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