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
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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
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61 \newrgbcolor{hred}{1.0 0.0 0.0}
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64 \newcommand{\headdiplom}{
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67 \begin{minipage}{14cm}
75 \newcommand{\headphd}{
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78 \begin{minipage}{14cm}
88 \extraslideheight{10in}
93 % specify width and height
98 \def\slidetopmargin{-0.15cm}
100 \newcommand{\ham}{\mathcal{H}}
101 \newcommand{\pot}{\mathcal{V}}
102 \newcommand{\foo}{\mathcal{U}}
103 \newcommand{\vir}{\mathcal{W}}
106 \renewcommand\labelitemii{{\color{gray}$\bullet$}}
109 \renewcommand{\phi}{\varphi}
112 \newcommand{\RM}[1]{\MakeUppercase{\romannumeral #1{}}}
115 \newrgbcolor{si-yellow}{.6 .6 0}
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118 \newrgbcolor{hlbb}{0.825 0.88 0.968}
119 \newrgbcolor{lachs}{1.0 .93 .81}
122 \newcommand{\si}{Si$_{\text{i}}${}}
123 \newcommand{\ci}{C$_{\text{i}}${}}
124 \newcommand{\cs}{C$_{\text{sub}}${}}
125 \newcommand{\degc}[1]{\unit[#1]{$^{\circ}$C}{}}
126 \newcommand{\distn}[1]{\unit[#1]{nm}{}}
127 \newcommand{\dista}[1]{\unit[#1]{\AA}{}}
128 \newcommand{\perc}[1]{\unit[#1]{\%}{}}
130 % no vertical centering
141 A B C D E F G H G F E D C B A
156 Atomistic simulation studies\\[0.2cm]
162 \textsc{Frank Zirkelbach}
166 Application talk at the Max Planck Institute for Solid State Research
170 Stuttgart, November 2011
175 % no vertical centering
183 % Phase diagram of the C/Si system\\
188 \begin{minipage}{6.5cm}
189 \includegraphics[width=6.5cm]{si-c_phase.eps}
192 R. I. Scace and G. A. Slack, J. Chem. Phys. 30, 1551 (1959)
195 \begin{pspicture}(0,0)(0,0)
196 \psellipse[linecolor=blue,linewidth=0.1cm](3.55,4.0)(0.5,2.9)
199 \begin{minipage}{6cm}
200 {\bf Phase diagram of the C/Si system}\\[0.2cm]
201 {\color{blue}Stoichiometric composition}
203 \item only chemical stable compound
204 \item wide band gap semiconductor\\
205 \underline{silicon carbide}, SiC
211 % motivation / properties / applications of silicon carbide
219 \begin{pspicture}(0,0)(13.5,5)
221 \psframe*[linecolor=hb](-0.2,0)(12.9,5)
223 \pspolygon[linecolor=hlbb,fillcolor=hlbb,fillstyle=solid](5.2,1)(6.5,1)(6.5,3)(5.2,3)
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226 \rput[lt](0,4.6){\color{gray}PROPERTIES}
228 \rput[lt](0.3,4){wide band gap}
229 \rput[lt](0.3,3.5){high electric breakdown field}
230 \rput[lt](0.3,3){good electron mobility}
231 \rput[lt](0.3,2.5){high electron saturation drift velocity}
232 \rput[lt](0.3,2){high thermal conductivity}
234 \rput[lt](0.3,1.5){hard and mechanically stable}
235 \rput[lt](0.3,1){chemically inert}
237 \rput[lt](0.3,0.5){radiation hardness}
239 \rput[rt](12.7,4.6){\color{gray}APPLICATIONS}
241 \rput[rt](12.5,3.85){high-temperature, high power}
242 \rput[rt](12.5,3.5){and high-frequency}
243 \rput[rt](12.5,3.15){electronic and optoelectronic devices}
245 \rput[rt](12.5,2.35){material suitable for extreme conditions}
246 \rput[rt](12.5,2){microelectromechanical systems}
247 \rput[rt](12.5,1.65){abrasives, cutting tools, heating elements}
249 \rput[rt](12.5,0.85){first wall reactor material, detectors}
250 \rput[rt](12.5,0.5){and electronic devices for space}
254 \begin{picture}(0,0)(5,-162)
255 \includegraphics[height=2.2cm]{3C_SiC_bs.eps}
257 \begin{picture}(0,0)(-120,-162)
258 \includegraphics[height=2.2cm]{nasa_600c_led.eps}
260 \begin{picture}(0,0)(-270,-162)
261 \includegraphics[height=2.2cm]{6h-sic_3c-sic.eps}
264 \begin{picture}(0,0)(10,65)
265 \includegraphics[height=2.8cm]{sic_switch.eps}
267 %\begin{picture}(0,0)(-243,65)
268 \begin{picture}(0,0)(-110,65)
269 \includegraphics[height=2.8cm]{ise_99.eps}
271 %\begin{picture}(0,0)(-135,65)
272 \begin{picture}(0,0)(-100,65)
273 \includegraphics[height=1.2cm]{infineon_schottky.eps}
275 \begin{picture}(0,0)(-233,65)
276 \includegraphics[height=2.8cm]{solar_car.eps}
286 Polytypes of SiC\\[0.4cm]
289 \includegraphics[width=3.8cm]{cubic_hex.eps}\\
290 \begin{minipage}{1.9cm}
291 {\tiny cubic (twist)}
293 \begin{minipage}{2.9cm}
294 {\tiny hexagonal (no twist)}
297 \begin{picture}(0,0)(-150,0)
298 \includegraphics[width=7cm]{polytypes.eps}
305 \begin{tabular}{l c c c c c c}
307 & 3C-SiC & 4H-SiC & 6H-SiC & Si & GaN & Diamond\\
309 Hardness [Mohs] & \multicolumn{3}{c}{------ 9.6 ------}& 6.5 & - & 10 \\
310 Band gap [eV] & 2.36 & 3.23 & 3.03 & 1.12 & 3.39 & 5.5 \\
311 Break down field [$10^6$ V/cm] & 4 & 3 & 3.2 & 0.6 & 5 & 10 \\
312 Saturation drift velocity [$10^7$ cm/s] & 2.5 & 2.0 & 2.0 & 1 & 2.7 & 2.7 \\
313 Electron mobility [cm$^2$/Vs] & 800 & 900 & 400 & 1100 & 900 & 2200 \\
314 Hole mobility [cm$^2$/Vs] & 320 & 120 & 90 & 420 & 150 & 1600 \\
315 Thermal conductivity [W/cmK] & 5.0 & 4.9 & 4.9 & 1.5 & 1.3 & 22 \\
319 \begin{pspicture}(0,0)(0,0)
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322 \begin{pspicture}(0,0)(0,0)
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325 \begin{pspicture}(0,0)(0,0)
326 \psellipse[linecolor=red](10.45,0.45)(0.4,0.2)
336 Fabrication of silicon carbide
345 \emph{Silicon carbide --- Born from the stars, perfected on earth.}
351 SiC thin films by MBE \& CVD
353 \item Much progress achieved in homo/heteroepitaxial SiC thin film growth
354 \item \underline{Commercially available} semiconductor power devices based on
355 \underline{\foreignlanguage{greek}{a}-SiC}
356 \item Production of favored \underline{3C-SiC} material
357 \underline{less advanced}
358 \item Quality and size not yet sufficient
360 \begin{picture}(0,0)(-310,-20)
361 \includegraphics[width=2.0cm]{cree.eps}
366 Alternative approach:
367 Ion beam synthesis (IBS) of burried 3C-SiC layers in Si\hkl(1 0 0)
374 \begin{minipage}{3.15cm}
376 \includegraphics[width=3cm]{imp.eps}\\
382 \begin{minipage}{3.15cm}
384 \includegraphics[width=3cm]{annealing.eps}\\
386 \unit[12]{h} annealing at \degc{1200}
391 \begin{minipage}{5.5cm}
392 \includegraphics[width=5.8cm]{ibs_3c-sic.eps}\\[-0.2cm]
395 XTEM: single crystalline 3C-SiC in Si\hkl(1 0 0)
407 Systematic investigation of C implantations into Si
413 \includegraphics[width=0.75\textwidth]{imp_inv.eps}
429 \includegraphics[width=0.75\textwidth]{imp_inv.eps}
432 \begin{pspicture}(0,0)(0,0)
433 \rput(6.0,7.0){\rnode{init}{\psframebox[fillstyle=gradient,gradbegin=hred,gradend=white,gradlines=1000,gradmidpoint=1.0,linestyle=none]{
434 \begin{minipage}{11cm}
435 {\color{black}Diploma thesis}\\
436 \underline{Monte Carlo} simulation modeling the selforganization process\\
437 leading to periodic arrays of nanometric amorphous SiC precipitates
441 \begin{pspicture}(0,0)(0,0)
442 \rput(6.0,-0.5){\rnode{init}{\psframebox[fillstyle=gradient,gradbegin=hblue,gradend=white,gradmidpoint=1.0,gradlines=1000,linestyle=none]{
