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)(0,-162)
 202 \includegraphics[height=2.0cm]{3C_SiC_bs.eps}
 203 \end{picture}
 204 \begin{picture}(0,0)(-130,-162)
 205 \includegraphics[height=2.0cm]{nasa_600c_led.eps}
 206 \end{picture}
 207 \begin{picture}(0,0)(-295,-162)
 208 \includegraphics[height=2.0cm]{6h-sic_3c-sic.eps}
 209 \end{picture}
 210 %%%%
 211 \begin{picture}(0,0)(5,65)
 212 \includegraphics[height=2.8cm]{sic_switch.eps}
 213 \end{picture}
 214 \begin{picture}(0,0)(-145,65)
 215 \includegraphics[height=2.8cm]{infineon_schottky.eps}
 216 \end{picture}
 217 \begin{picture}(0,0)(-260,65)
 218 \includegraphics[height=2.8cm]{ise_99.eps}
 219 \end{picture}
 220
 221 \end{slide}
 222
 223 % motivation
 224
 225 \begin{slide}
 226
 227  {\large\bf
 228   Polytypes of SiC
 229  }
 230
 231  \vspace{4cm}
 232
 233  \small
 234
 235 \begin{tabular}{l c c c c c c}
 236 \hline
 237  & 3C-SiC & 4H-SiC & 6H-SiC & Si & GaN & Diamond\\
 238 \hline
 239 Hardness [Mohs] & \multicolumn{3}{c}{------ 9.6 ------}& 6.5 & - & 10 \\
 240 Band gap [eV] & 2.36 & 3.23 & 3.03 & 1.12 & 3.39 & 5.5 \\
 241 Break down field [$10^6$ V/cm] & 4 & 3 & 3.2 & 0.6 & 5 & 10 \\
 242 Saturation drift velocity [$10^7$ cm/s] & 2.5 & 2.0 & 2.0 & 1 & 2.7 & 2.7 \\
 243 Electron mobility [cm$^2$/Vs] & 800 & 900 & 400 & 1100 & 900 & 2200 \\
 244 Hole mobility [cm$^2$/Vs] & 320 & 120 & 90 & 420 & 150 & 1600 \\
 245 Thermal conductivity [W/cmK] & 5.0 & 4.9 & 4.9 & 1.5 & 1.3 & 22 \\
 246 \hline
 247 \end{tabular}
 248
 249 {\tiny
 250  Values for $T=300$ K
 251 }
 252
 253 \begin{picture}(0,0)(-160,-155)
 254  \includegraphics[width=7cm]{polytypes.eps}
 255 \end{picture}
 256 \begin{picture}(0,0)(-10,-185)
 257  \includegraphics[width=3.8cm]{cubic_hex.eps}\\
 258 \end{picture}
 259 \begin{picture}(0,0)(-10,-175)
 260  {\tiny cubic (twist)}
 261 \end{picture}
 262 \begin{picture}(0,0)(-60,-175)
 263  {\tiny hexagonal (no twist)}
 264 \end{picture}
 265 \begin{pspicture}(0,0)(0,0)
 266 \psellipse[linecolor=green](5.7,3.03)(0.4,0.5)
 267 \end{pspicture}
 268 \begin{pspicture}(0,0)(0,0)
 269 \psellipse[linecolor=green](5.6,1.68)(0.4,0.2)
 270 \end{pspicture}
 271 \begin{pspicture}(0,0)(0,0)
 272 \psellipse[linecolor=red](10.7,1.13)(0.4,0.2)
 273 \end{pspicture}
 274
 275 \end{slide}
 276
 277 % fabrication
 278
 279 \begin{slide}
 280
 281  {\large\bf
 282   Fabrication of silicon carbide
 283  }
 284
 285  \small
 286
 287  \vspace{4pt}
 288
 289  SiC - \emph{Born from the stars, perfected on earth.}
 290
 291  IBS also here!
 292
 293  \vspace{4pt}
 294
 295  Conventional thin film SiC growth:
 296  \begin{itemize}
 297   \item \underline{Sublimation growth using the modified Lely method}
 298         \begin{itemize}
 299          \item SiC single-crystalline seed at $T=1800 \, ^{\circ} \text{C}$
 300          \item Surrounded by polycrystalline SiC in a graphite crucible\\
 301                at $T=2100-2400 \, ^{\circ} \text{C}$
 302          \item Deposition of supersaturated vapor on cooler seed crystal
 303         \end{itemize}
 304   \item \underline{Homoepitaxial growth using CVD}
 305         \begin{itemize}
 306          \item Step-controlled epitaxy on off-oriented 6H-SiC substrates
 307          \item C$_3$H$_8$/SiH$_4$/H$_2$ at $1100-1500 \, ^{\circ} \text{C}$
 308          \item Angle, temperature $\rightarrow$ 3C/6H/4H-SiC
 309         \end{itemize}
 310   \item \underline{Heteroepitaxial growth of 3C-SiC on Si using CVD/MBE}
 311         \begin{itemize}
 312          \item Two steps: carbonization and growth
 313          \item $T=650-1050 \, ^{\circ} \text{C}$
 314          \item SiC/Si lattice mismatch $\approx$ 20 \%
 315          \item Quality and size not yet sufficient
 316         \end{itemize}
 317  \end{itemize}
 318
 319  \begin{picture}(0,0)(-280,-65)
 320   \includegraphics[width=3.8cm]{6h-sic_3c-sic.eps}
 321  \end{picture}
 322  \begin{picture}(0,0)(-280,-55)
 323   \begin{minipage}{5cm}
 324   {\tiny
 325    NASA: 6H-SiC and 3C-SiC LED\\[-7pt]
 326    on 6H-SiC substrate
 327   }
 328   \end{minipage}
 329  \end{picture}
 330  \begin{picture}(0,0)(-265,-150)
 331   \includegraphics[width=2.4cm]{m_lely.eps}
 332  \end{picture}
 333  \begin{picture}(0,0)(-333,-175)
 334   \begin{minipage}{5cm}
 335   {\tiny
 336    1. Lid\\[-7pt]
 337    2. Heating\\[-7pt]
 338    3. Source\\[-7pt]
 339    4. Crucible\\[-7pt]
 340    5. Insulation\\[-7pt]
 341    6. Seed crystal
 342   }
 343   \end{minipage}
 344  \end{picture}
 345  \begin{picture}(0,0)(-230,-35)
 346  \framebox{
 347  {\footnotesize\color{blue}\bf Hex: micropipes along c-axis}
 348  }
 349  \end{picture}
 350  \begin{picture}(0,0)(-230,-10)
 351  \framebox{
 352  \begin{minipage}{3cm}
 353  {\footnotesize\color{blue}\bf 3C-SiC fabrication\\
 354                                less advanced}
 355  \end{minipage}
 356  }
 357  \end{picture}
 358
 359 \end{slide}
 360
 361 % contents
 362
 363 \begin{slide}
 364
 365 {\large\bf
 366  Outline
 367 }
 368
 369  \begin{itemize}
 370   \item Implantation of C in Si --- Overview of experimental observations
 371   \item Utilized simulation techniques and modeled problems
 372         \begin{itemize}
 373          \item {\color{blue}Diploma thesis}\\
 374                \underline{Monte Carlo} simulations
 375                modeling the selforganization process
 376                leading to periodic arrays of nanometric amorphous SiC
 377                precipitates
 378          \item {\color{blue}Doctoral studies}\\
 379                Classical potential \underline{molecular dynamics} simulations
 380                \ldots\\
 381                \underline{Density functional theory} calculations
 382                \ldots\\[0.2cm]
 383                \ldots on defects and SiC precipitation in Si
 384         \end{itemize}
 385   \item Summary / Conclusion / Outlook
 386  \end{itemize}
 387
 388 \end{slide}
 389
 390
 391
 392 \end{document}
 393
 394 \begin{slide}
 395
 396  {\large\bf
 397   Fabrication of silicon carbide
 398  }
 399
 400  \small
 401
 402  Alternative approach:
 403  Ion beam synthesis (IBS) of burried 3C-SiC layers in Si\hkl(1 0 0)
 404  \begin{itemize}
 405   \item \underline{Implantation step 1}\\
 406         180 keV C$^+$, $D=7.9\times 10^{17}$ cm$^{-2}$, $T_{\text{i}}=500\,^{\circ}\mathrm{C}$\\
 407         $\Rightarrow$ box-like distribution of equally sized
 408                        and epitactically oriented SiC precipitates
 409
 410   \item \underline{Implantation step 2}\\
 411         180 keV C$^+$, $D=0.6\times 10^{17}$ cm$^{-2}$, $T_{\text{i}}=250\,^{\circ}\mathrm{C}$\\
 412         $\Rightarrow$ destruction of SiC nanocrystals
 413                       in growing amorphous interface layers
 414   \item \underline{Annealing}\\
 415         $T=1250\,^{\circ}\mathrm{C}$, $t=10\,\text{h}$\\
 416         $\Rightarrow$ homogeneous, stoichiometric SiC layer
 417                       with sharp interfaces
 418  \end{itemize}
 419
 420  \begin{minipage}{6.3cm}
 421  \includegraphics[width=6cm]{ibs_3c-sic.eps}\\[-0.2cm]
