+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}[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.
+Todo: \hkl<1 1 0> to \hkl<1 0 0> and bond-centerd configuration (in progress).
+Todo: \hkl<1 1 0> to \hkl<0 -1 0> (rotation of the DB, in progress).
+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.
+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}[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}[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}[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.