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90 Atomistic simulation study\\[0.2cm]
91 of the SiC precipitation in Si
96 \textsc{F. Zirkelbach}
100 For the exchange among Paderborn and Augsburg
118 \begin{minipage}{6.5cm}
120 \item Start from scratch
121 \item $V_{xc}$: US LDA (out of ./pot directory)
122 \item $k$-points: Monkhorst $4\times 4\times 4$
123 \item Ionic relaxation
125 \item Conjugate gradient method
126 \item Scaling constant of 0.1 for forces
127 \item Default break condition ($0.1 \cdot 10^{-2}$ eV)
128 \item Maximum of 100 steps
132 \item No change in volume
136 \item Change of cell volume and shape\\
142 \begin{minipage}{6.0cm}
143 {\scriptsize\color{blue}
144 Example INCAR file (NVT):
147 System = C 100 interstitial in Si
156 {\scriptsize\color{red}
157 Example INCAR file (NPT):
160 System = C hexagonal interstitial in Si
176 Silicon bulk properties
181 Simulations (NPT, $\textrm{EDIFFG}=0.1\cdot 10^{-3}$ eV):
183 \item Supercell: $x_1=(0,0.5,0.5),\, x_2=(0.5,0,0.5),\, x_3=(0.5,0.5,0)$;
184 2 atoms (1 {\bf p}rimitive {\bf c}ell)
185 \item Supercell: $x_1=(0.5,-0.5,0),\, x_2=(0.5,0.5,0),\, x_3=(0,0,1)$;
187 \item Supercell: $x_1=(1,0,0),\, x_2=(0,1,0),\, x_3=(0,0,1)$;
189 \item Supercell: $x_1=(2,0,0),\, x_2=(0,2,0),\, x_3=(0,0,2)$;
192 \begin{minipage}{6cm}
193 Cohesive energy / Lattice constant:
195 \item $E_{\textrm{cut-off}}=150\, \textrm{eV}$: 5.955 eV / 5.378 \AA\\
196 $E_{\textrm{cut-off}}=300\, \textrm{eV}$: 5.975 eV / 5.387 \AA
197 \item $E_{\textrm{cut-off}}=150\, \textrm{eV}$: 5.989 eV / 5.356 \AA
198 \item $E_{\textrm{cut-off}}=150\, \textrm{eV}$: 5.955 eV / 5.380 \AA\\
199 $E_{\textrm{cut-off}}=200\, \textrm{eV}$: 5.972 eV / 5.388 \AA\\
200 $E_{\textrm{cut-off}}=250\, \textrm{eV}$: 5.975 eV / 5.389 \AA\\
201 $E_{\textrm{cut-off}}=300\, \textrm{eV}$: 5.975 eV / 5.389 \AA\\
202 $E_{\textrm{cut-off}}=300\, \textrm{eV}^{*}$: 5.975 eV / 5.390 \AA
203 \item $E_{\textrm{cut-off}}=300\, \textrm{eV}$: 5.977 eV / 5.389 \AA
206 \begin{minipage}{7cm}
207 \includegraphics[width=7cm]{si_lc_and_ce.ps}
208 \end{minipage}\\[0.3cm]
210 $^*$special settings (p. 138, VASP manual):
211 spin polarization, no symmetry, ...
219 Silicon bulk properties
223 \item Calculation of cohesive energies for different lattice constants
224 \item No ionic update
225 \item Tetrahedron method with Blöchl corrections for
226 the partial occupancies $f(\{\epsilon_{n{\bf k}}\})$
227 \item Supercell 3 (8 atoms, 4 primitive cells)
230 \begin{minipage}{6.5cm}
232 $E_{\textrm{cut-off}}=150$ eV\\
233 \includegraphics[width=6.5cm]{si_lc_fit.ps}
236 \begin{minipage}{6.5cm}
238 $E_{\textrm{cut-off}}=250$ eV\\
239 \includegraphics[width=6.5cm]{si_lc_fit_250.ps}
248 3C-SiC bulk properties\\[0.2cm]
251 \begin{minipage}{6.5cm}
252 \includegraphics[width=6.5cm]{sic_lc_and_ce2.ps}
254 \begin{minipage}{6.5cm}
255 \includegraphics[width=6.5cm]{sic_lc_and_ce.ps}
256 \end{minipage}\\[0.3cm]
258 \item Supercell 3 (4 primitive cells, 4+4 atoms)
259 \item Error in equilibrium lattice constant: {\color{green} $0.9\,\%$}
260 \item Error in cohesive energy: {\color{red} $31.6\,\%$}
268 3C-SiC bulk properties\\[0.2cm]
274 \item Calculation of cohesive energies for different lattice constants
275 \item No ionic update
276 \item Tetrahedron method with Blöchl corrections for
277 the partial occupancies $f(\{\epsilon_{n{\bf k}}\})$
280 \begin{minipage}{6.5cm}
282 Supercell 3, $4\times 4\times 4$ k-points\\
283 \includegraphics[width=6.5cm]{sic_lc_fit.ps}
286 \begin{minipage}{6.5cm}
289 Non-continuous energies\\
290 for $E_{\textrm{cut-off}}<1050\,\textrm{eV}$!\\
294 Does this matter in structural optimizaton simulations?
296 \item Derivative might be continuous
297 \item Similar lattice constants where derivative equals zero
308 3C-SiC bulk properties\\[0.2cm]
313 \begin{picture}(0,0)(-188,80)
314 %Supercell 1, $3\times 3\times 3$ k-points\\
315 \includegraphics[width=6.5cm]{sic_lc_fit_k3.ps}
318 \begin{minipage}{6.5cm}
320 \item Supercell 1 simulations
321 \item Variation of k-points
322 \item Continuous energies for
323 $E_{\textrm{cut-off}} > 550\,\textrm{eV}$
324 \item Critical $E_{\textrm{cut-off}}$ for
326 depending on supercell?
328 \end{minipage}\\[1.0cm]
329 \begin{minipage}{6.5cm}
331 \includegraphics[width=6.5cm]{sic_lc_fit_k5.ps}
334 \begin{minipage}{6.5cm}
336 \includegraphics[width=6.5cm]{sic_lc_fit_k7.ps}
348 {\bf\color{red} From now on ...}
350 {\small Energies used: free energy without entropy ($\sigma \rightarrow 0$)}
355 \item $E_{\textrm{free,sp}}$:
356 energy of spin polarized free atom
358 \item $k$-points: Monkhorst $1\times 1\times 1$
359 \item Symmetry switched off
360 \item Spin polarized calculation
361 \item Interpolation formula according to Vosko Wilk and Nusair
362 for the correlation part of the exchange correlation functional
363 \item Gaussian smearing for the partial occupancies
364 $f(\{\epsilon_{n{\bf k}}\})$
366 \item Magnetic mixing: AMIX = 0.2, BMIX = 0.0001
367 \item Supercell: one atom in cubic
368 $10\times 10\times 10$ \AA$^3$ box
371 $E_{\textrm{free,sp}}(\textrm{Si},{\color{green}250}\, \textrm{eV})=
372 -0.70036911\,\textrm{eV}$
375 $E_{\textrm{free,sp}}(\textrm{Si},{\color{red}650}\, \textrm{eV})=
376 -0.70021403\,\textrm{eV}$
379 $E_{\textrm{free,sp}}(\textrm{C},{\color{red}650}\, \textrm{eV})=
380 -1.3535731\,\textrm{eV}$
383 energy (non-polarized) of system of interest composed of\\
384 n atoms of type N, m atoms of type M, \ldots
390 E_{\textrm{coh}}=\frac{
391 -\Big(E(N_nM_m\ldots)-nE_{\textrm{free,sp}}(N)-mE_{\textrm{free,sp}}(M)
402 Calculation of the defect formation energy\\
407 {\color{blue}Method 1} (single species)
409 \item $E_{\textrm{coh}}^{\textrm{initial conf}}$:
410 cohesive energy per atom of the initial system
411 \item $E_{\textrm{coh}}^{\textrm{interstitial conf}}$:
412 cohesive energy per atom of the interstitial system
413 \item N: amount of atoms in the interstitial system
419 E_{\textrm{f}}=\Big(E_{\textrm{coh}}^{\textrm{interstitial conf}}
420 -E_{\textrm{coh}}^{\textrm{initial conf}}\Big) N
423 {\color{magenta}Method 2} (two and more species)
425 \item $E$: energy of the interstitial system
426 (with respect to the ground state of the free atoms!)
427 \item $N_{\text{Si}}$, $N_{\text{C}}$:
428 amount of Si and C atoms
429 \item $\mu_{\text{Si}}$, $\mu_{\text{C}}$:
430 chemical potential (cohesive energy) of Si and C
436 E_{\textrm{f}}=E-N_{\text{Si}}\mu_{\text{Si}}-N_{\text{C}}\mu_{\text{C}}
445 Used types of supercells\\
450 \begin{minipage}{4.3cm}
451 \includegraphics[width=4cm]{sc_type0.eps}\\[0.3cm]
452 \underline{Type 0}\\[0.2cm]
457 1 primitive cell / 2 atoms
459 \begin{minipage}{4.3cm}
460 \includegraphics[width=4cm]{sc_type1.eps}\\[0.3cm]
461 \underline{Type 1}\\[0.2cm]
466 2 primitive cells / 4 atoms
468 \begin{minipage}{4.3cm}
469 \includegraphics[width=4cm]{sc_type2.eps}\\[0.3cm]
470 \underline{Type 2}\\[0.2cm]
475 4 primitive cells / 8 atoms
476 \end{minipage}\\[0.4cm]
479 In the following these types of supercells are used and
480 are possibly scaled by integers in the different directions!
488 Silicon point defects\\
493 Influence of supercell size\\
494 \begin{minipage}{8cm}
495 \includegraphics[width=7.0cm]{si_self_int.ps}
497 \begin{minipage}{5cm}
498 $E_{\textrm{f}}^{\hkl<1 1 0>,\,32\textrm{pc}}=3.38\textrm{ eV}$\\
499 $E_{\textrm{f}}^{\textrm{tet},\,32\textrm{pc}}=3.41\textrm{ eV}$\\
500 $E_{\textrm{f}}^{\textrm{hex},\,32\textrm{pc}}=3.42\textrm{ eV}$\\
501 $E_{\textrm{f}}^{\textrm{vac},\,32\textrm{pc}}=3.51\textrm{ eV}$\\\\
502 $E_{\textrm{f}}^{\textrm{hex},\,54\textrm{pc}}=3.42\textrm{ eV}$\\
503 $E_{\textrm{f}}^{\textrm{tet},\,54\textrm{pc}}=3.45\textrm{ eV}$\\
504 $E_{\textrm{f}}^{\textrm{vac},\,54\textrm{pc}}=3.47\textrm{ eV}$\\
505 $E_{\textrm{f}}^{\hkl<1 1 0>,\,54\textrm{pc}}=3.48\textrm{ eV}$
508 Comparison with literature (PRL 88 235501 (2002)):\\[0.2cm]
509 \begin{minipage}{8cm}
512 \item $E_{\text{cut-off}}=35 / 25\text{ Ry}=476 / 340\text{ eV}$
513 \item 216 atom supercell
514 \item Gamma point only calculations
517 \begin{minipage}{5cm}
518 $E_{\textrm{f}}^{\hkl<1 1 0>}=3.31 / 2.88\textrm{ eV}$\\
519 $E_{\textrm{f}}^{\textrm{hex}}=3.31 / 2.87\textrm{ eV}$\\
520 $E_{\textrm{f}}^{\textrm{vac}}=3.17 / 3.56\textrm{ eV}$
529 Questions so far ...\\
532 What configuration to chose for C in Si simulations?
534 \item Switch to another method for the XC approximation (GGA, PAW)?
535 \item Reasonable cut-off energy
536 \item Switch off symmetry? (especially for defect simulations)
538 (Monkhorst? $\Gamma$-point only if cell is large enough?)
539 \item Switch to tetrahedron method or Gaussian smearing ($\sigma$?)
540 \item Size and type of supercell
542 \item connected to choice of $k$-point mesh?
543 \item hence also connected to choice of smearing method?
544 \item constraints can only be applied to the lattice vectors!
546 \item Use of real space projection operators?
555 Review (so far) ...\\
558 Smearing method for the partial occupancies $f(\{\epsilon_{n{\bf k}}\})$
561 \begin{minipage}{4.4cm}
562 \includegraphics[width=4.4cm]{sic_smear_k.ps}
564 \begin{minipage}{4.4cm}
565 \includegraphics[width=4.4cm]{c_smear_k.ps}
567 \begin{minipage}{4.3cm}
568 \includegraphics[width=4.4cm]{si_smear_k.ps}
569 \end{minipage}\\[0.3cm]
571 \item Convergence reached at $6\times 6\times 6$ k-point mesh
572 \item No difference between Gauss ($\sigma=0.05$)
573 and tetrahedron smearing method!
578 Gauss ($\sigma=0.05$) smearing
579 and $6\times 6\times 6$ Monkhorst $k$-point mesh used
588 Review (so far) ...\\
591 \underline{Symmetry (in defect simulations)}
595 difference in $1\times 1\times 1$ Type 2 defect calculations\\
597 Symmetry precission (SYMPREC) small enough\\
599 {\bf\color{blue}Symmetry switched on}\\
602 \underline{Real space projection}
605 Error in lattice constant of plain Si ($1\times 1\times 1$ Type 2):
607 Error in position of the \hkl<1 1 0> interstitital in Si
608 ($1\times 1\times 1$ Type 2):
612 Real space projection used for 'large supercell' simulations}
628 3C-SiC equilibrium lattice constant and free energy\\
629 \includegraphics[width=7cm]{plain_sic_lc.ps}\\
630 $\rightarrow$ Convergence reached at 650 eV\\[0.2cm]
636 650 eV used as energy cut-off
646 Not answered (so far) ...\\
668 Final parameter choice
673 \underline{Param 1}\\
674 My first choice. Used for more accurate calculations.
676 \item $6\times 6 \times 6$ Monkhorst k-point mesh
677 \item $E_{\text{cut-off}}=650\text{ eV}$
678 \item Gaussian smearing ($\sigma=0.05$)
682 \underline{Param 2}\\
683 After talking to the pros!
685 \item $\Gamma$-point only
686 \item $E_{\text{cut-off}}=xyz\text{ eV}$
687 \item Gaussian smearing ($\sigma=0.05$)
689 \item Real space projection (Auto, Medium) for 'large' simulations
693 In both parameter sets the ultra soft pseudo potential method
694 as well as the projector augmented wave method is used with both,
695 the LDA and GGA exchange correlation potential!
