posic/talks/mpi_app.tex

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   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
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  16
  17 \usepackage{fancyhdr}           % Headers and footers definitions
  18 \usepackage{fancyvrb}           % Fancy verbatim environments
  19 \usepackage{pstricks}           % PSTricks with the standard color package
  20
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  23
  24 %\usepackage{epic}
  25 %\usepackage{eepic}
  26
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  28
  29 \usepackage{graphicx}
  30 \graphicspath{{../img/}}
  31
  32 \usepackage{miller}
  33
  34 \usepackage[setpagesize=false]{hyperref}
  35
  36 % units
  37 \usepackage{units}
  38
  39 \usepackage{semcolor}
  40 \usepackage{semlayer}           % Seminar overlays
  41 \usepackage{slidesec}           % Seminar sections and list of slides
  42
  43 \input{seminar.bug}             % Official bugs corrections
  44 \input{seminar.bg2}             % Unofficial bugs corrections
  45
  46 \articlemag{1}
  47
  48 \special{landscape}
  49
  50 % font
  51 %\usepackage{cmbright}
  52 %\renewcommand{\familydefault}{\sfdefault}
  53 %\usepackage{mathptmx}
  54
  55 \usepackage{upgreek}
  56
  57 \begin{document}
  58
  59 \extraslideheight{10in}
  60 \slideframe{plain}
  61
  62 \pagestyle{empty}
  63
  64 % specify width and height
  65 \slidewidth 26.3cm
  66 \slideheight 19.9cm
  67
  68 % margin
  69 \def\slidetopmargin{-0.15cm}
  70
  71 \newcommand{\ham}{\mathcal{H}}
  72 \newcommand{\pot}{\mathcal{V}}
  73 \newcommand{\foo}{\mathcal{U}}
  74 \newcommand{\vir}{\mathcal{W}}
  75
  76 % itemize level ii
  77 \renewcommand\labelitemii{{\color{gray}$\bullet$}}
  78
  79 % nice phi
  80 \renewcommand{\phi}{\varphi}
  81
  82 % roman letters
  83 \newcommand{\RM}[1]{\MakeUppercase{\romannumeral #1{}}}
  84
  85 % colors
  86 \newrgbcolor{si-yellow}{.6 .6 0}
  87 \newrgbcolor{hb}{0.75 0.77 0.89}
  88 \newrgbcolor{lbb}{0.75 0.8 0.88}
  89 \newrgbcolor{hlbb}{0.825 0.88 0.968}
  90 \newrgbcolor{lachs}{1.0 .93 .81}
  91
  92 % shortcuts
  93 \newcommand{\si}{Si$_{\text{i}}${}}
  94 \newcommand{\ci}{C$_{\text{i}}${}}
  95 \newcommand{\cs}{C$_{\text{sub}}${}}
  96 \newcommand{\degc}[1]{\unit[#1]{$^{\circ}$C}{}}
  97 \newcommand{\distn}[1]{\unit[#1]{nm}{}}
  98 \newcommand{\dista}[1]{\unit[#1]{\AA}{}}
  99 \newcommand{\perc}[1]{\unit[#1]{\%}{}}
 100
 101 % no vertical centering
 102 %\centerslidesfalse
 103
 104 % layout check
 105 %\layout
 106 \begin{slide}
 107 \center
 108 {\Huge
 109 E\\
 110 F\\
 111 G\\
 112 A B C D E F G H G F E D C B A
 113 G\\
 114 F\\
 115 E\\
 116 }
 117 \end{slide}
 118
 119 % topic
 120
 121 \begin{slide}
 122 \begin{center}
 123
 124  \vspace{16pt}
 125
 126  {\LARGE\bf
 127   Atomistic simulation studies\\[0.2cm]
 128   in the C/Si system
 129  }
 130
 131  \vspace{48pt}
 132
 133  \textsc{Frank Zirkelbach}
 134
 135  \vspace{48pt}
 136
 137  Application talk at the Max Planck Institute for Solid State Research
 138
 139  \vspace{08pt}
 140
 141  Stuttgart, November 2011
 142
 143 \end{center}
 144 \end{slide}
 145
 146 \ifnum1=0
 147
 148 % intro
 149
 150 \begin{slide}
 151
 152 %{\large\bf
 153 % Phase diagram of the C/Si system\\
 154 %}
 155
 156 \vspace*{0.2cm}
 157
 158 \begin{minipage}{6.5cm}
 159 \includegraphics[width=6.5cm]{si-c_phase.eps}
 160 \begin{center}
 161 {\tiny
 162 R. I. Scace and G. A. Slack, J. Chem. Phys. 30, 1551 (1959)
 163 }
 164 \end{center}
 165 \begin{pspicture}(0,0)(0,0)
 166 \psellipse[linecolor=blue,linewidth=0.1cm](3.55,4.0)(0.5,2.9)
 167 \end{pspicture}
 168 \end{minipage}
 169 \begin{minipage}{6cm}
 170 {\bf Phase diagram of the C/Si system}\\[0.2cm]
 171 {\color{blue}Stoichiometric composition}
 172 \begin{itemize}
 173 \item only chemical stable compound
 174 \item wide band gap semiconductor\\
 175       \underline{silicon carbide}, SiC
 176 \end{itemize}
 177 \end{minipage}
 178
 179 \end{slide}
 180
 181 % motivation / properties / applications of silicon carbide
 182
 183 \begin{slide}
 184
 185 \small
 186
 187 \begin{pspicture}(0,0)(13.5,5)
 188
 189  \psframe*[linecolor=hb](-0.2,0)(12.9,5)
 190
 191  \pspolygon[linecolor=hlbb,fillcolor=hlbb,fillstyle=solid](5.2,1)(6.5,1)(6.5,3)(5.2,3)
 192  \pspolygon[linecolor=hlbb,fillcolor=hlbb,fillstyle=solid](6.4,0.5)(7.7,2)(7.7,2)(6.4,3.5)
 193
 194  \rput[lt](0,4.6){\color{gray}PROPERTIES}
 195
 196  \rput[lt](0.3,4){wide band gap}
 197  \rput[lt](0.3,3.5){high electric breakdown field}
 198  \rput[lt](0.3,3){good electron mobility}
 199  \rput[lt](0.3,2.5){high electron saturation drift velocity}
 200  \rput[lt](0.3,2){high thermal conductivity}
 201
 202  \rput[lt](0.3,1.5){hard and mechanically stable}
 203  \rput[lt](0.3,1){chemically inert}
 204
 205  \rput[lt](0.3,0.5){radiation hardness}
 206
 207  \rput[rt](12.7,4.6){\color{gray}APPLICATIONS}
 208
 209  \rput[rt](12.5,3.85){high-temperature, high power}
 210  \rput[rt](12.5,3.5){and high-frequency}
 211  \rput[rt](12.5,3.15){electronic and optoelectronic devices}
 212
 213  \rput[rt](12.5,2.35){material suitable for extreme conditions}
 214  \rput[rt](12.5,2){microelectromechanical systems}
 215  \rput[rt](12.5,1.65){abrasives, cutting tools, heating elements}
 216
 217  \rput[rt](12.5,0.85){first wall reactor material, detectors}
 218  \rput[rt](12.5,0.5){and electronic devices for space}
 219
 220 \end{pspicture}
 221
 222 \begin{picture}(0,0)(5,-162)
 223 \includegraphics[height=2.2cm]{3C_SiC_bs.eps}
 224 \end{picture}
 225 \begin{picture}(0,0)(-120,-162)
 226 \includegraphics[height=2.2cm]{nasa_600c_led.eps}
 227 \end{picture}
 228 \begin{picture}(0,0)(-270,-162)
 229 \includegraphics[height=2.2cm]{6h-sic_3c-sic.eps}
 230 \end{picture}
 231 %%%%
 232 \begin{picture}(0,0)(10,65)
 233 \includegraphics[height=2.8cm]{sic_switch.eps}
 234 \end{picture}
 235 %\begin{picture}(0,0)(-243,65)
 236 \begin{picture}(0,0)(-110,65)
 237 \includegraphics[height=2.8cm]{ise_99.eps}
 238 \end{picture}
 239 %\begin{picture}(0,0)(-135,65)
 240 \begin{picture}(0,0)(-100,65)
 241 \includegraphics[height=1.2cm]{infineon_schottky.eps}
 242 \end{picture}
 243 \begin{picture}(0,0)(-233,65)
 244 \includegraphics[height=2.8cm]{solar_car.eps}
 245 \end{picture}
 246
 247 \end{slide}
 248
 249 % motivation
 250
 251 \begin{slide}
 252
 253  {\large\bf
 254   Polytypes of SiC\\[0.4cm]
 255  }
 256
 257 \includegraphics[width=3.8cm]{cubic_hex.eps}\\
 258 \begin{minipage}{1.9cm}
 259 {\tiny cubic (twist)}
 260 \end{minipage}
 261 \begin{minipage}{2.9cm}
 262 {\tiny hexagonal (no twist)}
 263 \end{minipage}
 264
 265 \begin{picture}(0,0)(-150,0)
 266  \includegraphics[width=7cm]{polytypes.eps}
 267 \end{picture}
 268
 269 \vspace{0.6cm}
 270
 271 \footnotesize
 272
 273 \begin{tabular}{l c c c c c c}
 274 \hline
 275  & 3C-SiC & 4H-SiC & 6H-SiC & Si & GaN & Diamond\\
 276 \hline
 277 Hardness [Mohs] & \multicolumn{3}{c}{------ 9.6 ------}& 6.5 & - & 10 \\
 278 Band gap [eV] & 2.36 & 3.23 & 3.03 & 1.12 & 3.39 & 5.5 \\
 279 Break down field [$10^6$ V/cm] & 4 & 3 & 3.2 & 0.6 & 5 & 10 \\
 280 Saturation drift velocity [$10^7$ cm/s] & 2.5 & 2.0 & 2.0 & 1 & 2.7 & 2.7 \\
 281 Electron mobility [cm$^2$/Vs] & 800 & 900 & 400 & 1100 & 900 & 2200 \\
 282 Hole mobility [cm$^2$/Vs] & 320 & 120 & 90 & 420 & 150 & 1600 \\
 283 Thermal conductivity [W/cmK] & 5.0 & 4.9 & 4.9 & 1.5 & 1.3 & 22 \\
 284 \hline
 285 \end{tabular}
 286
 287 \begin{pspicture}(0,0)(0,0)
 288 \psellipse[linecolor=green](5.7,2.10)(0.4,0.5)
 289 \end{pspicture}
 290 \begin{pspicture}(0,0)(0,0)
 291 \psellipse[linecolor=green](5.6,0.92)(0.4,0.2)
 292 \end{pspicture}
 293 \begin{pspicture}(0,0)(0,0)
 294 \psellipse[linecolor=red](10.45,0.45)(0.4,0.2)
 295 \end{pspicture}
 296
 297 \end{slide}
 298
 299 \fi
 300 % fabrication
 301
 302 \begin{slide}
 303
 304  {\large\bf
 305   Fabrication of silicon carbide
 306  }
 307
 308  \small
 309
 310  \vspace{2pt}
 311
 312  {\color{gray}
 313  \emph{Silicon carbide --- Born from the stars, perfected on earth.}
 314  }
 315
 316 \vspace{2pt}
 317
 318 SiC thin film by MBE \& CVD
 319 \begin{itemize}
 320  \item Much progress achieved in homo/heteroepitaxial SiC thin film growth
 321  \item \underline{Commercially available} semiconductor power devices based on
 322        \underline{\foreignlanguage{greek}{a}-SiC}
 323  \item Production of favored \underline{3C-SiC} material
 324        \underline{less advanced}
 325  \item Quality and size not yet sufficient
 326 \end{itemize}
 327 \begin{picture}(0,0)(-310,-20)
 328   \includegraphics[width=2.0cm]{cree.eps}
 329 \end{picture}
 330
 331 Alternative method: Ion beam synthesis of SiC in Si
 332
 333  \begin{itemize}
 334   \item \underline{Sublimation growth using the modified Lely method}
 335         \begin{itemize}
 336          \item SiC single-crystalline seed at $T=1800 \, ^{\circ} \text{C}$
 337          \item Surrounded by polycrystalline SiC in a graphite crucible\\
 338                at $T=2100-2400 \, ^{\circ} \text{C}$
 339          \item Deposition of supersaturated vapor on cooler seed crystal
 340         \end{itemize}
 341   \item \underline{Homoepitaxial growth using CVD}
 342         \begin{itemize}
 343          \item Step-controlled epitaxy on off-oriented 6H-SiC substrates
 344          \item C$_3$H$_8$/SiH$_4$/H$_2$ at $1100-1500 \, ^{\circ} \text{C}$
 345          \item Angle, temperature $\rightarrow$ 3C/6H/4H-SiC
 346         \end{itemize}
 347   \item \underline{Heteroepitaxial growth of 3C-SiC on Si using CVD/MBE}
 348         \begin{itemize}
 349          \item Two steps: carbonization and growth
 350          \item $T=650-1050 \, ^{\circ} \text{C}$
 351          \item SiC/Si lattice mismatch $\approx$ 20 \%
 352          \item Quality and size not yet sufficient
 353         \end{itemize}
 354  \end{itemize}
 355
 356  \begin{picture}(0,0)(-280,-65)
 357   \includegraphics[width=3.8cm]{6h-sic_3c-sic.eps}
 358  \end{picture}
 359  \begin{picture}(0,0)(-280,-55)
 360   \begin{minipage}{5cm}
 361   {\tiny
 362    NASA: 6H-SiC and 3C-SiC LED\\[-7pt]
 363    on 6H-SiC substrate
 364   }
 365   \end{minipage}
 366  \end{picture}
 367  \begin{picture}(0,0)(-265,-150)
 368   \includegraphics[width=2.4cm]{m_lely.eps}
 369  \end{picture}
 370  \begin{picture}(0,0)(-333,-175)
 371   \begin{minipage}{5cm}
 372   {\tiny
 373    1. Lid\\[-7pt]
 374    2. Heating\\[-7pt]
 375    3. Source\\[-7pt]
 376    4. Crucible\\[-7pt]
 377    5. Insulation\\[-7pt]
 378    6. Seed crystal
 379   }
 380   \end{minipage}
 381  \end{picture}
 382  \begin{picture}(0,0)(-230,-35)
 383  \framebox{
 384  {\footnotesize\color{blue}\bf Hex: micropipes along c-axis}
 385  }
 386  \end{picture}
 387  \begin{picture}(0,0)(-230,-10)
 388  \framebox{
 389  \begin{minipage}{3cm}
 390  {\footnotesize\color{blue}\bf 3C-SiC fabrication\\
 391                                less advanced}
 392  \end{minipage}
 393  }
 394  \end{picture}
 395
 396 \end{slide}
 397
 398 \end{document}
 399 \ifnum1=0
 400
 401 % contents
 402
 403 \begin{slide}
 404
 405 {\large\bf
 406  Outline
 407 }
 408
 409  \begin{itemize}
 410   \item Implantation of C in Si --- Overview of experimental observations
 411   \item Utilized simulation techniques and modeled problems
 412         \begin{itemize}
 413          \item {\color{blue}Diploma thesis}\\
 414                \underline{Monte Carlo} simulations
 415                modeling the selforganization process
 416                leading to periodic arrays of nanometric amorphous SiC
 417                precipitates
 418          \item {\color{blue}Doctoral studies}\\
 419                Classical potential \underline{molecular dynamics} simulations
 420                \ldots\\
 421                \underline{Density functional theory} calculations
 422                \ldots\\[0.2cm]
 423                \ldots on defects and SiC precipitation in Si
 424         \end{itemize}
 425   \item Summary / Conclusion / Outlook
 426  \end{itemize}
 427
 428 \end{slide}
 429
 430
 431
 432 \end{document}
 433
 434 \begin{slide}
 435
 436  {\large\bf
 437   Fabrication of silicon carbide
 438  }
 439
 440  \small
 441
 442  Alternative approach:
 443  Ion beam synthesis (IBS) of burried 3C-SiC layers in Si\hkl(1 0 0)
 444  \begin{itemize}
 445   \item \underline{Implantation step 1}\\
 446         180 keV C$^+$, $D=7.9\times 10^{17}$ cm$^{-2}$, $T_{\text{i}}=500\,^{\circ}\mathrm{C}$\\
 447         $\Rightarrow$ box-like distribution of equally sized
 448                        and epitactically oriented SiC precipitates
 449
 450   \item \underline{Implantation step 2}\\
 451         180 keV C$^+$, $D=0.6\times 10^{17}$ cm$^{-2}$, $T_{\text{i}}=250\,^{\circ}\mathrm{C}$\\
 452         $\Rightarrow$ destruction of SiC nanocrystals
 453                       in growing amorphous interface layers
 454   \item \underline{Annealing}\\
 455         $T=1250\,^{\circ}\mathrm{C}$, $t=10\,\text{h}$\\
 456         $\Rightarrow$ homogeneous, stoichiometric SiC layer
 457                       with sharp interfaces
 458  \end{itemize}
 459
 460  \begin{minipage}{6.3cm}
 461  \includegraphics[width=6cm]{ibs_3c-sic.eps}\\[-0.2cm]
 462  {\tiny
 463   XTEM micrograph of single crystalline 3C-SiC in Si\hkl(1 0 0)
 464  }
 465  \end{minipage}
 466 \framebox{
 467  \begin{minipage}{6.3cm}
 468  \begin{center}
 469  {\color{blue}
 470   Precipitation mechanism not yet fully understood!
 471  }
 472  \renewcommand\labelitemi{$\Rightarrow$}
 473  \small
 474  \underline{Understanding the SiC precipitation}
 475  \begin{itemize}
 476   \item significant technological progress in SiC thin film formation
 477   \item perspectives for processes relying upon prevention of SiC precipitation
 478  \end{itemize}
 479  \end{center}
 480  \end{minipage}
 481 }
 482
 483 \end{slide}
 484
 485
 486 \begin{slide}
 487
 488  {\large\bf
 489   Supposed precipitation mechanism of SiC in Si
 490  }
 491
 492  \scriptsize
 493
 494  \vspace{0.1cm}
 495
 496  \begin{minipage}{3.8cm}
 497  Si \& SiC lattice structure\\[0.2cm]
 498  \includegraphics[width=3.5cm]{sic_unit_cell.eps}\\[-0.3cm]
 499  \hrule
 500  \end{minipage}
 501  \hspace{0.6cm}
 502  \begin{minipage}{3.8cm}
 503  \begin{center}
 504  \includegraphics[width=3.3cm]{tem_c-si-db.eps}
 505  \end{center}
 506  \end{minipage}
 507  \hspace{0.6cm}
 508  \begin{minipage}{3.8cm}
 509  \begin{center}
 510  \includegraphics[width=3.3cm]{tem_3c-sic.eps}
 511  \end{center}
 512  \end{minipage}
 513
 514  \begin{minipage}{4cm}
 515  \begin{center}
 516  C-Si dimers (dumbbells)\\[-0.1cm]
 517  on Si interstitial sites
 518  \end{center}
 519  \end{minipage}
 520  \hspace{0.2cm}
 521  \begin{minipage}{4.2cm}
 522  \begin{center}
 523  Agglomeration of C-Si dumbbells\\[-0.1cm]
 524  $\Rightarrow$ dark contrasts
 525  \end{center}
 526  \end{minipage}
 527  \hspace{0.2cm}
 528  \begin{minipage}{4cm}
 529  \begin{center}
 530  Precipitation of 3C-SiC in Si\\[-0.1cm]
 531  $\Rightarrow$ Moir\'e fringes\\[-0.1cm]
 532  \& release of Si self-interstitials
 533  \end{center}
 534  \end{minipage}
 535
 536  \begin{minipage}{3.8cm}
 537  \begin{center}
 538  \includegraphics[width=3.3cm]{sic_prec_seq_01.eps}
 539  \end{center}
 540  \end{minipage}
 541  \hspace{0.6cm}
 542  \begin{minipage}{3.8cm}
 543  \begin{center}
 544  \includegraphics[width=3.3cm]{sic_prec_seq_02.eps}
 545  \end{center}
 546  \end{minipage}
 547  \hspace{0.6cm}
 548  \begin{minipage}{3.8cm}
 549  \begin{center}
 550  \includegraphics[width=3.3cm]{sic_prec_seq_03.eps}
 551  \end{center}
 552  \end{minipage}
 553
 554 \begin{pspicture}(0,0)(0,0)
 555 \psline[linewidth=4pt]{->}(8.5,2)(9.0,2)
 556 \psellipse[linecolor=blue](11.5,5.8)(0.3,0.5)
 557 \rput{-20}{\psellipse[linecolor=blue](3.3,8.1)(0.3,0.5)}
 558 \psline[linewidth=4pt]{->}(4.0,2)(4.5,2)
 559 \rput(12.7,0.3){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
 560  $4a_{\text{Si}}=5a_{\text{SiC}}$
 561  }}}
 562 \rput(12.2,8){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
 563 \hkl(h k l) planes match
 564  }}}
 565 \rput(9.7,6.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
 566 r = 2 - 4 nm
 567  }}}
 568 \end{pspicture}
 569
 570 \end{slide}
 571
 572 \begin{slide}
 573
 574  {\large\bf
 575   Supposed precipitation mechanism of SiC in Si
 576  }
 577
 578  \scriptsize
 579
 580  \vspace{0.1cm}
 581
 582  \begin{minipage}{3.8cm}
 583  Si \& SiC lattice structure\\[0.2cm]
 584  \includegraphics[width=3.5cm]{sic_unit_cell.eps}\\[-0.3cm]
 585  \hrule
 586  \end{minipage}
 587  \hspace{0.6cm}
 588  \begin{minipage}{3.8cm}
 589  \begin{center}
 590  \includegraphics[width=3.3cm]{tem_c-si-db.eps}
 591  \end{center}
 592  \end{minipage}
 593  \hspace{0.6cm}
 594  \begin{minipage}{3.8cm}
 595  \begin{center}
 596  \includegraphics[width=3.3cm]{tem_3c-sic.eps}
 597  \end{center}
 598  \end{minipage}
 599
 600  \begin{minipage}{4cm}
 601  \begin{center}
 602  C-Si dimers (dumbbells)\\[-0.1cm]
 603  on Si interstitial sites
 604  \end{center}
 605  \end{minipage}
 606  \hspace{0.2cm}
 607  \begin{minipage}{4.2cm}
 608  \begin{center}
 609  Agglomeration of C-Si dumbbells\\[-0.1cm]
 610  $\Rightarrow$ dark contrasts
 611  \end{center}
 612  \end{minipage}
 613  \hspace{0.2cm}
 614  \begin{minipage}{4cm}
 615  \begin{center}
 616  Precipitation of 3C-SiC in Si\\[-0.1cm]
 617  $\Rightarrow$ Moir\'e fringes\\[-0.1cm]
 618  \& release of Si self-interstitials
 619  \end{center}
 620  \end{minipage}
 621
 622  \begin{minipage}{3.8cm}
 623  \begin{center}
 624  \includegraphics[width=3.3cm]{sic_prec_seq_01.eps}
 625  \end{center}
 626  \end{minipage}
 627  \hspace{0.6cm}
 628  \begin{minipage}{3.8cm}
 629  \begin{center}
 630  \includegraphics[width=3.3cm]{sic_prec_seq_02.eps}
 631  \end{center}
 632  \end{minipage}
 633  \hspace{0.6cm}
 634  \begin{minipage}{3.8cm}
 635  \begin{center}
 636  \includegraphics[width=3.3cm]{sic_prec_seq_03.eps}
 637  \end{center}
 638  \end{minipage}
 639
 640 \begin{pspicture}(0,0)(0,0)
 641 \psline[linewidth=4pt]{->}(8.5,2)(9.0,2)
 642 \psellipse[linecolor=blue](11.5,5.8)(0.3,0.5)
 643 \rput{-20}{\psellipse[linecolor=blue](3.3,8.1)(0.3,0.5)}
 644 \psline[linewidth=4pt]{->}(4.0,2)(4.5,2)
 645 \rput(12.7,0.3){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
 646  $4a_{\text{Si}}=5a_{\text{SiC}}$
 647  }}}
 648 \rput(12.2,8){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
 649 \hkl(h k l) planes match
 650  }}}
 651 \rput(9.7,6.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
 652 r = 2 - 4 nm
 653  }}}
 654 \rput(6.7,5.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=white]{
 655 \begin{minipage}{10cm}
 656 \small
 657 {\color{red}\bf Controversial views}
 658 \begin{itemize}
 659 \item Implantations at high T (Nejim et al.)
