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The wide band gap semiconductor (2.3 eV - 3.3 eV) exhibiting a high breakdown field, saturated electron drift velocity and thermal conductivity in conjunction with its unique thermal and mechanical stability as well as radiation hardness is a suitable material for high-temperature, high-frequency and high-power devices\cite{wesch96,morkoc94}, which are moreover deployable in harsh and radiation-hard environments\cite{capano97}.\r
% there are different polytpes with different properties and 3c-sic in special\r
SiC, which forms fourfold coordinated covalent bonds, tends to crystallize into many different modifications, which solely differ in the one-dimensional stacking sequence of identical, close-packed SiC bilayers\cite{fischer90}.\r
-Different polytypes exhibit different properties, in which the cubic phase (3C-SiC) shows increased values for the thermal conductivity and breakdown field compared to other polytypes\cite{wesch96}, which is thus most effective for high-performance electronic devices.\r
+Different polytypes exhibit different properties, in which the cubic phase (3C-SiC) shows increased values for the thermal conductivity and breakdown field compared to other polytypes\cite{wesch96}, which is, thus, most effective for high-performance electronic devices.\r
\r
% (thin films of) 3c-sic can be produced by ibs\r
Next to the fabrication of 3C-SiC layers by chemical vapor deposition (CVD) and molecular beam epitaxy (MBE) on hexagonal SiC\cite{powell90,fissel95,fissel95_apl} and Si\cite{nishino83,nishino87,kitabatake93,fissel95_apl} substrates, high-dose carbon implantation into crystalline silicon (c-Si) with subsequent or in situ annealing was found to result in SiC microcrystallites in Si\cite{borders71}.\r
However, only little is known on the SiC conversion in C implanted Si.\r
High resolution transmission electron microscopy (HREM) studies\cite{werner96,werner97,lindner99_2} suggest the formation of C-Si dimers (dumbbells) on regular Si lattice sites, which agglomerate into large clusters indicated by dark contrasts and otherwise undisturbed Si lattice fringes in HREM.\r
Once a critical radius of 2 nm to 4 nm is reached a topotactic transformation into a 3C-SiC precipitate occurs.\r
-The transformation is manifested by the disappearance of dark contrasts in favor of Moir\'e patterns due to the lattice mismatch of 20 \% of the 3C-SiC precipitate and c-Si.\r
-The insignificantly lower Si density of SiC ($\approx 4$ \%) compared to c-Si results in the emission of only a few excess Si atoms.\r
+The transformation is manifested by the disappearance of dark contrasts in favor of Moir\'e patterns due to the lattice mismatch of \unit[20]{\%} of the 3C-SiC precipitate and c-Si.\r
+The insignificantly lower Si density of SiC ($\approx \unit[4]{\%}$) compared to c-Si results in the emission of only a few excess Si atoms.\r
% motivation to understand the precipitation and link to atomistic simulations\r
A detailed understanding of the underlying processes will enable significant technological progress in 3C-SiC thin film formation and likewise offer perspectives for processes which rely upon prevention of precipitation events, e.g. the fabrication of strained pseudomorphic Si$_{1-y}$C$_y$ heterostructures\cite{strane96,laveant2002}.\r
\r
Atomistic simulations offer a powerful tool to study materials on a microscopic level providing detailed insight not accessible by experiment.\r
Relevant structures consisting of $\approx 10^4$ atoms for the nanocrystal and even more atoms for a reasonably sized Si host matrix are too large to be completely described by high accuracy quantum mechanical methods.\r
Directly modelling the dynamics of the processes mentioned above almost inevitably requires the atomic interaction to be described by less accurate though computationally more efficient classical potentials.\r
-The most common empirical potentials for covalent systems are the Stillinger-Weber\cite{stillinger85}, Brenner\cite{brenner90}, Tersoff\cite{tersoff_si3} and environment-dependent interatomic (EDIP)\cite{bazant96,bazant97,justo98} potential.\r
+The most common empirical potentials for covalent systems are the Stillinger-Weber\cite{stillinger85} (SW), Brenner\cite{brenner90}, Tersoff\cite{tersoff_si3} and environment-dependent interatomic (EDIP)\cite{bazant96,bazant97,justo98} potential.\r
Until recently\cite{lucas10}, a parametrization to describe the C-Si multicomponent system within the mentioned interaction models did only exist for the Tersoff\cite{tersoff_m} and related potentials.\r
Whether such potentials are appropriate for the description of the physical problem has, however, to be verified first by applying classical and quantum-mechanical methods to relevant processes that can be treated by both methods.\r
-\r
-\cite{mattoni02} <- good agreement ab initio and tersoff for ...\r
-\r
+By combination of empirical potential molecular dynamics (MD) and density functional theory (DFT) calculations the SW turned out to be best suited for simulations of dislocation nucleation processes\cite{godet03} and threshold displacement energy calculations\cite{holmstroem08} important in ion implantation.\r
+Also the Tersoff potential yields a qualitative agreement concerning the interaction of Si self-interstitials and substitutional C\cite{mattoni02}.\r
An extensive comparison\cite{balamane92} concludes that each potential has its strengths and limitations and none of them is clearly superior to others.\r
Despite their shortcomings these potentials are assumed to be reliable for large-scale simulations\cite{balamane92,huang95,godet03} on specific problems under investigation providing insight into phenomena that are otherwise not accessible by experimental or first principles methods.\r
+Remaining shortcomings have frequently been resolved by modifying the interaction\cite{tang95,mattoni2007} or extending it\cite{devanathan98_2} with data gained from ab initio calculations\cite{nordlund97}.\r
\r
-\cite{holmstroemXX} <- check for best empirical potential\r
-\r
-To overcome shortcomings, e.g. in simulations concerning ion-solid interaction during implantation ... potentials have been extended\cite{devanathan98_2} either by ab initio data\cite{nordlund97} to better describe the repulsive next-neighboured ....\r
-\cite{tang95} <- modified tersoff scaling cut-off\r
-\r
-In this work, the applicability of a Tersoff-like bond order potential\cite{albe_sic_pot} (claiming good ... nachschauen bei paper, dass dann doch edip nimmt) to some basic processes involved in the initially mentioned SiC precipitation mechanism is investigated by comparing results gained by classical and ab inito calculations.\r
-\r
-In the following a comparative investigation of density functional theory (DFT) studies and classical potential calculations of the structure, energetics and mobility of carbon defects in silicon is presented.\r
+In this work, the applicability of a Tersoff-like bond order potential\cite{albe_sic_pot} to some basic processes involved in the initially mentioned SiC precipitation mechanism is investigated by comparing results gained by classical and ab inito calculations.\r
+In the following a comparative investigation of density functional theory studies and classical potential calculations of the structure, energetics and mobility of carbon defects in silicon is presented.\r
\r
% --------------------------------------------------------------------------------\r
\section{Methodology}\r
% ----- DFT ------\r
-The first-principles DFT calculations were performed with the plane-wave based \r
-Vienna Ab-initio Simulation Package (VASP)\cite{kresse96}. The Kohn-Sham equations were solved \r
-using the generalized-gradient XC-functional approximation proposed by Perdew and \r
-Wang (GGA-PW91)\cite{perdew92}. \r
-The electron-ion interaction was described by the projector-augmented wave (PAW) method\cite{bloechel94,kresse99}. \r
-In the PAW data scalar relativistic corrections are contained. Throughout this work an \r
-energy cut-off of \unit[300]{eV} was used to expand the wave functions into the plane-wave basis. \r
-For the sampling of the Brillouin zone, only the $\Gamma$-point was used. \r
-The defect structures and the migration paths were modelled in cubic supercells containing 216 Si-atoms. Spin polarization has been \r
-fully accounted for. \r
+The first-principles DFT calculations are performed with the plane-wave based Vienna Ab-initio Simulation Package (VASP)\cite{kresse96}.\r
+The Kohn-Sham equations are solved using the generalized-gradient XC-functional approximation proposed by Perdew and Wang (GGA-PW91)\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 is restricted to the $\Gamma$-point.\r
+The defect structures and the migration paths are modelled in cubic supercells containing 216 Si atoms.\r
+The ions and cell shape are allowed to change in order to realize a constant pressure simulation.\r
+Spin polarization is fully accounted for.\r
\r
% ------ Albe potential ---------\r
%% Frank: Setup/short description of the potential ?\r
-For the calculations with the classical potentials... \r
-\r
+For the classical potential calculations a supercell of 9 Si lattice constants in each direction consisting of 5832 Si atoms is used.\r
+A Tersoff-like bond order potential by Erhart and Albe\cite{albe_sic_pot} is 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.\r
+Constant pressure simulations are realized by the Berendsen barostat\cite{brendsen84}.\r
+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.\r
\r
\section{Results}\r
\r
-After ion implantation, carbon interstitials are the most common defects in the silicon sample. Their mobility is the \r
-crucial quantity to be investigated. We thus started our comparative investigations by comparing the stability and the \r
-mobility of an isolated carbon interstitial in silicon bulk in the various possible geometries it can take. \r
+% ... wer sagt das ...\r
+%After ion implantation, carbon interstitials are the most common defects in the silicon sample.\r
+According to the assumed SiC precipitation model described in the introductary part, carbon interstitial defects form and agglomerate into large clusters.\r
+Thus, it is of crucial importance to investigate the various possible structures of carbon defects and the mobility of the lowest energy, hence most probable, defect configuration in crystalline silicon.\r
+%Their mobility is the crucial quantity to be investigated.\r
+%We thus started our comparative investigations by comparing the stability and the mobility of an isolated carbon interstitial in silicon bulk in the various possible geometries it can take.\r
\r
\subsection{Carbon interstitials in various geometries}\r
\r
-Several geometries have been calculated to be stable for the carbon interstitial. Fig.\ref{fig:interstitials} shows all these \r
-structures. However, there are some discrepancies between the results from classical potential calculations and those obtained \r
-from first principles. \r
-Table \ref{table:formation} summarizes the formation energies of the interstitial geometries for both methods used in this work \r
-and compares the results to literature values. (...check references for more data, ..) \r
-\r
-% Tables: like in the talk, but add further literature data and give the references/citations (also to bibliography \r
-% at the end!) \r
-%\begin{figure}\r
-%\includegraphics[width=1.0\columnwidth]{models.eps}\r
-%\caption{\label{fig:interstitials} Molecular model of the possible carbon interstitials. }\r
-%\end{figure} \r
-\r
-While the Albe potential predicts ... as stable, DFT does not. ...(further comparisons, trend "too high/low" E-formation,...)... \r
- Nevertheless, both methods predict the (110) dumb bell configuration to be the most stable... (?) \r
+Table~\ref{tab:defects} summarizes the formation energies of the interstitial configurations for the Erhart/Albe and VASP calculations performed in this work as well as further results from literature.\r
+The formation energies are defined in the same way as in the articles used for comparison\cite{tersoff90,dal_pino93} chosing SiC as a reservoir for the carbon impurity.\r
+Relaxed geometries are displayed in Fig.~\ref{fig:defects}.\r
+\begin{table}[th]\r
+\begin{tabular}{l c c c c c c}\r
+\hline\r
+\hline\r
+ & T & H & \hkl<1 0 0> DB & \hkl<1 1 0> DB & S & B \\\r
+\hline\r
+ Erhart/Albe & 6.09 & 9.05$^*$ & 3.88 & 5.18 & 0.75 & 5.59$^*$ \\\r
+ %VASP & unstable & unstable & 3.15 & 3.60 & 1.39 & 4.10 \\\r
+ VASP & unstable & unstable & 3.72 & 4.16 & 1.95 & 4.66 \\\r
+ Tersoff\cite{tersoff90} & 3.8 & 6.7 & 4.6 & 5.9 & 1.6 & 5.3 \\\r
+ ab initio & - & - & x & - & 1.89 \cite{dal_pino93} & x+2.1 \cite{capaz94} \\\r
+ more! & - & & & & & \\\r
+\hline\r
+\hline\r
+\end{tabular}\r
+\caption{Formation energies of carbon point defects in crystalline silicon determined by classical potential and ab initio methods. The formation energies are given in eV. T denotes the tetrahedral, H the hexagonal, B the bond-centered and S the substitutional interstitial configuration. The 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.}\r
+\label{tab:defects}\r
+\end{table}\r
+\begin{figure}\r
+\begin{minipage}[t]{0.32\columnwidth}\r
+\underline{Tetrahedral}\\\r
+\includegraphics[width=\columnwidth]{01.eps}\r
+\end{minipage}\r
+\begin{minipage}[t]{0.32\columnwidth}\r
+\underline{Hexagonal}\\\r
+\includegraphics[width=\columnwidth]{02.eps}\r
+\end{minipage}\r
+\begin{minipage}[t]{0.32\columnwidth}\r
+\underline{\hkl<1 0 0> dumbbell}\\\r
+\includegraphics[width=\columnwidth]{03.eps}\r
+\end{minipage}\\\r
+\begin{minipage}[t]{0.32\columnwidth}\r
+\underline{\hkl<1 1 0> dumbbell}\\\r
+\includegraphics[width=\columnwidth]{04.eps}\r
+\end{minipage}\r
+\begin{minipage}[t]{0.32\columnwidth}\r
+\underline{Substitutional}\\[0.05cm]\r
+\includegraphics[width=\columnwidth]{05.eps}\r
+\end{minipage}\r
+\begin{minipage}[t]{0.32\columnwidth}\r
+\underline{Bond-centered}\\\r
+\includegraphics[width=\columnwidth]{06.eps}\r
+\end{minipage}\r
+\caption{Configurations of carbon point defects in silicon. The silicon/carbon atoms and the bonds (only for the interstitial atom) are illustrated by yellow/grey spheres and blue lines. Bonds are drawn for atoms located within a certain distance and do not necessarily correspond to chemical bonds.}\r
+\label{fig:defects}\r
+\end{figure} \r
+\r
+However, there are some discrepancies between the results from classical potential calculations and those obtained from first principles.\r
+\r
+While the Erhart/Albe potential predicts ... as stable, DFT does not. ...(further comparisons, trend "too high/low" E-formation,...)... \r
