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}.
+\bibpunct{}{}{,}{n}{}{}
\begin{table}[tp]
\begin{center}
\begin{tabular}{l c c c c c}
\multicolumn{6}{c}{Other {\em ab initio} studies} \\
Ref. \cite{al-mushadani03} & 3.40 & 3.45 & - & - & 3.53 \\
Ref. \cite{leung99} & 3.31 & 3.31 & 3.43 & - & - \\
-% todo cite without []
\hline
\hline
\end{tabular}
\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}
+\bibpunct{[}{]}{,}{n}{}{}
\begin{figure}[tp]
\begin{center}
\begin{flushleft}
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{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.
+Among the established analytical potentials only the environment-dependent interatomic potential (EDIP) \cite{bazant97,justo98} and Stillinger-Weber \cite{stillinger85} potential reproduce the correct order in energy of the defects.
+However, these potentials show shortcomings concerning the description of other physical properties and are unable to describe the C-C and C-Si interaction.
In fact the EA potential calculations favor the tetrahedral defect configuration.
This limitation is assumed to arise due to the cut-off.
In the tetrahedral configuration the second neighbors are only slightly more distant than the first neighbors, which creates the particular problem.
While not completely rendering impossible further, more challenging empirical potential studies on large systems, the artifact has to be taken into account in the investigations of defect combinations later on in this chapter.
The hexagonal configuration is not stable opposed to results of the authors of the potential \cite{albe_sic_pot}.
-In the first two pico seconds, while kinetic energy is decoupled from the system, the \si{} seems to condense at the hexagonal site.
+In the first two picoseconds, while kinetic energy is decoupled from the system, the \si{} seems to condense at the hexagonal site.
The formation energy of \unit[4.48]{eV} is determined by this low kinetic energy configuration shortly before the relaxation process starts.
The \si{} atom then begins to slowly move towards an energetically more favorable position very close to the tetrahedral one but slightly displaced along the three coordinate axes.
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}.
To exclude failures in the implementation of the potential or the MD code itself the hexagonal defect structure was double-checked with the {\textsc parcas} MD code \cite{parcas_md}.
The respective relaxation energetics are likewise plotted and look similar to the energetics obtained by {\textsc posic}.
In fact, 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.
+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 tetrahedral 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.
+However, the energy barrier required for a transition into the tetrahedral configuration is small.
\begin{figure}[tp]
\begin{center}
\includegraphics[width=0.7\textwidth]{nhex_tet.ps}
\end{minipage}
\end{flushleft}
\end{center}
-\caption[Relaxed C point defect configurations obtained by classical potential calculations.]{Relaxed C point defect configurations obtained by classical potential calculations. The Si/C atoms and the bonds (only for the interstitial atom) are illustrated by yellow/grey spheres and blue lines.}
+\caption[Relaxed C point defect configurations obtained by classical potential calculations.]{Relaxed C point defect configurations obtained by classical potential calculations. The Si/C atoms and the bonds (only for the interstitial atom) are illustrated by yellow/gray spheres and blue lines.}
\label{fig:defects:c_conf}
\end{figure}
\cs{} occupying an already vacant Si lattice site, which is in fact not an interstitial defect, is found to be the lowest configuration in energy for all potential models.
-An experiemntal value of the formation energy of \cs{} was determined by a fit to solubility data yielding a concentration of $3.5 \times 10^{24} \exp{(-2.3\,\text{eV}/k_{\text{B}}T)} \text{ cm}^{-3}$ \cite{bean71}.
+An experimental value of the formation energy of \cs{} was determined by a fit to solubility data yielding a concentration of $3.5 \times 10^{24} \exp{(-2.3\,\text{eV}/k_{\text{B}}T)} \text{ cm}^{-3}$ \cite{bean71}.
However, there is no particular reason for treating the prefactor as a free parameter in the fit to the experimental data.
It is simply given by the atomic density of pure silicon, which is $5\times 10^{22}\text{ cm}^{-3}$.
-Tersoff \cite{tersoff90} and Dal Pino et al. \cite{dal_pino93} pointed out that by combining this prefactor with the calculated values for the energy of formation ranging from \unit[1.6-1.89]{eV} an excellent agreement with the experimental solubility data within the entire temeprature range of the experiment is obtained.
+Tersoff \cite{tersoff90} and Dal Pino et al. \cite{dal_pino93} pointed out that by combining this prefactor with the calculated values for the energy of formation ranging from \unit[1.6-1.89]{eV} an excellent agreement with the experimental solubility data within the entire temperature range of the experiment is obtained.
This reinterpretation of the solubility data, first proposed by Tersoff and later on reinforced by Dal~Pino~et~al. is in good agreement with the results of the quantum-mechanical calculations performed in this work.
Unfortunately the EA potential undervalues the formation energy roughly by a factor of two, which is a definite drawback of the potential.
-Except for Tersoff's results for the tedrahedral configuration, the \ci{} \hkl<1 0 0> DB is the energetically most favorable interstital configuration.
+Except for Tersoff's results for the tetrahedral configuration, the \ci{} \hkl<1 0 0> DB is the energetically most favorable interstitial configuration.
As mentioned above, the low energy of formation for the tetrahedral interstitial in the case of the Tersoff potential is believed to be an artifact of the abrupt cut-off set to \unit[2.5]{\AA} (see Ref. 11 and 13 in \cite{tersoff90}) and the real formation energy is, thus, supposed to be located between \unit[3-10]{eV}.
Keeping these considerations in mind, the \ci{} \hkl<1 0 0> DB is the most favorable interstitial configuration for all interaction models.
This finding is in agreement with several theoretical\cite{burnard93,leary97,dal_pino93,capaz94,jones04} and experimental\cite{watkins76,song90} investigations, which all predict this configuration to be the ground state.
Quantum-mechanical calculations reveal this configuration to be unstable, which is also reproduced by the EA potential.
In both cases a relaxation towards the \ci{} \hkl<1 0 0> DB configuration is observed.
Opposed to results of the first-principles calculations, Tersoff finds this configuration to be stable \cite{tersoff90}.
-In fact, the stability of the hexagonal interstitial could not be reproduced in simulations performed in this work using the unmodifed Tersoff potential parameters.
+In fact, the stability of the hexagonal interstitial could not be reproduced in simulations performed in this work using the unmodified Tersoff potential parameters.
Unfortunately, apart from the modified parameters, no more conditions specifying the relaxation process are given in Tersoff's study on C point defects in Si.
The tetrahedral is the second most unfavorable interstitial configuration using classical potentials if the abrupt cut-off effect of the Tersoff potential is taken into account.
Tersoff indeed predicts a metastable BC configuration.
However, it is not in the correct order and lower in energy than the \ci{} \hkl<1 1 0> DB.
Quantum-mechanical results of this configuration are discussed in more detail in section \ref{subsection:bc}.
-In another {\em ab inito} study, Capaz~et~al.~\cite{capaz94} in turn found the BC configuration to be an intermediate saddle point structure of a possible migration path, which is \unit[2.1]{eV} higher than the \ci{} \hkl<1 0 0> DB structure.
-This is assumed to be due to the neglection of the electron spin in these calculations.
+In another {\em ab initio} study, Capaz~et~al.~\cite{capaz94} in turn found the BC configuration to be an intermediate saddle point structure of a possible migration path, which is \unit[2.1]{eV} higher than the \ci{} \hkl<1 0 0> DB structure.
+This is assumed to be due to the neglecting of the electron spin in these calculations.
Another {\textsc vasp} calculation without fully accounting for the electron spin results in the smearing of a single electron over two non-degenerate states for the BC configuration.
This problem is resolved by spin polarized calculations resulting in a net spin of one accompanied by a reduction of the total energy by \unit[0.3]{eV} and the transformation into a metastable local minimum configuration.
It is worth to note that all other listed configurations are not affected by spin polarization.
To conclude, discrepancies between the results from classical potential calculations and those obtained from first principles are observed.
Within the classical potentials EA outperforms Tersoff and is, therefore, used for further studies.
Both methods (EA and DFT) predict the \ci{} \hkl<1 0 0> DB configuration to be most stable.
