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}
253 Polytypes of SiC\\[0.6cm]
258 \includegraphics[width=3.8cm]{cubic_hex.eps}\\
259 \begin{minipage}{1.9cm}
260 {\tiny cubic (twist)}
262 \begin{minipage}{2.9cm}
263 {\tiny hexagonal (no twist)}
266 \begin{picture}(0,0)(-150,0)
267 \includegraphics[width=7cm]{polytypes.eps}
274 \begin{tabular}{l c c c c c c}
276 & 3C-SiC & 4H-SiC & 6H-SiC & Si & GaN & Diamond\\
278 Hardness [Mohs] & \multicolumn{3}{c}{------ 9.6 ------}& 6.5 & - & 10 \\
279 Band gap [eV] & 2.36 & 3.23 & 3.03 & 1.12 & 3.39 & 5.5 \\
280 Break down field [$10^6$ V/cm] & 4 & 3 & 3.2 & 0.6 & 5 & 10 \\
281 Saturation drift velocity [$10^7$ cm/s] & 2.5 & 2.0 & 2.0 & 1 & 2.7 & 2.7 \\
282 Electron mobility [cm$^2$/Vs] & 800 & 900 & 400 & 1100 & 900 & 2200 \\
283 Hole mobility [cm$^2$/Vs] & 320 & 120 & 90 & 420 & 150 & 1600 \\
284 Thermal conductivity [W/cmK] & 5.0 & 4.9 & 4.9 & 1.5 & 1.3 & 22 \\
288 \begin{pspicture}(0,0)(0,0)
289 \psellipse[linecolor=green](5.7,2.05)(0.4,0.50)
291 \begin{pspicture}(0,0)(0,0)
292 \psellipse[linecolor=green](5.6,0.89)(0.4,0.20)
294 \begin{pspicture}(0,0)(0,0)
295 \psellipse[linecolor=red](10.45,0.42)(0.4,0.20)
306 Fabrication of silicon carbide
315 \emph{Silicon carbide --- Born from the stars, perfected on earth.}
321 SiC thin films by MBE \& CVD
323 \item Much progress achieved in homo/heteroepitaxial SiC thin film growth
324 \item \underline{Commercially available} semiconductor power devices based on
325 \underline{\foreignlanguage{greek}{a}-SiC}
326 \item Production of favored \underline{3C-SiC} material
327 \underline{less advanced}
328 \item Quality and size not yet sufficient
330 \begin{picture}(0,0)(-310,-20)
331 \includegraphics[width=2.0cm]{cree.eps}
339 {\footnotesize\color{black}
340 Mismatch in \underline{thermal expansion coeefficient}
341 and \underline{lattice parameter} w.r.t. substrate
348 {\bf Alternative approach}\\
349 Ion beam synthesis (IBS) of burried 3C-SiC layers in Si\hkl(1 0 0)
356 \begin{minipage}{3.15cm}
358 \includegraphics[width=3cm]{imp.eps}\\
364 \begin{minipage}{3.15cm}
366 \includegraphics[width=3cm]{annealing.eps}\\
368 Postannealing at $>$ \degc{1200}
373 \begin{minipage}{5.5cm}
376 No surface bending effects\\
377 High areal homogenity\\[0.1cm]
378 $\Downarrow$\\[0.1cm]
379 Synthesis of large area SiC films possible
390 IBS of epitaxial single crystalline 3C-SiC
399 \item \underline{Implantation step 1}\\[0.1cm]
400 Almost stoichiometric dose | \unit[180]{keV} | \degc{500}\\
401 $\Rightarrow$ Epitaxial {\color{blue}3C-SiC} layer \&
402 {\color{blue}precipitates}
403 \item \underline{Implantation step 2}\\[0.1cm]
404 Little remaining dose | \unit[180]{keV} | \degc{250}\\
406 Destruction/Amorphization of precipitates at layer interface
407 \item \underline{Annealing}\\[0.1cm]
408 \unit[10]{h} at \degc{1250}\\
409 $\Rightarrow$ Homogeneous 3C-SiC layer with sharp interfaces
413 \begin{minipage}{6.9cm}
414 \includegraphics[width=7cm]{ibs_3c-sic.eps}\\[-0.4cm]
417 XTEM: single crystalline 3C-SiC in Si\hkl(1 0 0)
421 \begin{minipage}{5cm}
422 \begin{pspicture}(0,0)(0,0)
424 \psframebox[fillstyle=solid,fillcolor=white,linecolor=blue,linestyle=solid]{
425 \begin{minipage}{5.3cm}
428 3C-SiC precipitation\\
429 not yet fully understood
433 \renewcommand\labelitemi{$\Rightarrow$}
434 Details of the SiC precipitation
436 \item significant technological progress\\
437 in SiC thin film formation
438 \item perspectives for processes relying\\
439 upon prevention of SiC precipitation
443 \rput(-6.8,5.5){\pnode{h0}}
444 \rput(-3.0,5.5){\pnode{h1}}
445 \ncline[linecolor=blue]{-}{h0}{h1}
446 \ncline[linecolor=blue]{->}{h1}{box}
458 Supposed precipitation mechanism of SiC in Si
466 \begin{minipage}{3.6cm}
468 Si \& SiC lattice structure\\[0.1cm]
469 \includegraphics[width=2.3cm]{sic_unit_cell.eps}
472 \begin{minipage}{1.7cm}
473 \underline{Silicon}\\
474 {\color{yellow}$\bullet$} Si | {\color{gray}$\bullet$} Si\\
475 $a=\unit[5.429]{\\A}$\\
476 $\rho^*_{\text{Si}}=\unit[100]{\%}$
478 \begin{minipage}{1.7cm}
479 \underline{Silicon carbide}\\
480 {\color{yellow}$\bullet$} Si | {\color{gray}$\bullet$} C\\
481 $a=\unit[4.359]{\\A}$\\
482 $\rho^*_{\text{Si}}=\unit[97]{\%}$
488 \begin{minipage}{4.1cm}
490 \includegraphics[width=3.3cm]{tem_c-si-db.eps}
494 \begin{minipage}{4.0cm}
496 \includegraphics[width=3.3cm]{tem_3c-sic.eps}
502 \begin{minipage}{4.0cm}
504 C-Si dimers (dumbbells)\\[-0.1cm]
509 \begin{minipage}{4.1cm}
511 Agglomeration of C-Si dumbbells\\[-0.1cm]
512 $\Rightarrow$ dark contrasts
516 \begin{minipage}{4.0cm}
518 Precipitation of 3C-SiC in Si\\[-0.1cm]
519 $\Rightarrow$ Moir\'e fringes\\[-0.1cm]
520 \& release of Si self-interstitials
526 \begin{minipage}{4.0cm}
528 \includegraphics[width=3.3cm]{sic_prec_seq_01.eps}
532 \begin{minipage}{4.1cm}
534 \includegraphics[width=3.3cm]{sic_prec_seq_02.eps}
538 \begin{minipage}{4.0cm}
540 \includegraphics[width=3.3cm]{sic_prec_seq_03.eps}
544 \begin{pspicture}(0,0)(0,0)
545 \psline[linewidth=2pt]{->}(8.3,2)(8.8,2)
546 \psellipse[linecolor=blue](11.1,6.0)(0.3,0.5)
547 \rput{-20}{\psellipse[linecolor=blue](3.1,8.2)(0.3,0.5)}
548 \psline[linewidth=2pt]{->}(3.9,2)(4.4,2)
549 \rput(11.8,0.3){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
550 $4a_{\text{Si}}=5a_{\text{SiC}}$
552 \rput(11.5,8){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
553 \hkl(h k l) planes match
555 \rput(8.5,6.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
566 Supposed precipitation mechanism of SiC in Si
574 \begin{minipage}{3.6cm}
576 Si \& SiC lattice structure\\[0.1cm]
577 \includegraphics[width=2.3cm]{sic_unit_cell.eps}
580 \begin{minipage}{1.7cm}
581 \underline{Silicon}\\
582 {\color{yellow}$\bullet$} Si | {\color{gray}$\bullet$} Si\\
583 $a=\unit[5.429]{\\A}$\\
584 $\rho^*_{\text{Si}}=\unit[100]{\%}$
586 \begin{minipage}{1.7cm}
587 \underline{Silicon carbide}\\
588 {\color{yellow}$\bullet$} Si | {\color{gray}$\bullet$} C\\
589 $a=\unit[4.359]{\\A}$\\
590 $\rho^*_{\text{Si}}=\unit[97]{\%}$
