posic/talks/mpi_app.tex

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  42 \usepackage{semlayer}           % Seminar overlays
  43 \usepackage{slidesec}           % Seminar sections and list of slides
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  81 \begin{document}
  82
  83 \extraslideheight{10in}
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  85
  86 \pagestyle{empty}
  87
  88 % specify width and height
  89 \slidewidth 26.3cm
  90 \slideheight 19.9cm
  91
  92 % margin
  93 \def\slidetopmargin{-0.15cm}
  94
  95 \newcommand{\ham}{\mathcal{H}}
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  99
 100 % itemize level ii
 101 \renewcommand\labelitemii{{\color{gray}$\bullet$}}
 102
 103 % nice phi
 104 \renewcommand{\phi}{\varphi}
 105
 106 % roman letters
 107 \newcommand{\RM}[1]{\MakeUppercase{\romannumeral #1{}}}
 108
 109 % colors
 110 \newrgbcolor{si-yellow}{.6 .6 0}
 111 \newrgbcolor{hb}{0.75 0.77 0.89}
 112 \newrgbcolor{lbb}{0.75 0.8 0.88}
 113 \newrgbcolor{hlbb}{0.825 0.88 0.968}
 114 \newrgbcolor{lachs}{1.0 .93 .81}
 115
 116 % shortcuts
 117 \newcommand{\si}{Si$_{\text{i}}${}}
 118 \newcommand{\ci}{C$_{\text{i}}${}}
 119 \newcommand{\cs}{C$_{\text{sub}}${}}
 120 \newcommand{\degc}[1]{\unit[#1]{$^{\circ}$C}{}}
 121 \newcommand{\distn}[1]{\unit[#1]{nm}{}}
 122 \newcommand{\dista}[1]{\unit[#1]{\AA}{}}
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 124
 125 % no vertical centering
 126 %\centerslidesfalse
 127
 128 % layout check
 129 %\layout
 130 \begin{slide}
 131 \center
 132 {\Huge
 133 E\\
 134 F\\
 135 G\\
 136 A B C D E F G H G F E D C B A
 137 G\\
 138 F\\
 139 E\\
 140 }
 141 \end{slide}
 142
 143 % topic
 144
 145 \begin{slide}
 146 \begin{center}
 147
 148  \vspace{16pt}
 149
 150  {\LARGE\bf
 151   Atomistic simulation studies\\[0.2cm]
 152   in the C/Si system
 153  }
 154
 155  \vspace{48pt}
 156
 157  \textsc{Frank Zirkelbach}
 158
 159  \vspace{48pt}
 160
 161  Application talk at the Max Planck Institute for Solid State Research
 162
 163  \vspace{08pt}
 164
 165  Stuttgart, November 2011
 166
 167 \end{center}
 168 \end{slide}
 169
 170 % no vertical centering
 171 \centerslidesfalse
 172
 173 \ifnum1=0
 174
 175 % intro
 176
 177 \begin{slide}
 178
 179 %{\large\bf
 180 % Phase diagram of the C/Si system\\
 181 %}
 182
 183 \vspace*{0.2cm}
 184
 185 \begin{minipage}{6.5cm}
 186 \includegraphics[width=6.5cm]{si-c_phase.eps}
 187 \begin{center}
 188 {\tiny
 189 R. I. Scace and G. A. Slack, J. Chem. Phys. 30, 1551 (1959)
 190 }
 191 \end{center}
 192 \begin{pspicture}(0,0)(0,0)
 193 \psellipse[linecolor=blue,linewidth=0.1cm](3.55,4.0)(0.5,2.9)
 194 \end{pspicture}
 195 \end{minipage}
 196 \begin{minipage}{6cm}
 197 {\bf Phase diagram of the C/Si system}\\[0.2cm]
 198 {\color{blue}Stoichiometric composition}
 199 \begin{itemize}
 200 \item only chemical stable compound
 201 \item wide band gap semiconductor\\
 202       \underline{silicon carbide}, SiC
 203 \end{itemize}
 204 \end{minipage}
 205
 206 \end{slide}
 207
 208 % motivation / properties / applications of silicon carbide
 209
 210 \begin{slide}
 211
 212 \vspace*{1.8cm}
 213
 214 \small
 215
 216 \begin{pspicture}(0,0)(13.5,5)
 217
 218  \psframe*[linecolor=hb](-0.2,0)(12.9,5)
 219
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 222
 223  \rput[lt](0,4.6){\color{gray}PROPERTIES}
 224
 225  \rput[lt](0.3,4){wide band gap}
 226  \rput[lt](0.3,3.5){high electric breakdown field}
 227  \rput[lt](0.3,3){good electron mobility}
 228  \rput[lt](0.3,2.5){high electron saturation drift velocity}
 229  \rput[lt](0.3,2){high thermal conductivity}
 230
 231  \rput[lt](0.3,1.5){hard and mechanically stable}
 232  \rput[lt](0.3,1){chemically inert}
 233
 234  \rput[lt](0.3,0.5){radiation hardness}
 235
 236  \rput[rt](12.7,4.6){\color{gray}APPLICATIONS}
 237
 238  \rput[rt](12.5,3.85){high-temperature, high power}
 239  \rput[rt](12.5,3.5){and high-frequency}
 240  \rput[rt](12.5,3.15){electronic and optoelectronic devices}
 241
 242  \rput[rt](12.5,2.35){material suitable for extreme conditions}
 243  \rput[rt](12.5,2){microelectromechanical systems}
 244  \rput[rt](12.5,1.65){abrasives, cutting tools, heating elements}
 245
 246  \rput[rt](12.5,0.85){first wall reactor material, detectors}
 247  \rput[rt](12.5,0.5){and electronic devices for space}
 248
 249 \end{pspicture}
 250
 251 \begin{picture}(0,0)(5,-162)
 252 \includegraphics[height=2.2cm]{3C_SiC_bs.eps}
 253 \end{picture}
 254 \begin{picture}(0,0)(-120,-162)
 255 \includegraphics[height=2.2cm]{nasa_600c_led.eps}
 256 \end{picture}
 257 \begin{picture}(0,0)(-270,-162)
 258 \includegraphics[height=2.2cm]{6h-sic_3c-sic.eps}
 259 \end{picture}
 260 %%%%
 261 \begin{picture}(0,0)(10,65)
 262 \includegraphics[height=2.8cm]{sic_switch.eps}
 263 \end{picture}
 264 %\begin{picture}(0,0)(-243,65)
 265 \begin{picture}(0,0)(-110,65)
 266 \includegraphics[height=2.8cm]{ise_99.eps}
 267 \end{picture}
 268 %\begin{picture}(0,0)(-135,65)
 269 \begin{picture}(0,0)(-100,65)
 270 \includegraphics[height=1.2cm]{infineon_schottky.eps}
 271 \end{picture}
 272 \begin{picture}(0,0)(-233,65)
 273 \includegraphics[height=2.8cm]{solar_car.eps}
 274 \end{picture}
 275
 276 \end{slide}
 277
 278 % motivation
 279
 280 \begin{slide}
 281
 282  {\large\bf
 283   Polytypes of SiC\\[0.4cm]
 284  }
 285
 286 \includegraphics[width=3.8cm]{cubic_hex.eps}\\
 287 \begin{minipage}{1.9cm}
 288 {\tiny cubic (twist)}
 289 \end{minipage}
 290 \begin{minipage}{2.9cm}
 291 {\tiny hexagonal (no twist)}
 292 \end{minipage}
 293
 294 \begin{picture}(0,0)(-150,0)
 295  \includegraphics[width=7cm]{polytypes.eps}
 296 \end{picture}
 297
 298 \vspace{0.6cm}
 299
 300 \footnotesize
 301
 302 \begin{tabular}{l c c c c c c}
 303 \hline
 304  & 3C-SiC & 4H-SiC & 6H-SiC & Si & GaN & Diamond\\
 305 \hline
 306 Hardness [Mohs] & \multicolumn{3}{c}{------ 9.6 ------}& 6.5 & - & 10 \\
 307 Band gap [eV] & 2.36 & 3.23 & 3.03 & 1.12 & 3.39 & 5.5 \\
 308 Break down field [$10^6$ V/cm] & 4 & 3 & 3.2 & 0.6 & 5 & 10 \\
 309 Saturation drift velocity [$10^7$ cm/s] & 2.5 & 2.0 & 2.0 & 1 & 2.7 & 2.7 \\
 310 Electron mobility [cm$^2$/Vs] & 800 & 900 & 400 & 1100 & 900 & 2200 \\
 311 Hole mobility [cm$^2$/Vs] & 320 & 120 & 90 & 420 & 150 & 1600 \\
 312 Thermal conductivity [W/cmK] & 5.0 & 4.9 & 4.9 & 1.5 & 1.3 & 22 \\
 313 \hline
 314 \end{tabular}
 315
 316 \begin{pspicture}(0,0)(0,0)
 317 \psellipse[linecolor=green](5.7,2.10)(0.4,0.5)
 318 \end{pspicture}
 319 \begin{pspicture}(0,0)(0,0)
 320 \psellipse[linecolor=green](5.6,0.92)(0.4,0.2)
 321 \end{pspicture}
 322 \begin{pspicture}(0,0)(0,0)
 323 \psellipse[linecolor=red](10.45,0.45)(0.4,0.2)
 324 \end{pspicture}
 325
 326 \end{slide}
 327
 328 % fabrication
 329
 330 \begin{slide}
 331
 332  {\large\bf
 333   Fabrication of silicon carbide
 334  }
 335
 336  \small
 337
 338  \vspace{2pt}
 339
 340 \begin{center}
 341  {\color{gray}
 342  \emph{Silicon carbide --- Born from the stars, perfected on earth.}
 343  }
 344 \end{center}
 345
 346 \vspace{2pt}
 347
 348 SiC thin films by MBE \& CVD
 349 \begin{itemize}
 350  \item Much progress achieved in homo/heteroepitaxial SiC thin film growth
 351  \item \underline{Commercially available} semiconductor power devices based on
 352        \underline{\foreignlanguage{greek}{a}-SiC}
 353  \item Production of favored \underline{3C-SiC} material
 354        \underline{less advanced}
 355  \item Quality and size not yet sufficient
 356 \end{itemize}
 357 \begin{picture}(0,0)(-310,-20)
 358   \includegraphics[width=2.0cm]{cree.eps}
 359 \end{picture}
 360
 361 \vspace{-0.2cm}
 362
 363 Alternative approach:
 364 Ion beam synthesis (IBS) of burried 3C-SiC layers in Si\hkl(1 0 0)
 365
 366 \vspace{0.2cm}
 367
 368 \scriptsize
 369
 370 \framebox{
 371 \begin{minipage}{3.15cm}
 372  \begin{center}
 373 \includegraphics[width=3cm]{imp.eps}\\
 374  {\tiny
 375   Carbon implantation
 376  }
 377  \end{center}
 378 \end{minipage}
 379 \begin{minipage}{3.15cm}
 380  \begin{center}
 381 \includegraphics[width=3cm]{annealing.eps}\\
 382  {\tiny
 383   \unit[12]{h} annealing at \degc{1200}
 384  }
 385  \end{center}
 386 \end{minipage}
 387 }
 388 \begin{minipage}{5.5cm}
 389  \includegraphics[width=5.8cm]{ibs_3c-sic.eps}\\[-0.2cm]
 390  \begin{center}
 391  {\tiny
 392   XTEM: single crystalline 3C-SiC in Si\hkl(1 0 0)
 393  }
 394  \end{center}
 395 \end{minipage}
 396
 397 \end{slide}
 398
 399 % contents
 400
 401 \begin{slide}
 402
 403 {\large\bf
 404  Systematic investigation of C implantations into Si
 405 }
 406
 407 \vspace{1.7cm}
 408 \begin{center}
 409 \hspace{-1.0cm}
 410 \includegraphics[width=0.75\textwidth]{imp_inv.eps}
 411 \end{center}
 412
 413 \end{slide}
 414
 415 % outline
 416
 417 \begin{slide}
 418
 419 {\large\bf
 420  Outline
 421 }
 422
 423 \vspace{1.7cm}
 424 \begin{center}
 425 \hspace{-1.0cm}
 426 \includegraphics[width=0.75\textwidth]{imp_inv.eps}
 427 \end{center}
 428
 429 \begin{pspicture}(0,0)(0,0)
 430 \rput(6.0,7.0){\rnode{init}{\psframebox[fillstyle=gradient,gradbegin=red,gradend=white,gradlines=1000,gradmidpoint=1.0,linestyle=none]{
 431 \begin{minipage}{11cm}
 432 {\color{black}Diploma thesis}\\
 433  \underline{Monte Carlo} simulation modeling the selforganization process\\
 434  leading to periodic arrays of nanometric amorphous SiC precipitates
 435 \end{minipage}
 436 }}}
 437 \end{pspicture}
 438 \begin{pspicture}(0,0)(0,0)
 439 \rput(6.0,-0.5){\rnode{init}{\psframebox[fillstyle=gradient,gradbegin=blue,gradend=white,gradmidpoint=1.0,gradlines=1000,linestyle=none]{
 440 \begin{minipage}{11cm}
 441 {\color{black}Doctoral studies}\\
 442  Classical potential \underline{molecular dynamics} simulations \ldots\\
 443  \underline{Density functional theory} calculations \ldots\\[0.2cm]
 444  \ldots on defect formation and SiC precipitation in Si
 445 \end{minipage}
 446 }}}
 447 \end{pspicture}
 448 \begin{pspicture}(0,0)(0,0)
 449 \psellipse[linecolor=red,linewidth=0.05cm](5,3.0)(0.8,1.0)
 450 \end{pspicture}
 451 \begin{pspicture}(0,0)(0,0)
 452 \psellipse[linecolor=blue,linewidth=0.05cm](8.2,3.2)(1.5,1.6)
 453 \end{pspicture}
 454
 455 \end{slide}
 456
 457 \begin{slide}
 458
 459 \headdiplom
 460 {\large\bf
 461  Selforganization of nanometric amorphous SiC lamellae
 462 }
 463
 464 \small
 465
 466 \vspace{0.2cm}
 467
 468 \begin{itemize}
 469  \item Regularly spaced, nanometric spherical\\
 470        and lamellar amorphous inclusions\\
 471        at the upper a/c interface
 472  \item Carbon accumulation\\
 473        in amorphous volumes
 474 \end{itemize}
 475
 476 \vspace{0.4cm}
 477
 478 \begin{minipage}{12cm}
 479 \includegraphics[width=9cm]{../../nlsop/img/k393abild1_e_l.eps}\\
 480 {\scriptsize
 481 XTEM bright-field, \unit[180]{keV} C$^+ \rightarrow$ Si,
 482 {\color{red}\underline{\degc{150}}},
 483 Dose: \unit[4.3 $\times 10^{17}$]{cm$^{-2}$}
 484 }
 485 \end{minipage}
 486
 487 \begin{picture}(0,0)(-182,-215)
 488 \begin{minipage}{6.5cm}
 489 \begin{center}
 490 \includegraphics[width=6.5cm]{../../nlsop/img/eftem.eps}\\[-0.2cm]
 491 {\scriptsize
 492 XTEM bright-field and respective EFTEM C map
 493 }
 494 \end{center}
 495 \end{minipage}
 496 \end{picture}
 497
 498 \end{slide}
 499
 500 \begin{slide}
 501
 502 \headdiplom
 503 {\large\bf
 504  Model displaying the formation of ordered lamellae
 505 }
 506
 507 \vspace{0.1cm}
 508
 509 \begin{center}
 510  \includegraphics[width=8.0cm]{../../nlsop/img/modell_ng_e.eps}
 511 \end{center}
 512
 513 \footnotesize
 514
 515 \begin{itemize}
 516 \item Supersaturation of C in c-Si\\
 517       $\rightarrow$ {\bf Carbon induced} nucleation of spherical
 518       SiC$_x$-precipitates
 519 \item High interfacial energy between 3C-SiC and c-Si\\
 520       $\rightarrow$ {\bf Amorphous} precipitates
 521 \item \unit[20-- 30]{\%} lower silicon density of a-SiC$_x$ compared to c-Si\\
 522       $\rightarrow$ {\bf Lateral strain} (black arrows)
 523 \item Implantation range near surface\\
 524       $\rightarrow$ {\bf Relaxation} of {\bf vertical strain component}
 525 \item Reduction of the carbon supersaturation in c-Si\\
 526       $\rightarrow$ {\bf Carbon diffusion} into amorphous volumina
 527       (white arrows)
 528 \item Remaining lateral strain\\
 529       $\rightarrow$ {\bf Strain enhanced} lateral amorphisation
 530 \item Absence of crystalline neighbours (structural information)\\
 531       $\rightarrow$ {\bf Stabilization} of amorphous inclusions
 532       {\bf against recrystallization}
 533 \end{itemize}
 534
 535 \end{slide}
 536
 537 \begin{slide}
 538
 539 \headdiplom
 540 {\large\bf
 541  Implementation of the Monte Carlo code
 542 }
 543
 544 \small
 545
 546 \begin{enumerate}
 547  \item \underline{Amorphization / Recrystallization}\\
 548        Ion collision in discretized target determined by random numbers
 549        distributed according to nuclear energy loss.
 550        Amorphization/recrystallization probability:
 551 \[
 552 p_{c \rightarrow a}(\vec{r}) = {\color{green} p_b} + {\color{blue} p_c c_C(\vec{r})} + {\color{red} \sum_{\textrm{amorphous neighbours}} \frac{p_s c_C(\vec{r'})}{(r-r')^2}}
 553 \]
 554 \begin{itemize}
 555  \item {\color{green} $p_b$} normal `ballistic' amorphization
 556  \item {\color{blue} $p_c$} carbon induced amorphization
 557  \item {\color{red} $p_s$} stress enhanced amorphization
 558 \end{itemize}
 559 \[
 560 p_{a \rightarrow c}(\vec r) = (1 - p_{c \rightarrow a}(\vec r)) \Big(1 - \frac{\sum_{direct \, neighbours} \delta (\vec{r'})}{6} \Big) \, \textrm{,}
 561 \]
 562 \[
 563 \delta (\vec r) = \left\{
 564 \begin{array}{ll}
 565         1 & \textrm{if volume at position $\vec r$ is amorphous} \\
 566         0 & \textrm{otherwise} \\
 567 \end{array}
 568 \right.
 569 \]
 570  \item \underline{Carbon incorporation}\\
 571        Incorporation volume determined according to implantation profile
 572  \item \underline{Diffusion / Sputtering}
 573        \begin{itemize}
 574         \item Transfer fraction of C atoms
 575               of crystalline into neighbored amorphous volumes
 576         \item Remove surface layer
 577        \end{itemize}
 578 \end{enumerate}
 579
 580 \end{slide}
 581
 582 \begin{slide}
 583
 584 \begin{minipage}{3.7cm}
 585 \begin{pspicture}(0,0)(0,0)
 586 \rput(1.7,0.2){\psframebox[fillstyle=gradient,gradbegin=red,gradend=white,gradlines=1000,gradangle=10,gradmidpoint=1,linestyle=none]{
 587 \begin{minipage}{3.7cm}
 588 \hfill
 589 \vspace{0.7cm}
 590 \end{minipage}
 591 }}
 592 \end{pspicture}
 593 {\large\bf
 594  Results
 595 }
 596
 597 \footnotesize
 598
 599 \vspace{1.2cm}
 600
 601 Evolution of the \ldots
 602 \begin{itemize}
 603  \item continuous\\
 604        amorphous layer
 605  \item a/c interface
 606  \item lamellar precipitates
 607 \end{itemize}
 608 \ldots reproduced!\\[1.4cm]
 609
 610 {\color{blue}
 611 \begin{center}
 612 Experiment \& simulation\\
 613 in good agreement\\[1.0cm]
 614
 615 Simulation is able to model the whole depth region\\[1.2cm]
 616 \end{center}
 617 }
 618
 619 \end{minipage}
 620 \begin{minipage}{0.5cm}
 621 \vfill
 622 \end{minipage}
 623 \begin{minipage}{8.0cm}
 624  \vspace{-0.3cm}
 625  \includegraphics[width=9cm]{../../nlsop/img/dosis_entwicklung_ng_e_1-2.eps}\\
 626  \includegraphics[width=9cm]{../../nlsop/img/dosis_entwicklung_ng_e2_2-2.eps}
 627 \end{minipage}
 628
 629 \end{slide}
 630
 631 \begin{slide}
 632
 633 \headdiplom
 634 {\large\bf
 635  Structural \& compositional details
 636 }
 637
 638 \begin{minipage}[t]{7.5cm}
 639 \includegraphics[height=6.5cm]{../../nlsop/img/ac_cconc_ver2_e.eps}\\
 640 \end{minipage}
 641 \begin{minipage}[t]{5.0cm}
 642 \includegraphics[height=6.5cm]{../../nlsop/img/97_98_e.eps}
 643 \end{minipage}
 644
 645 \footnotesize
 646
 647 \vspace{-0.1cm}
 648
 649 \begin{itemize}
 650  \item Fluctuation of C concentration in lamellae region
 651  \item \unit[8--10]{at.\%} C saturation limit
 652        within the respective conditions
 653  \item Complementarily arranged and alternating sequence of layers\\
 654        with a high and low amount of amorphous regions
 655  \item C accumulation in the amorphous phase / Origin of stress
 656 \end{itemize}
 657
 658 \begin{picture}(0,0)(-260,-50)
 659 \framebox{
 660 \begin{minipage}{3cm}
 661 \begin{center}
 662 {\color{blue}
 663 Precipitation process\\
 664 gets traceable\\
 665 by simulation!
 666 }
 667 \end{center}
 668 \end{minipage}
 669 }
 670 \end{picture}
 671
 672 \end{slide}
 673
 674 \begin{slide}
 675
 676 \headphd
 677 {\large\bf
 678  Formation of epitaxial single crystalline 3C-SiC
 679 }
 680
 681 \footnotesize
 682
 683 \vspace{0.2cm}
 684
 685 \begin{center}
 686 \begin{itemize}
 687  \item \underline{Implantation step 1}\\[0.1cm]
 688         Almost stoichiometric dose | \unit[180]{keV} | \degc{500}\\
 689         $\Rightarrow$ Epitaxial {\color{blue}3C-SiC} layer \&
 690         {\color{blue}precipitates}
 691  \item \underline{Implantation step 2}\\[0.1cm]
 692         Little remaining dose | \unit[180]{keV} | \degc{250}\\
 693         $\Rightarrow$
 694         Destruction/Amorphization of precipitates at layer interface
 695  \item \underline{Annealing}\\[0.1cm]
 696        \unit[10]{h} at \degc{1250}\\
 697        $\Rightarrow$ Homogeneous 3C-SiC layer with sharp interfaces
 698 \end{itemize}
 699 \end{center}
 700
 701 \begin{minipage}{7cm}
 702 \includegraphics[width=7cm]{ibs_3c-sic.eps}
 703 \end{minipage}
 704 \begin{minipage}{5cm}
 705 \begin{pspicture}(0,0)(0,0)
 706 \rnode{box}{
 707 \psframebox[fillstyle=solid,fillcolor=white,linecolor=blue,linestyle=solid]{
 708 \begin{minipage}{5.3cm}
 709  \begin{center}
 710  {\color{blue}
 711   3C-SiC precipitation\\
 712   not yet fully understood
 713  }
 714  \end{center}
 715  \vspace*{0.1cm}
 716  \renewcommand\labelitemi{$\Rightarrow$}
 717  Details of the SiC precipitation
 718  \begin{itemize}
 719   \item significant technological progress\\
 720         in SiC thin film formation
 721   \item perspectives for processes relying\\
 722         upon prevention of SiC precipitation
 723  \end{itemize}
 724 \end{minipage}
 725 }}
 726 \rput(-6.8,5.4){\pnode{h0}}
 727 \rput(-3.0,5.4){\pnode{h1}}
 728 \ncline[linecolor=blue]{-}{h0}{h1}
 729 \ncline[linecolor=blue]{->}{h1}{box}
 730 \end{pspicture}
 731 \end{minipage}
 732
 733 \end{slide}
 734
 735 \begin{slide}
 736
 737 \headphd
 738 {\large\bf
 739   Supposed precipitation mechanism of SiC in Si
 740 }
 741
 742  \scriptsize
 743
 744  \vspace{0.1cm}
 745
 746  \framebox{
 747  \begin{minipage}{3.6cm}
 748  \begin{center}
 749  Si \& SiC lattice structure\\[0.1cm]
 750  \includegraphics[width=2.3cm]{sic_unit_cell.eps}
 751  \end{center}
 752 {\tiny
 753  \begin{minipage}{1.7cm}
 754 \underline{Silicon}\\
 755 {\color{yellow}$\bullet$} Si | {\color{gray}$\bullet$} Si\\
 756 $a=\unit[5.429]{\\A}$\\
 757 $\rho^*_{\text{Si}}=\unit[100]{\%}$
 758  \end{minipage}
 759  \begin{minipage}{1.7cm}
 760 \underline{Silicon carbide}\\
 761 {\color{yellow}$\bullet$} Si | {\color{gray}$\bullet$} C\\
 762 $a=\unit[4.359]{\\A}$\\
 763 $\rho^*_{\text{Si}}=\unit[97]{\%}$
 764  \end{minipage}
 765 }
 766  \end{minipage}
 767  }
 768  \hspace{0.1cm}
 769  \begin{minipage}{4.1cm}
 770  \begin{center}
 771  \includegraphics[width=3.3cm]{tem_c-si-db.eps}
 772  \end{center}
 773  \end{minipage}
 774  \hspace{0.1cm}
 775  \begin{minipage}{4.0cm}
 776  \begin{center}
 777  \includegraphics[width=3.3cm]{tem_3c-sic.eps}
 778  \end{center}
 779  \end{minipage}
 780
 781  \vspace{0.1cm}
 782
 783  \begin{minipage}{4.0cm}
 784  \begin{center}
 785  C-Si dimers (dumbbells)\\[-0.1cm]
 786  on Si interstitial sites
 787  \end{center}
 788  \end{minipage}
 789  \hspace{0.1cm}
 790  \begin{minipage}{4.1cm}
 791  \begin{center}
 792  Agglomeration of C-Si dumbbells\\[-0.1cm]
 793  $\Rightarrow$ dark contrasts
 794  \end{center}
 795  \end{minipage}
 796  \hspace{0.1cm}
 797  \begin{minipage}{4.0cm}
 798  \begin{center}
 799  Precipitation of 3C-SiC in Si\\[-0.1cm]
 800  $\Rightarrow$ Moir\'e fringes\\[-0.1cm]
 801  \& release of Si self-interstitials
 802  \end{center}
 803  \end{minipage}
 804
 805  \vspace{0.1cm}
 806
 807  \begin{minipage}{4.0cm}
 808  \begin{center}
 809  \includegraphics[width=3.3cm]{sic_prec_seq_01.eps}
 810  \end{center}
 811  \end{minipage}
 812  \hspace{0.1cm}
 813  \begin{minipage}{4.1cm}
 814  \begin{center}
 815  \includegraphics[width=3.3cm]{sic_prec_seq_02.eps}
 816  \end{center}
 817  \end{minipage}
 818  \hspace{0.1cm}
 819  \begin{minipage}{4.0cm}
 820  \begin{center}
 821  \includegraphics[width=3.3cm]{sic_prec_seq_03.eps}
 822  \end{center}
 823  \end{minipage}
 824
 825 \begin{pspicture}(0,0)(0,0)
 826 \psline[linewidth=2pt]{->}(8.3,2)(8.8,2)
 827 \psellipse[linecolor=blue](11.1,6.0)(0.3,0.5)
 828 \rput{-20}{\psellipse[linecolor=blue](3.1,8.2)(0.3,0.5)}
 829 \psline[linewidth=2pt]{->}(3.9,2)(4.4,2)
 830 \rput(11.8,0.3){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
 831  $4a_{\text{Si}}=5a_{\text{SiC}}$
 832  }}}
 833 \rput(11.5,8){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
 834 \hkl(h k l) planes match
 835  }}}
 836 \rput(8.5,6.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
 837 r = \unit[2--4]{nm}
 838  }}}
 839 \end{pspicture}
 840
 841 \end{slide}
 842
 843 \begin{slide}
 844
 845 \headphd
 846 {\large\bf
 847  Supposed precipitation mechanism of SiC in Si
 848 }
 849
 850  \scriptsize
 851
 852  \vspace{0.1cm}
 853
 854  \framebox{
 855  \begin{minipage}{3.6cm}
 856  \begin{center}
 857  Si \& SiC lattice structure\\[0.1cm]
 858  \includegraphics[width=2.3cm]{sic_unit_cell.eps}
 859  \end{center}
 860 {\tiny
 861  \begin{minipage}{1.7cm}
 862 \underline{Silicon}\\
 863 {\color{yellow}$\bullet$} Si | {\color{gray}$\bullet$} Si\\
 864 $a=\unit[5.429]{\\A}$\\
 865 $\rho^*_{\text{Si}}=\unit[100]{\%}$
 866  \end{minipage}
 867  \begin{minipage}{1.7cm}
 868 \underline{Silicon carbide}\\
 869 {\color{yellow}$\bullet$} Si | {\color{gray}$\bullet$} C\\
 870 $a=\unit[4.359]{\\A}$\\
 871 $\rho^*_{\text{Si}}=\unit[97]{\%}$
 872  \end{minipage}
 873 }
 874  \end{minipage}
 875  }
 876  \hspace{0.1cm}
 877  \begin{minipage}{4.1cm}
 878  \begin{center}
 879  \includegraphics[width=3.3cm]{tem_c-si-db.eps}
 880  \end{center}
 881  \end{minipage}
 882  \hspace{0.1cm}
 883  \begin{minipage}{4.0cm}
 884  \begin{center}
 885  \includegraphics[width=3.3cm]{tem_3c-sic.eps}
 886  \end{center}
 887  \end{minipage}
 888
 889  \vspace{0.1cm}
 890
 891  \begin{minipage}{4.0cm}
 892  \begin{center}
 893  C-Si dimers (dumbbells)\\[-0.1cm]
 894  on Si interstitial sites
 895  \end{center}
 896  \end{minipage}
 897  \hspace{0.1cm}
 898  \begin{minipage}{4.1cm}
 899  \begin{center}
 900  Agglomeration of C-Si dumbbells\\[-0.1cm]
 901  $\Rightarrow$ dark contrasts
 902  \end{center}
 903  \end{minipage}
 904  \hspace{0.1cm}
 905  \begin{minipage}{4.0cm}
 906  \begin{center}
 907  Precipitation of 3C-SiC in Si\\[-0.1cm]
 908  $\Rightarrow$ Moir\'e fringes\\[-0.1cm]
 909  \& release of Si self-interstitials
 910  \end{center}
 911  \end{minipage}
 912
 913  \vspace{0.1cm}
 914
 915  \begin{minipage}{4.0cm}
 916  \begin{center}
 917  \includegraphics[width=3.3cm]{sic_prec_seq_01.eps}
 918  \end{center}
 919  \end{minipage}
 920  \hspace{0.1cm}
 921  \begin{minipage}{4.1cm}
 922  \begin{center}
 923  \includegraphics[width=3.3cm]{sic_prec_seq_02.eps}
 924  \end{center}
 925  \end{minipage}
 926  \hspace{0.1cm}
 927  \begin{minipage}{4.0cm}
 928  \begin{center}
 929  \includegraphics[width=3.3cm]{sic_prec_seq_03.eps}
 930  \end{center}
 931  \end{minipage}
 932
 933 \begin{pspicture}(0,0)(0,0)
 934 \psline[linewidth=2pt]{->}(8.3,2)(8.8,2)
 935 \psellipse[linecolor=blue](11.1,6.0)(0.3,0.5)
 936 \rput{-20}{\psellipse[linecolor=blue](3.1,8.2)(0.3,0.5)}
 937 \psline[linewidth=2pt]{->}(3.9,2)(4.4,2)
 938 \rput(11.8,0.3){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
 939  $4a_{\text{Si}}=5a_{\text{SiC}}$
 940  }}}
 941 \rput(11.5,8){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
 942 \hkl(h k l) planes match
 943  }}}
 944 \rput(8.5,6.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
 945 r = \unit[2--4]{nm}
 946  }}}
 947 % controversial view!
 948 \rput(6.5,5.0){\psframebox[fillstyle=solid,opacity=0.5,fillcolor=black]{
 949 \begin{minipage}{14cm}
 950 \hfill
 951 \vspace{12cm}
 952 \end{minipage}
 953 }}
 954 \rput(6.5,5.3){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=white,linewidth=0.1cm]{
 955 \begin{minipage}{10cm}
 956 \small
 957 \vspace*{0.2cm}
 958 \begin{center}
 959 {\color{gray}\bf Controversial findings}
 960 \end{center}
 961 \begin{itemize}
 962 \item High-temperature implantation {\tiny\color{gray}/Nejim~et~al./}
 963  \begin{itemize}
 964   \item C incorporated {\color{blue}substitutionally} on regular Si lattice sites
 965   \item \si{} reacting with further C in cleared volume
 966  \end{itemize}
 967 \item Annealing behavior {\tiny\color{gray}/Serre~et~al./}
 968  \begin{itemize}
 969   \item Room temperature implantation $\rightarrow$ high C diffusion
 970   \item Elevated temperature implantation $\rightarrow$ no C redistribution
 971  \end{itemize}
 972  $\Rightarrow$ mobile {\color{red}\ci} opposed to
 973  stable {\color{blue}\cs{}} configurations
 974 \item Strained silicon \& Si/SiC heterostructures
 975       {\tiny\color{gray}/Strane~et~al./Guedj~et~al./}
 976  \begin{itemize}
 977   \item {\color{blue}Coherent} SiC precipitates (tensile strain)
 978   \item Incoherent SiC (strain relaxation)
 979  \end{itemize}
 980 \end{itemize}
 981 \vspace{0.1cm}
 982 \begin{center}
 983 {\Huge${\lightning}$} \hspace{0.3cm}
 984 {\color{blue}\cs{}} --- vs --- {\color{red}\ci} \hspace{0.3cm}
 985 {\Huge${\lightning}$}
 986 \end{center}
 987 \vspace{0.2cm}
 988 \end{minipage}
 989  }}}
 990 \end{pspicture}
 991
 992 \end{slide}
 993
 994 \begin{slide}
 995
 996 \headphd
 997 {\large\bf
 998  Utilized computational methods
 999 }
1000
1001 \vspace{0.3cm}
1002
1003 \small
1004
1005 {\bf Molecular dynamics (MD)}\\[0.1cm]
1006 \scriptsize
1007 \begin{tabular}{| p{4.5cm} | p{7.5cm} |}
1008 \hline
1009 System of $N$ particles &
1010 $N=5832\pm 1$ (Defects), $N=238328+6000$ (Precipitation)\\
1011 Phase space propagation &
1012 Velocity Verlet | timestep: \unit[1]{fs} \\
1013 Analytical interaction potential &
1014 Tersoff-like {\color{red}short-range}, {\color{blue}bond order} potential
1015 (Erhart/Albe)
1016 $\displaystyle
1017 E = \frac{1}{2} \sum_{i \neq j} \pot_{ij}, \quad
1018     \pot_{ij} = {\color{red}f_C(r_{ij})}
1019     \left[ f_R(r_{ij}) + {\color{blue}b_{ij}} f_A(r_{ij}) \right]
1020 $\\
1021 Observables: time/ensemble averages &
1022 NpT (isothermal-isobaric) | Berendsen thermostat/barostat\\
1023 \hline
1024 \end{tabular}
1025
1026 \small
1027
1028 \vspace{0.3cm}
1029
1030 {\bf Density functional theory (DFT)}
1031
1032 \scriptsize
1033
1034 \begin{minipage}[t]{6cm}
1035 \begin{itemize}
1036  \item Hohenberg-Kohn theorem:\\
1037        $\Psi_0(r_1,r_2,\ldots,r_N)=\Psi[n_0(r)]$, $E_0=E[n_0]$
1038  \item Kohn-Sham approach:\\
1039        Single-particle effective theory
1040 \end{itemize}
1041 \hrule
1042 \begin{itemize}
1043 \item Code: \textsc{vasp}
1044 \item Plane wave basis set
1045 %$\displaystyle
1046 %\Phi_i=\sum_{|G+k|<G_{\text{cut}}} c_{i,k+G} \exp{\left(i(k+G)r\right)}
1047 %$\\
1048 %$\displaystyle
1049 %E_{\text{cut}}=\frac{\hbar^2}{2m}G^2_{\text{cut}}=\unit[300]{eV}
1050 %$
1051 \item Ultrasoft pseudopotential
1052 \item Exchange \& correlation: GGA
1053 \item Brillouin zone sampling: $\Gamma$-point
1054 \item Supercell: $N=216\pm2$
1055 \end{itemize}
1056 \end{minipage}
1057 \begin{minipage}{6cm}
1058 \begin{pspicture}(0,0)(0,0)
1059 \pscircle[fillcolor=yellow,fillstyle=solid,linestyle=none](3.5,-2.0){2.5}
1060 \rput(2.7,-0.7){\psframebox[fillstyle=solid,opacity=0.8,fillcolor=white]{
1061 $\displaystyle
1062 \left[ -\frac{\hbar^2}{2m}\nabla^2 + V_{\text{eff}}(r) - \epsilon_i \right] \Phi_i(r) = 0
1063 $
1064 }}
1065 \rput(5.2,-2.0){\psframebox[fillstyle=solid,opacity=0.8,fillcolor=white]{
1066 $\displaystyle
1067 n(r)=\sum_i^N|\Phi_i(r)|^2
1068 $
1069 }}
1070 \rput(3.0,-4.5){\psframebox[fillstyle=solid,opacity=0.8,fillcolor=white]{
1071 $\displaystyle
1072 V_{\text{eff}}(r)=V_{\text{ext}}(r)+\int\frac{e^2 n(r')}{|r-r'|}d^3r'
1073                  +V_{\text{XC}}[n(r)]
1074 $
1075 }}
1076 \psarcn[linewidth=0.07cm,linestyle=dashed]{->}(3.5,-2.0){2.5}{130}{15}
1077 \psarcn[linewidth=0.07cm,linestyle=dashed]{->}(3.5,-2.0){2.5}{230}{165}
1078 \psarcn[linewidth=0.07cm,linestyle=dashed]{->}(3.5,-2.0){2.5}{345}{310}
1079
1080 \end{pspicture}
1081 \end{minipage}
1082
1083 \end{slide}
1084
1085 \begin{slide}
1086
1087 \headphd
1088  {\large\bf
1089   Point defects \& defect migration
1090  }
1091
1092  \small
1093
1094  \vspace{0.2cm}
1095
1096 \begin{minipage}[b]{7.5cm}
1097 {\bf Defect structure}\\
1098   \begin{pspicture}(0,0)(7,4.4)
1099   \rput(3.5,3.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
1100    \parbox{7cm}{
1101    \begin{itemize}
1102     \item Creation of c-Si simulation volume
1103     \item Periodic boundary conditions
1104     \item $T=0\text{ K}$, $p=0\text{ bar}$
1105    \end{itemize}
1106   }}}}
1107 \rput(3.5,1.3){\rnode{insert}{\psframebox{
1108  \parbox{7cm}{
1109   \begin{center}
1110   Insertion of interstitial C/Si atoms
1111   \end{center}
1112   }}}}
1113   \rput(3.5,0.2){\rnode{cool}{\psframebox[fillstyle=solid,fillcolor=lbb]{
1114    \parbox{7cm}{
1115    \begin{center}
1116    Relaxation / structural energy minimization
1117    \end{center}
1118   }}}}
1119   \ncline[]{->}{init}{insert}
1120   \ncline[]{->}{insert}{cool}
1121  \end{pspicture}
1122 \end{minipage}
1123 \begin{minipage}[b]{4.5cm}
1124 \begin{center}
1125 \includegraphics[width=3.8cm]{unit_cell_e.eps}\\
1126 \end{center}
1127 \begin{minipage}{2.21cm}
1128 {\scriptsize
1129 {\color{red}$\bullet$} Tetrahedral\\[-0.1cm]
1130 {\color{green}$\bullet$} Hexagonal\\[-0.1cm]
1131 {\color{yellow}$\bullet$} \hkl<1 0 0> DB
1132 }
1133 \end{minipage}
1134 \begin{minipage}{2.21cm}
1135 {\scriptsize
1136 {\color{magenta}$\bullet$} \hkl<1 1 0> DB\\[-0.1cm]
1137 {\color{cyan}$\bullet$} Bond-centered\\[-0.1cm]
1138 {\color{black}$\bullet$} Vac. / Sub.
1139 }
1140 \end{minipage}
1141 \end{minipage}
1142
1143 \vspace{0.2cm}
1144
1145 \begin{minipage}[b]{6cm}
1146 {\bf Defect formation energy}\\
1147 \framebox{
1148 $E_{\text{f}}=E-\sum_i N_i\mu_i$}\\[0.1cm]
1149 Particle reservoir: Si \& SiC\\[0.2cm]
1150 {\bf Binding energy}\\
1151 \framebox{
1152 $
1153 E_{\text{b}}=
1154 E_{\text{f}}^{\text{comb}}-
1155 E_{\text{f}}^{1^{\text{st}}}-
1156 E_{\text{f}}^{2^{\text{nd}}}
1157 $
1158 }\\[0.1cm]
1159 \footnotesize
1160 $E_{\text{b}}<0$: energetically favorable configuration\\
1161 $E_{\text{b}}\rightarrow 0$: non-interacting, isolated defects\\
1162 \end{minipage}
1163 \begin{minipage}[b]{6cm}
1164 {\bf Migration barrier}
1165 \footnotesize
1166 \begin{itemize}
1167  \item Displace diffusing atom
1168  \item Constrain relaxation of (diffusing) atoms
1169  \item Record configurational energy
1170 \end{itemize}
1171 \begin{picture}(0,0)(-60,-33)
1172 \includegraphics[width=4.5cm]{crt_mod.eps}
1173 \end{picture}
1174 \end{minipage}
1175
1176 \end{slide}
1177
1178 \begin{slide}
1179
1180 \footnotesize
1181
1182 \headphd
1183 {\large\bf
1184  Si self-interstitial point defects in silicon\\[0.1cm]
1185 }
1186
1187 \begin{center}
1188 \begin{tabular}{l c c c c c}
1189 \hline
1190  $E_{\text{f}}$ [eV] & \hkl<1 1 0> DB & H & T & \hkl<1 0 0> DB & V \\
1191 \hline
1192  \textsc{vasp} & \underline{3.39} & 3.42 & 3.77 & 4.41 & 3.63 \\
1193  Erhart/Albe & 4.39 & 4.48$^*$ & \underline{3.40} & 5.42 & 3.13 \\
1194 \hline
1195 \end{tabular}\\[0.4cm]
1196 \end{center}
1197
1198 \begin{minipage}{3cm}
1199 \begin{center}
1200 \underline{Vacancy}\\
1201 \includegraphics[width=2.8cm]{si_pd_albe/vac.eps}
1202 \end{center}
1203 \end{minipage}
1204 \begin{minipage}{3cm}
1205 \begin{center}
1206 \underline{\hkl<1 1 0> DB}\\
1207 \includegraphics[width=2.8cm]{si_pd_albe/110_bonds.eps}
1208 \end{center}
1209 \end{minipage}
1210 \begin{minipage}{3cm}
1211 \begin{center}
1212 \underline{\hkl<1 0 0> DB}\\
1213 \includegraphics[width=2.8cm]{si_pd_albe/100_bonds.eps}
1214 \end{center}
1215 \end{minipage}
1216 \begin{minipage}{3cm}
1217 \begin{center}
1218 \underline{Tetrahedral}\\
1219 \includegraphics[width=2.8cm]{si_pd_albe/tet_bonds.eps}
1220 \end{center}
1221 \end{minipage}\\
1222
1223 \underline{Hexagonal} \hspace{2pt}
1224 \href{../video/si_self_int_hexa.avi}{$\rhd$}\\[0.1cm]
1225 \framebox{
1226 \begin{minipage}{2.7cm}
1227 $E_{\text{f}}^*=4.48\text{ eV}$\\
1228 \includegraphics[width=2.7cm]{si_pd_albe/hex_a_bonds.eps}
1229 \end{minipage}
1230 \begin{minipage}{0.4cm}
1231 \begin{center}
1232 $\Rightarrow$
1233 \end{center}
1234 \end{minipage}
1235 \begin{minipage}{2.7cm}
1236 $E_{\text{f}}=3.96\text{ eV}$\\
1237 \includegraphics[width=2.8cm]{si_pd_albe/hex_bonds.eps}
1238 \end{minipage}
1239 }
1240 \begin{minipage}{5.5cm}
1241 \begin{center}
1242 {\tiny nearly T $\rightarrow$ T}\\
1243 \end{center}
1244 \includegraphics[width=6.0cm]{nhex_tet.ps}
1245 \end{minipage}
1246
1247 \end{slide}
1248
1249 \begin{slide}
1250
1251 \footnotesize
1252
1253 \headphd
1254 {\large\bf
1255  C interstitial point defects in silicon\\
1256 }
1257
1258 \begin{tabular}{l c c c c c c r}
1259 \hline
1260  $E_{\text{f}}$ [eV] & T & H & \hkl<1 0 0> DB & \hkl<1 1 0> DB & S & B &
1261  {\color{black} \cs{} \& \si}\\
1262 \hline
1263  \textsc{vasp} & unstable & unstable & \underline{3.72} & 4.16 & 1.95 & 4.66 & {\color{green}4.17}\\
1264  Erhart/Albe & 6.09 & 9.05$^*$ & \underline{3.88} & 5.18 & {\color{red}0.75} & 5.59$^*$ & {\color{green}4.43} \\
1265 \hline
1266 \end{tabular}\\[0.1cm]
1267
1268 \framebox{
1269 \begin{minipage}{2.8cm}
1270 \underline{Hexagonal} \hspace{2pt}
1271 \href{../video/c_in_si_int_hexa.avi}{$\rhd$}\\
1272 $E_{\text{f}}^*=9.05\text{ eV}$\\
1273 \includegraphics[width=2.8cm]{c_pd_albe/hex_bonds.eps}
1274 \end{minipage}
1275 \begin{minipage}{0.4cm}
1276 \begin{center}
1277 $\Rightarrow$
1278 \end{center}
1279 \end{minipage}
1280 \begin{minipage}{2.8cm}
1281 \underline{\hkl<1 0 0>}\\
1282 $E_{\text{f}}=3.88\text{ eV}$\\
1283 \includegraphics[width=2.8cm]{c_pd_albe/100_bonds.eps}
1284 \end{minipage}
1285 }
1286 \begin{minipage}{1.4cm}
1287 \hfill
1288 \end{minipage}
1289 \begin{minipage}{3.0cm}
1290 \begin{flushright}
1291 \underline{Tetrahedral}\\
1292 \includegraphics[width=3.0cm]{c_pd_albe/tet_bonds.eps}
1293 \end{flushright}
1294 \end{minipage}
1295
1296 \framebox{
1297 \begin{minipage}{2.8cm}
1298 \underline{Bond-centered}\\
1299 $E_{\text{f}}^*=5.59\text{ eV}$\\
1300 \includegraphics[width=2.8cm]{c_pd_albe/bc_bonds.eps}
1301 \end{minipage}
1302 \begin{minipage}{0.4cm}
1303 \begin{center}
1304 $\Rightarrow$
1305 \end{center}
1306 \end{minipage}
1307 \begin{minipage}{2.8cm}
1308 \underline{\hkl<1 1 0> dumbbell}\\
1309 $E_{\text{f}}=5.18\text{ eV}$\\
1310 \includegraphics[width=2.8cm]{c_pd_albe/110_bonds.eps}
1311 \end{minipage}
1312 }
1313 \begin{minipage}{1.4cm}
1314 \hfill
1315 \end{minipage}
1316 \begin{minipage}{3.0cm}
1317 \begin{flushright}
1318 \underline{Substitutional}\\
1319 \includegraphics[width=3.0cm]{c_pd_albe/sub_bonds.eps}
1320 \end{flushright}
1321 \end{minipage}
1322
1323 \end{slide}
1324
1325 \begin{slide}
1326
1327 \headphd
1328 {\large\bf\boldmath
1329  C-Si dimer \& bond-centered interstitial configuration
1330 }
1331
1332 \footnotesize
1333
1334 \vspace{0.1cm}
1335
1336 \begin{minipage}[t]{4.1cm}
1337 {\bf\boldmath C \hkl<1 0 0> DB interstitial}\\[0.1cm]
1338 \begin{minipage}{2.0cm}
1339 \begin{center}
1340 \underline{Erhart/Albe}
1341 \includegraphics[width=2.0cm]{c_pd_albe/100_cmp.eps}
1342 \end{center}
1343 \end{minipage}
1344 \begin{minipage}{2.0cm}
1345 \begin{center}
1346 \underline{\textsc{vasp}}
1347 \includegraphics[width=2.0cm]{c_pd_vasp/100_cmp.eps}
1348 \end{center}
1349 \end{minipage}\\[0.2cm]
1350 Si-C-Si bond angle $\rightarrow$ \unit[180]{$^{\circ}$}\\
1351 $\Rightarrow$ $sp$ hybridization\\[0.1cm]
1352 Si-Si-Si bond angle $\rightarrow$ \unit[120]{$^{\circ}$}\\
1353 $\Rightarrow$ $sp^2$ hybridization
1354 \begin{center}
1355 \includegraphics[width=3.4cm]{c_pd_vasp/eden.eps}\\[-0.1cm]
1356 {\tiny Charge density isosurface}
1357 \end{center}
1358 \end{minipage}
1359 \begin{minipage}{0.2cm}
1360 \hfill
1361 \end{minipage}
1362 \begin{minipage}[t]{8.1cm}
1363 \begin{flushright}
1364 {\bf Bond-centered interstitial}\\[0.1cm]
1365 \begin{minipage}{4.4cm}
1366 %\scriptsize
1367 \begin{itemize}
1368  \item Linear Si-C-Si bond
1369  \item Si: one C \& 3 Si neighbours
1370  \item Spin polarized calculations
1371  \item No saddle point!\\
1372        Real local minimum!
1373 \end{itemize}
1374 \end{minipage}
1375 \begin{minipage}{2.7cm}
1376 %\includegraphics[width=2.8cm]{c_pd_vasp/bc_2333.eps}\\
1377 \vspace{0.2cm}
1378 \includegraphics[width=2.8cm]{c_pd_albe/bc_bonds.eps}\\
1379 \end{minipage}
1380
1381 \framebox{
1382  \tiny
1383  \begin{minipage}[t]{6.5cm}
1384   \begin{minipage}[t]{1.2cm}
1385   {\color{red}Si}\\
1386   {\tiny sp$^3$}\\[0.8cm]
1387   \underline{${\color{black}\uparrow}$}
1388   \underline{${\color{black}\uparrow}$}
1389   \underline{${\color{black}\uparrow}$}
1390   \underline{${\color{red}\uparrow}$}\\
1391   sp$^3$
1392   \end{minipage}
1393   \begin{minipage}[t]{1.4cm}
1394   \begin{center}
1395   {\color{red}M}{\color{blue}O}\\[0.8cm]
1396   \underline{${\color{blue}\uparrow}{\color{white}\downarrow}$}\\
1397   $\sigma_{\text{ab}}$\\[0.5cm]
1398   \underline{${\color{red}\uparrow}{\color{blue}\downarrow}$}\\
1399   $\sigma_{\text{b}}$
1400   \end{center}
1401   \end{minipage}
1402   \begin{minipage}[t]{1.0cm}
1403   \begin{center}
1404   {\color{blue}C}\\
1405   {\tiny sp}\\[0.2cm]
1406   \underline{${\color{white}\uparrow\uparrow}$}
1407   \underline{${\color{white}\uparrow\uparrow}$}\\
1408   2p\\[0.4cm]
1409   \underline{${\color{blue}\uparrow}{\color{blue}\downarrow}$}
1410   \underline{${\color{blue}\uparrow}{\color{blue}\downarrow}$}\\
1411   sp
1412   \end{center}
1413   \end{minipage}
1414   \begin{minipage}[t]{1.4cm}
1415   \begin{center}
1416   {\color{blue}M}{\color{green}O}\\[0.8cm]
1417   \underline{${\color{blue}\uparrow}{\color{white}\downarrow}$}\\
1418   $\sigma_{\text{ab}}$\\[0.5cm]
1419   \underline{${\color{green}\uparrow}{\color{blue}\downarrow}$}\\
1420   $\sigma_{\text{b}}$
1421   \end{center}
1422   \end{minipage}
1423   \begin{minipage}[t]{1.2cm}
1424   \begin{flushright}
1425   {\color{green}Si}\\
1426   {\tiny sp$^3$}\\[0.8cm]
1427   \underline{${\color{green}\uparrow}$}
1428   \underline{${\color{black}\uparrow}$}
1429   \underline{${\color{black}\uparrow}$}
1430   \underline{${\color{black}\uparrow}$}\\
1431   sp$^3$
1432   \end{flushright}
1433   \end{minipage}
1434  \end{minipage}
1435 }\\[0.4cm]
1436
1437 %\framebox{
1438 \begin{minipage}{3.0cm}
1439 %\scriptsize
1440 \underline{Charge density}\\
1441 {\color{gray}$\bullet$} Spin up\\
1442 {\color{green}$\bullet$} Spin down\\
1443 {\color{blue}$\bullet$} Resulting spin up\\
1444 {\color{yellow}$\bullet$} Si atoms\\
1445 {\color{red}$\bullet$} C atom
1446 \end{minipage}
1447 \begin{minipage}{3.6cm}
1448 \includegraphics[width=3.8cm]{c_100_mig_vasp/im_spin_diff.eps}
1449 \end{minipage}
1450 %}
1451
1452 \end{flushright}
1453
1454 \end{minipage}
1455 \begin{pspicture}(0,0)(0,0)
1456 \psline[linecolor=gray,linewidth=0.05cm](-7.8,-8.7)(-7.8,0)
1457 \end{pspicture}
1458
1459 \end{slide}
1460
1461 \begin{slide}
1462
1463 \headphd
1464 {\large\bf\boldmath
1465  C interstitial migration --- ab initio
1466 }
1467
1468 \scriptsize
1469
1470 \vspace{0.1cm}
1471
1472 \begin{minipage}{6.8cm}
1473 \framebox{\hkl[0 0 -1] $\rightarrow$ \hkl[0 0 1]}\\
1474 \begin{minipage}{2.0cm}
1475 \includegraphics[width=2.0cm]{c_pd_vasp/100_2333.eps}
1476 \end{minipage}
1477 \begin{minipage}{0.2cm}
1478 $\rightarrow$
1479 \end{minipage}
1480 \begin{minipage}{2.0cm}
1481 \includegraphics[width=2.0cm]{c_pd_vasp/bc_2333.eps}
1482 \end{minipage}
1483 \begin{minipage}{0.2cm}
1484 $\rightarrow$
1485 \end{minipage}
1486 \begin{minipage}{2.0cm}
1487 \includegraphics[width=2.0cm]{c_pd_vasp/100_next_2333.eps}
1488 \end{minipage}\\[0.1cm]
1489 Spin polarization\\
1490 $\Rightarrow$ BC configuration constitutes local minimum\\
1491 $\Rightarrow$ Migration barrier to reach BC | $\Delta E=\unit[1.2]{eV}$
1492 \end{minipage}
1493 \begin{minipage}{5.4cm}
1494 \includegraphics[width=6.0cm]{im_00-1_nosym_sp_fullct_thesis_vasp_s.ps}
1495 \end{minipage}\\[0.2cm]
1496 %\hrule
1497 %
1498 \begin{minipage}{6.8cm}
1499 \framebox{\hkl[0 0 -1] $\rightarrow$ \hkl[0 -1 0]}\\
1500 \begin{minipage}{2.0cm}
1501 \includegraphics[width=2.0cm]{c_pd_vasp/100_2333.eps}
1502 \end{minipage}
1503 \begin{minipage}{0.2cm}
1504 $\rightarrow$
1505 \end{minipage}
1506 \begin{minipage}{2.0cm}
1507 \includegraphics[width=2.0cm]{c_pd_vasp/00-1-0-10_2333.eps}
1508 \end{minipage}
1509 \begin{minipage}{0.2cm}
1510 $\rightarrow$
1511 \end{minipage}
1512 \begin{minipage}{2.0cm}
1513 \includegraphics[width=2.0cm]{c_pd_vasp/0-10_2333.eps}
1514 \end{minipage}\\[0.1cm]
1515 $\Delta E=\unit[0.9]{eV}$ | Experimental values: \unit[0.70--0.87]{eV}\\
1516 $\Rightarrow$ {\color{red}Migration mechanism identified!}\\
1517 Note: Change in orientation
1518 \end{minipage}
1519 \begin{minipage}{5.4cm}
1520 \includegraphics[width=6.0cm]{00-1_0-10_vasp_s.ps}
1521 \end{minipage}\\[0.1cm]
1522 %
1523 \begin{center}
1524 Reorientation pathway composed of two consecutive processes of the above type
1525 \end{center}
1526
1527 \end{slide}
1528
1529 \begin{slide}
1530
1531 \headphd
1532 {\large\bf\boldmath
1533  C interstitial migration --- analytical potential
1534 }
1535
1536 \scriptsize
1537
1538 \vspace{0.3cm}
1539
1540 \begin{minipage}[t]{6.0cm}
1541 {\bf\boldmath BC to \hkl[0 0 -1] transition}\\[0.2cm]
1542 \includegraphics[width=6.0cm]{bc_00-1_albe_s.ps}\\
1543 \begin{itemize}
1544  \item Lowermost migration barrier
1545  \item $\Delta E \approx \unit[2.2]{eV}$
1546  \item 2.4 times higher than ab initio result
1547  \item Different pathway
1548 \end{itemize}
1549 \end{minipage}
1550 \begin{minipage}[t]{0.2cm}
1551 \hfill
1552 \end{minipage}
1553 \begin{minipage}[t]{6.0cm}
1554 {\bf\boldmath Transition involving a \hkl<1 1 0> configuration}
1555 \vspace{0.1cm}
1556 \begin{itemize}
1557  \item Bond-centered configuration unstable\\
1558        $\rightarrow$ \ci{} \hkl<1 1 0> dumbbell
1559  \item Minima of the \hkl[0 0 -1] to \hkl[0 -1 0] transition\\
1560        $\rightarrow$ \ci{} \hkl<1 1 0> DB
1561 \end{itemize}
1562 \vspace{0.1cm}
1563 \includegraphics[width=6.0cm]{00-1_110_0-10_mig_albe.ps}
1564 \begin{itemize}
1565  \item $\Delta E \approx \unit[2.2]{eV} \text{ \& } \unit[0.9]{eV}$
1566  \item 2.4 -- 3.4 times higher than ab initio result
1567  \item After all: Change of the DB orientation
1568 \end{itemize}
1569 \end{minipage}
1570
1571 \vspace{0.5cm}
1572 \begin{center}
1573 {\color{red}\bf Drastically overestimated diffusion barrier}
1574 \end{center}
1575
1576 \begin{pspicture}(0,0)(0,0)
1577 \psline[linewidth=0.05cm,linecolor=gray](6.1,1.0)(6.1,9.3)
1578 \end{pspicture}
1579
1580 \end{slide}
1581
1582 % continue here
1583 \fi
1584
1585 \begin{slide}
1586
1587 \headphd
1588 {\large\bf\boldmath
1589  Defect combinations
1590 }
1591
1592 \small
1593
1594 \vspace{0.2cm}
1595
1596 \begin{minipage}{9cm}
1597 {\scriptsize
1598 Combinations of an initially created \ci{} \hkl[0 0 -1] DB\\
1599 \begin{tabular}{l c c c c c c}
1600 \hline
1601  $E_{\text{b}}$ [eV] & 1 & 2 & 3 & 4 & 5 & R\\
1602  \hline
1603  \hkl[0 0 -1] & {\color{red}-0.08} & -1.15 & {\color{red}-0.08} & 0.04 & -1.66 & -0.19\\
1604  \hkl[0 0 1] & 0.34 & 0.004 & -2.05 & 0.26 & -1.53 & -0.19\\
1605  \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}\\
1606  \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}\\
1607  \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}\\
1608  \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}\\
1609  \hline
1610  C$_{\text{sub}}$ & 0.26 & -0.51 & -0.93 & -0.15 & 0.49 & -0.05\\
1611  Vacancy & -5.39 ($\rightarrow$ C$_{\text{sub}}$) & -0.59 & -3.14 & -0.54 & -0.50 & -0.31\\
1612 \hline
1613 \end{tabular}
1614 }
1615 \end{minipage}
1616 \begin{minipage}{3cm}
1617 \includegraphics[width=3.5cm]{comb_pos.eps}
1618 \end{minipage}
1619
1620 \vspace*{0.3cm}
1621
1622 \footnotesize
1623
1624 \begin{minipage}[t]{3.2cm}
1625 \underline{\hkl[1 0 0] at position 1}\\[0.1cm]
1626 \includegraphics[width=2.8cm]{00-1dc/2-25.eps}
1627 \end{minipage}
1628 \begin{minipage}[t]{3.2cm}
1629 \underline{\hkl[0 -1 0] at position 1}\\[0.1cm]
1630 \includegraphics[width=2.8cm]{00-1dc/2-39.eps}
1631 \end{minipage}
1632 \begin{minipage}[t]{5.5cm}