443 \begin{minipage}{11cm}
444 {\color{black}Doctoral studies}\\
445 Classical potential \underline{molecular dynamics} simulations \ldots\\
446 \underline{Density functional theory} calculations \ldots\\[0.2cm]
447 \ldots on defect formation and SiC precipitation in Si
451 \begin{pspicture}(0,0)(0,0)
452 \psellipse[linecolor=red,linewidth=0.05cm](5,3.0)(0.8,1.0)
454 \begin{pspicture}(0,0)(0,0)
455 \psellipse[linecolor=blue,linewidth=0.05cm](8.2,3.2)(1.5,1.6)
464 Selforganization of nanometric amorphous SiC lamellae
472 \item Regularly spaced, nanometric spherical\\
473 and lamellar amorphous inclusions\\
474 at the upper a/c interface
475 \item Carbon accumulation\\
481 \begin{minipage}{12cm}
482 \includegraphics[width=9cm]{../../nlsop/img/k393abild1_e_l.eps}\\
484 XTEM bright-field, \unit[180]{keV} C$^+ \rightarrow$ Si,
485 {\color{red}\underline{\degc{150}}},
486 Dose: \unit[4.3 $\times 10^{17}$]{cm$^{-2}$}
490 \begin{picture}(0,0)(-182,-215)
491 \begin{minipage}{6.5cm}
493 \includegraphics[width=6.5cm]{../../nlsop/img/eftem.eps}\\[-0.2cm]
495 XTEM bright-field and respective EFTEM C map
507 Model displaying the formation of ordered lamellae
513 \includegraphics[width=8.0cm]{../../nlsop/img/modell_ng_e.eps}
519 \item Supersaturation of C in c-Si\\
520 $\rightarrow$ {\bf Carbon induced} nucleation of spherical
522 \item High interfacial energy between 3C-SiC and c-Si\\
523 $\rightarrow$ {\bf Amorphous} precipitates
524 \item \unit[20-- 30]{\%} lower silicon density of a-SiC$_x$ compared to c-Si\\
525 $\rightarrow$ {\bf Lateral strain} (black arrows)
526 \item Implantation range near surface\\
527 $\rightarrow$ {\bf Relaxation} of {\bf vertical strain component}
528 \item Reduction of the carbon supersaturation in c-Si\\
529 $\rightarrow$ {\bf Carbon diffusion} into amorphous volumina
531 \item Remaining lateral strain\\
532 $\rightarrow$ {\bf Strain enhanced} lateral amorphisation
533 \item Absence of crystalline neighbours (structural information)\\
534 $\rightarrow$ {\bf Stabilization} of amorphous inclusions
535 {\bf against recrystallization}
544 Implementation of the Monte Carlo code
550 \item \underline{Amorphization / Recrystallization}\\
551 Ion collision in discretized target determined by random numbers
552 distributed according to nuclear energy loss.
553 Amorphization/recrystallization probability:
555 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}}
558 \item {\color{green} $p_b$} normal `ballistic' amorphization
559 \item {\color{blue} $p_c$} carbon induced amorphization
560 \item {\color{red} $p_s$} stress enhanced amorphization
563 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{,}
566 \delta (\vec r) = \left\{
568 1 & \textrm{if volume at position $\vec r$ is amorphous} \\
569 0 & \textrm{otherwise} \\
573 \item \underline{Carbon incorporation}\\
574 Incorporation volume determined according to implantation profile
575 \item \underline{Diffusion / Sputtering}
577 \item Transfer fraction of C atoms
578 of crystalline into neighbored amorphous volumes
579 \item Remove surface layer
587 \begin{minipage}{3.7cm}
588 \begin{pspicture}(0,0)(0,0)
589 \rput(1.7,0.2){\psframebox[fillstyle=gradient,gradbegin=hred,gradend=white,gradlines=1000,gradangle=10,gradmidpoint=1,linestyle=none]{
590 \begin{minipage}{3.7cm}
604 Evolution of the \ldots
609 \item lamellar precipitates
611 \ldots reproduced!\\[1.4cm]
615 Experiment \& simulation\\
616 in good agreement\\[1.0cm]
618 Simulation is able to model the whole depth region\\[1.2cm]
623 \begin{minipage}{0.5cm}
626 \begin{minipage}{8.0cm}
628 \includegraphics[width=9cm]{../../nlsop/img/dosis_entwicklung_ng_e_1-2.eps}\\
629 \includegraphics[width=9cm]{../../nlsop/img/dosis_entwicklung_ng_e2_2-2.eps}
638 Structural \& compositional details
641 \begin{minipage}[t]{7.5cm}
642 \includegraphics[height=6.5cm]{../../nlsop/img/ac_cconc_ver2_e.eps}\\
644 \begin{minipage}[t]{5.0cm}
645 \includegraphics[height=6.5cm]{../../nlsop/img/97_98_e.eps}
653 \item Fluctuation of C concentration in lamellae region
654 \item \unit[8--10]{at.\%} C saturation limit
655 within the respective conditions
656 \item Complementarily arranged and alternating sequence of layers\\
657 with a high and low amount of amorphous regions
658 \item C accumulation in the amorphous phase / Origin of stress
661 \begin{picture}(0,0)(-260,-50)
663 \begin{minipage}{3cm}
666 Precipitation process\\
681 Formation of epitaxial single crystalline 3C-SiC
690 \item \underline{Implantation step 1}\\[0.1cm]
691 Almost stoichiometric dose | \unit[180]{keV} | \degc{500}\\
692 $\Rightarrow$ Epitaxial {\color{blue}3C-SiC} layer \&
693 {\color{blue}precipitates}
694 \item \underline{Implantation step 2}\\[0.1cm]
695 Little remaining dose | \unit[180]{keV} | \degc{250}\\
697 Destruction/Amorphization of precipitates at layer interface
698 \item \underline{Annealing}\\[0.1cm]
699 \unit[10]{h} at \degc{1250}\\
700 $\Rightarrow$ Homogeneous 3C-SiC layer with sharp interfaces
704 \begin{minipage}{7cm}
705 \includegraphics[width=7cm]{ibs_3c-sic.eps}
707 \begin{minipage}{5cm}
708 \begin{pspicture}(0,0)(0,0)
710 \psframebox[fillstyle=solid,fillcolor=white,linecolor=blue,linestyle=solid]{
711 \begin{minipage}{5.3cm}
714 3C-SiC precipitation\\
715 not yet fully understood
719 \renewcommand\labelitemi{$\Rightarrow$}
720 Details of the SiC precipitation
722 \item significant technological progress\\
723 in SiC thin film formation
724 \item perspectives for processes relying\\
725 upon prevention of SiC precipitation
729 \rput(-6.8,5.4){\pnode{h0}}
730 \rput(-3.0,5.4){\pnode{h1}}
731 \ncline[linecolor=blue]{-}{h0}{h1}
732 \ncline[linecolor=blue]{->}{h1}{box}
742 Supposed precipitation mechanism of SiC in Si
750 \begin{minipage}{3.6cm}
752 Si \& SiC lattice structure\\[0.1cm]
753 \includegraphics[width=2.3cm]{sic_unit_cell.eps}
756 \begin{minipage}{1.7cm}
757 \underline{Silicon}\\
758 {\color{yellow}$\bullet$} Si | {\color{gray}$\bullet$} Si\\
759 $a=\unit[5.429]{\\A}$\\
760 $\rho^*_{\text{Si}}=\unit[100]{\%}$
762 \begin{minipage}{1.7cm}
763 \underline{Silicon carbide}\\
764 {\color{yellow}$\bullet$} Si | {\color{gray}$\bullet$} C\\
765 $a=\unit[4.359]{\\A}$\\
766 $\rho^*_{\text{Si}}=\unit[97]{\%}$
772 \begin{minipage}{4.1cm}
774 \includegraphics[width=3.3cm]{tem_c-si-db.eps}
778 \begin{minipage}{4.0cm}
780 \includegraphics[width=3.3cm]{tem_3c-sic.eps}
786 \begin{minipage}{4.0cm}
788 C-Si dimers (dumbbells)\\[-0.1cm]
789 on Si interstitial sites
793 \begin{minipage}{4.1cm}
795 Agglomeration of C-Si dumbbells\\[-0.1cm]
796 $\Rightarrow$ dark contrasts
800 \begin{minipage}{4.0cm}
802 Precipitation of 3C-SiC in Si\\[-0.1cm]
803 $\Rightarrow$ Moir\'e fringes\\[-0.1cm]
804 \& release of Si self-interstitials
810 \begin{minipage}{4.0cm}
812 \includegraphics[width=3.3cm]{sic_prec_seq_01.eps}
816 \begin{minipage}{4.1cm}
818 \includegraphics[width=3.3cm]{sic_prec_seq_02.eps}
822 \begin{minipage}{4.0cm}
824 \includegraphics[width=3.3cm]{sic_prec_seq_03.eps}
828 \begin{pspicture}(0,0)(0,0)
829 \psline[linewidth=2pt]{->}(8.3,2)(8.8,2)
830 \psellipse[linecolor=blue](11.1,6.0)(0.3,0.5)
831 \rput{-20}{\psellipse[linecolor=blue](3.1,8.2)(0.3,0.5)}
832 \psline[linewidth=2pt]{->}(3.9,2)(4.4,2)
833 \rput(11.8,0.3){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
834 $4a_{\text{Si}}=5a_{\text{SiC}}$
836 \rput(11.5,8){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
837 \hkl(h k l) planes match
839 \rput(8.5,6.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
850 Supposed precipitation mechanism of SiC in Si
858 \begin{minipage}{3.6cm}
860 Si \& SiC lattice structure\\[0.1cm]
861 \includegraphics[width=2.3cm]{sic_unit_cell.eps}
864 \begin{minipage}{1.7cm}
865 \underline{Silicon}\\
866 {\color{yellow}$\bullet$} Si | {\color{gray}$\bullet$} Si\\
867 $a=\unit[5.429]{\\A}$\\
868 $\rho^*_{\text{Si}}=\unit[100]{\%}$
870 \begin{minipage}{1.7cm}
871 \underline{Silicon carbide}\\
872 {\color{yellow}$\bullet$} Si | {\color{gray}$\bullet$} C\\
873 $a=\unit[4.359]{\\A}$\\
874 $\rho^*_{\text{Si}}=\unit[97]{\%}$
880 \begin{minipage}{4.1cm}
882 \includegraphics[width=3.3cm]{tem_c-si-db.eps}
886 \begin{minipage}{4.0cm}
888 \includegraphics[width=3.3cm]{tem_3c-sic.eps}
894 \begin{minipage}{4.0cm}
896 C-Si dimers (dumbbells)\\[-0.1cm]
897 on Si interstitial sites
901 \begin{minipage}{4.1cm}
903 Agglomeration of C-Si dumbbells\\[-0.1cm]
904 $\Rightarrow$ dark contrasts
908 \begin{minipage}{4.0cm}
910 Precipitation of 3C-SiC in Si\\[-0.1cm]
911 $\Rightarrow$ Moir\'e fringes\\[-0.1cm]
912 \& release of Si self-interstitials
918 \begin{minipage}{4.0cm}
920 \includegraphics[width=3.3cm]{sic_prec_seq_01.eps}
924 \begin{minipage}{4.1cm}
926 \includegraphics[width=3.3cm]{sic_prec_seq_02.eps}
930 \begin{minipage}{4.0cm}
932 \includegraphics[width=3.3cm]{sic_prec_seq_03.eps}
936 \begin{pspicture}(0,0)(0,0)
937 \psline[linewidth=2pt]{->}(8.3,2)(8.8,2)
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939 \rput{-20}{\psellipse[linecolor=blue](3.1,8.2)(0.3,0.5)}
940 \psline[linewidth=2pt]{->}(3.9,2)(4.4,2)
941 \rput(11.8,0.3){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
942 $4a_{\text{Si}}=5a_{\text{SiC}}$
944 \rput(11.5,8){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
945 \hkl(h k l) planes match
947 \rput(8.5,6.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
950 % controversial view!
951 \rput(6.5,5.0){\psframebox[fillstyle=solid,opacity=0.5,fillcolor=black]{
952 \begin{minipage}{14cm}
957 \rput(6.5,5.3){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=white,linewidth=0.1cm]{
958 \begin{minipage}{10cm}
962 {\color{gray}\bf Controversial findings}
965 \item High-temperature implantation {\tiny\color{gray}/Nejim~et~al./}
967 \item C incorporated {\color{blue}substitutionally} on regular Si lattice sites
968 \item \si{} reacting with further C in cleared volume
970 \item Annealing behavior {\tiny\color{gray}/Serre~et~al./}
972 \item Room temperature implantation $\rightarrow$ high C diffusion
973 \item Elevated temperature implantation $\rightarrow$ no C redistribution
975 $\Rightarrow$ mobile {\color{red}\ci} opposed to
976 stable {\color{blue}\cs{}} configurations
977 \item Strained silicon \& Si/SiC heterostructures
978 {\tiny\color{gray}/Strane~et~al./Guedj~et~al./}
980 \item {\color{blue}Coherent} SiC precipitates (tensile strain)
981 \item Incoherent SiC (strain relaxation)
986 {\Huge${\lightning}$} \hspace{0.3cm}
987 {\color{blue}\cs{}} --- vs --- {\color{red}\ci} \hspace{0.3cm}
988 {\Huge${\lightning}$}
1001 Utilized computational methods
1008 {\bf Molecular dynamics (MD)}\\[0.1cm]
1010 \begin{tabular}{| p{4.5cm} | p{7.5cm} |}
1012 System of $N$ particles &
1013 $N=5832\pm 1$ (Defects), $N=238328+6000$ (Precipitation)\\
1014 Phase space propagation &
1015 Velocity Verlet | timestep: \unit[1]{fs} \\
1016 Analytical interaction potential &
1017 Tersoff-like {\color{red}short-range}, {\color{blue}bond order} potential
1020 E = \frac{1}{2} \sum_{i \neq j} \pot_{ij}, \quad
1021 \pot_{ij} = {\color{red}f_C(r_{ij})}
1022 \left[ f_R(r_{ij}) + {\color{blue}b_{ij}} f_A(r_{ij}) \right]
1024 Observables: time/ensemble averages &
1025 NpT (isothermal-isobaric) | Berendsen thermostat/barostat\\
1033 {\bf Density functional theory (DFT)}
1037 \begin{minipage}[t]{6cm}
1039 \item Hohenberg-Kohn theorem:\\
1040 $\Psi_0(r_1,r_2,\ldots,r_N)=\Psi[n_0(r)]$, $E_0=E[n_0]$
1041 \item Kohn-Sham approach:\\
1042 Single-particle effective theory
1046 \item Code: \textsc{vasp}
1047 \item Plane wave basis set
1049 %\Phi_i=\sum_{|G+k|<G_{\text{cut}}} c_{i,k+G} \exp{\left(i(k+G)r\right)}
1052 %E_{\text{cut}}=\frac{\hbar^2}{2m}G^2_{\text{cut}}=\unit[300]{eV}
1054 \item Ultrasoft pseudopotential
1055 \item Exchange \& correlation: GGA
1056 \item Brillouin zone sampling: $\Gamma$-point
1057 \item Supercell: $N=216\pm2$
1060 \begin{minipage}{6cm}
1061 \begin{pspicture}(0,0)(0,0)
1062 \pscircle[fillcolor=yellow,fillstyle=solid,linestyle=none](3.5,-2.0){2.5}
1063 \rput(2.7,-0.7){\psframebox[fillstyle=solid,opacity=0.8,fillcolor=white]{
1065 \left[ -\frac{\hbar^2}{2m}\nabla^2 + V_{\text{eff}}(r) - \epsilon_i \right] \Phi_i(r) = 0
1068 \rput(5.2,-2.0){\psframebox[fillstyle=solid,opacity=0.8,fillcolor=white]{
1070 n(r)=\sum_i^N|\Phi_i(r)|^2
1073 \rput(3.0,-4.5){\psframebox[fillstyle=solid,opacity=0.8,fillcolor=white]{
1075 V_{\text{eff}}(r)=V_{\text{ext}}(r)+\int\frac{e^2 n(r')}{|r-r'|}d^3r'
1076 +V_{\text{XC}}[n(r)]
1079 \psarcn[linewidth=0.07cm,linestyle=dashed]{->}(3.5,-2.0){2.5}{130}{15}
1080 \psarcn[linewidth=0.07cm,linestyle=dashed]{->}(3.5,-2.0){2.5}{230}{165}
1081 \psarcn[linewidth=0.07cm,linestyle=dashed]{->}(3.5,-2.0){2.5}{345}{310}
1092 Point defects \& defect migration
1099 \begin{minipage}[b]{7.5cm}
1100 {\bf Defect structure}\\
1101 \begin{pspicture}(0,0)(7,4.4)
1102 \rput(3.5,3.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
1105 \item Creation of c-Si simulation volume
1106 \item Periodic boundary conditions
1107 \item $T=0\text{ K}$, $p=0\text{ bar}$
1110 \rput(3.5,1.3){\rnode{insert}{\psframebox{
1113 Insertion of interstitial C/Si atoms
1116 \rput(3.5,0.2){\rnode{cool}{\psframebox[fillstyle=solid,fillcolor=lbb]{
1119 Relaxation / structural energy minimization
1122 \ncline[]{->}{init}{insert}
1123 \ncline[]{->}{insert}{cool}
1126 \begin{minipage}[b]{4.5cm}
1128 \includegraphics[width=3.8cm]{unit_cell_e.eps}\\
1130 \begin{minipage}{2.21cm}
1132 {\color{red}$\bullet$} Tetrahedral\\[-0.1cm]
1133 {\color{green}$\bullet$} Hexagonal\\[-0.1cm]
1134 {\color{yellow}$\bullet$} \hkl<1 0 0> DB
1137 \begin{minipage}{2.21cm}
1139 {\color{magenta}$\bullet$} \hkl<1 1 0> DB\\[-0.1cm]
1140 {\color{cyan}$\bullet$} Bond-centered\\[-0.1cm]
1141 {\color{black}$\bullet$} Vac. / Sub.
