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73 Molecular dynamics simulation study\\
74 of the silicon carbide precipitation process
79 \textsc{\small \underline{F. Zirkelbach}$^1$, J. K. N. Lindner$^1$,
80 K. Nordlund$^2$, B. Stritzker$^1$}\\
84 \begin{minipage}{2.0cm}
86 \includegraphics[height=1.6cm]{uni-logo.eps}
89 \begin{minipage}{8.0cm}
92 $^1$ Experimentalphysik IV, Institut f"ur Physik,\\
93 Universit"at Augsburg, Universit"atsstr. 1,\\
94 D-86135 Augsburg, Germany
98 \begin{minipage}{2.3cm}
100 \includegraphics[height=1.5cm]{Lehrstuhl-Logo.eps}
106 \begin{minipage}{4.0cm}
108 \includegraphics[height=1.6cm]{logo_eng.eps}
111 \begin{minipage}{8.0cm}
114 $^2$ Accelerator Laboratory, Department of Physical Sciences,\\
115 University of Helsinki, Pietari Kalmink. 2,\\
116 00014 Helsinki, Finland
129 Molecular dynamics simulation study\\
130 of the silicon carbide precipitation process
143 \item Motivation / Introduction
144 \item Molecular dynamics simulation details
146 \item Integrator, potential, ensemble control
147 \item Simulation sequence
149 \item Simulation results
151 \item Interstitials in silicon
152 \item SiC-precipitation experiments
154 \item Conclusion / Outlook
163 Motivation / Introduction
168 Reasons for investigating C in Si:
171 \item 3C-SiC wide band gap semiconductor formation
172 \item Strained Si (no precipitation wanted!)
179 \begin{minipage}{8cm}
183 \item {\color{yellow}fcc} $+$
184 \item {\color{gray}fcc shifted $1/4$ of volume diagonal}
186 \item Lattice constants: $4a_{Si}\approx5a_{SiC}$
187 \item Silicon density:
189 \frac{n_{SiC}}{n_{Si}}=
190 \frac{4/a_{SiC}^3}{8/a_{Si}^3}=
191 \frac{5^3}{2\cdot4^3}={\color{cyan}97,66}\,\%
196 \begin{minipage}{4cm}
197 \includegraphics[width=4cm]{sic_unit_cell.eps}
206 Motivation / Introduction
212 Supposed conversion mechanism of heavily carbon doped Si into SiC:
216 \begin{minipage}{3.8cm}
217 \includegraphics[width=3.7cm]{sic_prec_seq_01.eps}
220 \begin{minipage}{3.8cm}
221 \includegraphics[width=3.7cm]{sic_prec_seq_02.eps}
224 \begin{minipage}{3.8cm}
225 \includegraphics[width=3.7cm]{sic_prec_seq_03.eps}
230 \begin{minipage}{3.8cm}
231 Formation of C-Si dumbbells on regular c-Si lattice sites
234 \begin{minipage}{3.8cm}
235 Agglomeration into large clusters (embryos)\\
238 \begin{minipage}{3.8cm}
239 Precipitation of 3C-SiC + Creation of interstitials\\
244 Experimentally observed:
246 \item Minimal diameter of precipitation: 4 - 5 nm
247 \item (hkl)-planes identical for Si and SiC
262 \item Microscopic description of N particle system
263 \item Analytical interaction potential
264 \item Hamilton's equations of motion as propagation rule\\
265 in 6N-dimensional phase space
266 \item Observables obtained by time average
273 \item Integrator: Velocity Verlet, timestep: $1\, fs$
274 \item Ensemble control: NVT, Berendsen thermostat, $\tau=100.0$
275 \item Potential: Tersoff-like bond order potential\\
277 E = \frac{1}{2} \sum_{i \neq j} \pot_{ij}, \quad
278 \pot_{ij} = f_C(r_{ij}) \left[ f_R(r_{ij}) + b_{ij} f_A(r_{ij}) \right]
281 {\scriptsize P. Erhart und K. Albe. Phys. Rev. B 71 (2005) 035211}
285 \begin{picture}(0,0)(-240,-70)
286 \includegraphics[width=5cm]{tersoff_angle.eps}
