posic/talks/mpi_app.tex

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   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{layout}
  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 % no vertical centering
 147 \centerslidesfalse
 148
 149 \ifnum1=0
 150
 151 % intro
 152
 153 \begin{slide}
 154
 155 %{\large\bf
 156 % Phase diagram of the C/Si system\\
 157 %}
 158
 159 \vspace*{0.2cm}
 160
 161 \begin{minipage}{6.5cm}
 162 \includegraphics[width=6.5cm]{si-c_phase.eps}
 163 \begin{center}
 164 {\tiny
 165 R. I. Scace and G. A. Slack, J. Chem. Phys. 30, 1551 (1959)
 166 }
 167 \end{center}
 168 \begin{pspicture}(0,0)(0,0)
 169 \psellipse[linecolor=blue,linewidth=0.1cm](3.55,4.0)(0.5,2.9)
 170 \end{pspicture}
 171 \end{minipage}
 172 \begin{minipage}{6cm}
 173 {\bf Phase diagram of the C/Si system}\\[0.2cm]
 174 {\color{blue}Stoichiometric composition}
 175 \begin{itemize}
 176 \item only chemical stable compound
 177 \item wide band gap semiconductor\\
 178       \underline{silicon carbide}, SiC
 179 \end{itemize}
 180 \end{minipage}
 181
 182 \end{slide}
 183
 184 % motivation / properties / applications of silicon carbide
 185
 186 \begin{slide}
 187
 188 \small
 189
 190 \begin{pspicture}(0,0)(13.5,5)
 191
 192  \psframe*[linecolor=hb](-0.2,0)(12.9,5)
 193
 194  \pspolygon[linecolor=hlbb,fillcolor=hlbb,fillstyle=solid](5.2,1)(6.5,1)(6.5,3)(5.2,3)
 195  \pspolygon[linecolor=hlbb,fillcolor=hlbb,fillstyle=solid](6.4,0.5)(7.7,2)(7.7,2)(6.4,3.5)
 196
 197  \rput[lt](0,4.6){\color{gray}PROPERTIES}
 198
 199  \rput[lt](0.3,4){wide band gap}
 200  \rput[lt](0.3,3.5){high electric breakdown field}
 201  \rput[lt](0.3,3){good electron mobility}
 202  \rput[lt](0.3,2.5){high electron saturation drift velocity}
 203  \rput[lt](0.3,2){high thermal conductivity}
 204
 205  \rput[lt](0.3,1.5){hard and mechanically stable}
 206  \rput[lt](0.3,1){chemically inert}
 207
 208  \rput[lt](0.3,0.5){radiation hardness}
 209
 210  \rput[rt](12.7,4.6){\color{gray}APPLICATIONS}
 211
 212  \rput[rt](12.5,3.85){high-temperature, high power}
 213  \rput[rt](12.5,3.5){and high-frequency}
 214  \rput[rt](12.5,3.15){electronic and optoelectronic devices}
 215
 216  \rput[rt](12.5,2.35){material suitable for extreme conditions}
 217  \rput[rt](12.5,2){microelectromechanical systems}
 218  \rput[rt](12.5,1.65){abrasives, cutting tools, heating elements}
 219
 220  \rput[rt](12.5,0.85){first wall reactor material, detectors}
 221  \rput[rt](12.5,0.5){and electronic devices for space}
 222
 223 \end{pspicture}
 224
 225 \begin{picture}(0,0)(5,-162)
 226 \includegraphics[height=2.2cm]{3C_SiC_bs.eps}
 227 \end{picture}
 228 \begin{picture}(0,0)(-120,-162)
 229 \includegraphics[height=2.2cm]{nasa_600c_led.eps}
 230 \end{picture}
 231 \begin{picture}(0,0)(-270,-162)
 232 \includegraphics[height=2.2cm]{6h-sic_3c-sic.eps}
 233 \end{picture}
 234 %%%%
 235 \begin{picture}(0,0)(10,65)
 236 \includegraphics[height=2.8cm]{sic_switch.eps}
 237 \end{picture}
 238 %\begin{picture}(0,0)(-243,65)
 239 \begin{picture}(0,0)(-110,65)
 240 \includegraphics[height=2.8cm]{ise_99.eps}
 241 \end{picture}
 242 %\begin{picture}(0,0)(-135,65)
 243 \begin{picture}(0,0)(-100,65)
 244 \includegraphics[height=1.2cm]{infineon_schottky.eps}
 245 \end{picture}
 246 \begin{picture}(0,0)(-233,65)
 247 \includegraphics[height=2.8cm]{solar_car.eps}
 248 \end{picture}
 249
 250 \end{slide}
 251
 252 % motivation
 253
 254 \begin{slide}
 255
 256  {\large\bf
 257   Polytypes of SiC\\[0.4cm]
 258  }
 259
 260 \includegraphics[width=3.8cm]{cubic_hex.eps}\\
 261 \begin{minipage}{1.9cm}
 262 {\tiny cubic (twist)}
 263 \end{minipage}
 264 \begin{minipage}{2.9cm}
 265 {\tiny hexagonal (no twist)}
 266 \end{minipage}
 267
 268 \begin{picture}(0,0)(-150,0)
 269  \includegraphics[width=7cm]{polytypes.eps}
 270 \end{picture}
 271
 272 \vspace{0.6cm}
 273
 274 \footnotesize
 275
 276 \begin{tabular}{l c c c c c c}
 277 \hline
 278  & 3C-SiC & 4H-SiC & 6H-SiC & Si & GaN & Diamond\\
 279 \hline
 280 Hardness [Mohs] & \multicolumn{3}{c}{------ 9.6 ------}& 6.5 & - & 10 \\
 281 Band gap [eV] & 2.36 & 3.23 & 3.03 & 1.12 & 3.39 & 5.5 \\
 282 Break down field [$10^6$ V/cm] & 4 & 3 & 3.2 & 0.6 & 5 & 10 \\
 283 Saturation drift velocity [$10^7$ cm/s] & 2.5 & 2.0 & 2.0 & 1 & 2.7 & 2.7 \\
 284 Electron mobility [cm$^2$/Vs] & 800 & 900 & 400 & 1100 & 900 & 2200 \\
 285 Hole mobility [cm$^2$/Vs] & 320 & 120 & 90 & 420 & 150 & 1600 \\
 286 Thermal conductivity [W/cmK] & 5.0 & 4.9 & 4.9 & 1.5 & 1.3 & 22 \\
 287 \hline
 288 \end{tabular}
 289
 290 \begin{pspicture}(0,0)(0,0)
 291 \psellipse[linecolor=green](5.7,2.10)(0.4,0.5)
 292 \end{pspicture}
 293 \begin{pspicture}(0,0)(0,0)
 294 \psellipse[linecolor=green](5.6,0.92)(0.4,0.2)
 295 \end{pspicture}
 296 \begin{pspicture}(0,0)(0,0)
 297 \psellipse[linecolor=red](10.45,0.45)(0.4,0.2)
 298 \end{pspicture}
 299
 300 \end{slide}
 301
 302 % fabrication
 303
 304 \begin{slide}
 305
 306  {\large\bf
 307   Fabrication of silicon carbide
 308  }
 309
 310  \small
 311
 312  \vspace{2pt}
 313
 314 \begin{center}
 315  {\color{gray}
 316  \emph{Silicon carbide --- Born from the stars, perfected on earth.}
 317  }
 318 \end{center}
 319
 320 \vspace{2pt}
 321
 322 SiC thin films by MBE \& CVD
 323 \begin{itemize}
 324  \item Much progress achieved in homo/heteroepitaxial SiC thin film growth
 325  \item \underline{Commercially available} semiconductor power devices based on
 326        \underline{\foreignlanguage{greek}{a}-SiC}
 327  \item Production of favored \underline{3C-SiC} material
 328        \underline{less advanced}
 329  \item Quality and size not yet sufficient
 330 \end{itemize}
 331 \begin{picture}(0,0)(-310,-20)
 332   \includegraphics[width=2.0cm]{cree.eps}
 333 \end{picture}
 334
 335 \vspace{-0.2cm}
 336
 337 Alternative approach:
 338 Ion beam synthesis (IBS) of burried 3C-SiC layers in Si\hkl(1 0 0)
 339
 340 \vspace{0.2cm}
 341
 342 \scriptsize
 343
 344 \framebox{
 345 \begin{minipage}{3.15cm}
 346  \begin{center}
 347 \includegraphics[width=3cm]{imp.eps}\\
 348  {\tiny
 349   Carbon implantation
 350  }
 351  \end{center}
 352 \end{minipage}
 353 \begin{minipage}{3.15cm}
 354  \begin{center}
 355 \includegraphics[width=3cm]{annealing.eps}\\
 356  {\tiny
 357   \unit[12]{h} annealing at \degc{1200}
 358  }
 359  \end{center}
 360 \end{minipage}
 361 }
 362 \begin{minipage}{5.5cm}
 363  \includegraphics[width=5.8cm]{ibs_3c-sic.eps}\\[-0.2cm]
 364  \begin{center}
 365  {\tiny
 366   XTEM: single crystalline 3C-SiC in Si\hkl(1 0 0)
 367  }
 368  \end{center}
 369 \end{minipage}
 370
 371 \end{slide}
 372
 373 % contents
 374
 375 \begin{slide}
 376
 377 {\large\bf
 378  Systematic investigation of C implantations into Si
 379 }
 380
 381 \vspace{1.7cm}
 382 \begin{center}
 383 \hspace{-1.0cm}
 384 \includegraphics[width=0.75\textwidth]{imp_inv.eps}
 385 \end{center}
 386
 387 \end{slide}
 388
 389 % outline
 390
 391 \begin{slide}
 392
 393 {\large\bf
 394  Outline
 395 }
 396
 397 \vspace{1.7cm}
 398 \begin{center}
 399 \hspace{-1.0cm}
 400 \includegraphics[width=0.75\textwidth]{imp_inv.eps}
 401 \end{center}
 402
 403 \begin{pspicture}(0,0)(0,0)
 404 \rput(6.0,7.0){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=white]{
 405 \begin{minipage}{11cm}
 406 {\color{red}Diploma thesis}\\
 407  \underline{Monte Carlo} simulation modeling the selforganization process\\
 408  leading to periodic arrays of nanometric amorphous SiC precipitates
 409 \end{minipage}
 410 }}}
 411 \end{pspicture}
 412 \begin{pspicture}(0,0)(0,0)
 413 \rput(6.0,-0.5){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=white]{
 414 \begin{minipage}{11cm}
 415 {\color{blue}Doctoral studies}\\
 416  Classical potential \underline{molecular dynamics} simulations \ldots\\
 417  \underline{Density functional theory} calculations \ldots\\[0.2cm]
 418  \ldots on defect formation and SiC precipitation in Si
 419 \end{minipage}
 420 }}}
 421 \end{pspicture}
 422 \begin{pspicture}(0,0)(0,0)
 423 \psellipse[linecolor=red,linewidth=0.05cm](5,3.0)(0.8,1.0)
 424 \end{pspicture}
 425 \begin{pspicture}(0,0)(0,0)
 426 \psellipse[linecolor=blue,linewidth=0.05cm](8.2,3.2)(1.5,1.6)
 427 \end{pspicture}
 428
 429 \end{slide}
 430
 431 \begin{slide}
 432
 433 {\large\bf
 434  Selforganization of nanometric amorphous SiC lamellae
 435 }
 436
 437 \small
 438
 439 \vspace{0.2cm}
 440
 441 \begin{itemize}
 442  \item Regularly spaced, nanometric spherical\\
 443        and lamellar amorphous inclusions\\
 444        at the upper a/c interface
 445  \item Carbon accumulation\\
 446        in amorphous volumes
 447 \end{itemize}
 448
 449 \vspace{0.4cm}
 450
 451 \begin{minipage}{12cm}
 452 \includegraphics[width=9cm]{../../nlsop/img/k393abild1_e_l.eps}\\
 453 {\scriptsize
 454 XTEM bright-field, \unit[180]{keV} C$^+ \rightarrow$ Si, \degc{150},
 455 Dose: \unit[4.3 $\times 10^{17}$]{cm$^{-2}$}
 456 }
 457 \end{minipage}
 458
 459 \begin{picture}(0,0)(-182,-215)
 460 \begin{minipage}{6.5cm}
 461 \begin{center}
 462 \includegraphics[width=6.5cm]{../../nlsop/img/eftem.eps}\\[-0.2cm]
