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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
223 \psframebox[linecolor=blue,linewidth=0.05cm]{
224 \begin{minipage}{4.5cm}
227 3C-SiC precipitation\\
228 not yet fully understood
233 \begin{minipage}{5.0cm}
234 \includegraphics[width=5.5cm]{ibs_3c-sic.eps}\\[-0.4cm]
237 XTEM: single crystalline 3C-SiC in Si\hkl(1 0 0)
240 \end{minipage}\\[0.3cm]
242 \psframebox[fillstyle=solid,fillcolor=hb]{
243 \begin{minipage}{12.1cm}
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
263 Supposed precipitation mechanism of SiC in Si
271 \begin{minipage}{3.6cm}
273 Si \& SiC lattice structure\\[0.1cm]
274 \includegraphics[width=2.3cm]{sic_unit_cell.eps}
277 \begin{minipage}{1.7cm}
278 \underline{Silicon}\\
279 {\color{yellow}$\bullet$} Si | {\color{gray}$\bullet$} Si\\
280 $a=\unit[5.429]{\\A}$\\
281 $\rho^*_{\text{Si}}=\unit[100]{\%}$
283 \begin{minipage}{1.7cm}
284 \underline{Silicon carbide}\\
285 {\color{yellow}$\bullet$} Si | {\color{gray}$\bullet$} C\\
286 $a=\unit[4.359]{\\A}$\\
287 $\rho^*_{\text{Si}}=\unit[97]{\%}$
293 \begin{minipage}{4.1cm}
295 \includegraphics[width=3.3cm]{tem_c-si-db.eps}
299 \begin{minipage}{4.0cm}
301 \includegraphics[width=3.3cm]{tem_3c-sic.eps}
307 \begin{minipage}{4.0cm}
309 C-Si dimers (dumbbells)\\[-0.1cm]
314 \begin{minipage}{4.1cm}
316 Agglomeration of C-Si dumbbells\\[-0.1cm]
317 $\Rightarrow$ dark contrasts
321 \begin{minipage}{4.0cm}
323 Precipitation of 3C-SiC in Si\\[-0.1cm]
324 $\Rightarrow$ Moir\'e fringes\\[-0.1cm]
325 \& release of Si self-interstitials
331 \begin{minipage}{4.0cm}
333 \includegraphics[width=3.3cm]{sic_prec_seq_01.eps}
337 \begin{minipage}{4.1cm}
339 \includegraphics[width=3.3cm]{sic_prec_seq_02.eps}
343 \begin{minipage}{4.0cm}
345 \includegraphics[width=3.3cm]{sic_prec_seq_03.eps}
349 \begin{pspicture}(0,0)(0,0)
350 \psline[linewidth=2pt]{->}(8.3,2)(8.8,2)
351 \psellipse[linecolor=blue](11.1,6.0)(0.3,0.5)
352 \rput{-20}{\psellipse[linecolor=blue](3.1,8.2)(0.3,0.5)}
353 \psline[linewidth=2pt]{->}(3.9,2)(4.4,2)
354 \rput(11.8,0.3){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
355 $4a_{\text{Si}}=5a_{\text{SiC}}$
357 \rput(11.5,8){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
358 \hkl(h k l) planes match
360 \rput(8.5,6.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
371 Supposed precipitation mechanism of SiC in Si
379 \begin{minipage}{3.6cm}
381 Si \& SiC lattice structure\\[0.1cm]
382 \includegraphics[width=2.3cm]{sic_unit_cell.eps}
385 \begin{minipage}{1.7cm}
386 \underline{Silicon}\\
387 {\color{yellow}$\bullet$} Si | {\color{gray}$\bullet$} Si\\
388 $a=\unit[5.429]{\\A}$\\
389 $\rho^*_{\text{Si}}=\unit[100]{\%}$
391 \begin{minipage}{1.7cm}
392 \underline{Silicon carbide}\\
393 {\color{yellow}$\bullet$} Si | {\color{gray}$\bullet$} C\\
394 $a=\unit[4.359]{\\A}$\\
395 $\rho^*_{\text{Si}}=\unit[97]{\%}$
401 \begin{minipage}{4.1cm}
403 \includegraphics[width=3.3cm]{tem_c-si-db.eps}
407 \begin{minipage}{4.0cm}
409 \includegraphics[width=3.3cm]{tem_3c-sic.eps}
415 \begin{minipage}{4.0cm}
417 C-Si dimers (dumbbells)\\[-0.1cm]
418 on Si interstitial sites
422 \begin{minipage}{4.1cm}
424 Agglomeration of C-Si dumbbells\\[-0.1cm]
425 $\Rightarrow$ dark contrasts
429 \begin{minipage}{4.0cm}
431 Precipitation of 3C-SiC in Si\\[-0.1cm]
432 $\Rightarrow$ Moir\'e fringes\\[-0.1cm]
433 \& release of Si self-interstitials
439 \begin{minipage}{4.0cm}
441 \includegraphics[width=3.3cm]{sic_prec_seq_01.eps}
445 \begin{minipage}{4.1cm}
447 \includegraphics[width=3.3cm]{sic_prec_seq_02.eps}
451 \begin{minipage}{4.0cm}
453 \includegraphics[width=3.3cm]{sic_prec_seq_03.eps}
457 \begin{pspicture}(0,0)(0,0)
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461 \psline[linewidth=2pt]{->}(3.9,2)(4.4,2)
462 \rput(11.8,0.3){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
463 $4a_{\text{Si}}=5a_{\text{SiC}}$
465 \rput(11.5,8){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
466 \hkl(h k l) planes match
468 \rput(8.5,6.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
471 % controversial view!
472 \rput(6.5,5.0){\psframebox[fillstyle=solid,opacity=0.5,fillcolor=black]{
473 \begin{minipage}{14cm}
478 \rput(6.5,5.3){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=white,linewidth=0.1cm]{
479 \begin{minipage}{10cm}
483 {\color{gray}\bf Controversial findings}
486 \item High-temperature implantation {\tiny\color{gray}/Nejim~et~al./}
488 \item {\color{blue}Substitutionally} incorporated C on regular Si lattice sites
489 \item \si{} reacting with further C in cleared volume
491 \item Annealing behavior {\tiny\color{gray}/Serre~et~al./}
493 \item Room temperature implantation $\rightarrow$ high C diffusion
494 \item Elevated temperature implantation $\rightarrow$ no C redistribution
496 $\Rightarrow$ mobile {\color{red}\ci} opposed to
497 stable {\color{blue}\cs{}} configurations
498 \item Strained Si$_{1-y}$C$_y$/Si heterostructures
499 {\tiny\color{gray}/Strane~et~al./Guedj~et~al./}
501 \item Initial {\color{blue}coherent} SiC structures (tensile strain)
502 \item Incoherent SiC nanocrystals (strain relaxation)
507 {\Huge${\lightning}$} \hspace{0.3cm}
508 {\color{blue}\cs{}} --- vs --- {\color{red}\ci} \hspace{0.3cm}
509 {\Huge${\lightning}$}
522 Utilized computational methods
529 {\bf Molecular dynamics (MD)}\\[0.1cm]
531 \begin{tabular}{| p{4.5cm} | p{7.5cm} |}
533 System of $N$ particles &
534 $N=5832\pm 1$ (Defects), $N=238328+6000$ (Precipitation)\\
535 Phase space propagation &
536 Velocity Verlet | timestep: \unit[1]{fs} \\
537 Analytical interaction potential &
538 Tersoff-like {\color{red}short-range}, {\color{blue}bond order} potential
541 E = \frac{1}{2} \sum_{i \neq j} \pot_{ij}, \quad
542 \pot_{ij} = {\color{red}f_C(r_{ij})}
543 \left[ f_R(r_{ij}) + {\color{blue}b_{ij}} f_A(r_{ij}) \right]
545 Observables: time/ensemble averages &
546 NpT (isothermal-isobaric) | Berendsen thermostat/barostat\\
554 {\bf Density functional theory (DFT)}
558 \begin{minipage}[t]{6cm}
560 \item Hohenberg-Kohn theorem:\\
561 $\Psi_0(r_1,r_2,\ldots,r_N)=\Psi[n_0(r)]$, $E_0=E[n_0]$
562 \item Kohn-Sham approach:\\
563 Single-particle effective theory
567 \item Code: \textsc{vasp}
568 \item Plane wave basis set | $E_{\text{cut}}=\unit[300]{eV}$
570 %\Phi_i=\sum_{|G+k|<G_{\text{cut}}} c_{i,k+G} \exp{\left(i(k+G)r\right)}
573 %E_{\text{cut}}=\frac{\hbar^2}{2m}G^2_{\text{cut}}=\unit[300]{eV}
575 \item Ultrasoft pseudopotential
576 \item Exchange \& correlation: GGA
577 \item Brillouin zone sampling: $\Gamma$-point
578 \item Supercell: $N=216\pm2$
581 \begin{minipage}{6cm}
582 \begin{pspicture}(0,0)(0,0)
583 \pscircle[fillcolor=yellow,fillstyle=solid,linestyle=none](3.5,-2.0){2.5}
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586 \left[ -\frac{\hbar^2}{2m}\nabla^2 + V_{\text{eff}}(r) - \epsilon_i \right] \Phi_i(r) = 0
589 \rput(5.2,-2.0){\psframebox[fillstyle=solid,opacity=0.8,fillcolor=white]{
591 n(r)=\sum_i^N|\Phi_i(r)|^2
594 \rput(3.0,-4.5){\psframebox[fillstyle=solid,opacity=0.8,fillcolor=white]{
596 V_{\text{eff}}(r)=V_{\text{ext}}(r)+\int\frac{e^2 n(r')}{|r-r'|}d^3r'
600 \psarcn[linewidth=0.07cm,linestyle=dashed]{->}(3.5,-2.0){2.5}{130}{15}
601 \psarcn[linewidth=0.07cm,linestyle=dashed]{->}(3.5,-2.0){2.5}{230}{165}
602 \psarcn[linewidth=0.07cm,linestyle=dashed]{->}(3.5,-2.0){2.5}{345}{310}
613 Point defects \& defect migration
620 \begin{minipage}[b]{7.5cm}
621 {\bf Defect structure}\\
622 \begin{pspicture}(0,0)(7,4.4)
623 \rput(3.5,3.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
626 \item Creation of c-Si simulation volume
627 \item Periodic boundary conditions
628 \item $T=0\text{ K}$, $p=0\text{ bar}$
631 \rput(3.5,1.3){\rnode{insert}{\psframebox{
634 Insertion of interstitial C/Si atoms
637 \rput(3.5,0.2){\rnode{cool}{\psframebox[fillstyle=solid,fillcolor=lbb]{
640 Relaxation / structural energy minimization
643 \ncline[]{->}{init}{insert}
644 \ncline[]{->}{insert}{cool}
647 \begin{minipage}[b]{4.5cm}
649 \includegraphics[width=3.8cm]{unit_cell_e.eps}\\
651 \begin{minipage}{2.21cm}
653 {\color{red}$\bullet$} Tetrahedral\\[-0.1cm]
654 {\color{green}$\bullet$} Hexagonal\\[-0.1cm]
655 {\color{yellow}$\bullet$} \hkl<1 0 0> DB
658 \begin{minipage}{2.21cm}
660 {\color{magenta}$\bullet$} \hkl<1 1 0> DB\\[-0.1cm]
661 {\color{cyan}$\bullet$} Bond-centered\\[-0.1cm]
662 {\color{black}$\bullet$} Vac. / Sub.
669 \begin{minipage}[t]{6cm}
670 {\bf Defect formation energy}\\
672 $E_{\text{f}}=E-\sum_i N_i\mu_i$}\\[0.5cm]
673 %Particle reservoir: Si \& SiC\\[0.2cm]
674 {\bf Binding energy}\\
678 E_{\text{f}}^{\text{comb}}-
679 E_{\text{f}}^{1^{\text{st}}}-
680 E_{\text{f}}^{2^{\text{nd}}}
684 $E_{\text{b}}<0$: energetically favorable configuration\\
685 $E_{\text{b}}\rightarrow 0$: non-interacting, isolated defects\\
687 \begin{minipage}[t]{6cm}
689 {\bf Migration barrier}
692 \item Displace diffusing atom
693 \item Constrain relaxation of (diffusing) atoms
694 \item Record configurational energy
696 \begin{picture}(0,0)(-60,-33)
697 \includegraphics[width=4.5cm]{crt_mod.eps}
709 C interstitial point defects in silicon\\
712 \begin{tabular}{l c c c c c c r}
714 $E_{\text{f}}$ [eV] & T & H & \hkl<1 0 0> DB & \hkl<1 1 0> DB & S & B &
715 {\color{black} \cs{} \& \si}\\
717 \textsc{vasp} & unstable & unstable & \underline{3.72} & 4.16 & 1.95 & 4.66 & {\color{green}4.17}\\
718 Erhart/Albe & 6.09 & 9.05$^*$ & \underline{3.88} & 5.18 & {\color{red}0.75} & 5.59$^*$ & {\color{green}4.43} \\
720 \end{tabular}\\[0.1cm]
723 \begin{minipage}{2.8cm}
724 \underline{Hexagonal} \hspace{2pt}
725 \href{../video/c_in_si_int_hexa.avi}{$\rhd$}\\
726 $E_{\text{f}}^*=9.05\text{ eV}$\\
727 \includegraphics[width=2.8cm]{c_pd_albe/hex_bonds.eps}
729 \begin{minipage}{0.4cm}
734 \begin{minipage}{2.8cm}
735 \underline{\hkl<1 0 0>}\\
736 $E_{\text{f}}=3.88\text{ eV}$\\
737 \includegraphics[width=2.8cm]{c_pd_albe/100_bonds.eps}
740 \begin{minipage}{1.4cm}
743 \begin{minipage}{3.0cm}
745 \underline{Tetrahedral}\\