704 Properties of Si, C and SiC using the new parameters\\
707 $2\times 2\times 2$ Type 2 supercell, Param 1, LDA, US PP\\[0.2cm]
708 \begin{tabular}{|l|l|l|l|}
710 & c-Si & c-C (diamond) & 3C-SiC \\
712 Lattice constant [\AA] & 5.389 & 3.527 & 4.319 \\
713 Expt. [\AA] & 5.429 & 3.567 & 4.359 \\
714 Error [\%] & {\color{green}0.7} & {\color{green}1.1} & {\color{green}0.9} \\
716 Cohesive energy [eV] & -5.277 & -8.812 & -7.318 \\
717 Expt. [eV] & -4.63 & -7.374 & -6.340 \\
718 Error [\%] & {\color{red}14.0} & {\color{red}19.5} & {\color{red}15.4} \\
722 \begin{minipage}{10cm}
723 $2\times 2\times 2$ Type 2 supercell, 3C-SiC, Param 1\\[0.2cm]
724 \begin{tabular}{|l|l|l|l|}
726 & {\color{magenta}US PP, GGA} & PAW, LDA & PAW, GGA \\
728 Lattice constant [\AA] & 4.370 & 4.330 & 4.379 \\
729 Error [\%] & {\color{green}0.3} & {\color{green}0.7} & {\color{green}0.5} \\
731 Cohesive energy [eV] & -6.426 & -7.371 & -6.491 \\
732 Error [\%] & {\color{green}1.4} & {\color{red}16.3} & {\color{green}2.4} \\
736 \begin{minipage}{3cm}
738 \begin{tabular}{|l|l|}
743 {\color{green}0.5} & {\color{green}0.01} \\
746 {\color{green}0.8} & {\color{orange}4.5} \\
756 Energy cut-off for $\Gamma$-point only caclulations
759 $2\times 2\times 2$ Type 2 supercell, Param 2, US PP, LDA, 3C-SiC\\[0.2cm]
760 \includegraphics[width=5.5cm]{sic_32pc_gamma_cutoff.ps}
761 \includegraphics[width=5.5cm]{sic_32pc_gamma_cutoff_lc.ps}\\
762 $\Rightarrow$ Use 300 eV as energy cut-off?\\[0.2cm]
763 $2\times 2\times 2$ Type 2 supercell, Param 2, 300 eV, US PP, GGA\\[0.2cm]
765 \begin{minipage}{10cm}
766 \begin{tabular}{|l|l|l|l|}
768 & c-Si & c-C (diamond) & 3C-SiC \\
770 Lattice constant [\AA] & 5.470 & 3.569 & 4.364 \\
771 Error [\%] & {\color{green}0.8} & {\color{green}0.1} & {\color{green}0.1} \\
773 Cohesive energy [eV] & -4.488 & -7.612 & -6.359 \\
774 Error [\%] & {\color{orange}3.1} & {\color{orange}3.2} & {\color{green}0.3} \\
778 \begin{minipage}{2cm}
780 ${\color{green}\surd}$
789 C \hkl<1 0 0> interstitial migration along \hkl<1 1 0>
795 \begin{minipage}[t]{4.2cm}
796 \underline{Starting configuration}\\
797 \includegraphics[width=4cm]{c_100_mig/start.eps}
799 \begin{minipage}[t]{4.0cm}
801 $\Delta x=\frac{1}{4}a_{\text{Si}}=1.357\text{ \AA}$\\
802 $\Delta y=\frac{1}{4}a_{\text{Si}}=1.357\text{ \AA}$\\
803 $\Delta z=0.325\text{ \AA}$\\
805 \begin{minipage}[t]{4.2cm}
806 \underline{{\bf Expected} final configuration}\\
807 \includegraphics[width=4cm]{c_100_mig/final.eps}\\
809 \begin{minipage}{6cm}
811 \item Fix border atoms of the simulation cell
812 \item Constraints and displacement of the C atom:
814 \item along {\color{green}\hkl<1 1 0> direction}\\
815 displaced by {\color{green} $\frac{1}{10}(\Delta x,\Delta y)$}
816 \item C atom {\color{red}entirely fixed in position}\\
818 {\color{red}$\frac{1}{10}(\Delta x,\Delta y,\Delta z)$}
820 \item Berendsen thermostat applied
822 {\bf\color{blue}Expected configuration not obtained!}
824 \begin{minipage}{0.5cm}
827 \begin{minipage}{6cm}
828 \includegraphics[width=6.0cm]{c_100_110mig_01_albe.ps}
836 C \hkl<1 0 0> interstitial migration along \hkl<1 1 0>
842 \begin{minipage}{3.2cm}
843 \includegraphics[width=3cm]{c_100_mig/fixmig_50.eps}
848 \begin{minipage}{3.2cm}
849 \includegraphics[width=3cm]{c_100_mig/fixmig_80.eps}
854 \begin{minipage}{3.2cm}
855 \includegraphics[width=3cm]{c_100_mig/fixmig_90.eps}
860 \begin{minipage}{3.2cm}
861 \includegraphics[width=3cm]{c_100_mig/fixmig_99.eps}
869 \item Why is the expected configuration not obtained?
870 \item How to find a migration path preceding to the expected configuration?
875 \item Simple: it is not the right migration path!
877 \item (Surrounding) atoms settle into a local minimum configuration
878 \item A possibly existing more favorable configuration is not achieved
880 \item \begin{itemize}
881 \item Search global minimum in each step (by simulated annealing)\\
883 Loss of the correct energy needed for migration
884 \item Smaller displacements\\
885 A more favorable configuration might be achieved
886 possibly preceding to the expected configuration
896 C \hkl<1 0 0> interstitial migration along \hkl<1 1 0>
900 Displacement step size decreased to
901 $\frac{1}{100} (\Delta x,\Delta y)$\\[0.2cm]
903 \begin{minipage}{7.5cm}
904 Result: (Video \href{../video/c_in_si_smig_albe.avi}{$\rhd_{\text{local}}$ } $|$
905 \href{http://www.physik.uni-augsburg.de/~zirkelfr/download/posic/c_in_si_smig_albe.avi}{$\rhd_{\text{remote url}}$})
907 \item Expected final configuration not obtained
908 \item Bonds to neighboured silicon atoms persist
909 \item C and neighboured Si atoms move along the direction of displacement
910 \item Even the bond to the lower left silicon atom persists
913 Obviously: overestimated bond strength
916 \begin{minipage}{5cm}
917 \includegraphics[width=6cm]{c_100_110smig_01_albe.ps}
918 \end{minipage}\\[0.4cm]
919 New approach to find the migration path:\\
921 Place interstitial carbon atom at the respective coordinates
922 into a perfect c-Si matrix!
