2 \documentclass[landscape,semhelv]{seminar}
5 \usepackage[german]{babel}
6 \usepackage[latin1]{inputenc}
7 \usepackage[T1]{fontenc}
12 \usepackage{calc} % Simple computations with LaTeX variables
13 \usepackage{caption} % Improved captions
14 \usepackage{fancybox} % To have several backgrounds
16 \usepackage{fancyhdr} % Headers and footers definitions
17 \usepackage{fancyvrb} % Fancy verbatim environments
18 \usepackage{pstricks} % PSTricks with the standard color package
24 \graphicspath{{../img/}}
26 \usepackage[setpagesize=false]{hyperref}
29 \usepackage{semlayer} % Seminar overlays
30 \usepackage{slidesec} % Seminar sections and list of slides
32 \input{seminar.bug} % Official bugs corrections
33 \input{seminar.bg2} % Unofficial bugs corrections
41 \extraslideheight{10in}
46 % specify width and height
50 % shift it into visual area properly
51 \def\slideleftmargin{3.3cm}
52 \def\slidetopmargin{0.6cm}
54 \newcommand{\ham}{\mathcal{H}}
55 \newcommand{\pot}{\mathcal{V}}
56 \newcommand{\foo}{\mathcal{U}}
57 \newcommand{\vir}{\mathcal{W}}
60 \renewcommand\labelitemii{{\color{gray}$\bullet$}}
70 Molecular dynamics simulation study\\
71 of the silicon carbide precipitation process
76 \textsc{\small \underline{F. Zirkelbach}$^1$, J. K. N. Lindner$^1$,
77 K. Nordlund$^2$, B. Stritzker$^1$}\\
81 \begin{minipage}{2.0cm}
83 \includegraphics[height=1.6cm]{uni-logo.eps}
86 \begin{minipage}{8.0cm}
89 $^1$ Experimentalphysik IV, Institut f"ur Physik,\\
90 Universit"at Augsburg, Universit"atsstr. 1,\\
91 D-86135 Augsburg, Germany
95 \begin{minipage}{2.3cm}
97 \includegraphics[height=1.5cm]{Lehrstuhl-Logo.eps}
103 \begin{minipage}{4.0cm}
105 \includegraphics[height=1.6cm]{logo_eng.eps}
108 \begin{minipage}{8.0cm}
111 $^2$ Accelerator Laboratory, Department of Physical Sciences,\\
112 University of Helsinki, Pietari Kalmink. 2,\\
113 00014 Helsinki, Finland
126 Molecular dynamics simulation study\\
127 of the silicon carbide precipitation process
140 \item Motivation / Introduction
141 \item Molecular dynamics simulation details
143 \item Integrator, potential, ensemble control
144 \item Simulation sequence
146 \item Simulation results
148 \item Interstitials in silicon
149 \item SiC-precipitation experiments
151 \item Conclusion / Outlook
160 Motivation / Introduction
165 Reasons for investigating C in Si:
168 \item 3C-SiC wide band gap semiconductor formation
169 \item Strained Si (no precipitation wanted!)
176 \begin{minipage}{8cm}
180 \item {\color{yellow}fcc} $+$
181 \item {\color{gray}fcc shifted $1/4$ of volume diagonal}
183 \item Lattice constants: $4a_{Si}\approx5a_{SiC}$
184 \item Silicon density:
186 \frac{n_{SiC}}{n_{Si}}=
187 \frac{4/a_{SiC}^3}{8/a_{Si}^3}=
188 \frac{5^3}{2\cdot4^3}={\color{cyan}97,66}\,\%
193 \begin{minipage}{4cm}
194 \includegraphics[width=4cm]{sic_unit_cell.eps}
203 Motivation / Introduction
209 Supposed conversion mechanism of heavily carbon doped Si into SiC:
213 \begin{minipage}{3.8cm}
214 \includegraphics[width=3.7cm]{sic_prec_seq_01.eps}
217 \begin{minipage}{3.8cm}
218 \includegraphics[width=3.7cm]{sic_prec_seq_02.eps}
221 \begin{minipage}{3.8cm}
222 \includegraphics[width=3.7cm]{sic_prec_seq_03.eps}
227 \begin{minipage}{3.8cm}
228 Formation of C-Si dumbbells on regular c-Si lattice sites
231 \begin{minipage}{3.8cm}
232 Agglomeration into large clusters (embryos)\\
235 \begin{minipage}{3.8cm}
236 Precipitation of 3C-SiC + Creation of interstitials\\
241 Experimentally observed:
243 \item Minimal diameter of precipitation: 4 - 5 nm
244 \item (hkl)-planes identical for Si and SiC
259 \item Microscopic description of N particle system
260 \item Analytical interaction potential
261 \item Hamilton's equations of motion as propagation rule\\
262 in 6N-dimensional phase space
263 \item Observables obtained by time average
270 \item Integrator: Velocity Verlet, timestep: $1\, fs$
271 \item Ensemble control: NVT, Berendsen thermostat, $\tau=100.0$
272 \item Potential: Tersoff-like bond order potential\\
274 E = \frac{1}{2} \sum_{i \neq j} \pot_{ij}, \quad
275 \pot_{ij} = f_C(r_{ij}) \left[ f_R(r_{ij}) + b_{ij} f_A(r_{ij}) \right]
278 {\scriptsize P. Erhart und K. Albe. Phys. Rev. B 71 (2005) 035211}
282 \begin{picture}(0,0)(-240,-70)
283 \includegraphics[width=5cm]{tersoff_angle.eps}
296 Interstitial experiments:
301 \item Initial configuration: $9\times9\times9$ unit cells Si
302 \item Periodic boundary conditions
304 \item Insertion of Si / C atom at
306 \item $(0,0,0)$ $\rightarrow$ {\color{red}tetrahedral}
307 \item $(-1/8,-1/8,1/8)$ $\rightarrow$ {\color{green}hexagonal}
308 \item $(-1/8,-1/8,-1/4)$, $(-1/4,-1/4,-1/4)$\\
309 $\rightarrow$ {\color{yellow}110 dumbbell}
310 \item random positions (critical distance check)
312 \item Relaxation time: $2\, ps$
313 \item Optional heating-up
316 \begin{picture}(0,0)(-210,-45)
317 \includegraphics[width=6cm]{unit_cell.eps}
330 SiC precipitation experiments:
332 \begin{pspicture}(0,0)(12,8)
334 \rput(4.5,6.5){\rnode{init}{\psframebox{\parbox{7cm}{
336 \item Initial configuration: $31\times31\times31$ unit cells Si
337 \item Periodic boundary conditions
338 \item $T=450\, ^{\circ}C$
339 \item Equilibration of $E_{kin}$ and $E_{pot}$ for $600\, fs$
342 \rput(4.5,4.5){\rnode{tc1}{\psframebox[fillstyle=solid,fillcolor=red]{
343 $T=450\pm 1\, ^{\circ}C$}}}
344 \rput(7,3.5){\rnode{insert}{\psframebox[fillstyle=solid,fillcolor=red]{
346 Insertion of 10 atoms\\
347 at random positions}}}}
348 \rput(2,3.5){\rnode{adj1}{\psframebox[fillstyle=solid,fillcolor=red]{
350 Adjusting temperature\\
351 for another $100\, fs$}}}}
352 \rput(7,2.5){\rnode{nc}{\psframebox[fillstyle=solid,fillcolor=red]{
354 \rput(4.5,2){\rnode{tc2}{\psframebox[fillstyle=solid,fillcolor=cyan]{
356 \rput(7,1){\rnode{td}{\psframebox[fillstyle=solid,fillcolor=cyan]{
357 $T_{set}:=T_{set}-1\, ^{\circ}C$}}}
358 \rput(2,1){\rnode{adj2}{\psframebox[fillstyle=solid,fillcolor=cyan]{
360 Adjusting temperature\\
361 for another $50\, fs$}}}}
362 \rput(7,0){\rnode{tc3}{\psframebox[fillstyle=solid,fillcolor=cyan]{
363 $T_{set}=20\, ^{\circ}C$}}}
364 \rput(10,0){\rnode{end}{\psframebox{End}}}
366 \rput(7,4.5){\pnode{tc1-h}}
367 \rput(2,4.5){\pnode{tc1-hh}}
368 \rput(4.5,2.5){\pnode{nc-h}}
369 \rput(9,2.5){\pnode{nc-hh}}
370 \rput(9,2){\pnode{tc2-h}}
371 \rput(2,2){\pnode{tc2-hh}}
372 \rput(4.5,0){\pnode{tc3-h}}
374 \ncline[]{->}{init}{tc1}
375 \ncline[]{->}{adj1}{tc1}
376 \ncline[]{->}{insert}{nc}
377 \ncline[]{->}{adj2}{tc2}
378 \ncline[]{->}{tc2}{td}
380 \ncline[]{->}{td}{tc3}
381 \ncline[]{->}{tc3}{end}
383 % lines using help nodes
384 \ncline[]{tc1}{tc1-h}
386 \ncline[]{->}{tc1-h}{insert}
387 \ncline[]{tc1}{tc1-hh}
389 \ncline[]{->}{tc1-hh}{adj1}
392 \ncline[]{->}{nc-h}{tc1}
394 \ncline[]{-}{nc-hh}{tc2-h}
395 \ncline[]{->}{tc2-h}{tc2}
396 \lput*{0}{yes, {\footnotesize $T_{set}:=450\, ^{\circ}C$}}
397 \ncline[]{tc2}{tc2-hh}
399 \ncline[]{->}{tc2-hh}{adj2}
400 \ncline[]{tc3}{tc3-h}
402 \ncline[]{->}{tc3-h}{tc2}
404 \psframe[fillstyle=solid,fillcolor=white](9.5,1.3)(13.5,5.3)
405 \psframe[fillstyle=solid,fillcolor=lightgray](10,1.8)(13,4.8)
406 \psframe[fillstyle=solid,fillcolor=gray](10.5,2.3)(12.5,4.3)
407 \rput(9.75,3){\pnode{ins1}}
408 \rput(10.25,3.3){\pnode{ins2}}
409 \rput(10.75,3.6){\pnode{ins3}}
410 \ncline[]{-}{insert}{ins1}
411 \ncline[]{-}{insert}{ins2}
412 \ncline[]{-}{insert}{ins3}
413 \psframe[fillstyle=solid,fillcolor=white](9.5,7.6)(13.5,8.1)
414 \psframe[fillstyle=solid,fillcolor=lightgray](9.5,6.8)(13.5,7.3)
415 \psframe[fillstyle=solid,fillcolor=gray](9.5,6)(13.5,6.5)
416 \rput(11.5,7.85){{\tiny Simulation volume:
417 $31\times31\times31\, a^3_{Si}$}}
418 \rput(11.5,7.05){{\tiny Volume of minimal SiC precipitation}}
419 \rput(11.5,6.25){{\tiny Volume of necessary amount of Si}}
430 Si self-interstitial experiments:
435 \item $r_{cutoff}^{Si-Si}=2.96>\frac{5.43}{2}$
436 \item Bond length near $r_{cutoff} \Rightarrow$ small bond strength
444 \begin{minipage}[t]{4.0cm}
445 \underline{Tetrahedral}
447 \item $E_f=3.41\, eV$
448 \item essentialy tetrahedral\\
453 \begin{minipage}[t]{4.0cm}
454 \underline{110 dumbbell}
456 \item $E_f=4.39\, eV$
457 \item essentially 4 bonds
461 \begin{minipage}[t]{4.0cm}
462 \underline{Hexagonal}
464 \item $E_f^{\star}\approx4.48\, eV$
471 \begin{minipage}[t]{4.3cm}
472 \includegraphics[width=3.8cm]{si_self_int_tetra_0.eps}
474 \begin{minipage}[t]{4.3cm}
475 \includegraphics[width=3.8cm]{si_self_int_dumbbell_0.eps}
477 \begin{minipage}[t]{4.3cm}
478 \includegraphics[width=3.8cm]{si_self_int_hexa_0.eps}
480 \href{../video/si_self_int_hexa.avi}{$\rhd$}
494 Si self-interstitial \underline{random insertion} experiments:
500 \begin{minipage}[t]{4.0cm}
502 \item $E_f=3.97\, eV$
503 \item 3 identical weak bonds
504 \item displaced in volume\\ diagonal
508 \begin{minipage}[t]{4.0cm}
510 \item $E_f=3.75\, eV$
511 \item 4 identical weak bonds
512 \item displaced in plane\\ diagonal
516 \begin{minipage}[t]{4.0cm}
518 \item $E_f=3.56\, eV$
519 \item single weak bond
520 \item displaced along\\ $x$-direction
521 \item closest to tetrahedral\\ configuration
527 \begin{minipage}{4.3cm}
528 \includegraphics[width=3.8cm]{si_self_int_rand_397_0.eps}
530 \begin{minipage}{4.3cm}
531 \includegraphics[width=3.8cm]{si_self_int_rand_375_0.eps}
533 \begin{minipage}{4.3cm}
534 \includegraphics[width=3.8cm]{si_self_int_rand_356_0.eps}
541 {\bf Note:} Displacements relative to tetrahedral configuration
555 Carbon interstitial experiments:
561 \begin{minipage}[t]{4.0cm}
562 \underline{Tetrahedral}
564 \item $E_f=2.67\, eV$
565 \item tetrahedral bond
569 \begin{minipage}[t]{4.0cm}
570 \underline{110 dumbbell}
572 \item $E_f=1.76\, eV$
573 \item C forms 3 bonds
577 \begin{minipage}[t]{4.0cm}
578 \underline{Hexagonal}
580 \item $E_f^{\star}\approx5.6\, eV$
587 \begin{minipage}[t]{4.3cm}
588 \includegraphics[width=3.8cm]{c_in_si_int_tetra_0.eps}
590 \begin{minipage}[t]{4.3cm}
591 \includegraphics[width=3.8cm]{c_in_si_int_dumbbell_0.eps}
593 \begin{minipage}[t]{4.3cm}
594 \includegraphics[width=3.8cm]{c_in_si_int_hexa_0.eps}
596 \href{../video/c_in_si_int_hexa.avi}{$\rhd$}
610 Carbon \underline{random insertion} experiments:
614 High probability for dumbbell configurations!\\
616 \includegraphics[width=3.0cm]{c_in_si_int_001db_0.eps}
617 \includegraphics[width=3.0cm]{c_in_si_int_rand_162_0.eps}
618 \includegraphics[width=3.0cm]{c_in_si_int_rand_239_0.eps}
619 \includegraphics[width=3.0cm]{c_in_si_int_rand_341_0.eps}
629 SiC-precipitation experiments: