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18 \usepackage{fancyvrb} % Fancy verbatim environments
19 \usepackage{pstricks} % PSTricks with the standard color package
28 \graphicspath{{../img/}}
32 \usepackage[setpagesize=false]{hyperref}
38 \usepackage{semlayer} % Seminar overlays
39 \usepackage{slidesec} % Seminar sections and list of slides
41 \input{seminar.bug} % Official bugs corrections
42 \input{seminar.bg2} % Unofficial bugs corrections
49 %\usepackage{cmbright}
50 %\renewcommand{\familydefault}{\sfdefault}
51 %\usepackage{mathptmx}
57 \extraslideheight{10in}
62 % specify width and height
66 % shift it into visual area properly
67 \def\slideleftmargin{3.3cm}
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70 \newcommand{\ham}{\mathcal{H}}
71 \newcommand{\pot}{\mathcal{V}}
72 \newcommand{\foo}{\mathcal{U}}
73 \newcommand{\vir}{\mathcal{W}}
76 \renewcommand\labelitemii{{\color{gray}$\bullet$}}
79 \renewcommand{\phi}{\varphi}
82 \newcommand{\RM}[1]{\MakeUppercase{\romannumeral #1{}}}
85 \newrgbcolor{si-yellow}{.6 .6 0}
86 \newrgbcolor{hb}{0.75 0.77 0.89}
87 \newrgbcolor{lbb}{0.75 0.8 0.88}
88 \newrgbcolor{hlbb}{0.825 0.88 0.968}
89 \newrgbcolor{lachs}{1.0 .93 .81}
92 \newcommand{\si}{Si$_{\text{i}}${}}
93 \newcommand{\ci}{C$_{\text{i}}${}}
94 \newcommand{\cs}{C$_{\text{sub}}${}}
95 \newcommand{\degc}[1]{\unit[#1]{$^{\circ}$C}{}}
96 \newcommand{\distn}[1]{\unit[#1]{nm}{}}
97 \newcommand{\dista}[1]{\unit[#1]{\AA}{}}
98 \newcommand{\perc}[1]{\unit[#1]{\%}{}}
108 Atomistic simulation study on the silicon carbide precipitation
114 \textsc{F. Zirkelbach}
118 Yet another seminar talk
122 Augsburg, 26. Mai 2011
127 % motivation / properties / applications of silicon carbide
132 \begin{pspicture}(0,0)(13.5,5)
136 \psframe*[linecolor=hb](0,0)(13.5,5)
138 \pspolygon[linecolor=hlbb,fillcolor=hlbb,fillstyle=solid](5.5,1)(7,1)(7,3)(5.5,3)
139 \pspolygon[linecolor=hlbb,fillcolor=hlbb,fillstyle=solid](6.75,0.5)(8,2)(8,2)(6.75,3.5)
141 \rput[lt](0.2,4.6){\color{gray}PROPERTIES}
143 \rput[lt](0.5,4){wide band gap}
144 \rput[lt](0.5,3.5){high electric breakdown field}
145 \rput[lt](0.5,3){good electron mobility}
146 \rput[lt](0.5,2.5){high electron saturation drift velocity}
147 \rput[lt](0.5,2){high thermal conductivity}
149 \rput[lt](0.5,1.5){hard and mechanically stable}
150 \rput[lt](0.5,1){chemically inert}
152 \rput[lt](0.5,0.5){radiation hardness}
154 \rput[rt](13.3,4.6){\color{gray}APPLICATIONS}
156 \rput[rt](13,3.85){high-temperature, high power}
157 \rput[rt](13,3.5){and high-frequency}
158 \rput[rt](13,3.15){electronic and optoelectronic devices}
160 \rput[rt](13,2.35){material suitable for extreme conditions}
161 \rput[rt](13,2){microelectromechanical systems}
162 \rput[rt](13,1.65){abrasives, cutting tools, heating elements}
164 \rput[rt](13,0.85){first wall reactor material, detectors}
165 \rput[rt](13,0.5){and electronic devices for space}
169 \begin{picture}(0,0)(-3,68)
170 \includegraphics[width=2.6cm]{wide_band_gap.eps}
172 \begin{picture}(0,0)(-285,-162)
173 \includegraphics[width=3.38cm]{sic_led.eps}
175 \begin{picture}(0,0)(-195,-162)
176 \includegraphics[width=2.8cm]{6h-sic_3c-sic.eps}
178 \begin{picture}(0,0)(-313,65)
179 \includegraphics[width=2.2cm]{infineon_schottky.eps}
181 \begin{picture}(0,0)(-220,65)
182 \includegraphics[width=2.9cm]{sic_wechselrichter_ise.eps}
184 \begin{picture}(0,0)(0,-160)
185 \includegraphics[width=3.0cm]{sic_proton.eps}
187 \begin{picture}(0,0)(-60,65)
188 \includegraphics[width=3.4cm]{sic_switch.eps}
205 \begin{tabular}{l c c c c c c}
207 & 3C-SiC & 4H-SiC & 6H-SiC & Si & GaN & Diamond\\
209 Hardness [Mohs] & \multicolumn{3}{c}{------ 9.6 ------}& 6.5 & - & 10 \\
210 Band gap [eV] & 2.36 & 3.23 & 3.03 & 1.12 & 3.39 & 5.5 \\
211 Break down field [$10^6$ V/cm] & 4 & 3 & 3.2 & 0.6 & 5 & 10 \\
212 Saturation drift velocity [$10^7$ cm/s] & 2.5 & 2.0 & 2.0 & 1 & 2.7 & 2.7 \\
213 Electron mobility [cm$^2$/Vs] & 800 & 900 & 400 & 1100 & 900 & 2200 \\
214 Hole mobility [cm$^2$/Vs] & 320 & 120 & 90 & 420 & 150 & 1600 \\
215 Thermal conductivity [W/cmK] & 5.0 & 4.9 & 4.9 & 1.5 & 1.3 & 22 \\
223 \begin{picture}(0,0)(-160,-155)
224 \includegraphics[width=7cm]{polytypes.eps}
226 \begin{picture}(0,0)(-10,-185)
227 \includegraphics[width=3.8cm]{cubic_hex.eps}\\
229 \begin{picture}(0,0)(-10,-175)
230 {\tiny cubic (twist)}
232 \begin{picture}(0,0)(-60,-175)
233 {\tiny hexagonal (no twist)}
235 \begin{pspicture}(0,0)(0,0)
236 \psellipse[linecolor=green](5.7,3.03)(0.4,0.5)
238 \begin{pspicture}(0,0)(0,0)
239 \psellipse[linecolor=green](5.6,1.68)(0.4,0.2)
241 \begin{pspicture}(0,0)(0,0)
242 \psellipse[linecolor=red](10.7,1.13)(0.4,0.2)
250 Fabrication of silicon carbide
257 SiC - \emph{Born from the stars, perfected on earth.}
261 Conventional thin film SiC growth:
263 \item \underline{Sublimation growth using the modified Lely method}
265 \item SiC single-crystalline seed at $T=1800 \, ^{\circ} \text{C}$
266 \item Surrounded by polycrystalline SiC in a graphite crucible\\
267 at $T=2100-2400 \, ^{\circ} \text{C}$
268 \item Deposition of supersaturated vapor on cooler seed crystal
270 \item \underline{Homoepitaxial growth using CVD}
272 \item Step-controlled epitaxy on off-oriented 6H-SiC substrates
273 \item C$_3$H$_8$/SiH$_4$/H$_2$ at $1100-1500 \, ^{\circ} \text{C}$
274 \item Angle, temperature $\rightarrow$ 3C/6H/4H-SiC
276 \item \underline{Heteroepitaxial growth of 3C-SiC on Si using CVD/MBE}
278 \item Two steps: carbonization and growth
279 \item $T=650-1050 \, ^{\circ} \text{C}$
280 \item SiC/Si lattice mismatch $\approx$ 20 \%
281 \item Quality and size not yet sufficient
285 \begin{picture}(0,0)(-280,-65)
286 \includegraphics[width=3.8cm]{6h-sic_3c-sic.eps}
288 \begin{picture}(0,0)(-280,-55)
289 \begin{minipage}{5cm}
291 NASA: 6H-SiC and 3C-SiC LED\\[-7pt]
296 \begin{picture}(0,0)(-265,-150)
297 \includegraphics[width=2.4cm]{m_lely.eps}
299 \begin{picture}(0,0)(-333,-175)
300 \begin{minipage}{5cm}
306 5. Insulation\\[-7pt]
311 \begin{picture}(0,0)(-230,-35)
313 {\footnotesize\color{blue}\bf Hex: micropipes along c-axis}
316 \begin{picture}(0,0)(-230,-10)
318 \begin{minipage}{3cm}
319 {\footnotesize\color{blue}\bf 3C-SiC fabrication\\
330 Fabrication of silicon carbide
335 Alternative approach:
336 Ion beam synthesis (IBS) of burried 3C-SiC layers in Si\hkl(1 0 0)
338 \item \underline{Implantation step 1}\\
339 180 keV C$^+$, $D=7.9\times 10^{17}$ cm$^{-2}$, $T_{\text{i}}=500\,^{\circ}\mathrm{C}$\\
340 $\Rightarrow$ box-like distribution of equally sized
341 and epitactically oriented SiC precipitates
343 \item \underline{Implantation step 2}\\
344 180 keV C$^+$, $D=0.6\times 10^{17}$ cm$^{-2}$, $T_{\text{i}}=250\,^{\circ}\mathrm{C}$\\
345 $\Rightarrow$ destruction of SiC nanocrystals
346 in growing amorphous interface layers
347 \item \underline{Annealing}\\
348 $T=1250\,^{\circ}\mathrm{C}$, $t=10\,\text{h}$\\
349 $\Rightarrow$ homogeneous, stoichiometric SiC layer
350 with sharp interfaces
353 \begin{minipage}{6.3cm}
354 \includegraphics[width=6cm]{ibs_3c-sic.eps}\\[-0.2cm]
356 XTEM micrograph of single crystalline 3C-SiC in Si\hkl(1 0 0)
360 \begin{minipage}{6.3cm}
363 Precipitation mechanism not yet fully understood!
365 \renewcommand\labelitemi{$\Rightarrow$}
367 \underline{Understanding the SiC precipitation}
369 \item significant technological progress in SiC thin film formation
370 \item perspectives for processes relying upon prevention of SiC precipitation
387 \item Supposed precipitation mechanism of SiC in Si
388 \item Utilized simulation techniques
390 \item Molecular dynamics (MD) simulations
391 \item Density functional theory (DFT) calculations
393 \item C and Si self-interstitial point defects in silicon
394 \item Silicon carbide precipitation simulations
395 \item Summary / Conclusion / Outlook
403 Supposed precipitation mechanism of SiC in Si
410 \begin{minipage}{3.8cm}
411 Si \& SiC lattice structure\\[0.2cm]
412 \includegraphics[width=3.5cm]{sic_unit_cell.eps}\\[-0.3cm]
416 \begin{minipage}{3.8cm}
418 \includegraphics[width=3.3cm]{tem_c-si-db.eps}
422 \begin{minipage}{3.8cm}
424 \includegraphics[width=3.3cm]{tem_3c-sic.eps}
428 \begin{minipage}{4cm}
430 C-Si dimers (dumbbells)\\[-0.1cm]
431 on Si interstitial sites
435 \begin{minipage}{4.2cm}
437 Agglomeration of C-Si dumbbells\\[-0.1cm]
438 $\Rightarrow$ dark contrasts
442 \begin{minipage}{4cm}
444 Precipitation of 3C-SiC in Si\\[-0.1cm]
445 $\Rightarrow$ Moir\'e fringes\\[-0.1cm]
446 \& release of Si self-interstitials
450 \begin{minipage}{3.8cm}
452 \includegraphics[width=3.3cm]{sic_prec_seq_01.eps}
456 \begin{minipage}{3.8cm}
458 \includegraphics[width=3.3cm]{sic_prec_seq_02.eps}
462 \begin{minipage}{3.8cm}
464 \includegraphics[width=3.3cm]{sic_prec_seq_03.eps}
468 \begin{pspicture}(0,0)(0,0)
469 \psline[linewidth=4pt]{->}(8.5,2)(9.0,2)
470 \psellipse[linecolor=blue](11.5,5.8)(0.3,0.5)
471 \rput{-20}{\psellipse[linecolor=blue](3.3,8.1)(0.3,0.5)}
472 \psline[linewidth=4pt]{->}(4.0,2)(4.5,2)
473 \rput(12.7,0.3){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
474 $4a_{\text{Si}}=5a_{\text{SiC}}$
476 \rput(12.2,8){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
477 \hkl(h k l) planes match
479 \rput(9.7,6.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
489 Molecular dynamics (MD) simulations
498 \item Microscopic description of N particle system
499 \item Analytical interaction potential
500 \item Numerical integration using Newtons equation of motion\\
501 as a propagation rule in 6N-dimensional phase space
502 \item Observables obtained by time and/or ensemble averages
504 {\bf Details of the simulation:}
506 \item Integration: Velocity Verlet, timestep: $1\text{ fs}$
507 \item Ensemble: NpT (isothermal-isobaric)
509 \item Berendsen thermostat:
510 $\tau_{\text{T}}=100\text{ fs}$
511 \item Berendsen barostat:\\
512 $\tau_{\text{P}}=100\text{ fs}$,
513 $\beta^{-1}=100\text{ GPa}$
515 \item Erhart/Albe potential: Tersoff-like bond order potential
518 E = \frac{1}{2} \sum_{i \neq j} \pot_{ij}, \quad
519 \pot_{ij} = f_C(r_{ij}) \left[ f_R(r_{ij}) + b_{ij} f_A(r_{ij}) \right]
523 \begin{picture}(0,0)(-230,-30)
524 \includegraphics[width=5cm]{tersoff_angle.eps}
532 Density functional theory (DFT) calculations
537 Basic ingredients necessary for DFT
540 \item \underline{Hohenberg-Kohn theorem} - ground state density $n_0(r)$ ...
542 \item ... uniquely determines the ground state potential
544 \item ... minimizes the systems total energy
546 \item \underline{Born-Oppenheimer}
547 - $N$ moving electrons in an external potential of static nuclei
549 H\Psi = \left[-\sum_i^N \frac{\hbar^2}{2m}\nabla_i^2
550 +\sum_i^N V_{\text{ext}}(r_i)
551 +\sum_{i<j}^N V_{e-e}(r_i,r_j)\right]\Psi=E\Psi
553 \item \underline{Effective potential}
554 - averaged electrostatic potential \& exchange and correlation
556 V_{\text{eff}}(r)=V_{\text{ext}}(r)+\int\frac{e^2 n(r')}{|r-r'|}d^3r'
559 \item \underline{Kohn-Sham system}
560 - Schr\"odinger equation of N non-interacting particles
562 \left[ -\frac{\hbar^2}{2m}\nabla^2 + V_{\text{eff}}(r) \right] \Phi_i(r)
567 n(r)=\sum_i^N|\Phi_i(r)|^2
569 \item \underline{Self-consistent solution}\\
570 $n(r)$ depends on $\Phi_i$, which depend on $V_{\text{eff}}$,
571 which in turn depends on $n(r)$
572 \item \underline{Variational principle}
573 - minimize total energy with respect to $n(r)$
581 Density functional theory (DFT) calculations
588 Details of applied DFT calculations in this work
591 \item \underline{Exchange correlation functional}
592 - approximations for the inhomogeneous electron gas
594 \item LDA: $E_{\text{XC}}^{\text{LDA}}[n]=\int \epsilon_{\text{XC}}(n)n(r)d^3r$
595 \item GGA: $E_{\text{XC}}^{\text{GGA}}[n]=\int \epsilon_{\text{XC}}(n,\nabla n)n(r)d^3r$
597 \item \underline{Plane wave basis set}
598 - approximation of the wavefunction $\Phi_i$ by plane waves $\phi_j$
601 \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}}
602 \qquad ({\color{blue}300\text{ eV}})
604 \item \underline{Brillouin zone sampling} -
605 {\color{blue}$\Gamma$-point only} calculations
606 \item \underline{Pseudo potential}
607 - consider only the valence electrons
608 \item \underline{Code} - VASP 4.6
613 MD and structural optimization
616 \item MD integration: Gear predictor corrector algorithm
617 \item Pressure control: Parrinello-Rahman pressure control
618 \item Structural optimization: Conjugate gradient method
621 \begin{pspicture}(0,0)(0,0)
622 \psellipse[linecolor=blue](1.5,6.75)(0.5,0.3)
630 C and Si self-interstitial point defects in silicon
637 \begin{minipage}{8cm}
639 \begin{pspicture}(0,0)(7,5)
640 \rput(3.5,4){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
643 \item Creation of c-Si simulation volume
644 \item Periodic boundary conditions
645 \item $T=0\text{ K}$, $p=0\text{ bar}$
648 \rput(3.5,2.1){\rnode{insert}{\psframebox{
651 Insertion of interstitial C/Si atoms
654 \rput(3.5,1){\rnode{cool}{\psframebox[fillstyle=solid,fillcolor=lbb]{
657 Relaxation / structural energy minimization
660 \ncline[]{->}{init}{insert}
661 \ncline[]{->}{insert}{cool}
664 \begin{minipage}{5cm}
665 \includegraphics[width=5cm]{unit_cell_e.eps}\\
668 \begin{minipage}{9cm}
669 \begin{tabular}{l c c}
671 & size [unit cells] & \# atoms\\
673 VASP & $3\times 3\times 3$ & $216\pm 1$ \\
674 Erhart/Albe & $9\times 9\times 9$ & $5832\pm 1$\\
678 \begin{minipage}{4cm}
679 {\color{red}$\bullet$} Tetrahedral\\
680 {\color{green}$\bullet$} Hexagonal\\
681 {\color{yellow}$\bullet$} \hkl<1 0 0> dumbbell\\
682 {\color{magenta}$\bullet$} \hkl<1 1 0> dumbbell\\
683 {\color{cyan}$\bullet$} Bond-centered\\
684 {\color{black}$\bullet$} Vacancy / Substitutional
693 \begin{minipage}{9.5cm}
696 Si self-interstitial point defects in silicon\\
699 \begin{tabular}{l c c c c c}
701 $E_{\text{f}}$ [eV] & \hkl<1 1 0> DB & H & T & \hkl<1 0 0> DB & V \\
703 VASP & \underline{3.39} & 3.42 & 3.77 & 4.41 & 3.63 \\
704 Erhart/Albe & 4.39 & 4.48$^*$ & \underline{3.40} & 5.42 & 3.13 \\
706 \end{tabular}\\[0.2cm]
708 \begin{minipage}{4.7cm}
709 \includegraphics[width=4.7cm]{e_kin_si_hex.ps}
711 \begin{minipage}{4.7cm}
713 {\tiny nearly T $\rightarrow$ T}\\
715 \includegraphics[width=4.7cm]{nhex_tet.ps}
718 \underline{Hexagonal} \hspace{2pt}
719 \href{../video/si_self_int_hexa.avi}{$\rhd$}\\[0.1cm]
721 \begin{minipage}{2.7cm}
722 $E_{\text{f}}^*=4.48\text{ eV}$\\
723 \includegraphics[width=2.7cm]{si_pd_albe/hex_a.eps}
725 \begin{minipage}{0.4cm}
730 \begin{minipage}{2.7cm}
731 $E_{\text{f}}=3.96\text{ eV}$\\
732 \includegraphics[width=2.8cm]{si_pd_albe/hex.eps}
735 \begin{minipage}{2.9cm}
737 \underline{Vacancy}\\
738 \includegraphics[width=3.0cm]{si_pd_albe/vac.eps}
743 \begin{minipage}{3.5cm}
746 \underline{\hkl<1 1 0> dumbbell}\\
747 \includegraphics[width=3.0cm]{si_pd_albe/110.eps}\\
748 \underline{Tetrahedral}\\
749 \includegraphics[width=3.0cm]{si_pd_albe/tet.eps}\\
750 \underline{\hkl<1 0 0> dumbbell}\\
751 \includegraphics[width=3.0cm]{si_pd_albe/100.eps}
763 C interstitial point defects in silicon\\[-0.1cm]
766 \begin{tabular}{l c c c c c c r}
768 $E_{\text{f}}$ & T & H & \hkl<1 0 0> DB & \hkl<1 1 0> DB & S & B & \cs{} \& \si\\
770 VASP & unstable & unstable & \underline{3.72} & 4.16 & 1.95 & 4.66 & {\color{green}4.17}\\
771 Erhart/Albe MD & 6.09 & 9.05$^*$ & \underline{3.88} & 5.18 & {\color{red}0.75} & 5.59$^*$ & {\color{green}4.43} \\
773 \end{tabular}\\[0.1cm]
776 \begin{minipage}{2.7cm}
777 \underline{Hexagonal} \hspace{2pt}
778 \href{../video/c_in_si_int_hexa.avi}{$\rhd$}\\
779 $E_{\text{f}}^*=9.05\text{ eV}$\\
780 \includegraphics[width=2.7cm]{c_pd_albe/hex.eps}
782 \begin{minipage}{0.4cm}
