posic/talks/mpi_app.tex

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   9 \usepackage{amsmath}
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  18 \usepackage{fancyvrb}           % Fancy verbatim environments
  19 \usepackage{pstricks}           % PSTricks with the standard color package
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  23
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  29 \usepackage{graphicx}
  30 \graphicspath{{../img/}}
  31
  32 \usepackage{miller}
  33
  34 \usepackage[setpagesize=false]{hyperref}
  35
  36 % units
  37 \usepackage{units}
  38
  39 \usepackage{semcolor}
  40 \usepackage{semlayer}           % Seminar overlays
  41 \usepackage{slidesec}           % Seminar sections and list of slides
  42
  43 \input{seminar.bug}             % Official bugs corrections
  44 \input{seminar.bg2}             % Unofficial bugs corrections
  45
  46 \articlemag{1}
  47
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  49
  50 % font
  51 %\usepackage{cmbright}
  52 %\renewcommand{\familydefault}{\sfdefault}
  53 %\usepackage{mathptmx}
  54
  55 \usepackage{upgreek}
  56
  57 \begin{document}
  58
  59 \extraslideheight{10in}
  60 \slideframe{none}
  61
  62 \pagestyle{empty}
  63
  64 % specify width and height
  65 \slidewidth 27.7cm
  66 \slideheight 19.1cm
  67
  68 % shift it into visual area properly
  69 \def\slideleftmargin{3.3cm}
  70 \def\slidetopmargin{0.6cm}
  71
  72 \newcommand{\ham}{\mathcal{H}}
  73 \newcommand{\pot}{\mathcal{V}}
  74 \newcommand{\foo}{\mathcal{U}}
  75 \newcommand{\vir}{\mathcal{W}}
  76
  77 % itemize level ii
  78 \renewcommand\labelitemii{{\color{gray}$\bullet$}}
  79
  80 % nice phi
  81 \renewcommand{\phi}{\varphi}
  82
  83 % roman letters
  84 \newcommand{\RM}[1]{\MakeUppercase{\romannumeral #1{}}}
  85
  86 % colors
  87 \newrgbcolor{si-yellow}{.6 .6 0}
  88 \newrgbcolor{hb}{0.75 0.77 0.89}
  89 \newrgbcolor{lbb}{0.75 0.8 0.88}
  90 \newrgbcolor{hlbb}{0.825 0.88 0.968}
  91 \newrgbcolor{lachs}{1.0 .93 .81}
  92
  93 % shortcuts
  94 \newcommand{\si}{Si$_{\text{i}}${}}
  95 \newcommand{\ci}{C$_{\text{i}}${}}
  96 \newcommand{\cs}{C$_{\text{sub}}${}}
  97 \newcommand{\degc}[1]{\unit[#1]{$^{\circ}$C}{}}
  98 \newcommand{\distn}[1]{\unit[#1]{nm}{}}
  99 \newcommand{\dista}[1]{\unit[#1]{\AA}{}}
 100 \newcommand{\perc}[1]{\unit[#1]{\%}{}}
 101
 102 %\layout
 103
 104 % no vertical centering
 105 %\centerslidesfalse
 106
 107
 108 % topic
 109
 110 \begin{slide}
 111 \begin{center}
 112
 113  \vspace{16pt}
 114
 115  {\LARGE\bf
 116   Atomistic simulation studies\\[0.2cm]
 117   in the C/Si system
 118  }
 119
 120  \vspace{48pt}
 121
 122  \textsc{Frank Zirkelbach}
 123
 124  \vspace{48pt}
 125
 126  Application talk at the Max Planck Institute for Solid State Research
 127
 128  \vspace{08pt}
 129
 130  Stuttgart, November 2011
 131
 132 \end{center}
 133 \end{slide}
 134
 135 % intro
 136
 137 \begin{slide}
 138
 139 %{\large\bf
 140 % Phase diagram of the C/Si system\\
 141 %}
 142
 143 \vspace*{0.2cm}
 144
 145 \begin{minipage}{7cm}
 146 \includegraphics[width=6.5cm]{si-c_phase.eps}
 147 \begin{center}
 148 {\tiny
 149 R. I. Scace and G. A. Slack, J. Chem. Phys. 30, 1551 (1959)
 150 }
 151 \end{center}
 152 \begin{pspicture}(0,0)(0,0)
 153 \psellipse[linecolor=blue,linewidth=0.1cm](3.55,4.0)(0.5,2.9)
 154 \end{pspicture}
 155 \end{minipage}
 156 \begin{minipage}{6cm}
 157 {\bf Phase diagram of the C/Si system}\\[0.2cm]
 158 {\color{blue}Stoichiometric composition}
 159 \begin{itemize}
 160 \item only chemical stable compound
 161 \item wide band gap semiconductor\\
 162       \underline{silicon carbide}, SiC
 163 \end{itemize}
 164 \end{minipage}
 165
 166 \end{slide}
 167
 168 % motivation / properties / applications of silicon carbide
 169
 170 \begin{slide}
 171
 172 \small
 173
 174 \begin{pspicture}(0,0)(13.5,5)
 175
 176  \psframe*[linecolor=hb](0,0)(13.5,5)
 177
 178  \pspolygon[linecolor=hlbb,fillcolor=hlbb,fillstyle=solid](5.5,1)(7,1)(7,3)(5.5,3)
 179  \pspolygon[linecolor=hlbb,fillcolor=hlbb,fillstyle=solid](6.75,0.5)(8,2)(8,2)(6.75,3.5)
 180
 181  \rput[lt](0.2,4.6){\color{gray}PROPERTIES}
 182
 183  \rput[lt](0.5,4){wide band gap}
 184  \rput[lt](0.5,3.5){high electric breakdown field}
 185  \rput[lt](0.5,3){good electron mobility}
 186  \rput[lt](0.5,2.5){high electron saturation drift velocity}
 187  \rput[lt](0.5,2){high thermal conductivity}
 188
 189  \rput[lt](0.5,1.5){hard and mechanically stable}
 190  \rput[lt](0.5,1){chemically inert}
 191
 192  \rput[lt](0.5,0.5){radiation hardness}
 193
 194  \rput[rt](13.3,4.6){\color{gray}APPLICATIONS}
 195
 196  \rput[rt](13,3.85){high-temperature, high power}
 197  \rput[rt](13,3.5){and high-frequency}
 198  \rput[rt](13,3.15){electronic and optoelectronic devices}
 199
 200  \rput[rt](13,2.35){material suitable for extreme conditions}
 201  \rput[rt](13,2){microelectromechanical systems}
 202  \rput[rt](13,1.65){abrasives, cutting tools, heating elements}
 203
 204  \rput[rt](13,0.85){first wall reactor material, detectors}
 205  \rput[rt](13,0.5){and electronic devices for space}
 206
 207 \end{pspicture}
 208
 209 \begin{picture}(0,0)(0,-162)
 210 \includegraphics[height=2.0cm]{3C_SiC_bs.eps}
 211 \end{picture}
 212 \begin{picture}(0,0)(-130,-162)
 213 \includegraphics[height=2.0cm]{nasa_600c_led.eps}
 214 \end{picture}
 215 \begin{picture}(0,0)(-295,-162)
 216 \includegraphics[height=2.0cm]{6h-sic_3c-sic.eps}
 217 \end{picture}
 218 %%%%
 219 \begin{picture}(0,0)(5,65)
 220 \includegraphics[height=2.8cm]{sic_switch.eps}
 221 \end{picture}
 222 \begin{picture}(0,0)(-145,65)
 223 \includegraphics[height=2.8cm]{infineon_schottky.eps}
 224 \end{picture}
 225 \begin{picture}(0,0)(-260,65)
 226 \includegraphics[height=2.8cm]{ise_99.eps}
 227 \end{picture}
 228
 229 \end{slide}
 230
 231 % motivation
 232
 233 \begin{slide}
 234
 235  {\large\bf
 236   Polytypes of SiC
 237  }
 238
 239  \vspace{4cm}
 240
 241  \small
 242
 243 \begin{tabular}{l c c c c c c}
 244 \hline
 245  & 3C-SiC & 4H-SiC & 6H-SiC & Si & GaN & Diamond\\
 246 \hline
 247 Hardness [Mohs] & \multicolumn{3}{c}{------ 9.6 ------}& 6.5 & - & 10 \\
 248 Band gap [eV] & 2.36 & 3.23 & 3.03 & 1.12 & 3.39 & 5.5 \\
 249 Break down field [$10^6$ V/cm] & 4 & 3 & 3.2 & 0.6 & 5 & 10 \\
 250 Saturation drift velocity [$10^7$ cm/s] & 2.5 & 2.0 & 2.0 & 1 & 2.7 & 2.7 \\
 251 Electron mobility [cm$^2$/Vs] & 800 & 900 & 400 & 1100 & 900 & 2200 \\
 252 Hole mobility [cm$^2$/Vs] & 320 & 120 & 90 & 420 & 150 & 1600 \\
 253 Thermal conductivity [W/cmK] & 5.0 & 4.9 & 4.9 & 1.5 & 1.3 & 22 \\
 254 \hline
 255 \end{tabular}
 256
 257 {\tiny
 258  Values for $T=300$ K
 259 }
 260
 261 \begin{picture}(0,0)(-160,-155)
 262  \includegraphics[width=7cm]{polytypes.eps}
 263 \end{picture}
 264 \begin{picture}(0,0)(-10,-185)
 265  \includegraphics[width=3.8cm]{cubic_hex.eps}\\
 266 \end{picture}
 267 \begin{picture}(0,0)(-10,-175)
 268  {\tiny cubic (twist)}
 269 \end{picture}
 270 \begin{picture}(0,0)(-60,-175)
 271  {\tiny hexagonal (no twist)}
 272 \end{picture}
 273 \begin{pspicture}(0,0)(0,0)
 274 \psellipse[linecolor=green](5.7,3.03)(0.4,0.5)
 275 \end{pspicture}
 276 \begin{pspicture}(0,0)(0,0)
 277 \psellipse[linecolor=green](5.6,1.68)(0.4,0.2)
 278 \end{pspicture}
 279 \begin{pspicture}(0,0)(0,0)
 280 \psellipse[linecolor=red](10.7,1.13)(0.4,0.2)