 422  {\tiny
 423   XTEM micrograph of single crystalline 3C-SiC in Si\hkl(1 0 0)
 424  }
 425  \end{minipage}
 426 \framebox{
 427  \begin{minipage}{6.3cm}
 428  \begin{center}
 429  {\color{blue}
 430   Precipitation mechanism not yet fully understood!
 431  }
 432  \renewcommand\labelitemi{$\Rightarrow$}
 433  \small
 434  \underline{Understanding the SiC precipitation}
 435  \begin{itemize}
 436   \item significant technological progress in SiC thin film formation
 437   \item perspectives for processes relying upon prevention of SiC precipitation
 438  \end{itemize}
 439  \end{center}
 440  \end{minipage}
 441 }
 442
 443 \end{slide}
 444
 445
 446 \begin{slide}
 447
 448  {\large\bf
 449   Supposed precipitation mechanism of SiC in Si
 450  }
 451
 452  \scriptsize
 453
 454  \vspace{0.1cm}
 455
 456  \begin{minipage}{3.8cm}
 457  Si \& SiC lattice structure\\[0.2cm]
 458  \includegraphics[width=3.5cm]{sic_unit_cell.eps}\\[-0.3cm]
 459  \hrule
 460  \end{minipage}
 461  \hspace{0.6cm}
 462  \begin{minipage}{3.8cm}
 463  \begin{center}
 464  \includegraphics[width=3.3cm]{tem_c-si-db.eps}
 465  \end{center}
 466  \end{minipage}
 467  \hspace{0.6cm}
 468  \begin{minipage}{3.8cm}
 469  \begin{center}
 470  \includegraphics[width=3.3cm]{tem_3c-sic.eps}
 471  \end{center}
 472  \end{minipage}
 473
 474  \begin{minipage}{4cm}
 475  \begin{center}
 476  C-Si dimers (dumbbells)\\[-0.1cm]
 477  on Si interstitial sites
 478  \end{center}
 479  \end{minipage}
 480  \hspace{0.2cm}
 481  \begin{minipage}{4.2cm}
 482  \begin{center}
 483  Agglomeration of C-Si dumbbells\\[-0.1cm]
 484  $\Rightarrow$ dark contrasts
 485  \end{center}
 486  \end{minipage}
 487  \hspace{0.2cm}
 488  \begin{minipage}{4cm}
 489  \begin{center}
 490  Precipitation of 3C-SiC in Si\\[-0.1cm]
 491  $\Rightarrow$ Moir\'e fringes\\[-0.1cm]
 492  \& release of Si self-interstitials
 493  \end{center}
 494  \end{minipage}
 495
 496  \begin{minipage}{3.8cm}
 497  \begin{center}
 498  \includegraphics[width=3.3cm]{sic_prec_seq_01.eps}
 499  \end{center}
 500  \end{minipage}
 501  \hspace{0.6cm}
 502  \begin{minipage}{3.8cm}
 503  \begin{center}
 504  \includegraphics[width=3.3cm]{sic_prec_seq_02.eps}
 505  \end{center}
 506  \end{minipage}
 507  \hspace{0.6cm}
 508  \begin{minipage}{3.8cm}
 509  \begin{center}
 510  \includegraphics[width=3.3cm]{sic_prec_seq_03.eps}
 511  \end{center}
 512  \end{minipage}
 513
 514 \begin{pspicture}(0,0)(0,0)
 515 \psline[linewidth=4pt]{->}(8.5,2)(9.0,2)
 516 \psellipse[linecolor=blue](11.5,5.8)(0.3,0.5)
 517 \rput{-20}{\psellipse[linecolor=blue](3.3,8.1)(0.3,0.5)}
 518 \psline[linewidth=4pt]{->}(4.0,2)(4.5,2)
 519 \rput(12.7,0.3){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
 520  $4a_{\text{Si}}=5a_{\text{SiC}}$
 521  }}}
 522 \rput(12.2,8){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
 523 \hkl(h k l) planes match
 524  }}}
 525 \rput(9.7,6.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
 526 r = 2 - 4 nm
 527  }}}
 528 \end{pspicture}
 529
 530 \end{slide}
 531
 532 \begin{slide}
 533
 534  {\large\bf
 535   Supposed precipitation mechanism of SiC in Si
 536  }
 537
 538  \scriptsize
 539
 540  \vspace{0.1cm}
 541
 542  \begin{minipage}{3.8cm}
 543  Si \& SiC lattice structure\\[0.2cm]
 544  \includegraphics[width=3.5cm]{sic_unit_cell.eps}\\[-0.3cm]
 545  \hrule
 546  \end{minipage}
 547  \hspace{0.6cm}
 548  \begin{minipage}{3.8cm}
 549  \begin{center}
 550  \includegraphics[width=3.3cm]{tem_c-si-db.eps}
 551  \end{center}
 552  \end{minipage}
 553  \hspace{0.6cm}
 554  \begin{minipage}{3.8cm}
 555  \begin{center}
 556  \includegraphics[width=3.3cm]{tem_3c-sic.eps}
 557  \end{center}
 558  \end{minipage}
 559
 560  \begin{minipage}{4cm}
 561  \begin{center}
 562  C-Si dimers (dumbbells)\\[-0.1cm]
 563  on Si interstitial sites
 564  \end{center}
 565  \end{minipage}
 566  \hspace{0.2cm}
 567  \begin{minipage}{4.2cm}
 568  \begin{center}
 569  Agglomeration of C-Si dumbbells\\[-0.1cm]
 570  $\Rightarrow$ dark contrasts
 571  \end{center}
 572  \end{minipage}
 573  \hspace{0.2cm}
 574  \begin{minipage}{4cm}
 575  \begin{center}
 576  Precipitation of 3C-SiC in Si\\[-0.1cm]
 577  $\Rightarrow$ Moir\'e fringes\\[-0.1cm]
 578  \& release of Si self-interstitials
 579  \end{center}
 580  \end{minipage}
 581
 582  \begin{minipage}{3.8cm}
 583  \begin{center}
 584  \includegraphics[width=3.3cm]{sic_prec_seq_01.eps}
 585  \end{center}
 586  \end{minipage}
 587  \hspace{0.6cm}
 588  \begin{minipage}{3.8cm}
 589  \begin{center}
 590  \includegraphics[width=3.3cm]{sic_prec_seq_02.eps}
 591  \end{center}
 592  \end{minipage}
 593  \hspace{0.6cm}
 594  \begin{minipage}{3.8cm}
 595  \begin{center}
 596  \includegraphics[width=3.3cm]{sic_prec_seq_03.eps}
 597  \end{center}
 598  \end{minipage}
 599
 600 \begin{pspicture}(0,0)(0,0)
 601 \psline[linewidth=4pt]{->}(8.5,2)(9.0,2)
 602 \psellipse[linecolor=blue](11.5,5.8)(0.3,0.5)
 603 \rput{-20}{\psellipse[linecolor=blue](3.3,8.1)(0.3,0.5)}
 604 \psline[linewidth=4pt]{->}(4.0,2)(4.5,2)
 605 \rput(12.7,0.3){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
 606  $4a_{\text{Si}}=5a_{\text{SiC}}$
 607  }}}
 608 \rput(12.2,8){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
 609 \hkl(h k l) planes match
 610  }}}
 611 \rput(9.7,6.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
 612 r = 2 - 4 nm
 613  }}}
 614 \rput(6.7,5.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=white]{
 615 \begin{minipage}{10cm}
 616 \small
 617 {\color{red}\bf Controversial views}
 618 \begin{itemize}
 619 \item Implantations at high T (Nejim et al.)
 620  \begin{itemize}
 621   \item Topotactic transformation based on \cs
 622   \item \si{} as supply reacting with further C in cleared volume
 623  \end{itemize}
 624 \item Annealing behavior (Serre et al.)
 625  \begin{itemize}
 626   \item Room temperature implants $\rightarrow$ highly mobile C
 627   \item Elevated T implants $\rightarrow$ no/low C redistribution/migration\\
 628         (indicate stable \cs{} configurations)
 629  \end{itemize}
 630 \item Strained silicon \& Si/SiC heterostructures
 631  \begin{itemize}
 632   \item Coherent SiC precipitates (tensile strain)
 633   \item Incoherent SiC (strain relaxation)
 634  \end{itemize}
 635 \end{itemize}
 636 \end{minipage}
 637  }}}
 638 \end{pspicture}
 639
 640 \end{slide}
 641
 642 \begin{slide}
 643
 644  {\large\bf
 645   Molecular dynamics (MD) simulations
 646  }
 647
 648  \vspace{12pt}
 649
 650  \small
 651
 652  {\bf MD basics:}
 653  \begin{itemize}
 654   \item Microscopic description of N particle system
 655   \item Analytical interaction potential
 656   \item Numerical integration using Newtons equation of motion\\
 657         as a propagation rule in 6N-dimensional phase space
 658   \item Observables obtained by time and/or ensemble averages
 659  \end{itemize}
 660  {\bf Details of the simulation:}
 661  \begin{itemize}
 662   \item Integration: Velocity Verlet, timestep: $1\text{ fs}$
 663   \item Ensemble: NpT (isothermal-isobaric)
 664         \begin{itemize}
 665          \item Berendsen thermostat:
 666                $\tau_{\text{T}}=100\text{ fs}$
 667          \item Berendsen barostat:\\
 668                $\tau_{\text{P}}=100\text{ fs}$,
 669                $\beta^{-1}=100\text{ GPa}$
 670         \end{itemize}
 671   \item Erhart/Albe potential: Tersoff-like bond order potential
 672   \vspace*{12pt}
 673         \[
 674         E = \frac{1}{2} \sum_{i \neq j} \pot_{ij}, \quad
 675         \pot_{ij} = {\color{red}f_C(r_{ij})}
 676         \left[ f_R(r_{ij}) + {\color{blue}b_{ij}} f_A(r_{ij}) \right]
 677         \]
 678  \end{itemize}
 679
 680  \begin{picture}(0,0)(-230,-30)
 681   \includegraphics[width=5cm]{tersoff_angle.eps}
 682  \end{picture}
 683
 684 \end{slide}
 685
 686 \begin{slide}
 687
 688  {\large\bf
 689   Density functional theory (DFT) calculations
 690  }
 691
 692  \small
 693
 694  Basic ingredients necessary for DFT
 695
 696  \begin{itemize}
 697   \item \underline{Hohenberg-Kohn theorem} - ground state density $n_0(r)$ ...
 698         \begin{itemize}
 699          \item ... uniquely determines the ground state potential
 700                / wavefunctions
 701          \item ... minimizes the systems total energy
 702         \end{itemize}
 703   \item \underline{Born-Oppenheimer}
 704         - $N$ moving electrons in an external potential of static nuclei
 705 \[
 706 H\Psi = \left[-\sum_i^N \frac{\hbar^2}{2m}\nabla_i^2
 707               +\sum_i^N V_{\text{ext}}(r_i)
 708               +\sum_{i<j}^N V_{e-e}(r_i,r_j)\right]\Psi=E\Psi
 709 \]
 710   \item \underline{Effective potential}
 711         - averaged electrostatic potential \& exchange and correlation
 712 \[
 713 V_{\text{eff}}(r)=V_{\text{ext}}(r)+\int\frac{e^2 n(r')}{|r-r'|}d^3r'
 714                  +V_{\text{XC}}[n(r)]
 715 \]
 716   \item \underline{Kohn-Sham system}
 717         - Schr\"odinger equation of N non-interacting particles
 718 \[
 719 \left[ -\frac{\hbar^2}{2m}\nabla^2 + V_{\text{eff}}(r) \right] \Phi_i(r)
 720 =\epsilon_i\Phi_i(r)
 721 \quad
 722 \Rightarrow
 723 \quad
 724 n(r)=\sum_i^N|\Phi_i(r)|^2
 725 \]
 726   \item \underline{Self-consistent solution}\\
 727 $n(r)$ depends on $\Phi_i$, which depend on $V_{\text{eff}}$,
 728 which in turn depends on $n(r)$
 729   \item \underline{Variational principle}
 730         - minimize total energy with respect to $n(r)$
 731  \end{itemize}
 732
 733 \end{slide}
 734
 735 \begin{slide}
 736
 737  {\large\bf
 738   Density functional theory (DFT) calculations
 739  }
 740
 741  \small
 742
 743  \vspace*{0.2cm}
 744
 745  Details of applied DFT calculations in this work
 746
 747  \begin{itemize}
 748   \item \underline{Exchange correlation functional}
 749         - approximations for the inhomogeneous electron gas
 750         \begin{itemize}
 751          \item LDA: $E_{\text{XC}}^{\text{LDA}}[n]=\int \epsilon_{\text{XC}}(n)n(r)d^3r$
 752          \item GGA: $E_{\text{XC}}^{\text{GGA}}[n]=\int \epsilon_{\text{XC}}(n,\nabla n)n(r)d^3r$
 753         \end{itemize}
 754   \item \underline{Plane wave basis set}
 755         - approximation of the wavefunction $\Phi_i$ by plane waves $\phi_j$
 756 \[
 757 \rightarrow
 758 \text{Fourier series: } \Phi_i=\sum_{|G+k|<G_{\text{cut}}} c_j^i \phi_j(r), \quad E_{\text{cut}}=\frac{\hbar^2}{2m}G^2_{\text{cut}}
 759 \qquad ({\color{blue}300\text{ eV}})
 760 \]
 761   \item \underline{Brillouin zone sampling} -
 762         {\color{blue}$\Gamma$-point only} calculations
 763   \item \underline{Pseudo potential}
 764         - consider only the valence electrons
 765   \item \underline{Code} - VASP 4.6
 766  \end{itemize}
 767
 768  \vspace*{0.2cm}
 769
 770  MD and structural optimization
 771
 772  \begin{itemize}
 773   \item MD integration: Gear predictor corrector algorithm
 774   \item Pressure control: Parrinello-Rahman pressure control
 775   \item Structural optimization: Conjugate gradient method
 776  \end{itemize}
 777
 778 \begin{pspicture}(0,0)(0,0)
 779 \psellipse[linecolor=blue](1.5,6.75)(0.5,0.3)
 780 \end{pspicture}
 781
 782 \end{slide}
 783
 784 \begin{slide}
 785
 786  {\large\bf
 787   C and Si self-interstitial point defects in silicon
 788  }
 789
 790  \small
 791
 792  \vspace*{0.3cm}
 793
 794 \begin{minipage}{8cm}
 795 Procedure:\\[0.3cm]
 796   \begin{pspicture}(0,0)(7,5)
 797   \rput(3.5,4){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
 798    \parbox{7cm}{
 799    \begin{itemize}
 800     \item Creation of c-Si simulation volume
 801     \item Periodic boundary conditions
 802     \item $T=0\text{ K}$, $p=0\text{ bar}$
 803    \end{itemize}
 804   }}}}
 805 \rput(3.5,2.1){\rnode{insert}{\psframebox{
 806  \parbox{7cm}{
 807   \begin{center}
 808   Insertion of interstitial C/Si atoms
 809   \end{center}
 810   }}}}
 811   \rput(3.5,1){\rnode{cool}{\psframebox[fillstyle=solid,fillcolor=lbb]{
 812    \parbox{7cm}{
 813    \begin{center}
 814    Relaxation / structural energy minimization
 815    \end{center}
 816   }}}}
 817   \ncline[]{->}{init}{insert}
 818   \ncline[]{->}{insert}{cool}
 819  \end{pspicture}
 820 \end{minipage}
 821 \begin{minipage}{5cm}
 822   \includegraphics[width=5cm]{unit_cell_e.eps}\\
 823 \end{minipage}
 824
 825 \begin{minipage}{9cm}
 826  \begin{tabular}{l c c}
 827  \hline
 828  & size [unit cells] & \# atoms\\
 829 \hline
 830 VASP & $3\times 3\times 3$ & $216\pm 1$ \\
 831 Erhart/Albe & $9\times 9\times 9$ & $5832\pm 1$\\
 832 \hline
 833  \end{tabular}
 834 \end{minipage}
 835 \begin{minipage}{4cm}
 836 {\color{red}$\bullet$} Tetrahedral\\
 837 {\color{green}$\bullet$} Hexagonal\\
 838 {\color{yellow}$\bullet$} \hkl<1 0 0> dumbbell\\
 839 {\color{magenta}$\bullet$} \hkl<1 1 0> dumbbell\\
 840 {\color{cyan}$\bullet$} Bond-centered\\
 841 {\color{black}$\bullet$} Vacancy / Substitutional
 842 \end{minipage}
 843
 844 \end{slide}
 845
 846 \begin{slide}
 847
 848  \footnotesize
 849
 850 \begin{minipage}{9.5cm}
 851
 852  {\large\bf
 853   Si self-interstitial point defects in silicon\\
 854  }
 855
 856 \begin{tabular}{l c c c c c}
 857 \hline
 858  $E_{\text{f}}$ [eV] & \hkl<1 1 0> DB & H & T & \hkl<1 0 0> DB & V \\
 859 \hline
 860  VASP & \underline{3.39} & 3.42 & 3.77 & 4.41 & 3.63 \\
 861  Erhart/Albe & 4.39 & 4.48$^*$ & \underline{3.40} & 5.42 & 3.13 \\
 862 \hline
 863 \end{tabular}\\[0.2cm]
 864
 865 \begin{minipage}{4.7cm}
 866 \includegraphics[width=4.7cm]{e_kin_si_hex.ps}
 867 \end{minipage}
 868 \begin{minipage}{4.7cm}
 869 \begin{center}
 870 {\tiny nearly T $\rightarrow$ T}\\
 871 \end{center}
 872 \includegraphics[width=4.7cm]{nhex_tet.ps}
 873 \end{minipage}\\
 874
 875 \underline{Hexagonal} \hspace{2pt}
 876 \href{../video/si_self_int_hexa.avi}{$\rhd$}\\[0.1cm]
 877 \framebox{
 878 \begin{minipage}{2.7cm}
 879 $E_{\text{f}}^*=4.48\text{ eV}$\\
 880 \includegraphics[width=2.7cm]{si_pd_albe/hex_a.eps}
 881 \end{minipage}
 882 \begin{minipage}{0.4cm}
 883 \begin{center}
 884 $\Rightarrow$
 885 \end{center}
 886 \end{minipage}
 887 \begin{minipage}{2.7cm}
 888 $E_{\text{f}}=3.96\text{ eV}$\\
 889 \includegraphics[width=2.8cm]{si_pd_albe/hex.eps}
 890 \end{minipage}
 891 }
 892 \begin{minipage}{2.9cm}
 893 \begin{flushright}
 894 \underline{Vacancy}\\
 895 \includegraphics[width=3.0cm]{si_pd_albe/vac.eps}
 896 \end{flushright}
 897 \end{minipage}
 898
 899 \end{minipage}
 900 \begin{minipage}{3.5cm}
 901
 902 \begin{flushright}
 903 \underline{\hkl<1 1 0> dumbbell}\\
 904 \includegraphics[width=3.0cm]{si_pd_albe/110.eps}\\
 905 \underline{Tetrahedral}\\
 906 \includegraphics[width=3.0cm]{si_pd_albe/tet.eps}\\
 907 \underline{\hkl<1 0 0> dumbbell}\\
 908 \includegraphics[width=3.0cm]{si_pd_albe/100.eps}
 909 \end{flushright}
 910
 911 \end{minipage}
 912
 913 \end{slide}
 914
 915 \begin{slide}
 916
 917 \footnotesize
 918
 919  {\large\bf
 920   C interstitial point defects in silicon\\[-0.1cm]
 921  }
 922
 923 \begin{tabular}{l c c c c c c r}
 924 \hline
 925  $E_{\text{f}}$ & T & H & \hkl<1 0 0> DB & \hkl<1 1 0> DB & S & B & \cs{} \& \si\\
 926 \hline
 927  VASP & unstable & unstable & \underline{3.72} & 4.16 & 1.95 & 4.66 & {\color{green}4.17}\\
 928  Erhart/Albe MD & 6.09 & 9.05$^*$ & \underline{3.88} & 5.18 & {\color{red}0.75} & 5.59$^*$ & {\color{green}4.43} \\
 929 \hline
 930 \end{tabular}\\[0.1cm]
 931
 932 \framebox{
 933 \begin{minipage}{2.7cm}
 934 \underline{Hexagonal} \hspace{2pt}
 935 \href{../video/c_in_si_int_hexa.avi}{$\rhd$}\\
 936 $E_{\text{f}}^*=9.05\text{ eV}$\\
 937 \includegraphics[width=2.7cm]{c_pd_albe/hex.eps}
 938 \end{minipage}
 939 \begin{minipage}{0.4cm}
 940 \begin{center}
 941 $\Rightarrow$
 942 \end{center}
 943 \end{minipage}
 944 \begin{minipage}{2.7cm}
 945 \underline{\hkl<1 0 0>}\\
 946 $E_{\text{f}}=3.88\text{ eV}$\\
 947 \includegraphics[width=2.7cm]{c_pd_albe/100.eps}
 948 \end{minipage}
 949 }
 950 \begin{minipage}{2cm}
 951 \hfill
 952 \end{minipage}
 953 \begin{minipage}{3cm}
 954 \begin{flushright}
 955 \underline{Tetrahedral}\\
 956 \includegraphics[width=3.0cm]{c_pd_albe/tet.eps}
 957 \end{flushright}
 958 \end{minipage}
 959
 960 \framebox{
 961 \begin{minipage}{2.7cm}
 962 \underline{Bond-centered}\\
 963 $E_{\text{f}}^*=5.59\text{ eV}$\\
 964 \includegraphics[width=2.7cm]{c_pd_albe/bc.eps}
 965 \end{minipage}
 966 \begin{minipage}{0.4cm}
 967 \begin{center}
 968 $\Rightarrow$
 969 \end{center}
 970 \end{minipage}
 971 \begin{minipage}{2.7cm}
 972 \underline{\hkl<1 1 0> dumbbell}\\
 973 $E_{\text{f}}=5.18\text{ eV}$\\
 974 \includegraphics[width=2.7cm]{c_pd_albe/110.eps}
 975 \end{minipage}
 976 }
 977 \begin{minipage}{2cm}
 978 \hfill
 979 \end{minipage}
 980 \begin{minipage}{3cm}
 981 \begin{flushright}
 982 \underline{Substitutional}\\
 983 \includegraphics[width=3.0cm]{c_pd_albe/sub.eps}
 984 \end{flushright}
 985 \end{minipage}
 986
 987 \end{slide}
 988
 989 \begin{slide}
 990
 991 \footnotesize
 992
 993  {\large\bf\boldmath
 994   C \hkl<1 0 0> dumbbell interstitial configuration\\
 995  }
 996
 997 {\tiny
 998 \begin{tabular}{l c c c c c c c c}
 999 \hline
1000  Distances [nm] & $r(1C)$ & $r(2C)$ & $r(3C)$ & $r(12)$ & $r(13)$ & $r(34)$ & $r(23)$ & $r(25)$ \\
1001 \hline
1002 Erhart/Albe & 0.175 & 0.329 & 0.186 & 0.226 & 0.300 & 0.343 & 0.423 & 0.425 \\
1003 VASP & 0.174 & 0.341 & 0.182 & 0.229 & 0.286 & 0.347 & 0.422 & 0.417 \\
1004 \hline
1005 \end{tabular}\\[0.2cm]
1006 \begin{tabular}{l c c c c }
1007 \hline
1008  Angles [$^{\circ}$] & $\theta_1$ & $\theta_2$ & $\theta_3$ & $\theta_4$ \\
1009 \hline
1010 Erhart/Albe & 140.2 & 109.9 & 134.4 & 112.8 \\
1011 VASP & 130.7 & 114.4 & 146.0 & 107.0 \\
1012 \hline
1013 \end{tabular}\\[0.2cm]
1014 \begin{tabular}{l c c c}
1015 \hline
1016  Displacements [nm]& $a$ & $b$ & $|a|+|b|$ \\
1017 \hline
1018 Erhart/Albe & 0.084 & -0.091 & 0.175 \\
1019 VASP & 0.109 & -0.065 & 0.174 \\
1020 \hline
1021 \end{tabular}\\[0.6cm]
1022 }
1023
1024 \begin{minipage}{3.0cm}
1025 \begin{center}
1026 \underline{Erhart/Albe}
1027 \includegraphics[width=3.0cm]{c_pd_albe/100_cmp.eps}
1028 \end{center}
1029 \end{minipage}
1030 \begin{minipage}{3.0cm}
1031 \begin{center}
1032 \underline{VASP}
1033 \includegraphics[width=3.0cm]{c_pd_vasp/100_cmp.eps}
1034 \end{center}
1035 \end{minipage}\\
1036
1037 \begin{picture}(0,0)(-185,10)
1038 \includegraphics[width=6.8cm]{100-c-si-db_cmp.eps}
1039 \end{picture}
1040 \begin{picture}(0,0)(-280,-150)
1041 \includegraphics[width=3.3cm]{c_pd_vasp/eden.eps}
1042 \end{picture}
1043
1044 \begin{pspicture}(0,0)(0,0)
1045 \psellipse[linecolor=green](5.18,5.92)(0.5,0.3)
1046 \psellipse[linecolor=red](3.45,5.92)(1.0,0.4)
1047 \psellipse[linecolor=blue](2.7,6.92)(0.9,0.2)
1048 \psellipse[linecolor=blue](4.65,6.92)(0.9,0.2)
1049 \end{pspicture}
1050
1051 \end{slide}
1052
1053 \begin{slide}
1054
1055 \small
1056
1057 \begin{minipage}{8.5cm}
1058
1059  {\large\bf
1060   Bond-centered interstitial configuration\\[-0.1cm]
1061  }
1062
1063 \begin{minipage}{3.0cm}
1064 \includegraphics[width=2.8cm]{c_pd_vasp/bc_2333.eps}\\
1065 \end{minipage}
1066 \begin{minipage}{5.2cm}
1067 \begin{itemize}
1068  \item Linear Si-C-Si bond
1069  \item Si: one C \& 3 Si neighbours
1070  \item Spin polarized calculations
1071  \item No saddle point!\\
1072        Real local minimum!
1073 \end{itemize}
1074 \end{minipage}
1075
1076 \framebox{
1077  \tiny
1078  \begin{minipage}[t]{6.5cm}
1079   \begin{minipage}[t]{1.2cm}
1080   {\color{red}Si}\\
1081   {\tiny sp$^3$}\\[0.8cm]
1082   \underline{${\color{black}\uparrow}$}
1083   \underline{${\color{black}\uparrow}$}
1084   \underline{${\color{black}\uparrow}$}
1085   \underline{${\color{red}\uparrow}$}\\
1086   sp$^3$
1087   \end{minipage}
1088   \begin{minipage}[t]{1.4cm}
1089   \begin{center}
1090   {\color{red}M}{\color{blue}O}\\[0.8cm]
1091   \underline{${\color{blue}\uparrow}{\color{white}\downarrow}$}\\
1092   $\sigma_{\text{ab}}$\\[0.5cm]
1093   \underline{${\color{red}\uparrow}{\color{blue}\downarrow}$}\\
1094   $\sigma_{\text{b}}$
1095   \end{center}
1096   \end{minipage}
1097   \begin{minipage}[t]{1.0cm}
1098   \begin{center}
1099   {\color{blue}C}\\
1100   {\tiny sp}\\[0.2cm]
1101   \underline{${\color{white}\uparrow\uparrow}$}
1102   \underline{${\color{white}\uparrow\uparrow}$}\\
1103   2p\\[0.4cm]
1104   \underline{${\color{blue}\uparrow}{\color{blue}\downarrow}$}
1105   \underline{${\color{blue}\uparrow}{\color{blue}\downarrow}$}\\
1106   sp
1107   \end{center}
1108   \end{minipage}
1109   \begin{minipage}[t]{1.4cm}
1110   \begin{center}
1111   {\color{blue}M}{\color{green}O}\\[0.8cm]
1112   \underline{${\color{blue}\uparrow}{\color{white}\downarrow}$}\\
1113   $\sigma_{\text{ab}}$\\[0.5cm]
1114   \underline{${\color{green}\uparrow}{\color{blue}\downarrow}$}\\
1115   $\sigma_{\text{b}}$
1116   \end{center}
1117   \end{minipage}
1118   \begin{minipage}[t]{1.2cm}
1119   \begin{flushright}
1120   {\color{green}Si}\\
1121   {\tiny sp$^3$}\\[0.8cm]
1122   \underline{${\color{green}\uparrow}$}
1123   \underline{${\color{black}\uparrow}$}
1124   \underline{${\color{black}\uparrow}$}
1125   \underline{${\color{black}\uparrow}$}\\
1126   sp$^3$
1127   \end{flushright}
1128   \end{minipage}
1129  \end{minipage}
1130 }\\[0.1cm]
1131
1132 \framebox{
1133 \begin{minipage}{4.5cm}
1134 \includegraphics[width=4cm]{c_100_mig_vasp/im_spin_diff.eps}
1135 \end{minipage}
1136 \begin{minipage}{3.5cm}
1137 {\color{gray}$\bullet$} Spin up\\
1138 {\color{green}$\bullet$} Spin down\\
1139 {\color{blue}$\bullet$} Resulting spin up\\
1140 {\color{yellow}$\bullet$} Si atoms\\
1141 {\color{red}$\bullet$} C atom
1142 \end{minipage}
1143 }
1144
1145 \end{minipage}
1146 \begin{minipage}{4.2cm}
1147 \begin{flushright}
1148 \includegraphics[width=4.3cm]{c_pd_vasp/bc_2333_ksl.ps}\\
1149 {\color{green}$\Box$} {\tiny unoccupied}\\
1150 {\color{red}$\bullet$} {\tiny occupied}
1151 \end{flushright}
1152 \end{minipage}
1153
1154 \end{slide}
1155
1156 \begin{slide}
1157
1158  {\large\bf\boldmath
1159   Migration of the C \hkl<1 0 0> dumbbell interstitial
1160  }
1161
1162 \scriptsize
1163
1164  {\small Investigated pathways}
1165
1166 \begin{minipage}{8.5cm}
1167 \begin{minipage}{8.3cm}
1168 \underline{\hkl<0 0 -1> $\rightarrow$ \hkl<0 0 1>}\\
1169 \begin{minipage}{2.4cm}
1170 \includegraphics[width=2.4cm]{c_pd_vasp/100_2333.eps}
1171 \end{minipage}
1172 \begin{minipage}{0.4cm}
1173 $\rightarrow$
1174 \end{minipage}
1175 \begin{minipage}{2.4cm}
1176 \includegraphics[width=2.4cm]{c_pd_vasp/bc_2333.eps}
1177 \end{minipage}
1178 \begin{minipage}{0.4cm}
1179 $\rightarrow$
1180 \end{minipage}
1181 \begin{minipage}{2.4cm}
1182 \includegraphics[width=2.4cm]{c_pd_vasp/100_next_2333.eps}
1183 \end{minipage}
1184 \end{minipage}\\
1185 \begin{minipage}{8.3cm}
1186 \underline{\hkl<0 0 -1> $\rightarrow$ \hkl<0 -1 0>}\\
1187 \begin{minipage}{2.4cm}
1188 \includegraphics[width=2.4cm]{c_pd_vasp/100_2333.eps}
1189 \end{minipage}
1190 \begin{minipage}{0.4cm}
1191 $\rightarrow$
1192 \end{minipage}
1193 \begin{minipage}{2.4cm}
1194 \includegraphics[width=2.4cm]{c_pd_vasp/00-1-0-10_2333.eps}
1195 \end{minipage}
1196 \begin{minipage}{0.4cm}
1197 $\rightarrow$
1198 \end{minipage}
1199 \begin{minipage}{2.4cm}
1200 \includegraphics[width=2.4cm]{c_pd_vasp/0-10_2333.eps}
1201 \end{minipage}
1202 \end{minipage}\\
1203 \begin{minipage}{8.3cm}
1204 \underline{\hkl<0 0 -1> $\rightarrow$ \hkl<0 -1 0> (in place)}\\
1205 \begin{minipage}{2.4cm}
1206 \includegraphics[width=2.4cm]{c_pd_vasp/100_2333.eps}
1207 \end{minipage}
1208 \begin{minipage}{0.4cm}
1209 $\rightarrow$
1210 \end{minipage}
1211 \begin{minipage}{2.4cm}
1212 \includegraphics[width=2.4cm]{c_pd_vasp/00-1_ip0-10_2333.eps}
1213 \end{minipage}
1214 \begin{minipage}{0.4cm}
1215 $\rightarrow$
1216 \end{minipage}
1217 \begin{minipage}{2.4cm}
1218 \includegraphics[width=2.4cm]{c_pd_vasp/0-10_ip_2333.eps}
1219 \end{minipage}
1220 \end{minipage}
1221 \end{minipage}
1222 \framebox{
1223 \begin{minipage}{4.2cm}
1224  {\small Constrained relaxation\\
1225          technique (CRT) method}\\
1226 \includegraphics[width=4cm]{crt_orig.eps}
1227 \begin{itemize}
1228  \item Constrain diffusing atom
1229  \item Static constraints
1230 \end{itemize}
1231 \vspace*{0.3cm}
1232  {\small Modifications}\\
1233 \includegraphics[width=4cm]{crt_mod.eps}
1234 \begin{itemize}
1235  \item Constrain all atoms
1236  \item Update individual\\
1237        constraints
1238 \end{itemize}
1239 \end{minipage}
1240 }
1241
1242 \end{slide}
1243
1244 \begin{slide}
1245
1246  {\large\bf\boldmath
1247   Migration of the C \hkl<1 0 0> dumbbell interstitial
1248  }
1249
1250 \scriptsize
1251
1252 \framebox{
1253 \begin{minipage}{5.9cm}
1254 \begin{flushleft}
1255 \includegraphics[width=5.8cm]{im_00-1_nosym_sp_fullct_thesis.ps}\\[0.45cm]
1256 \end{flushleft}
1257 \begin{center}
1258 \begin{picture}(0,0)(60,0)
1259 \includegraphics[width=1cm]{vasp_mig/00-1.eps}
1260 \end{picture}
1261 \begin{picture}(0,0)(-5,0)
1262 \includegraphics[width=1cm]{vasp_mig/bc_00-1_sp.eps}
1263 \end{picture}
1264 \begin{picture}(0,0)(-55,0)
1265 \includegraphics[width=1cm]{vasp_mig/bc.eps}
1266 \end{picture}
1267 \begin{picture}(0,0)(12.5,10)
1268 \includegraphics[width=1cm]{110_arrow.eps}
1269 \end{picture}
1270 \begin{picture}(0,0)(90,0)
1271 \includegraphics[height=0.9cm]{001_arrow.eps}
1272 \end{picture}
1273 \end{center}
1274 \vspace*{0.35cm}
1275 \end{minipage}
1276 }
1277 \begin{minipage}{0.3cm}
1278 \hfill
1279 \end{minipage}
1280 \framebox{
1281 \begin{minipage}{5.9cm}
1282 \begin{flushright}
1283 \includegraphics[width=5.9cm]{vasp_mig/00-1_0-10_nosym_sp_fullct.ps}\\[0.5cm]
1284 \end{flushright}
1285 \begin{center}
1286 \begin{picture}(0,0)(60,0)
1287 \includegraphics[width=1cm]{vasp_mig/00-1_a.eps}
1288 \end{picture}
1289 \begin{picture}(0,0)(5,0)
1290 \includegraphics[width=1cm]{vasp_mig/00-1_0-10_sp.eps}
1291 \end{picture}
1292 \begin{picture}(0,0)(-55,0)
1293 \includegraphics[width=1cm]{vasp_mig/0-10.eps}
1294 \end{picture}
1295 \begin{picture}(0,0)(12.5,10)
1296 \includegraphics[width=1cm]{100_arrow.eps}
1297 \end{picture}
1298 \begin{picture}(0,0)(90,0)
1299 \includegraphics[height=0.9cm]{001_arrow.eps}
1300 \end{picture}
1301 \end{center}
1302 \vspace*{0.3cm}
1303 \end{minipage}\\
1304 }
1305
1306 \vspace*{0.05cm}
1307
1308 \framebox{
1309 \begin{minipage}{5.9cm}
1310 \begin{flushleft}
1311 \includegraphics[width=5.9cm]{vasp_mig/00-1_ip0-10_nosym_sp_fullct.ps}\\[0.6cm]
1312 \end{flushleft}
1313 \begin{center}
1314 \begin{picture}(0,0)(60,0)
1315 \includegraphics[width=0.9cm]{vasp_mig/00-1_b.eps}
1316 \end{picture}
1317 \begin{picture}(0,0)(10,0)
1318 \includegraphics[width=0.9cm]{vasp_mig/00-1_ip0-10_sp.eps}
1319 \end{picture}
1320 \begin{picture}(0,0)(-60,0)
1321 \includegraphics[width=0.9cm]{vasp_mig/0-10_b.eps}
1322 \end{picture}
1323 \begin{picture}(0,0)(12.5,10)
1324 \includegraphics[width=1cm]{100_arrow.eps}
1325 \end{picture}
1326 \begin{picture}(0,0)(90,0)
1327 \includegraphics[height=0.9cm]{001_arrow.eps}
1328 \end{picture}
1329 \end{center}
1330 \vspace*{0.3cm}
1331 \end{minipage}
1332 }
1333 \begin{minipage}{0.3cm}
1334 \hfill
1335 \end{minipage}
1336 \begin{minipage}{6.5cm}
1337 VASP results
1338 \begin{itemize}
1339  \item Energetically most favorable path
1340        \begin{itemize}
1341         \item Path 2
1342         \item Activation energy: $\approx$ 0.9 eV
1343         \item Experimental values: 0.73 ... 0.87 eV
1344        \end{itemize}
1345        $\Rightarrow$ {\color{blue}Diffusion} path identified!
