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32 \graphicspath{{../img/}}
36 \usepackage[setpagesize=false]{hyperref}
42 \usepackage{semlayer} % Seminar overlays
43 \usepackage{slidesec} % Seminar sections and list of slides
45 \input{seminar.bug} % Official bugs corrections
46 \input{seminar.bg2} % Unofficial bugs corrections
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71 % specify width and height
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79 \newcommand{\pot}{\mathcal{V}}
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84 \renewcommand\labelitemii{{\color{gray}$\bullet$}}
87 \renewcommand{\phi}{\varphi}
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101 \begin{pspicture}(0,0)(0,0)
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}{}}
116 \newcommand{\distn}[1]{\unit[#1]{nm}{}}
117 \newcommand{\dista}[1]{\unit[#1]{\AA}{}}
118 \newcommand{\perc}[1]{\unit[#1]{\%}{}}
120 % no vertical centering
132 A B C D E F G H G F E D C B A
150 Atomistic simulation study on silicon carbide\\[0.2cm]
151 precipitation in silicon\\
158 \textsc{Frank Zirkelbach}
162 Defense of doctor's thesis
171 % no vertical centering
174 % skip for preparation
179 % motivation / properties / applications of silicon carbide
187 \begin{pspicture}(0,0)(13.5,5)
189 \psframe*[linecolor=hb](-0.2,0)(12.9,5)
191 \pspolygon[linecolor=hlbb,fillcolor=hlbb,fillstyle=solid](5.2,1)(6.5,1)(6.5,3)(5.2,3)
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194 \rput[lt](0,4.6){\color{gray}PROPERTIES}
196 \rput[lt](0.3,4){wide band gap}
197 \rput[lt](0.3,3.5){high electric breakdown field}
198 \rput[lt](0.3,3){good electron mobility}
199 \rput[lt](0.3,2.5){high electron saturation drift velocity}
200 \rput[lt](0.3,2){high thermal conductivity}
202 \rput[lt](0.3,1.5){hard and mechanically stable}
203 \rput[lt](0.3,1){chemically inert}
205 \rput[lt](0.3,0.5){radiation hardness}
207 \rput[rt](12.7,4.6){\color{gray}APPLICATIONS}
209 \rput[rt](12.5,3.85){high-temperature, high power}
210 \rput[rt](12.5,3.5){and high-frequency}
211 \rput[rt](12.5,3.15){electronic and optoelectronic devices}
213 \rput[rt](12.5,2.35){material suitable for extreme conditions}
214 \rput[rt](12.5,2){microelectromechanical systems}
215 \rput[rt](12.5,1.65){abrasives, cutting tools, heating elements}
217 \rput[rt](12.5,0.85){first wall reactor material, detectors}
218 \rput[rt](12.5,0.5){and electronic devices for space}
222 \begin{picture}(0,0)(5,-162)
223 \includegraphics[height=2.2cm]{3C_SiC_bs.eps}
225 \begin{picture}(0,0)(-120,-162)
226 \includegraphics[height=2.2cm]{nasa_600c_led.eps}
228 \begin{picture}(0,0)(-270,-162)
229 \includegraphics[height=2.2cm]{6h-sic_3c-sic.eps}
232 \begin{picture}(0,0)(10,65)
233 \includegraphics[height=2.8cm]{sic_switch.eps}
235 %\begin{picture}(0,0)(-243,65)
236 \begin{picture}(0,0)(-110,65)
237 \includegraphics[height=2.8cm]{ise_99.eps}
239 %\begin{picture}(0,0)(-135,65)
240 \begin{picture}(0,0)(-100,65)
241 \includegraphics[height=1.2cm]{infineon_schottky.eps}
243 \begin{picture}(0,0)(-233,65)
244 \includegraphics[height=2.8cm]{solar_car.eps}
256 Fabrication of silicon carbide
265 \emph{Silicon carbide --- Born from the stars, perfected on earth.}
271 SiC thin films by MBE \& CVD
273 \item Much progress achieved in homo/heteroepitaxial SiC thin film growth
274 \item \underline{Commercially available} semiconductor power devices based on
275 \underline{\foreignlanguage{greek}{a}-SiC}
276 \item Production of favored \underline{3C-SiC} material
277 \underline{less advanced}
278 \item Quality and size not yet sufficient
280 \begin{picture}(0,0)(-310,-20)
281 \includegraphics[width=2.0cm]{cree.eps}
289 %{\footnotesize\color{black}
290 % Mismatch in \underline{thermal expansion coeefficient}
291 % and \underline{lattice parameter} w.r.t. substrate
298 {\bf Alternative approach}\\
299 Ion beam synthesis (IBS) of burried 3C-SiC layers in Si\hkl(1 0 0)
306 \begin{minipage}{3.15cm}
308 \includegraphics[width=3cm]{imp.eps}\\
314 \begin{minipage}{3.15cm}
316 \includegraphics[width=3cm]{annealing.eps}\\
318 Postannealing at $>$ \degc{1200}
323 \begin{minipage}{5.5cm}
324 \includegraphics[width=5.8cm]{ibs_3c-sic.eps}\\[-0.2cm]
327 XTEM: single crystalline 3C-SiC in Si\hkl(1 0 0)
332 %\begin{minipage}{5.5cm}
335 %No surface bending effects\\
336 %High areal homogenity\\[0.1cm]
337 %$\Downarrow$\\[0.1cm]
338 %Synthesis of large area SiC films possible
350 IBS of epitaxial single crystalline 3C-SiC
359 \item \underline{Implantation step 1}\\[0.1cm]
360 Almost stoichiometric dose | \unit[180]{keV} | \degc{500}\\
361 $\Rightarrow$ Epitaxial {\color{blue}3C-SiC} layer \&
362 {\color{blue}precipitates}
363 \item \underline{Implantation step 2}\\[0.1cm]
364 Low remaining amount of dose | \unit[180]{keV} | \degc{250}\\
366 Destruction/Amorphization of precipitates at layer interface
367 \item \underline{Annealing}\\[0.1cm]
368 \unit[10]{h} at \degc{1250}\\
369 $\Rightarrow$ Homogeneous 3C-SiC layer with sharp interfaces
373 \begin{minipage}{6.9cm}
374 \includegraphics[width=7cm]{ibs_3c-sic.eps}\\[-0.4cm]
377 XTEM: single crystalline 3C-SiC in Si\hkl(1 0 0)
381 \begin{minipage}{5cm}
382 \begin{pspicture}(0,0)(0,0)
384 \psframebox[fillstyle=solid,fillcolor=white,linecolor=blue,linestyle=solid]{
385 \begin{minipage}{5.3cm}
388 3C-SiC precipitation\\
389 not yet fully understood
393 \renewcommand\labelitemi{$\Rightarrow$}
394 Details of the SiC precipitation
396 \item significant technological progress\\
397 in SiC thin film formation
398 \item perspectives for processes relying\\
399 upon prevention of SiC precipitation
403 \rput(-6.8,5.5){\pnode{h0}}
404 \rput(-3.0,5.5){\pnode{h1}}
405 \ncline[linecolor=blue]{-}{h0}{h1}
406 \ncline[linecolor=blue]{->}{h1}{box}
416 Supposed precipitation mechanism of SiC in Si
424 \begin{minipage}{3.6cm}
426 Si \& SiC lattice structure\\[0.1cm]
427 \includegraphics[width=2.3cm]{sic_unit_cell.eps}
430 \begin{minipage}{1.7cm}
431 \underline{Silicon}\\
432 {\color{yellow}$\bullet$} Si | {\color{gray}$\bullet$} Si\\
433 $a=\unit[5.429]{\\A}$\\
434 $\rho^*_{\text{Si}}=\unit[100]{\%}$
436 \begin{minipage}{1.7cm}
437 \underline{Silicon carbide}\\
438 {\color{yellow}$\bullet$} Si | {\color{gray}$\bullet$} C\\
439 $a=\unit[4.359]{\\A}$\\
440 $\rho^*_{\text{Si}}=\unit[97]{\%}$
446 \begin{minipage}{4.1cm}
448 \includegraphics[width=3.3cm]{tem_c-si-db.eps}
452 \begin{minipage}{4.0cm}
454 \includegraphics[width=3.3cm]{tem_3c-sic.eps}
460 \begin{minipage}{4.0cm}
462 C-Si dimers (dumbbells)\\[-0.1cm]
467 \begin{minipage}{4.1cm}
469 Agglomeration of C-Si dumbbells\\[-0.1cm]
470 $\Rightarrow$ dark contrasts
474 \begin{minipage}{4.0cm}
476 Precipitation of 3C-SiC in Si\\[-0.1cm]
477 $\Rightarrow$ Moir\'e fringes\\[-0.1cm]
478 \& release of Si self-interstitials
484 \begin{minipage}{4.0cm}
486 \includegraphics[width=3.3cm]{sic_prec_seq_01.eps}
490 \begin{minipage}{4.1cm}
492 \includegraphics[width=3.3cm]{sic_prec_seq_02.eps}
496 \begin{minipage}{4.0cm}
498 \includegraphics[width=3.3cm]{sic_prec_seq_03.eps}
502 \begin{pspicture}(0,0)(0,0)
503 \psline[linewidth=2pt]{->}(8.3,2)(8.8,2)
504 \psellipse[linecolor=blue](11.1,6.0)(0.3,0.5)
505 \rput{-20}{\psellipse[linecolor=blue](3.1,8.2)(0.3,0.5)}
506 \psline[linewidth=2pt]{->}(3.9,2)(4.4,2)
507 \rput(11.8,0.3){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
508 $4a_{\text{Si}}=5a_{\text{SiC}}$
510 \rput(11.5,8){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
511 \hkl(h k l) planes match
513 \rput(8.5,6.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
524 Supposed precipitation mechanism of SiC in Si
532 \begin{minipage}{3.6cm}
534 Si \& SiC lattice structure\\[0.1cm]
535 \includegraphics[width=2.3cm]{sic_unit_cell.eps}
538 \begin{minipage}{1.7cm}
539 \underline{Silicon}\\
540 {\color{yellow}$\bullet$} Si | {\color{gray}$\bullet$} Si\\
541 $a=\unit[5.429]{\\A}$\\
542 $\rho^*_{\text{Si}}=\unit[100]{\%}$
544 \begin{minipage}{1.7cm}
545 \underline{Silicon carbide}\\
546 {\color{yellow}$\bullet$} Si | {\color{gray}$\bullet$} C\\
547 $a=\unit[4.359]{\\A}$\\
548 $\rho^*_{\text{Si}}=\unit[97]{\%}$
554 \begin{minipage}{4.1cm}
556 \includegraphics[width=3.3cm]{tem_c-si-db.eps}
560 \begin{minipage}{4.0cm}
562 \includegraphics[width=3.3cm]{tem_3c-sic.eps}
568 \begin{minipage}{4.0cm}
570 C-Si dimers (dumbbells)\\[-0.1cm]
571 on Si interstitial sites
575 \begin{minipage}{4.1cm}
577 Agglomeration of C-Si dumbbells\\[-0.1cm]
578 $\Rightarrow$ dark contrasts
582 \begin{minipage}{4.0cm}
584 Precipitation of 3C-SiC in Si\\[-0.1cm]
585 $\Rightarrow$ Moir\'e fringes\\[-0.1cm]
586 \& release of Si self-interstitials
592 \begin{minipage}{4.0cm}
594 \includegraphics[width=3.3cm]{sic_prec_seq_01.eps}
598 \begin{minipage}{4.1cm}
600 \includegraphics[width=3.3cm]{sic_prec_seq_02.eps}
604 \begin{minipage}{4.0cm}
606 \includegraphics[width=3.3cm]{sic_prec_seq_03.eps}
610 \begin{pspicture}(0,0)(0,0)
611 \psline[linewidth=2pt]{->}(8.3,2)(8.8,2)
612 \psellipse[linecolor=blue](11.1,6.0)(0.3,0.5)
613 \rput{-20}{\psellipse[linecolor=blue](3.1,8.2)(0.3,0.5)}
614 \psline[linewidth=2pt]{->}(3.9,2)(4.4,2)
615 \rput(11.8,0.3){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
616 $4a_{\text{Si}}=5a_{\text{SiC}}$
618 \rput(11.5,8){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
619 \hkl(h k l) planes match
621 \rput(8.5,6.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
624 % controversial view!
625 \rput(6.5,5.0){\psframebox[fillstyle=solid,opacity=0.5,fillcolor=black]{
626 \begin{minipage}{14cm}
631 \rput(6.5,5.3){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=white,linewidth=0.1cm]{
632 \begin{minipage}{10cm}
636 {\color{gray}\bf Controversial findings}
639 \item High-temperature implantation {\tiny\color{gray}/Nejim~et~al./}
641 \item {\color{blue}Substitutionally} incorporated C on regular Si lattice sites
642 \item \si{} reacting with further C in cleared volume
644 \item Annealing behavior {\tiny\color{gray}/Serre~et~al./}
646 \item Room temperature implantation $\rightarrow$ high C diffusion
647 \item Elevated temperature implantation $\rightarrow$ no C redistribution
649 $\Rightarrow$ mobile {\color{red}\ci} opposed to
650 stable {\color{blue}\cs{}} configurations
651 \item Strained silicon \& Si$_{1-y}$C$_y$ heterostructures
652 {\tiny\color{gray}/Strane~et~al./Guedj~et~al./}
654 \item Initial {\color{blue}coherent} SiC precipitates (tensile strain)
655 \item Incoherent SiC (strain relaxation)
660 {\Huge${\lightning}$} \hspace{0.3cm}
661 {\color{blue}\cs{}} --- vs --- {\color{red}\ci} \hspace{0.3cm}
662 {\Huge${\lightning}$}
682 \item Introduction / Motivation
683 \item Assumed SiC precipitation mechanisms / Controversy
685 \item Utilized simulation techniques
687 \item Molecular dynamics (MD) simulations
688 \item Density functional theory (DFT) calculations
690 \item Simulation results
692 \item C and Si self-interstitial point defects in silicon
693 \item Silicon carbide precipitation simulations
695 \item Summary / Conclusion
704 Utilized computational methods
711 {\bf Molecular dynamics (MD)}\\[0.1cm]
713 \begin{tabular}{| p{4.5cm} | p{7.5cm} |}
715 System of $N$ particles &
716 $N=5832\pm 1$ (Defects), $N=238328+6000$ (Precipitation)\\
717 Phase space propagation &
718 Velocity Verlet | timestep: \unit[1]{fs} \\
719 Analytical interaction potential &
720 Tersoff-like {\color{red}short-range}, {\color{blue}bond order} potential
723 E = \frac{1}{2} \sum_{i \neq j} \pot_{ij}, \quad
724 \pot_{ij} = {\color{red}f_C(r_{ij})}
725 \left[ f_R(r_{ij}) + {\color{blue}b_{ij}} f_A(r_{ij}) \right]
727 Observables: time/ensemble averages &
728 NpT (isothermal-isobaric) | Berendsen thermostat/barostat\\
736 {\bf Density functional theory (DFT)}
740 \begin{minipage}[t]{6cm}
742 \item Hohenberg-Kohn theorem:\\
743 $\Psi_0(r_1,r_2,\ldots,r_N)=\Psi[n_0(r)]$, $E_0=E[n_0]$
744 \item Kohn-Sham approach:\\
745 Single-particle effective theory
749 \item Code: \textsc{vasp}
750 \item Plane wave basis set | $E_{\text{cut}}=\unit[300]{eV}$
752 %\Phi_i=\sum_{|G+k|<G_{\text{cut}}} c_{i,k+G} \exp{\left(i(k+G)r\right)}
755 %E_{\text{cut}}=\frac{\hbar^2}{2m}G^2_{\text{cut}}=\unit[300]{eV}
757 \item Ultrasoft pseudopotential
758 \item Exchange \& correlation: GGA
759 \item Brillouin zone sampling: $\Gamma$-point
760 \item Supercell: $N=216\pm2$
763 \begin{minipage}{6cm}
764 \begin{pspicture}(0,0)(0,0)
765 \pscircle[fillcolor=yellow,fillstyle=solid,linestyle=none](3.5,-2.0){2.5}
766 \rput(2.7,-0.7){\psframebox[fillstyle=solid,opacity=0.8,fillcolor=white]{
768 \left[ -\frac{\hbar^2}{2m}\nabla^2 + V_{\text{eff}}(r) - \epsilon_i \right] \Phi_i(r) = 0
771 \rput(5.2,-2.0){\psframebox[fillstyle=solid,opacity=0.8,fillcolor=white]{
773 n(r)=\sum_i^N|\Phi_i(r)|^2
776 \rput(3.0,-4.5){\psframebox[fillstyle=solid,opacity=0.8,fillcolor=white]{
778 V_{\text{eff}}(r)=V_{\text{ext}}(r)+\int\frac{e^2 n(r')}{|r-r'|}d^3r'
782 \psarcn[linewidth=0.07cm,linestyle=dashed]{->}(3.5,-2.0){2.5}{130}{15}
783 \psarcn[linewidth=0.07cm,linestyle=dashed]{->}(3.5,-2.0){2.5}{230}{165}
784 \psarcn[linewidth=0.07cm,linestyle=dashed]{->}(3.5,-2.0){2.5}{345}{310}
795 Point defects \& defect migration
802 \begin{minipage}[b]{7.5cm}
803 {\bf Defect structure}\\
804 \begin{pspicture}(0,0)(7,4.4)
805 \rput(3.5,3.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
808 \item Creation of c-Si simulation volume
809 \item Periodic boundary conditions
810 \item $T=0\text{ K}$, $p=0\text{ bar}$
813 \rput(3.5,1.3){\rnode{insert}{\psframebox{
816 Insertion of interstitial C/Si atoms
819 \rput(3.5,0.2){\rnode{cool}{\psframebox[fillstyle=solid,fillcolor=lbb]{
822 Relaxation / structural energy minimization
825 \ncline[]{->}{init}{insert}
826 \ncline[]{->}{insert}{cool}
829 \begin{minipage}[b]{4.5cm}
831 \includegraphics[width=3.8cm]{unit_cell_e.eps}\\
833 \begin{minipage}{2.21cm}
835 {\color{red}$\bullet$} Tetrahedral\\[-0.1cm]
836 {\color{green}$\bullet$} Hexagonal\\[-0.1cm]
837 {\color{yellow}$\bullet$} \hkl<1 0 0> DB
840 \begin{minipage}{2.21cm}
842 {\color{magenta}$\bullet$} \hkl<1 1 0> DB\\[-0.1cm]
843 {\color{cyan}$\bullet$} Bond-centered\\[-0.1cm]
844 {\color{black}$\bullet$} Vac. / Sub.
851 \begin{minipage}[b]{6cm}
852 {\bf Defect formation energy}\\
854 $E_{\text{f}}=E-\sum_i N_i\mu_i$}\\[0.1cm]
855 Particle reservoir: Si \& SiC\\[0.2cm]
856 {\bf Binding energy}\\
860 E_{\text{f}}^{\text{comb}}-
861 E_{\text{f}}^{1^{\text{st}}}-
862 E_{\text{f}}^{2^{\text{nd}}}
866 $E_{\text{b}}<0$: energetically favorable configuration\\
867 $E_{\text{b}}\rightarrow 0$: non-interacting, isolated defects\\
869 \begin{minipage}[b]{6cm}
870 {\bf Migration barrier}
873 \item Displace diffusing atom
874 \item Constrain relaxation of (diffusing) atoms
875 \item Record configurational energy
877 \begin{picture}(0,0)(-60,-33)
878 \includegraphics[width=4.5cm]{crt_mod.eps}
890 C interstitial point defects in silicon\\
893 \begin{tabular}{l c c c c c c r}
895 $E_{\text{f}}$ [eV] & T & H & \hkl<1 0 0> DB & \hkl<1 1 0> DB & S & B &
896 {\color{black} \cs{} \& \si}\\
898 \textsc{vasp} & unstable & unstable & \underline{3.72} & 4.16 & 1.95 & 4.66 & {\color{green}4.17}\\
899 Erhart/Albe & 6.09 & 9.05$^*$ & \underline{3.88} & 5.18 & {\color{red}0.75} & 5.59$^*$ & {\color{green}4.43} \\
901 \end{tabular}\\[0.1cm]
904 \begin{minipage}{2.8cm}
905 \underline{Hexagonal} \hspace{2pt}
906 \href{../video/c_in_si_int_hexa.avi}{$\rhd$}\\
907 $E_{\text{f}}^*=9.05\text{ eV}$\\
908 \includegraphics[width=2.8cm]{c_pd_albe/hex_bonds.eps}
910 \begin{minipage}{0.4cm}
915 \begin{minipage}{2.8cm}
916 \underline{\hkl<1 0 0>}\\
917 $E_{\text{f}}=3.88\text{ eV}$\\
918 \includegraphics[width=2.8cm]{c_pd_albe/100_bonds.eps}
921 \begin{minipage}{1.4cm}
924 \begin{minipage}{3.0cm}
926 \underline{Tetrahedral}\\
927 \includegraphics[width=3.0cm]{c_pd_albe/tet_bonds.eps}
932 \begin{minipage}{2.8cm}
933 \underline{Bond-centered}\\
934 $E_{\text{f}}^*=5.59\text{ eV}$\\
935 \includegraphics[width=2.8cm]{c_pd_albe/bc_bonds.eps}
937 \begin{minipage}{0.4cm}
942 \begin{minipage}{2.8cm}
943 \underline{\hkl<1 1 0> dumbbell}\\
944 $E_{\text{f}}=5.18\text{ eV}$\\
945 \includegraphics[width=2.8cm]{c_pd_albe/110_bonds.eps}
948 \begin{minipage}{1.4cm}
951 \begin{minipage}{3.0cm}
953 \underline{Substitutional}\\
954 \includegraphics[width=3.0cm]{c_pd_albe/sub_bonds.eps}
964 C interstitial migration --- ab initio
971 \begin{minipage}{6.8cm}
972 \framebox{\hkl[0 0 -1] $\rightarrow$ \hkl[0 0 1]}\\
973 \begin{minipage}{2.0cm}
974 \includegraphics[width=2.0cm]{c_pd_vasp/100_2333.eps}
976 \begin{minipage}{0.2cm}
979 \begin{minipage}{2.0cm}
980 \includegraphics[width=2.0cm]{c_pd_vasp/bc_2333.eps}
982 \begin{minipage}{0.2cm}
985 \begin{minipage}{2.0cm}
986 \includegraphics[width=2.0cm]{c_pd_vasp/100_next_2333.eps}
987 \end{minipage}\\[0.1cm]
989 $\Rightarrow$ Sufficient to consider \hkl[00-1] to BC transition\\
990 $\Rightarrow$ Migration barrier to reach BC | $\Delta E=\unit[1.2]{eV}$
992 \begin{minipage}{5.4cm}
993 \includegraphics[width=6.0cm]{im_00-1_nosym_sp_fullct_thesis_vasp_s.ps}
994 %\end{minipage}\\[0.2cm]
995 \end{minipage}\\[0.4cm]
998 \begin{minipage}{6.8cm}
999 \framebox{\hkl[0 0 -1] $\rightarrow$ \hkl[0 -1 0]}\\
1000 \begin{minipage}{2.0cm}
1001 \includegraphics[width=2.0cm]{c_pd_vasp/100_2333.eps}
1003 \begin{minipage}{0.2cm}
1006 \begin{minipage}{2.0cm}
1007 \includegraphics[width=2.0cm]{c_pd_vasp/00-1-0-10_2333.eps}
1009 \begin{minipage}{0.2cm}
1012 \begin{minipage}{2.0cm}
1013 \includegraphics[width=2.0cm]{c_pd_vasp/0-10_2333.eps}
1014 \end{minipage}\\[0.1cm]
1015 $\Delta E=\unit[0.9]{eV}$ | Experimental values: \unit[0.70--0.87]{eV}\\
1016 $\Rightarrow$ {\color{red}Migration mechanism identified!}\\
1017 Note: Change in orientation
1019 \begin{minipage}{5.4cm}
1020 \includegraphics[width=6.0cm]{00-1_0-10_vasp_s.ps}
1021 \end{minipage}\\[0.1cm]
1024 %Reorientation pathway composed of two consecutive processes of the above type
1033 C interstitial migration --- analytical potential
1040 \begin{minipage}[t]{6.0cm}
1041 {\bf\boldmath BC to \hkl[0 0 -1] transition}\\[0.2cm]
1042 \includegraphics[width=6.0cm]{bc_00-1_albe_s.ps}\\
1044 \item Lowermost migration barrier
1045 \item $\Delta E \approx \unit[2.2]{eV}$
1046 \item 2.4 times higher than ab initio result
1047 \item Different pathway
1050 \begin{minipage}[t]{0.2cm}
1053 \begin{minipage}[t]{6.0cm}
1054 {\bf\boldmath Transition involving a \hkl<1 1 0> configuration}
1057 \item Bond-centered configuration unstable\\
1058 $\rightarrow$ \ci{} \hkl<1 1 0> dumbbell
1059 \item Minima of the \hkl[0 0 -1] to \hkl[0 -1 0] transition\\
1060 $\rightarrow$ \ci{} \hkl<1 1 0> DB
1063 \includegraphics[width=6.0cm]{00-1_110_0-10_mig_albe.ps}
1065 \item $\Delta E \approx \unit[2.2]{eV} \text{ \& } \unit[0.9]{eV}$
1066 \item 2.4 -- 3.4 times higher than ab initio result
1067 \item After all: Change of the DB orientation
1073 {\color{red}\bf Drastically overestimated diffusion barrier}
1076 \begin{pspicture}(0,0)(0,0)
1077 \psline[linewidth=0.05cm,linecolor=gray](6.1,1.0)(6.1,9.3)
1086 Defect combinations --- ab inito
1093 \begin{minipage}{9cm}
1095 Summary of combinations}\\[0.1cm]
1097 \begin{tabular}{l c c c c c c}
1099 $E_{\text{b}}$ [eV] & 1 & 2 & 3 & 4 & 5 & R\\
1101 \hkl[0 0 -1] & {\color{red}-0.08} & -1.15 & {\color{red}-0.08} & 0.04 & -1.66 & -0.19\\
1102 \hkl[0 0 1] & 0.34 & 0.004 & -2.05 & 0.26 & -1.53 & -0.19\\
1103 \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}\\
1104 \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}\\
1105 \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}\\
1106 \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}\\
1108 C$_{\text{sub}}$ & 0.26 & -0.51 & -0.93 & -0.15 & 0.49 & -0.05\\
1109 Vacancy & -5.39 ($\rightarrow$ C$_{\text{sub}}$) & -0.59 & -3.14 & -0.54 & -0.50 & -0.31\\
1116 $E_{\text{b}}$ explainable by stress compensation / increase
1120 \begin{minipage}{3cm}
1121 \includegraphics[width=3.5cm]{comb_pos.eps}
1126 {\bf\boldmath Combinations of \hkl<1 0 0>-type interstitials}\\[0.2cm]
1127 \begin{minipage}[t]{3.2cm}
1128 \underline{\hkl[1 0 0] at position 1}\\[0.1cm]
1129 \includegraphics[width=2.8cm]{00-1dc/2-25.eps}
1131 \begin{minipage}[t]{3.0cm}
1132 \underline{\hkl[0 -1 0] at position 1}\\[0.1cm]
1133 \includegraphics[width=2.8cm]{00-1dc/2-39.eps}
1135 \begin{minipage}[t]{6.1cm}
1138 \item \ci{} agglomeration energetically favorable
1139 \item Most favorable: C clustering\\
1140 {\color{red}However \ldots}\\
1141 \ldots high migration barrier ($>4\,\text{eV}$)\\
1143 $4\times{\color{cyan}[-2.25]}$ versus
1144 $2\times{\color{orange}[-2.39]}$
1147 {\color{blue}\ci{} agglomeration / no C clustering}
1164 \begin{minipage}{9cm}
1166 Summary of combinations}\\[0.1cm]
1168 \begin{tabular}{l c c c c c c}
1170 $E_{\text{b}}$ [eV] & 1 & 2 & 3 & 4 & 5 & R\\
1172 \hkl[0 0 -1] & {\color{red}-0.08} & -1.15 & {\color{red}-0.08} & 0.04 & -1.66 & -0.19\\
1173 \hkl[0 0 1] & 0.34 & 0.004 & -2.05 & 0.26 & -1.53 & -0.19\\
1174 \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}\\
1175 \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}\\
1176 \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}\\
1177 \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}\\
1179 C$_{\text{sub}}$ & 0.26 & -0.51 & -0.93 & -0.15 & 0.49 & -0.05\\
1180 Vacancy & -5.39 ($\rightarrow$ C$_{\text{sub}}$) & -0.59 & -3.14 & -0.54 & -0.50 & -0.31\\
1187 $E_{\text{b}}$ explainable by stress compensation / increase
1191 \begin{minipage}{3cm}
1192 \includegraphics[width=3.5cm]{comb_pos.eps}
1197 {\bf\boldmath Combinations of \hkl<1 0 0>-type interstitials}\\[0.2cm]
1198 \begin{minipage}[t]{3.2cm}
1199 \underline{\hkl[1 0 0] at position 1}\\[0.1cm]
1200 \includegraphics[width=2.8cm]{00-1dc/2-25.eps}
1202 \begin{minipage}[t]{3.0cm}
1203 \underline{\hkl[0 -1 0] at position 1}\\[0.1cm]
1204 \includegraphics[width=2.8cm]{00-1dc/2-39.eps}
1206 \begin{minipage}[t]{6.1cm}
1209 \item \ci{} agglomeration energetically favorable
1210 \item Most favorable: C clustering\\
1211 {\color{red}However \ldots}\\
1212 \ldots high migration barrier ($>4\,\text{eV}$)\\
1214 $4\times{\color{cyan}[-2.25]}$ versus
1215 $2\times{\color{orange}[-2.39]}$
1218 {\color{blue}\ci{} agglomeration / no C clustering}
1223 \begin{pspicture}(0,0)(0,0)
1224 \rput(6.5,5.0){\psframebox[fillstyle=solid,opacity=0.5,fillcolor=black]{
1225 \begin{minipage}{14cm}
1230 \rput(6.5,5.3){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=white,linewidth=0.1cm]{
1231 \begin{minipage}{8cm}
1235 Interaction along \hkl[1 1 0]
1236 \includegraphics[width=7cm]{db_along_110_cc.ps}
1248 Defect combinations of C-Si dimers and vacancies
1254 \begin{minipage}[b]{2.6cm}
1256 \underline{V at 2: $E_{\text{b}}=-0.59\text{ eV}$}\\[0.1cm]
1257 \includegraphics[width=2.5cm]{00-1dc/0-59.eps}
1260 \begin{minipage}[b]{7cm}
1263 \begin{minipage}[b]{2.6cm}
1265 \underline{V at 3, $E_{\text{b}}=-3.14\text{ eV}$}\\[0.1cm]
1266 \includegraphics[width=2.5cm]{00-1dc/3-14.eps}
1268 \end{minipage}\\[0.2cm]
1270 \begin{minipage}{6.5cm}
1271 \includegraphics[width=6.0cm]{059-539.ps}
1273 \begin{minipage}{5.7cm}
1274 \includegraphics[width=6.0cm]{314-539.ps}
1277 \begin{pspicture}(0,0)(0,0)
1278 \psline[linewidth=0.05cm,linecolor=gray](6.3,9.0)(6.3,0)
1280 \rput(6.3,7.0){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=white,linewidth=0.05cm,linecolor=gray]{
1281 \begin{minipage}{6.5cm}
1283 IBS: Impinging C creates V \& far away \si\\[0.3cm]
1284 Low migration barrier towards C$_{\text{sub}}$\\
1286 High barrier for reverse process\\[0.3cm]
1288 High probability of stable C$_{\text{sub}}$ configuration
1301 Combinations of substitutional C and Si self-interstitials
1308 \begin{minipage}{6.2cm}
1310 {\bf\boldmath C$_{\text{sub}}$ - \si{} \hkl<1 1 0> interaction}
1312 \item Most favorable: \cs{} along \hkl<1 1 0> chain of \si{}
1313 \item Less favorable than ground-state \ci{} \hkl<1 0 0> DB
1314 \item Interaction drops quickly to zero\\
1315 $\rightarrow$ low capture radius
1319 \begin{minipage}{0.2cm}
1322 \begin{minipage}{6.0cm}
1324 {\bf Transition from the ground state}
1326 \item Low transition barrier
1327 \item Barrier smaller than \ci{} migration barrier
1328 \item Low \si{} migration barrier (\unit[0.67]{eV})\\
1329 $\rightarrow$ Separation of \cs{} \& \si{} most probable
1332 \end{minipage}\\[0.3cm]
1334 \begin{minipage}{6.0cm}
1335 \includegraphics[width=6.0cm]{c_sub_si110.ps}
1337 \begin{minipage}{0.4cm}
1340 \begin{minipage}{6.0cm}
1342 \includegraphics[width=6.0cm]{162-097.ps}
1346 \begin{pspicture}(0,0)(0,0)
1347 \psline[linewidth=0.05cm,linecolor=gray](6.5,0)(6.5,7.5)
1348 \rput(6.5,-0.7){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=white,linewidth=0.05cm,linecolor=blue]{
1349 \begin{minipage}{8cm}
1353 \cs{} \& \si{} instead of thermodynamic ground state\\[0.1cm]
1354 IBS --- process far from equilibrium\\
1367 Combinations of substitutional C and Si self-interstitials
1374 \begin{minipage}{6.2cm}
1376 {\bf\boldmath C$_{\text{sub}}$ - \si{} \hkl<1 1 0> interaction}
1378 \item Most favorable: \cs{} along \hkl<1 1 0> chain of \si{}
1379 \item Less favorable than ground-state \ci{} \hkl<1 0 0> DB
1380 \item Interaction drops quickly to zero\\
1381 $\rightarrow$ low capture radius
1385 \begin{minipage}{0.2cm}
1388 \begin{minipage}{6.0cm}
1390 {\bf Transition from the ground state}
1392 \item Low transition barrier
1393 \item Barrier smaller than \ci{} migration barrier
1394 \item Low \si{} migration barrier (\unit[0.67]{eV})\\
1395 $\rightarrow$ Separation of \cs{} \& \si{} most probable
1398 \end{minipage}\\[0.3cm]
1400 \begin{minipage}{6.0cm}
1401 \includegraphics[width=6.0cm]{c_sub_si110.ps}
1403 \begin{minipage}{0.4cm}
1406 \begin{minipage}{6.0cm}
1408 \includegraphics[width=6.0cm]{162-097.ps}
1412 \begin{pspicture}(0,0)(0,0)
1413 \psline[linewidth=0.05cm,linecolor=gray](6.5,0)(6.5,7.5)
1414 \rput(6.5,-0.7){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=white,linewidth=0.05cm,linecolor=blue]{
1415 \begin{minipage}{8cm}
1419 \cs{} \& \si{} instead of thermodynamic ground state\\[0.1cm]
1420 IBS --- process far from equilibrium\\
1428 \begin{pspicture}(0,0)(0,0)
1429 \rput(6.5,5.0){\psframebox[fillstyle=solid,opacity=0.5,fillcolor=black]{
1430 \begin{minipage}{14cm}
1435 \rput(6.5,4.3){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=white,linewidth=0.1cm]{
1436 \begin{minipage}{11cm}
1440 Ab initio MD at \degc{900}\\[0.4cm]
1441 \begin{minipage}{5.4cm}
1443 \includegraphics[width=4.3cm]{md01_bonds.eps}\\
1446 \begin{minipage}{5.4cm}
1448 \includegraphics[width=4.3cm]{md02_bonds.eps}\\
1450 \end{minipage}\\[0.5cm]
1452 Contribution of entropy to structural formation\\[0.1cm]
1465 Silicon carbide precipitation simulations
1475 \begin{pspicture}(0,0)(12,6.5)
1477 \rput(3.5,5.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
1480 \item Create c-Si volume
1481 \item Periodc boundary conditions
1482 \item Set requested $T$ and $p=0\text{ bar}$
1483 \item Equilibration of $E_{\text{kin}}$ and $E_{\text{pot}}$
1486 \rput(3.5,2.7){\rnode{insert}{\psframebox[fillstyle=solid,fillcolor=lachs]{
1488 Insertion of C atoms at constant T
1490 \item total simulation volume {\pnode{in1}}
1491 \item volume of minimal SiC precipitate size {\pnode{in2}}
1492 \item volume consisting of Si atoms to form a minimal {\pnode{in3}}\\
1496 \rput(3.5,1){\rnode{cool}{\psframebox[fillstyle=solid,fillcolor=lbb]{
1498 Run for 100 ps followed by cooling down to $20\, ^{\circ}\textrm{C}$
