\begin{frontmatter}
-\title{Monte Carlo simulation study\\of a selforganization process\\leading to
+\title{Monte Carlo simulation study\\of a selforganisation process\\leading to
ordered precipitate structures}
\author[augsburg]{F. Zirkelbach\corauthref{cor}}
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.
+Periodically arranged, self-organised, 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.
+A Monte Carlo simulation code, which focuses on high-fluence carbon implantations into silicon, is able to reproduce experimentally observed nanolamella distributions as well as the formation of continuous amorphous layers.
+By means of simulation the selforganisation 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 ordered lamellar structures.
\end{abstract}
\begin{keyword}
-Monte Carlo simulation; Self-organization; Precipitation; Amorphization;
+Monte Carlo simulation; Self-organisation; Precipitation; Amorphization;
Nanostructures; Ion irradiation\\
\PACS 02.70.Uu; 61.72.Tt; 81.16.Rf
\end{keyword}
\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.
+Precipitates resulting from 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 [1-3].
+An inevitable condition for the material to exhibit this special self-organised 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.
+A model describing the process is implemented in a simulation code, focussing on high-fluence carbon implantations into silicon.
+Simulation results are compared to experimental data and a recipe for the fabrication of broad distributions of lamellar ordered structures is proposed.
\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.
+High-fluence carbon implantations in silicon at temperatures between $150$ and $400 \, ^{\circ} \mathrm{C}$ with an energy of $180 \, keV$ result in an amorphous buried $SiC_x$ layer along with ordered spherical and lamellar amorphous $SiC_x$ inclusions at the upper layer interface [4] (Fig. 1).
+A model [5] explaining the evolution of ordered lamellae with increasing amount of implanted carbon is schematically represented in Fig. 2.
-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.
+With increasing fluence the silicon gets supersaturated with carbon atoms which results in the nucleation of spherical $SiC_x$ precipitates.
+By the precipitation into the amorphous $SiC_x$ (a-$SiC_x$) phase an enormous interfacial energy can be saved, which would be required for cubic $SiC$ ($3C-SiC$) in crystalline silicon (c-$Si$) [6], originating from a $20 \, \%$ lattice mismatch.
+Since amorphous silicon (a-$Si$) is not stable against ion beam induced epitaxial recrystallization at temperatures above $130 \, ^{\circ} \mathrm{C}$ at the present low atomic displacement rates [7], the existence of the amorphous precipitates must be due to the accumulation of carbon (carbon induced amorphization), which stabilizes the amorphous phase [8].
+In fact, energy filtered XTEM studies [9] reveal 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.
+The $Si$ atomic density of a-$SiC$ is about $20$ to $30 \, \%$ lower than the one of $3C-SiC$ [10,11].
+Thus, a corresponding density reduction is also assumed for substoichiometric a-$SiC_x$ compared to c-$Si$.
+Therefore the amorphous volumes tend to expand and as a result, compressive stress is exerted on the $Si$ host lattice, represented by black arrows in Fig 2.
+This stress may relax in the vertical direction since the process occurs near the target surface.
+Upon continued ion irradiation volumes between amorphous inclusions will more likely turn into an amorphous state as the stress disturbs the rearrangement of atoms on regular lattice sites (stress enhanced amorphisation).
+In contrast, randomly formed amorphous precipitates (ballistic amorphisation) with low concentrations of carbon in a crystalline neighbourhood are likely to recrystallise under 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.
+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, as represented by the white arrows in Fig. 2.
+As a consequence the amorphous volumes accumulate carbon, which stabilizes them against recrystallisation and promotes the strain supported formation of additional a-$SiC_x$ in their lateral vicinity.
\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.
+For the Monte Carlo simulation the target is divided into cells with a cube length of $3 \, nm$.
+Each cell is either in a crystalline or amorphous state and stores the local carbon concentration.
+The simulation starts with a complete crystalline target and zero carbon concentration.
-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
+The simulation algorithm consists of three parts.
+In a first amorphisation/recrystallisation step random numbers are computed to specify the volume at position $\vec{r}$ in which a collision occurs.
+Two uniformly distributed random numbers $x$ and $y$ are generated to determine the lateral position of $\vec{r}$.
+Using the rejection method a random number $z$ specifying the depth coordinate of $\vec{r}$ is distributed according to the nuclear stopping power profile which, as will be seen below, is identical to the number of collisions caused by the ions per depth.
+The local amorphization or recrystallization probability is computed as detailed below 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 number of steps of cells in which collisions are caused by one ion, gained from {\em TRIM} [12] 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 {\em TRIM} implantation profile.
+In a last step the carbon diffusion, controlled by two simulation parameters $d_v$ and $d_r$, as well as sputtering, controlled by the parameter $n$ are treated.
+Every $d_v$ simulation steps, a fraction $d_r$ of the amount of carbon in crystalline volumes gets transfered to an amorphous neighbour in order to allow for a reduction of the supersaturation of carbon in crystalline volumes.
+Every $n$ steps a crystalline, carbon-free layer is inserted at the bottom of the cell array while the first layer is removed, where $n$ results from a RBS derived [5] sputter rate.
+
+In order to calculate the amorphisation probability, three factors have to be taken into account corresponding to our model.
+In the simulation, each of these mechanisms contributes to a local amorphisation probability of the cell at $\vec{r}$.
+The strength of each mechanism is controlled by simulation parameters.
+The local amorphisation 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
+The normal (ballistic) amorphisation is controlled by $p_b$ and is set constant.
+This choice is justified by analysing {\em TRIM} collision data that show identical depth profiles for the number of collisions per depth and the nuclear stopping power.
+Thus, on average an ion is loosing a constant energy per collision.