1148 \begin{minipage}[b]{6cm}
1149 {\bf Defect formation energy}\\
1151 $E_{\text{f}}=E-\sum_i N_i\mu_i$}\\[0.1cm]
1152 Particle reservoir: Si \& SiC\\[0.2cm]
1153 {\bf Binding energy}\\
1157 E_{\text{f}}^{\text{comb}}-
1158 E_{\text{f}}^{1^{\text{st}}}-
1159 E_{\text{f}}^{2^{\text{nd}}}
1163 $E_{\text{b}}<0$: energetically favorable configuration\\
1164 $E_{\text{b}}\rightarrow 0$: non-interacting, isolated defects\\
1166 \begin{minipage}[b]{6cm}
1167 {\bf Migration barrier}
1170 \item Displace diffusing atom
1171 \item Constrain relaxation of (diffusing) atoms
1172 \item Record configurational energy
1174 \begin{picture}(0,0)(-60,-33)
1175 \includegraphics[width=4.5cm]{crt_mod.eps}
1187 Si self-interstitial point defects in silicon\\[0.1cm]
1191 \begin{tabular}{l c c c c c}
1193 $E_{\text{f}}$ [eV] & \hkl<1 1 0> DB & H & T & \hkl<1 0 0> DB & V \\
1195 \textsc{vasp} & \underline{3.39} & 3.42 & 3.77 & 4.41 & 3.63 \\
1196 Erhart/Albe & 4.39 & 4.48$^*$ & \underline{3.40} & 5.42 & 3.13 \\
1198 \end{tabular}\\[0.4cm]
1201 \begin{minipage}{3cm}
1203 \underline{Vacancy}\\
1204 \includegraphics[width=2.8cm]{si_pd_albe/vac.eps}
1207 \begin{minipage}{3cm}
1209 \underline{\hkl<1 1 0> DB}\\
1210 \includegraphics[width=2.8cm]{si_pd_albe/110_bonds.eps}
1213 \begin{minipage}{3cm}
1215 \underline{\hkl<1 0 0> DB}\\
1216 \includegraphics[width=2.8cm]{si_pd_albe/100_bonds.eps}
1219 \begin{minipage}{3cm}
1221 \underline{Tetrahedral}\\
1222 \includegraphics[width=2.8cm]{si_pd_albe/tet_bonds.eps}
1226 \underline{Hexagonal} \hspace{2pt}
1227 \href{../video/si_self_int_hexa.avi}{$\rhd$}\\[0.1cm]
1229 \begin{minipage}{2.7cm}
1230 $E_{\text{f}}^*=4.48\text{ eV}$\\
1231 \includegraphics[width=2.7cm]{si_pd_albe/hex_a_bonds.eps}
1233 \begin{minipage}{0.4cm}
1238 \begin{minipage}{2.7cm}
1239 $E_{\text{f}}=3.96\text{ eV}$\\
1240 \includegraphics[width=2.8cm]{si_pd_albe/hex_bonds.eps}
1243 \begin{minipage}{5.5cm}
1245 {\tiny nearly T $\rightarrow$ T}\\
1247 \includegraphics[width=6.0cm]{nhex_tet.ps}
1258 C interstitial point defects in silicon\\
1261 \begin{tabular}{l c c c c c c r}
1263 $E_{\text{f}}$ [eV] & T & H & \hkl<1 0 0> DB & \hkl<1 1 0> DB & S & B &
1264 {\color{black} \cs{} \& \si}\\
1266 \textsc{vasp} & unstable & unstable & \underline{3.72} & 4.16 & 1.95 & 4.66 & {\color{green}4.17}\\
1267 Erhart/Albe & 6.09 & 9.05$^*$ & \underline{3.88} & 5.18 & {\color{red}0.75} & 5.59$^*$ & {\color{green}4.43} \\
1269 \end{tabular}\\[0.1cm]
1272 \begin{minipage}{2.8cm}
1273 \underline{Hexagonal} \hspace{2pt}
1274 \href{../video/c_in_si_int_hexa.avi}{$\rhd$}\\
1275 $E_{\text{f}}^*=9.05\text{ eV}$\\
1276 \includegraphics[width=2.8cm]{c_pd_albe/hex_bonds.eps}
1278 \begin{minipage}{0.4cm}
1283 \begin{minipage}{2.8cm}
1284 \underline{\hkl<1 0 0>}\\
1285 $E_{\text{f}}=3.88\text{ eV}$\\
1286 \includegraphics[width=2.8cm]{c_pd_albe/100_bonds.eps}
1289 \begin{minipage}{1.4cm}
1292 \begin{minipage}{3.0cm}
1294 \underline{Tetrahedral}\\
1295 \includegraphics[width=3.0cm]{c_pd_albe/tet_bonds.eps}
1300 \begin{minipage}{2.8cm}
1301 \underline{Bond-centered}\\
1302 $E_{\text{f}}^*=5.59\text{ eV}$\\
1303 \includegraphics[width=2.8cm]{c_pd_albe/bc_bonds.eps}
1305 \begin{minipage}{0.4cm}
1310 \begin{minipage}{2.8cm}
1311 \underline{\hkl<1 1 0> dumbbell}\\
1312 $E_{\text{f}}=5.18\text{ eV}$\\
1313 \includegraphics[width=2.8cm]{c_pd_albe/110_bonds.eps}
1316 \begin{minipage}{1.4cm}
1319 \begin{minipage}{3.0cm}
1321 \underline{Substitutional}\\
1322 \includegraphics[width=3.0cm]{c_pd_albe/sub_bonds.eps}
1332 C-Si dimer \& bond-centered interstitial configuration
1339 \begin{minipage}[t]{4.1cm}
1340 {\bf\boldmath C \hkl<1 0 0> DB interstitial}\\[0.1cm]
1341 \begin{minipage}{2.0cm}
1343 \underline{Erhart/Albe}
1344 \includegraphics[width=2.0cm]{c_pd_albe/100_cmp.eps}
1347 \begin{minipage}{2.0cm}
1349 \underline{\textsc{vasp}}
1350 \includegraphics[width=2.0cm]{c_pd_vasp/100_cmp.eps}
1352 \end{minipage}\\[0.2cm]
1353 Si-C-Si bond angle $\rightarrow$ \unit[180]{$^{\circ}$}\\
1354 $\Rightarrow$ $sp$ hybridization\\[0.1cm]
1355 Si-Si-Si bond angle $\rightarrow$ \unit[120]{$^{\circ}$}\\
1356 $\Rightarrow$ $sp^2$ hybridization
1358 \includegraphics[width=3.4cm]{c_pd_vasp/eden.eps}\\[-0.1cm]
1359 {\tiny Charge density isosurface}
1362 \begin{minipage}{0.2cm}
1365 \begin{minipage}[t]{8.1cm}
1367 {\bf Bond-centered interstitial}\\[0.1cm]
1368 \begin{minipage}{4.4cm}
1371 \item Linear Si-C-Si bond
1372 \item Si: one C \& 3 Si neighbours
1373 \item Spin polarized calculations
1374 \item No saddle point!\\
1378 \begin{minipage}{2.7cm}
1379 %\includegraphics[width=2.8cm]{c_pd_vasp/bc_2333.eps}\\
1381 \includegraphics[width=2.8cm]{c_pd_albe/bc_bonds.eps}\\
1386 \begin{minipage}[t]{6.5cm}
1387 \begin{minipage}[t]{1.2cm}
1389 {\tiny sp$^3$}\\[0.8cm]
1390 \underline{${\color{black}\uparrow}$}
1391 \underline{${\color{black}\uparrow}$}
1392 \underline{${\color{black}\uparrow}$}
1393 \underline{${\color{red}\uparrow}$}\\
1396 \begin{minipage}[t]{1.4cm}
1398 {\color{red}M}{\color{blue}O}\\[0.8cm]
1399 \underline{${\color{blue}\uparrow}{\color{white}\downarrow}$}\\
1400 $\sigma_{\text{ab}}$\\[0.5cm]
1401 \underline{${\color{red}\uparrow}{\color{blue}\downarrow}$}\\
1405 \begin{minipage}[t]{1.0cm}
1409 \underline{${\color{white}\uparrow\uparrow}$}
1410 \underline{${\color{white}\uparrow\uparrow}$}\\
1412 \underline{${\color{blue}\uparrow}{\color{blue}\downarrow}$}
1413 \underline{${\color{blue}\uparrow}{\color{blue}\downarrow}$}\\
1417 \begin{minipage}[t]{1.4cm}
1419 {\color{blue}M}{\color{green}O}\\[0.8cm]
1420 \underline{${\color{blue}\uparrow}{\color{white}\downarrow}$}\\
1421 $\sigma_{\text{ab}}$\\[0.5cm]
1422 \underline{${\color{green}\uparrow}{\color{blue}\downarrow}$}\\
1426 \begin{minipage}[t]{1.2cm}
1429 {\tiny sp$^3$}\\[0.8cm]
1430 \underline{${\color{green}\uparrow}$}
1431 \underline{${\color{black}\uparrow}$}
1432 \underline{${\color{black}\uparrow}$}
1433 \underline{${\color{black}\uparrow}$}\\
1441 \begin{minipage}{3.0cm}
1443 \underline{Charge density}\\
1444 {\color{gray}$\bullet$} Spin up\\
1445 {\color{green}$\bullet$} Spin down\\
1446 {\color{blue}$\bullet$} Resulting spin up\\
1447 {\color{yellow}$\bullet$} Si atoms\\
1448 {\color{red}$\bullet$} C atom
1450 \begin{minipage}{3.6cm}
1451 \includegraphics[width=3.8cm]{c_100_mig_vasp/im_spin_diff.eps}
1458 \begin{pspicture}(0,0)(0,0)
1459 \psline[linecolor=gray,linewidth=0.05cm](-7.8,-8.7)(-7.8,0)
1468 C interstitial migration --- ab initio
1475 \begin{minipage}{6.8cm}
1476 \framebox{\hkl[0 0 -1] $\rightarrow$ \hkl[0 0 1]}\\
1477 \begin{minipage}{2.0cm}
1478 \includegraphics[width=2.0cm]{c_pd_vasp/100_2333.eps}
1480 \begin{minipage}{0.2cm}
1483 \begin{minipage}{2.0cm}
1484 \includegraphics[width=2.0cm]{c_pd_vasp/bc_2333.eps}
1486 \begin{minipage}{0.2cm}
1489 \begin{minipage}{2.0cm}
1490 \includegraphics[width=2.0cm]{c_pd_vasp/100_next_2333.eps}
1491 \end{minipage}\\[0.1cm]
1493 $\Rightarrow$ BC configuration constitutes local minimum\\
1494 $\Rightarrow$ Migration barrier to reach BC | $\Delta E=\unit[1.2]{eV}$
1496 \begin{minipage}{5.4cm}
1497 \includegraphics[width=6.0cm]{im_00-1_nosym_sp_fullct_thesis_vasp_s.ps}
1498 \end{minipage}\\[0.2cm]
1501 \begin{minipage}{6.8cm}
1502 \framebox{\hkl[0 0 -1] $\rightarrow$ \hkl[0 -1 0]}\\
1503 \begin{minipage}{2.0cm}
1504 \includegraphics[width=2.0cm]{c_pd_vasp/100_2333.eps}
1506 \begin{minipage}{0.2cm}
1509 \begin{minipage}{2.0cm}
1510 \includegraphics[width=2.0cm]{c_pd_vasp/00-1-0-10_2333.eps}
1512 \begin{minipage}{0.2cm}
1515 \begin{minipage}{2.0cm}
1516 \includegraphics[width=2.0cm]{c_pd_vasp/0-10_2333.eps}
1517 \end{minipage}\\[0.1cm]
1518 $\Delta E=\unit[0.9]{eV}$ | Experimental values: \unit[0.70--0.87]{eV}\\
1519 $\Rightarrow$ {\color{red}Migration mechanism identified!}\\
1520 Note: Change in orientation
1522 \begin{minipage}{5.4cm}
1523 \includegraphics[width=6.0cm]{00-1_0-10_vasp_s.ps}
1524 \end{minipage}\\[0.1cm]
1527 Reorientation pathway composed of two consecutive processes of the above type
1536 C interstitial migration --- analytical potential
1543 \begin{minipage}[t]{6.0cm}
1544 {\bf\boldmath BC to \hkl[0 0 -1] transition}\\[0.2cm]
1545 \includegraphics[width=6.0cm]{bc_00-1_albe_s.ps}\\
1547 \item Lowermost migration barrier
1548 \item $\Delta E \approx \unit[2.2]{eV}$
1549 \item 2.4 times higher than ab initio result
1550 \item Different pathway
1553 \begin{minipage}[t]{0.2cm}
1556 \begin{minipage}[t]{6.0cm}
1557 {\bf\boldmath Transition involving a \hkl<1 1 0> configuration}
1560 \item Bond-centered configuration unstable\\
1561 $\rightarrow$ \ci{} \hkl<1 1 0> dumbbell
1562 \item Minima of the \hkl[0 0 -1] to \hkl[0 -1 0] transition\\
1563 $\rightarrow$ \ci{} \hkl<1 1 0> DB
1566 \includegraphics[width=6.0cm]{00-1_110_0-10_mig_albe.ps}
1568 \item $\Delta E \approx \unit[2.2]{eV} \text{ \& } \unit[0.9]{eV}$
1569 \item 2.4 -- 3.4 times higher than ab initio result
1570 \item After all: Change of the DB orientation
1576 {\color{red}\bf Drastically overestimated diffusion barrier}
1579 \begin{pspicture}(0,0)(0,0)
1580 \psline[linewidth=0.05cm,linecolor=gray](6.1,1.0)(6.1,9.3)
1596 \begin{minipage}{9cm}
1598 Summary of combinations}\\[0.1cm]
1600 \begin{tabular}{l c c c c c c}
1602 $E_{\text{b}}$ [eV] & 1 & 2 & 3 & 4 & 5 & R\\
1604 \hkl[0 0 -1] & {\color{red}-0.08} & -1.15 & {\color{red}-0.08} & 0.04 & -1.66 & -0.19\\
1605 \hkl[0 0 1] & 0.34 & 0.004 & -2.05 & 0.26 & -1.53 & -0.19\\
1606 \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}\\
1607 \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}\\
1608 \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}\\
1609 \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}\\
1611 C$_{\text{sub}}$ & 0.26 & -0.51 & -0.93 & -0.15 & 0.49 & -0.05\\
1612 Vacancy & -5.39 ($\rightarrow$ C$_{\text{sub}}$) & -0.59 & -3.14 & -0.54 & -0.50 & -0.31\\
1619 $E_{\text{b}}$ explainable by stress compensation / increase
1623 \begin{minipage}{3cm}
1624 \includegraphics[width=3.5cm]{comb_pos.eps}
1629 {\bf\boldmath Combinations of \hkl<1 0 0>-type interstitials}\\[0.2cm]
1630 \begin{minipage}[t]{3.2cm}
1631 \underline{\hkl[1 0 0] at position 1}\\[0.1cm]
1632 \includegraphics[width=2.8cm]{00-1dc/2-25.eps}
1634 \begin{minipage}[t]{3.0cm}
1635 \underline{\hkl[0 -1 0] at position 1}\\[0.1cm]
1636 \includegraphics[width=2.8cm]{00-1dc/2-39.eps}
1638 \begin{minipage}[t]{6.1cm}
1641 \item \ci{} agglomeration energetically favorable
1642 \item Most favorable: C clustering\\
1643 {\color{red}However \ldots}\\
1644 \ldots high migration barrier ($>4\,\text{eV}$)\\
1646 $4\times{\color{cyan}[-2.25]}$ versus
1647 $2\times{\color{orange}[-2.39]}$
1650 {\color{blue}\ci{} agglomeration / no C clustering}
1667 \begin{minipage}{9cm}
1669 Summary of combinations}\\[0.1cm]
1671 \begin{tabular}{l c c c c c c}
1673 $E_{\text{b}}$ [eV] & 1 & 2 & 3 & 4 & 5 & R\\
1675 \hkl[0 0 -1] & {\color{red}-0.08} & -1.15 & {\color{red}-0.08} & 0.04 & -1.66 & -0.19\\
1676 \hkl[0 0 1] & 0.34 & 0.004 & -2.05 & 0.26 & -1.53 & -0.19\\
1677 \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}\\
1678 \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}\\
1679 \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}\\
1680 \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}\\
1682 C$_{\text{sub}}$ & 0.26 & -0.51 & -0.93 & -0.15 & 0.49 & -0.05\\
1683 Vacancy & -5.39 ($\rightarrow$ C$_{\text{sub}}$) & -0.59 & -3.14 & -0.54 & -0.50 & -0.31\\
1690 $E_{\text{b}}$ explainable by stress compensation / increase