1500 \ncline[]{->}{init}{insert}
1501 \ncline[]{->}{insert}{cool}
1502 \psframe[fillstyle=solid,fillcolor=white](7.3,0.7)(12.8,6.3)
1503 \rput(7.6,6){\footnotesize $V_1$}
1504 \psframe[fillstyle=solid,fillcolor=lightgray](8.7,2)(11.6,5)
1505 \rput(8.9,4.85){\tiny $V_2$}
1506 \psframe[fillstyle=solid,fillcolor=gray](8.95,2.25)(11.35,4.75)
1507 \rput(9.25,4.45){\footnotesize $V_3$}
1508 \rput(7.9,3.2){\pnode{ins1}}
1509 \rput(8.92,2.8){\pnode{ins2}}
1510 \rput(10.8,2.4){\pnode{ins3}}
1511 \ncline[]{->}{in1}{ins1}
1512 \ncline[]{->}{in2}{ins2}
1513 \ncline[]{->}{in3}{ins3}
1523 \begin{minipage}{5.7cm}
1525 \item Amount of C atoms: 6000\\
1526 ($r_{\text{prec}}\approx 3.1\text{ nm}$, IBS: \unit[2--4]{nm})
1527 \item Simulation volume: $31^3$ Si unit cells\\
1531 \begin{minipage}{0.3cm}
1535 \begin{minipage}{6.0cm}
1536 Restricted to classical potential caclulations\\
1537 $\rightarrow$ Low C diffusion / overestimated barrier\\
1538 $\rightarrow$ Consider $V_2$ and $V_3$
1540 % \item $V_2$ and $V_3$ considered due to expected low C diffusion
1551 Silicon carbide precipitation simulations at \degc{450} as in IBS
1556 \begin{minipage}{6.3cm}
1557 \hspace*{-0.4cm}\includegraphics[width=6.5cm]{sic_prec_450_si-c.ps}\\
1558 \hspace*{-0.4cm}\includegraphics[width=6.5cm]{sic_prec_450_si-si_c-c.ps}
1561 \begin{minipage}{6.1cm}
1563 \underline{Low C concentration --- {\color{red}$V_1$}}\\[0.1cm]
1564 \hkl<1 0 0> C-Si dumbbell dominated structure
1566 \item Si-C bumbs around \unit[0.19]{nm}
1567 \item C-C peak at \unit[0.31]{nm} (expected in 3C-SiC):\\
1568 concatenated differently oriented \ci{} DBs
1569 \item Si-Si NN distance stretched to \unit[0.3]{nm}
1571 \begin{pspicture}(0,0)(6.0,1.0)
1572 \rput(3.2,0.5){\psframebox[linewidth=0.03cm,linecolor=blue]{
1573 \begin{minipage}{6cm}
1575 Formation of \ci{} dumbbells\\
1576 C atoms in proper 3C-SiC distance first
1579 \end{pspicture}\\[0.1cm]
1580 \underline{High C concentration --- {\color{green}$V_2$}/{\color{blue}$V_3$}}
1582 \item High amount of strongly bound C-C bonds
1583 \item Increased defect \& damage density\\
1584 $\rightarrow$ Arrangements hard to categorize and trace
1585 \item Only short range order observable
1587 \begin{pspicture}(0,0)(6.0,0.8)
1588 \rput(3.2,0.5){\psframebox[linewidth=0.03cm,linecolor=blue]{
1589 \begin{minipage}{6cm}
1591 Amorphous SiC-like phase
1594 \end{pspicture}\\[0.3cm]
1595 \begin{pspicture}(0,0)(6.0,2.0)
1596 \rput(3.2,1.0){\psframebox[linewidth=0.05cm,linecolor=white]{
1597 \begin{minipage}{6cm}
1611 Silicon carbide precipitation simulations at \degc{450} as in IBS
1616 \begin{minipage}{6.3cm}
1617 \hspace*{-0.4cm}\includegraphics[width=6.5cm]{sic_prec_450_si-c.ps}\\
1618 \hspace*{-0.4cm}\includegraphics[width=6.5cm]{sic_prec_450_si-si_c-c.ps}
1621 \begin{minipage}{6.1cm}
1623 \underline{Low C concentration --- {\color{red}$V_1$}}\\[0.1cm]
1624 \hkl<1 0 0> C-Si dumbbell dominated structure
1626 \item Si-C bumbs around \unit[0.19]{nm}
1627 \item C-C peak at \unit[0.31]{nm} (expected in 3C-SiC):\\
1628 concatenated differently oriented \ci{} DBs
1629 \item Si-Si NN distance stretched to \unit[0.3]{nm}
1631 \begin{pspicture}(0,0)(6.0,1.0)
1632 \rput(3.2,0.5){\psframebox[linewidth=0.03cm,linecolor=blue]{
1633 \begin{minipage}{6cm}
1635 Formation of \ci{} dumbbells\\
1636 C atoms in proper 3C-SiC distance first
1639 \end{pspicture}\\[0.1cm]
1640 \underline{High C concentration --- {\color{green}$V_2$}/{\color{blue}$V_3$}}
1642 \item High amount of strongly bound C-C bonds
1643 \item Increased defect \& damage density\\
1644 $\rightarrow$ Arrangements hard to categorize and trace
1645 \item Only short range order observable
1647 \begin{pspicture}(0,0)(6.0,0.8)
1648 \rput(3.2,0.5){\psframebox[linewidth=0.03cm,linecolor=blue]{
1649 \begin{minipage}{6cm}
1651 Amorphous SiC-like phase
1654 \end{pspicture}\\[0.3cm]
1655 \begin{pspicture}(0,0)(6.0,2.0)
1656 \rput(3.2,1.0){\psframebox[linewidth=0.05cm,linecolor=black]{
1657 \begin{minipage}{6cm}
1660 {\bf\color{red}3C-SiC formation fails to appear}\\[0.3cm]
1661 \begin{minipage}{0.8cm}
1662 {\bf\boldmath $V_1$:}
1664 \begin{minipage}{5.1cm}
1665 Formation of \ci{} indeed occurs\\
1666 Agllomeration not observed
1667 \end{minipage}\\[0.3cm]
1668 \begin{minipage}{0.8cm}
1669 {\bf\boldmath $V_{2,3}$:}
1671 \begin{minipage}{5.1cm}
1672 Amorphous SiC-like structure\\
1673 (not expected at \degc{450})\\[0.05cm]
1674 No rearrangement/transition into 3C-SiC
1675 \end{minipage}\\[0.1cm]
1687 Limitations of MD and short range potentials
1694 {\bf Time scale problem of MD}\\[0.2cm]
1695 Precise integration \& thermodynamic sampling\\
1696 $\Rightarrow$ $\Delta t \ll \left( \max{\omega} \right)^{-1}$,
1697 $\omega$: vibrational mode\\
1698 $\Rightarrow$ {\color{red}\underline{Slow}} phase space propagation\\[0.2cm]
1699 Several local minima separated by large energy barriers\\
1700 $\Rightarrow$ Transition event corresponds to a multiple
1701 of vibrational periods\\
1702 $\Rightarrow$ Phase transition consists of {\color{red}\underline{many}}
1703 infrequent transition events\\[0.2cm]
1704 {\color{blue}Accelerated methods:}
1705 \underline{Temperature accelerated} MD (TAD), self-guided MD \ldots
1709 {\bf Limitations related to the short range potential}\\[0.2cm]
1710 Cut-off function limits interaction to next neighbours\\
1711 $\Rightarrow$ Overestimated diffusion barrier (factor: 2.4--3.4)
1715 {\bf Approach to the (twofold) problem}\\[0.2cm]
1716 Increased temperature simulations without TAD corrections\\
1717 Accelerated methods or higher time scales exclusively not sufficient!
1719 \begin{pspicture}(0,0)(0,0)
1720 \rput(4.0,2.8){\psframebox[linewidth=0.07cm,linecolor=red]{
1721 \begin{minipage}{7.5cm}
1724 Potential enhanced slow phase space propagation
1727 \rput(11.3,7.5){\psframebox[linewidth=0.03cm,linecolor=blue]{
1728 \begin{minipage}{2.7cm}
1732 thermodynamic sampling
1735 \psline[linewidth=0.03cm,linecolor=blue]{<-}(11.3,7.0)(11.0,5.7)
1736 \rput(10.85,2.6){\psframebox[linewidth=0.03cm,linecolor=blue]{
1737 \begin{minipage}{3.6cm}
1740 \underline{IBS}\\[0.1cm]
1741 3C-SiC also observed for higher T\\[0.1cm]
1742 Higher T inside sample\\[0.1cm]
1743 Structural evolution vs.\\
1744 equilibrium properties
1747 \psline[linewidth=0.03cm,linecolor=blue]{->}(10.85,1.75)(9.0,1.0)
1756 Increased temperature simulations --- $V_1$
1761 \begin{minipage}{6.2cm}
1762 \hspace*{-0.4cm}\includegraphics[width=6.5cm]{tot_pc_thesis.ps}
1765 \begin{minipage}{6.2cm}
1766 \includegraphics[width=6.5cm]{tot_pc3_thesis.ps}
1769 \begin{minipage}{6.2cm}
1770 \hspace*{-0.4cm}\includegraphics[width=6.5cm]{tot_pc2_thesis.ps}
1773 \begin{minipage}{6.3cm}
1775 \underline{Si-C bonds:}
1777 \item Vanishing cut-off artifact (above $1650\,^{\circ}\mathrm{C}$)
1778 \item Structural change: C-Si \hkl<1 0 0> $\rightarrow$ C$_{\text{sub}}$
1780 \underline{Si-Si bonds:}
1781 {\color{blue}Si-C$_{\text{sub}}$-Si} along \hkl<1 1 0>
1782 ($\rightarrow$ 0.325 nm)\\[0.1cm]
1783 \underline{C-C bonds:}
1785 \item C-C next neighbour pairs reduced (mandatory)
1786 \item Peak at 0.3 nm slightly shifted
1788 \item C-Si \hkl<1 0 0> combinations (dashed arrows)\\
1789 $\rightarrow$ C-Si \hkl<1 0 0> \& C$_{\text{sub}}$
1791 $\rightarrow$ pure {\color{blue}C$_{\text{sub}}$ combinations}
1793 \item Range [|-$\downarrow$]:
1794 {\color{blue}C$_{\text{sub}}$ \& C$_{\text{sub}}$
1795 with nearby Si$_{\text{I}}$}
1806 Increased temperature simulations --- $V_1$
1811 \begin{minipage}{6.2cm}
1812 \hspace*{-0.4cm}\includegraphics[width=6.5cm]{tot_pc_thesis.ps}
1815 \begin{minipage}{6.2cm}
1816 \includegraphics[width=6.5cm]{tot_pc3_thesis.ps}
1819 \begin{minipage}{6.2cm}
1820 \hspace*{-0.4cm}\includegraphics[width=6.5cm]{tot_pc2_thesis.ps}
1823 \begin{minipage}{6.3cm}
1825 \underline{Si-C bonds:}
1827 \item Vanishing cut-off artifact (above $1650\,^{\circ}\mathrm{C}$)
1828 \item Structural change: C-Si \hkl<1 0 0> $\rightarrow$ C$_{\text{sub}}$
1830 \underline{Si-Si bonds:}
1831 {\color{blue}Si-C$_{\text{sub}}$-Si} along \hkl<1 1 0>
1832 ($\rightarrow$ 0.325 nm)\\[0.1cm]
1833 \underline{C-C bonds:}
1835 \item C-C next neighbour pairs reduced (mandatory)
1836 \item Peak at 0.3 nm slightly shifted
1838 \item C-Si \hkl<1 0 0> combinations (dashed arrows)\\
1839 $\rightarrow$ C-Si \hkl<1 0 0> \& C$_{\text{sub}}$
1841 $\rightarrow$ pure {\color{blue}C$_{\text{sub}}$ combinations}
1843 \item Range [|-$\downarrow$]:
1844 {\color{blue}C$_{\text{sub}}$ \& C$_{\text{sub}}$
1845 with nearby Si$_{\text{I}}$}
1851 \begin{pspicture}(0,0)(0,0)
1852 \rput(6.5,5.0){\psframebox[fillstyle=solid,opacity=0.5,fillcolor=black]{
1853 \begin{minipage}{14cm}
1858 \rput(6.5,5.0){\psframebox[fillstyle=solid,fillcolor=white,linewidth=0.1cm]{
1859 \begin{minipage}{9cm}
1863 {\color{gray}\bf Conclusions on SiC precipitation}\\[0.1cm]
1864 {\Huge$\lightning$} {\color{red}\ci{}} --- vs --- {\color{blue}\cs{}} {\Huge$\lightning$}\\
1867 \item Stretched coherent SiC structures\\
1868 $\Rightarrow$ Precipitation process involves {\color{blue}\cs}
1871 \item Vehicle to rearrange \cs --- [\cs{} \& \si{} $\leftrightarrow$ \ci]
1872 \item Building block for surrounding Si host \& further SiC
1873 \item Strain compensation \ldots\\
1874 \ldots Si/SiC interface\\
1875 \ldots within stretched coherent SiC structure
1877 \item Explains annealing behavior of high/low T C implantations
1879 \item Low T: highly mobile {\color{red}\ci}
1880 \item High T: stable configurations of {\color{blue}\cs}
1885 \psframebox[linecolor=blue,linewidth=0.05cm]{
1886 \begin{minipage}{7cm}
1888 Precipitation mechanism involving \cs\\
1889 High T $\leftrightarrow$ IBS conditions far from equilibrium\\
1899 % skip high c conc results
1905 Increased temperature simulations at high C concentration
1910 \begin{minipage}{6.0cm}
1911 \includegraphics[width=6.4cm]{12_pc_thesis.ps}
1913 \begin{minipage}{6.0cm}
1914 \includegraphics[width=6.4cm]{12_pc_c_thesis.ps}
1922 \begin{minipage}[t]{6.0cm}
1923 0.186 nm: Si-C pairs $\uparrow$\\
1924 (as expected in 3C-SiC)\\[0.2cm]
1925 0.282 nm: Si-C-C\\[0.2cm]
1926 $\approx$0.35 nm: C-Si-Si
1929 \begin{minipage}{0.2cm}
1933 \begin{minipage}[t]{6.0cm}
1934 0.15 nm: C-C pairs $\uparrow$\\
1935 (as expected in graphite/diamond)\\[0.2cm]
1936 0.252 nm: C-C-C (2$^{\text{nd}}$ NN for diamond)\\[0.2cm]
1937 0.31 nm: shifted towards 0.317 nm $\rightarrow$ C-Si-C
1942 \item Decreasing cut-off artifact
1943 \item {\color{red}Amorphous} SiC-like phase remains
1944 \item High amount of {\color{red}damage} \& alignement to c-Si host matrix lost
1945 \item Slightly sharper peaks $\Rightarrow$ indicate slight {\color{blue}acceleration of dynamics} due to temperature
1954 High C \& small $V$ \& short $t$
1957 Slow restructuring due to strong C-C bonds
1960 High C \& low T implants
1978 Summary and Conclusions
1986 \begin{minipage}{12.3cm}
1991 \item Point defects excellently / fairly well described
1993 \item Identified \ci{} migration path
1994 \item EA drastically overestimates the diffusion barrier
1996 \item Combinations of defects
1998 \item Agglomeration of point defects energetically favorable
1999 \item C$_{\text{sub}}$ favored conditions (conceivable in IBS)
2000 \item \ci{} \hkl<1 0 0> $\leftrightarrow$ \cs{} \& \si{} \hkl<1 1 0>\\
2001 Low barrier (\unit[0.77]{eV}) \& low capture radius
2008 \begin{minipage}[t]{12.3cm}
2009 \underline{Pecipitation simulations}
2011 \item Problem of potential enhanced slow phase space propagation
2012 \item Low T $\rightarrow$ C-Si \hkl<1 0 0> dumbbell dominated structure
2013 \item High T $\rightarrow$ C$_{\text{sub}}$ dominated structure
2014 \item High T necessary to simulate IBS conditions (far from equilibrium)
2015 \item \cs{} involved in the precipitation process at elevated temperatures
2016 \item \si{}: vehicle to form \cs{} \& supply of Si \& stress compensation
2017 (stretched SiC, interface)
2024 \framebox{IBS: 3C-SiC precipitation occurs by successive agglomeration of \cs{}}
2043 \underline{Augsburg}
2045 \item Prof. B. Stritzker
2049 \underline{Helsinki}
2051 \item Prof. K. Nordlund
2056 \item Bayerische Forschungsstiftung
2059 \underline{Paderborn}
2061 \item Prof. J. Lindner
2062 \item Prof. G. Schmidt
2070 \normalsize\bf Thank you for your attention!
