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 (DAS MAL DURCHRECHNEN).
+However, the energy barrier is small.
+Todo: Check!
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.
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 ...
+Todo: Plot the electron density for these types of defect to derive conclusions of existing bonds?
\section{Carbon related point defects}
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 the trajectory of the carbon atom is no longer proceeding in the \hkl(1 1 0) plane.
As a last migration path, the defect is only changing its orientation.
The silicon dumbbell partner remains.
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.
+Todo: \hkl<1 1 0> to \hkl<1 0 0> and bond-centerd configuration (in progress).
+Todo: \hkl<1 1 0> to \hkl<0 -1 0> (rotation of the DB, in progress).
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.
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.
+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.
-In figure ... results of the \hkl<0 0 -1> to \hkl<0 0 1> migration are presented.
+\begin{figure}[h]
+\begin{center}
+\includegraphics[width=13cm]{im_00-1_nosym_sp_fullct_thesis.ps}\\[0.5cm]
+\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}
+\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.
+
+\begin{figure}[h]
+\begin{center}
+\includegraphics[width=13cm]{im_00-1_nosym_sp_fullct_thesis.ps}\\[0.5cm]
+\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}
+\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}
\section{Combination of point defects}
--- /dev/null
+\chapter{Modifications to the VASP code}
+\label{app:patch_vasp}
+
+\section{Description}
+The modifications to the VASP code allow to rotate all atom coordinates individually in the particle position evaluation routine of VASP.
+In that way constraints for every atom can be applied independently of the chosen basis.
+A patch against version 4.6 of the VASP code containing these modifications is available for download\footnote{http://www.physik.uni-augsburg.de/\~{}zirkelfr/download/posic/sd\_rot\_all-atoms.patch}.
+
+\section{Usage}
+
+Since this feature only makes sense in selective dynamics mode, it can be switched on by adding the word {\em Transformed} in front of the {\em selective dynamics} switch.
+This feature only works in direct mode.
+Two values of angles need to be added after the extra flags of each atom.
+The first angle corresponds to the rotation of the basis about the $z$-axis.
+The second angle determines the rotation about the transformed $x$-axis, $x'$.
+All values have to be supplied in degrees.
+All these information is given in the POSCAR file as can be seen in the follwing example:
+\begin{verbatim}
+cubic diamond
+ 5.48000000000000
+ 2.9909698580839312 0.0039546630279804 -0.0039658085666586
+ 0.0039548953566878 2.9909698596656376 -0.0039660323646892
+ -0.0039680658132861 -0.0039674231313905 2.9909994291263242
+ 216 1
+Transformed selective dynamics
+Direct
+ 0.994174 0.994174 -0.000408732 T F T 45 36.5145
+ 0.182792 0.182792 0.981597 T F T -135 -5.95043
+ ...
+ 0.119896 0.119896 0.0385525 T F T -135 21.8036
+\end{verbatim}
+In case of the first atom the basis is transformed by a rotation of $45^{\circ}$ and $36.5145^{\circ}$ about the $z$ and $x'$ axis.
+Relaxation of this atom is constrained to the $y''$-axis.
+
projects / lectures / latex.git / commitdiff