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\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,\\
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.
+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.
\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.
+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 processes, e.g. for the fabrication of strained silicon.
Epitaxial growth of 3C-SiC films is achieved either by ion implantation or chemical vapour deposition techniques.
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{figure}[!h]
\begin{center}
\begin{minipage}{5.5cm}
\includegraphics[width=5cm]{sic_prec_seq_01.eps}
\begin{minipage}{5.5cm}
\includegraphics[width=5cm]{sic_prec_seq_03.eps}
\end{minipage}
- \caption{foo}
+ \caption{Schematic of the supposed conversion mechanism of highly C doped Si into SiC. C is represented by red dots, Si by black dots and residual Si atoms by white dots with black border.}
\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.
+There is a supposed conversion mechanism of heavily carbon doped Si into SiC.
+Fig. 1 schematically displays the mechanism.
+As indicated by high resolution transmission microscopy \cite{} 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.
\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 \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 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 \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.eps}
+ \caption{Distinguished interstitial configurations.}
+ \end{center}
+ \label{}
+\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 $(0,0,0)$ (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.
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 $30\, ps$.
-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 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.
\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{}.
+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.
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.
+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}
+ \label{}
+\end{figure}
+The influence of interstitials on the diffusion of a single carbon atom is displayed in Fig. 4.
+\ldots
-\section*{Conclusion}
+\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.
+
+\bibliography{../../bibdb/bibdb}
\end{document}
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