posic/talks/dpg_2008.tex

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  31 \usepackage{semcolor}
  32 \usepackage{semlayer}           % Seminar overlays
  33 \usepackage{slidesec}           % Seminar sections and list of slides
  34
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  42 \begin{document}
  43
  44 \extraslideheight{10in}
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  48
  49 % specify width and height
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  61
  62 % itemize level ii
  63 \renewcommand\labelitemii{{\color{gray}$\bullet$}}
  64
  65 % topic
  66
  67 \begin{slide}
  68 \begin{center}
  69
  70  \vspace{16pt}
  71
  72  {\LARGE\bf
  73   Molecular dynamics simulation study\\
  74   of the silicon carbide precipitation process
  75  }
  76
  77  \vspace{24pt}
  78
  79  \textsc{\small \underline{F. Zirkelbach}$^1$, J. K. N. Lindner$^1$,
  80          K. Nordlund$^2$, B. Stritzker$^1$}\\
  81
  82  \vspace{32pt}
  83
  84  \begin{minipage}{2.0cm}
  85   \begin{center}
  86   \includegraphics[height=1.6cm]{uni-logo.eps}
  87   \end{center}
  88  \end{minipage}
  89  \begin{minipage}{8.0cm}
  90   \begin{center}
  91    {\footnotesize
  92     $^1$ Experimentalphysik IV, Institut f"ur Physik,\\
  93          Universit"at Augsburg, Universit"atsstr. 1,\\
  94          D-86135 Augsburg, Germany
  95    }
  96   \end{center}
  97  \end{minipage}
  98  \begin{minipage}{2.3cm}
  99   \begin{center}
 100   \includegraphics[height=1.5cm]{Lehrstuhl-Logo.eps}
 101   \end{center}
 102  \end{minipage}
 103
 104  \vspace{16pt}
 105
 106  \begin{minipage}{4.0cm}
 107   \begin{center}
 108   \includegraphics[height=1.6cm]{logo_eng.eps}
 109   \end{center}
 110  \end{minipage}
 111  \begin{minipage}{8.0cm}
 112   \begin{center}
 113   {\footnotesize
 114    $^2$ Accelerator Laboratory, Department of Physical Sciences,\\
 115    University of Helsinki, Pietari Kalmink. 2,\\
 116    00014 Helsinki, Finland
 117   }
 118   \end{center}
 119  \end{minipage}
 120 \end{center}
 121 \end{slide}
 122
 123 % contents
 124
 125 \begin{slide}
 126
 127  \begin{center}
 128  {\bf
 129   Molecular dynamics simulation study\\
 130   of the silicon carbide precipitation process
 131  }
 132  \end{center}
 133
 134  \vspace{16pt}
 135
 136  {\large\bf
 137   Outline
 138  }
 139
 140  \vspace{16pt}
 141
 142  \begin{itemize}
 143   \item Motivation / Introduction
 144   \item Molecular dynamics simulation details
 145         \begin{itemize}
 146          \item Integrator, potential, ensemble control
 147          \item Simulation sequence
 148         \end{itemize}
 149   \item Simulation results
 150         \begin{itemize}
 151          \item Interstitials in silicon
 152          \item SiC-precipitation experiments
 153         \end{itemize}
 154   \item Conclusion / Outlook
 155  \end{itemize}
 156 \end{slide}
 157
 158 % start of contents
 159
 160 \begin{slide}
 161
 162  {\large\bf
 163   Motivation / Introduction
 164  }
 165
 166  \vspace{16pt}
 167
 168  Reasons for investigating C in Si:
 169
 170  \begin{itemize}
 171   \item 3C-SiC wide band gap semiconductor formation
 172   \item Strained Si (no precipitation wanted!)
 173  \end{itemize}
 174
 175  \vspace{16pt}
 176
 177  Si / 3C-SiC facts:
 178
 179  \begin{minipage}{8cm}
 180  \begin{itemize}
 181   \item Unit cell:
 182         \begin{itemize}
 183          \item {\color{yellow}fcc} $+$
 184          \item {\color{gray}fcc shifted $1/4$ of volume diagonal}
 185         \end{itemize}
 186   \item Lattice constants: $4a_{Si}\approx5a_{SiC}$
 187   \item Silicon density:
 188         \[
 189         \frac{n_{SiC}}{n_{Si}}=
 190         \frac{4/a_{SiC}^3}{8/a_{Si}^3}=
 191         \frac{5^3}{2\cdot4^3}={\color{cyan}97,66}\,\%
 192         \]
 193  \end{itemize}
 194  \end{minipage}
 195  \hspace{8pt}
 196  \begin{minipage}{4cm}
 197  \includegraphics[width=4cm]{sic_unit_cell.eps}
 198  \end{minipage}
 199
 200 \end{slide}
 201
 202  \small
 203 \begin{slide}
 204
 205  {\large\bf
 206   Motivation / Introduction
 207  }
 208
 209  \small
 210  \vspace{6pt}
 211
 212  Supposed conversion mechanism of heavily carbon doped Si into SiC:
 213
 214  \vspace{8pt}
 215
 216  \begin{minipage}{3.8cm}
 217  \includegraphics[width=3.7cm]{sic_prec_seq_01.eps}
 218  \end{minipage}
 219  \hspace{0.6cm}
 220  \begin{minipage}{3.8cm}
 221  \includegraphics[width=3.7cm]{sic_prec_seq_02.eps}
 222  \end{minipage}
 223  \hspace{0.6cm}
 224  \begin{minipage}{3.8cm}
 225  \includegraphics[width=3.7cm]{sic_prec_seq_03.eps}
 226  \end{minipage}
 227
 228  \vspace{8pt}
 229
 230  \begin{minipage}{3.8cm}
 231  Formation of C-Si dumbbells on regular c-Si lattice sites
 232  \end{minipage}
 233  \hspace{0.6cm}
 234  \begin{minipage}{3.8cm}
 235  Agglomeration into large clusters (embryos)\\
 236  \end{minipage}
 237  \hspace{0.6cm}
 238  \begin{minipage}{3.8cm}
 239  Precipitation of 3C-SiC + Creation of interstitials\\
 240  \end{minipage}
 241
 242  \vspace{12pt}
 243
 244  Experimentally observed:
 245  \begin{itemize}
 246   \item Minimal diameter of precipitation: 4 - 5 nm
 247   \item (hkl)-planes identical for Si and SiC
 248  \end{itemize}
 249
 250 \end{slide}
 251
 252 \begin{slide}
 253
 254  {\large\bf
 255   Simulation details
 256  }
 257
 258  \vspace{12pt}
 259
 260  MD basics:
 261  \begin{itemize}
 262   \item Microscopic description of N particle system
 263   \item Analytical interaction potential
 264   \item Hamilton's equations of motion as propagation rule\\
 265         in 6N-dimensional phase space
 266   \item Observables obtained by time average
 267  \end{itemize}
 268
 269  \vspace{12pt}
 270
 271  Application details:
 272  \begin{itemize}
 273   \item Integrator: Velocity Verlet, timestep: $1\, fs$
 274   \item Ensemble control: NVT, Berendsen thermostat, $\tau=100.0$
 275   \item Potential: Tersoff-like bond order potential\\
 276         \[
 277         E = \frac{1}{2} \sum_{i \neq j} \pot_{ij}, \quad
 278         \pot_{ij} = f_C(r_{ij}) \left[ f_R(r_{ij}) + b_{ij} f_A(r_{ij}) \right]
 279         \]
 280         \begin{center}
 281         {\scriptsize P. Erhart und K. Albe. Phys. Rev. B 71 (2005) 035211}
 282         \end{center}
 283  \end{itemize}
 284
 285  \begin{picture}(0,0)(-240,-70)
 286   \includegraphics[width=5cm]{tersoff_angle.eps}
 287  \end{picture}
 288
 289 \end{slide}
 290
 291 \begin{slide}
 292
 293  {\large\bf
 294   Simulation details
 295  }
 296
 297  \vspace{20pt}
 298
 299  Interstitial experiments:
 300
 301  \vspace{12pt}
 302
 303  \begin{itemize}
 304   \item Initial configuration: $9\times9\times9$ unit cells Si
 305   \item Periodic boundary conditions
 306   \item $T=0 \, K$
 307   \item Insertion of Si / C atom at
 308         \begin{itemize}
 309          \item $(0,0,0)$ $\rightarrow$ {\color{red}tetrahedral}
 310          \item $(-1/8,-1/8,1/8)$ $\rightarrow$ {\color{green}hexagonal}
 311          \item $(-1/8,-1/8,-1/4)$, $(-1/4,-1/4,-1/4)$\\
 312                $\rightarrow$ {\color{yellow}110 dumbbell}
 313          \item random positions (critical distance check)
 314         \end{itemize}
 315   \item Relaxation time: $2\, ps$
 316   \item Optional heating-up
 317  \end{itemize}
 318
 319  \begin{picture}(0,0)(-210,-45)
 320   \includegraphics[width=6cm]{unit_cell.eps}
 321  \end{picture}
 322
 323 \end{slide}
 324
 325 \begin{slide}
 326
 327  {\large\bf
 328   Simulation details
 329  }
 330
 331  \small
 332
 333  SiC precipitation experiments:
 334
 335  \begin{pspicture}(0,0)(12,8)
 336   % nodes
 337   \rput(4.5,6.5){\rnode{init}{\psframebox{\parbox{7cm}{
 338    \begin{itemize}
 339     \item Initial configuration: $31\times31\times31$ unit cells Si
 340     \item Periodic boundary conditions
 341     \item $T=450\, ^{\circ}C$
 342     \item Equilibration of $E_{kin}$ and $E_{pot}$ for $600\, fs$
 343    \end{itemize}
 344   }}}}
 345   \rput(4.5,4.5){\rnode{tc1}{\psframebox[fillstyle=solid,fillcolor=red]{
 346                              $T=450\pm 1\, ^{\circ}C$}}}
 347   \rput(7,3.5){\rnode{insert}{\psframebox[fillstyle=solid,fillcolor=red]{
 348                               \parbox{3cm}{
 349                               Insertion of 10 atoms\\
 350                               at random positions}}}}
 351   \rput(2,3.5){\rnode{adj1}{\psframebox[fillstyle=solid,fillcolor=red]{
 352                             \parbox{3.5cm}{
 353                             Adjusting temperature\\
 354                             for another $100\, fs$}}}}
 355   \rput(7,2.5){\rnode{nc}{\psframebox[fillstyle=solid,fillcolor=red]{
 356                           $N_{atoms}=6000$}}}
 357   \rput(4.5,2){\rnode{tc2}{\psframebox[fillstyle=solid,fillcolor=cyan]{
 358                            $T=T_{set}$}}}
 359   \rput(7,1){\rnode{td}{\psframebox[fillstyle=solid,fillcolor=cyan]{
 360                         $T_{set}:=T_{set}-1\, ^{\circ}C$}}}
 361   \rput(2,1){\rnode{adj2}{\psframebox[fillstyle=solid,fillcolor=cyan]{
 362                           \parbox{3.5cm}{
 363                           Adjusting temperature\\
 364                           for another $50\, fs$}}}}
 365   \rput(7,0){\rnode{tc3}{\psframebox[fillstyle=solid,fillcolor=cyan]{
 366                          $T_{set}=20\, ^{\circ}C$}}}
 367   \rput(10,0){\rnode{end}{\psframebox{End}}}
 368   % help nodes
 369   \rput(7,4.5){\pnode{tc1-h}}
 370   \rput(2,4.5){\pnode{tc1-hh}}
 371   \rput(4.5,2.5){\pnode{nc-h}}
 372   \rput(9,2.5){\pnode{nc-hh}}
 373   \rput(9,2){\pnode{tc2-h}}
 374   \rput(2,2){\pnode{tc2-hh}}
 375   \rput(4.5,0){\pnode{tc3-h}}
 376   % direct lines
 377   \ncline[]{->}{init}{tc1}
 378   \ncline[]{->}{adj1}{tc1}
 379   \ncline[]{->}{insert}{nc}
 380   \ncline[]{->}{adj2}{tc2}
 381   \ncline[]{->}{tc2}{td}
 382   \lput*{0}{yes}
 383   \ncline[]{->}{td}{tc3}
 384   \ncline[]{->}{tc3}{end}
 385   \lput*{0}{yes}
 386   % lines using help nodes
 387   \ncline[]{tc1}{tc1-h}
 388   \lput*{0}{yes}
 389   \ncline[]{->}{tc1-h}{insert}
 390   \ncline[]{tc1}{tc1-hh}
 391   \lput*{0}{no}
 392   \ncline[]{->}{tc1-hh}{adj1}
 393   \ncline[]{nc}{nc-h}
 394   \lput*{0}{no}
 395   \ncline[]{->}{nc-h}{tc1}
 396   \ncline[]{nc}{nc-hh}
 397   \ncline[]{-}{nc-hh}{tc2-h}
 398   \ncline[]{->}{tc2-h}{tc2}
 399   \lput*{0}{yes, {\footnotesize $T_{set}:=450\, ^{\circ}C$}}
 400   \ncline[]{tc2}{tc2-hh}
 401   \lput*{0}{no}
 402   \ncline[]{->}{tc2-hh}{adj2}
 403   \ncline[]{tc3}{tc3-h}
 404   \lput*{0}{no}
 405   \ncline[]{->}{tc3-h}{tc2}
 406   % insertion volumes
 407   \psframe[fillstyle=solid,fillcolor=white](9.5,1.3)(13.5,5.3)
 408   \psframe[fillstyle=solid,fillcolor=lightgray](10,1.8)(13,4.8)
 409   \psframe[fillstyle=solid,fillcolor=gray](10.5,2.3)(12.5,4.3)
 410   \rput(9.75,3){\pnode{ins1}}
 411   \rput(10.25,3.3){\pnode{ins2}}
 412   \rput(10.75,3.6){\pnode{ins3}}
 413   \ncline[]{-}{insert}{ins1}
 414   \ncline[]{-}{insert}{ins2}
 415   \ncline[]{-}{insert}{ins3}
 416   \psframe[fillstyle=solid,fillcolor=white](9.5,7.6)(13.5,8.1)
 417   \psframe[fillstyle=solid,fillcolor=lightgray](9.5,6.8)(13.5,7.3)
 418   \psframe[fillstyle=solid,fillcolor=gray](9.5,6)(13.5,6.5)
 419   \rput(11.5,7.85){{\tiny Simulation volume:
 420                           $31\times31\times31\, a^3_{Si}$}}
 421   \rput(11.5,7.05){{\tiny Volume of minimal SiC precipitation}}
 422   \rput(11.5,6.25){{\tiny Volume of necessary amount of Si}}
 423  \end{pspicture}
 424
 425 \end{slide}
 426
 427 \begin{slide}
 428
 429  {\large\bf
 430   Results
 431  }
 432
 433  Si self-interstitial experiments:
 434
 435  {\footnotesize
 436  {\bf Note:}
 437  \begin{itemize}
 438   \item $r_{cutoff}^{Si-Si}=2.96>\frac{5.43}{2}$
 439   \item Bond length near $r_{cutoff} \Rightarrow$ small bond strength
 440  \end{itemize}
 441  }
 442
 443  \vspace{8pt}
 444
 445  \small
 446
 447  \begin{minipage}[t]{4.0cm}
 448  \underline{Tetrahedral}
 449  \begin{itemize}
 450   \item $E_f=3.41\, eV$
 451   \item essentialy tetrahedral\\
 452         bonds
 453  \end{itemize}
 454  \end{minipage}
 455  \hspace{0.3cm}
 456  \begin{minipage}[t]{4.0cm}
 457  \underline{110 dumbbell}
 458  \begin{itemize}
 459   \item $E_f=4.39\, eV$
 460   \item essentially 4 bonds
 461  \end{itemize}
 462  \end{minipage}
 463  \hspace{0.3cm}
 464  \begin{minipage}[t]{4.0cm}
 465  \underline{Hexagonal}
 466  \begin{itemize}
 467   \item $E_f^{\star}\approx4.48\, eV$
 468   \item unstable!
 469  \end{itemize}
 470  \end{minipage}
 471
 472  \vspace{8pt}
 473
 474  \begin{minipage}[t]{4.3cm}
 475  \includegraphics[width=3.8cm]{si_self_int_tetra_0.eps}
 476  \end{minipage}
 477  \begin{minipage}[t]{4.3cm}
 478  \includegraphics[width=3.8cm]{si_self_int_dumbbell_0.eps}
 479  \end{minipage}
 480  \begin{minipage}[t]{4.3cm}
 481  \includegraphics[width=3.8cm]{si_self_int_hexa_0.eps}
 482  \begin{center}
 483  \href{../video/si_self_int_hexa.avi}{$\rhd$}
 484  \end{center}
 485  \end{minipage}
 486
 487 \end{slide}
 488
 489 \begin{slide}
 490
 491  {\large\bf
 492   Results
 493  }
 494
 495  \vspace{8pt}
 496
 497  Si self-interstitial \underline{random insertion} experiments:
 498
 499  \small
 500
 501  \vspace{8pt}
 502
 503  \begin{minipage}[t]{4.0cm}
 504  \begin{itemize}
 505   \item $E_f=3.97\, eV$
 506   \item 3 identical weak bonds
 507   \item displaced in volume\\ diagonal
 508  \end{itemize}
 509  \end{minipage}
 510  \hspace{0.3cm}
 511  \begin{minipage}[t]{4.0cm}
 512  \begin{itemize}
 513   \item $E_f=3.75\, eV$
 514   \item 4 identical weak bonds
 515   \item displaced in plane\\ diagonal
 516  \end{itemize}
 517  \end{minipage}
 518  \hspace{0.3cm}
 519  \begin{minipage}[t]{4.0cm}
 520  \begin{itemize}
 521   \item $E_f=3.56\, eV$
 522   \item single weak bond
 523   \item displaced along\\ $x$-direction
 524   \item closest to tetrahedral\\ configuration
 525  \end{itemize}
 526  \end{minipage}
 527
 528  \vspace{8pt}
 529
 530  \begin{minipage}{4.3cm}
 531  \includegraphics[width=3.8cm]{si_self_int_rand_397_0.eps}
 532  \end{minipage}
 533  \begin{minipage}{4.3cm}
 534  \includegraphics[width=3.8cm]{si_self_int_rand_375_0.eps}
 535  \end{minipage}
 536  \begin{minipage}{4.3cm}
 537  \includegraphics[width=3.8cm]{si_self_int_rand_356_0.eps}
 538  \end{minipage}
 539
 540  \vspace{8pt}
 541
 542  \begin{center}
 543   {\footnotesize
 544    {\bf Note:} Displacements relative to tetrahedral configuration
 545   }
 546  \end{center}
 547
 548 \end{slide}
 549
 550 \begin{slide}
 551
 552  {\large\bf
 553   Results
 554  }
 555
 556  \vspace{8pt}
 557
 558  Carbon interstitial experiments:
 559
 560  \vspace{12pt}
 561
 562  \small
 563
 564  \begin{minipage}[t]{4.0cm}
 565  \underline{Tetrahedral}
 566  \begin{itemize}
 567   \item $E_f=2.67\, eV$
 568   \item tetrahedral bond
 569  \end{itemize}
 570  \end{minipage}
 571  \hspace{0.3cm}
 572  \begin{minipage}[t]{4.0cm}
 573  \underline{110 dumbbell}
 574  \begin{itemize}
 575   \item $E_f=1.76\, eV$
 576   \item C forms 3 bonds
 577  \end{itemize}
 578  \end{minipage}
 579  \hspace{0.3cm}
 580  \begin{minipage}[t]{4.0cm}
 581  \underline{Hexagonal}
 582  \begin{itemize}
 583   \item $E_f^{\star}\approx5.6\, eV$
 584   \item unstable!
 585  \end{itemize}
 586  \end{minipage}
 587
 588  \vspace{8pt}
 589
 590  \begin{minipage}[t]{4.3cm}
 591  \includegraphics[width=3.8cm]{c_in_si_int_tetra_0.eps}
 592  \end{minipage}
 593  \begin{minipage}[t]{4.3cm}
 594  \includegraphics[width=3.8cm]{c_in_si_int_dumbbell_0.eps}
 595  \end{minipage}
 596  \begin{minipage}[t]{4.3cm}
 597  \includegraphics[width=3.8cm]{c_in_si_int_hexa_0.eps}
 598  \begin{center}
 599  \href{../video/c_in_si_int_hexa.avi}{$\rhd$}
 600  \end{center}
 601  \end{minipage}
 602
 603 \end{slide}
 604
 605 \begin{slide}
 606
 607  {\large\bf
 608   Results
 609  }
 610
 611  %\vspace{8pt}
 612
 613  Carbon \underline{random insertion} experiments:
 614
 615  %\vspace{8pt}
 616
 617  \footnotesize
 618
 619  \begin{minipage}[c]{6.3cm}
 620   \begin{minipage}{3.4cm}
 621    \includegraphics[width=3.3cm]{c_in_si_int_001db_0.eps}
 622   \end{minipage}
 623   \begin{minipage}{2.5cm}
 624    \begin{itemize}
 625     \item $E_f=0.47\, eV$
 626     \item 001 dumbbell
 627    \end{itemize}
 628   \end{minipage}
 629  \end{minipage}
 630  \begin{minipage}[c]{6.3cm}
 631   \begin{minipage}{3.4cm}
 632    \includegraphics[width=3.3cm]{c_in_si_int_rand_162_0.eps}
 633   \end{minipage}
 634   \begin{minipage}{2.8cm}
 635    \begin{itemize}
 636     \item $E_f=1.62\, eV$
 637     \item 3 weak + strong bonds
 638    \end{itemize}
 639   \end{minipage}
 640  \end{minipage}
 641
 642  \begin{minipage}[c]{6.3cm}
 643   \begin{minipage}{3.4cm}
 644    \includegraphics[width=3.3cm]{c_in_si_int_rand_239_0.eps}
 645   \end{minipage}
 646   \begin{minipage}{2.5cm}
 647    \begin{itemize}
 648     \item $E_f=2.39\, eV$
 649    \end{itemize}
 650    \begin{center}
 651     \href{../video/c_in_si_int_rand_239.avi}{$\rhd$}
 652    \end{center}
 653   \end{minipage}
 654  \end{minipage}
 655  \begin{minipage}[c]{6.3cm}
 656   \begin{minipage}{3.4cm}
 657    \includegraphics[width=3.3cm]{c_in_si_int_rand_341_0.eps}
 658   \end{minipage}
 659   \begin{minipage}{2.8cm}
 660    \begin{itemize}
 661     \item $E_f=3.41\, eV$
 662    \end{itemize}
 663    \begin{center}
 664     \href{../video/c_in_si_int_rand_341.avi}{$\rhd$}
 665    \end{center}
 666   \end{minipage}
 667  \end{minipage}
 668
 669  \vspace{4pt}
 670
 671  \begin{center}
 672  {\bf Note:} High probability for 110 dumbbell ($1.76\, eV$) configurations!
 673  \end{center}
 674
 675 \end{slide}
 676
 677 \begin{slide}
 678
 679  {\large\bf
 680   Results
 681  }
 682
 683  SiC-precipitation experiments:
 684
 685  \begin{minipage}[t]{6.3cm}
 686   %\input{../plot/sic_prec}
 687   \includegraphics[width=6.0cm]{../plot/sic_prec_energy.ps}
 688   \includegraphics[width=6.0cm]{../plot/sic_prec_temp.ps}
 689  \end{minipage}
 690  \begin{minipage}[t]{6cm}
 691   \includegraphics[width=6.0cm]{../plot/sic_pc.ps}
 692   \includegraphics[width=6.0cm]{../plot/sic_prec_pc.ps}
 693  \end{minipage}
 694
 695 \end{slide}
 696
 697 \begin{slide}
 698
 699  {\large\bf
 700   Conclusion / Outlook
 701  }
 702
 703 \vspace{24pt}
 704
 705 \begin{itemize}
 706  \item Importance of understanding C in Si
 707  \item Interstitial configurations in silicon using the Albe potential
 708  \item Indication of SiC precipitation
 709 \end{itemize}
 710
 711 \vspace{16pt}
 712
 713 \begin{itemize}
 714  \item Displacement and stress calculations
 715  \item Diffusion dependence of temperature and carbon concentration
 716  \item Analyzing results of the precipitation simulation runs
 717  \item Analyzing self-designed Si/SiC interface
 718 \end{itemize}
 719
 720 \end{slide}
 721
 722 \end{document}
 723