For investigating the \si{} structures a Si atom is inserted or removed according to Fig. \ref{fig:basics:ins_pos} of section \ref{section:basics:defects}.
The formation energies of \si{} configurations are listed in Table \ref{tab:defects:si_self} for both methods used in this work as well as results obtained by other {\em ab initio} studies \cite{al-mushadani03,leung99}.
-\begin{table}[t]
+\begin{table}[tp]
\begin{center}
\begin{tabular}{l c c c c c}
\hline
\caption[Formation energies of Si self-interstitials in crystalline Si determined by classical potential MD and DFT calculations.]{Formation energies of Si self-interstitials in crystalline Si determined by classical potential MD and DFT calculations. The formation energies are given in eV. T denotes the tetrahedral and H the hexagonal interstitial configuration. V corresponds to the vacancy configuration. Dumbbell configurations are abbreviated by DB. Formation energies for unstable configurations are marked by an asterisk and determined by using the low kinetic energy configuration shortly before the relaxation into the more favorable configuration starts.}
\label{tab:defects:si_self}
\end{table}
-\begin{figure}[t]
+\begin{figure}[tp]
\begin{center}
\begin{flushleft}
\begin{minipage}{5cm}
The formation energy of \unit[3.96]{eV} for this type of interstitial is equal to the result for the hexagonal one in the original work \cite{albe_sic_pot}.
Obviously the authors did not carefully check the relaxed results assuming a hexagonal configuration.
In Fig. \ref{fig:defects:kin_si_hex} the relaxation process is shown on the basis of the kinetic energy plot.
-\begin{figure}[t]
+\begin{figure}[tp]
\begin{center}
\includegraphics[width=0.7\textwidth]{e_kin_si_hex.ps}
\end{center}
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 fundamental problems of analytical potential models for describing defect structures.
However, the energy barrier is small.
-\begin{figure}[!ht]
+\begin{figure}[tp]
\begin{center}
\includegraphics[width=0.7\textwidth]{nhex_tet.ps}
\end{center}
The same applies to the bonds between the interstitial and the upper two atoms in the \si{} \hkl<1 1 0> DB configuration.
A more detailed description of the chemical bonding is achieved through quantum-mechanical calculations by investigating the accumulation of negative charge between the nuclei.
-%\clearpage{}
-
\section{Carbon point defects in silicon}
\subsection{Defect structures in a nutshell}
The type of reservoir of the C impurity to determine the formation energy of the defect is chosen to be SiC.
This is consistent with the methods used in the articles \cite{tersoff90,dal_pino93}, which the results are compared to in the following.
Hence, the chemical potential of Si and C is determined by the cohesive energy of Si and SiC as discussed in section \ref{section:basics:defects}.
-\begin{table}[t]
+\begin{table}[tp]
\begin{center}
\begin{tabular}{l c c c c c c}
\hline
\hline
& T & H & \hkl<1 0 0> DB & \hkl<1 1 0> DB & S & BC \\
\hline
-\multicolumn{6}{c}{Present study} \\
+Present study & & & & & & \\
{\textsc posic} & 6.09 & 9.05$^*$ & 3.88 & 5.18 & 0.75 & 5.59$^*$ \\
{\textsc vasp} & Unstable & Unstable & 3.72 & 4.16 & 1.95 & 4.66 \\
-\multicolumn{6}{c}{Other studies} \\
+Other studies & & & & & & \\
Tersoff \cite{tersoff90} & 3.8 & 6.7 & 4.6 & 5.9 & 1.6 & 5.3 \\
{\em Ab initio} \cite{dal_pino93,capaz94} & - & - & x & - & 1.89 & x+2.1 \\
\hline
\caption[Formation energies of C point defects in c-Si determined by classical potential MD and DFT calculations.]{Formation energies of C point defects in c-Si determined by classical potential MD and DFT calculations. The formation energies are given in eV. T denotes the tetrahedral, H the hexagonal and BC the bond-centered interstitial configuration. S corresponds to the substitutional interstitial configuration. The dumbbell configurations are abbreviated by DB. Formation energies for unstable configurations are marked by an asterisk and are determined by using the low kinetic energy configuration shortly before the relaxation into the more favorable configuration starts.}
\label{tab:defects:c_ints}
\end{table}
-\begin{figure}[t]
+\begin{figure}[tp]
\begin{center}
\begin{flushleft}
\begin{minipage}{4cm}
Fig. \ref{fig:defects:100db_cmp} schematically shows the \ci{} \hkl<1 0 0> DB structure and Table \ref{tab:defects:100db_cmp} lists the details of the atomic displacements, distances and bond angles obtained by classical potential and quantum-mechanical calculations.
For comparison, the obtained structures for both methods are visualized in Fig. \ref{fig:defects:100db_vis_cmp}.
-\begin{figure}[ht]
+\begin{figure}[tp]
\begin{center}
\includegraphics[width=12cm]{100-c-si-db_cmp.eps}
\end{center}
\caption[Sketch of the \ci{} \hkl<1 0 0> dumbbell structure.]{Sketch of the \ci{} \hkl<1 0 0> dumbbell structure. Atomic displacements, distances and bond angles are listed in Table \ref{tab:defects:100db_cmp}.}
\label{fig:defects:100db_cmp}
\end{figure}
-\begin{table}[ht]
+\begin{table}[tp]
\begin{center}
Displacements\\
\begin{tabular}{l c c c c c c c c c}
\caption[Atomic displacements, distances and bond angles of the \ci{} \hkl<1 0 0> DB structure obtained by {\textsc posic} and {\textsc vasp} calculations.]{Atomic displacements, distances and bond angles of the \ci{} \hkl<1 0 0> DB structure obtained by {\textsc posic} and {\textsc vasp} calculations. The displacements and distances are given in nm and the angles are given in degrees. Displacements, distances and angles are schematically displayed in Fig. \ref{fig:defects:100db_cmp}. In addition, the equilibrium lattice constant for crystalline Si is listed.}
\label{tab:defects:100db_cmp}
\end{table}
-\begin{figure}[ht]
+\begin{figure}[tp]
\begin{center}
\begin{minipage}{6cm}
\begin{center}
\caption{Comparison of the \ci{} \hkl<1 0 0> DB structures obtained by {\textsc posic} and {\textsc vasp} calculations.}
\label{fig:defects:100db_vis_cmp}
\end{figure}
-\begin{figure}[ht]
+\begin{figure}[tp]
\begin{center}
\includegraphics[height=10cm]{c_pd_vasp/eden.eps}
\includegraphics[height=12cm]{c_pd_vasp/100_2333_ksl.ps}
\subsection{Bond-centered interstitial configuration}
\label{subsection:bc}
-\begin{figure}[ht]
+\begin{figure}[tp]
\begin{center}
\begin{minipage}{8cm}
\includegraphics[width=8cm]{c_pd_vasp/bc_2333.eps}\\
In addition, the energy level diagram shows a net amount of two spin up electrons.
% todo smaller images, therefore add mo image
-\clearpage{}
-
-% todo migration of \si{}!
-
\section{Migration of the carbon interstitial}
\label{subsection:100mig}
The stable defect geometries have been discussed in the previous subsection.
In the following, the problem of interstitial C migration in Si is considered.
Since the \ci{} \hkl<1 0 0> DB is the most probable, hence, most important configuration, the migration of this defect atom from one site of the Si host lattice to a neighboring site is in the focus of investigation.
-\begin{figure}[ht]
+\begin{figure}[tp]
\begin{center}
\begin{minipage}{15cm}
\underline{\hkl<0 0 -1> $\rightarrow$ \hkl<0 0 1>}\\
\subsection{Migration paths obtained by first-principles calculations}
-\begin{figure}[ht]
+\begin{figure}[tp]
\begin{center}
\includegraphics[width=13cm]{im_00-1_nosym_sp_fullct_thesis.ps}\\[1.5cm]
\begin{picture}(0,0)(150,0)
This amount of energy is needed to break the bond of the C atom to the Si atom at the bottom left.
In a second process \unit[0.25]{eV} of energy are needed for the system to revert into a \hkl<1 0 0> configuration.
-\begin{figure}[ht]
+\begin{figure}[tp]
\begin{center}
\includegraphics[width=0.7\textwidth]{00-1_0-10_vasp_s.ps}
%\includegraphics[width=13cm]{vasp_mig/00-1_0-10_nosym_sp_fullct.ps}\\[1.6cm]
Fig. \ref{fig:defects:00-1_0-10_mig} shows the migration barrier and structures of the \ci{} \hkl<0 0 -1> to \hkl<0 -1 0> DB transition.
The resulting migration barrier of approximately \unit[0.9]{eV} is very close to the experimentally obtained values of \unit[0.70]{eV} \cite{lindner06}, \unit[0.73]{eV} \cite{song90} and \unit[0.87]{eV} \cite{tipping87}.
-\begin{figure}[ht]
+\begin{figure}[tp]
\begin{center}
\includegraphics[width=13cm]{vasp_mig/00-1_ip0-10_nosym_sp_fullct.ps}\\[1.8cm]
\begin{picture}(0,0)(140,0)
Nevertheless, the theoretical description performed in this work is improved compared to a former study \cite{capaz94}, which underestimates the experimental value by \unit[35]{\%}.
In addition, it is finally shown that the BC configuration, for which spin polarized calculations are necessary, constitutes a real local minimum instead of a saddle point configuration due to the presence of restoring forces for displacements in migration direction.
-\begin{figure}[ht]
+\begin{figure}[tp]
\begin{center}
\includegraphics[width=0.7\textwidth]{vasp_mig/110_mig_vasp.ps}
%\begin{picture}(0,0)(140,0)
%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}[ht]
+%\begin{figure}[tp]
%\begin{center}
%\includegraphics[width=13cm]{vasp_mig/00-1_0-10_nosym_sp_cmp.ps}
%\end{center}
\subsection{Migration described by classical potential calculations}
\label{subsection:defects:mig_classical}
-\begin{figure}[ht]
+\begin{figure}[tp]
\begin{center}
\includegraphics[width=0.7\textwidth]{bc_00-1_albe_s.ps}
%\includegraphics[width=13cm]{bc_00-1.ps}\\[5.6cm]
Assuming equal preexponential factors for both diffusion steps, the total probability of diffusion is given by $\exp\left((2.2\,\text{eV}+0.5\,\text{eV})/k_{\text{B}}T\right)$.
