\begin{document}
-% ref mod: no capital letters (by editors)
\title{Combined ab initio and classical potential simulation study on the silicon carbide precipitation in silicon}
\author{F. Zirkelbach}
\author{B. Stritzker}
% --------------------------------------------------------------------------------
\section{Introduction}
-% TOOD: redo complete intro!
-
The wide band gap semiconductor silicon carbide (SiC) is well known for its outstanding physical and chemical properties.
The high breakdown field, saturated electron drift velocity and thermal conductivity in conjunction with the unique thermal and mechanical stability as well as radiation hardness makes SiC a suitable material for high-temperature, high-frequency and high-power devices operational in harsh and radiation-hard environments\cite{edgar92,morkoc94,wesch96,capano97,park98}.
Different modifications of SiC exist, which solely differ in the one-dimensional stacking sequence of identical, close-packed SiC bilayers\cite{fischer90}.
However, only little is known about the SiC conversion in C implanted Si.
\begin{figure}
\begin{center}
-\subfigure[]{\label{fig:hrem:c-si}\includegraphics[width=0.48\columnwidth]{../img/tem_c-si-db.eps}}
-\subfigure[]{\label{fig:hrem:sic}\includegraphics[width=0.48\columnwidth]{../img/tem_3c-sic.eps}}
+\subfigure[]{\label{fig:hrem:c-si}\includegraphics[width=0.48\columnwidth]{tem_c-si-db.eps}}
+\subfigure[]{\label{fig:hrem:sic}\includegraphics[width=0.48\columnwidth]{tem_3c-sic.eps}}
\end{center}
\caption{High resolution transmission electron microscopy (HREM) micrographs\cite{lindner99_2} of agglomerates of C-Si dimers showing dark contrasts and otherwise undisturbed Si lattice fringes (a) and equally sized Moir\'e patterns indicating 3C-SiC precipitates (b).}
\label{fig:hrem}
These two different mechanisms of precipitation might be attributed to the respective method of fabrication.
While in CVD and MBE surface effects need to be taken into account, SiC formation during IBS takes place in the bulk of the Si crystal.
However, in another IBS study Nejim et~al.\cite{nejim95} propose a topotactic transformation that is likewise based on the formation of substitutional C.
-%The formation of substitutional C, however, is accompanied by Si self-interstitial atoms that previously occupied the lattice sites and concurrently by a reduction of volume due to the lower lattice constant of SiC compared to Si.
The formation of substitutional C, however, is accompanied by Si self-interstitial atoms that previously occupied the lattice sites and a concurrent reduction of volume due to the lower lattice constant of SiC compared to Si.
Both processes are believed to compensate one another.
All these potentials are short range potentials employing a cut-off function, which drops the atomic interaction to zero in between the first and second nearest neighbor distance.
In a combined ab initio and empirical potential study it was shown that the Tersoff potential properly describes binding energies of combinations of C defects in Si\cite{mattoni2002}.
However, investigations of brittleness in covalent materials\cite{mattoni2007} identified the short range character of these potentials to be responsible for overestimated forces necessary to snap the bond of two neighbored atoms.
-%In a previous study\cite{zirkelbach10a} we approved explicitly the influence on the migration barrier for C diffusion in Si.
-In a previous study\cite{zirkelbach10a} we determined the influence on the migration barrier for C diffusion in Si.
+In a previous study\cite{zirkelbach10} we determined the influence on the migration barrier for C diffusion in Si.
Using the Erhart/Albe (EA) potential\cite{albe_sic_pot}, an overestimated barrier height compared to ab initio calculations and experiment is obtained.
A proper description of C diffusion, however, is crucial for the problem under study.
The C interstitial defect with the lowest energy of formation has been found to be the C-Si \hkl<1 0 0> interstitial dumbbell (C$_{\text{i}}$ \hkl<1 0 0> DB), which, thus, constitutes the ground state of an additional C impurity in otherwise perfect c-Si.
This finding is in agreement with several theoretical\cite{burnard93,leary97,dal_pino93,capaz94} and experimental\cite{watkins76,song90} investigations.
Astonishingly EA and DFT predict almost equal formation energies.
-There are, however, geometric differences with regard to the DB position within the tetrahedron spanned by the four neighbored Si atoms, as already reported in a previous study\cite{zirkelbach10a}.
+There are, however, geometric differences with regard to the DB position within the tetrahedron spanned by the four neighbored Si atoms, as already reported in a previous study\cite{zirkelbach10}.
Since the energetic description is considered more important than the structural description, minor discrepancies of the latter are assumed non-problematic.
The second most favorable configuration is the C$_{\text{i}}$ \hkl<1 1 0> DB followed by the C$_{\text{i}}$ bond-centered (BC) configuration.
For both configurations EA overestimates the energy of formation by approximately \unit[1]{eV} compared to DFT.
Thus, nearly the same difference in energy has been observed for these configurations in both methods.
However, we have found the BC configuration to constitute a saddle point within the EA description relaxing into the \hkl<1 1 0> configuration.
Due to the high formation energy of the BC defect resulting in a low probability of occurrence of this defect, the wrong description is not posing a serious limitation of the EA potential.
-A more detailed discussion of C defects in Si modeled by EA and DFT including further defect configurations can be found in our recently published article\cite{zirkelbach10a}.
+A more detailed discussion of C defects in Si modeled by EA and DFT including further defect configurations can be found in our recently published article\cite{zirkelbach10}.
Regarding intrinsic defects in Si, both methods predict energies of formation that are within the same order of magnitude.
However discrepancies exist.
Indeed, an increase of the cut-off results in increased values of the formation energies\cite{albe_sic_pot}, which is most significant for the tetrahedral configuration.
The same issue has already been discussed by Tersoff\cite{tersoff90} with regard to the description of the tetrahedral C defect using his potential.
While not completely rendering impossible further, more challenging empirical potential studies on large systems, the artifact has to be taken into account in the following investigations of defect combinations.
-%This artifact does not necessarily render impossible further challenging empirical potential studies on large systems.
-%However, it has to be taken into account in the following investigations of defect combinations.
\subsection{Formation energies of C$_{\text{i}}$ and C$_{\text{s}}$-Si$_{\text{i}}$}
-As has been shown in a previous study\cite{zirkelbach10b}, the energetically most favorable configuration of C$_{\text{s}}$ and Si$_{\text{i}}$ is obtained for C$_{\text{s}}$ located at the neighbored lattice site along the \hkl<1 1 0> bond chain of a Si$_{\text{i}}$ \hkl<1 1 0> DB.
+As has been shown in a previous study\cite{zirkelbach11a}, the energetically most favorable configuration of C$_{\text{s}}$ and Si$_{\text{i}}$ is obtained for C$_{\text{s}}$ located at the neighbored lattice site along the \hkl<1 1 0> bond chain of a Si$_{\text{i}}$ \hkl<1 1 0> DB.
However, the energy of formation is slightly higher than that of the C$_{\text{i}}$ \hkl<1 0 0> DB, which constitutes the ground state for a C impurity introduced into otherwise perfect c-Si.
For a possible clarification of the controversial views on the participation of C$_{\text{s}}$ in the precipitation mechanism by classical potential simulations, test calculations need to ensure the proper description of the relative formation energies of combined structures of C$_{\text{s}}$ and Si$_{\text{i}}$ compared to C$_{\text{i}}$.
Combined structures of C$_{\text{s}}$ and Si$_{\text{i}}$ T are energetically less favorable than the ground state C$_{\text{i}}$ \hkl<1 0 0> DB configuration.
With increasing separation distance the energies of formation decrease.
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 C$_{\text{s}}$ and Si$_{\text{i}}$ according to quantum-mechanical calculations\cite{zirkelbach10b}, 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.
+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 C$_{\text{s}}$ and Si$_{\text{i}}$ according to quantum-mechanical calculations\cite{zirkelbach11a}, 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.