787 \begin{minipage}{2.7cm}
788 \underline{\hkl<1 0 0>}\\
789 $E_{\text{f}}=3.88\text{ eV}$\\
790 \includegraphics[width=2.7cm]{c_pd_albe/100.eps}
793 \begin{minipage}{2cm}
796 \begin{minipage}{3cm}
798 \underline{Tetrahedral}\\
799 \includegraphics[width=3.0cm]{c_pd_albe/tet.eps}
804 \begin{minipage}{2.7cm}
805 \underline{Bond-centered}\\
806 $E_{\text{f}}^*=5.59\text{ eV}$\\
807 \includegraphics[width=2.7cm]{c_pd_albe/bc.eps}
809 \begin{minipage}{0.4cm}
814 \begin{minipage}{2.7cm}
815 \underline{\hkl<1 1 0> dumbbell}\\
816 $E_{\text{f}}=5.18\text{ eV}$\\
817 \includegraphics[width=2.7cm]{c_pd_albe/110.eps}
820 \begin{minipage}{2cm}
823 \begin{minipage}{3cm}
825 \underline{Substitutional}\\
826 \includegraphics[width=3.0cm]{c_pd_albe/sub.eps}
837 C \hkl<1 0 0> dumbbell interstitial configuration\\
841 \begin{tabular}{l c c c c c c c c}
843 Distances [nm] & $r(1C)$ & $r(2C)$ & $r(3C)$ & $r(12)$ & $r(13)$ & $r(34)$ & $r(23)$ & $r(25)$ \\
845 Erhart/Albe & 0.175 & 0.329 & 0.186 & 0.226 & 0.300 & 0.343 & 0.423 & 0.425 \\
846 VASP & 0.174 & 0.341 & 0.182 & 0.229 & 0.286 & 0.347 & 0.422 & 0.417 \\
848 \end{tabular}\\[0.2cm]
849 \begin{tabular}{l c c c c }
851 Angles [$^{\circ}$] & $\theta_1$ & $\theta_2$ & $\theta_3$ & $\theta_4$ \\
853 Erhart/Albe & 140.2 & 109.9 & 134.4 & 112.8 \\
854 VASP & 130.7 & 114.4 & 146.0 & 107.0 \\
856 \end{tabular}\\[0.2cm]
857 \begin{tabular}{l c c c}
859 Displacements [nm]& $a$ & $b$ & $|a|+|b|$ \\
861 Erhart/Albe & 0.084 & -0.091 & 0.175 \\
862 VASP & 0.109 & -0.065 & 0.174 \\
864 \end{tabular}\\[0.6cm]
867 \begin{minipage}{3.0cm}
869 \underline{Erhart/Albe}
870 \includegraphics[width=3.0cm]{c_pd_albe/100_cmp.eps}
873 \begin{minipage}{3.0cm}
876 \includegraphics[width=3.0cm]{c_pd_vasp/100_cmp.eps}
880 \begin{picture}(0,0)(-185,10)
881 \includegraphics[width=6.8cm]{100-c-si-db_cmp.eps}
883 \begin{picture}(0,0)(-280,-150)
884 \includegraphics[width=3.3cm]{c_pd_vasp/eden.eps}
887 \begin{pspicture}(0,0)(0,0)
888 \psellipse[linecolor=green](5.18,5.92)(0.5,0.3)
889 \psellipse[linecolor=red](3.45,5.92)(1.0,0.4)
890 \psellipse[linecolor=blue](2.7,6.92)(0.9,0.2)
891 \psellipse[linecolor=blue](4.65,6.92)(0.9,0.2)
900 \begin{minipage}{8.5cm}
903 Bond-centered interstitial configuration\\[-0.1cm]
906 \begin{minipage}{3.0cm}
907 \includegraphics[width=2.8cm]{c_pd_vasp/bc_2333.eps}\\
909 \begin{minipage}{5.2cm}
911 \item Linear Si-C-Si bond
912 \item Si: one C \& 3 Si neighbours
913 \item Spin polarized calculations
914 \item No saddle point!\\
921 \begin{minipage}[t]{6.5cm}
922 \begin{minipage}[t]{1.2cm}
924 {\tiny sp$^3$}\\[0.8cm]
925 \underline{${\color{black}\uparrow}$}
926 \underline{${\color{black}\uparrow}$}
927 \underline{${\color{black}\uparrow}$}
928 \underline{${\color{red}\uparrow}$}\\
931 \begin{minipage}[t]{1.4cm}
933 {\color{red}M}{\color{blue}O}\\[0.8cm]
934 \underline{${\color{blue}\uparrow}{\color{white}\downarrow}$}\\
935 $\sigma_{\text{ab}}$\\[0.5cm]
936 \underline{${\color{red}\uparrow}{\color{blue}\downarrow}$}\\
940 \begin{minipage}[t]{1.0cm}
944 \underline{${\color{white}\uparrow\uparrow}$}
945 \underline{${\color{white}\uparrow\uparrow}$}\\
947 \underline{${\color{blue}\uparrow}{\color{blue}\downarrow}$}
948 \underline{${\color{blue}\uparrow}{\color{blue}\downarrow}$}\\
952 \begin{minipage}[t]{1.4cm}
954 {\color{blue}M}{\color{green}O}\\[0.8cm]
955 \underline{${\color{blue}\uparrow}{\color{white}\downarrow}$}\\
956 $\sigma_{\text{ab}}$\\[0.5cm]
957 \underline{${\color{green}\uparrow}{\color{blue}\downarrow}$}\\
961 \begin{minipage}[t]{1.2cm}
964 {\tiny sp$^3$}\\[0.8cm]
965 \underline{${\color{green}\uparrow}$}
966 \underline{${\color{black}\uparrow}$}
967 \underline{${\color{black}\uparrow}$}
968 \underline{${\color{black}\uparrow}$}\\
976 \begin{minipage}{4.5cm}
977 \includegraphics[width=4cm]{c_100_mig_vasp/im_spin_diff.eps}
979 \begin{minipage}{3.5cm}
980 {\color{gray}$\bullet$} Spin up\\
981 {\color{green}$\bullet$} Spin down\\
982 {\color{blue}$\bullet$} Resulting spin up\\
983 {\color{yellow}$\bullet$} Si atoms\\
984 {\color{red}$\bullet$} C atom
989 \begin{minipage}{4.2cm}
991 \includegraphics[width=4.3cm]{c_pd_vasp/bc_2333_ksl.ps}\\
992 {\color{green}$\Box$} {\tiny unoccupied}\\
993 {\color{red}$\bullet$} {\tiny occupied}
1002 Migration of the C \hkl<1 0 0> dumbbell interstitial
1007 {\small Investigated pathways}
1009 \begin{minipage}{8.5cm}
1010 \begin{minipage}{8.3cm}
1011 \underline{\hkl<0 0 -1> $\rightarrow$ \hkl<0 0 1>}\\
1012 \begin{minipage}{2.4cm}
1013 \includegraphics[width=2.4cm]{c_pd_vasp/100_2333.eps}
1015 \begin{minipage}{0.4cm}
1018 \begin{minipage}{2.4cm}
1019 \includegraphics[width=2.4cm]{c_pd_vasp/bc_2333.eps}
1021 \begin{minipage}{0.4cm}
1024 \begin{minipage}{2.4cm}
1025 \includegraphics[width=2.4cm]{c_pd_vasp/100_next_2333.eps}
1028 \begin{minipage}{8.3cm}
1029 \underline{\hkl<0 0 -1> $\rightarrow$ \hkl<0 -1 0>}\\
1030 \begin{minipage}{2.4cm}
1031 \includegraphics[width=2.4cm]{c_pd_vasp/100_2333.eps}
1033 \begin{minipage}{0.4cm}
1036 \begin{minipage}{2.4cm}
1037 \includegraphics[width=2.4cm]{c_pd_vasp/00-1-0-10_2333.eps}
1039 \begin{minipage}{0.4cm}
1042 \begin{minipage}{2.4cm}
1043 \includegraphics[width=2.4cm]{c_pd_vasp/0-10_2333.eps}
1046 \begin{minipage}{8.3cm}
1047 \underline{\hkl<0 0 -1> $\rightarrow$ \hkl<0 -1 0> (in place)}\\
1048 \begin{minipage}{2.4cm}
1049 \includegraphics[width=2.4cm]{c_pd_vasp/100_2333.eps}
1051 \begin{minipage}{0.4cm}
1054 \begin{minipage}{2.4cm}
1055 \includegraphics[width=2.4cm]{c_pd_vasp/00-1_ip0-10_2333.eps}
1057 \begin{minipage}{0.4cm}
1060 \begin{minipage}{2.4cm}
1061 \includegraphics[width=2.4cm]{c_pd_vasp/0-10_ip_2333.eps}
1066 \begin{minipage}{4.2cm}
1067 {\small Constrained relaxation\\
1068 technique (CRT) method}\\
1069 \includegraphics[width=4cm]{crt_orig.eps}
1071 \item Constrain diffusing atom
1072 \item Static constraints
1075 {\small Modifications}\\
1076 \includegraphics[width=4cm]{crt_mod.eps}
1078 \item Constrain all atoms
1079 \item Update individual\\
1090 Migration of the C \hkl<1 0 0> dumbbell interstitial
1096 \begin{minipage}{5.9cm}
1098 \includegraphics[width=5.8cm]{im_00-1_nosym_sp_fullct_thesis.ps}\\[0.45cm]
1101 \begin{picture}(0,0)(60,0)
1102 \includegraphics[width=1cm]{vasp_mig/00-1.eps}
1104 \begin{picture}(0,0)(-5,0)
1105 \includegraphics[width=1cm]{vasp_mig/bc_00-1_sp.eps}
1107 \begin{picture}(0,0)(-55,0)
1108 \includegraphics[width=1cm]{vasp_mig/bc.eps}
1110 \begin{picture}(0,0)(12.5,10)
1111 \includegraphics[width=1cm]{110_arrow.eps}
1113 \begin{picture}(0,0)(90,0)
1114 \includegraphics[height=0.9cm]{001_arrow.eps}
1120 \begin{minipage}{0.3cm}
1124 \begin{minipage}{5.9cm}
1126 \includegraphics[width=5.9cm]{vasp_mig/00-1_0-10_nosym_sp_fullct.ps}\\[0.5cm]
1129 \begin{picture}(0,0)(60,0)
1130 \includegraphics[width=1cm]{vasp_mig/00-1_a.eps}
1132 \begin{picture}(0,0)(5,0)
1133 \includegraphics[width=1cm]{vasp_mig/00-1_0-10_sp.eps}
1135 \begin{picture}(0,0)(-55,0)
1136 \includegraphics[width=1cm]{vasp_mig/0-10.eps}
1138 \begin{picture}(0,0)(12.5,10)
1139 \includegraphics[width=1cm]{100_arrow.eps}
1141 \begin{picture}(0,0)(90,0)
1142 \includegraphics[height=0.9cm]{001_arrow.eps}
1152 \begin{minipage}{5.9cm}
1154 \includegraphics[width=5.9cm]{vasp_mig/00-1_ip0-10_nosym_sp_fullct.ps}\\[0.6cm]
1157 \begin{picture}(0,0)(60,0)
1158 \includegraphics[width=0.9cm]{vasp_mig/00-1_b.eps}
1160 \begin{picture}(0,0)(10,0)
1161 \includegraphics[width=0.9cm]{vasp_mig/00-1_ip0-10_sp.eps}
1163 \begin{picture}(0,0)(-60,0)
1164 \includegraphics[width=0.9cm]{vasp_mig/0-10_b.eps}
1166 \begin{picture}(0,0)(12.5,10)
1167 \includegraphics[width=1cm]{100_arrow.eps}
1169 \begin{picture}(0,0)(90,0)
1170 \includegraphics[height=0.9cm]{001_arrow.eps}
1176 \begin{minipage}{0.3cm}
1179 \begin{minipage}{6.5cm}
1182 \item Energetically most favorable path
1185 \item Activation energy: $\approx$ 0.9 eV
1186 \item Experimental values: 0.73 ... 0.87 eV
1188 $\Rightarrow$ {\color{blue}Diffusion} path identified!
1189 \item Reorientation (path 3)
1191 \item More likely composed of two consecutive steps of type 2
1192 \item Experimental values: 0.77 ... 0.88 eV
1194 $\Rightarrow$ {\color{blue}Reorientation} transition identified!
1203 Migration of the C \hkl<1 0 0> dumbbell interstitial
1210 \begin{minipage}{6.5cm}
1213 \begin{minipage}[t]{5.9cm}
1215 \includegraphics[width=5.9cm]{bc_00-1.ps}\\[2.35cm]
1218 \begin{pspicture}(0,0)(0,0)
1219 \psframe[linecolor=red,fillstyle=none](-2.8,1.35)(3.3,2.7)
1221 \begin{picture}(0,0)(60,-50)
1222 \includegraphics[width=1cm]{albe_mig/bc_00-1_red_00.eps}
1224 \begin{picture}(0,0)(5,-50)
1225 \includegraphics[width=1cm]{albe_mig/bc_00-1_red_01.eps}
1227 \begin{picture}(0,0)(-55,-50)
1228 \includegraphics[width=1cm]{albe_mig/bc_00-1_red_02.eps}
1230 \begin{picture}(0,0)(12.5,-40)
1231 \includegraphics[width=1cm]{110_arrow.eps}
1233 \begin{picture}(0,0)(90,-45)
1234 \includegraphics[height=0.9cm]{001_arrow.eps}
1236 \begin{pspicture}(0,0)(0,0)
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1243 \includegraphics[width=0.9cm]{albe_mig/bc_00-1_02.eps}
1245 \begin{picture}(0,0)(-5,-15)
1246 \includegraphics[width=0.9cm]{albe_mig/bc_00-1_03.eps}
1248 \begin{picture}(0,0)(-55,-15)
1249 \includegraphics[width=0.9cm]{albe_mig/bc_00-1_04.eps}
1251 \begin{picture}(0,0)(12.5,-5)
1252 \includegraphics[width=1cm]{100_arrow.eps}
1254 \begin{picture}(0,0)(90,-15)
1255 \includegraphics[height=0.9cm]{010_arrow.eps}
1261 \begin{minipage}{5.9cm}
1264 \item Lowest activation energy: $\approx$ 2.2 eV
1265 \item 2.4 times higher than VASP
1266 \item Different pathway
1271 \begin{minipage}{6.5cm}
1274 \begin{minipage}{5.9cm}
1276 %\includegraphics[width=5.9cm]{00-1_0-10.ps}\\[0.75cm]
1279 %\begin{pspicture}(0,0)(0,0)
1280 %\psframe[linecolor=red,fillstyle=none](-2.8,-0.25)(3.3,1.1)
1282 %\begin{picture}(0,0)(60,-5)
1283 %\includegraphics[width=0.9cm]{albe_mig/00-1_0-10_red_00.eps}
1285 %\begin{picture}(0,0)(0,-5)
1286 %\includegraphics[width=0.9cm]{albe_mig/00-1_0-10_red_min.eps}
1288 %\begin{picture}(0,0)(-55,-5)
1289 %\includegraphics[width=0.9cm]{albe_mig/00-1_0-10_red_03.eps}
1291 %\begin{picture}(0,0)(12.5,5)
1292 %\includegraphics[width=1cm]{100_arrow.eps}
1294 %\begin{picture}(0,0)(90,0)
1295 %\includegraphics[height=0.9cm]{001_arrow.eps}
1303 %\begin{minipage}{5.9cm}
1304 \includegraphics[width=5.9cm]{00-1_110_0-10_mig_albe.ps}
1308 \begin{minipage}{5.9cm}
1309 Transition involving \ci{} \hkl<1 1 0>
1311 \item Bond-centered configuration unstable\\
1312 $\rightarrow$ \ci{} \hkl<1 1 0> dumbbell