1346  \item Reorientation (path 3)
1347        \begin{itemize}
1348         \item More likely composed of two consecutive steps of type 2
1349         \item Experimental values: 0.77 ... 0.88 eV
1350        \end{itemize}
1351        $\Rightarrow$ {\color{blue}Reorientation} transition identified!
1352 \end{itemize}
1353 \end{minipage}
1354
1355 \end{slide}
1356
1357 \begin{slide}
1358
1359  {\large\bf\boldmath
1360   Migration of the C \hkl<1 0 0> dumbbell interstitial
1361  }
1362
1363 \scriptsize
1364
1365  \vspace{0.1cm}
1366
1367 \begin{minipage}{6.5cm}
1368
1369 \framebox{
1370 \begin{minipage}[t]{5.9cm}
1371 \begin{flushleft}
1372 \includegraphics[width=5.9cm]{bc_00-1.ps}\\[2.35cm]
1373 \end{flushleft}
1374 \begin{center}
1375 \begin{pspicture}(0,0)(0,0)
1376 \psframe[linecolor=red,fillstyle=none](-2.8,1.35)(3.3,2.7)
1377 \end{pspicture}
1378 \begin{picture}(0,0)(60,-50)
1379 \includegraphics[width=1cm]{albe_mig/bc_00-1_red_00.eps}
1380 \end{picture}
1381 \begin{picture}(0,0)(5,-50)
1382 \includegraphics[width=1cm]{albe_mig/bc_00-1_red_01.eps}
1383 \end{picture}
1384 \begin{picture}(0,0)(-55,-50)
1385 \includegraphics[width=1cm]{albe_mig/bc_00-1_red_02.eps}
1386 \end{picture}
1387 \begin{picture}(0,0)(12.5,-40)
1388 \includegraphics[width=1cm]{110_arrow.eps}
1389 \end{picture}
1390 \begin{picture}(0,0)(90,-45)
1391 \includegraphics[height=0.9cm]{001_arrow.eps}
1392 \end{picture}\\
1393 \begin{pspicture}(0,0)(0,0)
1394 \psframe[linecolor=blue,fillstyle=none](-2.8,0)(3.3,1.6)
1395 \end{pspicture}
1396 \begin{picture}(0,0)(60,-15)
1397 \includegraphics[width=0.9cm]{albe_mig/bc_00-1_01.eps}
1398 \end{picture}
1399 \begin{picture}(0,0)(35,-15)
1400 \includegraphics[width=0.9cm]{albe_mig/bc_00-1_02.eps}
1401 \end{picture}
1402 \begin{picture}(0,0)(-5,-15)
1403 \includegraphics[width=0.9cm]{albe_mig/bc_00-1_03.eps}
1404 \end{picture}
1405 \begin{picture}(0,0)(-55,-15)
1406 \includegraphics[width=0.9cm]{albe_mig/bc_00-1_04.eps}
1407 \end{picture}
1408 \begin{picture}(0,0)(12.5,-5)
1409 \includegraphics[width=1cm]{100_arrow.eps}
1410 \end{picture}
1411 \begin{picture}(0,0)(90,-15)
1412 \includegraphics[height=0.9cm]{010_arrow.eps}
1413 \end{picture}
1414 \end{center}
1415 \end{minipage}
1416 }\\[0.1cm]
1417
1418 \begin{minipage}{5.9cm}
1419 Erhart/Albe results
1420 \begin{itemize}
1421  \item Lowest activation energy: $\approx$ 2.2 eV
1422  \item 2.4 times higher than VASP
1423  \item Different pathway
1424 \end{itemize}
1425 \end{minipage}
1426
1427 \end{minipage}
1428 \begin{minipage}{6.5cm}
1429
1430 \framebox{
1431 \begin{minipage}{5.9cm}
1432 %\begin{flushright}
1433 %\includegraphics[width=5.9cm]{00-1_0-10.ps}\\[0.75cm]
1434 %\end{flushright}
1435 %\begin{center}
1436 %\begin{pspicture}(0,0)(0,0)
1437 %\psframe[linecolor=red,fillstyle=none](-2.8,-0.25)(3.3,1.1)
1438 %\end{pspicture}
1439 %\begin{picture}(0,0)(60,-5)
1440 %\includegraphics[width=0.9cm]{albe_mig/00-1_0-10_red_00.eps}
1441 %\end{picture}
1442 %\begin{picture}(0,0)(0,-5)
1443 %\includegraphics[width=0.9cm]{albe_mig/00-1_0-10_red_min.eps}
1444 %\end{picture}
1445 %\begin{picture}(0,0)(-55,-5)
1446 %\includegraphics[width=0.9cm]{albe_mig/00-1_0-10_red_03.eps}
1447 %\end{picture}
1448 %\begin{picture}(0,0)(12.5,5)
1449 %\includegraphics[width=1cm]{100_arrow.eps}
1450 %\end{picture}
1451 %\begin{picture}(0,0)(90,0)
1452 %\includegraphics[height=0.9cm]{001_arrow.eps}
1453 %\end{picture}
1454 %\end{center}
1455 %\vspace{0.2cm}
1456 %\end{minipage}
1457 %}\\[0.2cm]
1458 %
1459 %\framebox{
1460 %\begin{minipage}{5.9cm}
1461 \includegraphics[width=5.9cm]{00-1_110_0-10_mig_albe.ps}
1462 \end{minipage}
1463 }\\[0.1cm]
1464
1465 \begin{minipage}{5.9cm}
1466 Transition involving \ci{} \hkl<1 1 0>
1467 \begin{itemize}
1468  \item Bond-centered configuration unstable\\
1469        $\rightarrow$ \ci{} \hkl<1 1 0> dumbbell
1470  \item Transition minima of path 2 \& 3\\
1471        $\rightarrow$ \ci{} \hkl<1 1 0> dumbbell
1472  \item Activation energy: $\approx$ 2.2 eV \& 0.9 eV
1473  \item 2.4 - 3.4 times higher than VASP
1474  \item Rotation of dumbbell orientation
1475 \end{itemize}
1476 \vspace{0.1cm}
1477 \begin{center}
1478 {\color{blue}Overestimated diffusion barrier}
1479 \end{center}
1480 \end{minipage}
1481
1482 \end{minipage}
1483
1484 \end{slide}
1485
1486 \begin{slide}
1487
1488  {\large\bf\boldmath
1489   Combinations with a C-Si \hkl<1 0 0>-type interstitial
1490  }
1491
1492 \small
1493
1494 \vspace*{0.1cm}
1495
1496 Binding energy:
1497 $
1498 E_{\text{b}}=
1499 E_{\text{f}}^{\text{defect combination}}-
1500 E_{\text{f}}^{\text{C \hkl<0 0 -1> dumbbell}}-
1501 E_{\text{f}}^{\text{2nd defect}}
1502 $
1503
1504 \vspace*{0.1cm}
1505
1506 {\scriptsize
1507 \begin{tabular}{l c c c c c c}
1508 \hline
1509  $E_{\text{b}}$ [eV] & 1 & 2 & 3 & 4 & 5 & R\\
1510  \hline
1511  \hkl<0 0 -1> & {\color{red}-0.08} & -1.15 & {\color{red}-0.08} & 0.04 & -1.66 & -0.19\\
1512  \hkl<0 0 1> & 0.34 & 0.004 & -2.05 & 0.26 & -1.53 & -0.19\\
1513  \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}\\
1514  \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}\\
1515  \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}\\
1516  \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}\\
1517  \hline
1518  C substitutional (C$_{\text{S}}$) & 0.26 & -0.51 & -0.93 & -0.15 & 0.49 & -0.05\\
1519  Vacancy & -5.39 ($\rightarrow$ C$_{\text{S}}$) & -0.59 & -3.14 & -0.54 & -0.50 & -0.31\\
1520 \hline
1521 \end{tabular}
1522 }
1523
1524 \vspace*{0.3cm}
1525
1526 \footnotesize
1527
1528 \begin{minipage}[t]{3.8cm}
1529 \underline{\hkl<1 0 0> at position 1}\\[0.1cm]
1530 \includegraphics[width=3.5cm]{00-1dc/2-25.eps}
1531 \end{minipage}
1532 \begin{minipage}[t]{3.5cm}
1533 \underline{\hkl<0 -1 0> at position 1}\\[0.1cm]
1534 \includegraphics[width=3.2cm]{00-1dc/2-39.eps}
1535 \end{minipage}
1536 \begin{minipage}[t]{5.5cm}
1537 \begin{itemize}
1538  \item $E_{\text{b}}=0$ $\Leftrightarrow$ non-interacting defects\\
1539        $E_{\text{b}} \rightarrow 0$ for increasing distance (R)
1540  \item Stress compensation / increase
1541  \item Unfavored: antiparallel orientations
1542  \item Indication of energetically favored\\
1543        agglomeration
1544  \item Most favorable: C clustering
1545  \item However: High barrier ($>4\,\text{eV}$)
1546  \item $4\times{\color{cyan}-2.25}$ versus $2\times{\color{orange}-2.39}$
1547        (Entropy)
1548 \end{itemize}
1549 \end{minipage}
1550
1551 \begin{picture}(0,0)(-295,-130)
1552 \includegraphics[width=3.5cm]{comb_pos.eps}
1553 \end{picture}
1554
1555 \end{slide}
1556
1557 \begin{slide}
1558
1559  {\large\bf\boldmath
1560   Combinations of C-Si \hkl<1 0 0>-type interstitials
1561  }
1562
1563 \small
1564
1565 \vspace*{0.1cm}
1566
1567 Energetically most favorable combinations along \hkl<1 1 0>
1568
1569 \vspace*{0.1cm}
1570
1571 {\scriptsize
1572 \begin{tabular}{l c c c c c c}
1573 \hline
1574  & 1 & 2 & 3 & 4 & 5 & 6\\
1575 \hline
1576 $E_{\text{b}}$ [eV] & -2.39 & -1.88 & -0.59 & -0.31 & -0.24 & -0.21 \\
1577 C-C distance [\AA] & 1.4 & 4.6 & 6.5 & 8.6 & 10.5 & 10.8 \\
1578 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>\\
1579 \hline
1580 \end{tabular}
1581 }
1582
1583 \vspace*{0.3cm}
1584
1585 \begin{minipage}{7.0cm}
1586 \includegraphics[width=7cm]{db_along_110_cc.ps}
1587 \end{minipage}
1588 \begin{minipage}{6.0cm}
1589 \begin{itemize}
1590  \item Interaction proportional to reciprocal cube of C-C distance
1591  \item Saturation in the immediate vicinity
1592  \renewcommand\labelitemi{$\Rightarrow$}
1593  \item Agglomeration of \ci{} expected
1594  \item Absence of C clustering
1595 \end{itemize}
1596 \begin{center}
1597 {\color{blue}
1598  Consisten with initial precipitation model
1599 }
1600 \end{center}
1601 \end{minipage}
1602
1603 \vspace{0.2cm}
1604
1605 \end{slide}
1606
1607 \begin{slide}
1608
1609  {\large\bf\boldmath
1610   Combinations of substitutional C and \hkl<1 1 0> Si self-interstitials
1611  }
1612
1613  \scriptsize
1614
1615 %\begin{center}
1616 %\begin{minipage}{3.2cm}
1617 %\includegraphics[width=3cm]{sub_110_combo.eps}
1618 %\end{minipage}
1619 %\begin{minipage}{7.8cm}
1620 %\begin{tabular}{l c c c c c c}
1621 %\hline
1622 %C$_{\text{sub}}$ & \hkl<1 1 0> & \hkl<-1 1 0> & \hkl<0 1 1> & \hkl<0 -1 1> &
1623 %                   \hkl<1 0 1> & \hkl<-1 0 1> \\
1624 %\hline
1625 %1 & \RM{1} & \RM{3} & \RM{3} & \RM{1} & \RM{3} & \RM{1} \\
1626 %2 & \RM{2} & A & A & \RM{2} & C & \RM{5} \\
1627 %3 & \RM{3} & \RM{1} & \RM{3} & \RM{1} & \RM{1} & \RM{3} \\
1628 %4 & \RM{4} & B & D & E & E & D \\
1629 %5 & \RM{5} & C & A & \RM{2} & A & \RM{2} \\
1630 %\hline
1631 %\end{tabular}
1632 %\end{minipage}
1633 %\end{center}
1634
1635 %\begin{center}
1636 %\begin{tabular}{l c c c c c c c c c c}
1637 %\hline
1638 %Conf & \RM{1} & \RM{2} & \RM{3} & \RM{4} & \RM{5} & A & B & C & D & E \\
1639 %\hline
1640 %$E_{\text{f}}$ [eV]& 4.37 & 5.26 & 5.57 & 5.37 & 5.12 & 5.10 & 5.32 & 5.28 & 5.39 & 5.32 \\
1641 %$E_{\text{b}}$ [eV] & -0.97 & -0.08 & 0.22 & -0.02 & -0.23 & -0.25 & -0.02 & -0.06 & 0.05 & -0.03 \\
1642 %$r$ [nm] & 0.292 & 0.394 & 0.241 & 0.453 & 0.407 & 0.408 & 0.452 & 0.392 & 0.456 & 0.453\\
1643 %\hline
1644 %\end{tabular}
1645 %\end{center}
1646
1647 \begin{minipage}{6.0cm}
1648 \includegraphics[width=5.8cm]{c_sub_si110.ps}
1649 \end{minipage}
1650 \begin{minipage}{7cm}
1651 \scriptsize
1652 \begin{itemize}
1653  \item IBS: C may displace Si\\
1654        $\Rightarrow$ C$_{\text{sub}}$ + \hkl<1 1 0> Si self-interstitial
1655  \item Assumption:\\
1656        \hkl<1 1 0>-type $\rightarrow$ favored combination
1657  \renewcommand\labelitemi{$\Rightarrow$}
1658  \item Most favorable: \cs{} along \hkl<1 1 0> chain \si{}
1659  \item Less favorable than C-Si \hkl<1 0 0> dumbbell
1660  \item Interaction drops quickly to zero\\
1661        $\rightarrow$ low capture radius
1662 \end{itemize}
1663 \begin{center}
1664  {\color{blue}
1665  IBS process far from equilibrium\\
1666  \cs{} \& \si{} instead of thermodynamic ground state
1667  }
1668 \end{center}
1669 \end{minipage}
1670
1671 \begin{minipage}{6.5cm}
1672 \includegraphics[width=6.0cm]{162-097.ps}
1673 \begin{itemize}
1674  \item Low migration barrier
1675 \end{itemize}
1676 \end{minipage}
1677 \begin{minipage}{6.5cm}
1678 \begin{center}
1679 Ab initio MD at \degc{900}\\
1680 \includegraphics[width=3.3cm]{md_vasp_01.eps}
1681 $t=\unit[2230]{fs}$\\
1682 \includegraphics[width=3.3cm]{md_vasp_02.eps}
1683 $t=\unit[2900]{fs}$
1684 \end{center}
1685 {\color{blue}
1686 Contribution of entropy to structural formation
1687 }
1688 \end{minipage}
1689
1690 \end{slide}
1691
1692 \begin{slide}
1693
1694  {\large\bf\boldmath
1695   Migration in C-Si \hkl<1 0 0> and vacancy combinations
1696  }
1697
1698  \footnotesize
1699
1700 \vspace{0.1cm}
1701
1702 \begin{minipage}[t]{3cm}
1703 \underline{Pos 2, $E_{\text{b}}=-0.59\text{ eV}$}\\
1704 \includegraphics[width=2.8cm]{00-1dc/0-59.eps}
1705 \end{minipage}
1706 \begin{minipage}[t]{7cm}
1707 \vspace{0.2cm}
1708 \begin{center}
1709  Low activation energies\\
1710  High activation energies for reverse processes\\
1711  $\Downarrow$\\
1712  {\color{blue}C$_{\text{sub}}$ very stable}\\
1713 \vspace*{0.1cm}
1714  \hrule
1715 \vspace*{0.1cm}
1716  Without nearby \hkl<1 1 0> Si self-interstitial (IBS)\\
1717  $\Downarrow$\\
1718  {\color{blue}Formation of SiC by successive substitution by C}
1719
1720 \end{center}
1721 \end{minipage}
1722 \begin{minipage}[t]{3cm}
1723 \underline{Pos 3, $E_{\text{b}}=-3.14\text{ eV}$}\\
1724 \includegraphics[width=2.8cm]{00-1dc/3-14.eps}
1725 \end{minipage}
1726
1727
1728 \framebox{
1729 \begin{minipage}{5.9cm}
1730 \includegraphics[width=5.9cm]{vasp_mig/comb_mig_3-2_vac_fullct.ps}\\[0.6cm]
1731 \begin{center}
1732 \begin{picture}(0,0)(70,0)
1733 \includegraphics[width=1.4cm]{vasp_mig/comb_2-1_init.eps}
1734 \end{picture}
1735 \begin{picture}(0,0)(30,0)
1736 \includegraphics[width=1.4cm]{vasp_mig/comb_2-1_seq_03.eps}
1737 \end{picture}
1738 \begin{picture}(0,0)(-10,0)
1739 \includegraphics[width=1.4cm]{vasp_mig/comb_2-1_seq_06.eps}
1740 \end{picture}
1741 \begin{picture}(0,0)(-48,0)
1742 \includegraphics[width=1.4cm]{vasp_mig/comb_2-1_final.eps}
1743 \end{picture}
1744 \begin{picture}(0,0)(12.5,5)
1745 \includegraphics[width=1cm]{100_arrow.eps}
1746 \end{picture}
1747 \begin{picture}(0,0)(97,-10)
1748 \includegraphics[height=0.9cm]{001_arrow.eps}
1749 \end{picture}
1750 \end{center}
1751 \vspace{0.1cm}
1752 \end{minipage}
1753 }
1754 \begin{minipage}{0.3cm}
1755 \hfill
1756 \end{minipage}
1757 \framebox{
1758 \begin{minipage}{5.9cm}
1759 \includegraphics[width=5.9cm]{vasp_mig/comb_mig_4-2_vac_fullct.ps}\\[0.1cm]
1760 \begin{center}
1761 \begin{picture}(0,0)(60,0)
1762 \includegraphics[width=0.9cm]{vasp_mig/comb_3-1_init.eps}
1763 \end{picture}
1764 \begin{picture}(0,0)(25,0)
1765 \includegraphics[width=0.9cm]{vasp_mig/comb_3-1_seq_03.eps}
1766 \end{picture}
1767 \begin{picture}(0,0)(-20,0)
1768 \includegraphics[width=0.9cm]{vasp_mig/comb_3-1_seq_07.eps}
1769 \end{picture}
1770 \begin{picture}(0,0)(-55,0)
1771 \includegraphics[width=0.9cm]{vasp_mig/comb_3-1_final.eps}
1772 \end{picture}
1773 \begin{picture}(0,0)(12.5,5)
1774 \includegraphics[width=1cm]{100_arrow.eps}
1775 \end{picture}
1776 \begin{picture}(0,0)(95,0)
1777 \includegraphics[height=0.9cm]{001_arrow.eps}
1778 \end{picture}
1779 \end{center}
1780 \vspace{0.1cm}
1781 \end{minipage}
1782 }
1783
1784 \end{slide}
1785
1786 \begin{slide}
1787
1788  {\large\bf
1789   Conclusion of defect / migration / combined defect simulations
1790  }
1791
1792  \footnotesize
1793
1794 \vspace*{0.1cm}
1795
1796 Defect structures
1797 \begin{itemize}
1798  \item Accurately described by quantum-mechanical simulations
1799  \item Less accurate description by classical potential simulations
1800  \item Underestimated formation energy of \cs{} by classical approach
1801  \item Both methods predict same ground state: \ci{} \hkl<1 0 0> dumbbell
1802 \end{itemize}
1803
1804 Migration
1805 \begin{itemize}
1806  \item C migration pathway in Si identified
1807  \item Consistent with reorientation and diffusion experiments
1808 \end{itemize}
1809 \begin{itemize}
1810  \item Different path and ...