1694 \begin{minipage}{3cm}
1695 \includegraphics[width=3.5cm]{comb_pos.eps}
1700 {\bf\boldmath Combinations of \hkl<1 0 0>-type interstitials}\\[0.2cm]
1701 \begin{minipage}[t]{3.2cm}
1702 \underline{\hkl[1 0 0] at position 1}\\[0.1cm]
1703 \includegraphics[width=2.8cm]{00-1dc/2-25.eps}
1705 \begin{minipage}[t]{3.0cm}
1706 \underline{\hkl[0 -1 0] at position 1}\\[0.1cm]
1707 \includegraphics[width=2.8cm]{00-1dc/2-39.eps}
1709 \begin{minipage}[t]{6.1cm}
1712 \item \ci{} agglomeration energetically favorable
1713 \item Most favorable: C clustering\\
1714 {\color{red}However \ldots}\\
1715 \ldots high migration barrier ($>4\,\text{eV}$)\\
1717 $4\times{\color{cyan}[-2.25]}$ versus
1718 $2\times{\color{orange}[-2.39]}$
1721 {\color{blue}\ci{} agglomeration / no C clustering}
1726 \begin{pspicture}(0,0)(0,0)
1727 \rput(6.5,5.0){\psframebox[fillstyle=solid,opacity=0.5,fillcolor=black]{
1728 \begin{minipage}{14cm}
1733 \rput(6.5,5.3){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=white,linewidth=0.1cm]{
1734 \begin{minipage}{8cm}
1738 Interaction along \hkl[1 1 0]
1739 \includegraphics[width=7cm]{db_along_110_cc.ps}
1751 Defect combinations of C-Si dimers and vacancies
1757 \begin{minipage}[b]{2.6cm}
1759 \underline{V at 2: $E_{\text{b}}=-0.59\text{ eV}$}\\[0.1cm]
1760 \includegraphics[width=2.5cm]{00-1dc/0-59.eps}
1763 \begin{minipage}[b]{7cm}
1766 \begin{minipage}[b]{2.6cm}
1768 \underline{V at 3, $E_{\text{b}}=-3.14\text{ eV}$}\\[0.1cm]
1769 \includegraphics[width=2.5cm]{00-1dc/3-14.eps}
1771 \end{minipage}\\[0.2cm]
1773 \begin{minipage}{6.5cm}
1774 \includegraphics[width=6.0cm]{059-539.ps}
1776 \begin{minipage}{5.7cm}
1777 \includegraphics[width=6.0cm]{314-539.ps}
1780 \begin{pspicture}(0,0)(0,0)
1781 \psline[linewidth=0.05cm,linecolor=gray](6.3,9.0)(6.3,0)
1783 \rput(6.3,7.0){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=white,linewidth=0.05cm,linecolor=gray]{
1784 \begin{minipage}{6.5cm}
1786 IBS: Impinging C creates V \& far away \si\\[0.3cm]
1787 Low migration barrier towards C$_{\text{sub}}$\\
1789 High barrier for reverse process\\[0.3cm]
1791 High probability of stable C$_{\text{sub}}$ configuration
1804 Combinations of substitutional C and Si self-interstitials
1811 \begin{minipage}{6.2cm}
1813 {\bf\boldmath C$_{\text{sub}}$ - \si{} \hkl<1 1 0> interaction}
1815 \item Most favorable: \cs{} along \hkl<1 1 0> chain of \si{}
1816 \item Less favorable than ground-state \ci{} \hkl<1 0 0> DB
1817 \item Interaction drops quickly to zero\\
1818 $\rightarrow$ low capture radius
1822 \begin{minipage}{0.2cm}
1825 \begin{minipage}{6.0cm}
1827 {\bf Transition from the ground state}
1829 \item Low transition barrier
1830 \item Barrier smaller than \ci{} migration barrier
1831 \item Low \si{} migration barrier (\unit[0.67]{eV})\\
1832 $\rightarrow$ Separation of \cs{} \& \si{} most probable
1835 \end{minipage}\\[0.3cm]
1837 \begin{minipage}{6.0cm}
1838 \includegraphics[width=6.0cm]{c_sub_si110.ps}
1840 \begin{minipage}{0.4cm}
1843 \begin{minipage}{6.0cm}
1845 \includegraphics[width=6.0cm]{162-097.ps}
1849 \begin{pspicture}(0,0)(0,0)
1850 \psline[linewidth=0.05cm,linecolor=gray](6.5,0)(6.5,7.5)
1851 \rput(6.5,-0.7){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=white,linewidth=0.05cm,linecolor=blue]{
1852 \begin{minipage}{8cm}
1856 \cs{} \& \si{} instead of thermodynamic ground state\\[0.1cm]
1857 IBS --- process far from equilibrium\\
1870 Combinations of substitutional C and Si self-interstitials
1877 \begin{minipage}{6.2cm}
1879 {\bf\boldmath C$_{\text{sub}}$ - \si{} \hkl<1 1 0> interaction}
1881 \item Most favorable: \cs{} along \hkl<1 1 0> chain of \si{}
1882 \item Less favorable than ground-state \ci{} \hkl<1 0 0> DB
1883 \item Interaction drops quickly to zero\\
1884 $\rightarrow$ low capture radius
1888 \begin{minipage}{0.2cm}
1891 \begin{minipage}{6.0cm}
1893 {\bf Transition from the ground state}
1895 \item Low transition barrier
1896 \item Barrier smaller than \ci{} migration barrier
1897 \item Low \si{} migration barrier (\unit[0.67]{eV})\\
1898 $\rightarrow$ Separation of \cs{} \& \si{} most probable
1901 \end{minipage}\\[0.3cm]
1903 \begin{minipage}{6.0cm}
1904 \includegraphics[width=6.0cm]{c_sub_si110.ps}
1906 \begin{minipage}{0.4cm}
1909 \begin{minipage}{6.0cm}
1911 \includegraphics[width=6.0cm]{162-097.ps}
1915 \begin{pspicture}(0,0)(0,0)
1916 \psline[linewidth=0.05cm,linecolor=gray](6.5,0)(6.5,7.5)
1917 \rput(6.5,-0.7){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=white,linewidth=0.05cm,linecolor=blue]{
1918 \begin{minipage}{8cm}
1922 \cs{} \& \si{} instead of thermodynamic ground state\\[0.1cm]
1923 IBS --- process far from equilibrium\\
1931 \begin{pspicture}(0,0)(0,0)
1932 \rput(6.5,5.0){\psframebox[fillstyle=solid,opacity=0.5,fillcolor=black]{
1933 \begin{minipage}{14cm}
1938 \rput(6.5,4.3){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=white,linewidth=0.1cm]{
1939 \begin{minipage}{11cm}
1943 Ab initio MD at \degc{900}\\[0.4cm]
1944 \begin{minipage}{5.4cm}
1946 \includegraphics[width=4.3cm]{md01_bonds.eps}\\
1949 \begin{minipage}{5.4cm}
1951 \includegraphics[width=4.3cm]{md02_bonds.eps}\\
1953 \end{minipage}\\[0.5cm]
1955 Contribution of entropy to structural formation\\[0.1cm]
1968 Silicon carbide precipitation simulations
1978 \begin{pspicture}(0,0)(12,6.5)
1980 \rput(3.5,5.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
1983 \item Create c-Si volume
1984 \item Periodc boundary conditions
1985 \item Set requested $T$ and $p=0\text{ bar}$
1986 \item Equilibration of $E_{\text{kin}}$ and $E_{\text{pot}}$
1989 \rput(3.5,2.7){\rnode{insert}{\psframebox[fillstyle=solid,fillcolor=lachs]{
1991 Insertion of C atoms at constant T
1993 \item total simulation volume {\pnode{in1}}
1994 \item volume of minimal SiC precipitate size {\pnode{in2}}
1995 \item volume consisting of Si atoms to form a minimal {\pnode{in3}}\\
1999 \rput(3.5,1){\rnode{cool}{\psframebox[fillstyle=solid,fillcolor=lbb]{
2001 Run for 100 ps followed by cooling down to $20\, ^{\circ}\textrm{C}$
2003 \ncline[]{->}{init}{insert}
2004 \ncline[]{->}{insert}{cool}
2005 \psframe[fillstyle=solid,fillcolor=white](7.3,0.7)(12.8,6.3)
2006 \rput(7.6,6){\footnotesize $V_1$}
2007 \psframe[fillstyle=solid,fillcolor=lightgray](8.7,2)(11.6,5)
2008 \rput(8.9,4.85){\tiny $V_2$}
2009 \psframe[fillstyle=solid,fillcolor=gray](8.95,2.25)(11.35,4.75)
2010 \rput(9.25,4.45){\footnotesize $V_3$}
2011 \rput(7.9,3.2){\pnode{ins1}}
2012 \rput(8.92,2.8){\pnode{ins2}}
2013 \rput(10.8,2.4){\pnode{ins3}}
2014 \ncline[]{->}{in1}{ins1}
2015 \ncline[]{->}{in2}{ins2}
2016 \ncline[]{->}{in3}{ins3}
2026 \begin{minipage}{5.7cm}
2028 \item Amount of C atoms: 6000\\
2029 ($r_{\text{prec}}\approx 3.1\text{ nm}$, IBS: \unit[2--4]{nm})
2030 \item Simulation volume: $31^3$ Si unit cells\\
2034 \begin{minipage}{0.3cm}
2038 \begin{minipage}{6.0cm}
2039 Restricted to classical potential caclulations\\
2040 $\rightarrow$ Low C diffusion / overestimated barrier\\
2041 $\rightarrow$ Consider $V_2$ and $V_3$
2043 % \item $V_2$ and $V_3$ considered due to expected low C diffusion
2054 Silicon carbide precipitation simulations at \degc{450} as in IBS
2059 \begin{minipage}{6.3cm}
2060 \hspace*{-0.4cm}\includegraphics[width=6.5cm]{sic_prec_450_si-c.ps}\\
2061 \hspace*{-0.4cm}\includegraphics[width=6.5cm]{sic_prec_450_si-si_c-c.ps}
2064 \begin{minipage}{6.1cm}
2066 \underline{Low C concentration --- {\color{red}$V_1$}}\\[0.1cm]
2067 \hkl<1 0 0> C-Si dumbbell dominated structure
2069 \item Si-C bumbs around \unit[0.19]{nm}
2070 \item C-C peak at \unit[0.31]{nm} (expected in 3C-SiC):\\
2071 concatenated differently oriented \ci{} DBs
2072 \item Si-Si NN distance stretched to \unit[0.3]{nm}
2074 \begin{pspicture}(0,0)(6.0,1.0)
2075 \rput(3.2,0.5){\psframebox[linewidth=0.03cm,linecolor=blue]{
2076 \begin{minipage}{6cm}
2078 Formation of \ci{} dumbbells\\
2079 C atoms in proper 3C-SiC distance first
2082 \end{pspicture}\\[0.1cm]
2083 \underline{High C concentration --- {\color{green}$V_2$}/{\color{blue}$V_3$}}
2085 \item High amount of strongly bound C-C bonds
2086 \item Increased defect \& damage density\\
2087 $\rightarrow$ Arrangements hard to categorize and trace
2088 \item Only short range order observable
2090 \begin{pspicture}(0,0)(6.0,0.8)
2091 \rput(3.2,0.5){\psframebox[linewidth=0.03cm,linecolor=blue]{
2092 \begin{minipage}{6cm}
2094 Amorphous SiC-like phase
2097 \end{pspicture}\\[0.3cm]
2098 \begin{pspicture}(0,0)(6.0,2.0)
2099 \rput(3.2,1.0){\psframebox[linewidth=0.05cm,linecolor=white]{
2100 \begin{minipage}{6cm}
2114 Silicon carbide precipitation simulations at \degc{450} as in IBS
2119 \begin{minipage}{6.3cm}
2120 \hspace*{-0.4cm}\includegraphics[width=6.5cm]{sic_prec_450_si-c.ps}\\
2121 \hspace*{-0.4cm}\includegraphics[width=6.5cm]{sic_prec_450_si-si_c-c.ps}
2124 \begin{minipage}{6.1cm}
2126 \underline{Low C concentration --- {\color{red}$V_1$}}\\[0.1cm]
2127 \hkl<1 0 0> C-Si dumbbell dominated structure
2129 \item Si-C bumbs around \unit[0.19]{nm}
2130 \item C-C peak at \unit[0.31]{nm} (expected in 3C-SiC):\\
2131 concatenated differently oriented \ci{} DBs
2132 \item Si-Si NN distance stretched to \unit[0.3]{nm}
2134 \begin{pspicture}(0,0)(6.0,1.0)
2135 \rput(3.2,0.5){\psframebox[linewidth=0.03cm,linecolor=blue]{
2136 \begin{minipage}{6cm}
2138 Formation of \ci{} dumbbells\\
2139 C atoms in proper 3C-SiC distance first
2142 \end{pspicture}\\[0.1cm]
2143 \underline{High C concentration --- {\color{green}$V_2$}/{\color{blue}$V_3$}}
2145 \item High amount of strongly bound C-C bonds
2146 \item Increased defect \& damage density\\
2147 $\rightarrow$ Arrangements hard to categorize and trace
2148 \item Only short range order observable
2150 \begin{pspicture}(0,0)(6.0,0.8)
2151 \rput(3.2,0.5){\psframebox[linewidth=0.03cm,linecolor=blue]{
2152 \begin{minipage}{6cm}
2154 Amorphous SiC-like phase
2157 \end{pspicture}\\[0.3cm]
2158 \begin{pspicture}(0,0)(6.0,2.0)
2159 \rput(3.2,1.0){\psframebox[linewidth=0.05cm,linecolor=black]{
2160 \begin{minipage}{6cm}
2163 {\bf\color{red}3C-SiC formation fails to appear}\\[0.3cm]
2164 \begin{minipage}{0.8cm}
2165 {\bf\boldmath $V_1$:}
2167 \begin{minipage}{5.1cm}
2168 Formation of \ci{} indeed occurs\\
2169 Agllomeration not observed
2170 \end{minipage}\\[0.3cm]
2171 \begin{minipage}{0.8cm}
2172 {\bf\boldmath $V_{2,3}$:}
2174 \begin{minipage}{5.1cm}
2175 Amorphous SiC-like structure\\
2176 (not expected at \degc{450})\\[0.05cm]
2177 No rearrangement/transition into 3C-SiC
2178 \end{minipage}\\[0.1cm]
2190 Limitations of MD and short range potentials
2197 {\bf Time scale problem of MD}\\[0.2cm]
2198 Precise integration \& thermodynamic sampling\\
2199 $\Rightarrow$ $\Delta t \ll \left( \max{\omega} \right)^{-1}$,
2200 $\omega$: vibrational mode\\
2201 $\Rightarrow$ {\color{red}\underline{Slow}} phase space propagation\\[0.2cm]
2202 Several local minima separated by large energy barriers\\
2203 $\Rightarrow$ Transition event corresponds to a multiple
2204 of vibrational periods\\
2205 $\Rightarrow$ Phase transition consists of {\color{red}\underline{many}}
2206 infrequent transition events\\[0.2cm]
2207 {\color{blue}Accelerated methods:}
2208 \underline{Temperature accelerated} MD (TAD), self-guided MD \ldots
2212 {\bf Limitations related to the short range potential}\\[0.2cm]
2213 Cut-off function limits interaction to next neighbours\\
2214 $\Rightarrow$ Overestimated unphysical high forces of next neighbours
2219 {\bf Approach to the (twofold) problem}\\[0.2cm]
