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102 \rput(6.0,0.2){\psframebox[fillstyle=gradient,gradbegin=hb,gradend=white,gradlines=1000,gradmidpoint=1,linestyle=none]{
103 \begin{minipage}{14cm}
112 \newcommand{\si}{Si$_{\text{i}}${}}
113 \newcommand{\ci}{C$_{\text{i}}${}}
114 \newcommand{\cs}{C$_{\text{sub}}${}}
115 \newcommand{\degc}[1]{\unit[#1]{$^{\circ}$C}{}}
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120 % no vertical centering
132 A B C D E F G H G F E D C B A
151 First-principles and empirical\\[0.2cm]
152 potential simulation study of intrinsic\\[0.2cm]
153 and carbon-related defects in silicon
161 \underline{F. Zirkelbach} $\color{gray}\bullet$ B. Stritzker\\
164 Experimentalphysik IV, Universit\"at Augsburg, 86135 Augsburg, Germany
170 Department of Physics, University of Helsinki, 00014 Helsinki, Finland
173 W. G. Schmidt $\color{gray}\bullet$ E. Rauls $\color{gray}\bullet$
177 Department Physik, Universit\"at Paderborn, 33095 Paderborn, Germany
183 E-MRS Spring Meeting, Strasbourg, 17.05.2012
188 % no vertical centering
191 % skip for preparation
200 Motivation \& Outline
206 Ion beam synthesis (IBS) of epitaxial single crystalline 3C-SiC
211 \begin{minipage}{7.0cm}
214 \item \underline{Implantation}\\[0.1cm]
215 Stoichiometric dose | \unit[180]{keV} | \degc{500}\\
216 $\Rightarrow$ Epitaxial {\color{blue}3C-SiC} layer \&
217 {\color{blue}precipitates}
218 \item \underline{Annealing}\\[0.1cm]
219 \unit[10]{h} at \degc{1250}\\
220 $\Rightarrow$ Homogeneous 3C-SiC layer
225 \begin{minipage}{4.5cm}
228 3C-SiC precipitation\\
229 not yet fully understood
235 \begin{minipage}{5.0cm}
236 \includegraphics[width=5.5cm]{ibs_3c-sic.eps}\\[-0.4cm]
239 XTEM: single crystalline 3C-SiC in Si\hkl(1 0 0)
242 \end{minipage}\\[0.2cm]
249 \item Assumed SiC precipitation mechanisms / Controversy
250 \item Utilized simulation techniques
251 \item C and Si self-interstitial point defects in silicon
252 \item Silicon carbide precipitation simulations
261 Supposed precipitation mechanism of SiC in Si
269 \begin{minipage}{3.6cm}
271 Si \& SiC lattice structure\\[0.1cm]
272 \includegraphics[width=2.3cm]{sic_unit_cell.eps}
275 \begin{minipage}{1.7cm}
276 \underline{Silicon}\\
277 {\color{yellow}$\bullet$} Si | {\color{gray}$\bullet$} Si\\
278 $a=\unit[5.429]{\\A}$\\
279 $\rho^*_{\text{Si}}=\unit[100]{\%}$
281 \begin{minipage}{1.7cm}
282 \underline{Silicon carbide}\\
283 {\color{yellow}$\bullet$} Si | {\color{gray}$\bullet$} C\\
284 $a=\unit[4.359]{\\A}$\\
285 $\rho^*_{\text{Si}}=\unit[97]{\%}$
291 \begin{minipage}{4.1cm}
293 \includegraphics[width=3.3cm]{tem_c-si-db.eps}
297 \begin{minipage}{4.0cm}
299 \includegraphics[width=3.3cm]{tem_3c-sic.eps}
305 \begin{minipage}{4.0cm}
307 C-Si dimers (dumbbells)\\[-0.1cm]
312 \begin{minipage}{4.1cm}
314 Agglomeration of C-Si dumbbells\\[-0.1cm]
315 $\Rightarrow$ dark contrasts
319 \begin{minipage}{4.0cm}
321 Precipitation of 3C-SiC in Si\\[-0.1cm]
322 $\Rightarrow$ Moir\'e fringes\\[-0.1cm]
323 \& release of Si self-interstitials
329 \begin{minipage}{4.0cm}
331 \includegraphics[width=3.3cm]{sic_prec_seq_01.eps}
335 \begin{minipage}{4.1cm}
337 \includegraphics[width=3.3cm]{sic_prec_seq_02.eps}
341 \begin{minipage}{4.0cm}
343 \includegraphics[width=3.3cm]{sic_prec_seq_03.eps}
347 \begin{pspicture}(0,0)(0,0)
348 \psline[linewidth=2pt]{->}(8.3,2)(8.8,2)
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350 \rput{-20}{\psellipse[linecolor=blue](3.1,8.2)(0.3,0.5)}
351 \psline[linewidth=2pt]{->}(3.9,2)(4.4,2)
352 \rput(11.8,0.3){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
353 $4a_{\text{Si}}=5a_{\text{SiC}}$
355 \rput(11.5,8){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
356 \hkl(h k l) planes match
358 \rput(8.5,6.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
369 Supposed precipitation mechanism of SiC in Si
377 \begin{minipage}{3.6cm}
379 Si \& SiC lattice structure\\[0.1cm]
380 \includegraphics[width=2.3cm]{sic_unit_cell.eps}
383 \begin{minipage}{1.7cm}
384 \underline{Silicon}\\
385 {\color{yellow}$\bullet$} Si | {\color{gray}$\bullet$} Si\\
386 $a=\unit[5.429]{\\A}$\\
387 $\rho^*_{\text{Si}}=\unit[100]{\%}$
389 \begin{minipage}{1.7cm}
390 \underline{Silicon carbide}\\
391 {\color{yellow}$\bullet$} Si | {\color{gray}$\bullet$} C\\
392 $a=\unit[4.359]{\\A}$\\
393 $\rho^*_{\text{Si}}=\unit[97]{\%}$
399 \begin{minipage}{4.1cm}
401 \includegraphics[width=3.3cm]{tem_c-si-db.eps}
405 \begin{minipage}{4.0cm}
407 \includegraphics[width=3.3cm]{tem_3c-sic.eps}
413 \begin{minipage}{4.0cm}
415 C-Si dimers (dumbbells)\\[-0.1cm]
416 on Si interstitial sites
420 \begin{minipage}{4.1cm}
422 Agglomeration of C-Si dumbbells\\[-0.1cm]
423 $\Rightarrow$ dark contrasts
427 \begin{minipage}{4.0cm}
429 Precipitation of 3C-SiC in Si\\[-0.1cm]
430 $\Rightarrow$ Moir\'e fringes\\[-0.1cm]
431 \& release of Si self-interstitials
437 \begin{minipage}{4.0cm}
439 \includegraphics[width=3.3cm]{sic_prec_seq_01.eps}
443 \begin{minipage}{4.1cm}
445 \includegraphics[width=3.3cm]{sic_prec_seq_02.eps}
449 \begin{minipage}{4.0cm}
451 \includegraphics[width=3.3cm]{sic_prec_seq_03.eps}
455 \begin{pspicture}(0,0)(0,0)
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459 \psline[linewidth=2pt]{->}(3.9,2)(4.4,2)
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461 $4a_{\text{Si}}=5a_{\text{SiC}}$
463 \rput(11.5,8){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
464 \hkl(h k l) planes match
466 \rput(8.5,6.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
469 % controversial view!
