+For investigating the \si{} structures a Si atom is inserted or removed according to Fig. \ref{fig:basics:ins_pos} of section \ref{section:basics:defects}.
+The formation energies of \si{} configurations are listed in Table \ref{tab:defects:si_self} for both methods used in this work as well as results obtained by other {\em ab initio} studies \cite{al-mushadani03,leung99}.
+\begin{table}[t]
+\begin{center}
+\begin{tabular}{l c c c c c}
+\hline
+\hline
+ & \hkl<1 1 0> DB & H & T & \hkl<1 0 0> DB & V \\
+\hline
+\multicolumn{6}{c}{Present study} \\
+{\textsc vasp} & 3.39 & 3.42 & 3.77 & 4.41 & 3.63 \\
+{\textsc posic} & 4.39 & 4.48$^*$ & 3.40 & 5.42 & 3.13 \\
+\multicolumn{6}{c}{Other {\em ab initio} studies} \\
+Ref. \cite{al-mushadani03} & 3.40 & 3.45 & - & - & 3.53 \\
+Ref. \cite{leung99} & 3.31 & 3.31 & 3.43 & - & - \\
+% todo cite without []
+\hline
+\hline
+\end{tabular}
+\end{center}
+\caption[Formation energies of Si self-interstitials in crystalline Si determined by classical potential MD and DFT calculations.]{Formation energies of Si self-interstitials in crystalline Si determined by classical potential MD and DFT calculations. The formation energies are given in eV. T denotes the tetrahedral and H the hexagonal interstitial configuration. V corresponds to the vacancy configuration. 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:si_self}
+\end{table}
+\begin{figure}[t]
+\begin{center}
+\begin{flushleft}
+\begin{minipage}{5cm}
+\underline{Tetrahedral}\\
+$E_{\text{f}}=3.40\,\text{eV}$\\
+\includegraphics[width=4.0cm]{si_pd_albe/tet.eps}
+\end{minipage}
+\begin{minipage}{10cm}
+\underline{Hexagonal}\\[0.1cm]
+\begin{minipage}{4cm}
+$E_{\text{f}}^*=4.48\,\text{eV}$\\
+\includegraphics[width=4.0cm]{si_pd_albe/hex_a.eps}
+\end{minipage}
+\begin{minipage}{0.8cm}
+\begin{center}
+$\Rightarrow$
+\end{center}
+\end{minipage}
+\begin{minipage}{4cm}
+$E_{\text{f}}=3.96\,\text{eV}$\\
+\includegraphics[width=4.0cm]{si_pd_albe/hex.eps}
+\end{minipage}
+\end{minipage}\\[0.2cm]
+\begin{minipage}{5cm}
+\underline{\hkl<1 0 0> dumbbell}\\
+$E_{\text{f}}=5.42\,\text{eV}$\\
+\includegraphics[width=4.0cm]{si_pd_albe/100.eps}
+\end{minipage}
+\begin{minipage}{5cm}
+\underline{\hkl<1 1 0> dumbbell}\\
+$E_{\text{f}}=4.39\,\text{eV}$\\
+\includegraphics[width=4.0cm]{si_pd_albe/110.eps}
+\end{minipage}
+\begin{minipage}{5cm}
+\underline{Vacancy}\\
+$E_{\text{f}}=3.13\,\text{eV}$\\
+\includegraphics[width=4.0cm]{si_pd_albe/vac.eps}
+\end{minipage}
+\end{flushleft}
+%\hrule
+\end{center}
+\caption[Relaxed Si self-interstitial defect configurations obtained by classical potential calculations.]{Relaxed Si self-interstitial defect configurations obtained by classical potential calculations. The Si atoms and the bonds (only for the interstitial atom) are illustrated by yellow spheres and blue lines.}
+\label{fig:defects:conf}
+\end{figure}
+The final configurations obtained after relaxation are presented in Fig. \ref{fig:defects:conf}.
+The displayed structures are the results of the classical potential simulations.
+
+There are differences between the various results of the quantum-mechanical calculations but the consensus view is that the \hkl<1 1 0> dumbbell (DB) followed by the hexagonal and tetrahedral defect is the lowest in energy.
+This is nicely reproduced by the DFT calculations performed in this work.
+
+It has turned out to be very difficult to capture the results of quantum-mechanical calculations in analytical potential models.
+Among the established analytical potentials only the EDIP \cite{bazant97,justo98} and Stillinger-Weber \cite{stillinger85} potential reproduce the correct order in energy of the defects.
+However, these potenitals show shortcomings concerning the description of other physical properties and are unable to describe the C-C and C-Si interaction.
+In fact the EA potential calculations favor the tetrahedral defect configuration.
+This limitation is assumed to arise due to the cut-off.
+In the tetrahedral configuration the second neighbors are only slightly more distant than the first neighbors, which creates the particular problem.
+Indeed, an increase of the cut-off results in increased values of the formation energies \cite{albe_sic_pot}, which is most significant for the tetrahedral configuration.
+The same issue has already been discussed by Tersoff \cite{tersoff90} with regard to the description of the tetrahedral C defect using his potential.
+While not completely rendering impossible further, more challenging empirical potential studies on large systems, the artifact has to be taken into account in the investigations of defect combinations later on in this chapter.
+
+The hexagonal configuration is not stable opposed to results of the authors of the potential \cite{albe_sic_pot}.
+In the first two pico seconds, while kinetic energy is decoupled from the system, the \si{} seems to condense at the hexagonal site.
+The formation energy of \unit[4.48]{eV} is determined by this low kinetic energy configuration shortly before the relaxation process starts.
+The \si{} atom then begins to slowly move towards an energetically more favorable position very close to the tetrahedral one but slightly displaced along the three coordinate axes.
+The formation energy of \unit[3.96]{eV} for this type of interstitial is equal to the result for the hexagonal one in the original work \cite{albe_sic_pot}.
+Obviously the authors did not carefully check the relaxed results assuming a hexagonal configuration.
+In Fig. \ref{fig:defects:kin_si_hex} the relaxation process is shown on the basis of the kinetic energy plot.
+\begin{figure}[t]
+\begin{center}
+\includegraphics[width=0.7\textwidth]{e_kin_si_hex.ps}
+\end{center}
+\caption{Kinetic energy plot of the relaxation process of the hexagonal silicon self-interstitial defect simulation using the EA potential.}
+\label{fig:defects:kin_si_hex}
+\end{figure}
+To exclude failures in the implementation of the potential or the MD code itself the hexagonal defect structure was double-checked with the {\textsc parcas} MD code \cite{parcas_md}.
+The respective relaxation energetics are likewise plotted and look similar to the energetics obtained by {\textsc posic}.
+In fact, 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 fundamental problems of analytical potential models for describing defect structures.
+However, the energy barrier is small.
+\begin{figure}[ht]
+\begin{center}
+\includegraphics[width=0.7\textwidth]{nhex_tet.ps}
+\end{center}
+\caption{Migration barrier of the tetrahedral Si self-interstitial slightly displaced along all three coordinate axes into the exact tetrahedral configuration using classical potential calculations.}
+\label{fig:defects:nhex_tet_mig}
+\end{figure}
+This is exemplified in Fig. \ref{fig:defects:nhex_tet_mig}, which shows the change in configurational energy during the migration of the interstitial displaced along all three coordinate axes into the tetrahedral configuration.
+The barrier is smaller than \unit[0.2]{eV}.
+Hence, these artifacts have a negligible influence in finite temperature simulations.
+
+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 length of these bonds are, however, close to the cut-off range and thus are weak interactions not constituting actual chemical 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.
+
+
+
+
+\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}.
+Again, the displayed structures are the results obtained by the classical potential calculations.
+The type of reservoir of the C impurity to determine the formation energy of the defect is chosen to be SiC.
+This is consistent with the methods used in the articles \cite{tersoff90,dal_pino93}, which the results are compared to in the following.
+Hence, the chemical potential of Si and C is determined by the cohesive energy of Si and SiC as discussed in section \ref{section:basics:defects}.
+\begin{table}[t]
+\begin{center}
+\begin{tabular}{l c c c c c c}
+\hline
+\hline
+ & T & H & \hkl<1 0 0> DB & \hkl<1 1 0> DB & S & BC \\
+\hline
+\multicolumn{6}{c}{Present study} \\
+ {\textsc posic} & 6.09 & 9.05$^*$ & 3.88 & 5.18 & 0.75 & 5.59$^*$ \\
+ {\textsc vasp} & Unstable & Unstable & 3.72 & 4.16 & 1.95 & 4.66 \\
+\multicolumn{6}{c}{Other studies} \\
+ Tersoff \cite{tersoff90} & 3.8 & 6.7 & 4.6 & 5.9 & 1.6 & 5.3 \\
+ {\em Ab initio} \cite{dal_pino93,capaz94} & - & - & x & - & 1.89 & x+2.1 \\
+\hline
+\hline
+\end{tabular}
+\end{center}
+\caption[Formation energies of C point defects in c-Si determined by classical potential MD and DFT calculations.]{Formation energies of C point defects in c-Si determined by classical potential MD and DFT calculations. The formation energies are given in eV. T denotes the tetrahedral, H the hexagonal and BC the bond-centered interstitial configuration. S corresponds to the substitutional interstitial configuration. The dumbbell configurations are abbreviated by DB. Formation energies for unstable configurations are marked by an asterisk and are 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}[t]
+\begin{center}
+\begin{flushleft}
+\begin{minipage}{4cm}
+\underline{Hexagonal}\\
+$E_{\text{f}}^*=9.05\,\text{eV}$\\
+\includegraphics[width=4.0cm]{c_pd_albe/hex.eps}
+\end{minipage}
+\begin{minipage}{0.8cm}
+\begin{center}
+$\Rightarrow$
+\end{center}
+\end{minipage}
+\begin{minipage}{4cm}
+\underline{\hkl<1 0 0>}\\
+$E_{\text{f}}=3.88\,\text{eV}$\\
+\includegraphics[width=4.0cm]{c_pd_albe/100.eps}
+\end{minipage}
+\begin{minipage}{0.5cm}
+\hfill
+\end{minipage}
+\begin{minipage}{5cm}
+\underline{Tetrahedral}\\
+$E_{\text{f}}=6.09\,\text{eV}$\\
+\includegraphics[width=4.0cm]{c_pd_albe/tet.eps}
+\end{minipage}\\[0.2cm]
+\begin{minipage}{4cm}
+\underline{Bond-centered}\\
+$E_{\text{f}}^*=5.59\,\text{eV}$\\
+\includegraphics[width=4.0cm]{c_pd_albe/bc.eps}
+\end{minipage}
+\begin{minipage}{0.8cm}
+\begin{center}
+$\Rightarrow$
+\end{center}
+\end{minipage}
+\begin{minipage}{4cm}
+\underline{\hkl<1 1 0> dumbbell}\\
+$E_{\text{f}}=5.18\,\text{eV}$\\
+\includegraphics[width=4.0cm]{c_pd_albe/110.eps}
+\end{minipage}
+\begin{minipage}{0.5cm}
+\hfill
+\end{minipage}
+\begin{minipage}{5cm}
+\underline{Substitutional}\\
+$E_{\text{f}}=0.75\,\text{eV}$\\
+\includegraphics[width=4.0cm]{c_pd_albe/sub.eps}
+\end{minipage}
+\end{flushleft}
+\end{center}
+\caption[Relaxed C point defect configurations obtained by classical potential calculations.]{Relaxed C point defect configurations obtained by classical potential calculations. The Si/C atoms and the bonds (only for the interstitial atom) are illustrated by yellow/grey spheres and blue lines.}
+\label{fig:defects:c_conf}
+\end{figure}
+
+\cs{} occupying an already vacant Si lattice site, which is in fact not an interstitial defect, is found to be the lowest configuration in energy for all potential models.
+An experiemntal value of the formation energy of \cs{} was determined by a fit to solubility data yielding a concentration of $3.5 \times 10^{24} \exp{(-2.3\,\text{eV}/k_{\text{B}}T)} \text{ cm}^{-3}$ \cite{bean71}.
+However, there is no particular reason for treating the prefactor as a free parameter in the fit to the experimental data.
+It is simply given by the atomic density of pure silicon, which is $5\times 10^{22}\text{ cm}^{-3}$.
+Tersoff \cite{tersoff90} and Dal Pino et al. \cite{dal_pino93} pointed out that by combining this prefactor with the calculated values for the energy of formation ranging from \unit[1.6-1.89]{eV} an excellent agreement with the experimental solubility data within the entire temeprature range of the experiment is obtained.
+This reinterpretation of the solubility data, first proposed by Tersoff and later on reinforced by Dal~Pino~et~al. is in good agreement with the results of the quantum-mechanical calculations performed in this work.
+Unfortunately the EA potential undervalues the formation energy roughly by a factor of two, which is a definite drawback of the potential.
+
+Except for Tersoff's results for the tedrahedral configuration, the \ci{} \hkl<1 0 0> DB is the energetically most favorable interstital configuration.
+As mentioned above, the low energy of formation for the tetrahedral interstitial in the case of the Tersoff potential is believed to be an artifact of the abrupt cut-off set to \unit[2.5]{\AA} (see Ref. 11 and 13 in \cite{tersoff90}) and the real formation energy is, thus, supposed to be located between \unit[3-10]{eV}.
+Keeping these considerations in mind, the \ci{} \hkl<1 0 0> DB is the most favorable interstitial configuration for all interaction models.
+This finding is in agreement with several theoretical\cite{burnard93,leary97,dal_pino93,capaz94,jones04} and experimental\cite{watkins76,song90} investigations, which all predict this configuration to be the ground state.
+However, no energy of formation for this type of defect based on first-principles calculations has yet been explicitly stated in literature.
+The defect is frequently generated in the classical potential simulation runs, in which C is inserted at random positions in the c-Si matrix.
+In quantum-mechanical simulations the unstable tetrahedral and hexagonal configurations undergo a relaxation into the \ci{} \hkl<1 0 0> DB configuration.
+Thus, this configuration is of great importance and discussed in more detail in section \ref{subsection:100db}.
+It should be noted that EA and DFT predict almost equal formation energies.
+
+The highest energy is observed for the hexagonal interstitial configuration using classical potentials.
+Quantum-mechanical calculations reveal this configuration to be unstable, which is also reproduced by the EA potential.
+In both cases a relaxation towards the \ci{} \hkl<1 0 0> DB configuration is observed.
+Opposed to results of the first-principles calculations, Tersoff finds this configuration to be stable \cite{tersoff90}.
+In fact, the stability of the hexagonal interstitial could not be reproduced in simulations performed in this work using the unmodifed Tersoff potential parameters.
+Unfortunately, apart from the modified parameters, no more conditions specifying the relaxation process are given in Tersoff's study on C point defects in Si.
+
+The tetrahedral is the second most unfavorable interstitial configuration using classical potentials if the abrupt cut-off effect of the Tersoff potential is taken into account.
+Again, quantum-mechanical results reveal this configuration to be unstable.
+The fact that the tetrahedral and hexagonal configurations are the two most unstable configurations in classical potential calculations and, thus, are less likely to arise in MD simulations, acts in concert with the fact that these configurations are found to be unstable in the more accurate quantum-mechanical description.
+
+Just as for \si{}, a \ci{} \hkl<1 1 0> DB configuration exists.
+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 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} 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}
+
+As the \ci{} \hkl<1 0 0> DB constitutes the ground-state configuration of a C atom incorporated into otherwise perfect c-Si it is the most probable and, hence, one of the most important interstitial configurations of C in Si.
+The structure was initially suspected by IR local vibrational mode absorption \cite{bean70} and finally verified by electron paramegnetic resonance (EPR) \cite{watkins76} studies on irradiated Si substrates at low temperatures.
+
+Fig. \ref{fig:defects:100db_cmp} schematically shows the \ci{} \hkl<1 0 0> DB structure and Table \ref{tab:defects:100db_cmp} lists the details of the atomic displacements, distances and bond angles obtained by classical potential and quantum-mechanical calculations.
+For comparison, the obtained structures for both methods are visualized in Fig. \ref{fig:defects:100db_vis_cmp}.
+\begin{figure}[ht]
+\begin{center}
+\includegraphics[width=12cm]{100-c-si-db_cmp.eps}
+\end{center}
+\caption[Sketch of the \ci{} \hkl<1 0 0> dumbbell structure.]{Sketch of the \ci{} \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}[ht]
+\begin{center}
+Displacements\\
+\begin{tabular}{l c c c c c c c c c}
+\hline
+\hline
+ & & & & \multicolumn{3}{c}{Atom 2} & \multicolumn{3}{c}{Atom 3} \\
+ & $a$ & $b$ & $|a|+|b|$ & $\Delta x$ & $\Delta y$ & $\Delta z$ & $\Delta x$ & $\Delta y$ & $\Delta z$ \\
+\hline
+{\textsc posic} & 0.084 & -0.091 & 0.175 & -0.015 & -0.015 & -0.031 & -0.014 & 0.014 & 0.020 \\
+{\textsc vasp} & 0.109 & -0.065 & 0.174 & -0.011 & -0.011 & -0.024 & -0.014 & 0.014 & 0.025 \\
+\hline
+\hline
+\end{tabular}\\[0.5cm]
+\end{center}
+\begin{center}
+Distances\\
+\begin{tabular}{l c c c c c c c c r}
+\hline
+\hline
+ & $r(1C)$ & $r(2C)$ & $r(3C)$ & $r(12)$ & $r(13)$ & $r(34)$ & $r(23)$ & $r(25)$ & $a_{\text{Si}}^{\text{equi}}$\\
+\hline
+{\textsc posic} & 0.175 & 0.329 & 0.186 & 0.226 & 0.300 & 0.343 & 0.423 & 0.425 & 0.543 \\
+{\textsc vasp} & 0.174 & 0.341 & 0.182 & 0.229 & 0.286 & 0.347 & 0.422 & 0.417 & 0.548 \\
+\hline
+\hline
+\end{tabular}\\[0.5cm]
+\end{center}
+\begin{center}
+Angles\\
+\begin{tabular}{l c c c c }
+\hline
+\hline
+ & $\theta_1$ & $\theta_2$ & $\theta_3$ & $\theta_4$ \\
+\hline
+{\textsc posic} & 140.2 & 109.9 & 134.4 & 112.8 \\
+{\textsc vasp} & 130.7 & 114.4 & 146.0 & 107.0 \\
+\hline
+\hline
+\end{tabular}\\[0.5cm]
+\end{center}
+\caption[Atomic displacements, distances and bond angles of the \ci{} \hkl<1 0 0> DB structure obtained by {\textsc posic} and {\textsc vasp} calculations.]{Atomic displacements, distances and bond angles of the \ci{} \hkl<1 0 0> DB structure obtained by {\textsc posic} and {\textsc vasp} calculations. The displacements and distances are given in nm and the angles are given in degrees. Displacements, distances and angles are schematically displayed in Fig. \ref{fig:defects:100db_cmp}. In addition, the equilibrium lattice constant for crystalline Si is listed.}
+\label{tab:defects:100db_cmp}
+\end{table}
+\begin{figure}[ht]
+\begin{center}
+\begin{minipage}{6cm}
+\begin{center}
+\underline{\textsc posic}
+\includegraphics[width=5cm]{c_pd_albe/100_cmp.eps}
+\end{center}
+\end{minipage}
+\begin{minipage}{6cm}
+\begin{center}
+\underline{\textsc vasp}
+\includegraphics[width=5cm]{c_pd_vasp/100_cmp.eps}
+\end{center}
+\end{minipage}
+\end{center}
+\caption{Comparison of the \ci{} \hkl<1 0 0> DB structures obtained by {\textsc posic} and {\textsc vasp} calculations.}
+\label{fig:defects:100db_vis_cmp}
+\end{figure}
+\begin{figure}[ht]
+\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 \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 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.
