posic/talks/upb-ua-xc.tex

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  32 \usepackage{semlayer}           % Seminar overlays
  33 \usepackage{slidesec}           % Seminar sections and list of slides
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  48
  49 \begin{document}
  50
  51 \extraslideheight{10in}
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  64 \newcommand{\ham}{\mathcal{H}}
  65 \newcommand{\pot}{\mathcal{V}}
  66 \newcommand{\foo}{\mathcal{U}}
  67 \newcommand{\vir}{\mathcal{W}}
  68
  69 % itemize level ii
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  72 % colors
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  76 \newrgbcolor{lachs}{1.0 .93 .81}
  77
  78 % topic
  79
  80 \begin{slide}
  81 \begin{center}
  82
  83  \vspace{16pt}
  84
  85  {\LARGE\bf
  86   Atomistic simulation study\\[0.2cm]
  87   of the SiC precipitation in Si
  88  }
  89
  90  \vspace{48pt}
  91
  92  \textsc{F. Zirkelbach}
  93
  94  \vspace{48pt}
  95
  96  For the exchange among Paderborn and Augsburg
  97
  98  \vspace{08pt}
  99
 100  July 2009
 101
 102 \end{center}
 103 \end{slide}
 104
 105 % start of contents
 106
 107 \begin{slide}
 108
 109  {\large\bf
 110   VASP parameters
 111  }
 112
 113  \small
 114  \begin{minipage}{6.5cm}
 115  \begin{itemize}
 116   \item Start from scratch
 117   \item $V_{xc}$: US LDA (out of ./pot directory)
 118   \item $k$-points: Monkhorst $4\times 4\times 4$
 119   \item Ionic relaxation
 120         \begin{itemize}
 121          \item Conjugate gradient method
 122          \item Scaling constant of 0.1 for forces
 123          \item Default break condition ($0.1 \cdot 10^{-2}$ eV)
 124          \item Maximum of 100 steps
 125         \end{itemize}
 126         {\color{blue} NVT}:
 127         \begin{itemize}
 128          \item No change in volume
 129         \end{itemize}
 130         {\color{red} NPT}:
 131         \begin{itemize}
 132          \item Change of cell volume and shape\\
 133                allowed
 134         \end{itemize}
 135  \end{itemize}
 136  \end{minipage}
 137  \hspace*{0.5cm}
 138  \begin{minipage}{6.0cm}
 139 {\scriptsize\color{blue}
 140  Example INCAR file (NVT):
 141 }
 142 \begin{verbatim}
 143 System = C 100 interstitial in Si
 144
 145 ISTART = 0
 146
 147 NSW = 100
 148 IBRION = 2
 149 ISIF = 2
 150 POTIM = 0.1
 151 \end{verbatim}
 152 {\scriptsize\color{red}
 153  Example INCAR file (NPT):
 154 }
 155 \begin{verbatim}
 156 System = C hexagonal interstitial in Si
 157
 158 ISTART = 0
 159
 160 NSW = 100
 161 IBRION = 2
 162 ISIF = 3
 163 POTIM = 0.1
 164 \end{verbatim}
 165  \end{minipage}
 166
 167 \end{slide}
 168
 169 \begin{slide}
 170
 171  {\large\bf
 172   Silicon bulk properties
 173  }
 174
 175  \small
 176
 177  Simulations (NPT, $\textrm{EDIFFG}=0.1\cdot 10^{-3}$ eV):
 178  \begin{enumerate}
 179   \item Supercell: $x_1=(0,0.5,0.5),\, x_2=(0.5,0,0.5),\, x_3=(0.5,0.5,0)$;
 180         2 atoms (1 {\bf p}rimitive {\bf c}ell)
 181   \item Supercell: $x_1=(0.5,-0.5,0),\, x_2=(0.5,0.5,0),\, x_3=(0,0,1)$;
 182         4 atoms (2 pc)
 183   \item Supercell: $x_1=(1,0,0),\, x_2=(0,1,0),\, x_3=(0,0,1)$;
 184         8 atoms (4 pc)
 185   \item Supercell: $x_1=(2,0,0),\, x_2=(0,2,0),\, x_3=(0,0,2)$;
 186         64 atoms (32 pc)
 187  \end{enumerate}
 188  \begin{minipage}{6cm}
 189  Cohesive energy / Lattice constant:
 190  \begin{enumerate}
 191   \item $E_{\textrm{cut-off}}=150\, \textrm{eV}$: 5.955 eV / 5.378 \AA\\
 192         $E_{\textrm{cut-off}}=300\, \textrm{eV}$: 5.975 eV / 5.387 \AA
 193   \item $E_{\textrm{cut-off}}=150\, \textrm{eV}$: 5.989 eV / 5.356 \AA
 194   \item $E_{\textrm{cut-off}}=150\, \textrm{eV}$: 5.955 eV / 5.380 \AA\\
 195         $E_{\textrm{cut-off}}=200\, \textrm{eV}$: 5.972 eV / 5.388 \AA\\
 196         $E_{\textrm{cut-off}}=250\, \textrm{eV}$: 5.975 eV / 5.389 \AA\\
 197         $E_{\textrm{cut-off}}=300\, \textrm{eV}$: 5.975 eV / 5.389 \AA\\
 198         $E_{\textrm{cut-off}}=300\, \textrm{eV}^{*}$: 5.975 eV / 5.390 \AA
 199   \item $E_{\textrm{cut-off}}=300\, \textrm{eV}$: 5.977 eV / 5.389 \AA
 200  \end{enumerate}
 201  \end{minipage}
 202  \begin{minipage}{7cm}
 203  \includegraphics[width=7cm]{si_lc_and_ce.ps}
 204  \end{minipage}\\[0.3cm]
 205  {\scriptsize
 206   $^*$special settings (p. 138, VASP manual):
 207   spin polarization, no symmetry, ...
 208  }
 209
 210 \end{slide}
 211
 212 \begin{slide}
 213
 214  {\large\bf
 215   Silicon bulk properties
 216  }
 217
 218  \begin{itemize}
 219   \item Calculation of cohesive energies for different lattice constants
 220   \item No ionic update
 221   \item tetrahedron method with Blöchl corrections for
 222         the partial occupancies $f_{nk}$
 223   \item Supercell 3 (8 atoms, 4 primitive cells)
 224  \end{itemize}
 225  \vspace*{0.6cm}
 226  \begin{minipage}{6.5cm}
 227  \begin{center}
 228  $E_{\textrm{cut-off}}=150$ eV\\
 229  \includegraphics[width=6.5cm]{si_lc_fit.ps}
 230  \end{center}
 231  \end{minipage}
 232  \begin{minipage}{6.5cm}
 233  \begin{center}
 234  $E_{\textrm{cut-off}}=250$ eV\\
 235  \includegraphics[width=6.5cm]{si_lc_fit_250.ps}
 236  \end{center}
 237  \end{minipage}
 238
 239 \end{slide}
 240
 241 \begin{slide}
 242
 243  {\large\bf
 244   3C-SiC bulk properties\\[0.2cm]
 245  }
 246
 247  \begin{minipage}{6.5cm}
 248  \includegraphics[width=6.5cm]{sic_lc_and_ce2.ps}
 249  \end{minipage}
 250  \begin{minipage}{6.5cm}
 251  \includegraphics[width=6.5cm]{sic_lc_and_ce.ps}
 252  \end{minipage}\\[0.3cm]
 253  \begin{itemize}
 254   \item Supercell 3 (4 primitive cells, 4+4 atoms)
 255   \item Error in equilibrium lattice constant: {\color{green} $0.9\,\%$}
 256   \item Error in cohesive energy: {\color{red} $31.6\,\%$}
 257  \end{itemize}
 258
 259 \end{slide}
 260
 261 \begin{slide}
 262
 263  {\large\bf
 264   3C-SiC bulk properties\\[0.2cm]
 265  }
 266
 267  \small
 268
 269  \begin{itemize}
 270   \item Calculation of cohesive energies for different lattice constants
 271   \item No ionic update
 272   \item tetrahedron method with Blöchl corrections for
 273         the partial occupancies $f_{nk}$
 274  \end{itemize}
 275  \vspace*{0.6cm}
 276  \begin{minipage}{6.5cm}
 277  \begin{center}
 278  Supercell 3, $4\times 4\times 4$ k-points\\
 279  \includegraphics[width=6.5cm]{sic_lc_fit.ps}
 280  \end{center}
 281  \end{minipage}
 282  \begin{minipage}{6.5cm}
 283  \begin{center}
 284  {\color{red}
 285   Non-continuous energies\\
 286   for $E_{\textrm{cut-off}}<1050\,\textrm{eV}$!
 287  }
 288  \end{center}
 289  \end{minipage}
 290
 291 \end{slide}
 292
 293 \begin{slide}
 294
 295  {\large\bf
 296   3C-SiC bulk properties\\[0.2cm]
 297  }
 298
 299  \footnotesize
 300
 301 \begin{picture}(0,0)(-188,80)
 302  %Supercell 1, $3\times 3\times 3$ k-points\\
 303  \includegraphics[width=6.5cm]{sic_lc_fit_k3.ps}
 304 \end{picture}
 305
 306  \begin{minipage}{6.5cm}
 307  \begin{itemize}
 308   \item Supercell 1 simulations
 309   \item Variation of k-points
 310   \item Continuous energies for
 311         $E_{\textrm{cut-off}} > 550\,\textrm{eV}$
 312   \item Critical $E_{\textrm{cut-off}}$ for
 313         different k-points\\
 314         depending on supercell?
 315  \end{itemize}
 316  \end{minipage}\\[1.0cm]
 317  \begin{minipage}{6.5cm}
 318  \begin{center}
 319  \includegraphics[width=6.5cm]{sic_lc_fit_k5.ps}
 320  \end{center}
 321  \end{minipage}
 322  \begin{minipage}{6.5cm}
 323  \begin{center}
 324  \includegraphics[width=6.5cm]{sic_lc_fit_k7.ps}
 325  \end{center}
 326  \end{minipage}
 327
 328 \end{slide}
 329
 330 \begin{slide}
 331
 332  {\large\bf
 333   Cohesive energies
 334  }
 335
 336  {\bf\color{red} From now on ...}
 337
 338  {\small Energies used: free energy without entropy ($\sigma \rightarrow 0$)}
 339
 340  \small
 341
 342  \begin{itemize}
 343   \item $E_{\textrm{free,sp}}$:
 344         energy of spin polarized free atom
 345         \begin{itemize}
 346          \item $k$-points: Monkhorst $1\times 1\times 1$
 347          \item Symmetry switched off
 348          \item Spin polarized calculation
 349          \item Interpolation formula according to Vosko Wilk and Nusair
 350                for the correlation part of the exchange correlation functional
 351          \item Gaussian smearing for the partial occupancies $f_{nk}$
 352                ($\sigma=0.05$)
 353          \item Magnetic mixing: AMIX = 0.2, BMIX = 0.0001
 354          \item Supercell: one atom in cubic
 355                $10\times 10\times 10$ \AA$^3$ box
 356         \end{itemize}
 357         {\color{blue}
 358         $E_{\textrm{free,sp}}(\textrm{Si},250\, \textrm{eV})=
 359          -0.70036911\,\textrm{eV}$
 360         },
 361         {\color{gray}
 362         $E_{\textrm{free,sp}}(\textrm{C},xxx\, \textrm{eV})=
 363          yyy\,\textrm{eV}$
 364         }
 365   \item $E$:
 366         energy (non-polarized) of system of interest composed of\\
 367         n atoms of type N, m atoms of type M, \ldots
 368  \end{itemize}
 369  \vspace*{0.3cm}
 370  {\color{red}
 371  \[
 372  \Rightarrow
 373  E_{\textrm{coh}}=\frac{
 374  -\Big(E(N_nM_m\ldots)-nE_{\textrm{free,sp}}(N)-mE_{\textrm{free,sp}}(M)
 375  -\ldots\Big)}
 376  {n+m+\ldots}
 377  \]
 378  }
 379
 380 \end{slide}
 381
 382 \begin{slide}
 383
 384  {\large\bf
 385   Silicon point defects\\
 386  }
 387
 388  \small
 389
 390  Calculation of formation energy $E_{\textrm{f}}$
 391  \begin{itemize}
 392   \item $E_{\textrm{coh}}^{\textrm{initial conf}}$:
 393         cohesive energy per atom of the initial system
 394   \item $E_{\textrm{coh}}^{\textrm{interstitial conf}}$:
 395         cohesive energy per atom of the interstitial system
 396   \item N: amount of atoms in the interstitial system
 397  \end{itemize}
 398  \vspace*{0.2cm}
 399  {\color{blue}
 400  \[
 401  \Rightarrow
 402  E_{\textrm{f}}=\Big(E_{\textrm{coh}}^{\textrm{interstitial conf}}
 403                -E_{\textrm{coh}}^{\textrm{initial conf}}\Big) N
 404  \]
 405  }
 406
 407  \begin{center}
 408  \includegraphics[width=7.0cm]{si_self_int.ps}
 409  \end{center}
 410
 411 \end{slide}
 412
 413 \end{document}
 414