\chapter{Point defects in silicon}
-Given the conversion mechnism of SiC in crystalline silicon introduced in \ref{section:assumed_prec} the understanding of carbon and silicon interstitial point defects in c-Si is of great interest.
+Given the conversion mechnism of SiC in crystalline silicon introduced in section \ref{section:assumed_prec} the understanding of carbon and silicon interstitial point defects in c-Si is of great interest.
Both types of defects are examined in the following both by classical potential as well as density functional theory calculations.
In case of the classical potential calculations a simulation volume of nine silicon lattice constants in each direction is used.
Calculations are performed in an isothermal-isobaric NPT ensemble.
Coupling to the heat bath is achieved by the Berendsen thermostat with a time constant of 100 fs.
The temperature is set to zero Kelvin.
-Pressure is controlled by a Berendsen barostat again using a time constant of 100 fs and a bulk modulus of 100 GPa for silicon.
+Pressure is controlled by a Berendsen barostat \cite{berendsen84} again using a time constant of 100 fs and a bulk modulus of 100 GPa for silicon.
To exclude surface effects periodic boundary conditions are applied.
Due to the restrictions in computer time three silicon lattice constants in each direction are considered sufficiently large enough for DFT calculations.
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{}.
+The cell volume and shape is allowed to change using the pressure control algorithm of Parrinello and Rahman \cite{parrinello81}.
Periodic boundary conditions in each direction are applied.
All point defects are calculated for the neutral charge state.
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.
+Among the established analytical potentials only the EDIP \cite{bazant97,justo98} and Stillinger-Weber \cite{stillinger85} 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{albe_sic_pot}.
\caption{Kinetic energy plot of the relaxation process of the hexagonal silicon self-interstitial defect simulation using the Erhard/Albe classical potential.}
\label{fig:defects:kin_si_hex}
\end{figure}
-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{}.
+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{parcas_md}.
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.
-{\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.
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}
+\section[Migration of the carbon \hkl<1 0 0> interstitial]{Migration of the carbon \boldmath\hkl<1 0 0> interstitial}
\label{subsection:100mig}
In the following the problem of interstitial carbon migration in silicon is considered.
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.
+\begin{figure}[th!]
+\begin{center}
+\includegraphics[width=13cm]{vasp_mig/110_mig_vasp.ps}
+%\begin{picture}(0,0)(140,0)
+%\includegraphics[width=2.2cm]{vasp_mig/00-1_b.eps}
+%\end{picture}
+%\begin{picture}(0,0)(20,0)
+%\includegraphics[width=2.2cm]{vasp_mig/00-1_ip0-10_sp.eps}
+%\end{picture}
+%\begin{picture}(0,0)(-120,0)
+%\includegraphics[width=2.2cm]{vasp_mig/0-10_b.eps}
+%\end{picture}
+\end{center}
+\caption{Migration barriers of the \hkl<1 1 0> dumbbell to bond-centered (red), \hkl<0 0 -1> (green) and \hkl<0 -1 0> (in place, blue) C-Si dumbbell transition.}
+\label{fig:defects:110_mig_vasp}
+\end{figure}
+Further migration pathways in particular those occupying other defect configurations than the \hkl<1 0 0>-type either as a transition state or a final or starting configuration are totally conceivable.
+In order to find possible migration pathways that have an activation energy lower than the ones found up to now.
+The next energetically favorable defect configuration is the \hkl<1 1 0> C-Si dumbbell interstitial.
+Figure \ref{fig:defects:110_mig_vasp} shows the migration barrier of the \hkl<1 1 0> C-Si dumbbell to the bond-centered, \hkl<0 0 -1> and \hkl<0 -1 0> (in place) transition.
+Indeed less than 0.7 eV are necessary to turn a \hkl<0 -1 0>- to a \hkl<1 1 0>-type C-Si dumbbell interstitial.
+This transition is carried out in place, that is the Si dumbbell pair is not changed and both, the Si and C atom share the same lattice site.
+Thus, this transition does not contribute to long-range diffusion.
+Once the C atom resides in the \hkl<1 1 0> interstitial configuration it can migrate into the bond-centered configuration by employing approximately 0.95 eV of activation energy, which is only slightly higher than the activation energy needed for the \hkl<0 0 -1> to \hkl<0 -1 0> pathway shown in figure \ref{fig:defects:00-1_0-10_mig}.
+As already known from the migration of the \hkl<0 0 -1> to the bond-centered configuration as discussed in figure \ref{fig:defects:00-1_001_mig} another 0.25 eV are needed to turn back from the bond-centered to a \hkl<1 0 0>-type interstitial.
+However, due to the fact that this migration consists of three single transitions with the second one having an activation energy slightly higher than observed for the direct transition it is considered very unlikely to occur.
+The migration barrier of the \hkl<1 1 0> to \hkl<0 0 -1> transition, in which the C atom is changing its Si partner and, thus, moving to the neighboured lattice site is approximately 1.35 eV.
+During this transition the C atom is escaping the \hkl(1 1 0) plane approaching the final configuration on a curved path.
+This barrier is much higher than the ones found previously, which again make this transition very unlikely to occur.
+For this reason the assumption that C diffusion and reorientation is achieved by transitions of the type presented in figure \ref{fig:defects:00-1_0-10_mig} is reinforced.
+
+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 does not constitute an energetically more 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.
+Since the evaluation of the classical potential and force is less computationally intensive higher amounts of steps can be used.
+
+\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 relaxing into the \hkl<1 1 0> C-Si dumbbell interstitial configuration within this potential the low kinetic energy state is used as a starting configuration.