746 \includegraphics[width=3.0cm]{c_pd_albe/tet_bonds.eps}
751 \begin{minipage}{2.8cm}
752 \underline{Bond-centered}\\
753 $E_{\text{f}}^*=5.59\text{ eV}$\\
754 \includegraphics[width=2.8cm]{c_pd_albe/bc_bonds.eps}
756 \begin{minipage}{0.4cm}
761 \begin{minipage}{2.8cm}
762 \underline{\hkl<1 1 0> dumbbell}\\
763 $E_{\text{f}}=5.18\text{ eV}$\\
764 \includegraphics[width=2.8cm]{c_pd_albe/110_bonds.eps}
767 \begin{minipage}{1.4cm}
770 \begin{minipage}{3.0cm}
772 \underline{Substitutional}\\
773 \includegraphics[width=3.0cm]{c_pd_albe/sub_bonds.eps}
783 C interstitial migration
790 \begin{minipage}{6.8cm}
791 {\bf\underline{First-principles}} $\quad$ \hkl[0 0 -1] $\rightarrow$ \hkl[0 -1 0]\\
792 \begin{minipage}{2.0cm}
793 \includegraphics[width=2.0cm]{c_pd_vasp/100_2333.eps}
795 \begin{minipage}{0.2cm}
798 \begin{minipage}{2.0cm}
799 \includegraphics[width=2.0cm]{c_pd_vasp/00-1-0-10_2333.eps}
801 \begin{minipage}{0.2cm}
804 \begin{minipage}{2.0cm}
805 \includegraphics[width=2.0cm]{c_pd_vasp/0-10_2333.eps}
806 \end{minipage}\\[0.1cm]
807 $\Delta E=\unit[0.9]{eV}$ | Experimental values: \unit[0.70--0.87]{eV}\\
808 $\Rightarrow$ {\color{blue}Migration mechanism identified!}\\
809 Note: Change in orientation\\
811 \begin{minipage}{5.4cm}
812 \includegraphics[width=6.0cm]{00-1_0-10_vasp_s.ps}
813 \end{minipage}\\[0.5cm]
814 \begin{minipage}{6.8cm}
815 {\bf\underline{Empirical potential}} $\quad$
816 \hkl[0 0 -1] $\rightarrow$ \hkl[1 1 0] $\rightarrow$ \hkl[0 -1 0]\\[-0.1cm]
818 \item Transition involving \hkl[1 1 0] DB\\
819 (instability of BC configuration)
820 \item $\Delta E \approx \unit[2.2]{eV} \text{ \& } \unit[0.9]{eV}$
821 \item 2.4 -- 3.4 times higher than ab initio result
822 \item After all: Change of the DB orientation
826 {\color{red}Drastically overestimated diffusion barrier}
830 \begin{minipage}{5.4cm}
831 \includegraphics[width=6.0cm]{00-1_110_0-10_mig_albe.ps}
840 Defect combinations --- ab inito
847 \begin{minipage}{9cm}
849 Summary of combinations}\\[0.1cm]
851 \begin{tabular}{l c c c c c c}
853 $E_{\text{b}}$ [eV] & 1 & 2 & 3 & 4 & 5 & R\\
855 \hkl[0 0 -1] & {\color{red}-0.08} & -1.15 & {\color{red}-0.08} & 0.04 & -1.66 & -0.19\\
856 \hkl[0 0 1] & 0.34 & 0.004 & -2.05 & 0.26 & -1.53 & -0.19\\
857 \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}\\
858 \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}\\
859 \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}\\
860 \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}\\
862 C$_{\text{sub}}$ & 0.26 & -0.51 & -0.93 & -0.15 & 0.49 & -0.05\\
863 Vacancy & -5.39 ($\rightarrow$ C$_{\text{sub}}$) & -0.59 & -3.14 & -0.54 & -0.50 & -0.31\\
868 \begin{minipage}{3cm}
869 \includegraphics[width=3.5cm]{comb_pos.eps}
874 {\bf\boldmath Combinations of \hkl<1 0 0>-type interstitials}\\[0.2cm]
875 \begin{minipage}{6.1cm}
877 \item \ci{} agglomeration energetically favorable
878 \item Reduction of strain
879 \item Capture radius exceeding \unit[1]{nm}
880 \item Disappearance of attractive forces\\
881 between two lowest separations.
885 {\color{blue}\ci{} agglomeration / no C clustering}
889 \begin{picture}(0,0)(-180,-50)
890 \begin{minipage}{6.0cm}
891 \scriptsize\centering
892 Interaction along \hkl[1 1 0]\\
893 \includegraphics[width=6.2cm]{db_along_110_cc.ps}
903 Defect combinations of C-Si dimers and vacancies
909 \begin{minipage}[b]{2.6cm}
911 \underline{V at 2: $E_{\text{b}}=-0.59\text{ eV}$}\\[0.1cm]
912 \includegraphics[width=2.5cm]{00-1dc/0-59.eps}
915 \begin{minipage}[b]{7cm}
918 \begin{minipage}[b]{2.6cm}
920 \underline{V at 3, $E_{\text{b}}=-3.14\text{ eV}$}\\[0.1cm]
921 \includegraphics[width=2.5cm]{00-1dc/3-14.eps}
923 \end{minipage}\\[0.2cm]
925 \begin{minipage}{6.5cm}
926 \includegraphics[width=6.0cm]{059-539.ps}
928 \begin{minipage}{5.7cm}
929 \includegraphics[width=6.0cm]{314-539.ps}
932 \begin{pspicture}(0,0)(0,0)
933 \psline[linewidth=0.05cm,linecolor=gray](6.3,9.0)(6.3,0)
935 \rput(6.3,7.0){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=white,linewidth=0.05cm,linecolor=gray]{
936 \begin{minipage}{6.5cm}
938 IBS: Impinging C creates V \& far away \si\\[0.3cm]
939 Low migration barrier towards C$_{\text{sub}}$\\
941 High barrier for reverse process\\[0.3cm]
943 High probability of stable C$_{\text{sub}}$ configuration
956 Combinations of substitutional C and Si self-interstitials
963 \begin{minipage}{6.2cm}
965 {\bf\boldmath C$_{\text{sub}}$ - \si{} \hkl<1 1 0> interaction}
967 \item Most favorable: \cs{} along \hkl<1 1 0> chain of \si{}
968 \item Less favorable than ground-state \ci{} \hkl<1 0 0> DB
969 \item Interaction drops quickly to zero\\
970 $\rightarrow$ low capture radius
974 \begin{minipage}{0.2cm}
977 \begin{minipage}{6.0cm}
979 {\bf Transition from the ground state}
981 \item Low transition barrier
982 \item Barrier smaller than \ci{} migration barrier
983 \item Low \si{} migration barrier (\unit[0.67]{eV})\\
984 $\rightarrow$ Separation of \cs{} \& \si{} most probable
987 \end{minipage}\\[0.3cm]
989 \begin{minipage}{6.0cm}
990 \includegraphics[width=6.0cm]{c_sub_si110.ps}
992 \begin{minipage}{0.4cm}
995 \begin{minipage}{6.0cm}
997 \includegraphics[width=6.0cm]{162-097.ps}
1001 \begin{pspicture}(0,0)(0,0)
1002 \psline[linewidth=0.05cm,linecolor=gray](6.5,0)(6.5,7.5)
1003 \rput(6.5,-0.7){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=white,linewidth=0.05cm,linecolor=blue]{
1004 \begin{minipage}{8cm}
1008 \cs{} \& \si{} instead of thermodynamic ground state\\[0.1cm]
1009 IBS --- process far from equilibrium\\
1022 Combinations of substitutional C and Si self-interstitials
1029 \begin{minipage}{6.2cm}
1031 {\bf\boldmath C$_{\text{sub}}$ - \si{} \hkl<1 1 0> interaction}
1033 \item Most favorable: \cs{} along \hkl<1 1 0> chain of \si{}
1034 \item Less favorable than ground-state \ci{} \hkl<1 0 0> DB
1035 \item Interaction drops quickly to zero\\
1036 $\rightarrow$ low capture radius
1040 \begin{minipage}{0.2cm}
1043 \begin{minipage}{6.0cm}
1045 {\bf Transition from the ground state}
1047 \item Low transition barrier
1048 \item Barrier smaller than \ci{} migration barrier
1049 \item Low \si{} migration barrier (\unit[0.67]{eV})\\
1050 $\rightarrow$ Separation of \cs{} \& \si{} most probable
1053 \end{minipage}\\[0.3cm]
1055 \begin{minipage}{6.0cm}
1056 \includegraphics[width=6.0cm]{c_sub_si110.ps}
1058 \begin{minipage}{0.4cm}
1061 \begin{minipage}{6.0cm}
1063 \includegraphics[width=6.0cm]{162-097.ps}
1067 \begin{pspicture}(0,0)(0,0)
1068 \psline[linewidth=0.05cm,linecolor=gray](6.5,0)(6.5,7.5)
1069 \rput(6.5,-0.7){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=white,linewidth=0.05cm,linecolor=blue]{
1070 \begin{minipage}{8cm}
1074 \cs{} \& \si{} instead of thermodynamic ground state\\[0.1cm]
1075 IBS --- process far from equilibrium\\
1083 \begin{pspicture}(0,0)(0,0)
1084 \rput(6.5,5.0){\psframebox[fillstyle=solid,opacity=0.5,fillcolor=black]{
1085 \begin{minipage}{14cm}
1090 \rput(6.5,4.3){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=white,linewidth=0.1cm]{
1091 \begin{minipage}{11cm}
1095 Ab initio MD at \degc{900}\\[0.4cm]
1096 \begin{minipage}{5.4cm}
1098 \includegraphics[width=4.3cm]{md01_bonds.eps}\\
1101 \begin{minipage}{5.4cm}
1103 \includegraphics[width=4.3cm]{md02_bonds.eps}\\
1105 \end{minipage}\\[0.5cm]
1107 Contribution of entropy to structural formation\\[0.1cm]
1120 Silicon carbide precipitation simulations
1130 \begin{pspicture}(0,0)(12,6.5)
1132 \rput(3.5,5.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
1135 \item Create c-Si volume
1136 \item Periodc boundary conditions
1137 \item Set requested $T$ and $p=0\text{ bar}$
1138 \item Equilibration of $E_{\text{kin}}$ and $E_{\text{pot}}$
1141 \rput(3.5,2.7){\rnode{insert}{\psframebox[fillstyle=solid,fillcolor=lachs]{
1143 Insertion of C atoms at constant T
1145 \item total simulation volume {\pnode{in1}}
1146 \item volume of minimal SiC precipitate size {\pnode{in2}}
1147 %\item volume consisting of Si atoms to form a minimal {\pnode{in3}}\\
1148 \item volume containing Si atoms to form a minimal {\pnode{in3}}\\
1152 \rput(3.5,1){\rnode{cool}{\psframebox[fillstyle=solid,fillcolor=lbb]{
1154 Run for 100 ps followed by cooling down to $20\, ^{\circ}\textrm{C}$
1156 \ncline[]{->}{init}{insert}
1157 \ncline[]{->}{insert}{cool}
1158 \psframe[fillstyle=solid,fillcolor=white](7.3,0.7)(12.8,6.3)
1159 \rput(7.6,6){\footnotesize $V_1$}
1160 \psframe[fillstyle=solid,fillcolor=lightgray](8.7,2)(11.6,5)
1161 \rput(8.9,4.85){\tiny $V_2$}
1162 \psframe[fillstyle=solid,fillcolor=gray](8.95,2.25)(11.35,4.75)
1163 \rput(9.25,4.45){\footnotesize $V_3$}
1164 \rput(7.9,3.2){\pnode{ins1}}
1165 \rput(8.92,2.8){\pnode{ins2}}
1166 \rput(10.8,2.4){\pnode{ins3}}
1167 \ncline[]{->}{in1}{ins1}
1168 \ncline[]{->}{in2}{ins2}
1169 \ncline[]{->}{in3}{ins3}
1179 \begin{minipage}{5.7cm}
1181 \item Amount of C atoms: 6000\\
1182 ($r_{\text{prec}}\approx 3.1\text{ nm}$, IBS: \unit[2--4]{nm})
1183 \item Simulation volume: $31^3$ Si unit cells\\
1187 \begin{minipage}{0.3cm}
1191 \begin{minipage}{6.0cm}
1192 Restricted to classical potential caclulations\\
1193 $\rightarrow$ Low C diffusion / overestimated barrier\\
1194 $\rightarrow$ Consider $V_2$ and $V_3$
1196 % \item $V_2$ and $V_3$ considered due to expected low C diffusion
1207 Silicon carbide precipitation simulations
1212 \begin{minipage}{6.3cm}
1213 \hspace*{-0.4cm}\includegraphics[width=6.5cm]{sic_prec_450_si-c.ps}\\
1214 \hspace*{-0.4cm}\includegraphics[width=6.5cm]{tot_pc_thesis.ps}
1217 \begin{minipage}{6.1cm}
1219 \underline{Temperature as used in IBS (\degc{450})}\\[0.2cm]
1220 \ci{} \hkl<1 0 0> dumbbell dominated structure\\
1221 \begin{pspicture}(0,0)(6.0,1.0)
1222 \rput(2.75,0.4){\psframebox[linewidth=0.05cm,linecolor=black]{
1223 \begin{minipage}{5cm}
1226 {\color{blue}Formation of \ci{} DBs}\\
1227 {\color{red}No agllomeration / precipitation}
1230 \end{pspicture}\\[0.1cm]
1233 \item Time scale problem of MD\\
1234 $\Rightarrow$ slow phase space propagation
1235 \item Short range potential\\
1236 $\Rightarrow$ overestimated diffusion barrier
1239 \underline{Increased temperatures}\\[0.2cm]
1240 \cs{} dominated structure\\
1241 \begin{pspicture}(0,0)(6.0,1.0)
1242 \rput(2.75,0.4){\psframebox[linewidth=0.05cm,linecolor=black]{
1243 \begin{minipage}{5cm}
1246 Si-{\color{blue}C$_{\text{sub}}$}-Si along \hkl<1 1 0>\\
1247 {\color{blue}\cs}-Si-{\color{blue}\cs} \& nearby \si
1250 \end{pspicture}\\[0.1cm]
1253 \item Stretched coherent SiC structures\\
1254 $\Rightarrow$ \cs{} involved in precipitation mechanism
1255 \item Reduction in strain by \si{}
1267 Summary and Conclusions
1273 \item First-principles investigation of defect combinations
1274 and mobilities in Si
1275 \item Empirical potential MD simulations on SiC precipitation in Si
1280 \psframebox[linecolor=hb,fillstyle=solid,fillcolor=hb]{
1281 \begin{minipage}{12cm}
1282 Conclusions on SiC precipitation $\qquad$
1283 {\Huge$\lightning$} {\color{red}\ci{}} --- vs --- {\color{blue}\cs{}} {\Huge$\lightning$}
1286 \item \cs{} involved in the precipitation mechanism
1287 \item Role of the \si{}
1289 \item Vehicle to rearrange \cs --- [\cs{} \& \si{} $\leftrightarrow$ \ci]
1290 \item Building block for surrounding Si host \& further SiC
1291 \item Strain compensation \ldots\\
1292 \ldots Si/SiC interface\\
1293 \ldots within stretched coherent SiC structure
1304 \item High T $\leftrightarrow$ IBS conditions far from equilibrium
1322 \begin{minipage}[t]{6cm}
1323 \underline{Augsburg}
1325 \item Prof. B. Stritzker
1328 \underline{Helsinki}
1330 \item Prof. K. Nordlund
1335 \item Bayerische Forschungsstiftung
1338 \begin{minipage}[t]{6cm}
1339 \underline{Paderborn}
1341 \item Prof. J. Lindner
1342 \item Prof. G. Schmidt
1351 \LARGE\bf Thank you for your attention!
