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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 Calculation of the defect formation energy\\
403 {\color{blue}Method 1} (single species)
405 \item $E_{\textrm{coh}}^{\textrm{initial conf}}$:
406 cohesive energy per atom of the initial system
407 \item $E_{\textrm{coh}}^{\textrm{interstitial conf}}$:
408 cohesive energy per atom of the interstitial system
409 \item N: amount of atoms in the interstitial system
415 E_{\textrm{f}}=\Big(E_{\textrm{coh}}^{\textrm{interstitial conf}}
416 -E_{\textrm{coh}}^{\textrm{initial conf}}\Big) N
419 {\color{magenta}Method 2} (two and more species)
421 \item $E$: energy of the interstitial system
422 (with respect to the ground state of the free atoms!)
423 \item $N_{\text{Si}}$, $N_{\text{C}}$:
424 amount of Si and C atoms
425 \item $\mu_{\text{Si}}$, $\mu_{\text{C}}$:
426 chemical potential (cohesive energy) of Si and C
432 E_{\textrm{f}}=E-N_{\text{Si}}\mu_{\text{Si}}-N_{\text{C}}\mu_{\text{C}}
441 Used types of supercells\\
446 \begin{minipage}{4.3cm}
447 \includegraphics[width=4cm]{sc_type0.eps}\\[0.3cm]
448 \underline{Type 0}\\[0.2cm]
453 1 primitive cell / 2 atoms
455 \begin{minipage}{4.3cm}
456 \includegraphics[width=4cm]{sc_type1.eps}\\[0.3cm]
457 \underline{Type 1}\\[0.2cm]
462 2 primitive cells / 4 atoms
464 \begin{minipage}{4.3cm}
465 \includegraphics[width=4cm]{sc_type2.eps}\\[0.3cm]
466 \underline{Type 2}\\[0.2cm]
471 4 primitive cells / 8 atoms
472 \end{minipage}\\[0.4cm]
475 In the following these types of supercells are used and
476 are possibly scaled by integers in the different directions!
484 Silicon point defects\\
489 Influence of supercell size\\
490 \begin{minipage}{8cm}
491 \includegraphics[width=7.0cm]{si_self_int.ps}
493 \begin{minipage}{5cm}
494 $E_{\textrm{f}}^{\textrm{110},\,32\textrm{pc}}=3.38\textrm{ eV}$\\
495 $E_{\textrm{f}}^{\textrm{tet},\,32\textrm{pc}}=3.41\textrm{ eV}$\\
496 $E_{\textrm{f}}^{\textrm{hex},\,32\textrm{pc}}=3.42\textrm{ eV}$\\
497 $E_{\textrm{f}}^{\textrm{vac},\,32\textrm{pc}}=3.51\textrm{ eV}$\\\\
498 $E_{\textrm{f}}^{\textrm{hex},\,54\textrm{pc}}=3.42\textrm{ eV}$\\
499 $E_{\textrm{f}}^{\textrm{tet},\,54\textrm{pc}}=3.45\textrm{ eV}$\\
500 $E_{\textrm{f}}^{\textrm{vac},\,54\textrm{pc}}=3.47\textrm{ eV}$\\
501 $E_{\textrm{f}}^{\textrm{110},\,54\textrm{pc}}=3.48\textrm{ eV}$
504 Comparison with literature (PRL 88 235501 (2002)):\\[0.2cm]
505 \begin{minipage}{8cm}
508 \item $E_{\text{cut-off}}=35 / 25\text{ Ry}=476 / 340\text{ eV}$
509 \item 216 atom supercell
510 \item Gamma point only calculations
513 \begin{minipage}{5cm}
514 $E_{\textrm{f}}^{\textrm{110}}=3.31 / 2.88\textrm{ eV}$\\
515 $E_{\textrm{f}}^{\textrm{hex}}=3.31 / 2.87\textrm{ eV}$\\
516 $E_{\textrm{f}}^{\textrm{vac}}=3.17 / 3.56\textrm{ eV}$
525 Questions so far ...\\
528 What configuration to chose for C in Si simulations?
530 \item Switch to another method for the XC approximation (GGA, PAW)?
531 \item Reasonable cut-off energy
532 \item Switch off symmetry? (especially for defect simulations)
534 (Monkhorst? $\Gamma$-point only if cell is large enough?)
535 \item Switch to tetrahedron method or Gaussian smearing ($\sigma$?)
536 \item Size and type of supercell
538 \item connected to choice of $k$-point mesh?
539 \item hence also connected to choice of smearing method?
540 \item constraints can only be applied to the lattice vectors!
542 \item Use of real space projection operators?
551 Review (so far) ...\\
554 Smearing method for the partial occupancies $f(\{\epsilon_{n{\bf k}}\})$
557 \begin{minipage}{4.4cm}
558 \includegraphics[width=4.4cm]{sic_smear_k.ps}
560 \begin{minipage}{4.4cm}
561 \includegraphics[width=4.4cm]{c_smear_k.ps}
563 \begin{minipage}{4.3cm}
564 \includegraphics[width=4.4cm]{si_smear_k.ps}
565 \end{minipage}\\[0.3cm]
567 \item Convergence reached at $6\times 6\times 6$ k-point mesh
568 \item No difference between Gauss ($\sigma=0.05$)
569 and tetrahedron smearing method!
574 Gauss ($\sigma=0.05$) smearing
575 and $6\times 6\times 6$ Monkhorst $k$-point mesh used
584 Review (so far) ...\\
587 \underline{Symmetry (in defect simulations)}
591 difference in $1\times 1\times 1$ Type 2 defect calculations\\
593 Symmetry precission (SYMPREC) small enough\\
595 {\bf\color{blue}Symmetry switched on}\\
598 \underline{Real space projection}
601 Error in lattice constant of plain Si ($1\times 1\times 1$ Type 2):
603 Error in position of the 110 interstitital in Si ($1\times 1\times 1$ Type 2):
607 Real space projection used for 'large supercell' simulations}
623 3C-SiC equilibrium lattice constant and free energy\\
624 \includegraphics[width=7cm]{plain_sic_lc.ps}\\
625 $\rightarrow$ Convergence reached at 650 eV\\[0.2cm]
631 650 eV used as energy cut-off
641 Not answered (so far) ...\\
663 Final parameter choice
668 \underline{Param 1}\\
669 My first choice. Used for more accurate calculations.
671 \item $6\times 6 \times 6$ Monkhorst k-point mesh
672 \item $E_{\text{cut-off}}=650\text{ eV}$
673 \item Gaussian smearing ($\sigma=0.05$)
677 \underline{Param 2}\\
678 After talking to the pros!
680 \item $\Gamma$-point only
681 \item $E_{\text{cut-off}}=xyz\text{ eV}$
682 \item Gaussian smearing ($\sigma=0.05$)
684 \item Real space projection (Auto, Medium) for 'large' simulations
688 In both parameter sets the ultra soft pseudo potential method
689 as well as the projector augmented wave method is used with both,
690 the LDA and GGA exchange correlation potential!
699 Properties of Si, C and SiC using the new parameters\\
702 $2\times 2\times 2$ Type 2 supercell, Param 1, LDA, US PP\\[0.2cm]
703 \begin{tabular}{|l|l|l|l|}
705 & c-Si & c-C (diamond) & 3C-SiC \\
707 Lattice constant [\AA] & 5.389 & 3.527 & 4.319 \\
708 Expt. [\AA] & 5.429 & 3.567 & 4.359 \\
709 Error [\%] & {\color{green}0.7} & {\color{green}1.1} & {\color{green}0.9} \\
711 Cohesive energy [eV] & -5.277 & -8.812 & -7.318 \\
712 Expt. [eV] & -4.63 & -7.374 & -6.340 \\
713 Error [\%] & {\color{red}14.0} & {\color{red}19.5} & {\color{red}15.4} \\
717 \begin{minipage}{10cm}
718 $2\times 2\times 2$ Type 2 supercell, 3C-SiC, Param 1\\[0.2cm]
719 \begin{tabular}{|l|l|l|l|}
721 & {\color{magenta}US PP, GGA} & PAW, LDA & PAW, GGA \\
723 Lattice constant [\AA] & 4.370 & 4.330 & 4.379 \\
724 Error [\%] & {\color{green}0.3} & {\color{green}0.7} & {\color{green}0.5} \\
726 Cohesive energy [eV] & -6.426 & -7.371 & -6.491 \\
727 Error [\%] & {\color{green}1.4} & {\color{red}16.3} & {\color{green}2.4} \\
731 \begin{minipage}{3cm}
733 \begin{tabular}{|l|l|}
738 {\color{green}0.5} & {\color{green}0.01} \\
741 {\color{green}0.8} & {\color{orange}4.5} \\
751 Energy cut-off for $\Gamma$-point only caclulations
754 $2\times 2\times 2$ Type 2 supercell, Param 2, US PP, LDA, 3C-SiC\\[0.2cm]
755 \includegraphics[width=5.5cm]{sic_32pc_gamma_cutoff.ps}
756 \includegraphics[width=5.5cm]{sic_32pc_gamma_cutoff_lc.ps}\\
757 $\Rightarrow$ Use 300 eV as energy cut-off?\\[0.2cm]
758 $2\times 2\times 2$ Type 2 supercell, Param 2, 300 eV, US PP, GGA\\[0.2cm]
760 \begin{minipage}{10cm}
761 \begin{tabular}{|l|l|l|l|}
763 & c-Si & c-C (diamond) & 3C-SiC \\
765 Lattice constant [\AA] & 5.470 & 3.569 & 4.364 \\
766 Error [\%] & {\color{green}0.8} & {\color{green}0.1} & {\color{green}0.1} \\
768 Cohesive energy [eV] & -4.488 & -7.612 & -6.359 \\
769 Error [\%] & {\color{orange}3.1} & {\color{orange}3.2} & {\color{green}0.3} \\
773 \begin{minipage}{2cm}
775 ${\color{green}\surd}$
784 C 100 interstitial migration along 110 in c-Si (Albe potential)
789 \begin{minipage}[t]{4.2cm}
790 \underline{Starting configuration}\\
791 \includegraphics[width=4cm]{c_100_mig/start.eps}
793 \begin{minipage}[t]{4.0cm}
795 $\Delta x=\frac{1}{4}a_{\text{Si}}=1.357\text{ \AA}$\\
796 $\Delta y=\frac{1}{4}a_{\text{Si}}=1.357\text{ \AA}$\\
797 $\Delta z=0.325\text{ \AA}$\\
799 \begin{minipage}[t]{4.2cm}
800 \underline{{\bf Expected} final configuration}\\
801 \includegraphics[width=4cm]{c_100_mig/final.eps}\\
803 \begin{minipage}{6cm}
805 \item Fix border atoms of the simulation cell
806 \item Constraints and displacement of the C atom:
808 \item along {\color{green}110 direction}\\
809 displaced by {\color{green} $\frac{1}{10}(\Delta x,\Delta y)$}
810 \item C atom {\color{red}entirely fixed in position}\\
812 {\color{red}$\frac{1}{10}(\Delta x,\Delta y,\Delta z)$}
814 \item Berendsen thermostat applied
816 {\bf\color{blue}Expected configuration not obtained!}
818 \begin{minipage}{0.5cm}
821 \begin{minipage}{6cm}
822 \includegraphics[width=6.0cm]{c_100_110mig_01_albe.ps}
830 C 100 interstitial migration along 110 in c-Si (Albe potential)
835 \begin{minipage}{3.2cm}
836 \includegraphics[width=3cm]{c_100_mig/fixmig_50.eps}
841 \begin{minipage}{3.2cm}
842 \includegraphics[width=3cm]{c_100_mig/fixmig_80.eps}
847 \begin{minipage}{3.2cm}
848 \includegraphics[width=3cm]{c_100_mig/fixmig_90.eps}
853 \begin{minipage}{3.2cm}
854 \includegraphics[width=3cm]{c_100_mig/fixmig_99.eps}
862 \item Why is the expected configuration not obtained?
863 \item How to find a migration path preceding to the expected configuration?
868 \item Simple: it is not the right migration path!
870 \item (Surrounding) atoms settle into a local minimum configuration
871 \item A possibly existing more favorable configuration is not achieved
873 \item \begin{itemize}
874 \item Search global minimum in each step (by simulated annealing)\\
876 Loss of the correct energy needed for migration
877 \item Smaller displacements\\
878 A more favorable configuration might be achieved
879 possibly preceding to the expected configuration
889 C 100 interstitial migration along 110 in c-Si (Albe potential)\\
892 Displacement step size decreased to
893 $\frac{1}{100} (\Delta x,\Delta y)$\\[0.2cm]
895 \begin{minipage}{7.5cm}
896 Result: (Video \href{../video/c_in_si_smig_albe.avi}{$\rhd_{\text{local}}$ } $|$
897 \href{http://www.physik.uni-augsburg.de/~zirkelfr/download/posic/c_in_si_smig_albe.avi}{$\rhd_{\text{remote url}}$})
899 \item Expected final configuration not obtained
900 \item Bonds to neighboured silicon atoms persist
901 \item C and neighboured Si atoms move along the direction of displacement
902 \item Even the bond to the lower left silicon atom persists
905 Obviously: overestimated bond strength
908 \begin{minipage}{5cm}
909 \includegraphics[width=6cm]{c_100_110smig_01_albe.ps}
910 \end{minipage}\\[0.4cm]
911 New approach to find the migration path:\\
913 Place interstitial carbon atom at the respective coordinates
914 into a perfect c-Si matrix!
922 C 100 interstitial migration along 110 in c-Si (Albe potential)
925 {\color{blue}New approach:}\\
926 Place interstitial carbon atom at the respective coordinates
927 into a perfect c-Si matrix!\\
928 {\color{blue}Problem:}\\
929 Too high forces due to the small distance of the C atom to the Si
930 atom sharing the lattice site.\\
931 {\color{blue}Solution:}
933 \item {\color{red}Slightly displace the Si atom}
934 (Video \href{../video/c_in_si_pmig_albe.avi}{$\rhd_{\text{local}}$ } $|$
935 \href{http://www.physik.uni-augsburg.de/~zirkelfr/download/posic/c_in_si_pmig_albe.avi}{$\rhd_{\text{remote url}}$})
936 \item {\color{green}Immediately quench the system}
937 (Video \href{../video/c_in_si_pqmig_albe.avi}{$\rhd_{\text{local}}$ } $|$
938 \href{http://www.physik.uni-augsburg.de/~zirkelfr/download/posic/c_in_si_pqmig_albe.avi}{$\rhd_{\text{remote url}}$})
941 \begin{minipage}{6.5cm}
942 \includegraphics[width=6cm]{c_100_110pqmig_01_albe.ps}
944 \begin{minipage}{6cm}
946 \item Jump in energy corresponds to the abrupt
947 structural change (as seen in the videos)
948 \item Due to the abrupt changes in structure and energy
949 this is {\color{red}not} the correct migration path and energy!?!
958 C 100 interstitial migration along 110 in c-Si (VASP)
963 {\color{blue}Method:}
965 \item Place interstitial carbon atom at the respective coordinates
967 \item 110 direction fixed for the C atom
968 \item $4\times 4\times 3$ Type 1, $198+1$ atoms
969 \item Atoms with $x=0$ or $y=0$ or $z=0$ fixed
971 {\color{blue}Results:}
972 (Video \href{../video/c_in_si_pmig_vasp.avi}{$\rhd_{\text{local}}$ } $|$
973 \href{http://www.physik.uni-augsburg.de/~zirkelfr/download/posic/c_in_si_pmig_vasp.avi}{$\rhd_{\text{remote url}}$})\\
974 \begin{minipage}{7cm}
975 \includegraphics[width=7cm]{c_100_110pmig_01_vasp.ps}
977 \begin{minipage}{5.5cm}
979 \item Characteristics nearly equal to classical calulations
980 \item Approximately half of the classical energy
984 Preliminary: 20 \% calculation still running