 281 \end{pspicture}
 282
 283 \end{slide}
 284
 285 % fabrication
 286
 287 \begin{slide}
 288
 289  {\large\bf
 290   Fabrication of silicon carbide
 291  }
 292
 293  \small
 294
 295  \vspace{4pt}
 296
 297  SiC - \emph{Born from the stars, perfected on earth.}
 298
 299  IBS also here!
 300
 301  \vspace{4pt}
 302
 303  Conventional thin film SiC growth:
 304  \begin{itemize}
 305   \item \underline{Sublimation growth using the modified Lely method}
 306         \begin{itemize}
 307          \item SiC single-crystalline seed at $T=1800 \, ^{\circ} \text{C}$
 308          \item Surrounded by polycrystalline SiC in a graphite crucible\\
 309                at $T=2100-2400 \, ^{\circ} \text{C}$
 310          \item Deposition of supersaturated vapor on cooler seed crystal
 311         \end{itemize}
 312   \item \underline{Homoepitaxial growth using CVD}
 313         \begin{itemize}
 314          \item Step-controlled epitaxy on off-oriented 6H-SiC substrates
 315          \item C$_3$H$_8$/SiH$_4$/H$_2$ at $1100-1500 \, ^{\circ} \text{C}$
 316          \item Angle, temperature $\rightarrow$ 3C/6H/4H-SiC
 317         \end{itemize}
 318   \item \underline{Heteroepitaxial growth of 3C-SiC on Si using CVD/MBE}
 319         \begin{itemize}
 320          \item Two steps: carbonization and growth
 321          \item $T=650-1050 \, ^{\circ} \text{C}$
 322          \item SiC/Si lattice mismatch $\approx$ 20 \%
 323          \item Quality and size not yet sufficient
 324         \end{itemize}
 325  \end{itemize}
 326
 327  \begin{picture}(0,0)(-280,-65)
 328   \includegraphics[width=3.8cm]{6h-sic_3c-sic.eps}
 329  \end{picture}
 330  \begin{picture}(0,0)(-280,-55)
 331   \begin{minipage}{5cm}
 332   {\tiny
 333    NASA: 6H-SiC and 3C-SiC LED\\[-7pt]
 334    on 6H-SiC substrate
 335   }
 336   \end{minipage}
 337  \end{picture}
 338  \begin{picture}(0,0)(-265,-150)
 339   \includegraphics[width=2.4cm]{m_lely.eps}
 340  \end{picture}
 341  \begin{picture}(0,0)(-333,-175)
 342   \begin{minipage}{5cm}
 343   {\tiny
 344    1. Lid\\[-7pt]
 345    2. Heating\\[-7pt]
 346    3. Source\\[-7pt]
 347    4. Crucible\\[-7pt]
 348    5. Insulation\\[-7pt]
 349    6. Seed crystal
 350   }
 351   \end{minipage}
 352  \end{picture}
 353  \begin{picture}(0,0)(-230,-35)
 354  \framebox{
 355  {\footnotesize\color{blue}\bf Hex: micropipes along c-axis}
 356  }
 357  \end{picture}
 358  \begin{picture}(0,0)(-230,-10)
 359  \framebox{
 360  \begin{minipage}{3cm}
 361  {\footnotesize\color{blue}\bf 3C-SiC fabrication\\
 362                                less advanced}
 363  \end{minipage}
 364  }
 365  \end{picture}
 366
 367 \end{slide}
 368
 369 % contents
 370
 371 \begin{slide}
 372
 373 {\large\bf
 374  Outline
 375 }
 376
 377  \begin{itemize}
 378   \item Implantation of C in Si --- Overview of experimental observations
 379   \item Utilized simulation techniques and modeled problems
 380         \begin{itemize}
 381          \item {\color{blue}Diploma thesis}\\
 382                \underline{Monte Carlo} simulations
 383                modeling the selforganization process
 384                leading to periodic arrays of nanometric amorphous SiC
 385                precipitates
 386          \item {\color{blue}Doctoral studies}\\
 387                Classical potential \underline{molecular dynamics} simulations
 388                \ldots\\
 389                \underline{Density functional theory} calculations
 390                \ldots\\[0.2cm]
 391                \ldots on defects and SiC precipitation in Si
 392         \end{itemize}
 393   \item Summary / Conclusion / Outlook
 394  \end{itemize}
 395
 396 \end{slide}
 397
 398
 399
 400 \end{document}
 401
 402 \begin{slide}
 403
 404  {\large\bf
 405   Fabrication of silicon carbide
 406  }
 407
 408  \small
 409
 410  Alternative approach:
 411  Ion beam synthesis (IBS) of burried 3C-SiC layers in Si\hkl(1 0 0)
 412  \begin{itemize}
 413   \item \underline{Implantation step 1}\\
 414         180 keV C$^+$, $D=7.9\times 10^{17}$ cm$^{-2}$, $T_{\text{i}}=500\,^{\circ}\mathrm{C}$\\
 415         $\Rightarrow$ box-like distribution of equally sized
 416                        and epitactically oriented SiC precipitates
 417
 418   \item \underline{Implantation step 2}\\
 419         180 keV C$^+$, $D=0.6\times 10^{17}$ cm$^{-2}$, $T_{\text{i}}=250\,^{\circ}\mathrm{C}$\\
 420         $\Rightarrow$ destruction of SiC nanocrystals
 421                       in growing amorphous interface layers
 422   \item \underline{Annealing}\\
 423         $T=1250\,^{\circ}\mathrm{C}$, $t=10\,\text{h}$\\
 424         $\Rightarrow$ homogeneous, stoichiometric SiC layer
 425                       with sharp interfaces
 426  \end{itemize}
 427
 428  \begin{minipage}{6.3cm}
 429  \includegraphics[width=6cm]{ibs_3c-sic.eps}\\[-0.2cm]
 430  {\tiny
 431   XTEM micrograph of single crystalline 3C-SiC in Si\hkl(1 0 0)
 432  }
 433  \end{minipage}
 434 \framebox{
 435  \begin{minipage}{6.3cm}
 436  \begin{center}
 437  {\color{blue}
 438   Precipitation mechanism not yet fully understood!
 439  }
 440  \renewcommand\labelitemi{$\Rightarrow$}
 441  \small
 442  \underline{Understanding the SiC precipitation}
 443  \begin{itemize}
 444   \item significant technological progress in SiC thin film formation
 445   \item perspectives for processes relying upon prevention of SiC precipitation
 446  \end{itemize}
 447  \end{center}
 448  \end{minipage}
 449 }
 450
 451 \end{slide}
 452
 453
 454 \begin{slide}
 455
 456  {\large\bf
 457   Supposed precipitation mechanism of SiC in Si
 458  }
 459
 460  \scriptsize
 461
 462  \vspace{0.1cm}
 463
 464  \begin{minipage}{3.8cm}
 465  Si \& SiC lattice structure\\[0.2cm]
 466  \includegraphics[width=3.5cm]{sic_unit_cell.eps}\\[-0.3cm]
 467  \hrule
 468  \end{minipage}
 469  \hspace{0.6cm}
 470  \begin{minipage}{3.8cm}
 471  \begin{center}
 472  \includegraphics[width=3.3cm]{tem_c-si-db.eps}
 473  \end{center}
 474  \end{minipage}
 475  \hspace{0.6cm}
 476  \begin{minipage}{3.8cm}
 477  \begin{center}
 478  \includegraphics[width=3.3cm]{tem_3c-sic.eps}
 479  \end{center}
 480  \end{minipage}
 481
 482  \begin{minipage}{4cm}
 483  \begin{center}
 484  C-Si dimers (dumbbells)\\[-0.1cm]
 485  on Si interstitial sites
 486  \end{center}
 487  \end{minipage}
 488  \hspace{0.2cm}
 489  \begin{minipage}{4.2cm}
 490  \begin{center}
 491  Agglomeration of C-Si dumbbells\\[-0.1cm]
 492  $\Rightarrow$ dark contrasts
 493  \end{center}
 494  \end{minipage}
 495  \hspace{0.2cm}
 496  \begin{minipage}{4cm}
 497  \begin{center}
 498  Precipitation of 3C-SiC in Si\\[-0.1cm]
 499  $\Rightarrow$ Moir\'e fringes\\[-0.1cm]
 500  \& release of Si self-interstitials
 501  \end{center}
 502  \end{minipage}
 503
 504  \begin{minipage}{3.8cm}
 505  \begin{center}
 506  \includegraphics[width=3.3cm]{sic_prec_seq_01.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_02.eps}
 513  \end{center}
 514  \end{minipage}
 515  \hspace{0.6cm}
 516  \begin{minipage}{3.8cm}
 517  \begin{center}
 518  \includegraphics[width=3.3cm]{sic_prec_seq_03.eps}
 519  \end{center}
 520  \end{minipage}
 521
 522 \begin{pspicture}(0,0)(0,0)
 523 \psline[linewidth=4pt]{->}(8.5,2)(9.0,2)
 524 \psellipse[linecolor=blue](11.5,5.8)(0.3,0.5)
 525 \rput{-20}{\psellipse[linecolor=blue](3.3,8.1)(0.3,0.5)}
 526 \psline[linewidth=4pt]{->}(4.0,2)(4.5,2)
 527 \rput(12.7,0.3){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
 528  $4a_{\text{Si}}=5a_{\text{SiC}}$
 529  }}}
 530 \rput(12.2,8){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
 531 \hkl(h k l) planes match
 532  }}}
 533 \rput(9.7,6.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
 534 r = 2 - 4 nm
 535  }}}
 536 \end{pspicture}
 537
 538 \end{slide}
 539
 540 \begin{slide}
 541
 542  {\large\bf
 543   Supposed precipitation mechanism of SiC in Si
 544  }
 545
 546  \scriptsize
 547
 548  \vspace{0.1cm}
 549
 550  \begin{minipage}{3.8cm}
 551  Si \& SiC lattice structure\\[0.2cm]
 552  \includegraphics[width=3.5cm]{sic_unit_cell.eps}\\[-0.3cm]
 553  \hrule
 554  \end{minipage}
 555  \hspace{0.6cm}
 556  \begin{minipage}{3.8cm}
 557  \begin{center}
 558  \includegraphics[width=3.3cm]{tem_c-si-db.eps}
 559  \end{center}
 560  \end{minipage}
 561  \hspace{0.6cm}
 562  \begin{minipage}{3.8cm}
 563  \begin{center}
 564  \includegraphics[width=3.3cm]{tem_3c-sic.eps}
 565  \end{center}
 566  \end{minipage}
 567
 568  \begin{minipage}{4cm}
 569  \begin{center}
 570  C-Si dimers (dumbbells)\\[-0.1cm]
 571  on Si interstitial sites
 572  \end{center}
 573  \end{minipage}
 574  \hspace{0.2cm}
 575  \begin{minipage}{4.2cm}
 576  \begin{center}
 577  Agglomeration of C-Si dumbbells\\[-0.1cm]
 578  $\Rightarrow$ dark contrasts
 579  \end{center}
 580  \end{minipage}
 581  \hspace{0.2cm}
 582  \begin{minipage}{4cm}
 583  \begin{center}
 584  Precipitation of 3C-SiC in Si\\[-0.1cm]
 585  $\Rightarrow$ Moir\'e fringes\\[-0.1cm]
 586  \& release of Si self-interstitials
 587  \end{center}
 588  \end{minipage}
 589
 590  \begin{minipage}{3.8cm}
 591  \begin{center}
 592  \includegraphics[width=3.3cm]{sic_prec_seq_01.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_02.eps}
 599  \end{center}
 600  \end{minipage}
 601  \hspace{0.6cm}
 602  \begin{minipage}{3.8cm}
 603  \begin{center}
 604  \includegraphics[width=3.3cm]{sic_prec_seq_03.eps}
 605  \end{center}
 606  \end{minipage}
 607
 608 \begin{pspicture}(0,0)(0,0)
 609 \psline[linewidth=4pt]{->}(8.5,2)(9.0,2)
 610 \psellipse[linecolor=blue](11.5,5.8)(0.3,0.5)
 611 \rput{-20}{\psellipse[linecolor=blue](3.3,8.1)(0.3,0.5)}
 612 \psline[linewidth=4pt]{->}(4.0,2)(4.5,2)
 613 \rput(12.7,0.3){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
 614  $4a_{\text{Si}}=5a_{\text{SiC}}$
 615  }}}
 616 \rput(12.2,8){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
 617 \hkl(h k l) planes match
 618  }}}
 619 \rput(9.7,6.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
 620 r = 2 - 4 nm
 621  }}}
 622 \rput(6.7,5.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=white]{
 623 \begin{minipage}{10cm}
 624 \small
 625 {\color{red}\bf Controversial views}
 626 \begin{itemize}
 627 \item Implantations at high T (Nejim et al.)
 628  \begin{itemize}
 629   \item Topotactic transformation based on \cs
 630   \item \si{} as supply reacting with further C in cleared volume
 631  \end{itemize}
 632 \item Annealing behavior (Serre et al.)