1366 Polytypes of SiC\\[0.6cm]
1371 \includegraphics[width=3.8cm]{cubic_hex.eps}\\
1372 \begin{minipage}{1.9cm}
1373 {\tiny cubic (twist)}
1375 \begin{minipage}{2.9cm}
1376 {\tiny hexagonal (no twist)}
1379 \begin{picture}(0,0)(-150,0)
1380 \includegraphics[width=7cm]{polytypes.eps}
1387 \begin{tabular}{l c c c c c c}
1389 & 3C-SiC & 4H-SiC & 6H-SiC & Si & GaN & Diamond\\
1391 Hardness [Mohs] & \multicolumn{3}{c}{------ 9.6 ------}& 6.5 & - & 10 \\
1392 Band gap [eV] & 2.36 & 3.23 & 3.03 & 1.12 & 3.39 & 5.5 \\
1393 Break down field [$10^6$ V/cm] & 4 & 3 & 3.2 & 0.6 & 5 & 10 \\
1394 Saturation drift velocity [$10^7$ cm/s] & 2.5 & 2.0 & 2.0 & 1 & 2.7 & 2.7 \\
1395 Electron mobility [cm$^2$/Vs] & 800 & 900 & 400 & 1100 & 900 & 2200 \\
1396 Hole mobility [cm$^2$/Vs] & 320 & 120 & 90 & 420 & 150 & 1600 \\
1397 Thermal conductivity [W/cmK] & 5.0 & 4.9 & 4.9 & 1.5 & 1.3 & 22 \\
1401 \begin{pspicture}(0,0)(0,0)
1402 \psellipse[linecolor=green](5.7,2.05)(0.4,0.50)
1404 \begin{pspicture}(0,0)(0,0)
1405 \psellipse[linecolor=green](5.6,0.89)(0.4,0.20)
1407 \begin{pspicture}(0,0)(0,0)
1408 \psellipse[linecolor=red](10.45,0.42)(0.4,0.20)
1417 IBS of epitaxial single crystalline 3C-SiC
1426 \item \underline{Implantation step 1}\\[0.1cm]
1427 Almost stoichiometric dose | \unit[180]{keV} | \degc{500}\\
1428 $\Rightarrow$ Epitaxial {\color{blue}3C-SiC} layer \&
1429 {\color{blue}precipitates}
1430 \item \underline{Implantation step 2}\\[0.1cm]
1431 Low remaining amount of dose | \unit[180]{keV} | \degc{250}\\
1433 Destruction/Amorphization of precipitates at layer interface
1434 \item \underline{Annealing}\\[0.1cm]
1435 \unit[10]{h} at \degc{1250}\\
1436 $\Rightarrow$ Homogeneous 3C-SiC layer with sharp interfaces
1440 \begin{minipage}{6.9cm}
1441 \includegraphics[width=7cm]{ibs_3c-sic.eps}\\[-0.4cm]
1444 XTEM: single crystalline 3C-SiC in Si\hkl(1 0 0)
1448 \begin{minipage}{5cm}
1450 \begin{pspicture}(0,0)(0,0)
1452 \psframebox[fillstyle=solid,fillcolor=white,linecolor=blue,linestyle=solid]{
1453 \begin{minipage}{3.3cm}
1456 3C-SiC precipitation\\
1457 not yet fully understood
1461 % \renewcommand\labelitemi{$\Rightarrow$}
1462 % Details of the SiC precipitation
1464 % \item significant technological progress\\
1465 % in SiC thin film formation
1466 % \item perspectives for processes relying\\
1467 % upon prevention of SiC precipitation
1471 \rput(-5.3,5.5){\pnode{h0}}
1472 \rput(-1.95,5.5){\pnode{h1}}
1473 \ncline[linecolor=blue]{-}{h0}{h1}
1474 \ncline[linecolor=blue]{->}{h1}{box}
1487 Si self-interstitial point defects in silicon\\[0.1cm]
1491 \begin{tabular}{l c c c c c}
1493 $E_{\text{f}}$ [eV] & \hkl<1 1 0> DB & H & T & \hkl<1 0 0> DB & V \\
1495 \textsc{vasp} & \underline{3.39} & 3.42 & 3.77 & 4.41 & 3.63 \\
1496 Erhart/Albe & 4.39 & 4.48$^*$ & \underline{3.40} & 5.42 & 3.13 \\
1498 \end{tabular}\\[0.4cm]
1501 \begin{minipage}{3cm}
1503 \underline{Vacancy}\\
1504 \includegraphics[width=2.8cm]{si_pd_albe/vac.eps}
1507 \begin{minipage}{3cm}
1509 \underline{\hkl<1 1 0> DB}\\
1510 \includegraphics[width=2.8cm]{si_pd_albe/110_bonds.eps}
1513 \begin{minipage}{3cm}
1515 \underline{\hkl<1 0 0> DB}\\
1516 \includegraphics[width=2.8cm]{si_pd_albe/100_bonds.eps}
1519 \begin{minipage}{3cm}
1521 \underline{Tetrahedral}\\
1522 \includegraphics[width=2.8cm]{si_pd_albe/tet_bonds.eps}
1526 \underline{Hexagonal} \hspace{2pt}
1527 \href{../video/si_self_int_hexa.avi}{$\rhd$}\\[0.1cm]
1529 \begin{minipage}{2.7cm}
1530 $E_{\text{f}}^*=4.48\text{ eV}$\\
1531 \includegraphics[width=2.7cm]{si_pd_albe/hex_a_bonds.eps}
1533 \begin{minipage}{0.4cm}
1538 \begin{minipage}{2.7cm}
1539 $E_{\text{f}}=3.96\text{ eV}$\\
1540 \includegraphics[width=2.8cm]{si_pd_albe/hex_bonds.eps}
1543 \begin{minipage}{5.5cm}
1545 {\tiny nearly T $\rightarrow$ T}\\
1547 \includegraphics[width=6.0cm]{nhex_tet.ps}
1556 C-Si dimer \& bond-centered interstitial configuration
1563 \begin{minipage}[t]{4.1cm}
1564 {\bf\boldmath C \hkl<1 0 0> DB interstitial}\\[0.1cm]
1565 \begin{minipage}{2.0cm}
1567 \underline{Erhart/Albe}
1568 \includegraphics[width=2.0cm]{c_pd_albe/100_cmp.eps}
1571 \begin{minipage}{2.0cm}
1573 \underline{\textsc{vasp}}
1574 \includegraphics[width=2.0cm]{c_pd_vasp/100_cmp.eps}
1576 \end{minipage}\\[0.2cm]
1577 Si-C-Si bond angle $\rightarrow$ \unit[180]{$^{\circ}$}\\
1578 $\Rightarrow$ $sp$ hybridization\\[0.1cm]
1579 Si-Si-Si bond angle $\rightarrow$ \unit[120]{$^{\circ}$}\\
1580 $\Rightarrow$ $sp^2$ hybridization
