2 %\documentclass[landscape,semhelv,draft]{seminar}
3 \documentclass[landscape,semhelv]{seminar}
6 \usepackage[greek,german]{babel}
7 \usepackage[latin1]{inputenc}
8 \usepackage[T1]{fontenc}
13 \usepackage{calc} % Simple computations with LaTeX variables
14 \usepackage{caption} % Improved captions
15 \usepackage{fancybox} % To have several backgrounds
17 \usepackage{fancyhdr} % Headers and footers definitions
18 \usepackage{fancyvrb} % Fancy verbatim environments
19 \usepackage{pstricks} % PSTricks with the standard color package
30 \graphicspath{{../img/}}
34 \usepackage[setpagesize=false]{hyperref}
40 \usepackage{semlayer} % Seminar overlays
41 \usepackage{slidesec} % Seminar sections and list of slides
43 \input{seminar.bug} % Official bugs corrections
44 \input{seminar.bg2} % Unofficial bugs corrections
51 %\usepackage{cmbright}
52 %\renewcommand{\familydefault}{\sfdefault}
53 %\usepackage{mathptmx}
59 \extraslideheight{10in}
64 % specify width and height
69 \def\slidetopmargin{-0.15cm}
71 \newcommand{\ham}{\mathcal{H}}
72 \newcommand{\pot}{\mathcal{V}}
73 \newcommand{\foo}{\mathcal{U}}
74 \newcommand{\vir}{\mathcal{W}}
77 \renewcommand\labelitemii{{\color{gray}$\bullet$}}
80 \renewcommand{\phi}{\varphi}
83 \newcommand{\RM}[1]{\MakeUppercase{\romannumeral #1{}}}
86 \newrgbcolor{si-yellow}{.6 .6 0}
87 \newrgbcolor{hb}{0.75 0.77 0.89}
88 \newrgbcolor{lbb}{0.75 0.8 0.88}
89 \newrgbcolor{hlbb}{0.825 0.88 0.968}
90 \newrgbcolor{lachs}{1.0 .93 .81}
93 \newcommand{\si}{Si$_{\text{i}}${}}
94 \newcommand{\ci}{C$_{\text{i}}${}}
95 \newcommand{\cs}{C$_{\text{sub}}${}}
96 \newcommand{\degc}[1]{\unit[#1]{$^{\circ}$C}{}}
97 \newcommand{\distn}[1]{\unit[#1]{nm}{}}
98 \newcommand{\dista}[1]{\unit[#1]{\AA}{}}
99 \newcommand{\perc}[1]{\unit[#1]{\%}{}}
101 % no vertical centering
112 A B C D E F G H G F E D C B A
127 Atomistic simulation studies\\[0.2cm]
133 \textsc{Frank Zirkelbach}
137 Application talk at the Max Planck Institute for Solid State Research
141 Stuttgart, November 2011
146 % no vertical centering
156 % Phase diagram of the C/Si system\\
161 \begin{minipage}{6.5cm}
162 \includegraphics[width=6.5cm]{si-c_phase.eps}
165 R. I. Scace and G. A. Slack, J. Chem. Phys. 30, 1551 (1959)
168 \begin{pspicture}(0,0)(0,0)
169 \psellipse[linecolor=blue,linewidth=0.1cm](3.55,4.0)(0.5,2.9)
172 \begin{minipage}{6cm}
173 {\bf Phase diagram of the C/Si system}\\[0.2cm]
174 {\color{blue}Stoichiometric composition}
176 \item only chemical stable compound
177 \item wide band gap semiconductor\\
178 \underline{silicon carbide}, SiC
184 % motivation / properties / applications of silicon carbide
192 \begin{pspicture}(0,0)(13.5,5)
194 \psframe*[linecolor=hb](-0.2,0)(12.9,5)
196 \pspolygon[linecolor=hlbb,fillcolor=hlbb,fillstyle=solid](5.2,1)(6.5,1)(6.5,3)(5.2,3)
197 \pspolygon[linecolor=hlbb,fillcolor=hlbb,fillstyle=solid](6.4,0.5)(7.7,2)(7.7,2)(6.4,3.5)
199 \rput[lt](0,4.6){\color{gray}PROPERTIES}
201 \rput[lt](0.3,4){wide band gap}
202 \rput[lt](0.3,3.5){high electric breakdown field}
203 \rput[lt](0.3,3){good electron mobility}
204 \rput[lt](0.3,2.5){high electron saturation drift velocity}
205 \rput[lt](0.3,2){high thermal conductivity}
207 \rput[lt](0.3,1.5){hard and mechanically stable}
208 \rput[lt](0.3,1){chemically inert}
210 \rput[lt](0.3,0.5){radiation hardness}
212 \rput[rt](12.7,4.6){\color{gray}APPLICATIONS}
214 \rput[rt](12.5,3.85){high-temperature, high power}
215 \rput[rt](12.5,3.5){and high-frequency}
216 \rput[rt](12.5,3.15){electronic and optoelectronic devices}
218 \rput[rt](12.5,2.35){material suitable for extreme conditions}
219 \rput[rt](12.5,2){microelectromechanical systems}
220 \rput[rt](12.5,1.65){abrasives, cutting tools, heating elements}
222 \rput[rt](12.5,0.85){first wall reactor material, detectors}
223 \rput[rt](12.5,0.5){and electronic devices for space}
227 \begin{picture}(0,0)(5,-162)
228 \includegraphics[height=2.2cm]{3C_SiC_bs.eps}
230 \begin{picture}(0,0)(-120,-162)
231 \includegraphics[height=2.2cm]{nasa_600c_led.eps}
233 \begin{picture}(0,0)(-270,-162)
234 \includegraphics[height=2.2cm]{6h-sic_3c-sic.eps}
237 \begin{picture}(0,0)(10,65)
238 \includegraphics[height=2.8cm]{sic_switch.eps}
240 %\begin{picture}(0,0)(-243,65)
241 \begin{picture}(0,0)(-110,65)
242 \includegraphics[height=2.8cm]{ise_99.eps}
244 %\begin{picture}(0,0)(-135,65)
245 \begin{picture}(0,0)(-100,65)
246 \includegraphics[height=1.2cm]{infineon_schottky.eps}
248 \begin{picture}(0,0)(-233,65)
249 \includegraphics[height=2.8cm]{solar_car.eps}
259 Polytypes of SiC\\[0.4cm]
262 \includegraphics[width=3.8cm]{cubic_hex.eps}\\
263 \begin{minipage}{1.9cm}
264 {\tiny cubic (twist)}
266 \begin{minipage}{2.9cm}
267 {\tiny hexagonal (no twist)}
270 \begin{picture}(0,0)(-150,0)
271 \includegraphics[width=7cm]{polytypes.eps}
278 \begin{tabular}{l c c c c c c}
280 & 3C-SiC & 4H-SiC & 6H-SiC & Si & GaN & Diamond\\
282 Hardness [Mohs] & \multicolumn{3}{c}{------ 9.6 ------}& 6.5 & - & 10 \\
283 Band gap [eV] & 2.36 & 3.23 & 3.03 & 1.12 & 3.39 & 5.5 \\
284 Break down field [$10^6$ V/cm] & 4 & 3 & 3.2 & 0.6 & 5 & 10 \\
285 Saturation drift velocity [$10^7$ cm/s] & 2.5 & 2.0 & 2.0 & 1 & 2.7 & 2.7 \\
286 Electron mobility [cm$^2$/Vs] & 800 & 900 & 400 & 1100 & 900 & 2200 \\
287 Hole mobility [cm$^2$/Vs] & 320 & 120 & 90 & 420 & 150 & 1600 \\
288 Thermal conductivity [W/cmK] & 5.0 & 4.9 & 4.9 & 1.5 & 1.3 & 22 \\
292 \begin{pspicture}(0,0)(0,0)
293 \psellipse[linecolor=green](5.7,2.10)(0.4,0.5)
295 \begin{pspicture}(0,0)(0,0)
296 \psellipse[linecolor=green](5.6,0.92)(0.4,0.2)
298 \begin{pspicture}(0,0)(0,0)
299 \psellipse[linecolor=red](10.45,0.45)(0.4,0.2)
309 Fabrication of silicon carbide
318 \emph{Silicon carbide --- Born from the stars, perfected on earth.}
324 SiC thin films by MBE \& CVD
326 \item Much progress achieved in homo/heteroepitaxial SiC thin film growth
327 \item \underline{Commercially available} semiconductor power devices based on
328 \underline{\foreignlanguage{greek}{a}-SiC}
329 \item Production of favored \underline{3C-SiC} material
330 \underline{less advanced}
331 \item Quality and size not yet sufficient
333 \begin{picture}(0,0)(-310,-20)
334 \includegraphics[width=2.0cm]{cree.eps}
339 Alternative approach:
340 Ion beam synthesis (IBS) of burried 3C-SiC layers in Si\hkl(1 0 0)
347 \begin{minipage}{3.15cm}
349 \includegraphics[width=3cm]{imp.eps}\\
355 \begin{minipage}{3.15cm}
357 \includegraphics[width=3cm]{annealing.eps}\\
359 \unit[12]{h} annealing at \degc{1200}
364 \begin{minipage}{5.5cm}
365 \includegraphics[width=5.8cm]{ibs_3c-sic.eps}\\[-0.2cm]
368 XTEM: single crystalline 3C-SiC in Si\hkl(1 0 0)
380 Systematic investigation of C implantations into Si
386 \includegraphics[width=0.75\textwidth]{imp_inv.eps}
402 \includegraphics[width=0.75\textwidth]{imp_inv.eps}
405 \begin{pspicture}(0,0)(0,0)
406 \rput(6.0,7.0){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=white]{
407 \begin{minipage}{11cm}
408 {\color{red}Diploma thesis}\\
409 \underline{Monte Carlo} simulation modeling the selforganization process\\
410 leading to periodic arrays of nanometric amorphous SiC precipitates
414 \begin{pspicture}(0,0)(0,0)
415 \rput(6.0,-0.5){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=white]{
416 \begin{minipage}{11cm}
417 {\color{blue}Doctoral studies}\\
418 Classical potential \underline{molecular dynamics} simulations \ldots\\
