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33 \usepackage{slidesec} % Seminar sections and list of slides
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86 Atomistic simulation study\\[0.2cm]
87 of the SiC precipitation in Si
92 \textsc{F. Zirkelbach}
96 For the exchange among Paderborn and Augsburg
114 \begin{minipage}{6.5cm}
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
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
128 \item No change in volume
132 \item Change of cell volume and shape\\
138 \begin{minipage}{6.0cm}
139 {\scriptsize\color{blue}
140 Example INCAR file (NVT):
143 System = C 100 interstitial in Si
152 {\scriptsize\color{red}
153 Example INCAR file (NPT):
156 System = C hexagonal interstitial in Si
172 Silicon bulk properties
177 Simulations (NPT, $\textrm{EDIFFG}=0.1\cdot 10^{-3}$ eV):
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: $x_1=(0.5,-0.5,0),\, x_2=(0.5,0.5,0),\, x_3=(0,0,1)$;
183 \item Supercell: $x_1=(1,0,0),\, x_2=(0,1,0),\, x_3=(0,0,1)$;
185 \item Supercell: $x_1=(2,0,0),\, x_2=(0,2,0),\, x_3=(0,0,2)$;
188 \begin{minipage}{6cm}
189 Cohesive energy / Lattice constant:
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
202 \begin{minipage}{7cm}
203 \includegraphics[width=7cm]{si_lc_and_ce.ps}
204 \end{minipage}\\[0.3cm]
206 $^*$special settings (p. 138, VASP manual):
207 spin polarization, no symmetry, ...
215 Silicon bulk properties
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 $f(\{\epsilon_{n{\bf k}}\})$
223 \item Supercell 3 (8 atoms, 4 primitive cells)
226 \begin{minipage}{6.5cm}
228 $E_{\textrm{cut-off}}=150$ eV\\
229 \includegraphics[width=6.5cm]{si_lc_fit.ps}
232 \begin{minipage}{6.5cm}
234 $E_{\textrm{cut-off}}=250$ eV\\
235 \includegraphics[width=6.5cm]{si_lc_fit_250.ps}
244 3C-SiC bulk properties\\[0.2cm]
247 \begin{minipage}{6.5cm}
248 \includegraphics[width=6.5cm]{sic_lc_and_ce2.ps}
250 \begin{minipage}{6.5cm}
251 \includegraphics[width=6.5cm]{sic_lc_and_ce.ps}
252 \end{minipage}\\[0.3cm]
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} $31.6\,\%$}
264 3C-SiC bulk properties\\[0.2cm]
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 $f(\{\epsilon_{n{\bf k}}\})$
276 \begin{minipage}{6.5cm}
278 Supercell 3, $4\times 4\times 4$ k-points\\
279 \includegraphics[width=6.5cm]{sic_lc_fit.ps}
282 \begin{minipage}{6.5cm}
285 Non-continuous energies\\
286 for $E_{\textrm{cut-off}}<1050\,\textrm{eV}$!\\
290 Does this matter in structural optimizaton simulations?
292 \item Derivative might be continuous
293 \item Similar lattice constants where derivative equals zero
304 3C-SiC bulk properties\\[0.2cm]
309 \begin{picture}(0,0)(-188,80)
310 %Supercell 1, $3\times 3\times 3$ k-points\\
311 \includegraphics[width=6.5cm]{sic_lc_fit_k3.ps}
314 \begin{minipage}{6.5cm}
316 \item Supercell 1 simulations
317 \item Variation of k-points
318 \item Continuous energies for
319 $E_{\textrm{cut-off}} > 550\,\textrm{eV}$
320 \item Critical $E_{\textrm{cut-off}}$ for
322 depending on supercell?
324 \end{minipage}\\[1.0cm]
325 \begin{minipage}{6.5cm}
327 \includegraphics[width=6.5cm]{sic_lc_fit_k5.ps}
330 \begin{minipage}{6.5cm}
332 \includegraphics[width=6.5cm]{sic_lc_fit_k7.ps}
344 {\bf\color{red} From now on ...}
346 {\small Energies used: free energy without entropy ($\sigma \rightarrow 0$)}
351 \item $E_{\textrm{free,sp}}$:
352 energy of spin polarized free atom
354 \item $k$-points: Monkhorst $1\times 1\times 1$
355 \item Symmetry switched off
356 \item Spin polarized calculation
357 \item Interpolation formula according to Vosko Wilk and Nusair
358 for the correlation part of the exchange correlation functional
359 \item Gaussian smearing for the partial occupancies
360 $f(\{\epsilon_{n{\bf k}}\})$
362 \item Magnetic mixing: AMIX = 0.2, BMIX = 0.0001
363 \item Supercell: one atom in cubic
364 $10\times 10\times 10$ \AA$^3$ box
367 $E_{\textrm{free,sp}}(\textrm{Si},{\color{green}250}\, \textrm{eV})=
368 -0.70036911\,\textrm{eV}$
371 $E_{\textrm{free,sp}}(\textrm{Si},{\color{red}650}\, \textrm{eV})=
372 -0.70021403\,\textrm{eV}$
375 $E_{\textrm{free,sp}}(\textrm{C},{\color{red}650}\, \textrm{eV})=
376 -1.3535731\,\textrm{eV}$
379 energy (non-polarized) of system of interest composed of\\
380 n atoms of type N, m atoms of type M, \ldots
386 E_{\textrm{coh}}=\frac{
387 -\Big(E(N_nM_m\ldots)-nE_{\textrm{free,sp}}(N)-mE_{\textrm{free,sp}}(M)
398 Used types of supercells\\
403 \begin{minipage}{4.3cm}
404 \includegraphics[width=4cm]{sc_type0.eps}\\[0.3cm]
405 \underline{Type 0}\\[0.2cm]
410 1 primitive cell / 2 atoms
412 \begin{minipage}{4.3cm}
413 \includegraphics[width=4cm]{sc_type1.eps}\\[0.3cm]
414 \underline{Type 1}\\[0.2cm]
419 2 primitive cells / 4 atoms
421 \begin{minipage}{4.3cm}
422 \includegraphics[width=4cm]{sc_type2.eps}\\[0.3cm]
423 \underline{Type 2}\\[0.2cm]
428 4 primitive cells / 8 atoms
429 \end{minipage}\\[0.4cm]
432 In the following these types of supercells are used and
433 are possibly scaled by integers in the different directions!
441 Silicon point defects\\
446 Calculation of formation energy $E_{\textrm{f}}$
448 \item $E_{\textrm{coh}}^{\textrm{initial conf}}$:
449 cohesive energy per atom of the initial system
450 \item $E_{\textrm{coh}}^{\textrm{interstitial conf}}$:
451 cohesive energy per atom of the interstitial system
452 \item N: amount of atoms in the interstitial system
458 E_{\textrm{f}}=\Big(E_{\textrm{coh}}^{\textrm{interstitial conf}}
459 -E_{\textrm{coh}}^{\textrm{initial conf}}\Big) N
462 Influence of supercell size\\
463 \begin{minipage}{8cm}
464 \includegraphics[width=7.0cm]{si_self_int.ps}
466 \begin{minipage}{5cm}
467 $E_{\textrm{f}}^{\textrm{110},\,{\color{red}32}\textrm{pc}}=3.38\textrm{ eV}$\\
468 $E_{\textrm{f}}^{\textrm{hex},\,54\textrm{pc}}=3.42\textrm{ eV}$\\
469 $E_{\textrm{f}}^{\textrm{tet},\,54\textrm{pc}}=3.45\textrm{ eV}$\\
470 $E_{\textrm{f}}^{\textrm{vac},\,54\textrm{pc}}=3.47\textrm{ eV}$
478 Questions so far ...\\
481 What configuration to chose for C in Si simulations?
483 \item Switch to another method for the XC approximation (GGA, PAW)?
484 \item Reasonable cut-off energy
485 \item Switch off symmetry? (especially for defect simulations)
487 (Monkhorst? $\Gamma$-point only if cell is large enough?)
488 \item Switch to tetrahedron method or Gaussian smearing ($\sigma$?)
489 \item Size and type of supercell
491 \item connected to choice of $k$-point mesh?
492 \item hence also connected to choice of smearing method?
493 \item constraints can only be applied to the lattice vectors!
495 \item Use of real space projection operators?
504 Review (so far) ...\\
507 Smearing method for the partial occupancies $f(\{\epsilon_{n{\bf k}}\})$
511 \item $1\times 1\times 1$ Type 0 simulations
513 \item No difference in tetrahedron method and Gauss smearing
516 \item $1\times 1\times 1$ Type 2 simulations
518 \item Again, no difference in tetrahedron method and Gauss smearing
523 {\LARGE\bf\color{red}
524 More simulations running ...
532 Review (so far) ...\\
535 Symmetry (in defect simulations)
537 {\LARGE\bf\color{red}
538 Simulations running ...
546 Review (so far) ...\\
549 Real space projection
556 Review (so far) ...\\
566 Review (so far) ...\\
569 Size and type of supercell
576 Not answered (so far) ...\\