posic/talks/mpi_app.tex

   1 \pdfoutput=0
   2 %\documentclass[landscape,semhelv,draft]{seminar}
   3 \documentclass[landscape,semhelv]{seminar}
   4
   5 \usepackage{verbatim}
   6 \usepackage[greek,german]{babel}
   7 \usepackage[latin1]{inputenc}
   8 \usepackage[T1]{fontenc}
   9 \usepackage{amsmath}
  10 \usepackage{latexsym}
  11 \usepackage{ae}
  12
  13 \usepackage{calc}               % Simple computations with LaTeX variables
  14 \usepackage{caption}            % Improved captions
  15 \usepackage{fancybox}           % To have several backgrounds
  16
  17 \usepackage{fancyhdr}           % Headers and footers definitions
  18 \usepackage{fancyvrb}           % Fancy verbatim environments
  19 \usepackage{pstricks}           % PSTricks with the standard color package
  20
  21 \usepackage{pstricks}
  22 \usepackage{pst-node}
  23
  24 %\usepackage{epic}
  25 %\usepackage{eepic}
  26
  27 \usepackage{graphicx}
  28 \graphicspath{{../img/}}
  29
  30 \usepackage{miller}
  31
  32 \usepackage[setpagesize=false]{hyperref}
  33
  34 % units
  35 \usepackage{units}
  36
  37 \usepackage{semcolor}
  38 \usepackage{semlayer}           % Seminar overlays
  39 \usepackage{slidesec}           % Seminar sections and list of slides
  40
  41 \input{seminar.bug}             % Official bugs corrections
  42 \input{seminar.bg2}             % Unofficial bugs corrections
  43
  44 \articlemag{1}
  45
  46 \special{landscape}
  47
  48 % font
  49 %\usepackage{cmbright}
  50 %\renewcommand{\familydefault}{\sfdefault}
  51 %\usepackage{mathptmx}
  52
  53 \usepackage{upgreek}
  54
  55 \begin{document}
  56
  57 \extraslideheight{10in}
  58 \slideframe{none}
  59
  60 \pagestyle{empty}
  61
  62 % specify width and height
  63 \slidewidth 27.7cm
  64 \slideheight 19.1cm
  65
  66 % shift it into visual area properly
  67 \def\slideleftmargin{3.3cm}
  68 \def\slidetopmargin{0.6cm}
  69
  70 \newcommand{\ham}{\mathcal{H}}
  71 \newcommand{\pot}{\mathcal{V}}
  72 \newcommand{\foo}{\mathcal{U}}
  73 \newcommand{\vir}{\mathcal{W}}
  74
  75 % itemize level ii
  76 \renewcommand\labelitemii{{\color{gray}$\bullet$}}
  77
  78 % nice phi
  79 \renewcommand{\phi}{\varphi}
  80
  81 % roman letters
  82 \newcommand{\RM}[1]{\MakeUppercase{\romannumeral #1{}}}
  83
  84 % colors
  85 \newrgbcolor{si-yellow}{.6 .6 0}
  86 \newrgbcolor{hb}{0.75 0.77 0.89}
  87 \newrgbcolor{lbb}{0.75 0.8 0.88}
  88 \newrgbcolor{hlbb}{0.825 0.88 0.968}
  89 \newrgbcolor{lachs}{1.0 .93 .81}
  90
  91 % shortcuts
  92 \newcommand{\si}{Si$_{\text{i}}${}}
  93 \newcommand{\ci}{C$_{\text{i}}${}}
  94 \newcommand{\cs}{C$_{\text{sub}}${}}
  95 \newcommand{\degc}[1]{\unit[#1]{$^{\circ}$C}{}}
  96 \newcommand{\distn}[1]{\unit[#1]{nm}{}}
  97 \newcommand{\dista}[1]{\unit[#1]{\AA}{}}
  98 \newcommand{\perc}[1]{\unit[#1]{\%}{}}
  99
 100 % topic
 101
 102 \begin{slide}
 103 \begin{center}
 104
 105  \vspace{16pt}
 106
 107  {\LARGE\bf
 108   Atomistic simulation studies\\[0.2cm]
 109   in the C/Si system
 110  }
 111
 112  \vspace{48pt}
 113
 114  \textsc{Frank Zirkelbach}
 115
 116  \vspace{48pt}
 117
 118  Application talk at the Max Planck Institute for Solid State Research
 119
 120  \vspace{08pt}
 121
 122  Stuttgart, November 2011
 123
 124 \end{center}
 125 \end{slide}
 126
 127 % intro
 128
 129 \begin{slide}
 130
 131 {\large\bf
 132  Introduction --- The C/Si system\\
 133 }
 134
 135 \begin{center}
 136 \includegraphics[width=6.3cm]{si-c_phase.eps}\\
 137 {\tiny
 138 R. I. Scace and G. A. Slack, J. Chem. Phys. 30, 1551 (1959)
 139 }
 140 \end{center}
 141 \begin{pspicture}(0,0)(0,0)
 142 \psellipse[linecolor=red,linewidth=0.1cm](6.95,3.95)(0.5,2.8)
 143 \end{pspicture}
 144
 145 \end{slide}
 146
 147 \end{document}
 148 \ifnum1=0
 149
 150 % motivation / properties / applications of silicon carbide
 151
 152 \begin{slide}
 153
 154 \small
 155
 156 \begin{pspicture}(0,0)(13.5,5)
 157
 158  \psframe*[linecolor=hb](0,0)(13.5,5)
 159
 160  \pspolygon[linecolor=hlbb,fillcolor=hlbb,fillstyle=solid](5.5,1)(7,1)(7,3)(5.5,3)
 161  \pspolygon[linecolor=hlbb,fillcolor=hlbb,fillstyle=solid](6.75,0.5)(8,2)(8,2)(6.75,3.5)
 162
 163  \rput[lt](0.2,4.6){\color{gray}PROPERTIES}
 164
 165  \rput[lt](0.5,4){wide band gap}
 166  \rput[lt](0.5,3.5){high electric breakdown field}
 167  \rput[lt](0.5,3){good electron mobility}
 168  \rput[lt](0.5,2.5){high electron saturation drift velocity}
 169  \rput[lt](0.5,2){high thermal conductivity}
 170
 171  \rput[lt](0.5,1.5){hard and mechanically stable}
 172  \rput[lt](0.5,1){chemically inert}
 173
 174  \rput[lt](0.5,0.5){radiation hardness}
 175
 176  \rput[rt](13.3,4.6){\color{gray}APPLICATIONS}
 177
 178  \rput[rt](13,3.85){high-temperature, high power}
 179  \rput[rt](13,3.5){and high-frequency}
 180  \rput[rt](13,3.15){electronic and optoelectronic devices}
 181
 182  \rput[rt](13,2.35){material suitable for extreme conditions}
 183  \rput[rt](13,2){microelectromechanical systems}
 184  \rput[rt](13,1.65){abrasives, cutting tools, heating elements}
 185
 186  \rput[rt](13,0.85){first wall reactor material, detectors}
 187  \rput[rt](13,0.5){and electronic devices for space}
 188
 189 \end{pspicture}
 190
 191 \begin{picture}(0,0)(-3,68)
 192 \includegraphics[width=2.6cm]{wide_band_gap.eps}
 193 \end{picture}
 194 \begin{picture}(0,0)(-285,-162)
 195 \includegraphics[width=3.38cm]{sic_led.eps}
 196 \end{picture}
 197 \begin{picture}(0,0)(-195,-162)
 198 \includegraphics[width=2.8cm]{6h-sic_3c-sic.eps}
 199 \end{picture}
 200 \begin{picture}(0,0)(-313,65)
 201 \includegraphics[width=2.2cm]{infineon_schottky.eps}
 202 \end{picture}
 203 \begin{picture}(0,0)(-220,65)
 204 \includegraphics[width=2.9cm]{sic_wechselrichter_ise.eps}
 205 \end{picture}
 206 \begin{picture}(0,0)(0,-160)
 207 \includegraphics[width=3.0cm]{sic_proton.eps}
 208 \end{picture}
 209 \begin{picture}(0,0)(-60,65)
 210 \includegraphics[width=3.4cm]{sic_switch.eps}
 211 \end{picture}
 212
 213 \end{slide}
 214
 215
 216 % contents
 217
 218 \begin{slide}
 219
 220 {\large\bf
 221  Outline
 222 }
 223
 224  \begin{itemize}
 225   \item Implantation of C in Si --- Overview of experimental observations
 226   \item Utilized simulation techniques and modeled problems
 227         \begin{itemize}
 228          \item {\color{blue}Diploma thesis}\\
 229                \underline{Monte Carlo} simulations
 230                modeling the selforganization process
 231                leading to periodic arrays of nanometric amorphous SiC
 232                precipitates
 233          \item {\color{blue}Doctoral studies}\\
 234                Classical potential \underline{molecular dynamics} simulations
 235                \ldots\\
 236                \underline{Density functional theory} calculations
 237                \ldots\\[0.2cm]
 238                \ldots on defects and SiC precipitation in Si
 239         \end{itemize}
 240   \item Summary / Conclusion / Outlook
 241  \end{itemize}
 242
 243 \end{slide}
 244
 245
 246
 247 \end{document}
 248
 249 \ifnum1=0
 250
 251
 252 % start of contents
 253
 254 \begin{slide}
 255
 256  {\large\bf
 257   Polytypes of SiC
 258  }
 259
 260  \vspace{4cm}
 261
 262  \small
 263
 264 \begin{tabular}{l c c c c c c}
 265 \hline
 266  & 3C-SiC & 4H-SiC & 6H-SiC & Si & GaN & Diamond\\
 267 \hline
 268 Hardness [Mohs] & \multicolumn{3}{c}{------ 9.6 ------}& 6.5 & - & 10 \\
 269 Band gap [eV] & 2.36 & 3.23 & 3.03 & 1.12 & 3.39 & 5.5 \\
 270 Break down field [$10^6$ V/cm] & 4 & 3 & 3.2 & 0.6 & 5 & 10 \\
 271 Saturation drift velocity [$10^7$ cm/s] & 2.5 & 2.0 & 2.0 & 1 & 2.7 & 2.7 \\
 272 Electron mobility [cm$^2$/Vs] & 800 & 900 & 400 & 1100 & 900 & 2200 \\
 273 Hole mobility [cm$^2$/Vs] & 320 & 120 & 90 & 420 & 150 & 1600 \\
 274 Thermal conductivity [W/cmK] & 5.0 & 4.9 & 4.9 & 1.5 & 1.3 & 22 \\
 275 \hline
 276 \end{tabular}
 277
 278 {\tiny
 279  Values for $T=300$ K
 280 }
 281
 282 \begin{picture}(0,0)(-160,-155)
 283  \includegraphics[width=7cm]{polytypes.eps}
 284 \end{picture}
 285 \begin{picture}(0,0)(-10,-185)
 286  \includegraphics[width=3.8cm]{cubic_hex.eps}\\
 287 \end{picture}
 288 \begin{picture}(0,0)(-10,-175)
 289  {\tiny cubic (twist)}
 290 \end{picture}
 291 \begin{picture}(0,0)(-60,-175)
 292  {\tiny hexagonal (no twist)}
 293 \end{picture}
 294 \begin{pspicture}(0,0)(0,0)
 295 \psellipse[linecolor=green](5.7,3.03)(0.4,0.5)
 296 \end{pspicture}
 297 \begin{pspicture}(0,0)(0,0)
 298 \psellipse[linecolor=green](5.6,1.68)(0.4,0.2)
 299 \end{pspicture}
 300 \begin{pspicture}(0,0)(0,0)
 301 \psellipse[linecolor=red](10.7,1.13)(0.4,0.2)
 302 \end{pspicture}
 303
 304 \end{slide}
 305
 306 \begin{slide}
 307
 308  {\large\bf
 309   Fabrication of silicon carbide
 310  }
 311
 312  \small
 313
 314  \vspace{4pt}
 315
 316  SiC - \emph{Born from the stars, perfected on earth.}
 317
 318  \vspace{4pt}
 319
 320  Conventional thin film SiC growth:
 321  \begin{itemize}
 322   \item \underline{Sublimation growth using the modified Lely method}
 323         \begin{itemize}
 324          \item SiC single-crystalline seed at $T=1800 \, ^{\circ} \text{C}$
 325          \item Surrounded by polycrystalline SiC in a graphite crucible\\
 326                at $T=2100-2400 \, ^{\circ} \text{C}$
 327          \item Deposition of supersaturated vapor on cooler seed crystal
 328         \end{itemize}
 329   \item \underline{Homoepitaxial growth using CVD}
 330         \begin{itemize}
 331          \item Step-controlled epitaxy on off-oriented 6H-SiC substrates
 332          \item C$_3$H$_8$/SiH$_4$/H$_2$ at $1100-1500 \, ^{\circ} \text{C}$
 333          \item Angle, temperature $\rightarrow$ 3C/6H/4H-SiC
 334         \end{itemize}
 335   \item \underline{Heteroepitaxial growth of 3C-SiC on Si using CVD/MBE}
 336         \begin{itemize}
 337          \item Two steps: carbonization and growth
 338          \item $T=650-1050 \, ^{\circ} \text{C}$
 339          \item SiC/Si lattice mismatch $\approx$ 20 \%
 340          \item Quality and size not yet sufficient
 341         \end{itemize}
 342  \end{itemize}
 343
 344  \begin{picture}(0,0)(-280,-65)
 345   \includegraphics[width=3.8cm]{6h-sic_3c-sic.eps}
 346  \end{picture}
 347  \begin{picture}(0,0)(-280,-55)
 348   \begin{minipage}{5cm}
 349   {\tiny
 350    NASA: 6H-SiC and 3C-SiC LED\\[-7pt]
 351    on 6H-SiC substrate
 352   }
 353   \end{minipage}
 354  \end{picture}
 355  \begin{picture}(0,0)(-265,-150)
 356   \includegraphics[width=2.4cm]{m_lely.eps}
 357  \end{picture}
 358  \begin{picture}(0,0)(-333,-175)
 359   \begin{minipage}{5cm}
 360   {\tiny
 361    1. Lid\\[-7pt]
 362    2. Heating\\[-7pt]
 363    3. Source\\[-7pt]
 364    4. Crucible\\[-7pt]
 365    5. Insulation\\[-7pt]
 366    6. Seed crystal
 367   }
 368   \end{minipage}
 369  \end{picture}
 370  \begin{picture}(0,0)(-230,-35)
 371  \framebox{
 372  {\footnotesize\color{blue}\bf Hex: micropipes along c-axis}
 373  }
 374  \end{picture}
 375  \begin{picture}(0,0)(-230,-10)
 376  \framebox{
 377  \begin{minipage}{3cm}
 378  {\footnotesize\color{blue}\bf 3C-SiC fabrication\\
 379                                less advanced}
 380  \end{minipage}
 381  }
 382  \end{picture}
 383
 384 \end{slide}
 385
 386 \begin{slide}
 387
 388  {\large\bf
 389   Fabrication of silicon carbide
 390  }
 391
 392  \small
 393
 394  Alternative approach:
 395  Ion beam synthesis (IBS) of burried 3C-SiC layers in Si\hkl(1 0 0)
 396  \begin{itemize}
 397   \item \underline{Implantation step 1}\\
 398         180 keV C$^+$, $D=7.9\times 10^{17}$ cm$^{-2}$, $T_{\text{i}}=500\,^{\circ}\mathrm{C}$\\
 399         $\Rightarrow$ box-like distribution of equally sized
 400                        and epitactically oriented SiC precipitates
 401
 402   \item \underline{Implantation step 2}\\
 403         180 keV C$^+$, $D=0.6\times 10^{17}$ cm$^{-2}$, $T_{\text{i}}=250\,^{\circ}\mathrm{C}$\\
 404         $\Rightarrow$ destruction of SiC nanocrystals
 405                       in growing amorphous interface layers
 406   \item \underline{Annealing}\\
 407         $T=1250\,^{\circ}\mathrm{C}$, $t=10\,\text{h}$\\
 408         $\Rightarrow$ homogeneous, stoichiometric SiC layer
 409                       with sharp interfaces
 410  \end{itemize}
 411
 412  \begin{minipage}{6.3cm}
 413  \includegraphics[width=6cm]{ibs_3c-sic.eps}\\[-0.2cm]
 414  {\tiny
 415   XTEM micrograph of single crystalline 3C-SiC in Si\hkl(1 0 0)
 416  }
 417  \end{minipage}
 418 \framebox{
 419  \begin{minipage}{6.3cm}
 420  \begin{center}
 421  {\color{blue}
 422   Precipitation mechanism not yet fully understood!
 423  }
 424  \renewcommand\labelitemi{$\Rightarrow$}
 425  \small
 426  \underline{Understanding the SiC precipitation}
 427  \begin{itemize}
 428   \item significant technological progress in SiC thin film formation
 429   \item perspectives for processes relying upon prevention of SiC precipitation
 430  \end{itemize}
 431  \end{center}
 432  \end{minipage}
 433 }
 434
 435 \end{slide}
 436
 437
 438 \begin{slide}
 439
 440  {\large\bf
 441   Supposed precipitation mechanism of SiC in Si
 442  }
 443
 444  \scriptsize
 445
 446  \vspace{0.1cm}
 447
 448  \begin{minipage}{3.8cm}
 449  Si \& SiC lattice structure\\[0.2cm]
 450  \includegraphics[width=3.5cm]{sic_unit_cell.eps}\\[-0.3cm]
 451  \hrule
 452  \end{minipage}
 453  \hspace{0.6cm}
 454  \begin{minipage}{3.8cm}
 455  \begin{center}
 456  \includegraphics[width=3.3cm]{tem_c-si-db.eps}
 457  \end{center}
 458  \end{minipage}
 459  \hspace{0.6cm}
 460  \begin{minipage}{3.8cm}
 461  \begin{center}
 462  \includegraphics[width=3.3cm]{tem_3c-sic.eps}
 463  \end{center}
 464  \end{minipage}
 465
 466  \begin{minipage}{4cm}
 467  \begin{center}
 468  C-Si dimers (dumbbells)\\[-0.1cm]
 469  on Si interstitial sites
 470  \end{center}
 471  \end{minipage}
 472  \hspace{0.2cm}
 473  \begin{minipage}{4.2cm}
 474  \begin{center}
 475  Agglomeration of C-Si dumbbells\\[-0.1cm]
 476  $\Rightarrow$ dark contrasts
 477  \end{center}
 478  \end{minipage}
 479  \hspace{0.2cm}
 480  \begin{minipage}{4cm}
 481  \begin{center}
 482  Precipitation of 3C-SiC in Si\\[-0.1cm]
 483  $\Rightarrow$ Moir\'e fringes\\[-0.1cm]
 484  \& release of Si self-interstitials
 485  \end{center}
 486  \end{minipage}
 487
 488  \begin{minipage}{3.8cm}
 489  \begin{center}
 490  \includegraphics[width=3.3cm]{sic_prec_seq_01.eps}
 491  \end{center}
 492  \end{minipage}
 493  \hspace{0.6cm}
 494  \begin{minipage}{3.8cm}
 495  \begin{center}
 496  \includegraphics[width=3.3cm]{sic_prec_seq_02.eps}
 497  \end{center}
 498  \end{minipage}
 499  \hspace{0.6cm}
 500  \begin{minipage}{3.8cm}
 501  \begin{center}
 502  \includegraphics[width=3.3cm]{sic_prec_seq_03.eps}
 503  \end{center}
 504  \end{minipage}
 505
 506 \begin{pspicture}(0,0)(0,0)
 507 \psline[linewidth=4pt]{->}(8.5,2)(9.0,2)
 508 \psellipse[linecolor=blue](11.5,5.8)(0.3,0.5)
 509 \rput{-20}{\psellipse[linecolor=blue](3.3,8.1)(0.3,0.5)}
 510 \psline[linewidth=4pt]{->}(4.0,2)(4.5,2)
 511 \rput(12.7,0.3){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
 512  $4a_{\text{Si}}=5a_{\text{SiC}}$
 513  }}}
 514 \rput(12.2,8){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
 515 \hkl(h k l) planes match
 516  }}}
 517 \rput(9.7,6.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
 518 r = 2 - 4 nm
 519  }}}
 520 \end{pspicture}
 521
 522 \end{slide}
 523
 524 \begin{slide}
 525
 526  {\large\bf
 527   Supposed precipitation mechanism of SiC in Si
 528  }
 529
 530  \scriptsize
 531
 532  \vspace{0.1cm}
 533
 534  \begin{minipage}{3.8cm}
 535  Si \& SiC lattice structure\\[0.2cm]
 536  \includegraphics[width=3.5cm]{sic_unit_cell.eps}\\[-0.3cm]
 537  \hrule
 538  \end{minipage}
 539  \hspace{0.6cm}
 540  \begin{minipage}{3.8cm}
 541  \begin{center}
 542  \includegraphics[width=3.3cm]{tem_c-si-db.eps}
 543  \end{center}
 544  \end{minipage}
 545  \hspace{0.6cm}
 546  \begin{minipage}{3.8cm}
 547  \begin{center}
 548  \includegraphics[width=3.3cm]{tem_3c-sic.eps}
 549  \end{center}
 550  \end{minipage}
 551
 552  \begin{minipage}{4cm}
 553  \begin{center}
 554  C-Si dimers (dumbbells)\\[-0.1cm]
 555  on Si interstitial sites
 556  \end{center}
 557  \end{minipage}
 558  \hspace{0.2cm}
 559  \begin{minipage}{4.2cm}
 560  \begin{center}
 561  Agglomeration of C-Si dumbbells\\[-0.1cm]
 562  $\Rightarrow$ dark contrasts
 563  \end{center}
 564  \end{minipage}
 565  \hspace{0.2cm}
 566  \begin{minipage}{4cm}
 567  \begin{center}
 568  Precipitation of 3C-SiC in Si\\[-0.1cm]
 569  $\Rightarrow$ Moir\'e fringes\\[-0.1cm]
 570  \& release of Si self-interstitials
 571  \end{center}
 572  \end{minipage}
 573
 574  \begin{minipage}{3.8cm}
 575  \begin{center}
 576  \includegraphics[width=3.3cm]{sic_prec_seq_01.eps}
 577  \end{center}
 578  \end{minipage}
 579  \hspace{0.6cm}
 580  \begin{minipage}{3.8cm}
 581  \begin{center}
 582  \includegraphics[width=3.3cm]{sic_prec_seq_02.eps}
 583  \end{center}
 584  \end{minipage}
 585  \hspace{0.6cm}
 586  \begin{minipage}{3.8cm}
 587  \begin{center}
 588  \includegraphics[width=3.3cm]{sic_prec_seq_03.eps}
 589  \end{center}
 590  \end{minipage}
 591
 592 \begin{pspicture}(0,0)(0,0)
 593 \psline[linewidth=4pt]{->}(8.5,2)(9.0,2)
 594 \psellipse[linecolor=blue](11.5,5.8)(0.3,0.5)
 595 \rput{-20}{\psellipse[linecolor=blue](3.3,8.1)(0.3,0.5)}
 596 \psline[linewidth=4pt]{->}(4.0,2)(4.5,2)
 597 \rput(12.7,0.3){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
 598  $4a_{\text{Si}}=5a_{\text{SiC}}$
 599  }}}
 600 \rput(12.2,8){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
 601 \hkl(h k l) planes match
 602  }}}
 603 \rput(9.7,6.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
 604 r = 2 - 4 nm
 605  }}}
 606 \rput(6.7,5.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=white]{
 607 \begin{minipage}{10cm}
 608 \small
 609 {\color{red}\bf Controversial views}
 610 \begin{itemize}
 611 \item Implantations at high T (Nejim et al.)
 612  \begin{itemize}
 613   \item Topotactic transformation based on \cs
 614   \item \si{} as supply reacting with further C in cleared volume
 615  \end{itemize}
 616 \item Annealing behavior (Serre et al.)
