Abstract
}
-We developed a Monte-Carlo-Simulation code based on a simple model that tries to explain the selforganization process leading to periodic arrays of nanometric amorphous precipitates by ion irradiation. In the present work we focus on high-dose carbon implantation into silicon. Due to the compressive stress caused by amorphous $SiC_x$ on the $Si$ host lattice, which is relaxing in vertical direction as this process occurs near the target surface, preferential amorphization of the stressed regions between amorphous inclusions during continued implantation is taking place. This, together with the diffusion of carbon into the amorphous volumes, to reduce the carbon supersaturation in the crystalline volumes leads to a uniform configuration of amorphous, lamella preciptates with high carbon concentration. The simulation is able to reproduce results gained by cross-sectional TEM meassurements of high-dose carbon implanted silicon. Adjusting the simulation parameters we found a configuration matching the depth distribution and the average length of these amorphous arrays. Furthermore conditions can be specified as a necessity for the selforganization process and information about the configuration in the layers of the target, which is not easily measurable is obtained.
+We developed a Monte-Carlo-Simulation code based on a simple model that tries to explain the selforganization process leading to periodic arrays of nanometric amorphous precipitates by ion irradiation. In the present work we focus on high-dose carbon implantation into silicon. Due to the compressive stress caused by amorphous $SiC_x$ on the $Si$ host lattice, which is relaxing in vertical direction as this process occurs near the target surface, preferential amorphization of the stressed regions between amorphous inclusions during continued implantation is taking place. This, together with the diffusion of carbon into the amorphous volumes, to reduce the carbon supersaturation in the crystalline volumes leads to a uniform configuration of amorphous, lamellar preciptates with high carbon concentration. The simulation is able to reproduce results gained by cross-sectional TEM measurements of high-dose carbon implanted silicon. Adjusting the simulation parameters we found a configuration matching the depth distribution and the average length of these amorphous arrays. Furthermore conditions can be specified as a necessity for the selforganization process and information about the configuration in the layers of the target, which is not easily measurable is obtained.
\newpage
\section{Introduction}
-Formation of nanometric selforganized ordered amorphous lamella precipitates is observed at certain conditions at high-dose implantation of impurity atoms. The present work focuses on high-dose carbon implantation into silicon. Typical doses are $1-10 \times 10^{17} cm^{-2}$ with an ion energy of $180 keV$. Temperatures below $400 \, ^{\circ} \mathrm{C}$ are needed. A model describing the selforganization process will be introduced, followed by a review of the implementation of the simulation code. Results of the Monte-Carlo-Simulation will be compared to cross-sectional TEM measurements. Necessary conditions for observing lamella precipitates are named and some additional, difficult to measure information like the carbon distribution and amorphous/crystalline structure in the layers of the target were obtained.
+Formation of nanometric selforganized ordered amorphous, lamellar precipitates is observed at certain conditions at high-dose implantation of impurity atoms. The present work focuses on high-dose carbon implantation into silicon. Typical doses are $1-10 \times 10^{17} cm^{-2}$ with an ion energy of $180 keV$. Temperatures below $400 \, ^{\circ} \mathrm{C}$ are needed. A model describing the selforganization process will be introduced, followed by a review of the implementation of the simulation code. Results of the Monte-Carlo-Simulation will be compared to cross-sectional TEM measurements. Necessary conditions for observing lamellar precipitates are named and some additional, difficult to measure information like the carbon distribution and amorphous/crystalline structure in the layers of the target were obtained.
\newpage
Figure \ref{3} shows the target devided into $64 \times 64 \times 100$ volumes with a side length of $3 \, nm$. Each of it has a state (crystalline/amorphous) and keeps the local carbon concentration. The simulation algorithm consists of three parts, the amorphization/recrystallization process, the carbon-incorporation process and finally the diffusion process.
-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 of that volume $p_{c \rightarrow a}$, 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.
+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.
+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 lamella structure is starting in ($300 \, nm$) 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 lamella selforganized structures can be reproduced by the simulation.
+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 lamella structures can be specified. Figure \ref{5} shows two identical simulation cycles with diffusion in $z$-direction switched off and on. The lamella structures only appear when diffusion in $z$-direction \ldots
+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.
-TODO:\\
-- diffusion rate -> depth of lamella structures\\
-- complementary arrays of c/a precipitates for z and z+1\\
-- evt FFT bilder\\
-- summary\\
+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{Conclusion}
+\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{figure}[!h]
\begin{center}
-\includegraphics[width=15cm]{mit_ohne_diff.eps}
-\caption[Identical simulation cycles, with diffusion switched off (left) and on (right)]{} \label{5}
+\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}
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