started with 110 - x mig (albe)

[lectures/latex.git] / posic / thesis / defects.tex

\chapter{Point defects in silicon}

Given the conversion mechnism of SiC in crystalline silicon introduced in section \ref{section:assumed_prec} the understanding of carbon and silicon interstitial point defects in c-Si is of great interest.
Both types of defects are examined in the following both by classical potential as well as density functional theory calculations.

In case of the classical potential calculations a simulation volume of nine silicon lattice constants in each direction is used.
Calculations are performed in an isothermal-isobaric NPT ensemble.
Coupling to the heat bath is achieved by the Berendsen thermostat with a time constant of 100 fs.
The temperature is set to zero Kelvin.
Pressure is controlled by a Berendsen barostat \cite{berendsen84} again using a time constant of 100 fs and a bulk modulus of 100 GPa for silicon.
To exclude surface effects periodic boundary conditions are applied.

Due to the restrictions in computer time three silicon lattice constants in each direction are considered sufficiently large enough for DFT calculations.
The ions are relaxed by a conjugate gradient method.
The cell volume and shape is allowed to change using the pressure control algorithm of Parrinello and Rahman \cite{parrinello81}.
Periodic boundary conditions in each direction are applied.
All point defects are calculated for the neutral charge state.

\begin{figure}[th]
\begin{center}
\includegraphics[width=9cm]{unit_cell_e.eps}
\end{center}
\caption[Insertion positions for the tetrahedral ({\color{red}$\bullet$}), hexagonal  ({\color{green}$\bullet$}), \hkl<1 0 0> dumbbell ({\color{yellow}$\bullet$}), \hkl<1 1 0> dumbbell ({\color{magenta}$\bullet$}) and bond-centered ({\color{cyan}$\bullet$}) interstitial configuration.]{Insertion positions for the tetrahedral ({\color{red}$\bullet$}), hexagonal  ({\color{green}$\bullet$}), \hkl<1 0 0> dumbbell ({\color{yellow}$\bullet$}), \hkl<1 1 0> dumbbell ({\color{magenta}$\bullet$}) and bond-centered ({\color{cyan}$\bullet$}) interstitial configuration. The black dots ({\color{black}$\bullet$}) correspond to the silicon atoms and the blue lines ({\color{blue}-}) indicate the covalent bonds of the perfect c-Si structure.}
\label{fig:defects:ins_pos}
\end{figure}

The interstitial atom positions are displayed in figure \ref{fig:defects:ins_pos}.
In seperated simulation runs the silicon or carbon atom is inserted at the
\begin{itemize}
 \item tetrahedral, $\vec{r}=(0,0,0)$, ({\color{red}$\bullet$})
 \item hexagonal, $\vec{r}=(-1/8,-1/8,1/8)$, ({\color{green}$\bullet$})
 \item nearly \hkl<1 0 0> dumbbell, $\vec{r}=(-1/4,-1/4,-1/8)$, ({\color{yellow}$\bullet$})
 \item nearly \hkl<1 1 0> dumbbell, $\vec{r}=(-1/8,-1/8,-1/4)$, ({\color{magenta}$\bullet$})
 \item bond-centered, $\vec{r}=(-1/8,-1/8,-3/8)$, ({\color{cyan}$\bullet$})
\end{itemize}
interstitial position.
For the dumbbell configurations the nearest silicon atom is displaced by $(0,0,-1/8)$ and $(-1/8,-1/8,0)$ respectively of the unit cell length to avoid too high forces.
A vacancy or a substitutional atom is realized by removing one silicon atom and switching the type of one silicon atom respectively.

From an energetic point of view the free energy of formation $E_{\text{f}}$ is suitable for the characterization of defect structures.
For defect configurations consisting of a single atom species the formation energy is defined as
\begin{equation}
E_{\text{f}}=\left(E_{\text{coh}}^{\text{defect}}
                  -E_{\text{coh}}^{\text{defect-free}}\right)N
\label{eq:defects:ef1}
\end{equation}
where $N$ and $E_{\text{coh}}^{\text{defect}}$ are the number of atoms and the cohesive energy per atom in the defect configuration and $E_{\text{coh}}^{\text{defect-free}}$ is the cohesive energy per atom of the defect-free structure.
The formation energy of defects consisting of two or more atom species is defined as
\begin{equation}
E_{\text{f}}=E-\sum_i N_i\mu_i
\label{eq:defects:ef2}
\end{equation}
where $E$ is the free energy of the interstitial system and $N_i$ and $\mu_i$ are the amount of atoms and the chemical potential of species $i$.
The chemical potential is determined by the cohesive energy of the structure of the specific type in equilibrium at zero Kelvin.
For a defect configuration of a single atom species equation \eqref{eq:defects:ef2} is equivalent to equation \eqref{eq:defects:ef1}.

\section{Silicon self-interstitials}

Point defects in silicon have been extensively studied, both experimentally and theoretically \cite{fahey89,leung99}.
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}[th]
\begin{center}
\begin{tabular}{l c c c c c}
\hline
\hline
 & T & H & \hkl<1 0 0> DB & \hkl<1 1 0> DB & V \\
\hline
 Erhard/Albe MD & 3.40 & 4.48$^*$ & 5.42 & 4.39 & 3.13 \\
 VASP & 3.77 & 3.42 & 4.41 & 3.39 & 3.63 \\
 LDA \cite{leung99} & 3.43 & 3.31 & - & 3.31 & - \\
 GGA \cite{leung99} & 4.07 & 3.80 & - & 3.84 & - \\
\hline
\hline
\end{tabular}
\end{center}
\caption[Formation energies of silicon self-interstitials in crystalline silicon determined by classical potential molecular dynamics and density functional calculations.]{Formation energies of silicon self-interstitials 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 V the vacancy 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:si_self}
\end{table}
The final configurations obtained after relaxation are presented in figure \ref{fig:defects:conf}.
\begin{figure}[t!h!]
\begin{center}
%\hrule
%\vspace*{0.2cm}
%\begin{flushleft}
%\begin{minipage}{5cm}
%\underline{\hkl<1 1 0> dumbbell}\\
%$E_{\text{f}}=3.39\text{ eV}$\\
%\includegraphics[width=3.0cm]{si_pd_vasp/110_2333.eps}
%\end{minipage}
%\begin{minipage}{5cm}
%\underline{Hexagonal}\\
%$E_{\text{f}}=3.42\text{ eV}$\\
%\includegraphics[width=3.0cm]{si_pd_vasp/hex_2333.eps}
%\end{minipage}
%\begin{minipage}{5cm}
%\underline{Tetrahedral}\\
%$E_{\text{f}}=3.77\text{ eV}$\\
%\includegraphics[width=3.0cm]{si_pd_vasp/tet_2333.eps}
%\end{minipage}\\[0.2cm]
%\begin{minipage}{5cm}
%\underline{\hkl<1 0 0> dumbbell}\\
%$E_{\text{f}}=4.41\text{ eV}$\\
%\includegraphics[width=3.0cm]{si_pd_vasp/100_2333.eps}
%\end{minipage}
%\begin{minipage}{5cm}
%\underline{Vacancy}\\
%$E_{\text{f}}=3.63\text{ eV}$\\
%\includegraphics[width=3.0cm]{si_pd_vasp/vac_2333.eps}
%\end{minipage}
%\begin{minipage}{5cm}
%\begin{center}
%VASP\\
%calculations\\
%\end{center}
%\end{minipage}
%\end{flushleft}
%\vspace*{0.2cm}
%\hrule
\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 silicon self-interstitial defect configurations obtained by classical potential calculations.]{Relaxed silicon self-interstitial defect configurations obtained by classical potential calculations. The silicon atoms and the bonds (only for the interstitial atom) are illustrated by yellow spheres and blue lines.}
\label{fig:defects:conf}
\end{figure}

There are differences between the various results of the quantum-mechanical calculations but the consensus view is that the \hkl<1 1 0> dumbbell 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 Erhard/Albe potential calculations favor the tetrahedral defect configuration.
The hexagonal configuration is not stable opposed to results of the authors of the potential \cite{albe_sic_pot}.
In the first two pico seconds while kinetic energy is decoupled from the system the Si interstitial seems to condense at the hexagonal site.
The formation energy of 4.48 eV is determined by this low kinetic energy configuration shortly before the relaxation process starts.
The Si interstitial 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 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}[th]
\begin{center}
\includegraphics[width=10cm]{e_kin_si_hex.ps}
\end{center}
\caption{Kinetic energy plot of the relaxation process of the hexagonal silicon self-interstitial defect simulation using the Erhard/Albe classical potential.}
\label{fig:defects:kin_si_hex}
\end{figure}
To exclude failures in the implementation of the potential or the MD code itself the hexagonal defect structure was double-checked with the PARCAS MD code \cite{parcas_md}.
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.
\begin{figure}[th]
\begin{center}
\includegraphics[width=12cm]{nhex_tet.ps}
\end{center}
\caption{Migration barrier of the tetrahedral Si self-interstitial slightly displaced along all three coordinate axes into the exact tetrahedral configuration using classical potential calculations.}
\label{fig:defects:nhex_tet_mig}
\end{figure}
This is exemplified in figure \ref{fig:defects:nhex_tet_mig}, which shows the change in potential energy during the migration of the interstitial displaced along all three coordinate axes into the tetrahedral configuration.
The technique used to obtain the migration data is explained in a later section (\ref{subsection:100mig}).
The barrier is less than 0.2 eV.
Hence these artifacts should have a negligent influence in finite temperature simulations.

The bond-centered configuration is unstable and the \hkl<1 0 0> dumbbell interstitial is the most unfavorable configuration for both, the Erhard/Albe and VASP calculations.

In the case of the classical potential simulations bonds between atoms are displayed if there is an interaction according to the potential model, that is if the distance of two atoms is within the cutoff region $S_{ij}$ introduced in equation \eqref{eq:basics:fc}.
For the tetrahedral and the slightly displaced configurations four bonds to the atoms located in the center of the planes of the unit cell exist in addition to the four tetrahedral bonds.
The length of these bonds are, however, close to the cutoff 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 \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.
{\color{red}Todo: Plot the electron density for these types of defect to derive conclusions of existing bonds?}

\section{Carbon related point defects}

Carbon is a common and technologically important impurity in silicon.
Concentrations as high as $10^{18}\text{ cm}^{-3}$ occur in Czochralski-grown silicon samples.
It is well established that carbon and other isovalent impurities prefer to dissolve substitutionally in silicon.
However, radiation damage can generate carbon interstitials \cite{watkins76} which have enough mobility at room temeprature to migrate and form defect complexes.

Formation energies of the most common carbon point defects in crystalline silicon are summarized in table \ref{tab:defects:c_ints} and the relaxed configurations obtained by classical potential calculations visualized in figure \ref{fig:defects:c_conf}.
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}[th]
\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 & B \\
\hline
 Erhard/Albe MD & 6.09 & 9.05$^*$ & 3.88 & 5.18 & 0.75 & 5.59$^*$ \\
 %VASP & unstable & unstable & 3.15 & 3.60 & 1.39 & 4.10 \\
 VASP & unstable & unstable & 3.72 & 4.16 & 1.95 & 4.66 \\
 Tersoff \cite{tersoff90} & 3.8 & 6.7 & 4.6 & 5.9 & 1.6 & 5.3 \\
 ab initio & - & - & x & - & 1.89 \cite{dal_pino93} & x+2.1 \cite{capaz94} \\
\hline
\hline
\end{tabular}
\end{center}
\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}[th]
\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 carbon point defect configurations obtained by classical potential calculations.]{Relaxed carbon point defect configurations obtained by classical potential calculations. The silicon/carbon 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}

Substitutional carbon in silicon is found to be the lowest configuration in energy for all potential models.
An experiemntal value of the formation energy of substitutional carbon 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 1.6 to 1.89 eV an excellent agreement with the experimental solubility data within the entire temeprature range of the experiment is obtained.
This reinterpretation of the solubility data, first proposed by Tersoff and later on reinforced by Dal Pino et al. is in good agreement with the results of the quantum-mechanical calculations performed in this work.
Unfortunately the Erhard/Albe potential undervalues the formation energy roughly by a factor of two.

Except for Tersoff's tedrahedral configuration results the \hkl<1 0 0> dumbbell is the energetically most favorable interstital configuration.
The low energy of formation for the tetrahedral interstitial in the case of the Tersoff potential is believed to be an artifact of the abrupt cutoff set to 2.5 \AA{} (see ref. 11 and 13 in \cite{tersoff90}) and the real formation energy is, thus, supposed to be located between 3 and 10 eV.
Keeping these considerations in mind, the \hkl<1 0 0> dumbbell is the most favorable interstitial configuration for all interaction models.
In addition to the theoretical results compared to in table \ref{tab:defects:c_ints} there is experimental evidence of the existence of this configuration \cite{watkins76}.
It is frequently generated in the classical potential simulation runs in which carbon 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 \hkl<1 0 0> dumbbell configuration.
Thus, this configuration is of great importance and discussed in more detail in section \ref{subsection:100db}.

The highest energy is observed for the hexagonal interstitial configuration using classical potentials.
Quantum-mechanical calculations reveal this configuration to be unstable, which is also reproduced by the Erhard/Albe potential.
In both cases a relaxation towards the \hkl<1 0 0> dumbbell configuration is observed.
In fact the stability of the hexagonal interstitial could not be reproduced in simulations performed in this work using the unmodifed Tersoff potential parameters.
Unfortunately, apart from the modified parameters, no more conditions specifying the relaxation process are given in Tersoff's study on carbon point defects in silicon \cite{tersoff90}.

The tetrahedral is the second most unfavorable interstitial configuration using classical potentials and keeping in mind the abrupt cutoff effect in the case of the Tersoff potential as discussed earlier.
Again, quantum-mechanical results reveal this configuration unstable.
The fact that the tetrahedral and hexagonal configurations are the two most unstable configurations in classical potential calculations and, thus, are less likely to arise in MD simulations acts in concert with the fact that these configurations are found to be unstable in the more accurate quantum-mechanical calculations.

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.

The bond-centered configuration is unstable for the Erhard/Albe potential.
The system moves into the \hkl<1 1 0> interstitial configuration.
This, like in the hexagonal case, is also true for the unmodified Tersoff potential and the given relaxation conditions.
Quantum-mechanical results of this configuration are discussed in more detail in section \ref{subsection:bc}.
In another ab inito study Capaz et al. \cite{capaz94} determined this configuration as an intermediate saddle point structure of a possible migration path, which is 2.1 eV higher than the \hkl<1 0 0> dumbbell configuration.
In calculations performed in this work the bond-centered configuration in fact is a real local minimum and an energy barrier is needed to reach this configuration starting from the \hkl<1 0 0> dumbbell configuration, which is discussed in section \ref{subsection:100mig}.

\subsection[\hkl<1 0 0> dumbbell interstitial configuration]{\boldmath\hkl<1 0 0> dumbbell interstitial configuration}
\label{subsection:100db}

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 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, distances and bond angles are listed in table \ref{tab:defects:100db_cmp}.}
\label{fig:defects:100db_cmp}
\end{figure}
%
\begin{table}[th]
\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
Erhard/Albe & 0.084 & -0.091 & 0.175 & -0.015 & -0.015 & -0.031 & -0.014 & 0.014 & 0.020 \\
VASP & 0.109 & -0.065 & 0.174 & -0.011 & -0.011 & -0.024 & -0.014 & 0.014 & 0.025 \\
\hline
\hline
\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
Erhard/Albe & 0.175 & 0.329 & 0.186 & 0.226 & 0.300 & 0.343 & 0.423 & 0.425 & 0.543 \\
VASP & 0.174 & 0.341 & 0.182 & 0.229 & 0.286 & 0.347 & 0.422 & 0.417 & 0.548 \\
\hline
\hline
\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
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, 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, 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.
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]{Migration of the carbon \boldmath\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.

Since the starting and final structure, which are both local minima of the potential energy surface, are known, the aim is to find the minimum energy path from one local minimum to the other one.
One method to find a minimum energy path is to move the diffusing atom stepwise from the starting to the final position and only allow relaxation in the plane perpendicular to the direction of the vector connecting its starting and final position.
No constraints are applied to the remaining atoms in order to allow relaxation of the surrounding lattice.
To prevent the remaining lattice to migrate according to the displacement of the defect an atom far away from the defect region is fixed in all three coordinate directions.
However, it turned out, that this method tremendously failed applying it to the present migration pathways and structures.
Abrupt changes in structure and free energy occured among relaxed structures of two successive displacement steps.
For some structures even the expected final configurations were never obtained.
Thus, the method mentioned above was adjusted adding further constraints in order to obtain smooth transitions, either in energy as well as structure is concerned.
In this new method all atoms are stepwise displaced towards their final positions.
Relaxation of each individual atom is only allowed in the plane perpendicular to the last individual displacement vector.
The modifications used to add this feature to the VASP code and a short instruction on how to use it can be found in appendix \ref{app:patch_vasp}.
Due to these constraints obtained activation energies can effectively be higher.
{\color{red}Todo: To refine the migration barrier one has to find the saddle point structure and recalculate the free energy of this configuration with a reduced set of constraints.}

\subsection{Migration barriers obtained by quantum-mechanical calculations}

In the following migration barriers are investigated using quantum-mechanical calculations.
The amount of simulated atoms is the same as for the investigation of the point defect structures.
Due to the time necessary for computing only ten displacement steps are used.

\begin{figure}[t!h!]
\begin{center}
\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.

{\color{red}TODO: Stress out that this is a promising result excellently matching experimental observations.}
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.

\begin{figure}[th!]
\begin{center}
\includegraphics[width=13cm]{vasp_mig/110_mig_vasp.ps}
%\begin{picture}(0,0)(140,0)
%\includegraphics[width=2.2cm]{vasp_mig/00-1_b.eps}
%\end{picture}
%\begin{picture}(0,0)(20,0)
%\includegraphics[width=2.2cm]{vasp_mig/00-1_ip0-10_sp.eps}
%\end{picture}
%\begin{picture}(0,0)(-120,0)
%\includegraphics[width=2.2cm]{vasp_mig/0-10_b.eps}
%\end{picture}
\end{center}
\caption{Migration barriers of the \hkl<1 1 0> dumbbell to bond-centered (red), \hkl<0 0 -1> (green) and \hkl<0 -1 0> (in place, blue) C-Si dumbbell transition.}
\label{fig:defects:110_mig_vasp}
\end{figure}
Further migration pathways in particular those occupying other defect configurations than the \hkl<1 0 0>-type either as a transition state or a final or starting configuration are totally conceivable.
This is investigated in the following in order to find possible migration pathways that have an activation energy lower than the ones found up to now.
The next energetically favorable defect configuration is the \hkl<1 1 0> C-Si dumbbell interstitial.
Figure \ref{fig:defects:110_mig_vasp} shows the migration barrier of the \hkl<1 1 0> C-Si dumbbell to the bond-centered, \hkl<0 0 -1> and \hkl<0 -1 0> (in place) transition.
Indeed less than 0.7 eV are necessary to turn a \hkl<0 -1 0>- to a \hkl<1 1 0>-type C-Si dumbbell interstitial.
This transition is carried out in place, that is the Si dumbbell pair is not changed and both, the Si and C atom share the initial lattice site.
Thus, this transition does not contribute to long-range diffusion.
Once the C atom resides in the \hkl<1 1 0> interstitial configuration it can migrate into the bond-centered configuration by employing approximately 0.95 eV of activation energy, which is only slightly higher than the activation energy needed for the \hkl<0 0 -1> to \hkl<0 -1 0> pathway shown in figure \ref{fig:defects:00-1_0-10_mig}.
As already known from the migration of the \hkl<0 0 -1> to the bond-centered configuration as discussed in figure \ref{fig:defects:00-1_001_mig} another 0.25 eV are needed to turn back from the bond-centered to a \hkl<1 0 0>-type interstitial.
However, due to the fact that this migration consists of three single transitions with the second one having an activation energy slightly higher than observed for the direct transition it is considered very unlikely to occur.
The migration barrier of the \hkl<1 1 0> to \hkl<0 0 -1> transition, in which the C atom is changing its Si partner and, thus, moving to the neighboured lattice site is approximately 1.35 eV.
During this transition the C atom is escaping the \hkl(1 1 0) plane approaching the final configuration on a curved path.
This barrier is much higher than the ones found previously, which again make this transition very unlikely to occur.
For this reason the assumption that C diffusion and reorientation is achieved by transitions of the type presented in figure \ref{fig:defects:00-1_0-10_mig} is reinforced.

As mentioned earlier the procedure to obtain the migration barriers differs from the usually applied procedure in two ways.
Firstly constraints to move along the displacement direction are applied on all atoms instead of solely constraining the diffusing atom.
Secondly the constrainted directions are not kept constant to the initial displacement direction.
Instead they are updated for every displacement step.
These modifications to the usual procedure are applied to avoid abrupt changes in structure and free energy on the one hand and to make sure the expected final configuration is reached on the other hand.
Due to applying updated constraints on all atoms the obtained migration barriers and pathes might be overestimated and misguided.
To reinforce the applicability of the employed technique the obtained activation energies and migration pathes for the \hkl<0 0 -1> to \hkl<0 -1 0> transition are compared to two further migration calculations, which do not update the constrainted direction and which only apply updated constraints on three selected atoms, that is the diffusing C atom and the Si dumbbell pair in the initial and final configuration.
Results are presented in figure \ref{fig:defects:00-1_0-10_cmp}.
\begin{figure}[th!]
\begin{center}
\includegraphics[width=13cm]{vasp_mig/00-1_0-10_nosym_sp_cmp.ps}
\end{center}
\caption[Comparison of three different techniques for obtaining migration barriers and pathways applied to the \hkl<0 0 -1> to \hkl<0 -1 0> C-Si dumbbell transition.]{Comparison of three different techniques for obtaining migration barriers and pathways applied to the \hkl<0 0 -1> to \hkl<0 -1 0> C-Si dumbbell transition.}
\label{fig:defects:00-1_0-10_cmp}
\end{figure}
The method without updating the constraints but still applying them to all atoms shows a delayed crossing of the saddle point.
This is understandable since the update results in a more aggressive advance towards the final configuration.
In any case the barrier obtained is slightly higher, which means that it does not constitute an energetically more favorable pathway.
The method in which the constraints are only applied to the diffusing C atom and two Si atoms, ... {\color{red}in progress} ...

\subsection{Migration barriers obtained by classical potential calculations}

The same method for obtaining migration barriers and the same suggested pathways are applied to calculations employing the classical Erhard/Albe potential.
Since the evaluation of the classical potential and force is less computationally intensive higher amounts of steps can be used.
The time constant $\tau$ for the Berendsen thermostat is set to 1 fs in order to have direct velocity scaling and with the temperature set to zero Kelvin perform a steepest descent minimazation to drive the system into a local minimum.
However, in some cases  a time constant of 100 fs resuls in lower barriers and, thus, is shown whenever appropriate.

\begin{figure}[th!]
\begin{center}
\includegraphics[width=13cm]{bc_00-1.ps}\\[5.6cm]
\begin{pspicture}(0,0)(0,0)
\psframe[linecolor=red,fillstyle=none](-7,2.7)(7.2,6)
\end{pspicture}
\begin{picture}(0,0)(140,-100)
\includegraphics[width=2.4cm]{albe_mig/bc_00-1_red_00.eps}
\end{picture}
\begin{picture}(0,0)(10,-100)
\includegraphics[width=2.4cm]{albe_mig/bc_00-1_red_01.eps}
\end{picture}
\begin{picture}(0,0)(-120,-100)
\includegraphics[width=2.4cm]{albe_mig/bc_00-1_red_02.eps}
\end{picture}
\begin{picture}(0,0)(25,-80)
\includegraphics[width=2.5cm]{110_arrow.eps}
\end{picture}
\begin{picture}(0,0)(215,-100)
\includegraphics[height=2.2cm]{001_arrow.eps}
\end{picture}\\
\begin{pspicture}(0,0)(0,0)
\psframe[linecolor=blue,fillstyle=none](-7,-0.5)(7.2,2.8)
\end{pspicture}
\begin{picture}(0,0)(160,-10)
\includegraphics[width=2.2cm]{albe_mig/bc_00-1_01.eps}
\end{picture}
\begin{picture}(0,0)(100,-10)
\includegraphics[width=2.2cm]{albe_mig/bc_00-1_02.eps}
\end{picture}
\begin{picture}(0,0)(10,-10)
\includegraphics[width=2.2cm]{albe_mig/bc_00-1_03.eps}
\end{picture}
\begin{picture}(0,0)(-120,-10)
\includegraphics[width=2.2cm]{albe_mig/bc_00-1_04.eps}
\end{picture}
\begin{picture}(0,0)(25,10)
\includegraphics[width=2.5cm]{100_arrow.eps}
\end{picture}
\begin{picture}(0,0)(215,-10)
\includegraphics[height=2.2cm]{010_arrow.eps}
\end{picture}
\end{center}
\caption{Migration barrier and structures of the bond-centered to \hkl<0 0 -1> dumbbell transition using the classical Erhard/Albe potential.}
\label{fig:defects:cp_bc_00-1_mig}
% red: ./visualize -w 640 -h 480 -d saves/c_in_si_mig_bc_00-1_s20 -nll -0.56 -0.56 -0.7 -fur 0.2 0.2 0.0 -c 0.75 -1.25 -0.25 -L -0.25 -0.25 -0.25 -r 0.6 -B 0.1
% blue: ./visualize -w 640 -h 480 -d saves/c_in_si_mig_bc_00-1_s20_tr100/ -nll -0.56 -0.56 -0.7 -fur 0.2 0.2 0.0 -c 0.0 -0.25 1.0 -L 0.0 -0.25 -0.25 -r 0.6 -B 0.1
\end{figure}
Figure \ref{fig:defects:cp_bc_00-1_mig} shows the migration barrier iand 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 0 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.

