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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}{7.7cm}
1180 \includegraphics[width=7cm]{100-c-si-db_light.eps}
1183 \begin{minipage}{4.5cm}
1184 \begin{tabular}{|l|l|l|}
1188 \underline{VASP} & & \\
1189 fractional & 0.1969 & 0.1211 \\
1190 in \AA & 1.08 & 0.66 \\
1192 \underline{Albe} & & \\
1193 fractional & 0.1547 & 0.1676 \\
1194 in \AA & 0.84 & 0.91 \\
1200 Qualitative {\color{red}and} quantitative {\color{red}difference}!
1208 Again: C \hkl<1 0 0> interstitial migration (VASP)
1211 $\hkl<0 0 -1> \rightarrow \hkl<0 0 1>$ migration:
1215 \begin{minipage}[t]{4.1cm}
1216 \underline{Starting configuration}\\
1217 \includegraphics[height=3.2cm]{c_100_mig_vasp/start.eps}
1219 $E_{\text{f}}=3.15 \text{ eV}$
1222 \begin{minipage}[t]{4.1cm}
1223 \underline{Intermediate configuration}\\
1224 \includegraphics[height=3.2cm]{c_100_mig_vasp/00-1_001_im.eps}
1226 $E_{\text{f}}=4.41 \text{ eV}$
1229 \begin{minipage}[t]{4.1cm}
1230 \underline{Final configuration}\\
1231 \includegraphics[height=3.2cm]{c_100_mig_vasp/final.eps}
1233 $E_{\text{f}}=3.17 \text{ eV}$
1235 \end{minipage}\\[0.4cm]
1237 \Rightarrow \Delta E_{\text{f}} = E_{\text{mig}} = 1.26 \text{ eV}
1240 Unexpected \& ({\color{red}more} or {\color{orange}less}) fatal:
1242 \renewcommand\labelitemi{{\color{orange}$\bullet$}}
1243 \item Difference in formation energy (0.02 eV)
1244 of the initial and final configuration
1245 \renewcommand\labelitemi{{\color{red}$\bullet$}}
1246 \item Huge discrepancy (0.3 - 0.4 eV) to the migration barrier
1247 of Type 1 (198+1 atoms) calculations
1248 \renewcommand\labelitemi{{\color{black}$\bullet$}}
1256 Again: C \hkl<1 0 0> interstitial migration (VASP)
1259 $\hkl<0 0 -1> \rightarrow \hkl<0 -1 0>$ migration:
1263 \begin{minipage}[t]{4.1cm}
1264 \underline{Starting configuration}\\
1265 \includegraphics[height=3.2cm]{c_100_mig_vasp/start.eps}
1267 $E_{\text{f}}=3.154 \text{ eV}$
1270 \begin{minipage}[t]{4.1cm}
1271 \underline{Intermediate configuration}\\
1274 $E_{\text{f}}=?.?? \text{ eV}$
1277 \begin{minipage}[t]{4.1cm}
1278 \underline{Final configuration}\\
1279 \includegraphics[height=3.2cm]{c_100_mig_vasp/0-10.eps}
1281 $E_{\text{f}}=3.157 \text{ eV}$
1283 \end{minipage}\\[0.4cm]
1285 \Rightarrow \Delta E_{\text{f}} = E_{\text{mig}} = ?.?? \text{ eV}
1288 Unexpected \& ({\color{red}more} or {\color{orange}less}) fatal:
1290 \renewcommand\labelitemi{{\color{orange}$\bullet$}}
1291 \item Difference in formation energy (0.02 eV)
1292 of the initial and final configuration
1293 \renewcommand\labelitemi{{\color{red}$\bullet$}}
1294 \item Huge discrepancy (0.3 - 0.4 eV) to the migration barrier
1295 of Type 1 (198+1 atoms) calculations
1296 \renewcommand\labelitemi{{\color{black}$\bullet$}}
1304 Molecular dynamics simulations (VASP)
1314 Density Functional Theory
1317 Hohenberg-Kohn theorem