The ions are relaxed by a conjugate gradient method.
The cell volume and shape is allowed to change using the pressure control algorithm of Parinello and Rahman \cite{}.
Periodic boundary conditions in each direction are applied.
+All point defects are calculated for the neutral charge state.
-\begin{figure}
+\begin{figure}[h]
\begin{center}
-\includegraphics[width=10cm]{unit_cell_e.eps}
+\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}$\bullet$}) and \hkl<1 1 0> dumbbell ({\color{magenta}$\bullet$}) interstitial configurations.}
+\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 Fig. \ref{fig:defects:ins_pos}.
-In seperated simulation runs the silicon or carbon atom is inserted at the tetrahedral $(0,0,0)$ ({\color{red}$\bullet$}), the hexagonal $(-1/8,-1/8,1/8)$ ({\color{green}$\bullet$}), the nearly \hkl<1 0 0> dumbbell $(-1/4,-1/4,-1/8)$ ({\color{yellow}$\bullet$}) and the nearly \hkl<1 1 0> dumbbell $(-1/8,-1/8,-1/4)$ ({\color{magenta}$\bullet$}) 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 to high forces.
+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, $\vec{p}=(0,0,0)$, ({\color{red}$\bullet$})
+ \item hexagonal, $\vec{p}=(-1/8,-1/8,1/8)$, ({\color{green}$\bullet$})
+ \item nearly \hkl<1 0 0> dumbbell, $\vec{p}=(-1/4,-1/4,-1/8)$, ({\color{yellow}$\bullet$})
+ \item nearly \hkl<1 1 0> dumbbell, $\vec{p}=(-1/8,-1/8,-1/4)$, ({\color{magenta}$\bullet$})
+ \item bond-centered, $\vec{p}=(-1/8,-1/8,-3/8)$, ({\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.
\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.
-Evtl Paper mit Ef rauskramen lenen schreiben ...
-Defects consisting of two or more atom species ...
+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 \ref{eq:defects:ef2} is equivalent to equation \ref{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}[h]
+\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 & unstable & 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.}
+\label{tab:defects:si_self}
+\end{table}
+The final configurations obtained after relaxation are presented in figure \ref{fig:defects:conf}.
+\begin{figure}[h]
+\begin{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=4.0cm]{si_pd_vasp/110_2333.eps}
+\end{minipage}
+\begin{minipage}{5cm}
+\underline{Hexagonal}\\
+$E_{\text{f}}=3.42\text{ eV}$\\
+\includegraphics[width=4.0cm]{si_pd_vasp/hex_2333.eps}
+\end{minipage}
+\begin{minipage}{5cm}
+\underline{Tetrahedral}\\
+$E_{\text{f}}=3.77\text{ eV}$\\
+\includegraphics[width=4.0cm]{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=4.0cm]{si_pd_vasp/100_2333.eps}
+\end{minipage}
+\begin{minipage}{5cm}
+\underline{Vacancy}\\
+$E_{\text{f}}=3.63\text{ eV}$\\
+\includegraphics[width=4.0cm]{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{\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{Hexagonal}\\
+$E_{\text{f}}=3.96\text{ eV}$\\
+\includegraphics[width=4.0cm]{si_pd_albe/hex.eps}
+\end{minipage}
+\begin{minipage}{5cm}
+\underline{Tetrahedral}\\
+$E_{\text{f}}=3.40\text{ eV}$\\
+\includegraphics[width=4.0cm]{si_pd_albe/tet.eps}
+\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{Vacancy}\\
+$E_{\text{f}}=3.13\text{ eV}$\\
+\includegraphics[width=4.0cm]{si_pd_albe/vac.eps}
+\end{minipage}
+\begin{minipage}{5cm}
+\begin{center}
+Erhard/Albe potential\\
+calculations\\
+\end{center}
+\end{minipage}
+\end{flushleft}
+\hrule
+\end{center}
+\caption[Relaxed silicon self-interstitial defect configurations.]{Relaxed silicon self-interstitial defect configurations. 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 consesus 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{} and Stillinger-Weber \cite{} 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{}.
+The Si interstitial atom moves 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{}.
+Obviously the authors did not carefully check the relaxed results assuming a hexagonal configuration.
+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{}.
+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 artefact of the analytical potential without physical authenticity revealing basic problems of analytical potential models for describing defect structures.
+However, the energy barrier is small (DAS MAL DURCHRECHNEN).
+Hence these artefacts 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 cut-off 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 cut-off range and thus are weak interactions not constituting actual chemical bonds.
+The same applies to the bonds between the interstitial and the upper two atoms in the \hkl<1 1 0> dumbbell configuration.
+
+A more detailed description of the chemical bonding is achieved by quantum-mechanical calculations by investigating the accumulation of negative charge between the nuclei.
+Todo: Plot the electron density for these types of defect to derive conclusions of existing bonds ...
+
\section{Carbon related point defects}
+
+
\section[Migration of the carbon \hkl<1 0 0> interstitial]{\boldmath Migration of the carbon \hkl<1 0 0> interstitial}
\section{Combination of point defects}
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