 463 {\scriptsize
 464 XTEM bright-field and respective EFTEM C map
 465 }
 466 \end{center}
 467 \end{minipage}
 468 \end{picture}
 469
 470 \end{slide}
 471
 472 % continue here
 473 \fi
 474
 475 \begin{slide}
 476
 477 {\large\bf
 478  Model displaying the formation of ordered lamellae
 479 }
 480
 481 \end{slide}
 482
 483 \end{document}
 484 \ifnum1=0
 485
 486 \begin{slide}
 487
 488 {\large\bf
 489  Model displaying the formation of ordered lamellae
 490 }
 491
 492 \framebox{
 493  \begin{minipage}{6.3cm}
 494  \begin{center}
 495  {\color{blue}
 496   Precipitation mechanism not yet fully understood!
 497  }
 498  \renewcommand\labelitemi{$\Rightarrow$}
 499  \small
 500  \underline{Understanding the SiC precipitation}
 501  \begin{itemize}
 502   \item significant technological progress in SiC thin film formation
 503   \item perspectives for processes relying upon prevention of SiC precipitation
 504  \end{itemize}
 505  \end{center}
 506  \end{minipage}
 507 }
 508
 509 \end{slide}
 510
 511 \begin{slide}
 512
 513  {\large\bf
 514   Supposed precipitation mechanism of SiC in Si
 515  }
 516
 517  \scriptsize
 518
 519  \vspace{0.1cm}
 520
 521  \begin{minipage}{3.8cm}
 522  Si \& SiC lattice structure\\[0.2cm]
 523  \includegraphics[width=3.5cm]{sic_unit_cell.eps}\\[-0.3cm]
 524  \hrule
 525  \end{minipage}
 526  \hspace{0.6cm}
 527  \begin{minipage}{3.8cm}
 528  \begin{center}
 529  \includegraphics[width=3.3cm]{tem_c-si-db.eps}
 530  \end{center}
 531  \end{minipage}
 532  \hspace{0.6cm}
 533  \begin{minipage}{3.8cm}
 534  \begin{center}
 535  \includegraphics[width=3.3cm]{tem_3c-sic.eps}
 536  \end{center}
 537  \end{minipage}
 538
 539  \begin{minipage}{4cm}
 540  \begin{center}
 541  C-Si dimers (dumbbells)\\[-0.1cm]
 542  on Si interstitial sites
 543  \end{center}
 544  \end{minipage}
 545  \hspace{0.2cm}
 546  \begin{minipage}{4.2cm}
 547  \begin{center}
 548  Agglomeration of C-Si dumbbells\\[-0.1cm]
 549  $\Rightarrow$ dark contrasts
 550  \end{center}
 551  \end{minipage}
 552  \hspace{0.2cm}
 553  \begin{minipage}{4cm}
 554  \begin{center}
 555  Precipitation of 3C-SiC in Si\\[-0.1cm]
 556  $\Rightarrow$ Moir\'e fringes\\[-0.1cm]
 557  \& release of Si self-interstitials
 558  \end{center}
 559  \end{minipage}
 560
 561  \begin{minipage}{3.8cm}
 562  \begin{center}
 563  \includegraphics[width=3.3cm]{sic_prec_seq_01.eps}
 564  \end{center}
 565  \end{minipage}
 566  \hspace{0.6cm}
 567  \begin{minipage}{3.8cm}
 568  \begin{center}
 569  \includegraphics[width=3.3cm]{sic_prec_seq_02.eps}
 570  \end{center}
 571  \end{minipage}
 572  \hspace{0.6cm}
 573  \begin{minipage}{3.8cm}
 574  \begin{center}
 575  \includegraphics[width=3.3cm]{sic_prec_seq_03.eps}
 576  \end{center}
 577  \end{minipage}
 578
 579 \begin{pspicture}(0,0)(0,0)
 580 \psline[linewidth=4pt]{->}(8.5,2)(9.0,2)
 581 \psellipse[linecolor=blue](11.5,5.8)(0.3,0.5)
 582 \rput{-20}{\psellipse[linecolor=blue](3.3,8.1)(0.3,0.5)}
 583 \psline[linewidth=4pt]{->}(4.0,2)(4.5,2)
 584 \rput(12.7,0.3){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
 585  $4a_{\text{Si}}=5a_{\text{SiC}}$
 586  }}}
 587 \rput(12.2,8){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
 588 \hkl(h k l) planes match
 589  }}}
 590 \rput(9.7,6.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
 591 r = 2 - 4 nm
 592  }}}
 593 \end{pspicture}
 594
 595 \end{slide}
 596
 597 \begin{slide}
 598
 599  {\large\bf
 600   Supposed precipitation mechanism of SiC in Si
 601  }
 602
 603  \scriptsize
 604
 605  \vspace{0.1cm}
 606
 607  \begin{minipage}{3.8cm}
 608  Si \& SiC lattice structure\\[0.2cm]
 609  \includegraphics[width=3.5cm]{sic_unit_cell.eps}\\[-0.3cm]
 610  \hrule
 611  \end{minipage}
 612  \hspace{0.6cm}
 613  \begin{minipage}{3.8cm}
 614  \begin{center}
 615  \includegraphics[width=3.3cm]{tem_c-si-db.eps}
 616  \end{center}
 617  \end{minipage}
 618  \hspace{0.6cm}
 619  \begin{minipage}{3.8cm}
 620  \begin{center}
 621  \includegraphics[width=3.3cm]{tem_3c-sic.eps}
 622  \end{center}
 623  \end{minipage}
 624
 625  \begin{minipage}{4cm}
 626  \begin{center}
 627  C-Si dimers (dumbbells)\\[-0.1cm]
 628  on Si interstitial sites
 629  \end{center}
 630  \end{minipage}
 631  \hspace{0.2cm}
 632  \begin{minipage}{4.2cm}
 633  \begin{center}
 634  Agglomeration of C-Si dumbbells\\[-0.1cm]
 635  $\Rightarrow$ dark contrasts
 636  \end{center}
 637  \end{minipage}
 638  \hspace{0.2cm}
 639  \begin{minipage}{4cm}
 640  \begin{center}
 641  Precipitation of 3C-SiC in Si\\[-0.1cm]
 642  $\Rightarrow$ Moir\'e fringes\\[-0.1cm]
 643  \& release of Si self-interstitials
 644  \end{center}
 645  \end{minipage}
 646
 647  \begin{minipage}{3.8cm}
 648  \begin{center}
 649  \includegraphics[width=3.3cm]{sic_prec_seq_01.eps}
 650  \end{center}
 651  \end{minipage}
 652  \hspace{0.6cm}
 653  \begin{minipage}{3.8cm}
 654  \begin{center}
 655  \includegraphics[width=3.3cm]{sic_prec_seq_02.eps}
 656  \end{center}
 657  \end{minipage}
 658  \hspace{0.6cm}
 659  \begin{minipage}{3.8cm}
 660  \begin{center}
 661  \includegraphics[width=3.3cm]{sic_prec_seq_03.eps}
 662  \end{center}
 663  \end{minipage}
 664
 665 \begin{pspicture}(0,0)(0,0)
 666 \psline[linewidth=4pt]{->}(8.5,2)(9.0,2)
 667 \psellipse[linecolor=blue](11.5,5.8)(0.3,0.5)
 668 \rput{-20}{\psellipse[linecolor=blue](3.3,8.1)(0.3,0.5)}
 669 \psline[linewidth=4pt]{->}(4.0,2)(4.5,2)
 670 \rput(12.7,0.3){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
 671  $4a_{\text{Si}}=5a_{\text{SiC}}$
 672  }}}
 673 \rput(12.2,8){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
 674 \hkl(h k l) planes match
 675  }}}
 676 \rput(9.7,6.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
 677 r = 2 - 4 nm
 678  }}}
 679 \rput(6.7,5.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=white]{
 680 \begin{minipage}{10cm}
 681 \small
 682 {\color{red}\bf Controversial views}
 683 \begin{itemize}
 684 \item Implantations at high T (Nejim et al.)
 685  \begin{itemize}
 686   \item Topotactic transformation based on \cs
 687   \item \si{} as supply reacting with further C in cleared volume
 688  \end{itemize}
 689 \item Annealing behavior (Serre et al.)
 690  \begin{itemize}
 691   \item Room temperature implants $\rightarrow$ highly mobile C
 692   \item Elevated T implants $\rightarrow$ no/low C redistribution/migration\\
 693         (indicate stable \cs{} configurations)
 694  \end{itemize}
 695 \item Strained silicon \& Si/SiC heterostructures
 696  \begin{itemize}
 697   \item Coherent SiC precipitates (tensile strain)
 698   \item Incoherent SiC (strain relaxation)
 699  \end{itemize}
 700 \end{itemize}
 701 \end{minipage}
 702  }}}
 703 \end{pspicture}
 704
 705 \end{slide}
 706
 707 \begin{slide}
 708
 709  {\large\bf
 710   Molecular dynamics (MD) simulations
 711  }
 712
 713  \vspace{12pt}
 714
 715  \small
 716
 717  {\bf MD basics:}
 718  \begin{itemize}
 719   \item Microscopic description of N particle system
 720   \item Analytical interaction potential
 721   \item Numerical integration using Newtons equation of motion\\
 722         as a propagation rule in 6N-dimensional phase space
 723   \item Observables obtained by time and/or ensemble averages
 724  \end{itemize}
 725  {\bf Details of the simulation:}
 726  \begin{itemize}
 727   \item Integration: Velocity Verlet, timestep: $1\text{ fs}$
 728   \item Ensemble: NpT (isothermal-isobaric)
 729         \begin{itemize}
 730          \item Berendsen thermostat:
 731                $\tau_{\text{T}}=100\text{ fs}$
 732          \item Berendsen barostat:\\
 733                $\tau_{\text{P}}=100\text{ fs}$,
 734                $\beta^{-1}=100\text{ GPa}$
 735         \end{itemize}
 736   \item Erhart/Albe potential: Tersoff-like bond order potential
 737   \vspace*{12pt}
 738         \[
 739         E = \frac{1}{2} \sum_{i \neq j} \pot_{ij}, \quad
 740         \pot_{ij} = {\color{red}f_C(r_{ij})}
 741         \left[ f_R(r_{ij}) + {\color{blue}b_{ij}} f_A(r_{ij}) \right]
 742         \]
 743  \end{itemize}
 744
 745  \begin{picture}(0,0)(-230,-30)
 746   \includegraphics[width=5cm]{tersoff_angle.eps}
 747  \end{picture}
 748
 749 \end{slide}
 750
 751 \begin{slide}
 752
 753  {\large\bf
 754   Density functional theory (DFT) calculations
 755  }
 756
 757  \small
 758
 759  Basic ingredients necessary for DFT
 760
 761  \begin{itemize}
 762   \item \underline{Hohenberg-Kohn theorem} - ground state density $n_0(r)$ ...