-Also the remaining defects and their energetical order are described fairly well.
+Also the remaining defects and their energetic order are described fairly well.
It is thus concluded that, so far, modeling of the SiC precipitation by the EA potential might lead to trustable results.
\subsection[C \hkl<1 0 0> dumbbell interstitial configuration]{\boldmath C \hkl<1 0 0> dumbbell interstitial configuration}
\label{subsection:100db}
As the \ci{} \hkl<1 0 0> DB constitutes the ground-state configuration of a C atom incorporated into otherwise perfect c-Si it is the most probable and, hence, one of the most important interstitial configurations of C in Si.
-The structure was initially suspected by IR local vibrational mode absorption \cite{bean70} and finally verified by electron paramegnetic resonance (EPR) \cite{watkins76} studies on irradiated Si substrates at low temperatures.
+The structure was initially suspected by IR local vibrational mode absorption \cite{bean70} and finally verified by electron paramagnetic resonance (EPR) \cite{watkins76} studies on irradiated Si substrates at low temperatures.
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}.
\includegraphics[height=10cm]{c_pd_vasp/eden.eps}
\includegraphics[height=12cm]{c_pd_vasp/100_2333_ksl.ps}
\end{center}
-\caption[Charge density isosurface and Kohn-Sham levels of the \ci{} \hkl<1 0 0> DB structure obtained by {\textsc vasp} calculations.]{Charge density isosurface and Kohn-Sham levels of the \ci{} \hkl<1 0 0> DB structure obtained by {\textsc vasp} calculations. Yellow and grey spheres correspond to Si and C atoms. The blue surface is the charge density isosurface. In the energy level diagram red and green lines and dots mark occupied and unoccupied states.}
+\caption[Charge density isosurface and Kohn-Sham levels of the \ci{} \hkl<1 0 0> DB structure obtained by {\textsc vasp} calculations.]{Charge density isosurface and Kohn-Sham levels of the \ci{} \hkl<1 0 0> DB structure obtained by {\textsc vasp} calculations. Yellow and gray spheres correspond to Si and C atoms. The blue surface is the charge density isosurface. In the energy level diagram red and green lines and dots mark occupied and unoccupied states.}
\label{img:defects:charge_den_and_ksl}
\end{figure}
The Si atom labeled '1' and the C atom compose the DB structure.
There is no magnetization density.
An acceptor level arises at approximately $E_v+0.35\,\text{eV}$ while a band gap of about \unit[0.75]{eV} can be estimated from the Kohn-Sham level diagram for plain Si.
However, strictly speaking, the Kohn-Sham levels and orbitals do not have a direct physical meaning and, thus, these values have to be taken with care.
-% todo band gap problem
+% todo - band gap problem, skip it?
\subsection{Bond-centered interstitial configuration}
\label{subsection:bc}
\begin{figure}[tp]
\begin{center}
\begin{minipage}{8cm}
-\includegraphics[width=8cm]{c_pd_vasp/bc_2333.eps}\\
+\begin{center}
+\includegraphics[width=6cm]{c_pd_vasp/bc_2333.eps}\\
+\vspace*{0.2cm}
\hrule
\vspace*{0.2cm}
-\includegraphics[width=8cm]{c_100_mig_vasp/im_spin_diff.eps}
+\includegraphics[width=6cm]{c_100_mig_vasp/im_spin_diff.eps}
+\vspace*{0.2cm}
+\framebox{
+ \footnotesize
+ \begin{minipage}[t]{7.5cm}
+ \begin{minipage}[t]{1.4cm}
+ {\color{red}Si}\\
+ {\tiny sp$^3$}\\[0.8cm]
+ \underline{${\color{black}\uparrow}$}
+ \underline{${\color{black}\uparrow}$}
+ \underline{${\color{black}\uparrow}$}
+ \underline{${\color{red}\uparrow}$}\\
+ sp$^3$
+ \end{minipage}
+ \begin{minipage}[t]{1.6cm}
+ \begin{center}
+ {\color{red}M}{\color{blue}O}\\[0.8cm]
+ \underline{${\color{blue}\uparrow}{\color{white}\downarrow}$}\\
+ $\sigma_{\text{ab}}$\\[0.5cm]
+ \underline{${\color{red}\uparrow}{\color{blue}\downarrow}$}\\
+ $\sigma_{\text{b}}$
+ \end{center}
+ \end{minipage}
+ \begin{minipage}[t]{1.2cm}
+ \begin{center}
+ {\color{blue}C}\\
+ {\tiny sp}\\[0.2cm]
+ \underline{${\color{white}\uparrow\uparrow}$}
+ \underline{${\color{white}\uparrow\uparrow}$}\\
+ 2p\\[0.4cm]
+ \underline{${\color{blue}\uparrow}{\color{blue}\downarrow}$}
+ \underline{${\color{blue}\uparrow}{\color{blue}\downarrow}$}\\
+ sp
+ \end{center}
+ \end{minipage}
+ \begin{minipage}[t]{1.6cm}
+ \begin{center}
+ {\color{blue}M}{\color{green}O}\\[0.8cm]
+ \underline{${\color{blue}\uparrow}{\color{white}\downarrow}$}\\
+ $\sigma_{\text{ab}}$\\[0.5cm]
+ \underline{${\color{green}\uparrow}{\color{blue}\downarrow}$}\\
+ $\sigma_{\text{b}}$
+ \end{center}
+ \end{minipage}
+ \begin{minipage}[t]{1.4cm}
+ \begin{flushright}
+ {\color{green}Si}\\
+ {\tiny sp$^3$}\\[0.8cm]
+ \underline{${\color{green}\uparrow}$}
+ \underline{${\color{black}\uparrow}$}
+ \underline{${\color{black}\uparrow}$}
+ \underline{${\color{black}\uparrow}$}\\
+ sp$^3$
+ \end{flushright}
+ \end{minipage}
+ \end{minipage}
+}
+\end{center}
\end{minipage}
\begin{minipage}{7cm}
\includegraphics[width=7cm]{c_pd_vasp/bc_2333_ksl.ps}
\end{minipage}
\end{center}
-\caption[Structure, charge density isosurface and Kohn-Sham level diagram of the bond-centered interstitial configuration.]{Structure, charge density isosurface and Kohn-Sham level diagram of the bond-centered interstitial configuration. Gray, green and blue surfaces mark the charge density of spin up, spin down and the resulting spin up electrons in the charge density isosurface, in which the carbon atom is represented by a red sphere. In the energy level diagram red and green lines mark occupied and unoccupied states.}
+\caption[Structure, charge density isosurface, molecular orbital diagram and Kohn-Sham level diagram of the bond-centered interstitial configuration.]{Structure, charge density, molecular orbital diagram isosurface and Kohn-Sham level diagram of the bond-centered interstitial configuration. Gray, green and blue surfaces mark the charge density of spin up, spin down and the resulting spin up electrons in the charge density isosurface, in which the carbon atom is represented by a red sphere. In the energy level diagram red and green lines mark occupied and unoccupied states.}
\label{img:defects:bc_conf}
\end{figure}
-In the BC insterstitial configuration the interstitial atom is located inbetween two next neighbored Si atoms forming linear bonds.
-In a previous study this configuration was found to constitute an intermediate saddle point configuration determining the migration barrier of one possibe migration path of a \ci{} \hkl<1 0 0> DB configuration into an equivalent one \cite{capaz94}.
+In the BC interstitial configuration the interstitial atom is located in between two next neighbored Si atoms forming linear bonds.
+In a previous study this configuration was found to constitute an intermediate saddle point configuration determining the migration barrier of one possible migration path of a \ci{} \hkl<1 0 0> DB configuration into an equivalent one \cite{capaz94}.
This is in agreement with results of the EA potential simulations, which reveal this configuration to be unstable relaxing into the \ci{} \hkl<1 1 0> configuration.
However, this fact could not be reproduced by spin polarized {\textsc vasp} calculations performed in this work.
Present results suggest this configuration to correspond to a real local minimum.
-In fact, an additional barrier has to be passed to reach this configuration starting from the \ci{} \hkl<1 0 0> interstitital configuration, which is investigated in section \ref{subsection:100mig}.