2080 Polytypes of SiC\\[0.6cm]
2085 \includegraphics[width=3.8cm]{cubic_hex.eps}\\
2086 \begin{minipage}{1.9cm}
2087 {\tiny cubic (twist)}
2089 \begin{minipage}{2.9cm}
2090 {\tiny hexagonal (no twist)}
2093 \begin{picture}(0,0)(-150,0)
2094 \includegraphics[width=7cm]{polytypes.eps}
2101 \begin{tabular}{l c c c c c c}
2103 & 3C-SiC & 4H-SiC & 6H-SiC & Si & GaN & Diamond\\
2105 Hardness [Mohs] & \multicolumn{3}{c}{------ 9.6 ------}& 6.5 & - & 10 \\
2106 Band gap [eV] & 2.36 & 3.23 & 3.03 & 1.12 & 3.39 & 5.5 \\
2107 Break down field [$10^6$ V/cm] & 4 & 3 & 3.2 & 0.6 & 5 & 10 \\
2108 Saturation drift velocity [$10^7$ cm/s] & 2.5 & 2.0 & 2.0 & 1 & 2.7 & 2.7 \\
2109 Electron mobility [cm$^2$/Vs] & 800 & 900 & 400 & 1100 & 900 & 2200 \\
2110 Hole mobility [cm$^2$/Vs] & 320 & 120 & 90 & 420 & 150 & 1600 \\
2111 Thermal conductivity [W/cmK] & 5.0 & 4.9 & 4.9 & 1.5 & 1.3 & 22 \\
2115 \begin{pspicture}(0,0)(0,0)
2116 \psellipse[linecolor=green](5.7,2.05)(0.4,0.50)
2118 \begin{pspicture}(0,0)(0,0)
2119 \psellipse[linecolor=green](5.6,0.89)(0.4,0.20)
2121 \begin{pspicture}(0,0)(0,0)
2122 \psellipse[linecolor=red](10.45,0.42)(0.4,0.20)
2133 Si self-interstitial point defects in silicon\\[0.1cm]
2137 \begin{tabular}{l c c c c c}
2139 $E_{\text{f}}$ [eV] & \hkl<1 1 0> DB & H & T & \hkl<1 0 0> DB & V \\
2141 \textsc{vasp} & \underline{3.39} & 3.42 & 3.77 & 4.41 & 3.63 \\
2142 Erhart/Albe & 4.39 & 4.48$^*$ & \underline{3.40} & 5.42 & 3.13 \\
2144 \end{tabular}\\[0.4cm]
2147 \begin{minipage}{3cm}
2149 \underline{Vacancy}\\
2150 \includegraphics[width=2.8cm]{si_pd_albe/vac.eps}
2153 \begin{minipage}{3cm}
2155 \underline{\hkl<1 1 0> DB}\\
2156 \includegraphics[width=2.8cm]{si_pd_albe/110_bonds.eps}
2159 \begin{minipage}{3cm}
2161 \underline{\hkl<1 0 0> DB}\\
2162 \includegraphics[width=2.8cm]{si_pd_albe/100_bonds.eps}
2165 \begin{minipage}{3cm}
2167 \underline{Tetrahedral}\\
2168 \includegraphics[width=2.8cm]{si_pd_albe/tet_bonds.eps}
2172 \underline{Hexagonal} \hspace{2pt}
2173 \href{../video/si_self_int_hexa.avi}{$\rhd$}\\[0.1cm]
2175 \begin{minipage}{2.7cm}
2176 $E_{\text{f}}^*=4.48\text{ eV}$\\
2177 \includegraphics[width=2.7cm]{si_pd_albe/hex_a_bonds.eps}
2179 \begin{minipage}{0.4cm}
2184 \begin{minipage}{2.7cm}
2185 $E_{\text{f}}=3.96\text{ eV}$\\
2186 \includegraphics[width=2.8cm]{si_pd_albe/hex_bonds.eps}
2189 \begin{minipage}{5.5cm}
2191 {\tiny nearly T $\rightarrow$ T}\\
2193 \includegraphics[width=6.0cm]{nhex_tet.ps}
2202 C-Si dimer \& bond-centered interstitial configuration
2209 \begin{minipage}[t]{4.1cm}
2210 {\bf\boldmath C \hkl<1 0 0> DB interstitial}\\[0.1cm]
2211 \begin{minipage}{2.0cm}
2213 \underline{Erhart/Albe}
2214 \includegraphics[width=2.0cm]{c_pd_albe/100_cmp.eps}
2217 \begin{minipage}{2.0cm}
2219 \underline{\textsc{vasp}}
2220 \includegraphics[width=2.0cm]{c_pd_vasp/100_cmp.eps}
2222 \end{minipage}\\[0.2cm]
2223 Si-C-Si bond angle $\rightarrow$ \unit[180]{$^{\circ}$}\\
2224 $\Rightarrow$ $sp$ hybridization\\[0.1cm]
2225 Si-Si-Si bond angle $\rightarrow$ \unit[120]{$^{\circ}$}\\
2226 $\Rightarrow$ $sp^2$ hybridization
2228 \includegraphics[width=3.4cm]{c_pd_vasp/eden.eps}\\[-0.1cm]
2229 {\tiny Charge density isosurface}
2232 \begin{minipage}{0.2cm}
2235 \begin{minipage}[t]{8.1cm}
2237 {\bf Bond-centered interstitial}\\[0.1cm]
2238 \begin{minipage}{4.4cm}
2241 \item Linear Si-C-Si bond
2242 \item Si: one C \& 3 Si neighbours
2243 \item Spin polarized calculations
2244 \item No saddle point!\\
2248 \begin{minipage}{2.7cm}
2249 %\includegraphics[width=2.8cm]{c_pd_vasp/bc_2333.eps}\\
2251 \includegraphics[width=2.8cm]{c_pd_albe/bc_bonds.eps}\\
2256 \begin{minipage}[t]{6.5cm}
2257 \begin{minipage}[t]{1.2cm}
2259 {\tiny sp$^3$}\\[0.8cm]
2260 \underline{${\color{black}\uparrow}$}
2261 \underline{${\color{black}\uparrow}$}
2262 \underline{${\color{black}\uparrow}$}
2263 \underline{${\color{red}\uparrow}$}\\
2266 \begin{minipage}[t]{1.4cm}
2268 {\color{red}M}{\color{blue}O}\\[0.8cm]
2269 \underline{${\color{blue}\uparrow}{\color{white}\downarrow}$}\\
2270 $\sigma_{\text{ab}}$\\[0.5cm]
2271 \underline{${\color{red}\uparrow}{\color{blue}\downarrow}$}\\
2275 \begin{minipage}[t]{1.0cm}
2279 \underline{${\color{white}\uparrow\uparrow}$}
2280 \underline{${\color{white}\uparrow\uparrow}$}\\
2282 \underline{${\color{blue}\uparrow}{\color{blue}\downarrow}$}
2283 \underline{${\color{blue}\uparrow}{\color{blue}\downarrow}$}\\
2287 \begin{minipage}[t]{1.4cm}
2289 {\color{blue}M}{\color{green}O}\\[0.8cm]
2290 \underline{${\color{blue}\uparrow}{\color{white}\downarrow}$}\\
2291 $\sigma_{\text{ab}}$\\[0.5cm]
2292 \underline{${\color{green}\uparrow}{\color{blue}\downarrow}$}\\
2296 \begin{minipage}[t]{1.2cm}
2299 {\tiny sp$^3$}\\[0.8cm]
2300 \underline{${\color{green}\uparrow}$}
2301 \underline{${\color{black}\uparrow}$}
2302 \underline{${\color{black}\uparrow}$}
2303 \underline{${\color{black}\uparrow}$}\\
2311 \begin{minipage}{3.0cm}
2313 \underline{Charge density}\\
2314 {\color{gray}$\bullet$} Spin up\\
2315 {\color{green}$\bullet$} Spin down\\
2316 {\color{blue}$\bullet$} Resulting spin up\\
2317 {\color{yellow}$\bullet$} Si atoms\\
2318 {\color{red}$\bullet$} C atom
2320 \begin{minipage}{3.6cm}
2321 \includegraphics[width=3.8cm]{c_100_mig_vasp/im_spin_diff.eps}
2328 \begin{pspicture}(0,0)(0,0)
2329 \psline[linecolor=gray,linewidth=0.05cm](-7.8,-8.7)(-7.8,0)