\end{minipage}
\begin{minipage}{4cm}
\underline{\hkl<1 0 0>}\\
-$E_{\text{f}}=3.96\text{ eV}$\\
+$E_{\text{f}}=3.88\text{ eV}$\\
\includegraphics[width=4.0cm]{c_pd_albe/100.eps}
\end{minipage}
\begin{minipage}{0.5cm}
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.
-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.
+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 system moves into the \hkl<1 1 0> interstitial configuration.
As the \hkl<1 0 0> dumbbell interstitial is the lowest configuration in energy it is the most probable hence important interstitial configuration of carbon in silicon.
It was first identified by infra-red (IR) spectroscopy \cite{bean70} and later on by electron paramagnetic resonance (EPR) spectroscopy \cite{watkins76}.
-Figure \ref{fig:defects:100db_cmp} schematically shows the \hkl<1 0 0> dumbbell structure and table \ref{tab:defects:100db_cmp} lists the details of displacements obtained by analytical potential and quantum-mechanical calculations.
-\begin{figure}[h]
+Figure \ref{fig:defects:100db_cmp} schematically shows the \hkl<1 0 0> dumbbell structure and table \ref{tab:defects:100db_cmp} lists the details of the atomic displacements, distances and bond angles obtained by analytical potential and quantum-mechanical calculations.
+For comparison, the obtained structures for both methods visualized out of the atomic position data are presented in figure \ref{fig:defects:100db_vis_cmp}.
+\begin{figure}[th]
\begin{center}
\includegraphics[width=12cm]{100-c-si-db_cmp.eps}
\end{center}
-\caption[Sketch of the \hkl<1 0 0> dumbbell structure.]{Sketch of the \hkl<1 0 0> dumbbell structure. Atomic displacements and distances are listed in table \ref{tab:defects:100db_cmp}.}
+\caption[Sketch of the \hkl<1 0 0> dumbbell structure.]{Sketch of the \hkl<1 0 0> dumbbell structure. Atomic displacements, distances and bond angles are listed in table \ref{tab:defects:100db_cmp}.}
\label{fig:defects:100db_cmp}
\end{figure}
%
-\begin{table}[h]
+\begin{table}[th]
\begin{center}
Displacements\\
\begin{tabular}{l c c c c c c c c c}
\hline
& $\theta_1$ & $\theta_2$ & $\theta_3$ & $\theta_4$ \\
\hline
-Erhard/Albe & 0.175 & 0.329 & 0.186 & 0.226 \\
-VASP & 0.174 & 0.341 & 0.182 & 0.229 \\
+Erhard/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 and distances of the \hkl<1 0 0> dumbbell structure obtained by the Erhard/Albe potential and VASP calculations.]{Atomic displacements and distances 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 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 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.}
\label{tab:defects:100db_cmp}
\end{table}
+\begin{figure}[t!h!]
+\begin{center}
+\begin{minipage}{6cm}
+\begin{center}
+\underline{Erhard/Albe}
+\includegraphics[width=5cm]{c_pd_albe/100_cmp.eps}
+\end{center}
+\end{minipage}
+\begin{minipage}{6cm}
+\begin{center}
+\underline{VASP}
+\includegraphics[width=5cm]{c_pd_vasp/100_cmp.eps}
+\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.}
+\label{fig:defects:100db_vis_cmp}
+\end{figure}
+\begin{figure}[th]
+\begin{center}
+\includegraphics[height=10cm]{c_pd_vasp/eden.eps}
+\includegraphics[height=12cm]{c_pd_vasp/100_2333_ksl.ps}
+\end{center}
+\caption[Charge density isosurface and Kohn-Sham levels of the C \hkl<1 0 0> dumbbell structure obtained by VASP calculations.]{Charge density isosurface and Kohn-Sham levels of the C \hkl<1 0 0> dumbbell structure obtained by VASP calculations. Yellow and grey spheres correspond to silicon and carbon 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 silicon atom numbered '1' and the C atom compose the dumbbell structure.
They share the lattice site which is indicated by the dashed red circle and which they are displaced from by length $a$ and $b$ respectively.
