Interactions of various combinations of defects have been characterized including a couple of selected migration pathways within these configurations.\r
Almost all of the investigated pairs of defects tend to agglomerate allowing for a reduction in strain.\r
The formation of structures involving strong carbon-carbon bonds was found to occur very unlikely.\r
-In contrast substitutional carbon was found to occur in all probability.\r
+In contrast, substitutional carbon was found to occur in all probability.\r
A long range capture radius has been found for pairs of interstitial carbon as well as interstitial carbon and vacancies.\r
A rather small capture radius has been identified for substitutional carbon and silicon self-interstitials.\r
Based on these results conclusions regarding the precipitation mechanism of silicon carbide in bulk silicon are derived and its conformability to experimental findings is discussed.\r
The highly mobile C interstitials agglomerate into large clusters followed by the formation of incoherent 3C-SiC nanocrystallites once a critical size of the cluster is reached.\r
In contrast, investigations of the precipitation in strained Si$_{1-y}$C$_y$/Si heterostructures formed by molecular beam epitaxy (MBE)\cite{strane94,guedj98} suggest an initial coherent precipitation by an agglomeration of substitutional instead of interstitial C followed by a loss of coherency once the increasing strain energy surpasses the interfacial energy of the incoherent 3C-SiC precipitate and c-Si.\r
These two different mechanisms of precipitation might be attributed to the respective method of fabrication, i.e. whether it occurs inside the Si bulk or on a Si surface.\r
-However, in another IBS study Nejim et al. propose a topotactic transformation remaining structure and orientation that is likewise based on the formation of substitutional C and a concurrent reaction of the excess Si self-interstitials with further implanted C atoms in the initial state\cite{nejim95}.\r
+However, in another IBS study Nejim et al. propose a topotactic transformation remaining structure and orientation that is likewise based on the formation of substitutional C and a concurrent reaction of the excess Si self-interstitials with further implanted C atoms\cite{nejim95}.\r
Solving this controversy and understanding the effective underlying processes will enable significant technological progress in 3C-SiC thin film formation driving the superior polytype for potential applications in high-performance electronic device production\cite{wesch96}.\r
\r
Atomistic simulations offer a powerful tool of investigation providing detailed insight not accessible by experiment.\r
\section{Methodology}\r
\r
The first principles DFT calculations were performed with the plane-wave based Vienna Ab-initio Simulation Package (VASP)\cite{kresse96}.\r
-The Kohn-Sham equations were solved using the generalized-gradient XC-functional approximation proposed by Perdew and Wang (GGA-PW91)\cite{perdew86,perdew92}.\r
+The Kohn-Sham equations were solved using the generalized-gradient exchange-correlation (XC) functional approximation proposed by Perdew and Wang\cite{perdew86,perdew92}.\r
The electron-ion interaction is described by norm-conserving ultra-soft pseudopotentials\cite{hamann79} as implemented in VASP\cite{vanderbilt90}.\r
Throughout this work an energy cut-off of \unit[300]{eV} was used to expand the wave functions into the plane-wave basis.\r
Sampling of the Brillouin zone was restricted to the $\Gamma$-point.\r
-The defect structures and the migration paths were modelled in cubic supercells containing $216\pm2$ Si atoms.\r
+The defect structures and the migration paths were modelled in cubic supercells with a side length of \unit[1.6]{nm} containing $216$ Si atoms.\r
The ions and cell shape were allowed to change in order to realize a constant pressure simulation.\r
Ionic relaxation was realized by the conjugate gradient algorithm.\r
Spin polarization has been fully accounted for.\r
\underline{C$_{\text{i}}$ bond-centered}\\\r
\includegraphics[width=\columnwidth]{cbc.eps}\r
\end{minipage}\r
-\caption{Configurations of silicon and carbon point defects in silicon. Silicon and carbon atoms are illustrated by yellow and grey spheres respectively. Blue lines are bonds drawn whenever considered appropriate to ease identifying defect structures for the reader. Dumbbell configurations are abbreviated by DB.}\r
+\caption{Configurations of silicon and carbon point defects in silicon. Silicon and carbon atoms are illustrated by yellow and gray spheres respectively. Bonds are drawn whenever considered appropriate to ease identifying defect structures for the reader. Dumbbell configurations are abbreviated by DB.}\r
\label{fig:sep_def}\r
\end{figure}\r
\begin{table*}\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 of \unit[0.3]{eV} due to the spin, the difference we found is much smaller (\unit[0.9]{eV}), which would nicely compare to experimental findings $(\unit[0.70-0.87]{eV})$\cite{lindner06,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} 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$_{\text{i}}$ \hkl[0 0 -1] DB migrates into a C$_{\text{i}}$ \hkl[0 -1 0] DB located at the next neighbored Si lattice site in \hkl[1 1 -1] direction.\r
+Indeed Capaz et al. propose another path and find it to be the lowest in energy\cite{capaz94}, in which a C$_{\text{i}}$ \hkl[0 0 -1] DB migrates to a C$_{\text{i}}$ \hkl[0 -1 0] DB located at the next neighbored Si lattice site in \hkl[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$_{\text{i}}$ \hkl<1 0 0> 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 localized in caps at the four surrounding Si atoms directed towards the vacant site and in two caps at each of the two DB atoms perpendicularly aligned to the bonds to the other two Si atoms respectively.\r
+For the vacancy the net spin up electron density is localized in caps at the four surrounding Si atoms directed towards the vacant site.\r
+In the Si$_{\text{i}}$ \hkl<1 0 0> DB configuration the net spin up density is localized in two caps at each of the two DB atoms perpendicularly 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 of \unit[0.67]{eV} was found for the Si$_{\text{i}}$ \hkl[0 1 -1] to \hkl[1 1 0] DB configuration located at the next neighbored Si lattice site in \hkl[1 1 -1] direction.\r
+Concerning the mobility of the ground state Si$_{\text{i}}$, an activation energy of \unit[0.67]{eV} was found for the transition of the Si$_{\text{i}}$ \hkl[0 1 -1] to the \hkl[1 1 0] DB located at the next neighbored Si lattice site in \hkl[1 1 -1] direction.\r
Further investigations revealed a barrier of \unit[0.94]{eV} for the Si$_{\text{i}}$ \hkl[1 1 0] DB to Si$_{\text{i}}$ H, \unit[0.53]{eV} for the Si$_{\text{i}}$ \hkl[1 1 0] DB to Si$_{\text{i}}$ T and \unit[0.35]{eV} for the Si$_{\text{i}}$ H to Si$_{\text{i}}$ T transition.\r
The obtained values are within the same order of magnitude than values derived from other ab initio studies\cite{bloechl93,sahli05}.\r
-% look for values in literature for neutraly charged Si_i diffusion\r
-% T seems to constitute a saddle point according to migration calculations\r
\r
\subsection{Pairs of C$_{\text{i}}$}\r
\r
\hkl[1 0 0] & -2.25 & -2.16 & -0.10 & -0.27 & -1.38 & -0.06\\\r
\end{tabular}\r
\end{ruledtabular}\r
-\caption{Binding energies of C$_{\text{i}}$ \hkl<1 0 0>-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$_{\text{i}}$ \hkl[0 0 -1] 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 $\frac{a_{\text{Si}}}{2}\hkl[3 2 3]$ relative to the initial defect position, which is the maximum realizable distance due to periodic boundary conditions.}\r
+\caption{Binding energies of C$_{\text{i}}$ \hkl<1 0 0>-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$_{\text{i}}$ \hkl[0 0 -1] 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 $\frac{a_{\text{Si}}}{2}\hkl[3 2 3]$ relative to the initial defect position, which is the maximum realizable distance ($\approx \unit[1.3]{nm}$) 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
\r
In summary, C and Si point defects in Si, combinations of these defects and diffusion processes within such configurations have been investigated.\r
It is shown that C interstitials in Si tend to agglomerate, which is mainly driven by a reduction of strain.\r
-Investigations of migration pathways, however, allow to conclude that C clustering fails to appear by thermally activated processes due to high activation energies of the the respective diffusion processes.\r
+Investigations of migration pathways, however, allow to conclude that C clustering fails to appear by thermally activated processes due to high activation energies of the respective diffusion processes.\r
A highly attractive interaction and a large capture radius has been identified for the C$_{\text{i}}$ \hkl<1 0 0> DB and the vacancy indicating a high probability for the formation of C$_{\text{s}}$.\r
In contrast, a rapidly decreasing interaction with respect to the separation distance has been identified for C$_{\text{s}}$ and a Si$_{\text{i}}$ \hkl<1 1 0> DB resulting in a low probability of defects exhibiting respective separations to transform into the C$_{\text{i}}$ \hkl<1 0 0> DB, which constitutes the ground state configuration for a C atom introduced into otherwise perfect Si. \r
Based on these findings conclusions on basic processes involved in the SiC precipitation in bulk Si are drawn.\r
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