Thus, the activation energy should be located within the range of \unit[2.2-2.7]{eV}.
-\begin{figure}[ht]
+\begin{figure}[tp]
\begin{center}
\includegraphics[width=13cm]{00-1_0-10.ps}\\[2.4cm]
\begin{pspicture}(0,0)(0,0)
% red: ./visualize -w 640 -h 480 -d saves/c_in_si_mig_00-1_0-10_s20 -nll -0.56 -0.56 -0.8 -fur 0.3 0.2 0 -c -0.125 -1.7 0.7 -L -0.125 -0.25 -0.25 -r 0.6 -B 0.1
\label{fig:defects:cp_00-1_0-10_mig}
\end{figure}
-\begin{figure}[ht]
+\begin{figure}[tp]
\begin{center}
\includegraphics[width=0.7\textwidth]{00-1_ip0-10.ps}
\end{center}
Further relaxation of the BC configuration results in the \ci{} \hkl<1 1 0> configuration.
Even the last two pathways show configurations almost identical to the \ci{} \hkl<1 1 0> configuration, which constitute local minima within the pathways.
Thus, migration pathways involving the \ci{} \hkl<1 1 0> DB configuration as a starting or final configuration are further investigated.
-\begin{figure}[ht]
+\begin{figure}[tp]
\begin{center}
\includegraphics[width=0.7\textwidth]{110_mig.ps}
\end{center}
Thus the just proposed migration path, which involves the \hkl<1 1 0> interstitial configuration, becomes even more probable than the initially porposed path, which involves the BC configuration that is, in fact, unstable.
Due to these findings, the respective path is proposed to constitute the diffusion-describing path.
The evolution of structure and configurational energy is displayed again in Fig. \ref{fig:defects:involve110}.
-\begin{figure}[ht]
+\begin{figure}[tp]
\begin{center}
\includegraphics[width=0.7\textwidth]{00-1_110_0-10_mig_albe.ps}
\end{center}
The probability of already rare diffusion events is further decreased for this reason.
However, agglomeration of C and diffusion of Si self-interstitials are an important part of the proposed SiC precipitation mechanism.
Thus, a serious limitation that has to be taken into account for appropriately modeling the C/Si system using the otherwise quite promising EA potential is revealed.
-Possible workarounds are discussed in more detail in section \ref{subsection:md:limit}.
-
-\clearpage{}
+Possible workarounds are discussed in more detail in section \ref{section:md:limit}.
\section{Combination of point defects}
The study proceeds with a structural and energetic investigation of pairs of the ground-state and, thus, most probable defect configurations that are believed to be fundamental in the Si to SiC conversion.
Investigations are restricted to quantum-mechanical calculations.
-\begin{figure}[t]
+\begin{figure}[tp]
\begin{center}
\subfigure[]{\label{fig:defects:combos_ci}\includegraphics[width=0.3\textwidth]{combos_ci.eps}}
\hspace{0.5cm}
\label{subsection:defects:c-si_comb}
\ci{} pairs of the \hkl<1 0 0>-type are investigated in the first part.
-\begin{table}[ht]
+\begin{table}[tp]
\begin{center}
\begin{tabular}{l c c c c c c}
\hline
Antiparallel orientations of the second defect, i.e. \hkl[0 0 1] for positions located below the \hkl(0 0 1) plane with respect to the initial one (positions 1, 2 and 4) form the energetically most unfavorable configurations.
In contrast, the parallel and particularly the twisted orientations constitute energetically favorable configurations, in which a vast reduction of strain is enabled by combination of these defects.
-\begin{figure}[ht]
+\begin{figure}[tp]
\begin{center}
\subfigure[\underline{$E_{\text{b}}=-2.25\,\text{eV}$}]{\label{fig:defects:225}\includegraphics[width=0.3\textwidth]{00-1dc/2-25.eps}}
\hspace{0.5cm}
Thus, particularly at high temepratures that cause an increase of the entropic contribution, this structure constitutes a serious opponent to the ground state.
In fact, following results on migration simulations will reinforce the assumption of a low probability for C clustering by thermally activated processes.
-\begin{figure}[ht]
+\begin{figure}[tp]
\begin{center}
\subfigure[\underline{$E_{\text{b}}=-2.16\,\text{eV}$}]{\label{fig:defects:216}\includegraphics[width=0.25\textwidth]{00-1dc/2-16.eps}}
\hspace{0.2cm}
Nonparallel orientations, i.e. the \hkl[0 1 0], \hkl[0 -1 0] and its equivalents, result in binding energies of \unit[-0.12]{eV} and \unit[-0.27]{eV}, thus, constituting energetically favorable configurations.
The reduction of strain energy is higher in the second case, where the C atom of the second DB is placed in the direction pointing away from the initial C atom.
-\begin{figure}[ht]
+\begin{figure}[tp]
\begin{center}
\subfigure[\underline{$E_{\text{b}}=-1.53\,\text{eV}$}]{\label{fig:defects:153}\includegraphics[width=0.25\textwidth]{00-1dc/1-53.eps}}
\hspace{0.7cm}
The interaction of \ci{} \hkl<1 0 0> DBs is investigated along the \hkl[1 1 0] bond chain assuming a possible reorientation of the DB atom at each position to minimize its configurational energy.
