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\begin{document}
% header
\textsc{\Large F. Zirkelbach$^1$, J. K. N. Lindner$^1$,
K. Nordlund$^2$, B. Stritzker$^1$}\\
\vspace{16pt}
- $^1$ Experimentalphysik IV, Institut f"ur Physik, Universit"at Augsburg,\\
- Universit"atsstr. 1, D-86135 Augsburg, Germany\\
+ $^1$ Experimentalphysik IV, Institut f\"ur Physik, Universit\"at Augsburg,\\
+ Universit\"atsstr. 1, D-86135 Augsburg, Germany\\
\vspace{16pt}
$^2$ Accelerator Laboratory, Department of Physical Sciences,
University of Helsinki,\\
<frank.zirkelbach@physik.uni-augsburg.de>}
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+%\vspace{24pt}
\section*{Abstract}
-The precipitation process of silicon carbide in heavily carbon doped silicon is not yet understood for the most part.
-High resolution transmission electron microscopy indicates that in a first step carbon atoms form $C-Si$ dumbbells on regular $Si$ lattice sites which agglomerate into large clusters.
-In a second step, when the cluster size reaches a radius of a few $nm$, the high interfacial energy due to the $SiC$/$Si$ lattice misfit of almost $20 \, \%$ is overcome and the precipitation occurs.
-By simulation details of the precipitation process can be obtained on the atomic level.
-A newly parametrized Tersoff like bond-order potential is used to model the system appropriately.
+The precipitation process of silicon carbide in heavily carbon doped silicon is not yet fully understood.
+High resolution transmission electron microscopy observations suggest that in a first step carbon atoms form C-Si dumbbells on regular Si lattice sites which agglomerate into large clusters.
+In a second step, when the cluster size reaches a radius of a few $nm$, the high interfacial energy due to the SiC/Si lattice misfit of almost 20\% is overcome and the precipitation occurs.
+By simulation, details of the precipitation process can be obtained on the atomic level.
+A newly parametrized Tersoff-like bond order potential is used to model the system appropriately.
First results gained by molecular dynamics simulations using this potential are presented.
-The influence of the amount and placement of inserted carbon atoms on the defect formation and structural changes is discussed.
-Furthermore a minimal carbon concentration necessary for precipitation is examined by simulation.
\\\\
-{\bf Keywords:} Silicon carbide, Nucleation, Molecular dynamics simulation.
+{\bf Keywords:} Silicon, carbon, silicon carbide, nucleation, defect formation,
+ molecular dynamics simulations
\section*{Introduction}
-Understanding the precipitation process of cubic silicon carbide (3C-SiC) in heavily carbon doped silicon (Si) will enable significant technological progress in thin film formation of an important wide band gap semiconductor material.
-On the other hand it will likewise offer perspectives for processes which rely upon prevention of precipitation processes, e.g. for the fabrication of strained silicon.
+Understanding the precipitation process of cubic silicon carbide (3C-SiC) in heavily carbon doped silicon will enable significant technological progress in thin film formation of the important wide band gap semiconductor material SiC \cite{edgar92}.
+On the other hand it will likewise offer perspectives for processes which rely upon prevention of precipitation events, e.g. the fabrication of strained, pseudomorphic $\text{Si}_{1-y}\text{C}_y$ heterostructures \cite{}.
-Epitaxial growth of 3C-SiC films is achieved either by ion implantation or chemical vapour deposition techniques.
-Surface effects dominate the CVD process while for the implantation process carbon is introduced into bulk silicon.
-This work tries to realize conditions which hold for the ion implantation process.
+Epitaxial growth of 3C-SiC films is achieved either by ion beam synthesis (IBS) \cite{lindner02} and chemical vapour deposition (CVD) or molecular beam epitaxy (MBE) techniques.
+While in CVD and MBE surface effects need to be takein into account, SiC formation during IBS takes place in the bulk of the Si crystal.
+In the present work the simulation tries to realize conditions which hold for the ion implantation process.
-First of all a suitable model is considered.
-After that the realization by simulation is discussed.
-First results gained by simulation are presented in a next step.
-Finally these results are outlined and conclusions are infered.
+First of all a picture of the supposed precipitation event is presented.
+Afterwards the applied simulation sequences are discussed.
+Finally first results gained by simulation are presented.
\section*{Supposed conversion mechanism}
-Silicon (Si) nucleates in diamond structure.
-Contains of two fcc lattices, on displaced one quarter of volume diagonal compared to the first.
-3C-SiC nucleates in zincblende structure where the shifted fcc lattice sites are composed of carbon atoms.
-The length of four lattice constants of Si is approximately equal to the length of five 3C-SiC lattice constants ($4a_{Si}\approx 5a_{3C-SiC}$), which means that there is a lattice misfit of almost 20\%.