+Instead, each of the DB atoms forms threefold coordinated bonds, which are located in a plane.
+One bond is formed to the other DB atom.
+The other two bonds are bonds to the two Si edge atoms located in the opposite direction of the DB atom.
+The distance of the two DB atoms is almost the same for both types of calculations.
+However, in the case of the {\textsc vasp} calculation, the DB structure is pushed upwards compared to the results using the EA potential.
+This is easily identified by comparing the values for $a$ and $b$ and the two structures in Fig. \ref{fig:defects:100db_vis_cmp}.
+Thus, the angles of bonds of the Si DB atom ($\theta_1$ and $\theta_2$) are closer to $120^{\circ}$ signifying the predominance of $sp^2$ hybridization.
+On the other hand, the C atom forms an almost collinear bond ($\theta_3$) with the two Si edge atoms implying the predominance of $sp$ bonding.
+This is supported by the image of the charge density isosurface in Fig. \ref{img:defects:charge_den_and_ksl}.
+The two lower Si atoms are $sp^3$ hybridized and form $\sigma$ bonds to the Si DB atom.
+The same is true for the upper two Si atoms and the C DB atom.
+In addition the DB atoms form $\pi$ bonds.
+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 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}
+
+\begin{figure}[ht]
+\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 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 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 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{}
+
+% todo migration of \si{}!
+
+\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}
+\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{Conceivable migration pathways among two \ci{} \hkl<1 0 0> DB configurations.}
+\label{img:defects:c_mig_path}
+\end{figure}
+Three different migration paths are accounted in this work, which are displayed in Fig. \ref{img:defects:c_mig_path}.
+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, 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.
+
+\subsection{Migration paths obtained by first-principles calculations}
+
+\begin{figure}[ht]
+\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> 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 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[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> 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}
+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[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[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 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 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=13cm]{vasp_mig/110_mig_vasp.ps}
+%\begin{picture}(0,0)(140,0)
+%\includegraphics[width=2.2cm]{vasp_mig/00-1_b.eps}
+%\end{picture}
+%\begin{picture}(0,0)(20,0)
+%\includegraphics[width=2.2cm]{vasp_mig/00-1_ip0-10_sp.eps}
+%\end{picture}
+%\begin{picture}(0,0)(-120,0)
+%\includegraphics[width=2.2cm]{vasp_mig/0-10_b.eps}
+%\end{picture}
+\end{center}
+\caption{Migration barriers of the \hkl<1 1 0> 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.
+% 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.
+Indeed less than 0.7 eV are necessary to turn a \hkl<0 -1 0>- to a \hkl<1 1 0>-type C-Si dumbbell interstitial.
+This transition is carried out in place, that is the Si dumbbell pair is not changed and both, the Si and C atom share the initial lattice site.
+Thus, this transition does not contribute to long-range diffusion.