+In fact, an additional barrier has to be passed to reach this configuration starting from the \ci{} \hkl<1 0 0> interstitial configuration, which is investigated in section \ref{subsection:100mig}.
After slightly displacing the C atom along the \hkl[1 0 0] (equivalent to a displacement along \hkl[0 1 0]), \hkl[0 0 1], \hkl[0 0 -1] and \hkl[1 -1 0] direction the distorted structures relax back into the BC configuration.
As will be shown in subsequent migration simulations the same would happen to structures where the C atom is displaced along the migration direction, which approximately is the \hkl[1 1 0] direction.
These relaxations indicate that the BC configuration is a real local minimum instead of an assumed saddle point configuration.
This is supported by the charge density isosurface and the Kohn-Sham levels in Fig. \ref{img:defects:bc_conf}.
The blue torus, which reinforces 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.
-% todo smaller images, therefore add mo image
\section{Migration of the carbon interstitial}
\label{subsection:100mig}
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}[tp]
\begin{center}
+%
\begin{minipage}{15cm}
-\underline{\hkl<0 0 -1> $\rightarrow$ \hkl<0 0 1>}\\
+\centering
+\framebox{\hkl<0 0 -1> $\rightarrow$ \hkl<0 0 1>}\\
\begin{minipage}{4.5cm}
\includegraphics[width=4.5cm]{c_pd_vasp/100_2333.eps}
\end{minipage}
\begin{minipage}{4.5cm}
\includegraphics[width=4.5cm]{c_pd_vasp/100_next_2333.eps}
\end{minipage}
-\end{minipage}\\
+\end{minipage}\\[0.5cm]
+%
\begin{minipage}{15cm}
-\underline{\hkl<0 0 -1> $\rightarrow$ \hkl<0 -1 0>}\\
+\centering
+\framebox{\hkl<0 0 -1> $\rightarrow$ \hkl<0 -1 0>}\\
\begin{minipage}{4.5cm}
\includegraphics[width=4.5cm]{c_pd_vasp/100_2333.eps}
\end{minipage}
\begin{minipage}{4.5cm}
\includegraphics[width=4.5cm]{c_pd_vasp/0-10_2333.eps}
\end{minipage}
-\end{minipage}\\
+\end{minipage}\\[0.5cm]
+%
\begin{minipage}{15cm}
-\underline{\hkl<0 0 -1> $\rightarrow$ \hkl<0 -1 0> (in place)}\\
+\centering
+\framebox{\hkl<0 0 -1> $\rightarrow$ \hkl<0 -1 0> (in place)}\\
\begin{minipage}{4.5cm}
\includegraphics[width=4.5cm]{c_pd_vasp/100_2333.eps}
\end{minipage}
\begin{figure}[tp]
\begin{center}
-\includegraphics[width=13cm]{im_00-1_nosym_sp_fullct_thesis.ps}\\[1.5cm]
-\begin{picture}(0,0)(150,0)
-\includegraphics[width=2.5cm]{vasp_mig/00-1.eps}
-\end{picture}
-\begin{picture}(0,0)(-10,0)
-\includegraphics[width=2.5cm]{vasp_mig/bc_00-1_sp.eps}
-\end{picture}
-\begin{picture}(0,0)(-120,0)
-\includegraphics[width=2.5cm]{vasp_mig/bc.eps}
-\end{picture}
-\begin{picture}(0,0)(25,20)
-\includegraphics[width=2.5cm]{110_arrow.eps}
-\end{picture}
-\begin{picture}(0,0)(200,0)
-\includegraphics[height=2.2cm]{001_arrow.eps}
-\end{picture}
+\includegraphics[width=0.7\textwidth]{im_00-1_nosym_sp_fullct_thesis_vasp_s.ps}
\end{center}
-\caption[Migration barrier and structures of the \hkl<0 0 -1> DB (left) to BC (right) transition.]{Migration barrier and structures of the \hkl<0 0 -1> DB (left) to BC (right) transition. Bonds of the C atom are illustrated by blue lines.}
+\caption[Migration barrier and structures of the {\hkl[0 0 -1]} DB to BC transition.]{Migration barrier and structures of the \hkl<0 0 -1> DB (left) to BC (right) transition. Bonds of the C atom are illustrated by blue lines.}
\label{fig:defects:00-1_001_mig}
\end{figure}
In Fig. \ref{fig:defects:00-1_001_mig} results of the \hkl<0 0 -1> to \hkl<0 0 1> migration fully described by the migration of the \hkl<0 0 -1> to the BC configuration is displayed.
\begin{center}
\includegraphics[width=0.7\textwidth]{00-1_0-10_vasp_s.ps}
\end{center}
-\caption[Migration barrier and structures of the \hkl<0 0 -1> DB (left) to the \hkl<0 -1 0> DB (right) transition.]{Migration barrier and structures of the \hkl[0 0 -1] DB (left) to the \hkl[0 -1 0] DB (right) transition. Bonds of the C atom are illustrated by blue lines. {\color{red} Prototype design, adjust related figures!}}
-% todo read above caption! enable [] hkls in short caption
+\caption[Migration barrier and structures of the {\hkl[0 0 -1]} DB to the {\hkl[0 -1 0]} DB transition.]{Migration barrier and structures of the \hkl[0 0 -1] DB (left) to the \hkl[0 -1 0] DB (right) transition. Bonds of the C atom are illustrated by blue lines.}
\label{fig:defects:00-1_0-10_mig}
\end{figure}
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.
\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)
-\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}
-\begin{picture}(0,0)(25,20)
-\includegraphics[width=2.5cm]{100_arrow.eps}
-\end{picture}
-\begin{picture}(0,0)(200,0)
-\includegraphics[height=2.2cm]{001_arrow.eps}
-\end{picture}
+\includegraphics[width=0.7\textwidth]{00-1_ip0-10_nosym_sp_fullct_vasp_s.ps}
\end{center}
-\caption[Reorientation barrier and structures of the \hkl<0 0 -1> DB (left) to the \hkl<0 -1 0> DB (right) transition in place.]{Reorientation barrier and structures of the \hkl<0 0 -1> DB (left) to the \hkl<0 -1 0> DB (right) transition in place. Bonds of the carbon atoms are illustrated by blue lines.}
+\caption[Reorientation barrier and structures of the {\hkl[0 0 -1]} DB to the {\hkl[0 -1 0]} DB transition in place.]{Reorientation barrier and structures of the \hkl[0 0 -1] DB (left) to the \hkl[0 -1 0] DB (right) transition in place. Bonds of the carbon atoms are illustrated by blue lines.}
\label{fig:defects:00-1_0-10_ip_mig}
\end{figure}
The third migration path, in which the DB is changing its orientation, is shown in Fig. \ref{fig:defects:00-1_0-10_ip_mig}.
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 C interstitial in Si explaining both, annealing and reorientation experiments.
The activation energy of roughly \unit[0.9]{eV} nicely compares to experimental values reinforcing the correct identification of the C-Si DB diffusion mechanism.
-Slightly increased values compared to experiment might be due to the tightend constraints applied in the modified CRT approach.
+Slightly increased values compared to experiment might be due to the tightened constraints applied in the modified CRT approach.
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.
%\includegraphics[width=2.2cm]{vasp_mig/0-10_b.eps}
%\end{picture}
\end{center}
-\caption{Migration barriers of the \hkl<1 1 0> DB to BC (blue), \hkl<0 0 -1> (green) and \hkl<0 -1 0> (in place, red) C-Si DB transition.}
+\caption[{Migration barriers of the \hkl[1 1 0] DB to BC, \hkl[0 0 -1] and \hkl[0 -1 0] (in place) C-Si DB transition.}]{Migration barriers of the \hkl[1 1 0] DB to BC (blue), \hkl[0 0 -1] (green) and \hkl[0 -1 0] (in place, red) C-Si DB 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.
%\includegraphics[height=2.2cm]{010_arrow.eps}
%\end{picture}
\end{center}
-\caption[Migration barrier and structures of the \ci{} BC to \hkl<0 0 -1> DB transition using the classical EA potential.]{Migration barrier and structures of the \ci{} BC to \hkl[0 0 -1] DB transition using the classical EA potential. Two migration pathways are obtained for different time constants of the Berendsen thermostat. The lowest activation energy is \unit[2.2]{eV}. {\color{red} Prototype design, adjust related figures!}}
+\caption[Migration barrier and structures of the \ci{} BC to {\hkl[0 0 -1]} DB transition using the classical EA potential.]{Migration barrier and structures of the \ci{} BC to \hkl[0 0 -1] DB transition using the classical EA potential. Two migration pathways are obtained for different time constants of the Berendsen thermostat. The lowest activation energy is \unit[2.2]{eV}.}
\label{fig:defects:cp_bc_00-1_mig}
% red: ./visualize -w 640 -h 480 -d saves/c_in_si_mig_bc_00-1_s20 -nll -0.56 -0.56 -0.7 -fur 0.2 0.2 0.0 -c 0.75 -1.25 -0.25 -L -0.25 -0.25 -0.25 -r 0.6 -B 0.1
% blue: ./visualize -w 640 -h 480 -d saves/c_in_si_mig_bc_00-1_s20_tr100/ -nll -0.56 -0.56 -0.7 -fur 0.2 0.2 0.0 -c 0.0 -0.25 1.0 -L 0.0 -0.25 -0.25 -r 0.6 -B 0.1
\end{figure}
-Fig. \ref{fig:defects:cp_bc_00-1_mig} shows the evolution of structure and energy along the \ci{} BC to \hkl<0 0 -1> DB transition.
-Since the \ci{} BC configuration is unstable relaxing into the \hkl<1 1 0> DB configuration within this potential, the low kinetic energy state is used as a starting configuration.
-Two different pathways are obtained for different time constants of the Berendse
-n thermostat.
+Fig. \ref{fig:defects:cp_bc_00-1_mig} shows the evolution of structure and energy along the \ci{} BC to \hkl[0 0 -1] DB transition.
+Since the \ci{} BC configuration is unstable relaxing into the \hkl[1 1 0] DB configuration within this potential, the low kinetic energy state is used as a starting configuration.
+Two different pathways are obtained for different time constants of the Berendsen thermostat.
With a time constant of \unit[1]{fs} the C atom resides in the \hkl(1 1 0) plane
resulting in a migration barrier of \unit[2.4]{eV}.
However, weaker coupling to the heat bath realized by an increase of the time constant to \unit[100]{fs} enables the C atom to move out of the \hkl(1 1 0) plane already at the beginning, which is accompanied by a reduction in energy, approaching the final configuration on a curved path.
The energy barrier of this path is \unit[0.2]{eV} lower in energy than the direct migration within the \hkl(1 1 0) plane.
However, the investigated pathways cover an activation energy approximately twice as high as the one obtained by quantum-mechanical calculations.
-If the entire transition of the \hkl<0 0 -1> into the \hkl<0 0 1> configuration is considered a two step process passing the intermediate BC configuration, an additional activation energy of \unit[0.5]{eV} is necessary to escape the BC towards the \hkl<0 0 1> configuration.
+If the entire transition of the \hkl[0 0 -1] into the \hkl[0 0 1] configuration is considered a two step process passing the intermediate BC configuration, an additional activation energy of \unit[0.5]{eV} is necessary to escape the BC towards the \hkl[0 0 1] configuration.
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}[tp]
\begin{center}
-\includegraphics[width=13cm]{00-1_0-10.ps}\\[2.4cm]
-\begin{pspicture}(0,0)(0,0)
-\psframe[linecolor=red,fillstyle=none](-6,-0.5)(7.2,2.8)
-\end{pspicture}
-\begin{picture}(0,0)(130,-10)
-\includegraphics[width=2.2cm]{albe_mig/00-1_0-10_red_00.eps}
-\end{picture}
-\begin{picture}(0,0)(0,-10)
-\includegraphics[width=2.2cm]{albe_mig/00-1_0-10_red_min.eps}
-\end{picture}
-\begin{picture}(0,0)(-120,-10)
-\includegraphics[width=2.2cm]{albe_mig/00-1_0-10_red_03.eps}
-\end{picture}
-\begin{picture}(0,0)(25,10)
-\includegraphics[width=2.5cm]{100_arrow.eps}
-\end{picture}
-\begin{picture}(0,0)(185,-10)
-\includegraphics[height=2.2cm]{001_arrow.eps}
-\end{picture}
+\includegraphics[width=0.7\textwidth]{00-1_0-10_albe_s.ps}
\end{center}
-\caption{Migration barrier and structures of the \ci{} \hkl<0 0 -1> to \hkl<0 -1 0> DB transition using the classical EA potential.}
+\caption{Migration barrier and structures of the \ci{} \hkl[0 0 -1] to \hkl[0 -1 0] DB transition using the classical EA potential.}
% 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{center}
\includegraphics[width=0.7\textwidth]{00-1_ip0-10.ps}
\end{center}
-\caption{Reorientation barrier of the \ci{} \hkl<0 0 -1> to \hkl<0 -1 0> DB transition in place using the classical EA potential.}
+\caption{Reorientation barrier of the \ci{} \hkl[0 0 -1] to \hkl[0 -1 0] DB transition in place using the classical EA potential.}
\label{fig:defects:cp_00-1_ip0-10_mig}
\end{figure}
-Figures \ref{fig:defects:cp_00-1_0-10_mig} and \ref{fig:defects:cp_00-1_ip0-10_mig} show the migration barriers of the \ci{} \hkl<0 0 -1> to \hkl<0 -1 0> DB transition.
+Figures \ref{fig:defects:cp_00-1_0-10_mig} and \ref{fig:defects:cp_00-1_ip0-10_mig} show the migration barriers of the \ci{} \hkl[0 0 -1] to \hkl[0 -1 0] DB transition.
In the first case, the transition involves a change in the lattice site of the C atom whereas in the second case, a reorientation at the same lattice site takes place.
-In the first case, the pathways for the two different time cosntants look similar.
-A local minimum exists inbetween two peaks of the graph.
-The corresponding configuration, which is illustrated for the results obtained for a time constant of \unit[1]{fs}, looks similar to the \ci{} \hkl<1 1 0> configuration.
+In the first case, the pathways for the two different time constants look similar.
+A local minimum exists in between two peaks of the graph.
+The corresponding configuration, which is illustrated for the results obtained for a time constant of \unit[1]{fs}, looks similar to the \ci{} \hkl[1 1 0] configuration.
Indeed, this configuration is obtained by relaxation simulations without constraints of configurations near the minimum.
Activation energies of roughly \unit[2.8]{eV} and \unit[2.7]{eV} are needed for migration.
-The \ci{} \hkl<1 1 0> configuration seems to play a decisive role in all migration pathways in the classical potential calculations.
+The \ci{} \hkl[1 1 0] configuration seems to play a decisive role in all migration pathways in the classical potential calculations.
As mentioned above, the starting configuration of the first migration path, i.e. the BC configuration, is fixed to be a transition point but in fact is unstable.
-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.
+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}[tp]
\begin{center}
\includegraphics[width=0.7\textwidth]{110_mig.ps}
\end{center}
-\caption[Migration barriers of the \ci{} \hkl<1 1 0> DB to BC (blue), \hkl<0 0 -1> (green) and \hkl<0 -1 0> (in place, red) transition.]{Migration barriers of the \ci{} \hkl<1 1 0> DB to BC (blue), \hkl<0 0 -1> (green) and \hkl<0 -1 0> (in place, red) transition. Solid lines show results for a time constant of \unit[1]{fs} and dashed lines correspond to simulations employing a time constant of \unit[100]{fs}.}
+\caption[{Migration barriers of the \ci{} \hkl[1 1 0] DB to BC, \hkl[0 0 -1] and \hkl[0 -1 0] (in place) transition.}]{Migration barriers of the \ci{} \hkl[1 1 0] DB to BC (blue), \hkl[0 0 -1] (green) and \hkl[0 -1 0] (in place, red) transition. Solid lines show results for a time constant of \unit[1]{fs} and dashed lines correspond to simulations employing a time constant of \unit[100]{fs}.}
\label{fig:defects:110_mig}
\end{figure}
-Fig. \ref{fig:defects:110_mig} shows migration barriers of the \ci{} \hkl<1 1 0> DB to \hkl<0 0 -1>, \hkl<0 -1 0> (in place) and BC configuration.
+Fig. \ref{fig:defects:110_mig} shows migration barriers of the \ci{} \hkl[1 1 0] DB to \hkl[0 0 -1], \hkl[0 -1 0] (in place) and BC configuration.
As expected there is no maximum for the transition into the BC configuration.
-As mentioned earlier the BC configuration itself constitutes a saddle point configuration relaxing into the energetically more favorable \hkl<1 1 0> DB configuration.
-An activation energy of \unit[2.2]{eV} is necessary to reorientate the \hkl<0 0 -1> into the \hkl<1 1 0> DB configuration, which is \unit[1.3]{eV} higher in energy.
-Residing in this state another \unit[0.90]{eV} is enough to make the C atom form a \hkl<0 0 -1> DB configuration with the Si atom of the neighbored lattice site.
-In contrast to quantum-mechanical calculations, in which the direct transition is the energetically most favorable transition and the transition composed of the intermmediate migration steps is very unlikely to occur, the just presented pathway is much more conceivable in classical potential simulations, since the energetically most favorable transition found so far is likewise composed of two migration steps with activation energies of \unit[2.2]{eV} and \unit[0.5]{eV}, for which the intermediate state is the BC configuration, which is unstable.
-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.
+As mentioned earlier, the BC configuration itself constitutes a saddle point configuration relaxing into the energetically more favorable \hkl[1 1 0] DB configuration.
+An activation energy of \unit[2.2]{eV} is necessary to reorientate the \hkl[0 0 -1] into the \hkl[1 1 0] DB configuration, which is \unit[1.3]{eV} higher in energy.
+Residing in this state another \unit[0.90]{eV} is enough to make the C atom form a \hkl[0 0 -1] DB configuration with the Si atom of the neighbored lattice site.
+In contrast to quantum-mechanical calculations, in which the direct transition is the energetically most favorable transition and the transition composed of the intermediate migration steps is very unlikely to occur, the just presented pathway is much more conceivable in classical potential simulations, since the energetically most favorable transition found so far is likewise composed of two migration steps with activation energies of \unit[2.2]{eV} and \unit[0.5]{eV}, for which the intermediate state is the BC configuration, which is unstable.
+Thus the just proposed migration path, which involves the \hkl[1 1 0] interstitial configuration, becomes even more probable than the initially proposed 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}[tp]
\begin{center}
\includegraphics[width=0.7\textwidth]{00-1_110_0-10_mig_albe.ps}
\end{center}
-\caption[Migration barrier and structures of the \ci{} \hkl<0 0 -1> (left) to the \hkl<0 -1 0> DB (right) transition involving the \hkl<1 1 0> DB (center) configuration.]{Migration barrier and structures of the \ci{} \hkl[0 0 -1] (left) to the \hkl[0 -1 0] DB (right) transition involving the \hkl[1 1 0] DB (center) configuration. Migration simulations are performed utilizing time constants of \unit[1]{fs} (solid line) and \unit[100]{fs} (dashed line) for the Berendsen thermostat.}
+\caption[Migration barrier and structures of the \ci{} {\hkl[0 0 -1]} to the {\hkl[0 -1 0]} DB transition involving the {\hkl[1 1 0]} DB configuration.]{Migration barrier and structures of the \ci{} \hkl[0 0 -1] (left) to the \hkl[0 -1 0] DB (right) transition involving the \hkl[1 1 0] DB (center) configuration. Migration simulations are performed utilizing time constants of \unit[1]{fs} (solid line) and \unit[100]{fs} (dashed line) for the Berendsen thermostat.}
\label{fig:defects:involve110}
\end{figure}
Approximately \unit[2.2]{eV} are needed to turn the \ci{} \hkl[0 0 -1] into the \hkl[1 1 0] DB located at the neighbored lattice site in \hkl[1 1 -1] direction.
\hspace{0.5cm}
\subfigure[]{\label{fig:defects:combos_si}\includegraphics[width=0.3\textwidth]{combos.eps}}
\end{center}
-\caption{Position of the initial \ci{} \hkl[0 0 -1] DB (I) (a) and of the lattice site chosen for the initial \si{} \hkl<1 1 0> DB (\si) (b). Lattice sites for the second defect used for investigating defect pairs are numbered from 1 to 5. For black/red/blue numbers, one/two/four possible atom(s) exist for the second defect to create equivalent defect combinations.}
+\caption[Position of the initial \ci{} {\hkl[0 0 -1]} DB and of the lattice site chosen for the initial \si{} \hkl<1 1 0> DB.]{Position of the initial \ci{} \hkl[0 0 -1] DB (I) (a) and of the lattice site chosen for the initial \si{} \hkl<1 1 0> DB (\si) (b). Lattice sites for the second defect used for investigating defect pairs are numbered from 1 to 5. For black/red/blue numbers, one/two/four possible atom(s) exist for the second defect to create equivalent defect combinations.}
\label{fig:defects:combos}
\end{figure}
Fig.~\ref{fig:defects:combos} schematically displays the initial \ci{} \hkl[0 0 -1] DB structure (Fig.~\ref{fig:defects:combos_ci}) as well as the lattice site chosen for the initial \si{} \hkl<1 1 0> DB (Fig.~\ref{fig:defects:combos_si}) and various positions for the second defect (1-5) that are used for investigating defect pairs.
\hspace{0.5cm}
\subfigure[\underline{$E_{\text{b}}=-2.39\,\text{eV}$}]{\label{fig:defects:239}\includegraphics[width=0.3\textwidth]{00-1dc/2-39.eps}}
\end{center}
-\caption{Relaxed structures of defect combinations obtained by creating \hkl[1 0 0] (a) and \hkl[0 -1 0] (b) DBs at position 1.}
+\caption[Relaxed structures of defect combinations obtained by creating {\hkl[1 0 0]} and {\hkl[0 -1 0]} DBs at position 1.]{Relaxed structures of defect combinations obtained by creating \hkl[1 0 0] (a) and \hkl[0 -1 0] (b) DBs at position 1.}
\label{fig:defects:comb_db_01}
\end{figure}
Mattoni~et~al. \cite{mattoni2002} predict the ground-state configuration of \ci{} \hkl<1 0 0>-type defect pairs for a \hkl[1 0 0] or equivalently a \hkl[0 1 0] defect created at position 1 with both defects basically maintaining the as-isolated DB structure, resulting in a binding energy of \unit[-2.1]{eV}.
The initial configuration is still evident in the relaxed configuration.
The two \ci{} atoms form a strong C-C bond, which is responsible for the large gain in energy resulting in a binding energy of \unit[-2.39]{eV}.
This bond has a length of \unit[1.38]{\AA} close to the next neighbor distance in diamond or graphite, which is approximately \unit[1.54]{\AA}.
-The minimum of the binding energy observed for this configuration suggests prefered C clustering as a competing mechnism to the \ci{} DB interstitial agglomeration inevitable for the SiC precipitation.
+The minimum of the binding energy observed for this configuration suggests preferred C clustering as a competing mechanism to the \ci{} DB interstitial agglomeration inevitable for the SiC precipitation.
However, the second most favorable configuration ($E_{\text{f}}=-2.25\,\text{eV}$) is represented four times, i.e. two times more often than the ground-state configuration, within the systematically investigated configuration space.
-Thus, particularly at high temepratures that cause an increase of the entropic contribution, this structure constitutes a serious opponent to the ground state.
+Thus, particularly at high temperatures 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}[tp]
\hspace{0.2cm}
\subfigure[\underline{$E_{\text{b}}=-2.05\,\text{eV}$}]{\label{fig:defects:205}\includegraphics[width=0.25\textwidth]{00-1dc/2-05.eps}}
\end{center}
-\caption{Relaxed structures of defect combinations obtained by creating \hkl[1 0 0] (a) and \hkl[0 1 0] (b) DBs at position 2 and a \hkl[0 0 1] (c) DB at position 3.}
+\caption[Relaxed structures of defect combinations obtained by creating {\hkl[1 0 0]} and {\hkl[0 1 0]} DBs at position 2 and a {\hkl[0 0 1]} DB at position 3.]{Relaxed structures of defect combinations obtained by creating \hkl[1 0 0] (a) and \hkl[0 1 0] (b) DBs at position 2 and a \hkl[0 0 1] (c) DB at position 3.}
\label{fig:defects:comb_db_02}
\end{figure}
Fig.~\ref{fig:defects:comb_db_02} shows the next three energetically favorable configurations.
The C atoms are spaced by \unit[3.14]{\AA}, which is very close to the expected C-C next neighbor distance of \unit[3.08]{\AA} in SiC.
Figure \ref{fig:defects:205} displays the results of a \hkl[0 0 1] DB inserted at position 3.
The binding energy is \unit[-2.05]{eV}.
-Both DBs are tilted along the same direction remaining parallely aligned and the second DB is pushed downwards in such a way, that the four DB atoms form a rhomboid.
+Both DBs are tilted along the same direction remaining aligned in parallel and the second DB is pushed downwards in such a way, that the four DB atoms form a rhomboid.
Both C atoms form tetrahedral bonds to four Si atoms.
However, Si atom number 1 and number 3, which are bound to the second \ci{} atom are also bound to the initial C atom.
These four atoms of the rhomboid reside in a plane and, thus, do not match the situation in SiC.
This low interaction is due to the large distance and a missing direct connection by bonds along a chain in the crystallographic \hkl<1 1 0> direction.
Both, C and Si atoms of the DBs form threefold coordinated bonds to their neighbors.
The energetically most unfavorable configuration ($E_{\text{b}}=0.26\,\text{eV}$) is obtained for the \ci{} \hkl[0 0 1] DB, which is oppositely orientated with respect to the initial one.
-A DB taking the same orientation as the initial one is less unfavorble ($E_{\text{b}}=0.04\,\text{eV}$).
+A DB taking the same orientation as the initial one is less unfavorable ($E_{\text{b}}=0.04\,\text{eV}$).
Both configurations are unfavorable compared to far-off, isolated DBs.
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.
\hspace{0.7cm}
\subfigure[\underline{$E_{\text{b}}=-1.38\,\text{eV}$}]{\label{fig:defects:138}\includegraphics[width=0.25\textwidth]{00-1dc/1-38.eps}}
\end{center}
-\caption{Relaxed structures of defect combinations obtained by creating \hkl[0 0 1] (a), \hkl[0 0 -1] (b), \hkl[0 -1 0] (c) and \hkl[1 0 0] (d) DBs at position 5.}
+\caption[Relaxed structures of defect combinations obtained by creating {\hkl[0 0 1]}, {\hkl[0 0 -1]}, {\hkl[0 -1 0]} and {\hkl[1 0 0]} DBs at position 5.]{Relaxed structures of defect combinations obtained by creating \hkl[0 0 1] (a), \hkl[0 0 -1] (b), \hkl[0 -1 0] (c) and \hkl[1 0 0] (d) DBs at position 5.}
\label{fig:defects:comb_db_03}
\end{figure}
-Energetically beneficial configurations of defect combinations are observed for interstititals of all orientations placed at position 5, a position two bonds away from the initial interstitial along the \hkl[1 1 0] direction.
+Energetically beneficial configurations of defect combinations are observed for interstitials 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 combinations are displayed in Fig. \ref{fig:defects:comb_db_03}.
Fig. \ref{fig:defects:153} and \ref{fig:defects:166} show the relaxed structures of \hkl[0 0 1] and \hkl[0 0 -1] DBs.
The upper DB atoms are pushed towards each other forming fourfold coordinated bonds.
\begin{center}
\includegraphics[width=0.7\textwidth]{db_along_110_cc_n.ps}
\end{center}
-\caption[Minimum binding energy of DB combinations separated along \hkl<1 1 0> with respect to the C-C distance.]{Minimum binding energy of dumbbell combinations separated along \hkl[1 1 0] with respect to the C-C distance. The blue line is a guide for the eye and the green curve corresponds to the most suitable fit function consisting of all but the first data point.}
+\caption[Minimum binding energy of DB combinations separated along {\hkl[1 1 0]} with respect to the C-C distance.]{Minimum binding energy of dumbbell combinations separated along \hkl[1 1 0] with respect to the C-C distance. The blue line is a guide for the eye and the green curve corresponds to the most suitable fit function consisting of all but the first data point.}
\label{fig:defects:comb_db110}
\end{figure}
The binding energy of these configurations with respect to the C-C distance is plotted in Fig.~\ref{fig:defects:comb_db110}.
\begin{center}
\includegraphics[width=0.7\textwidth]{036-239.ps}
\end{center}
-\caption{Migration barrier and structures of the transition of a C$_{\text{i}}$ \hkl[-1 0 0] DB at position 2 (left) into a C$_{\text{i}}$ \hkl[0 -1 0] DB at position 1 (right). An activation energy of \unit[1.2]{eV} is observed.}
+\caption[Migration barrier and structures of the transition of a C$_{\text{i}}$ {\hkl[-1 0 0]} DB at position 2 into a C$_{\text{i}}$ {\hkl[0 -1 0]} DB at position 1.]{Migration barrier and structures of the transition of a C$_{\text{i}}$ \hkl[-1 0 0] DB at position 2 (left) into a C$_{\text{i}}$ \hkl[0 -1 0] DB at position 1 (right). An activation energy of \unit[1.2]{eV} is observed.}
\label{fig:036-239}
\end{figure}
Since thermally activated C clustering is, thus, only possible by traversing energetically unfavored configurations, extensive C clustering is not expected.
\begin{center}
\includegraphics[width=0.7\textwidth]{188-225.ps}
\end{center}
-\caption{Migration barrier and structures of the transition of a C$_{\text{i}}$ \hkl[0 -1 0] DB at position 5 (left) into a C$_{\text{i}}$ \hkl[1 0 0] DB at position 1 (right). An activation energy of \unit[0.47]{eV} is observed.}
+\caption[Migration barrier and structures of the transition of a C$_{\text{i}}$ {\hkl[0 -1 0]} DB at position 5 into a C$_{\text{i}}$ {\hkl[1 0 0]} DB at position 1.]{Migration barrier and structures of the transition of a C$_{\text{i}}$ \hkl[0 -1 0] DB at position 5 (left) into a C$_{\text{i}}$ \hkl[1 0 0] DB at position 1 (right). An activation energy of \unit[0.47]{eV} is observed.}
\label{fig:188-225}
\end{figure}
Finally, as already mentioned above, this type of defect pair is represented two times more often than the ground-state configuration.
\hline
\end{tabular}
\end{center}
-\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.}
+\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}.]{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}[tp]
\begin{center}
\includegraphics[width=0.7\textwidth]{093-095.ps}
\end{center}
-\caption{Migration barrier and structures of the transition of the initial \ci{} \hkl[0 0 -1] DB and C$_{\text{s}}$ at position 3 (left) into a configuration of a twofold coordinated Si$_{\text{i}}$ located in between two C$_{\text{s}}$ atoms occupying the lattice sites of the initial DB and position 3 (right). An activation energy of \unit[0.44]{eV} is observed.}
+\caption[Migration barrier and structures of the transition of the initial \ci{} {\hkl[0 0 -1]} DB and C$_{\text{s}}$ at position 3 into a configuration of a twofold coordinated Si$_{\text{i}}$ located in between two C$_{\text{s}}$ atoms occupying the lattice sites of the initial DB and position 3.]{Migration barrier and structures of the transition of the initial \ci{} \hkl[0 0 -1] DB and C$_{\text{s}}$ at position 3 (left) into a configuration of a twofold coordinated Si$_{\text{i}}$ located in between two C$_{\text{s}}$ atoms occupying the lattice sites of the initial DB and position 3 (right). An activation energy of \unit[0.44]{eV} is observed.}
\label{fig:093-095}
\end{figure}
Configuration A consists of a C$_{\text{i}}$ \hkl[0 0 -1] DB with threefold coordinated Si and C DB atoms slightly disturbed by the C$_{\text{s}}$ at position 3, facing the Si DB atom as a neighbor.
\begin{center}
\includegraphics[width=0.7\textwidth]{comb_mig_026-128_vasp.ps}
\end{center}
-\caption{Migration barrier and structures of the transition of the initial C$_{\text{i}}$ \hkl[0 0 -1] DB and C$_{\text{s}}$ at position 1 (left) into a C-C \hkl[1 0 0] DB occupying the lattice site at position 1 (right). An activation energy of \unit[0.1]{eV} is observed.}
+\caption[Migration barrier and structures of the transition of the initial C$_{\text{i}}$ {\hkl[0 0 -1]} DB and C$_{\text{s}}$ at position 1 into a C-C {\hkl[1 0 0]} DB occupying the lattice site at position 1.]{Migration barrier and structures of the transition of the initial C$_{\text{i}}$ \hkl[0 0 -1] DB and C$_{\text{s}}$ at position 1 (left) into a C-C \hkl[1 0 0] DB occupying the lattice site at position 1 (right). An activation energy of \unit[0.1]{eV} is observed.}
\label{fig:026-128}
\end{figure}
Configuration $\alpha$ is similar to configuration A, except that the C$_{\text{s}}$ atom at position 1 is facing the C DB atom as a neighbor resulting in the formation of a strong C-C bond and a much more noticeable perturbation of the DB structure.
\hspace{0.2cm}
\subfigure[\underline{$E_{\text{b}}=0.49\,\text{eV}$}]{\label{fig:defects:049}\includegraphics[width=0.25\textwidth]{00-1dc/0-49.eps}}
\end{center}
-\caption{Relaxed structures of defect combinations obtained by creating \cs{} at positions 2 (a), 4 (b) and 5 (c) in the \ci{} \hkl[0 0 -1] DB configuration.}
+\caption[Relaxed structures of defect combinations obtained by creating \cs{} at positions 2, 4 and 5 in the \ci{} {\hkl[0 0 -1]} DB configuration.]{Relaxed structures of defect combinations obtained by creating \cs{} at positions 2 (a), 4 (b) and 5 (c) in the \ci{} \hkl[0 0 -1] DB configuration.}
\label{fig_defects:245csub}
\end{figure}
Fig.~\ref{fig_defects:245csub} lists the remaining configurations and binding energies of the relaxed structures obtained by creating a \cs{} at positions 2, 4 and 5 in the \ci{} \hkl[0 0 -1] DB configuration.
\hline
\end{tabular}
\end{center}
-\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.}
+\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}.]{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}[tp]
\hspace{0.7cm}
\subfigure[\underline{$E_{\text{b}}=-0.50\,\text{eV}$}]{\label{fig:defects:050}\includegraphics[width=0.25\textwidth]{00-1dc/0-50.eps}}
\end{center}
-\caption{Relaxed structures of defect combinations obtained by creating a vacancy at positions 2 (a), 3 (b), 4 (c) and 5 (d).}
+\caption[Relaxed structures of defect combinations obtained by creating a vacancy at positions 2, 3, 4 and 5.]{Relaxed structures of defect combinations obtained by creating a vacancy 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} shows the associated configurations.
The DB structure shifts towards the position of the vacancy, which replaces the Si atom usually bound to and at the same time strained by the \si{} DB atom.
Due to the displacement into the \hkl[1 -1 0] direction the bond of the DB Si atom to the Si atom on the top left breaks and instead forms a bond to the Si atom located in \hkl[1 -1 1] direction, which is not shown in Fig.~\ref{fig:defects:314}.
A binding energy of \unit[-3.14]{eV} is obtained for this structure composing another energetically favorable configuration.
-A vacancy ctreated at position 2 enables the relaxation of Si atom number 1 mainly in \hkl[0 0 -1] direction.
+A vacancy created at position 2 enables the relaxation of Si atom number 1 mainly in \hkl[0 0 -1] direction.
The bond to Si atom number 5 breaks.
Hence, the \si{} DB atom is not only displaced along \hkl[0 0 -1] but also and to a greater extent in \hkl[1 1 0] direction.
The C atom is slightly displaced in \hkl[0 1 -1] direction.
Fig.~\ref{fig:defects:050} shows the relaxed structure of a vacancy created at position 5.
The Si DB atom is largely displaced along \hkl[1 1 0] and somewhat less along \hkl[0 0 -1], which corresponds to the direction towards the vacancy.
The \si DB atom approaches Si atom 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 DB itself.
+Indeed, a non-zero charge density is observed in between these two atoms exhibiting a cylinder-like shape superposed with the charge density known from the DB itself.
Strain reduced by this huge displacement is partially absorbed by tensile strain on Si atom number 1 originating from attractive forces of the C atom and the vacancy.
A binding energy of \unit[-0.50]{eV} is observed.
\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.}
+\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 into a C$_{\text{s}}$ configuration.]{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}[tp]
\begin{center}
\includegraphics[width=0.7\textwidth]{059-539.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 2 (left) into a C$_{\text{s}}$ configuration (right). An activation energy of \unit[0.6]{eV} is observed.}
+\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 2 into a C$_{\text{s}}$ configuration.]{Migration barrier and structures of the transition of the initial C$_{\text{i}}$ \hkl[0 0 -1] DB and a V created at position 2 (left) into a C$_{\text{s}}$ configuration (right). An activation energy of \unit[0.6]{eV} is observed.}
\label{fig:059-539}
\end{figure}
Activation energies as low as \unit[0.1]{eV} and \unit[0.6]{eV} are observed.
A net amount of five Si-Si and one Si-C bond are additionally formed during transition.
An activation energy of \unit[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 \unit[5.4]{eV} are needed, which make this mechanism very unprobable.
+To reverse this process \unit[5.4]{eV} are needed, which make this mechanism very improbable.
%
The migration path is best described by the reverse process.
Starting at \unit[100]{\%}, energy is needed to break the bonds of Si atom 1 to its neighbored Si atoms as well as the bond of the C atom to Si atom number 5.
With regard to the IBS process, in which highly energetic C atoms enter the Si target being able to kick out Si atoms from their lattice sites, such configurations are absolutely conceivable and a significant influence on the precipitation process might be attributed to them.
Thus, combinations of \cs{} and an additional \si{} are examined in the following.
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.
+For the following study the same type of self-interstitial is assumed to provide the energetically most favorable configuration in combination with \cs.
\begin{table}[tp]
\begin{center}
\hline
\end{tabular}
\end{center}
-\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.}
+\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}.]{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}[tp]
%
In the same way the energetically most unfavorable configuration can be explained, which is configuration \RM{3}.
The \cs{} is located next to the lattice site shared by the \si{} \hkl[1 1 0] DB in \hkl[1 -1 1] direction.
-Thus, the compressive stress along \hkl[1 1 0] of the \si{} \hkl[1 1 0] DB is not compensated but intensified by the tensile stress of the \cs{} atom, which is no longer loacted along the direction of stress.
+Thus, the compressive stress along \hkl[1 1 0] of the \si{} \hkl[1 1 0] DB is not compensated but intensified by the tensile stress of the \cs{} atom, which is no longer located along the direction of stress.
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{center}
\includegraphics[width=0.7\textwidth]{162-097.ps}
\end{center}
-\caption{Migration barrier and structures of the transition of a \hkl[1 1 0] Si$_{\text{i}}$ DB next to C$_{\text{s}}$ (right) into the C$_{\text{i}}$ \hkl[0 0 -1] DB configuration (left). An activation energy of \unit[0.12]{eV} and \unit[0.77]{eV} for the reverse process is observed.}
+\caption[Migration barrier and structures of the transition of a {\hkl[1 1 0]} Si$_{\text{i}}$ DB next to C$_{\text{s}}$ into the C$_{\text{i}}$ {\hkl[0 0 -1]} DB configuration.]{Migration barrier and structures of the transition of a \hkl[1 1 0] Si$_{\text{i}}$ DB next to C$_{\text{s}}$ (right) into the C$_{\text{i}}$ \hkl[0 0 -1] DB configuration (left). An activation energy of \unit[0.12]{eV} and \unit[0.77]{eV} for the reverse process is observed.}
\label{fig:162-097}
\end{figure}
An activation energy as low as \unit[0.12]{eV} is necessary for the migration into the ground-state configuration.
\begin{center}
\includegraphics[width=0.7\textwidth]{c_sub_si110.ps}
\end{center}
-\caption{Binding energies of combinations of a C$_{\text{s}}$ and a Si$_{\text{i}}$ DB with respect to the separation distance. The interaction strength of the defect pairs are well approximated by a Lennard-Jones 6-12 potential, which is used for curve fitting.}
+\caption[Binding energies of combinations of a C$_{\text{s}}$ and a Si$_{\text{i}}$ DB with respect to the separation distance.]{Binding energies of combinations of a C$_{\text{s}}$ and a Si$_{\text{i}}$ DB with respect to the separation distance. The interaction strength of the defect pairs are well approximated by a Lennard-Jones 6-12 potential, which is used for curve fitting.}
\label{fig:dc_si-s}
\end{figure}
Fig.~\ref{fig:dc_si-s} shows the binding energies of pairs of C$_{\text{s}}$ and a Si$_{\text{i}}$ \hkl<1 1 0> DB with respect to the separation distance.
\begin{figure}[tp]
\begin{center}
\begin{minipage}{0.40\textwidth}
-\includegraphics[width=\columnwidth]{md_vasp_01.eps}
+\includegraphics[width=\columnwidth]{md01_bonds.eps}
\end{minipage}
\hspace{1cm}
\begin{minipage}{0.40\textwidth}
-\includegraphics[width=\columnwidth]{md_vasp_02.eps}
+\includegraphics[width=\columnwidth]{md02_bonds.eps}
\end{minipage}\\
\begin{minipage}{0.40\textwidth}
\begin{center}
\end{center}
\end{minipage}
\end{center}
-\caption{Atomic configurations of an ab initio molecular dynamics run at \unit[900]{$^{\circ}$C} starting from a configuration of C$_{\text{s}}$ located next to a Si$_{\text{i}}$ \hkl[1 1 0] DB (atoms 1 and 2). Equal atoms are marked by equal numbers. Bonds are drawn for substantial atoms only.}
+\caption[Atomic configurations of an ab initio molecular dynamics run at {\unit[900]{$^{\circ}$C}} starting from a configuration of C$_{\text{s}}$ located next to a Si$_{\text{i}}$ {\hkl[1 1 0]} DB.]{Atomic configurations of an ab initio molecular dynamics run at \unit[900]{$^{\circ}$C} starting from a configuration of C$_{\text{s}}$ located next to a Si$_{\text{i}}$ \hkl[1 1 0] DB (atoms 1 and 2). Equal atoms are marked by equal numbers. For substantial atoms, bonds are drawn in red color.}
\label{fig:defects:md}
\end{figure}
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.
-% link to migration of \si{}!
+\section{Mobility of the silicon self-interstitial}
+
The possibility for separated configurations of \cs{} and \si{} becomes even more likely if one of the constituents exhibits a low barrier of migration.
In this case, the \si{} is assumed to constitute the mobile defect compared to the stable \cs{} atom.
Thus, migration paths of \si{} are investigated in the following excursus.
-Acoording to Fig.~\ref{fig:defects:si_mig1}, an activation energy of \unit[0.67]{eV} is necessary for the transition of the \si{} \hkl[0 -1 1] to \hkl[1 1 0] DB located at the neighbored Si lattice site in \hkl[1 1 -1] direction.
+According to Fig.~\ref{fig:defects:si_mig1}, an activation energy of \unit[0.67]{eV} is necessary for the transition of the \si{} \hkl[0 -1 1] to \hkl[1 1 0] DB located at the neighbored Si lattice site in \hkl[1 1 -1] direction.
\begin{figure}[tp]
\begin{center}
\includegraphics[width=0.7\textwidth]{si_110_110_mig_02_conf.ps}
\end{center}
-\caption[Migration barrier and structures of the \si{} \hkl<1 1 0> DB.]{Migration barrier and structures of the \si{} \hkl[0 -1 1] DB (left) to the \hkl[1 1 0] DB (right) transition. Bonds are illustrated by blue lines.}
-% todo read above caption! enable [] hkls in short caption
+\caption[Migration barrier and structures of the \si{} {\hkl[0 -1 1]} DB to the {\hkl[1 1 0]} DB transition.]{Migration barrier and structures of the \si{} \hkl[0 -1 1] DB (left) to the \hkl[1 1 0] DB (right) transition. Bonds are illustrated by blue lines.}
\label{fig:defects:si_mig1}
\end{figure}
The barrier, which is even lower than the one for \ci{}, indeed indicates highly mobile \si.
\begin{center}
\includegraphics[width=0.7\textwidth]{si_mig_rest.ps}
\end{center}
-\caption{Migration barrier of the \si{} \hkl[1 1 0] DB into the hexagonal (H) and tetrahedral (T) configuration as well as the hexagonal \si{} to tetrahedral \si{} transition.}
-% todo read above caption! enable [] hkls in short caption
+\caption[Migration barrier of the \si{} {\hkl[1 1 0]} DB into the hexagonal and tetrahedral configuration as well as the hexagonal \si{} to tetrahedral \si{} transition.]{Migration barrier of the \si{} \hkl[1 1 0] DB into the hexagonal (H) and tetrahedral (T) configuration as well as the hexagonal \si{} to tetrahedral \si{} transition.}
\label{fig:defects:si_mig2}
\end{figure}
The obtained activation energies are of the same order of magnitude than values derived from other ab initio studies \cite{bloechl93,sahli05}.
However, even for non-interacting defects, the energy of formation, which is then given by the sum of the formation energies of the separated defects (\unit[4.15]{eV}) is still higher than that of the C$_{\text{i}}$ \hkl<1 0 0> DB.
Unexpectedly, the structure of a Si$_{\text{i}}$ \hkl<1 1 0> DB and a neighbored C$_{\text{s}}$, which is the most favored configuration of a C$_{\text{s}}$ and Si$_{\text{i}}$ DB according to quantum-mechanical calculations, likewise constitutes an energetically favorable configuration within the EA description, which is even preferred over the two least separated configurations of C$_{\text{s}}$ and Si$_{\text{i}}$ T.
This is attributed to an effective reduction in strain enabled by the respective combination.
-Quantum-mechanical results reveal a more favorable energy of fomation for the C$_{\text{s}}$ and Si$_{\text{i}}$ T (a) configuration.
+Quantum-mechanical results reveal a more favorable energy of formation for the C$_{\text{s}}$ and Si$_{\text{i}}$ T (a) configuration.
However, this configuration is unstable involving a structural transition into the C$_{\text{i}}$ \hkl<1 1 0> DB interstitial, thus, not maintaining the tetrahedral Si nor the \cs{} defect.
Thus, the underestimated energy of formation of C$_{\text{s}}$ within the EA calculation does not pose a serious limitation in the present context.
\section{Conclusions concerning the SiC conversion mechanism}
+\ifnum1=0
+
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
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.
+However, focusing 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.
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.
+Depending on orientation, energetically favorable configurations are found, in which these two interstitials are located close together instead of the occurrence 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> 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.
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
+\fi
+
+% todo - sync with conclusion chapter
-These findings allow to draw conclusions on the mechanisms involved in the process of SiC conversion in Si.
+These findings allow to draw conclusions on the mechanisms involved in the process of SiC conversion in Si, which is elaborated in more detail within the comprehensive description in chapter~\ref{chapter:summary}.
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.
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$_{\text{s}}$.
+To conclude, the available results suggest precipitation 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}.
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.
+Until then, however, these regions could either be 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$_{\text{s}}$.
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.
+It is easily understandable that the mismatch in alignment 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.
+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 alignment would be mandatory assuming the model of precipitation by C-Si dumbbell agglomeration.
\fi
projects / lectures / latex.git / blobdiff