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5 \usepackage[german]{babel}
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24 \begin{minipage} {.13\textwidth}
25 \includegraphics[height=11cm]{uni-logo.eps}
27 \begin{minipage} {.73\textwidth}
28 \centerline{{\Huge \bfseries Monte Carlo simulation study of a selforganisation}}
29 \centerline{{\Huge \bfseries process leading to ordered precipitate structures}}
31 \centerline{\huge\textsc {\underline{F.~Zirkelbach}}, M.~H"aberlen,
32 J.~K.~N.~Lindner, B.~Stritzker}
34 \centerline{\Large Institut f"ur Physik, Universit"at Augsburg,
35 D-86135 Augsburg, Germany}
37 \begin{minipage} {.13\textwidth}
38 \includegraphics[height=10cm]{Lehrstuhl-Logo.eps}
49 Experimentally observerd seflorganisation process at high-dose carbon
50 implantations under certain implantation conditions.}
52 \item Spherical and lamellar amorphous inclusions at the upper
55 \includegraphics[width=20cm]{k393abild1_e.eps}
57 Cross section TEM image:\\
58 $180 \, keV$ $C^+ \rightarrow Si$,
59 $T=150 \, ^{\circ} \mathrm{C}$,
60 Dose: $4.3 \times 10^{17} \, cm^{-2}$\\
61 black/white: crystalline/amorphous material\\
62 L: amorphous lamellae, S: spherical amorphous inclusions
63 \item Carbon accumulation in amorphous volumes
65 \includegraphics[width=20cm]{eftem.eps}
67 Brightfield TEM and respective EFTEM image:\\
68 $180 \, keV$ $C^+ \rightarrow Si$,
69 $T=200 \, ^{\circ} \mathrm{C}$,
70 Dose: $4.3 \times 10^{17} \, cm^{-2}$\\
71 yellow/blue: high/low concentrations of carbon
74 Observed for a number of ion/target combinations for which the
75 material undergoes drastic density change upon amorphisation.}\\
77 A. H. van Ommen, Nucl. Instr. and Meth. B 39 (1989) 194.\\
78 E. D. Specht et al., Nucl. Instr. and Meth. B 84 (1994) 323.\\
79 M. Ishimaru et al., Nucl. Instr. and Meth. B 166-167 (2000) 390.}
85 Model schematically displaying the formation of ordered lamellae
86 with increasing dose.}
89 \includegraphics[width=20cm]{modell_ng_e.eps}
92 \item supersaturation of $C$ in $c-Si$\\
93 $\rightarrow$ {\bf carbon induced} nucleation of spherical
95 \item high interfacial energy between $3C-SiC$ and $c-Si$\\
96 $\rightarrow$ {\bf amourphous} precipitates
97 \item $20 - 30\,\%$ lower silicon density of $a-SiC_x$ compared to $c-Si$\\
98 $\rightarrow$ {\bf lateral strain} (black arrows)
99 \item implantation range near surface\\
100 $\rightarrow$ {\bf ralaxation} of {\bf vertical strain component}
101 \item reduction of the carbon supersaturation in $c-Si$\\
102 $\rightarrow$ {\bf carbon diffusion} into amorphous volumina
104 \item remaining lateral strain\\
105 $\rightarrow$ {\bf strain induced} lateral amorphisation
110 \section*{Simulation}
111 \begin{minipage}[t]{0.5\textwidth}
112 {\bf Discretisation of the target}
114 \includegraphics[width=12cm]{gitter_e.eps}
118 \item divided into cells with a cube length of $3 \, nm$
119 \item periodic boundary conditions in $x$,$y$-direction
122 \begin{minipage}[t]{0.5\textwidth}
123 {\bf TRIM collision statstics}
125 \includegraphics[width=12cm]{trim_coll_e.eps}
128 \item[] $\Rightarrow$ identical depth profiles for
130 collisions per depth and nuclear stopping power
131 \item[] $\Rightarrow$ mean constant energy loss per
142 \section*{Simulation algorithm}
144 The simulation algorithm consists of the following three parts looped
145 $s$ times corresponding to a dose $D=s/(64\times64\times(3 \, nm)^2)$:}
146 \subsection*{1. Amorphisation/Recrystallisation}
148 \item random numbers distributed according to
149 the nuclear energy loss to determine the
150 volume in which a collision occurs
151 \item compute local probability for amorphisation:\\
154 \centerline{\fcolorbox[rgb]{0.,0.,0.}{1.,1.,.8}{
155 \begin{minipage}{20cm}
157 p_{c \rightarrow a}(\vec{r}) = {\color{green} p_b} + {\color{blue} p_c c_C(\vec{r})} + {\color{red} \sum_{\textrm{amorphous neighbours}} \frac{p_s c_C(\vec{r'})}{(r-r')^2}}
162 and recrystallisation:\\
165 \centerline{\fcolorbox[rgb]{0.,0.,0.}{1.,1.,.8}{
