\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}
\hline
\hline
& & & & \multicolumn{3}{c}{Atom 2} & \multicolumn{3}{c}{Atom 3} \\
-and bond angles & $a$ & $b$ & $|a|+|b|$ & $\Delta x$ & $\Delta y$ & $\Delta z$ & $\Delta x$ & $\Delta y$ & $\Delta z$ \\
+ & $a$ & $b$ & $|a|+|b|$ & $\Delta x$ & $\Delta y$ & $\Delta z$ & $\Delta x$ & $\Delta y$ & $\Delta z$ \\
\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 \\
\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.
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.
-Thus, the angles of bonds of the silicon dumbbell atom are clos to $120^{\circ}$ signifying the predominance of $sp^2$ hybridization.
-On the other hand, the carbon atom forms an almost colinear bond with the two silicon edge atoms implying the predominance of $p$ and $sp$ bonding.
-This is in figure \ref{fig:defects:100db_vis_cmp} as well as by the ...
+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}.
+The carbon atom has been displaced along the axes ... relaxation back ... indicating a real local minimum.
+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, illsutrates 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.
+
\section{Combination of point defects}
projects / lectures / latex.git / blobdiff