Therefore, the binding energies of the energetically most favorable configurations with the second DB located along the \hkl[1 1 0] direction and resulting C-C distances of the relaxed structures are summarized in Table~\ref{tab:defects:comb_db110}.
-\begin{table}[ht]
+\begin{table}[tp]
\begin{center}
\begin{tabular}{l c c c c c c}
\hline
\label{tab:defects:comb_db110}
\end{table}
%
-\begin{figure}[ht]
+\begin{figure}[tp]
\begin{center}
\includegraphics[width=0.7\textwidth]{db_along_110_cc_n.ps}
\end{center}
Low barriers are only identified for transitions starting from energetically less favorable configurations, e.g. the configuration of a \hkl[-1 0 0] DB located at position 2 (\unit[-0.36]{eV}).
Starting from this configuration, an activation energy of only \unit[1.2]{eV} is necessary for the transition into the ground state configuration.
The corresponding migration energies and atomic configurations are displayed in Fig.~\ref{fig:036-239}.
-\begin{figure}[ht]
+\begin{figure}[tp]
\begin{center}
\includegraphics[width=0.7\textwidth]{036-239.ps}
\end{center}
First of all, it constitutes the second most energetically favorable structure.
Secondly, a migration path with a barrier as low as \unit[0.47]{eV} exists starting from a configuration of largely separated defects exhibiting a low binding energy (\unit[-1.88]{eV}).
The migration barrier and corresponding structures are shown in Fig.~\ref{fig:188-225}.
-\begin{figure}[ht]
+\begin{figure}[tp]
\begin{center}
\includegraphics[width=0.7\textwidth]{188-225.ps}
\end{center}
\subsection[Combinations of the \ci{} \hkl<1 0 0> and \cs{} type]{\boldmath Combinations of the \ci{} \hkl<1 0 0> and \cs{} type}
\label{subsection:defects:c-csub}
-\begin{table}[ht]
+\begin{table}[tp]
\begin{center}
\begin{tabular}{c c c c c c}
\hline
\caption{Binding energies of combinations of the \ci{} \hkl[0 0 -1] defect with a \cs{} atom located at positions 1 to 5 according to Fig.~\ref{fig:defects:combos_ci}. R corresponds to the position located at $\frac{a_{\text{Si}}}{2}\hkl[3 2 3]$ relative to the initial defect position, which is the maximum realizable distance due to periodic boundary conditions.}
\label{tab:defects:c-s}
\end{table}
-%\begin{figure}[ht]
+%\begin{figure}[tp]
%\begin{center}
%\begin{minipage}[t]{5cm}
%a) \underline{Pos: 1, $E_{\text{b}}=0.26\text{ eV}$}
%\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}[ht]
+%\begin{figure}[tp]
%\begin{center}
%\begin{minipage}[t]{7cm}
%a) \underline{Pos: 2, $E_{\text{b}}=-0.51\text{ eV}$}
% A B
%./visualize_contcar -w 640 -h 480 -d results/c_00-1_c3_csub_B -nll -0.20 -0.4 -0.1 -fur 0.9 0.6 0.9 -c 0.5 -1.5 0.375 -L 0.5 0 0.3 -r 0.6 -A -1 2.465
-\begin{figure}[ht]
+\begin{figure}[tp]
\begin{center}
\includegraphics[width=0.7\textwidth]{093-095.ps}
\end{center}
% not satisfactory!
% a b
-\begin{figure}[ht]
+\begin{figure}[tp]
\begin{center}
\includegraphics[width=0.7\textwidth]{026-128.ps}
\end{center}
As mentioned earlier, these two lower Si atoms indeed experience tensile strain along the \hkl[1 1 0] bond chain, however, additional compressive strain along \hkl[0 0 1] exists.
The latter is partially compensated by the C$_{\text{s}}$ atom.
Yet less of compensation is realized if C$_{\text{s}}$ is located at position 4 due to a larger separation although both bottom Si atoms of the DB structure are indirectly affected, i.e. each of them is connected by another Si atom to the C atom enabling the reduction of strain along \hkl[0 0 1].
-\begin{figure}[ht]
+\begin{figure}[tp]
\begin{center}
\subfigure[\underline{$E_{\text{b}}=-0.51\,\text{eV}$}]{\label{fig:defects:051}\includegraphics[width=0.25\textwidth]{00-1dc/0-51.eps}}
\hspace{0.2cm}
In the last section, configurations of a C$_{\text{i}}$ DB with C$_{\text{s}}$ occupying a vacant site have been investigated.
Additionally, configurations might arise in IBS, in which the impinging C atom creates a vacant site near a C$_{\text{i}}$ DB, but does not occupy it.
These structures are investigated in the following.
+
Resulting binding energies of a C$_{\text{i}}$ DB and a nearby vacancy are listed in the second row of Table~\ref{tab:defects:c-v}.
-\begin{table}[ht]
+\begin{table}[tp]
\begin{center}
\begin{tabular}{c c c c c c}
\hline
\caption{Binding energies of combinations of the \ci{} \hkl[0 0 -1] defect with a vacancy located at positions 1 to 5 according to Fig.~\ref{fig:defects:combos_ci}. R corresponds to the position located at $\frac{a_{\text{Si}}}{2}\hkl[3 2 3]$ relative to the initial defect position, which is the maximum realizable distance due to periodic boundary conditions.}
\label{tab:defects:c-v}
\end{table}
-\begin{figure}[ht]
+\begin{figure}[tp]
\begin{center}
\subfigure[\underline{$E_{\text{b}}=-0.59\,\text{eV}$}]{\label{fig:defects:059}\includegraphics[width=0.25\textwidth]{00-1dc/0-59.eps}}
\hspace{0.7cm}
A binding energy of \unit[-0.50]{eV} is observed.
The migration pathways of configuration \ref{fig:defects:314} and \ref{fig:defects:059} into the ground-state configuration, i.e. the \cs{} configuration, are shown in Fig.~\ref{fig:314-539} and \ref{fig:059-539} respectively.
-\begin{figure}[ht]
+\begin{figure}[tp]
\begin{center}
\includegraphics[width=0.7\textwidth]{314-539.ps}
\end{center}
\caption{Migration barrier and structures of the transition of the initial C$_{\text{i}}$ \hkl[0 0 -1] DB and a V created at position 3 (left) into a C$_{\text{s}}$ configuration (right). An activation energy of \unit[0.1]{eV} is observed.}
\label{fig:314-539}
\end{figure}
-\begin{figure}[ht]
+\begin{figure}[tp]
\begin{center}
\includegraphics[width=0.7\textwidth]{059-539.ps}
\end{center}
If the vacancy is created at position 1 the system will end up in a configuration of C$_{\text{s}}$ anyways.
Based on these results, a high probability for the formation of C$_{\text{s}}$ must be concluded.
-%\clearpage{}
-
\subsection{Combinations of \si{} and \cs}
\label{subsection:si-cs}
The ground-state of a single \si{} was found to be the \si{} \hkl<1 1 0> DB configuration.
For the follwoing study the same type of self-interstitial is assumed to provide the energetically most favorable configuration in combination with \cs.
-\begin{table}[ht]
+\begin{table}[tp]
\begin{center}
\begin{tabular}{l c c c c c c}
\hline
\caption{Equivalent configurations labeled \RM{1}-\RM{10} of \hkl<1 1 0>-type Si$_{\text{i}}$ DBs created at position I and C$_{\text{s}}$ created at positions 1 to 5 according to Fig.~\ref{fig:defects:combos_si}. The respective orientation of the Si$_{\text{i}}$ DB is given in the first row.}
\label{tab:defects:comb_csub_si110}
\end{table}
-\begin{table}[ht]
+\begin{table}[tp]
\begin{center}
\begin{tabular}{l c c c c c c c c c c}
\hline
However, even configuration \RM{1} is energetically less favorable than the \hkl<1 0 0> C$_{\text{i}}$ DB, which, thus, remains the ground state of a C atom introduced into otherwise perfect c-Si.
The transition involving the latter two configurations is shown in Fig.~\ref{fig:162-097}.
-\begin{figure}[ht]
+\begin{figure}[tp]
\begin{center}
\includegraphics[width=0.7\textwidth]{162-097.ps}
\end{center}
However, only \unit[0.77]{eV} are needed for the reverse process, i.e. the formation of C$_{\text{s}}$ and a Si$_{\text{i}}$ DB out of the ground state.
Due to the low activation energy this process must be considered to be activated without much effort either thermally or by introduced energy of the implantation process.
-\begin{figure}[ht]
+\begin{figure}[tp]
\begin{center}
\includegraphics[width=0.7\textwidth]{c_sub_si110.ps}
\end{center}
At higher temperatures, the contribution of entropy to structural formation increases, which might result in a spatial separation even for defects located within the capture radius.
Indeed, an {em ab initio} MD run at \unit[900]{$^{\circ}$C} starting from configuration \RM{1}, which -- based on the above findings -- is assumed to recombine into the ground state configuration, results in a separation of the C$_{\text{s}}$ and Si$_{\text{i}}$ DB by more than 4 neighbor distances realized in a repeated migration mechanism of annihilating and arising Si$_{\text{i}}$ DBs.
The atomic configurations for two different points in time are shown in Fig.~\ref{fig:defects:md}.
-\begin{figure}[ht]
+\begin{figure}[tp]
\begin{center}
\begin{minipage}{0.40\textwidth}
-\includegraphics[width=\columnwidth]{md01.eps}
+\includegraphics[width=\columnwidth]{md_vasp_01.eps}
\end{minipage}
\hspace{1cm}
\begin{minipage}{0.40\textwidth}
-\includegraphics[width=\columnwidth]{md02.eps}\\
+\includegraphics[width=\columnwidth]{md_vasp_02.eps}
\end{minipage}\\
\begin{minipage}{0.40\textwidth}
\begin{center}
Si atoms 1 and 2, which form the initial DB, occupy Si lattice sites in the final configuration while Si atom 3 is transferred from a regular lattice site into the interstitial lattice.
These results support the above assumptions of an increased entropic contribution to structural formation involving C$_{\text{s}}$ to a greater extent.
+
+% todo migration of \si{}!
+
+
% kept for nostalgical reason!
%\section{Migration in systems of combined defects}
-%\begin{figure}[ht]
+%\begin{figure}[tp]
%\begin{center}
%\includegraphics[width=13cm]{vasp_mig/comb_mig_3-2_vac_fullct.ps}\\[2.0cm]
%\begin{picture}(0,0)(170,0)
%\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}[ht]
+%\begin{figure}[tp]
%\begin{center}
%\includegraphics[width=13cm]{vasp_mig/comb_mig_4-2_vac_fullct.ps}\\[1.0cm]
%\begin{picture}(0,0)(150,0)
%\label{fig:defects:comb_mig_02}
%\end{figure}
-\clearpage{}
-
\section{Applicability: Competition of \ci{} and \cs-\si{}}
\label{section:ea_app}
This is particularly important since the energy of formation of C$_{\text{s}}$ is drastically underestimated by the EA potential.
A possible occurrence of C$_{\text{s}}$ could then be attributed to a lower energy of formation of the C$_{\text{s}}$-Si$_{\text{i}}$ combination due to the low formation energy of C$_{\text{s}}$, which is obviously wrong.
-Since quantum-mechanical calculations reveal the Si$_{\text{i}}$ \hkl<1 1 0> DB as the ground-state configuration of Si$_{\text{i}}$ in Si, it was assumed to provide the energetically most favorable configuration in combination with C$_{\text{s}}$ in the calculations carried out in section \ref{subsection:cs-si}.
+Since quantum-mechanical calculations reveal the Si$_{\text{i}}$ \hkl<1 1 0> DB as the ground-state configuration of Si$_{\text{i}}$ in Si, it was assumed to provide the energetically most favorable configuration in combination with C$_{\text{s}}$ in the calculations carried out in section \ref{subsection:si-cs}.
Empirical potentials, however, predict Si$_{\text{i}}$ T to be the energetically most favorable configuration.
Thus, investigations of the relative energies of formation of defect pairs need to include combinations of C$_{\text{s}}$ with Si$_{\text{i}}$ T.
Results of {\em ab initio} and classical potential calculations are summarized in Table~\ref{tab:defect_combos}.
-\begin{table}[t]
+\begin{table}[tp]
\begin{center}
\begin{tabular}{l c c c}
\hline
\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, in which the threefold coordinated carbon and silicon atom share a usual silicon lattice site.
-This supports the assumption of C-Si \hkl<1 0 0>-type dumbbel interstitial formation in the first steps of the IBS process as proposed by the precipitation model introduced in section \ref{section:assumed_prec}.
-
-Migration simulations reveal this 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.
-This enables possible migration of the defects to form defect agglomerates as demanded by the model.
-Unfortunately classical potential simulations show tremendously overestimated migration barriers indicating a possible failure of the necessary agglomeration of such defects.
-
-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.
+Obtained results for separated point defects in Si are in good agreement to previous theoretical work on this subject, both for intrinsic defects \cite{leung99,al-mushadani03} as well as for C point defects \cite{dal_pino93,capaz94}.
+The ground-state configurations of these defects, i.e. the Si$_{\text{i}}$ \hkl<1 1 0> and C$_{\text{i}}$ \hkl<1 0 0> DB, are reproduced and compare well to previous findings of theoretical investigations on Si$_{\text{i}}$ \cite{leung99,al-mushadani03} as well as theoretical \cite{dal_pino93,capaz94,burnard93,leary97,jones04} and experimental \cite{watkins76,song90} studies on C$_{\text{i}}$.
+A quantitatively improved activation energy of \unit[0.9]{eV} for a qualitatively equal migration path based on studies by Capaz et.~al.~\cite{capaz94} to experimental values \cite{song90,lindner06,tipping87} ranging from \unit[0.70-0.87]{eV} reinforce their derived mechanism of diffusion for C$_{\text{i}}$ in Si
+However, it turns out that the BC configuration is not a saddle point configuration as proposed by Capaz et~al.~\cite{capaz94} but constitutes a real local minimum if the electron spin is properly accounted for.
+A net magnetization of two electrons, which is already clear by simple molecular orbital theory considerations on the bonding of the $sp$ hybridized C atom, is settled.
+By investigating the charge density isosurface it turns out that the two resulting spin up electrons are localized in a torus around the C atom.
+With an activation energy of \unit[0.9]{eV} the C$_{\text{i}}$ carbon interstitial can be expected to be highly mobile at prevailing temperatures in the process under investigation, i.e. IBS.
+Since the \ci{} \hkl<1 0 0> DB is the ground-state configuration and highly mobile, possible migration of these DBs to form defect agglomerates, as demanded by the model introduced in section \ref{section:assumed_prec}, is considered possible.
+
+Unfortunately the description of the same processes fails if classical potential methods are used.
+Already the geometry of the most stable DB configuration differs considerably from that obtained by first-principles calculations.
+The classical approach is unable to reproduce the correct character of bonding due to the deficiency of quantum-mechanical effects in the potential.
+Nevertheless, both methods predict the same type of interstitial as the ground-state configuration and also the order in energy of the remaining defects is reproduced fairly well.
+From this, a description of defect structures by classical potentials looks promising.
+%
+However, focussing on the description of diffusion processes the situation changes completely.
+Qualitative and quantitative differences exist.
+First of all, a different pathway is suggested as the lowest energy path, which again might be attributed to the absence of quantum-mechanical effects in the classical interaction model.