930 C \hkl<1 0 0> interstitial migration along \hkl<1 1 0>
934 {\color{blue}New approach:}\\
935 Place interstitial carbon atom at the respective coordinates
936 into a perfect c-Si matrix!\\
937 {\color{blue}Problem:}\\
938 Too high forces due to the small distance of the C atom to the Si
939 atom sharing the lattice site.\\
940 {\color{blue}Solution:}
942 \item {\color{red}Slightly displace the Si atom}
943 (Video \href{../video/c_in_si_pmig_albe.avi}{$\rhd_{\text{local}}$ } $|$
944 \href{http://www.physik.uni-augsburg.de/~zirkelfr/download/posic/c_in_si_pmig_albe.avi}{$\rhd_{\text{remote url}}$})
945 \item {\color{green}Immediately quench the system}
946 (Video \href{../video/c_in_si_pqmig_albe.avi}{$\rhd_{\text{local}}$ } $|$
947 \href{http://www.physik.uni-augsburg.de/~zirkelfr/download/posic/c_in_si_pqmig_albe.avi}{$\rhd_{\text{remote url}}$})
950 \begin{minipage}{6.5cm}
951 \includegraphics[width=6cm]{c_100_110pqmig_01_albe.ps}
953 \begin{minipage}{6cm}
955 \item Jump in energy corresponds to the abrupt
956 structural change (as seen in the videos)
957 \item Due to the abrupt changes in structure and energy
958 this is {\color{red}not} the correct migration path and energy!?!
967 C \hkl<1 0 0> interstitial migration along \hkl<1 1 0> in c-Si (VASP)
972 {\color{blue}Method:}
974 \item Place interstitial carbon atom at the respective coordinates
976 \item \hkl<1 1 0> direction fixed for the C atom
977 \item $4\times 4\times 3$ Type 1, $198+1$ atoms
978 \item Atoms with $x=0$ or $y=0$ or $z=0$ fixed
980 {\color{blue}Results:}
981 (Video \href{../video/c_in_si_pmig_vasp.avi}{$\rhd_{\text{local}}$ } $|$
982 \href{http://www.physik.uni-augsburg.de/~zirkelfr/download/posic/c_in_si_pmig_vasp.avi}{$\rhd_{\text{remote url}}$})\\
983 \begin{minipage}{7cm}
984 \includegraphics[width=7cm]{c_100_110pmig_01_vasp.ps}
986 \begin{minipage}{5.5cm}
988 \item Characteristics nearly equal to classical calulations
989 \item Approximately half of the classical energy
999 C \hkl<1 0 0> interstitial migration along \hkl<1 1 0> in c-Si (VASP)
1004 {\color{blue}Method:}
1006 \item Continue with atomic positions of the last run
1007 \item Displace the C atom in \hkl<1 1 0> direction
1008 \item \hkl<1 1 0> direction fixed for the C atom
1009 \item $4\times 4\times 3$ Type 1, $198+1$ atoms
1010 \item Atoms with $x=0$ or $y=0$ or $z=0$ fixed
1012 {\color{blue}Results:}
1013 (Video \href{../video/c_in_si_smig_vasp.avi}{$\rhd_{\text{local}}$ } $|$
1014 \href{http://www.physik.uni-augsburg.de/~zirkelfr/download/posic/c_in_si_smig_vasp.avi}{$\rhd_{\text{remote url}}$})\\
1015 \includegraphics[width=7cm]{c_100_110mig_01_vasp.ps}
1022 Again: C \hkl<1 0 0> interstitial migration
1027 {\color{blue}The applied methods:}
1031 \item Start in relaxed \hkl<1 0 0> interstitial configuration
1032 \item Displace C atom along \hkl<1 1 0> direction
1033 \item Relaxation (Berendsen thermostat)
1034 \item Continue with configuration of the last run
1038 \item Place interstitial carbon at the respective coordinates
1039 into the perfect Si matrix
1040 \item Quench the system
1043 {\color{blue}In both methods:}
1045 \item Fixed border atoms
1046 \item Applied \hkl<1 1 0> constraint for the C atom
1048 {\color{red}Pitfalls} and {\color{green}refinements}:
1050 \item {\color{red}Fixed border atoms} $\rightarrow$
1051 Relaxation of stress not possible\\
1053 {\color{green}Fix only one Si atom} (the one furthermost to the defect)
1054 \item {\color{red}\hkl<1 1 0> constraint not sufficient}\\
1055 $\Rightarrow$ {\color{green}Apply 11x constraint}
1056 (connecting line of initial and final C positions)
1064 Again: C \hkl<1 0 0> interstitial migration (Albe)
1067 Constraint applied by modifying the Velocity Verlet algorithm
1069 {\color{blue}Results:}
1070 (Video \href{../video/c_in_si_fmig_albe.avi}{$\rhd_{\text{local}}$ } $|$
1071 \href{http://www.physik.uni-augsburg.de/~zirkelfr/download/posic/c_in_si_fmig_albe.avi}{$\rhd_{\text{remote url}}$})\\
1072 \begin{minipage}{6.3cm}
1073 \includegraphics[width=6cm]{c_100_110fmig_01_albe.ps}
1075 \begin{minipage}{6cm}
1077 Again there are jumps in energy corresponding to abrupt
1078 structural changes as seen in the video
1082 \item Expected final configuration not obtained
1083 \item Bonds to neighboured silicon atoms persist
1084 \item C and neighboured Si atoms move along the direction of displacement
1085 \item Even the bond to the lower left silicon atom persists
1093 Again: C \hkl<1 0 0> interstitial migration (VASP)
1096 Transformation for the Type 2 supercell
1100 \begin{minipage}[t]{4.2cm}
1101 \underline{Starting configuration}\\
1102 \includegraphics[width=3cm]{c_100_mig_vasp/start.eps}
1104 \begin{minipage}[t]{4.0cm}
1106 $\Delta x=1.367\text{ \AA}$\\
1107 $\Delta y=1.367\text{ \AA}$\\
1108 $\Delta z=0.787\text{ \AA}$\\
1110 \begin{minipage}[t]{4.2cm}
1111 \underline{{\bf Expected} final configuration}\\
1112 \includegraphics[width=3cm]{c_100_mig_vasp/final.eps}\\
1114 \begin{minipage}{6.2cm}
1119 \beta=\arctan\frac{\Delta z}{\sqrt{2}\Delta x}=22.165^{\circ}
1122 \begin{minipage}{6.2cm}
1123 Length of migration path:
1125 l=\sqrt{\Delta x^2+\Delta y^2+\Delta z^2}=2.087\text{ \AA}
1127 \end{minipage}\\[0.3cm]
1128 Transformation of basis:
1130 T=ABA^{-1}A=AB \textrm{, mit }
1131 A=\left(\begin{array}{ccc}
1132 \cos\alpha & -\sin\alpha & 0\\
1133 \sin\alpha & \cos\alpha & 0\\
1137 B=\left(\begin{array}{ccc}
1139 0 & \cos\beta & \sin\beta \\
1140 0 & -\sin\beta & \cos\beta
1143 Atom coordinates transformed by: $T^{-1}=B^{-1}A^{-1}$
1150 Again: C \hkl<1 0 0> interstitial migration\\
1153 {\color{blue}Reminder:}\\
1154 Transformation needed since in VASP constraints can only be applied to
1155 the basis vectors!\\
1156 {\color{red}Problem:} (stupid me!)\\
1157 Transformation of supercell 'destroys' the correct periodicity!\\
1158 {\color{green}Solution:}\\
1159 Find a supercell with one basis vector forming the correct constraint\\
1160 {\color{red}Problem:}\\
1161 Hard to find a supercell for the $22.165^{\circ}$ rotation\\
1163 Another method to {\color{blue}\underline{estimate}} the migration energy:
1165 \item Assume an intermediate saddle point configuration during migration
1166 \item Determine the energy of the saddle point configuration
1167 \item Substract the saddle point configuration energy by
1168 the energy of the initial (final) defect configuration
1177 The C \hkl<1 0 0> defect configuration
1180 Needed so often for input configurations ...\\[0.8cm]
1181 \begin{minipage}{7.0cm}
1182 \includegraphics[width=6.5cm]{100-c-si-db_light.eps}\\
1183 Qualitative {\color{red}and} quantitative {\color{red}difference}!
1185 \begin{minipage}{5.5cm}
1188 \begin{tabular}{|l|l|l|}
1192 \underline{VASP} & & \\
1193 fractional & 0.1969 & 0.1211 \\
1194 in \AA & 1.08 & 0.66 \\
1196 \underline{Albe} & & \\
1197 fractional & 0.1547 & 0.1676 \\
1198 in \AA & 0.84 & 0.91 \\
1200 \end{tabular}\\[0.2cm]
1201 {\scriptsize\underline{PC (Vasp)}}
1202 \includegraphics[width=6.1cm]{c_100_pc_vasp.ps}
1211 Again: C \hkl<1 0 0> interstitial migration (VASP)
1214 $\hkl<0 0 -1> \rightarrow \hkl<0 0 1>$ migration
1215 ($3\times 3\times 3$ Type 2):
1219 \begin{minipage}[t]{4.1cm}
1220 \underline{Starting configuration}\\
1221 \includegraphics[height=3.2cm]{c_100_mig_vasp/start.eps}
1223 $E_{\text{f}}=3.15 \text{ eV}$
1226 \begin{minipage}[t]{4.1cm}
1227 \underline{Intermediate configuration}\\
1228 \includegraphics[height=3.2cm]{c_100_mig_vasp/00-1_001_im.eps}
1230 $E_{\text{f}}=4.41 \text{ eV}$
1233 \begin{minipage}[t]{4.1cm}
1234 \underline{Final configuration}\\
1235 \includegraphics[height=3.2cm]{c_100_mig_vasp/final.eps}
1237 $E_{\text{f}}=3.17 \text{ eV}$
1239 \end{minipage}\\[0.4cm]
1241 \Rightarrow \Delta E_{\text{f}} = E_{\text{mig}} = 1.26 \text{ eV}
1244 Unexpected \& ({\color{red}more} or {\color{orange}less}) fatal:
1246 \renewcommand\labelitemi{{\color{orange}$\bullet$}}
1247 \item Difference in formation energy (0.02 eV)
1248 of the initial and final configuration
1249 \renewcommand\labelitemi{{\color{red}$\bullet$}}
1250 \item Huge discrepancy (0.3 - 0.4 eV) to the migration barrier
1251 of Type 1 (198+1 atoms) calculations
1252 \renewcommand\labelitemi{{\color{black}$\bullet$}}
1260 Again: C \hkl<1 0 0> interstitial migration (VASP)
1263 $\hkl<0 0 -1> \rightarrow \hkl<0 -1 0>$ migration
1264 ($3\times 3\times 3$ Type 2):
1268 \begin{minipage}[t]{4.1cm}
1269 \underline{Starting configuration}\\
1270 \includegraphics[height=3.2cm]{c_100_mig_vasp/start.eps}
1272 $E_{\text{f}}=3.154 \text{ eV}$
1275 \begin{minipage}[t]{4.1cm}
1276 \underline{Intermediate configuration}\\
1279 $E_{\text{f}}=?.?? \text{ eV}$
1282 \begin{minipage}[t]{4.1cm}
1283 \underline{Final configuration}\\
1284 \includegraphics[height=3.2cm]{c_100_mig_vasp/0-10.eps}
1286 $E_{\text{f}}=3.157 \text{ eV}$
1288 \end{minipage}\\[0.4cm]
