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64 Molecular dynamics simulation study\\
65 of the silicon carbide precipitation process
70 \textsc{\small \underline{F. Zirkelbach}$^1$, J. K. N. Lindner$^1$,
71 K. Nordlund$^2$, B. Stritzker$^1$}\\
75 \begin{minipage}{2.0cm}
77 \includegraphics[height=1.6cm]{uni-logo.eps}
80 \begin{minipage}{8.0cm}
83 $^1$ Experimentalphysik IV, Institut f"ur Physik,\\
84 Universit"at Augsburg, Universit"atsstr. 1,\\
85 D-86135 Augsburg, Germany
89 \begin{minipage}{2.3cm}
91 \includegraphics[height=1.5cm]{Lehrstuhl-Logo.eps}
97 \begin{minipage}{4.0cm}
99 \includegraphics[height=1.6cm]{logo_eng.eps}
102 \begin{minipage}{8.0cm}
105 $^2$ Accelerator Laboratory, Department of Physical Sciences,\\
106 University of Helsinki, Pietari Kalmink. 2,\\
107 00014 Helsinki, Finland
120 Molecular dynamics simulation study\\
121 of the silicon carbide precipitation process
134 \item Motivation / Introduction
135 \item Molecular dynamics simulation details
137 \item Integrator, potential, ensemble control
138 \item Simulation sequence
140 \item Simulation results
142 \item Interstitials in silicon
143 \item SiC-precipitation experiments
145 \item Conclusion / Outlook
154 Motivation / Introduction
159 Reasons for investigating C in Si:
162 \item 3C-SiC wide band gap semiconductor formation
163 \item Strained Si (no precipitation wanted!)
170 \begin{minipage}{8cm}
174 \item {\color{yellow}fcc} $+$
175 \item {\color{gray}fcc shifted $1/4$ of volume diagonal}
177 \item Lattice constants: $4a_{Si}\approx5a_{SiC}$
178 \item Silicon density:
180 \frac{n_{SiC}}{n_{Si}}=
181 \frac{4/a_{SiC}^3}{8/a_{Si}^3}=
182 \frac{5^3}{2\cdot4^3}={\color{cyan}97,66}\,\%
187 \begin{minipage}{4cm}
188 \includegraphics[width=4cm]{sic_unit_cell.eps}
197 Motivation / Introduction
203 Supposed mechanism of the conversion of heavily carbon doped Si into SiC:
207 \begin{minipage}{3.8cm}
208 \includegraphics[width=3.7cm]{sic_prec_seq_01.eps}
211 \begin{minipage}{3.8cm}
212 \includegraphics[width=3.7cm]{sic_prec_seq_02.eps}
215 \begin{minipage}{3.8cm}
216 \includegraphics[width=3.7cm]{sic_prec_seq_03.eps}
221 \begin{minipage}{3.8cm}
222 Formation of C-Si dumbbells on regular c-Si lattice sites
225 \begin{minipage}{3.8cm}
226 Agglomeration into large clusters (embryos)\\
229 \begin{minipage}{3.8cm}
230 Precipitation of 3C-SiC + Creation of interstitials\\
235 Experimentally observed:
237 \item Minimal diameter of precipitation: 4 - 5 nm
238 \item (hkl)-planes identical for Si and SiC
253 \item Microscopic description of N particle system
254 \item Analytical interaction potential
255 \item Hamilton's equations of motion as propagation rule\\
256 in 6N-dimensional phase space
257 \item Observables obtained by time average
264 \item Integrator: Velocity Verlet, timestep: $1\, fs$
265 \item Ensemble control: NVT, Berendsen thermostat, $\tau=100.0$
266 \item Potential: Tersoff-like bond order potential\\
268 E = \frac{1}{2} \sum_{i \neq j} \pot_{ij}, \quad
269 \pot_{ij} = f_C(r_{ij}) \left[ f_R(r_{ij}) + b_{ij} f_A(r_{ij}) \right]
