% ------ Albe potential ---------
For the classical potential calculations, a supercell of 9 Si lattice constants in each direction consisting of 5832 Si atoms has been used.
-A Tersoff-like bond order potential by Erhart and Albe (EA)\cite{albe_sic_pot} has been utilized, which accounts for nearest neighbour interactions only realized by a cut-off function dropping the interaction to zero in between the first and second next neighbour distance.
+A Tersoff-like bond order potential by Erhart and Albe (EA)\cite{albe_sic_pot} has been utilized, which accounts for nearest neighbor interactions only realized by a cut-off function dropping the interaction to zero in between the first and second next neighbor distance.
Constant pressure simulations are realized by the Berendsen barostat\cite{berendsen84}.
Structural relaxation in the MD run is achieved by the verlocity verlet algorithm\cite{verlet67} and the Berendsen thermostat\cite{berendsen84} with a time constant of \unit[1]{fs} resulting in direct velocity scaling and the temperature set to zero Kelvin.
This finding is in agreement with several theoretical\cite{burnard93,leary97,dal_pino93,capaz94} and experimental\cite{watkins76,song90} investigations.
Tersoff as well, considers C$_{\text{i}}$ to be the ground state configuration and believes an artifact due to the abrupt C-Si cut-off used in the potential to be responsible for the small value of the tetrahedral formation energy\cite{tersoff90}.
It should be noted that EA and DFT predict almost equal formation energies.
-However, there is a qualitative difference: while the C-Si distance of the dumbbell atoms is almost equal for both methods, the position along $\langle0 0 1\rangle$ of the dumbbell inside the tetrahedron spanned by the four next neighboured Si atoms differs significantly.
+However, there is a qualitative difference: while the C-Si distance of the dumbbell atoms is almost equal for both methods, the position along $\langle0 0 1\rangle$ of the dumbbell inside the tetrahedron spanned by the four next neighbored Si atoms differs significantly.
The dumbbell based on the EA potential is almost centered around the regular Si lattice site as can be seen in Fig.~\ref{fig:defects} whereas for DFT calculations it is translated upwards with the C atom forming an almost collinear bond to the two Si atoms of the top face of the tetrahedron and the bond angle of the Si dumbbell atom to the two bottom face Si atoms approaching \unit[120]{$^\circ$}.
% maybe transfer to discussion chapter later
This indicates predominant sp and sp$^2$ hybridization for the C and Si dumbbell atom respectively.
A measure for the mobility of the interstitial carbon is the activation energy for the migration path from one stable position to another.
The stable defect geometries have been discussed in the previous subsection.
-In the following the migration of the most stable configuration, i.e. C$_{\text{i}}$, from one site of the Si host lattice to a neighbored site has been investigated by both, EA and DFT calculations utilizing the constraint conjugate gradient relaxation technique (CRT)\cite{kaukonen98}.
+In the following the migration of the most stable configuration, i.e. C$_{\text{i}}$, from one site of the Si host lattice to a neighboring site has been investigated by both, EA and DFT calculations utilizing the constraint conjugate gradient relaxation technique (CRT)\cite{kaukonen98}.
Three migration pathways are investigated.
-The starting configuration for all pathways was the $\langle0 0 -1\rangle$ dumbbell interstitial configuration.
-In path~1 and 2 the final configuration is a $\langle0 0 1\rangle$ and $\langle0 -1 0\rangle$ dumbbell interstitial respectively, located at the next neighboured Si lattice site displaced by $\frac{a_{\text{Si}}}{4}[1 1 -1]$, where $a_{\text{Si}}$ is the Si lattice constant.
+The starting configuration for all pathways was the $[0 0 -1]$ dumbbell interstitial configuration.
+In path~1 and 2 the final configuration is a $[0 0 1]$ and $[0 -1 0]$ dumbbell interstitial respectively, located at the next neighbored Si lattice site displaced by $\frac{a_{\text{Si}}}{4}[1 1 -1]$, where $a_{\text{Si}}$ is the Si lattice constant.
In path~1 the C atom resides in the $(1 1 0)$ plane crossing the BC configuration whereas in path~2 the C atom moves out of the $(1 1 0)$ plane.
-Path 3 ends in a $\langle0 -1 0\rangle$ configuration at the initial lattice site and, for this reason, corresponds to a reorientation of the dumbbell, a process not contributing to long range diffusion.
+Path 3 ends in a $[0 -1 0]$ configuration at the initial lattice site and, for this reason, corresponds to a reorientation of the dumbbell, a process not contributing to long range diffusion.
\begin{figure}
\begin{center}
%\includegraphics[width=\columnwidth]{path2_vasp.ps}
\includegraphics[width=\columnwidth]{path2_vasp_s.ps}
\end{center}
-\caption{Migration barrier and structures of the $\langle0 0 -1\rangle$ dumbbell (left) to the $\langle0 -1 0\rangle$ dumbbell (right) transition as obtained by first principles methods. The activation energy of \unit[0.9]{eV} agrees well with experimental findings of \unit[0.73]{eV}\cite{song90} and \unit[0.87]{eV}\cite{tipping87}.}
+\caption{Migration barrier and structures of the $\langle0 0 -1\rangle$ dumbbell (left) to the $\langle0 -1 0\rangle$ dumbbell (right) transition as obtained by first principles methods. The activation energy of \unit[0.9]{eV} agrees well with experimental findings of \unit[0.70]{eV}\cite{lindner06}, \unit[0.73]{eV}\cite{song90} and \unit[0.87]{eV}\cite{tipping87}.}
\label{fig:vasp_mig}
\end{figure}
-The lowest energy path (path~2) as detected by the first principles approach is illustrated in Fig.~\ref{fig:vasp_mig}, in which the $\langle0 0 -1\rangle$ dumbbell migrates towards the next neighboured Si atom escaping the $(1 1 0)$ plane forming a $\langle0 -1 0\rangle$ dumbbell.
-The activation energy of \unit[0.9]{eV} excellently agrees with experimental findings ranging from \unit[0.73]{eV}\cite{song90} to \unit[0.87]{eV}\cite{tipping87}.
+The lowest energy path (path~2) as detected by the first principles approach is illustrated in Fig.~\ref{fig:vasp_mig}, in which the $\langle0 0 -1\rangle$ dumbbell migrates towards the next neighbored Si atom escaping the $(1 1 0)$ plane forming a $\langle0 -1 0\rangle$ dumbbell.
+The activation energy of \unit[0.9]{eV} excellently agrees with experimental findings ranging from \unit[0.70]{eV} to \unit[0.87]{eV}\cite{lindner06,song90,tipping87}.
\begin{figure}
\begin{center}
The first principles results are in good agreement to previous work on this subject\cite{burnard93,leary97,dal_pino93,capaz94}.
The C-Si $\langle1 0 0\rangle$ dumbbell interstitial is found to be the ground state configuration of a C defect in Si.
-The lowest migration path already proposed by Capaz et~al.\cite{capaz94} is reinforced by an additional improvement of the quantitative conformance of the barrier height calculated in this work (\unit[0.9]{eV}) with experimentally observed values (\unit[0.73]{eV} -- \unit[0.87]{eV})\cite{song90,tipping87}.
+The lowest migration path already proposed by Capaz et~al.\cite{capaz94} is reinforced by an additional improvement of the quantitative conformance of the barrier height calculated in this work (\unit[0.9]{eV}) with experimentally observed values (\unit[0.70]{eV} -- \unit[0.87]{eV})\cite{lindner06,song90,tipping87}.
However, it turns out that the bond-centered configuration is not a saddle point configuration as proposed by Capaz et~al.\cite{capaz94} but constitutes a real local minimum if the electron spin is properly accounted for.
-A net magnetization of two electrons, which is already clear by simple molecular orbital theory considerations on the bonding of the sp hybridized C atom, adjusts.
+%A net magnetization of two electrons, which is already clear by simple molecular orbital theory considerations on the bonding of the sp hybridized C atom, adjusts.
+A net magnetization of two electrons, which is already clear by simple molecular orbital theory considerations on the bonding of the sp hybridized C atom, is settled.
+% J mod end
By investigating the charge density isosurface it turns out that the two resulting spin up electrons are localized in a torus around the C atom.
With an activation energy of \unit[0.9]{eV} the C$_{\text{i}}$ carbon interstitial can be expected to be highly mobile at prevailing temperatures in the process under investigation, i.e. IBS.
% KN mod end
Already the geometry of the most stable dumbbell configuration differs considerably from that obtained by first principles calculations.
%Obviously the classical approach is unable to reproduce the correct character of bonding due to a too short treatment of quantum-mechanical effects in the potential.
-The classical approach is unable to reproduce the correct character of bonding due to no treatment of quantum-mechanical effects in the potential.
+%The classical approach is unable to reproduce the correct character of bonding due to no treatment of quantum-mechanical effects in the potential.
+The classical approach is unable to reproduce the correct character of bonding due to the deficiency of quantum-mechanical effects in the potential.
+% J mod end
% KN mod end
-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.
+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 is changing 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 compared to the more accurate quantum-mechanical methods and experimental findings.
This is attributed to the sharp cut-off of the short range potential.
-%As already pointed out in a previous study\cite{mattoni2007} the short cut-off is responsible for overestimated and unphysical high forces of next neighboured atoms.
-As already pointed out in a previous study\cite{mattoni2007} the short cut-off is responsible for overestimated and unphysical high forces of next neighbour atoms.
+%As already pointed out in a previous study\cite{mattoni2007} the short cut-off is responsible for overestimated and unphysical high forces of next neighbored atoms.
+As already pointed out in a previous study\cite{mattoni2007} the short cut-off is responsible for overestimated and unphysical high forces of next neighbor atoms.
% KN mod end
The overestimated migration barrier, however, affects the diffusion behavior of the C interstitials.
By this artifact the mobility of the C atoms is tremendously decreased resulting in an inaccurate description or even absence of the dumbbell agglomeration as proposed by the precipitation model.
Classical potentials, however, fail to describe the selected processes.
This has been shown to have two reasons, i.e. the overestimated barrier of migration due to the artificial interaction cut-off on the one hand, and on the other hand the lack of quantum-mechanical effects which are crucial in the problem under study.
%Further investigations, i.e. the structure and energetics of defect combinations, still small enough to be treated by DFT have been accomplished in order to draw conclusions regarding the precipitation mechanism, will be published elsewhere.
-In order to get more insight on the SiC precipitation mechanism, further ab initio calculations are currently ivestigated.
+In order to get more insight on the SiC precipitation mechanism, further ab initio calculations are currently investigated.
% ----------------------------------------------------
\section*{Acknowledgment}
M.~Kaukonen, P.~K.~Sitch, G.~Jungnickel, R.~M.~Nieminen, S.~P{\"o}ykk{\"o}, D.~Porezag, and Th.~Frauenheim,
\newblock Phys. Rev. B {\bf 57}, 9965 (1998).
+\bibitem{lindner06}
+J.~K.~N.~Lindner, M.~H{\"a}berlen, G.~Thorwarth, and B.~Stritzker,
+\newblock Materials Science and Engineering: C {\bf 26}, 857 (2006).
+
\bibitem{tipping87}
A.~K. Tipping and R.~C. Newman,
\newblock Semiconductor Science and Technology {\bf 2}, 315 (1987).
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