2 \documentclass[landscape,semhelv]{seminar}
5 \usepackage[german]{babel}
6 \usepackage[latin1]{inputenc}
7 \usepackage[T1]{fontenc}
11 \usepackage{calc} % Simple computations with LaTeX variables
12 \usepackage{caption} % Improved captions
13 \usepackage{fancybox} % To have several backgrounds
15 \usepackage{fancyhdr} % Headers and footers definitions
16 \usepackage{fancyvrb} % Fancy verbatim environments
17 \usepackage{pstricks} % PSTricks with the standard color package
20 \graphicspath{{../img/}}
23 \usepackage{semlayer} % Seminar overlays
24 \usepackage{slidesec} % Seminar sections and list of slides
26 \input{seminar.bug} % Official bugs corrections
27 \input{seminar.bg2} % Unofficial bugs corrections
35 \extraslideheight{10in}
38 % specify width and height
42 % shift it into visual area properly
43 \def\slideleftmargin{3.3cm}
44 \def\slidetopmargin{0.0cm}
46 \newcommand{\ham}{\mathcal{H}}
47 \newcommand{\pot}{\mathcal{V}}
48 \newcommand{\foo}{\mathcal{U}}
49 \newcommand{\vir}{\mathcal{W}}
52 \renewcommand\labelitemii{{\color{gray}$\bullet$}}
62 Molecular dynamics simulation study\\
63 of the silicon carbide precipitation process
68 \textsc{\small \underline{F. Zirkelbach}$^1$, J. K. N. Lindner$^1$,
69 K. Nordlund$^2$, B. Stritzker$^1$}\\
73 \begin{minipage}{2.0cm}
75 \includegraphics[height=1.6cm]{uni-logo.eps}
78 \begin{minipage}{8.0cm}
81 $^1$ Experimentalphysik IV, Institut f"ur Physik,\\
82 Universit"at Augsburg, Universit"atsstr. 1,\\
83 D-86135 Augsburg, Germany
87 \begin{minipage}{2.3cm}
89 \includegraphics[height=1.5cm]{Lehrstuhl-Logo.eps}
95 \begin{minipage}{4.0cm}
97 \includegraphics[height=1.6cm]{logo_eng.eps}
100 \begin{minipage}{8.0cm}
103 $^2$ Accelerator Laboratory, Department of Physical Sciences,\\
104 University of Helsinki, Pietari Kalmink. 2,\\
105 00014 Helsinki, Finland
118 Molecular dynamics simulation study\\
119 of the silicon carbide precipitation process
132 \item Motivation / Introduction
133 \item Molecular dynamics simulation details
135 \item Integrator, potential, ensemble control
136 \item Simulation sequence
138 \item Results gained by simulation
140 \item Interstitials in silicon
141 \item $SiC$-precipitation experiments
143 \item Conclusion / Outlook
152 Motivation / Introduction
158 Supposed mechanism of the conversion of heavily carbon doped Si into SiC:
162 \begin{minipage}{3.8cm}
163 \includegraphics[width=3.7cm]{sic_prec_seq_01.eps}
166 \begin{minipage}{3.8cm}
167 \includegraphics[width=3.7cm]{sic_prec_seq_02.eps}
170 \begin{minipage}{3.8cm}
171 \includegraphics[width=3.7cm]{sic_prec_seq_03.eps}
176 \begin{minipage}{3.8cm}
177 Formation of C-Si dumbbells on regular c-Si lattice sites
180 \begin{minipage}{3.8cm}
181 Agglomeration into large clusters (embryos)\\
184 \begin{minipage}{3.8cm}
185 Precipitation of 3C-SiC + Creation of interstitials\\
189 \[5a_{SiC}=4a_{Si} \quad \Rightarrow \quad
190 \frac{n_{SiC}}{n_{Si}}=\frac{\frac{4}{a_{SiC}^3}}{\frac{8}{a_{Si}^3}}=
191 \frac{5^3}{2\cdot4^3}=97,66\%
195 Experimentally observed minimal diameter of precipitation: 4 - 5 nm
207 \item Microscopic description of N particle system
208 \item Analytical interaction potential