 617  \begin{itemize}
 618   \item Room temperature implants $\rightarrow$ highly mobile C
 619   \item Elevated T implants $\rightarrow$ no/low C redistribution/migration\\
 620         (indicate stable \cs{} configurations)
 621  \end{itemize}
 622 \item Strained silicon \& Si/SiC heterostructures
 623  \begin{itemize}
 624   \item Coherent SiC precipitates (tensile strain)
 625   \item Incoherent SiC (strain relaxation)
 626  \end{itemize}
 627 \end{itemize}
 628 \end{minipage}
 629  }}}
 630 \end{pspicture}
 631
 632 \end{slide}
 633
 634 \begin{slide}
 635
 636  {\large\bf
 637   Molecular dynamics (MD) simulations
 638  }
 639
 640  \vspace{12pt}
 641
 642  \small
 643
 644  {\bf MD basics:}
 645  \begin{itemize}
 646   \item Microscopic description of N particle system
 647   \item Analytical interaction potential
 648   \item Numerical integration using Newtons equation of motion\\
 649         as a propagation rule in 6N-dimensional phase space
 650   \item Observables obtained by time and/or ensemble averages
 651  \end{itemize}
 652  {\bf Details of the simulation:}
 653  \begin{itemize}
 654   \item Integration: Velocity Verlet, timestep: $1\text{ fs}$
 655   \item Ensemble: NpT (isothermal-isobaric)
 656         \begin{itemize}
 657          \item Berendsen thermostat:
 658                $\tau_{\text{T}}=100\text{ fs}$
 659          \item Berendsen barostat:\\
 660                $\tau_{\text{P}}=100\text{ fs}$,
 661                $\beta^{-1}=100\text{ GPa}$
 662         \end{itemize}
 663   \item Erhart/Albe potential: Tersoff-like bond order potential
 664   \vspace*{12pt}
 665         \[
 666         E = \frac{1}{2} \sum_{i \neq j} \pot_{ij}, \quad
 667         \pot_{ij} = {\color{red}f_C(r_{ij})}
 668         \left[ f_R(r_{ij}) + {\color{blue}b_{ij}} f_A(r_{ij}) \right]
 669         \]
 670  \end{itemize}
 671
 672  \begin{picture}(0,0)(-230,-30)
 673   \includegraphics[width=5cm]{tersoff_angle.eps}
 674  \end{picture}
 675
 676 \end{slide}
 677
 678 \begin{slide}
 679
 680  {\large\bf
 681   Density functional theory (DFT) calculations
 682  }
 683
 684  \small
 685
 686  Basic ingredients necessary for DFT
 687
 688  \begin{itemize}
 689   \item \underline{Hohenberg-Kohn theorem} - ground state density $n_0(r)$ ...
 690         \begin{itemize}
 691          \item ... uniquely determines the ground state potential
 692                / wavefunctions
 693          \item ... minimizes the systems total energy
 694         \end{itemize}
 695   \item \underline{Born-Oppenheimer}
 696         - $N$ moving electrons in an external potential of static nuclei
 697 \[
 698 H\Psi = \left[-\sum_i^N \frac{\hbar^2}{2m}\nabla_i^2
 699               +\sum_i^N V_{\text{ext}}(r_i)
 700               +\sum_{i<j}^N V_{e-e}(r_i,r_j)\right]\Psi=E\Psi
 701 \]
 702   \item \underline{Effective potential}
 703         - averaged electrostatic potential \& exchange and correlation
 704 \[
 705 V_{\text{eff}}(r)=V_{\text{ext}}(r)+\int\frac{e^2 n(r')}{|r-r'|}d^3r'
 706                  +V_{\text{XC}}[n(r)]
 707 \]
 708   \item \underline{Kohn-Sham system}
 709         - Schr\"odinger equation of N non-interacting particles
 710 \[
 711 \left[ -\frac{\hbar^2}{2m}\nabla^2 + V_{\text{eff}}(r) \right] \Phi_i(r)
 712 =\epsilon_i\Phi_i(r)
 713 \quad
 714 \Rightarrow
 715 \quad
 716 n(r)=\sum_i^N|\Phi_i(r)|^2
 717 \]
 718   \item \underline{Self-consistent solution}\\
 719 $n(r)$ depends on $\Phi_i$, which depend on $V_{\text{eff}}$,
 720 which in turn depends on $n(r)$
 721   \item \underline{Variational principle}
 722         - minimize total energy with respect to $n(r)$
 723  \end{itemize}
 724
 725 \end{slide}
 726
 727 \begin{slide}
 728
 729  {\large\bf
 730   Density functional theory (DFT) calculations
 731  }
 732
 733  \small
 734
 735  \vspace*{0.2cm}
 736
 737  Details of applied DFT calculations in this work
 738
 739  \begin{itemize}
 740   \item \underline{Exchange correlation functional}
 741         - approximations for the inhomogeneous electron gas
 742         \begin{itemize}
 743          \item LDA: $E_{\text{XC}}^{\text{LDA}}[n]=\int \epsilon_{\text{XC}}(n)n(r)d^3r$
 744          \item GGA: $E_{\text{XC}}^{\text{GGA}}[n]=\int \epsilon_{\text{XC}}(n,\nabla n)n(r)d^3r$
 745         \end{itemize}
 746   \item \underline{Plane wave basis set}
 747         - approximation of the wavefunction $\Phi_i$ by plane waves $\phi_j$
 748 \[
 749 \rightarrow
 750 \text{Fourier series: } \Phi_i=\sum_{|G+k|<G_{\text{cut}}} c_j^i \phi_j(r), \quad E_{\text{cut}}=\frac{\hbar^2}{2m}G^2_{\text{cut}}
 751 \qquad ({\color{blue}300\text{ eV}})
 752 \]
 753   \item \underline{Brillouin zone sampling} -
 754         {\color{blue}$\Gamma$-point only} calculations
 755   \item \underline{Pseudo potential}
 756         - consider only the valence electrons
 757   \item \underline{Code} - VASP 4.6
 758  \end{itemize}
 759
 760  \vspace*{0.2cm}
 761
 762  MD and structural optimization
 763
 764  \begin{itemize}
 765   \item MD integration: Gear predictor corrector algorithm
 766   \item Pressure control: Parrinello-Rahman pressure control
 767   \item Structural optimization: Conjugate gradient method
 768  \end{itemize}
 769
 770 \begin{pspicture}(0,0)(0,0)
 771 \psellipse[linecolor=blue](1.5,6.75)(0.5,0.3)
 772 \end{pspicture}
 773
 774 \end{slide}
 775
 776 \begin{slide}
 777
 778  {\large\bf
 779   C and Si self-interstitial point defects in silicon
 780  }
 781
 782  \small
 783
 784  \vspace*{0.3cm}
 785
 786 \begin{minipage}{8cm}
 787 Procedure:\\[0.3cm]
 788   \begin{pspicture}(0,0)(7,5)
 789   \rput(3.5,4){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
 790    \parbox{7cm}{
 791    \begin{itemize}
 792     \item Creation of c-Si simulation volume
 793     \item Periodic boundary conditions
 794     \item $T=0\text{ K}$, $p=0\text{ bar}$
 795    \end{itemize}
 796   }}}}
 797 \rput(3.5,2.1){\rnode{insert}{\psframebox{
 798  \parbox{7cm}{
 799   \begin{center}
 800   Insertion of interstitial C/Si atoms
 801   \end{center}
 802   }}}}
 803   \rput(3.5,1){\rnode{cool}{\psframebox[fillstyle=solid,fillcolor=lbb]{
 804    \parbox{7cm}{
 805    \begin{center}
 806    Relaxation / structural energy minimization
 807    \end{center}
 808   }}}}
 809   \ncline[]{->}{init}{insert}
 810   \ncline[]{->}{insert}{cool}
 811  \end{pspicture}
 812 \end{minipage}
 813 \begin{minipage}{5cm}
 814   \includegraphics[width=5cm]{unit_cell_e.eps}\\
 815 \end{minipage}
 816
 817 \begin{minipage}{9cm}
 818  \begin{tabular}{l c c}
 819  \hline
 820  & size [unit cells] & \# atoms\\
 821 \hline
 822 VASP & $3\times 3\times 3$ & $216\pm 1$ \\
 823 Erhart/Albe & $9\times 9\times 9$ & $5832\pm 1$\\
 824 \hline
 825  \end{tabular}
 826 \end{minipage}
 827 \begin{minipage}{4cm}
 828 {\color{red}$\bullet$} Tetrahedral\\
 829 {\color{green}$\bullet$} Hexagonal\\
 830 {\color{yellow}$\bullet$} \hkl<1 0 0> dumbbell\\
 831 {\color{magenta}$\bullet$} \hkl<1 1 0> dumbbell\\
 832 {\color{cyan}$\bullet$} Bond-centered\\
 833 {\color{black}$\bullet$} Vacancy / Substitutional
 834 \end{minipage}
 835
 836 \end{slide}
 837
 838 \begin{slide}
 839
 840  \footnotesize
 841
 842 \begin{minipage}{9.5cm}
 843
 844  {\large\bf
 845   Si self-interstitial point defects in silicon\\
 846  }
 847
 848 \begin{tabular}{l c c c c c}
 849 \hline
 850  $E_{\text{f}}$ [eV] & \hkl<1 1 0> DB & H & T & \hkl<1 0 0> DB & V \\
 851 \hline
 852  VASP & \underline{3.39} & 3.42 & 3.77 & 4.41 & 3.63 \\
 853  Erhart/Albe & 4.39 & 4.48$^*$ & \underline{3.40} & 5.42 & 3.13 \\
 854 \hline
 855 \end{tabular}\\[0.2cm]
 856
 857 \begin{minipage}{4.7cm}
 858 \includegraphics[width=4.7cm]{e_kin_si_hex.ps}
 859 \end{minipage}
 860 \begin{minipage}{4.7cm}
 861 \begin{center}
 862 {\tiny nearly T $\rightarrow$ T}\\
 863 \end{center}
 864 \includegraphics[width=4.7cm]{nhex_tet.ps}
 865 \end{minipage}\\
 866
 867 \underline{Hexagonal} \hspace{2pt}
 868 \href{../video/si_self_int_hexa.avi}{$\rhd$}\\[0.1cm]
 869 \framebox{
 870 \begin{minipage}{2.7cm}
 871 $E_{\text{f}}^*=4.48\text{ eV}$\\
 872 \includegraphics[width=2.7cm]{si_pd_albe/hex_a.eps}
 873 \end{minipage}
 874 \begin{minipage}{0.4cm}
 875 \begin{center}
 876 $\Rightarrow$
 877 \end{center}
 878 \end{minipage}
 879 \begin{minipage}{2.7cm}
 880 $E_{\text{f}}=3.96\text{ eV}$\\
 881 \includegraphics[width=2.8cm]{si_pd_albe/hex.eps}
 882 \end{minipage}
 883 }
 884 \begin{minipage}{2.9cm}
 885 \begin{flushright}
 886 \underline{Vacancy}\\
 887 \includegraphics[width=3.0cm]{si_pd_albe/vac.eps}
 888 \end{flushright}
 889 \end{minipage}
 890
 891 \end{minipage}
 892 \begin{minipage}{3.5cm}
 893
 894 \begin{flushright}
 895 \underline{\hkl<1 1 0> dumbbell}\\
 896 \includegraphics[width=3.0cm]{si_pd_albe/110.eps}\\
 897 \underline{Tetrahedral}\\
 898 \includegraphics[width=3.0cm]{si_pd_albe/tet.eps}\\
 899 \underline{\hkl<1 0 0> dumbbell}\\
 900 \includegraphics[width=3.0cm]{si_pd_albe/100.eps}
 901 \end{flushright}
 902
 903 \end{minipage}
 904
 905 \end{slide}
 906
 907 \begin{slide}
 908
 909 \footnotesize
 910
 911  {\large\bf
 912   C interstitial point defects in silicon\\[-0.1cm]
 913  }
 914
 915 \begin{tabular}{l c c c c c c r}
 916 \hline
 917  $E_{\text{f}}$ & T & H & \hkl<1 0 0> DB & \hkl<1 1 0> DB & S & B & \cs{} \& \si\\
 918 \hline
 919  VASP & unstable & unstable & \underline{3.72} & 4.16 & 1.95 & 4.66 & {\color{green}4.17}\\
 920  Erhart/Albe MD & 6.09 & 9.05$^*$ & \underline{3.88} & 5.18 & {\color{red}0.75} & 5.59$^*$ & {\color{green}4.43} \\
 921 \hline
 922 \end{tabular}\\[0.1cm]
 923
 924 \framebox{
 925 \begin{minipage}{2.7cm}
 926 \underline{Hexagonal} \hspace{2pt}
 927 \href{../video/c_in_si_int_hexa.avi}{$\rhd$}\\
 928 $E_{\text{f}}^*=9.05\text{ eV}$\\
 929 \includegraphics[width=2.7cm]{c_pd_albe/hex.eps}
 930 \end{minipage}
 931 \begin{minipage}{0.4cm}
 932 \begin{center}
 933 $\Rightarrow$
 934 \end{center}
 935 \end{minipage}
 936 \begin{minipage}{2.7cm}
 937 \underline{\hkl<1 0 0>}\\
 938 $E_{\text{f}}=3.88\text{ eV}$\\
 939 \includegraphics[width=2.7cm]{c_pd_albe/100.eps}
 940 \end{minipage}
 941 }
 942 \begin{minipage}{2cm}
 943 \hfill
 944 \end{minipage}
 945 \begin{minipage}{3cm}
 946 \begin{flushright}
 947 \underline{Tetrahedral}\\
 948 \includegraphics[width=3.0cm]{c_pd_albe/tet.eps}
 949 \end{flushright}
 950 \end{minipage}
 951
 952 \framebox{
 953 \begin{minipage}{2.7cm}
 954 \underline{Bond-centered}\\
 955 $E_{\text{f}}^*=5.59\text{ eV}$\\
 956 \includegraphics[width=2.7cm]{c_pd_albe/bc.eps}
 957 \end{minipage}
 958 \begin{minipage}{0.4cm}
 959 \begin{center}
 960 $\Rightarrow$
 961 \end{center}
 962 \end{minipage}
 963 \begin{minipage}{2.7cm}
 964 \underline{\hkl<1 1 0> dumbbell}\\
 965 $E_{\text{f}}=5.18\text{ eV}$\\
 966 \includegraphics[width=2.7cm]{c_pd_albe/110.eps}
 967 \end{minipage}
 968 }
 969 \begin{minipage}{2cm}
 970 \hfill
 971 \end{minipage}
 972 \begin{minipage}{3cm}
 973 \begin{flushright}
 974 \underline{Substitutional}\\
 975 \includegraphics[width=3.0cm]{c_pd_albe/sub.eps}
 976 \end{flushright}
 977 \end{minipage}
 978
 979 \end{slide}
 980
 981 \begin{slide}
 982
 983 \footnotesize
 984
 985  {\large\bf\boldmath
 986   C \hkl<1 0 0> dumbbell interstitial configuration\\
 987  }
 988
 989 {\tiny
 990 \begin{tabular}{l c c c c c c c c}
 991 \hline
 992  Distances [nm] & $r(1C)$ & $r(2C)$ & $r(3C)$ & $r(12)$ & $r(13)$ & $r(34)$ & $r(23)$ & $r(25)$ \\
 993 \hline
 994 Erhart/Albe & 0.175 & 0.329 & 0.186 & 0.226 & 0.300 & 0.343 & 0.423 & 0.425 \\
 995 VASP & 0.174 & 0.341 & 0.182 & 0.229 & 0.286 & 0.347 & 0.422 & 0.417 \\
 996 \hline
 997 \end{tabular}\\[0.2cm]
 998 \begin{tabular}{l c c c c }
 999 \hline
1000  Angles [$^{\circ}$] & $\theta_1$ & $\theta_2$ & $\theta_3$ & $\theta_4$ \\
1001 \hline
1002 Erhart/Albe & 140.2 & 109.9 & 134.4 & 112.8 \\
1003 VASP & 130.7 & 114.4 & 146.0 & 107.0 \\
1004 \hline
1005 \end{tabular}\\[0.2cm]
1006 \begin{tabular}{l c c c}
1007 \hline
1008  Displacements [nm]& $a$ & $b$ & $|a|+|b|$ \\
1009 \hline
1010 Erhart/Albe & 0.084 & -0.091 & 0.175 \\
1011 VASP & 0.109 & -0.065 & 0.174 \\
1012 \hline
1013 \end{tabular}\\[0.6cm]
1014 }
1015
1016 \begin{minipage}{3.0cm}
1017 \begin{center}
1018 \underline{Erhart/Albe}
1019 \includegraphics[width=3.0cm]{c_pd_albe/100_cmp.eps}
1020 \end{center}
1021 \end{minipage}
1022 \begin{minipage}{3.0cm}
1023 \begin{center}
1024 \underline{VASP}
1025 \includegraphics[width=3.0cm]{c_pd_vasp/100_cmp.eps}
1026 \end{center}
1027 \end{minipage}\\
1028
1029 \begin{picture}(0,0)(-185,10)
1030 \includegraphics[width=6.8cm]{100-c-si-db_cmp.eps}
1031 \end{picture}
1032 \begin{picture}(0,0)(-280,-150)
1033 \includegraphics[width=3.3cm]{c_pd_vasp/eden.eps}
1034 \end{picture}
1035
1036 \begin{pspicture}(0,0)(0,0)
1037 \psellipse[linecolor=green](5.18,5.92)(0.5,0.3)
1038 \psellipse[linecolor=red](3.45,5.92)(1.0,0.4)
1039 \psellipse[linecolor=blue](2.7,6.92)(0.9,0.2)
1040 \psellipse[linecolor=blue](4.65,6.92)(0.9,0.2)
1041 \end{pspicture}
1042
1043 \end{slide}
1044
1045 \begin{slide}
1046
1047 \small
1048
1049 \begin{minipage}{8.5cm}
1050
1051  {\large\bf
1052   Bond-centered interstitial configuration\\[-0.1cm]
1053  }
1054
1055 \begin{minipage}{3.0cm}
1056 \includegraphics[width=2.8cm]{c_pd_vasp/bc_2333.eps}\\
1057 \end{minipage}
1058 \begin{minipage}{5.2cm}
1059 \begin{itemize}
1060  \item Linear Si-C-Si bond
1061  \item Si: one C \& 3 Si neighbours
1062  \item Spin polarized calculations
1063  \item No saddle point!\\
1064        Real local minimum!
1065 \end{itemize}
1066 \end{minipage}
1067
1068 \framebox{
1069  \tiny
1070  \begin{minipage}[t]{6.5cm}
1071   \begin{minipage}[t]{1.2cm}
1072   {\color{red}Si}\\
1073   {\tiny sp$^3$}\\[0.8cm]
1074   \underline{${\color{black}\uparrow}$}
1075   \underline{${\color{black}\uparrow}$}
1076   \underline{${\color{black}\uparrow}$}
1077   \underline{${\color{red}\uparrow}$}\\
1078   sp$^3$
1079   \end{minipage}
1080   \begin{minipage}[t]{1.4cm}
1081   \begin{center}
1082   {\color{red}M}{\color{blue}O}\\[0.8cm]
1083   \underline{${\color{blue}\uparrow}{\color{white}\downarrow}$}\\
1084   $\sigma_{\text{ab}}$\\[0.5cm]
1085   \underline{${\color{red}\uparrow}{\color{blue}\downarrow}$}\\
1086   $\sigma_{\text{b}}$
1087   \end{center}
1088   \end{minipage}
1089   \begin{minipage}[t]{1.0cm}
1090   \begin{center}
1091   {\color{blue}C}\\
1092   {\tiny sp}\\[0.2cm]
1093   \underline{${\color{white}\uparrow\uparrow}$}
1094   \underline{${\color{white}\uparrow\uparrow}$}\\
1095   2p\\[0.4cm]
1096   \underline{${\color{blue}\uparrow}{\color{blue}\downarrow}$}
1097   \underline{${\color{blue}\uparrow}{\color{blue}\downarrow}$}\\
1098   sp
1099   \end{center}
1100   \end{minipage}
1101   \begin{minipage}[t]{1.4cm}
1102   \begin{center}
1103   {\color{blue}M}{\color{green}O}\\[0.8cm]
1104   \underline{${\color{blue}\uparrow}{\color{white}\downarrow}$}\\
1105   $\sigma_{\text{ab}}$\\[0.5cm]
1106   \underline{${\color{green}\uparrow}{\color{blue}\downarrow}$}\\
1107   $\sigma_{\text{b}}$
1108   \end{center}
1109   \end{minipage}
1110   \begin{minipage}[t]{1.2cm}
1111   \begin{flushright}
1112   {\color{green}Si}\\
1113   {\tiny sp$^3$}\\[0.8cm]
1114   \underline{${\color{green}\uparrow}$}
1115   \underline{${\color{black}\uparrow}$}
1116   \underline{${\color{black}\uparrow}$}
1117   \underline{${\color{black}\uparrow}$}\\
1118   sp$^3$
1119   \end{flushright}
1120   \end{minipage}
1121  \end{minipage}
1122 }\\[0.1cm]
1123
1124 \framebox{
1125 \begin{minipage}{4.5cm}
1126 \includegraphics[width=4cm]{c_100_mig_vasp/im_spin_diff.eps}
1127 \end{minipage}
1128 \begin{minipage}{3.5cm}
1129 {\color{gray}$\bullet$} Spin up\\
1130 {\color{green}$\bullet$} Spin down\\
1131 {\color{blue}$\bullet$} Resulting spin up\\
1132 {\color{yellow}$\bullet$} Si atoms\\
1133 {\color{red}$\bullet$} C atom
1134 \end{minipage}
1135 }
1136
1137 \end{minipage}
1138 \begin{minipage}{4.2cm}
1139 \begin{flushright}
1140 \includegraphics[width=4.3cm]{c_pd_vasp/bc_2333_ksl.ps}\\
1141 {\color{green}$\Box$} {\tiny unoccupied}\\
1142 {\color{red}$\bullet$} {\tiny occupied}
1143 \end{flushright}
1144 \end{minipage}
1145
1146 \end{slide}
1147
1148 \begin{slide}
1149
1150  {\large\bf\boldmath
1151   Migration of the C \hkl<1 0 0> dumbbell interstitial
1152  }
1153
1154 \scriptsize
1155
1156  {\small Investigated pathways}
1157
1158 \begin{minipage}{8.5cm}
1159 \begin{minipage}{8.3cm}
1160 \underline{\hkl<0 0 -1> $\rightarrow$ \hkl<0 0 1>}\\
1161 \begin{minipage}{2.4cm}
1162 \includegraphics[width=2.4cm]{c_pd_vasp/100_2333.eps}
1163 \end{minipage}
1164 \begin{minipage}{0.4cm}
1165 $\rightarrow$
1166 \end{minipage}
1167 \begin{minipage}{2.4cm}
1168 \includegraphics[width=2.4cm]{c_pd_vasp/bc_2333.eps}
1169 \end{minipage}
1170 \begin{minipage}{0.4cm}
1171 $\rightarrow$
1172 \end{minipage}
1173 \begin{minipage}{2.4cm}
1174 \includegraphics[width=2.4cm]{c_pd_vasp/100_next_2333.eps}
1175 \end{minipage}
1176 \end{minipage}\\
1177 \begin{minipage}{8.3cm}
1178 \underline{\hkl<0 0 -1> $\rightarrow$ \hkl<0 -1 0>}\\
1179 \begin{minipage}{2.4cm}
1180 \includegraphics[width=2.4cm]{c_pd_vasp/100_2333.eps}
1181 \end{minipage}
1182 \begin{minipage}{0.4cm}
1183 $\rightarrow$
1184 \end{minipage}
1185 \begin{minipage}{2.4cm}
1186 \includegraphics[width=2.4cm]{c_pd_vasp/00-1-0-10_2333.eps}
1187 \end{minipage}
1188 \begin{minipage}{0.4cm}
1189 $\rightarrow$
1190 \end{minipage}
1191 \begin{minipage}{2.4cm}
1192 \includegraphics[width=2.4cm]{c_pd_vasp/0-10_2333.eps}
1193 \end{minipage}
1194 \end{minipage}\\
1195 \begin{minipage}{8.3cm}
1196 \underline{\hkl<0 0 -1> $\rightarrow$ \hkl<0 -1 0> (in place)}\\
1197 \begin{minipage}{2.4cm}
1198 \includegraphics[width=2.4cm]{c_pd_vasp/100_2333.eps}
1199 \end{minipage}
1200 \begin{minipage}{0.4cm}
1201 $\rightarrow$
1202 \end{minipage}
1203 \begin{minipage}{2.4cm}
1204 \includegraphics[width=2.4cm]{c_pd_vasp/00-1_ip0-10_2333.eps}
1205 \end{minipage}
1206 \begin{minipage}{0.4cm}
1207 $\rightarrow$
1208 \end{minipage}
