%\end{center}
%\end{figure}
-As a result of supersaturation of carbon atoms in silicon at high concentrations there is a nucleation of spherical $SiC_x$ precipitates. Carbon implantations at much higher implantation temperatures usually lead to the precipitation of cubic $SiC$ ($3C-SiC$, $a=0.436 \, nm$). The lattice misfit of almost $20\%$ of $3C-SiC$ causes a large interfacial energy with the crystalline $Si$ matrix \cite{6}. This energy could be reduced if one of the phases exists in the amorphous state. Energy filtered XTEM studies in \cite{4} have revealed that the amorphous phase is more carbon-rich than the crystalline surrounding. In addition, annealing experiments have shown that the amorphous phase is stable against crystallization at temperatures far above the recrystallization temperatures of amorphous $Si$. Prolonged annealing at $900 \, ^{\circ} \mathrm{C}$ turns the lamellae into ordered chains of amrphous and crystalline ($3C-SiC$) nanoprecipitates \cite{5} demonstrating again the carbon-rich nature of amorphous inclusions. Since at the implantation conditions chosen, pure $a-Si$ would recrystallize by ion beam induced crystallization \cite{7}, it is understandable that it is the carbon-rich side of the two phases which occurs in the amorphous state in the present phase separation process.
+As a result of supersaturation of carbon atoms in silicon at high concentrations there is a nucleation of spherical $SiC_x$ precipitates. Carbon implantations at much higher implantation temperatures usually lead to the precipitation of cubic $SiC$ ($3C-SiC$, $a=0.436 \, nm$). The lattice misfit of almost $20\%$ of $3C-SiC$ causes a large interfacial energy with the crystalline $Si$ matrix \cite{6}. This energy could be reduced if one of the phases exists in the amorphous state. Energy filtered XTEM studies in \cite{4} have revealed that the amorphous phase is more carbon-rich than the crystalline surrounding. In addition, annealing experiments have shown that the amorphous phase is stable against crystallization at temperatures far above the recrystallization temperatures of amorphous $Si$. Prolonged annealing at $900 \, ^{\circ} \mathrm{C}$ turns the lamellae into ordered chains of amrphous and crystalline ($3C-SiC$) nanoprecipitates \cite{7} demonstrating again the carbon-rich nature of amorphous inclusions. Since at the implantation conditions chosen, pure $a-Si$ would recrystallize by ion beam induced crystallization \cite{8}, it is understandable that it is the carbon-rich side of the two phases which occurs in the amorphous state in the present phase separation process.
Stoichiometric $SiC$ has a smaller atomic density than $c-Si$. A reduced density is also assumed for substoichiometric $a-SiC_x$. Hence the amorphous $SiC_x$ tends to expand and as a result compressive stress is applied on the $Si$ host lattice. As the process occurs near the target surface, stress is relaxing in vertical direction and there is mainly lateral stress remaining. Thus volumes between amorphous inclusions will more likely turn into an amorphous phase as the stress hampers the rearrangement of atoms on regular lattice sites. In contrast $a-Si$ volumes located in a crystalline neighbourhood will recrystallize in all probability. Carbon is assumed to diffuse from the crystalline to the amorphous volumes in order to reduce the supersaturation of carbon in the crystalline interstices. As a consequence the amorphous volumes accumulate carbon.
\newpage
\section{Simulation}
-Before discussing the implementation some assumptions and approximations have to be made. Figure \ref{trim} shows the stopping powers and carbon concentration profile calculated by TRIM \cite{8}. The depth region we are interested in is between $0-300 \, nm$ (furtheron called simulation window), the region between the target surface and the beginning of the continuous amorphous $SiC_x$ layer at the implantation conditions of Figure \ref{xtem}. The nuclear stopping power and the implantation profile can be approximated by a linear function of depth within the simulation window.
+Before discussing the implementation some assumptions and approximations have to be made. Figure \ref{trim} shows the stopping powers and carbon concentration profile calculated by TRIM \cite{9}. The depth region we are interested in is between $0-300 \, nm$ (furtheron called simulation window), the region between the target surface and the beginning of the continuous amorphous $SiC_x$ layer at the implantation conditions of Figure \ref{xtem}. The nuclear stopping power and the implantation profile can be approximated by a linear function of depth within the simulation window.
The target is devided into $64 \times 64 \times 100$ cells with a side length of $3 \, nm$. Ech of it has a state (crystalline/amorphous) and keeps the local carbon concentration. The cell is addressed by a position vector $\vec r = (k,l,m)$, where $k$, $l$, $m$ are integers.
The simulation algorithm consists of three parts, the amorphization/re-crystallization process, the carbon incorporation and finally the carbon diffusion.