299 Interstitial experiments:
304 \item Initial configuration: $9\times9\times9$ unit cells Si
305 \item Periodic boundary conditions
307 \item Insertion of Si / C atom at
309 \item $(0,0,0)$ $\rightarrow$ {\color{red}tetrahedral}
310 \item $(-1/8,-1/8,1/8)$ $\rightarrow$ {\color{green}hexagonal}
311 \item $(-1/8,-1/8,-1/4)$, $(-1/4,-1/4,-1/4)$\\
312 $\rightarrow$ {\color{yellow}110 dumbbell}
313 \item random positions (critical distance check)
315 \item Relaxation time: $2\, ps$
316 \item Optional heating-up
319 \begin{picture}(0,0)(-210,-45)
320 \includegraphics[width=6cm]{unit_cell.eps}
333 SiC precipitation experiments:
335 \begin{pspicture}(0,0)(12,8)
337 \rput(4.5,6.5){\rnode{init}{\psframebox{\parbox{7cm}{
339 \item Initial configuration: $31\times31\times31$ unit cells Si
340 \item Periodic boundary conditions
341 \item $T=450\, ^{\circ}C$
342 \item Equilibration of $E_{kin}$ and $E_{pot}$ for $600\, fs$
345 \rput(4.5,4.5){\rnode{tc1}{\psframebox[fillstyle=solid,fillcolor=red]{
346 $T=450\pm 1\, ^{\circ}C$}}}
347 \rput(7,3.5){\rnode{insert}{\psframebox[fillstyle=solid,fillcolor=red]{
349 Insertion of 10 atoms\\
350 at random positions}}}}
351 \rput(2,3.5){\rnode{adj1}{\psframebox[fillstyle=solid,fillcolor=red]{
353 Adjusting temperature\\
354 for another $100\, fs$}}}}
355 \rput(7,2.5){\rnode{nc}{\psframebox[fillstyle=solid,fillcolor=red]{
357 \rput(4.5,2){\rnode{tc2}{\psframebox[fillstyle=solid,fillcolor=cyan]{
359 \rput(7,1){\rnode{td}{\psframebox[fillstyle=solid,fillcolor=cyan]{
360 $T_{set}:=T_{set}-1\, ^{\circ}C$}}}
361 \rput(2,1){\rnode{adj2}{\psframebox[fillstyle=solid,fillcolor=cyan]{
363 Adjusting temperature\\
364 for another $50\, fs$}}}}
365 \rput(7,0){\rnode{tc3}{\psframebox[fillstyle=solid,fillcolor=cyan]{
366 $T_{set}=20\, ^{\circ}C$}}}
367 \rput(10,0){\rnode{end}{\psframebox{End}}}
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374 \rput(2,2){\pnode{tc2-hh}}
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399 \lput*{0}{yes, {\footnotesize $T_{set}:=450\, ^{\circ}C$}}
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405 \ncline[]{->}{tc3-h}{tc2}
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419 \rput(11.5,7.85){{\tiny Simulation volume:
420 $31\times31\times31\, a^3_{Si}$}}
421 \rput(11.5,7.05){{\tiny Volume of minimal SiC precipitation}}
422 \rput(11.5,6.25){{\tiny Volume of necessary amount of Si}}
433 Si self-interstitial experiments:
438 \item $r_{cutoff}^{Si-Si}=2.96>\frac{5.43}{2}$
439 \item Bond length near $r_{cutoff} \Rightarrow$ small bond strength
447 \begin{minipage}[t]{4.0cm}
448 \underline{Tetrahedral}
450 \item $E_f=3.41\, eV$
451 \item essentialy tetrahedral\\
456 \begin{minipage}[t]{4.0cm}
457 \underline{110 dumbbell}
459 \item $E_f=4.39\, eV$
460 \item essentially 4 bonds
464 \begin{minipage}[t]{4.0cm}
465 \underline{Hexagonal}
467 \item $E_f^{\star}\approx4.48\, eV$
474 \begin{minipage}[t]{4.3cm}
475 \includegraphics[width=3.8cm]{si_self_int_tetra_0.eps}
477 \begin{minipage}[t]{4.3cm}
478 \includegraphics[width=3.8cm]{si_self_int_dumbbell_0.eps}
480 \begin{minipage}[t]{4.3cm}
481 \includegraphics[width=3.8cm]{si_self_int_hexa_0.eps}
483 \href{../video/si_self_int_hexa.avi}{$\rhd$}
