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67 Molecular dynamics simulation study\\
68 of the silicon carbide precipitation process
73 \textsc{\small \underline{F. Zirkelbach}$^1$, J. K. N. Lindner$^1$,
74 K. Nordlund$^2$, B. Stritzker$^1$}\\
78 \begin{minipage}{2.0cm}
80 \includegraphics[height=1.6cm]{uni-logo.eps}
83 \begin{minipage}{8.0cm}
86 $^1$ Experimentalphysik IV, Institut f"ur Physik,\\
87 Universit"at Augsburg, Universit"atsstr. 1,\\
88 D-86135 Augsburg, Germany
92 \begin{minipage}{2.3cm}
94 \includegraphics[height=1.5cm]{Lehrstuhl-Logo.eps}
100 \begin{minipage}{4.0cm}
102 \includegraphics[height=1.6cm]{logo_eng.eps}
105 \begin{minipage}{8.0cm}
108 $^2$ Accelerator Laboratory, Department of Physical Sciences,\\
109 University of Helsinki, Pietari Kalmink. 2,\\
110 00014 Helsinki, Finland
123 Molecular dynamics simulation study\\
124 of the silicon carbide precipitation process
137 \item Motivation / Introduction
138 \item Molecular dynamics simulation details
140 \item Integrator, potential, ensemble control
141 \item Simulation sequence
143 \item Simulation results
145 \item Interstitials in silicon
146 \item SiC-precipitation experiments
148 \item Conclusion / Outlook
157 Motivation / Introduction
162 Reasons for investigating C in Si:
165 \item 3C-SiC wide band gap semiconductor formation
166 \item Strained Si (no precipitation wanted!)
173 \begin{minipage}{8cm}
177 \item {\color{yellow}fcc} $+$
178 \item {\color{gray}fcc shifted $1/4$ of volume diagonal}
180 \item Lattice constants: $4a_{Si}\approx5a_{SiC}$
181 \item Silicon density:
183 \frac{n_{SiC}}{n_{Si}}=
184 \frac{4/a_{SiC}^3}{8/a_{Si}^3}=
185 \frac{5^3}{2\cdot4^3}={\color{cyan}97,66}\,\%
190 \begin{minipage}{4cm}
191 \includegraphics[width=4cm]{sic_unit_cell.eps}
200 Motivation / Introduction
206 Supposed conversion mechanism of heavily carbon doped Si into SiC:
210 \begin{minipage}{3.8cm}
211 \includegraphics[width=3.7cm]{sic_prec_seq_01.eps}
214 \begin{minipage}{3.8cm}
215 \includegraphics[width=3.7cm]{sic_prec_seq_02.eps}
218 \begin{minipage}{3.8cm}
219 \includegraphics[width=3.7cm]{sic_prec_seq_03.eps}
224 \begin{minipage}{3.8cm}
225 Formation of C-Si dumbbells on regular c-Si lattice sites
228 \begin{minipage}{3.8cm}
229 Agglomeration into large clusters (embryos)\\
232 \begin{minipage}{3.8cm}
233 Precipitation of 3C-SiC + Creation of interstitials\\
238 Experimentally observed:
240 \item Minimal diameter of precipitation: 4 - 5 nm
241 \item (hkl)-planes identical for Si and SiC
256 \item Microscopic description of N particle system
257 \item Analytical interaction potential
258 \item Hamilton's equations of motion as propagation rule\\
259 in 6N-dimensional phase space
260 \item Observables obtained by time average
267 \item Integrator: Velocity Verlet, timestep: $1\, fs$
268 \item Ensemble control: NVT, Berendsen thermostat, $\tau=100.0$
269 \item Potential: Tersoff-like bond order potential\\
271 E = \frac{1}{2} \sum_{i \neq j} \pot_{ij}, \quad
272 \pot_{ij} = f_C(r_{ij}) \left[ f_R(r_{ij}) + b_{ij} f_A(r_{ij}) \right]
275 {\scriptsize P. Erhart und K. Albe. Phys. Rev. B 71 (2005) 035211}
279 \begin{picture}(0,0)(-240,-70)
280 \includegraphics[width=5cm]{tersoff_angle.eps}
293 Interstitial experiments:
298 \item Initial configuration: $9\times9\times9$ unit cells Si
299 \item Periodic boundary conditions
301 \item Insertion of Si / C atom at