+The carbon induced amorphisation is proportional to the local amount of carbon $c_C(\vec{r})$ and controlled by weight factor $p_c$.
+The stress enhanced amorphisation is weighted by $p_s$.
+The forces originating from the amorphous volumes $\vec{r'}$ in the vicinity of $\vec{r}$ are assumed to be proportional to the amount of carbon $c_C(\vec{r'})$ in the neighbour cell.
+The sum is limited to volumes located in the same layer because of of stress relaxation towards the surface. Since the stress amplitude is decreasing with the square of the distance $r-r'$, a cutoff radius is used in the simulation.
+If an amorphous volume is hit by collisions, a recrystallisation 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}
\[
\delta (\vec r) = \left\{
\begin{array}{ll}
- 1 & \textrm{volume at position $\vec r$ amorphous} \\
+ 1 & \textrm{if the cell at position $\vec r$ is amorphous} \\
0 & \textrm{otherwise} \\
\end{array}
\right.
\]
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.
+First versions of this simulation just covered the limited depth region of the target in which selforganisation is observed [13,14].
+As can be seen in Fig. 3, the new version of the simulation code is able to model the whole depth region affected by the irradiation process and properly describes the fluence dependence of the amorphous phase formation.
+In Fig 3 a) only isolated amorphous cells exist in the simulation and cross-section transmission electron microscopy (XTEM) shows dark contrasts, corresponding to highly distorted regions caused by defects.
+XTEM at higher magnification [9] shows the existence of amorphous inclusions which are $3 \, nm$ in size.
+For a fluence of $2.1 \times 10^{17} cm^{-2}$ a continuous amorphous layer is formed (Fig. 3b).
+The simulation shows a broader continuous layer than observed experimentally.
+However dark contrasts below the continuous layer in the XTEM image of Fig. 3b) indicate a high concentration of defects and amorphous inclusions in this depth zone.
+The continuous amorphous layer together with the region showing the dark contrast has essentially the same thickness as the simulated continuous layer.
+For higher fluences (Fig. 3c) and d)) experimental and simulated data correspond to a high degree.
+The thickness of the continuous amorphous layer increases with increasing fluence.
+Next to the upper crystalline/amorphous interface, nanometric lamellar inclusions are formed which get more defined with increasing fluence, reflecting the progress of selforganisation.
+The difference in depth throughought all images is due to a deeper maximum of the used {\em SRIM} implantation profile compared to older, more accurate {\em TRIM} versions.
+
+By simulation it is possible to determine the carbon concentration in crystalline and amorphous volumes.
+This is shown in Fig. 4.
+Lamellae exist between $350$ and $400 \, nm$ and cause a fluctuation in the carbon concentration.
+This is due to the carbon diffusion, which is of great importance for the ordering process, as already pointed out in [13,14], and the complementarily arranged and alternating sequence of layers with high and low amount of amorphous regions.
+In addition, a saturation limit of carbon in c-$Si$ under the given implantation conditions can be identified between $8$ and $10 \, at. \%$, the maxima of carbon concentration in crystalline volumes.
+
+Based on above results a recipe is proposed to create thick layers with lamellar structure which might be favourable for applications.
+The starting point is a crystalline silcon target with a nearly constant carbon concentration of $10 \, at.\%$ in a $500 \, nm$ thick surface layer. This can possibly be achieved by multiple energy ($180$ to $10 \, keV$) carbon implantation at a temperature of $500 \, ^{\circ} \mathrm{C}$, preventing amorphisation [5].
+In a second step the target is irradiated at $150 \, ^{\circ} \mathrm{C}$ with $2 \, MeV$ $C^+$ ions, which have a nearly constant energy loss in the top $500 \, nm$ and do not significantly change the carbon concentration here.
+The result is displayed in Fig. 5.
+Already ordered structures appear after $100 \times 10^6$ steps corresponding to a fluence of $D=2.7 \times 10^{17} cm^{-2}$ and get more defined with increasing fluence.
+According to recent studies [15] these structures are expected to be the starting point for materials showing strong photoluminescence.
\section{Summary and conclusion}
-Ion irradiation of solids at certain implantation conditions may result in a regular ordered formation of amorphous precipitates.
+Ion irradiation of solids at certain implantation conditions may result in the formation of regularly ordered 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.
+Detailed information like the amount of carbon in amorphous and crystalline volumes is gained, shedding light on the selforganisation process.
+Finally a technique is proposed to produce thick films of ordered lamellar nanostructures.
\begin{thebibliography}{20}
\bibitem{ommen} A. H. van Ommen, Nucl. Instr. and Meth. B 39 (1989) 194.
\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}
+%\listoffigures
+
\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 Schematic explaining the selforganisation 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]{}
+\caption[1]{}
\end{center}
\label{img:tem}
\end{figure}
\begin{figure}[!h]
\begin{center}
\includegraphics[width=14cm]{modell_ng_e.eps}
-\caption[foo]{}
+\caption[2]{}
\end{center}
\label{img:model}
\end{figure}
\begin{figure}[!h]
\begin{center}
\includegraphics[width=14cm]{dosis_entwicklung_all_e.eps}
-\caption[foo]{}
+\caption[3]{}
\end{center}
\label{img:dose_cmp}
\end{figure}
\begin{figure}[!h]
\begin{center}
\includegraphics[width=14cm]{ac_cconc_ver2_e.eps}
-\caption[foo]{}
+\caption[4]{}
\end{center}
\label{img:carbon_distr}
\end{figure}
\begin{figure}[!h]
\begin{center}
\includegraphics[width=14cm]{multiple_impl_e.eps}
-\caption[foo]{}
+\caption[5]{}
\end{center}
\label{img:broad_lam}
\end{figure}
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