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88 Atomistic simulation study\\[0.2cm]
89 of the SiC precipitation in Si
94 \textsc{F. Zirkelbach}
98 For the exchange among Paderborn and Augsburg
116 \begin{minipage}{6.5cm}
118 \item Start from scratch
119 \item $V_{xc}$: US LDA (out of ./pot directory)
120 \item $k$-points: Monkhorst $4\times 4\times 4$
121 \item Ionic relaxation
123 \item Conjugate gradient method
124 \item Scaling constant of 0.1 for forces
125 \item Default break condition ($0.1 \cdot 10^{-2}$ eV)
126 \item Maximum of 100 steps
130 \item No change in volume
134 \item Change of cell volume and shape\\
140 \begin{minipage}{6.0cm}
141 {\scriptsize\color{blue}
142 Example INCAR file (NVT):
145 System = C 100 interstitial in Si
154 {\scriptsize\color{red}
155 Example INCAR file (NPT):
158 System = C hexagonal interstitial in Si
174 Silicon bulk properties
179 Simulations (NPT, $\textrm{EDIFFG}=0.1\cdot 10^{-3}$ eV):
181 \item Supercell: $x_1=(0,0.5,0.5),\, x_2=(0.5,0,0.5),\, x_3=(0.5,0.5,0)$;
182 2 atoms (1 {\bf p}rimitive {\bf c}ell)
183 \item Supercell: $x_1=(0.5,-0.5,0),\, x_2=(0.5,0.5,0),\, x_3=(0,0,1)$;
185 \item Supercell: $x_1=(1,0,0),\, x_2=(0,1,0),\, x_3=(0,0,1)$;
187 \item Supercell: $x_1=(2,0,0),\, x_2=(0,2,0),\, x_3=(0,0,2)$;
190 \begin{minipage}{6cm}
191 Cohesive energy / Lattice constant:
193 \item $E_{\textrm{cut-off}}=150\, \textrm{eV}$: 5.955 eV / 5.378 \AA\\
194 $E_{\textrm{cut-off}}=300\, \textrm{eV}$: 5.975 eV / 5.387 \AA
195 \item $E_{\textrm{cut-off}}=150\, \textrm{eV}$: 5.989 eV / 5.356 \AA
196 \item $E_{\textrm{cut-off}}=150\, \textrm{eV}$: 5.955 eV / 5.380 \AA\\
197 $E_{\textrm{cut-off}}=200\, \textrm{eV}$: 5.972 eV / 5.388 \AA\\
198 $E_{\textrm{cut-off}}=250\, \textrm{eV}$: 5.975 eV / 5.389 \AA\\
199 $E_{\textrm{cut-off}}=300\, \textrm{eV}$: 5.975 eV / 5.389 \AA\\
200 $E_{\textrm{cut-off}}=300\, \textrm{eV}^{*}$: 5.975 eV / 5.390 \AA
201 \item $E_{\textrm{cut-off}}=300\, \textrm{eV}$: 5.977 eV / 5.389 \AA
204 \begin{minipage}{7cm}
205 \includegraphics[width=7cm]{si_lc_and_ce.ps}
206 \end{minipage}\\[0.3cm]
208 $^*$special settings (p. 138, VASP manual):
209 spin polarization, no symmetry, ...
217 Silicon bulk properties
221 \item Calculation of cohesive energies for different lattice constants
222 \item No ionic update
223 \item Tetrahedron method with Blöchl corrections for
224 the partial occupancies $f(\{\epsilon_{n{\bf k}}\})$
225 \item Supercell 3 (8 atoms, 4 primitive cells)
228 \begin{minipage}{6.5cm}
230 $E_{\textrm{cut-off}}=150$ eV\\
231 \includegraphics[width=6.5cm]{si_lc_fit.ps}
234 \begin{minipage}{6.5cm}
236 $E_{\textrm{cut-off}}=250$ eV\\
237 \includegraphics[width=6.5cm]{si_lc_fit_250.ps}
246 3C-SiC bulk properties\\[0.2cm]
249 \begin{minipage}{6.5cm}
250 \includegraphics[width=6.5cm]{sic_lc_and_ce2.ps}
252 \begin{minipage}{6.5cm}
253 \includegraphics[width=6.5cm]{sic_lc_and_ce.ps}
254 \end{minipage}\\[0.3cm]
256 \item Supercell 3 (4 primitive cells, 4+4 atoms)
257 \item Error in equilibrium lattice constant: {\color{green} $0.9\,\%$}
258 \item Error in cohesive energy: {\color{red} $31.6\,\%$}
266 3C-SiC bulk properties\\[0.2cm]
272 \item Calculation of cohesive energies for different lattice constants
273 \item No ionic update
274 \item Tetrahedron method with Blöchl corrections for
275 the partial occupancies $f(\{\epsilon_{n{\bf k}}\})$
278 \begin{minipage}{6.5cm}
280 Supercell 3, $4\times 4\times 4$ k-points\\
281 \includegraphics[width=6.5cm]{sic_lc_fit.ps}
284 \begin{minipage}{6.5cm}
287 Non-continuous energies\\
288 for $E_{\textrm{cut-off}}<1050\,\textrm{eV}$!\\
292 Does this matter in structural optimizaton simulations?
294 \item Derivative might be continuous
295 \item Similar lattice constants where derivative equals zero
306 3C-SiC bulk properties\\[0.2cm]
311 \begin{picture}(0,0)(-188,80)
312 %Supercell 1, $3\times 3\times 3$ k-points\\
313 \includegraphics[width=6.5cm]{sic_lc_fit_k3.ps}
316 \begin{minipage}{6.5cm}
318 \item Supercell 1 simulations
319 \item Variation of k-points
320 \item Continuous energies for
321 $E_{\textrm{cut-off}} > 550\,\textrm{eV}$
322 \item Critical $E_{\textrm{cut-off}}$ for
324 depending on supercell?
326 \end{minipage}\\[1.0cm]
327 \begin{minipage}{6.5cm}
329 \includegraphics[width=6.5cm]{sic_lc_fit_k5.ps}
332 \begin{minipage}{6.5cm}
334 \includegraphics[width=6.5cm]{sic_lc_fit_k7.ps}
346 {\bf\color{red} From now on ...}
348 {\small Energies used: free energy without entropy ($\sigma \rightarrow 0$)}
353 \item $E_{\textrm{free,sp}}$:
354 energy of spin polarized free atom
356 \item $k$-points: Monkhorst $1\times 1\times 1$
357 \item Symmetry switched off
358 \item Spin polarized calculation
359 \item Interpolation formula according to Vosko Wilk and Nusair
360 for the correlation part of the exchange correlation functional
361 \item Gaussian smearing for the partial occupancies
362 $f(\{\epsilon_{n{\bf k}}\})$
364 \item Magnetic mixing: AMIX = 0.2, BMIX = 0.0001
365 \item Supercell: one atom in cubic
366 $10\times 10\times 10$ \AA$^3$ box
369 $E_{\textrm{free,sp}}(\textrm{Si},{\color{green}250}\, \textrm{eV})=
370 -0.70036911\,\textrm{eV}$
373 $E_{\textrm{free,sp}}(\textrm{Si},{\color{red}650}\, \textrm{eV})=
374 -0.70021403\,\textrm{eV}$
377 $E_{\textrm{free,sp}}(\textrm{C},{\color{red}650}\, \textrm{eV})=
378 -1.3535731\,\textrm{eV}$
381 energy (non-polarized) of system of interest composed of\\
382 n atoms of type N, m atoms of type M, \ldots
388 E_{\textrm{coh}}=\frac{
389 -\Big(E(N_nM_m\ldots)-nE_{\textrm{free,sp}}(N)-mE_{\textrm{free,sp}}(M)
400 Calculation of the defect formation energy\\
405 {\color{blue}Method 1} (single species)
407 \item $E_{\textrm{coh}}^{\textrm{initial conf}}$:
408 cohesive energy per atom of the initial system
409 \item $E_{\textrm{coh}}^{\textrm{interstitial conf}}$:
410 cohesive energy per atom of the interstitial system
411 \item N: amount of atoms in the interstitial system
417 E_{\textrm{f}}=\Big(E_{\textrm{coh}}^{\textrm{interstitial conf}}
418 -E_{\textrm{coh}}^{\textrm{initial conf}}\Big) N
421 {\color{magenta}Method 2} (two and more species)
423 \item $E$: energy of the interstitial system
424 (with respect to the ground state of the free atoms!)
425 \item $N_{\text{Si}}$, $N_{\text{C}}$:
426 amount of Si and C atoms
427 \item $\mu_{\text{Si}}$, $\mu_{\text{C}}$:
428 chemical potential (cohesive energy) of Si and C
434 E_{\textrm{f}}=E-N_{\text{Si}}\mu_{\text{Si}}-N_{\text{C}}\mu_{\text{C}}
443 Used types of supercells\\
448 \begin{minipage}{4.3cm}
449 \includegraphics[width=4cm]{sc_type0.eps}\\[0.3cm]
450 \underline{Type 0}\\[0.2cm]
455 1 primitive cell / 2 atoms
457 \begin{minipage}{4.3cm}
458 \includegraphics[width=4cm]{sc_type1.eps}\\[0.3cm]
459 \underline{Type 1}\\[0.2cm]
464 2 primitive cells / 4 atoms
466 \begin{minipage}{4.3cm}
467 \includegraphics[width=4cm]{sc_type2.eps}\\[0.3cm]
468 \underline{Type 2}\\[0.2cm]
473 4 primitive cells / 8 atoms
474 \end{minipage}\\[0.4cm]
477 In the following these types of supercells are used and
478 are possibly scaled by integers in the different directions!
486 Silicon point defects\\
491 Influence of supercell size\\
492 \begin{minipage}{8cm}
493 \includegraphics[width=7.0cm]{si_self_int.ps}
495 \begin{minipage}{5cm}
496 $E_{\textrm{f}}^{\hkl<1 1 0>,\,32\textrm{pc}}=3.38\textrm{ eV}$\\
497 $E_{\textrm{f}}^{\textrm{tet},\,32\textrm{pc}}=3.41\textrm{ eV}$\\
498 $E_{\textrm{f}}^{\textrm{hex},\,32\textrm{pc}}=3.42\textrm{ eV}$\\
499 $E_{\textrm{f}}^{\textrm{vac},\,32\textrm{pc}}=3.51\textrm{ eV}$\\\\
500 $E_{\textrm{f}}^{\textrm{hex},\,54\textrm{pc}}=3.42\textrm{ eV}$\\
501 $E_{\textrm{f}}^{\textrm{tet},\,54\textrm{pc}}=3.45\textrm{ eV}$\\
502 $E_{\textrm{f}}^{\textrm{vac},\,54\textrm{pc}}=3.47\textrm{ eV}$\\
503 $E_{\textrm{f}}^{\hkl<1 1 0>,\,54\textrm{pc}}=3.48\textrm{ eV}$
506 Comparison with literature (PRL 88 235501 (2002)):\\[0.2cm]
507 \begin{minipage}{8cm}
510 \item $E_{\text{cut-off}}=35 / 25\text{ Ry}=476 / 340\text{ eV}$
511 \item 216 atom supercell
512 \item Gamma point only calculations
515 \begin{minipage}{5cm}
516 $E_{\textrm{f}}^{\hkl<1 1 0>}=3.31 / 2.88\textrm{ eV}$\\
517 $E_{\textrm{f}}^{\textrm{hex}}=3.31 / 2.87\textrm{ eV}$\\
518 $E_{\textrm{f}}^{\textrm{vac}}=3.17 / 3.56\textrm{ eV}$
527 Questions so far ...\\
530 What configuration to chose for C in Si simulations?
532 \item Switch to another method for the XC approximation (GGA, PAW)?
533 \item Reasonable cut-off energy
534 \item Switch off symmetry? (especially for defect simulations)
536 (Monkhorst? $\Gamma$-point only if cell is large enough?)
537 \item Switch to tetrahedron method or Gaussian smearing ($\sigma$?)
538 \item Size and type of supercell
540 \item connected to choice of $k$-point mesh?
541 \item hence also connected to choice of smearing method?
542 \item constraints can only be applied to the lattice vectors!
544 \item Use of real space projection operators?
553 Review (so far) ...\\
556 Smearing method for the partial occupancies $f(\{\epsilon_{n{\bf k}}\})$
559 \begin{minipage}{4.4cm}
560 \includegraphics[width=4.4cm]{sic_smear_k.ps}
562 \begin{minipage}{4.4cm}
563 \includegraphics[width=4.4cm]{c_smear_k.ps}
565 \begin{minipage}{4.3cm}
566 \includegraphics[width=4.4cm]{si_smear_k.ps}
567 \end{minipage}\\[0.3cm]
569 \item Convergence reached at $6\times 6\times 6$ k-point mesh
570 \item No difference between Gauss ($\sigma=0.05$)
571 and tetrahedron smearing method!
576 Gauss ($\sigma=0.05$) smearing
577 and $6\times 6\times 6$ Monkhorst $k$-point mesh used
586 Review (so far) ...\\
589 \underline{Symmetry (in defect simulations)}
593 difference in $1\times 1\times 1$ Type 2 defect calculations\\
595 Symmetry precission (SYMPREC) small enough\\
597 {\bf\color{blue}Symmetry switched on}\\
600 \underline{Real space projection}
603 Error in lattice constant of plain Si ($1\times 1\times 1$ Type 2):
605 Error in position of the \hkl<1 1 0> interstitital in Si
606 ($1\times 1\times 1$ Type 2):
610 Real space projection used for 'large supercell' simulations}
626 3C-SiC equilibrium lattice constant and free energy\\
627 \includegraphics[width=7cm]{plain_sic_lc.ps}\\
628 $\rightarrow$ Convergence reached at 650 eV\\[0.2cm]
634 650 eV used as energy cut-off
644 Not answered (so far) ...\\
666 Final parameter choice
671 \underline{Param 1}\\
672 My first choice. Used for more accurate calculations.
674 \item $6\times 6 \times 6$ Monkhorst k-point mesh
675 \item $E_{\text{cut-off}}=650\text{ eV}$
676 \item Gaussian smearing ($\sigma=0.05$)
680 \underline{Param 2}\\
681 After talking to the pros!
683 \item $\Gamma$-point only
684 \item $E_{\text{cut-off}}=xyz\text{ eV}$
685 \item Gaussian smearing ($\sigma=0.05$)
687 \item Real space projection (Auto, Medium) for 'large' simulations
691 In both parameter sets the ultra soft pseudo potential method
692 as well as the projector augmented wave method is used with both,
693 the LDA and GGA exchange correlation potential!
702 Properties of Si, C and SiC using the new parameters\\
705 $2\times 2\times 2$ Type 2 supercell, Param 1, LDA, US PP\\[0.2cm]
706 \begin{tabular}{|l|l|l|l|}
708 & c-Si & c-C (diamond) & 3C-SiC \\
710 Lattice constant [\AA] & 5.389 & 3.527 & 4.319 \\
711 Expt. [\AA] & 5.429 & 3.567 & 4.359 \\
712 Error [\%] & {\color{green}0.7} & {\color{green}1.1} & {\color{green}0.9} \\
714 Cohesive energy [eV] & -5.277 & -8.812 & -7.318 \\
715 Expt. [eV] & -4.63 & -7.374 & -6.340 \\
716 Error [\%] & {\color{red}14.0} & {\color{red}19.5} & {\color{red}15.4} \\
720 \begin{minipage}{10cm}
721 $2\times 2\times 2$ Type 2 supercell, 3C-SiC, Param 1\\[0.2cm]
722 \begin{tabular}{|l|l|l|l|}
724 & {\color{magenta}US PP, GGA} & PAW, LDA & PAW, GGA \\
726 Lattice constant [\AA] & 4.370 & 4.330 & 4.379 \\
727 Error [\%] & {\color{green}0.3} & {\color{green}0.7} & {\color{green}0.5} \\
729 Cohesive energy [eV] & -6.426 & -7.371 & -6.491 \\
730 Error [\%] & {\color{green}1.4} & {\color{red}16.3} & {\color{green}2.4} \\
734 \begin{minipage}{3cm}
736 \begin{tabular}{|l|l|}
741 {\color{green}0.5} & {\color{green}0.01} \\
744 {\color{green}0.8} & {\color{orange}4.5} \\
754 Energy cut-off for $\Gamma$-point only caclulations
757 $2\times 2\times 2$ Type 2 supercell, Param 2, US PP, LDA, 3C-SiC\\[0.2cm]
758 \includegraphics[width=5.5cm]{sic_32pc_gamma_cutoff.ps}
759 \includegraphics[width=5.5cm]{sic_32pc_gamma_cutoff_lc.ps}\\
760 $\Rightarrow$ Use 300 eV as energy cut-off?\\[0.2cm]
761 $2\times 2\times 2$ Type 2 supercell, Param 2, 300 eV, US PP, GGA\\[0.2cm]
763 \begin{minipage}{10cm}
764 \begin{tabular}{|l|l|l|l|}
766 & c-Si & c-C (diamond) & 3C-SiC \\
768 Lattice constant [\AA] & 5.470 & 3.569 & 4.364 \\
769 Error [\%] & {\color{green}0.8} & {\color{green}0.1} & {\color{green}0.1} \\
771 Cohesive energy [eV] & -4.488 & -7.612 & -6.359 \\
772 Error [\%] & {\color{orange}3.1} & {\color{orange}3.2} & {\color{green}0.3} \\
776 \begin{minipage}{2cm}
778 ${\color{green}\surd}$
787 C \hkl<1 0 0> interstitial migration along \hkl<1 1 0>
793 \begin{minipage}[t]{4.2cm}
794 \underline{Starting configuration}\\
795 \includegraphics[width=4cm]{c_100_mig/start.eps}
797 \begin{minipage}[t]{4.0cm}
799 $\Delta x=\frac{1}{4}a_{\text{Si}}=1.357\text{ \AA}$\\
800 $\Delta y=\frac{1}{4}a_{\text{Si}}=1.357\text{ \AA}$\\
801 $\Delta z=0.325\text{ \AA}$\\
803 \begin{minipage}[t]{4.2cm}
804 \underline{{\bf Expected} final configuration}\\
805 \includegraphics[width=4cm]{c_100_mig/final.eps}\\
807 \begin{minipage}{6cm}
809 \item Fix border atoms of the simulation cell
810 \item Constraints and displacement of the C atom:
812 \item along {\color{green}\hkl<1 1 0> direction}\\
813 displaced by {\color{green} $\frac{1}{10}(\Delta x,\Delta y)$}
814 \item C atom {\color{red}entirely fixed in position}\\
816 {\color{red}$\frac{1}{10}(\Delta x,\Delta y,\Delta z)$}
818 \item Berendsen thermostat applied
820 {\bf\color{blue}Expected configuration not obtained!}
822 \begin{minipage}{0.5cm}
825 \begin{minipage}{6cm}
826 \includegraphics[width=6.0cm]{c_100_110mig_01_albe.ps}
834 C \hkl<1 0 0> interstitial migration along \hkl<1 1 0>
840 \begin{minipage}{3.2cm}
841 \includegraphics[width=3cm]{c_100_mig/fixmig_50.eps}
846 \begin{minipage}{3.2cm}
847 \includegraphics[width=3cm]{c_100_mig/fixmig_80.eps}
852 \begin{minipage}{3.2cm}
853 \includegraphics[width=3cm]{c_100_mig/fixmig_90.eps}
858 \begin{minipage}{3.2cm}
859 \includegraphics[width=3cm]{c_100_mig/fixmig_99.eps}
867 \item Why is the expected configuration not obtained?
868 \item How to find a migration path preceding to the expected configuration?
873 \item Simple: it is not the right migration path!
875 \item (Surrounding) atoms settle into a local minimum configuration
876 \item A possibly existing more favorable configuration is not achieved
878 \item \begin{itemize}
879 \item Search global minimum in each step (by simulated annealing)\\
881 Loss of the correct energy needed for migration
882 \item Smaller displacements\\
883 A more favorable configuration might be achieved
884 possibly preceding to the expected configuration
894 C \hkl<1 0 0> interstitial migration along \hkl<1 1 0>
898 Displacement step size decreased to
899 $\frac{1}{100} (\Delta x,\Delta y)$\\[0.2cm]
901 \begin{minipage}{7.5cm}
902 Result: (Video \href{../video/c_in_si_smig_albe.avi}{$\rhd_{\text{local}}$ } $|$
903 \href{http://www.physik.uni-augsburg.de/~zirkelfr/download/posic/c_in_si_smig_albe.avi}{$\rhd_{\text{remote url}}$})
905 \item Expected final configuration not obtained
906 \item Bonds to neighboured silicon atoms persist
907 \item C and neighboured Si atoms move along the direction of displacement
908 \item Even the bond to the lower left silicon atom persists
911 Obviously: overestimated bond strength
914 \begin{minipage}{5cm}
915 \includegraphics[width=6cm]{c_100_110smig_01_albe.ps}
916 \end{minipage}\\[0.4cm]
917 New approach to find the migration path:\\
919 Place interstitial carbon atom at the respective coordinates
920 into a perfect c-Si matrix!
928 C \hkl<1 0 0> interstitial migration along \hkl<1 1 0>
932 {\color{blue}New approach:}\\
933 Place interstitial carbon atom at the respective coordinates
934 into a perfect c-Si matrix!\\
935 {\color{blue}Problem:}\\
936 Too high forces due to the small distance of the C atom to the Si
937 atom sharing the lattice site.\\
938 {\color{blue}Solution:}
940 \item {\color{red}Slightly displace the Si atom}
941 (Video \href{../video/c_in_si_pmig_albe.avi}{$\rhd_{\text{local}}$ } $|$
942 \href{http://www.physik.uni-augsburg.de/~zirkelfr/download/posic/c_in_si_pmig_albe.avi}{$\rhd_{\text{remote url}}$})
943 \item {\color{green}Immediately quench the system}
944 (Video \href{../video/c_in_si_pqmig_albe.avi}{$\rhd_{\text{local}}$ } $|$
945 \href{http://www.physik.uni-augsburg.de/~zirkelfr/download/posic/c_in_si_pqmig_albe.avi}{$\rhd_{\text{remote url}}$})
948 \begin{minipage}{6.5cm}
949 \includegraphics[width=6cm]{c_100_110pqmig_01_albe.ps}
951 \begin{minipage}{6cm}
953 \item Jump in energy corresponds to the abrupt
954 structural change (as seen in the videos)
955 \item Due to the abrupt changes in structure and energy
956 this is {\color{red}not} the correct migration path and energy!?!
965 C \hkl<1 0 0> interstitial migration along \hkl<1 1 0> in c-Si (VASP)
970 {\color{blue}Method:}
972 \item Place interstitial carbon atom at the respective coordinates
974 \item \hkl<1 1 0> direction fixed for the C atom
975 \item $4\times 4\times 3$ Type 1, $198+1$ atoms
976 \item Atoms with $x=0$ or $y=0$ or $z=0$ fixed
978 {\color{blue}Results:}
979 (Video \href{../video/c_in_si_pmig_vasp.avi}{$\rhd_{\text{local}}$ } $|$
980 \href{http://www.physik.uni-augsburg.de/~zirkelfr/download/posic/c_in_si_pmig_vasp.avi}{$\rhd_{\text{remote url}}$})\\
981 \begin{minipage}{7cm}
982 \includegraphics[width=7cm]{c_100_110pmig_01_vasp.ps}
984 \begin{minipage}{5.5cm}
986 \item Characteristics nearly equal to classical calulations
987 \item Approximately half of the classical energy
997 C \hkl<1 0 0> interstitial migration along \hkl<1 1 0> in c-Si (VASP)
1002 {\color{blue}Method:}
1004 \item Continue with atomic positions of the last run
1005 \item Displace the C atom in \hkl<1 1 0> direction
1006 \item \hkl<1 1 0> direction fixed for the C atom
1007 \item $4\times 4\times 3$ Type 1, $198+1$ atoms
1008 \item Atoms with $x=0$ or $y=0$ or $z=0$ fixed
1010 {\color{blue}Results:}
1011 (Video \href{../video/c_in_si_smig_vasp.avi}{$\rhd_{\text{local}}$ } $|$
1012 \href{http://www.physik.uni-augsburg.de/~zirkelfr/download/posic/c_in_si_smig_vasp.avi}{$\rhd_{\text{remote url}}$})\\
1013 \includegraphics[width=7cm]{c_100_110mig_01_vasp.ps}
1020 Again: C \hkl<1 0 0> interstitial migration
1025 {\color{blue}The applied methods:}
1029 \item Start in relaxed \hkl<1 0 0> interstitial configuration
1030 \item Displace C atom along \hkl<1 1 0> direction
1031 \item Relaxation (Berendsen thermostat)
1032 \item Continue with configuration of the last run
1036 \item Place interstitial carbon at the respective coordinates
1037 into the perfect Si matrix
1038 \item Quench the system
1041 {\color{blue}In both methods:}
1043 \item Fixed border atoms
1044 \item Applied \hkl<1 1 0> constraint for the C atom
1046 {\color{red}Pitfalls} and {\color{green}refinements}:
1048 \item {\color{red}Fixed border atoms} $\rightarrow$
1049 Relaxation of stress not possible\\
1051 {\color{green}Fix only one Si atom} (the one furthermost to the defect)
1052 \item {\color{red}\hkl<1 1 0> constraint not sufficient}\\
1053 $\Rightarrow$ {\color{green}Apply 11x constraint}
1054 (connecting line of initial and final C positions)
1062 Again: C \hkl<1 0 0> interstitial migration (Albe)
1065 Constraint applied by modifying the Velocity Verlet algorithm
1067 {\color{blue}Results:}
1068 (Video \href{../video/c_in_si_fmig_albe.avi}{$\rhd_{\text{local}}$ } $|$
1069 \href{http://www.physik.uni-augsburg.de/~zirkelfr/download/posic/c_in_si_fmig_albe.avi}{$\rhd_{\text{remote url}}$})\\
1070 \begin{minipage}{6.3cm}
1071 \includegraphics[width=6cm]{c_100_110fmig_01_albe.ps}
1073 \begin{minipage}{6cm}
1075 Again there are jumps in energy corresponding to abrupt
1076 structural changes as seen in the video
1080 \item Expected final configuration not obtained
1081 \item Bonds to neighboured silicon atoms persist
1082 \item C and neighboured Si atoms move along the direction of displacement
1083 \item Even the bond to the lower left silicon atom persists
1091 Again: C \hkl<1 0 0> interstitial migration (VASP)
1094 Transformation for the Type 2 supercell
1098 \begin{minipage}[t]{4.2cm}
1099 \underline{Starting configuration}\\
1100 \includegraphics[width=3cm]{c_100_mig_vasp/start.eps}
1102 \begin{minipage}[t]{4.0cm}
1104 $\Delta x=1.367\text{ \AA}$\\
1105 $\Delta y=1.367\text{ \AA}$\\
1106 $\Delta z=0.787\text{ \AA}$\\
1108 \begin{minipage}[t]{4.2cm}
1109 \underline{{\bf Expected} final configuration}\\
1110 \includegraphics[width=3cm]{c_100_mig_vasp/final.eps}\\
1112 \begin{minipage}{6.2cm}
1117 \beta=\arctan\frac{\Delta z}{\sqrt{2}\Delta x}=22.165^{\circ}
1120 \begin{minipage}{6.2cm}
1121 Length of migration path:
1123 l=\sqrt{\Delta x^2+\Delta y^2+\Delta z^2}=2.087\text{ \AA}
1125 \end{minipage}\\[0.3cm]
1126 Transformation of basis:
1128 T=ABA^{-1}A=AB \textrm{, mit }
1129 A=\left(\begin{array}{ccc}
1130 \cos\alpha & -\sin\alpha & 0\\
1131 \sin\alpha & \cos\alpha & 0\\
1135 B=\left(\begin{array}{ccc}
1137 0 & \cos\beta & \sin\beta \\
1138 0 & -\sin\beta & \cos\beta
1141 Atom coordinates transformed by: $T^{-1}=B^{-1}A^{-1}$
1148 Again: C \hkl<1 0 0> interstitial migration\\
1151 {\color{blue}Reminder:}\\
1152 Transformation needed since in VASP constraints can only be applied to
1153 the basis vectors!\\
1154 {\color{red}Problem:} (stupid me!)\\
1155 Transformation of supercell 'destroys' the correct periodicity!\\
1156 {\color{green}Solution:}\\
1157 Find a supercell with one basis vector forming the correct constraint\\
1158 {\color{red}Problem:}\\
1159 Hard to find a supercell for the $22.165^{\circ}$ rotation\\
1161 Another method to {\color{blue}\underline{estimate}} the migration energy:
1163 \item Assume an intermediate saddle point configuration during migration
1164 \item Determine the energy of the saddle point configuration
1165 \item Substract the saddle point configuration energy by
1166 the energy of the initial (final) defect configuration
1175 The C \hkl<1 0 0> defect configuration
1178 Needed so often for input configurations ...\\[0.8cm]
1179 \begin{minipage}{7.0cm}
1180 \includegraphics[width=6.5cm]{100-c-si-db_light.eps}\\
1181 Qualitative {\color{red}and} quantitative {\color{red}difference}!
1183 \begin{minipage}{5.5cm}
1186 \begin{tabular}{|l|l|l|}
1190 \underline{VASP} & & \\
1191 fractional & 0.1969 & 0.1211 \\
1192 in \AA & 1.08 & 0.66 \\
1194 \underline{Albe} & & \\
1195 fractional & 0.1547 & 0.1676 \\
1196 in \AA & 0.84 & 0.91 \\
1198 \end{tabular}\\[0.2cm]
1199 {\scriptsize\underline{PC (Vasp)}}
1200 \includegraphics[width=6.1cm]{c_100_pc_vasp.ps}
1209 Again: C \hkl<1 0 0> interstitial migration (VASP)
1212 $\hkl<0 0 -1> \rightarrow \hkl<0 0 1>$ migration
1213 ($3\times 3\times 3$ Type 2):
1217 \begin{minipage}[t]{4.1cm}
1218 \underline{Starting configuration}\\
1219 \includegraphics[height=3.2cm]{c_100_mig_vasp/start.eps}
1221 $E_{\text{f}}=3.15 \text{ eV}$
1224 \begin{minipage}[t]{4.1cm}
1225 \underline{Intermediate configuration}\\
1226 \includegraphics[height=3.2cm]{c_100_mig_vasp/00-1_001_im.eps}
1228 $E_{\text{f}}=4.41 \text{ eV}$
1231 \begin{minipage}[t]{4.1cm}
1232 \underline{Final configuration}\\
1233 \includegraphics[height=3.2cm]{c_100_mig_vasp/final.eps}
1235 $E_{\text{f}}=3.17 \text{ eV}$
1237 \end{minipage}\\[0.4cm]
1239 \Rightarrow \Delta E_{\text{f}} = E_{\text{mig}} = 1.26 \text{ eV}
1242 Unexpected \& ({\color{red}more} or {\color{orange}less}) fatal:
1244 \renewcommand\labelitemi{{\color{orange}$\bullet$}}
1245 \item Difference in formation energy (0.02 eV)
1246 of the initial and final configuration
1247 \renewcommand\labelitemi{{\color{red}$\bullet$}}
1248 \item Huge discrepancy (0.3 - 0.4 eV) to the migration barrier
1249 of Type 1 (198+1 atoms) calculations
1250 \renewcommand\labelitemi{{\color{black}$\bullet$}}
1258 Again: C \hkl<1 0 0> interstitial migration (VASP)
1261 $\hkl<0 0 -1> \rightarrow \hkl<0 -1 0>$ migration
1262 ($3\times 3\times 3$ Type 2):
1266 \begin{minipage}[t]{4.1cm}
1267 \underline{Starting configuration}\\
1268 \includegraphics[height=3.2cm]{c_100_mig_vasp/start.eps}
1270 $E_{\text{f}}=3.154 \text{ eV}$
1273 \begin{minipage}[t]{4.1cm}
1274 \underline{Intermediate configuration}\\
1277 $E_{\text{f}}=?.?? \text{ eV}$
1280 \begin{minipage}[t]{4.1cm}
1281 \underline{Final configuration}\\
1282 \includegraphics[height=3.2cm]{c_100_mig_vasp/0-10.eps}
1284 $E_{\text{f}}=3.157 \text{ eV}$
1286 \end{minipage}\\[0.4cm]
1288 \Rightarrow \Delta E_{\text{f}} = E_{\text{mig}} = ?.?? \text{ eV}
1293 Intermediate configuration {\color{red}not found} by now!
1301 C in Si interstitial configurations (VASP)
1304 Check of Kohn-Sham eigenvalues\\
1308 \begin{minipage}{6cm}
1309 \hkl<1 0 0> interstitial\\
1311 \begin{minipage}{6cm}
1312 Saddle point configuration\\
1314 \underline{$4\times 4\times 3$ Type 1 - fixed border atoms}\\
1315 \begin{minipage}{6cm}
1316 385: 4.8567 - 2.00000\\
1317 386: 4.9510 - 2.00000\\
1318 387: 5.3437 - 0.00000\\
1319 388: 5.4930 - 0.00000
1321 \begin{minipage}{6cm}
1322 385: 4.8694 - 2.00000\\
1323 386: {\color{red}4.9917} - 1.92603\\
1324 387: {\color{red}5.1181} - 0.07397\\
1325 388: 5.4541 - 0.00000
1326 \end{minipage}\\[0.2cm]
1327 \underline{$4\times 4\times 3$ Type 1 - no constraints}\\
1328 \begin{minipage}{6cm}
1329 385: 4.8586 - 2.00000\\
1330 386: 4.9458 - 2.00000\\
1331 387: 5.3358 - 0.00000\\
1332 388: 5.4915 - 0.00000
1334 \begin{minipage}{6cm}
1335 385: 4.8693 - 2.00000\\
1336 386: {\color{red}4.9879} - 1.92065\\
1337 387: {\color{red}5.1120} - 0.07935\\
1338 388: 5.4544 - 0.00000
1339 \end{minipage}\\[0.2cm]
1340 \underline{$3\times 3\times 3$ Type 2 - no constraints}\\
1341 \begin{minipage}{6cm}
1342 433: 4.8054 - 2.00000\\
1343 434: 4.9027 - 2.00000\\
1344 435: 5.2543 - 0.00000\\
1345 436: 5.5718 - 0.00000
1347 \begin{minipage}{6cm}
1348 433: 4.8160 - 2.00000\\
1349 434: {\color{green}5.0109} - 1.00264\\
1350 435: {\color{green}5.0111} - 0.99736\\
1351 436: 5.5364 - 0.00000
1359 Once again: C \hkl<1 0 0> interstitial migration (VASP)
1364 \item Start in fully relaxed (assumed) saddle point configuration
1365 \item Move towards \hkl<1 0 0> configuration using updated values
1366 for $\Delta x$, $\Delta y$ and $\Delta z$ (CRT)
1367 \item \hkl<1 1 0> constraints applied, 1 Si atom fixed
1368 \item $4\times 4\times 3$ Type 1 supercell
1373 \begin{minipage}{6.2cm}
1374 \includegraphics[width=6.0cm]{c_100_110sp-i_vasp.ps}
1376 \begin{minipage}{6.2cm}
1377 \includegraphics[width=6.0cm]{c_100_110sp-i_rc_vasp.ps}
1380 Reaction coordinate:
1381 $r_{i+1}=r_i+\sum_{\text{atoms j}} \left| r_{j,i+1}-r_{j,i} \right|$
1388 Investigation of the migration path along \hkl<1 1 0> (VASP)
1393 \underline{Minimum:}\\
1394 \begin{minipage}{4cm}
1395 \includegraphics[width=3.5cm]{c_100_mig_vasp/110_c-si_split.eps}
1397 \begin{minipage}{8cm}
1399 \item Starting conf: 35 \% displacement results (1443)
1400 \item \hkl<1 1 0> constraint disabled
1406 \item C-Si \hkl<1 1 0> split interstitial
1407 \item Stable configuration
1408 \item $E_{\text{f}}=4.13\text{ eV}$
1410 \end{minipage}\\[0.1cm]
1412 \underline{Maximum:}\\
1413 \begin{minipage}{6cm}
1415 \includegraphics[width=2.3cm]{c_100_mig_vasp/100-110_01.eps}
1416 \includegraphics[width=2.3cm]{c_100_mig_vasp/100-110_02.eps}\\
1417 20 \% $\rightarrow$ 25 \%\\
1418 Breaking of Si-C bond
1421 \begin{minipage}{6cm}
1422 \includegraphics[width=6.2cm]{c_100_110sp-i_upd_vasp.ps}
1430 Displacing the \hkl<1 1 0> Si-C split along \hkl<1 -1 0> (VASP)
1435 $4\times 4\times 3$ Type 1 supercell
1437 \underline{Structures:}
1439 \begin{minipage}[t]{4.1cm}
1440 \includegraphics[height=3.0cm]{c_100_mig_vasp/start.eps}\\
1441 \hkl<0 0 -1> dumbbell\\
1442 $E_{\text{f}}={\color{orange}3.2254}\text{ eV}$
1444 \begin{minipage}[t]{4.1cm}
1445 \includegraphics[height=3.0cm]{c_100_mig_vasp/110_c-si_split.eps}\\
1446 Assumed \hkl<1 1 0> C-Si split\\
1447 $E_{\text{f}}=4.1314\text{ eV}$
1449 \begin{minipage}[t]{4.1cm}
1450 \includegraphics[height=3.0cm]{c_100_mig_vasp/110_dis_0-10.eps}\\
1451 First guess: \hkl<0 -1 0> dumbbell\\
1452 {\color{red}but:} $E_{\text{f}}={\color{orange}2.8924}\text{ eV}$\\
1456 \underline{Occupancies:}
1460 \begin{minipage}{4.1cm}
1461 385: 4.8586 - 2.00000\\
1462 386: 4.9458 - 2.00000\\
1463 387: 5.3358 - 0.00000\\
1464 388: 5.4915 - 0.00000
1467 \begin{minipage}{4.1cm}
1468 385: 4.7790 - 2.00000\\
1469 386: 4.8797 - 1.99964\\
1470 387: 5.1321 - 0.00036\\
1471 388: 5.4711 - 0.00000
1474 \begin{minipage}{4.1cm}
1475 385: 4.7670 - 2.00000\\
1476 386: 4.9190 - 2.00000\\
1477 387: 5.2886 - 0.00000\\
1478 388: 5.4849 - 0.00000
1485 {\color{red}? ! ? ! ? ! ? ! ?}
1493 Defect configurations in $4\times 4\times 3$ Type 1 supercells revisited
1498 \begin{tabular}{l|p{2.5cm}|p{2.5cm}|p{4cm}|}
1499 & \hkl<0 0 -1> interstitial
1500 & local minimum\newline
1501 \hkl<1 1 0> C-Si split
1502 & intermediate configuration\newline
1503 (bond centered conf)\\
1505 default & $E_{\text{f}}=3.3254\text{ eV}$\newline
1507 386: 4.9458 - 2.00000\newline
1508 387: 5.3358 - 0.00000}
1509 & $E_{\text{f}}=4.1314\text{ eV}$\newline
1511 386: 4.8797 - 1.99964\newline
1512 387: 5.1321 - 0.00036}
1513 & $E_{\text{f}}=4.2434\text{ eV}$\newline
1515 386: 4.9879 - 1.92065\newline
1516 387: 5.1120 - 0.07935} \\
1518 No symmetry & $E_{\text{f}}=3.3154\text{ eV}$\newline
1520 386: 4.9456 - 2.00000\newline
1521 387: 5.3366 - 0.00000}
1522 & $E_{\text{f}}=4.1314\text{ eV}$\newline
1524 386: 4.8798 - 1.99961\newline
1525 387: 5.1307 - 0.00039}
1526 & $E_{\text{f}}=4.2454\text{ eV}$\newline
1528 386: 4.9841 - 1.92147\newline
1529 387: 5.1085 - 0.07853} \\
1531 $+$ spin polarized & $E_{\text{f}}=3.3154\text{ eV}$\newline
1534 386: 4.9449 - 1.00000\newline
1535 387: 5.3365 - 0.00000\newline%
1538 386: 4.9449 - 1.00000\newline
1539 387: 5.3365 - 0.00000}}
1540 & $E_{\text{f}}={\color{red}4.1314}\text{ eV}$\newline
1543 386: 4.8799 - 0.99980\newline
1544 387: 5.1307 - 0.00020\newline%
1547 386: 4.8799 - 0.99980\newline
1548 387: 5.1306 - 0.00020}}
1549 & $E_{\text{f}}=4.0254\text{ eV}$\newline
1552 387: 4.8581 - 1.00000\newline
1553 388: 5.4662 - 0.00000\newline%
1556 385: 4.8620 - 1.00000\newline
1557 386: 5.2951 - 0.00000}} \\
1559 $+$ spin difference 2 & $E_{\text{f}}=3.6394\text{ eV}$\newline
1562 387: 5.2704 - 0.99891\newline
1563 388: 5.4886 - 0.00095\newline
1564 389: 5.5094 - 0.00011\newline
1565 390: 5.5206 - 0.00003\newline%
1568 385: 4.8565 - 0.98603\newline
1569 386: 5.0119 - 0.01397}}
1571 & $E_{\text{f}}=4.0254\text{ eV}$\newline
1574 387: 4.8578 - 1.00000\newline
1575 388: 5.4661 - 0.00000\newline%
1578 385: 4.8618 - 1.00000\newline
1579 386: 5.2950 - 0.00000}} \\
1588 C \hkl<1 0 0> interstitial migration (VASP)
1593 \begin{minipage}{6.2cm}
1595 \item $3\times 3\times 3$ Type 2 supercell
1596 \item \hkl<1 1 0> constraints applied
1597 (\href{http://www.physik.uni-augsburg.de/~zirkelfr/download/posic/sd_rot.patch}{Patch})
1598 \item Move from \hkl<1 0 0> towards\\
1599 bond centered configuration
1601 \underline{Sd Rot usage (POSCAR):}
1609 Transformed selective dynamics
1614 Only works in direct mode!\\
1615 $z,x'$-axis rotation: $45.0^{\circ}$, $0.0^{\circ}$
1617 \begin{minipage}{6.2cm}
1618 \includegraphics[width=6cm]{c_100_110sp-i_2333_vasp.ps}
1619 \includegraphics[width=6cm]{c_100_110sp-i_2333_rc_vasp.ps}
1623 Next: Migration calculation in 2333 using CRT
1624 (\hkl<0 0 -1> $\rightarrow$ \hkl<0 0 1> and \hkl<0 -1 0>)
1632 \hkl<0 0 -1> to \hkl <0 0 1> migration
1633 in the $3\times 3\times 3$ Type 2 supercell
1641 \hkl<0 0 -1> to \hkl <0 -1 0> migration
1642 in the $3\times 3\times 3$ Type 2 supercell
1650 Defect configurations in $3\times 3\times 3$ Type 2 supercells revisited
1658 Combination of defects
1661 TODO: introduce some Si self-interstitials and C interstitials before\\
1662 BUT: Concentrate on 100 C interstitial combinations and 100 C + vacancy\\
1664 Agglomeration of 100 defects energetically favorable?
1671 Molecular dynamics simulations (VASP)
1674 2 C atoms in $2\times 2\times 2$ Type 2 supercell at $450\,^{\circ}\text{C}$
1678 \begin{minipage}{7.6cm}
1679 Radial distribution\\
1680 \includegraphics[width=7.6cm]{md_02c_2222si_pc.ps}
1682 \begin{minipage}{5.0cm}
1685 $t_1=50$ ps to $t_2=50.93$ ps
1690 \item $<(x(t)-x(0))^2>$ hard to determine due to missing info of
1692 \item No jumps recognized in the
1693 Video \href{../video/md_02c_2222si_vasp.avi}{$\rhd_{\text{local}}$ } $|$
1694 \href{http://www.physik.uni-augsburg.de/~zirkelfr/download/posic/md_02c_2222si_vasp.avi}{$\rhd_{\text{remote url}}$}
1702 Molecular dynamics simulations (VASP)
1705 10 C atoms in $3\times 3\times 3$ Type 2 supercell at $450\,^{\circ}\text{C}$
1709 \begin{minipage}{7.2cm}
1710 Radial distribution (PC averaged over 1 ps)\\
1711 \includegraphics[width=7.0cm]{md_10c_2333si_pc_vasp.ps}
1713 \begin{minipage}{5.0cm}
1714 \includegraphics[width=6.0cm]{md_10c_2333si_pcc_vasp.ps}
1717 (Video \href{../video/md_10c_2333si_vasp.avi}{$\rhd_{\text{local}}$ } $|$
1718 \href{http://www.physik.uni-augsburg.de/~zirkelfr/download/posic/md_10c_2333si_vasp.avi}{$\rhd_{\text{remote url}}$})
1720 \item $<(x(t)-x(0))^2>$ hard to determine due to missing info of
1722 \item Agglomeration of C? (Video)
1730 Molecular dynamics simulations (VASP)
1733 1 C atom in $3\times 3\times 3$ Type 2 supercell at $900\,^{\circ}\text{C}$
1742 Molecular dynamics simulations (VASP)
1745 10 C atoms in $3\times 3\times 3$ Type 2 supercell at $900\,^{\circ}\text{C}$
1754 Density Functional Theory
1757 Hohenberg-Kohn theorem
1769 Transition state theory\\