1633 \begin{itemize}
1634  \item $E_{\text{b}}=0$ $\Leftrightarrow$ non-interacting defects\\
1635        $E_{\text{b}} \rightarrow 0$ for increasing distance (R)
1636  \item Stress compensation / increase
1637  \item Unfavored: antiparallel orientations
1638  \item Indication of energetically favored\\
1639        agglomeration
1640  \item Most favorable: C clustering
1641  \item However: High barrier ($>4\,\text{eV}$)
1642  \item $4\times{\color{cyan}-2.25}$ versus $2\times{\color{orange}-2.39}$
1643        (Entropy)
1644 \end{itemize}
1645 \end{minipage}
1646
1647
1648 \end{slide}
1649
1650 \end{document}
1651 \ifnum1=0
1652
1653 \begin{slide}
1654
1655  {\large\bf\boldmath
1656   Combinations of C-Si \hkl<1 0 0>-type interstitials
1657  }
1658
1659 \small
1660
1661 \vspace*{0.1cm}
1662
1663 Energetically most favorable combinations along \hkl<1 1 0>
1664
1665 \vspace*{0.1cm}
1666
1667 {\scriptsize
1668 \begin{tabular}{l c c c c c c}
1669 \hline
1670  & 1 & 2 & 3 & 4 & 5 & 6\\
1671 \hline
1672 $E_{\text{b}}$ [eV] & -2.39 & -1.88 & -0.59 & -0.31 & -0.24 & -0.21 \\
1673 C-C distance [\AA] & 1.4 & 4.6 & 6.5 & 8.6 & 10.5 & 10.8 \\
1674 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>\\
1675 \hline
1676 \end{tabular}
1677 }
1678
1679 \vspace*{0.3cm}
1680
1681 \begin{minipage}{7.0cm}
1682 \includegraphics[width=7cm]{db_along_110_cc.ps}
1683 \end{minipage}
1684 \begin{minipage}{6.0cm}
1685 \begin{itemize}
1686  \item Interaction proportional to reciprocal cube of C-C distance
1687  \item Saturation in the immediate vicinity
1688  \renewcommand\labelitemi{$\Rightarrow$}
1689  \item Agglomeration of \ci{} expected
1690  \item Absence of C clustering
1691 \end{itemize}
1692 \begin{center}
1693 {\color{blue}
1694  Consisten with initial precipitation model
1695 }
1696 \end{center}
1697 \end{minipage}
1698
1699 \vspace{0.2cm}
1700
1701 \end{slide}
1702
1703 \begin{slide}
1704
1705  {\large\bf\boldmath
1706   Combinations of substitutional C and \hkl<1 1 0> Si self-interstitials
1707  }
1708
1709  \scriptsize
1710
1711 %\begin{center}
1712 %\begin{minipage}{3.2cm}
1713 %\includegraphics[width=3cm]{sub_110_combo.eps}
1714 %\end{minipage}
1715 %\begin{minipage}{7.8cm}
1716 %\begin{tabular}{l c c c c c c}
1717 %\hline
1718 %C$_{\text{sub}}$ & \hkl<1 1 0> & \hkl<-1 1 0> & \hkl<0 1 1> & \hkl<0 -1 1> &
1719 %                   \hkl<1 0 1> & \hkl<-1 0 1> \\
1720 %\hline
1721 %1 & \RM{1} & \RM{3} & \RM{3} & \RM{1} & \RM{3} & \RM{1} \\
1722 %2 & \RM{2} & A & A & \RM{2} & C & \RM{5} \\
1723 %3 & \RM{3} & \RM{1} & \RM{3} & \RM{1} & \RM{1} & \RM{3} \\
1724 %4 & \RM{4} & B & D & E & E & D \\
1725 %5 & \RM{5} & C & A & \RM{2} & A & \RM{2} \\
1726 %\hline
1727 %\end{tabular}
1728 %\end{minipage}
1729 %\end{center}
1730
1731 %\begin{center}
1732 %\begin{tabular}{l c c c c c c c c c c}
1733 %\hline
1734 %Conf & \RM{1} & \RM{2} & \RM{3} & \RM{4} & \RM{5} & A & B & C & D & E \\
1735 %\hline
1736 %$E_{\text{f}}$ [eV]& 4.37 & 5.26 & 5.57 & 5.37 & 5.12 & 5.10 & 5.32 & 5.28 & 5.39 & 5.32 \\
1737 %$E_{\text{b}}$ [eV] & -0.97 & -0.08 & 0.22 & -0.02 & -0.23 & -0.25 & -0.02 & -0.06 & 0.05 & -0.03 \\
1738 %$r$ [nm] & 0.292 & 0.394 & 0.241 & 0.453 & 0.407 & 0.408 & 0.452 & 0.392 & 0.456 & 0.453\\
1739 %\hline
1740 %\end{tabular}
1741 %\end{center}
1742
1743 \begin{minipage}{6.0cm}
1744 \includegraphics[width=5.8cm]{c_sub_si110.ps}
1745 \end{minipage}
1746 \begin{minipage}{7cm}
1747 \scriptsize
1748 \begin{itemize}
1749  \item IBS: C may displace Si\\
1750        $\Rightarrow$ C$_{\text{sub}}$ + \hkl<1 1 0> Si self-interstitial
1751  \item Assumption:\\
1752        \hkl<1 1 0>-type $\rightarrow$ favored combination
1753  \renewcommand\labelitemi{$\Rightarrow$}
1754  \item Most favorable: \cs{} along \hkl<1 1 0> chain \si{}
1755  \item Less favorable than C-Si \hkl<1 0 0> dumbbell
1756  \item Interaction drops quickly to zero\\
1757        $\rightarrow$ low capture radius
1758 \end{itemize}
1759 \begin{center}
1760  {\color{blue}
1761  IBS process far from equilibrium\\
1762  \cs{} \& \si{} instead of thermodynamic ground state
1763  }
1764 \end{center}
1765 \end{minipage}
1766
1767 \begin{minipage}{6.5cm}
1768 \includegraphics[width=6.0cm]{162-097.ps}
1769 \begin{itemize}
1770  \item Low migration barrier
1771 \end{itemize}
1772 \end{minipage}
1773 \begin{minipage}{6.5cm}
1774 \begin{center}
1775 Ab initio MD at \degc{900}\\
1776 \includegraphics[width=3.3cm]{md_vasp_01.eps}
1777 $t=\unit[2230]{fs}$\\
1778 \includegraphics[width=3.3cm]{md_vasp_02.eps}
1779 $t=\unit[2900]{fs}$
1780 \end{center}
1781 {\color{blue}
1782 Contribution of entropy to structural formation
1783 }
1784 \end{minipage}
1785
1786 \end{slide}
1787
1788 \begin{slide}
1789
1790  {\large\bf\boldmath
1791   Migration in C-Si \hkl<1 0 0> and vacancy combinations
1792  }
1793
1794  \footnotesize
1795
1796 \vspace{0.1cm}
1797
1798 \begin{minipage}[t]{3cm}
1799 \underline{Pos 2, $E_{\text{b}}=-0.59\text{ eV}$}\\
1800 \includegraphics[width=2.8cm]{00-1dc/0-59.eps}
1801 \end{minipage}
1802 \begin{minipage}[t]{7cm}
1803 \vspace{0.2cm}
1804 \begin{center}
1805  Low activation energies\\
1806  High activation energies for reverse processes\\
1807  $\Downarrow$\\
1808  {\color{blue}C$_{\text{sub}}$ very stable}\\
1809 \vspace*{0.1cm}
1810  \hrule
1811 \vspace*{0.1cm}
1812  Without nearby \hkl<1 1 0> Si self-interstitial (IBS)\\
1813  $\Downarrow$\\
1814  {\color{blue}Formation of SiC by successive substitution by C}
1815
1816 \end{center}
1817 \end{minipage}
1818 \begin{minipage}[t]{3cm}
1819 \underline{Pos 3, $E_{\text{b}}=-3.14\text{ eV}$}\\
1820 \includegraphics[width=2.8cm]{00-1dc/3-14.eps}
1821 \end{minipage}
1822
1823
1824 \framebox{
1825 \begin{minipage}{5.9cm}
1826 \includegraphics[width=5.9cm]{vasp_mig/comb_mig_3-2_vac_fullct.ps}\\[0.6cm]
1827 \begin{center}
1828 \begin{picture}(0,0)(70,0)
1829 \includegraphics[width=1.4cm]{vasp_mig/comb_2-1_init.eps}
1830 \end{picture}
1831 \begin{picture}(0,0)(30,0)
1832 \includegraphics[width=1.4cm]{vasp_mig/comb_2-1_seq_03.eps}
1833 \end{picture}
1834 \begin{picture}(0,0)(-10,0)
1835 \includegraphics[width=1.4cm]{vasp_mig/comb_2-1_seq_06.eps}
1836 \end{picture}
1837 \begin{picture}(0,0)(-48,0)
1838 \includegraphics[width=1.4cm]{vasp_mig/comb_2-1_final.eps}
1839 \end{picture}
1840 \begin{picture}(0,0)(12.5,5)
1841 \includegraphics[width=1cm]{100_arrow.eps}
1842 \end{picture}
1843 \begin{picture}(0,0)(97,-10)
1844 \includegraphics[height=0.9cm]{001_arrow.eps}
1845 \end{picture}
1846 \end{center}
1847 \vspace{0.1cm}
1848 \end{minipage}
1849 }
1850 \begin{minipage}{0.3cm}
1851 \hfill
1852 \end{minipage}
1853 \framebox{
1854 \begin{minipage}{5.9cm}
1855 \includegraphics[width=5.9cm]{vasp_mig/comb_mig_4-2_vac_fullct.ps}\\[0.1cm]
1856 \begin{center}
1857 \begin{picture}(0,0)(60,0)
1858 \includegraphics[width=0.9cm]{vasp_mig/comb_3-1_init.eps}
1859 \end{picture}
1860 \begin{picture}(0,0)(25,0)
1861 \includegraphics[width=0.9cm]{vasp_mig/comb_3-1_seq_03.eps}
1862 \end{picture}
1863 \begin{picture}(0,0)(-20,0)
1864 \includegraphics[width=0.9cm]{vasp_mig/comb_3-1_seq_07.eps}
1865 \end{picture}
1866 \begin{picture}(0,0)(-55,0)
1867 \includegraphics[width=0.9cm]{vasp_mig/comb_3-1_final.eps}
1868 \end{picture}
1869 \begin{picture}(0,0)(12.5,5)
1870 \includegraphics[width=1cm]{100_arrow.eps}
1871 \end{picture}
1872 \begin{picture}(0,0)(95,0)
1873 \includegraphics[height=0.9cm]{001_arrow.eps}
1874 \end{picture}
1875 \end{center}
1876 \vspace{0.1cm}
1877 \end{minipage}
1878 }
1879
1880 \end{slide}
1881
1882 \begin{slide}
1883
1884  {\large\bf
1885   Conclusion of defect / migration / combined defect simulations
1886  }
1887
1888  \footnotesize
1889
1890 \vspace*{0.1cm}
1891
1892 Defect structures
1893 \begin{itemize}
1894  \item Accurately described by quantum-mechanical simulations
1895  \item Less accurate description by classical potential simulations
1896  \item Underestimated formation energy of \cs{} by classical approach
1897  \item Both methods predict same ground state: \ci{} \hkl<1 0 0> dumbbell
1898 \end{itemize}
1899
1900 Migration
1901 \begin{itemize}
1902  \item C migration pathway in Si identified
1903  \item Consistent with reorientation and diffusion experiments
1904 \end{itemize}
1905 \begin{itemize}
1906  \item Different path and ...
1907  \item overestimated barrier by classical potential calculations
1908 \end{itemize}
1909
1910 Concerning the precipitation mechanism
1911 \begin{itemize}
1912  \item Agglomeration of C-Si dumbbells energetically favorable
1913        (stress compensation)
1914  \item C-Si indeed favored compared to
1915        C$_{\text{sub}}$ \& \hkl<1 1 0> Si self-interstitial
1916  \item Possible low interaction capture radius of
1917        C$_{\text{sub}}$ \& \hkl<1 1 0> Si self-interstitial
1918  \item Low barrier for
1919        \ci{} \hkl<1 0 0> $\rightarrow$ \cs{} \& \si{} \hkl<1 1 0>
1920  \item In absence of nearby \hkl<1 1 0> Si self-interstitial:
1921        C-Si \hkl<1 0 0> + Vacancy $\rightarrow$ C$_{\text{sub}}$ (SiC)
1922 \end{itemize}
1923 \begin{center}
1924 {\color{blue}Results suggest increased participation of \cs}
1925 \end{center}
1926
1927 \end{slide}
1928
1929 \begin{slide}
1930
1931  {\large\bf
1932   Silicon carbide precipitation simulations
1933  }
1934
1935  \small
1936
1937 {\scriptsize
1938  \begin{pspicture}(0,0)(12,6.5)
1939   % nodes
1940   \rput(3.5,5.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
1941    \parbox{7cm}{
1942    \begin{itemize}
1943     \item Create c-Si volume
1944     \item Periodc boundary conditions
1945     \item Set requested $T$ and $p=0\text{ bar}$
1946     \item Equilibration of $E_{\text{kin}}$ and $E_{\text{pot}}$
1947    \end{itemize}
1948   }}}}
1949   \rput(3.5,2.7){\rnode{insert}{\psframebox[fillstyle=solid,fillcolor=lachs]{
1950    \parbox{7cm}{
1951    Insertion of C atoms at constant T
1952    \begin{itemize}
1953     \item total simulation volume {\pnode{in1}}
1954     \item volume of minimal SiC precipitate {\pnode{in2}}
1955     \item volume consisting of Si atoms to form a minimal {\pnode{in3}}\\
1956           precipitate
1957    \end{itemize}
1958   }}}}
1959   \rput(3.5,1){\rnode{cool}{\psframebox[fillstyle=solid,fillcolor=lbb]{
1960    \parbox{7.0cm}{
1961    Run for 100 ps followed by cooling down to $20\, ^{\circ}\textrm{C}$
1962   }}}}
1963   \ncline[]{->}{init}{insert}
1964   \ncline[]{->}{insert}{cool}
1965   \psframe[fillstyle=solid,fillcolor=white](7.5,0.7)(13.5,6.3)
1966   \rput(7.8,6){\footnotesize $V_1$}
1967   \psframe[fillstyle=solid,fillcolor=lightgray](9,2)(12,5)
1968   \rput(9.2,4.85){\tiny $V_2$}
1969   \psframe[fillstyle=solid,fillcolor=gray](9.25,2.25)(11.75,4.75)
1970   \rput(9.55,4.45){\footnotesize $V_3$}
1971   \rput(7.9,3.2){\pnode{ins1}}
1972   \rput(9.22,2.8){\pnode{ins2}}
1973   \rput(11.0,2.4){\pnode{ins3}}
1974   \ncline[]{->}{in1}{ins1}
1975   \ncline[]{->}{in2}{ins2}
1976   \ncline[]{->}{in3}{ins3}
1977  \end{pspicture}
1978 }
1979
1980 \begin{itemize}
1981  \item Restricted to classical potential simulations
1982  \item $V_2$ and $V_3$ considered due to low diffusion
1983  \item Amount of C atoms: 6000
1984        ($r_{\text{prec}}\approx 3.1\text{ nm}$, IBS: 2 ... 4 nm)
1985  \item Simulation volume: $31\times 31\times 31$ unit cells
1986        (238328 Si atoms)
1987 \end{itemize}
1988
1989 \end{slide}
1990
1991 \begin{slide}
1992
1993  {\large\bf\boldmath
1994   Silicon carbide precipitation simulations at $450\,^{\circ}\mathrm{C}$ as in IBS
1995  }
1996
1997  \small
1998
1999 \begin{minipage}{6.5cm}
2000 \includegraphics[width=6.4cm]{sic_prec_450_si-si_c-c.ps}
2001 \end{minipage}
2002 \begin{minipage}{6.5cm}
2003 \includegraphics[width=6.4cm]{sic_prec_450_energy.ps}
2004 \end{minipage}
2005
2006 \begin{minipage}{6.5cm}
2007 \includegraphics[width=6.4cm]{sic_prec_450_si-c.ps}
2008 \end{minipage}
2009 \begin{minipage}{6.5cm}
2010 \scriptsize
2011 \underline{Low C concentration ($V_1$)}\\
2012 \hkl<1 0 0> C-Si dumbbell dominated structure
2013 \begin{itemize}
2014  \item Si-C bumbs around 0.19 nm
2015  \item C-C peak at 0.31 nm (as expected in 3C-SiC):\\
2016        concatenated dumbbells of various orientation
2017  \item Si-Si NN distance stretched to 0.3 nm
2018 \end{itemize}
2019 {\color{blue}$\Rightarrow$ C atoms in proper 3C-SiC distance first}\\
2020 \underline{High C concentration ($V_2$, $V_3$)}\\
2021 High amount of strongly bound C-C bonds\\
2022 Defect density $\uparrow$ $\Rightarrow$ considerable amount of damage\\
2023 Only short range order observable\\
2024 {\color{blue}$\Rightarrow$ amorphous SiC-like phase}
2025 \end{minipage}
2026
2027 \end{slide}
2028
2029 \begin{slide}
2030
2031  {\large\bf\boldmath
2032   Silicon carbide precipitation simulations at $450\,^{\circ}\mathrm{C}$ as in IBS
2033  }
2034
2035  \small
2036
2037 \begin{minipage}{6.5cm}
2038 \includegraphics[width=6.4cm]{sic_prec_450_si-si_c-c.ps}
2039 \end{minipage}
2040 \begin{minipage}{6.5cm}
2041 \includegraphics[width=6.4cm]{sic_prec_450_energy.ps}
2042 \end{minipage}
2043
2044 \begin{minipage}{6.5cm}
2045 \includegraphics[width=6.4cm]{sic_prec_450_si-c.ps}
2046 \end{minipage}
2047 \begin{minipage}{6.5cm}
2048 \scriptsize
2049 \underline{Low C concentration ($V_1$)}\\
2050 \hkl<1 0 0> C-Si dumbbell dominated structure
2051 \begin{itemize}
2052  \item Si-C bumbs around 0.19 nm
2053  \item C-C peak at 0.31 nm (as expected in 3C-SiC):\\
2054        concatenated dumbbells of various orientation
2055  \item Si-Si NN distance stretched to 0.3 nm
2056 \end{itemize}
2057 {\color{blue}$\Rightarrow$ C atoms in proper 3C-SiC distance first}\\
2058 \underline{High C concentration ($V_2$, $V_3$)}\\
2059 High amount of strongly bound C-C bonds\\
2060 Defect density $\uparrow$ $\Rightarrow$ considerable amount of damage\\
2061 Only short range order observable\\
2062 {\color{blue}$\Rightarrow$ amorphous SiC-like phase}
2063 \end{minipage}
2064
2065 \begin{pspicture}(0,0)(0,0)
2066 \rput(6.7,5.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=white]{
2067 \begin{minipage}{10cm}
2068 \small
2069 {\color{red}\bf 3C-SiC formation fails to appear}
2070 \begin{itemize}
2071 \item Low C concentration simulations
2072  \begin{itemize}
2073   \item Formation of \ci{} indeed occurs
2074   \item Agllomeration not observed
2075  \end{itemize}
2076 \item High C concentration simulations
2077  \begin{itemize}
2078   \item Amorphous SiC-like structure\\
2079         (not expected at prevailing temperatures)
2080   \item Rearrangement and transition into 3C-SiC structure missing
2081  \end{itemize}
2082 \end{itemize}
2083 \end{minipage}
2084  }}}
2085 \end{pspicture}
2086
2087 \end{slide}
2088
2089 \begin{slide}
2090
2091  {\large\bf
2092   Limitations of molecular dynamics and short range potentials
2093  }
2094
2095 \footnotesize
2096
2097 \vspace{0.2cm}
2098
2099 \underline{Time scale problem of MD}\\[0.2cm]
2100 Minimize integration error\\
2101 $\Rightarrow$ discretization considerably smaller than
2102               reciprocal of fastest vibrational mode\\[0.1cm]
2103 Order of fastest vibrational mode: $10^{13} - 10^{14}\text{ Hz}$\\
2104 $\Rightarrow$ suitable choice of time step:
2105               $\tau=1\text{ fs}=10^{-15}\text{ s}$\\
2106 $\Rightarrow$ {\color{red}\underline{slow}} phase space propagation\\[0.1cm]
2107 Several local minima in energy surface separated by large energy barriers\\
2108 $\Rightarrow$ transition event corresponds to a multiple
2109               of vibrational periods\\
2110 $\Rightarrow$ phase transition made up of {\color{red}\underline{many}}
2111               infrequent transition events\\[0.1cm]
2112 {\color{blue}Accelerated methods:}
2113 \underline{Temperature accelerated} MD (TAD), self-guided MD \ldots
2114
2115 \vspace{0.3cm}
2116
2117 \underline{Limitations related to the short range potential}\\[0.2cm]
2118 Cut-off function pushing forces and energies to zero between 1$^{\text{st}}$
2119 and 2$^{\text{nd}}$ next neighbours\\
2120 $\Rightarrow$ overestimated unphysical high forces of next neighbours
2121
2122 \vspace{0.3cm}
2123
2124 \framebox{
2125 \color{red}
2126 Potential enhanced problem of slow phase space propagation
2127 }
2128
2129 \vspace{0.3cm}
2130
2131 \underline{Approach to the (twofold) problem}\\[0.2cm]
2132 Increased temperature simulations without TAD corrections\\
2133 (accelerated methods or higher time scales exclusively not sufficient)
2134
2135 \begin{picture}(0,0)(-260,-30)
2136 \framebox{
2137 \begin{minipage}{4.2cm}
2138 \tiny
2139 \begin{center}
2140 \vspace{0.03cm}
2141 \underline{IBS}
2142 \end{center}
2143 \begin{itemize}
2144 \item 3C-SiC also observed for higher T
2145 \item higher T inside sample
2146 \item structural evolution vs.\\
2147       equilibrium properties
2148 \end{itemize}
2149 \end{minipage}
2150 }
2151 \end{picture}
2152
2153 \begin{picture}(0,0)(-305,-155)
2154 \framebox{
2155 \begin{minipage}{2.5cm}
2156 \tiny
2157 \begin{center}
2158 retain proper\\
2159 thermodynmic sampling
2160 \end{center}
2161 \end{minipage}
2162 }
2163 \end{picture}
2164
2165 \end{slide}
2166
2167 \begin{slide}
2168
2169  {\large\bf
2170   Increased temperature simulations at low C concentration
2171  }
2172
2173 \small
2174
2175 \begin{minipage}{6.5cm}
2176 \includegraphics[width=6.4cm]{tot_pc_thesis.ps}
2177 \end{minipage}
2178 \begin{minipage}{6.5cm}
2179 \includegraphics[width=6.4cm]{tot_pc3_thesis.ps}
2180 \end{minipage}
2181
2182 \begin{minipage}{6.5cm}
2183 \includegraphics[width=6.4cm]{tot_pc2_thesis.ps}
2184 \end{minipage}
2185 \begin{minipage}{6.5cm}
2186 \scriptsize
2187  \underline{Si-C bonds:}
2188  \begin{itemize}
2189   \item Vanishing cut-off artifact (above $1650\,^{\circ}\mathrm{C}$)
2190   \item Structural change: C-Si \hkl<1 0 0> $\rightarrow$ C$_{\text{sub}}$
2191  \end{itemize}
2192  \underline{Si-Si bonds:}
2193  {\color{blue}Si-C$_{\text{sub}}$-Si} along \hkl<1 1 0>
2194  ($\rightarrow$ 0.325 nm)\\[0.1cm]
2195  \underline{C-C bonds:}
2196  \begin{itemize}
2197   \item C-C next neighbour pairs reduced (mandatory)
2198   \item Peak at 0.3 nm slightly shifted
2199         \begin{itemize}
2200          \item C-Si \hkl<1 0 0> combinations (dashed arrows)\\
2201                $\rightarrow$ C-Si \hkl<1 0 0> \& C$_{\text{sub}}$
2202                combinations (|)\\
2203                $\rightarrow$ pure {\color{blue}C$_{\text{sub}}$ combinations}
2204                ($\downarrow$)
2205          \item Range [|-$\downarrow$]:
2206                {\color{blue}C$_{\text{sub}}$ \& C$_{\text{sub}}$
2207                with nearby Si$_{\text{I}}$}
2208         \end{itemize}
2209  \end{itemize}
2210 \end{minipage}
2211
2212 \begin{picture}(0,0)(-330,-74)
2213 \color{blue}
2214 \framebox{
2215 \begin{minipage}{1.6cm}
2216 \tiny
2217 \begin{center}
2218 stretched SiC\\[-0.1cm]
2219 in c-Si
2220 \end{center}
2221 \end{minipage}
2222 }
2223 \end{picture}
2224
2225 \end{slide}
2226
2227 \begin{slide}
2228
2229  {\large\bf
2230   Increased temperature simulations at low C concentration
2231  }
2232
2233 \small
2234
2235 \begin{minipage}{6.5cm}
2236 \includegraphics[width=6.4cm]{tot_pc_thesis.ps}
2237 \end{minipage}
2238 \begin{minipage}{6.5cm}
2239 \includegraphics[width=6.4cm]{tot_pc3_thesis.ps}
2240 \end{minipage}
2241
2242 \begin{minipage}{6.5cm}
2243 \includegraphics[width=6.4cm]{tot_pc2_thesis.ps}
2244 \end{minipage}
2245 \begin{minipage}{6.5cm}
2246 \scriptsize
2247  \underline{Si-C bonds:}
2248  \begin{itemize}
2249   \item Vanishing cut-off artifact (above $1650\,^{\circ}\mathrm{C}$)
2250   \item Structural change: C-Si \hkl<1 0 0> $\rightarrow$ C$_{\text{sub}}$
2251  \end{itemize}
2252  \underline{Si-Si bonds:}
2253  {\color{blue}Si-C$_{\text{sub}}$-Si} along \hkl<1 1 0>
2254  ($\rightarrow$ 0.325 nm)\\[0.1cm]
2255  \underline{C-C bonds:}
2256  \begin{itemize}
2257   \item C-C next neighbour pairs reduced (mandatory)
2258   \item Peak at 0.3 nm slightly shifted
2259         \begin{itemize}
2260          \item C-Si \hkl<1 0 0> combinations (dashed arrows)\\
2261                $\rightarrow$ C-Si \hkl<1 0 0> \& C$_{\text{sub}}$
2262                combinations (|)\\
2263                $\rightarrow$ pure {\color{blue}C$_{\text{sub}}$ combinations}
2264                ($\downarrow$)
2265          \item Range [|-$\downarrow$]:
2266                {\color{blue}C$_{\text{sub}}$ \& C$_{\text{sub}}$
2267                with nearby Si$_{\text{I}}$}
2268         \end{itemize}
2269  \end{itemize}
2270 \end{minipage}
2271
2272 %\begin{picture}(0,0)(-330,-74)
2273 %\color{blue}
2274 %\framebox{
2275 %\begin{minipage}{1.6cm}
2276 %\tiny
2277 %\begin{center}
2278 %stretched SiC\\[-0.1cm]
2279 %in c-Si
2280 %\end{center}
2281 %\end{minipage}
2282 %}
2283 %\end{picture}
2284
2285 \begin{pspicture}(0,0)(0,0)
2286 \rput(6.7,5.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=white]{
2287 \begin{minipage}{10cm}
2288 \small
2289 {\color{blue}\bf Stretched SiC in c-Si}
2290 \begin{itemize}
2291 \item Consistent to precipitation model involving \cs{}
2292 \item Explains annealing behavior of high/low T C implants
2293       \begin{itemize}
2294        \item Low T: highly mobiel \ci{}
2295        \item High T: stable configurations of \cs{}
2296       \end{itemize}
2297 \end{itemize}
2298 $\Rightarrow$ High T $\leftrightarrow$ IBS conditions far from equilibrium\\
2299 $\Rightarrow$ Precipitation mechanism involving \cs{}
2300 \end{minipage}
2301  }}}
2302 \end{pspicture}
2303
2304 \end{slide}
2305
2306 \begin{slide}
2307
2308  {\large\bf
2309   Increased temperature simulations at high C concentration
2310  }
2311
2312 \footnotesize
2313
2314 \begin{minipage}{6.5cm}
2315 \includegraphics[width=6.4cm]{12_pc_thesis.ps}
2316 \end{minipage}
2317 \begin{minipage}{6.5cm}
2318 \includegraphics[width=6.4cm]{12_pc_c_thesis.ps}
2319 \end{minipage}
2320
2321 \vspace{0.1cm}
2322
2323 \scriptsize
2324
2325 \framebox{
2326 \begin{minipage}[t]{6.0cm}
2327 0.186 nm: Si-C pairs $\uparrow$\\
2328 (as expected in 3C-SiC)\\[0.2cm]
2329 0.282 nm: Si-C-C\\[0.2cm]
2330 $\approx$0.35 nm: C-Si-Si
2331 \end{minipage}
2332 }
2333 \begin{minipage}{0.2cm}
2334 \hfill
2335 \end{minipage}
2336 \framebox{
2337 \begin{minipage}[t]{6.0cm}
2338 0.15 nm: C-C pairs $\uparrow$\\
2339 (as expected in graphite/diamond)\\[0.2cm]
2340 0.252 nm: C-C-C (2$^{\text{nd}}$ NN for diamond)\\[0.2cm]
2341 0.31 nm: shifted towards 0.317 nm $\rightarrow$ C-Si-C
2342 \end{minipage}
2343 }
2344
2345 \begin{itemize}
2346 \item Decreasing cut-off artifact
2347 \item {\color{red}Amorphous} SiC-like phase remains
2348 \item High amount of {\color{red}damage} \& alignement to c-Si host matrix lost
2349 \item Slightly sharper peaks $\Rightarrow$ indicate slight {\color{blue}acceleration of dynamics} due to temperature
2350 \end{itemize}
2351
2352 \vspace{-0.1cm}
2353
2354 \begin{center}
2355 {\color{blue}
2356 \framebox{
2357 {\color{black}
2358 High C \& small $V$ \& short $t$
2359 $\Rightarrow$
2360 }
2361 Slow restructuring due to strong C-C bonds
2362 {\color{black}
2363 $\Leftarrow$
2364 High C \& low T implants
2365 }
2366 }
2367 }
2368 \end{center}
2369
2370 \end{slide}
2371
2372 \begin{slide}
2373
2374  {\large\bf
2375   Summary and Conclusions
2376  }
2377
2378  \scriptsize
2379
2380 %\vspace{0.1cm}
2381
2382 \framebox{
2383 \begin{minipage}[t]{12.9cm}
2384  \underline{Pecipitation simulations}
2385  \begin{itemize}
2386   \item High C concentration $\rightarrow$ amorphous SiC like phase
2387   \item Problem of potential enhanced slow phase space propagation
2388   \item Low T $\rightarrow$ C-Si \hkl<1 0 0> dumbbell dominated structure
2389   \item High T $\rightarrow$ C$_{\text{sub}}$ dominated structure
2390   \item High T necessary to simulate IBS conditions (far from equilibrium)
2391   \item Precipitation by successive agglomeration of \cs (epitaxy)
2392   \item \si{}: vehicle to form \cs{} \& supply of Si \& stress compensation
2393         (stretched SiC, interface)
2394  \end{itemize}
2395 \end{minipage}
2396 }
2397
2398 %\vspace{0.1cm}
2399
2400 \framebox{
2401 \begin{minipage}{12.9cm}
2402  \underline{Defects}
2403  \begin{itemize}
2404    \item DFT / EA
2405         \begin{itemize}
2406          \item Point defects excellently / fairly well described
2407                by DFT / EA
2408          \item C$_{\text{sub}}$ drastically underestimated by EA
2409          \item EA predicts correct ground state:
2410                C$_{\text{sub}}$ \& \si{} $>$ \ci{}
2411          \item Identified migration path explaining
2412                diffusion and reorientation experiments by DFT
2413          \item EA fails to describe \ci{} migration:
2414                Wrong path \& overestimated barrier
2415         \end{itemize}
2416    \item Combinations of defects
2417          \begin{itemize}
2418           \item Agglomeration of point defects energetically favorable
2419                 by compensation of stress
2420           \item Formation of C-C unlikely
2421           \item C$_{\text{sub}}$ favored conditions (conceivable in IBS)
2422           \item \ci{} \hkl<1 0 0> $\leftrightarrow$ \cs{} \& \si{} \hkl<1 1 0>\\
2423                 Low barrier (\unit[0.77]{eV}) \& low capture radius
2424         \end{itemize}
2425  \end{itemize}
2426 \end{minipage}
2427 }
2428
2429 \begin{center}
2430 {\color{blue}
2431 \framebox{Precipitation by successive agglomeration of \cs{}}
2432 }
2433 \end{center}
2434
2435 \end{slide}
2436
2437 \begin{slide}
2438
2439  {\large\bf
2440   Acknowledgements
2441  }
2442
2443  \vspace{0.1cm}
2444
2445  \small
2446
2447  Thanks to \ldots
2448
2449  \underline{Augsburg}
2450  \begin{itemize}
2451   \item Prof. B. Stritzker (accomodation at EP \RM{4})
2452   \item Ralf Utermann (EDV)
2453  \end{itemize}
2454
2455  \underline{Helsinki}
2456  \begin{itemize}
2457   \item Prof. K. Nordlund (MD)
2458  \end{itemize}
2459
2460  \underline{Munich}
2461  \begin{itemize}
2462   \item Bayerische Forschungsstiftung (financial support)
2463  \end{itemize}
2464
2465  \underline{Paderborn}
2466  \begin{itemize}
2467   \item Prof. J. Lindner (SiC)
2468   \item Prof. G. Schmidt (DFT + financial support)
2469   \item Dr. E. Rauls (DFT + SiC)
2470   \item Dr. S. Sanna (VASP)
2471  \end{itemize}
2472
2473 \vspace{0.2cm}
2474
2475 \begin{center}
2476 \framebox{
2477 \bf Thank you for your attention!
2478 }
2479 \end{center}
2480
2481 \end{slide}
2482
2483 \end{document}
2484
2485 \fi