+Secondly, the activation energy is overestimated by a factor of 2.4 to 3.5 compared to the more accurate quantum-mechanical methods and experimental findings.
+This is attributed to the sharp cut-off of the short range potential.
+As already pointed out in a previous study \cite{mattoni2007}, the short cut-off is responsible for overestimated and unphysical high forces of next neighbor atoms.
+The overestimated migration barrier, however, affects the diffusion behavior of the C interstitials.
+By this artifact, the mobility of the C atoms is tremendously decreased resulting in an inaccurate description or even absence of the DB agglomeration as proposed by one of the precipitation models.
+
+Quantum-mechanical investigations of two \ci{} of the \hkl<1 0 0>-type and varying separations and orientations state an attractive interaction between these interstitials.
+Obtained results for the most part compare well with results gained in previous studies \cite{leary97,capaz98,mattoni2002,liu02} and show an astonishingly good agreement with experiment \cite{song90}.
+%
+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.
-
-Next to the C-Si \hkl<1 0 0> dumbbell interstitial configuration, in which the C atom is sharing a Si lattice site with the corresponding Si atom the C atom could occupy the site of the Si atom, which in turn forms a Si self-interstitial.
-Combinations of substitutional C and a \hkl<1 1 0> Si self-interstitial, which is the ground state configuration for a Si self-interstitial and, thus, assumed to be the energetically most favorable configuration for combined structures, show formation energies 0.5 eV to 1.5 eV greater than that of the C-Si \hkl<1 0 0> interstitial configuration, which remains the energetically most favorable configuration.
-However, the binding energy of substitutional C and the Si self-interstitial quickly drops to zero already for short separations indicating a low interaction capture radius.
-Thus, due to missing attractive interaction forces driving the system to form C-Si \hkl<1 0 0> dumbbell interstitial complexes substitutional C, while thermodynamically not stable, constitutes a most likely configuration occuring in IBS, a process far from equlibrium.
-
-Due to the low interaction capture radius substitutional C can be treated independently of the existence of separated Si self-interstitials.
-This should be also true for combinations of C-Si interstitials next to a vacancy and a further separated Si self-interstitial excluded from treatment, which again is a conveivable configuration in IBS.
-By combination of a \hkl<1 0 0> dumbbell with a vacancy in the absence of the Si self-interstitial it is found that the configuration of substitutional carbon 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.
-
-While first results support the proposed precipitation model the latter suggest the formation of silicon carbide by succesive creation of substitutional carbon instead of the agglomeration of C-Si dumbbell interstitials followed by an abrupt transition.
-Prevailing conditions in the IBS process at elevated temperatures and the fact that IBS is a nonequilibrium process reinforce the possibility of formation of substitutional C instead of the thermodynamically stable C-Si dumbbell interstitials predicted by simulations at zero Kelvin.
+For dumbbells oriented along the \hkl<1 1 0> bond chain 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 \ci{} DB agglomeration.
+%
+The ground state configuration is found to consist of a C-C bond, which is responsible for the vast gain in energy.
+However, based on investigations of possible migration pathways, these structures are less likely to arise than structures, in which both C atoms are interconnected by another Si atom, which is due to high activation energies of the respective pathways or alternative pathways featuring less high activation energies, which, however, involve intermediate unfavorable configurations.
+Thus, agglomeration of C$_{\text{i}}$ is expected while the formation of C-C bonds is assumed to fail to appear by thermally activated diffusion processes.
+
+In contrast, C$_{\text{i}}$ and vacancies are found to efficiently react with each other exhibiting activation energies as low as \unit[0.1]{eV} and \unit[0.6]{eV} resulting in stable C$_{\text{s}}$ configurations.
+In addition, a highly attractive interaction exhibiting a large capture radius, effective independent of the orientation and the direction of separation of the defects, is observed.
+Accordingly, the formation of C$_{\text{s}}$ is very likely to occur.
+Comparatively high energies necessary for the reverse process reveal this configuration to be extremely stable.
+Thus, C interstitials and vacancies located close together are assumed to end up in such a configuration, in which the C atom is tetrahedrally coordinated and bound to four Si atoms as expected in SiC.
+
+Investigating configurations of C$_{\text{s}}$ and Si$_{\text{i}}$, formation energies higher than that of the C$_{\text{i}}$ \hkl<1 0 0> DB were obtained keeping up previously derived assumptions concerning the ground state of C$_{\text{i}}$ in otherwise perfect Si.
+However, a small capture radius is identified for the respective interaction that might prevent the recombination of defects exceeding a separation of \unit[0.6]{nm} into the ground state configuration.
+In addition, a rather small activation energy of \unit[0.77]{eV} allows for the formation of a C$_{\text{s}}$-Si$_{\text{i}}$ pair originating from the C$_{\text{i}}$ \hkl<1 0 0> DB structure by thermally activated processes.
+Thus, elevated temperatures might lead to thermodynamically unstable configurations of C$_{\text{s}}$ and a remaining Si atom in the near interstitial lattice, which is supported by the result of an {\em ab initio} molecular dynamics run.
+%Thus, due to missing attractive interaction forces driving the system to form C-Si \hkl<1 0 0> dumbbell interstitial complexes substitutional C, while thermodynamically not stable, constitutes a most likely configuration occuring in IBS, a process far from equlibrium.