303 \item $(0,0,0)$ $\rightarrow$ {\color{red}tetrahedral}
304 \item $(-1/8,-1/8,1/8)$ $\rightarrow$ {\color{green}hexagonal}
305 \item $(-1/8,-1/8,-1/4)$, $(-1/4,-1/4,-1/4)$\\
306 $\rightarrow$ {\color{yellow}110 dumbbell}
307 \item random positions (critical distance check)
309 \item Relaxation time: $2\, ps$
310 \item Optional heating-up
313 \begin{picture}(0,0)(-210,-45)
314 \includegraphics[width=6cm]{unit_cell.eps}
327 SiC precipitation experiments:
329 \begin{pspicture}(0,0)(12,8)
331 \rput(4.5,6.5){\rnode{init}{\psframebox{\parbox{7cm}{
333 \item Initial configuration: $31\times31\times31$ unit cells Si
334 \item Periodic boundary conditions
335 \item $T=450\, ^{\circ}C$
336 \item Equilibration of $E_{kin}$ and $E_{pot}$ for $600\, fs$
339 \rput(4.5,4.5){\rnode{tc1}{\psframebox[fillstyle=solid,fillcolor=red]{
340 $T=450\pm 1\, ^{\circ}C$}}}
341 \rput(7,3.5){\rnode{insert}{\psframebox[fillstyle=solid,fillcolor=red]{
343 Insertion of 10 atoms\\
344 at random positions}}}}
345 \rput(2,3.5){\rnode{adj1}{\psframebox[fillstyle=solid,fillcolor=red]{
347 Adjusting temperature\\
348 for another $100\, fs$}}}}
349 \rput(7,2.5){\rnode{nc}{\psframebox[fillstyle=solid,fillcolor=red]{
351 \rput(4.5,2){\rnode{tc2}{\psframebox[fillstyle=solid,fillcolor=cyan]{
353 \rput(7,1){\rnode{td}{\psframebox[fillstyle=solid,fillcolor=cyan]{
354 $T_{set}:=T_{set}-1\, ^{\circ}C$}}}
355 \rput(2,1){\rnode{adj2}{\psframebox[fillstyle=solid,fillcolor=cyan]{
357 Adjusting temperature\\
358 for another $50\, fs$}}}}
359 \rput(7,0){\rnode{tc3}{\psframebox[fillstyle=solid,fillcolor=cyan]{
360 $T_{set}=20\, ^{\circ}C$}}}
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413 \rput(11.5,7.85){{\tiny Simulation volume:
414 $31\times31\times31\, a^3_{Si}$}}
415 \rput(11.5,7.05){{\tiny Volume of minimal SiC precipitation}}
416 \rput(11.5,6.25){{\tiny Volume of necessary amount of Si}}
427 Si self-interstitial experiments:
432 \item $r_{cutoff}^{Si-Si}=2.96>\frac{5.43}{2}$
433 \item Bond length near $r_{cutoff} \Rightarrow$ small bond strength
441 \begin{minipage}[t]{4.0cm}
442 \underline{Tetrahedral}
444 \item $E_f=3.41\, eV$
445 \item essentialy tetrahedral\\
450 \begin{minipage}[t]{4.0cm}
451 \underline{110 dumbbell}
453 \item $E_f=4.39\, eV$
454 \item essentially 4 bonds
458 \begin{minipage}[t]{4.0cm}
459 \underline{Hexagonal}
461 \item $E_f^{\star}\approx4.48\, eV$
468 \begin{minipage}{4.3cm}
469 \includegraphics[width=3.8cm]{si_self_int_tetra_0.eps}
471 \begin{minipage}{4.3cm}
472 \includegraphics[width=3.8cm]{si_self_int_dumbbell_0.eps}
474 \begin{minipage}{4.3cm}
475 \includegraphics[width=3.8cm]{si_self_int_hexa_0.eps}
488 Si self-interstitial \underline{random insertion} experiments:
502 Carbon interstitial experiments:
508 \begin{minipage}[t]{4.0cm}
509 \underline{Tetrahedral}
511 \item $E_F=2.67\, eV$
512 \item tetrahedral bond
516 \begin{minipage}[t]{4.0cm}
517 \underline{110 dumbbell}
519 \item $E_F=1.76\, eV$
520 \item C forms 3 bonds
524 \begin{minipage}[t]{4.0cm}
525 \underline{Hexagonal}
527 \item $E_F^{\star}\approx5.6\, eV$
534 \begin{minipage}{4.3cm}
535 \includegraphics[width=3.8cm]{c_in_si_int_tetra_0.eps}
537 \begin{minipage}{4.3cm}
538 \includegraphics[width=3.8cm]{c_in_si_int_dumbbell_0.eps}
540 \begin{minipage}{4.3cm}
541 \includegraphics[width=3.8cm]{c_in_si_int_hexa_0.eps}
554 Carbon \underline{random insertion} experiments:
568 SiC-precipitation experiments: