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+%\usepackage{miller}\r
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\begin{document}\r
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%\title{Mobility of Carbon in Silicon -- a first principles study}\r
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Migration and recombination pathways have been ivestigated utilizing the constraint conjugate gradient relaxation technique (CRT)\cite{kaukonen98}.\r
The defect formation energy $E-N_{\text{Si}}\mu_{\text{Si}}-N_{\text{C}}\mu_{\text{C}}$ is defined by chosing SiC as a particle reservoir for the C impurity, i.e. the chemical potentials are determined by the cohesive energies of a perfect Si and SiC supercell after ionic relaxation.\r
+The binding energy of a defect pair is given by the difference of the formation energy of the complex and the sum of the two separated defect configurations, i.e. energetically favorable configurations show binding energies below zero and non-interacting isolated defects would result in a binding energy of zero.\r
\r
\section{Results}\r
\r
Next to individual Si$_{\text{i}}$, C$_{\text{i}}$, V and C$_{\text{s}}$ defects, combinations of these defects and their interaction are considered important for the problem under study.\r
In the following the structure and energetics of separated defects are presented.\r
The investigations proceed with pairs of the ground state and, thus, most probable defect configurations that are believed to be fundamental in the Si to SiC transition.\r
+Fig.~\ref{fig:combos} schematically displays the positions for the initial interstitial defect (Si$_{\text{i}}$/C$_{\text{i}}$) and the neighbured defect (1-5) used for investigating defect pairs.\r
+\begin{figure}\r
+\includegraphics[width=0.5\columnwidth]{cs.eps}\r
+\caption{Positions for the initial defect (Si$_{\text{i}}$/C$_{\text{i}}$) and the neighboured defect (1-5) used for investigating defect pairs.} \r
+\label{fig:combos}\r
+\end{figure}\r
\r
\subsection{Separated defects in silicon}\r
% we need both: Si self-int & C int ground state configuration (for combos)\r
\includegraphics[width=\columnwidth]{sivac.eps}\r
\end{minipage}\r
\begin{minipage}[t]{0.32\columnwidth}\r
-\underline{Substitutional}\\\r
+\underline{C$_{\text{s}}$}\\\r
\includegraphics[width=\columnwidth]{csub.eps}\r
\end{minipage}\\\r
\begin{minipage}[t]{0.32\columnwidth}\r
\label{table:sep_eof}\r
\end{table*}\r
Results obtained by the present study compare well with results from literature\cite{leung99,al-mushadani03,dal_pino93,capaz94}.\r
-Regarding intrinsic defects in Si, the $\langle 1 1 0 \rangle$ self-interstitial dumbbell is found to be the ground state configuration tersely followed by the hexagonal and tetrahedral configuration, which is the consensus view for Si$_{\text{i}}$\cite{leung99,al-mushadani03}.\r
-In the case of a C impurity, next to the C$_{\text{s}}$ configuration, in which a C atom occupies an already vacant Si lattice site, the C$_{\text{i}}$ $\langle 1 0 0 \rangle$ interstitial constitutes the energetically most favorable configuration, in which the C and Si dumbbell atoms share a regular Si lattice site.\r
+Regarding intrinsic defects in Si, the $\langle 1 1 0 \rangle$ self-interstitial dumbbell (Si$_{\text{i}}$ $\langle 1 1 0 \rangle$ DB) is found to be the ground state configuration tersely followed by the hexagonal and tetrahedral configuration, which is the consensus view for Si$_{\text{i}}$\cite{leung99,al-mushadani03}.\r
+In the case of a C impurity, next to the C$_{\text{s}}$ configuration, in which a C atom occupies an already vacant Si lattice site, the C$_{\text{i}}$ $\langle 1 0 0 \rangle$ interstitial dumbbell (C$_{\text{i}}$ $\langle 1 0 0 \rangle$ DB) constitutes the energetically most favorable configuration, in which the C and Si dumbbell atoms share a regular Si lattice site.\r
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 as the ground state.\r
However, to our best knowledge, no energy of formation for this type of defect based on first principles calculations has yet been explicitly stated in literature.\r
-Instead, Capaz et al.\cite{capaz94} give a relative ...\r
-... in a previous study\cite{zirkelbach10a}.\r
\r
-The ground state configurations of a Si self-interstitial and a C interstitial is the $\langle 1 1 0 \rangle$ and $\langle 1 0 0 \rangle$ dumbbell respectively.\r
+Instead, Capaz et al.\cite{capaz94}, investigating migration pathways of the C$_{\text{i}}$ $\langle 1 0 0 \rangle$ DB, find this defect to be \unit[2.1]{eV} lower in energy than the bond-centered (BC) configuration, which is claimed to constitute a saddle point configuration in the migration path within the $(1 1 0)$ plane and, thus, interpreted as the barrier of migration for the respective path.\r
+However, the present study indicates a local minimum state for the BC defect if spin polarized calculations are performed resulting in a net magnetization of two electrons localized in a torus around the C atom.\r
+Another DFT calculation without fully accounting for the electron spin results in the smearing of a single electron over two non-degenerate Kohn-Sham states and an increase of the total energy by \unit[0.3]{eV} for the BC configuration.\r
+Regardless of the rather small correction due to the spin, the difference we found is much smaller (\unit[0.9]{eV}), which would nicely compare to experimental findings $(\unit[0.73-0.87]{eV})$\cite{tipping87,song90} for the migration barrier.\r
+However, since the BC configuration constitutes a real local minimum another barrier exists which is about \unit[1.2]{eV} (\unit[0.9+0.3]{eV}) in height.\r
+Indeed Capaz et al. propose another path and find it to be the lowest in energy\cite{capaz94}, in which a C$_{\text{i}}$ $\langle 0 0 -1\rangle$ DB migrates into a C$_{\text{i}}$ $\langle 0 -1 0\rangle$ DB located at the next neighboured Si lattice site in $[1 1 -1]$ direction.\r
+Calculations in this work reinforce this path by an additional improvement of the quantitative conformance of the barrier height (\unit[0.9]{eV}) to experimental values.\r
+A more detailed description can be found in a previous study\cite{zirkelbach10a}.\r
+\r
+Next to the C BC configuration the vacancy and Si$_{\text{i}}$ $\langle 1 0 0\rangle$ DB have to be treated by taking into account the spin of the electrons.\r
+For the latter two the net spin up electron density is located in caps at the four surrounding Si atoms oriented towards the vacant site and in two caps at each of the two DB atoms perpendicular aligned to the bonds to the other two Si atoms respectively.\r
+No other configuration, within the ones that are mentioned, is affected.\r
+\r
+Concerning the mobility of the ground state Si$_{\text{i}}$, an activation energy shortly below \unit[0.7]{eV} was found for the migration of a Si$_{\text{i}}$ $\langle 0 1 -1\rangle$ into a $\langle 1 1 0\rangle$ DB configuration located at the next neighboured Si lattice site in $[1 1 -1]$ direction.\r
\r
-%% Kurze Beschreibung der Migration - auch wie im anderen paper, auch nur DFT. Und: \cite{zirkelbach10} ... to be published (2010). \r
+\subsection{Pairs of C$_{\text{i}}$}\r
\r
-\subsection{C$_I$ next to further C interstitials}\r
+C$_{\text{i}}$ pairs of the $\langle 1 0 0\rangle$-type have been considered in the first part.\r
+Table~\ref{table:dc_c-c} summarizes the binding energies obtained for configurations, in which an initial C$_{\text{i}}$ $\langle 0 0 -1\rangle$ DB located at position Si$_{\text{i}}$/C$_{\text{i}}$ is combined with a defect of the same type occupying various orientations at positions 1 to 5 (see Fig.~ref{fig:combos}).\r
+\begin{table}\r
+\begin{ruledtabular}\r
+\begin{tabular}{l c c c c c c }\r
+ & 1 & 2 & 3 & 4 & 5 & R \\\r
+\hline\r
+ $\langle 0 0 -1\rangle$ & -0.08 & -1.15 & -0.08 & 0.04 & -1.66 & -0.19\\\r
+ $\langle 0 0 1\rangle$ & 0.34 & 0.004 & -2.05 & 0.26 & -1.53 & -0.19\\\r
+ $\langle 0 -1 0\rangle$ & -2.39 & -0.17 & -0.10 & -0.27 & -1.88 & -0.05\\\r
+ $\langle 0 1 0\rangle$ & -2.25 & -1.90 & -2.25 & -0.12 & -1.38 & -0.06\\\r
+ $\langle -1 0 0\rangle$ & -2.39 & -0.36 & -2.25 & -0.12 & -1.88 & -0.05\\\r
+ $\langle 1 0 0\rangle$ & -2.25 & -2.16 & -0.10 & -0.27 & -1.38 & -0.06\\\r
+\end{tabular}\r
+\end{ruledtabular}\r
+\caption{Binding energies of C$_{\text{i}}$ $\langle 1 0 0\rangle$-type defect pairs in eV. Equivalent configurations exhibit equal energies. The first column lists the orientation of the defect, which is combined with the initial C$_{\text{i}}$ $\langle 0 0 -1\rangle$ dumbbell. The position index of the second defect is given in the first row according to Fig.~\ref{fig:combos}. R corresponds to the position located at $\frac{a_{\text{Si}}}{2} \langle3 2 3 \rangle$ relative to the initial defect position, which is the maximum realizable distance due to periodic boundary conditions.}\r
+\label{table:dc_c-c}\r
+\end{table}\r
+Most of the obtained configurations result in binding energies well below zero indicating a preferable agglomeration of these type of defects.\r
+For increasing distances of the defect pair the binding energy approaches to zero (R in Table~\ref{table:dc_c-c}) as it is expected for non-interacting isolated defects.\r
+Energetically favorable and unfavorable configurations can be explained by stress compensation and increase respectively, which is due to the resulting net strain of the respective configuration of the defect combination.\r
+Antiparallel orientations of the second defect ($\langle 0 0 1\rangle$) at positions located below the (001) plane with respect to the initial one (positions 1, 2 and 4) show the energetically most unfavorable configurations.\r
+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.\r
+\r
+In the energetically most favorable configuration, in which differently oriented next neighboured DBs with the two C atoms facing each other, a strong C-C bond has formed.\r
+Migration C-C ...\r
+% strange mig from -190 -> -2.39 (barrier > 4 eV)\r
+% C-C migration -> idea:\r
+% mig from low energy confs has extremely high barrier!\r
+% low barrier only from energetically less/unfavorable confs (?)! <- prove!\r
+% => low probability of C-C clustering ?!?\r
+\r
+Energetically most favorable orientations along $[1 1 0]$ direction ...\r
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\subsection{C$_I$ next to C$_{\text{s}}$}\r
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