1290 \Rightarrow \Delta E_{\text{f}} = E_{\text{mig}} = ?.?? \text{ eV}
1295 Intermediate configuration {\color{red}not found} by now!
1303 C in Si interstitial configurations (VASP)
1306 Check of Kohn-Sham eigenvalues\\
1310 \begin{minipage}{6cm}
1311 \hkl<1 0 0> interstitial\\
1313 \begin{minipage}{6cm}
1314 Saddle point configuration\\
1316 \underline{$4\times 4\times 3$ Type 1 - fixed border atoms}\\
1317 \begin{minipage}{6cm}
1318 385: 4.8567 - 2.00000\\
1319 386: 4.9510 - 2.00000\\
1320 387: 5.3437 - 0.00000\\
1321 388: 5.4930 - 0.00000
1323 \begin{minipage}{6cm}
1324 385: 4.8694 - 2.00000\\
1325 386: {\color{red}4.9917} - 1.92603\\
1326 387: {\color{red}5.1181} - 0.07397\\
1327 388: 5.4541 - 0.00000
1328 \end{minipage}\\[0.2cm]
1329 \underline{$4\times 4\times 3$ Type 1 - no constraints}\\
1330 \begin{minipage}{6cm}
1331 385: 4.8586 - 2.00000\\
1332 386: 4.9458 - 2.00000\\
1333 387: 5.3358 - 0.00000\\
1334 388: 5.4915 - 0.00000
1336 \begin{minipage}{6cm}
1337 385: 4.8693 - 2.00000\\
1338 386: {\color{red}4.9879} - 1.92065\\
1339 387: {\color{red}5.1120} - 0.07935\\
1340 388: 5.4544 - 0.00000
1341 \end{minipage}\\[0.2cm]
1342 \underline{$3\times 3\times 3$ Type 2 - no constraints}\\
1343 \begin{minipage}{6cm}
1344 433: 4.8054 - 2.00000\\
1345 434: 4.9027 - 2.00000\\
1346 435: 5.2543 - 0.00000\\
1347 436: 5.5718 - 0.00000
1349 \begin{minipage}{6cm}
1350 433: 4.8160 - 2.00000\\
1351 434: {\color{green}5.0109} - 1.00264\\
1352 435: {\color{green}5.0111} - 0.99736\\
1353 436: 5.5364 - 0.00000
1361 Once again: C \hkl<1 0 0> interstitial migration (VASP)
1366 \item Start in fully relaxed (assumed) saddle point configuration
1367 \item Move towards \hkl<1 0 0> configuration using updated values
1368 for $\Delta x$, $\Delta y$ and $\Delta z$ (CRT)
1369 \item \hkl<1 1 0> constraints applied, 1 Si atom fixed
1370 \item $4\times 4\times 3$ Type 1 supercell
1375 \begin{minipage}{6.2cm}
1376 \includegraphics[width=6.0cm]{c_100_110sp-i_vasp.ps}
1378 \begin{minipage}{6.2cm}
1379 \includegraphics[width=6.0cm]{c_100_110sp-i_rc_vasp.ps}
1382 Reaction coordinate:
1383 $r_{i+1}=r_i+\sum_{\text{atoms j}} \left| r_{j,i+1}-r_{j,i} \right|$
1390 Investigation of the migration path along \hkl<1 1 0> (VASP)
1395 \underline{Minimum:}\\
1396 \begin{minipage}{4cm}
1397 \includegraphics[width=3.5cm]{c_100_mig_vasp/110_c-si_split.eps}
1399 \begin{minipage}{8cm}
1401 \item Starting conf: 35 \% displacement results (1443)
1402 \item \hkl<1 1 0> constraint disabled
1408 \item C-Si \hkl<1 1 0> split interstitial
1409 \item Stable configuration
1410 \item $E_{\text{f}}=4.13\text{ eV}$
1412 \end{minipage}\\[0.1cm]
1414 \underline{Maximum:}\\
1415 \begin{minipage}{6cm}
1417 \includegraphics[width=2.3cm]{c_100_mig_vasp/100-110_01.eps}
1418 \includegraphics[width=2.3cm]{c_100_mig_vasp/100-110_02.eps}\\
1419 20 \% $\rightarrow$ 25 \%\\
1420 Breaking of Si-C bond
1423 \begin{minipage}{6cm}
1424 \includegraphics[width=6.2cm]{c_100_110sp-i_upd_vasp.ps}
1432 Displacing the \hkl<1 1 0> Si-C split along \hkl<1 -1 0> (VASP)
1437 $4\times 4\times 3$ Type 1 supercell
1439 \underline{Structures:}
1441 \begin{minipage}[t]{4.1cm}
1442 \includegraphics[height=3.0cm]{c_100_mig_vasp/start.eps}\\
1443 \hkl<0 0 -1> dumbbell\\
1444 $E_{\text{f}}={\color{orange}3.2254}\text{ eV}$
1446 \begin{minipage}[t]{4.1cm}
1447 \includegraphics[height=3.0cm]{c_100_mig_vasp/110_c-si_split.eps}\\
1448 Assumed \hkl<1 1 0> C-Si split\\
1449 $E_{\text{f}}=4.1314\text{ eV}$
1451 \begin{minipage}[t]{4.1cm}
1452 \includegraphics[height=3.0cm]{c_100_mig_vasp/110_dis_0-10.eps}\\
1453 First guess: \hkl<0 -1 0> dumbbell\\
1454 {\color{red}but:} $E_{\text{f}}={\color{orange}2.8924}\text{ eV}$\\
1458 \underline{Occupancies:}
1462 \begin{minipage}{4.1cm}
1463 385: 4.8586 - 2.00000\\
1464 386: 4.9458 - 2.00000\\
1465 387: 5.3358 - 0.00000\\
1466 388: 5.4915 - 0.00000
1469 \begin{minipage}{4.1cm}
1470 385: 4.7790 - 2.00000\\
1471 386: 4.8797 - 1.99964\\
1472 387: 5.1321 - 0.00036\\
1473 388: 5.4711 - 0.00000
1476 \begin{minipage}{4.1cm}
1477 385: 4.7670 - 2.00000\\
1478 386: 4.9190 - 2.00000\\
1479 387: 5.2886 - 0.00000\\
1480 388: 5.4849 - 0.00000
1487 {\color{red}? ! ? ! ? ! ? ! ?}
1495 Defect configurations in $4\times 4\times 3$ Type 1 supercells revisited
1500 \begin{tabular}{l|p{2.5cm}|p{2.5cm}|p{4cm}|}
1501 & \hkl<0 0 -1> interstitial
1502 & local minimum\newline
1503 \hkl<1 1 0> C-Si split
1504 & intermediate configuration\newline
1505 (bond centered conf)\\
1507 default & $E_{\text{f}}=3.3254\text{ eV}$\newline
1509 386: 4.9458 - 2.00000\newline
1510 387: 5.3358 - 0.00000}
1511 & $E_{\text{f}}=4.1314\text{ eV}$\newline
1513 386: 4.8797 - 1.99964\newline
1514 387: 5.1321 - 0.00036}
1515 & $E_{\text{f}}=4.2434\text{ eV}$\newline
1517 386: 4.9879 - 1.92065\newline
1518 387: 5.1120 - 0.07935} \\
1520 No symmetry & $E_{\text{f}}=3.3154\text{ eV}$\newline
1522 386: 4.9456 - 2.00000\newline
1523 387: 5.3366 - 0.00000}
1524 & $E_{\text{f}}=4.1314\text{ eV}$\newline
1526 386: 4.8798 - 1.99961\newline
1527 387: 5.1307 - 0.00039}
1528 & $E_{\text{f}}=4.2454\text{ eV}$\newline
1530 386: 4.9841 - 1.92147\newline
1531 387: 5.1085 - 0.07853} \\
1533 $+$ spin polarized & $E_{\text{f}}=3.3154\text{ eV}$\newline
1536 386: 4.9449 - 1.00000\newline
1537 387: 5.3365 - 0.00000\newline%
1540 386: 4.9449 - 1.00000\newline
1541 387: 5.3365 - 0.00000}}
1542 & $E_{\text{f}}={\color{red}4.1314}\text{ eV}$\newline
1545 386: 4.8799 - 0.99980\newline
1546 387: 5.1307 - 0.00020\newline%
1549 386: 4.8799 - 0.99980\newline
1550 387: 5.1306 - 0.00020}}
1551 & $E_{\text{f}}=4.0254\text{ eV}$\newline
1554 387: 4.8581 - 1.00000\newline
1555 388: 5.4662 - 0.00000\newline%
1558 385: 4.8620 - 1.00000\newline
1559 386: 5.2951 - 0.00000}} \\
1561 $+$ spin difference 2 & $E_{\text{f}}=3.6394\text{ eV}$\newline
1564 387: 5.2704 - 0.99891\newline
1565 388: 5.4886 - 0.00095\newline
1566 389: 5.5094 - 0.00011\newline
1567 390: 5.5206 - 0.00003\newline%
1570 385: 4.8565 - 0.98603\newline
1571 386: 5.0119 - 0.01397}}
1572 & Relaxation into\newline
1573 bond centered\newline
1574 configuration\newline
1576 & $E_{\text{f}}=4.0254\text{ eV}$\newline
1579 387: 4.8578 - 1.00000\newline
1580 388: 5.4661 - 0.00000\newline%
1583 385: 4.8618 - 1.00000\newline
1584 386: 5.2950 - 0.00000}} \\
1593 C \hkl<1 0 0> interstitial migration (VASP)
1598 \begin{minipage}{6.2cm}
1600 \item $3\times 3\times 3$ Type 2 supercell
1601 \item \hkl<1 1 0> constraints applied
1602 (\href{http://www.physik.uni-augsburg.de/~zirkelfr/download/posic/sd_rot.patch}{Patch})
1603 \item Move from \hkl<1 0 0> towards\\
1604 bond centered configuration
1606 \underline{Sd Rot usage (POSCAR):}
1614 Transformed selective dynamics
1619 Only works in direct mode!\\
1620 $z,x'$-axis rotation: $45.0^{\circ}$, $0.0^{\circ}$
1622 \begin{minipage}{6.2cm}
1623 \includegraphics[width=5cm]{c_100_110sp-i_2333_vasp.ps}
1624 \includegraphics[width=5cm]{c_100_110sp-i_2333_rc_vasp.ps}\\
1625 {\color{red}One fixed Si atom not enough!}\\
1626 Video: \href{../video/c_in_si_233_110mig_vasp.avi}{$\rhd_{\text{local}}$ } $|$
1627 \href{http://www.physik.uni-augsburg.de/~zirkelfr/download/posic/c_in_si_233_110mig_vasp.avi}{$\rhd_{\text{remote url}}$}\\
1631 Next: Migration calculation in 2333 using CRT
1632 (\hkl<0 0 -1> $\rightarrow$ \hkl<0 0 1> and \hkl<0 -1 0>)
1640 \hkl<0 0 -1> to \hkl <0 0 1> migration
1641 in the $3\times 3\times 3$ Type 2 supercell
1649 \hkl<0 0 -1> to \hkl <0 -1 0> migration
1650 in the $3\times 3\times 3$ Type 2 supercell
1658 Defect configurations in $3\times 3\times 3$ Type 2 supercells revisited\\
1663 \begin{tabular}{l|p{2.5cm}|p{2.5cm}|p{4cm}|}
1664 & \hkl<0 0 -1> interstitial
1665 & local minimum\newline
1666 \hkl<1 1 0> C-Si split
1667 & intermediate configuration\newline
1668 (bond centered conf)\\
1670 default & $E_{\text{f}}=3.15407\text{ eV}$\newline
1672 434: 4.9027 - 2.00000\newline
1673 435: 5.2543 - 0.00000}
1674 & $E_{\text{f}}=??\text{ eV}$\newline
1678 & $E_{\text{f}}=4.40907\text{ eV}$\newline
1680 434: 5.0109 - 1.00264\newline
1681 435: 5.0111 - 0.99736}\\
1683 No symmetry & $E_{\text{f}}=3.16107\text{ eV}$\newline
1685 434: 4.9032 - 2.00000\newline
1686 435: 5.2547 - 0.00000}
1687 & $E_{\text{f}}=??\text{ eV}$\newline
1691 & $E_{\text{f}}=4.41507\text{ eV}$\newline
1693 434: 5.0113 - 1.00140\newline
1694 435: 5.0114 - 0.99860} \\
1696 $+$ spin polarized & $E_{\text{f}}=3.16107\text{ eV}$\newline
1699 434: 4.9033 - 1.00000\newline
1700 435: 5.2544 - 0.00000\newline%
1703 434: 4.9035 - 1.00000\newline
1704 435: 5.2550 - 0.00000}}
1705 & $E_{\text{f}}=??\text{ eV}$\newline
1714 & $E_{\text{f}}=4.10307\text{ eV}$\newline
1717 435: 4.8118 - 1.00000\newline
1718 436: 5.5360 - 0.00000\newline%
1721 433: 4.8151 - 1.00000\newline
1722 434: 5.3475 - 0.00000}} \\
1730 {\color{blue}Tracer:}\\
1731 Whenever there is smearing of electrons over two or more energy levels\\
1732 $\Rightarrow$ use spin polarized calculations!
1739 Bond centered configuration revisited ($3\times 3\times 3$ Type 2)
1742 Spin polarized calculations
1745 \begin{minipage}[t]{5.8cm}
1746 \underline{Kohn-Sham eigenvalues}\\
1747 \begin{minipage}{2.8cm}
1749 430: 4.2639 - 1\newline
1750 431: 4.7332 - 1\newline
1751 432: 4.7354 - 1\newline
1752 433: 4.7700 - 1\newline
1753 434: 4.8116 - 1\newline
1754 435: 4.8118 - 1\newline
1755 436: 5.5360 - 0\newline
1758 \begin{minipage}{2.8cm}
1760 430: 4.2682 - 1\newline
1761 431: 4.7738 - 1\newline
1762 432: 4.8150 - 1\newline
1763 433: 4.8151 - 1\newline
1764 434: 5.3475 - 0\newline
1765 435: 5.3476 - 0\newline
1766 436: 5.5455 - 0\newline
1770 \begin{minipage}[t]{6.5cm}
1771 \underline{MO diagram}\\
1772 \begin{minipage}[t]{1.2cm}
1774 {\tiny sp$^3$}\\[0.8cm]
1775 \underline{${\color{red}\uparrow}$}
1776 \underline{${\color{red}\uparrow}$}
1777 \underline{${\color{red}\uparrow}$}
1778 \underline{${\color{red}\uparrow}$}\\
1781 \begin{minipage}[t]{1.4cm}
1783 {\color{red}M}{\color{blue}O}\\[1.0cm]
1784 \underline{${\color{blue}\uparrow}{\color{white}\downarrow}$}\\
1785 $\sigma_{\text{ab}}$\\[0.5cm]
1786 \underline{${\color{red}\uparrow}{\color{blue}\downarrow}$}\\
1790 \begin{minipage}[t]{1.0cm}
1794 \underline{${\color{white}\uparrow\uparrow}$}
1795 \underline{${\color{white}\uparrow\uparrow}$}\\
1797 \underline{${\color{blue}\uparrow}{\color{blue}\downarrow}$}
1798 \underline{${\color{blue}\uparrow}{\color{blue}\downarrow}$}\\
1802 \begin{minipage}[t]{1.4cm}
1804 {\color{blue}M}{\color{green}O}\\[1.0cm]
1805 \underline{${\color{blue}\uparrow}{\color{white}\downarrow}$}\\
1806 $\sigma_{\text{ab}}$\\[0.5cm]
1807 \underline{${\color{green}\uparrow}{\color{blue}\downarrow}$}\\
1811 \begin{minipage}[t]{1.2cm}
1814 {\tiny sp$^3$}\\[0.8cm]
1815 \underline{${\color{green}\uparrow}$}
1816 \underline{${\color{green}\uparrow}$}
1817 \underline{${\color{green}\uparrow}$}
1818 \underline{${\color{green}\uparrow}$}\\
1830 \hkl<0 0 -1> configuration revisited ($3\times 3\times 3$ Type 2)
1833 Spin polarized calculations
1836 \begin{minipage}[t]{5.8cm}
1837 \underline{Kohn-Sham eigenvalues}\\
1838 \begin{minipage}{2.8cm}
1840 430: 4.3317 - 1\newline
1841 431: 4.7418 - 1\newline
1842 432: 4.8014 - 1\newline
1843 433: 4.8060 - 1\newline
1844 434: 4.9033 - 1\newline
1845 435: 5.2544 - 0\newline
1846 436: 5.5723 - 0\newline
1849 \begin{minipage}{2.8cm}
1851 430: 4.3317 - 1\newline
1852 431: 4.7420 - 1\newline
1853 432: 4.8013 - 1\newline
1854 433: 4.8059 - 1\newline
1855 434: 4.9035 - 1\newline
1856 435: 5.2550 - 0\newline
1857 436: 5.5724 - 0\newline
1861 \begin{minipage}[t]{6.5cm}
1862 \underline{MO diagram}\\
1863 \begin{minipage}[t]{1.2cm}
1865 {\tiny sp$^3$}\\[0.8cm]
1866 \underline{${\color{red}\uparrow}$}
1867 \underline{${\color{red}\uparrow}$}
1868 \underline{${\color{red}\uparrow}$}
1869 \underline{${\color{red}\uparrow}$}\\
1872 \begin{minipage}[t]{1.4cm}
1874 {\color{red}M}{\color{blue}O}\\[1.0cm]
1875 \underline{${\color{blue}\uparrow}{\color{white}\downarrow}$}\\
1876 $\sigma_{\text{ab}}$\\[0.5cm]
1877 \underline{${\color{red}\uparrow}{\color{blue}\downarrow}$}\\
1881 \begin{minipage}[t]{1.0cm}
1885 \underline{${\color{white}\uparrow\uparrow}$}
1886 \underline{${\color{white}\uparrow\uparrow}$}\\
1888 \underline{${\color{blue}\uparrow}{\color{blue}\downarrow}$}
1889 \underline{${\color{blue}\uparrow}{\color{blue}\downarrow}$}\\
1893 \begin{minipage}[t]{1.4cm}
1895 {\color{blue}M}{\color{green}O}\\[1.0cm]
1896 \underline{${\color{blue}\uparrow}{\color{white}\downarrow}$}\\
1897 $\sigma_{\text{ab}}$\\[0.5cm]
1898 \underline{${\color{green}\uparrow}{\color{blue}\downarrow}$}\\
1902 \begin{minipage}[t]{1.2cm}
1905 {\tiny sp$^3$}\\[0.8cm]
1906 \underline{${\color{green}\uparrow}$}
1907 \underline{${\color{green}\uparrow}$}
1908 \underline{${\color{green}\uparrow}$}
1909 \underline{${\color{green}\uparrow}$}\\
1921 Combination of defects
1924 TODO: introduce some Si self-interstitials and C interstitials before\\
1925 BUT: Concentrate on 100 C interstitial combinations and 100 C + vacancy\\
1927 Agglomeration of 100 defects energetically favorable?
1934 Combination of defects
1938 \item Supercell: $3\times 3\times 3$ Type 2
1939 \item Starting configuration: \hkl<0 0 -1> C-Si interstitial
1940 \item Energies: $E_{\text{f}}$ of the interstitial combinations in eV
1943 \underline{Along \hkl<1 1 0>:}
1945 \begin{tabular}{|l|l|l|l|l|}
1948 \backslashbox{2nd interstitial}{Distance $[\frac{a}{4}]$}
1950 & \hkl<1 1 -1> & \hkl<2 2 0> & \hkl<3 3 -1> & \hkl<4 4 0>\\
1952 \hkl<0 0 -1> & 6.23514 & 4.65014 & 5.97314 & in progress\\
1954 \hkl<0 0 1> & TODO & TODO & 6.53614 & TODO \\
1956 \hkl<1 0 0>, \hkl<0 1 0> & TODO & TODO & TODO & TODO\\
1958 \hkl<-1 0 0>, \hkl<0 -1 0> & TODO & TODO & TODO & TODO\\
1967 Molecular dynamics simulations (VASP)
1970 2 C atoms in $2\times 2\times 2$ Type 2 supercell at $450\,^{\circ}\text{C}$
1974 \begin{minipage}{7.6cm}
1975 Radial distribution\\
1976 \includegraphics[width=7.6cm]{md_02c_2222si_pc.ps}
1978 \begin{minipage}{5.0cm}
1981 $t_1=50$ ps to $t_2=50.93$ ps
1986 \item $<(x(t)-x(0))^2>$ hard to determine due to missing info of
1988 \item No jumps recognized in the
1989 Video \href{../video/md_02c_2222si_vasp.avi}{$\rhd_{\text{local}}$ } $|$
1990 \href{http://www.physik.uni-augsburg.de/~zirkelfr/download/posic/md_02c_2222si_vasp.avi}{$\rhd_{\text{remote url}}$}
1998 Molecular dynamics simulations (VASP)
2001 10 C atoms in $3\times 3\times 3$ Type 2 supercell at $450\,^{\circ}\text{C}$
2005 \begin{minipage}{7.2cm}
2006 Radial distribution (PC averaged over 1 ps)\\
2007 \includegraphics[width=7.0cm]{md_10c_2333si_pc_vasp.ps}
2009 \begin{minipage}{5.0cm}
2010 \includegraphics[width=6.0cm]{md_10c_2333si_pcc_vasp.ps}
2013 (Video \href{../video/md_10c_2333si_vasp.avi}{$\rhd_{\text{local}}$ } $|$
2014 \href{http://www.physik.uni-augsburg.de/~zirkelfr/download/posic/md_10c_2333si_vasp.avi}{$\rhd_{\text{remote url}}$})
2016 \item $<(x(t)-x(0))^2>$ hard to determine due to missing info of
2018 \item Agglomeration of C? (Video)
2026 Molecular dynamics simulations (VASP)
2029 1 C atom in $3\times 3\times 3$ Type 2 supercell at $900\,^{\circ}\text{C}$
2038 Molecular dynamics simulations (VASP)
2041 10 C atoms in $3\times 3\times 3$ Type 2 supercell at $900\,^{\circ}\text{C}$
2050 Density Functional Theory
2053 Hohenberg-Kohn theorem
2065 Transition state theory\\