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
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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 \newcommand{\headdiplom}{
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61 \rput(6.0,0.2){\psframebox[fillstyle=gradient,gradbegin=red,gradend=white,gradlines=1000,gradmidpoint=1,linestyle=none]{
62 \begin{minipage}{14cm}
70 \newcommand{\headphd}{
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73 \begin{minipage}{14cm}
83 \extraslideheight{10in}
88 % specify width and height
93 \def\slidetopmargin{-0.15cm}
95 \newcommand{\ham}{\mathcal{H}}
96 \newcommand{\pot}{\mathcal{V}}
97 \newcommand{\foo}{\mathcal{U}}
98 \newcommand{\vir}{\mathcal{W}}
101 \renewcommand\labelitemii{{\color{gray}$\bullet$}}
104 \renewcommand{\phi}{\varphi}
107 \newcommand{\RM}[1]{\MakeUppercase{\romannumeral #1{}}}
110 \newrgbcolor{si-yellow}{.6 .6 0}
111 \newrgbcolor{hb}{0.75 0.77 0.89}
112 \newrgbcolor{lbb}{0.75 0.8 0.88}
113 \newrgbcolor{hlbb}{0.825 0.88 0.968}
114 \newrgbcolor{lachs}{1.0 .93 .81}
117 \newcommand{\si}{Si$_{\text{i}}${}}
118 \newcommand{\ci}{C$_{\text{i}}${}}
119 \newcommand{\cs}{C$_{\text{sub}}${}}
120 \newcommand{\degc}[1]{\unit[#1]{$^{\circ}$C}{}}
121 \newcommand{\distn}[1]{\unit[#1]{nm}{}}
122 \newcommand{\dista}[1]{\unit[#1]{\AA}{}}
123 \newcommand{\perc}[1]{\unit[#1]{\%}{}}
125 % no vertical centering
136 A B C D E F G H G F E D C B A
151 Atomistic simulation studies\\[0.2cm]
157 \textsc{Frank Zirkelbach}
161 Application talk at the Max Planck Institute for Solid State Research
165 Stuttgart, November 2011
170 % no vertical centering
180 % Phase diagram of the C/Si system\\
185 \begin{minipage}{6.5cm}
186 \includegraphics[width=6.5cm]{si-c_phase.eps}
189 R. I. Scace and G. A. Slack, J. Chem. Phys. 30, 1551 (1959)
192 \begin{pspicture}(0,0)(0,0)
193 \psellipse[linecolor=blue,linewidth=0.1cm](3.55,4.0)(0.5,2.9)
196 \begin{minipage}{6cm}
197 {\bf Phase diagram of the C/Si system}\\[0.2cm]
198 {\color{blue}Stoichiometric composition}
200 \item only chemical stable compound
201 \item wide band gap semiconductor\\
202 \underline{silicon carbide}, SiC
208 % motivation / properties / applications of silicon carbide
216 \begin{pspicture}(0,0)(13.5,5)
218 \psframe*[linecolor=hb](-0.2,0)(12.9,5)
220 \pspolygon[linecolor=hlbb,fillcolor=hlbb,fillstyle=solid](5.2,1)(6.5,1)(6.5,3)(5.2,3)
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223 \rput[lt](0,4.6){\color{gray}PROPERTIES}
225 \rput[lt](0.3,4){wide band gap}
226 \rput[lt](0.3,3.5){high electric breakdown field}
227 \rput[lt](0.3,3){good electron mobility}
228 \rput[lt](0.3,2.5){high electron saturation drift velocity}
229 \rput[lt](0.3,2){high thermal conductivity}
231 \rput[lt](0.3,1.5){hard and mechanically stable}
232 \rput[lt](0.3,1){chemically inert}
234 \rput[lt](0.3,0.5){radiation hardness}
236 \rput[rt](12.7,4.6){\color{gray}APPLICATIONS}
238 \rput[rt](12.5,3.85){high-temperature, high power}
239 \rput[rt](12.5,3.5){and high-frequency}
240 \rput[rt](12.5,3.15){electronic and optoelectronic devices}
242 \rput[rt](12.5,2.35){material suitable for extreme conditions}
243 \rput[rt](12.5,2){microelectromechanical systems}
244 \rput[rt](12.5,1.65){abrasives, cutting tools, heating elements}
246 \rput[rt](12.5,0.85){first wall reactor material, detectors}
247 \rput[rt](12.5,0.5){and electronic devices for space}
251 \begin{picture}(0,0)(5,-162)
252 \includegraphics[height=2.2cm]{3C_SiC_bs.eps}
254 \begin{picture}(0,0)(-120,-162)
255 \includegraphics[height=2.2cm]{nasa_600c_led.eps}
257 \begin{picture}(0,0)(-270,-162)
258 \includegraphics[height=2.2cm]{6h-sic_3c-sic.eps}
261 \begin{picture}(0,0)(10,65)
262 \includegraphics[height=2.8cm]{sic_switch.eps}
264 %\begin{picture}(0,0)(-243,65)
265 \begin{picture}(0,0)(-110,65)
266 \includegraphics[height=2.8cm]{ise_99.eps}
268 %\begin{picture}(0,0)(-135,65)
269 \begin{picture}(0,0)(-100,65)
270 \includegraphics[height=1.2cm]{infineon_schottky.eps}
272 \begin{picture}(0,0)(-233,65)
273 \includegraphics[height=2.8cm]{solar_car.eps}
283 Polytypes of SiC\\[0.4cm]
286 \includegraphics[width=3.8cm]{cubic_hex.eps}\\
287 \begin{minipage}{1.9cm}
288 {\tiny cubic (twist)}
290 \begin{minipage}{2.9cm}
291 {\tiny hexagonal (no twist)}
294 \begin{picture}(0,0)(-150,0)
295 \includegraphics[width=7cm]{polytypes.eps}
302 \begin{tabular}{l c c c c c c}
304 & 3C-SiC & 4H-SiC & 6H-SiC & Si & GaN & Diamond\\
306 Hardness [Mohs] & \multicolumn{3}{c}{------ 9.6 ------}& 6.5 & - & 10 \\
307 Band gap [eV] & 2.36 & 3.23 & 3.03 & 1.12 & 3.39 & 5.5 \\
308 Break down field [$10^6$ V/cm] & 4 & 3 & 3.2 & 0.6 & 5 & 10 \\
309 Saturation drift velocity [$10^7$ cm/s] & 2.5 & 2.0 & 2.0 & 1 & 2.7 & 2.7 \\
310 Electron mobility [cm$^2$/Vs] & 800 & 900 & 400 & 1100 & 900 & 2200 \\
311 Hole mobility [cm$^2$/Vs] & 320 & 120 & 90 & 420 & 150 & 1600 \\
312 Thermal conductivity [W/cmK] & 5.0 & 4.9 & 4.9 & 1.5 & 1.3 & 22 \\
316 \begin{pspicture}(0,0)(0,0)
317 \psellipse[linecolor=green](5.7,2.10)(0.4,0.5)
319 \begin{pspicture}(0,0)(0,0)
320 \psellipse[linecolor=green](5.6,0.92)(0.4,0.2)
322 \begin{pspicture}(0,0)(0,0)
323 \psellipse[linecolor=red](10.45,0.45)(0.4,0.2)
333 Fabrication of silicon carbide
342 \emph{Silicon carbide --- Born from the stars, perfected on earth.}
348 SiC thin films by MBE \& CVD
350 \item Much progress achieved in homo/heteroepitaxial SiC thin film growth
351 \item \underline{Commercially available} semiconductor power devices based on
352 \underline{\foreignlanguage{greek}{a}-SiC}
353 \item Production of favored \underline{3C-SiC} material
354 \underline{less advanced}
355 \item Quality and size not yet sufficient
357 \begin{picture}(0,0)(-310,-20)
358 \includegraphics[width=2.0cm]{cree.eps}
363 Alternative approach:
364 Ion beam synthesis (IBS) of burried 3C-SiC layers in Si\hkl(1 0 0)
371 \begin{minipage}{3.15cm}
373 \includegraphics[width=3cm]{imp.eps}\\
379 \begin{minipage}{3.15cm}
381 \includegraphics[width=3cm]{annealing.eps}\\
383 \unit[12]{h} annealing at \degc{1200}
388 \begin{minipage}{5.5cm}
389 \includegraphics[width=5.8cm]{ibs_3c-sic.eps}\\[-0.2cm]
392 XTEM: single crystalline 3C-SiC in Si\hkl(1 0 0)
404 Systematic investigation of C implantations into Si
410 \includegraphics[width=0.75\textwidth]{imp_inv.eps}
426 \includegraphics[width=0.75\textwidth]{imp_inv.eps}
429 \begin{pspicture}(0,0)(0,0)
430 \rput(6.0,7.0){\rnode{init}{\psframebox[fillstyle=gradient,gradbegin=red,gradend=white,gradlines=1000,gradmidpoint=1.0,linestyle=none]{
431 \begin{minipage}{11cm}
432 {\color{black}Diploma thesis}\\
433 \underline{Monte Carlo} simulation modeling the selforganization process\\
434 leading to periodic arrays of nanometric amorphous SiC precipitates
438 \begin{pspicture}(0,0)(0,0)
439 \rput(6.0,-0.5){\rnode{init}{\psframebox[fillstyle=gradient,gradbegin=blue,gradend=white,gradmidpoint=1.0,gradlines=1000,linestyle=none]{
440 \begin{minipage}{11cm}
441 {\color{black}Doctoral studies}\\
442 Classical potential \underline{molecular dynamics} simulations \ldots\\
443 \underline{Density functional theory} calculations \ldots\\[0.2cm]
444 \ldots on defect formation and SiC precipitation in Si
448 \begin{pspicture}(0,0)(0,0)
449 \psellipse[linecolor=red,linewidth=0.05cm](5,3.0)(0.8,1.0)
451 \begin{pspicture}(0,0)(0,0)
452 \psellipse[linecolor=blue,linewidth=0.05cm](8.2,3.2)(1.5,1.6)
461 Selforganization of nanometric amorphous SiC lamellae
469 \item Regularly spaced, nanometric spherical\\
470 and lamellar amorphous inclusions\\
471 at the upper a/c interface
472 \item Carbon accumulation\\
478 \begin{minipage}{12cm}
479 \includegraphics[width=9cm]{../../nlsop/img/k393abild1_e_l.eps}\\
481 XTEM bright-field, \unit[180]{keV} C$^+ \rightarrow$ Si,
482 {\color{red}\underline{\degc{150}}},
483 Dose: \unit[4.3 $\times 10^{17}$]{cm$^{-2}$}
487 \begin{picture}(0,0)(-182,-215)
488 \begin{minipage}{6.5cm}
490 \includegraphics[width=6.5cm]{../../nlsop/img/eftem.eps}\\[-0.2cm]
492 XTEM bright-field and respective EFTEM C map
504 Model displaying the formation of ordered lamellae
510 \includegraphics[width=8.0cm]{../../nlsop/img/modell_ng_e.eps}
516 \item Supersaturation of C in c-Si\\
517 $\rightarrow$ {\bf Carbon induced} nucleation of spherical
519 \item High interfacial energy between 3C-SiC and c-Si\\
520 $\rightarrow$ {\bf Amorphous} precipitates
521 \item \unit[20-- 30]{\%} lower silicon density of a-SiC$_x$ compared to c-Si\\
522 $\rightarrow$ {\bf Lateral strain} (black arrows)
523 \item Implantation range near surface\\
524 $\rightarrow$ {\bf Relaxation} of {\bf vertical strain component}
525 \item Reduction of the carbon supersaturation in c-Si\\
526 $\rightarrow$ {\bf Carbon diffusion} into amorphous volumina
528 \item Remaining lateral strain\\
529 $\rightarrow$ {\bf Strain enhanced} lateral amorphisation
530 \item Absence of crystalline neighbours (structural information)\\
531 $\rightarrow$ {\bf Stabilization} of amorphous inclusions
532 {\bf against recrystallization}
541 Implementation of the Monte Carlo code
547 \item \underline{Amorphization / Recrystallization}\\
548 Ion collision in discretized target determined by random numbers
549 distributed according to nuclear energy loss.