1811  \item overestimated barrier by classical potential calculations
1812 \end{itemize}
1813
1814 Concerning the precipitation mechanism
1815 \begin{itemize}
1816  \item Agglomeration of C-Si dumbbells energetically favorable
1817        (stress compensation)
1818  \item C-Si indeed favored compared to
1819        C$_{\text{sub}}$ \& \hkl<1 1 0> Si self-interstitial
1820  \item Possible low interaction capture radius of
1821        C$_{\text{sub}}$ \& \hkl<1 1 0> Si self-interstitial
1822  \item Low barrier for
1823        \ci{} \hkl<1 0 0> $\rightarrow$ \cs{} \& \si{} \hkl<1 1 0>
1824  \item In absence of nearby \hkl<1 1 0> Si self-interstitial:
1825        C-Si \hkl<1 0 0> + Vacancy $\rightarrow$ C$_{\text{sub}}$ (SiC)
1826 \end{itemize}
1827 \begin{center}
1828 {\color{blue}Results suggest increased participation of \cs}
1829 \end{center}
1830
1831 \end{slide}
1832
1833 \begin{slide}
1834
1835  {\large\bf
1836   Silicon carbide precipitation simulations
1837  }
1838
1839  \small
1840
1841 {\scriptsize
1842  \begin{pspicture}(0,0)(12,6.5)
1843   % nodes
1844   \rput(3.5,5.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
1845    \parbox{7cm}{
1846    \begin{itemize}
1847     \item Create c-Si volume
1848     \item Periodc boundary conditions
1849     \item Set requested $T$ and $p=0\text{ bar}$
1850     \item Equilibration of $E_{\text{kin}}$ and $E_{\text{pot}}$
1851    \end{itemize}
1852   }}}}
1853   \rput(3.5,2.7){\rnode{insert}{\psframebox[fillstyle=solid,fillcolor=lachs]{
1854    \parbox{7cm}{
1855    Insertion of C atoms at constant T
1856    \begin{itemize}
1857     \item total simulation volume {\pnode{in1}}
1858     \item volume of minimal SiC precipitate {\pnode{in2}}
1859     \item volume consisting of Si atoms to form a minimal {\pnode{in3}}\\
1860           precipitate
1861    \end{itemize}
1862   }}}}
1863   \rput(3.5,1){\rnode{cool}{\psframebox[fillstyle=solid,fillcolor=lbb]{
1864    \parbox{7.0cm}{
1865    Run for 100 ps followed by cooling down to $20\, ^{\circ}\textrm{C}$
1866   }}}}
1867   \ncline[]{->}{init}{insert}
1868   \ncline[]{->}{insert}{cool}
1869   \psframe[fillstyle=solid,fillcolor=white](7.5,0.7)(13.5,6.3)
1870   \rput(7.8,6){\footnotesize $V_1$}
1871   \psframe[fillstyle=solid,fillcolor=lightgray](9,2)(12,5)
1872   \rput(9.2,4.85){\tiny $V_2$}
1873   \psframe[fillstyle=solid,fillcolor=gray](9.25,2.25)(11.75,4.75)
1874   \rput(9.55,4.45){\footnotesize $V_3$}
1875   \rput(7.9,3.2){\pnode{ins1}}
1876   \rput(9.22,2.8){\pnode{ins2}}
1877   \rput(11.0,2.4){\pnode{ins3}}
1878   \ncline[]{->}{in1}{ins1}
1879   \ncline[]{->}{in2}{ins2}
1880   \ncline[]{->}{in3}{ins3}
1881  \end{pspicture}
1882 }
1883
1884 \begin{itemize}
1885  \item Restricted to classical potential simulations
1886  \item $V_2$ and $V_3$ considered due to low diffusion
1887  \item Amount of C atoms: 6000
1888        ($r_{\text{prec}}\approx 3.1\text{ nm}$, IBS: 2 ... 4 nm)
1889  \item Simulation volume: $31\times 31\times 31$ unit cells
1890        (238328 Si atoms)
1891 \end{itemize}
1892
1893 \end{slide}
1894
1895 \begin{slide}
1896
1897  {\large\bf\boldmath
1898   Silicon carbide precipitation simulations at $450\,^{\circ}\mathrm{C}$ as in IBS
1899  }
1900
1901  \small
1902
1903 \begin{minipage}{6.5cm}
1904 \includegraphics[width=6.4cm]{sic_prec_450_si-si_c-c.ps}
1905 \end{minipage}
1906 \begin{minipage}{6.5cm}
1907 \includegraphics[width=6.4cm]{sic_prec_450_energy.ps}
1908 \end{minipage}
1909
1910 \begin{minipage}{6.5cm}
1911 \includegraphics[width=6.4cm]{sic_prec_450_si-c.ps}
1912 \end{minipage}
1913 \begin{minipage}{6.5cm}
1914 \scriptsize
1915 \underline{Low C concentration ($V_1$)}\\
1916 \hkl<1 0 0> C-Si dumbbell dominated structure
1917 \begin{itemize}
1918  \item Si-C bumbs around 0.19 nm
1919  \item C-C peak at 0.31 nm (as expected in 3C-SiC):\\
1920        concatenated dumbbells of various orientation
1921  \item Si-Si NN distance stretched to 0.3 nm
1922 \end{itemize}
1923 {\color{blue}$\Rightarrow$ C atoms in proper 3C-SiC distance first}\\
1924 \underline{High C concentration ($V_2$, $V_3$)}\\
1925 High amount of strongly bound C-C bonds\\
1926 Defect density $\uparrow$ $\Rightarrow$ considerable amount of damage\\
1927 Only short range order observable\\
1928 {\color{blue}$\Rightarrow$ amorphous SiC-like phase}
1929 \end{minipage}
1930
1931 \end{slide}
1932
1933 \begin{slide}
1934
1935  {\large\bf\boldmath
1936   Silicon carbide precipitation simulations at $450\,^{\circ}\mathrm{C}$ as in IBS
1937  }
1938
1939  \small
1940
1941 \begin{minipage}{6.5cm}
1942 \includegraphics[width=6.4cm]{sic_prec_450_si-si_c-c.ps}
1943 \end{minipage}
1944 \begin{minipage}{6.5cm}
1945 \includegraphics[width=6.4cm]{sic_prec_450_energy.ps}
1946 \end{minipage}
1947
1948 \begin{minipage}{6.5cm}
1949 \includegraphics[width=6.4cm]{sic_prec_450_si-c.ps}
1950 \end{minipage}
1951 \begin{minipage}{6.5cm}
1952 \scriptsize
1953 \underline{Low C concentration ($V_1$)}\\
1954 \hkl<1 0 0> C-Si dumbbell dominated structure
1955 \begin{itemize}
1956  \item Si-C bumbs around 0.19 nm
1957  \item C-C peak at 0.31 nm (as expected in 3C-SiC):\\
1958        concatenated dumbbells of various orientation
1959  \item Si-Si NN distance stretched to 0.3 nm
1960 \end{itemize}
1961 {\color{blue}$\Rightarrow$ C atoms in proper 3C-SiC distance first}\\
1962 \underline{High C concentration ($V_2$, $V_3$)}\\
1963 High amount of strongly bound C-C bonds\\
1964 Defect density $\uparrow$ $\Rightarrow$ considerable amount of damage\\
1965 Only short range order observable\\
1966 {\color{blue}$\Rightarrow$ amorphous SiC-like phase}
1967 \end{minipage}
1968
1969 \begin{pspicture}(0,0)(0,0)
1970 \rput(6.7,5.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=white]{
1971 \begin{minipage}{10cm}
1972 \small
1973 {\color{red}\bf 3C-SiC formation fails to appear}
1974 \begin{itemize}
1975 \item Low C concentration simulations
1976  \begin{itemize}
1977   \item Formation of \ci{} indeed occurs
1978   \item Agllomeration not observed
1979  \end{itemize}
1980 \item High C concentration simulations
1981  \begin{itemize}
1982   \item Amorphous SiC-like structure\\
1983         (not expected at prevailing temperatures)
1984   \item Rearrangement and transition into 3C-SiC structure missing
1985  \end{itemize}
1986 \end{itemize}
1987 \end{minipage}
1988  }}}
1989 \end{pspicture}
1990
1991 \end{slide}
1992
1993 \begin{slide}
1994
1995  {\large\bf
1996   Limitations of molecular dynamics and short range potentials
1997  }
1998
1999 \footnotesize
2000
2001 \vspace{0.2cm}
2002
2003 \underline{Time scale problem of MD}\\[0.2cm]
2004 Minimize integration error\\
2005 $\Rightarrow$ discretization considerably smaller than
2006               reciprocal of fastest vibrational mode\\[0.1cm]
2007 Order of fastest vibrational mode: $10^{13} - 10^{14}\text{ Hz}$\\
2008 $\Rightarrow$ suitable choice of time step:
2009               $\tau=1\text{ fs}=10^{-15}\text{ s}$\\
2010 $\Rightarrow$ {\color{red}\underline{slow}} phase space propagation\\[0.1cm]
2011 Several local minima in energy surface separated by large energy barriers\\
2012 $\Rightarrow$ transition event corresponds to a multiple
2013               of vibrational periods\\
2014 $\Rightarrow$ phase transition made up of {\color{red}\underline{many}}
2015               infrequent transition events\\[0.1cm]
2016 {\color{blue}Accelerated methods:}
2017 \underline{Temperature accelerated} MD (TAD), self-guided MD \ldots
2018
2019 \vspace{0.3cm}
2020
2021 \underline{Limitations related to the short range potential}\\[0.2cm]
2022 Cut-off function pushing forces and energies to zero between 1$^{\text{st}}$
2023 and 2$^{\text{nd}}$ next neighbours\\
2024 $\Rightarrow$ overestimated unphysical high forces of next neighbours
2025
2026 \vspace{0.3cm}
2027
2028 \framebox{
2029 \color{red}
2030 Potential enhanced problem of slow phase space propagation
2031 }
2032
2033 \vspace{0.3cm}
2034
2035 \underline{Approach to the (twofold) problem}\\[0.2cm]
2036 Increased temperature simulations without TAD corrections\\
2037 (accelerated methods or higher time scales exclusively not sufficient)
2038
2039 \begin{picture}(0,0)(-260,-30)
2040 \framebox{
2041 \begin{minipage}{4.2cm}
2042 \tiny
2043 \begin{center}
2044 \vspace{0.03cm}
2045 \underline{IBS}
2046 \end{center}
2047 \begin{itemize}
2048 \item 3C-SiC also observed for higher T
2049 \item higher T inside sample
2050 \item structural evolution vs.\\
2051       equilibrium properties
2052 \end{itemize}
2053 \end{minipage}
2054 }
2055 \end{picture}
2056
2057 \begin{picture}(0,0)(-305,-155)
2058 \framebox{
2059 \begin{minipage}{2.5cm}
2060 \tiny
2061 \begin{center}
2062 retain proper\\
2063 thermodynmic sampling
2064 \end{center}
2065 \end{minipage}
2066 }
2067 \end{picture}
2068
2069 \end{slide}
2070
2071 \begin{slide}
2072
2073  {\large\bf
2074   Increased temperature simulations at low C concentration
2075  }
2076
2077 \small
2078
2079 \begin{minipage}{6.5cm}
2080 \includegraphics[width=6.4cm]{tot_pc_thesis.ps}
2081 \end{minipage}
2082 \begin{minipage}{6.5cm}
