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\begin{center}
 {\LARGE {\bf Molecular dynamics simulation
              of defect formation and precipitation
	      in heavily carbon doped silicon
              }\\}
 \vspace{16pt}
 \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\\
 \vspace{16pt}
 $^2$ Accelerator Laboratory, Department of Physical Sciences,
 University of Helsinki,\\
 Pietari Kalmink. 2, 00014 Helsinki, Finland\\
 \vspace{16pt}
 {\scriptsize Corresponding author: Frank Zirkelbach
                              <frank.zirkelbach@physik.uni-augsburg.de>}
\end{center}

%\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 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.
\\\\
{\bf Keywords:} Silicon, Carbon, Silicon carbide, Nucleation, Diffusion, Defect formation.

\section*{Introduction}
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 an important wide band gap semiconductor material.
On the other hand it will likewise offer perspectives for processes which rely upon prevention of precipitation events, e.g. the fabrication of strained silicon.

Epitaxial growth of 3C-SiC films is achieved either by ion implantation or chemical vapour deposition (CVD) techniques.
While CVD is governed by surface effects carbon is directly introduced into bulk silicon for the implantation process.
In the present work the simulation runs try to realize conditions which hold for the ion implantation process.

First of all a picture of the supposed precipitation event is worked out.
Afterwards the realization via simulation is discussed.
In a next step first results gained by simulation are presented.

\section*{Supposed conversion mechanism}
Silicon nucleates in diamond structure.
This structure is composed of two fcc lattices which are displaced by one quarter of the volume diagonal.
3C-SiC nucleates in zincblende structure in which atoms of one fcc lattice are substituted by carbon atoms.
The length of four lattice constants of Si is approximately equal to the length of five 3C-SiC lattice constants ($4a_{\text{Si}}\approx 5a_{\text{3C-SiC}}$) resultings in a lattice misfit of almost 20\%.
Due to this the silicon density of 3C-SiC is slightly lower than the one of Si.

\begin{figure}[!h]
 \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{Schematic of the supposed conversion mechanism of highly C (${\color{red}\bullet}$) doped Si (${\color{black}\bullet}$) into SiC ($_{\color{black}\bullet}^{{\color{red}\bullet}}$) 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{}.
Fig. 1 schematically displays the mechanism.
As indicated by high resolution transmission microscopy \cite{} introduced carbon atoms (${\color{red}\bullet}$) form C-Si (${\color{black}\bullet}\,{\color{red}\bullet}$) dumbbells on regular Si (${\color{black}\bullet}$) 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 ($_{\color{black}\bullet}^{{\color{red}\bullet}}$) residual silicon atoms ($\circ$) 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}
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 \cite{verlet67} 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 \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 \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.

\begin{figure}[!h]
 \begin{center}
 \includegraphics[width=8cm]{unit_cell.eps}
 \caption{Insertion positions for the tetrahedral (${\color{red}\bullet}$), hexagonal (${\color{green}\bullet}$) and <110> dumbbell (${\color{purple}\bullet}$) interstitial configuration.}
 \end{center}
\end{figure}
To investigate the intesrtitial 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 $(0,0,0)$ (${\color{red}\bullet}$), hexagonal $(-1/8,-1/8,1/8)$ (${\color{green}\bullet}$), supposed dumbbell $(-1/8,-1/8)$ (${\color{purple}\bullet}$) 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 $(-1/4,-1/4,-1/4)$ ($\circ$).
The introduced kinetic energy is scaled out by a relaxation time 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 $30\, ps$.

For the simulations aiming to reproduce a precipitation process the simulation 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.
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 $20\, ^{\circ} \textrm{C}$.

\section*{Results}

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 \cite{albe_sic_pot}.
The atom moves towards a 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 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 \cite{watkins76} of the existence of this configuration.
This type of configuration is frequently observed for the random insertion runs.

\begin{figure}[!h]
 \begin{center}
 \includegraphics[width=8cm]{../plot/foo150.ps}
 \caption{Diffusion constants}
 \end{center}
\end{figure}
The influence of interstitials on the diffusion of a single carbon atom is displayed in Fig. 4.
\ldots


\begin{figure}[!h]
 \begin{center}
 \begin{minipage}{8.25cm}
 \includegraphics[width=8cm]{../plot/foo150.ps}
 \end{minipage}
 \begin{minipage}{8.25cm}
 \includegraphics[width=8cm]{../plot/foo_end.ps}
 \end{minipage}
 \caption{Pair correlation functions for C-C and Si-C bonds.
          Carbon atoms are introduced into the whole simulation volume (red), the region which corresponds to the size of a minimal SiC precipitation (green) and the volume which contains the necessary amount of silicon for a minimal precipitation (blue).}
 \end{center}
\end{figure}
Fig. 5 shows results of the simulation runs targeting the observation of a precipitation event.
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.

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