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{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 % Phase diagram of the C/Si system\\
 133 %}
 134
 135 \vspace*{0.2cm}
 136
 137 \begin{minipage}{7cm}
 138 \includegraphics[width=6.5cm]{si-c_phase.eps}
 139 \begin{center}
 140 {\tiny
 141 R. I. Scace and G. A. Slack, J. Chem. Phys. 30, 1551 (1959)
 142 }
 143 \end{center}
 144 \begin{pspicture}(0,0)(0,0)
 145 \psellipse[linecolor=blue,linewidth=0.1cm](3.55,4.0)(0.5,2.9)
 146 \end{pspicture}
 147 \end{minipage}
 148 \begin{minipage}{6cm}
 149 {\bf Phase diagram of the C/Si system}\\[0.2cm]
 150 {\color{blue}Stoichiometric composition}
 151 \begin{itemize}
 152 \item only chemical stable compound
 153 \item wide band gap semiconductor\\
 154       \underline{silicon carbide}, SiC
 155 \end{itemize}
 156 \end{minipage}
 157
 158 \end{slide}
 159
 160 % motivation / properties / applications of silicon carbide
 161
 162 \begin{slide}
 163
 164 \small
 165
 166 \begin{pspicture}(0,0)(13.5,5)
 167
 168  \psframe*[linecolor=hb](0,0)(13.5,5)
 169
 170  \pspolygon[linecolor=hlbb,fillcolor=hlbb,fillstyle=solid](5.5,1)(7,1)(7,3)(5.5,3)
 171  \pspolygon[linecolor=hlbb,fillcolor=hlbb,fillstyle=solid](6.75,0.5)(8,2)(8,2)(6.75,3.5)
 172
 173  \rput[lt](0.2,4.6){\color{gray}PROPERTIES}
 174
 175  \rput[lt](0.5,4){wide band gap}
 176  \rput[lt](0.5,3.5){high electric breakdown field}
 177  \rput[lt](0.5,3){good electron mobility}
 178  \rput[lt](0.5,2.5){high electron saturation drift velocity}
 179  \rput[lt](0.5,2){high thermal conductivity}
 180
 181  \rput[lt](0.5,1.5){hard and mechanically stable}
 182  \rput[lt](0.5,1){chemically inert}
 183
 184  \rput[lt](0.5,0.5){radiation hardness}
 185
 186  \rput[rt](13.3,4.6){\color{gray}APPLICATIONS}
 187
 188  \rput[rt](13,3.85){high-temperature, high power}
 189  \rput[rt](13,3.5){and high-frequency}
 190  \rput[rt](13,3.15){electronic and optoelectronic devices}
 191
 192  \rput[rt](13,2.35){material suitable for extreme conditions}
 193  \rput[rt](13,2){microelectromechanical systems}
 194  \rput[rt](13,1.65){abrasives, cutting tools, heating elements}
 195
 196  \rput[rt](13,0.85){first wall reactor material, detectors}
 197  \rput[rt](13,0.5){and electronic devices for space}
 198
 199 \end{pspicture}
 200
 201 \begin{picture}(0,0)(-3,68)
 202 \includegraphics[width=2.6cm]{wide_band_gap.eps}
 203 \end{picture}
 204 \begin{picture}(0,0)(-285,-162)
 205 \includegraphics[width=3.38cm]{sic_led.eps}
 206 \end{picture}
 207 \begin{picture}(0,0)(-195,-162)
 208 \includegraphics[width=2.8cm]{6h-sic_3c-sic.eps}
 209 \end{picture}
 210 \begin{picture}(0,0)(-313,65)
 211 \includegraphics[width=2.2cm]{infineon_schottky.eps}
 212 \end{picture}
 213 \begin{picture}(0,0)(-220,65)
 214 \includegraphics[width=2.9cm]{sic_wechselrichter_ise.eps}
 215 \end{picture}
 216 \begin{picture}(0,0)(0,-160)
 217 \includegraphics[width=3.0cm]{sic_proton.eps}
 218 \end{picture}
 219 \begin{picture}(0,0)(-60,65)
 220 \includegraphics[width=3.4cm]{sic_switch.eps}
 221 \end{picture}
 222
 223 \end{slide}
 224
 225 % motivation
 226
 227 \begin{slide}
 228
 229  {\large\bf
 230   Polytypes of SiC
 231  }
 232
 233  \vspace{4cm}
 234
 235  \small
 236
 237 \begin{tabular}{l c c c c c c}
 238 \hline
 239  & 3C-SiC & 4H-SiC & 6H-SiC & Si & GaN & Diamond\\
 240 \hline
 241 Hardness [Mohs] & \multicolumn{3}{c}{------ 9.6 ------}& 6.5 & - & 10 \\
 242 Band gap [eV] & 2.36 & 3.23 & 3.03 & 1.12 & 3.39 & 5.5 \\
 243 Break down field [$10^6$ V/cm] & 4 & 3 & 3.2 & 0.6 & 5 & 10 \\
 244 Saturation drift velocity [$10^7$ cm/s] & 2.5 & 2.0 & 2.0 & 1 & 2.7 & 2.7 \\
 245 Electron mobility [cm$^2$/Vs] & 800 & 900 & 400 & 1100 & 900 & 2200 \\
 246 Hole mobility [cm$^2$/Vs] & 320 & 120 & 90 & 420 & 150 & 1600 \\
 247 Thermal conductivity [W/cmK] & 5.0 & 4.9 & 4.9 & 1.5 & 1.3 & 22 \\
 248 \hline
 249 \end{tabular}
 250
 251 {\tiny
 252  Values for $T=300$ K
 253 }
 254
 255 \begin{picture}(0,0)(-160,-155)
 256  \includegraphics[width=7cm]{polytypes.eps}
 257 \end{picture}
 258 \begin{picture}(0,0)(-10,-185)
 259  \includegraphics[width=3.8cm]{cubic_hex.eps}\\
 260 \end{picture}
 261 \begin{picture}(0,0)(-10,-175)
 262  {\tiny cubic (twist)}
 263 \end{picture}
 264 \begin{picture}(0,0)(-60,-175)
 265  {\tiny hexagonal (no twist)}
 266 \end{picture}
 267 \begin{pspicture}(0,0)(0,0)
 268 \psellipse[linecolor=green](5.7,3.03)(0.4,0.5)
 269 \end{pspicture}
 270 \begin{pspicture}(0,0)(0,0)
 271 \psellipse[linecolor=green](5.6,1.68)(0.4,0.2)
 272 \end{pspicture}
 273 \begin{pspicture}(0,0)(0,0)
 274 \psellipse[linecolor=red](10.7,1.13)(0.4,0.2)
 275 \end{pspicture}
 276
 277 \end{slide}
 278
 279 % fabrication
 280
 281 \begin{slide}
 282
 283  {\large\bf
 284   Fabrication of silicon carbide
 285  }
 286
 287  \small
 288
 289  \vspace{4pt}
 290
 291  SiC - \emph{Born from the stars, perfected on earth.}
 292
 293  IBS also here!
 294
 295  \vspace{4pt}
 296
 297  Conventional thin film SiC growth:
 298  \begin{itemize}
 299   \item \underline{Sublimation growth using the modified Lely method}
 300         \begin{itemize}
 301          \item SiC single-crystalline seed at $T=1800 \, ^{\circ} \text{C}$
 302          \item Surrounded by polycrystalline SiC in a graphite crucible\\
 303                at $T=2100-2400 \, ^{\circ} \text{C}$
 304          \item Deposition of supersaturated vapor on cooler seed crystal
 305         \end{itemize}
 306   \item \underline{Homoepitaxial growth using CVD}
 307         \begin{itemize}
 308          \item Step-controlled epitaxy on off-oriented 6H-SiC substrates
 309          \item C$_3$H$_8$/SiH$_4$/H$_2$ at $1100-1500 \, ^{\circ} \text{C}$
 310          \item Angle, temperature $\rightarrow$ 3C/6H/4H-SiC
 311         \end{itemize}
 312   \item \underline{Heteroepitaxial growth of 3C-SiC on Si using CVD/MBE}
 313         \begin{itemize}
 314          \item Two steps: carbonization and growth
 315          \item $T=650-1050 \, ^{\circ} \text{C}$
 316          \item SiC/Si lattice mismatch $\approx$ 20 \%
 317          \item Quality and size not yet sufficient
 318         \end{itemize}
 319  \end{itemize}
 320
 321  \begin{picture}(0,0)(-280,-65)
 322   \includegraphics[width=3.8cm]{6h-sic_3c-sic.eps}
 323  \end{picture}
 324  \begin{picture}(0,0)(-280,-55)
 325   \begin{minipage}{5cm}
 326   {\tiny
 327    NASA: 6H-SiC and 3C-SiC LED\\[-7pt]
 328    on 6H-SiC substrate
 329   }
 330   \end{minipage}
 331  \end{picture}
 332  \begin{picture}(0,0)(-265,-150)
 333   \includegraphics[width=2.4cm]{m_lely.eps}
 334  \end{picture}
 335  \begin{picture}(0,0)(-333,-175)
 336   \begin{minipage}{5cm}
 337   {\tiny
 338    1. Lid\\[-7pt]
 339    2. Heating\\[-7pt]
 340    3. Source\\[-7pt]
 341    4. Crucible\\[-7pt]
 342    5. Insulation\\[-7pt]
 343    6. Seed crystal
 344   }
 345   \end{minipage}
 346  \end{picture}
 347  \begin{picture}(0,0)(-230,-35)
 348  \framebox{
 349  {\footnotesize\color{blue}\bf Hex: micropipes along c-axis}
 350  }
 351  \end{picture}
 352  \begin{picture}(0,0)(-230,-10)
 353  \framebox{
 354  \begin{minipage}{3cm}
 355  {\footnotesize\color{blue}\bf 3C-SiC fabrication\\
 356                                less advanced}
 357  \end{minipage}
 358  }
 359  \end{picture}
 360
 361 \end{slide}
 362
 363 % contents
 364
 365 \begin{slide}
 366
 367 {\large\bf
 368  Outline
 369 }
 370
 371  \begin{itemize}
 372   \item Implantation of C in Si --- Overview of experimental observations
 373   \item Utilized simulation techniques and modeled problems
 374         \begin{itemize}
 375          \item {\color{blue}Diploma thesis}\\
 376                \underline{Monte Carlo} simulations
 377                modeling the selforganization process
 378                leading to periodic arrays of nanometric amorphous SiC
 379                precipitates
 380          \item {\color{blue}Doctoral studies}\\
 381                Classical potential \underline{molecular dynamics} simulations
 382                \ldots\\
 383                \underline{Density functional theory} calculations
 384                \ldots\\[0.2cm]
 385                \ldots on defects and SiC precipitation in Si
 386         \end{itemize}
 387   \item Summary / Conclusion / Outlook
 388  \end{itemize}
 389
 390 \end{slide}
 391
 392
 393
 394 \end{document}
 395
 396 \begin{slide}
 397
 398  {\large\bf
 399   Fabrication of silicon carbide
 400  }
 401
 402  \small
 403
 404  Alternative approach:
 405  Ion beam synthesis (IBS) of burried 3C-SiC layers in Si\hkl(1 0 0)
 406  \begin{itemize}
 407   \item \underline{Implantation step 1}\\
 408         180 keV C$^+$, $D=7.9\times 10^{17}$ cm$^{-2}$, $T_{\text{i}}=500\,^{\circ}\mathrm{C}$\\
 409         $\Rightarrow$ box-like distribution of equally sized
 410                        and epitactically oriented SiC precipitates
 411
 412   \item \underline{Implantation step 2}\\
 413         180 keV C$^+$, $D=0.6\times 10^{17}$ cm$^{-2}$, $T_{\text{i}}=250\,^{\circ}\mathrm{C}$\\
 414         $\Rightarrow$ destruction of SiC nanocrystals
 415                       in growing amorphous interface layers
 416   \item \underline{Annealing}\\
 417         $T=1250\,^{\circ}\mathrm{C}$, $t=10\,\text{h}$\\
 418         $\Rightarrow$ homogeneous, stoichiometric SiC layer
 419                       with sharp interfaces
 420  \end{itemize}
 421
 422  \begin{minipage}{6.3cm}
 423  \includegraphics[width=6cm]{ibs_3c-sic.eps}\\[-0.2cm]
 424  {\tiny
 425   XTEM micrograph of single crystalline 3C-SiC in Si\hkl(1 0 0)
 426  }
 427  \end{minipage}
 428 \framebox{
 429  \begin{minipage}{6.3cm}
 430  \begin{center}
 431  {\color{blue}
 432   Precipitation mechanism not yet fully understood!
 433  }
 434  \renewcommand\labelitemi{$\Rightarrow$}
 435  \small
 436  \underline{Understanding the SiC precipitation}
 437  \begin{itemize}
 438   \item significant technological progress in SiC thin film formation
 439   \item perspectives for processes relying upon prevention of SiC precipitation
 440  \end{itemize}
 441  \end{center}
 442  \end{minipage}
 443 }
 444
 445 \end{slide}
 446
 447
 448 \begin{slide}
 449
 450  {\large\bf
 451   Supposed precipitation mechanism of SiC in Si
 452  }
 453
 454  \scriptsize
 455
 456  \vspace{0.1cm}
 457
 458  \begin{minipage}{3.8cm}
 459  Si \& SiC lattice structure\\[0.2cm]
 460  \includegraphics[width=3.5cm]{sic_unit_cell.eps}\\[-0.3cm]
 461  \hrule
 462  \end{minipage}
 463  \hspace{0.6cm}
 464  \begin{minipage}{3.8cm}
 465  \begin{center}
 466  \includegraphics[width=3.3cm]{tem_c-si-db.eps}
 467  \end{center}
 468  \end{minipage}
 469  \hspace{0.6cm}
 470  \begin{minipage}{3.8cm}
 471  \begin{center}
 472  \includegraphics[width=3.3cm]{tem_3c-sic.eps}
 473  \end{center}
 474  \end{minipage}
 475
 476  \begin{minipage}{4cm}
 477  \begin{center}
 478  C-Si dimers (dumbbells)\\[-0.1cm]
 479  on Si interstitial sites
 480  \end{center}
 481  \end{minipage}
 482  \hspace{0.2cm}
 483  \begin{minipage}{4.2cm}
 484  \begin{center}
 485  Agglomeration of C-Si dumbbells\\[-0.1cm]
 486  $\Rightarrow$ dark contrasts
 487  \end{center}
 488  \end{minipage}
 489  \hspace{0.2cm}
 490  \begin{minipage}{4cm}
 491  \begin{center}
 492  Precipitation of 3C-SiC in Si\\[-0.1cm]
 493  $\Rightarrow$ Moir\'e fringes\\[-0.1cm]
 494  \& release of Si self-interstitials
 495  \end{center}
 496  \end{minipage}
 497
 498  \begin{minipage}{3.8cm}
 499  \begin{center}
 500  \includegraphics[width=3.3cm]{sic_prec_seq_01.eps}
 501  \end{center}
 502  \end{minipage}
 503  \hspace{0.6cm}
 504  \begin{minipage}{3.8cm}
 505  \begin{center}
 506  \includegraphics[width=3.3cm]{sic_prec_seq_02.eps}
 507  \end{center}
 508  \end{minipage}
 509  \hspace{0.6cm}
 510  \begin{minipage}{3.8cm}
 511  \begin{center}
 512  \includegraphics[width=3.3cm]{sic_prec_seq_03.eps}
 513  \end{center}
 514  \end{minipage}
 515
 516 \begin{pspicture}(0,0)(0,0)
 517 \psline[linewidth=4pt]{->}(8.5,2)(9.0,2)
 518 \psellipse[linecolor=blue](11.5,5.8)(0.3,0.5)
 519 \rput{-20}{\psellipse[linecolor=blue](3.3,8.1)(0.3,0.5)}
 520 \psline[linewidth=4pt]{->}(4.0,2)(4.5,2)
 521 \rput(12.7,0.3){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
 522  $4a_{\text{Si}}=5a_{\text{SiC}}$
 523  }}}
 524 \rput(12.2,8){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
 525 \hkl(h k l) planes match
 526  }}}
 527 \rput(9.7,6.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
 528 r = 2 - 4 nm
 529  }}}
 530 \end{pspicture}
 531
 532 \end{slide}
 533
 534 \begin{slide}
 535
 536  {\large\bf
 537   Supposed precipitation mechanism of SiC in Si
 538  }
 539
 540  \scriptsize
 541
 542  \vspace{0.1cm}
 543
 544  \begin{minipage}{3.8cm}
 545  Si \& SiC lattice structure\\[0.2cm]
 546  \includegraphics[width=3.5cm]{sic_unit_cell.eps}\\[-0.3cm]
 547  \hrule
 548  \end{minipage}
 549  \hspace{0.6cm}
 550  \begin{minipage}{3.8cm}
 551  \begin{center}
 552  \includegraphics[width=3.3cm]{tem_c-si-db.eps}
 553  \end{center}
 554  \end{minipage}
 555  \hspace{0.6cm}
 556  \begin{minipage}{3.8cm}
 557  \begin{center}
 558  \includegraphics[width=3.3cm]{tem_3c-sic.eps}
 559  \end{center}
 560  \end{minipage}
 561
 562  \begin{minipage}{4cm}
 563  \begin{center}
 564  C-Si dimers (dumbbells)\\[-0.1cm]
 565  on Si interstitial sites
 566  \end{center}
 567  \end{minipage}
 568  \hspace{0.2cm}
 569  \begin{minipage}{4.2cm}
 570  \begin{center}
 571  Agglomeration of C-Si dumbbells\\[-0.1cm]
 572  $\Rightarrow$ dark contrasts
 573  \end{center}
 574  \end{minipage}
 575  \hspace{0.2cm}
 576  \begin{minipage}{4cm}
 577  \begin{center}
 578  Precipitation of 3C-SiC in Si\\[-0.1cm]
 579  $\Rightarrow$ Moir\'e fringes\\[-0.1cm]
 580  \& release of Si self-interstitials
 581  \end{center}
 582  \end{minipage}
 583
 584  \begin{minipage}{3.8cm}
 585  \begin{center}
 586  \includegraphics[width=3.3cm]{sic_prec_seq_01.eps}
 587  \end{center}
 588  \end{minipage}
 589  \hspace{0.6cm}
 590  \begin{minipage}{3.8cm}
 591  \begin{center}
 592  \includegraphics[width=3.3cm]{sic_prec_seq_02.eps}
 593  \end{center}
 594  \end{minipage}
 595  \hspace{0.6cm}
 596  \begin{minipage}{3.8cm}
 597  \begin{center}
 598  \includegraphics[width=3.3cm]{sic_prec_seq_03.eps}
 599  \end{center}
 600  \end{minipage}
 601
 602 \begin{pspicture}(0,0)(0,0)
 603 \psline[linewidth=4pt]{->}(8.5,2)(9.0,2)
 604 \psellipse[linecolor=blue](11.5,5.8)(0.3,0.5)
 605 \rput{-20}{\psellipse[linecolor=blue](3.3,8.1)(0.3,0.5)}
 606 \psline[linewidth=4pt]{->}(4.0,2)(4.5,2)
 607 \rput(12.7,0.3){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
 608  $4a_{\text{Si}}=5a_{\text{SiC}}$
 609  }}}
 610 \rput(12.2,8){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
 611 \hkl(h k l) planes match
 612  }}}
 613 \rput(9.7,6.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
 614 r = 2 - 4 nm
 615  }}}
 616 \rput(6.7,5.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=white]{
 617 \begin{minipage}{10cm}
 618 \small
 619 {\color{red}\bf Controversial views}
 620 \begin{itemize}
 621 \item Implantations at high T (Nejim et al.)
 622  \begin{itemize}
 623   \item Topotactic transformation based on \cs
 624   \item \si{} as supply reacting with further C in cleared volume
 625  \end{itemize}
 626 \item Annealing behavior (Serre et al.)
 627  \begin{itemize}
 628   \item Room temperature implants $\rightarrow$ highly mobile C
 629   \item Elevated T implants $\rightarrow$ no/low C redistribution/migration\\
 630         (indicate stable \cs{} configurations)
 631  \end{itemize}
 632 \item Strained silicon \& Si/SiC heterostructures
 633  \begin{itemize}
 634   \item Coherent SiC precipitates (tensile strain)
 635   \item Incoherent SiC (strain relaxation)
 636  \end{itemize}
 637 \end{itemize}
 638 \end{minipage}
 639  }}}
 640 \end{pspicture}
 641
 642 \end{slide}
 643
 644 \begin{slide}
 645
 646  {\large\bf
 647   Molecular dynamics (MD) simulations
 648  }
 649
 650  \vspace{12pt}
 651
 652  \small
 653
 654  {\bf MD basics:}
 655  \begin{itemize}
 656   \item Microscopic description of N particle system
 657   \item Analytical interaction potential
 658   \item Numerical integration using Newtons equation of motion\\
 659         as a propagation rule in 6N-dimensional phase space
 660   \item Observables obtained by time and/or ensemble averages
 661  \end{itemize}
 662  {\bf Details of the simulation:}
 663  \begin{itemize}
 664   \item Integration: Velocity Verlet, timestep: $1\text{ fs}$
 665   \item Ensemble: NpT (isothermal-isobaric)
 666         \begin{itemize}
 667          \item Berendsen thermostat:
 668                $\tau_{\text{T}}=100\text{ fs}$
 669          \item Berendsen barostat:\\
 670                $\tau_{\text{P}}=100\text{ fs}$,
 671                $\beta^{-1}=100\text{ GPa}$
 672         \end{itemize}
 673   \item Erhart/Albe potential: Tersoff-like bond order potential
 674   \vspace*{12pt}
 675         \[
 676         E = \frac{1}{2} \sum_{i \neq j} \pot_{ij}, \quad
 677         \pot_{ij} = {\color{red}f_C(r_{ij})}
 678         \left[ f_R(r_{ij}) + {\color{blue}b_{ij}} f_A(r_{ij}) \right]
 679         \]
 680  \end{itemize}
 681
 682  \begin{picture}(0,0)(-230,-30)
 683   \includegraphics[width=5cm]{tersoff_angle.eps}
 684  \end{picture}
 685
 686 \end{slide}
 687
 688 \begin{slide}
 689
 690  {\large\bf
 691   Density functional theory (DFT) calculations
 692  }
 693
 694  \small
 695
 696  Basic ingredients necessary for DFT
 697
 698  \begin{itemize}
 699   \item \underline{Hohenberg-Kohn theorem} - ground state density $n_0(r)$ ...