-For the amorphization/recrystallization process random values are computed to specify the volume which is hit by an impinging carbon ion. Two random numbers $x,y \in [0,1]$ are generated and mapped to the coordinates $k,l$ using a uniform probability distribution, $p(x)dx=dx \textrm{, } p(y)dy=dy$. A random number $z$ corresponding to the $m$ coordinate is distributed according to the linear approximated nuclear stopping power, $p(z)dz=(s z+s_0)dz$, where $s$ and $s_0$ are simulation parameters describing the nuclear energy loss. After calculating the local probability of amorphization $p_{c \rightarrow a}(k,l,m)$ of the selected volume another random number determines depending on the current status whether the volume turns amorphous, recrystallizes or remains unchanged. This step is looped for the average hits per ion in the simulation window as extracted from TRIM \cite{8} collision data.
+For the amorphization/recrystallization process random values are computed to specify the volume which is hit by an impinging carbon ion. Two random numbers $x,y \in [0,1]$ are generated and mapped to the coordinates $k,l$ using a uniform probability distribution, $p(x)dx=dx \textrm{, } p(y)dy=dy$. A random number $z$ corresponding to the $m$ coordinate is distributed according to the linear approximated nuclear stopping power, $p(z)dz=(s z+s_0)dz$, where $s$ and $s_0$ are simulation parameters describing the nuclear energy loss. After calculating the local probability of amorphization $p_{c \rightarrow a}(k,l,m)$ of the selected volume another random number determines depending on the current status whether the volume turns amorphous, recrystallizes or remains unchanged. This step is looped for the average hits per ion in the simulation window as extracted from TRIM \cite{9} collision data.
In the same manner random coordinates are determined to select the cell where the carbon ion gets incorporated. In this step the probability distribution describing the stopping power profile is replaced by a distribution for the linearly approximated concentration profile. The local carbon concentration in the selected cell is increased.
-Following carbon incorporation carbon diffusion is considered in order to allow a reduction of the supersaturation of carbon in the crystalline volumes. This is done by a simple diffusion algorithm in which the concentration difference for each two neighbouring cells is considered and partially balanced according to a given diffusion rate $d_r$ (simulation parameter). This time consuming diffusion process is repeated after each $d_v$ (simulation parameter) impinging ions. A switch is implemented to exclude diffusion in $z$-direction. As in experimental studies diffusional broadening of carbon concentration profiles has not been observed even at significantly higher implantation temperatures where no amorphous phase is formed \cite{9} diffusion among crystalline volumes is assumed to be zero in the following simulations.
+Following carbon incorporation carbon diffusion is considered in order to allow a reduction of the supersaturation of carbon in the crystalline volumes. This is done by a simple diffusion algorithm in which the concentration difference for each two neighbouring cells is considered and partially balanced according to a given diffusion rate $d_r$ (simulation parameter). This time consuming diffusion process is repeated after each $d_v$ (simulation parameter) impinging ions. A switch is implemented to exclude diffusion in $z$-direction. As in experimental studies diffusional broadening of carbon concentration profiles has not been observed even at significantly higher implantation temperatures where no amorphous phase is formed \cite{10} diffusion among crystalline volumes is assumed to be zero in the following simulations.
\newpage
\bibitem{4} J.K.N. Lindner, Appl. Phys. A 77 (2003) 27-38
\bibitem{5} J.K.N. Lindner, M. Häberlen, M. Schmidt, W. Attenberger, B. Stritzker, Nucl. Instr. and Meth. B 186 (2000) 206-211
\bibitem{6} W.J. Taylor, T.Y. Tan, U.Gösele, Appl. Phys. Lett. 62 (1993) 3336
- \bibitem{7} J. Linnross, R.G. Elliman, W.L. Brown, J. Mater, Res. 3 (1988) 1208
- \bibitem{8} SRIM2000 Version of the TRIM program described by J.F. Ziegler, J.P. Biersack, U. Littmark in: The Stopping and Range of Ions in Matter, vol. 1, Pergamon Press, New York, 1985
- \bibitem{9} J.K.N. Lindner, W. Reiber, B. Stritzker, Mater. Sci. Forum Vols. 264-268 (1998) 215-218
+ \bibitem{7} M. Häberlen, J.K.N. Lindner, B. Stritzker, Nucl. Instr. and Meth. B 206 (2003) 916-921
+ \bibitem{8} J. Linnross, R.G. Elliman, W.L. Brown, J. Mater, Res. 3 (1988) 1208
+ \bibitem{9} SRIM2000 Version of the TRIM program described by J.F. Ziegler, J.P. Biersack, U. Littmark in: The Stopping and Range of Ions in Matter, vol. 1, Pergamon Press, New York, 1985
+ \bibitem{10} J.K.N. Lindner, W. Reiber, B. Stritzker, Mater. Sci. Forum Vols. 264-268 (1998) 215-218
\end{thebibliography}
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