 763         \begin{itemize}
 764          \item ... uniquely determines the ground state potential
 765                / wavefunctions
 766          \item ... minimizes the systems total energy
 767         \end{itemize}
 768   \item \underline{Born-Oppenheimer}
 769         - $N$ moving electrons in an external potential of static nuclei
 770 \[
 771 H\Psi = \left[-\sum_i^N \frac{\hbar^2}{2m}\nabla_i^2
 772               +\sum_i^N V_{\text{ext}}(r_i)
 773               +\sum_{i<j}^N V_{e-e}(r_i,r_j)\right]\Psi=E\Psi
 774 \]
 775   \item \underline{Effective potential}
 776         - averaged electrostatic potential \& exchange and correlation
 777 \[
 778 V_{\text{eff}}(r)=V_{\text{ext}}(r)+\int\frac{e^2 n(r')}{|r-r'|}d^3r'
 779                  +V_{\text{XC}}[n(r)]
 780 \]
 781   \item \underline{Kohn-Sham system}
 782         - Schr\"odinger equation of N non-interacting particles
 783 \[
 784 \left[ -\frac{\hbar^2}{2m}\nabla^2 + V_{\text{eff}}(r) \right] \Phi_i(r)
 785 =\epsilon_i\Phi_i(r)
 786 \quad
 787 \Rightarrow
 788 \quad
 789 n(r)=\sum_i^N|\Phi_i(r)|^2
 790 \]
 791   \item \underline{Self-consistent solution}\\
 792 $n(r)$ depends on $\Phi_i$, which depend on $V_{\text{eff}}$,
 793 which in turn depends on $n(r)$
 794   \item \underline{Variational principle}
 795         - minimize total energy with respect to $n(r)$
 796  \end{itemize}
 797
 798 \end{slide}
 799
 800 \begin{slide}
 801
 802  {\large\bf
 803   Density functional theory (DFT) calculations
 804  }
 805
 806  \small
 807
 808  \vspace*{0.2cm}
 809
 810  Details of applied DFT calculations in this work
 811
 812  \begin{itemize}
 813   \item \underline{Exchange correlation functional}
 814         - approximations for the inhomogeneous electron gas
 815         \begin{itemize}
 816          \item LDA: $E_{\text{XC}}^{\text{LDA}}[n]=\int \epsilon_{\text{XC}}(n)n(r)d^3r$
 817          \item GGA: $E_{\text{XC}}^{\text{GGA}}[n]=\int \epsilon_{\text{XC}}(n,\nabla n)n(r)d^3r$
 818         \end{itemize}
 819   \item \underline{Plane wave basis set}
 820         - approximation of the wavefunction $\Phi_i$ by plane waves $\phi_j$
 821 \[
 822 \rightarrow
 823 \text{Fourier series: } \Phi_i=\sum_{|G+k|<G_{\text{cut}}} c_j^i \phi_j(r), \quad E_{\text{cut}}=\frac{\hbar^2}{2m}G^2_{\text{cut}}
 824 \qquad ({\color{blue}300\text{ eV}})
 825 \]
 826   \item \underline{Brillouin zone sampling} -
 827         {\color{blue}$\Gamma$-point only} calculations
 828   \item \underline{Pseudo potential}
 829         - consider only the valence electrons
 830   \item \underline{Code} - VASP 4.6
 831  \end{itemize}
 832
 833  \vspace*{0.2cm}
 834
 835  MD and structural optimization
 836
 837  \begin{itemize}
 838   \item MD integration: Gear predictor corrector algorithm
 839   \item Pressure control: Parrinello-Rahman pressure control
 840   \item Structural optimization: Conjugate gradient method
 841  \end{itemize}
 842
 843 \begin{pspicture}(0,0)(0,0)
 844 \psellipse[linecolor=blue](1.5,6.75)(0.5,0.3)
 845 \end{pspicture}
 846
 847 \end{slide}
 848
 849 \begin{slide}
 850
 851  {\large\bf
 852   C and Si self-interstitial point defects in silicon
 853  }
 854
 855  \small
 856
 857  \vspace*{0.3cm}
 858
 859 \begin{minipage}{8cm}
 860 Procedure:\\[0.3cm]
 861   \begin{pspicture}(0,0)(7,5)
 862   \rput(3.5,4){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
 863    \parbox{7cm}{
 864    \begin{itemize}
 865     \item Creation of c-Si simulation volume
 866     \item Periodic boundary conditions
 867     \item $T=0\text{ K}$, $p=0\text{ bar}$
 868    \end{itemize}
 869   }}}}
 870 \rput(3.5,2.1){\rnode{insert}{\psframebox{
 871  \parbox{7cm}{
 872   \begin{center}
 873   Insertion of interstitial C/Si atoms
 874   \end{center}
 875   }}}}
 876   \rput(3.5,1){\rnode{cool}{\psframebox[fillstyle=solid,fillcolor=lbb]{
 877    \parbox{7cm}{
 878    \begin{center}
 879    Relaxation / structural energy minimization
 880    \end{center}
 881   }}}}
 882   \ncline[]{->}{init}{insert}
 883   \ncline[]{->}{insert}{cool}
 884  \end{pspicture}
 885 \end{minipage}
 886 \begin{minipage}{5cm}
 887   \includegraphics[width=5cm]{unit_cell_e.eps}\\
 888 \end{minipage}
 889
 890 \begin{minipage}{9cm}
 891  \begin{tabular}{l c c}
 892  \hline
 893  & size [unit cells] & \# atoms\\
 894 \hline
 895 VASP & $3\times 3\times 3$ & $216\pm 1$ \\
 896 Erhart/Albe & $9\times 9\times 9$ & $5832\pm 1$\\
 897 \hline
 898  \end{tabular}
 899 \end{minipage}
 900 \begin{minipage}{4cm}
 901 {\color{red}$\bullet$} Tetrahedral\\
 902 {\color{green}$\bullet$} Hexagonal\\
 903 {\color{yellow}$\bullet$} \hkl<1 0 0> dumbbell\\
 904 {\color{magenta}$\bullet$} \hkl<1 1 0> dumbbell\\
 905 {\color{cyan}$\bullet$} Bond-centered\\
 906 {\color{black}$\bullet$} Vacancy / Substitutional
 907 \end{minipage}
 908
 909 \end{slide}
 910
 911 \begin{slide}
 912
 913  \footnotesize
 914
 915 \begin{minipage}{9.5cm}
 916
 917  {\large\bf
 918   Si self-interstitial point defects in silicon\\
 919  }
 920
 921 \begin{tabular}{l c c c c c}
 922 \hline
 923  $E_{\text{f}}$ [eV] & \hkl<1 1 0> DB & H & T & \hkl<1 0 0> DB & V \\
 924 \hline
 925  VASP & \underline{3.39} & 3.42 & 3.77 & 4.41 & 3.63 \\
 926  Erhart/Albe & 4.39 & 4.48$^*$ & \underline{3.40} & 5.42 & 3.13 \\
 927 \hline
 928 \end{tabular}\\[0.2cm]
 929
 930 \begin{minipage}{4.7cm}
 931 \includegraphics[width=4.7cm]{e_kin_si_hex.ps}
 932 \end{minipage}
 933 \begin{minipage}{4.7cm}
 934 \begin{center}
 935 {\tiny nearly T $\rightarrow$ T}\\
 936 \end{center}
 937 \includegraphics[width=4.7cm]{nhex_tet.ps}
 938 \end{minipage}\\
 939
 940 \underline{Hexagonal} \hspace{2pt}
 941 \href{../video/si_self_int_hexa.avi}{$\rhd$}\\[0.1cm]
 942 \framebox{
 943 \begin{minipage}{2.7cm}
 944 $E_{\text{f}}^*=4.48\text{ eV}$\\
 945 \includegraphics[width=2.7cm]{si_pd_albe/hex_a.eps}
 946 \end{minipage}
 947 \begin{minipage}{0.4cm}
 948 \begin{center}
 949 $\Rightarrow$
 950 \end{center}
 951 \end{minipage}
 952 \begin{minipage}{2.7cm}
 953 $E_{\text{f}}=3.96\text{ eV}$\\
 954 \includegraphics[width=2.8cm]{si_pd_albe/hex.eps}
 955 \end{minipage}
 956 }
 957 \begin{minipage}{2.9cm}
 958 \begin{flushright}
 959 \underline{Vacancy}\\
 960 \includegraphics[width=3.0cm]{si_pd_albe/vac.eps}
 961 \end{flushright}
 962 \end{minipage}
 963
 964 \end{minipage}
 965 \begin{minipage}{3.5cm}
 966
 967 \begin{flushright}
 968 \underline{\hkl<1 1 0> dumbbell}\\
 969 \includegraphics[width=3.0cm]{si_pd_albe/110.eps}\\
 970 \underline{Tetrahedral}\\
 971 \includegraphics[width=3.0cm]{si_pd_albe/tet.eps}\\
 972 \underline{\hkl<1 0 0> dumbbell}\\
 973 \includegraphics[width=3.0cm]{si_pd_albe/100.eps}
 974 \end{flushright}
 975
 976 \end{minipage}
 977
 978 \end{slide}
 979
 980 \begin{slide}
 981
 982 \footnotesize
 983
 984  {\large\bf
 985   C interstitial point defects in silicon\\[-0.1cm]
 986  }
 987
 988 \begin{tabular}{l c c c c c c r}
 989 \hline
 990  $E_{\text{f}}$ & T & H & \hkl<1 0 0> DB & \hkl<1 1 0> DB & S & B & \cs{} \& \si\\
 991 \hline
 992  VASP & unstable & unstable & \underline{3.72} & 4.16 & 1.95 & 4.66 & {\color{green}4.17}\\
 993  Erhart/Albe MD & 6.09 & 9.05$^*$ & \underline{3.88} & 5.18 & {\color{red}0.75} & 5.59$^*$ & {\color{green}4.43} \\
 994 \hline
 995 \end{tabular}\\[0.1cm]
 996
 997 \framebox{
 998 \begin{minipage}{2.7cm}
 999 \underline{Hexagonal} \hspace{2pt}
1000 \href{../video/c_in_si_int_hexa.avi}{$\rhd$}\\
1001 $E_{\text{f}}^*=9.05\text{ eV}$\\
1002 \includegraphics[width=2.7cm]{c_pd_albe/hex.eps}
1003 \end{minipage}
1004 \begin{minipage}{0.4cm}
1005 \begin{center}
1006 $\Rightarrow$
1007 \end{center}
1008 \end{minipage}
1009 \begin{minipage}{2.7cm}
1010 \underline{\hkl<1 0 0>}\\
1011 $E_{\text{f}}=3.88\text{ eV}$\\
1012 \includegraphics[width=2.7cm]{c_pd_albe/100.eps}
1013 \end{minipage}
1014 }
1015 \begin{minipage}{2cm}
1016 \hfill
1017 \end{minipage}
1018 \begin{minipage}{3cm}
1019 \begin{flushright}
1020 \underline{Tetrahedral}\\
1021 \includegraphics[width=3.0cm]{c_pd_albe/tet.eps}
1022 \end{flushright}
1023 \end{minipage}
1024
1025 \framebox{
1026 \begin{minipage}{2.7cm}
1027 \underline{Bond-centered}\\
1028 $E_{\text{f}}^*=5.59\text{ eV}$\\
1029 \includegraphics[width=2.7cm]{c_pd_albe/bc.eps}
1030 \end{minipage}
1031 \begin{minipage}{0.4cm}
1032 \begin{center}
1033 $\Rightarrow$
1034 \end{center}
1035 \end{minipage}
1036 \begin{minipage}{2.7cm}
1037 \underline{\hkl<1 1 0> dumbbell}\\
1038 $E_{\text{f}}=5.18\text{ eV}$\\
1039 \includegraphics[width=2.7cm]{c_pd_albe/110.eps}
1040 \end{minipage}
1041 }
1042 \begin{minipage}{2cm}
1043 \hfill
1044 \end{minipage}
1045 \begin{minipage}{3cm}
1046 \begin{flushright}
1047 \underline{Substitutional}\\
1048 \includegraphics[width=3.0cm]{c_pd_albe/sub.eps}
1049 \end{flushright}
1050 \end{minipage}
1051
1052 \end{slide}
1053
1054 \begin{slide}
1055
1056 \footnotesize
1057
1058  {\large\bf\boldmath
1059   C \hkl<1 0 0> dumbbell interstitial configuration\\
1060  }
1061
1062 {\tiny
1063 \begin{tabular}{l c c c c c c c c}
1064 \hline
1065  Distances [nm] & $r(1C)$ & $r(2C)$ & $r(3C)$ & $r(12)$ & $r(13)$ & $r(34)$ & $r(23)$ & $r(25)$ \\
1066 \hline
1067 Erhart/Albe & 0.175 & 0.329 & 0.186 & 0.226 & 0.300 & 0.343 & 0.423 & 0.425 \\
1068 VASP & 0.174 & 0.341 & 0.182 & 0.229 & 0.286 & 0.347 & 0.422 & 0.417 \\
1069 \hline
1070 \end{tabular}\\[0.2cm]
1071 \begin{tabular}{l c c c c }
1072 \hline
1073  Angles [$^{\circ}$] & $\theta_1$ & $\theta_2$ & $\theta_3$ & $\theta_4$ \\
1074 \hline
1075 Erhart/Albe & 140.2 & 109.9 & 134.4 & 112.8 \\
1076 VASP & 130.7 & 114.4 & 146.0 & 107.0 \\
1077 \hline
1078 \end{tabular}\\[0.2cm]
1079 \begin{tabular}{l c c c}
1080 \hline
1081  Displacements [nm]& $a$ & $b$ & $|a|+|b|$ \\
1082 \hline
1083 Erhart/Albe & 0.084 & -0.091 & 0.175 \\
1084 VASP & 0.109 & -0.065 & 0.174 \\
1085 \hline
1086 \end{tabular}\\[0.6cm]
1087 }
1088
1089 \begin{minipage}{3.0cm}
1090 \begin{center}
1091 \underline{Erhart/Albe}
1092 \includegraphics[width=3.0cm]{c_pd_albe/100_cmp.eps}
1093 \end{center}
1094 \end{minipage}
1095 \begin{minipage}{3.0cm}
1096 \begin{center}
1097 \underline{VASP}
1098 \includegraphics[width=3.0cm]{c_pd_vasp/100_cmp.eps}
1099 \end{center}
1100 \end{minipage}\\
1101
1102 \begin{picture}(0,0)(-185,10)
1103 \includegraphics[width=6.8cm]{100-c-si-db_cmp.eps}
1104 \end{picture}
1105 \begin{picture}(0,0)(-280,-150)
1106 \includegraphics[width=3.3cm]{c_pd_vasp/eden.eps}
1107 \end{picture}
1108
1109 \begin{pspicture}(0,0)(0,0)
1110 \psellipse[linecolor=green](5.18,5.92)(0.5,0.3)
1111 \psellipse[linecolor=red](3.45,5.92)(1.0,0.4)
1112 \psellipse[linecolor=blue](2.7,6.92)(0.9,0.2)
1113 \psellipse[linecolor=blue](4.65,6.92)(0.9,0.2)
1114 \end{pspicture}
1115
1116 \end{slide}
1117
1118 \begin{slide}
1119
1120 \small
1121
1122 \begin{minipage}{8.5cm}
1123
1124  {\large\bf
1125   Bond-centered interstitial configuration\\[-0.1cm]
1126  }
1127
1128 \begin{minipage}{3.0cm}
1129 \includegraphics[width=2.8cm]{c_pd_vasp/bc_2333.eps}\\
1130 \end{minipage}
1131 \begin{minipage}{5.2cm}
1132 \begin{itemize}
1133  \item Linear Si-C-Si bond
1134  \item Si: one C \& 3 Si neighbours
1135  \item Spin polarized calculations
1136  \item No saddle point!\\
1137        Real local minimum!
1138 \end{itemize}
1139 \end{minipage}
1140
1141 \framebox{
1142  \tiny
1143  \begin{minipage}[t]{6.5cm}
1144   \begin{minipage}[t]{1.2cm}
1145   {\color{red}Si}\\
1146   {\tiny sp$^3$}\\[0.8cm]
1147   \underline{${\color{black}\uparrow}$}
1148   \underline{${\color{black}\uparrow}$}
1149   \underline{${\color{black}\uparrow}$}
1150   \underline{${\color{red}\uparrow}$}\\
1151   sp$^3$
1152   \end{minipage}
1153   \begin{minipage}[t]{1.4cm}
1154   \begin{center}
1155   {\color{red}M}{\color{blue}O}\\[0.8cm]
1156   \underline{${\color{blue}\uparrow}{\color{white}\downarrow}$}\\
1157   $\sigma_{\text{ab}}$\\[0.5cm]
1158   \underline{${\color{red}\uparrow}{\color{blue}\downarrow}$}\\
1159   $\sigma_{\text{b}}$
1160   \end{center}
1161   \end{minipage}
1162   \begin{minipage}[t]{1.0cm}
1163   \begin{center}
1164   {\color{blue}C}\\
1165   {\tiny sp}\\[0.2cm]
1166   \underline{${\color{white}\uparrow\uparrow}$}
1167   \underline{${\color{white}\uparrow\uparrow}$}\\
1168   2p\\[0.4cm]
1169   \underline{${\color{blue}\uparrow}{\color{blue}\downarrow}$}
1170   \underline{${\color{blue}\uparrow}{\color{blue}\downarrow}$}\\
1171   sp
1172   \end{center}
1173   \end{minipage}
1174   \begin{minipage}[t]{1.4cm}
1175   \begin{center}
1176   {\color{blue}M}{\color{green}O}\\[0.8cm]
1177   \underline{${\color{blue}\uparrow}{\color{white}\downarrow}$}\\
1178   $\sigma_{\text{ab}}$\\[0.5cm]
1179   \underline{${\color{green}\uparrow}{\color{blue}\downarrow}$}\\
1180   $\sigma_{\text{b}}$
1181   \end{center}
1182   \end{minipage}
1183   \begin{minipage}[t]{1.2cm}
1184   \begin{flushright}
1185   {\color{green}Si}\\
1186   {\tiny sp$^3$}\\[0.8cm]
1187   \underline{${\color{green}\uparrow}$}
1188   \underline{${\color{black}\uparrow}$}
1189   \underline{${\color{black}\uparrow}$}
1190   \underline{${\color{black}\uparrow}$}\\
1191   sp$^3$
1192   \end{flushright}
1193   \end{minipage}
1194  \end{minipage}
1195 }\\[0.1cm]
1196
1197 \framebox{
1198 \begin{minipage}{4.5cm}
1199 \includegraphics[width=4cm]{c_100_mig_vasp/im_spin_diff.eps}
1200 \end{minipage}
1201 \begin{minipage}{3.5cm}
1202 {\color{gray}$\bullet$} Spin up\\
1203 {\color{green}$\bullet$} Spin down\\
1204 {\color{blue}$\bullet$} Resulting spin up\\
1205 {\color{yellow}$\bullet$} Si atoms\\
1206 {\color{red}$\bullet$} C atom
1207 \end{minipage}
1208 }
1209
1210 \end{minipage}
1211 \begin{minipage}{4.2cm}
1212 \begin{flushright}
1213 \includegraphics[width=4.3cm]{c_pd_vasp/bc_2333_ksl.ps}\\
1214 {\color{green}$\Box$} {\tiny unoccupied}\\
1215 {\color{red}$\bullet$} {\tiny occupied}
1216 \end{flushright}
1217 \end{minipage}
1218
1219 \end{slide}
1220
1221 \begin{slide}
1222
1223  {\large\bf\boldmath
1224   Migration of the C \hkl<1 0 0> dumbbell interstitial
1225  }
1226
1227 \scriptsize
1228
1229  {\small Investigated pathways}
1230
1231 \begin{minipage}{8.5cm}
1232 \begin{minipage}{8.3cm}
1233 \underline{\hkl<0 0 -1> $\rightarrow$ \hkl<0 0 1>}\\
1234 \begin{minipage}{2.4cm}
1235 \includegraphics[width=2.4cm]{c_pd_vasp/100_2333.eps}
1236 \end{minipage}
1237 \begin{minipage}{0.4cm}
1238 $\rightarrow$
1239 \end{minipage}
1240 \begin{minipage}{2.4cm}
1241 \includegraphics[width=2.4cm]{c_pd_vasp/bc_2333.eps}
1242 \end{minipage}
1243 \begin{minipage}{0.4cm}
1244 $\rightarrow$
1245 \end{minipage}
1246 \begin{minipage}{2.4cm}
1247 \includegraphics[width=2.4cm]{c_pd_vasp/100_next_2333.eps}
1248 \end{minipage}
1249 \end{minipage}\\
1250 \begin{minipage}{8.3cm}
1251 \underline{\hkl<0 0 -1> $\rightarrow$ \hkl<0 -1 0>}\\
1252 \begin{minipage}{2.4cm}
1253 \includegraphics[width=2.4cm]{c_pd_vasp/100_2333.eps}
1254 \end{minipage}
1255 \begin{minipage}{0.4cm}
1256 $\rightarrow$
1257 \end{minipage}
1258 \begin{minipage}{2.4cm}
1259 \includegraphics[width=2.4cm]{c_pd_vasp/00-1-0-10_2333.eps}
1260 \end{minipage}
1261 \begin{minipage}{0.4cm}
1262 $\rightarrow$
1263 \end{minipage}
1264 \begin{minipage}{2.4cm}
1265 \includegraphics[width=2.4cm]{c_pd_vasp/0-10_2333.eps}
1266 \end{minipage}
1267 \end{minipage}\\
1268 \begin{minipage}{8.3cm}
1269 \underline{\hkl<0 0 -1> $\rightarrow$ \hkl<0 -1 0> (in place)}\\
1270 \begin{minipage}{2.4cm}
1271 \includegraphics[width=2.4cm]{c_pd_vasp/100_2333.eps}
1272 \end{minipage}
1273 \begin{minipage}{0.4cm}
1274 $\rightarrow$
1275 \end{minipage}
1276 \begin{minipage}{2.4cm}
1277 \includegraphics[width=2.4cm]{c_pd_vasp/00-1_ip0-10_2333.eps}
1278 \end{minipage}
1279 \begin{minipage}{0.4cm}
1280 $\rightarrow$
1281 \end{minipage}
1282 \begin{minipage}{2.4cm}
1283 \includegraphics[width=2.4cm]{c_pd_vasp/0-10_ip_2333.eps}
1284 \end{minipage}
1285 \end{minipage}
1286 \end{minipage}
1287 \framebox{
1288 \begin{minipage}{4.2cm}
1289  {\small Constrained relaxation\\
1290          technique (CRT) method}\\
1291 \includegraphics[width=4cm]{crt_orig.eps}
1292 \begin{itemize}
1293  \item Constrain diffusing atom
1294  \item Static constraints
1295 \end{itemize}
1296 \vspace*{0.3cm}
1297  {\small Modifications}\\
1298 \includegraphics[width=4cm]{crt_mod.eps}
1299 \begin{itemize}
1300  \item Constrain all atoms
1301  \item Update individual\\
1302        constraints
1303 \end{itemize}
1304 \end{minipage}
1305 }
1306
1307 \end{slide}
1308
1309 \begin{slide}
1310
1311  {\large\bf\boldmath
1312   Migration of the C \hkl<1 0 0> dumbbell interstitial
1313  }
1314
1315 \scriptsize
1316
1317 \framebox{
1318 \begin{minipage}{5.9cm}
1319 \begin{flushleft}
1320 \includegraphics[width=5.8cm]{im_00-1_nosym_sp_fullct_thesis.ps}\\[0.45cm]
1321 \end{flushleft}
1322 \begin{center}
1323 \begin{picture}(0,0)(60,0)
1324 \includegraphics[width=1cm]{vasp_mig/00-1.eps}