419 \underline{Density functional theory} calculations \ldots\\[0.2cm]
420 \ldots on defect formation and SiC precipitation in Si
424 \begin{pspicture}(0,0)(0,0)
425 \psellipse[linecolor=red,linewidth=0.05cm](5,3.0)(0.8,1.0)
427 \begin{pspicture}(0,0)(0,0)
428 \psellipse[linecolor=blue,linewidth=0.05cm](8.2,3.2)(1.5,1.6)
436 Selforganization of nanometric amorphous SiC lamellae
444 \item Regularly spaced, nanometric spherical\\
445 and lamellar amorphous inclusions\\
446 at the upper a/c interface
447 \item Carbon accumulation\\
453 \begin{minipage}{12cm}
454 \includegraphics[width=9cm]{../../nlsop/img/k393abild1_e_l.eps}\\
456 XTEM bright-field, \unit[180]{keV} C$^+ \rightarrow$ Si, \degc{150},
457 Dose: \unit[4.3 $\times 10^{17}$]{cm$^{-2}$}
461 \begin{picture}(0,0)(-182,-215)
462 \begin{minipage}{6.5cm}
464 \includegraphics[width=6.5cm]{../../nlsop/img/eftem.eps}\\[-0.2cm]
466 XTEM bright-field and respective EFTEM C map
477 Model displaying the formation of ordered lamellae
483 \includegraphics[width=8.0cm]{../../nlsop/img/modell_ng_e.eps}
489 \item Supersaturation of C in c-Si\\
490 $\rightarrow$ {\bf Carbon induced} nucleation of spherical
492 \item High interfacial energy between 3C-SiC and c-Si\\
493 $\rightarrow$ {\bf Amorphous} precipitates
494 \item \unit[20-- 30]{\%} lower silicon density of a-SiC$_x$ compared to c-Si\\
495 $\rightarrow$ {\bf Lateral strain} (black arrows)
496 \item Implantation range near surface\\
497 $\rightarrow$ {\bf Relaxation} of {\bf vertical strain component}
498 \item Reduction of the carbon supersaturation in c-Si\\
499 $\rightarrow$ {\bf Carbon diffusion} into amorphous volumina
501 \item Remaining lateral strain\\
502 $\rightarrow$ {\bf Strain enhanced} lateral amorphisation
503 \item Absence of crystalline neighbours (structural information)\\
504 $\rightarrow$ {\bf Stabilization} of amorphous inclusions
505 {\bf against recrystallization}
513 Implementation of the Monte Carlo code
519 \item \underline{Amorphization / Recrystallization}\\
520 Ion collision in discretized target determined by random numbers
521 distributed according to nuclear energy loss.
522 Amorphization/recrystallization probability:
524 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}}
527 \item {\color{green} $p_b$} normal `ballistic' amorphization
528 \item {\color{blue} $p_c$} carbon induced amorphization
529 \item {\color{red} $p_s$} stress enhanced amorphization
532 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{,}
535 \delta (\vec r) = \left\{
537 1 & \textrm{if volume at position $\vec r$ is amorphous} \\
538 0 & \textrm{otherwise} \\
542 \item \underline{Carbon incorporation}\\
543 Incorporation volume determined according to implantation profile
544 \item \underline{Diffusion / Sputtering}
546 \item Transfer fraction of C atoms
547 of crystalline into neighbored amorphous volumes
548 \item Remove surface layer
556 \begin{minipage}{3.7cm}
565 Evolution of the \ldots
570 \item lamella precipitates
572 \ldots reproduced!\\[1.5cm]
576 Experiment \& simulation\\
577 in good agreement\\[1.0cm]
579 Simulation is able to model the whole depth region\\[1.0cm]
584 \begin{minipage}{0.4cm}
587 \begin{minipage}{8.0cm}
589 \includegraphics[width=9cm]{../../nlsop/img/dosis_entwicklung_ng_e_1-2.eps}\\
590 \includegraphics[width=9cm]{../../nlsop/img/dosis_entwicklung_ng_e2_2-2.eps}
598 Structural \& compositional details
601 \begin{minipage}[t]{7.5cm}
602 \includegraphics[height=6.5cm]{../../nlsop/img/ac_cconc_ver2_e.eps}\\
604 \begin{minipage}[t]{5.0cm}
605 \includegraphics[height=6.5cm]{../../nlsop/img/97_98_e.eps}
613 \item Fluctuation of C concentration in lamellae region
614 \item \unit[8--10]{at.\%} C saturation limit
615 within the respective conditions
616 \item Complementarily arranged and alternating sequence of layers\\
617 with a high and low amount of amorphous regions
618 \item C accumulation in the amorphous phase / Origin of stress
621 \begin{picture}(0,0)(-265,-30)
623 \begin{minipage}{3cm}
626 Precipitation process\\
648 Model displaying the formation of ordered lamellae
652 \begin{minipage}{6.3cm}
655 Precipitation mechanism not yet fully understood!
657 \renewcommand\labelitemi{$\Rightarrow$}
659 \underline{Understanding the SiC precipitation}
661 \item significant technological progress in SiC thin film formation
662 \item perspectives for processes relying upon prevention of SiC precipitation
673 Supposed precipitation mechanism of SiC in Si
680 \begin{minipage}{3.8cm}
681 Si \& SiC lattice structure\\[0.2cm]
682 \includegraphics[width=3.5cm]{sic_unit_cell.eps}\\[-0.3cm]
686 \begin{minipage}{3.8cm}
688 \includegraphics[width=3.3cm]{tem_c-si-db.eps}
692 \begin{minipage}{3.8cm}
694 \includegraphics[width=3.3cm]{tem_3c-sic.eps}
698 \begin{minipage}{4cm}
700 C-Si dimers (dumbbells)\\[-0.1cm]
701 on Si interstitial sites
705 \begin{minipage}{4.2cm}
707 Agglomeration of C-Si dumbbells\\[-0.1cm]
708 $\Rightarrow$ dark contrasts
712 \begin{minipage}{4cm}
714 Precipitation of 3C-SiC in Si\\[-0.1cm]
715 $\Rightarrow$ Moir\'e fringes\\[-0.1cm]
716 \& release of Si self-interstitials
720 \begin{minipage}{3.8cm}
722 \includegraphics[width=3.3cm]{sic_prec_seq_01.eps}
726 \begin{minipage}{3.8cm}
728 \includegraphics[width=3.3cm]{sic_prec_seq_02.eps}
732 \begin{minipage}{3.8cm}
734 \includegraphics[width=3.3cm]{sic_prec_seq_03.eps}
738 \begin{pspicture}(0,0)(0,0)
739 \psline[linewidth=4pt]{->}(8.5,2)(9.0,2)
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741 \rput{-20}{\psellipse[linecolor=blue](3.3,8.1)(0.3,0.5)}
742 \psline[linewidth=4pt]{->}(4.0,2)(4.5,2)
743 \rput(12.7,0.3){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
744 $4a_{\text{Si}}=5a_{\text{SiC}}$
746 \rput(12.2,8){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
747 \hkl(h k l) planes match
749 \rput(9.7,6.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
759 Supposed precipitation mechanism of SiC in Si
766 \begin{minipage}{3.8cm}
767 Si \& SiC lattice structure\\[0.2cm]
768 \includegraphics[width=3.5cm]{sic_unit_cell.eps}\\[-0.3cm]
772 \begin{minipage}{3.8cm}
774 \includegraphics[width=3.3cm]{tem_c-si-db.eps}
778 \begin{minipage}{3.8cm}
780 \includegraphics[width=3.3cm]{tem_3c-sic.eps}
784 \begin{minipage}{4cm}
786 C-Si dimers (dumbbells)\\[-0.1cm]
787 on Si interstitial sites
791 \begin{minipage}{4.2cm}
793 Agglomeration of C-Si dumbbells\\[-0.1cm]
794 $\Rightarrow$ dark contrasts
798 \begin{minipage}{4cm}
800 Precipitation of 3C-SiC in Si\\[-0.1cm]
801 $\Rightarrow$ Moir\'e fringes\\[-0.1cm]
802 \& release of Si self-interstitials
806 \begin{minipage}{3.8cm}
808 \includegraphics[width=3.3cm]{sic_prec_seq_01.eps}
812 \begin{minipage}{3.8cm}
814 \includegraphics[width=3.3cm]{sic_prec_seq_02.eps}
818 \begin{minipage}{3.8cm}
820 \includegraphics[width=3.3cm]{sic_prec_seq_03.eps}
824 \begin{pspicture}(0,0)(0,0)
825 \psline[linewidth=4pt]{->}(8.5,2)(9.0,2)
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830 $4a_{\text{Si}}=5a_{\text{SiC}}$
832 \rput(12.2,8){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
833 \hkl(h k l) planes match
835 \rput(9.7,6.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
838 \rput(6.7,5.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=white]{
839 \begin{minipage}{10cm}
841 {\color{red}\bf Controversial views}
843 \item Implantations at high T (Nejim et al.)
845 \item Topotactic transformation based on \cs
846 \item \si{} as supply reacting with further C in cleared volume
848 \item Annealing behavior (Serre et al.)
850 \item Room temperature implants $\rightarrow$ highly mobile C
851 \item Elevated T implants $\rightarrow$ no/low C redistribution/migration\\
852 (indicate stable \cs{} configurations)
854 \item Strained silicon \& Si/SiC heterostructures
856 \item Coherent SiC precipitates (tensile strain)
857 \item Incoherent SiC (strain relaxation)
869 Molecular dynamics (MD) simulations
878 \item Microscopic description of N particle system
879 \item Analytical interaction potential
880 \item Numerical integration using Newtons equation of motion\\
881 as a propagation rule in 6N-dimensional phase space
882 \item Observables obtained by time and/or ensemble averages
884 {\bf Details of the simulation:}
886 \item Integration: Velocity Verlet, timestep: $1\text{ fs}$
887 \item Ensemble: NpT (isothermal-isobaric)
889 \item Berendsen thermostat:
890 $\tau_{\text{T}}=100\text{ fs}$
891 \item Berendsen barostat:\\
892 $\tau_{\text{P}}=100\text{ fs}$,
893 $\beta^{-1}=100\text{ GPa}$
895 \item Erhart/Albe potential: Tersoff-like bond order potential
898 E = \frac{1}{2} \sum_{i \neq j} \pot_{ij}, \quad
899 \pot_{ij} = {\color{red}f_C(r_{ij})}
900 \left[ f_R(r_{ij}) + {\color{blue}b_{ij}} f_A(r_{ij}) \right]
904 \begin{picture}(0,0)(-230,-30)
905 \includegraphics[width=5cm]{tersoff_angle.eps}
913 Density functional theory (DFT) calculations
918 Basic ingredients necessary for DFT
921 \item \underline{Hohenberg-Kohn theorem} - ground state density $n_0(r)$ ...
923 \item ... uniquely determines the ground state potential
925 \item ... minimizes the systems total energy
927 \item \underline{Born-Oppenheimer}
928 - $N$ moving electrons in an external potential of static nuclei
930 H\Psi = \left[-\sum_i^N \frac{\hbar^2}{2m}\nabla_i^2
931 +\sum_i^N V_{\text{ext}}(r_i)
932 +\sum_{i<j}^N V_{e-e}(r_i,r_j)\right]\Psi=E\Psi
934 \item \underline{Effective potential}
935 - averaged electrostatic potential \& exchange and correlation
937 V_{\text{eff}}(r)=V_{\text{ext}}(r)+\int\frac{e^2 n(r')}{|r-r'|}d^3r'
940 \item \underline{Kohn-Sham system}
941 - Schr\"odinger equation of N non-interacting particles
943 \left[ -\frac{\hbar^2}{2m}\nabla^2 + V_{\text{eff}}(r) \right] \Phi_i(r)
948 n(r)=\sum_i^N|\Phi_i(r)|^2
950 \item \underline{Self-consistent solution}\\
951 $n(r)$ depends on $\Phi_i$, which depend on $V_{\text{eff}}$,
952 which in turn depends on $n(r)$
953 \item \underline{Variational principle}
954 - minimize total energy with respect to $n(r)$
962 Density functional theory (DFT) calculations
969 Details of applied DFT calculations in this work
972 \item \underline{Exchange correlation functional}
973 - approximations for the inhomogeneous electron gas
975 \item LDA: $E_{\text{XC}}^{\text{LDA}}[n]=\int \epsilon_{\text{XC}}(n)n(r)d^3r$
976 \item GGA: $E_{\text{XC}}^{\text{GGA}}[n]=\int \epsilon_{\text{XC}}(n,\nabla n)n(r)d^3r$
978 \item \underline{Plane wave basis set}
979 - approximation of the wavefunction $\Phi_i$ by plane waves $\phi_j$
982 \text{Fourier series: } \Phi_i=\sum_{|G+k|<G_{\text{cut}}} c_j^i \phi_j(r), \quad E_{\text{cut}}=\frac{\hbar^2}{2m}G^2_{\text{cut}}
983 \qquad ({\color{blue}300\text{ eV}})
985 \item \underline{Brillouin zone sampling} -
986 {\color{blue}$\Gamma$-point only} calculations
987 \item \underline{Pseudo potential}
988 - consider only the valence electrons
989 \item \underline{Code} - VASP 4.6
994 MD and structural optimization
997 \item MD integration: Gear predictor corrector algorithm
998 \item Pressure control: Parrinello-Rahman pressure control
999 \item Structural optimization: Conjugate gradient method
1002 \begin{pspicture}(0,0)(0,0)
1003 \psellipse[linecolor=blue](1.5,6.75)(0.5,0.3)
1011 C and Si self-interstitial point defects in silicon
1018 \begin{minipage}{8cm}
1020 \begin{pspicture}(0,0)(7,5)
1021 \rput(3.5,4){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
1024 \item Creation of c-Si simulation volume
1025 \item Periodic boundary conditions
1026 \item $T=0\text{ K}$, $p=0\text{ bar}$
1029 \rput(3.5,2.1){\rnode{insert}{\psframebox{
1032 Insertion of interstitial C/Si atoms
1035 \rput(3.5,1){\rnode{cool}{\psframebox[fillstyle=solid,fillcolor=lbb]{
1038 Relaxation / structural energy minimization
1041 \ncline[]{->}{init}{insert}
1042 \ncline[]{->}{insert}{cool}
1045 \begin{minipage}{5cm}
1046 \includegraphics[width=5cm]{unit_cell_e.eps}\\
1049 \begin{minipage}{9cm}
1050 \begin{tabular}{l c c}
1052 & size [unit cells] & \# atoms\\
1054 VASP & $3\times 3\times 3$ & $216\pm 1$ \\
1055 Erhart/Albe & $9\times 9\times 9$ & $5832\pm 1$\\
1059 \begin{minipage}{4cm}
1060 {\color{red}$\bullet$} Tetrahedral\\
1061 {\color{green}$\bullet$} Hexagonal\\
1062 {\color{yellow}$\bullet$} \hkl<1 0 0> dumbbell\\
1063 {\color{magenta}$\bullet$} \hkl<1 1 0> dumbbell\\
1064 {\color{cyan}$\bullet$} Bond-centered\\
1065 {\color{black}$\bullet$} Vacancy / Substitutional
1074 \begin{minipage}{9.5cm}
1077 Si self-interstitial point defects in silicon\\
1080 \begin{tabular}{l c c c c c}
1082 $E_{\text{f}}$ [eV] & \hkl<1 1 0> DB & H & T & \hkl<1 0 0> DB & V \\
1084 VASP & \underline{3.39} & 3.42 & 3.77 & 4.41 & 3.63 \\
1085 Erhart/Albe & 4.39 & 4.48$^*$ & \underline{3.40} & 5.42 & 3.13 \\
1087 \end{tabular}\\[0.2cm]
1089 \begin{minipage}{4.7cm}
1090 \includegraphics[width=4.7cm]{e_kin_si_hex.ps}
1092 \begin{minipage}{4.7cm}
1094 {\tiny nearly T $\rightarrow$ T}\\
1096 \includegraphics[width=4.7cm]{nhex_tet.ps}
1099 \underline{Hexagonal} \hspace{2pt}
1100 \href{../video/si_self_int_hexa.avi}{$\rhd$}\\[0.1cm]
1102 \begin{minipage}{2.7cm}
1103 $E_{\text{f}}^*=4.48\text{ eV}$\\
1104 \includegraphics[width=2.7cm]{si_pd_albe/hex_a.eps}
1106 \begin{minipage}{0.4cm}
1111 \begin{minipage}{2.7cm}
1112 $E_{\text{f}}=3.96\text{ eV}$\\
1113 \includegraphics[width=2.8cm]{si_pd_albe/hex.eps}
1116 \begin{minipage}{2.9cm}
1118 \underline{Vacancy}\\
1119 \includegraphics[width=3.0cm]{si_pd_albe/vac.eps}
1124 \begin{minipage}{3.5cm}
1127 \underline{\hkl<1 1 0> dumbbell}\\
1128 \includegraphics[width=3.0cm]{si_pd_albe/110.eps}\\
1129 \underline{Tetrahedral}\\
1130 \includegraphics[width=3.0cm]{si_pd_albe/tet.eps}\\
