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\begin{document}

\hyphenation{TRIM}

\begin{frontmatter}

\title{Monte Carlo simulation study\\of a selforganization process\\leading to
       ordered precipitate structures}

\author[augsburg]{F. Zirkelbach\corauthref{cor}}
\author[augsburg]{M. Häberlen}
\author[augsburg]{J. K. N. Lindner}
\author[augsburg]{B. Stritzker}

\corauth[cor]{Corresponding author.\\
              Tel.: +49-821-5983008; fax: +49-821-5983425.\\
              E-mail address: frank.zirkelbach@physik.uni-augsburg.de (Frank
              Zirkelbach)}

\address[augsburg]{Universität Augsburg, Institut für Physik,
                   Universitätsstrasse 1,\\D-86135 Augsburg, Germany}

\begin{abstract}
Periodically arranged, self-organized, nanometric, amorphous precipitates have been observed after high-fluence ion implantations into solids for a number of ion/target combinations at certain implantation conditions.
A model describing the ordering process based on compressive stress exerted by the amorphous inclusions as a result of the density change upon amorphization is introduced.
A Monte Carlo simulation code, which focuses on high fluence carbon implantation into silicon, is able to reproduce experimentally observed results.
By means of simulation the selforganization process gets traceable and detailed information about the compositional and structural state during the ordering process is obtained.
Based on simulation results, a recipe is proposed for producing broad distributions of lamellar ordered structures.
\end{abstract}

\begin{keyword}
Monte Carlo simulation; Self-organization; Precipitation; Amorphization;
Nanostructures; Ion irradiation\\
\PACS 02.70.Uu; 61.72.Tt; 81.16.Rf
\end{keyword}

\end{frontmatter}

\section{Introduction}

Precipitates, as a result of high-fluence ion implantation into solids, are usually statistically arranged and have a broad size distribution.
However, the formation of ordered, lamellar inclusions has been observed for a number of ion/target combinations at certain implantation conditions \cite{ommen,specht,ishimaru}.
An inevitable condition for the material to observe this special self-organized arrangement is a largely reduced density of host atoms in the amorphous phase compared to the crystalline host lattice.
As a consequence stress is exerted by the amorphous inclusions which is responsible for the ordering process.
A model to describe the process is introduced.
The implementation of a simulation code based on that model is discussed.
Simulation results are compared to experimental data, focussing on high-fluence carbon implantations into silicon.
Finally a guideline for fabrication of broad ditributions of lamellar ordered structures is suggested.

\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 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.
Since amorphous silicon ($a-Si$) is not stable against ion beam induced epitaxial recrystallization at temperatures above $130 \, ^{\circ} \mathrm{C}$ \cite{ibic}, the existence of the amorphous precipitates must be due to the accumulation of carbon (carbon induced amorphization), which stabilizes the amorphous phase \cite{ap_stab}.
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 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.

Since the solid solubility of carbon in $c-Si$ is essentially zero, once formed, $a-SiC_x$ inclusions serve as diffusional sinks for excess carbon atoms in the $c-Si$ phase, represented by the white arrows in Fig. \ref{img:model}.
As a consequence the amorphous volumes accumulate carbon enhancing the selforganization process.

\section{Simulation}

For the Monte Carlo simulation the target is devided into cells with a side length of $3 \, nm$.
Each cell has a crystalline or amorphous state and stores the local carbon concentration.
It is addressed by a position vector $\vec{r} = (k,l,m)$ where $k$, $l$ and $m$ are integers.
The simulation starts with a complete crytsalline target and zero carbon inside.

The model proposes three mechanisms of amorphization.
In the simulation, each of this mechanisms contributes to a local amorphization probability of cell $\vec{r}$.
The influence of the mechanisms are controlled by simulation parameters.
The local amorphization probability at volume $\vec{r}$ is calculated by
\begin{equation}
p_{c \rightarrow a}(\vec{r}) = p_b + p_c c_C(\vec{r}) + \sum_{\textrm{amorphous neighbours}} \frac{p_s c_C(\vec{r'})}{(r-r')^2} \textrm{ .}
\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 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 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}
p_{a \rightarrow c}(\vec r) = (1 - p_{c \rightarrow a}(\vec r)) \Big(1 - \frac{\sum_{direct \, neighbours} \delta (\vec{r'})}{6} \Big) \, \textrm{,}
\end{equation}
\[
\delta (\vec r) = \left\{
\begin{array}{ll}
        1 & \textrm{volume at position $\vec r$ amorphous} \\
        0 & \textrm{otherwise} \\
\end{array}
\right.
\]
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.
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 $l$.
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{ishimaru} M. Ishimaru,  R. M. Dickerson, K. E. Sickafus, Nucl. Instr. and Meth. B 166-167 (2000) 390.
\bibitem{lamellar_inclusions} J. K. N. Lindner, M. Häberlen, M. Schmidt, W. Attenberger, B. Stritzker, Nucl. Instr. Meth. B 186 (2002) 206.
\bibitem{model_joerg} J. K. N. Lindner, Nucl. Instr. Meth. B 178 (2001) 44.
\bibitem{int_eng} W. J. Taylor, T. Y. Tan, U. Gösele, Appl. Phys. Lett. 62 (1993) 3336.
\bibitem{ibic} J. Linnross, R. G. Elliman, W. L. Brown, J. Matter. Res. 3 (1988) 1208.
\bibitem{ap_stab} E. F. Kennedy, L. Csepregi, J. W. Mayer, J. Appl. Phys. 48 (1977) 4241.
\bibitem{eftem_maik} M. Häberlen, Bildung und Ausheilverhalten nanometrischer amorpher Einschlüsse in Kohlenstoff-implantierten Silizium, Diploma thesis, Augsburg, 2002 (in Germany).
\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 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 $10 \, keV$. $100 \times 10^6$ simulation steps correspond to a fluence of $2.7 \times 10^{17} \, cm^{-2}$.
\end{enumerate}

%\listoffigures

\newpage
\section*{Figures}

\begin{figure}[!h]
\begin{center}
\includegraphics[width=14cm]{k393abild1_e.eps}
\caption[foo]{}
\end{center}
\label{img:tem}
\end{figure}

\newpage
\begin{figure}[!h]
\begin{center}
\includegraphics[width=14cm]{modell_ng_e.eps}
\caption[foo]{}
\end{center}
\label{img:model}
\end{figure}

\newpage
\begin{figure}[!h]
\begin{center}
\includegraphics[width=14cm]{dosis_entwicklung_all_e.eps}
\caption[foo]{}
\end{center}
\label{img:dose_cmp}
\end{figure}

\newpage
\begin{figure}[!h]
\begin{center}
\includegraphics[width=14cm]{ac_cconc_ver2_e.eps}
\caption[foo]{}
\end{center}
\label{img:carbon_distr}
\end{figure}

\newpage
\begin{figure}[!h]
\begin{center}
\includegraphics[width=14cm]{multiple_impl_e.eps}
\caption[foo]{}
\end{center}
\label{img:broad_lam}
\end{figure}

\end{document}
