-Figure ...
-c-sub:
-position 5: the sub and the db both pull the the bottom si atoms in concerning \hkl<1 1 0> direction.
-Tensile strain which explains the binding energy.
-lowest energy observed at position 3.
-sub is located in top of the initial db.
-in contrast to the latter case, strain compensation occurs.
-position 2 and 4 the sub is (zwar) located (unter) db but due to the configuration not that much strain arises, since ...
-at position 1, c-c bond is formed, like in graphit or diamond.
-both c atoms are pushed towards each other resulting in high displacements and high strain energy in the near surrounding of the si crystal, which perfectly explains the high energy of ... eV.
-
-The creation of the vacancy at position 1 ... c interstitital moves to acancy position ending up in a configuration of a substitutional carbon which explains the highbinding energy.
-At position 3 a great amount of strain energy is reduced, since the the vacancy replaces a silicon atom usually bond to and thus starined by the silicon dumbbell atom.
-db moves towards the vacancy in \hkl<1 -1 0> direction.
-Vac at position 2 and 4 have similar results.
-Less strain is reduced, since the displacement of the bottom silicon atom, whcih would be directly bond to the silicon atom replaced by the vacancy, is less.
-In the second case, there is even less strain reduction since the second next neighbour is replaced by the vacancy.
-A symmetric configuration is expected, but it is not!
-jahn-Teller distortion ... check this!
-In both cases the db is tilted in such a way, that the carbon atom moves towards the vacancy.
-At position 5 the silicon dumbbell atom moves in \hkl<1 1 0> direction, the same direction where the vacancy is located.
-Strain reducde by this is partialy absorbed by strain originating from the fact that si atom bound to and pulled by the carbon atom is also pulled by the vacancy.
-
-CHECK C-C DIST AND SI-C DIST !!! of all!!!
-
-{\color{red}Todo: Jahn-Teller distortion (vacancy) $\rightarrow$ actually three possibilities? Due to the initial defect symmetries are broken. It should have relaxed into the minumum energy configuration!?}
-Once a vacancy exists the minimal e conf is the c sub conf and ofcourse necessary for formation of SiC.
-The question is whether the migration into this conf is possible.
-Fig shows the migration of the 2 and 3 conf into the c sub conf.
-Low migration barriers, which means that SiC will modt probably form ... and so on ...
-
-{\color{red}Todo: Si int and C sub ...}
-The existance of a vacancy is most often accompanied by an interstitial.
-The silicon interstitital might diffuse to the surface or recombine with other vacancy defects and tus is out of the interested simulation region.
-However, investigation of near by vacancy, Si and C interstititla is necessary, too.
-As for the ground state of the single Si self-int a 110 this is also assumed as the lowest possibility in combination with other defects, which is a cruel assumption!!!
+\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{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)!
+}
+
+\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=3cm]{vasp_mig/comb_2-1_init.eps}
+\end{picture}
+\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=3cm]{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 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}
+\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/int ...?}