1131 \underline{\hkl<1 0 0> dumbbell}\\
1132 \includegraphics[width=3.0cm]{si_pd_albe/100.eps}
1144 C interstitial point defects in silicon\\[-0.1cm]
1147 \begin{tabular}{l c c c c c c r}
1149 $E_{\text{f}}$ & T & H & \hkl<1 0 0> DB & \hkl<1 1 0> DB & S & B & \cs{} \& \si\\
1151 VASP & unstable & unstable & \underline{3.72} & 4.16 & 1.95 & 4.66 & {\color{green}4.17}\\
1152 Erhart/Albe MD & 6.09 & 9.05$^*$ & \underline{3.88} & 5.18 & {\color{red}0.75} & 5.59$^*$ & {\color{green}4.43} \\
1154 \end{tabular}\\[0.1cm]
1157 \begin{minipage}{2.7cm}
1158 \underline{Hexagonal} \hspace{2pt}
1159 \href{../video/c_in_si_int_hexa.avi}{$\rhd$}\\
1160 $E_{\text{f}}^*=9.05\text{ eV}$\\
1161 \includegraphics[width=2.7cm]{c_pd_albe/hex.eps}
1163 \begin{minipage}{0.4cm}
1168 \begin{minipage}{2.7cm}
1169 \underline{\hkl<1 0 0>}\\
1170 $E_{\text{f}}=3.88\text{ eV}$\\
1171 \includegraphics[width=2.7cm]{c_pd_albe/100.eps}
1174 \begin{minipage}{2cm}
1177 \begin{minipage}{3cm}
1179 \underline{Tetrahedral}\\
1180 \includegraphics[width=3.0cm]{c_pd_albe/tet.eps}
1185 \begin{minipage}{2.7cm}
1186 \underline{Bond-centered}\\
1187 $E_{\text{f}}^*=5.59\text{ eV}$\\
1188 \includegraphics[width=2.7cm]{c_pd_albe/bc.eps}
1190 \begin{minipage}{0.4cm}
1195 \begin{minipage}{2.7cm}
1196 \underline{\hkl<1 1 0> dumbbell}\\
1197 $E_{\text{f}}=5.18\text{ eV}$\\
1198 \includegraphics[width=2.7cm]{c_pd_albe/110.eps}
1201 \begin{minipage}{2cm}
1204 \begin{minipage}{3cm}
1206 \underline{Substitutional}\\
1207 \includegraphics[width=3.0cm]{c_pd_albe/sub.eps}
1218 C \hkl<1 0 0> dumbbell interstitial configuration\\
1222 \begin{tabular}{l c c c c c c c c}
1224 Distances [nm] & $r(1C)$ & $r(2C)$ & $r(3C)$ & $r(12)$ & $r(13)$ & $r(34)$ & $r(23)$ & $r(25)$ \\
1226 Erhart/Albe & 0.175 & 0.329 & 0.186 & 0.226 & 0.300 & 0.343 & 0.423 & 0.425 \\
1227 VASP & 0.174 & 0.341 & 0.182 & 0.229 & 0.286 & 0.347 & 0.422 & 0.417 \\
1229 \end{tabular}\\[0.2cm]
1230 \begin{tabular}{l c c c c }
1232 Angles [$^{\circ}$] & $\theta_1$ & $\theta_2$ & $\theta_3$ & $\theta_4$ \\
1234 Erhart/Albe & 140.2 & 109.9 & 134.4 & 112.8 \\
1235 VASP & 130.7 & 114.4 & 146.0 & 107.0 \\
1237 \end{tabular}\\[0.2cm]
1238 \begin{tabular}{l c c c}
1240 Displacements [nm]& $a$ & $b$ & $|a|+|b|$ \\
1242 Erhart/Albe & 0.084 & -0.091 & 0.175 \\
1243 VASP & 0.109 & -0.065 & 0.174 \\
1245 \end{tabular}\\[0.6cm]
1248 \begin{minipage}{3.0cm}
1250 \underline{Erhart/Albe}
1251 \includegraphics[width=3.0cm]{c_pd_albe/100_cmp.eps}
1254 \begin{minipage}{3.0cm}
1257 \includegraphics[width=3.0cm]{c_pd_vasp/100_cmp.eps}
1261 \begin{picture}(0,0)(-185,10)
1262 \includegraphics[width=6.8cm]{100-c-si-db_cmp.eps}
1264 \begin{picture}(0,0)(-280,-150)
1265 \includegraphics[width=3.3cm]{c_pd_vasp/eden.eps}
1268 \begin{pspicture}(0,0)(0,0)
1269 \psellipse[linecolor=green](5.18,5.92)(0.5,0.3)
1270 \psellipse[linecolor=red](3.45,5.92)(1.0,0.4)
1271 \psellipse[linecolor=blue](2.7,6.92)(0.9,0.2)
1272 \psellipse[linecolor=blue](4.65,6.92)(0.9,0.2)
1281 \begin{minipage}{8.5cm}
1284 Bond-centered interstitial configuration\\[-0.1cm]
1287 \begin{minipage}{3.0cm}
1288 \includegraphics[width=2.8cm]{c_pd_vasp/bc_2333.eps}\\
1290 \begin{minipage}{5.2cm}
1292 \item Linear Si-C-Si bond
1293 \item Si: one C \& 3 Si neighbours
1294 \item Spin polarized calculations
1295 \item No saddle point!\\
1302 \begin{minipage}[t]{6.5cm}
1303 \begin{minipage}[t]{1.2cm}
1305 {\tiny sp$^3$}\\[0.8cm]
1306 \underline{${\color{black}\uparrow}$}
1307 \underline{${\color{black}\uparrow}$}
1308 \underline{${\color{black}\uparrow}$}
1309 \underline{${\color{red}\uparrow}$}\\
1312 \begin{minipage}[t]{1.4cm}
1314 {\color{red}M}{\color{blue}O}\\[0.8cm]
1315 \underline{${\color{blue}\uparrow}{\color{white}\downarrow}$}\\
1316 $\sigma_{\text{ab}}$\\[0.5cm]
1317 \underline{${\color{red}\uparrow}{\color{blue}\downarrow}$}\\
1321 \begin{minipage}[t]{1.0cm}
1325 \underline{${\color{white}\uparrow\uparrow}$}
1326 \underline{${\color{white}\uparrow\uparrow}$}\\
1328 \underline{${\color{blue}\uparrow}{\color{blue}\downarrow}$}
1329 \underline{${\color{blue}\uparrow}{\color{blue}\downarrow}$}\\
1333 \begin{minipage}[t]{1.4cm}
1335 {\color{blue}M}{\color{green}O}\\[0.8cm]
1336 \underline{${\color{blue}\uparrow}{\color{white}\downarrow}$}\\
1337 $\sigma_{\text{ab}}$\\[0.5cm]
1338 \underline{${\color{green}\uparrow}{\color{blue}\downarrow}$}\\
1342 \begin{minipage}[t]{1.2cm}
1345 {\tiny sp$^3$}\\[0.8cm]
1346 \underline{${\color{green}\uparrow}$}
1347 \underline{${\color{black}\uparrow}$}
1348 \underline{${\color{black}\uparrow}$}
1349 \underline{${\color{black}\uparrow}$}\\
1357 \begin{minipage}{4.5cm}
1358 \includegraphics[width=4cm]{c_100_mig_vasp/im_spin_diff.eps}
1360 \begin{minipage}{3.5cm}
1361 {\color{gray}$\bullet$} Spin up\\
1362 {\color{green}$\bullet$} Spin down\\
1363 {\color{blue}$\bullet$} Resulting spin up\\
1364 {\color{yellow}$\bullet$} Si atoms\\
1365 {\color{red}$\bullet$} C atom
1370 \begin{minipage}{4.2cm}
1372 \includegraphics[width=4.3cm]{c_pd_vasp/bc_2333_ksl.ps}\\
1373 {\color{green}$\Box$} {\tiny unoccupied}\\
1374 {\color{red}$\bullet$} {\tiny occupied}
1383 Migration of the C \hkl<1 0 0> dumbbell interstitial
1388 {\small Investigated pathways}
1390 \begin{minipage}{8.5cm}
1391 \begin{minipage}{8.3cm}
1392 \underline{\hkl<0 0 -1> $\rightarrow$ \hkl<0 0 1>}\\
1393 \begin{minipage}{2.4cm}
1394 \includegraphics[width=2.4cm]{c_pd_vasp/100_2333.eps}
1396 \begin{minipage}{0.4cm}
1399 \begin{minipage}{2.4cm}
1400 \includegraphics[width=2.4cm]{c_pd_vasp/bc_2333.eps}
1402 \begin{minipage}{0.4cm}
1405 \begin{minipage}{2.4cm}
1406 \includegraphics[width=2.4cm]{c_pd_vasp/100_next_2333.eps}
1409 \begin{minipage}{8.3cm}
1410 \underline{\hkl<0 0 -1> $\rightarrow$ \hkl<0 -1 0>}\\
1411 \begin{minipage}{2.4cm}
1412 \includegraphics[width=2.4cm]{c_pd_vasp/100_2333.eps}
1414 \begin{minipage}{0.4cm}
1417 \begin{minipage}{2.4cm}
1418 \includegraphics[width=2.4cm]{c_pd_vasp/00-1-0-10_2333.eps}
1420 \begin{minipage}{0.4cm}
1423 \begin{minipage}{2.4cm}
1424 \includegraphics[width=2.4cm]{c_pd_vasp/0-10_2333.eps}
1427 \begin{minipage}{8.3cm}
1428 \underline{\hkl<0 0 -1> $\rightarrow$ \hkl<0 -1 0> (in place)}\\
1429 \begin{minipage}{2.4cm}
1430 \includegraphics[width=2.4cm]{c_pd_vasp/100_2333.eps}
1432 \begin{minipage}{0.4cm}
1435 \begin{minipage}{2.4cm}
