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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 \usepackage{semcolor}
35 \usepackage{semlayer}           % Seminar overlays
36 \usepackage{slidesec}           % Seminar sections and list of slides
37
38 \input{seminar.bug}             % Official bugs corrections
39 \input{seminar.bg2}             % Unofficial bugs corrections
40
41 \articlemag{1}
42
43 \special{landscape}
44
45 % font
46 %\usepackage{cmbright}
47 %\renewcommand{\familydefault}{\sfdefault}
48 %\usepackage{mathptmx}
49
50 \usepackage{upgreek}
51
52 \begin{document}
53
54 \extraslideheight{10in}
55 \slideframe{none}
56
57 \pagestyle{empty}
58
59 % specify width and height
60 \slidewidth 27.7cm 
61 \slideheight 19.1cm 
62
63 % shift it into visual area properly
64 \def\slideleftmargin{3.3cm}
65 \def\slidetopmargin{0.6cm}
66
67 \newcommand{\ham}{\mathcal{H}}
68 \newcommand{\pot}{\mathcal{V}}
69 \newcommand{\foo}{\mathcal{U}}
70 \newcommand{\vir}{\mathcal{W}}
71
72 % itemize level ii
73 \renewcommand\labelitemii{{\color{gray}$\bullet$}}
74
75 % nice phi
76 \renewcommand{\phi}{\varphi}
77
78 % roman letters
79 \newcommand{\RM}[1]{\MakeUppercase{\romannumeral #1{}}}
80
81 % colors
82 \newrgbcolor{si-yellow}{.6 .6 0}
83 \newrgbcolor{hb}{0.75 0.77 0.89}
84 \newrgbcolor{lbb}{0.75 0.8 0.88}
85 \newrgbcolor{hlbb}{0.825 0.88 0.968}
86 \newrgbcolor{lachs}{1.0 .93 .81}
87
88 % topic
89
90 \begin{slide}
91 \begin{center}
92
93  \vspace{16pt}
94
95  {\LARGE\bf
96   Atomistic simulation study of the silicon carbide precipitation
97   in silicon
98  }
99
100  \vspace{48pt}
101
102  \textsc{F. Zirkelbach}
103
104  \vspace{48pt}
105
106  Lehrstuhlseminar
107
108  \vspace{08pt}
109
110  17. Juni 2010
111
112 \end{center}
113 \end{slide}
114
115 % motivation / properties / applications of silicon carbide
116 \begin{slide}
117
118 \small
119
120 \begin{pspicture}(0,0)(13.5,5)
121
122
123
124  \psframe*[linecolor=hb](0,0)(13.5,5)
125
126  \pspolygon[linecolor=hlbb,fillcolor=hlbb,fillstyle=solid](5.5,1)(7,1)(7,3)(5.5,3)
127  \pspolygon[linecolor=hlbb,fillcolor=hlbb,fillstyle=solid](6.75,0.5)(8,2)(8,2)(6.75,3.5)
128
129  \rput[lt](0.2,4.6){\color{gray}PROPERTIES}
130
131  \rput[lt](0.5,4){wide band gap}
132  \rput[lt](0.5,3.5){high electric breakdown field}
133  \rput[lt](0.5,3){good electron mobility}
134  \rput[lt](0.5,2.5){high electron saturation drift velocity}
135  \rput[lt](0.5,2){high thermal conductivity}
136
137  \rput[lt](0.5,1.5){hard and mechanically stable}
138  \rput[lt](0.5,1){chemically inert}
139
140  \rput[lt](0.5,0.5){radiation hardness}
141
142  \rput[rt](13.3,4.6){\color{gray}APPLICATIONS}
143
144  \rput[rt](13,3.85){high-temperature, high power}
145  \rput[rt](13,3.5){and high-frequency}
146  \rput[rt](13,3.15){electronic and optoelectronic devices}
147
148  \rput[rt](13,2.35){material suitable for extreme conditions}
149  \rput[rt](13,2){microelectromechanical systems}
150  \rput[rt](13,1.65){abrasives, cutting tools, heating elements}
151
152  \rput[rt](13,0.85){first wall reactor material, detectors}
153  \rput[rt](13,0.5){and electronic devices for space}
154
155 \end{pspicture}
156
157 \begin{picture}(0,0)(-3,68)
158 \includegraphics[width=2.6cm]{wide_band_gap.eps}
159 \end{picture}
160 \begin{picture}(0,0)(-285,-162)
161 \includegraphics[width=3.38cm]{sic_led.eps}
162 \end{picture}
163 \begin{picture}(0,0)(-195,-162)
164 \includegraphics[width=2.8cm]{6h-sic_3c-sic.eps}
165 \end{picture}
166 \begin{picture}(0,0)(-313,65)
167 \includegraphics[width=2.2cm]{infineon_schottky.eps}
168 \end{picture}
169 \begin{picture}(0,0)(-220,65)
170 \includegraphics[width=2.9cm]{sic_wechselrichter_ise.eps}
171 \end{picture}
172 \begin{picture}(0,0)(0,-160)
173 \includegraphics[width=3.0cm]{sic_proton.eps}
174 \end{picture}
175 \begin{picture}(0,0)(-60,65)
176 \includegraphics[width=3.4cm]{sic_switch.eps}
177 \end{picture}
178
179 \end{slide}
180
181 % contents
182
183 \begin{slide}
184
185 {\large\bf
186  Outline
187 }
188
189  \begin{itemize}
190   \item Polytyps and fabrication of silicon carbide
191   \item Supposed precipitation mechanism of SiC in Si
192   \item Utilized simulation techniques
193         \begin{itemize}
194          \item Molecular dynamics (MD) simulations
195          \item Density functional theory (DFT) calculations
196         \end{itemize}
197   \item C and Si self-interstitial point defects in silicon
198   \item Silicon carbide precipitation simulations
199   \item Investigation of a silicon carbide precipitate in silicon
200   \item Summary / Conclusion / Outlook
201  \end{itemize}
202
203 \end{slide}
204
205 % start of contents
206
207 \begin{slide}
208
209  {\large\bf
210   Polytypes of SiC
211  }
212
213  \vspace{4cm}
214
215  \small
216
217 \begin{tabular}{l c c c c c c}
218 \hline
219  & 3C-SiC & 4H-SiC & 6H-SiC & Si & GaN & Diamond\\
220 \hline
221 Hardness [Mohs] & \multicolumn{3}{c}{------ 9.6 ------}& 6.5 & - & 10 \\
222 Band gap [eV] & 2.36 & 3.23 & 3.03 & 1.12 & 3.39 & 5.5 \\
223 Break down field [$10^6$ V/cm] & 4 & 3 & 3.2 & 0.6 & 5 & 10 \\
224 Saturation drift velocity [$10^7$ cm/s] & 2.5 & 2.0 & 2.0 & 1 & 2.7 & 2.7 \\
225 Electron mobility [cm$^2$/Vs] & 800 & 900 & 400 & 1100 & 900 & 2200 \\
226 Hole mobility [cm$^2$/Vs] & 320 & 120 & 90 & 420 & 150 & 1600 \\
227 Thermal conductivity [W/cmK] & 5.0 & 4.9 & 4.9 & 1.5 & 1.3 & 22 \\
228 \hline
229 \end{tabular}
230
231 {\tiny
232  Values for $T=300$ K
233 }
234
235 \begin{picture}(0,0)(-160,-155)
236  \includegraphics[width=7cm]{polytypes.eps}
237 \end{picture}
238 \begin{picture}(0,0)(-10,-185)
239  \includegraphics[width=3.8cm]{cubic_hex.eps}\\
240 \end{picture}
241 \begin{picture}(0,0)(-10,-175)
242  {\tiny cubic (twist)}
243 \end{picture}
244 \begin{picture}(0,0)(-60,-175)
245  {\tiny hexagonal (no twist)}
246 \end{picture}
247 \begin{pspicture}(0,0)(0,0)
248 \psellipse[linecolor=green](5.7,3.03)(0.4,0.5)
249 \end{pspicture}
250 \begin{pspicture}(0,0)(0,0)
251 \psellipse[linecolor=green](5.6,1.68)(0.4,0.2)
252 \end{pspicture}
253 \begin{pspicture}(0,0)(0,0)
254 \psellipse[linecolor=red](10.7,1.13)(0.4,0.2)
255 \end{pspicture}
256
257 \end{slide}
258
259 \begin{slide}
260
261  {\large\bf
262   Fabrication of silicon carbide
263  }
264
265  \small
266  
267  \vspace{4pt}
268
269  SiC - \emph{Born from the stars, perfected on earth.}
270  
271  \vspace{4pt}
272
273  Conventional thin film SiC growth:
274  \begin{itemize}
275   \item \underline{Sublimation growth using the modified Lely method}
276         \begin{itemize}
277          \item SiC single-crystalline seed at $T=1800 \, ^{\circ} \text{C}$
278          \item Surrounded by polycrystalline SiC in a graphite crucible\\
279                at $T=2100-2400 \, ^{\circ} \text{C}$
280          \item Deposition of supersaturated vapor on cooler seed crystal
281         \end{itemize}
282   \item \underline{Homoepitaxial growth using CVD}
283         \begin{itemize}
284          \item Step-controlled epitaxy on off-oriented 6H-SiC substrates
285          \item C$_3$H$_8$/SiH$_4$/H$_2$ at $1100-1500 \, ^{\circ} \text{C}$
286          \item Angle, temperature $\rightarrow$ 3C/6H/4H-SiC
287          \item High quality but limited in size of substrates
288         \end{itemize}
289   \item \underline{Heteroepitaxial growth of 3C-SiC on Si using CVD/MBE}
290         \begin{itemize}
291          \item Two steps: carbonization and growth
292          \item $T=650-1050 \, ^{\circ} \text{C}$
293          \item Quality and size not yet sufficient
294         \end{itemize}
295  \end{itemize}
296
297  \begin{picture}(0,0)(-280,-65)
298   \includegraphics[width=3.8cm]{6h-sic_3c-sic.eps}
299  \end{picture}
300  \begin{picture}(0,0)(-280,-55)
301   \begin{minipage}{5cm}
302   {\tiny
303    NASA: 6H-SiC and 3C-SiC LED\\[-7pt]
304    on 6H-SiC substrate
305   }
306   \end{minipage}
307  \end{picture}
308  \begin{picture}(0,0)(-265,-150)
309   \includegraphics[width=2.4cm]{m_lely.eps}
310  \end{picture}
311  \begin{picture}(0,0)(-333,-175)
312   \begin{minipage}{5cm}
313   {\tiny
314    1. Lid\\[-7pt]
315    2. Heating\\[-7pt]
316    3. Source\\[-7pt]
317    4. Crucible\\[-7pt]
318    5. Insulation\\[-7pt]
319    6. Seed crystal
320   }
321   \end{minipage}
322  \end{picture}
323  \begin{picture}(0,0)(-230,-35)
324  \framebox{
325  {\footnotesize\color{blue}\bf Hex: micropipes along c-axis}
326  }
327  \end{picture}
328  \begin{picture}(0,0)(-230,-10)
329  \framebox{
330  \begin{minipage}{3cm}
331  {\footnotesize\color{blue}\bf 3C-SiC fabrication\\
332                                less advanced}
333  \end{minipage}
334  }
335  \end{picture}
336
337 \end{slide}
338
339 \begin{slide}
340
341  {\large\bf
342   Fabrication of silicon carbide
343  }
344
345  \small
346
347  Alternative approach:
348  Ion beam synthesis (IBS) of burried 3C-SiC layers in Si\hkl(1 0 0)
349  \begin{itemize}
350   \item \underline{Implantation step 1}\\
351         180 keV C$^+$, $D=7.9\times 10^{17}$ cm$^{-2}$, $T_{\text{i}}=500\,^{\circ}\mathrm{C}$\\
352         $\Rightarrow$ box-like distribution of equally sized
353                        and epitactically oriented SiC precipitates
354                        
355   \item \underline{Implantation step 2}\\
356         180 keV C$^+$, $D=0.6\times 10^{17}$ cm$^{-2}$, $T_{\text{i}}=250\,^{\circ}\mathrm{C}$\\
357         $\Rightarrow$ destruction of SiC nanocrystals
358                       in growing amorphous interface layers
359   \item \underline{Annealing}\\
360         $T=1250\,^{\circ}\mathrm{C}$, $t=10\,\text{h}$\\
361         $\Rightarrow$ homogeneous, stoichiometric SiC layer
362                       with sharp interfaces
363  \end{itemize}
364
365  \begin{minipage}{6.3cm}
366  \includegraphics[width=6cm]{ibs_3c-sic.eps}\\[-0.2cm]
367  {\tiny
368   XTEM micrograph of single crystalline 3C-SiC in Si\hkl(1 0 0)
369  }
370  \end{minipage}
371 \framebox{
372  \begin{minipage}{6.3cm}
373  \begin{center}
374  {\color{blue}
375   Precipitation mechanism not yet fully understood!
376  }
377  \renewcommand\labelitemi{$\Rightarrow$}
378  \small
379  \underline{Understanding the SiC precipitation}
380  \begin{itemize}
381   \item significant technological progress in SiC thin film formation
382   \item perspectives for processes relying upon prevention of SiC precipitation
383  \end{itemize}
384  \end{center}
385  \end{minipage}
386 }
387  
388 \end{slide}
389
390 \begin{slide}
391
392  {\large\bf
393   Supposed precipitation mechanism of SiC in Si
394  }
395
396  \scriptsize
397
398  \vspace{0.1cm}
399
400  \begin{minipage}{3.8cm}
401  Si \& SiC lattice structure\\[0.2cm]
402  \includegraphics[width=3.5cm]{sic_unit_cell.eps}\\[-0.3cm]
403  \hrule
404  \end{minipage}
405  \hspace{0.6cm}
406  \begin{minipage}{3.8cm}
407  \begin{center}
408  \includegraphics[width=3.3cm]{tem_c-si-db.eps}
409  \end{center}
410  \end{minipage}
411  \hspace{0.6cm}
412  \begin{minipage}{3.8cm}
413  \begin{center}
414  \includegraphics[width=3.3cm]{tem_3c-sic.eps}
415  \end{center}
416  \end{minipage}
417
418  \begin{minipage}{4cm}
419  \begin{center}
420  C-Si dimers (dumbbells)\\[-0.1cm]
421  on Si interstitial sites
422  \end{center}
423  \end{minipage}
424  \hspace{0.2cm}
425  \begin{minipage}{4.2cm}
426  \begin{center}
427  Agglomeration of C-Si dumbbells\\[-0.1cm]
428  $\Rightarrow$ dark contrasts
429  \end{center}
430  \end{minipage}
431  \hspace{0.2cm}
432  \begin{minipage}{4cm}
433  \begin{center}
434  Precipitation of 3C-SiC in Si\\[-0.1cm]
435  $\Rightarrow$ Moir\'e fringes\\[-0.1cm]
436  \& release of Si self-interstitials
437  \end{center}
438  \end{minipage}
439
440  \begin{minipage}{3.8cm}
441  \begin{center}
442  \includegraphics[width=3.3cm]{sic_prec_seq_01.eps}
443  \end{center}
444  \end{minipage}
445  \hspace{0.6cm}
446  \begin{minipage}{3.8cm}
447  \begin{center}
448  \includegraphics[width=3.3cm]{sic_prec_seq_02.eps}
449  \end{center}
450  \end{minipage}
451  \hspace{0.6cm}
452  \begin{minipage}{3.8cm}
453  \begin{center}
454  \includegraphics[width=3.3cm]{sic_prec_seq_03.eps}
455  \end{center}
456  \end{minipage}
457
458 \begin{pspicture}(0,0)(0,0)
459 \psline[linewidth=4pt]{->}(8.5,2)(9.0,2)
460 \psellipse[linecolor=blue](11.5,5.8)(0.3,0.5)
461 \rput{-20}{\psellipse[linecolor=blue](3.3,8.1)(0.3,0.5)}
462 \psline[linewidth=4pt]{->}(4.0,2)(4.5,2)
463 \rput(12.7,0.3){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
464  $4a_{\text{Si}}=5a_{\text{SiC}}$
465  }}}
466 \rput(12.2,8){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
467 \hkl(h k l) planes match
468  }}}
469 \end{pspicture}
470
471 \end{slide}
472
473 \begin{slide}
474
475  {\large\bf
476   Molecular dynamics (MD) simulations
477  }
478
479  \vspace{12pt}
480
481  \small
482
483  {\bf MD basics:}
484  \begin{itemize}
485   \item Microscopic description of N particle system
486   \item Analytical interaction potential
487   \item Numerical integration using Newtons equation of motion\\
488         as a propagation rule in 6N-dimensional phase space
489   \item Observables obtained by time and/or ensemble averages
490  \end{itemize}
491  {\bf Details of the simulation:}
492  \begin{itemize}
493   \item Integration: Velocity Verlet, timestep: $1\text{ fs}$
494   \item Ensemble: NpT (isothermal-isobaric)
495         \begin{itemize}
496          \item Berendsen thermostat:
497                $\tau_{\text{T}}=100\text{ fs}$
498          \item Berendsen barostat:\\
499                $\tau_{\text{P}}=100\text{ fs}$,
500                $\beta^{-1}=100\text{ GPa}$
501         \end{itemize}
502   \item Erhart/Albe potential: Tersoff-like bond order potential
503   \vspace*{12pt}
504         \[
505         E = \frac{1}{2} \sum_{i \neq j} \pot_{ij}, \quad
506         \pot_{ij} = f_C(r_{ij}) \left[ f_R(r_{ij}) + b_{ij} f_A(r_{ij}) \right]
507         \]
508  \end{itemize}
509
510  \begin{picture}(0,0)(-230,-30)
511   \includegraphics[width=5cm]{tersoff_angle.eps} 
512  \end{picture}
513  
514 \end{slide}
515
516 \begin{slide}
517
518  {\large\bf
519   Density functional theory (DFT) calculations
520  }
521
522  \small
523
524  Basic ingredients necessary for DFT
525
526  \begin{itemize}
527   \item \underline{Hohenberg-Kohn theorem} - ground state density $n_0(r)$ ...
528         \begin{itemize}
529          \item ... uniquely determines the ground state potential
530                / wavefunctions
531          \item ... minimizes the systems total energy
532         \end{itemize}
533   \item \underline{Born-Oppenheimer}
534         - $N$ moving electrons in an external potential of static nuclei
535 \[
536 H\Psi = \left[-\sum_i^N \frac{\hbar^2}{2m}\nabla_i^2
537               +\sum_i^N V_{\text{ext}}(r_i)
538               +\sum_{i<j}^N V_{e-e}(r_i,r_j)\right]\Psi=E\Psi
539 \]
540   \item \underline{Effective potential}
541         - averaged electrostatic potential \& exchange and correlation
542 \[
543 V_{\text{eff}}(r)=V_{\text{ext}}(r)+\int\frac{e^2 n(r')}{|r-r'|}d^3r'
544                  +V_{\text{XC}}[n(r)]
545 \]
546   \item \underline{Kohn-Sham system}
547         - Schr\"odinger equation of N non-interacting particles
548 \[
549 \left[ -\frac{\hbar^2}{2m}\nabla^2 + V_{\text{eff}}(r) \right] \Phi_i(r)
550 =\epsilon_i\Phi_i(r)
551 \quad
552 \Rightarrow
553 \quad
554 n(r)=\sum_i^N|\Phi_i(r)|^2
555 \]
556   \item \underline{Self-consistent solution}\\
557 $n(r)$ depends on $\Phi_i$, which depend on $V_{\text{eff}}$,
558 which in turn depends on $n(r)$
559   \item \underline{Variational principle}
560         - minimize total energy with respect to $n(r)$
561  \end{itemize}
562
563 \end{slide}
564
565 \begin{slide}
566
567  {\large\bf
568   Density functional theory (DFT) calculations
569  }
570
571  \small
572
573  \vspace*{0.2cm}
574
575  Details of applied DFT calculations in this work
576
577  \begin{itemize}
578   \item \underline{Exchange correlation functional}
579         - approximations for the inhomogeneous electron gas
580         \begin{itemize}
581          \item LDA: $E_{\text{XC}}^{\text{LDA}}[n]=\int \epsilon_{\text{XC}}(n)n(r)d^3r$
582          \item GGA: $E_{\text{XC}}^{\text{GGA}}[n]=\int \epsilon_{\text{XC}}(n,\nabla n)n(r)d^3r$
583         \end{itemize}
584   \item \underline{Plane wave basis set}
585         - approximation of the wavefunction $\Phi_i$ by plane waves $\phi_j$
586 \[
587 \rightarrow
588 \text{Fourier series: } \Phi_i=\sum_{|G+k|<G_{\text{cut}}} c_j^i \phi_j(r), \quad E_{\text{cut}}=\frac{\hbar^2}{2m}G^2_{\text{cut}}
589 \qquad ({\color{blue}300\text{ eV}})
590 \]
591   \item \underline{Brillouin zone sampling} -
592         {\color{blue}$\Gamma$-point only} calculations
593   \item \underline{Pseudo potential} 
594         - consider only the valence electrons
595   \item \underline{Code} - VASP 4.6
596  \end{itemize}
597
598  \vspace*{0.2cm}
599
600  MD and structural optimization
601
602  \begin{itemize}
603   \item MD integration: Gear predictor corrector algorithm
604   \item Pressure control: Parrinello-Rahman pressure control
605   \item Structural optimization: Conjugate gradient method
606  \end{itemize}
607
608 \begin{pspicture}(0,0)(0,0)
609 \psellipse[linecolor=blue](1.5,6.75)(0.5,0.3)
610 \end{pspicture}
611
612 \end{slide}
613
614 \begin{slide}
615
616  {\large\bf
617   C and Si self-interstitial point defects in silicon
618  }
619
620  \small
621
622  \vspace*{0.3cm}
623
624 \begin{minipage}{8cm}
625 Procedure:\\[0.3cm]
626   \begin{pspicture}(0,0)(7,5)
627   \rput(3.5,4){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
628    \parbox{7cm}{
629    \begin{itemize}
630     \item Creation of c-Si simulation volume
631     \item Periodic boundary conditions
632     \item $T=0\text{ K}$, $p=0\text{ bar}$
633    \end{itemize}
634   }}}}
635 \rput(3.5,2.1){\rnode{insert}{\psframebox{
636  \parbox{7cm}{
637   \begin{center}
638   Insertion of interstitial C/Si atoms
639   \end{center}
640   }}}}
641   \rput(3.5,1){\rnode{cool}{\psframebox[fillstyle=solid,fillcolor=lbb]{
642    \parbox{7cm}{
643    \begin{center}
644    Relaxation / structural energy minimization
645    \end{center}
646   }}}}
647   \ncline[]{->}{init}{insert}
648   \ncline[]{->}{insert}{cool}
649  \end{pspicture}
650 \end{minipage}
651 \begin{minipage}{5cm}
652   \includegraphics[width=5cm]{unit_cell_e.eps}\\
653 \end{minipage}
654
655 \begin{minipage}{9cm}
656  \begin{tabular}{l c c}
657  \hline
658  & size [unit cells] & \# atoms\\
659 \hline
660 VASP & $3\times 3\times 3$ & $216\pm 1$ \\
661 Erhart/Albe & $9\times 9\times 9$ & $5832\pm 1$\\
662 \hline
663  \end{tabular}
664 \end{minipage}
665 \begin{minipage}{4cm}
666 {\color{red}$\bullet$} Tetrahedral\\
667 {\color{green}$\bullet$} Hexagonal\\
668 {\color{yellow}$\bullet$} \hkl<1 0 0> dumbbell\\
669 {\color{magenta}$\bullet$} \hkl<1 1 0> dumbbell\\
670 {\color{cyan}$\bullet$} Bond-centered\\
671 {\color{black}$\bullet$} Vacancy / Substitutional
672 \end{minipage}
673
674 \end{slide}
675
676 \begin{slide}
677
678  \footnotesize
679
680 \begin{minipage}{9.5cm}
681
682  {\large\bf
683   Si self-interstitial point defects in silicon\\
684  }
685
686 \begin{tabular}{l c c c c c}
687 \hline
688  $E_{\text{f}}$ [eV] & \hkl<1 1 0> DB & H & T & \hkl<1 0 0> DB & V \\
689 \hline
690  VASP & \underline{3.39} & 3.42 & 3.77 & 4.41 & 3.63 \\
691  Erhart/Albe & 4.39 & 4.48$^*$ & \underline{3.40} & 5.42 & 3.13 \\
692 \hline
693 \end{tabular}\\[0.2cm]
694
695 \begin{minipage}{4.7cm}
696 \includegraphics[width=4.7cm]{e_kin_si_hex.ps}
697 \end{minipage}
698 \begin{minipage}{4.7cm}
699 \begin{center}
700 {\tiny nearly T $\rightarrow$ T}\\
701 \end{center}
702 \includegraphics[width=4.7cm]{nhex_tet.ps}
703 \end{minipage}\\
704
705 \underline{Hexagonal} \hspace{2pt}
706 \href{../video/si_self_int_hexa.avi}{$\rhd$}\\[0.1cm]
707 \framebox{
708 \begin{minipage}{2.7cm}
709 $E_{\text{f}}^*=4.48\text{ eV}$\\
710 \includegraphics[width=2.7cm]{si_pd_albe/hex_a.eps}
711 \end{minipage}
712 \begin{minipage}{0.4cm}
713 \begin{center}
714 $\Rightarrow$
715 \end{center}
716 \end{minipage}
717 \begin{minipage}{2.7cm}
718 $E_{\text{f}}=3.96\text{ eV}$\\
719 \includegraphics[width=2.8cm]{si_pd_albe/hex.eps}
720 \end{minipage}
721 }
722 \begin{minipage}{2.9cm}
723 \begin{flushright}
724 \underline{Vacancy}\\
725 \includegraphics[width=3.0cm]{si_pd_albe/vac.eps}
726 \end{flushright}
727 \end{minipage}
728
729 \end{minipage}
730 \begin{minipage}{3.5cm}
731
732 \begin{flushright}
733 \underline{\hkl<1 1 0> dumbbell}\\
734 \includegraphics[width=3.0cm]{si_pd_albe/110.eps}\\
735 \underline{Tetrahedral}\\
736 \includegraphics[width=3.0cm]{si_pd_albe/tet.eps}\\
737 \underline{\hkl<1 0 0> dumbbell}\\
738 \includegraphics[width=3.0cm]{si_pd_albe/100.eps}
739 \end{flushright}
740
741 \end{minipage}
742
743 \end{slide}
744
745 \begin{slide}
746
747 \footnotesize
748
749  {\large\bf
750   C interstitial point defects in silicon\\[-0.1cm]
751  }
752
753 \begin{tabular}{l c c c c c c}
754 \hline
755  $E_{\text{f}}$ & T & H & \hkl<1 0 0> DB & \hkl<1 1 0> DB & S & B \\
756 \hline
757  VASP & unstable & unstable & \underline{3.72} & 4.16 & 1.95 & 4.66 \\
758  Erhart/Albe MD & 6.09 & 9.05$^*$ & \underline{3.88} & 5.18 & 0.75 & 5.59$^*$ \\
759 \hline
760 \end{tabular}\\[0.1cm]
761
762 \framebox{
763 \begin{minipage}{2.7cm}
764 \underline{Hexagonal} \hspace{2pt}
765 \href{../video/c_in_si_int_hexa.avi}{$\rhd$}\\
766 $E_{\text{f}}^*=9.05\text{ eV}$\\
767 \includegraphics[width=2.7cm]{c_pd_albe/hex.eps}
768 \end{minipage}
769 \begin{minipage}{0.4cm}
770 \begin{center}
771 $\Rightarrow$
772 \end{center}
773 \end{minipage}
774 \begin{minipage}{2.7cm}
775 \underline{\hkl<1 0 0>}\\
776 $E_{\text{f}}=3.88\text{ eV}$\\
777 \includegraphics[width=2.7cm]{c_pd_albe/100.eps}
778 \end{minipage}
779 }
780 \begin{minipage}{2cm}
781 \hfill
782 \end{minipage}
783 \begin{minipage}{3cm}
784 \begin{flushright}
785 \underline{Tetrahedral}\\
786 \includegraphics[width=3.0cm]{c_pd_albe/tet.eps}
787 \end{flushright}
788 \end{minipage}
789
790 \framebox{
791 \begin{minipage}{2.7cm}
792 \underline{Bond-centered}\\
793 $E_{\text{f}}^*=5.59\text{ eV}$\\
794 \includegraphics[width=2.7cm]{c_pd_albe/bc.eps}
795 \end{minipage}
796 \begin{minipage}{0.4cm}
797 \begin{center}
798 $\Rightarrow$
799 \end{center}
800 \end{minipage}
801 \begin{minipage}{2.7cm}
802 \underline{\hkl<1 1 0> dumbbell}\\
803 $E_{\text{f}}=5.18\text{ eV}$\\
804 \includegraphics[width=2.7cm]{c_pd_albe/110.eps}
805 \end{minipage}
806 }
807 \begin{minipage}{2cm}
808 \hfill
809 \end{minipage}
810 \begin{minipage}{3cm}
811 \begin{flushright}
812 \underline{Substitutional}\\
813 \includegraphics[width=3.0cm]{c_pd_albe/sub.eps}
814 \end{flushright}
815 \end{minipage}
816
817 \end{slide}
818
819 \begin{slide}
820
821 \footnotesize
822
823  {\large\bf\boldmath
824   C \hkl<1 0 0> dumbbell interstitial configuration\\
825  }
826
827 {\tiny
828 \begin{tabular}{l c c c c c c c c}
829 \hline
830  Distances [nm] & $r(1C)$ & $r(2C)$ & $r(3C)$ & $r(12)$ & $r(13)$ & $r(34)$ & $r(23)$ & $r(25)$ \\
831 \hline
832 Erhart/Albe & 0.175 & 0.329 & 0.186 & 0.226 & 0.300 & 0.343 & 0.423 & 0.425 \\
833 VASP & 0.174 & 0.341 & 0.182 & 0.229 & 0.286 & 0.347 & 0.422 & 0.417 \\
834 \hline
835 \end{tabular}\\[0.2cm]
836 \begin{tabular}{l c c c c }
837 \hline
838  Angles [$^{\circ}$] & $\theta_1$ & $\theta_2$ & $\theta_3$ & $\theta_4$ \\
839 \hline
840 Erhart/Albe & 140.2 & 109.9 & 134.4 & 112.8 \\
841 VASP & 130.7 & 114.4 & 146.0 & 107.0 \\
842 \hline
843 \end{tabular}\\[0.2cm]
844 \begin{tabular}{l c c c}
845 \hline
846  Displacements [nm]& $a$ & $b$ & $|a|+|b|$ \\
847 \hline
848 Erhart/Albe & 0.084 & -0.091 & 0.175 \\
849 VASP & 0.109 & -0.065 & 0.174 \\
850 \hline
851 \end{tabular}\\[0.6cm]
852 }
853
854 \begin{minipage}{3.0cm}
855 \begin{center}
856 \underline{Erhart/Albe}
857 \includegraphics[width=3.0cm]{c_pd_albe/100_cmp.eps}
858 \end{center}
859 \end{minipage}
860 \begin{minipage}{3.0cm}
861 \begin{center}
862 \underline{VASP}
863 \includegraphics[width=3.0cm]{c_pd_vasp/100_cmp.eps}
864 \end{center}
865 \end{minipage}\\
866
867 \begin{picture}(0,0)(-185,10)
868 \includegraphics[width=6.8cm]{100-c-si-db_cmp.eps}
869 \end{picture}
870 \begin{picture}(0,0)(-280,-150)
871 \includegraphics[width=3.3cm]{c_pd_vasp/eden.eps}
872 \end{picture}
873
874 \begin{pspicture}(0,0)(0,0)
875 \psellipse[linecolor=green](5.18,5.92)(0.5,0.3)
876 \psellipse[linecolor=red](3.45,5.92)(1.0,0.4)
877 \psellipse[linecolor=blue](2.7,6.92)(0.9,0.2)
878 \psellipse[linecolor=blue](4.65,6.92)(0.9,0.2)
879 \end{pspicture}
880
881 \end{slide}
882
883 \begin{slide}
884
885 \small
886
887 \begin{minipage}{8.5cm}
888
889  {\large\bf
890   Bond-centered interstitial configuration\\[-0.1cm]
891  }
892
893 \begin{minipage}{3.0cm}
894 \includegraphics[width=2.8cm]{c_pd_vasp/bc_2333.eps}\\
895 \end{minipage}
896 \begin{minipage}{5.2cm}
897 \begin{itemize}
898  \item Linear Si-C-Si bond
899  \item Si: one C \& 3 Si neighbours
900  \item Spin polarized calculations
901  \item No saddle point!\\
902        Real local minimum!
903 \end{itemize}
904 \end{minipage}
905
906 \framebox{
907  \tiny
908  \begin{minipage}[t]{6.5cm}
909   \begin{minipage}[t]{1.2cm}
910   {\color{red}Si}\\
911   {\tiny sp$^3$}\\[0.8cm]
912   \underline{${\color{black}\uparrow}$}
913   \underline{${\color{black}\uparrow}$}
914   \underline{${\color{black}\uparrow}$}
915   \underline{${\color{red}\uparrow}$}\\
916   sp$^3$
917   \end{minipage}
918   \begin{minipage}[t]{1.4cm}
919   \begin{center}
920   {\color{red}M}{\color{blue}O}\\[0.8cm]
921   \underline{${\color{blue}\uparrow}{\color{white}\downarrow}$}\\
922   $\sigma_{\text{ab}}$\\[0.5cm]
923   \underline{${\color{red}\uparrow}{\color{blue}\downarrow}$}\\
924   $\sigma_{\text{b}}$
925   \end{center}
926   \end{minipage}
927   \begin{minipage}[t]{1.0cm}
928   \begin{center}
929   {\color{blue}C}\\
930   {\tiny sp}\\[0.2cm]
931   \underline{${\color{white}\uparrow\uparrow}$}
932   \underline{${\color{white}\uparrow\uparrow}$}\\
933   2p\\[0.4cm]
934   \underline{${\color{blue}\uparrow}{\color{blue}\downarrow}$}
935   \underline{${\color{blue}\uparrow}{\color{blue}\downarrow}$}\\
936   sp
937   \end{center}
938   \end{minipage}
939   \begin{minipage}[t]{1.4cm}
940   \begin{center}
941   {\color{blue}M}{\color{green}O}\\[0.8cm]
942   \underline{${\color{blue}\uparrow}{\color{white}\downarrow}$}\\
943   $\sigma_{\text{ab}}$\\[0.5cm]
944   \underline{${\color{green}\uparrow}{\color{blue}\downarrow}$}\\
945   $\sigma_{\text{b}}$
946   \end{center}
947   \end{minipage}
948   \begin{minipage}[t]{1.2cm}
949   \begin{flushright}
950   {\color{green}Si}\\
951   {\tiny sp$^3$}\\[0.8cm]
952   \underline{${\color{green}\uparrow}$}
953   \underline{${\color{black}\uparrow}$}
954   \underline{${\color{black}\uparrow}$}
955   \underline{${\color{black}\uparrow}$}\\
956   sp$^3$
957   \end{flushright}
958   \end{minipage}
959  \end{minipage}
960 }\\[0.1cm]
961
962 \framebox{
963 \begin{minipage}{4.5cm}
964 \includegraphics[width=4cm]{c_100_mig_vasp/im_spin_diff.eps}
965 \end{minipage}
966 \begin{minipage}{3.5cm}
967 {\color{gray}$\bullet$} Spin up\\
968 {\color{green}$\bullet$} Spin down\\
969 {\color{blue}$\bullet$} Resulting spin up\\
970 {\color{yellow}$\bullet$} Si atoms\\
971 {\color{red}$\bullet$} C atom
972 \end{minipage}
973 }
974
975 \end{minipage}
976 \begin{minipage}{4.2cm}
977 \begin{flushright}
978 \includegraphics[width=4.3cm]{c_pd_vasp/bc_2333_ksl.ps}\\
979 {\color{green}$\Box$} {\tiny unoccupied}\\
980 {\color{red}$\bullet$} {\tiny occupied}
981 \end{flushright}
982 \end{minipage}
983
984 \end{slide}
985
986 \begin{slide}
987
988  {\large\bf\boldmath
989   Migration of the C \hkl<1 0 0> dumbbell interstitial
990  }
991
992 \scriptsize
993
994  {\small Investigated pathways}
995
996 \begin{minipage}{8.5cm}
997 \begin{minipage}{8.3cm}
998 \underline{\hkl<0 0 -1> $\rightarrow$ \hkl<0 0 1>}\\
999 \begin{minipage}{2.4cm}
1000 \includegraphics[width=2.4cm]{c_pd_vasp/100_2333.eps}
1001 \end{minipage}
1002 \begin{minipage}{0.4cm}
1003 $\rightarrow$
1004 \end{minipage}
1005 \begin{minipage}{2.4cm}
1006 \includegraphics[width=2.4cm]{c_pd_vasp/bc_2333.eps}
1007 \end{minipage}
1008 \begin{minipage}{0.4cm}
1009 $\rightarrow$
1010 \end{minipage}
1011 \begin{minipage}{2.4cm}
1012 \includegraphics[width=2.4cm]{c_pd_vasp/100_next_2333.eps}
1013 \end{minipage}
1014 \end{minipage}\\
1015 \begin{minipage}{8.3cm}
1016 \underline{\hkl<0 0 -1> $\rightarrow$ \hkl<0 -1 0>}\\
1017 \begin{minipage}{2.4cm}
1018 \includegraphics[width=2.4cm]{c_pd_vasp/100_2333.eps}
1019 \end{minipage}
1020 \begin{minipage}{0.4cm}
1021 $\rightarrow$
1022 \end{minipage}
1023 \begin{minipage}{2.4cm}
1024 \includegraphics[width=2.4cm]{c_pd_vasp/00-1-0-10_2333.eps}
1025 \end{minipage}
1026 \begin{minipage}{0.4cm}
1027 $\rightarrow$
1028 \end{minipage}
1029 \begin{minipage}{2.4cm}
1030 \includegraphics[width=2.4cm]{c_pd_vasp/0-10_2333.eps}
1031 \end{minipage}
1032 \end{minipage}\\
1033 \begin{minipage}{8.3cm}
1034 \underline{\hkl<0 0 -1> $\rightarrow$ \hkl<0 -1 0> (in place)}\\
1035 \begin{minipage}{2.4cm}
1036 \includegraphics[width=2.4cm]{c_pd_vasp/100_2333.eps}
1037 \end{minipage}
1038 \begin{minipage}{0.4cm}
1039 $\rightarrow$
1040 \end{minipage}
1041 \begin{minipage}{2.4cm}
1042 \includegraphics[width=2.4cm]{c_pd_vasp/00-1_ip0-10_2333.eps}
1043 \end{minipage}
1044 \begin{minipage}{0.4cm}
1045 $\rightarrow$
1046 \end{minipage}
1047 \begin{minipage}{2.4cm}
1048 \includegraphics[width=2.4cm]{c_pd_vasp/0-10_ip_2333.eps}
1049 \end{minipage}
1050 \end{minipage}
1051 \end{minipage}
1052 \framebox{
1053 \begin{minipage}{4.2cm}
1054  {\small Constrained relaxation\\
1055          technique (CRT) method}\\
1056 \includegraphics[width=4cm]{crt_orig.eps}
1057 \begin{itemize}
1058  \item Constrain diffusing atom
1059  \item Static constraints 
1060 \end{itemize}
1061 \vspace*{0.3cm}
1062  {\small Modifications}\\
1063 \includegraphics[width=4cm]{crt_mod.eps}
1064 \begin{itemize}
1065  \item Constrain all atoms
1066  \item Update individual\\
1067        constraints
1068 \end{itemize}
1069 \end{minipage}
1070 }
1071
1072 \end{slide}
1073
1074 \begin{slide}
1075
1076  {\large\bf\boldmath
1077   Migration of the C \hkl<1 0 0> dumbbell interstitial
1078  }
1079
1080 \scriptsize
1081
1082 \framebox{
1083 \begin{minipage}{5.9cm}
1084 \begin{flushleft}
1085 \includegraphics[width=5.8cm]{im_00-1_nosym_sp_fullct_thesis.ps}\\[0.45cm]
1086 \end{flushleft}
1087 \begin{center}
1088 \begin{picture}(0,0)(60,0)
1089 \includegraphics[width=1cm]{vasp_mig/00-1.eps}
1090 \end{picture}
1091 \begin{picture}(0,0)(-5,0)
1092 \includegraphics[width=1cm]{vasp_mig/bc_00-1_sp.eps}
1093 \end{picture}
1094 \begin{picture}(0,0)(-55,0)
1095 \includegraphics[width=1cm]{vasp_mig/bc.eps}
1096 \end{picture}
1097 \begin{picture}(0,0)(12.5,10)
1098 \includegraphics[width=1cm]{110_arrow.eps}
1099 \end{picture}
1100 \begin{picture}(0,0)(90,0)
1101 \includegraphics[height=0.9cm]{001_arrow.eps}
1102 \end{picture}
1103 \end{center}
1104 \vspace*{0.35cm}
1105 \end{minipage}
1106 }
1107 \begin{minipage}{0.3cm}
1108 \hfill
1109 \end{minipage}
1110 \framebox{
1111 \begin{minipage}{5.9cm}
1112 \begin{flushright}
1113 \includegraphics[width=5.9cm]{vasp_mig/00-1_0-10_nosym_sp_fullct.ps}\\[0.5cm]
1114 \end{flushright}
1115 \begin{center}
1116 \begin{picture}(0,0)(60,0)
1117 \includegraphics[width=1cm]{vasp_mig/00-1_a.eps}
1118 \end{picture}
1119 \begin{picture}(0,0)(5,0)
1120 \includegraphics[width=1cm]{vasp_mig/00-1_0-10_sp.eps}
1121 \end{picture}
1122 \begin{picture}(0,0)(-55,0)
1123 \includegraphics[width=1cm]{vasp_mig/0-10.eps}
1124 \end{picture}
1125 \begin{picture}(0,0)(12.5,10)
1126 \includegraphics[width=1cm]{100_arrow.eps}
1127 \end{picture}
1128 \begin{picture}(0,0)(90,0)
1129 \includegraphics[height=0.9cm]{001_arrow.eps}
1130 \end{picture}
1131 \end{center}
1132 \vspace*{0.3cm}
1133 \end{minipage}\\
1134 }
1135
1136 \vspace*{0.05cm}
1137
1138 \framebox{
1139 \begin{minipage}{5.9cm}
1140 \begin{flushleft}
1141 \includegraphics[width=5.9cm]{vasp_mig/00-1_ip0-10_nosym_sp_fullct.ps}\\[0.6cm]
1142 \end{flushleft}
1143 \begin{center}
1144 \begin{picture}(0,0)(60,0)
1145 \includegraphics[width=0.9cm]{vasp_mig/00-1_b.eps}
1146 \end{picture}
1147 \begin{picture}(0,0)(10,0)
1148 \includegraphics[width=0.9cm]{vasp_mig/00-1_ip0-10_sp.eps}
1149 \end{picture}
1150 \begin{picture}(0,0)(-60,0)
1151 \includegraphics[width=0.9cm]{vasp_mig/0-10_b.eps}
1152 \end{picture}
1153 \begin{picture}(0,0)(12.5,10)
1154 \includegraphics[width=1cm]{100_arrow.eps}
1155 \end{picture}
1156 \begin{picture}(0,0)(90,0)
1157 \includegraphics[height=0.9cm]{001_arrow.eps}
1158 \end{picture}
1159 \end{center}
1160 \vspace*{0.3cm}
1161 \end{minipage}
1162 }
1163 \begin{minipage}{0.3cm}
1164 \hfill
1165 \end{minipage}
1166 \begin{minipage}{6.5cm}
1167 VASP results
1168 \begin{itemize}
1169  \item Energetically most favorable path
1170        \begin{itemize}
1171         \item Path 2
1172         \item Activation energy: $\approx$ 0.9 eV 
1173         \item Experimental values: 0.73 ... 0.87 eV
1174        \end{itemize}
1175        $\Rightarrow$ {\color{blue}Diffusion} path identified!
1176  \item Reorientation (path 3)
1177        \begin{itemize}
1178         \item More likely composed of two consecutive steps of type 2
1179         \item Experimental values: 0.77 ... 0.88 eV
1180        \end{itemize}
1181        $\Rightarrow$ {\color{blue}Reorientation} transition identified!
1182 \end{itemize}
1183 \end{minipage}
1184
1185 \end{slide}
1186
1187 \begin{slide}
1188
1189  {\large\bf\boldmath
1190   Migration of the C \hkl<1 0 0> dumbbell interstitial
1191  }
1192
1193 \scriptsize
1194
1195 \begin{minipage}{6.5cm}
1196
1197 \framebox{
1198 \begin{minipage}{5.9cm}
1199 \begin{flushleft}
1200 \includegraphics[width=5.9cm]{bc_00-1.ps}\\[2.35cm]
1201 \end{flushleft}
1202 \begin{center}
1203 \begin{pspicture}(0,0)(0,0)
1204 \psframe[linecolor=red,fillstyle=none](-2.8,1.35)(3.3,2.7)
1205 \end{pspicture}
1206 \begin{picture}(0,0)(60,-50)
1207 \includegraphics[width=1cm]{albe_mig/bc_00-1_red_00.eps}
1208 \end{picture}
1209 \begin{picture}(0,0)(5,-50)
1210 \includegraphics[width=1cm]{albe_mig/bc_00-1_red_01.eps}
1211 \end{picture}
1212 \begin{picture}(0,0)(-55,-50)
1213 \includegraphics[width=1cm]{albe_mig/bc_00-1_red_02.eps}
1214 \end{picture}
1215 \begin{picture}(0,0)(12.5,-40)
1216 \includegraphics[width=1cm]{110_arrow.eps}
1217 \end{picture}
1218 \begin{picture}(0,0)(90,-45)
1219 \includegraphics[height=0.9cm]{001_arrow.eps}
1220 \end{picture}\\
1221 \begin{pspicture}(0,0)(0,0)
1222 \psframe[linecolor=blue,fillstyle=none](-2.8,0)(3.3,1.6)
1223 \end{pspicture}
1224 \begin{picture}(0,0)(60,-15)
1225 \includegraphics[width=0.9cm]{albe_mig/bc_00-1_01.eps}
1226 \end{picture}
1227 \begin{picture}(0,0)(35,-15)
1228 \includegraphics[width=0.9cm]{albe_mig/bc_00-1_02.eps}
1229 \end{picture}
1230 \begin{picture}(0,0)(-5,-15)
1231 \includegraphics[width=0.9cm]{albe_mig/bc_00-1_03.eps}
1232 \end{picture}
1233 \begin{picture}(0,0)(-55,-15)
1234 \includegraphics[width=0.9cm]{albe_mig/bc_00-1_04.eps}
1235 \end{picture}
1236 \begin{picture}(0,0)(12.5,-5)
1237 \includegraphics[width=1cm]{100_arrow.eps}
1238 \end{picture}
1239 \begin{picture}(0,0)(90,-15)
1240 \includegraphics[height=0.9cm]{010_arrow.eps}
1241 \end{picture}
1242 \end{center}
1243 \end{minipage}
1244 }\\[0.1cm]
1245
1246 \begin{minipage}{5.9cm}
1247 Erhart/Albe results
1248 \begin{itemize}
1249  \item Lowest activation energy: $\approx$ 2.2 eV
1250  \item 2.4 times higher than VASP
1251  \item Different pathway
1252  \item Transition minima ($\rightarrow$ \hkl<1 1 0> dumbbell)
1253 \end{itemize}
1254 \end{minipage}
1255
1256 \end{minipage}
1257 \begin{minipage}{6.5cm}
1258
1259 \framebox{
1260 \begin{minipage}{5.9cm}
1261 \begin{flushright}
1262 \includegraphics[width=5.9cm]{00-1_0-10.ps}\\[0.75cm]
1263 \end{flushright}
1264 \begin{center}
1265 \begin{pspicture}(0,0)(0,0)
1266 \psframe[linecolor=red,fillstyle=none](-2.8,-0.25)(3.3,1.1)
1267 \end{pspicture}
1268 \begin{picture}(0,0)(60,-5)
1269 \includegraphics[width=0.9cm]{albe_mig/00-1_0-10_red_00.eps}
1270 \end{picture}
1271 \begin{picture}(0,0)(0,-5)
1272 \includegraphics[width=0.9cm]{albe_mig/00-1_0-10_red_min.eps}
1273 \end{picture}
1274 \begin{picture}(0,0)(-55,-5)
1275 \includegraphics[width=0.9cm]{albe_mig/00-1_0-10_red_03.eps}
1276 \end{picture}
1277 \begin{picture}(0,0)(12.5,5)
1278 \includegraphics[width=1cm]{100_arrow.eps}
1279 \end{picture}
1280 \begin{picture}(0,0)(90,0)
1281 \includegraphics[height=0.9cm]{001_arrow.eps}
1282 \end{picture}
1283 \end{center}
1284 \vspace{0.2cm}
1285 \end{minipage}
1286 }\\[0.2cm]
1287
1288 \framebox{
1289 \begin{minipage}{5.9cm}
1290 \includegraphics[width=5.9cm]{00-1_ip0-10.ps}
1291 \end{minipage}
1292 }
1293
1294 \end{minipage}
1295
1296 \end{slide}
1297
1298 \begin{slide}
1299
1300  {\large\bf\boldmath
1301   Migrations involving the C \hkl<1 1 0> dumbbell interstitial
1302  }
1303
1304 \small
1305
1306 \vspace*{0.1cm}
1307
1308 VASP
1309
1310 \begin{minipage}{6.0cm}
1311 \includegraphics[width=6cm]{vasp_mig/110_mig_vasp.ps}
1312 \end{minipage}
1313 \begin{minipage}{7cm}
1314 \underline{Alternative pathway and energies [eV]}\\[0.1cm]
1315 \hkl<0 -1 0> $\stackrel{0.7}{{\color{red}\longrightarrow}}$
1316 \hkl<1 1 0> $\stackrel{0.95}{{\color{blue}\longrightarrow}}$
1317 BC $\stackrel{0.25}{\longrightarrow}$ \hkl<0 0 -1>\\[0.3cm]
1318 Composed of three single transitions\\[0.3cm]
1319 Activation energy of second transition slightly\\
1320 higher than direct transition (path 2)\\[0.3cm]
1321 $\Rightarrow$ very unlikely to happen
1322 \end{minipage}\\[0.2cm]
1323
1324 Erhart/Albe
1325
1326 \begin{minipage}{6.0cm}
1327 \includegraphics[width=6cm]{110_mig.ps}
1328 \end{minipage}
1329 \begin{minipage}{7cm}
1330 \underline{Alternative pathway and energies [eV]}\\[0.1cm]
1331 \hkl<0 0 -1> $\stackrel{2.2}{{\color{green}\longrightarrow}}$
1332 \hkl<1 1 0> $\stackrel{0.9}{{\color{red}\longrightarrow}}$
1333 \hkl<0 0 -1>\\[0.3cm]
1334 Composed of two single transitions\\[0.3cm]
1335 Compared to direct transition: (2.2 eV \& 0.5 eV)\\[0.3cm]
1336 $\Rightarrow$ more readily constituting a probable transition
1337 \end{minipage}
1338
1339 \end{slide}
1340
1341 \begin{slide}
1342
1343  {\large\bf\boldmath
1344   Combinations with a C-Si \hkl<1 0 0>-type interstitial
1345  }
1346
1347 \small
1348
1349 \vspace*{0.1cm}
1350
1351 Binding energy: 
1352 $
1353 E_{\text{b}}=
1354 E_{\text{f}}^{\text{defect combination}}-
1355 E_{\text{f}}^{\text{C \hkl<0 0 -1> dumbbell}}-
1356 E_{\text{f}}^{\text{2nd defect}}
1357 $
1358
1359 \vspace*{0.1cm}
1360
1361 {\scriptsize
1362 \begin{tabular}{l c c c c c c}
1363 \hline
1364  $E_{\text{b}}$ [eV] & 1 & 2 & 3 & 4 & 5 & R\\
1365  \hline
1366  \hkl<0 0 -1> & {\color{red}-0.08} & -1.15 & {\color{red}-0.08} & 0.04 & -1.66 & -0.19\\
1367  \hkl<0 0 1> & 0.34 & 0.004 & -2.05 & 0.26 & -1.53 & -0.19\\
1368  \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}\\
1369  \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}\\
1370  \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}\\
1371  \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}\\
1372  \hline
1373  C substitutional (C$_{\text{S}}$) & 0.26 & -0.51 & -0.93 & -0.15 & 0.49 & -0.05\\
1374  Vacancy & -5.39 ($\rightarrow$ C$_{\text{S}}$) & -0.59 & -3.14 & -0.54 & -0.50 & -0.31\\
1375 \hline
1376 \end{tabular}
1377 }
1378
1379 \vspace*{0.3cm}
1380
1381 \footnotesize
1382
1383 \begin{minipage}[t]{3.8cm}
1384 \underline{\hkl<1 0 0> at position 1}\\[0.1cm]
1385 \includegraphics[width=3.5cm]{00-1dc/2-25.eps}
1386 \end{minipage}
1387 \begin{minipage}[t]{3.5cm}
1388 \underline{\hkl<0 -1 0> at position 1}\\[0.1cm]
1389 \includegraphics[width=3.2cm]{00-1dc/2-39.eps}
1390 \end{minipage}
1391 \begin{minipage}[t]{5.5cm}
1392 \begin{itemize}
1393  \item Restricted to VASP simulations
1394  \item $E_{\text{b}}=0$ for isolated non-interacting defects
1395  \item $E_{\text{b}} \rightarrow 0$ for increasing distance (R)
1396  \item Stress compensation / increase
1397  \item Most favorable: C clustering
1398  \item Unfavored: antiparallel orientations
1399  \item Indication of energetically favored\\
1400        agglomeration
1401 \end{itemize}
1402 \end{minipage}
1403
1404 \begin{picture}(0,0)(-295,-130)
1405 \includegraphics[width=3.5cm]{comb_pos.eps}
1406 \end{picture}
1407
1408 \end{slide}
1409
1410 \begin{slide}
1411
1412  {\large\bf\boldmath
1413   Combinations of C-Si \hkl<1 0 0>-type interstitials
1414  }
1415
1416 \small
1417
1418 \vspace*{0.1cm}
1419
1420 Energetically most favorable combinations along \hkl<1 1 0>
1421
1422 \vspace*{0.1cm}
1423
1424 {\scriptsize
1425 \begin{tabular}{l c c c c c c}
1426 \hline
1427  & 1 & 2 & 3 & 4 & 5 & 6\\
1428 \hline
1429 $E_{\text{b}}$ [eV] & -2.39 & -1.88 & -0.59 & -0.31 & -0.24 & -0.21 \\
1430 C-C distance [\AA] & 1.4 & 4.6 & 6.5 & 8.6 & 10.5 & 10.8 \\
1431 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>\\
1432 \hline
1433 \end{tabular}
1434 }
1435
1436 \vspace*{0.3cm}
1437
1438 \begin{minipage}{7.0cm}
1439 \includegraphics[width=7cm]{db_along_110_cc.ps}
1440 \end{minipage}
1441 \begin{minipage}{6.0cm}
1442 \begin{center}
1443 {\color{blue}
1444  Interaction proportional to reciprocal cube of C-C distance
1445 }\\[0.2cm]
1446  Saturation in the immediate vicinity
1447 \end{center}
1448 \end{minipage}
1449
1450 \vspace{0.2cm}
1451
1452 \end{slide}
1453
1454 \begin{slide}
1455
1456  {\large\bf\boldmath
1457   Combinations of substitutional C and \hkl<1 1 0> Si self-interstitials
1458  }
1459
1460  \scriptsize
1461
1462 \begin{center}
1463 \begin{minipage}{3.2cm}
1464 \includegraphics[width=3cm]{sub_110_combo.eps}
1465 \end{minipage}
1466 \begin{minipage}{7.8cm}
1467 \begin{tabular}{l c c c c c c}
1468 \hline
1469 C$_{\text{sub}}$ & \hkl<1 1 0> & \hkl<-1 1 0> & \hkl<0 1 1> & \hkl<0 -1 1> &
1470                    \hkl<1 0 1> & \hkl<-1 0 1> \\
1471 \hline
1472 1 & \RM{1} & \RM{3} & \RM{3} & \RM{1} & \RM{3} & \RM{1} \\
1473 2 & \RM{2} & A & A & \RM{2} & C & \RM{5} \\
1474 3 & \RM{3} & \RM{1} & \RM{3} & \RM{1} & \RM{1} & \RM{3} \\
1475 4 & \RM{4} & B & D & E & E & D \\
1476 5 & \RM{5} & C & A & \RM{2} & A & \RM{2} \\
1477 \hline
1478 \end{tabular}
1479 \end{minipage}
1480 \end{center}
1481
1482 \begin{center}
1483 \begin{tabular}{l c c c c c c c c c c}
1484 \hline
1485 Conf & \RM{1} & \RM{2} & \RM{3} & \RM{4} & \RM{5} & A & B & C & D & E \\
1486 \hline
1487 $E_{\text{f}}$ [eV]& 4.37 & 5.26 & 5.57 & 5.37 & 5.12 & 5.10 & 5.32 & 5.28 & 5.39 & 5.32 \\
1488 $E_{\text{b}}$ [eV] & -0.97 & -0.08 & 0.22 & -0.02 & -0.23 & -0.25 & -0.02 & -0.06 & 0.05 & -0.03 \\
1489 $r$ [nm] & 0.292 & 0.394 & 0.241 & 0.453 & 0.407 & 0.408 & 0.452 & 0.392 & 0.456 & 0.453\\
1490 \hline
1491 \end{tabular}
1492 \end{center}
1493
1494 \begin{minipage}{6.0cm}
1495 \includegraphics[width=5.8cm]{c_sub_si110.ps}
1496 \end{minipage}
1497 \begin{minipage}{7cm}
1498 \small
1499 \begin{itemize}
1500  \item IBS: C may displace Si\\
1501        $\Rightarrow$ C$_{\text{sub}}$ + \hkl<1 1 0> Si self-interstitial
1502  \item Assumption:\\
1503        \hkl<1 1 0>-type $\rightarrow$ favored combination
1504  \renewcommand\labelitemi{$\Rightarrow$}
1505  \item Less favorable than C-Si \hkl<1 0 0> dumbbell\\
1506        ($E_{\text{f}}=3.88\text{ eV}$)
1507  \item Interaction drops quickly to zero\\
1508        (low interaction capture radius)
1509 \end{itemize}
1510 \end{minipage}
1511
1512 \end{slide}
1513
1514 \begin{slide}
1515
1516  {\large\bf\boldmath
1517   Migration in C-Si \hkl<1 0 0> and vacancy combinations
1518  }
1519
1520  \footnotesize
1521
1522 \vspace{0.1cm}
1523
1524 \begin{minipage}[t]{3cm}
1525 \underline{Pos 2, $E_{\text{b}}=-0.59\text{ eV}$}\\
1526 \includegraphics[width=2.8cm]{00-1dc/0-59.eps}
1527 \end{minipage}
1528 \begin{minipage}[t]{7cm}
1529 \vspace{0.2cm}
1530 \begin{center}
1531  Low activation energies\\
1532  High activation energies for reverse processes\\
1533  $\Downarrow$\\
1534  {\color{blue}C$_{\text{sub}}$ very stable}\\
1535 \vspace*{0.1cm}
1536  \hrule
1537 \vspace*{0.1cm}
1538  Without nearby \hkl<1 1 0> Si self-interstitial (IBS)\\
1539  $\Downarrow$\\
1540  {\color{blue}Formation of SiC by successive substitution by C}
1541
1542 \end{center}
1543 \end{minipage}
1544 \begin{minipage}[t]{3cm}
1545 \underline{Pos 3, $E_{\text{b}}=-3.14\text{ eV}$}\\
1546 \includegraphics[width=2.8cm]{00-1dc/3-14.eps}
1547 \end{minipage}
1548
1549
1550 \framebox{
1551 \begin{minipage}{5.9cm}
1552 \includegraphics[width=5.9cm]{vasp_mig/comb_mig_3-2_vac_fullct.ps}\\[0.6cm]
1553 \begin{center}
1554 \begin{picture}(0,0)(70,0)
1555 \includegraphics[width=1.4cm]{vasp_mig/comb_2-1_init.eps}
1556 \end{picture}
1557 \begin{picture}(0,0)(30,0)
1558 \includegraphics[width=1.4cm]{vasp_mig/comb_2-1_seq_03.eps}
1559 \end{picture}
1560 \begin{picture}(0,0)(-10,0)
1561 \includegraphics[width=1.4cm]{vasp_mig/comb_2-1_seq_06.eps}
1562 \end{picture}
1563 \begin{picture}(0,0)(-48,0)
1564 \includegraphics[width=1.4cm]{vasp_mig/comb_2-1_final.eps}
1565 \end{picture}
1566 \begin{picture}(0,0)(12.5,5)
1567 \includegraphics[width=1cm]{100_arrow.eps}
1568 \end{picture}
1569 \begin{picture}(0,0)(97,-10)
1570 \includegraphics[height=0.9cm]{001_arrow.eps}
1571 \end{picture}
1572 \end{center}
1573 \vspace{0.1cm}
1574 \end{minipage}
1575 }
1576 \begin{minipage}{0.3cm}
1577 \hfill
1578 \end{minipage}
1579 \framebox{
1580 \begin{minipage}{5.9cm}
1581 \includegraphics[width=5.9cm]{vasp_mig/comb_mig_4-2_vac_fullct.ps}\\[0.1cm]
1582 \begin{center}
1583 \begin{picture}(0,0)(60,0)
1584 \includegraphics[width=0.9cm]{vasp_mig/comb_3-1_init.eps}
1585 \end{picture}
1586 \begin{picture}(0,0)(25,0)
1587 \includegraphics[width=0.9cm]{vasp_mig/comb_3-1_seq_03.eps}
1588 \end{picture}
1589 \begin{picture}(0,0)(-20,0)
1590 \includegraphics[width=0.9cm]{vasp_mig/comb_3-1_seq_07.eps}
1591 \end{picture}
1592 \begin{picture}(0,0)(-55,0)
1593 \includegraphics[width=0.9cm]{vasp_mig/comb_3-1_final.eps}
1594 \end{picture}
1595 \begin{picture}(0,0)(12.5,5)
1596 \includegraphics[width=1cm]{100_arrow.eps}
1597 \end{picture}
1598 \begin{picture}(0,0)(95,0)
1599 \includegraphics[height=0.9cm]{001_arrow.eps}
1600 \end{picture}
1601 \end{center}
1602 \vspace{0.1cm}
1603 \end{minipage}
1604 }
1605
1606 \end{slide}
1607
1608 \begin{slide}
1609
1610  {\large\bf
1611   Conclusion of defect / migration / combined defect simulations
1612  }
1613
1614  \small
1615
1616 \vspace*{0.1cm}
1617
1618 Defect structures
1619 \begin{itemize}
1620  \item Accurately described by quantum-mechanical simulations
1621  \item Less correct description by classical potential simulations
1622 \end{itemize}
1623 \vspace*{0.2cm}
1624 \begin{itemize}
1625  \item Consistent with solubility data of C in Si
1626  \item \hkl<1 0 0> C-Si dumbbell interstitial ground state configuration
1627  \item Consistent with reorientation and diffusion experiments
1628  \item C migration pathway in Si identified
1629 \end{itemize} 
1630
1631 \vspace*{0.2cm}
1632
1633 Concerning the precipitation mechanism
1634 \begin{itemize}
1635  \item Agglomeration of C-Si dumbbells energetically favorable
1636  \item C-Si indeed favored compared to
1637        C$_{\text{sub}}$ \& \hkl<1 1 0> Si self-interstitial
1638  \item Possible low interaction capture radius of
1639        C$_{\text{sub}}$ \& \hkl<1 1 0> Si self-interstitial
1640  \item In absence of nearby \hkl<1 1 0> Si self-interstitial:
1641        C-Si \hkl<1 0 0> + Vacancy $\rightarrow$ C$_{\text{sub}}$ (SiC)
1642 \end{itemize} 
1643
1644 \vspace*{0.1cm}
1645 \begin{center}
1646 {\color{blue}Some results point to a different precipitation mechanism!}
1647 \end{center}
1648
1649 \end{slide}
1650
1651 \begin{slide}
1652
1653  {\large\bf
1654   Silicon carbide precipitation simulations
1655  }
1656
1657  \small
1658
1659 {\scriptsize
1660  \begin{pspicture}(0,0)(12,6.5)
1661   % nodes
1662   \rput(3.5,5.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
1663    \parbox{7cm}{
1664    \begin{itemize}
1665     \item Create c-Si volume
1666     \item Periodc boundary conditions
1667     \item Set requested $T$ and $p=0\text{ bar}$
1668     \item Equilibration of $E_{\text{kin}}$ and $E_{\text{pot}}$
1669    \end{itemize}
1670   }}}}
1671   \rput(3.5,2.7){\rnode{insert}{\psframebox[fillstyle=solid,fillcolor=lachs]{
1672    \parbox{7cm}{
1673    Insertion of C atoms at constant T
1674    \begin{itemize}
1675     \item total simulation volume {\pnode{in1}}
1676     \item volume of minimal SiC precipitate {\pnode{in2}}
1677     \item volume consisting of Si atoms to form a minimal {\pnode{in3}}\\
1678           precipitate
1679    \end{itemize} 
1680   }}}}
1681   \rput(3.5,1){\rnode{cool}{\psframebox[fillstyle=solid,fillcolor=lbb]{
1682    \parbox{7.0cm}{
1683    Run for 100 ps followed by cooling down to $20\, ^{\circ}\textrm{C}$
1684   }}}}
1685   \ncline[]{->}{init}{insert}
1686   \ncline[]{->}{insert}{cool}
1687   \psframe[fillstyle=solid,fillcolor=white](7.5,0.7)(13.5,6.3)
1688   \rput(7.8,6){\footnotesize $V_1$}
1689   \psframe[fillstyle=solid,fillcolor=lightgray](9,2)(12,5)
1690   \rput(9.2,4.85){\tiny $V_2$}
1691   \psframe[fillstyle=solid,fillcolor=gray](9.25,2.25)(11.75,4.75)
1692   \rput(9.55,4.45){\footnotesize $V_3$}
1693   \rput(7.9,3.2){\pnode{ins1}}
1694   \rput(9.22,2.8){\pnode{ins2}}
1695   \rput(11.0,2.4){\pnode{ins3}}
1696   \ncline[]{->}{in1}{ins1}
1697   \ncline[]{->}{in2}{ins2}
1698   \ncline[]{->}{in3}{ins3}
1699  \end{pspicture}
1700 }
1701
1702 \begin{itemize}
1703  \item Restricted to classical potential simulations
1704  \item $V_2$ and $V_3$ considered due to low diffusion
1705  \item Amount of C atoms: 6000
1706        ($r_{\text{prec}}\approx 3.1\text{ nm}$, IBS: 2 ... 4 nm)
1707  \item Simulation volume: $31\times 31\times 31$ unit cells
1708        (238328 Si atoms)
1709 \end{itemize}
1710
1711 \end{slide}
1712
1713 \begin{slide}
1714
1715  {\large\bf\boldmath
1716   Silicon carbide precipitation simulations at $450\,^{\circ}\mathrm{C}$ as in IBS
1717  }
1718
1719  \small
1720
1721 \begin{minipage}{6.5cm}
1722 \includegraphics[width=6.4cm]{sic_prec_450_si-si_c-c.ps}
1723 \end{minipage} 
1724 \begin{minipage}{6.5cm}
1725 \includegraphics[width=6.4cm]{sic_prec_450_energy.ps}
1726 \end{minipage} 
1727
1728 \begin{minipage}{6.5cm}
1729 \includegraphics[width=6.4cm]{sic_prec_450_si-c.ps}
1730 \end{minipage} 
1731 \begin{minipage}{6.5cm}
1732 \scriptsize
1733 \underline{Low C concentration ($V_1$)}\\
1734 \hkl<1 0 0> C-Si dumbbell dominated structure
1735 \begin{itemize}
1736  \item Si-C bumbs around 0.19 nm
1737  \item C-C peak at 0.31 nm (as expected in 3C-SiC):\\
1738        concatenated dumbbells of various orientation
1739  \item Si-Si NN distance stretched to 0.3 nm
1740 \end{itemize}
1741 {\color{blue}$\Rightarrow$ C atoms in proper 3C-SiC distance first}\\
1742 \underline{High C concentration ($V_2$, $V_3$)}\\
1743 High amount of strongly bound C-C bonds\\
1744 Defect density $\uparrow$ $\Rightarrow$ considerable amount of damage\\
1745 Only short range order observable\\
1746 {\color{blue}$\Rightarrow$ amorphous SiC-like phase}
1747 \end{minipage} 
1748
1749 \end{slide}
1750
1751 \begin{slide}
1752
1753  {\large\bf
1754   Limitations of molecular dynamics and short range potentials
1755  }
1756
1757 \footnotesize
1758
1759 \vspace{0.2cm}
1760
1761 \underline{Time scale problem of MD}\\[0.2cm]
1762 Minimize integration error\\
1763 $\Rightarrow$ discretization considerably smaller than
1764               reciprocal of fastest vibrational mode\\[0.1cm]
1765 Order of fastest vibrational mode: $10^{13} - 10^{14}\text{ Hz}$\\
1766 $\Rightarrow$ suitable choice of time step:
1767               $\tau=1\text{ fs}=10^{-15}\text{ s}$\\
1768 $\Rightarrow$ {\color{red}\underline{slow}} phase space propagation\\[0.1cm]
1769 Several local minima in energy surface separated by large energy barriers\\
1770 $\Rightarrow$ transition event corresponds to a multiple
1771               of vibrational periods\\
1772 $\Rightarrow$ phase transition made up of {\color{red}\underline{many}}
1773               infrequent transition events\\[0.1cm]
1774 {\color{blue}Accelerated methods:}
1775 \underline{Temperature accelerated} MD (TAD), self-guided MD \ldots
1776
1777 \vspace{0.3cm}
1778
1779 \underline{Limitations related to the short range potential}\\[0.2cm]
1780 Cut-off function pushing forces and energies to zero between 1$^{\text{st}}$
1781 and 2$^{\text{nd}}$ next neighbours\\
1782 $\Rightarrow$ overestimated unphysical high forces of next neighbours
1783
1784 \vspace{0.3cm}
1785
1786 \framebox{
1787 \color{red}
1788 Potential enhanced problem of slow phase space propagation
1789 }
1790
1791 \vspace{0.3cm}
1792
1793 \underline{Approach to the (twofold) problem}\\[0.2cm]
1794 Increased temperature simulations without TAD corrections\\
1795 (accelerated methods or higher time scales exclusively not sufficient)
1796
1797 \begin{picture}(0,0)(-260,-30)
1798 \framebox{
1799 \begin{minipage}{4.2cm}
1800 \tiny
1801 \begin{center}
1802 \vspace{0.03cm}
1803 \underline{IBS}
1804 \end{center}
1805 \begin{itemize}
1806 \item 3C-SiC also observed for higher T
1807 \item higher T inside sample
1808 \item structural evolution vs.\\
1809       equilibrium properties
1810 \end{itemize}
1811 \end{minipage}
1812 }
1813 \end{picture}
1814
1815 \begin{picture}(0,0)(-305,-155)
1816 \framebox{
1817 \begin{minipage}{2.5cm}
1818 \tiny
1819 \begin{center}
1820 retain proper\\
1821 thermodynmic sampling
1822 \end{center}
1823 \end{minipage}
1824 }
1825 \end{picture}
1826
1827 \end{slide}
1828
1829 \begin{slide}
1830
1831  {\large\bf
1832   Increased temperature simulations at low C concentration
1833  }
1834
1835 \small
1836
1837 \begin{minipage}{6.5cm}
1838 \includegraphics[width=6.4cm]{tot_pc_thesis.ps}
1839 \end{minipage}
1840 \begin{minipage}{6.5cm}
1841 \includegraphics[width=6.4cm]{tot_pc3_thesis.ps}
1842 \end{minipage}
1843
1844 \begin{minipage}{6.5cm}
1845 \includegraphics[width=6.4cm]{tot_pc2_thesis.ps}
1846 \end{minipage}
1847 \begin{minipage}{6.5cm}
1848 \scriptsize
1849  \underline{Si-C bonds:}
1850  \begin{itemize}
1851   \item Vanishing cut-off artifact (above $1650\,^{\circ}\mathrm{C}$)
1852   \item Structural change: C-Si \hkl<1 0 0> $\rightarrow$ C$_{\text{sub}}$
1853  \end{itemize}
1854  \underline{Si-Si bonds:}
1855  {\color{blue}Si-C$_{\text{sub}}$-Si} along \hkl<1 1 0>
1856  ($\rightarrow$ 0.325 nm)\\[0.1cm]
1857  \underline{C-C bonds:}
1858  \begin{itemize}
1859   \item C-C next neighbour pairs reduced (mandatory)
1860   \item Peak at 0.3 nm slightly shifted
1861         \begin{itemize}
1862          \item C-Si \hkl<1 0 0> combinations (dashed arrows)\\
1863                $\rightarrow$ C-Si \hkl<1 0 0> \& C$_{\text{sub}}$
1864                combinations (|)\\
1865                $\rightarrow$ pure {\color{blue}C$_{\text{sub}}$ combinations}
1866                ($\downarrow$)
1867          \item Range [|-$\downarrow$]:
1868                {\color{blue}C$_{\text{sub}}$ \& C$_{\text{sub}}$
1869                with nearby Si$_{\text{I}}$}
1870         \end{itemize}
1871  \end{itemize}
1872 \end{minipage}
1873
1874 \begin{picture}(0,0)(-330,-74)
1875 \color{blue}
1876 \framebox{
1877 \begin{minipage}{1.6cm}
1878 \tiny
1879 \begin{center}
1880 stretched SiC\\[-0.1cm]
1881 in c-Si
1882 \end{center}
1883 \end{minipage}
1884 }
1885 \end{picture}
1886
1887 \end{slide}
1888
1889 \begin{slide}
1890
1891  {\large\bf
1892   Increased temperature simulations at high C concentration
1893  }
1894
1895 \footnotesize
1896
1897 \begin{minipage}{6.5cm}
1898 \includegraphics[width=6.4cm]{12_pc_thesis.ps}
1899 \end{minipage}
1900 \begin{minipage}{6.5cm}
1901 \includegraphics[width=6.4cm]{12_pc_c_thesis.ps}
1902 \end{minipage}
1903
1904 \begin{center}
1905 Decreasing cut-off artifact\\
1906 High amount of {\color{red}damage} \& alignement to c-Si host matrix lost
1907 $\Rightarrow$ hard to categorize
1908 \end{center}
1909
1910 \vspace{0.1cm}
1911
1912 \framebox{
1913 \begin{minipage}[t]{6.0cm}
1914 0.186 nm: Si-C pairs $\uparrow$\\
1915 (as expected in 3C-SiC)\\[0.2cm]
1916 0.282 nm: Si-C-C\\[0.2cm]
1917 $\approx$0.35 nm: C-Si-Si
1918 \end{minipage}
1919 }
1920 \begin{minipage}{0.2cm}
1921 \hfill
1922 \end{minipage}
1923 \framebox{
1924 \begin{minipage}[t]{6.0cm}
1925 0.15 nm: C-C pairs $\uparrow$\\
1926 (as expected in graphite/diamond)\\[0.2cm]
1927 0.252 nm: C-C-C (2$^{\text{nd}}$ NN for diamond)\\[0.2cm]
1928 0.31 nm: shifted towards 0.317 nm $\rightarrow$ C-Si-C
1929 \end{minipage}
1930 }
1931
1932 \vspace{0.1cm}
1933
1934 \begin{center}
1935 {\color{red}Amorphous} SiC-like phase remains\\
1936 Slightly sharper peaks
1937 $\Rightarrow$ indicate slight {\color{blue}acceleration of dynamics}
1938 due to temperature\\[0.1cm]
1939 \framebox{
1940 \bf
1941 Continue with higher temperatures and longer time scales
1942 }
1943 \end{center}
1944
1945 \end{slide}
1946
1947 \begin{slide}
1948
1949  {\large\bf
1950   Valuation of a practicable temperature limit
1951  }
1952
1953  \small
1954
1955 \vspace{0.1cm}
1956
1957 \begin{center}
1958 \framebox{
1959 {\color{blue}
1960 Recrystallization is a hard task!
1961 $\Rightarrow$ Avoid melting!
1962 }
1963 }
1964 \end{center}
1965  
1966 \vspace{0.1cm}
1967
1968 \footnotesize
1969
1970 \begin{minipage}{7.5cm}
1971 \includegraphics[width=7cm]{fe_and_t.ps}
1972 \end{minipage}
1973 \begin{minipage}{5.5cm}
1974 \underline{Melting does not occur instantly after}\\
1975 \underline{exceeding the melting point $T_{\text{m}}=2450\text{ K}$}
1976 \begin{itemize}
1977 \item required transition enthalpy
1978 \item hysterisis behaviour
1979 \end{itemize}
1980 \underline{Heating up c-Si by 1 K/ps}
1981 \begin{itemize}
1982 \item transition occurs at $\approx$ 3125 K
1983 \item $\Delta E=0.58\text{ eV/atom}=55.7\text{ kJ/mole}$\\
1984       (literature: 50.2 kJ/mole)
1985 \end{itemize}
1986 \end{minipage}
1987
1988 \vspace{0.1cm}
1989
1990 \framebox{
1991 \begin{minipage}{4cm}
1992 Initially chosen temperatures:\\
1993 $1.0 - 1.2 \cdot T_{\text{m}}$
1994 \end{minipage}
1995 }
1996 \begin{minipage}{3cm}
1997 \begin{center}
1998 $\Longrightarrow$
1999 \end{center}
2000 \end{minipage}
2001 \framebox{
2002 \begin{minipage}{5cm}
2003 Introduced C (defects)\\
2004 $\rightarrow$ reduction of transition point\\
2005 $\rightarrow$ melting already at $T_{\text{m}}$
2006 \end{minipage}
2007 }
2008
2009 \vspace{0.4cm}
2010
2011 \begin{center}
2012 \framebox{
2013 {\color{blue}
2014 Maximum temperature used: $0.95\cdot T_{\text{m}}$
2015 }
2016 }
2017 \end{center}
2018
2019 \end{slide}
2020
2021 \begin{slide}
2022
2023  {\large\bf
2024   Long time scale simulations at maximum temperature
2025  }
2026
2027 \small
2028
2029 \vspace{0.1cm}
2030  
2031 \underline{Differences}
2032 \begin{itemize}
2033  \item Temperature set to $0.95 \cdot T_{\text{m}}$
2034  \item Cubic insertion volume $\Rightarrow$ spherical insertion volume
2035  \item Amount of C atoms: 6000 $\rightarrow$ 5500
2036        $\Leftrightarrow r_{\text{prec}}=0.3\text{ nm}$
2037  \item Simulation volume: 21 unit cells of c-Si in each direction
2038 \end{itemize}
2039
2040 \footnotesize
2041
2042 \vspace{0.3cm}
2043
2044 \begin{minipage}[t]{4.5cm}
2045 \begin{center}
2046 \underline{Low C concentration, Si-C}
2047 \includegraphics[width=4.5cm]{c_in_si_95_v1_si-c.ps}\\
2048 Sharper peaks!
2049 \end{center}
2050 \end{minipage}
2051 \begin{minipage}[t]{4.5cm}
2052 \begin{center}
2053 \underline{Low C concentration, C-C}
2054 \includegraphics[width=4.5cm]{c_in_si_95_v1_c-c.ps}\\
2055 Sharper peaks!\\
2056 No C agglomeration!
2057 \end{center}
2058 \end{minipage}
2059 \begin{minipage}[t]{4cm}
2060 \begin{center}
2061 \underline{High C concentration}
2062 \includegraphics[width=4.5cm]{c_in_si_95_v2.ps}\\
2063 No significant changes
2064 \end{center}
2065 \end{minipage}
2066
2067 \begin{center}
2068 \framebox{
2069 Long time scales and high temperatures most probably not sufficient enough!
2070 }
2071 \end{center}
2072
2073 \end{slide}
2074
2075 \begin{slide}
2076
2077  {\large\bf
2078   Investigation of a silicon carbide precipitate in silicon
2079  }
2080
2081  \footnotesize
2082
2083 \vspace{0.2cm}
2084
2085 \framebox{
2086 \scriptsize
2087 \begin{minipage}{5.3cm}
2088 \[
2089 \frac{8}{a_{\text{Si}}^3}(
2090 \underbrace{21^3 a_{\text{Si}}^3}_{=V}
2091 -\frac{4}{3}\pi x^3)+
2092 \underbrace{\frac{4}{y^3}\frac{4}{3}\pi x^3}_{\stackrel{!}{=}5500}
2093 =21^3\cdot 8
2094 \]
2095 \[
2096 \Downarrow
2097 \]
2098 \[
2099 \frac{8}{a_{\text{Si}}^3}\frac{4}{3}\pi x^3=5500
2100 \Rightarrow x = \left(\frac{5500 \cdot 3}{32 \pi} \right)^{1/3}a_{\text{Si}}
2101 \]
2102 \[
2103 y=\left(\frac{1}{2} \right)^{1/3}a_{\text{Si}}
2104 \]
2105 \end{minipage}
2106 }
2107 \begin{minipage}{0.3cm}
2108 \hfill
2109 \end{minipage}
2110 \begin{minipage}{7.0cm}
2111 \underline{Construction}
2112 \begin{itemize}
2113  \item Simulation volume: 21$^3$ unit cells of c-Si
2114  \item Spherical topotactically aligned precipitate\\
2115        $r=3.0\text{ nm}$ $\Leftrightarrow$ $\approx$ 5500 C atoms
2116  \item Create c-Si but skipped inside sphere of radius $x$
2117  \item Create 3C-SiC inside sphere of radius $x$\\
2118        and lattice constant $y$
2119  \item Strong coupling to heat bath ($T=20\,^{\circ}\mathrm{C}$)
2120 \end{itemize}
2121 \end{minipage}
2122
2123 \vspace{0.3cm}
2124
2125 \begin{minipage}{6.2cm}
2126 \includegraphics[width=6cm]{pc_0.ps}
2127 \end{minipage}
2128 \begin{minipage}{6.8cm}
2129 \underline{Results}
2130 \begin{itemize}
2131  \item Slight increase of c-Si lattice constant!
2132  \item C-C peaks (imply same distanced Si-Si peaks)
2133        \begin{itemize}
2134         \item New peak at 0.307 nm: 2$^{\text{nd}}$ NN in 3C-SiC
2135         \item Bumps ({\color{green}$\downarrow$}):
2136               4$^{\text{th}}$ and 6$^{\text{th}}$ NN
2137        \end{itemize}
2138  \item 3C-SiC lattice constant: 4.34 \AA (bulk: 4.36 \AA)\\
2139        $\rightarrow$ compressed precipitate
2140  \item Interface tension:\\
2141        20.15 eV/nm$^2$ or $3.23 \times 10^{-4}$ J/cm$^2$\\
2142        (literature: $2 - 8 \times 10^{-4}$ J/cm$^2$)
2143 \end{itemize}
2144 \end{minipage}
2145
2146 \end{slide}
2147
2148 \begin{slide}
2149
2150  {\large\bf
2151   Investigation of a silicon carbide precipitate in silicon
2152  }
2153
2154  \footnotesize
2155
2156 \begin{minipage}{7cm}
2157 \underline{Appended annealing steps}
2158 \begin{itemize}
2159  \item artificially constructed interface\\
2160        $\rightarrow$ allow for rearrangement of interface atoms
2161  \item check SiC stability
2162 \end{itemize}
2163 \underline{Temperature schedule}
2164 \begin{itemize}
2165  \item rapidly heat up structure up to $2050\,^{\circ}\mathrm{C}$\\
2166        (75 K/ps)
2167  \item slow heating up to $1.2\cdot T_{\text{m}}=2940\text{ K}$
2168        by 1 K/ps\\
2169        $\rightarrow$ melting at around 2840 K
2170        (\href{../video/sic_prec_120.avi}{$\rhd$})
2171  \item cooling down structure at 100 \% $T_{\text{m}}$ (1 K/ps)\\
2172        $\rightarrow$ no energetically more favorable struture
2173 \end{itemize}
2174 \end{minipage}
2175 \begin{minipage}{6cm}
2176 \includegraphics[width=6.7cm]{fe_and_t_sic.ps}
2177 \end{minipage}
2178
2179 \begin{minipage}{4cm}
2180 \includegraphics[width=4cm]{sic_prec/melt_01.eps}
2181 \end{minipage}
2182 \begin{minipage}{0.4cm}
2183 $\rightarrow$
2184 \end{minipage}
2185 \begin{minipage}{4cm}
2186 \includegraphics[width=4cm]{sic_prec/melt_02.eps}
2187 \end{minipage}
2188 \begin{minipage}{0.4cm}
2189 $\rightarrow$
2190 \end{minipage}
2191 \begin{minipage}{4cm}
2192 \includegraphics[width=4cm]{sic_prec/melt_03.eps}
2193 \end{minipage}
2194
2195 \end{slide}
2196
2197 \begin{slide}
2198
2199  {\large\bf
2200   Summary / Conclusion / Outlook
2201  }
2202
2203  \scriptsize
2204
2205 \vspace{0.1cm}
2206
2207 \framebox{
2208 \begin{minipage}{12.9cm}
2209  \underline{Defects}
2210  \begin{itemize}
2211   \item Summary \& conclusion
2212         \begin{itemize}
2213          \item Point defects excellently / fairly well described
2214                by QM / classical potential simulations
2215          \item Identified migration path explaining
2216                diffusion and reorientation experiments
2217          \item Agglomeration of point defects energetically favorable
2218          \item C$_{\text{sub}}$ favored conditions (conceivable in IBS)
2219         \end{itemize}
2220   \item Todo
2221         \begin{itemize}
2222          \item Discussions concerning interpretation of QM results (Paderborn)
2223          \item Compare migration barrier of
2224                \hkl<1 1 0> Si and C-Si \hkl<1 0 0> dumbbell
2225          \item Combination: Vacancy \& \hkl<1 1 0> Si self-interstitial \&
2226                             C-Si \hkl<1 0 0> dumbbell (IBS)
2227         \end{itemize}
2228  \end{itemize}
2229 \end{minipage}
2230 }
2231
2232 \vspace{0.2cm}
2233
2234 \framebox{
2235 \begin{minipage}[t]{6.2cm}
2236  \underline{Pecipitation simulations}
2237  \begin{itemize}
2238   \item Summary \& conclusion
2239         \begin{itemize}
2240          \item Low T
2241                $\rightarrow$ C-Si \hkl<1 0 0> dumbbell\\
2242                dominated structure
2243          \item High T $\rightarrow$ C$_{\text{sub}}$ dominated structure
2244          \item High C concentration\\
2245                $\rightarrow$ amorphous SiC like phase
2246         \end{itemize}
2247   \item Todo
2248         \begin{itemize}
2249          \item Accelerated method: self-guided MD
2250          \item Activation relaxation technique
2251          \item Constrainted transition path
2252         \end{itemize}
2253  \end{itemize}
2254 \end{minipage}
2255 }
2256 \framebox{
2257 \begin{minipage}[t]{6.2cm}
2258  \underline{Constructed 3C-SiC precipitate}
2259  \begin{itemize}
2260   \item Summary \& conclusion
2261         \begin{itemize}
2262          \item Small / stable / compressed 3C-SiC\\
2263                precipitate in slightly stretched\\
2264                c-Si matrix
2265          \item Interface tension matches experiemnts
2266         \end{itemize}
2267   \item Todo
2268         \begin{itemize}
2269          \item Try to improve interface
2270          \item Precipitates of different size
2271         \end{itemize}
2272  \end{itemize}
2273 \end{minipage}
2274 }
2275
2276  \small
2277
2278 \end{slide}
2279
2280 \begin{slide}
2281
2282  {\large\bf
2283   Acknowledgements
2284  }
2285
2286  \vspace{0.1cm}
2287
2288  \small
2289
2290  Thanks to \ldots
2291
2292  \underline{Augsburg}
2293  \begin{itemize}
2294   \item Prof. B. Stritzker (accepting a simulator at EP \RM{4})
2295   \item Ralf Utermann (EDV)
2296  \end{itemize}
2297  
2298  \underline{Helsinki}
2299  \begin{itemize}
2300   \item Prof. K. Nordlund (MD)
2301  \end{itemize}
2302  
2303  \underline{Munich}
2304  \begin{itemize}
2305   \item Bayerische Forschungsstiftung (financial support)
2306  \end{itemize}
2307  
2308  \underline{Paderborn}
2309  \begin{itemize}
2310   \item Prof. J. Lindner (SiC)
2311   \item Prof. G. Schmidt (DFT + financial support)
2312   \item Dr. E. Rauls (DFT + SiC)
2313  \end{itemize}
2314
2315 \vspace{0.2cm}
2316
2317 \begin{center}
2318 \framebox{
2319 \bf Thank you for your attention!
2320 }
2321 \end{center}
2322
2323 \end{slide}
2324
2325 \end{document}