However, this configuration is unstable involving a structural transition into the C$_{\text{i}}$ \hkl<1 1 0> interstitial, thus, not maintaining the tetrahedral Si nor the substitutional C defect.
\label{subsection:cmob}
To accurately model the SiC precipitation, which involves the agglomeration of C, a proper description of the migration process of the C impurity is required.
-As shown in a previous study\cite{zirkelbach10a}, quantum-mechanical results properly describe the C$_{\text{i}}$ \hkl<1 0 0> DB diffusion resulting in a migration barrier height of \unit[0.90]{eV}, excellently matching experimental values of \unit[0.70-0.87]{eV}\cite{lindner06,tipping87,song90} and, for this reason, reinforcing the respective migration path as already proposed by Capaz et~al.\cite{capaz94}.
+As shown in a previous study\cite{zirkelbach10}, quantum-mechanical results properly describe the C$_{\text{i}}$ \hkl<1 0 0> DB diffusion resulting in a migration barrier height of \unit[0.90]{eV}, excellently matching experimental values of \unit[0.70-0.87]{eV}\cite{lindner06,tipping87,song90} and, for this reason, reinforcing the respective migration path as already proposed by Capaz et~al.\cite{capaz94}.
During transition a C$_{\text{i}}$ \hkl[0 0 -1] DB migrates towards a C$_{\text{i}}$ \hkl[0 -1 0] DB located at the neighbored lattice site in \hkl[1 1 -1] direction.
However, it turned out that the description fails if the EA potential is used, which overestimates the migration barrier (\unit[2.2]{eV}) by a factor of 2.4.
In addition a different diffusion path is found to exhibit the lowest migration barrier.
\end{figure}
Approximately \unit[2.24]{eV} are needed to turn the C$_{\text{i}}$ \hkl[0 0 -1] DB into the C$_{\text{i}}$ \hkl[1 1 0] DB located at the neighbored lattice site in \hkl[1 1 -1] direction.
Another barrier of \unit[0.90]{eV} exists for the rotation into the C$_{\text{i}}$ \hkl[0 -1 0] DB configuration for the path obtained with a time constant of \unit[100]{fs} for the Berendsen thermostat.
-Roughly the same amount would be necessary to excite the C$_{\text{i}}$ \hkl[1 1 0] DB to the BC configuration (\unit[0.40]{eV}) and a successive migration into the \hkl[0 0 1] DB configuration (\unit[0.50]{eV}) as displayed in our previous study\cite{zirkelbach10a}.
+Roughly the same amount would be necessary to excite the C$_{\text{i}}$ \hkl[1 1 0] DB to the BC configuration (\unit[0.40]{eV}) and a successive migration into the \hkl[0 0 1] DB configuration (\unit[0.50]{eV}) as displayed in our previous study\cite{zirkelbach10}.
The former diffusion process, however, would more nicely agree with the ab initio path, since the migration is accompanied by a rotation of the DB orientation.
By considering a two step process and assuming equal preexponential factors for both diffusion steps, the probability of the total diffusion event is given by $\exp(\frac{\unit[2.24]{eV}+\unit[0.90]{eV}}{k_{\text{B}}T})$, which corresponds to a single diffusion barrier that is 3.5 times higher than the barrier obtained by ab initio calculations.
\begin{figure}
\begin{center}
\subfigure[]{\label{fig:450:a}
-\includegraphics[width=\columnwidth]{../img/sic_prec_450_si-si_c-c.ps}
+\includegraphics[width=\columnwidth]{sic_prec_450_si-si_c-c.ps}
}
\subfigure[]{\label{fig:450:b}
-\includegraphics[width=\columnwidth]{../img/sic_prec_450_si-c.ps}
+\includegraphics[width=\columnwidth]{sic_prec_450_si-c.ps}
}
\end{center}
\caption{Radial distribution function for C-C and Si-Si (Fig.~\ref{fig:450:a}) as well as Si-C (Fig.~\ref{fig:450:b}) pairs for C inserted at \unit[450]{$^{\circ}$C}. In the latter case the resulting C-Si distances for a C$_{\text{i}}$ \hkl<1 0 0> DB are given additionally and the Si-C cut-off distance is marked by an arrow. Insets in Fig.~\ref{fig:450:a} show magnified regions of the respective distribution functions.}
\begin{figure}
\begin{center}
\subfigure[]{\label{fig:tot:si-c}
-\includegraphics[width=\columnwidth]{../img/tot_pc_thesis.ps}
+\includegraphics[width=\columnwidth]{tot_pc_thesis.ps}
}
\subfigure[]{\label{fig:tot:si-si}
-\includegraphics[width=\columnwidth]{../img/tot_pc3_thesis.ps}
+\includegraphics[width=\columnwidth]{tot_pc3_thesis.ps}
}
\subfigure[]{\label{fig:tot:c-c}
-\includegraphics[width=\columnwidth]{../img/tot_pc2_thesis.ps}
+\includegraphics[width=\columnwidth]{tot_pc2_thesis.ps}
}
\end{center}
\caption{Radial distribution function for Si-C (Fig.~\ref{fig:tot:si-c}), Si-Si (Fig.~\ref{fig:tot:si-si}) and C-C (Fig.~\ref{fig:tot:c-c}) pairs for the C insertion into $V_1$ at elevated temperatures. For the Si-C distribution resulting Si-C distances of a C$_{\text{s}}$ configuration are plotted. In the C-C distribution dashed arrows mark C-C distances occurring from C$_{\text{i}}$ \hkl<1 0 0> DB combinations, solid arrows mark C-C distances of pure C$_{\text{s}}$ combinations and the dashed line marks C-C distances of a C$_{\text{i}}$ and C$_{\text{s}}$ combination.}
\begin{figure}
\begin{center}
\subfigure[]{\label{fig:v2:si-c}
-\includegraphics[width=\columnwidth]{../img/12_pc_thesis.ps}
+\includegraphics[width=\columnwidth]{12_pc_thesis.ps}
}
\subfigure[]{\label{fig:v2:c-c}
-\includegraphics[width=\columnwidth]{../img/12_pc_c_thesis.ps}
+\includegraphics[width=\columnwidth]{12_pc_c_thesis.ps}
}
\end{center}
\caption{Radial distribution function for Si-C (Fig.~\ref{fig:v2:si-c}) and C-C (Fig.~\ref{fig:v2:c-c}) pairs for the C insertion into $V_2$ at elevated temperatures. Arrows mark the respective cut-off distances.}
\end{figure}
\begin{figure}
\begin{center}
-\includegraphics[width=\columnwidth]{../img/2050.eps}
+\includegraphics[width=\columnwidth]{2050.eps}
\end{center}
\caption{Cross section along the \hkl(1 -1 0) plane of the atomic structure of the high concentration simulation for a C insertion temperature of \unit[2050]{$^{\circ}$C}.}
\label{fig:v2as}
\section{Discussion and Summary}
Investigations are targeted at the initially stated controversy of SiC precipitation, i.e. whether precipitation occurs abruptly after enough C$_{\text{i}}$ agglomerated or after a successive agglomeration of C$_{\text{s}}$ on usual Si lattice sites (and Si$_{\text{i}}$) followed by a contraction into incoherent SiC.
-Results of a previous ab initio study on defects and defect combinations in C implanted Si\cite{zirkelbach10b} suggest C$_{\text{s}}$ to play a decisive role in the precipitation of SiC in Si.
+Results of a previous ab initio study on defects and defect combinations in C implanted Si\cite{zirkelbach11a} suggest C$_{\text{s}}$ to play a decisive role in the precipitation of SiC in Si.
To support previous assumptions MD simulations, which are capable of modeling the necessary amount of atoms, i.e. the precipitate and the surrounding c-Si structure, have been employed in the current study.
-In a previous comparative study\cite{zirkelbach10a} we have shown that the utilized empirical potential fails to describe some selected processes.