 660  \begin{itemize}
 661   \item Topotactic transformation based on \cs
 662   \item \si{} as supply reacting with further C in cleared volume
 663  \end{itemize}
 664 \item Annealing behavior (Serre et al.)
 665  \begin{itemize}
 666   \item Room temperature implants $\rightarrow$ highly mobile C
 667   \item Elevated T implants $\rightarrow$ no/low C redistribution/migration\\
 668         (indicate stable \cs{} configurations)
 669  \end{itemize}
 670 \item Strained silicon \& Si/SiC heterostructures
 671  \begin{itemize}
 672   \item Coherent SiC precipitates (tensile strain)
 673   \item Incoherent SiC (strain relaxation)
 674  \end{itemize}
 675 \end{itemize}
 676 \end{minipage}
 677  }}}
 678 \end{pspicture}
 679
 680 \end{slide}
 681
 682 \begin{slide}
 683
 684  {\large\bf
 685   Molecular dynamics (MD) simulations
 686  }
 687
 688  \vspace{12pt}
 689
 690  \small
 691
 692  {\bf MD basics:}
 693  \begin{itemize}
 694   \item Microscopic description of N particle system
 695   \item Analytical interaction potential
 696   \item Numerical integration using Newtons equation of motion\\
 697         as a propagation rule in 6N-dimensional phase space
 698   \item Observables obtained by time and/or ensemble averages
 699  \end{itemize}
 700  {\bf Details of the simulation:}
 701  \begin{itemize}
 702   \item Integration: Velocity Verlet, timestep: $1\text{ fs}$
 703   \item Ensemble: NpT (isothermal-isobaric)
 704         \begin{itemize}
 705          \item Berendsen thermostat:
 706                $\tau_{\text{T}}=100\text{ fs}$
 707          \item Berendsen barostat:\\
 708                $\tau_{\text{P}}=100\text{ fs}$,
 709                $\beta^{-1}=100\text{ GPa}$
 710         \end{itemize}
 711   \item Erhart/Albe potential: Tersoff-like bond order potential
 712   \vspace*{12pt}
 713         \[
 714         E = \frac{1}{2} \sum_{i \neq j} \pot_{ij}, \quad
 715         \pot_{ij} = {\color{red}f_C(r_{ij})}
 716         \left[ f_R(r_{ij}) + {\color{blue}b_{ij}} f_A(r_{ij}) \right]
 717         \]
 718  \end{itemize}
 719
 720  \begin{picture}(0,0)(-230,-30)
 721   \includegraphics[width=5cm]{tersoff_angle.eps}
 722  \end{picture}
 723
 724 \end{slide}
 725
 726 \begin{slide}
 727
 728  {\large\bf
 729   Density functional theory (DFT) calculations
 730  }
 731
 732  \small
 733
 734  Basic ingredients necessary for DFT
 735
 736  \begin{itemize}
 737   \item \underline{Hohenberg-Kohn theorem} - ground state density $n_0(r)$ ...
 738         \begin{itemize}
 739          \item ... uniquely determines the ground state potential
 740                / wavefunctions
 741          \item ... minimizes the systems total energy
 742         \end{itemize}
 743   \item \underline{Born-Oppenheimer}
 744         - $N$ moving electrons in an external potential of static nuclei
 745 \[
 746 H\Psi = \left[-\sum_i^N \frac{\hbar^2}{2m}\nabla_i^2
 747               +\sum_i^N V_{\text{ext}}(r_i)
 748               +\sum_{i<j}^N V_{e-e}(r_i,r_j)\right]\Psi=E\Psi
 749 \]
 750   \item \underline{Effective potential}
 751         - averaged electrostatic potential \& exchange and correlation
 752 \[
 753 V_{\text{eff}}(r)=V_{\text{ext}}(r)+\int\frac{e^2 n(r')}{|r-r'|}d^3r'
 754                  +V_{\text{XC}}[n(r)]
 755 \]
 756   \item \underline{Kohn-Sham system}
 757         - Schr\"odinger equation of N non-interacting particles
 758 \[
 759 \left[ -\frac{\hbar^2}{2m}\nabla^2 + V_{\text{eff}}(r) \right] \Phi_i(r)
 760 =\epsilon_i\Phi_i(r)
 761 \quad
 762 \Rightarrow
 763 \quad
 764 n(r)=\sum_i^N|\Phi_i(r)|^2
 765 \]
 766   \item \underline{Self-consistent solution}\\
 767 $n(r)$ depends on $\Phi_i$, which depend on $V_{\text{eff}}$,
 768 which in turn depends on $n(r)$
 769   \item \underline{Variational principle}
 770         - minimize total energy with respect to $n(r)$
 771  \end{itemize}
 772
 773 \end{slide}
 774
 775 \begin{slide}
 776
 777  {\large\bf
 778   Density functional theory (DFT) calculations
 779  }
 780
 781  \small
 782
 783  \vspace*{0.2cm}
 784
 785  Details of applied DFT calculations in this work
 786
 787  \begin{itemize}
 788   \item \underline{Exchange correlation functional}
 789         - approximations for the inhomogeneous electron gas
 790         \begin{itemize}
 791          \item LDA: $E_{\text{XC}}^{\text{LDA}}[n]=\int \epsilon_{\text{XC}}(n)n(r)d^3r$
 792          \item GGA: $E_{\text{XC}}^{\text{GGA}}[n]=\int \epsilon_{\text{XC}}(n,\nabla n)n(r)d^3r$
 793         \end{itemize}
 794   \item \underline{Plane wave basis set}
 795         - approximation of the wavefunction $\Phi_i$ by plane waves $\phi_j$
 796 \[
 797 \rightarrow
 798 \text{Fourier series: } \Phi_i=\sum_{|G+k|<G_{\text{cut}}} c_j^i \phi_j(r), \quad E_{\text{cut}}=\frac{\hbar^2}{2m}G^2_{\text{cut}}
 799 \qquad ({\color{blue}300\text{ eV}})
 800 \]
 801   \item \underline{Brillouin zone sampling} -
 802         {\color{blue}$\Gamma$-point only} calculations
 803   \item \underline{Pseudo potential}
 804         - consider only the valence electrons
 805   \item \underline{Code} - VASP 4.6
 806  \end{itemize}
 807
 808  \vspace*{0.2cm}
 809
 810  MD and structural optimization
 811
 812  \begin{itemize}
 813   \item MD integration: Gear predictor corrector algorithm
 814   \item Pressure control: Parrinello-Rahman pressure control
 815   \item Structural optimization: Conjugate gradient method
 816  \end{itemize}
 817
 818 \begin{pspicture}(0,0)(0,0)
 819 \psellipse[linecolor=blue](1.5,6.75)(0.5,0.3)
 820 \end{pspicture}
 821
 822 \end{slide}
 823
 824 \begin{slide}
 825
 826  {\large\bf
 827   C and Si self-interstitial point defects in silicon
 828  }
 829
 830  \small
 831
 832  \vspace*{0.3cm}
 833
 834 \begin{minipage}{8cm}
 835 Procedure:\\[0.3cm]
 836   \begin{pspicture}(0,0)(7,5)
 837   \rput(3.5,4){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
 838    \parbox{7cm}{
 839    \begin{itemize}
 840     \item Creation of c-Si simulation volume
 841     \item Periodic boundary conditions
 842     \item $T=0\text{ K}$, $p=0\text{ bar}$
 843    \end{itemize}
 844   }}}}
 845 \rput(3.5,2.1){\rnode{insert}{\psframebox{
 846  \parbox{7cm}{
 847   \begin{center}
 848   Insertion of interstitial C/Si atoms
 849   \end{center}
 850   }}}}
 851   \rput(3.5,1){\rnode{cool}{\psframebox[fillstyle=solid,fillcolor=lbb]{
 852    \parbox{7cm}{
 853    \begin{center}
 854    Relaxation / structural energy minimization
 855    \end{center}
 856   }}}}
 857   \ncline[]{->}{init}{insert}
 858   \ncline[]{->}{insert}{cool}
 859  \end{pspicture}
 860 \end{minipage}
 861 \begin{minipage}{5cm}
 862   \includegraphics[width=5cm]{unit_cell_e.eps}\\
 863 \end{minipage}
 864
 865 \begin{minipage}{9cm}
 866  \begin{tabular}{l c c}
 867  \hline
 868  & size [unit cells] & \# atoms\\
 869 \hline
 870 VASP & $3\times 3\times 3$ & $216\pm 1$ \\
 871 Erhart/Albe & $9\times 9\times 9$ & $5832\pm 1$\\
 872 \hline
 873  \end{tabular}
 874 \end{minipage}
 875 \begin{minipage}{4cm}
 876 {\color{red}$\bullet$} Tetrahedral\\
 877 {\color{green}$\bullet$} Hexagonal\\
 878 {\color{yellow}$\bullet$} \hkl<1 0 0> dumbbell\\
 879 {\color{magenta}$\bullet$} \hkl<1 1 0> dumbbell\\
 880 {\color{cyan}$\bullet$} Bond-centered\\
 881 {\color{black}$\bullet$} Vacancy / Substitutional
 882 \end{minipage}
 883
 884 \end{slide}
 885
 886 \begin{slide}
 887
 888  \footnotesize
 889
 890 \begin{minipage}{9.5cm}
 891
 892  {\large\bf
 893   Si self-interstitial point defects in silicon\\
 894  }
 895
 896 \begin{tabular}{l c c c c c}
 897 \hline
 898  $E_{\text{f}}$ [eV] & \hkl<1 1 0> DB & H & T & \hkl<1 0 0> DB & V \\
 899 \hline
 900  VASP & \underline{3.39} & 3.42 & 3.77 & 4.41 & 3.63 \\
 901  Erhart/Albe & 4.39 & 4.48$^*$ & \underline{3.40} & 5.42 & 3.13 \\
 902 \hline
 903 \end{tabular}\\[0.2cm]
 904
 905 \begin{minipage}{4.7cm}
 906 \includegraphics[width=4.7cm]{e_kin_si_hex.ps}
 907 \end{minipage}
 908 \begin{minipage}{4.7cm}
 909 \begin{center}
 910 {\tiny nearly T $\rightarrow$ T}\\
 911 \end{center}
 912 \includegraphics[width=4.7cm]{nhex_tet.ps}
 913 \end{minipage}\\
 914
 915 \underline{Hexagonal} \hspace{2pt}
 916 \href{../video/si_self_int_hexa.avi}{$\rhd$}\\[0.1cm]
 917 \framebox{
 918 \begin{minipage}{2.7cm}
 919 $E_{\text{f}}^*=4.48\text{ eV}$\\
 920 \includegraphics[width=2.7cm]{si_pd_albe/hex_a.eps}
 921 \end{minipage}
 922 \begin{minipage}{0.4cm}
 923 \begin{center}
 924 $\Rightarrow$
 925 \end{center}
 926 \end{minipage}
 927 \begin{minipage}{2.7cm}
 928 $E_{\text{f}}=3.96\text{ eV}$\\
 929 \includegraphics[width=2.8cm]{si_pd_albe/hex.eps}
 930 \end{minipage}
 931 }
 932 \begin{minipage}{2.9cm}
 933 \begin{flushright}
 934 \underline{Vacancy}\\
 935 \includegraphics[width=3.0cm]{si_pd_albe/vac.eps}
 936 \end{flushright}
 937 \end{minipage}
 938
 939 \end{minipage}
 940 \begin{minipage}{3.5cm}
 941
 942 \begin{flushright}
 943 \underline{\hkl<1 1 0> dumbbell}\\
 944 \includegraphics[width=3.0cm]{si_pd_albe/110.eps}\\
 945 \underline{Tetrahedral}\\
 946 \includegraphics[width=3.0cm]{si_pd_albe/tet.eps}\\
