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}
\begin{minipage}{7.5cm}
Thus, $E_{\text{b}}$ can be best thought of a binding energy, which is required to bring the defects to infinite separation.
In fact, a \hkl<0 0 -1> dumbbell interstitial created at position R with a distance of $\frac{a_{\text{Si}}}{2}\hkl<3 2 3>$ ($\approx 12.8$ \AA) from the initial one results in an energy as low as -0.19 eV.
There is still a low interaction which is due to the equal orientation of the defects.
-By changing the orientation of the second dumbbell interstitial to the \hkl<0 -1 0>-type the interaction is even mor reduced resulting in an energy of $E_{\text{b}}=-0.05\text{ eV}$ for a distance, which is the maximum that can be realized due to periodic boundary conditions.
+By changing the orientation of the second dumbbell interstitial to the \hkl<0 -1 0>-type the interaction is even more reduced resulting in an energy of $E_{\text{b}}=-0.05\text{ eV}$ for a distance, which is the maximum that can be realized due to periodic boundary conditions.
The energies obtained in the R column of table \ref{eq:defects:e_of_comb} are used as a reference to identify, whether less distanced defects of the same type are favorable or unfavorable compared to the far-off located defect.
Configurations wih energies greater than zero or the reference value are energetically unfavorable and expose a repulsive interaction.
These configurations are unlikely to arise or to persist for non-zero temperatures.
The binding energy is -2.05 eV.
Both dumbbells are tilted along the same direction remaining parallely aligned and the second dumbbell is pushed downwards in such a way, that the four dumbbell atoms form a rhomboid.
Both carbon atoms form tetrahedral bonds to four silicon atoms.
-However, silicon atom 1 and 3, which are bond to the second carbon dumbbell interstitial are also bond to the initial carbon atom.
+However, silicon atom 1 and 3, which are bound to the second carbon dumbbell interstitial are also bound to the initial carbon atom.
These four atoms of the rhomboid reside in a plane and, thus, do not match the situation in silicon carbide.
The carbon atoms have a distance of 2.75 \AA.
In figure \ref{fig:defects:comb_db_02} b) a second \hkl<0 1 0> dumbbell is constructed at position 2.
\end{minipage}
\end{center}
\caption{Relaxed structures of defect complexes obtained by creating a a) \hkl<0 0 1>, a b) \hkl<0 0 -1>, a c) \hkl<0 -1 0> and a d) \hkl<1 0 0> dumbbell at position 5.}
-\label{fig:defects:comb_db_05}
+\label{fig:defects:comb_db_03}
\end{figure}
Energetically beneficial configurations of defect complexes are observed for second interstititals of all orientations placed at position 5, a position two bonds away from the initial interstitial along the \hkl<1 1 0> direction.
-Relaxed structures of these complexes are displayed in figure \ref{fig:defects:comb_db_05}.
-Figure \ref{fig:defects:comb_db_05} a) and b) show the relaxed structures of \hkl<0 0 1> and \hkl<0 0 -1> dumbbells.
+Relaxed structures of these complexes are displayed in figure \ref{fig:defects:comb_db_03}.
+Figure \ref{fig:defects:comb_db_03} a) and b) show the relaxed structures of \hkl<0 0 1> and \hkl<0 0 -1> dumbbells.
The upper dumbbell atoms are pushed towards each other forming fourfold coordinated bonds.
While the displacements of the silicon atoms in case b) are symmetric to the \hkl(1 1 0) plane, in case a) the silicon atom of the initial dumbbel is pushed a little further in the direction of the carbon atom of the second dumbbell than the carbon atom is pushed towards the silicon atom.
The bottom atoms of the dumbbells remain in threefold coordination.
The symmetric configuration is energetically more favorable ($E_{\text{b}}=-1.66\text{ eV}$) since the displacements of the atoms is less than in the antiparallel case ($E_{\text{b}}=-1.53\text{ eV}$).
-In figure \ref{fig:defects:comb_db_05} c) and d) the nonparallel orientations, namely the \hkl<0 -1 0> and \hkl<1 0 0> dumbbells are shown.
+In figure \ref{fig:defects:comb_db_03} c) and d) the nonparallel orientations, namely the \hkl<0 -1 0> and \hkl<1 0 0> dumbbells are shown.
Binding energies of -1.88 eV and -1.38 eV are obtained for the relaxed structures.
In both cases the silicon atom of the initial interstitial is pulled towards the near by atom of the second dumbbell so that both atoms form fourfold coordinated bonds to their next neighbours.
In case c) it is the carbon and in case d) the silicon atom of the second interstitial which forms the additional bond with the silicon atom of the initial interstitial.
Figure \ref{fig:defects:comb_db110} shows the corresponding plot of the data including a cubic spline interplation and a suitable fitting curve.
The funtion found most suitable for curve fitting is $f(x)=a/x^3$ comprising the single fit parameter $a$.
Thus, far-off located dumbbells show an interaction proportional to the reciprocal cube of the distance and the amount of bonds along \hkl<1 1 0> respectively.
-This behavior is no longer valid for the immediate vicinity revealed by the saturating binding energy of a second dumbbell at position 1, which was ignored in the fitting procedure.
+This behavior is no longer valid for the immediate vicinity revealed by the saturating binding energy of a second dumbbell at position 1, which is ignored in the fitting procedure.
+
+\begin{figure}[t!h!]
+\begin{center}
+\begin{minipage}[t]{5cm}
+a) \underline{Pos: 1, $E_{\text{b}}=0.26\text{ eV}$}
+\begin{center}
+\includegraphics[width=4.8cm]{00-1dc/0-26.eps}
+\end{center}
+\end{minipage}
+\begin{minipage}[t]{5cm}
+b) \underline{Pos: 3, $E_{\text{b}}=-0.93\text{ eV}$}
+\begin{center}
+\includegraphics[width=4.8cm]{00-1dc/0-93.eps}
+\end{center}
+\end{minipage}
+\begin{minipage}[t]{5cm}
+c) \underline{Pos: 5, $E_{\text{b}}=0.49\text{ eV}$}
+\begin{center}
+\includegraphics[width=4.8cm]{00-1dc/0-49.eps}
+\end{center}
+\end{minipage}
+\end{center}
+\caption{Relaxed structures of defect complexes obtained by creating a carbon substitutional at position 1 (a)), 3 (b)) and 5 (c)).}
+\label{fig:defects:comb_db_04}
+\end{figure}
+\begin{figure}[t!h!]
+\begin{center}
+\begin{minipage}[t]{7cm}
+a) \underline{Pos: 2, $E_{\text{b}}=-0.51\text{ eV}$}
+\begin{center}
+\includegraphics[width=6cm]{00-1dc/0-51.eps}
+\end{center}
+\end{minipage}
+\begin{minipage}[t]{7cm}
+b) \underline{Pos: 4, $E_{\text{b}}=-0.15\text{ eV}$}
+\begin{center}
+\includegraphics[width=6cm]{00-1dc/0-15.eps}
+\end{center}
+\end{minipage}
+\end{center}
+\caption{Relaxed structures of defect complexes obtained by creating a carbon substitutional at position 2 (a)) and 4 (b)).}
+\label{fig:defects:comb_db_05}
+\end{figure}
+The second part of table \ref{tab:defects:e_of_comb} lists the energetic results of substitutional carbon and vacancy combinations with the initial \hkl<0 0 -1> dumbbell.
+Figures \ref{fig:defects:comb_db_04} and \ref{fig:defects:comb_db_05} show relaxed structures of substitutional carbon in combination with the \hkl<0 0 -1> dumbbell for several positions.
+In figure \ref{fig:defects:comb_db_04} positions 1 (a)), 3 (b)) and 5 (c)) are displayed.
+A substituted carbon atom at position 5 results in an energetically extremely unfavorable configuration.
+Both carbon atoms, the substitutional and the dumbbell atom, pull silicon atom number 1 towards their own location regarding the \hkl<1 1 0> direction.
+Due to this a large amount of tensile strain energy is needed, which explains the high positive value of 0.49 eV.
+The lowest binding energy is observed for a substitutional carbon atom inserted at position 3.
+The substitutional carbon atom is located above the dumbbell substituting a silicon atom which would usually be bound to and displaced along \hkl<0 0 1> and \hkl<1 1 0> by the silicon dumbbell atom.
+In contrast to the previous configuration strain compensation occurs resulting in a binding energy as low as -0.93 eV.
+Substitutional carbon at position 2 and 4, visualized in figure \ref{fig:defects:comb_db_05}, is located below the initial dumbbell.
+Silicon atom number 1, which is bound to the interstitial carbon atom is displaced along \hkl<0 0 -1> and \hkl<-1 -1 0>.
+In case a) only the first displacement is compensated by the substitutional carbon atom.
+This results in a somewhat higher binding energy of -0.51 eV.
+The binding energy gets even higher in case b) ($E_{\text{b}}=-0.15\text{ eV}$), in which the substitutional carbon is located further away from the initial dumbbell.
+In both cases, silicon atom number 1 is displaced in such a way, that the bond to silicon atom number 5 vanishes.
+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.
+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.
+
+\begin{figure}[t!h!]
+\begin{center}
+\begin{minipage}[t]{7cm}
+a) \underline{Pos: 2, $E_{\text{b}}=-0.59\text{ eV}$}
+\begin{center}
+\includegraphics[width=6.0cm]{00-1dc/0-59.eps}
+\end{center}
+\end{minipage}
+\begin{minipage}[t]{7cm}
+b) \underline{Pos: 3, $E_{\text{b}}=-3.14\text{ eV}$}
+\begin{center}
+\includegraphics[width=6.0cm]{00-1dc/3-14.eps}
+\end{center}
+\end{minipage}\\[0.2cm]
+\begin{minipage}[t]{7cm}
+c) \underline{Pos: 4, $E_{\text{b}}=-0.54\text{ eV}$}
+\begin{center}
+\includegraphics[width=6.0cm]{00-1dc/0-54.eps}
+\end{center}
+\end{minipage}
+\begin{minipage}[t]{7cm}
+d) \underline{Pos: 5, $E_{\text{b}}=-0.50\text{ eV}$}
+\begin{center}
+\includegraphics[width=6.0cm]{00-1dc/0-50.eps}
+\end{center}
+\end{minipage}
+\end{center}
+\caption{Relaxed structures of defect complexes obtained by creating vacancies at positions 2 (a)), 3 (b)), 4 (c)) and 5 (d)).}
+\label{fig:defects:comb_db_06}
+\end{figure}
+Figure \ref{fig:defects:comb_db_06} displays relaxed structures of vacancies in combination with the \hkl<0 0 -1> dumbbell interstital.
+The creation of a vacancy at position 1 results in a configuration of substitutional carbon on a silicon lattice site and no other remaining defects.
+The carbon dumbbell atom moves to position 1 where the vacancy is created and the silicon dumbbell atom recaptures the dumbbell lattice site.
+With a binding energy of -5.39 eV, this is the energetically most favorable configuration observed.
+A great amount of strain energy is reduced by removing the silicon atom at position 3, which is illustrated in figure \ref{fig:defects:comb_db_06} b).
+The dumbbell structure shifts towards the position of the vacancy which replaces the silicon atom usually bound to and at the same time strained by the silicon dumbbell atom.
+Due to the displacement into the \hkl<1 -1 0> direction the bond of the dumbbell silicon atom to the silicon atom on the top left breaks and instead forms a bond to the silicon atom located in \hkl<1 -1 1> direction which is not shown in the figure.
+A binding energy of -3.14 eV is obtained for this structure composing another energetically favorable configuration.
+A vacancy ctreated at position 2 enables a relaxation of the silicon atom number 1 mainly in \hkl<0 0 -1> direction.
+The bond to silicon atom number 5 breaks.
+Hence, the silicon dumbbell atom is not only displaced along \hkl<0 0 -1> but also and to a greater extent in \hkl<1 1 0> direction.
+The carbon atom is slightly displaced in \hkl<0 1 -1> direction.
+A binding energy of -0.59 eV indicates the occurrence of much less strain reduction compared to that in the latter configuration.
+Evidently this is due to a smaller displacement of silicon atom number 1, which would be directly bound to the replaced silicon atom at position 2.
+In the case of a vacancy created at position 4, even a slightly higher binding energy of -0.54 eV is observed, while the silicon atom at the bottom left, which is bound to the carbon dumbbell atom, is vastly displaced along \hkl<1 0 -1>.
+However the displacement of the carbon atom along \hkl<0 0 -1> is less than it is in the preceding configuration.
+Although expected due to the symmetric initial configuration silicon atom number 1 is not displaced correspondingly and also the silicon dumbbell atom is displaced to a greater extent in \hkl<-1 0 0> than in \hkl<0 -1 0> direction.
+The symmetric configuration is, thus, assumed to constitute a local maximum, which is driven into the present state by the conjugate gradient method used for relaxation.
+Figure \ref{fig:defects:comb_db_06} d) shows the relaxed structure of a vacancy created at position 5.
+The silicon dumbbell atom is largely displaced along \hkl<1 1 0> and somewaht less along \hkl<0 0 -1>, which corresponds to the direction towards the vacancy.
+The silicon dumbbell atom approaches silicon number 1.
+Indeed a non-zero charge density is observed inbetween these two atoms exhibiting a cylinder-like shape superposed with the charge density known from the dumbbell itself.
+Strain reduced by this huge displacement is partially absorbed by tensile strain on silicon atom number 1 originating from attractive forces of the carbon atom and the vacancy.
+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{red}Todo: DB mig along 110?}
+{\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)!
+}
-{\color{red}Todo: Si int and C sub ...}
+\section{Migration in systems of combined defects}
+
+During carbon implantation into crystalline silicon the energetic carbon atoms may kick out silicon atoms from their lattice sites.
+A vacancy accompanied by a silicon self-interstitial is generated.
+The silicon self-interstitial may migrate to the surface or recombine with other vacancies.
+Once a vacancy and a carbon interstitial defect exist the energetically most favorable configuration is the configuration of a substitutional carbon atom, that is the carbon atom occupying the vacant site.
+In addition, it is a conceivable configuration the system might experience during the silicon carbide precipitation process.
+Energies needed to overcome the migration barrier of the transformation into this configuration enable predictions concerning the feasibility of a silicon carbide conversion mechanism derived from these microscopic processes.
+This is especially important for the case, in which the vacancy is created at position 3, as discussed in the last section and figure \ref{fig:defects:comb_db_06} b).
+Due to the low binding energy this configuration might constitute a trap, which it is hard to escape from.
+However, migration simulations show that only a low amount of energy is necessary to transform the system into the energetically most favorable configuration.
+\begin{figure}[!t!h]
+\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}
+\end{picture}
+\begin{picture}(0,0)(60,0)
+\includegraphics[width=3.5cm]{vasp_mig/comb_2-1_seq.eps}
+\end{picture}
+\begin{picture}(0,0)(-120,0)
+\includegraphics[width=3.5cm]{vasp_mig/comb_2-1_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 vacancy-interstitial combinations 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 ...
+
+
+{\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: Model of kick-out and kick-in mechnism?}
-{\color{red}Todo: Jahn-Teller distortion (vacancy) $\rightarrow$ actually three possibilities! :(}
+
+\section{Conclusions for SiC preciptation}
+
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