1582 \includegraphics[width=3.4cm]{c_pd_vasp/eden.eps}\\[-0.1cm]
1583 {\tiny Charge density isosurface}
1586 \begin{minipage}{0.2cm}
1589 \begin{minipage}[t]{8.1cm}
1591 {\bf Bond-centered interstitial}\\[0.1cm]
1592 \begin{minipage}{4.4cm}
1595 \item Linear Si-C-Si bond
1596 \item Si: one C \& 3 Si neighbours
1597 \item Spin polarized calculations
1598 \item No saddle point!\\
1602 \begin{minipage}{2.7cm}
1603 %\includegraphics[width=2.8cm]{c_pd_vasp/bc_2333.eps}\\
1605 \includegraphics[width=2.8cm]{c_pd_albe/bc_bonds.eps}\\
1610 \begin{minipage}[t]{6.5cm}
1611 \begin{minipage}[t]{1.2cm}
1613 {\tiny sp$^3$}\\[0.8cm]
1614 \underline{${\color{black}\uparrow}$}
1615 \underline{${\color{black}\uparrow}$}
1616 \underline{${\color{black}\uparrow}$}
1617 \underline{${\color{red}\uparrow}$}\\
1620 \begin{minipage}[t]{1.4cm}
1622 {\color{red}M}{\color{blue}O}\\[0.8cm]
1623 \underline{${\color{blue}\uparrow}{\color{white}\downarrow}$}\\
1624 $\sigma_{\text{ab}}$\\[0.5cm]
1625 \underline{${\color{red}\uparrow}{\color{blue}\downarrow}$}\\
1629 \begin{minipage}[t]{1.0cm}
1633 \underline{${\color{white}\uparrow\uparrow}$}
1634 \underline{${\color{white}\uparrow\uparrow}$}\\
1636 \underline{${\color{blue}\uparrow}{\color{blue}\downarrow}$}
1637 \underline{${\color{blue}\uparrow}{\color{blue}\downarrow}$}\\
1641 \begin{minipage}[t]{1.4cm}
1643 {\color{blue}M}{\color{green}O}\\[0.8cm]
1644 \underline{${\color{blue}\uparrow}{\color{white}\downarrow}$}\\
1645 $\sigma_{\text{ab}}$\\[0.5cm]
1646 \underline{${\color{green}\uparrow}{\color{blue}\downarrow}$}\\
1650 \begin{minipage}[t]{1.2cm}
1653 {\tiny sp$^3$}\\[0.8cm]
1654 \underline{${\color{green}\uparrow}$}
1655 \underline{${\color{black}\uparrow}$}
1656 \underline{${\color{black}\uparrow}$}
1657 \underline{${\color{black}\uparrow}$}\\
1665 \begin{minipage}{3.0cm}
1667 \underline{Charge density}\\
1668 {\color{gray}$\bullet$} Spin up\\
1669 {\color{green}$\bullet$} Spin down\\
1670 {\color{blue}$\bullet$} Resulting spin up\\
1671 {\color{yellow}$\bullet$} Si atoms\\
1672 {\color{red}$\bullet$} C atom
1674 \begin{minipage}{3.6cm}
1675 \includegraphics[width=3.8cm]{c_100_mig_vasp/im_spin_diff.eps}
1682 \begin{pspicture}(0,0)(0,0)
1683 \psline[linecolor=gray,linewidth=0.05cm](-7.8,-8.7)(-7.8,0)
1692 C interstitial migration --- ab initio
1699 \begin{minipage}{6.8cm}
1700 \framebox{\hkl[0 0 -1] $\rightarrow$ \hkl[0 0 1]}\\
1701 \begin{minipage}{2.0cm}
1702 \includegraphics[width=2.0cm]{c_pd_vasp/100_2333.eps}
1704 \begin{minipage}{0.2cm}
1707 \begin{minipage}{2.0cm}
1708 \includegraphics[width=2.0cm]{c_pd_vasp/bc_2333.eps}
1710 \begin{minipage}{0.2cm}
1713 \begin{minipage}{2.0cm}
1714 \includegraphics[width=2.0cm]{c_pd_vasp/100_next_2333.eps}
1715 \end{minipage}\\[0.1cm]
1717 $\Rightarrow$ Sufficient to consider \hkl[00-1] to BC transition\\
1718 $\Rightarrow$ Migration barrier to reach BC | $\Delta E=\unit[1.2]{eV}$
1720 \begin{minipage}{5.4cm}
1721 \includegraphics[width=6.0cm]{im_00-1_nosym_sp_fullct_thesis_vasp_s.ps}
1722 %\end{minipage}\\[0.2cm]
1723 \end{minipage}\\[0.4cm]
1726 \begin{minipage}{6.8cm}
1727 \framebox{\hkl[0 0 -1] $\rightarrow$ \hkl[0 -1 0]}\\
1728 \begin{minipage}{2.0cm}
1729 \includegraphics[width=2.0cm]{c_pd_vasp/100_2333.eps}
1731 \begin{minipage}{0.2cm}
1734 \begin{minipage}{2.0cm}
1735 \includegraphics[width=2.0cm]{c_pd_vasp/00-1-0-10_2333.eps}
1737 \begin{minipage}{0.2cm}
1740 \begin{minipage}{2.0cm}
1741 \includegraphics[width=2.0cm]{c_pd_vasp/0-10_2333.eps}
1742 \end{minipage}\\[0.1cm]
1743 $\Delta E=\unit[0.9]{eV}$ | Experimental values: \unit[0.70--0.87]{eV}\\
1744 $\Rightarrow$ {\color{red}Migration mechanism identified!}\\
1745 Note: Change in orientation
1747 \begin{minipage}{5.4cm}
1748 \includegraphics[width=6.0cm]{00-1_0-10_vasp_s.ps}
1749 \end{minipage}\\[0.1cm]
1752 %Reorientation pathway composed of two consecutive processes of the above type
1761 C interstitial migration --- analytical potential
1767 \begin{minipage}[t]{6.0cm}
1768 {\bf\boldmath BC to \hkl[0 0 -1] transition}\\[0.2cm]
1769 \includegraphics[width=6.0cm]{bc_00-1_albe_s.ps}\\
1771 \item Lowermost migration barrier
1772 \item $\Delta E \approx \unit[2.2]{eV}$
1773 \item 2.4 times higher than ab initio result
1774 \item Different pathway
1777 \begin{minipage}[t]{0.2cm}
1780 \begin{minipage}[t]{6.0cm}
1781 {\bf\boldmath Transition involving a \hkl<1 1 0> configuration}
1784 \item Bond-centered configuration unstable\\
1785 $\rightarrow$ \ci{} \hkl<1 1 0> dumbbell
1786 \item Minimum of the \hkl[0 0 -1] to \hkl[0 -1 0] transition\\
1787 $\rightarrow$ \ci{} \hkl<1 1 0> DB
1790 \includegraphics[width=6.0cm]{00-1_110_0-10_mig_albe.ps}
1792 \item $\Delta E \approx \unit[2.2]{eV} \text{ \& } \unit[0.9]{eV}$
1793 \item 2.4 -- 3.4 times higher than ab initio result
1794 \item After all: Change of the DB orientation
1800 {\color{red}\bf Drastically overestimated diffusion barrier}
1803 \begin{pspicture}(0,0)(0,0)
1804 \psline[linewidth=0.05cm,linecolor=gray](6.1,1.0)(6.1,9.3)
1813 Silicon carbide precipitation simulations at \degc{450} as in IBS
1818 \begin{minipage}{6.3cm}
1819 \hspace*{-0.4cm}\includegraphics[width=6.5cm]{sic_prec_450_si-c.ps}\\
1820 \hspace*{-0.4cm}\includegraphics[width=6.5cm]{sic_prec_450_si-si_c-c.ps}
1823 \begin{minipage}{6.1cm}
1825 \underline{Low C concentration --- {\color{red}$V_1$}}\\[0.1cm]
1826 \ci{} \hkl<1 0 0> dumbbell dominated structure
1828 \item Si-C bumbs around \unit[0.19]{nm}
1829 \item C-C peak at \unit[0.31]{nm} (expected in 3C-SiC):\\
1830 concatenated differently oriented \ci{} DBs
1831 \item Si-Si NN distance stretched to \unit[0.3]{nm}
1833 \begin{pspicture}(0,0)(6.0,1.0)
1834 \rput(3.2,0.5){\psframebox[linewidth=0.03cm,linecolor=blue]{
1835 \begin{minipage}{6cm}
1837 Formation of \ci{} dumbbells\\
1838 C atoms separated as expected in 3C-SiC
1841 \end{pspicture}\\[0.1cm]
1842 \underline{High C concentration --- {\color{green}$V_2$}/{\color{blue}$V_3$}}
1844 \item High amount of strongly bound C-C bonds
1845 \item Increased defect \& damage density\\
1846 $\rightarrow$ Arrangements hard to categorize and trace
1847 \item Only short range order observable
1849 \begin{pspicture}(0,0)(6.0,0.8)
1850 \rput(3.2,0.5){\psframebox[linewidth=0.03cm,linecolor=blue]{
1851 \begin{minipage}{6cm}
1853 Amorphous SiC-like phase
1856 \end{pspicture}\\[0.3cm]
1857 \begin{pspicture}(0,0)(6.0,2.0)
1858 \rput(3.2,1.0){\psframebox[linewidth=0.05cm,linecolor=black]{
1859 \begin{minipage}{6cm}
1862 {\bf\color{red}Formation of 3C-SiC fails to appear}\\[0.3cm]
1863 \begin{minipage}{0.8cm}
1864 {\bf\boldmath $V_1$:}
1866 \begin{minipage}{5.1cm}
1867 Formation of \ci{} indeed occurs\\
1868 Agllomeration not observed
1869 \end{minipage}\\[0.3cm]
1870 \begin{minipage}{0.8cm}
1871 {\bf\boldmath $V_{2,3}$:}
1873 \begin{minipage}{5.1cm}
1874 Amorphous SiC-like structure\\
1875 (not expected at \degc{450})\\[0.05cm]
1876 No rearrangement/transition into 3C-SiC
1877 \end{minipage}\\[0.1cm]
1889 Increased temperature simulations --- $V_1$
1894 \begin{minipage}{6.2cm}
1895 \hspace*{-0.4cm}\includegraphics[width=6.5cm]{tot_pc_thesis.ps}
1898 \begin{minipage}{6.2cm}
1899 \includegraphics[width=6.5cm]{tot_pc3_thesis.ps}
1902 \begin{minipage}{6.2cm}
1903 \hspace*{-0.4cm}\includegraphics[width=6.5cm]{tot_pc2_thesis.ps}
1906 \begin{minipage}{6.3cm}
1908 \underline{Si-C bonds:}
1910 \item Vanishing cut-off artifact (above $1650\,^{\circ}\mathrm{C}$)
1911 \item Structural change: \ci{} \hkl<1 0 0> DB $\rightarrow$
1914 \underline{Si-Si bonds:}
1915 {\color{blue}Si-C$_{\text{sub}}$-Si} along \hkl<1 1 0>
1916 ($\rightarrow$ 0.325 nm)\\[0.1cm]
1917 \underline{C-C bonds:}
1919 \item C-C next neighbour pairs reduced (mandatory)
1920 \item Peak at 0.3 nm slightly shifted\\[0.05cm]
1921 $\searrow$ \ci{} combinations (dashed arrows)\\
1922 $\nearrow$ \ci{} \hkl<1 0 0> \& {\color{blue}\cs{} combinations} (|)\\
1923 $\nearrow$ \ci{} pure \cs{} combinations ($\downarrow$)\\[0.05cm]
1924 Range [|-$\downarrow$]: {\color{blue}\cs{} \& \cs{} with nearby \si}
1933 Increased temperature simulations at high C concentration
1938 \begin{minipage}{6.0cm}
1939 \includegraphics[width=6.4cm]{12_pc_thesis.ps}
1941 \begin{minipage}{6.0cm}
1942 \includegraphics[width=6.4cm]{12_pc_c_thesis.ps}
1950 \begin{minipage}[t]{5.5cm}
1951 0.186 nm: Si-C pairs $\uparrow$\\
1952 (as expected in 3C-SiC)\\[0.2cm]
1953 0.282 nm: Si-C-C\\[0.2cm]
1954 $\approx$0.35 nm: C-Si-Si
1957 \begin{minipage}{0.1cm}
1961 \begin{minipage}[t]{5.9cm}
1962 0.15 nm: C-C pairs $\uparrow$\\
1963 (as expected in graphite/diamond)\\[0.2cm]
1964 0.252 nm: C-C-C (2$^{\text{nd}}$ NN for diamond)\\[0.2cm]
1965 0.31 nm: shifted towards 0.317 nm $\rightarrow$ C-Si-C
1970 \item Decreasing cut-off artifact
1971 \item {\color{red}Amorphous} SiC-like phase remains
1972 \item High amount of {\color{red}damage} \& alignement to c-Si host matrix lost
1973 \item Slightly sharper peaks $\Rightarrow$ indicate slight {\color{blue}acceleration of dynamics} due to temperature
