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\begin{document}
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<frank.zirkelbach@physik.uni-augsburg.de>}
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\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.
+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 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 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 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.
-
-\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 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}
-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.
-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 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 deployed.
+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 $\tau_T=100\, fs$.
\begin{figure}[!h]
\begin{center}
- \includegraphics[width=8cm]{unit_cell.eps}
- \caption{Distinguished interstitial configurations.}
+ \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}
- \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.
+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 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 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 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 $V_1$, 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 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 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 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 not stable.
-The interstitial atom moves to the more favorable <100> dumbbell position, which has a formation energy of $0.5\, eV$.
+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.
-This type of configuration is frequently observed for the random insertion runs.
+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]{../plot/foo150.ps}
- \caption{Diffusion constants}
+ \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}
- \label{}
\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}
+% \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}
- \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).}
+ \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. 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.
+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 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.
+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.
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