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27 \graphicspath{{../img/}}
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32 \usepackage{semlayer} % Seminar overlays
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 $f_{nk}$
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 $f_{nk}$
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 $E_{\textrm{cut-off}}<1050\,\textrm{eV}$!
296 3C-SiC bulk properties\\[0.2cm]
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}
306 \begin{minipage}{6.5cm}
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
314 depending on supercell?
316 \end{minipage}\\[1.0cm]
317 \begin{minipage}{6.5cm}
319 \includegraphics[width=6.5cm]{sic_lc_fit_k5.ps}
322 \begin{minipage}{6.5cm}
324 \includegraphics[width=6.5cm]{sic_lc_fit_k7.ps}
336 {\bf\color{red} From now on ...}
338 {\small Energies used: free energy without entropy ($\sigma \rightarrow 0$)}
343 \item $E_{\textrm{free,sp}}$:
344 energy of spin polarized free atom
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 $f_{nk}$
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
358 $E_{\textrm{free,sp}}(\textrm{Si},250\, \textrm{eV})=
359 -0.70036911\,\textrm{eV}$
362 $E_{\textrm{free,sp}}(\textrm{C},xxx\, \textrm{eV})=
366 energy (non-polarized) of system of interest composed of\\
367 n atoms of type N, m atoms of type M, \ldots
373 E_{\textrm{coh}}=\frac{
374 -\Big(E(N_nM_m\ldots)-nE_{\textrm{free,sp}}(N)-mE_{\textrm{free,sp}}(M)
385 Used types of supercells\\
390 \begin{minipage}{4.3cm}
391 \includegraphics[width=4cm]{sc_type0.eps}\\[0.3cm]
392 \underline{Type 0}\\[0.2cm]
397 1 primitive cell / 2 atoms
399 \begin{minipage}{4.3cm}
400 \includegraphics[width=4cm]{sc_type1.eps}\\[0.3cm]
401 \underline{Type 1}\\[0.2cm]
406 2 primitive cells / 4 atoms
408 \begin{minipage}{4.3cm}
409 \includegraphics[width=4cm]{sc_type2.eps}\\[0.3cm]
410 \underline{Type 2}\\[0.2cm]
415 4 primitive cells / 8 atoms
416 \end{minipage}\\[0.4cm]
419 In the following these types of supercells are used and
420 are possibly scaled by integers in the different directions!
428 Silicon point defects\\
433 Calculation of formation energy $E_{\textrm{f}}$
435 \item $E_{\textrm{coh}}^{\textrm{initial conf}}$:
436 cohesive energy per atom of the initial system
437 \item $E_{\textrm{coh}}^{\textrm{interstitial conf}}$:
438 cohesive energy per atom of the interstitial system
439 \item N: amount of atoms in the interstitial system
445 E_{\textrm{f}}=\Big(E_{\textrm{coh}}^{\textrm{interstitial conf}}
446 -E_{\textrm{coh}}^{\textrm{initial conf}}\Big) N
449 Influence of supercell size\\
450 \begin{minipage}{8cm}
451 \includegraphics[width=7.0cm]{si_self_int.ps}
453 \begin{minipage}{5cm}
454 $E_{\textrm{f}}^{\textrm{110},\,32\textrm{pc}}=3.38\textrm{ eV}$\\
455 $E_{\textrm{f}}^{\textrm{tet},\,32\textrm{pc}}=3.41\textrm{ eV}$\\
456 $E_{\textrm{f}}^{\textrm{hex},\,32\textrm{pc}}=3.42\textrm{ eV}$\\
457 $E_{\textrm{f}}^{\textrm{vac},\,32\textrm{pc}}=3.51\textrm{ eV}$
465 Questions so far ...\\
468 What configuration to chose for C in Si simulations?
470 \item Switch to another method for the XC approximation (GGA, PAW)?
471 \item Reasonable cut-off energy
472 \item Switch off symmetry? (especially for defect simulations)
474 (Monkhorst? $\Gamma$-point only if cell is large enough?)
475 \item Switch to tetrahedron method or Gaussian smearing ($\sigma$?)
476 \item Size and type of supercell
478 \item connected to choice of $k$-point mesh?
479 \item hence also connected to choice of smearing method?
480 \item constraints can only be applied to the lattice vectors!
490 Review (so far) ...\\