+Once the C atom resides in the \hkl<1 1 0> interstitial configuration it can migrate into the bond-centered configuration by employing approximately 0.95 eV of activation energy, which is only slightly higher than the activation energy needed for the \hkl<0 0 -1> to \hkl<0 -1 0> pathway shown in figure \ref{fig:defects:00-1_0-10_mig}.
+As already known from the migration of the \hkl<0 0 -1> to the bond-centered configuration as discussed in figure \ref{fig:defects:00-1_001_mig} another 0.25 eV are needed to turn back from the bond-centered to a \hkl<1 0 0>-type interstitial.
+However, due to the fact that this migration consists of three single transitions with the second one having an activation energy slightly higher than observed for the direct transition it is considered very unlikely to occur.
+The migration barrier of the \hkl<1 1 0> to \hkl<0 0 -1> transition, in which the C atom is changing its Si partner and, thus, moving to the 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.
+
+As mentioned earlier the procedure to obtain the migration barriers differs from the usually applied procedure in two ways.
+Firstly constraints to move along the displacement direction are applied on all atoms instead of solely constraining the diffusing atom.
+Secondly the constrainted directions are not kept constant to the initial displacement direction.
+Instead they are updated for every displacement step.
+These modifications to the usual procedure are applied to avoid abrupt changes in structure and free energy on the one hand and to make sure the expected final configuration is reached on the other hand.
+Due to applying updated constraints on all atoms the obtained migration barriers and pathes might be overestimated and misguided.
+To reinforce the applicability of the employed technique the obtained activation energies and migration pathes for the \hkl<0 0 -1> to \hkl<0 -1 0> transition are compared to two further migration calculations, which do not update the constrainted direction and which only apply updated constraints on three selected atoms, that is the diffusing C atom and the Si dumbbell pair in the initial and final configuration.
+Results are presented in figure \ref{fig:defects:00-1_0-10_cmp}.
+\begin{figure}[ht]
+\begin{center}
+\includegraphics[width=13cm]{vasp_mig/00-1_0-10_nosym_sp_cmp.ps}
+\end{center}
+\caption[Comparison of three different techniques for obtaining migration barriers and pathways applied to the \hkl<0 0 -1> to \hkl<0 -1 0> C-Si dumbbell transition.]{Comparison of three different techniques for obtaining migration barriers and pathways applied to the \hkl<0 0 -1> to \hkl<0 -1 0> C-Si dumbbell transition.}
+\label{fig:defects:00-1_0-10_cmp}
+\end{figure}
+The method without updating the constraints but still applying them to all atoms shows a delayed crossing of the saddle point.
+This is understandable since the update results in a more aggressive advance towards the final configuration.
+In any case the barrier obtained is slightly higher, which means that it does not constitute an energetically more favorable pathway.
+The method in which the constraints are only applied to the diffusing C atom and two Si atoms, ... {\color{red}Todo: does not work!} ...
+
+\subsection{Migration barriers obtained by classical potential calculations}
+\label{subsection:defects:mig_classical}
+
+The same method for obtaining migration barriers and the same suggested pathways are applied to calculations employing the classical EA potential.
+Since the evaluation of the classical potential and force is less computationally intensive higher amounts of steps can be used.
+The time constant $\tau$ for the Berendsen thermostat is set to 1 fs in order to have direct velocity scaling and with the temperature set to zero Kelvin perform a steepest descent minimazation to drive the system into a local minimum.
+However, in some cases a time constant of 100 fs resuls in lower barriers and, thus, is shown whenever appropriate.
+
+\begin{figure}[ht]
+\begin{center}
+\includegraphics[width=13cm]{bc_00-1.ps}\\[5.6cm]
+\begin{pspicture}(0,0)(0,0)
+\psframe[linecolor=red,fillstyle=none](-7,2.7)(7.2,6)
+\end{pspicture}
+\begin{picture}(0,0)(140,-100)
+\includegraphics[width=2.4cm]{albe_mig/bc_00-1_red_00.eps}
+\end{picture}
+\begin{picture}(0,0)(10,-100)
+\includegraphics[width=2.4cm]{albe_mig/bc_00-1_red_01.eps}
+\end{picture}
+\begin{picture}(0,0)(-120,-100)
+\includegraphics[width=2.4cm]{albe_mig/bc_00-1_red_02.eps}
+\end{picture}
+\begin{picture}(0,0)(25,-80)
+\includegraphics[width=2.5cm]{110_arrow.eps}
+\end{picture}
+\begin{picture}(0,0)(215,-100)
+\includegraphics[height=2.2cm]{001_arrow.eps}
+\end{picture}\\
+\begin{pspicture}(0,0)(0,0)
+\psframe[linecolor=blue,fillstyle=none](-7,-0.5)(7.2,2.8)
+\end{pspicture}
+\begin{picture}(0,0)(160,-10)
+\includegraphics[width=2.2cm]{albe_mig/bc_00-1_01.eps}
+\end{picture}
+\begin{picture}(0,0)(100,-10)
+\includegraphics[width=2.2cm]{albe_mig/bc_00-1_02.eps}
+\end{picture}
+\begin{picture}(0,0)(10,-10)
+\includegraphics[width=2.2cm]{albe_mig/bc_00-1_03.eps}
+\end{picture}
+\begin{picture}(0,0)(-120,-10)
+\includegraphics[width=2.2cm]{albe_mig/bc_00-1_04.eps}
+\end{picture}
+\begin{picture}(0,0)(25,10)
+\includegraphics[width=2.5cm]{100_arrow.eps}
+\end{picture}
+\begin{picture}(0,0)(215,-10)
+\includegraphics[height=2.2cm]{010_arrow.eps}
+\end{picture}
+\end{center}
+\caption{Migration barrier and structures of the bond-centered to \hkl<0 0 -1> dumbbell transition using the classical EA potential.}
+\label{fig:defects:cp_bc_00-1_mig}
+% red: ./visualize -w 640 -h 480 -d saves/c_in_si_mig_bc_00-1_s20 -nll -0.56 -0.56 -0.7 -fur 0.2 0.2 0.0 -c 0.75 -1.25 -0.25 -L -0.25 -0.25 -0.25 -r 0.6 -B 0.1
+% blue: ./visualize -w 640 -h 480 -d saves/c_in_si_mig_bc_00-1_s20_tr100/ -nll -0.56 -0.56 -0.7 -fur 0.2 0.2 0.0 -c 0.0 -0.25 1.0 -L 0.0 -0.25 -0.25 -r 0.6 -B 0.1
+\end{figure}
+Figure \ref{fig:defects:cp_bc_00-1_mig} shows the migration barrier and corresponding structures of the bond-centered to \hkl<0 0 -1> dumbbell transition.
+Since the bond-centered configuration is unstable relaxing into the \hkl<1 1 0> C-Si dumbbell interstitial configuration within this potential the low kinetic energy state is used as a starting configuration.
+Depending on the time constant activation energies of 2.4 eV and 2.2 eV respectively are obtained.
+The migration path obtained by simulations with a time constant of 1 fs remains in the \hkl(1 1 0) plane.
+Using 100 fs as a time constant the C atom breaks out of the \hkl(1 1 0) plane already at the beginning of the migration accompanied by a reduction in energy.
+The energy barrier of this path is 0.2 eV lower in energy than the direct migration within the \hkl(1 1 0) plane.
+However, the investigated pathways cover an activation energy approximately twice as high as the one obtained by quantum-mechanical calculations.
+For the entire transition of the \hkl<0 0 -1> into the \hkl<0 0 1> configuration by passing the bond-centered configuration an additional activation energy of 0.5 eV is necessary to escape from the bond-centered and reach the \hkl<0 0 1> configuration.