 700         \begin{itemize}
 701          \item ... uniquely determines the ground state potential
 702                / wavefunctions
 703          \item ... minimizes the systems total energy
 704         \end{itemize}
 705   \item \underline{Born-Oppenheimer}
 706         - $N$ moving electrons in an external potential of static nuclei
 707 \[
 708 H\Psi = \left[-\sum_i^N \frac{\hbar^2}{2m}\nabla_i^2
 709               +\sum_i^N V_{\text{ext}}(r_i)
 710               +\sum_{i<j}^N V_{e-e}(r_i,r_j)\right]\Psi=E\Psi
 711 \]
 712   \item \underline{Effective potential}
 713         - averaged electrostatic potential \& exchange and correlation
 714 \[
 715 V_{\text{eff}}(r)=V_{\text{ext}}(r)+\int\frac{e^2 n(r')}{|r-r'|}d^3r'
 716                  +V_{\text{XC}}[n(r)]
 717 \]
 718   \item \underline{Kohn-Sham system}
 719         - Schr\"odinger equation of N non-interacting particles
 720 \[
 721 \left[ -\frac{\hbar^2}{2m}\nabla^2 + V_{\text{eff}}(r) \right] \Phi_i(r)
 722 =\epsilon_i\Phi_i(r)
 723 \quad
 724 \Rightarrow
 725 \quad
 726 n(r)=\sum_i^N|\Phi_i(r)|^2
 727 \]
 728   \item \underline{Self-consistent solution}\\
 729 $n(r)$ depends on $\Phi_i$, which depend on $V_{\text{eff}}$,
 730 which in turn depends on $n(r)$
 731   \item \underline{Variational principle}
 732         - minimize total energy with respect to $n(r)$
 733  \end{itemize}
 734
 735 \end{slide}
 736
 737 \begin{slide}
 738
 739  {\large\bf
 740   Density functional theory (DFT) calculations
 741  }
 742
 743  \small
 744
 745  \vspace*{0.2cm}
 746
 747  Details of applied DFT calculations in this work
 748
 749  \begin{itemize}
 750   \item \underline{Exchange correlation functional}
 751         - approximations for the inhomogeneous electron gas
 752         \begin{itemize}
 753          \item LDA: $E_{\text{XC}}^{\text{LDA}}[n]=\int \epsilon_{\text{XC}}(n)n(r)d^3r$
 754          \item GGA: $E_{\text{XC}}^{\text{GGA}}[n]=\int \epsilon_{\text{XC}}(n,\nabla n)n(r)d^3r$
 755         \end{itemize}
 756   \item \underline{Plane wave basis set}
 757         - approximation of the wavefunction $\Phi_i$ by plane waves $\phi_j$
 758 \[
 759 \rightarrow
 760 \text{Fourier series: } \Phi_i=\sum_{|G+k|<G_{\text{cut}}} c_j^i \phi_j(r), \quad E_{\text{cut}}=\frac{\hbar^2}{2m}G^2_{\text{cut}}
 761 \qquad ({\color{blue}300\text{ eV}})
 762 \]
 763   \item \underline{Brillouin zone sampling} -
 764         {\color{blue}$\Gamma$-point only} calculations
 765   \item \underline{Pseudo potential}
 766         - consider only the valence electrons
 767   \item \underline{Code} - VASP 4.6
 768  \end{itemize}
 769
 770  \vspace*{0.2cm}
 771
 772  MD and structural optimization
 773
 774  \begin{itemize}
 775   \item MD integration: Gear predictor corrector algorithm
 776   \item Pressure control: Parrinello-Rahman pressure control
 777   \item Structural optimization: Conjugate gradient method
 778  \end{itemize}
 779
 780 \begin{pspicture}(0,0)(0,0)
 781 \psellipse[linecolor=blue](1.5,6.75)(0.5,0.3)
 782 \end{pspicture}
 783
 784 \end{slide}
 785
 786 \begin{slide}
 787
 788  {\large\bf
 789   C and Si self-interstitial point defects in silicon
 790  }
 791
 792  \small
 793
 794  \vspace*{0.3cm}
 795
 796 \begin{minipage}{8cm}
 797 Procedure:\\[0.3cm]
 798   \begin{pspicture}(0,0)(7,5)
 799   \rput(3.5,4){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
 800    \parbox{7cm}{
 801    \begin{itemize}
 802     \item Creation of c-Si simulation volume
 803     \item Periodic boundary conditions
 804     \item $T=0\text{ K}$, $p=0\text{ bar}$
 805    \end{itemize}
 806   }}}}
 807 \rput(3.5,2.1){\rnode{insert}{\psframebox{
 808  \parbox{7cm}{
 809   \begin{center}
 810   Insertion of interstitial C/Si atoms
 811   \end{center}
 812   }}}}
 813   \rput(3.5,1){\rnode{cool}{\psframebox[fillstyle=solid,fillcolor=lbb]{
 814    \parbox{7cm}{
 815    \begin{center}
 816    Relaxation / structural energy minimization
 817    \end{center}
 818   }}}}
 819   \ncline[]{->}{init}{insert}
 820   \ncline[]{->}{insert}{cool}
 821  \end{pspicture}
 822 \end{minipage}
 823 \begin{minipage}{5cm}
 824   \includegraphics[width=5cm]{unit_cell_e.eps}\\
 825 \end{minipage}
 826
 827 \begin{minipage}{9cm}
 828  \begin{tabular}{l c c}
 829  \hline
 830  & size [unit cells] & \# atoms\\
 831 \hline
 832 VASP & $3\times 3\times 3$ & $216\pm 1$ \\
 833 Erhart/Albe & $9\times 9\times 9$ & $5832\pm 1$\\
 834 \hline
 835  \end{tabular}
 836 \end{minipage}
 837 \begin{minipage}{4cm}
 838 {\color{red}$\bullet$} Tetrahedral\\
 839 {\color{green}$\bullet$} Hexagonal\\
 840 {\color{yellow}$\bullet$} \hkl<1 0 0> dumbbell\\
 841 {\color{magenta}$\bullet$} \hkl<1 1 0> dumbbell\\
 842 {\color{cyan}$\bullet$} Bond-centered\\
 843 {\color{black}$\bullet$} Vacancy / Substitutional
 844 \end{minipage}
 845
 846 \end{slide}
 847
 848 \begin{slide}
 849
 850  \footnotesize
 851
 852 \begin{minipage}{9.5cm}
 853
 854  {\large\bf
 855   Si self-interstitial point defects in silicon\\
 856  }
 857
 858 \begin{tabular}{l c c c c c}
 859 \hline
 860  $E_{\text{f}}$ [eV] & \hkl<1 1 0> DB & H & T & \hkl<1 0 0> DB & V \\
 861 \hline
 862  VASP & \underline{3.39} & 3.42 & 3.77 & 4.41 & 3.63 \\
 863  Erhart/Albe & 4.39 & 4.48$^*$ & \underline{3.40} & 5.42 & 3.13 \\
 864 \hline
 865 \end{tabular}\\[0.2cm]
 866
 867 \begin{minipage}{4.7cm}
 868 \includegraphics[width=4.7cm]{e_kin_si_hex.ps}
 869 \end{minipage}
 870 \begin{minipage}{4.7cm}
 871 \begin{center}
 872 {\tiny nearly T $\rightarrow$ T}\\
 873 \end{center}
 874 \includegraphics[width=4.7cm]{nhex_tet.ps}
 875 \end{minipage}\\
 876
 877 \underline{Hexagonal} \hspace{2pt}
 878 \href{../video/si_self_int_hexa.avi}{$\rhd$}\\[0.1cm]
 879 \framebox{
 880 \begin{minipage}{2.7cm}
 881 $E_{\text{f}}^*=4.48\text{ eV}$\\
 882 \includegraphics[width=2.7cm]{si_pd_albe/hex_a.eps}
 883 \end{minipage}
 884 \begin{minipage}{0.4cm}
 885 \begin{center}
 886 $\Rightarrow$
 887 \end{center}
 888 \end{minipage}
 889 \begin{minipage}{2.7cm}
 890 $E_{\text{f}}=3.96\text{ eV}$\\
 891 \includegraphics[width=2.8cm]{si_pd_albe/hex.eps}
 892 \end{minipage}
 893 }
 894 \begin{minipage}{2.9cm}
 895 \begin{flushright}
 896 \underline{Vacancy}\\
 897 \includegraphics[width=3.0cm]{si_pd_albe/vac.eps}
 898 \end{flushright}
 899 \end{minipage}
 900
 901 \end{minipage}
 902 \begin{minipage}{3.5cm}
 903
 904 \begin{flushright}
 905 \underline{\hkl<1 1 0> dumbbell}\\
 906 \includegraphics[width=3.0cm]{si_pd_albe/110.eps}\\
 907 \underline{Tetrahedral}\\
 908 \includegraphics[width=3.0cm]{si_pd_albe/tet.eps}\\
 909 \underline{\hkl<1 0 0> dumbbell}\\
 910 \includegraphics[width=3.0cm]{si_pd_albe/100.eps}
 911 \end{flushright}
 912
 913 \end{minipage}
 914
 915 \end{slide}
 916
 917 \begin{slide}
 918
 919 \footnotesize
 920
 921  {\large\bf
 922   C interstitial point defects in silicon\\[-0.1cm]
 923  }
 924
 925 \begin{tabular}{l c c c c c c r}
 926 \hline
 927  $E_{\text{f}}$ & T & H & \hkl<1 0 0> DB & \hkl<1 1 0> DB & S & B & \cs{} \& \si\\
 928 \hline
 929  VASP & unstable & unstable & \underline{3.72} & 4.16 & 1.95 & 4.66 & {\color{green}4.17}\\
 930  Erhart/Albe MD & 6.09 & 9.05$^*$ & \underline{3.88} & 5.18 & {\color{red}0.75} & 5.59$^*$ & {\color{green}4.43} \\
 931 \hline
 932 \end{tabular}\\[0.1cm]
 933
 934 \framebox{
 935 \begin{minipage}{2.7cm}
 936 \underline{Hexagonal} \hspace{2pt}
 937 \href{../video/c_in_si_int_hexa.avi}{$\rhd$}\\
 938 $E_{\text{f}}^*=9.05\text{ eV}$\\
 939 \includegraphics[width=2.7cm]{c_pd_albe/hex.eps}
 940 \end{minipage}
 941 \begin{minipage}{0.4cm}
 942 \begin{center}
 943 $\Rightarrow$
 944 \end{center}
 945 \end{minipage}
 946 \begin{minipage}{2.7cm}
 947 \underline{\hkl<1 0 0>}\\
 948 $E_{\text{f}}=3.88\text{ eV}$\\
 949 \includegraphics[width=2.7cm]{c_pd_albe/100.eps}
 950 \end{minipage}
 951 }
 952 \begin{minipage}{2cm}
 953 \hfill
 954 \end{minipage}
 955 \begin{minipage}{3cm}
 956 \begin{flushright}
 957 \underline{Tetrahedral}\\
 958 \includegraphics[width=3.0cm]{c_pd_albe/tet.eps}
 959 \end{flushright}
 960 \end{minipage}
 961
 962 \framebox{
 963 \begin{minipage}{2.7cm}
 964 \underline{Bond-centered}\\
 965 $E_{\text{f}}^*=5.59\text{ eV}$\\
 966 \includegraphics[width=2.7cm]{c_pd_albe/bc.eps}
 967 \end{minipage}
 968 \begin{minipage}{0.4cm}
 969 \begin{center}
 970 $\Rightarrow$
 971 \end{center}
 972 \end{minipage}
 973 \begin{minipage}{2.7cm}
 974 \underline{\hkl<1 1 0> dumbbell}\\
 975 $E_{\text{f}}=5.18\text{ eV}$\\
 976 \includegraphics[width=2.7cm]{c_pd_albe/110.eps}
 977 \end{minipage}
 978 }
 979 \begin{minipage}{2cm}
 980 \hfill
 981 \end{minipage}
 982 \begin{minipage}{3cm}
 983 \begin{flushright}
 984 \underline{Substitutional}\\
 985 \includegraphics[width=3.0cm]{c_pd_albe/sub.eps}
 986 \end{flushright}
 987 \end{minipage}
 988
 989 \end{slide}
 990
 991 \begin{slide}
 992
 993 \footnotesize
 994
 995  {\large\bf\boldmath
 996   C \hkl<1 0 0> dumbbell interstitial configuration\\
 997  }
 998
 999 {\tiny
1000 \begin{tabular}{l c c c c c c c c}
1001 \hline
1002  Distances [nm] & $r(1C)$ & $r(2C)$ & $r(3C)$ & $r(12)$ & $r(13)$ & $r(34)$ & $r(23)$ & $r(25)$ \\
1003 \hline
1004 Erhart/Albe & 0.175 & 0.329 & 0.186 & 0.226 & 0.300 & 0.343 & 0.423 & 0.425 \\
1005 VASP & 0.174 & 0.341 & 0.182 & 0.229 & 0.286 & 0.347 & 0.422 & 0.417 \\
1006 \hline
1007 \end{tabular}\\[0.2cm]
1008 \begin{tabular}{l c c c c }
1009 \hline
1010  Angles [$^{\circ}$] & $\theta_1$ & $\theta_2$ & $\theta_3$ & $\theta_4$ \\
1011 \hline
1012 Erhart/Albe & 140.2 & 109.9 & 134.4 & 112.8 \\
1013 VASP & 130.7 & 114.4 & 146.0 & 107.0 \\
1014 \hline
1015 \end{tabular}\\[0.2cm]
1016 \begin{tabular}{l c c c}
1017 \hline
1018  Displacements [nm]& $a$ & $b$ & $|a|+|b|$ \\
1019 \hline
1020 Erhart/Albe & 0.084 & -0.091 & 0.175 \\
1021 VASP & 0.109 & -0.065 & 0.174 \\
1022 \hline
1023 \end{tabular}\\[0.6cm]
1024 }
1025
1026 \begin{minipage}{3.0cm}
1027 \begin{center}
1028 \underline{Erhart/Albe}
1029 \includegraphics[width=3.0cm]{c_pd_albe/100_cmp.eps}
1030 \end{center}
1031 \end{minipage}
1032 \begin{minipage}{3.0cm}
1033 \begin{center}
1034 \underline{VASP}
1035 \includegraphics[width=3.0cm]{c_pd_vasp/100_cmp.eps}
1036 \end{center}
1037 \end{minipage}\\
1038
1039 \begin{picture}(0,0)(-185,10)
1040 \includegraphics[width=6.8cm]{100-c-si-db_cmp.eps}
1041 \end{picture}
1042 \begin{picture}(0,0)(-280,-150)
1043 \includegraphics[width=3.3cm]{c_pd_vasp/eden.eps}
1044 \end{picture}
1045
1046 \begin{pspicture}(0,0)(0,0)
1047 \psellipse[linecolor=green](5.18,5.92)(0.5,0.3)
1048 \psellipse[linecolor=red](3.45,5.92)(1.0,0.4)
1049 \psellipse[linecolor=blue](2.7,6.92)(0.9,0.2)
1050 \psellipse[linecolor=blue](4.65,6.92)(0.9,0.2)
1051 \end{pspicture}
1052
1053 \end{slide}
1054
1055 \begin{slide}
1056
1057 \small
1058
1059 \begin{minipage}{8.5cm}
1060
1061  {\large\bf
1062   Bond-centered interstitial configuration\\[-0.1cm]
1063  }
1064
1065 \begin{minipage}{3.0cm}
1066 \includegraphics[width=2.8cm]{c_pd_vasp/bc_2333.eps}\\
1067 \end{minipage}
1068 \begin{minipage}{5.2cm}
1069 \begin{itemize}
1070  \item Linear Si-C-Si bond
1071  \item Si: one C \& 3 Si neighbours
1072  \item Spin polarized calculations
1073  \item No saddle point!\\
1074        Real local minimum!
1075 \end{itemize}
1076 \end{minipage}
1077
1078 \framebox{
1079  \tiny
1080  \begin{minipage}[t]{6.5cm}
1081   \begin{minipage}[t]{1.2cm}
1082   {\color{red}Si}\\
1083   {\tiny sp$^3$}\\[0.8cm]
1084   \underline{${\color{black}\uparrow}$}
1085   \underline{${\color{black}\uparrow}$}
1086   \underline{${\color{black}\uparrow}$}
1087   \underline{${\color{red}\uparrow}$}\\
1088   sp$^3$
1089   \end{minipage}
1090   \begin{minipage}[t]{1.4cm}
1091   \begin{center}
1092   {\color{red}M}{\color{blue}O}\\[0.8cm]
1093   \underline{${\color{blue}\uparrow}{\color{white}\downarrow}$}\\
1094   $\sigma_{\text{ab}}$\\[0.5cm]
1095   \underline{${\color{red}\uparrow}{\color{blue}\downarrow}$}\\
1096   $\sigma_{\text{b}}$
1097   \end{center}
1098   \end{minipage}
1099   \begin{minipage}[t]{1.0cm}
1100   \begin{center}
1101   {\color{blue}C}\\
1102   {\tiny sp}\\[0.2cm]
1103   \underline{${\color{white}\uparrow\uparrow}$}
1104   \underline{${\color{white}\uparrow\uparrow}$}\\
1105   2p\\[0.4cm]
1106   \underline{${\color{blue}\uparrow}{\color{blue}\downarrow}$}
1107   \underline{${\color{blue}\uparrow}{\color{blue}\downarrow}$}\\
1108   sp
1109   \end{center}
1110   \end{minipage}
1111   \begin{minipage}[t]{1.4cm}
1112   \begin{center}
1113   {\color{blue}M}{\color{green}O}\\[0.8cm]
1114   \underline{${\color{blue}\uparrow}{\color{white}\downarrow}$}\\
1115   $\sigma_{\text{ab}}$\\[0.5cm]
1116   \underline{${\color{green}\uparrow}{\color{blue}\downarrow}$}\\
1117   $\sigma_{\text{b}}$
1118   \end{center}
1119   \end{minipage}
1120   \begin{minipage}[t]{1.2cm}
1121   \begin{flushright}
1122   {\color{green}Si}\\
1123   {\tiny sp$^3$}\\[0.8cm]
1124   \underline{${\color{green}\uparrow}$}
1125   \underline{${\color{black}\uparrow}$}
1126   \underline{${\color{black}\uparrow}$}
1127   \underline{${\color{black}\uparrow}$}\\
1128   sp$^3$
1129   \end{flushright}
1130   \end{minipage}
1131  \end{minipage}
1132 }\\[0.1cm]
1133
1134 \framebox{
1135 \begin{minipage}{4.5cm}
1136 \includegraphics[width=4cm]{c_100_mig_vasp/im_spin_diff.eps}
1137 \end{minipage}
1138 \begin{minipage}{3.5cm}
1139 {\color{gray}$\bullet$} Spin up\\
1140 {\color{green}$\bullet$} Spin down\\
1141 {\color{blue}$\bullet$} Resulting spin up\\
1142 {\color{yellow}$\bullet$} Si atoms\\
1143 {\color{red}$\bullet$} C atom
1144 \end{minipage}
1145 }
1146
1147 \end{minipage}
1148 \begin{minipage}{4.2cm}
1149 \begin{flushright}
1150 \includegraphics[width=4.3cm]{c_pd_vasp/bc_2333_ksl.ps}\\
1151 {\color{green}$\Box$} {\tiny unoccupied}\\
1152 {\color{red}$\bullet$} {\tiny occupied}
1153 \end{flushright}
1154 \end{minipage}
1155
1156 \end{slide}
1157
1158 \begin{slide}
1159
1160  {\large\bf\boldmath
1161   Migration of the C \hkl<1 0 0> dumbbell interstitial
1162  }
1163
1164 \scriptsize
1165
1166  {\small Investigated pathways}
1167
1168 \begin{minipage}{8.5cm}
1169 \begin{minipage}{8.3cm}
1170 \underline{\hkl<0 0 -1> $\rightarrow$ \hkl<0 0 1>}\\
1171 \begin{minipage}{2.4cm}
1172 \includegraphics[width=2.4cm]{c_pd_vasp/100_2333.eps}
1173 \end{minipage}
1174 \begin{minipage}{0.4cm}
1175 $\rightarrow$
1176 \end{minipage}
1177 \begin{minipage}{2.4cm}
1178 \includegraphics[width=2.4cm]{c_pd_vasp/bc_2333.eps}
1179 \end{minipage}
1180 \begin{minipage}{0.4cm}
1181 $\rightarrow$
1182 \end{minipage}
1183 \begin{minipage}{2.4cm}
1184 \includegraphics[width=2.4cm]{c_pd_vasp/100_next_2333.eps}
1185 \end{minipage}
1186 \end{minipage}\\
1187 \begin{minipage}{8.3cm}
1188 \underline{\hkl<0 0 -1> $\rightarrow$ \hkl<0 -1 0>}\\
1189 \begin{minipage}{2.4cm}
1190 \includegraphics[width=2.4cm]{c_pd_vasp/100_2333.eps}
1191 \end{minipage}
1192 \begin{minipage}{0.4cm}
1193 $\rightarrow$
1194 \end{minipage}
1195 \begin{minipage}{2.4cm}
1196 \includegraphics[width=2.4cm]{c_pd_vasp/00-1-0-10_2333.eps}
1197 \end{minipage}
1198 \begin{minipage}{0.4cm}
1199 $\rightarrow$
1200 \end{minipage}
1201 \begin{minipage}{2.4cm}
1202 \includegraphics[width=2.4cm]{c_pd_vasp/0-10_2333.eps}
1203 \end{minipage}
1204 \end{minipage}\\
1205 \begin{minipage}{8.3cm}
1206 \underline{\hkl<0 0 -1> $\rightarrow$ \hkl<0 -1 0> (in place)}\\
1207 \begin{minipage}{2.4cm}
1208 \includegraphics[width=2.4cm]{c_pd_vasp/100_2333.eps}
1209 \end{minipage}
1210 \begin{minipage}{0.4cm}
1211 $\rightarrow$
1212 \end{minipage}
1213 \begin{minipage}{2.4cm}
1214 \includegraphics[width=2.4cm]{c_pd_vasp/00-1_ip0-10_2333.eps}
1215 \end{minipage}
1216 \begin{minipage}{0.4cm}
1217 $\rightarrow$
1218 \end{minipage}
1219 \begin{minipage}{2.4cm}
1220 \includegraphics[width=2.4cm]{c_pd_vasp/0-10_ip_2333.eps}
1221 \end{minipage}
1222 \end{minipage}
1223 \end{minipage}
1224 \framebox{
1225 \begin{minipage}{4.2cm}
1226  {\small Constrained relaxation\\
1227          technique (CRT) method}\\
1228 \includegraphics[width=4cm]{crt_orig.eps}
1229 \begin{itemize}
1230  \item Constrain diffusing atom
1231  \item Static constraints
1232 \end{itemize}
1233 \vspace*{0.3cm}
1234  {\small Modifications}\\
1235 \includegraphics[width=4cm]{crt_mod.eps}
1236 \begin{itemize}
1237  \item Constrain all atoms
1238  \item Update individual\\
1239        constraints
1240 \end{itemize}
1241 \end{minipage}
1242 }
1243
1244 \end{slide}
1245
1246 \begin{slide}
1247
1248  {\large\bf\boldmath
1249   Migration of the C \hkl<1 0 0> dumbbell interstitial
1250  }
1251
1252 \scriptsize
1253
1254 \framebox{
1255 \begin{minipage}{5.9cm}
1256 \begin{flushleft}
1257 \includegraphics[width=5.8cm]{im_00-1_nosym_sp_fullct_thesis.ps}\\[0.45cm]
1258 \end{flushleft}
1259 \begin{center}
1260 \begin{picture}(0,0)(60,0)
1261 \includegraphics[width=1cm]{vasp_mig/00-1.eps}
1262 \end{picture}
1263 \begin{picture}(0,0)(-5,0)
1264 \includegraphics[width=1cm]{vasp_mig/bc_00-1_sp.eps}
1265 \end{picture}
1266 \begin{picture}(0,0)(-55,0)
1267 \includegraphics[width=1cm]{vasp_mig/bc.eps}
1268 \end{picture}
1269 \begin{picture}(0,0)(12.5,10)
1270 \includegraphics[width=1cm]{110_arrow.eps}
1271 \end{picture}
1272 \begin{picture}(0,0)(90,0)
1273 \includegraphics[height=0.9cm]{001_arrow.eps}
1274 \end{picture}
1275 \end{center}
1276 \vspace*{0.35cm}
1277 \end{minipage}
1278 }
1279 \begin{minipage}{0.3cm}
1280 \hfill
1281 \end{minipage}
1282 \framebox{
1283 \begin{minipage}{5.9cm}
1284 \begin{flushright}
1285 \includegraphics[width=5.9cm]{vasp_mig/00-1_0-10_nosym_sp_fullct.ps}\\[0.5cm]
1286 \end{flushright}
1287 \begin{center}
1288 \begin{picture}(0,0)(60,0)
1289 \includegraphics[width=1cm]{vasp_mig/00-1_a.eps}
1290 \end{picture}
1291 \begin{picture}(0,0)(5,0)
1292 \includegraphics[width=1cm]{vasp_mig/00-1_0-10_sp.eps}
1293 \end{picture}
1294 \begin{picture}(0,0)(-55,0)
1295 \includegraphics[width=1cm]{vasp_mig/0-10.eps}
1296 \end{picture}
1297 \begin{picture}(0,0)(12.5,10)
1298 \includegraphics[width=1cm]{100_arrow.eps}
1299 \end{picture}
1300 \begin{picture}(0,0)(90,0)
1301 \includegraphics[height=0.9cm]{001_arrow.eps}
1302 \end{picture}
1303 \end{center}
1304 \vspace*{0.3cm}
1305 \end{minipage}\\
1306 }
1307
1308 \vspace*{0.05cm}
1309
1310 \framebox{
1311 \begin{minipage}{5.9cm}
1312 \begin{flushleft}
1313 \includegraphics[width=5.9cm]{vasp_mig/00-1_ip0-10_nosym_sp_fullct.ps}\\[0.6cm]
1314 \end{flushleft}
1315 \begin{center}
1316 \begin{picture}(0,0)(60,0)
1317 \includegraphics[width=0.9cm]{vasp_mig/00-1_b.eps}
1318 \end{picture}
1319 \begin{picture}(0,0)(10,0)
1320 \includegraphics[width=0.9cm]{vasp_mig/00-1_ip0-10_sp.eps}
1321 \end{picture}
1322 \begin{picture}(0,0)(-60,0)
1323 \includegraphics[width=0.9cm]{vasp_mig/0-10_b.eps}
1324 \end{picture}
1325 \begin{picture}(0,0)(12.5,10)
1326 \includegraphics[width=1cm]{100_arrow.eps}
1327 \end{picture}
1328 \begin{picture}(0,0)(90,0)
1329 \includegraphics[height=0.9cm]{001_arrow.eps}
1330 \end{picture}
1331 \end{center}
1332 \vspace*{0.3cm}
1333 \end{minipage}
1334 }
1335 \begin{minipage}{0.3cm}
1336 \hfill
1337 \end{minipage}
1338 \begin{minipage}{6.5cm}
1339 VASP results
1340 \begin{itemize}
1341  \item Energetically most favorable path
1342        \begin{itemize}
1343         \item Path 2
1344         \item Activation energy: $\approx$ 0.9 eV
1345         \item Experimental values: 0.73 ... 0.87 eV
1346        \end{itemize}
1347        $\Rightarrow$ {\color{blue}Diffusion} path identified!
1348  \item Reorientation (path 3)
1349        \begin{itemize}
1350         \item More likely composed of two consecutive steps of type 2
1351         \item Experimental values: 0.77 ... 0.88 eV
1352        \end{itemize}
1353        $\Rightarrow$ {\color{blue}Reorientation} transition identified!
1354 \end{itemize}
1355 \end{minipage}
1356
1357 \end{slide}
1358
1359 \begin{slide}
1360
1361  {\large\bf\boldmath
1362   Migration of the C \hkl<1 0 0> dumbbell interstitial
1363  }
1364
1365 \scriptsize
1366
1367  \vspace{0.1cm}
1368
1369 \begin{minipage}{6.5cm}
1370
1371 \framebox{
1372 \begin{minipage}[t]{5.9cm}
1373 \begin{flushleft}
1374 \includegraphics[width=5.9cm]{bc_00-1.ps}\\[2.35cm]
1375 \end{flushleft}
1376 \begin{center}
1377 \begin{pspicture}(0,0)(0,0)
1378 \psframe[linecolor=red,fillstyle=none](-2.8,1.35)(3.3,2.7)
1379 \end{pspicture}
1380 \begin{picture}(0,0)(60,-50)
1381 \includegraphics[width=1cm]{albe_mig/bc_00-1_red_00.eps}
1382 \end{picture}
1383 \begin{picture}(0,0)(5,-50)
1384 \includegraphics[width=1cm]{albe_mig/bc_00-1_red_01.eps}
1385 \end{picture}
1386 \begin{picture}(0,0)(-55,-50)
1387 \includegraphics[width=1cm]{albe_mig/bc_00-1_red_02.eps}
1388 \end{picture}
1389 \begin{picture}(0,0)(12.5,-40)
1390 \includegraphics[width=1cm]{110_arrow.eps}
1391 \end{picture}
1392 \begin{picture}(0,0)(90,-45)
1393 \includegraphics[height=0.9cm]{001_arrow.eps}
1394 \end{picture}\\
1395 \begin{pspicture}(0,0)(0,0)
1396 \psframe[linecolor=blue,fillstyle=none](-2.8,0)(3.3,1.6)
1397 \end{pspicture}
1398 \begin{picture}(0,0)(60,-15)
1399 \includegraphics[width=0.9cm]{albe_mig/bc_00-1_01.eps}
1400 \end{picture}
1401 \begin{picture}(0,0)(35,-15)
1402 \includegraphics[width=0.9cm]{albe_mig/bc_00-1_02.eps}
1403 \end{picture}
1404 \begin{picture}(0,0)(-5,-15)
1405 \includegraphics[width=0.9cm]{albe_mig/bc_00-1_03.eps}
1406 \end{picture}
1407 \begin{picture}(0,0)(-55,-15)
1408 \includegraphics[width=0.9cm]{albe_mig/bc_00-1_04.eps}
1409 \end{picture}
1410 \begin{picture}(0,0)(12.5,-5)
1411 \includegraphics[width=1cm]{100_arrow.eps}
1412 \end{picture}
1413 \begin{picture}(0,0)(90,-15)
1414 \includegraphics[height=0.9cm]{010_arrow.eps}
1415 \end{picture}
1416 \end{center}
1417 \end{minipage}
1418 }\\[0.1cm]
1419
1420 \begin{minipage}{5.9cm}
1421 Erhart/Albe results
1422 \begin{itemize}
1423  \item Lowest activation energy: $\approx$ 2.2 eV
1424  \item 2.4 times higher than VASP
1425  \item Different pathway
1426 \end{itemize}
1427 \end{minipage}
1428
1429 \end{minipage}
1430 \begin{minipage}{6.5cm}
1431
1432 \framebox{
1433 \begin{minipage}{5.9cm}
1434 %\begin{flushright}
1435 %\includegraphics[width=5.9cm]{00-1_0-10.ps}\\[0.75cm]
1436 %\end{flushright}
1437 %\begin{center}
1438 %\begin{pspicture}(0,0)(0,0)
1439 %\psframe[linecolor=red,fillstyle=none](-2.8,-0.25)(3.3,1.1)
1440 %\end{pspicture}
1441 %\begin{picture}(0,0)(60,-5)
1442 %\includegraphics[width=0.9cm]{albe_mig/00-1_0-10_red_00.eps}
1443 %\end{picture}
1444 %\begin{picture}(0,0)(0,-5)
1445 %\includegraphics[width=0.9cm]{albe_mig/00-1_0-10_red_min.eps}
1446 %\end{picture}
1447 %\begin{picture}(0,0)(-55,-5)
1448 %\includegraphics[width=0.9cm]{albe_mig/00-1_0-10_red_03.eps}
1449 %\end{picture}
1450 %\begin{picture}(0,0)(12.5,5)
1451 %\includegraphics[width=1cm]{100_arrow.eps}
1452 %\end{picture}
1453 %\begin{picture}(0,0)(90,0)
1454 %\includegraphics[height=0.9cm]{001_arrow.eps}
1455 %\end{picture}
1456 %\end{center}
1457 %\vspace{0.2cm}
1458 %\end{minipage}
1459 %}\\[0.2cm]
1460 %
1461 %\framebox{
1462 %\begin{minipage}{5.9cm}
1463 \includegraphics[width=5.9cm]{00-1_110_0-10_mig_albe.ps}
1464 \end{minipage}
1465 }\\[0.1cm]
1466
1467 \begin{minipage}{5.9cm}
1468 Transition involving \ci{} \hkl<1 1 0>
1469 \begin{itemize}
1470  \item Bond-centered configuration unstable\\
1471        $\rightarrow$ \ci{} \hkl<1 1 0> dumbbell
1472  \item Transition minima of path 2 \& 3\\
1473        $\rightarrow$ \ci{} \hkl<1 1 0> dumbbell
1474  \item Activation energy: $\approx$ 2.2 eV \& 0.9 eV
1475  \item 2.4 - 3.4 times higher than VASP
1476  \item Rotation of dumbbell orientation
1477 \end{itemize}
1478 \vspace{0.1cm}
1479 \begin{center}
1480 {\color{blue}Overestimated diffusion barrier}
1481 \end{center}
1482 \end{minipage}
1483
1484 \end{minipage}
1485
1486 \end{slide}
1487
1488 \begin{slide}
1489
1490  {\large\bf\boldmath
1491   Combinations with a C-Si \hkl<1 0 0>-type interstitial
1492  }
1493
1494 \small
1495
1496 \vspace*{0.1cm}
1497
1498 Binding energy:
1499 $
1500 E_{\text{b}}=
1501 E_{\text{f}}^{\text{defect combination}}-
1502 E_{\text{f}}^{\text{C \hkl<0 0 -1> dumbbell}}-
1503 E_{\text{f}}^{\text{2nd defect}}
1504 $
1505
1506 \vspace*{0.1cm}
1507
1508 {\scriptsize
1509 \begin{tabular}{l c c c c c c}
1510 \hline
1511  $E_{\text{b}}$ [eV] & 1 & 2 & 3 & 4 & 5 & R\\
1512  \hline
1513  \hkl<0 0 -1> & {\color{red}-0.08} & -1.15 & {\color{red}-0.08} & 0.04 & -1.66 & -0.19\\
1514  \hkl<0 0 1> & 0.34 & 0.004 & -2.05 & 0.26 & -1.53 & -0.19\\
1515  \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}\\
1516  \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}\\
1517  \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}\\
1518  \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}\\
1519  \hline
1520  C substitutional (C$_{\text{S}}$) & 0.26 & -0.51 & -0.93 & -0.15 & 0.49 & -0.05\\
1521  Vacancy & -5.39 ($\rightarrow$ C$_{\text{S}}$) & -0.59 & -3.14 & -0.54 & -0.50 & -0.31\\
1522 \hline
1523 \end{tabular}
1524 }
1525
1526 \vspace*{0.3cm}
1527
1528 \footnotesize
1529
1530 \begin{minipage}[t]{3.8cm}
1531 \underline{\hkl<1 0 0> at position 1}\\[0.1cm]
1532 \includegraphics[width=3.5cm]{00-1dc/2-25.eps}
1533 \end{minipage}
1534 \begin{minipage}[t]{3.5cm}
1535 \underline{\hkl<0 -1 0> at position 1}\\[0.1cm]
1536 \includegraphics[width=3.2cm]{00-1dc/2-39.eps}
1537 \end{minipage}
1538 \begin{minipage}[t]{5.5cm}
1539 \begin{itemize}
1540  \item $E_{\text{b}}=0$ $\Leftrightarrow$ non-interacting defects\\
1541        $E_{\text{b}} \rightarrow 0$ for increasing distance (R)
1542  \item Stress compensation / increase
1543  \item Unfavored: antiparallel orientations
1544  \item Indication of energetically favored\\
1545        agglomeration
1546  \item Most favorable: C clustering
1547  \item However: High barrier ($>4\,\text{eV}$)
1548  \item $4\times{\color{cyan}-2.25}$ versus $2\times{\color{orange}-2.39}$
1549        (Entropy)
1550 \end{itemize}
1551 \end{minipage}
1552
1553 \begin{picture}(0,0)(-295,-130)
1554 \includegraphics[width=3.5cm]{comb_pos.eps}
1555 \end{picture}
1556
1557 \end{slide}
1558
1559 \begin{slide}
1560
1561  {\large\bf\boldmath
1562   Combinations of C-Si \hkl<1 0 0>-type interstitials
1563  }
1564
1565 \small
1566
1567 \vspace*{0.1cm}
1568
1569 Energetically most favorable combinations along \hkl<1 1 0>
1570
1571 \vspace*{0.1cm}
1572
1573 {\scriptsize
1574 \begin{tabular}{l c c c c c c}
1575 \hline
1576  & 1 & 2 & 3 & 4 & 5 & 6\\
1577 \hline
1578 $E_{\text{b}}$ [eV] & -2.39 & -1.88 & -0.59 & -0.31 & -0.24 & -0.21 \\
1579 C-C distance [\AA] & 1.4 & 4.6 & 6.5 & 8.6 & 10.5 & 10.8 \\
1580 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>\\
1581 \hline
1582 \end{tabular}
1583 }
1584
1585 \vspace*{0.3cm}
1586
1587 \begin{minipage}{7.0cm}
1588 \includegraphics[width=7cm]{db_along_110_cc.ps}
1589 \end{minipage}
1590 \begin{minipage}{6.0cm}
1591 \begin{itemize}
1592  \item Interaction proportional to reciprocal cube of C-C distance
1593  \item Saturation in the immediate vicinity
1594  \renewcommand\labelitemi{$\Rightarrow$}
1595  \item Agglomeration of \ci{} expected
1596  \item Absence of C clustering
1597 \end{itemize}
1598 \begin{center}
1599 {\color{blue}
1600  Consisten with initial precipitation model
1601 }
1602 \end{center}
1603 \end{minipage}
1604
1605 \vspace{0.2cm}
1606
1607 \end{slide}
1608
1609 \begin{slide}
1610
1611  {\large\bf\boldmath
1612   Combinations of substitutional C and \hkl<1 1 0> Si self-interstitials
1613  }
1614
1615  \scriptsize
1616
1617 %\begin{center}
1618 %\begin{minipage}{3.2cm}
1619 %\includegraphics[width=3cm]{sub_110_combo.eps}
1620 %\end{minipage}
1621 %\begin{minipage}{7.8cm}
1622 %\begin{tabular}{l c c c c c c}
1623 %\hline
1624 %C$_{\text{sub}}$ & \hkl<1 1 0> & \hkl<-1 1 0> & \hkl<0 1 1> & \hkl<0 -1 1> &
1625 %                   \hkl<1 0 1> & \hkl<-1 0 1> \\
1626 %\hline
1627 %1 & \RM{1} & \RM{3} & \RM{3} & \RM{1} & \RM{3} & \RM{1} \\
1628 %2 & \RM{2} & A & A & \RM{2} & C & \RM{5} \\
1629 %3 & \RM{3} & \RM{1} & \RM{3} & \RM{1} & \RM{1} & \RM{3} \\
1630 %4 & \RM{4} & B & D & E & E & D \\
1631 %5 & \RM{5} & C & A & \RM{2} & A & \RM{2} \\
1632 %\hline
1633 %\end{tabular}
1634 %\end{minipage}
1635 %\end{center}
1636
1637 %\begin{center}
1638 %\begin{tabular}{l c c c c c c c c c c}
1639 %\hline
1640 %Conf & \RM{1} & \RM{2} & \RM{3} & \RM{4} & \RM{5} & A & B & C & D & E \\
1641 %\hline
1642 %$E_{\text{f}}$ [eV]& 4.37 & 5.26 & 5.57 & 5.37 & 5.12 & 5.10 & 5.32 & 5.28 & 5.39 & 5.32 \\
1643 %$E_{\text{b}}$ [eV] & -0.97 & -0.08 & 0.22 & -0.02 & -0.23 & -0.25 & -0.02 & -0.06 & 0.05 & -0.03 \\
1644 %$r$ [nm] & 0.292 & 0.394 & 0.241 & 0.453 & 0.407 & 0.408 & 0.452 & 0.392 & 0.456 & 0.453\\
1645 %\hline
1646 %\end{tabular}
1647 %\end{center}
1648
1649 \begin{minipage}{6.0cm}
1650 \includegraphics[width=5.8cm]{c_sub_si110.ps}
1651 \end{minipage}
1652 \begin{minipage}{7cm}
1653 \scriptsize
1654 \begin{itemize}
1655  \item IBS: C may displace Si\\
1656        $\Rightarrow$ C$_{\text{sub}}$ + \hkl<1 1 0> Si self-interstitial
1657  \item Assumption:\\
1658        \hkl<1 1 0>-type $\rightarrow$ favored combination
1659  \renewcommand\labelitemi{$\Rightarrow$}
1660  \item Most favorable: \cs{} along \hkl<1 1 0> chain \si{}
1661  \item Less favorable than C-Si \hkl<1 0 0> dumbbell
1662  \item Interaction drops quickly to zero\\
1663        $\rightarrow$ low capture radius
1664 \end{itemize}
1665 \begin{center}
1666  {\color{blue}
1667  IBS process far from equilibrium\\
1668  \cs{} \& \si{} instead of thermodynamic ground state
1669  }
1670 \end{center}
1671 \end{minipage}
1672
1673 \begin{minipage}{6.5cm}
1674 \includegraphics[width=6.0cm]{162-097.ps}
1675 \begin{itemize}
1676  \item Low migration barrier
1677 \end{itemize}
1678 \end{minipage}
1679 \begin{minipage}{6.5cm}
1680 \begin{center}
1681 Ab initio MD at \degc{900}\\
1682 \includegraphics[width=3.3cm]{md_vasp_01.eps}
1683 $t=\unit[2230]{fs}$\\
1684 \includegraphics[width=3.3cm]{md_vasp_02.eps}
1685 $t=\unit[2900]{fs}$
1686 \end{center}
1687 {\color{blue}
1688 Contribution of entropy to structural formation
1689 }
1690 \end{minipage}
1691
1692 \end{slide}
1693
1694 \begin{slide}
1695
1696  {\large\bf\boldmath
1697   Migration in C-Si \hkl<1 0 0> and vacancy combinations
1698  }
1699
1700  \footnotesize
1701
1702 \vspace{0.1cm}
1703
1704 \begin{minipage}[t]{3cm}
1705 \underline{Pos 2, $E_{\text{b}}=-0.59\text{ eV}$}\\
1706 \includegraphics[width=2.8cm]{00-1dc/0-59.eps}
1707 \end{minipage}
1708 \begin{minipage}[t]{7cm}
1709 \vspace{0.2cm}
1710 \begin{center}
1711  Low activation energies\\
1712  High activation energies for reverse processes\\
1713  $\Downarrow$\\
1714  {\color{blue}C$_{\text{sub}}$ very stable}\\
1715 \vspace*{0.1cm}
1716  \hrule
1717 \vspace*{0.1cm}
1718  Without nearby \hkl<1 1 0> Si self-interstitial (IBS)\\
1719  $\Downarrow$\\
1720  {\color{blue}Formation of SiC by successive substitution by C}
1721
1722 \end{center}
1723 \end{minipage}
1724 \begin{minipage}[t]{3cm}
1725 \underline{Pos 3, $E_{\text{b}}=-3.14\text{ eV}$}\\
1726 \includegraphics[width=2.8cm]{00-1dc/3-14.eps}
1727 \end{minipage}
1728
1729
1730 \framebox{
1731 \begin{minipage}{5.9cm}
1732 \includegraphics[width=5.9cm]{vasp_mig/comb_mig_3-2_vac_fullct.ps}\\[0.6cm]
1733 \begin{center}
1734 \begin{picture}(0,0)(70,0)
1735 \includegraphics[width=1.4cm]{vasp_mig/comb_2-1_init.eps}
1736 \end{picture}
1737 \begin{picture}(0,0)(30,0)
1738 \includegraphics[width=1.4cm]{vasp_mig/comb_2-1_seq_03.eps}
1739 \end{picture}
1740 \begin{picture}(0,0)(-10,0)
1741 \includegraphics[width=1.4cm]{vasp_mig/comb_2-1_seq_06.eps}
1742 \end{picture}
1743 \begin{picture}(0,0)(-48,0)
1744 \includegraphics[width=1.4cm]{vasp_mig/comb_2-1_final.eps}
1745 \end{picture}
1746 \begin{picture}(0,0)(12.5,5)
1747 \includegraphics[width=1cm]{100_arrow.eps}
1748 \end{picture}
1749 \begin{picture}(0,0)(97,-10)
1750 \includegraphics[height=0.9cm]{001_arrow.eps}
1751 \end{picture}
1752 \end{center}
1753 \vspace{0.1cm}
1754 \end{minipage}
1755 }
1756 \begin{minipage}{0.3cm}
1757 \hfill
1758 \end{minipage}
1759 \framebox{
1760 \begin{minipage}{5.9cm}
1761 \includegraphics[width=5.9cm]{vasp_mig/comb_mig_4-2_vac_fullct.ps}\\[0.1cm]
1762 \begin{center}
1763 \begin{picture}(0,0)(60,0)
1764 \includegraphics[width=0.9cm]{vasp_mig/comb_3-1_init.eps}
1765 \end{picture}
1766 \begin{picture}(0,0)(25,0)
1767 \includegraphics[width=0.9cm]{vasp_mig/comb_3-1_seq_03.eps}
1768 \end{picture}
1769 \begin{picture}(0,0)(-20,0)
1770 \includegraphics[width=0.9cm]{vasp_mig/comb_3-1_seq_07.eps}
1771 \end{picture}
1772 \begin{picture}(0,0)(-55,0)
1773 \includegraphics[width=0.9cm]{vasp_mig/comb_3-1_final.eps}
1774 \end{picture}
1775 \begin{picture}(0,0)(12.5,5)
1776 \includegraphics[width=1cm]{100_arrow.eps}
1777 \end{picture}
1778 \begin{picture}(0,0)(95,0)
1779 \includegraphics[height=0.9cm]{001_arrow.eps}
1780 \end{picture}
1781 \end{center}
1782 \vspace{0.1cm}
1783 \end{minipage}
1784 }
1785
1786 \end{slide}
1787
1788 \begin{slide}
1789
1790  {\large\bf
1791   Conclusion of defect / migration / combined defect simulations
1792  }
1793
1794  \footnotesize
1795
1796 \vspace*{0.1cm}
1797
1798 Defect structures
1799 \begin{itemize}
1800  \item Accurately described by quantum-mechanical simulations
1801  \item Less accurate description by classical potential simulations
1802  \item Underestimated formation energy of \cs{} by classical approach
1803  \item Both methods predict same ground state: \ci{} \hkl<1 0 0> dumbbell
1804 \end{itemize}
1805
1806 Migration
1807 \begin{itemize}
1808  \item C migration pathway in Si identified
1809  \item Consistent with reorientation and diffusion experiments
1810 \end{itemize}
1811 \begin{itemize}
1812  \item Different path and ...
1813  \item overestimated barrier by classical potential calculations
1814 \end{itemize}
1815
1816 Concerning the precipitation mechanism
1817 \begin{itemize}
1818  \item Agglomeration of C-Si dumbbells energetically favorable
1819        (stress compensation)
1820  \item C-Si indeed favored compared to
1821        C$_{\text{sub}}$ \& \hkl<1 1 0> Si self-interstitial
1822  \item Possible low interaction capture radius of
1823        C$_{\text{sub}}$ \& \hkl<1 1 0> Si self-interstitial
1824  \item Low barrier for
1825        \ci{} \hkl<1 0 0> $\rightarrow$ \cs{} \& \si{} \hkl<1 1 0>
1826  \item In absence of nearby \hkl<1 1 0> Si self-interstitial:
1827        C-Si \hkl<1 0 0> + Vacancy $\rightarrow$ C$_{\text{sub}}$ (SiC)
1828 \end{itemize}
1829 \begin{center}
1830 {\color{blue}Results suggest increased participation of \cs}
1831 \end{center}
1832
1833 \end{slide}
1834
1835 \begin{slide}
1836
1837  {\large\bf
1838   Silicon carbide precipitation simulations
1839  }
1840
1841  \small
1842
1843 {\scriptsize
1844  \begin{pspicture}(0,0)(12,6.5)
1845   % nodes
1846   \rput(3.5,5.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
1847    \parbox{7cm}{
1848    \begin{itemize}
1849     \item Create c-Si volume
1850     \item Periodc boundary conditions
1851     \item Set requested $T$ and $p=0\text{ bar}$
1852     \item Equilibration of $E_{\text{kin}}$ and $E_{\text{pot}}$
1853    \end{itemize}
1854   }}}}
1855   \rput(3.5,2.7){\rnode{insert}{\psframebox[fillstyle=solid,fillcolor=lachs]{
1856    \parbox{7cm}{
1857    Insertion of C atoms at constant T
1858    \begin{itemize}
1859     \item total simulation volume {\pnode{in1}}
1860     \item volume of minimal SiC precipitate {\pnode{in2}}
1861     \item volume consisting of Si atoms to form a minimal {\pnode{in3}}\\
1862           precipitate
1863    \end{itemize}
1864   }}}}
1865   \rput(3.5,1){\rnode{cool}{\psframebox[fillstyle=solid,fillcolor=lbb]{
1866    \parbox{7.0cm}{
1867    Run for 100 ps followed by cooling down to $20\, ^{\circ}\textrm{C}$
1868   }}}}
1869   \ncline[]{->}{init}{insert}
1870   \ncline[]{->}{insert}{cool}
1871   \psframe[fillstyle=solid,fillcolor=white](7.5,0.7)(13.5,6.3)
1872   \rput(7.8,6){\footnotesize $V_1$}
1873   \psframe[fillstyle=solid,fillcolor=lightgray](9,2)(12,5)
1874   \rput(9.2,4.85){\tiny $V_2$}
1875   \psframe[fillstyle=solid,fillcolor=gray](9.25,2.25)(11.75,4.75)
1876   \rput(9.55,4.45){\footnotesize $V_3$}
1877   \rput(7.9,3.2){\pnode{ins1}}
1878   \rput(9.22,2.8){\pnode{ins2}}
1879   \rput(11.0,2.4){\pnode{ins3}}
1880   \ncline[]{->}{in1}{ins1}
1881   \ncline[]{->}{in2}{ins2}
1882   \ncline[]{->}{in3}{ins3}
1883  \end{pspicture}
1884 }
1885
1886 \begin{itemize}
1887  \item Restricted to classical potential simulations
1888  \item $V_2$ and $V_3$ considered due to low diffusion
1889  \item Amount of C atoms: 6000
1890        ($r_{\text{prec}}\approx 3.1\text{ nm}$, IBS: 2 ... 4 nm)
1891  \item Simulation volume: $31\times 31\times 31$ unit cells
1892        (238328 Si atoms)
1893 \end{itemize}
1894
1895 \end{slide}
1896
1897 \begin{slide}
1898
1899  {\large\bf\boldmath
1900   Silicon carbide precipitation simulations at $450\,^{\circ}\mathrm{C}$ as in IBS
1901  }
1902
1903  \small
1904
1905 \begin{minipage}{6.5cm}
1906 \includegraphics[width=6.4cm]{sic_prec_450_si-si_c-c.ps}
1907 \end{minipage}
1908 \begin{minipage}{6.5cm}
1909 \includegraphics[width=6.4cm]{sic_prec_450_energy.ps}
1910 \end{minipage}
1911
1912 \begin{minipage}{6.5cm}
1913 \includegraphics[width=6.4cm]{sic_prec_450_si-c.ps}
1914 \end{minipage}
1915 \begin{minipage}{6.5cm}
1916 \scriptsize
1917 \underline{Low C concentration ($V_1$)}\\
1918 \hkl<1 0 0> C-Si dumbbell dominated structure
1919 \begin{itemize}
1920  \item Si-C bumbs around 0.19 nm
1921  \item C-C peak at 0.31 nm (as expected in 3C-SiC):\\
1922        concatenated dumbbells of various orientation
1923  \item Si-Si NN distance stretched to 0.3 nm
1924 \end{itemize}
1925 {\color{blue}$\Rightarrow$ C atoms in proper 3C-SiC distance first}\\
1926 \underline{High C concentration ($V_2$, $V_3$)}\\
1927 High amount of strongly bound C-C bonds\\
1928 Defect density $\uparrow$ $\Rightarrow$ considerable amount of damage\\
1929 Only short range order observable\\
1930 {\color{blue}$\Rightarrow$ amorphous SiC-like phase}
1931 \end{minipage}
1932
1933 \end{slide}
1934
1935 \begin{slide}
1936
1937  {\large\bf\boldmath
1938   Silicon carbide precipitation simulations at $450\,^{\circ}\mathrm{C}$ as in IBS
1939  }
1940
1941  \small
1942
1943 \begin{minipage}{6.5cm}
1944 \includegraphics[width=6.4cm]{sic_prec_450_si-si_c-c.ps}
1945 \end{minipage}
1946 \begin{minipage}{6.5cm}
1947 \includegraphics[width=6.4cm]{sic_prec_450_energy.ps}
1948 \end{minipage}
1949
1950 \begin{minipage}{6.5cm}
1951 \includegraphics[width=6.4cm]{sic_prec_450_si-c.ps}
1952 \end{minipage}
1953 \begin{minipage}{6.5cm}
1954 \scriptsize
1955 \underline{Low C concentration ($V_1$)}\\
1956 \hkl<1 0 0> C-Si dumbbell dominated structure
1957 \begin{itemize}
1958  \item Si-C bumbs around 0.19 nm
1959  \item C-C peak at 0.31 nm (as expected in 3C-SiC):\\
1960        concatenated dumbbells of various orientation
1961  \item Si-Si NN distance stretched to 0.3 nm
1962 \end{itemize}
1963 {\color{blue}$\Rightarrow$ C atoms in proper 3C-SiC distance first}\\
1964 \underline{High C concentration ($V_2$, $V_3$)}\\
1965 High amount of strongly bound C-C bonds\\
1966 Defect density $\uparrow$ $\Rightarrow$ considerable amount of damage\\
1967 Only short range order observable\\
1968 {\color{blue}$\Rightarrow$ amorphous SiC-like phase}
1969 \end{minipage}
1970
1971 \begin{pspicture}(0,0)(0,0)
1972 \rput(6.7,5.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=white]{
1973 \begin{minipage}{10cm}
1974 \small
1975 {\color{red}\bf 3C-SiC formation fails to appear}
1976 \begin{itemize}
1977 \item Low C concentration simulations
1978  \begin{itemize}
1979   \item Formation of \ci{} indeed occurs
1980   \item Agllomeration not observed
1981  \end{itemize}
1982 \item High C concentration simulations
1983  \begin{itemize}
1984   \item Amorphous SiC-like structure\\
1985         (not expected at prevailing temperatures)
1986   \item Rearrangement and transition into 3C-SiC structure missing
1987  \end{itemize}
1988 \end{itemize}
1989 \end{minipage}
1990  }}}
1991 \end{pspicture}
1992
1993 \end{slide}
1994
1995 \begin{slide}
1996
1997  {\large\bf
1998   Limitations of molecular dynamics and short range potentials
1999  }
2000
2001 \footnotesize
2002
2003 \vspace{0.2cm}
2004
2005 \underline{Time scale problem of MD}\\[0.2cm]
2006 Minimize integration error\\
2007 $\Rightarrow$ discretization considerably smaller than
2008               reciprocal of fastest vibrational mode\\[0.1cm]
2009 Order of fastest vibrational mode: $10^{13} - 10^{14}\text{ Hz}$\\
2010 $\Rightarrow$ suitable choice of time step:
2011               $\tau=1\text{ fs}=10^{-15}\text{ s}$\\
2012 $\Rightarrow$ {\color{red}\underline{slow}} phase space propagation\\[0.1cm]
2013 Several local minima in energy surface separated by large energy barriers\\
2014 $\Rightarrow$ transition event corresponds to a multiple
2015               of vibrational periods\\
2016 $\Rightarrow$ phase transition made up of {\color{red}\underline{many}}
2017               infrequent transition events\\[0.1cm]
2018 {\color{blue}Accelerated methods:}
2019 \underline{Temperature accelerated} MD (TAD), self-guided MD \ldots
2020
2021 \vspace{0.3cm}
2022
2023 \underline{Limitations related to the short range potential}\\[0.2cm]
2024 Cut-off function pushing forces and energies to zero between 1$^{\text{st}}$
2025 and 2$^{\text{nd}}$ next neighbours\\
2026 $\Rightarrow$ overestimated unphysical high forces of next neighbours
2027
2028 \vspace{0.3cm}
2029
2030 \framebox{
2031 \color{red}
2032 Potential enhanced problem of slow phase space propagation
2033 }
2034
2035 \vspace{0.3cm}
2036
2037 \underline{Approach to the (twofold) problem}\\[0.2cm]
2038 Increased temperature simulations without TAD corrections\\
2039 (accelerated methods or higher time scales exclusively not sufficient)
2040
2041 \begin{picture}(0,0)(-260,-30)
2042 \framebox{
2043 \begin{minipage}{4.2cm}
2044 \tiny
2045 \begin{center}
2046 \vspace{0.03cm}
2047 \underline{IBS}
2048 \end{center}
2049 \begin{itemize}
2050 \item 3C-SiC also observed for higher T
2051 \item higher T inside sample
2052 \item structural evolution vs.\\
2053       equilibrium properties
2054 \end{itemize}
2055 \end{minipage}
2056 }
2057 \end{picture}
2058
2059 \begin{picture}(0,0)(-305,-155)
2060 \framebox{
2061 \begin{minipage}{2.5cm}
2062 \tiny
2063 \begin{center}
2064 retain proper\\
2065 thermodynmic sampling
2066 \end{center}
2067 \end{minipage}
2068 }
2069 \end{picture}
2070
2071 \end{slide}
2072
2073 \begin{slide}
2074
2075  {\large\bf
2076   Increased temperature simulations at low C concentration
2077  }
2078
2079 \small
2080
2081 \begin{minipage}{6.5cm}
2082 \includegraphics[width=6.4cm]{tot_pc_thesis.ps}
2083 \end{minipage}
2084 \begin{minipage}{6.5cm}