272 {\scriptsize P. Erhart und K. Albe. Phys. Rev. B 71 (2005) 035211}
276 \begin{picture}(0,0)(-240,-70)
277 \includegraphics[width=5cm]{tersoff_angle.eps}
290 Interstitial experiments:
295 \item Initial configuration: $9\times9\times9$ unit cells Si
296 \item Periodic boundary conditions
298 \item Insertion of Si / C atom at
300 \item $(0,0,0)$ $\rightarrow$ {\color{red}tetrahedral}
301 \item $(-1/8,-1/8,1/8)$ $\rightarrow$ {\color{green}hexagonal}
302 \item $(-1/8,-1/8,-1/4)$, $(-1/4,-1/4,-1/4)$\\
303 $\rightarrow$ {\color{yellow}110 dumbbell}
304 \item random positions (critical distance check)
306 \item Relaxation time: $2\, ps$
307 \item Optional heating-up
310 \begin{picture}(0,0)(-210,-45)
311 \includegraphics[width=6cm]{unit_cell.eps}
324 SiC precipitation experiments:
326 \item Initial configuration: $31\times31\times31$ unit cells Si
327 \item Periodic boundary conditions
328 \item $T=450\, ^{\circ}C$
329 \item Steady state time: $600\, fs$
330 \item C insertion steps:
332 \item If $T=450\pm 1\, ^{\circ}C$:\\
333 Insertion of 10 atoms at random positions within $V_{ins}$
334 \item Otherwise: Annealing for another $100\, fs$
336 \item Annealing: ($T_a: 450\rightarrow 20 \, ^{\circ}C$)
338 \item If $T=T_a$: Decrease $T_a$ by $1\, ^{\circ}C$
339 \item Otherwise: Annealing for another $50\, fs$
345 \item $V_{ins}$: total simulation volume $V$
346 \item $V_{ins}$: $12\times12\times12$ SiC unit cells
347 ($\sim$ volume of minimal SiC precipitation)
348 \item $V_{ins}$: $9\times9\times9$ SiC unit cells
349 ($\sim$ volume of necessary amount of Si)
360 Si self-interstitial experiments:
365 \item $r_{cutoff}^{Si-Si}=2.96>\frac{5.43}{2}$
366 \item Bond length near $r_{cutoff} \Rightarrow$ small bond strength
374 \begin{minipage}[t]{4.0cm}
375 \underline{Tetrahedral}
377 \item $E_F=3.41\, eV$
378 \item essentialy tetrahedral\\
383 \begin{minipage}[t]{4.0cm}
384 \underline{110 dumbbell}
386 \item $E_F=4.39\, eV$
387 \item essentially 4 bonds
391 \begin{minipage}[t]{4.0cm}
392 \underline{Hexagonal}
394 \item $E_F^{\star}\approx4.48\, eV$
401 \begin{minipage}{4.3cm}
402 \includegraphics[width=3.8cm]{si_self_int_tetra_0.eps}
404 \begin{minipage}{4.3cm}
405 \includegraphics[width=3.8cm]{si_self_int_dumbbell_0.eps}
407 \begin{minipage}{4.3cm}
408 \includegraphics[width=3.8cm]{si_self_int_hexa_0.eps}
421 Si self-interstitial \underline{random insertion} experiments:
435 Carbon interstitial experiments:
441 \begin{minipage}[t]{4.0cm}
442 \underline{Tetrahedral}
444 \item $E_F=2.67\, eV$
445 \item tetrahedral bond
449 \begin{minipage}[t]{4.0cm}
450 \underline{110 dumbbell}
452 \item $E_F=1.76\, eV$
453 \item C forms 3 bonds
457 \begin{minipage}[t]{4.0cm}
458 \underline{Hexagonal}
460 \item $E_F^{\star}\approx5.6\, eV$
467 \begin{minipage}{4.3cm}
468 \includegraphics[width=3.8cm]{c_in_si_int_tetra_0.eps}
470 \begin{minipage}{4.3cm}
471 \includegraphics[width=3.8cm]{c_in_si_int_dumbbell_0.eps}
473 \begin{minipage}{4.3cm}
474 \includegraphics[width=3.8cm]{c_in_si_int_hexa_0.eps}
487 Carbon \underline{random insertion} experiments:
501 SiC-precipitation experiments: