\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.
\label{fig:defects:conf}
\end{figure}
-There are differences between the various results of the quantum-mechanical calculations but the consesus view is that the \hkl<1 1 0> dumbbell followed by the hexagonal and tetrahedral defect is the lowest in energy.
+There are differences between the various results of the quantum-mechanical calculations but the consensus view is that the \hkl<1 1 0> dumbbell followed by the hexagonal and tetrahedral defect is the lowest in energy.
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.
In case of \ref{fig:defects:comb_db_04} a) the carbon atoms form a bond with a distance of 1.5 \AA, which is close to the C-C distance expected in diamond or graphit.
Both carbon atoms are highly attracted by each other resulting in large displacements and high strain energy in the surrounding.
A binding energy of 0.26 eV is observed.
-Substitutional carbon at positions 2, 3 and 5 are the energetically most favorable configurations and constitute promising starting points for SiC precipitation.
+Substitutional carbon at positions 2, 3 and 4 are the energetically most favorable configurations and constitute promising starting points for SiC precipitation.
On the one hand, C-C distances around 3.1 \AA{} exist for substitution positions 2 and 3, which are close to the C-C distance expected in silicon carbide.
-On the other hand stretched silicon carbide is obtained by the transition of the silicon dumbbell atom into a silicon self-interstitial located somewhere in the silicon host matrix and th etransition of the carbon dumbbell atom into another substitutional atom occupying the dumbbell lattice site.
+On the other hand stretched silicon carbide is obtained by the transition of the silicon dumbbell atom into a silicon self-interstitial located somewhere in the silicon host matrix and the transition of the carbon dumbbell atom into another substitutional atom occupying the dumbbell lattice site.
\begin{figure}[t!h!]
\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 ...?
+{\color{blue}Todo: Si int + vac and C sub/int ...?
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)!
}
\begin{center}
\includegraphics[width=13cm]{vasp_mig/comb_mig_3-2_vac_fullct.ps}\\[2.0cm]
\begin{picture}(0,0)(170,0)
-\includegraphics[width=3.5cm]{vasp_mig/comb_2-1_init.eps}
+\includegraphics[width=3cm]{vasp_mig/comb_2-1_init.eps}
\end{picture}
-\begin{picture}(0,0)(60,0)
-\includegraphics[width=3.5cm]{vasp_mig/comb_2-1_seq.eps}
+\begin{picture}(0,0)(80,0)
+\includegraphics[width=3cm]{vasp_mig/comb_2-1_seq_03.eps}
+\end{picture}
+\begin{picture}(0,0)(-10,0)
+\includegraphics[width=3cm]{vasp_mig/comb_2-1_seq_06.eps}
\end{picture}
\begin{picture}(0,0)(-120,0)
-\includegraphics[width=3.5cm]{vasp_mig/comb_2-1_final.eps}
+\includegraphics[width=3cm]{vasp_mig/comb_2-1_final.eps}
\end{picture}
\begin{picture}(0,0)(25,20)
\includegraphics[width=2.5cm]{100_arrow.eps}
\includegraphics[height=2.2cm]{001_arrow.eps}
\end{picture}
\end{center}
-\caption{Transition vacancy-interstitial combinations into the configuration of substitutional carbon.}
+\caption{Transition of the configuration of the C-Si dumbbell interstitial in combination with a vacancy created at position 2 into the configuration of substitutional carbon.}
\label{fig:defects:comb_mig_01}
\end{figure}
-Figure \ref{fig:defects:comb_mig_01} shows the migration barriers and structures for transitions of the vacancy-interstitial configurations examined in figure \ref{fig:defects:comb_db_06} a) and b) into the configuration of substitutional carbon.
-
-
-
-Low migration barriers, which means that SiC will modt probably form ... and so on ...