497 Si self-interstitial \underline{random insertion} experiments:
503 \begin{minipage}[t]{4.0cm}
505 \item $E_f=3.97\, eV$
506 \item 3 identical weak bonds
507 \item displaced in volume\\ diagonal
511 \begin{minipage}[t]{4.0cm}
513 \item $E_f=3.75\, eV$
514 \item 4 identical weak bonds
515 \item displaced in plane\\ diagonal
519 \begin{minipage}[t]{4.0cm}
521 \item $E_f=3.56\, eV$
522 \item single weak bond
523 \item displaced along\\ $x$-direction
524 \item closest to tetrahedral\\ configuration
530 \begin{minipage}{4.3cm}
531 \includegraphics[width=3.8cm]{si_self_int_rand_397_0.eps}
533 \begin{minipage}{4.3cm}
534 \includegraphics[width=3.8cm]{si_self_int_rand_375_0.eps}
536 \begin{minipage}{4.3cm}
537 \includegraphics[width=3.8cm]{si_self_int_rand_356_0.eps}
544 {\bf Note:} Displacements relative to tetrahedral configuration
558 Carbon interstitial experiments:
564 \begin{minipage}[t]{4.0cm}
565 \underline{Tetrahedral}
567 \item $E_f=2.67\, eV$
568 \item tetrahedral bond
572 \begin{minipage}[t]{4.0cm}
573 \underline{110 dumbbell}
575 \item $E_f=1.76\, eV$
576 \item C forms 3 bonds
580 \begin{minipage}[t]{4.0cm}
581 \underline{Hexagonal}
583 \item $E_f^{\star}\approx5.6\, eV$
590 \begin{minipage}[t]{4.3cm}
591 \includegraphics[width=3.8cm]{c_in_si_int_tetra_0.eps}
593 \begin{minipage}[t]{4.3cm}
594 \includegraphics[width=3.8cm]{c_in_si_int_dumbbell_0.eps}
596 \begin{minipage}[t]{4.3cm}
597 \includegraphics[width=3.8cm]{c_in_si_int_hexa_0.eps}
599 \href{../video/c_in_si_int_hexa.avi}{$\rhd$}
613 Carbon \underline{random insertion} experiments:
619 \begin{minipage}[c]{6.3cm}
620 \begin{minipage}{3.4cm}
621 \includegraphics[width=3.3cm]{c_in_si_int_001db_0.eps}
623 \begin{minipage}{2.5cm}
625 \item $E_f=0.47\, eV$
630 \begin{minipage}[c]{6.3cm}
631 \begin{minipage}{3.4cm}
632 \includegraphics[width=3.3cm]{c_in_si_int_rand_162_0.eps}
634 \begin{minipage}{2.8cm}
636 \item $E_f=1.62\, eV$
637 \item 3 weak + strong bonds
642 \begin{minipage}[c]{6.3cm}
643 \begin{minipage}{3.4cm}
644 \includegraphics[width=3.3cm]{c_in_si_int_rand_239_0.eps}
646 \begin{minipage}{2.5cm}
648 \item $E_f=2.39\, eV$
651 \href{../video/c_in_si_int_rand_239.avi}{$\rhd$}
655 \begin{minipage}[c]{6.3cm}
656 \begin{minipage}{3.4cm}
657 \includegraphics[width=3.3cm]{c_in_si_int_rand_341_0.eps}
659 \begin{minipage}{2.8cm}
661 \item $E_f=3.41\, eV$
664 \href{../video/c_in_si_int_rand_341.avi}{$\rhd$}
672 {\bf Note:} High probability for 110 dumbbell ($1.76\, eV$) configurations!
683 SiC-precipitation experiments:
685 \begin{minipage}[t]{6.3cm}
686 %\input{../plot/sic_prec}
687 \includegraphics[width=6.0cm]{../plot/sic_prec_energy.ps}
688 \includegraphics[width=6.0cm]{../plot/sic_prec_temp.ps}
690 \begin{minipage}[t]{6cm}
691 \includegraphics[width=6.0cm]{../plot/sic_pc.ps}
692 \includegraphics[width=6.0cm]{../plot/sic_prec_pc.ps}
706 \item Importance of understanding C in Si
707 \item Interstitial configurations in silicon using the Albe potential
708 \item Indication of SiC precipitation
714 \item Displacement and stress calculations
715 \item Diffusion dependence of temperature and carbon concentration
716 \item Analyzing results of the precipitation simulation runs
717 \item Analyzing self-designed Si/SiC interface