 947 \underline{\hkl<1 0 0> dumbbell}\\
 948 \includegraphics[width=3.0cm]{si_pd_albe/100.eps}
 949 \end{flushright}
 950
 951 \end{minipage}
 952
 953 \end{slide}
 954
 955 \begin{slide}
 956
 957 \footnotesize
 958
 959  {\large\bf
 960   C interstitial point defects in silicon\\[-0.1cm]
 961  }
 962
 963 \begin{tabular}{l c c c c c c r}
 964 \hline
 965  $E_{\text{f}}$ & T & H & \hkl<1 0 0> DB & \hkl<1 1 0> DB & S & B & \cs{} \& \si\\
 966 \hline
 967  VASP & unstable & unstable & \underline{3.72} & 4.16 & 1.95 & 4.66 & {\color{green}4.17}\\
 968  Erhart/Albe MD & 6.09 & 9.05$^*$ & \underline{3.88} & 5.18 & {\color{red}0.75} & 5.59$^*$ & {\color{green}4.43} \\
 969 \hline
 970 \end{tabular}\\[0.1cm]
 971
 972 \framebox{
 973 \begin{minipage}{2.7cm}
 974 \underline{Hexagonal} \hspace{2pt}
 975 \href{../video/c_in_si_int_hexa.avi}{$\rhd$}\\
 976 $E_{\text{f}}^*=9.05\text{ eV}$\\
 977 \includegraphics[width=2.7cm]{c_pd_albe/hex.eps}
 978 \end{minipage}
 979 \begin{minipage}{0.4cm}
 980 \begin{center}
 981 $\Rightarrow$
 982 \end{center}
 983 \end{minipage}
 984 \begin{minipage}{2.7cm}
 985 \underline{\hkl<1 0 0>}\\
 986 $E_{\text{f}}=3.88\text{ eV}$\\
 987 \includegraphics[width=2.7cm]{c_pd_albe/100.eps}
 988 \end{minipage}
 989 }
 990 \begin{minipage}{2cm}
 991 \hfill
 992 \end{minipage}
 993 \begin{minipage}{3cm}
 994 \begin{flushright}
 995 \underline{Tetrahedral}\\
 996 \includegraphics[width=3.0cm]{c_pd_albe/tet.eps}
 997 \end{flushright}
 998 \end{minipage}
 999
1000 \framebox{
1001 \begin{minipage}{2.7cm}
1002 \underline{Bond-centered}\\
1003 $E_{\text{f}}^*=5.59\text{ eV}$\\
1004 \includegraphics[width=2.7cm]{c_pd_albe/bc.eps}
1005 \end{minipage}
1006 \begin{minipage}{0.4cm}
1007 \begin{center}
1008 $\Rightarrow$
1009 \end{center}
1010 \end{minipage}
1011 \begin{minipage}{2.7cm}
1012 \underline{\hkl<1 1 0> dumbbell}\\
1013 $E_{\text{f}}=5.18\text{ eV}$\\
1014 \includegraphics[width=2.7cm]{c_pd_albe/110.eps}
1015 \end{minipage}
1016 }
1017 \begin{minipage}{2cm}
1018 \hfill
1019 \end{minipage}
1020 \begin{minipage}{3cm}
1021 \begin{flushright}
1022 \underline{Substitutional}\\
1023 \includegraphics[width=3.0cm]{c_pd_albe/sub.eps}
1024 \end{flushright}
1025 \end{minipage}
1026
1027 \end{slide}
1028
1029 \begin{slide}
1030
1031 \footnotesize
1032
1033  {\large\bf\boldmath
1034   C \hkl<1 0 0> dumbbell interstitial configuration\\
1035  }
1036
1037 {\tiny
1038 \begin{tabular}{l c c c c c c c c}
1039 \hline
1040  Distances [nm] & $r(1C)$ & $r(2C)$ & $r(3C)$ & $r(12)$ & $r(13)$ & $r(34)$ & $r(23)$ & $r(25)$ \\
1041 \hline
1042 Erhart/Albe & 0.175 & 0.329 & 0.186 & 0.226 & 0.300 & 0.343 & 0.423 & 0.425 \\
1043 VASP & 0.174 & 0.341 & 0.182 & 0.229 & 0.286 & 0.347 & 0.422 & 0.417 \\
1044 \hline
1045 \end{tabular}\\[0.2cm]
1046 \begin{tabular}{l c c c c }
1047 \hline
1048  Angles [$^{\circ}$] & $\theta_1$ & $\theta_2$ & $\theta_3$ & $\theta_4$ \\
1049 \hline
1050 Erhart/Albe & 140.2 & 109.9 & 134.4 & 112.8 \\
1051 VASP & 130.7 & 114.4 & 146.0 & 107.0 \\
1052 \hline
1053 \end{tabular}\\[0.2cm]
1054 \begin{tabular}{l c c c}
1055 \hline
1056  Displacements [nm]& $a$ & $b$ & $|a|+|b|$ \\
1057 \hline
1058 Erhart/Albe & 0.084 & -0.091 & 0.175 \\
1059 VASP & 0.109 & -0.065 & 0.174 \\
1060 \hline
1061 \end{tabular}\\[0.6cm]
1062 }
1063
1064 \begin{minipage}{3.0cm}
1065 \begin{center}
1066 \underline{Erhart/Albe}
1067 \includegraphics[width=3.0cm]{c_pd_albe/100_cmp.eps}
1068 \end{center}
1069 \end{minipage}
1070 \begin{minipage}{3.0cm}
1071 \begin{center}
1072 \underline{VASP}
1073 \includegraphics[width=3.0cm]{c_pd_vasp/100_cmp.eps}
1074 \end{center}
1075 \end{minipage}\\
1076
1077 \begin{picture}(0,0)(-185,10)
1078 \includegraphics[width=6.8cm]{100-c-si-db_cmp.eps}
1079 \end{picture}
1080 \begin{picture}(0,0)(-280,-150)
1081 \includegraphics[width=3.3cm]{c_pd_vasp/eden.eps}
1082 \end{picture}
1083
1084 \begin{pspicture}(0,0)(0,0)
1085 \psellipse[linecolor=green](5.18,5.92)(0.5,0.3)
1086 \psellipse[linecolor=red](3.45,5.92)(1.0,0.4)
1087 \psellipse[linecolor=blue](2.7,6.92)(0.9,0.2)
1088 \psellipse[linecolor=blue](4.65,6.92)(0.9,0.2)
1089 \end{pspicture}
1090
1091 \end{slide}
1092
1093 \begin{slide}
1094
1095 \small
1096
1097 \begin{minipage}{8.5cm}
1098
1099  {\large\bf
1100   Bond-centered interstitial configuration\\[-0.1cm]
1101  }
1102
1103 \begin{minipage}{3.0cm}
1104 \includegraphics[width=2.8cm]{c_pd_vasp/bc_2333.eps}\\
1105 \end{minipage}
1106 \begin{minipage}{5.2cm}
1107 \begin{itemize}
1108  \item Linear Si-C-Si bond
1109  \item Si: one C \& 3 Si neighbours
1110  \item Spin polarized calculations
1111  \item No saddle point!\\
1112        Real local minimum!
1113 \end{itemize}
1114 \end{minipage}
1115
1116 \framebox{
1117  \tiny
1118  \begin{minipage}[t]{6.5cm}
1119   \begin{minipage}[t]{1.2cm}
1120   {\color{red}Si}\\
1121   {\tiny sp$^3$}\\[0.8cm]
1122   \underline{${\color{black}\uparrow}$}
1123   \underline{${\color{black}\uparrow}$}
1124   \underline{${\color{black}\uparrow}$}
1125   \underline{${\color{red}\uparrow}$}\\
1126   sp$^3$
1127   \end{minipage}
1128   \begin{minipage}[t]{1.4cm}
1129   \begin{center}
1130   {\color{red}M}{\color{blue}O}\\[0.8cm]
1131   \underline{${\color{blue}\uparrow}{\color{white}\downarrow}$}\\
1132   $\sigma_{\text{ab}}$\\[0.5cm]
1133   \underline{${\color{red}\uparrow}{\color{blue}\downarrow}$}\\
1134   $\sigma_{\text{b}}$
1135   \end{center}
1136   \end{minipage}
1137   \begin{minipage}[t]{1.0cm}
1138   \begin{center}
1139   {\color{blue}C}\\
1140   {\tiny sp}\\[0.2cm]
1141   \underline{${\color{white}\uparrow\uparrow}$}
1142   \underline{${\color{white}\uparrow\uparrow}$}\\
1143   2p\\[0.4cm]
1144   \underline{${\color{blue}\uparrow}{\color{blue}\downarrow}$}
1145   \underline{${\color{blue}\uparrow}{\color{blue}\downarrow}$}\\
1146   sp
1147   \end{center}
1148   \end{minipage}
1149   \begin{minipage}[t]{1.4cm}
1150   \begin{center}
1151   {\color{blue}M}{\color{green}O}\\[0.8cm]
1152   \underline{${\color{blue}\uparrow}{\color{white}\downarrow}$}\\
1153   $\sigma_{\text{ab}}$\\[0.5cm]
1154   \underline{${\color{green}\uparrow}{\color{blue}\downarrow}$}\\
1155   $\sigma_{\text{b}}$
1156   \end{center}
1157   \end{minipage}
1158   \begin{minipage}[t]{1.2cm}
1159   \begin{flushright}
1160   {\color{green}Si}\\
1161   {\tiny sp$^3$}\\[0.8cm]
1162   \underline{${\color{green}\uparrow}$}
1163   \underline{${\color{black}\uparrow}$}
1164   \underline{${\color{black}\uparrow}$}
1165   \underline{${\color{black}\uparrow}$}\\
1166   sp$^3$
1167   \end{flushright}
1168   \end{minipage}
1169  \end{minipage}
1170 }\\[0.1cm]
1171
1172 \framebox{
1173 \begin{minipage}{4.5cm}
1174 \includegraphics[width=4cm]{c_100_mig_vasp/im_spin_diff.eps}
1175 \end{minipage}
1176 \begin{minipage}{3.5cm}
1177 {\color{gray}$\bullet$} Spin up\\
1178 {\color{green}$\bullet$} Spin down\\
1179 {\color{blue}$\bullet$} Resulting spin up\\
1180 {\color{yellow}$\bullet$} Si atoms\\
1181 {\color{red}$\bullet$} C atom
1182 \end{minipage}
1183 }
1184
1185 \end{minipage}
1186 \begin{minipage}{4.2cm}
1187 \begin{flushright}
1188 \includegraphics[width=4.3cm]{c_pd_vasp/bc_2333_ksl.ps}\\
1189 {\color{green}$\Box$} {\tiny unoccupied}\\
1190 {\color{red}$\bullet$} {\tiny occupied}
1191 \end{flushright}
1192 \end{minipage}
1193
1194 \end{slide}
1195
1196 \begin{slide}
1197
1198  {\large\bf\boldmath
1199   Migration of the C \hkl<1 0 0> dumbbell interstitial
1200  }
1201
1202 \scriptsize
1203
1204  {\small Investigated pathways}
1205
1206 \begin{minipage}{8.5cm}
1207 \begin{minipage}{8.3cm}
1208 \underline{\hkl<0 0 -1> $\rightarrow$ \hkl<0 0 1>}\\
1209 \begin{minipage}{2.4cm}
1210 \includegraphics[width=2.4cm]{c_pd_vasp/100_2333.eps}
1211 \end{minipage}
1212 \begin{minipage}{0.4cm}
1213 $\rightarrow$
1214 \end{minipage}
1215 \begin{minipage}{2.4cm}
1216 \includegraphics[width=2.4cm]{c_pd_vasp/bc_2333.eps}
1217 \end{minipage}
1218 \begin{minipage}{0.4cm}
1219 $\rightarrow$
1220 \end{minipage}
1221 \begin{minipage}{2.4cm}
1222 \includegraphics[width=2.4cm]{c_pd_vasp/100_next_2333.eps}
1223 \end{minipage}
1224 \end{minipage}\\
1225 \begin{minipage}{8.3cm}
1226 \underline{\hkl<0 0 -1> $\rightarrow$ \hkl<0 -1 0>}\\
1227 \begin{minipage}{2.4cm}
1228 \includegraphics[width=2.4cm]{c_pd_vasp/100_2333.eps}
1229 \end{minipage}
1230 \begin{minipage}{0.4cm}
1231 $\rightarrow$
1232 \end{minipage}
1233 \begin{minipage}{2.4cm}
1234 \includegraphics[width=2.4cm]{c_pd_vasp/00-1-0-10_2333.eps}
1235 \end{minipage}
1236 \begin{minipage}{0.4cm}
1237 $\rightarrow$
1238 \end{minipage}
1239 \begin{minipage}{2.4cm}
1240 \includegraphics[width=2.4cm]{c_pd_vasp/0-10_2333.eps}
1241 \end{minipage}
1242 \end{minipage}\\
1243 \begin{minipage}{8.3cm}
1244 \underline{\hkl<0 0 -1> $\rightarrow$ \hkl<0 -1 0> (in place)}\\
1245 \begin{minipage}{2.4cm}
1246 \includegraphics[width=2.4cm]{c_pd_vasp/100_2333.eps}
1247 \end{minipage}
1248 \begin{minipage}{0.4cm}
1249 $\rightarrow$
1250 \end{minipage}
1251 \begin{minipage}{2.4cm}
1252 \includegraphics[width=2.4cm]{c_pd_vasp/00-1_ip0-10_2333.eps}
1253 \end{minipage}
1254 \begin{minipage}{0.4cm}
1255 $\rightarrow$
1256 \end{minipage}
1257 \begin{minipage}{2.4cm}
1258 \includegraphics[width=2.4cm]{c_pd_vasp/0-10_ip_2333.eps}
1259 \end{minipage}
1260 \end{minipage}
1261 \end{minipage}
1262 \framebox{
1263 \begin{minipage}{4.2cm}
1264  {\small Constrained relaxation\\
1265          technique (CRT) method}\\
1266 \includegraphics[width=4cm]{crt_orig.eps}
1267 \begin{itemize}
1268  \item Constrain diffusing atom
1269  \item Static constraints
1270 \end{itemize}
1271 \vspace*{0.3cm}
1272  {\small Modifications}\\
1273 \includegraphics[width=4cm]{crt_mod.eps}
1274 \begin{itemize}
1275  \item Constrain all atoms
1276  \item Update individual\\
1277        constraints
1278 \end{itemize}
1279 \end{minipage}
1280 }
1281
1282 \end{slide}
1283
1284 \begin{slide}
1285
1286  {\large\bf\boldmath
1287   Migration of the C \hkl<1 0 0> dumbbell interstitial
1288  }
1289
1290 \scriptsize
1291
1292 \framebox{
1293 \begin{minipage}{5.9cm}
1294 \begin{flushleft}
1295 \includegraphics[width=5.8cm]{im_00-1_nosym_sp_fullct_thesis.ps}\\[0.45cm]
1296 \end{flushleft}
1297 \begin{center}
1298 \begin{picture}(0,0)(60,0)
1299 \includegraphics[width=1cm]{vasp_mig/00-1.eps}
1300 \end{picture}
1301 \begin{picture}(0,0)(-5,0)
1302 \includegraphics[width=1cm]{vasp_mig/bc_00-1_sp.eps}
1303 \end{picture}
1304 \begin{picture}(0,0)(-55,0)
1305 \includegraphics[width=1cm]{vasp_mig/bc.eps}
1306 \end{picture}
1307 \begin{picture}(0,0)(12.5,10)
1308 \includegraphics[width=1cm]{110_arrow.eps}
1309 \end{picture}
1310 \begin{picture}(0,0)(90,0)
1311 \includegraphics[height=0.9cm]{001_arrow.eps}
1312 \end{picture}
1313 \end{center}
1314 \vspace*{0.35cm}
1315 \end{minipage}
1316 }
1317 \begin{minipage}{0.3cm}
1318 \hfill
1319 \end{minipage}
1320 \framebox{
1321 \begin{minipage}{5.9cm}
1322 \begin{flushright}
1323 \includegraphics[width=5.9cm]{vasp_mig/00-1_0-10_nosym_sp_fullct.ps}\\[0.5cm]
1324 \end{flushright}
1325 \begin{center}
1326 \begin{picture}(0,0)(60,0)
1327 \includegraphics[width=1cm]{vasp_mig/00-1_a.eps}
1328 \end{picture}
1329 \begin{picture}(0,0)(5,0)
1330 \includegraphics[width=1cm]{vasp_mig/00-1_0-10_sp.eps}
1331 \end{picture}
1332 \begin{picture}(0,0)(-55,0)
1333 \includegraphics[width=1cm]{vasp_mig/0-10.eps}
1334 \end{picture}
1335 \begin{picture}(0,0)(12.5,10)
1336 \includegraphics[width=1cm]{100_arrow.eps}
1337 \end{picture}
1338 \begin{picture}(0,0)(90,0)
1339 \includegraphics[height=0.9cm]{001_arrow.eps}
1340 \end{picture}
1341 \end{center}
1342 \vspace*{0.3cm}
1343 \end{minipage}\\
1344 }
1345
1346 \vspace*{0.05cm}
1347
1348 \framebox{
1349 \begin{minipage}{5.9cm}
1350 \begin{flushleft}
1351 \includegraphics[width=5.9cm]{vasp_mig/00-1_ip0-10_nosym_sp_fullct.ps}\\[0.6cm]
1352 \end{flushleft}
1353 \begin{center}
1354 \begin{picture}(0,0)(60,0)
1355 \includegraphics[width=0.9cm]{vasp_mig/00-1_b.eps}
1356 \end{picture}
1357 \begin{picture}(0,0)(10,0)
1358 \includegraphics[width=0.9cm]{vasp_mig/00-1_ip0-10_sp.eps}
1359 \end{picture}
1360 \begin{picture}(0,0)(-60,0)
1361 \includegraphics[width=0.9cm]{vasp_mig/0-10_b.eps}
1362 \end{picture}
1363 \begin{picture}(0,0)(12.5,10)
1364 \includegraphics[width=1cm]{100_arrow.eps}
1365 \end{picture}
1366 \begin{picture}(0,0)(90,0)
1367 \includegraphics[height=0.9cm]{001_arrow.eps}
1368 \end{picture}
1369 \end{center}
1370 \vspace*{0.3cm}
1371 \end{minipage}
1372 }
1373 \begin{minipage}{0.3cm}
1374 \hfill
1375 \end{minipage}
1376 \begin{minipage}{6.5cm}
1377 VASP results
1378 \begin{itemize}
1379  \item Energetically most favorable path
1380        \begin{itemize}
1381         \item Path 2
1382         \item Activation energy: $\approx$ 0.9 eV
1383         \item Experimental values: 0.73 ... 0.87 eV
1384        \end{itemize}
1385        $\Rightarrow$ {\color{blue}Diffusion} path identified!
1386  \item Reorientation (path 3)
1387        \begin{itemize}
1388         \item More likely composed of two consecutive steps of type 2
1389         \item Experimental values: 0.77 ... 0.88 eV
1390        \end{itemize}
1391        $\Rightarrow$ {\color{blue}Reorientation} transition identified!
1392 \end{itemize}
1393 \end{minipage}
1394
1395 \end{slide}
1396
1397 \begin{slide}
1398
1399  {\large\bf\boldmath
1400   Migration of the C \hkl<1 0 0> dumbbell interstitial
1401  }
1402
1403 \scriptsize
1404
1405  \vspace{0.1cm}
1406
1407 \begin{minipage}{6.5cm}
1408
1409 \framebox{
1410 \begin{minipage}[t]{5.9cm}
1411 \begin{flushleft}
1412 \includegraphics[width=5.9cm]{bc_00-1.ps}\\[2.35cm]
1413 \end{flushleft}
1414 \begin{center}
1415 \begin{pspicture}(0,0)(0,0)
1416 \psframe[linecolor=red,fillstyle=none](-2.8,1.35)(3.3,2.7)
1417 \end{pspicture}
1418 \begin{picture}(0,0)(60,-50)
1419 \includegraphics[width=1cm]{albe_mig/bc_00-1_red_00.eps}
1420 \end{picture}
1421 \begin{picture}(0,0)(5,-50)
1422 \includegraphics[width=1cm]{albe_mig/bc_00-1_red_01.eps}
1423 \end{picture}
1424 \begin{picture}(0,0)(-55,-50)
1425 \includegraphics[width=1cm]{albe_mig/bc_00-1_red_02.eps}
1426 \end{picture}
1427 \begin{picture}(0,0)(12.5,-40)
1428 \includegraphics[width=1cm]{110_arrow.eps}
1429 \end{picture}
1430 \begin{picture}(0,0)(90,-45)
1431 \includegraphics[height=0.9cm]{001_arrow.eps}
1432 \end{picture}\\
1433 \begin{pspicture}(0,0)(0,0)
1434 \psframe[linecolor=blue,fillstyle=none](-2.8,0)(3.3,1.6)
1435 \end{pspicture}
1436 \begin{picture}(0,0)(60,-15)
1437 \includegraphics[width=0.9cm]{albe_mig/bc_00-1_01.eps}
1438 \end{picture}
1439 \begin{picture}(0,0)(35,-15)
1440 \includegraphics[width=0.9cm]{albe_mig/bc_00-1_02.eps}
1441 \end{picture}
1442 \begin{picture}(0,0)(-5,-15)
1443 \includegraphics[width=0.9cm]{albe_mig/bc_00-1_03.eps}
1444 \end{picture}
1445 \begin{picture}(0,0)(-55,-15)
1446 \includegraphics[width=0.9cm]{albe_mig/bc_00-1_04.eps}
1447 \end{picture}
1448 \begin{picture}(0,0)(12.5,-5)
1449 \includegraphics[width=1cm]{100_arrow.eps}
1450 \end{picture}
1451 \begin{picture}(0,0)(90,-15)
1452 \includegraphics[height=0.9cm]{010_arrow.eps}
1453 \end{picture}
1454 \end{center}
1455 \end{minipage}
1456 }\\[0.1cm]
1457
1458 \begin{minipage}{5.9cm}
1459 Erhart/Albe results
1460 \begin{itemize}
1461  \item Lowest activation energy: $\approx$ 2.2 eV
1462  \item 2.4 times higher than VASP
1463  \item Different pathway
1464 \end{itemize}
1465 \end{minipage}
1466
1467 \end{minipage}
1468 \begin{minipage}{6.5cm}
1469
1470 \framebox{
1471 \begin{minipage}{5.9cm}
1472 %\begin{flushright}
1473 %\includegraphics[width=5.9cm]{00-1_0-10.ps}\\[0.75cm]
1474 %\end{flushright}
1475 %\begin{center}
1476 %\begin{pspicture}(0,0)(0,0)
1477 %\psframe[linecolor=red,fillstyle=none](-2.8,-0.25)(3.3,1.1)
1478 %\end{pspicture}
1479 %\begin{picture}(0,0)(60,-5)
1480 %\includegraphics[width=0.9cm]{albe_mig/00-1_0-10_red_00.eps}
1481 %\end{picture}
1482 %\begin{picture}(0,0)(0,-5)
1483 %\includegraphics[width=0.9cm]{albe_mig/00-1_0-10_red_min.eps}
1484 %\end{picture}
1485 %\begin{picture}(0,0)(-55,-5)
1486 %\includegraphics[width=0.9cm]{albe_mig/00-1_0-10_red_03.eps}
1487 %\end{picture}
1488 %\begin{picture}(0,0)(12.5,5)
1489 %\includegraphics[width=1cm]{100_arrow.eps}
1490 %\end{picture}
1491 %\begin{picture}(0,0)(90,0)
1492 %\includegraphics[height=0.9cm]{001_arrow.eps}
1493 %\end{picture}
1494 %\end{center}
1495 %\vspace{0.2cm}
1496 %\end{minipage}
1497 %}\\[0.2cm]
1498 %
1499 %\framebox{
1500 %\begin{minipage}{5.9cm}
1501 \includegraphics[width=5.9cm]{00-1_110_0-10_mig_albe.ps}
1502 \end{minipage}
1503 }\\[0.1cm]
1504
1505 \begin{minipage}{5.9cm}
1506 Transition involving \ci{} \hkl<1 1 0>
1507 \begin{itemize}
1508  \item Bond-centered configuration unstable\\
1509        $\rightarrow$ \ci{} \hkl<1 1 0> dumbbell
1510  \item Transition minima of path 2 \& 3\\
1511        $\rightarrow$ \ci{} \hkl<1 1 0> dumbbell
1512  \item Activation energy: $\approx$ 2.2 eV \& 0.9 eV
1513  \item 2.4 - 3.4 times higher than VASP
1514  \item Rotation of dumbbell orientation
1515 \end{itemize}
1516 \vspace{0.1cm}
1517 \begin{center}
1518 {\color{blue}Overestimated diffusion barrier}
1519 \end{center}
1520 \end{minipage}
1521
1522 \end{minipage}
1523
1524 \end{slide}
1525
1526 \begin{slide}
1527
1528  {\large\bf\boldmath
1529   Combinations with a C-Si \hkl<1 0 0>-type interstitial
1530  }
1531
1532 \small
1533
1534 \vspace*{0.1cm}
1535
1536 Binding energy:
1537 $
1538 E_{\text{b}}=
1539 E_{\text{f}}^{\text{defect combination}}-
1540 E_{\text{f}}^{\text{C \hkl<0 0 -1> dumbbell}}-
1541 E_{\text{f}}^{\text{2nd defect}}
1542 $
1543
1544 \vspace*{0.1cm}
1545
1546 {\scriptsize
1547 \begin{tabular}{l c c c c c c}
1548 \hline
1549  $E_{\text{b}}$ [eV] & 1 & 2 & 3 & 4 & 5 & R\\
1550  \hline
1551  \hkl<0 0 -1> & {\color{red}-0.08} & -1.15 & {\color{red}-0.08} & 0.04 & -1.66 & -0.19\\
1552  \hkl<0 0 1> & 0.34 & 0.004 & -2.05 & 0.26 & -1.53 & -0.19\\
1553  \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}\\
1554  \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}\\
1555  \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}\\
1556  \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}\\
1557  \hline
1558  C substitutional (C$_{\text{S}}$) & 0.26 & -0.51 & -0.93 & -0.15 & 0.49 & -0.05\\
1559  Vacancy & -5.39 ($\rightarrow$ C$_{\text{S}}$) & -0.59 & -3.14 & -0.54 & -0.50 & -0.31\\
1560 \hline
1561 \end{tabular}
1562 }
1563
1564 \vspace*{0.3cm}
1565
1566 \footnotesize
1567
1568 \begin{minipage}[t]{3.8cm}
1569 \underline{\hkl<1 0 0> at position 1}\\[0.1cm]
1570 \includegraphics[width=3.5cm]{00-1dc/2-25.eps}
1571 \end{minipage}
1572 \begin{minipage}[t]{3.5cm}
1573 \underline{\hkl<0 -1 0> at position 1}\\[0.1cm]
1574 \includegraphics[width=3.2cm]{00-1dc/2-39.eps}
1575 \end{minipage}
1576 \begin{minipage}[t]{5.5cm}
1577 \begin{itemize}
1578  \item $E_{\text{b}}=0$ $\Leftrightarrow$ non-interacting defects\\
1579        $E_{\text{b}} \rightarrow 0$ for increasing distance (R)
1580  \item Stress compensation / increase
1581  \item Unfavored: antiparallel orientations
1582  \item Indication of energetically favored\\
1583        agglomeration
1584  \item Most favorable: C clustering
1585  \item However: High barrier ($>4\,\text{eV}$)
1586  \item $4\times{\color{cyan}-2.25}$ versus $2\times{\color{orange}-2.39}$
1587        (Entropy)
1588 \end{itemize}
1589 \end{minipage}
1590
1591 \begin{picture}(0,0)(-295,-130)
1592 \includegraphics[width=3.5cm]{comb_pos.eps}
1593 \end{picture}
1594
1595 \end{slide}
1596
1597 \begin{slide}
1598
1599  {\large\bf\boldmath
1600   Combinations of C-Si \hkl<1 0 0>-type interstitials
1601  }
1602
1603 \small
1604
1605 \vspace*{0.1cm}
1606
1607 Energetically most favorable combinations along \hkl<1 1 0>
1608
1609 \vspace*{0.1cm}
1610
1611 {\scriptsize
1612 \begin{tabular}{l c c c c c c}
1613 \hline
1614  & 1 & 2 & 3 & 4 & 5 & 6\\
1615 \hline
1616 $E_{\text{b}}$ [eV] & -2.39 & -1.88 & -0.59 & -0.31 & -0.24 & -0.21 \\
1617 C-C distance [\AA] & 1.4 & 4.6 & 6.5 & 8.6 & 10.5 & 10.8 \\
1618 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>\\
1619 \hline
1620 \end{tabular}
1621 }
1622
1623 \vspace*{0.3cm}
1624
1625 \begin{minipage}{7.0cm}
1626 \includegraphics[width=7cm]{db_along_110_cc.ps}
1627 \end{minipage}
1628 \begin{minipage}{6.0cm}
1629 \begin{itemize}
1630  \item Interaction proportional to reciprocal cube of C-C distance
1631  \item Saturation in the immediate vicinity
1632  \renewcommand\labelitemi{$\Rightarrow$}
1633  \item Agglomeration of \ci{} expected
1634  \item Absence of C clustering
1635 \end{itemize}
1636 \begin{center}
1637 {\color{blue}
1638  Consisten with initial precipitation model
1639 }
1640 \end{center}
1641 \end{minipage}
1642
1643 \vspace{0.2cm}
1644
1645 \end{slide}
1646
1647 \begin{slide}
1648
1649  {\large\bf\boldmath
1650   Combinations of substitutional C and \hkl<1 1 0> Si self-interstitials
1651  }
1652
1653  \scriptsize
1654
1655 %\begin{center}
1656 %\begin{minipage}{3.2cm}
1657 %\includegraphics[width=3cm]{sub_110_combo.eps}
1658 %\end{minipage}
1659 %\begin{minipage}{7.8cm}
1660 %\begin{tabular}{l c c c c c c}
1661 %\hline
1662 %C$_{\text{sub}}$ & \hkl<1 1 0> & \hkl<-1 1 0> & \hkl<0 1 1> & \hkl<0 -1 1> &
1663 %                   \hkl<1 0 1> & \hkl<-1 0 1> \\
1664 %\hline
1665 %1 & \RM{1} & \RM{3} & \RM{3} & \RM{1} & \RM{3} & \RM{1} \\
1666 %2 & \RM{2} & A & A & \RM{2} & C & \RM{5} \\
1667 %3 & \RM{3} & \RM{1} & \RM{3} & \RM{1} & \RM{1} & \RM{3} \\
1668 %4 & \RM{4} & B & D & E & E & D \\
1669 %5 & \RM{5} & C & A & \RM{2} & A & \RM{2} \\
1670 %\hline
1671 %\end{tabular}
1672 %\end{minipage}
1673 %\end{center}
1674
1675 %\begin{center}
1676 %\begin{tabular}{l c c c c c c c c c c}
1677 %\hline
1678 %Conf & \RM{1} & \RM{2} & \RM{3} & \RM{4} & \RM{5} & A & B & C & D & E \\
1679 %\hline
1680 %$E_{\text{f}}$ [eV]& 4.37 & 5.26 & 5.57 & 5.37 & 5.12 & 5.10 & 5.32 & 5.28 & 5.39 & 5.32 \\
1681 %$E_{\text{b}}$ [eV] & -0.97 & -0.08 & 0.22 & -0.02 & -0.23 & -0.25 & -0.02 & -0.06 & 0.05 & -0.03 \\
1682 %$r$ [nm] & 0.292 & 0.394 & 0.241 & 0.453 & 0.407 & 0.408 & 0.452 & 0.392 & 0.456 & 0.453\\
1683 %\hline
1684 %\end{tabular}
1685 %\end{center}
1686
1687 \begin{minipage}{6.0cm}
1688 \includegraphics[width=5.8cm]{c_sub_si110.ps}
1689 \end{minipage}
1690 \begin{minipage}{7cm}
1691 \scriptsize
1692 \begin{itemize}
1693  \item IBS: C may displace Si\\
1694        $\Rightarrow$ C$_{\text{sub}}$ + \hkl<1 1 0> Si self-interstitial
1695  \item Assumption:\\
1696        \hkl<1 1 0>-type $\rightarrow$ favored combination
1697  \renewcommand\labelitemi{$\Rightarrow$}
1698  \item Most favorable: \cs{} along \hkl<1 1 0> chain \si{}
1699  \item Less favorable than C-Si \hkl<1 0 0> dumbbell
1700  \item Interaction drops quickly to zero\\
1701        $\rightarrow$ low capture radius
1702 \end{itemize}
1703 \begin{center}
1704  {\color{blue}
1705  IBS process far from equilibrium\\
1706  \cs{} \& \si{} instead of thermodynamic ground state
1707  }
1708 \end{center}
1709 \end{minipage}
1710
1711 \begin{minipage}{6.5cm}
1712 \includegraphics[width=6.0cm]{162-097.ps}
1713 \begin{itemize}
1714  \item Low migration barrier
1715 \end{itemize}
1716 \end{minipage}
1717 \begin{minipage}{6.5cm}
1718 \begin{center}
1719 Ab initio MD at \degc{900}\\
1720 \includegraphics[width=3.3cm]{md_vasp_01.eps}
1721 $t=\unit[2230]{fs}$\\
1722 \includegraphics[width=3.3cm]{md_vasp_02.eps}
1723 $t=\unit[2900]{fs}$
1724 \end{center}
1725 {\color{blue}
1726 Contribution of entropy to structural formation
1727 }
1728 \end{minipage}
1729
1730 \end{slide}
1731
1732 \begin{slide}
1733
1734  {\large\bf\boldmath
1735   Migration in C-Si \hkl<1 0 0> and vacancy combinations
1736  }
1737
1738  \footnotesize
1739
1740 \vspace{0.1cm}
1741
1742 \begin{minipage}[t]{3cm}
1743 \underline{Pos 2, $E_{\text{b}}=-0.59\text{ eV}$}\\
1744 \includegraphics[width=2.8cm]{00-1dc/0-59.eps}
1745 \end{minipage}
1746 \begin{minipage}[t]{7cm}
1747 \vspace{0.2cm}
1748 \begin{center}
1749  Low activation energies\\
1750  High activation energies for reverse processes\\
1751  $\Downarrow$\\
1752  {\color{blue}C$_{\text{sub}}$ very stable}\\
1753 \vspace*{0.1cm}
1754  \hrule
1755 \vspace*{0.1cm}
1756  Without nearby \hkl<1 1 0> Si self-interstitial (IBS)\\
1757  $\Downarrow$\\
1758  {\color{blue}Formation of SiC by successive substitution by C}
1759
1760 \end{center}
1761 \end{minipage}
1762 \begin{minipage}[t]{3cm}
1763 \underline{Pos 3, $E_{\text{b}}=-3.14\text{ eV}$}\\
1764 \includegraphics[width=2.8cm]{00-1dc/3-14.eps}
1765 \end{minipage}
1766
1767
1768 \framebox{
1769 \begin{minipage}{5.9cm}
1770 \includegraphics[width=5.9cm]{vasp_mig/comb_mig_3-2_vac_fullct.ps}\\[0.6cm]
1771 \begin{center}
1772 \begin{picture}(0,0)(70,0)
1773 \includegraphics[width=1.4cm]{vasp_mig/comb_2-1_init.eps}
1774 \end{picture}
1775 \begin{picture}(0,0)(30,0)
1776 \includegraphics[width=1.4cm]{vasp_mig/comb_2-1_seq_03.eps}
1777 \end{picture}
1778 \begin{picture}(0,0)(-10,0)
1779 \includegraphics[width=1.4cm]{vasp_mig/comb_2-1_seq_06.eps}
1780 \end{picture}
1781 \begin{picture}(0,0)(-48,0)
1782 \includegraphics[width=1.4cm]{vasp_mig/comb_2-1_final.eps}
1783 \end{picture}
1784 \begin{picture}(0,0)(12.5,5)
1785 \includegraphics[width=1cm]{100_arrow.eps}
1786 \end{picture}
1787 \begin{picture}(0,0)(97,-10)
1788 \includegraphics[height=0.9cm]{001_arrow.eps}
1789 \end{picture}
1790 \end{center}
1791 \vspace{0.1cm}
1792 \end{minipage}
1793 }
1794 \begin{minipage}{0.3cm}
1795 \hfill
1796 \end{minipage}
1797 \framebox{
1798 \begin{minipage}{5.9cm}
1799 \includegraphics[width=5.9cm]{vasp_mig/comb_mig_4-2_vac_fullct.ps}\\[0.1cm]
1800 \begin{center}
1801 \begin{picture}(0,0)(60,0)
1802 \includegraphics[width=0.9cm]{vasp_mig/comb_3-1_init.eps}
1803 \end{picture}
1804 \begin{picture}(0,0)(25,0)
1805 \includegraphics[width=0.9cm]{vasp_mig/comb_3-1_seq_03.eps}
1806 \end{picture}
1807 \begin{picture}(0,0)(-20,0)
1808 \includegraphics[width=0.9cm]{vasp_mig/comb_3-1_seq_07.eps}
1809 \end{picture}
1810 \begin{picture}(0,0)(-55,0)
1811 \includegraphics[width=0.9cm]{vasp_mig/comb_3-1_final.eps}
1812 \end{picture}
1813 \begin{picture}(0,0)(12.5,5)
1814 \includegraphics[width=1cm]{100_arrow.eps}
1815 \end{picture}
1816 \begin{picture}(0,0)(95,0)
1817 \includegraphics[height=0.9cm]{001_arrow.eps}
1818 \end{picture}
1819 \end{center}
1820 \vspace{0.1cm}
1821 \end{minipage}
1822 }
1823
1824 \end{slide}
1825
1826 \begin{slide}
1827
1828  {\large\bf
1829   Conclusion of defect / migration / combined defect simulations
1830  }
1831
1832  \footnotesize
1833
1834 \vspace*{0.1cm}
1835
1836 Defect structures
1837 \begin{itemize}
1838  \item Accurately described by quantum-mechanical simulations
1839  \item Less accurate description by classical potential simulations
1840  \item Underestimated formation energy of \cs{} by classical approach
1841  \item Both methods predict same ground state: \ci{} \hkl<1 0 0> dumbbell
1842 \end{itemize}
1843
1844 Migration
1845 \begin{itemize}
1846  \item C migration pathway in Si identified
1847  \item Consistent with reorientation and diffusion experiments
1848 \end{itemize}
1849 \begin{itemize}
1850  \item Different path and ...
1851  \item overestimated barrier by classical potential calculations
1852 \end{itemize}
1853
1854 Concerning the precipitation mechanism
1855 \begin{itemize}
1856  \item Agglomeration of C-Si dumbbells energetically favorable
1857        (stress compensation)
1858  \item C-Si indeed favored compared to
1859        C$_{\text{sub}}$ \& \hkl<1 1 0> Si self-interstitial
1860  \item Possible low interaction capture radius of
1861        C$_{\text{sub}}$ \& \hkl<1 1 0> Si self-interstitial
1862  \item Low barrier for
1863        \ci{} \hkl<1 0 0> $\rightarrow$ \cs{} \& \si{} \hkl<1 1 0>
1864  \item In absence of nearby \hkl<1 1 0> Si self-interstitial:
1865        C-Si \hkl<1 0 0> + Vacancy $\rightarrow$ C$_{\text{sub}}$ (SiC)
1866 \end{itemize}
1867 \begin{center}
1868 {\color{blue}Results suggest increased participation of \cs}
1869 \end{center}
1870
1871 \end{slide}
1872
1873 \begin{slide}
1874
1875  {\large\bf
1876   Silicon carbide precipitation simulations
1877  }
1878
1879  \small
1880
1881 {\scriptsize
1882  \begin{pspicture}(0,0)(12,6.5)
1883   % nodes
1884   \rput(3.5,5.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
1885    \parbox{7cm}{
1886    \begin{itemize}
1887     \item Create c-Si volume
1888     \item Periodc boundary conditions
1889     \item Set requested $T$ and $p=0\text{ bar}$
1890     \item Equilibration of $E_{\text{kin}}$ and $E_{\text{pot}}$
1891    \end{itemize}
1892   }}}}
1893   \rput(3.5,2.7){\rnode{insert}{\psframebox[fillstyle=solid,fillcolor=lachs]{
1894    \parbox{7cm}{
1895    Insertion of C atoms at constant T
1896    \begin{itemize}
1897     \item total simulation volume {\pnode{in1}}
1898     \item volume of minimal SiC precipitate {\pnode{in2}}
1899     \item volume consisting of Si atoms to form a minimal {\pnode{in3}}\\
1900           precipitate
1901    \end{itemize}
1902   }}}}
1903   \rput(3.5,1){\rnode{cool}{\psframebox[fillstyle=solid,fillcolor=lbb]{
1904    \parbox{7.0cm}{
1905    Run for 100 ps followed by cooling down to $20\, ^{\circ}\textrm{C}$
1906   }}}}
1907   \ncline[]{->}{init}{insert}
1908   \ncline[]{->}{insert}{cool}
1909   \psframe[fillstyle=solid,fillcolor=white](7.5,0.7)(13.5,6.3)
1910   \rput(7.8,6){\footnotesize $V_1$}
1911   \psframe[fillstyle=solid,fillcolor=lightgray](9,2)(12,5)
1912   \rput(9.2,4.85){\tiny $V_2$}
1913   \psframe[fillstyle=solid,fillcolor=gray](9.25,2.25)(11.75,4.75)
1914   \rput(9.55,4.45){\footnotesize $V_3$}
1915   \rput(7.9,3.2){\pnode{ins1}}
1916   \rput(9.22,2.8){\pnode{ins2}}
1917   \rput(11.0,2.4){\pnode{ins3}}
1918   \ncline[]{->}{in1}{ins1}
1919   \ncline[]{->}{in2}{ins2}
1920   \ncline[]{->}{in3}{ins3}
1921  \end{pspicture}
1922 }
1923
1924 \begin{itemize}
1925  \item Restricted to classical potential simulations
1926  \item $V_2$ and $V_3$ considered due to low diffusion
1927  \item Amount of C atoms: 6000
1928        ($r_{\text{prec}}\approx 3.1\text{ nm}$, IBS: 2 ... 4 nm)
1929  \item Simulation volume: $31\times 31\times 31$ unit cells
1930        (238328 Si atoms)
1931 \end{itemize}
1932
1933 \end{slide}
1934
1935 \begin{slide}
1936
1937  {\large\bf\boldmath
1938   Silicon carbide precipitation simulations at $450\,^{\circ}\mathrm{C}$ as in IBS
1939  }
1940
1941  \small
1942
1943 \begin{minipage}{6.5cm}
1944 \includegraphics[width=6.4cm]{sic_prec_450_si-si_c-c.ps}
1945 \end{minipage}
1946 \begin{minipage}{6.5cm}
1947 \includegraphics[width=6.4cm]{sic_prec_450_energy.ps}
1948 \end{minipage}
1949
1950 \begin{minipage}{6.5cm}
1951 \includegraphics[width=6.4cm]{sic_prec_450_si-c.ps}
1952 \end{minipage}
1953 \begin{minipage}{6.5cm}
1954 \scriptsize
1955 \underline{Low C concentration ($V_1$)}\\
1956 \hkl<1 0 0> C-Si dumbbell dominated structure
1957 \begin{itemize}
1958  \item Si-C bumbs around 0.19 nm
1959  \item C-C peak at 0.31 nm (as expected in 3C-SiC):\\
1960        concatenated dumbbells of various orientation
1961  \item Si-Si NN distance stretched to 0.3 nm
1962 \end{itemize}
1963 {\color{blue}$\Rightarrow$ C atoms in proper 3C-SiC distance first}\\
1964 \underline{High C concentration ($V_2$, $V_3$)}\\
1965 High amount of strongly bound C-C bonds\\
1966 Defect density $\uparrow$ $\Rightarrow$ considerable amount of damage\\
1967 Only short range order observable\\
1968 {\color{blue}$\Rightarrow$ amorphous SiC-like phase}
1969 \end{minipage}
1970
1971 \end{slide}
1972
1973 \begin{slide}
1974
1975  {\large\bf\boldmath
1976   Silicon carbide precipitation simulations at $450\,^{\circ}\mathrm{C}$ as in IBS
1977  }
1978
1979  \small
1980
1981 \begin{minipage}{6.5cm}
1982 \includegraphics[width=6.4cm]{sic_prec_450_si-si_c-c.ps}
1983 \end{minipage}
1984 \begin{minipage}{6.5cm}
1985 \includegraphics[width=6.4cm]{sic_prec_450_energy.ps}
1986 \end{minipage}
1987
1988 \begin{minipage}{6.5cm}
1989 \includegraphics[width=6.4cm]{sic_prec_450_si-c.ps}
1990 \end{minipage}
1991 \begin{minipage}{6.5cm}
1992 \scriptsize
1993 \underline{Low C concentration ($V_1$)}\\
1994 \hkl<1 0 0> C-Si dumbbell dominated structure
1995 \begin{itemize}
1996  \item Si-C bumbs around 0.19 nm
1997  \item C-C peak at 0.31 nm (as expected in 3C-SiC):\\
1998        concatenated dumbbells of various orientation
1999  \item Si-Si NN distance stretched to 0.3 nm
2000 \end{itemize}
2001 {\color{blue}$\Rightarrow$ C atoms in proper 3C-SiC distance first}\\
2002 \underline{High C concentration ($V_2$, $V_3$)}\\
2003 High amount of strongly bound C-C bonds\\
2004 Defect density $\uparrow$ $\Rightarrow$ considerable amount of damage\\
2005 Only short range order observable\\
2006 {\color{blue}$\Rightarrow$ amorphous SiC-like phase}
2007 \end{minipage}
2008
2009 \begin{pspicture}(0,0)(0,0)
2010 \rput(6.7,5.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=white]{
2011 \begin{minipage}{10cm}
2012 \small
2013 {\color{red}\bf 3C-SiC formation fails to appear}
2014 \begin{itemize}
2015 \item Low C concentration simulations
2016  \begin{itemize}
2017   \item Formation of \ci{} indeed occurs
2018   \item Agllomeration not observed
2019  \end{itemize}
2020 \item High C concentration simulations
2021  \begin{itemize}
2022   \item Amorphous SiC-like structure\\
2023         (not expected at prevailing temperatures)
2024   \item Rearrangement and transition into 3C-SiC structure missing
2025  \end{itemize}
2026 \end{itemize}
2027 \end{minipage}
2028  }}}
2029 \end{pspicture}
2030
2031 \end{slide}
2032
2033 \begin{slide}
2034
2035  {\large\bf
2036   Limitations of molecular dynamics and short range potentials
2037  }
2038
2039 \footnotesize
2040
2041 \vspace{0.2cm}
2042
2043 \underline{Time scale problem of MD}\\[0.2cm]
2044 Minimize integration error\\
2045 $\Rightarrow$ discretization considerably smaller than
2046               reciprocal of fastest vibrational mode\\[0.1cm]
2047 Order of fastest vibrational mode: $10^{13} - 10^{14}\text{ Hz}$\\
2048 $\Rightarrow$ suitable choice of time step:
2049               $\tau=1\text{ fs}=10^{-15}\text{ s}$\\
2050 $\Rightarrow$ {\color{red}\underline{slow}} phase space propagation\\[0.1cm]
2051 Several local minima in energy surface separated by large energy barriers\\
2052 $\Rightarrow$ transition event corresponds to a multiple
2053               of vibrational periods\\
2054 $\Rightarrow$ phase transition made up of {\color{red}\underline{many}}
2055               infrequent transition events\\[0.1cm]
2056 {\color{blue}Accelerated methods:}
2057 \underline{Temperature accelerated} MD (TAD), self-guided MD \ldots
2058
2059 \vspace{0.3cm}
2060
2061 \underline{Limitations related to the short range potential}\\[0.2cm]
2062 Cut-off function pushing forces and energies to zero between 1$^{\text{st}}$
2063 and 2$^{\text{nd}}$ next neighbours\\
2064 $\Rightarrow$ overestimated unphysical high forces of next neighbours
2065
2066 \vspace{0.3cm}
2067
2068 \framebox{
2069 \color{red}
2070 Potential enhanced problem of slow phase space propagation
2071 }
2072
2073 \vspace{0.3cm}
2074
2075 \underline{Approach to the (twofold) problem}\\[0.2cm]
2076 Increased temperature simulations without TAD corrections\\
2077 (accelerated methods or higher time scales exclusively not sufficient)
2078
2079 \begin{picture}(0,0)(-260,-30)
2080 \framebox{
2081 \begin{minipage}{4.2cm}
2082 \tiny
2083 \begin{center}
2084 \vspace{0.03cm}
2085 \underline{IBS}
2086 \end{center}
2087 \begin{itemize}
2088 \item 3C-SiC also observed for higher T
2089 \item higher T inside sample
2090 \item structural evolution vs.\\
2091       equilibrium properties
2092 \end{itemize}
2093 \end{minipage}
2094 }
2095 \end{picture}
2096
2097 \begin{picture}(0,0)(-305,-155)
2098 \framebox{
2099 \begin{minipage}{2.5cm}
2100 \tiny
2101 \begin{center}
2102 retain proper\\
2103 thermodynmic sampling
2104 \end{center}
2105 \end{minipage}
2106 }
2107 \end{picture}
2108
2109 \end{slide}
2110
2111 \begin{slide}
2112
2113  {\large\bf
2114   Increased temperature simulations at low C concentration
2115  }
2116
2117 \small
2118
2119 \begin{minipage}{6.5cm}
2120 \includegraphics[width=6.4cm]{tot_pc_thesis.ps}
2121 \end{minipage}
2122 \begin{minipage}{6.5cm}
2123 \includegraphics[width=6.4cm]{tot_pc3_thesis.ps}
2124 \end{minipage}
2125
2126 \begin{minipage}{6.5cm}
2127 \includegraphics[width=6.4cm]{tot_pc2_thesis.ps}
2128 \end{minipage}
2129 \begin{minipage}{6.5cm}
2130 \scriptsize
2131  \underline{Si-C bonds:}
2132  \begin{itemize}
2133   \item Vanishing cut-off artifact (above $1650\,^{\circ}\mathrm{C}$)
2134   \item Structural change: C-Si \hkl<1 0 0> $\rightarrow$ C$_{\text{sub}}$
2135  \end{itemize}
2136  \underline{Si-Si bonds:}
2137  {\color{blue}Si-C$_{\text{sub}}$-Si} along \hkl<1 1 0>
2138  ($\rightarrow$ 0.325 nm)\\[0.1cm]
2139  \underline{C-C bonds:}
2140  \begin{itemize}
2141   \item C-C next neighbour pairs reduced (mandatory)
2142   \item Peak at 0.3 nm slightly shifted
2143         \begin{itemize}
2144          \item C-Si \hkl<1 0 0> combinations (dashed arrows)\\
2145                $\rightarrow$ C-Si \hkl<1 0 0> \& C$_{\text{sub}}$
2146                combinations (|)\\
2147                $\rightarrow$ pure {\color{blue}C$_{\text{sub}}$ combinations}
2148                ($\downarrow$)
2149          \item Range [|-$\downarrow$]:
2150                {\color{blue}C$_{\text{sub}}$ \& C$_{\text{sub}}$
2151                with nearby Si$_{\text{I}}$}
2152         \end{itemize}
2153  \end{itemize}
2154 \end{minipage}
2155
2156 \begin{picture}(0,0)(-330,-74)
2157 \color{blue}
2158 \framebox{
2159 \begin{minipage}{1.6cm}
2160 \tiny
2161 \begin{center}
2162 stretched SiC\\[-0.1cm]
2163 in c-Si
2164 \end{center}
2165 \end{minipage}
2166 }
2167 \end{picture}
2168
2169 \end{slide}
2170
2171 \begin{slide}
2172
2173  {\large\bf
2174   Increased temperature simulations at low C concentration
2175  }
2176
2177 \small
2178
2179 \begin{minipage}{6.5cm}
2180 \includegraphics[width=6.4cm]{tot_pc_thesis.ps}
2181 \end{minipage}
2182 \begin{minipage}{6.5cm}
2183 \includegraphics[width=6.4cm]{tot_pc3_thesis.ps}
2184 \end{minipage}
2185
2186 \begin{minipage}{6.5cm}
2187 \includegraphics[width=6.4cm]{tot_pc2_thesis.ps}
2188 \end{minipage}
2189 \begin{minipage}{6.5cm}
2190 \scriptsize
2191  \underline{Si-C bonds:}
2192  \begin{itemize}
2193   \item Vanishing cut-off artifact (above $1650\,^{\circ}\mathrm{C}$)
2194   \item Structural change: C-Si \hkl<1 0 0> $\rightarrow$ C$_{\text{sub}}$
2195  \end{itemize}
2196  \underline{Si-Si bonds:}
2197  {\color{blue}Si-C$_{\text{sub}}$-Si} along \hkl<1 1 0>
2198  ($\rightarrow$ 0.325 nm)\\[0.1cm]
2199  \underline{C-C bonds:}
2200  \begin{itemize}
2201   \item C-C next neighbour pairs reduced (mandatory)
2202   \item Peak at 0.3 nm slightly shifted
2203         \begin{itemize}
2204          \item C-Si \hkl<1 0 0> combinations (dashed arrows)\\
2205                $\rightarrow$ C-Si \hkl<1 0 0> \& C$_{\text{sub}}$
2206                combinations (|)\\
2207                $\rightarrow$ pure {\color{blue}C$_{\text{sub}}$ combinations}
2208                ($\downarrow$)
2209          \item Range [|-$\downarrow$]:
2210                {\color{blue}C$_{\text{sub}}$ \& C$_{\text{sub}}$
2211                with nearby Si$_{\text{I}}$}
2212         \end{itemize}
2213  \end{itemize}
2214 \end{minipage}
2215
2216 %\begin{picture}(0,0)(-330,-74)
2217 %\color{blue}
2218 %\framebox{
2219 %\begin{minipage}{1.6cm}
2220 %\tiny
2221 %\begin{center}
2222 %stretched SiC\\[-0.1cm]
2223 %in c-Si
2224 %\end{center}
2225 %\end{minipage}
2226 %}
2227 %\end{picture}
2228
2229 \begin{pspicture}(0,0)(0,0)
2230 \rput(6.7,5.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=white]{
2231 \begin{minipage}{10cm}
2232 \small
2233 {\color{blue}\bf Stretched SiC in c-Si}
2234 \begin{itemize}
2235 \item Consistent to precipitation model involving \cs{}
2236 \item Explains annealing behavior of high/low T C implants
2237       \begin{itemize}
2238        \item Low T: highly mobiel \ci{}
2239        \item High T: stable configurations of \cs{}
2240       \end{itemize}
2241 \end{itemize}
2242 $\Rightarrow$ High T $\leftrightarrow$ IBS conditions far from equilibrium\\
2243 $\Rightarrow$ Precipitation mechanism involving \cs{}
2244 \end{minipage}
2245  }}}
2246 \end{pspicture}
2247
2248 \end{slide}
2249
2250 \begin{slide}
2251
2252  {\large\bf
2253   Increased temperature simulations at high C concentration
2254  }
2255
2256 \footnotesize
2257
2258 \begin{minipage}{6.5cm}
2259 \includegraphics[width=6.4cm]{12_pc_thesis.ps}
2260 \end{minipage}
2261 \begin{minipage}{6.5cm}
2262 \includegraphics[width=6.4cm]{12_pc_c_thesis.ps}
2263 \end{minipage}
2264
2265 \vspace{0.1cm}
2266
2267 \scriptsize
2268
2269 \framebox{
2270 \begin{minipage}[t]{6.0cm}
2271 0.186 nm: Si-C pairs $\uparrow$\\
2272 (as expected in 3C-SiC)\\[0.2cm]
2273 0.282 nm: Si-C-C\\[0.2cm]
2274 $\approx$0.35 nm: C-Si-Si
2275 \end{minipage}
2276 }
2277 \begin{minipage}{0.2cm}
2278 \hfill
2279 \end{minipage}
2280 \framebox{
2281 \begin{minipage}[t]{6.0cm}
2282 0.15 nm: C-C pairs $\uparrow$\\
2283 (as expected in graphite/diamond)\\[0.2cm]
2284 0.252 nm: C-C-C (2$^{\text{nd}}$ NN for diamond)\\[0.2cm]
2285 0.31 nm: shifted towards 0.317 nm $\rightarrow$ C-Si-C
2286 \end{minipage}
2287 }
2288
2289 \begin{itemize}
2290 \item Decreasing cut-off artifact
2291 \item {\color{red}Amorphous} SiC-like phase remains
2292 \item High amount of {\color{red}damage} \& alignement to c-Si host matrix lost
2293 \item Slightly sharper peaks $\Rightarrow$ indicate slight {\color{blue}acceleration of dynamics} due to temperature
2294 \end{itemize}
2295
2296 \vspace{-0.1cm}
2297
2298 \begin{center}
2299 {\color{blue}
2300 \framebox{
2301 {\color{black}
2302 High C \& small $V$ \& short $t$
2303 $\Rightarrow$
2304 }
2305 Slow restructuring due to strong C-C bonds
2306 {\color{black}
2307 $\Leftarrow$
2308 High C \& low T implants
2309 }
2310 }
2311 }
2312 \end{center}
2313
2314 \end{slide}
2315
2316 \begin{slide}
2317
2318  {\large\bf
2319   Summary and Conclusions
2320  }
2321
2322  \scriptsize
2323
2324 %\vspace{0.1cm}
2325
2326 \framebox{
2327 \begin{minipage}[t]{12.9cm}
2328  \underline{Pecipitation simulations}
2329  \begin{itemize}
2330   \item High C concentration $\rightarrow$ amorphous SiC like phase
2331   \item Problem of potential enhanced slow phase space propagation
2332   \item Low T $\rightarrow$ C-Si \hkl<1 0 0> dumbbell dominated structure
2333   \item High T $\rightarrow$ C$_{\text{sub}}$ dominated structure
2334   \item High T necessary to simulate IBS conditions (far from equilibrium)
2335   \item Precipitation by successive agglomeration of \cs (epitaxy)
2336   \item \si{}: vehicle to form \cs{} \& supply of Si \& stress compensation
2337         (stretched SiC, interface)
2338  \end{itemize}
2339 \end{minipage}
2340 }
2341
2342 %\vspace{0.1cm}
2343
2344 \framebox{
2345 \begin{minipage}{12.9cm}
2346  \underline{Defects}
2347  \begin{itemize}
2348    \item DFT / EA
2349         \begin{itemize}
2350          \item Point defects excellently / fairly well described
2351                by DFT / EA
2352          \item C$_{\text{sub}}$ drastically underestimated by EA
2353          \item EA predicts correct ground state:
2354                C$_{\text{sub}}$ \& \si{} $>$ \ci{}
2355          \item Identified migration path explaining
2356                diffusion and reorientation experiments by DFT
2357          \item EA fails to describe \ci{} migration:
2358                Wrong path \& overestimated barrier
2359         \end{itemize}
2360    \item Combinations of defects
2361          \begin{itemize}
2362           \item Agglomeration of point defects energetically favorable
2363                 by compensation of stress
2364           \item Formation of C-C unlikely
2365           \item C$_{\text{sub}}$ favored conditions (conceivable in IBS)
2366           \item \ci{} \hkl<1 0 0> $\leftrightarrow$ \cs{} \& \si{} \hkl<1 1 0>\\
2367                 Low barrier (\unit[0.77]{eV}) \& low capture radius
2368         \end{itemize}
2369  \end{itemize}
2370 \end{minipage}
2371 }
2372
2373 \begin{center}
2374 {\color{blue}
2375 \framebox{Precipitation by successive agglomeration of \cs{}}
2376 }
2377 \end{center}
2378
2379 \end{slide}
2380
2381 \begin{slide}
2382
2383  {\large\bf
2384   Acknowledgements
2385  }
2386
2387  \vspace{0.1cm}
2388
2389  \small
2390
2391  Thanks to \ldots
2392
2393  \underline{Augsburg}
2394  \begin{itemize}
2395   \item Prof. B. Stritzker (accomodation at EP \RM{4})
2396   \item Ralf Utermann (EDV)
2397  \end{itemize}
2398
2399  \underline{Helsinki}
2400  \begin{itemize}
2401   \item Prof. K. Nordlund (MD)
2402  \end{itemize}
2403
2404  \underline{Munich}
2405  \begin{itemize}
2406   \item Bayerische Forschungsstiftung (financial support)
2407  \end{itemize}
2408
2409  \underline{Paderborn}
2410  \begin{itemize}
2411   \item Prof. J. Lindner (SiC)
2412   \item Prof. G. Schmidt (DFT + financial support)
2413   \item Dr. E. Rauls (DFT + SiC)
2414   \item Dr. S. Sanna (VASP)
2415  \end{itemize}
2416
2417 \vspace{0.2cm}
2418
2419 \begin{center}
2420 \framebox{
2421 \bf Thank you for your attention!
2422 }
2423 \end{center}
2424
2425 \end{slide}
2426
2427 \end{document}
2428
2429 \fi