209 \item Hamilton's equations of motion as propagation rule\\
210 in 6N-dimemnsional phase space
211 \item Observables obtained by time average
218 \item Integrator: velocity verlet, timestep: $1\, fs$
219 \item Ensemble control: NVT, Berendsen thermostat, $\tau=100.0$
220 \item Potential: Tersoff-like bond order potential\\
222 E = \frac{1}{2} \sum_{i \neq j} \pot_{ij}, \quad
223 \pot_{ij} = f_C(r_{ij}) \left[ f_R(r_{ij}) + b_{ij} f_A(r_{ij}) \right]
226 {\scriptsize P. Erhart und K. Albe. Phys. Rev. B 71 (2005) 035211}
240 Interstitial experiments:
245 \item Initial configuration: $9\times9\times9$ unit cells Si
246 \item Periodic boundary conditions
248 \item Insertion of Si / C atom at
250 \item $(0,0,0)$ $\rightarrow$ {\color{red}tetrahedral}
251 \item $(-1/8,-1/8,1/8)$ $\rightarrow$ {\color{green}hexagonal}
252 \item $(-1/8,-1/8,-1/4)$, $(-1/4,-1/4,-1/4)$
253 $\rightarrow$ {\color{yellow}110 dumbbell}
254 \item random positions (critical distance check)
256 \item Relaxation time: $2\, ps$
259 \begin{picture}(0,0)(-210,-65)
260 \includegraphics[width=6cm]{unit_cell.eps}
273 SiC precipitation experiments:
275 \item Initial configuration: $31\times31\times31$ unit cells Si
276 \item Periodic boundary conditions
277 \item $T=450\, ^{\circ}C$
278 \item Steady state time: $600\, fs$
279 \item C insertion steps:
281 \item If $T=450\pm 1\, ^{\circ}C$:\\
282 Insertion of 10 atoms at random positions within $V_{ins}$
283 \item Otherwise: Annealing for another $100\, fs$
285 \item Annealing: ($T_a: 450\rightarrow 20 \, ^{\circ}C$)
287 \item If $T=T_a$: Decrease $T_a$ by $1\, ^{\circ}C$
288 \item Otherwise: Annealing for another $50\, fs$
294 \item $V_{ins}$: total simulation volume $V$
295 \item $V_{ins}$: $12\times12\times12$ SiC unit cells
296 ($\sim$ volume of minimal SiC precipitation)
297 \item $V_{ins}$: $9\times9\times9$ SiC unit cells
298 ($\sim$ volume of necessary amount of Si)
309 Si self-interstitial experiments:
315 \begin{minipage}[t]{4.0cm}
316 \underline{Tetrahedral}
318 \item $E_F=3.41\, eV$
319 \item $r_{cutoff}^{Si-Si}=2.96>\frac{5.43}{2}$
323 \begin{minipage}[t]{4.0cm}
324 \underline{110 dumbbell}
326 \item $E_F=4.39\, eV$
327 \item Small bond strength
331 \begin{minipage}[t]{4.0cm}
332 \underline{Hexagonal}
334 \item $E_F=4.48\, eV$
335 \item unstable for $T\ne 0\,K$
341 \begin{minipage}{4.3cm}
342 \includegraphics[width=3.8cm]{si_self_int_tetra_0.eps}
344 \begin{minipage}{4.3cm}
345 \includegraphics[width=3.8cm]{si_self_int_dumbbell_0.eps}
347 \begin{minipage}{4.3cm}
348 \includegraphics[width=3.8cm]{si_self_int_hexa_0.eps}
367 Carbon interstitial experiments:
373 \begin{minipage}[t]{4.0cm}
374 \underline{Tetrahedral}
376 \item $E_F=2.67\, eV$
380 \begin{minipage}[t]{4.0cm}
381 \underline{110 dumbbell}
383 \item $E_F=1.76\, eV$
387 \begin{minipage}[t]{4.0cm}
388 \underline{Hexagonal}
396 \begin{minipage}{4.3cm}
397 \includegraphics[width=3.8cm]{c_in_si_int_tetra_0.eps}
399 \begin{minipage}{4.3cm}
400 \includegraphics[width=3.8cm]{c_in_si_int_dumbbell_0.eps}
402 \begin{minipage}{4.3cm}
403 %\includegraphics[width=3.8cm]{si_self_int_hexa_0.eps}
414 SiC-precipitation experiments: