\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}.
+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}
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.
+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 (DAS MAL DURCHRECHNEN).
-Hence these artefacts should have a negligent influence in finite temperature simulations.
+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.
\end{minipage}
\begin{minipage}{4cm}
\underline{\hkl<1 0 0>}\\
-$E_{\text{f}}=3.96\text{ eV}$\\
+$E_{\text{f}}=3.88\text{ eV}$\\
\includegraphics[width=4.0cm]{c_pd_albe/100.eps}
\end{minipage}
\begin{minipage}{0.5cm}
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.
+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.
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 displacements obtained by analytical potential and quantum-mechanical calculations.
+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}[h]
\begin{center}
-\includegraphics[width=10cm]{100-c-si-db_cmp.eps}
+\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}
-\caption[Sketch of the \hkl<1 0 0> dumbbell structure.]{Sketch of the \hkl<1 0 0> dumbbell structure. Atomic displacements and distances are listed in table \ref{tab:defects:100db_cmp}.}
\end{figure}
%
\begin{table}[h]
\begin{center}
+Displacements\\
\begin{tabular}{l c c c c c c c c c}
\hline
\hline
& $a$ & $b$ & $|a|+|b|$ & $\Delta x$ & $\Delta y$ & $\Delta z$ & $\Delta x$ & $\Delta y$ & $\Delta z$ \\
\hline
Erhard/Albe & 0.084 & -0.091 & 0.175 & -0.015 & -0.015 & -0.031 & -0.014 & 0.014 & 0.020 \\
-VASP & & & & & & & & & \\
+VASP & 0.109 & -0.065 & 0.174 & -0.011 & -0.011 & -0.024 & -0.014 & 0.014 & 0.025 \\
\hline
\hline
-\end{tabular}
+\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}
-\begin{tabular}{l c c c c c c c c}
+Angles\\
+\begin{tabular}{l c c c c }
\hline
\hline
- & $r(1C)$ & $r(2C)$ & $r(3C)$ & $r(12)$ & $r(13)$ & $r(34)$ & $r(23)$ & $r(25)$\\
+ & $\theta_1$ & $\theta_2$ & $\theta_3$ & $\theta_4$ \\
\hline
-Erhard/Albe & & & & & & & \\
-VASP & & & & & & & \\
+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}
-\caption[Atomic displacements and distances of the \hkl<1 0 0> dumbbell structure obtained by the Erhard/Albe potential and VASP calculations.]{Atomic displacements and distances 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 schematically displayed in figure \ref{fig:defects:100db_cmp}.}
\end{table}
+\begin{figure}[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}[h]
+\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}[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]{\boldmath Migration of the carbon \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}[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}
+\label{img:defects:c_mig_path}
+\caption{Migration pathways of the carbon \hkl<1 0 0> interstitial dumbbell in silicon.}
+\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.
+The new silicon dumbbell partner is the one located at $\frac{a}{4}\hkl<1 1 -1>$ compared to the initial one.
+The carbon atom resides in the \hkl(1 1 0) plane along the path.
+As a last migration path, the \hkl<0 0 -1>
+
+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 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.
+
+Different approaches were used to compute migration paths and energies.
+CRT erklaeren
+Nur C atom constrainted
+Problem, 1) relaxation of host matrix 2) abrubt changes in energy and configuration
+as can be seen in ...
+Thus, 'all atom CRT' but! relaxation perpendicular to delta step ...
+
+Results and comparison with diffusion experiments.
+
\section{Combination of point defects}
projects / lectures / latex.git / blobdiff