+In a previous comparative study\cite{zirkelbach10} we have shown that the utilized empirical potential fails to describe some selected processes.
Thus, limitations of the employed potential have been further investigated and taken into account in the present study.
We focussed on two major shortcomings: the overestimated activation energy and the improper description of intrinsic and C point defects in Si.
Overestimated forces between nearest neighbor atoms that are expected for short range potentials\cite{mattoni2007} have been confirmed to influence the C$_{\text{i}}$ diffusion.
-The migration barrier was estimated to be larger by a factor of 2.4 to 3.5 compared to highly accurate quantum-mechanical calculations\cite{zirkelbach10a}.
+The migration barrier was estimated to be larger by a factor of 2.4 to 3.5 compared to highly accurate quantum-mechanical calculations\cite{zirkelbach10}.
Concerning point defects, the drastically underestimated formation energy of C$_{\text{s}}$ and deficiency in the description of the Si$_{\text{i}}$ ground state necessitated further investigations on structures that are considered important for the problem under study.
It turned out that the EA potential still favors a C$_{\text{i}}$ \hkl<1 0 0> DB over a C$_{\text{s}}$-Si$_{\text{i}}$ configuration, which, thus, does not constitute any limitation for the simulations aiming to resolve the present controversy of the proposed SiC precipitation models.
However, we observed a phase transition of the C$_{\text{i}}$-dominated into a clearly C$_{\text{s}}$-dominated structure.
The amount of substitutionally occupied C atoms increases with increasing temperature.
Entropic contributions are assumed to be responsible for these structures at elevated temperatures that deviate from the ground state at 0 K.
-Indeed, in a previous ab initio MD simulation\cite{zirkelbach10b} performed at \unit[900]{$^{\circ}$C} we observed the departing of a Si$_{\text{i}}$ \hkl<1 1 0> DB located next to a C$_{\text{s}}$ atom instead of a recombination into the ground state configuration, i.e. a C$_{\text{i}}$ \hkl<1 0 0> DB.
+Indeed, in a previous ab initio MD simulation\cite{zirkelbach11a} performed at \unit[900]{$^{\circ}$C} we observed the departing of a Si$_{\text{i}}$ \hkl<1 1 0> DB located next to a C$_{\text{s}}$ atom instead of a recombination into the ground state configuration, i.e. a C$_{\text{i}}$ \hkl<1 0 0> DB.
% postannealing less efficient than hot implantation
Experimental studies revealed increased implantation temperatures to be more efficient than postannealing methods for the formation of topotactically aligned precipitates\cite{eichhorn02}.
The latter has been directly identified in the present simulation study, i.e. structures of two C$_{\text{s}}$ atoms and Si$_{\text{i}}$ located in the vicinity.
It is, thus, concluded that precipitation occurs by successive agglomeration of C$_{\text{s}}$ as already proposed by Nejim et~al.\cite{nejim95}.
-This agrees well with a previous ab initio study on defects in C implanted Si\cite{zirkelbach10b}, which showed C$_{\text{s}}$ to occur in all probability.
+This agrees well with a previous ab initio study on defects in C implanted Si\cite{zirkelbach11a}, which showed C$_{\text{s}}$ to occur in all probability.
However, agglomeration and rearrangement is enabled by mobile C$_{\text{i}}$, which has to be present at the same time and is formed by recombination of C$_{\text{s}}$ and Si$_{\text{i}}$.
In contrast to assumptions of an abrupt precipitation of an agglomerate of C$_{\text{i}}$\cite{werner96,werner97,eichhorn99,lindner99_2,koegler03}, however, structural evolution is believed to occur by a successive occupation of usual Si lattice sites with substitutional C.
This mechanism satisfies the experimentally observed alignment of the \hkl(h k l) planes of the precipitate and the substrate, whereas there is no obvious reason for the topotactic orientation of an agglomerate consisting exclusively of C-Si dimers, which would necessarily involve a much more profound change in structure for the transition into SiC.
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\end{document}
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