1436 \includegraphics[width=2.4cm]{c_pd_vasp/00-1_ip0-10_2333.eps}
1438 \begin{minipage}{0.4cm}
1441 \begin{minipage}{2.4cm}
1442 \includegraphics[width=2.4cm]{c_pd_vasp/0-10_ip_2333.eps}
1447 \begin{minipage}{4.2cm}
1448 {\small Constrained relaxation\\
1449 technique (CRT) method}\\
1450 \includegraphics[width=4cm]{crt_orig.eps}
1452 \item Constrain diffusing atom
1453 \item Static constraints
1456 {\small Modifications}\\
1457 \includegraphics[width=4cm]{crt_mod.eps}
1459 \item Constrain all atoms
1460 \item Update individual\\
1471 Migration of the C \hkl<1 0 0> dumbbell interstitial
1477 \begin{minipage}{5.9cm}
1479 \includegraphics[width=5.8cm]{im_00-1_nosym_sp_fullct_thesis.ps}\\[0.45cm]
1482 \begin{picture}(0,0)(60,0)
1483 \includegraphics[width=1cm]{vasp_mig/00-1.eps}
1485 \begin{picture}(0,0)(-5,0)
1486 \includegraphics[width=1cm]{vasp_mig/bc_00-1_sp.eps}
1488 \begin{picture}(0,0)(-55,0)
1489 \includegraphics[width=1cm]{vasp_mig/bc.eps}
1491 \begin{picture}(0,0)(12.5,10)
1492 \includegraphics[width=1cm]{110_arrow.eps}
1494 \begin{picture}(0,0)(90,0)
1495 \includegraphics[height=0.9cm]{001_arrow.eps}
1501 \begin{minipage}{0.3cm}
1505 \begin{minipage}{5.9cm}
1507 \includegraphics[width=5.9cm]{vasp_mig/00-1_0-10_nosym_sp_fullct.ps}\\[0.5cm]
1510 \begin{picture}(0,0)(60,0)
1511 \includegraphics[width=1cm]{vasp_mig/00-1_a.eps}
1513 \begin{picture}(0,0)(5,0)
1514 \includegraphics[width=1cm]{vasp_mig/00-1_0-10_sp.eps}
1516 \begin{picture}(0,0)(-55,0)
1517 \includegraphics[width=1cm]{vasp_mig/0-10.eps}
1519 \begin{picture}(0,0)(12.5,10)
1520 \includegraphics[width=1cm]{100_arrow.eps}
1522 \begin{picture}(0,0)(90,0)
1523 \includegraphics[height=0.9cm]{001_arrow.eps}
1533 \begin{minipage}{5.9cm}
1535 \includegraphics[width=5.9cm]{vasp_mig/00-1_ip0-10_nosym_sp_fullct.ps}\\[0.6cm]
1538 \begin{picture}(0,0)(60,0)
1539 \includegraphics[width=0.9cm]{vasp_mig/00-1_b.eps}
1541 \begin{picture}(0,0)(10,0)
1542 \includegraphics[width=0.9cm]{vasp_mig/00-1_ip0-10_sp.eps}
1544 \begin{picture}(0,0)(-60,0)
1545 \includegraphics[width=0.9cm]{vasp_mig/0-10_b.eps}
1547 \begin{picture}(0,0)(12.5,10)
1548 \includegraphics[width=1cm]{100_arrow.eps}
1550 \begin{picture}(0,0)(90,0)
1551 \includegraphics[height=0.9cm]{001_arrow.eps}
1557 \begin{minipage}{0.3cm}
1560 \begin{minipage}{6.5cm}
1563 \item Energetically most favorable path
1566 \item Activation energy: $\approx$ 0.9 eV
1567 \item Experimental values: 0.73 ... 0.87 eV
1569 $\Rightarrow$ {\color{blue}Diffusion} path identified!
1570 \item Reorientation (path 3)
1572 \item More likely composed of two consecutive steps of type 2
1573 \item Experimental values: 0.77 ... 0.88 eV
1575 $\Rightarrow$ {\color{blue}Reorientation} transition identified!
1584 Migration of the C \hkl<1 0 0> dumbbell interstitial
1591 \begin{minipage}{6.5cm}
1594 \begin{minipage}[t]{5.9cm}
1596 \includegraphics[width=5.9cm]{bc_00-1.ps}\\[2.35cm]
1599 \begin{pspicture}(0,0)(0,0)
1600 \psframe[linecolor=red,fillstyle=none](-2.8,1.35)(3.3,2.7)
1602 \begin{picture}(0,0)(60,-50)
1603 \includegraphics[width=1cm]{albe_mig/bc_00-1_red_00.eps}
1605 \begin{picture}(0,0)(5,-50)
1606 \includegraphics[width=1cm]{albe_mig/bc_00-1_red_01.eps}
1608 \begin{picture}(0,0)(-55,-50)
1609 \includegraphics[width=1cm]{albe_mig/bc_00-1_red_02.eps}
1611 \begin{picture}(0,0)(12.5,-40)
1612 \includegraphics[width=1cm]{110_arrow.eps}
1614 \begin{picture}(0,0)(90,-45)
1615 \includegraphics[height=0.9cm]{001_arrow.eps}
1617 \begin{pspicture}(0,0)(0,0)
1618 \psframe[linecolor=blue,fillstyle=none](-2.8,0)(3.3,1.6)
1620 \begin{picture}(0,0)(60,-15)
1621 \includegraphics[width=0.9cm]{albe_mig/bc_00-1_01.eps}
1623 \begin{picture}(0,0)(35,-15)
1624 \includegraphics[width=0.9cm]{albe_mig/bc_00-1_02.eps}
1626 \begin{picture}(0,0)(-5,-15)
1627 \includegraphics[width=0.9cm]{albe_mig/bc_00-1_03.eps}
1629 \begin{picture}(0,0)(-55,-15)
1630 \includegraphics[width=0.9cm]{albe_mig/bc_00-1_04.eps}
1632 \begin{picture}(0,0)(12.5,-5)
1633 \includegraphics[width=1cm]{100_arrow.eps}
1635 \begin{picture}(0,0)(90,-15)
1636 \includegraphics[height=0.9cm]{010_arrow.eps}
1642 \begin{minipage}{5.9cm}
1645 \item Lowest activation energy: $\approx$ 2.2 eV
1646 \item 2.4 times higher than VASP
1647 \item Different pathway
1652 \begin{minipage}{6.5cm}
1655 \begin{minipage}{5.9cm}
1657 %\includegraphics[width=5.9cm]{00-1_0-10.ps}\\[0.75cm]
1660 %\begin{pspicture}(0,0)(0,0)
1661 %\psframe[linecolor=red,fillstyle=none](-2.8,-0.25)(3.3,1.1)
1663 %\begin{picture}(0,0)(60,-5)
1664 %\includegraphics[width=0.9cm]{albe_mig/00-1_0-10_red_00.eps}
1666 %\begin{picture}(0,0)(0,-5)
1667 %\includegraphics[width=0.9cm]{albe_mig/00-1_0-10_red_min.eps}
1669 %\begin{picture}(0,0)(-55,-5)
1670 %\includegraphics[width=0.9cm]{albe_mig/00-1_0-10_red_03.eps}
1672 %\begin{picture}(0,0)(12.5,5)
1673 %\includegraphics[width=1cm]{100_arrow.eps}
1675 %\begin{picture}(0,0)(90,0)
1676 %\includegraphics[height=0.9cm]{001_arrow.eps}
1684 %\begin{minipage}{5.9cm}
1685 \includegraphics[width=5.9cm]{00-1_110_0-10_mig_albe.ps}
1689 \begin{minipage}{5.9cm}
1690 Transition involving \ci{} \hkl<1 1 0>
1692 \item Bond-centered configuration unstable\\
1693 $\rightarrow$ \ci{} \hkl<1 1 0> dumbbell
1694 \item Transition minima of path 2 \& 3\\
1695 $\rightarrow$ \ci{} \hkl<1 1 0> dumbbell
1696 \item Activation energy: $\approx$ 2.2 eV \& 0.9 eV
1697 \item 2.4 - 3.4 times higher than VASP
1698 \item Rotation of dumbbell orientation
1702 {\color{blue}Overestimated diffusion barrier}
1713 Combinations with a C-Si \hkl<1 0 0>-type interstitial
1723 E_{\text{f}}^{\text{defect combination}}-
1724 E_{\text{f}}^{\text{C \hkl<0 0 -1> dumbbell}}-
1725 E_{\text{f}}^{\text{2nd defect}}
1731 \begin{tabular}{l c c c c c c}
1733 $E_{\text{b}}$ [eV] & 1 & 2 & 3 & 4 & 5 & R\\
1735 \hkl<0 0 -1> & {\color{red}-0.08} & -1.15 & {\color{red}-0.08} & 0.04 & -1.66 & -0.19\\
1736 \hkl<0 0 1> & 0.34 & 0.004 & -2.05 & 0.26 & -1.53 & -0.19\\
1737 \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}\\
1738 \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}\\
1739 \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}\\
1740 \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}\\
1742 C substitutional (C$_{\text{S}}$) & 0.26 & -0.51 & -0.93 & -0.15 & 0.49 & -0.05\\
1743 Vacancy & -5.39 ($\rightarrow$ C$_{\text{S}}$) & -0.59 & -3.14 & -0.54 & -0.50 & -0.31\\
1752 \begin{minipage}[t]{3.8cm}
1753 \underline{\hkl<1 0 0> at position 1}\\[0.1cm]
1754 \includegraphics[width=3.5cm]{00-1dc/2-25.eps}
1756 \begin{minipage}[t]{3.5cm}
1757 \underline{\hkl<0 -1 0> at position 1}\\[0.1cm]
1758 \includegraphics[width=3.2cm]{00-1dc/2-39.eps}
1760 \begin{minipage}[t]{5.5cm}
1762 \item $E_{\text{b}}=0$ $\Leftrightarrow$ non-interacting defects\\
1763 $E_{\text{b}} \rightarrow 0$ for increasing distance (R)
1764 \item Stress compensation / increase
1765 \item Unfavored: antiparallel orientations
1766 \item Indication of energetically favored\\
1768 \item Most favorable: C clustering
1769 \item However: High barrier ($>4\,\text{eV}$)
1770 \item $4\times{\color{cyan}-2.25}$ versus $2\times{\color{orange}-2.39}$
1775 \begin{picture}(0,0)(-295,-130)
1776 \includegraphics[width=3.5cm]{comb_pos.eps}
1784 Combinations of C-Si \hkl<1 0 0>-type interstitials
1791 Energetically most favorable combinations along \hkl<1 1 0>
1796 \begin{tabular}{l c c c c c c}
1798 & 1 & 2 & 3 & 4 & 5 & 6\\
1800 $E_{\text{b}}$ [eV] & -2.39 & -1.88 & -0.59 & -0.31 & -0.24 & -0.21 \\
1801 C-C distance [\AA] & 1.4 & 4.6 & 6.5 & 8.6 & 10.5 & 10.8 \\
1802 Type & \hkl<-1 0 0> & \hkl<1 0 0> & \hkl<1 0 0> & \hkl<1 0 0> & \hkl<1 0 0> & \hkl<1 0 0>, \hkl<0 -1 0>\\
1809 \begin{minipage}{7.0cm}
1810 \includegraphics[width=7cm]{db_along_110_cc.ps}
1812 \begin{minipage}{6.0cm}
1814 \item Interaction proportional to reciprocal cube of C-C distance
1815 \item Saturation in the immediate vicinity
1816 \renewcommand\labelitemi{$\Rightarrow$}
1817 \item Agglomeration of \ci{} expected
1818 \item Absence of C clustering
1822 Consisten with initial precipitation model
1834 Combinations of substitutional C and \hkl<1 1 0> Si self-interstitials
1840 %\begin{minipage}{3.2cm}
1841 %\includegraphics[width=3cm]{sub_110_combo.eps}
1843 %\begin{minipage}{7.8cm}
1844 %\begin{tabular}{l c c c c c c}
1846 %C$_{\text{sub}}$ & \hkl<1 1 0> & \hkl<-1 1 0> & \hkl<0 1 1> & \hkl<0 -1 1> &
1847 % \hkl<1 0 1> & \hkl<-1 0 1> \\
1849 %1 & \RM{1} & \RM{3} & \RM{3} & \RM{1} & \RM{3} & \RM{1} \\
1850 %2 & \RM{2} & A & A & \RM{2} & C & \RM{5} \\
1851 %3 & \RM{3} & \RM{1} & \RM{3} & \RM{1} & \RM{1} & \RM{3} \\
1852 %4 & \RM{4} & B & D & E & E & D \\
1853 %5 & \RM{5} & C & A & \RM{2} & A & \RM{2} \\
1860 %\begin{tabular}{l c c c c c c c c c c}
1862 %Conf & \RM{1} & \RM{2} & \RM{3} & \RM{4} & \RM{5} & A & B & C & D & E \\
1864 %$E_{\text{f}}$ [eV]& 4.37 & 5.26 & 5.57 & 5.37 & 5.12 & 5.10 & 5.32 & 5.28 & 5.39 & 5.32 \\
1865 %$E_{\text{b}}$ [eV] & -0.97 & -0.08 & 0.22 & -0.02 & -0.23 & -0.25 & -0.02 & -0.06 & 0.05 & -0.03 \\
1866 %$r$ [nm] & 0.292 & 0.394 & 0.241 & 0.453 & 0.407 & 0.408 & 0.452 & 0.392 & 0.456 & 0.453\\
1871 \begin{minipage}{6.0cm}
1872 \includegraphics[width=5.8cm]{c_sub_si110.ps}
1874 \begin{minipage}{7cm}
1877 \item IBS: C may displace Si\\
1878 $\Rightarrow$ C$_{\text{sub}}$ + \hkl<1 1 0> Si self-interstitial
1880 \hkl<1 1 0>-type $\rightarrow$ favored combination
1881 \renewcommand\labelitemi{$\Rightarrow$}
1882 \item Most favorable: \cs{} along \hkl<1 1 0> chain \si{}
1883 \item Less favorable than C-Si \hkl<1 0 0> dumbbell
1884 \item Interaction drops quickly to zero\\
1885 $\rightarrow$ low capture radius
1889 IBS process far from equilibrium\\
1890 \cs{} \& \si{} instead of thermodynamic ground state
1895 \begin{minipage}{6.5cm}
1896 \includegraphics[width=6.0cm]{162-097.ps}
1898 \item Low migration barrier
1901 \begin{minipage}{6.5cm}
1903 Ab initio MD at \degc{900}\\
1904 \includegraphics[width=3.3cm]{md_vasp_01.eps}
1905 $t=\unit[2230]{fs}$\\
1906 \includegraphics[width=3.3cm]{md_vasp_02.eps}
1910 Contribution of entropy to structural formation
1919 Migration in C-Si \hkl<1 0 0> and vacancy combinations
1926 \begin{minipage}[t]{3cm}
1927 \underline{Pos 2, $E_{\text{b}}=-0.59\text{ eV}$}\\
1928 \includegraphics[width=2.8cm]{00-1dc/0-59.eps}
1930 \begin{minipage}[t]{7cm}
1933 Low activation energies\\
1934 High activation energies for reverse processes\\
1936 {\color{blue}C$_{\text{sub}}$ very stable}\\
1940 Without nearby \hkl<1 1 0> Si self-interstitial (IBS)\\
1942 {\color{blue}Formation of SiC by successive substitution by C}
1946 \begin{minipage}[t]{3cm}
1947 \underline{Pos 3, $E_{\text{b}}=-3.14\text{ eV}$}\\
1948 \includegraphics[width=2.8cm]{00-1dc/3-14.eps}
1953 \begin{minipage}{5.9cm}
1954 \includegraphics[width=5.9cm]{vasp_mig/comb_mig_3-2_vac_fullct.ps}\\[0.6cm]
1956 \begin{picture}(0,0)(70,0)
1957 \includegraphics[width=1.4cm]{vasp_mig/comb_2-1_init.eps}
1959 \begin{picture}(0,0)(30,0)
1960 \includegraphics[width=1.4cm]{vasp_mig/comb_2-1_seq_03.eps}
1962 \begin{picture}(0,0)(-10,0)
1963 \includegraphics[width=1.4cm]{vasp_mig/comb_2-1_seq_06.eps}
1965 \begin{picture}(0,0)(-48,0)
1966 \includegraphics[width=1.4cm]{vasp_mig/comb_2-1_final.eps}
1968 \begin{picture}(0,0)(12.5,5)
1969 \includegraphics[width=1cm]{100_arrow.eps}
1971 \begin{picture}(0,0)(97,-10)
1972 \includegraphics[height=0.9cm]{001_arrow.eps}
1978 \begin{minipage}{0.3cm}
1982 \begin{minipage}{5.9cm}
1983 \includegraphics[width=5.9cm]{vasp_mig/comb_mig_4-2_vac_fullct.ps}\\[0.1cm]
1985 \begin{picture}(0,0)(60,0)
1986 \includegraphics[width=0.9cm]{vasp_mig/comb_3-1_init.eps}
1988 \begin{picture}(0,0)(25,0)
1989 \includegraphics[width=0.9cm]{vasp_mig/comb_3-1_seq_03.eps}
1991 \begin{picture}(0,0)(-20,0)
1992 \includegraphics[width=0.9cm]{vasp_mig/comb_3-1_seq_07.eps}
1994 \begin{picture}(0,0)(-55,0)
1995 \includegraphics[width=0.9cm]{vasp_mig/comb_3-1_final.eps}
1997 \begin{picture}(0,0)(12.5,5)
1998 \includegraphics[width=1cm]{100_arrow.eps}
2000 \begin{picture}(0,0)(95,0)
2001 \includegraphics[height=0.9cm]{001_arrow.eps}
2013 Conclusion of defect / migration / combined defect simulations
2022 \item Accurately described by quantum-mechanical simulations
2023 \item Less accurate description by classical potential simulations
2024 \item Underestimated formation energy of \cs{} by classical approach
2025 \item Both methods predict same ground state: \ci{} \hkl<1 0 0> dumbbell
2030 \item C migration pathway in Si identified
2031 \item Consistent with reorientation and diffusion experiments
2034 \item Different path and ...
2035 \item overestimated barrier by classical potential calculations
2038 Concerning the precipitation mechanism
2040 \item Agglomeration of C-Si dumbbells energetically favorable
2041 (stress compensation)
2042 \item C-Si indeed favored compared to
2043 C$_{\text{sub}}$ \& \hkl<1 1 0> Si self-interstitial
2044 \item Possible low interaction capture radius of
2045 C$_{\text{sub}}$ \& \hkl<1 1 0> Si self-interstitial
2046 \item Low barrier for
2047 \ci{} \hkl<1 0 0> $\rightarrow$ \cs{} \& \si{} \hkl<1 1 0>
2048 \item In absence of nearby \hkl<1 1 0> Si self-interstitial:
2049 C-Si \hkl<1 0 0> + Vacancy $\rightarrow$ C$_{\text{sub}}$ (SiC)
2052 {\color{blue}Results suggest increased participation of \cs}
2060 Silicon carbide precipitation simulations
2066 \begin{pspicture}(0,0)(12,6.5)
2068 \rput(3.5,5.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
2071 \item Create c-Si volume
2072 \item Periodc boundary conditions
2073 \item Set requested $T$ and $p=0\text{ bar}$
2074 \item Equilibration of $E_{\text{kin}}$ and $E_{\text{pot}}$
2077 \rput(3.5,2.7){\rnode{insert}{\psframebox[fillstyle=solid,fillcolor=lachs]{
2079 Insertion of C atoms at constant T
2081 \item total simulation volume {\pnode{in1}}
2082 \item volume of minimal SiC precipitate {\pnode{in2}}
2083 \item volume consisting of Si atoms to form a minimal {\pnode{in3}}\\
2087 \rput(3.5,1){\rnode{cool}{\psframebox[fillstyle=solid,fillcolor=lbb]{
2089 Run for 100 ps followed by cooling down to $20\, ^{\circ}\textrm{C}$
2091 \ncline[]{->}{init}{insert}
2092 \ncline[]{->}{insert}{cool}
2093 \psframe[fillstyle=solid,fillcolor=white](7.5,0.7)(13.5,6.3)
2094 \rput(7.8,6){\footnotesize $V_1$}
2095 \psframe[fillstyle=solid,fillcolor=lightgray](9,2)(12,5)
2096 \rput(9.2,4.85){\tiny $V_2$}
2097 \psframe[fillstyle=solid,fillcolor=gray](9.25,2.25)(11.75,4.75)
2098 \rput(9.55,4.45){\footnotesize $V_3$}
2099 \rput(7.9,3.2){\pnode{ins1}}
2100 \rput(9.22,2.8){\pnode{ins2}}
2101 \rput(11.0,2.4){\pnode{ins3}}
2102 \ncline[]{->}{in1}{ins1}
2103 \ncline[]{->}{in2}{ins2}
2104 \ncline[]{->}{in3}{ins3}
2109 \item Restricted to classical potential simulations
2110 \item $V_2$ and $V_3$ considered due to low diffusion
2111 \item Amount of C atoms: 6000
2112 ($r_{\text{prec}}\approx 3.1\text{ nm}$, IBS: 2 ... 4 nm)
2113 \item Simulation volume: $31\times 31\times 31$ unit cells
2122 Silicon carbide precipitation simulations at $450\,^{\circ}\mathrm{C}$ as in IBS
2127 \begin{minipage}{6.5cm}
2128 \includegraphics[width=6.4cm]{sic_prec_450_si-si_c-c.ps}
2130 \begin{minipage}{6.5cm}
2131 \includegraphics[width=6.4cm]{sic_prec_450_energy.ps}
2134 \begin{minipage}{6.5cm}
2135 \includegraphics[width=6.4cm]{sic_prec_450_si-c.ps}
2137 \begin{minipage}{6.5cm}
2139 \underline{Low C concentration ($V_1$)}\\
2140 \hkl<1 0 0> C-Si dumbbell dominated structure
2142 \item Si-C bumbs around 0.19 nm
2143 \item C-C peak at 0.31 nm (as expected in 3C-SiC):\\
2144 concatenated dumbbells of various orientation
2145 \item Si-Si NN distance stretched to 0.3 nm
2147 {\color{blue}$\Rightarrow$ C atoms in proper 3C-SiC distance first}\\
2148 \underline{High C concentration ($V_2$, $V_3$)}\\
2149 High amount of strongly bound C-C bonds\\
2150 Defect density $\uparrow$ $\Rightarrow$ considerable amount of damage\\
2151 Only short range order observable\\
2152 {\color{blue}$\Rightarrow$ amorphous SiC-like phase}
2160 Silicon carbide precipitation simulations at $450\,^{\circ}\mathrm{C}$ as in IBS
2165 \begin{minipage}{6.5cm}
2166 \includegraphics[width=6.4cm]{sic_prec_450_si-si_c-c.ps}
2168 \begin{minipage}{6.5cm}
2169 \includegraphics[width=6.4cm]{sic_prec_450_energy.ps}
2172 \begin{minipage}{6.5cm}
2173 \includegraphics[width=6.4cm]{sic_prec_450_si-c.ps}
2175 \begin{minipage}{6.5cm}
2177 \underline{Low C concentration ($V_1$)}\\
2178 \hkl<1 0 0> C-Si dumbbell dominated structure
2180 \item Si-C bumbs around 0.19 nm
2181 \item C-C peak at 0.31 nm (as expected in 3C-SiC):\\
2182 concatenated dumbbells of various orientation
2183 \item Si-Si NN distance stretched to 0.3 nm
2185 {\color{blue}$\Rightarrow$ C atoms in proper 3C-SiC distance first}\\
2186 \underline{High C concentration ($V_2$, $V_3$)}\\
2187 High amount of strongly bound C-C bonds\\
2188 Defect density $\uparrow$ $\Rightarrow$ considerable amount of damage\\
2189 Only short range order observable\\
2190 {\color{blue}$\Rightarrow$ amorphous SiC-like phase}
2193 \begin{pspicture}(0,0)(0,0)
2194 \rput(6.7,5.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=white]{
2195 \begin{minipage}{10cm}
2197 {\color{red}\bf 3C-SiC formation fails to appear}
2199 \item Low C concentration simulations
2201 \item Formation of \ci{} indeed occurs
2202 \item Agllomeration not observed
2204 \item High C concentration simulations
2206 \item Amorphous SiC-like structure\\
2207 (not expected at prevailing temperatures)
2208 \item Rearrangement and transition into 3C-SiC structure missing
2220 Limitations of molecular dynamics and short range potentials
2227 \underline{Time scale problem of MD}\\[0.2cm]
2228 Minimize integration error\\
2229 $\Rightarrow$ discretization considerably smaller than
2230 reciprocal of fastest vibrational mode\\[0.1cm]
2231 Order of fastest vibrational mode: $10^{13} - 10^{14}\text{ Hz}$\\
2232 $\Rightarrow$ suitable choice of time step:
2233 $\tau=1\text{ fs}=10^{-15}\text{ s}$\\
2234 $\Rightarrow$ {\color{red}\underline{slow}} phase space propagation\\[0.1cm]
2235 Several local minima in energy surface separated by large energy barriers\\
2236 $\Rightarrow$ transition event corresponds to a multiple
2237 of vibrational periods\\
2238 $\Rightarrow$ phase transition made up of {\color{red}\underline{many}}
2239 infrequent transition events\\[0.1cm]
2240 {\color{blue}Accelerated methods:}
2241 \underline{Temperature accelerated} MD (TAD), self-guided MD \ldots
2245 \underline{Limitations related to the short range potential}\\[0.2cm]
2246 Cut-off function pushing forces and energies to zero between 1$^{\text{st}}$
2247 and 2$^{\text{nd}}$ next neighbours\\
2248 $\Rightarrow$ overestimated unphysical high forces of next neighbours
2254 Potential enhanced problem of slow phase space propagation
2259 \underline{Approach to the (twofold) problem}\\[0.2cm]
2260 Increased temperature simulations without TAD corrections\\
