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\title{Modelling of a selforganization process leading to periodic arrays of nanometric amorphous precipitates by ion irradiation}

\author{F. Zirkelbach, M. Häberlen, J.K.N. Lindner and B. Stritzker}
%\from{Institute of Physics, University of Augsburg, Universitätsstrasse 1, D-86135 Augsburg, Germany}

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\begin{center}
 {\Large\bf
  Modelling of a selforganization process leading to periodic arrays of nanometric amorphous precipitates by ion irradiation
  \par
 }
\end{center}

\begin{center}
 {\large
  F. Zirkelbach, M. Häberlen, J. K. N. Lindner and B. Stritzker
  \par
 }
\end{center}

\begin{center}
 {\scriptsize
  Institute of Physics, University of Augsburg, Universitätsstrasse 1, D-86135 Augsburg, Germany
  \par
 }
\end{center}

\vspace{12pt}

\linebreak[2]
{\bf
 Abstract
}

Ion irradiation of materials which undergo a drastic density change upon amorphization have been shown to exhibit selforganized nanometric structures of the amorphous phase in the crystalline host lattice. In order to better understand that process, a Monte-Carlo-Simulation code based on a simple model is developed. In the present work we focus on high-dose carbon implantation into silicon. The simulation is able to reproduce results gained by cross-sectional TEM measurements of high-dose carbon implanted silicon. Conditions can be specified as a necessity for the selforganization process and compositional and structural information is obtained about the state during the ordering process which is difficult to be measured.

\newpage

\section{Introduction}
Formation of nanometric, selforganized, amorphous, lamellar precipitates, as a result of high-dose implantation of impurity ions is observed at certain conditions for various ion/target combinations \cite{1}\cite{2}\cite{3}. The present work focuses on high-dose carbon implantation into silicon. This is surprising, since high-dose implantation, commonly used to create burried compound layers for semiconductor devices, usually result in the formation of unordered ensembles of precipitates with a brought size distribution \cite{4}. Typical doses are $1-10 \times 10^{17} cm^{-2}$ with an ion energy of $180 \, keV$ and substrate temperatures below $400 \, ^{\circ} \mathrm{C}$. An example of such a lamellar structure is given in the cross sectional transmission electron microscopy (XTEM) in Figure \ref{xtem}. A model describing the selforganization process is introduced in this article. This model is used to implement a Monte-Carlo-Simulation code which reproduces the amorphization and precipitation process. Simulation results will be compared with XTEM measurements. Necessary conditions for creating lamellar precipitates are identified and some additional, difficult to measure information, like the carbon distribution and amorphous/crystalline structure in the layers of the target are obtained.

\newpage

\section{Model}
A model describing the formation of nanometric, selforganized, regularly arranged, amorphous $SiC_x$ inclusions was introduced in \cite{5}. Figure \ref{model} shows the evolution into ordered lamellae with increasing dose.

%\begin{figure}[!h]
%\begin{center}
%\includegraphics[width=13cm]{model1_.eps}
%\caption{Rough model explaining the selforganization of amorphous $SiC_x$ precipitates and the evolution into ordered lamellae with increasing dose} \label{1}
%\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.

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.

The target is devided into $64 \times 64 \times 100$ cells with a side length of $3 \, nmm$. Ech of it has a state (crystalline/amorphous) and keeps the local carbon concentration. The cell is addressed by a position vector $r^{\to}=(x,y,z)$, where $x$, $y$, $z$ are integers.

The probability of amorphization is assumed to be proportional to the nuclear stopping power. A local probability of amorphization at any point in the target is composed of three contributions, the ballistic amorphization, a carbon-induced and a stress-induced amorphization. The ballistic amorphization is proportional to the nuclear stopping power, as mentioned before. The carbon-induced amorphization is a linear function of the local carbon concentration. The stress-induced amorphization is proportional to the compressive stress originating from the amorphous volumes in the vicinity, the stress amplitude decreasing with the square of distance $d=|r^{\to}-r^{\to}|$. Thus the probability of a crystalline volume getting amorphous can be calculated as
\[
 p_{c \rightarrow a}(r^{\to}) = p_{b} + p_{c} \times c_{carbon}(r^{\to}) + \sum_{amorphous \,  neighbours} \frac{p_{s} \times c_{carbon}(r^{\to})}{d^2}
\]
with $p_{b}$, $p_{c}$ and $p_{s}$ being simulation parameters to weight the three different mechanisms of amorphization. The probability $p_{a \rightarrow c}$ of an amorphous volume to turn crystalline should behave contrary to $p_{c \rightarrow a}$ and is thus assumed as $p_{a \rightarrow c} = 1 - p_{c \rightarrow a}$.

The simulation algorithm consists of three parts, the amorphization/recrystallization process, the carbon-incorporation and finally the carbon diffusion.

For the amorphization/recrystallization process, random coordinates are computed to specify the volume which is hit by an implanted carbon ion. The two random numbers corresponding to the $x$ and $y$ coordinates are generated with a uniform probability distribution, $p(x)dx=dx \textrm{, } p(y)dy=dy$. The random number corresponding to the $z$ coordinate is distributed according to the linear approximated nuclear stopping power, $p(z)dz=(a_{el} \times z+b_{el})dz$, where $a_{el}$ and $b_{el}$ are simulation parameters describing the nuclear energy loss. After calculating the local probability of amorphization $p_{c \rightarrow a}$ of that volume, another random number decides, depending on the current state, whether the volume gets amorphous or recrystallized. This step is looped for the average hit per ion in the simulation window, counted by TRIM collision data.

In an analogous manner random coordinates (expect the $z$ coordinate being distributed according the linear approximated implantation profile) are obtained to acquire the volume where the carbon ion gets stock and the local carbon concentration increases.

Finally a standard diffusion algorithm is started, so the supersaturation of carbon in the crystalline volumes can be reduced. This process adds a few simulation parameters, the diffusion velocity, the diffusion rate and a switch whether to do diffusion in $z$-direction or not. Notice that there is no diffusion among crystalline volumes.

\newpage

\section{Results}
Figure \ref{4} shows a comparison of a simulation result and a cross-sectional TEM snapshot of $180 \, keV$ implanted carbon in silicon at $150 \,^{\circ} \mathrm{C}$ with $4.3 \times 10^{17} cm^{-2}$. The depth the lamellar structure is starting in ($200 \, nm$) and also the average length of these precipitates complies to that one of the experimental data. The arrays are ordered in uniform intervals. It can be seen that lamellar, selforganized structures can be reproduced by the simulation.

Furthermore conditions for observing lamellar structures can be specified. Figure \ref{5} shows two identical simulation cycles with diffusion in $z$-direction switched off and on. The lamellar structures only appear with diffusion in $z$-direction enabled. Amorphous volumes deprive the neighboring crystalline layers of carbon so the probability of amorphization is increasing locally while decreasing in the adjoining layers. This fortifies the formation of lamellar precipitates and proves the diffusion in $z$-direction to be a must for the selforganization process.

In Figure \ref{6} two simulation results with different diffusion rates are compared. Higher diffusion rates cause a larger depth domain of lamellar structure as higher diffusion rates result in amorphous volumes holding plenty of carbon which is increasing the local amorhization probability. For low diffusion rates lamellar structures stabilize considerably lower as of the increasing presence of carbon with depth due to the carbon implantation profile.

Complementary arrays of crystalline/amorphous domains are observed looking at successive layers $z$ and $z+1$ as shown in Figure \ref{7}. Again the diffusion of carbon into the amorphous volumes is responsible for the complementary arrangement. In fact the two lower figures displaying the carbon distribution of layer $z$ and $z+1$ show that nearly all the carbon is located in the amorphous precipitates.

%Finally fourier transformation was applied on experimental XTEM measurements and simulatin results. \ldots

\newpage

\section{Summary and conclusion}
A simple model explaining the selforganization process of lamellar, amorphous precipitates was introduced. In addition the implementation of that model to reasonable simulation code was discussed. This simulation code is able to reproduce experimental results. Furthermore the formation of these lamellar structures get traceable by the simulation code. Necessary conditions, i.e. diffusion in $z$-direction can be stated. We found the diffusion rate to influence the depth distribution of lamellar precipitates. Not easily measurable information is gained by the simulation like the complementary configuration of amorphous and crystalline arrays in successive layers.

\newpage

\begin{thebibliography}{20}
 \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} 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} M. Häberlen, J.K.N. Lindner, B. Stritzker, Nucl. Instr. and Meth. B 216 (2004) 36-40.
\end{thebibliography}

\newpage

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\newpage

\begin{figure}[!h]
\begin{center}
\includegraphics[width=17cm]{k393abild1_e.eps}
\caption[Cross-sectional TEM image of $180 \, keV C^+$ in $Si$ with a dose of $4.3 \times 10^{17} \, cm^{-2}$ and substrate temperature of $150 \,^{\circ} \mathrm{C}$. and substrate temperature of $150 \,^{\circ} \mathrm{C}$ Lamellar and spharical amorphous inclusions are marked by $L$ and $S$.]{} \label{xtem}
\end{center}
\end{figure}

\begin{figure}[!h]
\begin{center}
\includegraphics[width=17cm]{model1_e.eps}
\caption[Rough model explaining the selforganization of amorphous $SiC_x$ precipitates and the evolution into ordered lamellae with increasing dose.]{} \label{model}
\end{center}
\end{figure}

\begin{figure}[!h]
\begin{center}
\includegraphics[width=14cm]{2pTRIM180C.eps}
\caption[Stopping powers and concentration profile calculated by TRIM.]{} \label{trim}
\end{center}
\end{figure}

\begin{figure}[!h]
\begin{center}
\includegraphics[width=15cm]{if_cmp2_e.eps}
\caption[Comparison of a simulation result and a cross-sectional TEM snapshot of $180 \, keV$ implanted carbon in silicon at $150 \,^{\circ} \mathrm{C}$ with $4.3 \times 10^{17} cm^{-2}$.]{} \label{4}
\end{center}
\end{figure}

\begin{figure}[!h]
\begin{center}
\includegraphics[width=8cm]{mit_ohne_diff_e.eps}
\caption[Identical simulation cycles with diffusion switched off (left) and on (right)]{} \label{5}
\end{center}
\end{figure}

\begin{figure}[!h]
\begin{center}
\includegraphics[width=8cm]{high_low_ac-diff_e.eps}
\caption[Identical simulation cycles with different diffusion rates]{} \label{6}
\end{center}
\end{figure}

\begin{figure}[!h]
\begin{center}
\includegraphics[width=10cm]{all_z-z_plus_1.eps}
\caption[Amorphous/crystalline and carbon distribution view of two successive layers]{} \label{7}
\end{center}
\end{figure}

\end{document}