+
+
+\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 investigated in the following.
-The focus is on combinations of the \hkl<0 0 -1> dumbbell interstitial with a second defect.
+The structural and energetic properties of combinations of point defects are examined in the following.
+Investigations are restricted to quantum-mechanical calculations.
+
+\subsection[Combinations with a C-Si \hkl<1 0 0>-type interstitial]{\boldmath Combinations with a C-Si \hkl<1 0 0>-type interstitial}
+
+This section focuses on combinations of the \hkl<0 0 -1> dumbbell interstitial with a second defect.
The second defect is either another \hkl<1 0 0>-type interstitial occupying different orientations, a vacany or a substitutional carbon atom.
Several distances of the two defects are examined.
-Investigations are restricted to quantum-mechanical calculations.
\begin{figure}[th]
\begin{center}
A binding energy of -0.50 eV is observed.
{\color{red}Todo: Jahn-Teller distortion (vacancy) $\rightarrow$ actually three possibilities. Due to the initial defect, symmetries are broken. The system should have relaxed into the minumum energy configuration!?}
-{\color{blue}Todo: Si int + vac and C sub ...?
-Investigation of vacancy, Si and C interstitital.
-As for the ground state of the single Si self-int, a 110 is also assumed as the lowest possibility in combination with other defects (which is a cruel assumption)!
+\subsection{Combinations of Si self-interstitials and substitutional carbon}
+
+{\color{blue}TODO: Explain why this might be important.}
+The ground state of a single Si self-interstitial was found to be the Si \hkl<1 1 0> self-interstitial configuration.
+For the follwoing study the same type of self-interstitial is assumed to provide the energetically most favorable configuration in combination with a C substitutional.
+
+\begin{table}[ht!]
+\begin{center}
+\begin{tabular}{l c c c c c c}
+\hline
+\hline
+C$_{\text{sub}}$ & \hkl<1 1 0> & \hkl<-1 1 0> & \hkl<0 1 1> & \hkl<0 -1 1> &
+ \hkl<1 0 1> & \hkl<-1 0 1> \\
+\hline
+1 & \RM{1} & \RM{3} & \RM{3} & \RM{1} & \RM{3} & \RM{1} \\
+2 & \RM{2} & A & A & \RM{2} & C & \RM{5} \\
+3 & \RM{3} & \RM{1} & \RM{3} & \RM{1} & \RM{1} & \RM{3} \\
+4 & \RM{4} & B & D & E & F & D \\
+5 & \RM{5} & C & A & \RM{2} & A & \RM{2} \\
+\hline
+\hline
+\end{tabular}
+\end{center}
+\caption{Equivalent configurations of \hkl<1 1 0>-type Si self-interstitials created at position I of figure \ref{fig:defects:pos_of_comb} and substitutional C created at positions 1 to 5.}
+\label{tab:defects:comb_csub_si110}
+\end{table}
+\begin{table}[ht!]
+\begin{center}
+\begin{tabular}{l c c c c c c c c c c c}
+\hline
+\hline
+Conf & \RM{1} & \RM{2} & \RM{3} & \RM{4} & \RM{5} & A & B & C & D & E &F\\
+\hline
+$E_{\text{f}}$ [eV]& 4.37 & 5.26 & 5.56 & 5.32 & 5.12 & 5.10 & 5.32 & 5.28 & 5.39 & 5.32 & 5.32 \\
+$E_{\text{b}}$ [eV] & -0.97 & -0.08 & 0.22 & -0.03 & -0.23 & -0.25 & -0.02 & -0.06 & 0.05 & -0.03 & -0.03 \\
+\hline
+\hline
+\end{tabular}
+\end{center}
+\caption{Formation $E_{\text{f}}$ and binding $E_{\text{b}}$ energies in eV of the combinational substitutional C and Si self-interstitial configurations as defined in table \ref{tab:defects:comb_csub_si110}.}
+\label{tab:defects:comb_csub_si110_energy}
+\end{table}
+Table \ref{tab:defects:comb_csub_si110} shows equivalent configurations of \hkl<1 1 0>-type Si self-interstitials and substitutional C.
+The notation of figure \ref{fig:defects:pos_of_comb} is used with the six possible Si self-interstitials created at the usual C-Si dumbbell position.
+Substitutional C is created at positions 1 to 5.
+
+{\color{blue}TODO:
+Results of energies ...
+Thus ...
}
\section{Migration in systems of combined defects}
{\color{red}Todo: DB mig along 110 (at the starting of this section)?}
-{\color{red}Todo: Migration of Si int + vac and C sub ...?}
+{\color{red}Todo: Migration of Si int + vac and C sub/int ...?}
{\color{red}Todo: Model of kick-out and kick-in mechnism?}
Thus, carbon interstitials and vacancies located close together are assumed to end up in such a configuration in which the carbon atom is tetrahedrally coordinated and bound to four silicon atoms as expected in silicon carbide.
In contrast to the above, this would suggest a silicon carbide precipitation by succesive creation of substitutional carbon instead of the agglomeration of C-Si dumbbell interstitials followed by an abrupt precipitation.
+{\color{red}Todo:
+C atoms in 100 DB tightend in 110 direction, which is important for stress compensation in some combined constellations.
+}
+{\color{red}Todo:
+Most of the combinations should only be thought of intermediate configurations, which need to be transformed in the later SiC precipitation process.
+}
+{\color{red}Todo:
+Better structure, better language, better methodology!
+}
+{\color{red}Todo:
+Fit of lennard-jones an other rep + attr potentials in 110 interaction data!
+}
+
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