1980 High C \& small $V$ \& short $t$
1983 \begin{minipage}{4cm}
1985 Slow structural evolution due to strong C-C bonds
1990 High C \& low T implants
2001 Long time scale simulations at maximum temperature
2008 \underline{Differences}
2010 \item Temperature set to $0.95 \cdot T_{\text{m}}$
2011 \item Cubic insertion volume $\Rightarrow$ spherical insertion volume
2012 \item Amount of C atoms: 6000 $\rightarrow$ 5500
2013 $\Leftrightarrow r_{\text{prec}}=0.3\text{ nm}$
2014 \item Simulation volume: 21 unit cells of c-Si in each direction
2021 \begin{minipage}[t]{4.3cm}
2023 \underline{Low C concentration, Si-C}
2024 \includegraphics[width=4.3cm]{c_in_si_95_v1_si-c.ps}\\
2028 \begin{minipage}[t]{4.3cm}
2030 \underline{Low C concentration, C-C}
2031 \includegraphics[width=4.3cm]{c_in_si_95_v1_c-c.ps}\\
2036 \begin{minipage}[t]{3.4cm}
2038 \underline{High C concentration}
2039 \includegraphics[width=4.3cm]{c_in_si_95_v2.ps}\\
2040 No significant changes\\
2041 iC-Si-Si $\uparrow$\\
2048 Long time scales and high temperatures most probably not sufficient enough!
2057 Investigation of a silicon carbide precipitate in silicon
2066 \begin{minipage}{5.3cm}
2068 \frac{8}{a_{\text{Si}}^3}(
2069 \underbrace{21^3 a_{\text{Si}}^3}_{=V}
2070 -\frac{4}{3}\pi x^3)+
2071 \underbrace{\frac{4}{y^3}\frac{4}{3}\pi x^3}_{\stackrel{!}{=}5500}
2078 \frac{8}{a_{\text{Si}}^3}\frac{4}{3}\pi x^3=5500
2079 \Rightarrow x = \left(\frac{5500 \cdot 3}{32 \pi} \right)^{1/3}a_{\text{Si}}
2082 y=\left(\frac{1}{2} \right)^{1/3}a_{\text{Si}}
2086 \begin{minipage}{0.1cm}
2089 \begin{minipage}{6.3cm}
2090 \underline{Construction}
2092 \item Simulation volume: 21$^3$ unit cells of c-Si
2093 \item Spherical topotactically aligned precipitate\\
2094 $r=3.0\text{ nm}$ $\Leftrightarrow$ $\approx$ 5500 C atoms
2095 \item Create c-Si but skipped inside sphere\\
2097 \item Create 3C-SiC inside sphere of radius $x$\\
2098 and lattice constant $y$
2099 \item Strong coupling to heat bath ($T=20\,^{\circ}\mathrm{C}$)
2105 \begin{minipage}{6.0cm}
2106 \includegraphics[width=6cm]{pc_0.ps}
2108 \begin{minipage}{6.1cm}
2111 \item Slight increase of c-Si lattice constant!
2113 (imply same distanced Si-Si peaks)
2115 \item New peak at 0.307 nm: 2$^{\text{nd}}$ NN in 3C-SiC
2116 \item Bumps ({\color{green}$\downarrow$}):
2117 4$^{\text{th}}$ and 6$^{\text{th}}$ NN
2119 \item 3C-SiC lattice constant: 4.34 \AA (bulk: 4.36 \AA)\\
2120 $\rightarrow$ compressed precipitate
2121 \item Interface tension:\\
2122 20.15 eV/nm$^2$ or $3.23 \times 10^{-4}$ J/cm$^2$\\
2123 (literature: $2 - 8 \times 10^{-4}$ J/cm$^2$)
2132 Investigation of a silicon carbide precipitate in silicon
2137 \begin{minipage}{7cm}
2138 \underline{Appended annealing steps}
2140 \item artificially constructed interface\\
2141 $\rightarrow$ allow for rearrangement of interface atoms
2142 \item check SiC stability
2144 \underline{Temperature schedule}
2146 \item rapidly heat up structure up to $2050\,^{\circ}\mathrm{C}$\\
2148 \item slow heating up to $1.2\cdot T_{\text{m}}=2940\text{ K}$
2150 $\rightarrow$ melting at around 2840 K
2151 (\href{../video/sic_prec_120.avi}{$\rhd$})
2152 \item cooling down structure at 100 \% $T_{\text{m}}$ (1 K/ps)\\
2153 $\rightarrow$ no energetically more favorable struture
2156 \begin{minipage}{5cm}
2157 \includegraphics[width=5.5cm]{fe_and_t_sic.ps}
2160 \begin{minipage}{4cm}
2161 \includegraphics[width=4cm]{sic_prec/melt_01.eps}
2163 \begin{minipage}{0.2cm}
2166 \begin{minipage}{4cm}
2167 \includegraphics[width=4cm]{sic_prec/melt_02.eps}
2169 \begin{minipage}{0.2cm}
2172 \begin{minipage}{3.7cm}
2173 \includegraphics[width=4cm]{sic_prec/melt_03.eps}
2188 Equilibrium lattice constants and cohesive energies
2190 \begin{tabular}{l r c c c c c}
2193 & & USPP, LDA & USPP, GGA & PAW, LDA & PAW, GGA & Exp. \\
2195 Si (dia) & $a$ [\AA] & 5.389 & 5.455 & - & - & 5.429 \\
2196 & $\Delta_a$ [\%] & \unit[{\color{green}0.7}]{\%} & \unit[{\color{green}0.5}]{\%} & - & - & - \\
2197 & $E_{\text{coh}}$ [eV] & -5.277 & -4.591 & - & - & -4.63 \\
2198 & $\Delta_E$ [\%] & \unit[{\color{red}14.0}]{\%} & \unit[{\color{green}0.8}]{\%} & - & - & - \\
2200 C (dia) & $a$ [\AA] & 3.527 & 3.567 & - & - & 3.567 \\
2201 & $\Delta_a$ [\%] & \unit[{\color{green}1.1}]{\%} & \unit[{\color{green}0.01}]{\%} & - & - & - \\
2202 & $E_{\text{coh}}$ [eV] & -8.812 & -7.703 & - & - & -7.374 \\
2203 & $\Delta_E$ [\%] & \unit[{\color{red}19.5}]{\%} & \unit[{\color{orange}4.5}]{\%} & - & - & - \\
2205 3C-SiC & $a$ [\AA] & 4.319 & 4.370 & 4.330 & 4.379 & 4.359 \\
2206 & $\Delta_a$ [\%] & \unit[{\color{green}0.9}]{\%} & \unit[{\color{green}0.3}]{\%} & \unit[{\color{green}0.7}]{\%} & \unit[{\color{green}0.5}]{\%} & - \\
2207 & $E_{\text{coh}}$ [eV] & -7.318 & -6.426 & -7.371 & -6.491 & -6.340 \\
2208 & $\Delta_E$ [\%] & \unit[{\color{red}15.4}]{\%} & \unit[{\color{green}1.4}]{\%} & \unit[{\color{red}16.3}]{\%} & \unit[{\color{orange}2.4}]{\%} & - \\
2215 \begin{minipage}{7cm}
2217 \begin{tabular}{l c c c}
2220 & Si (dia) & C (dia) & 3C-SiC \\
2222 $a$ [\AA] & 5.458 & 3.562 & 4.365 \\
2223 $\Delta_a$ [\%] & 0.5 & 0.1 & 0.1 \\
2225 $E_{\text{coh}}$ [eV] & -4.577 & -7.695 & -6.419 \\
2226 $\Delta_E$ [\%] & 1.1 & 4.4 & 1.2 \\
2232 \begin{minipage}{5cm}
2233 $\leftarrow$ entire parameter set
2246 \begin{minipage}{6cm}
2248 \includegraphics[width=6cm]{sic_32pc_gamma_cutoff_lc.ps}
2251 \begin{minipage}{6cm}
2253 Lattice constants with respect to the PW cut-off energy
2257 \begin{minipage}{6cm}
2259 \includegraphics[width=6cm]{si_self_int_thesis.ps}
2262 \begin{minipage}{6cm}
2264 Defect formation energy with respect to the size of the supercell\\[0.1cm]