+\r
+Nevertheless, both methods predict the \hkl<1 0 0> dumbbell configuration to be most stable.\r
\r
\r
\subsection{Mobility}\r
\r
% ----------------------------------------------------\r
\section*{Acknowledgment}\r
-The calculations were done using grants of computer time from the \r
-Paderborn Center for Parallel Computing (PC$^2$) and the \r
-H\"ochstleistungs-Rechenzentrum Stuttgart. The Deutsche \r
-Forschungsgemeinschaft is acknowledged for financial support.\r
+%The calculations were done using grants of computer time from the \r
+%Paderborn Center for Parallel Computing (PC$^2$) and the \r
+%H\"ochstleistungs-Rechenzentrum Stuttgart. The Deutsche \r
+%Forschungsgemeinschaft is acknowledged for financial support.\r
+One of us (F.Z.) wants to acknowledge financial support by the Bayerische Forschungsstiftung (DPA-61/05).\r
\r
% --------------------------------- references -------------------\r
-\begin{thebibliography}{99}\r
-\bibitem{kresse96} G. Kresse and J. Furthm\"uller, \r
- Comput. Mater. Sci. {\bf 6}, 15 (1996). \r
-\bibitem{perdew92} J. P. Perdew, J. A. Chevary, S. H. Vosko, K. A. Jackson, M. R. Pederson, D. J. Singh and C. Fiolhais, \r
- Phys. Rev. B {\bf 46}, 6671 (1992). \r
-\bibitem{ceperley80} D. M. Ceperley and B. J. Alder, \r
- Phys. Rev. Lett. {\bf 45}, 556 (1980). \r
-\bibitem{perdew81} J. P. Perdew and A. Zunger, \r
- Phys. Rev. B {\bf 23}, 5048 (1981). \r
-\bibitem{bloechel94} P. E. Bl\"ochl, \r
- Phys. Rev. B {\bf 50}, 17953 (1994).\r
-\bibitem{kresse99} G. Kresse and D. Joubert, \r
- Phys. Rev. B {\bf 59}, 1758 (1999).\r
-\bibitem{monk76} H. J. Monkhorst and J. D. Pack, \r
- Phys. Rev. B {\bf 13}, 5188 (1976). \r
-\bibitem{albe} Albe potential\r
-\bibitem{stillinger} Stillinger-Weber potential \r
-\bibitem{joannopoulos} Joannopoulos\r
-\bibitem{xyz} who else? \r
-\bibitem{rauls03a} E. Rauls, A. Gali, P. De´ak, and Th. Frauenheim, Phys. Rev. B, 68, 155208 (2003).\r
-\bibitem{rauls03b} E. Rauls, U. Gerstmann, H. Overhof, and Th. Frauenheim, Physica B, Vols. 340-342, p. 184-189 (2003). \r
-\bibitem{gerstmann03} U. Gerstmann, E. Rauls, Th. Frauenheim, and H. Overhof, Phys. Rev. B, 67, 205202, (2003).\r
-\r
-\end{thebibliography}\r
+\bibliography{../../bibdb/bibdb}{}\r
+%\begin{thebibliography}{99}\r
+%\bibitem{kresse96} G. Kresse and J. Furthm\"uller, \r
+% Comput. Mater. Sci. {\bf 6}, 15 (1996). \r
+%\bibitem{perdew92} J. P. Perdew, J. A. Chevary, S. H. Vosko, K. A. Jackson, M. R. Pederson, D. J. Singh and C. Fiolhais, \r
+% Phys. Rev. B {\bf 46}, 6671 (1992). \r
+%\bibitem{ceperley80} D. M. Ceperley and B. J. Alder, \r
+% Phys. Rev. Lett. {\bf 45}, 556 (1980). \r
+%\bibitem{perdew81} J. P. Perdew and A. Zunger, \r
+% Phys. Rev. B {\bf 23}, 5048 (1981). \r
+%\bibitem{bloechel94} P. E. Bl\"ochl, \r
+% Phys. Rev. B {\bf 50}, 17953 (1994).\r
+%\bibitem{kresse99} G. Kresse and D. Joubert, \r
+% Phys. Rev. B {\bf 59}, 1758 (1999).\r
+%\bibitem{monk76} H. J. Monkhorst and J. D. Pack, \r
+% Phys. Rev. B {\bf 13}, 5188 (1976). \r
+%\bibitem{albe} Albe potential\r
+%\bibitem{stillinger} Stillinger-Weber potential \r
+%\bibitem{joannopoulos} Joannopoulos\r
+%\bibitem{xyz} who else? \r
+%\bibitem{rauls03a} E. Rauls, A. Gali, P. De´ak, and Th. Frauenheim, Phys. Rev. B, 68, 155208 (2003).\r
+%\bibitem{rauls03b} E. Rauls, U. Gerstmann, H. Overhof, and Th. Frauenheim, Physica B, Vols. 340-342, p. 184-189 (2003). \r
+%\bibitem{gerstmann03} U. Gerstmann, E. Rauls, Th. Frauenheim, and H. Overhof, Phys. Rev. B, 67, 205202, (2003).\r
+%\end{thebibliography}\r
\r
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\end{document}\r
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