2261 (accelerated methods or higher time scales exclusively not sufficient)
2263 \begin{picture}(0,0)(-260,-30)
2265 \begin{minipage}{4.2cm}
2272 \item 3C-SiC also observed for higher T
2273 \item higher T inside sample
2274 \item structural evolution vs.\\
2275 equilibrium properties
2281 \begin{picture}(0,0)(-305,-155)
2283 \begin{minipage}{2.5cm}
2287 thermodynmic sampling
2298 Increased temperature simulations at low C concentration
2303 \begin{minipage}{6.5cm}
2304 \includegraphics[width=6.4cm]{tot_pc_thesis.ps}
2306 \begin{minipage}{6.5cm}
2307 \includegraphics[width=6.4cm]{tot_pc3_thesis.ps}
2310 \begin{minipage}{6.5cm}
2311 \includegraphics[width=6.4cm]{tot_pc2_thesis.ps}
2313 \begin{minipage}{6.5cm}
2315 \underline{Si-C bonds:}
2317 \item Vanishing cut-off artifact (above $1650\,^{\circ}\mathrm{C}$)
2318 \item Structural change: C-Si \hkl<1 0 0> $\rightarrow$ C$_{\text{sub}}$
2320 \underline{Si-Si bonds:}
2321 {\color{blue}Si-C$_{\text{sub}}$-Si} along \hkl<1 1 0>
2322 ($\rightarrow$ 0.325 nm)\\[0.1cm]
2323 \underline{C-C bonds:}
2325 \item C-C next neighbour pairs reduced (mandatory)
2326 \item Peak at 0.3 nm slightly shifted
2328 \item C-Si \hkl<1 0 0> combinations (dashed arrows)\\
2329 $\rightarrow$ C-Si \hkl<1 0 0> \& C$_{\text{sub}}$
2331 $\rightarrow$ pure {\color{blue}C$_{\text{sub}}$ combinations}
2333 \item Range [|-$\downarrow$]:
2334 {\color{blue}C$_{\text{sub}}$ \& C$_{\text{sub}}$
2335 with nearby Si$_{\text{I}}$}
2340 \begin{picture}(0,0)(-330,-74)
2343 \begin{minipage}{1.6cm}
2346 stretched SiC\\[-0.1cm]
2358 Increased temperature simulations at low C concentration
2363 \begin{minipage}{6.5cm}
2364 \includegraphics[width=6.4cm]{tot_pc_thesis.ps}
2366 \begin{minipage}{6.5cm}
2367 \includegraphics[width=6.4cm]{tot_pc3_thesis.ps}
2370 \begin{minipage}{6.5cm}
2371 \includegraphics[width=6.4cm]{tot_pc2_thesis.ps}
2373 \begin{minipage}{6.5cm}
2375 \underline{Si-C bonds:}
2377 \item Vanishing cut-off artifact (above $1650\,^{\circ}\mathrm{C}$)
2378 \item Structural change: C-Si \hkl<1 0 0> $\rightarrow$ C$_{\text{sub}}$
2380 \underline{Si-Si bonds:}
2381 {\color{blue}Si-C$_{\text{sub}}$-Si} along \hkl<1 1 0>
2382 ($\rightarrow$ 0.325 nm)\\[0.1cm]
2383 \underline{C-C bonds:}
2385 \item C-C next neighbour pairs reduced (mandatory)
2386 \item Peak at 0.3 nm slightly shifted
2388 \item C-Si \hkl<1 0 0> combinations (dashed arrows)\\
2389 $\rightarrow$ C-Si \hkl<1 0 0> \& C$_{\text{sub}}$
2391 $\rightarrow$ pure {\color{blue}C$_{\text{sub}}$ combinations}
2393 \item Range [|-$\downarrow$]:
2394 {\color{blue}C$_{\text{sub}}$ \& C$_{\text{sub}}$
2395 with nearby Si$_{\text{I}}$}
2400 %\begin{picture}(0,0)(-330,-74)
2403 %\begin{minipage}{1.6cm}
2406 %stretched SiC\\[-0.1cm]
2413 \begin{pspicture}(0,0)(0,0)
2414 \rput(6.7,5.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=white]{
2415 \begin{minipage}{10cm}
2417 {\color{blue}\bf Stretched SiC in c-Si}
2419 \item Consistent to precipitation model involving \cs{}
2420 \item Explains annealing behavior of high/low T C implants
2422 \item Low T: highly mobiel \ci{}
2423 \item High T: stable configurations of \cs{}
2426 $\Rightarrow$ High T $\leftrightarrow$ IBS conditions far from equilibrium\\
2427 $\Rightarrow$ Precipitation mechanism involving \cs{}
2437 Increased temperature simulations at high C concentration
2442 \begin{minipage}{6.5cm}
2443 \includegraphics[width=6.4cm]{12_pc_thesis.ps}
2445 \begin{minipage}{6.5cm}
2446 \includegraphics[width=6.4cm]{12_pc_c_thesis.ps}
2454 \begin{minipage}[t]{6.0cm}
2455 0.186 nm: Si-C pairs $\uparrow$\\
2456 (as expected in 3C-SiC)\\[0.2cm]
2457 0.282 nm: Si-C-C\\[0.2cm]
2458 $\approx$0.35 nm: C-Si-Si
2461 \begin{minipage}{0.2cm}
2465 \begin{minipage}[t]{6.0cm}
2466 0.15 nm: C-C pairs $\uparrow$\\
2467 (as expected in graphite/diamond)\\[0.2cm]
2468 0.252 nm: C-C-C (2$^{\text{nd}}$ NN for diamond)\\[0.2cm]
2469 0.31 nm: shifted towards 0.317 nm $\rightarrow$ C-Si-C
2474 \item Decreasing cut-off artifact
2475 \item {\color{red}Amorphous} SiC-like phase remains
2476 \item High amount of {\color{red}damage} \& alignement to c-Si host matrix lost
2477 \item Slightly sharper peaks $\Rightarrow$ indicate slight {\color{blue}acceleration of dynamics} due to temperature
2486 High C \& small $V$ \& short $t$
2489 Slow restructuring due to strong C-C bonds
2492 High C \& low T implants
2503 Summary and Conclusions
2511 \begin{minipage}[t]{12.9cm}
2512 \underline{Pecipitation simulations}
2514 \item High C concentration $\rightarrow$ amorphous SiC like phase
2515 \item Problem of potential enhanced slow phase space propagation
2516 \item Low T $\rightarrow$ C-Si \hkl<1 0 0> dumbbell dominated structure
2517 \item High T $\rightarrow$ C$_{\text{sub}}$ dominated structure
2518 \item High T necessary to simulate IBS conditions (far from equilibrium)
2519 \item Precipitation by successive agglomeration of \cs (epitaxy)
2520 \item \si{}: vehicle to form \cs{} \& supply of Si \& stress compensation
2521 (stretched SiC, interface)
2529 \begin{minipage}{12.9cm}
2534 \item Point defects excellently / fairly well described
2536 \item C$_{\text{sub}}$ drastically underestimated by EA
2537 \item EA predicts correct ground state:
2538 C$_{\text{sub}}$ \& \si{} $>$ \ci{}
2539 \item Identified migration path explaining
2540 diffusion and reorientation experiments by DFT
2541 \item EA fails to describe \ci{} migration:
2542 Wrong path \& overestimated barrier
2544 \item Combinations of defects
2546 \item Agglomeration of point defects energetically favorable
2547 by compensation of stress
2548 \item Formation of C-C unlikely
2549 \item C$_{\text{sub}}$ favored conditions (conceivable in IBS)
2550 \item \ci{} \hkl<1 0 0> $\leftrightarrow$ \cs{} \& \si{} \hkl<1 1 0>\\
2551 Low barrier (\unit[0.77]{eV}) \& low capture radius
2559 \framebox{Precipitation by successive agglomeration of \cs{}}
2577 \underline{Augsburg}
2579 \item Prof. B. Stritzker (accomodation at EP \RM{4})
2580 \item Ralf Utermann (EDV)
2583 \underline{Helsinki}
2585 \item Prof. K. Nordlund (MD)
2590 \item Bayerische Forschungsstiftung (financial support)
2593 \underline{Paderborn}
2595 \item Prof. J. Lindner (SiC)
2596 \item Prof. G. Schmidt (DFT + financial support)
2597 \item Dr. E. Rauls (DFT + SiC)
2598 \item Dr. S. Sanna (VASP)
2605 \bf Thank you for your attention!