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3 \documentclass[landscape,semhelv]{seminar}
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8 \usepackage[T1]{fontenc}
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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}
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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
153 % Phase diagram of the C/Si system\\
158 \begin{minipage}{6.5cm}
159 \includegraphics[width=6.5cm]{si-c_phase.eps}
162 R. I. Scace and G. A. Slack, J. Chem. Phys. 30, 1551 (1959)
165 \begin{pspicture}(0,0)(0,0)
166 \psellipse[linecolor=blue,linewidth=0.1cm](3.55,4.0)(0.5,2.9)
169 \begin{minipage}{6cm}
170 {\bf Phase diagram of the C/Si system}\\[0.2cm]
171 {\color{blue}Stoichiometric composition}
173 \item only chemical stable compound
174 \item wide band gap semiconductor\\
175 \underline{silicon carbide}, SiC
181 % motivation / properties / applications of silicon carbide
187 \begin{pspicture}(0,0)(13.5,5)
189 \psframe*[linecolor=hb](-0.2,0)(12.9,5)
191 \pspolygon[linecolor=hlbb,fillcolor=hlbb,fillstyle=solid](5.2,1)(6.5,1)(6.5,3)(5.2,3)
192 \pspolygon[linecolor=hlbb,fillcolor=hlbb,fillstyle=solid](6.4,0.5)(7.7,2)(7.7,2)(6.4,3.5)
194 \rput[lt](0,4.6){\color{gray}PROPERTIES}
196 \rput[lt](0.3,4){wide band gap}
197 \rput[lt](0.3,3.5){high electric breakdown field}
198 \rput[lt](0.3,3){good electron mobility}
199 \rput[lt](0.3,2.5){high electron saturation drift velocity}
200 \rput[lt](0.3,2){high thermal conductivity}
202 \rput[lt](0.3,1.5){hard and mechanically stable}
203 \rput[lt](0.3,1){chemically inert}
205 \rput[lt](0.3,0.5){radiation hardness}
207 \rput[rt](12.7,4.6){\color{gray}APPLICATIONS}
209 \rput[rt](12.5,3.85){high-temperature, high power}
210 \rput[rt](12.5,3.5){and high-frequency}
211 \rput[rt](12.5,3.15){electronic and optoelectronic devices}
213 \rput[rt](12.5,2.35){material suitable for extreme conditions}
214 \rput[rt](12.5,2){microelectromechanical systems}
215 \rput[rt](12.5,1.65){abrasives, cutting tools, heating elements}
217 \rput[rt](12.5,0.85){first wall reactor material, detectors}
218 \rput[rt](12.5,0.5){and electronic devices for space}
222 \begin{picture}(0,0)(5,-162)
223 \includegraphics[height=2.2cm]{3C_SiC_bs.eps}
225 \begin{picture}(0,0)(-120,-162)
226 \includegraphics[height=2.2cm]{nasa_600c_led.eps}
228 \begin{picture}(0,0)(-270,-162)
229 \includegraphics[height=2.2cm]{6h-sic_3c-sic.eps}
232 \begin{picture}(0,0)(10,65)
233 \includegraphics[height=2.8cm]{sic_switch.eps}
235 %\begin{picture}(0,0)(-243,65)
236 \begin{picture}(0,0)(-110,65)
237 \includegraphics[height=2.8cm]{ise_99.eps}
239 %\begin{picture}(0,0)(-135,65)
240 \begin{picture}(0,0)(-100,65)
241 \includegraphics[height=1.2cm]{infineon_schottky.eps}
243 \begin{picture}(0,0)(-233,65)
244 \includegraphics[height=2.8cm]{solar_car.eps}
254 Polytypes of SiC\\[0.4cm]
257 \includegraphics[width=3.8cm]{cubic_hex.eps}\\
258 \begin{minipage}{1.9cm}
259 {\tiny cubic (twist)}
261 \begin{minipage}{2.9cm}
262 {\tiny hexagonal (no twist)}
265 \begin{picture}(0,0)(-150,0)
266 \includegraphics[width=7cm]{polytypes.eps}
273 \begin{tabular}{l c c c c c c}
275 & 3C-SiC & 4H-SiC & 6H-SiC & Si & GaN & Diamond\\
277 Hardness [Mohs] & \multicolumn{3}{c}{------ 9.6 ------}& 6.5 & - & 10 \\
278 Band gap [eV] & 2.36 & 3.23 & 3.03 & 1.12 & 3.39 & 5.5 \\
279 Break down field [$10^6$ V/cm] & 4 & 3 & 3.2 & 0.6 & 5 & 10 \\
280 Saturation drift velocity [$10^7$ cm/s] & 2.5 & 2.0 & 2.0 & 1 & 2.7 & 2.7 \\
281 Electron mobility [cm$^2$/Vs] & 800 & 900 & 400 & 1100 & 900 & 2200 \\
282 Hole mobility [cm$^2$/Vs] & 320 & 120 & 90 & 420 & 150 & 1600 \\
283 Thermal conductivity [W/cmK] & 5.0 & 4.9 & 4.9 & 1.5 & 1.3 & 22 \\
287 \begin{pspicture}(0,0)(0,0)
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290 \begin{pspicture}(0,0)(0,0)
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293 \begin{pspicture}(0,0)(0,0)
294 \psellipse[linecolor=red](10.45,0.45)(0.4,0.2)
305 Fabrication of silicon carbide
314 \emph{Silicon carbide --- Born from the stars, perfected on earth.}
320 SiC thin film by MBE \& CVD
322 \item Much progress achieved in homo/heteroepitaxial SiC thin film growth
323 \item \underline{Commercially available} semiconductor power devices based on
324 \underline{\foreignlanguage{greek}{a}-SiC}
325 \item Production of favored \underline{3C-SiC} material
326 \underline{less advanced}
327 \item Quality and size not yet sufficient
329 \begin{picture}(0,0)(-310,-20)
330 \includegraphics[width=2.0cm]{cree.eps}
335 Alternative approach:
336 Ion beam synthesis (IBS) of burried 3C-SiC layers in Si\hkl(1 0 0)
340 \begin{minipage}{6.5cm}
342 \item \underline{Implantation step 1}\\
343 180 keV C$^+$, $D=7.9\times 10^{17}$ cm$^{-2}$, $T_{\text{i}}=500\,^{\circ}\mathrm{C}$\\[0.1cm]
344 Box-like distribution of equally sized \&\\
345 epitaxially oriented SiC precipitates
347 \item \underline{Implantation step 2}\\
348 180 keV C$^+$, $D=0.6\times 10^{17}$ cm$^{-2}$, $T_{\text{i}}=250\,^{\circ}\mathrm{C}$\\[0.1cm]
349 Destruction of SiC nanocrystals\\
350 in growing amorphous interface layers
351 \item \underline{Annealing}\\
352 $T=1250\,^{\circ}\mathrm{C}$, $t=10\,\text{h}$\\[0.1cm]
353 Homogeneous, stoichiometric SiC layer\\
354 with sharp interfaces
357 \begin{minipage}{0.3cm}
360 \begin{minipage}{5.5cm}
361 \includegraphics[width=5.8cm]{ibs_3c-sic.eps}\\[-0.2cm]
