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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 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.
-\\\\
-{\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 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 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 taken 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 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 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 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 \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 \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_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 $(0,0,0)$ (${\color{red}\triangleleft}$), hexagonal $(-1/8,-1/8,1/8)$ (${\color{green}\triangleright}$), nearby dumbbell $(-1/8,-1/8,-1/4)$ (${\color{magenta}\Box}$) 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.
-%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 $30\, ps$.
-
-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 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 $V_1$, the volume corresponding to the size of a minimal SiC precipitate $V_2$ and the volume containing the amount of silicon necessary for the formation of such a minimal precipitate $V_3$ are examined.
-The two latter ones are accomplished since no long range diffusion of the carbon atoms is expected at this temperature.
-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 \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 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 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> dumbbell 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 unstable.
-The interstitial atom moves to the more favorable <100> dumbbell position which has a formation energy of $0.5\, eV$.
-The interstitial configuration is shown in Fig. 2.
-There is experimental evidence \cite{watkins76} of the existence of this configuration.
-It is frequently generated and has the lowest formation energy of all the defects observed in all the simulation runs in which carbon is inserted at random positions.
-Fig. 3 schematically displays the <110> dumbbell configuration including the displacements relative to their initial positions and resulting new Si-Si and C-Si pair distances.
-
-\begin{figure}[!h]
- \begin{center}
- \includegraphics[width=8cm]{c_in_si_int_001db_0.eps}
- \caption{Position of a <100> dumbbell carbon interstitial in silicon.
-          Only bonds of the carbon interstitial atom are shown.}
- \end{center}
-\end{figure}
-\begin{figure}[!h]
- \begin{center}
- \includegraphics[width=16cm]{100-c-si-db.eps}
- \caption{Schematic of the <100> C-Si dumbbell configuration.
-          Displacements of the atoms relative to their initial position are given.
-	  The displacement of the carbon atom is relative to the initial position of atom 1.
-	  New resulting Si-Si and C-Si pair distances for the atoms shown in the schematic and the distances to Si' atoms outside of the displayed region are recorded.}
- \end{center}
-\end{figure}
-
-%\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]{pc_si-c_c-c.ps}
- \caption{Pair correlation functions for Si-C and C-C bonds.
-          Carbon atoms are introduced into the whole simulation volume $V_1$, the region which corresponds to the size of a minimal SiC precipitate $V_2$ and the volume which contains the necessary amount of silicon for such a minimal precipitate $V_2$ respectively.}
- \end{center}
-\end{figure}
-\begin{figure}[!h]
- \begin{center}
- \includegraphics[width=12cm]{pc_si-si.ps}
- \caption{Si-Si pair correlation function for pure Si and Si with 3000 inserted C atoms.
-          The inset shows a magnified region between 0.28 and 0.36 nm.}
- \end{center}
-\end{figure}
-Fig. 4 shows resulting pair correlation functions of the simulation runs targeting the observation of precipitation events.
-The contributions of Si-C and C-C pairs are presented separately each of them displaying the pair correlation for the three different volumes $V_1$, $V_2$ and $V_3$ (as explained above) exposed to carbon insertion.
-Results show no signigicant difference between $V_1$ and $V_2$.
-Si-Si pairs for the case of 3000 inserted C atoms inserted into $V_2$ and a reference function for pure Si are displayed in Fig. 5.
-
-The amount of C-C bonds for $V_1$ are much smaller than for $V_2$ and $V_3$ since carbon atoms are spread over the total simulation volume which means that there are only 0.2 carbon atoms per silicon unit cell on average.
-The first C-C peak appears at about 0.15 nm.
-This is comparable to the nearest neighbour distance for graphite or diamond.
-It is assumed that these carbon atoms form strong C-C bonds, which is supported by a decrease of the total energy during carbon insertion for the $V_2$ and $V_3$ in contrast to the $V_3$ simulation.
-
-The peak at 0.31 nm perfectly matches the distance of two carbon atoms in the SiC lattice which in SiC is also expected for the Si-Si bonds.
-After insertion of carbon atoms the Si-Si pair correlation function in fact shows non-zero values in the range of the C-C peak width while the amount of Si pairs at the regular distances at 0.24 and 0.38 nm decreases.
-However no clear peak is observed and random analyses of configurations in which distances around 0.3 nm appear, i.e. visualization of such atom pairs, identify <100> C-Si dumbbells to be responsible for stretching the Si-Si next neighbour distance for low concentrations of carbon, i.e. for the $V_1$ and early stages of $V_2$ and $V_3$ simulation runs.
-This excellently agrees with the calculation for a single <100> dumbbell ($r(13)$ in Fig. 4).
-For higher carbon concentrations the defect concentration is likewise increased and a considerable amount of damage is introduced into the inserted volume.
-Damage and superposition of defects generate new displacement arrangements which become hard to categorize and trace and obviously lead to a broader distribution of pair distances.
-The slightly higher amount and intense increase of Si-Si pairs at distances smaller 0.31 nm is probably due to the Si-Si cutoff radius of 0.296 nm.
-The cutoff function causes artificial forces pushing the Si atoms out of the cutoff region.
-By again visualizing the C-C atom pairs with distances of 0.31 nm concatenated, differently oriented <100> dumbbell interstitials are frequently observed.
-Since dumbbells of this type with different orientations are perpendicularly aligned the C atoms are displaced along the plane diagonal of the original lattice.
-One might now assume for the precipitation process, that C atoms are arrenged first and at a later point pull the Si atoms into the right configuration.
-In this way the hkl planes of the SiC in Si and the Si matrix would have equal orientations which is supported by experimental transmission electron microscopy data \cite{}.
-\\\\
-Ab hier weiter ...
-\\\\
-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 explained.
-The <100> C-Si dumbbel is reproduced and is the energetically most favorable configuration observed by simulation.
-%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}