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1 \pdfoutput=0
2 %\documentclass[landscape,semhelv,draft]{seminar}
3 \documentclass[landscape,semhelv]{seminar}
4
5 \usepackage{verbatim}
6 \usepackage[greek,german]{babel}
7 \usepackage[latin1]{inputenc}
8 \usepackage[T1]{fontenc}
9 \usepackage{amsmath}
10 \usepackage{latexsym}
11 \usepackage{ae}
12
13 \usepackage{calc}               % Simple computations with LaTeX variables
14 \usepackage{caption}            % Improved captions
15 \usepackage{fancybox}           % To have several backgrounds
16
17 \usepackage{fancyhdr}           % Headers and footers definitions
18 \usepackage{fancyvrb}           % Fancy verbatim environments
19 \usepackage{pstricks}           % PSTricks with the standard color package
20
21 \usepackage{pstricks}
22 \usepackage{pst-node}
23
24 %\usepackage{epic}
25 %\usepackage{eepic}
26
27 \usepackage{graphicx}
28 \graphicspath{{../img/}}
29
30 \usepackage{miller}
31
32 \usepackage[setpagesize=false]{hyperref}
33
34 % units
35 \usepackage{units}
36
37 \usepackage{semcolor}
38 \usepackage{semlayer}           % Seminar overlays
39 \usepackage{slidesec}           % Seminar sections and list of slides
40
41 \input{seminar.bug}             % Official bugs corrections
42 \input{seminar.bg2}             % Unofficial bugs corrections
43
44 \articlemag{1}
45
46 \special{landscape}
47
48 % font
49 %\usepackage{cmbright}
50 %\renewcommand{\familydefault}{\sfdefault}
51 %\usepackage{mathptmx}
52
53 \usepackage{upgreek}
54
55 \begin{document}
56
57 \extraslideheight{10in}
58 \slideframe{none}
59
60 \pagestyle{empty}
61
62 % specify width and height
63 \slidewidth 27.7cm 
64 \slideheight 19.1cm 
65
66 % shift it into visual area properly
67 \def\slideleftmargin{3.3cm}
68 \def\slidetopmargin{0.6cm}
69
70 \newcommand{\ham}{\mathcal{H}}
71 \newcommand{\pot}{\mathcal{V}}
72 \newcommand{\foo}{\mathcal{U}}
73 \newcommand{\vir}{\mathcal{W}}
74
75 % itemize level ii
76 \renewcommand\labelitemii{{\color{gray}$\bullet$}}
77
78 % nice phi
79 \renewcommand{\phi}{\varphi}
80
81 % roman letters
82 \newcommand{\RM}[1]{\MakeUppercase{\romannumeral #1{}}}
83
84 % colors
85 \newrgbcolor{si-yellow}{.6 .6 0}
86 \newrgbcolor{hb}{0.75 0.77 0.89}
87 \newrgbcolor{lbb}{0.75 0.8 0.88}
88 \newrgbcolor{hlbb}{0.825 0.88 0.968}
89 \newrgbcolor{lachs}{1.0 .93 .81}
90
91 % shortcuts
92 \newcommand{\si}{Si$_{\text{i}}${}}
93 \newcommand{\ci}{C$_{\text{i}}${}}
94 \newcommand{\cs}{C$_{\text{sub}}${}}
95 \newcommand{\degc}[1]{\unit[#1]{$^{\circ}$C}{}}
96 \newcommand{\distn}[1]{\unit[#1]{nm}{}}
97 \newcommand{\dista}[1]{\unit[#1]{\AA}{}}
98 \newcommand{\perc}[1]{\unit[#1]{\%}{}}
99
100 % topic
101
102 \begin{slide}
103 \begin{center}
104
105  \vspace{16pt}
106
107  {\LARGE\bf
108   Atomistic simulation study on the silicon carbide precipitation
109   in silicon
110  }
111
112  \vspace{48pt}
113
114  \textsc{F. Zirkelbach}
115
116  \vspace{48pt}
117
118 Yet another seminar talk
119
120  \vspace{08pt}
121
122  Augsburg, 26. Mai 2011
123
124 \end{center}
125 \end{slide}
126
127 % motivation / properties / applications of silicon carbide
128 \begin{slide}
129
130 \small
131
132 \begin{pspicture}(0,0)(13.5,5)
133
134
135
136  \psframe*[linecolor=hb](0,0)(13.5,5)
137
138  \pspolygon[linecolor=hlbb,fillcolor=hlbb,fillstyle=solid](5.5,1)(7,1)(7,3)(5.5,3)
139  \pspolygon[linecolor=hlbb,fillcolor=hlbb,fillstyle=solid](6.75,0.5)(8,2)(8,2)(6.75,3.5)
140
141  \rput[lt](0.2,4.6){\color{gray}PROPERTIES}
142
143  \rput[lt](0.5,4){wide band gap}
144  \rput[lt](0.5,3.5){high electric breakdown field}
145  \rput[lt](0.5,3){good electron mobility}
146  \rput[lt](0.5,2.5){high electron saturation drift velocity}
147  \rput[lt](0.5,2){high thermal conductivity}
148
149  \rput[lt](0.5,1.5){hard and mechanically stable}
150  \rput[lt](0.5,1){chemically inert}
151
152  \rput[lt](0.5,0.5){radiation hardness}
153
154  \rput[rt](13.3,4.6){\color{gray}APPLICATIONS}
155
156  \rput[rt](13,3.85){high-temperature, high power}
157  \rput[rt](13,3.5){and high-frequency}
158  \rput[rt](13,3.15){electronic and optoelectronic devices}
159
160  \rput[rt](13,2.35){material suitable for extreme conditions}
161  \rput[rt](13,2){microelectromechanical systems}
162  \rput[rt](13,1.65){abrasives, cutting tools, heating elements}
163
164  \rput[rt](13,0.85){first wall reactor material, detectors}
165  \rput[rt](13,0.5){and electronic devices for space}
166
167 \end{pspicture}
168
169 \begin{picture}(0,0)(-3,68)
170 \includegraphics[width=2.6cm]{wide_band_gap.eps}
171 \end{picture}
172 \begin{picture}(0,0)(-285,-162)
173 \includegraphics[width=3.38cm]{sic_led.eps}
174 \end{picture}
175 \begin{picture}(0,0)(-195,-162)
176 \includegraphics[width=2.8cm]{6h-sic_3c-sic.eps}
177 \end{picture}
178 \begin{picture}(0,0)(-313,65)
179 \includegraphics[width=2.2cm]{infineon_schottky.eps}
180 \end{picture}
181 \begin{picture}(0,0)(-220,65)
182 \includegraphics[width=2.9cm]{sic_wechselrichter_ise.eps}
183 \end{picture}
184 \begin{picture}(0,0)(0,-160)
185 \includegraphics[width=3.0cm]{sic_proton.eps}
186 \end{picture}
187 \begin{picture}(0,0)(-60,65)
188 \includegraphics[width=3.4cm]{sic_switch.eps}
189 \end{picture}
190
191 \end{slide}
192
193 % start of contents
194
195 \begin{slide}
196
197  {\large\bf
198   Polytypes of SiC
199  }
200
201  \vspace{4cm}
202
203  \small
204
205 \begin{tabular}{l c c c c c c}
206 \hline
207  & 3C-SiC & 4H-SiC & 6H-SiC & Si & GaN & Diamond\\
208 \hline
209 Hardness [Mohs] & \multicolumn{3}{c}{------ 9.6 ------}& 6.5 & - & 10 \\
210 Band gap [eV] & 2.36 & 3.23 & 3.03 & 1.12 & 3.39 & 5.5 \\
211 Break down field [$10^6$ V/cm] & 4 & 3 & 3.2 & 0.6 & 5 & 10 \\
212 Saturation drift velocity [$10^7$ cm/s] & 2.5 & 2.0 & 2.0 & 1 & 2.7 & 2.7 \\
213 Electron mobility [cm$^2$/Vs] & 800 & 900 & 400 & 1100 & 900 & 2200 \\
214 Hole mobility [cm$^2$/Vs] & 320 & 120 & 90 & 420 & 150 & 1600 \\
215 Thermal conductivity [W/cmK] & 5.0 & 4.9 & 4.9 & 1.5 & 1.3 & 22 \\
216 \hline
217 \end{tabular}
218
219 {\tiny
220  Values for $T=300$ K
221 }
222
223 \begin{picture}(0,0)(-160,-155)
224  \includegraphics[width=7cm]{polytypes.eps}
225 \end{picture}
226 \begin{picture}(0,0)(-10,-185)
227  \includegraphics[width=3.8cm]{cubic_hex.eps}\\
228 \end{picture}
229 \begin{picture}(0,0)(-10,-175)
230  {\tiny cubic (twist)}
231 \end{picture}
232 \begin{picture}(0,0)(-60,-175)
233  {\tiny hexagonal (no twist)}
234 \end{picture}
235 \begin{pspicture}(0,0)(0,0)
236 \psellipse[linecolor=green](5.7,3.03)(0.4,0.5)
237 \end{pspicture}
238 \begin{pspicture}(0,0)(0,0)
239 \psellipse[linecolor=green](5.6,1.68)(0.4,0.2)
240 \end{pspicture}
241 \begin{pspicture}(0,0)(0,0)
242 \psellipse[linecolor=red](10.7,1.13)(0.4,0.2)
243 \end{pspicture}
244
245 \end{slide}
246
247 \begin{slide}
248
249  {\large\bf
250   Fabrication of silicon carbide
251  }
252
253  \small
254  
255  \vspace{4pt}
256
257  SiC - \emph{Born from the stars, perfected on earth.}
258  
259  \vspace{4pt}
260
261  Conventional thin film SiC growth:
262  \begin{itemize}
263   \item \underline{Sublimation growth using the modified Lely method}
264         \begin{itemize}
265          \item SiC single-crystalline seed at $T=1800 \, ^{\circ} \text{C}$
266          \item Surrounded by polycrystalline SiC in a graphite crucible\\
267                at $T=2100-2400 \, ^{\circ} \text{C}$
268          \item Deposition of supersaturated vapor on cooler seed crystal
269         \end{itemize}
270   \item \underline{Homoepitaxial growth using CVD}
271         \begin{itemize}
272          \item Step-controlled epitaxy on off-oriented 6H-SiC substrates
273          \item C$_3$H$_8$/SiH$_4$/H$_2$ at $1100-1500 \, ^{\circ} \text{C}$
274          \item Angle, temperature $\rightarrow$ 3C/6H/4H-SiC
275         \end{itemize}
276   \item \underline{Heteroepitaxial growth of 3C-SiC on Si using CVD/MBE}
277         \begin{itemize}
278          \item Two steps: carbonization and growth
279          \item $T=650-1050 \, ^{\circ} \text{C}$
280          \item SiC/Si lattice mismatch $\approx$ 20 \%
281          \item Quality and size not yet sufficient
282         \end{itemize}
283  \end{itemize}
284
285  \begin{picture}(0,0)(-280,-65)
286   \includegraphics[width=3.8cm]{6h-sic_3c-sic.eps}
287  \end{picture}
288  \begin{picture}(0,0)(-280,-55)
289   \begin{minipage}{5cm}
290   {\tiny
291    NASA: 6H-SiC and 3C-SiC LED\\[-7pt]
292    on 6H-SiC substrate
293   }
294   \end{minipage}
295  \end{picture}
296  \begin{picture}(0,0)(-265,-150)
297   \includegraphics[width=2.4cm]{m_lely.eps}
298  \end{picture}
299  \begin{picture}(0,0)(-333,-175)
300   \begin{minipage}{5cm}
301   {\tiny
302    1. Lid\\[-7pt]
303    2. Heating\\[-7pt]
304    3. Source\\[-7pt]
305    4. Crucible\\[-7pt]
306    5. Insulation\\[-7pt]
307    6. Seed crystal
308   }
309   \end{minipage}
310  \end{picture}
311  \begin{picture}(0,0)(-230,-35)
312  \framebox{
313  {\footnotesize\color{blue}\bf Hex: micropipes along c-axis}
314  }
315  \end{picture}
316  \begin{picture}(0,0)(-230,-10)
317  \framebox{
318  \begin{minipage}{3cm}
319  {\footnotesize\color{blue}\bf 3C-SiC fabrication\\
320                                less advanced}
321  \end{minipage}
322  }
323  \end{picture}
324
325 \end{slide}
326
327 \begin{slide}
328
329  {\large\bf
330   Fabrication of silicon carbide
331  }
332
333  \small
334
335  Alternative approach:
336  Ion beam synthesis (IBS) of burried 3C-SiC layers in Si\hkl(1 0 0)
337  \begin{itemize}
338   \item \underline{Implantation step 1}\\
339         180 keV C$^+$, $D=7.9\times 10^{17}$ cm$^{-2}$, $T_{\text{i}}=500\,^{\circ}\mathrm{C}$\\
340         $\Rightarrow$ box-like distribution of equally sized
341                        and epitactically oriented SiC precipitates
342                        
343   \item \underline{Implantation step 2}\\
344         180 keV C$^+$, $D=0.6\times 10^{17}$ cm$^{-2}$, $T_{\text{i}}=250\,^{\circ}\mathrm{C}$\\
345         $\Rightarrow$ destruction of SiC nanocrystals
346                       in growing amorphous interface layers
347   \item \underline{Annealing}\\
348         $T=1250\,^{\circ}\mathrm{C}$, $t=10\,\text{h}$\\
349         $\Rightarrow$ homogeneous, stoichiometric SiC layer
350                       with sharp interfaces
351  \end{itemize}
352
353  \begin{minipage}{6.3cm}
354  \includegraphics[width=6cm]{ibs_3c-sic.eps}\\[-0.2cm]
355  {\tiny
356   XTEM micrograph of single crystalline 3C-SiC in Si\hkl(1 0 0)
357  }
358  \end{minipage}
359 \framebox{
360  \begin{minipage}{6.3cm}
361  \begin{center}
362  {\color{blue}
363   Precipitation mechanism not yet fully understood!
364  }
365  \renewcommand\labelitemi{$\Rightarrow$}
366  \small
367  \underline{Understanding the SiC precipitation}
368  \begin{itemize}
369   \item significant technological progress in SiC thin film formation
370   \item perspectives for processes relying upon prevention of SiC precipitation
371  \end{itemize}
372  \end{center}
373  \end{minipage}
374 }
375  
376 \end{slide}
377
378 % contents
379
380 \begin{slide}
381
382 {\large\bf
383  Outline
384 }
385
386  \begin{itemize}
387   \item Supposed precipitation mechanism of SiC in Si
388   \item Utilized simulation techniques
389         \begin{itemize}
390          \item Molecular dynamics (MD) simulations
391          \item Density functional theory (DFT) calculations
392         \end{itemize}
393   \item C and Si self-interstitial point defects in silicon
394   \item Silicon carbide precipitation simulations
395   \item Summary / Conclusion / Outlook
396  \end{itemize}
397
398 \end{slide}
399
400
401 \begin{slide}
402
403  {\large\bf
404   Supposed precipitation mechanism of SiC in Si
405  }
406
407  \scriptsize
408
409  \vspace{0.1cm}
410
411  \begin{minipage}{3.8cm}
412  Si \& SiC lattice structure\\[0.2cm]
413  \includegraphics[width=3.5cm]{sic_unit_cell.eps}\\[-0.3cm]
414  \hrule
415  \end{minipage}
416  \hspace{0.6cm}
417  \begin{minipage}{3.8cm}
418  \begin{center}
419  \includegraphics[width=3.3cm]{tem_c-si-db.eps}
420  \end{center}
421  \end{minipage}
422  \hspace{0.6cm}
423  \begin{minipage}{3.8cm}
424  \begin{center}
425  \includegraphics[width=3.3cm]{tem_3c-sic.eps}
426  \end{center}
427  \end{minipage}
428
429  \begin{minipage}{4cm}
430  \begin{center}
431  C-Si dimers (dumbbells)\\[-0.1cm]
432  on Si interstitial sites
433  \end{center}
434  \end{minipage}
435  \hspace{0.2cm}
436  \begin{minipage}{4.2cm}
437  \begin{center}
438  Agglomeration of C-Si dumbbells\\[-0.1cm]
439  $\Rightarrow$ dark contrasts
440  \end{center}
441  \end{minipage}
442  \hspace{0.2cm}
443  \begin{minipage}{4cm}
444  \begin{center}
445  Precipitation of 3C-SiC in Si\\[-0.1cm]
446  $\Rightarrow$ Moir\'e fringes\\[-0.1cm]
447  \& release of Si self-interstitials
448  \end{center}
449  \end{minipage}
450
451  \begin{minipage}{3.8cm}
452  \begin{center}
453  \includegraphics[width=3.3cm]{sic_prec_seq_01.eps}
454  \end{center}
455  \end{minipage}
456  \hspace{0.6cm}
457  \begin{minipage}{3.8cm}
458  \begin{center}
459  \includegraphics[width=3.3cm]{sic_prec_seq_02.eps}
460  \end{center}
461  \end{minipage}
462  \hspace{0.6cm}
463  \begin{minipage}{3.8cm}
464  \begin{center}
465  \includegraphics[width=3.3cm]{sic_prec_seq_03.eps}
466  \end{center}
467  \end{minipage}
468
469 \begin{pspicture}(0,0)(0,0)
470 \psline[linewidth=4pt]{->}(8.5,2)(9.0,2)
471 \psellipse[linecolor=blue](11.5,5.8)(0.3,0.5)
472 \rput{-20}{\psellipse[linecolor=blue](3.3,8.1)(0.3,0.5)}
473 \psline[linewidth=4pt]{->}(4.0,2)(4.5,2)
474 \rput(12.7,0.3){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
475  $4a_{\text{Si}}=5a_{\text{SiC}}$
476  }}}
477 \rput(12.2,8){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
478 \hkl(h k l) planes match
479  }}}
480 \rput(9.7,6.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
481 r = 2 - 4 nm
482  }}}
483 \end{pspicture}
484
485 \end{slide}
486
487 \begin{slide}
488
489  {\large\bf
490   Supposed precipitation mechanism of SiC in Si
491  }
492
493  \scriptsize
494
495  \vspace{0.1cm}
496
497  \begin{minipage}{3.8cm}
498  Si \& SiC lattice structure\\[0.2cm]
499  \includegraphics[width=3.5cm]{sic_unit_cell.eps}\\[-0.3cm]
500  \hrule
501  \end{minipage}
502  \hspace{0.6cm}
503  \begin{minipage}{3.8cm}
504  \begin{center}
505  \includegraphics[width=3.3cm]{tem_c-si-db.eps}
506  \end{center}
507  \end{minipage}
508  \hspace{0.6cm}
509  \begin{minipage}{3.8cm}
510  \begin{center}
511  \includegraphics[width=3.3cm]{tem_3c-sic.eps}
512  \end{center}
513  \end{minipage}
514
515  \begin{minipage}{4cm}
516  \begin{center}
517  C-Si dimers (dumbbells)\\[-0.1cm]
518  on Si interstitial sites
519  \end{center}
520  \end{minipage}
521  \hspace{0.2cm}
522  \begin{minipage}{4.2cm}
523  \begin{center}
524  Agglomeration of C-Si dumbbells\\[-0.1cm]
525  $\Rightarrow$ dark contrasts
526  \end{center}
527  \end{minipage}
528  \hspace{0.2cm}
529  \begin{minipage}{4cm}
530  \begin{center}
531  Precipitation of 3C-SiC in Si\\[-0.1cm]
532  $\Rightarrow$ Moir\'e fringes\\[-0.1cm]
533  \& release of Si self-interstitials
534  \end{center}
535  \end{minipage}
536
537  \begin{minipage}{3.8cm}
538  \begin{center}
539  \includegraphics[width=3.3cm]{sic_prec_seq_01.eps}
540  \end{center}
541  \end{minipage}
542  \hspace{0.6cm}
543  \begin{minipage}{3.8cm}
544  \begin{center}
545  \includegraphics[width=3.3cm]{sic_prec_seq_02.eps}
546  \end{center}
547  \end{minipage}
548  \hspace{0.6cm}
549  \begin{minipage}{3.8cm}
550  \begin{center}
551  \includegraphics[width=3.3cm]{sic_prec_seq_03.eps}
552  \end{center}
553  \end{minipage}
554
555 \begin{pspicture}(0,0)(0,0)
556 \psline[linewidth=4pt]{->}(8.5,2)(9.0,2)
557 \psellipse[linecolor=blue](11.5,5.8)(0.3,0.5)
558 \rput{-20}{\psellipse[linecolor=blue](3.3,8.1)(0.3,0.5)}
559 \psline[linewidth=4pt]{->}(4.0,2)(4.5,2)
560 \rput(12.7,0.3){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
561  $4a_{\text{Si}}=5a_{\text{SiC}}$
562  }}}
563 \rput(12.2,8){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
564 \hkl(h k l) planes match
565  }}}
566 \rput(9.7,6.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
567 r = 2 - 4 nm
568  }}}
569 \rput(6.7,5.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=white]{
570 \begin{minipage}{10cm}
571 \small
572 {\color{red}\bf Controversial views}
573 \begin{itemize}
574 \item Implantations at high T (Nejim et al.)
575  \begin{itemize}
576   \item Topotactic transformation based on \cs
577   \item \si{} as supply reacting with further C in cleared volume
578  \end{itemize}
579 \item Annealing behavior (Serre et al.)
580  \begin{itemize}
581   \item Room temperature implants $\rightarrow$ highly mobile C
582   \item Elevated T implants $\rightarrow$ no/low C redistribution/migration\\
583         (indicate stable \cs{} configurations)
584  \end{itemize}
585 \item Strained silicon \& Si/SiC heterostructures
586  \begin{itemize}
587   \item Coherent SiC precipitates (tensile strain)
588   \item Incoherent SiC (strain relaxation)
589  \end{itemize}
590 \end{itemize}
591 \end{minipage}
592  }}}
593 \end{pspicture}
594
595 \end{slide}
596
597 \begin{slide}
598
599  {\large\bf
600   Molecular dynamics (MD) simulations
601  }
602
603  \vspace{12pt}
604
605  \small
606
607  {\bf MD basics:}
608  \begin{itemize}
609   \item Microscopic description of N particle system
610   \item Analytical interaction potential
611   \item Numerical integration using Newtons equation of motion\\
612         as a propagation rule in 6N-dimensional phase space
613   \item Observables obtained by time and/or ensemble averages
614  \end{itemize}
615  {\bf Details of the simulation:}
616  \begin{itemize}
617   \item Integration: Velocity Verlet, timestep: $1\text{ fs}$
618   \item Ensemble: NpT (isothermal-isobaric)
619         \begin{itemize}
620          \item Berendsen thermostat:
621                $\tau_{\text{T}}=100\text{ fs}$
622          \item Berendsen barostat:\\
623                $\tau_{\text{P}}=100\text{ fs}$,
624                $\beta^{-1}=100\text{ GPa}$
625         \end{itemize}
626   \item Erhart/Albe potential: Tersoff-like bond order potential
627   \vspace*{12pt}
628         \[
629         E = \frac{1}{2} \sum_{i \neq j} \pot_{ij}, \quad
630         \pot_{ij} = {\color{red}f_C(r_{ij})}
631         \left[ f_R(r_{ij}) + {\color{blue}b_{ij}} f_A(r_{ij}) \right]
632         \]
633  \end{itemize}
634
635  \begin{picture}(0,0)(-230,-30)
636   \includegraphics[width=5cm]{tersoff_angle.eps} 
637  \end{picture}
638  
639 \end{slide}
640
641 \begin{slide}
642
643  {\large\bf
644   Density functional theory (DFT) calculations
645  }
646
647  \small
648
649  Basic ingredients necessary for DFT
650
651  \begin{itemize}
652   \item \underline{Hohenberg-Kohn theorem} - ground state density $n_0(r)$ ...
653         \begin{itemize}
654          \item ... uniquely determines the ground state potential
655                / wavefunctions
656          \item ... minimizes the systems total energy
657         \end{itemize}
658   \item \underline{Born-Oppenheimer}
659         - $N$ moving electrons in an external potential of static nuclei
660 \[
661 H\Psi = \left[-\sum_i^N \frac{\hbar^2}{2m}\nabla_i^2
662               +\sum_i^N V_{\text{ext}}(r_i)
663               +\sum_{i<j}^N V_{e-e}(r_i,r_j)\right]\Psi=E\Psi
664 \]
665   \item \underline{Effective potential}
666         - averaged electrostatic potential \& exchange and correlation
667 \[
668 V_{\text{eff}}(r)=V_{\text{ext}}(r)+\int\frac{e^2 n(r')}{|r-r'|}d^3r'
669                  +V_{\text{XC}}[n(r)]
670 \]
671   \item \underline{Kohn-Sham system}
672         - Schr\"odinger equation of N non-interacting particles
673 \[
674 \left[ -\frac{\hbar^2}{2m}\nabla^2 + V_{\text{eff}}(r) \right] \Phi_i(r)
675 =\epsilon_i\Phi_i(r)
676 \quad
677 \Rightarrow
678 \quad
679 n(r)=\sum_i^N|\Phi_i(r)|^2
680 \]
681   \item \underline{Self-consistent solution}\\
682 $n(r)$ depends on $\Phi_i$, which depend on $V_{\text{eff}}$,
683 which in turn depends on $n(r)$
684   \item \underline{Variational principle}
685         - minimize total energy with respect to $n(r)$
686  \end{itemize}
687
688 \end{slide}
689
690 \begin{slide}
691
692  {\large\bf
693   Density functional theory (DFT) calculations
694  }
695
696  \small
697
698  \vspace*{0.2cm}
699
700  Details of applied DFT calculations in this work
701
702  \begin{itemize}
703   \item \underline{Exchange correlation functional}
704         - approximations for the inhomogeneous electron gas
705         \begin{itemize}
706          \item LDA: $E_{\text{XC}}^{\text{LDA}}[n]=\int \epsilon_{\text{XC}}(n)n(r)d^3r$
707          \item GGA: $E_{\text{XC}}^{\text{GGA}}[n]=\int \epsilon_{\text{XC}}(n,\nabla n)n(r)d^3r$
708         \end{itemize}
709   \item \underline{Plane wave basis set}
710         - approximation of the wavefunction $\Phi_i$ by plane waves $\phi_j$
711 \[
712 \rightarrow
713 \text{Fourier series: } \Phi_i=\sum_{|G+k|<G_{\text{cut}}} c_j^i \phi_j(r), \quad E_{\text{cut}}=\frac{\hbar^2}{2m}G^2_{\text{cut}}
714 \qquad ({\color{blue}300\text{ eV}})
715 \]
716   \item \underline{Brillouin zone sampling} -
717         {\color{blue}$\Gamma$-point only} calculations
718   \item \underline{Pseudo potential} 
719         - consider only the valence electrons
720   \item \underline{Code} - VASP 4.6
721  \end{itemize}
722
723  \vspace*{0.2cm}
724
725  MD and structural optimization
726
727  \begin{itemize}
728   \item MD integration: Gear predictor corrector algorithm
729   \item Pressure control: Parrinello-Rahman pressure control
730   \item Structural optimization: Conjugate gradient method
731  \end{itemize}
732
733 \begin{pspicture}(0,0)(0,0)
734 \psellipse[linecolor=blue](1.5,6.75)(0.5,0.3)
735 \end{pspicture}
736
737 \end{slide}
738
739 \begin{slide}
740
741  {\large\bf
742   C and Si self-interstitial point defects in silicon
743  }
744
745  \small
746
747  \vspace*{0.3cm}
748
749 \begin{minipage}{8cm}
750 Procedure:\\[0.3cm]
751   \begin{pspicture}(0,0)(7,5)
752   \rput(3.5,4){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
753    \parbox{7cm}{
754    \begin{itemize}
755     \item Creation of c-Si simulation volume
756     \item Periodic boundary conditions
757     \item $T=0\text{ K}$, $p=0\text{ bar}$
758    \end{itemize}
759   }}}}
760 \rput(3.5,2.1){\rnode{insert}{\psframebox{
761  \parbox{7cm}{
762   \begin{center}
763   Insertion of interstitial C/Si atoms
764   \end{center}
765   }}}}
766   \rput(3.5,1){\rnode{cool}{\psframebox[fillstyle=solid,fillcolor=lbb]{
767    \parbox{7cm}{
768    \begin{center}
769    Relaxation / structural energy minimization
770    \end{center}
771   }}}}
772   \ncline[]{->}{init}{insert}
773   \ncline[]{->}{insert}{cool}
774  \end{pspicture}
775 \end{minipage}
776 \begin{minipage}{5cm}
777   \includegraphics[width=5cm]{unit_cell_e.eps}\\
778 \end{minipage}
779
780 \begin{minipage}{9cm}
781  \begin{tabular}{l c c}
782  \hline
783  & size [unit cells] & \# atoms\\
784 \hline
785 VASP & $3\times 3\times 3$ & $216\pm 1$ \\
786 Erhart/Albe & $9\times 9\times 9$ & $5832\pm 1$\\
787 \hline
788  \end{tabular}
789 \end{minipage}
790 \begin{minipage}{4cm}
791 {\color{red}$\bullet$} Tetrahedral\\
792 {\color{green}$\bullet$} Hexagonal\\
793 {\color{yellow}$\bullet$} \hkl<1 0 0> dumbbell\\
794 {\color{magenta}$\bullet$} \hkl<1 1 0> dumbbell\\
795 {\color{cyan}$\bullet$} Bond-centered\\
796 {\color{black}$\bullet$} Vacancy / Substitutional
797 \end{minipage}
798
799 \end{slide}
800
801 \begin{slide}
802
803  \footnotesize
804
805 \begin{minipage}{9.5cm}
806
807  {\large\bf
808   Si self-interstitial point defects in silicon\\
809  }
810
811 \begin{tabular}{l c c c c c}
812 \hline
813  $E_{\text{f}}$ [eV] & \hkl<1 1 0> DB & H & T & \hkl<1 0 0> DB & V \\
814 \hline
815  VASP & \underline{3.39} & 3.42 & 3.77 & 4.41 & 3.63 \\
816  Erhart/Albe & 4.39 & 4.48$^*$ & \underline{3.40} & 5.42 & 3.13 \\
817 \hline
818 \end{tabular}\\[0.2cm]
819
820 \begin{minipage}{4.7cm}
821 \includegraphics[width=4.7cm]{e_kin_si_hex.ps}
822 \end{minipage}
823 \begin{minipage}{4.7cm}
824 \begin{center}
825 {\tiny nearly T $\rightarrow$ T}\\
826 \end{center}
827 \includegraphics[width=4.7cm]{nhex_tet.ps}
828 \end{minipage}\\
829
830 \underline{Hexagonal} \hspace{2pt}
831 \href{../video/si_self_int_hexa.avi}{$\rhd$}\\[0.1cm]
832 \framebox{
833 \begin{minipage}{2.7cm}
834 $E_{\text{f}}^*=4.48\text{ eV}$\\
835 \includegraphics[width=2.7cm]{si_pd_albe/hex_a.eps}
836 \end{minipage}
837 \begin{minipage}{0.4cm}
838 \begin{center}
839 $\Rightarrow$
840 \end{center}
841 \end{minipage}
842 \begin{minipage}{2.7cm}
843 $E_{\text{f}}=3.96\text{ eV}$\\
844 \includegraphics[width=2.8cm]{si_pd_albe/hex.eps}
845 \end{minipage}
846 }
847 \begin{minipage}{2.9cm}
848 \begin{flushright}
849 \underline{Vacancy}\\
850 \includegraphics[width=3.0cm]{si_pd_albe/vac.eps}
851 \end{flushright}
852 \end{minipage}
853
854 \end{minipage}
855 \begin{minipage}{3.5cm}
856
857 \begin{flushright}
858 \underline{\hkl<1 1 0> dumbbell}\\
859 \includegraphics[width=3.0cm]{si_pd_albe/110.eps}\\
860 \underline{Tetrahedral}\\
861 \includegraphics[width=3.0cm]{si_pd_albe/tet.eps}\\
862 \underline{\hkl<1 0 0> dumbbell}\\
863 \includegraphics[width=3.0cm]{si_pd_albe/100.eps}
864 \end{flushright}
865
866 \end{minipage}
867
868 \end{slide}
869
870 \begin{slide}
871
872 \footnotesize
873
874  {\large\bf
875   C interstitial point defects in silicon\\[-0.1cm]
876  }
877
878 \begin{tabular}{l c c c c c c r}
879 \hline
880  $E_{\text{f}}$ & T & H & \hkl<1 0 0> DB & \hkl<1 1 0> DB & S & B & \cs{} \& \si\\
881 \hline
882  VASP & unstable & unstable & \underline{3.72} & 4.16 & 1.95 & 4.66 & {\color{green}4.17}\\
883  Erhart/Albe MD & 6.09 & 9.05$^*$ & \underline{3.88} & 5.18 & {\color{red}0.75} & 5.59$^*$ & {\color{green}4.43} \\
884 \hline
885 \end{tabular}\\[0.1cm]
886
887 \framebox{
888 \begin{minipage}{2.7cm}
889 \underline{Hexagonal} \hspace{2pt}
890 \href{../video/c_in_si_int_hexa.avi}{$\rhd$}\\
891 $E_{\text{f}}^*=9.05\text{ eV}$\\
892 \includegraphics[width=2.7cm]{c_pd_albe/hex.eps}
893 \end{minipage}
894 \begin{minipage}{0.4cm}
895 \begin{center}
896 $\Rightarrow$
897 \end{center}
898 \end{minipage}
899 \begin{minipage}{2.7cm}
900 \underline{\hkl<1 0 0>}\\
901 $E_{\text{f}}=3.88\text{ eV}$\\
902 \includegraphics[width=2.7cm]{c_pd_albe/100.eps}
903 \end{minipage}
904 }
905 \begin{minipage}{2cm}
906 \hfill
907 \end{minipage}
908 \begin{minipage}{3cm}
909 \begin{flushright}
910 \underline{Tetrahedral}\\
911 \includegraphics[width=3.0cm]{c_pd_albe/tet.eps}
912 \end{flushright}
913 \end{minipage}
914
915 \framebox{
916 \begin{minipage}{2.7cm}
917 \underline{Bond-centered}\\
918 $E_{\text{f}}^*=5.59\text{ eV}$\\
919 \includegraphics[width=2.7cm]{c_pd_albe/bc.eps}
920 \end{minipage}
921 \begin{minipage}{0.4cm}
922 \begin{center}
923 $\Rightarrow$
924 \end{center}
925 \end{minipage}
926 \begin{minipage}{2.7cm}
927 \underline{\hkl<1 1 0> dumbbell}\\
928 $E_{\text{f}}=5.18\text{ eV}$\\
929 \includegraphics[width=2.7cm]{c_pd_albe/110.eps}
930 \end{minipage}
931 }
932 \begin{minipage}{2cm}
933 \hfill
934 \end{minipage}
935 \begin{minipage}{3cm}
936 \begin{flushright}
937 \underline{Substitutional}\\
938 \includegraphics[width=3.0cm]{c_pd_albe/sub.eps}
939 \end{flushright}
940 \end{minipage}
941
942 \end{slide}
943
944 \begin{slide}
945
946 \footnotesize
947
948  {\large\bf\boldmath
949   C \hkl<1 0 0> dumbbell interstitial configuration\\
950  }
951
952 {\tiny
953 \begin{tabular}{l c c c c c c c c}
954 \hline
955  Distances [nm] & $r(1C)$ & $r(2C)$ & $r(3C)$ & $r(12)$ & $r(13)$ & $r(34)$ & $r(23)$ & $r(25)$ \\
956 \hline
957 Erhart/Albe & 0.175 & 0.329 & 0.186 & 0.226 & 0.300 & 0.343 & 0.423 & 0.425 \\
958 VASP & 0.174 & 0.341 & 0.182 & 0.229 & 0.286 & 0.347 & 0.422 & 0.417 \\
959 \hline
960 \end{tabular}\\[0.2cm]
961 \begin{tabular}{l c c c c }
962 \hline
963  Angles [$^{\circ}$] & $\theta_1$ & $\theta_2$ & $\theta_3$ & $\theta_4$ \\
964 \hline
965 Erhart/Albe & 140.2 & 109.9 & 134.4 & 112.8 \\
966 VASP & 130.7 & 114.4 & 146.0 & 107.0 \\
967 \hline
968 \end{tabular}\\[0.2cm]
969 \begin{tabular}{l c c c}
970 \hline
971  Displacements [nm]& $a$ & $b$ & $|a|+|b|$ \\
972 \hline
973 Erhart/Albe & 0.084 & -0.091 & 0.175 \\
974 VASP & 0.109 & -0.065 & 0.174 \\
975 \hline
976 \end{tabular}\\[0.6cm]
977 }
978
979 \begin{minipage}{3.0cm}
980 \begin{center}
981 \underline{Erhart/Albe}
982 \includegraphics[width=3.0cm]{c_pd_albe/100_cmp.eps}
983 \end{center}
984 \end{minipage}
985 \begin{minipage}{3.0cm}
986 \begin{center}
987 \underline{VASP}
988 \includegraphics[width=3.0cm]{c_pd_vasp/100_cmp.eps}
989 \end{center}
990 \end{minipage}\\
991
992 \begin{picture}(0,0)(-185,10)
993 \includegraphics[width=6.8cm]{100-c-si-db_cmp.eps}
994 \end{picture}
995 \begin{picture}(0,0)(-280,-150)
996 \includegraphics[width=3.3cm]{c_pd_vasp/eden.eps}
997 \end{picture}
998
999 \begin{pspicture}(0,0)(0,0)
1000 \psellipse[linecolor=green](5.18,5.92)(0.5,0.3)
1001 \psellipse[linecolor=red](3.45,5.92)(1.0,0.4)
1002 \psellipse[linecolor=blue](2.7,6.92)(0.9,0.2)
1003 \psellipse[linecolor=blue](4.65,6.92)(0.9,0.2)
1004 \end{pspicture}
1005
1006 \end{slide}
1007
1008 \begin{slide}
1009
1010 \small
1011
1012 \begin{minipage}{8.5cm}
1013
1014  {\large\bf
1015   Bond-centered interstitial configuration\\[-0.1cm]
1016  }
1017
1018 \begin{minipage}{3.0cm}
1019 \includegraphics[width=2.8cm]{c_pd_vasp/bc_2333.eps}\\
1020 \end{minipage}
1021 \begin{minipage}{5.2cm}
1022 \begin{itemize}
1023  \item Linear Si-C-Si bond
1024  \item Si: one C \& 3 Si neighbours
1025  \item Spin polarized calculations
1026  \item No saddle point!\\
1027        Real local minimum!
1028 \end{itemize}
1029 \end{minipage}
1030
1031 \framebox{
1032  \tiny
1033  \begin{minipage}[t]{6.5cm}
1034   \begin{minipage}[t]{1.2cm}
1035   {\color{red}Si}\\
1036   {\tiny sp$^3$}\\[0.8cm]
1037   \underline{${\color{black}\uparrow}$}
1038   \underline{${\color{black}\uparrow}$}
1039   \underline{${\color{black}\uparrow}$}
1040   \underline{${\color{red}\uparrow}$}\\
1041   sp$^3$
1042   \end{minipage}
1043   \begin{minipage}[t]{1.4cm}
1044   \begin{center}
1045   {\color{red}M}{\color{blue}O}\\[0.8cm]
1046   \underline{${\color{blue}\uparrow}{\color{white}\downarrow}$}\\
1047   $\sigma_{\text{ab}}$\\[0.5cm]
1048   \underline{${\color{red}\uparrow}{\color{blue}\downarrow}$}\\
1049   $\sigma_{\text{b}}$
1050   \end{center}
1051   \end{minipage}
1052   \begin{minipage}[t]{1.0cm}
1053   \begin{center}
1054   {\color{blue}C}\\
1055   {\tiny sp}\\[0.2cm]
1056   \underline{${\color{white}\uparrow\uparrow}$}
1057   \underline{${\color{white}\uparrow\uparrow}$}\\
1058   2p\\[0.4cm]
1059   \underline{${\color{blue}\uparrow}{\color{blue}\downarrow}$}
1060   \underline{${\color{blue}\uparrow}{\color{blue}\downarrow}$}\\
1061   sp
1062   \end{center}
1063   \end{minipage}
1064   \begin{minipage}[t]{1.4cm}
1065   \begin{center}
1066   {\color{blue}M}{\color{green}O}\\[0.8cm]
1067   \underline{${\color{blue}\uparrow}{\color{white}\downarrow}$}\\
1068   $\sigma_{\text{ab}}$\\[0.5cm]
1069   \underline{${\color{green}\uparrow}{\color{blue}\downarrow}$}\\
1070   $\sigma_{\text{b}}$
1071   \end{center}
1072   \end{minipage}
1073   \begin{minipage}[t]{1.2cm}
1074   \begin{flushright}
1075   {\color{green}Si}\\
1076   {\tiny sp$^3$}\\[0.8cm]
1077   \underline{${\color{green}\uparrow}$}
1078   \underline{${\color{black}\uparrow}$}
1079   \underline{${\color{black}\uparrow}$}
1080   \underline{${\color{black}\uparrow}$}\\
1081   sp$^3$
1082   \end{flushright}
1083   \end{minipage}
1084  \end{minipage}
1085 }\\[0.1cm]
1086
1087 \framebox{
1088 \begin{minipage}{4.5cm}
1089 \includegraphics[width=4cm]{c_100_mig_vasp/im_spin_diff.eps}
1090 \end{minipage}
1091 \begin{minipage}{3.5cm}
1092 {\color{gray}$\bullet$} Spin up\\
1093 {\color{green}$\bullet$} Spin down\\
1094 {\color{blue}$\bullet$} Resulting spin up\\
1095 {\color{yellow}$\bullet$} Si atoms\\
1096 {\color{red}$\bullet$} C atom
1097 \end{minipage}
1098 }
1099
1100 \end{minipage}
1101 \begin{minipage}{4.2cm}
1102 \begin{flushright}
1103 \includegraphics[width=4.3cm]{c_pd_vasp/bc_2333_ksl.ps}\\
1104 {\color{green}$\Box$} {\tiny unoccupied}\\
1105 {\color{red}$\bullet$} {\tiny occupied}
1106 \end{flushright}
1107 \end{minipage}
1108
1109 \end{slide}
1110
1111 \begin{slide}
1112
1113  {\large\bf\boldmath
1114   Migration of the C \hkl<1 0 0> dumbbell interstitial
1115  }
1116
1117 \scriptsize
1118
1119  {\small Investigated pathways}
1120
1121 \begin{minipage}{8.5cm}
1122 \begin{minipage}{8.3cm}
1123 \underline{\hkl<0 0 -1> $\rightarrow$ \hkl<0 0 1>}\\
1124 \begin{minipage}{2.4cm}
1125 \includegraphics[width=2.4cm]{c_pd_vasp/100_2333.eps}
1126 \end{minipage}
1127 \begin{minipage}{0.4cm}
1128 $\rightarrow$
1129 \end{minipage}
1130 \begin{minipage}{2.4cm}
1131 \includegraphics[width=2.4cm]{c_pd_vasp/bc_2333.eps}
1132 \end{minipage}
1133 \begin{minipage}{0.4cm}
1134 $\rightarrow$
1135 \end{minipage}
1136 \begin{minipage}{2.4cm}
1137 \includegraphics[width=2.4cm]{c_pd_vasp/100_next_2333.eps}
1138 \end{minipage}
1139 \end{minipage}\\
1140 \begin{minipage}{8.3cm}
1141 \underline{\hkl<0 0 -1> $\rightarrow$ \hkl<0 -1 0>}\\
1142 \begin{minipage}{2.4cm}
1143 \includegraphics[width=2.4cm]{c_pd_vasp/100_2333.eps}
1144 \end{minipage}
1145 \begin{minipage}{0.4cm}
1146 $\rightarrow$
1147 \end{minipage}
1148 \begin{minipage}{2.4cm}
1149 \includegraphics[width=2.4cm]{c_pd_vasp/00-1-0-10_2333.eps}
1150 \end{minipage}
1151 \begin{minipage}{0.4cm}
1152 $\rightarrow$
1153 \end{minipage}
1154 \begin{minipage}{2.4cm}
1155 \includegraphics[width=2.4cm]{c_pd_vasp/0-10_2333.eps}
1156 \end{minipage}
1157 \end{minipage}\\
1158 \begin{minipage}{8.3cm}
1159 \underline{\hkl<0 0 -1> $\rightarrow$ \hkl<0 -1 0> (in place)}\\
1160 \begin{minipage}{2.4cm}
1161 \includegraphics[width=2.4cm]{c_pd_vasp/100_2333.eps}
1162 \end{minipage}
1163 \begin{minipage}{0.4cm}
1164 $\rightarrow$
1165 \end{minipage}
1166 \begin{minipage}{2.4cm}
1167 \includegraphics[width=2.4cm]{c_pd_vasp/00-1_ip0-10_2333.eps}
1168 \end{minipage}
1169 \begin{minipage}{0.4cm}
1170 $\rightarrow$
1171 \end{minipage}
1172 \begin{minipage}{2.4cm}
1173 \includegraphics[width=2.4cm]{c_pd_vasp/0-10_ip_2333.eps}
1174 \end{minipage}
1175 \end{minipage}
1176 \end{minipage}
1177 \framebox{
1178 \begin{minipage}{4.2cm}
1179  {\small Constrained relaxation\\
1180          technique (CRT) method}\\
1181 \includegraphics[width=4cm]{crt_orig.eps}
1182 \begin{itemize}
1183  \item Constrain diffusing atom
1184  \item Static constraints 
1185 \end{itemize}
1186 \vspace*{0.3cm}
1187  {\small Modifications}\\
1188 \includegraphics[width=4cm]{crt_mod.eps}
1189 \begin{itemize}
1190  \item Constrain all atoms
1191  \item Update individual\\
1192        constraints
1193 \end{itemize}
1194 \end{minipage}
1195 }
1196
1197 \end{slide}
1198
1199 \begin{slide}
1200
1201  {\large\bf\boldmath
1202   Migration of the C \hkl<1 0 0> dumbbell interstitial
1203  }
1204
1205 \scriptsize
1206
1207 \framebox{
1208 \begin{minipage}{5.9cm}
1209 \begin{flushleft}
1210 \includegraphics[width=5.8cm]{im_00-1_nosym_sp_fullct_thesis.ps}\\[0.45cm]
1211 \end{flushleft}
1212 \begin{center}
1213 \begin{picture}(0,0)(60,0)
1214 \includegraphics[width=1cm]{vasp_mig/00-1.eps}
1215 \end{picture}
1216 \begin{picture}(0,0)(-5,0)
1217 \includegraphics[width=1cm]{vasp_mig/bc_00-1_sp.eps}
1218 \end{picture}
1219 \begin{picture}(0,0)(-55,0)
1220 \includegraphics[width=1cm]{vasp_mig/bc.eps}
1221 \end{picture}
1222 \begin{picture}(0,0)(12.5,10)
1223 \includegraphics[width=1cm]{110_arrow.eps}
1224 \end{picture}
1225 \begin{picture}(0,0)(90,0)
1226 \includegraphics[height=0.9cm]{001_arrow.eps}
1227 \end{picture}
1228 \end{center}
1229 \vspace*{0.35cm}
1230 \end{minipage}
1231 }
1232 \begin{minipage}{0.3cm}
1233 \hfill
1234 \end{minipage}
1235 \framebox{
1236 \begin{minipage}{5.9cm}
1237 \begin{flushright}
1238 \includegraphics[width=5.9cm]{vasp_mig/00-1_0-10_nosym_sp_fullct.ps}\\[0.5cm]
1239 \end{flushright}
1240 \begin{center}
1241 \begin{picture}(0,0)(60,0)
1242 \includegraphics[width=1cm]{vasp_mig/00-1_a.eps}
1243 \end{picture}
1244 \begin{picture}(0,0)(5,0)
1245 \includegraphics[width=1cm]{vasp_mig/00-1_0-10_sp.eps}
1246 \end{picture}
1247 \begin{picture}(0,0)(-55,0)
1248 \includegraphics[width=1cm]{vasp_mig/0-10.eps}
1249 \end{picture}
1250 \begin{picture}(0,0)(12.5,10)
1251 \includegraphics[width=1cm]{100_arrow.eps}
1252 \end{picture}
1253 \begin{picture}(0,0)(90,0)
1254 \includegraphics[height=0.9cm]{001_arrow.eps}
1255 \end{picture}
1256 \end{center}
1257 \vspace*{0.3cm}
1258 \end{minipage}\\
1259 }
1260
1261 \vspace*{0.05cm}
1262
1263 \framebox{
1264 \begin{minipage}{5.9cm}
1265 \begin{flushleft}
1266 \includegraphics[width=5.9cm]{vasp_mig/00-1_ip0-10_nosym_sp_fullct.ps}\\[0.6cm]
1267 \end{flushleft}
1268 \begin{center}
1269 \begin{picture}(0,0)(60,0)
1270 \includegraphics[width=0.9cm]{vasp_mig/00-1_b.eps}
1271 \end{picture}
1272 \begin{picture}(0,0)(10,0)
1273 \includegraphics[width=0.9cm]{vasp_mig/00-1_ip0-10_sp.eps}
1274 \end{picture}
1275 \begin{picture}(0,0)(-60,0)
1276 \includegraphics[width=0.9cm]{vasp_mig/0-10_b.eps}
1277 \end{picture}
1278 \begin{picture}(0,0)(12.5,10)
1279 \includegraphics[width=1cm]{100_arrow.eps}
1280 \end{picture}
1281 \begin{picture}(0,0)(90,0)
1282 \includegraphics[height=0.9cm]{001_arrow.eps}
1283 \end{picture}
1284 \end{center}
1285 \vspace*{0.3cm}
1286 \end{minipage}
1287 }
1288 \begin{minipage}{0.3cm}
1289 \hfill
1290 \end{minipage}
1291 \begin{minipage}{6.5cm}
1292 VASP results
1293 \begin{itemize}
1294  \item Energetically most favorable path
1295        \begin{itemize}
1296         \item Path 2
1297         \item Activation energy: $\approx$ 0.9 eV 
1298         \item Experimental values: 0.73 ... 0.87 eV
1299        \end{itemize}
1300        $\Rightarrow$ {\color{blue}Diffusion} path identified!
1301  \item Reorientation (path 3)
1302        \begin{itemize}
1303         \item More likely composed of two consecutive steps of type 2
1304         \item Experimental values: 0.77 ... 0.88 eV
1305        \end{itemize}
1306        $\Rightarrow$ {\color{blue}Reorientation} transition identified!
1307 \end{itemize}
1308 \end{minipage}
1309
1310 \end{slide}
1311
1312 \begin{slide}
1313
1314  {\large\bf\boldmath
1315   Migration of the C \hkl<1 0 0> dumbbell interstitial
1316  }
1317
1318 \scriptsize
1319
1320  \vspace{0.1cm}
1321
1322 \begin{minipage}{6.5cm}
1323
1324 \framebox{
1325 \begin{minipage}[t]{5.9cm}
1326 \begin{flushleft}
1327 \includegraphics[width=5.9cm]{bc_00-1.ps}\\[2.35cm]
1328 \end{flushleft}
1329 \begin{center}
1330 \begin{pspicture}(0,0)(0,0)
1331 \psframe[linecolor=red,fillstyle=none](-2.8,1.35)(3.3,2.7)
1332 \end{pspicture}
1333 \begin{picture}(0,0)(60,-50)
1334 \includegraphics[width=1cm]{albe_mig/bc_00-1_red_00.eps}
1335 \end{picture}
1336 \begin{picture}(0,0)(5,-50)
1337 \includegraphics[width=1cm]{albe_mig/bc_00-1_red_01.eps}
1338 \end{picture}
1339 \begin{picture}(0,0)(-55,-50)
1340 \includegraphics[width=1cm]{albe_mig/bc_00-1_red_02.eps}
1341 \end{picture}
1342 \begin{picture}(0,0)(12.5,-40)
1343 \includegraphics[width=1cm]{110_arrow.eps}
1344 \end{picture}
1345 \begin{picture}(0,0)(90,-45)
1346 \includegraphics[height=0.9cm]{001_arrow.eps}
1347 \end{picture}\\
1348 \begin{pspicture}(0,0)(0,0)
1349 \psframe[linecolor=blue,fillstyle=none](-2.8,0)(3.3,1.6)
1350 \end{pspicture}
1351 \begin{picture}(0,0)(60,-15)
1352 \includegraphics[width=0.9cm]{albe_mig/bc_00-1_01.eps}
1353 \end{picture}
1354 \begin{picture}(0,0)(35,-15)
1355 \includegraphics[width=0.9cm]{albe_mig/bc_00-1_02.eps}
1356 \end{picture}
1357 \begin{picture}(0,0)(-5,-15)
1358 \includegraphics[width=0.9cm]{albe_mig/bc_00-1_03.eps}
1359 \end{picture}
1360 \begin{picture}(0,0)(-55,-15)
1361 \includegraphics[width=0.9cm]{albe_mig/bc_00-1_04.eps}
1362 \end{picture}
1363 \begin{picture}(0,0)(12.5,-5)
1364 \includegraphics[width=1cm]{100_arrow.eps}
1365 \end{picture}
1366 \begin{picture}(0,0)(90,-15)
1367 \includegraphics[height=0.9cm]{010_arrow.eps}
1368 \end{picture}
1369 \end{center}
1370 \end{minipage}
1371 }\\[0.1cm]
1372
1373 \begin{minipage}{5.9cm}
1374 Erhart/Albe results
1375 \begin{itemize}
1376  \item Lowest activation energy: $\approx$ 2.2 eV
1377  \item 2.4 times higher than VASP
1378  \item Different pathway
1379 \end{itemize}
1380 \end{minipage}
1381
1382 \end{minipage}
1383 \begin{minipage}{6.5cm}
1384
1385 \framebox{
1386 \begin{minipage}{5.9cm}
1387 %\begin{flushright}
1388 %\includegraphics[width=5.9cm]{00-1_0-10.ps}\\[0.75cm]
1389 %\end{flushright}
1390 %\begin{center}
1391 %\begin{pspicture}(0,0)(0,0)
1392 %\psframe[linecolor=red,fillstyle=none](-2.8,-0.25)(3.3,1.1)
1393 %\end{pspicture}
1394 %\begin{picture}(0,0)(60,-5)
1395 %\includegraphics[width=0.9cm]{albe_mig/00-1_0-10_red_00.eps}
1396 %\end{picture}
1397 %\begin{picture}(0,0)(0,-5)
1398 %\includegraphics[width=0.9cm]{albe_mig/00-1_0-10_red_min.eps}
1399 %\end{picture}
1400 %\begin{picture}(0,0)(-55,-5)
1401 %\includegraphics[width=0.9cm]{albe_mig/00-1_0-10_red_03.eps}
1402 %\end{picture}
1403 %\begin{picture}(0,0)(12.5,5)
1404 %\includegraphics[width=1cm]{100_arrow.eps}
1405 %\end{picture}
1406 %\begin{picture}(0,0)(90,0)
1407 %\includegraphics[height=0.9cm]{001_arrow.eps}
1408 %\end{picture}
1409 %\end{center}
1410 %\vspace{0.2cm}
1411 %\end{minipage}
1412 %}\\[0.2cm]
1413 %
1414 %\framebox{
1415 %\begin{minipage}{5.9cm}
1416 \includegraphics[width=5.9cm]{00-1_110_0-10_mig_albe.ps}
1417 \end{minipage}
1418 }\\[0.1cm]
1419
1420 \begin{minipage}{5.9cm}
1421 Transition involving \ci{} \hkl<1 1 0>
1422 \begin{itemize}
1423  \item Bond-centered configuration unstable\\
1424        $\rightarrow$ \ci{} \hkl<1 1 0> dumbbell
1425  \item Transition minima of path 2 \& 3\\
1426        $\rightarrow$ \ci{} \hkl<1 1 0> dumbbell
1427  \item Activation energy: $\approx$ 2.2 eV \& 0.9 eV
1428  \item 2.4 - 3.4 times higher than VASP
1429  \item Rotation of dumbbell orientation
1430 \end{itemize}
1431 \vspace{0.1cm}
1432 \begin{center}
1433 {\color{blue}Overestimated diffusion barrier}
1434 \end{center}
1435 \end{minipage}
1436
1437 \end{minipage}
1438
1439 \end{slide}
1440
1441 \begin{slide}
1442
1443  {\large\bf\boldmath
1444   Combinations with a C-Si \hkl<1 0 0>-type interstitial
1445  }
1446
1447 \small
1448
1449 \vspace*{0.1cm}
1450
1451 Binding energy: 
1452 $
1453 E_{\text{b}}=
1454 E_{\text{f}}^{\text{defect combination}}-
1455 E_{\text{f}}^{\text{C \hkl<0 0 -1> dumbbell}}-
1456 E_{\text{f}}^{\text{2nd defect}}
1457 $
1458
1459 \vspace*{0.1cm}
1460
1461 {\scriptsize
1462 \begin{tabular}{l c c c c c c}
1463 \hline
1464  $E_{\text{b}}$ [eV] & 1 & 2 & 3 & 4 & 5 & R\\
1465  \hline
1466  \hkl<0 0 -1> & {\color{red}-0.08} & -1.15 & {\color{red}-0.08} & 0.04 & -1.66 & -0.19\\
1467  \hkl<0 0 1> & 0.34 & 0.004 & -2.05 & 0.26 & -1.53 & -0.19\\
1468  \hkl<0 -1 0> & {\color{orange}-2.39} & -0.17 & {\color{green}-0.10} & {\color{blue}-0.27} & {\color{magenta}-1.88} & {\color{gray}-0.05}\\
1469  \hkl<0 1 0> & {\color{cyan}-2.25} & -1.90 & {\color{cyan}-2.25} & {\color{purple}-0.12} & {\color{violet}-1.38} & {\color{yellow}-0.06}\\
1470  \hkl<-1 0 0> & {\color{orange}-2.39} & -0.36 & {\color{cyan}-2.25} & {\color{purple}-0.12} & {\color{magenta}-1.88} & {\color{gray}-0.05}\\
1471  \hkl<1 0 0> & {\color{cyan}-2.25} & -2.16 & {\color{green}-0.10} & {\color{blue}-0.27} & {\color{violet}-1.38} & {\color{yellow}-0.06}\\
1472  \hline
1473  C substitutional (C$_{\text{S}}$) & 0.26 & -0.51 & -0.93 & -0.15 & 0.49 & -0.05\\
1474  Vacancy & -5.39 ($\rightarrow$ C$_{\text{S}}$) & -0.59 & -3.14 & -0.54 & -0.50 & -0.31\\
1475 \hline
1476 \end{tabular}
1477 }
1478
1479 \vspace*{0.3cm}
1480
1481 \footnotesize
1482
1483 \begin{minipage}[t]{3.8cm}
1484 \underline{\hkl<1 0 0> at position 1}\\[0.1cm]
1485 \includegraphics[width=3.5cm]{00-1dc/2-25.eps}
1486 \end{minipage}
1487 \begin{minipage}[t]{3.5cm}
1488 \underline{\hkl<0 -1 0> at position 1}\\[0.1cm]
1489 \includegraphics[width=3.2cm]{00-1dc/2-39.eps}
1490 \end{minipage}
1491 \begin{minipage}[t]{5.5cm}
1492 \begin{itemize}
1493  \item $E_{\text{b}}=0$ $\Leftrightarrow$ non-interacting defects\\
1494        $E_{\text{b}} \rightarrow 0$ for increasing distance (R)
1495  \item Stress compensation / increase
1496  \item Unfavored: antiparallel orientations
1497  \item Indication of energetically favored\\
1498        agglomeration
1499  \item Most favorable: C clustering
1500  \item However: High barrier ($>4\,\text{eV}$)
1501  \item $4\times{\color{cyan}-2.25}$ versus $2\times{\color{orange}-2.39}$
1502        (Entropy)
1503 \end{itemize}
1504 \end{minipage}
1505
1506 \begin{picture}(0,0)(-295,-130)
1507 \includegraphics[width=3.5cm]{comb_pos.eps}
1508 \end{picture}
1509
1510 \end{slide}
1511
1512 \begin{slide}
1513
1514  {\large\bf\boldmath
1515   Combinations of C-Si \hkl<1 0 0>-type interstitials
1516  }
1517
1518 \small
1519
1520 \vspace*{0.1cm}
1521
1522 Energetically most favorable combinations along \hkl<1 1 0>
1523
1524 \vspace*{0.1cm}
1525
1526 {\scriptsize
1527 \begin{tabular}{l c c c c c c}
1528 \hline
1529  & 1 & 2 & 3 & 4 & 5 & 6\\
1530 \hline
1531 $E_{\text{b}}$ [eV] & -2.39 & -1.88 & -0.59 & -0.31 & -0.24 & -0.21 \\
1532 C-C distance [\AA] & 1.4 & 4.6 & 6.5 & 8.6 & 10.5 & 10.8 \\
1533 Type & \hkl<-1 0 0> & \hkl<1 0 0> & \hkl<1 0 0> & \hkl<1 0 0> & \hkl<1 0 0> & \hkl<1 0 0>, \hkl<0 -1 0>\\
1534 \hline
1535 \end{tabular}
1536 }
1537
1538 \vspace*{0.3cm}
1539
1540 \begin{minipage}{7.0cm}
1541 \includegraphics[width=7cm]{db_along_110_cc.ps}
1542 \end{minipage}
1543 \begin{minipage}{6.0cm}
1544 \begin{itemize}
1545  \item Interaction proportional to reciprocal cube of C-C distance
1546  \item Saturation in the immediate vicinity
1547  \renewcommand\labelitemi{$\Rightarrow$}
1548  \item Agglomeration of \ci{} expected
1549  \item Absence of C clustering
1550 \end{itemize}
1551 \begin{center}
1552 {\color{blue}
1553  Consisten with initial precipitation model
1554 }
1555 \end{center}
1556 \end{minipage}
1557
1558 \vspace{0.2cm}
1559
1560 \end{slide}
1561
1562 \begin{slide}
1563
1564  {\large\bf\boldmath
1565   Combinations of substitutional C and \hkl<1 1 0> Si self-interstitials
1566  }
1567
1568  \scriptsize
1569
1570 %\begin{center}
1571 %\begin{minipage}{3.2cm}
1572 %\includegraphics[width=3cm]{sub_110_combo.eps}
1573 %\end{minipage}
1574 %\begin{minipage}{7.8cm}
1575 %\begin{tabular}{l c c c c c c}
1576 %\hline
1577 %C$_{\text{sub}}$ & \hkl<1 1 0> & \hkl<-1 1 0> & \hkl<0 1 1> & \hkl<0 -1 1> &
1578 %                   \hkl<1 0 1> & \hkl<-1 0 1> \\
1579 %\hline
1580 %1 & \RM{1} & \RM{3} & \RM{3} & \RM{1} & \RM{3} & \RM{1} \\
1581 %2 & \RM{2} & A & A & \RM{2} & C & \RM{5} \\
1582 %3 & \RM{3} & \RM{1} & \RM{3} & \RM{1} & \RM{1} & \RM{3} \\
1583 %4 & \RM{4} & B & D & E & E & D \\
1584 %5 & \RM{5} & C & A & \RM{2} & A & \RM{2} \\
1585 %\hline
1586 %\end{tabular}
1587 %\end{minipage}
1588 %\end{center}
1589
1590 %\begin{center}
1591 %\begin{tabular}{l c c c c c c c c c c}
1592 %\hline
1593 %Conf & \RM{1} & \RM{2} & \RM{3} & \RM{4} & \RM{5} & A & B & C & D & E \\
1594 %\hline
1595 %$E_{\text{f}}$ [eV]& 4.37 & 5.26 & 5.57 & 5.37 & 5.12 & 5.10 & 5.32 & 5.28 & 5.39 & 5.32 \\
1596 %$E_{\text{b}}$ [eV] & -0.97 & -0.08 & 0.22 & -0.02 & -0.23 & -0.25 & -0.02 & -0.06 & 0.05 & -0.03 \\
1597 %$r$ [nm] & 0.292 & 0.394 & 0.241 & 0.453 & 0.407 & 0.408 & 0.452 & 0.392 & 0.456 & 0.453\\
1598 %\hline
1599 %\end{tabular}
1600 %\end{center}
1601
1602 \begin{minipage}{6.0cm}
1603 \includegraphics[width=5.8cm]{c_sub_si110.ps}
1604 \end{minipage}
1605 \begin{minipage}{7cm}
1606 \scriptsize
1607 \begin{itemize}
1608  \item IBS: C may displace Si\\
1609        $\Rightarrow$ C$_{\text{sub}}$ + \hkl<1 1 0> Si self-interstitial
1610  \item Assumption:\\
1611        \hkl<1 1 0>-type $\rightarrow$ favored combination
1612  \renewcommand\labelitemi{$\Rightarrow$}
1613  \item Most favorable: \cs{} along \hkl<1 1 0> chain \si{}
1614  \item Less favorable than C-Si \hkl<1 0 0> dumbbell
1615  \item Interaction drops quickly to zero\\
1616        $\rightarrow$ low capture radius
1617 \end{itemize}
1618 \begin{center}
1619  {\color{blue}
1620  IBS process far from equilibrium\\
1621  \cs{} \& \si{} instead of thermodynamic ground state
1622  }
1623 \end{center}
1624 \end{minipage}
1625
1626 \begin{minipage}{6.5cm}
1627 \includegraphics[width=6.0cm]{162-097.ps}
1628 \begin{itemize}
1629  \item Low migration barrier
1630 \end{itemize}
1631 \end{minipage}
1632 \begin{minipage}{6.5cm}
1633 \begin{center}
1634 Ab initio MD at \degc{900}\\
1635 \includegraphics[width=3.3cm]{md_vasp_01.eps}
1636 $t=\unit[2230]{fs}$\\
1637 \includegraphics[width=3.3cm]{md_vasp_02.eps}
1638 $t=\unit[2900]{fs}$
1639 \end{center}
1640 {\color{blue}
1641 Contribution of entropy to structural formation
1642 }
1643 \end{minipage}
1644
1645 \end{slide}
1646
1647 \begin{slide}
1648
1649  {\large\bf\boldmath
1650   Migration in C-Si \hkl<1 0 0> and vacancy combinations
1651  }
1652
1653  \footnotesize
1654
1655 \vspace{0.1cm}
1656
1657 \begin{minipage}[t]{3cm}
1658 \underline{Pos 2, $E_{\text{b}}=-0.59\text{ eV}$}\\
1659 \includegraphics[width=2.8cm]{00-1dc/0-59.eps}
1660 \end{minipage}
1661 \begin{minipage}[t]{7cm}
1662 \vspace{0.2cm}
1663 \begin{center}
1664  Low activation energies\\
1665  High activation energies for reverse processes\\
1666  $\Downarrow$\\
1667  {\color{blue}C$_{\text{sub}}$ very stable}\\
1668 \vspace*{0.1cm}
1669  \hrule
1670 \vspace*{0.1cm}
1671  Without nearby \hkl<1 1 0> Si self-interstitial (IBS)\\
1672  $\Downarrow$\\
1673  {\color{blue}Formation of SiC by successive substitution by C}
1674
1675 \end{center}
1676 \end{minipage}
1677 \begin{minipage}[t]{3cm}
1678 \underline{Pos 3, $E_{\text{b}}=-3.14\text{ eV}$}\\
1679 \includegraphics[width=2.8cm]{00-1dc/3-14.eps}
1680 \end{minipage}
1681
1682
1683 \framebox{
1684 \begin{minipage}{5.9cm}
1685 \includegraphics[width=5.9cm]{vasp_mig/comb_mig_3-2_vac_fullct.ps}\\[0.6cm]
1686 \begin{center}
1687 \begin{picture}(0,0)(70,0)
1688 \includegraphics[width=1.4cm]{vasp_mig/comb_2-1_init.eps}
1689 \end{picture}
1690 \begin{picture}(0,0)(30,0)
1691 \includegraphics[width=1.4cm]{vasp_mig/comb_2-1_seq_03.eps}
1692 \end{picture}
1693 \begin{picture}(0,0)(-10,0)
1694 \includegraphics[width=1.4cm]{vasp_mig/comb_2-1_seq_06.eps}
1695 \end{picture}
1696 \begin{picture}(0,0)(-48,0)
1697 \includegraphics[width=1.4cm]{vasp_mig/comb_2-1_final.eps}
1698 \end{picture}
1699 \begin{picture}(0,0)(12.5,5)
1700 \includegraphics[width=1cm]{100_arrow.eps}
1701 \end{picture}
1702 \begin{picture}(0,0)(97,-10)
1703 \includegraphics[height=0.9cm]{001_arrow.eps}
1704 \end{picture}
1705 \end{center}
1706 \vspace{0.1cm}
1707 \end{minipage}
1708 }
1709 \begin{minipage}{0.3cm}
1710 \hfill
1711 \end{minipage}
1712 \framebox{
1713 \begin{minipage}{5.9cm}
1714 \includegraphics[width=5.9cm]{vasp_mig/comb_mig_4-2_vac_fullct.ps}\\[0.1cm]
1715 \begin{center}
1716 \begin{picture}(0,0)(60,0)
1717 \includegraphics[width=0.9cm]{vasp_mig/comb_3-1_init.eps}
1718 \end{picture}
1719 \begin{picture}(0,0)(25,0)
1720 \includegraphics[width=0.9cm]{vasp_mig/comb_3-1_seq_03.eps}
1721 \end{picture}
1722 \begin{picture}(0,0)(-20,0)
1723 \includegraphics[width=0.9cm]{vasp_mig/comb_3-1_seq_07.eps}
1724 \end{picture}
1725 \begin{picture}(0,0)(-55,0)
1726 \includegraphics[width=0.9cm]{vasp_mig/comb_3-1_final.eps}
1727 \end{picture}
1728 \begin{picture}(0,0)(12.5,5)
1729 \includegraphics[width=1cm]{100_arrow.eps}
1730 \end{picture}
1731 \begin{picture}(0,0)(95,0)
1732 \includegraphics[height=0.9cm]{001_arrow.eps}
1733 \end{picture}
1734 \end{center}
1735 \vspace{0.1cm}
1736 \end{minipage}
1737 }
1738
1739 \end{slide}
1740
1741 \begin{slide}
1742
1743  {\large\bf
1744   Conclusion of defect / migration / combined defect simulations
1745  }
1746
1747  \footnotesize
1748
1749 \vspace*{0.1cm}
1750
1751 Defect structures
1752 \begin{itemize}
1753  \item Accurately described by quantum-mechanical simulations
1754  \item Less accurate description by classical potential simulations
1755  \item Underestimated formation energy of \cs{} by classical approach
1756  \item Both methods predict same ground state: \ci{} \hkl<1 0 0> dumbbell
1757 \end{itemize}
1758
1759 Migration
1760 \begin{itemize}
1761  \item C migration pathway in Si identified
1762  \item Consistent with reorientation and diffusion experiments
1763 \end{itemize} 
1764 \begin{itemize}
1765  \item Different path and ...
1766  \item overestimated barrier by classical potential calculations
1767 \end{itemize} 
1768
1769 Concerning the precipitation mechanism
1770 \begin{itemize}
1771  \item Agglomeration of C-Si dumbbells energetically favorable
1772        (stress compensation)
1773  \item C-Si indeed favored compared to
1774        C$_{\text{sub}}$ \& \hkl<1 1 0> Si self-interstitial
1775  \item Possible low interaction capture radius of
1776        C$_{\text{sub}}$ \& \hkl<1 1 0> Si self-interstitial
1777  \item Low barrier for
1778        \ci{} \hkl<1 0 0> $\rightarrow$ \cs{} \& \si{} \hkl<1 1 0>
1779  \item In absence of nearby \hkl<1 1 0> Si self-interstitial:
1780        C-Si \hkl<1 0 0> + Vacancy $\rightarrow$ C$_{\text{sub}}$ (SiC)
1781 \end{itemize} 
1782 \begin{center}
1783 {\color{blue}Results suggest increased participation of \cs}
1784 \end{center}
1785
1786 \end{slide}
1787
1788 \begin{slide}
1789
1790  {\large\bf
1791   Silicon carbide precipitation simulations
1792  }
1793
1794  \small
1795
1796 {\scriptsize
1797  \begin{pspicture}(0,0)(12,6.5)
1798   % nodes
1799   \rput(3.5,5.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
1800    \parbox{7cm}{
1801    \begin{itemize}
1802     \item Create c-Si volume
1803     \item Periodc boundary conditions
1804     \item Set requested $T$ and $p=0\text{ bar}$
1805     \item Equilibration of $E_{\text{kin}}$ and $E_{\text{pot}}$
1806    \end{itemize}
1807   }}}}
1808   \rput(3.5,2.7){\rnode{insert}{\psframebox[fillstyle=solid,fillcolor=lachs]{
1809    \parbox{7cm}{
1810    Insertion of C atoms at constant T
1811    \begin{itemize}
1812     \item total simulation volume {\pnode{in1}}
1813     \item volume of minimal SiC precipitate {\pnode{in2}}
1814     \item volume consisting of Si atoms to form a minimal {\pnode{in3}}\\
1815           precipitate
1816    \end{itemize} 
1817   }}}}
1818   \rput(3.5,1){\rnode{cool}{\psframebox[fillstyle=solid,fillcolor=lbb]{
1819    \parbox{7.0cm}{
1820    Run for 100 ps followed by cooling down to $20\, ^{\circ}\textrm{C}$
1821   }}}}
1822   \ncline[]{->}{init}{insert}
1823   \ncline[]{->}{insert}{cool}
1824   \psframe[fillstyle=solid,fillcolor=white](7.5,0.7)(13.5,6.3)
1825   \rput(7.8,6){\footnotesize $V_1$}
1826   \psframe[fillstyle=solid,fillcolor=lightgray](9,2)(12,5)
1827   \rput(9.2,4.85){\tiny $V_2$}
1828   \psframe[fillstyle=solid,fillcolor=gray](9.25,2.25)(11.75,4.75)
1829   \rput(9.55,4.45){\footnotesize $V_3$}
1830   \rput(7.9,3.2){\pnode{ins1}}
1831   \rput(9.22,2.8){\pnode{ins2}}
1832   \rput(11.0,2.4){\pnode{ins3}}
1833   \ncline[]{->}{in1}{ins1}
1834   \ncline[]{->}{in2}{ins2}
1835   \ncline[]{->}{in3}{ins3}
1836  \end{pspicture}
1837 }
1838
1839 \begin{itemize}
1840  \item Restricted to classical potential simulations
1841  \item $V_2$ and $V_3$ considered due to low diffusion
1842  \item Amount of C atoms: 6000
1843        ($r_{\text{prec}}\approx 3.1\text{ nm}$, IBS: 2 ... 4 nm)
1844  \item Simulation volume: $31\times 31\times 31$ unit cells
1845        (238328 Si atoms)
1846 \end{itemize}
1847
1848 \end{slide}
1849
1850 \begin{slide}
1851
1852  {\large\bf\boldmath
1853   Silicon carbide precipitation simulations at $450\,^{\circ}\mathrm{C}$ as in IBS
1854  }
1855
1856  \small
1857
1858 \begin{minipage}{6.5cm}
1859 \includegraphics[width=6.4cm]{sic_prec_450_si-si_c-c.ps}
1860 \end{minipage} 
1861 \begin{minipage}{6.5cm}
1862 \includegraphics[width=6.4cm]{sic_prec_450_energy.ps}
1863 \end{minipage} 
1864
1865 \begin{minipage}{6.5cm}
1866 \includegraphics[width=6.4cm]{sic_prec_450_si-c.ps}
1867 \end{minipage} 
1868 \begin{minipage}{6.5cm}
1869 \scriptsize
1870 \underline{Low C concentration ($V_1$)}\\
1871 \hkl<1 0 0> C-Si dumbbell dominated structure
1872 \begin{itemize}
1873  \item Si-C bumbs around 0.19 nm
1874  \item C-C peak at 0.31 nm (as expected in 3C-SiC):\\
1875        concatenated dumbbells of various orientation
1876  \item Si-Si NN distance stretched to 0.3 nm
1877 \end{itemize}
1878 {\color{blue}$\Rightarrow$ C atoms in proper 3C-SiC distance first}\\
1879 \underline{High C concentration ($V_2$, $V_3$)}\\
1880 High amount of strongly bound C-C bonds\\
1881 Defect density $\uparrow$ $\Rightarrow$ considerable amount of damage\\
1882 Only short range order observable\\
1883 {\color{blue}$\Rightarrow$ amorphous SiC-like phase}
1884 \end{minipage} 
1885
1886 \end{slide}
1887
1888 \begin{slide}
1889
1890  {\large\bf\boldmath
1891   Silicon carbide precipitation simulations at $450\,^{\circ}\mathrm{C}$ as in IBS
1892  }
1893
1894  \small
1895
1896 \begin{minipage}{6.5cm}
1897 \includegraphics[width=6.4cm]{sic_prec_450_si-si_c-c.ps}
1898 \end{minipage} 
1899 \begin{minipage}{6.5cm}
1900 \includegraphics[width=6.4cm]{sic_prec_450_energy.ps}
1901 \end{minipage} 
1902
1903 \begin{minipage}{6.5cm}
1904 \includegraphics[width=6.4cm]{sic_prec_450_si-c.ps}
1905 \end{minipage} 
1906 \begin{minipage}{6.5cm}
1907 \scriptsize
1908 \underline{Low C concentration ($V_1$)}\\
1909 \hkl<1 0 0> C-Si dumbbell dominated structure
1910 \begin{itemize}
1911  \item Si-C bumbs around 0.19 nm
1912  \item C-C peak at 0.31 nm (as expected in 3C-SiC):\\
1913        concatenated dumbbells of various orientation
1914  \item Si-Si NN distance stretched to 0.3 nm
1915 \end{itemize}
1916 {\color{blue}$\Rightarrow$ C atoms in proper 3C-SiC distance first}\\
1917 \underline{High C concentration ($V_2$, $V_3$)}\\
1918 High amount of strongly bound C-C bonds\\
1919 Defect density $\uparrow$ $\Rightarrow$ considerable amount of damage\\
1920 Only short range order observable\\
1921 {\color{blue}$\Rightarrow$ amorphous SiC-like phase}
1922 \end{minipage} 
1923
1924 \begin{pspicture}(0,0)(0,0)
1925 \rput(6.7,5.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=white]{
1926 \begin{minipage}{10cm}
1927 \small
1928 {\color{red}\bf 3C-SiC formation fails to appear}
1929 \begin{itemize}
1930 \item Low C concentration simulations
1931  \begin{itemize}
1932   \item Formation of \ci{} indeed occurs
1933   \item Agllomeration not observed
1934  \end{itemize}
1935 \item High C concentration simulations
1936  \begin{itemize}
1937   \item Amorphous SiC-like structure\\
1938         (not expected at prevailing temperatures)
1939   \item Rearrangement and transition into 3C-SiC structure missing
1940  \end{itemize}
1941 \end{itemize}
1942 \end{minipage}
1943  }}}
1944 \end{pspicture}
1945
1946 \end{slide}
1947
1948 \begin{slide}
1949
1950  {\large\bf
1951   Limitations of molecular dynamics and short range potentials
1952  }
1953
1954 \footnotesize
1955
1956 \vspace{0.2cm}
1957
1958 \underline{Time scale problem of MD}\\[0.2cm]
1959 Minimize integration error\\
1960 $\Rightarrow$ discretization considerably smaller than
1961               reciprocal of fastest vibrational mode\\[0.1cm]
1962 Order of fastest vibrational mode: $10^{13} - 10^{14}\text{ Hz}$\\
1963 $\Rightarrow$ suitable choice of time step:
1964               $\tau=1\text{ fs}=10^{-15}\text{ s}$\\
1965 $\Rightarrow$ {\color{red}\underline{slow}} phase space propagation\\[0.1cm]
1966 Several local minima in energy surface separated by large energy barriers\\
1967 $\Rightarrow$ transition event corresponds to a multiple
1968               of vibrational periods\\
1969 $\Rightarrow$ phase transition made up of {\color{red}\underline{many}}
1970               infrequent transition events\\[0.1cm]
1971 {\color{blue}Accelerated methods:}
1972 \underline{Temperature accelerated} MD (TAD), self-guided MD \ldots
1973
1974 \vspace{0.3cm}
1975
1976 \underline{Limitations related to the short range potential}\\[0.2cm]
1977 Cut-off function pushing forces and energies to zero between 1$^{\text{st}}$
1978 and 2$^{\text{nd}}$ next neighbours\\
1979 $\Rightarrow$ overestimated unphysical high forces of next neighbours
1980
1981 \vspace{0.3cm}
1982
1983 \framebox{
1984 \color{red}
1985 Potential enhanced problem of slow phase space propagation
1986 }
1987
1988 \vspace{0.3cm}
1989
1990 \underline{Approach to the (twofold) problem}\\[0.2cm]
1991 Increased temperature simulations without TAD corrections\\
1992 (accelerated methods or higher time scales exclusively not sufficient)
1993
1994 \begin{picture}(0,0)(-260,-30)
1995 \framebox{
1996 \begin{minipage}{4.2cm}
1997 \tiny
1998 \begin{center}
1999 \vspace{0.03cm}
2000 \underline{IBS}
2001 \end{center}
2002 \begin{itemize}
2003 \item 3C-SiC also observed for higher T
2004 \item higher T inside sample
2005 \item structural evolution vs.\\
2006       equilibrium properties
2007 \end{itemize}
2008 \end{minipage}
2009 }
2010 \end{picture}
2011
2012 \begin{picture}(0,0)(-305,-155)
2013 \framebox{
2014 \begin{minipage}{2.5cm}
2015 \tiny
2016 \begin{center}
2017 retain proper\\
2018 thermodynmic sampling
2019 \end{center}
2020 \end{minipage}
2021 }
2022 \end{picture}
2023
2024 \end{slide}
2025
2026 \begin{slide}
2027
2028  {\large\bf
2029   Increased temperature simulations at low C concentration
2030  }
2031
2032 \small
2033
2034 \begin{minipage}{6.5cm}
2035 \includegraphics[width=6.4cm]{tot_pc_thesis.ps}
2036 \end{minipage}
2037 \begin{minipage}{6.5cm}
2038 \includegraphics[width=6.4cm]{tot_pc3_thesis.ps}
2039 \end{minipage}
2040
2041 \begin{minipage}{6.5cm}
2042 \includegraphics[width=6.4cm]{tot_pc2_thesis.ps}
2043 \end{minipage}
2044 \begin{minipage}{6.5cm}
2045 \scriptsize
2046  \underline{Si-C bonds:}
2047  \begin{itemize}
2048   \item Vanishing cut-off artifact (above $1650\,^{\circ}\mathrm{C}$)
2049   \item Structural change: C-Si \hkl<1 0 0> $\rightarrow$ C$_{\text{sub}}$
2050  \end{itemize}
2051  \underline{Si-Si bonds:}
2052  {\color{blue}Si-C$_{\text{sub}}$-Si} along \hkl<1 1 0>
2053  ($\rightarrow$ 0.325 nm)\\[0.1cm]
2054  \underline{C-C bonds:}
2055  \begin{itemize}
2056   \item C-C next neighbour pairs reduced (mandatory)
2057   \item Peak at 0.3 nm slightly shifted
2058         \begin{itemize}
2059          \item C-Si \hkl<1 0 0> combinations (dashed arrows)\\
2060                $\rightarrow$ C-Si \hkl<1 0 0> \& C$_{\text{sub}}$
2061                combinations (|)\\
2062                $\rightarrow$ pure {\color{blue}C$_{\text{sub}}$ combinations}
2063                ($\downarrow$)
2064          \item Range [|-$\downarrow$]:
2065                {\color{blue}C$_{\text{sub}}$ \& C$_{\text{sub}}$
2066                with nearby Si$_{\text{I}}$}
2067         \end{itemize}
2068  \end{itemize}
2069 \end{minipage}
2070
2071 \begin{picture}(0,0)(-330,-74)
2072 \color{blue}
2073 \framebox{
2074 \begin{minipage}{1.6cm}
2075 \tiny
2076 \begin{center}
2077 stretched SiC\\[-0.1cm]
2078 in c-Si
2079 \end{center}
2080 \end{minipage}
2081 }
2082 \end{picture}
2083
2084 \end{slide}
2085
2086 \begin{slide}
2087
2088  {\large\bf
2089   Increased temperature simulations at low C concentration
2090  }
2091
2092 \small
2093
2094 \begin{minipage}{6.5cm}
2095 \includegraphics[width=6.4cm]{tot_pc_thesis.ps}
2096 \end{minipage}
2097 \begin{minipage}{6.5cm}
2098 \includegraphics[width=6.4cm]{tot_pc3_thesis.ps}
2099 \end{minipage}
2100
2101 \begin{minipage}{6.5cm}
2102 \includegraphics[width=6.4cm]{tot_pc2_thesis.ps}
2103 \end{minipage}
2104 \begin{minipage}{6.5cm}
2105 \scriptsize
2106  \underline{Si-C bonds:}
2107  \begin{itemize}
2108   \item Vanishing cut-off artifact (above $1650\,^{\circ}\mathrm{C}$)
2109   \item Structural change: C-Si \hkl<1 0 0> $\rightarrow$ C$_{\text{sub}}$
2110  \end{itemize}
2111  \underline{Si-Si bonds:}
2112  {\color{blue}Si-C$_{\text{sub}}$-Si} along \hkl<1 1 0>
2113  ($\rightarrow$ 0.325 nm)\\[0.1cm]
2114  \underline{C-C bonds:}
2115  \begin{itemize}
2116   \item C-C next neighbour pairs reduced (mandatory)
2117   \item Peak at 0.3 nm slightly shifted
2118         \begin{itemize}
2119          \item C-Si \hkl<1 0 0> combinations (dashed arrows)\\
2120                $\rightarrow$ C-Si \hkl<1 0 0> \& C$_{\text{sub}}$
2121                combinations (|)\\
2122                $\rightarrow$ pure {\color{blue}C$_{\text{sub}}$ combinations}
2123                ($\downarrow$)
2124          \item Range [|-$\downarrow$]:
2125                {\color{blue}C$_{\text{sub}}$ \& C$_{\text{sub}}$
2126                with nearby Si$_{\text{I}}$}
2127         \end{itemize}
2128  \end{itemize}
2129 \end{minipage}
2130
2131 %\begin{picture}(0,0)(-330,-74)
2132 %\color{blue}
2133 %\framebox{
2134 %\begin{minipage}{1.6cm}
2135 %\tiny
2136 %\begin{center}
2137 %stretched SiC\\[-0.1cm]
2138 %in c-Si
2139 %\end{center}
2140 %\end{minipage}
2141 %}
2142 %\end{picture}
2143
2144 \begin{pspicture}(0,0)(0,0)
2145 \rput(6.7,5.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=white]{
2146 \begin{minipage}{10cm}
2147 \small
2148 {\color{blue}\bf Stretched SiC in c-Si}
2149 \begin{itemize}
2150 \item Consistent to precipitation model involving \cs{}
2151 \item Explains annealing behavior of high/low T C implants
2152       \begin{itemize}
2153        \item Low T: highly mobiel \ci{}
2154        \item High T: stable configurations of \cs{}
2155       \end{itemize}
2156 \end{itemize}
2157 $\Rightarrow$ High T $\leftrightarrow$ IBS conditions far from equilibrium\\
2158 $\Rightarrow$ Precipitation mechanism involving \cs{}
2159 \end{minipage}
2160  }}}
2161 \end{pspicture}
2162
2163 \end{slide}
2164
2165 \begin{slide}
2166
2167  {\large\bf
2168   Increased temperature simulations at high C concentration
2169  }
2170
2171 \footnotesize
2172
2173 \begin{minipage}{6.5cm}
2174 \includegraphics[width=6.4cm]{12_pc_thesis.ps}
2175 \end{minipage}
2176 \begin{minipage}{6.5cm}
2177 \includegraphics[width=6.4cm]{12_pc_c_thesis.ps}
2178 \end{minipage}
2179
2180 \vspace{0.1cm}
2181
2182 \scriptsize
2183
2184 \framebox{
2185 \begin{minipage}[t]{6.0cm}
2186 0.186 nm: Si-C pairs $\uparrow$\\
2187 (as expected in 3C-SiC)\\[0.2cm]
2188 0.282 nm: Si-C-C\\[0.2cm]
2189 $\approx$0.35 nm: C-Si-Si
2190 \end{minipage}
2191 }
2192 \begin{minipage}{0.2cm}
2193 \hfill
2194 \end{minipage}
2195 \framebox{
2196 \begin{minipage}[t]{6.0cm}
2197 0.15 nm: C-C pairs $\uparrow$\\
2198 (as expected in graphite/diamond)\\[0.2cm]
2199 0.252 nm: C-C-C (2$^{\text{nd}}$ NN for diamond)\\[0.2cm]
2200 0.31 nm: shifted towards 0.317 nm $\rightarrow$ C-Si-C
2201 \end{minipage}
2202 }
2203
2204 \begin{itemize}
2205 \item Decreasing cut-off artifact
2206 \item {\color{red}Amorphous} SiC-like phase remains
2207 \item High amount of {\color{red}damage} \& alignement to c-Si host matrix lost
2208 \item Slightly sharper peaks $\Rightarrow$ indicate slight {\color{blue}acceleration of dynamics} due to temperature
2209 \end{itemize}
2210
2211 \vspace{-0.1cm}
2212
2213 \begin{center}
2214 {\color{blue}
2215 \framebox{
2216 {\color{black}
2217 High C \& small $V$ \& short $t$
2218 $\Rightarrow$
2219 }
2220 Slow restructuring due to strong C-C bonds
2221 {\color{black}
2222 $\Leftarrow$
2223 High C \& low T implants
2224 }
2225 }
2226 }
2227 \end{center}
2228
2229 \end{slide}
2230
2231 \begin{slide}
2232
2233  {\large\bf
2234   Summary and Conclusions
2235  }
2236
2237  \scriptsize
2238
2239 %\vspace{0.1cm}
2240
2241 \framebox{
2242 \begin{minipage}[t]{12.9cm}
2243  \underline{Pecipitation simulations}
2244  \begin{itemize}
2245   \item High C concentration $\rightarrow$ amorphous SiC like phase
2246   \item Problem of potential enhanced slow phase space propagation
2247   \item Low T $\rightarrow$ C-Si \hkl<1 0 0> dumbbell dominated structure
2248   \item High T $\rightarrow$ C$_{\text{sub}}$ dominated structure
2249   \item High T necessary to simulate IBS conditions (far from equilibrium)
2250   \item Precipitation by successive agglomeration of \cs (epitaxy)
2251   \item \si{}: vehicle to form \cs{} \& supply of Si \& stress compensation
2252         (stretched SiC, interface)
2253  \end{itemize}
2254 \end{minipage}
2255 }
2256
2257 %\vspace{0.1cm}
2258
2259 \framebox{
2260 \begin{minipage}{12.9cm}
2261  \underline{Defects}
2262  \begin{itemize}
2263    \item DFT / EA
2264         \begin{itemize}
2265          \item Point defects excellently / fairly well described
2266                by DFT / EA
2267          \item C$_{\text{sub}}$ drastically underestimated by EA
2268          \item EA predicts correct ground state:
2269                C$_{\text{sub}}$ \& \si{} $>$ \ci{}
2270          \item Identified migration path explaining
2271                diffusion and reorientation experiments by DFT
2272          \item EA fails to describe \ci{} migration:
2273                Wrong path \& overestimated barrier
2274         \end{itemize}
2275    \item Combinations of defects
2276          \begin{itemize}
2277           \item Agglomeration of point defects energetically favorable
2278                 by compensation of stress
2279           \item Formation of C-C unlikely
2280           \item C$_{\text{sub}}$ favored conditions (conceivable in IBS)
2281           \item \ci{} \hkl<1 0 0> $\leftrightarrow$ \cs{} \& \si{} \hkl<1 1 0>\\
2282                 Low barrier (\unit[0.77]{eV}) \& low capture radius
2283         \end{itemize}
2284  \end{itemize}
2285 \end{minipage}
2286 }
2287
2288 \begin{center}
2289 {\color{blue}
2290 \framebox{Precipitation by successive agglomeration of \cs{}}
2291 }
2292 \end{center}
2293
2294 \end{slide}
2295
2296 \begin{slide}
2297
2298  {\large\bf
2299   Acknowledgements
2300  }
2301
2302  \vspace{0.1cm}
2303
2304  \small
2305
2306  Thanks to \ldots
2307
2308  \underline{Augsburg}
2309  \begin{itemize}
2310   \item Prof. B. Stritzker (accomodation at EP \RM{4})
2311   \item Ralf Utermann (EDV)
2312  \end{itemize}
2313  
2314  \underline{Helsinki}
2315  \begin{itemize}
2316   \item Prof. K. Nordlund (MD)
2317  \end{itemize}
2318  
2319  \underline{Munich}
2320  \begin{itemize}
2321   \item Bayerische Forschungsstiftung (financial support)
2322  \end{itemize}
2323  
2324  \underline{Paderborn}
2325  \begin{itemize}
2326   \item Prof. J. Lindner (SiC)
2327   \item Prof. G. Schmidt (DFT + financial support)
2328   \item Dr. E. Rauls (DFT + SiC)
2329   \item Dr. S. Sanna (VASP)
2330  \end{itemize}
2331
2332 \vspace{0.2cm}
2333
2334 \begin{center}
2335 \framebox{
2336 \bf Thank you for your attention!
2337 }
2338 \end{center}
2339
2340 \end{slide}
2341
2342 \end{document}