364 XTEM: single crystalline 3C-SiC in Si\hkl(1 0 0)
370 \begin{minipage}{6.3cm}
373 Precipitation mechanism not yet fully understood!
375 \renewcommand\labelitemi{$\Rightarrow$}
377 \underline{Understanding the SiC precipitation}
379 \item significant technological progress in SiC thin film formation
380 \item perspectives for processes relying upon prevention of SiC precipitation
397 \item Implantation of C in Si --- Overview of experimental observations
398 \item Utilized simulation techniques and modeled problems
400 \item {\color{blue}Diploma thesis}\\
401 \underline{Monte Carlo} simulations
402 modeling the selforganization process
403 leading to periodic arrays of nanometric amorphous SiC
405 \item {\color{blue}Doctoral studies}\\
406 Classical potential \underline{molecular dynamics} simulations
408 \underline{Density functional theory} calculations
410 \ldots on defects and SiC precipitation in Si
412 \item Summary / Conclusion / Outlook
425 Supposed precipitation mechanism of SiC in Si
432 \begin{minipage}{3.8cm}
433 Si \& SiC lattice structure\\[0.2cm]
434 \includegraphics[width=3.5cm]{sic_unit_cell.eps}\\[-0.3cm]
438 \begin{minipage}{3.8cm}
440 \includegraphics[width=3.3cm]{tem_c-si-db.eps}
444 \begin{minipage}{3.8cm}
446 \includegraphics[width=3.3cm]{tem_3c-sic.eps}
450 \begin{minipage}{4cm}
452 C-Si dimers (dumbbells)\\[-0.1cm]
453 on Si interstitial sites
457 \begin{minipage}{4.2cm}
459 Agglomeration of C-Si dumbbells\\[-0.1cm]
460 $\Rightarrow$ dark contrasts
464 \begin{minipage}{4cm}
466 Precipitation of 3C-SiC in Si\\[-0.1cm]
467 $\Rightarrow$ Moir\'e fringes\\[-0.1cm]
468 \& release of Si self-interstitials
472 \begin{minipage}{3.8cm}
474 \includegraphics[width=3.3cm]{sic_prec_seq_01.eps}
478 \begin{minipage}{3.8cm}
480 \includegraphics[width=3.3cm]{sic_prec_seq_02.eps}
484 \begin{minipage}{3.8cm}
486 \includegraphics[width=3.3cm]{sic_prec_seq_03.eps}
490 \begin{pspicture}(0,0)(0,0)
491 \psline[linewidth=4pt]{->}(8.5,2)(9.0,2)
492 \psellipse[linecolor=blue](11.5,5.8)(0.3,0.5)
493 \rput{-20}{\psellipse[linecolor=blue](3.3,8.1)(0.3,0.5)}
494 \psline[linewidth=4pt]{->}(4.0,2)(4.5,2)
495 \rput(12.7,0.3){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
496 $4a_{\text{Si}}=5a_{\text{SiC}}$
498 \rput(12.2,8){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
499 \hkl(h k l) planes match
501 \rput(9.7,6.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
511 Supposed precipitation mechanism of SiC in Si
518 \begin{minipage}{3.8cm}
519 Si \& SiC lattice structure\\[0.2cm]
520 \includegraphics[width=3.5cm]{sic_unit_cell.eps}\\[-0.3cm]
524 \begin{minipage}{3.8cm}
526 \includegraphics[width=3.3cm]{tem_c-si-db.eps}
530 \begin{minipage}{3.8cm}
532 \includegraphics[width=3.3cm]{tem_3c-sic.eps}
536 \begin{minipage}{4cm}
538 C-Si dimers (dumbbells)\\[-0.1cm]
539 on Si interstitial sites
543 \begin{minipage}{4.2cm}
545 Agglomeration of C-Si dumbbells\\[-0.1cm]
546 $\Rightarrow$ dark contrasts
550 \begin{minipage}{4cm}
552 Precipitation of 3C-SiC in Si\\[-0.1cm]
553 $\Rightarrow$ Moir\'e fringes\\[-0.1cm]
554 \& release of Si self-interstitials
558 \begin{minipage}{3.8cm}
560 \includegraphics[width=3.3cm]{sic_prec_seq_01.eps}
564 \begin{minipage}{3.8cm}
566 \includegraphics[width=3.3cm]{sic_prec_seq_02.eps}
570 \begin{minipage}{3.8cm}
572 \includegraphics[width=3.3cm]{sic_prec_seq_03.eps}
576 \begin{pspicture}(0,0)(0,0)
577 \psline[linewidth=4pt]{->}(8.5,2)(9.0,2)
578 \psellipse[linecolor=blue](11.5,5.8)(0.3,0.5)
579 \rput{-20}{\psellipse[linecolor=blue](3.3,8.1)(0.3,0.5)}
580 \psline[linewidth=4pt]{->}(4.0,2)(4.5,2)
581 \rput(12.7,0.3){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
582 $4a_{\text{Si}}=5a_{\text{SiC}}$
584 \rput(12.2,8){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
585 \hkl(h k l) planes match
587 \rput(9.7,6.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
590 \rput(6.7,5.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=white]{
591 \begin{minipage}{10cm}
593 {\color{red}\bf Controversial views}
595 \item Implantations at high T (Nejim et al.)
597 \item Topotactic transformation based on \cs
598 \item \si{} as supply reacting with further C in cleared volume
600 \item Annealing behavior (Serre et al.)
602 \item Room temperature implants $\rightarrow$ highly mobile C
603 \item Elevated T implants $\rightarrow$ no/low C redistribution/migration\\
604 (indicate stable \cs{} configurations)
606 \item Strained silicon \& Si/SiC heterostructures
608 \item Coherent SiC precipitates (tensile strain)
609 \item Incoherent SiC (strain relaxation)
621 Molecular dynamics (MD) simulations
630 \item Microscopic description of N particle system
631 \item Analytical interaction potential
632 \item Numerical integration using Newtons equation of motion\\
633 as a propagation rule in 6N-dimensional phase space
634 \item Observables obtained by time and/or ensemble averages
636 {\bf Details of the simulation:}
638 \item Integration: Velocity Verlet, timestep: $1\text{ fs}$
639 \item Ensemble: NpT (isothermal-isobaric)
641 \item Berendsen thermostat:
642 $\tau_{\text{T}}=100\text{ fs}$
643 \item Berendsen barostat:\\
644 $\tau_{\text{P}}=100\text{ fs}$,
645 $\beta^{-1}=100\text{ GPa}$
647 \item Erhart/Albe potential: Tersoff-like bond order potential
650 E = \frac{1}{2} \sum_{i \neq j} \pot_{ij}, \quad
651 \pot_{ij} = {\color{red}f_C(r_{ij})}
652 \left[ f_R(r_{ij}) + {\color{blue}b_{ij}} f_A(r_{ij}) \right]
656 \begin{picture}(0,0)(-230,-30)
657 \includegraphics[width=5cm]{tersoff_angle.eps}
665 Density functional theory (DFT) calculations
670 Basic ingredients necessary for DFT
673 \item \underline{Hohenberg-Kohn theorem} - ground state density $n_0(r)$ ...
