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3 \usepackage{graphicx,amsmath,amssymb}
4 \graphicspath{{../img/}}
5 \usepackage[german]{babel}
9 \hyphenation{pho-to-lu-mi-nescence in-clu-sions}
11 % Fliessenden Hintergrund von RGB-Farbe 1. .98 .98 nach 1. .85 .85
12 % und wieder nach 1. .98 .98 (1. .85 .85 wird nach 0.1=10% des Hinter-
14 % Achtung Werte unter .8 verbrauchen zu viel Tinte!!!
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21 % Groesse der einzelnen Spalten als Anteil der Gesamt-Textbreite
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27 \begin{minipage} {.13\textwidth}
28 \includegraphics[height=11cm]{uni-logo.eps}
30 \begin{minipage} {.73\textwidth}
31 \centerline{{\Huge \bfseries Monte Carlo simulation study of a selforganisation}}
32 \centerline{{\Huge \bfseries process leading to ordered precipitate structures}}
34 \centerline{\huge\textsc {\underline{F.~Zirkelbach}}, M.~H"aberlen,
35 J.~K.~N.~Lindner, B.~Stritzker}
37 \centerline{\Large Institut f"ur Physik, Universit"at Augsburg,
38 D-86135 Augsburg, Germany}
40 \begin{minipage} {.13\textwidth}
41 \includegraphics[height=10cm]{Lehrstuhl-Logo.eps}
52 Experimentally observed selforganisation process at high-dose carbon
53 implantations under certain implantation conditions.}
55 \item Regularly spaced, nanometric spherical and lamellar
56 amorphous inclusions at the upper a/c interface
58 \includegraphics[width=20cm]{k393abild1_e.eps}
60 Cross-section TEM bright-field images:\\
61 $180 \, keV$ $C^+ \rightarrow Si$,
62 $T_i=150 \, ^{\circ} \mathrm{C}$,
63 Dose: $4.3 \times 10^{17} \, cm^{-2}$\\
64 Amorphous inclusions appear white on darker backgrounds\\
65 L: amorphous lamellae, S: spherical amorphous inclusions
66 \item Carbon accumulation in amorphous volumes
68 \includegraphics[width=20cm]{eftem.eps}
70 Bright-field TEM image and respective EFTEM $C$ map:\\
71 $180 \, keV$ $C^+ \rightarrow Si$,
72 $T_i=200 \, ^{\circ} \mathrm{C}$,
73 Dose: $4.3 \times 10^{17} \, cm^{-2}$\\
74 yellow/blue: high/low concentrations of carbon
77 Similarly ordered precipitate nanostructures also
78 observed for a number of ion/target combinations for which the
79 material undergoes drastic density change upon amorphisation.}\\
81 A. H. van Ommen, Nucl. Instr. and Meth. B 39 (1989) 194.\\
82 E. D. Specht et al., Nucl. Instr. and Meth. B 84 (1994) 323.\\
83 M. Ishimaru et al., Nucl. Instr. and Meth. B 166-167 (2000) 390.}
89 Model schematically displaying the formation of ordered lamellae
90 with increasing dose.}
93 \includegraphics[width=20cm]{modell_ng_e.eps}
96 \item Supersaturation of $C$ in $c-Si$\\
97 $\rightarrow$ {\bf Carbon induced} nucleation of spherical
99 \item High interfacial energy between $3C-SiC$ and $c-Si$\\
100 $\rightarrow$ {\bf Amorphous} precipitates
101 \item $20 - 30\,\%$ lower silicon density of $a-SiC_x$ compared to $c-Si$\\
102 $\rightarrow$ {\bf Lateral strain} (black arrows)
103 \item Implantation range near surface\\
104 $\rightarrow$ {\bf Relaxation} of {\bf vertical strain component}
105 \item Reduction of the carbon supersaturation in $c-Si$\\
106 $\rightarrow$ {\bf Carbon diffusion} into amorphous volumina
108 \item Remaining lateral strain\\
109 $\rightarrow$ {\bf Strain enhanced} lateral amorphisation
110 \item Absence of crystalline neighbours (structural information)\\
111 $\rightarrow$ {\bf Stabilisation} of amorphous inclusions
112 {\bf against recrystallisation}
117 \section*{Simulation}
118 \begin{minipage}[t]{0.5\textwidth}
119 {\bf Discretisation of the target}
121 \includegraphics[width=12cm]{gitter_e.eps}
125 \item divided into cells with a cube length of $3 \, nm$
126 \item periodic boundary conditions in $x$,$y$-direction
129 \begin{minipage}[t]{0.5\textwidth}
130 {\bf TRIM collision statistics}
132 \includegraphics[width=12cm]{trim_coll_e.eps}
135 \item[] $\Rightarrow$ identical depth profiles for
137 collisions per depth and nuclear stopping power
138 \item[] $\Rightarrow$ mean constant energy loss per
149 \section*{Simulation algorithm}
151 The simulation algorithm consists of the following three parts looped
152 $s$ times corresponding to a dose
153 $D=s/(64\times64\times(3 \, nm)^2)$:}\\
154 \begin{minipage}{0.10\textwidth}
155 \begin{picture}(0,0)(0,600)
156 \includegraphics[height=40.0cm]{loop-arrow_ver2.eps}
159 \begin{minipage}{0.90\textwidth}
161 \subsection*{1. Amorphisation/Recrystallisation}
163 \item random numbers distributed according to
164 the nuclear energy loss to determine the
165 volume in which a collision occurs
166 \item compute local probability for amorphisation:\\
169 \centerline{\fcolorbox[rgb]{0.,0.,0.}{1.,1.,.8}{
170 \begin{minipage}{20cm}
172 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}}
177 and recrystallisation:\\
180 \centerline{\fcolorbox[rgb]{0.,0.,0.}{1.,1.,.8}{
181 \begin{minipage}{20cm}
183 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{,}
186 \delta (\vec r) = \left\{
188 1 & \textrm{if volume at position $\vec r$ is amorphous} \\
189 0 & \textrm{otherwise} \\
196 \item loop for the mean amount of hits by the ion
198 Three contributions to the amorphisation process controlled by:
200 \item {\color{green} $p_b$} normal 'ballistic' amorphisation
201 \item {\color{blue} $p_c$} carbon induced amorphisation
202 \item {\color{red} $p_s$} stress enhanced amorphisation
204 \subsection*{2. Carbon incorporation}
206 \item random numbers distributed according to
207 the implantation profile to determine the
209 \item increase the amount of carbon atoms in
212 \subsection*{3. Diffusion/Sputtering}
214 Simulation parameters $d_v$, $d_r$ and $n$ control the
215 diffusion and sputtering process.}
217 \item every $d_v$ steps transfer of a fraction $d_r$
218 of carbon atoms from crystalline volumina to
219 an amorphous neighbour volume
220 \item remove $3 \, nm$ surface layer after $n$ loops,
221 shift remaining cells $3 \, nm$ up and insert
222 an empty, crystalline $3 \, nm$ bottom layer
228 \section*{Comparison of experiment and simulation}
230 \includegraphics[width=25cm]{dosis_entwicklung_ng_e_1-2.eps}
233 \includegraphics[width=25cm]{dosis_entwicklung_ng_e_2-2.eps}
235 Simulation parameters:\\
236 $p_b=0.01$, $p_c=0.001 \times (3 \, nm)^3$,
237 $p_s=0.0001 \times (3 \, nm)^5$, $d_r=0.05$, $d_v=1 \times 10^6$.
238 \\[0.7cm]{\bf Conclusion:}
240 \item Simulation in good agreement with experimentally observed
241 formation and growth of the continuous amorphous layer
242 \item Lamellar precipitates and their evolution at the upper
243 a/c interface with increasing dose is reproduced
245 {\bf\color{red} Simulation is able to model the whole
246 depth region affected by the
253 \section*{Structural/compositional information}
254 \begin{minipage}[t]{0.57\textwidth}
255 \includegraphics[height=15cm=]{ac_cconc_ver2_e.eps}
256 \begin{minipage}[t]{0.9\textwidth}
259 \item Fluctuation of the carbon concentration in the
260 region of the lamellae
261 \item Saturation limit of carbon in c-$Si$ under given
262 implantation conditions between $8$ and
267 \begin{minipage}[t]{0.43\textwidth}
268 \includegraphics[height=15cm]{97_98_ng_e.eps}
269 %\includegraphics[height=13cm]{gitter_e.eps}
270 %\includegraphics[height=15cm=]{test_foo.eps}
272 \item Complementarily arranged and alternating sequence
273 of layers with high and low amount of amorphous
275 \item Carbon accumulation in the amorphous phase
281 \section*{Recipe for thick films of ordered lamellae}
282 \begin{minipage}{0.33\textwidth}
283 {\bf Prerequisites:}\\
284 Crystalline silicon target with a nearly constant carbon
285 concentration at $10 \, at. \%$ in a $500 \, nm$ thick
288 \begin{minipage}{0.65\textwidth}
290 \includegraphics[width=15cm]{multiple_impl_cp_e.eps}
295 \item Multiple energy ($180$-$10 \, keV$) $C^+$ $\rightarrow$
297 \item $T_i=500 \, ^{\circ} \mathrm{C}$, to prevent amorphisation
300 {\bf Stirring up:}\\[0.5cm]
301 $2 \, MeV$ $C^+$ $\rightarrow$ $Si$ irradiation step at
302 $150 \, ^{\circ} \mathrm{C}$
304 \item This does not significantly change the carbon
305 concentration in the top $500 \, nm$
306 \item Nearly constant nuclear energy loss in the top $700 \, nm$
313 \includegraphics[width=25cm]{multiple_impl_e_ver2.eps}
316 \item Already ordered structures after $100 \times 10^6$ steps
317 corresponding to a dose of $D=2.7 \times 10^{17} cm^{-2}$
318 \item More defined structures with increasing dose
320 {\bf\color{blue} Starting point for materials showing strong
322 {\scriptsize Dihu Chen et al. Opt. Mater. 23 (2003) 65.}
326 \section*{Conclusions}
328 \item Observation of selforganised nanometric
329 precipitates by ion irradiation
330 \item Model proposed describing the selforganisation
332 \item Model implemented in a Monte Carlo simulation code
333 \item Modelling of the complete depth region affected
334 by the irradiation process
335 \item Simulation is able to reproduce entire amorphous
337 \item Precipitation process gets traceable by simulation
338 \item Detailed structural/compositional information
339 available by simulation
340 \item Recipe proposed for the formation of thick films
341 of lamellar structure
346 %\section*{Literature}
349 F. Zirkelbach, M. H"aberlen, J. K. N. Lindner,
350 B. Stritzker. Comp. Mater. Sci. 33 (2005) 310.\\
351 F. Zirkelbach, M. H"aberlen, J. K. N. Lindner,
352 B. Stritzker. Nucl. Instr. and Meth. B 242 (2006) 679.}