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}
1329 C-Si dimer \& bond-centered interstitial configuration
1336 \begin{minipage}[t]{4.1cm}
1337 {\bf\boldmath C \hkl<1 0 0> DB interstitial}\\[0.1cm]
1338 \begin{minipage}{2.0cm}
1340 \underline{Erhart/Albe}
1341 \includegraphics[width=2.0cm]{c_pd_albe/100_cmp.eps}
1344 \begin{minipage}{2.0cm}
1346 \underline{\textsc{vasp}}
1347 \includegraphics[width=2.0cm]{c_pd_vasp/100_cmp.eps}
1349 \end{minipage}\\[0.2cm]
1350 Si-C-Si bond angle $\rightarrow$ \unit[180]{$^{\circ}$}\\
1351 $\Rightarrow$ $sp$ hybridization\\[0.1cm]
1352 Si-Si-Si bond angle $\rightarrow$ \unit[120]{$^{\circ}$}\\
1353 $\Rightarrow$ $sp^2$ hybridization
1355 \includegraphics[width=3.4cm]{c_pd_vasp/eden.eps}\\[-0.1cm]
1356 {\tiny Charge density isosurface}
1359 \begin{minipage}{0.2cm}
1362 \begin{minipage}[t]{8.1cm}
1364 {\bf Bond-centered interstitial}\\[0.1cm]
1365 \begin{minipage}{4.4cm}
1368 \item Linear Si-C-Si bond
1369 \item Si: one C \& 3 Si neighbours
1370 \item Spin polarized calculations
1371 \item No saddle point!\\
1375 \begin{minipage}{2.7cm}
1376 %\includegraphics[width=2.8cm]{c_pd_vasp/bc_2333.eps}\\
1378 \includegraphics[width=2.8cm]{c_pd_albe/bc_bonds.eps}\\
1383 \begin{minipage}[t]{6.5cm}
1384 \begin{minipage}[t]{1.2cm}
1386 {\tiny sp$^3$}\\[0.8cm]
1387 \underline{${\color{black}\uparrow}$}
1388 \underline{${\color{black}\uparrow}$}
1389 \underline{${\color{black}\uparrow}$}
1390 \underline{${\color{red}\uparrow}$}\\
1393 \begin{minipage}[t]{1.4cm}
1395 {\color{red}M}{\color{blue}O}\\[0.8cm]
1396 \underline{${\color{blue}\uparrow}{\color{white}\downarrow}$}\\
1397 $\sigma_{\text{ab}}$\\[0.5cm]
1398 \underline{${\color{red}\uparrow}{\color{blue}\downarrow}$}\\
1402 \begin{minipage}[t]{1.0cm}
1406 \underline{${\color{white}\uparrow\uparrow}$}
1407 \underline{${\color{white}\uparrow\uparrow}$}\\
1409 \underline{${\color{blue}\uparrow}{\color{blue}\downarrow}$}
1410 \underline{${\color{blue}\uparrow}{\color{blue}\downarrow}$}\\
1414 \begin{minipage}[t]{1.4cm}
1416 {\color{blue}M}{\color{green}O}\\[0.8cm]
1417 \underline{${\color{blue}\uparrow}{\color{white}\downarrow}$}\\
1418 $\sigma_{\text{ab}}$\\[0.5cm]
1419 \underline{${\color{green}\uparrow}{\color{blue}\downarrow}$}\\
1423 \begin{minipage}[t]{1.2cm}
1426 {\tiny sp$^3$}\\[0.8cm]
1427 \underline{${\color{green}\uparrow}$}
1428 \underline{${\color{black}\uparrow}$}
1429 \underline{${\color{black}\uparrow}$}
1430 \underline{${\color{black}\uparrow}$}\\
1438 \begin{minipage}{3.0cm}
1440 \underline{Charge density}\\
1441 {\color{gray}$\bullet$} Spin up\\
1442 {\color{green}$\bullet$} Spin down\\
1443 {\color{blue}$\bullet$} Resulting spin up\\
1444 {\color{yellow}$\bullet$} Si atoms\\
1445 {\color{red}$\bullet$} C atom
1447 \begin{minipage}{3.6cm}
1448 \includegraphics[width=3.8cm]{c_100_mig_vasp/im_spin_diff.eps}
1455 \begin{pspicture}(0,0)(0,0)
1456 \psline[linecolor=gray,linewidth=0.05cm](-7.8,-8.7)(-7.8,0)
1465 C interstitial migration --- ab initio
1472 \begin{minipage}{6.8cm}
1473 \framebox{\hkl[0 0 -1] $\rightarrow$ \hkl[0 0 1]}\\
1474 \begin{minipage}{2.0cm}
1475 \includegraphics[width=2.0cm]{c_pd_vasp/100_2333.eps}
1477 \begin{minipage}{0.2cm}
1480 \begin{minipage}{2.0cm}
1481 \includegraphics[width=2.0cm]{c_pd_vasp/bc_2333.eps}
1483 \begin{minipage}{0.2cm}
1486 \begin{minipage}{2.0cm}
1487 \includegraphics[width=2.0cm]{c_pd_vasp/100_next_2333.eps}
1488 \end{minipage}\\[0.1cm]
1490 $\Rightarrow$ BC configuration constitutes local minimum\\
1491 $\Rightarrow$ Migration barrier to reach BC | $\Delta E=\unit[1.2]{eV}$
1493 \begin{minipage}{5.4cm}
1494 \includegraphics[width=6.0cm]{im_00-1_nosym_sp_fullct_thesis_vasp_s.ps}
1495 \end{minipage}\\[0.2cm]
1498 \begin{minipage}{6.8cm}
1499 \framebox{\hkl[0 0 -1] $\rightarrow$ \hkl[0 -1 0]}\\
1500 \begin{minipage}{2.0cm}
1501 \includegraphics[width=2.0cm]{c_pd_vasp/100_2333.eps}
1503 \begin{minipage}{0.2cm}
1506 \begin{minipage}{2.0cm}
1507 \includegraphics[width=2.0cm]{c_pd_vasp/00-1-0-10_2333.eps}
1509 \begin{minipage}{0.2cm}
1512 \begin{minipage}{2.0cm}
1513 \includegraphics[width=2.0cm]{c_pd_vasp/0-10_2333.eps}
1514 \end{minipage}\\[0.1cm]
1515 $\Delta E=\unit[0.9]{eV}$ | Experimental values: \unit[0.70--0.87]{eV}\\
1516 $\Rightarrow$ {\color{red}Migration mechanism identified!}\\
1517 Note: Change in orientation
1519 \begin{minipage}{5.4cm}
1520 \includegraphics[width=6.0cm]{00-1_0-10_vasp_s.ps}
1521 \end{minipage}\\[0.1cm]
1524 Reorientation pathway composed of two consecutive processes of the above type
1533 C interstitial migration --- analytical potential
1540 \begin{minipage}[t]{6.0cm}
1541 {\bf\boldmath BC to \hkl[0 0 -1] transition}\\[0.2cm]
1542 \includegraphics[width=6.0cm]{bc_00-1_albe_s.ps}\\
1544 \item Lowermost migration barrier
1545 \item $\Delta E \approx \unit[2.2]{eV}$
1546 \item 2.4 times higher than ab initio result
1547 \item Different pathway
1550 \begin{minipage}[t]{0.2cm}
1553 \begin{minipage}[t]{6.0cm}
1554 {\bf\boldmath Transition involving a \hkl<1 1 0> configuration}
1557 \item Bond-centered configuration unstable\\
1558 $\rightarrow$ \ci{} \hkl<1 1 0> dumbbell
1559 \item Minima of the \hkl[0 0 -1] to \hkl[0 -1 0] transition\\
1560 $\rightarrow$ \ci{} \hkl<1 1 0> DB
1563 \includegraphics[width=6.0cm]{00-1_110_0-10_mig_albe.ps}
1565 \item $\Delta E \approx \unit[2.2]{eV} \text{ \& } \unit[0.9]{eV}$
1566 \item 2.4 -- 3.4 times higher than ab initio result
1567 \item After all: Change of the DB orientation
1573 {\color{red}\bf Drastically overestimated diffusion barrier}
1576 \begin{pspicture}(0,0)(0,0)
1577 \psline[linewidth=0.05cm,linecolor=gray](6.1,1.0)(6.1,9.3)
1593 \begin{minipage}{9cm}
1595 Summary of combinations}\\[0.1cm]
1597 \begin{tabular}{l c c c c c c}
1599 $E_{\text{b}}$ [eV] & 1 & 2 & 3 & 4 & 5 & R\\
1601 \hkl[0 0 -1] & {\color{red}-0.08} & -1.15 & {\color{red}-0.08} & 0.04 & -1.66 & -0.19\\
1602 \hkl[0 0 1] & 0.34 & 0.004 & -2.05 & 0.26 & -1.53 & -0.19\\
1603 \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}\\
1604 \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}\\
1605 \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}\\
1606 \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}\\
1608 C$_{\text{sub}}$ & 0.26 & -0.51 & -0.93 & -0.15 & 0.49 & -0.05\\
1609 Vacancy & -5.39 ($\rightarrow$ C$_{\text{sub}}$) & -0.59 & -3.14 & -0.54 & -0.50 & -0.31\\
1616 $E_{\text{b}}$ explainable by stress compensation / increase
1620 \begin{minipage}{3cm}
1621 \includegraphics[width=3.5cm]{comb_pos.eps}
1626 {\bf\boldmath Combinations of \hkl<1 0 0>-type interstitials}\\[0.2cm]
1627 \begin{minipage}[t]{3.2cm}
1628 \underline{\hkl[1 0 0] at position 1}\\[0.1cm]
1629 \includegraphics[width=2.8cm]{00-1dc/2-25.eps}
1631 \begin{minipage}[t]{3.0cm}
1632 \underline{\hkl[0 -1 0] at position 1}\\[0.1cm]
1633 \includegraphics[width=2.8cm]{00-1dc/2-39.eps}
1635 \begin{minipage}[t]{6.1cm}
1638 \item \ci{} agglomeration energetically favorable
1639 \item Most favorable: C clustering\\
1640 {\color{red}However \ldots}\\
1641 \ldots high migration barrier ($>4\,\text{eV}$)\\
1643 $4\times{\color{cyan}[-2.25]}$ versus
1644 $2\times{\color{orange}[-2.39]}$
1647 {\color{blue}\ci{} agglomeration / no C clustering}
1664 \begin{minipage}{9cm}
1666 Summary of combinations}\\[0.1cm]
1668 \begin{tabular}{l c c c c c c}
1670 $E_{\text{b}}$ [eV] & 1 & 2 & 3 & 4 & 5 & R\\
1672 \hkl[0 0 -1] & {\color{red}-0.08} & -1.15 & {\color{red}-0.08} & 0.04 & -1.66 & -0.19\\
1673 \hkl[0 0 1] & 0.34 & 0.004 & -2.05 & 0.26 & -1.53 & -0.19\\
1674 \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}\\
1675 \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}\\
1676 \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}\\
1677 \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}\\
1679 C$_{\text{sub}}$ & 0.26 & -0.51 & -0.93 & -0.15 & 0.49 & -0.05\\
1680 Vacancy & -5.39 ($\rightarrow$ C$_{\text{sub}}$) & -0.59 & -3.14 & -0.54 & -0.50 & -0.31\\
1687 $E_{\text{b}}$ explainable by stress compensation / increase
1691 \begin{minipage}{3cm}
1692 \includegraphics[width=3.5cm]{comb_pos.eps}
1697 {\bf\boldmath Combinations of \hkl<1 0 0>-type interstitials}\\[0.2cm]
1698 \begin{minipage}[t]{3.2cm}
1699 \underline{\hkl[1 0 0] at position 1}\\[0.1cm]
1700 \includegraphics[width=2.8cm]{00-1dc/2-25.eps}
1702 \begin{minipage}[t]{3.0cm}
1703 \underline{\hkl[0 -1 0] at position 1}\\[0.1cm]
1704 \includegraphics[width=2.8cm]{00-1dc/2-39.eps}
1706 \begin{minipage}[t]{6.1cm}
1709 \item \ci{} agglomeration energetically favorable
1710 \item Most favorable: C clustering\\
1711 {\color{red}However \ldots}\\
1712 \ldots high migration barrier ($>4\,\text{eV}$)\\
1714 $4\times{\color{cyan}[-2.25]}$ versus
1715 $2\times{\color{orange}[-2.39]}$
1718 {\color{blue}\ci{} agglomeration / no C clustering}
1723 \begin{pspicture}(0,0)(0,0)
1724 \rput(6.5,5.0){\psframebox[fillstyle=solid,opacity=0.5,fillcolor=black]{
1725 \begin{minipage}{14cm}
1730 \rput(6.5,5.3){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=white,linewidth=0.1cm]{
1731 \begin{minipage}{8cm}
1735 Interaction along \hkl[1 1 0]
1736 \includegraphics[width=7cm]{db_along_110_cc.ps}
1748 Defect combinations of C-Si dimers and vacancies
1754 \begin{minipage}[b]{2.6cm}
1756 \underline{V at 2: $E_{\text{b}}=-0.59\text{ eV}$}\\[0.1cm]
1757 \includegraphics[width=2.5cm]{00-1dc/0-59.eps}
1760 \begin{minipage}[b]{7cm}
1763 \begin{minipage}[b]{2.6cm}
1765 \underline{V at 3, $E_{\text{b}}=-3.14\text{ eV}$}\\[0.1cm]
1766 \includegraphics[width=2.5cm]{00-1dc/3-14.eps}
1768 \end{minipage}\\[0.2cm]
1770 \begin{minipage}{6.5cm}
1771 \includegraphics[width=6.0cm]{059-539.ps}
1773 \begin{minipage}{5.7cm}
1774 \includegraphics[width=6.0cm]{314-539.ps}
1777 \begin{pspicture}(0,0)(0,0)
1778 \psline[linewidth=0.05cm,linecolor=gray](6.3,9.0)(6.3,0)
1780 \rput(6.3,7.0){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=white,linewidth=0.05cm,linecolor=gray]{
1781 \begin{minipage}{6.5cm}
1783 IBS: Impinging C creates V \& far away \si\\[0.3cm]
1784 Low migration barrier towards C$_{\text{sub}}$\\
1786 High barrier for reverse process\\[0.3cm]
1788 High probability of stable C$_{\text{sub}}$ configuration
1805 Combinations of substitutional C and Si self-interstitials
1810 \begin{minipage}{6.0cm}
1811 \includegraphics[width=5.8cm]{c_sub_si110.ps}
1813 \begin{minipage}{6.3cm}
1816 \item IBS: C may displace Si\\
1817 $\Rightarrow$ C$_{\text{sub}}$ + \hkl<1 1 0> Si self-interstitial
1819 \hkl<1 1 0>-type $\rightarrow$ favored combination
1820 \renewcommand\labelitemi{$\Rightarrow$}
1821 \item Most favorable: \cs{} along \hkl<1 1 0> chain \si{}
1822 \item Less favorable than C-Si \hkl<1 0 0> dumbbell
1823 \item Interaction drops quickly to zero\\
1824 $\rightarrow$ low capture radius
1828 IBS process far from equilibrium\\
1829 \cs{} \& \si{} instead of thermodynamic ground state
1834 \begin{minipage}{6.0cm}
1835 \includegraphics[width=6.0cm]{162-097.ps}
1837 \begin{minipage}{6.2cm}
1839 Ab initio MD at \degc{900}\\
1840 \includegraphics[width=3.3cm]{md_vasp_01.eps}
1841 $t=\unit[2230]{fs}$\\
1842 \includegraphics[width=3.3cm]{md_vasp_02.eps}
1846 Contribution of entropy to structural formation
1858 Conclusion of defect / migration / combined defect simulations
1867 \item Accurately described by quantum-mechanical simulations
1868 \item Less accurate description by classical potential simulations
1869 \item Underestimated formation energy of \cs{} by classical approach
1870 \item Both methods predict same ground state: \ci{} \hkl<1 0 0> dumbbell
1875 \item C migration pathway in Si identified
1876 \item Consistent with reorientation and diffusion experiments
1879 \item Different path and ...
1880 \item overestimated barrier by classical potential calculations
1883 Concerning the precipitation mechanism
1885 \item Agglomeration of C-Si dumbbells energetically favorable
1886 (stress compensation)
1887 \item C-Si indeed favored compared to
1888 C$_{\text{sub}}$ \& \hkl<1 1 0> Si self-interstitial
1889 \item Possible low interaction capture radius of
1890 C$_{\text{sub}}$ \& \hkl<1 1 0> Si self-interstitial
1891 \item Low barrier for
1892 \ci{} \hkl<1 0 0> $\rightarrow$ \cs{} \& \si{} \hkl<1 1 0>
1893 \item In absence of nearby \hkl<1 1 0> Si self-interstitial:
1894 C-Si \hkl<1 0 0> + Vacancy $\rightarrow$ C$_{\text{sub}}$ (SiC)
1897 {\color{blue}Results suggest increased participation of \cs}
1905 Silicon carbide precipitation simulations
1911 \begin{pspicture}(0,0)(12,6.5)
1913 \rput(3.5,5.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
1916 \item Create c-Si volume
1917 \item Periodc boundary conditions
1918 \item Set requested $T$ and $p=0\text{ bar}$
1919 \item Equilibration of $E_{\text{kin}}$ and $E_{\text{pot}}$
1922 \rput(3.5,2.7){\rnode{insert}{\psframebox[fillstyle=solid,fillcolor=lachs]{
1924 Insertion of C atoms at constant T
1926 \item total simulation volume {\pnode{in1}}
1927 \item volume of minimal SiC precipitate {\pnode{in2}}
1928 \item volume consisting of Si atoms to form a minimal {\pnode{in3}}\\
1932 \rput(3.5,1){\rnode{cool}{\psframebox[fillstyle=solid,fillcolor=lbb]{
1934 Run for 100 ps followed by cooling down to $20\, ^{\circ}\textrm{C}$
1936 \ncline[]{->}{init}{insert}
1937 \ncline[]{->}{insert}{cool}
1938 \psframe[fillstyle=solid,fillcolor=white](7.5,0.7)(13.5,6.3)
1939 \rput(7.8,6){\footnotesize $V_1$}
1940 \psframe[fillstyle=solid,fillcolor=lightgray](9,2)(12,5)
1941 \rput(9.2,4.85){\tiny $V_2$}
1942 \psframe[fillstyle=solid,fillcolor=gray](9.25,2.25)(11.75,4.75)
1943 \rput(9.55,4.45){\footnotesize $V_3$}
1944 \rput(7.9,3.2){\pnode{ins1}}
1945 \rput(9.22,2.8){\pnode{ins2}}
1946 \rput(11.0,2.4){\pnode{ins3}}
1947 \ncline[]{->}{in1}{ins1}
1948 \ncline[]{->}{in2}{ins2}
1949 \ncline[]{->}{in3}{ins3}
1954 \item Restricted to classical potential simulations
1955 \item $V_2$ and $V_3$ considered due to low diffusion
1956 \item Amount of C atoms: 6000
1957 ($r_{\text{prec}}\approx 3.1\text{ nm}$, IBS: 2 ... 4 nm)
1958 \item Simulation volume: $31\times 31\times 31$ unit cells
1967 Silicon carbide precipitation simulations at $450\,^{\circ}\mathrm{C}$ as in IBS
1972 \begin{minipage}{6.5cm}
1973 \includegraphics[width=6.4cm]{sic_prec_450_si-si_c-c.ps}
1975 \begin{minipage}{6.5cm}
1976 \includegraphics[width=6.4cm]{sic_prec_450_energy.ps}
1979 \begin{minipage}{6.5cm}
1980 \includegraphics[width=6.4cm]{sic_prec_450_si-c.ps}
1982 \begin{minipage}{6.5cm}
1984 \underline{Low C concentration ($V_1$)}\\
1985 \hkl<1 0 0> C-Si dumbbell dominated structure
1987 \item Si-C bumbs around 0.19 nm
1988 \item C-C peak at 0.31 nm (as expected in 3C-SiC):\\
1989 concatenated dumbbells of various orientation
1990 \item Si-Si NN distance stretched to 0.3 nm
1992 {\color{blue}$\Rightarrow$ C atoms in proper 3C-SiC distance first}\\
1993 \underline{High C concentration ($V_2$, $V_3$)}\\
1994 High amount of strongly bound C-C bonds\\
1995 Defect density $\uparrow$ $\Rightarrow$ considerable amount of damage\\
1996 Only short range order observable\\
1997 {\color{blue}$\Rightarrow$ amorphous SiC-like phase}
2005 Silicon carbide precipitation simulations at $450\,^{\circ}\mathrm{C}$ as in IBS
2010 \begin{minipage}{6.5cm}
2011 \includegraphics[width=6.4cm]{sic_prec_450_si-si_c-c.ps}
2013 \begin{minipage}{6.5cm}
2014 \includegraphics[width=6.4cm]{sic_prec_450_energy.ps}
2017 \begin{minipage}{6.5cm}
2018 \includegraphics[width=6.4cm]{sic_prec_450_si-c.ps}
2020 \begin{minipage}{6.5cm}
2022 \underline{Low C concentration ($V_1$)}\\
2023 \hkl<1 0 0> C-Si dumbbell dominated structure
2025 \item Si-C bumbs around 0.19 nm
2026 \item C-C peak at 0.31 nm (as expected in 3C-SiC):\\
2027 concatenated dumbbells of various orientation
2028 \item Si-Si NN distance stretched to 0.3 nm
2030 {\color{blue}$\Rightarrow$ C atoms in proper 3C-SiC distance first}\\
2031 \underline{High C concentration ($V_2$, $V_3$)}\\
2032 High amount of strongly bound C-C bonds\\
2033 Defect density $\uparrow$ $\Rightarrow$ considerable amount of damage\\
2034 Only short range order observable\\
2035 {\color{blue}$\Rightarrow$ amorphous SiC-like phase}
2038 \begin{pspicture}(0,0)(0,0)
2039 \rput(6.7,5.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=white]{
2040 \begin{minipage}{10cm}
2042 {\color{red}\bf 3C-SiC formation fails to appear}
2044 \item Low C concentration simulations
2046 \item Formation of \ci{} indeed occurs
2047 \item Agllomeration not observed
2049 \item High C concentration simulations
2051 \item Amorphous SiC-like structure\\
2052 (not expected at prevailing temperatures)
2053 \item Rearrangement and transition into 3C-SiC structure missing
2065 Limitations of molecular dynamics and short range potentials
2072 \underline{Time scale problem of MD}\\[0.2cm]
2073 Minimize integration error\\
2074 $\Rightarrow$ discretization considerably smaller than
2075 reciprocal of fastest vibrational mode\\[0.1cm]
2076 Order of fastest vibrational mode: $10^{13} - 10^{14}\text{ Hz}$\\
2077 $\Rightarrow$ suitable choice of time step:
2078 $\tau=1\text{ fs}=10^{-15}\text{ s}$\\
2079 $\Rightarrow$ {\color{red}\underline{slow}} phase space propagation\\[0.1cm]
2080 Several local minima in energy surface separated by large energy barriers\\
2081 $\Rightarrow$ transition event corresponds to a multiple
2082 of vibrational periods\\
2083 $\Rightarrow$ phase transition made up of {\color{red}\underline{many}}
2084 infrequent transition events\\[0.1cm]
2085 {\color{blue}Accelerated methods:}
2086 \underline{Temperature accelerated} MD (TAD), self-guided MD \ldots
2090 \underline{Limitations related to the short range potential}\\[0.2cm]
2091 Cut-off function pushing forces and energies to zero between 1$^{\text{st}}$
2092 and 2$^{\text{nd}}$ next neighbours\\
2093 $\Rightarrow$ overestimated unphysical high forces of next neighbours
2099 Potential enhanced problem of slow phase space propagation
2104 \underline{Approach to the (twofold) problem}\\[0.2cm]
2105 Increased temperature simulations without TAD corrections\\
2106 (accelerated methods or higher time scales exclusively not sufficient)
2108 \begin{picture}(0,0)(-260,-30)
2110 \begin{minipage}{4.2cm}
2117 \item 3C-SiC also observed for higher T
2118 \item higher T inside sample
2119 \item structural evolution vs.\\
2120 equilibrium properties
2126 \begin{picture}(0,0)(-305,-155)
2128 \begin{minipage}{2.5cm}
2132 thermodynmic sampling
2143 Increased temperature simulations at low C concentration
2148 \begin{minipage}{6.5cm}
2149 \includegraphics[width=6.4cm]{tot_pc_thesis.ps}
2151 \begin{minipage}{6.5cm}
2152 \includegraphics[width=6.4cm]{tot_pc3_thesis.ps}
2155 \begin{minipage}{6.5cm}
2156 \includegraphics[width=6.4cm]{tot_pc2_thesis.ps}
2158 \begin{minipage}{6.5cm}
2160 \underline{Si-C bonds:}
2162 \item Vanishing cut-off artifact (above $1650\,^{\circ}\mathrm{C}$)
2163 \item Structural change: C-Si \hkl<1 0 0> $\rightarrow$ C$_{\text{sub}}$
2165 \underline{Si-Si bonds:}
2166 {\color{blue}Si-C$_{\text{sub}}$-Si} along \hkl<1 1 0>
2167 ($\rightarrow$ 0.325 nm)\\[0.1cm]
2168 \underline{C-C bonds:}
2170 \item C-C next neighbour pairs reduced (mandatory)
2171 \item Peak at 0.3 nm slightly shifted
2173 \item C-Si \hkl<1 0 0> combinations (dashed arrows)\\
2174 $\rightarrow$ C-Si \hkl<1 0 0> \& C$_{\text{sub}}$
2176 $\rightarrow$ pure {\color{blue}C$_{\text{sub}}$ combinations}
2178 \item Range [|-$\downarrow$]:
2179 {\color{blue}C$_{\text{sub}}$ \& C$_{\text{sub}}$
2180 with nearby Si$_{\text{I}}$}
2185 \begin{picture}(0,0)(-330,-74)
2188 \begin{minipage}{1.6cm}
2191 stretched SiC\\[-0.1cm]
2203 Increased temperature simulations at low C concentration
2208 \begin{minipage}{6.5cm}
2209 \includegraphics[width=6.4cm]{tot_pc_thesis.ps}
2211 \begin{minipage}{6.5cm}
2212 \includegraphics[width=6.4cm]{tot_pc3_thesis.ps}
2215 \begin{minipage}{6.5cm}
2216 \includegraphics[width=6.4cm]{tot_pc2_thesis.ps}
2218 \begin{minipage}{6.5cm}
2220 \underline{Si-C bonds:}
2222 \item Vanishing cut-off artifact (above $1650\,^{\circ}\mathrm{C}$)
2223 \item Structural change: C-Si \hkl<1 0 0> $\rightarrow$ C$_{\text{sub}}$
2225 \underline{Si-Si bonds:}
2226 {\color{blue}Si-C$_{\text{sub}}$-Si} along \hkl<1 1 0>
2227 ($\rightarrow$ 0.325 nm)\\[0.1cm]
2228 \underline{C-C bonds:}
2230 \item C-C next neighbour pairs reduced (mandatory)
2231 \item Peak at 0.3 nm slightly shifted
2233 \item C-Si \hkl<1 0 0> combinations (dashed arrows)\\
2234 $\rightarrow$ C-Si \hkl<1 0 0> \& C$_{\text{sub}}$
2236 $\rightarrow$ pure {\color{blue}C$_{\text{sub}}$ combinations}
2238 \item Range [|-$\downarrow$]:
2239 {\color{blue}C$_{\text{sub}}$ \& C$_{\text{sub}}$
2240 with nearby Si$_{\text{I}}$}
2245 %\begin{picture}(0,0)(-330,-74)
2248 %\begin{minipage}{1.6cm}
2251 %stretched SiC\\[-0.1cm]
2258 \begin{pspicture}(0,0)(0,0)
2259 \rput(6.7,5.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=white]{
2260 \begin{minipage}{10cm}
2262 {\color{blue}\bf Stretched SiC in c-Si}
2264 \item Consistent to precipitation model involving \cs{}
2265 \item Explains annealing behavior of high/low T C implants
2267 \item Low T: highly mobiel \ci{}
2268 \item High T: stable configurations of \cs{}
2271 $\Rightarrow$ High T $\leftrightarrow$ IBS conditions far from equilibrium\\
2272 $\Rightarrow$ Precipitation mechanism involving \cs{}
2282 Increased temperature simulations at high C concentration
2287 \begin{minipage}{6.5cm}
2288 \includegraphics[width=6.4cm]{12_pc_thesis.ps}
2290 \begin{minipage}{6.5cm}
2291 \includegraphics[width=6.4cm]{12_pc_c_thesis.ps}
2299 \begin{minipage}[t]{6.0cm}
2300 0.186 nm: Si-C pairs $\uparrow$\\
2301 (as expected in 3C-SiC)\\[0.2cm]
2302 0.282 nm: Si-C-C\\[0.2cm]
2303 $\approx$0.35 nm: C-Si-Si
2306 \begin{minipage}{0.2cm}
2310 \begin{minipage}[t]{6.0cm}
2311 0.15 nm: C-C pairs $\uparrow$\\
2312 (as expected in graphite/diamond)\\[0.2cm]
2313 0.252 nm: C-C-C (2$^{\text{nd}}$ NN for diamond)\\[0.2cm]
2314 0.31 nm: shifted towards 0.317 nm $\rightarrow$ C-Si-C
2319 \item Decreasing cut-off artifact
2320 \item {\color{red}Amorphous} SiC-like phase remains
2321 \item High amount of {\color{red}damage} \& alignement to c-Si host matrix lost
2322 \item Slightly sharper peaks $\Rightarrow$ indicate slight {\color{blue}acceleration of dynamics} due to temperature
2331 High C \& small $V$ \& short $t$
2334 Slow restructuring due to strong C-C bonds
2337 High C \& low T implants
2348 Summary and Conclusions
2356 \begin{minipage}[t]{12.9cm}
2357 \underline{Pecipitation simulations}
2359 \item High C concentration $\rightarrow$ amorphous SiC like phase
2360 \item Problem of potential enhanced slow phase space propagation
2361 \item Low T $\rightarrow$ C-Si \hkl<1 0 0> dumbbell dominated structure
2362 \item High T $\rightarrow$ C$_{\text{sub}}$ dominated structure
2363 \item High T necessary to simulate IBS conditions (far from equilibrium)
2364 \item Precipitation by successive agglomeration of \cs (epitaxy)
2365 \item \si{}: vehicle to form \cs{} \& supply of Si \& stress compensation
2366 (stretched SiC, interface)
2374 \begin{minipage}{12.9cm}
2379 \item Point defects excellently / fairly well described
2381 \item C$_{\text{sub}}$ drastically underestimated by EA
2382 \item EA predicts correct ground state:
2383 C$_{\text{sub}}$ \& \si{} $>$ \ci{}
2384 \item Identified migration path explaining
2385 diffusion and reorientation experiments by DFT
2386 \item EA fails to describe \ci{} migration:
2387 Wrong path \& overestimated barrier
2389 \item Combinations of defects
2391 \item Agglomeration of point defects energetically favorable
2392 by compensation of stress
2393 \item Formation of C-C unlikely
2394 \item C$_{\text{sub}}$ favored conditions (conceivable in IBS)
2395 \item \ci{} \hkl<1 0 0> $\leftrightarrow$ \cs{} \& \si{} \hkl<1 1 0>\\
2396 Low barrier (\unit[0.77]{eV}) \& low capture radius
2404 \framebox{Precipitation by successive agglomeration of \cs{}}
2422 \underline{Augsburg}
2424 \item Prof. B. Stritzker (accomodation at EP \RM{4})
2425 \item Ralf Utermann (EDV)
2428 \underline{Helsinki}
2430 \item Prof. K. Nordlund (MD)
2435 \item Bayerische Forschungsstiftung (financial support)
2438 \underline{Paderborn}
2440 \item Prof. J. Lindner (SiC)
2441 \item Prof. G. Schmidt (DFT + financial support)
2442 \item Dr. E. Rauls (DFT + SiC)
2443 \item Dr. S. Sanna (VASP)
2450 \bf Thank you for your attention!