1325 \end{picture}
1326 \begin{picture}(0,0)(-5,0)
1327 \includegraphics[width=1cm]{vasp_mig/bc_00-1_sp.eps}
1328 \end{picture}
1329 \begin{picture}(0,0)(-55,0)
1330 \includegraphics[width=1cm]{vasp_mig/bc.eps}
1331 \end{picture}
1332 \begin{picture}(0,0)(12.5,10)
1333 \includegraphics[width=1cm]{110_arrow.eps}
1334 \end{picture}
1335 \begin{picture}(0,0)(90,0)
1336 \includegraphics[height=0.9cm]{001_arrow.eps}
1337 \end{picture}
1338 \end{center}
1339 \vspace*{0.35cm}
1340 \end{minipage}
1341 }
1342 \begin{minipage}{0.3cm}
1343 \hfill
1344 \end{minipage}
1345 \framebox{
1346 \begin{minipage}{5.9cm}
1347 \begin{flushright}
1348 \includegraphics[width=5.9cm]{vasp_mig/00-1_0-10_nosym_sp_fullct.ps}\\[0.5cm]
1349 \end{flushright}
1350 \begin{center}
1351 \begin{picture}(0,0)(60,0)
1352 \includegraphics[width=1cm]{vasp_mig/00-1_a.eps}
1353 \end{picture}
1354 \begin{picture}(0,0)(5,0)
1355 \includegraphics[width=1cm]{vasp_mig/00-1_0-10_sp.eps}
1356 \end{picture}
1357 \begin{picture}(0,0)(-55,0)
1358 \includegraphics[width=1cm]{vasp_mig/0-10.eps}
1359 \end{picture}
1360 \begin{picture}(0,0)(12.5,10)
1361 \includegraphics[width=1cm]{100_arrow.eps}
1362 \end{picture}
1363 \begin{picture}(0,0)(90,0)
1364 \includegraphics[height=0.9cm]{001_arrow.eps}
1365 \end{picture}
1366 \end{center}
1367 \vspace*{0.3cm}
1368 \end{minipage}\\
1369 }
1370
1371 \vspace*{0.05cm}
1372
1373 \framebox{
1374 \begin{minipage}{5.9cm}
1375 \begin{flushleft}
1376 \includegraphics[width=5.9cm]{vasp_mig/00-1_ip0-10_nosym_sp_fullct.ps}\\[0.6cm]
1377 \end{flushleft}
1378 \begin{center}
1379 \begin{picture}(0,0)(60,0)
1380 \includegraphics[width=0.9cm]{vasp_mig/00-1_b.eps}
1381 \end{picture}
1382 \begin{picture}(0,0)(10,0)
1383 \includegraphics[width=0.9cm]{vasp_mig/00-1_ip0-10_sp.eps}
1384 \end{picture}
1385 \begin{picture}(0,0)(-60,0)
1386 \includegraphics[width=0.9cm]{vasp_mig/0-10_b.eps}
1387 \end{picture}
1388 \begin{picture}(0,0)(12.5,10)
1389 \includegraphics[width=1cm]{100_arrow.eps}
1390 \end{picture}
1391 \begin{picture}(0,0)(90,0)
1392 \includegraphics[height=0.9cm]{001_arrow.eps}
1393 \end{picture}
1394 \end{center}
1395 \vspace*{0.3cm}
1396 \end{minipage}
1397 }
1398 \begin{minipage}{0.3cm}
1399 \hfill
1400 \end{minipage}
1401 \begin{minipage}{6.5cm}
1402 VASP results
1403 \begin{itemize}
1404  \item Energetically most favorable path
1405        \begin{itemize}
1406         \item Path 2
1407         \item Activation energy: $\approx$ 0.9 eV
1408         \item Experimental values: 0.73 ... 0.87 eV
1409        \end{itemize}
1410        $\Rightarrow$ {\color{blue}Diffusion} path identified!
1411  \item Reorientation (path 3)
1412        \begin{itemize}
1413         \item More likely composed of two consecutive steps of type 2
1414         \item Experimental values: 0.77 ... 0.88 eV
1415        \end{itemize}
1416        $\Rightarrow$ {\color{blue}Reorientation} transition identified!
1417 \end{itemize}
1418 \end{minipage}
1419
1420 \end{slide}
1421
1422 \begin{slide}
1423
1424  {\large\bf\boldmath
1425   Migration of the C \hkl<1 0 0> dumbbell interstitial
1426  }
1427
1428 \scriptsize
1429
1430  \vspace{0.1cm}
1431
1432 \begin{minipage}{6.5cm}
1433
1434 \framebox{
1435 \begin{minipage}[t]{5.9cm}
1436 \begin{flushleft}
1437 \includegraphics[width=5.9cm]{bc_00-1.ps}\\[2.35cm]
1438 \end{flushleft}
1439 \begin{center}
1440 \begin{pspicture}(0,0)(0,0)
1441 \psframe[linecolor=red,fillstyle=none](-2.8,1.35)(3.3,2.7)
1442 \end{pspicture}
1443 \begin{picture}(0,0)(60,-50)
1444 \includegraphics[width=1cm]{albe_mig/bc_00-1_red_00.eps}
1445 \end{picture}
1446 \begin{picture}(0,0)(5,-50)
1447 \includegraphics[width=1cm]{albe_mig/bc_00-1_red_01.eps}
1448 \end{picture}
1449 \begin{picture}(0,0)(-55,-50)
1450 \includegraphics[width=1cm]{albe_mig/bc_00-1_red_02.eps}
1451 \end{picture}
1452 \begin{picture}(0,0)(12.5,-40)
1453 \includegraphics[width=1cm]{110_arrow.eps}
1454 \end{picture}
1455 \begin{picture}(0,0)(90,-45)
1456 \includegraphics[height=0.9cm]{001_arrow.eps}
1457 \end{picture}\\
1458 \begin{pspicture}(0,0)(0,0)
1459 \psframe[linecolor=blue,fillstyle=none](-2.8,0)(3.3,1.6)
1460 \end{pspicture}
1461 \begin{picture}(0,0)(60,-15)
1462 \includegraphics[width=0.9cm]{albe_mig/bc_00-1_01.eps}
1463 \end{picture}
1464 \begin{picture}(0,0)(35,-15)
1465 \includegraphics[width=0.9cm]{albe_mig/bc_00-1_02.eps}
1466 \end{picture}
1467 \begin{picture}(0,0)(-5,-15)
1468 \includegraphics[width=0.9cm]{albe_mig/bc_00-1_03.eps}
1469 \end{picture}
1470 \begin{picture}(0,0)(-55,-15)
1471 \includegraphics[width=0.9cm]{albe_mig/bc_00-1_04.eps}
1472 \end{picture}
1473 \begin{picture}(0,0)(12.5,-5)
1474 \includegraphics[width=1cm]{100_arrow.eps}
1475 \end{picture}
1476 \begin{picture}(0,0)(90,-15)
1477 \includegraphics[height=0.9cm]{010_arrow.eps}
1478 \end{picture}
1479 \end{center}
1480 \end{minipage}
1481 }\\[0.1cm]
1482
1483 \begin{minipage}{5.9cm}
1484 Erhart/Albe results
1485 \begin{itemize}
1486  \item Lowest activation energy: $\approx$ 2.2 eV
1487  \item 2.4 times higher than VASP
1488  \item Different pathway
1489 \end{itemize}
1490 \end{minipage}
1491
1492 \end{minipage}
1493 \begin{minipage}{6.5cm}
1494
1495 \framebox{
1496 \begin{minipage}{5.9cm}
1497 %\begin{flushright}
1498 %\includegraphics[width=5.9cm]{00-1_0-10.ps}\\[0.75cm]
1499 %\end{flushright}
1500 %\begin{center}
1501 %\begin{pspicture}(0,0)(0,0)
1502 %\psframe[linecolor=red,fillstyle=none](-2.8,-0.25)(3.3,1.1)
1503 %\end{pspicture}
1504 %\begin{picture}(0,0)(60,-5)
1505 %\includegraphics[width=0.9cm]{albe_mig/00-1_0-10_red_00.eps}
1506 %\end{picture}
1507 %\begin{picture}(0,0)(0,-5)
1508 %\includegraphics[width=0.9cm]{albe_mig/00-1_0-10_red_min.eps}
1509 %\end{picture}
1510 %\begin{picture}(0,0)(-55,-5)
1511 %\includegraphics[width=0.9cm]{albe_mig/00-1_0-10_red_03.eps}
1512 %\end{picture}
1513 %\begin{picture}(0,0)(12.5,5)
1514 %\includegraphics[width=1cm]{100_arrow.eps}
1515 %\end{picture}
1516 %\begin{picture}(0,0)(90,0)
1517 %\includegraphics[height=0.9cm]{001_arrow.eps}
1518 %\end{picture}
1519 %\end{center}
1520 %\vspace{0.2cm}
1521 %\end{minipage}
1522 %}\\[0.2cm]
1523 %
1524 %\framebox{
1525 %\begin{minipage}{5.9cm}
1526 \includegraphics[width=5.9cm]{00-1_110_0-10_mig_albe.ps}
1527 \end{minipage}
1528 }\\[0.1cm]
1529
1530 \begin{minipage}{5.9cm}
1531 Transition involving \ci{} \hkl<1 1 0>
1532 \begin{itemize}
1533  \item Bond-centered configuration unstable\\
1534        $\rightarrow$ \ci{} \hkl<1 1 0> dumbbell
1535  \item Transition minima of path 2 \& 3\\
1536        $\rightarrow$ \ci{} \hkl<1 1 0> dumbbell
1537  \item Activation energy: $\approx$ 2.2 eV \& 0.9 eV
1538  \item 2.4 - 3.4 times higher than VASP
1539  \item Rotation of dumbbell orientation
1540 \end{itemize}
1541 \vspace{0.1cm}
1542 \begin{center}
1543 {\color{blue}Overestimated diffusion barrier}
1544 \end{center}
1545 \end{minipage}
1546
1547 \end{minipage}
1548
1549 \end{slide}
1550
1551 \begin{slide}
1552
1553  {\large\bf\boldmath
1554   Combinations with a C-Si \hkl<1 0 0>-type interstitial
1555  }
1556
1557 \small
1558
1559 \vspace*{0.1cm}
1560
1561 Binding energy:
1562 $
1563 E_{\text{b}}=
1564 E_{\text{f}}^{\text{defect combination}}-
1565 E_{\text{f}}^{\text{C \hkl<0 0 -1> dumbbell}}-
1566 E_{\text{f}}^{\text{2nd defect}}
1567 $
1568
1569 \vspace*{0.1cm}
1570
1571 {\scriptsize
1572 \begin{tabular}{l c c c c c c}
1573 \hline
1574  $E_{\text{b}}$ [eV] & 1 & 2 & 3 & 4 & 5 & R\\
1575  \hline
1576  \hkl<0 0 -1> & {\color{red}-0.08} & -1.15 & {\color{red}-0.08} & 0.04 & -1.66 & -0.19\\
1577  \hkl<0 0 1> & 0.34 & 0.004 & -2.05 & 0.26 & -1.53 & -0.19\\
1578  \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}\\
1579  \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}\\
1580  \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}\\
1581  \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}\\
1582  \hline
1583  C substitutional (C$_{\text{S}}$) & 0.26 & -0.51 & -0.93 & -0.15 & 0.49 & -0.05\\
1584  Vacancy & -5.39 ($\rightarrow$ C$_{\text{S}}$) & -0.59 & -3.14 & -0.54 & -0.50 & -0.31\\
1585 \hline
1586 \end{tabular}
1587 }
1588
1589 \vspace*{0.3cm}
1590
1591 \footnotesize
1592
1593 \begin{minipage}[t]{3.8cm}
1594 \underline{\hkl<1 0 0> at position 1}\\[0.1cm]
1595 \includegraphics[width=3.5cm]{00-1dc/2-25.eps}
1596 \end{minipage}
1597 \begin{minipage}[t]{3.5cm}
1598 \underline{\hkl<0 -1 0> at position 1}\\[0.1cm]
1599 \includegraphics[width=3.2cm]{00-1dc/2-39.eps}
