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)
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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)
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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${\lightning}$}
998 Utilized computational methods
1005 {\bf Molecular dynamics (MD)}\\[0.1cm]
1007 \begin{tabular}{| p{4.5cm} | p{7.5cm} |}
1009 System of $N$ particles &
1010 $N=5832\pm 1$ (Defects), $N=238328+6000$ (Precipitation)\\
1011 Phase space propagation &
1012 Velocity Verlet | timestep: \unit[1]{fs} \\
1013 Analytical interaction potential &
1014 Tersoff-like {\color{red}short-range}, {\color{blue}bond order} potential
1017 E = \frac{1}{2} \sum_{i \neq j} \pot_{ij}, \quad
1018 \pot_{ij} = {\color{red}f_C(r_{ij})}
1019 \left[ f_R(r_{ij}) + {\color{blue}b_{ij}} f_A(r_{ij}) \right]
1021 Observables: time/ensemble averages &
1022 NpT (isothermal-isobaric) | Berendsen thermostat/barostat\\
1030 {\bf Density functional theory (DFT)}
1034 \begin{minipage}[t]{6cm}
1036 \item Hohenberg-Kohn theorem:\\
1037 $\Psi_0(r_1,r_2,\ldots,r_N)=\Psi[n_0(r)]$, $E_0=E[n_0]$
1038 \item Kohn-Sham approach:\\
1039 Single-particle effective theory
1043 \item Code: \textsc{vasp}
1044 \item Plane wave basis set
1046 %\Phi_i=\sum_{|G+k|<G_{\text{cut}}} c_{i,k+G} \exp{\left(i(k+G)r\right)}
1049 %E_{\text{cut}}=\frac{\hbar^2}{2m}G^2_{\text{cut}}=\unit[300]{eV}
1051 \item Ultrasoft pseudopotential
1052 \item Exchange \& correlation: GGA
1053 \item Brillouin zone sampling: $\Gamma$-point
1054 \item Supercell: $N=216\pm2$
1057 \begin{minipage}{6cm}
1058 \begin{pspicture}(0,0)(0,0)
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1060 \rput(2.7,-0.7){\psframebox[fillstyle=solid,opacity=0.8,fillcolor=white]{
1062 \left[ -\frac{\hbar^2}{2m}\nabla^2 + V_{\text{eff}}(r) - \epsilon_i \right] \Phi_i(r) = 0
1065 \rput(5.2,-2.0){\psframebox[fillstyle=solid,opacity=0.8,fillcolor=white]{
1067 n(r)=\sum_i^N|\Phi_i(r)|^2
1070 \rput(3.0,-4.5){\psframebox[fillstyle=solid,opacity=0.8,fillcolor=white]{
1072 V_{\text{eff}}(r)=V_{\text{ext}}(r)+\int\frac{e^2 n(r')}{|r-r'|}d^3r'
1073 +V_{\text{XC}}[n(r)]
1076 \psarcn[linewidth=0.07cm,linestyle=dashed]{->}(3.5,-2.0){2.5}{130}{15}
1077 \psarcn[linewidth=0.07cm,linestyle=dashed]{->}(3.5,-2.0){2.5}{230}{165}
1078 \psarcn[linewidth=0.07cm,linestyle=dashed]{->}(3.5,-2.0){2.5}{345}{310}
1089 Point defects \& defect migration
1096 \begin{minipage}[b]{7.5cm}
1097 {\bf Defect structure}\\
1098 \begin{pspicture}(0,0)(7,4.4)
1099 \rput(3.5,3.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
1102 \item Creation of c-Si simulation volume
1103 \item Periodic boundary conditions
1104 \item $T=0\text{ K}$, $p=0\text{ bar}$
1107 \rput(3.5,1.3){\rnode{insert}{\psframebox{
1110 Insertion of interstitial C/Si atoms
1113 \rput(3.5,0.2){\rnode{cool}{\psframebox[fillstyle=solid,fillcolor=lbb]{
1116 Relaxation / structural energy minimization
1119 \ncline[]{->}{init}{insert}
1120 \ncline[]{->}{insert}{cool}
1123 \begin{minipage}[b]{4.5cm}
1125 \includegraphics[width=3.8cm]{unit_cell_e.eps}\\
1127 \begin{minipage}{2.21cm}
1129 {\color{red}$\bullet$} Tetrahedral\\[-0.1cm]
1130 {\color{green}$\bullet$} Hexagonal\\[-0.1cm]
1131 {\color{yellow}$\bullet$} \hkl<1 0 0> DB
1134 \begin{minipage}{2.21cm}
1136 {\color{magenta}$\bullet$} \hkl<1 1 0> DB\\[-0.1cm]
1137 {\color{cyan}$\bullet$} Bond-centered\\[-0.1cm]
1138 {\color{black}$\bullet$} Vac. / Sub.
1145 \begin{minipage}[b]{6cm}
1146 {\bf Defect formation energy}\\
1148 $E_{\text{f}}=E-\sum_i N_i\mu_i$}\\[0.1cm]
1149 Particle reservoir: Si \& SiC\\[0.2cm]
1150 {\bf Binding energy}\\
1154 E_{\text{f}}^{\text{comb}}-
1155 E_{\text{f}}^{1^{\text{st}}}-
1156 E_{\text{f}}^{2^{\text{nd}}}
1160 $E_{\text{b}}<0$: energetically favorable configuration\\
1161 $E_{\text{b}}\rightarrow 0$: non-interacting, isolated defects\\
1163 \begin{minipage}[b]{6cm}
1164 {\bf Migration barrier}
1167 \item Displace diffusing atom
1168 \item Constrain relaxation of (diffusing) atoms
1169 \item Record configurational energy
1171 \begin{picture}(0,0)(-60,-33)
1172 \includegraphics[width=4.5cm]{crt_mod.eps}
1184 Si self-interstitial point defects in silicon\\[0.1cm]
1188 \begin{tabular}{l c c c c c}
1190 $E_{\text{f}}$ [eV] & \hkl<1 1 0> DB & H & T & \hkl<1 0 0> DB & V \\
1192 \textsc{vasp} & \underline{3.39} & 3.42 & 3.77 & 4.41 & 3.63 \\
1193 Erhart/Albe & 4.39 & 4.48$^*$ & \underline{3.40} & 5.42 & 3.13 \\
1195 \end{tabular}\\[0.4cm]
1198 \begin{minipage}{3cm}
1200 \underline{Vacancy}\\
1201 \includegraphics[width=2.8cm]{si_pd_albe/vac.eps}
1204 \begin{minipage}{3cm}
1206 \underline{\hkl<1 1 0> DB}\\
1207 \includegraphics[width=2.8cm]{si_pd_albe/110_bonds.eps}
1210 \begin{minipage}{3cm}
1212 \underline{\hkl<1 0 0> DB}\\
1213 \includegraphics[width=2.8cm]{si_pd_albe/100_bonds.eps}
1216 \begin{minipage}{3cm}
1218 \underline{Tetrahedral}\\
1219 \includegraphics[width=2.8cm]{si_pd_albe/tet_bonds.eps}
1223 \underline{Hexagonal} \hspace{2pt}
1224 \href{../video/si_self_int_hexa.avi}{$\rhd$}\\[0.1cm]
1226 \begin{minipage}{2.7cm}
1227 $E_{\text{f}}^*=4.48\text{ eV}$\\
1228 \includegraphics[width=2.7cm]{si_pd_albe/hex_a_bonds.eps}
1230 \begin{minipage}{0.4cm}
1235 \begin{minipage}{2.7cm}
1236 $E_{\text{f}}=3.96\text{ eV}$\\
1237 \includegraphics[width=2.8cm]{si_pd_albe/hex_bonds.eps}
1240 \begin{minipage}{5.5cm}
1242 {\tiny nearly T $\rightarrow$ T}\\
1244 \includegraphics[width=6.0cm]{nhex_tet.ps}
1255 C interstitial point defects in silicon\\
1258 \begin{tabular}{l c c c c c c r}
1260 $E_{\text{f}}$ [eV] & T & H & \hkl<1 0 0> DB & \hkl<1 1 0> DB & S & B &
1261 {\color{black} \cs{} \& \si}\\
1263 \textsc{vasp} & unstable & unstable & \underline{3.72} & 4.16 & 1.95 & 4.66 & {\color{green}4.17}\\
1264 Erhart/Albe & 6.09 & 9.05$^*$ & \underline{3.88} & 5.18 & {\color{red}0.75} & 5.59$^*$ & {\color{green}4.43} \\
1266 \end{tabular}\\[0.1cm]
1269 \begin{minipage}{2.8cm}
1270 \underline{Hexagonal} \hspace{2pt}
1271 \href{../video/c_in_si_int_hexa.avi}{$\rhd$}\\
1272 $E_{\text{f}}^*=9.05\text{ eV}$\\
1273 \includegraphics[width=2.8cm]{c_pd_albe/hex_bonds.eps}
1275 \begin{minipage}{0.4cm}
1280 \begin{minipage}{2.8cm}
1281 \underline{\hkl<1 0 0>}\\
1282 $E_{\text{f}}=3.88\text{ eV}$\\
1283 \includegraphics[width=2.8cm]{c_pd_albe/100_bonds.eps}
1286 \begin{minipage}{1.4cm}
1289 \begin{minipage}{3.0cm}
1291 \underline{Tetrahedral}\\
1292 \includegraphics[width=3.0cm]{c_pd_albe/tet_bonds.eps}
1297 \begin{minipage}{2.8cm}
1298 \underline{Bond-centered}\\
1299 $E_{\text{f}}^*=5.59\text{ eV}$\\
1300 \includegraphics[width=2.8cm]{c_pd_albe/bc_bonds.eps}
1302 \begin{minipage}{0.4cm}
1307 \begin{minipage}{2.8cm}
1308 \underline{\hkl<1 1 0> dumbbell}\\
1309 $E_{\text{f}}=5.18\text{ eV}$\\
1310 \includegraphics[width=2.8cm]{c_pd_albe/110_bonds.eps}
1313 \begin{minipage}{1.4cm}
1316 \begin{minipage}{3.0cm}
1318 \underline{Substitutional}\\
1319 \includegraphics[width=3.0cm]{c_pd_albe/sub_bonds.eps}
1332 C-Si dimer \& bond-centered interstitial configuration
1339 \begin{minipage}[t]{4.1cm}
1340 {\bf\boldmath C \hkl<1 0 0> DB interstitial}\\[0.1cm]
1341 \begin{minipage}{2.0cm}
1343 \underline{Erhart/Albe}
1344 \includegraphics[width=2.0cm]{c_pd_albe/100_cmp.eps}
1347 \begin{minipage}{2.0cm}
1349 \underline{\textsc{vasp}}
1350 \includegraphics[width=2.0cm]{c_pd_vasp/100_cmp.eps}
1352 \end{minipage}\\[0.2cm]
1353 Si-C-Si bond angle $\rightarrow$ \unit[180]{$^{\circ}$}\\
1354 $\Rightarrow$ $sp$ hybridization\\[0.1cm]
1355 Si-Si-Si bond angle $\rightarrow$ \unit[120]{$^{\circ}$}\\
1356 $\Rightarrow$ $sp^2$ hybridization
1358 \includegraphics[width=3.4cm]{c_pd_vasp/eden.eps}\\[-0.1cm]
1359 {\tiny Charge density isosurface}
1362 \begin{minipage}{0.2cm}
1365 \begin{minipage}[t]{8.1cm}
1367 {\bf Bond-centered interstitial}\\[0.1cm]
1368 \begin{minipage}{4.4cm}
1371 \item Linear Si-C-Si bond
1372 \item Si: one C \& 3 Si neighbours
1373 \item Spin polarized calculations
1374 \item No saddle point!\\
1378 \begin{minipage}{2.7cm}
1379 \includegraphics[width=2.8cm]{c_pd_vasp/bc_2333.eps}\\
1384 \begin{minipage}[t]{6.5cm}
1385 \begin{minipage}[t]{1.2cm}
1387 {\tiny sp$^3$}\\[0.8cm]
1388 \underline{${\color{black}\uparrow}$}
1389 \underline{${\color{black}\uparrow}$}
1390 \underline{${\color{black}\uparrow}$}
1391 \underline{${\color{red}\uparrow}$}\\
1394 \begin{minipage}[t]{1.4cm}
1396 {\color{red}M}{\color{blue}O}\\[0.8cm]
1397 \underline{${\color{blue}\uparrow}{\color{white}\downarrow}$}\\
1398 $\sigma_{\text{ab}}$\\[0.5cm]
1399 \underline{${\color{red}\uparrow}{\color{blue}\downarrow}$}\\
1403 \begin{minipage}[t]{1.0cm}
1407 \underline{${\color{white}\uparrow\uparrow}$}
1408 \underline{${\color{white}\uparrow\uparrow}$}\\
1410 \underline{${\color{blue}\uparrow}{\color{blue}\downarrow}$}
1411 \underline{${\color{blue}\uparrow}{\color{blue}\downarrow}$}\\
1415 \begin{minipage}[t]{1.4cm}
1417 {\color{blue}M}{\color{green}O}\\[0.8cm]
1418 \underline{${\color{blue}\uparrow}{\color{white}\downarrow}$}\\
1419 $\sigma_{\text{ab}}$\\[0.5cm]
1420 \underline{${\color{green}\uparrow}{\color{blue}\downarrow}$}\\
1424 \begin{minipage}[t]{1.2cm}
1427 {\tiny sp$^3$}\\[0.8cm]
1428 \underline{${\color{green}\uparrow}$}
1429 \underline{${\color{black}\uparrow}$}
1430 \underline{${\color{black}\uparrow}$}
1431 \underline{${\color{black}\uparrow}$}\\
1439 \begin{minipage}{2.8cm}
1442 {\color{gray}$\bullet$} Spin up\\
1443 {\color{green}$\bullet$} Spin down\\
1444 {\color{blue}$\bullet$} Resulting spin up\\
1445 {\color{yellow}$\bullet$} Si atoms\\
1446 {\color{red}$\bullet$} C atom
1448 \begin{minipage}{3.6cm}
1449 \includegraphics[width=3.8cm]{c_100_mig_vasp/im_spin_diff.eps}
1465 Migration of the C \hkl<1 0 0> dumbbell interstitial
1470 {\small Investigated pathways}
1472 \begin{minipage}{8.5cm}
1473 \begin{minipage}{8.3cm}
1474 \underline{\hkl<0 0 -1> $\rightarrow$ \hkl<0 0 1>}\\
1475 \begin{minipage}{2.4cm}
1476 \includegraphics[width=2.4cm]{c_pd_vasp/100_2333.eps}
1478 \begin{minipage}{0.4cm}
1481 \begin{minipage}{2.4cm}
1482 \includegraphics[width=2.4cm]{c_pd_vasp/bc_2333.eps}
1484 \begin{minipage}{0.4cm}
1487 \begin{minipage}{2.4cm}
1488 \includegraphics[width=2.4cm]{c_pd_vasp/100_next_2333.eps}
1491 \begin{minipage}{8.3cm}
1492 \underline{\hkl<0 0 -1> $\rightarrow$ \hkl<0 -1 0>}\\
1493 \begin{minipage}{2.4cm}
1494 \includegraphics[width=2.4cm]{c_pd_vasp/100_2333.eps}
1496 \begin{minipage}{0.4cm}
1499 \begin{minipage}{2.4cm}
1500 \includegraphics[width=2.4cm]{c_pd_vasp/00-1-0-10_2333.eps}
1502 \begin{minipage}{0.4cm}
1505 \begin{minipage}{2.4cm}
1506 \includegraphics[width=2.4cm]{c_pd_vasp/0-10_2333.eps}
1509 \begin{minipage}{8.3cm}
1510 \underline{\hkl<0 0 -1> $\rightarrow$ \hkl<0 -1 0> (in place)}\\
1511 \begin{minipage}{2.4cm}
1512 \includegraphics[width=2.4cm]{c_pd_vasp/100_2333.eps}
1514 \begin{minipage}{0.4cm}
1517 \begin{minipage}{2.4cm}
1518 \includegraphics[width=2.4cm]{c_pd_vasp/00-1_ip0-10_2333.eps}
1520 \begin{minipage}{0.4cm}
1523 \begin{minipage}{2.4cm}
1524 \includegraphics[width=2.4cm]{c_pd_vasp/0-10_ip_2333.eps}
1534 Migration of the C \hkl<1 0 0> dumbbell interstitial
1540 \begin{minipage}{5.9cm}
1542 \includegraphics[width=5.8cm]{im_00-1_nosym_sp_fullct_thesis.ps}\\[0.45cm]
1545 \begin{picture}(0,0)(60,0)
1546 \includegraphics[width=1cm]{vasp_mig/00-1.eps}
1548 \begin{picture}(0,0)(-5,0)
1549 \includegraphics[width=1cm]{vasp_mig/bc_00-1_sp.eps}
1551 \begin{picture}(0,0)(-55,0)
1552 \includegraphics[width=1cm]{vasp_mig/bc.eps}
1554 \begin{picture}(0,0)(12.5,10)
1555 \includegraphics[width=1cm]{110_arrow.eps}
1557 \begin{picture}(0,0)(90,0)
1558 \includegraphics[height=0.9cm]{001_arrow.eps}
1564 \begin{minipage}{0.3cm}
1568 \begin{minipage}{5.9cm}
1570 \includegraphics[width=5.9cm]{vasp_mig/00-1_0-10_nosym_sp_fullct.ps}\\[0.5cm]
1573 \begin{picture}(0,0)(60,0)
1574 \includegraphics[width=1cm]{vasp_mig/00-1_a.eps}
1576 \begin{picture}(0,0)(5,0)
1577 \includegraphics[width=1cm]{vasp_mig/00-1_0-10_sp.eps}
1579 \begin{picture}(0,0)(-55,0)
1580 \includegraphics[width=1cm]{vasp_mig/0-10.eps}
1582 \begin{picture}(0,0)(12.5,10)
1583 \includegraphics[width=1cm]{100_arrow.eps}
1585 \begin{picture}(0,0)(90,0)
1586 \includegraphics[height=0.9cm]{001_arrow.eps}
1596 \begin{minipage}{5.9cm}
1598 \includegraphics[width=5.9cm]{vasp_mig/00-1_ip0-10_nosym_sp_fullct.ps}\\[0.6cm]
1601 \begin{picture}(0,0)(60,0)
1602 \includegraphics[width=0.9cm]{vasp_mig/00-1_b.eps}
1604 \begin{picture}(0,0)(10,0)
1605 \includegraphics[width=0.9cm]{vasp_mig/00-1_ip0-10_sp.eps}
1607 \begin{picture}(0,0)(-60,0)
1608 \includegraphics[width=0.9cm]{vasp_mig/0-10_b.eps}
1610 \begin{picture}(0,0)(12.5,10)
1611 \includegraphics[width=1cm]{100_arrow.eps}
1613 \begin{picture}(0,0)(90,0)
1614 \includegraphics[height=0.9cm]{001_arrow.eps}
1620 \begin{minipage}{0.3cm}
1623 \begin{minipage}{6.5cm}
1626 \item Energetically most favorable path
1629 \item Activation energy: $\approx$ 0.9 eV
1630 \item Experimental values: 0.73 ... 0.87 eV
1632 $\Rightarrow$ {\color{blue}Diffusion} path identified!
1633 \item Reorientation (path 3)
1635 \item More likely composed of two consecutive steps of type 2
1636 \item Experimental values: 0.77 ... 0.88 eV
1638 $\Rightarrow$ {\color{blue}Reorientation} transition identified!
1647 Migration of the C \hkl<1 0 0> dumbbell interstitial
1654 \begin{minipage}{6.5cm}
1657 \begin{minipage}[t]{5.9cm}
1659 \includegraphics[width=5.9cm]{bc_00-1.ps}\\[2.35cm]
1662 \begin{pspicture}(0,0)(0,0)
1663 \psframe[linecolor=red,fillstyle=none](-2.8,1.35)(3.3,2.7)
1665 \begin{picture}(0,0)(60,-50)
1666 \includegraphics[width=1cm]{albe_mig/bc_00-1_red_00.eps}
1668 \begin{picture}(0,0)(5,-50)
1669 \includegraphics[width=1cm]{albe_mig/bc_00-1_red_01.eps}
1671 \begin{picture}(0,0)(-55,-50)
1672 \includegraphics[width=1cm]{albe_mig/bc_00-1_red_02.eps}
1674 \begin{picture}(0,0)(12.5,-40)
1675 \includegraphics[width=1cm]{110_arrow.eps}
1677 \begin{picture}(0,0)(90,-45)
1678 \includegraphics[height=0.9cm]{001_arrow.eps}
1680 \begin{pspicture}(0,0)(0,0)
1681 \psframe[linecolor=blue,fillstyle=none](-2.8,0)(3.3,1.6)
1683 \begin{picture}(0,0)(60,-15)
1684 \includegraphics[width=0.9cm]{albe_mig/bc_00-1_01.eps}
1686 \begin{picture}(0,0)(35,-15)
1687 \includegraphics[width=0.9cm]{albe_mig/bc_00-1_02.eps}
1689 \begin{picture}(0,0)(-5,-15)
1690 \includegraphics[width=0.9cm]{albe_mig/bc_00-1_03.eps}
1692 \begin{picture}(0,0)(-55,-15)
1693 \includegraphics[width=0.9cm]{albe_mig/bc_00-1_04.eps}
1695 \begin{picture}(0,0)(12.5,-5)
1696 \includegraphics[width=1cm]{100_arrow.eps}
1698 \begin{picture}(0,0)(90,-15)
1699 \includegraphics[height=0.9cm]{010_arrow.eps}
1705 \begin{minipage}{5.9cm}
1708 \item Lowest activation energy: $\approx$ 2.2 eV
1709 \item 2.4 times higher than VASP
1710 \item Different pathway
1715 \begin{minipage}{6.5cm}
1718 \begin{minipage}{5.9cm}
1720 %\includegraphics[width=5.9cm]{00-1_0-10.ps}\\[0.75cm]
1723 %\begin{pspicture}(0,0)(0,0)
1724 %\psframe[linecolor=red,fillstyle=none](-2.8,-0.25)(3.3,1.1)
1726 %\begin{picture}(0,0)(60,-5)
1727 %\includegraphics[width=0.9cm]{albe_mig/00-1_0-10_red_00.eps}
1729 %\begin{picture}(0,0)(0,-5)
1730 %\includegraphics[width=0.9cm]{albe_mig/00-1_0-10_red_min.eps}
1732 %\begin{picture}(0,0)(-55,-5)
1733 %\includegraphics[width=0.9cm]{albe_mig/00-1_0-10_red_03.eps}
1735 %\begin{picture}(0,0)(12.5,5)
1736 %\includegraphics[width=1cm]{100_arrow.eps}
1738 %\begin{picture}(0,0)(90,0)
1739 %\includegraphics[height=0.9cm]{001_arrow.eps}
1747 %\begin{minipage}{5.9cm}
1748 \includegraphics[width=5.9cm]{00-1_110_0-10_mig_albe.ps}
1752 \begin{minipage}{5.9cm}
1753 Transition involving \ci{} \hkl<1 1 0>
1755 \item Bond-centered configuration unstable\\
1756 $\rightarrow$ \ci{} \hkl<1 1 0> dumbbell
1757 \item Transition minima of path 2 \& 3\\
1758 $\rightarrow$ \ci{} \hkl<1 1 0> dumbbell
1759 \item Activation energy: $\approx$ 2.2 eV \& 0.9 eV
1760 \item 2.4 - 3.4 times higher than VASP
1761 \item Rotation of dumbbell orientation
1765 {\color{blue}Overestimated diffusion barrier}
1776 Combinations with a C-Si \hkl<1 0 0>-type interstitial
1786 E_{\text{f}}^{\text{defect combination}}-
1787 E_{\text{f}}^{\text{C \hkl<0 0 -1> dumbbell}}-
1788 E_{\text{f}}^{\text{2nd defect}}
1794 \begin{tabular}{l c c c c c c}
1796 $E_{\text{b}}$ [eV] & 1 & 2 & 3 & 4 & 5 & R\\
1798 \hkl<0 0 -1> & {\color{red}-0.08} & -1.15 & {\color{red}-0.08} & 0.04 & -1.66 & -0.19\\
1799 \hkl<0 0 1> & 0.34 & 0.004 & -2.05 & 0.26 & -1.53 & -0.19\\
1800 \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}\\
1801 \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}\\
1802 \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}\\
1803 \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}\\
1805 C substitutional (C$_{\text{S}}$) & 0.26 & -0.51 & -0.93 & -0.15 & 0.49 & -0.05\\
1806 Vacancy & -5.39 ($\rightarrow$ C$_{\text{S}}$) & -0.59 & -3.14 & -0.54 & -0.50 & -0.31\\
1815 \begin{minipage}[t]{3.8cm}
1816 \underline{\hkl<1 0 0> at position 1}\\[0.1cm]
1817 \includegraphics[width=3.5cm]{00-1dc/2-25.eps}
1819 \begin{minipage}[t]{3.5cm}
1820 \underline{\hkl<0 -1 0> at position 1}\\[0.1cm]
1821 \includegraphics[width=3.2cm]{00-1dc/2-39.eps}
1823 \begin{minipage}[t]{5.5cm}
1825 \item $E_{\text{b}}=0$ $\Leftrightarrow$ non-interacting defects\\
1826 $E_{\text{b}} \rightarrow 0$ for increasing distance (R)
1827 \item Stress compensation / increase
1828 \item Unfavored: antiparallel orientations
1829 \item Indication of energetically favored\\
1831 \item Most favorable: C clustering
1832 \item However: High barrier ($>4\,\text{eV}$)
1833 \item $4\times{\color{cyan}-2.25}$ versus $2\times{\color{orange}-2.39}$
1838 \begin{picture}(0,0)(-295,-130)
1839 \includegraphics[width=3.5cm]{comb_pos.eps}
1847 Combinations of C-Si \hkl<1 0 0>-type interstitials
1854 Energetically most favorable combinations along \hkl<1 1 0>
1859 \begin{tabular}{l c c c c c c}
1861 & 1 & 2 & 3 & 4 & 5 & 6\\
1863 $E_{\text{b}}$ [eV] & -2.39 & -1.88 & -0.59 & -0.31 & -0.24 & -0.21 \\
1864 C-C distance [\AA] & 1.4 & 4.6 & 6.5 & 8.6 & 10.5 & 10.8 \\
1865 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>\\
1872 \begin{minipage}{7.0cm}
1873 \includegraphics[width=7cm]{db_along_110_cc.ps}
1875 \begin{minipage}{6.0cm}
1877 \item Interaction proportional to reciprocal cube of C-C distance
1878 \item Saturation in the immediate vicinity
1879 \renewcommand\labelitemi{$\Rightarrow$}
1880 \item Agglomeration of \ci{} expected
1881 \item Absence of C clustering
1885 Consisten with initial precipitation model
1897 Combinations of substitutional C and \hkl<1 1 0> Si self-interstitials
1903 %\begin{minipage}{3.2cm}
1904 %\includegraphics[width=3cm]{sub_110_combo.eps}
1906 %\begin{minipage}{7.8cm}
1907 %\begin{tabular}{l c c c c c c}
1909 %C$_{\text{sub}}$ & \hkl<1 1 0> & \hkl<-1 1 0> & \hkl<0 1 1> & \hkl<0 -1 1> &
1910 % \hkl<1 0 1> & \hkl<-1 0 1> \\
1912 %1 & \RM{1} & \RM{3} & \RM{3} & \RM{1} & \RM{3} & \RM{1} \\
1913 %2 & \RM{2} & A & A & \RM{2} & C & \RM{5} \\
1914 %3 & \RM{3} & \RM{1} & \RM{3} & \RM{1} & \RM{1} & \RM{3} \\
1915 %4 & \RM{4} & B & D & E & E & D \\
1916 %5 & \RM{5} & C & A & \RM{2} & A & \RM{2} \\
1923 %\begin{tabular}{l c c c c c c c c c c}
1925 %Conf & \RM{1} & \RM{2} & \RM{3} & \RM{4} & \RM{5} & A & B & C & D & E \\
1927 %$E_{\text{f}}$ [eV]& 4.37 & 5.26 & 5.57 & 5.37 & 5.12 & 5.10 & 5.32 & 5.28 & 5.39 & 5.32 \\
1928 %$E_{\text{b}}$ [eV] & -0.97 & -0.08 & 0.22 & -0.02 & -0.23 & -0.25 & -0.02 & -0.06 & 0.05 & -0.03 \\
1929 %$r$ [nm] & 0.292 & 0.394 & 0.241 & 0.453 & 0.407 & 0.408 & 0.452 & 0.392 & 0.456 & 0.453\\
1934 \begin{minipage}{6.0cm}
1935 \includegraphics[width=5.8cm]{c_sub_si110.ps}
1937 \begin{minipage}{7cm}
1940 \item IBS: C may displace Si\\
1941 $\Rightarrow$ C$_{\text{sub}}$ + \hkl<1 1 0> Si self-interstitial
1943 \hkl<1 1 0>-type $\rightarrow$ favored combination
1944 \renewcommand\labelitemi{$\Rightarrow$}
1945 \item Most favorable: \cs{} along \hkl<1 1 0> chain \si{}
1946 \item Less favorable than C-Si \hkl<1 0 0> dumbbell
1947 \item Interaction drops quickly to zero\\
1948 $\rightarrow$ low capture radius
1952 IBS process far from equilibrium\\
1953 \cs{} \& \si{} instead of thermodynamic ground state
1958 \begin{minipage}{6.5cm}
1959 \includegraphics[width=6.0cm]{162-097.ps}
1961 \item Low migration barrier
1964 \begin{minipage}{6.5cm}
1966 Ab initio MD at \degc{900}\\
1967 \includegraphics[width=3.3cm]{md_vasp_01.eps}
1968 $t=\unit[2230]{fs}$\\
1969 \includegraphics[width=3.3cm]{md_vasp_02.eps}
1973 Contribution of entropy to structural formation
1982 Migration in C-Si \hkl<1 0 0> and vacancy combinations
1989 \begin{minipage}[t]{3cm}
1990 \underline{Pos 2, $E_{\text{b}}=-0.59\text{ eV}$}\\
1991 \includegraphics[width=2.8cm]{00-1dc/0-59.eps}
1993 \begin{minipage}[t]{7cm}
1996 Low activation energies\\
1997 High activation energies for reverse processes\\
1999 {\color{blue}C$_{\text{sub}}$ very stable}\\
2003 Without nearby \hkl<1 1 0> Si self-interstitial (IBS)\\
2005 {\color{blue}Formation of SiC by successive substitution by C}
2009 \begin{minipage}[t]{3cm}
2010 \underline{Pos 3, $E_{\text{b}}=-3.14\text{ eV}$}\\
2011 \includegraphics[width=2.8cm]{00-1dc/3-14.eps}
2016 \begin{minipage}{5.9cm}
2017 \includegraphics[width=5.9cm]{vasp_mig/comb_mig_3-2_vac_fullct.ps}\\[0.6cm]
2019 \begin{picture}(0,0)(70,0)
2020 \includegraphics[width=1.4cm]{vasp_mig/comb_2-1_init.eps}
2022 \begin{picture}(0,0)(30,0)
2023 \includegraphics[width=1.4cm]{vasp_mig/comb_2-1_seq_03.eps}
2025 \begin{picture}(0,0)(-10,0)
2026 \includegraphics[width=1.4cm]{vasp_mig/comb_2-1_seq_06.eps}
2028 \begin{picture}(0,0)(-48,0)
2029 \includegraphics[width=1.4cm]{vasp_mig/comb_2-1_final.eps}
2031 \begin{picture}(0,0)(12.5,5)
2032 \includegraphics[width=1cm]{100_arrow.eps}
2034 \begin{picture}(0,0)(97,-10)
2035 \includegraphics[height=0.9cm]{001_arrow.eps}
2041 \begin{minipage}{0.3cm}
2045 \begin{minipage}{5.9cm}
2046 \includegraphics[width=5.9cm]{vasp_mig/comb_mig_4-2_vac_fullct.ps}\\[0.1cm]
2048 \begin{picture}(0,0)(60,0)
2049 \includegraphics[width=0.9cm]{vasp_mig/comb_3-1_init.eps}
2051 \begin{picture}(0,0)(25,0)
2052 \includegraphics[width=0.9cm]{vasp_mig/comb_3-1_seq_03.eps}
2054 \begin{picture}(0,0)(-20,0)
2055 \includegraphics[width=0.9cm]{vasp_mig/comb_3-1_seq_07.eps}
2057 \begin{picture}(0,0)(-55,0)
2058 \includegraphics[width=0.9cm]{vasp_mig/comb_3-1_final.eps}
2060 \begin{picture}(0,0)(12.5,5)
2061 \includegraphics[width=1cm]{100_arrow.eps}
2063 \begin{picture}(0,0)(95,0)
2064 \includegraphics[height=0.9cm]{001_arrow.eps}
2076 Conclusion of defect / migration / combined defect simulations
2085 \item Accurately described by quantum-mechanical simulations
2086 \item Less accurate description by classical potential simulations
2087 \item Underestimated formation energy of \cs{} by classical approach
2088 \item Both methods predict same ground state: \ci{} \hkl<1 0 0> dumbbell
2093 \item C migration pathway in Si identified
2094 \item Consistent with reorientation and diffusion experiments
2097 \item Different path and ...
2098 \item overestimated barrier by classical potential calculations
2101 Concerning the precipitation mechanism
2103 \item Agglomeration of C-Si dumbbells energetically favorable
2104 (stress compensation)
2105 \item C-Si indeed favored compared to
2106 C$_{\text{sub}}$ \& \hkl<1 1 0> Si self-interstitial
2107 \item Possible low interaction capture radius of
2108 C$_{\text{sub}}$ \& \hkl<1 1 0> Si self-interstitial
2109 \item Low barrier for
2110 \ci{} \hkl<1 0 0> $\rightarrow$ \cs{} \& \si{} \hkl<1 1 0>
2111 \item In absence of nearby \hkl<1 1 0> Si self-interstitial:
2112 C-Si \hkl<1 0 0> + Vacancy $\rightarrow$ C$_{\text{sub}}$ (SiC)
2115 {\color{blue}Results suggest increased participation of \cs}
2123 Silicon carbide precipitation simulations
2129 \begin{pspicture}(0,0)(12,6.5)
2131 \rput(3.5,5.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
2134 \item Create c-Si volume
2135 \item Periodc boundary conditions
2136 \item Set requested $T$ and $p=0\text{ bar}$
2137 \item Equilibration of $E_{\text{kin}}$ and $E_{\text{pot}}$
2140 \rput(3.5,2.7){\rnode{insert}{\psframebox[fillstyle=solid,fillcolor=lachs]{
2142 Insertion of C atoms at constant T
2144 \item total simulation volume {\pnode{in1}}
2145 \item volume of minimal SiC precipitate {\pnode{in2}}
2146 \item volume consisting of Si atoms to form a minimal {\pnode{in3}}\\
2150 \rput(3.5,1){\rnode{cool}{\psframebox[fillstyle=solid,fillcolor=lbb]{
2152 Run for 100 ps followed by cooling down to $20\, ^{\circ}\textrm{C}$
2154 \ncline[]{->}{init}{insert}
2155 \ncline[]{->}{insert}{cool}
2156 \psframe[fillstyle=solid,fillcolor=white](7.5,0.7)(13.5,6.3)
2157 \rput(7.8,6){\footnotesize $V_1$}
2158 \psframe[fillstyle=solid,fillcolor=lightgray](9,2)(12,5)
2159 \rput(9.2,4.85){\tiny $V_2$}
2160 \psframe[fillstyle=solid,fillcolor=gray](9.25,2.25)(11.75,4.75)
2161 \rput(9.55,4.45){\footnotesize $V_3$}
2162 \rput(7.9,3.2){\pnode{ins1}}
2163 \rput(9.22,2.8){\pnode{ins2}}
2164 \rput(11.0,2.4){\pnode{ins3}}
2165 \ncline[]{->}{in1}{ins1}
2166 \ncline[]{->}{in2}{ins2}
2167 \ncline[]{->}{in3}{ins3}
2172 \item Restricted to classical potential simulations
2173 \item $V_2$ and $V_3$ considered due to low diffusion
2174 \item Amount of C atoms: 6000
2175 ($r_{\text{prec}}\approx 3.1\text{ nm}$, IBS: 2 ... 4 nm)
2176 \item Simulation volume: $31\times 31\times 31$ unit cells
2185 Silicon carbide precipitation simulations at $450\,^{\circ}\mathrm{C}$ as in IBS
2190 \begin{minipage}{6.5cm}
2191 \includegraphics[width=6.4cm]{sic_prec_450_si-si_c-c.ps}
2193 \begin{minipage}{6.5cm}
2194 \includegraphics[width=6.4cm]{sic_prec_450_energy.ps}
2197 \begin{minipage}{6.5cm}
2198 \includegraphics[width=6.4cm]{sic_prec_450_si-c.ps}
2200 \begin{minipage}{6.5cm}
2202 \underline{Low C concentration ($V_1$)}\\
2203 \hkl<1 0 0> C-Si dumbbell dominated structure
2205 \item Si-C bumbs around 0.19 nm
2206 \item C-C peak at 0.31 nm (as expected in 3C-SiC):\\
2207 concatenated dumbbells of various orientation
2208 \item Si-Si NN distance stretched to 0.3 nm
2210 {\color{blue}$\Rightarrow$ C atoms in proper 3C-SiC distance first}\\
2211 \underline{High C concentration ($V_2$, $V_3$)}\\
2212 High amount of strongly bound C-C bonds\\
2213 Defect density $\uparrow$ $\Rightarrow$ considerable amount of damage\\
2214 Only short range order observable\\
2215 {\color{blue}$\Rightarrow$ amorphous SiC-like phase}
2223 Silicon carbide precipitation simulations at $450\,^{\circ}\mathrm{C}$ as in IBS
2228 \begin{minipage}{6.5cm}
2229 \includegraphics[width=6.4cm]{sic_prec_450_si-si_c-c.ps}
2231 \begin{minipage}{6.5cm}
2232 \includegraphics[width=6.4cm]{sic_prec_450_energy.ps}
2235 \begin{minipage}{6.5cm}
2236 \includegraphics[width=6.4cm]{sic_prec_450_si-c.ps}
2238 \begin{minipage}{6.5cm}
2240 \underline{Low C concentration ($V_1$)}\\
2241 \hkl<1 0 0> C-Si dumbbell dominated structure
2243 \item Si-C bumbs around 0.19 nm
2244 \item C-C peak at 0.31 nm (as expected in 3C-SiC):\\
2245 concatenated dumbbells of various orientation
2246 \item Si-Si NN distance stretched to 0.3 nm
2248 {\color{blue}$\Rightarrow$ C atoms in proper 3C-SiC distance first}\\
2249 \underline{High C concentration ($V_2$, $V_3$)}\\
2250 High amount of strongly bound C-C bonds\\
2251 Defect density $\uparrow$ $\Rightarrow$ considerable amount of damage\\
2252 Only short range order observable\\
2253 {\color{blue}$\Rightarrow$ amorphous SiC-like phase}
2256 \begin{pspicture}(0,0)(0,0)
2257 \rput(6.7,5.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=white]{
2258 \begin{minipage}{10cm}
2260 {\color{red}\bf 3C-SiC formation fails to appear}
2262 \item Low C concentration simulations
2264 \item Formation of \ci{} indeed occurs
2265 \item Agllomeration not observed
2267 \item High C concentration simulations
2269 \item Amorphous SiC-like structure\\
2270 (not expected at prevailing temperatures)
2271 \item Rearrangement and transition into 3C-SiC structure missing
2283 Limitations of molecular dynamics and short range potentials
2290 \underline{Time scale problem of MD}\\[0.2cm]
2291 Minimize integration error\\
2292 $\Rightarrow$ discretization considerably smaller than
2293 reciprocal of fastest vibrational mode\\[0.1cm]
2294 Order of fastest vibrational mode: $10^{13} - 10^{14}\text{ Hz}$\\
2295 $\Rightarrow$ suitable choice of time step:
2296 $\tau=1\text{ fs}=10^{-15}\text{ s}$\\
2297 $\Rightarrow$ {\color{red}\underline{slow}} phase space propagation\\[0.1cm]
2298 Several local minima in energy surface separated by large energy barriers\\
2299 $\Rightarrow$ transition event corresponds to a multiple
2300 of vibrational periods\\
2301 $\Rightarrow$ phase transition made up of {\color{red}\underline{many}}
2302 infrequent transition events\\[0.1cm]
2303 {\color{blue}Accelerated methods:}
2304 \underline{Temperature accelerated} MD (TAD), self-guided MD \ldots
2308 \underline{Limitations related to the short range potential}\\[0.2cm]
2309 Cut-off function pushing forces and energies to zero between 1$^{\text{st}}$
2310 and 2$^{\text{nd}}$ next neighbours\\
2311 $\Rightarrow$ overestimated unphysical high forces of next neighbours
2317 Potential enhanced problem of slow phase space propagation
2322 \underline{Approach to the (twofold) problem}\\[0.2cm]
2323 Increased temperature simulations without TAD corrections\\
2324 (accelerated methods or higher time scales exclusively not sufficient)
2326 \begin{picture}(0,0)(-260,-30)
2328 \begin{minipage}{4.2cm}
2335 \item 3C-SiC also observed for higher T
2336 \item higher T inside sample
2337 \item structural evolution vs.\\
2338 equilibrium properties
2344 \begin{picture}(0,0)(-305,-155)
2346 \begin{minipage}{2.5cm}
2350 thermodynmic sampling
2361 Increased temperature simulations at low C concentration
2366 \begin{minipage}{6.5cm}
2367 \includegraphics[width=6.4cm]{tot_pc_thesis.ps}
2369 \begin{minipage}{6.5cm}
2370 \includegraphics[width=6.4cm]{tot_pc3_thesis.ps}
2373 \begin{minipage}{6.5cm}
2374 \includegraphics[width=6.4cm]{tot_pc2_thesis.ps}
2376 \begin{minipage}{6.5cm}
2378 \underline{Si-C bonds:}
2380 \item Vanishing cut-off artifact (above $1650\,^{\circ}\mathrm{C}$)
2381 \item Structural change: C-Si \hkl<1 0 0> $\rightarrow$ C$_{\text{sub}}$
2383 \underline{Si-Si bonds:}
2384 {\color{blue}Si-C$_{\text{sub}}$-Si} along \hkl<1 1 0>
2385 ($\rightarrow$ 0.325 nm)\\[0.1cm]
2386 \underline{C-C bonds:}
2388 \item C-C next neighbour pairs reduced (mandatory)
2389 \item Peak at 0.3 nm slightly shifted
2391 \item C-Si \hkl<1 0 0> combinations (dashed arrows)\\
2392 $\rightarrow$ C-Si \hkl<1 0 0> \& C$_{\text{sub}}$
2394 $\rightarrow$ pure {\color{blue}C$_{\text{sub}}$ combinations}
2396 \item Range [|-$\downarrow$]:
2397 {\color{blue}C$_{\text{sub}}$ \& C$_{\text{sub}}$
2398 with nearby Si$_{\text{I}}$}
2403 \begin{picture}(0,0)(-330,-74)
2406 \begin{minipage}{1.6cm}
2409 stretched SiC\\[-0.1cm]
2421 Increased temperature simulations at low C concentration
2426 \begin{minipage}{6.5cm}
2427 \includegraphics[width=6.4cm]{tot_pc_thesis.ps}
2429 \begin{minipage}{6.5cm}
2430 \includegraphics[width=6.4cm]{tot_pc3_thesis.ps}
2433 \begin{minipage}{6.5cm}
2434 \includegraphics[width=6.4cm]{tot_pc2_thesis.ps}
2436 \begin{minipage}{6.5cm}
2438 \underline{Si-C bonds:}
2440 \item Vanishing cut-off artifact (above $1650\,^{\circ}\mathrm{C}$)
2441 \item Structural change: C-Si \hkl<1 0 0> $\rightarrow$ C$_{\text{sub}}$
2443 \underline{Si-Si bonds:}
2444 {\color{blue}Si-C$_{\text{sub}}$-Si} along \hkl<1 1 0>
2445 ($\rightarrow$ 0.325 nm)\\[0.1cm]
2446 \underline{C-C bonds:}
2448 \item C-C next neighbour pairs reduced (mandatory)
2449 \item Peak at 0.3 nm slightly shifted
2451 \item C-Si \hkl<1 0 0> combinations (dashed arrows)\\
2452 $\rightarrow$ C-Si \hkl<1 0 0> \& C$_{\text{sub}}$
2454 $\rightarrow$ pure {\color{blue}C$_{\text{sub}}$ combinations}
2456 \item Range [|-$\downarrow$]:
2457 {\color{blue}C$_{\text{sub}}$ \& C$_{\text{sub}}$
2458 with nearby Si$_{\text{I}}$}
2463 %\begin{picture}(0,0)(-330,-74)
2466 %\begin{minipage}{1.6cm}
2469 %stretched SiC\\[-0.1cm]
2476 \begin{pspicture}(0,0)(0,0)
2477 \rput(6.7,5.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=white]{
2478 \begin{minipage}{10cm}
2480 {\color{blue}\bf Stretched SiC in c-Si}
2482 \item Consistent to precipitation model involving \cs{}
2483 \item Explains annealing behavior of high/low T C implants
2485 \item Low T: highly mobiel \ci{}
2486 \item High T: stable configurations of \cs{}
2489 $\Rightarrow$ High T $\leftrightarrow$ IBS conditions far from equilibrium\\
2490 $\Rightarrow$ Precipitation mechanism involving \cs{}
2500 Increased temperature simulations at high C concentration
2505 \begin{minipage}{6.5cm}
2506 \includegraphics[width=6.4cm]{12_pc_thesis.ps}
2508 \begin{minipage}{6.5cm}
2509 \includegraphics[width=6.4cm]{12_pc_c_thesis.ps}
2517 \begin{minipage}[t]{6.0cm}
2518 0.186 nm: Si-C pairs $\uparrow$\\
2519 (as expected in 3C-SiC)\\[0.2cm]
2520 0.282 nm: Si-C-C\\[0.2cm]
2521 $\approx$0.35 nm: C-Si-Si
2524 \begin{minipage}{0.2cm}
2528 \begin{minipage}[t]{6.0cm}
2529 0.15 nm: C-C pairs $\uparrow$\\
2530 (as expected in graphite/diamond)\\[0.2cm]
2531 0.252 nm: C-C-C (2$^{\text{nd}}$ NN for diamond)\\[0.2cm]
2532 0.31 nm: shifted towards 0.317 nm $\rightarrow$ C-Si-C
2537 \item Decreasing cut-off artifact
2538 \item {\color{red}Amorphous} SiC-like phase remains
2539 \item High amount of {\color{red}damage} \& alignement to c-Si host matrix lost
2540 \item Slightly sharper peaks $\Rightarrow$ indicate slight {\color{blue}acceleration of dynamics} due to temperature
2549 High C \& small $V$ \& short $t$
2552 Slow restructuring due to strong C-C bonds
2555 High C \& low T implants
2566 Summary and Conclusions
2574 \begin{minipage}[t]{12.9cm}
2575 \underline{Pecipitation simulations}
2577 \item High C concentration $\rightarrow$ amorphous SiC like phase
2578 \item Problem of potential enhanced slow phase space propagation
2579 \item Low T $\rightarrow$ C-Si \hkl<1 0 0> dumbbell dominated structure
2580 \item High T $\rightarrow$ C$_{\text{sub}}$ dominated structure
2581 \item High T necessary to simulate IBS conditions (far from equilibrium)
2582 \item Precipitation by successive agglomeration of \cs (epitaxy)
2583 \item \si{}: vehicle to form \cs{} \& supply of Si \& stress compensation
2584 (stretched SiC, interface)
2592 \begin{minipage}{12.9cm}
2597 \item Point defects excellently / fairly well described
2599 \item C$_{\text{sub}}$ drastically underestimated by EA
2600 \item EA predicts correct ground state:
2601 C$_{\text{sub}}$ \& \si{} $>$ \ci{}
2602 \item Identified migration path explaining
2603 diffusion and reorientation experiments by DFT
2604 \item EA fails to describe \ci{} migration:
2605 Wrong path \& overestimated barrier
2607 \item Combinations of defects
2609 \item Agglomeration of point defects energetically favorable
2610 by compensation of stress
2611 \item Formation of C-C unlikely
2612 \item C$_{\text{sub}}$ favored conditions (conceivable in IBS)
2613 \item \ci{} \hkl<1 0 0> $\leftrightarrow$ \cs{} \& \si{} \hkl<1 1 0>\\
2614 Low barrier (\unit[0.77]{eV}) \& low capture radius
2622 \framebox{Precipitation by successive agglomeration of \cs{}}
2640 \underline{Augsburg}
2642 \item Prof. B. Stritzker (accomodation at EP \RM{4})
2643 \item Ralf Utermann (EDV)
2646 \underline{Helsinki}
2648 \item Prof. K. Nordlund (MD)
2653 \item Bayerische Forschungsstiftung (financial support)
2656 \underline{Paderborn}
2658 \item Prof. J. Lindner (SiC)
2659 \item Prof. G. Schmidt (DFT + financial support)
2660 \item Dr. E. Rauls (DFT + SiC)
2661 \item Dr. S. Sanna (VASP)
2668 \bf Thank you for your attention!