The atoms no longer have four tetrahedral bonds to the silicon atoms located on the alternating opposite edges of the cube.
-Instead, each of the dumbbell atoms forms threefold coordinated bonds, whcih are located in a plane.
+Instead, each of the dumbbell atoms forms threefold coordinated bonds, which are located in a plane.
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.
-Angles ...
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.
+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.
+This is supported by the image of the charge density isosurface in figure \ref{img:defects:charge_den_and_ksl}.
+The two lower Si atoms are $sp^3$ hybridised and form $\sigma$ bonds to the silicon dumbbell atom.
+The same is true for the upper two silicon atoms and the C dumbbell atom.
+In addition the dumbbell atoms form $\pi$ bonds.
+However, due to the increased electronegativity of the carbon atom the electron density is attracted by and thus localized around the carbon 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 0.75 eV can be estimated from the Kohn-Sham level diagram for plain silicon.
\subsection{Bond-centered interstitial configuration}
\label{subsection:bc}
+\begin{figure}[t!h!]
+\begin{center}
+\begin{minipage}{8cm}
+\includegraphics[width=8cm]{c_pd_vasp/bc_2333.eps}\\
+\hrule
+\vspace*{0.2cm}
+\includegraphics[width=8cm]{c_100_mig_vasp/im_spin_diff.eps}
+\end{minipage}
+\begin{minipage}{7cm}
+\includegraphics[width=7cm]{c_pd_vasp/bc_2333_ksl.ps}
+\end{minipage}
+\end{center}
+\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 Erhard/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}.
+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.
+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.
+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}
\label{subsection:100mig}
+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.
+
+\begin{figure}[t!h!]
+\begin{center}
+\begin{minipage}{15cm}
+\underline{\hkl<0 0 -1> $\rightarrow$ \hkl<0 0 1>}\\
+\begin{minipage}{4.5cm}
+\includegraphics[width=4.5cm]{c_pd_vasp/100_2333.eps}
+\end{minipage}
+\begin{minipage}{0.5cm}
+$\rightarrow$
+\end{minipage}
+\begin{minipage}{4.5cm}
+\includegraphics[width=4.5cm]{c_pd_vasp/bc_2333.eps}
+\end{minipage}
+\begin{minipage}{0.5cm}
+$\rightarrow$
+\end{minipage}
+\begin{minipage}{4.5cm}
+\includegraphics[width=4.5cm]{c_pd_vasp/100_next_2333.eps}
+\end{minipage}
+\end{minipage}\\
+\begin{minipage}{15cm}
+\underline{\hkl<0 0 -1> $\rightarrow$ \hkl<0 -1 0>}\\
+\begin{minipage}{4.5cm}
+\includegraphics[width=4.5cm]{c_pd_vasp/100_2333.eps}
+\end{minipage}
+\begin{minipage}{0.5cm}
+$\rightarrow$
+\end{minipage}
+\begin{minipage}{4.5cm}
+\includegraphics[width=4.5cm]{c_pd_vasp/00-1-0-10_2333.eps}
+\end{minipage}
+\begin{minipage}{0.5cm}
+$\rightarrow$
+\end{minipage}
+\begin{minipage}{4.5cm}
+\includegraphics[width=4.5cm]{c_pd_vasp/0-10_2333.eps}
+\end{minipage}
+\end{minipage}\\
+\begin{minipage}{15cm}
+\underline{\hkl<0 0 -1> $\rightarrow$ \hkl<0 -1 0> (in place)}\\
+\begin{minipage}{4.5cm}
+\includegraphics[width=4.5cm]{c_pd_vasp/100_2333.eps}
+\end{minipage}
+\begin{minipage}{0.5cm}
+$\rightarrow$
+\end{minipage}
+\begin{minipage}{4.5cm}
+\includegraphics[width=4.5cm]{c_pd_vasp/00-1_ip0-10_2333.eps}
+\end{minipage}
+\begin{minipage}{0.5cm}
+$\rightarrow$
+\end{minipage}
+\begin{minipage}{4.5cm}
+\includegraphics[width=4.5cm]{c_pd_vasp/0-10_ip_2333.eps}
+\end{minipage}
+\end{minipage}
+\end{center}
+\caption{Migration pathways of the carbon \hkl<1 0 0> interstitial dumbbell in silicon.}
+\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.