1313 \item Transition minima of path 2 \& 3\\
1314 $\rightarrow$ \ci{} \hkl<1 1 0> dumbbell
1315 \item Activation energy: $\approx$ 2.2 eV \& 0.9 eV
1316 \item 2.4 - 3.4 times higher than VASP
1317 \item Rotation of dumbbell orientation
1328 Combinations with a C-Si \hkl<1 0 0>-type interstitial
1338 E_{\text{f}}^{\text{defect combination}}-
1339 E_{\text{f}}^{\text{C \hkl<0 0 -1> dumbbell}}-
1340 E_{\text{f}}^{\text{2nd defect}}
1346 \begin{tabular}{l c c c c c c}
1348 $E_{\text{b}}$ [eV] & 1 & 2 & 3 & 4 & 5 & R\\
1350 \hkl<0 0 -1> & {\color{red}-0.08} & -1.15 & {\color{red}-0.08} & 0.04 & -1.66 & -0.19\\
1351 \hkl<0 0 1> & 0.34 & 0.004 & -2.05 & 0.26 & -1.53 & -0.19\\
1352 \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}\\
1353 \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}\\
1354 \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}\\
1355 \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}\\
1357 C substitutional (C$_{\text{S}}$) & 0.26 & -0.51 & -0.93 & -0.15 & 0.49 & -0.05\\
1358 Vacancy & -5.39 ($\rightarrow$ C$_{\text{S}}$) & -0.59 & -3.14 & -0.54 & -0.50 & -0.31\\
1367 \begin{minipage}[t]{3.8cm}
1368 \underline{\hkl<1 0 0> at position 1}\\[0.1cm]
1369 \includegraphics[width=3.5cm]{00-1dc/2-25.eps}
1371 \begin{minipage}[t]{3.5cm}
1372 \underline{\hkl<0 -1 0> at position 1}\\[0.1cm]
1373 \includegraphics[width=3.2cm]{00-1dc/2-39.eps}
1375 \begin{minipage}[t]{5.5cm}
1377 \item Restricted to VASP simulations
1378 \item $E_{\text{b}}=0$ for isolated non-interacting defects
1379 \item $E_{\text{b}} \rightarrow 0$ for increasing distance (R)
1380 \item Stress compensation / increase
1381 \item Most favorable: C clustering
1382 \item Unfavored: antiparallel orientations
1383 \item Indication of energetically favored\\
1388 \begin{picture}(0,0)(-295,-130)
1389 \includegraphics[width=3.5cm]{comb_pos.eps}
1397 Combinations of C-Si \hkl<1 0 0>-type interstitials
1404 Energetically most favorable combinations along \hkl<1 1 0>
1409 \begin{tabular}{l c c c c c c}
1411 & 1 & 2 & 3 & 4 & 5 & 6\\
1413 $E_{\text{b}}$ [eV] & -2.39 & -1.88 & -0.59 & -0.31 & -0.24 & -0.21 \\
1414 C-C distance [\AA] & 1.4 & 4.6 & 6.5 & 8.6 & 10.5 & 10.8 \\
1415 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>\\
1422 \begin{minipage}{7.0cm}
1423 \includegraphics[width=7cm]{db_along_110_cc.ps}
1425 \begin{minipage}{6.0cm}
1428 Interaction proportional to reciprocal cube of C-C distance
1430 Saturation in the immediate vicinity
1441 Combinations of substitutional C and \hkl<1 1 0> Si self-interstitials
1447 \begin{minipage}{3.2cm}
1448 \includegraphics[width=3cm]{sub_110_combo.eps}
1450 \begin{minipage}{7.8cm}
1451 \begin{tabular}{l c c c c c c}
1453 C$_{\text{sub}}$ & \hkl<1 1 0> & \hkl<-1 1 0> & \hkl<0 1 1> & \hkl<0 -1 1> &
1454 \hkl<1 0 1> & \hkl<-1 0 1> \\
1456 1 & \RM{1} & \RM{3} & \RM{3} & \RM{1} & \RM{3} & \RM{1} \\
1457 2 & \RM{2} & A & A & \RM{2} & C & \RM{5} \\
1458 3 & \RM{3} & \RM{1} & \RM{3} & \RM{1} & \RM{1} & \RM{3} \\
1459 4 & \RM{4} & B & D & E & E & D \\
1460 5 & \RM{5} & C & A & \RM{2} & A & \RM{2} \\
1467 \begin{tabular}{l c c c c c c c c c c}
1469 Conf & \RM{1} & \RM{2} & \RM{3} & \RM{4} & \RM{5} & A & B & C & D & E \\
1471 $E_{\text{f}}$ [eV]& 4.37 & 5.26 & 5.57 & 5.37 & 5.12 & 5.10 & 5.32 & 5.28 & 5.39 & 5.32 \\
1472 $E_{\text{b}}$ [eV] & -0.97 & -0.08 & 0.22 & -0.02 & -0.23 & -0.25 & -0.02 & -0.06 & 0.05 & -0.03 \\
1473 $r$ [nm] & 0.292 & 0.394 & 0.241 & 0.453 & 0.407 & 0.408 & 0.452 & 0.392 & 0.456 & 0.453\\
1478 \begin{minipage}{6.0cm}
1479 \includegraphics[width=5.8cm]{c_sub_si110.ps}
1481 \begin{minipage}{7cm}
1484 \item IBS: C may displace Si\\
1485 $\Rightarrow$ C$_{\text{sub}}$ + \hkl<1 1 0> Si self-interstitial
1487 \hkl<1 1 0>-type $\rightarrow$ favored combination
1488 \renewcommand\labelitemi{$\Rightarrow$}
1489 \item Less favorable than C-Si \hkl<1 0 0> dumbbell\\
1490 ($E_{\text{f}}=3.88\text{ eV}$)
1491 \item Interaction drops quickly to zero\\
1492 (low interaction capture radius)
1501 Migration in C-Si \hkl<1 0 0> and vacancy combinations
1508 \begin{minipage}[t]{3cm}
1509 \underline{Pos 2, $E_{\text{b}}=-0.59\text{ eV}$}\\
1510 \includegraphics[width=2.8cm]{00-1dc/0-59.eps}
1512 \begin{minipage}[t]{7cm}
1515 Low activation energies\\
1516 High activation energies for reverse processes\\
1518 {\color{blue}C$_{\text{sub}}$ very stable}\\
1522 Without nearby \hkl<1 1 0> Si self-interstitial (IBS)\\
1524 {\color{blue}Formation of SiC by successive substitution by C}
1528 \begin{minipage}[t]{3cm}
1529 \underline{Pos 3, $E_{\text{b}}=-3.14\text{ eV}$}\\
1530 \includegraphics[width=2.8cm]{00-1dc/3-14.eps}
1535 \begin{minipage}{5.9cm}
1536 \includegraphics[width=5.9cm]{vasp_mig/comb_mig_3-2_vac_fullct.ps}\\[0.6cm]
1538 \begin{picture}(0,0)(70,0)
1539 \includegraphics[width=1.4cm]{vasp_mig/comb_2-1_init.eps}
1541 \begin{picture}(0,0)(30,0)
1542 \includegraphics[width=1.4cm]{vasp_mig/comb_2-1_seq_03.eps}
1544 \begin{picture}(0,0)(-10,0)
1545 \includegraphics[width=1.4cm]{vasp_mig/comb_2-1_seq_06.eps}
1547 \begin{picture}(0,0)(-48,0)
1548 \includegraphics[width=1.4cm]{vasp_mig/comb_2-1_final.eps}
1550 \begin{picture}(0,0)(12.5,5)
1551 \includegraphics[width=1cm]{100_arrow.eps}
1553 \begin{picture}(0,0)(97,-10)
