\section{Model}
High-fluence carbon implantations at temperatures between $150$ and $400 \, ^{\circ} \mathrm{C}$ with an energy of $180 \, keV$ result in an amorphous $SiC_x$ layer along with spherical and lamellar amorphous $SiC_x$ inclusions at the layer interface \cite{lamellar_inclusions}, as can be ssen in Fig. \ref{img:tem}.
-A model is proposed in \cite{model_joerg}, which is schematically displayed in Fig. \ref{img:model}, showing the evolution into ordered lamellae with increasingamount of implanted carbon.
+A model is proposed in \cite{model_joerg}, which is schematically displayed in Fig. \ref{img:model}, showing the evolution into ordered lamellae with increasing amount of implanted carbon.
With increasing fluence the silcon is supersaturated of carbon atoms which results in a nucleation of spherical $SiC_x$ precipitates.
By the precipitation into the amorphous $SiC_x$ ($a-SiC_x$) phase an enormous interfacial energy \cite{int_eng} required for cubic $SiC$ ($3C-SiC$, $a=0.536 \, nm$) in crystalline silicon ($c-Si$, $a=0.543 \, nm$) due to a $20 \, \%$ lattice mismatch can be saved.
In fact, energy filtered XTEM studies \cite{eftem_maik} revealed the carbon-rich nature of the precipitates.
The $Si$ atomic density of $a-SiC$ is about $20$ to $30 \, \%$ lower compared to $3C-SiC$ \cite{si_dens1,si_dens2}.
-The same is assumed for substoicheometrc $a-SiC_x$ compared to $c-Si$.
+The same is assumed for substoicheometric $a-SiC_x$ compared to $c-Si$.
Therefor the amorphous volumes tend to expand and as a result compressive stress - which is relaxing in the vertical direction since the process occurs near the target surface - is applied on the $Si$ host lattice, represented by black arrows in Fig \ref{img:model}.
Volumes between amorphous inclusions will more likely turn into an amorphous state as the stress aggravates the rearrangement of atoms on regular lattice sites (stress enhanced amorphization).
In contrast, randomly originated amorphous precipitates (ballistic amorphization) located in a crystalline neighbourhood not containing high amounts of carbon will recrystallize in all probability under the present implantation conditions.
\end{equation}
The ballistic amorphization is constant and controlled by $p_b$.
-This choice is justified by analysing {\em TRIM} \cite{trim} collision data that show a mean constant energy loss per collision of an ion.
+This choice is justified by analysing {\em TRIM} \cite{trim} collision data that show an identical behaviour of the graph displaying the amounts of collisions per depth and the nuclear stopping power.
+Thus an ion is losing a mean constant energy per collision.
The carbon induced amorphization is proportional to the local amount of carbon $c_C(\vec{r})$ and controlled by the simulation parameter $p_c$.
The stress enhanced amorphization is controlled by $p_s$.
-The forces originating from the amorphous volumes $\vec{r}'$ in the vicinity are assumed to be proportional to the amount of carbon $c_C(\vec{r}')$.
+The forces originating from the amorphous volumes $\vec{r'}$ in the vicinity are assumed to be proportional to the amount of carbon $c_C(\vec{r'})$.
The sum is just taken over volumes located in the layer and since the stress amplitude is decreasing with the square of the distance $r-r'$ a cutoff radius is used in the simulation.
In case of an amorphous volume, a recrystallization probability is given by
\begin{equation}
which is basically $1$ minus the amorphization probability and a term taking into account the crystalline neighbourhood which is needed for epitaxial recrystallization.
The simulation algorithm consists of three parts.
-The first part is the amorphization/recrystallization step.
-Random values are computed to specify the volume $\vec{r}$ which is hit by an impinging carbon ion.
+In a first amorphization/recrystallization step random values are computed to specify the volume $\vec{r}$ which is hit by an impinging carbon ion.
Two uniformly distributed random numbers $x$ and $y$ are mapped to the coordinates $k$ and $l$.
-A random number $z$ corresponding to the depth coordinate $m$ is distributed according to the nuclear stopping power gained by {\em TRIM} using the rejection method.
-
-Again analysing {\em TRIM} collision data shows the nuclear stopping power being essentially
-The amorphization or recrystallization probability is computed and another random number decides whether there is amorphization or recrystallization or
+Using the rejection method a random number $z$ corresponding to the depth coordinate $m$ is distributed according to the nuclear stopping power which, as seen above, is identical to the amount of collisions caused be the ions per depth.
+The local amorphization or recrystallization probability is computed and another random number between $0$ and $1$ decides whether there is amorphization or recrystallization or the state of that volume is unchanged.
+This step is repeated for the mean amount of volumes in which collisions are caused by an ion, again gained by {\em TRIM} collision data.
+In a second step, the ion gets incorporated in the target at randomly chosen coordinates with the depth coordinate being distributed according to the implantation profile.
+In a last step the diffusion, controlled by the simulation parameters $d_v$ and $d_r$, and sputtering, controlled by the parameter $n$ are treated.
+Every $d_v$ simulation steps $d_r$ of the amount of carbon in crystalline volumes gets transfered to an amorphous neighbour in order to alollow a reduction of the supersaturation of carbon in crystalline volumes.
+Every $n$ steps a crystalline, carbon less layer is inserted at maximum depth while the first layer gets lost.
+The sputter rate $S$, derived from RBS meassurements \cite{sputter}, is connected to $n$ by
+\begin{equation}
+S = \frac{(3 \, nm)^2 X Y}{n} \textrm{ .}
+\end{equation}
\section{Results}
+While first versions of this simulation, just covering a limit depth region of the target in which selforganization is observed, have already been discussed in \cite{me1,me2}, only results of the new version, which is able to model the whole depth region affected by the irradiation process, will be presented.
+
+A set of simulation parameters exists to properly describe the fluence dependent formation of the amorphous phase, as can be seen in Fig \ref{img:dose_cmp}.
+\ldots
+
+By simulation it is possible to determine the carbon concentration in crystalline, amorphous and both volumes.
+Fig. \ref{img:carbon_distr} \ldots
+
+Based on simulation runs a recipe is proposed to create broad distributions of lamellar structure.
+The starting point is a crystalline silcon target with a nearly constant carbon concentration of $10 \, at.\%$ starting from the surfcae downto $500 \, nm$, which can be achieved by multiple carbon implantation steps with energies between $180$ and $10 \, keV$ at a temperature $T=500 \, ^{\circ} \mathrm{C}$ to prevent amorphization \cite{sputter}.
+In a second step the target is irradiated with $2 \, MeV$ $C^+$ ions, which have a nearly constant energy loss and an essentially zero implantation profile in the affected depth region.
+The result is displayed in Fig. \ref{img:broad_lam}, showing already ordered structures after $s=100 \times 10^6$ steps corresponding to a fluence of $D=2.7 \times 10^{17} cm^{-2}$.
+The structure gets more defined with increasing fluence.
+According to recent studies \cite{photo} these structures are the starting point for materials showing high photoluminescence.
\section{Summary and conclusion}
+Ion irradiation of solids at certain implantation conditions may result in a regular ordered formation of amorphous precipitates.
+The ordering process can be understood by the presented model, which is able to reproduce experimental observations by means of a Monte Carlo simulation code.
+Detailed information, like the distribution of carbon located in amorphous and crystalline volumes, is gained again shedding light on the selforganization process.
+Finally a technique is proposed to produce broad distributions of lamellar ordered structures.
+
\begin{thebibliography}{20}
\bibitem{ommen} A. H. van Ommen, Nucl. Instr. and Meth. B 39 (1989) 194.
\bibitem{specht} E. D. Specht, D. A. Walko, S. J. Zinkle, Nucl. Instr. and Meth. B 84 (2000) 390.
\bibitem{si_dens1} L. L. Horton, J. Bentley, L. Romana, A. Perez, C. J. McHargue, J. C. McCallum, Nucl. Instr. Meth. B 65 (1992) 345.
\bibitem{si_dens2} 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, Eur. Mater. Res. Soc. Symp. Proc. 65, Part 1, Elsevier,Amsterdam, 1997,p. 114.
\bibitem{trim} J. F. Ziegler, J. P. Biersack, U. Littmark, The Stopping and Range of Ions in Solids, Pergamon, New York, 1985.
+\bibitem{sputter} J. K. N. Lindner, Nucl. Instr. Meth. B 178 (2001) 44.
+\bibitem{me1} F. Zirkelbach, M. Häberlen, J. K. N. Lindner, B. Stritzker, Comp. Matter. Sci. 33 (2005) 310.
+\bibitem{me2} F. Zirkelbach, M. Häberlen, J. K. N. Lindner, B. Stritzker, Nucl. Instr. Meth. B 242 (2006) 679.
+\bibitem{photo} D. Chen, Z. M. Liao, L. Wang, H. Z. Wang, F. Zhao, W. Y. Cheung, S. P. Wong, Opt. Mater. 23 (2003) 65.
\end{thebibliography}
\newpage
-
\section*{Figure Captions}
\begin{enumerate}
\item Cross-sectional transmission electron microscopy (XTEM) image of a $Si(100)$ sample implanted with $180 \, keV$ $C^+$ ions at a fluence of $4.3 \times 10^{17} \, cm^{-2}$ and a substrate temperature of $150 \, ^{\circ} \mathrm{C}$. Lamellar and spherical amorphous inclusions at the interface of the continuous amorphous layer are marked by L and S.
\item Schematic explaining the selforganization of amorphous $SiC_x$ precipitates and the evolution into ordered lamellae with increasing fluence (see text).
-\item Comparison of simulation results and XTEM images ($180 \, keV$ $C^+$ implantation into silicon at $150 \, ^{\circ} mathrm{C}$) for several fluence. Amorphous cells are white. Simulation parameters: $p_b=0.01$, $p_c=0.001$, $p_s=0.0001$, $d_r=0.05$, $d_v=1 \times 10^6$.
+\item Comparison of simulation results and XTEM images ($180 \, keV$ $C^+$ implantation into silicon at $150 \, ^{\circ} \mathrm{C}$) for several fluence. Amorphous cells are white. Simulation parameters: $p_b=0.01$, $p_c=0.001$, $p_s=0.0001$, $d_r=0.05$, $d_v=1 \times 10^6$.
\item Amorphous cell distribution and corresponding carbon implantation profile. The implantation profile shows the mean amount of carbon in amorphous and crystalline volumes as well as the sum for a fluence of $4.3 \times 10^{17} \, cm^{-2}$.
-\item Simulation result for a $2 \, MeV$ $C^+$ irradiation into silicon doped with $10 \, at. \%$ carbon by multiple implantation steps between $180$ and $10 \, keV$. $20 \times 10^6$ simulation steps correspond to a fluence of $0.54 \times 10^{17} \, cm^{-2}$.
+\item Simulation result for a $2 \, MeV$ $C^+$ irradiation into silicon doped with $10 \, at. \%$ carbon by multiple implantation steps between $180$ and $10 \, keV$. $100 \times 10^6$ simulation steps correspond to a fluence of $2.7 \times 10^{17} \, cm^{-2}$.
\end{enumerate}
+%\listoffigures
+
\newpage
\section*{Figures}
projects / lectures / latex.git / commitdiff