166 \begin{minipage}{20cm}
168 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{,}
171 \delta (\vec r) = \left\{
173 1 & \textrm{if volume at position $\vec r$ is amorphous} \\
174 0 & \textrm{otherwise} \\
181 \item loop for the mean amount of hits by the ion
183 Three contributions to the amorphisation process controlled by:
185 \item {\color{green} $p_b$} normal 'ballistic' amorphisation
186 \item {\color{blue} $p_c$} carbon induced amorphisation
187 \item {\color{red} $p_s$} stress enhanced amorphisation
189 \subsection*{2. Carbon incorporation}
191 \item random numbers distributed according to
192 the implantation profile to determine the
194 \item increase the amount of carbon atoms in
197 \subsection*{3. Diffusion/Sputtering}
199 \item every $d_v$ steps transfer $d_r$ of the
200 carbon atoms of crystalline volumina to
201 an amorphous neighbour volume
202 \item do the sputter routine after $n$ steps
203 corresponding to $3 \, nm$ of substrat
207 Simulation parameters $d_v$, $d_r$ and $n$ control the
208 diffusion and sputtering process.}
212 \section*{Comparison of experiment and simulation}
214 \includegraphics[width=25cm]{dosis_entwicklung_ng_e_1-2.eps}
217 \includegraphics[width=25cm]{dosis_entwicklung_ng_e_2-2.eps}
219 Simulation parameters:\\
220 $p_b=0.01$, $p_c=0.001$, $p_s=0.0001$, $d_r=0.05$, $d_v=1 \times 10^6$.
221 \\[0.7cm]{\bf Conclusion:}
223 \item Essentially conforming formation and growth of the
224 continuous amorphous layer
225 \item Lamellar precipitates and their evolution at the upper
226 a/c interface with increasing dose is reproduced
228 {\bf\color{red} Simulation is able to model the whole
229 depth region affected by the
236 \section*{Structural/compositional information}
237 \begin{minipage}[t]{0.57\textwidth}
238 \includegraphics[height=15cm=]{ac_cconc_ver2_e.eps}
240 \item Fluctuation of the carbon concentration in the
241 region of the lamellae
242 \item Saturation limit of carbon in c-$Si$ under given
243 implantation conditions between $8$ and
247 \begin{minipage}[t]{0.43\textwidth}
248 \includegraphics[height=15cm]{97_98_ng_e.eps}
250 \item Complementarily arranged and alternating sequence
251 of layers with high and low amount of amorphous
253 \item Carbon accumulation in the amorphous phase
260 Thick films of ordered lamellar structure}
261 \begin{minipage}{0.33\textwidth}
262 {\bf Prerequisites:}\\
263 Crystalline silicon target with a nearly constant carbon
264 concentration at $10 \, at. \%$ in a $500 \, nm$ thick
267 \begin{minipage}{0.65\textwidth}
269 \includegraphics[width=15cm]{multiple_impl_cp_e.eps}
274 \item multiple energy ($180$-$10 \, keV$) $C^+$ $\rightarrow$
276 \item $T=500 \, ^{\circ} \mathrm{C}$, to prevent amorphisation
279 {\bf Stiring up:}\\[0.5cm]
280 2nd $2 \, MeV$ $C^+$ $\rightarrow$ $Si$ implantation step at
281 $150 \, ^{\circ} \mathrm{C}$
283 \item This does not significantly change the carbon
284 concentration in the top $500 \, nm$
285 \item Nearly constant energy loss in the affected depth region
291 \includegraphics[width=25cm]{multiple_impl_e.eps}
294 \item Already ordered structures after $100 \times 10^6$ steps
295 corresponding to a dose of $D=2.7 \times 10^{17} cm^{-2}$
296 \item More defined structures with increasing dose
298 {\bf\color{blue} Starting point for materials showing strong\\
300 {\scriptsize Dihu Chen et al. Opt. Mater. 23 (2003) 65.}
304 \section*{Conclusions}
306 \item Observation of self-organised nanometric
307 precipitates by ion irradiation
308 \item Model proposed describing the seoforganisation
310 \item Model implemented to a Monte Carlo simulation code
311 \item Simulation is able to reproduce experimental
313 \item Modelling of the complete depth region affected
314 by the irradiation process
315 \item Precipitation process gets traceable by simulation
316 \item Detailed structural/compositional information
317 available by simulation
318 \item Recipe proposed for the formation of broad
319 distributions of lamellar structure