1209 \begin{minipage}{2.4cm}
1210 \includegraphics[width=2.4cm]{c_pd_vasp/0-10_ip_2333.eps}
1211 \end{minipage}
1212 \end{minipage}
1213 \end{minipage}
1214 \framebox{
1215 \begin{minipage}{4.2cm}
1216  {\small Constrained relaxation\\
1217          technique (CRT) method}\\
1218 \includegraphics[width=4cm]{crt_orig.eps}
1219 \begin{itemize}
1220  \item Constrain diffusing atom
1221  \item Static constraints
1222 \end{itemize}
1223 \vspace*{0.3cm}
1224  {\small Modifications}\\
1225 \includegraphics[width=4cm]{crt_mod.eps}
1226 \begin{itemize}
1227  \item Constrain all atoms
1228  \item Update individual\\
1229        constraints
1230 \end{itemize}
1231 \end{minipage}
1232 }
1233
1234 \end{slide}
1235
1236 \begin{slide}
1237
1238  {\large\bf\boldmath
1239   Migration of the C \hkl<1 0 0> dumbbell interstitial
1240  }
1241
1242 \scriptsize
1243
1244 \framebox{
1245 \begin{minipage}{5.9cm}
1246 \begin{flushleft}
1247 \includegraphics[width=5.8cm]{im_00-1_nosym_sp_fullct_thesis.ps}\\[0.45cm]
1248 \end{flushleft}
1249 \begin{center}
1250 \begin{picture}(0,0)(60,0)
1251 \includegraphics[width=1cm]{vasp_mig/00-1.eps}
1252 \end{picture}
1253 \begin{picture}(0,0)(-5,0)
1254 \includegraphics[width=1cm]{vasp_mig/bc_00-1_sp.eps}
1255 \end{picture}
1256 \begin{picture}(0,0)(-55,0)
1257 \includegraphics[width=1cm]{vasp_mig/bc.eps}
1258 \end{picture}
1259 \begin{picture}(0,0)(12.5,10)
1260 \includegraphics[width=1cm]{110_arrow.eps}
1261 \end{picture}
1262 \begin{picture}(0,0)(90,0)
1263 \includegraphics[height=0.9cm]{001_arrow.eps}
1264 \end{picture}
1265 \end{center}
1266 \vspace*{0.35cm}
1267 \end{minipage}
1268 }
1269 \begin{minipage}{0.3cm}
1270 \hfill
1271 \end{minipage}
1272 \framebox{
1273 \begin{minipage}{5.9cm}
1274 \begin{flushright}
1275 \includegraphics[width=5.9cm]{vasp_mig/00-1_0-10_nosym_sp_fullct.ps}\\[0.5cm]
1276 \end{flushright}
1277 \begin{center}
1278 \begin{picture}(0,0)(60,0)
1279 \includegraphics[width=1cm]{vasp_mig/00-1_a.eps}
1280 \end{picture}
1281 \begin{picture}(0,0)(5,0)
1282 \includegraphics[width=1cm]{vasp_mig/00-1_0-10_sp.eps}
1283 \end{picture}
1284 \begin{picture}(0,0)(-55,0)
1285 \includegraphics[width=1cm]{vasp_mig/0-10.eps}
1286 \end{picture}
1287 \begin{picture}(0,0)(12.5,10)
1288 \includegraphics[width=1cm]{100_arrow.eps}
1289 \end{picture}
1290 \begin{picture}(0,0)(90,0)
1291 \includegraphics[height=0.9cm]{001_arrow.eps}
1292 \end{picture}
1293 \end{center}
1294 \vspace*{0.3cm}
1295 \end{minipage}\\
1296 }
1297
1298 \vspace*{0.05cm}
1299
1300 \framebox{
1301 \begin{minipage}{5.9cm}
1302 \begin{flushleft}
1303 \includegraphics[width=5.9cm]{vasp_mig/00-1_ip0-10_nosym_sp_fullct.ps}\\[0.6cm]
1304 \end{flushleft}
1305 \begin{center}
1306 \begin{picture}(0,0)(60,0)
1307 \includegraphics[width=0.9cm]{vasp_mig/00-1_b.eps}
1308 \end{picture}
1309 \begin{picture}(0,0)(10,0)
1310 \includegraphics[width=0.9cm]{vasp_mig/00-1_ip0-10_sp.eps}
1311 \end{picture}
1312 \begin{picture}(0,0)(-60,0)
1313 \includegraphics[width=0.9cm]{vasp_mig/0-10_b.eps}
1314 \end{picture}
1315 \begin{picture}(0,0)(12.5,10)
1316 \includegraphics[width=1cm]{100_arrow.eps}
1317 \end{picture}
1318 \begin{picture}(0,0)(90,0)
1319 \includegraphics[height=0.9cm]{001_arrow.eps}
1320 \end{picture}
1321 \end{center}
1322 \vspace*{0.3cm}
1323 \end{minipage}
1324 }
1325 \begin{minipage}{0.3cm}
1326 \hfill
1327 \end{minipage}
1328 \begin{minipage}{6.5cm}
1329 VASP results
1330 \begin{itemize}
1331  \item Energetically most favorable path
1332        \begin{itemize}
1333         \item Path 2
1334         \item Activation energy: $\approx$ 0.9 eV
1335         \item Experimental values: 0.73 ... 0.87 eV
1336        \end{itemize}
1337        $\Rightarrow$ {\color{blue}Diffusion} path identified!
1338  \item Reorientation (path 3)
1339        \begin{itemize}
1340         \item More likely composed of two consecutive steps of type 2
1341         \item Experimental values: 0.77 ... 0.88 eV
1342        \end{itemize}
1343        $\Rightarrow$ {\color{blue}Reorientation} transition identified!
1344 \end{itemize}
1345 \end{minipage}
1346
1347 \end{slide}
1348
1349 \begin{slide}
1350
1351  {\large\bf\boldmath
1352   Migration of the C \hkl<1 0 0> dumbbell interstitial
1353  }
1354
1355 \scriptsize
1356
1357  \vspace{0.1cm}
1358
1359 \begin{minipage}{6.5cm}
1360
1361 \framebox{
1362 \begin{minipage}[t]{5.9cm}
1363 \begin{flushleft}
1364 \includegraphics[width=5.9cm]{bc_00-1.ps}\\[2.35cm]
1365 \end{flushleft}
1366 \begin{center}
1367 \begin{pspicture}(0,0)(0,0)
1368 \psframe[linecolor=red,fillstyle=none](-2.8,1.35)(3.3,2.7)
1369 \end{pspicture}
1370 \begin{picture}(0,0)(60,-50)
1371 \includegraphics[width=1cm]{albe_mig/bc_00-1_red_00.eps}
1372 \end{picture}
1373 \begin{picture}(0,0)(5,-50)
1374 \includegraphics[width=1cm]{albe_mig/bc_00-1_red_01.eps}
1375 \end{picture}
1376 \begin{picture}(0,0)(-55,-50)
1377 \includegraphics[width=1cm]{albe_mig/bc_00-1_red_02.eps}
1378 \end{picture}
1379 \begin{picture}(0,0)(12.5,-40)
1380 \includegraphics[width=1cm]{110_arrow.eps}
1381 \end{picture}
1382 \begin{picture}(0,0)(90,-45)
1383 \includegraphics[height=0.9cm]{001_arrow.eps}
1384 \end{picture}\\
1385 \begin{pspicture}(0,0)(0,0)
1386 \psframe[linecolor=blue,fillstyle=none](-2.8,0)(3.3,1.6)
1387 \end{pspicture}
1388 \begin{picture}(0,0)(60,-15)
1389 \includegraphics[width=0.9cm]{albe_mig/bc_00-1_01.eps}
1390 \end{picture}
1391 \begin{picture}(0,0)(35,-15)
1392 \includegraphics[width=0.9cm]{albe_mig/bc_00-1_02.eps}
1393 \end{picture}
1394 \begin{picture}(0,0)(-5,-15)
1395 \includegraphics[width=0.9cm]{albe_mig/bc_00-1_03.eps}
1396 \end{picture}
1397 \begin{picture}(0,0)(-55,-15)
1398 \includegraphics[width=0.9cm]{albe_mig/bc_00-1_04.eps}
1399 \end{picture}
1400 \begin{picture}(0,0)(12.5,-5)
1401 \includegraphics[width=1cm]{100_arrow.eps}
1402 \end{picture}
1403 \begin{picture}(0,0)(90,-15)
1404 \includegraphics[height=0.9cm]{010_arrow.eps}
1405 \end{picture}
1406 \end{center}
1407 \end{minipage}
1408 }\\[0.1cm]
1409
1410 \begin{minipage}{5.9cm}
1411 Erhart/Albe results
1412 \begin{itemize}
1413  \item Lowest activation energy: $\approx$ 2.2 eV
1414  \item 2.4 times higher than VASP
1415  \item Different pathway
1416 \end{itemize}
1417 \end{minipage}
1418
1419 \end{minipage}
1420 \begin{minipage}{6.5cm}
1421
1422 \framebox{
1423 \begin{minipage}{5.9cm}
1424 %\begin{flushright}
1425 %\includegraphics[width=5.9cm]{00-1_0-10.ps}\\[0.75cm]
1426 %\end{flushright}
1427 %\begin{center}
1428 %\begin{pspicture}(0,0)(0,0)
1429 %\psframe[linecolor=red,fillstyle=none](-2.8,-0.25)(3.3,1.1)
1430 %\end{pspicture}
1431 %\begin{picture}(0,0)(60,-5)
1432 %\includegraphics[width=0.9cm]{albe_mig/00-1_0-10_red_00.eps}
1433 %\end{picture}
1434 %\begin{picture}(0,0)(0,-5)
1435 %\includegraphics[width=0.9cm]{albe_mig/00-1_0-10_red_min.eps}
1436 %\end{picture}
1437 %\begin{picture}(0,0)(-55,-5)
1438 %\includegraphics[width=0.9cm]{albe_mig/00-1_0-10_red_03.eps}
1439 %\end{picture}
1440 %\begin{picture}(0,0)(12.5,5)
1441 %\includegraphics[width=1cm]{100_arrow.eps}
1442 %\end{picture}
1443 %\begin{picture}(0,0)(90,0)
1444 %\includegraphics[height=0.9cm]{001_arrow.eps}
1445 %\end{picture}
1446 %\end{center}
1447 %\vspace{0.2cm}
1448 %\end{minipage}
1449 %}\\[0.2cm]
1450 %
1451 %\framebox{
1452 %\begin{minipage}{5.9cm}
1453 \includegraphics[width=5.9cm]{00-1_110_0-10_mig_albe.ps}
1454 \end{minipage}
1455 }\\[0.1cm]
1456
1457 \begin{minipage}{5.9cm}
1458 Transition involving \ci{} \hkl<1 1 0>
1459 \begin{itemize}
1460  \item Bond-centered configuration unstable\\
1461        $\rightarrow$ \ci{} \hkl<1 1 0> dumbbell
1462  \item Transition minima of path 2 \& 3\\
1463        $\rightarrow$ \ci{} \hkl<1 1 0> dumbbell
1464  \item Activation energy: $\approx$ 2.2 eV \& 0.9 eV
1465  \item 2.4 - 3.4 times higher than VASP
1466  \item Rotation of dumbbell orientation
1467 \end{itemize}
1468 \vspace{0.1cm}
1469 \begin{center}
1470 {\color{blue}Overestimated diffusion barrier}
1471 \end{center}
1472 \end{minipage}
1473
1474 \end{minipage}
1475
1476 \end{slide}
1477
1478 \begin{slide}
1479
1480  {\large\bf\boldmath
1481   Combinations with a C-Si \hkl<1 0 0>-type interstitial
1482  }
1483
1484 \small
1485
1486 \vspace*{0.1cm}
1487
1488 Binding energy:
1489 $
1490 E_{\text{b}}=
1491 E_{\text{f}}^{\text{defect combination}}-
1492 E_{\text{f}}^{\text{C \hkl<0 0 -1> dumbbell}}-
1493 E_{\text{f}}^{\text{2nd defect}}
1494 $
1495
1496 \vspace*{0.1cm}
1497
1498 {\scriptsize
1499 \begin{tabular}{l c c c c c c}
1500 \hline
1501  $E_{\text{b}}$ [eV] & 1 & 2 & 3 & 4 & 5 & R\\
1502  \hline
1503  \hkl<0 0 -1> & {\color{red}-0.08} & -1.15 & {\color{red}-0.08} & 0.04 & -1.66 & -0.19\\
1504  \hkl<0 0 1> & 0.34 & 0.004 & -2.05 & 0.26 & -1.53 & -0.19\\
1505  \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}\\
1506  \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}\\
1507  \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}\\
1508  \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}\\
1509  \hline
1510  C substitutional (C$_{\text{S}}$) & 0.26 & -0.51 & -0.93 & -0.15 & 0.49 & -0.05\\
1511  Vacancy & -5.39 ($\rightarrow$ C$_{\text{S}}$) & -0.59 & -3.14 & -0.54 & -0.50 & -0.31\\
1512 \hline
1513 \end{tabular}
1514 }
1515
1516 \vspace*{0.3cm}
1517
1518 \footnotesize
1519
1520 \begin{minipage}[t]{3.8cm}
1521 \underline{\hkl<1 0 0> at position 1}\\[0.1cm]
1522 \includegraphics[width=3.5cm]{00-1dc/2-25.eps}
1523 \end{minipage}
1524 \begin{minipage}[t]{3.5cm}
1525 \underline{\hkl<0 -1 0> at position 1}\\[0.1cm]
1526 \includegraphics[width=3.2cm]{00-1dc/2-39.eps}
1527 \end{minipage}
1528 \begin{minipage}[t]{5.5cm}
1529 \begin{itemize}
1530  \item $E_{\text{b}}=0$ $\Leftrightarrow$ non-interacting defects\\
1531        $E_{\text{b}} \rightarrow 0$ for increasing distance (R)
1532  \item Stress compensation / increase
1533  \item Unfavored: antiparallel orientations
1534  \item Indication of energetically favored\\
1535        agglomeration
1536  \item Most favorable: C clustering
1537  \item However: High barrier ($>4\,\text{eV}$)
1538  \item $4\times{\color{cyan}-2.25}$ versus $2\times{\color{orange}-2.39}$
1539        (Entropy)
1540 \end{itemize}
1541 \end{minipage}
1542
1543 \begin{picture}(0,0)(-295,-130)
1544 \includegraphics[width=3.5cm]{comb_pos.eps}
1545 \end{picture}
1546
1547 \end{slide}
1548
1549 \begin{slide}
1550
1551  {\large\bf\boldmath
1552   Combinations of C-Si \hkl<1 0 0>-type interstitials
1553  }
1554
1555 \small
1556
1557 \vspace*{0.1cm}
1558
1559 Energetically most favorable combinations along \hkl<1 1 0>
1560
1561 \vspace*{0.1cm}
1562
1563 {\scriptsize
1564 \begin{tabular}{l c c c c c c}
1565 \hline
1566  & 1 & 2 & 3 & 4 & 5 & 6\\
1567 \hline
1568 $E_{\text{b}}$ [eV] & -2.39 & -1.88 & -0.59 & -0.31 & -0.24 & -0.21 \\
1569 C-C distance [\AA] & 1.4 & 4.6 & 6.5 & 8.6 & 10.5 & 10.8 \\
1570 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>\\
1571 \hline
1572 \end{tabular}
1573 }
1574
1575 \vspace*{0.3cm}
1576
1577 \begin{minipage}{7.0cm}
1578 \includegraphics[width=7cm]{db_along_110_cc.ps}
1579 \end{minipage}
1580 \begin{minipage}{6.0cm}
1581 \begin{itemize}
1582  \item Interaction proportional to reciprocal cube of C-C distance
1583  \item Saturation in the immediate vicinity
1584  \renewcommand\labelitemi{$\Rightarrow$}
1585  \item Agglomeration of \ci{} expected
1586  \item Absence of C clustering
1587 \end{itemize}
1588 \begin{center}
1589 {\color{blue}
1590  Consisten with initial precipitation model
1591 }
1592 \end{center}
1593 \end{minipage}
1594
1595 \vspace{0.2cm}
1596
1597 \end{slide}
1598
1599 \begin{slide}
1600
1601  {\large\bf\boldmath
1602   Combinations of substitutional C and \hkl<1 1 0> Si self-interstitials
1603  }
1604
1605  \scriptsize
1606
1607 %\begin{center}
1608 %\begin{minipage}{3.2cm}
1609 %\includegraphics[width=3cm]{sub_110_combo.eps}
1610 %\end{minipage}
1611 %\begin{minipage}{7.8cm}
1612 %\begin{tabular}{l c c c c c c}
1613 %\hline
1614 %C$_{\text{sub}}$ & \hkl<1 1 0> & \hkl<-1 1 0> & \hkl<0 1 1> & \hkl<0 -1 1> &
1615 %                   \hkl<1 0 1> & \hkl<-1 0 1> \\
1616 %\hline
1617 %1 & \RM{1} & \RM{3} & \RM{3} & \RM{1} & \RM{3} & \RM{1} \\
1618 %2 & \RM{2} & A & A & \RM{2} & C & \RM{5} \\
1619 %3 & \RM{3} & \RM{1} & \RM{3} & \RM{1} & \RM{1} & \RM{3} \\
1620 %4 & \RM{4} & B & D & E & E & D \\
1621 %5 & \RM{5} & C & A & \RM{2} & A & \RM{2} \\
1622 %\hline
1623 %\end{tabular}
1624 %\end{minipage}
1625 %\end{center}
1626
1627 %\begin{center}
1628 %\begin{tabular}{l c c c c c c c c c c}
1629 %\hline
1630 %Conf & \RM{1} & \RM{2} & \RM{3} & \RM{4} & \RM{5} & A & B & C & D & E \\
1631 %\hline
1632 %$E_{\text{f}}$ [eV]& 4.37 & 5.26 & 5.57 & 5.37 & 5.12 & 5.10 & 5.32 & 5.28 & 5.39 & 5.32 \\
1633 %$E_{\text{b}}$ [eV] & -0.97 & -0.08 & 0.22 & -0.02 & -0.23 & -0.25 & -0.02 & -0.06 & 0.05 & -0.03 \\
1634 %$r$ [nm] & 0.292 & 0.394 & 0.241 & 0.453 & 0.407 & 0.408 & 0.452 & 0.392 & 0.456 & 0.453\\
1635 %\hline
1636 %\end{tabular}
1637 %\end{center}
1638
1639 \begin{minipage}{6.0cm}
1640 \includegraphics[width=5.8cm]{c_sub_si110.ps}
1641 \end{minipage}
1642 \begin{minipage}{7cm}
1643 \scriptsize
1644 \begin{itemize}
1645  \item IBS: C may displace Si\\
1646        $\Rightarrow$ C$_{\text{sub}}$ + \hkl<1 1 0> Si self-interstitial
1647  \item Assumption:\\
1648        \hkl<1 1 0>-type $\rightarrow$ favored combination
1649  \renewcommand\labelitemi{$\Rightarrow$}
1650  \item Most favorable: \cs{} along \hkl<1 1 0> chain \si{}
1651  \item Less favorable than C-Si \hkl<1 0 0> dumbbell
1652  \item Interaction drops quickly to zero\\
1653        $\rightarrow$ low capture radius
1654 \end{itemize}
1655 \begin{center}
1656  {\color{blue}
1657  IBS process far from equilibrium\\
1658  \cs{} \& \si{} instead of thermodynamic ground state
1659  }
1660 \end{center}
1661 \end{minipage}
1662
1663 \begin{minipage}{6.5cm}
1664 \includegraphics[width=6.0cm]{162-097.ps}
1665 \begin{itemize}
1666  \item Low migration barrier
1667 \end{itemize}
1668 \end{minipage}
1669 \begin{minipage}{6.5cm}
1670 \begin{center}
1671 Ab initio MD at \degc{900}\\
1672 \includegraphics[width=3.3cm]{md_vasp_01.eps}
1673 $t=\unit[2230]{fs}$\\
1674 \includegraphics[width=3.3cm]{md_vasp_02.eps}
1675 $t=\unit[2900]{fs}$
1676 \end{center}
1677 {\color{blue}
1678 Contribution of entropy to structural formation
1679 }
1680 \end{minipage}
1681
1682 \end{slide}
1683
1684 \begin{slide}
1685
1686  {\large\bf\boldmath
1687   Migration in C-Si \hkl<1 0 0> and vacancy combinations
1688  }
1689
1690  \footnotesize
1691
1692 \vspace{0.1cm}
1693
1694 \begin{minipage}[t]{3cm}
1695 \underline{Pos 2, $E_{\text{b}}=-0.59\text{ eV}$}\\
1696 \includegraphics[width=2.8cm]{00-1dc/0-59.eps}
1697 \end{minipage}
1698 \begin{minipage}[t]{7cm}
1699 \vspace{0.2cm}
1700 \begin{center}
1701  Low activation energies\\
1702  High activation energies for reverse processes\\
1703  $\Downarrow$\\
1704  {\color{blue}C$_{\text{sub}}$ very stable}\\
1705 \vspace*{0.1cm}
1706  \hrule
1707 \vspace*{0.1cm}
1708  Without nearby \hkl<1 1 0> Si self-interstitial (IBS)\\
1709  $\Downarrow$\\
1710  {\color{blue}Formation of SiC by successive substitution by C}
1711
1712 \end{center}
1713 \end{minipage}
1714 \begin{minipage}[t]{3cm}
1715 \underline{Pos 3, $E_{\text{b}}=-3.14\text{ eV}$}\\
1716 \includegraphics[width=2.8cm]{00-1dc/3-14.eps}
1717 \end{minipage}
1718
1719
1720 \framebox{
1721 \begin{minipage}{5.9cm}
1722 \includegraphics[width=5.9cm]{vasp_mig/comb_mig_3-2_vac_fullct.ps}\\[0.6cm]
1723 \begin{center}
1724 \begin{picture}(0,0)(70,0)
1725 \includegraphics[width=1.4cm]{vasp_mig/comb_2-1_init.eps}
1726 \end{picture}
1727 \begin{picture}(0,0)(30,0)
1728 \includegraphics[width=1.4cm]{vasp_mig/comb_2-1_seq_03.eps}
1729 \end{picture}
1730 \begin{picture}(0,0)(-10,0)
1731 \includegraphics[width=1.4cm]{vasp_mig/comb_2-1_seq_06.eps}
1732 \end{picture}
1733 \begin{picture}(0,0)(-48,0)
1734 \includegraphics[width=1.4cm]{vasp_mig/comb_2-1_final.eps}
1735 \end{picture}
1736 \begin{picture}(0,0)(12.5,5)
1737 \includegraphics[width=1cm]{100_arrow.eps}
1738 \end{picture}
1739 \begin{picture}(0,0)(97,-10)
1740 \includegraphics[height=0.9cm]{001_arrow.eps}
1741 \end{picture}
1742 \end{center}
1743 \vspace{0.1cm}
1744 \end{minipage}
1745 }
1746 \begin{minipage}{0.3cm}
1747 \hfill
1748 \end{minipage}
1749 \framebox{
1750 \begin{minipage}{5.9cm}
1751 \includegraphics[width=5.9cm]{vasp_mig/comb_mig_4-2_vac_fullct.ps}\\[0.1cm]
1752 \begin{center}
1753 \begin{picture}(0,0)(60,0)
1754 \includegraphics[width=0.9cm]{vasp_mig/comb_3-1_init.eps}
1755 \end{picture}
1756 \begin{picture}(0,0)(25,0)
1757 \includegraphics[width=0.9cm]{vasp_mig/comb_3-1_seq_03.eps}
1758 \end{picture}
1759 \begin{picture}(0,0)(-20,0)
1760 \includegraphics[width=0.9cm]{vasp_mig/comb_3-1_seq_07.eps}
1761 \end{picture}
1762 \begin{picture}(0,0)(-55,0)
1763 \includegraphics[width=0.9cm]{vasp_mig/comb_3-1_final.eps}
1764 \end{picture}
1765 \begin{picture}(0,0)(12.5,5)
1766 \includegraphics[width=1cm]{100_arrow.eps}
1767 \end{picture}
1768 \begin{picture}(0,0)(95,0)
1769 \includegraphics[height=0.9cm]{001_arrow.eps}
1770 \end{picture}
1771 \end{center}
1772 \vspace{0.1cm}
1773 \end{minipage}
1774 }
1775
1776 \end{slide}
1777
1778 \begin{slide}
1779
1780  {\large\bf
1781   Conclusion of defect / migration / combined defect simulations
1782  }
1783
1784  \footnotesize
1785
1786 \vspace*{0.1cm}
1787
1788 Defect structures
1789 \begin{itemize}
1790  \item Accurately described by quantum-mechanical simulations
1791  \item Less accurate description by classical potential simulations
1792  \item Underestimated formation energy of \cs{} by classical approach
1793  \item Both methods predict same ground state: \ci{} \hkl<1 0 0> dumbbell
1794 \end{itemize}
1795
1796 Migration
1797 \begin{itemize}
1798  \item C migration pathway in Si identified
1799  \item Consistent with reorientation and diffusion experiments
1800 \end{itemize}
1801 \begin{itemize}
1802  \item Different path and ...
1803  \item overestimated barrier by classical potential calculations
1804 \end{itemize}
1805
1806 Concerning the precipitation mechanism
1807 \begin{itemize}
1808  \item Agglomeration of C-Si dumbbells energetically favorable
1809        (stress compensation)
1810  \item C-Si indeed favored compared to
1811        C$_{\text{sub}}$ \& \hkl<1 1 0> Si self-interstitial
1812  \item Possible low interaction capture radius of
1813        C$_{\text{sub}}$ \& \hkl<1 1 0> Si self-interstitial
1814  \item Low barrier for
1815        \ci{} \hkl<1 0 0> $\rightarrow$ \cs{} \& \si{} \hkl<1 1 0>
1816  \item In absence of nearby \hkl<1 1 0> Si self-interstitial:
1817        C-Si \hkl<1 0 0> + Vacancy $\rightarrow$ C$_{\text{sub}}$ (SiC)
1818 \end{itemize}
1819 \begin{center}
1820 {\color{blue}Results suggest increased participation of \cs}
1821 \end{center}
1822
1823 \end{slide}
1824
1825 \begin{slide}
1826
1827  {\large\bf
1828   Silicon carbide precipitation simulations
1829  }
1830
1831  \small
1832
1833 {\scriptsize
1834  \begin{pspicture}(0,0)(12,6.5)
1835   % nodes
1836   \rput(3.5,5.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
1837    \parbox{7cm}{
1838    \begin{itemize}
1839     \item Create c-Si volume
1840     \item Periodc boundary conditions
1841     \item Set requested $T$ and $p=0\text{ bar}$
1842     \item Equilibration of $E_{\text{kin}}$ and $E_{\text{pot}}$
1843    \end{itemize}
1844   }}}}
1845   \rput(3.5,2.7){\rnode{insert}{\psframebox[fillstyle=solid,fillcolor=lachs]{
1846    \parbox{7cm}{
1847    Insertion of C atoms at constant T
1848    \begin{itemize}
1849     \item total simulation volume {\pnode{in1}}
1850     \item volume of minimal SiC precipitate {\pnode{in2}}
1851     \item volume consisting of Si atoms to form a minimal {\pnode{in3}}\\
1852           precipitate
1853    \end{itemize}
1854   }}}}
1855   \rput(3.5,1){\rnode{cool}{\psframebox[fillstyle=solid,fillcolor=lbb]{
1856    \parbox{7.0cm}{
1857    Run for 100 ps followed by cooling down to $20\, ^{\circ}\textrm{C}$
1858   }}}}
1859   \ncline[]{->}{init}{insert}
1860   \ncline[]{->}{insert}{cool}
1861   \psframe[fillstyle=solid,fillcolor=white](7.5,0.7)(13.5,6.3)
1862   \rput(7.8,6){\footnotesize $V_1$}
1863   \psframe[fillstyle=solid,fillcolor=lightgray](9,2)(12,5)
1864   \rput(9.2,4.85){\tiny $V_2$}
1865   \psframe[fillstyle=solid,fillcolor=gray](9.25,2.25)(11.75,4.75)
1866   \rput(9.55,4.45){\footnotesize $V_3$}
1867   \rput(7.9,3.2){\pnode{ins1}}
1868   \rput(9.22,2.8){\pnode{ins2}}
1869   \rput(11.0,2.4){\pnode{ins3}}
1870   \ncline[]{->}{in1}{ins1}
1871   \ncline[]{->}{in2}{ins2}
1872   \ncline[]{->}{in3}{ins3}
1873  \end{pspicture}
1874 }
1875
1876 \begin{itemize}
1877  \item Restricted to classical potential simulations
1878  \item $V_2$ and $V_3$ considered due to low diffusion
1879  \item Amount of C atoms: 6000
1880        ($r_{\text{prec}}\approx 3.1\text{ nm}$, IBS: 2 ... 4 nm)
1881  \item Simulation volume: $31\times 31\times 31$ unit cells
1882        (238328 Si atoms)
1883 \end{itemize}
1884
1885 \end{slide}
1886
1887 \begin{slide}
1888
1889  {\large\bf\boldmath
1890   Silicon carbide precipitation simulations at $450\,^{\circ}\mathrm{C}$ as in IBS
1891  }
1892
1893  \small
1894
1895 \begin{minipage}{6.5cm}
1896 \includegraphics[width=6.4cm]{sic_prec_450_si-si_c-c.ps}
1897 \end{minipage}
1898 \begin{minipage}{6.5cm}
1899 \includegraphics[width=6.4cm]{sic_prec_450_energy.ps}
1900 \end{minipage}
1901
1902 \begin{minipage}{6.5cm}
1903 \includegraphics[width=6.4cm]{sic_prec_450_si-c.ps}
1904 \end{minipage}
1905 \begin{minipage}{6.5cm}
1906 \scriptsize
1907 \underline{Low C concentration ($V_1$)}\\
1908 \hkl<1 0 0> C-Si dumbbell dominated structure
1909 \begin{itemize}
1910  \item Si-C bumbs around 0.19 nm
1911  \item C-C peak at 0.31 nm (as expected in 3C-SiC):\\
1912        concatenated dumbbells of various orientation
1913  \item Si-Si NN distance stretched to 0.3 nm
1914 \end{itemize}
1915 {\color{blue}$\Rightarrow$ C atoms in proper 3C-SiC distance first}\\
1916 \underline{High C concentration ($V_2$, $V_3$)}\\
1917 High amount of strongly bound C-C bonds\\
1918 Defect density $\uparrow$ $\Rightarrow$ considerable amount of damage\\
1919 Only short range order observable\\
1920 {\color{blue}$\Rightarrow$ amorphous SiC-like phase}
1921 \end{minipage}
1922
1923 \end{slide}
1924
1925 \begin{slide}
1926
1927  {\large\bf\boldmath
1928   Silicon carbide precipitation simulations at $450\,^{\circ}\mathrm{C}$ as in IBS
1929  }
1930
1931  \small
1932
1933 \begin{minipage}{6.5cm}
1934 \includegraphics[width=6.4cm]{sic_prec_450_si-si_c-c.ps}
1935 \end{minipage}
1936 \begin{minipage}{6.5cm}
1937 \includegraphics[width=6.4cm]{sic_prec_450_energy.ps}
1938 \end{minipage}
1939
1940 \begin{minipage}{6.5cm}
1941 \includegraphics[width=6.4cm]{sic_prec_450_si-c.ps}
1942 \end{minipage}
1943 \begin{minipage}{6.5cm}
1944 \scriptsize
1945 \underline{Low C concentration ($V_1$)}\\
1946 \hkl<1 0 0> C-Si dumbbell dominated structure
1947 \begin{itemize}
1948  \item Si-C bumbs around 0.19 nm
1949  \item C-C peak at 0.31 nm (as expected in 3C-SiC):\\
1950        concatenated dumbbells of various orientation
1951  \item Si-Si NN distance stretched to 0.3 nm
1952 \end{itemize}
1953 {\color{blue}$\Rightarrow$ C atoms in proper 3C-SiC distance first}\\
1954 \underline{High C concentration ($V_2$, $V_3$)}\\
1955 High amount of strongly bound C-C bonds\\
1956 Defect density $\uparrow$ $\Rightarrow$ considerable amount of damage\\
1957 Only short range order observable\\
1958 {\color{blue}$\Rightarrow$ amorphous SiC-like phase}
1959 \end{minipage}
1960
1961 \begin{pspicture}(0,0)(0,0)
1962 \rput(6.7,5.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=white]{
1963 \begin{minipage}{10cm}
1964 \small
1965 {\color{red}\bf 3C-SiC formation fails to appear}
1966 \begin{itemize}
1967 \item Low C concentration simulations
1968  \begin{itemize}
1969   \item Formation of \ci{} indeed occurs
1970   \item Agllomeration not observed
1971  \end{itemize}
1972 \item High C concentration simulations
1973  \begin{itemize}
1974   \item Amorphous SiC-like structure\\
1975         (not expected at prevailing temperatures)
1976   \item Rearrangement and transition into 3C-SiC structure missing
1977  \end{itemize}
1978 \end{itemize}
1979 \end{minipage}
1980  }}}
1981 \end{pspicture}
1982
1983 \end{slide}
1984
1985 \begin{slide}
1986
1987  {\large\bf
1988   Limitations of molecular dynamics and short range potentials
1989  }
1990
1991 \footnotesize
1992
1993 \vspace{0.2cm}
1994
1995 \underline{Time scale problem of MD}\\[0.2cm]
1996 Minimize integration error\\
1997 $\Rightarrow$ discretization considerably smaller than
1998               reciprocal of fastest vibrational mode\\[0.1cm]
1999 Order of fastest vibrational mode: $10^{13} - 10^{14}\text{ Hz}$\\
2000 $\Rightarrow$ suitable choice of time step:
2001               $\tau=1\text{ fs}=10^{-15}\text{ s}$\\
2002 $\Rightarrow$ {\color{red}\underline{slow}} phase space propagation\\[0.1cm]
2003 Several local minima in energy surface separated by large energy barriers\\
2004 $\Rightarrow$ transition event corresponds to a multiple
2005               of vibrational periods\\
2006 $\Rightarrow$ phase transition made up of {\color{red}\underline{many}}
2007               infrequent transition events\\[0.1cm]
2008 {\color{blue}Accelerated methods:}
2009 \underline{Temperature accelerated} MD (TAD), self-guided MD \ldots
2010
2011 \vspace{0.3cm}
2012
2013 \underline{Limitations related to the short range potential}\\[0.2cm]
2014 Cut-off function pushing forces and energies to zero between 1$^{\text{st}}$
2015 and 2$^{\text{nd}}$ next neighbours\\
2016 $\Rightarrow$ overestimated unphysical high forces of next neighbours
2017
2018 \vspace{0.3cm}
2019
2020 \framebox{
2021 \color{red}
2022 Potential enhanced problem of slow phase space propagation
2023 }
2024
2025 \vspace{0.3cm}
2026
2027 \underline{Approach to the (twofold) problem}\\[0.2cm]
2028 Increased temperature simulations without TAD corrections\\
2029 (accelerated methods or higher time scales exclusively not sufficient)
2030
2031 \begin{picture}(0,0)(-260,-30)
2032 \framebox{
2033 \begin{minipage}{4.2cm}
2034 \tiny
2035 \begin{center}
2036 \vspace{0.03cm}
2037 \underline{IBS}
2038 \end{center}
2039 \begin{itemize}
2040 \item 3C-SiC also observed for higher T
2041 \item higher T inside sample
2042 \item structural evolution vs.\\
2043       equilibrium properties
2044 \end{itemize}
2045 \end{minipage}
2046 }
2047 \end{picture}
2048
2049 \begin{picture}(0,0)(-305,-155)
2050 \framebox{
2051 \begin{minipage}{2.5cm}
2052 \tiny
2053 \begin{center}
2054 retain proper\\
2055 thermodynmic sampling
2056 \end{center}
2057 \end{minipage}
2058 }
2059 \end{picture}
2060
2061 \end{slide}
2062
2063 \begin{slide}
2064
2065  {\large\bf
2066   Increased temperature simulations at low C concentration
2067  }
2068
2069 \small
2070
2071 \begin{minipage}{6.5cm}
2072 \includegraphics[width=6.4cm]{tot_pc_thesis.ps}
2073 \end{minipage}
2074 \begin{minipage}{6.5cm}