675 \item ... uniquely determines the ground state potential
677 \item ... minimizes the systems total energy
679 \item \underline{Born-Oppenheimer}
680 - $N$ moving electrons in an external potential of static nuclei
682 H\Psi = \left[-\sum_i^N \frac{\hbar^2}{2m}\nabla_i^2
683 +\sum_i^N V_{\text{ext}}(r_i)
684 +\sum_{i<j}^N V_{e-e}(r_i,r_j)\right]\Psi=E\Psi
686 \item \underline{Effective potential}
687 - averaged electrostatic potential \& exchange and correlation
689 V_{\text{eff}}(r)=V_{\text{ext}}(r)+\int\frac{e^2 n(r')}{|r-r'|}d^3r'
692 \item \underline{Kohn-Sham system}
693 - Schr\"odinger equation of N non-interacting particles
695 \left[ -\frac{\hbar^2}{2m}\nabla^2 + V_{\text{eff}}(r) \right] \Phi_i(r)
700 n(r)=\sum_i^N|\Phi_i(r)|^2
702 \item \underline{Self-consistent solution}\\
703 $n(r)$ depends on $\Phi_i$, which depend on $V_{\text{eff}}$,
704 which in turn depends on $n(r)$
705 \item \underline{Variational principle}
706 - minimize total energy with respect to $n(r)$
714 Density functional theory (DFT) calculations
721 Details of applied DFT calculations in this work
724 \item \underline{Exchange correlation functional}
725 - approximations for the inhomogeneous electron gas
727 \item LDA: $E_{\text{XC}}^{\text{LDA}}[n]=\int \epsilon_{\text{XC}}(n)n(r)d^3r$
728 \item GGA: $E_{\text{XC}}^{\text{GGA}}[n]=\int \epsilon_{\text{XC}}(n,\nabla n)n(r)d^3r$
730 \item \underline{Plane wave basis set}
731 - approximation of the wavefunction $\Phi_i$ by plane waves $\phi_j$
734 \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}}
735 \qquad ({\color{blue}300\text{ eV}})
737 \item \underline{Brillouin zone sampling} -
738 {\color{blue}$\Gamma$-point only} calculations
739 \item \underline{Pseudo potential}
740 - consider only the valence electrons
741 \item \underline{Code} - VASP 4.6
746 MD and structural optimization
749 \item MD integration: Gear predictor corrector algorithm
750 \item Pressure control: Parrinello-Rahman pressure control
751 \item Structural optimization: Conjugate gradient method
754 \begin{pspicture}(0,0)(0,0)
755 \psellipse[linecolor=blue](1.5,6.75)(0.5,0.3)
763 C and Si self-interstitial point defects in silicon
770 \begin{minipage}{8cm}
772 \begin{pspicture}(0,0)(7,5)
773 \rput(3.5,4){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
776 \item Creation of c-Si simulation volume
777 \item Periodic boundary conditions
778 \item $T=0\text{ K}$, $p=0\text{ bar}$
781 \rput(3.5,2.1){\rnode{insert}{\psframebox{
784 Insertion of interstitial C/Si atoms
787 \rput(3.5,1){\rnode{cool}{\psframebox[fillstyle=solid,fillcolor=lbb]{
790 Relaxation / structural energy minimization
793 \ncline[]{->}{init}{insert}
794 \ncline[]{->}{insert}{cool}
797 \begin{minipage}{5cm}
798 \includegraphics[width=5cm]{unit_cell_e.eps}\\
801 \begin{minipage}{9cm}
802 \begin{tabular}{l c c}
804 & size [unit cells] & \# atoms\\
806 VASP & $3\times 3\times 3$ & $216\pm 1$ \\
807 Erhart/Albe & $9\times 9\times 9$ & $5832\pm 1$\\
811 \begin{minipage}{4cm}
812 {\color{red}$\bullet$} Tetrahedral\\
813 {\color{green}$\bullet$} Hexagonal\\
814 {\color{yellow}$\bullet$} \hkl<1 0 0> dumbbell\\
815 {\color{magenta}$\bullet$} \hkl<1 1 0> dumbbell\\
816 {\color{cyan}$\bullet$} Bond-centered\\
817 {\color{black}$\bullet$} Vacancy / Substitutional
826 \begin{minipage}{9.5cm}
829 Si self-interstitial point defects in silicon\\
832 \begin{tabular}{l c c c c c}
834 $E_{\text{f}}$ [eV] & \hkl<1 1 0> DB & H & T & \hkl<1 0 0> DB & V \\
836 VASP & \underline{3.39} & 3.42 & 3.77 & 4.41 & 3.63 \\
837 Erhart/Albe & 4.39 & 4.48$^*$ & \underline{3.40} & 5.42 & 3.13 \\
839 \end{tabular}\\[0.2cm]
841 \begin{minipage}{4.7cm}
842 \includegraphics[width=4.7cm]{e_kin_si_hex.ps}
844 \begin{minipage}{4.7cm}
846 {\tiny nearly T $\rightarrow$ T}\\
848 \includegraphics[width=4.7cm]{nhex_tet.ps}
851 \underline{Hexagonal} \hspace{2pt}
852 \href{../video/si_self_int_hexa.avi}{$\rhd$}\\[0.1cm]
854 \begin{minipage}{2.7cm}
855 $E_{\text{f}}^*=4.48\text{ eV}$\\
856 \includegraphics[width=2.7cm]{si_pd_albe/hex_a.eps}
858 \begin{minipage}{0.4cm}
863 \begin{minipage}{2.7cm}
864 $E_{\text{f}}=3.96\text{ eV}$\\
865 \includegraphics[width=2.8cm]{si_pd_albe/hex.eps}
868 \begin{minipage}{2.9cm}
870 \underline{Vacancy}\\
871 \includegraphics[width=3.0cm]{si_pd_albe/vac.eps}
876 \begin{minipage}{3.5cm}
879 \underline{\hkl<1 1 0> dumbbell}\\
880 \includegraphics[width=3.0cm]{si_pd_albe/110.eps}\\
881 \underline{Tetrahedral}\\
882 \includegraphics[width=3.0cm]{si_pd_albe/tet.eps}\\
883 \underline{\hkl<1 0 0> dumbbell}\\
884 \includegraphics[width=3.0cm]{si_pd_albe/100.eps}
896 C interstitial point defects in silicon\\[-0.1cm]
899 \begin{tabular}{l c c c c c c r}
901 $E_{\text{f}}$ & T & H & \hkl<1 0 0> DB & \hkl<1 1 0> DB & S & B & \cs{} \& \si\\
903 VASP & unstable & unstable & \underline{3.72} & 4.16 & 1.95 & 4.66 & {\color{green}4.17}\\
904 Erhart/Albe MD & 6.09 & 9.05$^*$ & \underline{3.88} & 5.18 & {\color{red}0.75} & 5.59$^*$ & {\color{green}4.43} \\
906 \end{tabular}\\[0.1cm]
909 \begin{minipage}{2.7cm}
910 \underline{Hexagonal} \hspace{2pt}
911 \href{../video/c_in_si_int_hexa.avi}{$\rhd$}\\
912 $E_{\text{f}}^*=9.05\text{ eV}$\\
913 \includegraphics[width=2.7cm]{c_pd_albe/hex.eps}
915 \begin{minipage}{0.4cm}
920 \begin{minipage}{2.7cm}
921 \underline{\hkl<1 0 0>}\\
922 $E_{\text{f}}=3.88\text{ eV}$\\
923 \includegraphics[width=2.7cm]{c_pd_albe/100.eps}
926 \begin{minipage}{2cm}
929 \begin{minipage}{3cm}
931 \underline{Tetrahedral}\\
932 \includegraphics[width=3.0cm]{c_pd_albe/tet.eps}
937 \begin{minipage}{2.7cm}
938 \underline{Bond-centered}\\
939 $E_{\text{f}}^*=5.59\text{ eV}$\\
940 \includegraphics[width=2.7cm]{c_pd_albe/bc.eps}
942 \begin{minipage}{0.4cm}
947 \begin{minipage}{2.7cm}
948 \underline{\hkl<1 1 0> dumbbell}\\
949 $E_{\text{f}}=5.18\text{ eV}$\\
950 \includegraphics[width=2.7cm]{c_pd_albe/110.eps}
953 \begin{minipage}{2cm}
956 \begin{minipage}{3cm}
958 \underline{Substitutional}\\
959 \includegraphics[width=3.0cm]{c_pd_albe/sub.eps}
970 C \hkl<1 0 0> dumbbell interstitial configuration\\
974 \begin{tabular}{l c c c c c c c c}
976 Distances [nm] & $r(1C)$ & $r(2C)$ & $r(3C)$ & $r(12)$ & $r(13)$ & $r(34)$ & $r(23)$ & $r(25)$ \\
978 Erhart/Albe & 0.175 & 0.329 & 0.186 & 0.226 & 0.300 & 0.343 & 0.423 & 0.425 \\
979 VASP & 0.174 & 0.341 & 0.182 & 0.229 & 0.286 & 0.347 & 0.422 & 0.417 \\
981 \end{tabular}\\[0.2cm]
982 \begin{tabular}{l c c c c }
984 Angles [$^{\circ}$] & $\theta_1$ & $\theta_2$ & $\theta_3$ & $\theta_4$ \\
986 Erhart/Albe & 140.2 & 109.9 & 134.4 & 112.8 \\
987 VASP & 130.7 & 114.4 & 146.0 & 107.0 \\
989 \end{tabular}\\[0.2cm]
990 \begin{tabular}{l c c c}
992 Displacements [nm]& $a$ & $b$ & $|a|+|b|$ \\
994 Erhart/Albe & 0.084 & -0.091 & 0.175 \\
995 VASP & 0.109 & -0.065 & 0.174 \\
997 \end{tabular}\\[0.6cm]
1000 \begin{minipage}{3.0cm}
1002 \underline{Erhart/Albe}
1003 \includegraphics[width=3.0cm]{c_pd_albe/100_cmp.eps}
1006 \begin{minipage}{3.0cm}
1009 \includegraphics[width=3.0cm]{c_pd_vasp/100_cmp.eps}
1013 \begin{picture}(0,0)(-185,10)
1014 \includegraphics[width=6.8cm]{100-c-si-db_cmp.eps}
1016 \begin{picture}(0,0)(-280,-150)
1017 \includegraphics[width=3.3cm]{c_pd_vasp/eden.eps}
1020 \begin{pspicture}(0,0)(0,0)
1021 \psellipse[linecolor=green](5.18,5.92)(0.5,0.3)
1022 \psellipse[linecolor=red](3.45,5.92)(1.0,0.4)
1023 \psellipse[linecolor=blue](2.7,6.92)(0.9,0.2)
1024 \psellipse[linecolor=blue](4.65,6.92)(0.9,0.2)
1033 \begin{minipage}{8.5cm}
1036 Bond-centered interstitial configuration\\[-0.1cm]
1039 \begin{minipage}{3.0cm}
1040 \includegraphics[width=2.8cm]{c_pd_vasp/bc_2333.eps}\\
1042 \begin{minipage}{5.2cm}
1044 \item Linear Si-C-Si bond
1045 \item Si: one C \& 3 Si neighbours
1046 \item Spin polarized calculations
1047 \item No saddle point!\\
1054 \begin{minipage}[t]{6.5cm}
1055 \begin{minipage}[t]{1.2cm}
1057 {\tiny sp$^3$}\\[0.8cm]
1058 \underline{${\color{black}\uparrow}$}
1059 \underline{${\color{black}\uparrow}$}
1060 \underline{${\color{black}\uparrow}$}
1061 \underline{${\color{red}\uparrow}$}\\
1064 \begin{minipage}[t]{1.4cm}
1066 {\color{red}M}{\color{blue}O}\\[0.8cm]
1067 \underline{${\color{blue}\uparrow}{\color{white}\downarrow}$}\\
1068 $\sigma_{\text{ab}}$\\[0.5cm]
1069 \underline{${\color{red}\uparrow}{\color{blue}\downarrow}$}\\
1073 \begin{minipage}[t]{1.0cm}
1077 \underline{${\color{white}\uparrow\uparrow}$}
1078 \underline{${\color{white}\uparrow\uparrow}$}\\
1080 \underline{${\color{blue}\uparrow}{\color{blue}\downarrow}$}
1081 \underline{${\color{blue}\uparrow}{\color{blue}\downarrow}$}\\
1085 \begin{minipage}[t]{1.4cm}
1087 {\color{blue}M}{\color{green}O}\\[0.8cm]
1088 \underline{${\color{blue}\uparrow}{\color{white}\downarrow}$}\\
1089 $\sigma_{\text{ab}}$\\[0.5cm]
1090 \underline{${\color{green}\uparrow}{\color{blue}\downarrow}$}\\
1094 \begin{minipage}[t]{1.2cm}
1097 {\tiny sp$^3$}\\[0.8cm]
1098 \underline{${\color{green}\uparrow}$}
1099 \underline{${\color{black}\uparrow}$}
1100 \underline{${\color{black}\uparrow}$}
1101 \underline{${\color{black}\uparrow}$}\\
1109 \begin{minipage}{4.5cm}
1110 \includegraphics[width=4cm]{c_100_mig_vasp/im_spin_diff.eps}
1112 \begin{minipage}{3.5cm}
1113 {\color{gray}$\bullet$} Spin up\\
1114 {\color{green}$\bullet$} Spin down\\
1115 {\color{blue}$\bullet$} Resulting spin up\\
1116 {\color{yellow}$\bullet$} Si atoms\\
1117 {\color{red}$\bullet$} C atom
1122 \begin{minipage}{4.2cm}
1124 \includegraphics[width=4.3cm]{c_pd_vasp/bc_2333_ksl.ps}\\
1125 {\color{green}$\Box$} {\tiny unoccupied}\\
1126 {\color{red}$\bullet$} {\tiny occupied}
1135 Migration of the C \hkl<1 0 0> dumbbell interstitial
1140 {\small Investigated pathways}
1142 \begin{minipage}{8.5cm}
1143 \begin{minipage}{8.3cm}
1144 \underline{\hkl<0 0 -1> $\rightarrow$ \hkl<0 0 1>}\\
1145 \begin{minipage}{2.4cm}
1146 \includegraphics[width=2.4cm]{c_pd_vasp/100_2333.eps}
1148 \begin{minipage}{0.4cm}
1151 \begin{minipage}{2.4cm}
1152 \includegraphics[width=2.4cm]{c_pd_vasp/bc_2333.eps}
1154 \begin{minipage}{0.4cm}
1157 \begin{minipage}{2.4cm}
1158 \includegraphics[width=2.4cm]{c_pd_vasp/100_next_2333.eps}
1161 \begin{minipage}{8.3cm}
1162 \underline{\hkl<0 0 -1> $\rightarrow$ \hkl<0 -1 0>}\\
1163 \begin{minipage}{2.4cm}
1164 \includegraphics[width=2.4cm]{c_pd_vasp/100_2333.eps}
1166 \begin{minipage}{0.4cm}
1169 \begin{minipage}{2.4cm}
1170 \includegraphics[width=2.4cm]{c_pd_vasp/00-1-0-10_2333.eps}
1172 \begin{minipage}{0.4cm}
1175 \begin{minipage}{2.4cm}
1176 \includegraphics[width=2.4cm]{c_pd_vasp/0-10_2333.eps}
1179 \begin{minipage}{8.3cm}
1180 \underline{\hkl<0 0 -1> $\rightarrow$ \hkl<0 -1 0> (in place)}\\
1181 \begin{minipage}{2.4cm}
1182 \includegraphics[width=2.4cm]{c_pd_vasp/100_2333.eps}
1184 \begin{minipage}{0.4cm}
1187 \begin{minipage}{2.4cm}
1188 \includegraphics[width=2.4cm]{c_pd_vasp/00-1_ip0-10_2333.eps}
1190 \begin{minipage}{0.4cm}
1193 \begin{minipage}{2.4cm}
1194 \includegraphics[width=2.4cm]{c_pd_vasp/0-10_ip_2333.eps}
1199 \begin{minipage}{4.2cm}
1200 {\small Constrained relaxation\\
1201 technique (CRT) method}\\
1202 \includegraphics[width=4cm]{crt_orig.eps}
1204 \item Constrain diffusing atom
1205 \item Static constraints
1208 {\small Modifications}\\
1209 \includegraphics[width=4cm]{crt_mod.eps}
1211 \item Constrain all atoms
1212 \item Update individual\\
1223 Migration of the C \hkl<1 0 0> dumbbell interstitial
1229 \begin{minipage}{5.9cm}
1231 \includegraphics[width=5.8cm]{im_00-1_nosym_sp_fullct_thesis.ps}\\[0.45cm]
1234 \begin{picture}(0,0)(60,0)
1235 \includegraphics[width=1cm]{vasp_mig/00-1.eps}
1237 \begin{picture}(0,0)(-5,0)
1238 \includegraphics[width=1cm]{vasp_mig/bc_00-1_sp.eps}
1240 \begin{picture}(0,0)(-55,0)
1241 \includegraphics[width=1cm]{vasp_mig/bc.eps}
1243 \begin{picture}(0,0)(12.5,10)
1244 \includegraphics[width=1cm]{110_arrow.eps}
1246 \begin{picture}(0,0)(90,0)
1247 \includegraphics[height=0.9cm]{001_arrow.eps}
1253 \begin{minipage}{0.3cm}
1257 \begin{minipage}{5.9cm}
1259 \includegraphics[width=5.9cm]{vasp_mig/00-1_0-10_nosym_sp_fullct.ps}\\[0.5cm]
1262 \begin{picture}(0,0)(60,0)
1263 \includegraphics[width=1cm]{vasp_mig/00-1_a.eps}
1265 \begin{picture}(0,0)(5,0)
1266 \includegraphics[width=1cm]{vasp_mig/00-1_0-10_sp.eps}
1268 \begin{picture}(0,0)(-55,0)
1269 \includegraphics[width=1cm]{vasp_mig/0-10.eps}
1271 \begin{picture}(0,0)(12.5,10)
1272 \includegraphics[width=1cm]{100_arrow.eps}
1274 \begin{picture}(0,0)(90,0)
1275 \includegraphics[height=0.9cm]{001_arrow.eps}
1285 \begin{minipage}{5.9cm}
1287 \includegraphics[width=5.9cm]{vasp_mig/00-1_ip0-10_nosym_sp_fullct.ps}\\[0.6cm]
1290 \begin{picture}(0,0)(60,0)
1291 \includegraphics[width=0.9cm]{vasp_mig/00-1_b.eps}
1293 \begin{picture}(0,0)(10,0)
1294 \includegraphics[width=0.9cm]{vasp_mig/00-1_ip0-10_sp.eps}
1296 \begin{picture}(0,0)(-60,0)
1297 \includegraphics[width=0.9cm]{vasp_mig/0-10_b.eps}
1299 \begin{picture}(0,0)(12.5,10)
1300 \includegraphics[width=1cm]{100_arrow.eps}
1302 \begin{picture}(0,0)(90,0)
1303 \includegraphics[height=0.9cm]{001_arrow.eps}
1309 \begin{minipage}{0.3cm}
1312 \begin{minipage}{6.5cm}
1315 \item Energetically most favorable path
1318 \item Activation energy: $\approx$ 0.9 eV
1319 \item Experimental values: 0.73 ... 0.87 eV
1321 $\Rightarrow$ {\color{blue}Diffusion} path identified!
1322 \item Reorientation (path 3)
1324 \item More likely composed of two consecutive steps of type 2
1325 \item Experimental values: 0.77 ... 0.88 eV
1327 $\Rightarrow$ {\color{blue}Reorientation} transition identified!
1336 Migration of the C \hkl<1 0 0> dumbbell interstitial
1343 \begin{minipage}{6.5cm}
1346 \begin{minipage}[t]{5.9cm}
1348 \includegraphics[width=5.9cm]{bc_00-1.ps}\\[2.35cm]
1351 \begin{pspicture}(0,0)(0,0)
1352 \psframe[linecolor=red,fillstyle=none](-2.8,1.35)(3.3,2.7)
1354 \begin{picture}(0,0)(60,-50)
1355 \includegraphics[width=1cm]{albe_mig/bc_00-1_red_00.eps}
1357 \begin{picture}(0,0)(5,-50)
1358 \includegraphics[width=1cm]{albe_mig/bc_00-1_red_01.eps}
1360 \begin{picture}(0,0)(-55,-50)
1361 \includegraphics[width=1cm]{albe_mig/bc_00-1_red_02.eps}
1363 \begin{picture}(0,0)(12.5,-40)
1364 \includegraphics[width=1cm]{110_arrow.eps}
1366 \begin{picture}(0,0)(90,-45)
1367 \includegraphics[height=0.9cm]{001_arrow.eps}
1369 \begin{pspicture}(0,0)(0,0)
1370 \psframe[linecolor=blue,fillstyle=none](-2.8,0)(3.3,1.6)
1372 \begin{picture}(0,0)(60,-15)
1373 \includegraphics[width=0.9cm]{albe_mig/bc_00-1_01.eps}
1375 \begin{picture}(0,0)(35,-15)
1376 \includegraphics[width=0.9cm]{albe_mig/bc_00-1_02.eps}
1378 \begin{picture}(0,0)(-5,-15)
1379 \includegraphics[width=0.9cm]{albe_mig/bc_00-1_03.eps}
1381 \begin{picture}(0,0)(-55,-15)
1382 \includegraphics[width=0.9cm]{albe_mig/bc_00-1_04.eps}
1384 \begin{picture}(0,0)(12.5,-5)
1385 \includegraphics[width=1cm]{100_arrow.eps}
1387 \begin{picture}(0,0)(90,-15)
1388 \includegraphics[height=0.9cm]{010_arrow.eps}
1394 \begin{minipage}{5.9cm}
1397 \item Lowest activation energy: $\approx$ 2.2 eV
1398 \item 2.4 times higher than VASP
1399 \item Different pathway
1404 \begin{minipage}{6.5cm}
1407 \begin{minipage}{5.9cm}
1409 %\includegraphics[width=5.9cm]{00-1_0-10.ps}\\[0.75cm]
1412 %\begin{pspicture}(0,0)(0,0)
1413 %\psframe[linecolor=red,fillstyle=none](-2.8,-0.25)(3.3,1.1)
1415 %\begin{picture}(0,0)(60,-5)
1416 %\includegraphics[width=0.9cm]{albe_mig/00-1_0-10_red_00.eps}
1418 %\begin{picture}(0,0)(0,-5)
1419 %\includegraphics[width=0.9cm]{albe_mig/00-1_0-10_red_min.eps}
1421 %\begin{picture}(0,0)(-55,-5)
1422 %\includegraphics[width=0.9cm]{albe_mig/00-1_0-10_red_03.eps}
1424 %\begin{picture}(0,0)(12.5,5)
1425 %\includegraphics[width=1cm]{100_arrow.eps}
1427 %\begin{picture}(0,0)(90,0)
1428 %\includegraphics[height=0.9cm]{001_arrow.eps}
1436 %\begin{minipage}{5.9cm}
1437 \includegraphics[width=5.9cm]{00-1_110_0-10_mig_albe.ps}
1441 \begin{minipage}{5.9cm}
1442 Transition involving \ci{} \hkl<1 1 0>
1444 \item Bond-centered configuration unstable\\
1445 $\rightarrow$ \ci{} \hkl<1 1 0> dumbbell
1446 \item Transition minima of path 2 \& 3\\
1447 $\rightarrow$ \ci{} \hkl<1 1 0> dumbbell
1448 \item Activation energy: $\approx$ 2.2 eV \& 0.9 eV
1449 \item 2.4 - 3.4 times higher than VASP
1450 \item Rotation of dumbbell orientation
1454 {\color{blue}Overestimated diffusion barrier}
1465 Combinations with a C-Si \hkl<1 0 0>-type interstitial
1475 E_{\text{f}}^{\text{defect combination}}-
1476 E_{\text{f}}^{\text{C \hkl<0 0 -1> dumbbell}}-
1477 E_{\text{f}}^{\text{2nd defect}}
1483 \begin{tabular}{l c c c c c c}
1485 $E_{\text{b}}$ [eV] & 1 & 2 & 3 & 4 & 5 & R\\
1487 \hkl<0 0 -1> & {\color{red}-0.08} & -1.15 & {\color{red}-0.08} & 0.04 & -1.66 & -0.19\\
1488 \hkl<0 0 1> & 0.34 & 0.004 & -2.05 & 0.26 & -1.53 & -0.19\\
1489 \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}\\
1490 \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}\\
1491 \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}\\
1492 \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}\\
1494 C substitutional (C$_{\text{S}}$) & 0.26 & -0.51 & -0.93 & -0.15 & 0.49 & -0.05\\
1495 Vacancy & -5.39 ($\rightarrow$ C$_{\text{S}}$) & -0.59 & -3.14 & -0.54 & -0.50 & -0.31\\
1504 \begin{minipage}[t]{3.8cm}
1505 \underline{\hkl<1 0 0> at position 1}\\[0.1cm]
1506 \includegraphics[width=3.5cm]{00-1dc/2-25.eps}
1508 \begin{minipage}[t]{3.5cm}
1509 \underline{\hkl<0 -1 0> at position 1}\\[0.1cm]
1510 \includegraphics[width=3.2cm]{00-1dc/2-39.eps}
1512 \begin{minipage}[t]{5.5cm}
1514 \item $E_{\text{b}}=0$ $\Leftrightarrow$ non-interacting defects\\
1515 $E_{\text{b}} \rightarrow 0$ for increasing distance (R)
1516 \item Stress compensation / increase
1517 \item Unfavored: antiparallel orientations
1518 \item Indication of energetically favored\\
1520 \item Most favorable: C clustering
1521 \item However: High barrier ($>4\,\text{eV}$)
1522 \item $4\times{\color{cyan}-2.25}$ versus $2\times{\color{orange}-2.39}$
1527 \begin{picture}(0,0)(-295,-130)
1528 \includegraphics[width=3.5cm]{comb_pos.eps}
1536 Combinations of C-Si \hkl<1 0 0>-type interstitials
1543 Energetically most favorable combinations along \hkl<1 1 0>
1548 \begin{tabular}{l c c c c c c}
1550 & 1 & 2 & 3 & 4 & 5 & 6\\
1552 $E_{\text{b}}$ [eV] & -2.39 & -1.88 & -0.59 & -0.31 & -0.24 & -0.21 \\
1553 C-C distance [\AA] & 1.4 & 4.6 & 6.5 & 8.6 & 10.5 & 10.8 \\
1554 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>\\
1561 \begin{minipage}{7.0cm}
1562 \includegraphics[width=7cm]{db_along_110_cc.ps}
1564 \begin{minipage}{6.0cm}
1566 \item Interaction proportional to reciprocal cube of C-C distance
1567 \item Saturation in the immediate vicinity
1568 \renewcommand\labelitemi{$\Rightarrow$}
1569 \item Agglomeration of \ci{} expected
1570 \item Absence of C clustering
1574 Consisten with initial precipitation model
1586 Combinations of substitutional C and \hkl<1 1 0> Si self-interstitials
1592 %\begin{minipage}{3.2cm}
1593 %\includegraphics[width=3cm]{sub_110_combo.eps}
1595 %\begin{minipage}{7.8cm}
1596 %\begin{tabular}{l c c c c c c}
1598 %C$_{\text{sub}}$ & \hkl<1 1 0> & \hkl<-1 1 0> & \hkl<0 1 1> & \hkl<0 -1 1> &
1599 % \hkl<1 0 1> & \hkl<-1 0 1> \\
1601 %1 & \RM{1} & \RM{3} & \RM{3} & \RM{1} & \RM{3} & \RM{1} \\
1602 %2 & \RM{2} & A & A & \RM{2} & C & \RM{5} \\
1603 %3 & \RM{3} & \RM{1} & \RM{3} & \RM{1} & \RM{1} & \RM{3} \\
1604 %4 & \RM{4} & B & D & E & E & D \\
1605 %5 & \RM{5} & C & A & \RM{2} & A & \RM{2} \\
1612 %\begin{tabular}{l c c c c c c c c c c}
1614 %Conf & \RM{1} & \RM{2} & \RM{3} & \RM{4} & \RM{5} & A & B & C & D & E \\
1616 %$E_{\text{f}}$ [eV]& 4.37 & 5.26 & 5.57 & 5.37 & 5.12 & 5.10 & 5.32 & 5.28 & 5.39 & 5.32 \\
1617 %$E_{\text{b}}$ [eV] & -0.97 & -0.08 & 0.22 & -0.02 & -0.23 & -0.25 & -0.02 & -0.06 & 0.05 & -0.03 \\
1618 %$r$ [nm] & 0.292 & 0.394 & 0.241 & 0.453 & 0.407 & 0.408 & 0.452 & 0.392 & 0.456 & 0.453\\
1623 \begin{minipage}{6.0cm}
1624 \includegraphics[width=5.8cm]{c_sub_si110.ps}
1626 \begin{minipage}{7cm}
1629 \item IBS: C may displace Si\\
1630 $\Rightarrow$ C$_{\text{sub}}$ + \hkl<1 1 0> Si self-interstitial
1632 \hkl<1 1 0>-type $\rightarrow$ favored combination
1633 \renewcommand\labelitemi{$\Rightarrow$}
1634 \item Most favorable: \cs{} along \hkl<1 1 0> chain \si{}
1635 \item Less favorable than C-Si \hkl<1 0 0> dumbbell
1636 \item Interaction drops quickly to zero\\
1637 $\rightarrow$ low capture radius
1641 IBS process far from equilibrium\\
1642 \cs{} \& \si{} instead of thermodynamic ground state
1647 \begin{minipage}{6.5cm}
1648 \includegraphics[width=6.0cm]{162-097.ps}
1650 \item Low migration barrier
1653 \begin{minipage}{6.5cm}
1655 Ab initio MD at \degc{900}\\
1656 \includegraphics[width=3.3cm]{md_vasp_01.eps}
1657 $t=\unit[2230]{fs}$\\
1658 \includegraphics[width=3.3cm]{md_vasp_02.eps}
1662 Contribution of entropy to structural formation
1671 Migration in C-Si \hkl<1 0 0> and vacancy combinations
1678 \begin{minipage}[t]{3cm}
1679 \underline{Pos 2, $E_{\text{b}}=-0.59\text{ eV}$}\\
1680 \includegraphics[width=2.8cm]{00-1dc/0-59.eps}
1682 \begin{minipage}[t]{7cm}
1685 Low activation energies\\
1686 High activation energies for reverse processes\\
1688 {\color{blue}C$_{\text{sub}}$ very stable}\\
1692 Without nearby \hkl<1 1 0> Si self-interstitial (IBS)\\
1694 {\color{blue}Formation of SiC by successive substitution by C}
1698 \begin{minipage}[t]{3cm}
1699 \underline{Pos 3, $E_{\text{b}}=-3.14\text{ eV}$}\\
1700 \includegraphics[width=2.8cm]{00-1dc/3-14.eps}
1705 \begin{minipage}{5.9cm}
1706 \includegraphics[width=5.9cm]{vasp_mig/comb_mig_3-2_vac_fullct.ps}\\[0.6cm]
1708 \begin{picture}(0,0)(70,0)
1709 \includegraphics[width=1.4cm]{vasp_mig/comb_2-1_init.eps}
1711 \begin{picture}(0,0)(30,0)
1712 \includegraphics[width=1.4cm]{vasp_mig/comb_2-1_seq_03.eps}
1714 \begin{picture}(0,0)(-10,0)
1715 \includegraphics[width=1.4cm]{vasp_mig/comb_2-1_seq_06.eps}
1717 \begin{picture}(0,0)(-48,0)
1718 \includegraphics[width=1.4cm]{vasp_mig/comb_2-1_final.eps}
1720 \begin{picture}(0,0)(12.5,5)
1721 \includegraphics[width=1cm]{100_arrow.eps}
1723 \begin{picture}(0,0)(97,-10)
1724 \includegraphics[height=0.9cm]{001_arrow.eps}
1730 \begin{minipage}{0.3cm}
1734 \begin{minipage}{5.9cm}
1735 \includegraphics[width=5.9cm]{vasp_mig/comb_mig_4-2_vac_fullct.ps}\\[0.1cm]
1737 \begin{picture}(0,0)(60,0)
1738 \includegraphics[width=0.9cm]{vasp_mig/comb_3-1_init.eps}
1740 \begin{picture}(0,0)(25,0)
1741 \includegraphics[width=0.9cm]{vasp_mig/comb_3-1_seq_03.eps}
1743 \begin{picture}(0,0)(-20,0)
1744 \includegraphics[width=0.9cm]{vasp_mig/comb_3-1_seq_07.eps}
1746 \begin{picture}(0,0)(-55,0)
1747 \includegraphics[width=0.9cm]{vasp_mig/comb_3-1_final.eps}
1749 \begin{picture}(0,0)(12.5,5)
1750 \includegraphics[width=1cm]{100_arrow.eps}
1752 \begin{picture}(0,0)(95,0)
1753 \includegraphics[height=0.9cm]{001_arrow.eps}
1765 Conclusion of defect / migration / combined defect simulations
1774 \item Accurately described by quantum-mechanical simulations
1775 \item Less accurate description by classical potential simulations
1776 \item Underestimated formation energy of \cs{} by classical approach
1777 \item Both methods predict same ground state: \ci{} \hkl<1 0 0> dumbbell
1782 \item C migration pathway in Si identified
1783 \item Consistent with reorientation and diffusion experiments
1786 \item Different path and ...
1787 \item overestimated barrier by classical potential calculations
1790 Concerning the precipitation mechanism
1792 \item Agglomeration of C-Si dumbbells energetically favorable
1793 (stress compensation)
1794 \item C-Si indeed favored compared to
1795 C$_{\text{sub}}$ \& \hkl<1 1 0> Si self-interstitial
1796 \item Possible low interaction capture radius of
1797 C$_{\text{sub}}$ \& \hkl<1 1 0> Si self-interstitial
1798 \item Low barrier for
1799 \ci{} \hkl<1 0 0> $\rightarrow$ \cs{} \& \si{} \hkl<1 1 0>
1800 \item In absence of nearby \hkl<1 1 0> Si self-interstitial:
1801 C-Si \hkl<1 0 0> + Vacancy $\rightarrow$ C$_{\text{sub}}$ (SiC)
1804 {\color{blue}Results suggest increased participation of \cs}
1812 Silicon carbide precipitation simulations
1818 \begin{pspicture}(0,0)(12,6.5)
1820 \rput(3.5,5.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
1823 \item Create c-Si volume
1824 \item Periodc boundary conditions
1825 \item Set requested $T$ and $p=0\text{ bar}$
1826 \item Equilibration of $E_{\text{kin}}$ and $E_{\text{pot}}$
1829 \rput(3.5,2.7){\rnode{insert}{\psframebox[fillstyle=solid,fillcolor=lachs]{
1831 Insertion of C atoms at constant T
1833 \item total simulation volume {\pnode{in1}}
1834 \item volume of minimal SiC precipitate {\pnode{in2}}
1835 \item volume consisting of Si atoms to form a minimal {\pnode{in3}}\\
1839 \rput(3.5,1){\rnode{cool}{\psframebox[fillstyle=solid,fillcolor=lbb]{
1841 Run for 100 ps followed by cooling down to $20\, ^{\circ}\textrm{C}$
1843 \ncline[]{->}{init}{insert}
1844 \ncline[]{->}{insert}{cool}
1845 \psframe[fillstyle=solid,fillcolor=white](7.5,0.7)(13.5,6.3)
1846 \rput(7.8,6){\footnotesize $V_1$}
1847 \psframe[fillstyle=solid,fillcolor=lightgray](9,2)(12,5)
1848 \rput(9.2,4.85){\tiny $V_2$}
1849 \psframe[fillstyle=solid,fillcolor=gray](9.25,2.25)(11.75,4.75)
1850 \rput(9.55,4.45){\footnotesize $V_3$}
1851 \rput(7.9,3.2){\pnode{ins1}}
1852 \rput(9.22,2.8){\pnode{ins2}}
1853 \rput(11.0,2.4){\pnode{ins3}}
1854 \ncline[]{->}{in1}{ins1}
1855 \ncline[]{->}{in2}{ins2}
1856 \ncline[]{->}{in3}{ins3}
1861 \item Restricted to classical potential simulations
1862 \item $V_2$ and $V_3$ considered due to low diffusion
1863 \item Amount of C atoms: 6000
1864 ($r_{\text{prec}}\approx 3.1\text{ nm}$, IBS: 2 ... 4 nm)
1865 \item Simulation volume: $31\times 31\times 31$ unit cells
1874 Silicon carbide precipitation simulations at $450\,^{\circ}\mathrm{C}$ as in IBS
1879 \begin{minipage}{6.5cm}
1880 \includegraphics[width=6.4cm]{sic_prec_450_si-si_c-c.ps}
1882 \begin{minipage}{6.5cm}
1883 \includegraphics[width=6.4cm]{sic_prec_450_energy.ps}
1886 \begin{minipage}{6.5cm}
1887 \includegraphics[width=6.4cm]{sic_prec_450_si-c.ps}
1889 \begin{minipage}{6.5cm}
1891 \underline{Low C concentration ($V_1$)}\\
1892 \hkl<1 0 0> C-Si dumbbell dominated structure
1894 \item Si-C bumbs around 0.19 nm
1895 \item C-C peak at 0.31 nm (as expected in 3C-SiC):\\
1896 concatenated dumbbells of various orientation
1897 \item Si-Si NN distance stretched to 0.3 nm
1899 {\color{blue}$\Rightarrow$ C atoms in proper 3C-SiC distance first}\\
1900 \underline{High C concentration ($V_2$, $V_3$)}\\
1901 High amount of strongly bound C-C bonds\\
1902 Defect density $\uparrow$ $\Rightarrow$ considerable amount of damage\\
1903 Only short range order observable\\
1904 {\color{blue}$\Rightarrow$ amorphous SiC-like phase}
1912 Silicon carbide precipitation simulations at $450\,^{\circ}\mathrm{C}$ as in IBS
1917 \begin{minipage}{6.5cm}
1918 \includegraphics[width=6.4cm]{sic_prec_450_si-si_c-c.ps}
1920 \begin{minipage}{6.5cm}
1921 \includegraphics[width=6.4cm]{sic_prec_450_energy.ps}
1924 \begin{minipage}{6.5cm}
1925 \includegraphics[width=6.4cm]{sic_prec_450_si-c.ps}
1927 \begin{minipage}{6.5cm}
1929 \underline{Low C concentration ($V_1$)}\\
1930 \hkl<1 0 0> C-Si dumbbell dominated structure
1932 \item Si-C bumbs around 0.19 nm
1933 \item C-C peak at 0.31 nm (as expected in 3C-SiC):\\
1934 concatenated dumbbells of various orientation
1935 \item Si-Si NN distance stretched to 0.3 nm
1937 {\color{blue}$\Rightarrow$ C atoms in proper 3C-SiC distance first}\\
1938 \underline{High C concentration ($V_2$, $V_3$)}\\
1939 High amount of strongly bound C-C bonds\\
1940 Defect density $\uparrow$ $\Rightarrow$ considerable amount of damage\\
1941 Only short range order observable\\
1942 {\color{blue}$\Rightarrow$ amorphous SiC-like phase}
1945 \begin{pspicture}(0,0)(0,0)
1946 \rput(6.7,5.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=white]{
1947 \begin{minipage}{10cm}
1949 {\color{red}\bf 3C-SiC formation fails to appear}
1951 \item Low C concentration simulations
1953 \item Formation of \ci{} indeed occurs
1954 \item Agllomeration not observed
1956 \item High C concentration simulations
1958 \item Amorphous SiC-like structure\\
1959 (not expected at prevailing temperatures)
1960 \item Rearrangement and transition into 3C-SiC structure missing
1972 Limitations of molecular dynamics and short range potentials
1979 \underline{Time scale problem of MD}\\[0.2cm]
1980 Minimize integration error\\
1981 $\Rightarrow$ discretization considerably smaller than
1982 reciprocal of fastest vibrational mode\\[0.1cm]
1983 Order of fastest vibrational mode: $10^{13} - 10^{14}\text{ Hz}$\\
1984 $\Rightarrow$ suitable choice of time step:
1985 $\tau=1\text{ fs}=10^{-15}\text{ s}$\\
1986 $\Rightarrow$ {\color{red}\underline{slow}} phase space propagation\\[0.1cm]
1987 Several local minima in energy surface separated by large energy barriers\\
1988 $\Rightarrow$ transition event corresponds to a multiple
1989 of vibrational periods\\
1990 $\Rightarrow$ phase transition made up of {\color{red}\underline{many}}
1991 infrequent transition events\\[0.1cm]
1992 {\color{blue}Accelerated methods:}
1993 \underline{Temperature accelerated} MD (TAD), self-guided MD \ldots
1997 \underline{Limitations related to the short range potential}\\[0.2cm]
1998 Cut-off function pushing forces and energies to zero between 1$^{\text{st}}$
1999 and 2$^{\text{nd}}$ next neighbours\\
2000 $\Rightarrow$ overestimated unphysical high forces of next neighbours
2006 Potential enhanced problem of slow phase space propagation
2011 \underline{Approach to the (twofold) problem}\\[0.2cm]
2012 Increased temperature simulations without TAD corrections\\