 633  \begin{itemize}
 634   \item Room temperature implants $\rightarrow$ highly mobile C
 635   \item Elevated T implants $\rightarrow$ no/low C redistribution/migration\\
 636         (indicate stable \cs{} configurations)
 637  \end{itemize}
 638 \item Strained silicon \& Si/SiC heterostructures
 639  \begin{itemize}
 640   \item Coherent SiC precipitates (tensile strain)
 641   \item Incoherent SiC (strain relaxation)
 642  \end{itemize}
 643 \end{itemize}
 644 \end{minipage}
 645  }}}
 646 \end{pspicture}
 647
 648 \end{slide}
 649
 650 \begin{slide}
 651
 652  {\large\bf
 653   Molecular dynamics (MD) simulations
 654  }
 655
 656  \vspace{12pt}
 657
 658  \small
 659
 660  {\bf MD basics:}
 661  \begin{itemize}
 662   \item Microscopic description of N particle system
 663   \item Analytical interaction potential
 664   \item Numerical integration using Newtons equation of motion\\
 665         as a propagation rule in 6N-dimensional phase space
 666   \item Observables obtained by time and/or ensemble averages
 667  \end{itemize}
 668  {\bf Details of the simulation:}
 669  \begin{itemize}
 670   \item Integration: Velocity Verlet, timestep: $1\text{ fs}$
 671   \item Ensemble: NpT (isothermal-isobaric)
 672         \begin{itemize}
 673          \item Berendsen thermostat:
 674                $\tau_{\text{T}}=100\text{ fs}$
 675          \item Berendsen barostat:\\
 676                $\tau_{\text{P}}=100\text{ fs}$,
 677                $\beta^{-1}=100\text{ GPa}$
 678         \end{itemize}
 679   \item Erhart/Albe potential: Tersoff-like bond order potential
 680   \vspace*{12pt}
 681         \[
 682         E = \frac{1}{2} \sum_{i \neq j} \pot_{ij}, \quad
 683         \pot_{ij} = {\color{red}f_C(r_{ij})}
 684         \left[ f_R(r_{ij}) + {\color{blue}b_{ij}} f_A(r_{ij}) \right]
 685         \]
 686  \end{itemize}
 687
 688  \begin{picture}(0,0)(-230,-30)
 689   \includegraphics[width=5cm]{tersoff_angle.eps}
 690  \end{picture}
 691
 692 \end{slide}
 693
 694 \begin{slide}
 695
 696  {\large\bf
 697   Density functional theory (DFT) calculations
 698  }
 699
 700  \small
 701
 702  Basic ingredients necessary for DFT
 703
 704  \begin{itemize}
 705   \item \underline{Hohenberg-Kohn theorem} - ground state density $n_0(r)$ ...
 706         \begin{itemize}
 707          \item ... uniquely determines the ground state potential
 708                / wavefunctions
 709          \item ... minimizes the systems total energy
 710         \end{itemize}
 711   \item \underline{Born-Oppenheimer}
 712         - $N$ moving electrons in an external potential of static nuclei
 713 \[
 714 H\Psi = \left[-\sum_i^N \frac{\hbar^2}{2m}\nabla_i^2
 715               +\sum_i^N V_{\text{ext}}(r_i)
 716               +\sum_{i<j}^N V_{e-e}(r_i,r_j)\right]\Psi=E\Psi
 717 \]
 718   \item \underline{Effective potential}
 719         - averaged electrostatic potential \& exchange and correlation
 720 \[
 721 V_{\text{eff}}(r)=V_{\text{ext}}(r)+\int\frac{e^2 n(r')}{|r-r'|}d^3r'
 722                  +V_{\text{XC}}[n(r)]
 723 \]
 724   \item \underline{Kohn-Sham system}
 725         - Schr\"odinger equation of N non-interacting particles
 726 \[
 727 \left[ -\frac{\hbar^2}{2m}\nabla^2 + V_{\text{eff}}(r) \right] \Phi_i(r)
 728 =\epsilon_i\Phi_i(r)
 729 \quad
 730 \Rightarrow
 731 \quad
 732 n(r)=\sum_i^N|\Phi_i(r)|^2
 733 \]
 734   \item \underline{Self-consistent solution}\\
 735 $n(r)$ depends on $\Phi_i$, which depend on $V_{\text{eff}}$,
 736 which in turn depends on $n(r)$
 737   \item \underline{Variational principle}
 738         - minimize total energy with respect to $n(r)$
 739  \end{itemize}
 740
 741 \end{slide}
 742
 743 \begin{slide}
 744
 745  {\large\bf
 746   Density functional theory (DFT) calculations
 747  }
 748
 749  \small
 750
 751  \vspace*{0.2cm}
 752
 753  Details of applied DFT calculations in this work
 754
 755  \begin{itemize}
 756   \item \underline{Exchange correlation functional}
 757         - approximations for the inhomogeneous electron gas
 758         \begin{itemize}
 759          \item LDA: $E_{\text{XC}}^{\text{LDA}}[n]=\int \epsilon_{\text{XC}}(n)n(r)d^3r$
 760          \item GGA: $E_{\text{XC}}^{\text{GGA}}[n]=\int \epsilon_{\text{XC}}(n,\nabla n)n(r)d^3r$
 761         \end{itemize}
 762   \item \underline{Plane wave basis set}
 763         - approximation of the wavefunction $\Phi_i$ by plane waves $\phi_j$
 764 \[
 765 \rightarrow
 766 \text{Fourier series: } \Phi_i=\sum_{|G+k|<G_{\text{cut}}} c_j^i \phi_j(r), \quad E_{\text{cut}}=\frac{\hbar^2}{2m}G^2_{\text{cut}}
 767 \qquad ({\color{blue}300\text{ eV}})
 768 \]
 769   \item \underline{Brillouin zone sampling} -
 770         {\color{blue}$\Gamma$-point only} calculations
 771   \item \underline{Pseudo potential}
 772         - consider only the valence electrons
 773   \item \underline{Code} - VASP 4.6
 774  \end{itemize}
 775
 776  \vspace*{0.2cm}
 777
 778  MD and structural optimization
 779
 780  \begin{itemize}
 781   \item MD integration: Gear predictor corrector algorithm
 782   \item Pressure control: Parrinello-Rahman pressure control
 783   \item Structural optimization: Conjugate gradient method
 784  \end{itemize}
 785
 786 \begin{pspicture}(0,0)(0,0)
 787 \psellipse[linecolor=blue](1.5,6.75)(0.5,0.3)
 788 \end{pspicture}
 789
 790 \end{slide}
 791
 792 \begin{slide}
 793
 794  {\large\bf
 795   C and Si self-interstitial point defects in silicon
 796  }
 797
 798  \small
 799
 800  \vspace*{0.3cm}
 801
 802 \begin{minipage}{8cm}
 803 Procedure:\\[0.3cm]
 804   \begin{pspicture}(0,0)(7,5)
 805   \rput(3.5,4){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
 806    \parbox{7cm}{
 807    \begin{itemize}
 808     \item Creation of c-Si simulation volume
 809     \item Periodic boundary conditions
 810     \item $T=0\text{ K}$, $p=0\text{ bar}$
 811    \end{itemize}
 812   }}}}
 813 \rput(3.5,2.1){\rnode{insert}{\psframebox{
 814  \parbox{7cm}{
 815   \begin{center}
 816   Insertion of interstitial C/Si atoms
 817   \end{center}
 818   }}}}
 819   \rput(3.5,1){\rnode{cool}{\psframebox[fillstyle=solid,fillcolor=lbb]{
 820    \parbox{7cm}{
 821    \begin{center}
 822    Relaxation / structural energy minimization
 823    \end{center}
 824   }}}}
 825   \ncline[]{->}{init}{insert}
 826   \ncline[]{->}{insert}{cool}
 827  \end{pspicture}
 828 \end{minipage}
 829 \begin{minipage}{5cm}
 830   \includegraphics[width=5cm]{unit_cell_e.eps}\\
 831 \end{minipage}
 832
 833 \begin{minipage}{9cm}
 834  \begin{tabular}{l c c}
 835  \hline
 836  & size [unit cells] & \# atoms\\
 837 \hline
 838 VASP & $3\times 3\times 3$ & $216\pm 1$ \\
 839 Erhart/Albe & $9\times 9\times 9$ & $5832\pm 1$\\
 840 \hline
 841  \end{tabular}
 842 \end{minipage}
 843 \begin{minipage}{4cm}
 844 {\color{red}$\bullet$} Tetrahedral\\
 845 {\color{green}$\bullet$} Hexagonal\\
 846 {\color{yellow}$\bullet$} \hkl<1 0 0> dumbbell\\
 847 {\color{magenta}$\bullet$} \hkl<1 1 0> dumbbell\\
 848 {\color{cyan}$\bullet$} Bond-centered\\
 849 {\color{black}$\bullet$} Vacancy / Substitutional
 850 \end{minipage}
 851
 852 \end{slide}
 853
 854 \begin{slide}
 855
 856  \footnotesize
 857
 858 \begin{minipage}{9.5cm}
 859
 860  {\large\bf
 861   Si self-interstitial point defects in silicon\\
 862  }
 863
 864 \begin{tabular}{l c c c c c}
 865 \hline
 866  $E_{\text{f}}$ [eV] & \hkl<1 1 0> DB & H & T & \hkl<1 0 0> DB & V \\
 867 \hline
 868  VASP & \underline{3.39} & 3.42 & 3.77 & 4.41 & 3.63 \\
 869  Erhart/Albe & 4.39 & 4.48$^*$ & \underline{3.40} & 5.42 & 3.13 \\
 870 \hline
 871 \end{tabular}\\[0.2cm]
 872
 873 \begin{minipage}{4.7cm}
 874 \includegraphics[width=4.7cm]{e_kin_si_hex.ps}
 875 \end{minipage}
 876 \begin{minipage}{4.7cm}
 877 \begin{center}
 878 {\tiny nearly T $\rightarrow$ T}\\
 879 \end{center}
 880 \includegraphics[width=4.7cm]{nhex_tet.ps}
 881 \end{minipage}\\
 882
 883 \underline{Hexagonal} \hspace{2pt}
 884 \href{../video/si_self_int_hexa.avi}{$\rhd$}\\[0.1cm]
 885 \framebox{
 886 \begin{minipage}{2.7cm}
 887 $E_{\text{f}}^*=4.48\text{ eV}$\\
 888 \includegraphics[width=2.7cm]{si_pd_albe/hex_a.eps}
 889 \end{minipage}
 890 \begin{minipage}{0.4cm}
 891 \begin{center}
 892 $\Rightarrow$
 893 \end{center}
 894 \end{minipage}
 895 \begin{minipage}{2.7cm}
 896 $E_{\text{f}}=3.96\text{ eV}$\\
 897 \includegraphics[width=2.8cm]{si_pd_albe/hex.eps}
 898 \end{minipage}
 899 }
 900 \begin{minipage}{2.9cm}
 901 \begin{flushright}
 902 \underline{Vacancy}\\
 903 \includegraphics[width=3.0cm]{si_pd_albe/vac.eps}
 904 \end{flushright}
 905 \end{minipage}
 906
 907 \end{minipage}
 908 \begin{minipage}{3.5cm}
 909
 910 \begin{flushright}
 911 \underline{\hkl<1 1 0> dumbbell}\\
 912 \includegraphics[width=3.0cm]{si_pd_albe/110.eps}\\
 913 \underline{Tetrahedral}\\
 914 \includegraphics[width=3.0cm]{si_pd_albe/tet.eps}\\
 915 \underline{\hkl<1 0 0> dumbbell}\\
 916 \includegraphics[width=3.0cm]{si_pd_albe/100.eps}
 917 \end{flushright}
 918
 919 \end{minipage}
 920
 921 \end{slide}
 922
 923 \begin{slide}
 924
 925 \footnotesize
 926
 927  {\large\bf
 928   C interstitial point defects in silicon\\[-0.1cm]
 929  }
 930
 931 \begin{tabular}{l c c c c c c r}
 932 \hline
 933  $E_{\text{f}}$ & T & H & \hkl<1 0 0> DB & \hkl<1 1 0> DB & S & B & \cs{} \& \si\\
 934 \hline
 935  VASP & unstable & unstable & \underline{3.72} & 4.16 & 1.95 & 4.66 & {\color{green}4.17}\\
 936  Erhart/Albe MD & 6.09 & 9.05$^*$ & \underline{3.88} & 5.18 & {\color{red}0.75} & 5.59$^*$ & {\color{green}4.43} \\
 937 \hline
 938 \end{tabular}\\[0.1cm]
 939
 940 \framebox{
 941 \begin{minipage}{2.7cm}
 942 \underline{Hexagonal} \hspace{2pt}
 943 \href{../video/c_in_si_int_hexa.avi}{$\rhd$}\\
 944 $E_{\text{f}}^*=9.05\text{ eV}$\\
 945 \includegraphics[width=2.7cm]{c_pd_albe/hex.eps}
 946 \end{minipage}
 947 \begin{minipage}{0.4cm}
 948 \begin{center}
 949 $\Rightarrow$
 950 \end{center}
 951 \end{minipage}
 952 \begin{minipage}{2.7cm}
 953 \underline{\hkl<1 0 0>}\\
 954 $E_{\text{f}}=3.88\text{ eV}$\\
 955 \includegraphics[width=2.7cm]{c_pd_albe/100.eps}
 956 \end{minipage}
 957 }
 958 \begin{minipage}{2cm}
 959 \hfill
 960 \end{minipage}
 961 \begin{minipage}{3cm}
 962 \begin{flushright}
 963 \underline{Tetrahedral}\\
 964 \includegraphics[width=3.0cm]{c_pd_albe/tet.eps}
 965 \end{flushright}
 966 \end{minipage}
 967
 968 \framebox{
 969 \begin{minipage}{2.7cm}
 970 \underline{Bond-centered}\\
 971 $E_{\text{f}}^*=5.59\text{ eV}$\\
 972 \includegraphics[width=2.7cm]{c_pd_albe/bc.eps}
 973 \end{minipage}
 974 \begin{minipage}{0.4cm}
 975 \begin{center}
 976 $\Rightarrow$
 977 \end{center}
 978 \end{minipage}
 979 \begin{minipage}{2.7cm}
 980 \underline{\hkl<1 1 0> dumbbell}\\
 981 $E_{\text{f}}=5.18\text{ eV}$\\
 982 \includegraphics[width=2.7cm]{c_pd_albe/110.eps}
 983 \end{minipage}
 984 }
 985 \begin{minipage}{2cm}
 986 \hfill
 987 \end{minipage}
 988 \begin{minipage}{3cm}
 989 \begin{flushright}
 990 \underline{Substitutional}\\
 991 \includegraphics[width=3.0cm]{c_pd_albe/sub.eps}
 992 \end{flushright}
 993 \end{minipage}
 994
 995 \end{slide}
 996
 997 \begin{slide}
 998
 999 \footnotesize
1000
1001  {\large\bf\boldmath
1002   C \hkl<1 0 0> dumbbell interstitial configuration\\
1003  }
1004
1005 {\tiny
1006 \begin{tabular}{l c c c c c c c c}
1007 \hline
1008  Distances [nm] & $r(1C)$ & $r(2C)$ & $r(3C)$ & $r(12)$ & $r(13)$ & $r(34)$ & $r(23)$ & $r(25)$ \\
1009 \hline
1010 Erhart/Albe & 0.175 & 0.329 & 0.186 & 0.226 & 0.300 & 0.343 & 0.423 & 0.425 \\
1011 VASP & 0.174 & 0.341 & 0.182 & 0.229 & 0.286 & 0.347 & 0.422 & 0.417 \\
1012 \hline
1013 \end{tabular}\\[0.2cm]
1014 \begin{tabular}{l c c c c }
1015 \hline
1016  Angles [$^{\circ}$] & $\theta_1$ & $\theta_2$ & $\theta_3$ & $\theta_4$ \\
1017 \hline
1018 Erhart/Albe & 140.2 & 109.9 & 134.4 & 112.8 \\
1019 VASP & 130.7 & 114.4 & 146.0 & 107.0 \\
1020 \hline
1021 \end{tabular}\\[0.2cm]
1022 \begin{tabular}{l c c c}
1023 \hline
1024  Displacements [nm]& $a$ & $b$ & $|a|+|b|$ \\
1025 \hline
1026 Erhart/Albe & 0.084 & -0.091 & 0.175 \\
1027 VASP & 0.109 & -0.065 & 0.174 \\
1028 \hline
1029 \end{tabular}\\[0.6cm]
1030 }
1031
1032 \begin{minipage}{3.0cm}
1033 \begin{center}
1034 \underline{Erhart/Albe}
1035 \includegraphics[width=3.0cm]{c_pd_albe/100_cmp.eps}
1036 \end{center}
1037 \end{minipage}
1038 \begin{minipage}{3.0cm}
1039 \begin{center}
1040 \underline{VASP}
1041 \includegraphics[width=3.0cm]{c_pd_vasp/100_cmp.eps}
1042 \end{center}
1043 \end{minipage}\\
1044
1045 \begin{picture}(0,0)(-185,10)
1046 \includegraphics[width=6.8cm]{100-c-si-db_cmp.eps}
1047 \end{picture}
1048 \begin{picture}(0,0)(-280,-150)
1049 \includegraphics[width=3.3cm]{c_pd_vasp/eden.eps}
1050 \end{picture}
1051
1052 \begin{pspicture}(0,0)(0,0)
1053 \psellipse[linecolor=green](5.18,5.92)(0.5,0.3)
1054 \psellipse[linecolor=red](3.45,5.92)(1.0,0.4)
1055 \psellipse[linecolor=blue](2.7,6.92)(0.9,0.2)
1056 \psellipse[linecolor=blue](4.65,6.92)(0.9,0.2)
1057 \end{pspicture}
1058
1059 \end{slide}
1060
1061 \begin{slide}
1062
1063 \small
1064
1065 \begin{minipage}{8.5cm}
1066
1067  {\large\bf
1068   Bond-centered interstitial configuration\\[-0.1cm]
1069  }
1070
1071 \begin{minipage}{3.0cm}
1072 \includegraphics[width=2.8cm]{c_pd_vasp/bc_2333.eps}\\
1073 \end{minipage}
1074 \begin{minipage}{5.2cm}
1075 \begin{itemize}
1076  \item Linear Si-C-Si bond
1077  \item Si: one C \& 3 Si neighbours
1078  \item Spin polarized calculations
1079  \item No saddle point!\\
1080        Real local minimum!
1081 \end{itemize}
1082 \end{minipage}
1083
1084 \framebox{
1085  \tiny
1086  \begin{minipage}[t]{6.5cm}
1087   \begin{minipage}[t]{1.2cm}
1088   {\color{red}Si}\\
1089   {\tiny sp$^3$}\\[0.8cm]
1090   \underline{${\color{black}\uparrow}$}
1091   \underline{${\color{black}\uparrow}$}
1092   \underline{${\color{black}\uparrow}$}
1093   \underline{${\color{red}\uparrow}$}\\
1094   sp$^3$
1095   \end{minipage}
1096   \begin{minipage}[t]{1.4cm}
1097   \begin{center}
1098   {\color{red}M}{\color{blue}O}\\[0.8cm]
1099   \underline{${\color{blue}\uparrow}{\color{white}\downarrow}$}\\
1100   $\sigma_{\text{ab}}$\\[0.5cm]
1101   \underline{${\color{red}\uparrow}{\color{blue}\downarrow}$}\\
1102   $\sigma_{\text{b}}$
1103   \end{center}
1104   \end{minipage}
1105   \begin{minipage}[t]{1.0cm}
1106   \begin{center}
1107   {\color{blue}C}\\
1108   {\tiny sp}\\[0.2cm]
1109   \underline{${\color{white}\uparrow\uparrow}$}
1110   \underline{${\color{white}\uparrow\uparrow}$}\\
1111   2p\\[0.4cm]
1112   \underline{${\color{blue}\uparrow}{\color{blue}\downarrow}$}
1113   \underline{${\color{blue}\uparrow}{\color{blue}\downarrow}$}\\
1114   sp
1115   \end{center}
1116   \end{minipage}
1117   \begin{minipage}[t]{1.4cm}
1118   \begin{center}
1119   {\color{blue}M}{\color{green}O}\\[0.8cm]
1120   \underline{${\color{blue}\uparrow}{\color{white}\downarrow}$}\\
1121   $\sigma_{\text{ab}}$\\[0.5cm]
1122   \underline{${\color{green}\uparrow}{\color{blue}\downarrow}$}\\
1123   $\sigma_{\text{b}}$
1124   \end{center}
1125   \end{minipage}
1126   \begin{minipage}[t]{1.2cm}
1127   \begin{flushright}
1128   {\color{green}Si}\\
1129   {\tiny sp$^3$}\\[0.8cm]
1130   \underline{${\color{green}\uparrow}$}
1131   \underline{${\color{black}\uparrow}$}
1132   \underline{${\color{black}\uparrow}$}
1133   \underline{${\color{black}\uparrow}$}\\
1134   sp$^3$
1135   \end{flushright}
1136   \end{minipage}
1137  \end{minipage}
1138 }\\[0.1cm]
1139
1140 \framebox{
1141 \begin{minipage}{4.5cm}
1142 \includegraphics[width=4cm]{c_100_mig_vasp/im_spin_diff.eps}
1143 \end{minipage}
1144 \begin{minipage}{3.5cm}
1145 {\color{gray}$\bullet$} Spin up\\
1146 {\color{green}$\bullet$} Spin down\\
1147 {\color{blue}$\bullet$} Resulting spin up\\
1148 {\color{yellow}$\bullet$} Si atoms\\
1149 {\color{red}$\bullet$} C atom
1150 \end{minipage}
1151 }
1152
1153 \end{minipage}
1154 \begin{minipage}{4.2cm}
1155 \begin{flushright}
1156 \includegraphics[width=4.3cm]{c_pd_vasp/bc_2333_ksl.ps}\\
1157 {\color{green}$\Box$} {\tiny unoccupied}\\
1158 {\color{red}$\bullet$} {\tiny occupied}
1159 \end{flushright}
1160 \end{minipage}
1161
1162 \end{slide}
1163
1164 \begin{slide}
1165
1166  {\large\bf\boldmath
1167   Migration of the C \hkl<1 0 0> dumbbell interstitial
1168  }
1169
1170 \scriptsize
1171
1172  {\small Investigated pathways}
1173
1174 \begin{minipage}{8.5cm}
1175 \begin{minipage}{8.3cm}
1176 \underline{\hkl<0 0 -1> $\rightarrow$ \hkl<0 0 1>}\\
1177 \begin{minipage}{2.4cm}
1178 \includegraphics[width=2.4cm]{c_pd_vasp/100_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/bc_2333.eps}
1185 \end{minipage}
1186 \begin{minipage}{0.4cm}
1187 $\rightarrow$
1188 \end{minipage}
1189 \begin{minipage}{2.4cm}
1190 \includegraphics[width=2.4cm]{c_pd_vasp/100_next_2333.eps}
1191 \end{minipage}
1192 \end{minipage}\\
1193 \begin{minipage}{8.3cm}
1194 \underline{\hkl<0 0 -1> $\rightarrow$ \hkl<0 -1 0>}\\
1195 \begin{minipage}{2.4cm}
1196 \includegraphics[width=2.4cm]{c_pd_vasp/100_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/00-1-0-10_2333.eps}
1203 \end{minipage}
1204 \begin{minipage}{0.4cm}
1205 $\rightarrow$
1206 \end{minipage}
1207 \begin{minipage}{2.4cm}
1208 \includegraphics[width=2.4cm]{c_pd_vasp/0-10_2333.eps}
1209 \end{minipage}
1210 \end{minipage}\\
1211 \begin{minipage}{8.3cm}
1212 \underline{\hkl<0 0 -1> $\rightarrow$ \hkl<0 -1 0> (in place)}\\
1213 \begin{minipage}{2.4cm}
1214 \includegraphics[width=2.4cm]{c_pd_vasp/100_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/00-1_ip0-10_2333.eps}
1221 \end{minipage}
1222 \begin{minipage}{0.4cm}
1223 $\rightarrow$
1224 \end{minipage}
1225 \begin{minipage}{2.4cm}
1226 \includegraphics[width=2.4cm]{c_pd_vasp/0-10_ip_2333.eps}
1227 \end{minipage}
1228 \end{minipage}
1229 \end{minipage}
1230 \framebox{
1231 \begin{minipage}{4.2cm}
1232  {\small Constrained relaxation\\
1233          technique (CRT) method}\\
1234 \includegraphics[width=4cm]{crt_orig.eps}
1235 \begin{itemize}
1236  \item Constrain diffusing atom
1237  \item Static constraints
1238 \end{itemize}
1239 \vspace*{0.3cm}
1240  {\small Modifications}\\
1241 \includegraphics[width=4cm]{crt_mod.eps}
1242 \begin{itemize}
1243  \item Constrain all atoms
1244  \item Update individual\\
1245        constraints
1246 \end{itemize}
1247 \end{minipage}
1248 }
1249
1250 \end{slide}
1251
1252 \begin{slide}
1253
1254  {\large\bf\boldmath
1255   Migration of the C \hkl<1 0 0> dumbbell interstitial
1256  }
1257
1258 \scriptsize
1259
1260 \framebox{
1261 \begin{minipage}{5.9cm}
1262 \begin{flushleft}
1263 \includegraphics[width=5.8cm]{im_00-1_nosym_sp_fullct_thesis.ps}\\[0.45cm]
1264 \end{flushleft}
1265 \begin{center}
1266 \begin{picture}(0,0)(60,0)
1267 \includegraphics[width=1cm]{vasp_mig/00-1.eps}
1268 \end{picture}
1269 \begin{picture}(0,0)(-5,0)
1270 \includegraphics[width=1cm]{vasp_mig/bc_00-1_sp.eps}
1271 \end{picture}
1272 \begin{picture}(0,0)(-55,0)
1273 \includegraphics[width=1cm]{vasp_mig/bc.eps}
1274 \end{picture}
1275 \begin{picture}(0,0)(12.5,10)
1276 \includegraphics[width=1cm]{110_arrow.eps}
1277 \end{picture}
1278 \begin{picture}(0,0)(90,0)
1279 \includegraphics[height=0.9cm]{001_arrow.eps}
1280 \end{picture}
1281 \end{center}
1282 \vspace*{0.35cm}
1283 \end{minipage}
1284 }
1285 \begin{minipage}{0.3cm}
1286 \hfill
1287 \end{minipage}
1288 \framebox{
1289 \begin{minipage}{5.9cm}
1290 \begin{flushright}
1291 \includegraphics[width=5.9cm]{vasp_mig/00-1_0-10_nosym_sp_fullct.ps}\\[0.5cm]
1292 \end{flushright}
1293 \begin{center}
1294 \begin{picture}(0,0)(60,0)
1295 \includegraphics[width=1cm]{vasp_mig/00-1_a.eps}
1296 \end{picture}
1297 \begin{picture}(0,0)(5,0)
1298 \includegraphics[width=1cm]{vasp_mig/00-1_0-10_sp.eps}
1299 \end{picture}
1300 \begin{picture}(0,0)(-55,0)
1301 \includegraphics[width=1cm]{vasp_mig/0-10.eps}
1302 \end{picture}
1303 \begin{picture}(0,0)(12.5,10)
1304 \includegraphics[width=1cm]{100_arrow.eps}
1305 \end{picture}
1306 \begin{picture}(0,0)(90,0)
1307 \includegraphics[height=0.9cm]{001_arrow.eps}
1308 \end{picture}
1309 \end{center}
1310 \vspace*{0.3cm}
1311 \end{minipage}\\
1312 }
1313
1314 \vspace*{0.05cm}
1315
1316 \framebox{
1317 \begin{minipage}{5.9cm}
1318 \begin{flushleft}
1319 \includegraphics[width=5.9cm]{vasp_mig/00-1_ip0-10_nosym_sp_fullct.ps}\\[0.6cm]
1320 \end{flushleft}
1321 \begin{center}
1322 \begin{picture}(0,0)(60,0)
1323 \includegraphics[width=0.9cm]{vasp_mig/00-1_b.eps}
1324 \end{picture}
1325 \begin{picture}(0,0)(10,0)
1326 \includegraphics[width=0.9cm]{vasp_mig/00-1_ip0-10_sp.eps}
1327 \end{picture}
1328 \begin{picture}(0,0)(-60,0)
1329 \includegraphics[width=0.9cm]{vasp_mig/0-10_b.eps}
1330 \end{picture}
1331 \begin{picture}(0,0)(12.5,10)
1332 \includegraphics[width=1cm]{100_arrow.eps}
1333 \end{picture}
1334 \begin{picture}(0,0)(90,0)
1335 \includegraphics[height=0.9cm]{001_arrow.eps}
1336 \end{picture}
1337 \end{center}
1338 \vspace*{0.3cm}
1339 \end{minipage}
1340 }
1341 \begin{minipage}{0.3cm}
1342 \hfill
1343 \end{minipage}
1344 \begin{minipage}{6.5cm}
1345 VASP results
1346 \begin{itemize}
1347  \item Energetically most favorable path
1348        \begin{itemize}
1349         \item Path 2
1350         \item Activation energy: $\approx$ 0.9 eV
1351         \item Experimental values: 0.73 ... 0.87 eV
1352        \end{itemize}
1353        $\Rightarrow$ {\color{blue}Diffusion} path identified!
1354  \item Reorientation (path 3)
1355        \begin{itemize}
1356         \item More likely composed of two consecutive steps of type 2
1357         \item Experimental values: 0.77 ... 0.88 eV
1358        \end{itemize}
1359        $\Rightarrow$ {\color{blue}Reorientation} transition identified!
1360 \end{itemize}
1361 \end{minipage}
1362
1363 \end{slide}
1364
1365 \begin{slide}
1366
1367  {\large\bf\boldmath
1368   Migration of the C \hkl<1 0 0> dumbbell interstitial
1369  }
1370
1371 \scriptsize
1372
1373  \vspace{0.1cm}
1374
1375 \begin{minipage}{6.5cm}
1376
1377 \framebox{
1378 \begin{minipage}[t]{5.9cm}
1379 \begin{flushleft}
1380 \includegraphics[width=5.9cm]{bc_00-1.ps}\\[2.35cm]
1381 \end{flushleft}
1382 \begin{center}
1383 \begin{pspicture}(0,0)(0,0)
1384 \psframe[linecolor=red,fillstyle=none](-2.8,1.35)(3.3,2.7)
1385 \end{pspicture}
1386 \begin{picture}(0,0)(60,-50)
1387 \includegraphics[width=1cm]{albe_mig/bc_00-1_red_00.eps}
1388 \end{picture}
1389 \begin{picture}(0,0)(5,-50)
1390 \includegraphics[width=1cm]{albe_mig/bc_00-1_red_01.eps}
1391 \end{picture}
1392 \begin{picture}(0,0)(-55,-50)
1393 \includegraphics[width=1cm]{albe_mig/bc_00-1_red_02.eps}
1394 \end{picture}
1395 \begin{picture}(0,0)(12.5,-40)
1396 \includegraphics[width=1cm]{110_arrow.eps}
1397 \end{picture}
1398 \begin{picture}(0,0)(90,-45)
1399 \includegraphics[height=0.9cm]{001_arrow.eps}
1400 \end{picture}\\
1401 \begin{pspicture}(0,0)(0,0)
1402 \psframe[linecolor=blue,fillstyle=none](-2.8,0)(3.3,1.6)
1403 \end{pspicture}
1404 \begin{picture}(0,0)(60,-15)
1405 \includegraphics[width=0.9cm]{albe_mig/bc_00-1_01.eps}
1406 \end{picture}
1407 \begin{picture}(0,0)(35,-15)
1408 \includegraphics[width=0.9cm]{albe_mig/bc_00-1_02.eps}
1409 \end{picture}
1410 \begin{picture}(0,0)(-5,-15)
1411 \includegraphics[width=0.9cm]{albe_mig/bc_00-1_03.eps}
1412 \end{picture}
1413 \begin{picture}(0,0)(-55,-15)
1414 \includegraphics[width=0.9cm]{albe_mig/bc_00-1_04.eps}
1415 \end{picture}
1416 \begin{picture}(0,0)(12.5,-5)
1417 \includegraphics[width=1cm]{100_arrow.eps}
1418 \end{picture}
1419 \begin{picture}(0,0)(90,-15)
1420 \includegraphics[height=0.9cm]{010_arrow.eps}
1421 \end{picture}
1422 \end{center}
1423 \end{minipage}
1424 }\\[0.1cm]
1425
1426 \begin{minipage}{5.9cm}
1427 Erhart/Albe results
1428 \begin{itemize}
1429  \item Lowest activation energy: $\approx$ 2.2 eV
1430  \item 2.4 times higher than VASP
1431  \item Different pathway
1432 \end{itemize}
1433 \end{minipage}
1434
1435 \end{minipage}
1436 \begin{minipage}{6.5cm}
1437
1438 \framebox{
1439 \begin{minipage}{5.9cm}
1440 %\begin{flushright}
1441 %\includegraphics[width=5.9cm]{00-1_0-10.ps}\\[0.75cm]
1442 %\end{flushright}
1443 %\begin{center}
1444 %\begin{pspicture}(0,0)(0,0)
1445 %\psframe[linecolor=red,fillstyle=none](-2.8,-0.25)(3.3,1.1)
1446 %\end{pspicture}
1447 %\begin{picture}(0,0)(60,-5)
1448 %\includegraphics[width=0.9cm]{albe_mig/00-1_0-10_red_00.eps}
1449 %\end{picture}
1450 %\begin{picture}(0,0)(0,-5)
1451 %\includegraphics[width=0.9cm]{albe_mig/00-1_0-10_red_min.eps}
1452 %\end{picture}
1453 %\begin{picture}(0,0)(-55,-5)
1454 %\includegraphics[width=0.9cm]{albe_mig/00-1_0-10_red_03.eps}
1455 %\end{picture}
1456 %\begin{picture}(0,0)(12.5,5)
1457 %\includegraphics[width=1cm]{100_arrow.eps}
1458 %\end{picture}
1459 %\begin{picture}(0,0)(90,0)
1460 %\includegraphics[height=0.9cm]{001_arrow.eps}
1461 %\end{picture}
1462 %\end{center}
1463 %\vspace{0.2cm}
1464 %\end{minipage}
1465 %}\\[0.2cm]
1466 %
1467 %\framebox{
1468 %\begin{minipage}{5.9cm}
1469 \includegraphics[width=5.9cm]{00-1_110_0-10_mig_albe.ps}
1470 \end{minipage}
1471 }\\[0.1cm]
1472
1473 \begin{minipage}{5.9cm}
1474 Transition involving \ci{} \hkl<1 1 0>
1475 \begin{itemize}
1476  \item Bond-centered configuration unstable\\
1477        $\rightarrow$ \ci{} \hkl<1 1 0> dumbbell
1478  \item Transition minima of path 2 \& 3\\
1479        $\rightarrow$ \ci{} \hkl<1 1 0> dumbbell
1480  \item Activation energy: $\approx$ 2.2 eV \& 0.9 eV
1481  \item 2.4 - 3.4 times higher than VASP
1482  \item Rotation of dumbbell orientation
1483 \end{itemize}
1484 \vspace{0.1cm}
1485 \begin{center}
1486 {\color{blue}Overestimated diffusion barrier}
1487 \end{center}
1488 \end{minipage}
1489
1490 \end{minipage}
1491
1492 \end{slide}
1493
1494 \begin{slide}
1495
1496  {\large\bf\boldmath
1497   Combinations with a C-Si \hkl<1 0 0>-type interstitial
1498  }
1499
1500 \small
1501
1502 \vspace*{0.1cm}
1503
1504 Binding energy:
1505 $
1506 E_{\text{b}}=
1507 E_{\text{f}}^{\text{defect combination}}-
1508 E_{\text{f}}^{\text{C \hkl<0 0 -1> dumbbell}}-
1509 E_{\text{f}}^{\text{2nd defect}}
1510 $
1511
1512 \vspace*{0.1cm}
1513
1514 {\scriptsize
1515 \begin{tabular}{l c c c c c c}
1516 \hline
1517  $E_{\text{b}}$ [eV] & 1 & 2 & 3 & 4 & 5 & R\\
1518  \hline
1519  \hkl<0 0 -1> & {\color{red}-0.08} & -1.15 & {\color{red}-0.08} & 0.04 & -1.66 & -0.19\\
1520  \hkl<0 0 1> & 0.34 & 0.004 & -2.05 & 0.26 & -1.53 & -0.19\\
1521  \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}\\
1522  \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}\\
1523  \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}\\
1524  \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}\\
1525  \hline
1526  C substitutional (C$_{\text{S}}$) & 0.26 & -0.51 & -0.93 & -0.15 & 0.49 & -0.05\\
1527  Vacancy & -5.39 ($\rightarrow$ C$_{\text{S}}$) & -0.59 & -3.14 & -0.54 & -0.50 & -0.31\\
1528 \hline
1529 \end{tabular}
1530 }
1531
1532 \vspace*{0.3cm}
1533
1534 \footnotesize
1535
1536 \begin{minipage}[t]{3.8cm}
1537 \underline{\hkl<1 0 0> at position 1}\\[0.1cm]
1538 \includegraphics[width=3.5cm]{00-1dc/2-25.eps}
1539 \end{minipage}
1540 \begin{minipage}[t]{3.5cm}
1541 \underline{\hkl<0 -1 0> at position 1}\\[0.1cm]
1542 \includegraphics[width=3.2cm]{00-1dc/2-39.eps}
1543 \end{minipage}
1544 \begin{minipage}[t]{5.5cm}
1545 \begin{itemize}
1546  \item $E_{\text{b}}=0$ $\Leftrightarrow$ non-interacting defects\\
1547        $E_{\text{b}} \rightarrow 0$ for increasing distance (R)
1548  \item Stress compensation / increase
1549  \item Unfavored: antiparallel orientations
1550  \item Indication of energetically favored\\
1551        agglomeration
1552  \item Most favorable: C clustering
1553  \item However: High barrier ($>4\,\text{eV}$)
1554  \item $4\times{\color{cyan}-2.25}$ versus $2\times{\color{orange}-2.39}$
1555        (Entropy)
1556 \end{itemize}
1557 \end{minipage}
1558
1559 \begin{picture}(0,0)(-295,-130)
1560 \includegraphics[width=3.5cm]{comb_pos.eps}
1561 \end{picture}
1562
1563 \end{slide}
1564
1565 \begin{slide}
1566
1567  {\large\bf\boldmath
1568   Combinations of C-Si \hkl<1 0 0>-type interstitials
1569  }
1570
1571 \small
1572
1573 \vspace*{0.1cm}
1574
1575 Energetically most favorable combinations along \hkl<1 1 0>
1576
1577 \vspace*{0.1cm}
1578
1579 {\scriptsize
1580 \begin{tabular}{l c c c c c c}
1581 \hline
1582  & 1 & 2 & 3 & 4 & 5 & 6\\
1583 \hline
1584 $E_{\text{b}}$ [eV] & -2.39 & -1.88 & -0.59 & -0.31 & -0.24 & -0.21 \\
1585 C-C distance [\AA] & 1.4 & 4.6 & 6.5 & 8.6 & 10.5 & 10.8 \\
1586 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>\\
1587 \hline
1588 \end{tabular}
1589 }
1590
1591 \vspace*{0.3cm}
1592
1593 \begin{minipage}{7.0cm}
1594 \includegraphics[width=7cm]{db_along_110_cc.ps}
1595 \end{minipage}
1596 \begin{minipage}{6.0cm}
1597 \begin{itemize}
1598  \item Interaction proportional to reciprocal cube of C-C distance
1599  \item Saturation in the immediate vicinity
1600  \renewcommand\labelitemi{$\Rightarrow$}
1601  \item Agglomeration of \ci{} expected
1602  \item Absence of C clustering
1603 \end{itemize}
1604 \begin{center}
1605 {\color{blue}
1606  Consisten with initial precipitation model
1607 }
1608 \end{center}
1609 \end{minipage}
1610
1611 \vspace{0.2cm}
1612
1613 \end{slide}
1614
1615 \begin{slide}
1616
1617  {\large\bf\boldmath
1618   Combinations of substitutional C and \hkl<1 1 0> Si self-interstitials
1619  }
1620
1621  \scriptsize
1622
1623 %\begin{center}
1624 %\begin{minipage}{3.2cm}
1625 %\includegraphics[width=3cm]{sub_110_combo.eps}
1626 %\end{minipage}
1627 %\begin{minipage}{7.8cm}
1628 %\begin{tabular}{l c c c c c c}
1629 %\hline
1630 %C$_{\text{sub}}$ & \hkl<1 1 0> & \hkl<-1 1 0> & \hkl<0 1 1> & \hkl<0 -1 1> &
1631 %                   \hkl<1 0 1> & \hkl<-1 0 1> \\
1632 %\hline
1633 %1 & \RM{1} & \RM{3} & \RM{3} & \RM{1} & \RM{3} & \RM{1} \\
1634 %2 & \RM{2} & A & A & \RM{2} & C & \RM{5} \\
1635 %3 & \RM{3} & \RM{1} & \RM{3} & \RM{1} & \RM{1} & \RM{3} \\
1636 %4 & \RM{4} & B & D & E & E & D \\
1637 %5 & \RM{5} & C & A & \RM{2} & A & \RM{2} \\
1638 %\hline
1639 %\end{tabular}
1640 %\end{minipage}
1641 %\end{center}
1642
1643 %\begin{center}
1644 %\begin{tabular}{l c c c c c c c c c c}
1645 %\hline
1646 %Conf & \RM{1} & \RM{2} & \RM{3} & \RM{4} & \RM{5} & A & B & C & D & E \\
1647 %\hline
1648 %$E_{\text{f}}$ [eV]& 4.37 & 5.26 & 5.57 & 5.37 & 5.12 & 5.10 & 5.32 & 5.28 & 5.39 & 5.32 \\
1649 %$E_{\text{b}}$ [eV] & -0.97 & -0.08 & 0.22 & -0.02 & -0.23 & -0.25 & -0.02 & -0.06 & 0.05 & -0.03 \\
1650 %$r$ [nm] & 0.292 & 0.394 & 0.241 & 0.453 & 0.407 & 0.408 & 0.452 & 0.392 & 0.456 & 0.453\\
1651 %\hline
1652 %\end{tabular}
1653 %\end{center}
1654
1655 \begin{minipage}{6.0cm}
1656 \includegraphics[width=5.8cm]{c_sub_si110.ps}
1657 \end{minipage}
1658 \begin{minipage}{7cm}
1659 \scriptsize
1660 \begin{itemize}
1661  \item IBS: C may displace Si\\
1662        $\Rightarrow$ C$_{\text{sub}}$ + \hkl<1 1 0> Si self-interstitial
1663  \item Assumption:\\
1664        \hkl<1 1 0>-type $\rightarrow$ favored combination
1665  \renewcommand\labelitemi{$\Rightarrow$}
1666  \item Most favorable: \cs{} along \hkl<1 1 0> chain \si{}
1667  \item Less favorable than C-Si \hkl<1 0 0> dumbbell
1668  \item Interaction drops quickly to zero\\
1669        $\rightarrow$ low capture radius
1670 \end{itemize}
1671 \begin{center}
1672  {\color{blue}
1673  IBS process far from equilibrium\\
1674  \cs{} \& \si{} instead of thermodynamic ground state
1675  }
1676 \end{center}
1677 \end{minipage}
1678
1679 \begin{minipage}{6.5cm}
1680 \includegraphics[width=6.0cm]{162-097.ps}
1681 \begin{itemize}
1682  \item Low migration barrier
1683 \end{itemize}
1684 \end{minipage}
1685 \begin{minipage}{6.5cm}
1686 \begin{center}
1687 Ab initio MD at \degc{900}\\
1688 \includegraphics[width=3.3cm]{md_vasp_01.eps}
1689 $t=\unit[2230]{fs}$\\
1690 \includegraphics[width=3.3cm]{md_vasp_02.eps}
1691 $t=\unit[2900]{fs}$
1692 \end{center}
1693 {\color{blue}
1694 Contribution of entropy to structural formation
1695 }
1696 \end{minipage}
1697
1698 \end{slide}
1699
1700 \begin{slide}
1701
1702  {\large\bf\boldmath
1703   Migration in C-Si \hkl<1 0 0> and vacancy combinations
1704  }
1705
1706  \footnotesize
1707
1708 \vspace{0.1cm}
1709
1710 \begin{minipage}[t]{3cm}
1711 \underline{Pos 2, $E_{\text{b}}=-0.59\text{ eV}$}\\
1712 \includegraphics[width=2.8cm]{00-1dc/0-59.eps}
1713 \end{minipage}
1714 \begin{minipage}[t]{7cm}
1715 \vspace{0.2cm}
1716 \begin{center}
1717  Low activation energies\\
1718  High activation energies for reverse processes\\
1719  $\Downarrow$\\
1720  {\color{blue}C$_{\text{sub}}$ very stable}\\
1721 \vspace*{0.1cm}
1722  \hrule
1723 \vspace*{0.1cm}
1724  Without nearby \hkl<1 1 0> Si self-interstitial (IBS)\\
1725  $\Downarrow$\\
1726  {\color{blue}Formation of SiC by successive substitution by C}
1727
1728 \end{center}
1729 \end{minipage}
1730 \begin{minipage}[t]{3cm}
1731 \underline{Pos 3, $E_{\text{b}}=-3.14\text{ eV}$}\\
1732 \includegraphics[width=2.8cm]{00-1dc/3-14.eps}
1733 \end{minipage}
1734
1735
1736 \framebox{
1737 \begin{minipage}{5.9cm}
1738 \includegraphics[width=5.9cm]{vasp_mig/comb_mig_3-2_vac_fullct.ps}\\[0.6cm]
1739 \begin{center}
1740 \begin{picture}(0,0)(70,0)
1741 \includegraphics[width=1.4cm]{vasp_mig/comb_2-1_init.eps}
1742 \end{picture}
1743 \begin{picture}(0,0)(30,0)
1744 \includegraphics[width=1.4cm]{vasp_mig/comb_2-1_seq_03.eps}
1745 \end{picture}
1746 \begin{picture}(0,0)(-10,0)
1747 \includegraphics[width=1.4cm]{vasp_mig/comb_2-1_seq_06.eps}
1748 \end{picture}
1749 \begin{picture}(0,0)(-48,0)
1750 \includegraphics[width=1.4cm]{vasp_mig/comb_2-1_final.eps}
1751 \end{picture}
1752 \begin{picture}(0,0)(12.5,5)
1753 \includegraphics[width=1cm]{100_arrow.eps}
1754 \end{picture}
1755 \begin{picture}(0,0)(97,-10)
1756 \includegraphics[height=0.9cm]{001_arrow.eps}
1757 \end{picture}
1758 \end{center}
1759 \vspace{0.1cm}
1760 \end{minipage}
1761 }
1762 \begin{minipage}{0.3cm}
1763 \hfill
1764 \end{minipage}
1765 \framebox{
1766 \begin{minipage}{5.9cm}
1767 \includegraphics[width=5.9cm]{vasp_mig/comb_mig_4-2_vac_fullct.ps}\\[0.1cm]
1768 \begin{center}
1769 \begin{picture}(0,0)(60,0)
1770 \includegraphics[width=0.9cm]{vasp_mig/comb_3-1_init.eps}
1771 \end{picture}
1772 \begin{picture}(0,0)(25,0)
1773 \includegraphics[width=0.9cm]{vasp_mig/comb_3-1_seq_03.eps}
1774 \end{picture}
1775 \begin{picture}(0,0)(-20,0)
1776 \includegraphics[width=0.9cm]{vasp_mig/comb_3-1_seq_07.eps}
1777 \end{picture}
1778 \begin{picture}(0,0)(-55,0)
1779 \includegraphics[width=0.9cm]{vasp_mig/comb_3-1_final.eps}
1780 \end{picture}
1781 \begin{picture}(0,0)(12.5,5)
1782 \includegraphics[width=1cm]{100_arrow.eps}
1783 \end{picture}
1784 \begin{picture}(0,0)(95,0)
1785 \includegraphics[height=0.9cm]{001_arrow.eps}
1786 \end{picture}
1787 \end{center}
1788 \vspace{0.1cm}
1789 \end{minipage}
1790 }
1791
1792 \end{slide}
1793
1794 \begin{slide}
1795
1796  {\large\bf
1797   Conclusion of defect / migration / combined defect simulations
1798  }
1799
1800  \footnotesize
1801
1802 \vspace*{0.1cm}
1803
1804 Defect structures
1805 \begin{itemize}
1806  \item Accurately described by quantum-mechanical simulations
1807  \item Less accurate description by classical potential simulations
1808  \item Underestimated formation energy of \cs{} by classical approach
1809  \item Both methods predict same ground state: \ci{} \hkl<1 0 0> dumbbell
1810 \end{itemize}
1811
1812 Migration
1813 \begin{itemize}
1814  \item C migration pathway in Si identified
1815  \item Consistent with reorientation and diffusion experiments
1816 \end{itemize}
1817 \begin{itemize}
1818  \item Different path and ...
1819  \item overestimated barrier by classical potential calculations
1820 \end{itemize}
1821
1822 Concerning the precipitation mechanism
1823 \begin{itemize}
1824  \item Agglomeration of C-Si dumbbells energetically favorable
1825        (stress compensation)
1826  \item C-Si indeed favored compared to
1827        C$_{\text{sub}}$ \& \hkl<1 1 0> Si self-interstitial
1828  \item Possible low interaction capture radius of
1829        C$_{\text{sub}}$ \& \hkl<1 1 0> Si self-interstitial
1830  \item Low barrier for
1831        \ci{} \hkl<1 0 0> $\rightarrow$ \cs{} \& \si{} \hkl<1 1 0>
1832  \item In absence of nearby \hkl<1 1 0> Si self-interstitial:
1833        C-Si \hkl<1 0 0> + Vacancy $\rightarrow$ C$_{\text{sub}}$ (SiC)
1834 \end{itemize}
1835 \begin{center}
1836 {\color{blue}Results suggest increased participation of \cs}
1837 \end{center}
1838
1839 \end{slide}
1840
1841 \begin{slide}
1842
1843  {\large\bf
1844   Silicon carbide precipitation simulations
1845  }
1846
1847  \small
1848
1849 {\scriptsize
1850  \begin{pspicture}(0,0)(12,6.5)
1851   % nodes
1852   \rput(3.5,5.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
1853    \parbox{7cm}{
1854    \begin{itemize}
1855     \item Create c-Si volume
1856     \item Periodc boundary conditions
1857     \item Set requested $T$ and $p=0\text{ bar}$
1858     \item Equilibration of $E_{\text{kin}}$ and $E_{\text{pot}}$
1859    \end{itemize}
1860   }}}}
1861   \rput(3.5,2.7){\rnode{insert}{\psframebox[fillstyle=solid,fillcolor=lachs]{
1862    \parbox{7cm}{
1863    Insertion of C atoms at constant T
1864    \begin{itemize}
1865     \item total simulation volume {\pnode{in1}}
1866     \item volume of minimal SiC precipitate {\pnode{in2}}
1867     \item volume consisting of Si atoms to form a minimal {\pnode{in3}}\\
1868           precipitate
1869    \end{itemize}
1870   }}}}
1871   \rput(3.5,1){\rnode{cool}{\psframebox[fillstyle=solid,fillcolor=lbb]{
1872    \parbox{7.0cm}{
1873    Run for 100 ps followed by cooling down to $20\, ^{\circ}\textrm{C}$
1874   }}}}
1875   \ncline[]{->}{init}{insert}
1876   \ncline[]{->}{insert}{cool}
1877   \psframe[fillstyle=solid,fillcolor=white](7.5,0.7)(13.5,6.3)
1878   \rput(7.8,6){\footnotesize $V_1$}
1879   \psframe[fillstyle=solid,fillcolor=lightgray](9,2)(12,5)
1880   \rput(9.2,4.85){\tiny $V_2$}
1881   \psframe[fillstyle=solid,fillcolor=gray](9.25,2.25)(11.75,4.75)
1882   \rput(9.55,4.45){\footnotesize $V_3$}
1883   \rput(7.9,3.2){\pnode{ins1}}
1884   \rput(9.22,2.8){\pnode{ins2}}
1885   \rput(11.0,2.4){\pnode{ins3}}
1886   \ncline[]{->}{in1}{ins1}
1887   \ncline[]{->}{in2}{ins2}
1888   \ncline[]{->}{in3}{ins3}
1889  \end{pspicture}
1890 }
1891
1892 \begin{itemize}
1893  \item Restricted to classical potential simulations
1894  \item $V_2$ and $V_3$ considered due to low diffusion
1895  \item Amount of C atoms: 6000
1896        ($r_{\text{prec}}\approx 3.1\text{ nm}$, IBS: 2 ... 4 nm)
1897  \item Simulation volume: $31\times 31\times 31$ unit cells
1898        (238328 Si atoms)
1899 \end{itemize}
1900
1901 \end{slide}
1902
1903 \begin{slide}
1904
1905  {\large\bf\boldmath
1906   Silicon carbide precipitation simulations at $450\,^{\circ}\mathrm{C}$ as in IBS
1907  }
1908
1909  \small
1910
1911 \begin{minipage}{6.5cm}
1912 \includegraphics[width=6.4cm]{sic_prec_450_si-si_c-c.ps}
1913 \end{minipage}
1914 \begin{minipage}{6.5cm}
1915 \includegraphics[width=6.4cm]{sic_prec_450_energy.ps}
1916 \end{minipage}
1917
1918 \begin{minipage}{6.5cm}
1919 \includegraphics[width=6.4cm]{sic_prec_450_si-c.ps}
1920 \end{minipage}
1921 \begin{minipage}{6.5cm}
1922 \scriptsize
1923 \underline{Low C concentration ($V_1$)}\\
1924 \hkl<1 0 0> C-Si dumbbell dominated structure
1925 \begin{itemize}
1926  \item Si-C bumbs around 0.19 nm
1927  \item C-C peak at 0.31 nm (as expected in 3C-SiC):\\
1928        concatenated dumbbells of various orientation
1929  \item Si-Si NN distance stretched to 0.3 nm
1930 \end{itemize}
1931 {\color{blue}$\Rightarrow$ C atoms in proper 3C-SiC distance first}\\
1932 \underline{High C concentration ($V_2$, $V_3$)}\\
1933 High amount of strongly bound C-C bonds\\
1934 Defect density $\uparrow$ $\Rightarrow$ considerable amount of damage\\
1935 Only short range order observable\\
1936 {\color{blue}$\Rightarrow$ amorphous SiC-like phase}
1937 \end{minipage}
1938
1939 \end{slide}
1940
1941 \begin{slide}
1942
1943  {\large\bf\boldmath
1944   Silicon carbide precipitation simulations at $450\,^{\circ}\mathrm{C}$ as in IBS
1945  }
1946
1947  \small
1948
1949 \begin{minipage}{6.5cm}
1950 \includegraphics[width=6.4cm]{sic_prec_450_si-si_c-c.ps}
1951 \end{minipage}
1952 \begin{minipage}{6.5cm}
1953 \includegraphics[width=6.4cm]{sic_prec_450_energy.ps}
1954 \end{minipage}
1955
1956 \begin{minipage}{6.5cm}
1957 \includegraphics[width=6.4cm]{sic_prec_450_si-c.ps}
1958 \end{minipage}
1959 \begin{minipage}{6.5cm}
1960 \scriptsize
1961 \underline{Low C concentration ($V_1$)}\\
1962 \hkl<1 0 0> C-Si dumbbell dominated structure
1963 \begin{itemize}
1964  \item Si-C bumbs around 0.19 nm
1965  \item C-C peak at 0.31 nm (as expected in 3C-SiC):\\
1966        concatenated dumbbells of various orientation
1967  \item Si-Si NN distance stretched to 0.3 nm
1968 \end{itemize}
1969 {\color{blue}$\Rightarrow$ C atoms in proper 3C-SiC distance first}\\
1970 \underline{High C concentration ($V_2$, $V_3$)}\\
1971 High amount of strongly bound C-C bonds\\
1972 Defect density $\uparrow$ $\Rightarrow$ considerable amount of damage\\
1973 Only short range order observable\\
1974 {\color{blue}$\Rightarrow$ amorphous SiC-like phase}
1975 \end{minipage}
1976
1977 \begin{pspicture}(0,0)(0,0)
1978 \rput(6.7,5.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=white]{
1979 \begin{minipage}{10cm}
1980 \small
1981 {\color{red}\bf 3C-SiC formation fails to appear}
1982 \begin{itemize}
1983 \item Low C concentration simulations
1984  \begin{itemize}
1985   \item Formation of \ci{} indeed occurs
1986   \item Agllomeration not observed
1987  \end{itemize}
1988 \item High C concentration simulations
1989  \begin{itemize}
1990   \item Amorphous SiC-like structure\\
1991         (not expected at prevailing temperatures)
1992   \item Rearrangement and transition into 3C-SiC structure missing
1993  \end{itemize}
1994 \end{itemize}
1995 \end{minipage}
1996  }}}
1997 \end{pspicture}
1998
1999 \end{slide}
2000
2001 \begin{slide}
2002
2003  {\large\bf
2004   Limitations of molecular dynamics and short range potentials
2005  }
2006
2007 \footnotesize
2008
2009 \vspace{0.2cm}
2010
2011 \underline{Time scale problem of MD}\\[0.2cm]
2012 Minimize integration error\\
2013 $\Rightarrow$ discretization considerably smaller than
2014               reciprocal of fastest vibrational mode\\[0.1cm]
2015 Order of fastest vibrational mode: $10^{13} - 10^{14}\text{ Hz}$\\
2016 $\Rightarrow$ suitable choice of time step:
2017               $\tau=1\text{ fs}=10^{-15}\text{ s}$\\
2018 $\Rightarrow$ {\color{red}\underline{slow}} phase space propagation\\[0.1cm]
2019 Several local minima in energy surface separated by large energy barriers\\
2020 $\Rightarrow$ transition event corresponds to a multiple
2021               of vibrational periods\\
2022 $\Rightarrow$ phase transition made up of {\color{red}\underline{many}}
2023               infrequent transition events\\[0.1cm]
2024 {\color{blue}Accelerated methods:}
2025 \underline{Temperature accelerated} MD (TAD), self-guided MD \ldots
2026
2027 \vspace{0.3cm}
2028
2029 \underline{Limitations related to the short range potential}\\[0.2cm]
2030 Cut-off function pushing forces and energies to zero between 1$^{\text{st}}$
2031 and 2$^{\text{nd}}$ next neighbours\\
2032 $\Rightarrow$ overestimated unphysical high forces of next neighbours
2033
2034 \vspace{0.3cm}
2035
2036 \framebox{
2037 \color{red}
2038 Potential enhanced problem of slow phase space propagation
2039 }
2040
2041 \vspace{0.3cm}
2042
2043 \underline{Approach to the (twofold) problem}\\[0.2cm]
2044 Increased temperature simulations without TAD corrections\\
2045 (accelerated methods or higher time scales exclusively not sufficient)
2046
2047 \begin{picture}(0,0)(-260,-30)
2048 \framebox{
2049 \begin{minipage}{4.2cm}
2050 \tiny
2051 \begin{center}
2052 \vspace{0.03cm}
2053 \underline{IBS}
2054 \end{center}
2055 \begin{itemize}
2056 \item 3C-SiC also observed for higher T
2057 \item higher T inside sample
2058 \item structural evolution vs.\\
2059       equilibrium properties
2060 \end{itemize}
2061 \end{minipage}
2062 }
2063 \end{picture}
2064
2065 \begin{picture}(0,0)(-305,-155)
2066 \framebox{
2067 \begin{minipage}{2.5cm}
2068 \tiny
2069 \begin{center}
2070 retain proper\\
2071 thermodynmic sampling
2072 \end{center}
2073 \end{minipage}
2074 }
2075 \end{picture}
2076
2077 \end{slide}
2078
2079 \begin{slide}
2080
2081  {\large\bf
2082   Increased temperature simulations at low C concentration
2083  }
2084
2085 \small
2086
2087 \begin{minipage}{6.5cm}
2088 \includegraphics[width=6.4cm]{tot_pc_thesis.ps}
2089 \end{minipage}
2090 \begin{minipage}{6.5cm}
