-\documentclass[12pt,a4paper,twoside]{article}
+\documentclass[12pt,a4paper,oneside]{article}
\usepackage{verbatim}
\usepackage[english]{babel}
\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. Stritzkeri}
+\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}
\hyphenation{}
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
\section{Model}
-A model describing the formation of nanometric selforganized ordered amorphous $SiC_x$ inclusions with increasing dose was introduced in \cite{1}. Figure \ref{1} shows the evolution into ordered lamellae with increasing dose.
+A model describing the formation of nanometric selforganized ordered amorphous $SiC_x$ inclusions was introduced in \cite{1}\cite{2}. Figure \ref{1} shows the evolution into ordered lamellae with increasing dose.
%\begin{figure}[!h]
%\begin{center}
%\end{center}
%\end{figure}
-As a result of the supersaturation of carbon atoms in silicon there is a nucleation of spherical $SiC_x$-precipitates. The almost $20\%$ lattice misfit of the diamond lattice of crystalline silicon ($c-Si$, $a=0.543 \, nm$) to the cubic polytype of $SiC$ ($3C-SiC$, $a=0.436 \, nm$) causes a large interfacial energy, which could be reduced if one of the participants exists in the amorphous phase. It has been shown \cite{1} that $SiC$ turns into the amorphous phase. In fact, amorphous silicon ($a-Si$) would recrystallize under the granted conditions due to ion beam induced recrystallization. Stoichiometric $SiC$ has a smaller atomic density than $c-Si$. The same is assumed for substoichiometric $a-SiC_x$. Hence the amorphous $SiC_x$ is anxious to expand, and as a result compressive stress is applied on the $Si$ host lattice. As the process occurs near the targets surface, the stress is relaxing in vertical direction and there is just lateral stress remaining. Thus volumes between amorphous inclusions will more likely turn into amorphous phase, as the stress aggravates the reassembly of the atoms on their lattice site, while amorphous volumes located in a crystalline neighbourhood will recrystallize in all probability. In addition carbon diffuses to the amorphous volumes in order to reduce the supersaturation of carbon in the crystalline volumes. As a consequence the amorphous volumes hold plenty of carbon.
+As a result of the supersaturation of carbon atoms in silicon there is a nucleation of spherical $SiC_x$-precipitates. The almost $20\%$ lattice misfit of the diamond lattice of crystalline silicon ($c-Si$, $a=0.543 \, nm$) to the cubic polytype of $SiC$ ($3C-SiC$, $a=0.436 \, nm$) causes a large interfacial energy, which could be reduced if one of the participants exists in the amorphous phase. It has been shown \cite{1} that $SiC$ turns into the amorphous phase. In fact, amorphous silicon ($a-Si$) would recrystallize under the granted conditions due to ion beam induced recrystallization. Stoichiometric $SiC$ has a smaller atomic density than $c-Si$. The same is 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 targets surface, the stress is relaxing in vertical direction and there is just lateral stress remaining. Thus volumes between amorphous inclusions will more likely turn into amorphous phase, as the stress hampers the reassembly of the atoms on their lattice site, while amorphous volumes located in a crystalline neighbourhood will recrystallize in all probability. In addition carbon diffuses to the amorphous volumes in order to reduce the supersaturation of carbon in the crystalline volumes. As a consequence the amorphous volumes hold plenty of carbon.
\newpage
\section{Simulation}
-Before discussing the implementation some assumptions and approximations have to be done. Figure \ref{2} shows the stopping powers and carbon concentration profile calculated by TRIM. The depth region we are interested in is between $0-300 \, nm$, the region between surface of the target and start of the continued amorphous $SiC_x$ layer (from now on called simulation window), where the nuclear stopping power and the implantation profile can be approximated by a linear function of depth. Furthermore the probability of amorphization is assumed to be proportional to the nuclear stopping power. A local probability of amorphization somewhere in the target is composed of three parts, the ballistic, carbon-induced and 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 apllied by the amorphous neighbours. Thus the probability of a crystalline volume getting amorphous can be calculated as follows,
+Before discussing the implementation some assumptions and approximations have to be made. Figure \ref{2} shows the stopping powers and carbon concentration profile calculated by TRIM. The depth region we are interested in is between $0-300 \, nm$, the region between the surface of the target and the beginning of the continuous amorphous $SiC_x$ layer (from now on called simulation window), where the nuclear stopping power and the implantation profile can be approximated by a linear function of depth. Furthermore the probability of amorphization is assumed to be proportional to the nuclear stopping power. A local probability of amorphization somewhere in the target is composed of three parts, the ballistic, carbon-induced and 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 applied by the amorphous neighbours. Thus the probability of a crystalline volume getting amorphous can be calculated as follows,
\[
p_{c \rightarrow a} = b_{ap} + a_{cp} \times c^{local}_{carbon} + \sum_{amorphous \, neighbours} \frac{a_{ap} \times c_{carbon}}{distance^2}
\]
-with $b_{ap}$, $a_{cp}$ and $a_{ap}$ being parameters of the simulation to weight the three diffrent ways of amorphization. The probability of an amorphous volume turning crystalline should behave contrary to $p_{c \rightarrow a}$ and thus is assumed to $p_{a \rightarrow c} = 1 - p_{c \rightarrow a}$.
+with $b_{ap}$, $a_{cp}$ and $a_{ap}$ being parameters of the simulation to weight the three different ways of amorphization. The probability of an amorphous volume turning crystalline should behave contrary to $p_{c \rightarrow a}$ and thus is assumed to $p_{a \rightarrow c} = 1 - p_{c \rightarrow a}$.
-For the simulation, the target is devided into $50 \times 50 \times 100$ 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-implanting process and finaly the diffusion process.
+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.
-In the first part random coordinates according to the nuclear stopping power are computed to specify the volume which is hit by an implanted carbon ion. After calculating the local probability of amorphization, another random number decides about the state of the volume. 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.
-After that random coordinates according to the implantation profile are obtained to acquire the volume where the carbon ion gets stock and the local carbon concentration increases.
+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 paramters, 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 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.
-\section{Conclusion}
+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{figure}[!h]
\begin{center}
-\includegraphics[width=13cm]{model1_e.eps}
+\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{1}
\end{center}
\end{figure}
\begin{figure}[!h]
\begin{center}
-\includegraphics[width=10cm]{2pTRIM180C.eps}
+\includegraphics[width=14cm]{2pTRIM180C.eps}
\caption[Stopping powers and concentration profile calculated by TRIM]{} \label{2}
\end{center}
\end{figure}
+\begin{figure}[!h]
+\begin{center}
+\includegraphics[width=15cm]{gitter_e.eps}
+\caption[Target devided into $64 \times 64 \times 100$ volumes with a side length of $3 \, nm$ holding state and carbon concentration]{} \label{3}
+\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}
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