2083 \includegraphics[width=6.4cm]{tot_pc3_thesis.ps}
2084 \end{minipage}
2085
2086 \begin{minipage}{6.5cm}
2087 \includegraphics[width=6.4cm]{tot_pc2_thesis.ps}
2088 \end{minipage}
2089 \begin{minipage}{6.5cm}
2090 \scriptsize
2091  \underline{Si-C bonds:}
2092  \begin{itemize}
2093   \item Vanishing cut-off artifact (above $1650\,^{\circ}\mathrm{C}$)
2094   \item Structural change: C-Si \hkl<1 0 0> $\rightarrow$ C$_{\text{sub}}$
2095  \end{itemize}
2096  \underline{Si-Si bonds:}
2097  {\color{blue}Si-C$_{\text{sub}}$-Si} along \hkl<1 1 0>
2098  ($\rightarrow$ 0.325 nm)\\[0.1cm]
2099  \underline{C-C bonds:}
2100  \begin{itemize}
2101   \item C-C next neighbour pairs reduced (mandatory)
2102   \item Peak at 0.3 nm slightly shifted
2103         \begin{itemize}
2104          \item C-Si \hkl<1 0 0> combinations (dashed arrows)\\
2105                $\rightarrow$ C-Si \hkl<1 0 0> \& C$_{\text{sub}}$
2106                combinations (|)\\
2107                $\rightarrow$ pure {\color{blue}C$_{\text{sub}}$ combinations}
2108                ($\downarrow$)
2109          \item Range [|-$\downarrow$]:
2110                {\color{blue}C$_{\text{sub}}$ \& C$_{\text{sub}}$
2111                with nearby Si$_{\text{I}}$}
2112         \end{itemize}
2113  \end{itemize}
2114 \end{minipage}
2115
2116 \begin{picture}(0,0)(-330,-74)
2117 \color{blue}
2118 \framebox{
2119 \begin{minipage}{1.6cm}
2120 \tiny
2121 \begin{center}
2122 stretched SiC\\[-0.1cm]
2123 in c-Si
2124 \end{center}
2125 \end{minipage}
2126 }
2127 \end{picture}
2128
2129 \end{slide}
2130
2131 \begin{slide}
2132
2133  {\large\bf
2134   Increased temperature simulations at low C concentration
2135  }
2136
2137 \small
2138
2139 \begin{minipage}{6.5cm}
2140 \includegraphics[width=6.4cm]{tot_pc_thesis.ps}
2141 \end{minipage}
2142 \begin{minipage}{6.5cm}
2143 \includegraphics[width=6.4cm]{tot_pc3_thesis.ps}
2144 \end{minipage}
2145
2146 \begin{minipage}{6.5cm}
2147 \includegraphics[width=6.4cm]{tot_pc2_thesis.ps}
2148 \end{minipage}
2149 \begin{minipage}{6.5cm}
2150 \scriptsize
2151  \underline{Si-C bonds:}
2152  \begin{itemize}
2153   \item Vanishing cut-off artifact (above $1650\,^{\circ}\mathrm{C}$)
2154   \item Structural change: C-Si \hkl<1 0 0> $\rightarrow$ C$_{\text{sub}}$
2155  \end{itemize}
2156  \underline{Si-Si bonds:}
2157  {\color{blue}Si-C$_{\text{sub}}$-Si} along \hkl<1 1 0>
2158  ($\rightarrow$ 0.325 nm)\\[0.1cm]
2159  \underline{C-C bonds:}
2160  \begin{itemize}
2161   \item C-C next neighbour pairs reduced (mandatory)
2162   \item Peak at 0.3 nm slightly shifted
2163         \begin{itemize}
2164          \item C-Si \hkl<1 0 0> combinations (dashed arrows)\\
2165                $\rightarrow$ C-Si \hkl<1 0 0> \& C$_{\text{sub}}$
2166                combinations (|)\\
2167                $\rightarrow$ pure {\color{blue}C$_{\text{sub}}$ combinations}
2168                ($\downarrow$)
2169          \item Range [|-$\downarrow$]:
2170                {\color{blue}C$_{\text{sub}}$ \& C$_{\text{sub}}$
2171                with nearby Si$_{\text{I}}$}
2172         \end{itemize}
2173  \end{itemize}
2174 \end{minipage}
2175
2176 %\begin{picture}(0,0)(-330,-74)
2177 %\color{blue}
2178 %\framebox{
2179 %\begin{minipage}{1.6cm}
2180 %\tiny
2181 %\begin{center}
2182 %stretched SiC\\[-0.1cm]
2183 %in c-Si
2184 %\end{center}
2185 %\end{minipage}
2186 %}
2187 %\end{picture}
2188
2189 \begin{pspicture}(0,0)(0,0)
2190 \rput(6.7,5.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=white]{
2191 \begin{minipage}{10cm}
2192 \small
2193 {\color{blue}\bf Stretched SiC in c-Si}
2194 \begin{itemize}
2195 \item Consistent to precipitation model involving \cs{}
2196 \item Explains annealing behavior of high/low T C implants
2197       \begin{itemize}
2198        \item Low T: highly mobiel \ci{}
2199        \item High T: stable configurations of \cs{}
2200       \end{itemize}
2201 \end{itemize}
2202 $\Rightarrow$ High T $\leftrightarrow$ IBS conditions far from equilibrium\\
2203 $\Rightarrow$ Precipitation mechanism involving \cs{}
2204 \end{minipage}
2205  }}}
2206 \end{pspicture}
2207
2208 \end{slide}
2209
2210 \begin{slide}
2211
2212  {\large\bf
2213   Increased temperature simulations at high C concentration
2214  }
2215
2216 \footnotesize
2217
2218 \begin{minipage}{6.5cm}
2219 \includegraphics[width=6.4cm]{12_pc_thesis.ps}
2220 \end{minipage}
2221 \begin{minipage}{6.5cm}
2222 \includegraphics[width=6.4cm]{12_pc_c_thesis.ps}
2223 \end{minipage}
2224
2225 \vspace{0.1cm}
2226
2227 \scriptsize
2228
2229 \framebox{
2230 \begin{minipage}[t]{6.0cm}
2231 0.186 nm: Si-C pairs $\uparrow$\\
2232 (as expected in 3C-SiC)\\[0.2cm]
2233 0.282 nm: Si-C-C\\[0.2cm]
2234 $\approx$0.35 nm: C-Si-Si
2235 \end{minipage}
2236 }
2237 \begin{minipage}{0.2cm}
2238 \hfill
2239 \end{minipage}
2240 \framebox{
2241 \begin{minipage}[t]{6.0cm}
2242 0.15 nm: C-C pairs $\uparrow$\\
2243 (as expected in graphite/diamond)\\[0.2cm]
2244 0.252 nm: C-C-C (2$^{\text{nd}}$ NN for diamond)\\[0.2cm]
2245 0.31 nm: shifted towards 0.317 nm $\rightarrow$ C-Si-C
2246 \end{minipage}
2247 }
2248
2249 \begin{itemize}
2250 \item Decreasing cut-off artifact
2251 \item {\color{red}Amorphous} SiC-like phase remains
2252 \item High amount of {\color{red}damage} \& alignement to c-Si host matrix lost
2253 \item Slightly sharper peaks $\Rightarrow$ indicate slight {\color{blue}acceleration of dynamics} due to temperature
2254 \end{itemize}
2255
2256 \vspace{-0.1cm}
2257
2258 \begin{center}
2259 {\color{blue}
2260 \framebox{
2261 {\color{black}
2262 High C \& small $V$ \& short $t$
2263 $\Rightarrow$
2264 }
2265 Slow restructuring due to strong C-C bonds
2266 {\color{black}
2267 $\Leftarrow$
2268 High C \& low T implants
2269 }
2270 }
2271 }
2272 \end{center}
2273
2274 \end{slide}
2275
2276 \begin{slide}
2277
2278  {\large\bf
2279   Summary and Conclusions
2280  }
2281
2282  \scriptsize
2283
2284 %\vspace{0.1cm}
2285
2286 \framebox{
2287 \begin{minipage}[t]{12.9cm}
2288  \underline{Pecipitation simulations}
2289  \begin{itemize}
2290   \item High C concentration $\rightarrow$ amorphous SiC like phase
2291   \item Problem of potential enhanced slow phase space propagation
2292   \item Low T $\rightarrow$ C-Si \hkl<1 0 0> dumbbell dominated structure
2293   \item High T $\rightarrow$ C$_{\text{sub}}$ dominated structure
2294   \item High T necessary to simulate IBS conditions (far from equilibrium)
2295   \item Precipitation by successive agglomeration of \cs (epitaxy)
2296   \item \si{}: vehicle to form \cs{} \& supply of Si \& stress compensation
2297         (stretched SiC, interface)
2298  \end{itemize}
2299 \end{minipage}
2300 }
2301
2302 %\vspace{0.1cm}
2303
2304 \framebox{
2305 \begin{minipage}{12.9cm}
2306  \underline{Defects}
2307  \begin{itemize}
2308    \item DFT / EA
2309         \begin{itemize}
2310          \item Point defects excellently / fairly well described
2311                by DFT / EA
2312          \item C$_{\text{sub}}$ drastically underestimated by EA
2313          \item EA predicts correct ground state:
2314                C$_{\text{sub}}$ \& \si{} $>$ \ci{}
2315          \item Identified migration path explaining
2316                diffusion and reorientation experiments by DFT
2317          \item EA fails to describe \ci{} migration:
2318                Wrong path \& overestimated barrier
2319         \end{itemize}
2320    \item Combinations of defects
2321          \begin{itemize}
2322           \item Agglomeration of point defects energetically favorable
2323                 by compensation of stress
2324           \item Formation of C-C unlikely
2325           \item C$_{\text{sub}}$ favored conditions (conceivable in IBS)
2326           \item \ci{} \hkl<1 0 0> $\leftrightarrow$ \cs{} \& \si{} \hkl<1 1 0>\\
2327                 Low barrier (\unit[0.77]{eV}) \& low capture radius
2328         \end{itemize}
2329  \end{itemize}
2330 \end{minipage}
2331 }
2332
2333 \begin{center}
2334 {\color{blue}
2335 \framebox{Precipitation by successive agglomeration of \cs{}}
2336 }
2337 \end{center}
2338
2339 \end{slide}
2340
2341 \begin{slide}
2342
2343  {\large\bf
2344   Acknowledgements
2345  }
2346
2347  \vspace{0.1cm}
2348
2349  \small
2350
2351  Thanks to \ldots
2352
2353  \underline{Augsburg}
2354  \begin{itemize}
2355   \item Prof. B. Stritzker (accomodation at EP \RM{4})
2356   \item Ralf Utermann (EDV)
2357  \end{itemize}
2358
2359  \underline{Helsinki}
2360  \begin{itemize}
2361   \item Prof. K. Nordlund (MD)
2362  \end{itemize}
2363
2364  \underline{Munich}
2365  \begin{itemize}
2366   \item Bayerische Forschungsstiftung (financial support)
2367  \end{itemize}
2368
2369  \underline{Paderborn}
2370  \begin{itemize}
2371   \item Prof. J. Lindner (SiC)
2372   \item Prof. G. Schmidt (DFT + financial support)
2373   \item Dr. E. Rauls (DFT + SiC)
2374   \item Dr. S. Sanna (VASP)
2375  \end{itemize}
2376
2377 \vspace{0.2cm}
2378
2379 \begin{center}
2380 \framebox{
2381 \bf Thank you for your attention!
2382 }
2383 \end{center}
2384
2385 \end{slide}
2386
2387 \end{document}
2388
2389 \fi