\begin{figure}[th!]
\begin{center}
\includegraphics[width=13cm]{00-1_0-10.ps}\\[2.4cm]
\begin{pspicture}(0,0)(0,0)
\psframe[linecolor=red,fillstyle=none](-6,-0.5)(7.2,2.8)
\end{pspicture}
\begin{picture}(0,0)(130,-10)
\includegraphics[width=2.2cm]{albe_mig/00-1_0-10_red_00.eps}
\end{picture}
\begin{picture}(0,0)(0,-10)
\includegraphics[width=2.2cm]{albe_mig/00-1_0-10_red_min.eps}
\end{picture}
\begin{picture}(0,0)(-120,-10)
\includegraphics[width=2.2cm]{albe_mig/00-1_0-10_red_03.eps}
\end{picture}
\begin{picture}(0,0)(25,10)
\includegraphics[width=2.5cm]{100_arrow.eps}
\end{picture}
\begin{picture}(0,0)(185,-10)
\includegraphics[height=2.2cm]{001_arrow.eps}
\end{picture}
\end{center}
\caption{Migration barrier and structures of the \hkl<0 0 -1> to \hkl<0 -1 0> C-Si dumbbell transition using the classical Erhard/Albe potential.}
% red: ./visualize -w 640 -h 480 -d saves/c_in_si_mig_00-1_0-10_s20 -nll -0.56 -0.56 -0.8 -fur 0.3 0.2 0 -c -0.125 -1.7 0.7 -L -0.125 -0.25 -0.25 -r 0.6 -B 0.1
\label{fig:defects:cp_00-1_0-10_mig}
\end{figure}
\begin{figure}[th!]
\begin{center}
\includegraphics[width=13cm]{00-1_ip0-10.ps}
\end{center}
\caption{Migration barrier of the \hkl<0 0 -1> to \hkl<0 -1 0> C-Si dumbbell transition in place using the classical Erhard/Albe potential.}
\label{fig:defects:cp_00-1_ip0-10_mig}
\end{figure}
Figure \ref{fig:defects:cp_00-1_0-10_mig} and \ref{fig:defects:cp_00-1_ip0-10_mig} show the migration barriers of \hkl<0 0 -1> to \hkl<0 -1 0> C-Si dumbbell transition, with a transition of the C atom to the neighboured lattice site in the first case and a reorientation within the same lattice site in the latter case.
Both pathways look similar.
A local minimum exists inbetween two peaks of the graph.
The corresponding configuration, which is illustrated for the migration simulation with a time constant of 1 fs, looks similar to the \hkl<1 1 0> configuration.
Indeed, this configuration is obtained by relaxation simulations without constraints of configurations near the minimum.
Activation energies of roughly 2.8 eV and 2.7 eV respectively are needed for migration.

The \hkl<1 1 0> configuration seems to play a decisive role in all migration pathways.
In the first migration path it is the configuration resulting from a further relaxation of the rather unstable bond-centered configuration, which is fixed to be a transition point.
The last two  pathways show configurations almost identical to the \hkl<1 1 0> configuration, which constitute a local minimum within the pathway.
Thus, migration pathways with the \hkl<1 1 0> C-Si dumbbell interstitial configuration as a starting or final configuration are further investigated.
Figure ... shows ...


The
... diffusion ...
... indicate a problem that is formulated and discussed in more detail in section ...


\section{Combination of point defects}

The structural and energetic properties of combinations of point defects are examined in the following.
Investigations are restricted to quantum-mechanical calculations.

\subsection[Combinations with a C-Si \hkl<1 0 0>-type interstitial]{\boldmath Combinations with a C-Si \hkl<1 0 0>-type interstitial}

This section focuses on combinations of the \hkl<0 0 -1> dumbbell interstitial with a second defect.
The second defect is either another \hkl<1 0 0>-type interstitial occupying different orientations, a vacany or a substitutional carbon atom.
Several distances of the two defects are examined.

\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 more 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 bound to the second carbon dumbbell interstitial are also bound 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_03}
\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_03}.
Figure \ref{fig:defects:comb_db_03} 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_03} c) and d) the nonparallel orientations, namely the \hkl<0 -1 0> and \hkl<1 0 0> dumbbells are shown.
Binding energies of -1.88 eV and -1.38 eV are obtained for the relaxed structures.
In both cases the silicon atom of the initial interstitial is pulled towards the near by atom of the second dumbbell so that both atoms form fourfold coordinated bonds to their next neighbours.
In 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 is ignored in the fitting procedure.

\begin{figure}[t!h!]
\begin{center}
\begin{minipage}[t]{5cm}
a) \underline{Pos: 1, $E_{\text{b}}=0.26\text{ eV}$}
\begin{center}
\includegraphics[width=4.8cm]{00-1dc/0-26.eps}
\end{center}
\end{minipage}
\begin{minipage}[t]{5cm}
b) \underline{Pos: 3, $E_{\text{b}}=-0.93\text{ eV}$}
\begin{center}
\includegraphics[width=4.8cm]{00-1dc/0-93.eps}
\end{center}
\end{minipage}
\begin{minipage}[t]{5cm}
c) \underline{Pos: 5, $E_{\text{b}}=0.49\text{ eV}$}
\begin{center}
\includegraphics[width=4.8cm]{00-1dc/0-49.eps}
\end{center}
\end{minipage}
\end{center}
\caption{Relaxed structures of defect complexes obtained by creating a carbon substitutional at position 1 (a)), 3 (b)) and 5 (c)).}
\label{fig:defects:comb_db_04}
\end{figure}
\begin{figure}[t!h!]
\begin{center}
\begin{minipage}[t]{7cm}
a) \underline{Pos: 2, $E_{\text{b}}=-0.51\text{ eV}$}
\begin{center}
\includegraphics[width=6cm]{00-1dc/0-51.eps}
\end{center}
\end{minipage}
\begin{minipage}[t]{7cm}
b) \underline{Pos: 4, $E_{\text{b}}=-0.15\text{ eV}$}
\begin{center}
\includegraphics[width=6cm]{00-1dc/0-15.eps}
\end{center}
\end{minipage}
\end{center}
\caption{Relaxed structures of defect complexes obtained by creating a carbon substitutional at position 2 (a)) and 4 (b)).}
\label{fig:defects:comb_db_05}
\end{figure}
The second part of table \ref{tab:defects:e_of_comb} lists the energetic results of substitutional carbon and vacancy combinations with the initial \hkl<0 0 -1> dumbbell.
Figures \ref{fig:defects:comb_db_04} and \ref{fig:defects:comb_db_05} show relaxed structures of substitutional carbon in combination with the \hkl<0 0 -1> dumbbell for several positions.
In figure \ref{fig:defects:comb_db_04} positions 1 (a)), 3 (b)) and 5 (c)) are displayed.
A substituted carbon atom at position 5 results in an energetically extremely unfavorable configuration.
Both carbon atoms, the substitutional and the dumbbell atom, pull silicon atom number 1 towards their own location regarding the \hkl<1 1 0> direction.
Due to this a large amount of tensile strain energy is needed, which explains the high positive value of 0.49 eV.
The lowest binding energy is observed for a substitutional carbon atom inserted at position 3.
The substitutional carbon atom is located above the dumbbell substituting a silicon atom which would usually be bound to and displaced along \hkl<0 0 1> and \hkl<1 1 0> by the silicon dumbbell atom.
In contrast to the previous configuration strain compensation occurs resulting in a binding energy as low as -0.93 eV.
Substitutional carbon at position 2 and 4, visualized in figure \ref{fig:defects:comb_db_05}, is located below the initial dumbbell.
Silicon atom number 1, which is bound to the interstitial carbon atom is displaced along \hkl<0 0 -1> and \hkl<-1 -1 0>.
In case a) only the first displacement is compensated by the substitutional carbon atom.
This results in a somewhat higher binding energy of -0.51 eV.
The binding energy gets even higher in case b) ($E_{\text{b}}=-0.15\text{ eV}$), in which the substitutional carbon is located further away from the initial dumbbell.
In both cases, silicon atom number 1 is displaced in such a way, that the bond to silicon atom number 5 vanishes.
In case of \ref{fig:defects:comb_db_04} a) the carbon atoms form a bond with a distance of 1.5 \AA, which is close to the C-C distance expected in diamond or graphit.
Both carbon atoms are highly attracted by each other resulting in large displacements and high strain energy in the surrounding.
A binding energy of 0.26 eV is observed.
Substitutional carbon at positions 2, 3 and 4 are the energetically most favorable configurations and constitute promising starting points for SiC precipitation.
On the one hand, C-C distances around 3.1 \AA{} exist for substitution positions 2 and 3, which are close to the C-C distance expected in silicon carbide.
On the other hand stretched silicon carbide is obtained by the transition of the silicon dumbbell atom into a silicon self-interstitial located somewhere in the silicon host matrix and the transition of the carbon dumbbell atom into another substitutional atom occupying the dumbbell lattice site.

\begin{figure}[t!h!]
\begin{center}
\begin{minipage}[t]{7cm}
a) \underline{Pos: 2, $E_{\text{b}}=-0.59\text{ eV}$}
\begin{center}
\includegraphics[width=6.0cm]{00-1dc/0-59.eps}
\end{center}
\end{minipage}
\begin{minipage}[t]{7cm}
b) \underline{Pos: 3, $E_{\text{b}}=-3.14\text{ eV}$}
\begin{center}
\includegraphics[width=6.0cm]{00-1dc/3-14.eps}
\end{center}
\end{minipage}\\[0.2cm]
\begin{minipage}[t]{7cm}
c) \underline{Pos: 4, $E_{\text{b}}=-0.54\text{ eV}$}
\begin{center}
\includegraphics[width=6.0cm]{00-1dc/0-54.eps}
\end{center}
\end{minipage}
\begin{minipage}[t]{7cm}
d) \underline{Pos: 5, $E_{\text{b}}=-0.50\text{ eV}$}
\begin{center}
\includegraphics[width=6.0cm]{00-1dc/0-50.eps}
\end{center}
\end{minipage}
\end{center}
\caption{Relaxed structures of defect complexes obtained by creating vacancies at positions 2 (a)), 3 (b)), 4 (c)) and 5 (d)).}
\label{fig:defects:comb_db_06}
\end{figure}
Figure \ref{fig:defects:comb_db_06} displays relaxed structures of vacancies in combination with the \hkl<0 0 -1> dumbbell interstital.
The creation of a vacancy at position 1 results in a configuration of substitutional carbon on a silicon lattice site and no other remaining defects.
The carbon dumbbell atom moves to position 1 where the vacancy is created and the silicon dumbbell atom recaptures the dumbbell lattice site.
With a binding energy of -5.39 eV, this is the energetically most favorable configuration observed.
A great amount of strain energy is reduced by removing the silicon atom at position 3, which is illustrated in figure \ref{fig:defects:comb_db_06} b).
The dumbbell structure shifts towards the position of the vacancy which replaces the silicon atom usually bound to and at the same time strained by the silicon dumbbell atom.
Due to the displacement into the \hkl<1 -1 0> direction the bond of the dumbbell silicon atom to the silicon atom on the top left breaks and instead forms a bond to the silicon atom located in \hkl<1 -1 1> direction which is not shown in the figure.
A binding energy of -3.14 eV is obtained for this structure composing another energetically favorable configuration.
A vacancy ctreated at position 2 enables a relaxation of the silicon atom number 1 mainly in \hkl<0 0 -1> direction.
The bond to silicon atom number 5 breaks.
Hence, the silicon dumbbell atom is not only displaced along \hkl<0 0 -1> but also and to a greater extent in \hkl<1 1 0> direction.
The carbon atom is slightly displaced in \hkl<0 1 -1> direction.
A binding energy of -0.59 eV indicates the occurrence of much less strain reduction compared to that in the latter configuration.
Evidently this is due to a smaller displacement of silicon atom number 1, which would be directly bound to the replaced silicon atom at position 2.
In the case of a vacancy created at position 4, even a slightly higher binding energy of -0.54 eV is observed, while the silicon atom at the bottom left, which is bound to the carbon dumbbell atom, is vastly displaced along \hkl<1 0 -1>.
However the displacement of the carbon atom along \hkl<0 0 -1> is less than it is in the preceding configuration.
Although expected due to the symmetric initial configuration silicon atom number 1 is not displaced correspondingly and also the silicon dumbbell atom is displaced to a greater extent in \hkl<-1 0 0> than in \hkl<0 -1 0> direction.
The symmetric configuration is, thus, assumed to constitute a local maximum, which is driven into the present state by the conjugate gradient method used for relaxation.
Figure \ref{fig:defects:comb_db_06} d) shows the relaxed structure of a vacancy created at position 5.
The silicon dumbbell atom is largely displaced along \hkl<1 1 0> and somewaht less along \hkl<0 0 -1>, which corresponds to the direction towards the vacancy.
The silicon dumbbell atom approaches silicon number 1.
Indeed a non-zero charge density is observed inbetween these two atoms exhibiting a cylinder-like shape superposed with the charge density known from the dumbbell itself.
Strain reduced by this huge displacement is partially absorbed by tensile strain on silicon atom number 1 originating from attractive forces of the carbon atom and the vacancy.
A binding energy of -0.50 eV is observed.
{\color{red}Todo: Jahn-Teller distortion (vacancy) $\rightarrow$ actually three possibilities. Due to the initial defect, symmetries are broken. The system should have relaxed into the minumum energy configuration!?}

\subsection{Combinations of Si self-interstitials and substitutional carbon}

{\color{blue}TODO: Explain why this might be important.}
The ground state of a single Si self-interstitial was found to be the Si \hkl<1 1 0> self-interstitial configuration.
For the follwoing study the same type of self-interstitial is assumed to provide the energetically most favorable configuration in combination with a C substitutional.

\begin{table}[ht!]
\begin{center}
\begin{tabular}{l c c c c c c}
\hline
\hline
C$_{\text{sub}}$ & \hkl<1 1 0> & \hkl<-1 1 0> & \hkl<0 1 1> & \hkl<0 -1 1> &
                   \hkl<1 0 1> & \hkl<-1 0 1> \\
\hline
1 & \RM{1} & \RM{3} & \RM{3} & \RM{1} & \RM{3} & \RM{1} \\
2 & \RM{2} & A & A & \RM{2} & C & \RM{5} \\
3 & \RM{3} & \RM{1} & \RM{3} & \RM{1} & \RM{1} & \RM{3} \\
4 & \RM{4} & B & D & E & F & D \\
5 & \RM{5} & C & A & \RM{2} & A & \RM{2} \\
\hline
\hline
\end{tabular}
\end{center}
\caption{Equivalent configurations of \hkl<1 1 0>-type Si self-interstitials created at position I of figure \ref{fig:defects:pos_of_comb} and substitutional C created at positions 1 to 5.}
\label{tab:defects:comb_csub_si110}
\end{table}
\begin{table}[ht!]
\begin{center}
\begin{tabular}{l c c c c c c c c c c c}
\hline
\hline
Conf & \RM{1} & \RM{2} & \RM{3} & \RM{4} & \RM{5} & A & B & C & D & E &F\\
\hline
$E_{\text{f}}$ [eV]& 4.37 & 5.26 & 5.56 & 5.32 & 5.12 & 5.10 & 5.32 & 5.28 & 5.39 & 5.32 & 5.32 \\
$E_{\text{b}}$ [eV] & -0.97 & -0.08 & 0.22 & -0.03 & -0.23 & -0.25 & -0.02 & -0.06 & 0.05 & -0.03 & -0.03 \\
\hline
\hline
\end{tabular}
\end{center}
\caption{Formation $E_{\text{f}}$ and binding $E_{\text{b}}$ energies in eV of the combinational substitutional C and Si self-interstitial configurations as defined in table \ref{tab:defects:comb_csub_si110}.}
\label{tab:defects:comb_csub_si110_energy}
\end{table}
Table \ref{tab:defects:comb_csub_si110} shows equivalent configurations of \hkl<1 1 0>-type Si self-interstitials and substitutional C.
The notation of figure \ref{fig:defects:pos_of_comb} is used with the six possible Si self-interstitials created at the usual C-Si dumbbell position.
Substitutional C is created at positions 1 to 5.

{\color{blue}TODO:
Results of energies ...
Thus ...
}

\section{Migration in systems of combined defects}

During carbon implantation into crystalline silicon the energetic carbon atoms may kick out silicon atoms from their lattice sites.
A vacancy accompanied by a silicon self-interstitial is generated.
The silicon self-interstitial may migrate to the surface or recombine with other vacancies.
Once a vacancy and a carbon interstitial defect exist the energetically most favorable configuration is the configuration of a substitutional carbon atom, that is the carbon atom occupying the vacant site.
In addition, it is a conceivable configuration the system might experience during the silicon carbide precipitation process.
Energies needed to overcome the migration barrier of the transformation into this configuration enable predictions concerning the feasibility of a silicon carbide conversion mechanism derived from these microscopic processes.
This is especially important for the case, in which the vacancy is created at position 3, as discussed in the last section and figure \ref{fig:defects:comb_db_06} b).
Due to the low binding energy this configuration might constitute a trap, which it is hard to escape from.
However, migration simulations show that only a low amount of energy is necessary to transform the system into the energetically most favorable configuration.
\begin{figure}[!t!h]
\begin{center}
\includegraphics[width=13cm]{vasp_mig/comb_mig_3-2_vac_fullct.ps}\\[2.0cm]
\begin{picture}(0,0)(170,0)
\includegraphics[width=3cm]{vasp_mig/comb_2-1_init.eps}
\end{picture}
\begin{picture}(0,0)(80,0)
\includegraphics[width=3cm]{vasp_mig/comb_2-1_seq_03.eps}
\end{picture}
\begin{picture}(0,0)(-10,0)
\includegraphics[width=3cm]{vasp_mig/comb_2-1_seq_06.eps}
\end{picture}
\begin{picture}(0,0)(-120,0)
\includegraphics[width=3cm]{vasp_mig/comb_2-1_final.eps}
\end{picture}
\begin{picture}(0,0)(25,20)
\includegraphics[width=2.5cm]{100_arrow.eps}
\end{picture}
\begin{picture}(0,0)(230,0)
\includegraphics[height=2.2cm]{001_arrow.eps}
\end{picture}
\end{center}
\caption{Transition of the configuration of the C-Si dumbbell interstitial in combination with a vacancy created at position 2 into the configuration of substitutional carbon.}
\label{fig:defects:comb_mig_01}
\end{figure}
\begin{figure}[!t!h]
\begin{center}
\includegraphics[width=13cm]{vasp_mig/comb_mig_4-2_vac_fullct.ps}\\[1.0cm]
\begin{picture}(0,0)(150,0)
\includegraphics[width=2cm]{vasp_mig/comb_3-1_init.eps}
\end{picture}
\begin{picture}(0,0)(60,0)
\includegraphics[width=2cm]{vasp_mig/comb_3-1_seq_03.eps}
\end{picture}
\begin{picture}(0,0)(-45,0)
\includegraphics[width=2cm]{vasp_mig/comb_3-1_seq_07.eps}
\end{picture}
\begin{picture}(0,0)(-130,0)
\includegraphics[width=2cm]{vasp_mig/comb_3-1_final.eps}
\end{picture}
\begin{picture}(0,0)(25,20)
\includegraphics[width=2.5cm]{100_arrow.eps}
\end{picture}
\begin{picture}(0,0)(230,0)
\includegraphics[height=2.2cm]{001_arrow.eps}
\end{picture}
\end{center}
\caption{Transition of the configuration of the C-Si dumbbell interstitial in combination with a vacancy created at position 3 into the configuration of substitutional carbon.}
\label{fig:defects:comb_mig_02}
\end{figure}
Figure \ref{fig:defects:comb_mig_01} and \ref{fig:defects:comb_mig_02} show the migration barriers and structures for transitions of the vacancy-interstitial configurations examined in figure \ref{fig:defects:comb_db_06} a) and b) into a configuration of substitutional carbon.

In the first case the focus is on the migration of silicon atom number 1 towards the vacant site created at position 2, while the carbon atom substitutes the site of the migrating silicon atom.
An energy of 0.6 eV necessary to overcome the migration barrier is found.
This energy is low enough to constitute a feasible mechanism in SiC precipitation.
To reverse this process 5.4 eV are needed, which make this mechanism very unprobable.
The migration path is best described by the reverse process.
Starting at 100 \% energy is needed to break the bonds of silicon atom 1 to its neighboured silicon atoms and that of the carbon atom to silicon atom number 5.
At a displacement of 60 \% these bonds are broken.
Due to this and due to the formation of new bonds, that is the bond of silicon atom number 1 to silicon atom number 5 and the bond of the carbon atom to its siliocn neighbour in the bottom left, a less steep increase of free energy is observed.
At a displacement of approximately 30 \% the bond of silicon atom number 1 to the just recently created siliocn atom is broken up again, which explains the repeated boost in energy.
Finally the system gains energy relaxing into the configuration of zero displacement.

Due to the low binding energy observed, the configuration of the vacancy created at position 3 is assumed to be stable against transition.
However, a relatively simple migration path exists, which intuitively seems to be a low energy process.
The migration path and the corresponding differences in free energy are displayed in figure \ref{fig:defects:comb_mig_02}.
In fact, migration simulations yield a barrier as low as 0.1 eV.
This energy is needed to tilt the dumbbell as the displayed structure at 30 \% displacement shows.
Once this barrier is overcome, the carbon atom forms a bond to the top left silicon atom and the interstitial silicon atom capturing the vacant site is forming new tetrahedral bonds to its neighboured silicon atoms.
These new bonds and the relaxation into the substitutional carbon configuration are responsible for the gain free energy.
For the reverse process approximately 2.4 eV are nedded, which is 24 times higher than the forward process.
Thus, substitutional carbon is assumed to be stable in contrast to the C-Si dumbbell interstitial located next to a vacancy.
 
{\color{red}Todo: DB mig along 110 (at the starting of this section)?}

{\color{red}Todo: Migration of Si int + vac and C sub/int ...?}

{\color{red}Todo: Model of kick-out and kick-in mechnism?}

\section{Conclusions concerning the SiC conversion mechanism}

The ground state configuration of a carbon interstitial in crystalline siliocn is found to be the C-Si \hkl<1 0 0> dumbbell interstitial configuration.
The threefold coordinated carbon and silicon atom share a usual silicon lattice site.
Migration simulations reveal the carbon interstitial to be mobile at prevailing implantation temperatures requireing an activation energy of approximately 0.9 eV for migration as well as reorientation processes.

Investigations of two carbon interstitials of the \hkl<1 0 0>-type and varying separations and orientations state an attractive interaction between these interstitials.
Depending on orientation, energetically favorable configurations are found in which these two interstitials are located close together instead of the occurernce of largely separated and isolated defects.
This is due to strain compensation enabled by the combination of such defects in certain orientations.
For dumbbells oriented along the \hkl<1 1 0> direction and the assumption that there is the possibility of free orientation, an interaction energy proportional to the reciprocal cube of the distance in the far field regime is found.
These findings support the assumption of the C-Si dumbbell agglomeration proposed by the precipitation model introduced in section \ref{section:assumed_prec}.

By combination of the \hkl<1 0 0> dumbbell with a vacancy it is found that the configuration of substitutional carbon arising by the carbon interstitial atom occupying the vacant site is the energetically most favorable configuration.
Low migration barriers are necessary to obtain this configuration and in contrast comparatively high activation energies necessary for the reverse process.
Thus, carbon interstitials and vacancies located close together are assumed to end up in such a configuration in which the carbon atom is tetrahedrally coordinated and bound to four silicon atoms as expected in silicon carbide.
In contrast to the above, this would suggest a silicon carbide precipitation by succesive creation of substitutional carbon instead of the agglomeration of C-Si dumbbell interstitials followed by an abrupt precipitation.

{\color{red}Todo:
C atoms in 100 DB tightend in 110 direction, which is important for stress compensation in some combined constellations.
}
{\color{red}Todo:
Most of the combinations should only be thought of intermediate configurations, which need to be transformed in the later SiC precipitation process.
}
{\color{red}Todo:
Better structure, better language, better methodology!
}
{\color{red}Todo:
Fit of lennard-jones an other rep + attr potentials in 110 interaction data!
}