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
1801 Combinations of substitutional C and Si self-interstitials
1808 \begin{minipage}{6.2cm}
1810 {\bf\boldmath C$_{\text{sub}}$ - \si{} \hkl<1 1 0> interaction}
1812 \item Most favorable: \cs{} along \hkl<1 1 0> chain of \si{}
1813 \item Less favorable than ground-state \ci{} \hkl<1 0 0> DB
1814 \item Interaction drops quickly to zero\\
1815 $\rightarrow$ low capture radius
1819 \begin{minipage}{0.2cm}
1822 \begin{minipage}{6.0cm}
1824 {\bf Transition from the ground state}
1826 \item Low transition barrier
1827 \item Barrier smaller than \ci{} migration barrier
1828 \item Low \si{} migration barrier (\unit[0.67]{eV})\\
1829 $\rightarrow$ Separation of \cs{} \& \si{} most probable
1832 \end{minipage}\\[0.3cm]
1834 \begin{minipage}{6.0cm}
1835 \includegraphics[width=6.0cm]{c_sub_si110.ps}
1837 \begin{minipage}{0.4cm}
1840 \begin{minipage}{6.0cm}
1842 \includegraphics[width=6.0cm]{162-097.ps}
1846 \begin{pspicture}(0,0)(0,0)
1847 \psline[linewidth=0.05cm,linecolor=gray](6.5,0)(6.5,7.5)
1848 \rput(6.5,-0.7){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=white,linewidth=0.05cm,linecolor=blue]{
1849 \begin{minipage}{8cm}
1853 \cs{} \& \si{} instead of thermodynamic ground state\\[0.1cm]
1854 IBS --- process far from equilibrium\\
1867 Combinations of substitutional C and Si self-interstitials
1874 \begin{minipage}{6.2cm}
1876 {\bf\boldmath C$_{\text{sub}}$ - \si{} \hkl<1 1 0> interaction}
1878 \item Most favorable: \cs{} along \hkl<1 1 0> chain of \si{}
1879 \item Less favorable than ground-state \ci{} \hkl<1 0 0> DB
1880 \item Interaction drops quickly to zero\\
1881 $\rightarrow$ low capture radius
1885 \begin{minipage}{0.2cm}
1888 \begin{minipage}{6.0cm}
1890 {\bf Transition from the ground state}
1892 \item Low transition barrier
1893 \item Barrier smaller than \ci{} migration barrier
1894 \item Low \si{} migration barrier (\unit[0.67]{eV})\\
1895 $\rightarrow$ Separation of \cs{} \& \si{} most probable
1898 \end{minipage}\\[0.3cm]
1900 \begin{minipage}{6.0cm}
1901 \includegraphics[width=6.0cm]{c_sub_si110.ps}
1903 \begin{minipage}{0.4cm}
1906 \begin{minipage}{6.0cm}
1908 \includegraphics[width=6.0cm]{162-097.ps}
1912 \begin{pspicture}(0,0)(0,0)
1913 \psline[linewidth=0.05cm,linecolor=gray](6.5,0)(6.5,7.5)
1914 \rput(6.5,-0.7){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=white,linewidth=0.05cm,linecolor=blue]{
1915 \begin{minipage}{8cm}
1919 \cs{} \& \si{} instead of thermodynamic ground state\\[0.1cm]
1920 IBS --- process far from equilibrium\\
1928 \begin{pspicture}(0,0)(0,0)
1929 \rput(6.5,5.0){\psframebox[fillstyle=solid,opacity=0.5,fillcolor=black]{
1930 \begin{minipage}{14cm}
1935 \rput(6.5,4.3){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=white,linewidth=0.1cm]{
1936 \begin{minipage}{11cm}
1940 Ab initio MD at \degc{900}\\[0.4cm]
1941 \begin{minipage}{5.4cm}
1943 \includegraphics[width=4.3cm]{md01_bonds.eps}\\
1946 \begin{minipage}{5.4cm}
1948 \includegraphics[width=4.3cm]{md02_bonds.eps}\\
1950 \end{minipage}\\[0.5cm]
1952 Contribution of entropy to structural formation\\[0.1cm]
1965 Silicon carbide precipitation simulations
1975 \begin{pspicture}(0,0)(12,6.5)
1977 \rput(3.5,5.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
1980 \item Create c-Si volume
1981 \item Periodc boundary conditions
1982 \item Set requested $T$ and $p=0\text{ bar}$
1983 \item Equilibration of $E_{\text{kin}}$ and $E_{\text{pot}}$
1986 \rput(3.5,2.7){\rnode{insert}{\psframebox[fillstyle=solid,fillcolor=lachs]{
1988 Insertion of C atoms at constant T
1990 \item total simulation volume {\pnode{in1}}
1991 \item volume of minimal SiC precipitate size {\pnode{in2}}
1992 \item volume consisting of Si atoms to form a minimal {\pnode{in3}}\\
1996 \rput(3.5,1){\rnode{cool}{\psframebox[fillstyle=solid,fillcolor=lbb]{
1998 Run for 100 ps followed by cooling down to $20\, ^{\circ}\textrm{C}$
2000 \ncline[]{->}{init}{insert}
2001 \ncline[]{->}{insert}{cool}
2002 \psframe[fillstyle=solid,fillcolor=white](7.3,0.7)(12.8,6.3)
2003 \rput(7.6,6){\footnotesize $V_1$}
2004 \psframe[fillstyle=solid,fillcolor=lightgray](8.7,2)(11.6,5)
2005 \rput(8.9,4.85){\tiny $V_2$}
2006 \psframe[fillstyle=solid,fillcolor=gray](8.95,2.25)(11.35,4.75)
2007 \rput(9.25,4.45){\footnotesize $V_3$}
2008 \rput(7.9,3.2){\pnode{ins1}}
2009 \rput(8.92,2.8){\pnode{ins2}}
2010 \rput(10.8,2.4){\pnode{ins3}}
2011 \ncline[]{->}{in1}{ins1}
2012 \ncline[]{->}{in2}{ins2}
2013 \ncline[]{->}{in3}{ins3}
2023 \begin{minipage}{5.7cm}
2025 \item Amount of C atoms: 6000\\
2026 ($r_{\text{prec}}\approx 3.1\text{ nm}$, IBS: \unit[2--4]{nm})
2027 \item Simulation volume: $31^3$ Si unit cells\\
2031 \begin{minipage}{0.3cm}
2035 \begin{minipage}{6.0cm}
2036 Restricted to classical potential caclulations\\
2037 $\rightarrow$ Low C diffusion / overestimated barrier\\
2038 $\rightarrow$ Consider $V_2$ and $V_3$
2040 % \item $V_2$ and $V_3$ considered due to expected low C diffusion
2051 Silicon carbide precipitation simulations at \degc{450} as in IBS
2056 \begin{minipage}{6.3cm}
2057 \hspace*{-0.4cm}\includegraphics[width=6.5cm]{sic_prec_450_si-c.ps}\\
2058 \hspace*{-0.4cm}\includegraphics[width=6.5cm]{sic_prec_450_si-si_c-c.ps}
2061 \begin{minipage}{6.1cm}
2063 \underline{Low C concentration --- {\color{red}$V_1$}}\\[0.1cm]
2064 \hkl<1 0 0> C-Si dumbbell dominated structure
2066 \item Si-C bumbs around \unit[0.19]{nm}
2067 \item C-C peak at \unit[0.31]{nm} (expected in 3C-SiC):\\
2068 concatenated differently oriented \ci{} DBs
2069 \item Si-Si NN distance stretched to \unit[0.3]{nm}
2071 \begin{pspicture}(0,0)(6.0,1.0)
2072 \rput(3.2,0.5){\psframebox[linewidth=0.03cm,linecolor=blue]{
2073 \begin{minipage}{6cm}
2075 Formation of \ci{} dumbbells\\
2076 C atoms in proper 3C-SiC distance first
2079 \end{pspicture}\\[0.1cm]
2080 \underline{High C concentration --- {\color{green}$V_2$}/{\color{blue}$V_3$}}
2082 \item High amount of strongly bound C-C bonds
2083 \item Increased defect \& damage density\\
2084 $\rightarrow$ Arrangements hard to categorize and trace
2085 \item Only short range order observable
2087 \begin{pspicture}(0,0)(6.0,0.8)
2088 \rput(3.2,0.5){\psframebox[linewidth=0.03cm,linecolor=blue]{
2089 \begin{minipage}{6cm}
2091 Amorphous SiC-like phase
2094 \end{pspicture}\\[0.3cm]
2095 \begin{pspicture}(0,0)(6.0,2.0)
2096 \rput(3.2,1.0){\psframebox[linewidth=0.05cm,linecolor=white]{
2097 \begin{minipage}{6cm}
2111 Silicon carbide precipitation simulations at \degc{450} as in IBS
2116 \begin{minipage}{6.3cm}
2117 \hspace*{-0.4cm}\includegraphics[width=6.5cm]{sic_prec_450_si-c.ps}\\
2118 \hspace*{-0.4cm}\includegraphics[width=6.5cm]{sic_prec_450_si-si_c-c.ps}
2121 \begin{minipage}{6.1cm}
2123 \underline{Low C concentration --- {\color{red}$V_1$}}\\[0.1cm]
2124 \hkl<1 0 0> C-Si dumbbell dominated structure
2126 \item Si-C bumbs around \unit[0.19]{nm}
2127 \item C-C peak at \unit[0.31]{nm} (expected in 3C-SiC):\\
2128 concatenated differently oriented \ci{} DBs
2129 \item Si-Si NN distance stretched to \unit[0.3]{nm}
2131 \begin{pspicture}(0,0)(6.0,1.0)
2132 \rput(3.2,0.5){\psframebox[linewidth=0.03cm,linecolor=blue]{
2133 \begin{minipage}{6cm}
2135 Formation of \ci{} dumbbells\\
2136 C atoms in proper 3C-SiC distance first
2139 \end{pspicture}\\[0.1cm]
2140 \underline{High C concentration --- {\color{green}$V_2$}/{\color{blue}$V_3$}}
2142 \item High amount of strongly bound C-C bonds
2143 \item Increased defect \& damage density\\
2144 $\rightarrow$ Arrangements hard to categorize and trace
2145 \item Only short range order observable
2147 \begin{pspicture}(0,0)(6.0,0.8)
2148 \rput(3.2,0.5){\psframebox[linewidth=0.03cm,linecolor=blue]{
2149 \begin{minipage}{6cm}
2151 Amorphous SiC-like phase
2154 \end{pspicture}\\[0.3cm]
2155 \begin{pspicture}(0,0)(6.0,2.0)
2156 \rput(3.2,1.0){\psframebox[linewidth=0.05cm,linecolor=black]{
2157 \begin{minipage}{6cm}
2160 {\bf\color{red}3C-SiC formation fails to appear}\\[0.3cm]
2161 \begin{minipage}{0.8cm}
2162 {\bf\boldmath $V_1$:}
2164 \begin{minipage}{5.1cm}
2165 Formation of \ci{} indeed occurs\\
2166 Agllomeration not observed
2167 \end{minipage}\\[0.3cm]
2168 \begin{minipage}{0.8cm}
2169 {\bf\boldmath $V_{2,3}$:}
2171 \begin{minipage}{5.1cm}
2172 Amorphous SiC-like structure\\
2173 (not expected at \degc{450})\\[0.05cm]
2174 No rearrangement/transition into 3C-SiC
2175 \end{minipage}\\[0.1cm]
2187 Limitations of MD and short range potentials
2194 {\bf Time scale problem of MD}\\[0.2cm]
2195 Precise integration \& thermodynamic sampling\\
2196 $\Rightarrow$ $\Delta t \ll \left( \max{\omega} \right)^{-1}$,
2197 $\omega$: vibrational mode\\
2198 $\Rightarrow$ {\color{red}\underline{Slow}} phase space propagation\\[0.2cm]
2199 Several local minima separated by large energy barriers\\
2200 $\Rightarrow$ Transition event corresponds to a multiple
2201 of vibrational periods\\
2202 $\Rightarrow$ Phase transition consists of {\color{red}\underline{many}}
2203 infrequent transition events\\[0.2cm]
2204 {\color{blue}Accelerated methods:}
2205 \underline{Temperature accelerated} MD (TAD), self-guided MD \ldots
2209 {\bf Limitations related to the short range potential}\\[0.2cm]
2210 Cut-off function limits interaction to next neighbours\\
2211 $\Rightarrow$ Overestimated unphysical high forces of next neighbours
2215 {\bf Approach to the (twofold) problem}\\[0.2cm]
2216 Increased temperature simulations without TAD corrections\\
2217 Accelerated methods or higher time scales exclusively not sufficient!
2219 \begin{pspicture}(0,0)(0,0)
2220 \rput(4.0,2.8){\psframebox[linewidth=0.07cm,linecolor=red]{
2221 \begin{minipage}{7.5cm}
2224 Potential enhanced slow phase space propagation
2227 \rput(11.3,7.5){\psframebox[linewidth=0.03cm,linecolor=blue]{
2228 \begin{minipage}{2.7cm}
2232 thermodynamic sampling
2235 \psline[linewidth=0.03cm,linecolor=blue]{<-}(11.3,7.0)(11.0,5.7)
2236 \rput(10.85,2.6){\psframebox[linewidth=0.03cm,linecolor=blue]{
2237 \begin{minipage}{3.6cm}
2240 \underline{IBS}\\[0.1cm]
2241 3C-SiC also observed for higher T\\[0.1cm]
2242 Higher T inside sample\\[0.1cm]
2243 Structural evolution vs.\\
2244 equilibrium properties
2247 \psline[linewidth=0.03cm,linecolor=blue]{->}(10.85,1.75)(9.0,1.0)
2259 Increased temperature simulations --- $V_1$
2264 \begin{minipage}{6.2cm}
2265 \hspace*{-0.4cm}\includegraphics[width=6.5cm]{tot_pc_thesis.ps}
2268 \begin{minipage}{6.2cm}
2269 \includegraphics[width=6.5cm]{tot_pc3_thesis.ps}
2272 \begin{minipage}{6.2cm}
2273 \hspace*{-0.4cm}\includegraphics[width=6.5cm]{tot_pc2_thesis.ps}
2276 \begin{minipage}{6.3cm}
2278 \underline{Si-C bonds:}
2280 \item Vanishing cut-off artifact (above $1650\,^{\circ}\mathrm{C}$)
2281 \item Structural change: C-Si \hkl<1 0 0> $\rightarrow$ C$_{\text{sub}}$
2283 \underline{Si-Si bonds:}
2284 {\color{blue}Si-C$_{\text{sub}}$-Si} along \hkl<1 1 0>
2285 ($\rightarrow$ 0.325 nm)\\[0.1cm]
2286 \underline{C-C bonds:}
2288 \item C-C next neighbour pairs reduced (mandatory)
2289 \item Peak at 0.3 nm slightly shifted
2291 \item C-Si \hkl<1 0 0> combinations (dashed arrows)\\
2292 $\rightarrow$ C-Si \hkl<1 0 0> \& C$_{\text{sub}}$
2294 $\rightarrow$ pure {\color{blue}C$_{\text{sub}}$ combinations}
2296 \item Range [|-$\downarrow$]:
2297 {\color{blue}C$_{\text{sub}}$ \& C$_{\text{sub}}$
2298 with nearby Si$_{\text{I}}$}
2309 Increased temperature simulations --- $V_1$
2314 \begin{minipage}{6.2cm}
2315 \hspace*{-0.4cm}\includegraphics[width=6.5cm]{tot_pc_thesis.ps}
2318 \begin{minipage}{6.2cm}
2319 \includegraphics[width=6.5cm]{tot_pc3_thesis.ps}
2322 \begin{minipage}{6.2cm}
2323 \hspace*{-0.4cm}\includegraphics[width=6.5cm]{tot_pc2_thesis.ps}
2326 \begin{minipage}{6.3cm}
2328 \underline{Si-C bonds:}
2330 \item Vanishing cut-off artifact (above $1650\,^{\circ}\mathrm{C}$)
2331 \item Structural change: C-Si \hkl<1 0 0> $\rightarrow$ C$_{\text{sub}}$
2333 \underline{Si-Si bonds:}
2334 {\color{blue}Si-C$_{\text{sub}}$-Si} along \hkl<1 1 0>
2335 ($\rightarrow$ 0.325 nm)\\[0.1cm]
2336 \underline{C-C bonds:}
2338 \item C-C next neighbour pairs reduced (mandatory)
2339 \item Peak at 0.3 nm slightly shifted
2341 \item C-Si \hkl<1 0 0> combinations (dashed arrows)\\
2342 $\rightarrow$ C-Si \hkl<1 0 0> \& C$_{\text{sub}}$
2344 $\rightarrow$ pure {\color{blue}C$_{\text{sub}}$ combinations}
2346 \item Range [|-$\downarrow$]:
2347 {\color{blue}C$_{\text{sub}}$ \& C$_{\text{sub}}$
2348 with nearby Si$_{\text{I}}$}
2354 \begin{pspicture}(0,0)(0,0)
2355 \rput(6.5,5.0){\psframebox[fillstyle=solid,opacity=0.5,fillcolor=black]{
2356 \begin{minipage}{14cm}
2361 \rput(6.5,5.0){\psframebox[fillstyle=solid,fillcolor=white,linewidth=0.1cm]{
2362 \begin{minipage}{9cm}
2365 {\color{blue}\bf Stretched structures of SiC in c-Si}
2367 \item Consistent to precipitation model involving \cs{}
2368 \item Explains annealing behavior of high/low T C implants
2370 \item Low T: highly mobiel \ci{}
2371 \item High T: stable configurations of \cs{}
2374 $\Rightarrow$ High T $\leftrightarrow$ IBS conditions far from equilibrium\\
2375 $\Rightarrow$ Precipitation mechanism involving \cs{}
2386 Increased temperature simulations at high C concentration
2391 \begin{minipage}{6.5cm}
2392 \includegraphics[width=6.4cm]{12_pc_thesis.ps}
2394 \begin{minipage}{6.5cm}
2395 \includegraphics[width=6.4cm]{12_pc_c_thesis.ps}
2403 \begin{minipage}[t]{6.0cm}
2404 0.186 nm: Si-C pairs $\uparrow$\\
2405 (as expected in 3C-SiC)\\[0.2cm]
2406 0.282 nm: Si-C-C\\[0.2cm]
2407 $\approx$0.35 nm: C-Si-Si
2410 \begin{minipage}{0.2cm}
2414 \begin{minipage}[t]{6.0cm}
2415 0.15 nm: C-C pairs $\uparrow$\\
2416 (as expected in graphite/diamond)\\[0.2cm]
2417 0.252 nm: C-C-C (2$^{\text{nd}}$ NN for diamond)\\[0.2cm]
2418 0.31 nm: shifted towards 0.317 nm $\rightarrow$ C-Si-C
2423 \item Decreasing cut-off artifact
2424 \item {\color{red}Amorphous} SiC-like phase remains
2425 \item High amount of {\color{red}damage} \& alignement to c-Si host matrix lost
2426 \item Slightly sharper peaks $\Rightarrow$ indicate slight {\color{blue}acceleration of dynamics} due to temperature
2435 High C \& small $V$ \& short $t$
2438 Slow restructuring due to strong C-C bonds
2441 High C \& low T implants
2455 Summary and Conclusions
2463 \begin{minipage}[t]{12.9cm}
2464 \underline{Pecipitation simulations}
2466 \item High C concentration $\rightarrow$ amorphous SiC like phase
2467 \item Problem of potential enhanced slow phase space propagation
2468 \item Low T $\rightarrow$ C-Si \hkl<1 0 0> dumbbell dominated structure
2469 \item High T $\rightarrow$ C$_{\text{sub}}$ dominated structure
2470 \item High T necessary to simulate IBS conditions (far from equilibrium)
2471 \item Precipitation by successive agglomeration of \cs (epitaxy)
2472 \item \si{}: vehicle to form \cs{} \& supply of Si \& stress compensation
2473 (stretched SiC, interface)
2481 \begin{minipage}{12.9cm}
2486 \item Point defects excellently / fairly well described
2488 \item C$_{\text{sub}}$ drastically underestimated by EA
2489 \item EA predicts correct ground state:
2490 C$_{\text{sub}}$ \& \si{} $>$ \ci{}
2491 \item Identified migration path explaining
2492 diffusion and reorientation experiments by DFT
2493 \item EA fails to describe \ci{} migration:
2494 Wrong path \& overestimated barrier
2496 \item Combinations of defects
2498 \item Agglomeration of point defects energetically favorable
2499 by compensation of stress
2500 \item Formation of C-C unlikely
2501 \item C$_{\text{sub}}$ favored conditions (conceivable in IBS)
2502 \item \ci{} \hkl<1 0 0> $\leftrightarrow$ \cs{} \& \si{} \hkl<1 1 0>\\
2503 Low barrier (\unit[0.77]{eV}) \& low capture radius
2511 \framebox{Precipitation by successive agglomeration of \cs{}}
2529 \underline{Augsburg}
2531 \item Prof. B. Stritzker (accomodation at EP \RM{4})
2532 \item Ralf Utermann (EDV)
2535 \underline{Helsinki}
2537 \item Prof. K. Nordlund (MD)
2542 \item Bayerische Forschungsstiftung (financial support)
2545 \underline{Paderborn}
2547 \item Prof. J. Lindner (SiC)
2548 \item Prof. G. Schmidt (DFT + financial support)
2549 \item Dr. E. Rauls (DFT + SiC)
2550 \item Dr. S. Sanna (VASP)
2557 \bf Thank you for your attention!