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3 \usepackage{graphicx,amsmath,amssymb}
4 \graphicspath{{../img/}}
5 \usepackage[german]{babel}
9 \hyphenation{pho-to-lu-mi-nescence}
11 % Fliessenden Hintergrund von RGB-Farbe 1. .98 .98 nach 1. .85 .85
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14 % Achtung Werte unter .8 verbrauchen zu viel Tinte!!!
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20 % Groesse der einzelnen Spalten als Anteil der Gesamt-Textbreite
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26 \begin{minipage} {.13\textwidth}
27 \includegraphics[height=11cm]{uni-logo.eps}
29 \begin{minipage} {.73\textwidth}
30 \centerline{{\Huge \bfseries Monte Carlo simulation study of a selforganisation}}
31 \centerline{{\Huge \bfseries process leading to ordered precipitate structures}}
33 \centerline{\huge\textsc {\underline{F.~Zirkelbach}}, M.~H"aberlen,
34 J.~K.~N.~Lindner, B.~Stritzker}
36 \centerline{\Large Institut f"ur Physik, Universit"at Augsburg,
37 D-86135 Augsburg, Germany}
39 \begin{minipage} {.13\textwidth}
40 \includegraphics[height=10cm]{Lehrstuhl-Logo.eps}
51 Experimentally observerd seflorganisation process at high-dose carbon
52 implantations under certain implantation conditions.}
54 \item Regularly spaced, nanometric spherical and lamellar
55 amorphous inclusions at the upper a/c interface
57 \includegraphics[width=20cm]{k393abild1_e.eps}
59 Cross-section TEM bright-field image:\\
60 $180 \, keV$ $C^+ \rightarrow Si$,
61 $T_i=150 \, ^{\circ} \mathrm{C}$,
62 Dose: $4.3 \times 10^{17} \, cm^{-2}$\\
63 Amorphous inclusions appear white on darker backgrounds\\
64 L: amorphous lamellae, S: spherical amorphous inclusions
65 \item Carbon accumulation in amorphous volumes
67 \includegraphics[width=20cm]{eftem.eps}
69 Bright-field TEM image and respective EFTEM $C$ map:\\
70 $180 \, keV$ $C^+ \rightarrow Si$,
71 $T_i=200 \, ^{\circ} \mathrm{C}$,
72 Dose: $4.3 \times 10^{17} \, cm^{-2}$\\
73 yellow/blue: high/low concentrations of carbon
76 Similarly ordered precipitate nanostructures also
77 observed for a number of ion/target combinations for which the
78 material undergoes drastic density change upon amorphisation.}\\
80 A. H. van Ommen, Nucl. Instr. and Meth. B 39 (1989) 194.\\
81 E. D. Specht et al., Nucl. Instr. and Meth. B 84 (1994) 323.\\
82 M. Ishimaru et al., Nucl. Instr. and Meth. B 166-167 (2000) 390.}
88 Model schematically displaying the formation of ordered lamellae
89 with increasing dose.}
92 \includegraphics[width=20cm]{modell_ng_e.eps}
95 \item Supersaturation of $C$ in $c-Si$\\
96 $\rightarrow$ {\bf Carbon induced} nucleation of spherical
98 \item High interfacial energy between $3C-SiC$ and $c-Si$\\
99 $\rightarrow$ {\bf Amourphous} precipitates
100 \item $20 - 30\,\%$ lower silicon density of $a-SiC_x$ compared to $c-Si$\\
101 $\rightarrow$ {\bf Lateral strain} (black arrows)
102 \item Implantation range near surface\\
103 $\rightarrow$ {\bf Ralaxation} of {\bf vertical strain component}
104 \item Reduction of the carbon supersaturation in $c-Si$\\
105 $\rightarrow$ {\bf Carbon diffusion} into amorphous volumina
107 \item Remaining lateral strain\\
108 $\rightarrow$ {\bf Strain enhanced} lateral amorphisation
109 \item Absence of crystalline neighbours (structural information)\\
110 $\rightarrow$ {\bf Stabilisation} of amorphous inclusions
111 {\bf against recrystallisation}
116 \section*{Simulation}
117 \begin{minipage}[t]{0.5\textwidth}
118 {\bf Discretisation of the target}
120 \includegraphics[width=12cm]{gitter_e.eps}
124 \item divided into cells with a cube length of $3 \, nm$
125 \item periodic boundary conditions in $x$,$y$-direction
128 \begin{minipage}[t]{0.5\textwidth}
129 {\bf TRIM collision statstics}
131 \includegraphics[width=12cm]{trim_coll_e.eps}
134 \item[] $\Rightarrow$ identical depth profiles for
136 collisions per depth and nuclear stopping power
137 \item[] $\Rightarrow$ mean constant energy loss per
148 \section*{Simulation algorithm}
150 The simulation algorithm consists of the following three parts looped
151 $s$ times corresponding to a dose
152 $D=s/(64\times64\times(3 \, nm)^2)$:}
153 \subsection*{1. Amorphisation/Recrystallisation}
155 \item random numbers distributed according to
156 the nuclear energy loss to determine the
157 volume in which a collision occurs
158 \item compute local probability for amorphisation:\\
161 \centerline{\fcolorbox[rgb]{0.,0.,0.}{1.,1.,.8}{
162 \begin{minipage}{20cm}
164 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}}
169 and recrystallisation:\\
172 \centerline{\fcolorbox[rgb]{0.,0.,0.}{1.,1.,.8}{
173 \begin{minipage}{20cm}
175 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{,}
178 \delta (\vec r) = \left\{
180 1 & \textrm{if volume at position $\vec r$ is amorphous} \\
181 0 & \textrm{otherwise} \\
188 \item loop for the mean amount of hits by the ion
190 Three contributions to the amorphisation process controlled by:
192 \item {\color{green} $p_b$} normal 'ballistic' amorphisation
193 \item {\color{blue} $p_c$} carbon induced amorphisation
194 \item {\color{red} $p_s$} stress enhanced amorphisation
196 \subsection*{2. Carbon incorporation}
198 \item random numbers distributed according to
199 the implantation profile to determine the
201 \item increase the amount of carbon atoms in
204 \subsection*{3. Diffusion/Sputtering}
206 \item every $d_v$ steps transfer of a fraction $d_r$
207 of carbon atoms from crystalline volumina to
208 an amorphous neighbour volume
209 \item remove $3 \, nm$ surface layer after $n$ loops,
210 shift remaining cells $3 \, nm$ up and insert
211 an empty, crystalline $3 \, nm$ bottom layer
213 \begin{picture}(0,0)(+40,-32)
214 \includegraphics[height=39.2cm]{loop-arrow.eps}
217 Simulation parameters $d_v$, $d_r$ and $n$ control the
218 diffusion and sputtering process.}
222 \section*{Comparison of experiment and simulation}
224 \includegraphics[width=25cm]{dosis_entwicklung_ng_e_1-2.eps}
227 \includegraphics[width=25cm]{dosis_entwicklung_ng_e_2-2.eps}
229 Simulation parameters:\\
230 $p_b=0.01$, $p_c=0.001 \times (3 \, nm)^3$,
231 $p_s=0.0001 \times (3 \, nm)^5$, $d_r=0.05$, $d_v=1 \times 10^6$.
232 \\[0.7cm]{\bf Conclusion:}
234 \item Simulation in good agreement with experimentally observed
235 formation and growth of the continuous amorphous layer
236 \item Lamellar precipitates and their evolution at the upper
237 a/c interface with increasing dose is reproduced
239 {\bf\color{red} Simulation is able to model the whole
240 depth region affected by the
247 \section*{Structural/compositional information}
248 \begin{minipage}[t]{0.57\textwidth}
249 \includegraphics[height=15cm=]{ac_cconc_ver2_e.eps}
251 \item Fluctuation of the carbon concentration in the
252 region of the lamellae
253 \item Saturation limit of carbon in c-$Si$ under given
254 implantation conditions between $8$ and
258 \begin{minipage}[t]{0.43\textwidth}
259 \includegraphics[height=15cm]{97_98_ng_e.eps}
260 %\includegraphics[height=13cm]{gitter_e.eps}
261 %\includegraphics[height=15cm=]{test_foo.eps}
263 \item Complementarily arranged and alternating sequence
264 of layers with high and low amount of amorphous
266 \item Carbon accumulation in the amorphous phase
272 \section*{Recipe for thick films of ordered lamellae}
273 \begin{minipage}{0.33\textwidth}
274 {\bf Prerequisites:}\\
275 Crystalline silicon target with a nearly constant carbon
276 concentration at $10 \, at. \%$ in a $500 \, nm$ thick
279 \begin{minipage}{0.65\textwidth}
281 \includegraphics[width=15cm]{multiple_impl_cp_e.eps}
286 \item Multiple energy ($180$-$10 \, keV$) $C^+$ $\rightarrow$
288 \item $T_i=500 \, ^{\circ} \mathrm{C}$, to prevent amorphisation
291 {\bf Stirring up:}\\[0.5cm]
292 $2 \, MeV$ $C^+$ $\rightarrow$ $Si$ irradiation step at
293 $150 \, ^{\circ} \mathrm{C}$
295 \item This does not significantly change the carbon
296 concentration in the top $500 \, nm$
297 \item Nearly constant nuclear energy loss in the top $700 \, nm$
304 \includegraphics[width=25cm]{multiple_impl_e_ver2.eps}
307 \item Already ordered structures after $100 \times 10^6$ steps
308 corresponding to a dose of $D=2.7 \times 10^{17} cm^{-2}$
309 \item More defined structures with increasing dose
311 {\bf\color{blue} Starting point for materials showing strong
313 {\scriptsize Dihu Chen et al. Opt. Mater. 23 (2003) 65.}
317 \section*{Conclusions}
319 \item Observation of selforganised nanometric
320 precipitates by ion irradiation
321 \item Model proposed describing the selforganisation
323 \item Model implemented in a Monte Carlo simulation code
324 \item Modelling of the complete depth region affected
325 by the irradiation process
326 \item Simulation is able to reproduce entire amorphous
328 \item Precipitation process gets traceable by simulation
329 \item Detailed structural/compositional information
330 available by simulation
331 \item Recipe proposed for the formation of thick films
332 of lamellar structure
337 \section*{Publications}
339 F. Zirkelbach, M. H"aberlen, J. K. N. Lindner,
340 B. Stritzker. Comp. Mater. Sci. 33 (2005) 310.\\
341 F. Zirkelbach, M. H"aberlen, J. K. N. Lindner,
342 B. Stritzker. Nucl. Instr. and Meth. B 242 (2006) 679.}