1554 \includegraphics[height=0.9cm]{001_arrow.eps}
1560 \begin{minipage}{0.3cm}
1564 \begin{minipage}{5.9cm}
1565 \includegraphics[width=5.9cm]{vasp_mig/comb_mig_4-2_vac_fullct.ps}\\[0.1cm]
1567 \begin{picture}(0,0)(60,0)
1568 \includegraphics[width=0.9cm]{vasp_mig/comb_3-1_init.eps}
1570 \begin{picture}(0,0)(25,0)
1571 \includegraphics[width=0.9cm]{vasp_mig/comb_3-1_seq_03.eps}
1573 \begin{picture}(0,0)(-20,0)
1574 \includegraphics[width=0.9cm]{vasp_mig/comb_3-1_seq_07.eps}
1576 \begin{picture}(0,0)(-55,0)
1577 \includegraphics[width=0.9cm]{vasp_mig/comb_3-1_final.eps}
1579 \begin{picture}(0,0)(12.5,5)
1580 \includegraphics[width=1cm]{100_arrow.eps}
1582 \begin{picture}(0,0)(95,0)
1583 \includegraphics[height=0.9cm]{001_arrow.eps}
1595 Conclusion of defect / migration / combined defect simulations
1604 \item Accurately described by quantum-mechanical simulations
1605 \item Less accurate description by classical potential simulations
1606 \item Underestimated formation energy of \cs{} by classical approach
1607 \item Both methods predict same ground state: \ci{} \hkl<1 0 0> dumbbell
1612 \item C migration pathway in Si identified
1613 \item Consistent with reorientation and diffusion experiments
1616 \item Different path and ...
1617 \item overestimated barrier by classical potential calculations
1620 Concerning the precipitation mechanism
1622 \item Agglomeration of C-Si dumbbells energetically favorable
1623 (stress compensation)
1624 \item C-Si indeed favored compared to
1625 C$_{\text{sub}}$ \& \hkl<1 1 0> Si self-interstitial
1626 \item Possible low interaction capture radius of
1627 C$_{\text{sub}}$ \& \hkl<1 1 0> Si self-interstitial
1628 \item Low barrier for
1629 \ci{} \hkl<1 0 0> $\rightarrow$ \cs{} \& \si{} \hkl<1 1 0>
1630 \item In absence of nearby \hkl<1 1 0> Si self-interstitial:
1631 C-Si \hkl<1 0 0> + Vacancy $\rightarrow$ C$_{\text{sub}}$ (SiC)
1634 {\color{blue}Results suggest increased participation of \cs}
1642 Silicon carbide precipitation simulations
1648 \begin{pspicture}(0,0)(12,6.5)
1650 \rput(3.5,5.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
1653 \item Create c-Si volume
1654 \item Periodc boundary conditions
1655 \item Set requested $T$ and $p=0\text{ bar}$
1656 \item Equilibration of $E_{\text{kin}}$ and $E_{\text{pot}}$
1659 \rput(3.5,2.7){\rnode{insert}{\psframebox[fillstyle=solid,fillcolor=lachs]{
1661 Insertion of C atoms at constant T
1663 \item total simulation volume {\pnode{in1}}
1664 \item volume of minimal SiC precipitate {\pnode{in2}}
1665 \item volume consisting of Si atoms to form a minimal {\pnode{in3}}\\
1669 \rput(3.5,1){\rnode{cool}{\psframebox[fillstyle=solid,fillcolor=lbb]{
1671 Run for 100 ps followed by cooling down to $20\, ^{\circ}\textrm{C}$
1673 \ncline[]{->}{init}{insert}
1674 \ncline[]{->}{insert}{cool}
1675 \psframe[fillstyle=solid,fillcolor=white](7.5,0.7)(13.5,6.3)
1676 \rput(7.8,6){\footnotesize $V_1$}
1677 \psframe[fillstyle=solid,fillcolor=lightgray](9,2)(12,5)
1678 \rput(9.2,4.85){\tiny $V_2$}
1679 \psframe[fillstyle=solid,fillcolor=gray](9.25,2.25)(11.75,4.75)
1680 \rput(9.55,4.45){\footnotesize $V_3$}
1681 \rput(7.9,3.2){\pnode{ins1}}
1682 \rput(9.22,2.8){\pnode{ins2}}
1683 \rput(11.0,2.4){\pnode{ins3}}
1684 \ncline[]{->}{in1}{ins1}
1685 \ncline[]{->}{in2}{ins2}
1686 \ncline[]{->}{in3}{ins3}
1691 \item Restricted to classical potential simulations
1692 \item $V_2$ and $V_3$ considered due to low diffusion
1693 \item Amount of C atoms: 6000
1694 ($r_{\text{prec}}\approx 3.1\text{ nm}$, IBS: 2 ... 4 nm)
1695 \item Simulation volume: $31\times 31\times 31$ unit cells
1704 Silicon carbide precipitation simulations at $450\,^{\circ}\mathrm{C}$ as in IBS
1709 \begin{minipage}{6.5cm}
1710 \includegraphics[width=6.4cm]{sic_prec_450_si-si_c-c.ps}
1712 \begin{minipage}{6.5cm}
1713 \includegraphics[width=6.4cm]{sic_prec_450_energy.ps}
1716 \begin{minipage}{6.5cm}
1717 \includegraphics[width=6.4cm]{sic_prec_450_si-c.ps}
1719 \begin{minipage}{6.5cm}
1721 \underline{Low C concentration ($V_1$)}\\
1722 \hkl<1 0 0> C-Si dumbbell dominated structure
1724 \item Si-C bumbs around 0.19 nm
1725 \item C-C peak at 0.31 nm (as expected in 3C-SiC):\\
1726 concatenated dumbbells of various orientation
1727 \item Si-Si NN distance stretched to 0.3 nm
1729 {\color{blue}$\Rightarrow$ C atoms in proper 3C-SiC distance first}\\
1730 \underline{High C concentration ($V_2$, $V_3$)}\\
1731 High amount of strongly bound C-C bonds\\
1732 Defect density $\uparrow$ $\Rightarrow$ considerable amount of damage\\
1733 Only short range order observable\\
1734 {\color{blue}$\Rightarrow$ amorphous SiC-like phase}
1742 Limitations of molecular dynamics and short range potentials
1749 \underline{Time scale problem of MD}\\[0.2cm]
1750 Minimize integration error\\
1751 $\Rightarrow$ discretization considerably smaller than
1752 reciprocal of fastest vibrational mode\\[0.1cm]
1753 Order of fastest vibrational mode: $10^{13} - 10^{14}\text{ Hz}$\\
1754 $\Rightarrow$ suitable choice of time step:
1755 $\tau=1\text{ fs}=10^{-15}\text{ s}$\\
1756 $\Rightarrow$ {\color{red}\underline{slow}} phase space propagation\\[0.1cm]
1757 Several local minima in energy surface separated by large energy barriers\\
1758 $\Rightarrow$ transition event corresponds to a multiple