+
+% todo
+% maybe move above stuff to conclusion chapter, at least shorten!
+% see remember in sic chapter
+
+These findings allow to draw conclusions on the mechanisms involved in the process of SiC conversion in Si.
+Agglomeration of C$_{\text{i}}$ is energetically favored and enabled by a low activation energy for migration.
+Although ion implantation is a process far from thermodynamic equilibrium, which might result in phases not described by the Si/C phase diagram, i.e. a C phase in Si, high activation energies are believed to be responsible for a low probability of the formation of C-C clusters.
+
+In the context of the initially stated controversy present in the precipitation model, these findings suggest an increased participation of C$_{\text{s}}$ already in the initial stage due to its high probability of incidence.
+In addition, thermally activated, C$_{\text{i}}$ might turn into C$_{\text{s}}$.
+The associated emission of Si$_{\text{i}}$ serves two needs: as a vehicle for other C$_{\text{s}}$ atoms and as a supply of Si atoms needed elsewhere to form the SiC structure.
+As for the vehicle, Si$_{\text{i}}$ is believed to react with C$_{\text{s}}$ turning it into highly mobile C$_{\text{i}}$ again, allowing for the rearrangement of the C atom.
+The rearrangement is crucial to end up in a configuration of C atoms only occupying substitutionally the lattice sites of one of the two fcc lattices that build up the diamond lattice.
+On the other hand, the conversion of some region of Si into SiC by \cs{} is accompanied by a reduction of the volume since SiC exhibits a \unit[20]{\%} smaller lattice constant than Si.
+The reduction in volume is compensated by excess Si$_{\text{i}}$ serving as building blocks for the surrounding Si host or a further formation of SiC.
+
+To conclude, precipitation occurs by successive agglomeration of C$_{\text{s}}$.
+However, the agglomeration and rearrangement of C$_{\text{s}}$ is only possible by mobile C$_{\text{i}}$, which has to be present at the same time.
+Accordingly, the process is governed by both, C$_{\text{s}}$ accompanied by Si$_{\text{i}}$ as well as C$_{\text{i}}$.
+It is worth to mention that there is no contradiction to results of the HREM studies \cite{werner96,werner97,eichhorn99,lindner99_2,koegler03}.
+Regions showing dark contrasts in an otherwise undisturbed Si lattice are attributed to C atoms in the interstitial lattice.
+However, there is no particular reason for the C species to reside in the interstitial lattice.
+Contrasts are also assumed for Si$_{\text{i}}$.
+Once precipitation occurs, regions of dark contrasts disappear in favor of Moir\'e patterns indicating 3C-SiC in c-Si due to the mismatch in the lattice constant.
+Until then, however, these regions are either composed of stretched coherent SiC and interstitials or of already contracted incoherent SiC surrounded by Si and interstitials, where the latter is too small to be detected in HREM.
+In both cases Si$_{\text{i}}$ might be attributed a third role, which is the partial compensation of tensile strain that is present either in the stretched SiC or at the interface of the contracted SiC and the Si host.
+
+Furthermore, the experimentally observed alignment of the \hkl(h k l) planes of the precipitate and the substrate is satisfied by the mechanism of successive positioning of C$_{\text{s}}$.
+In contrast, there is no obvious reason for the topotactic orientation of an agglomerate consisting exclusively of C-Si dimers, which would necessarily involve a much more profound change in structure for the transition into SiC.
+
+%Prevailing conditions in the IBS process at elevated temperatures and the fact that IBS is a nonequilibrium process reinforce the possibility of formation of substitutional C instead of the thermodynamically stable C-Si dumbbell interstitials predicted by simulations at zero Kelvin.
\label{section:defects:noneq_process_02}
-{\color{blue}
+\ifnum1=0
+
In addition, there are experimental findings, which might be exploited to reinforce the non-validity of the proposed precipitation model.
High resolution TEM shows equal orientation of \hkl(h k l) planes of the c-Si host matrix and the 3C-SiC precipitate.
+
Formation of 3C-SiC realized by successive formation of substitutional C, in which the atoms belonging to one of the two fcc lattices are substituted by C atoms perfectly conserves the \hkl(h k l) planes of the initial c-Si diamond lattice.
+
Silicon self-interstitials consecutively created to the same degree are able to diffuse into the c-Si host one after another.
+
Investigated combinations of C interstitials, however, result in distorted configurations, in which C atoms, which at some point will form SiC, are no longer aligned to the host.
+
It is easily understandable that the mismatch in alignement will increase with increasing defect density.
+
In addition, the amount of Si self-interstitials equal to the amount of agglomerated C atoms would be released all of a sudden probably not being able to diffuse into the c-Si host matrix without damaging the Si surrounding or the precipitate itself.
+
In addition, IBS results in the formation of the cubic polytype of SiC only.
+
As this result conforms well with the model of precipitation by substitutional C there is no obvious reason why hexagonal polytypes should not be able to form or an equal alignement would be mandatory assuming the model of precipitation by C-Si dumbbell agglomeration.
-}
-{\color{red}Todo: C mobility higher than Si mobility? -> substitutional C is more likely to arise, since it migrates 'faster' to vacant sites?}
+\fi
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