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.
-{\color{red}Todo: Check!}
+\begin{figure}[th]
+\begin{center}
+\includegraphics[width=12cm]{nhex_tet.ps}
+\end{center}
+\caption{Migration barrier of the tetrahedral Si self-interstitial slightly displaced along all three coordinate axes into the exact tetrahedral configuration using classical potential calculations.}
+\label{fig:defects:nhex_tet_mig}
+\end{figure}
+This is exemplified in figure \ref{fig:defects:nhex_tet_mig}, which shows the change in potential energy during the migration of the interstitial displaced along all three coordinate axes into the tetrahedral configuration.
+The technique used to obtain the migration data is explained in a later section (\ref{subsection:100mig}).
+The barrier is less than 0.2 eV.
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.
Thus, it is not responsible for long-range migration.
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.
-{\color{red}Todo: Comparison with classical potential simulations or explanation to only focus on ab initio calculations.}
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.
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.
+The amount of simulated atoms is the same as for the investigation of the point defect structures.
+Due to the time necessary for computing only ten displacement steps are used.
+
\begin{figure}[t!h!]
\begin{center}
\includegraphics[width=13cm]{im_00-1_nosym_sp_fullct_thesis.ps}\\[1.5cm]
Experimentally measured activation energies for reorientation range from 0.77 eV to 0.88 eV \cite{watkins76,song90}.
Thus, this pathway is more likely to be composed of two consecutive steps of the second path.
+{\color{red}TODO: Stress out that this is a promising result excellently matching experimental observations.}
Since the activation energy of the first and last migration path is much greater than the experimental value, the second path is identified to be responsible as a migration path for the most likely carbon interstitial in silicon explaining both, annealing and reorientation experiments.
The activation energy of roughly 0.9 eV nicely compares to experimental values.
The theoretical description performed in this work is improved compared to a former study \cite{capaz94}, which underestimates the experimental value by 35 \%.
In addition the bond-ceneterd configuration, for which spin polarized calculations are necessary, is found to be a real local minimum instead of a saddle point configuration.
-\section{Combination of adjacent point defects}
+As mentioned earlier the procedure to obtain the migration barriers differs from the usually applied procedure in two ways.
+Firstly constraints to move along the displacement direction are applied on all atoms instead of solely constraining the diffusing atom.
+Secondly the constrainted directions are not kept constant to the initial displacement direction.
+Instead they are updated for every displacement step.
+These modifications to the usual procedure are applied to avoid abrupt changes in structure and free energy on the one hand and to make sure the expected final configuration is reached on the other hand.
+Due to applying updated constraints on all atoms the obtained migration barriers and pathes might be overestimated and misguided.
+To reinforce the applicability of the employed technique the obtained activation energies and migration pathes for the \hkl<0 0 -1> to \hkl<0 -1 0> transition are compared to two further migration calculations, which do not update the constrainted direction and which only apply updated constraints on three selected atoms, that is the diffusing C atom and the Si dumbbell pair in the initial and final configuration.
+Results are presented in figure \ref{fig:defects:00-1_0-10_cmp}.
+\begin{figure}[th!]
+\begin{center}
+\includegraphics[width=13cm]{vasp_mig/00-1_0-10_nosym_sp_cmp.ps}
+\end{center}
+\caption[Comparison of three different techniques for obtaining migration barriers and pathways applied to the \hkl<0 0 -1> to \hkl<0 -1 0> C-Si dumbbell transition.]{Comparison of three different techniques for obtaining migration barriers and pathways applied to the \hkl<0 0 -1> to \hkl<0 -1 0> C-Si dumbbell transition.}
+\label{fig:defects:00-1_0-10_cmp}
+\end{figure}
+The method without updating the constraints but still applying them to all atoms shows a delayed crossing of the saddle point.
+This is understandable since the update results in a more aggressive advance towards the final configuration.
+In any case the barrier obtained is slightly higher, which means that it is not the energetically most favorable pathway.
+The method in which the constraints are only applied to the diffusing C atom and two Si atoms, ... {\color{red}in progress} ...
+
+\subsection{Migration barriers obtained by classical potential calculations}
+
+The same method for obtaining migration barriers and the same suggested pathways are applied to calculations employing the classical Erhard/Albe potential.
+
+\begin{figure}[th!]
+\begin{center}
+\includegraphics[width=13cm]{bc_00-1.ps}
+\end{center}
+\caption{Migration barrier of the bond-centered to \hkl<0 0 -1> dumbbell transition using the classical Erhard/Albe potential.}
+\label{fig:defects:cp_bc_00-1_mig}
+\end{figure}
+Figure \ref{fig:defects:cp_bc_00-1_mig} shows the migration barrier of the bond-centered to \hkl<0 0 -1> dumbbell transition.
+Since the bond-centered configuration is unstable within this potential the low kinetic energy state is used as a starting configuration.
+This would relax towards the \hkl<1 1 0> C-Si interstitial.
+
+\begin{figure}[th!]
+\begin{center}
+\includegraphics[width=13cm]{00-1_0-10.ps}
+\end{center}
+\caption{Migration barrier of the \hkl<0 0 -1> \hkl<0 -1 0> C-Si dumbbell transition using the classical Erhard/Albe potential.}
+\label{fig:defects:cp_00-1_0-10_mig}
+\end{figure}
+Figure \ref{fig:defects:cp_00-1_0-10_mig} shows the migration barrier of the \hkl<0 0 -1> to \hkl<0 -1 0> C-Si dumbbell transition.
+After the first maximum the system relaxes to a configuration similar to the \hkl<1 1 0> C-Si dumbbell configuration.
+
+\begin{figure}[th!]
+\begin{center}
+\includegraphics[width=13cm]{00-1_ip0-10.ps}
+\end{center}
+\caption{Migration barrier of the \hkl<0 0 -1> to \hkl<0 -1 0> C-Si dumbbell transition in place using the classical Erhard/Albe potential.}
+\label{fig:defects:cp_00-1_ip0-10_mig}
+\end{figure}
+Figure \ref{fig:defects:cp_00-1_ip0-10_mig} shows the migration barrier of the \hkl<0 0 -1> to \hkl<0 -1 0> C-Si dumbbell transition in place.
+
+\section{Combination of point defects}
The structural and energetic properties of combinations of point defects are examined in the following.
Investigations are restricted to quantum-mechanical calculations.
projects / lectures / latex.git / commitdiff