-Due to this the silicon density of 3C-SiC is slightly lower than the one of silicon.
-
-There is a supposed conversion mechanism of heavily carbon doped Si into SiC.
-Fig. 1 schematically displays the mechanism.
-\begin{figure}
- \begin{center}
- \begin{minipage}{5.5cm}
- \includegraphics[width=5cm]{sic_prec_seq_01.eps}
- \end{minipage}
- \begin{minipage}{5.5cm}
- \includegraphics[width=5cm]{sic_prec_seq_02.eps}
- \end{minipage}
- \begin{minipage}{5.5cm}
- \includegraphics[width=5cm]{sic_prec_seq_03.eps}
- \end{minipage}
- \caption{foo}
- \end{center}
-\end{figure}
-As indicated by high resolution transmission microscopy \ref{hrem_ind} introduced carbon atoms (red dots) form C-Si dumbbells on regular Si (black dots) lattice sites.
-The dumbbells agglomerate int large clusters, so called embryos.
-Finally, when the cluster size reaches a critical radius of 2 to 4 nm, the high interfacial energy due to the lattice misfit is overcome and the precipitation occurs.
-Due to the slightly lower silicon density of 3C-SiC residual silicon atoms exist.
-The residual atoms will most probably end up as self interstitials in the silicon matrix since there is more space than in 3C-SiC.
-
-Taking this into account, it is important to understand both, the configuration and dynamics of carbon interstitials in silicon and silicon self interstitials.
-Additionaly the influence of interstitials on atomic diffusion is investigated.
-
-\section*{Simulation}
+Silicon has diamond structure and thus is composed of two fcc lattices which are displaced by one quarter of the volume diagonal.
+3C-SiC grows in zincblende structure, i.e. is also composed of two fcc lattices out of which one is occupied by Si the other by C atoms.
+The length of four lattice constants of Si is approximately equal to the length of five 3C-SiC lattice constants ($4a_{\text{Si}}\approx 5a_{\text{3C-SiC}}$) resulting in a lattice misfit of almost 20\%.
+Due to this the silicon atomic density of 3C-SiC is slightly lower than the one of pure Si.
+
+%\begin{figure}[!h]
+% \begin{center}
+% \begin{minipage}{5.5cm}
+% \includegraphics[width=5cm]{sic_prec_seq_01_s.eps}
+% \end{minipage}
+% \begin{minipage}{5.5cm}
+% \includegraphics[width=5cm]{sic_prec_seq_02_s.eps}
+% \end{minipage}
+% \begin{minipage}{5.5cm}
+% \includegraphics[width=5cm]{sic_prec_seq_03_s.eps}
+% \end{minipage}
+% \caption{Schematic of the supposed conversion mechanism of highly C (${\color{red}\Box}$) doped Si (${\color{black}\bullet}$) into SiC ($_{\color{black}\bullet}^{{\color{red}\Box}}$) and residual Si atoms ($\circ$). The figure shows the dumbbell formation (left), the agglomeration into clusters (middle) and the situation after precipitation (right).}
+% \end{center}
+%\end{figure}
+There is a supposed conversion mechanism of heavily carbon doped Si into SiC \cite{werner97}.
+As concluded by high resolution transmission electron microscopy \cite{werber97,} introduced carbon atoms form C-Si dumbbells on regular Si lattice sites.
+The dumbbells agglomerate into large clusters, called embryos.
+Finally, when the cluster size reaches a critical radius of 2 to 4 nm, the high interfacial energy due to the 3C-SiC/Si lattice misfit is overcome and precipitation occurs.
+Due to the slightly lower silicon density of 3C-SiC excessive silicon atoms exist which will most probably end up as self-interstitials in the silicon matrix since there is more space than in 3C-SiC.
+
+Thus, in addition to the precipitation event itself, knowledge of C and Si interstitials in Si are of great interest in order to investigate the precipitation of heavily C doped Si into SiC.
+%Additionaly the influence of interstitials on atomic diffusion is investigated.
+
+\section*{Simulation sequences}
A molecular dynamics simulation approach is used to examine the steps involved in the precipitation process.
-For integrating the equations of motion the velocity verlet algorithm \ref{} with a timestep of 1 fs is deployed.
-The interaction of the silicon and carbon atoms is realized by a newly parametrized Tersoff like bond order potential \ref{}.
+For integrating the equations of motion the velocity verlet algorithm \cite{verlet67} with a timestep of $1\, fs$ is adopted.
+The interaction of the silicon and carbon atoms is realized by a newly parametrized Tersoff-like bond order potential \cite{albe_sic_pot}.
Since temperature and pressure of the system is kept constant in experiment the isothermal-isobaric NPT ensemble is chosen for the simulation.
-Coupling to the heat bath is achieved by the Berendsen thermostat \ref{} with a time constant $\tau_T=100\, fs$.
-The pressure is scaled by the Berendsen barostat \ref{} again using a timeconstant of $\tau_P=100\, fs$ and a bulk modulus of $100\, GPa$ for silicon.
+Coupling to the heat bath is achieved by the Berendsen thermostat \cite{berendsen84} with a time constant $\tau_T=100\, fs$.
+The pressure is scaled by the Berendsen barostat \cite{berendsen84} again using a timeconstant of $\tau_P=100\, fs$ and a bulk modulus of $100\, GPa$ for silicon.
To exclude surface effects periodic boundary conditions are applied.
-To investigate the intesrtitial configurations of C and Si in Si, a simulation volume of 9 silicon unit cells is each direction used.
+\begin{figure}[!h]
+ \begin{center}
+ \includegraphics[width=8cm]{unit_cell_s.eps}
+ \caption{Insertion positions for the tetrahedral (${\color{red}\triangleleft}$), hexagonal (${\color{green}\triangleright}$) and <110> dumbbell (${\color{magenta}\Box}$) interstitial configuration.}
+ \end{center}
+\end{figure}
+To investigate the interstitial configurations of C and Si in Si, a simulation volume of 9 silicon unit cells in each direction is used.
The temperature is set to $T=0\, K$.
-The insertion positions are illustrated in Fig 2.
-In separated simulation runs a carbon and a silicon atom respectively is inserted at the tetrahedral $(0,0,0)$ (red), hexagonal $(-1/8,-1/8,1/8)$ (green), supposed dumbbell $(-1/8,-1/8)$ (purple) and at random positions (in units of the silicon lattice constant) where the origin is located in the middle of the unit cell.
-In order to avoid too high kinetic energies in the case of the dumbbell configuration the nearest silicon neighbour atom is shifted to $(-1/4,-1/4,-1/4)$ (dashed border).
-The introduced kinetic energy is scaled out by a relaxation time of $2\, ps$.
+The insertion positions are illustrated in Fig. 2.
+In separated simulation runs a carbon and a silicon atom respectively is inserted at the tetrahedral $(0,0,0)$ (${\color{red}\triangleleft}$), hexagonal $(-1/8,-1/8,1/8)$ (${\color{green}\triangleright}$), supposed dumbbell $(-1/8,-1/8,-1/4)$ (${\color{magenta}\bullet}$) and at random positions (in units of the silicon lattice constant) where the origin is located in the centre of the unit cell.
+In order to avoid too high potential energies in the case of the dumbbell configuration the nearest silicon neighbour atom is shifted to $(-3/8,-3/8,-1/4)$ ($\circ$).
+The energy introduced into the system is scaled out within a relaxation phase of $2\, ps$.
The same volume is used to investigate diffusion.
-A certain amount of silicon atoms are inserted at random positions in a centered region of $11 \,\textrm{\AA}$ in each direction.
-The insertion is taking place step by step in order to maintain a constant system temeprature.
-Finally a carbon atom is inserted at a random position in the unit cell located in the middle of the simulation volume.
-The simulation continues for another $30\, ps$.
+Different amounts of silicon atoms are inserted at random positions within a centered region of $11 \,\textrm{\AA}$ in each direction.
+Insertion events are carried out step by step maintaining a constant system temperature of $450\, ^{\circ} \textrm{C}$.
+Finally a single carbon atom is inserted at a random position within the unit cell located in the middle of the simulation volume.
+The simulation is proceeded for another $30\, ps$.
-The sequence of the simulations aiming to reproduce a precipitation process is indicated in Fig 3.
-The size of the simulation volume is 31 silicon lattice constants in each direction.
+For the simulations aiming to reproduce a precipitation process the volume is 31 silicon lattice constants in each direction.
The system temperature is set to $450\, ^{\circ} \textrm{C}$.
-$6000$ carbon atoms (the amount necessary to form a minimal 3C-SiC precipitation) are consecutively inserted in a way to keep constant the system temperature.
+$6000$ carbon atoms (the amount necessary to form a 3C-SiC precipitate with a radius of 3 nm) are consecutively inserted in a way to keep constant the system temperature.
Precipitation is examined for three insertion volumes which differ in size.
-The whole simulation volume, the volume corresponding to a minimal SiC precipitation volume and the volume containing the necessary amount of silicon to form such a precipitation.
-After the insertion procedure the system is cooled down to $20\, ^{\circ} \textrm{C}$.
+The whole simulation volume, the volume corresponding to the size of a minimal SiC precipitate and the volume containing the amount of silicon necessary for the formation of such a minimal precipitate.
+Following the insertion procedure the system is cooled down to $20\, ^{\circ} \textrm{C}$.
\section*{Results}
-The tetrahedral and the <110> dumbbell self interstitial configurations can be reproduced as observed in \ref{}.
+The tetrahedral and the <110> dumbbell self-interstitial configurations can be reproduced as observed in \cite{albe_sic_pot}.
The formation energies are $3.4\, eV$ and $4.4\, eV$ respectively.
-However the hexagonal one is not stable opposed to what is presented in \ref{}.
-The atom moves towards a energetically more favorable position very close to the tetrahedral one but slightly displaced along the three coordinate axes.
+However the hexagonal one is not stable opposed to what is presented in \cite{albe_sic_pot}.
+The atom moves towards an energetically more favorable position very close to the tetrahedral one but slightly displaced along the three coordinate axes.
The formation energy of $4.0\, eV$ of this type of interstitial equals the result obtained in the reference for the hexagonal one.
-The same type of interstitial is observed within the random insertion runs.
-Variations exist where the displacement is along two axes ($E_f=3.8\, eV$) or along one axis ($E_f=3.6\, eV$) succesively approximating the tetrahedral configuration and formation energy.
+The same type of interstitial may arise using random insertions.
+In addition variations exist in which the displacement is only along two axes ($E_f=3.8\, eV$) or along a single axis ($E_f=3.6\, eV$) succesively approximating the tetrahedral configuration and formation energy.
The tetrahedral and <110> dumbbel carbon interstitial configurations are stable.
The formation energies are $2.7\, eV$ and $1.8\, eV$ respectively.
-Again the hexagonal one is found to be not stable.
-The interstitial atom moves to the more favorable <100> dumbbell position, which has a formation energy of $0.5\, eV$.
-There is experimental evidence \ref{} of the existence of this configuration.
-This type of configuration is frequently observed for the random insertion runs.
+Again the hexagonal one is found to be unstable.
+The interstitial atom moves to the more favorable <100> dumbbell position which has a formation energy of $0.5\, eV$.
+There is experimental evidence \cite{watkins76} of the existence of this configuration.
+This type of configuration is frequently observed for the random insertion runs and is assumed to be the lowest in energy.
-
-
-\section*{Conclusion}
+\begin{figure}[!h]
+ \begin{center}
+ \includegraphics[width=12cm]{../plot/diff_dep.ps}
+ \caption{Diffusion coefficients of a single carbon atom for different amount of Si selft interstitials}
+ \end{center}
+\end{figure}
+The influence of Si self-interstitials on the diffusion of a single carbon atom is displayed in Fig. 3.
+Diffusion coefficients for different amount of Si self-interstitials are shown.
+A slight increase is first observed in the case of 30 interstitial atoms.
+Further increasing the amount of interstitials leads to a tremendous decay of the diffusion coeeficient.
+Generally there is no long range diffusion of the carbon atom for a temperature of $450\, ^{\circ} \textrm{C}$.
+The maximal displacement of the carbon atom relativ to its insertion position is between 0.5 and 0.7 \AA.
+
+\begin{figure}[!h]
+ \begin{center}
+ \includegraphics[width=12cm]{../plot/foo_end.ps}
+ \includegraphics[width=12cm]{../plot/foo150.ps}
+ \caption{Pair correlation functions for Si-C and C-C bonds.
+ Carbon atoms are introduced into the whole simulation volume ({\color{red}-}), the region which corresponds to the size of a minimal SiC precipitate ({\color{green}-}) and the volume which contains the necessary amount of silicon for such a minimal precipitate ({\color{blue}-}).}
+ \end{center}
+\end{figure}
+Fig. 4 shows results of the simulation runs targeting the observation of precipitation events.
+The C-C pair correlation function suggests carbon nucleation for the simulation runs where carbon was inserted into the two smaller regions.
+The peak at $1.5\, \textrm{\AA}$ fits quite well the next neighbour distance of diamond.
+On the other hand the Si-C pair correlation function indicates formation of SiC bonds with an increased crystallinity for the simulation in which carbon is inserted into the whole simulation volume.
+There is more carbon forming Si-C bonds than C-C bonds.
+This gives suspect to the competition of Si-C and C-C bond formation in which the predominance of either of them depends on the method handling carbon insertion.
+
+\section*{Summary}
+The supposed conversion mechanism of heavily carbon doped silicon into silicon carbide is presented.
+Molecular dynamics simulation sequences to investigate interstitial configurations, the influence of interstitials on the atomic diffusion and the precipitation of SiC are proposed.
+The <100> C-Si dumbbel is reproducable by simulation and is the energetically most favorable configuration.
+The influence of silicon self-interstitials on the diffusion of a single carbon atom is demonstrated.
+Two competing bond formations, either Si-C or C-C, seem to coexist, where the strength of either of them depends on the size of the region in which carbon is introduced.
+
+\bibliography{../../bibdb/bibdb}
\end{document}
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