2085 \includegraphics[width=6.4cm]{tot_pc3_thesis.ps}
2086 \end{minipage}
2087
2088 \begin{minipage}{6.5cm}
2089 \includegraphics[width=6.4cm]{tot_pc2_thesis.ps}
2090 \end{minipage}
2091 \begin{minipage}{6.5cm}
2092 \scriptsize
2093  \underline{Si-C bonds:}
2094  \begin{itemize}
2095   \item Vanishing cut-off artifact (above $1650\,^{\circ}\mathrm{C}$)
2096   \item Structural change: C-Si \hkl<1 0 0> $\rightarrow$ C$_{\text{sub}}$
2097  \end{itemize}
2098  \underline{Si-Si bonds:}
2099  {\color{blue}Si-C$_{\text{sub}}$-Si} along \hkl<1 1 0>
2100  ($\rightarrow$ 0.325 nm)\\[0.1cm]
2101  \underline{C-C bonds:}
2102  \begin{itemize}
2103   \item C-C next neighbour pairs reduced (mandatory)
2104   \item Peak at 0.3 nm slightly shifted
2105         \begin{itemize}
2106          \item C-Si \hkl<1 0 0> combinations (dashed arrows)\\
2107                $\rightarrow$ C-Si \hkl<1 0 0> \& C$_{\text{sub}}$
2108                combinations (|)\\
2109                $\rightarrow$ pure {\color{blue}C$_{\text{sub}}$ combinations}
2110                ($\downarrow$)
2111          \item Range [|-$\downarrow$]:
2112                {\color{blue}C$_{\text{sub}}$ \& C$_{\text{sub}}$
2113                with nearby Si$_{\text{I}}$}
2114         \end{itemize}
2115  \end{itemize}
2116 \end{minipage}
2117
2118 \begin{picture}(0,0)(-330,-74)
2119 \color{blue}
2120 \framebox{
2121 \begin{minipage}{1.6cm}
2122 \tiny
2123 \begin{center}
2124 stretched SiC\\[-0.1cm]
2125 in c-Si
2126 \end{center}
2127 \end{minipage}
2128 }
2129 \end{picture}
2130
2131 \end{slide}
2132
2133 \begin{slide}
2134
2135  {\large\bf
2136   Increased temperature simulations at low C concentration
2137  }
2138
2139 \small
2140
2141 \begin{minipage}{6.5cm}
2142 \includegraphics[width=6.4cm]{tot_pc_thesis.ps}
2143 \end{minipage}
2144 \begin{minipage}{6.5cm}
2145 \includegraphics[width=6.4cm]{tot_pc3_thesis.ps}
2146 \end{minipage}
2147
2148 \begin{minipage}{6.5cm}
2149 \includegraphics[width=6.4cm]{tot_pc2_thesis.ps}
2150 \end{minipage}
2151 \begin{minipage}{6.5cm}
2152 \scriptsize
2153  \underline{Si-C bonds:}
2154  \begin{itemize}
2155   \item Vanishing cut-off artifact (above $1650\,^{\circ}\mathrm{C}$)
2156   \item Structural change: C-Si \hkl<1 0 0> $\rightarrow$ C$_{\text{sub}}$
2157  \end{itemize}
2158  \underline{Si-Si bonds:}
2159  {\color{blue}Si-C$_{\text{sub}}$-Si} along \hkl<1 1 0>
2160  ($\rightarrow$ 0.325 nm)\\[0.1cm]
2161  \underline{C-C bonds:}
2162  \begin{itemize}
2163   \item C-C next neighbour pairs reduced (mandatory)
2164   \item Peak at 0.3 nm slightly shifted
2165         \begin{itemize}
2166          \item C-Si \hkl<1 0 0> combinations (dashed arrows)\\
2167                $\rightarrow$ C-Si \hkl<1 0 0> \& C$_{\text{sub}}$
2168                combinations (|)\\
2169                $\rightarrow$ pure {\color{blue}C$_{\text{sub}}$ combinations}
2170                ($\downarrow$)
2171          \item Range [|-$\downarrow$]:
2172                {\color{blue}C$_{\text{sub}}$ \& C$_{\text{sub}}$
2173                with nearby Si$_{\text{I}}$}
2174         \end{itemize}
2175  \end{itemize}
2176 \end{minipage}
2177
2178 %\begin{picture}(0,0)(-330,-74)
2179 %\color{blue}
2180 %\framebox{
2181 %\begin{minipage}{1.6cm}
2182 %\tiny
2183 %\begin{center}
2184 %stretched SiC\\[-0.1cm]
2185 %in c-Si
2186 %\end{center}
2187 %\end{minipage}
2188 %}
2189 %\end{picture}
2190
2191 \begin{pspicture}(0,0)(0,0)
2192 \rput(6.7,5.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=white]{
2193 \begin{minipage}{10cm}
2194 \small
2195 {\color{blue}\bf Stretched SiC in c-Si}
2196 \begin{itemize}
2197 \item Consistent to precipitation model involving \cs{}
2198 \item Explains annealing behavior of high/low T C implants
2199       \begin{itemize}
2200        \item Low T: highly mobiel \ci{}
2201        \item High T: stable configurations of \cs{}
2202       \end{itemize}
2203 \end{itemize}
2204 $\Rightarrow$ High T $\leftrightarrow$ IBS conditions far from equilibrium\\
2205 $\Rightarrow$ Precipitation mechanism involving \cs{}
2206 \end{minipage}
2207  }}}
2208 \end{pspicture}
2209
2210 \end{slide}
2211
2212 \begin{slide}
2213
2214  {\large\bf
2215   Increased temperature simulations at high C concentration
2216  }
2217
2218 \footnotesize
2219
2220 \begin{minipage}{6.5cm}
2221 \includegraphics[width=6.4cm]{12_pc_thesis.ps}
2222 \end{minipage}
2223 \begin{minipage}{6.5cm}
2224 \includegraphics[width=6.4cm]{12_pc_c_thesis.ps}
2225 \end{minipage}
2226
2227 \vspace{0.1cm}
2228
2229 \scriptsize
2230
2231 \framebox{
2232 \begin{minipage}[t]{6.0cm}
2233 0.186 nm: Si-C pairs $\uparrow$\\
2234 (as expected in 3C-SiC)\\[0.2cm]
2235 0.282 nm: Si-C-C\\[0.2cm]
2236 $\approx$0.35 nm: C-Si-Si
2237 \end{minipage}
2238 }
2239 \begin{minipage}{0.2cm}
2240 \hfill
2241 \end{minipage}
2242 \framebox{
2243 \begin{minipage}[t]{6.0cm}
2244 0.15 nm: C-C pairs $\uparrow$\\
2245 (as expected in graphite/diamond)\\[0.2cm]
2246 0.252 nm: C-C-C (2$^{\text{nd}}$ NN for diamond)\\[0.2cm]
2247 0.31 nm: shifted towards 0.317 nm $\rightarrow$ C-Si-C
2248 \end{minipage}
2249 }
2250
2251 \begin{itemize}
2252 \item Decreasing cut-off artifact
2253 \item {\color{red}Amorphous} SiC-like phase remains
2254 \item High amount of {\color{red}damage} \& alignement to c-Si host matrix lost
2255 \item Slightly sharper peaks $\Rightarrow$ indicate slight {\color{blue}acceleration of dynamics} due to temperature
2256 \end{itemize}
2257
2258 \vspace{-0.1cm}
2259
2260 \begin{center}
2261 {\color{blue}
2262 \framebox{
2263 {\color{black}
2264 High C \& small $V$ \& short $t$
2265 $\Rightarrow$
2266 }
2267 Slow restructuring due to strong C-C bonds
2268 {\color{black}
2269 $\Leftarrow$
2270 High C \& low T implants
2271 }
2272 }
2273 }
2274 \end{center}
2275
2276 \end{slide}
2277
2278 \begin{slide}
2279
2280  {\large\bf
2281   Summary and Conclusions
2282  }
2283
2284  \scriptsize
2285
2286 %\vspace{0.1cm}
2287
2288 \framebox{
2289 \begin{minipage}[t]{12.9cm}
2290  \underline{Pecipitation simulations}
2291  \begin{itemize}
2292   \item High C concentration $\rightarrow$ amorphous SiC like phase
2293   \item Problem of potential enhanced slow phase space propagation
2294   \item Low T $\rightarrow$ C-Si \hkl<1 0 0> dumbbell dominated structure
2295   \item High T $\rightarrow$ C$_{\text{sub}}$ dominated structure
2296   \item High T necessary to simulate IBS conditions (far from equilibrium)
2297   \item Precipitation by successive agglomeration of \cs (epitaxy)
2298   \item \si{}: vehicle to form \cs{} \& supply of Si \& stress compensation
2299         (stretched SiC, interface)
2300  \end{itemize}
2301 \end{minipage}
2302 }
2303
2304 %\vspace{0.1cm}
2305
2306 \framebox{
2307 \begin{minipage}{12.9cm}
2308  \underline{Defects}
2309  \begin{itemize}
2310    \item DFT / EA
2311         \begin{itemize}
2312          \item Point defects excellently / fairly well described
2313                by DFT / EA
2314          \item C$_{\text{sub}}$ drastically underestimated by EA
2315          \item EA predicts correct ground state:
2316                C$_{\text{sub}}$ \& \si{} $>$ \ci{}
2317          \item Identified migration path explaining
2318                diffusion and reorientation experiments by DFT
2319          \item EA fails to describe \ci{} migration:
2320                Wrong path \& overestimated barrier
2321         \end{itemize}
2322    \item Combinations of defects
2323          \begin{itemize}
2324           \item Agglomeration of point defects energetically favorable
2325                 by compensation of stress
2326           \item Formation of C-C unlikely
2327           \item C$_{\text{sub}}$ favored conditions (conceivable in IBS)
2328           \item \ci{} \hkl<1 0 0> $\leftrightarrow$ \cs{} \& \si{} \hkl<1 1 0>\\
2329                 Low barrier (\unit[0.77]{eV}) \& low capture radius
2330         \end{itemize}
2331  \end{itemize}
2332 \end{minipage}
2333 }
2334
2335 \begin{center}
2336 {\color{blue}
2337 \framebox{Precipitation by successive agglomeration of \cs{}}
2338 }
2339 \end{center}
2340
2341 \end{slide}
2342
2343 \begin{slide}
2344
2345  {\large\bf
2346   Acknowledgements
2347  }
2348
2349  \vspace{0.1cm}
2350
2351  \small
2352
2353  Thanks to \ldots
2354
2355  \underline{Augsburg}
2356  \begin{itemize}
2357   \item Prof. B. Stritzker (accomodation at EP \RM{4})
2358   \item Ralf Utermann (EDV)
2359  \end{itemize}
2360
2361  \underline{Helsinki}
2362  \begin{itemize}
2363   \item Prof. K. Nordlund (MD)
2364  \end{itemize}
2365
2366  \underline{Munich}
2367  \begin{itemize}
2368   \item Bayerische Forschungsstiftung (financial support)
2369  \end{itemize}
2370
2371  \underline{Paderborn}
2372  \begin{itemize}
2373   \item Prof. J. Lindner (SiC)
2374   \item Prof. G. Schmidt (DFT + financial support)
2375   \item Dr. E. Rauls (DFT + SiC)
2376   \item Dr. S. Sanna (VASP)
2377  \end{itemize}
2378
2379 \vspace{0.2cm}
2380
2381 \begin{center}
2382 \framebox{
2383 \bf Thank you for your attention!
2384 }
2385 \end{center}
2386
2387 \end{slide}
2388
2389 \end{document}
2390
2391 \fi