2075 \includegraphics[width=6.4cm]{tot_pc3_thesis.ps}
2076 \end{minipage}
2077
2078 \begin{minipage}{6.5cm}
2079 \includegraphics[width=6.4cm]{tot_pc2_thesis.ps}
2080 \end{minipage}
2081 \begin{minipage}{6.5cm}
2082 \scriptsize
2083  \underline{Si-C bonds:}
2084  \begin{itemize}
2085   \item Vanishing cut-off artifact (above $1650\,^{\circ}\mathrm{C}$)
2086   \item Structural change: C-Si \hkl<1 0 0> $\rightarrow$ C$_{\text{sub}}$
2087  \end{itemize}
2088  \underline{Si-Si bonds:}
2089  {\color{blue}Si-C$_{\text{sub}}$-Si} along \hkl<1 1 0>
2090  ($\rightarrow$ 0.325 nm)\\[0.1cm]
2091  \underline{C-C bonds:}
2092  \begin{itemize}
2093   \item C-C next neighbour pairs reduced (mandatory)
2094   \item Peak at 0.3 nm slightly shifted
2095         \begin{itemize}
2096          \item C-Si \hkl<1 0 0> combinations (dashed arrows)\\
2097                $\rightarrow$ C-Si \hkl<1 0 0> \& C$_{\text{sub}}$
2098                combinations (|)\\
2099                $\rightarrow$ pure {\color{blue}C$_{\text{sub}}$ combinations}
2100                ($\downarrow$)
2101          \item Range [|-$\downarrow$]:
2102                {\color{blue}C$_{\text{sub}}$ \& C$_{\text{sub}}$
2103                with nearby Si$_{\text{I}}$}
2104         \end{itemize}
2105  \end{itemize}
2106 \end{minipage}
2107
2108 \begin{picture}(0,0)(-330,-74)
2109 \color{blue}
2110 \framebox{
2111 \begin{minipage}{1.6cm}
2112 \tiny
2113 \begin{center}
2114 stretched SiC\\[-0.1cm]
2115 in c-Si
2116 \end{center}
2117 \end{minipage}
2118 }
2119 \end{picture}
2120
2121 \end{slide}
2122
2123 \begin{slide}
2124
2125  {\large\bf
2126   Increased temperature simulations at low C concentration
2127  }
2128
2129 \small
2130
2131 \begin{minipage}{6.5cm}
2132 \includegraphics[width=6.4cm]{tot_pc_thesis.ps}
2133 \end{minipage}
2134 \begin{minipage}{6.5cm}
2135 \includegraphics[width=6.4cm]{tot_pc3_thesis.ps}
2136 \end{minipage}
2137
2138 \begin{minipage}{6.5cm}
2139 \includegraphics[width=6.4cm]{tot_pc2_thesis.ps}
2140 \end{minipage}
2141 \begin{minipage}{6.5cm}
2142 \scriptsize
2143  \underline{Si-C bonds:}
2144  \begin{itemize}
2145   \item Vanishing cut-off artifact (above $1650\,^{\circ}\mathrm{C}$)
2146   \item Structural change: C-Si \hkl<1 0 0> $\rightarrow$ C$_{\text{sub}}$
2147  \end{itemize}
2148  \underline{Si-Si bonds:}
2149  {\color{blue}Si-C$_{\text{sub}}$-Si} along \hkl<1 1 0>
2150  ($\rightarrow$ 0.325 nm)\\[0.1cm]
2151  \underline{C-C bonds:}
2152  \begin{itemize}
2153   \item C-C next neighbour pairs reduced (mandatory)
2154   \item Peak at 0.3 nm slightly shifted
2155         \begin{itemize}
2156          \item C-Si \hkl<1 0 0> combinations (dashed arrows)\\
2157                $\rightarrow$ C-Si \hkl<1 0 0> \& C$_{\text{sub}}$
2158                combinations (|)\\
2159                $\rightarrow$ pure {\color{blue}C$_{\text{sub}}$ combinations}
2160                ($\downarrow$)
2161          \item Range [|-$\downarrow$]:
2162                {\color{blue}C$_{\text{sub}}$ \& C$_{\text{sub}}$
2163                with nearby Si$_{\text{I}}$}
2164         \end{itemize}
2165  \end{itemize}
2166 \end{minipage}
2167
2168 %\begin{picture}(0,0)(-330,-74)
2169 %\color{blue}
2170 %\framebox{
2171 %\begin{minipage}{1.6cm}
2172 %\tiny
2173 %\begin{center}
2174 %stretched SiC\\[-0.1cm]
2175 %in c-Si
2176 %\end{center}
2177 %\end{minipage}
2178 %}
2179 %\end{picture}
2180
2181 \begin{pspicture}(0,0)(0,0)
2182 \rput(6.7,5.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=white]{
2183 \begin{minipage}{10cm}
2184 \small
2185 {\color{blue}\bf Stretched SiC in c-Si}
2186 \begin{itemize}
2187 \item Consistent to precipitation model involving \cs{}
2188 \item Explains annealing behavior of high/low T C implants
2189       \begin{itemize}
2190        \item Low T: highly mobiel \ci{}
2191        \item High T: stable configurations of \cs{}
2192       \end{itemize}
2193 \end{itemize}
2194 $\Rightarrow$ High T $\leftrightarrow$ IBS conditions far from equilibrium\\
2195 $\Rightarrow$ Precipitation mechanism involving \cs{}
2196 \end{minipage}
2197  }}}
2198 \end{pspicture}
2199
2200 \end{slide}
2201
2202 \begin{slide}
2203
2204  {\large\bf
2205   Increased temperature simulations at high C concentration
2206  }
2207
2208 \footnotesize
2209
2210 \begin{minipage}{6.5cm}
2211 \includegraphics[width=6.4cm]{12_pc_thesis.ps}
2212 \end{minipage}
2213 \begin{minipage}{6.5cm}
2214 \includegraphics[width=6.4cm]{12_pc_c_thesis.ps}
2215 \end{minipage}
2216
2217 \vspace{0.1cm}
2218
2219 \scriptsize
2220
2221 \framebox{
2222 \begin{minipage}[t]{6.0cm}
2223 0.186 nm: Si-C pairs $\uparrow$\\
2224 (as expected in 3C-SiC)\\[0.2cm]
2225 0.282 nm: Si-C-C\\[0.2cm]
2226 $\approx$0.35 nm: C-Si-Si
2227 \end{minipage}
2228 }
2229 \begin{minipage}{0.2cm}
2230 \hfill
2231 \end{minipage}
2232 \framebox{
2233 \begin{minipage}[t]{6.0cm}
2234 0.15 nm: C-C pairs $\uparrow$\\
2235 (as expected in graphite/diamond)\\[0.2cm]
2236 0.252 nm: C-C-C (2$^{\text{nd}}$ NN for diamond)\\[0.2cm]
2237 0.31 nm: shifted towards 0.317 nm $\rightarrow$ C-Si-C
2238 \end{minipage}
2239 }
2240
2241 \begin{itemize}
2242 \item Decreasing cut-off artifact
2243 \item {\color{red}Amorphous} SiC-like phase remains
2244 \item High amount of {\color{red}damage} \& alignement to c-Si host matrix lost
2245 \item Slightly sharper peaks $\Rightarrow$ indicate slight {\color{blue}acceleration of dynamics} due to temperature
2246 \end{itemize}
2247
2248 \vspace{-0.1cm}
2249
2250 \begin{center}
2251 {\color{blue}
2252 \framebox{
2253 {\color{black}
2254 High C \& small $V$ \& short $t$
2255 $\Rightarrow$
2256 }
2257 Slow restructuring due to strong C-C bonds
2258 {\color{black}
2259 $\Leftarrow$
2260 High C \& low T implants
2261 }
2262 }
2263 }
2264 \end{center}
2265
2266 \end{slide}
2267
2268 \begin{slide}
2269
2270  {\large\bf
2271   Summary and Conclusions
2272  }
2273
2274  \scriptsize
2275
2276 %\vspace{0.1cm}
2277
2278 \framebox{
2279 \begin{minipage}[t]{12.9cm}
2280  \underline{Pecipitation simulations}
2281  \begin{itemize}
2282   \item High C concentration $\rightarrow$ amorphous SiC like phase
2283   \item Problem of potential enhanced slow phase space propagation
2284   \item Low T $\rightarrow$ C-Si \hkl<1 0 0> dumbbell dominated structure
2285   \item High T $\rightarrow$ C$_{\text{sub}}$ dominated structure
2286   \item High T necessary to simulate IBS conditions (far from equilibrium)
2287   \item Precipitation by successive agglomeration of \cs (epitaxy)
2288   \item \si{}: vehicle to form \cs{} \& supply of Si \& stress compensation
2289         (stretched SiC, interface)
2290  \end{itemize}
2291 \end{minipage}
2292 }
2293
2294 %\vspace{0.1cm}
2295
2296 \framebox{
2297 \begin{minipage}{12.9cm}
2298  \underline{Defects}
2299  \begin{itemize}
2300    \item DFT / EA
2301         \begin{itemize}
2302          \item Point defects excellently / fairly well described
2303                by DFT / EA
2304          \item C$_{\text{sub}}$ drastically underestimated by EA
2305          \item EA predicts correct ground state:
2306                C$_{\text{sub}}$ \& \si{} $>$ \ci{}
2307          \item Identified migration path explaining
2308                diffusion and reorientation experiments by DFT
2309          \item EA fails to describe \ci{} migration:
2310                Wrong path \& overestimated barrier
2311         \end{itemize}
2312    \item Combinations of defects
2313          \begin{itemize}
2314           \item Agglomeration of point defects energetically favorable
2315                 by compensation of stress
2316           \item Formation of C-C unlikely
2317           \item C$_{\text{sub}}$ favored conditions (conceivable in IBS)
2318           \item \ci{} \hkl<1 0 0> $\leftrightarrow$ \cs{} \& \si{} \hkl<1 1 0>\\
2319                 Low barrier (\unit[0.77]{eV}) \& low capture radius
2320         \end{itemize}
2321  \end{itemize}
2322 \end{minipage}
2323 }
2324
2325 \begin{center}
2326 {\color{blue}
2327 \framebox{Precipitation by successive agglomeration of \cs{}}
2328 }
2329 \end{center}
2330
2331 \end{slide}
2332
2333 \begin{slide}
2334
2335  {\large\bf
2336   Acknowledgements
2337  }
2338
2339  \vspace{0.1cm}
2340
2341  \small
2342
2343  Thanks to \ldots
2344
2345  \underline{Augsburg}
2346  \begin{itemize}
2347   \item Prof. B. Stritzker (accomodation at EP \RM{4})
2348   \item Ralf Utermann (EDV)
2349  \end{itemize}
2350
2351  \underline{Helsinki}
2352  \begin{itemize}
2353   \item Prof. K. Nordlund (MD)
2354  \end{itemize}
2355
2356  \underline{Munich}
2357  \begin{itemize}
2358   \item Bayerische Forschungsstiftung (financial support)
2359  \end{itemize}
2360
2361  \underline{Paderborn}
2362  \begin{itemize}
2363   \item Prof. J. Lindner (SiC)
2364   \item Prof. G. Schmidt (DFT + financial support)
2365   \item Dr. E. Rauls (DFT + SiC)
2366   \item Dr. S. Sanna (VASP)
2367  \end{itemize}
2368
2369 \vspace{0.2cm}
2370
2371 \begin{center}
2372 \framebox{
2373 \bf Thank you for your attention!
2374 }
2375 \end{center}
2376
2377 \end{slide}
2378
2379 \end{document}
2380
2381 \fi