+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{13}, diffusion among crystalline volumes is assumed to be zero in the following simulations.
+
+\newpage
+
+\section{Results}
+Figure \ref{c-xtem} shows a comparison of a simulation result and a XTEM bright-field image of silicon implanted at $150 \,^{\circ} \mathrm{C}$ with $180 \, keV$ $C^+$ ions at a dose of $4.3 \times 10^{17} \, cm^{-2}$. Significant lamellar structure formation is observed in the depth interval between $200$ and $300 \, nm$ (Figure 4(b)). This is nicely reproduced by the simulation result shown in Figure 4(a). Even the average length of the precipitates complies to the experimental data. The lamellae are arranged in uniform intervals. Obviously the simulation is able to reproduce lamellar, selforganized structures.
+
+Simulations with different model parameters allow to specify conditions for observing lamellar structures. First runs with a simplified version of the program have shown that it is essential to assume low amorphization probabilities to avoid early complete amorphization of the whole cell ensemble. Instead small amorphization parameters $p_b$, $p_c$, $p_s$ and a large number of simulation steps are required to observe lamellar structures. This finding is in agreement to the fact that due to the low nuclear energy deposition of the light carbon ions in silicon, amorphization would not be expected at all at these elevated target temperatures \cite{4} and thus carbon mediated amorphization has to be taken into account to explain the amorphization process.
+
+Figure \ref{zdiff} shows the results of two identical simulation runs with diffusion in $z$-direction switched off and on. The lamellar structures only appear if diffusion in $z$-direction is enabled. Amorphous volumes denude the neighbouring crystalline layers of carbon. In consequence the stability of such cells against recrystallization is enhanced, the probability to amorphize crystalline cells in the same depth is increased due to the stress term and the amorphization in the carbon denuded cells and their lateral vicinity is decreased. This fortifies the formation of lamellar precipitates. The result highlights the importance of the diffusion in $z$-direction for the selforganization process.
+
+In Figure \ref{diffrate} two simulation results with different diffusion rates are compared. Higher diffusion rates cause a larger depth domain of lamellar structure. This can be understood since higher diffusion rates result in amorphous volumes holding more carbon which consequently stabilizes the amorphous state. In case of slower diffusion rates (Figure 6(b)) the redistribution of carbon is too slow to allow for an effective agglomeration of carbon atoms in amorphous cells to stabilize the amorphous state against recrystallization. This results in a smaller total amount of amorphous material in Figure 6(a) compared to Figure 6(b). The stabilization occurs only at a depth larger than approximately $240 \, nm$ where the total concentration of carbon is high enough. The sufficient stabilization of amorphous volumes in this deeper depth zone enables also the more effective contribution of the stress mediated amorphization.
+
+The influence of the stress term $p_s$ is considered in Figure \ref{stress}. For otherwise the same conditions as in Figure 6(b) calculations with decreased $p_s$ in Figure 7(c),(b),(a) show a systematically reduced extension of the lamellae zone. The mean diameter of amorphous lamellae decreases with decreasing $p_s$. Both observations support the assumption of stress mediated amorphization as a mechanism contributing to lamella formation.
+
+Figure \ref{compl-str} shows the extension of amorphous lamellae in plane view for two consecutive slices $m$ and $m+1$ of the ensemble. It is obvious that amorphous and crystalline lamellae have a complementary arrangement in neighbouring slices (Figure 8(a),(b)) which again is a result of the carbon accumulation in the amorphous lamellae. This can be clearly seen by comparison with the corresponding carbon maps in Figure 8(c),(d).
+
+\newpage
+
+\section{Summary and conclusion}
+A simple model explaining the selforganization process of lamellar, amorphous precipitates during high-dose ion implantation is introduced. The implementation of the model in a simulation code is described. The simulation is able to reproduce the experimentally observed formation of lamellae. The evolution of these lamellar structures gets traceable by the simulation. The weight of different mechanisms which contribute to the selforganization process is explored by variation of simulation parameters. It is found that diffusion in $z$-direction and stress mediated amorphization are necessary to create ordered arrays of amorphous, lamellar precipitates. Thus by simulation, information is gained about the selforganization process which is not easily accessible by experimental techniques.
+
+\newpage
+
+\begin{thebibliography}{10}
+ \bibitem{1} L.L. Snead, S.J. Zinkle, J.C. Hay, M.C. Osborne, Nucl. Instr. and Meth. B 141 (1998) 123.
+ \bibitem{2} A.H. van Ommen, Nucl. Instr. and Meth. B 39 (1989) 194.
+ \bibitem{3} M. Ishimaru, R.M. Dickerson, K.E. Sickafus, Nucl. Instr. and Meth. B 166-167 (2000) 390.
+ \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} M. Häberlen, J.K.N. Lindner, B. Stritzker, to be published.
+ \bibitem{8} M. Häberlen, J.K.N. Lindner, B. Stritzker, Nucl. Instr. and Meth. B 206 (2003) 916-921.
+ \bibitem{9} J. Linnross, R.G. Elliman, W.L. Brown, J. Mater, Res. 3 (1988) 1208.
+ \bibitem{10} L. L. Horton, J. Bentley, L. Romana, A. Perez, C.J. McHargue, J.C. McCallum, Nucl. Intr. and Meth. B 65 (1992) 345.
+ \bibitem{11} W. Skorupa, V. Heera, Y. Pacaud, H. Weishart, in: F. Priolo, J.K.N. Lindner, A. Nylandsted Larsen, J.M. Poate (Eds.), New Trends in Ion Beam Processing of Materials, Europ. Mater. Res. Soc. Symp. Proc. 65, Part 1, Elsevier, Amsterdam, 1997, p. 114.
+ \bibitem{12} 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{13} J.K.N. Lindner, W. Reiber, B. Stritzker, Mater. Sci. Forum Vols. 264-268 (1998) 215-218.