+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.
+{\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.}
+
+\begin{figure}[t!h!]
+\begin{center}
+\includegraphics[width=13cm]{im_00-1_nosym_sp_fullct_thesis.ps}\\[1.5cm]
+\begin{picture}(0,0)(150,0)
+\includegraphics[width=2.5cm]{vasp_mig/00-1.eps}
+\end{picture}
+\begin{picture}(0,0)(-10,0)
+\includegraphics[width=2.5cm]{vasp_mig/bc_00-1_sp.eps}
+\end{picture}
+\begin{picture}(0,0)(-120,0)
+\includegraphics[width=2.5cm]{vasp_mig/bc.eps}
+\end{picture}
+\begin{picture}(0,0)(25,20)
+\includegraphics[width=2.5cm]{110_arrow.eps}
+\end{picture}
+\begin{picture}(0,0)(200,0)
+\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.}
+\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.
+
+\begin{figure}[t!h!]
+\begin{center}
+\includegraphics[width=13cm]{vasp_mig/00-1_0-10_nosym_sp_fullct.ps}\\[1.6cm]
+\begin{picture}(0,0)(140,0)
+\includegraphics[width=2.5cm]{vasp_mig/00-1_a.eps}
+\end{picture}
+\begin{picture}(0,0)(20,0)
+\includegraphics[width=2.5cm]{vasp_mig/00-1_0-10_sp.eps}
+\end{picture}
+\begin{picture}(0,0)(-120,0)
+\includegraphics[width=2.5cm]{vasp_mig/0-10.eps}
+\end{picture}
+\begin{picture}(0,0)(25,20)
+\includegraphics[width=2.5cm]{100_arrow.eps}
+\end{picture}
+\begin{picture}(0,0)(200,0)
+\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.}
+\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}.
+
+\begin{figure}[t!h!]
+\begin{center}
+\includegraphics[width=13cm]{vasp_mig/00-1_ip0-10_nosym_sp_fullct.ps}\\[1.8cm]
+\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}
+\begin{picture}(0,0)(25,20)
+\includegraphics[width=2.5cm]{100_arrow.eps}
+\end{picture}
+\begin{picture}(0,0)(200,0)
+\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.}
+\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}.
+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 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.
+
\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 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}
+\begin{minipage}{7.5cm}
+\includegraphics[width=7cm]{comb_pos.eps}
+% ./visualize_contcar -w 640 -h 480 -d results/.../CONTCAR -nll -0.20 -0.20 -0.6 -fur 1.2 1.2 0.6 -c 0.5 -1.5 0.3 -L 0.5 0 0 -r 0.6 -m 3.0 0.0 0.0 0.0 3.0 0.0 0.0 0.0 3.0 -A -1 2.465
+\end{minipage}
+\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:
+\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]
+ $\frac{1}{2}\hkl<0 -1 -1>$, $\frac{1}{2}\hkl<-1 0 -1>$
+ \item $\frac{1}{4}\hkl<1 -1 1>$, $\frac{1}{4}\hkl<-1 1 1>$
+ \item $\frac{1}{4}\hkl<-1 1 -3>$, $\frac{1}{4}\hkl<1 -1 -3>$
+ \item $\frac{1}{2}\hkl<-1 -1 0>$, $\frac{1}{2}\hkl<1 1 0>$
+\end{enumerate}
+\end{minipage}\\
+\begin{picture}(0,0)(190,20)
+\includegraphics[width=2.3cm]{100_arrow.eps}
+\end{picture}
+\begin{picture}(0,0)(220,0)
+\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.}
+\label{fig:defects:pos_of_comb}
+\end{figure}
+\begin{table}[th]
+\begin{center}
+\begin{tabular}{l c c c c c c}
+\hline
+\hline
+ & 1 & 2 & 3 & 4 & 5 & R\\
+ \hline
+ \hkl<0 0 -1> & {\color{red}-0.08} & -1.15 & {\color{red}-0.08} & 0.04 & -1.66 & -0.19\\
+ \hkl<0 0 1> & 0.34 & 0.004 & -2.05 & 0.26 & -1.53 & -0.19\\
+ \hkl<0 -1 0> & {\color{orange}-2.39} & -0.17 & {\color{green}-0.10} & {\color{blue}-0.27} & {\color{magenta}-1.88} & {\color{gray}-0.05}\\
+ \hkl<0 1 0> & {\color{cyan}-2.25} & -1.90 & {\color{cyan}-2.25} & {\color{purple}-0.12} & {\color{violet}-1.38} & {\color{yellow}-0.06}\\
+ \hkl<-1 0 0> & {\color{orange}-2.39} & -0.36 & {\color{cyan}-2.25} & {\color{purple}-0.12} & {\color{magenta}-1.88} & {\color{gray}-0.05}\\
+ \hkl<1 0 0> & {\color{cyan}-2.25} & -2.16 & {\color{green}-0.10} & {\color{blue}-0.27} & {\color{violet}-1.38} & {\color{yellow}-0.06}\\
+ \hline
+ C substitutional (C$_{\text{S}}$) & 0.26 & -0.51 & -0.93 & -0.15 & 0.49 & -0.05\\
+ Vacancy & -5.39 ($\rightarrow$ C$_{\text{S}}$) & -0.59 & -3.14 & -0.54 & -0.50 & -0.31\\
+\hline
+\hline
+\end{tabular}
+\end{center}
+\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.
+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}
+E_{\text{b}}=
+E_{\text{f}}^{\text{defect combination}}-
+E_{\text{f}}^{\text{C \hkl<0 0 -1> dumbbell}}-
+E_{\text{f}}^{\text{2nd defect}}
+\label{eq:defects:e_of_comb}
+\end{equation}
+with $E_{\text{f}}^{\text{defect combination}}$ being the formation energy of the defect combination, $E_{\text{f}}^{\text{C \hkl<0 0 -1> dumbbell}}$ being the formation energy of the C \hkl<0 0 -1> dumbbell interstitial defect and $E_{\text{f}}^{\text{2nd defect}}$ being the formation energy of the second defect.
+For defects far away from each other the formation energy of the defect combination should approximately become the sum of the formation energies of the individual defects without an interaction resulting in $E_{\text{b}}=0$.
+Thus, $E_{\text{b}}$ can be best thought of a binding energy, which is required to bring the defects to infinite separation.
+In fact, a \hkl<0 0 -1> dumbbell interstitial created at position R with a distance of $\frac{a_{\text{Si}}}{2}\hkl<3 2 3>$ ($\approx 12.8$ \AA) from the initial one results in an energy as low as -0.19 eV.
+There is still a low interaction which is due to the equal orientation of the defects.
+By changing the orientation of the second dumbbell interstitial to the \hkl<0 -1 0>-type the interaction is even mor reduced resulting in an energy of $E_{\text{b}}=-0.05\text{ eV}$ for a distance, which is the maximum that can be realized due to periodic boundary conditions.
+The energies obtained in the R column of table \ref{eq:defects:e_of_comb} are used as a reference to identify, whether less distanced defects of the same type are favorable or unfavorable compared to the far-off located defect.
+Configurations wih energies greater than zero or the reference value are energetically unfavorable and expose a repulsive interaction.
+These configurations are unlikely to arise or to persist for non-zero temperatures.
+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.
+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}[t!h!]
+\begin{center}
+\begin{minipage}[t]{7cm}
+a) \underline{$E_{\text{b}}=-2.25\text{ eV}$}
+\begin{center}
+\includegraphics[width=6cm]{00-1dc/2-25.eps}
+\end{center}
+\end{minipage}
+\begin{minipage}[t]{7cm}
+b) \underline{$E_{\text{b}}=-2.39\text{ eV}$}
+\begin{center}
+\includegraphics[width=6cm]{00-1dc/2-39.eps}
+\end{center}
+\end{minipage}
+\end{center}
+\caption{Relaxed structures of defect complexes obtained by creating a) \hkl<1 0 0> and b) \hkl<0 -1 0> dumbbels at position 1.}
+\label{fig:defects:comb_db_01}
+\end{figure}
+Figure \ref{fig:defects:comb_db_01} shows the structure of these two configurations.
+The displayed configurations are realized by creating a \hkl<1 0 0> (a)) and \hkl<0 -1 0> (b)) dumbbell at position 1.
+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.
+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?}
+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 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.
+{\color{red}Todo: Is this conf really benificial for SiC prec?}
+
+\begin{figure}[t!h!]
+\begin{center}
+\begin{minipage}[t]{5cm}
+a) \underline{$E_{\text{b}}=-2.16\text{ eV}$}
+\begin{center}
+\includegraphics[width=4.8cm]{00-1dc/2-16.eps}
+\end{center}
+\end{minipage}
+\begin{minipage}[t]{5cm}
+b) \underline{$E_{\text{b}}=-1.90\text{ eV}$}
+\begin{center}
+\includegraphics[width=4.8cm]{00-1dc/1-90.eps}
+\end{center}
+\end{minipage}
+\begin{minipage}[t]{5cm}
+c) \underline{$E_{\text{b}}=-2.05\text{ eV}$}
+\begin{center}
+\includegraphics[width=4.8cm]{00-1dc/2-05.eps}
+\end{center}
+\end{minipage}
+\end{center}
+\caption{Relaxed structures of defect complexes obtained by creating a a) \hkl<1 0 0> and b) \hkl<0 1 0> dumbbell at position 2 and a c) \hkl<0 0 1> dumbbel at position 3.}
+\label{fig:defects:comb_db_02}
+\end{figure}
+Figure \ref{fig:defects:comb_db_02} shows the next three most energetically favorable configurations.
+The relaxed configuration obtained by creating a second \hkl<1 0 0> dumbbell at position 2 is shown in figure \ref{fig:defects:comb_db_02} a).
+A binding energy of -2.16 eV is observed.
+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.
+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.
+Both carbon atoms form tetrahedral bonds to four silicon atoms.
+However, silicon atom 1 and 3, which are bond to the second carbon dumbbell interstitial are also bond to the initial carbon atom.
+These four atoms of the rhomboid reside in a plane and, thus, do not match the situation in silicon carbide.
+The carbon atoms have a distance of 2.75 \AA.
+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.
+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.
+
+Configurations obtained by adding a second dumbbell interstitial at position 4 are characterized by minimal changes from their initial creation condition during relaxation.
+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.
+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.
+Nonparallel orientations, that is the \hkl<0 1 0>, \hkl<0 -1 0> and its equivalents, result in binding energies of -0.12 eV and -0.27 eV, thus, constituting energetically favorable configurations.
+The reduction of strain energy is higher in the second case where the carbon atom of the second dumbbell is placed in the direction pointing away from the initial carbon atom.
+
+\begin{figure}[t!h!]
+\begin{center}
+\begin{minipage}[t]{7cm}
+a) \underline{$E_{\text{b}}=-1.53\text{ eV}$}
+\begin{center}
+\includegraphics[width=6.0cm]{00-1dc/1-53.eps}
+\end{center}
+\end{minipage}
+\begin{minipage}[t]{7cm}
+b) \underline{$E_{\text{b}}=-1.66\text{ eV}$}
+\begin{center}
+\includegraphics[width=6.0cm]{00-1dc/1-66.eps}
+\end{center}
+\end{minipage}\\[0.2cm]
+\begin{minipage}[t]{7cm}
+c) \underline{$E_{\text{b}}=-1.88\text{ eV}$}
+\begin{center}
+\includegraphics[width=6.0cm]{00-1dc/1-88.eps}
+\end{center}
+\end{minipage}
+\begin{minipage}[t]{7cm}
+d) \underline{$E_{\text{b}}=-1.38\text{ eV}$}
+\begin{center}
+\includegraphics[width=6.0cm]{00-1dc/1-38.eps}
+\end{center}
+\end{minipage}
+\end{center}
+\caption{Relaxed structures of defect complexes obtained by creating a a) \hkl<0 0 1>, a b) \hkl<0 0 -1>, a c) \hkl<0 -1 0> and a d) \hkl<1 0 0> dumbbell at position 5.}
+\label{fig:defects:comb_db_05}
+\end{figure}
+Energetically beneficial configurations of defect complexes are observed for second interstititals of all orientations placed at position 5, a position two bonds away from the initial interstitial along the \hkl<1 1 0> direction.
+Relaxed structures of these complexes are displayed in figure \ref{fig:defects:comb_db_05}.
+Figure \ref{fig:defects:comb_db_05} a) and b) show the relaxed structures of \hkl<0 0 1> and \hkl<0 0 -1> dumbbells.
+The upper dumbbell atoms are pushed towards each other forming fourfold coordinated bonds.
+While the displacements of the silicon atoms in case b) are symmetric to the \hkl(1 1 0) plane, in case a) the silicon atom of the initial dumbbel is pushed a little further in the direction of the carbon atom of the second dumbbell than the carbon atom is pushed towards the silicon atom.
+The bottom atoms of the dumbbells remain in threefold coordination.
+The symmetric configuration is energetically more favorable ($E_{\text{b}}=-1.66\text{ eV}$) since the displacements of the atoms is less than in the antiparallel case ($E_{\text{b}}=-1.53\text{ eV}$).
+In figure \ref{fig:defects:comb_db_05} 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 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).
+The far-off atom of the second dumbbell resides in threefold coordination.
+
+Assuming that it is possible for the system to minimize free energy by an in place reorientation of the dumbbell at any position the minimum energy orientation of dumbbells along the \hkl<1 1 0> direction and the resulting C-C distance is shown in table \ref{tab:defects:comb_db110}.
+\begin{table}[t!h!]
+\begin{center}
+\begin{tabular}{l c c c c c c}
+\hline
+\hline
+ & 1 & 2 & 3 & 4 & 5 & 6\\
+\hline
+$E_{\text{b}}$ [eV] & -2.39 & -1.88 & -0.59 & -0.31 & -0.24 & -0.21 \\
+C-C distance [\AA] & 1.4 & 4.6 & 6.5 & 8.6 & 10.5 & 10.8 \\
+Type & \hkl<-1 0 0> & \hkl<1 0 0> & \hkl<1 0 0> & \hkl<1 0 0> & \hkl<1 0 0> & \hkl<1 0 0>, \hkl<0 -1 0>\\
+\hline
+\hline
+\end{tabular}
+\end{center}
+\caption{Binding energy and type of the minimum energy configuration of an additional dumbbell with respect to the separation distance in bonds along the \hkl<1 1 0> direction and the C-C distance.}
+\label{tab:defects:comb_db110}
+\end{table}
+\begin{figure}[t!h!]
+\begin{center}
+\includegraphics[width=12.5cm]{db_along_110.ps}\\
+\includegraphics[width=12.5cm]{db_along_110_cc.ps}
+\end{center}
+\caption{Minimum binding energy of dumbbell combinations with respect to the separation distance in bonds along \hkl<1 1 0> and C-C distance.}
+\label{fig:defects:comb_db110}
+\end{figure}
+Figure \ref{fig:defects:comb_db110} shows the corresponding plot of the data including a cubic spline interplation and a suitable fitting curve.
+The funtion found most suitable for curve fitting is $f(x)=a/x^3$ comprising the single fit parameter $a$.
+Thus, far-off located dumbbells show an interaction proportional to the reciprocal cube of the distance and the amount of bonds along \hkl<1 1 0> respectively.
+This behavior is no longer valid for the immediate vicinity revealed by the saturating binding energy of a second dumbbell at position 1, which was ignored in the fitting procedure.
+
+{\color{red}Todo: DB mig along 110?}
+
+{\color{red}Todo: Si int and C sub ...}
+
+{\color{red}Todo: Model of kick-out and kick-in mechnism?}
+
+{\color{red}Todo: Jahn-Teller distortion (vacancy) $\rightarrow$ actually three possibilities! :(}
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