2 %\documentclass[landscape,semhelv,draft]{seminar}
3 \documentclass[landscape,semhelv]{seminar}
6 \usepackage[greek,german]{babel}
7 \usepackage[latin1]{inputenc}
8 \usepackage[T1]{fontenc}
14 \usepackage{calc} % Simple computations with LaTeX variables
15 \usepackage{caption} % Improved captions
16 \usepackage{fancybox} % To have several backgrounds
18 \usepackage{fancyhdr} % Headers and footers definitions
19 \usepackage{fancyvrb} % Fancy verbatim environments
20 \usepackage{pstricks} % PSTricks with the standard color package
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
53 %\usepackage{cmbright}
54 %\renewcommand{\familydefault}{\sfdefault}
55 %\usepackage{mathptmx}
59 %\newrgbcolor{hred}{0.9 0.13 0.13}
60 %\newrgbcolor{hblue}{0.13 0.13 0.9}
61 \newrgbcolor{hred}{1.0 0.0 0.0}
62 \newrgbcolor{hblue}{0.0 0.0 1.0}
66 \extraslideheight{10in}
71 % specify width and height
76 \def\slidetopmargin{-0.15cm}
78 \newcommand{\ham}{\mathcal{H}}
79 \newcommand{\pot}{\mathcal{V}}
80 \newcommand{\foo}{\mathcal{U}}
81 \newcommand{\vir}{\mathcal{W}}
84 \renewcommand\labelitemii{{\color{gray}$\bullet$}}
87 \renewcommand{\phi}{\varphi}
90 \newcommand{\RM}[1]{\MakeUppercase{\romannumeral #1{}}}
93 \newrgbcolor{si-yellow}{.6 .6 0}
94 \newrgbcolor{hb}{0.75 0.77 0.89}
95 \newrgbcolor{lbb}{0.75 0.8 0.88}
96 \newrgbcolor{hlbb}{0.825 0.88 0.968}
97 \newrgbcolor{lachs}{1.0 .93 .81}
100 \newcommand{\headphd}{
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
131 A B C D E F G H G F E D C B A
148 Atomistic simulation study on silicon carbide\\[0.2cm]
149 precipitation in silicon\\
156 \textsc{Frank Zirkelbach}
160 Defense of doctor's thesis
169 % no vertical centering
172 % skip for preparation
177 % motivation / properties / applications of silicon carbide
185 \begin{pspicture}(0,0)(13.5,5)
187 \psframe*[linecolor=hb](-0.2,0)(12.9,5)
189 \pspolygon[linecolor=hlbb,fillcolor=hlbb,fillstyle=solid](5.2,1)(6.5,1)(6.5,3)(5.2,3)
190 \pspolygon[linecolor=hlbb,fillcolor=hlbb,fillstyle=solid](6.4,0.5)(7.7,2)(7.7,2)(6.4,3.5)
192 \rput[lt](0,4.6){\color{gray}PROPERTIES}
194 \rput[lt](0.3,4){wide band gap}
195 \rput[lt](0.3,3.5){high electric breakdown field}
196 \rput[lt](0.3,3){good electron mobility}
197 \rput[lt](0.3,2.5){high electron saturation drift velocity}
198 \rput[lt](0.3,2){high thermal conductivity}
200 \rput[lt](0.3,1.5){hard and mechanically stable}
201 \rput[lt](0.3,1){chemically inert}
203 \rput[lt](0.3,0.5){radiation hardness}
205 \rput[rt](12.7,4.6){\color{gray}APPLICATIONS}
207 \rput[rt](12.5,3.85){high-temperature, high power}
208 \rput[rt](12.5,3.5){and high-frequency}
209 \rput[rt](12.5,3.15){electronic and optoelectronic devices}
211 \rput[rt](12.5,2.35){material suitable for extreme conditions}
212 \rput[rt](12.5,2){microelectromechanical systems}
213 \rput[rt](12.5,1.65){abrasives, cutting tools, heating elements}
215 \rput[rt](12.5,0.85){first wall reactor material, detectors}
216 \rput[rt](12.5,0.5){and electronic devices for space}
220 \begin{picture}(0,0)(5,-162)
221 \includegraphics[height=2.2cm]{3C_SiC_bs.eps}
223 \begin{picture}(0,0)(-120,-162)
224 \includegraphics[height=2.2cm]{nasa_600c_led.eps}
226 \begin{picture}(0,0)(-270,-162)
227 \includegraphics[height=2.2cm]{6h-sic_3c-sic.eps}
230 \begin{picture}(0,0)(10,65)
231 \includegraphics[height=2.8cm]{sic_switch.eps}
233 %\begin{picture}(0,0)(-243,65)
234 \begin{picture}(0,0)(-110,65)
235 \includegraphics[height=2.8cm]{ise_99.eps}
237 %\begin{picture}(0,0)(-135,65)
238 \begin{picture}(0,0)(-100,65)
239 \includegraphics[height=1.2cm]{infineon_schottky.eps}
241 \begin{picture}(0,0)(-233,65)
242 \includegraphics[height=2.8cm]{solar_car.eps}
252 Polytypes of SiC\\[0.6cm]
257 \includegraphics[width=3.8cm]{cubic_hex.eps}\\
258 \begin{minipage}{1.9cm}
259 {\tiny cubic (twist)}
261 \begin{minipage}{2.9cm}
262 {\tiny hexagonal (no twist)}
265 \begin{picture}(0,0)(-150,0)
266 \includegraphics[width=7cm]{polytypes.eps}
273 \begin{tabular}{l c c c c c c}
275 & 3C-SiC & 4H-SiC & 6H-SiC & Si & GaN & Diamond\\
277 Hardness [Mohs] & \multicolumn{3}{c}{------ 9.6 ------}& 6.5 & - & 10 \\
278 Band gap [eV] & 2.36 & 3.23 & 3.03 & 1.12 & 3.39 & 5.5 \\
279 Break down field [$10^6$ V/cm] & 4 & 3 & 3.2 & 0.6 & 5 & 10 \\
280 Saturation drift velocity [$10^7$ cm/s] & 2.5 & 2.0 & 2.0 & 1 & 2.7 & 2.7 \\
281 Electron mobility [cm$^2$/Vs] & 800 & 900 & 400 & 1100 & 900 & 2200 \\
282 Hole mobility [cm$^2$/Vs] & 320 & 120 & 90 & 420 & 150 & 1600 \\
283 Thermal conductivity [W/cmK] & 5.0 & 4.9 & 4.9 & 1.5 & 1.3 & 22 \\
287 \begin{pspicture}(0,0)(0,0)
288 \psellipse[linecolor=green](5.7,2.10)(0.4,0.5)
290 \begin{pspicture}(0,0)(0,0)
291 \psellipse[linecolor=green](5.6,0.92)(0.4,0.2)
293 \begin{pspicture}(0,0)(0,0)
294 \psellipse[linecolor=red](10.45,0.45)(0.4,0.2)
307 Fabrication of silicon carbide
316 \emph{Silicon carbide --- Born from the stars, perfected on earth.}
322 SiC thin films by MBE \& CVD
324 \item Much progress achieved in homo/heteroepitaxial SiC thin film growth
325 \item \underline{Commercially available} semiconductor power devices based on
326 \underline{\foreignlanguage{greek}{a}-SiC}
327 \item Production of favored \underline{3C-SiC} material
328 \underline{less advanced}
329 \item Quality and size not yet sufficient
331 \begin{picture}(0,0)(-310,-20)
332 \includegraphics[width=2.0cm]{cree.eps}
334 {\color{red}\scriptsize Mismatch in thermal expansion coeefficient
335 and lattice paramater}
339 {\bf Alternative approach}\\
340 Ion beam synthesis (IBS) of burried 3C-SiC layers in Si\hkl(1 0 0)
347 \begin{minipage}{3.15cm}
349 \includegraphics[width=3cm]{imp.eps}\\
355 \begin{minipage}{3.15cm}
357 \includegraphics[width=3cm]{annealing.eps}\\
359 Postannealing at $>$ \degc{1200}
364 \begin{minipage}{5.5cm}
367 No surface bending effects\\
368 $\Rightarrow$ Synthesis of large area SiC films possible
379 IBS of epitaxial single crystalline 3C-SiC
388 \item \underline{Implantation step 1}\\[0.1cm]
389 Almost stoichiometric dose | \unit[180]{keV} | \degc{500}\\
390 $\Rightarrow$ Epitaxial {\color{blue}3C-SiC} layer \&
391 {\color{blue}precipitates}
392 \item \underline{Implantation step 2}\\[0.1cm]
393 Little remaining dose | \unit[180]{keV} | \degc{250}\\
395 Destruction/Amorphization of precipitates at layer interface
396 \item \underline{Annealing}\\[0.1cm]
397 \unit[10]{h} at \degc{1250}\\
398 $\Rightarrow$ Homogeneous 3C-SiC layer with sharp interfaces
402 \begin{minipage}{6.9cm}
403 \includegraphics[width=7cm]{ibs_3c-sic.eps}\\[-0.4cm]
406 XTEM: single crystalline 3C-SiC in Si\hkl(1 0 0)
410 \begin{minipage}{5cm}
411 \begin{pspicture}(0,0)(0,0)
413 \psframebox[fillstyle=solid,fillcolor=white,linecolor=blue,linestyle=solid]{
414 \begin{minipage}{5.3cm}
417 3C-SiC precipitation\\
418 not yet fully understood
422 \renewcommand\labelitemi{$\Rightarrow$}
423 Details of the SiC precipitation
425 \item significant technological progress\\
426 in SiC thin film formation
427 \item perspectives for processes relying\\
428 upon prevention of SiC precipitation
432 \rput(-6.8,5.5){\pnode{h0}}
433 \rput(-3.0,5.5){\pnode{h1}}
434 \ncline[linecolor=blue]{-}{h0}{h1}
435 \ncline[linecolor=blue]{->}{h1}{box}
455 \item Supposed precipitation mechanism of SiC in Si
456 \item Utilized simulation techniques
458 \item Molecular dynamics (MD) simulations
459 \item Density functional theory (DFT) calculations
461 \item C and Si self-interstitial point defects in silicon
462 \item Silicon carbide precipitation simulations
463 \item Summary / Conclusion
472 Supposed precipitation mechanism of SiC in Si
480 \begin{minipage}{3.6cm}
482 Si \& SiC lattice structure\\[0.1cm]
483 \includegraphics[width=2.3cm]{sic_unit_cell.eps}
486 \begin{minipage}{1.7cm}
487 \underline{Silicon}\\
488 {\color{yellow}$\bullet$} Si | {\color{gray}$\bullet$} Si\\
489 $a=\unit[5.429]{\\A}$\\
490 $\rho^*_{\text{Si}}=\unit[100]{\%}$
492 \begin{minipage}{1.7cm}
493 \underline{Silicon carbide}\\
494 {\color{yellow}$\bullet$} Si | {\color{gray}$\bullet$} C\\
495 $a=\unit[4.359]{\\A}$\\
496 $\rho^*_{\text{Si}}=\unit[97]{\%}$
502 \begin{minipage}{4.1cm}
504 \includegraphics[width=3.3cm]{tem_c-si-db.eps}
508 \begin{minipage}{4.0cm}
510 \includegraphics[width=3.3cm]{tem_3c-sic.eps}
516 \begin{minipage}{4.0cm}
518 C-Si dimers (dumbbells)\\[-0.1cm]
519 on Si interstitial sites
523 \begin{minipage}{4.1cm}
525 Agglomeration of C-Si dumbbells\\[-0.1cm]
526 $\Rightarrow$ dark contrasts
530 \begin{minipage}{4.0cm}
532 Precipitation of 3C-SiC in Si\\[-0.1cm]
533 $\Rightarrow$ Moir\'e fringes\\[-0.1cm]
534 \& release of Si self-interstitials
540 \begin{minipage}{4.0cm}
542 \includegraphics[width=3.3cm]{sic_prec_seq_01.eps}
546 \begin{minipage}{4.1cm}
548 \includegraphics[width=3.3cm]{sic_prec_seq_02.eps}
552 \begin{minipage}{4.0cm}
554 \includegraphics[width=3.3cm]{sic_prec_seq_03.eps}
558 \begin{pspicture}(0,0)(0,0)
559 \psline[linewidth=2pt]{->}(8.3,2)(8.8,2)
560 \psellipse[linecolor=blue](11.1,6.0)(0.3,0.5)
561 \rput{-20}{\psellipse[linecolor=blue](3.1,8.2)(0.3,0.5)}
562 \psline[linewidth=2pt]{->}(3.9,2)(4.4,2)
563 \rput(11.8,0.3){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
564 $4a_{\text{Si}}=5a_{\text{SiC}}$
566 \rput(11.5,8){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
567 \hkl(h k l) planes match
569 \rput(8.5,6.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
580 Supposed precipitation mechanism of SiC in Si
588 \begin{minipage}{3.6cm}
590 Si \& SiC lattice structure\\[0.1cm]
591 \includegraphics[width=2.3cm]{sic_unit_cell.eps}
594 \begin{minipage}{1.7cm}
595 \underline{Silicon}\\
596 {\color{yellow}$\bullet$} Si | {\color{gray}$\bullet$} Si\\
597 $a=\unit[5.429]{\\A}$\\
598 $\rho^*_{\text{Si}}=\unit[100]{\%}$
600 \begin{minipage}{1.7cm}
601 \underline{Silicon carbide}\\
602 {\color{yellow}$\bullet$} Si | {\color{gray}$\bullet$} C\\
603 $a=\unit[4.359]{\\A}$\\
604 $\rho^*_{\text{Si}}=\unit[97]{\%}$
610 \begin{minipage}{4.1cm}
612 \includegraphics[width=3.3cm]{tem_c-si-db.eps}
616 \begin{minipage}{4.0cm}
618 \includegraphics[width=3.3cm]{tem_3c-sic.eps}
624 \begin{minipage}{4.0cm}
626 C-Si dimers (dumbbells)\\[-0.1cm]
627 on Si interstitial sites
631 \begin{minipage}{4.1cm}
633 Agglomeration of C-Si dumbbells\\[-0.1cm]
634 $\Rightarrow$ dark contrasts
638 \begin{minipage}{4.0cm}
640 Precipitation of 3C-SiC in Si\\[-0.1cm]
641 $\Rightarrow$ Moir\'e fringes\\[-0.1cm]
642 \& release of Si self-interstitials
648 \begin{minipage}{4.0cm}
650 \includegraphics[width=3.3cm]{sic_prec_seq_01.eps}
654 \begin{minipage}{4.1cm}
656 \includegraphics[width=3.3cm]{sic_prec_seq_02.eps}
660 \begin{minipage}{4.0cm}
662 \includegraphics[width=3.3cm]{sic_prec_seq_03.eps}
666 \begin{pspicture}(0,0)(0,0)
667 \psline[linewidth=2pt]{->}(8.3,2)(8.8,2)
668 \psellipse[linecolor=blue](11.1,6.0)(0.3,0.5)
669 \rput{-20}{\psellipse[linecolor=blue](3.1,8.2)(0.3,0.5)}
670 \psline[linewidth=2pt]{->}(3.9,2)(4.4,2)
671 \rput(11.8,0.3){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
672 $4a_{\text{Si}}=5a_{\text{SiC}}$
674 \rput(11.5,8){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
675 \hkl(h k l) planes match
677 \rput(8.5,6.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
680 % controversial view!
681 \rput(6.5,5.0){\psframebox[fillstyle=solid,opacity=0.5,fillcolor=black]{
682 \begin{minipage}{14cm}
687 \rput(6.5,5.3){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=white,linewidth=0.1cm]{
688 \begin{minipage}{10cm}
692 {\color{gray}\bf Controversial findings}
695 \item High-temperature implantation {\tiny\color{gray}/Nejim~et~al./}
697 \item C incorporated {\color{blue}substitutionally} on regular Si lattice sites
698 \item \si{} reacting with further C in cleared volume
700 \item Annealing behavior {\tiny\color{gray}/Serre~et~al./}
702 \item Room temperature implantation $\rightarrow$ high C diffusion
703 \item Elevated temperature implantation $\rightarrow$ no C redistribution
705 $\Rightarrow$ mobile {\color{red}\ci} opposed to
706 stable {\color{blue}\cs{}} configurations
707 \item Strained silicon \& Si/SiC heterostructures
708 {\tiny\color{gray}/Strane~et~al./Guedj~et~al./}
710 \item {\color{blue}Coherent} SiC precipitates (tensile strain)
711 \item Incoherent SiC (strain relaxation)
716 {\Huge${\lightning}$} \hspace{0.3cm}
717 {\color{blue}\cs{}} --- vs --- {\color{red}\ci} \hspace{0.3cm}
718 {\Huge${\lightning}$}
731 Utilized computational methods
738 {\bf Molecular dynamics (MD)}\\[0.1cm]
740 \begin{tabular}{| p{4.5cm} | p{7.5cm} |}
742 System of $N$ particles &
743 $N=5832\pm 1$ (Defects), $N=238328+6000$ (Precipitation)\\
744 Phase space propagation &
745 Velocity Verlet | timestep: \unit[1]{fs} \\
746 Analytical interaction potential &
747 Tersoff-like {\color{red}short-range}, {\color{blue}bond order} potential
750 E = \frac{1}{2} \sum_{i \neq j} \pot_{ij}, \quad
751 \pot_{ij} = {\color{red}f_C(r_{ij})}
752 \left[ f_R(r_{ij}) + {\color{blue}b_{ij}} f_A(r_{ij}) \right]
754 Observables: time/ensemble averages &
755 NpT (isothermal-isobaric) | Berendsen thermostat/barostat\\
763 {\bf Density functional theory (DFT)}
767 \begin{minipage}[t]{6cm}
769 \item Hohenberg-Kohn theorem:\\
770 $\Psi_0(r_1,r_2,\ldots,r_N)=\Psi[n_0(r)]$, $E_0=E[n_0]$
771 \item Kohn-Sham approach:\\
772 Single-particle effective theory
776 \item Code: \textsc{vasp}
777 \item Plane wave basis set
779 %\Phi_i=\sum_{|G+k|<G_{\text{cut}}} c_{i,k+G} \exp{\left(i(k+G)r\right)}
782 %E_{\text{cut}}=\frac{\hbar^2}{2m}G^2_{\text{cut}}=\unit[300]{eV}
784 \item Ultrasoft pseudopotential
785 \item Exchange \& correlation: GGA
786 \item Brillouin zone sampling: $\Gamma$-point
787 \item Supercell: $N=216\pm2$
790 \begin{minipage}{6cm}
791 \begin{pspicture}(0,0)(0,0)
792 \pscircle[fillcolor=yellow,fillstyle=solid,linestyle=none](3.5,-2.0){2.5}
793 \rput(2.7,-0.7){\psframebox[fillstyle=solid,opacity=0.8,fillcolor=white]{
795 \left[ -\frac{\hbar^2}{2m}\nabla^2 + V_{\text{eff}}(r) - \epsilon_i \right] \Phi_i(r) = 0
798 \rput(5.2,-2.0){\psframebox[fillstyle=solid,opacity=0.8,fillcolor=white]{
800 n(r)=\sum_i^N|\Phi_i(r)|^2
803 \rput(3.0,-4.5){\psframebox[fillstyle=solid,opacity=0.8,fillcolor=white]{
805 V_{\text{eff}}(r)=V_{\text{ext}}(r)+\int\frac{e^2 n(r')}{|r-r'|}d^3r'
809 \psarcn[linewidth=0.07cm,linestyle=dashed]{->}(3.5,-2.0){2.5}{130}{15}
810 \psarcn[linewidth=0.07cm,linestyle=dashed]{->}(3.5,-2.0){2.5}{230}{165}
811 \psarcn[linewidth=0.07cm,linestyle=dashed]{->}(3.5,-2.0){2.5}{345}{310}
822 Point defects \& defect migration
829 \begin{minipage}[b]{7.5cm}
830 {\bf Defect structure}\\
831 \begin{pspicture}(0,0)(7,4.4)
832 \rput(3.5,3.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
835 \item Creation of c-Si simulation volume
836 \item Periodic boundary conditions
837 \item $T=0\text{ K}$, $p=0\text{ bar}$
840 \rput(3.5,1.3){\rnode{insert}{\psframebox{
843 Insertion of interstitial C/Si atoms
846 \rput(3.5,0.2){\rnode{cool}{\psframebox[fillstyle=solid,fillcolor=lbb]{
849 Relaxation / structural energy minimization
852 \ncline[]{->}{init}{insert}
853 \ncline[]{->}{insert}{cool}
856 \begin{minipage}[b]{4.5cm}
858 \includegraphics[width=3.8cm]{unit_cell_e.eps}\\
860 \begin{minipage}{2.21cm}
862 {\color{red}$\bullet$} Tetrahedral\\[-0.1cm]
863 {\color{green}$\bullet$} Hexagonal\\[-0.1cm]
864 {\color{yellow}$\bullet$} \hkl<1 0 0> DB
867 \begin{minipage}{2.21cm}
869 {\color{magenta}$\bullet$} \hkl<1 1 0> DB\\[-0.1cm]
870 {\color{cyan}$\bullet$} Bond-centered\\[-0.1cm]
871 {\color{black}$\bullet$} Vac. / Sub.
878 \begin{minipage}[b]{6cm}
879 {\bf Defect formation energy}\\
881 $E_{\text{f}}=E-\sum_i N_i\mu_i$}\\[0.1cm]
882 Particle reservoir: Si \& SiC\\[0.2cm]
883 {\bf Binding energy}\\
887 E_{\text{f}}^{\text{comb}}-
888 E_{\text{f}}^{1^{\text{st}}}-
889 E_{\text{f}}^{2^{\text{nd}}}
893 $E_{\text{b}}<0$: energetically favorable configuration\\
894 $E_{\text{b}}\rightarrow 0$: non-interacting, isolated defects\\
896 \begin{minipage}[b]{6cm}
897 {\bf Migration barrier}
900 \item Displace diffusing atom
901 \item Constrain relaxation of (diffusing) atoms
902 \item Record configurational energy
904 \begin{picture}(0,0)(-60,-33)
905 \includegraphics[width=4.5cm]{crt_mod.eps}
917 Si self-interstitial point defects in silicon\\[0.1cm]
921 \begin{tabular}{l c c c c c}
923 $E_{\text{f}}$ [eV] & \hkl<1 1 0> DB & H & T & \hkl<1 0 0> DB & V \\
925 \textsc{vasp} & \underline{3.39} & 3.42 & 3.77 & 4.41 & 3.63 \\
926 Erhart/Albe & 4.39 & 4.48$^*$ & \underline{3.40} & 5.42 & 3.13 \\
928 \end{tabular}\\[0.4cm]
931 \begin{minipage}{3cm}
933 \underline{Vacancy}\\
934 \includegraphics[width=2.8cm]{si_pd_albe/vac.eps}
937 \begin{minipage}{3cm}
939 \underline{\hkl<1 1 0> DB}\\
940 \includegraphics[width=2.8cm]{si_pd_albe/110_bonds.eps}
943 \begin{minipage}{3cm}
945 \underline{\hkl<1 0 0> DB}\\
946 \includegraphics[width=2.8cm]{si_pd_albe/100_bonds.eps}
949 \begin{minipage}{3cm}
951 \underline{Tetrahedral}\\
952 \includegraphics[width=2.8cm]{si_pd_albe/tet_bonds.eps}
956 \underline{Hexagonal} \hspace{2pt}
957 \href{../video/si_self_int_hexa.avi}{$\rhd$}\\[0.1cm]
959 \begin{minipage}{2.7cm}
960 $E_{\text{f}}^*=4.48\text{ eV}$\\
961 \includegraphics[width=2.7cm]{si_pd_albe/hex_a_bonds.eps}
963 \begin{minipage}{0.4cm}
968 \begin{minipage}{2.7cm}
969 $E_{\text{f}}=3.96\text{ eV}$\\
970 \includegraphics[width=2.8cm]{si_pd_albe/hex_bonds.eps}
973 \begin{minipage}{5.5cm}
975 {\tiny nearly T $\rightarrow$ T}\\
977 \includegraphics[width=6.0cm]{nhex_tet.ps}
988 C interstitial point defects in silicon\\
991 \begin{tabular}{l c c c c c c r}
993 $E_{\text{f}}$ [eV] & T & H & \hkl<1 0 0> DB & \hkl<1 1 0> DB & S & B &
994 {\color{black} \cs{} \& \si}\\
996 \textsc{vasp} & unstable & unstable & \underline{3.72} & 4.16 & 1.95 & 4.66 & {\color{green}4.17}\\
997 Erhart/Albe & 6.09 & 9.05$^*$ & \underline{3.88} & 5.18 & {\color{red}0.75} & 5.59$^*$ & {\color{green}4.43} \\
999 \end{tabular}\\[0.1cm]
1002 \begin{minipage}{2.8cm}
1003 \underline{Hexagonal} \hspace{2pt}
1004 \href{../video/c_in_si_int_hexa.avi}{$\rhd$}\\
1005 $E_{\text{f}}^*=9.05\text{ eV}$\\
1006 \includegraphics[width=2.8cm]{c_pd_albe/hex_bonds.eps}
1008 \begin{minipage}{0.4cm}
1013 \begin{minipage}{2.8cm}
1014 \underline{\hkl<1 0 0>}\\
1015 $E_{\text{f}}=3.88\text{ eV}$\\
1016 \includegraphics[width=2.8cm]{c_pd_albe/100_bonds.eps}
1019 \begin{minipage}{1.4cm}
1022 \begin{minipage}{3.0cm}
1024 \underline{Tetrahedral}\\
1025 \includegraphics[width=3.0cm]{c_pd_albe/tet_bonds.eps}
1030 \begin{minipage}{2.8cm}
1031 \underline{Bond-centered}\\
1032 $E_{\text{f}}^*=5.59\text{ eV}$\\
1033 \includegraphics[width=2.8cm]{c_pd_albe/bc_bonds.eps}
1035 \begin{minipage}{0.4cm}
1040 \begin{minipage}{2.8cm}
1041 \underline{\hkl<1 1 0> dumbbell}\\
1042 $E_{\text{f}}=5.18\text{ eV}$\\
1043 \includegraphics[width=2.8cm]{c_pd_albe/110_bonds.eps}
1046 \begin{minipage}{1.4cm}
1049 \begin{minipage}{3.0cm}
1051 \underline{Substitutional}\\
1052 \includegraphics[width=3.0cm]{c_pd_albe/sub_bonds.eps}
1062 C-Si dimer \& bond-centered interstitial configuration
1069 \begin{minipage}[t]{4.1cm}
1070 {\bf\boldmath C \hkl<1 0 0> DB interstitial}\\[0.1cm]
1071 \begin{minipage}{2.0cm}
1073 \underline{Erhart/Albe}
1074 \includegraphics[width=2.0cm]{c_pd_albe/100_cmp.eps}
1077 \begin{minipage}{2.0cm}
1079 \underline{\textsc{vasp}}
1080 \includegraphics[width=2.0cm]{c_pd_vasp/100_cmp.eps}
1082 \end{minipage}\\[0.2cm]
1083 Si-C-Si bond angle $\rightarrow$ \unit[180]{$^{\circ}$}\\
1084 $\Rightarrow$ $sp$ hybridization\\[0.1cm]
1085 Si-Si-Si bond angle $\rightarrow$ \unit[120]{$^{\circ}$}\\
1086 $\Rightarrow$ $sp^2$ hybridization
1088 \includegraphics[width=3.4cm]{c_pd_vasp/eden.eps}\\[-0.1cm]
1089 {\tiny Charge density isosurface}
1092 \begin{minipage}{0.2cm}
1095 \begin{minipage}[t]{8.1cm}
1097 {\bf Bond-centered interstitial}\\[0.1cm]
1098 \begin{minipage}{4.4cm}
1101 \item Linear Si-C-Si bond
1102 \item Si: one C \& 3 Si neighbours
1103 \item Spin polarized calculations
1104 \item No saddle point!\\
1108 \begin{minipage}{2.7cm}
1109 %\includegraphics[width=2.8cm]{c_pd_vasp/bc_2333.eps}\\
1111 \includegraphics[width=2.8cm]{c_pd_albe/bc_bonds.eps}\\
1116 \begin{minipage}[t]{6.5cm}
1117 \begin{minipage}[t]{1.2cm}
1119 {\tiny sp$^3$}\\[0.8cm]
1120 \underline{${\color{black}\uparrow}$}
1121 \underline{${\color{black}\uparrow}$}
1122 \underline{${\color{black}\uparrow}$}
1123 \underline{${\color{red}\uparrow}$}\\
1126 \begin{minipage}[t]{1.4cm}
1128 {\color{red}M}{\color{blue}O}\\[0.8cm]
1129 \underline{${\color{blue}\uparrow}{\color{white}\downarrow}$}\\
1130 $\sigma_{\text{ab}}$\\[0.5cm]
1131 \underline{${\color{red}\uparrow}{\color{blue}\downarrow}$}\\
1135 \begin{minipage}[t]{1.0cm}
1139 \underline{${\color{white}\uparrow\uparrow}$}
1140 \underline{${\color{white}\uparrow\uparrow}$}\\
1142 \underline{${\color{blue}\uparrow}{\color{blue}\downarrow}$}
1143 \underline{${\color{blue}\uparrow}{\color{blue}\downarrow}$}\\
1147 \begin{minipage}[t]{1.4cm}
1149 {\color{blue}M}{\color{green}O}\\[0.8cm]
1150 \underline{${\color{blue}\uparrow}{\color{white}\downarrow}$}\\
1151 $\sigma_{\text{ab}}$\\[0.5cm]
1152 \underline{${\color{green}\uparrow}{\color{blue}\downarrow}$}\\
1156 \begin{minipage}[t]{1.2cm}
1159 {\tiny sp$^3$}\\[0.8cm]
1160 \underline{${\color{green}\uparrow}$}
1161 \underline{${\color{black}\uparrow}$}
1162 \underline{${\color{black}\uparrow}$}
1163 \underline{${\color{black}\uparrow}$}\\
1171 \begin{minipage}{3.0cm}
1173 \underline{Charge density}\\
1174 {\color{gray}$\bullet$} Spin up\\
1175 {\color{green}$\bullet$} Spin down\\
1176 {\color{blue}$\bullet$} Resulting spin up\\
1177 {\color{yellow}$\bullet$} Si atoms\\
1178 {\color{red}$\bullet$} C atom
1180 \begin{minipage}{3.6cm}
1181 \includegraphics[width=3.8cm]{c_100_mig_vasp/im_spin_diff.eps}
1188 \begin{pspicture}(0,0)(0,0)
1189 \psline[linecolor=gray,linewidth=0.05cm](-7.8,-8.7)(-7.8,0)
1198 C interstitial migration --- ab initio
1205 \begin{minipage}{6.8cm}
1206 \framebox{\hkl[0 0 -1] $\rightarrow$ \hkl[0 0 1]}\\
1207 \begin{minipage}{2.0cm}
1208 \includegraphics[width=2.0cm]{c_pd_vasp/100_2333.eps}
1210 \begin{minipage}{0.2cm}
1213 \begin{minipage}{2.0cm}
1214 \includegraphics[width=2.0cm]{c_pd_vasp/bc_2333.eps}
1216 \begin{minipage}{0.2cm}
1219 \begin{minipage}{2.0cm}
1220 \includegraphics[width=2.0cm]{c_pd_vasp/100_next_2333.eps}
1221 \end{minipage}\\[0.1cm]
1223 $\Rightarrow$ BC configuration constitutes local minimum\\
1224 $\Rightarrow$ Migration barrier to reach BC | $\Delta E=\unit[1.2]{eV}$
1226 \begin{minipage}{5.4cm}
1227 \includegraphics[width=6.0cm]{im_00-1_nosym_sp_fullct_thesis_vasp_s.ps}
1228 \end{minipage}\\[0.2cm]
1231 \begin{minipage}{6.8cm}
1232 \framebox{\hkl[0 0 -1] $\rightarrow$ \hkl[0 -1 0]}\\
1233 \begin{minipage}{2.0cm}
1234 \includegraphics[width=2.0cm]{c_pd_vasp/100_2333.eps}
1236 \begin{minipage}{0.2cm}
1239 \begin{minipage}{2.0cm}
1240 \includegraphics[width=2.0cm]{c_pd_vasp/00-1-0-10_2333.eps}
1242 \begin{minipage}{0.2cm}
1245 \begin{minipage}{2.0cm}
1246 \includegraphics[width=2.0cm]{c_pd_vasp/0-10_2333.eps}
1247 \end{minipage}\\[0.1cm]
1248 $\Delta E=\unit[0.9]{eV}$ | Experimental values: \unit[0.70--0.87]{eV}\\
1249 $\Rightarrow$ {\color{red}Migration mechanism identified!}\\
1250 Note: Change in orientation
1252 \begin{minipage}{5.4cm}
1253 \includegraphics[width=6.0cm]{00-1_0-10_vasp_s.ps}
1254 \end{minipage}\\[0.1cm]
1257 Reorientation pathway composed of two consecutive processes of the above type
1266 C interstitial migration --- analytical potential
1273 \begin{minipage}[t]{6.0cm}
1274 {\bf\boldmath BC to \hkl[0 0 -1] transition}\\[0.2cm]
1275 \includegraphics[width=6.0cm]{bc_00-1_albe_s.ps}\\
1277 \item Lowermost migration barrier
1278 \item $\Delta E \approx \unit[2.2]{eV}$
1279 \item 2.4 times higher than ab initio result
1280 \item Different pathway
1283 \begin{minipage}[t]{0.2cm}
1286 \begin{minipage}[t]{6.0cm}
1287 {\bf\boldmath Transition involving a \hkl<1 1 0> configuration}
1290 \item Bond-centered configuration unstable\\
1291 $\rightarrow$ \ci{} \hkl<1 1 0> dumbbell
1292 \item Minima of the \hkl[0 0 -1] to \hkl[0 -1 0] transition\\
1293 $\rightarrow$ \ci{} \hkl<1 1 0> DB
1296 \includegraphics[width=6.0cm]{00-1_110_0-10_mig_albe.ps}
1298 \item $\Delta E \approx \unit[2.2]{eV} \text{ \& } \unit[0.9]{eV}$
1299 \item 2.4 -- 3.4 times higher than ab initio result
1300 \item After all: Change of the DB orientation
1306 {\color{red}\bf Drastically overestimated diffusion barrier}
1309 \begin{pspicture}(0,0)(0,0)
1310 \psline[linewidth=0.05cm,linecolor=gray](6.1,1.0)(6.1,9.3)
1326 \begin{minipage}{9cm}
1328 Summary of combinations}\\[0.1cm]
1330 \begin{tabular}{l c c c c c c}
1332 $E_{\text{b}}$ [eV] & 1 & 2 & 3 & 4 & 5 & R\\
1334 \hkl[0 0 -1] & {\color{red}-0.08} & -1.15 & {\color{red}-0.08} & 0.04 & -1.66 & -0.19\\
1335 \hkl[0 0 1] & 0.34 & 0.004 & -2.05 & 0.26 & -1.53 & -0.19\\
1336 \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}\\
1337 \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}\\
1338 \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}\\
1339 \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}\\
1341 C$_{\text{sub}}$ & 0.26 & -0.51 & -0.93 & -0.15 & 0.49 & -0.05\\
1342 Vacancy & -5.39 ($\rightarrow$ C$_{\text{sub}}$) & -0.59 & -3.14 & -0.54 & -0.50 & -0.31\\
1349 $E_{\text{b}}$ explainable by stress compensation / increase
1353 \begin{minipage}{3cm}
1354 \includegraphics[width=3.5cm]{comb_pos.eps}
1359 {\bf\boldmath Combinations of \hkl<1 0 0>-type interstitials}\\[0.2cm]
1360 \begin{minipage}[t]{3.2cm}
1361 \underline{\hkl[1 0 0] at position 1}\\[0.1cm]
1362 \includegraphics[width=2.8cm]{00-1dc/2-25.eps}
1364 \begin{minipage}[t]{3.0cm}
1365 \underline{\hkl[0 -1 0] at position 1}\\[0.1cm]
1366 \includegraphics[width=2.8cm]{00-1dc/2-39.eps}
1368 \begin{minipage}[t]{6.1cm}
1371 \item \ci{} agglomeration energetically favorable
1372 \item Most favorable: C clustering\\
1373 {\color{red}However \ldots}\\
1374 \ldots high migration barrier ($>4\,\text{eV}$)\\
1376 $4\times{\color{cyan}[-2.25]}$ versus
1377 $2\times{\color{orange}[-2.39]}$
1380 {\color{blue}\ci{} agglomeration / no C clustering}
1397 \begin{minipage}{9cm}
1399 Summary of combinations}\\[0.1cm]
1401 \begin{tabular}{l c c c c c c}
1403 $E_{\text{b}}$ [eV] & 1 & 2 & 3 & 4 & 5 & R\\
1405 \hkl[0 0 -1] & {\color{red}-0.08} & -1.15 & {\color{red}-0.08} & 0.04 & -1.66 & -0.19\\
1406 \hkl[0 0 1] & 0.34 & 0.004 & -2.05 & 0.26 & -1.53 & -0.19\\
1407 \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}\\
1408 \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}\\
1409 \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}\\
1410 \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}\\
1412 C$_{\text{sub}}$ & 0.26 & -0.51 & -0.93 & -0.15 & 0.49 & -0.05\\
1413 Vacancy & -5.39 ($\rightarrow$ C$_{\text{sub}}$) & -0.59 & -3.14 & -0.54 & -0.50 & -0.31\\
1420 $E_{\text{b}}$ explainable by stress compensation / increase
1424 \begin{minipage}{3cm}
1425 \includegraphics[width=3.5cm]{comb_pos.eps}
1430 {\bf\boldmath Combinations of \hkl<1 0 0>-type interstitials}\\[0.2cm]
1431 \begin{minipage}[t]{3.2cm}
1432 \underline{\hkl[1 0 0] at position 1}\\[0.1cm]
1433 \includegraphics[width=2.8cm]{00-1dc/2-25.eps}
1435 \begin{minipage}[t]{3.0cm}
1436 \underline{\hkl[0 -1 0] at position 1}\\[0.1cm]
1437 \includegraphics[width=2.8cm]{00-1dc/2-39.eps}
1439 \begin{minipage}[t]{6.1cm}
1442 \item \ci{} agglomeration energetically favorable
1443 \item Most favorable: C clustering\\
1444 {\color{red}However \ldots}\\
1445 \ldots high migration barrier ($>4\,\text{eV}$)\\
1447 $4\times{\color{cyan}[-2.25]}$ versus
1448 $2\times{\color{orange}[-2.39]}$
1451 {\color{blue}\ci{} agglomeration / no C clustering}
1456 \begin{pspicture}(0,0)(0,0)
1457 \rput(6.5,5.0){\psframebox[fillstyle=solid,opacity=0.5,fillcolor=black]{
1458 \begin{minipage}{14cm}
1463 \rput(6.5,5.3){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=white,linewidth=0.1cm]{
1464 \begin{minipage}{8cm}
1468 Interaction along \hkl[1 1 0]
1469 \includegraphics[width=7cm]{db_along_110_cc.ps}
1481 Defect combinations of C-Si dimers and vacancies
1487 \begin{minipage}[b]{2.6cm}
1489 \underline{V at 2: $E_{\text{b}}=-0.59\text{ eV}$}\\[0.1cm]
1490 \includegraphics[width=2.5cm]{00-1dc/0-59.eps}
1493 \begin{minipage}[b]{7cm}
1496 \begin{minipage}[b]{2.6cm}
1498 \underline{V at 3, $E_{\text{b}}=-3.14\text{ eV}$}\\[0.1cm]
1499 \includegraphics[width=2.5cm]{00-1dc/3-14.eps}
1501 \end{minipage}\\[0.2cm]
1503 \begin{minipage}{6.5cm}
1504 \includegraphics[width=6.0cm]{059-539.ps}
1506 \begin{minipage}{5.7cm}
1507 \includegraphics[width=6.0cm]{314-539.ps}
1510 \begin{pspicture}(0,0)(0,0)
1511 \psline[linewidth=0.05cm,linecolor=gray](6.3,9.0)(6.3,0)
1513 \rput(6.3,7.0){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=white,linewidth=0.05cm,linecolor=gray]{
1514 \begin{minipage}{6.5cm}
1516 IBS: Impinging C creates V \& far away \si\\[0.3cm]
1517 Low migration barrier towards C$_{\text{sub}}$\\
1519 High barrier for reverse process\\[0.3cm]
1521 High probability of stable C$_{\text{sub}}$ configuration
1534 Combinations of substitutional C and Si self-interstitials
1541 \begin{minipage}{6.2cm}
1543 {\bf\boldmath C$_{\text{sub}}$ - \si{} \hkl<1 1 0> interaction}
1545 \item Most favorable: \cs{} along \hkl<1 1 0> chain of \si{}
1546 \item Less favorable than ground-state \ci{} \hkl<1 0 0> DB
1547 \item Interaction drops quickly to zero\\
1548 $\rightarrow$ low capture radius
1552 \begin{minipage}{0.2cm}
1555 \begin{minipage}{6.0cm}
1557 {\bf Transition from the ground state}
1559 \item Low transition barrier
1560 \item Barrier smaller than \ci{} migration barrier
1561 \item Low \si{} migration barrier (\unit[0.67]{eV})\\
1562 $\rightarrow$ Separation of \cs{} \& \si{} most probable
1565 \end{minipage}\\[0.3cm]
1567 \begin{minipage}{6.0cm}
1568 \includegraphics[width=6.0cm]{c_sub_si110.ps}
1570 \begin{minipage}{0.4cm}
1573 \begin{minipage}{6.0cm}
1575 \includegraphics[width=6.0cm]{162-097.ps}
1579 \begin{pspicture}(0,0)(0,0)
1580 \psline[linewidth=0.05cm,linecolor=gray](6.5,0)(6.5,7.5)
1581 \rput(6.5,-0.7){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=white,linewidth=0.05cm,linecolor=blue]{
1582 \begin{minipage}{8cm}
1586 \cs{} \& \si{} instead of thermodynamic ground state\\[0.1cm]
1587 IBS --- process far from equilibrium\\
1600 Combinations of substitutional C and Si self-interstitials
1607 \begin{minipage}{6.2cm}
1609 {\bf\boldmath C$_{\text{sub}}$ - \si{} \hkl<1 1 0> interaction}
1611 \item Most favorable: \cs{} along \hkl<1 1 0> chain of \si{}
1612 \item Less favorable than ground-state \ci{} \hkl<1 0 0> DB
1613 \item Interaction drops quickly to zero\\
1614 $\rightarrow$ low capture radius
1618 \begin{minipage}{0.2cm}
1621 \begin{minipage}{6.0cm}
1623 {\bf Transition from the ground state}
1625 \item Low transition barrier
1626 \item Barrier smaller than \ci{} migration barrier
1627 \item Low \si{} migration barrier (\unit[0.67]{eV})\\
1628 $\rightarrow$ Separation of \cs{} \& \si{} most probable
1631 \end{minipage}\\[0.3cm]
1633 \begin{minipage}{6.0cm}
1634 \includegraphics[width=6.0cm]{c_sub_si110.ps}
1636 \begin{minipage}{0.4cm}
1639 \begin{minipage}{6.0cm}
1641 \includegraphics[width=6.0cm]{162-097.ps}
1645 \begin{pspicture}(0,0)(0,0)
1646 \psline[linewidth=0.05cm,linecolor=gray](6.5,0)(6.5,7.5)
1647 \rput(6.5,-0.7){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=white,linewidth=0.05cm,linecolor=blue]{
1648 \begin{minipage}{8cm}
1652 \cs{} \& \si{} instead of thermodynamic ground state\\[0.1cm]
1653 IBS --- process far from equilibrium\\
1661 \begin{pspicture}(0,0)(0,0)
1662 \rput(6.5,5.0){\psframebox[fillstyle=solid,opacity=0.5,fillcolor=black]{
1663 \begin{minipage}{14cm}
1668 \rput(6.5,4.3){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=white,linewidth=0.1cm]{
1669 \begin{minipage}{11cm}
1673 Ab initio MD at \degc{900}\\[0.4cm]
1674 \begin{minipage}{5.4cm}
1676 \includegraphics[width=4.3cm]{md01_bonds.eps}\\
1679 \begin{minipage}{5.4cm}
1681 \includegraphics[width=4.3cm]{md02_bonds.eps}\\
1683 \end{minipage}\\[0.5cm]
1685 Contribution of entropy to structural formation\\[0.1cm]
1698 Silicon carbide precipitation simulations
1708 \begin{pspicture}(0,0)(12,6.5)
1710 \rput(3.5,5.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
1713 \item Create c-Si volume
1714 \item Periodc boundary conditions
1715 \item Set requested $T$ and $p=0\text{ bar}$
1716 \item Equilibration of $E_{\text{kin}}$ and $E_{\text{pot}}$
1719 \rput(3.5,2.7){\rnode{insert}{\psframebox[fillstyle=solid,fillcolor=lachs]{
1721 Insertion of C atoms at constant T
1723 \item total simulation volume {\pnode{in1}}
1724 \item volume of minimal SiC precipitate size {\pnode{in2}}
1725 \item volume consisting of Si atoms to form a minimal {\pnode{in3}}\\
1729 \rput(3.5,1){\rnode{cool}{\psframebox[fillstyle=solid,fillcolor=lbb]{
1731 Run for 100 ps followed by cooling down to $20\, ^{\circ}\textrm{C}$
1733 \ncline[]{->}{init}{insert}
1734 \ncline[]{->}{insert}{cool}
1735 \psframe[fillstyle=solid,fillcolor=white](7.3,0.7)(12.8,6.3)
1736 \rput(7.6,6){\footnotesize $V_1$}
1737 \psframe[fillstyle=solid,fillcolor=lightgray](8.7,2)(11.6,5)
1738 \rput(8.9,4.85){\tiny $V_2$}
1739 \psframe[fillstyle=solid,fillcolor=gray](8.95,2.25)(11.35,4.75)
1740 \rput(9.25,4.45){\footnotesize $V_3$}
1741 \rput(7.9,3.2){\pnode{ins1}}
1742 \rput(8.92,2.8){\pnode{ins2}}
1743 \rput(10.8,2.4){\pnode{ins3}}
1744 \ncline[]{->}{in1}{ins1}
1745 \ncline[]{->}{in2}{ins2}
1746 \ncline[]{->}{in3}{ins3}
1756 \begin{minipage}{5.7cm}
1758 \item Amount of C atoms: 6000\\
1759 ($r_{\text{prec}}\approx 3.1\text{ nm}$, IBS: \unit[2--4]{nm})
1760 \item Simulation volume: $31^3$ Si unit cells\\
1764 \begin{minipage}{0.3cm}
1768 \begin{minipage}{6.0cm}
1769 Restricted to classical potential caclulations\\
1770 $\rightarrow$ Low C diffusion / overestimated barrier\\
1771 $\rightarrow$ Consider $V_2$ and $V_3$
1773 % \item $V_2$ and $V_3$ considered due to expected low C diffusion
1784 Silicon carbide precipitation simulations at \degc{450} as in IBS
1789 \begin{minipage}{6.3cm}
1790 \hspace*{-0.4cm}\includegraphics[width=6.5cm]{sic_prec_450_si-c.ps}\\
1791 \hspace*{-0.4cm}\includegraphics[width=6.5cm]{sic_prec_450_si-si_c-c.ps}
1794 \begin{minipage}{6.1cm}
1796 \underline{Low C concentration --- {\color{red}$V_1$}}\\[0.1cm]
1797 \hkl<1 0 0> C-Si dumbbell dominated structure
1799 \item Si-C bumbs around \unit[0.19]{nm}
1800 \item C-C peak at \unit[0.31]{nm} (expected in 3C-SiC):\\
1801 concatenated differently oriented \ci{} DBs
1802 \item Si-Si NN distance stretched to \unit[0.3]{nm}
1804 \begin{pspicture}(0,0)(6.0,1.0)
1805 \rput(3.2,0.5){\psframebox[linewidth=0.03cm,linecolor=blue]{
1806 \begin{minipage}{6cm}
1808 Formation of \ci{} dumbbells\\
1809 C atoms in proper 3C-SiC distance first
1812 \end{pspicture}\\[0.1cm]
1813 \underline{High C concentration --- {\color{green}$V_2$}/{\color{blue}$V_3$}}
1815 \item High amount of strongly bound C-C bonds
1816 \item Increased defect \& damage density\\
1817 $\rightarrow$ Arrangements hard to categorize and trace
1818 \item Only short range order observable
1820 \begin{pspicture}(0,0)(6.0,0.8)
1821 \rput(3.2,0.5){\psframebox[linewidth=0.03cm,linecolor=blue]{
1822 \begin{minipage}{6cm}
1824 Amorphous SiC-like phase
1827 \end{pspicture}\\[0.3cm]
1828 \begin{pspicture}(0,0)(6.0,2.0)
1829 \rput(3.2,1.0){\psframebox[linewidth=0.05cm,linecolor=white]{
1830 \begin{minipage}{6cm}
1844 Silicon carbide precipitation simulations at \degc{450} as in IBS
1849 \begin{minipage}{6.3cm}
1850 \hspace*{-0.4cm}\includegraphics[width=6.5cm]{sic_prec_450_si-c.ps}\\
1851 \hspace*{-0.4cm}\includegraphics[width=6.5cm]{sic_prec_450_si-si_c-c.ps}
1854 \begin{minipage}{6.1cm}
1856 \underline{Low C concentration --- {\color{red}$V_1$}}\\[0.1cm]
1857 \hkl<1 0 0> C-Si dumbbell dominated structure
1859 \item Si-C bumbs around \unit[0.19]{nm}
1860 \item C-C peak at \unit[0.31]{nm} (expected in 3C-SiC):\\
1861 concatenated differently oriented \ci{} DBs
1862 \item Si-Si NN distance stretched to \unit[0.3]{nm}
1864 \begin{pspicture}(0,0)(6.0,1.0)
1865 \rput(3.2,0.5){\psframebox[linewidth=0.03cm,linecolor=blue]{
1866 \begin{minipage}{6cm}
1868 Formation of \ci{} dumbbells\\
1869 C atoms in proper 3C-SiC distance first
1872 \end{pspicture}\\[0.1cm]
1873 \underline{High C concentration --- {\color{green}$V_2$}/{\color{blue}$V_3$}}
1875 \item High amount of strongly bound C-C bonds
1876 \item Increased defect \& damage density\\
1877 $\rightarrow$ Arrangements hard to categorize and trace
1878 \item Only short range order observable
1880 \begin{pspicture}(0,0)(6.0,0.8)
1881 \rput(3.2,0.5){\psframebox[linewidth=0.03cm,linecolor=blue]{
1882 \begin{minipage}{6cm}
1884 Amorphous SiC-like phase
1887 \end{pspicture}\\[0.3cm]
1888 \begin{pspicture}(0,0)(6.0,2.0)
1889 \rput(3.2,1.0){\psframebox[linewidth=0.05cm,linecolor=black]{
1890 \begin{minipage}{6cm}
1893 {\bf\color{red}3C-SiC formation fails to appear}\\[0.3cm]
1894 \begin{minipage}{0.8cm}
1895 {\bf\boldmath $V_1$:}
1897 \begin{minipage}{5.1cm}
1898 Formation of \ci{} indeed occurs\\
1899 Agllomeration not observed
1900 \end{minipage}\\[0.3cm]
1901 \begin{minipage}{0.8cm}
1902 {\bf\boldmath $V_{2,3}$:}
1904 \begin{minipage}{5.1cm}
1905 Amorphous SiC-like structure\\
1906 (not expected at \degc{450})\\[0.05cm]
1907 No rearrangement/transition into 3C-SiC
1908 \end{minipage}\\[0.1cm]
1920 Limitations of MD and short range potentials
1927 {\bf Time scale problem of MD}\\[0.2cm]
1928 Precise integration \& thermodynamic sampling\\
1929 $\Rightarrow$ $\Delta t \ll \left( \max{\omega} \right)^{-1}$,
1930 $\omega$: vibrational mode\\
1931 $\Rightarrow$ {\color{red}\underline{Slow}} phase space propagation\\[0.2cm]
1932 Several local minima separated by large energy barriers\\
1933 $\Rightarrow$ Transition event corresponds to a multiple
1934 of vibrational periods\\
1935 $\Rightarrow$ Phase transition consists of {\color{red}\underline{many}}
1936 infrequent transition events\\[0.2cm]
1937 {\color{blue}Accelerated methods:}
1938 \underline{Temperature accelerated} MD (TAD), self-guided MD \ldots
1942 {\bf Limitations related to the short range potential}\\[0.2cm]
1943 Cut-off function limits interaction to next neighbours\\
1944 $\Rightarrow$ Overestimated unphysical high forces of next neighbours
1949 {\bf Approach to the (twofold) problem}\\[0.2cm]
1950 Increased temperature simulations without TAD corrections\\
1951 Accelerated methods or higher time scales exclusively not sufficient!
1953 \begin{pspicture}(0,0)(0,0)
1954 \rput(4.0,2.8){\psframebox[linewidth=0.07cm,linecolor=red]{
1955 \begin{minipage}{7.5cm}
1958 Potential enhanced slow phase space propagation
1961 \rput(11.3,7.5){\psframebox[linewidth=0.03cm,linecolor=blue]{
1962 \begin{minipage}{2.7cm}
1966 thermodynamic sampling
1969 \psline[linewidth=0.03cm,linecolor=blue]{<-}(11.3,7.0)(11.0,5.7)
1970 \rput(10.85,2.6){\psframebox[linewidth=0.03cm,linecolor=blue]{
1971 \begin{minipage}{3.6cm}
1974 \underline{IBS}\\[0.1cm]
1975 3C-SiC also observed for higher T\\[0.1cm]
1976 Higher T inside sample\\[0.1cm]
1977 Structural evolution vs.\\
1978 equilibrium properties
1981 \psline[linewidth=0.03cm,linecolor=blue]{->}(10.85,1.75)(9.0,1.0)
1990 Increased temperature simulations --- $V_1$
1995 \begin{minipage}{6.2cm}
1996 \hspace*{-0.4cm}\includegraphics[width=6.5cm]{tot_pc_thesis.ps}
1999 \begin{minipage}{6.2cm}
2000 \includegraphics[width=6.5cm]{tot_pc3_thesis.ps}
2003 \begin{minipage}{6.2cm}
2004 \hspace*{-0.4cm}\includegraphics[width=6.5cm]{tot_pc2_thesis.ps}
2007 \begin{minipage}{6.3cm}
2009 \underline{Si-C bonds:}
2011 \item Vanishing cut-off artifact (above $1650\,^{\circ}\mathrm{C}$)
2012 \item Structural change: C-Si \hkl<1 0 0> $\rightarrow$ C$_{\text{sub}}$
2014 \underline{Si-Si bonds:}
2015 {\color{blue}Si-C$_{\text{sub}}$-Si} along \hkl<1 1 0>
2016 ($\rightarrow$ 0.325 nm)\\[0.1cm]
2017 \underline{C-C bonds:}
2019 \item C-C next neighbour pairs reduced (mandatory)
2020 \item Peak at 0.3 nm slightly shifted
2022 \item C-Si \hkl<1 0 0> combinations (dashed arrows)\\
2023 $\rightarrow$ C-Si \hkl<1 0 0> \& C$_{\text{sub}}$
2025 $\rightarrow$ pure {\color{blue}C$_{\text{sub}}$ combinations}
2027 \item Range [|-$\downarrow$]:
2028 {\color{blue}C$_{\text{sub}}$ \& C$_{\text{sub}}$
2029 with nearby Si$_{\text{I}}$}
2040 Increased temperature simulations --- $V_1$
2045 \begin{minipage}{6.2cm}
2046 \hspace*{-0.4cm}\includegraphics[width=6.5cm]{tot_pc_thesis.ps}
2049 \begin{minipage}{6.2cm}
2050 \includegraphics[width=6.5cm]{tot_pc3_thesis.ps}
2053 \begin{minipage}{6.2cm}
2054 \hspace*{-0.4cm}\includegraphics[width=6.5cm]{tot_pc2_thesis.ps}
2057 \begin{minipage}{6.3cm}
2059 \underline{Si-C bonds:}
2061 \item Vanishing cut-off artifact (above $1650\,^{\circ}\mathrm{C}$)
2062 \item Structural change: C-Si \hkl<1 0 0> $\rightarrow$ C$_{\text{sub}}$
2064 \underline{Si-Si bonds:}
2065 {\color{blue}Si-C$_{\text{sub}}$-Si} along \hkl<1 1 0>
2066 ($\rightarrow$ 0.325 nm)\\[0.1cm]
2067 \underline{C-C bonds:}
2069 \item C-C next neighbour pairs reduced (mandatory)
2070 \item Peak at 0.3 nm slightly shifted
2072 \item C-Si \hkl<1 0 0> combinations (dashed arrows)\\
2073 $\rightarrow$ C-Si \hkl<1 0 0> \& C$_{\text{sub}}$
2075 $\rightarrow$ pure {\color{blue}C$_{\text{sub}}$ combinations}
2077 \item Range [|-$\downarrow$]:
2078 {\color{blue}C$_{\text{sub}}$ \& C$_{\text{sub}}$
2079 with nearby Si$_{\text{I}}$}
2085 \begin{pspicture}(0,0)(0,0)
2086 \rput(6.5,5.0){\psframebox[fillstyle=solid,opacity=0.5,fillcolor=black]{
2087 \begin{minipage}{14cm}
2092 \rput(6.5,5.0){\psframebox[fillstyle=solid,fillcolor=white,linewidth=0.1cm]{
2093 \begin{minipage}{9cm}
2097 {\color{gray}\bf Conclusions on SiC precipitation}\\[0.1cm]
2098 {\Huge$\lightning$} {\color{red}\ci{}} --- vs --- {\color{blue}\cs{}} {\Huge$\lightning$}\\
2101 \item Stretched coherent SiC structures\\
2102 $\Rightarrow$ Precipitation process involves {\color{blue}\cs}
2103 \item Explains annealing behavior of high/low T C implantations
2105 \item Low T: highly mobile {\color{red}\ci}
2106 \item High T: stable configurations of {\color{blue}\cs}
2110 \item Vehicle to rearrange \cs --- [\cs{} \& \si{} $\leftrightarrow$ \ci]
2111 \item Building block for surrounding Si host \& further SiC
2112 \item Strain compensation \ldots\\
2113 \ldots Si/SiC interface\\
2114 \ldots within stretched coherent SiC structure
2119 \psframebox[linecolor=blue,linewidth=0.05cm]{
2120 \begin{minipage}{7cm}
2122 Precipitation mechanism involving \cs\\
2123 High T $\leftrightarrow$ IBS conditions far from equilibrium\\
2133 % skip high T / C conc ... only here!
2139 Increased temperature simulations at high C concentration
2144 \begin{minipage}{6.5cm}
2145 \includegraphics[width=6.4cm]{12_pc_thesis.ps}
2147 \begin{minipage}{6.5cm}
2148 \includegraphics[width=6.4cm]{12_pc_c_thesis.ps}
2156 \begin{minipage}[t]{6.0cm}
2157 0.186 nm: Si-C pairs $\uparrow$\\
2158 (as expected in 3C-SiC)\\[0.2cm]
2159 0.282 nm: Si-C-C\\[0.2cm]
2160 $\approx$0.35 nm: C-Si-Si
2163 \begin{minipage}{0.2cm}
2167 \begin{minipage}[t]{6.0cm}
2168 0.15 nm: C-C pairs $\uparrow$\\
2169 (as expected in graphite/diamond)\\[0.2cm]
2170 0.252 nm: C-C-C (2$^{\text{nd}}$ NN for diamond)\\[0.2cm]
2171 0.31 nm: shifted towards 0.317 nm $\rightarrow$ C-Si-C
2176 \item Decreasing cut-off artifact
2177 \item {\color{red}Amorphous} SiC-like phase remains
2178 \item High amount of {\color{red}damage} \& alignement to c-Si host matrix lost
2179 \item Slightly sharper peaks $\Rightarrow$ indicate slight {\color{blue}acceleration of dynamics} due to temperature
2188 High C \& small $V$ \& short $t$
2191 Slow restructuring due to strong C-C bonds
2194 High C \& low T implants
2202 % skipped high T / C conc
2213 \begin{pspicture}(0,0)(12,1.0)
2214 \psframebox[fillstyle=gradient,gradbegin=hred,gradend=white,gradlines=1000,gradmidpoint=1.0,linestyle=none]{
2215 \begin{minipage}{11cm}
2216 {\color{black}Diploma thesis}\\
2217 \underline{Monte Carlo} simulation modeling the selforganization process\\
2218 leading to periodic arrays of nanometric amorphous SiC precipitates
2221 \end{pspicture}\\[0.4cm]
2222 \begin{pspicture}(0,0)(12,2)
2223 \psframebox[fillstyle=gradient,gradbegin=hblue,gradend=white,gradmidpoint=1.0,gradlines=1000,linestyle=none]{
2224 \begin{minipage}{11cm}
2225 {\color{black}Doctoral studies}\\
2226 Classical potential \underline{molecular dynamics} simulations \ldots\\
2227 \underline{Density functional theory} calculations \ldots\\[0.2cm]
2228 \ldots on defect formation and SiC precipitation in Si
2231 \end{pspicture}\\[0.5cm]
2232 \begin{pspicture}(0,0)(12,3)
2233 \psframebox[fillstyle=solid,fillcolor=white,linestyle=solid]{
2234 \begin{minipage}{11cm}
2236 {\color{black}\bf How to proceed \ldots}\\[0.1cm]
2237 MC $\rightarrow$ empirical potential MD $\rightarrow$ Ground-state DFT \ldots
2239 \renewcommand\labelitemi{$\ldots$}
2240 \item beyond LDA/GGA methods \& ground-state DFT
2242 Investigation of structure \& structural evolution \ldots
2244 \renewcommand\labelitemi{$\ldots$}
2245 \item electronic/optical properties
2246 \item electronic correlations
2247 \item non-equilibrium systems
2251 \end{pspicture}\\[0.5cm]
2267 \underline{Augsburg}
2269 \item Prof. B. Stritzker (accomodation at EP \RM{4})
2270 \item Ralf Utermann (EDV)
2273 \underline{Helsinki}
2275 \item Prof. K. Nordlund (MD)
2280 \item Bayerische Forschungsstiftung (financial support)
2283 \underline{Paderborn}
2285 \item Prof. J. Lindner (SiC)
2286 \item Prof. G. Schmidt (DFT + financial support)
2287 \item Dr. E. Rauls (DFT + SiC)
2290 \underline{Stuttgart}
2293 \bf Thank you for your attention / invitation!