596 \begin{minipage}{4.1cm}
598 \includegraphics[width=3.3cm]{tem_c-si-db.eps}
602 \begin{minipage}{4.0cm}
604 \includegraphics[width=3.3cm]{tem_3c-sic.eps}
610 \begin{minipage}{4.0cm}
612 C-Si dimers (dumbbells)\\[-0.1cm]
613 on Si interstitial sites
617 \begin{minipage}{4.1cm}
619 Agglomeration of C-Si dumbbells\\[-0.1cm]
620 $\Rightarrow$ dark contrasts
624 \begin{minipage}{4.0cm}
626 Precipitation of 3C-SiC in Si\\[-0.1cm]
627 $\Rightarrow$ Moir\'e fringes\\[-0.1cm]
628 \& release of Si self-interstitials
634 \begin{minipage}{4.0cm}
636 \includegraphics[width=3.3cm]{sic_prec_seq_01.eps}
640 \begin{minipage}{4.1cm}
642 \includegraphics[width=3.3cm]{sic_prec_seq_02.eps}
646 \begin{minipage}{4.0cm}
648 \includegraphics[width=3.3cm]{sic_prec_seq_03.eps}
652 \begin{pspicture}(0,0)(0,0)
653 \psline[linewidth=2pt]{->}(8.3,2)(8.8,2)
654 \psellipse[linecolor=blue](11.1,6.0)(0.3,0.5)
655 \rput{-20}{\psellipse[linecolor=blue](3.1,8.2)(0.3,0.5)}
656 \psline[linewidth=2pt]{->}(3.9,2)(4.4,2)
657 \rput(11.8,0.3){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
658 $4a_{\text{Si}}=5a_{\text{SiC}}$
660 \rput(11.5,8){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
661 \hkl(h k l) planes match
663 \rput(8.5,6.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
666 % controversial view!
667 \rput(6.5,5.0){\psframebox[fillstyle=solid,opacity=0.5,fillcolor=black]{
668 \begin{minipage}{14cm}
673 \rput(6.5,5.3){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=white,linewidth=0.1cm]{
674 \begin{minipage}{10cm}
678 {\color{gray}\bf Controversial findings}
681 \item High-temperature implantation {\tiny\color{gray}/Nejim~et~al./}
683 \item C incorporated {\color{blue}substitutionally} on regular Si lattice sites
684 \item \si{} reacting with further C in cleared volume
686 \item Annealing behavior {\tiny\color{gray}/Serre~et~al./}
688 \item Room temperature implantation $\rightarrow$ high C diffusion
689 \item Elevated temperature implantation $\rightarrow$ no C redistribution
691 $\Rightarrow$ mobile {\color{red}\ci} opposed to
692 stable {\color{blue}\cs{}} configurations
693 \item Strained silicon \& Si/SiC heterostructures
694 {\tiny\color{gray}/Strane~et~al./Guedj~et~al./}
696 \item {\color{blue}Coherent} SiC precipitates (tensile strain)
697 \item Incoherent SiC (strain relaxation)
702 {\Huge${\lightning}$} \hspace{0.3cm}
703 {\color{blue}\cs{}} --- vs --- {\color{red}\ci} \hspace{0.3cm}
704 {\Huge${\lightning}$}
722 \item Introduction / Motivation
723 \item Assumed SiC precipitation mechanisms / Controversy
725 \item Utilized simulation techniques
727 \item Molecular dynamics (MD) simulations
728 \item Density functional theory (DFT) calculations
730 \item Simulation results
732 \item C and Si self-interstitial point defects in silicon
733 \item Silicon carbide precipitation simulations
735 \item Summary / Conclusion
744 Utilized computational methods
751 {\bf Molecular dynamics (MD)}\\[0.1cm]
753 \begin{tabular}{| p{4.5cm} | p{7.5cm} |}
755 System of $N$ particles &
756 $N=5832\pm 1$ (Defects), $N=238328+6000$ (Precipitation)\\
757 Phase space propagation &
758 Velocity Verlet | timestep: \unit[1]{fs} \\
759 Analytical interaction potential &
760 Tersoff-like {\color{red}short-range}, {\color{blue}bond order} potential
763 E = \frac{1}{2} \sum_{i \neq j} \pot_{ij}, \quad
764 \pot_{ij} = {\color{red}f_C(r_{ij})}
765 \left[ f_R(r_{ij}) + {\color{blue}b_{ij}} f_A(r_{ij}) \right]
767 Observables: time/ensemble averages &
768 NpT (isothermal-isobaric) | Berendsen thermostat/barostat\\
776 {\bf Density functional theory (DFT)}
780 \begin{minipage}[t]{6cm}
782 \item Hohenberg-Kohn theorem:\\
783 $\Psi_0(r_1,r_2,\ldots,r_N)=\Psi[n_0(r)]$, $E_0=E[n_0]$
784 \item Kohn-Sham approach:\\
785 Single-particle effective theory
789 \item Code: \textsc{vasp}
790 \item Plane wave basis set | $E_{\text{cut}}=\unit[300]{eV}$
792 %\Phi_i=\sum_{|G+k|<G_{\text{cut}}} c_{i,k+G} \exp{\left(i(k+G)r\right)}
795 %E_{\text{cut}}=\frac{\hbar^2}{2m}G^2_{\text{cut}}=\unit[300]{eV}
797 \item Ultrasoft pseudopotential
798 \item Exchange \& correlation: GGA
799 \item Brillouin zone sampling: $\Gamma$-point
800 \item Supercell: $N=216\pm2$
803 \begin{minipage}{6cm}
804 \begin{pspicture}(0,0)(0,0)
805 \pscircle[fillcolor=yellow,fillstyle=solid,linestyle=none](3.5,-2.0){2.5}
806 \rput(2.7,-0.7){\psframebox[fillstyle=solid,opacity=0.8,fillcolor=white]{
808 \left[ -\frac{\hbar^2}{2m}\nabla^2 + V_{\text{eff}}(r) - \epsilon_i \right] \Phi_i(r) = 0
811 \rput(5.2,-2.0){\psframebox[fillstyle=solid,opacity=0.8,fillcolor=white]{
813 n(r)=\sum_i^N|\Phi_i(r)|^2
816 \rput(3.0,-4.5){\psframebox[fillstyle=solid,opacity=0.8,fillcolor=white]{
818 V_{\text{eff}}(r)=V_{\text{ext}}(r)+\int\frac{e^2 n(r')}{|r-r'|}d^3r'
822 \psarcn[linewidth=0.07cm,linestyle=dashed]{->}(3.5,-2.0){2.5}{130}{15}
823 \psarcn[linewidth=0.07cm,linestyle=dashed]{->}(3.5,-2.0){2.5}{230}{165}
824 \psarcn[linewidth=0.07cm,linestyle=dashed]{->}(3.5,-2.0){2.5}{345}{310}
835 Point defects \& defect migration
842 \begin{minipage}[b]{7.5cm}
843 {\bf Defect structure}\\
844 \begin{pspicture}(0,0)(7,4.4)
845 \rput(3.5,3.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
848 \item Creation of c-Si simulation volume
849 \item Periodic boundary conditions
850 \item $T=0\text{ K}$, $p=0\text{ bar}$
853 \rput(3.5,1.3){\rnode{insert}{\psframebox{
856 Insertion of interstitial C/Si atoms
859 \rput(3.5,0.2){\rnode{cool}{\psframebox[fillstyle=solid,fillcolor=lbb]{
862 Relaxation / structural energy minimization
865 \ncline[]{->}{init}{insert}
866 \ncline[]{->}{insert}{cool}
869 \begin{minipage}[b]{4.5cm}
871 \includegraphics[width=3.8cm]{unit_cell_e.eps}\\
873 \begin{minipage}{2.21cm}
875 {\color{red}$\bullet$} Tetrahedral\\[-0.1cm]
876 {\color{green}$\bullet$} Hexagonal\\[-0.1cm]
877 {\color{yellow}$\bullet$} \hkl<1 0 0> DB
880 \begin{minipage}{2.21cm}
882 {\color{magenta}$\bullet$} \hkl<1 1 0> DB\\[-0.1cm]
883 {\color{cyan}$\bullet$} Bond-centered\\[-0.1cm]
884 {\color{black}$\bullet$} Vac. / Sub.
891 \begin{minipage}[b]{6cm}
892 {\bf Defect formation energy}\\
894 $E_{\text{f}}=E-\sum_i N_i\mu_i$}\\[0.1cm]
895 Particle reservoir: Si \& SiC\\[0.2cm]
896 {\bf Binding energy}\\
900 E_{\text{f}}^{\text{comb}}-
901 E_{\text{f}}^{1^{\text{st}}}-
902 E_{\text{f}}^{2^{\text{nd}}}
906 $E_{\text{b}}<0$: energetically favorable configuration\\
907 $E_{\text{b}}\rightarrow 0$: non-interacting, isolated defects\\
909 \begin{minipage}[b]{6cm}
910 {\bf Migration barrier}
913 \item Displace diffusing atom
914 \item Constrain relaxation of (diffusing) atoms
915 \item Record configurational energy
917 \begin{picture}(0,0)(-60,-33)
918 \includegraphics[width=4.5cm]{crt_mod.eps}
930 Si self-interstitial point defects in silicon\\[0.1cm]
934 \begin{tabular}{l c c c c c}
936 $E_{\text{f}}$ [eV] & \hkl<1 1 0> DB & H & T & \hkl<1 0 0> DB & V \\
938 \textsc{vasp} & \underline{3.39} & 3.42 & 3.77 & 4.41 & 3.63 \\
939 Erhart/Albe & 4.39 & 4.48$^*$ & \underline{3.40} & 5.42 & 3.13 \\
941 \end{tabular}\\[0.4cm]
944 \begin{minipage}{3cm}
946 \underline{Vacancy}\\
947 \includegraphics[width=2.8cm]{si_pd_albe/vac.eps}
950 \begin{minipage}{3cm}
952 \underline{\hkl<1 1 0> DB}\\
953 \includegraphics[width=2.8cm]{si_pd_albe/110_bonds.eps}
956 \begin{minipage}{3cm}
958 \underline{\hkl<1 0 0> DB}\\
959 \includegraphics[width=2.8cm]{si_pd_albe/100_bonds.eps}
962 \begin{minipage}{3cm}
964 \underline{Tetrahedral}\\
965 \includegraphics[width=2.8cm]{si_pd_albe/tet_bonds.eps}
969 \underline{Hexagonal} \hspace{2pt}
970 \href{../video/si_self_int_hexa.avi}{$\rhd$}\\[0.1cm]
972 \begin{minipage}{2.7cm}
973 $E_{\text{f}}^*=4.48\text{ eV}$\\
974 \includegraphics[width=2.7cm]{si_pd_albe/hex_a_bonds.eps}
976 \begin{minipage}{0.4cm}
981 \begin{minipage}{2.7cm}
982 $E_{\text{f}}=3.96\text{ eV}$\\
983 \includegraphics[width=2.8cm]{si_pd_albe/hex_bonds.eps}
986 \begin{minipage}{5.5cm}
988 {\tiny nearly T $\rightarrow$ T}\\
990 \includegraphics[width=6.0cm]{nhex_tet.ps}
1001 C interstitial point defects in silicon\\
1004 \begin{tabular}{l c c c c c c r}
1006 $E_{\text{f}}$ [eV] & T & H & \hkl<1 0 0> DB & \hkl<1 1 0> DB & S & B &
1007 {\color{black} \cs{} \& \si}\\
1009 \textsc{vasp} & unstable & unstable & \underline{3.72} & 4.16 & 1.95 & 4.66 & {\color{green}4.17}\\
1010 Erhart/Albe & 6.09 & 9.05$^*$ & \underline{3.88} & 5.18 & {\color{red}0.75} & 5.59$^*$ & {\color{green}4.43} \\
1012 \end{tabular}\\[0.1cm]
1015 \begin{minipage}{2.8cm}
1016 \underline{Hexagonal} \hspace{2pt}
1017 \href{../video/c_in_si_int_hexa.avi}{$\rhd$}\\
1018 $E_{\text{f}}^*=9.05\text{ eV}$\\
1019 \includegraphics[width=2.8cm]{c_pd_albe/hex_bonds.eps}
1021 \begin{minipage}{0.4cm}
1026 \begin{minipage}{2.8cm}
1027 \underline{\hkl<1 0 0>}\\
1028 $E_{\text{f}}=3.88\text{ eV}$\\
1029 \includegraphics[width=2.8cm]{c_pd_albe/100_bonds.eps}
1032 \begin{minipage}{1.4cm}
1035 \begin{minipage}{3.0cm}
1037 \underline{Tetrahedral}\\
1038 \includegraphics[width=3.0cm]{c_pd_albe/tet_bonds.eps}
1043 \begin{minipage}{2.8cm}
1044 \underline{Bond-centered}\\
1045 $E_{\text{f}}^*=5.59\text{ eV}$\\
1046 \includegraphics[width=2.8cm]{c_pd_albe/bc_bonds.eps}
1048 \begin{minipage}{0.4cm}
1053 \begin{minipage}{2.8cm}
1054 \underline{\hkl<1 1 0> dumbbell}\\
1055 $E_{\text{f}}=5.18\text{ eV}$\\
1056 \includegraphics[width=2.8cm]{c_pd_albe/110_bonds.eps}
1059 \begin{minipage}{1.4cm}
1062 \begin{minipage}{3.0cm}
1064 \underline{Substitutional}\\
1065 \includegraphics[width=3.0cm]{c_pd_albe/sub_bonds.eps}
1075 C-Si dimer \& bond-centered interstitial configuration
1082 \begin{minipage}[t]{4.1cm}
1083 {\bf\boldmath C \hkl<1 0 0> DB interstitial}\\[0.1cm]
1084 \begin{minipage}{2.0cm}
1086 \underline{Erhart/Albe}
1087 \includegraphics[width=2.0cm]{c_pd_albe/100_cmp.eps}
1090 \begin{minipage}{2.0cm}
1092 \underline{\textsc{vasp}}
1093 \includegraphics[width=2.0cm]{c_pd_vasp/100_cmp.eps}
1095 \end{minipage}\\[0.2cm]
1096 Si-C-Si bond angle $\rightarrow$ \unit[180]{$^{\circ}$}\\
1097 $\Rightarrow$ $sp$ hybridization\\[0.1cm]
1098 Si-Si-Si bond angle $\rightarrow$ \unit[120]{$^{\circ}$}\\
1099 $\Rightarrow$ $sp^2$ hybridization
1101 \includegraphics[width=3.4cm]{c_pd_vasp/eden.eps}\\[-0.1cm]
1102 {\tiny Charge density isosurface}
1105 \begin{minipage}{0.2cm}
1108 \begin{minipage}[t]{8.1cm}
1110 {\bf Bond-centered interstitial}\\[0.1cm]
1111 \begin{minipage}{4.4cm}
1114 \item Linear Si-C-Si bond
1115 \item Si: one C \& 3 Si neighbours
1116 \item Spin polarized calculations
1117 \item No saddle point!\\
1121 \begin{minipage}{2.7cm}
1122 %\includegraphics[width=2.8cm]{c_pd_vasp/bc_2333.eps}\\
1124 \includegraphics[width=2.8cm]{c_pd_albe/bc_bonds.eps}\\
1129 \begin{minipage}[t]{6.5cm}
1130 \begin{minipage}[t]{1.2cm}
1132 {\tiny sp$^3$}\\[0.8cm]
1133 \underline{${\color{black}\uparrow}$}
1134 \underline{${\color{black}\uparrow}$}
1135 \underline{${\color{black}\uparrow}$}
1136 \underline{${\color{red}\uparrow}$}\\
1139 \begin{minipage}[t]{1.4cm}
1141 {\color{red}M}{\color{blue}O}\\[0.8cm]
1142 \underline{${\color{blue}\uparrow}{\color{white}\downarrow}$}\\
1143 $\sigma_{\text{ab}}$\\[0.5cm]
1144 \underline{${\color{red}\uparrow}{\color{blue}\downarrow}$}\\
1148 \begin{minipage}[t]{1.0cm}
1152 \underline{${\color{white}\uparrow\uparrow}$}
1153 \underline{${\color{white}\uparrow\uparrow}$}\\
1155 \underline{${\color{blue}\uparrow}{\color{blue}\downarrow}$}
1156 \underline{${\color{blue}\uparrow}{\color{blue}\downarrow}$}\\
1160 \begin{minipage}[t]{1.4cm}
1162 {\color{blue}M}{\color{green}O}\\[0.8cm]
1163 \underline{${\color{blue}\uparrow}{\color{white}\downarrow}$}\\
1164 $\sigma_{\text{ab}}$\\[0.5cm]
1165 \underline{${\color{green}\uparrow}{\color{blue}\downarrow}$}\\
1169 \begin{minipage}[t]{1.2cm}
1172 {\tiny sp$^3$}\\[0.8cm]
1173 \underline{${\color{green}\uparrow}$}
1174 \underline{${\color{black}\uparrow}$}
1175 \underline{${\color{black}\uparrow}$}
1176 \underline{${\color{black}\uparrow}$}\\
1184 \begin{minipage}{3.0cm}
1186 \underline{Charge density}\\
1187 {\color{gray}$\bullet$} Spin up\\
1188 {\color{green}$\bullet$} Spin down\\
1189 {\color{blue}$\bullet$} Resulting spin up\\
1190 {\color{yellow}$\bullet$} Si atoms\\
1191 {\color{red}$\bullet$} C atom
1193 \begin{minipage}{3.6cm}
1194 \includegraphics[width=3.8cm]{c_100_mig_vasp/im_spin_diff.eps}
1201 \begin{pspicture}(0,0)(0,0)
1202 \psline[linecolor=gray,linewidth=0.05cm](-7.8,-8.7)(-7.8,0)
1211 C interstitial migration --- ab initio
1218 \begin{minipage}{6.8cm}
1219 \framebox{\hkl[0 0 -1] $\rightarrow$ \hkl[0 0 1]}\\
1220 \begin{minipage}{2.0cm}
1221 \includegraphics[width=2.0cm]{c_pd_vasp/100_2333.eps}
1223 \begin{minipage}{0.2cm}
1226 \begin{minipage}{2.0cm}
1227 \includegraphics[width=2.0cm]{c_pd_vasp/bc_2333.eps}
1229 \begin{minipage}{0.2cm}
1232 \begin{minipage}{2.0cm}
1233 \includegraphics[width=2.0cm]{c_pd_vasp/100_next_2333.eps}
1234 \end{minipage}\\[0.1cm]
1236 $\Rightarrow$ BC configuration constitutes local minimum\\
1237 $\Rightarrow$ Migration barrier to reach BC | $\Delta E=\unit[1.2]{eV}$
1239 \begin{minipage}{5.4cm}
1240 \includegraphics[width=6.0cm]{im_00-1_nosym_sp_fullct_thesis_vasp_s.ps}
1241 \end{minipage}\\[0.2cm]
1244 \begin{minipage}{6.8cm}
1245 \framebox{\hkl[0 0 -1] $\rightarrow$ \hkl[0 -1 0]}\\
1246 \begin{minipage}{2.0cm}
1247 \includegraphics[width=2.0cm]{c_pd_vasp/100_2333.eps}
1249 \begin{minipage}{0.2cm}
1252 \begin{minipage}{2.0cm}
1253 \includegraphics[width=2.0cm]{c_pd_vasp/00-1-0-10_2333.eps}
1255 \begin{minipage}{0.2cm}
1258 \begin{minipage}{2.0cm}
1259 \includegraphics[width=2.0cm]{c_pd_vasp/0-10_2333.eps}
1260 \end{minipage}\\[0.1cm]
1261 $\Delta E=\unit[0.9]{eV}$ | Experimental values: \unit[0.70--0.87]{eV}\\
1262 $\Rightarrow$ {\color{red}Migration mechanism identified!}\\
1263 Note: Change in orientation
1265 \begin{minipage}{5.4cm}
1266 \includegraphics[width=6.0cm]{00-1_0-10_vasp_s.ps}
1267 \end{minipage}\\[0.1cm]
1270 Reorientation pathway composed of two consecutive processes of the above type
1279 C interstitial migration --- analytical potential
1286 \begin{minipage}[t]{6.0cm}
1287 {\bf\boldmath BC to \hkl[0 0 -1] transition}\\[0.2cm]
1288 \includegraphics[width=6.0cm]{bc_00-1_albe_s.ps}\\
1290 \item Lowermost migration barrier
1291 \item $\Delta E \approx \unit[2.2]{eV}$
1292 \item 2.4 times higher than ab initio result
1293 \item Different pathway
1296 \begin{minipage}[t]{0.2cm}
1299 \begin{minipage}[t]{6.0cm}
1300 {\bf\boldmath Transition involving a \hkl<1 1 0> configuration}
1303 \item Bond-centered configuration unstable\\
1304 $\rightarrow$ \ci{} \hkl<1 1 0> dumbbell
1305 \item Minima of the \hkl[0 0 -1] to \hkl[0 -1 0] transition\\
1306 $\rightarrow$ \ci{} \hkl<1 1 0> DB
1309 \includegraphics[width=6.0cm]{00-1_110_0-10_mig_albe.ps}
1311 \item $\Delta E \approx \unit[2.2]{eV} \text{ \& } \unit[0.9]{eV}$
1312 \item 2.4 -- 3.4 times higher than ab initio result
1313 \item After all: Change of the DB orientation
1319 {\color{red}\bf Drastically overestimated diffusion barrier}
1322 \begin{pspicture}(0,0)(0,0)
1323 \psline[linewidth=0.05cm,linecolor=gray](6.1,1.0)(6.1,9.3)
1339 \begin{minipage}{9cm}
1341 Summary of combinations}\\[0.1cm]
1343 \begin{tabular}{l c c c c c c}
1345 $E_{\text{b}}$ [eV] & 1 & 2 & 3 & 4 & 5 & R\\
1347 \hkl[0 0 -1] & {\color{red}-0.08} & -1.15 & {\color{red}-0.08} & 0.04 & -1.66 & -0.19\\
1348 \hkl[0 0 1] & 0.34 & 0.004 & -2.05 & 0.26 & -1.53 & -0.19\\
1349 \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}\\
1350 \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}\\
1351 \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}\\
1352 \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}\\
1354 C$_{\text{sub}}$ & 0.26 & -0.51 & -0.93 & -0.15 & 0.49 & -0.05\\
1355 Vacancy & -5.39 ($\rightarrow$ C$_{\text{sub}}$) & -0.59 & -3.14 & -0.54 & -0.50 & -0.31\\
1362 $E_{\text{b}}$ explainable by stress compensation / increase
1366 \begin{minipage}{3cm}
1367 \includegraphics[width=3.5cm]{comb_pos.eps}
1372 {\bf\boldmath Combinations of \hkl<1 0 0>-type interstitials}\\[0.2cm]
1373 \begin{minipage}[t]{3.2cm}
1374 \underline{\hkl[1 0 0] at position 1}\\[0.1cm]
1375 \includegraphics[width=2.8cm]{00-1dc/2-25.eps}
1377 \begin{minipage}[t]{3.0cm}
1378 \underline{\hkl[0 -1 0] at position 1}\\[0.1cm]
1379 \includegraphics[width=2.8cm]{00-1dc/2-39.eps}
1381 \begin{minipage}[t]{6.1cm}
1384 \item \ci{} agglomeration energetically favorable
1385 \item Most favorable: C clustering\\
1386 {\color{red}However \ldots}\\
1387 \ldots high migration barrier ($>4\,\text{eV}$)\\
1389 $4\times{\color{cyan}[-2.25]}$ versus
1390 $2\times{\color{orange}[-2.39]}$
1393 {\color{blue}\ci{} agglomeration / no C clustering}
1410 \begin{minipage}{9cm}
1412 Summary of combinations}\\[0.1cm]
1414 \begin{tabular}{l c c c c c c}
1416 $E_{\text{b}}$ [eV] & 1 & 2 & 3 & 4 & 5 & R\\
1418 \hkl[0 0 -1] & {\color{red}-0.08} & -1.15 & {\color{red}-0.08} & 0.04 & -1.66 & -0.19\\
1419 \hkl[0 0 1] & 0.34 & 0.004 & -2.05 & 0.26 & -1.53 & -0.19\\
1420 \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}\\
1421 \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}\\
1422 \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}\\
1423 \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}\\
1425 C$_{\text{sub}}$ & 0.26 & -0.51 & -0.93 & -0.15 & 0.49 & -0.05\\
1426 Vacancy & -5.39 ($\rightarrow$ C$_{\text{sub}}$) & -0.59 & -3.14 & -0.54 & -0.50 & -0.31\\
1433 $E_{\text{b}}$ explainable by stress compensation / increase
1437 \begin{minipage}{3cm}
1438 \includegraphics[width=3.5cm]{comb_pos.eps}
1443 {\bf\boldmath Combinations of \hkl<1 0 0>-type interstitials}\\[0.2cm]
1444 \begin{minipage}[t]{3.2cm}
1445 \underline{\hkl[1 0 0] at position 1}\\[0.1cm]
1446 \includegraphics[width=2.8cm]{00-1dc/2-25.eps}
1448 \begin{minipage}[t]{3.0cm}
1449 \underline{\hkl[0 -1 0] at position 1}\\[0.1cm]
1450 \includegraphics[width=2.8cm]{00-1dc/2-39.eps}
1452 \begin{minipage}[t]{6.1cm}
1455 \item \ci{} agglomeration energetically favorable
1456 \item Most favorable: C clustering\\
1457 {\color{red}However \ldots}\\
1458 \ldots high migration barrier ($>4\,\text{eV}$)\\
1460 $4\times{\color{cyan}[-2.25]}$ versus
1461 $2\times{\color{orange}[-2.39]}$
1464 {\color{blue}\ci{} agglomeration / no C clustering}
1469 \begin{pspicture}(0,0)(0,0)
1470 \rput(6.5,5.0){\psframebox[fillstyle=solid,opacity=0.5,fillcolor=black]{
1471 \begin{minipage}{14cm}
1476 \rput(6.5,5.3){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=white,linewidth=0.1cm]{
1477 \begin{minipage}{8cm}
1481 Interaction along \hkl[1 1 0]
1482 \includegraphics[width=7cm]{db_along_110_cc.ps}
1494 Defect combinations of C-Si dimers and vacancies
1500 \begin{minipage}[b]{2.6cm}
1502 \underline{V at 2: $E_{\text{b}}=-0.59\text{ eV}$}\\[0.1cm]
1503 \includegraphics[width=2.5cm]{00-1dc/0-59.eps}
1506 \begin{minipage}[b]{7cm}
1509 \begin{minipage}[b]{2.6cm}
1511 \underline{V at 3, $E_{\text{b}}=-3.14\text{ eV}$}\\[0.1cm]
1512 \includegraphics[width=2.5cm]{00-1dc/3-14.eps}
1514 \end{minipage}\\[0.2cm]
1516 \begin{minipage}{6.5cm}
1517 \includegraphics[width=6.0cm]{059-539.ps}
1519 \begin{minipage}{5.7cm}
1520 \includegraphics[width=6.0cm]{314-539.ps}
1523 \begin{pspicture}(0,0)(0,0)
1524 \psline[linewidth=0.05cm,linecolor=gray](6.3,9.0)(6.3,0)
1526 \rput(6.3,7.0){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=white,linewidth=0.05cm,linecolor=gray]{
1527 \begin{minipage}{6.5cm}
1529 IBS: Impinging C creates V \& far away \si\\[0.3cm]
1530 Low migration barrier towards C$_{\text{sub}}$\\
1532 High barrier for reverse process\\[0.3cm]
1534 High probability of stable C$_{\text{sub}}$ configuration
1547 Combinations of substitutional C and Si self-interstitials
1554 \begin{minipage}{6.2cm}
1556 {\bf\boldmath C$_{\text{sub}}$ - \si{} \hkl<1 1 0> interaction}
1558 \item Most favorable: \cs{} along \hkl<1 1 0> chain of \si{}
1559 \item Less favorable than ground-state \ci{} \hkl<1 0 0> DB
1560 \item Interaction drops quickly to zero\\
1561 $\rightarrow$ low capture radius
1565 \begin{minipage}{0.2cm}
1568 \begin{minipage}{6.0cm}
1570 {\bf Transition from the ground state}
1572 \item Low transition barrier
1573 \item Barrier smaller than \ci{} migration barrier
1574 \item Low \si{} migration barrier (\unit[0.67]{eV})\\
1575 $\rightarrow$ Separation of \cs{} \& \si{} most probable
1578 \end{minipage}\\[0.3cm]
1580 \begin{minipage}{6.0cm}
1581 \includegraphics[width=6.0cm]{c_sub_si110.ps}
1583 \begin{minipage}{0.4cm}
1586 \begin{minipage}{6.0cm}
1588 \includegraphics[width=6.0cm]{162-097.ps}
1592 \begin{pspicture}(0,0)(0,0)
1593 \psline[linewidth=0.05cm,linecolor=gray](6.5,0)(6.5,7.5)
1594 \rput(6.5,-0.7){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=white,linewidth=0.05cm,linecolor=blue]{
1595 \begin{minipage}{8cm}
1599 \cs{} \& \si{} instead of thermodynamic ground state\\[0.1cm]
1600 IBS --- process far from equilibrium\\
1613 Combinations of substitutional C and Si self-interstitials
1620 \begin{minipage}{6.2cm}
1622 {\bf\boldmath C$_{\text{sub}}$ - \si{} \hkl<1 1 0> interaction}
1624 \item Most favorable: \cs{} along \hkl<1 1 0> chain of \si{}
1625 \item Less favorable than ground-state \ci{} \hkl<1 0 0> DB
1626 \item Interaction drops quickly to zero\\
1627 $\rightarrow$ low capture radius
1631 \begin{minipage}{0.2cm}
1634 \begin{minipage}{6.0cm}
1636 {\bf Transition from the ground state}
1638 \item Low transition barrier
1639 \item Barrier smaller than \ci{} migration barrier
1640 \item Low \si{} migration barrier (\unit[0.67]{eV})\\
1641 $\rightarrow$ Separation of \cs{} \& \si{} most probable
1644 \end{minipage}\\[0.3cm]
1646 \begin{minipage}{6.0cm}
1647 \includegraphics[width=6.0cm]{c_sub_si110.ps}
1649 \begin{minipage}{0.4cm}
1652 \begin{minipage}{6.0cm}
1654 \includegraphics[width=6.0cm]{162-097.ps}
1658 \begin{pspicture}(0,0)(0,0)
1659 \psline[linewidth=0.05cm,linecolor=gray](6.5,0)(6.5,7.5)
1660 \rput(6.5,-0.7){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=white,linewidth=0.05cm,linecolor=blue]{
1661 \begin{minipage}{8cm}
1665 \cs{} \& \si{} instead of thermodynamic ground state\\[0.1cm]
1666 IBS --- process far from equilibrium\\
1674 \begin{pspicture}(0,0)(0,0)
1675 \rput(6.5,5.0){\psframebox[fillstyle=solid,opacity=0.5,fillcolor=black]{
1676 \begin{minipage}{14cm}
1681 \rput(6.5,4.3){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=white,linewidth=0.1cm]{
1682 \begin{minipage}{11cm}
1686 Ab initio MD at \degc{900}\\[0.4cm]
1687 \begin{minipage}{5.4cm}
1689 \includegraphics[width=4.3cm]{md01_bonds.eps}\\
1692 \begin{minipage}{5.4cm}
1694 \includegraphics[width=4.3cm]{md02_bonds.eps}\\
1696 \end{minipage}\\[0.5cm]
1698 Contribution of entropy to structural formation\\[0.1cm]
1714 Silicon carbide precipitation simulations
1724 \begin{pspicture}(0,0)(12,6.5)
1726 \rput(3.5,5.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
1729 \item Create c-Si volume
1730 \item Periodc boundary conditions
1731 \item Set requested $T$ and $p=0\text{ bar}$
1732 \item Equilibration of $E_{\text{kin}}$ and $E_{\text{pot}}$
1735 \rput(3.5,2.7){\rnode{insert}{\psframebox[fillstyle=solid,fillcolor=lachs]{
1737 Insertion of C atoms at constant T
1739 \item total simulation volume {\pnode{in1}}
1740 \item volume of minimal SiC precipitate size {\pnode{in2}}
1741 \item volume consisting of Si atoms to form a minimal {\pnode{in3}}\\
1745 \rput(3.5,1){\rnode{cool}{\psframebox[fillstyle=solid,fillcolor=lbb]{
1747 Run for 100 ps followed by cooling down to $20\, ^{\circ}\textrm{C}$
1749 \ncline[]{->}{init}{insert}
1750 \ncline[]{->}{insert}{cool}
1751 \psframe[fillstyle=solid,fillcolor=white](7.3,0.7)(12.8,6.3)
1752 \rput(7.6,6){\footnotesize $V_1$}
1753 \psframe[fillstyle=solid,fillcolor=lightgray](8.7,2)(11.6,5)
1754 \rput(8.9,4.85){\tiny $V_2$}
1755 \psframe[fillstyle=solid,fillcolor=gray](8.95,2.25)(11.35,4.75)
1756 \rput(9.25,4.45){\footnotesize $V_3$}
1757 \rput(7.9,3.2){\pnode{ins1}}
1758 \rput(8.92,2.8){\pnode{ins2}}
1759 \rput(10.8,2.4){\pnode{ins3}}
1760 \ncline[]{->}{in1}{ins1}
1761 \ncline[]{->}{in2}{ins2}
1762 \ncline[]{->}{in3}{ins3}
1772 \begin{minipage}{5.7cm}
1774 \item Amount of C atoms: 6000\\
1775 ($r_{\text{prec}}\approx 3.1\text{ nm}$, IBS: \unit[2--4]{nm})
1776 \item Simulation volume: $31^3$ Si unit cells\\
1780 \begin{minipage}{0.3cm}
1784 \begin{minipage}{6.0cm}
1785 Restricted to classical potential caclulations\\
1786 $\rightarrow$ Low C diffusion / overestimated barrier\\
1787 $\rightarrow$ Consider $V_2$ and $V_3$
1789 % \item $V_2$ and $V_3$ considered due to expected low C diffusion
1800 Silicon carbide precipitation simulations at \degc{450} as in IBS
1805 \begin{minipage}{6.3cm}
1806 \hspace*{-0.4cm}\includegraphics[width=6.5cm]{sic_prec_450_si-c.ps}\\
1807 \hspace*{-0.4cm}\includegraphics[width=6.5cm]{sic_prec_450_si-si_c-c.ps}
1810 \begin{minipage}{6.1cm}
1812 \underline{Low C concentration --- {\color{red}$V_1$}}\\[0.1cm]
1813 \hkl<1 0 0> C-Si dumbbell dominated structure
1815 \item Si-C bumbs around \unit[0.19]{nm}
1816 \item C-C peak at \unit[0.31]{nm} (expected in 3C-SiC):\\
1817 concatenated differently oriented \ci{} DBs
1818 \item Si-Si NN distance stretched to \unit[0.3]{nm}
1820 \begin{pspicture}(0,0)(6.0,1.0)
1821 \rput(3.2,0.5){\psframebox[linewidth=0.03cm,linecolor=blue]{
1822 \begin{minipage}{6cm}
1824 Formation of \ci{} dumbbells\\
1825 C atoms in proper 3C-SiC distance first
1828 \end{pspicture}\\[0.1cm]
1829 \underline{High C concentration --- {\color{green}$V_2$}/{\color{blue}$V_3$}}
1831 \item High amount of strongly bound C-C bonds
1832 \item Increased defect \& damage density\\
1833 $\rightarrow$ Arrangements hard to categorize and trace
1834 \item Only short range order observable
1836 \begin{pspicture}(0,0)(6.0,0.8)
1837 \rput(3.2,0.5){\psframebox[linewidth=0.03cm,linecolor=blue]{
1838 \begin{minipage}{6cm}
1840 Amorphous SiC-like phase
1843 \end{pspicture}\\[0.3cm]
1844 \begin{pspicture}(0,0)(6.0,2.0)
1845 \rput(3.2,1.0){\psframebox[linewidth=0.05cm,linecolor=white]{
1846 \begin{minipage}{6cm}
1860 Silicon carbide precipitation simulations at \degc{450} as in IBS
1865 \begin{minipage}{6.3cm}
1866 \hspace*{-0.4cm}\includegraphics[width=6.5cm]{sic_prec_450_si-c.ps}\\
1867 \hspace*{-0.4cm}\includegraphics[width=6.5cm]{sic_prec_450_si-si_c-c.ps}
1870 \begin{minipage}{6.1cm}
1872 \underline{Low C concentration --- {\color{red}$V_1$}}\\[0.1cm]
1873 \hkl<1 0 0> C-Si dumbbell dominated structure
1875 \item Si-C bumbs around \unit[0.19]{nm}
1876 \item C-C peak at \unit[0.31]{nm} (expected in 3C-SiC):\\
1877 concatenated differently oriented \ci{} DBs
1878 \item Si-Si NN distance stretched to \unit[0.3]{nm}
1880 \begin{pspicture}(0,0)(6.0,1.0)
1881 \rput(3.2,0.5){\psframebox[linewidth=0.03cm,linecolor=blue]{
1882 \begin{minipage}{6cm}
1884 Formation of \ci{} dumbbells\\
1885 C atoms in proper 3C-SiC distance first
1888 \end{pspicture}\\[0.1cm]
1889 \underline{High C concentration --- {\color{green}$V_2$}/{\color{blue}$V_3$}}
1891 \item High amount of strongly bound C-C bonds
1892 \item Increased defect \& damage density\\
1893 $\rightarrow$ Arrangements hard to categorize and trace
1894 \item Only short range order observable
1896 \begin{pspicture}(0,0)(6.0,0.8)
1897 \rput(3.2,0.5){\psframebox[linewidth=0.03cm,linecolor=blue]{
1898 \begin{minipage}{6cm}
1900 Amorphous SiC-like phase
1903 \end{pspicture}\\[0.3cm]
1904 \begin{pspicture}(0,0)(6.0,2.0)
1905 \rput(3.2,1.0){\psframebox[linewidth=0.05cm,linecolor=black]{
1906 \begin{minipage}{6cm}
1909 {\bf\color{red}3C-SiC formation fails to appear}\\[0.3cm]
1910 \begin{minipage}{0.8cm}
1911 {\bf\boldmath $V_1$:}
1913 \begin{minipage}{5.1cm}
1914 Formation of \ci{} indeed occurs\\
1915 Agllomeration not observed
1916 \end{minipage}\\[0.3cm]
1917 \begin{minipage}{0.8cm}
1918 {\bf\boldmath $V_{2,3}$:}
1920 \begin{minipage}{5.1cm}
1921 Amorphous SiC-like structure\\
1922 (not expected at \degc{450})\\[0.05cm]
1923 No rearrangement/transition into 3C-SiC
1924 \end{minipage}\\[0.1cm]
1936 Limitations of MD and short range potentials
1943 {\bf Time scale problem of MD}\\[0.2cm]
1944 Precise integration \& thermodynamic sampling\\
1945 $\Rightarrow$ $\Delta t \ll \left( \max{\omega} \right)^{-1}$,
1946 $\omega$: vibrational mode\\
1947 $\Rightarrow$ {\color{red}\underline{Slow}} phase space propagation\\[0.2cm]
1948 Several local minima separated by large energy barriers\\
1949 $\Rightarrow$ Transition event corresponds to a multiple
1950 of vibrational periods\\
1951 $\Rightarrow$ Phase transition consists of {\color{red}\underline{many}}
1952 infrequent transition events\\[0.2cm]
1953 {\color{blue}Accelerated methods:}
1954 \underline{Temperature accelerated} MD (TAD), self-guided MD \ldots
1958 {\bf Limitations related to the short range potential}\\[0.2cm]
1959 Cut-off function limits interaction to next neighbours\\
1960 $\Rightarrow$ Overestimated unphysical high forces of next neighbours
1965 {\bf Approach to the (twofold) problem}\\[0.2cm]
1966 Increased temperature simulations without TAD corrections\\
1967 Accelerated methods or higher time scales exclusively not sufficient!
1969 \begin{pspicture}(0,0)(0,0)
1970 \rput(4.0,2.8){\psframebox[linewidth=0.07cm,linecolor=red]{
1971 \begin{minipage}{7.5cm}
1974 Potential enhanced slow phase space propagation
1977 \rput(11.3,7.5){\psframebox[linewidth=0.03cm,linecolor=blue]{
1978 \begin{minipage}{2.7cm}
1982 thermodynamic sampling
1985 \psline[linewidth=0.03cm,linecolor=blue]{<-}(11.3,7.0)(11.0,5.7)
1986 \rput(10.85,2.6){\psframebox[linewidth=0.03cm,linecolor=blue]{
1987 \begin{minipage}{3.6cm}
1990 \underline{IBS}\\[0.1cm]
1991 3C-SiC also observed for higher T\\[0.1cm]
1992 Higher T inside sample\\[0.1cm]
1993 Structural evolution vs.\\
1994 equilibrium properties
1997 \psline[linewidth=0.03cm,linecolor=blue]{->}(10.85,1.75)(9.0,1.0)
2006 Increased temperature simulations --- $V_1$
2011 \begin{minipage}{6.2cm}
2012 \hspace*{-0.4cm}\includegraphics[width=6.5cm]{tot_pc_thesis.ps}
2015 \begin{minipage}{6.2cm}
2016 \includegraphics[width=6.5cm]{tot_pc3_thesis.ps}
2019 \begin{minipage}{6.2cm}
2020 \hspace*{-0.4cm}\includegraphics[width=6.5cm]{tot_pc2_thesis.ps}
2023 \begin{minipage}{6.3cm}
2025 \underline{Si-C bonds:}
2027 \item Vanishing cut-off artifact (above $1650\,^{\circ}\mathrm{C}$)
2028 \item Structural change: C-Si \hkl<1 0 0> $\rightarrow$ C$_{\text{sub}}$
2030 \underline{Si-Si bonds:}
2031 {\color{blue}Si-C$_{\text{sub}}$-Si} along \hkl<1 1 0>
2032 ($\rightarrow$ 0.325 nm)\\[0.1cm]
2033 \underline{C-C bonds:}
2035 \item C-C next neighbour pairs reduced (mandatory)
2036 \item Peak at 0.3 nm slightly shifted
2038 \item C-Si \hkl<1 0 0> combinations (dashed arrows)\\
2039 $\rightarrow$ C-Si \hkl<1 0 0> \& C$_{\text{sub}}$
2041 $\rightarrow$ pure {\color{blue}C$_{\text{sub}}$ combinations}
2043 \item Range [|-$\downarrow$]:
2044 {\color{blue}C$_{\text{sub}}$ \& C$_{\text{sub}}$
2045 with nearby Si$_{\text{I}}$}
2056 Increased temperature simulations --- $V_1$
2061 \begin{minipage}{6.2cm}
2062 \hspace*{-0.4cm}\includegraphics[width=6.5cm]{tot_pc_thesis.ps}
2065 \begin{minipage}{6.2cm}
2066 \includegraphics[width=6.5cm]{tot_pc3_thesis.ps}
2069 \begin{minipage}{6.2cm}
2070 \hspace*{-0.4cm}\includegraphics[width=6.5cm]{tot_pc2_thesis.ps}
2073 \begin{minipage}{6.3cm}
2075 \underline{Si-C bonds:}
2077 \item Vanishing cut-off artifact (above $1650\,^{\circ}\mathrm{C}$)
2078 \item Structural change: C-Si \hkl<1 0 0> $\rightarrow$ C$_{\text{sub}}$
2080 \underline{Si-Si bonds:}
2081 {\color{blue}Si-C$_{\text{sub}}$-Si} along \hkl<1 1 0>
2082 ($\rightarrow$ 0.325 nm)\\[0.1cm]
2083 \underline{C-C bonds:}
2085 \item C-C next neighbour pairs reduced (mandatory)
2086 \item Peak at 0.3 nm slightly shifted
2088 \item C-Si \hkl<1 0 0> combinations (dashed arrows)\\
2089 $\rightarrow$ C-Si \hkl<1 0 0> \& C$_{\text{sub}}$
2091 $\rightarrow$ pure {\color{blue}C$_{\text{sub}}$ combinations}
2093 \item Range [|-$\downarrow$]:
2094 {\color{blue}C$_{\text{sub}}$ \& C$_{\text{sub}}$
2095 with nearby Si$_{\text{I}}$}
2101 \begin{pspicture}(0,0)(0,0)
2102 \rput(6.5,5.0){\psframebox[fillstyle=solid,opacity=0.5,fillcolor=black]{
2103 \begin{minipage}{14cm}
2108 \rput(6.5,5.0){\psframebox[fillstyle=solid,fillcolor=white,linewidth=0.1cm]{
2109 \begin{minipage}{9cm}
2113 {\color{gray}\bf Conclusions on SiC precipitation}\\[0.1cm]
2114 {\Huge$\lightning$} {\color{red}\ci{}} --- vs --- {\color{blue}\cs{}} {\Huge$\lightning$}\\
2117 \item Stretched coherent SiC structures\\
2118 $\Rightarrow$ Precipitation process involves {\color{blue}\cs}
2119 \item Explains annealing behavior of high/low T C implantations
2121 \item Low T: highly mobile {\color{red}\ci}
2122 \item High T: stable configurations of {\color{blue}\cs}
2126 \item Vehicle to rearrange \cs --- [\cs{} \& \si{} $\leftrightarrow$ \ci]
2127 \item Building block for surrounding Si host \& further SiC
2128 \item Strain compensation \ldots\\
2129 \ldots Si/SiC interface\\
2130 \ldots within stretched coherent SiC structure
2135 \psframebox[linecolor=blue,linewidth=0.05cm]{
2136 \begin{minipage}{7cm}
2138 Precipitation mechanism involving \cs\\
2139 High T $\leftrightarrow$ IBS conditions far from equilibrium\\
2152 Increased temperature simulations at high C concentration
2157 \begin{minipage}{6.5cm}
2158 \includegraphics[width=6.4cm]{12_pc_thesis.ps}
2160 \begin{minipage}{6.5cm}
2161 \includegraphics[width=6.4cm]{12_pc_c_thesis.ps}
2169 \begin{minipage}[t]{6.0cm}
2170 0.186 nm: Si-C pairs $\uparrow$\\
2171 (as expected in 3C-SiC)\\[0.2cm]
2172 0.282 nm: Si-C-C\\[0.2cm]
2173 $\approx$0.35 nm: C-Si-Si
2176 \begin{minipage}{0.2cm}
2180 \begin{minipage}[t]{6.0cm}
2181 0.15 nm: C-C pairs $\uparrow$\\
2182 (as expected in graphite/diamond)\\[0.2cm]
2183 0.252 nm: C-C-C (2$^{\text{nd}}$ NN for diamond)\\[0.2cm]
2184 0.31 nm: shifted towards 0.317 nm $\rightarrow$ C-Si-C
2189 \item Decreasing cut-off artifact
2190 \item {\color{red}Amorphous} SiC-like phase remains
2191 \item High amount of {\color{red}damage} \& alignement to c-Si host matrix lost
2192 \item Slightly sharper peaks $\Rightarrow$ indicate slight {\color{blue}acceleration of dynamics} due to temperature
2201 High C \& small $V$ \& short $t$
2204 Slow restructuring due to strong C-C bonds
2207 High C \& low T implants
2219 Summary / Conclusions
2224 \begin{pspicture}(0,0)(12,1.0)
2225 \psframebox[fillstyle=gradient,gradbegin=hred,gradend=white,gradlines=1000,gradmidpoint=1.0,linestyle=none]{
2226 \begin{minipage}{11cm}
2227 {\color{black}Diploma thesis}\\
2228 \underline{Monte Carlo} simulation modeling the selforganization process\\
2229 leading to periodic arrays of nanometric amorphous SiC precipitates
2232 \end{pspicture}\\[0.4cm]
2233 \begin{pspicture}(0,0)(12,2)
2234 \psframebox[fillstyle=gradient,gradbegin=hblue,gradend=white,gradmidpoint=1.0,gradlines=1000,linestyle=none]{
2235 \begin{minipage}{11cm}
2236 {\color{black}Doctoral studies}\\
2237 Classical potential \underline{molecular dynamics} simulations \ldots\\
2238 \underline{Density functional theory} calculations \ldots\\[0.2cm]
2239 \ldots on defect formation and SiC precipitation in Si
2242 \end{pspicture}\\[0.5cm]
2243 \begin{pspicture}(0,0)(12,3)
2244 \psframebox[fillstyle=solid,fillcolor=white,linestyle=solid]{
2245 \begin{minipage}{11cm}
2247 {\color{black}\bf How to proceed \ldots}\\[0.1cm]
2248 MC $\rightarrow$ empirical potential MD $\rightarrow$ Ground-state DFT \ldots
2250 \renewcommand\labelitemi{$\ldots$}
2251 \item beyond LDA/GGA methods \& ground-state DFT
2253 Investigation of structure \& structural evolution \ldots
2255 \renewcommand\labelitemi{$\ldots$}
2256 \item electronic/optical properties
2257 \item electronic correlations
2258 \item non-equilibrium systems
2262 \end{pspicture}\\[0.5cm]
2278 \underline{Augsburg}
2280 \item Prof. B. Stritzker (accomodation at EP \RM{4})
2281 \item Ralf Utermann (EDV)
2284 \underline{Berlin/Brandenburg}
2286 \item PD Volker Eyert (Ref)
2289 \underline{Helsinki}
2291 \item Prof. K. Nordlund (MD)
2296 \item Bayerische Forschungsstiftung (financial support)
2299 \underline{Paderborn}
2301 \item Prof. J. Lindner (SiC)
2302 \item Prof. G. Schmidt (DFT + financial support)
2303 \item Dr. E. Rauls (DFT + SiC)
2308 \bf Thank you for your attention!