2220 Increased temperature simulations without TAD corrections\\
2221 Accelerated methods or higher time scales exclusively not sufficient!
2223 \begin{pspicture}(0,0)(0,0)
2224 \rput(4.0,2.8){\psframebox[linewidth=0.07cm,linecolor=red]{
2225 \begin{minipage}{7.5cm}
2228 Potential enhanced slow phase space propagation
2231 \rput(11.3,7.5){\psframebox[linewidth=0.03cm,linecolor=blue]{
2232 \begin{minipage}{2.7cm}
2236 thermodynamic sampling
2239 \psline[linewidth=0.03cm,linecolor=blue]{<-}(11.3,7.0)(11.0,5.7)
2240 \rput(10.85,2.6){\psframebox[linewidth=0.03cm,linecolor=blue]{
2241 \begin{minipage}{3.6cm}
2244 \underline{IBS}\\[0.1cm]
2245 3C-SiC also observed for higher T\\[0.1cm]
2246 Higher T inside sample\\[0.1cm]
2247 Structural evolution vs.\\
2248 equilibrium properties
2251 \psline[linewidth=0.03cm,linecolor=blue]{->}(10.85,1.75)(9.0,1.0)
2260 Increased temperature simulations --- $V_1$
2265 \begin{minipage}{6.2cm}
2266 \hspace*{-0.4cm}\includegraphics[width=6.5cm]{tot_pc_thesis.ps}
2269 \begin{minipage}{6.2cm}
2270 \includegraphics[width=6.5cm]{tot_pc3_thesis.ps}
2273 \begin{minipage}{6.2cm}
2274 \hspace*{-0.4cm}\includegraphics[width=6.5cm]{tot_pc2_thesis.ps}
2277 \begin{minipage}{6.3cm}
2279 \underline{Si-C bonds:}
2281 \item Vanishing cut-off artifact (above $1650\,^{\circ}\mathrm{C}$)
2282 \item Structural change: C-Si \hkl<1 0 0> $\rightarrow$ C$_{\text{sub}}$
2284 \underline{Si-Si bonds:}
2285 {\color{blue}Si-C$_{\text{sub}}$-Si} along \hkl<1 1 0>
2286 ($\rightarrow$ 0.325 nm)\\[0.1cm]
2287 \underline{C-C bonds:}
2289 \item C-C next neighbour pairs reduced (mandatory)
2290 \item Peak at 0.3 nm slightly shifted
2292 \item C-Si \hkl<1 0 0> combinations (dashed arrows)\\
2293 $\rightarrow$ C-Si \hkl<1 0 0> \& C$_{\text{sub}}$
2295 $\rightarrow$ pure {\color{blue}C$_{\text{sub}}$ combinations}
2297 \item Range [|-$\downarrow$]:
2298 {\color{blue}C$_{\text{sub}}$ \& C$_{\text{sub}}$
2299 with nearby Si$_{\text{I}}$}
2310 Increased temperature simulations --- $V_1$
2315 \begin{minipage}{6.2cm}
2316 \hspace*{-0.4cm}\includegraphics[width=6.5cm]{tot_pc_thesis.ps}
2319 \begin{minipage}{6.2cm}
2320 \includegraphics[width=6.5cm]{tot_pc3_thesis.ps}
2323 \begin{minipage}{6.2cm}
2324 \hspace*{-0.4cm}\includegraphics[width=6.5cm]{tot_pc2_thesis.ps}
2327 \begin{minipage}{6.3cm}
2329 \underline{Si-C bonds:}
2331 \item Vanishing cut-off artifact (above $1650\,^{\circ}\mathrm{C}$)
2332 \item Structural change: C-Si \hkl<1 0 0> $\rightarrow$ C$_{\text{sub}}$
2334 \underline{Si-Si bonds:}
2335 {\color{blue}Si-C$_{\text{sub}}$-Si} along \hkl<1 1 0>
2336 ($\rightarrow$ 0.325 nm)\\[0.1cm]
2337 \underline{C-C bonds:}
2339 \item C-C next neighbour pairs reduced (mandatory)
2340 \item Peak at 0.3 nm slightly shifted
2342 \item C-Si \hkl<1 0 0> combinations (dashed arrows)\\
2343 $\rightarrow$ C-Si \hkl<1 0 0> \& C$_{\text{sub}}$
2345 $\rightarrow$ pure {\color{blue}C$_{\text{sub}}$ combinations}
2347 \item Range [|-$\downarrow$]:
2348 {\color{blue}C$_{\text{sub}}$ \& C$_{\text{sub}}$
2349 with nearby Si$_{\text{I}}$}
2355 \begin{pspicture}(0,0)(0,0)
2356 \rput(6.5,5.0){\psframebox[fillstyle=solid,opacity=0.5,fillcolor=black]{
2357 \begin{minipage}{14cm}
2362 \rput(6.5,5.0){\psframebox[fillstyle=solid,fillcolor=white,linewidth=0.1cm]{
2363 \begin{minipage}{9cm}
2367 {\color{gray}\bf Conclusions on SiC precipitation}\\[0.1cm]
2368 {\Huge$\lightning$} {\color{red}\ci{}} --- vs --- {\color{blue}\cs{}} {\Huge$\lightning$}\\
2371 \item Stretched coherent SiC structures\\
2372 $\Rightarrow$ Precipitation process involves {\color{blue}\cs}
2373 \item Explains annealing behavior of high/low T C implantations
2375 \item Low T: highly mobile {\color{red}\ci}
2376 \item High T: stable configurations of {\color{blue}\cs}
2380 \item Vehicle to rearrange \cs --- [\cs{} \& \si{} $\leftrightarrow$ \ci]
2381 \item Building block for surrounding Si host \& further SiC
2382 \item Strain compensation \ldots\\
2383 \ldots Si/SiC interface\\
2384 \ldots within stretched coherent SiC structure
2389 \psframebox[linecolor=blue,linewidth=0.05cm]{
2390 \begin{minipage}{7cm}
2392 Precipitation mechanism involving \cs\\
2393 High T $\leftrightarrow$ IBS conditions far from equilibrium\\
2403 % skip high T / C conc ... only here!
2409 Increased temperature simulations at high C concentration
2414 \begin{minipage}{6.5cm}
2415 \includegraphics[width=6.4cm]{12_pc_thesis.ps}
2417 \begin{minipage}{6.5cm}
2418 \includegraphics[width=6.4cm]{12_pc_c_thesis.ps}
2426 \begin{minipage}[t]{6.0cm}
2427 0.186 nm: Si-C pairs $\uparrow$\\
2428 (as expected in 3C-SiC)\\[0.2cm]
2429 0.282 nm: Si-C-C\\[0.2cm]
2430 $\approx$0.35 nm: C-Si-Si
2433 \begin{minipage}{0.2cm}
2437 \begin{minipage}[t]{6.0cm}
2438 0.15 nm: C-C pairs $\uparrow$\\
2439 (as expected in graphite/diamond)\\[0.2cm]
2440 0.252 nm: C-C-C (2$^{\text{nd}}$ NN for diamond)\\[0.2cm]
2441 0.31 nm: shifted towards 0.317 nm $\rightarrow$ C-Si-C
2446 \item Decreasing cut-off artifact
2447 \item {\color{red}Amorphous} SiC-like phase remains
2448 \item High amount of {\color{red}damage} \& alignement to c-Si host matrix lost
2449 \item Slightly sharper peaks $\Rightarrow$ indicate slight {\color{blue}acceleration of dynamics} due to temperature
2458 High C \& small $V$ \& short $t$
2461 Slow restructuring due to strong C-C bonds
2464 High C \& low T implants
2472 % skipped high T / C conc
2483 \begin{pspicture}(0,0)(12,1.0)
2484 \psframebox[fillstyle=gradient,gradbegin=hred,gradend=white,gradlines=1000,gradmidpoint=1.0,linestyle=none]{
2485 \begin{minipage}{11cm}
2486 {\color{black}Diploma thesis}\\
2487 \underline{Monte Carlo} simulation modeling the selforganization process\\
2488 leading to periodic arrays of nanometric amorphous SiC precipitates
2491 \end{pspicture}\\[0.4cm]
2492 \begin{pspicture}(0,0)(12,2)
2493 \psframebox[fillstyle=gradient,gradbegin=hblue,gradend=white,gradmidpoint=1.0,gradlines=1000,linestyle=none]{
2494 \begin{minipage}{11cm}
2495 {\color{black}Doctoral studies}\\
2496 Classical potential \underline{molecular dynamics} simulations \ldots\\
2497 \underline{Density functional theory} calculations \ldots\\[0.2cm]
2498 \ldots on defect formation and SiC precipitation in Si
2501 \end{pspicture}\\[0.5cm]
2502 \begin{pspicture}(0,0)(12,3)
2503 \psframebox[fillstyle=solid,fillcolor=white,linestyle=solid]{
2504 \begin{minipage}{11cm}
2506 {\color{black}\bf How to proceed \ldots}\\[0.1cm]
2507 MC $\rightarrow$ empirical potential MD $\rightarrow$ Ground-state DFT \ldots
2509 \renewcommand\labelitemi{$\ldots$}
2510 \item beyond LDA/GGA methods \& ground-state DFT
2512 Investigation of structure \& structural evolution \ldots
2514 \renewcommand\labelitemi{$\ldots$}
2515 \item electronic/optical properties
2516 \item electronic correlations
2517 \item non-equilibrium systems
2521 \end{pspicture}\\[0.5cm]
2537 \underline{Augsburg}
2539 \item Prof. B. Stritzker (accomodation at EP \RM{4})
2540 \item Ralf Utermann (EDV)
2543 \underline{Helsinki}
2545 \item Prof. K. Nordlund (MD)
2550 \item Bayerische Forschungsstiftung (financial support)
2553 \underline{Paderborn}
2555 \item Prof. J. Lindner (SiC)
2556 \item Prof. G. Schmidt (DFT + financial support)
2557 \item Dr. E. Rauls (DFT + SiC)
2560 \underline{Stuttgart}
2563 \bf Thank you for your attention / invitation!