2013 (accelerated methods or higher time scales exclusively not sufficient)
2015 \begin{picture}(0,0)(-260,-30)
2017 \begin{minipage}{4.2cm}
2024 \item 3C-SiC also observed for higher T
2025 \item higher T inside sample
2026 \item structural evolution vs.\\
2027 equilibrium properties
2033 \begin{picture}(0,0)(-305,-155)
2035 \begin{minipage}{2.5cm}
2039 thermodynmic sampling
2050 Increased temperature simulations at low C concentration
2055 \begin{minipage}{6.5cm}
2056 \includegraphics[width=6.4cm]{tot_pc_thesis.ps}
2058 \begin{minipage}{6.5cm}
2059 \includegraphics[width=6.4cm]{tot_pc3_thesis.ps}
2062 \begin{minipage}{6.5cm}
2063 \includegraphics[width=6.4cm]{tot_pc2_thesis.ps}
2065 \begin{minipage}{6.5cm}
2067 \underline{Si-C bonds:}
2069 \item Vanishing cut-off artifact (above $1650\,^{\circ}\mathrm{C}$)
2070 \item Structural change: C-Si \hkl<1 0 0> $\rightarrow$ C$_{\text{sub}}$
2072 \underline{Si-Si bonds:}
2073 {\color{blue}Si-C$_{\text{sub}}$-Si} along \hkl<1 1 0>
2074 ($\rightarrow$ 0.325 nm)\\[0.1cm]
2075 \underline{C-C bonds:}
2077 \item C-C next neighbour pairs reduced (mandatory)
2078 \item Peak at 0.3 nm slightly shifted
2080 \item C-Si \hkl<1 0 0> combinations (dashed arrows)\\
2081 $\rightarrow$ C-Si \hkl<1 0 0> \& C$_{\text{sub}}$
2083 $\rightarrow$ pure {\color{blue}C$_{\text{sub}}$ combinations}
2085 \item Range [|-$\downarrow$]:
2086 {\color{blue}C$_{\text{sub}}$ \& C$_{\text{sub}}$
2087 with nearby Si$_{\text{I}}$}
2092 \begin{picture}(0,0)(-330,-74)
2095 \begin{minipage}{1.6cm}
2098 stretched SiC\\[-0.1cm]
2110 Increased temperature simulations at low C concentration
2115 \begin{minipage}{6.5cm}
2116 \includegraphics[width=6.4cm]{tot_pc_thesis.ps}
2118 \begin{minipage}{6.5cm}
2119 \includegraphics[width=6.4cm]{tot_pc3_thesis.ps}
2122 \begin{minipage}{6.5cm}
2123 \includegraphics[width=6.4cm]{tot_pc2_thesis.ps}
2125 \begin{minipage}{6.5cm}
2127 \underline{Si-C bonds:}
2129 \item Vanishing cut-off artifact (above $1650\,^{\circ}\mathrm{C}$)
2130 \item Structural change: C-Si \hkl<1 0 0> $\rightarrow$ C$_{\text{sub}}$
2132 \underline{Si-Si bonds:}
2133 {\color{blue}Si-C$_{\text{sub}}$-Si} along \hkl<1 1 0>
2134 ($\rightarrow$ 0.325 nm)\\[0.1cm]
2135 \underline{C-C bonds:}
2137 \item C-C next neighbour pairs reduced (mandatory)
2138 \item Peak at 0.3 nm slightly shifted
2140 \item C-Si \hkl<1 0 0> combinations (dashed arrows)\\
2141 $\rightarrow$ C-Si \hkl<1 0 0> \& C$_{\text{sub}}$
2143 $\rightarrow$ pure {\color{blue}C$_{\text{sub}}$ combinations}
2145 \item Range [|-$\downarrow$]:
2146 {\color{blue}C$_{\text{sub}}$ \& C$_{\text{sub}}$
2147 with nearby Si$_{\text{I}}$}
2152 %\begin{picture}(0,0)(-330,-74)
2155 %\begin{minipage}{1.6cm}
2158 %stretched SiC\\[-0.1cm]
2165 \begin{pspicture}(0,0)(0,0)
2166 \rput(6.7,5.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=white]{
2167 \begin{minipage}{10cm}
2169 {\color{blue}\bf Stretched SiC in c-Si}
2171 \item Consistent to precipitation model involving \cs{}
2172 \item Explains annealing behavior of high/low T C implants
2174 \item Low T: highly mobiel \ci{}
2175 \item High T: stable configurations of \cs{}
2178 $\Rightarrow$ High T $\leftrightarrow$ IBS conditions far from equilibrium\\
2179 $\Rightarrow$ Precipitation mechanism involving \cs{}
2189 Increased temperature simulations at high C concentration
2194 \begin{minipage}{6.5cm}
2195 \includegraphics[width=6.4cm]{12_pc_thesis.ps}
2197 \begin{minipage}{6.5cm}
2198 \includegraphics[width=6.4cm]{12_pc_c_thesis.ps}
2206 \begin{minipage}[t]{6.0cm}
2207 0.186 nm: Si-C pairs $\uparrow$\\
2208 (as expected in 3C-SiC)\\[0.2cm]
2209 0.282 nm: Si-C-C\\[0.2cm]
2210 $\approx$0.35 nm: C-Si-Si
2213 \begin{minipage}{0.2cm}
2217 \begin{minipage}[t]{6.0cm}
2218 0.15 nm: C-C pairs $\uparrow$\\
2219 (as expected in graphite/diamond)\\[0.2cm]
2220 0.252 nm: C-C-C (2$^{\text{nd}}$ NN for diamond)\\[0.2cm]
2221 0.31 nm: shifted towards 0.317 nm $\rightarrow$ C-Si-C
2226 \item Decreasing cut-off artifact
2227 \item {\color{red}Amorphous} SiC-like phase remains
2228 \item High amount of {\color{red}damage} \& alignement to c-Si host matrix lost
2229 \item Slightly sharper peaks $\Rightarrow$ indicate slight {\color{blue}acceleration of dynamics} due to temperature
2238 High C \& small $V$ \& short $t$
2241 Slow restructuring due to strong C-C bonds
2244 High C \& low T implants
2255 Summary and Conclusions
2263 \begin{minipage}[t]{12.9cm}
2264 \underline{Pecipitation simulations}
2266 \item High C concentration $\rightarrow$ amorphous SiC like phase
2267 \item Problem of potential enhanced slow phase space propagation
2268 \item Low T $\rightarrow$ C-Si \hkl<1 0 0> dumbbell dominated structure
2269 \item High T $\rightarrow$ C$_{\text{sub}}$ dominated structure
2270 \item High T necessary to simulate IBS conditions (far from equilibrium)
2271 \item Precipitation by successive agglomeration of \cs (epitaxy)
2272 \item \si{}: vehicle to form \cs{} \& supply of Si \& stress compensation
2273 (stretched SiC, interface)
2281 \begin{minipage}{12.9cm}
2286 \item Point defects excellently / fairly well described
2288 \item C$_{\text{sub}}$ drastically underestimated by EA
2289 \item EA predicts correct ground state:
2290 C$_{\text{sub}}$ \& \si{} $>$ \ci{}
2291 \item Identified migration path explaining
2292 diffusion and reorientation experiments by DFT
2293 \item EA fails to describe \ci{} migration:
2294 Wrong path \& overestimated barrier
2296 \item Combinations of defects
2298 \item Agglomeration of point defects energetically favorable
2299 by compensation of stress
2300 \item Formation of C-C unlikely
2301 \item C$_{\text{sub}}$ favored conditions (conceivable in IBS)
2302 \item \ci{} \hkl<1 0 0> $\leftrightarrow$ \cs{} \& \si{} \hkl<1 1 0>\\
2303 Low barrier (\unit[0.77]{eV}) \& low capture radius
2311 \framebox{Precipitation by successive agglomeration of \cs{}}
2329 \underline{Augsburg}
2331 \item Prof. B. Stritzker (accomodation at EP \RM{4})
2332 \item Ralf Utermann (EDV)
2335 \underline{Helsinki}
2337 \item Prof. K. Nordlund (MD)
2342 \item Bayerische Forschungsstiftung (financial support)
2345 \underline{Paderborn}
2347 \item Prof. J. Lindner (SiC)
2348 \item Prof. G. Schmidt (DFT + financial support)
2349 \item Dr. E. Rauls (DFT + SiC)
2350 \item Dr. S. Sanna (VASP)
2357 \bf Thank you for your attention!