1600 \end{minipage}
1601 \begin{minipage}[t]{5.5cm}
1602 \begin{itemize}
1603  \item $E_{\text{b}}=0$ $\Leftrightarrow$ non-interacting defects\\
1604        $E_{\text{b}} \rightarrow 0$ for increasing distance (R)
1605  \item Stress compensation / increase
1606  \item Unfavored: antiparallel orientations
1607  \item Indication of energetically favored\\
1608        agglomeration
1609  \item Most favorable: C clustering
1610  \item However: High barrier ($>4\,\text{eV}$)
1611  \item $4\times{\color{cyan}-2.25}$ versus $2\times{\color{orange}-2.39}$
1612        (Entropy)
1613 \end{itemize}
1614 \end{minipage}
1615
1616 \begin{picture}(0,0)(-295,-130)
1617 \includegraphics[width=3.5cm]{comb_pos.eps}
1618 \end{picture}
1619
1620 \end{slide}
1621
1622 \begin{slide}
1623
1624  {\large\bf\boldmath
1625   Combinations of C-Si \hkl<1 0 0>-type interstitials
1626  }
1627
1628 \small
1629
1630 \vspace*{0.1cm}
1631
1632 Energetically most favorable combinations along \hkl<1 1 0>
1633
1634 \vspace*{0.1cm}
1635
1636 {\scriptsize
1637 \begin{tabular}{l c c c c c c}
1638 \hline
1639  & 1 & 2 & 3 & 4 & 5 & 6\\
1640 \hline
1641 $E_{\text{b}}$ [eV] & -2.39 & -1.88 & -0.59 & -0.31 & -0.24 & -0.21 \\
1642 C-C distance [\AA] & 1.4 & 4.6 & 6.5 & 8.6 & 10.5 & 10.8 \\
1643 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>\\
1644 \hline
1645 \end{tabular}
1646 }
1647
1648 \vspace*{0.3cm}
1649
1650 \begin{minipage}{7.0cm}
1651 \includegraphics[width=7cm]{db_along_110_cc.ps}
1652 \end{minipage}
1653 \begin{minipage}{6.0cm}
1654 \begin{itemize}
1655  \item Interaction proportional to reciprocal cube of C-C distance
1656  \item Saturation in the immediate vicinity
1657  \renewcommand\labelitemi{$\Rightarrow$}
1658  \item Agglomeration of \ci{} expected
1659  \item Absence of C clustering
1660 \end{itemize}
1661 \begin{center}
1662 {\color{blue}
1663  Consisten with initial precipitation model
1664 }
1665 \end{center}
1666 \end{minipage}
1667
1668 \vspace{0.2cm}
1669
1670 \end{slide}
1671
1672 \begin{slide}
1673
1674  {\large\bf\boldmath
1675   Combinations of substitutional C and \hkl<1 1 0> Si self-interstitials
1676  }
1677
1678  \scriptsize
1679
1680 %\begin{center}
1681 %\begin{minipage}{3.2cm}
1682 %\includegraphics[width=3cm]{sub_110_combo.eps}
1683 %\end{minipage}
1684 %\begin{minipage}{7.8cm}
1685 %\begin{tabular}{l c c c c c c}
1686 %\hline
1687 %C$_{\text{sub}}$ & \hkl<1 1 0> & \hkl<-1 1 0> & \hkl<0 1 1> & \hkl<0 -1 1> &
1688 %                   \hkl<1 0 1> & \hkl<-1 0 1> \\
1689 %\hline
1690 %1 & \RM{1} & \RM{3} & \RM{3} & \RM{1} & \RM{3} & \RM{1} \\
1691 %2 & \RM{2} & A & A & \RM{2} & C & \RM{5} \\
1692 %3 & \RM{3} & \RM{1} & \RM{3} & \RM{1} & \RM{1} & \RM{3} \\
1693 %4 & \RM{4} & B & D & E & E & D \\
1694 %5 & \RM{5} & C & A & \RM{2} & A & \RM{2} \\
1695 %\hline
1696 %\end{tabular}
1697 %\end{minipage}
1698 %\end{center}
1699
1700 %\begin{center}
1701 %\begin{tabular}{l c c c c c c c c c c}
1702 %\hline
1703 %Conf & \RM{1} & \RM{2} & \RM{3} & \RM{4} & \RM{5} & A & B & C & D & E \\
1704 %\hline
1705 %$E_{\text{f}}$ [eV]& 4.37 & 5.26 & 5.57 & 5.37 & 5.12 & 5.10 & 5.32 & 5.28 & 5.39 & 5.32 \\
1706 %$E_{\text{b}}$ [eV] & -0.97 & -0.08 & 0.22 & -0.02 & -0.23 & -0.25 & -0.02 & -0.06 & 0.05 & -0.03 \\
1707 %$r$ [nm] & 0.292 & 0.394 & 0.241 & 0.453 & 0.407 & 0.408 & 0.452 & 0.392 & 0.456 & 0.453\\
1708 %\hline
1709 %\end{tabular}
1710 %\end{center}
1711
1712 \begin{minipage}{6.0cm}
1713 \includegraphics[width=5.8cm]{c_sub_si110.ps}
1714 \end{minipage}
1715 \begin{minipage}{7cm}
1716 \scriptsize
1717 \begin{itemize}
1718  \item IBS: C may displace Si\\
1719        $\Rightarrow$ C$_{\text{sub}}$ + \hkl<1 1 0> Si self-interstitial
1720  \item Assumption:\\
1721        \hkl<1 1 0>-type $\rightarrow$ favored combination
1722  \renewcommand\labelitemi{$\Rightarrow$}
1723  \item Most favorable: \cs{} along \hkl<1 1 0> chain \si{}
1724  \item Less favorable than C-Si \hkl<1 0 0> dumbbell
1725  \item Interaction drops quickly to zero\\
1726        $\rightarrow$ low capture radius
1727 \end{itemize}
1728 \begin{center}
1729  {\color{blue}
1730  IBS process far from equilibrium\\
1731  \cs{} \& \si{} instead of thermodynamic ground state
1732  }
1733 \end{center}
1734 \end{minipage}
1735
1736 \begin{minipage}{6.5cm}
1737 \includegraphics[width=6.0cm]{162-097.ps}
1738 \begin{itemize}
1739  \item Low migration barrier
1740 \end{itemize}
1741 \end{minipage}
1742 \begin{minipage}{6.5cm}
1743 \begin{center}
1744 Ab initio MD at \degc{900}\\
1745 \includegraphics[width=3.3cm]{md_vasp_01.eps}
1746 $t=\unit[2230]{fs}$\\
1747 \includegraphics[width=3.3cm]{md_vasp_02.eps}
1748 $t=\unit[2900]{fs}$
1749 \end{center}
1750 {\color{blue}
1751 Contribution of entropy to structural formation
1752 }
1753 \end{minipage}
1754
1755 \end{slide}
1756
1757 \begin{slide}
1758
1759  {\large\bf\boldmath
1760   Migration in C-Si \hkl<1 0 0> and vacancy combinations
1761  }
1762
1763  \footnotesize
1764
1765 \vspace{0.1cm}
1766
1767 \begin{minipage}[t]{3cm}
1768 \underline{Pos 2, $E_{\text{b}}=-0.59\text{ eV}$}\\
1769 \includegraphics[width=2.8cm]{00-1dc/0-59.eps}
1770 \end{minipage}
1771 \begin{minipage}[t]{7cm}
1772 \vspace{0.2cm}
1773 \begin{center}
1774  Low activation energies\\
1775  High activation energies for reverse processes\\
1776  $\Downarrow$\\
1777  {\color{blue}C$_{\text{sub}}$ very stable}\\
1778 \vspace*{0.1cm}
1779  \hrule
1780 \vspace*{0.1cm}
1781  Without nearby \hkl<1 1 0> Si self-interstitial (IBS)\\
1782  $\Downarrow$\\
1783  {\color{blue}Formation of SiC by successive substitution by C}
1784
1785 \end{center}
1786 \end{minipage}
1787 \begin{minipage}[t]{3cm}
1788 \underline{Pos 3, $E_{\text{b}}=-3.14\text{ eV}$}\\
1789 \includegraphics[width=2.8cm]{00-1dc/3-14.eps}
1790 \end{minipage}
1791
1792
1793 \framebox{
1794 \begin{minipage}{5.9cm}
1795 \includegraphics[width=5.9cm]{vasp_mig/comb_mig_3-2_vac_fullct.ps}\\[0.6cm]
1796 \begin{center}
1797 \begin{picture}(0,0)(70,0)
1798 \includegraphics[width=1.4cm]{vasp_mig/comb_2-1_init.eps}
1799 \end{picture}
1800 \begin{picture}(0,0)(30,0)
1801 \includegraphics[width=1.4cm]{vasp_mig/comb_2-1_seq_03.eps}
1802 \end{picture}
1803 \begin{picture}(0,0)(-10,0)
1804 \includegraphics[width=1.4cm]{vasp_mig/comb_2-1_seq_06.eps}
1805 \end{picture}
1806 \begin{picture}(0,0)(-48,0)
1807 \includegraphics[width=1.4cm]{vasp_mig/comb_2-1_final.eps}
1808 \end{picture}
1809 \begin{picture}(0,0)(12.5,5)
1810 \includegraphics[width=1cm]{100_arrow.eps}
1811 \end{picture}
1812 \begin{picture}(0,0)(97,-10)
1813 \includegraphics[height=0.9cm]{001_arrow.eps}
1814 \end{picture}
1815 \end{center}
1816 \vspace{0.1cm}
1817 \end{minipage}
1818 }
1819 \begin{minipage}{0.3cm}
1820 \hfill
1821 \end{minipage}
1822 \framebox{
1823 \begin{minipage}{5.9cm}
1824 \includegraphics[width=5.9cm]{vasp_mig/comb_mig_4-2_vac_fullct.ps}\\[0.1cm]
1825 \begin{center}
1826 \begin{picture}(0,0)(60,0)
1827 \includegraphics[width=0.9cm]{vasp_mig/comb_3-1_init.eps}
1828 \end{picture}
1829 \begin{picture}(0,0)(25,0)
1830 \includegraphics[width=0.9cm]{vasp_mig/comb_3-1_seq_03.eps}
1831 \end{picture}
1832 \begin{picture}(0,0)(-20,0)
1833 \includegraphics[width=0.9cm]{vasp_mig/comb_3-1_seq_07.eps}
1834 \end{picture}
1835 \begin{picture}(0,0)(-55,0)
1836 \includegraphics[width=0.9cm]{vasp_mig/comb_3-1_final.eps}
1837 \end{picture}
1838 \begin{picture}(0,0)(12.5,5)
1839 \includegraphics[width=1cm]{100_arrow.eps}
1840 \end{picture}
1841 \begin{picture}(0,0)(95,0)
1842 \includegraphics[height=0.9cm]{001_arrow.eps}
1843 \end{picture}
1844 \end{center}
1845 \vspace{0.1cm}
1846 \end{minipage}
1847 }
1848
1849 \end{slide}
1850
1851 \begin{slide}
1852
1853  {\large\bf
1854   Conclusion of defect / migration / combined defect simulations
1855  }
1856
1857  \footnotesize
1858
1859 \vspace*{0.1cm}
1860
1861 Defect structures
1862 \begin{itemize}
1863  \item Accurately described by quantum-mechanical simulations
1864  \item Less accurate description by classical potential simulations
1865  \item Underestimated formation energy of \cs{} by classical approach
1866  \item Both methods predict same ground state: \ci{} \hkl<1 0 0> dumbbell
1867 \end{itemize}
1868
1869 Migration
1870 \begin{itemize}
1871  \item C migration pathway in Si identified
1872  \item Consistent with reorientation and diffusion experiments
1873 \end{itemize}
1874 \begin{itemize}
1875  \item Different path and ...