2091 \includegraphics[width=6.4cm]{tot_pc3_thesis.ps}
2092 \end{minipage}
2093
2094 \begin{minipage}{6.5cm}
2095 \includegraphics[width=6.4cm]{tot_pc2_thesis.ps}
2096 \end{minipage}
2097 \begin{minipage}{6.5cm}
2098 \scriptsize
2099  \underline{Si-C bonds:}
2100  \begin{itemize}
2101   \item Vanishing cut-off artifact (above $1650\,^{\circ}\mathrm{C}$)
2102   \item Structural change: C-Si \hkl<1 0 0> $\rightarrow$ C$_{\text{sub}}$
2103  \end{itemize}
2104  \underline{Si-Si bonds:}
2105  {\color{blue}Si-C$_{\text{sub}}$-Si} along \hkl<1 1 0>
2106  ($\rightarrow$ 0.325 nm)\\[0.1cm]
2107  \underline{C-C bonds:}
2108  \begin{itemize}
2109   \item C-C next neighbour pairs reduced (mandatory)
2110   \item Peak at 0.3 nm slightly shifted
2111         \begin{itemize}
2112          \item C-Si \hkl<1 0 0> combinations (dashed arrows)\\
2113                $\rightarrow$ C-Si \hkl<1 0 0> \& C$_{\text{sub}}$
2114                combinations (|)\\
2115                $\rightarrow$ pure {\color{blue}C$_{\text{sub}}$ combinations}
2116                ($\downarrow$)
2117          \item Range [|-$\downarrow$]:
2118                {\color{blue}C$_{\text{sub}}$ \& C$_{\text{sub}}$
2119                with nearby Si$_{\text{I}}$}
2120         \end{itemize}
2121  \end{itemize}
2122 \end{minipage}
2123
2124 \begin{picture}(0,0)(-330,-74)
2125 \color{blue}
2126 \framebox{
2127 \begin{minipage}{1.6cm}
2128 \tiny
2129 \begin{center}
2130 stretched SiC\\[-0.1cm]
2131 in c-Si
2132 \end{center}
2133 \end{minipage}
2134 }
2135 \end{picture}
2136
2137 \end{slide}
2138
2139 \begin{slide}
2140
2141  {\large\bf
2142   Increased temperature simulations at low C concentration
2143  }
2144
2145 \small
2146
2147 \begin{minipage}{6.5cm}
2148 \includegraphics[width=6.4cm]{tot_pc_thesis.ps}
2149 \end{minipage}
2150 \begin{minipage}{6.5cm}
2151 \includegraphics[width=6.4cm]{tot_pc3_thesis.ps}
2152 \end{minipage}
2153
2154 \begin{minipage}{6.5cm}
2155 \includegraphics[width=6.4cm]{tot_pc2_thesis.ps}
2156 \end{minipage}
2157 \begin{minipage}{6.5cm}
2158 \scriptsize
2159  \underline{Si-C bonds:}
2160  \begin{itemize}
2161   \item Vanishing cut-off artifact (above $1650\,^{\circ}\mathrm{C}$)
2162   \item Structural change: C-Si \hkl<1 0 0> $\rightarrow$ C$_{\text{sub}}$
2163  \end{itemize}
2164  \underline{Si-Si bonds:}
2165  {\color{blue}Si-C$_{\text{sub}}$-Si} along \hkl<1 1 0>
2166  ($\rightarrow$ 0.325 nm)\\[0.1cm]
2167  \underline{C-C bonds:}
2168  \begin{itemize}
2169   \item C-C next neighbour pairs reduced (mandatory)
2170   \item Peak at 0.3 nm slightly shifted
2171         \begin{itemize}
2172          \item C-Si \hkl<1 0 0> combinations (dashed arrows)\\
2173                $\rightarrow$ C-Si \hkl<1 0 0> \& C$_{\text{sub}}$
2174                combinations (|)\\
2175                $\rightarrow$ pure {\color{blue}C$_{\text{sub}}$ combinations}
2176                ($\downarrow$)
2177          \item Range [|-$\downarrow$]:
2178                {\color{blue}C$_{\text{sub}}$ \& C$_{\text{sub}}$
2179                with nearby Si$_{\text{I}}$}
2180         \end{itemize}
2181  \end{itemize}
2182 \end{minipage}
2183
2184 %\begin{picture}(0,0)(-330,-74)
2185 %\color{blue}
2186 %\framebox{
2187 %\begin{minipage}{1.6cm}
2188 %\tiny
2189 %\begin{center}
2190 %stretched SiC\\[-0.1cm]
2191 %in c-Si
2192 %\end{center}
2193 %\end{minipage}
2194 %}
2195 %\end{picture}
2196
2197 \begin{pspicture}(0,0)(0,0)
2198 \rput(6.7,5.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=white]{
2199 \begin{minipage}{10cm}
2200 \small
2201 {\color{blue}\bf Stretched SiC in c-Si}
2202 \begin{itemize}
2203 \item Consistent to precipitation model involving \cs{}
2204 \item Explains annealing behavior of high/low T C implants
2205       \begin{itemize}
2206        \item Low T: highly mobiel \ci{}
2207        \item High T: stable configurations of \cs{}
2208       \end{itemize}
2209 \end{itemize}
2210 $\Rightarrow$ High T $\leftrightarrow$ IBS conditions far from equilibrium\\
2211 $\Rightarrow$ Precipitation mechanism involving \cs{}
2212 \end{minipage}
2213  }}}
2214 \end{pspicture}
2215
2216 \end{slide}
2217
2218 \begin{slide}
2219
2220  {\large\bf
2221   Increased temperature simulations at high C concentration
2222  }
2223
2224 \footnotesize
2225
2226 \begin{minipage}{6.5cm}
2227 \includegraphics[width=6.4cm]{12_pc_thesis.ps}
2228 \end{minipage}
2229 \begin{minipage}{6.5cm}
2230 \includegraphics[width=6.4cm]{12_pc_c_thesis.ps}
2231 \end{minipage}
2232
2233 \vspace{0.1cm}
2234
2235 \scriptsize
2236
2237 \framebox{
2238 \begin{minipage}[t]{6.0cm}
2239 0.186 nm: Si-C pairs $\uparrow$\\
2240 (as expected in 3C-SiC)\\[0.2cm]
2241 0.282 nm: Si-C-C\\[0.2cm]
2242 $\approx$0.35 nm: C-Si-Si
2243 \end{minipage}
2244 }
2245 \begin{minipage}{0.2cm}
2246 \hfill
2247 \end{minipage}
2248 \framebox{
2249 \begin{minipage}[t]{6.0cm}
2250 0.15 nm: C-C pairs $\uparrow$\\
2251 (as expected in graphite/diamond)\\[0.2cm]
2252 0.252 nm: C-C-C (2$^{\text{nd}}$ NN for diamond)\\[0.2cm]
2253 0.31 nm: shifted towards 0.317 nm $\rightarrow$ C-Si-C
2254 \end{minipage}
2255 }
2256
2257 \begin{itemize}
2258 \item Decreasing cut-off artifact
2259 \item {\color{red}Amorphous} SiC-like phase remains
2260 \item High amount of {\color{red}damage} \& alignement to c-Si host matrix lost
2261 \item Slightly sharper peaks $\Rightarrow$ indicate slight {\color{blue}acceleration of dynamics} due to temperature
2262 \end{itemize}
2263
2264 \vspace{-0.1cm}
2265
2266 \begin{center}
2267 {\color{blue}
2268 \framebox{
2269 {\color{black}
2270 High C \& small $V$ \& short $t$
2271 $\Rightarrow$
2272 }
2273 Slow restructuring due to strong C-C bonds
2274 {\color{black}
2275 $\Leftarrow$
2276 High C \& low T implants
2277 }
2278 }
2279 }
2280 \end{center}
2281
2282 \end{slide}
2283
2284 \begin{slide}
2285
2286  {\large\bf
2287   Summary and Conclusions
2288  }
2289
2290  \scriptsize
2291
2292 %\vspace{0.1cm}
2293
2294 \framebox{
2295 \begin{minipage}[t]{12.9cm}
2296  \underline{Pecipitation simulations}
2297  \begin{itemize}
2298   \item High C concentration $\rightarrow$ amorphous SiC like phase
2299   \item Problem of potential enhanced slow phase space propagation
2300   \item Low T $\rightarrow$ C-Si \hkl<1 0 0> dumbbell dominated structure
2301   \item High T $\rightarrow$ C$_{\text{sub}}$ dominated structure
2302   \item High T necessary to simulate IBS conditions (far from equilibrium)
2303   \item Precipitation by successive agglomeration of \cs (epitaxy)
2304   \item \si{}: vehicle to form \cs{} \& supply of Si \& stress compensation
2305         (stretched SiC, interface)
2306  \end{itemize}
2307 \end{minipage}
2308 }
2309
2310 %\vspace{0.1cm}
2311
2312 \framebox{
2313 \begin{minipage}{12.9cm}
2314  \underline{Defects}
2315  \begin{itemize}
2316    \item DFT / EA
2317         \begin{itemize}
2318          \item Point defects excellently / fairly well described
2319                by DFT / EA
2320          \item C$_{\text{sub}}$ drastically underestimated by EA
2321          \item EA predicts correct ground state:
2322                C$_{\text{sub}}$ \& \si{} $>$ \ci{}
2323          \item Identified migration path explaining
2324                diffusion and reorientation experiments by DFT
2325          \item EA fails to describe \ci{} migration:
2326                Wrong path \& overestimated barrier
2327         \end{itemize}
2328    \item Combinations of defects
2329          \begin{itemize}
2330           \item Agglomeration of point defects energetically favorable
2331                 by compensation of stress
2332           \item Formation of C-C unlikely
2333           \item C$_{\text{sub}}$ favored conditions (conceivable in IBS)
2334           \item \ci{} \hkl<1 0 0> $\leftrightarrow$ \cs{} \& \si{} \hkl<1 1 0>\\
2335                 Low barrier (\unit[0.77]{eV}) \& low capture radius
2336         \end{itemize}
2337  \end{itemize}
2338 \end{minipage}
2339 }
2340
2341 \begin{center}
2342 {\color{blue}
2343 \framebox{Precipitation by successive agglomeration of \cs{}}
2344 }
2345 \end{center}
2346
2347 \end{slide}
2348
2349 \begin{slide}
2350
2351  {\large\bf
2352   Acknowledgements
2353  }
2354
2355  \vspace{0.1cm}
2356
2357  \small
2358
2359  Thanks to \ldots
2360
2361  \underline{Augsburg}
2362  \begin{itemize}
2363   \item Prof. B. Stritzker (accomodation at EP \RM{4})
2364   \item Ralf Utermann (EDV)
2365  \end{itemize}
2366
2367  \underline{Helsinki}
2368  \begin{itemize}
2369   \item Prof. K. Nordlund (MD)
2370  \end{itemize}
2371
2372  \underline{Munich}
2373  \begin{itemize}
2374   \item Bayerische Forschungsstiftung (financial support)
2375  \end{itemize}
2376
2377  \underline{Paderborn}
2378  \begin{itemize}
2379   \item Prof. J. Lindner (SiC)
2380   \item Prof. G. Schmidt (DFT + financial support)
2381   \item Dr. E. Rauls (DFT + SiC)
2382   \item Dr. S. Sanna (VASP)
2383  \end{itemize}
2384
2385 \vspace{0.2cm}
2386
2387 \begin{center}
2388 \framebox{
2389 \bf Thank you for your attention!
2390 }
2391 \end{center}
2392
2393 \end{slide}
2394
2395 \end{document}
2396
2397 \fi