+\begin{figure}[!t!h]
+\begin{center}
+\includegraphics[width=13cm]{vasp_mig/comb_mig_4-2_vac_fullct.ps}\\[1.0cm]
+\begin{picture}(0,0)(150,0)
+\includegraphics[width=2cm]{vasp_mig/comb_3-1_init.eps}
+\end{picture}
+\begin{picture}(0,0)(60,0)
+\includegraphics[width=2cm]{vasp_mig/comb_3-1_seq_03.eps}
+\end{picture}
+\begin{picture}(0,0)(-45,0)
+\includegraphics[width=2cm]{vasp_mig/comb_3-1_seq_07.eps}
+\end{picture}
+\begin{picture}(0,0)(-130,0)
+\includegraphics[width=2cm]{vasp_mig/comb_3-1_final.eps}
+\end{picture}
+\begin{picture}(0,0)(25,20)
+\includegraphics[width=2.5cm]{100_arrow.eps}
+\end{picture}
+\begin{picture}(0,0)(230,0)
+\includegraphics[height=2.2cm]{001_arrow.eps}
+\end{picture}
+\end{center}
+\caption{Transition of the configuration of the C-Si dumbbell interstitial in combination with a vacancy created at position 3 into the configuration of substitutional carbon.}
+\label{fig:defects:comb_mig_02}
+\end{figure}
+Figure \ref{fig:defects:comb_mig_01} and \ref{fig:defects:comb_mig_02} show the migration barriers and structures for transitions of the vacancy-interstitial configurations examined in figure \ref{fig:defects:comb_db_06} a) and b) into a configuration of substitutional carbon.
+In the first case the focus is on the migration of silicon atom number 1 towards the vacant site created at position 2, while the carbon atom substitutes the site of the migrating silicon atom.
+An energy of 0.6 eV necessary to overcome the migration barrier is found.
+This energy is low enough to constitute a feasible mechanism in SiC precipitation.
+To reverse this process 5.4 eV are needed, which make this mechanism very unprobable.
+The migration path is best described by the reverse process.
+Starting at 100 \% energy is needed to break the bonds of silicon atom 1 to its neighboured silicon atoms and that of the carbon atom to silicon atom number 5.
+At a displacement of 60 \% these bonds are broken.
+Due to this and due to the formation of new bonds, that is the bond of silicon atom number 1 to silicon atom number 5 and the bond of the carbon atom to its siliocn neighbour in the bottom left, a less steep increase of free energy is observed.
+At a displacement of approximately 30 \% the bond of silicon atom number 1 to the just recently created siliocn atom is broken up again, which explains the repeated boost in energy.
+Finally the system gains energy relaxing into the configuration of zero displacement.
+Due to the low binding energy observed, the configuration of the vacancy created at position 3 is assumed to be stable against transition.
+However, a relatively simple migration path exists, which intuitively seems to be a low energy process.
+The migration path and the corresponding differences in free energy are displayed in figure \ref{fig:defects:comb_mig_02}.
+In fact, migration simulations yield a barrier as low as 0.1 eV.
+This energy is needed to tilt the dumbbell as the displayed structure at 30 \% displacement shows.
+Once this barrier is overcome, the carbon atom forms a bond to the top left silicon atom and the interstitial silicon atom capturing the vacant site is forming new tetrahedral bonds to its neighboured silicon atoms.
+These new bonds and the relaxation into the substitutional carbon configuration are responsible for the gain free energy.
+For the reverse process approximately 2.4 eV are nedded, which is 24 times higher than the forward process.
+Thus, substitutional carbon is assumed to be stable in contrast to the C-Si dumbbell interstitial located next to a vacancy.
+
{\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?}
+\section{Conclusions concerning the SiC conversion mechanism}
+
+The ground state configuration of a carbon interstitial in crystalline siliocn is found to be the C-Si \hkl<1 0 0> dumbbell interstitial configuration.
+The threefold coordinated carbon and silicon atom share a usual silicon lattice site.
+Migration simulations reveal the carbon interstitial to be mobile at prevailing implantation temperatures requireing an activation energy of approximately 0.9 eV for migration as well as reorientation processes.
+
+Investigations of two carbon interstitials of the \hkl<1 0 0>-type and varying separations and orientations state an attractive interaction between these interstitials.
+Depending on orientation, energetically favorable configurations are found in which these two interstitials are located close together instead of the occurernce of largely separated and isolated defects.
+This is due to strain compensation enabled by the combination of such defects in certain orientations.
+For dumbbells oriented along the \hkl<1 1 0> direction and the assumption that there is the possibility of free orientation, an interaction energy proportional to the reciprocal cube of the distance in the far field regime is found.
+These findings support the assumption of the C-Si dumbbell agglomeration proposed by the precipitation model introduced in section \ref{section:assumed_prec}.
-\section{Conclusions for SiC preciptation}
+By combination of the \hkl<1 0 0> dumbbell with a vacancy it is found that the configuration of substitutional carbon arising by the carbon interstitial atom occupying the vacant site is the energetically most favorable configuration.
+Low migration barriers are necessary to obtain this configuration and in contrast comparatively high activation energies necessary for the reverse process.
+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.
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