1759 of vibrational periods\\
1760 $\Rightarrow$ phase transition made up of {\color{red}\underline{many}}
1761 infrequent transition events\\[0.1cm]
1762 {\color{blue}Accelerated methods:}
1763 \underline{Temperature accelerated} MD (TAD), self-guided MD \ldots
1767 \underline{Limitations related to the short range potential}\\[0.2cm]
1768 Cut-off function pushing forces and energies to zero between 1$^{\text{st}}$
1769 and 2$^{\text{nd}}$ next neighbours\\
1770 $\Rightarrow$ overestimated unphysical high forces of next neighbours
1776 Potential enhanced problem of slow phase space propagation
1781 \underline{Approach to the (twofold) problem}\\[0.2cm]
1782 Increased temperature simulations without TAD corrections\\
1783 (accelerated methods or higher time scales exclusively not sufficient)
1785 \begin{picture}(0,0)(-260,-30)
1787 \begin{minipage}{4.2cm}
1794 \item 3C-SiC also observed for higher T
1795 \item higher T inside sample
1796 \item structural evolution vs.\\
1797 equilibrium properties
1803 \begin{picture}(0,0)(-305,-155)
1805 \begin{minipage}{2.5cm}
1809 thermodynmic sampling
1820 Increased temperature simulations at low C concentration
1825 \begin{minipage}{6.5cm}
1826 \includegraphics[width=6.4cm]{tot_pc_thesis.ps}
1828 \begin{minipage}{6.5cm}
1829 \includegraphics[width=6.4cm]{tot_pc3_thesis.ps}
1832 \begin{minipage}{6.5cm}
1833 \includegraphics[width=6.4cm]{tot_pc2_thesis.ps}
1835 \begin{minipage}{6.5cm}
1837 \underline{Si-C bonds:}
1839 \item Vanishing cut-off artifact (above $1650\,^{\circ}\mathrm{C}$)
1840 \item Structural change: C-Si \hkl<1 0 0> $\rightarrow$ C$_{\text{sub}}$
1842 \underline{Si-Si bonds:}
1843 {\color{blue}Si-C$_{\text{sub}}$-Si} along \hkl<1 1 0>
1844 ($\rightarrow$ 0.325 nm)\\[0.1cm]
1845 \underline{C-C bonds:}
1847 \item C-C next neighbour pairs reduced (mandatory)
1848 \item Peak at 0.3 nm slightly shifted
1850 \item C-Si \hkl<1 0 0> combinations (dashed arrows)\\
1851 $\rightarrow$ C-Si \hkl<1 0 0> \& C$_{\text{sub}}$
1853 $\rightarrow$ pure {\color{blue}C$_{\text{sub}}$ combinations}
1855 \item Range [|-$\downarrow$]:
1856 {\color{blue}C$_{\text{sub}}$ \& C$_{\text{sub}}$
1857 with nearby Si$_{\text{I}}$}
1862 \begin{picture}(0,0)(-330,-74)
1865 \begin{minipage}{1.6cm}
1868 stretched SiC\\[-0.1cm]
1880 Increased temperature simulations at high C concentration
1885 \begin{minipage}{6.5cm}
1886 \includegraphics[width=6.4cm]{12_pc_thesis.ps}
1888 \begin{minipage}{6.5cm}
1889 \includegraphics[width=6.4cm]{12_pc_c_thesis.ps}
1893 Decreasing cut-off artifact\\
1894 High amount of {\color{red}damage} \& alignement to c-Si host matrix lost
1895 $\Rightarrow$ hard to categorize
1901 \begin{minipage}[t]{6.0cm}
1902 0.186 nm: Si-C pairs $\uparrow$\\
1903 (as expected in 3C-SiC)\\[0.2cm]
1904 0.282 nm: Si-C-C\\[0.2cm]
1905 $\approx$0.35 nm: C-Si-Si
1908 \begin{minipage}{0.2cm}
1912 \begin{minipage}[t]{6.0cm}
1913 0.15 nm: C-C pairs $\uparrow$\\
1914 (as expected in graphite/diamond)\\[0.2cm]
1915 0.252 nm: C-C-C (2$^{\text{nd}}$ NN for diamond)\\[0.2cm]
1916 0.31 nm: shifted towards 0.317 nm $\rightarrow$ C-Si-C
1923 {\color{red}Amorphous} SiC-like phase remains\\
1924 Slightly sharper peaks
1925 $\Rightarrow$ indicate slight {\color{blue}acceleration of dynamics}
1926 due to temperature\\[0.1cm]
1929 Actual SiC precipitation not accessible by MD
1938 Summary and Conclusions
1946 \begin{minipage}{12.9cm}
1951 \item Point defects excellently / fairly well described
1953 \item C$_{\text{sub}}$ drastically underestimated by EA
1954 \item EA predicts correct ground state:
1955 C$_{\text{sub}}$ \& \si{} $>$ \ci{}
1956 \item Identified migration path explaining
1957 diffusion and reorientation experiments by DFT
1958 \item EA fails to describe \ci{} migration:
1959 Wrong path \& overestimated barrier
1961 \item Combinations of defects
1963 \item Agglomeration of point defects energetically favorable
1964 by compensation of stress
1965 \item Formation of C-C unlikely
1966 \item C$_{\text{sub}}$ favored conditions (conceivable in IBS)
1967 \item \ci{} \hkl<1 0 0> $\leftrightarrow$ \cs{} \& \si{} \hkl<1 1 0>\\
1968 Low barrier (\unit[0.77]{eV}) \& low capture radius
1977 \begin{minipage}[t]{12.9cm}
1978 \underline{Pecipitation simulations}
1980 \item High C concentration $\rightarrow$ amorphous SiC like phase
1981 \item Problem of potential enhanced slow phase space propagation
1982 \item Low T $\rightarrow$ C-Si \hkl<1 0 0> dumbbell dominated structure
1983 \item High T $\rightarrow$ C$_{\text{sub}}$ dominated structure
1984 \item High T necessary to simulate IBS conditions (far from equilibrium)
1985 \item Precipitation by successive agglomeration of \cs (epitaxy)
1986 \item \si{}: vehicle to form \cs{} \& supply of Si \& stress compensation
1987 (stretched SiC, interface)
2006 \underline{Augsburg}
2008 \item Prof. B. Stritzker (accomodation at EP \RM{4})
2009 \item Ralf Utermann (EDV)
2012 \underline{Helsinki}
2014 \item Prof. K. Nordlund (MD)
2019 \item Bayerische Forschungsstiftung (financial support)
2022 \underline{Paderborn}
2024 \item Prof. J. Lindner (SiC)
2025 \item Prof. G. Schmidt (DFT + financial support)
2026 \item Dr. E. Rauls (DFT + SiC)
2027 \item Dr. S. Sanna (VASP)
2034 \bf Thank you for your attention!