1876  \item overestimated barrier by classical potential calculations
1877 \end{itemize}
1878
1879 Concerning the precipitation mechanism
1880 \begin{itemize}
1881  \item Agglomeration of C-Si dumbbells energetically favorable
1882        (stress compensation)
1883  \item C-Si indeed favored compared to
1884        C$_{\text{sub}}$ \& \hkl<1 1 0> Si self-interstitial
1885  \item Possible low interaction capture radius of
1886        C$_{\text{sub}}$ \& \hkl<1 1 0> Si self-interstitial
1887  \item Low barrier for
1888        \ci{} \hkl<1 0 0> $\rightarrow$ \cs{} \& \si{} \hkl<1 1 0>
1889  \item In absence of nearby \hkl<1 1 0> Si self-interstitial:
1890        C-Si \hkl<1 0 0> + Vacancy $\rightarrow$ C$_{\text{sub}}$ (SiC)
1891 \end{itemize}
1892 \begin{center}
1893 {\color{blue}Results suggest increased participation of \cs}
1894 \end{center}
1895
1896 \end{slide}
1897
1898 \begin{slide}
1899
1900  {\large\bf
1901   Silicon carbide precipitation simulations
1902  }
1903
1904  \small
1905
1906 {\scriptsize
1907  \begin{pspicture}(0,0)(12,6.5)
1908   % nodes
1909   \rput(3.5,5.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
1910    \parbox{7cm}{
1911    \begin{itemize}
1912     \item Create c-Si volume
1913     \item Periodc boundary conditions
1914     \item Set requested $T$ and $p=0\text{ bar}$
1915     \item Equilibration of $E_{\text{kin}}$ and $E_{\text{pot}}$
1916    \end{itemize}
1917   }}}}
1918   \rput(3.5,2.7){\rnode{insert}{\psframebox[fillstyle=solid,fillcolor=lachs]{
1919    \parbox{7cm}{
1920    Insertion of C atoms at constant T
1921    \begin{itemize}
1922     \item total simulation volume {\pnode{in1}}
1923     \item volume of minimal SiC precipitate {\pnode{in2}}
1924     \item volume consisting of Si atoms to form a minimal {\pnode{in3}}\\
1925           precipitate
1926    \end{itemize}
1927   }}}}
1928   \rput(3.5,1){\rnode{cool}{\psframebox[fillstyle=solid,fillcolor=lbb]{
1929    \parbox{7.0cm}{
1930    Run for 100 ps followed by cooling down to $20\, ^{\circ}\textrm{C}$
1931   }}}}
1932   \ncline[]{->}{init}{insert}
1933   \ncline[]{->}{insert}{cool}
1934   \psframe[fillstyle=solid,fillcolor=white](7.5,0.7)(13.5,6.3)
1935   \rput(7.8,6){\footnotesize $V_1$}
1936   \psframe[fillstyle=solid,fillcolor=lightgray](9,2)(12,5)
1937   \rput(9.2,4.85){\tiny $V_2$}
1938   \psframe[fillstyle=solid,fillcolor=gray](9.25,2.25)(11.75,4.75)
1939   \rput(9.55,4.45){\footnotesize $V_3$}
1940   \rput(7.9,3.2){\pnode{ins1}}
1941   \rput(9.22,2.8){\pnode{ins2}}
1942   \rput(11.0,2.4){\pnode{ins3}}
1943   \ncline[]{->}{in1}{ins1}
1944   \ncline[]{->}{in2}{ins2}
1945   \ncline[]{->}{in3}{ins3}
1946  \end{pspicture}
1947 }
1948
1949 \begin{itemize}
1950  \item Restricted to classical potential simulations
1951  \item $V_2$ and $V_3$ considered due to low diffusion
1952  \item Amount of C atoms: 6000
1953        ($r_{\text{prec}}\approx 3.1\text{ nm}$, IBS: 2 ... 4 nm)
1954  \item Simulation volume: $31\times 31\times 31$ unit cells
1955        (238328 Si atoms)
1956 \end{itemize}
1957
1958 \end{slide}
1959
1960 \begin{slide}
1961
1962  {\large\bf\boldmath
1963   Silicon carbide precipitation simulations at $450\,^{\circ}\mathrm{C}$ as in IBS
1964  }
1965
1966  \small
1967
1968 \begin{minipage}{6.5cm}
1969 \includegraphics[width=6.4cm]{sic_prec_450_si-si_c-c.ps}
1970 \end{minipage}
1971 \begin{minipage}{6.5cm}
1972 \includegraphics[width=6.4cm]{sic_prec_450_energy.ps}
1973 \end{minipage}
1974
1975 \begin{minipage}{6.5cm}
1976 \includegraphics[width=6.4cm]{sic_prec_450_si-c.ps}
1977 \end{minipage}
1978 \begin{minipage}{6.5cm}
1979 \scriptsize
1980 \underline{Low C concentration ($V_1$)}\\
1981 \hkl<1 0 0> C-Si dumbbell dominated structure
1982 \begin{itemize}
1983  \item Si-C bumbs around 0.19 nm
1984  \item C-C peak at 0.31 nm (as expected in 3C-SiC):\\
1985        concatenated dumbbells of various orientation
1986  \item Si-Si NN distance stretched to 0.3 nm
1987 \end{itemize}
1988 {\color{blue}$\Rightarrow$ C atoms in proper 3C-SiC distance first}\\
1989 \underline{High C concentration ($V_2$, $V_3$)}\\
1990 High amount of strongly bound C-C bonds\\
1991 Defect density $\uparrow$ $\Rightarrow$ considerable amount of damage\\
1992 Only short range order observable\\
1993 {\color{blue}$\Rightarrow$ amorphous SiC-like phase}
1994 \end{minipage}
1995
1996 \end{slide}
1997
1998 \begin{slide}
1999
2000  {\large\bf\boldmath
2001   Silicon carbide precipitation simulations at $450\,^{\circ}\mathrm{C}$ as in IBS
2002  }
2003
2004  \small
2005
2006 \begin{minipage}{6.5cm}
2007 \includegraphics[width=6.4cm]{sic_prec_450_si-si_c-c.ps}
2008 \end{minipage}
2009 \begin{minipage}{6.5cm}
2010 \includegraphics[width=6.4cm]{sic_prec_450_energy.ps}
2011 \end{minipage}
2012
2013 \begin{minipage}{6.5cm}
2014 \includegraphics[width=6.4cm]{sic_prec_450_si-c.ps}
2015 \end{minipage}
2016 \begin{minipage}{6.5cm}
2017 \scriptsize
2018 \underline{Low C concentration ($V_1$)}\\
2019 \hkl<1 0 0> C-Si dumbbell dominated structure
2020 \begin{itemize}
2021  \item Si-C bumbs around 0.19 nm
2022  \item C-C peak at 0.31 nm (as expected in 3C-SiC):\\
2023        concatenated dumbbells of various orientation
2024  \item Si-Si NN distance stretched to 0.3 nm
2025 \end{itemize}
2026 {\color{blue}$\Rightarrow$ C atoms in proper 3C-SiC distance first}\\
2027 \underline{High C concentration ($V_2$, $V_3$)}\\
2028 High amount of strongly bound C-C bonds\\
2029 Defect density $\uparrow$ $\Rightarrow$ considerable amount of damage\\
2030 Only short range order observable\\
2031 {\color{blue}$\Rightarrow$ amorphous SiC-like phase}
2032 \end{minipage}
2033
2034 \begin{pspicture}(0,0)(0,0)
2035 \rput(6.7,5.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=white]{
2036 \begin{minipage}{10cm}
2037 \small
2038 {\color{red}\bf 3C-SiC formation fails to appear}
2039 \begin{itemize}
2040 \item Low C concentration simulations
2041  \begin{itemize}
2042   \item Formation of \ci{} indeed occurs
2043   \item Agllomeration not observed
2044  \end{itemize}
2045 \item High C concentration simulations
2046  \begin{itemize}
2047   \item Amorphous SiC-like structure\\
2048         (not expected at prevailing temperatures)
2049   \item Rearrangement and transition into 3C-SiC structure missing
2050  \end{itemize}
2051 \end{itemize}
2052 \end{minipage}
2053  }}}
2054 \end{pspicture}
2055
2056 \end{slide}
2057
2058 \begin{slide}
2059
2060  {\large\bf
2061   Limitations of molecular dynamics and short range potentials
2062  }
2063
2064 \footnotesize
2065
2066 \vspace{0.2cm}
2067
2068 \underline{Time scale problem of MD}\\[0.2cm]
2069 Minimize integration error\\
2070 $\Rightarrow$ discretization considerably smaller than
2071               reciprocal of fastest vibrational mode\\[0.1cm]
2072 Order of fastest vibrational mode: $10^{13} - 10^{14}\text{ Hz}$\\
2073 $\Rightarrow$ suitable choice of time step:
2074               $\tau=1\text{ fs}=10^{-15}\text{ s}$\\
2075 $\Rightarrow$ {\color{red}\underline{slow}} phase space propagation\\[0.1cm]
2076 Several local minima in energy surface separated by large energy barriers\\
2077 $\Rightarrow$ transition event corresponds to a multiple
2078               of vibrational periods\\
2079 $\Rightarrow$ phase transition made up of {\color{red}\underline{many}}
2080               infrequent transition events\\[0.1cm]
2081 {\color{blue}Accelerated methods:}
2082 \underline{Temperature accelerated} MD (TAD), self-guided MD \ldots
2083
2084 \vspace{0.3cm}
2085
2086 \underline{Limitations related to the short range potential}\\[0.2cm]
2087 Cut-off function pushing forces and energies to zero between 1$^{\text{st}}$
2088 and 2$^{\text{nd}}$ next neighbours\\
2089 $\Rightarrow$ overestimated unphysical high forces of next neighbours
2090
2091 \vspace{0.3cm}
2092
2093 \framebox{
2094 \color{red}
2095 Potential enhanced problem of slow phase space propagation
2096 }
2097
2098 \vspace{0.3cm}
2099
2100 \underline{Approach to the (twofold) problem}\\[0.2cm]
2101 Increased temperature simulations without TAD corrections\\
2102 (accelerated methods or higher time scales exclusively not sufficient)
2103
2104 \begin{picture}(0,0)(-260,-30)
2105 \framebox{
2106 \begin{minipage}{4.2cm}
2107 \tiny
2108 \begin{center}
2109 \vspace{0.03cm}
2110 \underline{IBS}
2111 \end{center}
2112 \begin{itemize}
2113 \item 3C-SiC also observed for higher T
2114 \item higher T inside sample
2115 \item structural evolution vs.\\
2116       equilibrium properties
2117 \end{itemize}
2118 \end{minipage}
2119 }
2120 \end{picture}
2121
2122 \begin{picture}(0,0)(-305,-155)
2123 \framebox{
2124 \begin{minipage}{2.5cm}
2125 \tiny
2126 \begin{center}
2127 retain proper\\
2128 thermodynmic sampling
2129 \end{center}
2130 \end{minipage}
2131 }
2132 \end{picture}
2133
2134 \end{slide}
2135
2136 \begin{slide}
2137
2138  {\large\bf
2139   Increased temperature simulations at low C concentration
2140  }
2141
2142 \small
2143
2144 \begin{minipage}{6.5cm}
2145 \includegraphics[width=6.4cm]{tot_pc_thesis.ps}
2146 \end{minipage}
2147 \begin{minipage}{6.5cm}
