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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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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 \item 3C-SiC wide band gap semiconductor formation
170 Motivation / Introduction
176 Supposed mechanism of the conversion of heavily carbon doped Si into SiC:
180 \begin{minipage}{3.8cm}
181 \includegraphics[width=3.7cm]{sic_prec_seq_01.eps}
184 \begin{minipage}{3.8cm}
185 \includegraphics[width=3.7cm]{sic_prec_seq_02.eps}
188 \begin{minipage}{3.8cm}
189 \includegraphics[width=3.7cm]{sic_prec_seq_03.eps}
194 \begin{minipage}{3.8cm}
195 Formation of C-Si dumbbells on regular c-Si lattice sites
198 \begin{minipage}{3.8cm}
199 Agglomeration into large clusters (embryos)\\
202 \begin{minipage}{3.8cm}
203 Precipitation of 3C-SiC + Creation of interstitials\\
208 \textrm{Silicon density: } \quad
209 5a_{SiC}=4a_{Si} \quad \Rightarrow \quad
210 \frac{n_{SiC}}{n_{Si}}=\frac{\frac{4}{a_{SiC}^3}}{\frac{8}{a_{Si}^3}}=
211 \frac{5^3}{2\cdot4^3}={\color{cyan}97,66}\,\%
215 Experimentally observed minimal diameter of precipitation: 4 - 5 nm
227 \item Microscopic description of N particle system
228 \item Analytical interaction potential
229 \item Hamilton's equations of motion as propagation rule\\
230 in 6N-dimemnsional phase space
231 \item Observables obtained by time average
238 \item Integrator: velocity verlet, timestep: $1\, fs$
239 \item Ensemble control: NVT, Berendsen thermostat, $\tau=100.0$
240 \item Potential: Tersoff-like bond order potential\\
242 E = \frac{1}{2} \sum_{i \neq j} \pot_{ij}, \quad
243 \pot_{ij} = f_C(r_{ij}) \left[ f_R(r_{ij}) + b_{ij} f_A(r_{ij}) \right]
246 {\scriptsize P. Erhart und K. Albe. Phys. Rev. B 71 (2005) 035211}
260 Interstitial experiments:
265 \item Initial configuration: $9\times9\times9$ unit cells Si
266 \item Periodic boundary conditions
268 \item Insertion of Si / C atom at
270 \item $(0,0,0)$ $\rightarrow$ {\color{red}tetrahedral}
271 \item $(-1/8,-1/8,1/8)$ $\rightarrow$ {\color{green}hexagonal}
272 \item $(-1/8,-1/8,-1/4)$, $(-1/4,-1/4,-1/4)$
273 $\rightarrow$ {\color{yellow}110 dumbbell}
274 \item random positions (critical distance check)
276 \item Relaxation time: $2\, ps$
277 \item Optional heating-up
280 \begin{picture}(0,0)(-210,-85)
281 \includegraphics[width=6cm]{unit_cell.eps}
294 SiC precipitation experiments:
296 \item Initial configuration: $31\times31\times31$ unit cells Si
297 \item Periodic boundary conditions
298 \item $T=450\, ^{\circ}C$
299 \item Steady state time: $600\, fs$
300 \item C insertion steps:
302 \item If $T=450\pm 1\, ^{\circ}C$:\\
303 Insertion of 10 atoms at random positions within $V_{ins}$
304 \item Otherwise: Annealing for another $100\, fs$
306 \item Annealing: ($T_a: 450\rightarrow 20 \, ^{\circ}C$)
308 \item If $T=T_a$: Decrease $T_a$ by $1\, ^{\circ}C$
309 \item Otherwise: Annealing for another $50\, fs$
315 \item $V_{ins}$: total simulation volume $V$
316 \item $V_{ins}$: $12\times12\times12$ SiC unit cells
317 ($\sim$ volume of minimal SiC precipitation)
318 \item $V_{ins}$: $9\times9\times9$ SiC unit cells
319 ($\sim$ volume of necessary amount of Si)
330 Si self-interstitial experiments:
335 \item $r_{cutoff}^{Si-Si}=2.96>\frac{5.43}{2}$
336 \item Bond length near $r_{cutoff} \Rightarrow$ small bond strength
344 \begin{minipage}[t]{4.0cm}
345 \underline{Tetrahedral}
347 \item $E_F=3.41\, eV$
348 \item essentialy tetrahedral\\
353 \begin{minipage}[t]{4.0cm}
354 \underline{110 dumbbell}
356 \item $E_F=4.39\, eV$
357 \item essentially 4 bonds
361 \begin{minipage}[t]{4.0cm}
362 \underline{Hexagonal}
364 \item $E_F^{\star}\approx4.48\, eV$
371 \begin{minipage}{4.3cm}
372 \includegraphics[width=3.8cm]{si_self_int_tetra_0.eps}
374 \begin{minipage}{4.3cm}
375 \includegraphics[width=3.8cm]{si_self_int_dumbbell_0.eps}
377 \begin{minipage}{4.3cm}
378 \includegraphics[width=3.8cm]{si_self_int_hexa_0.eps}
391 Si self-interstitial \underline{random insertion} experiments:
405 Carbon interstitial experiments:
411 \begin{minipage}[t]{4.0cm}
412 \underline{Tetrahedral}
414 \item $E_F=2.67\, eV$
415 \item tetrahedral bond
419 \begin{minipage}[t]{4.0cm}
420 \underline{110 dumbbell}
422 \item $E_F=1.76\, eV$
423 \item C forms 3 bonds
427 \begin{minipage}[t]{4.0cm}
428 \underline{Hexagonal}
430 \item $E_F^{\star}\approx5.6\, eV$
437 \begin{minipage}{4.3cm}
438 \includegraphics[width=3.8cm]{c_in_si_int_tetra_0.eps}
440 \begin{minipage}{4.3cm}
441 \includegraphics[width=3.8cm]{c_in_si_int_dumbbell_0.eps}
443 \begin{minipage}{4.3cm}
444 \includegraphics[width=3.8cm]{c_in_si_int_hexa_0.eps}
457 Carbon \underline{random insertion} experiments:
471 SiC-precipitation experiments: