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}.
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.
-\begin{figure}[t!h!]
-\begin{center}
-\begin{minipage}{6cm}
-\underline{Original}\\
-\includegraphics[width=6cm]{crt_orig.eps}
-\end{minipage}
-\begin{minipage}{1cm}
-\hfill
-\end{minipage}
-\begin{minipage}{6cm}
-\underline{Modified}\\
-\includegraphics[width=6cm]{crt_mod.eps}
-\end{minipage}
-\end{center}
-\caption{Schematic of the constrained relaxation technique (CRT) (left) and of the modified version (right) used to obtain migration pathways and corresponding activation energies.}
-\label{fig:defects:crt}
-\end{figure}
-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.
-This is called the constrained relaxation technique (CRT), which is schematically displayed in the left part of figure \ref{fig:defects:crt}.
-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, as displayed in the right part of figure \ref{fig:defects:crt}.
-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.
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