2148 \includegraphics[width=6.4cm]{tot_pc3_thesis.ps}
2149 \end{minipage}
2150
2151 \begin{minipage}{6.5cm}
2152 \includegraphics[width=6.4cm]{tot_pc2_thesis.ps}
2153 \end{minipage}
2154 \begin{minipage}{6.5cm}
2155 \scriptsize
2156  \underline{Si-C bonds:}
2157  \begin{itemize}
2158   \item Vanishing cut-off artifact (above $1650\,^{\circ}\mathrm{C}$)
2159   \item Structural change: C-Si \hkl<1 0 0> $\rightarrow$ C$_{\text{sub}}$
2160  \end{itemize}
2161  \underline{Si-Si bonds:}
2162  {\color{blue}Si-C$_{\text{sub}}$-Si} along \hkl<1 1 0>
2163  ($\rightarrow$ 0.325 nm)\\[0.1cm]
2164  \underline{C-C bonds:}
2165  \begin{itemize}
2166   \item C-C next neighbour pairs reduced (mandatory)
2167   \item Peak at 0.3 nm slightly shifted
2168         \begin{itemize}
2169          \item C-Si \hkl<1 0 0> combinations (dashed arrows)\\
2170                $\rightarrow$ C-Si \hkl<1 0 0> \& C$_{\text{sub}}$
2171                combinations (|)\\
2172                $\rightarrow$ pure {\color{blue}C$_{\text{sub}}$ combinations}
2173                ($\downarrow$)
2174          \item Range [|-$\downarrow$]:
2175                {\color{blue}C$_{\text{sub}}$ \& C$_{\text{sub}}$
2176                with nearby Si$_{\text{I}}$}
2177         \end{itemize}
2178  \end{itemize}
2179 \end{minipage}
2180
2181 \begin{picture}(0,0)(-330,-74)
2182 \color{blue}
2183 \framebox{
2184 \begin{minipage}{1.6cm}
2185 \tiny
2186 \begin{center}
2187 stretched SiC\\[-0.1cm]
2188 in c-Si
2189 \end{center}
2190 \end{minipage}
2191 }
2192 \end{picture}
2193
2194 \end{slide}
2195
2196 \begin{slide}
2197
2198  {\large\bf
2199   Increased temperature simulations at low C concentration
2200  }
2201
2202 \small
2203
2204 \begin{minipage}{6.5cm}
2205 \includegraphics[width=6.4cm]{tot_pc_thesis.ps}
2206 \end{minipage}
2207 \begin{minipage}{6.5cm}
2208 \includegraphics[width=6.4cm]{tot_pc3_thesis.ps}
2209 \end{minipage}
2210
2211 \begin{minipage}{6.5cm}
2212 \includegraphics[width=6.4cm]{tot_pc2_thesis.ps}
2213 \end{minipage}
2214 \begin{minipage}{6.5cm}
2215 \scriptsize
2216  \underline{Si-C bonds:}
2217  \begin{itemize}
2218   \item Vanishing cut-off artifact (above $1650\,^{\circ}\mathrm{C}$)
2219   \item Structural change: C-Si \hkl<1 0 0> $\rightarrow$ C$_{\text{sub}}$
2220  \end{itemize}
2221  \underline{Si-Si bonds:}
2222  {\color{blue}Si-C$_{\text{sub}}$-Si} along \hkl<1 1 0>
2223  ($\rightarrow$ 0.325 nm)\\[0.1cm]
2224  \underline{C-C bonds:}
2225  \begin{itemize}
2226   \item C-C next neighbour pairs reduced (mandatory)
2227   \item Peak at 0.3 nm slightly shifted
2228         \begin{itemize}
2229          \item C-Si \hkl<1 0 0> combinations (dashed arrows)\\
2230                $\rightarrow$ C-Si \hkl<1 0 0> \& C$_{\text{sub}}$
2231                combinations (|)\\
2232                $\rightarrow$ pure {\color{blue}C$_{\text{sub}}$ combinations}
2233                ($\downarrow$)
2234          \item Range [|-$\downarrow$]:
2235                {\color{blue}C$_{\text{sub}}$ \& C$_{\text{sub}}$
2236                with nearby Si$_{\text{I}}$}
2237         \end{itemize}
2238  \end{itemize}
2239 \end{minipage}
2240
2241 %\begin{picture}(0,0)(-330,-74)
2242 %\color{blue}
2243 %\framebox{
2244 %\begin{minipage}{1.6cm}
2245 %\tiny
2246 %\begin{center}
2247 %stretched SiC\\[-0.1cm]
2248 %in c-Si
2249 %\end{center}
2250 %\end{minipage}
2251 %}
2252 %\end{picture}
2253
2254 \begin{pspicture}(0,0)(0,0)
2255 \rput(6.7,5.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=white]{
2256 \begin{minipage}{10cm}
2257 \small
2258 {\color{blue}\bf Stretched SiC in c-Si}
2259 \begin{itemize}
2260 \item Consistent to precipitation model involving \cs{}
2261 \item Explains annealing behavior of high/low T C implants
2262       \begin{itemize}
2263        \item Low T: highly mobiel \ci{}
2264        \item High T: stable configurations of \cs{}
2265       \end{itemize}
2266 \end{itemize}
2267 $\Rightarrow$ High T $\leftrightarrow$ IBS conditions far from equilibrium\\
2268 $\Rightarrow$ Precipitation mechanism involving \cs{}
2269 \end{minipage}
2270  }}}
2271 \end{pspicture}
2272
2273 \end{slide}
2274
2275 \begin{slide}
2276
2277  {\large\bf
2278   Increased temperature simulations at high C concentration
2279  }
2280
2281 \footnotesize
2282
2283 \begin{minipage}{6.5cm}
2284 \includegraphics[width=6.4cm]{12_pc_thesis.ps}
2285 \end{minipage}
2286 \begin{minipage}{6.5cm}
2287 \includegraphics[width=6.4cm]{12_pc_c_thesis.ps}
2288 \end{minipage}
2289
2290 \vspace{0.1cm}
2291
2292 \scriptsize
2293
2294 \framebox{
2295 \begin{minipage}[t]{6.0cm}
2296 0.186 nm: Si-C pairs $\uparrow$\\
2297 (as expected in 3C-SiC)\\[0.2cm]
2298 0.282 nm: Si-C-C\\[0.2cm]
2299 $\approx$0.35 nm: C-Si-Si
2300 \end{minipage}
2301 }
2302 \begin{minipage}{0.2cm}
2303 \hfill
2304 \end{minipage}
2305 \framebox{
2306 \begin{minipage}[t]{6.0cm}
2307 0.15 nm: C-C pairs $\uparrow$\\
2308 (as expected in graphite/diamond)\\[0.2cm]
2309 0.252 nm: C-C-C (2$^{\text{nd}}$ NN for diamond)\\[0.2cm]
2310 0.31 nm: shifted towards 0.317 nm $\rightarrow$ C-Si-C
2311 \end{minipage}
2312 }
2313
2314 \begin{itemize}
2315 \item Decreasing cut-off artifact
2316 \item {\color{red}Amorphous} SiC-like phase remains
2317 \item High amount of {\color{red}damage} \& alignement to c-Si host matrix lost
2318 \item Slightly sharper peaks $\Rightarrow$ indicate slight {\color{blue}acceleration of dynamics} due to temperature
2319 \end{itemize}
2320
2321 \vspace{-0.1cm}
2322
2323 \begin{center}
2324 {\color{blue}
2325 \framebox{
2326 {\color{black}
2327 High C \& small $V$ \& short $t$
2328 $\Rightarrow$
2329 }
2330 Slow restructuring due to strong C-C bonds
2331 {\color{black}
2332 $\Leftarrow$
2333 High C \& low T implants
2334 }
2335 }
2336 }
2337 \end{center}
2338
2339 \end{slide}
2340
2341 \begin{slide}
2342
2343  {\large\bf
2344   Summary and Conclusions
2345  }
2346
2347  \scriptsize
2348
2349 %\vspace{0.1cm}
2350
2351 \framebox{
2352 \begin{minipage}[t]{12.9cm}
2353  \underline{Pecipitation simulations}
2354  \begin{itemize}
2355   \item High C concentration $\rightarrow$ amorphous SiC like phase
2356   \item Problem of potential enhanced slow phase space propagation
2357   \item Low T $\rightarrow$ C-Si \hkl<1 0 0> dumbbell dominated structure
2358   \item High T $\rightarrow$ C$_{\text{sub}}$ dominated structure
2359   \item High T necessary to simulate IBS conditions (far from equilibrium)
2360   \item Precipitation by successive agglomeration of \cs (epitaxy)
2361   \item \si{}: vehicle to form \cs{} \& supply of Si \& stress compensation
2362         (stretched SiC, interface)
2363  \end{itemize}
2364 \end{minipage}
2365 }
2366
2367 %\vspace{0.1cm}
2368
2369 \framebox{
2370 \begin{minipage}{12.9cm}
2371  \underline{Defects}
2372  \begin{itemize}
2373    \item DFT / EA
2374         \begin{itemize}
2375          \item Point defects excellently / fairly well described
2376                by DFT / EA
2377          \item C$_{\text{sub}}$ drastically underestimated by EA
2378          \item EA predicts correct ground state:
2379                C$_{\text{sub}}$ \& \si{} $>$ \ci{}
2380          \item Identified migration path explaining
2381                diffusion and reorientation experiments by DFT
2382          \item EA fails to describe \ci{} migration:
2383                Wrong path \& overestimated barrier
2384         \end{itemize}
2385    \item Combinations of defects
2386          \begin{itemize}
2387           \item Agglomeration of point defects energetically favorable
2388                 by compensation of stress
2389           \item Formation of C-C unlikely
2390           \item C$_{\text{sub}}$ favored conditions (conceivable in IBS)
2391           \item \ci{} \hkl<1 0 0> $\leftrightarrow$ \cs{} \& \si{} \hkl<1 1 0>\\
2392                 Low barrier (\unit[0.77]{eV}) \& low capture radius
2393         \end{itemize}
2394  \end{itemize}
2395 \end{minipage}
2396 }
2397
2398 \begin{center}
2399 {\color{blue}
2400 \framebox{Precipitation by successive agglomeration of \cs{}}
2401 }
2402 \end{center}
2403
2404 \end{slide}
2405
2406 \begin{slide}
2407
2408  {\large\bf
2409   Acknowledgements
2410  }
2411
2412  \vspace{0.1cm}
2413
2414  \small
2415
2416  Thanks to \ldots
2417
2418  \underline{Augsburg}
2419  \begin{itemize}
2420   \item Prof. B. Stritzker (accomodation at EP \RM{4})
2421   \item Ralf Utermann (EDV)
2422  \end{itemize}
2423
2424  \underline{Helsinki}
2425  \begin{itemize}
2426   \item Prof. K. Nordlund (MD)
2427  \end{itemize}
2428
2429  \underline{Munich}
2430  \begin{itemize}
2431   \item Bayerische Forschungsstiftung (financial support)
2432  \end{itemize}
2433
2434  \underline{Paderborn}
2435  \begin{itemize}
2436   \item Prof. J. Lindner (SiC)
2437   \item Prof. G. Schmidt (DFT + financial support)
2438   \item Dr. E. Rauls (DFT + SiC)
2439   \item Dr. S. Sanna (VASP)
2440  \end{itemize}
2441
2442 \vspace{0.2cm}
2443
2444 \begin{center}
2445 \framebox{
2446 \bf Thank you for your attention!
2447 }
2448 \end{center}
2449
2450 \end{slide}
2451
2452 \end{document}
2453
2454 \fi