550 Amorphization/recrystallization probability:
552 p_{c \rightarrow a}(\vec{r}) = {\color{green} p_b} + {\color{blue} p_c c_C(\vec{r})} + {\color{red} \sum_{\textrm{amorphous neighbours}} \frac{p_s c_C(\vec{r'})}{(r-r')^2}}
555 \item {\color{green} $p_b$} normal `ballistic' amorphization
556 \item {\color{blue} $p_c$} carbon induced amorphization
557 \item {\color{red} $p_s$} stress enhanced amorphization
560 p_{a \rightarrow c}(\vec r) = (1 - p_{c \rightarrow a}(\vec r)) \Big(1 - \frac{\sum_{direct \, neighbours} \delta (\vec{r'})}{6} \Big) \, \textrm{,}
563 \delta (\vec r) = \left\{
565 1 & \textrm{if volume at position $\vec r$ is amorphous} \\
566 0 & \textrm{otherwise} \\
570 \item \underline{Carbon incorporation}\\
571 Incorporation volume determined according to implantation profile
572 \item \underline{Diffusion / Sputtering}
574 \item Transfer fraction of C atoms
575 of crystalline into neighbored amorphous volumes
576 \item Remove surface layer
584 \begin{minipage}{3.7cm}
585 \begin{pspicture}(0,0)(0,0)
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587 \begin{minipage}{3.7cm}
601 Evolution of the \ldots
606 \item lamellar precipitates
608 \ldots reproduced!\\[1.4cm]
612 Experiment \& simulation\\
613 in good agreement\\[1.0cm]
615 Simulation is able to model the whole depth region\\[1.2cm]
620 \begin{minipage}{0.5cm}
623 \begin{minipage}{8.0cm}
625 \includegraphics[width=9cm]{../../nlsop/img/dosis_entwicklung_ng_e_1-2.eps}\\
626 \includegraphics[width=9cm]{../../nlsop/img/dosis_entwicklung_ng_e2_2-2.eps}
635 Structural \& compositional details
638 \begin{minipage}[t]{7.5cm}
639 \includegraphics[height=6.5cm]{../../nlsop/img/ac_cconc_ver2_e.eps}\\
641 \begin{minipage}[t]{5.0cm}
642 \includegraphics[height=6.5cm]{../../nlsop/img/97_98_e.eps}
650 \item Fluctuation of C concentration in lamellae region
651 \item \unit[8--10]{at.\%} C saturation limit
652 within the respective conditions
653 \item Complementarily arranged and alternating sequence of layers\\
654 with a high and low amount of amorphous regions
655 \item C accumulation in the amorphous phase / Origin of stress
658 \begin{picture}(0,0)(-260,-50)
660 \begin{minipage}{3cm}
663 Precipitation process\\
678 Formation of epitaxial single crystalline 3C-SiC
687 \item \underline{Implantation step 1}\\[0.1cm]
688 Almost stoichiometric dose | \unit[180]{keV} | \degc{500}\\
689 $\Rightarrow$ Epitaxial {\color{blue}3C-SiC} layer \&
690 {\color{blue}precipitates}
691 \item \underline{Implantation step 2}\\[0.1cm]
692 Little remaining dose | \unit[180]{keV} | \degc{250}\\
694 Destruction/Amorphization of precipitates at layer interface
695 \item \underline{Annealing}\\[0.1cm]
696 \unit[10]{h} at \degc{1250}\\
697 $\Rightarrow$ Homogeneous 3C-SiC layer with sharp interfaces
701 \begin{minipage}{7cm}
702 \includegraphics[width=7cm]{ibs_3c-sic.eps}
704 \begin{minipage}{5cm}
705 \begin{pspicture}(0,0)(0,0)
707 \psframebox[fillstyle=solid,fillcolor=white,linecolor=blue,linestyle=solid]{
708 \begin{minipage}{5.3cm}
711 3C-SiC precipitation\\
712 not yet fully understood
716 \renewcommand\labelitemi{$\Rightarrow$}
717 Details of the SiC precipitation
719 \item significant technological progress\\
720 in SiC thin film formation
721 \item perspectives for processes relying\\
722 upon prevention of SiC precipitation
726 \rput(-6.8,5.4){\pnode{h0}}
727 \rput(-3.0,5.4){\pnode{h1}}
728 \ncline[linecolor=blue]{-}{h0}{h1}
729 \ncline[linecolor=blue]{->}{h1}{box}
739 Supposed precipitation mechanism of SiC in Si
747 \begin{minipage}{3.6cm}
749 Si \& SiC lattice structure\\[0.1cm]
750 \includegraphics[width=2.3cm]{sic_unit_cell.eps}
753 \begin{minipage}{1.7cm}
754 \underline{Silicon}\\
755 {\color{yellow}$\bullet$} Si | {\color{gray}$\bullet$} Si\\
756 $a=\unit[5.429]{\\A}$\\
757 $\rho^*_{\text{Si}}=\unit[100]{\%}$
759 \begin{minipage}{1.7cm}
760 \underline{Silicon carbide}\\
761 {\color{yellow}$\bullet$} Si | {\color{gray}$\bullet$} C\\
762 $a=\unit[4.359]{\\A}$\\
763 $\rho^*_{\text{Si}}=\unit[97]{\%}$
769 \begin{minipage}{4.1cm}
771 \includegraphics[width=3.3cm]{tem_c-si-db.eps}
775 \begin{minipage}{4.0cm}
777 \includegraphics[width=3.3cm]{tem_3c-sic.eps}
783 \begin{minipage}{4.0cm}
785 C-Si dimers (dumbbells)\\[-0.1cm]
786 on Si interstitial sites
790 \begin{minipage}{4.1cm}
792 Agglomeration of C-Si dumbbells\\[-0.1cm]
793 $\Rightarrow$ dark contrasts
797 \begin{minipage}{4.0cm}
799 Precipitation of 3C-SiC in Si\\[-0.1cm]
800 $\Rightarrow$ Moir\'e fringes\\[-0.1cm]
801 \& release of Si self-interstitials
807 \begin{minipage}{4.0cm}
809 \includegraphics[width=3.3cm]{sic_prec_seq_01.eps}
813 \begin{minipage}{4.1cm}
815 \includegraphics[width=3.3cm]{sic_prec_seq_02.eps}
819 \begin{minipage}{4.0cm}
821 \includegraphics[width=3.3cm]{sic_prec_seq_03.eps}
825 \begin{pspicture}(0,0)(0,0)
826 \psline[linewidth=2pt]{->}(8.3,2)(8.8,2)
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828 \rput{-20}{\psellipse[linecolor=blue](3.1,8.2)(0.3,0.5)}
829 \psline[linewidth=2pt]{->}(3.9,2)(4.4,2)
830 \rput(11.8,0.3){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
831 $4a_{\text{Si}}=5a_{\text{SiC}}$
833 \rput(11.5,8){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
834 \hkl(h k l) planes match
836 \rput(8.5,6.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
847 Supposed precipitation mechanism of SiC in Si
855 \begin{minipage}{3.6cm}
857 Si \& SiC lattice structure\\[0.1cm]
858 \includegraphics[width=2.3cm]{sic_unit_cell.eps}
861 \begin{minipage}{1.7cm}
862 \underline{Silicon}\\
863 {\color{yellow}$\bullet$} Si | {\color{gray}$\bullet$} Si\\
864 $a=\unit[5.429]{\\A}$\\
865 $\rho^*_{\text{Si}}=\unit[100]{\%}$
867 \begin{minipage}{1.7cm}
868 \underline{Silicon carbide}\\
869 {\color{yellow}$\bullet$} Si | {\color{gray}$\bullet$} C\\
870 $a=\unit[4.359]{\\A}$\\
871 $\rho^*_{\text{Si}}=\unit[97]{\%}$
877 \begin{minipage}{4.1cm}
879 \includegraphics[width=3.3cm]{tem_c-si-db.eps}
883 \begin{minipage}{4.0cm}
885 \includegraphics[width=3.3cm]{tem_3c-sic.eps}
891 \begin{minipage}{4.0cm}
893 C-Si dimers (dumbbells)\\[-0.1cm]
894 on Si interstitial sites
898 \begin{minipage}{4.1cm}
900 Agglomeration of C-Si dumbbells\\[-0.1cm]
901 $\Rightarrow$ dark contrasts
905 \begin{minipage}{4.0cm}
907 Precipitation of 3C-SiC in Si\\[-0.1cm]
908 $\Rightarrow$ Moir\'e fringes\\[-0.1cm]
909 \& release of Si self-interstitials
915 \begin{minipage}{4.0cm}
917 \includegraphics[width=3.3cm]{sic_prec_seq_01.eps}
921 \begin{minipage}{4.1cm}
923 \includegraphics[width=3.3cm]{sic_prec_seq_02.eps}
927 \begin{minipage}{4.0cm}
929 \includegraphics[width=3.3cm]{sic_prec_seq_03.eps}
933 \begin{pspicture}(0,0)(0,0)
934 \psline[linewidth=2pt]{->}(8.3,2)(8.8,2)
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936 \rput{-20}{\psellipse[linecolor=blue](3.1,8.2)(0.3,0.5)}
937 \psline[linewidth=2pt]{->}(3.9,2)(4.4,2)
938 \rput(11.8,0.3){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
939 $4a_{\text{Si}}=5a_{\text{SiC}}$
941 \rput(11.5,8){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
942 \hkl(h k l) planes match
944 \rput(8.5,6.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
947 % controversial view!
948 \rput(6.5,5.0){\psframebox[fillstyle=solid,opacity=0.5,fillcolor=black]{
949 \begin{minipage}{14cm}
954 \rput(6.5,5.3){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=white,linewidth=0.1cm]{
955 \begin{minipage}{10cm}
959 {\color{gray}\bf Controversial findings}
962 \item High-temperature implantation {\tiny\color{gray}/Nejim~et~al./}
964 \item C incorporated {\color{blue}substitutionally} on regular Si lattice sites
965 \item \si{} reacting with further C in cleared volume
967 \item Annealing behavior {\tiny\color{gray}/Serre~et~al./}
969 \item Room temperature implantation $\rightarrow$ high C diffusion
970 \item Elevated temperature implantation $\rightarrow$ no C redistribution
972 $\Rightarrow$ mobile {\color{red}\ci} opposed to
973 stable {\color{blue}\cs{}} configurations
974 \item Strained silicon \& Si/SiC heterostructures
975 {\tiny\color{gray}/Strane~et~al./Guedj~et~al./}
977 \item {\color{blue}Coherent} SiC precipitates (tensile strain)
978 \item Incoherent SiC (strain relaxation)
983 {\Huge${\lightning}$} \hspace{0.3cm}
984 {\color{blue}\cs{}} --- vs --- {\color{red}\ci} \hspace{0.3cm}
985 {\Huge${\lightning}$}
1001 Utilized computational methods
1008 {\bf Molecular dynamics (MD)}\\
1010 \begin{tabular}{p{4.5cm} p{7.5cm}}
1013 System of $N$ particles &
1014 $N=5832\pm 1$ (Defects), $N=238328+6000$ (Precipitation)\\
1016 Phase space propagation &
1017 Velocity Verlet | timestep: \unit[1]{fs} \\
1019 Analytical interaction potential &
1020 Tersoff-like {\color{red}short-range}, {\color{blue}bond order} potential
1023 E = \frac{1}{2} \sum_{i \neq j} \pot_{ij}, \quad
1024 \pot_{ij} = {\color{red}f_C(r_{ij})}
1025 \left[ f_R(r_{ij}) + {\color{blue}b_{ij}} f_A(r_{ij}) \right]
1028 Observables: time/ensemble averages &
1029 NpT (isothermal-isobaric) | Berendsen thermostat/barostat\\
1037 {\bf Density functional theory (DFT)}
1041 \begin{minipage}[t]{6cm}
1044 \item Born-Oppenheimer approximation:\\
1045 Decouple electronic \& ionic motion
1046 \item Hohenberg-Kohn theorem:\\
1047 $n_0(r) \stackrel{\text{uniquely}}{\rightarrow}$
1048 $V_0$ / $H$ / $\Phi_i$ / \underline{$E_0$}
1052 \item Code: \textsc{vasp}
1053 \item Plane wave basis set $\{\phi_j\}$\\[0.1cm]
1055 \Phi_i=\sum_{|G+k|<G_{\text{cut}}} c_j^i \phi_j(r)
1058 E_{\text{cut}}=\frac{\hbar^2}{2m}G^2_{\text{cut}}=\unit[300]{eV}
1060 \item Ultrasoft pseudopotential
1061 \item Brillouin zone sampling: $\Gamma$-point
1064 \begin{minipage}[t]{6cm}
1067 \left[ -\frac{\hbar^2}{2m}\nabla^2 + V_{\text{eff}}(r) - \epsilon_i \right] \Phi_i(r) = 0
1070 n(r)=\sum_i^N|\Phi_i(r)|^2
1073 V_{\text{eff}}(r)=V_{\text{ext}}(r)+\int\frac{e^2 n(r')}{|r-r'|}d^3r'
1074 +V_{\text{XC}}[n(r)]
1088 Density functional theory (DFT) calculations
1091 Basic ingredients necessary for DFT
1094 \item \underline{Hohenberg-Kohn theorem} - ground state density $n_0(r)$ ...
1096 \item ... uniquely determines the ground state potential
1098 \item ... minimizes the systems total energy
1100 \item \underline{Born-Oppenheimer}
1101 - $N$ moving electrons in an external potential of static nuclei
1103 H\Psi = \left[-\sum_i^N \frac{\hbar^2}{2m}\nabla_i^2
1104 +\sum_i^N V_{\text{ext}}(r_i)
1105 +\sum_{i<j}^N V_{e-e}(r_i,r_j)\right]\Psi=E\Psi
1107 \item \underline{Effective potential}
1108 - averaged electrostatic potential \& exchange and correlation
1110 V_{\text{eff}}(r)=V_{\text{ext}}(r)+\int\frac{e^2 n(r')}{|r-r'|}d^3r'
1111 +V_{\text{XC}}[n(r)]
1113 \item \underline{Kohn-Sham system}
1114 - Schr\"odinger equation of N non-interacting particles
1116 \left[ -\frac{\hbar^2}{2m}\nabla^2 + V_{\text{eff}}(r) \right] \Phi_i(r)
1117 =\epsilon_i\Phi_i(r)
1121 n(r)=\sum_i^N|\Phi_i(r)|^2
1123 \item \underline{Self-consistent solution}\\
1124 $n(r)$ depends on $\Phi_i$, which depend on $V_{\text{eff}}$,
1125 which in turn depends on $n(r)$
1126 \item \underline{Variational principle}
1127 - minimize total energy with respect to $n(r)$
1135 Density functional theory (DFT) calculations
1142 Details of applied DFT calculations in this work
1145 \item \underline{Exchange correlation functional}
1146 - approximations for the inhomogeneous electron gas
1148 \item LDA: $E_{\text{XC}}^{\text{LDA}}[n]=\int \epsilon_{\text{XC}}(n)n(r)d^3r$
1149 \item GGA: $E_{\text{XC}}^{\text{GGA}}[n]=\int \epsilon_{\text{XC}}(n,\nabla n)n(r)d^3r$
1151 \item \underline{Plane wave basis set}
1152 - approximation of the wavefunction $\Phi_i$ by plane waves $\phi_j$
1153 \item \underline{Brillouin zone sampling} -
1154 {\color{blue}$\Gamma$-point only} calculations
1155 \item \underline{Pseudo potential}
1156 - consider only the valence electrons
1157 \item \underline{Code} - VASP 4.6
1162 MD and structural optimization
1165 \item MD integration: Gear predictor corrector algorithm
1166 \item Pressure control: Parrinello-Rahman pressure control
1167 \item Structural optimization: Conjugate gradient method
1170 \begin{pspicture}(0,0)(0,0)
1171 \psellipse[linecolor=blue](1.5,6.75)(0.5,0.3)
1179 C and Si self-interstitial point defects in silicon
1186 \begin{minipage}{8cm}
1188 \begin{pspicture}(0,0)(7,5)
1189 \rput(3.5,4){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
1192 \item Creation of c-Si simulation volume
1193 \item Periodic boundary conditions
1194 \item $T=0\text{ K}$, $p=0\text{ bar}$
1197 \rput(3.5,2.1){\rnode{insert}{\psframebox{
1200 Insertion of interstitial C/Si atoms
1203 \rput(3.5,1){\rnode{cool}{\psframebox[fillstyle=solid,fillcolor=lbb]{
1206 Relaxation / structural energy minimization
1209 \ncline[]{->}{init}{insert}
1210 \ncline[]{->}{insert}{cool}
1213 \begin{minipage}{5cm}
1214 \includegraphics[width=5cm]{unit_cell_e.eps}\\
1217 \begin{minipage}{9cm}
1218 \begin{tabular}{l c c}
1220 & size [unit cells] & \# atoms\\
1222 VASP & $3\times 3\times 3$ & $216\pm 1$ \\
1223 Erhart/Albe & $9\times 9\times 9$ & $5832\pm 1$\\
1227 \begin{minipage}{4cm}
1228 {\color{red}$\bullet$} Tetrahedral\\
1229 {\color{green}$\bullet$} Hexagonal\\
1230 {\color{yellow}$\bullet$} \hkl<1 0 0> dumbbell\\
1231 {\color{magenta}$\bullet$} \hkl<1 1 0> dumbbell\\
1232 {\color{cyan}$\bullet$} Bond-centered\\
1233 {\color{black}$\bullet$} Vacancy / Substitutional
1242 \begin{minipage}{9.5cm}
1245 Si self-interstitial point defects in silicon\\
1248 \begin{tabular}{l c c c c c}
1250 $E_{\text{f}}$ [eV] & \hkl<1 1 0> DB & H & T & \hkl<1 0 0> DB & V \\
1252 VASP & \underline{3.39} & 3.42 & 3.77 & 4.41 & 3.63 \\
1253 Erhart/Albe & 4.39 & 4.48$^*$ & \underline{3.40} & 5.42 & 3.13 \\
1255 \end{tabular}\\[0.2cm]
1257 \begin{minipage}{4.7cm}
1258 \includegraphics[width=4.7cm]{e_kin_si_hex.ps}
1260 \begin{minipage}{4.7cm}
1262 {\tiny nearly T $\rightarrow$ T}\\
1264 \includegraphics[width=4.7cm]{nhex_tet.ps}
1267 \underline{Hexagonal} \hspace{2pt}
1268 \href{../video/si_self_int_hexa.avi}{$\rhd$}\\[0.1cm]
1270 \begin{minipage}{2.7cm}
1271 $E_{\text{f}}^*=4.48\text{ eV}$\\
1272 \includegraphics[width=2.7cm]{si_pd_albe/hex_a.eps}
1274 \begin{minipage}{0.4cm}
1279 \begin{minipage}{2.7cm}
1280 $E_{\text{f}}=3.96\text{ eV}$\\
1281 \includegraphics[width=2.8cm]{si_pd_albe/hex.eps}
1284 \begin{minipage}{2.9cm}
1286 \underline{Vacancy}\\
1287 \includegraphics[width=3.0cm]{si_pd_albe/vac.eps}
1292 \begin{minipage}{3.5cm}
1295 \underline{\hkl<1 1 0> dumbbell}\\
1296 \includegraphics[width=3.0cm]{si_pd_albe/110.eps}\\
1297 \underline{Tetrahedral}\\
1298 \includegraphics[width=3.0cm]{si_pd_albe/tet.eps}\\
1299 \underline{\hkl<1 0 0> dumbbell}\\
1300 \includegraphics[width=3.0cm]{si_pd_albe/100.eps}
1312 C interstitial point defects in silicon\\[-0.1cm]
1315 \begin{tabular}{l c c c c c c r}
1317 $E_{\text{f}}$ & T & H & \hkl<1 0 0> DB & \hkl<1 1 0> DB & S & B & \cs{} \& \si\\
1319 VASP & unstable & unstable & \underline{3.72} & 4.16 & 1.95 & 4.66 & {\color{green}4.17}\\
1320 Erhart/Albe MD & 6.09 & 9.05$^*$ & \underline{3.88} & 5.18 & {\color{red}0.75} & 5.59$^*$ & {\color{green}4.43} \\
1322 \end{tabular}\\[0.1cm]
1325 \begin{minipage}{2.7cm}
1326 \underline{Hexagonal} \hspace{2pt}
1327 \href{../video/c_in_si_int_hexa.avi}{$\rhd$}\\
1328 $E_{\text{f}}^*=9.05\text{ eV}$\\
1329 \includegraphics[width=2.7cm]{c_pd_albe/hex.eps}
1331 \begin{minipage}{0.4cm}
1336 \begin{minipage}{2.7cm}
1337 \underline{\hkl<1 0 0>}\\
1338 $E_{\text{f}}=3.88\text{ eV}$\\
1339 \includegraphics[width=2.7cm]{c_pd_albe/100.eps}
1342 \begin{minipage}{2cm}
1345 \begin{minipage}{3cm}
1347 \underline{Tetrahedral}\\
1348 \includegraphics[width=3.0cm]{c_pd_albe/tet.eps}
1353 \begin{minipage}{2.7cm}
1354 \underline{Bond-centered}\\
1355 $E_{\text{f}}^*=5.59\text{ eV}$\\
1356 \includegraphics[width=2.7cm]{c_pd_albe/bc.eps}
1358 \begin{minipage}{0.4cm}
1363 \begin{minipage}{2.7cm}
1364 \underline{\hkl<1 1 0> dumbbell}\\
1365 $E_{\text{f}}=5.18\text{ eV}$\\
1366 \includegraphics[width=2.7cm]{c_pd_albe/110.eps}
1369 \begin{minipage}{2cm}
1372 \begin{minipage}{3cm}
1374 \underline{Substitutional}\\
1375 \includegraphics[width=3.0cm]{c_pd_albe/sub.eps}
1386 C \hkl<1 0 0> dumbbell interstitial configuration\\
1390 \begin{tabular}{l c c c c c c c c}
1392 Distances [nm] & $r(1C)$ & $r(2C)$ & $r(3C)$ & $r(12)$ & $r(13)$ & $r(34)$ & $r(23)$ & $r(25)$ \\
1394 Erhart/Albe & 0.175 & 0.329 & 0.186 & 0.226 & 0.300 & 0.343 & 0.423 & 0.425 \\
1395 VASP & 0.174 & 0.341 & 0.182 & 0.229 & 0.286 & 0.347 & 0.422 & 0.417 \\
1397 \end{tabular}\\[0.2cm]
1398 \begin{tabular}{l c c c c }
1400 Angles [$^{\circ}$] & $\theta_1$ & $\theta_2$ & $\theta_3$ & $\theta_4$ \\
1402 Erhart/Albe & 140.2 & 109.9 & 134.4 & 112.8 \\
1403 VASP & 130.7 & 114.4 & 146.0 & 107.0 \\
1405 \end{tabular}\\[0.2cm]
1406 \begin{tabular}{l c c c}
1408 Displacements [nm]& $a$ & $b$ & $|a|+|b|$ \\
1410 Erhart/Albe & 0.084 & -0.091 & 0.175 \\
1411 VASP & 0.109 & -0.065 & 0.174 \\
1413 \end{tabular}\\[0.6cm]
1416 \begin{minipage}{3.0cm}
1418 \underline{Erhart/Albe}
1419 \includegraphics[width=3.0cm]{c_pd_albe/100_cmp.eps}
1422 \begin{minipage}{3.0cm}
1425 \includegraphics[width=3.0cm]{c_pd_vasp/100_cmp.eps}
1429 \begin{picture}(0,0)(-185,10)
1430 \includegraphics[width=6.8cm]{100-c-si-db_cmp.eps}
1432 \begin{picture}(0,0)(-280,-150)
1433 \includegraphics[width=3.3cm]{c_pd_vasp/eden.eps}
1436 \begin{pspicture}(0,0)(0,0)
1437 \psellipse[linecolor=green](5.18,5.92)(0.5,0.3)
1438 \psellipse[linecolor=red](3.45,5.92)(1.0,0.4)
1439 \psellipse[linecolor=blue](2.7,6.92)(0.9,0.2)
1440 \psellipse[linecolor=blue](4.65,6.92)(0.9,0.2)
1449 \begin{minipage}{8.5cm}
1452 Bond-centered interstitial configuration\\[-0.1cm]
1455 \begin{minipage}{3.0cm}
1456 \includegraphics[width=2.8cm]{c_pd_vasp/bc_2333.eps}\\
1458 \begin{minipage}{5.2cm}
1460 \item Linear Si-C-Si bond
1461 \item Si: one C \& 3 Si neighbours
1462 \item Spin polarized calculations
1463 \item No saddle point!\\
1470 \begin{minipage}[t]{6.5cm}
1471 \begin{minipage}[t]{1.2cm}
1473 {\tiny sp$^3$}\\[0.8cm]
1474 \underline{${\color{black}\uparrow}$}
1475 \underline{${\color{black}\uparrow}$}
1476 \underline{${\color{black}\uparrow}$}
1477 \underline{${\color{red}\uparrow}$}\\
1480 \begin{minipage}[t]{1.4cm}
1482 {\color{red}M}{\color{blue}O}\\[0.8cm]
1483 \underline{${\color{blue}\uparrow}{\color{white}\downarrow}$}\\
1484 $\sigma_{\text{ab}}$\\[0.5cm]
1485 \underline{${\color{red}\uparrow}{\color{blue}\downarrow}$}\\
1489 \begin{minipage}[t]{1.0cm}
1493 \underline{${\color{white}\uparrow\uparrow}$}
1494 \underline{${\color{white}\uparrow\uparrow}$}\\
1496 \underline{${\color{blue}\uparrow}{\color{blue}\downarrow}$}
1497 \underline{${\color{blue}\uparrow}{\color{blue}\downarrow}$}\\
1501 \begin{minipage}[t]{1.4cm}
1503 {\color{blue}M}{\color{green}O}\\[0.8cm]
1504 \underline{${\color{blue}\uparrow}{\color{white}\downarrow}$}\\
1505 $\sigma_{\text{ab}}$\\[0.5cm]
1506 \underline{${\color{green}\uparrow}{\color{blue}\downarrow}$}\\
1510 \begin{minipage}[t]{1.2cm}
1513 {\tiny sp$^3$}\\[0.8cm]
1514 \underline{${\color{green}\uparrow}$}
1515 \underline{${\color{black}\uparrow}$}
1516 \underline{${\color{black}\uparrow}$}
1517 \underline{${\color{black}\uparrow}$}\\
1525 \begin{minipage}{4.5cm}
1526 \includegraphics[width=4cm]{c_100_mig_vasp/im_spin_diff.eps}
1528 \begin{minipage}{3.5cm}
1529 {\color{gray}$\bullet$} Spin up\\
1530 {\color{green}$\bullet$} Spin down\\
1531 {\color{blue}$\bullet$} Resulting spin up\\
1532 {\color{yellow}$\bullet$} Si atoms\\
1533 {\color{red}$\bullet$} C atom
1538 \begin{minipage}{4.2cm}
1540 \includegraphics[width=4.3cm]{c_pd_vasp/bc_2333_ksl.ps}\\
1541 {\color{green}$\Box$} {\tiny unoccupied}\\
1542 {\color{red}$\bullet$} {\tiny occupied}
1551 Migration of the C \hkl<1 0 0> dumbbell interstitial
1556 {\small Investigated pathways}
1558 \begin{minipage}{8.5cm}
1559 \begin{minipage}{8.3cm}
1560 \underline{\hkl<0 0 -1> $\rightarrow$ \hkl<0 0 1>}\\
1561 \begin{minipage}{2.4cm}
1562 \includegraphics[width=2.4cm]{c_pd_vasp/100_2333.eps}
1564 \begin{minipage}{0.4cm}
1567 \begin{minipage}{2.4cm}
1568 \includegraphics[width=2.4cm]{c_pd_vasp/bc_2333.eps}
1570 \begin{minipage}{0.4cm}
1573 \begin{minipage}{2.4cm}
1574 \includegraphics[width=2.4cm]{c_pd_vasp/100_next_2333.eps}
1577 \begin{minipage}{8.3cm}
1578 \underline{\hkl<0 0 -1> $\rightarrow$ \hkl<0 -1 0>}\\
1579 \begin{minipage}{2.4cm}
1580 \includegraphics[width=2.4cm]{c_pd_vasp/100_2333.eps}
1582 \begin{minipage}{0.4cm}
1585 \begin{minipage}{2.4cm}
1586 \includegraphics[width=2.4cm]{c_pd_vasp/00-1-0-10_2333.eps}
1588 \begin{minipage}{0.4cm}
1591 \begin{minipage}{2.4cm}
1592 \includegraphics[width=2.4cm]{c_pd_vasp/0-10_2333.eps}
1595 \begin{minipage}{8.3cm}
1596 \underline{\hkl<0 0 -1> $\rightarrow$ \hkl<0 -1 0> (in place)}\\
1597 \begin{minipage}{2.4cm}
1598 \includegraphics[width=2.4cm]{c_pd_vasp/100_2333.eps}
1600 \begin{minipage}{0.4cm}
1603 \begin{minipage}{2.4cm}
1604 \includegraphics[width=2.4cm]{c_pd_vasp/00-1_ip0-10_2333.eps}
1606 \begin{minipage}{0.4cm}
1609 \begin{minipage}{2.4cm}
1610 \includegraphics[width=2.4cm]{c_pd_vasp/0-10_ip_2333.eps}
1615 \begin{minipage}{4.2cm}
1616 {\small Constrained relaxation\\
1617 technique (CRT) method}\\
1618 \includegraphics[width=4cm]{crt_orig.eps}
1620 \item Constrain diffusing atom
1621 \item Static constraints
1624 {\small Modifications}\\
1625 \includegraphics[width=4cm]{crt_mod.eps}
1627 \item Constrain all atoms
1628 \item Update individual\\
1639 Migration of the C \hkl<1 0 0> dumbbell interstitial
1645 \begin{minipage}{5.9cm}
1647 \includegraphics[width=5.8cm]{im_00-1_nosym_sp_fullct_thesis.ps}\\[0.45cm]
1650 \begin{picture}(0,0)(60,0)
1651 \includegraphics[width=1cm]{vasp_mig/00-1.eps}
1653 \begin{picture}(0,0)(-5,0)
1654 \includegraphics[width=1cm]{vasp_mig/bc_00-1_sp.eps}
1656 \begin{picture}(0,0)(-55,0)
1657 \includegraphics[width=1cm]{vasp_mig/bc.eps}
1659 \begin{picture}(0,0)(12.5,10)
1660 \includegraphics[width=1cm]{110_arrow.eps}
1662 \begin{picture}(0,0)(90,0)
1663 \includegraphics[height=0.9cm]{001_arrow.eps}
1669 \begin{minipage}{0.3cm}
1673 \begin{minipage}{5.9cm}
1675 \includegraphics[width=5.9cm]{vasp_mig/00-1_0-10_nosym_sp_fullct.ps}\\[0.5cm]
1678 \begin{picture}(0,0)(60,0)
1679 \includegraphics[width=1cm]{vasp_mig/00-1_a.eps}
1681 \begin{picture}(0,0)(5,0)
1682 \includegraphics[width=1cm]{vasp_mig/00-1_0-10_sp.eps}
1684 \begin{picture}(0,0)(-55,0)
1685 \includegraphics[width=1cm]{vasp_mig/0-10.eps}
1687 \begin{picture}(0,0)(12.5,10)
1688 \includegraphics[width=1cm]{100_arrow.eps}
1690 \begin{picture}(0,0)(90,0)
1691 \includegraphics[height=0.9cm]{001_arrow.eps}
1701 \begin{minipage}{5.9cm}
1703 \includegraphics[width=5.9cm]{vasp_mig/00-1_ip0-10_nosym_sp_fullct.ps}\\[0.6cm]
1706 \begin{picture}(0,0)(60,0)
1707 \includegraphics[width=0.9cm]{vasp_mig/00-1_b.eps}
1709 \begin{picture}(0,0)(10,0)
1710 \includegraphics[width=0.9cm]{vasp_mig/00-1_ip0-10_sp.eps}
1712 \begin{picture}(0,0)(-60,0)
1713 \includegraphics[width=0.9cm]{vasp_mig/0-10_b.eps}
1715 \begin{picture}(0,0)(12.5,10)
1716 \includegraphics[width=1cm]{100_arrow.eps}
1718 \begin{picture}(0,0)(90,0)
1719 \includegraphics[height=0.9cm]{001_arrow.eps}
1725 \begin{minipage}{0.3cm}
1728 \begin{minipage}{6.5cm}
1731 \item Energetically most favorable path
1734 \item Activation energy: $\approx$ 0.9 eV
1735 \item Experimental values: 0.73 ... 0.87 eV
1737 $\Rightarrow$ {\color{blue}Diffusion} path identified!
1738 \item Reorientation (path 3)
1740 \item More likely composed of two consecutive steps of type 2
1741 \item Experimental values: 0.77 ... 0.88 eV
1743 $\Rightarrow$ {\color{blue}Reorientation} transition identified!
1752 Migration of the C \hkl<1 0 0> dumbbell interstitial
1759 \begin{minipage}{6.5cm}
1762 \begin{minipage}[t]{5.9cm}
1764 \includegraphics[width=5.9cm]{bc_00-1.ps}\\[2.35cm]
1767 \begin{pspicture}(0,0)(0,0)
1768 \psframe[linecolor=red,fillstyle=none](-2.8,1.35)(3.3,2.7)
1770 \begin{picture}(0,0)(60,-50)
1771 \includegraphics[width=1cm]{albe_mig/bc_00-1_red_00.eps}
1773 \begin{picture}(0,0)(5,-50)
1774 \includegraphics[width=1cm]{albe_mig/bc_00-1_red_01.eps}
1776 \begin{picture}(0,0)(-55,-50)
1777 \includegraphics[width=1cm]{albe_mig/bc_00-1_red_02.eps}
1779 \begin{picture}(0,0)(12.5,-40)
1780 \includegraphics[width=1cm]{110_arrow.eps}
1782 \begin{picture}(0,0)(90,-45)
1783 \includegraphics[height=0.9cm]{001_arrow.eps}
1785 \begin{pspicture}(0,0)(0,0)
1786 \psframe[linecolor=blue,fillstyle=none](-2.8,0)(3.3,1.6)
1788 \begin{picture}(0,0)(60,-15)
1789 \includegraphics[width=0.9cm]{albe_mig/bc_00-1_01.eps}
1791 \begin{picture}(0,0)(35,-15)
1792 \includegraphics[width=0.9cm]{albe_mig/bc_00-1_02.eps}
1794 \begin{picture}(0,0)(-5,-15)
1795 \includegraphics[width=0.9cm]{albe_mig/bc_00-1_03.eps}
1797 \begin{picture}(0,0)(-55,-15)
1798 \includegraphics[width=0.9cm]{albe_mig/bc_00-1_04.eps}
1800 \begin{picture}(0,0)(12.5,-5)
1801 \includegraphics[width=1cm]{100_arrow.eps}
1803 \begin{picture}(0,0)(90,-15)
1804 \includegraphics[height=0.9cm]{010_arrow.eps}
1810 \begin{minipage}{5.9cm}
1813 \item Lowest activation energy: $\approx$ 2.2 eV
1814 \item 2.4 times higher than VASP
1815 \item Different pathway
1820 \begin{minipage}{6.5cm}
1823 \begin{minipage}{5.9cm}
1825 %\includegraphics[width=5.9cm]{00-1_0-10.ps}\\[0.75cm]
1828 %\begin{pspicture}(0,0)(0,0)
1829 %\psframe[linecolor=red,fillstyle=none](-2.8,-0.25)(3.3,1.1)
1831 %\begin{picture}(0,0)(60,-5)
1832 %\includegraphics[width=0.9cm]{albe_mig/00-1_0-10_red_00.eps}
1834 %\begin{picture}(0,0)(0,-5)
1835 %\includegraphics[width=0.9cm]{albe_mig/00-1_0-10_red_min.eps}
1837 %\begin{picture}(0,0)(-55,-5)
1838 %\includegraphics[width=0.9cm]{albe_mig/00-1_0-10_red_03.eps}
1840 %\begin{picture}(0,0)(12.5,5)
1841 %\includegraphics[width=1cm]{100_arrow.eps}
1843 %\begin{picture}(0,0)(90,0)
1844 %\includegraphics[height=0.9cm]{001_arrow.eps}
1852 %\begin{minipage}{5.9cm}
1853 \includegraphics[width=5.9cm]{00-1_110_0-10_mig_albe.ps}
1857 \begin{minipage}{5.9cm}
1858 Transition involving \ci{} \hkl<1 1 0>
1860 \item Bond-centered configuration unstable\\
1861 $\rightarrow$ \ci{} \hkl<1 1 0> dumbbell
1862 \item Transition minima of path 2 \& 3\\
1863 $\rightarrow$ \ci{} \hkl<1 1 0> dumbbell
1864 \item Activation energy: $\approx$ 2.2 eV \& 0.9 eV
1865 \item 2.4 - 3.4 times higher than VASP
1866 \item Rotation of dumbbell orientation
1870 {\color{blue}Overestimated diffusion barrier}
1881 Combinations with a C-Si \hkl<1 0 0>-type interstitial
1891 E_{\text{f}}^{\text{defect combination}}-
1892 E_{\text{f}}^{\text{C \hkl<0 0 -1> dumbbell}}-
1893 E_{\text{f}}^{\text{2nd defect}}
1899 \begin{tabular}{l c c c c c c}
1901 $E_{\text{b}}$ [eV] & 1 & 2 & 3 & 4 & 5 & R\\
1903 \hkl<0 0 -1> & {\color{red}-0.08} & -1.15 & {\color{red}-0.08} & 0.04 & -1.66 & -0.19\\
1904 \hkl<0 0 1> & 0.34 & 0.004 & -2.05 & 0.26 & -1.53 & -0.19\\
1905 \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}\\
1906 \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}\\
1907 \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}\\
1908 \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}\\
1910 C substitutional (C$_{\text{S}}$) & 0.26 & -0.51 & -0.93 & -0.15 & 0.49 & -0.05\\
1911 Vacancy & -5.39 ($\rightarrow$ C$_{\text{S}}$) & -0.59 & -3.14 & -0.54 & -0.50 & -0.31\\
1920 \begin{minipage}[t]{3.8cm}
1921 \underline{\hkl<1 0 0> at position 1}\\[0.1cm]
1922 \includegraphics[width=3.5cm]{00-1dc/2-25.eps}
1924 \begin{minipage}[t]{3.5cm}
1925 \underline{\hkl<0 -1 0> at position 1}\\[0.1cm]
1926 \includegraphics[width=3.2cm]{00-1dc/2-39.eps}
1928 \begin{minipage}[t]{5.5cm}
1930 \item $E_{\text{b}}=0$ $\Leftrightarrow$ non-interacting defects\\
1931 $E_{\text{b}} \rightarrow 0$ for increasing distance (R)
1932 \item Stress compensation / increase
1933 \item Unfavored: antiparallel orientations
1934 \item Indication of energetically favored\\
1936 \item Most favorable: C clustering
1937 \item However: High barrier ($>4\,\text{eV}$)
1938 \item $4\times{\color{cyan}-2.25}$ versus $2\times{\color{orange}-2.39}$
1943 \begin{picture}(0,0)(-295,-130)
1944 \includegraphics[width=3.5cm]{comb_pos.eps}
1952 Combinations of C-Si \hkl<1 0 0>-type interstitials
1959 Energetically most favorable combinations along \hkl<1 1 0>
1964 \begin{tabular}{l c c c c c c}
1966 & 1 & 2 & 3 & 4 & 5 & 6\\
1968 $E_{\text{b}}$ [eV] & -2.39 & -1.88 & -0.59 & -0.31 & -0.24 & -0.21 \\
1969 C-C distance [\AA] & 1.4 & 4.6 & 6.5 & 8.6 & 10.5 & 10.8 \\
1970 Type & \hkl<-1 0 0> & \hkl<1 0 0> & \hkl<1 0 0> & \hkl<1 0 0> & \hkl<1 0 0> & \hkl<1 0 0>, \hkl<0 -1 0>\\
1977 \begin{minipage}{7.0cm}
1978 \includegraphics[width=7cm]{db_along_110_cc.ps}
1980 \begin{minipage}{6.0cm}
1982 \item Interaction proportional to reciprocal cube of C-C distance
1983 \item Saturation in the immediate vicinity
1984 \renewcommand\labelitemi{$\Rightarrow$}
1985 \item Agglomeration of \ci{} expected
1986 \item Absence of C clustering
1990 Consisten with initial precipitation model
2002 Combinations of substitutional C and \hkl<1 1 0> Si self-interstitials
2008 %\begin{minipage}{3.2cm}
2009 %\includegraphics[width=3cm]{sub_110_combo.eps}
2011 %\begin{minipage}{7.8cm}
2012 %\begin{tabular}{l c c c c c c}
2014 %C$_{\text{sub}}$ & \hkl<1 1 0> & \hkl<-1 1 0> & \hkl<0 1 1> & \hkl<0 -1 1> &
2015 % \hkl<1 0 1> & \hkl<-1 0 1> \\
2017 %1 & \RM{1} & \RM{3} & \RM{3} & \RM{1} & \RM{3} & \RM{1} \\
2018 %2 & \RM{2} & A & A & \RM{2} & C & \RM{5} \\
2019 %3 & \RM{3} & \RM{1} & \RM{3} & \RM{1} & \RM{1} & \RM{3} \\
2020 %4 & \RM{4} & B & D & E & E & D \\
2021 %5 & \RM{5} & C & A & \RM{2} & A & \RM{2} \\
2028 %\begin{tabular}{l c c c c c c c c c c}
2030 %Conf & \RM{1} & \RM{2} & \RM{3} & \RM{4} & \RM{5} & A & B & C & D & E \\
2032 %$E_{\text{f}}$ [eV]& 4.37 & 5.26 & 5.57 & 5.37 & 5.12 & 5.10 & 5.32 & 5.28 & 5.39 & 5.32 \\
2033 %$E_{\text{b}}$ [eV] & -0.97 & -0.08 & 0.22 & -0.02 & -0.23 & -0.25 & -0.02 & -0.06 & 0.05 & -0.03 \\
2034 %$r$ [nm] & 0.292 & 0.394 & 0.241 & 0.453 & 0.407 & 0.408 & 0.452 & 0.392 & 0.456 & 0.453\\
2039 \begin{minipage}{6.0cm}
2040 \includegraphics[width=5.8cm]{c_sub_si110.ps}
2042 \begin{minipage}{7cm}
2045 \item IBS: C may displace Si\\
2046 $\Rightarrow$ C$_{\text{sub}}$ + \hkl<1 1 0> Si self-interstitial
2048 \hkl<1 1 0>-type $\rightarrow$ favored combination
2049 \renewcommand\labelitemi{$\Rightarrow$}
2050 \item Most favorable: \cs{} along \hkl<1 1 0> chain \si{}
2051 \item Less favorable than C-Si \hkl<1 0 0> dumbbell
2052 \item Interaction drops quickly to zero\\
2053 $\rightarrow$ low capture radius
2057 IBS process far from equilibrium\\
2058 \cs{} \& \si{} instead of thermodynamic ground state
2063 \begin{minipage}{6.5cm}
2064 \includegraphics[width=6.0cm]{162-097.ps}
2066 \item Low migration barrier
2069 \begin{minipage}{6.5cm}
2071 Ab initio MD at \degc{900}\\
2072 \includegraphics[width=3.3cm]{md_vasp_01.eps}
2073 $t=\unit[2230]{fs}$\\
2074 \includegraphics[width=3.3cm]{md_vasp_02.eps}
2078 Contribution of entropy to structural formation
2087 Migration in C-Si \hkl<1 0 0> and vacancy combinations
2094 \begin{minipage}[t]{3cm}
2095 \underline{Pos 2, $E_{\text{b}}=-0.59\text{ eV}$}\\
2096 \includegraphics[width=2.8cm]{00-1dc/0-59.eps}
2098 \begin{minipage}[t]{7cm}
2101 Low activation energies\\
2102 High activation energies for reverse processes\\
2104 {\color{blue}C$_{\text{sub}}$ very stable}\\
2108 Without nearby \hkl<1 1 0> Si self-interstitial (IBS)\\
2110 {\color{blue}Formation of SiC by successive substitution by C}
2114 \begin{minipage}[t]{3cm}
2115 \underline{Pos 3, $E_{\text{b}}=-3.14\text{ eV}$}\\
2116 \includegraphics[width=2.8cm]{00-1dc/3-14.eps}
2121 \begin{minipage}{5.9cm}
2122 \includegraphics[width=5.9cm]{vasp_mig/comb_mig_3-2_vac_fullct.ps}\\[0.6cm]
2124 \begin{picture}(0,0)(70,0)
2125 \includegraphics[width=1.4cm]{vasp_mig/comb_2-1_init.eps}
2127 \begin{picture}(0,0)(30,0)
2128 \includegraphics[width=1.4cm]{vasp_mig/comb_2-1_seq_03.eps}
2130 \begin{picture}(0,0)(-10,0)
2131 \includegraphics[width=1.4cm]{vasp_mig/comb_2-1_seq_06.eps}
2133 \begin{picture}(0,0)(-48,0)
2134 \includegraphics[width=1.4cm]{vasp_mig/comb_2-1_final.eps}
2136 \begin{picture}(0,0)(12.5,5)
2137 \includegraphics[width=1cm]{100_arrow.eps}
2139 \begin{picture}(0,0)(97,-10)
2140 \includegraphics[height=0.9cm]{001_arrow.eps}
2146 \begin{minipage}{0.3cm}
2150 \begin{minipage}{5.9cm}
2151 \includegraphics[width=5.9cm]{vasp_mig/comb_mig_4-2_vac_fullct.ps}\\[0.1cm]
2153 \begin{picture}(0,0)(60,0)
2154 \includegraphics[width=0.9cm]{vasp_mig/comb_3-1_init.eps}
2156 \begin{picture}(0,0)(25,0)
2157 \includegraphics[width=0.9cm]{vasp_mig/comb_3-1_seq_03.eps}
2159 \begin{picture}(0,0)(-20,0)
2160 \includegraphics[width=0.9cm]{vasp_mig/comb_3-1_seq_07.eps}
2162 \begin{picture}(0,0)(-55,0)
2163 \includegraphics[width=0.9cm]{vasp_mig/comb_3-1_final.eps}
2165 \begin{picture}(0,0)(12.5,5)
2166 \includegraphics[width=1cm]{100_arrow.eps}
2168 \begin{picture}(0,0)(95,0)
2169 \includegraphics[height=0.9cm]{001_arrow.eps}
2181 Conclusion of defect / migration / combined defect simulations
2190 \item Accurately described by quantum-mechanical simulations
2191 \item Less accurate description by classical potential simulations
2192 \item Underestimated formation energy of \cs{} by classical approach
2193 \item Both methods predict same ground state: \ci{} \hkl<1 0 0> dumbbell
2198 \item C migration pathway in Si identified
2199 \item Consistent with reorientation and diffusion experiments
2202 \item Different path and ...
2203 \item overestimated barrier by classical potential calculations
2206 Concerning the precipitation mechanism
2208 \item Agglomeration of C-Si dumbbells energetically favorable
2209 (stress compensation)
2210 \item C-Si indeed favored compared to
2211 C$_{\text{sub}}$ \& \hkl<1 1 0> Si self-interstitial
2212 \item Possible low interaction capture radius of
2213 C$_{\text{sub}}$ \& \hkl<1 1 0> Si self-interstitial
2214 \item Low barrier for
2215 \ci{} \hkl<1 0 0> $\rightarrow$ \cs{} \& \si{} \hkl<1 1 0>
2216 \item In absence of nearby \hkl<1 1 0> Si self-interstitial:
2217 C-Si \hkl<1 0 0> + Vacancy $\rightarrow$ C$_{\text{sub}}$ (SiC)
2220 {\color{blue}Results suggest increased participation of \cs}
2228 Silicon carbide precipitation simulations
2234 \begin{pspicture}(0,0)(12,6.5)
2236 \rput(3.5,5.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
2239 \item Create c-Si volume
2240 \item Periodc boundary conditions
2241 \item Set requested $T$ and $p=0\text{ bar}$
2242 \item Equilibration of $E_{\text{kin}}$ and $E_{\text{pot}}$
2245 \rput(3.5,2.7){\rnode{insert}{\psframebox[fillstyle=solid,fillcolor=lachs]{
2247 Insertion of C atoms at constant T
2249 \item total simulation volume {\pnode{in1}}
2250 \item volume of minimal SiC precipitate {\pnode{in2}}
2251 \item volume consisting of Si atoms to form a minimal {\pnode{in3}}\\
2255 \rput(3.5,1){\rnode{cool}{\psframebox[fillstyle=solid,fillcolor=lbb]{
2257 Run for 100 ps followed by cooling down to $20\, ^{\circ}\textrm{C}$
2259 \ncline[]{->}{init}{insert}
2260 \ncline[]{->}{insert}{cool}
2261 \psframe[fillstyle=solid,fillcolor=white](7.5,0.7)(13.5,6.3)
2262 \rput(7.8,6){\footnotesize $V_1$}
2263 \psframe[fillstyle=solid,fillcolor=lightgray](9,2)(12,5)
2264 \rput(9.2,4.85){\tiny $V_2$}
2265 \psframe[fillstyle=solid,fillcolor=gray](9.25,2.25)(11.75,4.75)
2266 \rput(9.55,4.45){\footnotesize $V_3$}
2267 \rput(7.9,3.2){\pnode{ins1}}
2268 \rput(9.22,2.8){\pnode{ins2}}
2269 \rput(11.0,2.4){\pnode{ins3}}
2270 \ncline[]{->}{in1}{ins1}
2271 \ncline[]{->}{in2}{ins2}
2272 \ncline[]{->}{in3}{ins3}
2277 \item Restricted to classical potential simulations
2278 \item $V_2$ and $V_3$ considered due to low diffusion
2279 \item Amount of C atoms: 6000
2280 ($r_{\text{prec}}\approx 3.1\text{ nm}$, IBS: 2 ... 4 nm)
2281 \item Simulation volume: $31\times 31\times 31$ unit cells
2290 Silicon carbide precipitation simulations at $450\,^{\circ}\mathrm{C}$ as in IBS
2295 \begin{minipage}{6.5cm}
2296 \includegraphics[width=6.4cm]{sic_prec_450_si-si_c-c.ps}
2298 \begin{minipage}{6.5cm}
2299 \includegraphics[width=6.4cm]{sic_prec_450_energy.ps}
2302 \begin{minipage}{6.5cm}
2303 \includegraphics[width=6.4cm]{sic_prec_450_si-c.ps}
2305 \begin{minipage}{6.5cm}
2307 \underline{Low C concentration ($V_1$)}\\
2308 \hkl<1 0 0> C-Si dumbbell dominated structure
2310 \item Si-C bumbs around 0.19 nm
2311 \item C-C peak at 0.31 nm (as expected in 3C-SiC):\\
2312 concatenated dumbbells of various orientation
2313 \item Si-Si NN distance stretched to 0.3 nm
2315 {\color{blue}$\Rightarrow$ C atoms in proper 3C-SiC distance first}\\
2316 \underline{High C concentration ($V_2$, $V_3$)}\\
2317 High amount of strongly bound C-C bonds\\
2318 Defect density $\uparrow$ $\Rightarrow$ considerable amount of damage\\
2319 Only short range order observable\\
2320 {\color{blue}$\Rightarrow$ amorphous SiC-like phase}
2328 Silicon carbide precipitation simulations at $450\,^{\circ}\mathrm{C}$ as in IBS
2333 \begin{minipage}{6.5cm}
2334 \includegraphics[width=6.4cm]{sic_prec_450_si-si_c-c.ps}
2336 \begin{minipage}{6.5cm}
2337 \includegraphics[width=6.4cm]{sic_prec_450_energy.ps}
2340 \begin{minipage}{6.5cm}
2341 \includegraphics[width=6.4cm]{sic_prec_450_si-c.ps}
2343 \begin{minipage}{6.5cm}
2345 \underline{Low C concentration ($V_1$)}\\
2346 \hkl<1 0 0> C-Si dumbbell dominated structure
2348 \item Si-C bumbs around 0.19 nm
2349 \item C-C peak at 0.31 nm (as expected in 3C-SiC):\\
2350 concatenated dumbbells of various orientation
2351 \item Si-Si NN distance stretched to 0.3 nm
2353 {\color{blue}$\Rightarrow$ C atoms in proper 3C-SiC distance first}\\
2354 \underline{High C concentration ($V_2$, $V_3$)}\\
2355 High amount of strongly bound C-C bonds\\
2356 Defect density $\uparrow$ $\Rightarrow$ considerable amount of damage\\
2357 Only short range order observable\\
2358 {\color{blue}$\Rightarrow$ amorphous SiC-like phase}
2361 \begin{pspicture}(0,0)(0,0)
2362 \rput(6.7,5.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=white]{
2363 \begin{minipage}{10cm}
2365 {\color{red}\bf 3C-SiC formation fails to appear}
2367 \item Low C concentration simulations
2369 \item Formation of \ci{} indeed occurs
2370 \item Agllomeration not observed
2372 \item High C concentration simulations
2374 \item Amorphous SiC-like structure\\
2375 (not expected at prevailing temperatures)
2376 \item Rearrangement and transition into 3C-SiC structure missing
2388 Limitations of molecular dynamics and short range potentials
2395 \underline{Time scale problem of MD}\\[0.2cm]
2396 Minimize integration error\\
2397 $\Rightarrow$ discretization considerably smaller than
2398 reciprocal of fastest vibrational mode\\[0.1cm]
2399 Order of fastest vibrational mode: $10^{13} - 10^{14}\text{ Hz}$\\
2400 $\Rightarrow$ suitable choice of time step:
2401 $\tau=1\text{ fs}=10^{-15}\text{ s}$\\
2402 $\Rightarrow$ {\color{red}\underline{slow}} phase space propagation\\[0.1cm]
2403 Several local minima in energy surface separated by large energy barriers\\
2404 $\Rightarrow$ transition event corresponds to a multiple
2405 of vibrational periods\\
2406 $\Rightarrow$ phase transition made up of {\color{red}\underline{many}}
2407 infrequent transition events\\[0.1cm]
2408 {\color{blue}Accelerated methods:}
2409 \underline{Temperature accelerated} MD (TAD), self-guided MD \ldots
2413 \underline{Limitations related to the short range potential}\\[0.2cm]
2414 Cut-off function pushing forces and energies to zero between 1$^{\text{st}}$
2415 and 2$^{\text{nd}}$ next neighbours\\
2416 $\Rightarrow$ overestimated unphysical high forces of next neighbours
2422 Potential enhanced problem of slow phase space propagation
2427 \underline{Approach to the (twofold) problem}\\[0.2cm]
2428 Increased temperature simulations without TAD corrections\\
2429 (accelerated methods or higher time scales exclusively not sufficient)
2431 \begin{picture}(0,0)(-260,-30)
2433 \begin{minipage}{4.2cm}
2440 \item 3C-SiC also observed for higher T
2441 \item higher T inside sample
2442 \item structural evolution vs.\\
2443 equilibrium properties
2449 \begin{picture}(0,0)(-305,-155)
2451 \begin{minipage}{2.5cm}
2455 thermodynmic sampling
2466 Increased temperature simulations at low C concentration
2471 \begin{minipage}{6.5cm}
2472 \includegraphics[width=6.4cm]{tot_pc_thesis.ps}
2474 \begin{minipage}{6.5cm}
2475 \includegraphics[width=6.4cm]{tot_pc3_thesis.ps}
2478 \begin{minipage}{6.5cm}
2479 \includegraphics[width=6.4cm]{tot_pc2_thesis.ps}
2481 \begin{minipage}{6.5cm}
2483 \underline{Si-C bonds:}
2485 \item Vanishing cut-off artifact (above $1650\,^{\circ}\mathrm{C}$)
2486 \item Structural change: C-Si \hkl<1 0 0> $\rightarrow$ C$_{\text{sub}}$
2488 \underline{Si-Si bonds:}
2489 {\color{blue}Si-C$_{\text{sub}}$-Si} along \hkl<1 1 0>
2490 ($\rightarrow$ 0.325 nm)\\[0.1cm]
2491 \underline{C-C bonds:}
2493 \item C-C next neighbour pairs reduced (mandatory)
2494 \item Peak at 0.3 nm slightly shifted
2496 \item C-Si \hkl<1 0 0> combinations (dashed arrows)\\
2497 $\rightarrow$ C-Si \hkl<1 0 0> \& C$_{\text{sub}}$
2499 $\rightarrow$ pure {\color{blue}C$_{\text{sub}}$ combinations}
2501 \item Range [|-$\downarrow$]:
2502 {\color{blue}C$_{\text{sub}}$ \& C$_{\text{sub}}$
2503 with nearby Si$_{\text{I}}$}
2508 \begin{picture}(0,0)(-330,-74)
2511 \begin{minipage}{1.6cm}
2514 stretched SiC\\[-0.1cm]
2526 Increased temperature simulations at low C concentration
2531 \begin{minipage}{6.5cm}
2532 \includegraphics[width=6.4cm]{tot_pc_thesis.ps}
2534 \begin{minipage}{6.5cm}
2535 \includegraphics[width=6.4cm]{tot_pc3_thesis.ps}
2538 \begin{minipage}{6.5cm}
2539 \includegraphics[width=6.4cm]{tot_pc2_thesis.ps}
2541 \begin{minipage}{6.5cm}
2543 \underline{Si-C bonds:}
2545 \item Vanishing cut-off artifact (above $1650\,^{\circ}\mathrm{C}$)
2546 \item Structural change: C-Si \hkl<1 0 0> $\rightarrow$ C$_{\text{sub}}$
2548 \underline{Si-Si bonds:}
2549 {\color{blue}Si-C$_{\text{sub}}$-Si} along \hkl<1 1 0>
2550 ($\rightarrow$ 0.325 nm)\\[0.1cm]
2551 \underline{C-C bonds:}
2553 \item C-C next neighbour pairs reduced (mandatory)
2554 \item Peak at 0.3 nm slightly shifted
2556 \item C-Si \hkl<1 0 0> combinations (dashed arrows)\\
2557 $\rightarrow$ C-Si \hkl<1 0 0> \& C$_{\text{sub}}$
2559 $\rightarrow$ pure {\color{blue}C$_{\text{sub}}$ combinations}
2561 \item Range [|-$\downarrow$]:
2562 {\color{blue}C$_{\text{sub}}$ \& C$_{\text{sub}}$
2563 with nearby Si$_{\text{I}}$}
2568 %\begin{picture}(0,0)(-330,-74)
2571 %\begin{minipage}{1.6cm}
2574 %stretched SiC\\[-0.1cm]
2581 \begin{pspicture}(0,0)(0,0)
2582 \rput(6.7,5.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=white]{
2583 \begin{minipage}{10cm}
2585 {\color{blue}\bf Stretched SiC in c-Si}
2587 \item Consistent to precipitation model involving \cs{}
2588 \item Explains annealing behavior of high/low T C implants
2590 \item Low T: highly mobiel \ci{}
2591 \item High T: stable configurations of \cs{}
2594 $\Rightarrow$ High T $\leftrightarrow$ IBS conditions far from equilibrium\\
2595 $\Rightarrow$ Precipitation mechanism involving \cs{}
2605 Increased temperature simulations at high C concentration
2610 \begin{minipage}{6.5cm}
2611 \includegraphics[width=6.4cm]{12_pc_thesis.ps}
2613 \begin{minipage}{6.5cm}
2614 \includegraphics[width=6.4cm]{12_pc_c_thesis.ps}
2622 \begin{minipage}[t]{6.0cm}
2623 0.186 nm: Si-C pairs $\uparrow$\\
2624 (as expected in 3C-SiC)\\[0.2cm]
2625 0.282 nm: Si-C-C\\[0.2cm]
2626 $\approx$0.35 nm: C-Si-Si
2629 \begin{minipage}{0.2cm}
2633 \begin{minipage}[t]{6.0cm}
2634 0.15 nm: C-C pairs $\uparrow$\\
2635 (as expected in graphite/diamond)\\[0.2cm]
2636 0.252 nm: C-C-C (2$^{\text{nd}}$ NN for diamond)\\[0.2cm]
2637 0.31 nm: shifted towards 0.317 nm $\rightarrow$ C-Si-C
2642 \item Decreasing cut-off artifact
2643 \item {\color{red}Amorphous} SiC-like phase remains
2644 \item High amount of {\color{red}damage} \& alignement to c-Si host matrix lost
2645 \item Slightly sharper peaks $\Rightarrow$ indicate slight {\color{blue}acceleration of dynamics} due to temperature
2654 High C \& small $V$ \& short $t$
2657 Slow restructuring due to strong C-C bonds
2660 High C \& low T implants
2671 Summary and Conclusions
2679 \begin{minipage}[t]{12.9cm}
2680 \underline{Pecipitation simulations}
2682 \item High C concentration $\rightarrow$ amorphous SiC like phase
2683 \item Problem of potential enhanced slow phase space propagation
2684 \item Low T $\rightarrow$ C-Si \hkl<1 0 0> dumbbell dominated structure
2685 \item High T $\rightarrow$ C$_{\text{sub}}$ dominated structure
2686 \item High T necessary to simulate IBS conditions (far from equilibrium)
2687 \item Precipitation by successive agglomeration of \cs (epitaxy)
2688 \item \si{}: vehicle to form \cs{} \& supply of Si \& stress compensation
2689 (stretched SiC, interface)
2697 \begin{minipage}{12.9cm}
2702 \item Point defects excellently / fairly well described
2704 \item C$_{\text{sub}}$ drastically underestimated by EA
2705 \item EA predicts correct ground state:
2706 C$_{\text{sub}}$ \& \si{} $>$ \ci{}
2707 \item Identified migration path explaining
2708 diffusion and reorientation experiments by DFT
2709 \item EA fails to describe \ci{} migration:
2710 Wrong path \& overestimated barrier
2712 \item Combinations of defects
2714 \item Agglomeration of point defects energetically favorable
2715 by compensation of stress
2716 \item Formation of C-C unlikely
2717 \item C$_{\text{sub}}$ favored conditions (conceivable in IBS)
2718 \item \ci{} \hkl<1 0 0> $\leftrightarrow$ \cs{} \& \si{} \hkl<1 1 0>\\
2719 Low barrier (\unit[0.77]{eV}) \& low capture radius
2727 \framebox{Precipitation by successive agglomeration of \cs{}}
2745 \underline{Augsburg}
2747 \item Prof. B. Stritzker (accomodation at EP \RM{4})
2748 \item Ralf Utermann (EDV)
2751 \underline{Helsinki}
2753 \item Prof. K. Nordlund (MD)
2758 \item Bayerische Forschungsstiftung (financial support)
2761 \underline{Paderborn}
2763 \item Prof. J. Lindner (SiC)
2764 \item Prof. G. Schmidt (DFT + financial support)
2765 \item Dr. E. Rauls (DFT + SiC)
2766 \item Dr. S. Sanna (VASP)
2773 \bf Thank you for your attention!