470 \rput(6.5,5.0){\psframebox[fillstyle=solid,opacity=0.5,fillcolor=black]{
471 \begin{minipage}{14cm}
476 \rput(6.5,5.3){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=white,linewidth=0.1cm]{
477 \begin{minipage}{10cm}
481 {\color{gray}\bf Controversial findings}
484 \item High-temperature implantation {\tiny\color{gray}/Nejim~et~al./}
486 \item {\color{blue}Substitutionally} incorporated C on regular Si lattice sites
487 \item \si{} reacting with further C in cleared volume
489 \item Annealing behavior {\tiny\color{gray}/Serre~et~al./}
491 \item Room temperature implantation $\rightarrow$ high C diffusion
492 \item Elevated temperature implantation $\rightarrow$ no C redistribution
494 $\Rightarrow$ mobile {\color{red}\ci} opposed to
495 stable {\color{blue}\cs{}} configurations
496 \item Strained Si$_{1-y}$C$_y$/Si heterostructures
497 {\tiny\color{gray}/Strane~et~al./Guedj~et~al./}
499 \item Initial {\color{blue}coherent} SiC structures (tensile strain)
500 \item Incoherent SiC nanocrystals (strain relaxation)
505 {\Huge${\lightning}$} \hspace{0.3cm}
506 {\color{blue}\cs{}} --- vs --- {\color{red}\ci} \hspace{0.3cm}
507 {\Huge${\lightning}$}
520 Utilized computational methods
527 {\bf Molecular dynamics (MD)}\\[0.1cm]
529 \begin{tabular}{| p{4.5cm} | p{7.5cm} |}
531 System of $N$ particles &
532 $N=5832\pm 1$ (Defects), $N=238328+6000$ (Precipitation)\\
533 Phase space propagation &
534 Velocity Verlet | timestep: \unit[1]{fs} \\
535 Analytical interaction potential &
536 Tersoff-like {\color{red}short-range}, {\color{blue}bond order} potential
539 E = \frac{1}{2} \sum_{i \neq j} \pot_{ij}, \quad
540 \pot_{ij} = {\color{red}f_C(r_{ij})}
541 \left[ f_R(r_{ij}) + {\color{blue}b_{ij}} f_A(r_{ij}) \right]
543 Observables: time/ensemble averages &
544 NpT (isothermal-isobaric) | Berendsen thermostat/barostat\\
552 {\bf Density functional theory (DFT)}
556 \begin{minipage}[t]{6cm}
558 \item Hohenberg-Kohn theorem:\\
559 $\Psi_0(r_1,r_2,\ldots,r_N)=\Psi[n_0(r)]$, $E_0=E[n_0]$
560 \item Kohn-Sham approach:\\
561 Single-particle effective theory
565 \item Code: \textsc{vasp}
566 \item Plane wave basis set | $E_{\text{cut}}=\unit[300]{eV}$
568 %\Phi_i=\sum_{|G+k|<G_{\text{cut}}} c_{i,k+G} \exp{\left(i(k+G)r\right)}
571 %E_{\text{cut}}=\frac{\hbar^2}{2m}G^2_{\text{cut}}=\unit[300]{eV}
573 \item Ultrasoft pseudopotential
574 \item Exchange \& correlation: GGA
575 \item Brillouin zone sampling: $\Gamma$-point
576 \item Supercell: $N=216\pm2$
579 \begin{minipage}{6cm}
580 \begin{pspicture}(0,0)(0,0)
581 \pscircle[fillcolor=yellow,fillstyle=solid,linestyle=none](3.5,-2.0){2.5}
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584 \left[ -\frac{\hbar^2}{2m}\nabla^2 + V_{\text{eff}}(r) - \epsilon_i \right] \Phi_i(r) = 0
587 \rput(5.2,-2.0){\psframebox[fillstyle=solid,opacity=0.8,fillcolor=white]{
589 n(r)=\sum_i^N|\Phi_i(r)|^2
592 \rput(3.0,-4.5){\psframebox[fillstyle=solid,opacity=0.8,fillcolor=white]{
594 V_{\text{eff}}(r)=V_{\text{ext}}(r)+\int\frac{e^2 n(r')}{|r-r'|}d^3r'
598 \psarcn[linewidth=0.07cm,linestyle=dashed]{->}(3.5,-2.0){2.5}{130}{15}
599 \psarcn[linewidth=0.07cm,linestyle=dashed]{->}(3.5,-2.0){2.5}{230}{165}
600 \psarcn[linewidth=0.07cm,linestyle=dashed]{->}(3.5,-2.0){2.5}{345}{310}
611 Point defects \& defect migration
618 \begin{minipage}[b]{7.5cm}
619 {\bf Defect structure}\\
620 \begin{pspicture}(0,0)(7,4.4)
621 \rput(3.5,3.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
624 \item Creation of c-Si simulation volume
625 \item Periodic boundary conditions
626 \item $T=0\text{ K}$, $p=0\text{ bar}$
629 \rput(3.5,1.3){\rnode{insert}{\psframebox{
632 Insertion of interstitial C/Si atoms
635 \rput(3.5,0.2){\rnode{cool}{\psframebox[fillstyle=solid,fillcolor=lbb]{
638 Relaxation / structural energy minimization
641 \ncline[]{->}{init}{insert}
642 \ncline[]{->}{insert}{cool}
645 \begin{minipage}[b]{4.5cm}
647 \includegraphics[width=3.8cm]{unit_cell_e.eps}\\
649 \begin{minipage}{2.21cm}
651 {\color{red}$\bullet$} Tetrahedral\\[-0.1cm]
652 {\color{green}$\bullet$} Hexagonal\\[-0.1cm]
653 {\color{yellow}$\bullet$} \hkl<1 0 0> DB
656 \begin{minipage}{2.21cm}
658 {\color{magenta}$\bullet$} \hkl<1 1 0> DB\\[-0.1cm]
659 {\color{cyan}$\bullet$} Bond-centered\\[-0.1cm]
660 {\color{black}$\bullet$} Vac. / Sub.
667 \begin{minipage}[t]{6cm}
668 {\bf Defect formation energy}\\
670 $E_{\text{f}}=E-\sum_i N_i\mu_i$}\\[0.5cm]
671 %Particle reservoir: Si \& SiC\\[0.2cm]
672 {\bf Binding energy}\\
676 E_{\text{f}}^{\text{comb}}-
677 E_{\text{f}}^{1^{\text{st}}}-
678 E_{\text{f}}^{2^{\text{nd}}}
682 $E_{\text{b}}<0$: energetically favorable configuration\\
683 $E_{\text{b}}\rightarrow 0$: non-interacting, isolated defects\\
685 \begin{minipage}[t]{6cm}
687 {\bf Migration barrier}
690 \item Displace diffusing atom
691 \item Constrain relaxation of (diffusing) atoms
692 \item Record configurational energy
694 \begin{picture}(0,0)(-60,-33)
695 \includegraphics[width=4.5cm]{crt_mod.eps}
707 C interstitial point defects in silicon\\
710 \begin{tabular}{l c c c c c c r}
712 $E_{\text{f}}$ [eV] & T & H & \hkl<1 0 0> DB & \hkl<1 1 0> DB & S & B &
713 {\color{black} \cs{} \& \si}\\
715 \textsc{vasp} & unstable & unstable & \underline{3.72} & 4.16 & 1.95 & 4.66 & {\color{green}4.17}\\
716 Erhart/Albe & 6.09 & 9.05$^*$ & \underline{3.88} & 5.18 & {\color{red}0.75} & 5.59$^*$ & {\color{green}4.43} \\
718 \end{tabular}\\[0.1cm]
721 \begin{minipage}{2.8cm}
722 \underline{Hexagonal} \hspace{2pt}
723 \href{../video/c_in_si_int_hexa.avi}{$\rhd$}\\
724 $E_{\text{f}}^*=9.05\text{ eV}$\\
725 \includegraphics[width=2.8cm]{c_pd_albe/hex_bonds.eps}
727 \begin{minipage}{0.4cm}
732 \begin{minipage}{2.8cm}
733 \underline{\hkl<1 0 0>}\\
734 $E_{\text{f}}=3.88\text{ eV}$\\
735 \includegraphics[width=2.8cm]{c_pd_albe/100_bonds.eps}
738 \begin{minipage}{1.4cm}
741 \begin{minipage}{3.0cm}
743 \underline{Tetrahedral}\\
744 \includegraphics[width=3.0cm]{c_pd_albe/tet_bonds.eps}
749 \begin{minipage}{2.8cm}
750 \underline{Bond-centered}\\
751 $E_{\text{f}}^*=5.59\text{ eV}$\\
752 \includegraphics[width=2.8cm]{c_pd_albe/bc_bonds.eps}
754 \begin{minipage}{0.4cm}
759 \begin{minipage}{2.8cm}
760 \underline{\hkl<1 1 0> dumbbell}\\
761 $E_{\text{f}}=5.18\text{ eV}$\\
762 \includegraphics[width=2.8cm]{c_pd_albe/110_bonds.eps}
765 \begin{minipage}{1.4cm}
768 \begin{minipage}{3.0cm}
770 \underline{Substitutional}\\
771 \includegraphics[width=3.0cm]{c_pd_albe/sub_bonds.eps}
781 C interstitial migration
788 \begin{minipage}{6.8cm}
789 {\bf\underline{First-principles}} $\quad$ \hkl[0 0 -1] $\rightarrow$ \hkl[0 -1 0]\\
790 \begin{minipage}{2.0cm}
791 \includegraphics[width=2.0cm]{c_pd_vasp/100_2333.eps}
793 \begin{minipage}{0.2cm}
796 \begin{minipage}{2.0cm}
797 \includegraphics[width=2.0cm]{c_pd_vasp/00-1-0-10_2333.eps}
799 \begin{minipage}{0.2cm}
802 \begin{minipage}{2.0cm}
803 \includegraphics[width=2.0cm]{c_pd_vasp/0-10_2333.eps}
804 \end{minipage}\\[0.1cm]
805 $\Delta E=\unit[0.9]{eV}$ | Experimental values: \unit[0.70--0.87]{eV}\\
806 $\Rightarrow$ {\color{blue}Migration mechanism identified!}\\
807 Note: Change in orientation
809 \begin{minipage}{5.4cm}
810 \includegraphics[width=6.0cm]{00-1_0-10_vasp_s.ps}
811 \end{minipage}\\[0.4cm]
812 \begin{minipage}{6.8cm}
813 {\bf\underline{Empirical potential}} $\quad$
814 \hkl[0 0 -1] $\rightarrow$ \hkl[1 1 0] $\rightarrow$ \hkl[0 -1 0]\\
816 \item Transition involving \hkl[1 1 0] DB\\
817 (instability of BC configuration)
818 \item $\Delta E \approx \unit[2.2]{eV} \text{ \& } \unit[0.9]{eV}$
819 \item 2.4 -- 3.4 times higher than ab initio result
820 \item After all: Change of the DB orientation
824 {\color{red}Drastically overestimated diffusion barrier}
827 \begin{minipage}{5.4cm}
828 \includegraphics[width=6.0cm]{00-1_110_0-10_mig_albe.ps}
837 Defect combinations --- ab inito
844 \begin{minipage}{9cm}
846 Summary of combinations}\\[0.1cm]
848 \begin{tabular}{l c c c c c c}
850 $E_{\text{b}}$ [eV] & 1 & 2 & 3 & 4 & 5 & R\\
852 \hkl[0 0 -1] & {\color{red}-0.08} & -1.15 & {\color{red}-0.08} & 0.04 & -1.66 & -0.19\\
853 \hkl[0 0 1] & 0.34 & 0.004 & -2.05 & 0.26 & -1.53 & -0.19\\
854 \hkl[0 -1 0] & {\color{orange}-2.39} & -0.17 & {\color{green}-0.10} & {\color{blue}-0.27} & {\color{magenta}-1.88} & {\color{gray}-0.05}\\
855 \hkl[0 1 0] & {\color{cyan}-2.25} & -1.90 & {\color{cyan}-2.25} & {\color{purple}-0.12} & {\color{violet}-1.38} & {\color{yellow}-0.06}\\
856 \hkl[-1 0 0] & {\color{orange}-2.39} & -0.36 & {\color{cyan}-2.25} & {\color{purple}-0.12} & {\color{magenta}-1.88} & {\color{gray}-0.05}\\
857 \hkl[1 0 0] & {\color{cyan}-2.25} & -2.16 & {\color{green}-0.10} & {\color{blue}-0.27} & {\color{violet}-1.38} & {\color{yellow}-0.06}\\
859 C$_{\text{sub}}$ & 0.26 & -0.51 & -0.93 & -0.15 & 0.49 & -0.05\\
860 Vacancy & -5.39 ($\rightarrow$ C$_{\text{sub}}$) & -0.59 & -3.14 & -0.54 & -0.50 & -0.31\\
865 \begin{minipage}{3cm}
866 \includegraphics[width=3.5cm]{comb_pos.eps}
871 {\bf\boldmath Combinations of \hkl<1 0 0>-type interstitials}\\[0.2cm]
872 \begin{minipage}{6.1cm}
874 \item \ci{} agglomeration energetically favorable
875 \item Reduction of strain
876 \item Capture radius exceeding \unit[1]{nm}
877 \item Disappearance of attractive forces\\
878 between two lowest separations.
881 {\color{blue}\ci{} agglomeration / no C clustering}
885 \begin{picture}(0,0)(-180,-40)
886 \begin{minipage}{6.0cm}
887 \scriptsize\centering
888 Interaction along \hkl[1 1 0]\\
889 \includegraphics[width=6.2cm]{db_along_110_cc.ps}
899 Defect combinations of C-Si dimers and vacancies
905 \begin{minipage}[b]{2.6cm}
907 \underline{V at 2: $E_{\text{b}}=-0.59\text{ eV}$}\\[0.1cm]
908 \includegraphics[width=2.5cm]{00-1dc/0-59.eps}
911 \begin{minipage}[b]{7cm}
914 \begin{minipage}[b]{2.6cm}
916 \underline{V at 3, $E_{\text{b}}=-3.14\text{ eV}$}\\[0.1cm]
917 \includegraphics[width=2.5cm]{00-1dc/3-14.eps}
919 \end{minipage}\\[0.2cm]
921 \begin{minipage}{6.5cm}
922 \includegraphics[width=6.0cm]{059-539.ps}
924 \begin{minipage}{5.7cm}
925 \includegraphics[width=6.0cm]{314-539.ps}
928 \begin{pspicture}(0,0)(0,0)
929 \psline[linewidth=0.05cm,linecolor=gray](6.3,9.0)(6.3,0)
931 \rput(6.3,7.0){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=white,linewidth=0.05cm,linecolor=gray]{
932 \begin{minipage}{6.5cm}
934 IBS: Impinging C creates V \& far away \si\\[0.3cm]
935 Low migration barrier towards C$_{\text{sub}}$\\
937 High barrier for reverse process\\[0.3cm]
939 High probability of stable C$_{\text{sub}}$ configuration
952 Combinations of substitutional C and Si self-interstitials
959 \begin{minipage}{6.2cm}
961 {\bf\boldmath C$_{\text{sub}}$ - \si{} \hkl<1 1 0> interaction}
963 \item Most favorable: \cs{} along \hkl<1 1 0> chain of \si{}
964 \item Less favorable than ground-state \ci{} \hkl<1 0 0> DB
965 \item Interaction drops quickly to zero\\
966 $\rightarrow$ low capture radius
970 \begin{minipage}{0.2cm}
973 \begin{minipage}{6.0cm}
975 {\bf Transition from the ground state}
977 \item Low transition barrier
978 \item Barrier smaller than \ci{} migration barrier
979 \item Low \si{} migration barrier (\unit[0.67]{eV})\\
980 $\rightarrow$ Separation of \cs{} \& \si{} most probable
983 \end{minipage}\\[0.3cm]
985 \begin{minipage}{6.0cm}
986 \includegraphics[width=6.0cm]{c_sub_si110.ps}
988 \begin{minipage}{0.4cm}
991 \begin{minipage}{6.0cm}
993 \includegraphics[width=6.0cm]{162-097.ps}
997 \begin{pspicture}(0,0)(0,0)
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999 \rput(6.5,-0.7){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=white,linewidth=0.05cm,linecolor=blue]{
1000 \begin{minipage}{8cm}
1004 \cs{} \& \si{} instead of thermodynamic ground state\\[0.1cm]
1005 IBS --- process far from equilibrium\\
1018 Combinations of substitutional C and Si self-interstitials
1025 \begin{minipage}{6.2cm}
1027 {\bf\boldmath C$_{\text{sub}}$ - \si{} \hkl<1 1 0> interaction}
1029 \item Most favorable: \cs{} along \hkl<1 1 0> chain of \si{}
1030 \item Less favorable than ground-state \ci{} \hkl<1 0 0> DB
1031 \item Interaction drops quickly to zero\\
1032 $\rightarrow$ low capture radius
1036 \begin{minipage}{0.2cm}
1039 \begin{minipage}{6.0cm}
1041 {\bf Transition from the ground state}
1043 \item Low transition barrier
1044 \item Barrier smaller than \ci{} migration barrier
1045 \item Low \si{} migration barrier (\unit[0.67]{eV})\\
1046 $\rightarrow$ Separation of \cs{} \& \si{} most probable
1049 \end{minipage}\\[0.3cm]
1051 \begin{minipage}{6.0cm}
1052 \includegraphics[width=6.0cm]{c_sub_si110.ps}
1054 \begin{minipage}{0.4cm}
1057 \begin{minipage}{6.0cm}
1059 \includegraphics[width=6.0cm]{162-097.ps}
1063 \begin{pspicture}(0,0)(0,0)
1064 \psline[linewidth=0.05cm,linecolor=gray](6.5,0)(6.5,7.5)
1065 \rput(6.5,-0.7){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=white,linewidth=0.05cm,linecolor=blue]{
1066 \begin{minipage}{8cm}
1070 \cs{} \& \si{} instead of thermodynamic ground state\\[0.1cm]
1071 IBS --- process far from equilibrium\\
1079 \begin{pspicture}(0,0)(0,0)
1080 \rput(6.5,5.0){\psframebox[fillstyle=solid,opacity=0.5,fillcolor=black]{
1081 \begin{minipage}{14cm}
1086 \rput(6.5,4.3){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=white,linewidth=0.1cm]{
1087 \begin{minipage}{11cm}
1091 Ab initio MD at \degc{900}\\[0.4cm]
1092 \begin{minipage}{5.4cm}
1094 \includegraphics[width=4.3cm]{md01_bonds.eps}\\
1097 \begin{minipage}{5.4cm}
1099 \includegraphics[width=4.3cm]{md02_bonds.eps}\\
1101 \end{minipage}\\[0.5cm]
1103 Contribution of entropy to structural formation\\[0.1cm]
1118 Silicon carbide precipitation simulations
1128 \begin{pspicture}(0,0)(12,6.5)
1130 \rput(3.5,5.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
1133 \item Create c-Si volume
1134 \item Periodc boundary conditions
1135 \item Set requested $T$ and $p=0\text{ bar}$
1136 \item Equilibration of $E_{\text{kin}}$ and $E_{\text{pot}}$
1139 \rput(3.5,2.7){\rnode{insert}{\psframebox[fillstyle=solid,fillcolor=lachs]{
1141 Insertion of C atoms at constant T
1143 \item total simulation volume {\pnode{in1}}
1144 \item volume of minimal SiC precipitate size {\pnode{in2}}
1145 %\item volume consisting of Si atoms to form a minimal {\pnode{in3}}\\
1146 \item volume containing Si atoms to form a minimal {\pnode{in3}}\\
1150 \rput(3.5,1){\rnode{cool}{\psframebox[fillstyle=solid,fillcolor=lbb]{
1152 Run for 100 ps followed by cooling down to $20\, ^{\circ}\textrm{C}$
1154 \ncline[]{->}{init}{insert}
1155 \ncline[]{->}{insert}{cool}
1156 \psframe[fillstyle=solid,fillcolor=white](7.3,0.7)(12.8,6.3)
1157 \rput(7.6,6){\footnotesize $V_1$}
1158 \psframe[fillstyle=solid,fillcolor=lightgray](8.7,2)(11.6,5)
1159 \rput(8.9,4.85){\tiny $V_2$}
1160 \psframe[fillstyle=solid,fillcolor=gray](8.95,2.25)(11.35,4.75)
1161 \rput(9.25,4.45){\footnotesize $V_3$}
1162 \rput(7.9,3.2){\pnode{ins1}}
1163 \rput(8.92,2.8){\pnode{ins2}}
1164 \rput(10.8,2.4){\pnode{ins3}}
1165 \ncline[]{->}{in1}{ins1}
1166 \ncline[]{->}{in2}{ins2}
1167 \ncline[]{->}{in3}{ins3}
1177 \begin{minipage}{5.7cm}
1179 \item Amount of C atoms: 6000\\
1180 ($r_{\text{prec}}\approx 3.1\text{ nm}$, IBS: \unit[2--4]{nm})
1181 \item Simulation volume: $31^3$ Si unit cells\\
1185 \begin{minipage}{0.3cm}
1189 \begin{minipage}{6.0cm}
1190 Restricted to classical potential caclulations\\
1191 $\rightarrow$ Low C diffusion / overestimated barrier\\
1192 $\rightarrow$ Consider $V_2$ and $V_3$
1194 % \item $V_2$ and $V_3$ considered due to expected low C diffusion
1205 Silicon carbide precipitation simulations
1210 \begin{minipage}{6.3cm}
1211 \hspace*{-0.4cm}\includegraphics[width=6.5cm]{sic_prec_450_si-c.ps}\\
1212 \hspace*{-0.4cm}\includegraphics[width=6.5cm]{tot_pc_thesis.ps}
1215 \begin{minipage}{6.1cm}
1217 \underline{Temperature as used in IBS (\degc{450})}\\[0.2cm]
1218 \ci{} \hkl<1 0 0> dumbbell dominated structure\\
1219 \begin{pspicture}(0,0)(6.0,1.0)
1220 \rput(2.75,0.4){\psframebox[linewidth=0.05cm,linecolor=black]{
1221 \begin{minipage}{5cm}
1224 {\color{blue}Formation of \ci{} DBs}\\
1225 {\color{red}No agllomeration / precipitation}
1228 \end{pspicture}\\[0.1cm]
1231 \item Time scale problem of MD\\
1232 $\Rightarrow$ slow phase space propagation
1233 \item Short range potential\\
1234 $\Rightarrow$ overestimated diffusion barrier
1237 \underline{Increased temperatures}\\[0.2cm]
1238 \cs{} dominated structure\\
1239 \begin{pspicture}(0,0)(6.0,1.0)
1240 \rput(2.75,0.4){\psframebox[linewidth=0.05cm,linecolor=black]{
1241 \begin{minipage}{5cm}
1244 Si-{\color{blue}C$_{\text{sub}}$}-Si along \hkl<1 1 0>\\
1245 {\color{blue}\cs}-Si-{\color{blue}\cs} \& nearby \si
1248 \end{pspicture}\\[0.1cm]
1251 \item Stretched coherent SiC structures\\
1252 $\Rightarrow$ \cs{} involved in precipitation mechanism
1253 \item High T $\leftrightarrow$ non-equilibrium IBS conditions
1265 Summary and Conclusions
1271 \item First-principles investigation of defect combinations
1272 and mobilities in Si
1273 \item Empirical potential MD simulations on SiC prcipitation in Si
1278 \rput(6.5,-4.0){\psframebox[fillstyle=solid,fillcolor=white,linewidth=0.1cm]{
1279 \begin{minipage}{9cm}
1283 {\color{gray}\bf Conclusions on SiC precipitation}\\[0.1cm]
1284 {\Huge$\lightning$} {\color{red}\ci{}} --- vs --- {\color{blue}\cs{}} {\Huge$\lightning$}\\
1287 \item Stretched coherent SiC structures directly observed\\
1288 \psframebox[linecolor=blue,linewidth=0.05cm]{
1289 \begin{minipage}{7cm}
1291 \cs{} involved in the precipitation mechanism\\
1294 \item Emission of \si{} serves several needs:
1296 \item Vehicle to rearrange \cs --- [\cs{} \& \si{} $\leftrightarrow$ \ci]
1297 \item Building block for surrounding Si host \& further SiC
1298 \item Strain compensation \ldots\\
1299 \ldots Si/SiC interface\\
1300 \ldots within stretched coherent SiC structure
1302 \item Explains annealing behavior of high/low T C implantations
1304 \item Low T: highly mobile {\color{red}\ci}
1305 \item High T: stable configurations of {\color{blue}\cs}
1307 \psframebox[linecolor=blue,linewidth=0.05cm]{
1308 \begin{minipage}{7cm}
1310 High T $\leftrightarrow$ IBS conditions far from equilibrium\\
1333 \begin{minipage}[t]{6cm}
1334 \underline{Augsburg}
1336 \item Prof. B. Stritzker
1339 \underline{Helsinki}
1341 \item Prof. K. Nordlund
1346 \item Bayerische Forschungsstiftung
1349 \begin{minipage}[t]{6cm}
1350 \underline{Paderborn}
1352 \item Prof. J. Lindner
1353 \item Prof. G. Schmidt
1362 \LARGE\bf Thank you for your attention!
1380 Polytypes of SiC\\[0.6cm]
1385 \includegraphics[width=3.8cm]{cubic_hex.eps}\\
1386 \begin{minipage}{1.9cm}
1387 {\tiny cubic (twist)}
1389 \begin{minipage}{2.9cm}
1390 {\tiny hexagonal (no twist)}
1393 \begin{picture}(0,0)(-150,0)
1394 \includegraphics[width=7cm]{polytypes.eps}
1401 \begin{tabular}{l c c c c c c}
1403 & 3C-SiC & 4H-SiC & 6H-SiC & Si & GaN & Diamond\\
1405 Hardness [Mohs] & \multicolumn{3}{c}{------ 9.6 ------}& 6.5 & - & 10 \\
1406 Band gap [eV] & 2.36 & 3.23 & 3.03 & 1.12 & 3.39 & 5.5 \\
1407 Break down field [$10^6$ V/cm] & 4 & 3 & 3.2 & 0.6 & 5 & 10 \\
1408 Saturation drift velocity [$10^7$ cm/s] & 2.5 & 2.0 & 2.0 & 1 & 2.7 & 2.7 \\
1409 Electron mobility [cm$^2$/Vs] & 800 & 900 & 400 & 1100 & 900 & 2200 \\
1410 Hole mobility [cm$^2$/Vs] & 320 & 120 & 90 & 420 & 150 & 1600 \\
1411 Thermal conductivity [W/cmK] & 5.0 & 4.9 & 4.9 & 1.5 & 1.3 & 22 \\
1415 \begin{pspicture}(0,0)(0,0)
1416 \psellipse[linecolor=green](5.7,2.05)(0.4,0.50)
1418 \begin{pspicture}(0,0)(0,0)
1419 \psellipse[linecolor=green](5.6,0.89)(0.4,0.20)
1421 \begin{pspicture}(0,0)(0,0)
1422 \psellipse[linecolor=red](10.45,0.42)(0.4,0.20)
1433 Si self-interstitial point defects in silicon\\[0.1cm]
1437 \begin{tabular}{l c c c c c}
1439 $E_{\text{f}}$ [eV] & \hkl<1 1 0> DB & H & T & \hkl<1 0 0> DB & V \\
1441 \textsc{vasp} & \underline{3.39} & 3.42 & 3.77 & 4.41 & 3.63 \\
1442 Erhart/Albe & 4.39 & 4.48$^*$ & \underline{3.40} & 5.42 & 3.13 \\
1444 \end{tabular}\\[0.4cm]
1447 \begin{minipage}{3cm}
1449 \underline{Vacancy}\\
1450 \includegraphics[width=2.8cm]{si_pd_albe/vac.eps}
1453 \begin{minipage}{3cm}
1455 \underline{\hkl<1 1 0> DB}\\
1456 \includegraphics[width=2.8cm]{si_pd_albe/110_bonds.eps}
1459 \begin{minipage}{3cm}
1461 \underline{\hkl<1 0 0> DB}\\
1462 \includegraphics[width=2.8cm]{si_pd_albe/100_bonds.eps}
1465 \begin{minipage}{3cm}
1467 \underline{Tetrahedral}\\
1468 \includegraphics[width=2.8cm]{si_pd_albe/tet_bonds.eps}
1472 \underline{Hexagonal} \hspace{2pt}
1473 \href{../video/si_self_int_hexa.avi}{$\rhd$}\\[0.1cm]
1475 \begin{minipage}{2.7cm}
1476 $E_{\text{f}}^*=4.48\text{ eV}$\\
1477 \includegraphics[width=2.7cm]{si_pd_albe/hex_a_bonds.eps}
1479 \begin{minipage}{0.4cm}
1484 \begin{minipage}{2.7cm}
1485 $E_{\text{f}}=3.96\text{ eV}$\\
1486 \includegraphics[width=2.8cm]{si_pd_albe/hex_bonds.eps}
1489 \begin{minipage}{5.5cm}
1491 {\tiny nearly T $\rightarrow$ T}\\
1493 \includegraphics[width=6.0cm]{nhex_tet.ps}
1502 C-Si dimer \& bond-centered interstitial configuration
1509 \begin{minipage}[t]{4.1cm}
1510 {\bf\boldmath C \hkl<1 0 0> DB interstitial}\\[0.1cm]
1511 \begin{minipage}{2.0cm}
1513 \underline{Erhart/Albe}
1514 \includegraphics[width=2.0cm]{c_pd_albe/100_cmp.eps}
1517 \begin{minipage}{2.0cm}
1519 \underline{\textsc{vasp}}
1520 \includegraphics[width=2.0cm]{c_pd_vasp/100_cmp.eps}
1522 \end{minipage}\\[0.2cm]
1523 Si-C-Si bond angle $\rightarrow$ \unit[180]{$^{\circ}$}\\
1524 $\Rightarrow$ $sp$ hybridization\\[0.1cm]
1525 Si-Si-Si bond angle $\rightarrow$ \unit[120]{$^{\circ}$}\\
1526 $\Rightarrow$ $sp^2$ hybridization
1528 \includegraphics[width=3.4cm]{c_pd_vasp/eden.eps}\\[-0.1cm]
1529 {\tiny Charge density isosurface}
1532 \begin{minipage}{0.2cm}
1535 \begin{minipage}[t]{8.1cm}
1537 {\bf Bond-centered interstitial}\\[0.1cm]
1538 \begin{minipage}{4.4cm}
1541 \item Linear Si-C-Si bond
1542 \item Si: one C \& 3 Si neighbours
1543 \item Spin polarized calculations
1544 \item No saddle point!\\
1548 \begin{minipage}{2.7cm}
1549 %\includegraphics[width=2.8cm]{c_pd_vasp/bc_2333.eps}\\
1551 \includegraphics[width=2.8cm]{c_pd_albe/bc_bonds.eps}\\
1556 \begin{minipage}[t]{6.5cm}
1557 \begin{minipage}[t]{1.2cm}
1559 {\tiny sp$^3$}\\[0.8cm]
1560 \underline{${\color{black}\uparrow}$}
1561 \underline{${\color{black}\uparrow}$}
1562 \underline{${\color{black}\uparrow}$}
1563 \underline{${\color{red}\uparrow}$}\\
1566 \begin{minipage}[t]{1.4cm}
1568 {\color{red}M}{\color{blue}O}\\[0.8cm]
1569 \underline{${\color{blue}\uparrow}{\color{white}\downarrow}$}\\
1570 $\sigma_{\text{ab}}$\\[0.5cm]
1571 \underline{${\color{red}\uparrow}{\color{blue}\downarrow}$}\\
1575 \begin{minipage}[t]{1.0cm}
1579 \underline{${\color{white}\uparrow\uparrow}$}
1580 \underline{${\color{white}\uparrow\uparrow}$}\\
1582 \underline{${\color{blue}\uparrow}{\color{blue}\downarrow}$}
1583 \underline{${\color{blue}\uparrow}{\color{blue}\downarrow}$}\\
1587 \begin{minipage}[t]{1.4cm}
1589 {\color{blue}M}{\color{green}O}\\[0.8cm]
1590 \underline{${\color{blue}\uparrow}{\color{white}\downarrow}$}\\
1591 $\sigma_{\text{ab}}$\\[0.5cm]
1592 \underline{${\color{green}\uparrow}{\color{blue}\downarrow}$}\\
1596 \begin{minipage}[t]{1.2cm}
1599 {\tiny sp$^3$}\\[0.8cm]
1600 \underline{${\color{green}\uparrow}$}
1601 \underline{${\color{black}\uparrow}$}
1602 \underline{${\color{black}\uparrow}$}
1603 \underline{${\color{black}\uparrow}$}\\
1611 \begin{minipage}{3.0cm}
1613 \underline{Charge density}\\
1614 {\color{gray}$\bullet$} Spin up\\
1615 {\color{green}$\bullet$} Spin down\\
1616 {\color{blue}$\bullet$} Resulting spin up\\
1617 {\color{yellow}$\bullet$} Si atoms\\
1618 {\color{red}$\bullet$} C atom
1620 \begin{minipage}{3.6cm}
1621 \includegraphics[width=3.8cm]{c_100_mig_vasp/im_spin_diff.eps}
1628 \begin{pspicture}(0,0)(0,0)
1629 \psline[linecolor=gray,linewidth=0.05cm](-7.8,-8.7)(-7.8,0)
1637 Increased temperature simulations at high C concentration
1642 \begin{minipage}{6.0cm}
1643 \includegraphics[width=6.4cm]{12_pc_thesis.ps}
1645 \begin{minipage}{6.0cm}
1646 \includegraphics[width=6.4cm]{12_pc_c_thesis.ps}
1654 \begin{minipage}[t]{5.5cm}
1655 0.186 nm: Si-C pairs $\uparrow$\\
1656 (as expected in 3C-SiC)\\[0.2cm]
1657 0.282 nm: Si-C-C\\[0.2cm]
1658 $\approx$0.35 nm: C-Si-Si
1661 \begin{minipage}{0.1cm}
1665 \begin{minipage}[t]{5.9cm}
1666 0.15 nm: C-C pairs $\uparrow$\\
1667 (as expected in graphite/diamond)\\[0.2cm]
1668 0.252 nm: C-C-C (2$^{\text{nd}}$ NN for diamond)\\[0.2cm]
1669 0.31 nm: shifted towards 0.317 nm $\rightarrow$ C-Si-C
1674 \item Decreasing cut-off artifact
1675 \item {\color{red}Amorphous} SiC-like phase remains
1676 \item High amount of {\color{red}damage} \& alignement to c-Si host matrix lost
1677 \item Slightly sharper peaks $\Rightarrow$ indicate slight {\color{blue}acceleration of dynamics} due to temperature
1684 High C \& small $V$ \& short $t$
1687 \begin{minipage}{4cm}
1689 Slow structural evolution due to strong C-C bonds
1694 High C \& low T implants
1707 Valuation of a practicable temperature limit
1717 Recrystallization is a hard task!
1718 $\Rightarrow$ Avoid melting!
1727 \begin{minipage}{6.4cm}
1728 \includegraphics[width=6.4cm]{fe_and_t.ps}
1730 \begin{minipage}{5.7cm}
1731 \underline{Melting does not occur instantly after}\\
1732 \underline{exceeding the melting point $T_{\text{m}}=2450\text{ K}$}
1734 \item required transition enthalpy
1735 \item hysterisis behaviour
1737 \underline{Heating up c-Si by 1 K/ps}
1739 \item transition occurs at $\approx$ 3125 K
1740 \item $\Delta E=0.58\text{ eV/atom}=55.7\text{ kJ/mole}$\\
1741 (literature: 50.2 kJ/mole)
1748 \begin{minipage}{4cm}
1749 Initially chosen temperatures:\\
1750 $1.0 - 1.2 \cdot T_{\text{m}}$
1753 \begin{minipage}{2cm}
1759 \begin{minipage}{5cm}
1760 Introduced C (defects)\\
1761 $\rightarrow$ reduction of transition point\\
1762 $\rightarrow$ melting already at $T_{\text{m}}$
1771 Maximum temperature used: $0.95\cdot T_{\text{m}}$
1781 Long time scale simulations at maximum temperature
1788 \underline{Differences}
1790 \item Temperature set to $0.95 \cdot T_{\text{m}}$
1791 \item Cubic insertion volume $\Rightarrow$ spherical insertion volume
1792 \item Amount of C atoms: 6000 $\rightarrow$ 5500
1793 $\Leftrightarrow r_{\text{prec}}=0.3\text{ nm}$
1794 \item Simulation volume: 21 unit cells of c-Si in each direction
1801 \begin{minipage}[t]{4.3cm}
1803 \underline{Low C concentration, Si-C}
1804 \includegraphics[width=4.3cm]{c_in_si_95_v1_si-c.ps}\\
1808 \begin{minipage}[t]{4.3cm}
1810 \underline{Low C concentration, C-C}
1811 \includegraphics[width=4.3cm]{c_in_si_95_v1_c-c.ps}\\
1816 \begin{minipage}[t]{3.4cm}
1818 \underline{High C concentration}
1819 \includegraphics[width=4.3cm]{c_in_si_95_v2.ps}\\
1820 No significant changes\\
1821 iC-Si-Si $\uparrow$\\
1828 Long time scales and high temperatures most probably not sufficient enough!
1837 Investigation of a silicon carbide precipitate in silicon
1846 \begin{minipage}{5.3cm}
1848 \frac{8}{a_{\text{Si}}^3}(
1849 \underbrace{21^3 a_{\text{Si}}^3}_{=V}
1850 -\frac{4}{3}\pi x^3)+
1851 \underbrace{\frac{4}{y^3}\frac{4}{3}\pi x^3}_{\stackrel{!}{=}5500}
1858 \frac{8}{a_{\text{Si}}^3}\frac{4}{3}\pi x^3=5500
1859 \Rightarrow x = \left(\frac{5500 \cdot 3}{32 \pi} \right)^{1/3}a_{\text{Si}}
1862 y=\left(\frac{1}{2} \right)^{1/3}a_{\text{Si}}
1866 \begin{minipage}{0.1cm}
1869 \begin{minipage}{6.3cm}
1870 \underline{Construction}
1872 \item Simulation volume: 21$^3$ unit cells of c-Si
1873 \item Spherical topotactically aligned precipitate\\
1874 $r=3.0\text{ nm}$ $\Leftrightarrow$ $\approx$ 5500 C atoms
1875 \item Create c-Si but skipped inside sphere\\
1877 \item Create 3C-SiC inside sphere of radius $x$\\
1878 and lattice constant $y$
1879 \item Strong coupling to heat bath ($T=20\,^{\circ}\mathrm{C}$)
1885 \begin{minipage}{6.0cm}
1886 \includegraphics[width=6cm]{pc_0.ps}
1888 \begin{minipage}{6.1cm}
1891 \item Slight increase of c-Si lattice constant!
1893 (imply same distanced Si-Si peaks)
1895 \item New peak at 0.307 nm: 2$^{\text{nd}}$ NN in 3C-SiC
1896 \item Bumps ({\color{green}$\downarrow$}):
1897 4$^{\text{th}}$ and 6$^{\text{th}}$ NN
1899 \item 3C-SiC lattice constant: 4.34 \AA (bulk: 4.36 \AA)\\
1900 $\rightarrow$ compressed precipitate
1901 \item Interface tension:\\
1902 20.15 eV/nm$^2$ or $3.23 \times 10^{-4}$ J/cm$^2$\\
1903 (literature: $2 - 8 \times 10^{-4}$ J/cm$^2$)
1912 Investigation of a silicon carbide precipitate in silicon
1917 \begin{minipage}{7cm}
1918 \underline{Appended annealing steps}
1920 \item artificially constructed interface\\
1921 $\rightarrow$ allow for rearrangement of interface atoms
1922 \item check SiC stability
1924 \underline{Temperature schedule}
1926 \item rapidly heat up structure up to $2050\,^{\circ}\mathrm{C}$\\
1928 \item slow heating up to $1.2\cdot T_{\text{m}}=2940\text{ K}$
1930 $\rightarrow$ melting at around 2840 K
1931 (\href{../video/sic_prec_120.avi}{$\rhd$})
1932 \item cooling down structure at 100 \% $T_{\text{m}}$ (1 K/ps)\\
1933 $\rightarrow$ no energetically more favorable struture
1936 \begin{minipage}{5cm}
1937 \includegraphics[width=5.5cm]{fe_and_t_sic.ps}
1940 \begin{minipage}{4cm}
1941 \includegraphics[width=4cm]{sic_prec/melt_01.eps}
1943 \begin{minipage}{0.2cm}
1946 \begin{minipage}{4cm}
1947 \includegraphics[width=4cm]{sic_prec/melt_02.eps}
1949 \begin{minipage}{0.2cm}
1952 \begin{minipage}{3.7cm}
1953 \includegraphics[width=4cm]{sic_prec/melt_03.eps}
1968 Equilibrium lattice constants and cohesive energies
1970 \begin{tabular}{l r c c c c c}
1973 & & USPP, LDA & USPP, GGA & PAW, LDA & PAW, GGA & Exp. \\
1975 Si (dia) & $a$ [\AA] & 5.389 & 5.455 & - & - & 5.429 \\
1976 & $\Delta_a$ [\%] & \unit[{\color{green}0.7}]{\%} & \unit[{\color{green}0.5}]{\%} & - & - & - \\
1977 & $E_{\text{coh}}$ [eV] & -5.277 & -4.591 & - & - & -4.63 \\
1978 & $\Delta_E$ [\%] & \unit[{\color{red}14.0}]{\%} & \unit[{\color{green}0.8}]{\%} & - & - & - \\
1980 C (dia) & $a$ [\AA] & 3.527 & 3.567 & - & - & 3.567 \\
1981 & $\Delta_a$ [\%] & \unit[{\color{green}1.1}]{\%} & \unit[{\color{green}0.01}]{\%} & - & - & - \\
1982 & $E_{\text{coh}}$ [eV] & -8.812 & -7.703 & - & - & -7.374 \\
1983 & $\Delta_E$ [\%] & \unit[{\color{red}19.5}]{\%} & \unit[{\color{orange}4.5}]{\%} & - & - & - \\
1985 3C-SiC & $a$ [\AA] & 4.319 & 4.370 & 4.330 & 4.379 & 4.359 \\
1986 & $\Delta_a$ [\%] & \unit[{\color{green}0.9}]{\%} & \unit[{\color{green}0.3}]{\%} & \unit[{\color{green}0.7}]{\%} & \unit[{\color{green}0.5}]{\%} & - \\
1987 & $E_{\text{coh}}$ [eV] & -7.318 & -6.426 & -7.371 & -6.491 & -6.340 \\
1988 & $\Delta_E$ [\%] & \unit[{\color{red}15.4}]{\%} & \unit[{\color{green}1.4}]{\%} & \unit[{\color{red}16.3}]{\%} & \unit[{\color{orange}2.4}]{\%} & - \\
1995 \begin{minipage}{7cm}
1997 \begin{tabular}{l c c c}
2000 & Si (dia) & C (dia) & 3C-SiC \\
2002 $a$ [\AA] & 5.458 & 3.562 & 4.365 \\
2003 $\Delta_a$ [\%] & 0.5 & 0.1 & 0.1 \\
2005 $E_{\text{coh}}$ [eV] & -4.577 & -7.695 & -6.419 \\
2006 $\Delta_E$ [\%] & 1.1 & 4.4 & 1.2 \\
2012 \begin{minipage}{5cm}
2013 $\leftarrow$ entire parameter set
2026 \begin{minipage}{6cm}
2028 \includegraphics[width=6cm]{sic_32pc_gamma_cutoff_lc.ps}
2031 \begin{minipage}{6cm}
2033 Lattice constants with respect to the PW cut-off energy
2037 \begin{minipage}{6cm}
2039 \includegraphics[width=6cm]{si_self_int_thesis.ps}
2042 \begin{minipage}{6cm}
2044 Defect formation energy with respect to the size of the supercell\\[0.1cm]