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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}
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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\\
190 \textrm{Silicon density: } \quad
191 5a_{SiC}=4a_{Si} \quad \Rightarrow \quad
192 \frac{n_{SiC}}{n_{Si}}=\frac{\frac{4}{a_{SiC}^3}}{\frac{8}{a_{Si}^3}}=
193 \frac{5^3}{2\cdot4^3}={\color{cyan}97,66}\,\%
197 Experimentally observed minimal diameter of precipitation: 4 - 5 nm
209 \item Microscopic description of N particle system
210 \item Analytical interaction potential
211 \item Hamilton's equations of motion as propagation rule\\
212 in 6N-dimemnsional phase space
213 \item Observables obtained by time average
220 \item Integrator: velocity verlet, timestep: $1\, fs$
221 \item Ensemble control: NVT, Berendsen thermostat, $\tau=100.0$
222 \item Potential: Tersoff-like bond order potential\\
224 E = \frac{1}{2} \sum_{i \neq j} \pot_{ij}, \quad
225 \pot_{ij} = f_C(r_{ij}) \left[ f_R(r_{ij}) + b_{ij} f_A(r_{ij}) \right]
228 {\scriptsize P. Erhart und K. Albe. Phys. Rev. B 71 (2005) 035211}
242 Interstitial experiments:
247 \item Initial configuration: $9\times9\times9$ unit cells Si
248 \item Periodic boundary conditions
250 \item Insertion of Si / C atom at
252 \item $(0,0,0)$ $\rightarrow$ {\color{red}tetrahedral}
253 \item $(-1/8,-1/8,1/8)$ $\rightarrow$ {\color{green}hexagonal}
254 \item $(-1/8,-1/8,-1/4)$, $(-1/4,-1/4,-1/4)$
255 $\rightarrow$ {\color{yellow}110 dumbbell}
256 \item random positions (critical distance check)
258 \item Relaxation time: $2\, ps$
259 \item Optional heating-up
262 \begin{picture}(0,0)(-210,-85)
263 \includegraphics[width=6cm]{unit_cell.eps}
276 SiC precipitation experiments:
278 \item Initial configuration: $31\times31\times31$ unit cells Si
279 \item Periodic boundary conditions
280 \item $T=450\, ^{\circ}C$
281 \item Steady state time: $600\, fs$
282 \item C insertion steps:
284 \item If $T=450\pm 1\, ^{\circ}C$:\\
285 Insertion of 10 atoms at random positions within $V_{ins}$
286 \item Otherwise: Annealing for another $100\, fs$
288 \item Annealing: ($T_a: 450\rightarrow 20 \, ^{\circ}C$)
290 \item If $T=T_a$: Decrease $T_a$ by $1\, ^{\circ}C$
291 \item Otherwise: Annealing for another $50\, fs$
297 \item $V_{ins}$: total simulation volume $V$
298 \item $V_{ins}$: $12\times12\times12$ SiC unit cells
299 ($\sim$ volume of minimal SiC precipitation)
300 \item $V_{ins}$: $9\times9\times9$ SiC unit cells
301 ($\sim$ volume of necessary amount of Si)
312 Si self-interstitial experiments:
317 \item $r_{cutoff}^{Si-Si}=2.96>\frac{5.43}{2}$
318 \item Bond length near $r_{cutoff} \Rightarrow$ small bond strength
326 \begin{minipage}[t]{4.0cm}
327 \underline{Tetrahedral}
329 \item $E_F=3.41\, eV$
330 \item essentialy tetrahedral\\
335 \begin{minipage}[t]{4.0cm}
336 \underline{110 dumbbell}
338 \item $E_F=4.39\, eV$
339 \item essentially 4 bonds
343 \begin{minipage}[t]{4.0cm}
344 \underline{Hexagonal}
346 \item $E_F^{\star}=4.48\, eV$
353 \begin{minipage}{4.3cm}
354 \includegraphics[width=3.8cm]{si_self_int_tetra_0.eps}
356 \begin{minipage}{4.3cm}
357 \includegraphics[width=3.8cm]{si_self_int_dumbbell_0.eps}
359 \begin{minipage}{4.3cm}
360 \includegraphics[width=3.8cm]{si_self_int_hexa_0.eps}
373 Si self-interstitial \underline{random insertion} experiments:
387 Carbon interstitial experiments:
393 \begin{minipage}[t]{4.0cm}
394 \underline{Tetrahedral}
396 \item $E_F=2.67\, eV$
397 \item tetrahedral bond
401 \begin{minipage}[t]{4.0cm}
402 \underline{110 dumbbell}
404 \item $E_F=1.76\, eV$
405 \item C forms 3 bonds
409 \begin{minipage}[t]{4.0cm}
410 \underline{Hexagonal}
412 \item $E_F\sim5.6\, eV$
419 \begin{minipage}{4.3cm}
420 \includegraphics[width=3.8cm]{c_in_si_int_tetra_0.eps}
422 \begin{minipage}{4.3cm}
423 \includegraphics[width=3.8cm]{c_in_si_int_dumbbell_0.eps}
425 \begin{minipage}{4.3cm}
426 \includegraphics[width=3.8cm]{c_in_si_int_hexa_0.eps}
439 Carbon \underline{random insertion} experiments:
453 SiC-precipitation experiments: