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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
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 Amourphous} 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 statstics}
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 \subsection*{1. Amorphisation/Recrystallisation}
156 \item random numbers distributed according to
157 the nuclear energy loss to determine the
158 volume in which a collision occurs
159 \item compute local probability for amorphisation:\\
162 \centerline{\fcolorbox[rgb]{0.,0.,0.}{1.,1.,.8}{
163 \begin{minipage}{20cm}
165 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}}
170 and recrystallisation:\\
173 \centerline{\fcolorbox[rgb]{0.,0.,0.}{1.,1.,.8}{
174 \begin{minipage}{20cm}
176 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{,}
179 \delta (\vec r) = \left\{
181 1 & \textrm{if volume at position $\vec r$ is amorphous} \\
182 0 & \textrm{otherwise} \\
189 \item loop for the mean amount of hits by the ion
191 Three contributions to the amorphisation process controlled by:
193 \item {\color{green} $p_b$} normal 'ballistic' amorphisation
194 \item {\color{blue} $p_c$} carbon induced amorphisation
195 \item {\color{red} $p_s$} stress enhanced amorphisation
197 \subsection*{2. Carbon incorporation}
199 \item random numbers distributed according to
200 the implantation profile to determine the
202 \item increase the amount of carbon atoms in
205 \subsection*{3. Diffusion/Sputtering}
207 \item every $d_v$ steps transfer of a fraction $d_r$
208 of carbon atoms from crystalline volumina to
209 an amorphous neighbour volume
210 \item remove $3 \, nm$ surface layer after $n$ loops,
211 shift remaining cells $3 \, nm$ up and insert
212 an empty, crystalline $3 \, nm$ bottom layer
214 \begin{picture}(0,0)(+40,-32)
215 \includegraphics[height=39.2cm]{loop-arrow_ver2.eps}
218 Simulation parameters $d_v$, $d_r$ and $n$ control the
219 diffusion and sputtering process.}
223 \section*{Comparison of experiment and simulation}
225 \includegraphics[width=25cm]{dosis_entwicklung_ng_e_1-2.eps}
228 \includegraphics[width=25cm]{dosis_entwicklung_ng_e_2-2.eps}
230 Simulation parameters:\\
231 $p_b=0.01$, $p_c=0.001 \times (3 \, nm)^3$,
232 $p_s=0.0001 \times (3 \, nm)^5$, $d_r=0.05$, $d_v=1 \times 10^6$.
233 \\[0.7cm]{\bf Conclusion:}
235 \item Simulation in good agreement with experimentally observed
236 formation and growth of the continuous amorphous layer
237 \item Lamellar precipitates and their evolution at the upper
238 a/c interface with increasing dose is reproduced
240 {\bf\color{red} Simulation is able to model the whole
241 depth region affected by the
248 \section*{Structural/compositional information}
249 \begin{minipage}[t]{0.57\textwidth}
250 \includegraphics[height=15cm=]{ac_cconc_ver2_e.eps}
252 \item Fluctuation of the carbon concentration in the
253 region of the lamellae
254 \item Saturation limit of carbon in c-$Si$ under given
255 implantation conditions between $8$ and
259 \begin{minipage}[t]{0.43\textwidth}
260 \includegraphics[height=15cm]{97_98_ng_e.eps}
261 %\includegraphics[height=13cm]{gitter_e.eps}
262 %\includegraphics[height=15cm=]{test_foo.eps}
264 \item Complementarily arranged and alternating sequence
265 of layers with high and low amount of amorphous
267 \item Carbon accumulation in the amorphous phase
273 \section*{Recipe for thick films of ordered lamellae}
274 \begin{minipage}{0.33\textwidth}
275 {\bf Prerequisites:}\\
276 Crystalline silicon target with a nearly constant carbon
277 concentration at $10 \, at. \%$ in a $500 \, nm$ thick
280 \begin{minipage}{0.65\textwidth}
282 \includegraphics[width=15cm]{multiple_impl_cp_e.eps}
287 \item Multiple energy ($180$-$10 \, keV$) $C^+$ $\rightarrow$
289 \item $T_i=500 \, ^{\circ} \mathrm{C}$, to prevent amorphisation
292 {\bf Stirring up:}\\[0.5cm]
293 $2 \, MeV$ $C^+$ $\rightarrow$ $Si$ irradiation step at
294 $150 \, ^{\circ} \mathrm{C}$
296 \item This does not significantly change the carbon
297 concentration in the top $500 \, nm$
298 \item Nearly constant nuclear energy loss in the top $700 \, nm$
305 \includegraphics[width=25cm]{multiple_impl_e_ver2.eps}
308 \item Already ordered structures after $100 \times 10^6$ steps
309 corresponding to a dose of $D=2.7 \times 10^{17} cm^{-2}$
310 \item More defined structures with increasing dose
312 {\bf\color{blue} Starting point for materials showing strong
314 {\scriptsize Dihu Chen et al. Opt. Mater. 23 (2003) 65.}
318 \section*{Conclusions}
320 \item Observation of selforganised nanometric
321 precipitates by ion irradiation
322 \item Model proposed describing the selforganisation
324 \item Model implemented in a Monte Carlo simulation code
325 \item Modelling of the complete depth region affected
326 by the irradiation process
327 \item Simulation is able to reproduce entire amorphous
329 \item Precipitation process gets traceable by simulation
330 \item Detailed structural/compositional information
331 available by simulation
332 \item Recipe proposed for the formation of thick films
333 of lamellar structure
338 %\section*{Literature}
341 F. Zirkelbach, M. H"aberlen, J. K. N. Lindner,
342 B. Stritzker. Comp. Mater. Sci. 33 (2005) 310.\\
343 F. Zirkelbach, M. H"aberlen, J. K. N. Lindner,
344 B. Stritzker. Nucl. Instr. and Meth. B 242 (2006) 679.}